ALMOST RINGS AND PERFECTOID SPACES

OFER GABBER AND LORENZO RAMERO

September 10, 2023

Release 8

Ofer Gabber I.H.E.S. Le Bois-Marie 35, route de Chartres F-91440 Bures-sur-Yvette e-mail address: gabber@ihes.fr

Lorenzo Ramero Université de Lille I Laboratoire de Mathématiques F-59655 Villeneuve d’Ascq Cédex e-mail address: lorenzo.ramero@univ-lille.fr

web page: https://pro.univ-lille.fr/lorenzo-ramero

Acknowledgements We thank Yves André, Francesco Baldassarri, Niels Borne, Mateo Carmona, Kestutis Cesnavicius, Michel Emsalem, Pierre-Yves Gaillard, Lutz Geissler, W.-P. Heidorn, Laurent Moret-Bailly, Fabrice Orgogozo, Olaf Schnürer and Peter Scholze for pointing out some mistakes in earlier drafts, and for useful suggestions.

2 OFER GABBER AND LORENZO RAMERO

Contents

0. Introduction................................................................. 4
1. Basic category theory........................................................ 8
1.1. Categories, functors and natural transformations.............................. 8
1.2. Presheaves and limits....................................................... 22
1.3. Adjunctions and Kan extensions............................................. 35
1.4. Special properties of the categories of presheaves ............................. 45
1.5. Coinitial and cofinal functors................................................ 53
1.6. Localizations of categories.................................................. 64
2. 2-Category theory ........................................................... 72
2.1. 2-Categories and pseudo-functors............................................ 72
2.2. Pseudo-natural transformations and their modifications........................ 85
2.3. The formalism of base change............................................... 98
2.4. Adjunctions in 2-categories................................................. 109
2.5. 2-Limits and 2-colimits..................................................... 131
2.6. 2-Categorical Kan extensions ............................................... 141
3. Special categories ........................................................... 157
3.1. Fibrations................................................................. 157
3.2. Split fibrations and cartesian sections........................................ 176
3.3. 2-Fibrations............................................................... 182
3.4. Fibrations in groupoids..................................................... 199
3.5. Sieves and descent theory................................................... 201
3.6. Profinite groups and Galois categories....................................... 216
3.7. Tensor categories and abelian categories......................................224
4. Sites and topoi .............................................................. 242
4.1. Topologies and sites........................................................242
4.2. Continuous and cocontinuous functors....................................... 251
4.3. Morphisms of sites......................................................... 259
4.4. Topoi..................................................................... 273
4.5. Fibred sites................................................................284
4.6. Fibred topoi............................................................... 292
4.7. Localization and points of a topos ........................................... 307
4.8. Algebra on a topos......................................................... 323
4.9. Torsors on a topos ......................................................... 340
5. Stacks...................................................................... 347
5.1. Prestacks and stacks on a site ............................................... 347
5.2. Covering morphisms of prestacks............................................360
5.3. Local calculus of fractions.................................................. 368
5.4. Functorial properties of the categories of stacks............................... 373
5.5. Sheaves of categories.......................................................390
5.6. Stacks in groupoids and ind-finite stacks..................................... 405
5.7. Stacks on fibred sites and fibred topoi........................................ 420
6. Monoids and polyhedra ...................................................... 438
6.1. Monoids.................................................................. 438
6.2. Integral monoids........................................................... 455
6.3. Polyhedral cones...........................................................465
6.4. Fine and saturated monoids................................................. 478
6.5. Fans...................................................................... 497
6.6. Special subdivisions........................................................513

7. Homological algebra......................................................... 529
7.1. Complexes in an additive category........................................... 529
7.2. Filtered complexes and spectral sequences....................................541
7.3. Derived categories and derived functors...................................... 549
7.4. Simplicial objects.......................................................... 570
7.5. Simplicial sets............................................................. 596
7.6. Graded rings .............................................................. 629
7.7. Differential graded algebras................................................. 639
7.8. Koszul algebras and regular sequences....................................... 644
7.9. Filtered rings and Rees algebras............................................. 660
7.10. Some homotopical algebra.................................................667
7.11. Injective modules, flat modules and indecomposable modules................. 687
8. Complements of topology and topological algebra .............................. 697
8.1. Spectral spaces and constructible subsets..................................... 698
8.2. Topological groups.........................................................721
8.3. Topological rings and topological modules ................................... 731
8.4. Topologically local and topologically henselian rings.......................... 754
8.5. Graded structures on topological rings ....................................... 760
8.6. Homological algebra for topological modules................................. 766
9. Complements of commutative algebra ......................................... 780
9.1. Valuation theory........................................................... 780
9.2. Huber’s theory of the valuation spectrum..................................... 795
9.3. Witt vectors............................................................... 814
9.4. Fontaine rings............................................................. 839
9.5. Divided power modules and algebras ........................................ 849
9.6. Regular rings.............................................................. 876
9.7. Regular morphisms and excellent rings.......................................895
9.8. Normalization, weak normalization and p-root closure......................... 913
10. Cohomology and local cohomology of sheaves................................ 924
10.1. Cohomology of topoi and topological spaces ................................ 924
10.2. Čech cohomology ........................................................ 937
10.3. Quasi-coherent modules................................................... 951
10.4. Depth and cohomology with supports....................................... 961
10.5. Depth and associated primes............................................... 976
10.6. Cohomology of projective schemes......................................... 989
11. Duality theory..............................................................1007
11.1. Duality for quasi-coherent modules.........................................1007
11.2. Cousin complexes ........................................................1014
11.3. Duality over coherent schemes.............................................1024
11.4. Schemes over a valuation ring..............................................1045
11.5. Local duality.............................................................1062
11.6. Hochster’s theorem and Stanley’s theorem...................................1078
12. Logarithmic geometry ......................................................1091
12.1. Log topoi................................................................1091
12.2. Log schemes.............................................................1102
12.3. Logarithmic differentials and smooth morphisms ............................1117
12.4. Logarithmic blow up of a coherent ideal ....................................1131
12.5. Regular log schemes......................................................1147
12.6. Resolution of singularities of regular log schemes............................1160
12.7. Local properties of the fibres of a smooth morphism..........................1179

13. Étale coverings of schemes and log schemes ..................................1186
13.1. Acyclic morphisms of schemes.............................................1186
13.2. Local asphericity of smooth morphisms of schemes..........................1201
13.3. Étale coverings of log schemes.............................................1213
13.4. Local acyclicity of smooth morphisms of log schemes........................1234
14. The almost purity toolbox...................................................1247
14.1. Non-flat almost structures.................................................1247
14.2. Inverse systems of almost modules .........................................1275
14.3. Quasi-coherent sheaves of almost modules and almost rings...................1302
14.4. Almost pure pairs.........................................................1315
14.5. Normalized lengths.......................................................1320
14.6. Finite group actions on almost algebras.....................................1346
14.7. Almost Witt vectors.......................................................1354
14.8. Complements : locally measurable algebras.................................1368
15. Continuous valuations and adic spaces........................................1385
15.1. Formal schemes..........................................................1385
15.2. Analytically noetherian rings ..............................................1400
15.3. Continuous valuations.....................................................1414
15.4. Affinoid rings and affinoid schemes.........................................1424
15.5. Adic spaces..............................................................1438
15.6. Special loci of quasi-affinoid schemes......................................1461
15.7. Etale coverings of quasi-affinoid schemes...................................1466
16. Perfectoid rings and perfectoid spaces ........................................1479
16.1. Distinguished elements and transversal pairs.................................1479
16.2. P-rings ..................................................................1485
16.3. Perfectoid rings...........................................................1494
16.4. Homological theory of perfectoid rings .....................................1525
16.5. Perfectoid quasi-affinoid rings.............................................1545
16.6. Graded perfectoid rings ...................................................1573
16.7. Perfectoid spaces.........................................................1593
16.8. Almost purity ............................................................1611
16.9. Perfectoid Tate rings and perfectoid Abhyankar’s lemma......................1631
17. Applications...............................................................1662
17.1. Model algebras...........................................................1662
17.2. Almost purity : the log regular case.........................................1677
17.3. The direct summand conjecture ............................................1690
17.4. Diagonal idempotents of generically étale maps..............................1696
17.5. Big Cohen-Macaulay algebras .............................................1714
17.6. Monomial conjectures.....................................................1752
References......................................................................1754

0. Introduction

Both the focus of this monograph and its subject matter have evolved considerably in the last few years. On the one hand, the insistence on making the text self-contained (aside from a reduced canon of basic references, which should ideally contain only EGA and some parts of Bourbaki’s Éléments), has resulted in a rather weighty mass of material of independent interest, that is applied to, but is completely separate from almost ring theory, and whose relationship to p-adic Hodge theory is thus even more indirect. Rather than stemming from a well-thought-out plan, this part is the outcome of a haphazard process, lumbering between alternating phases of accretion and consolidation, with new topics piled up as dictated by need, or occasionally by whim, when we just branched out from the main flow to pursue a certain line of thought to its logical destination. Nevertheless, a few themes have spontaneously emerged, around which the originally amorphous magma has been able to settle, to the point where by now a distinct shape is finally discernible, and it is perhaps time to pause and take stock of its broad outlines.

Now then, we may distinguish :

• First of all, a rather thorough exposition of the foundations of logarithmic algebraic geometry, comprising chapters 6, 12 and 13. Inevitably, our treatment owes a lot to the works of Kato and his school : our contribution is foremost that of gathering and tidying up the subject, which until now was scattered in a disparate number of research articles, many of them still unpublished and even unfinished. A closer scrutiny would also reveal a few technical innovations that we hope will become standard issue of the working algebraic log-geometer : we may mention the systematic use of pointed monoids and pointed modules, the projective fan associated with a graded monoid, or a definition of α-flatness for log structures which refines and generalizes an older notion of “toric flatness”. Furthermore, we took the occasion to repair a few small (and not so small) mistakes and inaccuracies that we detected in the literature.

• Two other chapters are dedicated to local cohomology and Grothendieck’s duality theory. Early on, the emphasis here was on generalizations : especially, we were interested in removing from the theory the pervasive noetherian assumptions, to pave the way for our recasting of Faltings’s almost purity in the framework of valuation theory. Applications of local cohomology to non-archimedean analytic geometry furnished another influential motivation, though one that has remained, so far, hidden from view. More recently, the noetherian aspects have also become relevant to our project, and this latest release contains a detailed account of the most important properties of noetherian rings endowed with a dualizing complex. The latter, in turn, could be dealt with satisfactorily only after a thorough revisitation of the general theory of the dualizing complex, so that our chapters 10 and 11 can also be regarded as complementary to Conrad’s book [50] (dedicated to the trace morphism and the deeper aspects of duality) : totaling our respective efforts, it should eventually become possible to bypass entirely Hartshorne’s notes [87] which, as is well known, are wanting in many ways.

• Chapters 7 and 9 present (for the time being, anyway) a looser structure : a miscellanea of self-contained units devoted to more or less independent topics. However, there is at least one thread running through several sections, and whose stretch can be traced all the way back to the earliest beginnings of almost ring theory; it connects sections 7.4 and 7.5 – on simplicial homotopy theory – to a section 7.10 dedicated to homotopical algebra, then on to sections 9.6 and 9.7, which make extensive use of the cotangent complex to derive important characterizations of regular and excellent rings, including an up-to-date presentation of classical results due to André, extracted from his monograph [3], and from his paper [4] on localization of formal smoothness. This homotopical algebraic thread resurfaces again in section 14.1, but there we are already squarely into almost ring theory proper.

On the other hand, two recent notable developments are compelling a revision of our understanding of almost ring theory itself, and of its situation within commutative algebra and algebraic geometry at large :

• The first is Scholze’s PhD thesis [149] on perfectoid spaces, that contains both a maximal generalization of the almost purity theorem, and a major simplification of its proof, based on his “tilting” technique (and completely different from Faltings’s). However, the range of Scholze’s theory transcends the domain of p-adic Hodge theory (which was not even his original motivation) : to drive the point home, his thesis concludes with a clever application to the long standing weight monodromy conjecture, thus affording the unusual spectacle of a tool which was fashioned out of purely p-adic concerns, and ends up playing a crucial role in the solution of a purely ℓ-adic problem.

• The second spectacular development is Yves André’s proof of the direct summand conjecture ([5]); the latter is a deceptively simple assertion that has been a central problem in commutative algebra for the last thirty years : it asserts that every finite injective ring homomorphism f : A → B from a regular local ring A, admits a A-linear splitting. The relevance of almost purity to this question was first surmised by Paul Roberts in 2001 (after a talk by the second author at the University of Utah), and has been widely advertised by him ever since. André’s solution uses perfectoid techniques, and builds on earlier work by Bhargav Bhatt, who in [22] proved the conjecture in the case where A is essentially smooth over a mixed characteristic discrete valuation ring and f ⊗_{ℤ} ℚ is étale outside a relative normal crossings divisor of Spec A. Moreover, Bhatt has subsequently simplified some of André’s arguments and shown how the same method yields a more general “derived version” of the conjecture, for proper schemes over any regular ring : see [23].

We see then, that almost ring theory has emancipated itself from its former ancillary role in the exclusive service of p-adic Hodge theory, and is now elbowing out a niche in the wider ecosystem of algebraic geometry.

The present release completes the project announced in the introduction of the 6th release :

• First we introduce a class of topological rings that generalize the perfectoid rings of [149]; it is very easy to say what a (generalized) perfectoid 𝔽_{p}-algebra is : namely, it is just a perfect and complete topological 𝔽_{p}-algebra whose topology is linear, defined by an ideal of finite type. The general definition is somewhat more involved, but we prove the following characterization. For any perfectoid 𝔽_{p}-algebra E, we consider the ring of Witt vectors W (E), and we endow it with a natural topology, induced from that of E; then every perfectoid ring is a topological quotient of the form A := W (E)/\underline{a}W (E), for such a suitable E, and where \underline{a} ∈ W (E) is what we call a distinguished element : see definition 16.1.6. Moreover, just as in Scholze’s work, the perfectoid 𝔽_{p}-algebra E can be recovered from A via a tilting functor that establishes, more precisely, an equivalence between the category of all perfectoid rings and that of pairs (E, ℐ) consisting of a perfectoid 𝔽_{p}-algebra E and a principal ideal ℐ ⊂ W (E) generated by a distinguished element (as it is well known, this construction is rooted in Fontaine and Winterberger’s theory of the field of norms). The distinguished ideal ℐ represents an extra parameter that remains hidden in Scholze’s original approach : the reason is that he fixes from the start a base perfectoid field K, thereby implicitly fixing as well a distinguished element \underline{a} in the ring of Witt vectors of the tilt of K, and then every perfectoid ring in his work is supposed to be a K-algebra, which – from our viewpoint – amounts to restricting to perfectoid rings whose associated distinguished ideal is generated by \underline{a}. Having thus removed the parameter ℐ, he can then also do away with Witt vectors altogether, and the inverse to the tilting construction is obtained in [149] via a more abstract deformation theoretic argument. This route is precluded to us, so we rely instead on direct and rather concrete Witt vectors calculations. A similar strategy has been proposed in [114], and our viewpoint can indeed be described fairly as an interpolation of those of Scholze and Kedlaya-Liu, though we only studied [149] in detail.

• The first three sections of chapter 16 are devoted to exploring this new class of perfectoid rings and its manifold remarkable features. The rest of the chapter then merges the theory of perfectoid rings with Huber’s adic spaces, to forge the perfectoid spaces that are the main tool for our proof of almost purity, whose most general form is given by theorem 16.8.44and applies to formally perfectoid rings, i.e. to topological rings whose completion is perfectoid. The proof proceeds via several preliminary reductions : first, to the case of a perfectoid quasi-affinoid ring, covered by theorem 16.8.32, then – by exploiting the local geometry of adic spaces – to the case of a perfectoid valuation ring, which was treated already in our monograph [75]. What enables here this localization argument is a basic feature of the étale topology of arbitrary adic spaces : the fibred category of finite étale coverings of the affinoid subsets of an adic space is a stack. The latter result is in turn a special case of our theorem 15.7.6.

• We also include a detailed treatment of the foundations of the theory of adic spaces, that essentially follows [98], but contains some modest improvement : notably, the systematic use of analytically noetherian rings (borrowed from [71]) allows us to unify the two classes of topological rings that Huber dealt with separately in his work (the strongly noetherian rings and the f-adic rings with a noetherian ring of definition). We also point out a henselian variant of the structure sheaf that is available on the adic spectrum of any f-adic ring, with no restriction whatsoever.

• The last chapter proposes a few applications : in section 17.1 we introduce a class of model algebras over any rank one valuation ring K^{+} of mixed characteristic (0, p), and we show that when K^{+} is deeply ramified, such algebras are formally perfectoid rings for their p-adic topology; hence the theory of chapter 16 immediately yields an almost purity theorem for model algebras. Likewise, section 17.2 proves an almost purity theorem for certain very ramified towers of log-regular rings; again, after some preliminary reductions, the proof amounts to the observation that the inductive limits of such towers are formally perfectoid for their p-adic topology. These instances of almost purity were already contained in a previous draft of our work (Release 6), where they were proven by an extension of Faltings’s method, that relied on deep results from logarithmic algebraic geometry, and also entailed the construction of certain normalized lengths for torsion modules over model algebras, and respectively over the rings occurring in section 17.2. Neither of these two ingredients intervenes any longer in the new proofs; however, we have found worthwhile to explain how model algebras arise from suitable very ramified towers of log-smooth K^{+}-algebras, and we have also retained the construction of normalized lengths for model algebras and for limits of towers log-regular rings, since they are sufficiently interesting in their own right, and might be useful for other applications (normalized lengths for torsion K^{+}-modules are exploited in [150]).

Section 17.3 contains our account of André’s work on the direct summand conjecture, which we generalize to the case of a finite injective ring homomorphism A → B where A is logregular : see theorem 17.3.12, whose proof relies on a refinement of André’s perfectoid Abhyankar’s lemma, which is the main result of section 16.9. In a future release we shall present a corresponding logarithmic generalization of Bhatt’s derived version of the direct summand conjecture.

Section 17.4 considers a finite injective map f : A → B of noetherian rings, where A is again log-regular; if f is étale, B ⊗_{A} B has a canonical diagonal idempotent e_{f}, so called because its support is the open and closed diagonal in Spec B ×_{Spec A} Spec B. When f is only generically étale, e_{f} is only well defined on some localization B ⊗_{A} B[a^{−1}] (for some a ∈ A whose zero locus is nowhere dense in Spec A), and the problem we address, is to exhibit an explicit d ∈ B ⊗_{A} B such that de_{f} is integral, i.e. lies in the image of B ⊗_{A} B. This question has an easy answer when B is a projective A-module : indeed, in this case we have a well defined trace map B → A, whence a different ideal 𝒟 ⊂ B, and a standard calculation shows that (b⊗1)·e_{f} is integral for every b ∈ 𝒟. If B is not projective, the question seems to be much more difficult : our solution makes an essential use of perfectoid techniques, and especially the perfectoid Abhyankar’s lemma of section 16.9 : see theorem 17.4.19and remark 17.4.44.

Section 17.5 proves the existence and weak functoriality of big Cohen-Macaulay algebras, over any local noetherian ring A. The case where A is equicharacteristic was already known before André’s work, and is due to Hoschster and Huneke, but our construction is different from theirs, and yields actual functors on suitable subcategories : see remark 17.5.36(ii) and theorem 17.5.45. For the mixed characteristic case, we rely, as in [7], on the perfectoid Abhyankar’s lemma, but again, our construction differs from the one given in the latter work; also here we get good functorial properties, but the proofs are much harder and longer than for equicharacteristic rings. A crucial ingredient for both cases, is a simple construction, introduced in (17.5.2)

– and inspired by some ultrafilter techniques – whose remarkable property is to transform almost Cohen-Macaulay (resp. almost faithfully flat) algebras into actual Cohen-Macaulay (resp. faithfully flat) algebras. Lastly, weak functoriality is given by theorem 17.5.90.

The final section 17.6 explains how to apply the existence and weak functoriality of Cohen-Macaulay algebras, to deduce the monomial conjecture, first in its original form due to Hochster (theorem 17.6.1), and then in its more recent strong form (theorem 17.6.6).

1. Basic category theory

The purpose of this chapter is to fix some notation that shall stand throughout this work, and to collect, for ease of reference, a few well known generalities on categories and functors that are frequently used. Our main reference on general nonsense is the treatise [28], and another good reference is the more recent [110].

Sooner or later, any honest discussion of categories and topoi gets tangled up with some foundational issues revolving around the manipulation of large sets. For this reason, to be able to move on solid ground, it is essential to select from the outset a definite set-theoretical framework (among the several currently available), and stick to it unwaveringly.

Thus, throughout this work we will accept the so-called Zermelo-Fraenkel system of axioms for set theory. (In this version of set theory, everything is a set, and there is no primitive notion of class, in contrast to other axiomatisations.)

Additionally, following [8, Exp.I, §0], we shall assume that, for every set S, there exists a universe V such that S ∈ V. (For the notion of universe, the reader may also see [28, §1.1].)

Throughout this chapter, we fix some universe U such that ℕ ∈ U (where ℕ is the set of natural numbers; the latter condition is required, in order to be able to perform some standard set-theoretical operations without leaving U). A set S is U-small (resp. essentially U-small), if S ∈ U (resp. if S has the cardinality of a U-small set). If the context is not ambiguous, we shall just write small, instead of U-small.

1.1. Categories, functors and natural transformations. A category 𝒞 is the datum of a set Ob(𝒞) of objects and, for every A, B ∈ Ob(𝒞), a set of morphisms from A to B, denoted :

Hom_{𝒞} (A, B)

and as usual we write f : A → B to signify f ∈ Hom_{𝒞} (A, B). Furthermore, we set

Morph(𝒞) := {(A, B, f) | A, B ∈ Ob(𝒞), f ∈ Hom_{𝒞} (A, B)}.

For any \underline{f} := (A, B, f) ∈ Morph(𝒞), the object A is called the source of \underline{f}, and B is the target of \underline{f}. We also often use the notation

End_{𝒞} (A) := Hom_{𝒞} (A, A)

and the elements of End_{𝒞} (A) are called the endomorphisms of A in 𝒞. We say that a pair of elements (\underline{f}, \underline{g}) of Morph(𝒞) is composable if the target of \underline{f} equals the source of \underline{g}. Moreover, for every A, B, C ∈ Ob(𝒞) we have a composition law

Hom_{𝒞} (A, B) × Hom_{𝒞} (B, C) → Hom_{𝒞} (A, C) : (f, g) ↦ g ∘ f

fulfilling the following two standard axioms :

• For every A ∈ Ob(𝒞) there exists an identity endomorphism 1_{A} of A, such that

1_{A} ∘ f = f g ∘ 1_{A} = g for every B, C ∈ Ob(𝒞) and every f : B → A and g : A → C .

• The composition law is associative, i.e. we have

(h ∘ g) ∘ f = h ∘ (g ∘ f)

for every A, B, C, D ∈ Ob(𝒞) and every f : A → B, g : B → C and h : C → D. Clearly, it follows that (End_{𝒞} (A), ∘, 1_{A}) is a monoid, and we get a group :

Aut_{𝒞} (A) ⊂ End_{𝒞} (A)

of invertible endomorphisms, i.e. the automorphisms of the object A.

1.1.1. We say that the category 𝒞 is U-small (or just small), if both Ob(𝒞) and Morph(𝒞) are small sets. We say that 𝒞 has small Hom-sets if Hom_{𝒞} (A, B) ∈ U for every A, B ∈ Ob(𝒞).

A subcategory of 𝒞 is a category ℬ with Ob(ℬ) ⊂ Ob(𝒞) and Morph(ℬ) ⊂ Morph(𝒞).

The opposite category 𝒞^{o} is the category with Ob(𝒞^{o}) = Ob(𝒞), and such that :

Hom_{𝒞^{o}}(A, B) := Hom_{𝒞} (B, A) for every A, B ∈ Ob(𝒞)

(with composition law induced by that of 𝒞, in the obvious way). Given A ∈ Ob(𝒞), sometimes we denote by A^{o} the same object, viewed as an element of Ob(𝒞^{o}); likewise, given a morphism f : A → B in 𝒞, we write f^{o} for the corresponding morphism B^{o} → A^{o} in 𝒞^{o}.

1.1.2. A morphism f : A → B in 𝒞 is said to be a monomorphism if the induced map

Hom_{𝒞} (X, f) : Hom_{𝒞} (X, A) → Hom_{𝒞} (X, B) g ↦ f ∘ g

is injective, for every X ∈ Ob(𝒞). Dually, we say that f is an epimorphism if f^{o} is a monomorphism in 𝒞^{o}. Also, f is an isomorphism if there exists a morphism g : B → A such that g ∘ f = 1_{A} and f ∘ g = 1_{B}. Obviously, an isomorphism is both a monomorphism and an epimorphism. The converse does not necessarily hold, in an arbitrary category.

Two monomorphisms f : A → B and f′ : A′ → B are equivalent, if there exists an isomorphism h : A → A′ such that f = f′ ∘ h. A subobject of B is defined as an equivalence class of monomorphisms A → B. Dually, a quotient of B is a subobject of B^{o} in 𝒞^{o}.

One says that 𝒞 is well-powered if, for every A ∈ Ob(𝒞), the set :

Sub(A)

of all subobjects of A is essentially small. Dually, 𝒞 is co-well-powered, if 𝒞^{o} is well-powered.

1.1.3. Let 𝒜 and ℬ be any two categories; a functor F : 𝒜 → ℬ is a pair of maps

Ob(𝒜) → Ob(ℬ) Morph(𝒜) → Morph(ℬ)

both denoted also by F, such that • F assigns to any morphism f : A → A′ in 𝒜, a morphism F f : F A → F A′ in ℬ

• F 1_{A} = 1_{F A} for every A ∈ Ob(𝒜)

• F (g ∘ f) = F g ∘ F f for every A, A′, A′′ ∈ Ob(𝒜) and every pair of morphisms

f : A → A′, g : A′ → A′′ in 𝒜.

If F : 𝒜 → ℬ and G : ℬ → 𝒞 are any two functors, we get a composition

G ∘ F : 𝒜 → 𝒞

which is the functor whose maps on objects and morphisms are the compositions of the respective maps for F and G. We denote by

Fun(𝒜 , ℬ)

the set of all functors 𝒜 → ℬ. Moreover, any such F induces a functor F^{o} : 𝒜^{o} → ℬ^{o} with F^{o}A^{o} := (F A)^{o} and F^{o}f^{o} := (F f)^{o} for every A ∈ Ob(𝒜) and every f ∈ Morph(𝒜).

Definition 1.1.4. Let F : 𝒜 → ℬ be a functor.

(i) We say that F is faithful (resp. full, resp. fully faithful), if for every A, A′ ∈ Ob(𝒜) it induces injective (resp. surjective, resp. bijective) maps :

Hom_{𝒜} (A, A′) → Hom_{ℬ} (F A, F A′) : f ↦ F f.

(ii) We say that F reflects monomorphisms (resp. reflects epimorphisms, resp. is conservative) if the following holds. For every morphism f : A → A′ in 𝒜, if the morphism F f of ℬ is a monomorphism (resp. epimorphism, resp. isomorphism), then the same holds for f.

(iii) If 𝒜 is a subcategory of ℬ, and F is the natural inclusion functor, then F is obviously faithful, and we say that 𝒜 is a full subcategory of ℬ, if F is fully faithful.

(iv) The essential image of F is the full subcategory of ℬ whose objects are the objects of ℬ that are isomorphic to an object of the form F A, for some A ∈ Ob(𝒜). We say that F is essentially surjective if its essential image is ℬ.

(v) We say that F is an equivalence, if it is fully faithful and essentially surjective.

Remark 1.1.5. For later use, it is convenient to introduce the notion of n-faithful functor, for all integers n ≤ 2. Namely : if n < 0, every functor is n-faithful; a functor F : 𝒜 → ℬ (between any two categories 𝒜 and ℬ) is 0-faithful, if it is faithful; F is 1-faithful, if it is fully faithful; finally, we say that F is 2-faithful, if it is an equivalence.

Example 1.1.6. (i) The collection of all small categories, together with the functors between them, forms a category

U-Cat.

Unless we have to deal with more than one universe, we shall usually omit the prefix U, and write just Cat. It is easily seen that Cat is a category with small Hom-sets.

(ii) The category of all small sets shall be denoted U-Set or just Set, if there is no need to emphasize the chosen universe. There is a natural fully faithful embedding :

Set → Cat.

Indeed, to any set S one may assign its discrete category also denoted S, i.e. the unique category such that Ob(S) = S and Morph(S) = {(s, s, 1_{s}) | s ∈ S}. If S and S′ are two discrete categories, the datum of a functor S → S′ is clearly the same as a map of sets Ob(S) → Ob(S′).

Notice also the natural functor

Ob : Cat → Set 𝒞 ↦ Ob(𝒞)

that assigns to each functor F : 𝒞 → 𝒟 the underlying map Ob(𝒞) → Ob(𝒟) : C ↦ F C.

(iii) Recall that a preordered set is a pair (I, ≤) consisting of a set I and a binary relation ≤ on I which is reflexive and transitive. In this case, we also say that ≤ is a preordering on I. We say that (I, ≤) is a partially ordered set, if ≤ is also antisymmetric, i.e. if we have

(x ≤ y and y ≤ x) ⇒ x = y for every x, y ∈ I.

We say that (I, ≤) is a totally ordered set, if it is partially ordered and any two elements are comparable, i.e. for every x, y ∈ I we have either x ≤ y or y ≤ x. An order-preserving map f : (I, ≤) → (J, ≤) between preordered sets is a mapping f : I → J such that

x ≤ y ⇒ f(x) ≤ f(y) for every x, y ∈ I.

We denote by Preorder (resp. POSet) the category of small preordered (resp. partially ordered) sets, with morphisms given by the order-preserving maps. To any preordered set (I, ≤) one assigns a category whose set of objects is I, and whose morphisms are given as follows. For every i, j ∈ I, the set of morphisms i → j contains exactly one element when i ≤ j, and is empty otherwise. Clearly, this rule defines a fully faithful functor

Preorder → Cat.

Notice that if a category C lies in the essential image of this functor, then the same holds for C^{o}. Indeed, if C corresponds to the preordered set (I, ≤), then C^{o} corresponds to the preordered set (I^{o}, ≤) with I^{o} := I and x ≤ y in I^{o} if and only if y ≤ x in I, for every x, y ∈ I. Clearly (I, ≤) is a partially ordered set if and only if the same holds for (I^{o}, ≤).

1.1.7. Let 𝒜, ℬ be two categories, F, G : 𝒜 → ℬ two functors. A natural transformation

(1.1.8)  α : F ⇒ G

from F to G is a family of morphisms (α_{A} : F A → GA | A ∈ Ob(𝒜)) of ℬ such that, for every morphism f : A → B in 𝒜, the diagram :

(1.1.9)  \xymatrix{F A \ar[r]^{α_{A}} \ar[d]_{F f} & GA \ar[d]^{Gf} \\ F B \ar[r]^{α_{B}} & GB}

commutes. If α_{A} is an isomorphism for every A ∈ Ob(𝒜), we say that α is a natural isomorphism of functors. For instance, the rule that assigns to any object A the identity morphism 1_{F A} : F A → F A, defines a natural isomorphism 1_{F} : F ⇒ F. A natural transformation (1.1.8) is also indicated by a diagram of the type :

[figure: 𝒜; F; G; α; ℬ.]

1.1.10. The natural transformations between functors 𝒜 → ℬ can be composed; namely, if α : F ⇒ G and β : G ⇒ H are two such transformations, we obtain a natural transformation

β ⊙ α : F ⇒ H  by the rule : A ↦ β_{A} ∘ α_{A}  for every A ∈ Ob(𝒜) .

With this composition, Fun(𝒜 , ℬ) is the set of objects of a category which we shall denote

Fun(𝒜 , ℬ).

There is also a second composition law for natural transformations : if 𝒞 is another category, and we have a diagram of functors and natural transformations

[figure: 𝒜; F; G; α; ℬ; F′; G′; α′; 𝒞]

we get a natural transformation

α′ ∗α : F′ ∘F ⇒ G′ ∘G : A ↦ α′_{GA} ∘F′(α_{A}) = G′(α_{A})∘α′_{F A} for every A ∈ Ob(𝒜)

called the Godement product of α and α′ ([28, Prop.1.3.4]). Especially, if H : ℬ → 𝒞 (resp. H : 𝒞 → 𝒜) is any functor, we write H ∗ α (resp. α ∗ H) instead of 1_{H} ∗ α (resp. α ∗ 1_{H}).

Both composition laws are associative, i.e., if we have additional natural transformations

[figure: 𝒜; H; K; γ; ℬ; 𝒞; F′′; G′′; α′′; 𝒟]

then we get the identities

γ ⊙ (β ⊙ α) = (γ ⊙ β) ⊙ α α′′ ∗ (α′ ∗ α) = (α′′ ∗ α′) ∗ α.

Moreover, the composition laws are related as follows. Suppose that 𝒜, ℬ and 𝒞 are three categories, and we have a diagram of six functors and four natural transformations :

[figure: 𝒜; F_{1}; α_{1}; H_{1}; β_{1}; G_{1}; ℬ; F_{2}; α_{2}; H_{2}; β_{2}; G_{2}; 𝒞 .]

Then we have the identity :

(1.1.11)  (β_{2} ∗ β_{1}) ⊙ (α_{2} ∗ α_{1}) = (β_{2} ⊙ α_{2}) ∗ (β_{1} ⊙ α_{1}).

The proofs are left as exercises for the reader (see [28, Prop.1.3.5]).

Remark 1.1.12. (i) In the situation of (1.1.10), if 𝒜 and ℬ are small categories, the same holds for Fun(𝒜 , ℬ).

(ii) Also, if 𝒜 is small, ℬ has small Hom-sets and Ob(ℬ) ⊂ U, then Fun(𝒜 , ℬ) has small Hom-sets, and Ob(Fun(𝒜 , ℬ)) ⊂ U.

(iii) Assertion (ii) depends on our choices on how to encode arbitrary maps of sets : according to our (implicit) convention, a map of sets f : S → S′ is the graph Γ(f) ⊂ S × S′. This does not agree, e.g. with the definition found in Bourbaki’s treatise [36], where such a map f is the triple (S, S′, Γ(f)). With Bourbaki’s convention, assertion (ii) fails. Other references are not so explicit about their choices for encoding maps, but for instance the SGA4 treatise ([8], [9], [10]) appears to follow Bourbaki’s conventions, in view of [8, Exp.I, Rem.1.1.2], which states that Fun(𝒜 , ℬ) is not necessarily a subset of U, and Fun(𝒜 , ℬ) does not necessarily have small Hom-sets, under the assumptions of (ii). On the other hand, under the same assumptions, it is stated in [76, Ch.II, Prop.1] that Fun(𝒜 , ℬ) has small Hom-sets, so the set-theoretic conventions of the latter are not compatible with those of SGA4.

1.1.13. Adjoint pairs of functors. Let 𝒜 and ℬ be two categories, F : 𝒜 → ℬ and G : ℬ → 𝒜 two functors. We say that G is left adjoint to F if there exist bijections

ϑ_{A,B} : Hom_{𝒜} (GB, A) →^{∼} Hom_{ℬ} (B, F A) for every A ∈ Ob(𝒜) and B ∈ Ob(ℬ)

and these bijections are natural in both A and B, i.e.

ϑ_{A′B′}(g ∘f ∘Gh)=F g ∘ϑ_{AB}(f)∘h for all morphisms GB ⟶^{f}A⟶^{g}A′ in 𝒜 and B′ ⟶^{h}B in ℬ .

Then one also says that F is right adjoint to G, that (G, F) is an adjoint pair of functors, and that ϑ_{••} is an adjunction for the pair (G, F).

Especially, to any object B of ℬ (resp. A of 𝒜), the adjunction ϑ_{••} assigns a morphism ϑ_{GB,B}(1_{GB}) : B → F GB (resp. ϑ_{A,F A}^{−1}(1_{F A}) : GF A → A), whence a natural transformation

(1.1.14)  η : 1_{ℬ} ⇒ F ∘ G  (resp. ε : G ∘ F ⇒ 1_{𝒜} )

called the unit (resp. counit) of the adjunction. The naturality of η follows from the calculation:

F G(f) ∘ η_{B} = F G(f) ∘ ϑ_{GB,B}(1_{GB}) = ϑ_{GB,B′}(Gf ∘ 1_{GB}) = ϑ_{GB,B′}(1_{GB′} ∘ Gf) = η_{B′} ∘ f

for every morphism f : B → B′ in ℬ. A similar computation shows the naturality of ε. The naturality of ϑ_{••} implies that we have commutative diagrams

\xymatrix{B \ar[r]^{η_{B}} \ar[dr]_{ϑ_{A,B}(f)} & F GB \ar[d]^{F f} & GB \ar[r]^{Gg} \ar[dr]_{ϑ_{A,B}^{−1}(g)} & GF A \ar[d]^{ε_{A}} \\  & F A &  & A}

for every morphism f : GB → A in 𝒜 and g : B → F A in ℬ. Taking f = ε_{A} and g = η_{B}, we see that the unit and counit are related by the so-called triangular identities expressed by the commutative diagrams :

[figure: F; η∗F; 1_{F}; F GF; F ∗ε; G; G∗η; 1_{G}; GF G; ε∗G; F; G.]

Conversely, we have ([28, Th.3.1.5] or [110, Prop.1.5.4]) :

Proposition 1.1.15. Let F : 𝒜 → ℬ and G : ℬ → 𝒜 be two functors.

(i) If η, ε are natural transformations as in (1.1.14), fulfilling the triangular identities of (1.1.13), then there is a unique adjunction for the pair (G,F), with unit η and counit ε. (ii) Suppose that (G, F) is an adjoint pair, and η is the unit (resp. ε is the counit) of an adjunction for (G, F), then there exists a unique natural transformation ε (resp. η) as in (1.1.14), fulfilling the triangular identities of (1.1.13).

Proof. (i): Let A ∈ Ob(𝒜) and B ∈ Ob(ℬ) be any two objects; in light of the discussion of (1.1.13), we see that the sought natural bijection ϑ_{A,B} must be given by the rule :

f ↦ F f ∘ η_{B} for every f ∈ Hom_{𝒜} (GB, A)

and its inverse must be the mapping

g ↦ ε_{A} ∘ Gg for every g ∈ Hom_{ℬ} (B, F A).

So we come down to checking that the triangular identities imply that these rules do induce mutually inverse bijections on the respective Hom-sets. We leave the verification to the reader.

(ii) is clear from the explicit construction of ϑ_{••} in (i). □

Example 1.1.16. To every preordered set (ℱ , ≤) we may attach its partially ordered quotient

(ℱ/∼, ≤)

where ∼ denotes the equivalence relation such that x ∼ y if and only if x ≤ y and y ≤ x, for every x, y ∈ ℱ. The ordering on ℱ/∼ is the unique one such that the quotient map ℱ → ℱ/∼ defines a morphism

q_{ℱ} : (ℱ , ≤) → (ℱ/∼, ≤)

of preordered sets, and it is easily seen that the rule (ℱ , ≤) ↦ (ℱ/∼, ≤) defines a left adjoint to the inclusion functor

POSet → Preorder.

Moreover, the rule (ℱ , ≤) ↦ q_{ℱ} is a unit for this adjunction.

The following observations are borrowed from, and are further developed in [81, §I.6].

Remark 1.1.17. (i) Consider two adjoint pairs (G_{1}, F_{1}) and (G_{2}, F_{2}) :

\xymatrix{𝒜 \ar[r]^{F_{1}} & ℬ \ar[r]^{F_{2}} \ar[l]^{G_{1}} & 𝒞 \ar[l]^{G_{2}}}

and suppose that for i = 1, 2 we are given adjunctions ϑ_{i,••} for the pair (G_{i}, F_{i}). Then clearly (G_{1} ∘ G_{2}, F_{2} ∘ F_{1}) is an adjoint pair, and we get an induced adjunction for this pair, by the composition :

Hom_{𝒜} (G_{1}G_{2}C, A) ⟶^{ϑ_{1,A,G_{2}C}} Hom_{ℬ} (G_{2}C, F_{1}A) ⟶^{ϑ_{2,F_{1}A,C}} Hom_{ℬ} (C, F_{2}F_{1}A)

for every A ∈ Ob(𝒜) and C ∈ Ob(𝒞). We denote this adjunction by

(ϑ_{2} ∘ ϑ_{1})_{••}

and we call it the composition of the adjunctions ϑ_{1} and ϑ_{2}. If (η_{i}, ε_{i}) are the units and counits of ϑ_{i,••} (for i = 1, 2), then the unit and counit of (ϑ_{2} ∘ ϑ_{1})_{••} are respectively :

(F_{2} ∗ η_{1} ∗ G_{2}) ⊙ η_{2} and ε_{1} ⊙ (G_{1} ∗ ε_{2} ∗ F_{1}).

Moreover, suppose that \xymatrix{𝒞 \ar[r]^{F_{3}} & 𝒟 \ar[l]^{G_{3}}} is another adjoint pair of functors, and ϑ_{3,••} an adjunction

for this pair; with this notation, it is also then clear that

(ϑ_{3} ∘ (ϑ_{2} ∘ ϑ_{1}))_{••} = (ϑ_{3} ∘ (ϑ_{2} ∘ ϑ_{1}))_{••}.

(ii) Suppose that we have two pairs of adjoint functors and two natural transformations

\xymatrix{𝒜 \ar[r]^{F} & ℬ \ar[l]^{G} & 𝒜 \ar[r]^{F′} & ℬ \ar[l]^{G′}}  τ : F ⇒ F′  μ : G′ ⇒ G

and let us fix units and counits (η, ε) (resp. (η′, ε′)) for the adjoint pair (G, F) (resp. (G′, F′)). Then we obtain adjoint transformations

τ^{†} : G′ ⇒ G μ^{†} : F ⇒ F′

given by the compositions :

G′B ⟶^{G′(η_{B})} G′F GB ⟶^{G′(τ_{GB})} G′F′GB ⟶^{ε′_{GB}} GB  for every B ∈ Ob(ℬ)
F A ⟶^{η′_{F A}} F′G′F A ⟶^{F′(μ_{F A})} F′GF A ⟶^{F′(ε_{A})} F′A  for every A ∈ Ob(𝒜) .

We claim that (τ^{†})^{†} = τ and (μ^{†})^{†} = μ. Indeed, let ϑ_{••} (resp. ϑ′_{••}) be the adjunctions corresponding to (η, ε) (resp. to (η′, ε′)), and notice that

τ_{B}^{†} = ϑ_{GB,F GB}^{′−1}(τ_{GB}) ∘ G′(η_{B}) μ_{A}^{†} = F′(ε_{A}) ∘ ϑ′_{GF A,F A}(μ_{F A})

so we may compute :

\begin{matrix}(τ^{†})_{A}^{†} = & F′(ε_{A}) ∘ ϑ′_{GF A,F A}(ϑ_{GF A,F GF A}^{′−1}(τ_{GF A}) ∘ G′(η_{F A})) \\ = & ϑ′_{A,F A}(ε_{A} ∘ ϑ_{GF A,F GF A}^{′−1}(τ_{GF A}) ∘ G′(η_{F A})) \\ = & ϑ′_{A,F A}(ϑ_{A,F GF A}^{′−1}(F′(ε_{A}) ∘ τ_{GF A}) ∘ G′(η_{F A})) \\ = & ϑ′_{A,F A}(ϑ_{A,F GF A}^{′−1}(τ_{A} ∘ F (ε_{A})) ∘ G′(η_{F A})) \\ = & ϑ′_{A,F A}(ϑ_{A,F A}^{′−1}(τ_{A}) ∘ G′F (ε_{A}) ∘ G′(η_{F A})) \\ = & ϑ′_{A,F A}(ϑ_{A,F A}^{′−1}(τ_{A})) \\ = & τ_{A} \end{matrix}

where the second, third and fifth identities follow from the naturality of ϑ′_{••}, the fourth from the naturality of τ, and the sixth from the triangular identities of (1.1.13). We leave to the reader the similar calculation which gives the second identity. Hence the rule

τ ↦ τ^{†} := (τ, ϑ, ϑ′)^{†}

establishes a natural bijection from the set of natural transformations F ⇒ F′, to the set of natural transformations G′ ⇒ G. Notice that this correspondence depends not only on (G, F) and (G′, F′), but also on (η, ε) and (η′, ε′). Sometimes we denote this adjoint transformation also by (τ, η, η′)^{†}.

(iii) Moreover, using the triangular identities of (1.1.13), it is easily seen that the diagrams :

\xymatrix{G′ ∘ F \ar@{=>}[r]^{G′∗τ} \ar@{=>}[d]_{τ^{†}∗F} & G′ ∘ F′ \ar@{=>}[d]^{ε′} & 1_{ℬ} \ar@{=>}[r]^{η} \ar@{=>}[d]_{η′} & F ∘ G \ar@{=>}[d]^{τ∗G} \\ G ∘ F \ar@{=>}[r]^{ε} & 1_{𝒜} & F′ ∘ G′ \ar@{=>}[r]^{F′∗τ^{†}} & F′ ∘ G}

commute. Also, (τ, ϑ, ϑ′)^{†} is characterized as the unique natural transformation G′ ⇒ G such that the following diagram commutes for every A ∈ Ob(𝒜) and B ∈ Ob(ℬ) :

(1.1.18)  \xymatrix{Hom_{𝒜} (GB, A) \ar[r]^{ϑ_{A,B}} \ar[d]_{Hom_{𝒜} (τ_{B}^{†},A)} & Hom_{ℬ} (B, F A) \ar[d]^{Hom_{ℬ} (B,τ_{A})} \\ Hom_{𝒜} (G′B, A) \ar[r]^{ϑ′_{A,B}} & Hom_{ℬ} (B, F′A)}

Indeed, letting A := GB and recalling that ϑ_{GB,B}(1_{GB}) = η_{B}, we see easily that the commutativity of (1.1.18) determines uniquely τ^{†} (details left to the reader). Conversely, if τ^{†} is defined as in (i), we may compute, for every morphism f : GB → A in 𝒜 :

\begin{matrix}ϑ′_{A,B}(f ∘ τ_{B}^{†}) = & ϑ′_{A,B}(f ∘ ϑ_{GB,F GB}^{′−1}(τ_{GB}) ∘ G′(η_{B})) \\ = & ϑ′_{A,F GB}(f ∘ ϑ_{GB,F GB}^{′−1}(τ_{GB})) ∘ η_{B} \\ = & ϑ′_{A,F GB}(ϑ_{A,F GB}^{′−1}(F′f ∘ τ_{GB})) ∘ η_{B} \\ = & F′f ∘ τ_{GB} ∘ η_{B} \\ = & τ_{A} ∘ F f ∘ η_{B} \\ = & τ_{A} ∘ ϑ_{A,B}(f). \end{matrix}

(iv) Furthermore, suppose we have a third pair of adjoint functors

\xymatrix{𝒜 \ar[r]^{F′′} & ℬ \ar[l]^{G′′}} and a natural transformation ω : F′ ⇒ F′′

and let us fix an adjunction ϑ′′_{••} for the pair (G′′, F′′). Then we have :

(ω ⊙ τ, ϑ, ϑ′′)^{†} = (τ, ϑ, ϑ′)^{†} ⊙ (ω, ϑ′, ϑ′′)^{†}.

Indeed, this identity follows easily from the characterization of τ^{†}, ω^{†} and (ω ∘ τ)^{†} given in (iii) (details left to the reader).

(v) Lastly, in the situation of (i), suppose moreover that we have two other adjoint pairs

\xymatrix{𝒜 \ar[r]^{F′_{1}} & ℬ \ar[l]^{G′_{1}} \ar[r]^{F′_{2}} & 𝒞 \ar[l]^{G′_{2}}}  and natural transformations τ_{1} : F_{1} ⇒ F′_{1}  τ_{2} : F_{2} ⇒ F′_{2}

and for i = 1, 2, let us fix an adjunction ϑ′_{i,••} for (G′_{i}, F′_{i}). Then we get as in (ii) the natural transformations τ_{1}^{†} : G′_{1} ⇒ G_{1} and τ_{2}^{†} : G′_{2} ⇒ G_{2}, and we have the identity

(τ_{2} ∗ τ_{1}, ϑ_{2} ∘ ϑ_{1}, ϑ′_{2} ∘ ϑ′_{1})^{†} = (τ_{1}, ϑ_{1}, ϑ′_{1})^{†} ∗ (τ_{2}, ϑ_{2}, ϑ′_{2})^{†}.

Indeed, taking into account (iii) we get the commutative diagram

\xymatrix{Hom_{𝒜} (G_{1}G_{2}C, A) \ar[r]^{ϑ_{1,A,G_{2}C}} \ar[d]_{Hom_{𝒜} (τ_{1,G_{2}C}^{†},A)} & Hom_{ℬ} (G_{2}C, F_{1}A) \ar[r]^{ϑ_{2,F_{1}A,C}} \ar[d]_{Hom_{ℬ} (G_{2}C,τ_{1,C})} & Hom_{𝒞} (C, F_{2}F_{1}A) \ar[d]^{Hom_{𝒞} (C,F_{2}(τ_{1,C}))} \\ Hom_{𝒜} (G′_{1}G_{2}C, A) \ar[r]^{ϑ′_{1,A,G_{2}C}} \ar[d]_{Hom_{𝒜} (G′_{1}(τ_{1,C}^{†}),A)} & Hom_{ℬ} (G_{2}C, F′_{1}A) \ar[r]^{ϑ_{2,F′_{1}A,C}} \ar[d]_{Hom_{ℬ} (τ_{2,C}^{†},F′_{1}A)} & Hom_{𝒞} (C, F_{2}F′_{1}A) \ar[d]^{Hom_{𝒞} (C,τ_{2,F′_{1}C})} \\ Hom_{𝒜} (G′_{1}G′_{2}C, A) \ar[r]^{ϑ′_{1,A,G′_{2}C}} & Hom_{ℬ} (G′_{2}C, F′_{1}A) \ar[r]^{ϑ′_{2,F′_{1}A,C}} & Hom_{𝒞} (C, F′_{2}F′_{1}A)}

for every A ∈ Ob(𝒜) and C ∈ Ob(𝒞). The sought identity follows after invoking again (iii). (vi) Especially, taking into account the triangular identities of (1.1.13), it is easily seen that :

\begin{aligned}(1_{F_{2}}, ϑ_{2}, ϑ_{2})^{†} & = 1_{G_{2}} \\ (F_{2} ∗ τ_{1}, ϑ_{2} ∘ ϑ_{1}, ϑ_{2} ∘ ϑ′_{1})^{†} & = (τ_{1}, ϑ_{1}, ϑ′_{1})^{†} ∗ G_{2} \\ (τ_{2} ∗ F_{1}, ϑ_{2} ∘ ϑ_{1}, ϑ′_{2} ∘ ϑ_{1})^{†} & = G_{1} ∗ (τ_{2}, ϑ_{2}, ϑ′_{2})^{†}. \end{aligned}

Remark 1.1.19. (i) For any two categories 𝒞 , 𝒟 we have a natural isomorphism of categories

Fun(𝒞 , 𝒟)^{o} →^{∼} Fun(𝒞^{o}, 𝒟^{o})

that assigns to any functor H : 𝒞 → 𝒟 the opposite functor H^{o} : 𝒞^{o} → 𝒟^{o}, and to any natural transformation τ : H ⇒ K the opposite transformation τ^{o} : K^{o} ⇒ H^{o} such that τ_{C^{o}}^{o} := (τ_{C})^{o} for every C ∈ Ob(𝒞). Also, in the situation of (1.1.10), notice the identities

(β ⊙ α)^{o} = α^{o} ⊙ β^{o} and (α′ ∗ α)^{o} = α′^{o} ∗ α^{o}.

(ii) Let ℬ and ℰ be two other categories, f : ℬ → 𝒞 and g : 𝒟 → ℰ two functors; we get an induced functor

Fun(f, g) : Fun(𝒞 , 𝒟) → Fun(ℬ, ℰ)  H ↦ g ∘ H ∘ f  (α : H ⇒ K) ↦ g ∗ α ∗ f.

Likewise, if f′ : ℬ → 𝒞 and g′ : 𝒟 → ℰ are any two other functors, every pair of natural transformations γ : f ⇒ f′ and δ : g ⇒ g′ induces a transformation

Fun(γ, δ) : Fun(f, g) ⇒ Fun(f′, g′) H ↦ δ ∗ H ∗ γ.

We shall usually write Fun(f, 𝒟) (resp. Fun(𝒞 , g)) instead of Fun(f, 1_{𝒟} ) (resp. of Fun(1_{𝒞} , g)). Furthermore, in the situation of (1.1.10), notice the identities

Fun(β, 𝒟) ⊙ Fun(α, 𝒟) = Fun(β ⊙ α, 𝒟)
Fun(α, 𝒟) ∗ Fun(α′, 𝒟) = Fun(α′ ∗ α, 𝒟).

(iii) In the situation of (ii), suppose that the functor f admits a left adjoint g : 𝒞 → ℬ. Then f^{∗} := Fun(f, 𝒟) is left adjoint to g^{∗} := Fun(g, 𝒟). More precisely, let η be a unit and ε a counit for the adjoint pair (g, f). From the triangular identities (1.1.13) for (g, f), and taking into account (ii), we deduce commutative diagrams :

\xymatrix{g^{∗} \ar@{=>}[r]^{Fun(η,𝒟)∗g^{∗}} & g^{∗}f^{∗}g^{∗} \ar@{=>}[d]^{g^{∗}∗Fun(ε,𝒟)} & f^{∗} \ar@{=>}[r]^{f^{∗}∗Fun(η,𝒟)} & f^{∗}g^{∗}f^{∗} \ar@{=>}[d]^{Fun(ε,𝒟)∗f^{∗}} \\  & g^{∗} &  & f^{∗}}

which, in light of proposition 1.1.15(i), says that Fun(η, 𝒟) is a unit and Fun(ε, 𝒟) a counit for the adjoint pair (f^{∗}, g^{∗}).

(iv) Let F : 𝒞 → 𝒟 be a functor, and G : 𝒟 → 𝒞 a left adjoint to F. Then F^{o} is left adjoint to G^{o}. More precisely, let η be a unit and ε a counit for the adjoint pair (G, F); then it follows easily from (i) and proposition 1.1.15(i) that ε^{o} is a unit and η^{o} is a counit for the adjoint pair (F^{o}, G^{o}) : details left to the reader.

Proposition 1.1.20. Let F : 𝒜 → ℬ be a functor.

(i) The following conditions are equivalent : (a) F is fully faithful and has a fully faithful left adjoint.

(b) There exist a functor G : ℬ → 𝒜 and isomorphisms of functors

G ∘ F →^{∼} 1_{𝒜} 1_{ℬ} →^{∼} F ∘ G.

(c) F is an equivalence. (ii) Suppose that F admits a left adjoint G : ℬ → 𝒜, and let η : 1_{ℬ} ⇒ F ∘ G and ε : G ∘ F ⇒ 1_{𝒜} be a unit and respectively a counit for the adjoint pair (G, F). Then F (resp. G) is faithful if and only if ε_{X} is an epimorphism for every X ∈ Ob(𝒜) (resp. η_{Y} is a monomorphism for every Y ∈ Ob(ℬ)).

(iii) In the situation of (ii), the following conditions are equivalent : (a) F (resp. G) is fully faithful.

(b) The counit ε (resp. the unit η) is an isomorphism of functors.

(c) There exists an isomorphism of functors ε′ : G∘F →^{∼} 1_{𝒜} (resp. η′ : 1_{ℬ} →^{∼} F ∘G). Moreover, if (c) holds, there exists a unique adjunction ϑ′ for the pair (G, F) whose counit is ε′ (resp. whose unit is η′).

(iv) Suppose that F admits both a left adjoint G : ℬ → 𝒜 and a right adjoint H : ℬ → 𝒜. Then G is fully faithful if and only if H is fully faithful.

Proof. (ii): In view of remark 1.1.19(iv), it suffices to consider the assertion relative to ε_{•}. Thus, suppose that F f_{1} = F f_{2} for two morphisms f_{1}, f_{2} : X → X′. The naturality of ε_{•} yields the identity : ε_{X′} ∘ GF f_{i} = f_{i} ∘ ε_{X} for i = 1, 2, and if ε_{X} is an epimorphism, we deduce f_{1} = f_{2}. Conversely, suppose F is faithful and f_{1} ∘ ε_{X} = f_{2} ∘ ε_{X}; from the triangular identities we get :

F f_{i} = F (ε_{X′}) ∘ F GF (f_{i}) ∘ η_{F X} = F (f_{i} ∘ ε_{X}) ∘ η_{F X}

so F f_{1} = F f_{2} and therefore f_{1} = f_{2} which shows that ε_{X} is an epimorphism.

(iii): Again, remark 1.1.19(iv) reduces to considering the assertion for ε_{•}. We check first that (iii.a)⇒(iii.b) : indeed, if F is fully faithful, for every X ∈ Ob(𝒜), there exists a morphism β_{X} : X → GF X such that F β_{X} = η_{F X} : F X → F GF X. From the triangular identities of (1.1.13) we deduce that F (ε_{X} ∘ β_{X}) = F (ε_{X}) ∘ η_{F X} = 1_{F X}, whence ε_{X} ∘ β_{X} = 1_{X}, since F is faithful. Next, let ϑ be the unique adjunction for the pair (G, F) whose unit and counit are η and ε; by inspection of the explicit description of ϑ in the proof of proposition 1.1.15(i) we get

ϑ_{GF X,F X}(β_{X} ∘ ε_{X}) = F (β_{X} ∘ ε_{X}) ∘ η_{F X} = η_{F X} ∘ F (ε_{X}) ∘ η_{F X} = η_{F X} = ϑ_{GF X,F X}(1_{GF X})

whence β_{X} ∘ ε_{X} = 1_{GF X}. Obviously (iii.b)⇒(iii.c). Lastly, suppose that (iii.c) holds; for every X, Y ∈ Ob(𝒜) we deduce a bijection

ξ_{X,Y} : Hom_{𝒜} (X, Y) ⟶^{φ_{X,Y}} Hom_{𝒜} (GF X, Y) ⟶^{ϑ_{GX,Y}} Hom_{ℬ} (F X, F Y)

where φ_{X,Y} (f) := f ∘ ε′_{X} for every morphism f : X → Y in 𝒜. It is easily seen that ξ_{X,Y} is natural in both X and Y; especially, for every f ∈ Hom_{𝒜} (X, Y) we have :

ξ_{Y,Y} (1_{Y}) ∘ F f = ξ_{X,Y} (1_{Y} ∘ f) = ξ_{X,Y} (f ∘ 1_{X}) = F f ∘ ξ_{X,X}(1_{X}).

Letting X = Y, and taking f : X → X with ξ_{X,X}(f) = 1_{F X}, we deduce that ξ_{X,X}(1_{X}) is an isomorphism in ℬ, for every X ∈ Ob(𝒜). Since ξ_{X,Y} (f) = F f ∘ ξ_{X,X}(1_{X}) for every f ∈ Hom_{𝒜} (X, Y), we conclude that the rule f ↦ F f is a bijection Hom_{𝒜} (X, Y) →^{∼} Hom_{ℬ} (F X, F Y), whence (iii.a). Lastly, let us check the existence and uniqueness of the adjunction ϑ′ whose counit is ε′. To this aim, let ω : 1_{𝒜} →^{∼} 1_{𝒜} be the automorphism such that ω ∘ ε = ε′, and set η′ := (F ∗ ω^{−1} ∗ G) ∘ η. We compute :

\begin{aligned}(F ∗ ε′) ⊙ (η′ ∗ F) & = (F ∗ ω) ⊙ (F ∗ ε) ⊙ (F ∗ ω^{−1} ∗ GF) ⊙ (η ∗ F) \\ & = (F ∗ ω) ⊙ (F ∗ ω^{−1}) ⊙ (F ∗ ε) ⊙ (η ∗ F) = 1_{F} . \end{aligned}

Likewise we check that (ε′ ∗ G) ⊙ (G ∗ η′) = 1_{G}, whence the contention, by proposition 1.1.15.

(i): The equivalence (i.a)⇔(i.b) follows immediately from (iii). Moreover, if (i.b) holds, then for every Y ∈ Ob(ℬ) we have an isomorphism Y →^{∼} F GY, so F is essentially surjective. Since we have just seen that (i.b) implies that F is fully faithful, we deduce as well that (i.b)⇒(i.c).

(i.c)⇒(i.a): We construct a left adjoint G : ℬ → 𝒜 to F as follows. Since F is essentially surjective, for every B ∈ Ob(ℬ) we may find an object A ∈ Ob(𝒜) with an isomorphism η_{B} : B →^{∼} F A, and we set GB := A. Next, since F is fully faithful, for every morphism g : B → B′ in ℬ there exists a unique morphism f : GB → GB′ in 𝒜 which makes commute the diagram

\xymatrix{B \ar[r]^{g} \ar[d]_{ω_{B}} & B′ \ar[d]^{ω_{B′}} \\ F GB \ar[r]^{F f} & F GB′}

and we let Gg := f. It is easily seen that these rules yield a well defined functor G as sought, and η is then a natural isomorphism 1_{ℬ} ⇒ F G. Moreover from the full faithfulness of F we deduce that G is fully faithful as well (details left to the reader). To conclude we remark, more generally :

Claim 1.1.21. Let F : 𝒜 → ℬ and G : ℬ → 𝒜 be two functors, η : 1_{ℬ} ⇒ F G an isomorphism of functors, and suppose that F is fully faithful. Then there exists a unique adjunction ϑ for the pair (G, F) whose unit is η.

Proof of the claim. From the proof of proposition 1.1.15 we know that η determines ϑ, by the rule : ϑ_{A,B}(f) := F f ∘ η_{B} for every A ∈ Ob(𝒜), B ∈ Ob(ℬ) and every morphism f : GB → A in 𝒜. Conversely, our assumptions easily imply that this rule does yield a natural bijection Hom_{𝒜} (GB, A) →^{∼} Hom_{ℬ} (B, F A), whence the contention (details left to the reader). ♢

(iv): Let η : 1_{ℬ} ⇒ F G (resp. η′ : 1_{𝒜} ⇒ HF) be the unit and ε : GF ⇒ 1_{𝒜} (resp. ε′ : F H ⇒ 1_{ℬ} ) the counit of a given adjunction for the adjoint pair (G, F) (resp. (F, H)). By remark 1.1.19(iv) we may assume that H is fully faithful, and we show that the same holds for G. By (iii), this is the same as assuming that ε′ is an isomorphism, and we need to check that the same holds for η. To this aim, denote γ : F G ⇒ 1_{ℬ} the composition

F G ⟶^{(F G)∗ε′^{−1}} F GF H ⟶^{F ∗ε∗H} F H ⟶^{ε′} 1_{ℬ} .

We show that γ is inverse to η. Indeed we have

γ ⊙ η = ε′ ⊙ (F ∗ ε ∗ H) ⊙ (η ∗ F H) ⊙ ε′^{−1} = ε′ ⊙ ε′^{−1} = 1_{ℬ}

where the first identity holds by the naturality of η, and the second follows from the triangular identites of (1.1.13). Likewise, we have

\begin{matrix}η ⊙ γ = & (ε′ ∗ F G) ⊙ (F H ∗ η) ⊙ (F ∗ ε ∗ H) ⊙ (F G ∗ ε′^{−1}) \\ = & (ε′ ∗ F G) ⊙ (F ∗ ε ∗ HF G) ⊙ (F GF H ∗ η) ⊙ (F G ∗ ε′^{−1}) \\ = & (ε′ ∗ F G) ⊙ (F ∗ ε ∗ HF G) ⊙ (F G ∗ ε′^{−1} ∗ F G) ⊙ (F G ∗ η) \\ = & (ε′ ∗ F G) ⊙ (F ∗ ε ∗ HF G) ⊙ (F GF ∗ η′ ∗ G) ⊙ (F G ∗ η) \\ = & (ε′ ∗ F G) ⊙ (F ∗ η′ ∗ G) ⊙ (F ∗ ε ∗ G) ⊙ (F G ∗ η) \\ = & (1_{F} ∗ G) ⊙ (F ∗ 1_{G}) = 1_{F G} \end{matrix}

where the first and third identities follow from the naturality of ε′, the second and fifth from that of ε, the fourth and sixth from the triangular identities for the pairs (η′, ε′) and (η, ε). □

Definition 1.1.22. Let F : 𝒜 → ℬ be any equivalence of categories. A quasi-inverse for F is the datum of a functor G : ℬ → 𝒜 and an adjunction for the pair (G, F). Then we also say that F is a quasi-inverse for G.

Remark 1.1.23. It follows easily from proposition 1.1.20(i,iii) and claim 1.1.21, that a quasiinverse for an equivalence F : 𝒜 → ℬ is the same as the datum of a functor G : ℬ → 𝒜 and an isomorphism of functors 1_{ℬ} →^{∼} F G, and moreover any such G is also an equivalence. Taking into account remark 1.1.19(iv), we see that a quasi-inverse for F is also the same as the datum of such a G and an isomorphism of functors GF →^{∼} 1_{𝒜} .

1.1.24. Slice categories. A standard construction attaches to any X ∈ Ob(𝒞) a category :

𝒞 /X

as follows. The objects of 𝒞 /X are all the pairs (A, f) where A ∈ Ob(𝒞) and f : A → X is any morphism of 𝒞. For any two such objects (A, f), (B, g), the set Hom_{𝒞 /X} ((A, f), (B, g)) consists of all the commutative diagrams of morphisms of 𝒞 :

\xymatrix{A \ar[rr]^{h} \ar[dr]_{f} &  & B \ar[dl]^{g} \\  & X}

with composition of morphisms induced by the composition law of 𝒞. We denote sometimes such a morphism of 𝒞 /X by

h/X : (A, f) → (B, g).

An object (resp. a morphism) of 𝒞 /X is also called an X-object (resp. an X-morphism) of 𝒞. Dually, one defines

X/𝒞 := (𝒞^{o}/X^{o})^{o}

i.e. the objects of X/𝒞 are the pairs (A, f) with A ∈ Ob(𝒞) and f : X → A any morphism of 𝒞. We have an obvious faithful source functor

s_{X} : 𝒞 /X → 𝒞  (A, f) ↦ A  ((A, f) ⟶^{h/X} (B, g)) ↦ (A ⟶^{h} B)

and likewise one obtains a target functor

t_{X} := s_{X^{o}}^{o} : X/𝒞 → 𝒞 .

Moreover, any morphism f : X → Y in 𝒞 induces functors :

(1.1.25)  \begin{matrix}f_{∗} : 𝒞 /X → 𝒞 /Y : & (A, g : A → X) ↦ (A, f_{∗}g := f ∘ g : A → Y) \\ f^{∗} : Y/𝒞 → X/𝒞 : & (B, h : Y → B) ↦ (B, f^{∗}h := h ∘ f : X → B). \end{matrix}

Furthermore, given a functor F : 𝒞 → ℬ, any X ∈ Ob(𝒞) induces functors :

(1.1.26)  \begin{matrix}F_{|X} : 𝒞/X → ℬ/F X : & (A, g) ↦ (F A, F g) \\ _{X|}F : X/𝒞 → F X/ℬ : & (B, h) ↦ (F B, F h). \end{matrix}

1.1.27. The categories 𝒞 /X and X/𝒞 are special cases of the following more general construction. Let F : 𝒜 → ℬ be any functor. For any B ∈ Ob(ℬ), we define

F 𝒜 /B

as the category whose objects are all the pairs (A, f), where A ∈ Ob(𝒜) and f : F A → B is a morphism in ℬ. The morphisms g : (A, f) → (A′, f′) are the morphisms g : A → A′ in 𝒜 such that f′ ∘ F g = f. There are well-defined functors :

F/B : F 𝒜 /B → ℬ/B : (A, f) ↦ (F A, f) and s_{B} : F 𝒜 /B → 𝒜 : (A, f) ↦ A.

Dually, we define :

B/F 𝒜 := (F^{o}𝒜^{o}/B^{o})^{o}

and likewise one has natural functors :

B/F : B/F 𝒜 → B/ℬ and t_{B} : B/F 𝒜 → 𝒜 .

Any morphism g : B′ → B induces functors :

\begin{matrix}g/F 𝒜 : B/F 𝒜 → B′/F 𝒜 : & (A, f) ↦ (A, f ∘ g) \\ F 𝒜 /g : F 𝒜 /B′ → F 𝒜 /B : & (A, f) ↦ (A, g ∘ f). \end{matrix}

Obviously, the category 𝒞/X (resp. X/𝒞) is the same as 1_{𝒞} 𝒞 /X (resp. X/1_{𝒞} 𝒞).

1.1.28. In the situation of (1.1.27), the categories of the form F 𝒜 /B can be faithfully embedded in a single category F 𝒜 /ℬ. The latter is the category whose objects are all the triples (A, B, f), where A ∈ Ob(𝒜), B ∈ Ob(ℬ) are any two objects, and f : F A → B is any morphism of ℬ. If f : F A → B and f′ : F A′ → B′ are any two objects, the set

Hom_{F 𝒜 /ℬ} ((A, B, f), (A′, B′, f′))

consists of all pairs (g, g′) where g is a morphism in 𝒜 and g′ a morphism in ℬ, that make commute the diagram :

(1.1.29)  \xymatrix{F A \ar[r]^{f} \ar[d]_{F g} & B \ar[d]^{g′} \\ F A′ \ar[r]^{f′} & B′}

with composition of morphisms induced by the composition laws of 𝒜 and ℬ, in the obvious way. There are two natural source and target functors :

𝒜 ⟵^{s} F 𝒜 /ℬ ⟶^{t} ℬ

such that s(F A → B) := A, t(F A → B) := B for any object F A → B of F 𝒜 /ℬ, and s(g, g′) = g, t(g, g′) = g′. Dually, we let

ℬ/F 𝒜 := (F^{o}𝒜^{o}/ℬ^{o})^{o}

and the corresponding source and target functors are switched :

𝒜 ⟵^{t} ℬ/F 𝒜 ⟶^{s} ℬ.

1.1.30. For the special case of the identity endofunctor 1_{𝒞} of any category 𝒞, we obtain the category of arrows of 𝒞

Morph(𝒞) := 1_{𝒞} 𝒞 /𝒞 .

So, the set of objects of Morph(𝒞) is Morph(𝒞) (notation of (1.1)) and the morphisms are the commutative square diagrams in 𝒞. The functor s_{X} of (1.1.24) is the restriction of s : Morph(𝒞) → 𝒞 to the subcategory 𝒞/X, and likewise for t_{X}. Likewise, in the situation of (1.1.28), the target functor on F 𝒜 /ℬ and the source functor on ℬ/F 𝒜 factor through functors

F 𝒜 /ℬ ⟶^{T} Morph(ℬ) ⟵^{S} ℬ/F 𝒜

where T(A, B, f) := (F A, B, f) for every object (A, B, f) of F 𝒜 /ℬ, and T assigns to any morphism (g, g′) : (A, B, f) → (A′, B′, f′) the commutative square (1.1.29), regarded as a morphism (F A, B, f) → (F A′, B′, f′) in Morph(ℬ). Likewise one describes the functor S.

Notice also the natural transformation

[figure: Morph(𝒞); s; t; m; 𝒞]

where m(A, B, f) := f for every (A, B, f) ∈ Morph(𝒞). Furthermore, every functor F : ℬ → 𝒞 induces a functor

Morph(F) : Morph(ℬ) → Morph(𝒞)

which maps (A, B, f) ∈ Morph(ℬ) to (F A, F B, F f) ∈ Morph(𝒞) and which sends every commutative square diagram D in ℬ to the commutative square diagram F D in 𝒞.

Notice that a natural transformation α as in (1.1.8) is equivalent to the datum of a functor

α̃ : 𝒜 → Morph(ℬ)  such that s ∘ α̃ = F  and t ∘ α̃ = G.

Namely, one defines α̃ by the rule : A ↦ (F A, GA, α_{A}) for every A ∈ Ob(𝒜), and for every morphism f : A → B in 𝒜, one lets α̃(f) be the commutative square diagram (1.1.9).

1.1.31. Let 𝒜 , ℬ be two categories, F : 𝒜 → ℬ a functor, G : ℬ → 𝒜 a left adjoint for F, and ϑ an adjunction for the pair (G, F). Then for every A ∈ Ob(𝒜) the functor F_{|A} : 𝒜 /A → ℬ/F A of (1.1.26) admits a left adjoint that we denote

G_{|A} : ℬ/F A → 𝒜 /A.

Namely, to every (f : B → F A) ∈ Ob(ℬ/F A) we assign the object (ϑ_{AB}^{−1}(f) : GB → A) ∈ Ob(𝒜 /A), and to every morphism h/F A : (f : B → F A) → (f′ : B′ → F A) in ℬ/F A we assign the morphism Gh/A : ϑ_{AB}^{−1}(f) → ϑ_{AB′}^{−1}(f′) in 𝒜 /A. Indeed, ϑ induces an adjunction ϑ_{|A} for the pair (G_{|A}, F_{|A}) : to every (f : B → F A) ∈ Ob(ℬ/F A) and (g : A′ → A) ∈ Ob(𝒜 /A) we assign the bijection

(ϑ_{|A})_{g,f} : Hom_{𝒜 /A} (ϑ_{AB}^{−1}(f), g) →^{∼} Hom_{ℬ/F A} (f, F g) h/A ↦ ϑ_{GB,A′}(h)/F A.

Dually, since F^{o} is left adjoint to G^{o} (remark 1.1.19(iv)), we see that for every B ∈ Ob(ℬ) the functor_{B|}G : B/ℬ → GB/𝒜 of (1.1.26) admits the right adjoint

B|F : GB/𝒜 → B/ℬ  (GB ⟶^{f} A) ↦ (B ⟶^{ϑ_{AB}(f)} F A)  GB/h ↦ B/F h.

The detailed verifications shall be left to the reader. See example 1.2.28for a related result.

1.1.32. In the situation of (1.1.31) we get furthermore for every A ∈ Ob(𝒜) an isomorphism of categories :

ϑ^{A} : Gℬ/A →^{∼} ℬ/F A (f : GB → A) ↦ (ϑ_{AB}(f) : B → F A)

that assigns to every morphism g : (f : GB → A) → (f′ : GB′ → A) of Gℬ/A the morphism g : (ϑ_{AB}(f) : B → F A) → (ϑ_{AB′}(f′) : B′ → F A) of ℬ/F A. Also, for every morphism h : A → A′ of 𝒜, we get a commutative diagram of categories :

\xymatrix{Gℬ/A \ar[r]^{ϑ^{A}} \ar[d]_{Gℬ/h} & ℬ/F A \ar[d]^{(F h)_{∗}} \\ Gℬ/A′ \ar[r]^{ϑ^{A′}} & ℬ/F A′.}

Clearly, the isomorphisms ϑ^{A} are restrictions of a single isomorphism of categories :

Gℬ/𝒜 →^{∼} ℬ/F 𝒜 .

The detailed verifications shall be again left to the reader.

1.2. Presheaves and limits. A very important construction associated with every category 𝒞 is the category

𝒞_{U}^{∧} := Fun(𝒞^{o}, U-Set)

whose objects are called the U-presheaves on 𝒞 (notation of (1.1.10)). We usually drop the subscript U, unless we have to deal with more than one universe. If U′ is another universe with U ⊂ U′, the natural inclusion of categories :

(1.2.1)  𝒞_{U}^{∧} → 𝒞_{U′}^{∧}

is fully faithful (verification left to the reader). For every functor F : ℬ → 𝒞 and every natural transformation α : F ⇒ G we set

(1.2.2)  F_{U}^{∧} := Fun(F^{o}, Set) : 𝒞_{U}^{∧} → ℬ_{U}^{∧}  α_{U}^{∧} := Fun(α^{o}, Set) : G_{U}^{∧} ⇒ F_{U}^{∧}

(notation of remark 1.1.19(i,ii)). Again, we shall drop the subscript and write simply F^{∧} and α^{∧}, unless the omission of U may be a source of ambiguities. Clearly, for every pair of functors F_{1} : 𝒜 → ℬ and F_{2} : ℬ → 𝒞 we have

(F_{2} ∘ F_{1})^{∧} = F_{1}^{∧} ∘ F_{2}^{∧}.

Moreover, in the situation of (1.1.10), remark 1.1.19(i,iii) yields the identities

(1.2.3)  (β ⊙ α)^{∧} = α^{∧} ⊙ β^{∧}  and (α′ ∗ α)^{∧} = α^{∧} ∗ α′^{∧}.

1.2.4. If 𝒞 has small Hom-sets (see (1.1.1)), there is a natural functor

h_{𝒞} : 𝒞 → 𝒞^{∧}

called the Yoneda embedding, which assigns to every X ∈ Ob(𝒞) the functor

h_{X} : 𝒞^{o} → Set Y ↦ Hom_{𝒞} (Y, X) for every Y ∈ Ob(𝒞)

and to any morphism f : X → X′ in 𝒞, the natural transformation h_{f} : h_{X} ⇒ h_{X′} such that

h_{f,Y} (g) := f ∘ g for every Y ∈ Ob(𝒞) and every g ∈ Hom_{𝒞} (Y, X).

Definition 1.2.5. Let 𝒞 be a category, and V any universe such that 𝒞 has V-small Hom-sets. We say that an object F of 𝒞_{V}^{∧} is representable in 𝒞, if there exists an isomorphism

h_{X} →^{∼} F in 𝒞_{V}^{∧}

for some X ∈ Ob(𝒞), in which case we also say that X represents F. From the full faithfulness of (1.2.1), it follows that the representability of a presheaf is independent of the universe V.

Proposition 1.2.6 (Yoneda’s lemma). With the notation of (1.2.4), we have :

(i) The functor h_{𝒞} is fully faithful.

(ii) Moreover, for every F ∈ Ob(𝒞^{∧}) and every X ∈ Ob(𝒞) there is a natural bijection

F (X) →^{∼} Hom_{𝒞^{∧}}(h_{X}, F)

functorial in both X and F.

Proof. Clearly it suffices to check (ii). However, the sought bijection is obtained explicitly as follows. To a given a ∈ F (X), we assign the natural transformation τ_{a} : h_{X} ⇒ F such that

τ_{a,Y} (f) := F f(a) for every Y ∈ Ob(𝒞) and every f ∈ h_{X}(Y).

Conversely, to a given natural transformation τ : h_{X} ⇒ F we assign τ_{X}(1_{X}) ∈ F (X). It is easily seen that these rules establish mutually inverse bijections. The functoriality in F is immediate, and the functoriality in the argument X amounts to the commutativity of the diagram

\xymatrix{Hom_{𝒞^{∧}}(h_{X′}, F) \ar[r]^{∼} \ar[d]_{Hom_{𝒞^{∧}} (h_{φ},F)} & F (X′) \ar[d]^{F (φ)} \\ Hom_{𝒞^{∧}}(h_{X}, F) \ar[r]^{∼} & F (X)}

for every morphism φ : X → X′ in 𝒞 : the verification shall also be left to the reader. □

Example 1.2.7. (i) Every representable presheaf on the category Set admits a natural representing object. Indeed, let 𝟙 denote any set with one element (a canonical choice for 𝟙 is the set {∅}); then for every representable F ∈ Ob(Set^{∧}) we get a natural isomorphism

F →^{∼} h_{F (𝟙)}

as follows. For any set S, we have a natural bijection S →^{∼} Hom_{Set}(𝟙, S) that assigns to every s ∈ S the unique map 𝟙_{s} : 𝟙 → S whose image is {s}. We deduce a map

S × F (S) → F (𝟙) (s, a) ↦ F (𝟙_{s})(a)

which is the same as a map F (S) → Hom_{Set}(S, F (𝟙)) that realizes the sought isomorphism.

(ii) We may apply Yoneda’s lemma to prove the uniqueness of the (left or right) adjoint for a given functor. Indeed, let G : ℬ → 𝒜 be a functor between any two categories, suppose that F and F′ are both right adjoint to G, and fix adjunctions ϑ_{••} and ϑ′_{••} for F and respectively F′. Choose also a universe U such that both 𝒜 and ℬ are U-small. Then ϑ and ϑ′ can be regarded as isomorphisms from the functor G^{∧} ∘ h_{𝒜} to h_{ℬ} ∘ F and respectively to h_{ℬ} ∘ F′. In this situation, proposition 1.2.6(i) implies that there exists a unique isomorphism ω : F →^{∼} F′ which makes commute the diagram

\xymatrix{ & G^{∧} ∘ h_{𝒜} \ar[dl]_{ϑ} \ar[dr]^{ϑ′} \\ h_{ℬ} ∘ F \ar[rr]^{h_{ℬ} ∗ω} &  & h_{ℬ} ∘ F′.}

Taking into account remark 1.1.19(iv), a dual argument yields a corresponding uniqueness assertion for left adjoints.

Remark 1.2.8. (i) Let 𝒞 be any category, and F_{1}, F_{2} two representable presheaves on 𝒞, and pick X_{1}, X_{2} ∈ Ob(𝒞) with isomorphisms ω_{i} : h_{X_{i}} →^{∼} F_{i} (i = 1, 2). It follows from proposition 1.2.6(i) that for every morphism f : F_{1} → F_{2} in 𝒞^{∧} there exists a unique morphism φ : X_{1} → X_{2} in 𝒞 that makes commute the diagram

\xymatrix{h_{X_{1}} \ar[r]^{h_{φ}} \ar[d] & h_{X_{2}} \ar[d] \\ F_{1} \ar[r]^{f} & F_{2}.}

In this situation, we say that φ represents the morphism f.

(ii) Likewise, suppose that G : ℬ → 𝒞^{∧} is any functor. We say that G is representable if there exists a functor γ : ℬ → 𝒞 with an isomorphism G →^{∼} h_{𝒞} ∘ γ. It follows easily from (i) that G is representable if and only if G(B) is representable in 𝒞 for every B ∈ Ob(ℬ). Moreover, any two representatives for G are isomorphic in Fun(ℬ, 𝒞). Furthermore, if G, G′ : ℬ → 𝒞 are any two representable functors, and γ, γ′ : ℬ → 𝒞 two corresponding representing functors, then any natural transformation t : G ⇒ G′ is represented uniquely by a natural transformation τ : γ ⇒ γ′, by which we mean that t = h_{𝒞} ∗ τ.

1.2.9. Limits and colimits. We wish to explain some standard constructions of presheaves that are used pervasively throughout this work. Namely, let I be a small category, 𝒞 a category and X any object of 𝒞. We denote by c_{X} : I → 𝒞 the constant functor associated with X :

c_{X}(i) := X for every i ∈ Ob(I) c_{X}(φ) := 1_{X} for every φ ∈ Morph(I).

Any morphism f : X′ → X induces a natural transformation

c_{f} : c_{X′} ⇒ c_{X} by the rule : (c_{f})_{i} := f for every i ∈ I.

If F : I → 𝒞 is any functor, a cone of vertex X and basis F is any natural transformation c_{X} ⇒ F. Dually, a cocone with vertex X and basis F is a natural transformation F ⇒ c_{X}.

Definition 1.2.10. With the notation of (1.2.9), let V be any universe with U ⊂ V, such that 𝒞 has V-small Hom-sets, and let F : I → 𝒞 be any functor.

(i) The limit of F is the V-presheaf on 𝒞 denoted

lim_{I} F : 𝒞^{o} → V-Set

that assigns to every X ∈ Ob(𝒞), the set lim_{I} F (X) of all cones c_{X} ⇒ F. Any morphism f : X′ → X of 𝒞 induces the map

lim_{I} F (f) : lim_{I} F (X) → lim_{I} F (X′) τ ↦ τ ⊙ c_{f} for every τ : c_{X} ⇒ F.

Then we also say that I is the indexing category for the limit of F. (ii) Dually, the colimit of F is the V-presheaf on 𝒞^{o}

colim_{I} F := lim_{I^{o}} F^{o}.

Remark 1.2.11. Let 𝒞 , 𝒞′ be any two categories with V-small Hom-sets for some universe V such that U ⊂ V, and I, I′ two small categories.

(i) Any diagram of functors

I′ ⟶^{φ} I ⟶^{F} 𝒞 ⟶^{H} 𝒞′

induces a natural morphism

lim_{φ} H : lim_{I} F → H^{∧}(lim_{I′} H ∘ F ∘ φ) in 𝒞_{V}^{∧}

(notation of (1.2.2)) by ruling that

lim_{φ} H(X)(τ) := H ∗ τ ∗ φ for every X ∈ Ob(𝒞) and every cone τ : c_{X} ⇒ F

(notation of (1.1.10)). If ψ : I′′ → I′ and K : 𝒞′ → 𝒞′′ are any two other functors, with I′′ also small and 𝒞′′ with V-small Hom-sets, the resulting diagram in 𝒞_{V}^{∧} commutes :

(1.2.12)  \xymatrix{lim_{I} F \ar[d]_{lim_{φ} H} \ar[r]^{lim_{φ∘ψ} K∘H} & H^{∧} ∘ K^{∧}(lim_{I′′} K ∘ H ∘ F ∘ φ ∘ ψ) \\ H^{∧}(lim_{I′} H ∘ F ∘ φ). \ar@{-->}[ur]_{H^{∧}(lim_{ψ} K^{∧})}}

The reader may spell out the corresponding assertions for colimits.

(ii) Any diagram of functors and natural transformations

[figure: I′; φ_{1}; φ_{2}; α; I; F_{1}; F_{2}; g; 𝒞]

induces commutative diagrams

\xymatrix{lim_{I} F_{1} \ar[r]^{lim_{I} g} \ar[d]_{lim_{φ_{1}}1_{𝒞}} & lim_{I} F_{2} \ar[d]_{lim_{φ_{2}} 1_{𝒞}} & colim_{I} F_{2} \ar[r]^{colim_{I} g} \ar[d]^{colim_{φ_{2}} 1_{𝒞}} & colim_{I} F_{1} \ar[d]^{colim_{φ_{1}}1_{𝒞}} \\ lim_{I′}(F_{1}∘φ_{1}) \ar[r]^{lim_{I′}(g∗α)} & lim_{I′}(F_{2}∘φ_{2}) & colim_{I′}(F_{2}∘φ_{2}) \ar[r]^{colim_{I′}(g∗α)} & colim_{I′}(F_{1}∘φ_{1})}

where the top (resp. bottom) arrow of the left diagram is given by the rule :

τ ↦ g ⊙ τ ( resp. τ ↦ (g ∗ α) ⊙ τ )

for every X ∈ Ob(𝒞) and every cone τ : c_{X} ⇒ F_{1} (resp. τ : c_{X} ⇒ F_{1} ∘ φ_{1}). The top (resp. bottom) arrow of the right diagram is given by the rule

τ ↦ τ ⊙ g ( resp. τ ↦ τ ⊙ (g ∗ α) )

for every X ∈ Ob(𝒞) and every cocone τ : F_{2} ⇒ c_{X} (resp. τ : F_{2} ∘ φ_{1} ⇒ c_{X}).

(iii) Suppose that lim_{I} F is representable by L ∈ Ob(𝒞), so we have an isomorphism

(1.2.13)  ω : h_{L} →^{∼} lim_{I} F  in 𝒞_{V}^{∧} .

Notice that the cone τ := ω_{L}(1_{L}) determines ω: indeed the latter assigns, to every X ∈ Ob(𝒞), the bijection :

ω_{X} : Hom_{𝒞} (X, L) →^{∼} lim_{I} F (X) f ↦ τ ⊙ c_{f}.

Conversely, we say that a given τ : c_{L} ⇒ F is a universal cone, if the induced map ω_{X} is a bijection for every X ∈ Ob(𝒞), in which case the limit of F is representable by L. Clearly, the universal property for a cone is independent of the choice of auxiliary universe V.

Likewise, if colim_{I} F is representable by C^{o} ∈ Ob(𝒞^{o}), the choice of an isomorphism

(1.2.14)  h_{C^{o}} →^{∼} colim_{I} F  in (𝒞^{o})_{V}^{∧}

induces a universal cocone μ : F ⇒ c_{C}, which in turns determines (1.2.14) by the rule :

Hom_{𝒞} (C, X) →^{∼} colim_{I} F (X^{o}) f ↦ c_{f} ⊙ μ.

Moreover, with the notation of (ii), suppose that the limits (resp. colimits) of F_{1} and F_{2} are representable by objects L_{1} and L_{2} of 𝒞 (resp. C_{1}^{o} and C_{2}^{o} of 𝒞^{o}), and let us fix isomorphisms as (1.2.13) and (1.2.14) for F_{1} and F_{2}. Then :

• The limit of g is represented by a morphism L_{1} → L_{2} in 𝒞.

• The colimit of g is represented by a morphism C_{2}^{o} → C_{1}^{o} in 𝒞^{o}, i.e. by a morphism C_{1} → C_{2} in 𝒞 (see remark 1.2.8(ii)).

(iv) Every i ∈ I induces a morphism of presheaves

t_{i} : lim_{I} F → h_{F i}

that assigns to every X ∈ Ob(𝒞) and every cone τ : c_{X} ⇒ F the morphism τ_{i} : X → F i, regarded as an element of h_{F i}(X). We obtain in this way a natural cone

(1.2.15)  c_{lim_{I} F} ⇒ h_{𝒞} ∘ F  i ↦ t_{i}

where h_{𝒞} : 𝒞 → 𝒞_{V}^{∧} is the Yoneda embedding. We call (1.2.15) the tautological cone associated with F. We claim that the tautological cone is universal. Indeed, say that G is any presheaf on 𝒞, and τ : c_{G} ⇒ h_{𝒞} ∘ F any cone. We define a morphism f_{τ} : G → lim_{I} F as follows. To every i ∈ Ob(I), the cone τ attaches a morphism of presheaves τ_{i} : G → h_{F i}, which is the datum, for every X ∈ Ob(𝒞) of a map τ_{i}^{X} : G(X) → Hom_{𝒞} (X, F i), and for fixed s ∈ G(X), the system

τ_{•}^{X}(s) := (τ_{i}^{X}(s) : X → F i | X ∈ Ob(𝒞))

is a cone c_{X} ⇒ F. Then we let f_{τ}(X) : G(X) → lim_{I} F (X) be the map such that s ↦ τ_{•}^{X}(s) for every X ∈ Ob(𝒞) and every s ∈ G(X). The rule τ ↦ f_{τ} yields an inverse for the map

ω_{G} : Hom_{𝒞^{∧}}(G, lim_{I} F) → lim_{I} h_{𝒞} ∘ F

associated, as in (iii), with the cone (1.2.15), whence the claim : details left to the reader. Dually, with F we may also associate a tautological cocone

h_{𝒞^{o}} ∘ F^{o} ⇒ h_{colim_{I} F} in (𝒞^{o})_{V}^{∧}

which is a universal cocone.

Example 1.2.16. Let V be a universe containing U, and 𝒞 any category with V-small Hom-sets.

(i) For i = 1, 2, let f_{i} : A → B_{i} be two morphisms in 𝒞; the push-out or amalgamated sum of f_{1} and f_{2} is the colimit of the functor F : I → 𝒞, defined as follows. The set Ob(I) consists of three objects s, t_{1}, t_{2} and Morph(I) consists of two morphisms

t_{1} ⟵^{φ_{1}} s ⟶^{φ_{2}} t_{2}

(in addition to the identity morphisms of the objects of I); the functor is given by the rule : F s := A, F t_{i} := B_{i} and F φ_{i} := f_{i} (for i = 1, 2). For any C ∈ Ob(𝒞), a cocone F ⇒ c_{C} amounts to a pair of morphisms f′_{i} : B_{i} → C (i = 1, 2) such that f′_{1} ∘ f_{1} = f′_{2} ∘ f_{2}. If the cocone is universal (and thus, C represents the push-out of f_{1} and f_{2}), we say that the resulting commutative diagram :

\xymatrix{A \ar[r]^{f_{1}} \ar[d]_{f_{2}} & B_{1} \ar[d]^{f′_{1}} \\ B_{2} \ar[r]^{f′_{2}} & C}

is cocartesian. Dually one defines the fibre product or pull-back of two morphisms g_{i} : A_{i} → B (i = 1, 2). If D ∈ Ob(𝒞) represents this fibre product, a cone with vertex D is given by a pair of morphisms g′_{i} : D → A_{i} (i = 1, 2) such that g_{1} ∘ g′_{1} = g_{2} ∘ g′_{2}, and we say that the diagram :

\xymatrix{D \ar[r]^{g′_{1}} \ar[d]_{g′_{2}} & A_{1} \ar[d]^{g_{1}} \\ A_{2} \ar[r]^{g_{2}} & B}

is cartesian if this cone is universal. The push-out of f_{1} and f_{2} is a V-presheaf, and is usually just called the amalgamated sum of B_{1} and B_{2} over A, denoted

B_{1} ⨿_{(f_{1},f_{2})} B_{2} or simply B_{1} ⨿_{A} B_{2}

unless the notation gives rise to ambiguities. Likewise one writes

A_{1} ×_{(g_{1},g_{2})} A_{2} or just A_{1} ×_{B} A_{2}

for the fibre product of g_{1} and g_{2}, which is a V-presheaf as well.

(ii) Similarly, consider the category I′ with Ob(I′) = {s, t}, and with Morph(I′) consisting of two morphisms

\xymatrix{s \ar[r]^{ψ_{1}} \ar[r]_{ψ_{2}} & t}

(in addition to 1_{s} and 1_{t}). If A, B ∈ Ob(𝒞) are any two objects and f_{1}, f_{2} : A → B any two morphisms, we get a functor F′ : I′ → 𝒞 by the rule : s ↦ A, t ↦ B and ψ_{i} ↦ f_{i} for i = 1, 2. Then, the colimit of F′ is also called the coequalizer of f_{1} and f_{2}, and is sometimes denoted

Coequal(f_{1}, f_{2}).

Dually, the limit of F′ is also called the equalizer of f_{1} and f_{2}, sometimes denoted

Equal(g_{1}, g_{2}).

(iii) Let I ∈ U be any small set, and \underline{B} := (B_{i} | i ∈ I) any family of objects of 𝒞. We may regard I as a discrete category (see example 1.1.6(ii)), and then the rule i ↦ B_{i} yields a functor I → 𝒞, whose limit (resp. colimit) is called the product (resp. coproduct) of the family \underline{B}, and is denoted

∏_{i∈I} B_{i} (resp. ∐_{i∈I} B_{i} ).

If I = {1, 2} is a set with exactly two elements, we also write B_{1} × B_{2} (resp. B_{1} ⨿ B_{2}) for this limit (resp. colimit), and we call it sometimes a binary product (resp. a binary coproduct).

(iv) Let ℬ be any category, and suppose that a given X ∈ Ob(ℬ) represents the empty product in ℬ, i.e. the product of an empty family of objects of ℬ. This means precisely that Hom_{ℬ} (Y, X) consists of exactly one element, for every Y ∈ Ob(ℬ). Any such X is called a final object of ℬ.

Dually, we say that X is an initial object of ℬ, if Hom_{ℬ} (X, Y) consists of a single element, for every Y ∈ Ob(ℬ). Then an object of ℬ is initial if and only if it represents the empty coproduct in ℬ.

Moreover, we shall say that an object X of ℬ is disconnected, if there exist A, B ∈ Ob(ℬ), neither of which is an initial object of ℬ, and such that X represents the coproduct A ⨿ B (which, again, is well defined in any sufficiently large universe V). We say that X is connected, if X is not disconnected and is not an initial object of ℬ.

(v) For instance, the initial object of Set is the empty set, and any set 𝟙 with one element is a final object of Set. More generally, if ℬ is any category, the initial object of ℬ^{∧} is the presheaf ∅_{ℬ} such that ∅_{ℬ} (X) = ∅ for every X ∈ Ob(ℬ), and the presheaf 𝟙_{ℬ} such that 𝟙_{ℬ} (X) = 𝟙 for every X ∈ Ob(ℬ) is a final object.

1.2.17. Let 𝒞 be a category, f : A → B a morphism in 𝒞; as explained in example 1.2.16(i), the pair of morphisms B ⟵^{1_{B}} B ⟶^{1_{B}} B can be regarded as an element τ ∈ (B ⨿_{(f,f)} B)(B), which, by Yoneda’s lemma (proposition 1.2.6(ii)) corresponds to a morphism

π_{f}^{o} : h_{B} → B ⨿_{A} B in (𝒞^{o})^{∧} .

If B ⨿_{A} B is representable in 𝒞, this is the same as a morphism

π_{f} : B ⨿_{A} B → B in 𝒞

called the codiagonal of f. Dually, we have a natural morphism :

ι_{f} : h_{A} → A ×_{B} A in 𝒞^{∧}

which – in case A ×_{B} A is representable in 𝒞 – is the same as a diagonal morphism of f

ι_{f} : A → A ×_{B} A in 𝒞 .

Proposition 1.2.18. (i) With the notation of (1.2.17), the following conditions are equivalent : (a) f : A → B is a monomorphism in 𝒞

(b) ι_{f} : h_{A} → A ×_{B} A is an isomorphism in 𝒞^{∧}.

In case A ×_{B} A is representable in 𝒞, denote by A ⟵^{p_{1}} A ×_{B} A ⟶^{p_{2}} A the universal cone of this fibre product. Then (a) and (b) are moreover equivalent to :

(c) ι_{f} : A → A ×_{B} A is an isomorphism in 𝒞.

(d) ι_{f} : A → A ×_{B} A is an epimorphism in 𝒞.

(e) p_{1} = p_{2}. (ii) Dually, the following conditions are equivalent :

(a) f : A → B is an epimorphism in 𝒞.

(b) π_{f}^{o} : h_{B} → B ⨿_{A} B is an isomorphism in (𝒞^{o})^{∧}.

In case B ⨿_{A} B is representable in 𝒞, denote by B ⟶^{e_{1}} B ⨿_{A} B ⟵^{e_{2}} B the universal cocone of this amalgamated sum. Then (a) and (b) are moreover equivalent to :

(c) π_{f} : B ⨿_{A} B → B is an isomorphism in 𝒞.

(d) π_{f} : B ⨿_{A} B → B is a monomorphism in 𝒞.
(e) e_{1} = e_{2}.

Proof. (i.a)⇔(i.b): For every X ∈ Ob(𝒞) the map ι_{f,X} : h_{A}(X) → A ×_{B} A(X) is given by the rule : (X ⟶^{g} A) ↦ (A ⟵^{g} X ⟶^{g} A). Hence, ι_{f} is an isomorphism if and only if for every X ∈ Ob(𝒞) and every pair of morphisms g_{1}, g_{2} : X → A such that f ∘ g_{1} = f ∘ g_{2}, we have g_{1} = g_{2}. This means precisely that f is a monomorphism.

(i.b)⇔(i.c) is clear, since the Yoneda imbedding h_{𝒞} is fully faithful.

(i.c)⇒(i.d) is trivial.

(i.d)⇒(i.e): By definition, p_{1} ∘ ι_{f} = 1_{A} = p_{2} ∘ ι_{f}. If ι_{f} is an epimorphism, then p_{1} = p_{2}. (i.e)⇒(i.c): Set p := p_{1}; since p ∘ ι_{f} = 1_{A}, it suffices to show that ι_{f} ∘ p = 1_{A×_{B}A}. Due

to the universality of the cone A ⟵^{p_{1}} A ×_{B} A ⟶^{p_{2}} A, we are then reduced to checking that p_{1} ∘ ι_{f} ∘ p = p_{1} and p_{2} ∘ ι_{f} ∘ p = p_{2}, which is clear, since by assumption p = p_{2}.

Assertion (ii) follows from (i) by duality. □

Some special kinds of limits occur frequently in applications; we gather some of them in the following :

Definition 1.2.19. Let I be a category.

(i) We say that I is finite if both Ob(I) and Morph(I) are finite sets.

(ii) We say that I is connected, if Ob(I) ≠ ∅ and every two objects i, j can be connected by a finite sequence of morphisms in I :

(1.2.20)  i → k_{1} ← k_{2} → ⋯ ← k_{n} → j.

(iii) We say that I is directed, if for every i, j ∈ Ob(I) there exist k ∈ Ob(𝒞) and morphisms i → k ← j in I. We say that I is codirected, if I^{o} is directed.

(iv) We say that I is locally directed if, for every i ∈ Ob(I), the category i/I is directed. We say that I is locally codirected, if I^{o} is locally directed.

(v) We say that I is pseudo-filtered if it is locally directed, and the following coequalizing condition holds. For any i, j ∈ Ob(I), and any two morphisms f, g : i → j, there exist k ∈ Ob(I) and a morphism h : j → k such that h ∘ f = h ∘ g.

(vi) We say that I is filtered, if it is pseudo-filtered and connected. We say that I is cofiltered if I^{o} is filtered.

(vii) Let F : I → 𝒞 be a functor to any category 𝒞. Then lim_{I} F is a well defined Vpresheaf, for any sufficiently large universe V, and we say that the limit of F is small (resp. connected, resp. codirected, resp. locally codirected, resp. cofiltered, resp. finite), if I is small (resp. connected, resp. codirected, resp. locally codirected, resp. cofiltered, resp. finite). Dually, one defines small, connected, directed, locally directed, filtered and finite colimits.

(viii) We say that a category 𝒞 is complete (resp. finitely complete) if every small limit (resp. finite limit) of 𝒞 is representable in 𝒞. We say that 𝒞 is cocomplete (resp. finitely cocomplete) if every small colimit (resp. finite colimit) of 𝒞 is representable in 𝒞^{o}.

Remark 1.2.21. Let I be any category.

(i) If I is connected and locally directed, then I is directed. Indeed, let i, j be any two objects of I; we have to show that there exist k ∈ Ob(I) and morphisms a : i → k and b : j → k. However, by assumption there exists a sequence (1.2.20), for some n ∈ ℕ and objects k_{1}, …, k_{n} of I. Then, a simple induction reduces the assertion to the case where n = 2. But since I is locally directed, we may then find k ∈ Ob(I) and morphisms c : k_{1} → k, b : j → k, so it suffices to take a := c ∘ d, where d : i → k_{1} is the morphism appearing in (1.2.20). Dually, if I is connected and locally codirected, then I is codirected.

It follows that a category I is filtered if and only if it is directed, satisfies the coequalizing condition of definition 1.2.19(v), and its set of objects is non-empty.

(ii) Let I be a small category. It is easily seen that I is connected if and only if it is a connected object of the category Cat (see example 1.2.16(iv)). In general, there is a natural decomposition in Cat :

I →^{∼} ∐_{s∈S} I_{s}

where each I_{s} is a connected category, and S is a small set. We call {I_{s} | s ∈ S} the set of connected components of I, and we denote it

π_{0}(I).

Especially, every small pseudo-filtered category is a coproduct of a (small) family of filtered categories. This decomposition induces natural isomorphisms in 𝒜^{∧} :

colim_{I} F →^{∼} ∐_{s∈S} colim_{I_{s}} F ∘ e_{s}  lim_{I} F →^{∼} ∏_{s∈S} lim_{s∈S} F ∘ e_{s}

for any functor F : I → 𝒜, where e_{s} : I_{s} → I is the natural inclusion functor, for every s ∈ S. The details shall be left to the reader.

Proposition 1.2.22. For any category 𝒞 we have :

(i) 𝒞 is complete (resp. finitely complete) if and only if the following conditions hold : (a) The product of every small (resp. finite) family of objects is representable in 𝒞.

(b) All equalizers are representable in 𝒞. (ii) Dually, 𝒞 is cocomplete (resp. finitely cocomplete) if and only if the following holds : (a) The coproduct of any small (resp. finite) family of objects is representable in 𝒞.

(b) All coequalizers are representable in 𝒞.

Proof. Clearly it suffices to show (i). Now, conditions (a) and (b) obviously hold if 𝒞 is complete. Conversely, suppose that the conditions hold, let I be any small (resp. finite) category, and F : I → 𝒞 any functor. We regard Ob(I) (resp. Morph(I)) as a discrete subcategory of I (resp. of Morph(I) : see example 1.1.6(ii)), and we consider the diagram

\xymatrix{Morph(I) \ar[r]^{s} \ar[r]_{t} & Ob(I) \ar[r]^{ob} & I}

where s and t are respectively the restrictions of the functors s and t from (1.1.30), and ob is the inclusion functor. We let as well m : ob ∘ s ⇒ ob ∘ t be the restriction of the natural transformation m of (1.1.30). By assumption, we may find objects P, Q ∈ Ob(𝒞) with isomorphisms

ω_{P} : h_{P} →^{∼} L_{1} := lim_{Ob(I)} F ∘ ob  ω_{Q} : h_{Q} →^{∼} L_{2} := lim_{Morph(I)} F ∘ ob ∘ t  in 𝒞_{V}^{∧}

for any universe V such that 𝒞 has V-small Hom-sets. We define two morphisms of presheaves a, b : L_{1} → L_{2} by the rules :

a(μ) := μ ∗ t and b(μ) := (F ∗ m) ⊙ (μ ∗ s)

for every X ∈ Ob(𝒞) and every cone μ : c_{X} ⇒ F ∘ob. Then a and b are represented by unique morphisms α, β : P → Q in 𝒞 (see example 1.2.8(i)) such that

a ∘ ω_{P} = ω_{Q} ∘ h_{α} and b ∘ ω_{P} = ω_{Q} ∘ h_{β}.

Lastly, let E := Equal(α, β) (notation of example 1.2.16(ii)), and notice that the tautological cone for E yields a monomorphism E → h_{P}, so we may regard E as a subobject of h_{P}; it follows easily that ω_{P} restricts to an isomorphism between E ⊂ h_{P} and the presheaf E′ ⊂ L_{1} such that, for every X ∈ Ob(𝒞), the set E′(X) consists of all the cones μ : c_{X} ⇒ F ∘ ob with a(μ) = b(μ). A simple inspection shows that the latter condition describes precisely the cones c_{X} ⇒ F; on the other hand, by assumption E is representable in 𝒞, so the same holds for the limit of F. □

Example 1.2.23. (i) The category Set is complete and cocomplete. More precisely, for any small family of sets S_{•} := (S_{i} | i ∈ I) we have natural representatives for the product and coproduct of S_{•} : namely, for the product one may take the usual cartesian product of the sets S_{i}, and for the coproduct one may take the disjoint union

∐_{i∈I} S_{i} := ⋃_{i∈I} ({i} × S_{i}).

Likewise, we have natural representatives for the equalizer and coequalizer of any pair of maps f, g : S → S′ : namely, the equalizer of f and g is represented by the subset

Equal(f, g) := {s ∈ S | f(s) = g(s)}

and the coequalizer is represented by the quotient S′/ ∼, where ∼ is the smallest equivalence relation on S′ whose graph contains the subset {(f(s), g(s)) | s ∈ S}. Taking into account proposition 1.2.22(i,ii), we get natural representatives for all limits and colimits in Set. For instance, by inspecting the proof of proposition 1.2.22(i), we see that the colimit of a functor F : I → Set (for any small category I) is represented by the quotient

C(F) := ( ∐_{i∈Ob(I)} F i)/∼

where ∼ is the smallest equivalence relation such that (i, s) ∼ (j, F (φ)(s)) for every morphism φ : i → j in I and every s ∈ F i. In the special case where I is filtered, this equivalence relation can be described more explicitly : namely (i, s) ∼ (i′, s′) if and only if there exist j ∈ Ob(I) and morphisms φ : i → j, φ′ : i′ → j such that F (φ)(s) = F (φ′)(s′).

For a general small category I, to every i ∈ Ob(I) we may attach a natural map τ_{i} : F i → C(F) : namely, τ_{i}(s) is the class of (i, s) in C(F), for every s ∈ F i; then the system (τ_{i} | i ∈ Ob(I)) yields a universal cocone τ : F ⇒ c_{C(F)}, in the sense of remark 1.2.11(iii).

(ii) In view of (i) and example 1.2.7, we can also give a more compact description of the limit of any functor F : I → Set : namely, if 𝟙 is any fixed set with one element, the set L(F) := lim_{I} F (𝟙), consisting of all cones c_{𝟙} ⇒ F, is another natural representative for the limit of F. More precisely, we get a natural isomorphism

h_{L(F)} →^{∼} lim_{I} F in Set^{∧}

as follows. A map of sets S → L(F) amounts to a system of cones {τ_{s} : c_{𝟙} ⇒ F | s ∈ S}, and the latter is naturally identified with a cone c_{S} ⇒ F (details left to the reader). We easily deduce the following natural universal cone τ : c_{L(F)} ⇒ F. For every i ∈ Ob(I), the map τ_{i} : L(F) → F i assigns to any cone μ : c_{𝟙} ⇒ F the image of μ_{i} : 𝟙 → F i.

Example 1.2.24. (i) More generally, for every category 𝒞, the category 𝒞^{∧} is complete and cocomplete. Indeed, let I be any small category, and F : I → 𝒞^{∧} any functor. For every X ∈ Ob(𝒞) we obtain a functor F_{X} : I → Set such that i ↦ F i(X) for every i ∈ Ob(I) and (φ : i → j) ↦ (F φ(X) : F i(X) → F j(X)) for every morphism φ of I; we let LX be the natural representative for lim_{I} F_{X}, and τ^{X} : c_{LX} ⇒ F_{X} the corresponding universal cone, as described in example 1.2.23(ii). Every morphism f : X → Y in 𝒞 induces a natural transformation F_{f} : F_{Y} ⇒ F_{X} by the rule : i ↦ F i(f) : F_{Y} (i) → F_{X}(i), whence a natural transformation lim_{I} F_{f} : lim_{I} F_{Y} ⇒ lim_{I} F_{X} (remark 1.2.11(ii)) which is in turn represented by a unique map of sets Lf : LY → LX such that

τ^{X} ⊙ c_{Lf} = F_{f} ⊙ τ^{Y}

(remark 1.2.8(ii)). It is then easily seen that the rules : X ↦ LX and f ↦ Lf yield a well defined presheaf L on 𝒞. Also, for every i ∈ Ob(I) the rule X ↦ τ_{i}^{X} yields a morphism of presheaves τ_{i}^{•} : L → F i, and the rule i ↦ τ_{i}^{•} yields a cone τ : c_{L} ⇒ F. Now, let η : c_{G} ⇒ F be any cone with vertex G ∈ Ob(𝒞^{∧}). For every X ∈ Ob(𝒞) we deduce a cone η^{X} : c_{G,X} ⇒ F_{X} by the rule : i ↦ η_{i}(X) : GX = G_{X}(i) → F_{X}(i), whence a unique morphism t_{X} : GX → LX such that τ^{X} ⊙ c_{t_{X}} = η^{X}. With this notation, a direct inspection shows that the diagram

\xymatrix{c_{G,Y} \ar@{=>}[r]^{η^{Y}} \ar@{=>}[d]_{c_{G,f}} & F_{Y} \ar@{=>}[d]^{F_{f}} \\ c_{G,X} \ar@{=>}[r]^{η^{X}} & F_{X}}

commutes for every morphism f : X → Y of 𝒞. Thus, τ^{X} ⊙ c_{t_{X}∘G,f} = F_{f} ⊙ τ^{Y} ⊙ c_{t_{Y}} = τ^{X} ⊙ c_{Lf∘t_{Y}}, whence t_{X} ∘ Gf = Lf ∘ t_{Y}, by the universality of τ^{X}. In other words, the rule : X ↦ t_{X} yields a morphism t : G → L in 𝒞^{∧}, and by construction, it is clear that t is the unique such morphism with τ ⊙c_{t} = η. This shows that τ is a universal cone, whence the completeness of 𝒞^{∧}. A similar argument proves the cocompleteness of 𝒞^{∧} : the details are left to the reader.

(ii) Furthermore, let F : I → 𝒞 be a functor from a small category I, whose limit ℒ (which is an object of 𝒞_{V}^{∧} for a suitable universe V) is representable by some L ∈ Ob(𝒞). Let τ : c_{L} ⇒ F be a universal cone and denote by ω : h_{L} →^{∼} ℒ the isomorphism corresponding to τ, as in remark 1.2.11(iii).. Then the limit of h_{𝒞} ∘ F : I → 𝒞_{V}^{∧} is represented by h_{L} and h_{𝒞} ∗ τ : c_{h_{L}} ⇒ h_{𝒞} ∘ F is a universal cone. Indeed, let t : c_{ℒ} ⇒ h_{𝒞} ∘ F be the tautological cone (remark 1.2.11(iv)); a simple inspection shows that the resulting diagram

[figure: c_{h_{L}}; c_{ω}; h_{𝒞} ∗τ; c_{ℒ}; t; h_{𝒞} ∘ F]

commutes. On the other hand, we know that t is universal, so the same follows for h_{𝒞} ∗ τ.

Example 1.2.25. (i) Also the category Cat is complete and cocomplete. Indeed :

• If F : I → Cat is any functor from a small category I, the limit of F is representable by the category 𝒞 with Ob(𝒞) := lim_{I} Ob ∘ F, where Ob : Cat → Set is the functor of example 1.1.6(ii). Hence, an object of 𝒞 is a family (C_{i} | i ∈ Ob(I)) where C_{i} ∈ Ob(𝒞_{i}) for every i ∈ I, and F (φ)(C_{i}) = C_{j} for every φ : i → j in I. For any two objects C_{•} := (C_{i} | i ∈ I) and C′_{•} := (C′_{i} | i ∈ I), notice that the rule i ↦ Hom_{𝒞_{i}}(C_{i}, C′_{i}) defines a functor H_{C_{•},C′_{•}} : I → Set; namely, to every φ : i → j in I we assign the map H_{C_{•},C′_{•}}(i) → H_{C_{•},C′_{•}}(j) given by F (φ) : Morph(𝒞_{i}) → Morph(𝒞_{j}). Then we set

Hom_{𝒞} (C_{•}, C′_{•}) := lim_{I} H_{C_{•},C′_{•}}.

The composition of morphisms in 𝒞 is induced by the composition laws of the categories 𝒞_{i}, in the obvious way. The obvious projection functors 𝒞 → 𝒞_{i} yield a universal cone for lim_{I} F.

• Especially, for any pair of functors

𝒜 ⟶^{F} 𝒞 ⟵^{G} ℬ

the fibre product (in the category Cat) of F and G is represented by the category :

𝒜 ×_{(F,G)} ℬ

whose set of objects is Ob(𝒜) ×_{Ob(𝒞)} Ob(ℬ); the morphisms (A, B) → (A′, B′) are the pairs (f, g) where f : A → A′ (resp. g : B → B′) is a morphism in 𝒜 (resp. in ℬ), such that F f = Gg. If the notation is not ambiguous, we may also denote this category by 𝒜 ×_{𝒞} ℬ. In case ℬ is a subcategory of 𝒞 and G is the natural inclusion functor, we also write F^{−1}ℬ instead of 𝒜 ×_{𝒞} ℬ. The two obvious functors 𝒜 ← 𝒜 ×_{𝒞} ℬ → ℬ provide a universal cone.

• If (𝒞_{i} | i ∈ I) is any small family of categories, the coproduct 𝒞 := ⨿_{i∈I}𝒞_{i} is the category whose set of objects is the disjoint union ⨿_{i∈I}Ob(𝒞_{i}) (see example 1.2.23(i)). Hence, an object of 𝒞 is a pair (i, C), with i ∈ I and C ∈ Ob(𝒞_{i}). For any two such objects (i, C) and (i′, C′), we have

Hom_{𝒞} ((i, C), (i′, C′)) = \{ \begin{matrix}Hom_{𝒞_{i}}(C, C′) & if i = i′ \\ ∅ & otherwise. \end{matrix}

The obvious inclusion functors 𝒞_{i} → 𝒞 provide a universal cocone. The proof of the cocompleteness of Cat shall be postponed to example 1.5.20 (see [28, Prop.5.1.7] for a more constructive proof, and also example 1.5.10).

(ii) For any category 𝒜, the objects of the fibre product category Morph(𝒜)×_{(t,s)}Morph(𝒜) are just the composable pairs of morphisms of 𝒜 (notation of (1.1.30)), and the morphisms are the pairs of commutative square diagrams that have one vertical edge in common. Thus, we have a natural composition functor

c : Morph(𝒜) ×_{(t,s)} Morph(𝒜) → Morph(𝒜)

such that c(f, g) := g ∘ f for every composable pair of morphisms of 𝒜, and where c assigns to every pair of commutative square diagrams

\xymatrix{A \ar[r]^{f} \ar[d]_{h_{1}} & B \ar[r]^{g} \ar[d]_{h_{2}} & C \ar[d]^{h_{3}} \\ A′ \ar[r]^{f′} & B′ \ar[r]^{g′} & C′}

their composition, which is the square diagram whose top (resp. bottom) horizontal arrow is g ∘ f (resp. g′ ∘ f′) and whose vertical arrows are h_{1} and h_{3}.

(iii) We can use the constructions of (ii) to rephrase the definition of the composition laws for natural transformations. Indeed, let us return to the situation of (1.1.10), and denote by

α̃, β̃: 𝒜 → Morph(ℬ)

the functors attached to α and β as in (1.1.30). Clearly t ∘ α̃ = s ∘ β̃, whence a unique functor

(α̃, β̃) : 𝒜 → Morph(ℬ) ×_{(t,s)} Morph(ℬ)

whose composition with the first (second) projection to Morph(ℬ) equals α̃ (resp. β̃). Then it is easily seen that c ∘ (α̃, β̃) : 𝒜 → Morph(𝒜) is the (unique) functor associated with β ⊙ α.

The Godement product of two natural transformations α and α′ as in (1.1.10) can be similarly described : to this aim, notice that

(G′ ∗ α) ⊙ (α′ ∗ F) = α′ ∗ α = (α′ ∗ G) ⊙ (F′ ∗ α)

so we are reduced to exhibit the functors 𝒜 → Morph(𝒞) associated with G′ ∗ α and α′ ∗ F. However, it is easily seen that the latter are respectively Morph(G′) ∘ α̃ and α̃′ ∘ F (details left to the reader). These considerations shall be amplified in section 2.1, in order to construct analogous composition laws for pseudo-natural transformations : see remark 2.2.5.

Example 1.2.26. Let 𝒞 be any category, and φ : X → Y any morphism of 𝒞. The image of φ

Im(φ)

is defined the smallest of the subobjects Y′ → Y such that φ factors through a (necessarily unique) morphism X → Y′. If the image of φ exists, clearly a subobject Z′ → Y contains Im(φ) if and only if φ factors through Z′.

(i) In this generality, the image of φ does not necessarily exists; however, suppose that all equalizers in 𝒞 are representable, and the same for the limit (i.e. the intersection) of any system of subobjects of Y in 𝒞 (these limits are well defined U-presheaves, for a sufficiently large universe U). Then we claim that φ admits an image i : Z → Y, and the unique morphism p : X → Z such that φ = i ∘ p is an epimorphism. Indeed, let ℱ be the family of subobjects Y′ → Y such that φ factors through Y′; choose – for every equivalence class c ∈ ℱ – a representing monomorphism Y_{c} → Y, and let I be the full subcategory of 𝒞/Y such that Ob(I) = {Y_{c} → Y | c ∈ ℱ }. Then I is a U-small category, and we let i (any representing object for) the limit of the inclusion functor ι : I → 𝒞/Y. The resulting system of morphisms (X → Y_{c} | c ∈ ℱ) is a cone on the indexing category I, whence a unique morphism p : X → Z, as required. To show that p is an epimorphism, consider any two morphisms f, g : Z → Z′ (for any Z′ ∈ Ob(𝒞)) such that f ∘ p = g ∘ p, and let E ∈ Ob(𝒞) be any object representing the equalizer of f and g. Thus, the induced morphism ψ : E → Z a monomorphism, hence E is a subobject of Y, and φ still factors through E. By the minimality of Z, we deduce that ψ is an isomorphism, and therefore f = g, whence the claim.

(ii) Dually, if all coequalizers in 𝒞 are representable, and the same for the colimit of any system of quotients of X in 𝒞, then the family of all quotients of X through which φ factors admits a smallest element ψ′ : X → Z, so that a quotient Z′ of X maps to Z if and only if φ factors through Z′. Arguing as in (i), we see that the resulting morphism Z′ → Y is also a monomorphism. We call Z the coimage of φ, and we denote it Coim(φ).

(iii) If both the image and the coimage of φ exist in 𝒞, we get a natural factorization of φ

X → Im(φ) ⟶^{ω} Coim(φ) → Y.

But ω is not necessarily an isomorphism, in this generality.

Remark 1.2.27. (i) If all equalizers and all amalgamated sums are representable in the category 𝒞, then one can alternatively define the image of any morphism φ : X → Y of 𝒞 as the equalizer of the natural morphisms e_{1}, e_{2} : Y → Y ⨿_{X} Y. This definition does not necessarily agree with that of example 1.2.26(i), but if every monomorphism i : Z → Y of 𝒞 is regular, i.e. represents the equalizer of a pair of morphisms f, g : Y → Y′, then one can show that the two definitions coincide.

(ii) Dually, if all coequalizers and all fibre products are representable in 𝒞, then one has a (dual) alternative definition of coimages, which does not necessarily agree with that of example 1.2.26(ii), unless all the epimorphisms of 𝒞 are regular, i.e. unless all the monomorphisms of 𝒞 are regular.

(iii) Moreover, if 𝒞 is both finitely complete and finitely cocomplete, then these alternative images Im^{∗}(φ) and coimages Coim^{∗}(φ) as in (i) and (ii) are well defined for every morphism φ : X → Y, and we get a natural factorization of φ :

X → Coim^{∗}(φ) ⟶^{ω^{∗}} Im^{∗}(φ) → Y

where, again, ω^{∗} is not necessarily an isomorphism.

Example 1.2.28. (i) In the situation of (1.1.31), let η_{•} : 1_{ℬ} ⇒ F G and ε_{•} : GF ⇒ 1_{𝒜} be the unit and counit of the adjunction ϑ, and suppose that all the fibre products of ℬ are representable; then for every B ∈ Ob(ℬ) the functor G_{|B} : ℬ/B → 𝒜 /GB admits a right adjoint, denoted

F_{|B} : 𝒜 /GB → ℬ/B.

Indeed, for every object (g : A → GB) of 𝒜 /GB let us fix a cartesian diagram :

\xymatrix{F^{∗}A \ar[r]^{F^{∗}g} \ar[d]_{η_{A}^{∗}} & B \ar[d]^{η_{B}} \\ F A \ar[r]^{F g} & F GB.}

If h/GB : (g : A → GB) → (g′ : A′ → GB) is a morphism in 𝒜 /GB, the universal property of the fibre product yields a unique morphism F^{∗}h : F^{∗}A → F^{∗}A′ such that F^{∗}g′ ∘ F^{∗}h = F^{∗}g and F^{∗}g′ ∘ η_{A′}^{∗} = F h ∘ η_{A}^{∗}. With this notation, we set

F_{|B}(A, g) := (F^{∗}A, F^{∗}g) and F_{|B}(h/GB) := F^{∗}h/B : F_{|B}(A, g) → F_{|B}(A′, g′)

for every such (A, g) and h/GB. It is easily seen that these rules define a functor as sought. In order to check that F_{|B} is right adjoint to G_{|B}, consider any morphism

h/GB : G_{|B}(B′ ⟶^{f} B) → (A ⟶^{g} GB) in 𝒜 /GB .

Hence, h : GB′ → A is a morphism in 𝒜 with g ∘ h = Gf, and we notice that :

F g ∘ ϑ(h) = F g ∘ F h ∘ η_{B′} = F Gf ∘ η_{B′} = η_{B} ∘ f.

It follows that the pair (ϑ_{AB′}(h), f) determines a unique morphism k : B′ → F^{∗}A such that η_{A}^{∗}∘ k = ϑ_{AB′}(h) and F^{∗}g ∘ k = f. With this notation, we set (ϑ_{|B})_{g,f}(h/GB) := k/B : f → F^{∗}g in ℬ/B. Conversely, to every morphism k/B : (f : B′ → B) → F_{|B}(g : A → GB) in ℬ/B, we attach the morphism h := ϑ_{AB′}^{−1}(η_{A}^{∗} ∘ k) : GB′ → A. Let us show that h/GB : Gf → g is a morphism in 𝒜 /GB; recalling that F^{∗}g ∘ k = f, it suffices to compute :

g ∘ h = g ∘ ε_{A} ∘ G(η_{A}^{∗} ∘ k) = ε_{GB} ∘ GF g ∘ G(η_{A}^{∗} ∘ k) = ε_{GB} ∘ G(η_{B} ∘ F^{∗}g ∘ k) = Gf

where the last equality follows from the triangular identities of (1.1.13). Hence the map

(ϑ_{|B})_{g,f} : Hom_{𝒜 /GB} (G_{|B}f, g) → Hom_{ℬ/B} (f, F_{|B}g)

is a bijection with inverse given by the rule : k/B ↦ ϑ_{AB′}^{−1}(η_{A}^{∗} ∘ k)/GB. Indeed, by definition (ϑ_{|B})_{g,f}(ϑ_{AB′}^{−1}(η_{A}^{∗} ∘ k)/GB) is the morphism f → F_{|B}g of ℬ/B determined by the pair (ϑ_{AB′}ϑ_{AB′}^{−1}(η_{A}^{∗} ∘ k), f) = (η_{A}^{∗} ∘ k), f), which is just k/B, and on the other hand, ϑ_{AB′}^{−1}(η_{A}^{∗} ∘ (ϑ_{|B})_{g,f}(h/GB)) = ϑ_{AB′}^{−1}ϑ_{AB′}(h)/GB = h/GB, whence the contention. The naturality of (ϑ_{|B})_{g,f} with respect to g and f follows by a simple inspection : details left to the reader.

(ii) Dually, if all the coproducts of 𝒜 are representable, then the functor _{A|}F : A/𝒜 → F A/ℬ admits a left adjoint_{A|}G : F A/ℬ → A/𝒜, which the reader is invited to spell out.

(iii) Keep the situation of (i), and let F′ : ℬ → 𝒞 be another functor with a left adjoint G′ : 𝒞 → ℬ. Then we have an isomorphism of functors :

F′_{|C} ∘ F_{|G′C} →^{∼} (F′F)_{|C} for every C ∈ Ob(𝒞).

Indeed, clearly we have G_{|G′C} ∘G′_{C} = (GG′)_{|C}, so the assertion follows from example 1.2.7(ii).

1.3. Adjunctions and Kan extensions. Let φ : I → J be a functor between small categories. The calculation of limits indexed by I can be sometimes simplified by a Fubini-style technique of “integration along the fibres of φ”, which replaces any given functor F : I → 𝒞 (to any category 𝒞) with another functor J → 𝒞^{∧} whose limit is isomorphic to that of F. Namely, let V be a universe containing U, such that 𝒞 has V-small Hom-sets; notice first that by virtue of remark 1.2.11(ii), the limit construction yields a well defined functor

lim_{I} : Fun(I, 𝒞) → 𝒞_{V}^{∧} F ↦ lim_{I} F.

Moreover, if 𝒞 is complete (resp. if I is finite and 𝒞 is finitely complete), this functor is naturally isomorphic to the composition of the Yoneda embedding h_{𝒞} and a functor

Lim_{I} : Fun(I, 𝒞) → 𝒞

which is right adjoint to the constant functor

(1.3.1)  c : 𝒞 → Fun(I, 𝒞)  X ↦ c_{X}  (f : X → Y) ↦ (c_{f} : c_{X} ⇒ c_{Y})

(notation of (1.2.9)), and especially, it is well defined up to isomorphism (example 1.2.7(ii)). We define now as follows a functor

∫_{φ}^{∧} : Fun(I, 𝒞) → Fun(J, 𝒞_{V}^{∧}).

For every functor F : I → 𝒞, every j ∈ Ob(J), and every morphism g : j → j′ in J we set

∫_{φ}^{∧} F (j) := lim_{j/φI} F ∘ t_{j} ∫_{φ}^{∧} F (g) := lim_{g/φI} 1_{𝒞}

(notation of (1.1.27) and remark 1.2.11(i)). We may then state :

Proposition 1.3.2. With the notation of (1.3), the diagram

\xymatrix{Fun(I, 𝒞) \ar[r]^{^{lim^{I}}} \ar[d]_{_{∫{∧ \atop φ}}} & 𝒞_{V}^{∧} \\ Fun(J, 𝒞_{V}^{∧}) \ar[ur]_{_{Lim_{J}}}}

is essentially commutative, i.e. there is a natural isomorphism of functors : lim_{I} →^{∼} Lim_{J} ∘ ∫_{φ}^{∧}.

Proof. Notice that the functor Lim_{J} is well defined by virtue of example 1.2.24. Let F : I → 𝒞 be any functor; unwinding the definitions, we see that Lim_{J} ∘ ∫_{φ}^{∧} F is isomorphic to the presheaf that assigns to every X ∈ Ob(𝒞) the set of all compatible systems

τ_{•}^{•} := (τ_{•}^{j} : c_{X} ⇒ F ∘ t_{j} | j ∈ Ob(J))

such that τ_{•}^{j′} = τ_{•}^{j} ∗ (g/φI) for every morphism g : j → j′ in J. Explicitly, τ_{•}^{•} is the datum, for every (h : j′ → φi) ∈ Ob(J/φI), of a morphism τ_{h}^{j} : X → F i in 𝒞, such that

τ_{h∘g}^{j} = τ_{h}^{j′} and F f ∘ τ_{h}^{j} = τ_{φ(f)∘h}^{j}

for every g as above, and every morphism f : i → i′ in I. Clearly, every such system is determined by the family (\overline{τ}_{i} := τ_{1_{φi}}^{φi} | i ∈ Ob(I)), and the latter is a cone \overline{τ}_{•} : c_{X} ⇒ F. Conversely, given any cone μ_{•} : c_{X} ⇒ F, the family \overline{μ}_{•}^{•} := (μ_{•} ∗ t_{j} | j ∈ Ob(J)) is a compatible system of the foregoing type. A simple inspection shows that the rules τ_{•}^{•} ↦ \overline{τ}_{•} and μ_{•} ↦ \overline{μ}_{•}^{•} are mutually inverse, whence the proposition. □

Remark 1.3.3. (i) In the situation of (1.3), suppose moreover that 𝒞 is complete. Then ∫_{φ}^{∧} is isomorphic to the composition of the functor Fun(J, h_{𝒞} ) (notation of remark 1.1.19(ii) and (1.2.4)) and a functor

∫_{φ} : Fun(I, 𝒞) → Fun(J, 𝒞)

called the right Kan extension along φ. Taking into account example 1.2.24(ii), proposition 1.3.2implies that the resulting diagram

\xymatrix{Fun(I, 𝒞) \ar[r]^{^{Lim^{I}}} \ar[d]_{_{∫_{φ}}} & 𝒞 \\ Fun(J, 𝒞) \ar@{-->}[ur]_{_{Lim_{J}}}}

is also essentially commutative (details left to the reader).

(ii) By remark 1.2.11(ii), we may also define a colimit functor

colim_{I} : Fun(I, 𝒞) → 𝒞_{V}^{o∧o} F ↦ colim_{I} F

and notice that the isomorphism of remark 1.1.19(i) identifies this functor with the opposite of the functor lim_{I^{o}} defined on Fun(I^{o}, 𝒞_{V}^{o}). Consequently, if 𝒞 is cocomplete (resp. if I is finite and 𝒞 is finitely cocomplete), colim_{I} is isomorphic to the composition of h_{𝒞^{o}}^{o} and of a functor

Colim_{I} : Fun(I, 𝒞) → 𝒞

which is likewise identified with the opposite of the functor Lim_{I^{o}} : Fun(I^{o}, 𝒞^{o}) → 𝒞^{o}. Notice as well that Colim_{I} is left adjoint to the constant functor (1.3.1). Moreover, the opposite of the functor ∫_{φ^{o}}^{∧} on Fun(I^{o}, 𝒞^{o}) is identified with a functor

∫_{∧}^{φ} : Fun(I, 𝒞) → Fun(J, 𝒞_{V}^{o∧o}).

Explicitly, we have

∫_{∧}^{φ} F (j) = colim_{φI/j} F ∘ s_{j} for every functor F : I → 𝒞 and every j ∈ Ob(J)

(where s_{j} is the source functor) and we get an essentially commutative diagram :

\xymatrix{Fun(I, 𝒞) \ar[r]^{^{colim^{I}}} \ar[d]_{_{∫{φ \atop ∧}}} & 𝒞_{V}^{o∧o} \\ Fun(J, 𝒞_{V}^{o∧o}) \ar[ur]_{_{Colim_{J}}}}

Notice the natural identification : 𝒞_{V}^{o∧o} →^{∼} Fun(𝒞^{o}, V-Set^{o}), again due to remark 1.1.19(i). If 𝒞 is cocomplete, the opposite of the functor ∫_{φ^{o}} is naturally identified with a functor

∫ ^{φ} : Fun(I, 𝒞) → Fun(J, 𝒞)

called the left Kan extension along φ, fitting into the essentially commutative diagram :

\xymatrix{Fun(I, 𝒞) \ar[r]^{^{Colim^{I}}} \ar[d]_{_{∫_{φ}}} & 𝒞 \\ Fun(J, 𝒞). \ar[ur]_{_{Colim_{J}}}}

(iii) Notice that the functors ∫_{φ}^{∧} and ∫_{∧}^{φ} are defined more generally whenever I is small and J has small Hom-sets. Likewise, if 𝒞 is complete (resp. cocomplete) the right (resp. left) Kan extension along φ is well defined under the same weakened assumptions.

(iv) In the same vein, if the category j/φI (resp. φI/j) is finite for every j ∈ Ob(J), for the right (resp. left) Kan extension along φ to be well defined it suffices that 𝒞 is finitely complete (resp. finitely cocomplete).

Theorem 1.3.4. Let I, J and 𝒞 be categories fulfilling the conditions of either part (iii) or part (iv) of remark 1.3.3, and φ : I → J any functor. Then the right (resp. left) Kan extension along φ : I → J is right (resp. left) adjoint to the functor

Fun(φ, 𝒞) : Fun(J, 𝒞) → Fun(I, 𝒞) G ↦ G ∘ φ.

Proof. (Cp. [28, Th.3.7.2] or [110, Th.2.3.3]). Pick a universe V containing U, and such that 𝒞 has V-small Hom-sets. In light of Yoneda’s lemma (proposition 1.2.6(ii)) we get for every pair of functors F : I → 𝒞 and G : J → 𝒞 a natural bijection

Hom_{Fun(J,𝒞)}(G, ∫_{φ} F ) →^{∼} Hom_{Fun(J,𝒞_{V}^{∧})}(h_{𝒞} ∘ G, ∫_{φ}^{∧} F )

where h_{𝒞} : 𝒞 → 𝒞_{V}^{∧} is the Yoneda embedding of (1.2.4). Hence, in order to prove the assertion for right Kan extensions, it suffices to exhibit a natural bijection :

(1.3.5)  Hom_{Fun(I,𝒞)}(G ∘ φ, F) →^{∼} Hom_{Fun(J,𝒞_{V}^{∧})}(h_{𝒞} ∘ G, ∫_{φ}^{∧} F ).

Notice that, by virtue of proposition 1.2.6(ii), if X ∈ Ob(𝒞) and j ∈ Ob(J) are any two objects, a morphism h_{X} → ∫_{φ}^{∧} F (j) in 𝒞_{V}^{∧} is the same as the datum of a cone c_{X} ⇒ F ∘ t_{j}, where t_{j} : j/φI → I is the target functor. Now, let τ : G∘φ ⇒ F be any natural transformation; for every j ∈ Ob(J) and every (i, f : j → φ(i)) ∈ Ob(j/φI) we set

τ′_{j,(i,f)} := τ_{i} ∘ Gf : Gj → F i.

It is easily seen that, for fixed j ∈ Ob(J), the system (τ′_{j,(i,f)} | (i, f) ∈ Ob(j/φI)) gives a cone τ′_{j} : c_{Gj} ⇒ F ∘ t_{X}, i.e. a morphism h_{Gj} → ∫_{φ}^{∧} F (j) in 𝒞_{V}^{∧}, and the rule j ↦ τ′_{j} for every j ∈ Ob(J) yields a natural transformation τ′ : h_{𝒞} ∘ G ⇒ ∫_{φ}^{∧} F. Conversely, if μ : h_{𝒞} ∘G ⇒ ∫_{φ}^{∧} F is any such transformation, every j ∈ Ob(J) yields a cone μ_{j} : c_{Gj} ⇒ F ∘t_{j}, and we define a natural transformation μ′ : G ∘ φ ⇒ F by setting

μ′_{i} := μ_{φ(i),(i,1_{φ(i)})} : G ∘ φ(i) → F i for every i ∈ Ob(I).

The reader can check that (τ′)′ = τ and (μ′)′ = μ for every τ and μ as above, and clearly the resulting bijections (1.3.5) are functorial in both F and G. The assertion for left Kan extensions, in case 𝒞 is cocomplete, is dual to the foregoing, by virtue of remark 1.1.19(i). □

Remark 1.3.6. (i) Let F : ℬ → 𝒞 be a functor from a small category ℬ to any category 𝒞 with small Hom-sets; from theorem 1.3.4 we obtain both a left and a right adjoint for F_{U}^{∧} = Fun(F^{o}, U-Set), denoted respectively

F_{U!} : ℬ_{U}^{∧} → 𝒞_{U}^{∧} and F_{U∗} : ℬ_{U}^{∧} → 𝒞_{U}^{∧}.

As usual, we drop the subscript U, unless the omission may cause ambiguities.

(ii) With the notation of (i), notice that the diagram of functors :

(1.3.7)  \xymatrix{ℬ \ar[r]^{h_{ℬ}} \ar[d]_{F} & ℬ^{∧} \ar[d]^{F_{!}} \\ 𝒞 \ar[r]^{h_{𝒞}} & 𝒞^{∧}}

(whose horizontal arrows are the Yoneda embeddings) is essentially commutative, i.e. the two compositions F_{!} ∘ h_{ℬ} and h_{𝒞} ∘ F are isomorphic functors. Indeed – by proposition 1.2.6(ii) – for every B ∈ Ob(ℬ), the objects F_{!}h_{B} and h_{F B} both represent the functor

𝒞^{∧} → Set : φ ↦ φ(F B)

(the latter is a V-presheaf on 𝒞^{∧o}, for any universe V such that 𝒞^{∧} has small Hom-sets).

(iii) Moreover, taking into account example 1.2.23 we see that there are natural choices for the left and right adjoints of the functor F^{∧}. Namely :

• For any presheaf f on ℬ, the presheaf F_{!}f on 𝒞 can be given canonically by the rule :

(1.3.8)  C ↦ colim_{(C/F ℬ)^{o}} f ∘ t_{C}^{o}  for every C ∈ Ob(𝒞)

where the colimits in this expression denote (by abuse of notation) the natural representatives for the corresponding presheaves, as described in example 1.2.23(i). Therefore, every element of F_{!}f(C) is the equivalence class of a pair (ψ : C → F B, s), where ψ is any object of C/F ℬ and s ∈ fB is any element. If φ : C′ → C is any morphism of 𝒞, the induced map F_{!}f(φ) : F_{!}f(C) → F_{!}f(C′) assigns to any such pair the equivalence class of the pair (ψ ∘ φ : C′ → F B, s). Furthermore, the adjunction for the pair (F_{!}, F^{∧}) provided by the proof of theorem 1.3.4comes down to the natural bijection

Hom_{ℬ^{∧}} (f, F^{∧}g) →^{∼} Hom_{𝒞^{∧}}(F_{!}f, g) for every f ∈ Ob(ℬ^{∧}) and g ∈ Ob(𝒞^{∧})

which, to any morphism τ : f → F^{∧}g in ℬ^{∧} and to any C ∈ Ob(𝒞) assigns the map of sets F_{!}f(C) → gC given by the rule :

(ψ : C → F B, s) ↦ g(ψ) ∘ τ_{B}(s) for every (B, ψ) ∈ Ob(C/F ℬ) and every s ∈ fB .

The inverse to this bijection assigns to any morphism μ : F_{!}f → g and any B ∈ Ob(ℬ) the map f(B) → g(F B) given by the rule :

s ↦ μ_{F B}(1_{F B}, s) for every s ∈ f(B).

• And the presheaf F_{∗}f can be chosen canonically by the rule :

C ↦ lim_{(F ℬ/C)^{o}} f ∘ s_{C}^{o}(𝟙) φ ↦ lim_{(F ℬ/φ)^{o}} 1_{ℬ} (𝟙)

for every C ∈ Ob(𝒞) and every morphism φ : C′ → C in 𝒞 (notation of (1.1.27), and 𝟙 is any fixed set with one element). More explicitly, under this identification, the map F_{∗}f(φ) : F_{∗}f(C) → F_{∗}f(C′) is given by the rule :

(τ : c_{𝟙} ⇒ f ∘ s_{C}^{o}) ↦ (τ ∗ (F ℬ/φ)^{o} : c_{𝟙} ⇒ f ∘ s_{C′}^{o}).

(iv) Let V be a universe such that U ⊂ V. Notice that, with the canonical choices of (iii), we get a commutative diagram of categories

\xymatrix{𝒞_{U}^{∧} \ar[d] & ℬ_{U}^{∧} \ar[r]^{F_{U!}} \ar[l]_{F_{U∗}} \ar[d] & 𝒞_{U}^{∧} \ar[d] \\ 𝒞_{V}^{∧} & ℬ_{V}^{∧} \ar[r]^{F_{V!}} \ar[l]_{F_{V∗}} & 𝒞_{V}^{∧}}

whose vertical arrows are the inclusion functors.

(v) Lastly, with the canonical choices in (iii), we can make (ii) more precise : we get a natural identification

F_{!}h_{B} →^{∼} h_{F B} for every B ∈ Ob(ℬ) .

Namely, to any pair (ψ′ : Y → F B′, s : B′ → B) consisting of an object of Y/F ℬ and an element s ∈ h_{B}(B′), we attach the element β(ψ′, s) := F (s) ∘ ψ′ ∈ h_{F B}(Y), and it is easily seen that β(ψ′, s) depends only on the class of (ψ′, s) in F_{!}h_{B}(Y), whence the required functorial bijection (details left to the reader). Notice as well that – under these identifications – the unit of the adjunction for the pair (F_{!}, F^{∧}) explicited in (iii) becomes the natural morphism

h_{B} → F^{∧}h_{B} : (s : B′ → B) ↦ (F s : F B′ → F B) for every B ∈ Ob(B).

Example 1.3.9. Let 𝟙_{ℬ} be a final object of ℬ^{∧} (see example 1.2.16(v)); by specializing the explicit description of F_{!} provided by remark 1.3.6(iii), we get a natural isomorphism

F_{!}(𝟙_{ℬ} ) →^{∼} π_{0}(•/F ℬ) in 𝒞^{∧}

where π_{0}(•/F ℬ) is the presheaf that assigns to any C ∈ Ob(𝒞) the set π_{0}(C/F ℬ), and to any morphism φ : C′ → C in 𝒞 the map π_{0}(φ/F ℬ) : π_{0}(C/F ℬ) → π_{0}(C′/F ℬ) induced by φ/F ℬ in the obvious way (notation of (1.1.27) : we leave the verification to the reader).

1.3.10. For many questions, it is useful to know whether a given functor commutes with a certain limit, or more generally with a prescribed class of limits (or colimits), in the sense explained by the following :

Definition 1.3.11. Let I, 𝒞 and 𝒟 be any three categories, and F : I → 𝒞, f : 𝒞 → 𝒟 two functors. Pick also a universe V such that I, 𝒞 and 𝒟 are V-small.

(i) According to remark 1.2.11(i) we have a natural morphism in 𝒞_{V}^{∧}

(1.3.12)  lim_{I} F → f_{V}^{∧} lim_{I} (f ∘ F)

whence – by the explicit adjunction of remark 1.3.6(iii) – a morphism in 𝒟_{V}^{∧}

(1.3.13)  f_{V!}(lim_{I} F) → lim_{I} (f ∘ F).

Then, we say that f commutes with the limit of F, if (1.3.13) is an isomorphism. (ii) We say that f commutes with the colimit of F, if the dual condition holds, i.e. if f^{o}

commutes with the limit of F^{o}, so the natural morphism

f_{V!}^{o}(colim_{I} F) → colim_{I}(f ∘ F)

is an isomorphism. (iii) We say that the functor f is left exact, if f commutes with all finite limits. Dually, we

say that f is right exact if it commutes with all finite colimits. Finally, we say that f is exact if it is both left and right exact.

It follows easily from remark (1.3.6)(iv) that these definitions depend only on f and F, and not on the choice of the universe V.

Remark 1.3.14. (i) Keep the situation of definition 1.3.11. The most interesting case is that where the limit of F is representable, say by an object L of 𝒞, so we have an isomorphism β : h_{L} →^{∼} lim_{I} F. Indeed, in this case, by remark 1.3.6(v) we deduce a morphism

γ : h_{fL} →^{∼} f_{!}h_{L} →^{∼} f_{!} lim_{I} F → lim_{I}(f ∘ F)

that is an isomorphism if and only if f commutes with the limit of F, in which case we see that the limit of f ∘ F is representable by fL. More precisely, let τ := β_{L}(1_{L}) : c_{L} ⇒ F be the universal cone deduced from β; we claim that

γ_{fL}(1_{fL}) = f ∗ τ : c_{fL} ⇒ f ∘ F

so f commutes with the limit of F if and only if f ∗ τ is a universal cone for f ∘ F. Indeed, notice that the composition h_{L} → f_{V}^{∧} lim_{I}(f ∘ F) of β and (1.3.12) assigns, to any C ∈ Ob(𝒞) and any morphism s : C → L in 𝒞, the cone (f ∗ τ) ⊙ c_{fs} : c_{fC} ⇒ f ∘ F. In light of remark 1.3.6(iii), it follows that the adjoint morphism f_{!}h_{L} → lim_{I} f ∘ F assigns, to every D ∈ Ob(𝒟) and every pair (ψ : D → fC, s : C → L) the cone (f ∗ τ) ⊙ c_{fs} ⊙ c_{ψ} : c_{D} ⇒ f ∘ F. Lastly, notice that the isomorphism f_{!}h_{L} →^{∼} h_{fL} provided from remark 1.3.6(v) maps the pair (1_{fL}, 1_{L}) to 1_{fL} ∈ h_{fL}(fL); we conclude that γ_{fL}(1_{fL}) = (f ∗ τ) ⊙ c_{f1_{L}} ⊙ c_{1_{fL}} = f ∗ τ, as claimed.

(ii) Dually, suppose that the colimit of F is representable by an object C^{o} of 𝒞^{o}, and pick any universal cocone μ : F ⇒ c_{C}. Then f commutes with the colimit of F if and only if f ∗ μ is a universal cocone for f ∘ F.

Example 1.3.15. (i) Let 𝒞 be any category, V a universe such that 𝒞 has V-small Hom-sets, and f : 𝒞^{o} → V-Set a representable functor. Then it is easily seen that for every small category I, every functor F : I → 𝒞 and every universal cocone τ : F ⇒ c_{X}, the induced cone f ∗ τ^{o} : c_{fX} ⇒ f ∘ F^{o} is universal. In view of remark 1.3.14(i), we conclude that f commutes with all the representable limits of 𝒞^{o} (which are the representable colimits of 𝒞). On the other hand, if 𝒞 has small Hom-sets and all small products of 𝒞 are representable, then f commutes even with non-representable colimits of 𝒞 : see example 1.4.4(i).

(ii) Dually, if f : 𝒞 → Set is a functor representable by some object Y^{o} of the opposite category 𝒞^{o}, then f commutes with all representable limits of 𝒞, and even with non-representable limits, if all small coproducts of 𝒞 are representable and 𝒞 has small Hom-sets.

Example 1.3.16. (i) Let 𝒜 be any finitely complete category, and F : 𝒜 → ℬ any left exact functor. Then the category B/F 𝒜 is cofiltered, for every B ∈ Ob(ℬ). Indeed, if φ_{i} : B → F A_{i} (i = 1, 2) are any two objects of B/F 𝒜, the product A_{1} × A_{2} is representable by some object A of 𝒜, and a universal cone is given by a pair of morphisms (p_{i} : A → A_{i} | i = 1, 2); by remark 1.3.14(i), it follows that the pair (F p_{i} : F A → F A_{i} | i = 1, 2) is still a universal cone for the product F A_{1} × F A_{2}, whence a unique morphism φ : B → F A such that F p_{i} ∘ φ = φ_{i} for i = 1, 2. This shows that B/F 𝒜 is codirected. Moreover, if A is any final object of 𝒜, then F A is a final object of ℬ, hence Ob(B/F 𝒜) is non-empty. By remark 1.2.21(i), it then remains only to check that B/F 𝒜 satisfies the equalizing condition dual to that of definition 1.2.19(v). Namely, let φ_{1} and φ_{2} as in the foregoing, and suppose we have a pair of morphisms ψ, ψ′ : A_{1} → A_{2} such that F ψ∘φ_{1} = φ_{2} = F ψ′∘φ_{1}. The equalizer of ψ and ψ′ is representable by some object E of 𝒜, and a universal cone for E is given by a morphism β : E → A_{1} such that ψ ∘ β = ψ′ ∘ β; by remark 1.3.14(i), F E represents the equalizer of F ψ and F ψ′, and F β : F E → F A_{1} still yields a universal cone. There follows a unique morphism γ : B → F E in ℬ such that F β ∘ γ = φ_{1}, whence the claim.

(ii) Dually, if 𝒜 is finitely cocomplete and F is right exact, then the category F 𝒜 /B is filtered for every B ∈ Ob(ℬ).

Proposition 1.3.17. Let I, 𝒞, 𝒟 and ℰ be four categories, and F : I → 𝒞, f : 𝒞 → 𝒟 and g : 𝒟 → ℰ any three functors. The following holds :

(i) If f commutes with the limit of F, and g commutes with the limit of f ∘ F, then g ∘ f commutes with the limit of F.

(ii) Especially, if both f and g are left (resp. right) exact, then the same holds for g ∘ f.

Proof. Clearly it suffices to show (i), and after replacing U by a larger universe, we may assume that the four categories of the proposition are small. Set L := lim_{I} F, L′ := lim_{I}(f ∘ F), L′′ := lim_{I}(g ∘ f ∘ F), and denote

ω_{f,F} : L → f^{∧}L′ and ω_{g,f∘F} : L′ → g^{∧}L′′

the morphisms provided by remark 1.2.11(i). Let also (ε^{f}, η^{f}) (resp. (ε^{g}, η^{g})) be the units and counits associated with the explicit adjunction ϑ_{f} (resp ϑ_{g}) for the pair (f_{!}, f^{∧}) (resp. (g_{!}, g^{∧})) provided by remark 1.3.6(iii); by assumption, the adjoint morphisms

ω_{f,F}^{∗} := ε_{L′}^{f} ∘ (f_{!}ω_{f,F}) : f_{!}L → L′ and ω_{g,f∘F}^{∗} := ε_{L′′}^{g} ∘ (g_{!}ω_{g,f∘F}) : g_{!}L′ → L′′

are isomorphisms, so the same holds for ω_{g∘f,F}^{∗∗} := ω_{g,f∘F}^{∗} ∘ g_{!}ω_{f,F}^{∗} : g_{!} ∘ f_{!}L → L′′. However :

ω_{g∘f,F}^{∗∗} = ε_{L′′}^{g} ∘ g_{!}(ω_{g,f∘F} ∘ ω_{f,F}^{∗})

corresponds, under the adjunction ϑ_{g}, to the morphism ω_{g,f∘F} ∘ ω_{f,F}^{∗} : f_{!}L → g^{∧}L′′. The latter, in turns, corresponds – under the adjunction ϑ_{f} – to the morphism

ω_{g∘f,F} := f^{∧}(ω_{g,f∘F} ∘ ω_{f,F}^{∗}) ∘ η_{L}^{f} : L → f^{∧} ∘ g^{∧}L′′ = (g ∘ f)^{∧}L′′

and notice that f^{∧}(ω_{f,F}^{∗}) ∘ η_{L}^{f} = ω_{f,F}. Taking into account (1.2.12), we conclude that ω_{g∘f,F} is precisely the natural morphism denoted lim_{1_{I}}(g ∘ f) in remark 1.2.11(i).

Summing up, we have shown that, under the adjunction ϑ_{g} ∘ ϑ_{f} for the pair (g_{!} ∘ f_{!}, (g ∘ f)^{∧}) (notation of remark 1.1.17(i)), the morphism ω_{g∘f,F} corresponds to the isomorphism ω_{g∘f,F}^{∗∗}. Then the assertion follows from example 1.2.7(ii). □

Proposition 1.3.18. Let 𝒞 , 𝒟 be two categories, F : 𝒞 → 𝒟 a functor, and φ : X → Y a morphism in 𝒞. The following holds :

(i) If F commutes with pull backs and φ is a monomorphism, then F φ is a monomorphism. (ii) If F commutes with push-outs and φ is an epimorphism, then F φ is an epimorphism.

Proof. (i): After replacing U by a larger universe, we may assume that 𝒞 and 𝒟 have small Hom-sets. Notice then that the tautological cone of X ×_{Y} X amounts to a pair of morphisms

p_{1}, p_{2} : X ×_{Y} X → h_{X} in 𝒞^{∧}

such that h_{φ}∘p_{1} = h_{φ}∘p_{2} (see remark 1.2.11(iv)), and the diagonal morphism ι_{φ} : h_{X} → X ×_{Y} X (see (1.2.17)) is characterized as the unique morphism in 𝒞^{∧} such that p_{1}∘ι_{φ} = 1_{h_{X}} = p_{2}∘ι_{φ}. Likewise we can characterize ι_{F φ}. Now, by assumption there exist unique morphisms j and q_{i} for i = 1, 2, fitting into a commutative diagram

(1.3.19)  \xymatrix{F_{!}h_{X} \ar[r]^{F_{!}ι_{φ}} \ar[d] & F_{!}(X ×_{Y} X) \ar[r]^{F_{!}p_{i}} \ar[d] & F_{!}h_{X} \ar[d] \\ h_{F X} \ar[r]^{j} & F X ×_{F Y} F X \ar[r]^{q_{i}} & h_{F X}}

whose leftmost and rightmost vertical arrows are the natural identifications of remark 1.3.6(v), and whose central vertical arrow is the isomorphism given by definition 1.3.11(i). Taking into account proposition 1.2.18(i), we see that the assertion will follow, once we know that j = ι_{F φ}. However, the composition of the two top horizontal arrows equals 1_{F_{!}h_{X}}, so the composition of the bottom horizontal arrows is 1_{h_{F X}}, and thus we come down to showing :

Claim 1.3.20. q_{1} and q_{2} are the two morphisms defining the tautological cone for F X ×_{F Y} F X.

Proof of the claim. Notice that, for any Z ∈ Ob(𝒞), an element of X ×_{Y} X(Z) is the same as the datum of a pair of morphisms α, β : Z → X such that φ ∘ α = φ ∘ β. The natural morphism X ×_{Y} X → F^{∧}(F X ×_{F Y} F X) of (1.3.12) sends such a pair (α, β) to the pair (F α, F β).

Therefore, the central vertical arrow of (1.3.19) is the map that assigns to any W ∈ Ob(𝒟) and any pair (ψ : W → F Z, (α, β : Z → X)) the pair (F α ∘ ψ, F β ∘ ψ : W → F X). Then, q_{1} (resp. q_{2}) sends such a pair to F α ∘ ψ (resp. to F β ∘ ψ). On the other hand, F_{!}p_{1} (resp. F_{!}p_{2}) sends (ψ, (α, β)) to the pair (ψ, α) (resp. to (ψ, β)). Now the claim follows by inspecting the explicit definition of the rightmost vertical arrow of (1.3.19). ♢

(ii) follows by dualizing the foregoing argument (details left to the reader). □

Proposition 1.3.21. (i) Let 𝒞 , 𝒟 be two categories, F : 𝒞 → 𝒟 a left exact (resp. right exact) functor; suppose that 𝒞 is finitely complete (resp. finitely cocomplete), and consider the following conditions :

(a) F is conservative (see definition 1.1.4(ii))

(b) F is faithful

(c) F reflects epimorphisms (resp. monomorphisms)

(d) F reflects monomorphisms (resp. epimorphisms). Then (a)⇒(b)⇒(c)⇒(d).

(ii) Suppose moreover that the isomorphisms in 𝒞 are the morphisms that are both monomorphisms and epimorphisms. Then (c)⇒(a).

Proof. We prove the assertions in case f is left exact and 𝒞 is finitely complete; the assertions for the case where F is right exact and 𝒞 is finitely cocomplete will follow by considering F^{o} : 𝒞^{o} → 𝒟^{o}.

(i.a)⇒(i.b): Let f; g : A → B be two morphisms in 𝒞; by assumption, the equalizer of f and g is representable in 𝒞 by some object E, and the universal cone is the datum of a morphism h : E → A such that f ∘ h = g ∘ h. Moreover, F E represents the equalizer of F f and F g, and F h : F E → F A is a universal cone for this equalizer, by remark 1.3.14(i). Now, suppose that F f = F g; then F h is an isomorphism, hence the same holds for h, and therefore f = g.

(i.b)⇒(i.c): Let f : A → B and g, h : B → C be three morphisms in 𝒞 with g ∘ f = h ∘ f, and suppose that F f is an epimorphism; since F g ∘ F f = F h ∘ F f, it follows that F g = F h, and then g = h, since F is faithful. This shows that f is an epimorphism.

(i.c)⇒(i.d): Let f : A → B be a morphism in 𝒞 such that F f : F A → F B is a monomorphism; by assumption the fibre product A ×_{B} A is representable in 𝒞, and if p_{1}, p_{2} : A ×_{B} A → A are the natural projections, then F (A ×_{B} A) represents F A ×_{F B} F A, and F p_{1}, F p_{2} : F (A ×_{B} A) → F A provide a universal cone for this fibre product. It follows easily that if ι_{f} : A → A ×_{B} A is the diagonal of f, then F ι_{f} : F A → F (A ×_{B} A) is the diagonal of F f, hence F ι_{f} is an isomorphism in 𝒟, by proposition 1.2.18(i). By assumption, ι_{f} is then an epimorphism, so f is a monomorphism, again by proposition 1.2.18(i).

(ii): Let f : A → B be a morphism such that F f is an isomorphism; in particular, F f is an epimorphism, so by assumption f is an epimorphism. But F f is also a monomorphism, hence the same holds for f, because we know already that (i.c)⇒(i.d). Then f is an isomorphism, under our assumption on 𝒞. □

Proposition 1.3.22. Let 𝒞 , 𝒟 be any two categories, and f : 𝒞 → 𝒟 any functor.

(i) Suppose that (a) f commutes with equalizers and with products (resp. and with finite products)

(b) all small products (resp. all finite products) are representable in 𝒞.

Then f commutes with all limits of 𝒞 (resp. f is left exact). (ii) Dually, suppose that

(a) f commutes with equalizers and with coproducts (resp. and with finite coproducts)

(b) all small coproducts (resp. all finite coproducts) are representable in 𝒞.

Then f commutes with all colimits of 𝒞 (resp. f is right exact).

Proof. (i): With the notation of the proof of proposition 1.2.22(i), we have shown that the rule

(μ : c_{X} ⇒ F) ↦ ω_{P}^{−1}(μ ∗ ob) for every X ∈ Ob(𝒞) and μ ∈ lim_{I} F (X)

defines an isomorphism g : lim_{I} F →^{∼} E ⊂ h_{P}. Also, since by assumption f commutes with products (resp. with finite products), the isomorphisms ω_{P} and ω_{Q} induce isomorphisms

γ_{P} : h_{fP} →^{∼} lim_{Ob(I)} f ∘ F ∘ ob γ_{Q} : h_{fQ} →^{∼} lim_{Morph(I)} f ∘ F ∘ ob ∘ t

as explained in remark 1.3.14(i). We set E′ := Equal(fα, fβ), and we identify E′ as well with a subobject of h_{fP}, via the monomorphism E′ → h_{fP} provided by the tautological cone. Then we define likewise a morphism g′ : lim_{I}(f ∘ F) → E′ by the rule :

(μ : c_{Y} ⇒ f ∘ F) ↦ γ_{P}^{−1}(μ ∗ ob) for every Y ∈ Ob(𝒟) and μ ∈ lim_{I}(f ∘ F)(Y) .

Consider now the diagram of presheaves on 𝒟 :

(1.3.23)  \xymatrix{f_{!} lim_{I} F \ar[r]^{f_{!}g} \ar[d]_{d} & f_{!}E \ar[d]^{e} \\ lim_{I} f ∘ F \ar[r]^{g′} & E′}

whose vertical arrows are the morphisms (1.3.13).

Claim 1.3.24. Diagram (1.3.23) commutes.

Proof of the claim. Let τ : c_{P} ⇒ F ∘ ob be the universal cone arising from the isomorphism ω_{P}; according to remark 1.3.14(i), for every X ∈ Ob(𝒞) and every cone μ : c_{X} ⇒ F (resp. every Y ∈ Ob(𝒟) and every cone μ′ : c_{Y} ⇒ f ∘ F ∘ ob), the morphism g(μ) (resp. g′(μ′)) is characterized as the unique one such that

τ ⊙ c_{g(μ)} = μ ∗ ob (resp. (f ∗ τ) ⊙ c_{g′(μ′)} = μ′ ∗ ob ).

Now, let (ψ : Y → fX, μ : c_{X} ⇒ F) be the representative of any given element s ∈ f_{!} lim_{I} F (Y). Then f_{!}g(s) is represented by the pair (ψ, g(μ)), and

d(s) = (f ∗ μ) ⊙ c_{ψ} e(f_{!}g(s)) = f(g(μ)) ∘ ψ

(remark 1.3.6(iii)), and the foregoing yields the identities

(f ∗ τ) ⊙ c_{g′((f∗μ)∘c_{ψ})} = ((f ∗ μ) ⊙ c_{ψ}) ∗ ob
= (f ∗ μ ∗ ob) ⊙ c_{ψ}
= (f ∗ (τ ⊙ c_{g(μ)})) ⊙ c_{ψ}
= (f ∗ τ) ⊙ (f ∗ c_{g(μ)}) ⊙ c_{ψ}
= (f ∗ τ) ⊙ c_{f(g(μ))∘ψ}

whence e(f_{!}g(s)) = g′(d(s)), as claimed. ♢

Now, by assumption both f_{!}g and e are isomorphisms; in light of claim 1.3.24, it then suffices to check that g′ is an isomorphism. By unwinding the definitions, we see that the latter assertion amounts to the following. For every Y ∈ Ob(𝒟) he rule μ ↦ μ ∗ ob yields a bijection from the set of all cones μ : c_{Y} ⇒ f ∘ F onto the set of all cones λ : c_{Y} ⇒ f ∘ F ∘ ob such that λ ∗ t = ((f ∘ F) ∗ m) ⊙ (λ ∗ s). In turns, this follows by a direct inspection. □

Proposition 1.3.25. Let 𝒜 , ℬ be two categories, F : 𝒜 → ℬ a functor that admits a left adjoint G : ℬ → 𝒜. The following holds :

(i) If 𝒜 and ℬ are small, we have natural isomorphisms of functors

F^{∧} →^{∼} G_{∗} F_{!} →^{∼} G^{∧}.

(ii) If 𝒜 and ℬ have small Hom-sets, then for every small category I and every functor φ : I → 𝒜, there is a natural isomorphism of presheaves

G^{∧}(lim_{I} φ) →^{∼} lim_{I} (F ∘ φ).

(iii) F commutes with all limits of 𝒜.

(iv) Dually, G commutes with all colimits of ℬ.

(v) F transforms monomorphisms into monomorphisms, and G transforms epimorphisms into epimorphisms.

Proof. (i): In view of example 1.2.7(ii), the assertion means that G^{∧} is left adjoint to F^{∧}; however F^{o} is left adjoint to G^{o} (remark 1.1.19(iv)), so it suffices to apply remark 1.1.19(iii), which shows more precisely that ε^{∧} and η^{∧} are respectively the unit and counit of a unique adjunction for the pair (G^{∧}, F^{∧}).

(ii): We are easily reduced to the case where 𝒜 and ℬ are small. Set L := lim_{I} F ∘ φ. According to remark 1.2.11(i), we have a natural morphism ω : lim_{I} φ → F^{∧}L, to which the adjunction exhibited in the proof of (i) attaches a unique morphism ω^{∗} : G^{∧}(lim_{I} φ) → L. We shall show more precisely that ω^{∗} is an isomorphism of presheaves. Indeed, let B ∈ Ob(ℬ) be any object; by unwinding the definitions, we see that ω_{B}^{∗} is given by the rule

μ ↦ μ^{∗} := ((F ∗ μ) ⊙ c_{η_{B}} : c_{B} ⇒ F ∘ φ) for every cone μ : c_{GB} ⇒ φ

(notation of (1.2.9)). We provide an explicit inverse for the rule μ ↦ μ^{∗}, by setting

τ^{∗} := (ε ∗ φ) ⊙ (G ∗ τ) : c_{GB} ⇒ φ for every cone τ : c_{B} ⇒ F ∘ φ.

Clearly the rule τ ↦ τ^{∗} defines a morphism of presheaves L → G^{∧}(lim_{I} φ). Then, for every such τ the triangular identities of (1.1.13) give :

(τ^{∗})^{∗} = (F ∗ ((ε ∗ φ) ⊙ (G ∗ τ))) ⊙ c_{η_{B}}
= (F ∗ ε ∗ φ) ⊙ (F ∗ G ∗ τ) ⊙ c_{η_{B}}
= (F ∗ ε ∗ φ) ⊙ (η ∗ F ∗ φ) ⊙ τ
= τ.

Likewise one sees that (μ^{∗})^{∗} = μ for every cone μ : c_{GB} ⇒ φ, as needed.

(iii): Again, we can assume that 𝒜 and ℬ are small. We have just seen that the adjunction for the pair (G^{∧}, F^{∧}) exhibited in (i) transforms the morphism ω into an isomorphism ω^{∗}. By example 1.2.7(ii), it follows that any adjunction for the pair (F_{!}, F^{∧}) will also transform ω into an isomorphism, whence the contention. (Notice that F_{!} is – a priori – well defined only as a functor on V-presheaves, for some universe V such that 𝒜 and ℬ are small V-categories, but (i) implies that actually it is – up to natural isomorphism – already defined on U-presheaves.)

(iv) follows as usual, by considering the opposite categories and taking into account remark 1.1.19(iv). Lastly, (v) follows from (iii),(iv) and proposition 1.3.18. □

Let 𝒜 be a subcategory of a category ℬ, and φ : I → 𝒜 a functor from a small category I, whose limit is representable in 𝒜 by an object A. In this situation – and even in case 𝒜 is a full subcategory of ℬ – it is not necessarily true that A represents the limit of the composition I → ℬ of φ with the inclusion functor 𝒜 → ℬ. We have nevertheless the following result :

Corollary 1.3.26. Let 𝒜 , ℬ be two categories, F : 𝒜 → ℬ a fully faithful functor admitting a left adjoint G : ℬ → 𝒜, and φ : I → 𝒜 a functor from a small category I. We have :

(i) If the limit of F ∘ φ is representable by some L ∈ Ob(ℬ), then the limit of φ is representable by some object M ∈ Ob(𝒜).

(ii) In the situation of (i), there exist isomorphisms L →^{∼} F M in ℬ and M →^{∼} GL in 𝒜. (iii) Especially, if ℬ is complete, the same holds for 𝒜.

Proof. Let η : 1_{ℬ} ⇒ F ∘ G be a unit and ε : G ∘ F ⇒ 1_{𝒜} a counit for the pair (G, F), and τ : c_{L} ⇒ F ∘ φ a universal cone; set τ^{∗} := (ε ∗ φ) ⊙ (G ∗ τ) : c_{GL} ⇒ φ. There exists a unique morphism f : F GL → L in ℬ such that τ ⊙ c_{f} = F ∗ τ^{∗}. We have :

\begin{aligned}τ ⊙ c_{f∘η_{L}} = τ ⊙ c_{f} ⊙ c_{η_{L}} = (F ∗ τ^{∗}) ⊙ (η ∗ c_{L}) & = (F ∗ ε ∗ φ) ⊙ (F G ∗ τ) ⊙ (η ∗ c_{L}) \\ & = (F ∗ ε ∗ φ) ⊙ (η ∗ F φ) ⊙ τ = τ \end{aligned}

whence f ∘ η_{L} = 1_{L}, by the universality of τ. On the other hand, since F is fully faithful, there exists a morphism h : GL → GL in 𝒜 such that F h = η_{L} ∘ f : F GL → F GL. Now, the adjunction of the pair (G, F) assigns to h the morphism F h ∘ η_{L} = η_{L} ∘ f ∘ η_{L} = F (1_{GL}) ∘ η_{L} (see (1.1.13)), so we must have h = 1_{GL}, therefore f is an isomorphism. Lastly, since F is fully faithful, we deduce easily that GL represents the limit of F, whence the corollary. □

1.4. Special properties of the categories of presheaves. In this section we gather some results concerning the structure of categories of presheaves and the exactness properties of various natural functors between such categories. We begin with a corollary of proposition 1.3.25:

Corollary 1.4.1. Let 𝒜 and ℬ be two categories. We have :

(i) If 𝒜 is small and ℬ has small Hom-sets, the category Fun(𝒜 , ℬ) has small Hom-sets. (ii) Suppose that ℬ is complete or cocomplete. Then, for every A ∈ Ob(𝒜) the functor

ε_{A} : Fun(𝒜 , ℬ) → ℬ  F ↦ F A  (η : F ⇒ G) ↦ (η_{A} : F A → GA)

commutes with all colimits and with all limits. (iii) If ℬ is finitely complete (resp. finitely cocomplete) then ε_{A} commutes with finite limits

(resp. with finite colimits), for every A ∈ Ob(𝒜). (iv) If ℬ is complete (resp. cocomplete, resp. finitely complete, resp. finitely cocomplete),

the same holds for Fun(𝒜 , ℬ).

Proof. (i): If F, G : 𝒜 → ℬ are any two functors, then Hom_{Fun(𝒜 ,ℬ)} (F, G) is a subset of the product ∏_{A∈Ob(𝒜)} Hom_{ℬ} (F A, GA), hence it is a small set.

(iv): Let I be any small category, and

F : I → Fun(𝒜 , ℬ)  i ↦ (F_{i} : 𝒜 → ℬ)  (φ : i → j) ↦ (F_{φ} : F_{i} ⇒ F_{j})

any functor. For every A ∈ Ob(𝒜) we obtain a functor F_{A} : I → ℬ by the rule

F_{A}(i) := F_{i}A F_{A}(φ) := (F_{φ})_{A} : F_{i}A → F_{j}A

for every i, j ∈ Ob(I) and every φ ∈ Hom_{I}(i, j). If ℬ is complete, for every such A we may find LA ∈ Ob(ℬ) and a universal cone γ_{A} : c_{LA} ⇒ F_{A}. Next, if g : A → A′ is any morphism in 𝒜, we obtain a natural transformation F_{g} : F_{A} ⇒ F_{A′} by the rule

(F_{g})_{i} := F_{i}(g) : F_{i}A → F_{i}A′ for every i ∈ Ob(I)

and the limit of F_{g} is represented by a morphism Lg : LA → LA′ in ℬ such that

F_{g} ∘ γ_{A} = γ_{A′} ∘ c_{Lg}.

It is then easily seen that the rules A ↦ LA and g ↦ Lg amount to a well defined functor 𝒜 → ℬ that represents the limit of F : details left to the reader. Lastly, if ℬ is cocomplete (resp. finitely cocomplete), ℬ^{o} is complete (resp. finitely complete), so the same holds for Fun(𝒜^{o}, ℬ^{o}), by the foregoing; then Fun(𝒜 , ℬ) is cocomplete (resp. finitely complete), by remark 1.1.19(i).

(ii): Suppose that ℬ is complete (resp. cocomplete); in this case, the proof of (iv) already shows that ε_{A} commutes with all limits (resp. colimits). Hence, it remains only to check that ε_{A} commutes as well with all colimits (resp. limits). To this aim, we may assume that 𝒜 and ℬ are both small, and we denote by 1 the category with one object and one morphism; we let i_{A} : 1 → 𝒜 be the (unique) functor that sends the object of 1 to A. We have a commutative diagram of functors

[figure: Fun(𝒜 , ℬ); Fun(i_{A},ℬ); ε_{A}; Fun(1, ℬ); ℬ]

whose vertical arrow is an obvious isomorphism of categories. It then suffices to check that the functor Fun(i_{A}, ℬ) commutes with all colimits (resp. limits). However, by theorem 1.3.4, the functor Fun(i_{A}, ℬ) admits a right (resp. left) adjoint, so it suffices to invoke proposition 1.3.25(ii,iii).

(iii) follows likewise from the proof of (iv). □

Remark 1.4.2. (i) By inspecting the proof of theorem 1.3.4, it is easily seen that, more generally, the functors ε_{A} of corollary 1.4.1 commute with limits when ℬ is a category with representable coproducts. Dually, if all products of ℬ are representable, ε_{A} commutes with colimits.

(ii) In the same vein, suppose moreover that Hom_{𝒜} (A, A′) is a finite set for every A, A′ ∈ Ob(𝒜). Then, again by inspection of the proof of theorem 1.3.4 we see that for the functors ε_{A} to commute with limits (resp. colimits) it suffices that all finite coproducts (resp. finite products) of ℬ are representable.

(iii) Furthermore, the proof of corollary 1.4.1(iv) shows more precisely the following. If F : I → Fun(𝒜 , ℬ) is a functor from a small category I, and the limit (resp. colimit) of ε_{A} ∘F is representable in ℬ for every A ∈ Ob(𝒜), then the limit (resp. colimit) of F is representable in Fun(𝒜 , ℬ), and ε_{A} commutes with this limit (resp. colimit), for every A ∈ Ob(𝒜).

Corollary 1.4.3. Let 𝒞 be any small category, and f : F → G a morphism in 𝒞^{∧}. (i) 𝒞^{∧} has small Hom-sets and is both well-powered and co-well-powered (see (1.1.2)). (ii) For every X ∈ Ob(𝒞), the functor

𝒞^{∧} → Set : F ↦ F X (f : F → G) ↦ (f_{X} : F X → GX)

commutes with all limits and all colimits (in other words, the limits and colimits in 𝒞^{∧} are computed argumentwise).

(iii) f is a monomorphism (resp. an epimorphism) if and only if the map f_{X} : F X → GX

is injective (resp. surjective) for every X ∈ Ob(𝒞). (iv) f is an isomorphism if and only if it is both a monomorphism and an epimorphism.

(v) Let f(F) ∈ Ob(𝒞^{∧}) be the subobject of F with f(F)(X) := Im(f_{X} : F X → GX) for every X ∈ Ob(𝒞). Then f(F) = Im(f) = Coim(f) (see example 1.2.26).

(vi) The Yoneda embedding h_{𝒞} : 𝒞 → 𝒞^{∧} commutes with all representable limits of 𝒞. (vii) If 𝒟 is any category with small Hom-sets and φ : 𝒞 → 𝒟 any functor, then φ^{∧}

commutes with all limits and all colimits of 𝒟^{∧}, and φ_{!} (resp. φ_{∗}) commutes with all colimits (resp. with all limits) of 𝒞^{∧}.

Proof. (i): The first assertion is a special case of corollary 1.4.1(i). Next, to see that 𝒞^{∧} is wellpowered notice that for every presheaf F on 𝒞, and every X ∈ Ob(𝒞), the set of subsets of F (X) is small, and a subobject of F is just a compatible system of subsets F′(X) ⊂ F (X), for X ranging over the small set of objects of 𝒞. Likewise one sees that 𝒞^{∧} is co-well-powered.

(ii) is a special case of corollary 1.4.1(ii), but it is also already clear from example 1.2.24(i). (iii): It is easily seen that if f_{X} is injective (resp. surjective) for every X ∈ Ob(𝒞), then f

is a monomorphism (resp. an epimorphism). The converse follows from (ii) and proposition 1.3.18.

(iv) is an immediate consequence of (iii).

(v): It follows from (iii) that the induced morphism F → f(F) is a quotient of F, and the sequence F → f(F) → G is the unique factorization of f (up to unique isomorphism) as a composition of an epimorphism followed by a monomorphism, whence the assertion.

(vi) follows directly from example 1.2.24(ii) and remark 1.3.14(i).

(vii) is a special case of proposition 1.3.22(iii,iv). □

Example 1.4.4. Let 𝒞 be a category with small Hom-sets, and Y ∈ Ob(𝒞) any object.

(i) Suppose that all the products of small families of objects of 𝒞 are representable, and consider the functor

Y^{•} : Set^{o} → 𝒞^{∧}

which to any (U-small) set S assigns the product in 𝒞 of S copies of Y, i.e. the limit of the functor φ_{Y}^{S} : S → 𝒞 such that φ_{Y} (s) := Y for every s ∈ S (where S is regarded as a discrete category), and to any map of sets f : T → S, attaches the morphism of presheaves Y^{f} := lim_{f} φ_{Y}. The assumption on 𝒞 implies that Y^{•} is representable by a functor which we denote as well

Y^{•} : Set^{o} → 𝒞

(see remark 1.2.8(ii)). On the other hand, we have the functor h_{Y} : 𝒞^{o} → Set such that h_{Y} (X^{o}) := Hom_{𝒞} (X, Y) for every X ∈ Ob(𝒞), and notice the natural identifications :

Hom_{𝒞^{o}}((Y^{S})^{o}, X^{o}) = Hom_{𝒞} (X, Y^{S}) →^{∼} Hom_{Set}(S, h_{Y} (X^{o}))

which say that h_{Y} is right adjoint to the functor (Y^{•})^{o}. By proposition 1.3.25(iii) it follows that h_{Y} commutes with all small limits of 𝒞^{o}.

(ii) Likewise, if all the small coproducts of 𝒞 are representable, we may apply (i) to the functor h_{Y^{o}} : 𝒞 → Set. We find that h_{Y^{o}} commutes with all small limits of 𝒞.

(iii) Especially, let I, 𝒞 be two small categories, F : I → 𝒞 a functor, and G any presheaf on 𝒞. From (i),(ii), example 1.2.24(i) and corollary 1.4.3(vi) we get natural isomorphisms :

(1.4.5)  lim_{I^{o}} Hom_{𝒞^{∧}}(h_{𝒞}^{o} ∘ F^{o}, G) →^{∼} Hom_{𝒞^{∧}}(colim_{I} h_{𝒞} ∘ F, G)
(1.4.6)  lim_{I} Hom_{𝒞^{∧}}(G, h_{𝒞} ∘ F) →^{∼} Hom_{𝒞^{∧}}(G, lim_{I} F)

(more precisely, the limit on the left is represented by the set on the right, in both cases). Furthermore, since the tautological cone τ (resp. the tautological cocone μ) for F is universal, we get the universal cone Hom_{𝒞^{∧}}(τ, G) (resp. Hom_{𝒞^{∧}}(G, μ)) for the limit appearing in (1.4.5) (resp. in (1.4.6) : see remarks 1.2.11(iv)) and 1.3.14(i)).

1.4.7. It is an important fact that every presheaf on a small category is the colimit of a system of representable presheaves; more generally, for any category 𝒞 and every F ∈ Ob(𝒞^{∧}), let us consider the category of elements of F :

ℱ ib(F)

whose objects are the pairs (X, s) where X ∈ Ob(𝒞) and s ∈ F X (the notation shall be explained in (3.1.15)). The morphisms f : (X, s) → (Y, t) in ℱ ib(F) are the morphisms f : X → Y of 𝒞 such that F f(t) = s. We have an obvious source functor

s_{F} : ℱ ib(F) → 𝒞  (X, s) ↦ X  ((X, s) ⟶^{f} (Y, t)) ↦ (X ⟶^{f} Y).

Notice that if 𝒞 is small, the same holds for ℱ ib(F). Also, if 𝒞 has small Hom-sets, Yoneda’s lemma naturally identifies ℱ ib(F) with the category h_{𝒞} 𝒞 /F (notation of (1.1.27) and (1.2.4)); namely, a pair (X, s) corresponds to the unique morphism of presheaves

h_{(X,s)} : h_{X} → F such that h_{(X,s),X}(1_{X}) = s.

Moreover, in this case s_{F} is identified with the source functor s_{F} : h_{𝒞} 𝒞 /F → 𝒞 of (1.1.27). We have a natural cocone

h^{F} : h_{𝒞} ∘ s_{F} ⇒ c_{F} (X, s) ↦ h_{(X,s)}.

Lemma 1.4.8. With the notation of (1.4.7), for every category 𝒞 and every presheaf F on 𝒞, the cocone h^{F} is universal.

Proof. Let G be any presheaf on 𝒞. A natural transformation τ : h_{𝒞} ∘ s_{F} ⇒ c_{G} is a rule that assigns to every (X, s) ∈ Ob(h_{𝒞} 𝒞 /F) a morphism of presheaves h_{X} → G; the latter is naturally identified with an element τ_{(X,s)} ∈ GX. Thus, τ induces a map τ_{(X,•)} : F X → GX for every X ∈ Ob(𝒞), by the rule : (X, s) ↦ τ_{(X,s)}. Moreover, the naturality of τ implies that

Gf(τ_{(Y,s)}) = τ_{(X,s)} for every morphism f : (X, s) → (Y, t) in h_{𝒞} 𝒞 /F

(details left to the reader); i.e. the rule X ↦ τ_{(X,•)} for every X ∈ Ob(𝒞) yields a morphism of presheaves h^{τ} : F → G, and it is easily seen that τ = c_{h^{τ}} ⊙ h^{F}, whence the contention. □

Example 1.4.9. (i) Let 𝒞 be a small category, and 𝟙_{𝒞} a final object of 𝒞^{∧} (see example 1.2.16(v)); since the source functor s_{𝟙_{𝒞}} : h_{𝒞} 𝒞 /𝟙_{𝒞} → 𝒞 is an isomorphism of categories, we deduce from lemma 1.4.8a natural isomorphism

colim_{𝒞}h_{𝒞} →^{∼} 𝟙_{𝒞} in 𝒞^{∧} .

(ii) Let 𝒞 be a category with small Hom-sets, X ∈ Ob(𝒞), and h_{X} : 𝒞^{o} → Set the corresponding presheaf (notation of (1.2.4)). By inspecting the definition, we find that

ℱ ib(h_{X}) = 𝒞 /X

and under this identification, the source functor s_{h_{X}} of (1.4.7) corresponds to the source functor s_{X} of (1.1.24). So, lemma 1.4.8in this case shows that X represents in 𝒞 the colimit of h_{𝒞} ∘s_{X}.

(iii) More generally, if 𝒜 and ℬ are two categories, F : 𝒜 → ℬ any functor, we have

ℱ ib(h_{B} ∘ F) = F 𝒜 /B for every B ∈ Ob(ℬ)

(notation of (1.1.27)) and again, the respective source functors agree, under this identification. Hence, if 𝒜 is small and ℬ has small Hom-sets, then the presheaf h_{B} ∘ F on 𝒜 is isomorphic to the colimit of the functor h_{𝒜} ∘ s_{B} : F 𝒜 /B → 𝒜^{∧}.

1.4.10. We conclude this section with a few observations concerning presheaves on the slice categories of (1.1.24). Thus, let 𝒞 be a category, X any object of 𝒞, denote by s_{X} : 𝒞/X → 𝒞 the source functor of (1.1.24), and suppose that 𝒞 is V-small, for some universe V. To begin with, a direct inspection of (1.3.8) yields a natural isomorphism :

(1.4.11)  (s_{X})_{V!}F (Y) →^{∼} {(φ, a) | φ ∈ Hom_{𝒞} (Y, X), a ∈ F (φ)}

for every V-presheaf F on 𝒞/X and every Y ∈ Ob(𝒞). Indeed, from loc.cit. we see that every element of (s_{X})_{V!}F (Y) is the equivalence class of a datum

(ψ : Y → Z, α : Z → X, s ∈ F (α))

and such a datum is equivalent to (1_{Y} , α ∘ ψ : Y → X, F (ψ)(s) ∈ F (α ∘ ψ)). If ψ : Z → Y is any morphism of 𝒞, the corresponding map (s_{X})_{V!}F (Y) → (s_{X})_{V!}F (Z) is given by the rule :

(1.4.12)  (φ, a) ↦ (φ ∘ ψ, F (ψ)(a))  for every φ : Y → X and a ∈ F (φ).

Moreover, under this natural identification the adjoint pair ((s_{X})_{V!}, s_{X}^{∧}) assigns to any morphism t : F → s_{X}^{∧}G of presheaves on 𝒞/X the morphism t^{∗} : (s_{X})_{V!}F → G in 𝒞^{∧} given by the rule :

(φ, a) ↦ t(φ)(a) for every Y ∈ Ob(𝒞), φ : Y → X and a ∈ F (φ).

Proposition 1.4.13. Let 𝒞 be any category and X any object of 𝒞.

(i) The source functor s_{X} : 𝒞/X → 𝒞 commutes with all connected limits and with the colimit of every functor F : I → 𝒞/X such that colim_{I} s_{X} ∘ F is representable in 𝒞. (ii) Dually, the target functor t_{X} : X/𝒞 → 𝒞 commutes with all connected colimits and with the limit of every functor F:I →X/𝒞 such that lim_{I} t_{X} ∘F is representable in 𝒞.

(iii) If all the binary products of 𝒞 are representable, s_{X} admits a right adjoint, and there-

fore it commutes with all colimits of 𝒞/X. (iv) Dually, if all the binary coproducts of 𝒞 are representable, t_{X} admits a left adjoint,

and therefore it commutes with all limits of X/𝒞.

(v) If 𝒞 is complete (resp. cocomplete, resp. finitely complete, resp. finitely cocomplete), then the same holds for both 𝒞/X and X/𝒞.

(vi) Suppose that 𝒞 has small Hom-sets. Then the following holds : (a) The functor s_{X}^{∧} admits a left adjoint s_{X!} : (𝒞/X)^{∧} → 𝒞^{∧}.

(b) Moreover s_{X!} factors as the composition of an equivalence of categories

e_{X} : (𝒞/X)^{∧} → 𝒞^{∧}/h_{X}

and the source functor s_{h_{X}} : 𝒞^{∧}/h_{X} → 𝒞^{∧} (notation of (1.2.4)).

(c) s_{X!} commutes with fibre products and preserves monomorphisms.

Proof. (v): Suppose that 𝒞 is complete (resp. finitely complete); to show that the same holds for 𝒞/X, we may apply the criterion of proposition 1.2.22(i). Indeed, suppose that p_{•} := (p_{i} : Y_{i} → X | i ∈ I) is any family (resp. any finite family) of objects of 𝒞/X, indexed by a small set I, which we regard as a discrete category. We consider the category I_{X} whose set of objects is the disjoint union of I and {X}, and with

Morph(I_{X}) = Morph(I) ∪ {(X, X, 1_{X})} ∪ {(i, X, p_{i}) | i ∈ I}

so that X is the unique final object of I_{X}. Next, we define a functor F : I_{X} → 𝒞, by letting

F X := X  and F i := Y_{i} F p_{i} := p_{i}  for every i ∈ I.

Any universal cone τ : c_{L} ⇒ F yields a morphism τ_{X} : L → X, and it is easily seen that the object τ_{X} of 𝒞/X represents the product of the family p_{•}. Lastly, let p_{i} : Y_{i} → X (i = 1, 2) be two objects of 𝒞/X, and f, g : p_{1} → p_{2} two morphisms in 𝒞/X. Denote by E any object of 𝒞 representing the equalizer of s_{X}(f), s_{X}(g) : Y_{1} → Y_{2} in 𝒞, where s_{X} : 𝒞/X → 𝒞 is the source functor. The corresponding universal cone yields a morphism h : E → Y_{1} such that s_{X}(f) ∘ h = s_{X}(g) ∘ h, and it is easily seen that the object p_{1} ∘ h : E → X of 𝒞/X represents the equalizer of f and g (details left to the reader).

Next, suppose that 𝒞 is cocomplete (resp. finitely cocomplete), and let F : I → 𝒞/X be a functor from a small (resp. finite) category I. Let also C be any object of 𝒞 representing the colimit of s_{X} ∘ F; the functor F can be regarded as a cocone s_{X} ∘ F ⇒ c_{X}, which corresponds to a morphism τ : C → X, and it is easily seen that τ ∈ Ob(𝒞/X) represents the colimit of F.

The assertions for X/𝒞 follows as usual, by considering the opposite categories.

(vi.a): The discussion of (1.4.10) yields a left adjoint (s_{X})_{V!} to (s_{X})_{V}^{∧}, for any universe V such that 𝒞 has V-small Hom-sets. However, by inspecting (1.4.11), we see that if 𝒞 has small Hom-sets and F is a U-presheaf on 𝒞/X, then (s_{X})_{V!}F is a U-presheaf. Since the inclusion (𝒞/X)_{U}^{∧} → (𝒞/X)_{V}^{∧} is fully faithful, we deduce that the target of (1.4.11) can be used to define the sought left adjoint (s_{X})_{!} : (𝒞/X)_{U}^{∧} → 𝒞_{U}^{∧} to (s_{X})_{U}^{∧}.

(vi.b): Notice that the presheaf 𝟙_{𝒞 /X} := h_{1_{X}} is a final object of (𝒞 /X)^{∧} : indeed, for every object f : Y → X of 𝒞 /X we have 𝟙_{𝒞 /X} (f) = {f/X} (notation of (1.1.24)). Then remark 1.3.6(v) yields a natural isomorphism :

s_{X!}(𝟙_{𝒞 /X}) →^{∼} h_{X}

that identifies the unit of adjunction with the morphism of presheaves :

(1.4.14) 𝟙_{𝒞 /X} → s_{X}^{∧}(h_{X})  such that f/X ↦ f  for every f ∈ Ob(𝒞 /X).

Now, every presheaf F on 𝒞 /X admits a unique morphism F → 𝟙_{𝒞 /X}, whence a natural morphism s_{X!}F → h_{X}, so we see already that s_{X!} factors through s_{h_{X}} and a functor e_{X} as required; more precisely, (1.4.12), says that the morphism e_{X}(F) : s_{X!}F → h_{X} is given by the rule : (f, a) ↦ f for every Y ∈ Ob(𝒞) and every (f, a) ∈ s_{X!}F (Y). To check that e_{X} is an equivalence, we exhibit an explicit quasi-inverse : namely, to every object φ : F → h_{X} of 𝒞^{∧}/h_{X} we assign the presheaf

e′_{X}(φ) := s_{X}^{∧}(F) ×_{s_{X}^{∧}(h_{X})} 𝟙_{𝒞 /X}

defined via (1.4.14) and the morphism s_{X}^{∧}(φ) : s_{X}^{∧}(F) → s_{X}^{∧}(h_{X}). Now, let F be any presheaf on 𝒞 /X, and f : Y → X any object of 𝒞 /X; by unwinding the definitions, we see that

(e′_{X} ∘ e_{X}(F))(f) = s_{X!}F (Y) ×_{h_{X}(Y)} {f/X}

which is the set of all pairs (f, a) with a ∈ F f. Thus we get an obvious isomorphism of presheaves e′_{X} ∘ e_{X}(F) →^{∼} F, natural with respect to F. Likewise, for every object φ : G → h_{X} of 𝒞^{∧}/h_{X} and every Y ∈ Ob(𝒞) we get the map

(e_{X} ∘ e′_{X}(φ))_{Y} : s_{X!}(s_{X}^{∧}(F) ×_{s_{X}^{∧}(h_{X})} 𝟙_{𝒞 /X})(Y) → h_{X}(Y)

which is described as follows. First, s_{X!}(s_{X}^{∧}(F) ×_{s_{X}^{∧}(h_{X})} 𝟙_{𝒞 /X})(Y) is the set of all pairs (f, a) with f : Y → X a morphism of 𝒞, and a ∈ GY ×_{h_{X}(Y)} {f/X}; i.e. the pairs (f, a) with φ_{Y} (a) = f. Then, the map (e_{X} ∘ e′_{X}(φ))_{Y} is given by the rule : (f, a) ↦ f for every such pair (f, a). Again, we deduce an obvious isomorphism e_{X} ∘ e′_{X}(φ) →^{∼} φ of h_{X}-objects of 𝒞^{∧}, natural with respect to φ, as required.

(vi.c) follows from the explicit description of s_{X!} given in (1.4.10), together with corollary 1.4.3(ii,iii).

(i): The assertion for representable colimits follows from the proof of (v) (together with remark 1.3.14(i)). Next, let F : I → 𝒞/X be any functor from a connected small category I; taking into account examples 1.2.7(ii) and (1.4.10) it suffices to show that the natural morphism

((s_{X})_{V!} lim_{I} F)(Z) → (lim_{I} s_{X} ∘F)(Z) (φ : Z → X, τ : c_{φ} ⇒ F) ↦ (s_{X} ∗τ : c_{Z} ⇒ s_{X} ∘F)

is an isomorphism for every Z ∈ Ob(𝒞) and any sufficiently large universe V. Now, let τ : c_{Z} ⇒ s_{X} ∘ F be any cone (with vertex an arbitrary Z ∈ Ob(𝒞)); we remark :

Claim 1.4.15. For every i, j ∈ Ob(I) we have F i ∘ τ_{i} = F j ∘ τ_{j} : Z → X.

Proof of the claim. Since I is connected, a simple induction argument reduces to the case where there exists k ∈ Ob(I) with morphisms a : k → i and b : k → j. In this case, we may compute directly : F i ∘ τ_{i} = F i ∘ s_{X}(F a) ∘ τ_{k} = F_{k} ∘ τ_{k} = F j ∘ s_{X}(F b) ∘ τ_{k} = F j ∘ τ_{j}. ♢

Hence, pick any i ∈ Ob(I) (recall that this set is non-empty), and set φ := F i ∘ τ_{i}; claim 1.4.15 says that τ lifts uniquely to a cone τ_{/X} : c_{φ} ⇒ F, such that s_{X} ∗ τ_{/X} = τ, whence the contention.

(iv): Pick a universe V such that 𝒞 has V-small Hom-sets. For every Y ∈ Ob(𝒞), the functor

𝒞 → 𝒞_{V}^{∧} Y ↦ X ⨿ Y

is then representable by a functor 𝒞 → 𝒞, and we abuse notation, by denoting both functors in the same way. Moreover, the universal cocone for the coproduct consists of a pair of morphisms

X ⟶^{i_{X,Y}} X ⨿ Y ⟵^{i′_{X,Y}} Y whence a functor

i_{X} : 𝒞 → X/𝒞 : Y ↦ i_{X,Y} for every Y ∈ Ob(𝒞).

It is easily seen that i_{X} is left adjoint to t_{X} (details left to the reader). Thus, the assertion follows from proposition 1.3.25(i,iii).

(iii): Notice that the source functor s_{X} : 𝒞/X → 𝒞 equals t_{X^{o}}^{o} and the coproducts of 𝒞^{o} are representable; then the assertion follows from (iv) and remark 1.1.19(iv). The same argument shows that (i) implies (ii). □

Remark 1.4.16. Let 𝒞 and X be as in proposition 1.4.13(vi).

(i) The functor s_{X!} does not generally preserve final objects, hence it is not generally exact. (ii) The quasi-inverse to e_{X} constructed in the proof of proposition 1.4.13(vi.b), can be

described more compactly as the functor that assigns to any object g : F → h_{X} of 𝒞^{∧}/h_{X} the presheaf given by the rule :

(Y ⟶^{φ} X) ↦ Hom_{𝒞^{∧}/h_{X}} ((h_{Y} ⟶^{h_{φ}} h_{X}), g).

Example 1.4.17. (i) Let 𝒞 be a category whose binary products are representable, and pick a universe V such that 𝒞 has V-small Hom-sets. For every X ∈ Ob(𝒞), we have a functor

p_{X} : 𝒞 → 𝒞_{V}^{∧} Y ↦ X × Y

and the assumption on 𝒞 implies that p_{X} is representable by a functor π_{X} : 𝒞 → 𝒞 (remark 1.2.8(ii)). Let now F : I → 𝒞 be a functor from a small category I; we say that the colimit of F commutes with products, if π_{X} commutes with the colimit of F, for every X ∈ Ob(𝒞).

(ii) A more restricted class of colimits plays a role in applications. Namely, suppose now that all fibre products of 𝒞 are representable. Let F be as in (i), and consider any cone τ : F ⇒ c_{X}, for any X ∈ Ob(𝒞). Notice that τ is the same as the datum of a functor F_{τ} : I → 𝒞/X which lifts F, i.e. such that s_{X} ∘ F_{τ} = F, where s_{X} : 𝒞/X → 𝒞 is the source functor. Moreover, the assumption on 𝒞 implies that the products are representable in 𝒞/X, for every such X. In this situation, we shall then say that the colimit of F is universal, if the colimit of F_{τ} commutes with products, for every X and τ as above.

(iii) Suppose moreover that the colimit of F is representable by an object C^{o} of 𝒞^{o}, and fix a universal cocone μ : F ⇒ c_{C}. Then any τ as in (ii) corresponds to a unique morphism f : C → X, and μ lifts to a universal cocone μ_{τ} : F_{τ} ⇒ c_{f}. For every morphism g : Y → X in 𝒞 we may consider the functor

F ×_{(τ,g)} Y := s_{X} ∘ π_{g} ∘ F_{τ} : I → 𝒞

(that is, F ×_{(τ,g)} Y (i) represents F i ×_{(τ_{i},g)} Y for every i ∈ Ob(I)). Likewise, we abuse notation and write C ×_{(f,g)} Y to denote the representative s_{X} ∘ π_{g}(f) for this fibre product; taking into account proposition 1.4.13(v), we see that the colimit of F is universal if and only if the cocone

(s_{X} ∘ π_{g}) ∗ μ_{τ} : F ×_{(τ,g)} Y ⇒ c_{C×_{(f,g)}Y}

is universal for every such X, τ and g.

(iv) For instance, from the explicit descriptions of example 1.2.23(i) one may deduce that all colimits in Set are universal. In view of example 1.2.24(i) and corollary 1.4.3(ii), it follows easily that all colimits in ℬ^{∧} are universal, for any small category ℬ.

1.4.18. Let F : 𝒜 → ℬ be a functor from a small category 𝒜 to a category ℬ with small Hom-sets. By remark 1.3.6(ii), for every object A ∈ Ob(𝒜) we have a natural isomorphism h_{F A} →^{∼} F_{!}A; combining with example 1.2.28(i), we deduce that the adjoint pair of functors F^{∧} : ℬ^{∧} → 𝒜^{∧} and F_{!} : 𝒜^{∧} → ℬ^{∧} induces an adjoint pair :

(F^{∧})_{|h_{A}} : ℬ^{∧}/h_{F A} →^{∼} ℬ^{∧}/F_{!}h_{A} → 𝒜^{∧}/h_{A} (F_{!})_{|h_{A}} : 𝒜^{∧}/h_{A} → ℬ^{∧}/F_{!}h_{A} →^{∼} ℬ^{∧}/h_{F A}.

For every category I, let I_{∘} be the category such that • Ob(I_{∘}) is the disjoint union of Ob(I) with a set {i_{∘}} • i_{∘} is the unique final object of I_{∘}

• I is a full subcategory of I_{∘} (more precisely, the full subcategory of I_{∘} whose set of

objects is Ob(I) coincides with I). It is clear that these conditions determine uniquely I_{∘}. Moreover, for every category 𝒞 and every C ∈ Ob(𝒞), every functor φ : I → 𝒞 /C extends uniquely to a well defined functor φ_{∘} : I_{∘} → 𝒞 /C such that φ_{∘}(i_{∘}) = 1_{C}.

Proposition 1.4.19. (i) With the notation of (1.4.18), for every A ∈ Ob(𝒜) we have essentially commutative diagrams :

\xymatrix{(ℬ/F A)^{∧} \ar[d]_{e_{F A}} \ar[r]^{(F_{|A})^{∧}} & (𝒜 /A)^{∧} \ar[d]^{e_{A}} & (𝒜 /A)^{∧} \ar[r]^{(F_{|A})_{!}} \ar[d]_{e_{A}} & (ℬ/F A)^{∧} \ar[d]^{e_{F A}} \\ ℬ^{∧}/h_{F A} \ar[r]^{(F^{∧})_{|h_{A}}} & 𝒜^{∧}/h_{A} & 𝒜^{∧}/h_{A} \ar[r]^{(F_{!})_{|h_{A}}} & ℬ^{∧}/h_{F A}}

where e_{A} and e_{F A} are the equivalences of proposition 1.4.13(vi.b).

(ii) Let moreover φ : I → 𝒜 /A be any functor. Then F_{|A} commutes with the limit of φ if and only if F commutes with the limit of s_{A} ∘ φ_{∘} : I_{∘} → 𝒜.

Proof. (i): The assertion means that there exists an isomorphism of functors e_{A} ∘ (F_{|A})^{∧} →^{∼} (F^{∧})_{|h_{A}} ∘ e_{F A}, and likewise for the second diagram. Let φ : K → h_{F A} be any object of ℬ^{∧}/h_{F A}; with e′_{A} and e′_{F A} as in the proof of proposition 1.4.13(vi.b), we get :

(F_{|A})^{∧} ∘ e′_{F A}(φ) = (F_{|A})^{∧}(s_{F A}^{∧}K ×_{s_{F A}^{∧}h_{F A}} 𝟙_{ℬ/F A} )

→^{∼} (s_{F A} ∘ F_{|A})^{∧}K ×_{(s_{F A}∘F_{|A})^{∧}h_{F A}} 𝟙_{𝒜 /A}

→^{∼} s_{A}^{∧} ∘ F^{∧}K ×_{s_{A}^{∧}∘F^{∧}h_{F A}} 𝟙_{𝒜 /A}

→^{∼} e′_{A}(u^{∧}K ×_{F^{∧}h_{F A}} h_{A} → h_{A})

→^{∼} e′_{A} ∘ (F^{∧})_{|h_{A}}(φ)

whence the essential commutativity of the left square diagram. Since e_{A} and e_{F A} are equivalences, and since (F_{!})_{|h_{A}} is left adjoint to (F^{∧})_{|h_{A}}, we deduce the essential commutativity for the right square diagram as well.

(ii): Let j : I → I_{∘} be the inclusion functor; notice first that for every category 𝒞, every C ∈ Ob(𝒞) and every functor ψ : I → 𝒞 we have ψ = ψ_{∘}j, and the morphism of presheaves

lim_{j} 1_{𝒞} : lim_{I_{∘}} ψ_{∘} → lim_{I} ψ

is an isomorphism (notation of remark 1.2.11(i)). Moreover, it is clear that F_{|A}∘φ_{∘} = (F_{|A}∘φ)_{∘}. There follows a commutative diagram of presheaves :

\xymatrix{(F_{|A})_{!}(lim_{I_{∘}} φ_{∘}) \ar[r] \ar[d] & lim_{I_{∘}} F_{|A} ∘ φ_{∘} \ar[d] \\ (F_{|A})_{!}(lim_{I} φ) \ar[r] & lim_{I} F_{|A} ∘ φ.}

whose vertical arrows are isomorphisms, and whose horizontal arrows are the morphisms of definition 1.3.11(i). Especially, F_{|A} commutes with the limit of φ if and only if it commutes with the limit of φ_{∘}. We are thus reduced to checking the following more general :

Claim 1.4.20. Let J be a connected category, and ψ : J → 𝒜 /A a functor. Then F_{|A} commutes with the limit of ψ if and only if F commutes with the limit of s_{A} ∘ ψ.

Proof of the claim. Recall first that s_{F A!} = s_{h_{F A}} ∘ e_{F A}. Since s_{h_{F A}} is obviously conservative, and e_{F A} is an equivalence, it follows that s_{F A!} is conservative. Recall moreover that s_{A} and s_{F A} commute with every connected limit (proposition 1.4.13(i)); summing up, we see that F_{|A} commutes with the limit of ψ if and only if the morphism

s_{F A!} ∘ (F_{|A})_{!}(lim_{J} ψ) ⟶^{s_{F A!}(ω)} s_{F A!}(lim_{J} F_{|A} ∘ ψ) ⟶^{ω′} lim_{J} s_{F A} ∘ F_{|A} ∘ ψ

is an isomorphism, where ω : (F_{|A})_{!}(lim_{J} ψ) → lim_{J} F_{|A} ∘ ψ and ω′ are the morphisms of definition 1.3.11(i). But this composition is, up to isomorphism, the same as the morphism

(1.4.21)  (F ∘ s_{A})_{!}(lim_{J} ψ) → lim_{J} F ∘ s_{A} ∘ ψ

of definition 1.3.11(i), relative to the functor s_{F A} ∘F_{|A} = F ∘s_{A} and the limit of ψ (see the proof of proposition 1.3.17(i)). By the same token, (1.4.21) is also the composition ω′′′ ∘ F_{!}(ω′′) of the same type, where ω′′ : s_{A!}(lim_{J} ψ) → lim_{J} s_{A} ∘ ψ is an isomorphism (since J is connected), and ω′′′ : F_{!}(lim_{J} ψ) → lim_{J} F ∘ ψ is an isomorphism if and only if F commutes with J. □

1.5. Coinitial and cofinal functors. Let I be a small category, 𝒞 a category, and F : I → 𝒞 any functor. For the computation of the limit or colimit of F, it may sometimes be desirable to replace the indexing category I by simpler ones. That is, we would like to be able to detect whether a given functor φ : J → I induces an isomorphism from the colimit of F to that of F ∘ φ, and if possible, to construct useful functors of this type, to aid with the calculation of limits or colimits. Concerning the first aim, one has a general criterion, for which we shall need the following :

Definition 1.5.1. Let I, J be any two categories, and φ : J → I any functor.

(i) We say that φ is cofinal if the category i/φJ is connected, for every i ∈ Ob(I) (see definition 1.2.19(ii)).

(ii) We say that φ is coinitial if φ^{o} : J^{o} → I^{o} is cofinal.

(iii) If φ is the inclusion functor of a subcategory J of I, and φ is cofinal (resp. coinitial)

we also say that J is cofinal in I (resp. coinitial in I). (iv) We say that the category I is coinitially small (resp. cofinally small), if there exists a

coinitial (resp. cofinal) functor φ : J → I with J a small category. (See corollary 1.6.5.)

Proposition 1.5.2. Let φ : J → I be any functor between small categories. Then the following conditions are equivalent :

(a) For every category 𝒞 with small Hom-sets, and every functor F : I → 𝒞, the functor φ induces an isomorphism of presheaves on I^{o} (see remark 1.2.11(i)) :

(1.5.3)  colim_{φ} 1_{𝒞} : colim_{I} F →^{∼} colim_{J} F ∘ φ.

(b) The functor φ is cofinal.

Proof. (a)⇒(b): We fix i ∈ Ob(I) and we apply the definition to the functor

h_{i^{o}} : I → Set i′ ↦ Hom_{I}(i, i′)

whose colimit is a presheaf on the category Set^{o}. Then the assertion is an immediate consequence of the following :

Claim 1.5.4. For every i ∈ Ob(I) we have :

(i) The colimit of h_{i^{o}} is representable by a set with one element.

(ii) The colimit of h_{i^{o}} ∘ φ is representable by π_{0}(i/φJ) (notation of remark 1.2.21(ii)).

Proof of the claim. Notice that (i) is the special case of (ii) for φ = 1_{I}, since the category i/I admits an initial object, and hence is connected.

(ii): Let h_{I} : I → I^{∧} be the Yoneda embedding and choose a representative L ∈ Ob(I^{∧}) for the colimit of the functor h_{I} ∘ φ. By corollary 1.4.3(ii) and remark 1.3.14(i), the set L(i) represents the colimit of h_{i^{o}} ∘ φ. On the other hand, from remark 1.3.6(ii) we see that L represents as well the colimit of φ_{!} ∘ h_{J}. Moreover, recall that the colimit of h_{J} is represented by the coinitial object 𝟙_{J} of J^{∧} (example 1.4.9(i)); by corollary 1.4.3(vii) and remark 1.3.14(i) we deduce that L is isomorphic to φ_{!}(𝟙_{J}). Then the assertion follows from example 1.3.9. ♢

(b)⇒(a): Let F : I → 𝒞 be any functor; recall that the colimit of F is the functor L : 𝒞 → Set that assigns to any X ∈ Ob(𝒞) the set L(X) of all cocones F ⇒ c_{X}, i.e. all the compatible systems of morphisms (τ_{i} : F i → X | i ∈ Ob(I)) in 𝒞, such that

τ_{i′} ∘ F g = τ_{i} for every morphism g : i → i′ in I.

Likewise, the colimit of F ∘ φ is the functor L′ : 𝒞 → Set that assigns to any X ∈ Ob(𝒞) the set L′(X) of all compatible systems (μ_{j} : F φ(j) → X | j ∈ Ob(J)) in 𝒞, such that

μ_{j′} ∘ F φ(h) = μ_{j} for every morphism h : j → j′ in J.

Under these identifications, (1.5.3) is the map that assigns to a given τ as above, the compatible system τ ∗ φ, such that (τ ∗ φ)_{j} := τ_{φ(j)} for every j ∈ Ob(J). We have to check that the rule τ ↦ τ ∗ φ is a bijection β_{X} : L(X) →^{∼} L′(X). However, say that τ ∗ φ = τ′ ∗ φ for given τ, τ′ ∈ L(X), and consider any i ∈ Ob(I); since i/φJ is non-empty, we may find j ∈ Ob(J) and a morphism g : i → φ(j) in I, whence

τ_{i} = τ_{φ(j)} ∘ F g = τ′_{φ(j)} ∘ F g = τ′_{i}

hence β_{X} is injective. Next, let μ ∈ L′(X) be any compatible system; we consider the map

τ^{∗} : Ob(I/φJ) → Morph(𝒞) (g : i → φ(j)) ↦ μ_{j} ∘ F g

(notation of (1.1.28) and (1.1.30)). We claim that τ^{∗} factors through the map

Ob(I/φJ) → Ob(I) : (g : i → φ(j)) ↦ i.

Indeed, say that g : i → φ(j) and g′ : i → φ(j′) are any two morphisms in I; we have to show that τ_{g}^{∗} = τ_{g′}^{∗}. To this aim, since i/φJ is connected, we may assume that there exists a morphism h : g → g′ in i/φJ; then

τ_{g}^{∗} = μ_{j} ∘ F g = μ_{j′} ∘ F φ(h) ∘ F g = μ_{j′} ∘ F g′ = τ_{g′}^{∗}

as stated. Thus, let τ : Ob(I) → Morph(𝒞) be the resulting map; we claim that τ is an element of L(X). Indeed, say that h : i → i′ is any morphism in I, and pick any j, j′ ∈ Ob(J) with morphisms g : i → φ(j), g′ : i′ → φ(j′) in I; we have just seen that τ_{g′∘h}^{∗} = τ_{g}^{∗}, which translates as the identity : τ_{i} = τ_{i′} ∘ F h, whence the claim. Lastly, by construction we have β_{X}(τ) = μ, so β_{X} is also surjective. □

Remark 1.5.5. (i) If φ : I → J and ψ : J → K are cofinal functors, the same holds for ψ ∘ φ. For the proof, we may replace our universe U by a larger one, after which we may assume that I, J, K are small categories, and then the assertion follows directly from proposition 1.5.2.

(ii) Let φ : J → I be a functor between small categories. Then φ is coinitial if and only if it induces isomorphisms of presheaves on I

(1.5.6)  lim_{I} F →^{∼} lim_{J} F ∘ φ

for every category 𝒞 with small Hom-sets, and every functor F : I → 𝒞.

(iii) Let φ : J → I and F : I → 𝒞 be any two functors; suppose that the induced morphism of presheaves colim_{φ} 1_{𝒞} : colim_{I} F → colim_{J} F ∘ φ is an isomorphism, and let τ : F ⇒ c_{L} be a cocone. Then τ is universal if and only if the same holds for τ ∗ φ : F ∘ φ ⇒ c_{L}. Indeed, τ and τ ∗ φ determine morphisms of presheaves t : h_{L^{o}} → colim_{I} F and t′ : h_{L^{o}} → colim_{J} F ∘ φ on 𝒞^{o} that assign to every X ∈ Ob(𝒞) and every section f : L → X of h_{L^{o}}(X^{o}) the cocone τ ⊙ c_{f} ∈ colim_{I} F (X^{o}) and respectively the cocone (τ ∗ φ) ⊙ c_{f} ∈ colim_{J} F ∘ φ(X), and t (resp. t′) is an isomorphism if and only if τ (resp. τ ∗ φ) is universal. However, (colim_{φ} 1_{𝒞} ) ∘ t is the morphism of presheaves h_{L^{o}} →^{∼} colim_{J} F ∘ φ that assigns to every such X and f the cone (τ ⊙ c_{f}) ∗ φ = (τ ∗ φ) ⊙ c_{f}, ı.e. t′ = (colim_{φ} 1_{𝒞} ) ∘ t. So t is an isomorphism if and only if the same holds for t′, whence the contention. Likewise, if (1.5.6) is an isomorphism, a cone τ′ : c_{L′} ⇒ F is universal if and only if the same holds for τ′ ∗ φ : c_{L′} ⇒ F ∘ φ.

Lemma 1.5.7. Let I, J be two categories, and φ : J → I a functor. We have :

(i) If J is filtered, then φ is cofinal if and only if the following two conditions hold : (a) For every i ∈ Ob(I) there exist j ∈ Ob(J) and a morphism i → φ(j) in I.

(b) For every i ∈ Ob(I), every j ∈ Ob(J) and every pair of morphisms f, g : i → φ(j) in I, there exists a morphism h : j → j′ in J such that φ(h) ∘ f = φ(h) ∘ g.

(ii) If J is filtered and φ is cofinal, then I is filtered.

(iii) Suppose that I is filtered, and condition (i.a) is fulfilled. Then the following holds : (a) If φ is full, then φ is cofinal and J is directed.

(b) If φ is fully faithful, J is filtered.

Proof. (i): Indeed, suppose that the categories i/φJ are connected for every i ∈ Ob(I); then (a) clearly holds. To check that (b) holds as well, notice that, since J is filtered, i/φJ is locally directed for every i ∈ Ob(I) (details left to the reader), and therefore it is directed, by remark 1.2.21(i). This implies that, for given f, g as in (b), there exist j′ ∈ Ob(J) and morphisms h_{1}, h_{2} : j → j′ such that φ(h_{1}) ∘ f = φ(h_{2}) ∘ g. But since J is filtered, we may then find a morphism h′ : j′ → j′′ in J such that h′ ∘ h_{1} = h′ ∘ h_{2}, so (b) holds with h := h′ ∘ h_{1}.

Conversely, if (a) holds, then i/φJ is non-empty for every i ∈ Ob(I). Next, let g : i → φ(k) and g′ : i → φ(k′) be any two objects of i/φJ; since J is directed we may find morphisms h : k → j and h′ : k′ → j for some j ∈ Ob(J), whence objects φ(h) ∘ g, φ(h′) ∘ g′ : i → φ(j) of i/φJ, and using (b) we deduce that there exist an object g′′ : i → φ(j′) and morphisms g → g′′, g′ → g′′, i.e. i/φJ is directed.

(ii): Let us check that I is directed : if i and i′ are two objects of I, in view of condition (i.a), we may find j, j′ ∈ Ob(J) and morphisms f : i → φ(j), f′ : i′ → φ(j′) in I, and since J is directed, we have as well morphisms g : j → j′′ and g′ : j′ → j′′ in J, for some j′′ ∈ Ob(J), whence morphisms φ(g) ∘ f : i → φ(j′′) and φ(g′) ∘ f′ : i′ → φ(j′′) in I. By remark 1.2.21(i) it remains to check the coequalizing condition of definition 1.2.19(v), but the latter is an immediate consequence of conditions (i.a) and (i.b).

(iii.a): It has already been remarked that condition (i.a) says that the category i/φJ is nonempty for every i ∈ Ob(I). Next, let g : i → φ(k) and g′ : i → φ(k′) be two objects of i/φJ. Since I is filtered, we may find i′ ∈ Ob(I) and morphisms h : φ(k) → i′ and h′ : φ(k′) → i′ of I such that f := h∘g = h′ ∘g′, and since (i.a) holds, we may find k′′ ∈ Ob(J) with a morphism l : i′ → φ(k′′) in I; then, since φ is full, there exist morphisms t : k → k′′ and t′ : k′ → k′′ in J such that φ(t) = l ∘ h and φ(t′) = l ∘ h′. Then l ∘ f : i → φ(k′′) is an object of i/φJ with morphisms t : g → l ∘ f and t′ : g′ → l ∘ f, so i/φJ is directed, and this shows that φ is cofinal.

Next, let j, j′ ∈ Ob(J); since I is filtered, there exists i ∈ Ob(I) with morphisms g : φ(j) → i and g′ : φ(j′) → i in I. Since (i.a) holds, there exists a morphism h : i → φ(k) in I, and since φ is full, there exist morphisms t : j → k and t′ : j′ → k in J with φ(t) = h ∘ g and φ(t′) = h ∘ g′; this shows that J is directed.

(iii.b): Clearly the assumptions show that Ob(J) ≠ ∅, and we know already that J is directed, by (iii.a). Next, let g, g′ : j → j′ be two morphisms in J; since I is filtered, we have a morphism h : φ(j′) → i in I such that h ∘ φ(g) = h ∘ φ(g′), and since (i.a) holds, there exists j′′ ∈ Ob(J) with a morphism l : i → φ(j′′) in I. Since φ is full, there exists a morphism t : j′ → j′′ such that φ(t) = l ∘ h, so that φ(t ∘ g) = φ(t ∘ g′), and since φ is faithful, we must then have t ∘ g = t ∘ g′, so J is filtered, by remark 1.2.21(i). □

Remark 1.5.8. In light of lemma 1.5.7(i), we see that if I is a filtered (resp. cofiltered) preordered set and J is a subset, then J is cofinal (resp. coinitial) in I in the sense of [36, Ch.III, §8, n.7] if and only if the inclusion functor J → I is cofinal (resp. coinitial) in the sense of our definition 1.5.1(i,ii).

Example 1.5.9. (i) For instance, if I is a filtered category, and i is any object of I, the category i/I is again filtered, and the target functor t_{i} : i/I → I is cofinal; indeed, one checks easily that t_{i} fulfills conditions (a) and (b) of lemma 1.5.7(i). Dually, if I is cofiltered, the category I/i is again cofiltered, and the source functor s_{i} : I/i → I is coinitial.

(ii) If I admits a final object i_{0}, then the (full) subcategory J of I with Ob(J) = {i_{0}} fulfills conditions (a) and (b) of lemma 1.5.7(i), so the inclusion functor J → I is cofinal (and I is trivially filtered). Then, let F : I → 𝒞 be any functor, and τ : F ⇒ c_{L} a universal cocone; taking into account remark 1.5.5(iii), we see that F i_{0} represents the colimit of F, and τ_{i_{0}} : F i → L is an isomorphism in 𝒞. Dually, if i′_{0} is any initial object of I, and τ′ : c_{L′} ⇒ F any universal cone, then F i′_{0} represents the limit of F and τ_{i′_{0}} : L′ → F i′_{0} is an isomorphism.

Example 1.5.10. As an application, we give an explicit construction of the filtered colimits of Cat. Thus, let I be a small filtered category, and consider a functor

𝒞_{•} : I → Cat  i ↦ 𝒞_{i}  (φ : i → j) ↦ (𝒞_{φ} : 𝒞_{i} → 𝒞_{j}).

We deduce a functor Ob(𝒞_{•}) : I → Set that assigns to every i ∈ Ob(I) the set Ob(𝒞_{i}), and to every morphism φ : i → j in I the map Ob(𝒞_{i}) → Ob(𝒞_{j}) defined by 𝒞_{φ}. We set

L := colim_{I} Ob(𝒞_{•}).

According to example 1.2.23(i) – and since I is filtered – the elements of L are the equivalence classes [i, X] of pairs (i, X), where i ∈ Ob(I) and X ∈ Ob(𝒞_{i}), for the equivalence relation such that (i, X) ∼ (j, Y) if and only if there exist k ∈ Ob(I) and morphisms φ_{1} : i → k and φ_{2} : j → k such that 𝒞_{φ_{1}}X = 𝒞_{φ_{2}}Y. For every i ∈ Ob(I), let t_{i} : i/I → I be the target functor (see (1.1.24)); notice that for every pair of objects i, j ∈ Ob(I), the objects of the category (i, j)/I := i/I ×_{(t_{i},t_{j})} j/I are the pairs i ⟶^{φ_{1}} k ⟵^{φ_{2}} j of morphisms of I, and the morphisms

(1.5.11)  (i, j)/ν : (i ⟶^{φ_{1}} k ⟵^{φ_{2}} j) → (i ⟶^{φ′_{1}} k′ ⟵^{φ′_{2}} j)

are the morphisms ν : k → k′ of I such that ν ∘ φ_{r} = φ′_{r} for r = 1, 2. For every couple of pairs ((i, X), (j, Y)) as in the foregoing, we get a functor h(i, X, j, Y) : (i, j)/I → Set that assigns to every (i ⟶^{φ_{1}} k ⟵^{φ_{2}} j) ∈ Ob((i, j)/I) the set Hom_{𝒞_{k}} (𝒞_{φ_{1}}X, 𝒞_{φ_{2}}Y), and to every morphism (i, j)/ν as in (1.5.11) the induced map

Hom_{𝒞_{k}} (𝒞_{φ_{1}}X, 𝒞_{φ_{2}}Y) → Hom_{𝒞_{k′}} (𝒞_{φ′_{1}}X, 𝒞_{φ′_{2}}Y) f ↦ 𝒞_{ν}(f)

and we set

H(i, X, j, Y) := colim_{(i,j)/I} h(i, X, j, Y).

Let also t_{i,j} : (i, j)/I → I be the target functor given by the rules : (i ⟶^{φ_{1}} k ⟵^{φ_{2}} j) ↦ k, and (i, j)/ν ↦ ν; for every i, j, k ∈ Ob(I) we set as well (i, j, k)/I := (i, j)/I ×_{(t_{i,j},t_{k})} k/I. Explicitly, the objects of this category are all the systems φ_{•} := (φ_{1} : i → t, φ_{2} : j → t, φ_{3} : k → t) of morphisms of I with a common target t that we call the target of φ_{•}. The morphisms

(i, j, k)/ν : φ_{•} → φ′_{•}

are the morphisms ν of I with ν ∘ φ_{r} = φ′_{r} for r = 1, 2, 3. With this notation, for every third pair (k, Z) as in the foregoing, we get a system of composition maps

h(i, X, j, Y)(φ_{•}) × h(j, Y, k, Z)(φ_{•}) → h(i, X, k, Z)(φ_{•}) for every φ_{•} ∈ Ob((i, j, k)/I)

given by the composition law of the category 𝒞_{t}, where t is the target of φ_{•}. Clearly this system of maps is natural with respect to morphisms of (i, j, k)/I; moreover we have obvious projection functors (i, k)/I ← (i, j, k)/I → (i, j)/I and (i, j, k)/I → (j, k)/I, and using the criterion of lemma 1.5.7(i) it is easily seen all that these three functors are cofinal. Therefore, taking colimits over (i, j, k)/I of the foregoing compatible system of composition maps yields a well defined map

H(i, X, j, Y) × H(j, Y, k, Z) → H(i, X, k, Z) (f, g) ↦ g ∘ f

and a simple inspection of the construction shows that this composition law is associative : if h ∈ H(k, Z, t, W) is any other element, we have h ∘ (g ∘ f) = (h ∘ g) ∘ f (details left to the reader); likewise, the class of 1_{X} in H(i, X, i, X) yields a left and right unit for this composition law. Hence, let us choose for every A ∈ L a representative (i_{A}, X_{A}); we obtain a well defined category ℒ with Ob(ℒ) = L and with Hom_{ℒ} (A, B) := H(i_{A}, X_{A}, i_{B}, X_{B}) for every A, B ∈ L, with the composition law thus defined. Moreover, for every i ∈ I we obtain a well defined functor G_{i} : 𝒞_{i} → ℒ by setting GX := [i, X] for every X ∈ Ob(𝒞_{i}), and by assigning to every morphism X → Y of 𝒞_{i} its class in H(i, X, i, Y). Lastly, let F : 𝒞_{•} ⇒ 𝒟 be a cocone with basis 𝒞_{•} and whose vertex is any category 𝒟. The induced cocone Ob(𝒞_{•}) ⇒ Ob(𝒟) yields a well defined map L → Ob(𝒟). Moreover, for every two pairs (i, X) and (j, Y) and every object (i ⟶^{φ_{1}} k ⟵^{φ_{2}} j) of (i, j)/I, the functor F_{k} yields a map

Hom_{𝒞_{k}} (𝒞_{φ_{1}}X, 𝒞_{φ_{2}}Y) → Hom_{𝒟} (F_{k} ∘ 𝒞_{φ_{1}}X, F_{k} ∘ 𝒞_{φ_{2}}Y) = Hom_{𝒟} (F_{i}X, F_{j}Y)

which defines a cocone with vertex Hom_{𝒟} (F_{i}X, F_{j}Y) and for basis the functor h(i, X, j, Y), whence a well defined map

F_{i,X,j,Y} : H(i, X, j, Y) → Hom_{𝒟} (F_{i}X, F_{j}Y).

A simple inspection shows that F_{i,X,k,Z}(g ∘ f) = F_{j,Y,k,Z}(g) ∘ F_{i,X,j,Y} (f) for every f, g as in the foregoing (details left to the reader). We obtain therefore a well defined functor F : ℒ → 𝒟 by the rule : A ↦ F_{i_{A}}X_{A} and F_{AB}f := F_{i_{A},X_{A},i_{B},X_{B}}(f) for every A, B ∈ L and every f ∈ Hom_{ℒ} (A, B). By construction, we have F ∘ G_{i} = F_{i} for every i ∈ I, and F is clearly the unique functor fulfilling this system of identities, so the proof is concluded.

Remark 1.5.12. (i) Let I be a coinitially small category, 𝒞 a category with small Hom-sets, and F : I → 𝒞 any functor. By definition, there exists a coinitial functor φ : J → I with J a small category, and therefore we may define the presheaf

L_{F,φ} := lim_{J} F ∘ φ.

We claim that L_{F,φ} is independent - up to natural isomorphism - of the choice of φ. To see this, let us choose a universe U′ such that U ⊂ U′ and such that I is a small U′-category. By remark 1.5.5(i), the induced morphism of U′-presheaves

ω_{φ} : lim_{I} F → L_{F,φ}

is an isomorphism. Therefore, if φ′ : J′ → I is any other coinitial functor with J′ also small, we get an isomorphism of U′-presheaves ω_{φ′} ∘ ω_{φ}^{−1} : L_{F,φ} →^{∼} L_{F,φ′}. Since the inclusion (1.2.1) is fully faithful, the latter is actually an isomorphism of U-presheaves, as required.

(ii) In view of (i), we may define the limit of F as L_{F,φ}, for any choice of φ. This presheaf is therefore well defined, up to natural isomorphism, and we denote it as usual by

lim_{I} F.

We also say that lim_{I} F is a coinitially small limit of 𝒞. Of course, if I is already small, we can choose φ := 1_{I}, so the above notation is compatible with that of definition 1.2.10(i).

(iii) Dually, if I is cofinally small, we can define the colimit of F

colim_{I} F := colim_{J} F ∘ φ

for any choice of cofinal functor φ : J → I with J small, and again the resulting presheaf is well defined up to natural isomorphism, and we also say that it is a cofinally small colimit of 𝒞. Clearly, if 𝒞 is complete (resp. cocomplete), then every coinitially small (resp. cofinally small) limit is representable in 𝒞. As an application, we may prove Freyd’s adjoint functor theorem, which is the following partial converse of proposition 1.3.25(iii) :

Theorem 1.5.13. Let 𝒜 be a complete category, F : 𝒜 → ℬ a functor, and suppose that 𝒜 and ℬ have small Hom-sets (see (1.1.1)). The following conditions are equivalent :

(a) F admits a left adjoint.

(b) F commutes with all the limits of 𝒜, and every object B of ℬ admits a solution set, i.e. an essentially small subset S_{B} ⊂ Ob(𝒜) such that, for every A ∈ Ob(𝒜), every morphism f : B → F A admits a factorization of the form

f = F h ∘ g

for h : A′ → A a morphism in 𝒜 with A′ ∈ S_{B}, and g : B → F A′ a morphism in ℬ.

Proof. If F admits a left adjoint G, then F commutes with all limits of 𝒜, by proposition 1.3.25(iii), and it is easily seen that {GB} is a solution set for every object B ∈ Ob(ℬ).

Conversely, fix any B ∈ Ob(ℬ); under the stated assumptions, example 1.3.16(i) says that the category B/F 𝒜 is cofiltered. Denote by ℰ_{B} the full subcategory of B/F 𝒜 whose objects are all the morphisms B → F A with A ∈ S_{B}. Then condition (b) means that for every object X of B/F 𝒜 there exists an object X′ ∈ ℰ_{B} with a morphism X′ → X of B/F 𝒜. It follows easily that ℰ_{B} is cofiltered as well; then the inclusion functor ℰ_{B} → B/F 𝒜 is coinitial, by lemma 1.5.7(i), and ℰ_{B} is small, hence B/F 𝒜 is coinitially small. Following remark 1.5.12(ii), we may thus define

G′B := lim_{B/F 𝒜} t_{B} G′f := lim_{f/F 𝒜} 1_{𝒜}

for any B ∈ Ob(ℬ) and any morphism f : B′ → B (notation of (1.1.27)). Next, since 𝒜 is complete, G′ is representable by some functor G : ℬ → 𝒜 (remark 1.2.8(ii)), and we claim that G is the sought left adjoint. To check this assertion, we may replace the universe U by a larger one, after which we may assume that 𝒜 and ℬ are small, and therefore G′B is the usual limit of definition 1.2.10(i). We shall then exhibit explicit unit and counit for the adjoint pair (G, F), as follows. Let τ^{B} : c_{GB} ⇒ t_{B} be a universal cone; since F commutes with limits, the cone F ∗ τ^{B} : c_{F GB} ⇒ F ∘ t_{B} is still universal (remark 1.3.14(i)); on the other hand, the functor B/F : B/F 𝒜 → B/ℬ can be regarded as a cone B/F : c_{B} ⇒ F ∘ t_{B}, so there exists a unique morphism η_{B} : B → F GB such that B/F = (F ∗ τ^{B}) ⊙ c_{η_{B}}. Likewise, for any A ∈ Ob(𝒜) we get a morphism ε_{A} := τ_{(A,1_{F A})}^{F A} : GF A → A. It is easily seen that η (resp. ε) is a natural transformation 1_{ℬ} ⇒ F G (resp. GF ⇒ 1_{𝒜} ), and the identity (F ∗ ε) ⊙ (η ∗ F) = 1_{F} is immediate from the construction. Lastly, we fix B ∈ Ob(ℬ) and we check that (ε ∗ G)_{B} ∘ (G ∗ η)_{B} = 1_{G(B)}; to this aim, and due the universality of τ^{B}, it suffices to show that

τ_{ψ}^{B} = τ_{ψ}^{B} ∘ ε_{GB} ∘ G(η_{B}) for every object ψ : B → F A of B/F 𝒜 .

However, notice that the construction of Gf yields the identity

(1.5.14) τ_{ψ}^{B′} ∘ Gf = τ_{ψ∘f}^{B}  for all morphisms f : B → B′ and ψ : B′ → F A in 𝒞 .

Thus, we may compute :

τ_{ψ}^{B} ∘ ε_{GB} ∘ G(η_{B}) = τ_{ψ}^{B} ∘ τ_{1_{F GB}}^{F GB} ∘ G(η_{B}) = τ_{F (τ_{ψ}^{B})}^{F GB} ∘ G(η_{B}) = τ_{F (τ_{ψ}^{B})∘η_{B}}^{B} = τ_{ψ}^{B}

where the second equality holds by the naturality of τ^{F GB}, the third follows from (1.5.14), and the fourth comes from the construction of η_{B}. □

Of course, the “dual” of theorem 1.5.13yields a criterion for the existence of right adjoints.

Example 1.5.15. (i) Let J and K be any two small categories; set I := J ×K and let π : I → J be the projection functor. For every j ∈ Ob(J) we have an induced functor

ι_{j} : K → j/πI  k ↦ ((j, k), 1_{j})  (g : k → k′) ↦ (1_{j}, g)

and we claim that ι_{j} is coinitial (definition 1.5.1(ii)). Indeed, for any ((j_{0}, k_{0}), f_{0} : j → j_{0}) ∈ Ob(j/πI), the category ι_{j}K/((j_{0}, k_{0}), f_{0}) is isomorphic to K/k_{0}, which is obviously connected, whence the claim.

(ii) Moreover, every morphism f : j → j′ in J induces a natural transformation

β^{f} : ι_{j} ⇒ (f/πI) ∘ ι_{j′}

which assigns to every k ∈ Ob(K) the morphism (f, 1_{k}) : ((j, k), 1_{j}) → ((j′, k), f). Then, t_{j} ∗ β^{f} is a natural transformation t_{j} ∘ ι_{j} ⇒ t_{j} ∘ (f/πI) ∘ ι_{j′} = t_{j′} ∘ ι_{j′} and a direct inspection of the definitions yields a commutative diagram for every functor F : I → 𝒞

\xymatrix{∫_{π}^{∧} F (j) \ar[r]^{∫_{π}^{∧} F (f)} \ar[d] & ∫_{π}^{∧} F (j′) \ar[d] \\ lim_{K} F ∘ t_{j} ∘ ι_{j} \ar[r]^{lim_{K}(F ∗t_{j}∗β^{f})} & lim_{K} F ∘ t_{j′} ∘ ι_{j′}}

whose vertical arrows are the natural isomorphisms provided by (i) and remark 1.5.5(ii) and whose bottom arrow is given by remark 1.2.11(ii). In other words, the functor ∫_{π}^{∧} is naturally isomorphic to the one that assigns to every such F the functor

∫_{K}^{∧} F : J → 𝒞^{∧}  j ↦ lim_{K} F ∘ t_{j} ∘ ι_{j}  (f : j → j′) ↦ lim_{K}(F ∗ t_{j} ∗ β^{f}).

Lastly, proposition 1.3.2 can be restated in this case more synthetically, as the existence of a natural isomorphism in 𝒞^{∧}

lim_{J×K} F →^{∼} Lim_{J} ∫_{K}^{∧} F for every functor F : J × K → 𝒞 .

(iii) On the other hand, we may also apply the same considerations to the second projection functor I → K. Summing up, we deduce a natural isomorphism

Lim_{J} ∫_{K}^{∧} F →^{∼} Lim_{K} ∫_{J}^{∧} F for every functor F : J × K → 𝒞

which expresses the well known interchange property for double limits.

(iv) Dually, we obtain as well an interchange property for double colimits, that states the existence of natural isomorphisms

Colim_{J} ∫_{∧}^{K} F →^{∼} Colim_{J×K} F →^{∼} Colim_{K} ∫_{∧}^{J} F for every functor F : J × K → 𝒞

where ∫_{∧}^{K} is the opposite of the functor ∫_{K^{o}}^{∧} (and likewise for ∫_{∧}^{J} : details left to the reader).

Remark 1.5.16. (i) Keep the situation of example 1.5.15, and suppose moreover that 𝒞 is complete (resp. cocomplete). Then the functor ∫_{K}^{∧} (resp. ∫_{∧}^{K}) is isomorphic to the composition of Fun(J, h_{𝒞} ) (resp. of Fun(J, h_{𝒞^{o}}^{o})) and a functor

∫_{K} : Fun(I, 𝒞) → Fun(J, 𝒞) (resp. ∫ ^{K} : Fun(I, 𝒞) → Fun(J, 𝒞))

that is well defined up to isomorphism, and is isomorphic to ∫_{π} (resp. to ∫^{π}), so the interchange properties of example 1.5.15(iii,iv) can, in this case, be expressed also in terms of these new functors (details left to the reader).

(ii) Furthermore, if 𝒞 is both complete and cocomplete, we get a natural morphism

(1.5.17)  Colim_{J} ∫_{K} F → Lim_{K} ∫ ^{J} F  for every functor F : J × K → 𝒞

as follows. First, for every (j, k) ∈ Ob(J × K) we have a natural morphism

ω_{j,k} : ∫_{K} F (j) → F (j, k) → ∫ ^{J} F (k)

given by the choice of a universal cone and cocone for the limit and colimit that are represented respectively by the source and target of this morphism. By inspecting the constructions, we then check easily that, for fixed j ∈ Ob(J), the system (ω_{j,k} | k ∈ Ob(K)) is a cone with vertex ∫_{K} F (j); the choice of a universal cone for the functor ∫^{K} F then determines a unique morphism

ω_{j} : ∫_{K} F (j) → Lim_{J} ∫ ^{K} F for every j ∈ Ob(J).

In turns, the system (ω_{j} | j ∈ Ob(J)) is a cocone which determines (1.5.17), after fixing a universal cocone for the functor ∫_{K} F. Of course, (1.5.17) depends on the choices of all these universal cones and cocones, but two different sets of such choices will modify the morphism by left and right composition with uniquely determined isomorphisms.

(iii) The question rises, whether (1.5.17) is an isomorphism. This is not always the case. If (1.5.17) is an isomorphism for every functor F : J × K → 𝒞, we say that in the category 𝒞 the limits indexed by K commute with the colimits indexed by J.

(iv) For instance, if K is finite and J is filtered, then the limits indexed by K commute with the colimits indexed by J in the category Set ([28, Th.2.13.4]). We say briefly that the finite limits commute with the filtered colimits in Set.

Taking into account example 1.2.24(i), we deduce more generally that, for any category 𝒞, the finite limits in 𝒞^{∧} commute with all filtered colimits.

Here is another useful application to Kan extensions :

Proposition 1.5.18. In the situation of theorem 1.3.4, suppose that φ is fully faithful. Then the same holds for the right (resp. left) Kan extension along φ.

Proof. Consider the right Kan extension along φ (and hence, we assume that the relevant completeness conditions for 𝒞 as in remark 1.3.3(iii,iv) are fulfilled). Let F : I → 𝒞 be any functor, and to ease notation set K := ∫_{φ} F; by proposition 1.1.20(iii), it suffices to exhibit an isomorphism of functors ε : K ∘ φ →^{∼} F. Now, recall that Kj ∈ 𝒞 represents the limit of the functor F ∘ t_{j} : j/φI → 𝒞, for every j ∈ Ob(J), and the construction of K involves the choice of a universal cone τ^{j} : c_{Kj} ⇒ F ∘ t_{j} for every such j. However, notice that (i, 1_{φ(i)}) is an initial object of φ(i)/φI : indeed, let (i′, f : φ(i) → φ(i′)) be any other object; since φ is fully faithful, there exists a unique morphism g : i → i′ in J such that φ(g) = f, and then (g, 1_{f}) is the unique morphism (i, 1_{φ(i)}) → (i′, f) in φ(i)/φI. By example 1.5.9(ii), it follows that

ε_{i} := τ_{(i,1_{φ(i)})}^{φ(i)} : K ∘ φ(i) → F i

is an isomorphism for every i ∈ Ob(I). It remains to check the naturality of the rule : i ↦ ε_{i}. Hence, let g : i → i′ be any morphism of I; by construction, for every X ∈ Ob(𝒞) we have a commutative diagram of sets :

\xymatrix{h_{Kφ(i)}(X) \ar[r] \ar[d]_{h_{Kφ(g)}} & lim_{φ(i)/φI} F ∘ t_{φ(i)}(X) \ar[d] \\ h_{Kφ(i′)}(X) \ar[r] & lim_{φ(i′)/φI} F ∘ t_{φ(i′)}(X)}

where h denotes as usual the Yoneda embedding, and where the top and bottom horizontal arrows are the bijections induced by τ^{i} and τ^{i′} respectively. The right vertical arrow is the map that sends every cone β : c_{X} ⇒ F ∘ t_{φ(i)} to the cone β ∗ (φg/φI) : c_{X} ⇒ F ∘ t_{φ(i′)}, where φg/φI : φ(i′)/φI → φ(i)/φI is the functor defined as in (1.1.27). Taking X := Kφ(i), we see that the top horizontal (resp. left vertical) arrow maps 1_{Kφ(i)} to τ^{φ(i)} (resp. to Kφ(g)); then we get the identity :

τ^{φ(i′)} ⊙ c_{Kφ(g)} = τ^{φ(i)} ∗ (φg/φI)

whence ε^{i′} ∘ Kφ(g) = τ_{(i′,φ(g))}^{φ(i)} : Kφ(i) → F i′. Lastly, the morphism g induces a morphism

φ(i)/g : (i, 1_{φ(i)}) → (i′, φ(g)) of φ(i)/φI, whence the identity : τ_{(i′,φ(g))}^{φ(i)} = F g ∘ τ_{(i,1_{φ(i)})}^{φ(i)}. Summing up, we conclude that

ε^{i′} ∘ Kφ(g) = F g ∘ ε^{i}

as required. The assertion for left Kan extensions admits the dual proof. □

Corollary 1.5.19. Let ℬ, 𝒞 be two small categories, and F : ℬ → 𝒞 a functor. We have : (i) If ℬ is finitely complete and F is left exact, then F_{!} is exact. (ii) F is fully faithful ⇔ F_{!} is fully faithful ⇔ F_{∗} is fully faithful.

Proof. (i): Under these assumptions, the category Y/F ℬ is cofiltered for every Y ∈ Ob(𝒞) (example 1.3.16(i)), hence (Y/F ℬ)^{o} is filtered. However, the filtered colimits in the category Set commute with all finite limits (see remark 1.5.16(iv)), so the assertion follows from remark 1.3.6(iii) and proposition 1.3.25(iv).

(ii): By proposition 1.5.18 we know that if F is fully faithful, the same holds for F_{!} and F_{∗}. Also, by proposition 1.1.20(iv), F_{!} is fully faithful if and only if the same holds for F_{∗}. Lastly, if F_{!} is fully faithful, then (1.3.7) and the full faithfulness of the Yoneda embeddings, imply that F is fully faithful. □

Example 1.5.20. Let I be any small category; we wish to apply Freyd’s adjoint theorem 1.5.13 in order to exhibit a left adjoint

Colim_{I} : Fun(I, Cat) → Cat

for the corresponding constant functor c for the category Cat. In particular, this will prove that all the colimits indexed by I are representable in Cat, and since I is arbitrary, we will conclude that Cat is cocomplete. Now, we know already that Cat is complete (example 1.2.25(i)), and c commutes with all limits of Cat, by virtue of corollary 1.4.1(ii). It remains to check that every functor 𝒞_{•} : I → Cat admits a solution set S ⊂ Ob(Cat). To this aim, let 𝒟 := ⨿_{i∈Ob(I)}𝒞_{i} be the coproduct of the family of categories (𝒞_{i} | i ∈ Ob(I)), as in example 1.2.25(i); for every small category ℬ and every co-cone F_{•} : 𝒞_{•} ⇒ ℬ, let F : 𝒟 → ℬ be the resulting functor, and denote by Im(F) ⊂ ℬ the image of F, defined as in example 1.2.26 (it is easily seen that Cat is well-powered, so the image is well defined). The system of categories

ℱ := (Im(F) | ℬ ∈ Ob(Cat), F_{•} : 𝒞_{•} → ℬ)

is not small, but if we pick in ℱ a representative for each isomorphism class, we do get a small family (details left to the reader); it is easily seen that any such choice of representatives yields a solution set for 𝒞_{•}.

In applications, often the indexing category of a limit (or colimit) is a partially ordered set, and usually such limits are easier to handle than limits over general indexing categories. Now, given such a general indexing category I, one may try to find a cofinal functor J → I from a partially ordered set J. The following proposition – whose part (ii) is due to Deligne ([8, Exp.I, Prop.8.1.6]) – says that this can always be achieved.

Proposition 1.5.21. Let I be any category. The following holds :

(i) There exists a cofinal functor e : J → I, with J a partially ordered set.

(ii) If moreover I is filtered, we can find a cofinal functor e as in (i), with J filtered as well. (iii) In the situation of (i) or (ii), if furthermore I is small, we can take J to be also small.

Proof. (i): Let ℕ be the filtered category associated with the totally ordered set of natural numbers, and 𝒫_{0}(ℕ) the set of finite non-empty subsets of ℕ; we endow each Σ ∈ 𝒫_{0}(ℕ) with the restriction of the ordering of ℕ, so that Σ is a full subcategory of ℕ. If Σ′ ⊂ Σ are two elements of 𝒫_{0}(ℕ), and F : Σ → I is any functor, we let F_{|Σ′} : Σ′ → I be the restriction of F. We endow

J := ⋃_{Σ∈𝒫_{0}(ℕ)} Fun(Σ, I)

with the partial ordering such that :

(F′ : Σ′ → I) ≤ (F : Σ → I) ⇔ Σ′ ⊂ Σ and F′ = F_{|Σ′}.

We define a functor e : J → I as follows. For every (F : Σ → I) ∈ J, we let

e(F) := F (s) where s is the largest element of Σ.

For every pair (F′ : Σ′ → I) ≤ (F : Σ → I) of elements of J, let s ∈ Σ and s′ ∈ Σ′ be the respective maximal elements, and ψ_{s′,s} : s′ → s (resp. φ_{F′,F} : F′ → F) the corresponding unique morphism of Σ (resp. of J); then we let

e(φ_{F′,F}) := F (ψ_{s′,s}).

Let now i ∈ Ob(I); we need to check that the category i/eJ is connected. To this aim, let F_{i} : {0} → I be the unique functor such that F_{i}(0) := i; clearly (1_{i} : i → e(F_{i})) ∈ Ob(i/eJ), so Ob(i/eJ) is non-empty. Next, let (F : Σ → I) ∈ J and β : i → i′ := e(F) be any morphism of I; we construct step-by-step a chain of morphisms in i/eJ connecting (β, F) with (1_{i}, F_{i}). First, let s be the largest element of Σ; if s = 0, we take Δ := {0, 1}, and consider the functor G : Δ → I such that G(0) = G(1) := i′, and G(ψ_{0,1}) := 1_{i′}. Thus, F ≤ G, and (β : i → e(G)) ∈ Ob(i/eJ), so we may replace F by G, and assume from start that s > 0. Then we have F′ := F_{|{s}} ≤ F, and (β : i → e(F′)) ∈ Ob(i/eJ). Define F′′ : {0, s} → I by : F′′(0) = i, F′′(s) := i′, and F′′(ψ_{0,s}) := β; again, F′ ≤ F′′, and (β : i → e(F′′)) ∈ Ob(i/eJ). Lastly, F_{i} ≤ F′′, and this completes the sought chain of morphisms.

(ii): Let us say that I admits a largest object, if there exists i ∈ Ob(I) such that Hom_{I}(i′, i) ≠ ∅ for every i′ ∈ Ob(I). We notice :

Claim 1.5.22. Let j ∈ Ob(I) be any object, and (f_{t} : i_{t} → j | t = 1, …, n) any finite system of morphisms in I. If I does not have a largest object, there exists a morphism j → i in I such that i ≠ i_{t} for every t = 1, …, n.

Proof of the claim. Suppose the claim fails, in which case we may assume that j = i_{1}. Since I is filtered, for every k ∈ Ob(I) we may find morphisms g : k → k′ and j → k′, and the assumption implies that k′ = i_{n(k)} for some n(k) ∈ {1, …, n}. Set h := f_{n(k)} ∘ g, so h is a morphism k → i_{1}, and especially Hom_{I}(k, i_{1}) ≠ ∅ for every k ∈ Ob(I), which is absurd, since I does not have a largest object. ♢

Next, define the filtered category ℕ as in the proof of (i); we remark that if I is any filtered category, then ℕ × I is still filtered, and it does not admit a largest object; moreover, one checks easily that the projection functor ℕ × I → I fulfills conditions (a) and (b) of lemma 1.5.7(i), so it is cofinal. In light of remark 1.5.5(i), we may then replace I by ℕ × I, and assume from start that I does not have a largest object.

Let us say that a diagram of I is any pair D := (A, B) with A ⊂ Ob(I), B ⊂ Morph(I), and such that, for every f ∈ B, the source and target of f lie in A. We say that A (resp. B) is the set of objects (resp. morphisms) of D. An element e of A is said to be final in (A, B) if the following holds :

• for every i ∈ Ob(I), the set Hom_{I}(i, e) ∩ B contains exactly one element f_{i}^{D}

• f_{e}^{D} = 1_{e}, and for every morphism g : i → j in B we have f_{j}^{D} ∘ g = f_{i}^{D}. Denote by J the set consisting of all diagrams (A, B) which admit a unique final element e(A, B), and such that A and B are finite sets. We order J by inclusion, i.e. (A, B) ≤ (A′, B′) if and only if A ⊂ A′ and B ⊂ B′. Then, if D, D′ ∈ J and D ≤ D′, there exists a unique morphism e(D, D′) : e(D) → e(D′) in the set of morphisms of D′, and if D ≤ D′ ≤ D′′, then clearly e(D, D′′) = e(D′, D′′) ∘ e(D, D′). Thus, the rule D ↦ e(D) yields a well defined functor e : J → I, and we have to prove that e is cofinal.

To this aim, we check that conditions (a) and (b) of lemma 1.5.7(i) hold for e. Now, if i is any object of I, the diagram D_{i} := ({i}, {1_{i}}) lies in J, and 1_{i} : i → e(D_{i}) is a morphism in I, so condition (a) holds. Next, let D = (A, B) ∈ J be any diagram, i ∈ Ob(I) any object, and f, g : i → e(D) any two morphisms in I; we notice :

Claim 1.5.23. There exist i′ ∈ Ob(I)\A and a morphism h : e(D) → i′ such that h∘f = h∘g.

Proof of the claim. Since I is filtered, we can find a morphism h : e(D) → i′ that coequalizes f and g, and claim 1.5.22says that we may choose i ∉ A. ♢

Now, let i′ be as in claim 1.5.23, and D′ the diagram of I whose set of objects is A ∪ {i′}, and whose set of morphisms is B ∪ {h ∘ f_{j} | j ∈ A}. It is easily seen that D′ ∈ J, and by construction, e(D′) = i′ and e(D, D′) = h, so (b) holds.

It remains to show that J is filtered. However, say that D = (A, B) and D′ = (A′, B′) are any two elements of J; we notice :

Claim 1.5.24. There exist i ∈ Ob(I)\(A∪A′) and morphisms f : e(D) → i and f′ : e(D′) → i in I.

Proof of the claim. This is similar to the proof of claim 1.5.23: we can always find f : e(D) → i and f′ : e(D′) → i, and by claim 1.5.22we may arrange that i ∉ A ∪ A′. ♢

Hence, let f, f′ be as in claim 1.5.24, and denote D(f, f′) the diagram whose set of objects is A ∪ A′ ∪ {i}, and whose set of morphisms is B ∪ B′ ∪ {f ∘ f_{j}^{D} | j ∈ A} ∪ {f′ ∘ f_{j′}^{D′} | j′ ∈ A′}. Clearly, the assertion will follow from :

Claim 1.5.25. We can choose f and f′ so that D(f, f′) ∈ J.

Proof of the claim. It amounts to checking that, for suitable f and f′, we have f ∘ f_{j}^{D} = f′ ∘ f_{j}^{D} for every j ∈ A ∩ A′. However, if f and f′ as in claim 1.5.24, have been picked, a simple induction on the cardinality of A ∩ A′ shows that we may find h : i → i′ such that

h ∘ f ∘ f_{j}^{D} = h ∘ f′ ∘ f_{j}^{D} for every j ∈ A ∩ A′

and by claim 1.5.22, we may arrange that i′ ∉ A ∪ A′. Then the diagram D(h ∘ f, h ∘ f′) will do. ♢

Lastly, if I is small, clearly the same holds for J as exhibited in the proofs of (i) and (ii). □

Example 1.5.26. Let us say that a category I is countable if both Ob(I) and Morph(I) are countable sets. A simple inspection of the proof of proposition 1.5.21(ii) shows that if I is a filtered countable category, there exist a countable filtered partially ordered set J := {j_{n} | n ∈ ℕ} and a cofinal functor J → I. But it is easy to construct a cofinal functor f : ℕ → J (where ℕ is endowed with its standard total ordering) : indeed, we may choose f(0) := j_{0}, and then pick inductively for every n ∈ ℕ an element f(n + 1) ∈ J larger than j_{n+1} and f(n). Summing up, we see that for every countable filtered category I there exists a cofinal functor ℕ → I.

1.6. Localizations of categories. Given a category 𝒞 and an arbitrary set of morphisms Σ ⊂ Morph(𝒞), we wish to describe a general procedure that adds formally to 𝒞 an inverse f^{−1} for every f ∈ Σ. More details can be found in [28, Ch.5]. We begin with the following :

Definition 1.6.1. (i) A graph 𝒢 is the datum of • a set V (𝒢), whose elements are called the vertices of 𝒢

• for every A, B ∈ V (𝒢) a set 𝒢 (A, B), whose elements are called the arrows from A to B. If f ∈ 𝒢 (A, B), we say also that A is the source and B is the target of f.

(ii) A morphism of graphs F : 𝒢 → 𝒢′ consists of • A map of sets F : V (𝒢) → V (𝒢′)

• For every A, B ∈ V (𝒢), a map of sets 𝒢 (A, B) → 𝒢′(F A, F B) : f ↦ F f. (iii) A graph is small if V (𝒢) is a small set, and 𝒢 (A, B) is a small set for all A, B ∈ V (𝒢).

Obviously, morphisms of graphs can be composed. so we have a category

U-Graph

whose objects are all the small graphs. As usual, we shall omit the prefix U, unless we have to deal with more than one universe. Clearly, there is a natural functor

(1.6.2)  Cat → Graph

which assigns to any category 𝒞 its underlying graph, whose vertices are the objects of 𝒞, and whose arrows are the morphisms of 𝒞, and simply forgets the composition law of 𝒞. We are going to construct an explicit left adjoint for the functor (1.6.2). To this aim, let 𝒢 be any graph and n ∈ ℕ any integer; a path of length n in 𝒢 is a sequence

p := (A_{0}, f_{1}, A_{1}, f_{2}, …, A_{n})

alternating vertices A_{0}, …, A_{n} of 𝒢 and arrows f_{1}, …, f_{n}, such that the source and target of f_{i} are respectively A_{i−1} and A_{i} for every i = 1, …, n. Then A_{0} is called the source and A_{n} the target of p. Given paths p := (A_{0}, f_{1}, A_{1}, …, A_{n}) and p′ := (A_{n}, f_{n+1}, A_{n+1}, …, A_{n+m}) of lengths respectively n and m, and such that the target of p equals the source of p′, we obtain a path of length n + m, by the rule :

p′ ∘ p := (A_{0}, f_{1}, …, A_{n}, f_{n}, …, A_{n+m}).

Denote by P (𝒢) the set of all paths of 𝒢 (of arbitrary length); with this composition law, clearly P (𝒢) is the set of morphisms of a path category whose set of objects is V (𝒢) (for any A ∈ V (𝒢), the path (A) of length zero serves as the identity endomorphism of A). Also, if F : 𝒢 → 𝒢′ is any morphism of graphs, we have an induced functor

P (F) : (V (𝒢), P (𝒢)) → (V (𝒢′), P (𝒢′))

whose map on objects is F, and such that P (F)p := (F A_{0}, F f_{1}, …, F A_{n}) for every path p := (A_{0}, f_{1}, …, A_{n}) of 𝒢. We have thus defined a functor

(1.6.3)  Graph → Cat  𝒢 ↦ (V (𝒢), P (𝒢)).

Proposition 1.6.4. The functor (1.6.3) is left adjoint to the forgetful functor (1.6.2).

Proof. Indeed, given a graph 𝒢, a category 𝒞 and a morphism of graphs F : 𝒢 → 𝒞, define a functor G : (V (𝒢), P (𝒢)) → 𝒞 by the rule

GA := F A Gp := F f_{n} ∘ ⋯ F f_{1}

for every A ∈ V (𝒢) and every p := (A_{0}, f_{1}, …, A_{n}) ∈ P (𝒢). Clearly G is the unique functor whose restriction to 𝒢 agrees with F, whence the contention. □

Corollary 1.6.5. A category 𝒞 is coinitially (resp. cofinally) small if and only if it admits a small coinitial (resp. cofinal) subcategory.

Proof. Indeed, the condition is obviously sufficient. Conversely, suppose that φ : I → 𝒞 is a coinitial (resp. cofinal) functor, with I small; the datum 𝒢 := (φ(Ob(I)), φ(Morph(I))) is a subgraph of the category 𝒞 (see (1.6)), and the inclusion morphism into 𝒞 extends to a functor φ′ : I′ → 𝒞, where I′ is the path category of 𝒢 (proposition 1.6.4). It is then easily seen that the graph (φ(Ob(I′)), φ′(Morph(I′))) is actually a coinitial (resp. cofinal) small subcategory of 𝒞 (details left to the reader). □

The path category of a graph 𝒢 is a sort of free category generated by 𝒢. We explain next how to define equivalence relations on such a path category, and how to form the corresponding quotient categories. To this aim, let 𝒢 be any graph; we shall call a commutativity condition on 𝒢 any pair of paths of 𝒢 with the same sources and targets. A conditional graph is a pair (𝒢 , C) consisting of a graph 𝒢 and a set C of commutativity conditions on 𝒢. A morphism F : (𝒢 , C) → (𝒢′, C′) of conditional graphs is a morphism F : 𝒢 → 𝒢′ of graphs, which induces a map C → C′ : (p_{1}, p_{2}) ↦ (P (F)p_{1}, P (F)p_{2}). With this composition rule, the conditional graphs form also a category, and we say that a conditional graph is small if its underlying graph is small. We denote

U-CondGraph

the category of all small conditional graphs, and as usual we drop the subscript U, unless ambiguities are likely to arise from this omission.

1.6.6. We may attach to any small category 𝒞 a natural set of commutativity conditions; namely, one takes the set C_{𝒞} consisting of all pairs of paths ((A_{0}, f_{1}, …, A_{n}), (B_{0}, g_{1}, …, B_{m})) in 𝒞 such that A_{0} = B_{0}, A_{n} = B_{m}, and f_{n} ∘ ⋯ ∘ f_{1} = g_{m} ∘ ⋯ ∘ g_{1}. (Especially, for n = 1 and m = 0, we get the pairs ((A, 1_{A}, A), (A)), for A ranging over all the objects of 𝒞).

Clearly, every functor F : 𝒞 → 𝒞′ induces a map C_{𝒞} → C_{𝒞′}, whence a forgetful functor

(1.6.7)  Cat → CondGraph  𝒞 ↦ (𝒞 , C_{𝒞} ).

Proposition 1.6.8. The forgetful functor (1.6.7) admits a left adjoint.

Proof. Given a conditional graph (𝒢 , C), denote by 𝒫 the path category of 𝒢, and by 𝒬 the category such that

• Ob(𝒬) = V (𝒢)

• for every A, B ∈ Ob(𝒬), the set Hom_{𝒬} (A, B) consists of all pairs (p_{1}, p_{2}) ∈ P (𝒢) × P (𝒢) such that the source (resp. the target) of both p_{1} and p_{2} is A (resp. is B)

• the composition law of 𝒬 is given by the rule : (p_{1}, p_{2}) ∘ (p′_{1}, p′_{2}) := (p_{1} ∘ p′_{1}, p_{2} ∘ p′_{2})

for every pair of composable morphisms (p_{1}, p_{2}), (p′_{1}, p′_{2}) ∈ Morph(𝒬) and notice that C ⊂ Morph(𝒬). Consider the family S_{C} of all subcategories 𝒮 of 𝒬 such that • Ob(𝒮) = Ob(𝒬) and C ⊂ Morph(𝒮)

• Morph(𝒮) is an equivalence relation on P (𝒢) = Morph(𝒫). Clearly 𝒬 ∈ S_{C}, so S_{C} ≠ ∅, and S_{C} admits a smallest element, namely the subcategory ℛ whose set of objects is Ob(𝒬) and whose set of morphisms is ⋂_{𝒮 ∈S_{C}} Morph(𝒮). We may then form a category

𝒫/C

whose set of objects is Ob(𝒫) = V (𝒢), and such that

Hom_{𝒫/C} (A, B) := Hom_{𝒫} (A, B)/∼_{C}

where ∼_{C} is the equivalence relation induced by ℛ on Hom_{𝒫} (A, B). The composition law for morphisms is the unique one such that the resulting pair of maps

Ob(𝒫) → Ob(𝒫/C) Morph(𝒫) → Morph(𝒫/C)

(which is the identity map on objects, and the natural projection on morphisms) is a functor

𝒫 → 𝒫/C.

To check that this law is well defined, say that p, q : A → A′ and p′, q′ : A′ → A′′ are two pairs of morphisms in 𝒫, such that p ∼_{C} q and p′ ∼_{C} q′; since ℛ is a subcategory, it follows easily that p′ ∘ p ∼_{C} q′ ∘ p ∼_{C} q′ ∘ q, whence the claim. Lastly, let 𝒞 be any small category, and F : (𝒢 , C) → (𝒞 , C_{𝒞} ) a given morphism of conditional graphs; by proposition 1.6.4, the morphism F extends to a functor G : 𝒫 → 𝒞. Let 𝒮 ⊂ 𝒬 be the subcategory such that Ob(𝒮) = Ob(𝒬) and whose morphisms are all the pairs of paths (p, p′) such that (P (F)p, P (F)p′) ∈ C_{𝒞} ; by construction, 𝒮 ∈ S_{C}, so 𝒮 contains ℛ; it follows that G factors uniquely through 𝒫/C, whence the proposition. □

Theorem 1.6.9. Let 𝒞 be any category and Σ ⊂ Morph(𝒞) any subset. Then there exist a category 𝒞 [Σ^{−1}] and a localization functor

L : 𝒞 → 𝒞 [Σ^{−1}]

with the following properties :

(i) Lf is an isomorphism in 𝒞 [Σ^{−1}], for every f ∈ Σ.

(ii) For every category ℬ and any functor G : 𝒞 → ℬ such that Gf is an isomorphism in ℬ for every f ∈ Σ, there exists a unique functor G′ : 𝒞 [Σ^{−1}] → ℬ such that G = G′ ∘ L.

Proof. Pick any universe U′ containing U, and such that 𝒞 is U′-small; after replacing U by U′, we may assume that 𝒞 is small. We consider the graph 𝒢 such that

• V (𝒢) = Ob(𝒞)

• 𝒢 (A, B) is the disjoint union of Hom_{𝒞} (A, B) and Hom_{𝒞} (B, A) ∩ Σ, for every A, B ∈

V (𝒢).

For every f ∈ Hom_{𝒞} (B, A) ∩ Σ, we write f^{−1} : A → B for the corresponding arrow of 𝒢. Next, we endow 𝒢 with the set Θ consisting of the following commutativity conditions :

• For every A ∈ Ob(𝒞), the pair ((A, 1_{A}, A), (A))

• for every A, B, C ∈ Ob(𝒞), every f ∈ Hom_{𝒞} (A, B) and every g ∈ Hom_{𝒞} (B, C), the

pair ((A, f, B, g, C), (A, g ∘ f, C))

• the pairs ((A, f^{−1}, B, f, A), (A, 1_{A}, A)) and ((B, f, A, f^{−1}, B), (B, 1_{B}, B)), for every

A, B ∈ Ob(𝒞) and every f ∈ Hom_{𝒞} (B, A) ∩ Σ.

By proposition 1.6.8, the left adjoint of (1.6.7) maps the conditional graph (𝒢 , Θ) to a category which we denote 𝒞 [Σ^{−1}]. Now, suppose that G : 𝒞 → ℬ is a functor such that Gf is an isomorphism in ℬ, for every f ∈ Σ. We define a morphism of graphs H : 𝒢 → ℬ by the rule :

• H(A) := GA for every A ∈ V (𝒢)

• H(f) := Gf for every A, B ∈ V (𝒢) and every f ∈ Hom_{𝒞} (A, B)

• H(f) := (Gf)^{−1} for every A, B ∈ V (𝒢) and every f ∈ Hom_{𝒞} (B, A) ∩ Σ.

It is easily seen that H induces a morphism of conditional graphs

(𝒢 , Θ) → (ℬ, C_{ℬ} )

which, under the adjunction of proposition 1.6.8, corresponds to a unique functor 𝒞 [Σ^{−1}] → ℬ with the sought properties. □

Remark 1.6.10. (i) Clearly conditions (i) and (ii) of theorem 1.6.9 characterize 𝒞 [Σ^{−1}] up to isomorphism of categories. We call a localization of 𝒞 relative to Σ the datum of a functor G : 𝒞 → 𝒟 and an equivalence of categories H : 𝒞 [Σ^{−1}] → 𝒟 such that H ∘ L = G, where L : 𝒞 → 𝒞 [Σ^{−1}] is the localization functor of theorem 1.6.9.

(ii) For any subset Σ ⊂ Morph(𝒞), let us set Σ^{o} := {s^{o} | s ∈ Σ} ⊂ Morph(𝒞^{o}) (notation of (1.1.1)). In view of (i), we see that the localization functor 𝒞^{o} → 𝒞^{o}[Σ^{o−1}] extends uniquely to an isomorphism of categories :

𝒞 [Σ^{−1}]^{o} →^{∼} 𝒞^{o}[Σ^{o−1}].

Corollary 1.6.11. In the situation of theorem 1.6.9, the localization functor L induces a fully faithful functor

Fun(L, ℬ) : Fun(𝒜 [Σ^{−1}], ℬ) → Fun(𝒜 , ℬ) for every category ℬ .

Proof. Let F, G : 𝒜 [Σ^{−1}] → ℬ be two functors; recall that a natural transformation α : F ⇒ G is the same as the datum of a functor α̃ : 𝒜 [Σ^{−1}] → Morph(ℬ) with s∘α̃ = F and t∘α̃ = G (see (1.1.30)). Notice also that a morphism (g, g′) of Morph(ℬ) is invertible if and only if both s(g, g′) := g and t(g, g′) := g′ are invertible in ℬ. Combining with the universal property of F from theorem 1.6.9, we conclude that the datum of α is equivalent to that of a functor β̃: 𝒜 → Morph(ℬ) such that s ∘ β̃ = F ∘ L and t ∘ β̃ = G ∘ L; but the latter is the same as a natural transformation F ∘ L ⇒ G ∘ L, whence the contention. □

Example 1.6.12. Let 𝒞 be a category that admits either a final or an initial object, and let Σ be the set of all morphisms of 𝒞. Then 𝒞 [Σ^{−1}] is (isomorphic to) the category \overline{𝒞} whose set of objects is Ob(𝒞) and such that for every X, Y ∈ Ob(𝒞) the set Hom\overline{𝒞} (X, Y) contains exactly one element. Indeed, there exists a unique functor p : 𝒞 → \overline{𝒞} which is the identity on objects; since every morphism of \overline{𝒞} is an isomorphism, p factors uniquely through a functor 𝒞 [Σ^{−1}] → \overline{𝒞} that is still the identity on objects. On the other hand, say that X_{0} is a final object for 𝒞, and for every X ∈ Ob(𝒞), let t_{X} : X → X_{0} be the unique morphism; denote by [t_{X}] the class of t_{X} in 𝒞 [Σ^{−1}]. Then we have also a functor q : \overline{𝒞} → 𝒞 [Σ^{−1}] that is identity on objects, and sends every morphism X → Y of \overline{𝒞} to the morphism τ_{XY} := [t_{Y}]^{−1} ∘ [t_{X}] : X → Y of 𝒞 [Σ^{−1}]. Clearly p ∘ q = 1\overline{𝒞} , and to conclude, it suffices to check that if f : Y → X is any morphisms of 𝒞, then the class [f] : X → Y of f in 𝒞 [Σ^{−1}] coincides with τ_{XY}. But the latter assertion is clear, since t_{Y} ∘f = t_{X} in 𝒞. One argues likewise in case 𝒞 admits an initial object.

Proposition 1.6.13. Let 𝒞,𝒟 be two categories, F:𝒞 → 𝒟, G:𝒟 → 𝒞 two functors, and set

Σ_{F} := {f ∈ Morph(𝒞) | F f is an isomorphism}.

Suppose that G is fully faithful, and is either right or left adjoint to F. Then F factors uniquely through the localization L : 𝒞 → 𝒞 [Σ_{F}^{−1}] and an equivalence

F′ : 𝒞 [Σ_{F}^{−1}] →^{∼} 𝒟.

Proof. To ease notation, set 𝒞′ := 𝒞 [Σ_{F}^{−1}], and notice that Σ_{F^{o}} = Σ_{F}^{o} (notation of remark 1.6.10(ii)), whence a natural isomorphism of categories :

𝒞^{′o} →^{∼} 𝒞^{o}[Σ_{F^{o}}^{−1}]

that identifies L^{o} : 𝒞^{o} → 𝒞^{′o} with the localization 𝒞^{o} → 𝒞^{o}[Σ_{F^{o}}^{−1}]; moreover, recall that (F, G) is an adjoint pair if and only if the same holds for the pair (G^{o}, F^{o}) (remark 1.1.19(iv)). Thus, we may assume that G is right adjoint to F. Now, the existence and uniqueness of F′ is clear from the universal property of the localization; it remains to check that F′ is an equivalence. To this aim, set as well G′ := L ∘ G : 𝒟 → 𝒞′, and let (η, ε) be the unit and counit of an adjunction for the pair (F, G). Since F = F′ ∘ L, we get F′ ∘ G′ = F ∘ G, hence ε is also a natural transformation F′G′ ⇒ 1_{𝒟} . On the other hand, by corollary 1.6.11there exists a unique natural transformation

η′ : 1_{𝒞′} ⇒ G′F′ such that η′ ∗ L = L ∗ η.

Using the triangular identities of (1.1.13), we compute :

(G′ ∗ ε) ⊙ (η′ ∗ G′) = (LG ∗ ε) ⊙ (L ∗ η ∗ G) = L ∗ ((G ∗ ε) ⊙ (η ∗ G)) = L ∗ 1_{G} = 1_{G′}.

Likewise, let us show that (ε ∗ F′) ⊙ (F′ ∗ η′) = 1_{F′}; to this aim, again by virtue of corollary 1.6.11, it suffices to check that

((ε ∗ F′) ⊙ (F′ ∗ η′)) ∗ L = 1_{F′} ∗ L = 1_{F} .

However, the left-hand side equals (ε ∗ F′L) ⊙ (F′ ∗ η′ ∗ L) = (ε ∗ L) ⊙ (F′L ∗ η), so the sought identity follows again from (1.1.13). By proposition 1.1.15(i), it follows that (F′, G′) is an adjoint pair of functors, with unit η′ and counit ε. Lastly, since G is fully faithful, ε is an isomorphism of functors (proposition 1.1.20(i)); invoking again the triangular identities (1.1.13), we deduce that F ∗ η is an isomorphism of functors, hence η_{X} ∈ Σ_{F} for every X ∈ Ob(𝒞). Since Ob(𝒞) = Ob(𝒞′), it follows easily that η′ is an isomorphism of functors, whence the proposition. □

Definition 1.6.14. Let 𝒞 be a category, Σ ⊂ Morph(𝒞) a set of morphisms.

(i) We say that Σ admits a right calculus of fractions if the following conditions hold : (CF1) 1_{A} ∈ Σ, for every A ∈ Ob(𝒞). (CF2) For every s : A → B and t : B → C with s, t ∈ Σ, we have t ∘ s ∈ Σ as well. (CF3) For every morphism f : A → B in 𝒞 and every s : C → B in Σ, there exist

g : D → C in 𝒞 and t : D → A in Σ such that f ∘ t = s ∘ g. (CF4) If f, g : A → B are any two morphisms in 𝒞 such that s ∘ f = s ∘ g for some s : B → C in Σ, then there exists t : D → A in Σ such that f ∘ t = g ∘ t. (ii) We say that Σ admits a left calculus of fractions if the subset Σ^{o} := {s^{o} | s ∈ Σ} admits

a right calculus of fractions (notation of remark 1.6.10(ii)). (iii) For every A ∈ Ob(𝒞), let Σ_{A} be the full subcategory of 𝒞 /A whose objects are the

elements of Σ with target equal to A (notation of (1.1.24)). We say that Σ is right cofinally small, if Σ_{A} is cofinally small for every A ∈ Ob(𝒞) (see definition 1.5.1(iv)).

1.6.15. Let now 𝒞 and Σ be as in definition 1.6.14, and suppose that 𝒞 has small Hom-sets, and Σ admits a right calculus of fractions. For every A, B ∈ Ob(𝒞), let us consider the functor

H_{A,B} : Σ_{A}^{o} → Set (s : I → A) ↦ {(s, f) | f ∈ Hom_{𝒞} (I, B)}

where, for every morphism h/A : s → t in Σ_{A}, the mapping H_{A,B}(h/A) is given by the rule

H_{A,B}(t) → H_{A,B}(s) : (t, f) ↦ (s, f ∘ h).

Furthermore, we define a mapping

c_{s}^{A,B} : H_{A,B}(s) → Hom_{𝒞 [Σ^{−1}]}(A, B)  (s, f) ↦ f ∘ s^{−1}  for every s ∈ Ob(Σ_{A}).

Proposition 1.6.16. In the situation of (1.6.15), the following holds :

(i) For every A ∈ Ob(𝒞), the category Σ_{A}^{o} is filtered.

(ii) For every A, B ∈ Ob(𝒞), the rule :

s ↦ c_{s}^{A,B} for every s ∈ Ob(Σ_{A})

defines a universal cocone with basis H_{A,B} and vertex Hom_{𝒞 [Σ^{−1}]}(A, B). (iii) If Σ is right cofinally small, The category 𝒞 [Σ^{−1}] has small Hom-sets.

Proof. (i): First, Σ_{A}^{o} is not empty, due to (CF1). Next, say that s, s′ ∈ Ob(Σ_{A}), and denote by I and I′ the sources of s and respectively s′; due to (CF3), we may find I′′ ∈ Ob(𝒞), a morphism f : I′′ → I in 𝒞, and an element g : I′′ → I′ of Σ such that s ∘ f = s′ ∘ g. Then, (CF2) says that s′ ∘ g lies in Σ, and therefore it defines an object s′′ of Σ_{A}, with morphisms f : s → s′′ and g : s′ → s′′ in Σ_{A}^{o}. Lastly, say that f, f′ : s → s′ are any two morphisms in Σ_{A}^{o}, and denote by I′ the source of s′. By (CF4), we may find g : I′′ → I′ in Σ such that f ∘ g = f′ ∘ g; clearly s′ ∘ g defines a morphism h : s′ → s′′ in Σ_{A}^{o} such that h ∘ f = h ∘ f′. Now the assertion follows from remark 1.2.21(i).

(ii): Let s : I → A and s′ : I′ → A be two objects of Σ_{A}, and g : I′ → I a morphism s → s′ in Σ_{A}^{o}. Given f : I → B, set f′ := f ∘ g; we notice :

Claim 1.6.17. f ∘ s^{−1} = f′ ∘ s′^{−1} in 𝒞 [Σ^{−1}].

Proof of the claim. Due to (CF3), we may find I′′ ∈ Ob(𝒞), an element t : I′′ → I of Σ and a morphism t′ : I′′ → I′ such that s ∘ t = s′ ∘ t′. We compute :

s ∘ g ∘ t′ = s′ ∘ t′ = s ∘ t

whence, by (CF4), an element h : I′′′ → I′′ of Σ, such that g ∘ t′ ∘ h = t ∘ h. It follows that g ∘ t′ = t in 𝒞 [Σ^{−1}] and therefore

f ∘ t = f ∘ g ∘ t′ = f′ ∘ t′ in 𝒞 [Σ^{−1}].

Consequently :

f ∘ s^{−1} = f′ ∘ t′ ∘ t^{−1} ∘ s^{−1} = f′ ∘ s′^{−1} ∘ s ∘ t ∘ t^{−1} ∘ s^{−1} = f′ ∘ s′^{−1} in 𝒞 [Σ^{−1}]

as stated. ♢

Claim 1.6.17 says already that c_{•}^{A,B} is a well defined cocone. To check the universality property, we define a new category 𝒟 with Ob(𝒟) = Ob(𝒞) and such that Hom_{𝒟} (A, B) is the colimit of the functor H_{A,B}, for every A, B ∈ Ob(𝒞). From (i) and example 1.2.23(i) we see that this is the set of equivalence classes [s, f] of pairs (s, f) ∈ H_{A,B}(s) with s ranging over all objects of Σ_{A}; two such pairs (s, f) and (s′, f′) are equivalent, if there exist t ∈ Ob(Σ_{A}) and morphisms h : t → s, h′ : t → s′ in Σ_{A} such that f ∘h = f′∘h′. The composition of morphisms in 𝒟 is defined as follows. Let A, B, C ∈ Ob(𝒞) be any three objects, s : I → A, t : J → B any elements of Σ, and (s, f) ∈ H_{A,B}(s), (t, g) ∈ H_{B,C}(t); by (CF3) we find D ∈ Ob(𝒞), a morphism f′ : D → J in 𝒞 and an element t′ : D → I of Σ such that t ∘ f′ = f ∘ t′. We define

(1.6.18)  (t, g) ∘ (s, f) := [s ∘ t′, g ∘ f′] ∈ Hom_{𝒟} (A, C).

Let us check that this class does not depend on the choice of D, f′ and t′. Indeed, suppose that D′ ∈ Ob(𝒞) is another object and t′′ : D′ → I in Σ, f′′ : D′ → J in 𝒞 satisfy the condition t ∘ f′′ = f ∘ t′′; by (CF3) we may then find D′′ ∈ Ob(𝒞), h : D′′ → D in Σ and h′ : D′′ → D′ in 𝒞 such that t′ ∘ h = t′′ ∘ h′. We compute :

t ∘ f′ ∘ h = f ∘ t′ ∘ h = f ∘ t′′ ∘ h′ = t ∘ f′′ ∘ h′.

By (CF4), it follows that there exist D′′′ ∈ Ob(𝒞) and an element u : D′′′ → D′′ of Σ such that f′ ∘ h ∘ u = f′′ ∘ h′ ∘ u. By (CF2), we may then replace D′′ by D′′′ and h, h′ with h ∘ u and respectively h′ ∘ u, after which we may assume as well that f′′ ∘ h′ = f′ ∘ h. Finally, we find :

[s ∘ t′, g ∘ f′] = [s ∘ t′ ∘ h, g ∘ f′ ∘ h] = [s ∘ t′′ ∘ h′, g ∘ f′′ ∘ h′] = [s ∘ t′′, g ∘ f′′]

as required. It also follows that [s ∘ t′, g ∘ f′] depends only on [s, f] and [t, g]; indeed, say that u : I′ → I is any element of Σ, and set (s′, f′′) := (s∘u, f ∘u). Suppose also that f′′∘t′′ = t∘f′ for some t′′ : D → I′ in Σ; then (s′ ∘ t′′, g ∘ f′) = (s ∘ u ∘ t′′, g ∘ f′), and the foregoing shows that the latter pair is equivalent to (s ∘ t′, g ∘ f′), which shows the independence on the chosen representative for [s, f]; similarly one checks the independence on the representative for [t, g].

We next check the associativity of the composition law thus obtained. To this aim, say that A, B, C, D ∈ Ob(𝒞) are any four objects, s : I → A, t : J → B, u : K → C any three elements of Σ, and f : I → B, g : J → C, h : K → D any three morphisms of 𝒞. Choose E ∈ Ob(𝒞) with an element t′ : E → I of Σ and a morphism f′ : E → J of 𝒞 such that f ∘ t′ = t ∘ f′, and therefore

[t, g] ∘ [s, f] = [s ∘ t′, g ∘ f′].

Then, choose F ∈ Ob(𝒞) with u′ : F → J in Σ and g′ : F → K in 𝒞 such that g ∘ u′ = u ∘ g′, and therefore

[u, h] ∘ [t, g] = [t ∘ u′, h ∘ g′].

By (CF3) we may find G ∈ Ob(𝒞), u′′ : G → E in Σ and f′′ : G → F in 𝒞 such that u′ ∘ f′′ = f′ ∘ u′′. By (CF2), t′ ∘ u′′ lies in Σ, and therefore it is easily seen that the pair (s ∘ t′ ∘ u′′, h ∘ g′ ∘ f′′) represents both [t ∘ u′, h ∘ g′] ∘ [s, f] and [u, h] ∘ [s ∘ t′, g ∘ f′], whence the assertion. Let us also remark that – due to (CF1) – for every A ∈ Ob(𝒞), the class [1_{A}, 1_{A}] is the identity endomorphism of A, seen as an object of 𝒟. This completes the construction of the category 𝒟. Next, we claim that the system of cocones c_{•}^{••} yields a functor

F : 𝒟 → 𝒞 [Σ^{−1}]

whose map on objects is the identity mapping of Ob(𝒞), and such that

F [s, f] := f ∘ s^{−1} for every morphism [s, f] in 𝒟 .

Indeed, claim 1.6.17implies that the foregoing rule yields a well defined map

Hom_{𝒟} (A, B) → Hom_{𝒞 [Σ^{−1}]}(A, B) for every A, B ∈ Ob(𝒞).

It is also clear that F 1_{A} = 1_{A} for every A ∈ Ob(𝒞), hence it remains only to check that F respects the composition laws of the two categories; however, say that s, f, t, g are as in (1.6.18); then we have

(g ∘ f′) ∘ (s ∘ t′)^{−1} = (g ∘ t^{−1} ∘ f ∘ t′) ∘ t′^{−1} ∘ s^{−1} = (g ∘ t^{−1}) ∘ (f ∘ s^{−1}) in 𝒞 [Σ^{−1}]

whence the contention. Likewise, we have a functor

G : 𝒞 → 𝒟

which is the identity mapping on objects, and such that Gf := [1_{A}, f] for every A, B ∈ Ob(𝒞) and every morphism f : A → B in 𝒞. Moreover, the identities

[1_{I}, s] ∘ [s, 1_{I}] = [1_{A}, 1_{A}]  [s, 1_{I}] ∘ [1_{I}, s] = [1_{I}, 1_{I}]  for every s : I → A in Σ

show that F s is invertible in 𝒟, for every such s. By theorem 1.6.9, it follows that G factors through a unique functor G′ : 𝒞 [Σ^{−1}] → 𝒟, and a simple inspection shows that F ∘ G′ is the identity endofunctor of 𝒞 [Σ^{−1}]. Likewise, it is easily seen that G′ ∘ F = 1_{𝒟} , so F establishes an isomorphism of categories, and (ii) follows.

(iii) is an immediate consequence of (ii). □

Remark 1.6.19. Proposition 1.6.16(ii) means that c_{•}^{A,B} induces a natural identification

colim_{Σ_{A}^{o}} H_{A,B} →^{∼} Hom_{𝒞 [Σ^{−1}]}(A, B) for every A, B ∈ Ob(𝒞).

The dual of this assertion provides a corresponding computation of the morphisms in 𝒞 [Σ^{−1}] in terms of left fractions. Namely, suppose that Σ admits a left calculus of fractions; for every A ∈ Ob(𝒞) we let Σ^{A} be the full subcategory of A/𝒞 whose objects are the elements of Σ with source equal to A, and we say that Σ is left cofinally small if Σ^{A} is cofinally small for every such A. For every other object B of 𝒞 we consider the functor

H_{B,A} : Σ^{A} → Set (s : A → I) ↦ {(f, s) | f ∈ Hom_{𝒞} (B, I)}

where, for every morphism h : s → t in Σ^{A}, the mapping H_{B,A}(h) is given by the rule

H_{B,A}(s) → H_{B,A}(t) : (f, s) ↦ (h ∘ f, t).

Then Σ^{A} is a filtered category, and if Σ is left cofinally small, 𝒞 [Σ^{−1}] has small Hom-sets; more precisely, we have a natural identification :

colim_{Σ^{A}} H_{B,A} →^{∼} Hom_{𝒞 [Σ^{−1}]}(B, A) (f, s) ↦ s^{−1} ∘ f.

1.6.20. Let 𝒜 be a category, 𝒜_{0} a full subcategory of 𝒜, Σ ⊂ Ob(𝒜) any subset, and set

Σ_{0} := Σ ∩ Morph(𝒜_{0}).

Clearly, the inclusion functor i : 𝒜_{0} → 𝒜 extends uniquely to a functor

i[Σ_{0}^{−1}] : 𝒜_{0}[Σ_{0}^{−1}] → 𝒜 [Σ^{−1}].

Proposition 1.6.21. In the situation of (1.6.20), suppose that :

(a) For every A ∈ Ob(𝒜), A_{0} ∈ Ob(𝒜_{0}) and every morphism f : A → iA_{0} that lies in Σ, there exist A′_{0} ∈ Ob(𝒜_{0}) and a morphism g : iA′_{0} → A such that f ∘ g ∈ Σ.

(b) Σ admits a right calculus of fractions. Then the set Σ_{0} admits a right calculus of fractions in 𝒜_{0}, and the functor i[Σ_{0}^{−1}] is fully faithful.

Proof. As usual, after replacing our universe U by a larger one, we may assume that 𝒜 is small. Let us check that Σ_{0} admits a right calculus of fractions. (CF1) and (CF2) are obviously fulfilled by Σ_{0}. Next, suppose that we have A, B, C ∈ Ob(𝒜_{0}), D ∈ Ob(𝒜) and a commutative diagram in 𝒜

(1.6.22)  \xymatrix{D \ar[r]^{g} \ar[d]_{t} & iC \ar[d]^{is} \\ iA \ar[r]^{if} & iB}  with s, t ∈ Σ.

By (a), it follows that there exists a morphism u : iD_{0} → D for some D_{0} ∈ Ob(𝒜_{0}), such that t′ := t ∘ u : iD_{0} → iA lies in Σ_{0}, g′ := g ∘ u : iD_{0} → iC lies in Hom_{𝒜_{0}} (D_{0}, C), and clearly f ∘ t′ = s ∘ g′, whence (CF3). Lastly, suppose that A, B ∈ Ob(𝒜_{0}), D ∈ Ob(𝒜), and we have morphisms f, g : A → B and t : D → iA such that i(f) ∘ t = i(g) ∘ t and with t ∈ Σ. Pick u : iD_{0} → D as in the foregoing, and let again t′ := t ∘ u; then f ∘ t′ = g ∘ t′, and we have just seen that t′ ∈ Σ_{0}, whence (CF4).

Next, let us check that i[Σ_{0}^{−1}] is fully faithful. In light of propositions 1.6.16(ii) and 1.5.2, it suffices to show that the inclusion functor

(1.6.23)  Σ_{0,A}^{o} → Σ_{iA}^{o}

is cofinal for every A ∈ Ob(𝒜_{0}) (notation of (1.6.15)). To this aim, in view of proposition 1.6.16(i), it suffices to check that the functor (1.6.23) fulfills conditions (a) and (b) of lemma 1.5.7(i). However, condition (a) of loc.cit. translates directly as the assumption (a) of the proposition. Next, consider any pair of morphisms f, g : iD_{0} → D and any element s : D → iA in Σ such that s ∘ f = s ∘ g. By (CF4), there exists t : E → iD_{0} in Σ such that f ∘ t = g ∘ t, and then assumption (a) yields a morphism u : iE_{0} → E such that t′ := t ∘ u lies in Σ_{0}, and f ∘ i(t′) = g ∘ i(t′), which shows that condition (b) of lemma 1.5.7(iv) holds as well. □

Remark 1.6.24. Taking into account remarks 1.6.10(ii) and 1.6.19, we see that the dual of proposition 1.6.21holds as well; i.e. in the situation of (1.6.20), suppose that

(a) For every A ∈ Ob(𝒜), A_{0} ∈ Ob(𝒜_{0}) and every morphism f : iA_{0} → A that lies in Σ, there exist A′_{0} ∈ Ob(𝒜) and a morphism g : A → iA′_{0} such that g ∘ f ∈ Σ.

(b) Σ admits a left calculus of fractions. Then the set Σ_{0} admits a left calculus of fractions in 𝒜_{0}, and the functor i[Σ_{0}^{−1}] is fully faithful.

Proposition 1.6.25. Let 𝒞 be a category with small Hom-sets, Σ ⊂ Morph(𝒞) a set of morphisms admitting a right calculus of fractions, F : 𝒞 → 𝒞 [Σ^{−1}] the localization. We have:

(i) For every X ∈ Ob(𝒞 [Σ^{−1}]) the category X/F 𝒞 is cofiltered.

(ii) If 𝒞 is small, the functor F_{!} : 𝒞^{∧} → 𝒞 [Σ^{−1}]^{∧} is exact (notation of remark 1.3.6(i)).

Proof. (i): Let ((Y_{i}, f_{i} : X → F Y_{i}) | i = 1, 2) be a pair of objects of X/F 𝒞. For i = 1, 2, we may write f_{i} = g_{i} ∘ s_{i}^{−1} with some morphism s_{i} : Z_{i} → X in Σ and some morphism g_{i} : Z_{i} → Y_{i} in 𝒞. By condition (CF3) of definition 1.6.14 we may then find a morphism t_{1} : Z → Z_{1} in Σ and a morphism t_{2} : Z → Z_{2} in 𝒞 such that s := s_{1} ∘ t_{1} = s_{2} ∘ t_{2}, and notice that s ∈ Σ, by condition (CF2). For i = 1, 2, the composition t_{i} ∘ g_{i} then yields a morphism (Z, s^{−1} : X → F Z) → (Y_{i}, f_{i}) in X/F 𝒞. Next, let h_{1}, h_{2} : (Y_{1}, f_{1}) → (Y_{2}, f_{2}) be two morphisms in X/F 𝒞; this means that h_{i} : Y_{1} → Y_{2} is a morphism in 𝒞 for i = 1, 2, and F h_{1} ∘ f_{1} = F h_{2} ∘ f_{1} = f_{2} in 𝒞 [Σ^{−1}]. Thus, F (h_{1} ∘ g_{1}) = F (h_{2} ∘ g_{1}), and therefore there exists (u : Z′ → Z_{1}) ∈ Σ such that h_{1} ∘ g_{1} ∘ u = h_{2} ∘ g_{1} ∘ u in 𝒞. Then the composition g := g_{1} ∘ u yields a morphism g : (Z′, (s_{1} ∘ u)^{−1} : X → F Z′) → (Y_{1}, f_{1}) in X/F 𝒞 such that h_{1} ∘ g = h_{2} ∘ g. The assertion then follows from remark 1.2.21(i).

(ii): The assertion follows from (i), arguing as in the proof of corollary 1.5.19(i). □

2. 2-Category theory

In dealing with categories, the notion of equivalence is much more central than the notion of isomorphism. On the other hand, equivalence of categories is usually not preserved by the standard categorical operations discussed thus far. For instance, consider the following :

Example 2.0.1. Let 𝒞 be the category with Ob(𝒞) = {a, b}, and whose only morphisms are 1_{a}, 1_{b} and u : a → b, v : b → a. Then necessarily u ∘ v = 1_{b} and v ∘ u = 1_{a}. Let 𝒞_{a} (resp. 𝒞_{b}) be the unique subcategory of 𝒞 with Ob(𝒞_{a}) = {a} (resp. Ob(𝒞_{b}) = {b}). Clearly both inclusion functors 𝒞_{a} → 𝒞 ← 𝒞_{b} are equivalences. However, 𝒞_{a} ×_{𝒞} 𝒞_{b} is the empty category; especially, this fibre product is not equivalent to 𝒞 = 𝒞 ×_{𝒞} 𝒞.

It is therefore natural to seek a new framework for the manipulation of categories and functors “up to equivalences”, and thus more consonant with the very spirit of category theory. Precisely such a framework is provided by the theory of 2-categories, the subject of this chapter.

2.1. 2-Categories and pseudo-functors. The category Cat, together with the category structure on the sets Fun(−, −) (as in (1.1.10)), provides the first example of a 2-category. The latter is the datum of :

• A set Ob(𝒜), whose elements are called the objects of 𝒜.

• For every A, B ∈ Ob(𝒜), a category 𝒜 (A, B). The objects of 𝒜 (A, B) are called 1-cells or arrows, and are designated by the usual arrow notation f : A → B. Given f, g ∈ Ob(𝒜 (A, B)), we shall write f ⇒ g to denote a morphism from f to g in 𝒜 (A, B). Such morphisms are called 2-cells. The composition of 2-cells α : f ⇒ g and β : g ⇒ h shall be denoted by β ⊙ α : f ⇒ h.

• For every A, B, C ∈ Ob(𝒜), a composition bifunctor :

c_{ABC} : 𝒜 (A, B) × 𝒜 (B, C) → 𝒜 (A, C).

Given 1-cells A ⟶^{f} B ⟶^{g} C, we write g ∘ f := c_{ABC}(f, g).

Given two 2-cells α : f ⇒ g and β : h ⇒ k, respectively in 𝒜 (A, B) and 𝒜 (B, C), we use the notation

β ∗ α := c_{ABC}(α, β) : h ∘ f ⇒ k ∘ g.

Also, if h is any 1-cell of 𝒜 (B, C), we usually write h ∗ α instead of 1_{h} ∗ α. Likewise, we set β ∗ f := β ∗ 1_{f}, for every 1-cell f in 𝒜 (A, B).

• For every element A ∈ Ob(𝒜), a unit functor :

u_{A} : 1 → 𝒜 (A, A)

where 1 := (∗, 1_{∗}) is the final object of Cat. Hence u_{A} is the datum of an object:

1_{A} ∈ Ob(𝒜 (A, A))

and its identity endomorphism, which we shall denote by i_{A} : 1_{A} → 1_{A}. The bifunctors c_{ABC} are required to satisfy an associativity axiom, which says that the diagram:

[figure: 𝒜 (A, B) × 𝒜 (B, C) × 𝒜 (C, D); 1×c_{BCD}; c_{ABC}×1; 𝒜 (A, B) × 𝒜 (B, D); c_{ABD}; 𝒜 (A, C) × 𝒜 (C, D); c_{ACD}; 𝒜 (A, D)]

commutes for every A, B, C, D ∈ Ob(𝒜). Likewise, the functor u_{A} is required to satisfy a unit axiom; namely, the diagram :

[figure: 1 × 𝒜 (A, B); u_{A}×1_{𝒜 (A,B)}; 𝒜 (A, B); ∼; ∼; 𝒜 (A, B) × 1; 1_{𝒜 (A,B)} ×u_{B}; 𝒜 (A, A) × 𝒜 (A, B); c_{AAB}; 𝒜 (A, B); 𝒜 (A, B) × 𝒜 (B, B); c_{ABB}]

commutes for every A, B ∈ Ob(𝒜).

Remark 2.1.1. (i) In any 2-category, consider a diagram with two 2-cells :

[figure: A; f; g; α; B; f′; g′; α′; C.]

In this situation, we have the following commutation identity :

(α′ ∗ g) ⊙ (f′ ∗ α) = α′ ∗ α = (g′ ∗ α) ⊙ (α′ ∗ f)

which is sometimes useful, to perform certain verifications. To see this, notice that α′ ∗ α = (α′ ⊙ 1_{f′}) ∗ (1_{g} ⊙ α), whence the first stated identity, by functoriality of the composition for 2-cells. Likewise, we get α′ ∗ α = (1_{g′} ⊙ α′) ∗ (α ⊙ 1_{f}), whence the second identity.

(ii) As already announced, the category Cat becomes naturally a 2-category, by letting

Cat(𝒜 , ℬ) := Fun(𝒜 , ℬ) for every 𝒜 , ℬ ∈ Ob(Cat)

with composition bifunctor defined as in (1.1.10). Indeed, by spelling out the definitions, we see that the functoriality of the Godement product boils down precisely to the identity (1.1.11).

On the other hand, from every 2-category 𝒜 we obtain a category, simply by forgetting the 2-cells : the set of objects of this underlying category of 𝒜 is Ob(𝒜), and the morphisms are the 1-cells of 𝒜, with composition law induced by the composition bifunctor of 𝒜.

(iii) We shall say that a 2-category 𝒜 has U-small Hom-categories if 𝒜 (A, B) is a Usmall category for every A, B ∈ Ob(𝒜). We shall say that 𝒜 is U-small if it has U-small Hom-categories and Ob(𝒜) is U-small. As usual, the universe U will be dropped from the notation, unless there is a danger of ambiguities. For instance, the 2-category Cat has small Hom-categories.

2.1.2. If 𝒜 is a 2-category, we have three distinct constructions of opposite 2-categories associated with 𝒜. Namely we have :

• The 2-category 𝒜^{o} such that Ob(𝒜^{o}) := {A^{o} | A ∈ Ob(𝒜)} (cp. (1.1.1)) and with

𝒜^{o}(A^{o}, B^{o}) := 𝒜 (B, A) for every A^{o}, B^{o} ∈ Ob(𝒜^{o}) .

For any 1-cell f : B → A in 𝒜, we denote f^{o} : A^{o} → B^{o} the corresponding 1-cell in 𝒜^{o}. The composition bifunctor c_{•••}^{o} and unit functors u_{•}^{o} of 𝒜^{o} are given by the rules :

c_{A^{o}B^{o}C^{o}}^{o} := c_{CBA}  1_{A^{o}} := 1_{A}^{o}  for every A^{o}, B^{o}, C^{o} ∈ Ob(𝒜^{o}) .

Notice that to any 2-cell α : f ⇒ g in 𝒜 there corresponds a 2-cell α^{o} : f^{o} ⇒ g^{o} in 𝒜^{o}. The associativity and unit axioms for 𝒜^{o} follow formally from the corresponding properties for 𝒜.

• The 2-category^{o}𝒜 such that Ob(^{o}𝒜) := {^{o}A | A ∈ Ob(𝒜)} and with

o𝒜 (^{o}A,^{o}B) := 𝒜 (A, B)^{o} for every^{o}A,^{o}B ∈ Ob(^{o}𝒜) .

For any 1-cell f : A → B in 𝒜, we denote ^{o}f : ^{o}A → ^{o}B the corresponding 1-cell in o𝒜. The composition bifunctor^{o}c_{•••} and unit functors^{o}u_{•} of^{o}𝒜 are given by the rules :

oc_{^{o}A^{o}B^{o}C} := c_{ABC}^{o}  1_{^{o}A} := ^{o}1_{A}  for every^{o}A,^{o}B,^{o}C ∈ Ob(^{o}𝒜) .

Then to any 2-cell α : f ⇒ g in 𝒜 there corresponds a 2-cell^{o}α : ^{o}g ⇒ ^{o}f in^{o}𝒜. The associativity and unit axioms for^{o}𝒜 are also immediately deduced from those of 𝒜.

• Lastly, we may combine the two foregoing constructions to get the 2-category ^{o}𝒜^{o}, which inverts the direction of both 1-cells and 2-cells.

Definition 2.1.3. Let 𝒜 be any 2-category, and f : A → B any 1-cell of 𝒜.

(i) A 2-cell β : f ⇒ f′ of 𝒜 is invertible, if it is an isomorphism in the category 𝒜 (A, B). (ii) We say that the 1-cell g : B → A of 𝒜 is right adjoint to f, if there exist 2-cells

η : 1_{A} ⇒ g ∘ f and ε : f ∘ g ⇒ 1_{B} fulfilling the triangular identities

(g ∗ ε) ⊙ (η ∗ g) = 1_{g} (ε ∗ f) ⊙ (f ∗ η) = 1_{f}.

(see [28, Def.7.1.2]). In this case, we also say that g is right adjoint to f, and that (f, g) is an adjoint pair of 1-cells; moreover, we say that (η, ε) is an adjunction for (f, g). Then η is called the unit and ε the counit of the adjunction (η, ε).

(iii) We say that a 1-cell f : A → B of 𝒜 is an equivalence if there exist 1-cells g, h : B → A and invertible 2-cells 1_{A} ⇒ g ∘ f and f ∘ h ⇒ 1_{B}. In this case, we also say that g and h are quasi-inverse 1-cells for f.

Remark 2.1.4. (i) Let f, g and h be as in definition 2.1.3(iii), so we have invertible 2-cells η : 1_{A} ⇒ g ∘ f and ε : f ∘ h ⇒ 1_{B}. Then we have the invertible 2-cell β := (g ∗ ε) ⊙ (h ∗ η) : h ⇒ g, whence the invertible 2-cell ε ∘ (f ∗ β^{−1}) : f ∘ g ⇒ 1_{B}. In other words, we may as well assume that g = h in definition 2.1.3(iii).

(ii) A composition of equivalences is an equivalence. Indeed, let f_{1} : A_{1} → A_{2} and f_{2} : A_{2} → A_{3} be two equivalences, g_{1} : A_{2} → A_{1} (resp. g_{2} : A_{3} → A_{2}) a quasi-inverse for f_{1} (resp. for f_{2}) so that we have invertible 2-cells α_{i} : 1_{A_{i}} ⇒ g_{i} ∘ f_{i} and β_{i} : f_{i} ∘ g_{i} ⇒ 1_{A_{i+1}} for i = 1, 2. Then g := g_{1} ∘ g_{2} is a quasi-inverse for f := f_{2} ∘ f_{1}, since we have the invertible 2-cells (g_{1} ∗ α_{2} ∗ f_{1}) ⊙ α_{1} : 1_{A_{1}} ⇒ g ∘ f and β_{2} ⊙ (f_{2} ∗ β_{1} ∗ g_{2}) : f ∘ g ⇒ 1_{A_{3}}.

(iii) Let f, f′ : A → B be two 1-cells of 𝒜, and β : f →^{∼} f′ an invertible 2-cell. Then f is an equivalence if and only if the same holds for f′. Indeed, suppose we have a 1-cell g : B → A with an invertible 2-cells α : 1_{A} →^{∼} g ∘ f and α′ : f ∘ g →^{∼} 1_{B}; then we get the invertible 2-cells (g ∗ β) ⊙ α : 1_{A} →^{∼} g ∘ f′ and α′ ⊙ (β^{−1} ∗ g) : f′ ∘ g →^{∼} 1_{B}, so f′ is an equivalence.

(iv) Likewise, if (f, g) is an adjoint pair of 1-cells of 𝒜 and we have invertible 2-cells α : g →^{∼} g′ and β : f →^{∼} f′, then (f′, g′) is an adjoint pair of 1-cells. Indeed, by assumption we have 2-cells η : 1_{A} ⇒ g ∘ f and ε : f ∘ g ⇒ 1_{B} fulfilling the triangular identities of definition 2.1.3(ii); it suffices to check that η′ := (α ∗ β) ⊙ η and ε′ := ε ⊙ (β ∗ α)^{−1} are a unit and a counit for the pair (f′, g′). However, we have :

(g′ ∗ ε′) ⊙ (η′ ∗ g′) = (g′ ∗ (ε ⊙ (β ∗ α)^{−1})) ⊙ (((α ∗ β) ⊙ η) ∗ g′)
= (g′ ∗ ε) ⊙ (g′ ∗ β ∗ α)^{−1} ⊙ (α ∗ β ∗ g′) ⊙ (η ∗ g′)
= (g′ ∗ ε) ⊙ (α ∗ f ∗ α^{−1}) ⊙ (η ∗ g′)
= (g′ ∗ ε) ⊙ (α ∗ f ∗ g) ⊙ (g ∗ f ∗ α^{−1}) ⊙ (η ∗ g′)
= α ⊙ (g ∗ ε) ⊙ (η ∗ g) ⊙ α^{−1}
= α ⊙ α^{−1} = 1_{g′}

where the third, fourth and fifith equalities follow from remark 2.1.1(i). Likewise one checks the other required triangular identity.

(v) Furthermore, if f : A → B is a 1-cell in 𝒜 and g, g′ : B → A are both right adjoint to f, then we have an invertible 2-cell α : g →^{∼} g′. Indeed, let (η, ε) (resp. (η′, ε′)) be an adjunction for the pair (f, g) (resp. for the pair (f, g′)). We set

α := (g′ ∗ ε) ⊙ (η′ ∗ g) : g ⇒ g′ β := (g ∗ ε′) ⊙ (η ∗ g′) : g′ ⇒ g.

Applying repeatedly remark 2.1.1(i), we compute :

β ⊙ α = (g ∗ ε′) ⊙ (g ∗ f ∗ g′ ∗ ε) ⊙ (η ∗ g′ ∗ f ∗ g) ⊙ (η′ ∗ g)
= (g ∗ ε) ⊙ (g ∗ ε′ ∗ f ∗ g) ⊙ (g ∗ f ∗ η′ ∗ g) ⊙ (η ∗ g)
= (g ∗ ε) ⊙ (g ∗ ((ε′ ∗ f) ⊙ (f ∗ η′)) ∗ g) ⊙ (η ∗ g)
= (g ∗ ε) ⊙ (η ∗ g) = 1_{g}.

Likewise one checks that α ⊙ β = 1_{g′}, whence the contention.

2.1.5. Let 𝒜 be any 2-category; an oriented square in 𝒜 is a (not necessarily commutative) diagram of the type :

(2.1.6)  [figure: A; f; g; α; B; h; A′; f′; B′]

where f, f′g, h are any four 1-cells and α : h ∘ f ⇒ f′ ∘ g is any 2-cell of 𝒜. If α is an invertible 2-cell, we also say that the square (2.1.6) is essentially commutative. We say that α orients the square (2.1.6), and often we refer to such a square just by naming its orienting 2-cell; when dealing with complex diagrams, this shorthand may lead to unacceptable ambiguities, but in these cases usually one may resolve such ambiguities just by specifying two of the opposite sides of the square, for which we shall employ a fractional notation : thus, the square (2.1.6) shall be denoted, depending on the context, in either of the following three manners :

α  \frac{α}{f|f′}  \frac{α}{g|h}.

One basic operation consists in combining two squares that share one edge, to obtain a new square, essentially by omitting the common edge of the squares. Namely, suppose we have two squares α and β as in the diagram :

(2.1.7)  [figure: A; g; f; α; A′; f′; k; β; A′′; f′′; B; h; B′; i; B′′]

Then we say that α and β are vertically composable, and we define the square

β □ α  :  [figure: A; k∘g; f; γ; A′′; f′′; B; i∘h; B′′]  where γ := (β ∗ g) ⊙ (i ∗ α).

In this operation, we have joined to the square α of (2.1.6) another square β that shares the edge f′. Clearly, if β′ is another square that shares with α the edge f, the same operation can be performed to obtain the square α □ β′ which omits the edge f. On the other hand, the rule to join to α a square that shares one of the two remaining edges g or h, is slightly different. Namely, suppose we have the diagram of two squares :

[figure: A; f; g; α; B; h; i; γ; C; k; A′; f′; B′; i′; C′.]

Then we say that α and γ are horizontally composable, and we set

\frac{γ}{i|i′} ⊟ \frac{α}{f|f′} := \frac{(i′ ∗ α) ⊙ (γ ∗ f)}{i ∘ f|i′ ∘ f′}.

However, notice that the second join operation is turned into the first one, when we replace 𝒜 by its opposite 2-categories; namely, we have the identities :

(2.1.8)  (γ ⊟ α)^{o} = α^{o} □ γ^{o}  ^{o}(γ ⊟ α) = ^{o}γ □^{o}α

The following proposition establishes the two basic rules of the algebra of oriented squares.

Proposition 2.1.9. With the notation of (2.1.5), the following holds :

(i) The join operation is associative, i.e. for any diagram of three squares

[figure: A; g; f; α; A′; f′; k; β; A′′; f′′; l; γ; A′′′; f′′′; B; h; B′; i; B′′; m; B′′′]

we have

γ □ (β □ α) = (γ □ β) □ α.

(ii) For every (not necessarily commutative) diagram

[figure: A; f; h; α; B; k; g; β; C; i; A′; f′; h′; α′; B′; g′; k′; β′; C′; i′; A′′; f′′; B′′; g′′; C′′.]

we have

(β′ □ β) ⊟ (α′ □ α) = (β′ ⊟ α′) □ (β ⊟ α).

Proof. (i): We compute :

\begin{matrix}γ □ (β □ α) = & γ □ ((β ∗ g) ⊙ (i ∗ α)) \\ = & ((γ ∗ (k ∘ g)) ⊙ (m ∗ ((β ∗ g) ⊙ (i ∗ α))) \\ = & ((γ ∗ (k ∘ g)) ⊙ (m ∗ (β ∗ g)) ⊙ (m ∗ (i ∗ α)) \\ = & (((γ ∗ k) ⊙ (m ∗ β)) ∗ g) ⊙ ((m ∘ i) ∗ α) \\ = & (γ □ β) □ α. \end{matrix}

(ii): We compute :

\begin{matrix}(β′ □ β) ⊟ (α′ □ α) = & (g′′ ∗ ((α′ ∗ h) ⊙ (k′ ∗ α))) ⊙ (((β′ ∗ k) ⊙ (i′ ∗ β)) ∗ f) \\ = & (g′′ ∗ α′ ∗ h) ⊙ (g′′ ∗ k′ ∗ α) ⊙ (β′ ∗ k ∗ f) ⊙ (i′ ∗ β ∗ f) \\ = & (g′′ ∗ α′ ∗ h) ⊙ (β′ ∗ f ∗ h) ⊙ (i′ ∗ g′ ∗ α) ⊙ (i′ ∗ β ∗ f) \\ = & ((β′ ⊟ α′) ∗ h) ⊙ (i′ ∗ (β ⊟ α)) \\ = & (β′ ⊟ α′) □ (β ⊟ α) \end{matrix}

where the fourth equality follows from the commutation rule of remark 2.1.1(i). □

Remark 2.1.10. Proposition 2.1.5(i) establishes explicitly only the associativity of the join operation □, but in light of (2.1.8) we also get immediately the associativity of ⊟.

Example 2.1.11. (i) Let 𝒜 be any 2-category. We may construct a new 2-category

2-Morph(𝒜)

which is the 2-categorical counterpart of the category of arrows (see (1.1.30)). Namely : • the objects of 2-Morph(𝒜) are the 1-cells of 𝒜.

• If f : A → B and f′ : A′ → B′ are any two 1-cells of 𝒜, the arrows f → f′ in 2-Morph(𝒜) are all the oriented squares (2.1.6).

• For 1-cells (g_{1}, h_{1}, α_{1}), (g_{2}, h_{2}, α_{2}) : f → f′, the 2-cells (g_{1}, h_{1}, α_{1}) ⇒ (g_{2}, h_{2}, α_{2}) are all the pairs

(β : g_{1} ⇒ g_{2}, γ : h_{1} ⇒ h_{2})  such that  \frac{α_{1}}{h_{1}|g_{1}} ⊟ \frac{β}{g_{1}|g_{2}} = \frac{γ}{h_{1}|h_{2}} ⊟ \frac{α_{2}}{h_{2}|g_{2}} .

• For f as in the foregoing, the unit functor 1 → 2-Morph(f, f) is given by the 2-cell

(2.1.12)  [figure: A; f; 1_{A}; 1_{f}; B; 1_{B}; A; f; B]

and its identity endomorphism (i_{A}, i_{B}). For f and f′ as in the foregoing, the composition law in 2-Morph(𝒜)(f, f′) is given by the rule

(β′, γ′) ⊙ (β, γ) := (β′ ⊙ β, γ′ ⊙ γ)

for every composable pair ((β, γ), (β′, γ′)) of 2-cells in 2-Morph(𝒜). The composition bifunctor is given by the composition law for squares; i.e., given 1-cells f → f’ and f′ → f′′ in 2-Morph(𝒜) as in (2.1.7), we set

(β, k, i) ∘ (α, g, h) := (β □ α, k ∘ g, i ∘ h).

Lastly, given 2-cells (β, γ) in 2-Morph(𝒜)(f, f′) and (β′, γ′) in 2-Morph(𝒜)(f′, f′′) we set

(β′, γ′) ∗ (β, γ) := (β′ ∗ β, γ′ ∗ γ).

The functoriality and associativity properties of the composition laws for 2-cells are then immediate from the definitions. The associativity of the composition law for 1-cells holds by proposition 2.1.9(i).

(ii) Just as in (1.1.24), for every X ∈ Ob(𝒜) we may consider the 2-subcategories 𝒜 /X and X/𝒜 of 2-Morph(𝒜). Explicitly, the objects of 𝒜 /X are the 1-cells A → X of 𝒜. Given two such objects f, f′, the 1-cells f → f′ of 𝒜 /X are all the diagrams in 𝒜 of the type

[figure: A; f; g; α; X; A′; f′; X]

and for 1-cells (g_{1}, α_{1}), (g_{2}, α_{2}) : f → f′, the 2-cells of 𝒜 /X are the 2-cells β : g_{1} ⇒ g_{2} of 𝒜 such that (f′ ∗ β) ⊙ α_{1} = α_{2}. A similar description applies to X/𝒜.

(iii) To every 1-cell h : X → Y in 𝒜 we attach functors

h_{∗}^{Z} : 𝒜 (Z, X) → 𝒜 (Z, Y) for every Z ∈ Ob(𝒜)

by ruling that h_{∗}^{Z}(k) := h ∘ k and h_{∗}^{Z}(β) := h ∗ β for every 1-cell k : Z → X of 𝒜 and every morphism β : k ⇒ k′ in 𝒜 (Z, X). Let ν : h ⇒ h′ be any 2-cell in 𝒜; we claim that the rule

k ↦ (ν_{∗k}^{Z} := ν ∗ k : h_{∗}^{Z}(k) ⇒ h_{∗}^{′Z}(k)) for every (k : Z → X) ∈ Ob(𝒜 (Z, X))

defines a natural transformation

ν_{∗}^{Z} : h_{∗}^{Z} ⇒ h_{∗}^{′Z} for every Z ∈ Ob(𝒜) .

Indeed, let k, k′ : Z → X be any two objects of 𝒜 (Z, X), and μ : k ⇒ k′ a 2-cell; we need to show that the resulting diagram commutes in 𝒜 :

\xymatrix{h_{∗}^{Z}(k) \ar@{=>}[r]^{ν∗k} \ar@{=>}[d]_{h∗μ} & h_{∗}^{′Z}(k) \ar@{=>}[d]^{h′∗μ} \\ h_{∗}^{Z}(k′) \ar@{=>}[r]^{ν∗k′} & h_{∗}^{′Z}(k′)}

But this follows directly from remark 2.1.1(i). Likewise, we may attach to f the functors

h_{Z}^{∗} : 𝒜 (Y, Z) → 𝒜 (X, Z)  k ↦ k ∘ h  β ↦ β ∗ h  for every Z ∈ Ob(𝒜)

and then ν induces as well a natural transformation

ν_{Z}^{∗} : h_{Z}^{∗} ⇒ h_{Z}^{′∗}  k ↦ k ∗ ν  for every Z ∈ Ob(𝒜) .

We apply these constructions to prove the following result, which generalizes propositions 1.1.15(ii) and 1.1.20(i) to arbitrary 2-categories :

Lemma 2.1.13. Let 𝒜 be a 2-category, f : A → B, g : B → A two 1-cells in 𝒜. We have :

(i) Suppose that f is an equivalence, and g a quasi-inverse for f. Then, for every invertible 2-cell η : 1_{A} ⇒ g ∘ f there exists a unique invertible 2-cell ε : f ∘ g ⇒ 1_{B} such that (η, ε) is an adjunction for the pair (f, g).

(ii) Suppose that (f, g) is an adjoint pair, and that (η, ε) and (η′, ε′) are two adjunctions

for (f, g). Then η = η′ if and only if ε = ε′. (iii) The following conditions are equivalent :

(a) f is an equivalence.

(b) For every Z ∈ Ob(𝒜) the functor f_{Z}^{∗} : 𝒜 (B, Z) → 𝒜 (A, Z) is an equivalence.

(c) For every Z ∈ Ob(𝒜) the functor f_{∗}^{Z} : 𝒜 (Z, A) → 𝒜 (Z, B) is an equivalence.

Proof. (i): By assumption, there exists an invertible 2-cell ε′ : f ∘ g ⇒ 1_{B}. Now, since η and ε′ are invertible 2-cells, it is clear that η_{∗}^{Z} and ε_{∗}^{′Z} are isomorphisms of functors for every Z ∈ Ob(𝒜), so f_{∗}^{Z} and g_{∗}^{Z} are equivalences of categories (proposition 1.1.20(i)); therefore there exists a unique adjunction for the pair (f_{∗}^{Z}, g_{∗}^{Z}) whose unit is η_{∗}^{Z} (claim 1.1.21) and we denote by ε_{∗}^{Z} the unique counit for this adjunction. Then also ε_{∗}^{Z} is an isomorphism of functors (proposition 1.1.20(iii)), and we define the invertible 2-cell ε := ε_{1_{B}}^{B} : f ∘ g ⇒ 1_{B}. Lastly, the triangular identities (1.1.13) for the pair (η_{∗}^{Z}, ε_{∗}^{Z}) immediately imply the corresponding identities for (η, ε).

(ii): As in the foregoing, we obtain for every Z ∈ Ob(𝒜) an adjoint pair of functors (f_{∗}^{Z}, g_{∗}^{Z}), and two adjunctions (η_{∗}^{Z}, ε_{∗}^{Z}), (η_{∗}^{′Z}, ε_{∗}^{′Z}) for (f_{∗}^{Z}, g_{∗}^{Z}). Then proposition 1.1.15(ii) implies that η_{∗}^{Z} = η_{∗}^{′Z} if and only if ε_{∗}^{Z} = ε_{∗}^{′Z}. The assertion is an immediate consequence.

(iii.a)⇒(iii.b),(iii.c): Indeed, let g : B → A be a quasi-inverse for f; it is easily seen that the functor g_{Z}^{∗} is a quasi-inverse for f_{Z}^{∗} and g_{∗}^{Z} is a quasi-inverse for f_{∗}^{Z}.

(iii.b)⇒(iii.a): Since f_{A}^{∗} is an equivalence, we find a 1-cell g : B → A of 𝒜 with an isomorphism g ∘ f = f_{A}^{∗}(g) →^{∼} 1_{A}. There follows an isomorphism f_{B}^{∗}(f ∘ g) = f ∘ g ∘ f →^{∼} f = f_{B}^{∗}(1_{B}), and since f_{B}^{∗} is an equivalence, we deduce an isomorphism f ∘ g →^{∼} 1_{B}, whence the contention. Likewise one shows that (iii.c)⇒(iii.a), or alternatively one can deduce this implication from the foregoing one, by considering the opposite 2-categories. □

Definition 2.1.14. ([28, Def.7.5.1]) Let 𝒜 and ℬ be two 2-categories.

(i) A pseudo-functor F : 𝒜 → ℬ is the datum of :

• For every A ∈ Ob(𝒜), an object F A ∈ Ob(ℬ).

• For every A, B ∈ Ob(𝒜), a functor :

F_{AB} : 𝒜 (A, B) → ℬ(F A, F B).

We shall often omit the subscript, and write only F f instead of F_{AB}f : F A → F B, for a 1-cell f : A → B, and likewise for 2-cells.

• For every A, B, C ∈ Ob(𝒜), a natural isomorphism γ_{ABC} between two functors 𝒜 (A, B) × 𝒜 (B, C) → ℬ(F A, F C) as indicated by the (not necessarily commutative) diagram :

(2.1.15)  [figure: 𝒜 (A, B) × 𝒜 (B, C); c_{ABC}; F_{AB}×F_{BC}; γ_{ABC}; 𝒜 (A, C); F_{AC}; ℬ(F A, F B) × ℬ(F B, F C); c_{F A,F B,F C}; ℬ(F A, F C).]

To ease notation, for every (f, g) ∈ 𝒜 (A, B) × 𝒜 (B, C), we shall write γ_{f,g} instead of (γ_{ABC})_{(f,g)} : c_{F A,F B,F C}(F_{AB}f, F_{BC}g) ⇒ F_{AC}(c_{ABC}(f, g)).

• For every A ∈ Ob(𝒜), a natural isomorphism δ_{A} between functors 1 → ℬ(F A, F A), as indicated by the (not necessarily commutative) diagram :

(2.1.16)  [figure: 1; u_{A}; δ_{A}; 𝒜 (A, A); F_{AA}; 1; u_{F A}; ℬ(F A, F A).]

The system (δ_{•}, γ_{•••}) is called the coherence constraint for F. This datum is required to satisfy: • A composition axiom, which says that the diagram

\xymatrix{F h ∘ F g ∘ F f \ar@{=>}[r]^{F h∗γ_{f,g}} \ar@{=>}[d]_{γ_{g,h}∗F f} & F h ∘ F (g ∘ f) \ar@{=>}[d]^{γ_{g∘f,h}} \\ F (h ∘ g) ∘ F f \ar@{=>}[r]^{γ_{f,h∘g}} & F (h ∘ g ∘ f)}

commutes for every sequence of arrows A ⟶^{f} B ⟶^{g} C ⟶^{h} D in 𝒜. We will write

γ_{f,g,h} : F h ∘ F g ∘ F f ⇒ F (h ∘ g ∘ f)

for the common composition of these two pairs of cells.

• A unit axiom, which says that the diagrams :

\xymatrix{F f ∘ 1_{F A} \ar@{=>}[d]_{1_{F f}} \ar@{=>}[r]^{F f∗δ_{A}} & F f ∘ F 1_{A} \ar@{=>}[d]^{γ_{1_{A},f}} & 1_{F B} ∘ F f \ar@{=>}[d]_{1_{F f}} \ar@{=>}[r]^{δ_{B}∗F f} & F (1_{B}) ∘ F f \ar@{=>}[d]^{γ_{f,1_{B}}} \\ F f \ar@{=>}[r]^{1_{F f}} & F (f ∘ 1_{A}) & F f \ar@{=>}[r]^{1_{F f}} & F (1_{B} ∘ f)}

commute for every arrow f : A → B (where, to ease notation, we have written δ_{A} instead of (δ_{A})_{∗} : 1_{F A} ⇒ F_{AA}1_{A}, and likewise for δ_{B}).

(ii) A pseudo-functor F as in (i) is called strict if (2.1.15) and (2.1.16) both commute, and γ_{ABC} and δ_{A} are the identity natural transformations for every A, B, C ∈ Ob(𝒜).

Remark 2.1.17. With the notation of definition 2.1.14, we have :

(i) The functoriality of F_{AB}, for every A, B ∈ Ob(𝒜), comes down to the identities

F (1_{f}) = 1_{F f}  F (β ⊙ α) = F β ⊙ F α  for every 1-cell f and 2-cells α, β of 𝒜 (A, B) .

(ii) Likewise, the naturality of γ_{ABC} boils down to the commutativity of the diagram

\xymatrix{F g ∘ F f \ar@{=>}[r]^{γ_{f,g}} \ar@{=>}[d]_{F β∗F α} & F (g ∘ f) \ar@{=>}[d]^{F (β∗α)} \\ F g′ ∘ F f′ \ar@{=>}[r]^{γ_{f′,g′}} & F (g′ ∘ f′)}

for every 1-cells f, f′ : A → B, g, g′ : B → C and 2-cells α : f ⇒ f′, β : g ⇒ g′.

(iii) If h : C → D and h′ : C′ → D′ are any two other 1-cells, and μ : h ⇒ h′ is any 2-cell, it follows easily that the diagram

\xymatrix{F h ∘ F g ∘ F f \ar@{=>}[r]^{γ_{f,g,h}} \ar@{=>}[d]_{F μ∗F β∗F α} & F (h ∘ g ∘ f) \ar@{=>}[d]^{F (μ∗β∗α)} \\ F h′ ∘ F g′ ∘ F f′ \ar@{=>}[r]^{γ_{f′,g′,h′}} & F (h′ ∘ g′ ∘ f′)}

commutes : details left to the reader.

(iv) The pseudo-functor F induces opposite pseudo-functors

F^{o} : 𝒜^{o} → ℬ^{o} and ^{o}F : ^{o}𝒜 → ^{o}ℬ

(as well as the functor^{o}F^{o}, by combining the two operations). Namely, F^{o} is given by the rules

F^{o}(A^{o}) := (F A)^{o}  F^{o}(f^{o}) := (F f)^{o}  F^{o}(β^{o}) := (F β)^{o}

for every A ∈ Ob(𝒜), every 1-cell f and every 2-cell β of 𝒜. The coherence constraint of F^{o} is the pair (γ^{o}, δ^{o}) such that

γ_{g^{o},f^{o}}^{o} := (γ_{f,g})^{o} and δ_{A^{o}}^{o} := (δ_{A})^{o}

for every composable pair of 1-cells f, g of 𝒜, and every A ∈ Ob(𝒜). The composition and unit axioms for F^{o} follow immediately from the same for F. Likewise,^{o}F is given by the rules

oF (^{o}A) := ^{o}(F A)  ^{o}F (^{o}f) := ^{o}(F f)  ^{o}F (^{o}β) := ^{o}(F β)

for every A, f, β as in the foregoing. The coherence constraint for^{o}F is the pair (^{o}γ,^{o}δ) with

oγ_{^{o}f,^{o}g} := ^{o}(γ_{f,g})^{−1} and ^{o}δ_{^{o}A} := ^{o}(δ_{A})^{−1}

whose composition and unit axioms are again derived straightforwardly from the same for F.

(v) Let 𝒞 be a third 2-category, and G : ℬ → 𝒞 another pseudo-functor. We may define a composition G ∘ F : 𝒜 → ℬ, which is the pseudo-functor such that

G ∘ F (A) := G(F A)  and (G ∘ F)_{AB} := G_{F A,F B} ∘ F_{AB}  for every A, B ∈ Ob(𝒜) .

Denote by (δ^{F} , γ^{F}) and (δ^{G}, γ^{G}) the coherence constraints of F and respectively G; the coherence constraint of G ∘ F is then the pair (δ^{G∘F} , γ^{G∘F}) such that

δ_{A}^{G∘F} := G(δ_{A}^{F}) ⊙ δ_{F A}^{G} γ_{f,g}^{G∘F} := G(γ_{f,g}^{F}) ⊙ γ_{F f,F g}^{G}

for every A, B, C ∈ Ob(𝒜) and every 1-cell f of 𝒜 (A, B) and g of 𝒜 (B, C). Let us verify the composition axiom for γ^{G∘F} : given a composable sequence of 1-cells f, g, h, we have

\begin{matrix}γ_{g∘f,h}^{G∘F} ⊙ (GF h ∗ γ_{f,g}^{G∘F}) = & G(γ_{g∘f,h}^{F}) ⊙ γ_{F (g∘f),F h}^{G} ⊙ (GF h ∗ (G(γ_{f,g}^{F}) ⊙ γ_{F f,F g}^{G})) \\ = & G(γ_{g∘f,h}^{F}) ⊙ γ_{F (g∘f),F h}^{G} ⊙ (GF h ∗ G(γ_{f,g}^{F})) ⊙ (GF h ∗ γ_{F f,F g}^{G}) \\ = & G(γ_{g∘f,h}^{F}) ⊙ G(F h ∗ γ_{f,g}^{F}) ⊙ γ_{F g∘F f,F h}^{G} ⊙ (GF h ∗ γ_{F f,F g}^{G}) \\ = & G(γ_{g∘f,h}^{F} ⊙ (F h ∗ γ_{f,g}^{F})) ⊙ γ_{F f,F h∘F g}^{G} ⊙ (γ_{F g,F h}^{G} ∗ GF f) \\ = & G(γ_{f,h∘g}^{F} ⊙ (γ_{g,h}^{F} ∗ F f)) ⊙ γ_{F f,F h∘F g}^{G} ⊙ (γ_{F g,F h}^{G} ∗ GF f) \\ = & G(γ_{f,h∘g}^{F}) ⊙ G(γ_{g,h}^{F} ∗ F f) ⊙ γ_{F f,F h∘F g}^{G} ⊙ (γ_{F g,F h}^{G} ∗ GF f) \\ = & G(γ_{f,h∘g}^{F}) ⊙ γ_{F f,F (h∘g)}^{G} ⊙ (G(γ_{g,h}^{F}) ∗ GF f) ⊙ (γ_{F g,F h}^{G} ∗ GF f) \\ = & γ_{f,h∘g}^{G∘F} ⊙ ((G(γ_{g,h}^{F}) ⊙ γ_{F g,F h}^{G}) ∗ GF f) \\ = & γ_{f,h∘g}^{G∘F} ⊙ (γ_{g,h}^{G∘F} ∗ GF f) \end{matrix}

where the third and seventh equalities follow from (ii) (applied to G), the fourth from (i) and the composition axiom for G, the fifth from the composition axiom for F, the sixth from (i).

Next, to check the unit axiom, we compute :

\begin{matrix}γ_{1_{A},f}^{G∘F} ⊙ (GF f ∗ δ_{A}^{G∘F}) = & G(γ_{1_{A},f}^{F}) ⊙ γ_{F 1_{A},F f}^{G} ⊙ (GF f ∗ (G(δ_{A}^{F}) ⊙ δ_{F A}^{G})) \\ = & G(γ_{1_{A},f}^{F}) ⊙ γ_{F 1_{A},F f}^{G} ⊙ (GF f ∗ G(δ_{A}^{F})) ⊙ (GF f ∗ δ_{F A}^{G}) \\ = & G(γ_{1_{A},f}^{F}) ⊙ G(F f ∗ δ_{A}^{F}) ⊙ γ_{1_{F A},F f}^{G} ⊙ (GF f ∗ δ_{F A}^{G}) \\ = & G(γ_{1_{A},f}^{F} ⊙ (F f ∗ δ_{A}^{F})) \\ = & G(1_{F f}) \\ = & 1_{GF f} \end{matrix}

where the third equality follows from (ii), the fourth from (i) and the unit axiom for G, and the fifth from the unit axiom for F. Similarly one verifies the commutativity of the remaining diagram required for the unit axiom : details left to the reader.

(vi) Also, if H : 𝒞 → 𝒟 is any other pseudo-functor, then H ∘ (G ∘ F) = (H ∘ G) ∘ F. Indeed, taking into account (i) we may compute :

\begin{matrix}γ_{f,g}^{H∘(G∘F)} = & H(γ_{f,g}^{G∘F}) ⊙ γ_{G∘F f,G∘F g}^{H} \\ = & H ∘ G(γ_{f,g}^{F}) ⊙ H(γ_{F f,Gf}^{G}) ⊙ γ_{G∘F f,G∘F g}^{H} \\ = & H ∘ G(γ_{f,g}^{F}) ⊙ γ_{F f,Gf}^{H∘G} \\ = & γ_{f,g}^{(H∘G)∘F} . \end{matrix}

Similarly one checks that δ_{A}^{H∘(G∘F)} = δ_{A}^{(H∘G)∘F} for every A ∈ Ob(𝒜), whence the assertion. (vii) Notice as well that if F and G are strict, then the same holds for G ∘ F.

Remark 2.1.18. (i) Remark 2.1.17(v) shows that the set of all U-small 2-categories together with all pseudo-functors between them forms a category with small Hom-sets that we denote

U-2-Cat

(and as always, we shall usually drop the universe U from the notation). We do not know whether 2-Cat is a complete category : for instance, if F : 𝒜 → ℬ and F′ : 𝒜′ → ℬ are any two pseudo-functors, it is not clear whether the fibre product of F and F′ is always representable in 2-Cat. However, in case F and F′ are both strict, this fibre product is indeed representable by a 2-category that we denote

𝒜 ×_{(F,F′)} 𝒜′

(or simply 𝒜 ×_{ℬ} 𝒜′, if the notation is not ambiguous). Namely, the objects of 𝒜 ×_{ℬ} 𝒜′ are all the pairs (A, A′) consisting of objects A of 𝒜 and A′ of 𝒜′ such that F A = F′A′, and the Hom-categories are given by the rule :

𝒜 ×_{ℬ} 𝒜′((A_{1}, A′_{1}), (A_{2}, A′_{2})) := 𝒜 (A_{1}, A_{2}) ×_{ℬ(F A_{1},F A_{2})} 𝒜 (A_{2}, A′_{2})

for every pair of objects (A_{1}, A′_{1}), (A_{2}, A′_{2}) of 𝒜 ×_{ℬ} 𝒜′ (see example 1.2.25(i)). The composition laws and unit functors for 𝒜 ×_{ℬ} 𝒜′ are deduced from those of 𝒜 and 𝒜′, in the obvious fashion. A universal cone for this fibre product is provided by the two projections 𝒜 ← 𝒜 ×_{ℬ} 𝒜′ → 𝒜′, i.e. the strict pseudo-functors defined in the obvious fashion.

(ii) For instance, just as in (1.1.30), for every 2-category 𝒜 we have two natural source and target strict pseudo-functors :

𝒜 ⟵^{s} 2-Morph(𝒜) ⟶^{t} 𝒜

(notation of example 2.1.11(i)). Namely, s assigns to every 1-cell f : A → B the object A of 𝒜, to every diagram (2.1.6) the 1-cell g : s(f) → s(f′) of 𝒜, and to every 2-cell (β, γ) of 2-Morph(𝒜), the 2-cell β of 𝒜. The reader may likewise spell out the definition of t. Then we have as well a natural composition strict pseudo-functor

c : 2-Morph(𝒜) ×_{(t,s)} 2-Morph(𝒜) → 2-Morph(𝒜)

which assigns :

• to every pair of 1-cells (f, g) such that t(f) = s(g), the composition c(f, g) := g ∘ f • to every pair of 1-cells of 2-Morph(𝒜) :

(2.1.19)  [figure: A; f; h; α; B; k; g; β; C; i; A′; f′; B′; g′; C′]

the 1-cell of 2-Morph(𝒜)

c((α, h, k), (β, k, i)) := (β ⊟ α, h, i)

• for (f, g), (f′, g′) as in (2.1.19), and every 2-cells (φ_{1}, φ_{2}) : (α_{1}, h_{1}, k_{1}) ⇒ (α_{2}, h_{2}, k_{2}) in 2-Morph(f, f′) and (φ_{2}, φ_{3}) : (β_{1}, k_{1}, i_{1}) ⇒ (β_{2}, k_{2}, i_{2}) in 2-Morph(g, g′), the pair

c((φ_{1}, φ_{2}), (φ_{2}, φ_{3})) := (φ_{1}, φ_{3})

which is a well defined 2-cell c((α_{1}, h_{1}, k_{1}), (β_{1}, k_{1}, i_{1})) ⇒ c((α_{2}, h_{2}, k_{2}), (β_{2}, k_{2}, i_{2})) in 2-Morph(g ∘ f, g′ ∘ f′), since by assumption we have

\frac{α_{1}}{h_{1}|g_{1}} ⊟ \frac{φ_{1}}{g_{1}|g_{2}} = \frac{φ_{2}}{h_{1}|h_{2}} ⊟ \frac{α_{2}}{h_{2}|g_{2}} \frac{β_{1}}{k_{1}|h_{1}} ⊟ \frac{φ_{2}}{h_{1}|h_{2}} = \frac{φ_{3}}{k_{1}|k_{2}} ⊟ \frac{β_{2}}{k_{2}|h_{2}}

and therefore

\begin{matrix}(β_{1} ⊟ α_{1}) ⊟ φ_{1} = & β_{1} ⊟ (α_{1} ⊟ φ_{1}) \\ = & β_{1} ⊟ (φ_{2} ⊟ α_{2}) \\ = & (β_{1} ⊟ φ_{2}) ⊟ α_{2} \\ = & (φ_{3} ⊟ β_{2}) ⊟ α_{2} \\ = & φ_{3} ⊟ (β_{2} ⊟ α_{2}). \end{matrix}

With these rules, the functoriality of c for 2-cells is immediate. To check the functoriality of c for 1-cells, consider a diagram as in proposition 2.1.9(ii), and to ease notation, set \underline{α} := (α, h, k) and define likewise \underline{α}′, \underline{β} and \underline{β}′. Then proposition 2.1.9(ii) translates as the identity

c(\underline{α}′ ∘ \underline{α}, \underline{β}′ ∘ \underline{β}) = c(\underline{α}′, \underline{β}′) ∘ c(\underline{α}, \underline{β})

as required.

Example 2.1.20. (i) Let 𝒜, ℬ be two 2-categories, and F : 𝒜 → ℬ any pseudo-functor, with coherence constraint (γ^{F} , δ^{F}). Then F induces a pseudo-functor

2-Morph(F) : 2-Morph(𝒜) → 2-Morph(ℬ)

by the following rule. To every 1-cell f : A → B in 𝒜 we assign the 1-cell F f : F A → F B. To every diagram (2.1.6) we assign the diagram

[figure: F A; F f; F g; α^{F}; F B; F h; F A′; F f′; F B′]  where α^{F} := (γ_{g,f′}^{F})^{−1} ⊙ F α ⊙ γ_{f,h}^{F}

and to any 2-cell (β, δ) : (g_{1}, h_{1}, α_{1}) ⇒ (g_{2}, h_{2}, α_{2}) in 2-Morph(f, f′), we assign the pair (F β, F δ). Indeed, let us check that the latter is a well defined 2-cell in 2-Morph(F f, F f′); the assertion boils down to the identity :

(γ_{g_{2},f′}^{F})^{−1} ⊙ F α_{2} ⊙ γ_{f,h_{2}}^{F} ⊙ (F δ ∗ F f) = (F f′ ∗ F β) ⊙ (γ_{g_{1},f′})^{−1} ⊙ F α_{1} ⊙ γ_{f,h_{1}}^{F}.

However, from remark 2.1.17(ii) we get

(F f′ ∗ F β) ⊙ (γ_{g_{1},f′}^{F})^{−1} = (γ_{g_{2},f′}^{F})^{−1} ⊙ F (f′ ∗ β)

hence it suffices to check that

F α_{2} ⊙ γ_{f,h_{2}}^{F} ⊙ (F δ ∗ F f) = F (f′ ∗ β) ⊙ F α_{1} ⊙ γ_{f,h_{1}}^{F}.

But by assumption, we know that (f′ ∗ β) ⊙ α_{1} = α_{2} ⊙ (δ ∗ f), so – taking into account remark 2.1.17(i) – we are further reduced to showing that

γ_{f,h_{2}}^{F} ⊙ (F δ ∗ F f) = F (δ ∗ f) ⊙ γ_{f,h_{1}}^{F}

which follows again from remark 2.1.17(ii). Lastly, the coherence constraint of 2-Morph(F) assigns to any diagram (2.1.12) (resp. (2.1.7)) the pair

(2.1.21)  (δ_{A}^{F}, δ_{B}^{F})  (resp. (γ_{g,k}^{F}, γ_{h,i}^{F}) ).

Indeed, let us check that (δ_{A}^{F}, δ_{B}^{F}) is a well defined 2-cell in 2-Morph(F f, F f) : the assertion boils down to the identity

(F f ∗ δ_{A}^{F}) ⊙ 1_{f}^{F} = 1_{f}^{F} ⊙ (δ_{B}^{F} ∗ F f) where 1_{f}^{F} := (γ_{1_{A},f}^{F})^{−1} ⊙ γ_{f,1_{B}}^{F}

which in turns follows easily from the unit axiom for δ^{F}. Likewise, the assertion that (γ_{g,k}^{F}, γ_{h,i}^{F}) is a well defined 2-cell in 2-Morph(F f, F f′′) comes down to the identity

(2.1.22)  γ_{g,k}^{F} ⊟ (β^{F} □ α^{F}) = (β □ α)^{F} ⊟ γ_{h,i}^{F}.

However, by definition, the left-hand side of (2.1.22) equals

(F f′′ ∗γ_{g,k}^{F})⊙((γ_{k,f′′}^{F})^{−1} ∗F g)⊙(F β ∗F g)⊙(γ_{f′,i}^{F} ∗F g)⊙(F i∗(γ_{g,f′}^{F})^{−1})⊙(F i∗(F α⊙γ_{f,h}^{F}))

and the composition axiom for F yields the identities :

\begin{matrix}(F f′′ ∗ γ_{g,k}^{F}) ⊙ ((γ_{k,f′′}^{F})^{−1} ∗ F g) = & (γ_{k∘g,f′′}^{F})^{−1} ⊙ γ_{g,f′′∘k}^{F} \\ (γ_{f′,i}^{F} ∗ F g) ⊙ (F i ∗ (γ_{g,f′}^{F})^{−1}) = & (γ_{g,i∘f′}^{F})^{−1} ⊙ γ_{f′∘g,i}^{F}. \end{matrix}

So the left-hand side of (2.1.22) also equals

(γ_{k∘g,f′′}^{F})^{−1} ⊙ γ_{g,f′′∘k}^{F} ⊙ (F β ∗ F g) ⊙ (γ_{g,i∘f′}^{F})^{−1} ⊙ γ_{f′∘g,i}^{F} ⊙ (F i ∗ (F α ⊙ γ_{f,h}^{F})).

Next, remark 2.1.17(ii) yields the identities

\begin{matrix}(F β ∗ F g) ⊙ (γ_{g,i∘f′}^{F})^{−1} = & (γ_{g,f′′∘k}^{F})^{−1} ⊙ F (β ∗ g) \\ γ_{f′∘g,i}^{F} ⊙ (F i ∗ F α) = & F (i ∗ α) ⊙ γ_{h∘f,i}^{F}. \end{matrix}

So we have to show that the right-hand side of (2.1.22) equals

(γ_{k∘g,f′′}^{F})^{−1} ⊙ F (β ∗ g) ⊙ F (i ∗ α) ⊙ γ_{h∘f,i}^{F} ⊙ (F i ∗ γ_{f,h}^{F}).

But by unwinding the definition, we see that the right-hand side of (2.1.22) equals

(γ_{k∘g,f′′}^{F})^{−1} ⊙ F (β □ α) ⊙ γ_{f,i∘h}^{F} ⊙ (γ_{h,i}^{F} ∗ F f) = (γ_{k∘g,f′′}^{F})^{−1} ⊙ F (β □ α) ⊙ γ_{h∘f,i}^{F} ⊙ (F i ∗ γ_{f,h}^{F})

(where the equality holds by the composition axiom). Thus, we come down to checking that

F (β ∗ g) ⊙ F (i ∗ α) = F (β □ α)

which follows from remark 2.1.17(i). With these rules, the functoriality of 2-Morph(F) is clear from remark 2.1.17(i), and the naturality of (2.1.21), as well as the composition and unit axioms for the latter, follow from the respective properties of (γ^{F} , δ^{F}), by a direct inspection.

(ii) In the situation of (i), let f : A → B be any 1-cell of 𝒜. Then the unit axiom for the coherence constraint (γ^{F} , δ^{F}) implies the identity :

1_{f}^{F} = (F f ∗ δ_{A}^{F}) ⊙ ((δ_{B}^{F})^{−1} ∗ F f).

(iii) Let 𝒞 be a third 2-category, and G : ℬ → 𝒢 another pseudo-functor; then we have

2-Morph(G ∘ F) = 2-Morph(G) ∘ 2-Morph(F)

(details left to the reader).

Example 2.1.23. Let 𝒜 be any 2-category.

(i) We have a natural strict isomorphism of 2-categories

o(2-Morph(𝒜))^{o} →^{∼} 2-Morph(^{o}𝒜^{o})

that assigns :

• to every object f : A → B of 2-Morph(𝒜) the object^{o}f^{o} : ^{o}B^{o} → ^{o}A^{o}

• to every oriented square (2.1.6) the corresponding oriented square^{o}(2.1.6)^{o}, which is a 1-cell^{o}α^{o} : ^{o}f^{′o} → ^{o}f^{o} in 2-Morph(^{o}𝒜^{o})

• to every 2-cell (β, γ) in 2-Morph(𝒜)(f, f′) the pair (^{o}γ^{o},^{o}β^{o}) which is a 2-cell in

2-Morph(^{o}𝒜^{o})(^{o}f^{′o},^{o}f^{o}). (ii) We deduce from (i) a strict isomorphism of 2-categories

(2-Morph(𝒜^{o}))^{o} →^{∼} ^{o}(2-Morph(^{o}𝒜)).

These latter 2-categories can be described as follows. The objects are the same as those of 2-Morph(𝒜), and the 1-cells f → f′ are the oriented squares just like (2.1.6), but with reversed direction of the orienting arrow, i.e. α is replaced by a 2-cell f′ ∘ g ⇒ h ∘ f. Given two 1-cells (g_{1}, h_{1}, α_{1}), (g_{2}, h_{2}, α_{2}) : f → f′ in 2-Morph(𝒜^{o})^{o}(f, f′), the 2-cells (g_{1}, h_{1}, α_{1}) ⇒ (g_{2}, h_{2}, α_{2}) are still all the pairs (β : g_{1} ⇒ g_{2}, γ : h_{1} ⇒ h_{2}) such that α_{1} ⊟ β = γ ⊟ α_{2}, and the composition rule for such 2-cells is the same as in 2-Morph(𝒜). However, the composition law for 1-cells in 2-Morph(𝒜^{o})^{o} is given by the join operation ⊟ instead of □ : details left to the reader.

Example 2.1.24. We have a natural strict isomorphism of 2-categories :

(−)^{o} : ^{o}Cat →^{∼} Cat

that assigns to every small category 𝒞 the opposite category 𝒞^{o}, to every functor F : ℬ → 𝒞 the opposite functor F^{o} : ℬ^{o} → 𝒞^{o}, and to every natural transformation β : F ⇒ G the opposite transformation β^{o} : G^{o} ⇒ F^{o}.

2.2. Pseudo-natural transformations and their modifications. The following definition introduces the 2-categorical analogue of a natural transformation between functors.

Definition 2.2.1. Consider two pseudo-functors F, G : 𝒜 → ℬ between 2-categories 𝒜, ℬ. (i) A lax-natural transformation α : F ⇒ G is the datum of :

• For every object A of 𝒜, a 1-cell α_{A} : F A → GA.

• For every A, B ∈ Ob(𝒜), a natural transformation τ_{AB} between two functors 𝒜 (A, B) → ℬ(F A, GB), as shown by the (not necessarily commutative) diagram

(2.2.2)  [figure: 𝒜 (A, B); G_{AB}; F_{AB}; τ_{AB}; ℬ(F A, F B); H_{ℬ} (1_{F A},α_{B}); ℬ(GA, GB); H_{ℬ} (α_{A},1_{GB}); ℬ(F A, GB)]

where H_{ℬ} (1_{F A}, α_{B}) and H_{ℬ} (α_{A}, 1_{GB}) are as in example 2.2.7. The datum τ_{••} is called the coherence constraint for α, and is required to satisfy the following coherence axioms (in which we denote by (δ^{F} , γ^{F}) and (δ^{G}, γ^{G}) the coherence constraints for F and respectively G) :

• For every A ∈ Ob(𝒜), the following diagram commutes :

\xymatrix{α_{A} \ar@{=>}[r]^{1_{α_{A}}} \ar@{=>}[d]_{1_{α_{A}}} & 1_{GA} ∘ α_{A} \ar@{=>}[r]^{δ_{A}^{G}∗α_{A}} & G(1_{A}) ∘ α_{A} \ar@{=>}[d]^{τ_{1_{A}}} \\ α_{A} ∘ 1_{F A} \ar@{=>}[rr]^{α_{A}∗δ_{A}^{F}} &  & α_{A} ∘ F (1_{A}).}

• For each pair of 1-cells A ⟶^{f} B ⟶^{g} C in 𝒜, the following diagram commutes :

\xymatrix{Gg ∘ Gf ∘ α_{A} \ar@{=>}[r]^{Gg∗τ_{f}} \ar@{=>}[d]_{γ_{f,g}^{G}∗α_{A}} & Gg ∘ α_{B} ∘ F f \ar@{=>}[r]^{τ_{g}∗F f} & α_{C} ∘ F g ∘ F f \ar@{=>}[d]^{α_{C}∗γ_{f,g}^{F}} \\ G(g ∘ f) ∘ α_{A} \ar@{=>}[rr]^{τ_{g∘f}} &  & α_{C} ∘ F (g ∘ f).}

(ii) A lax-natural transformation α as in (i) is called a pseudo-natural transformation (resp. strict) if τ_{AB} is an isomorphism of functors (resp. if (2.2.2) commutes and τ_{AB} is the identity natural transformation) for every A, B ∈ Ob(𝒜).

(iii) A pseudo-natural transformation α as in (i) is called a pseudo-natural isomorphism if α_{A} is an invertible 1-cell, for every A ∈ Ob(𝒜).

Example 2.2.3. For any 2-category 𝒞, denote by

p_{𝒞} , p′_{𝒞} : 2-Morph(𝒞) ×_{(t,s)} 2-Morph(𝒞) → 2-Morph(𝒞)

the projections, i.e. the strict pseudo-functors that yield a universal cone for this fibre product, where t and s are the target and source pseudo-functors for 2-Morph(𝒞) as in remark 2.1.18(ii). Now, let 𝒜 , ℬ be two 2-categories, and F : 𝒜 → ℬ a pseudo-functor; it is easily seen that

t ∘ 2-Morph(F) ∘ p_{𝒜} = s ∘ 2-Morph(F) ∘ p′_{𝒜}

where 2-Morph(F) is defined as in example 2.1.20(i). There follows a unique pseudo-functor

2-Morph(F, F) : 2-Morph(𝒜) ×_{(t,s)} 2-Morph(𝒜) → 2-Morph(ℬ) ×_{(t,s)} 2-Morph(ℬ)

such that

p_{ℬ} ∘2-Morph(F, F) = 2-Morph(F)∘p_{𝒜} and p′_{ℬ} ∘2-Morph(F, F) = 2-Morph(F)∘p′_{𝒜}

and we claim that the coherence constraint γ^{F} of F can be viewed as a strict pseudo-natural isomorphism

[figure: 2-Morph(𝒜) ×_{(t,s)} 2-Morph(𝒜); 2-Morph(F,F); c; γ^{F}; 2-Morph(𝒜); 2-Morph(F); 2-Morph(ℬ) ×_{(t,s)} 2-Morph(ℬ); c; 2-Morph(ℬ).]

Namely, for every pair of composable 1-cells (f, g) of 𝒜, we regard γ_{f,g}^{F} : F g ∘ F f ⇒ F (g ∘ f) as a 1-cell of 2-Morph(ℬ). Then the strictness of γ^{F} boils down to the identity

γ_{f′,g′}^{F} □ (β^{F} ⊟ α^{F}) = (β ⊟ α)^{F} □ γ_{f,g}^{F}

for every diagram (2.1.19) in 𝒜. For the verification, notice that the associativity constraint for the functor (2-Morph(F^{o}))^{o} assigns to (2.1.19) the pair (γ_{f,g}^{o}, γ_{f′,g′}^{o})^{o} = (γ_{f,g}, γ_{f′,g′}) (see remark 2.1.17(iv) and example 2.1.20(i)), and then the foregoing identity just translates the assertion that (γ_{f,g}, γ_{f′,g′}) is a well defined 2-cell in 2-Morph(F^{o}h^{o}, F^{o}i^{o})^{o} (see example 2.1.23(ii)). The coherence axioms for γ^{F} follow easily, by unwinding the definitions.

Example 2.2.4. (i) Any category 𝒜 can be regarded as a 2-category in a natural way : namely, for any two objects A and B of 𝒜 one lets 𝒜 (A, B) be the discrete category Hom_{𝒜} (A, B); hence the only 2-cells of 𝒜 are the identities 1_{f} : f ⇒ f, for every morphism f : A → B. The composition bifunctor c_{ABC} is of course given (on 1-cells) by the composition law for morphisms of 𝒜. Likewise, the functor u_{A} assigns to every object A its identity endomorphism.

(ii) In the same vein, every functor between usual categories is a strict pseudo-functor between the corresponding 2-categories as in (i). Finally, every natural transformation of usual functors can be regarded naturally as a strict pseudo-natural transformation between the corresponding strict pseudo-functors.

Remark 2.2.5. With the notation of definition 2.2.1, we have :

(i) The coherence constraint of α can be regarded as a system of oriented squares

[figure: F A; α_{A}; F f; τ_{f}; GA; Gf; F B; α_{B}; GB] for every 1-cell f : A → B of 𝒜 .

Then α is a pseudo-natural transformation if and only if the orientation τ_{f} is invertible for every 1-cell f of 𝒜. The naturality of τ_{••} comes down to the commutativity of the diagram

\xymatrix{Gf ∘ α_{A} \ar@{=>}[r]^{τ_{f}} \ar@{=>}[d]_{Gβ∗α_{A}} & α_{B} ∘ F f \ar@{=>}[d]^{α_{B}∗F β} \\ Gf′ ∘ α_{A} \ar@{=>}[r]^{τ_{f′}} & α_{B} ∘ F f′}

for every A, B ∈ Ob(𝒜) and every 2-cell β : f ⇒ f′ in 𝒜 (A, B). The latter can be also be written as the identity

τ_{f} ⊟ F β = Gβ ⊟ τ_{f′}.

Likewise, the coherence axioms can be written as the identities

δ_{A}^{G} ⊟ τ_{1_{A}} = 1_{α_{A}} ⊟ δ_{A}^{F} (τ_{g} □ τ_{f}) ⊟ γ_{f,g}^{F} = γ_{f,g}^{G} ⊟ τ_{g∘f}

for every A ∈ Ob(𝒜) and every composable pair (f, g) of 1-cells of 𝒜.

(ii) The foregoing list of requirements becomes more intelligible, when we observe that a lax-natural transformation α : F ⇒ G is equivalent to the datum of a pseudo-functor

α̃ : 𝒜 → 2-Morph(ℬ)  such that s ∘ α̃ = F  and t ∘ α̃ = G

and α is a pseudo-natural transformation if and only if α̃(f) is an invertible 2-cell for every 1-cell f of 𝒜. Namely, in light of (i) we obtain such α̃ by setting

α̃(A) := α_{A}  α̃(f) := τ_{f}  α̃(β) := (F β, Gβ)

for every A ∈ Ob(𝒜), every 1-cell f of 𝒜, and every 2-cell β of 𝒜. The coherence constraint (δ, γ) of α̃ are defined by setting

δ_{A} := (δ_{A}^{F}, δ_{A}^{G}) γ_{f,g} := (γ_{f,g}^{F}, γ_{f,g}^{G})

for every A ∈ Ob(𝒜) and every composable pair (f, g) of 1-cells of 𝒜. Especially, α̃ is a strict pseudo-functor if and only if the same holds for both F and G.

(iii) If α : F ⇒ G and β : G ⇒ H are two lax-natural transformations, let α̃, β̃ : 𝒜 → 2-Morph(ℬ) be the associated pseudo-functors, as in (ii); then we may proceed as in example 1.2.25(iii) to define the composition

β ⊙ α : F ⇒ H.

Namely, it shall be the lax-natural transformation associated with the pseudo-functor

𝒜 ⟶^{(α̃,β̃)} 2-Morph(𝒜) ×_{(t,s)} 2-Morph(𝒜) ⟶^{c} 2-Morph(𝒜)

(notation of remark 2.1.18(ii)). Unwinding the definitions, we see that this is the lax-natural transformation given by the rule

A ↦ β_{A} ∘ α_{A} for every A ∈ Ob(𝒜)

and with coherence constraint τ_{••}^{β⊙α} given by the rule (notation of remark 2.1.18(ii)) :

(f : A → B) ↦ τ_{f}^{β} ⊟ τ_{f}^{α} for every 1-cell f of 𝒜

where τ_{••}^{α} (resp. τ_{••}^{β}) denotes the coherence constraint of α (resp. of β). Hence, if both α and β are pseudo-natural (resp. strict), the same holds for β ⊙ α. Since the composition law ⊟ is associative (proposition 2.1.9(i) and remark 2.1.10), we deduce that

λ ⊙ (β ⊙ α) = (λ ⊙ β) ⊙ α

for every lax-natural transformation λ : H ⇒ K (and any pseudo-functor K : 𝒜 → ℬ).

(iv) If 𝒞 is a third 2-category, F′, G′ : ℬ → 𝒞 two other pseudo-functors, and α′ : F′ ⇒ G′ another lax-natural transformation, we may likewise define the Godement products

α′ ∗ F : F′ ∘ F ⇒ G′ ∘ F G′ ∗ α : G′ ∘ F ⇒ G′ ∘ G.

Namely, they shall be the pseudo-natural transformations attached to the pseudo-functors

α̃′ ∘ F and respectively 2-Morph(G′) ∘ α̃

where α̃′ : ℬ → 2-Morph(𝒞) is the pseudo-functor associated with α′, as in (i), and we define 2-Morph(G′) as in example 2.1.20(i). If α (resp. α′) is pseudo-natural, the same holds for G′ ∗α (resp. for α′ ∗ F). In view of example 2.1.20(iii) and remark 2.1.17(vi), it is then clear that

α′ ∗ (F ∘ K) = (α′ ∗ F) ∗ K and (K′ ∘ G′) ∗ α = K′ ∗ (G′ ∗ α)

for any other 2-category 𝒟, and every pseudo-functors K : 𝒟 → 𝒜 and K′ : 𝒞 → 𝒟. Furthermore, if β is as in (ii), and β′ : G′ ⇒ H′ is another lax-natural transformation (to a target pseudo-functor H′ : ℬ → 𝒞), then we get straightforwardly :

(G′ ∗ α) ∗ K = G′ ∗ (α ∗ K) and (β′ ⊙ α′) ∗ F = (β′ ∗ F) ⊙ (α′ ∗ F).

On the other hand, example 2.2.3says that the coherence constraint γ^{G′} of G′ induces a pseudonatural isomorphism from the pseudo-functor associated with (G′ ∗ β) ⊙ (G′ ∗ α) to the pseudofunctor associated with G′ ∗ (β ⊙ α). Thus, if G′ is a strict pseudo-functor, then these two lax-natural transformations coincide, but in general they will be different.

(v) In the same vein, it is tempting to mimick example 1.2.25(iii), in order to define a full Godement product α′ ∗ α of any two lax-natural transformations as in (iv) : namely, one could for example declare this product to be (G′ ∗ α) ⊙ (α′ ∗ F). However, notice that this lax-natural transformation shall be, in general, different from (α′ ∗ G) ⊙ (F′ ∗ α), which would be another plausible definition. More precisely, we have a 2-cell

τ_{α_{A}}^{α′} : ((G′ ∗ α) ⊙ (α′ ∗ F))_{A} ⇒ ((α′ ∗ G) ⊙ (F′ ∗ α))_{A} for every A ∈ Ob(𝒜)

where τ_{••}^{α′} denotes the coherence constraint of α′. Thus, the two possible definition will in general coincide only in case α′ is a strict pseudo-natural transformation.

(vi) Taking into account remark 2.1.1(i) and the observations in (iv) and (v), we conclude that the datum of all small 2-categories, all pseudo-functors between such 2-categories, and all lax-natural (or pseudo-natural) transformations, does not form a 2-category, at least not with Godement products defined as in the foregoing. Rather, the pseudo-natural isomorphisms exhibited in (iv) hint at a higher order structure, whose full explication would require the introduction of a 3-categorical formalism that lies beyond the bounds of our treatise. Instead, we will venture only as far as needed to reach the notion of modification of pseudo-natural transformations (see definition 2.2.9), which will leave us just outside the threshold of 3-category theory, but still will provide a workable language for expressing the foregoing higher order compatibilities, and others as well of a similar nature, that will be encountered in this section.

(vii) Let us also mention that the datum of all small 2-categories, all strict pseudo-functors between such categories, and all strict pseudo-natural transformations of such pseudo-functors, does carry a natural 2-category structure, with the composition laws for pseudo-functors and pseudo-natural transformations defined above : the easy verification shall be left to the reader.

Example 2.2.6. (i) Consider two pseudo-functors F, G : 𝒜 → ℬ between 2-categories, and β : F ⇒ G a pseudo-natural transformation. Denote by (γ^{F} , δ^{F}) (resp. (γ^{G}, δ^{G}), resp. τ^{β}) the coherence constraint of F (resp. G, resp. β). We deduce from β pseudo-natural transformations

β^{o} : G^{o} ⇒ F^{o} ^{o}β : ^{o}F ⇒ ^{o}G

with coherence constraints given by the rules

\begin{matrix}(f^{o} : A^{o} → A^{′o}) & ↦ τ_{f^{o}}^{β^{o}} := (τ_{f}^{β −1})^{o} : F^{o}f^{o} ∘ β_{A^{′o}}^{o} ⇒ β_{A^{o}}^{o} ∘ G^{o}f^{o} \\ (^{o}f : ^{o}A → ^{o}A′) & ↦ τ_{f^{o}}^{^{o}β} := ^{o}(τ_{f}^{β −1}) : ^{o}G^{o}f ∘^{o}β_{^{o}A} ⇒ ^{o}β_{^{o}A′} ∘^{o}F^{o}f. \end{matrix}

Indeed, the coherence axioms for τ^{β} imply the identities

(τ_{1_{A}}^{β −1})^{o}⊟(δ_{A}^{G −1})^{o} = (δ_{A}^{F −1})^{o}⊟1_{β_{A}}^{o} (γ_{f,f′}^{F −1})^{o}⊟((τ_{f}^{β −1})^{o}□(τ_{f′}^{β −1})^{o}) = (τ_{f′∘f}^{β −1})^{o}⊟(γ_{f,f′}^{G −1})^{o}

whence :

(δ_{A}^{F})^{o} ⊟ (τ_{1_{A}}^{β −1})^{o} = 1_{β_{A}}^{o} ⊟ (δ_{A}^{G −1})^{o} ((τ_{f}^{β −1})^{o} □ (τ_{f′}^{β −1})^{o}) ⊟ (γ_{f,f′}^{G})^{o} = (γ_{f,f′}^{F})^{o} ⊟ (τ_{f′∘f}^{β −1})^{o}

which in view, of remark 2.1.17(iv), shows that τ_{•}^{β^{o}} fulfills the required coherence axioms. Likewise one verifies easily the naturality condition, whence the assertion. The corresponding verifications for^{o}β are similar, and shall be left to the reader.

(ii) By virtue of (i), one can say that – just as for usual natural transformation between functors – the datum of a pseudo-natural transformation β : F ⇒ G is equivalent to that of a pseudo-natural transformation β^{o} : F^{o} ⇒ G^{o}, and also to that of a pseudo-natural transformation ^{o}β : ^{o}F ⇒ ^{o}G. On the other hand, these equivalences does not extend to general lax-natural transformations. By inspecting the definition, we see that a lax-natural transformation α : ^{o}F ⇒ ^{o}G is the datum of a system of 1-cells (α_{A} : F A → GA | A ∈ Ob(𝒜)) together with oriented squares as in remark 2.2.5(i), but whose orientation is reversed, hence τ_{f} is a 2-cell α_{B} ∘ F f ⇒ Gf ∘ α_{A} in ℬ. The naturality and coherence conditions for such a system of oriented squares must be likewise suitably reoriented : the details shall be left to the reader. In some literature, such data (α_{•}, τ_{••}) are called oplax-natural transformations, but we shall not use this terminology.

Example 2.2.7. (i) Let 𝒞 be any 2-category, and pick a universe U such that 𝒞 is U-small; according to remark 2.1.18(i) the product 𝒞^{o} × 𝒞 is representable in U-2-Cat. Then we get the strict pseudo-functor

H_{𝒞} : 𝒞^{o} × 𝒞 → U-Cat

(for the natural 2-category structure on U-Cat discussed in remark 2.1.1(ii)) that assigns : • to every C, C′ ∈ Ob(𝒞), the category H_{𝒞} (C, C′) := 𝒞 (C, C′)

• to every pair of 1-cells f : C_{2} → C_{1} and f′ : C′_{1} → C′_{2} of 𝒞, the functor

H_{𝒞} (f, f′) : 𝒞 (C_{1}, C′_{1}) → 𝒞 (C_{2}, C′_{2}) g ↦ f′ ∘ g ∘ f α ↦ f′ ∗ α ∗ f

where g : C_{1} → C′_{1} is any 1-cell and α any 2-cell in 𝒞 (C_{1}, C′_{1})

• to every pair of 2-cells β : f ⇒ g, β′ : f′ ⇒ g′ (with 1-cells f, g : C_{2} → C_{1} and f′, g′ : C′_{1} → C′_{2}), the natural transformation

H_{𝒞} (β, β′) : H_{𝒞} (f, f′) ⇒ H_{𝒞} (g, g′) (h : C_{1} → C′_{1}) ↦ β′ ∗ h ∗ β.

(ii) If F : 𝒜 → 𝒞 and G : ℬ → 𝒞 are any two pseudo-functors, we also define

H_{ℬ} (F, G) := H_{ℬ} ∘ (F^{o} × G) : 𝒜^{o} × ℬ → U-Cat.

Likewise, if F′ : 𝒜 → 𝒞 and G′ : ℬ → 𝒞 is another pair of pseudo-functors, and α : F ⇒ F′, β : G ⇒ G′ two pseudo-natural transformations, we let

H_{ℬ} (α, β) := H_{ℬ} ∗ (α^{o} × β) : H_{ℬ} (F, G) ⇒ H_{ℬ} (F′, G′).

Example 2.2.8. (i) For every 2-category 𝒜 and every X ∈ Ob(𝒜), the restriction s_{X} : 𝒜 /X → 𝒜 (resp. t_{X} : X/𝒜 → 𝒜) of the pseudo-functor s (resp. t) of remark 2.1.18(ii) is a well defined strict pseudo-functor on the 2-category 𝒜 /X (resp. X/𝒜 of example 2.1.11(ii)). Every 1-cell h : X → X′ induces a strict pseudo-functor

h_{∗} : 𝒜 /X → 𝒜 /X′ (A ⟶^{g} X) ↦ (A ⟶^{h∘g} X′).

To every pair of objects f : A → X and f′ : A′ → X and every 1-cell (g, α) : f ⇒ f′ of 𝒜 /X, the pseudo-functor h_{∗} assigns the 1-cell (g, h ∗ β) : h_{∗}f ⇒ h_{∗}f′ of 𝒜 /X′; lastly, for every pair of 1-cells (g_{1}, α_{1}), (g_{2}, α_{2}) : f → f′ and every 2-cell β : g_{1} ⇒ g_{2}, we have h_{∗}β := h ∗ β.

Additionally, every 2-cell β : h ⇒ h′ between 1-cells h, h′ : X → X′ induces a strict pseudo-natural transformation

β_{∗} : h_{∗} ⇒ h′_{∗}

which assigns to every (A ⟶^{g} X) ∈ Ob(𝒜 /X), the 1-cell (1_{A}, β ∗ g) : h ∘ g → h′ ∘ g of 𝒜 /X′. Lastly, notice that^{o}(^{o}𝒜^{o}/^{o}X^{o})^{o} = X/𝒜; we get therefore a strict pseudo-functor

h^{∗} := ^{o}((^{o}h^{o})_{∗})^{o} : X′/𝒜 → X/𝒜 (X′ ⟶^{g} A) ↦ (X ⟶^{g∘h} A)

and the 2-cell β as in the foregoing induces as well a strict pseudo-natural transformation

β^{∗} := ^{o}((^{o}β^{o})_{∗})^{o} : h^{∗} ⇒ h^{′∗}.

(ii) More generally, if F : 𝒜 → ℬ is any pseudo-functor with coherence constraint (δ^{F} , γ^{F}), and B ∈ Ob(ℬ) any object, we may define a 2-category

F 𝒜 /B

whose objects are the pairs (A, f : F A → B) where A ∈ Ob(𝒜) and f is a 1-cell of ℬ. The 1cells (g, α) : (A, f) → (A′, f′) consist of a 1-cell g : A → A′ of 𝒜 and a 2-cell α : f ⇒ f′∘F g. For 1-cells (g_{1}, α_{1}), (g_{2}, α_{2}) : (A, f) → (A′, f′), the 2-cells β : (g_{1}, α_{1}) ⇒ (g_{2}, α_{2}) are the 2cells β : g_{1} ⇒ g_{2} such that (f′ ∗ F β) ⊙ α_{1} = α_{2}. The composition law for 1-cells assigns to

every pair of 1-cells (A, f) ⟶^{(g,α)} (A′, f′) ⟶^{(g′,α′)} (A′′, f′′) the 1-cell

(g′, α′) ∘ (g, α) := (g′ ∘ g, (\frac{α′}{f′|f′′} □ \frac{α}{f|f′} ) ⊟ \frac{γ_{g,g′}^{F}}{1_{F A}|1_{F A′′}} ).

Let us check the associativity property : let (A′′, f′′) ⟶^{(g′′,α′′)} (A′′′, f′′′) be another 1-cell; then

\begin{matrix}(g′′, α′′) ∘ ((g′, α′) ∘ (g, α)) = & (α′′ □ ((α′ □ α) ⊟ γ_{g,g′}^{F})) ⊟ γ_{g′∘g,g′′}^{F} \\ = & ((α′′ ⊟ 1_{F g′′}) □ ((α′ □ α) ⊟ γ_{g,g′}^{F})) ⊟ γ_{g′∘g,g′′}^{F} \\ = & (α′′ □ α′ □ α) ⊟ (1_{F g′′} □ γ_{g,g′}^{F}) ⊟ γ_{g′∘g,g′′}^{F}. \end{matrix}

On the other hand, a similar calculation yields :

((g′′, α′′) ∘ (g′, α′)) ∘ (g, α) = (α′′ □ α′ □ α) ⊟ (γ_{g′,g′′}^{F} □ 1_{F g}) ⊟ γ_{g,g′′∘g′}^{F}

and the coherence axiom for F translates as : (1_{F g′′} □γ_{g,g′}^{F})⊟γ_{g′∘g,g′′}^{F} = (γ_{g′,g′′}^{F} □1_{F g})⊟γ_{g,g′′∘g′}^{F}, whence the contention. Then it is easily seen that the identity 1-cell of any object (A, f) is given by the pair (1_{A}, δ_{A}^{F}) : details left to the reader. The composition laws for 2-cells in F 𝒜 /B are induced by those of 𝒜, in the obvious way; then the required associativity and functoriality properties follow straightforwardly. Lastly we define :

B/F 𝒜 := ^{o}(^{o}F^{o}(^{o}𝒜^{o})/^{o}B^{o})^{o}.

The source and target pseudo-functors of (i) generalize as well : we get strict pseudo-functors

\begin{aligned}s_{B} & : F 𝒜 /B → 𝒜 : (A, f : F A → B) ↦ A \\ t_{B} & : B/F 𝒜 → 𝒜 : (A, f : B → F A) ↦ A \end{aligned}

and every 1-cell h : B → B′ of ℬ induces strict pseudo-functors

\begin{aligned}h_{∗} & : F 𝒜 /B → F 𝒜 /B′ : (A, f : F A → B) ↦ (A, h ∘ f) \\ h^{∗} & : B′/F 𝒜 → B/F 𝒜 : (A, f : B′ → F A) ↦ (A, f ∘ h). \end{aligned}

Likewise, if h′ : B → B′ is another 1-cell of ℬ, every 2-cell β : h′ ⇒ h′ of ℬ′ induces strict pseudo-natural transformations :

\begin{aligned}β_{∗} & : h_{∗} ⇒ h′_{∗} : (A, f : F A → B) ↦ (1_{A}, β ∗ f) \\ β^{∗} & : h^{∗} ⇒ h^{′∗} : (A, f : B → F A) ↦ (1_{A}, f ∗ β). \end{aligned}

Definition 2.2.9. Consider two pseudo-functors F, G : 𝒜 → ℬ between 2-categories 𝒜, ℬ, and two lax-natural transformations α, β : F ⇒ G. A modification Ξ : α ⇝ β is a family

Ξ_{A} : α_{A} ⇒ β_{A} for every A ∈ Ob(𝒜)

of 2-cells of ℬ that satisfies the compatibility condition :

(Ξ_{A′} ∗ F f) ⊙ τ_{f}^{α} = τ_{f}^{β} ⊙ (Gf ∗ Ξ_{A}) every 1-cell f : A → A′ of 𝒜

where τ_{••}^{α} (resp. τ_{••}^{β}) denotes the coherence constraint for α (resp. for β).

Remark 2.2.10. In the situation of definition 2.2.9, let Ξ : α ⇝ β be any modification. (i) For any pair of 1-cells f, g : A → A′ of 𝒜, and any 2-cell γ : f ⇒ g we have the identity:

(Ξ_{A′} ∗ F γ) ⊙ τ_{f}^{α} = τ_{g}^{β} ⊙ (Gγ ∗ Ξ_{A}).

Indeed, we may compute :

\begin{matrix}(Ξ_{A′} ∗ F γ) ⊙ τ_{f}^{α} = & (β_{A′} ∗ F γ) ⊙ (Ξ_{A′} ∗ F f) ⊙ τ_{f}^{α} \\ = & (β_{A′} ∗ F γ) ⊙ τ_{f}^{β} ⊙ (Gf ∗ Ξ_{A}) \\ = & τ_{g}^{β} ⊙ (Gγ ∗ β_{A}) ⊙ (Gf ∗ Ξ_{A}) \\ = & τ_{g}^{β} ⊙ (Gγ ∗ Ξ_{A}) \end{matrix}

where the first and the last identities follow from remark 2.1.1(i), and the third one from remark 2.2.5(i). Notice also that the condition of definition 2.2.9can be restated as the identity

(2.2.11)  \frac{Ξ_{A′}}{α_{A′}|β_{A′}} □ \frac{τ_{f}^{α}}{α_{A}|α_{A′}} = \frac{τ_{f}^{β}}{β_{A}|β_{A′}} □ \frac{Ξ_{A}}{α_{A}|β_{A}} .

(ii) Let α̃, β̃ : 𝒜 → 2-Morph(ℬ) be the pseudo-functors attached to α and β as in remark 2.2.5(ii). By unwinding the definitions, it is easily seen that Ξ is equivalent to the datum of a strict pseudo-natural transformation

Ξ̃ : α̃ ⇒ β̃ such that s ∗ Ξ̃ = 1_{F} and t ∗ Ξ̃ = 1_{G}.

Namely, we obtain such a Ξ̃ by assigning, to every object A of 𝒜, the 1-cell of 2-Morph(ℬ) :

[figure: F A; α_{A}; Ξ_{A}; GA; F A; β_{A}; GA.]

(iii) If γ : F ⇒ G is any other lax-natural transformation, and Θ : β ⇝ γ any other modification, we can define the composition

Θ ⊙ Ξ : α ⇝ γ : A ↦ Θ_{A} ⊙ Ξ_{A} for every A ∈ Ob(𝒜).

In order to check that Θ ⊙ Ξ is indeed a well defined modification, we may notice that if Ξ̃ and Θ̃ are defined as in (ii), then Ξ ⊙ Θ is the modification corresponding to the pseudo-natural transformation

Θ̃ ⊙ Ξ̃ : 𝒜 → 2-Morph(ℬ)

with the composition rule ⊙ for pseudo-natural transformations given by remark 2.2.5(iii). It is clear that this composition law is associative for any triple of composable modifications.

(iv) Likewise, if H : 𝒜 → ℬ is another pseudo-functor, α′, β′ : G ⇒ H two other laxnatural transformations, and Ξ′ : α′ ⇝ β′ any modification, we can define the modification

Ξ′ ∗ Ξ : α′ ⊙ α ⇝ β′ ⊙ β  A ↦ Ξ′_{A} ∗ Ξ_{A}  for every A ∈ Ob(𝒜) .

To see that Ξ′ ∗ Ξ is well defined, consider – for any 1-cell f : A → A′ in 𝒜 – the diagram

[figure: F A; α_{A}; F f; τ_{f}^{α}; GA; τ_{f}^{α′}; α′_{A}; Gf; HA; Hf; F A′; α_{A′}; Ξ_{A′}; GA′; Ξ′_{A′}; α′_{A′}; HA′; F A′; β_{A′}; GA′; β′_{A′}; HA′.]

We compute :

(Ξ′_{A′} ∗ Ξ_{A′} ∗ F f) ⊙ τ_{f}^{α′∘α} = (Ξ′_{A′} ⊟ Ξ_{A′}) □ (τ_{f}^{α′} ⊟ τ_{f}^{α})
= (Ξ′_{A′} □ τ_{f}^{α′}) ⊟ (Ξ_{A′} □ τ_{f}^{α})
= (τ_{f}^{β′} □ Ξ′_{A}) ⊟ (τ_{f}^{β} □ Ξ_{A})
= (τ_{f}^{β′} ⊟ τ_{f}^{β}) □ (Ξ′_{A} ⊟ Ξ_{A})
= τ_{f}^{β′∘β} ⊙ (Hf ∗ Ξ′_{A} ∗ Ξ_{A})

where the second and fourth equalities follow from proposition 2.1.9(ii), and the third follows from (2.2.11). The contention follows. This operation is clearly associative. As usual, we write α ∗ Ξ and Ξ ∗ α′ instead of 1_{α} ∗ Ξ and respectively Ξ ∗ 1_{α′}, for any lax-natural transformations α : G ⇒ H and α′ : E ⇒ F.

(v) There is a third type of operation for modifications : namely, let 𝒞 be another 2-category and I : 𝒞 → 𝒜 any pseudo-functor; then we obtain the modification

Ξ ∘ I : α ∗ I ⇒ β ∗ I  C ↦ Ξ_{IC}  for every C ∈ Ob(𝒞).

To check that Ξ ∘ I is indeed a modification, it suffices to notice that the coherence constraint of α ∗ I is given by the system of 2-cells τ_{If}^{α}, for f ranging over the 1-cells of 𝒞, and correspondingly for the coherence constraint of β ∗ I.

(vi) Likewise, if K : ℬ → 𝒞 is any pseudo-functor, we get the modification

K ∘ Ξ : K ∗ α ⇝ K ∗ β  A ↦ K(Ξ_{A})  for every A ∈ Ob(𝒜) .

Indeed, notice that the coherence constraints of K∗α and K∗β attach to every 1-cell f : A → A′ in 𝒜 the 2-cells

τ_{f}^{K∗α} := γ_{F f,α_{A′}}^{K −1} ⊙ K(τ_{f}^{α}) ⊙ γ_{α_{A},Gf}^{K} τ_{f}^{K∗β} := γ_{F f,β_{A′}}^{K −1} ⊙ K(τ_{f}^{β}) ⊙ γ_{β_{A},Gf}^{K}.

Hence we may compute :

(K(Ξ_{A′}) ∗ KF f) ⊙ τ_{f}^{K∗α} = (K(Ξ_{A′}) ∗ KF f) ⊙ γ_{F f,α_{A′}}^{K −1} ⊙ K(τ_{f}^{α}) ⊙ γ_{α_{A},Gf}^{K}
= γ_{F f,β_{A′}}^{K −1} ⊙ K(Ξ_{A′} ∗ F f) ⊙ K(τ_{f}^{α}) ⊙ γ_{α_{A},Gf}^{K}
= γ_{F f,β_{A′}}^{K −1} ⊙ K((Ξ_{A′} ∗ F f) ⊙ τ_{f}^{α}) ⊙ γ_{α_{A},Gf}^{K}
= γ_{F f,β_{A′}}^{K −1} ⊙ K(τ_{f}^{β} ⊙ (Ξ_{A} ∗ Gf)) ⊙ γ_{α_{A},Gf}^{K}
= γ_{F f,β_{A′}}^{K −1} ⊙ K(τ_{f}^{β}) ⊙ K(Ξ_{A} ∗ Gf) ⊙ γ_{α_{A},Gf}^{K}
= γ_{F f,β_{A′}}^{K −1} ⊙ K(τ_{f}^{β}) ⊙ γ_{β_{A},Gf}^{K} ⊙ (K(Ξ_{A}) ∗ KGf)
= τ_{f}^{K∗β} ⊙ (K(Ξ_{A}) ∗ KGf)

where the second and sixth equality follows from remark 2.1.17(ii), and the third and fifth follow from remark 2.1.17(i).

(vii) Let Θ and Ξ be as in (iii), and I and K like in (v) and (vi); then it is easily seen that

(Θ ⊙ Ξ) ∘ I = (Θ ∘ I) ⊙ (Ξ ∘ I) K ∘ (Θ ⊙ Ξ) = (K ∘ Θ) ⊙ (K ∘ Ξ).

Indeed, the first identity is immediate, and the second follows directly from remark 2.1.17(i). Furthermore, if Ξ and Ξ′ are as in (iv), we have the identities

(Ξ′ ∗ Ξ) ∘ I = (Ξ′ ∘ I) ∗ (Ξ ∘ I) K ∘ (Ξ′ ∗ Ξ) ⊙ γ_{α,α′}^{K} = γ_{β,β′}^{K} ⊙ ((K ∘ Ξ′) ∗ (K ∘ Ξ)).

Indeed, the first identity is immediate, and the second follows directly from remark 2.1.17(ii).

Definition 2.2.12. Let 𝒜 and ℬ be two 2-categories.

(i) For every pair of pseudo-functors F, G : 𝒜 → ℬ we have a category

PsNat(F, G)

whose objects are the pseudo-natural transformations F ⇒ G, and whose morphisms are the modifications α ⇝ β between them, with the composition rule ⊙ given by remark 2.2.10(iii).

(ii) The pseudo-functors 𝒜 → ℬ are the objects of a 2-category :

PsFun(𝒜 , ℬ)

whose 1-cells F → G are the pseudo-natural transformations F ⇒ G of, and whose 2-cells are the modifications. For fixed F, G : 𝒜 → ℬ, the category structure on the set of 1-cells F → G is the one of PsNat(F, G). The composition functor

PsNat(F, G) × PsNat(G, H) → PsNat(F, H)

assigns, to any two modifications Ξ : α ⇝ β and Ξ′ : α′ ⇝ β′, the modification Ξ′ ∗ Ξ. The associativity and unit axioms for this composition functor follow by a simple inspection.

(iii) The strict pseudo-functors 𝒜 → ℬ are the objects of a 2-category :

stPsFun(𝒜 , ℬ)

defined as the sub-2-category of PsFun(𝒜 , ℬ) whose 1-cells are the strict pseudo-natural transformations, and whose 2-cells are all the modifications.

2.2.13. In the situation of definition 2.2.9, suppose that α and β are pseudo-natural; in view of example 2.2.6(i), the modification Ξ induces modifications :

\begin{aligned}Ξ^{o} & : α^{o} ⇝ β^{o}  A^{o} ↦ ((Ξ_{A}^{o} : α_{A^{o}}^{o} ⇒ β_{A^{o}}^{o}) \\ ^{o}Ξ & : ^{o}β ⇝ ^{o}α  ^{o}A ↦ (^{o}(Ξ_{A}) : ^{o}β_{^{o}A} ⇒ ^{o}α_{^{o}A}). \end{aligned}

The detailed verification shall be left to the reader. With the notation of definition 2.2.12, it follows that the rules : (F : 𝒜 → ℬ) ↦ (F^{o} : 𝒜^{o} → ℬ^{o}), (β : F ⇒ G) ↦ (β^{o} : F^{o} ⇒ G^{o}) and (Ξ : α ⇝ β) ↦ (Ξ^{o} : α^{o} ⇝ β^{o}) yield a strict isomorphism of 2-categories :

PsFun(𝒜 , ℬ)^{o} →^{∼} PsFun(𝒜^{o}, ℬ^{o}).

Likewise, the rules F ↦ ^{o}F, β ↦ ^{o}β and Ξ ↦ ^{o}Ξ yield a strict isomorphism of 2-categories :

oPsFun(𝒜 , ℬ) →^{∼} PsFun(^{o}𝒜 ,^{o}ℬ).

Remark 2.2.14. (i) The isomorphisms of the category PsFun(𝒜 , ℬ) are precisely the pseudonatural isomorphisms of pseudo-functors. Indeed, if α : F ⇒ G is such a pseudo-natural isomorphism, and τ is its coherence constraint, we obtain a pseudo-natural isomorphism α^{−1} : G ⇒ F by the rule that assigns to every A ∈ Ob(𝒜) the 1-cell (α_{A})^{−1} : GA → F A, and to every 1-cell f : A → A′ the oriented square

[figure: GA; α_{A}^{−1}; Gf; τ′_{f}; F A; F f; GA′; α_{A′}^{−1}; F A′] where τ′_{f} := α_{A′}^{−1} ∗ τ_{f}^{−1} ∗ α_{A}^{−1}.

Indeed, let us check the coherence axioms for α^{−1}; first, we compute, for every A ∈ Ob(𝒜) :

\begin{matrix}τ′_{1_{A}} ⊙ (δ_{A}^{F} ∗ α_{A}^{−1}) = (α_{A}^{−1} ∗ τ_{1_{A}}^{−1} ∗ α_{A}^{−1}) ⊙ (δ_{A}^{F} ∗ α_{A}^{−1}) = & ((α_{A}^{−1} ∗ τ_{1_{A}}^{−1}) ⊙ δ_{A}^{F}) ∗ α_{A}^{−1} \\ = & α_{A}^{−1} ∗ (τ_{1_{A}}^{−1} ⊙ (α_{A} ∗ δ_{A}^{F})) ∗ α_{A}^{−1} \\ = & α_{A}^{−1} ∗ (δ_{A}^{F} ∗ α_{A}) ∗ α_{A}^{−1} \\ = & α_{A}^{−1} ∗ δ_{A}^{F} \end{matrix}

which proves the first axiom. Next, for every pair of 1-cells A ⟶^{f} A′ ⟶^{g} A′′ of 𝒜 we have :

\begin{matrix}(τ′_{g} □ τ′_{f}) ⊟ γ_{f,g}^{G} = & (α_{A′′}^{−1} ∗ γ_{f,g}^{G}) ⊙ (α_{A′′}^{−1} ∗ τ_{g}^{−1} ∗ α_{A′}^{−1} ∗ Gf) ⊙ (F g ∗ α_{A′}^{−1} ∗ τ_{f}^{−1} ∗ α_{A}^{−1}) \\ = & (α_{A′′}^{−1} ∗ γ_{f,g}^{G}) ⊙ (α_{A′′}^{−1} ∗ Gg ∗ τ_{f}^{−1} ∗ α_{A}^{−1}) ⊙ (α_{A′′}^{−1} ∗ τ_{g}^{−1} ∗ F f ∗ α_{A}^{−1}) \\ = & α_{A′′}^{−1} ∗ (γ_{f,g}^{G} ⊙ (Gg ∗ τ_{f}^{−1} ∗ α_{A}^{−1}) ⊙ (τ_{g}^{−1} ∗ F f ∗ α_{A}^{−1})) \\ = & α_{A′′}^{−1} ∗ ((γ_{f,g}^{G} ∗ α_{A}) ⊙ (Gg ∗ τ_{f}^{−1}) ⊙ (τ_{g}^{−1} ∗ F f)) ∗ α_{A}^{−1} \end{matrix}

and on the other hand

γ_{f,g}^{F} ⊟ τ′_{g∘f} = (α_{A′′}^{−1} ∗ τ_{g∘f}^{−1} ∗ α_{A}^{−1}) ⊙ (γ_{f,g}^{F} ∗ α_{A}^{−1}) = α_{A′′}^{−1} ∗ (τ_{g∘f}^{−1} ⊙ (α_{A′′} ∗ γ_{f,g}^{F})) ∗ α_{A}^{−1}

so we are reduced to showing that

(γ_{f,g}^{G} ∗ α_{A}) ⊙ (Gg ∗ τ_{f}^{−1}) ⊙ (τ_{g}^{−1} ∗ F f) = τ_{g∘f}^{−1} ⊙ (α_{A′′} ∗ γ_{f,g}^{F})

which is equivalent to the second coherence axiom for α. Conversely, it is clear that every isomorphism in PsFun(𝒜 , ℬ) must be a pseudo-natural isomorphism.

(ii) In the same vein, with the notation of definition 2.2.12(i), it is easily seen that the isomorphisms in the category PsNat(F, G) are the modifications Ξ : α ⇝ β such that Ξ_{A} : α_{A} ⇒ β_{A} is an isomorphism in ℬ(F A, GA) for every A ∈ Ob(𝒜) : the details are left to the reader.

Example 2.2.15. (i) With the notation of remark 2.2.5(iv), we may now say that the coherence constraint γ^{G′} furnishes an invertible modification

γ_{α,β}^{G′} : (G′ ∗ β) ⊙ (G′ ∗ α) ⇝ G′ ∗ (β ⊙ α).

(ii) Likewise, let us check that the 2-cell of remark 2.2.5(v) yields a modification

τ_{α}^{α′} : (G′ ∗ α) ⊙ (α′ ∗ F) ⇝ (α′ ∗ G) ⊙ (F′ ∗ α)

which is invertible if α′ is pseudo-natural. To this aim, for any 1-cell f : A → A′ in 𝒜 we consider the diagrams :

[figure: F′F A; α′_{F A}; F′F f; τ_{F f}^{α′}; G′F A; (τ_{f}^{α})^{G′}; G′α_{A}; G′F f; G′GA; G′Gf; F′F A′; α′_{F A′}; F′α_{A′}; τ_{α_{A′}}^{α′}; G′F A′; 1_{G′α_{A′}}; G′α_{A′}; G′(α_{A′}); G′GA′; F′GA′; α′_{GA′}; G′GA′; G′GA′; F′F A; 1_{F′α_{A}}; F′F A; α′_{F A}; F′α_{A}; τ_{α_{A}}^{α′}; G′F A; G′α_{A}; F′F A; F′α_{A}; F′F f; (τ_{f}^{α})^{F′}; F′GA; α′_{GA}; F′Gf; τ_{Gf}^{α′}; G′GA; G′Gf; F′F A′; F′α_{A′}; F′GA′; α′_{GA′}; G′GA′]

(notation of example 2.1.20(i)) and notice that

\begin{matrix}(τ_{α_{A′}}^{α′} ∗ F′F f) ⊙ τ_{f}^{(G′∗α)⊙(α′∗F)} = λ := & (1_{G′α_{A′}} ⊟ τ_{α_{A′}}^{α′}) □ ((τ_{f}^{α})^{G′} ⊟ τ_{F f}^{α′}) \\ = & (1_{G′α_{A′}} □ (τ_{f}^{α})^{G′}) ⊟ (τ_{α_{A′}}^{α′} □ τ_{F f}^{α′}) \\ τ_{f}^{(α′∗G)⊙(F′∗α)} ⊙ (G′Gf ∗ τ_{α_{A}}^{α′}) = μ := & (τ_{Gf}^{α′} ⊟ (τ_{f}^{α′})^{F′}) □ (τ_{α_{A}}^{α′} ⊟ 1_{F′α_{A}}) \\ = & (τ_{Gf}^{α′} □ τ_{α_{A}}^{α′}) ⊟ ((τ_{f}^{α′})^{F′} □ 1_{F′α_{A}}). \end{matrix}

So we have only to check that

λ ⊟ \frac{1_{F′α_{A′}∘F′F f}}{1_{F′F A}|F′α_{A′}} = \frac{1_{G′Gf∘G′α_{A}}}{G′α_{A}|1_{G′GA}} ⊟ μ.

However, the definition of 2-Morph(G′) and the coherence axiom for α′ yield respectively

\begin{matrix}1_{G′α_{A′}} □ (τ_{f}^{α})^{G′} = & \frac{γ_{α_{A},Gf}^{G′}}{G′(α_{A})|1_{G′GA′}} ⊟ \frac{G′(τ_{f}^{α})}{1_{G′F A}|1_{G′GA′}} ⊟ \frac{(γ_{F f,α_{A′}}^{G′})^{−1}}{1_{G′F A}|1_{G′GA′}} \\ τ_{α_{A′}}^{α′} □ τ_{F f}^{α′} = & \frac{γ_{F f,α_{A′}}^{G′}}{1_{G′F A}|1_{G′GA′}} ⊟ \frac{τ_{α_{A′}∘F f}^{α′}}{α′_{F A}|α′_{GA′}} ⊟ \frac{(γ_{F f,α_{A′}}^{F′})^{−1}}{1_{F′F A}|1_{F′GA′}} \end{matrix}

therefore :

\begin{matrix}λ = & γ_{α_{A},Gf}^{G′} ⊟ G′(τ_{f}^{α}) ⊟ τ_{α_{A′}∘F f}^{α′} ⊟ (γ_{F f,α_{A′}}^{F′})^{−1} \\ = & γ_{α_{A},Gf}^{G′} ⊟ \frac{τ_{Gf∘α_{A}}^{α′}}{α′_{F A}|α′_{GA′}} ⊟ \frac{F′(τ_{f}^{α})}{1_{F′F A}|1_{F′GA′}} ⊟ (γ_{F f,α_{A′}}^{F′})^{−1} \\ = & \frac{1_{G′Gf∘G′α_{A}}}{G′α_{A}|1_{G′GA′}} ⊟ (τ_{Gf}^{α′} □ τ_{α_{A}}^{α′}) ⊟ \frac{(γ_{α_{A},Gf}^{F′})}{1_{F′F A}|1_{F′GA′}} ⊟ \frac{(γ_{α_{A},Gf}^{F′})^{−1}}{1_{F′F A}|1_{F′GA′}} ⊟ \frac{(τ_{f}^{α})^{F′}}{1_{F′F A}|1_{F′GA′}} \\ = & \frac{1_{G′Gf∘G′α_{A}}}{G′α_{A}|1_{G′GA′}} ⊟ (τ_{Gf}^{α′} □ τ_{α_{A}}^{α′}) ⊟ \frac{(τ_{f}^{α})^{F′}}{1_{F′F A}|1_{F′GA′}} \end{matrix}

where the second equality follows from remark 2.2.5(i) and the coherence axiom for α′. The sought identity follows straightforwardly.

(iii) For every 2-category 𝒜, we denote by i_{𝒜} the identity (strict) pseudo-natural transformation of 1_{𝒜} . In other words, i_{𝒜} assigns to every A ∈ Ob(𝒜) the 1-cell 1_{A}. Let ℬ, 𝒞 be any other 2-categories, and F : 𝒜 → ℬ, G : 𝒞 → 𝒜 any two pseudo-functors; clearly

i_{𝒜} ∗ G = 1_{G}.

On the other hand, F ∗ i_{𝒜} is a pseudo-natural transformation that assigns to every A ∈ Ob(𝒜) the 1-cell F 1_{A} : F A → F A. If (δ^{F} , γ^{F}) is the coherence constraint for F, the coherence constraint of F ∗ i_{𝒜} assigns to every 1-cell f : A → A′ of 𝒜 the 2-cell 1_{f}^{F} := γ_{f,1_{A′}}^{F −1} ⊙ γ_{1_{A},f}^{F}. Then, the unit axiom for F implies that the system of 2-cells (δ_{A}^{F} | A ∈ Ob(𝒜)) defines an invertible modification

δ^{F} : 1_{F} ⇝ F ∗ i_{𝒜} .

Remark 2.2.16. As already pointed out (remark 2.2.5(vi)), the datum of the category 2-Cat and the system of categories (PsFun(𝒜 , ℬ) | 𝒜 , ℬ ∈ Ob(2-Cat)) does not amount to a 2category. However, example 2.2.15 suggests a straightforward variation that does result in an interesting 2-category. Namely, for every pair of categories 𝒜 and ℬ, let us denote by

\overline{PsFun}(𝒜 , ℬ)

the category whose objects are the pseudo-functors 𝒜 → ℬ and whose morphisms are the equivalence classes [β] of pseudo-natural transformations β between such pseudo-functors, where we declare that two pseudo-natural transfomations α, β : F ⇒ G are equivalent if and only if there exists an invertible modification α ⇝ β. The composition law for such equivalence classes is given by the rule

[β] ⊙ [β′] := [β ⊙ β′]

for every two pseudo-natural transformations β : F ⇒ G, β′ : G ⇒ H of pseudo-functors F, G, H : 𝒜 → ℬ. Then, if 𝒞 is another 2-category, F′, G′ : ℬ → 𝒞 and α : F′ ⇒ G′ another pseudo-natural transformation, we set

[α] ∗ [β] := [(G′ ∗ β) ⊙ (α ∗ F)].

By virtue of example 2.2.15, this operation yields then a well defined functor

\overline{PsFun}(𝒜 , ℬ) × \overline{PsFun}(ℬ, 𝒞) → \overline{PsFun}(𝒜 , 𝒞)

that satisfies the associativity axiom for the composition bifunctor in a 2-category (see (2.1)). Thus, for every universe U′, the U′-small 2-categories are the objects of a 2-category

U′-\overline{2-Cat}

that we call the reduced 2-category of U′-small 2-categories, whose 1-cells are the pseudofunctors and whose Hom-categories are the foregoing categories \overline{PsFun}(−, −). As usual, when U′ = U we write simply \overline{2-Cat} for this 2-category.

Remark 2.2.17. (i) Let 𝒜 and ℬ be two 2-categories. Every pair of pseudo-functors F : 𝒜′ → 𝒜 and G : ℬ → ℬ′ induces a pseudo-functor

PsFun(F, G) : PsFun(𝒜 , ℬ) → PsFun(𝒜′, ℬ′) (H : 𝒜 → ℬ) ↦ G ∘ H ∘ F

that assigns to every pseudo-natural transformation β : H ⇒ H′ between pseudo-functors H, H′ : 𝒜 → ℬ the pseudo-natural transformation G ∗ β ∗ F : G ∘ H ∘ F ⇒ G ∘ H′ ∘ F, and to every modification Ξ : β ⇝ β′ between such pseudo-natural transformations β, β′, the modification G ∘ Ξ ∘ F : G ∗ β ∗ F ⇝ G ∗ β′ ∗ F. The coherence constraint of PsFun(F, G) is the pair (δ_{•}^{(F,G)}, γ_{••}^{(F,G)}) defined as follows :

δ_{H}^{(F,G)} := δ^{G} ∘ H ∘ F : 1_{GHF} ⇝ G ∗ 1_{HF} for every pseudo-functor H : 𝒜 → ℬ

where δ^{G} : 1_{G} ⇝ G ∗ i_{𝒜} is the invertible modification provided by example 2.2.15(iii), and

γ_{β,β′}^{(F,G)} := γ_{β,β′}^{G} ∘ F : (G ∗ β′ ∗ F) ⊙ (G ∗ β ∗ F) ⇝ G ∗ (β′ ⊙ β) ∗ F

for every pair of pseudo-natural transformations β : H ⇒ H′ and β′ : H′ ⇒ H′, where γ^{G}β, β′ is the invertible modification as in example 2.2.15(i). The verification is straightforward, in view of remark 2.2.10(vii). In case F = 1_{𝒜} (resp. G = ℬ), we also write PsFun(𝒜 , G) (resp. PsFun(F, ℬ)) for this pseudo-functor. Notice that PsFun(F, ℬ) is strict for every pseudofunctor F, whereas PsFun(𝒜 , G) is strict only if G is strict.

(ii) Every pseudo-natural transformation α : F ⇒ F′ between pseudo-functors F, F′ : 𝒜′ → 𝒜 induces a pseudo-natural transformation

PsFun(α, ℬ) : PsFun(F, ℬ) ⇒ PsFun(F′, ℬ) (H : 𝒜 → ℬ) ↦ (H∗α : H∘F ⇒ H∘F′)

whose coherence constraint assigns to every pseudo-natural transformation β : H ⇒ H′ the oriented square :

[figure: H ∘ F; H∗α; β∗F; (τ_{α}^{β})^{−1}; H ∘ F′; β∗F′; H′ ∘ F; H′∗α; H′ ∘ F′]

where τ_{α}^{β} is the invertible modification provided by example 2.2.15(ii). Indeed, the naturality of the rule β ↦ (τ_{α}^{β})^{−1} with respect to modifications Ξ : β ⇝ β′ follows directly from the compatibility condition for Ξ. To check the coherence axioms, notice first that τ_{α}^{1_{H}} = 1_{H∗α} for every pseudo-functor H : 𝒜 ⇒ ℬ. Lastly, if β : H ⇒ H′ and β′ : H′ ⇒ H′′ are two pseudo-natural transformations, we need to show the identity

(τ_{α}^{β′})^{−1} □ (τ_{α}^{β})^{−1} = (τ_{α}^{β′⊙β})^{−1}

or equivalently : τ_{α}^{β′} ⊟ τ_{α}^{β} = τ_{α}^{β′⊙β}, which follows by direct inspection.

(iii) Likewise, every pseudo-natural transformation λ : G ⇒ G′ between pseudo-functors G, G′ : ℬ → ℬ′ induces a pseudo-natural transformation

PsFun(𝒜 , λ) : PsFun(𝒜 , G) ⇒ PsFun(𝒜 , G′) (H : 𝒜 → ℬ) ↦ (λ∗H : G∘F ⇒ G′∘F)

whose coherence constraint assigns to every pseudo-natural transformation β : H ⇒ H′ the oriented square τ_{β}^{λ} provided by example 2.2.15(ii). Indeed, the naturality of the rule : β ↦ τ_{β}^{λ} with respect to modifications Ξ : β ⇝ β′ follows from the naturality of τ^{λ}, applied to the 2-cells Ξ_{A} : β_{A} → β′_{A}, for every A ∈ Ob(𝒜). The coherence condition for τ_{1_{H}}^{λ} follows from the coherence condition for τ_{HA}^{λ}, for every pseudo-functor H : 𝒜 → ℬ and every A ∈ Ob(𝒜). Likewise, the coherence condition relative to a composable pair of pseudo-natural transformations β : H ⇒ H′, β′ : H′ ⇒ H′′ follows from the coherence condition for τ^{λ}, relative to the composable pair of 1-cells β_{A}, β′_{A}, for every A ∈ Ob(𝒜).

(iv) Furthermore, for every F, F′ as in (ii), every modification Ξ : α ⇝ β between pseudonatural transformations α, β : F ⇒ F′ induces a modification

PsFun(Ξ, ℬ) : PsFun(α, ℬ) ⇝ PsFun(β, ℬ) (G : 𝒜 → ℬ) ↦ (G ∘ Ξ : G ∗ α ⇝ G ∗ β).

Indeed, the compatibility condition for this modification amounts to the identity :

(G′(Ξ_{X}) ∗ γ_{F X}) ⊙ (τ_{α_{X}}^{γ})^{−1} = (τ_{β_{X}}^{γ})^{−1} ⊙ (γ_{F′X} ∗ G(Ξ_{X})) for every X ∈ Ob(𝒜)

which follows easily from the naturality of τ^{γ}.

(v) Suppose moreover that ℬ is small; then we deduce a strict pseudo-functor

PsFun(−, ℬ) : \overline{2-Cat}^{o} → \overline{2-Cat} 𝒜 ↦ PsFun(𝒜 , ℬ)

acting on the 1-cells and 2-cells of \overline{2-Cat}^{o} as explicited in (i) and (ii). Indeed, if F : 𝒜′ → 𝒜 and F′ : 𝒜′′ → 𝒜′ are two pseudo-functors between small 2-categories, a simple inspection shows that PsFun(F′, ℬ) ∘ PsFun(F, ℬ) = PsFun(F ∘ F′, ℬ). Moreover, (iv) implies that if α, β : F ⇒ F′ are pseudo-natural transformations with [α] = [β] in \overline{2-Cat}, then [PsFun(α, ℬ)] = [PsFun(β, ℬ)]. Next, for every three pseudo-functors F, F′, F′′ : 𝒜′ → 𝒜 and every pseudo-natural transformations α : F ⇒ F′ and α′ : F′ ⇒ F′′, with the notation of remark 2.2.16, we need to check that :

(2.2.18)  [PsFun(α′, ℬ) ⊙ PsFun(α, ℬ)] = [PsFun(α′ ⊙ α, ℬ)].

Now, to every pseudo-functor G : 𝒜 → ℬ, the left-hand side of (2.2.18) assigns the class of (G∗α′)⊙(G∗α), whereas the right-hand side assigns the class of G∗(α′⊙α). These two pseudonatural transformations are related by the invertible modification γ_{α,α′}^{G} of example 2.2.15(i), so we come down to checking that the rule : G ↦ γ_{α,α′}^{G} defines an invertible modification PsFun(α′, ℬ) ⊙ PsFun(α, ℬ) ⇝ PsFun(α′ ⊙ α, ℬ). The latter assertion is an immediate consequence of the coherence axioms for τ^{β} : details left to the reader. Lastly, let us check that for every two pairs of pseudo-functors F_{1}, F′_{1} : 𝒜′ → 𝒜, F_{2}, F′_{2} : 𝒜′′ → 𝒜′ and every pseudo-natural transformations α_{1} : F_{1} ⇒ F′_{1}, α_{2} : F_{2} ⇒ F′_{2} we have :

(2.2.19)  [PsFun(α_{2}, ℬ)] ∗ [PsFun(α_{1}, ℬ)] = [PsFun([α_{1}] ∗ [α_{2}], ℬ)].

We may consider separately the case where F_{2} = F′_{2} and α_{2} = 1_{F_{2}}, and the case where F_{1} = F′_{1} and α_{1} = 1_{F_{1}}. In the first case, the left-hand side assigns to every G : 𝒜 → ℬ the pseudonatural transformation (G ∗ α_{1}) ∗ F_{2}, whereas the right-hand side assigns G ∗ (α_{1}) ∗ F_{2}; but these coincide, and likewise, the coherence constraints of both sides is given by the rule : β ↦ (τ_{α_{1}∗F_{2}}^{β})^{−1}. One argues similarly in the second case : details left to the reader.

2.3. The formalism of base change. Let 𝒜 be a 2-category, and A, B two objects of 𝒜. A link from A to B in 𝒜 is a datum

ℒ := (F, G, η, ε) : A → B

consisting of :

• 1-cells F : A → B and G : B → A

• 2-cells η : 1_{B} ⇒ F ∘ G and ε : G ∘ F ⇒ 1_{A}

such that η and ε are related by the triangular identities :

(F ∗ ε) ⊙ (η ∗ F) = 1_{F} and (ε ∗ G) ⊙ (G ∗ η) = 1_{G}.

Especially, (G, F) is an adjoint pair of 1-cells of 𝒜. We say that η and ε are the unit and counit of the link ℒ. We define a composition law for links as follows. With ℒ as in the foregoing, a third object C of 𝒜 and a second link ℒ′ := (F′, G′, η′, ε′) : B → C we let

ℒ′ ∘ ℒ := (F′ ∘ F, G ∘ G′, η^{ℒ′∘ℒ} , ε^{ℒ′∘ℒ} ) : A → C

where η^{ℒ′∘ℒ} := (F′ ∗ η ∗ G′) ⊙ η′ and ε^{ℒ′∘ℒ} := ε ⊙ (G ∗ ε′ ∗ F). We need to check that α := (F′F ∗ (ε ⊙ G ∗ ε′ ∗ F)) ⊙ ((F′ ∗ η ∗ G′ ⊙ η′) ∗ F′F) = 1_{F′F}. We compute :

α = (F′ ∗ ((F ∗ ε) ⊙ (F G ∗ ε′ ∗ F))) ⊙ (((F′ ∗ η ∗ G′F′) ⊙ (η′ ∗ F′)) ∗ F)
= (F′ ∗ ((F ∗ ε) ⊙ (η ∗ F))) ⊙ (((F′ ∗ ε′) ⊙ (η′ ∗ F′)) ∗ F)
= (F′ ∗ 1_{F}) ⊙ (1_{F′} ∗ F)
= 1_{F′F}

where the second equality holds by remark 2.1.1(i) and the third follows from the triangular identities for (η, ε) and (η′, ε′). Likewise one checks the second triangular identity for (η′′, ε′′) (the details are left to the reader).

Remark 2.3.1. (i) Notice that the composition law for links is associative; namely, in the situation of (2.3), if ℒ′′ := (F′′, G′′, η′′, ε′′) : C → D is another link, we have (ℒ′′∘ℒ′)∘ℒ = ℒ′′ ∘ (ℒ′ ∘ ℒ). Indeed, the assertion boils down to the identities :

\begin{aligned}(F′′ ∗ ((F′ ∗ η ∗ G′) ⊙ η′) ∗ G′′) ⊙ η′′ & = (F′′F′ ∗ η ∗ G′G′′) ⊙ (F′′ ∗ η′ ∗ G′′) ⊙ η′′ \\ ε ⊙ (G ∗ ε′ ∗ F) ⊙ (GG′ ∗ ε′′ ∗ F′F) & = ε ⊙ (G ∗ (ε′ ⊙ (G′ ∗ ε′′ ∗ F′)) ∗ F) \end{aligned}

which are both clear.

(ii) Notice also that every link ℒ := (F, G, η, ε) : A → B induces two opposite links

\begin{aligned}ℒ^{o} & := (G^{o}, F^{o}, η^{o}, ε^{o}) : A^{o} → B^{o} & in 𝒜^{o} \\ ^{o}ℒ & := (^{o}G,^{o}F,^{o}ε,^{o}η) : ^{o}B → ^{o}A & in^{o}𝒜 . \end{aligned}

2.3.2. Let ℒ , ℒ′ : A → B be two links, with ℒ = (F, G, η, ε) and ℒ′ = (F′, G′, η′, ε′); a transformation from ℒ to ℒ′, denoted

β : ℒ ⇒ ℒ′

is defined as a 2-cell β : F ⇒ F′. The standard operations on 2-cells can be upgraded easily to transformations of links :

• If ℒ′′ : A → B is another link and β′ : ℒ′ ⇒ ℒ′′ a second transformation, we get the transformation

β′ ⊙ β : ℒ ⇒ ℒ′′.

• For a diagram of links and transformations of links :

(2.3.3)  [figure: A; ℒ; ℒ′; β; B; 𝒫; 𝒫′; α; C]

we get the transformation

α ∗ β : 𝒫 ∘ ℒ ⇒ 𝒫′ ∘ ℒ′.

For α = 1_{𝒫} (resp. for β = 1_{ℒ} ) we also denote this transformation by 𝒫 ∗ β (resp. by α ∗ ℒ). Thus, the links of 𝒜 are the 1-cells of a 2-category

Link(𝒜)

whose objects are the objects of 𝒜 and whose 2-cells are the transformations of links, with the compositions laws for 1-cells and 2-cells given in the foregoing.

• For a transformation β : ℒ ⇒ ℒ′, we define an adjoint 2-cell of 𝒜 :

β^{†} := (ε′ ∗ G) ⊙ (G′ ∗ β ∗ G) ⊙ (G′ ∗ η) : G′ ⇒ G.

The operations of remark 2.3.1(ii) can be combined with this adjoint construction to produce natural isomorphisms of 2-categories; indeed, we have :

Proposition 2.3.4. For every 2-category 𝒜 we have strict isomorphisms of 2-categories :

oLink(𝒜) →^{∼} Link(𝒜^{o}) Link(𝒜)^{o} →^{∼} Link(^{o}𝒜).

Proof. Indeed, the first strict pseudo-functor is given by the rules :

A ↦ A^{o}  (ℒ : A → B) ↦ (ℒ^{o} : A^{o} → B^{o})  (β : ℒ ⇒ ℒ′) ↦ (β^{†o} : ℒ^{′o} ⇒ ℒ^{o})

and the second one is given by the rules :

A ↦ ^{o}A  (ℒ : A → B) ↦ (^{o}ℒ : ^{o}B → ^{o}A)  (β : ℒ ⇒ ℒ′) ↦ (^{o}β^{†} : ^{o}ℒ ⇒ ^{o}ℒ′)

for every object A, every 1-cell ℒ and every 2-cell β of Link(𝒜). The verification that these rules yield strict pseudo-functors comes down to the following :

Claim 2.3.5. (i) Let ℒ , ℒ′, ℒ′′ : A → B be three links of 𝒜, and β : ℒ ⇒ ℒ′, β′ : ℒ′ ⇒ ℒ′′ two transformations. Then we have β^{†} ⊙ β^{′†} = (β′ ⊙ β)^{†}.

(ii) In the situation of diagram (2.3.3) we have (α ∗ β)^{†} = β^{†} ∗ α^{†}.

Proof of the claim. Say that ℒ = (F, G, η, ε), ℒ′ = (F′, G′, η′, ε′), ℒ′′ = (F′′, G′′, η′′, ε′′). Recall that every 1-cell f : X → Y of 𝒜 induces functors

f_{∗}^{Z} : 𝒜 (Z, X) → 𝒜 (Z, Y)  (h : Z → X) ↦ f ∘ h  (ν : h ⇒ h′) ↦ f ∗ ν

for every Z ∈ Ob(𝒜), and every 2-cell λ : f ⇒ g of 𝒜 induces natural transformations

λ_{∗}^{Z} : f_{∗}^{Z} ⇒ g_{∗}^{Z} (h : Z → X) ↦ (λ ∗ h : f_{∗}^{Z}(h) ⇒ g_{∗}^{Z}(h))

(see the proof of lemma 2.1.13(i)). Then it is clear that for every Z ∈ Ob(𝒜), every link 𝒫 := (H, K, η^{𝒫} , ε^{𝒫} ) induces an adjoint pair of functors (H_{∗}^{Z}, K_{∗}^{Z}) with unit (η^{𝒫} )^{Z} and counit (ε^{𝒫} )^{Z}. With this notation, it is easily seen that

(2.3.6)  (β^{†})_{∗}^{Z} = (β_{∗}^{Z}, η_{∗}^{Z}, η_{∗}^{′Z})^{†}

where the right-hand side is the adjoint of the natural transformation β_{∗}^{Z}, defined as in remark 1.1.17(ii) : the details are left to the reader. Then both (i) and (ii) follow straightforwardly from remark 1.1.17(iv,v). ♢

Lastly, in order to see that these pseudo-functors are isomorphisms of 2-categories, it suffices to show more precisely that for every transformation of links β : ℒ ⇒ ℒ′ we have :

(β^{†})^{†} = β.

Again, by virtue of (2.3.6), the assertion is reduced to the corresponding identity for adjoints of natural transformations, and the latter is known by virtue of remark 1.1.17(ii). □

2.3.7. Notice that every 1-cell F : A → B of the 2-category 𝒜 induces a strict pseudo-functor

F_{∗} : 𝒜 /A → 𝒜 /B (g : X → A) ↦ (F ∘ g : X → B)

that maps every 1-cell (h, α) : (X ⟶^{g} A) → (Y ⟶^{g′} A) of 𝒜 /A to the 1-cell of 𝒜 /B

F ∗ (h, α) := (h, F ∗ α)

and every 2-cell β : (h_{1}, α_{1}) ⇒ (h_{2}, α_{2}) of 𝒜 /A to the 2-cell β : (h_{1}, F ∗ α_{1}) ⇒ (h_{2}, F ∗ α_{2}) of 𝒜 /B (notation of example 2.1.11(ii)). With this notation, we claim that every link ℒ := (F, G, η, ε) : A → B of 𝒜 induces an adjunction for the pair of functors F_{∗} : 𝒜 /A → 𝒜 /B and G_{∗} : 𝒜 /B → 𝒜 /A. Namely, we have a natural bijection :

ϑ_{g,f}^{ℒ} : 𝒜 /B(Y ⟶^{g} B, X ⟶^{F ∘f} B) →^{∼} 𝒜 /A(Y ⟶^{G∘g} A, X ⟶^{f} A) (h, α) ↦ (h, (ε∗f)⊙(G∗α))

whose inverse μ_{g,f}^{ℒ} : 𝒜 /A(G ∘ g, f) →^{∼} 𝒜 /B(g, F ∘ f) is given by the rule :

(k, β) ↦ (k, (F ∗ β) ⊙ (η ∗ g)).

The verification of the naturality of ϑ_{f,g}^{ℒ} with respect to f and g shall be left to the reader. Let us check that μ_{g,f}^{ℒ} ∘ ϑ_{g,f}^{ℒ} is the identity map on the set 𝒜 /B(g, F ∘ f) : indeed, for every 1-cell (h, α), this composition is computed as the composition of the following oriented squares :

[figure: Y; h; g; α; X; 1; f; X; f; B; A; F; ε; A; B; 1; B; G; η; A; F; B; B; B.]

Then the assertion follows easily, by applying the triangular identities for the pair (η, ε) and invoking as usual proposition 2.1.9. Likewise we check that ϑ_{g,f}^{ℒ} ∘ μ_{g,f}^{ℒ} is the identity on 𝒜 /A(G ∘ g, f) : the details shall be left to the reader.

2.3.8. Consider now an oriented square 𝒟 of links, together with its adjoint squares :

𝒟 : [figure: A′; ℒ′; ℳ′; β; B′; ℳ; A; ℒ; B]  𝒟^{†} : [figure: A^{′o}; ℒ^{′o}; ℳ^{′o}; β^{†o}; B^{′o}; ℳ^{o}; A^{o}; ℒ^{o}; B^{o}]  ^{†}𝒟 : [figure: ^{o}B; ^{o}ℒ; ^{o}ℳ; ^{o}β^{†}; ^{o}A; ^{o}ℳ′; ^{o}B′; ^{o}ℒ′; ^{o}A′]

i.e. the orientation β : ℳ ∘ ℒ′ ⇒ ℒ ∘ ℳ′ is a transformation of links, so that^{o}β^{†} is the adjoint transformation ^{o}ℒ′ ∘ ^{o}ℳ ⇒ ^{o}ℳ′ ∘ ^{o}ℒ, and likewise for β^{†o}. Say that ℒ = (L_{∗}, L^{∗}, η^{L}, ε^{ℒ} ), ℒ′ = (L′_{∗}, L^{′∗}, η^{ℒ′} , ε^{ℒ′} ), ℳ = (M_{∗}, M^{∗}, η^{M}, ε^{ℳ} ) and ℳ′ = (M′_{∗}, M^{′∗}, η^{ℳ′} , ε^{ℳ′} ). Notice that for horizontally composable oriented squares of links we have the identities :

(2.3.9)  (𝒟′ ⊟ 𝒟)^{†} = 𝒟^{′†} □ 𝒟^{†}  ^{†}(𝒟′ ⊟ 𝒟) = ^{†}𝒟 □^{†}𝒟′.

Thus, the operation 𝒟 ↦ 𝒟^{†} for oriented squares of links differs in this respect from the analogous operation for oriented squares in the 2-category 𝒜 : see (2.1.8).

With this notation, we attach to the oriented square 𝒟 the base change oriented square in 𝒜:

Υ(𝒟) : [figure: B′; M_{∗}; L^{′∗}; Υ(β); B; L^{∗}; A′; M′_{∗}; A]

whose orientation is the base change transformation

Υ(β) := (ε^{ℒ} ∗ M′_{∗}L^{′∗}) ⊙ (L^{∗} ∗ β ∗ L^{′∗}) ⊙ (L^{∗}M_{∗} ∗ η^{ℒ′} ) : L^{∗}M_{∗} ⇒ M′_{∗}L^{′∗}.

Proposition 2.3.10. With the notation of (2.3.8), we have :

Υ(𝒟)^{o} = Υ(𝒟^{†}) ^{o}Υ(𝒟) = Υ(^{†}𝒟).

Proof. The sought identity for Υ(^{†}𝒟) will be proven by inspecting the following diagram :

[figure: B′; M_{∗}; ε^{ℳ}; B; 1_{M^{∗}}; M^{∗}; B; 1_{M^{∗}}; M^{∗}; B; η^{ℳ}; M^{∗}; B; i_{B}; B; B′; L^{′∗}; 1_{L^{′∗}}; B′; L^{′∗}; 1_{L^{′∗}}; B′; L^{′∗}; η^{ℒ′}; B′; M_{∗}; 1_{M∗}; B; i_{B}; B; A′; i_{A′}; A′; 1_{M′_{∗}}; A′; L′_{∗}; M′_{∗}; β; B′; M_{∗}; M_{∗}; 1_{M∗}; B; i_{B}; B; A′; i_{A′}; A′; M′_{∗}; 1_{M∗}; A; L_{∗}; ε^{ℒ}; B; L^{∗}; 1_{L^{∗}}; B; L^{∗}; 1_{L^{∗}}; B; L^{∗}; A′; i_{A′}; A′; M′_{∗}; ε^{ℳ′}; A; M^{′∗}; 1_{M^{′∗}}; A; M^{′∗}; 1_{M^{′∗}}; A; M^{′∗}; η^{ℳ′}; A; A′; A′; A′; A′; A′; M′_{∗}; A.]

Indeed, since the unit η^{^{o}ℳ} of^{o}ℳ is^{o}(ε^{ℳ} ) and the counit ε^{^{o}ℳ′} of^{o}ℳ′ is^{o}(η^{ℳ′} ), we have :

\begin{aligned}Υ(^{†}𝒟) & = (ε^{^{o}ℳ′} ∗^{o}L^{∗} ^{o}M_{∗}) ⊙ (^{o}M′_{∗} ∗^{o}β^{†} ∗^{o}M_{∗}) ⊙ (^{o}M′_{∗} ^{o}L^{′∗} ∗ η^{^{o}ℳ} ) \\ & = ^{o}((M′_{∗}L^{′∗} ∗ ε^{ℳ} ) ⊙ (M′_{∗} ∗ β^{†} ∗ M_{∗}) ⊙ (η^{ℳ′} ∗ L^{∗}M_{∗})) \end{aligned}

and if η^{ℳ ∘ℒ′} is the unit of ℳ ∘ ℒ′ and ε^{ℒ ∘ℳ′} is the counit of ℒ ∘ ℳ′, we have

\begin{matrix}η^{ℳ ∘ℒ′} = (M_{∗} ∗ η^{ℒ′} ∗ M^{∗}) ⊙ η^{ℳ} ε^{ℒ ∘ℳ′} = ε^{ℳ′} ⊙ (M^{′∗} ∗ ε^{ℒ} ∗ M′_{∗}) \\ β^{†} = (ε^{ℒ ∘ℳ′} ∗ L^{′∗}M^{∗}) ⊙ (M^{′∗}L^{∗} ∗ β ∗ L^{′∗}M^{∗}) ⊙ (M^{′∗}L^{∗} ∗ η^{ℳ ∘ℒ′} ). \end{matrix}

Thus, if in the foregoing diagram we initially disregard the first and last columns, we find that the composition of the six remaining squares in the uppermost two rows equals precisely η^{ℳ ∘ℒ′} , and the composition of the six squares in the bottom two rows equals ε^{ℒ ∘ℳ′} . Therefore, if we further compose these two blocks of six squares with the central row of three squares, we get precisely β^{†}. And if we finally compose the result with the two omitted left and right columns, we then get ^{o}Υ(^{†}𝒟). But according to proposition 2.1.9 we may compose these squares in a different order : notice then that the composition of the five squares in the uppermost row equals 1_{M_{∗}}, and the composition of the five square in the bottom row equals 1_{M′_{∗}}. So, we may disregard these two rows; but in the remaining three central rows, we may likewise disregard everything except the central column, since the squares outside the central column are all identities. So, finally we are left with the vertical composition of the squares in the central column, and that gives precisely Υ(𝒟), as required. Lastly, we have :

\begin{aligned}Υ(𝒟^{†}) & = (ε^{ℳ^{o}} ∗ L^{′∗o} M_{∗}^{′o}) ⊙ (M_{∗}^{o} ∗ β^{†o} ∗ M_{∗}^{′o}) ⊙ (M_{∗}^{o} L^{∗o} ∗ η^{ℳ^{′o}} ) \\ & = ((M′_{∗}L^{′∗} ∗ ε^{ℳ} ) ⊙ (M′_{∗} ∗ β^{†} ∗ M_{∗}) ⊙ (η^{ℳ′} ∗ L^{∗}M_{∗}))^{o} \end{aligned}

so the sought identity for Υ(𝒟^{†}) follows from that for Υ(^{†}𝒟). □

Remark 2.3.11. In the situation of (2.3.8), suppose that β is an invertible 2-cell of 𝒜 and that both L_{∗} and L′_{∗} (resp. both M_{∗} and M′_{∗}) are equivalences in 𝒜. Then it follows easily from the definition (resp. and from proposition 2.3.10) that also Υ(β) is an invertible 2-cell.

Proposition 2.3.12. Consider two oriented squares of links :

𝒟 : [figure: A′; ℒ′; ℳ′; β; B′; ℳ; A; ℒ; B] 𝒟′ : [figure: C′; 𝒫′; 𝒬′; α; D′; 𝒬; C; 𝒫; D.]

(i) Suppose that A = C′, B = D′ and ℒ = 𝒫′, so that 𝒟′ □ 𝒟 is well defined. Then :

Υ(𝒟′ □ 𝒟) = Υ(𝒟′) ⊟ Υ(𝒟).

(ii) Suppose that B = C, B′ = C′ and ℳ = 𝒬′, so that 𝒟′ ⊟ 𝒟 is well defined. Then :

Υ(𝒟′ ⊟ 𝒟) = Υ(𝒟) □ Υ(𝒟′).

Proof. (i): Say that ℒ = (L_{∗}, L^{∗}, η^{ℒ} , ε^{ℒ} ), 𝒫 = (P_{∗}, P^{∗}, η^{𝒫} , ε^{𝒫} ), 𝒬 = (Q_{∗}, Q^{∗}, η^{𝒬} , ε^{𝒬} ), ℳ′ = (M′_{∗}, M^{′∗}, η^{ℳ′} , ε^{ℳ′} ), ℳ = (M_{∗}, M^{∗}, η^{ℳ} , ε^{ℳ} ) and 𝒬′ = (Q′_{∗}, Q^{′∗}, η^{𝒬′} , ε^{𝒬′} ). We consider the following diagram :

[figure: B′; L^{′∗}; η^{ℒ′}; A′; β; L′_{∗}; M′_{∗}; A; ε^{ℒ}; L_{∗}; A; 1_{Q^{′∗}}; Q^{′∗}; C; i_{C}; C; B′; i_{B′}; B′; M_{∗}; 1_{M∗}; B; L^{∗}; η^{ℒ}; A; Q^{′∗}; L_{∗}; α; C; P_{∗}; ε^{𝒫}; C; B′; B′; M_{∗}; B; B; Q_{∗}; D; P^{∗}; C]

and notice that the composition of the five squares of the top (resp. bottom) row equals Υ(𝒟) (resp. Υ(𝒟′)), so the composition of all the square in the diagram is precisely Υ(𝒟′) ⊟ Υ(𝒟). On the other hand, the composition of the two squares in the first (resp. second, resp. third, resp. fourth, resp. fifth) column from the left equals η^{ℒ′} (resp. β, resp. 1_{L_{∗}}, resp. α, resp. ε^{𝒫} ), and then the composition of these five columns also equals Υ(𝒟′ □ 𝒟), as stated.

(ii): We compute :

Υ(𝒟′ ⊟ 𝒟) = Υ(𝒟^{′††} ⊟ 𝒟^{††}) = Υ((𝒟^{′†} □ 𝒟^{†})^{†})  = Υ(𝒟^{′†} □ 𝒟^{†})^{o}
= (Υ(𝒟^{′†}) ⊟ Υ(𝒟^{†}))^{o}
= Υ(𝒟^{†})^{o} ⊟ Υ(𝒟^{′†})^{o}
= Υ(𝒟^{††}) ⊟ Υ(𝒟^{′††})
= Υ(𝒟) ⊟ Υ(𝒟′)

where the second equality follows from (2.3.9), the third and sixth follow from proposition 2.3.10, and the fourth follows from (i), and the fifth follows from (2.1.8). □

2.3.13. In the situation of (2.3.8), the links ℒ and ℒ′ induce adjunctions ϑ^{ℒ} and ϑ^{ℒ′} for the pairs of functors ((L^{∗})_{∗}, (L_{∗})_{∗}) and respectively ((L^{′∗})_{∗}, (L′_{∗})_{∗}), as in (2.3.7). The base change transformation Υ(β) relates these adjunctions, as explained by the following :

Proposition 2.3.14. With the notation of (2.3.13), for every (X ⟶^{f} A′) ∈ Ob(𝒜 /A′), every (Y ⟶^{g} B′) ∈ Ob(𝒜 /B′), and every 1-cell (h, α) : g → L′_{∗} ∘ f of 𝒜 /B′ we have the identity :

(M′_{∗} ∗ ϑ_{f,g}^{ℒ′}(h, α)) ∘ (1_{Y} , Υ(β) ∗ g) = ϑ_{M′_{∗}∘f,M_{∗}∘g}^{ℒ}((1_{X}, β ∗ f) ∘ (h, M_{∗} ∗ α)) in 𝒜 /A .

Proof. The left-hand side of the stated identity is computed as the composition of the following oriented squares :

[figure: Y; h; g; α; X; f; 1; X; f; 1; X; B′; A′; L′_{∗}; ε^{ℒ′}; A′; B′; 1; B′; L^{′∗}; η^{ℒ′}; A′; M′_{∗}; L′_{∗}; β; A; L_{∗}; ε^{ℒ}; A; B′; B′; B′; M_{∗}; B; L^{∗}; A.]

By applying as usual proposition 2.1.9, and the triangular identities for the pair (η^{ℒ′} , ε^{ℒ′} ), we easily see that the same composition of squares yields as well the right-hand side of the sought identity : the details are left to the reader. □

2.3.15. Recall that the links of 𝒜 are the objects of the 2-category 2-Morph(Link(𝒜)) : see example 2.1.11(i). An oriented square 𝒟 as in (2.3.8) yields a 1-cell in this 2-category

𝒟(β, ℳ , ℳ′) : ℒ′ → ℒ .

Given a pair of oriented squares 𝒟_{1}(β_{1}, ℳ_{1}, ℳ′_{1}), 𝒟_{2}(β_{2}, ℳ_{2}, ℳ′_{2}) : ℒ′ → ℒ, a 2-cell (α, α′) : 𝒟_{1} ⇒ 𝒟_{2} is the datum of a pair of 2-cells of the 2-category Link(𝒜)

(2.3.16) α : ℳ_{1} ⇒ ℳ_{2} α′ : ℳ′_{1} ⇒ ℳ′_{2} such that β_{2} ⊙ (α ∗ ℒ′) = (ℒ ∗ α′) ⊙ β_{1}.

Theorem 2.3.17. The rules :

ℒ := (L_{∗}, L^{∗}, η, ε) ↦ L^{∗o}  𝒟 ↦ Υ(𝒟)^{o}  (α, α′) ↦ (α^{o}, α^{′o})

for every link ℒ, every oriented square 𝒟 of Link(𝒜), and every 2-cell (α, α′) as in (2.3.15) define a strict pseudo-functor :

Υ_{𝒜} : 2-Morph(Link(𝒜))^{o} → 2-Morph(𝒜^{o}).

Proof. The functoriality of Υ for composition of 1-cells is already clear from proposition 2.3.12. Let us next check that for every 2-cell (α, α′) : 𝒟_{1} ⇒ 𝒟_{2} as in (2.3.15), the pair (α^{o}, α^{′o}) is a 2-cell Υ(𝒟_{1})^{o} ⇒ Υ(𝒟_{2})^{o} in 2-Morph(𝒜^{o}). We come down to showing the identity

(2.3.18)  Υ(β_{2})^{o} ⊙ (α^{o} ∗ L^{′∗o}) = (L^{′∗o} ∗ α′) ⊙ Υ(β_{1})^{o}.

To this aim, for every X ∈ Ob(𝒜) let 𝟙_{X} : X → X be the link (1_{X}, 1_{X}, i_{X}, i_{X}) (where i_{X} is the identity automorphism of 1_{X}); also, for i = 1, 2 let 𝒫_{i} := (P_{i∗} : X → Y, P_{i}^{∗}, η_{i}, ε_{i}) be a link of 𝒜, and τ : P_{1∗} ⇒ P_{2∗} a 2-cell of 𝒜. We consider the following oriented squares of Link(𝒜) and of 𝒜^{o} :

𝒟_{τ} : [figure: X; 𝟙_{X}; 𝒫_{2}; τ; X; 𝒫_{1}; Y; 𝟙_{Y}; Y] ℰ_{τ} : [figure: Y^{o}; 1_{Y^{o}}; P_{2∗}^{o}; τ^{o}; Y^{o}; P_{1∗}^{o}; X^{o}; 1_{X^{o}}; X^{o}.]

With this notation, by applying proposition 2.3.10it is easily seen that

Υ(𝒟_{τ})^{o} = ℰ_{τ}

and (2.3.18) is equivalent to the identity :

Υ(𝒟_{1})^{o} ⊟ ℰ_{α′} = ℰ_{α} ⊟ Υ(𝒟_{2})^{o}.

In view of (2.3.9) and proposition 2.3.12we are then reduced to checking that

𝒟_{1} ⊟ 𝒟_{α′} = 𝒟_{α} ⊟ 𝒟_{2}

which follows immediately from (2.3.16). Lastly, the functoriality of Υ for the two types of compositions of 2-cells is obvious from the definition. □

2.3.19. Resume the situation of (2.3.8), and notice that the units and counits for ℳ and ℳ′ do not play any role in the definition of Υ(β), and neither do the 1-cells M^{∗} and M^{′∗}. This leads to the following variant of theorem 2.3.17. We replace 2-Morph(Link(𝒜)) by the 2-category

wLink(𝒜)

of weak links in 𝒜, whose objects are still the links of 𝒜; for any two links ℒ := (L_{∗} : A → B, L^{∗}, η^{ℒ} , ε^{ℒ} ) and ℒ′ := (L′_{∗} : A′ → B′, L^{′∗}, η^{ℒ′} , ε^{ℒ′} ), the 1-cells ℒ → ℒ′ are the 1-cells (M_{∗}, M′_{∗}, β) : L_{∗} → L′_{∗} of the 2-category 2-Morph(𝒜), i.e. the oriented squares

𝒟 : [figure: A′; L′_{∗}; M′_{∗}; β; B′; M_{∗}; A; L_{∗}; B.]

The 2-cells of wLink(𝒜), and the composition laws for 1-cells and 2-cells are then the same as for 2-Morph(𝒜). For any such 1-cell 𝒟, we may then define the 2-cell Υ(β) and the oriented square Υ(𝒟) as in (2.3.8). A direct inspection shows that part (i) of proposition 2.3.12 still holds for weak links, with the same proof. The proof of part (ii) of proposition 2.3.12 does not apply to weak links, but we may check the assertion directly. Indeed, consider another 1-cell 𝒫′ := (P′_{∗}, P^{′∗}, η^{𝒫′} , ε^{𝒫′} ) → 𝒫 := (P_{∗}, P^{∗}, η^{𝒫} , ε^{𝒫} ) of wLink(𝒜) :

𝒟′ : [figure: B′; M_{∗}; Q_{∗}; α; C′; Q_{∗}; B; P_{∗}; C.]

Set :

\begin{matrix}X := & ε^{ℒ} ∗ M′_{∗}L^{′∗}P^{′∗} & & & Y := & L^{∗}P^{∗}Q_{∗} ∗ η^{𝒫′} \\ δ := & (P_{∗} ∗ β) ⊙ (α ∗ L′_{∗}) & α′ & := P^{∗} ∗ α ∗ P^{′∗} & β′ := & L^{∗} ∗ β ∗ L^{′∗}. \end{matrix}

By definition we have :

\begin{matrix}Υ(𝒟′ ⊟ 𝒟) = X ⊙(L^{∗}∗ε^{𝒫} ∗L_{∗}M′_{∗}L^{′∗}P^{′∗})⊙(L^{∗}P^{∗}∗δ∗L^{′∗}P^{′∗})⊙(L^{∗}P^{∗}Q_{∗}P′_{∗}∗η^{ℒ′} ∗P^{′∗})⊙Y \\ Υ(𝒟) □ Υ(𝒟′) = X ⊙ ((β′ ⊙ (L^{∗}M_{∗} ∗ η^{ℒ′} )) ∗ P^{′∗}) ⊙ (L^{∗} ∗ ((ε^{𝒫} ∗ M_{∗}P^{′∗}) ⊙ α′)) ⊙ Y. \end{matrix}

Hence we are reduced to checking that

Z := (ε^{𝒫} ∗ L_{∗}M′_{∗}L^{′∗}) ⊙ (P^{∗}∗δ∗L^{′∗}) ⊙ (P^{∗}Q_{∗}P′_{∗}∗η^{ℒ′} )

equals

Z′ := ((β ∗ L^{′∗}) ⊙ (M_{∗} ∗ η^{ℒ′} )) ⊙ (ε^{𝒫} ∗ M_{∗}) ⊙ (P^{∗} ∗ α).

The latter follows by applying repeatedly remark 2.1.1(i) : the details shall be left to the reader. With these preliminaries, we may then also repeat the proof of theorem 2.3.17 : summing up, we obtain a strict pseudo-functor :

Υ : wLink(𝒜)^{o} → 2-Morph(𝒜^{o})

given by the same rules as for the previously defined one on 2-Morph(Link(𝒜))^{o}.

Remark 2.3.20. In the situation of (2.3.8), notice that the base change oriented square Υ(𝒟) can be regarded as a 1-cell ℳ → ℳ′ in the 2-category of weak links. By the discussion of (2.3.19), the oriented square Υ(Υ(𝒟)) is then well defined. A straightforward computation that we shall leave to the reader yields the identity :

Υ(Υ(𝒟)) = β^{†}.

2.3.21. Let us consider two 1-cells in the 2-category of weak links in 𝒜 :

\begin{matrix}(M′_{1∗}, M_{1∗}, δ_{1}) : ℒ′_{1} := (L′_{1∗}, L_{1}^{′∗}, η^{ℒ′_{1}} , ε^{ℒ′_{1}} ) → ℒ_{1} := (L_{1∗}, L_{1}^{∗}, η^{ℒ_{1}} , ε^{ℒ_{1}} ) \\ (M′_{2∗}, M_{2∗}, δ_{2}) : ℒ′_{2} := (L′_{2∗}, L_{2}^{′∗}, η^{ℒ′_{2}} , ε^{ℒ′_{2}} ) → ℒ_{2} := (L_{2∗}, L_{2}^{∗}, η^{ℒ_{2}} , ε^{ℒ_{2}} ) \end{matrix}

and a diagram of 2-cells :

[figure: A′_{2}; L′_{2∗}; M′_{2∗}; α; B′_{2}; M_{2∗}; φ′; A′_{1}; L′_{1∗}; M′_{1∗}; Q′_{1∗}; δ_{1}; B′_{1}; M_{1∗}; Q′_{2∗}; φ; A_{1}; L_{1∗}; Q_{1∗}; β; B_{1}; Q_{2∗}; A_{2}; L_{2∗}; B_{2}]

so that α and β are 1-cells ℒ′_{1} → ℒ′_{2} and ℒ_{1} → ℒ_{2} in wLink(𝒜). We complete the diagram by adding as well the 2-cell δ_{2} (the reader should picture this as a cubical diagram whose faces are oriented by the given 2-cells), and we suppose moreover that the resulting diagram commutes on 2-cells, i.e. we have :

(δ_{2} □ α) ⊟ φ′ = φ ⊟ (β □ δ_{1}).

We may then regard α and β as 1-cells of wLink(𝒜) :

(Q′_{1∗}, Q′_{2∗}, α) : ℒ′_{1} → ℒ′_{2} (Q_{1∗}, Q_{2∗}, β) : ℒ_{1} → ℒ_{2}

and the pair (φ′, φ) as a 2-cell of wLink(𝒜) :

(φ′, φ) : (M′_{2∗}, M_{2∗}, δ_{2}) ∘ (Q′_{1∗}, Q′_{2∗}, α) ⇒ (Q_{1∗}, Q_{2∗}, β) ∘ (M′_{1∗}, M_{1∗}, δ_{1}).

The pseudo-functor Υ of (2.3.19) then yields the 2-cell :

(φ^{′o}, φ^{o}) : Υ(M′_{2∗}, M_{2∗}, δ_{2})^{o} ∘ Υ(Q′_{1∗}, Q′_{2∗}, α)^{o} ⇒ Υ(Q_{1∗}, Q_{2∗}, β)^{o} ∘ Υ(M′_{1∗}, M_{1∗}, δ_{1})^{o}

which in turns translates as the diagram of 2-cells

[figure: B′_{2}; M_{2∗}; L_{2}^{′∗}; φ; B_{2}; L_{2}^{∗}; Υ(α); B′_{1}; M_{1∗}; L_{1}^{′∗}; Q′_{2∗}; Υ(δ_{1}); B_{1}; L_{1}^{∗}; Q_{2∗}; Υ(β); A′_{1}; M′_{1∗}; Q′_{1∗}; φ′; A_{1}; Q_{1∗}; A′_{2}; M′_{2∗}; A_{2}]

completed to a cubical diagram, by adding in the 2-cell Υ(M′_{2∗}, M_{2∗}, δ_{2}), and commuting again on 2-cells, i.e. verifying the identity :

(Υ(β) ⊟ Υ(δ_{1})) □ φ = φ′ □ (Υ(δ_{2}) ⊟ Υ(α)).

Remark 2.3.22. (i) In the situation of (2.3.21), suppose that Υ(α), Υ(β), Υ(δ_{1}), φ and φ′ are invertible 2-cells of 𝒜. In this case, we deduce that the same holds for Υ(δ_{2}) ∗ Q′_{2}, and if Q′_{2} is an equivalence in 𝒜, we conclude easily that also Υ(δ_{2}) is an invertible 2-cell.

(ii) For instance, consider the case where L_{1∗} = L_{2∗}, L′_{1∗} = L′_{2∗}, M_{1∗} = M_{2∗}, M′_{1∗} = M′_{2∗} and where all the 1-cells Q_{1∗}, Q_{2∗}, Q′_{1∗}, Q′_{2∗} and all the 2-cells α, β, φ and φ′ are identities (and then δ_{1} = δ_{2}). In this case, applying (i) we conclude that Υ(δ_{1} : ℒ′_{1} → ℒ_{1}) is invertible if and only if the same holds for Υ(δ_{2} : ℒ′_{2} → ℒ_{2}).

(iii) We may also consider the variant where we invert the orientation of the 2-cells φ and φ′ (and keep unchanged the others); then the commutativity on 2-cells becomes the condition :

φ ⊟ (δ_{2} □ α) = (β □ δ_{1}) ⊟ φ′

so that the pair (φ, φ′) can be regarded as a 1-cell of wLink(𝒜) as in the foregoing, but with reversed direction. After applying Υ, we deduce the identity

φ′ □ (Υ(β) ⊟ Υ(δ_{1})) = (Υ(δ_{2}) ⊟ Υ(α)) □ φ.

Especially, (i) applies verbatim to this variant as well.

2.3.23. Lastly, let 𝒜 and ℬ be two 2-categories, and φ : 𝒜 → ℬ a pseudo-functor with coherence constraints (δ_{•}, γ_{•}). Then φ induces a pseudo-functor :

Link(φ) : Link(𝒜) → Link(ℬ) A ↦ φA

that assigns to every link ℒ := (F, G, η^{ℒ} , ε^{ℒ} ) : A → B of 𝒜 the datum

φ(ℒ) := (φF, φG, η^{φ(ℒ)} , ε^{φ(ℒ)} )

where η^{φ(ℒ)} and ε^{φ(ℒ)} are the unique 2-cells that make commute the diagrams

\xymatrix{1_{φB} \ar@{=>}[r]^{η^{φ(ℒ)}} \ar@{=>}[d]_{δ_{B}} & φF ∘ φG \ar@{=>}[d]^{γ_{F,G}} & φG ∘ φF \ar@{=>}[r]^{ε^{φ(ℒ)}} \ar@{=>}[d]_{γ_{G,F}} & 1_{φA} \ar@{=>}[d]^{δ_{A}} \\ φ1_{B} \ar@{=>}[r]^{φ(η^{ℒ} )} & φ(F ∘ G) & φ(G ∘ F) \ar@{=>}[r]^{φ(ε^{ℒ} )} & φ1_{A}.}

We leave to the reader the verification that φ(ℒ) is indeed a link φA → φB in ℬ. To every transformation of links β : ℒ ⇒ ℒ′, the pseudo-functor Link(φ) assigns the transformation φ(β) : φ(ℒ) ⇒ φ(ℒ′). The associativity constraint of Link(φ) assigns to every composable pair of morphisms of links ℒ := (F, G, η^{ℒ} , ε^{ℒ} ) : A → B, ℒ′ := (F′, G′, η^{ℒ′} , ε^{ℒ′} ) : B → C the 2-cell γ_{ℒ ,ℒ′}^{∗} := γ_{F,F′} : φ(ℒ′) ∘ φ(ℒ) ⇒ φ(ℒ′ ∘ ℒ). Likewise, the unit constraint of Link(φ) assigns to every object A of 𝒜 the 2-cell δ_{A}^{∗} := δ_{A}.

Proposition 2.3.24. In the situation of (2.3.23), the following diagram commutes :

\xymatrix{2-Morph(Link(𝒜))^{o} \ar[r]^{Υ_{𝒜}} \ar[d]_{2-Morph(Link(φ))^{o}} & 2-Morph(𝒜^{o}) \ar[d]^{2-Morph(φ^{o})} \\ 2-Morph(Link(ℬ))^{o} \ar[r]^{Υ_{ℬ}} & 2-Morph(ℬ^{o}).}

Proof. The commutativity on objects and 2-cells of 2-Morph(Link(𝒜))^{o} is immediate from the definition. To check commutativity on 1-cells, consider a diagram 𝒟 as in (2.3.8); we come down to verifying that the 2-cell of ℬ :

X := (ε^{φ(ℒ)} ∗ φM′_{∗}φL^{′∗}) ⊙ (φL^{∗} ∗ (γ_{M′_{∗},L_{∗}}^{−1} ⊙ φ(β) ⊙ γ_{L′_{∗},M_{∗}}) ∗ φL^{′∗}) ⊙ (φL^{∗}φM_{∗} ∗ η^{φ(ℒ′)} ) equals the 2-cell :

Y := γ_{M′_{∗},L^{′∗}}^{−1} ⊙ φ((ε^{ℒ} ∗ M′_{∗}L^{′∗}) ⊙ (L^{∗} ∗ β ∗ L^{′∗}) ⊙ (L^{∗}M_{∗} ∗ η^{ℒ′} )) ⊙ γ_{L^{∗},M_{∗}}.

To this aim, set

\begin{matrix}Z := & ((δ_{A}^{−1} ⊙ φ(ε^{ℒ} ) ⊙ γ_{L^{∗},L_{∗}}) ∗ φM′_{∗}) ⊙ (φL^{∗} ∗ γ_{M′_{∗},L_{∗}}^{−1}) \\ Z′ := & (γ_{L′_{∗},M_{∗}} ∗ φL^{′∗}) ⊙ (φM_{∗} ∗ (γ_{L′_{∗},L^{′∗}}^{−1} ⊙ φ(η^{ℒ′} ) ⊙ δ_{B′})) \end{matrix}

so that X = (Z ∗ φL^{′∗}) ⊙ X′ ⊙ (φL^{∗} ∗ Z′), with X′ := φL^{∗} ∗ φ(β) ∗ φL^{′∗}. We compute :

\begin{matrix}Z = & γ_{1_{A},M′_{∗}} ⊙ (φ(ε^{ℒ} ) ∗ φM′_{∗}) ⊙ (γ_{L^{∗},L_{∗}} ∗ φM′_{∗}) ⊙ (φL^{∗} ∗ γ_{M′_{∗},L_{∗}}^{−1}) \\ = & γ_{1_{A},M′_{∗}} ⊙ (φ(ε^{ℒ} ) ∗ φM′_{∗}) ⊙ γ_{L^{∗}L_{∗},M′_{∗}}^{−1} ⊙ γ_{L^{∗},L_{∗}M′_{∗}} \\ = & φ(ε^{ℒ} ∗ M′_{∗}) ⊙ γ_{L^{∗},L_{∗}M′_{∗}} \\ Z′ = & (γ_{L′_{∗},M_{∗}} ∗ φL^{′∗}) ⊙ (φM_{∗} ∗ γ_{L′_{∗},L^{′∗}}^{−1}) ⊙ (φM_{∗} ∗ φ(η^{ℒ′} )) ⊙ γ_{M_{∗},1_{B}}^{−1} \\ = & γ_{M_{∗}L′_{∗},L^{′∗}}^{−1} ⊙ γ_{M_{∗},L′_{∗}L^{′∗}} ⊙ (φM_{∗} ∗ φ(η^{ℒ′} )) ⊙ γ_{M_{∗},1_{B}}^{−1} \\ = & γ_{M_{∗}L′_{∗},L^{′∗}}^{−1} ⊙ φ(M_{∗} ∗ η^{ℒ′} ). \end{matrix}

On the other hand, we have :

\begin{matrix}γ_{M′_{∗},L^{′∗}}^{−1} ⊙ φ(ε^{ℒ} ∗ M′_{∗}L^{′∗}) = & (φ(ε^{ℒ} ∗ M′_{∗}) ∗ φL^{′∗}) ⊙ γ_{L^{∗}L_{∗}M′_{∗},L^{′∗}}^{−1} \\ φ(L^{∗}M_{∗} ∗ η^{ℒ′} ) ⊙ γ_{L^{∗},M_{∗}} = & γ_{L^{∗},M_{∗}L′_{∗}L^{′∗}} ⊙ (φL^{∗} ∗ φ(M_{∗} ∗ η^{ℒ′} )) \end{matrix}

so we are reduced to showing the identity :

(γ_{L^{∗},L_{∗}M′_{∗}} ∗ φL^{′∗}) ⊙ X′ ⊙ (φL^{∗} ∗ γ_{M_{∗}L′_{∗},L^{′∗}}^{−1}) = γ_{L^{∗}L_{∗}M′_{∗},L^{′∗}}^{−1} ⊙ φ(L^{∗} ∗ β ∗ L^{′∗}) ⊙ γ_{L^{∗},M_{∗}L′_{∗}L^{′∗}}.

The latter is the same as the identity : γ_{L^{∗},L_{∗}M′_{∗},L^{′∗}} ⊙ X′ = φ(L^{∗} ∗ β ∗ L^{′∗}) ⊙ γ_{L^{∗},M_{∗}L′_{∗},L^{′∗}}, which holds by virtue of remark 2.1.17(iii). □

2.3.25. We conclude with a further construction derived from the formalism of base change, that shall help us in the following section. First, to every 2-category 𝒜 we attach the 2-category

Equiv(𝒜)

defined as the sub-2-category of 2-Morph(𝒜) whose objects are the equivalences in 𝒜 (see definition 2.1.3(iii)); for any two equivalences f : X → Y and f′ : X′ → Y′ in 𝒜, the 1-cells f → f′ in Equiv(𝒜) are the essentially commutative oriented squares :

(2.3.26)  [figure: X; f; g′; β; Y; g; X′; f′; Y′]

and the 2-cells between such 1-cells are as in 2-Morph(𝒜). We define a pseudo-functor

L : Equiv(𝒜) → wLink(𝒜)

as follows. For any (f : X → Y) ∈ Ob(Equiv(𝒜)) we choose a quasi-inverse f^{†} : Y → X (see definition 2.1.3(iii)) and the unit η^{f} and counit ε^{f} of an adjunction for (f^{†}, f), and we set

L(f) := (f^{†}, f, η^{f}, ε^{f}) ∈ Ob(wLink(𝒜)).

Then every 1-cell (g, g′, β) : f → f′ as in (2.3.26) yields a 1-cell L(f) → L(f′) in wLink(𝒜) which we denote L(g, g′, β); likewise, L is the identity on 2-cells. Then of course the coherence constraints of L are given by identities, but L is not necessarily strict, since we do not necessarily have (g ∘ f)^{†} = f^{†} ∘ g^{†} for every f, g ∈ Ob(Equiv(𝒜)).

2.3.27. Moreover, we have a natural strict isomorphism of 2-categories :

Equiv(𝒜) →^{∼} Equiv(𝒜^{o})^{o}.

Namely, to every equivalence f : X → Y of 𝒜 we assign the equivalence f^{o} : Y^{o} → X^{o} of 𝒜^{o}, and to every 1-cell (2.3.26) we assign the following 1-cell f^{′o} → f^{o} of Equiv(𝒜^{o})^{o} :

[figure: Y^{′o}; f^{′o}; g^{o}; (β^{o})^{−1}; X^{′o}; g^{′o}; Y^{o}; f^{o}; X^{o}.]

Lastly, let (h, h′, β′) : f → f′ be another 1-cell of Equiv(𝒜), and (α_{1}, α_{2}) : (g, g′, β) ⇒ (h, h′, β′) any 2-cell; recall that the latter is a pair of 2-cells α_{1} : g′ ⇒ h′ and α_{2} : g ⇒ h of 𝒜 such that α_{1} ⊙ β = β′ ⊙ α_{2}. Then our strict isomorphism assigns to (α_{1}, α_{2}) the pair (α_{2}^{o}, α_{1}^{o}), which is easily seen to be a 2-cell (g^{′o}, g^{o}, (β^{o})^{−1}) ⇒ (h^{′o}, h^{o}, (β^{′o})^{−1}) of Equiv(𝒜^{o})^{o}.

Next, by composing the pseudo-functors L of (2.3.25) and Υ of (2.3.19) we get a pseudofunctor Υ^{o} ∘ L : Equiv(𝒜) → 2-Morph(𝒜^{o})^{o}, and taking into account remark 2.3.11 we see that Υ^{o} ∘ L factors through the inclusion strict pseudo-functor Equiv(𝒜^{o})^{o} → 2-Morph(𝒜^{o})^{o}. Therefore, we can further compose with the foregoing isomorphism to deduce a pseudo-functor

Υ̃ : Equiv(𝒜) → Equiv(𝒜)  f ↦ f^{†}  (g, g′, β) ↦ Υ(g, g′, β)^{−1}.

2.4. Adjunctions in 2-categories. We begin by introducing two constructions that will often allow us to reduce the proof of assertions concerning general pseudo-functors, to the special case of strict, or at least unital pseudo-functors, in the sense of definition 2.4.1.

Definition 2.4.1. Let 𝒜 and ℬ be two 2-categories, and F : 𝒜 → ℬ a pseudo-functor with coherence constraint (δ_{•}^{F} , γ_{••}^{F}). We say that F is unital if F 1_{A} = 1_{F A} and δ_{A}^{F} = i_{F A} for every A ∈ Ob(𝒜). Since the component δ_{•}^{F} is determined by the these conditions, we shall usually say that the coherence constraint of a unital pseudo-functor F is the component γ_{••}^{F}.

Remark 2.4.2. Let F : 𝒜 → ℬ be a unital pseudo-functor with coherence constraint γ^{F}.

(i) Directly from the composition axiom we see that for every pair of 1-cells A ⟶^{f} B ⟶^{g} C of 𝒜, the 2-cell γ_{f,g}^{F} equals 1_{F g} if f = 1_{A}, and equals 1_{F f} if g = 1_{B}.

(ii) Let G : 𝒜 → ℬ be another unital pseudo-functor, and α : F ⇒ G any pseudo-natural transformation with coherence constraint τ. From the first coherence axiom for α it is clear that τ_{1_{A}} = 1_{α_{A}} for every A ∈ Ob(𝒜).

Proposition 2.4.3. For every pseudo-functor F : 𝒜 → ℬ there exists a unital pseudo-functor F^{u} : 𝒜 → ℬ with a pseudo-natural isomorphism F →^{∼} F^{u}.

Proof. Let (δ^{F} , γ^{F}) be the coherence constraint of F; for every 1-cell f : A → B of 𝒜, we define the 2-cell of ℬ

ω_{f} := \{ \begin{matrix}1_{F f} & if f ≠ 1_{A} \\ δ_{A}^{F} & if f = 1_{A} \end{matrix} F^{u}f ⇒ F f.

We shall show more precisely, that there exists a unique well-defined unital pseudo-functor F^{u} : 𝒜 → ℬ with coherence constraint γ^{F^{u}} such that :

• F^{u}A = F A for every A ∈ Ob(𝒜)

• F^{u}f = F f for every A, B ∈ Ob(𝒜) every 1-cell f : A → B such that f ≠ 1_{A}
• for every pair of 1-cells A ⟶^{f} B ⟶^{g} C of 𝒜 we have
γ_{f,g}^{F^{u}} = ω_{g∘f}^{−1} ⊙ γ_{f,g}^{F} ⊙ (ω_{g} ∗ ω_{f})

• for every pair of 1-cells f, f′ : A → B of 𝒜 and every 2-cell β : f ⇒ f′, we have

F^{u}β = ω_{f′}^{−1} ⊙ F β ⊙ ω_{f}.

Indeed, the uniqueness of such F^{u} is obvious; it is also clear that for every A, B ∈ Ob(𝒜) the stated rules yield a well-defined functor F_{AB}^{u} : 𝒜 (A, B) → ℬ(F^{u}A, F^{u}B). For the naturality of γ^{F^{u}} consider 1-cells f, f′ : A → B and g, g′ : B → C in 𝒜 and 2-cells β_{1} : f ⇒ f′ and β_{2} : g ⇒ g′; we notice that

\begin{matrix}F^{u}β_{2} ∗ F^{u}β_{1} = & (ω_{g′}^{−1} ⊙ F β_{2} ⊙ ω_{g}) ∗ (ω_{f′}^{−1} ⊙ F β_{1} ⊙ ω_{f}) \\ = & (ω_{g′}^{−1} ∗ ω_{f′}^{−1}) ⊙ (F β_{2} ∗ F β_{1}) ⊙ (ω_{g} ∗ ω_{f}) \end{matrix}

whence

\begin{matrix}F^{u}(β_{2} ∗ β_{1}) ⊙ γ_{f,g}^{F^{u}} = & ω_{g′∘f′}^{−1} ⊙ F (β_{2} ∗ β_{1}) ⊙ γ_{f,g}^{F} ⊙ (ω_{g} ∗ ω_{f}) \\ = & ω_{g′∘f′}^{−1} ⊙ γ_{f′,g′}^{F} ⊙ (F β_{2} ∗ F β_{1}) ⊙ (ω_{g} ∗ ω_{f}) \\ = & γ_{f′,g′}^{F^{u}} ⊙ (F^{u}β_{2} ∗ F^{u}β_{1}) \end{matrix}

which is the contention, by remark 2.1.17(ii). Next, let us check the composition axiom for γ^{F^{u}}; we consider three 1-cells A ⟶^{f} B ⟶^{g} C ⟶^{h} D of 𝒜, and we need to show the identity

X := γ_{f,h∘g}^{F^{u}} ⊙ (γ_{g,h}^{F^{u}} ∗ F^{u}f) = Y := γ_{g∘f,h}^{F^{u}} ⊙ (F^{u}h ∗ γ_{f,g}^{F^{u}}).

We compute, on the one hand :

\begin{matrix}X = & ω_{h∘g∘f}^{−1} ⊙ γ_{f,h∘g}^{F} ⊙ (ω_{h∘g} ⊙ ω_{f}) ⊙ ((ω_{h∘g}^{−1} ⊙ γ_{g,h}^{F} ∗ (ω_{h} ∗ ω_{g})) ∗ F^{u}f) \\ = & ω_{h∘g∘f}^{−1} ⊙ γ_{f,h∘g}^{F} ⊙ (ω_{h∘g} ∗ ω_{f}) ⊙ (ω_{h∘g}^{−1} ∗ F^{u}f) ⊙ ((γ_{g,h}^{F} ⊙ (ω_{h} ∗ ω_{g})) ∗ F^{u}f) \\ = & ω_{h∘g∘f}^{−1} ⊙ γ_{f,h∘g}^{F} ⊙ (F (h ∘ g) ∗ ω_{f}) ⊙ ((γ_{g,h}^{F} ⊙ (ω_{h} ∗ ω_{g})) ∗ F^{u}f) \\ = & ω_{h∘g∘f}^{−1} ⊙ γ_{f,h∘g}^{F} ⊙ (F (h ∘ g) ∗ ω_{f}) ⊙ (γ_{g,h}^{F} ∗ F^{u}f) ⊙ (ω_{h} ∗ ω_{g} ∗ F^{u}f) \\ = & ω_{h∘g∘f}^{−1} ⊙ γ_{f,h∘g}^{F} ⊙ (γ_{g,h}^{F} ∗ F f) ⊙ (F h ∗ F g ∗ ω_{f}) ⊙ (ω_{h} ∗ ω_{g} ∗ F^{u}f) \\ = & ω_{h∘g∘f}^{−1} ⊙ γ_{f,h∘g}^{F} ⊙ (γ_{g,h}^{F} ∗ F f) ⊙ (ω_{h} ∗ ω_{g} ∗ ω_{f}) \\ = & ω_{h∘g∘f}^{−1} ⊙ γ_{g∘f,h}^{F} ⊙ (F h ∗ γ_{f,g}^{F}) ⊙ (ω_{h} ∗ ω_{g} ∗ ω_{f}) \end{matrix}

and on the other hand :

\begin{matrix}Y = & ω_{h∘g∘f}^{−1} ⊙ γ_{g∘f,h}^{F} ⊙ (ω_{h} ∗ ω_{g∘f}) ⊙ (F^{u}h ∗ (ω_{g∘f}^{−1} ⊙ γ_{f,g}^{F} ⊙ (ω_{g} ∗ ω_{f}))) \\ = & ω_{h∘g∘f}^{−1} ⊙ γ_{g∘f,h}^{F} ⊙ (ω_{h} ∗ ω_{g∘f}) ⊙ (F^{u}h ∗ ω_{g∘f}^{−1}) ⊙ (F^{u}h ∗ (γ_{f,g}^{F} ⊙ (ω_{g} ∗ ω_{f}))) \\ = & ω_{h∘g∘f}^{−1} ⊙ γ_{g∘f,h}^{F} ⊙ (ω_{h} ∗ F (g ∘ f)) ⊙ (F^{u}h ∗ γ_{f,g}^{F}) ⊙ (F^{u}h ∗ ω_{g} ∗ ω_{f}) \\ = & ω_{h∘g∘f}^{−1} ⊙ γ_{g∘f,h}^{F} ⊙ (F h ∗ γ_{f,g}^{F}) ⊙ (ω_{h} ∗ F g ∗ F h) ⊙ (F^{u}h ∗ ω_{g} ∗ ω_{f}) \\ = & ω_{h∘g∘f}^{−1} ⊙ γ_{g∘f,h}^{F} ⊙ (F h ∗ γ_{f,g}^{F}) ⊙ (ω_{h} ∗ ω_{g} ∗ ω_{f}) \end{matrix}

whence the contention. Lastly, the unit axiom can be verified by a simple inspection.

The sought pseudo-isomorphism is given by the rule that assigns α_{A} := 1_{F A} : F A → F^{u}A to every A ∈ Ob(𝒜), and the oriented square

[figure: F A; α_{A}; F f; ω_{f}; F^{u}A; F^{u}f; F B; α_{B}; F^{u}B] to every 1-cell f : A → B of 𝒜 .

The naturality of ω_{f} and the coherence axioms follow by a simple inspection. □

Remark 2.4.4. (i) For any two 2-categories 𝒜 and ℬ, let us denote by

uniPsFun(𝒜 , ℬ)

the sub-2-category of PsFun(𝒜 , ℬ) whose objects are the unital pseudo-functors 𝒜 → ℬ, and whose Hom-categories are given by the categories PsNat(F, G), for any two such unital pseudo-functors F, G. For every pseudo-functor F : 𝒜 → ℬ, let also α^{F} : F →^{∼} F^{u} be the pseudo-natural isomorphism furnished by proposition 2.4.3. It is easily seen that the rules F ↦ F^{u} for every pseudo-functor F : 𝒜 → ℬ, (β : F ⇒ G) ↦ β^{u} := α^{G} ⊙ β ⊙ (α^{F})^{−1} for every pseudo-natural transformation β, and (Ξ : β ⇝ β′) ↦ Ξ^{u} := α^{G} ∗ Ξ ∗ (α^{F})^{−1} for every modification Ξ, define a strict 2-equivalence of 2-categories

(−)^{u} : PsFun(𝒜 , ℬ) →^{∼} uniPsFun(𝒜 , ℬ)

(details left to the reader). Likewise, if i : uniPsFun(𝒜 , ℬ) → PsFun(𝒜 , ℬ) denotes the inclusion strict pseudo-functor, the rule : F ↦ α^{F} for every pseudo-functor F : 𝒜 → ℬ yields a strict pseudo-natural isomorphism of strict pseudo-functors :

(2.4.5)  1_{PsFun(𝒜 ,ℬ)} →^{∼} i ∘ (−)^{u}.

(ii) Let F : 𝒜 → ℬ and G : ℬ → 𝒞 be two pseudo-functors, α^{F} : F →^{∼} F^{u}, α^{G} : G →^{∼} G^{u}, α^{G∘F} : G ∘ F →^{∼} (G ∘ F)^{u} the corresponding pseudo-natural isomorphisms as in (i). Since α^{F} and α^{G} are not strict, we have distinct pseudo-natural isomorphisms

(α^{G} ∗ F^{u}) ⊙ (G ∗ α^{F}), (G^{u} ∗ α^{F}) ⊙ (α^{G} ∗ F) : G ∘ F →^{∼} G^{u} ∘ F^{u}

that are related by an invertible modification (see example 2.2.15(ii)). By composing with (α^{G∘F})^{−1}, we get therefore two pseudo-natural isomorphisms of unital pseudo-functors

(G ∘ F)^{u} →^{∼} G^{u} ∘ F^{u}

but in general these pseudo-functors are different.

Proposition 2.4.6. For every pseudo-functor F : 𝒜 → ℬ there exist :

(a) A 2-category 𝒜^{F} and strict pseudo-functors 𝒜 ⟵^{π^{F}} 𝒜^{F} ⟶^{F^{♭}} ℬ.
(b) With an isomorphism of pseudo-functors ω^{F} : F ∘ π^{F} →^{∼} F^{♭}.
(c) A pseudo-functor σ^{F} : 𝒜 → 𝒜^{F} such that :
π^{F} ∘ σ^{F} = 1_{𝒜}  F^{♭} ∘ σ^{F} = F  ω^{F} ∗ σ^{F} = 1_{F} .
(d) A pseudo-natural equivalence μ^{F} : σ^{F} ∘ π^{F} →^{∼} 1_{𝒜^{F}} .

Proof. We let 𝒜^{F} be the 2-category with Ob(𝒜^{F}) = Ob(𝒜), and whose 1-cells

(f, g, λ) : X → Y for every X, Y ∈ Ob(𝒜)

are all the data consisting of a 1-cell f : X → Y of 𝒜, a 1-cell g : F X → F Y of ℬ, and an invertible 2-cell λ : g →^{∼} F f. Given two such 1-cells (f, g, λ), (f′, g′, λ′) : X → Y, the 2-cells

(f, g, λ) ⇒ (f′, g′, λ′)

are the pairs (α, β) where α : f ⇒ f′ (resp. β : g ⇒ g′) is a 2-cell of 𝒜 (resp. of ℬ), such that

λ′ ⊙ β = F α ⊙ λ.

The composition of two 1-cells X ⟶^{(f_{1},g_{1},λ_{1})} Y ⟶^{(f_{2},g_{2},λ_{2})} Z is the 1-cell

(f_{2}, g_{2}, λ_{2}) ∘ (f_{1}, g_{1}, λ_{1}) := (f_{2} ∘ f_{1}, g_{2} ∘ g_{1}, γ_{f_{1},f_{2}}^{F} ⊙ (λ_{2} ∗ λ_{1}))

where (δ^{F} , γ^{F}) denotes the coherence constraint of the pseudo-functor F. The associativity of this composition law follows easily from the composition axiom for γ^{F} : the details are left to the reader. Especially, for every X ∈ Ob(𝒜) the triple (1_{X}, 1_{F X}, δ_{X}^{F}) is the identity 1-cell of X in 𝒜^{F}. Given three 1-cells (f, g, λ), (f′, g′, λ′), (f′′, g′′, λ′′) : X → Y, and two 2-cells (α, β) : (f, g, λ) ⇒ (f′, g′, λ′) and (α′, β′) : (f′, g′, λ′) ⇒ (f′′, g′′, λ′′), we let

(α′, β′) ⊙ (α, β) := (α′ ⊙ α, β′ ⊙ β)

which is a well defined 2-cell (f, g, λ) ⇒ (f′′, g′′, λ′′). Lastly, if (f_{1}, g_{1}, λ_{1}), (f′_{1}, g′_{1}, λ′_{1}) : X → Y and (f_{2}, g_{2}, λ_{2}), (f′_{2}, g′_{2}, λ′_{2}) : Y → Z are four given 1-cells, and (α_{1}, β_{1}) : (f_{1}, g_{1}, λ_{1}) ⇒ (f′_{1}, g′_{1}, λ′_{1}), (α_{2}, β_{2}) : (f_{2}, g_{2}, λ_{2}) ⇒ (f′_{2}, g′_{2}, λ′_{2}) are two 2-cells, we set :

(α_{2}, β_{2}) ∗ (α_{1}, β_{1}) := (α_{2} ∗ α_{1}, β_{2} ∗ β_{1})

which is a well defined 2-cell (f_{2}, g_{2}, λ_{2}) ∘ (f_{1}, g_{1}, λ_{1}) ⇒ (f′_{2}, g′_{2}, λ′_{2}) ∘ (f′_{1}, g′_{1}, λ′_{1}). The associativity of these two composition laws are obvious, and it is then easily seen that 𝒜^{F} is a well defined 2-category with such laws for 1-cells and 2-cells : the details are left to the reader.

The sought strict pseudo-functors F^{♭} and π^{F} are given by the rules :

F^{♭}X := F X  π^{F} X := X for every X ∈ Ob(𝒜^{F})
F^{♭}(f, g, λ) := g  π^{F} (f, g, λ) := f  for every 1-cell (f, g, λ)
F^{♭}(α, β) := β  π^{F} (α, β) := α  for every 2-cell (α, β).

The isomorphism ω^{F} is given by the rule : X ↦ 1_{F X} for every X ∈ Ob(𝒜^{F}), and its coherence costraint assigns to every 1-cell (f, g, λ) : X → Y the oriented square

[figure: F X; F f; λ; F X; g; F Y; F Y.]

Next, the pseudo-functor σ^{F} is given by the rules :

X ↦ X  f ↦ (f, F f, 1_{F f})  α ↦ (α, F α)

for every X ∈ Ob(𝒜), every 1-cell f, and every 2-cell α of 𝒜. The coherence constraints of σ^{F} are given by the rules :

X ↦ (i_{X}, δ_{X}^{F}) (f, f′) ↦ (1_{f′∘f}, γ_{f,f′}^{F})

for every X ∈ Ob(𝒜) and every composable pair of 1-cells (f, f′) of 𝒜. The required coherence axioms are easily verified, and the desired identities as in (c) follow straightforwardly : details left to the reader. Lastly, we define μ_{X}^{F} := 1_{σ^{F} X} = (1_{X}, 1_{F X}, δ_{X}^{F}) for every X ∈ Ob(𝒜^{F}). For every 1-cell (f, g, λ) : X → Y of 𝒜^{F}, notice that σ^{F} ∘ π^{F} (f, g, λ) = (f, F f, 1_{F f}); then the coherence constraint of μ^{F} assigns to (f, g, λ) the oriented square in 𝒜^{F} :

[figure: X; (1_{X},1_{F X},δ_{X}^{F}); (f,F f,1_{F f}); (1_{f} ,λ); X; (f,g,λ); Y; (1_{Y} ,1_{F Y} ,δ_{Y}^{F}); Y.]

A little diagram chase that we leave to the reader shows that these rules yield the sought pseudonatural equivalence. □

Proposition 2.4.7. (i) Let 𝒜 , ℬ be two 2-categories, F : 𝒜 → ℬ and G : ℬ → 𝒜 two pseudo-functors. Then there exist :

(a) Two 2-categories 𝒜^{F,G} and ℬ^{G,F}, and a diagram of strict pseudo-functors :

\xymatrix{𝒜 & 𝒜^{F,G} \ar[r]^{F^{♭}} \ar[l]_{π^{F,G}} & ℬ^{G,F} \ar[l]^{G^{♭}} \ar[r]^{π^{G,F}} & ℬ.}

(b) With two isomorphisms of pseudo-functors :

ω^{F,G} : F ∘ π^{F,G} →^{∼} π^{G,F} ∘ F^{♭}  and  ω^{G,F} : G ∘ π^{G,F} →^{∼} π^{F,G} ∘ G^{♭}.

(c) Two pseudo-functors σ^{F,G} : 𝒜 → 𝒜^{F,G} and σ^{G,F} : ℬ → ℬ^{G,F} such that :

\begin{aligned}π^{F,G} ∘ σ^{F,G} & = 1_{𝒜} & F^{♭} ∘ σ^{F,G} & = σ^{G,F} ∘ F & ω^{F,G} ∗ σ^{F,G} & = 1_{F} \\ π^{G,F} ∘ σ^{G,F} & = 1_{ℬ} & G^{♭} ∘ σ^{G,F} & = σ^{F,G} ∘ G & ω^{G,F} ∗ σ^{G,F} & = 1_{G}. \end{aligned}

(d) With two isomorphisms of pseudo-functors :

ψ^{F,G} : σ^{F,G} ∘ π^{F,G} →^{∼} 1_{𝒜^{F,G}} and ψ^{G,F} : σ^{G,F} ∘ π^{G,F} →^{∼} 1_{ℬ^{G,F}}

(ii) Suppose moreover that F and G are unital. Then we have as well :

G^{♭} ∗ ψ^{G,F} = (ψ^{F,G} ∗ G^{♭}) ⊙ (σ^{F,G} ∗ ω^{G,F}) F^{♭} ∗ ψ^{F,G} = (ψ^{G,F} ∗ F^{♭}) ⊙ (σ^{G,F} ∗ ω^{F,G}).

Proof. (i): To begin with, for every k, n ∈ ℕ let us set :

H_{2k} := G  H_{2k+1} := F  K_{0} := 1_{𝒜}  K_{n+1} := H_{n+1} ∘ H_{n−1} ∘ ⋯ ∘ H_{1}.

Then we let 𝒜^{F,G} be the 2-category with Ob(𝒜^{F,G}) = Ob(𝒜), and whose 1-cells

(f_{•}, λ_{•}) : X → Y for every X, Y ∈ Ob(𝒜)

are all the systems of 1-cells (f_{n} : K_{n}X → K_{n}Y | n ∈ ℕ), and of invertible 2-cells (λ_{n} : f_{n+1} →^{∼} H_{n+1}f_{n} | n ∈ ℕ). Given such 1-cells (f_{•}, λ_{•}), (f′_{•}, λ′_{•}) : X → Y, the 2-cells

α_{•} : (f_{•}, λ_{•}) ⇒ (f′_{•}, λ′_{•})

are the systems of 2-cells (α_{n} : f_{n} → f′_{n} | k ∈ ℕ) such that :

(2.4.8)  H_{n+1}(α_{n}) ⊙ λ_{n} = λ′_{n} ⊙ α_{n+1}  for every n ∈ ℕ.

The composition of two 1-cells X ⟶^{(f_{1,•},λ_{1,•})} Y ⟶^{(f_{2,•},λ_{2,•})} Z is the 1-cell

(f_{2,•}, λ_{1,•}) ∘ (f_{1,•}, λ_{1,•}) := (f_{2,n} ∘ f_{1,n}, γ_{f_{1,n},f_{2,n}}^{H_{n+1}} ⊙ (λ_{2,n} ∗ λ_{1,n}) | n ∈ ℕ)

where (δ^{H_{n}}, γ^{H_{n}}) denotes the coherence constraint of the pseudo-functor H_{n}, for every n ∈ ℕ. Just as in the proof of proposition 2.4.6, the associativity of this rule follows easily from the composition axioms for γ^{H_{n}}, and for every X ∈ Ob(𝒜), the identity 1-cell of X in 𝒜^{F,G} is the datum 1_{X}^{F,G} := (1_{K_{n}X}, δ_{K_{n}X}^{H_{n+1}} | n ∈ ℕ). Given three 1-cells (f_{•}, λ_{•}), (f′_{•}, λ′_{•}), (f′′_{•}, λ′′_{•}) : X → Y, and two 2-cells α_{•} : (f_{•}, λ_{•}) ⇒ (f′_{•}, λ′_{•}) and α′_{•} : (f′_{•}, λ′_{•}) ⇒ (f′′_{•}, λ′′_{•}), we let

α′_{•} ⊙ α_{•} := (α′_{n} ⊙ α_{n} | n ∈ ℕ)

which is a well defined 2-cell (f_{•}, λ_{•}) ⇒ (f′′_{•}, λ′′_{•}). Lastly, if (f_{1,•}, λ_{1,•}), (f′_{1,•}, λ′_{1,•}) : X → Y and (f_{2,•}, λ_{2,•}), (f′_{2,•}, λ′_{2,•}) : Y → Z are four given 1-cells, and α_{1,•} : (f_{1,•}, λ_{1,•}) ⇒ (f′_{1,•}, λ′_{1,•}), α_{2,•} : (f_{2,•}, λ_{2,•}) ⇒ (f′_{2,•}, λ′_{2,•}) are two 2-cells, we set :

α_{2,•} ∗ α_{1,•} := (α_{2,n} ∗ α_{1,n} | n ∈ ℕ)

which is a well defined 2-cell (f_{2,•}, λ_{2,•}) ∘ (f_{1,•}, λ_{1,•}) ⇒ (f′_{2,•}, λ′_{2,•}) ∘ (f′_{1,•}, λ′_{1,•}). The associativity of these two composition laws are obvious, and it is then easily seen that 𝒜^{F,G} is a well defined 2-category with such laws for 1-cells and 2-cells : the details are left to the reader.

We define ℬ^{G,F} by exactly the same rules, after swapping 𝒜 and F with respectively ℬ and G. Then the sought strict pseudo-functors F^{♭} and π^{F,G} are given by the rules :

F^{♭}X := F X  π^{F,G}X := X for every X ∈ Ob(𝒜^{F,G})
F^{♭}(f_{•}, λ_{•}) := (f_{n+1}, λ_{n+1} | n ∈ ℕ) π^{F,G}(f_{•}, λ_{•}) := f_{0}  for every 1-cell (f_{•}, λ_{•})
F^{♭}(α_{•}) := (α_{n+1} | n ∈ ℕ)  π^{F,G}(α_{•}) := α_{0}  for every 2-cell α_{•}

and again, G^{♭} and π^{G,F} are defined by the same rules, after swapping F and G.

The isomorphism ω^{F,G} is given by the rule : X ↦ 1_{F X} for every X ∈ Ob(𝒜^{F}), and its coherence costraint assigns to every 1-cell (f_{•}, λ_{•}) : X → Y the oriented square

[figure: F X; F f_{0}; λ_{0}; F X; f_{1}; F Y; F Y.]

Lastly, the pseudo-functor σ^{F,G} is given by the rules :

X ↦ X  f ↦ (K_{n}f, 1_{K_{n+1}f} | n ∈ ℕ)  α ↦ (K_{n}α | n ∈ ℕ)

for every X ∈ Ob(𝒜), every 1-cell f, and every 2-cell α of 𝒜. For every n ∈ ℕ, let (δ^{K_{n}}, γ^{K_{n}}) be the coherence constraint of K_{n}; the coherence constraint of σ^{F,G} is given by the rules :

X ↦ (δ_{X}^{K_{n}} | n ∈ ℕ) (f, f′) ↦ (γ_{f,f′}^{K_{n}} | n ∈ ℕ)

for every X ∈ Ob(𝒜) and every composable pair of 1-cells (f, f′) of 𝒜, where H. The required coherence axioms are verified as in the proof of the corresponding assertions of the proposition 2.4.6, and shall again be left to the reader. Likewise, one defines ω^{G,F} and σ^{G,F} by the same rules, after swapping the roles of F and G. Then the desired identities in (c) follow straightforwardly. It remains to exhibit an isomorphism of pseudo-functors ψ^{F,G} : σ^{F,G}∘π^{F,G} →^{∼} 1_{𝒜^{F,G}} . To this aim, notice that σ^{F,G} ∘ π^{F,G} is given by the rules :

X ↦ X  (f_{•}, λ_{•}) ↦ (K_{n}f_{0}, 1_{K_{n+1}f_{0}} | n ∈ ℕ)  α_{•} ↦ (K_{n}α_{0} | n ∈ ℕ)

for every object X, every 1-cell (f_{•}, λ_{•}) and every 2-cell α_{•} of 𝒜^{F,G}; then we let ψ_{X}^{F,G} := 1_{X}^{F,G} for every such X, and to every such (f_{•}, λ_{•}) we attach the coherence constraint

τ_{(f_{•},λ_{•}),•}^{ψ^{F,G}} : (f_{•}, λ_{•}) → (K_{n}f_{0}, 1_{K_{n+1}f_{0}} | n ∈ ℕ)

defined inductively by the rules :

τ_{(f_{•},λ_{•}),0}^{ψ^{F,G}} := 1_{f_{0}}  τ_{(f_{•},λ_{•}),n+1}^{ψ^{F,G}} := H_{n+1}(τ_{(f_{•},λ_{•}),n}^{ψ^{F,G}}) ⊙ λ_{n}  for every n ∈ ℕ.

The coherence axioms for τ^{ψ} amount to the identities :

\begin{aligned}τ_{1_{X}^{F,G},n}^{ψ^{F,G}} & = δ_{X}^{K_{n}} \\ A_{n} := γ_{f_{0},f′_{0}}^{K_{n}} ⊙ (τ_{(f′_{•},λ′_{•}),n}^{ψ^{F,G}} ∗ τ_{(f_{•},λ_{•}),n}^{ψ^{F,G}}) & = B_{n} := τ_{(f′′_{•},λ′′_{•}),n}^{ψ^{F,G}} \end{aligned}  for every n ∈ ℕ

for every object X of 𝒜 and every composable pair of 1-cells (f_{•}, λ_{•}) and (f′_{•}, λ′_{•}), with (f′′_{•}, λ′′_{•}) := (f′_{•}, λ′_{•}) ∘ (f_{•}, λ_{•}). The first identity is checked easily by induction on n, recalling that δ_{X}^{K_{n+1}} = H_{n+1}(δ_{X}^{K_{n}}) ⊙ δ_{K_{n}X}^{H_{n+1}} for every n ∈ ℕ. The second identity is obvious for n = 0. Suppose then that the second identity is known for some n ∈ ℕ; we compute :

A_{n+1} = H_{n+1}(γ_{f_{0},f′_{0}}^{K_{n}}) ⊙ γ_{K_{n}f_{0},K_{n}f′_{0}}^{H_{n+1}} ⊙ ((H_{n+1}(τ_{(f′_{•},λ′_{•}),n}^{ψ}) ⊙ λ′_{n}) ∗ (H_{n+1}(τ_{(f_{•},λ_{•}),n}^{ψ}) ⊙ λ_{n}))
= H_{n+1}(γ_{f_{0},f′_{0}}^{K_{n}}) ⊙ γ_{K_{n}f_{0},K_{n}f′_{0}}^{H_{n+1}} ⊙ (H_{n+1}(τ_{(f′_{•},λ′_{•}),n}^{ψ}) ∗ H_{n+1}(τ_{(f_{•},λ_{•}),n}^{ψ})) ⊙ (λ′_{n} ∗ λ_{n})
= H_{n+1}(γ_{f_{0},f′_{0}}^{K_{n}}) ⊙ H_{n+1}(τ_{(f′_{•},λ′_{•}),n}^{ψ} ∗ τ_{(f_{•},λ_{•}),n}^{ψ}) ⊙ γ_{f_{0},f′_{0}}^{H_{n+1}} ⊙ (λ′_{n} ∗ λ_{n})
= H_{n+1}(A_{n}) ⊙ λ′′_{n}
= H_{n+1}(B_{n}) ⊙ λ′′_{n}
= B_{n+1}

where the first equality follows after recalling that γ_{f_{0},f′_{0}}^{K_{n+1}} = H_{n+1}(γ_{f_{0},f′_{0}}^{K_{n}}) ⊙ γ_{K_{n}f_{0},K_{n}f′_{0}}^{H_{n+1}}. The same rules yield, mutatis mutandis, the sought isomorphism ψ^{G,F} : σ^{G,F} ∘ π^{G,F} →^{∼} 1_{ℬ^{G,F}} .

(ii): Let us set as well :

H′_{2k} := F  H′_{2k+1} := G  K′_{0} := 1_{ℬ}  K′_{n+1} := H′_{n+1} ∘ H′_{n−1} ∘ ⋯ ∘ H′_{1}.

With this notation, since F are unital, we have :

\begin{aligned}(G^{♭} ∗ ψ^{G,F})_{X} & = G^{♭}(1_{X}^{G,F}) = (1_{K′_{n+1}X}, i_{K′_{n+2}X} | n ∈ ℕ) \\ (ψ^{F,G} ∗ G^{♭})_{X} & = ψ_{GX}^{F,G} = (1_{K_{n}GX}, i_{K_{n+1}GX} | n ∈ ℕ) \\ (σ^{F,G} ∗ ω^{G,F})_{X} & = σ_{GX}^{F,G} = (1_{K_{n}GX}, i_{K_{n+1}GX} | n ∈ ℕ). \end{aligned}

Noticing that K′_{n+2} = K_{n+1}G and γ_{1_{KnGX},1_{KnGX}}^{H_{n+1}} = δ_{K_{n}GX}^{H_{n+1}} = i_{K_{n}GX}, we deduce :

(ψ^{F,G} ∗ G^{♭})_{X} ∘(σ^{F,G}∗ω^{G,F})_{X} = (G^{♭} ∗ ψ^{G,F})_{X} for every X ∈ Ob(ℬ^{G,F}) .

Next, the coherence constraint of σ^{F,G} ∗ ω^{G,F} is given by the rule :

(X ⟶^{(f_{•},λ_{•})} Y) ↦ ((γ_{Gf_{0},1_{GY}}^{K_{n}})^{−1} ⊙ K_{n}λ_{0} ⊙ γ_{1_{GX},f_{1}}^{K_{n}} | n ∈ℕ) = (K_{n}λ_{0} | n ∈ℕ)

whereas the ones of ψ^{F,G} ∗ G^{♭} and respectively G^{♭} ∗ ψ^{G,F} are defined by the rules :

(f_{•}, λ_{•}) ↦ (τ_{(f_{n+1},λ_{n+1} | n∈ℕ),n}^{ψ^{F,G}} | n ∈ ℕ) (f_{•}, λ_{•}) ↦ (τ_{(f_{•},λ_{•}),n+1}^{ψ^{G,F}} | n ∈ ℕ).

Hence we are reduced to showing that

K_{n}λ_{0} ⊙ τ_{(f_{n+1},λ_{n+1} | n∈ℕ),n}^{ψ^{F,G}} = τ_{(f_{•},λ_{•}),n+1}^{ψ^{G,F}} for every n ∈ ℕ.

The latter follows by an easy induction on n : details left to the reader. □

Definition 2.4.9. Let 𝒜, ℬ be two 2-categories, and F, F′ : 𝒜 → ℬ two pseudo-functors.

(i) We say that a pseudo-natural transformation F ⇒ F′ is a pseudo-natural equivalence if it is an equivalence in the 2-category PsFun(𝒜 , ℬ) (in the sense of definition 2.1.3(iii)).

(ii) We say that F is fully faithful (resp. strongly faithful) if for every A, A′ ∈ Ob(𝒜), the functor F_{AA′} : 𝒜 (A, A′) → ℬ(F A, F A′) is an equivalence (resp. is an isomorphism of categories).

(iii) We say that 𝒜 is a full sub-2-category (resp. a strong sub-2-category) of ℬ, if it is a sub-2-category and the inclusion pseudo-functor 𝒜 → ℬ is fully (resp. strongly) faithful.

(iv) We say that F is a 2-equivalence (resp. a strong 2-equivalence) from 𝒜 to ℬ if it is fully faithful (resp. strongly faithful), and for every B ∈ Ob(ℬ) there exists A ∈ Ob(𝒜) with an equivalence (resp. an isomorphism) F A → B.

Remark 2.4.10. Consider a square diagram of 2-categories and pseudo-functors

𝒟  :  \xymatrix{𝒜 \ar[r]^{F} \ar[d]_{H} & ℬ \ar[d]^{G} \\ 𝒞 \ar[r]^{K} & 𝒟.}

We shall say that 𝒟 is essentially commutative (resp. pseudo-commutative) if there exists an isomorphism of pseudo-functors (resp. a pseudo-natural equivalence) α : G ∘ F ⇒ K ∘ H. Even though the set of (small) 2-categories does not form a 2-category, the datum (𝒟, α) is a type of oriented square analogous to those contemplated in (2.1.5). Especially, two such data (𝒟, α) and (𝒟′, α′) can be composed if they share a side, in exactly the same way as for usual oriented square. Then, it is clear that a horizontal or vertical composition of essentially commutative (resp. pseudo-commutative) squares is again essentially commutative (resp. pseudo-commutative). On the other hand, the discussion of remark 2.2.5(iv,v) shows that the identities of proposition 2.1.9 hold only up to isomorphism of pseudo-natural transformations, i.e. up to invertible modifications.

Lemma 2.4.11. Let 𝒜 , ℬ be two 2-categories, F : 𝒜 → ℬ a pseudo-functor, and f : A → B a 1-cell in 𝒜. We have :

(i) If f admits a right adjoint g, then F f admits the right adjoint F g.

(ii) If f is an equivalence in 𝒜, then F f is an equivalence in ℬ, and if g is a quasi-inverse

for f, then F g is a quasi-inverse for F f. (iii) If F is fully faithful, then f is an equivalence if and only if F f is an equivalence in ℬ.

Proof. (i): Indeed, let ε : f ∘g ⇒ 1_{B} and η : 1_{A} ⇒ g ∘f be the unit and counit of an adjunction for the pair (f, g). We consider the 2-cells

ε′ := δ_{B}^{F −1} ⊙ F (ε) ⊙ γ_{g,f}^{F} : F f ∘ F g ⇒ 1_{F B} η′ := γ_{f,g}^{F −1} ⊙ F (η) ⊙ δ_{A}^{F} : 1_{F A} ⇒ F g ∘ F f

and we compute :

\begin{matrix}(F g ∗ F ε′)⊙(F η′ ∗ F g) = & (F g ∗ δ_{B}^{F})^{−1}⊙(F g ∗ F ε)⊙γ_{f∘g,g}^{F −1}⊙γ_{g,g∘f}^{F} ⊙(F η ∗ F g)⊙(δ_{A}^{F} ∗ F g) \\ = & (F g ∗ δ_{B}^{F})^{−1} ⊙ γ_{1_{B},g}^{F −1} ⊙ F (g ∗ ε) ⊙ F (η ∗ g) ⊙ γ_{g,1_{B}}^{F} ⊙ (δ_{A}^{F} ∗ F g) \\ = & 1_{F g}. \end{matrix}

Likewise we check that (F ε ∗ F f) ⊙ (F f ∗ F η) = 1_{F f} : the details are left to the reader.

(ii): It is clear from the proof of (i) that if ε and η are invertible 2-cells, the same holds for η′ and ε′, whence the contention.

(iii): Due to (ii), we may assume that F f is an equivalence in ℬ, and we check that f is an equivalence in 𝒜. To this aim, let g : F B → F A be a quasi-inverse for F f, and η : 1_{F B} → F f ∘ g and ε : g ∘ F f ⇒ 1_{F A} two invertible 2-cells. Since F_{AB} is an equivalence, we have a 1-cell h : B → A and an invertible 2-cell β : F h ⇒ g, whence the invertible 2-cell

\begin{matrix}ε′ := & δ_{A}^{F} ⊙ ε ⊙ (β ∗ F f) ⊙ γ_{f,h}^{F −1} : F (h ∘ f) ⇒ F 1_{A} \\ η′ := & η′ := 1γ_{h,f}^{F} ⊙ (F f ∗ β^{−1}) ⊙ η ⊙ δ_{B}^{F −1} : F 1_{B} ⇒ F (f ∘ h). \end{matrix}

Again, since F_{AB} is an equivalence, we deduce invertible 2-cells ε′′ : h ∘ f ⇒ 1_{A} and η′′ : 1_{B} ⇒ f ∘ h such that F ε′′ = ε′ and F η′′ = η′, whence the contention. □

Theorem 2.4.12. Let 𝒜 , ℬ be two 2-categories, F, G : 𝒜 → ℬ two pseudo-functors, and α : F ⇒ G a pseudo-natural transformation. The following conditions are equivalent :

(a) α is a pseudo-natural equivalence of pseudo-functors.

(b) The 1-cell α_{A} : F A → GA is an equivalence in ℬ, for every A ∈ Ob(𝒜).

Proof. Let 𝒜^{F} be the 2-category associated with F as in proposition 2.4.6, together with the strict pseudo-functors F^{♭} : 𝒜^{F} → ℬ and π^{F} : 𝒜^{F} → 𝒜, the pseudo-functor σ^{F} : 𝒜 → 𝒜^{F} and the pseudo-natural isomorphism ω^{F} : F ∘ π^{F} →^{∼} F^{♭}. Set α′ := (α ∗ π^{F}) ⊙ ω^{F −1} : F^{♭} ⇒ G ∘ π^{F}, and suppose that there exists a pseudo-natural transformation β′ : G ∘ π^{F} ⇒ F^{♭} with invertible modifications Ξ_{1} : 1_{F^{♭}} ⇝ β′ ⊙ α′ and Ξ_{2} : 1_{G∘π^{F}} ⇝ α′ ⊙ β′. Then notice that α′ ∗ σ^{F} = α, and set β := β′ ∗ π^{F}; we deduce invertible modifications Ξ_{1} ∘ π^{F} : 1_{F} ⇝ β ⊙ α and Ξ_{2} ∘ π^{F} : 1_{G} ⇝ α ⊙ β. Thus, we may replace 𝒜, F, G and α by 𝒜^{F}, F^{♭}, G ∘ π^{F} and α′, and assume from start that F is strict. Arguing likewise with 𝒜^{G} and G^{♭}, we may then reduce to the case where both F and G are strict.

(b)⇒(a): Notice that the pseudo-functor α̃ : 𝒜 → 2-Morph(ℬ) associated with α (see remark 2.2.5(ii)) factors through the inclusion pseudo-functor Equiv(ℬ) → 2-Morph(ℬ) (notation of (2.3.25)); then define β̃ as the composition

𝒜 \underrightarrow{α̃} Equiv(ℬ) \underrightarrow{Υ̃} Equiv(ℬ)

(notation of (2.3.27)). Notice as well that the coherence constraints of Υ̃ are given by identities, and α̃ is strict (since F and G are strict), therefore the coherence constraints of β̃ are given as well by identities, so β̃ is strict. Furthermore, a simple inspection shows that s ∘ β̃ = G and t∘β̃ = F, where s and t are the restrictions to Equiv(𝒜) of the source and target pseudo-functors of remark 2.1.18(ii). We conclude that β̃ is the pseudo-functor associated as in remark 2.2.5(ii) with a pseudo-natural transformation β : G ⇒ F. We can describe the coherence constraint τ^{β} of β as follows. Let τ^{α} be the coherence constraint of α, which assigns to every 1-cell f : A → B of 𝒜 the oriented square τ_{f}^{α} as in remark 2.2.5(i). Recall that the pseudo-functor L of (2.3.25) assigns to every 1-cell g : X → Y of ℬ a link L(g) := (g^{†}, g, η^{g}, ε^{g}) : X → Y; then τ_{f}^{α} yields the 1-cell L(τ_{f}^{α}) : L(α_{A}) → L(α_{B}) in the 2-category wLink(ℬ). With this notation, we have β_{A} := (α_{A})^{†} for every A ∈ Ob(𝒜), and τ^{β} assigns to f the oriented square

τ_{f}^{β}  :  [figure: GA; β_{A}; Gf; Υ(L(τ_{f}^{α}))^{−1}; F A; F f; GB; β_{B}; F B.]

To conclude, it remains to check that the systems of 2-cells :

λ_{A} := (ε^{α_{A}})^{−1}  and μ_{A} := (η^{α_{A}})^{−1}  for every A ∈ Ob(𝒜)

amount to two invertible modifications

λ_{•} : 1_{F} ⇝ β ⊙ α μ_{•} : α ⊙ β ⇝ 1_{G}.

However, if τ^{α⊙β} denotes the coherence constraint of α ⊙ β, the compatibility condition for μ_{•} comes down to the identity :

(2.4.13)  (μ_{B} ∗ Gf) ⊙ τ_{f}^{α⊙β} = Gf ∗ μ_{A}  for every 1-cell f : A → B of 𝒜 .

Let us then consider the diagram :

[figure: GA; β_{A}; 1_{β_{A}}; F A; 1_{F f}; F A; F f; 1_{F f}; F A; α_{A}; F f; τ_{f}^{α}; GA; Gf; GA; β_{A}; μ_{A}; F A; F f; α_{A}; L(τ_{f}^{α})^{−1}; F B; α_{B}; λ_{B}; F B; α_{B}; 1_{α_{B}}; GB; GA; Gf; 1_{Gf}; GA; Gf; Gf; 1_{Gf}; GB; β_{B}; μ_{B}; F B; α_{B}; α_{B}; 1_{α_{B}}; GB; GB; GB; GB; GB; GB.]

We see that the composition of the squares of the top row equals τ_{f}^{α} ∗ β_{A}, and the composition of the squares of the middle row equals α_{B} ∗ τ_{f}^{β}. Hence the composition of the squares of the top and middle rows equals τ_{f}^{α⊙β}, and by further composing with the squares of the bottom row we get the left hand-side of (2.4.13). On the other hand, by proposition 2.1.9 we can also compose the squares column by column : then notice that the composition of the squares in the third column from the left equals the identity of α_{B} ∘ F f. Thus, we may disregard this column, and then notice that the composition of the squares of the second column equals the inverse of the composition of the squares of the fourth column. So we may disregard all columns except the first from the left; but the composition of the squares of the latter equals the right-hand side of (2.4.13), as required. The verification for λ_{•} is similar, and shall be left to the reader.

(a)⇒(b) is immediate from the definitions. □

Corollary 2.4.14. Let 𝒜 , ℬ be two 2-categories, F, F′ : 𝒜 → ℬ two pseudo-functors, and α : F ⇒ F′ a pseudo-natural equivalence. We have :

(i) For every 2-category 𝒜′ and every pseudo-functor G : 𝒜′ → 𝒜, the composition α ∗G : F ∘G ⇒ F′ ∘G is a pseudo-natural equivalence (notation of remark 2.2.5(iv)).

(ii) For every 2-category ℬ′ and every pseudo-functor H : ℬ → ℬ′, the composition

H ∗ α : H ∘ F ⇒ H ∘ F′ is a pseudo-natural equivalence. (iii) If F′ : 𝒜 → ℬ is any other pseudo-functor, and α′ : F′ ⇒ F′′ any other pseudo-

natural equivalence, then the composition α′ ⊙ α : F ⇒ F′′ is a pseudo-natural equivalence as well (notation of remark 2.2.5(iii)).

Proof. Assertions (i) and (ii) follow immediately from theorem 2.4.12 and lemma 2.4.11(ii). Assertion (iii) follows from theorem 2.4.12and lemma 2.1.4(ii) : details left to the reader. □

Definition 2.4.15. Let 𝒜, ℬ be two 2-categories, F : 𝒜 → ℬ, G : ℬ → 𝒜 two pseudofunctors. We say that G is a right 2-adjoint to F (resp. a strong right 2-adjoint to F) if there exists a pseudo-natural equivalence (resp. a pseudo-natural isomorphism) of pseudo-functors

ϑ : H_{ℬ} (F, 1_{ℬ} ) ⇒ H_{𝒜} (1_{𝒜} , G)

(definition 2.4.9(i)). In this case, we say that (F, G) is a 2-adjoint pair (resp. a strong 2-adjoint pair) of pseudo-functors, and ϑ is a 2-adjunction (resp. a strong 2-adjunction) for (F, G).

Remark 2.4.16. According to theorem 2.4.12and remark 2.2.5(i), the datum of a 2-adjunction as in definition 2.4.15is equivalent to that of :

• a system of equivalences of categories

ϑ_{AB} : ℬ(F A, B) → 𝒜 (A, GB) for every A ∈ Ob(𝒜) and B ∈ Ob(ℬ)

• and for every pair of 1-cells f : A′ → A in 𝒜, g : B → B′ in ℬ, a natural isomorphism of functors

[figure: ℬ(F A, B); ϑ_{AB}; ℬ(F f,g); τ_{(f,g)}^{ϑ}; 𝒜 (A, GB); 𝒜 (f,Gg); ℬ(F A′, B′); ϑ_{A′B′}; 𝒜 (A′, GB′)]

• such that, for every 2-cells β : f ⇒ f′ in 𝒜 and λ : g ⇒ g′ in ℬ, we have the identity

(2.4.17)  τ_{(f,g)}^{ϑ} ⊟ H_{ℬ} (F β, λ) = H_{𝒜} (β, Gλ) ⊟ τ_{(f′,g′)}^{ϑ}

• and for every composable pairs of 1-cells A′′ ⟶^{f′} A′ ⟶^{f} A in 𝒜, and B ⟶^{g} B′ ⟶^{g′} B′′ in ℬ, we have the identities

\begin{matrix}(τ_{(f′,g′)}^{ϑ} □ τ_{(f,g)}^{ϑ}) ⊟ H_{ℬ} (γ_{f′,f}^{F}, 1_{g′∘g}) = & H_{𝒜} (1_{f∘f′}, γ_{g,g′}^{G}) ⊟ τ_{(f∘f′,g′∘g)}^{ϑ} \\ H_{𝒜} (i_{A}, δ_{B}^{G}) ⊟ τ_{(1_{A},1_{B})}^{ϑ} = & 1_{ϑ_{AB}} ⊟ H_{ℬ} (δ_{A}^{F}, i_{B}) \end{matrix}

where (δ^{F} , γ^{F}) (resp. (δ^{G}, γ^{G})) denote by coherence constraint for F (resp. for G).

We wish to attach to every 2-adjunction suitable units and counits, as for usual adjoint pairs of functors; we shall see that the triangular identities will have to be replaced by certain invertible modifications. We begin with a few auxiliary lemmata :

Lemma 2.4.18. Let 𝒜,ℬ be any two 2-categories, F, F′ : 𝒜 → ℬ two strict pseudo-functors, β : F ⇒ F′ a pseudo-natural transformation. We have :

(i) F induces a strict pseudo-natural transformation (notation of example 2.2.7(ii)) :

H_{F} : H_{𝒜} ⇒ H_{ℬ} (F, F) (A, A′) ↦ (F_{AA′} : 𝒜 (A, A′) → ℬ(F A, F A′)).

(ii) The coherence constraint τ^{β} of β induces an invertible modification

H_{β} : H_{ℬ} (β, 1_{F′}) ⊙ H_{F′} ⇝ H_{ℬ} (1_{F} , β) ⊙ H_{F}

that assigns to every object (A_{1}, A_{2}) of 𝒜^{o} × 𝒜 the natural transformation

τ_{A_{1}A_{2}}^{β} : H_{ℬ} (β_{A_{1}}, 1_{F′A_{2}}) ∘ F′_{A_{1}A_{2}} ⇒ H_{ℬ} (1_{F A_{1}}, β_{A_{2}}) ∘ F_{A_{1}A_{2}}.

(iii) If F is fully faithful, H_{F} is a pseudo-natural equivalence.

Proof. Assertion (i) is straightforward, and (iii) follows immediately from theorem 2.4.12. (ii): For every 1-cell (f_{1}, f_{2}) : (A_{1}, A_{2}) → (A′_{1}, A′_{2}) of 𝒜^{o} × 𝒜 we consider the diagrams

[figure: 𝒜 (A_{1}, A_{2}); H_{F′}; 𝒜 (f_{1},f_{2}); 1; ℬ(F′A_{1}, F′A_{2}); ℬ(β_{A_{1}},1_{F′A_{2}}); ℬ(F′f_{1},F′f_{2}); ℬ((τ_{f_{1}}^{β})^{−1},1_{F′f_{2}}); ℬ(F A_{1}, F′A_{2}); ℬ(F f_{1},F′f_{2}); 𝒜 (A′_{1}, A′_{2}); H_{F′}; τ_{A′_{1}A′_{2}}^{β}; ℬ(F′A′_{1}, F′A′_{2}); ℬ(β_{A′_{1}},1_{F′A′_{2}}); ℬ(F A′_{1}, F′A′_{2}); 𝒜 (A′_{1}, A′_{2}); H_{F}; ℬ(F A′_{1}, F A′_{2}); ℬ(1_{F A′_{1}},β_{A_{2}}); ℬ(F A′_{1}, F′A′_{2})]

[figure: 𝒜 (A_{1}, A_{2}); H_{F′}; τ_{A_{1}A_{2}}^{β}; ℬ(F′A_{1}, F′A_{2}); ℬ(β_{A_{1}},1_{F′A_{2}}); ℬ(F A_{1}, F′A_{2}); 𝒜 (A_{1}, A_{2}); H_{F}; 𝒜 (f_{1},f_{2}); 1; ℬ(F A_{1}, F A_{2}); ℬ(1_{F A_{1}},β_{A_{2}}); ℬ(F f_{1},F f_{2}); ℬ(1_{F f_{1}},τ_{f_{2}}^{β}); ℬ(F A_{1}, F′A_{2}); ℬ(F f_{1},F′f_{2}); 𝒜 (A′_{1}, A′_{2}); H_{F}; ℬ(F A′_{1}, F A′_{2}); ℬ(1_{F A′_{1}},β_{A′_{2}}); ℬ(F A′_{1}, F′A′_{2})]

and notice that the coherence constraint of H_{ℬ} (β, 1_{F′}) ⊙ H_{F′} is given by the compositions of oriented squares on the top row of the first diagram (see example 2.2.6). Likewise, the coherence constraint of H_{ℬ} (1_{F} , β)⊙H_{F} is given by the composition of the oriented squares on the bottom row of the second diagram. Therefore, we come down to checking that the composition of the oriented squares of the first diagram equals the composition of the oriented squares of the second diagram. The latter translates as the identity :

τ_{f_{2}∘t∘f_{1}}^{β} ⊙ (F′(f_{2} ∘ t) ∗ τ_{f_{1}}^{β −1}) = (τ_{f_{2}}^{β} ∗ F (t ∘ f_{1})) ⊙ (F′f_{2} ∗ τ_{t}^{β} ∗ F f_{1})

for every 1-cell t : A_{1} → A_{2} in 𝒜. However, the coherence axiom for β yields the identities :

\begin{aligned}(τ_{t}^{β} ∗ F f_{1}) ⊙ (F′t ∗ τ_{f_{1}}^{β}) & = τ_{t∘f_{1}}^{β} \\ (τ_{f_{2}}^{β} ∗ F (t ∘ f_{1})) ⊙ (F′f_{2} ∗ τ_{t∘f_{1}}^{β}) & = τ_{f_{2}∘t∘f_{1}}^{β}. \end{aligned}

The sought identity is an immediate consequence. □

2.4.19. Conversely, consider now two strict pseudo-functors F, G : 𝒜 → ℬ, and a pseudonatural transformation λ : H_{𝒜} ⇒ H_{ℬ} (F, G) with coherence constraint τ_{•}. We set

λ_{A}^{∨} := λ_{(A,A)}(1_{A}) : F A → GA for every A ∈ Ob(𝒜)

and for every 1-cell f : A → A′ in 𝒜 we let

τ_{f}^{∨} : Gf ∘ λ_{A}^{∨} ⇒ λ_{A′}^{∨} ∘ F f

be the unique 2-cell of ℬ that fits into the commutative diagram

[figure: Gf ∘ λ_{A}^{∨}; τ_{f}^{∨}; λ_{A′}^{∨} ∘ F f; Gf ∘ λ_{A}^{∨} ∘ F 1_{A}; τ_{(1_{A},f),1_{A}}; λ_{(A,A′)}(f); G1_{A′} ∘ λ_{A′}^{∨} ∘ F f.; τ_{(f,1_{A′}),1_{A′}}]

Lemma 2.4.20. With the notation of (2.4.19), the following holds :

(i) The system of 1-cells (λ_{A}^{∨} | A ∈ Ob(𝒜)) defines a pseudo-natural transformation

λ^{∨} : F ⇒ G

with coherence constraint given by the system of 2-cells τ_{•}^{∨}. (ii) H_{F}^{∨} = 1_{F}. (iii) Let μ : H_{𝒜} ⇒ H_{ℬ} (F, G) be another pseudo-natural transformations, and Θ : λ ⇝ μ

any modification. Then we have a modification Θ^{∨} : λ^{∨} ⇝ μ^{∨} given by the rule :

A ↦ (Θ_{A}^{∨} := Θ_{(A,A),1_{A}} : λ_{A}^{∨} ⇒ μ_{A}^{∨}) for every A ∈ Ob(𝒜) .

Proof. (i): Let f, f′ : A → A′ be two 1-cells in 𝒜; the naturality of τ_{•}^{∨} amounts to the commutativity of the diagram

\xymatrix{Gf ∘ λ_{A}^{∨} \ar@{=>}[r]^{τ_{(1_{A},f),1_{A}}} \ar@{=>}[d]_{Gα∗λ_{A}^{∨}} & λ_{(A,A′)}(f) \ar@{=>}[d]_{λ_{(A,A′)}(α)} & λ_{A′}^{∨} ∘ F f \ar@{=>}[l]_{τ_{(f,1_{A′}),1_{A′}}} \ar@{=>}[d]^{λ_{A′}^{∨}∗F α} \\ Gf′ ∘ λ_{A}^{∨} \ar@{=>}[r]^{τ_{(1_{A},f′),1_{A}}} & λ_{(A,A′)}(f′) & λ_{A′}^{∨} ∘F f′ \ar@{=>}[l]_{τ_{(f′,1_{A′}),1_{A′}}}}

for every 2-cell α : f ⇒ f′ in 𝒜. However, the commutativity of the two square subdiagrams follows by applying the naturality of τ to the 2-cells (i_{A}, α) and (α, i_{A′}) of 𝒜^{o} × 𝒜.

Let us check the coherence axioms for τ_{•}^{∨}. First, since F and G are strict, we need to show :

τ_{1_{A}}^{∨} = 1_{λ_{A}^{∨}} for every A ∈ Ob(𝒜)

which follows by a simple inspection. Next, let A ⟶^{f} A′ ⟶^{g} A′′ be a composable pair of 1-cell of 𝒜; we need to show that

(τ_{g}^{∨} ∗ F f) ⊙ (Gg ∗ τ_{f}^{∨}) = τ_{g∘f}^{∨}.

Now, the coherence axiom for τ_{•} yields the identity

τ_{(1_{A},g),f} ⊙ (Gg ∗ τ_{(1_{A},f),1_{A}}) = τ_{(1_{A},g∘f),1_{A}}.

Hence we are reduced to showing that

(τ_{g}^{∨} ∗ F f) ⊙ (Gg ∗ τ_{(f,1_{A′}),1_{A′}}^{−1}) = τ_{(g∘f,1_{A′′}),1_{A′′}}^{−1} ⊙ τ_{(1_{A},g),f}.

Next, again by the coherence axiom for τ_{•} we see that

τ_{(g∘f,1_{A′′}),1_{A′′}} = τ_{(f,1_{A′′}),g} ⊙ (τ_{(g,1_{A′′}),1_{A′′}} ∗ F f).

Hence we are further reduced to checking the identity :

(2.4.21)  τ_{(f,1_{A′′}),g} ⊙ (τ_{(1_{A′},g),1_{A′}} ∗ F f) = τ_{(1_{A},g),f} ⊙ (Gg ∗ τ_{(f,1_{A′}),1_{A′}}).

However, applying twice the coherence axiom for τ_{•} to the identities in 𝒜^{o} × 𝒜 :

(f, 1_{A′′}) ∘ (1_{A′}, g) = (f, g) = (1_{A}, g) ∘ (f, 1_{A′})

we see that both sides of (2.4.21) equal τ_{(f,g),1_{A′}}.

(ii) follows by a direct inspection of the definitions.

(iii): The assertion follows from the commutativity of the diagram :

\xymatrix{Gf ∘ λ_{A}^{∨} \ar@{=>}[r]^{τ_{(1_{A},f),1_{A}}^{λ}} \ar@{=>}[d]_{Gf∗Θ_{A}^{∨}} & λ_{(A,A′)}(f) \ar@{=>}[d]_{Θ_{(A,A′),f}} & λ_{A′}^{∨} ∘ F f. \ar@{=>}[d]^{Θ_{A′}^{∨}∗F f} \ar@{=>}[l]_{τ_{(f,1_{A′}),1_{A′}}^{λ}} \\ Gf ∘ μ_{A}^{∨} \ar@{=>}[r]^{τ_{(1_{A},f),1_{A}}^{μ}} & μ_{(A,A′)}(f) & μ_{A′}^{∨} ∘ F f \ar@{=>}[l]_{τ_{(f,1_{A′}),1_{A′}}^{μ}}}

which in turn follows directly from the definition of Θ. □

2.4.22. Let now F : 𝒜 → ℬ be a pseudo-functor, and G : ℬ → 𝒜 a right 2-adjoint for F, and ϑ a 2-adjunction for the pair (F, G) as in definition 2.4.15. We suppose moreover that F and G are strict; by lemma 2.1.13(i) we may find a pseudo-natural equivalence

ψ : H_{𝒜} (1_{𝒜} , G) ⇒ H_{ℬ} (F, 1_{ℬ} )

as well as invertible modifications

Ξ : 1_{H_{ℬ} (F,1_{ℬ} )} ⇝ ψ ⊙ ϑ Θ : ϑ ⊙ ψ ⇝ 1_{H_{𝒜} (1_{𝒜} ,G)}

such that

(ψ ∗ Θ) ⊙ (Ξ ∗ ψ) = 1_{ψ} and (Θ ∗ ϑ) ⊙ (ϑ ∗ Ξ) = 1_{ϑ}.

To ease notation, we shall drop the subscripts when referring to ϑ_{AB}(h) or ϑ_{AB}(μ) for any 1-cell h : F A → B or 2-cell μ : h ⇒ h′ in ℬ, and write simply ϑ(h) and ϑ(μ). Likewise, we shall write simply ψ(k) and ψ(ν) for any 1-cell k : A → GB and any 2-cell ν : k ⇒ k′ in 𝒜. Now, for every A ∈ Ob(𝒜) and B ∈ Ob(ℬ) let us set

η_{A} := ϑ(1_{F A}) : A → GF A ε_{B} := ψ(1_{GB}) : F GB → B.

For every 1-cell f : A → A′ in 𝒜, and every 1-cell g : B → B′ in ℬ we let

τ_{f}^{η} : GF f ∘ η_{A} ⇒ η_{A′} ∘ f and τ_{g}^{ε} : g ∘ ε_{B} ⇒ ε_{B′} ∘ F Gg

be the unique 2-cells in 𝒜 and respectively in ℬ that make commute the diagrams :

\xymatrix{GF f ∘ η_{A} \ar@{=>}[dr]_{τ_{(1_{A},F f),1_{F A}}^{ϑ}} \ar@{=>}[rr]^{τ_{f}^{η}} &  & η_{A′} ∘ f \ar@{=>}[dl]^{τ_{(f,1_{F A′}),1_{F A′}}^{ϑ}} & g ∘ ε_{B} \ar@{=>}[rr]^{τ_{g}^{ε}} \ar@{=>}[dr]_{τ_{(1_{GB} ,g),1_{GB}}^{ψ}} &  & ε_{B′} ∘ F Gg \ar@{=>}[dl]^{τ_{(Gg,1_{B′}),1_{GB′}}^{ψ}} \\  & ϑ(F f) &  &  & ψ(Gg)}

where, as usual, τ^{ψ} denotes the coherence constraint of ψ.

Lemma 2.4.23. With the notation of (2.4.22), we have :

(i) The system (η_{A} | A ∈ Ob(𝒜)) defines a pseudo-natural transformation

η : 1_{𝒜} ⇒ GF

whose coherence constraint is given by the system of 2-cells τ_{•}^{η}. (ii) The system (ε_{B} | B ∈ Ob(ℬ)) defines a pseudo-natural transformation

ε : F G ⇒ 1_{ℬ}

whose coherence constraint is given by the system of 2-cells τ_{•}^{ε}.

Proof. (i): Define the pseudo-natural transformation H_{F} : H_{𝒜} ⇒ H_{ℬ} (F, F) as in lemma 2.4.18(i), and set

λ := (ϑ ∗ (1_{𝒜^{o}} × F)) ⊙ H_{F} : H_{𝒜} ⇒ H_{𝒜} (1_{𝒜} , G ∘ F).

Lemma 2.4.20(i) yields a pseudo-natural transformation

λ^{∨} : 1_{𝒜} ⇒ G ∘ F

and it is easily seen that λ_{A}^{∨} = η_{A} for every A ∈ Ob(𝒜). It remains to check that τ^{η} agrees with the coherence constraint τ^{λ^{∨}} of λ^{∨}. However, denote by τ^{λ}, τ^{H_{F}} and τ^{ϑ∗(1_{𝒜} ×F)} the coherence constraints of respectively λ, H_{F} and ϑ ∗ (1_{𝒜} × F); by inspecting the definitions we find that

τ_{(1_{A},f),1_{A}}^{λ} = (ϑ ∗ (1_{𝒜} × F))_{A,A′}(τ_{(1_{A},f),1_{A}}^{H_{F}}) ⊙ τ_{(1_{A},f),1_{F A}}^{ϑ∗(1_{𝒜} ×F)} = τ_{(1_{A},F f),1_{F A}}^{ϑ}

and likewise : τ_{(f,1_{A′}),1_{A′}}^{λ} = τ_{(f,1_{F A′}),1_{F A′}}^{ϑ}. The assertion follows.

(ii): A similar calculation shows that ε = ((ψ ∗ (G × 1_{ℬ} )) ⊙ H_{G})^{∨}. □

We now turn to the case of an arbitrary 2-adjoint pair of pseudo-functors :

Theorem 2.4.24. Let F : 𝒜 → ℬ and G : ℬ → 𝒜 be two pseudo-functors. We have : (i) If (F, G) is a 2-adjoint pair, there exist pseudo-natural transformations

η : 1_{𝒜} ⇒ GF ε : F G ⇒ 1_{ℬ}

related by a pair of invertible modifications

Σ : (G ∗ ε) ⊙ (η ∗ G) ⇝ 1_{G} Σ′ : (ε ∗ F) ⊙ (F ∗ η) ⇝ 1_{F} .

We call η a unit and ε a counit for the 2-adjunction ϑ, and Σ and Σ′ the triangular modifications associated with (η, ε).

(ii) Conversely, the existence of pseudo-natural transformations ε, η and invertible modifications Σ, Σ′ as in (i), implies that G is right 2-adjoint to F.

Proof. After replacing U by a larger universe, we may assume that 𝒜 and ℬ are small; in this case, let us remark that the existence of pseudo-natural transformations η and ε related by triangular modifications as in (i) means precisely that (F, G) is an adjoint pair of 1-cells of the 2-category \overline{2-Cat} of remark 2.2.16.

Claim 2.4.25. We may assume that F and G are unital pseudo-functors.

Proof of the claim. Let F^{u} and G^{u} be the unital pseudo-functors associated with F and G, with pseudo-natural isomorphisms α^{F} : F ⇒ F^{u} and α^{G} : G ⇒ G^{u}, as in proposition 2.4.3. If ϑ is a 2-adjunction for (F, G), we get a unique pseudo-natural equivalence ϑ^{∗} fitting into the commutative diagram :

\xymatrix{H_{ℬ} (F, 1_{ℬ} ) \ar@{=>}[d]_{H_{ℬ} (α^{F} ,i_{ℬ} )} \ar@{=>}[r]^{ϑ} & H_{𝒜} (1_{𝒜} , G) \ar@{=>}[d]^{H_{𝒜} (i_{𝒜} ,α^{G})} \\ H_{ℬ} (F^{u}, 1_{ℬ} ) \ar@{=>}[r]^{ϑ^{∗}} & H_{𝒜} (1_{𝒜} , G^{u})}

(notation of example 2.2.7(ii); here i_{𝒜} is the identity transformation of the pseudo-functor 1_{𝒜} , and likewise for i_{ℬ} ). Then ϑ^{∗} is a 2-adjunction for the pair (F^{u}, G^{u}). By the same token, from a given 2-adjunction for the pair (F^{u}, G^{u}) we easily deduce a 2-adjunction for (F, G). Also, (F, G) is an adjoint pair of 1-cells in \overline{2-Cat} if and only if the same holds for the pair (F^{u}, G^{u}) (remark 2.1.4(iv)). Thus, it suffices to show the theorem for F^{u} and G^{u}. ♢

Thus, henceforth we assume that F and G are unital. Next we consider the 2-categories 𝒜^{F,G} and ℬ^{G,F} of proposition 2.4.7, as well as the strict pseudo-functors F^{♭}, G^{♭}, π^{F,G} and π^{G,F}, the pseudo-functors σ^{F,G} and σ^{G,F}, and the isomorphisms of pseudo-functors

ψ^{F,G} : σ^{F,G} ∘ π^{F,G} →^{∼} 1_{𝒜^{F,G}} ψ^{G,F} : σ^{G,F} ∘ π^{G,F} →^{∼} 1_{ℬ^{G,F}} .

Claim 2.4.26. π^{F,G} and π^{G,F} are 2-equivalences.

Proof of the claim. Since π^{F,G} is the identity on objects, it suffices to check that it is fully faithful. The latter is clear, since π^{F,G} ∘ σ^{F,G} = 1_{𝒜} and σ^{F,G} ∘ π^{F,G} is isomorphic to 1_{𝒜^{F,G}} (details left to the reader). The same argument applies to π^{G,F}. ♢ Claim 2.4.27. We may assume that F and G are strict pseudo-functors.

Proof of the claim. Let ϑ : H_{ℬ} (F, 1_{ℬ} ) →^{∼} H_{𝒜} (1_{𝒜} , G) be a 2-adjunction for the pair (F, G). By claim 2.4.26and lemma 2.4.18(iii), π^{F,G} and π^{G,F} induce pseudo-natural equivalences

H_{π^{F,G}} : H_{𝒜^{F,G}} →^{∼} H_{𝒜} (π^{F,G}, π^{F,G}) and H_{π^{G,F}} : H_{ℬ^{G,F}} →^{∼} H_{ℬ} (π^{G,F} , π^{G,F}).

Then we may find a pseudo-natural equivalence ϑ^{♭} : H_{ℬ^{G,F}} (F^{♭}, 1_{ℬ^{G,F}} ) →^{∼} H_{𝒜^{F,G}} (1_{𝒜^{F,G}} , G^{♭}) fitting into the essentially commutative diagram of pseudo-functors :

\xymatrix{H_{ℬ^{G,F}} (F^{♭}, 1_{ℬ^{G,F}} ) \ar[d]_{H_{π^{G,F}} ∗(F^{♭o}×1_{ℬ^{G,F}} )} \ar[r]^{ϑ^{♭}} & H_{𝒜^{F,G}} (1_{𝒜^{F,G}} , G^{♭}) \ar[d]^{H_{π^{F,G}}∗(1_{𝒜^{F,G}} ×G^{♭})} \\ H_{ℬ} (π^{G,F} ∘ F^{♭}, π^{G,F}) \ar[d]_{H_{ℬ} ∗(ω^{F,G}×1_{π^{G,F}})} & H_{𝒜} (π^{F,G}, π^{F,G} ∘ G^{♭}) \\ H_{ℬ} (F ∘ π^{F,G}, π^{G,F}) \ar[r]^{ϑ∗(π^{F,G}×π^{G,F})} & H_{𝒜} (π^{F,G}, G ∘ π^{G,F}). \ar[u]_{H_{𝒜} ∗(1_{π^{F,G}}×ω^{G,F})}}

Conversely, if ϑ^{♭} is a given 2-adjunction for the pair (F^{♭}, G^{♭}), then we deduce a pseudo-natural equivalence ϑ fitting into the essentially commutative diagram :

[figure: H_{ℬ} (F, 1_{ℬ} ); ϑ; H_{𝒜} (1_{𝒜} , G); H_{ℬ^{G,F}} (σ^{G,F} ∘ F, σ^{G,F}); H_{π^{G,F}} ∗((σ^{G,F} ∘F)^{o}×σ^{G,F}); H_{𝒜^{F,G}} (σ^{F,G}, σ^{F,G} ∘ G); H_{π^{F,G}}∗(σ^{F,G}×(σ^{F,G}∘G)); H_{ℬ^{G,F}} (F^{♭} ∘ σ^{F,G}, σ^{G,F}); ϑ^{♭}∗(σ^{F,G}×σ^{G,F}); H_{𝒜^{F,G}} (σ^{F,G}, G^{♭} ∘ σ^{G,F}).]

Thus (F, G) is a 2-adjoint pair if and only if the same holds for (F^{♭}, G^{♭}).

Likewise, if η : 1_{𝒜} ⇒ GF and η : F G ⇒ 1_{ℬ} are given pseudo-natural transformations fulfilling the triangular identities (G ∗ ε) ⊙ (η ∗ G) = 1_{G} and (ε ∗ F) ⊙ (F ∗ η) = 1_{F} in \overline{2-Cat}, there follow pseudo-natural transformations :

[figure: η^{♭} : 1_{𝒜^{F,G}}; ψ^{F,G −1}; σ^{F,G} ∘ π^{F,G}; σ^{F,G}∗η∗π^{F,G}; σ^{F,G} ∘ GF ∘ π^{F,G}; G^{♭}F^{♭}; G^{♭}F^{♭} ∘ σ^{F,G} ∘ π^{F,G}; G^{♭}F^{♭}∗ψ^{F,G}; G^{♭} ∘ σ^{G,F} ∘ F ∘ π^{F,G}; ε^{♭} : F^{♭}G^{♭}; F^{♭}G^{♭}∗ψ^{G,F −1}; F^{♭}G^{♭} ∘ σ^{G,F} ∘ π^{G,F}; F^{♭} ∘ σ^{F,G} ∘ G ∘ π^{G,F}; 1_{ℬ^{F,G}}; σ^{G,F} ∘ π^{G,F}; ψ^{G,F}; σ^{G,F} ∘ F G ∘ π^{G,F}; σ^{G,F} ∗ε∗π^{G,F}]

Set X := G^{♭} ∗ ψ^{G,F} and Y := ψ^{F,G} ∗ G^{♭}. By proposition 2.4.7(ii) we have :

Z := G^{♭}F^{♭} ∗ ((G^{♭} ∗ ψ^{G,F})^{−1} ⊙ (ψ^{F,G} ∗ G^{♭})) = (G^{♭}F^{♭}σ^{F,G} ∗ ω^{G,F})^{−1} = (σ^{F,G}GF ∗ ω^{G,F})^{−1}

whence : Z ⊙ (σ^{F,G} ∗ η ∗ π^{F,G} ∗ G^{♭}) = σ^{F,G} ∗ ((η ∗ Gπ^{G,F}) ⊙ ω^{G,F −1}). We deduce :

(G^{♭} ∗ ε^{♭}) ⊙ (η^{♭} ∗ G^{♭}) = X ⊙ (G^{♭}σ^{G,F} ∗ ε ∗ π^{G,F}) ⊙ Z ⊙ (σ^{F,G} ∗ η ∗ π^{F,G} ∗ G^{♭}) ⊙ Y^{−1}

= X ⊙(G^{♭}σ^{G,F} ∗ ε ∗ π^{G,F}) ⊙ (σ^{F,G} ∗ ((η ∗ Gπ^{G,F}) ⊙ ω^{G,F −1})) ⊙ Y^{−1}

= X ⊙ (σ^{F,G}G ∗ ε ∗ π^{G,F}) ⊙ (σ^{F,G} ∗ ((η ∗ Gπ^{G,F}) ⊙ ω^{G,F −1})) ⊙ Y^{−1}

= X ⊙ (σ^{F,G} ∗ ω^{G,F −1}) ⊙ Y^{−1}
= 1_{G^{♭}}

where the last identity follows again from proposition 2.4.7(ii). A similar computation that we leave to the reader shows that (ε^{♭} ∗F^{♭})⊙(F^{♭} ∗η^{♭}) = 1_{F^{♭}} in \overline{2-Cat}. Conversely, if given pseudonatural transformations η^{♭} : 1_{𝒜^{F,G}} ⇒ G^{♭}F^{♭} and ε^{♭} : F^{♭}G^{♭} ⇒ 1_{ℬ^{G,F}} satisfy the triangular identities, then we get pseudo-natural transformations

η := π^{F,G} ∗ η^{♭} ∗ σ^{F,G} : 1_{𝒜} ⇒ GF ε := π^{G,F} ∗ ε^{♭} ∗ σ^{G,F} : F G ⇒ 1_{ℬ}

that also fulfill the corresponding triangular identities (details left to the reader). Thus, it suffices to prove the theorem for F^{♭} and G^{♭}. ♢

Thus, henceforth we assume that F and G are strict. In order to exhibit Σ, we mimick the discussion of (1.1.13) : first, for every B ∈ Ob(ℬ) we have an invertible 2-cell

[figure: GB; (η∗G)_{B}; ϑ⊙ψ(1_{GB}); τ_{(1_{GB} ,ε_{B}),1_{F GB}}^{ϑ}; GF GB; (G∗ε)_{B}; GB]

where τ^{ϑ} denotes the coherence constraint of ϑ, and ψ is the pseudo-natural equivalence as in (2.4.22) used to define ε. Then, with the notation of (2.4.22), we set

Σ_{B} := Θ_{GB,B} ⊙ τ_{(1_{F GB},ε_{B}),1_{F GB}}^{ϑ} for every B ∈ Ob(ℬ) .

We have to check that the system (Σ_{B} | B ∈ Ob(ℬ)) is a modification (G ∗ ε) ⊙ (η ∗ G) ⇝ 1_{G}. However, set (notation of remark 2.2.10(iv,v) and lemma 2.4.18(i))

μ := ((ϑ ⊙ ψ) ∗ (G × 1_{ℬ} )) ⊙ H_{G} and Θ′ := (Θ ∘ (G × 1_{ℬ} )) ∗ H_{G} : μ ⇝ H_{G}.

Unwinding the definitions, we see that Θ′_{B,B′} = Θ_{GB,B′} ∗ G_{BB′} for every (B, B′) ∈ Ob(ℬ^{o} × ℬ); according to lemma 2.4.20(ii,iii) we then obtain an invertible modification

Θ^{′∨} : μ^{∨} ⇝ 1_{G} B ↦ Θ_{GB,B}.

We are thus reduced to showing that the system (τ_{(1_{GB},ε_{B}),1_{F GB}}^{ϑ} | B ∈ Ob(ℬ)) yields an invertible modification

Θ′′ : (G ∗ ε) ⊙ (η ∗ G) ⇝ μ^{∨}

since in this case we shall have Σ = Θ^{′∨} ⊙ Θ′′. However, by unwinding the definitions we see that the coherence constraint of μ^{∨} is the system of 2-cells

τ_{g}^{μ} := (τ_{(Gg,1_{B′}),ε_{B′}}^{ϑ})^{−1} ⊙ ϑ(τ_{g}^{ε}) ⊙ τ_{(1_{GB},g),ε_{B}}^{ϑ} for every 1-cell g : B → B′ of ℬ

whereas that of (G ∗ ε) ⊙ (η ∗ G) is given by the system of 2-cells

τ_{g}^{(G∗ε)⊙(η∗G)} := (G(ε_{B′}) ∗ τ_{Gg}^{η}) ⊙ (G(τ_{g}^{ε}) ∗ η_{GB}) for every g : B → B′

and we need to show the identities :

(τ_{(1_{GB′},ε_{B′}),1_{F GB′}}^{ϑ} ∗ Gg) ⊙ τ_{g}^{(G∗ε)⊙(η∗G)} = X := τ_{g}^{μ} ⊙ (Gg ∗ τ_{(1_{F GB},ε_{B}),1_{F GB}}^{ϑ})

for every 1-cell g : B → B′ of ℬ. We compute :

\begin{matrix}X = & (τ_{(Gg,1_{B′}),ε_{B′}}^{ϑ})^{−1} ⊙ ϑ(τ_{g}^{ε}) ⊙ τ_{(1_{GB},g∘ε_{B}),1_{F GB}}^{ϑ} \\ = & (τ_{(Gg,1_{B′}),ε_{B′}}^{ϑ})^{−1} ⊙ τ_{(1_{GB},ε_{B′}∘F Gg),1_{F GB}}^{ϑ} ⊙ (G(τ_{g}^{ε}) ∗ η_{GB}) \end{matrix}

where the first equality follows from the coherence axioms for τ^{ϑ}, and the second follows from (2.4.17). So, we are further reduced to checking the identities :

τ_{(Gg,1_{B′}),ε_{B′}}^{ϑ} ⊙ (τ_{(1_{GB′},ε_{B′}),1_{F GB′}}^{ϑ} ∗ Gg) ⊙ (G(ε_{B′}) ∗ τ_{Gg}^{η}) = τ_{(1_{GB},ε_{B′}∘F Gg),1_{F GB}}^{ϑ}.

However, by applying again the coherence axiom for τ^{ϑ} we see that

\begin{matrix}τ_{(1_{GB},ε_{B′}∘F Gg),1_{F GB}}^{ϑ} = & τ_{1_{GB},ε_{B′},F Gg}^{ϑ} ⊙ (G(ε_{B′}) ∗ τ_{(1_{GB},F Gg),1_{F GB}}^{ϑ}) \\ τ_{(Gg,1_{B′}),ε_{B′}}^{ϑ} ⊙ (τ_{(1_{GB′},ε_{B′}),1_{F GB′}}^{ϑ} ∗ Gg) = & τ_{(Gg,ε_{B′}),1_{F GB′}}^{ϑ} \end{matrix}

so we are further reduced to showing :

τ_{(Gg,ε_{B′}),1_{F GB′}}^{ϑ} = τ_{1_{GB},ε_{B′},F Gg}^{ϑ} ⊙ (G(ε_{B′}) ∗ τ_{(Gg,1_{F GB′}),1_{F GB′}}^{ϑ}).

But the latter follows by yet another application of the coherence axioms for τ^{ϑ}. This concludes the construction of Σ. Concerning Σ′, notice that for every A ∈ Ob(𝒜) and B ∈ Ob(ℬ)

𝒜^{o}(G^{o}B^{o}, A^{o}) = 𝒜 (A, GB) and ℬ^{o}(B^{o}, F^{o}A^{o}) = ℬ(F A, B)

and ψ can thus be regarded as a system of equivalences of categories ψ_{B^{o}A^{o}}^{o} : 𝒜^{o}(G^{o}B^{o}, A^{o}) → ℬ^{o}(B^{o}, F^{o}A^{o}). The latter amounts then to a pseudo-natural equivalence

ψ^{o} : 𝒜^{o}(G^{o}, 1_{𝒜^{o}} ) ⇒ ℬ^{o}(1_{ℬ^{o}} , F^{o})

with coherence constraint given by the system of isomorphisms of functors τ_{(g^{o},f^{o})}^{ψ^{o}} := τ_{(f,g)}^{ψ}, for every 2-cell f of 𝒜 and g of ℬ. Likewise, we may define the 2-adjunction ϑ^{o} for the pair (G^{o}, F^{o}), and a simple inspection shows that performing the foregoing constructions on this new pair of pseudo-natural equivalences amounts to swapping the roles of ε and η; then the modification Σ^{o} for the new pair of unit and counit will give the sought Σ′ for the original pair.

(ii): Given such η and ε, we set

\begin{matrix}ϑ := & H_{𝒜} (η, 1_{G}) ⊙ (H_{G} ∗ (F^{o} × 1_{ℬ} )) : H_{ℬ} (F, 1_{ℬ} ) ⇒ H_{𝒜} (1_{𝒜} , G) \\ ψ := & H_{ℬ} (1_{F} , ε) ⊙ (H_{F} ∗ (1_{𝒜}^{o} × G)) : H_{𝒜} (1_{𝒜} , G) ⇒ H_{ℬ} (F, 1_{ℬ} ). \end{matrix}

Now, example 2.2.15(ii) yields the following identity in \overline{2-Cat} :

Ξ : (H_{F} ∗ (1_{𝒜}^{o} × G)) ⊙ H_{𝒜} (η, 1_{G}) = (H_{ℬ} (F, F) ∗ (η^{o} × 1_{G})) ⊙ (H_{F} ∗ ((GF)^{o} × G))

and on the other hand, it is easily seen that

\begin{matrix}(H_{F} ∗((GF)^{o} × G)) ⊙ (H_{G}∗(F^{o} × 1_{ℬ} ))=H_{F G}∗(F^{o} × 1_{ℬ} ) : & H_{ℬ} (F, 1_{ℬ} )⇒H_{ℬ} (F GF, F G) \\ (H_{ℬ} (1_{F} , ε) ⊙ (H_{ℬ} (F, F) ∗ (η^{o}×1_{G})) = H_{ℬ} (F ∗ η, ε) : & H_{ℬ} (F GF, F G)⇒H_{ℬ} (F, 1_{ℬ} ) \end{matrix}

whence the identity in \overline{2-Cat} :

ψ⊙ϑ = H_{ℬ} (F ∗η, ε)⊙(H_{F G}∗(F^{o}×1_{ℬ} )) = H_{ℬ} (F ∗η, i_{ℬ} )⊙((H_{ℬ} (1_{F G}, ε)⊙H_{F G})∗(F^{o}×1_{ℬ} )).

But we have H_{ℬ} (1_{F G}, ε) ⊙ H_{F G} = H_{ℬ} (ε, i_{ℬ} ) in \overline{2-Cat}, by lemma 2.4.18(ii), so that :

ψ ⊙ ϑ = H_{ℬ} (F ∗ η, i_{ℬ} ) ⊙ (H_{ℬ} (ε, i_{ℬ} ) ∗ (F^{o} × 1_{ℬ} ))
= H_{ℬ} (F ∗ η, i_{ℬ} ) ⊙ H_{ℬ} (ε ∗ F, i_{ℬ} )
= H_{ℬ} ((ε ∗ F) ⊙ (F ∗ η), i_{ℬ} )
= H_{ℬ} (1_{F} , i_{ℬ} )
= 1_{H_{ℬ} (F,1_{ℬ} )}.

Arguing likewise, we obtain as well the identity ϑ ⊙ ψ = 1_{H_{𝒜} (1_{𝒜} ,G)} in \overline{2-Cat} : the details shall be left to the reader. Summing up, we have shown that ϑ is an equivalence in the 2-category PsFun(𝒜^{o} × ℬ, Cat) with quasi-inverse ψ, and the proof of the theorem is concluded. □

Remark 2.4.28. (i) Let (F : 𝒜 → ℬ, G : ℬ → 𝒜) be a 2-adjoint pair of pseudo-functors, and denote by η and ε the unit and the counit of a 2-adjunction for this pair, and Σ,Σ′ the associated triangular modifications as in theorem 2.4.24. There follow pseudo-natural transformations

η^{o} : G^{o}F^{o} ⇒ 1_{𝒜^{o}} ε^{o} : 1_{ℬ^{o}} ⇒ F^{o}G^{o}

and invertible modifications :

Σ^{o} : (η^{o} ∗ G^{o}) ⊙ (G^{o} ∗ ε^{o}) ⇝ 1_{G^{o}} Σ^{′o} : (F^{o} ∗ η^{o}) ⊙ (ε^{o} ∗ F^{o}) ⇝ 1_{F^{o}}.

In light of theorem 2.4.24, it follows that (G^{o}, F^{o}) is a 2-adjoint pair of pseudo-functors, and ε^{o} and η^{o} are respectively a unit and a counit of a 2-adjunction for this pair.

(ii) Likewise, we see that (^{o}F,^{o}G) is a 2-adjoint pair of pseudo-functors, and ^{o}η and ^{o}ε are the unit and respectively the counit of a 2-adjunction for this pair.

(iii) Moreover, the right 2-adjoint G of F is unique up to pseudo-natural equivalence of pseudo-functors. Indeed, we have already observed that the pair (F, G) is 2-adjoint if and only if it is an adjoint pair of 1-cells of the 2-category U′-\overline{2-Cat} (for a suitable universe U′); on the other hand, the pseudo-natural equivalences of pseudo-functors are precisely the invertible 2-cells of U′-\overline{2-Cat}, hence the assertion follows from remark 2.1.4(v).

Corollary 2.4.29. Let F : 𝒜 → ℬ be a pseudo-functor and G : ℬ → 𝒜 a right 2-adjoint for F. Let also η : 1_{𝒜} ⇒ GF and ε : F G ⇒ 1_{ℬ} be the unit and counit of a 2-adjunction ϑ for the pair (F, G). Then the following conditions are equivalent :

(a) F (resp. G) is fully faithful.

(b) η (resp. ε) is a pseudo-natural equivalence.

(c) There exists a pseudo-natural equivalence η′ : 1_{𝒜} →^{∼} GF (resp. ε′ : F G →^{∼} 1_{ℬ} ). Moreover, if (c) holds, there exists a pseudo-natural transformation ε′ (resp. η′) with invertible modifications (G ∗ ε′) ⊙ (η′ ∗ G) ⇝ 1_{G} and (ε′ ∗ F) ⊙ (F ∗ η′) ⇝ 1_{F}.

Proof. By inspecting the construction of η and ε, we are easily reduced to the case where 𝒜 and ℬ are small and F and G are strict. Also, by remark 2.4.28(i) it suffices to check the assertions for F, η and η′. Suppose now that (a) holds for F. Then the composition

λ_{A′A} : 𝒜 (A′, A) ⟶^{F_{A′A}} ℬ(F A′, F A) ⟶^{ϑ_{A′,F A}} 𝒜 (A′, GF A) for every A, A′ ∈ Ob(𝒜)

is an equivalence of categories, and η_{A} = λ_{AA}(1_{A}) for every A ∈ Ob(𝒜). Taking A′ := GF A, we find a 1-cell f : GF A → A with an invertible 2-cell λ_{GF A,A}(f) →^{∼} 1_{GF A}. Moreover, as explained in the proof of lemma 2.4.23(i), the rule : (A′, A) ↦ λ_{A′A} is pseudo-natural in both A and A′; thus, for every 1-cell h : A′ → A of 𝒜, the coherence constraint τ^{λ} of λ yields an isomorphism of functors :

τ_{(h,1_{A})}^{λ} : 𝒜 (h, 1_{GF A}) ∘ λ_{AA} →^{∼} λ_{GF A,A} ∘ 𝒜 (h, 1_{A})

and especially, we get as well an invertible 2-cell η_{A} ∘ f →^{∼} λ_{GF A,A}(f); summing up, we get an invertible 2-cell η_{A} ∘ f →^{∼} 1_{GF A}. On the other hand, the invertible modification (ε ∗ F) ⊙ (F ∗ η) ⇝ 1_{F} provided by theorem 2.4.24(i) yields an invertible 2-cell ε_{F A} ∘ F η_{A} →^{∼} 1_{F A}. We conclude that F η_{A} is an equivalence, and then the same holds for η_{A}, by virtue of lemma 2.4.11(iii). Combining with theorem 2.4.12, we see that (b) holds for η.

Next, obviously (b)⇒(c). Suppose then that η′ : 1_{𝒜} →^{∼} GF is a pseudo-natural equivalence, and denote also by i_{𝒜^{o}} the identity automorphism of 1_{𝒜^{o}} ; we choose a pseudo-natural equivalence ψ : H_{𝒜} (1_{𝒜} , G) →^{∼} H_{ℬ} (F, 1_{ℬ} ) and we set

ξ := (ψ ∗ (1_{𝒜^{o}} × F)) ⊙ (H_{𝒜} ∗ (i_{𝒜^{o}} × η′)) : H_{𝒜} →^{∼} H_{ℬ} (F, F).

Since ξ is pseudo-natural, for every 1-cell f : A′ → A in 𝒜, the coherence constraint τ^{ξ} of ξ yields invertible 2-cells :

(2.4.30)  F f ∘ ξ_{A′A′}(1_{A′}) ⟶^{τ_{(f,1_{A}),1_{A′}}^{ξ}} ξ_{A′A}(f) ⟵^{τ_{(1_{A′} ,f),1_{A}}^{ξ}} ξ_{AA}(1_{A}) ∘ F f.

Moreover, since ξ_{AA} : 𝒜 (A, A) →^{∼} ℬ(F A, F A) is an equivalence for every A ∈ Ob(𝒜), there exists a 1-cell h : A → A with an isomorphism ξ_{AA}(h) →^{∼} 1_{A}. Taking A′ = A and f := h in (2.4.30), we conclude that the 1-cell g_{A} := ξ_{AA}(1_{A}) is an equivalence in 𝒜 for every A ∈ Ob(𝒜). On the other hand, the naturality of τ^{ξ} easily implies that the rule :

(f : A′ → A) ↦ τ_{(f,1_{A}),1_{A}}^{ξ} : g_{A} ∘ F f →^{∼} ξ_{A′A}(f)

defines an isomorphism of functors 𝒜 (1_{A}, g_{A}) →^{∼} ξ_{A′A}. But since g_{A} is an equivalence, 𝒜 (1_{A}, g_{A}) is an equivalence of categories, and the same holds for ξ_{A′A}. It follows that the rule : f ↦ F f yields an equivalence F_{A′A} : 𝒜 (A′, A) →^{∼} ℬ(F A′, F A), i.e. (a) holds for F.

Lastly, for a given η′ as in (c), there exists a pseudo-natural equivalence ω : 1_{𝒜} →^{∼} 1_{𝒜} such that η ⊙ ω = η′ in \overline{2-Cat}. We set ε′ := ε ⊙ (F ∗ ω^{−1} ∗ G) : F G ⇒ 1_{ℬ} . We compute in \overline{2-Cat} :

(G ∗ ε′) ⊙ (η′ ∗ G) = (G ∗ ε) ⊙ (GF ∗ ω^{−1} ∗ G) ⊙ (η′ ∗ G)
= (G ∗ ε) ⊙ (GF ∗ ω^{−1} ∗ G) ⊙ (η ∗ G) ⊙ (ω ∗ G)
= (G ∗ ε) ⊙ (η ∗ G) ⊙ (ω^{−1} ∗ G) ⊙ (ω ∗ G)
= (G ∗ ε) ⊙ (η ∗ G)
= 1_{G}.

Likewise we get (ε′ ∗ F) ⊙ (F ∗ η′) = 1_{F} in \overline{2-Cat} : the details shall be left to the reader. □

Remark 2.4.31. Let F : 𝒜 → ℬ be a strict and strong 2-equivalence between two 2categories. Then there exists a strict and strong 2-equivalence G : ℬ → 𝒜 with strict pseudonatural isomorphisms η : 1_{ℬ} →^{∼} F ∘ G and ε : G ∘ F →^{∼} 1_{𝒜} . Indeed, for every B ∈ Ob(ℬ) let us pick GB ∈ Ob(𝒜) with an isomorphism η_{B} : B →^{∼} F GB; for every B, B′ ∈ Ob(ℬ) we then get an isomorphism of categories

𝒜 (GB, GB′) ⟶^{F_{GB,GB′}} ℬ(F GB, F GB′) ⟶^{ℬ(η_{B},η_{B′}^{−1})} ℬ(B, B′)

where ℬ(η_{B}, η_{B′}^{−1}) denotes the functor given by the rule : φ ↦ η_{B′}^{−1} ∘ φ ∘ η_{B} for every 1cell φ : F GB → F GB′ of ℬ, and β ↦ η_{B′}^{−1} ∗ β ∗ η_{B} for every 2-cell β between 1-cells φ, φ′ : F GB → F GB′. We let G_{B,B′} : ℬ(B, B′) → 𝒜 (GB, GB′) be the inverse of this isomorphism. Then the sought G is given by the rules : B ↦ GB for every B ∈ Ob(ℬ), and (B, B′) ↦ G_{B,B′} for every pair (B, B′) of objects of ℬ : the straightforward verification is left to the reader. It is also clear that the rule : B ↦ η_{B} for every B ∈ Ob(ℬ) defines a strict pseudo-natural isomorphism η. Lastly, we define ε by the rule : A ↦ F_{GF A,A}^{−1}(η_{F A}^{−1}) for every A ∈ Ob(𝒜). The following corollary extends this simple observation to every 2-equivalence.

Corollary 2.4.32. Let 𝒜 , ℬ be two 2-categories, F : 𝒜 → ℬ a pseudo-functor. We have :

(i) The following conditions are equivalent : (a) F is a 2-equivalence.

(b) There exists a pseudo-functor L : ℬ → 𝒜 with two pseudo-natural equivalences

η^{∗} : 1_{ℬ} ⇒ F ∘ L  and  ε^{∗} : L ∘ F ⇒ 1_{𝒜} .

We say that L is a pseudo-inverse for F. (ii) Suppose that F is a 2-equivalence. Then, for every L and η^{∗} as in (i) there exist a

pseudo-natural equivalence ε^{∗} : L ∘ F ⇒ 1_{𝒜} and invertible modifications

Θ : (F ∗ ε^{∗}) ⊙ (η^{∗} ∗ F) ⇝ 1_{F} Θ′ : (ε^{∗} ∗ L) ⊙ (L ∗ η^{∗}) ⇝ 1_{L}.

Proof. (i.b)⇒(i.a): The coherence constraint of ε^{∗} gives an isomorphism of functors

H_{𝒜} (1_{LF A}, ε_{A′}^{∗}) ∘ L_{F A,F A′} ∘ F_{AA′} ⇒ H_{𝒜} (ε_{A}^{∗}, 1_{A′}) for every A, A′ ∈ Ob(𝒜) .

However, it is easily seen that both H_{𝒜} (1_{LF A}, ε_{A′}) and H_{𝒜} (ε_{A}, 1_{A′}) are equivalences of categories (cp. the proof of lemma 2.1.13), hence the same holds for L_{F A,F A′} ∘ F_{AA′}. Arguing likewise with the coherence constraint for η^{∗}, we see as well that F_{LB,LB′} ∘ L_{BB′} is an equivalence for every B, B′ ∈ Ob(ℬ). It follows easily that F_{AA′} and L_{BB′} are faithful for every such A, A′ and B, B′; especially, L_{F A,F A′} and F_{LB,LB′} are both faithful, and therefore F_{AA′} and L_{BB′} must also be full. Lastly, let f : F A → F A′ be any 1-cell of ℬ; by the foregoing there exists a 1-cell g : A → A′ with an invertible 2-cell b : LF g ⇒ Lf, and since L_{F A,F A′} is fully faithful, we have b = Lc for an invertible 2-cell c : F g ⇒ f, which shows that F_{AA′} is essentially surjective, so it is an equivalence of categories. Furthermore, theorem 2.4.12says that the 1-cell η_{B}^{∗} : B → F LB is an equivalence for every B ∈ Ob(ℬ), so (i.a) holds.

(i.b)⇒(i.a): Without loss of generality, we may assume that both 𝒜 and ℬ are small 2categories. Now, consider the category 𝒜^{F} and the pseudo-functors F^{♭} : 𝒜^{F} → ℬ, π^{F} : 𝒜^{F} → 𝒜 and σ^{F} : 𝒜 → 𝒜^{F} provided by proposition 2.4.6; we remark that π^{F} is a 2equivalence, and σ^{F} is its pseudo-inverse : indeed, we have π^{F} ∘σ^{F} = 1_{𝒜} , and a pseudo-natural equivalence μ^{F} : σ^{F} ∘ π^{F} →^{∼} 1_{𝒜} , so the assertion follows from the foregoing. It is then clear that F is a 2-equivalence if and only if the same holds for F ∘ π^{F}, if and only if the same holds for F^{♭}. Now, suppose we have found a pseudo-functor L′ : ℬ → 𝒜^{F}, and pseudo-natural equivalences η^{∗∗} : 1_{ℬ} →^{∼} F^{♭} ∘ L′, ε^{∗∗} : L′ ∘ F^{♭} →^{∼} 1_{𝒜^{F}} ; then set L := π^{F} ∘ L′, and notice that we get pseudo-natural equivalences

(F^{♭} ∗ μ^{F} ∗ L′)^{−1} ⊙ η^{∗∗} : 1_{ℬ} →^{∼} F ∘ L π^{F} ∗ ε^{∗∗} ∗ σ^{F} : L ∘ F →^{∼} 1_{𝒜} .

We may thus replace F by F^{♭}, and assume from start that F is strict. Next, consider the pseudonatural transformation H_{F} : H_{𝒜} ⇒ H_{ℬ} (F, F) assigned to 1_{F} : F ⇒ F, as in lemma 2.4.18(i). Notice that our assumption on F means that H_{F,(A,A′)} is an equivalence in Cat, for every object (A, A′) of 𝒜^{o} × 𝒜, so theorem 2.4.12 implies that H_{F} is a pseudo-natural equivalence of pseudo-functors. We may therefore find a pseudo-natural equivalence G : H_{ℬ} (F, F) ⇒ H_{𝒜} , with coherence constraint τ^{G}, and an adjunction :

η : 1_{H_{𝒜}} ⇝ G ⊙ H_{F} ε : H_{F} ⊙ G ⇝ 1_{H_{ℬ} ∘(F,F)}

for the pair (H_{F} , G). Explicitly, G attaches to every A, A′ ∈ Ob(𝒜) a functor

G_{AA′} : ℬ(F A, F A′) → 𝒜 (A, A′)

which is an equivalence of categories, and η and ε are given by natural isomorphisms of functors

η_{AA′} : 1_{𝒜 (A,A′)} ⇒ G_{AA′} ∘ F_{AA′} ε_{AA′} : F_{AA′} ∘ G_{AA′} ⇒ 1_{ℬ(F A,F A′)} .

On the other hand, by assumption, for every B ∈ Ob(ℬ) we may find LB ∈ Ob(𝒜) with an equivalence f_{B} : B → F LB; then lemma 2.1.13(i) says that we may also find a 1-cell g_{B} : F LB → B and invertible 2-cells

η′_{B} : 1_{F LB} ⇒ f_{B} ∘ g_{B} ε′_{B} : g_{B} ∘ f_{B} ⇒ 1_{B}

that form an adjunction for the pair (g_{B}, f_{B}). To ease notation, we set T_{BB′} := ℬ(g_{B}, f_{B′}) for every B, B′ ∈ Ob(ℬ) and we define

L_{BB′} := G_{LB,LB′} ∘ T_{BB′} : ℬ(B, B′) → 𝒜 (LB, LB′).

We shall henceforth drop the subscripts from the notation for F, G, L, T, η and ε. We claim that L is a pseudo-functor ℬ → 𝒜, with coherence constraint given by the system of 2-cells in 𝒜

\begin{matrix}γ_{h,k}^{L} := & L(k ∗ ε′_{B′} ∗ h) ⊙ G(ε_{T k} ∗ ε_{T h}) ⊙ η_{Lk∘Lh} : Lk ∘ Lh ⇒ L(k ∘ h) \\ δ_{B}^{L} := & G(η′_{B}) ⊙ η_{1_{LB}} : 1_{LB} ⇒ L(1_{B}) \end{matrix}

for every B, B′, B′′ ∈ Ob(ℬ) and every pair of 1-cells h : B → B′ and k : B′ → B′′ in ℬ. Indeed, the naturality of the rule (h, k) ↦ γ_{h,k}^{L} is clear from the definition; to check the composition axiom, we consider another 1-cell l : B′′ → B′′′ in ℬ, and we need to show that

γ_{k∘h,l}^{L} ⊙ (Ll ∗ γ_{h,k}^{L}) = γ_{h,l∘k}^{L} ⊙ (γ_{l,k}^{L} ∗ Lh).

However, the naturality of η yields the identities :

\begin{matrix}η_{Ll∘L(k∘h)} ⊙ (Ll ∗ γ_{h,k}^{L}) = & GF (Ll ∗ γ_{h,k}^{L}) ⊙ η_{Ll∘Lk∘Lh} \\ η_{L(l∘k)∘Lh} ⊙ (γ_{k,l}^{L} ∗ Lh) = & GF (γ_{k,l}^{L} ∗ Lh) ⊙ η_{Ll∘Lk∘Lh}. \end{matrix}

To ease notation, set

X := (ε_{T (l∘k)} ∗ ε_{T h}) ⊙ F (γ_{k,l}^{L} ∗ Lh) Y := (ε_{T l} ∗ ε_{T (k∘h)}) ⊙ F (Ll ∗ γ_{h,k}^{L}).

We may compute :

\begin{matrix}X = & (ε_{T (l∘k)} ∗ ε_{T h}) ⊙ (F (L(l ∗ ε′_{B′′} ∗ k) ⊙ G(ε_{T l} ∗ ε_{T k}) ⊙ η_{Ll∘Lk}) ∗ F Lh) \\ = & (ε_{T (l∘k)} ∗ ε_{T h}) ⊙ (F (L(l ∗ ε′_{B′′} ∗ k) ⊙ G(ε_{T l} ∗ ε_{T k})) ∗ F Lh) ⊙ F (η_{Ll∘Lk} ∗ Lh) \\ = & (ε_{T (l∘k)} ∗ ε_{T h}) ⊙ (F G(T (l ∗ ε′_{B′′} ∗ k) ⊙ (ε_{T l} ∗ ε_{T k})) ∗ F Lh) ⊙ F (η_{Ll∘Lk} ∗ Lh) \\ = & ((T (l ∗ ε′_{B′′} ∗ k) ⊙ (ε_{T l} ∗ ε_{T k})) ∗ T h) ⊙ (ε_{F (Ll∘Lk)} ∗ ε_{T h}) ⊙ F (η_{Ll∘Lk} ∗ Lh) \\ = & ((T (l ∗ ε′_{B′′} ∗ k) ⊙ (ε_{T l} ∗ ε_{T k})) ∗ T h) ⊙ (F (Ll ∘ Lk) ∗ ε_{T h}) \\ = & ((T (l ∗ ε′_{B′′} ∗ k) ⊙ (ε_{T l} ∗ ε_{T k})) ∗ T h) ⊙ (F Ll ∗ F Lk ∗ ε_{T h}) \\ = & (T (l ∗ ε′_{B′′} ∗ k) ∗ T h) ⊙ (ε_{T l} ∗ ε_{T k} ∗ ε_{T h}) \end{matrix}

where the fourth equality follows from the naturality of ε, the fifth follows from the triangular identities (1.1.13) for the adjunction (η, ε), and the sixth and seventh follow from remark 2.1.1(i). Likewise, a similar calculation gives :

Y = (T l ∗ T (k ∗ ε′_{B′} ∗ h)) ⊙ (ε_{T l} ∗ ε_{T k} ∗ ε_{T h}).

To conclude, it suffices to notice the identity :

T (l ∗ k ∗ ε′_{B′} ∗ h) ⊙ (T (l ∗ ε′_{B′′} ∗ k) ∗ T h) = T (l ∗ ε′_{B′′} ∗ k ∗ h) ⊙ (T l ∗ T (k ∗ ε′_{B′} ∗ h)).

Next, let us show the unit axiom : we come down to checking the identity

γ_{1_{B},f}^{L} ⊙ (Lf ∗ δ_{B}^{L}) = 1_{Lf} for every 1-cell f : B → B′ in ℬ .

However,

\begin{matrix}γ_{1_{B},f}^{L} ⊙ (Lf ∗ δ_{B}^{L}) = & L(f ∗ ε′_{B}) ⊙ G(ε_{T f} ∗ ε_{T 1_{B}}) ⊙ η_{Lf∘L1_{B}} ⊙ (Lf ∗ δ_{B}^{L}) \\ = & L(f ∗ ε′_{B}) ⊙ G(ε_{T f} ∗ ε_{T 1_{B}}) ⊙ GF (Lf ∗ δ_{B}^{L}) ⊙ η_{Lf} \end{matrix}

by the naturality of η; taking into account the triangularity identities (1.1.13) for the adjunction (η, ε), we are then reduced to showing that

Z := T (f ∗ ε′_{B}) ⊙ (ε_{T f} ∗ ε_{T 1_{B}}) ⊙ F (Lf ∗ δ_{B}^{L}) = ε_{T f}.

However, from the triangular identities for the adjunction (η′_{B}, ε′_{B}), we easily obtain :

T (f ∗ ε′_{B}) = T f ∗ η_{B}^{′−1}

and on the other hand, the naturality of ε yields

η_{B}^{′−1} ⊙ ε_{T 1_{B}} = ε_{1_{F LB}} ⊙ F Gη_{B}^{′−1}

whence :

\begin{matrix}Z = & (ε_{T f} ∗ (ε_{1_{F LB}} ⊙ F Gη_{B}^{′−1})) ⊙ (F Lf ∗ F δ_{B}^{L}) \\ = & (ε_{T f} ∗ (ε_{1_{F LB}} ⊙ F Gη_{B}^{′−1})) ⊙ (F Lf ∗ F Gη′_{B}) ⊙ (F Lf ∗ F η_{1_{LB}}) \\ = & (ε_{T f} ∗ ε_{1_{F LB}}) ⊙ (F Lf ∗ F η_{1_{LB}}). \end{matrix}

From the triangular identities for (η, ε) we also see that F (η_{1_{LB}}) = ε_{F 1_{LB}}^{−1} = ε_{1_{F LB}}^{−1}, so finally :

Z = (ε_{T f} ∗ ε_{1_{F LB}}) ⊙ (F Lf ∗ ε_{1_{F LB}}^{−1}) = ε_{T f}

as required. A similar computation establishes the second identity for the unit axiom of L. This completes the construction of the pseudo-functor L. Next we set, for every B, B′ ∈ Ob(ℬ) and every 1-cell h : B → B′ in ℬ

\begin{matrix}η_{B}^{∗} := & f_{B} : B → F LB \\ τ_{h}^{∗} := & (f_{B′} ∗ h ∗ ε′_{B}) ⊙ (ε_{T h} ∗ f_{B}) : F L(h) ∘ f_{B} ⇒ f_{B′} ∘ h. \end{matrix}

Claim 2.4.33. The system of 1-cells η_{•}^{∗} defines a pseudo-natural equivalence η^{∗} : 1_{ℬ} ⇒ F L with coherence constraint given by the system of 2-cells τ_{•}^{∗}.

Proof of the claim. By remark 2.2.5(i), the naturality of τ^{∗} amounts to the identity

(f_{B′} ∗ β) ⊙ (f_{B′} ∗ h_{1} ∗ ε′_{B}) ⊙ (ε_{T h_{1}} ∗ f_{B}) = (f_{B′} ∗ h_{2} ∗ ε′_{B}) ⊙ (ε_{T h_{2}} ∗ f_{B}) ⊙ (F L(β) ∗ f_{B})

for every pair of 1-cells h_{1}, h_{2} : B → B′ and every 2-cell β : h_{1} ⇒ h_{2}. However, a simple inspection shows that

(f_{B′} ∗ β) ⊙ (f_{B′} ∗ h_{1} ∗ ε′_{B}) = (f_{B′} ∗ h_{2} ∗ ε′_{B}) ⊙ (T (β) ∗ f_{B})

so we are reduced to checking that

T (β) ⊙ ε_{T h_{1}} = ε_{T h_{2}} ⊙ F Lβ

which in turns follows from the naturality of ε. Next, let us check the coherence axioms for τ^{∗}. Denote by (δ^{F L}, γ^{F L}) the coherence constraint of F L (see remark 2.1.17(v)); for given 1-cells h : B → B′ and h′ : B′ → B′′ of ℬ we have to verify the identities

(2.4.34)  τ_{1_{B}}^{∗} ⊙ (δ_{B}^{F L} ∗ f_{B}) = 1_{f_{B}}  (τ_{h′}^{∗} ∗ h) ⊙ (F Lh′ ∗ τ_{h}^{∗}) = τ_{h′∘h}^{∗} ⊙ (γ_{h,h′}^{F L} ∗ f_{B}).

However, a direct inspection gives the identity

(τ_{h′}^{∗} ∗ h) ⊙ (F Lh′ ∗ τ_{h}^{∗}) = (f_{B′′} ∗ h′ ∗ ε′_{B′} ∗ h ∗ ε′_{B}) ⊙ (ε_{T h′} ∗ ε_{T h} ∗ f_{B})

so we are reduced to showing that

T (h′ ∗ ε′_{B′} ∗ h) ⊙ (ε_{T h′} ∗ ε_{T h}) = ε_{T (h′∘h)} ⊙ γ_{h,h′}^{F L}.

Now, we compute :

\begin{matrix}ε_{T (h′∘h)} ⊙ γ_{h,h′}^{F L} = & ε_{T (h′∘h)} ⊙ F (γ_{h,h′}^{L}) \\ = & ε_{T (h′∘h)} ⊙ F G(T (h′ ∗ ε′_{B′} ∗ h) ⊙ (ε_{T h′} ∗ ε_{T h})) ⊙ F (η_{Lh′∘Lh}) \\ = & T (h′ ∗ ε′_{B′} ∗ h) ⊙ (ε_{T h′} ∗ ε_{T h}) ⊙ ε_{F (Lh′∘Lh)} ⊙ F (η_{Lh′∘Lh}) \\ = & T (h′ ∗ ε′_{B′} ∗ h) ⊙ (ε_{T h′} ∗ ε_{T h}) \end{matrix}

where the third identity follows from the naturality of ε, and the fourth follows from the triangular identities (1.1.13) for the pair (η, ε). For the first identity (2.4.34), we notice that

τ_{1_{B}}^{∗} = (f_{B} ∗ ε′_{B}) ⊙ (ε_{T 1_{B}} ∗ f_{B}) = (η_{B}^{′−1} ∗ f_{B}) ⊙ (ε_{T 1_{B}} ∗ f_{B})

due to the triangular identities for the pair (η′, ε′), so we are reduced to checking that

ε_{T 1_{B}} ⊙ δ_{B}^{F L} = η′_{B}.

We compute :

ε_{T 1_{B}} ⊙ δ_{B}^{F L} = ε_{T 1_{B}} ⊙ F δ_{B}^{L} = ε_{T 1_{B}} ⊙ F (G(η′_{B}) ⊙ η_{1_{LB}}) = η′_{B} ⊙ ε_{1_{F LB}} ⊙ F (η_{1_{LB}}) = η′_{B}

which concludes the verification of the pseudo-naturality of η^{∗}. Lastly, η_{B}^{∗} is an equivalence for every B ∈ Ob(ℬ), so η^{∗} is a pseudo-natural equivalence, by theorem 2.4.12. ♢

Claim 2.4.35. For every A ∈ Ob(𝒜), the 1-cell G(g_{F A}) : LF A → A is an equivalence in 𝒜.

Proof of the claim. We have an invertible 2-cell ε_{g_{F A}} : F G(g_{F A}) ⇒ g_{F A}, and since g_{F A} is an equivalence, it follows that the same holds for F G(g_{F A}). Then the assertion follows from lemma 2.4.11(iii). ♢

A simple inspection shows that L_{BB′} is equivalence of categories, for every B, B′ ∈ Ob(ℬ). Taking into account claim 2.4.35, we deduce that L is a 2-equivalence; therefore, the foregoing discussion applies as well with F replaced by L, and yields another 2-equivalence M : 𝒜 → ℬ with a pseudo-natural equivalence η^{∗∗} : 1_{𝒜} ⇒ LM. Pick also pseudo-natural equivalences

μ^{∗} : F L ⇒ 1_{ℬ} and μ^{∗∗} : LM ⇒ 1_{𝒜}

representing the inverses of the 2-cells η^{∗}, η^{∗∗} in \overline{2-Cat}. We get a pseudo-natural equivalence

β := (μ^{∗} ∗ M) ⊙ (F ∗ η^{∗∗}) : F ⇒ M

(corollary 2.4.14), whence the sought pseudo-natural equivalence

ε^{∗} := μ^{∗∗} ⊙ (L ∗ β) : LF ⇒ 1_{𝒜} .

(ii): Assertion (i) says that F is a 2-equivalence if and only if it is an equivalence in the 2-category \overline{2-Cat}. Then the assertion follows immediately from lemma 2.1.13(i). □

2.5. 2-Limits and 2-colimits. Consider 2-categories 𝒜, ℬ. For every object B of ℬ, one may define the constant pseudo-functor with value B : this is the pseudo-functor

F_{B} : 𝒜 → ℬ  such that F_{B}(A) := B  F_{B}(f) := 1_{B}  F_{B}(α) := i_{B}

for every A ∈ Ob(𝒜), every 1-cell f, and every 2-cell α of 𝒜. The coherence constraint for F_{B} consists of identities. Given a pseudo-functor F : 𝒜 → ℬ, a pseudo-cone on F with vertex B is a pseudo-natural transformation F_{B} ⇒ F. Dually, a pseudo-cocone on F with vertex B is a pseudo-natural transformation F ⇒ F_{B}. Especially, every 1-cell f : B → B′ induces a pseudo-cone with vertex B :

F_{f} : F_{B} ⇒ F_{B′} : (F_{f})_{A} := f for every A ∈ Ob(𝒜)

whose coherence constraint consists of identities. Notice that F_{f} can also be viewed as a pseudococone with vertex B′. Also, every 2-cell β : f ⇒ f′ induces a modification

F_{β} : F_{f} ⇝ F_{f′} : A ↦ β for every A ∈ Ob(𝒜) .

Definition 2.5.1. Let I and ℬ be two 2-categories, F : I → ℬ a pseudo-functor.

(i) A 2-limit (resp. a strong 2-limit) of F is a pair :

2-lim_{I} F := (L, π)

consisting of an object L of ℬ and a pseudo-cone π : F_{L} ⇒ F, such that the functor :

ℬ(B, L) → PsNat(F_{B}, F) : f ↦ π ⊙ F_{f} (β : f ⇒ f′) ↦ (π ∗ F_{β} : F_{f} ⇝ F_{f′})

is an equivalence (resp. an isomorphism) of categories, for every B ∈ Ob(ℬ). In this case, we also say that π is a universal pseudo-cone (resp. a strong universal pseudo-cone).

(ii) Dually, a 2-colimit (resp. a strong 2-colimit) of F is a pair :

2-colim_{I} F := (L, π)

consisting of an object L of ℬ and a pseudo-cocone π : F ⇒ F_{L}, such that the functor:

ℬ(L, B) → PsNat(F, F_{B}) : f ↦ F_{f} ⊙ π (β : f ⇒ f′) ↦ (F_{β} ∗ π : F_{f} ⇝ F_{f′})

is an equivalence (resp. an isomorphism) of categories, for every B ∈ Ob(ℬ). In this case, we also say that π is a universal pseudo-cocone (resp. a strong universal pseudo-cocone).

(iii) We say that ℬ is 2-complete (resp. 2-cocomplete, resp. strongly 2-complete, resp. strongly 2-cocomplete) if, for every small 2-category I, every pseudo-functor I → ℬ admits a 2-limit (resp. a 2-colimit, resp. a strong 2-limit, resp. a strong 2-colimit).

(iv) Let 𝒞 be any other 2-category, G : ℬ → 𝒞 any pseudo-functor, and (L, π) a 2-limit (resp. a 2-colimit) for F. We say that G commutes with the 2-limit of F (resp. that G commutes with the 2-colimit of F) if the pseudo-cone G ∗ π : F_{GL} ⇒ G ∘ F (resp. the pseudo-cocone G ∗ π : G ∘ F ⇒ F_{GL}) is universal.

Remark 2.5.2. (i) As usual, if the 2-limit of a pseudo-functor exists, it is unique up to (nonunique) equivalence : if (L′, π′) is another 2-limit, there exist an equivalence h : L → L′ and an isomorphism β : π′ ⊙ F_{h} →^{∼} π; moreover, the pair (h, β) is unique up to unique isomorphism, in a suitable sense, that the reader may spell out, as an exercise. A similar remark holds for 2-colimits.

(ii) Likewise, in the situation of definition 2.5.1(iv), the commutation of G with the 2-limit or 2-colimit of F is an intrinsic property of G, i.e. is independent of the choice of a 2-limit or 2-colimit for F.

(iii) To be in keeping with the terminology of [28, Ch.VII], we should write pseudo-bilimit instead of 2-limit (and likewise for 2-colimit). The term “2-limit” denotes in loc.cit. a related notion, which is unique up to isomorphism, not just up to equivalence. However, the notion introduced in definition 2.5.1occurs more frequently in applications.

(iv) Let A_{•} := (A_{i} | i ∈ I) be any family of objects of the 2-category 𝒜, indexed by a small set I. We may regard A_{•} as a strict pseudo-functor from the discrete category I to 𝒜. Then a 2-limit (resp. a 2-colimit) of this pseudo-functor shall also be called a 2-product (resp. a 2-coproduct) of A_{•}.

(v) Let L be the small category with Ob(L) := {0, 1, 2}, and whose set of arrows consists of the identity morphisms, and two more arrows 1 → 0 and 2 → 0. An essentially commutative square (2.1.6) can be regarded as a pseudo-cone π with vertex A, on the functor F : L → 𝒜 such that F (0) := D, F (1) := B, F (2) := C, F (1 → 0) := h and F (2 → 0) := k. We say that (2.1.6) is 2-cartesian if (A, π) is a 2-limit of the functor F. This 2-limit shall be called the 2-fibre product of k and h, and shall be denoted

B ×_{(h,k)}^{2} C

or sometimes, just B ×^{2}_{D} C, if there is no danger of ambiguity.

(vi) In view of (2.2.13), it is easily seen that a pair (L, π) is a 2-colimit for the pseudo-functor F : I → ℬ if and only if the pair (L^{o}, π^{o}) is a 2-limit for the pseudo-functor F^{o} : I^{o} → ℬ^{o}, and if and only if (^{o}L,^{o}π) is a 2-limit for the pseudo-functor ^{o}F : ^{o}I → ^{o}ℬ. Especially, ℬ is 2-cocomplete if and only if ℬ^{o} is 2-complete, and if and only if^{o}ℬ is 2-cocomplete.

(vii) Let I and ℬ be two categories, which we regard as 2-categories with trivial 2-cells. Then every pseudo-functor F : I → ℬ is a functor, every pseudo-natural transformation F ⇒ G of functors F, G : I → ℬ is a natural transformation, and the category PsNat(F, G) is discrete. It follows easily that a 2-limit (L, π) of any such functor F : I → ℬ is a strong 2-limit, it represents also the (usual) limit of F, and the universal pseudo-cone π is also a universal cone. Likewise, any 2-colimit of F represents the colimit of F, and every universal pseudo-cocone is a universal cocone.

The following lemma 2.5.3indicates that the framework of 2-categories does indeed provide an adequate answer to the issues raised in the introduction of this chapter.

Lemma 2.5.3. Let I be a 2-category, F, G : I → ℬ two pseudo-functors, ω : F ⇒ G a pseudo-natural equivalence, and suppose that the 2-limit of F exists. Then the same holds for the 2-limit of G, and there is a natural equivalence in ℬ :

2-lim_{I} F →^{∼} 2-lim_{I} G.

More precisely, if (L, π) is a pair with L ∈ Ob(ℬ) and a pseudo-cone π : F_{L} ⇒ F representing the 2-limit of F, then the pair (L, ω ⊙ π) represents the 2-limit of G.

A dual assertion holds for 2-colimits.

Proof. It is easily seen that the rule α ↦ ω ⊙ α induces an equivalence of categories :

PsNat(F_{B}, F) → PsNat(F_{B}, G) for every B ∈ Ob(ℬ) .

The claim is an immediate consequence. □

2.5.4. Let 𝒞 be any 2-category with small Hom-categories. To every object X of 𝒞 we attach the strict pseudo-functor

h_{X} : 𝒞^{o} → Cat Y ↦ 𝒞 (Y, X) ⊂ Y/𝒞

that assigns to every 1-cell Y′ ⟶^{f} Y of 𝒞 the restriction 𝒞 (Y, X) → 𝒞 (Y′, X) of the strict pseudo-functor f^{∗} : Y/𝒞 → Y′/𝒞, and to every 2-cell β : f ⇒ f′ the restriction of the strict pseudo-natural transformation β^{∗} : f^{∗} ⇒ f^{′∗} (notation of example 2.2.8(i)).

Notice that h_{X^{o}} : 𝒞 → Cat is the strict pseudo-functor given by the rule : Y ↦ 𝒞 (X, Y) for every Y ∈ Ob(𝒞), and which assigns to the 1-cell f as in the foregoing the restriction of the strict pseudo-functor f_{∗} : 𝒞 /Y → 𝒞 /Y′, and to the 2-cell β the restriction of β_{∗} : f_{∗} ⇒ f′_{∗} (again, with the notation of example 2.2.8(i)).

Proposition 2.5.5. With the notation of (2.5.4), let I be another 2-category, F : I → 𝒞 a pseudo-functor, and L ∈ Ob(𝒞) any object. We have :

(i) For every pseudo-cone π : F_{L} ⇒ F, the following conditions are equivalent : (a) The pseudo-cone π is universal.

(b) For every X ∈ Ob(𝒞) the pseudo-cone h_{X^{o}} ∗ π : F_{𝒞 (X,L)} ⇒ h_{X^{o}} ∘ F is universal. (ii) For every pseudo-cocone π : F ⇒ F_{L}, the following conditions are equivalent :

(a) The pseudo-cocone π is universal.

(b) For every X ∈ Ob(𝒞) the pseudo-cone h_{X} ∗ π : F_{𝒞 (L,X)} ⇒ h_{X} ∘ F is universal.

Proof. (i.a)⇒(i.b): We have to check that for every small category ℬ, the functor

T : Fun(ℬ, 𝒞 (X, L)) → PsNat(F_{ℬ} , h_{X^{o}} ∗ π) (G : ℬ → 𝒞 (X, L)) ↦ (h_{X^{o}} ∗ π) ⊙ F_{G}

is an equivalence. To this aim, consider any pseudo-cone λ : F_{ℬ} ⇒ h_{X^{o}} ∗ π. Explicitly, λ is the datum of a system of functors (λ_{i} : ℬ → 𝒞 (X, F i) | i ∈ Ob(I)) and a system of natural isomorphisms of functors (τ_{φ}^{λ} : h_{X^{o}}(F φ) ∘ λ_{i} ⇒ λ_{j} | (i ⟶^{φ} j) ∈ Morph(I)) fulfilling the usual coherence axioms. To every B ∈ Ob(ℬ) we attach the system

λ_{•}(B) := (λ_{i}(B) : X → F i | i ∈ Ob((I)).

It is easily seen that λ_{•}(B) : F_{X} ⇒ F is a pseudo-cone with coherence constraint given by the system (τ_{φ,B}^{λ} : F (φ) ∘ λ_{i}(B) ⇒ λ_{j}(B) | (i ⟶^{φ} j) ∈ Morph(I)). Since (L, π) is universal, we may then find a 1-cell λ_{B}^{†} : X → L and an isomorphism of pseudo-cones

Ω_{B}^{λ} : π ⊙ F_{λ_{B}^{†}} ⇝ λ_{•}(B) for every B ∈ Ob(ℬ) .

Moreover, for every morphism f : B → B′ in ℬ, the system (λ_{i}(f) : λ_{i}(B) ⇒ λ_{i}(B′) | i ∈ Ob(I)) yields a modification λ_{•}(f) : λ_{•}(B) ⇝ λ_{•}(B′). Then, there exists a unique 2-cell in 𝒞

(2.5.6)  λ_{f}^{†} : λ_{B}^{†} ⇒ λ_{B′}^{†}  such that Ω_{B′}^{λ} ⊙ (π ∗ F_{λ_{f}^{†}}) = λ_{•}(f) ⊙ Ω_{B}^{λ}.

Clearly λ_{•}(g) ⊙ λ_{•}(f) = λ_{•}(g ∘ f) for every composable pair of morphisms B ⟶^{f} B′ ⟶^{g} B′′ of ℬ, whence λ_{g}^{†} ⊙ λ_{f}^{†} = λ_{g∘f}^{†}. We have thus associated with λ a well defined functor λ^{†} : ℬ → 𝒞 (X, L), together with an isomorphism

Ω_{•}^{λ} : (h_{X^{o}} ∗ π) ⊙ F_{λ^{†}} →^{∼} λ in PsNat(F_{ℬ} , h_{X^{o}} ∘ F).

This proves that T is essentially surjective. Next, let us check that T is faithful : indeed, consider two functors G, H : ℬ → 𝒞 (X, L) and two natural transformations α, β : G ⇒ H such that (h_{X^{o}} ∗ π) ⊙ F_{α} = (h_{X^{o}} ∗ π) ⊙ F_{β}; the latter identity means that

π_{i} ∗ α_{B} = π_{i} ∗ β_{B} : π_{i} ∘ GB ⇒ π_{i} ∘ HB for every i ∈ Ob(I) and every B ∈ Ob(ℬ) .

In other words, for every B ∈ Ob(ℬ), the morphisms of pseudo-cones π ∗ F_{α_{B}}, π ∗ F_{β_{B}} : π ⊙ F_{GB} ⇒ π ⊙ F_{HB} coincide; then the universality of π implies that α_{B} = β_{B} for every B ∈ Ob(ℬ), whence the assertion. Lastly, to check that T is full we consider, for G and H as in the foregoing, any modification Ξ : (h_{X^{o}} ∗ π) ⊙ F_{G} ⇝ (h_{X^{o}} ∗ π) ⊙ F_{H}; hence, Ξ amounts to a system of natural transformations (Ξ_{i} : π_{i,∗} ∘ G ⇒ π_{i,∗} ∘ H | i ∈ Ob(I)) related by the compatibility conditions :

Ξ_{j} = (τ_{φ∗}^{π} ∗ H) ⊙ ((F φ)_{∗} ∗ Ξ_{i}) for every 1-cell φ : i → j of I

where π_{i,∗} : 𝒞 (X, L) → 𝒞 (X, F i) and (F φ)_{∗} : 𝒞 (X, F i) → 𝒞 (X, F j) are the functors induced by π_{i} : L → F i and respectively F φ : F i → F j, and where τ_{φ∗}^{π} : (F φ)_{∗} ∘ π_{i∗} →^{∼} π_{j∗} denotes the isomorphism of functors induced by the coherence constraint τ^{π} of π. Then it is easily seen that for every B ∈ Ob(ℬ) the system

(Ξ_{i,B} : π_{i} ∘ GB ⇒ π_{i} ∘ HB | i ∈ Ob(I))

yields a morphism of pseudo-cones Ξ_{•,B} : π ∗ F_{GB} ⇝ π ∗ F_{HB}, and the naturality of Ξ_{i} gives the identities

(2.5.7) (π ∗ F_{Hf}) ⊙ Ξ_{•,B} = Ξ_{•,B′} ⊙ (π ∗ F_{Gf})  for every morphism f : B → B′ of ℬ .

By the universality of π, the morphisms Ξ_{•,B} corresponds to a unique 2-cell Ξ_{B}^{∗} : GB ⇒ HB, and (2.5.7) implies that (Ξ_{B}^{∗} | B ∈ Ob(ℬ)) is a well defined natural transformation Ξ^{∗} : G ⇒ H. By direct inspection, we see that (h_{X^{o}} ∗ π) ∗ F_{Ξ^{∗}} = Ξ, whence the contention.

(i.b)⇒(i.a): To any X ∈ Ob(𝒞) and any pseudo-cone φ : F_{X} ⇒ F we attach the pseudocone

φ_{∗} : F_{𝒞 (X,X)} ⇒ h_{X^{o}} ∘ F i ↦ (φ_{i∗} : 𝒞 (X, X) → 𝒞 (X, F i))

with coherence constraint given by the system of isomorphisms of functors

(τ_{ψ∗}^{φ} : (F ψ)_{∗} ∘ φ_{i∗} ⇒ φ_{j∗} | (i ⟶^{ψ} j) ∈ Morph(I))

where τ_{•}^{φ} denotes the coherence constraint of φ. By assumption, there exist a functor φ^{†} : 𝒞 (X, X) → 𝒞 (X, L) and an isomorphism of pseudo-cones ω : φ_{∗} →^{∼} (h_{X^{o}} ∗ π) ⊙ F_{φ^{†}}. Then it is easily seen that the system

(ω_{i,1_{X}} : φ_{i} ⇒ π_{i} ∘ φ^{†}(1_{X}) | i ∈ Ob(I))

yields an isomorphism of pseudo-cones φ →^{∼} π ⊙ F_{φ^{†}(1_{X})}. This shows that the functor

(2.5.8)  𝒞 (X, L) → PsNat(F_{X}, F)

as in definition 2.5.1(i) is essentially surjective. Next, in order to check that (2.5.8) is faithful, consider two 1-cells g, h : X → L in 𝒞 and two 2-cells α, β : g ⇒ h such that π_{i} ∗ α = π_{i} ∗ β for every i ∈ Ob(I); let also ℬ be the initial object of Cat, i.e. ℬ is a category with a unique object b and a unique morphism 1_{b}. We let G, H : ℬ → 𝒞 (X, L) be the functors such that Gb := g and Hb := h; then α and β correspond to natural transformations α′, β′ : G ⇒ H such that (h_{X^{o}}∗π)⊙F_{α′} = (h_{X^{o}}∗π)⊙F_{β′}, and the universality of h_{X^{o}}∗π implies that α′ = β′, whence α = β, as required. Lastly, for g, h, ℬ, G and H as in the foregoing, let Ξ : π ⊙ F_{g} ⇝ π ⊙ F_{h} be any modification; we deduce a modification Ξ′ : (h_{X^{o}} ∗ π) ⊙ F_{G} ⇝ (h_{X^{o}} ∗ π) ⊙ F_{H}, whence – by the universality of (h_{X^{o}} ∗ π) – a unique natural transformation β′ : G ⇒ H such that Ξ′ = (h_{X^{o}} ∗ π) ⊙ F_{β}. Then β′ is the datum of a 2-cell β : g ⇒ h such that Ξ = π ∗ F_{β}, which shows that the functor (2.5.8) is also full.

Assertion (ii) is easily deduced from (i), by considering the pseudo-cone π^{∘} : F_{L^{∘}} ⇒ F^{∘} (details left to the reader). □

Proposition 2.5.9. Let I, ℬ and 𝒞 be three 2-categories, G : ℬ → 𝒞, and H : 𝒞 → ℬ two pseudo-functors, and suppose that (H, G) is a 2-adjoint pair. We have :

(i) The pseudo-functor G commutes with the 2-limit of every pseudo-functor F : I → ℬ. (ii) Dually, H commutes with the 2-colimit of every pseudo-functor F′ : I → 𝒞.

Proof. (i): Let F : I → ℬ be any pseudo-functor, and (L, π) any 2-limit for F. We fix a unit and a counit (η, ε) for the 2-adjoint pair (H, G), and a pair of invertible modifications

Σ : (G ∗ ε) ⊙ (η ∗ G) ⇝ 1_{G} Σ′ : (ε ∗ H) ⊙ (H ∗ η) ⇝ 1_{H}.

For every Y ∈ Ob(𝒞) we consider the diagram of functors :

𝒟  :  \xymatrix{𝒞 (Y, GL) \ar[r]^{ϑ} \ar[d] & ℬ(HY, L) \ar[d] \\ PsNat(F_{Y} , GF) \ar[r]^{ϑ^{†}} & PsNat(F_{HY} , F)}

where ϑ is given by the rules :

(f : Y → GL) ↦ (ε_{L} ∘ Hf : HY → HGL) and (β : f ⇒ f′) ↦ ε_{L} ∗ Hβ

for every 1-cell f and every 2-cell β of 𝒞 (Y, GL). The functor ϑ^{†} assigns to every pseudonatural transformation ψ : F_{Y} ⇒ GF the pseudo-natural transformation (ε ∗ F) ⊙ (H ∗ ψ) : F_{HY} ⇒ F, and to every modification Ξ : ψ ⇝ ψ′ the modification (ε ∗ F) ∗ (H ∘ Ξ) (notation of remark 2.2.10(vi)). The left (resp. right) vertical arrow is the functor associated with the pseudo-cone G ∗ π (resp. π) as in definition 2.2.10(i). By inspecting the proof of theorem 2.2.10(ii), it is easily seen that ϑ is an equivalence, and by assumption, the same holds for the right vertical arrow. We notice :

Claim 2.5.10. Diagram 𝒟 is essentially commutative.

Proof of the claim. The composition of ϑ with the right vertical arrow assigns to every 1-cell f : Y → GL the pseudo-natural transformation π⊙F_{ε_{L}∘Hf} = π⊙F_{ε_{L}}⊙F_{Hf} = π⊙(ε∗F_{L})⊙F_{Hf}. On the other hand, the composition of ϑ^{†} with the left vertical arrow assigns to f the pseudonatural transformation (ε∗F)⊙(H ∗((G∗π)⊙F_{f})). However, we have invertible modifications:

Θ_{f} : (ε ∗ F) ⊙ (H ∗ ((G ∗ π) ⊙ F_{f})) ⇝ (ε ∗ F) ⊙ (HG ∗ π) ⊙ (H ∗ F_{f}) ⇝ π ⊙ (ε ∗ F_{L}) ⊙ F_{Hf}

for every such f (example 2.2.15(i,ii)). We need to check that the rule : f ↦ Θ_{f} yields a natural transformation; by unwinding the definitions, we come down to showing the commutativity of the diagram :

\xymatrix{ε_{F i} ∘ H(Gπ_{i} ∘ f) \ar@{=>}[d]_{ε_{F i}∗γ_{f,Gπ_{i}}^{H}} \ar@{=>}[r]^{ε_{F i}∗H(Gπ_{i}∗β)} & ε_{F i} ∘ H(Gπ_{i} ∘ f′) \ar@{=>}[d]^{ε_{F i}∗γ_{f′,Gπ_{i}}^{H}} \\ ε_{F i} ∘ HGπ_{i} ∘ Hf \ar@{=>}[r]^{ε_{F i}∗HGπ_{i}∗Hβ} \ar@{=>}[d]_{(τ_{π_{i}}^{ε})^{−1}∗Hf} & ε_{F i} ∘ HGπ_{i} ∘ Hf′ \ar@{=>}[d]^{(τ_{π_{i}}^{ε})^{−1}∗Hf′} \\ π_{i} ∘ ε_{L} ∘ Hf \ar@{=>}[r]^{(π_{i}∘ε_{L})∗Hβ} & π_{i} ∘ ε_{L} ∘ Hf′}

where (δ^{H}, γ^{H}) is the coherence constraint of H and τ^{ε} is the coherence constraint of ε. However, the commutativity of the top square subdiagram follows from the naturality of γ^{H} (remark 2.1.17(ii)). The commutativity of the bottom square subdiagram is obvious, since the compositions of both pairs of arrows equal (τ_{π_{i}}^{ε})^{−1} ∗ Hβ (remark 2.1.1(i)). ♢

In view of claim 2.5.10, we are reduced to checking that ϑ^{†} is an equivalence of categories. To this aim, let us consider the functor

μ : PsNat(F_{HY} , F) → PsNat(F_{Y} , GF) (φ : F_{HY} ⇒ F) ↦ (G ∗ φ) ⊙ F_{η_{Y}}

that assigns to every modification Θ : φ ⇝ φ′ between pseudo-cones φ, φ′ : F_{HY} ⇒ F the modification (G ∘ Θ) ∗ F_{η_{Y}}. We claim that μ is a quasi-inverse for ϑ^{†}. Indeed, for every pseudocone ψ : F_{Y} ⇒ GF we have μ∘ϑ^{†}(ψ) = (G∗((ε∗F)⊙(H ∗ψ)))⊙F_{η_{Y}}, and example 2.2.15(i) and remark 2.2.10(iv) yield an invertible modification

γ_{ε∗F,H∗ψ}^{G} ∗ F_{η_{Y}} : μ ∘ ϑ^{†}(ψ) ⇝ (G ∗ ε ∗ F) ⊙ (GH ∗ ψ) ⊙ F_{η_{Y}} .

Moreover, example 2.2.15(ii) and remark 2.2.10(iv) yield an invertible modification

(G ∗ ε ∗ F) ∗ τ_{ψ}^{η} : (G ∗ ε ∗ F) ⊙ (GH ∗ ψ) ⊙ F_{η_{Y}} ⇝ (G ∗ ε ∗ F) ⊙ (η ∗ GF) ⊙ ψ

which we may compose with the invertible modification

(Σ ∘ F) ∗ ψ : (G ∗ ε ∗ F) ⊙ (η ∗ GF) ⊙ ψ ⇝ 1_{GF} ⊙ ψ = ψ.

Summing up, we obtain an invertible modification

Λ_{ψ} : μ ∘ ϑ^{†}(ψ) ⇝ ψ i ↦ (Σ_{F i} ∗ ψ_{i}) ⊙ (G(ε_{F i}) ∗ τ_{ψ_{i}}^{η}) ⊙ (γ_{ε_{F i},Hψ_{i}}^{G} ∗ η_{Y})

and we need to check that the rule ψ ↦ Λ_{ψ} yields a natural transformation

Λ : μ ∘ ϑ^{†} ⇒ 1_{PsNat(F_{Y} ,GF)}.

Thus, let Ξ : ψ ⇝ ψ′ be any modification, and notice that μ ∘ ϑ^{†}(Ξ)_{i} = G(ε_{F i} ∗ H(Ξ_{i})) ∗ η_{Y} for every i ∈ Ob(I); we compute :

Ξ_{i} ⊙ Λ_{ψ,i} = (Σ_{F i} ∗ ψ′_{i}) ⊙ (G(ε_{F i}) ∗ η_{GF i} ∗ Ξ_{i}) ⊙ (G(ε_{F i}) ∗ τ_{ψ_{i}}^{η}) ⊙ (γ_{ε_{F i},Hψ_{i}}^{G} ∗ η_{Y})
= (Σ_{F i} ∗ ψ′_{i}) ⊙ (G(ε_{F i}) ∗ τ_{ψ′_{i}}^{η}) ⊙ (G(ε_{F i}) ∗ GH(Ξ_{i}) ∗ η_{Y}) ⊙ (γ_{ε_{F i},Hψ_{i}}^{G} ∗ η_{Y})
= (Σ_{F i} ∗ ψ′_{i}) ⊙ (G(ε_{F i}) ∗ τ_{ψ′_{i}}^{η}) ⊙ (γ_{ε_{F i},Hψ′_{i}}^{G} ∗ η_{Y}) ⊙ (G(ε_{F i} ∗ H(Ξ_{i})) ∗ η_{Y})
= Λ_{ψ′,i} ⊙ μ ∘ ϑ^{†}(Ξ)_{i}

where the first identity follows from remark 2.1.1(i), the second one from remark 2.2.5(i), and the third one from remark 2.1.17(ii). The assertion follows.

Lastly, we have ϑ^{†}∘μ(φ) = (ε∗F)⊙H∗((G∗φ)⊙F_{η_{Y}}) for every pseudo-cone φ : F_{HY} ⇒ F, and we get an invertible modification

Λ′_{φ} : ϑ^{†} ∘ μ(φ) ⇝ (ε ∗ F) ⊙ (HG ∗ φ) ⊙ F_{Hη_{Y}} ⇝ φ ⊙ F_{ε_{HY}} ⊙ F_{Hη_{Y}} ⇝ φ

as the composition of (ε∗F)∗(γ_{F_{η_{Y}} ,G∗φ}^{H})^{−1} from example 2.2.15(i), together with (τ_{φ}^{ε})^{−1}∗F_{Hη_{Y}} from example 2.2.15(ii) and φ ∗ F_{Σ′_{Y}}. An explicit calculation as in the foregoing shows that the rule φ ↦ Λ′_{φ} yields a natural transformation Λ : ϑ^{†} ∘ μ ⇒ 1_{PsNat(F_{HY} ,F)} (details left to the reader), and concludes the proof of (i).

Assertion (ii) follows from (i), by considering the opposite 2-categories : details left to the reader. □

Proposition 2.5.11. Let 𝒜 , ℬ be two 2-categories, F : 𝒜 → ℬ a fully faithful pseudo-functor that admits a left 2-adjoint G : ℬ → 𝒜. The following holds :

(i) For every small 2-category I and every pseudo-functor φ : I → 𝒜, if F ∘ φ : I → ℬ admits a 2-limit (L, π^{L}), then φ admits a 2-limit (M, π^{M}).

(ii) In the situation of (i), we have equivalences L →^{∼} F M in ℬ and M →^{∼} GL in 𝒜.

Proof. Let η : 1_{ℬ} ⇒ F G and ε : GF ⇒ 1_{𝒜} be a unit and a counit for the 2-adjoint pair (G, F), and set π^{GL} := (ε ∗ φ) ⊙ (G ∗ π^{L}) : F_{GL} ⇒ φ. By the universality of π^{L}, there exist a 1-cell f : F GL → L in ℬ and an isomorphism of pseudo-cones π^{L} ⊙ F_{f} →^{∼} F ∗ π^{GL}. Invoking example 2.2.15(i,ii) we deduce isomorphisms of pseudo-cones :

\begin{aligned}π^{L}⊙F_{f∘η_{L}} = π^{L}⊙F_{f} ⊙F_{η_{L}} →^{∼} (F ∗ π^{GL})⊙(η ∗ F_{L}) & →^{∼} (F ∗ ε ∗ φ)⊙(F G ∗ π^{L})⊙(η ∗ F_{L}) \\ & →^{∼} (F ∗ ε ∗ φ) ⊙ (η ∗ F φ) ∘ π^{L} →^{∼} π^{L} \end{aligned}

whence an isomorphism f ∘ η_{L} →^{∼} 1_{L} in ℬ, by universality of π^{L}. On the other hand, since F is fully faithful, there exists a 1-cell h : GL → GL in 𝒜 with an isomorphism F h →^{∼} η_{L} ∘ f : F GL → F GL. Notice the isomorphisms :

(2.5.12)  F h ∘ η_{L} →^{∼} η_{L} ∘ f ∘ η_{L} →^{∼} η_{L} →^{∼} F 1_{GL} ∘ η_{L}.

On the other hand, using the pseudo-naturality of ε and the triangular modifications for the pair (η, ε) we get isomorphisms :

ε_{GL} ∘ G(F h ∘ η_{L}) →^{∼} ε_{GL} ∘ GF h ∘ Gη_{L} →^{∼} h ∘ ε_{GL} ∘ Gη_{L} →^{∼} h

and similarly : ε_{GL} ∘ G(F 1_{GL} ∘ η_{L}) →^{∼} 1_{GL}. Combining with (2.5.12), we deduce an isomorphism h →^{∼} 1_{GL}, whence η_{L} ∘ f →^{∼} F 1_{GL} →^{∼} 1_{F GL}, which shows that f is an equivalence in ℬ. Especially, (ii) holds with M := GL, and it also follows that (F GL, F ∗ π^{GL}) is a 2-limit of F ∘ φ. It remains to check that (GL, π^{GL}) is a 2-limit of φ. To this aim, we consider for every A ∈ Ob(𝒜) the essentially commutative diagram of categories :

\xymatrix{𝒜 (A, GL) \ar[r] \ar[d]_{F_{A,GL}} & PsNat(F_{A}, φ) \ar[d]^{U} \\ ℬ(F A, F GL) \ar[r] & PsNat(F_{F A}, F ∘ φ)}

whose top (resp. bottom) horizontal arrow is the pseudo-functor induced by the pseudo-cone π^{GL} (resp. F ∗ π^{GL}) as in definition 2.5.1(i). The pseudo-functor U is given by the rules : β ↦ F ∗ β and Θ ↦ F ∗ Θ for every pseudo-cone β : F_{A} ⇒ φ and every modification of pseudo-cones Θ : β ⇝ β′. Since F is fully faithful, the functor F_{A,GL} is an equivalence, and the same holds for the bottom horizontal arrow. We are then reduced to checking that U is an equivalence. To this aim, recall that ε is a pseudo-natural equivalence (corollary 2.4.29) and fix a 1-cell ω_{A} : A → GF A in 𝒜 and an isomorphism

ρ_{A} : ε_{A} ∘ ω_{A} →^{∼} 1_{A}.

There follows an isomorphism F ε_{A} ∘ F ω_{A} →^{∼} 1_{F A}; on the other hand, the triangular modifications for (η, ε) give an isomorphism F ε_{A} ∘ η_{F A} →^{∼} 1_{F A}, so we get as well an isomorphism

ρ′_{A} : F ω_{A} →^{∼} η_{F A}.

We consider the pseudo-functor

V : PsNat(F_{F A}, F ∘ φ) → PsNat(F_{A}, φ) β ↦ (ε ∗ φ) ⊙ (G ∗ β) ⊙ F_{ω_{A}}

that assigns to every modification Θ : β ⇝ β′ of pseudo-cones β, β′ : F_{F A} ⇒ F ∘ φ the modification (ε ∗ φ) ∗ (G ∗ Θ) ∗ F_{ω_{A}}. We construct as follows isomorphisms of functors

Λ : V U →^{∼} 1_{PsNat(F_{A},φ)} Λ′ : UV →^{∼} 1_{PsNat(F_{F A},F ∘φ)}.

For every pseudo-cone β : F_{A} ⇒ φ we have V U(β) = (ε ∗ φ) ⊙ (GF ∗ β) ⊙ F_{ω_{A}}. By example 2.2.15(ii), we have an invertible modification

Ξ_{β} : β ⊙ (ε ∗ F_{A}) ⇝ (ε ∗ φ) ⊙ (GF ∗ β) i ↦ τ_{β_{i}}^{ε}

where τ^{ε} denotes the coherence constraint of ε. There follows an isomorphism Ξ_{β}^{−1} ∗ F_{ω_{A}} : V U(β) →^{∼} β ⊙ (ε ∗ F_{A}) ⊙ F_{ω_{A}} = β ⊙ F_{ε_{A}∘ω_{A}}. We compose the latter with the isomorphism β ∗ F_{ρ_{A}} : β ⊙ F_{ε_{A}∘ω_{A}} →^{∼} β to obtain the isomorphism Λ_{β} : V U(β) →^{∼} β. In order to check the naturality of the rule : β ↦ Λ_{β}, we come down to verifying the identity :

((ε ∗ φ) ∗ (GF ∗ Θ)) ⊙ Ξ_{β} = Ξ_{β′} ⊙ (Θ ∗ (ε ∗ F_{A}))

for every morphism of pseudo-cones Θ : β ⇝ β′. This follows from the naturality of τ^{ε}, applied to the system of 2-cells Θ_{i} : β_{i} ⇒ β′_{i} : details left to the reader. Lastly, for every pseudo-cone β : F_{F A} ⇒ F ∘ φ we have UV (β) = F ∗ ((ε ∗ φ) ⊙ (G ∗ β) ⊙ F_{ω_{A}}). By example 2.2.15(i) we have the invertible modification

Γ_{β} : (F ∗ ε ∗ φ) ⊙ (F G ∗ β) ⊙ F_{F ω_{A}} ⇝ F ∗ ((ε ∗ φ) ⊙ (G ∗ β) ⊙ F_{ω_{A}}) i ↦ γ_{ω_{A},Gβ_{i},ε_{φ(i)}}^{F}

where γ_{•••}^{F} is coherence constraint of F (for compositions of three 1-cells). We compose Γ_{β}^{−1} with the invertible modification ((F ∗ε∗φ)⊙(F G∗β))∗F_{ρ′_{A}} to get an isomorphism UV (β) →^{∼} (F ∗ ε ∗ φ) ⊙ (F G ∗ β) ⊙ (η ∗ F_{A}). Then, example 2.2.15(ii) yields the invertible modification

Ξ′_{β} : (F G ∗ β) ⊙ (η ∗ F_{A}) ⇝ (η ∗ F ∗ φ) ⊙ β i ↦ τ_{β_{i}}^{η}

where τ^{η} denotes the coherence constraint of η. On the other hand we have the triangular modification Σ : (F ∗ ε) ⊙ (η ∗ F) ⇝ 1_{F}, so we can compose further with

((Σ ∗ φ) ∗ β) ⊙ ((F ∗ ε ∗ φ) ∗ Ξ′_{β}) : (F ∗ ε ∗ φ) ⊙ (F G ∗ β) ⊙ (η ∗ F_{A}) →^{∼} β

to get the sought isomorphism Λ′_{β} : UV (β) →^{∼} β. Again, the naturality of the rule : β ↦ Λ′_{β} follows easily from the naturality of γ_{•••}^{F} and τ^{η}. □

2.5.13. Let I, J be two small 2-categories, 𝒞 another 2-category, and φ : J → I, F : I → 𝒞 two pseudo-functors. Let also (L, π) and (L′, π′) be 2-limits for F and respectively F ∘φ. Then there exist a 1-cell f : L → L′ of 𝒞 and an invertible modification

Θ : π ∗ φ ⇝ π′ ⊙ F_{f}.

Moreover, if f′ : L → L′ is another such 1-cell with an invertible modification Θ′ : π ∗ φ ⇝ π′ ⊙ F_{f′}, there exists a unique invertible 2-cell in 𝒞 :

β : f →^{∼} f′ such that F_{β} ⊙ Θ = Θ′.

In particular, f shall be an equivalence in 𝒞 if and only if the same holds for f′ (remark 2.1.4(iii)), so the property that f is an equivalence is an intrinsic feature of the pair (φ, F). Furthermore, Θ induces an essentially commutative diagram of categories :

\xymatrix{𝒞 (X, L) \ar[r]^{T} \ar[d]_{f_{∗}^{X}} & PsNat(F_{X}, F) \ar[d]^{S^{X}} \\ 𝒞 (X, L′) \ar[r]^{T′} & PsNat(F_{X}, F ∘ φ)}  for every X ∈ Ob(𝒞)

whose horizontal arrows are the equivalences of definition 2.5.1(i), where f_{∗}^{X} is defined as in example 2.1.11(iii), and S^{X} is given by the rules : (β : F_{X} ⇒ F) ↦ β ∗ φ and (Ξ : β ⇝ β′) ↦ Ξ ∘ φ. Then the required isomorphism of functors S^{X} ∘ T →^{∼} T′ ∘ f_{∗}^{X} is given by the rule : (g : X → L) ↦ Θ ∗ F_{g}. Hence, f is an equivalence in 𝒞 if and only if S^{X} is an equivalence of categories for every X ∈ Ob(𝒞).

Definition 2.5.14. Let I, J be two small 2-categories. We say that a pseudo-functor φ : J → I is 2-coinitial if for every 2-complete 2-category 𝒞 and every pseudo-functor F : I → 𝒞 the induced 1-cell f : L → L′ as in (2.5.13) is an equivalence. Likewise, we say that φ is 2-cofinal if φ^{o} : J^{o} → I^{o} is 2-coinitial.

Remark 2.5.15. Unlike as in the case for usual category, we do not provide a general characterization of 2-coinitial or 2-cofinal pseudo-functors, but we shall point out at least two simple, though useful, classes of such pseudo-functors. Such a characterization – in terms of simple connectivity of comma categories – can be found in [146, Prop.A.2.2], though under slightly different assumptions.

For our first example, let 𝟙 be a final object of Cat, i.e. a category with a unique object and a unique morphism (then the unique morphism is the identity of the unique object); let likewise 𝕀 be a final object of 2-Cat, i.e. a 2-category with a unique object, a unique 1-cell and a unique 2-cell. We say that a given object i_{0} of I is pseudo-initial (resp. pseudo-final) if the category I(i_{0}, i) (resp. I(i, i_{0})) is equivalent to 𝟙, for every i ∈ Ob(I).

Proposition 2.5.16. Let i_{0} be a pseudo-initial (resp. pseudo-final) object of I, and φ : 𝕀 → I the strict pseudo-functor that maps the object of 𝕀 to i_{0}. Then φ is 2-coinitial (resp. 2-cofinal).

Proof. Let i_{0} be pseudo-initial; we need to check that for every 2-complete 2-category 𝒞, every pseudo-functor F : I → 𝒞 and every 2-limit (L, π) of F, the pair (L, π ∗ φ) is a 2-limit of F ∘ φ, and notice that the latter condition means that π_{i_{0}} : L → L′ := F i_{0} is an equivalence. To this aim, it suffices to construct a universal pseudo-cone π′ : F_{L′} ⇒ F with π′_{i_{0}} = 1_{L′}. We may also assume that F is unital, by lemma 2.5.3and proposition 2.4.3. Thus, let i ∈ Ob(I) be any object; by assumption, there exists a 1-cell f_{i} : i_{0} → i in I, and we set π′_{i} := F f_{i} : L′ → F i. Naturally, we take f_{i_{0}} := 1_{i_{0}}, so that π′_{i_{0}} = 1_{L′}, since F is unital. Next, for every one-cell g : i → i′ in I we have by assumption a unique 2-cell β_{g} : g ∘f_{i} ⇒ f_{i′} in I, and it is easily seen that β_{g} must then be invertible (details left to the reader); we set τ_{g} := (F β_{g})⊙γ_{f_{i},g}^{F} : F f ∘π_{i} ⇒ π′_{i′}, where γ^{F} denotes the coherence constraint of F.

Claim 2.5.17. The rule : i ↦ π′_{i} defines a pseudo-cone π′ : F_{L′} ⇒ F with coherence constraint given by the system of 2-cells τ_{•}.

Proof of the claim. The uniqueness of β_{g} implies that β_{1_{i}} = 1_{f_{i}} for every i ∈ Ob(I) and

β_{h} ⊙ (h ∗ β_{g}) = β_{h∘g} for every pair of 1-cells i ⟶^{g} i′ ⟶^{h} i′′.

Since F is unital, it follows already that τ_{1_{i}} = 1_{π′_{i}} for every i ∈ Ob(I). It remains to check that

τ_{h} ⊙ (F h ∗ τ_{g}) = τ_{h∘g} ⊙ (γ_{g,h}^{F} ∗ π′_{i})

for every pair of 1-cells (g, h) as in the foregoing. We compute :

τ_{h} ⊙ (F h ∗ τ_{g}) = F β_{h} ⊙ γ_{f_{i′},h}^{F} ⊙ (F h ∗ F β_{g}) ⊙ (F h ∗ γ_{f_{i},g}^{F})
= F β_{h} ⊙ F (h ∗ β_{g}) ⊙ γ_{g∘f_{i},h}^{F} ⊙ (F h ∗ γ_{f_{i},g}^{F})
= F β_{h∘g} ⊙ γ_{g∘f_{i},h}^{F} ⊙ (F h ∗ γ_{f_{i},g}^{F})

whereas τ_{h∘g} ⊙ (γ_{g,h}^{F} ∗ π′_{i}) = F β_{g∘h} ⊙ γ_{f_{i},g}^{F} ⊙ (γ_{g,h}^{F} ∗ F f_{i}), so the sought identity follows from the composition axiom for γ^{F}. ♢

To conclude, we need to check that functor

𝒞 (X, L) → PsNat(c_{X}, F) t ↦ π′ ⊙ F_{t}

is an equivalence for every X ∈ Ob(𝒞). The faithfulness is clear, since π′_{i_{0}} = 1_{L′}. To check that the functor is full, consider two 1-cells t, t′ : X → L and a modification Ξ : π′ ⊙ F_{t} ⇝ π′ ⊙ F_{t′}; we set μ := Ξ_{i_{0}} : t ⇒ t′. Now, notice that β_{f_{i}} = 1_{f_{i}} for every i ∈ Ob(I), so that τ_{f_{i}} = 1_{F f_{i}}, since F is unital; then the compatibility condition for Ξ, relative to the 1-cell f_{i} : i_{0} → i yields:

Ξ_{i} = F f_{i} ∗ μ for every i ∈ Ob(I)

which shows that Ξ = π′ ∗ F_{μ}, as required. Lastly, let α : F_{X} ⇒ F be any pseudo-cone; set t := α_{i_{0}} : X → L′, and Ξ_{i} := τ_{f_{i}}^{α} : F f_{i} ∘t ⇒ α_{i} for every i ∈ Ob(I), where τ^{α} is the coherence constraint of α. To conclude, it suffices to check that the rule : i ↦ Ξ_{i} yields an invertible modification Ξ : π′ ⊙ F_{t} ⇝ α. This comes down to showing :

τ_{g}^{α} ⊙ (F g ∗ τ_{f_{i}}^{α}) = τ_{f_{i′}}^{α} ⊙ (τ_{g} ∗ α_{i_{0}}) for every 1-cell g : i → i′ of I.

However, the left-hand side of the latter identity equals τ_{g∘f_{i}}^{α} ⊙ (γ_{f_{i},g}^{F} ∗ α_{i_{0}}), by the coherence axiom for τ^{α}, so we are reduced to checking the identity :

τ_{g∘f_{i}}^{α} = τ_{f_{i′}}^{α} ⊙ (F β_{g} ∗ α_{i_{0}})

which holds by naturality of τ^{α}. □

2.5.18. For our second example, consider a small category I (which we regard as a 2-category with trivial 2-cells, as usual), and a full subcategory J of I, such that the inclusion functor φ : J → I admits a left adjoint σ : I → J with σ ∘ φ = 1_{J}.

Proposition 2.5.19. In the situation of (2.5.18), the pseudo-functor φ : J → I is 2-cofinal.

Proof. For every pseudo-functor F : I → 𝒞, every pseudo-cocone π : F ⇒ F_{L}, and every C ∈ Ob(𝒞) we have a commutative diagram of categories :

\xymatrix{ & 𝒞 (L, C) \ar[dl] \ar@{-->}[dr] \\ PsNat(F, F_{C}) \ar[rr]^{S_{C}} &  & PsNat(F ∘ φ, F_{C})}

whose downward arrows are the functors f ↦ π⊙F_{f} and f ↦ (π∗φ)⊙F_{f} of definition 2.5.1(ii), and where S_{C} is given by the rules : (β : F ⇒ F_{C}) ↦ β ∗ φ and (Ξ : β ⇝ β′) ↦ Ξ ∘ φ. We come down to checking that S_{C} is an equivalence, for every such F. To this aim, notice first that the adjoint pair (σ, φ) admits an adjunction whose counit σ ∘ φ ⇒ 1_{J} is the identity natural transformation (proposition 1.1.20(iii)); in view of the triangular identities of (1.1.13), it follows that the unit η : 1_{I} ⇒ φ ∘ σ fulfills as well the condition :

η ∗ φ = 1_{φ}.

We consider for every C ∈ Ob(𝒞) the functor

S′_{C} : PsNat(F ∘ φ, F_{C}) → PsNat(F, F_{C}) (β : F ∘ φ ⇒ F_{C}) ↦ (β ∗ σ) ⊙ (F ∗ η)

that assigns to every modification Ξ : β ⇝ β′ the modification (Ξ ∘ σ) ∗ (F ∗ η) : S′_{C}(β) ⇝ S′_{C}(β′). Notice that for every pseudo-natural transformation β : F ∘ φ ⇒ F_{C} we have

S_{C} ∘ S′_{C}(β) = ((β ∗ σ) ⊙ (F ∗ η)) ∗ φ = (β ∗ σ ∗ φ) ⊙ (F ∗ η ∗ φ) = β

and likewise, if β′ : F ∘ φ ⇒ F_{C} is another such pseudo-natural transformation, and Ξ : β ⇝ β′ is any modification, we get S_{C} ∘ S′_{C}(Ξ) = Ξ. On the other hand, for every pseudo-natural transformation β : F ⇒ F_{C}, example 2.2.15(ii) yields an invertible modification

τ_{η}^{β} : β = (F_{C} ∗ η) ⊙ (β ∗ 1_{I}) ⇝ (β ∗ φ ∗ σ) ⊙ (F ∗ η) = S′_{C} ∘ S_{C}(β).

To conclude the proof, it suffices to check that the rule β ↦ τ_{η}^{β} yields a natural transformation 1_{PsNat(F,F_{C})} ⇒ S′_{C} ∘ S_{C}. The latter comes down to the commutativity of the diagram of modifications :

[figure: β; τ_{η}^{β}; Ξ; (β ∗ φ ∗ σ) ⊙ (F ∗ η); (Ξ∘φ∘σ)∗(F ∗η); β′; τ_{η}^{β′}; (β′ ∗ φ ∗ σ) ⊙ (F ∗ η)]

for every modification Ξ : β ⇝ β′. This in turn amounts to the compatibility condition for Ξ, relative to the 1-cell η_{i} : i → φ ∘ σ(i), for every i ∈ Ob(I). □

2.6. 2-Categorical Kan extensions. Let I, J, 𝒞 be three 2-categories, φ : I → J a unital pseudo-functor, and suppose that 𝒞 is 2-complete, I is small and J has small Hom-categories. We define as follows a pseudo-functor

2-∫_{φ} : PsFun(I, 𝒞) → PsFun(J, 𝒞).

For every unital pseudo-functor F : I → 𝒞 and every j ∈ Ob(J), notice that the 2-category j/φI is small under our assumptions, and choose a 2-limit (L_{F,j}, π^{F,j}) of F ∘ t_{j} : j/φI → 𝒞 (notation of example 2.2.8(ii)). Then, for every 1-cell g : j → j′ in J there exist a 1-cell L_{F,g} : L_{F,j} → L_{F,j′} in 𝒞 and an invertible modification

Ω^{F,g} : π^{F,j} ∗ g^{∗} ⇝ π^{F,j′} ⊙ F_{L_{F,g}}

with the strict pseudo-functor g^{∗} : j′/φI → j/φI defined as in example 2.2.8(ii), and we set

(2.6.1)  2-∫_{φ} F (j) := L_{F,j}  2-∫_{φ} F (g) := L_{F,g}.

Obviously, for g = 1_{j}, we take L_{F,g} to be the identity 1-cell of L_{F,j}, and Ω^{F,g} the identity modification of π^{F,j}. If h : j′ → j′′ is another 1-cell in J, there exists a unique invertible 2-cell L_{F,g,h} : L_{F,h} ∘ L_{F,g} ⇒ L_{F,h∘g} in 𝒞 such that

(2.6.2)  (π^{F,j′′} ∗ F_{L_{F,g,h}}) ⊙ (Ω^{F,h} ∗ F_{L_{F,g}}) ⊙ (Ω^{F,g} ∘ h^{∗}) = Ω^{F,h∘g}.

Moreover, if g, g′ : j → j′ are two 1-cells of J, and β : g ⇒ g′ is a 2-cell, we set

π_{f}^{F,β} := τ_{β^{∗}(f)}^{F,j} for every (f : j′ → φ(i)) ∈ Ob(j′/φI)

where τ^{F,j} denotes the coherence constraint of π^{F,j}, and β^{∗} : g^{∗} ⇒ g^{′∗} is the strict pseudonatural transformation associated with β as in example 2.2.8(ii). Since F is unital, π_{f}^{F,β} is a 2-cell π_{g^{∗}(f)}^{F,j} ⇒ π_{g^{′∗}(f)}^{F,j} in 𝒞, between two 1-cells L_{F,j} → F i. We claim that the rule : (f : j′ → φ(i)) ↦ π_{f}^{F,β} defines an invertible modification π^{F,β} : π^{F,j} ∗ g ⇝ π^{F,j} ∗ g′. Indeed,

notice first that for every pair of 1-cells (i_{1}, g_{1}) ⟶^{(f,α)} (i_{2}, g_{2}) ⟶^{(f′,α′)} (i_{3}, g_{3}) of j/φ(I) we have by definition :

(2.6.3)  (f′, α′) ∘ (f, α) = α′ ⊙ (φ(f′) ∗ α) ⊙ ((γ_{f,f′}^{φ})^{−1} ∗ g_{1})  in j/φ(I).

Now, again using the unital assumption for F, the assertion comes down to the identity :

τ_{g^{′∗}(h,α)}^{F,j} ⊙ (F h ∗ τ_{β^{∗}(f)}^{F,j}) = τ_{β^{∗}(f′)}^{F,j} ⊙ τ_{g^{∗}(h,α)}^{F,j}

for every 1-cell (h, α) : (f : j′ → φ(i)) → (f′ : j′ → φ(i′)) of j′/φI. Let us also remark that β^{∗}(f′)∘g^{∗}(α, f) = g^{′∗}(α, f)∘β^{∗}(f), and β^{∗}(f) = (1_{i}, f ∗β) and likewise for β^{∗}(f′); since also φ is unital, we then have γ_{1_{i},h}^{φ} = 1_{h} = γ_{h,1_{i′}}^{φ}, and the sought identity follows from (2.6.3) and the coherence axiom for τ^{F,j}. Consequently, there exists a unique 2-cell L_{F,β} : L_{F,g} ⇒ L_{F,g′} in 𝒞 such that

(2.6.4)  Ω^{F,g′} ⊙ π^{F,β} = (π^{F,j′} ∗ F_{L_{F,β}}) ⊙ Ω^{F,g}

and we set

(2.6.5)  2-∫_{φ} F (β) := L_{F,β}.

Lemma 2.6.6. In the situation of (2.6), the system of invertible 2-cells L_{F,••} yields a coherence constraint for a unital pseudo-functor 2-∫_{φ} F : J → 𝒞 defined on objects and 1-cells by (2.6.1), and on 2-cells by (2.6.5).

Proof. Let β : g ⇒ g′ and β′ : g′ ⇒ g′′ be two 2-cells of J; a simple inspection shows that

β^{′∗}(f) ∘ β^{∗}(f) = (β′ ⊙ β)^{∗}(f) for every (f : j′ → φ(i)) ∈ Ob(j′/φI)

whence π^{F,β′} ⊙ π^{F,β} = π^{F,β′⊙β}. On the other hand, we have

\begin{matrix}(π^{F,j′} ∗ F_{L_{F,β′}}) ⊙ (π^{F,j′} ∗ F_{L_{F,β}}) ⊙ Ω^{F,g} = & (π^{F,j′} ∗ F_{L_{F,β′}}) ⊙ Ω^{F,g′} ⊙ π^{F,β} \\ = & Ω^{F,g′′} ⊙ π^{F,β′} ⊙ π^{F,β}. \end{matrix}

Summing up, we already see that 2-∫_{φ} F (β′) ⊙ 2-∫_{φ} F (β) = 2-∫_{φ} F (β′ ⊙ β). Moreover, notice that π^{F,1_{g}} is the identity automorphism of π^{F,j} ∗ g (remark 2.4.2(ii)), whence

2-∫_{φ} F (1_{g}) = 1_{L_{F,g}} for every 1-cell g of J.

Next, let g, g′ : j → j′ and h, h′ : j′ → j′′ be four 1-cells of J and β : g ⇒ g′, β : h ⇒ h′ two 2-cells; we need to check that

(2-∫_{φ} F (β′ ∗ β)) ⊙ L_{F,g,h} = L_{F,g′,h′} ⊙ (2-∫_{φ} F (β′) ∗ 2-∫_{φ} F (β))

and in light of the foregoing, we may assume that either β = 1_{g} or β′ = 1_{h}. In these cases, the assertion comes down to the identities :

\begin{matrix}L_{F,h∗β} ⊙ L_{F,g,h} = & L_{F,g′,h} ⊙ (L_{F,h} ∗ L_{F,β}) \\ L_{F,β′∗g} ⊙ L_{F,g,h} = & L_{F,g,h′} ⊙ (L_{F,β′} ∗ L_{F,g}). \end{matrix}

However, set X := (Ω^{F,h} ∗ F_{L_{F,g}}) ⊙ (Ω^{F,g} ∘ h^{∗}). It suffices to show that

Y := (π^{F,j′′} ∗ (F_{L_{F,g′,h}} ⊙ (F_{L_{F,h}} ∗ F_{L_{F,β}})) ⊙ X = Z := (π^{F,j′′} ∗ (F_{L_{F,h∗β}} ⊙ F_{L_{F,g,h}})) ⊙ X
Y′ := (π^{F,j′′} ∗ (F_{L_{F,g,h′}} ⊙ (F_{L_{F,β′}} ∗ F_{L_{F,g}})) ⊙ X = Z′ := (π^{F,j′′} ∗ (F_{L_{F,β′∗g}} ⊙ F_{L_{F,g,h}})) ⊙ X.

We compute :

Y = (π^{F,j′′} ∗ F_{L_{F,g′,h}}) ⊙ (Ω^{F,h} ∗ F_{L_{F,g′}}) ⊙ ((π^{F,j′} ∗ h^{∗}) ∗ F_{L_{F,β}}) ⊙ (Ω^{F,g} ∘ h^{∗})
= (π^{F,j′′} ∗ F_{L_{F,g′,h}}) ⊙ (Ω^{F,h} ∗ F_{L_{F,g′}}) ⊙ (Ω^{F,g′} ∘ h^{∗}) ⊙ (π^{F,β} ∘ h^{∗})
= Ω^{F,h∘g′} ⊙ (π^{F,β} ∘ h^{∗}).

and on the other hand, Z = (π^{F,j′′}∗F_{L_{F,h∗β}})⊙Ω^{F,h∘g} = Ω^{F,h∘g′}⊙π^{F,1_{h}∗β}. So, for the first identity we are reduced to showing that π^{F,β} ∘ h^{∗} = π^{F,1_{h}∗β}. The latter follows by direct inspection. Next, we compute :

Y′ = (π^{F,j′′} ∗ F_{L_{F,g,h′}}) ⊙ (Ω^{F,h′} ∗ F_{L_{F,g}}) ⊙ (π^{F,β′} ∗ F_{L_{F,g}}) ⊙ (Ω^{F,g} ∘ h^{∗})
= (Ω^{F,h′∘g}) ⊙ (Ω^{F,g} ∗ h^{′∗})^{−1} ⊙ (π^{F,β′} ∗ F_{L_{F,g}}) ⊙ (Ω^{F,g} ∘ h^{∗})

and on the other hand, Z′ = (π^{F,j′′} ∗ F_{L_{F,β′∗1g}}) ⊙ Ω^{F,h∘g} = Ω^{F,h′∘g} ⊙ π^{F,β′∗g}. So, for the second identity we are reduced to checking that

(Ω^{F,g} ∗ h^{′∗}) ⊙ π^{F,β′∗g} = (π^{F,β′} ∗ F_{L_{F,g}}) ⊙ (Ω^{F,g} ∘ h^{∗}).

But this identity follows from the compatibility condition for Ω^{F,g}. Lastly, let us check the composition and unit axioms for 2-∫_{φ} F. Hence, let j ⟶^{g} j′ ⟶^{h} j′′ ⟶^{k} j′′′ be three 1-cells of J; we need to prove that

L_{F,h∘g,k} ⊙ (L_{F,k} ∗ L_{F,g,h}) = L_{F,g,k∘h} ⊙ (L_{F,h,k} ∗ L_{F,g}) and L_{F,h,1_{j′}} = 1_{L_{F,h}} = L_{F,1_{j},h}.

The two identities that express the unit axiom follow by a simple inspection. For the composition axiom, set X := (Ω^{F,k} ∗ F_{L_{F,h}} ∗ F_{L_{F,g}}) ⊙ ((Ω^{F,h} ∘ k^{∗}) ∗ F_{L_{F,g}}) ⊙ (Ω^{F,g} ∘ h^{∗} ∘ k^{∗}). It suffices to show that

Y := (π^{F,j′′′}∗(F_{L_{F,h∘g,k}}⊙(F_{L_{F,k}}∗F_{L_{F,g,h}})))⊙X = Z := (π^{F,j′′′}∗(F_{L_{F,g,k∘h}}⊙(F_{L_{F,h,k}}∗F_{L_{F,g}})))⊙X.

We compute :

Y =(π^{F,j′′′}∗F_{L_{F,h∘g,k}})⊙(Ω^{F,k}∗F_{L_{F,h∘g}})⊙((π^{F,j′′}∗k^{∗})∗F_{L_{F,g,h}})⊙((Ω^{F,h}∘k^{∗})∗F_{L_{F,g}})⊙(Ω^{F,g}∘h^{∗}∘k^{∗})

=(π^{F,j′′′} ∗ F_{L_{F,h∘g,k}}) ⊙ (Ω^{F,k} ∗ F_{L_{F,h∘g}}) ⊙ (Ω^{F,h∘g} ∘ k^{∗})

=Ω^{F,k∘h∘g}

and a similar calculation yields as well Z = Ω^{F,k∘h∘g} : details left to the reader. □

Remark 2.6.7. In the situation of (2.6), suppose moreover that 𝒞 is strongly 2-complete. In this case we can choose for (L_{F,j}, π^{F,j}) a strong 2-limit of F ∘ t_{j}, for every j ∈ Ob(J). Then, we may also choose L_{F,g} for every 1-cell g : j → j′ in J such that π^{F,j} ∗ g^{∗} = π^{F,j′} ⊙ F_{L_{F,g}}, and take Ω^{F,g} to be the identity modification. With these choices, it follows easily that L_{F,g,h} is an identity 2-cell, for every composable pair of 1-cells g, h of J. We conclude that, in this situation, the pseudo-functor 2-∫_{φ} F of lemma 2.6.6shall be strict.

2.6.8. Keep the notation of (2.6), and let F, F′ : I → 𝒞 be two unital pseudo-functors, and β : F ⇒ F′ a pseudo-natural transformation. Then, for every j ∈ Ob(J) there exist a 1-cell L_{β,j} : L_{F,j} → L_{F′,j} and an invertible modification

Ω^{β,j} : (β ∗ t_{j}) ⊙ π^{F,j} ⇝ π^{F′,j} ⊙ F_{L_{β,j}}.

Evidently, we take L_{1_{F} ,j} := 1_{L_{F,j}} and Ω^{1_{F} ,j} := 1_{π^{F,j}} for every such F and j. If g : j → j′ is any 1-cell of J, there exists a unique invertible 2-cell L_{β,g} : L_{F′,g} ∘ L_{β,j} ⇒ L_{β,j′} ∘ L_{F,g} such that

(2.6.9) (π^{F′,j′} ∗ F_{L_{β,g}}) ⊙ (Ω^{F′,g} ∗ F_{L_{β,j}}) ⊙ (Ω^{β,j} ∘ g^{∗}) = (Ω^{β,j′} ∗ F_{L_{F,g}}) ⊙ ((β ∗ t_{j′}) ∗ Ω^{F,g}).

A direct inspection shows that L_{1_{F} ,g} = 1_{L_{F,g}} for every such F and g.

If β′ : F′ ⇒ F′′ is another pseudo-natural transformation of unital pseudo-functors, there exists a unique invertible 2-cell L_{β,β′,j} : L_{β′,j} ∘ L_{β,j} ⇒ L_{β′⊙β,j} such that

(2.6.10)  (π^{F′′,j} ∗ F_{L_{β,β′,j}}) ⊙ (Ω^{β′,j} ∗ F_{L_{β,j}}) ⊙ ((β′ ∗ t_{j}) ∗ Ω^{β,j}) = Ω^{β′⊙β,j}.

Lemma 2.6.11. In the situation of (2.6.8), the system of 2-cells (L_{β,g} | g : j → j′) yields a coherence constraint for a pseudo-natural transformation

2-∫_{φ} β : 2-∫_{φ} F ⇒ 2-∫_{φ} F′ j ↦ L_{β,j}.

Proof. Clearly L_{β,1_{j}} = 1_{L_{β,j}} for every j ∈ Ob(J). Next, let j ⟶^{g} j′ ⟶^{g′} j′′ be two 1-cells of J; we need to check the identity :

(L_{β,j′′} ∗ L_{F,g,g′}) ⊙ (L_{β,g′} ∗ L_{F,g}) ⊙ (L_{F′,g′} ∗ L_{β,g}) = L_{β,g′∘g} ⊙ (L_{F′,g,g′} ∗ L_{β,j}).

To this aim, set

X := (Ω^{F′,g′} ∗ F_{L_{F′,g}} ∗ F_{L_{β,j}}) ⊙ ((Ω^{F′,g} ∘ g^{′∗}) ∗ F_{L_{β,j}}) ⊙ (Ω^{β,j} ∘ g^{∗} ∘ g^{′∗})
Y := π^{F′,j′′} ∗ ((F_{L_{β,j′′}} ∗ F_{L_{F,g,g′}}) ⊙ (F_{L_{β,g′}} ∗ F_{L_{F,g}}))
Z := Y ⊙ (π^{F′,j′′} ∗ F_{L_{F′,g′}} ∗ F_{L_{β,g}}) ⊙ X
U := (π^{F′,j′′} ∗ (F_{L_{β,g′∘g}} ⊙ (F_{L_{F′,g,g′}} ∗ F_{L_{β,j}}))) ⊙ X.

It suffices to show that U = Z. We compute :

Z = Y ⊙(Ω^{F′,g′}∗F_{L_{β,j′}} ∗ F_{L_{F,g}})⊙((π^{F′,j′}∗ g^{′∗}) ∗ F_{L_{β,g}})⊙((Ω^{F′,g} ∘ g^{′∗}) ∗ F_{L_{β,j}})⊙(Ω^{β,j} ∘g^{∗}∘g^{′∗})

= Y ⊙ (Ω^{F′,g′} ∗ F_{L_{β,j′}} ∗ F_{L_{F,g}}) ⊙ ((Ω^{β,j′} ∘ g^{′∗}) ∗ F_{L_{F,g}}) ⊙ ((β ∗ t_{j′′})∗(Ω^{F,g} ∘ g^{′∗}))

= (π^{F′,j′′}∗F_{L_{β,j′′}∗L_{F,g,g′}})⊙(Ω^{β,j′′}∗F_{L_{F,g′}∗L_{F,g}})⊙((β∗t_{j′′}) ∗ Ω^{F,g′}∗F_{L_{F,g}})⊙((β∗t_{j′′})∗(Ω^{F,g}∘g^{′∗}))

= (Ω^{β,j′′}∗F_{L_{F,g′∘g}})⊙((β∗t_{j′′})∗π^{F,j′′}∗F_{L_{F,g,g′}})⊙((β∗t_{j′′})∗Ω^{F,g′}∗F_{L_{F,g}})⊙((β∗t_{j′′})∗(Ω^{F,g}∘g^{′∗}))

= (Ω^{β,j′′} ∗ F_{L_{F,g′∘g}}) ⊙ ((β ∗ t_{j′′}) ∗ Ω^{F,g′∘g})

and on the other hand U = (π^{F′,j′′} ∗ F_{L_{β,g′∘g}}) ⊙ (Ω^{F′,g′∘g} ∗ F_{L_{β,j}}) ⊙ (Ω^{β,j} ∘ g^{∗} ∘ g^{′∗}), whence the contention. Lastly, we check the naturality condition for L_{β,g} : let f, g : j → j′ be two 1-cells of J and α : f ⇒ g a 2-cell; we need to check the identity :

L_{β,g} ⊙ (L_{F′,α} ∗ L_{β,j}) = (L_{β,j′} ∗ L_{F,α}) ⊙ L_{β,f}.

To this aim, set X := (Ω^{F′,f} ∗ F_{L_{β,j}}) ⊙ (Ω^{β,j} ∘ f^{∗}); it suffices to show that

Y′ := (π^{F′,j′} ∗ (F_{L_{β,g}} ⊙ (F_{L_{F′,α}} ∗ F_{L_{β,j}}))) ⊙ X = Z′ := (π^{F′,j′} ∗ ((F_{L_{β,j′}} ∗ F_{L_{F,α}}) ⊙ F_{L_{β,f}})) ⊙ X.

Claim 2.6.12. (π^{F′,α} ∗ F_{L_{β,j}}) ⊙ (Ω^{β,j} ∘ f^{∗}) = (Ω^{β,j} ∘ g^{∗}) ⊙ ((β ∗ t_{j′}) ∗ π^{F,α}).

Proof of the claim. Recall that α induces a pseudo-natural transformation α^{∗} : f^{∗} ⇒ g^{∗}, and α^{∗}(h) = (1_{i}, α) : h ∘ f → h ∘ g for every (h : j′ → φ(i)) ∈ Ob(j′/φI). The compatibility condition of Ω^{β,j} then yields a commutative diagram

\xymatrix{β_{i} ∘ π_{f^{∗}(h)}^{F,j} \ar@{=>}[d]_{β_{i}∗π_{h}^{F,α}} \ar@{=>}[r]^{Ω_{f^{∗}(h)}^{β,j}} & π_{f^{∗}(h)}^{F′,j} ∘ L_{β,j} \ar@{=>}[d]^{π_{h}^{F′,α}} \\ β_{i} ∘ π_{g^{∗}(h)}^{F,j} \ar@{=>}[r]^{Ω_{g^{∗}(h)}^{β,j}} & π_{g^{∗}(h)}^{F′,j} ∘ L_{β,j}.}

The claim is an immediate consequence. ♢ Using claim 2.6.12, we may compute :

Y′ = (π^{F′,j′} ∗ F_{L_{β,g}}) ⊙ (Ω^{F′,g} ∗ F_{L_{β,j}}) ⊙ (π^{F′,α} ∗ F_{L_{β,j}}) ⊙ (Ω^{β,j} ∘ f^{∗})
= (π^{F′,j′} ∗ F_{L_{β,g}}) ⊙ (Ω^{F′,g} ∗ F_{L_{β,j}}) ⊙ (Ω^{β,j} ∘ g^{∗}) ⊙ ((β ∗ t_{j′}) ∗ π^{F,α})
= (Ω^{β,j′} ∗ F_{L_{F,g}}) ⊙ ((β ∗ t_{j′}) ∗ Ω^{F,g}) ⊙ ((β ∗ t_{j′}) ∗ π^{F,α})
= (Ω^{β,j′} ∗ F_{L_{F,g}}) ⊙ (β ∗ t_{j′} ∗ ((π^{F,j′} ∗ F_{L_{F,α}}) ⊙ Ω^{F,f}))

and on the other hand Z′ = (π^{F′,j′} ∗ (F_{L_{β,j′}} ∗ F_{L_{F,α}})) ⊙ (Ω^{β,j′} ∗ F_{L_{F,f}}) ⊙ ((β ∗ t_{j′}) ∗ Ω^{F,f}). So, we are reduced to proving that

(Ω^{β,j′} ∗ F_{L_{F,g}}) ⊙ (β ∗ t_{j′} ∗ π^{F,j′} ∗ F_{L_{F,α}}) = (π^{F′,j′} ∗ (F_{L_{β,j′}} ∗ F_{L_{F,α}})) ⊙ (Ω^{β,j′} ∗ F_{L_{F,f}}).

But the latter follows as usual from remark 2.1.1(i). □

Lemma 2.6.13. In the situation of (2.6.8), the system of 2-cells (L_{β,β′,j} | j ∈ Ob(J)) yields an invertible modification

L_{β,β′} : 2-∫_{φ} β′ ⊙ 2-∫_{φ} β ⇝ 2-∫_{φ} β′ ⊙ β.

Proof. Let g : j → j′ be any 1-cell of J; we need to show the identity

(L_{β,β′,j′} ∗ L_{F,g}) ⊙ (L_{β′,j′} ∗ L_{β,g}) ⊙ (L_{β′,g} ∗ L_{β,j}) = L_{β′⊙β,g} ⊙ (L_{F′′,g} ∗ L_{β,β′,j}).

To this aim, let us set X := ((β′ ∗ t_{j′}) ∗ Ω^{β,j′} ∗ F_{L_{F,g}})^{−1} ⊙ (Ω^{β′,j′} ∗ F_{L_{β,j′}} ∗ F_{L_{F,g}})^{−1} and

Y_{1} := X ⊙ (π^{F′′,j′} ∗ F_{L_{β′,j′}} ∗ F_{L_{β,g}})
Y_{2} := π^{F′′,j′} ∗ ((F_{L_{β′,g}} ∗ F_{L_{β,j}}) ⊙ (F_{L_{F′′,g}} ∗ F_{L_{β,β′,j}})^{−1})
Z := X ⊙ (π^{F′′,j′} ∗ ((F_{L_{β,β′,j′}} ∗ F_{L_{F,g}})^{−1} ⊙ F_{L_{β′⊙β,g}})).

It suffices to check that Y := Y_{1} ⊙ Y_{2} = Z. Set as well U := (((β′ ⊙ β) ∗ t_{j′}) ∗ Ω^{F,g}); we compute :

\begin{aligned}Z & = (Ω^{β′⊙β,j′} ∗ F_{L_{F,g}})^{−1} ⊙ (π^{F′′,j′} ∗ F_{L_{β′⊙β,g}}) \\ & = U ⊙ (Ω^{β′⊙β,j} ∘ g^{∗})^{−1} ⊙ (Ω^{F′′,g} ∗ F_{L_{β′⊙β,j}})^{−1}. \end{aligned}

On the other hand, set U′ := (Ω^{F′′,g} ∗ F_{L_{β′,j}})^{−1} ⊙ (π^{F′′,j′} ∗ F_{L_{β′,g}})^{−1}) ∗ F_{L_{β,j}}; then

Y = ((β′ ∗ t_{j′}) ∗ Ω^{β,j′} ∗ F_{L_{F,g}})^{−1}⊙((β′ ∗ t_{j′}) ∗ π^{F′,j′} ∗ F_{L_{β,g}})⊙(Ω^{β′,j′} ∗ F_{L_{F′,g}} ∗ F_{L_{β,j}})^{−1}⊙Y_{2}

= U ⊙ ((β′ ∗ t_{j′}) ∗ ((Ω^{β,j} ∘ g^{∗})^{−1} ⊙ (Ω^{F′,g} ∗ F_{L_{β,j}})^{−1}) ⊙ (Ω^{β′,j′} ∗ F_{L_{F′,g}} ∗ F_{L_{β,j}})^{−1} ⊙ Y_{2}

= U ⊙ ((β′ ∗ t_{j′}) ∗ (Ω^{β,j} ∘ g^{∗})^{−1}) ⊙ ((Ω^{β′,j} ∘ g^{∗})^{−1} ∗ F_{L_{β,j}}) ⊙ U′ ⊙ Y_{2}.

Therefore, we are reduced to checking the identity :

((π^{F′′,j} ∗ g^{∗}) ∗ F_{β,β′,j})^{−1} ⊙ (Ω^{F′′,g} ∗ F_{L_{β′⊙β,j}})^{−1} = U′ ⊙ Y_{2}.

However, we have :

((π^{F′′,j}∗g^{∗})∗F_{β,β′,j})^{−1}⊙(Ω^{F′′,g}∗F_{L_{β′⊙β,j}})^{−1} =(Ω^{F′′,g}∗F_{L_{β′,j}∗L_{β,j}})^{−1}⊙(π^{F′′,j′}∗F_{L_{F′′,g}} ∗F_{β,β′,j})^{−1}

so we are further reduced to showing that :

(π^{F′′,j′} ∗ F_{L_{F′′,g}} ∗ F_{β,β′,j})^{−1} = (π^{F′′,j′} ∗ F_{L_{β′,g}} ∗ F_{L_{β,j}})^{−1} ⊙ Y_{2}

which is obvious. □

2.6.14. Let F, F′ : I → 𝒞 be two unital pseudo-functors, β, β′ : F ⇒ F′ two pseudo-natural transformations, and Ξ : β ⇝ β′ a modification. Then for every j ∈ Ob(J) there exists a unique modification L_{Ξ,j} : L_{β,j} ⇝ L_{β′,j} such that

Ω^{β′,j} ⊙ ((Ξ ∘ t_{j}) ∗ π^{F,j}) = (π^{F′,j} ∗ F_{L_{Ξ,j}}) ⊙ Ω^{β,j}.

Lemma 2.6.15. In the situation of (2.6.14), the rule j ↦ L_{Ξ,j} defines a modification

2-∫_{φ} Ξ : 2-∫_{φ} β ⇝ 2-∫_{φ} β′.

Proof. Let g : j → j′ be any 1-cell of J; we need to check the identity :

(L_{Ξ,j′} ∗ L_{F,g}) ⊙ L_{β,g} = L_{β′,g} ⊙ (L_{F′,g} ∗ L_{Ξ,j})

and as usual, it suffices to show that

Y := π^{F′,j′} ∗ ((F_{L_{Ξ,j′}} ∗ F_{L_{F,g}}) ⊙ F_{L_{β,g}}) = Z := π^{F′,j′} ∗ (F_{L_{β′,g}} ⊙ (F_{L_{F′,g}} ∗ F_{L_{Ξ,j}})).

We compute :

Y =((Ω^{β′,j′} ⊙ ((Ξ ∘ t_{j′}) ∗ π^{F,j′}) ⊙ (Ω^{β,j′})^{−1}) ∗ F_{L_{F,g}}) ⊙ (π^{F′,j′} ∗ F_{L_{β,g}})

=((Ω^{β′,j′} ⊙((Ξ ∘ t_{j′}) ∗ π^{F,j′})) ∗ F_{L_{F,g}})⊙((β∗t_{j′}) ∗ Ω^{F,g})⊙(Ω^{β,j} ∘ g^{∗})^{−1}⊙(Ω^{F′,g} ∗ F_{L_{β,j}})^{−1}

=(Ω^{β′,j′}∗F_{L_{F,g}})⊙((β′ ∗ t_{j′})∗Ω^{F,g})⊙((Ξ ∘ t_{j′}) ∗ (π^{F,j} ∗g^{∗}))⊙(Ω^{β,j} ∘g^{∗})^{−1}⊙(Ω^{F′,g}∗F_{L_{β,j}})^{−1}

=(Ω^{β′,j′}∗F_{L_{F,g}})⊙((β′ ∗ t_{j′})∗Ω^{F,g})⊙(Ω^{β′,j} ∘g^{∗})^{−1}⊙((π^{F′,j} ∗ g^{∗}) ∗ F_{L_{Ξ,j}})⊙(Ω^{F′,g} ∗ F_{L_{β,j}})^{−1}

=(π^{F′,j′} ∗ F_{L_{β′,g}}) ⊙ (Ω^{F′,g} ∗ F_{L_{β′,j}}) ⊙ ((π^{F′,j} ∗ g^{∗}) ∗ F_{L_{Ξ,j}})⊙(Ω^{F′,g} ∗ F_{L_{β,j}})^{−1}

=(π^{F′,j′} ∗ F_{L_{β′,g}}) ⊙ (Ω^{F′,g} ∗ F_{L_{β′,j}}) ⊙ (Ω^{F′,g} ∗ F_{β′,j})^{−1} ⊙ (π^{F′,j′} ∗ F_{L_{F,g}} ∗ F_{Ξ,j})

=(π^{F′,j′} ∗ F_{L_{β′,g}}) ⊙ (π^{F′,j′} ∗ F_{L_{F,g}} ∗ F_{Ξ,j})

whence the contention. □

Proposition 2.6.16. Let I, J, 𝒞 be as in (2.6), and φ : I → J any pseudo-functor. The rules:

F ↦ 2-∫_{φ^{u}} F^{u}  β ↦ 2-∫_{φ^{u}} β^{u}  Ξ ↦ 2-∫_{φ^{u}} Ξ^{u}

for every pseudo-functor F : I → 𝒞, every pseudo-natural transformation β between pseudofunctors F, F′ : I → 𝒞, and every modification Ξ of such pseudo-natural transformations (notation of remark 2.4.4(i)) define a unital pseudo-functor

2-∫_{φ} : PsFun(I, 𝒞) → PsFun(J, 𝒞)

with coherence constraint given by the system of invertible modifications L_{β,β′} of lemma 2.6.13. We call this pseudo-functor the right 2-Kan extension along φ.

Proof. The sought functor shall be the composition of the strict 2-equivalence of remark 2.4.4(i) and a similar pseudo-functor uniPsFun(I, 𝒞) → uniPsFun(J, 𝒞) given by the foregoing rules; we are therefore reduced to prove the existence of the latter. Now, a direct inspection shows that L_{1_{F} ,β,j} = 1_{L_{β,j}} = L_{β,1_{F′},j} for every pair of unital pseudo-functors F, F′ : I → 𝒞, every β : F ⇒ F′ and every j ∈ Ob(J), whence the unit axiom for the system of modifications L_{β,β′}. In order to check the composition axiom, we need to show the identity :

L_{β′⊙β,β′′,j} ⊙ (L_{β′′,j} ∗ L_{β,β′,j}) = L_{β,β′′⊙β′,j} ⊙ (L_{β′,β′′,j} ∗ L_{β,j})

for every j ∈ Ob(J) and every three pseudo-natural transformations of unital pseudo-functors β : F ⇒ F′, β′ : F′ ⇒ F′′, β′′ : F′′ ⇒ F′′′. To this aim, set X := Ω^{β′′⊙β′⊙β,j}, as well as :

\begin{aligned}Y & := (π^{F′′′,j} ∗ ((F_{L_{β′′,j}} ∗ F_{L_{β,β′,j}})^{−1} ⊙ F_{L_{β′⊙β,β′′,j}}^{−1})) ⊙ X \\ Z & := (π^{F′′′,j} ∗ ((F_{L_{β′,β′′,j}} ∗ F_{L_{β,j}})^{−1} ⊙ F_{L_{β,β′′⊙β′,j}}^{−1})) ⊙ X. \end{aligned}

It suffices to prove that Y = Z. We compute :

Y = (π^{F′′′,j} ∗ F_{L_{β′′,j}} ∗ F_{L_{β,β′,j}})^{−1} ⊙ (Ω^{β′′,j} ∗ F_{L_{β′⊙β,j}}) ⊙ ((β′′ ∗ t_{j}) ∗ Ω^{β′⊙β,j})
= (Ω^{β′′,j} ∗ F_{L_{β′,j}} ∗ F_{L_{β,j}}) ⊙ ((β′′ ∗ t_{j}) ∗ π^{F′′,j} ∗ F_{L_{β,β′,j}})^{−1} ⊙ ((β′′ ∗ t_{j}) ∗ Ω^{β′⊙β,j})
= (Ω^{β′′,j} ∗ F_{L_{β′,j}} ∗ F_{L_{β,j}}) ⊙ ((β′′ ∗ t_{j}) ∗ ((Ω^{β′,j} ∗ F_{L_{β,j}}) ⊙ ((β′ ∗ t_{j}) ∗ Ω^{β,j})))

and likewise :

Z = (π^{F′′′,j} ∗ F_{L_{β′,β′′,j}} ∗ F_{L_{β,j}})^{−1} ⊙ (Ω^{β′′⊙β′,j} ∗ F_{L_{β,j}}) ⊙ ((β′′ ⊙ β′ ∗ t_{j}) ∗ Ω^{β,j})
= (((Ω^{β′′,j} ∗ F_{L_{β′,j}}) ⊙ ((β′′ ∗ t_{j}) ∗ Ω^{β′,j})) ∗ F_{L_{β,j}}) ⊙ ((β′′ ⊙ β′ ∗ t_{j}) ∗ Ω^{β,j})

whence the contention. Next, consider three pseudo-natural transformations β, β′, β′′ : F ⇒ F′ and two modifications Ξ : β ⇝ β′ and Ξ′ : β′ ⇝ β′′. For every j ∈ Ob(J) we have :

Ω^{β′′,j}⊙(((Ξ′⊙Ξ)∘t_{j})∗π^{F,j})=(π^{F′,j}∗F_{L_{Ξ′,j}})⊙Ω^{β′,j}⊙((Ξ∘t_{j})∗π^{F,j})=(π^{F′,j}∗F_{L_{Ξ′,j}}∗F_{L_{Ξ,j}})⊙Ω^{β,j}

whence L_{Ξ′⊙Ξ,j} = L_{Ξ′,j} ⊙ L_{Ξ,j}. This shows that 2-∫_{φ}(Ξ′ ⊙ Ξ) = 2-∫_{φ} Ξ′ ⊙ 2-∫_{φ} Ξ. Moreover, a simple inspection shows that L_{1_{β},j} = 1_{L_{β,j}} for every pseudo-natural transformation β and every j ∈ Ob(J), so 2-∫_{φ}(1_{β}) is the identity modification of 2-∫_{φ} β.

Lastly, let F, F′, F′′ : I → 𝒞 be three unital pseudo-functors, β, β : F ⇒ F′ and α, α′ : F′ ⇒ F′′ four pseudo-natural transformations, and Ξ : β ⇝ β′, Θ : α ⇝ α′ two modifications; we need to show that

L_{Θ∗Ξ,j} ⊙ L_{β,α,j} = L_{β′,α′,j} ⊙ (L_{Θ,j} ∗ L_{Ξ,j}) for every j ∈ Ob(J)

and by the foregoing, we may assume that either Ξ = 1_{β} or Θ = 1_{α}. Suppose first that Θ = 1_{α}, and let X := (Ω^{α,j} ∗ F_{β,j}) ⊙ ((α ∗ t_{j}) ∗ Ω^{β,j}); it suffices to prove that

Y := (π^{F′′,j} ∗ (F_{L_{α∗Ξ,j}} ⊙ F_{L_{β,α,j}})) ⊙ X = Z := (π^{F′′,j} ∗ (F_{L_{β′,α,j}} ⊙ (F_{L_{α,j}} ∗ F_{L_{Ξ,j}}))) ⊙ X.

We compute :

Z = (π^{F′′,j} ∗ F_{L_{β′,α,j}}) ⊙ (Ω^{α,j} ∗ F_{β′,j}) ⊙ ((α ∗ t_{j}) ∗ π^{F′,j} ∗ F_{L_{Ξ,j}}) ⊙ ((α ∗ t_{j}) ∗ Ω^{β,j})
= (π^{F′′,j} ∗ F_{L_{β′,α,j}}) ⊙ (Ω^{α,j} ∗ F_{β′,j}) ⊙ ((α ∗ t_{j}) ∗ (Ω^{β′,j} ⊙ ((Ξ ∘ t_{j}) ∗ π^{F,j})))
= Ω^{α⊙β′,j} ⊙ ((α ∗ t_{j}) ∗ ((Ξ ∘ t_{j}) ∗ π^{F,j})))

and on the other hand, Y = (π^{F′′,j} ∗ F_{L_{α∗Ξ,j}}) ⊙ Ω^{α⊙β}, whence the contention. A similar calculation, left to the reader, settles the case where Ξ = 1_{β}, and concludes the proof. □

2.6.17. On the other hand, if 𝒞 is 2-cocomplete, then 𝒞^{o} is 2-complete (remark 2.5.2(vi)), and in view of the strict isomorphisms of (2.2.13), we get a left 2-Kan extension along φ :

2-∫ ^{φ} : PsFun(I, 𝒞) → PsFun(J, 𝒞) F ↦ (2-∫_{φ^{o}} F^{o})^{o}.

Explicitly, for every j ∈ Ob(J), the object (2-∫^{φ} F)(j) represents the 2-colimit of the pseudofunctor F ∘^{o}s_{^{o}j} : ^{o}(^{o}φ(^{o}I)/^{o}j) → 𝒞.

Remark 2.6.18. (i) In the situation of (2.6.8), suppose that 𝒞 is strongly 2-complete, and choose strong 2-limits (L_{F,j}, π^{F,j}), identity modifications Ω^{F,g}, and 1-cells L_{F,g} as in remark 2.6.7. Then we may further choose L_{β,j} such that (β ∗ t_{j}) ⊙ π^{F,j} = π^{F′,j} ⊙ F_{L_{β,j}}, and let Ω^{β,j} as well be the corresponding identity modification. With such choices, it follows easily that L_{β,g} shall be an identity 2-cell, for every 1-cell g of J, and the same for L_{β,β′}. Therefore, in this situation the pseudo-natural transformation 2-∫_{φ} β of lemma 2.6.11 shall be strict as well, and moreover (2-∫_{φ} β′) ⊙ (2-∫_{φ} β) = 2-∫_{φ}(β′ ⊙ β).

(ii) We conclude that, with these choices, the pseudo-functor 2-∫_{φ} of proposition 2.6.16 factors through a strict pseudo-functor called the strong right 2-Kan extension along φ :

PsFun(I, 𝒞) → stPsFun(J, 𝒞)

(notation of definition 2.2.12(iii)). Likewise, if 𝒞 is strongly 2-cocomplete, we obtain as in (2.6.17), a strong left 2-Kan extension along φ which is another strict pseudo-functors with values in stPsFun(J, 𝒞), namely the opposite of the strong right 2-Kan extension along φ^{o}.

2.6.19. Keep the notation of (2.6), and suppose now that all the 2-cells of J are invertible. Under this assumption, we easily see that the rule : (i, f : j → φ(i)) ↦ f yields a pseudo-cone

φ_{j}^{∗} : F_{j} ⇒ φ ∘ t_{j} for every j ∈ Ob(J)

whose coherence constraint is given by the rule : (h, α) ↦ α (notation of example 2.2.8(ii); notice that this would not be a coherence constraint, if α were not invertible : details left to the reader). Moreover, we have :

(2.6.20)  φ_{j′}^{∗} ⊙ F_{g} = φ_{j}^{∗} ∗ g^{∗}  for every 1-cell g : j → j′ of J.

Furthermore, for every pair of 1-cells g, g′ : j → j′ and every 2-cell α : g ⇒ g′ we get a modification

φ_{j}^{∗} ∗ α^{∗} : φ_{j}^{∗} ∗ g^{∗} ⇝ φ_{j}^{∗} ∗ g^{′∗} (f : j′ → φ(i)) ↦ (α ∗ f : g^{∗}(f) ⇒ g^{′∗}(f)).

Let F : I → 𝒞 and G : J → 𝒞 be two unital pseudo-functors, and β : G ∘ φ ⇒ F a pseudonatural transformation; there exist a 1-cell β_{j}^{†} : Gj → L_{F,j} and an invertible modification

Θ^{β,j} : (β ∗ t_{j}) ⊙ (G ∗ φ_{j}^{∗}) ⇝ π^{F,j} ⊙ F_{β_{j}^{†}} for every j ∈ Ob(J).

Next, let g : j → j′ be any 1-cell of J, and notice that G ∗ φ_{j}^{∗} ∗ g^{∗} = G ∗ (φ_{j′}^{∗} ⊙ F_{g}), due to (2.6.20); according to example 2.2.15(i) the coherence constraint γ^{G} of G induces an invertible modification

Γ^{G,g} : (G ∗ φ_{j′}^{∗}) ⊙ F_{Gg} ⇝ G ∗ φ_{j}^{∗} ∗ g^{∗} (f : j′ → φ(i)) ↦ γ_{g,f}^{G}.

Then there exists a unique invertible 2-cell τ_{g}^{β^{†}} : L_{F,g} ∘ β_{j}^{†} ⇒ β_{j′}^{†} ∘ Gg such that :

(2.6.21) (π^{F,j′} ∗ F_{τ_{g}^{β^{†}}}) ⊙ (Ω^{F,g} ∗ F_{β_{j}^{†}}) ⊙ (Θ^{β,j} ∘ g^{∗}) ⊙ ((β ∗ t_{j′}) ∗ Γ^{G,g}) = Θ^{β,j′} ∗ F_{Gg}.

Lemma 2.6.22. With the notation of (2.6.19), the rule : j ↦ β_{j}^{†} yields a pseudo-natural transformation

β^{†} : G ⇒ 2-∫_{φ} F

with coherence constraint given by the system of 2-cells τ_{•}^{β^{†}}.

Proof. A simple inspection shows that τ_{1_{j}}^{β^{†}} = 1_{β_{j}^{†}} for every j ∈ Ob(J). Next, let g : j → g′ and g′ : j′ → j′′ be two 1-cells of J; we have to check the identity :

(β_{j′′}^{†} ∗ γ_{g,g′}^{G}) ⊙ (τ_{g′}^{β^{†}} ∗ Gg) ⊙ (L_{F,g′} ∗ τ_{g}^{β^{†}}) = τ_{g′∘g}^{β^{†}} ⊙ (L_{F,g,g′} ∗ β_{j}^{†}).

To this aim, set

X := (Θ^{β,j} ∘ g^{∗} ∘ g^{′∗}) ⊙ ((β ∗ t_{j′′}) ∗ (Γ^{G,g} ∘ g^{′∗})) ⊙ ((β ∗ t_{j′′}) ∗ Γ^{G,g′} ∗ F_{Gg})
X′ := (Ω^{F,g′} ∗ F_{L_{F,g}} ∗ F_{β_{j}^{†}}) ⊙ ((Ω^{F,g} ∘ g^{′∗}) ∗ F_{β_{j}^{†}}) ⊙ X
Y := (π^{F,j′′} ∗ ((F_{β_{j′′}^{†}} ∗ F_{γ_{g,g′}^{G}}) ⊙ (F_{τ_{g′}^{β^{†}}} ∗ F_{Gg}) ⊙ (F_{L_{F,g′}} ∗ F_{τ_{g}^{β^{†}}}))) ⊙ X′
Z := (π^{F,j′′} ∗ (F_{τ_{g′∘g}^{β^{†}}} ⊙ (F_{L_{F,g,g′}} ∗ F_{β_{j}^{†}}))) ⊙ X′.

ALMOST RINGS AND PERFECTOID SPACES 149

It suffices to show that Y = Z. Set as well Y′ := π^{F,j′′} ∗ ((F_{β_{j′′}^{†}} ∗ F_{γ_{g,g′}^{G}}) ⊙ (F_{τ_{g′}^{β^{†}}} ∗ F_{Gg})); we compute :

Y = Y′⊙(Ω^{F,g′} ∗ F_{β_{j′}^{†}} ∗ F_{Gg})⊙((π^{F,j′} ∗ g^{′∗}) ∗ F_{τ_{g}^{β^{†}}}) ⊙ ((Ω^{F,g} ∘ g^{′∗}) ∗ F_{β_{j}^{†}}) ⊙ X′
= Y′ ⊙ (Ω^{F,g′} ∗ F_{β_{j′}^{†}} ∗ F_{Gg}) ⊙ ((Θ^{β,j′} ∘ g^{′∗}) ∗ F_{Gg}) ⊙ ((β ∗ t_{j′′}) ∗ Γ^{G,g′} ∗ F_{Gg})
= (π^{F,j′′} ∗ F_{β_{j′′}^{†}} ∗ F_{γ_{g,g′}^{G}}) ⊙ (Θ^{β,j′′} ∗ F_{Gg′} ∗ F_{Gg})
= (Θ^{β,j′′} ∗ F_{G(g′∘g)}) ⊙ ((β ∗ t_{j′′}) ∗ (G ∗ φ_{j′′}^{∗}) ∗ F_{γ_{g,g′}^{G}})

whereas : Z = (π^{F,j′′} ∗ F_{τ_{g′∘g}^{β^{†}}}) ⊙ (Ω^{F,g′∘g} ∗ F_{β_{j}^{†}}) ⊙ X. So, we are reduced to checking that

(Γ^{G,g} ∘ g^{′∗}) ⊙ (Γ^{G,g′} ∗ F_{Gg}) = Γ^{G,g′∘g} ⊙ ((G ∗ φ_{j′′}^{∗}) ∗ F_{γ_{g,g′}^{G}}).

The latter follows from the composition axiom for γ^{G}. Lastly, the naturality of τ^{β^{†}} amounts to the identity :

τ_{g′}^{β^{†}} ⊙ (L_{F,α} ∗ β_{j}^{†}) = (β_{j′}^{†} ∗ Gα) ⊙ τ_{g}^{β^{†}} for every 2-cell α : g ⇒ g′ of J. To prove the latter, set X := (Ω^{F,g} ∗ F_{β_{j}^{†}}) ⊙ (Θ^{β,j} ∘ g^{∗}) ⊙ ((β ∗ t_{j′}) ∗ Γ^{G,g}); it suffices to show:

Y := (π^{F,j′} ∗ (F_{τ_{g′}^{β^{†}}} ⊙ (F_{L_{F,α}} ∗ F_{β_{j}^{†}}))) ⊙ X = Z := (π^{F,j′} ∗ ((F_{β_{j′}^{†}} ∗ F_{Gα}) ⊙ F_{τ_{g}^{β^{†}}})) ⊙ X. To this aim, we remark : Claim 2.6.23. (π^{F,α} ∗ F_{β_{j}^{†}}) ⊙ (Θ^{β,j} ∘ g^{∗}) = (Θ^{β,j} ∘ g^{′∗}) ⊙ ((β ∗ t_{j′}) ∗ (G ∘ (φ_{j}^{∗} ∗ α^{∗})). Proof of the claim. We have α^{∗}(f) = (1_{i}, α) : g ∘ f → g′ ∘ f for every (f : j′ → φ(i)) ∈ Ob(j′/φI). Now, let τ^{G∗φ_{j}^{∗}} be the coherence constraint of G ∗ φ_{j}^{∗}; since φ is unital, it is easily seen that τ_{α^{∗}(f)}^{G∗φ_{j}^{∗}} = G(f ∗ α) (details left to the reader). Then the compatibility condition of Θ^{β,j} yields the commutative diagram :

\xymatrix{β_{i} ∘ G(f ∘ g) \ar@{=>}[r]^{Θ_{g^{∗}(f)}^{β,j}} \ar@{=>}[d]_{β_{i}∗G(f∗α)} & π_{g^{∗}(f)}^{F,j} ∘ β_{j}^{†} \ar@{=>}[d]^{π^{F,α}} \\ β_{i} ∘ G(f ∘ g′) \ar@{=>}[r]^{Θ_{g^{′∗}(f)}^{β,j}} & π_{g^{′∗}(f)}^{F,j} ⊙ β_{j}^{†}}

whence the claim. ♢

Using claim 2.6.23we compute :

Y =(π^{F,j′} ∗ F_{τ_{g′}^{β^{†}}}) ⊙ ((Ω^{F,g′} ⊙ π^{F,α}) ∗ F_{β_{j}^{†}}) ⊙ (Θ^{β,j} ∘ g^{∗}) ⊙ ((β ∗ t_{j′}) ∗ Γ^{G,g})

=(π^{F,j′} ∗ F_{τ_{g′}^{β^{†}}})⊙(Ω^{F,g′} ∗ F_{β_{j}^{†}})⊙(Θ^{β,j} ∘ g^{′∗})⊙((β ∗ t_{j′}) ∗ (G ∘ (φ_{j}^{∗} ∗ α^{∗}))⊙((β ∗ t_{j′}) ∗ Γ^{G,g}) =(Θ^{β,j′} ∗ F_{Gg′}) ⊙ ((β ∗ t_{j′}) ∗ Γ^{G,g′})^{−1} ⊙ ((β ∗ t_{j′}) ∗ (G ∘ (φ_{j}^{∗} ∗ α^{∗})) ⊙ ((β ∗ t_{j′}) ∗ Γ^{G,g})

and on the other hand

Z = (π^{F,j′} ∗ F_{β_{j′}^{†}} ∗ F_{Gα}) ⊙ (Θ^{β,j′}∗F_{Gg}) = (Θ^{β,j′}∗F_{Gg′}) ⊙ ((β ∗ t_{j′}) ∗ (G ∗ φ_{j′}^{∗}) ∗ F_{Gα}) so we are reduced to showing that :

(G ∘ (φ_{j}^{∗} ∗ α^{∗})) ⊙ Γ^{G,g} = Γ^{G,g′} ⊙ ((G ∗ φ_{j′}^{∗}) ∗ F_{Gα}).

But the latter follows from the naturality condition for γ^{G}. □

Remark 2.6.24. In the situation of (2.6.19), suppose that 𝒞 is strongly 2-complete, so that we may take for 2-∫_{φ} the strong 2-right Kan extension of remark 2.6.18(ii). Then we may also choose β_{j}^{†} such that (β ∗ t_{j}) ⊙ (G ∗ φ_{j}^{∗}) = π^{F,j} ⊙ F_{β_{j}^{†}}, and we may let Θ^{β,j} be the corresponding identity modification. Suppose now additionally that G is a strict pseudo-functor; by inspecting (2.6.21), we deduce that with such choices, β^{†} is then a strict pseudo-natural transformation.

2.6.25. Keep the notation of (2.6.19), and let β, β′ : G ∘ φ ⇒ F be two pseudo-natural transformations and Ξ : β ⇝ β′ a modification. Then there exists for every j ∈ Ob(J) a unique 2-cell Ξ_{j}^{†} : β_{j}^{†} ⇒ β_{j}^{′†} such that

(2.6.26)  (π^{F,j} ∗ F_{Ξ_{j}^{†}}) ⊙ Θ^{β,j} = Θ^{β′,j} ⊙ ((Ξ ∘ t_{j}) ∗ (G ∗ φ_{j}^{∗})).

Lemma 2.6.27. In the situation of (2.6.25), the rule : j ↦ Ξ_{j}^{†} defines a modification

Ξ^{†} : β^{†} ⇝ β^{′†}.

Proof. Let g : j → j′ be any 1-cell of J; we need to show the identity :

τ_{g}^{β^{′†}} ⊙ (L_{F,g} ∗ Ξ_{j}^{†}) = (Ξ_{j′}^{†} ∗ Gg) ⊙ τ_{g}^{β^{†}}.

To this aim, set X := (Ω^{F,g} ∗ F_{β_{j}^{†}}) ⊙ (Θ^{β,j} ∘ g^{∗}) ⊙ ((β ∗ t_{j′}) ∗ Γ^{G,g}); it suffices to show that

Y := (π^{F,j′} ∗ (F_{τ_{g}^{β^{′†}}} ⊙ (F_{L_{F,g}} ∗ F_{Ξ_{j}^{†})})) ⊙ X = Z := (π^{F,j′} ∗ ((F_{Ξ_{j′}^{†}} ∗ F_{Gg}) ⊙ F_{τ_{g}^{β^{†}}})) ⊙ X.

We compute :

Y = (π^{F,j′} ∗ F_{τ_{g}^{β^{′†}}}) ⊙ (Ω^{F,g} ∗ F_{β_{j}^{′†}}) ⊙ ((π^{F,j} ∗ g^{∗}) ∗ F_{Ξ_{j}^{†}}) ⊙ (Θ^{β,j} ∘ g^{∗}) ⊙ ((β ∗ t_{j′}) ∗ Γ^{G,g})

= (π^{F,j′} ∗ F_{τ_{g}^{β^{′†}}})⊙(Ω^{F,g} ∗ F_{β_{j}^{′†}})⊙(Θ^{β′,j} ∘ g^{∗})⊙((Ξ ∘ t_{j′}) ∗ (G ∗ φ_{j}^{∗} ∗ g^{∗}))⊙((β ∗ t_{j′}) ∗ Γ^{G,g})

= (π^{F,j′} ∗F_{τ_{g}^{β^{′†}}})⊙(Ω^{F,g} ∗ F_{β_{j}^{′†}})⊙(Θ^{β′,j} ∘g^{∗})⊙((β′ ∗ t_{j′})∗Γ^{G,g})⊙((Ξ ∘ t_{j′}) ∗ (G ∗ φ_{j′}^{∗})∗F_{Gg})

= (Θ^{β′,j′} ∗ F_{Gg}) ⊙ ((Ξ ∘ t_{j′}) ∗ (G ∗ φ_{j′}^{∗}) ∗ F_{Gg}).

whereas : Z = (π^{F,j′} ∗ F_{Ξ_{j′}^{†}} ∗ F_{Gg}) ⊙ (Θ^{β,j′} ∗ F_{Gg}), whence the contention. □

Proposition 2.6.28. With the notation of lemmata 2.6.22 and 2.6.27, the rules : β ↦ β^{†} and Ξ ↦ Ξ^{†} define a functor

(−)_{F,G}^{†} : PsNat(G ∘ φ, F) → PsNat(G, 2-∫_{φ} F ).

Proof. A simple inspection shows that (1_{β})^{†} = 1_{β^{†}} for every pseudo-natural transformation β : G ∘ φ ⇒ F. It remains to check that (Ξ′ ⊙ Ξ)^{†} = Ξ^{′†} ⊙ Ξ for every three pseudo-natural transformations β, β′, β′′ : G ∘ φ ⇒ F and every pair of modifications Ξ : β ⇝ β′ and Ξ′ : β′ ⇝ β′′. However, we have :

(π^{F,j} ∗ F_{Ξ_{j}^{′†}} ∗ F_{Ξ_{j}^{†}}) ⊙ Θ^{β,j} = (π^{F,j} ∗ F_{Ξ_{j}^{′†}}) ⊙ Θ^{β′,j} ⊙ ((Ξ ∘ t_{j}) ∗ (G ∗ φ_{j}^{∗}))
= Θ^{β′′,j} ⊙ ((Ξ′ ∘ t_{j}) ∗ (G ∗ φ_{j}^{∗})) ⊙ ((Ξ ∘ t_{j}) ∗ (G ∗ φ_{j}^{∗}))
= Θ^{β′′,j} ⊙ (((Ξ′ ⊙ Ξ) ∘ t_{j}) ∗ (G ∗ φ_{j}^{∗}))

whence the contention. □

2.6.29. Keep the notation of (2.6.19). We set

ω_{i}^{F} := π_{(i,1_{φ(i)})}^{F,φ(i)} : L_{F,φ(i)} → F i  for every i ∈ Ob(I)
τ_{f}^{ω^{F}} := Ω_{(i′,1_{φ(i′)})}^{F,φ(f)} ⊙ τ_{(f,1_{φ(f)})}^{F,φ(i)} : F f ∘ ω_{i}^{F} ⇒ ω_{i′}^{F} ∘ L_{F,φ(f)} for every 1-cell f : i → i′ of I

where τ^{F,φ(i)} denotes the coherence constraint of π^{F,φ(i)}.

Lemma 2.6.30. The rule : i ↦ ω_{i}^{F} for every i ∈ Ob(I) defines a pseudo-natural transformation

ω^{F} : (2-∫_{φ} F ) ∘ φ ⇒ F

whose coherence constraint is given by the system of 2-cells τ_{•}^{ω^{F}}.

Proof. Let f, f′ : i → i′ be two 1-cells of I and α : f ⇒ f′ a 2-cell; for the naturality of τ^{ω^{F}} we need to check the identity :

τ_{f′}^{ω^{F}} ⊙ (F α ∗ ω_{i}^{F}) = (ω_{i′}^{F} ∗ L_{F,φ(α)}) ⊙ τ_{f}^{ω^{F}} .

However, notice that α induces two 1-cells (f, φ(α)), (f′, 1_{φ(f′)}) : (i, 1_{φ(i)}) → (i′, φ(f′)) and a 2-cell α : (f, φ(α)) ⇒ (f′, 1_{φ(f′)}) of φ(i)/φI. By naturality of τ^{F,φ(i)} we deduce the identity :

τ_{(f,φ(α))}^{F,φ(i)} = π_{(f′,1_{φ(f′)})}^{F,φ(i)} ⊙ (F α ∗ π_{(i,1_{φ(i)})}^{F,φ(i)})

whence :

τ_{f′}^{ω^{F}} ⊙ (F α ∗ ω_{i}^{F}) = Ω_{(i′,1_{φ(i′)})}^{F,φ(f′)} ⊙ τ_{(f,φ(α))}^{F,φ(i)}.

On the other hand, from (2.6.4) we get :

(ω_{i′}^{F} ∗ L_{F,φ(α)}) ⊙ τ_{f}^{ω^{F}} = Ω_{(i′,1_{φ(i′)})}^{F,φ(f′)} ⊙ π_{(i′,1_{φ(i′)})}^{F,φ(α)} ⊙ τ_{(f,1_{φ(f)})}^{F,φ(i)}.

So we are reduced to checking the identity : τ_{(f,φ(α))}^{F,φ(i)} = τ_{(1_{i′},φ(α))}^{F,φ(i)} ⊙ τ_{(f,1_{φ(f)})}^{F,φ(i)}. The latter follows from the coherence axiom for τ^{F,φ(i)}. Next, we check the coherence axioms for ω^{F}. To this aim, notice that for every i ∈ Ob(I) and f := 1_{i}, the two-cell τ_{(f,1_{φ(f)})}^{F,φ(i)} is the identity of the

1-cell π_{(i,1_{φ(i)})}^{F,φ(i)} (remark 2.4.2(ii)); it follows easily that τ_{1_{i}}^{ω^{F}} = 1_{ω_{i}^{F}}. Lastly, consider two 1-cells f : i → i′, f′ : i′ → i′′ of I, and set X := ω_{i′′}^{F} ∗ (L_{F,γ_{f,f′}^{φ}} ⊙ L_{F,φ(f),φ(f′)}); we need to show that

Y := X ⊙ (τ_{f′}^{ω^{F}} ∗ L_{F,φ(f)}) ⊙ (F f′ ∗ τ_{f}^{ω^{F}}) = Z := τ_{f′∘f}^{ω^{F}} ⊙ (γ_{f,f′}^{F} ∗ ω_{i}^{F})

where γ^{φ} and γ^{F} are the coherence constraints of φ and F. We compute :

Y = X ⊙ ((Ω_{(i′′,1_{φ(i′′)})}^{F,φ(f′)} ⊙ τ_{(f′,1_{φ(f′)})}^{F,φ(i′)}) ∗ L_{F,φ(f)}) ⊙ (F f′ ∗ (Ω_{(i′,1_{φ(i′)})}^{F,φ(f)} ⊙ τ_{(f,1_{φ(f)})}^{F,φ(i)}))
= X ⊙ (Ω_{(i′′,1_{φ(i′′)})}^{F,φ(f′)} ∗ L_{F,φ(f)}) ⊙ Ω_{(i′,φ(f′))}^{F,φ(f)} ⊙ τ_{(f′,1_{φ(f′)∘φ(f)})}^{F,φ(i)} ⊙ (F f′ ∗ τ_{(f,1_{φ(f)})}^{F,φ(i)})
= (ω_{i′′}^{F} ∗ L_{F,γ_{f,f′}^{φ}}) ⊙ Ω_{(i′′,1_{φ(i′′)})}^{F,φ(f′)∘φ(f)} ⊙ τ_{(f′,1_{φ(f′)∘φ(f)})}^{F,φ(i)} ⊙ (F f′ ∗ τ_{(f,1_{φ(f)})}^{F,φ(i)})
= Ω_{(i′′,1_{φ(i′′)})}^{F,φ(f′∘f)} ⊙ π_{(i′′,1_{φ(i′′)})}^{F,γ_{f,f′}^{φ}} ⊙ τ_{(f′,1_{φ(f′)∘φ(f)})}^{F,φ(i)} ⊙ (F f′ ∗ τ_{(f,1_{φ(f)})}^{F,φ(i)})
= Ω_{(i′′,1_{φ(i′′)})}^{F,φ(f′∘f)} ⊙ π_{(i′′,1_{φ(i′′)})}^{F,γ_{f,f′}^{φ}} ⊙ τ_{(f′∘f,(γ_{f,f′}^{φ})^{−1})}^{F,φ(i)} ⊙ (γ_{f,f′}^{F} ∗ π_{i,1_{φ(i)}}^{F,φ(i)})

where the second equality follows from the compatibility condition for Ω^{F,φ(f)}, the third follows from (2.6.2), the fourth follows from (2.6.4), and the fifth holds by virtue of the coherence axiom for τ^{F,φ(i)}, taking into account (2.6.3). So, we are reduced to checking that

π_{(i′′,1_{φ(i′′)})}^{F,γ_{f,f′}^{φ}} ⊙ τ_{(f′∘f,(γ_{f,f′}^{φ})^{−1}))}^{F,φ(i)} = τ_{(f′∘f,1_{φ(f′∘f)})}^{F,φ(i)}

152 OFER GABBER AND LORENZO RAMERO

which follows again from (2.6.3) and the coherence axiom for τ^{F,φ(i)}. □ 2.6.31. In view of lemma 2.6.30we may define a functor :

(−)_{F,G}^{‡} : PsNat(G, 2-∫_{φ} F ) → PsNat(G ∘ φ, F) β ↦ β^{‡} := ω^{F} ⊙ (β ∗ φ) attaching to every modification Ξ : β ⇝ β′ of pseudo-natural transformations β, β′ : G ⇒ 2-∫_{φ} F, the modification ω^{F} ∗ (Ξ ∘ φ). For any pseudo-natural transformation β : G ∘ φ ⇒ F we set

Λ_{i}^{β} := Θ_{(i,1_{φ(i)})}^{β,φ(i)} : β_{i} ⇒ (β^{†})_{i}^{‡} for every i ∈ Ob(I).

Lemma 2.6.32. The rule : i ↦ Λ_{i}^{β} defines an invertible modification

Λ^{β} : β ⇝ (β^{†})^{‡}

and the rule : β ↦ Λ^{β} defines an isomorphism of functors :

Λ : 1_{PsNat(G∘φ,F)} →^{∼} (−)_{F,G}^{‡} ∘ (−)_{F,G}^{†}.

Proof. Let f : i → i′ be any 1-cell of I; we need to check the identity : Y := (π_{(i′,1_{φ(i′)})}^{F,φ(i′)} ∗ τ_{φ(f)}^{β^{†}}) ⊙ (τ_{f}^{ω^{F}} ∗ β_{φ(i)}^{†}) ⊙ (F f ∗ Θ_{(i,1_{φ(i)})}^{β,φ(i)}) = Z := (Θ_{i′,1_{φ(i′)}}^{β,φ(i′)} ∗ Gφ(f)) ⊙ τ_{f}^{β} where τ^{β} denotes the coherence constraint of β. Let also τ^{(β∗t_{φ(i)})⊙(G∗φ_{φ(i)}^{∗})} be the coherence constraint of (β ∗ t_{φ(i)}) ⊙ (G ∗ φ_{φ(i)}^{∗}), and notice that τ_{(f,1_{φ(f)})}^{(β∗t_{φ(i)})⊙(G∗φ_{φ(i)}^{∗})} = τ_{f}^{β}. Combining with the compatibility condition for Θ^{β,φ(i)} relative to the 1-cell (f, 1_{φ(f)}) of φ(i)/φI, we get :

Y = (π_{(i′,1_{φ(i′)})}^{F,φ(i′)} ∗ τ_{φ(f)}^{β^{†}}) ⊙ (Ω_{(i′,1_{φ(i′)})}^{F,φ(f)} ∗ β_{φ(i)}^{†}) ⊙ (Θ_{(i′,φ(f))}^{β,φ(i)} ∗ Gφ(i)) ⊙ τ_{f}^{β}.

Then the sought identity follows from (2.6.21), applied with g := (f, 1_{φ(i)}), after noticing that Γ_{(i′,1_{φ(i′)})}^{G,g} = 1_{Gφ(f)}, since G is unital. Lastly, let β, β′ : G ⇒ 2-∫_{φ} F be two pseudo-natural transformations, and Ξ : β ⇝ β′ a modification; the naturality of Λ amounts to the identity :

(ω_{i}^{F} ∗ Ξ_{φ(i)}^{†}) ⊙ Λ_{i}^{β} = Λ_{i}^{β′} ⊙ Ξ_{i} for every i ∈ Ob(I)

which follows directly from (2.6.26). □ Lemma 2.6.33. In the situation of (2.6.19), for every j ∈ Ob(J) the rule

(i, j ⟶^{g} φ(i)) ↦ Δ_{(i,g)}^{F,j} := Ω_{(i,1_{φ(i)})}^{F,g} for every (i, g) ∈ Ob(j/φI)

defines an invertible modification Δ^{F,j} : π^{F,j} ⇝ (ω^{F} ∗ t_{j}) ⊙ ((2-∫_{φ} F ) ∗ φ_{j}^{∗}).

Proof. Let (f, α) : (i, g) → (i′, g′) be any 1-cell of j/φI; we need to check the identity : Ω_{(i′,1_{φ(i′)})}^{F,g′} ⊙ τ_{(f,α)}^{F,j} = Y := (π_{(i′,1_{φ(i′)})}^{F,φ(i′)} ∗ (L_{F,α} ⊙ L_{F,g,φ(f)})) ⊙ (τ_{f}^{ω^{F}} ∗ L_{F,g}) ⊙ (F f ∗ Ω_{(i,1_{φ(i)})}^{F,g}). We compute :

Y = (π_{(i′,1_{φ(i′)})}^{F,φ(i′)} ∗ (L_{F,α} ⊙ L_{F,g,φ(f)})) ⊙ ((Ω_{(i′,1_{φ(i′)})}^{F,φ(f)} ⊙ τ_{(f,1_{φ(f)})}^{F,φ(i)}) ∗ L_{F,g}) ⊙ (F f ∗ Ω_{(i,1_{φ(i)})}^{F,g})
= (π_{(i′,1_{φ(i′)})}^{F,φ(i′)} ∗ (L_{F,α} ⊙ L_{F,g,φ(f)})) ⊙ (Ω_{(i′,1_{φ(i′)})}^{F,φ(f)} ∗ L_{F,g}) ⊙ Ω_{(i′,φ(f))}^{F,g} ⊙ τ_{(f,g^{∗}(1_{φ(f)}))}^{F,j}
= (π_{(i′,1_{φ(i′)})}^{F,φ(i′)} ∗ L_{F,α}) ⊙ Ω_{(i′,1_{φ(i′)})}^{F,φ(f)∘g} ⊙ τ_{(f,g^{∗}(1_{φ(f)}))}^{F,j}
= Ω_{(i′,1_{φ(i′)})}^{F,g′} ⊙ π_{(i′,1_{φ(i′)})}^{F,α} ⊙ τ_{(f,g^{∗}(1_{φ(f)}))}^{F,j}

where the second equality holds by the compatibility condition of Ω^{F,g}, applied to the 1-cell (f, 1_{φ(f)}) : (i, 1_{φ(i)}) → (i′, φ(f)) of φ(i)/φI, the third equality follows from (2.6.2), and the fourth equality follows from (2.6.4). So we are reduced to checking that

τ_{(f,α)}^{F,j} = π_{(i′,1_{φ(i′)})}^{F,α} ⊙ τ_{(f,g^{∗}(1_{φ(f)}))}^{F,j}

which follows from the coherence condition for τ^{F,j}, relative to the composition of 1-cells

(i, g) ⟶^{(f,1_{φ(f)∘g})} (i′, φ(f) ∘ g) ⟶^{(1_{i′},α)} (i′, g′). □

2.6.34. Now, let β : G ⇒ 2-∫_{φ} F be a pseudo-natural transformation. For every j ∈ Ob(J) we get the 1-cell (β^{‡})_{j}^{†} : Gj → L_{F,j}, as well as the invertible modification

Θ^{β^{‡},j} : (ω^{F} ∗ t_{j}) ⊙ (β ∗ (φ ∘ t_{j})) ⊙ (G ∗ φ_{j}^{∗}) ⇝ π^{F,j} ⊙ F_{(β^{‡})_{j}^{†}}.

On the other hand, by example 2.2.15(ii) we have the invertible modification

Υ^{β,j} : ((2-∫_{φ} F ) ∗ φ_{j}^{∗}) ⊙ (β ∗ F_{j}) ⇝ (β ∗ (φ ∘ t_{j})) ⊙ (G ∗ φ_{j}^{∗}) (i, f : j → φ(i)) ↦ τ_{φ_{j,(i,f)}^{∗}}^{β}

where τ^{β} is the coherence constraint of β, and notice that β ∗ F_{j} = F_{β_{j}}. We set

Δ^{β,j} := Θ^{β^{‡,j}} ⊙ ((ω^{F} ∗ t_{j}) ∗ Υ^{β,j}) ⊙ (Δ^{F,j} ∗ F_{β_{j}}) : π^{F,j} ⊙ F_{β_{j}} ⇝ π^{F,j} ⊙ F_{(β^{‡})_{j}^{†}}.

There exists then a unique invertible 2-cell

Ψ_{j}^{β} : β_{j} ⇒ (β^{‡})_{j}^{†} such that π^{F,j} ∗ F_{Ψ_{j}^{β}} = Δ^{β,j}.

Lemma 2.6.35. The rule : j ↦ Ψ_{j}^{β} for every j ∈ Ob(J) yields an invertible modification

Ψ^{β} : β ⇝ (β^{‡})^{†}.

Proof. Let g : j → j′ be any 1-cell of J; we need to show that

τ_{g}^{(β^{‡})^{†}} ⊙ (L_{F,g} ∗ Ψ_{j}^{β}) = (Ψ_{j′}^{β} ∗ Gg) ⊙ τ_{g}^{β}

and recall that the coherence constraint τ_{g}^{(β^{‡})^{†}} of (β^{‡})^{†} is characterized by the identity :

(π^{F,j′} ∗ F_{τ_{g}^{(β^{‡})^{†}}}) ⊙ (Ω^{F,g} ∗ F_{(β^{‡})_{j}^{†}}) ⊙ (Θ^{β^{‡},j} ∘ g^{∗}) ⊙ ((β^{‡} ∗ t_{j′}) ∗ Γ^{G,g}) = Θ^{β^{‡},j′} ∗ F_{Gg}.

As usual, we reduce to checking the identity :

Y := π^{F,j′} ∗ (F_{τ_{g}^{(β^{‡})^{†}}} ⊙ (F_{L_{F,g}} ∗ F_{Ψ_{j}^{β}})) = Z := π^{F,j′} ∗ ((F_{Ψ_{j′}^{β}} ∗ F_{Gg}) ⊙ F_{τ_{g}^{β}}).

To ease notation, set H := 2-∫_{φ} F; we remark :

Claim 2.6.36. ((β∗φ∗t_{j′})∗Γ^{G,g})^{−1}⊙(Υ^{β,j}∘g^{∗}) = (Υ^{β,j′}∗F_{Gg})⊙((H∗φ_{j′}^{∗})∗F_{τ_{g}^{β}})⊙(Γ^{H,g}∗F_{β_{j}})^{−1}.

Proof of the claim. The left-hand side is the modification

(H∗φ_{j}^{∗}∗g^{∗})⊙F_{β_{j}} ⇝ (β∗(φ∘t_{j′}))⊙(G∗φ_{j′}^{∗})⊙F_{Gg} (i, h : j′ → φ(i)) ↦ τ_{h∘g}^{β}⊙(β_{φ(i)}∗γ_{g,h}^{G})^{−1}

and the coherence axiom for τ^{β} gives :

(β_{φ(i)} ∗ γ_{g,h}^{G})^{−1} ⊙ τ_{h∘g}^{β} = (τ_{h}^{β} ∗ Gg) ⊙ (L_{F,h} ∗ τ_{g}^{β}) ⊙ (L_{F,g,h} ∗ β_{j})^{−1}

where γ^{G} is the coherence constraint of G. The claim translates the latter identity. ♢

154 OFER GABBER AND LORENZO RAMERO

Now, let X := Θ^{β^{‡},j′} ∗ F_{Gg} and X′ := X ⊙ (ω^{F} ∗ t_{j′}) ∗ Υ^{β′,j} ∗ F_{Gg}. We compute :

Y = X ⊙ ((β^{‡} ∗ t_{j′}) ∗ Γ^{G,g})^{−1} ⊙ (Θ^{β^{‡},j} ∘ g^{∗})^{−1} ⊙ (Ω^{F,g} ∗ F_{(β^{‡})_{j}^{†}})^{−1} ⊙ (π^{F,j′} ∗ F_{L_{F,g}} ∗ F_{Ψ_{j}^{β}})

= X ⊙ ((β^{‡} ∗ t_{j′}) ∗ Γ^{G,g})^{−1} ⊙ (Θ^{β^{‡},j} ∘ g^{∗})^{−1} ⊙ ((π^{F,j} ∗ g^{∗}) ∗ F_{ψ_{j}^{β}}) ⊙ (Ω^{F,g} ∗ F_{β_{j}})^{−1} = X ⊙ ((β^{‡} ∗ t_{j′}) ∗ Γ^{G,g})^{−1} ⊙ (Θ^{β^{‡},j} ∘ g^{∗})^{−1} ⊙ (Δ^{β,j} ∘ g^{∗}) ⊙ (Ω^{F,g} ∗ F_{β_{j}})^{−1} = X ⊙((β^{‡} ∗ t_{j′}) ∗ Γ^{G,g})^{−1}⊙((ω^{F} ∗ t_{j′}) ∗ (Υ^{β,j} ∘ g^{∗}))⊙((Δ^{F,j} ∘ g^{∗}) ∗ F_{β_{j}})⊙(Ω^{F,g} ∗ F_{β_{j}})^{−1} = X′⊙((ω^{F}∗ t_{j′})∗(((H ∗φ_{j′}^{∗}) ∗ F_{τ_{g}^{β}})⊙(Γ^{H,g} ∗ F_{β_{j}})^{−1}))⊙((Δ^{F,j} ∘g^{∗}) ∗ F_{β_{j}})⊙(Ω^{F,g} ∗ F_{β_{j}})^{−1}

where the last equality follows from claim 2.6.36. On the other hand : Z = X′⊙(Δ^{F,j′}∗F_{β_{j′}} ∗F_{Gg})⊙(π^{F,j′}∗F_{τ_{g}^{β}}) = X′⊙((ω^{F} ∗t_{j′})∗(H∗φ_{j′}^{∗})∗F_{τ_{g}^{β}})⊙(Δ^{F,j′}∗F_{Hg}∗F_{β_{j}}) so we are reduced to showing that

Δ^{F,j} ∘ g^{∗} = ((ω^{F} ∗ t_{j′}) ∗ Γ^{H,g}) ⊙ (Δ^{F,j′} ∗ F_{Hg}) ⊙ Ω^{F,g}

which follows from (2.6.2). □ Proposition 2.6.37. For every unital pseudo-functors F : I → 𝒞 and G : J → 𝒞, the functors (−)_{F,G}^{†} of proposition 2.6.28and (−)_{F,G}^{‡} of (2.6.31) are equivalences. Proof. In view of lemmata 2.6.32and 2.6.35, it suffices to prove that the rule : β ↦ Ψ^{β} defines a natural transformation

Ψ : 1_{PsNat(G,2-∫_{φ} F)} ⇒ (−)_{F,G}^{†} ∘ (−)_{F,G}^{‡}.

Thus, let α, β : G ⇒ 2-∫_{φ} F be two pseudo-natural transformations, and Ξ : α ⇝ β a modification; we need to check the identity :

Ψ_{j}^{β} ⊙ Ξ_{j} = (ω^{F} ∗ (Ξ ∘ φ))_{j}^{†} ⊙ Ψ_{j}^{α} for every j ∈ Ob(J)

and as usual, it suffices to show that

Y := π^{F,j} ∗ (F_{Ψ_{j}^{β}} ⊙ F_{Ξ_{j}}) = Z := π^{F,j} ∗ (F_{(ω^{F} ∗(Ξ∘φ))_{j}^{†}} ⊙ F_{Ψ_{j}^{α}}).

To ease notation, set H := 2-∫_{φ} F; we remark : Claim 2.6.38. ((Ξ ∘ φ ∘ t_{j}) ∗ (G ∗ φ_{j}^{∗})) ⊙ Υ^{α,j} = Υ^{β,j} ⊙ ((H ∗ φ_{j}^{∗}) ∗ (Ξ ∘ F_{j})). Proof of the claim. The left-hand side is the modification :

(H ∗ φ_{j}^{∗}) ⊙ (α ∗ F_{j}) ⇝ (β ∗ (φ ∘ t_{j})) ⊙ (G ∗ φ_{j}^{∗}) (j, g ↦ φ(i)) ↦ (Ξ_{φ(i)} ∗ Gg) ⊙ τ_{g}^{α} where τ^{α} is the coherence constraint of α; then the compatibility condition for Ξ gives :

(Ξ_{φ(i)} ∗ Gg) ⊙ τ_{g}^{α} = τ_{g}^{β} ⊙ (Hg ∗ Ξ_{j})

and the claim translates the latter identity. ♢ We have : Y =Δ^{β,j}⊙(π^{F,j} ∗ F_{Ξ_{j}})=Θ^{β^{‡,j}}⊙((ω^{F} ∗ t_{j}) ∗ Υ^{β,j})⊙(Δ^{F,j} ∗ F_{β_{j}})⊙(π^{F,j} ∗ F_{Ξ_{j}}), and

Z = (π^{F,j} ∗ F_{(ω^{F} ∗(Ξ∘φ))_{j}^{†}}) ⊙ Δ_{j}^{α}
= (π^{F,j} ∗ F_{(ω^{F} ∗(Ξ∘φ))_{j}^{†}}) ⊙ Θ^{α^{‡,j}} ⊙ ((ω^{F} ∗ t_{j}) ∗ Υ^{α,j}) ⊙ (Δ^{F,j} ∗ F_{α_{j}})
= Θ^{β^{‡},j} ⊙ (((ω^{F} ∗ (Ξ ∘ φ)) ∘ t_{j}) ∗ (G ∗ φ_{j}^{∗})) ⊙ ((ω^{F} ∗ t_{j}) ∗ Υ^{α,j}) ⊙ (Δ^{F,j} ∗ F_{α_{j}})
= Θ^{β^{‡},j} ⊙ ((ω^{F} ∗ t_{j}) ∗ (Ξ ∘ φ ∘ t_{j}) ∗ (G ∗ φ_{j}^{∗})) ⊙ ((ω^{F} ∗ t_{j}) ∗ Υ^{α,j}) ⊙ (Δ^{F,j} ∗ F_{α_{j}})
= Θ^{β^{‡},j} ⊙ ((ω^{F} ∗ t_{j}) ∗ (Υ^{β,j} ⊙ ((H ∗ φ_{j}^{∗}) ∗ (Ξ ∘ F_{j})))) ⊙ (Δ^{F,j} ∗ F_{α_{j}})

where the last equality holds by claim 2.6.38. Thus, we are reduced to checking the identity :

Δ^{F,j} ∗ F_{β_{j}} ⊙ (π^{F,j} ∗ F_{Ξ_{j}}) = ((ω^{F} ∗ t_{j}) ∗ (H ∗ φ_{j}^{∗}) ∗ (Ξ ∘ F_{j})) ⊙ (Δ^{F,j} ∗ F_{α_{j}})

which follows as usual from remark 2.1.1(i). □

Theorem 2.6.39. Let I be a small 2-category, J a 2-category with small Hom-categories, φ : I → J any pseudo-functor, and suppose that all the 2-cells of J are invertible. Then for every 2-complete (resp. 2-cocomplete) 2-category 𝒞 the right (resp. left) 2-Kan extension along φ is a right (resp. left) 2-adjoint for the pseudo-functor

PsFun(φ, 𝒞) : PsFun(J, 𝒞) → PsFun(I, 𝒞).

Proof. Let us show first that the rule : (F, G) ↦ (−)_{F,G}^{‡} yields a pseudo-natural equivalence :

(−)^{‡} : PsNat(1_{uniPsFun(J,𝒞)}, 2-∫_{φ}) →^{∼} PsNat(uniPsFun(φ^{u}, 𝒞), 1_{uniPsFun(I,𝒞)})

(notation of remark 2.4.4(i)). To this aim let us remark :

Claim 2.6.40. Let F, F′ : I → 𝒞 be two unital pseudo-functors. Every pseudo-natural transformation μ : F ⇒ F′ induces an invertible modification :

Σ^{μ} : μ ⊙ ω^{F} ⇝ ω^{F′} ⊙ ((2-∫_{φ} μ) ∗ φ) i ↦ Ω_{(i,1_{φ(i)})}^{μ,φ(i)}.

Proof of the claim. Let f : i → i′ be any 1-cell of I, and τ^{μ} the coherence constraint of μ; set :

\begin{aligned}Y & := (π_{(i′,1_{φ(i′)})}^{F′,φ(i′)} ∗ L_{μ,φ(f)}) ⊙ (τ_{f}^{ω^{F′}} ∗ L_{μ,φ(i)}) ⊙ (F′f ∗ Ω_{(i,1_{φ(i)})}^{μ,φ(i)}) \\ Z & := (Ω_{(i′,1_{φ(i′)})}^{μ,φ(i′)} ∗ L_{F,φ(f)}) ⊙ (μ_{i′} ∗ τ_{f}^{ω^{F}}) ⊙ (τ_{f}^{μ} ∗ π_{(i,1_{φ(i)})}^{F,φ(i)}) \end{aligned}

so that we need to check the identity Y = Z. We compute :

\begin{aligned}Y & = (π_{(i′,1_{φ(i′)})}^{F′,φ(i′)} ∗ L_{μ,φ(f)}) ⊙ ((Ω_{(i′,1_{φ(i′)})}^{F′,φ(f)} ⊙ τ_{(f,1_{φ(f)})}^{F′,φ(i)}) ∗ L_{μ,φ(i)}) ⊙ (F′f ∗ Ω_{(i,1_{φ(i)})}^{μ,φ(i)}) \\ & = (π_{(i′,1_{φ(i′)})}^{F′,φ(i′)} ∗ L_{μ,φ(f)})⊙(Ω_{(i′,1_{φ(i′)})}^{F′,φ(f)} ∗ L_{μ,φ(i)})⊙Ω_{(i′,φ(f))}^{μ,φ(i)}⊙(μ_{i′} ∗ τ_{(f,1_{φ(f)})}^{F,φ(i)})⊙(τ_{f}^{μ} ∗ π_{(i,1_{φ(i)})}^{F,φ(i)}) \end{aligned}

where the second equality follows from the compatibility condition of Ω^{μ,φ(i)}, relative to the 1-cell (f, 1_{φ(f)}) : (i, 1_{φ(i)}) → (i′, φ(f)) of φ(i)/φI. We are thus reduced to showing that :

(π_{(i′,1_{φ(i′)})}^{F′,φ(i′)} ∗ L_{μ,φ(f)})⊙(Ω_{(i′,1_{φ(i′)})}^{F′,φ(f)} ∗ L_{μ,φ(i)})⊙Ω_{(i′,φ(f))}^{μ,φ(i)} = (Ω_{(i′,1_{φ(i′)})}^{μ,φ(i′)} ∗ L_{F,φ(f)})⊙(μ_{i′} ∗ Ω_{(i′,1_{φ(i′)})}^{F,φ(f)})

which follows from (2.6.9). ♢

Claim 2.6.41. Let F, F′ : I → 𝒞 and G, G′ : J → 𝒞 be four unital pseudo-functors, λ : G′ ⇒ G and μ : F ⇒ F′ two pseudo-natural transformations. The rule

β ↦ τ_{(λ,μ),β}^{‡} := Σ^{μ} ∗ ((β ⊙ λ) ∗ φ)

yields the orientation for an essentially commutative square diagram of functors :

[figure: PsNat(G, 2-∫_{φ} F); PsNat(λ,2-∫_{φ} μ); (−)_{F,G}^{‡}; τ_{(λ,μ)}^{‡}; PsNat(G ∘ φ, F); PsNat(λ∗φ,μ); PsNat(G′, 2-∫_{φ} F′); (−)_{F′,G′}^{‡}; PsNat(G′ ∘ φ, F′).]

Proof of the claim. Let β, β′ : G → 2-∫_{φ} F be two pseudo-natural transformations, and Ξ : β ⇝ β′ a modification; we need to check the identity :

τ_{(λ,μ),β′}^{‡} ⊙ (μ ∗ ω^{F} ∗ (Ξ ∘ φ) ∗ (λ ∗ φ)) = (ω^{F′} ∗ (((2-∫_{φ} μ) ∗ Ξ ∗ λ) ∘ φ)) ⊙ τ_{(λ,μ),β}^{‡}

and we come down to showing that :

(Σ^{μ} ∗ (β′ ∗ φ)) ⊙ (μ ∗ ω^{F} ∗ (Ξ ∘ φ)) = (ω^{F′} ∗ ((2-∫_{φ} μ) ∗ φ) ∗ (Ξ ∘ φ)) ⊙ (Σ^{μ} ∗ (β ∗ φ))

which follows from remark 2.1.1(i). ♢

To conclude the proof, it remains to check that the rule (λ, μ) ↦ τ_{(λ,μ)}^{‡} yields a coherence constraint for the sought pseudo-functor (−)^{‡}. For the first coherence axiom, notice that both the source and target of (−)^{‡} are themselves unital pseudo-functors, hence it suffices to check that τ_{(1_{G},1_{F})}^{‡} is the identity automorphism of (−)_{F,G}^{‡}, for every pair of unital pseudo-functors F : I → 𝒞 and G : J → 𝒞 (remark 2.4.2(ii)). We then come down to checking that Σ^{1_{F}} = 1_{ω^{F}} for every such F; the latter identity follows by a direct inspection of the definitions.

Next, consider unital pseudo-functors F, F′, F′′ : I → 𝒞, G, G′, G′′ : J → 𝒞 and two pairs of pseudo-natural transformations (λ′ : G′′ ⇒ G′, μ′ : F′ ⇒ F′′) and (λ : G′ ⇒ G, μ : F ⇒ F′); we need to check the identity :

((−)_{F′′,G′′}^{‡}∗PsNat(G′′,L_{μ,μ′}))⊙(τ_{(λ′,μ′)}^{‡}∗PsNat(λ, 2-∫_{φ}μ))⊙(PsNat(λ′∗φ, μ′)∗τ_{(λ,μ)}^{‡})=τ_{(λ′⊙λ,μ′⊙μ)}^{‡}.

Thus, let β : G ⇒ 2-∫_{φ} F be any pseudo-natural transformation, and set ρ := (β ⊙ λ ⊙ λ′) ∗ φ; the assertion comes down to the identity :

(ω^{F} ∗ (L_{μ,μ′} ∘ φ) ∗ ρ) ⊙ (Σ^{μ′} ∗ (((2-∫_{φ} μ) ∗ φ) ⊙ ρ))⊙(μ′ ∗ Σ^{μ} ∗ ρ) = Σ^{μ′⊙μ} ∗ ρ

so it suffices to show that :

(ω^{F} ∗ (L_{μ,μ′} ∘ φ)) ⊙ (Σ^{μ′} ∗ ((2-∫_{φ} μ) ∗ φ))⊙(μ′ ∗ Σ^{μ}) = Σ^{μ′⊙μ}

which holds by (2.6.10). Lastly, the source and target of (−)^{‡} are pseudo-functors

uniPsFun(J, 𝒞)^{o} × uniPsFun(I, 𝒞) → Cat.

On the other hand, the strict 2-equivalences (−)^{u} of remark 2.4.4(i) yield a strict 2-equivalence

U : PsFun(J, 𝒞)^{o} × PsFun(I, 𝒞) →^{∼} uniPsFun(J, 𝒞)^{o} × uniPsFun(I, 𝒞)

and the strict isomorphisms of pseudo-functors (2.4.5) yield pseudo-natural isomorphisms

PsNat(1_{uniPsFun(J,𝒞)}, 2-∫_{φ}) ∘ U →^{∼} PsNat(1_{PsFun(J,𝒞)}, 2-∫_{φ})
PsNat(uniPsFun(φ^{u}, 𝒞), 1_{uniPsFun(I,𝒞)}) ∘ U →^{∼} PsNat(PsFun(φ, 𝒞), 1_{PsFun(I,𝒞)}).

Summing up, we deduce the sought 2-adjunction between PsFun(φ, 𝒞) and 2-∫_{φ}.

The assertion concerning the left 2-Kan extension along φ is an immediate consequence, in view of remark 2.4.28(i). □

Corollary 2.6.42. In the situation of theorem 2.6.39, suppose that φ is fully faithful. Then the same holds for the pseudo-functor 2-∫_{φ} (resp. for 2-∫^{φ}).

Proof. As usual, it suffices to check the assertion concerning right 2-Kan extensions. To this aim, we notice :

Claim 2.6.43. Suppose that φ : I → J is fully faithful and that all the 2-cells of J are invertible. Then for every i ∈ Ob(I), the object (i, 1_{φ(i)}) is pseudo-initial in φ(i)/φI.

Proof of the claim. Let (i′, f) be any othe object of φ(i)/φI; by assumption, there exist a 1-cell g : i → i′ in I and an invertible 2-cell α : φ(g) →^{∼} f in J. Then we get the 1-cell

(i, 1_{φ(i)}) ⟶^{(g,1_{φ(g)})} (i′, φ(g)) ⟶^{(1_{i′},α)} (i′, f).

Next, let (t, α), (s, β) : (i, 1_{φ(i)}) → (i′, f) be two 1-cells in φ(i)/φI; by assumption β and α are invertible, hence we get the invertible 2-cell β^{−1} ⊙ α : φ(t) →^{∼} φ(s), and since φ is fully faithful, there exists a unique invertible 2-cell λ : t →^{∼} s in I such that φ(λ) = β^{−1} ⊙ α. We deduce an invertible 2-cell λ : (t, α) →^{∼} (s, β) in φ(i)/φI, and it is easily seen that this is the unique 2-cell from (t, α) to (s, β). The claim follows. ♢

Let now F : I → 𝒞 be any pseudo-functor, and for every j ∈ Ob(J), let (L_{F,j}, π^{F,j}) be a 2-limit of F ∘ t_{j}, as in (2.6); in view of claim 2.6.43and proposition 2.5.16, we deduce that the 1-cell π_{(i,1_{φ(i)})}^{F,φ(i)} : L_{F,j} → F i is an equivalence for every i ∈ Ob(I), and therefore the pseudonatural transformation ω^{F} of lemma 2.6.30 is a pseudo-natural equivalence. To conclude, it suffices now to invoke corollary 2.4.29. □

3. Special categories

3.1. Fibrations. Let φ : 𝒜 → ℬ be a functor, f : A′ → A a morphism in 𝒜. We say that f is φ-cartesian, or – slightly abusively – that f is ℬ-cartesian, if the induced commutative diagram of sets (notation of (1.1.25)) :

\xymatrix{Hom_{𝒜} (X, A′) \ar[r]^{f_{∗}} \ar[d]_{φ} & Hom_{𝒜} (X, A) \ar[d]^{φ} \\ Hom_{ℬ} (φX, φA′) \ar[r]^{(φf)_{∗}} & Hom_{ℬ} (φX, φA)}

is cartesian for every X ∈ Ob(𝒜). In this case, one also says that f is an inverse image of A over φf, or – slightly abusively – that A′ is an inverse image of A over φf.

Remark 3.1.1. Keep the notation of (3.1).

(i) It is easily seen that the composition of two ℬ-cartesian morphisms is ℬ-cartesian.

(ii) Let g : B′ → B and g′ : B′′ → B′ be two morphisms of ℬ, and f : A′ → A and f′′ : A′′ → A two ℬ-cartesian morphisms of 𝒜 such that φf = g and φf′′ = g ∘ g′. Then there exists a unique morphism f′ : A′′ → A′ of 𝒜 such that φf′ = g′ and f′′ = f ∘ f′, and this morphism is ℬ-cartesian : the detailed verification shall be left to the reader.

Definition 3.1.2. Let φ : 𝒜 → ℬ be a functor, and B any object of ℬ.

(i) The fibre of φ over B is the category whose objects are all the A ∈ Ob(𝒜) such that φA = B, and whose morphisms f : A′ → A are the elements of Hom_{𝒜} (A′, A) such that φf = 1_{B}. We denote this category by

φ^{−1}B or more simply by 𝒜_{B}

in the contexts where the latter notation does not give rise to ambiguities. We denote the natural faithful embedding of 𝒜_{B} into 𝒜 by :

ι_{B} : 𝒜_{B} → 𝒜 .

(ii) We say that φ is a fibration if, for every morphism g : B′ → B in ℬ, and every A ∈ Ob(𝒜_{B}), there exists an inverse image f : A′ → A of A over g. In this case, we also say that (𝒜 , φ) is a fibred ℬ-category, and φ is called the structure functor of 𝒜.

Example 3.1.3. Let ℬ, 𝒞 be any two categories, F : 𝒞 → ℬ a functor, X any object of ℬ.

(i) The source functor s_{X} : F 𝒞 /X → 𝒞 of (1.1.27) is a fibration, and all the morphisms in F 𝒞 /X are 𝒞 -cartesian. The easy verification shall be left to the reader.

(ii) The source functor s : ℬ/F 𝒞 → ℬ of (1.1.28) is a fibration. Namely, the s-cartesian morphisms are the commutative diagrams

\xymatrix{B \ar[r]^{f} \ar[d]_{g} & F C \ar[d]^{F g′} \\ B′ \ar[r]^{f′} & F C′}

such that g′ is an isomorphism in 𝒞. The fibre s^{−1}X is the category X/F 𝒞. As a special case, the source functor s : Morph(ℬ) → ℬ of (1.1.30) is a fibration.

(iii) Suppose that all fibre products are representable in ℬ. Then also the target functor t : Morph(ℬ) → ℬ is a fibration; more precisely, the t-cartesian morphisms are the square diagrams as in (ii) (with F := 1_{ℬ} ) which are cartesian (i.e. fibred). We have t^{−1}X = ℬ/X.

Definition 3.1.4. Let 𝒜, 𝒜′, and ℬ be three categories, φ : 𝒜 → ℬ and φ′ : 𝒜′ → ℬ two functors, and F : 𝒜 → 𝒜′ a ℬ-functor, i.e. F verifies the identity φ′ ∘ F = φ.

(i) We say that F is cartesian if it sends ℬ-cartesian morphisms of 𝒜 to ℬ-cartesian morphisms in 𝒜′. We denote by :

Cart_{ℬ} (𝒜 , 𝒜′)

the category whose objects are the cartesian ℬ-functors F : 𝒜 → 𝒜′, and whose morphisms are the natural ℬ-transformations, i.e. the natural transformations such that :

α : F ⇒ G such that φ′ ∗ α = 1_{φ}.

The composition law is the usual composition of natural transformations : (β, α) ↦ α ⊙ β. Notice that if 𝒜 and 𝒜′ are small, the same holds for Cart_{ℬ} (𝒜 , 𝒜′) : details left to the reader.

(ii) For any two other ℬ-categories 𝒞 → ℬ and 𝒞′ → ℬ, every pair of cartesian ℬ-functors H : 𝒞 → 𝒜 and K : 𝒜′ → 𝒞′ induces a functor :

Cart_{ℬ} (H, K) : Cart_{𝒜} (𝒜 , 𝒜′) → Cart_{ℬ} (𝒞 , 𝒞′) G ↦ K ∘ G ∘ H.

To any morphism α : G ⇒ G′ in Cart_{ℬ} (𝒜 , 𝒜′), the functor Cart_{ℬ} (H, K) assigns the natural transformation K ∗ α ∗ H : K ∘ G ∘ H ⇒ K ∘ G′ ∘ H. In case H = 1_{𝒜} (resp. K = 1_{𝒜′} ) we also denote this functor by Cart_{ℬ} (𝒜 , K) (resp. by Cart_{ℬ} (H, 𝒜′)).

Likewise, if H, H′ : 𝒞 → 𝒜 and K, K′ : 𝒜′ → 𝒞′ are four ℬ-cartesian functors, every pair of natural ℬ-transformations β : H ⇒ H′ and γ : K ⇒ K′ induces a natural transformation :

Cart_{ℬ} (β, γ) : Cart_{ℬ} (H, K) ⇒ Cart_{ℬ} (H′, K′) G ↦ γ ∗ G ∗ β

and again, if β = 1_{H} (resp. γ = 1_{K}) we also denote this natural transformation by Cart_{ℬ} (H, γ) (resp. Cart_{ℬ} (α, K)).

(iii) Let U, V be two universes, with U ⊂ V, and such that ℬ is V-small; we say that the fibration φ has essentially U-small fibres if φ^{−1}B is an essentially U-small category for every B ∈ Ob(ℬ). The V-small fibrations over ℬ with essentially U-small fibres form a 2-category

(U, V)-Fib(ℬ)

with Hom-category Cart_{ℬ} (𝒞 , 𝒞′), for every pair of fibrations 𝒞 → ℬ ← 𝒞′, and with composition functors given by (ii). When there is no danger of ambiguities, we shall often write U-Fib(ℬ), or even just Fib(ℬ) for this 2-category. Notice that if ℬ is essentially U-small, then (U, V)-Fib(ℬ) has essentially U-small Hom-categories.

Remark 3.1.5. Let ℬ and ℬ′ be two categories and ψ : ℬ′ → ℬ any functor.

(i) For every category 𝒜 and every functor φ : 𝒜 → ℬ, the cartesian morphisms of the projection φ′ : 𝒜 ×_{(φ,ψ)} ℬ′ → ℬ′ are the pairs (f, f′) where f is a cartesian morphism of 𝒜, f′ is a morphism of ℬ′, and φf = ψf′ (see example 1.2.25(i)). Moreover, the projection 𝒜 ×_{(φ,ψ)} ℬ′ → 𝒜 restricts to a natural isomorphism of fibre categories

φ^{′−1}B′ →^{∼} φ^{−1}(ψB′) for every B′ ∈ Ob(ℬ′) .

If φ is a fibration, the same then holds for φ′. Therefore, for every pair of universes U ⊂ V such that ℬ and ℬ′ are V-small, we get a well defined strict pseudo-functor

(U, V)-Fib(ψ)^{∗} : (U, V)-Fib(ℬ) → (U, V)-Fib(ℬ′) (𝒜 ⟶^{φ} ℬ) ↦ (𝒜 ×_{(φ,ψ)} ℬ′ ⟶^{φ′} ℬ′)

which assigns to every ℬ-cartesian functor F : 𝒜_{1} → 𝒜_{2} the ℬ′-cartesian functor F ×_{ℬ} ℬ′ : 𝒜_{1}×_{ℬ} ℬ′ → 𝒜_{2}×_{ℬ} ℬ′, and to any natural ℬ-transformation α : F_{1} ⇒ F_{2} between ℬ-cartesian functors F_{1}, F_{2} : 𝒜_{1} → 𝒜_{2} the induced ℬ′-transformation α ×_{ℬ} ℬ′ : F_{1} ×_{ℬ} ℬ′ ⇒ F_{2} ×_{ℬ} ℬ′. As usual, we often write U-Fib(ψ)^{∗}, or just Fib(ψ)^{∗} instead of (U, V)-Fib(ψ)^{∗}.

(ii) Suppose that ψ is a fibration with essentially U-small fibres; then for every fibration F′ : 𝒜′ → ℬ′ with essentially U-small fibres, the composition ψ ∘ F′ : 𝒜′ → ℬ is also a fibration with essentially U-small fibres, and in this case we get therefore also a well defined strict pseudo-functor

(U, V)-Fib(ℬ′) → (U, V)-Fib(ℬ) (F′ : 𝒜′ → ℬ′) ↦ (ψ ∘ F : 𝒜′ → ℬ).

(iii) For i = 1, 2, let 𝒜_{i} → ℬ be two fibrations; combining (i) and (ii) we see that the induced functor 𝒜_{1} ×_{ℬ} 𝒜_{2} → ℬ is also a fibration.

(iv) In the situation of (i), it is easily seen that Cart_{ℬ′} (ℬ′, 𝒜 ×_{(φ,ψ)} ℬ′) is naturally identified with the full subcategory of Cart_{ℬ} (ℬ′, 𝒜) whose objects are the ℬ-functors ℬ′ → 𝒜 that send every morphism of ℬ′ to a ℬ-cartesian morphism of 𝒜 : the detailed verification shall be left to the reader.

(v) Suppose that F : 𝒜 → 𝒜′ is a ℬ-equivalence, i.e. an equivalence in the 2-category V-Cat/ℬ (in the sense of definition 2.1.3(iii)). Then it is easily seen that F is ℬ-cartesian.

(vi) In the situation of (i), let ψ′ : ℬ′′ → ℬ′ be another functor between V-small categories; then we have a natural isomorphism of categories :

(U, V)-Fib(ψ′)^{∗} ∘ (U, V)-Fib(ψ)^{∗}(𝒜) →^{∼} (U, V)-Fib(ψ ∘ ψ′)^{∗}(𝒜)

that assigns to every object (A, B′, B′′) of (𝒜 ×_{(φ,ψ)} ℬ′) ×_{(φ′,ψ′)} ℬ′′ the object (A, B′′) of 𝒜 ×_{(φ,ψ∘ψ′)} ℬ′′, and it is likewise defined on morphisms. Clearly this isomorphism is strictly pseudo-natural with respect to 𝒜, so we get a strict pseudo-natural isomorphism of strict pseudo-functors :

γ_{ψ′,ψ}^{(U,V)-Fib} : (U, V)-Fib(ψ′)^{∗} ∘ (U, V)-Fib(ψ)^{∗} →^{∼} (U, V)-Fib(ψ ∘ ψ′)^{∗}.

3.1.6. Cleavages. Let φ : 𝒜 → ℬ be a fibration, and for every object (A, g : B → F A) of ℬ/φ𝒜 let us choose an inverse image of A over g :

g_{A} : g^{∗}A → A.

Then we claim that the rule (A, g) ↦ g_{A} extends uniquely to a functor

λ : ℬ/φ𝒜 → Morph(𝒜)

that makes commute the resulting diagram (notation of (1.1.30)) :

(3.1.7)  [figure: ℬ/φ𝒜; λ; S; Morph(𝒜); Morph(φ); Morph(ℬ).]

Indeed, since g_{A} is φ-cartesian, for every morphism (h, k) : (A′, B′ ⟶^{g′} φA′) → (A, B ⟶^{g} φA) of ℬ/φ𝒜 there exists a unique morphism f : g^{′∗}A′ → g^{∗}A that makes commute the diagram

(3.1.8)  \xymatrix{g^{′∗}A′ \ar[r]^{g′_{A′}} \ar[d]_{f} & A′ \ar[d]^{h} \\ g^{∗}A \ar[r]^{g_{A}} & A}  and such that φ(f) = k

and this square diagram can be regarded as a morphism (f, h) : g′_{A′} → g_{A} in Morph(𝒜) such that Morph(φ)(f, h) = (k, φ(h)) = S(h, k). The uniqueness of f easily implies that the rule (h, k) ↦ (f, h) yields a well defined functor as sought. We call a cleavage for φ any such functor (“clivage” in french); hence, the cleavages for φ are characterized as the functors λ that make commute (3.1.7) and that map every object of ℬ/φ𝒜 to a φ-cartesian morphism.

Notice that we may always choose a cleavage such that λ(A, 1_{F A}) = 1_{A} for every A ∈ Ob(𝒜). We call unital a cleavage fulfilling this condition.

Example 3.1.9. Let φ : 𝒜 → ℬ be any fibration. Then Morph(φ) : Morph(𝒜) → Morph(ℬ) is a fibration as well (notation of (1.1.30)). Indeed, pick a cleavage λ for φ; consider any object (A′ ⟶^{h} A) of Morph(𝒜) and a morphism (g′, g) : (B′ ⟶^{k} B) → (φA′ ⟶^{φ(h)} φA) of Morph(ℬ). By definition, g : B → φA and g′ : B′ → φA′ are morphisms of ℬ such that

φ(h) ∘ g′ = g ∘ k, hence (h, k) : (A′, B′ ⟶^{g′} φA′) → (A, B ⟶^{g} φA) is a morphism of ℬ/φ𝒜,

and (f, h) := λ(h, k) is a commutative square (3.1.8). With this notation, it is easily seen that (g^{′∗}A′ ⟶^{f} g^{∗}A) is an inverse image of (A′ ⟶^{h} A) over (g′, g) : the details are left to the reader.

3.1.10. The pseudo-functor associated with a cleavage. Let now λ be any cleavage for a given fibration φ : 𝒜 → ℬ; to any morphism g : B′ → B in ℬ we attach a functor

c_{g} : φ^{−1}B → φ^{−1}B′

as follows. First, we consider the functor

(−, g) : φ^{−1}B → B′/φ𝒜  A ↦ (A, g)  (A_{1} ⟶^{h} A_{2}) ↦ ((A_{1}, g) ⟶^{(h,1_{B′})} (A_{2}, g)).

Let also s : Morph(𝒜) → 𝒜 be the source functor (see (1.1.30)), and notice that the composition s ∘ λ ∘ (−, g) factors through φ^{−1}B′, and yields therefore the sought functor c_{g}. Moreover, the commutativity of the diagram (3.1.8) also means that the rule : A ↦ λ(A, g) defines a natural transformation

(3.1.11)  g_{•} : ι_{B′} ∘ c_{g} ⇒ ι_{B}

(notation of definition 3.1.2(i)). Notice especially the natural isomorphism of functors

δ_{B} : 1_{φ^{−1}B} ⇒ c_{1_{B}}  A ↦ ((1_{B})_{A}^{−1} : A →^{∼} (1_{B})^{∗}A)  for every B ∈ Ob(ℬ) .

Furthermore, let B′′ ⟶^{h} B′ ⟶^{g} B be two morphisms in ℬ, so we get the functors

c_{g} : φ^{−1}B → φ^{−1}B′  c_{h} : φ^{−1}B′ → φ^{−1}B′′  c_{g∘h} : φ^{−1}B → φ^{−1}B′′

as well as the natural transformations :

g_{•} : ι_{B′} ∘ c_{g} ⇒ ι_{B}  h_{•} : ι_{B′′} ∘ c_{h} ⇒ ι_{B′}  (g ∘ h)_{•} : ι_{B′′} ∘ c_{g∘h} ⇒ ι_{B}.

By inspecting the constructions, one easily finds a unique natural isomorphism :

γ_{(h,g)} : c_{h} ∘ c_{g} ⇒ c_{g∘h}

which fits into a commutative diagram :

\xymatrix{ι_{B′′} ∘ c_{h} ∘ c_{g} \ar@{=>}[r]^{h_{•}∗c_{g}} \ar@{=>}[d]_{ι_{B′′}∗γ_{(h,g)}} & ι_{B′} ∘ c_{g} \ar@{=>}[d]^{g_{•}} \\ ι_{B′′} ∘ c_{g∘h} \ar@{=>}[r]^{(g∘h)_{•}} & ι_{B}.}

Moreover, if k : B′′′ → B′′ is a third morphism of ℬ, we can compute :

(g ∘ h ∘ k)_{A} ∘ γ_{(k,g∘h),A} ∘ c_{k}(γ_{(h,g),A}) = (g ∘ h)_{A} ∘ k_{c_{g∘h}A} ∘ c_{k}(γ_{(h,g),A})
= (g ∘ h)_{A} ∘ γ_{(h,g),A} ∘ k_{c_{h}c_{g}A}
= g_{A} ∘ h_{c_{g}A} ∘ k_{c_{h}c_{g}A}
= g_{A} ∘ (h ∘ k)_{c_{g}A} ∘ γ_{(k,h),c_{g}A}
= (g ∘ h ∘ k)_{A} ∘ γ_{(h∘k,g),A} ∘ γ_{(k,h),c_{g}A}

for every A ∈ Ob(𝒜), and since (g ∘ h ∘ k)_{A} is ℬ-cartesian, we deduce :

(3.1.12)  γ_{(k,g∘h),A} ∘ c_{k}(γ_{(h,g),A}) = γ_{(h∘k,g),A} ∘ γ_{(k,h),c_{g}A}.

Likewise, for every morphism f : B′ → B of ℬ and every A ∈ Ob(φ^{−1}B) we have the commutative diagram :

\xymatrix{f^{∗}1_{B}^{∗}A \ar[r]^{c_{f} (1_{B})_{A}} \ar[d]_{f_{1^{∗}_{B} A}} & f^{∗}A \ar[d]^{f_{A}} & 1_{B′}^{∗}f^{∗}A \ar[l]_{(1_{B′})_{f^{∗}A}} \ar[d]^{γ_{(1_{B′} ,f),A}} \\ 1_{B}^{∗}A \ar[r]^{(1_{B})_{A}} & A & f^{∗}A \ar[l]_{f_{A}}}

which implies :

(3.1.13)  γ_{(f,1_{B}),A} = c_{f}((1_{B})_{A})  γ_{(1_{B′},f),A} = (1_{B′})_{c_{f} A}.

Suppose now that φ is an object of (U, V)-Fib, for some universe V, and denote by

(U, V)-Cat

the strong sub-2-category of V-Cat whose objects are the essentially U-small categories (see definition 2.4.9(iii)). Then the identities (3.1.12) and (3.1.13) mean that the rule which assigns to each B ∈ Ob(ℬ) the small category φ^{−1}B and to each morphism g in ℬ the functor c_{g} defines a pseudo-functor

c : ℬ^{o} → (U, V)-Cat

whose coherence constraint is the system of natural isomorphisms (δ_{•}, γ_{•}). Moreover, the system of inclusion functors (ι_{B} | B ∈ Ob(ℬ)) amounts to a pseudo-cocone

ι : c ⇒ F_{𝒜}

whose coherence constraint is given by the system of natural transformations (3.1.11). Notice that if λ is a unital cleavage, c will be a unital pseudo-functor. (Here we view ℬ^{o} as a 2category, as explained in example 2.2.4(i); also, the 2-category structure on V-Cat is the one of remark 2.1.1(ii)). We call c and ι respectively the pseudo-functor and the pseudo-cocone associated with the cleavage λ.

Example 3.1.14. As an application, we may generalize example 1.5.15 as follows. Let I, J be two small categories, φ : I → J a fibration, fix a cleavage λ for φ, and denote by c the associated pseudo-functor.

(i) First, we claim that for every j ∈ Ob(J) the functor (−, 1_{j}) : φ^{−1}j → j/φI is coinitial (notation of (3.1.10)). Indeed, for any (i_{0}, f : j → φi_{0}) ∈ Ob(j/φI) we have the cartesian morphism λ(i_{0}, f) : c_{f}(i_{0}) → i_{0}, and it is easily seen that the category (−, 1_{j})(φ^{−1}j)/(i_{o}, f) is isomorphic to (φ^{−1}j)/c_{f}(i_{0}), which is obviously connected, whence the claim.

(ii) Notice that the composition t_{j} ∘ (−, 1_{j}) : φ^{−1}j → I equals the inclusion functor ι_{j} (here t_{j} : j/φI → I is the target functor). Moreover, for every morphism f : j → j′ in J, the natural transformation f_{•} : ι_{j} ∘ c_{f} ⇒ ι_{j′} associated with λ induces a morphism

ω_{F}^{f} : lim_{φ^{−1}j} F ∘ ι_{j} ⟶^{lim_{c_{f}} 1_{F ∘ι_{j}}} lim_{φ^{−1}j′} F ∘ ι_{j} ∘ c_{f} ⟶^{lim_{φ^{−1}j′} F ∗f_{•}} lim_{φ^{−1}j′} F ∘ ι_{j′}

for every functor F : I → 𝒞, fitting into a commutative diagram

\xymatrix{∫_{φ}^{∧} F (j) \ar[r]^{∫_{φ}^{∧} F (f)} \ar[d] & ∫_{φ}^{∧} F (j′) \ar[d] \\ lim_{φ^{−1}j} F ∘ ι_{j} \ar[r]^{ω_{F}^{f}} & lim_{φ^{−1}j′} F ∘ ι_{j′}}

whose vertical arrows are the natural isomorphisms provided by (i) and remark 1.5.5(ii). In other words, ∫_{φ}^{∧} is naturally isomorphic to the functor that assigns to every such F the functor

∫_{λ}^{∧} F : J → 𝒞^{∧}  j ↦ lim_{φ^{−1}j} F ∘ ι_{j}  (f : j → j′) ↦ ω_{F}^{f} .

(iii) Furthermore, in this situation, proposition 1.3.2says that there is a natural isomorphism

lim_{I} F →^{∼} Lim_{J} ∫_{λ}^{∧} F in 𝒞^{∧}

for every functor F : I → 𝒞. If 𝒞 is complete, the functor ∫_{λ}^{∧} is isomorphic, by remark 1.5.16, to the composition of Fun(J, h_{𝒞} ) and a functor well defined up to isomorphism

∫_{λ} : Fun(I, 𝒞) → Fun(J, 𝒞) F ↦ (j ↦ Lim_{φ^{−1}j} F ∘ ι_{j})

and we can restate the foregoing “Fubini” isomorphism in terms of this latter functor.

(iv) Lastly, let G : I^{o} → 𝒞 be any functor. It follows from (ii) that the functor ∫_{∧}^{φ^{o}} G is naturally isomorphic to the functor

∫_{∧}^{λ} G : J^{o} → 𝒞^{o∧o}  j^{o} ↦ colim_{(φ^{−1}j)^{o}} G ∘ ι_{j}^{o}  (f^{o} : j^{′o} → j^{o}) ↦ ω_{G^{o}}^{f}

and we have a natural isomorphism

colim_{I^{o}} G →^{∼} colim_{J^{o}} ∫_{∧}^{λ} G in 𝒞^{o∧o} .

Again, if 𝒞 is cocomplete, the functor ∫_{∧}^{λ} is the composition of Fun(J^{o}, h_{𝒞^{o}}^{o}) and a functor

∫ ^{λ} : Fun(I^{o}, 𝒞) → Fun(J^{o}, 𝒞) G ↦ (j^{o} ↦ Colim_{(φ^{−1}j)^{o}} G ∘ ι_{j}^{o})

and we may state the foregoing “Fubini” isomorphism for colimits in terms of this latter functor.

3.1.15. Fibration associated with a presheaf. Let ℬ be a category, F a presheaf on ℬ. As in (1.4.7), we let ℱ ib(F) be the category of elements of F, and we notice that the source functor

s_{F} : ℱ ib(F) → ℬ

is a fibration. For every X ∈ Ob(ℬ), the fibre s_{F}^{−1}(X) is (naturally isomorphic to) the discrete category F X (see example 1.1.6(ii)). Notice also that every morphism in ℱ ib(F) is cartesian, and for every ℬ-category 𝒞, the category Cart_{ℬ} (𝒞 , ℱ ib(F)) is discrete.

For every pair of presheaves F, G on ℬ, we have a natural bijection:

Hom_{𝒞^{∧}}(F, G) →^{∼} Ob(Cart_{ℬ} (ℱ ib(F), ℱ ib(G)))

(which we can view as an isomorphism of discrete categories). Namely, to a morphism ψ : F → G one assigns the functor

ℱ ib(ψ) : ℱ ib(F) → ℱ ib(G) (X, s) ↦ (X, ψ_{X}(s)) for every (X, s) ∈ Ob(ℱ ib(F)) .

Example 3.1.16. Let U be a universe, and 𝒞 a category with U-small Hom-sets.

(i) We have already remarked in example 1.4.9(ii) the identification of 𝒞 -fibrations

ℱ ib(h_{X}) = 𝒞 /X for every X ∈ Ob(𝒞)

where h_{X} is the presheaf represented by X (see (1.2.4)), and 𝒞 /X is regarded as a 𝒞 -fibration as in example 3.1.3(i). Moreover, every morphism f : X → Y of 𝒞 induces a morphism of presheaves h_{f} : h_{Y} → h_{X}, and under the foregoing identification, ℱ ib(h_{f}) corresponds to the functor f_{∗} : 𝒞 /X → 𝒞 /Y of (1.1.25).

(ii) In the situation of (i), let u : 𝒞′ → 𝒞 be any functor; as in example 1.4.9(iii), by direct inspection we get a natural isomorphism of categories

Fib(u)^{∗}(𝒞 /X) →^{∼} ℱ ib(h_{X} ∘ u) = u𝒞′/X for every X ∈ Ob(𝒞)

(notation of remark 3.1.5(i)) which identifies the fibration Fib(u)^{∗}(𝒞 /X) → 𝒞′ with the source functor s_{X} : u𝒞′/X → 𝒞′ of (1.1.27). Likewise, for every morphism f : X → Y in 𝒞, the cartesian functor Fib(u)^{∗}(f_{∗}) : Fib(u)^{∗}(𝒞 /X) → Fib(u)^{∗}(𝒞 /Y) is identified with the functor u𝒞′/f : u𝒞′/X → u𝒞′/Y as in (1.1.27).

(iii) As a special case, let h_{𝒞} : 𝒞 → 𝒞_{U}^{∧} be the Yoneda embedding, and F any U-presheaf on 𝒞. As noted in (1.4.7), we have a natural isomorphism of categories h_{𝒞} 𝒞 /F →^{∼} ℱ ib(F), so (i) generalizes to a natural isomorphism of 𝒞 -fibrations :

ℱ ib(F) →^{∼} (V, V)-Fib(h_{𝒞} )^{∗}(𝒞_{U}^{∧}/F)

for any universe V with U ⊂ V, and such that 𝒞 is V-small.

3.1.17. Conversely, let φ : 𝒜 → ℬ be a fibration with discrete fibres, i.e. such that φ^{−1}B is a discrete small category for every B ∈ Ob(ℬ). Then it is easily seen that for every morphism g : B′ → B of ℬ, every A ∈ φ^{−1}B admits a unique inverse image over g. It follows that φ admits a unique cleavage. Moreover the associated pseudo-functor c : ℬ^{o} → Cat is strict, hence it may be regarded as a functor with values in the subcategory Set of Cat; i.e. c is a presheaf on ℬ, and a simple inspection shows that the fibration ℱ ib(c) is naturally isomorphic to φ. Summing up, we have obtained a fully faithful functor

ℱ ib : ℬ^{∧} → Fib(ℬ)

whose essential image is the full subcategory of Fib(ℬ) whose objects are the fibrations with discrete fibres. We shall show next how to extend this equivalence to the whole of Fib(ℬ).

3.1.18. Fibration associated with a pseudo-functor. Let ℬ be any V-small category, and c : ℬ^{o} → (U, V)Cat any pseudo-functor, with coherence constraint (δ^{c}, γ^{c}). We attach to c the ℬ-category

π^{c} : ℱ ib(c) → ℬ

such that Ob(ℱ ib(c)) := {(B, X) | B ∈ Ob(ℬ), X ∈ Ob(c_{B})}, and where

Hom_{ℱ ib(c)} ((B, X), (B′, Y)) := {(φ, f) | φ ∈ Hom_{ℬ} (B, B′), f ∈ Hom_{c_{B}}(X, c_{φ}Y)}

for every two objects (B, X), (B′, Y). The composition of two morphisms f : X → c_{φ}Y and g : Y → c_{ψ}Z is the pair (ψ ∘ φ, t), where t is the composition

X ⟶^{f} c_{φ}Y ⟶^{c_{φ}(g)} c_{φ}c_{ψ}Z ⟶^{γ_{(φ,ψ),Z}^{c}} c_{ψ∘φ}Z.

The unit axiom for δ^{c} implies easily that for every object (B, X) the morphism

(1_{B}, δ_{B,X}^{c} : X → c_{1_{B}}X)

is neutral for left and right composition with any other morphism of ℱ ib(c). Let us show the associativity : if h : Z → c_{λ}W is a third morphism, we need to verify the identity

γ_{(ψ∘φ,λ),W}^{c} ∘ c_{ψ∘φ}(h) ∘ γ_{(φ,ψ),Z}^{c} ∘ c_{φ}(g) ∘ f = γ_{(φ,λ∘ψ),W}^{c} ∘ c_{φ}(γ_{(ψ,λ),W}^{c} ) ∘ c_{φ}c_{ψ}(h) ∘ c_{φ}(g) ∘ f.

But we have c_{ψ∘φ}(h) ∘ γ_{(φ,ψ),Z}^{c} = γ_{(φ,ψ),c_{λ}W}^{c} ∘ c_{φ}c_{ψ}(h) by the naturality of γ_{(φ,ψ)}^{c}, hence we are reduced to checking that

γ_{(ψ∘φ,λ),W}^{c} ∘ γ_{(φ,ψ),c_{λ}W}^{c} = γ_{(φ,λ∘ψ),W}^{c} ∘ c_{φ}(γ_{(ψ,λ),W}^{c} )

which follows from the composition axioms for γ^{c}. The functor π^{c} is given by the rules : (B, X) ↦ B and (φ, f) ↦ φ for every object (B, X) and every morphism (φ, f) of ℱ ib(c).

3.1.19. Let d : ℬ^{o} → (U, V)-Cat be another pseudo-functor, with coherence constraint (δ^{d}, γ^{d}), and

ω : c ⇒ d

any pseudo-natural transformation, with coherence constraint τ^{ω}. We define a ℬ-functor

ℱ ib(ω) : ℱ ib(c) → ℱ ib(d)

by assigning to every object (B, X) of ℱ ib(c) the object (B, ω_{B}X) of ℱ ib(d), and to every morphism (φ, f) : (B, X) → (B′, Y) the morphism (φ, t), where t is the composition

ω_{B}X ⟶^{ω_{B}f} ω_{B} ∘ c_{φ}Y ⟶^{(τ_{φ,Y}^{ω} )^{−1}} d_{φ} ∘ ω_{B′}Y.

To check that these rules do yield a functor, consider any other morphism (ψ, g) : (B′, Y) → (B′′, Z) of ℱ ib(c); we need to show that

τ_{ψ∘φ,Z}^{ω −1} ∘ ω_{B}(γ_{(φ,ψ),Z}^{c} ∘ c_{φ}(g) ∘ f) = γ_{(φ,ψ),ω_{B′′}Z}^{d} ∘ d_{φ}(τ_{ψ,Z}^{ω −1} ∘ ω_{B′}(g)) ∘ τ_{φ,Y}^{ω −1} ∘ ω_{B}(f).

However, the naturality of τ_{φ}^{ω} implies that τ_{φ,c_{ψ}Z}^{ω −1} ∘ ω_{B}(c_{φ}g) = d_{φ}(ω_{B′}g) ∘ τ_{φ,Y}^{ω −1}, so we are reduced to checking that :

τ_{ψ∘φ,Z}^{ω −1} ∘ ω_{B}(γ_{(φ,ψ),Z}^{c}) = γ_{(φ,ψ),ω_{B′′}Z}^{d} ∘ d_{φ}(τ_{ψ,Z}^{ω −1}) ∘ τ_{φ,c_{ψ}Z}^{ω −1}.

But the latter follows directly from the coherence axioms for τ^{ω}. Likewise, the assertion that ℱ ib(ω) respects identity morphisms comes down to the equalities :

τ_{1_{B},X}^{ω −1} ∘ ω_{B}(δ_{B,X}^{c}) = δ_{B,ω_{B}X}^{d}

which again follows from the coherence axioms for τ^{ω}. Moreover, if e : ℬ^{o} → (U, V)-Cat is a third pseudo-functor, and

μ : d ⇒ e

a second pseudo-natural transformation, we have

ℱ ib(μ) ∘ ℱ ib(ω) = ℱ ib(μ ⊙ ω).

Indeed, if τ^{μ} and τ^{μ⊙ω} are the coherence constraints for μ and μ ⊙ ω, the assertion amounts to :

τ_{φ,ω_{B′}Y}^{μ −1} ∘ μ_{B}(τ_{φ,Y}^{ω})^{−1} ∘ μ_{B}(ω_{B}f) = τ_{B,Y}^{μ⊙ω −1} ∘ μ_{B}(ω_{B}f)

for every morphism (φ, f) : (B, X) → (B′, Y) in ℱ ib(c), which is clear from the definition of μ ⊙ ω.

Lemma 3.1.20. With the notation of (3.1.19), the following holds :

(i) A morphism (φ, f) : (B, X) → (B′, Y) of ℱ ib(c) is π^{c}-cartesian if and only if f : X → c_{φ}Y is an isomorphism in c_{B}.

(ii) The functor π^{c} is a fibration, and is endowed with a distinguished cleavage

λ^{∗} : ℬ/π^{c}ℱ ib(c) → Morph(ℱ ib(c)) (X, φ) ↦ (φ, 1_{c_{φ}X}).

(iii) The functor ℱ ib(ω) is ℬ-cartesian.

Proof. (i): In view of proposition 2.4.3and the discussion of (3.1.19), we may assume that λ is unital (details left to the reader). Now, suppose first that (φ, f) is ℬ-cartesian; then there exists a unique morphism (1_{B}, g) : (B, c_{φ}Y) → (B, X) such that (φ, f) ∘ (1_{𝔹}, g) = (φ, 1_{c_{φ}Y}). Since c is unital, we easily deduce that f ∘ g = 1_{c_{φ}Y}. Moreover, (1_{B}, g ∘ f) : (B, X) → (B, X) is a morphism of ℱ ib(c) such that (φ, f) ∘ (1_{B}, g ∘ f) = (φ, f) = (φ, f) ∘ (1_{B}, 1_{X}), whence g ∘ f = 1_{X}. Conversely, suppose that f is an isomorphism, and let ψ : C → B any morphism in ℬ, Z ∈ Ob(c_{C}) any object, and (φ ∘ ψ, h) : (C, Z) → (B′, Y) any morphism of ℱ ib(c); we let k : Z → c_{ψ}X be the composition

Z ⟶^{h} c_{φ∘ψ}Y ⟶^{γ_{(ψ,φ),Y}^{c −1}} c_{ψ}c_{φ}Y ⟶^{c_{φ}(f^{−1})} c_{ψ}X.

Then (ψ, k) : (C, Z) → (B, X) is the unique morphism of ℱ ib(c) such that (φ, f) ∘ (ψ, k) = (φ ∘ ψ, h); this shows that (φ, f) is ℬ-cartesian.

(ii) and (iii) are immediate consequences of (i). □

3.1.21. Lastly, consider pseudo-natural transformations

ω, ω′ : c ⇒ d and a modification Ξ : ω ⇝ ω′.

We attach to Ξ the natural transformation of functors

ℱ ib(Ξ) : ℱ ib(ω) ⇒ ℱ ib(ω′)

assigning to every object (B, X) of ℱ ib(c) the morphism (1_{B}, t) of ℱ ib(d), where t is the composition

ω_{B}X ⟶^{Ξ_{B,X}} ω′_{B}X ⟶^{δ_{B,ω′_{B} X}^{d}} d_{1_{B}}(ω′_{B}X).

In order to verify the naturality of ℱ ib(Ξ), it suffices to check the commutativity of the diagram:

\xymatrix{ω_{B}X \ar[r]^{Ξ_{B,X}} \ar[d]_{ω_{B}f} & ω′_{B}X \ar[r]^{δ_{B,ω′_{B} X}^{d}} \ar[d]_{ω′_{B}f} & d_{1_{B}}(ω′_{B}X) \ar[d]^{d_{1_{B}} (ω′_{B}f)} \\ ω_{B}c_{φ}Y \ar[r]^{Ξ_{B,cφY}} \ar[d]_{τ_{φ,Y}^{ω −1}} & ω′_{B}c_{φ}Y \ar[r]^{δ_{B,ω′_{B} cφY}^{d}} \ar[d]_{τ_{φ,Y}^{ω′ −1}} & d_{1_{B}}(ω′_{B}c_{φ}Y) \ar[d]^{γ_{(1_{B} ,φ),ω{′ \atop B′} Y}^{d} ∘d_{1_{B}} (τ_{φ,Y}^{ω′ −1})} \\ d_{φ}ω_{B′}Y \ar[r]^{d_{φ}(Ξ_{B′,Y} )} & d_{φ}ω′_{B′}Y \ar[r]^{γ_{(φ,1_{B}),ω{′ \atop B′} Y}^{d} ∘d_{φ}(δ_{B′,ω{′ \atop B′} Y}^{d} )} & d_{φ}ω′_{B′}Y}

for every morphism (φ, f) : (B, X) → (B′, Y) of ℱ ib(c). However, the commutativity of the two top squares follows from the naturality of Ξ_{B} and δ_{B}^{d}, and that of the left bottom square translates the compatibility condition for Ξ. Then, since

γ_{(1_{B},φ),ω′_{B′}Y}^{d} = δ_{B′,d_{φ}ω′_{B′}Y}^{d −1} and γ_{(φ,1_{B}),ω′_{B′}Y}^{d} ∘ d_{φ}(δ_{B′,ω′_{B′}Y}^{d} ) = 1_{d_{φ}ω′_{B′}Y}

we are reduced to checking the identity

d_{1_{B}}(τ_{φ,Y}^{ω′}) ∘ δ_{B′,d_{φ}ω′_{B′}Y}^{d} = δ_{B,ω′_{B}c_{φ}Y}^{d} ∘ τ_{φ,Y}^{ω′}

which follows from the naturality of δ_{B}^{d}. Furthermore, suppose we have a third pseudo-natural transformation

ω′′ : c ⇒ d and a modification Ξ′ : ω′ ⇝ ω′′.

Then we have as well

(3.1.22)  ℱ ib(Ξ′ ⊙ Ξ) = ℱ ib(Ξ′) ⊙ ℱ ib(Ξ).

Indeed, the assertion amounts to checking, for every object (B, X) of ℱ ib(c), the identity

γ_{(1_{B},1_{B}),ω′′_{B}X}^{d} ∘ d_{1_{B}}(δ_{B,ω′′_{B}X}^{d}) ∘ d_{1_{B}}(Ξ′_{B,X}) ∘ δ_{B,ω′_{B}X}^{d} ∘ Ξ_{B,X} = δ_{B,ω′′_{B}X}^{d} ∘ Ξ′_{B,X} ∘ Ξ_{B,X}

which is clear, since γ_{(1_{B},1_{B}),ω′′_{B}X}^{d} ∘ d_{1_{B}}(δ_{B,ω′′_{B}X}^{d}) = 1_{d_{1_{B}} ω′′_{B}X} by the unit axiom for δ^{d}, and d_{1_{B}}(Ξ′_{B,X}) ∘ δ_{B,ω′_{B}X}^{d} = δ_{B,ω′′_{B}X}^{d} ∘ Ξ′_{B,X}, by the naturality of δ_{B}^{d}. Likewise, suppose we have two pseudo-natural transformations

μ, μ′ : d ⇒ e and a modification Θ : μ ⇝ μ′.

Then we have as well

ℱ ib(Θ) ∗ ℱ ib(Ξ) = ℱ ib(Θ ∗ Ξ).

For the proof, in light of (3.1.22) we may assume that either Θ = 1_{μ} or Ξ = 1_{ω}. In the first case, the assertion comes down to the identity

(τ_{1_{B},ω′_{B}X}^{μ})^{−1} ∘ μ_{B}(δ_{B,ω′_{B}X}^{d} ∘ Ξ_{B,X}) = δ_{B,μ_{B}ω′_{B}X}^{d} ∘ μ_{B}(Ξ_{B,X})

which is equivalent to : μ_{B}(δ_{B,ω′_{B}X}^{d}) = τ_{1_{B},ω′_{B}X}^{μ} ∘ δ_{B,μ_{B}ω′_{B}X}^{d}, which in turns follows from the coherence axioms for τ^{μ}. The case where Ξ = 1_{ω} is clear by a simple inspection. Summing up, for every universe V, and every V-small category ℬ, we have obtained a strict pseudo-functor

ℱ ib_{ℬ} : PsFun(ℬ^{o}, (U, V)-Cat) → (U, V)-Fib(ℬ).

Remark 3.1.23. (i) Notice that the discussion of (3.1.19) requires the invertibility of the coherence constraint τ^{ω}, and therefore it does not apply to general lax-natural transformations ω : c ⇒ d. However, with the obvious changes, it does apply to lax-natural transformations ω : ^{o}c ⇒ ^{o}d (see example 2.2.6(ii)) : the details shall be left to the reader. From lemma 3.1.20 and its proof, we then easily see that the resulting functor ℱ ib(ω) : ℱ ib(c) → ℱ ib(d) will be cartesian if and only if ω is pseudo-natural.

(ii) Likewise, the discussion of (3.1.21) applies more generally to a modification Ξ : ω′ ⇝ ω between any pair of lax-natural transformations ω, ω′ : ^{o}c ⇒ ^{o}d; namely, any such modification induces a natural transformation ℱ ib(Ξ) : ℱ ib(ω) ⇒ ℱ ib(ω′).

Theorem 3.1.24. For every category ℬ, the pseudo-functor ℱ ib_{ℬ} is a strong 2-equivalence.

Proof. Let φ : 𝒜 → ℬ be any fibration with essentially small fibres; pick a unital cleavage λ for φ, and let c : ℬ^{o} → (U, V)-Cat be the pseudo-functor associated with λ.

Claim 3.1.25. There exists an isomorphism of ℬ-categories

ψ^{λ} : ℱ ib(c) →^{∼} 𝒜  (B, X) ↦ X  ((B, X) ⟶^{(g,f)} (B′, X′)) ↦ λ(X′, g) ∘ f.

Proof of the claim. Indeed, since c is unital, it is clear that ψ^{λ}(1_{(B,X)}) = 1_{ψ(B,X)} for every (B, X) ∈ Ob(ℱ ib(c)). Next, let (g, f) : (B, X) → (B′, X′) and (g′, f′) : (B′, X′) → (B′′, X′′) be two morphisms; then (g′, f′) ∘ (g, f) = (g′ ∘ g, γ_{(g,g′),X′′}^{c} ∘ c_{g}(f′) ∘ f), and on the other hand, we may compute

\begin{matrix}ψ^{λ}(g′, f′) ∘ ψ^{λ}(g, f) = & λ(X′′, g′) ∘ f′ ∘ λ(X′, g) ∘ f \\ = & λ(X′′, g′) ∘ λ(c_{g′}X′′, g) ∘ c_{g}(f′) ∘ f \\ = & λ(X′′, g′g) ∘ γ_{(g,g′),X′′}^{c} ∘ c_{g}(f′) ∘ f \\ = & ψ^{λ}((g′, f′) ∘ (g, f)) \end{matrix}

as required. Clearly ψ^{λ} is bijective on objects, and since λ(A, g) is a φ-cartesian morphism for every object (A, g) of ℬ/φ𝒜, it is easily seen that ψ^{λ} is fully faithful, hence it is an isomorphism of ℬ-categories. ♢

In view of claim 3.1.25, it remains only to check that for every pair of pseudo-functors c, d : ℬ^{o} → (U, V)-Cat the induced functor

(3.1.26)  PsNat(c, d) → Cart_{ℬ} (ℱ ib(c), ℱ ib(d))

is an isomorphism of categories, and due to proposition 2.4.3and the discussion of (3.1.19), we may assume that c and d are unital. Thus, let ω, ω′ : c ⇒ d be two pseudo-natural transformations, with respective coherence constraints τ^{ω} and τ^{ω′}, and suppose that ℱ ib(ω) = ℱ ib(ω′); then clearly ω_{B}X = ω′_{B}X for every object (B, X) of ℱ ib(c). Likewise, in light of remark 2.4.2(ii) we see that τ_{1_{B},X}^{ω} = τ_{1_{B},X}^{ω′} = 1_{ω_{B}X} for every such (B, X), whence ω_{B}f = ω′_{B}f for every B ∈ Ob(ℬ) and every morphism f of c_{B}. Then it is also clear that τ_{φ,Y}^{ω} = τ_{φ,Y}^{ω′} for every morphism φ : B → B′ of ℬ and every Y ∈ Ob(c_{B′}), i.e. ω = ω′.

Next, let F : ℱ ib(c) → ℱ ib(d) be any ℬ-cartesian functor; for every object (B, X) of ℱ ib(c) we define ω_{B}X as the unique object of d_{B} such that F (B, X) = (B, ω_{B}X). Likewise – and since d is unital – for every morphism of ℱ ib(c) of the form (1_{B}, f) : (B, X) → (B, Y) we may define ω_{B}f : ω_{B}X → ω_{B}Y so that F (1_{B}, f) = (1_{B}, ω_{B}f). Since both c and d are unital, it is easily seen that these rules yield a well defined functor ω_{B} : c_{B} → d_{B}. To define a coherence constraint τ^{ω} for this system (ω_{B} | B ∈ Ob(ℬ)), notice that for every morphism φ : B → B′ of ℬ and every Y ∈ Ob(c_{B′}) we have the ℬ-cartesian morphism (φ, 1_{c_{φ}Y}) : (B, c_{φ}Y) → (B′, Y) of ℱ ib(c), and let t : d_{φ} ∘ ω_{B′}Y → ω_{B} ∘ c_{φ}Y be the unique morphism of d_{B} such that F (φ, 1_{c_{φ}Y}) = (φ, t) : (B, ω_{B}c_{φ}Y) → (B′, ω_{B′}Y); in light of lemma 3.1.20(i) we see that t is an isomorphism of d_{B}, and we set τ_{φ,Y}^{ω} := t^{−1} : d_{φ}∘ω_{B′}Y → ω_{B} ∘c_{φ}Y. To check the naturality of the rule Y ↦ τ_{φ,Y}^{ω}, notice that every morphism f : X → Y in c_{B′} yields a commutative diagram :

\xymatrix{(B, c_{φ}X) \ar[r]^{(1_{B},c_{φ}f)} \ar[d]_{(φ,1_{cφX})} & (B, c_{φ}Y) \ar[d]^{(φ,1_{cφY} )} \\ (B′, X) \ar[r]^{(1_{B′},f)} & (B′, Y)}

in ℱ ib(c), whence the identity :

(φ, τ_{φ,Y}^{ω −1}) ∘ (1_{B}, ω_{B}c_{φ}f) = (1_{B′}, ω_{B′}f) ∘ (φ, τ_{φ,X}^{ω −1}) in ℱ ib(d)

which in turn translates as the identity τ_{φ,Y}^{ω −1} ∘ ω_{B}c_{φ}(f) = d_{φ}ω_{B′}(f) ∘ τ_{φ,X}^{ω −1} in d_{B}, as required. Lastly, we need to verify the coherence axioms for τ^{ω}. However, since c and d are unital, the first coherence axiom amounts to the identity

τ_{1_{B}}^{ω} = 1_{ω_{B}} for every B ∈ Ob(ℬ)

which is immediate from the construction of τ^{ω}. Next, let φ : B → B′ and ψ : B′ → B′′ be two morphisms of ℬ, and X any object of c_{B′′}; we notice the commutative diagram :

\xymatrix{(B, c_{φ}c_{ψ}X) \ar[d]_{(1_{B},γ_{(φ,ψ),X}^{c})} \ar[r]^{(φ,1_{cφc_{ψ}X})} & (B′, c_{ψ}X) \ar[d]^{(ψ,1_{c_{ψ}X})} \\ (B, c_{ψφ}X) \ar[r]^{(ψφ,1_{c_{ψφ}X})} & (B′′, X).}  in ℱ ib(c)

which, after applying the functor F, yields the identity in ℱ ib(d) :

(ψ, τ_{ψ,X}^{ω −1}) ∘ (φ, τ_{φ,c_{ψ}X}^{ω −1}) = (ψφ, τ_{ψφ,X}^{ω −1}) ∘ (1_{B}, ω_{B}(γ_{(φ,ψ),X}^{c})).

which in turn translates as the second coherence axiom. A direct inspection now gives :

ℱ ib(ω) = F.

Summing up, we have shown that (3.1.26) is bijective on objects. A simple inspection shows that (3.1.26) is also injective on morphisms. To conclude, consider therefore two pseudo-natural transformations ω, ω′ : c → d and a natural transformation ξ : ℱ ib(ω) ⇒ ℱ ib(ω′) such that π^{d} ∗ ξ = 1_{π^{c}}, and for every object (B, X) of ℱ ib(c), let Ξ_{B,X} : ω_{B}X → ω′_{B}X be the unique morphism of d_{B} such that ξ_{(B,X)} = (1_{B}, Ξ_{B,X}). We need to check that the rule: (B, X) ↦ Ξ_{B,X} yields a well defined modification ω ⇝ ω′, in which case we shall have ℱ ib(Ξ) = ξ, by inspection. However, for every B ∈ Ob(ℬ), the naturality of the rule : X ↦ Ξ_{B,X} follows from a simple inspection. It remains thus only to verify the compatibility condition for Ξ. To this aim, let φ : B → B′ be any morphism of ℬ, and Y any object of c_{B′}; the morphism (φ, 1_{c_{φ}Y}) : (B, c_{φ}Y) → (B′, Y) of ℱ ib(c) yields a commutative diagram

\xymatrix{(B, ω_{B}c_{φ}Y) \ar[r]^{ξ_{(B,cφY )}} \ar[d]_{(φ,τ_{φ,Y}^{ω −1})} & (B, ω′_{B}c_{φ}Y) \ar[d]^{(φ,τ_{φ,Y}^{ω′ −1})} \\ (B′, ω_{B′}Y) \ar[r]^{ξ_{(B′,Y )}} & (B′, ω′_{B′}Y)}  in ℱ ib(d)

which in turn is equivalent to the required identity : we leave the details to the reader. □

Remark 3.1.27. (i) Let φ : 𝒜 → ℬ be any fibration, and c the pseudo-functor associated with a given cleavage λ for φ. Claim 3.1.25exhibits a natural isomorphism ψ^{λ} : ℱ ib(c) →^{∼} 𝒜 and the fibration π^{c} : ℱ ib(c) → ℬ carries the distinguished cleavage λ^{∗} provided by lemma 3.1.20(ii). By inspecting the constructions, we find

ψ^{λ}(λ^{∗}(X, g)) = ψ^{λ}(g, 1_{c_{φ}X}) = λ(X, g)

for every morphism g : B′ → B of ℬ and every object X of φ^{−1}B, whence :

ψ^{λ} ∘ λ^{∗} = λ.

Denote by c^{∗} the pseudo-functor associated with λ^{∗}; it follows that the restrictions of ψ^{λ}

ψ_{|B}^{λ} : ℱ ib(c)_{B} →^{∼} 𝒜_{B} for every B ∈ Ob(ℬ)

define a strict pseudo-natural isomorphism of pseudo-functors

ψ_{|•}^{λ} : c^{∗} →^{∼} c.

(ii) Let ℬ, 𝒞 be two V-small categories (for some universe V containing U), ρ : 𝒞 → ℬ a functor, and c : ℬ^{o} → (U, V)-Cat a pseudo-functor. We have a natural commutative diagram

\xymatrix{ℱ ib(c ∘ ρ^{o}) \ar[r]^{ℱ ib(ρ)} \ar[d]_{π^{c∘ρ^{o}}} & ℱ ib(c) \ar[d]^{π^{c}} \\ 𝒞 \ar[r]^{ρ} & ℬ}

where ℱ ib(ρ) is the functor given by the rules : (C, X) ↦ (ρ(C), X) and (g, f) ↦ (ρ(g), f) for every object (C, X) and morphism (g, f) of ℱ ib(c ∘ ρ). This diagram induces a natural identification of fibrations over 𝒞 :

T_{c}^{ρ} : ℱ ib(c ∘ ρ^{o}) →^{∼} Fib(ρ)^{∗}(ℱ ib(c)) (C, X) ↦ (C, (ρ(C), X))

(notation of remark 3.1.5(i)). Especially, via this identification, c ∘ ρ^{o} is the pseudo-functor associated with a cleavage for Fib(ρ)^{∗}(ℱ ib(c)). By a simple inspection, it is easily seen that the rule : c ↦ T_{c}^{ρ} yields a strict pseudo-natural isomorphism of strict pseudo-functors :

[figure: PsFun(ℬ^{o}, (U, V)-Cat); ℱ ib_{ℬ}; PsFun(ρ^{o},Cat); T^{ρ}; (U, V)-Fib(ℬ); Fib(ρ)^{∗}; PsFun(𝒞^{o}, (U, V)-Cat); ℱ ib_{𝒞}; (U, V)-Fib(𝒞).]

(iii) Let 𝒜 ⟶^{φ} ℬ ⟵^{φ′} 𝒜′ be two fibrations, and G : 𝒜 → 𝒜′ a ℬ-cartesian functor; let us fix cleavages λ for 𝒜 and λ′ for 𝒜′, and let c and c′ be the respective associated pseudofunctors. By theorem 3.1.24, there exists a unique pseudo-natural transformation

c^{G} : c ⇒ c′

that makes commute the diagram

\xymatrix{ℱ ib(c) \ar[r]^{ψ^{λ}} \ar[d]_{ℱ ib(c^{G})} & 𝒜 \ar[d]^{G} \\ ℱ ib(c′) \ar[r]^{ψ^{^{λ}′}} & 𝒜′.}

By inspecting the constructions, we can extract the following explicit description of c^{G} : • for every B ∈ Ob(ℬ), the functor c_{B}^{G} : 𝒜_{B} → 𝒜′_{B} is the restriction of G

• for every morphism g : B → B′ in ℬ, the coherence constraint of c^{G} is the isomorphism of functors τ_{g} : c′_{g} ∘ c_{B}^{G} ⇒ c_{B′}^{G} ∘ c_{g} that assigns to every A ∈ Ob(𝒜_{B′}) the unique isomorphism of 𝒜′_{B}

τ_{g,A} : c′_{g}(GA) →^{∼} G(c_{g}A) such that G(λ(A, g)) ∘ τ_{g,A} = λ′(GA, g).

Corollary 3.1.28. Let φ : 𝒜 → ℬ and φ′ : 𝒜′ → ℬ be two fibrations, F : 𝒜 → 𝒜′ a cartesian ℬ-functor, and i ≤ 2 an integer. We have :

(i) The following conditions are equivalent :

(a) F is fibrewise i-faithful, i.e. the restriction 𝒜_{B} → 𝒜′_{B} of the functor F is i-faithful for every B ∈ Ob(ℬ) (see remark 1.1.5).

(b) F is i-faithful, and in case i = 2, it is even a ℬ-equivalence (see remark 3.1.5(v)). (ii) Let 𝒞 → ℬ be any fibration, and suppose that the conditions of (i) hold. Then :

(a) The functor Cart_{ℬ} (𝒞 , F) is i-faithful.

(b) If i = 2, also Cart_{ℬ} (F, 𝒞) is i-faithful.

Proof. (i): The implication (i.b)⇒(i.a) is trivial. For the converse, we consider first the case where i = 2 : by theorem 3.1.24 we may assume that 𝒜 = ℱ ib(c) and 𝒜′ = ℱ ib(c′) for (unital) pseudo-functors c, c′ : ℬ^{o} → (U, V)-Cat (with suitable universes U and V), and F = ℱ ib(ω) for a pseudo-natural transformation ω : c ⇒ c′. Then condition (a) means that ω_{B} : c_{B} → c′_{B} is an equivalence of categories for every B ∈ Ob(ℬ). By theorem 2.4.12, the latter holds if and only if ω is a pseudo-natural equivalence, in which case (i.b) follows.

Next, if i = 0, let A, A′ ∈ Ob(𝒜) and f_{1}, f_{2} : A → A′ be two morphisms of 𝒜 such that F f_{1} = F f_{2}; set B := φA and t := φ(f_{1}), and notice that φ(f_{2}) = φ′(F f_{2}) = φ′(F f_{1}) = t as well. Pick any cartesian morphism h : A′′ → A′ with φ(h) = t; then there exist unique morphisms g_{1}, g_{2} : A → A′′ in φ^{−1}B with h ∘ g_{i} = f_{i} for i = 1, 2. Therefore F h ∘ F g_{1} = F h ∘ F g_{2}, and since F is cartesian we deduce that F g_{1} = F g_{2}; by assumption, it follows that g_{1} = g_{2}, whence f_{1} = f_{2}, as required.

The case where i = 1 is similar : let A, A′ and B as in the foregoing, and f′ : F A → F A′ a morphism of 𝒜′; set t := φ′(f′), and pick a cartesian morphism h : A′′ → A′ with φ(h) = t; then F h is still cartesian, so there exists a unique morphism g′ : F A → F A′′ in φ^{′−1}B with F h ∘ g′ = f′. By assumption, g′ = F g for some morphism g : A → A′′ in φ^{−1}B, so that F (h ∘ g) = f′, as required.

(ii): we consider again first the case where i = 2 : then, notice that every ℬ-functor G : 𝒜′ → 𝒜 that is quasi-inverse to F is also ℬ-cartesian (remark 3.1.5(v)); both assertions (ii.a) and (ii.b) are immediate consequences.

Next, suppose that i = 0, and let G, G′ : 𝒞 → 𝒜 be two functors, α, β : G ⇒ G′ two natural transformations with F ∗ α = F ∗ β. The latter means that F (α_{C}) = F (β_{C}) for every C ∈ Ob(𝒞); by assumption, we then have α_{C} = β_{C} for every such C, as required.

Lastly, suppose that i = 1, and let α′ : F ∘ G ⇒ F ∘ G′ be a natural transformation; by assumption, there exists for every C ∈ Ob(𝒞) a morphism α_{C} : GC → G′C such that F (α_{C}) = α′_{C}. We claim that the rule : C ↦ α_{C} yields a natural transformation α : G ⇒ G′; indeed, the assertion amounts to the identity G′f ∘α_{C} = α_{C′} ∘Gf for every morphism f : C → C′ of 𝒞. The latter can be checked after composition with F, where it is clear. □

3.1.29. By remark 2.2.17(v), for every universe V with U ∈ V, we get a strict pseudo-functor

PsFun((−)^{o}, Cat) : ^{o}V-Cat^{o} → V-\overline{2-Cat}.

We wish now to show how, likewise, the rule ℬ ↦ (U, V)-Fib(ℬ) yields a well defined pseudofunctor. Indeed, remark 3.1.5 already says that every functor ρ : ℬ′ → ℬ between V-small categories induces a strict pseudo-functor (U, V)-Fib(ρ)^{∗} : (U, V)-Fib(ℬ) → (U, V)-Fib(ℬ′), and we have an obvious strict pseudo-natural isomorphism

δ_{ℬ}^{(U,V)-Fib} : 1_{(U,V)-Fib(ℬ)} →^{∼} (U, V)-Fib(1_{ℬ} )^{∗} for every small category ℬ .

Hence, let V′ be yet another universe, with V ∈ V′; combining with the strict pseudo-natural isomorphisms of remark 3.1.5(vi), we have a datum (δ_{•}^{(U,V)-Fib}, γ_{•,•}^{(U,V)-Fib}) fulfilling the unit and associativity axioms for a pseudo-functor

(U, V)-Fib : ^{o}V-Cat^{o} → V′-\overline{2-Cat}

for which we must however still assign the rule prescribing its action on natural transformations. To this aim, we choose for every ℬ ∈ Ob(V-Cat) and every V-small fibration F : 𝒜 → ℬ with essentially U-small fibres, a cleavage λ^{F}, and we denote by c^{F} : ℬ^{o} → (U, V)-Cat the corresponding pseudo-functor. With the notation of remark 3.1.27, we deduce for every functor ρ : ℬ′ → ℬ between V-small categories, a cartesian isomorphism of ℬ′-fibrations

T_{F}^{ρ} := (U, V)-Fib(ρ)^{∗}(ψ^{λ^{F}}) ∘ T_{c^{F}}^{ρ} : ℱ ib(c^{F} ∘ ρ^{o}) →^{∼} (U, V)-Fib(ρ)^{∗}(𝒜).

Let now ρ, ρ′ : ℬ′ → ℬ be two functors, and α : ρ ⇒ ρ′ a natural transformation. There exists a unique ℬ′-cartesian functor (U, V)-Fib(α)_{𝒜}^{∗} making commute the diagram :

\xymatrix{ℱ ib(c^{F} ∘ ρ^{′o}) \ar[r]^{ℱ ib(c^{F} ∗α^{o})} \ar[d]_{T_{F}^{ρ′}} & ℱ ib(c^{F} ∘ ρ^{o}) \ar[d]^{T_{F}^{ρ}} \\ (U, V)-Fib(ρ′)^{∗}𝒜 \ar[r]^{(U,V)-Fib(α)_{𝒜}^{∗}} & (U, V)-Fib(ρ)^{∗}𝒜 .}

To ease notation, throughout the rest of this section we shall write Fib instead of (U, V)-Fib. Next, let F′ : 𝒜′ → ℬ be another fibration with essentially U-small fibres, G : 𝒜 → 𝒜′ a cartesian functor between ℬ-fibrations, and define c^{G} : c^{F} ⇒ c^{F′} as in remark 3.1.27(iii); by example 2.2.15(ii), the coherence constraint τ^{G} of c^{G} defines an invertible modification :

τ_{α}^{G} : (c^{F′} ∗ α^{o}) ⊙ (c^{G} ∗ ρ^{′o}) ⇝ (c^{G} ∗ ρ^{o}) ⊙ (c^{F} ∗ α^{o}).

We consider the induced diagram of ℬ′-cartesian functors :

\xymatrix{ℱ ib(c^{F′} ∘ ρ^{′o}) \ar[rrr]^{ℱ ib(c^{F′}∗α^{o})} \ar[ddd]_{T_{F′}^{ρ′}} &  &  & ℱ ib(c^{F′} ∘ ρ^{o}) \ar[ddd]^{T_{F′}^{ρ}} \\  & ℱ ib(c^{F} ∘ ρ^{′o}) \ar@{-->}[ul]^{ℱ ib(c^{G}∗ρ^{′o})} \ar[r]^{ℱ ib(c^{F} ∗α^{o})} \ar[d]_{T_{F}^{ρ′}} & ℱ ib(c^{F} ∘ ρ^{o}) \ar[d]^{T_{F}^{ρ}} \ar@{-->}[ur]_{ℱ ib(c^{G}∗ρ^{o})} \\  & Fib(ρ′)^{∗}𝒜 \ar[r]^{Fib(α)_{𝒜}^{∗}} \ar[dl]_{Fib(ρ′)^{∗}(G)} & Fib(ρ)^{∗}𝒜 \ar@{-->}[dr]^{Fib(ρ)^{∗}(G)} \\ Fib(ρ′)^{∗}𝒜′ \ar[rrr]^{Fib(α)_{𝒜′}^{∗}} &  &  & Fib(ρ)^{∗}𝒜′}

whose inner and outer rectangular subdiagrams commute by definition, and whose left and right trapezoidal subdiagrams commute by the strict pseudo-naturality of T^{ρ} and T^{ρ′}. If we orient the upper trapezoidal subdiagram with the natural transformation ℱ ib(τ_{α}^{G})^{−1}, there exists therefore a unique natural transformation

τ_{G}^{α} : Fib(α)_{𝒜′}^{∗} ∘ Fib(ρ′)^{∗}(G) ⇒ Fib(ρ)^{∗}(G) ∘ Fib(α)_{𝒜}^{∗}

orienting the lower trapezoidal subdiagram, such that the resulting oriented cubical diagram (whose other four faces are oriented by identities) commutes on 2-cells, in the sense of (2.3.21); the latter means that :

T_{F′}^{ρ} ∗ ℱ ib(τ_{α}^{G})^{−1} = τ_{G}^{α} ∗ T_{F}^{ρ′}.

Lemma 3.1.30. With the notation of (3.1.29), the rule : 𝒜 ↦ Fib(α)_{𝒜}^{∗} defines a pseudonatural transformation Fib(α)^{∗} : Fib(ρ′)^{∗} ⇒ Fib(ρ)^{∗} whose coherence constraint is given by the system of natural transformations τ_{•}^{α}.

Proof. By inspecting the construction, we are easily reduced to checking that τ_{α}^{G′} ⊟ τ_{α}^{G} = τ_{α}^{G′∘G} for every pair of cartesian functors of ℬ-fibrations 𝒜 ⟶^{G} 𝒜′ ⟶^{G′} 𝒜′′. But this is clear, since c^{G′∘G} = c^{G′} ⊙ c^{G}. □

3.1.31. Next, let ρ′′ : ℬ′ → ℬ be another functor, and α′ : ρ′ ⇒ ρ′′ a second natural transformation. According to example 2.2.15(i), we have an invertible modification

γ_{α,α′}^{F} : (c^{F} ∗ α^{o}) ⊙ (c^{F} ∗ α^{′o}) ⇝ c^{F} ∗ (α′ ⊙ α)^{o}

induced by the coherence constraint γ_{••}^{F} of c^{F}. Whence, a unique natural transformation

Ξ_{𝒜}^{α,α′} : Fib(α)_{𝒜}^{∗} ∘ Fib(α′)_{𝒜}^{∗} ⇒ Fib(α′ ⊙ α)_{𝒜}^{∗} such that T_{F}^{ρ} ∗ ℱ ib(γ_{α,α′}^{F}) = Ξ_{𝒜}^{α,α′} ∗ T_{F}^{ρ′′}.

Lemma 3.1.32. With the notation of (3.1.31), the following holds :

(i) The rule : 𝒜 ↦ Ξ_{𝒜}^{α,α′} is a modification Ξ^{α,α′} : Fib(α)^{∗} ⊙ Fib(α′)^{∗} ⇝ Fib(α′⊙α)^{∗}.
(ii) Especially, we have Fib(α)^{∗} ⊙ Fib(α′)^{∗} = Fib(α′ ⊙ α)^{∗} in V′-\overline{2-Cat}.

Proof. Recall that the coherence constraint of Fib(α)^{∗} ⊙ Fib(α′)^{∗} assigns to every cartesian functor G : 𝒜 → 𝒜′ of ℬ-fibrations the natural transformation

β := (Fib(α)_{𝒜′}^{∗} ∗ τ_{G}^{α′}) ⊙ (τ_{G}^{α} ∗ Fib(α′)_{𝒜}^{∗} )

and we need to check the identity :

X := (Ξ_{𝒜′}^{α,α′} ∗ Fib(ρ′′)^{∗}(G)) ⊙ β = Y := τ_{G}^{α′⊙α} ⊙ (Fib(ρ)^{∗}(G) ∗ Ξ_{𝒜}^{α,α′}).

To this aim, it suffices to check that X ∗ T_{F}^{ρ′′} = Y ∗ T_{F}^{ρ′′}. We compute :

X∗T_{F}^{ρ′′} =(Ξ_{𝒜′}^{α,α′} ∗ Fib(ρ′′)^{∗}(G) ∗ T_{F}^{ρ′′}) ⊙ (β ∗ T_{F}^{ρ′′})

=(Ξ_{𝒜′}^{α,α′}∗T_{F′}^{ρ′′} ∗ℱ ib(c^{G}∗ρ^{′′o})) ⊙ (Fib(α)_{𝒜′}^{∗} ∗T_{F′}^{ρ′} ∗ℱ ib(τ_{α′}^{G})^{−1}) ⊙ (τ_{G}^{α}∗T_{F}^{ρ′} ∗ℱ ib(c^{F}∗α^{′o}))

=T_{F′}^{ρ} ∗ (ℱ ib(γ_{α,α′}^{F′} ∗ (c^{G} ∗ ρ^{′′o})) ⊙ ℱ ib((c^{F′} ∗ α^{o}) ∗ τ_{α′}^{G})^{−1} ⊙ ℱ ib(τ_{α}^{G} ∗ (c^{F} ∗ α^{′o}))^{−1})

and :

Y ∗ T_{F}^{ρ′′} = (T_{F′}^{ρ} ∗ ℱ ib(τ_{α′⊙α}^{G})^{−1}) ⊙ (Fib(ρ)^{∗}(G) ∗ T_{F}^{ρ} ∗ ℱ ib(γ_{α,α′}^{F}))
= (T_{F′}^{ρ} ∗ ℱ ib(τ_{α′⊙α}^{G})^{−1}) ⊙ (T_{F′}^{ρ} ∗ ℱ ib((c^{G} ∗ ρ^{o}) ∗ γ_{α,α′}^{F}).

Thus, we are reduced to showing the identity :

τ_{α′⊙α}^{G} ⊙ (γ_{α,α′}^{F′} ∗ (c^{G} ∗ ρ^{′′o})) = ((c^{G} ∗ ρ^{o}) ∗ γ_{α,α′}^{F}) ⊙ (τ_{α}^{G}∗(c^{F} ∗ α^{′o})) ⊙ ((c^{F′} ∗ α^{o})∗τ_{α′}^{G}).

which follows directly from the coherence axiom for τ^{G}. □

In light of lemma 3.1.32(ii), the construction of the pseudo-functor (U, V)-Fib of (3.1.29) shall be complete, once we have shown :

Lemma 3.1.33. Let μ, μ′ : ℬ′′ → ℬ′ and ρ, ρ′ : ℬ′ → ℬ be four functors, and α : ρ ⇒ ρ′, β : μ ⇒ μ′ two natural transformations. Then the following diagram commutes in V′-\overline{2-Cat} :

\xymatrix{Fib(μ′)^{∗} ∘ Fib(ρ′)^{∗} \ar@{=>}[r]^{Fib(β)^{∗}∗Fib(α)^{∗}} \ar@{=>}[d]_{γ_{μ′,ρ′}^{Fib}} & Fib(μ)^{∗} ∘ Fib(ρ)^{∗} \ar@{=>}[d]^{γ_{μ,ρ}^{Fib}} \\ Fib(ρ′ ∘ μ′)^{∗} \ar@{=>}[r]^{Fib(α∗β)^{∗}} & Fib(ρ ∘ μ)^{∗}.}

Proof. In view of lemma 3.1.32(ii), it suffices to consider separately the cases where μ = μ′ and β = 1_{μ}, and where ρ = ρ′ and α = 1_{ρ}. In the first case, we have to check the identities :

γ_{(μ,ρ),𝒜}^{Fib} ∘ Fib(μ)^{∗}(Fib(α)_{𝒜}^{∗} ) = Fib(α ∗ μ)_{𝒜}^{∗} ∘ γ_{(μ,ρ′),𝒜}^{Fib}
X := τ_{G}^{α∗μ} ∗ γ_{(μ,ρ′),𝒜}^{Fib} = Y := γ_{(μ,ρ),𝒜′}^{Fib} ∗ Fib(μ)^{∗}(τ_{G}^{α})

for every fibration F : 𝒜 → ℬ and every cartesian functor of ℬ-fibrations G : 𝒜 → 𝒜′. For the first identity, we consider the diagram :

\xymatrix{ℱ ib(c^{F} ∘ ρ^{′o} ∘ μ^{o}) \ar[r]^{ℱ ib(c^{F} ∗α^{o}∗μ^{o})} \ar[d]_{T_{c^{F} ∘ρ^{′o}}^{μ}} & ℱ ib(c^{F} ∘ ρ^{o} ∘ μ^{o}) \ar[d]^{T_{c^{F} ∘ρ^{o}}^{μ}} \\ Fib(μ)^{∗}ℱ ib(c^{F} ∘ ρ^{′o}) \ar[r]^{Fib(μ)^{∗}ℱ ib(c^{F} ∗α^{o})} \ar[d]_{Fib(μ)^{∗}(T_{F}^{ρ′})} & Fib(μ)^{∗}ℱ ib(c^{F} ∘ ρ^{o}) \ar[d]^{Fib(μ)^{∗}(T_{F}^{ρ})} \\ Fib(μ)^{∗}Fib(ρ′)^{∗}𝒜 \ar[r]^{Fib(μ)^{∗}Fib(α)_{𝒜}^{∗}} \ar[d]_{γ_{(μ,ρ′),𝒜}^{Fib}} & Fib(μ)^{∗}Fib(ρ)^{∗}𝒜 \ar[d]^{γ_{(μ,ρ),𝒜}^{Fib}} \\ Fib(ρ′ ∘ μ)^{∗}𝒜 \ar[r]^{Fib(α∗μ)_{𝒜}^{∗}} & Fib(ρ ∘ μ)^{∗}𝒜}

whose central square subdiagrams commutes by construction, and whose upper square subdiagram commutes by strict naturality of T^{μ}. Notice that the composition of the three left (resp. right) vertical arrows equals T_{F}^{ρ′∘μ} (resp. T_{F}^{ρ∘μ}); it then follows that also the external square diagram commutes. We conclude that the bottom square subdiagram commutes as well, whence the sought identity. For the second identity, set Z := Fib(μ)^{∗}(T_{F}^{ρ′}) ∘ T_{c^{F} ∘ρ^{′o}}^{μ}; it suffices to check that X ∗ Z = Y ∗ Z. We compute :

X ∗ Z = τ_{G}^{α∗μ} ∗ T_{F}^{ρ′∘μ} = T_{F′}^{ρ∘μ} ∗ ℱ ib(τ_{α∗μ}^{G})^{−1}
Y ∗ Z = γ_{(μ,ρ),𝒜′}^{Fib} ∗ Fib(μ)^{∗}(τ_{G}^{α} ∗ T_{F}^{ρ′}) ∗ T_{c^{F} ∘ρ^{′o}}^{μ}
= γ_{(μ,ρ),𝒜′}^{Fib} ∗ Fib(μ)^{∗}(T_{F′}^{ρ} ∗ ℱ ib(τ_{α}^{G})^{−1}) ∗ T_{c^{F} ∘ρ^{′o}}^{μ}
= T_{F′}^{ρ∘μ} ∗ T_{c^{F′}∘ρ^{o}}^{μ −1} ∗ Fib(μ)^{∗}ℱ ib(τ_{α}^{G})^{−1} ∗ T_{c^{F} ∘ρ^{′o}}^{μ}
= T_{F′}^{ρ∘μ} ∗ ℱ ib(τ_{α}^{G} ∗ μ^{o})

where the last equality holds by the strict pseudo-naturality of T^{μ}. So we are reduced to checking that τ_{α∗μ}^{G} = τ_{α}^{G} ∗ μ^{o}. The latter follows by direct inspection.

For the case where α = 1_{ρ}, let F : 𝒜 → ℬ be any fibration, set ρ^{∗}𝒜 := Fib(ρ)^{∗}𝒜, and let ρ^{∗}F : ρ^{∗}𝒜 → ℬ′ be the induced functor. There exists a unique pseudo-natural isomorphism

ω_{F,ρ} : c^{F} ∘ ρ^{o} →^{∼} c^{ρ^{∗}F}

that makes commute the diagram :

\xymatrix{ℱ ib(c^{F} ∘ ρ^{o}) \ar[dr]_{T_{F}^{ρ}} \ar[rr]^{ℱ ib(ω_{F,ρ})} &  & ℱ ib(c^{ρ^{∗}F}) \ar[dl]^{ψ^{^{λ}^{ρ^{∗}F}}} \\  & ρ^{∗}𝒜 .}

It follows easily that for every cartesian functor G : 𝒜 → 𝒜′ we have :

(3.1.34)  c^{Fib(ρ)^{∗}G} = ω_{F′,ρ} ⊙ (c^{G} ∗ ρ) ⊙ ω_{F,ρ}^{−1}

(details left to the reader). We consider the diagram :

\xymatrix{ℱ ib(c^{ρ^{∗}F} ∘μ^{′o}) \ar[rrr]^{ℱ ib(c^{ρ^{∗}F} ∗β^{o})} \ar@{-->}[dr]^{T_{ρ^{∗}F}^{μ′}} \ar[ddd]|{ℱ ib(ω_{F,ρ}∗μ^{′o})^{−1}} &  &  & ℱ ib(c^{ρ^{∗}F} ∘μ^{o}) \ar[dl]_{T_{ρ^{∗}F}^{μ}} \ar[ddd]|{ℱ ib(ω_{F,ρ}∗μ^{o})^{−1}} \\  & Fib(μ′)^{∗}(ρ^{∗}𝒜) \ar[r]^{Fib(β)_{ρ^{∗}𝒜}^{∗}} \ar[d]_{γ_{(μ′,ρ),𝒜}^{Fib}} & Fib(μ)^{∗}(ρ^{∗}𝒜) \ar[d]^{γ_{(μ,ρ),𝒜}^{Fib}} \\  & Fib(ρ ∘ μ′)^{∗}𝒜 \ar[r]^{Fib(ρ∗β)_{𝒜}^{∗}} & Fib(ρ ∘ μ)^{∗}𝒜 \\ ℱ ib(c^{F} ∘ρ^{o}∘μ^{′o}) \ar[rrr]^{ℱ ib(c^{F} ∗ρ^{o}∗β^{o})} \ar@{-->}[ur]^{T_{F}^{ρ∘μ′}} &  &  & ℱ ib(c^{F} ∘ρ^{o}∘μ^{o}) \ar[ul]_{T_{F}^{ρ∘μ}}}

whose upper and lower trapezoidal subdiagrams commute by construction; it is easily seen that the same holds for the right and left trapezoidal subdiagrams (details left to the reader). Moreover, by example 2.2.15(ii) there exists an invertible modification

τ_{β^{o}}^{ω_{F,ρ}^{−1}} : (ω_{F,ρ}^{−1} ∗ μ^{o}) ⊙ (c^{ρ^{∗}F} ∗ β^{o}) ⇝ ((c^{F} ∘ ρ^{o}) ∗ β^{o}) ⊙ (ω_{F,ρ}^{−1} ∗ μ^{′o}).

Hence, there exists a unique orientation

Ξ_{𝒜} : γ_{(μ,ρ),𝒜}^{Fib} ∘ Fib(β)_{ρ^{∗}𝒜}^{∗} ⇒ Fib(β ∗ ρ)_{𝒜}^{∗} ∘ γ_{(μ′,ρ),𝒜}^{Fib}

for the inner square subdiagram, such that :

Ξ_{𝒜} ∗ T_{ρ^{∗}F}^{μ′} = T_{F}^{ρ∘μ} ∗ ℱ ib(τ_{β^{o}}^{ω_{F,ρ}^{−1}}).

To conclude the proof, it suffices then to show :

Claim 3.1.35. The rule : 𝒜 ↦ Ξ_{𝒜} defines an invertible modification

Ξ : γ_{μ,ρ}^{Fib} ⊙ (Fib(β)^{∗} ∗ Fib(ρ)^{∗}) ⇝ Fib(β ∗ ρ)^{∗} ⊙ γ_{μ′,ρ}^{Fib}.

Proof of the claim. Notice that the coherence constraint of γ_{μ,ρ}^{Fib} ⊙ (Fib(β)^{∗} ∗ Fib(ρ)^{∗}) assigns to every cartesian functor G : 𝒜 → 𝒜′ of ℬ-fibrations the natural transformation X := γ_{(μ,ρ),𝒜′}^{Fib} ∗ τ_{Fib(ρ)^{∗}(G)}^{β}, and the coherence constraint of Fib(β ∗ ρ)^{∗} ⊙ γ_{μ′,ρ}^{Fib} assigns to G the natural transformation Y := τ_{G}^{ρ∗β} ∗ γ_{(μ′,ρ),𝒜}^{Fib} . Then the assertion comes down to the identity:

X′ := (Ξ_{𝒜′} ∗ (Fib(μ′)^{∗}Fib(ρ)^{∗}G)) ⊙ X = Y′ := Y ⊙ ((Fib(ρ ∘ μ)^{∗}G) ∗ Ξ_{𝒜} )

and it suffices to check that X′ ∗ T_{ρ^{∗}F}^{μ′} = Y′ ∗ T_{ρ^{∗}(F)}^{μ′}. We compute :

Y′∗T_{ρ^{∗}F}^{μ′} = (τ_{G}^{ρ∗β} ∗ T_{F}^{ρ∘μ′} ∗ ℱ ib(ω_{F,ρ} ∗ μ^{′o})^{−1}) ⊙ ((Fib(ρ ∘ μ)^{∗}G) ∗ T_{F}^{ρ∘μ} ∗ ℱ ib(τ_{β^{o}}^{ω_{F,ρ}^{−1}}))
= (T_{F′}^{ρ∘μ} ∗ ℱ ib(τ_{ρ∗β}^{G −1} ∗ (ω_{F,ρ} ∗ μ^{′o})^{−1})) ⊙ ((Fib(ρ ∘ μ)^{∗}G) ∗ T_{F}^{ρ∘μ} ∗ ℱ ib(τ_{β^{o}}^{ω_{F,ρ}^{−1}}))
= (T_{F′}^{ρ∘μ} ∗ ℱ ib(τ_{ρ∗β}^{G −1} ∗ (ω_{F,ρ} ∗ μ^{′o})^{−1}))⊙(T_{F′}^{ρ∘μ} ∗ ℱ ib(c^{G} ∗ ρ^{o} ∗ μ^{o}) ∗ ℱ ib(τ_{β^{o}}^{ω_{F,ρ}^{−1}}))
= T_{F′}^{ρ∘μ} ∗ ℱ ib((τ_{ρ∗β}^{G −1} ∗ (ω_{F,ρ} ∗ μ^{′o})^{−1}) ⊙ ((c^{G} ∗ ρ^{o} ∗ μ^{o}) ∗ τ_{β^{o}}^{ω_{F,ρ}^{−1}}))
X ∗ T_{ρ^{∗}F}^{μ′} = γ_{(μ,ρ),𝒜′}^{Fib} ∗ T_{ρ^{∗}F′}^{μ} ∗ ℱ ib(τ_{β}^{Fib(ρ)^{∗}G})^{−1}
= T_{F′}^{ρ∘μ} ∗ ℱ ib((ω_{F′,ρ} ∗ μ^{o})^{−1} ∗ (τ_{β}^{Fib(ρ)^{∗}G})^{−1})

and :

Ξ_{𝒜′} ∗ (Fib(μ′)^{∗}Fib(ρ)^{∗}G) ∗ T_{ρ^{∗}F}^{μ′} = Ξ_{𝒜′} ∗ T_{ρ^{∗}F′}^{μ′} ∗ ℱ ib(c^{Fib(ρ)^{∗}G} ∗ μ^{′o})
= T_{F′}^{ρ∘μ} ∗ ℱ ib(τ_{β^{o}}^{ω_{F′,ρ}^{−1}} ∗ (c^{Fib(ρ)^{∗}G} ∗ μ^{′o}))

so that X′ ∗ Z = T_{F′}^{ρ∘μ} ∗ ℱ ib((τ_{β^{o}}^{ω_{F′,ρ}^{−1}} ∗ (c^{Fib(ρ)^{∗}G} ∗ μ^{′o})) ⊙ ((ω_{F′,ρ} ∗ μ^{o})^{−1} ∗ (τ_{β}^{Fib(ρ)^{∗}G})^{−1})). We are thus reduced to checking that :

((c^{G}∗ρ^{o}∗μ^{o})∗τ_{β^{o}}^{ω_{F,ρ}^{−1}})⊙((ω_{F′,ρ}^{−1}∗μ^{o})∗τ_{β}^{Fib(ρ)^{∗}G}) = (τ_{ρ∗β}^{G} ∗(ω_{F,ρ}^{−1}∗μ^{′o}))⊙(τ_{β^{o}}^{ω_{F′,ρ}^{−1}} ∗(c^{Fib(ρ)^{∗}G}∗μ^{′o})).

The latter follows easily from (3.1.34) : details left to the reader. □

Corollary 3.1.36. We have a pseudo-natural equivalence

ℱ ib_{•} : PsFun((−)^{o}, Cat) ⇒ (U, V)-Fib 𝒞 ↦ ℱ ib_{𝒞}

whose coherence constraint attaches to every functor ρ of small categories the strict pseudonatural isomorphism (T^{ρ})^{−1}.

Proof. The required coherence axioms for the rule ρ ↦ (T^{ρ})^{−1} hold by direct inspection (details left to the reader). There remains therefore only to check the naturality of this rule, relative to natural transformations β : ρ ⇒ ρ′ between functors ρ, ρ′ : ℬ′ → ℬ of V-small categories. The assertion amounts to checking the following identity in the 2-category V-\overline{2-Cat} :

(3.1.37)  (T^{ρ})^{−1} ⊙ (Fib(β)^{∗} ∗ ℱ ib_{ℬ} ) = (ℱ ib_{ℬ′} ∗ PsFun(β^{o}, Cat)) ⊙ (T^{ρ′})^{−1}.

Thus, let d : ℬ^{o} → Cat be any pseudo-functor, and denote by F_{d} : ℱ ib(d) → ℬ the induced fibration. With the notation of (3.1.29), there exists a unique pseudo-natural isomorphism

ω^{d} : c^{F_{d}} →^{∼} d such that ℱ ib(ω^{d}) = ψ^{λ^{F_{d}}}.

We consider the diagram :

\xymatrix{ℱ ib(c^{F_{d}} ∘ ρ^{′o}) \ar[r]^{T_{F_{d}}^{ρ′}} \ar[d]_{ℱ ib(c^{F_{d}}∗β^{o})} & Fib(ρ′)^{∗}ℱ ib(d) \ar[r]^{(T_{d}^{ρ′})^{−1}} \ar[d]|{Fib(β)^{∗}ℱ ib(d)} & ℱ ib(d ∘ ρ^{′o}) \ar[d]^{ℱ ib(d∗β^{o})} \\ ℱ ib(c^{F_{d}} ∘ ρ^{o}) \ar[r]_{T_{F_{d}}^{ρ}} & Fib(ρ)^{∗}ℱ ib(d) \ar[r]_{(T_{d}^{ρ})^{−1}} & ℱ ib(d ∘ ρ^{′o})}

whose left square subdiagram commutes by construction. Invoking the strict pseudo-naturality of T^{ρ}, it is easily seen that the composition of the two bottom horizontal arrows agrees with ℱ ib(ω^{d} ∗ ρ^{o}); likewise, the composition of the two top horizontal arrows yields ℱ ib(ω^{d} ∗ ρ^{′o}). Now, according to example 2.2.15(ii), there exists an invertible modification :

τ_{β}^{ω^{d}} : (d ∗ β^{o}) ⊙ (ω^{d} ∗ ρ^{′o}) ⇝ (ω^{d} ∗ ρ^{o}) ⊙ (c^{F_{d}} ∗ β^{o}).

Hence, there exists a unique isomorphism of functors

ϑ_{d}^{β} : ℱ ib(d ∗ β^{o}) ∘ (T_{d}^{ρ′})^{−1} →^{∼} (T_{d}^{ρ})^{−1} ∘ Fib(ρ)^{∗}ℱ ib(d) such that ϑ_{d}^{β} ∗ T_{F_{d}}^{ρ′} = ℱ ib(τ_{β}^{ω^{d}})

and we are reduced to checking that the rule : d ↦ ϑ_{d}^{β} defines an invertible modification from the right to the left side of (3.1.37). Hence, let α : d ⇒ d′ be a pseudo-natural transformation, and define c^{ℱ ib(α)} : c^{F_{d}} ⇒ c^{F_{d′}} as in remark 3.1.27(iii); it is easily seen that

c^{ℱ ib(α)} = ω^{d′ −1} ⊙ α ⊙ ω^{d}.

Now, the coherence constraint of the left-hand side of (3.1.37) assigns to α the natural transformation X := (T_{d′}^{ρ})^{−1} ∗ τ_{ℱ ib(α)}^{β} , where τ_{ℱ ib(α)}^{β} is characterized by the identity :

τ_{ℱ ib(α)}^{β} ∗ T_{F_{d}}^{ρ′} = T_{F_{d′}}^{ρ} ∗ ℱ ib(τ_{β}^{ℱ ib(α)} )^{−1}

and τ_{β}^{ℱ ib(α)} : (c^{F_{d′}} ∗ β^{o}) ⊙ (c^{ℱ ib(α)} ∗ ρ^{′o}) ⇝ (c^{ℱ ib(α)} ∗ ρ^{o}) ⊙ (c^{F_{d}} ∗ β^{o}) is given as well by example 2.2.15(ii). According to remark 2.2.17(ii), the right-hand side of (3.1.37) assigns to α the natural transformation Y := ℱ ib(τ_{β}^{α})^{−1} ∗ (T_{d}^{ρ′})^{−1}, where again τ_{β}^{α} : (d′ ∗ β^{o}) ⊙ (α ∗ ρ^{′o}) ⇒ (α ∗ ρ^{o}) ⊙ (d ∗ β^{o}) is given by example 2.2.15(ii). We need then to check that :

Y′ := (ϑ_{d′}^{β} ∗ Fib(ρ′)^{∗}ℱ ib(α)) ⊙ Y = X′ := X ⊙ (ℱ ib(α ∗ ρ^{o}) ∗ ϑ_{d}^{β})

and it suffices to check that Y′ ∗ T_{F_{d}}^{ρ′} = X′ ∗ T_{F_{d}}^{ρ′}. We compute :

Y′ ∗ T_{F_{d}}^{ρ′} = (ϑ_{d′}^{β} ∗ T_{F′_{d}}^{ρ′} ∗ ℱ ib(c^{ℱ ib(α)} ∗ ρ^{′o})) ⊙ ℱ ib(τ_{β}^{α} ∗ (ω^{d} ∗ ρ^{′o}))^{−1}
= (ℱ ib(τ_{β}^{ω^{d′}}) ∗ ℱ ib(c^{ℱ ib(α)} ∗ ρ^{′o})) ⊙ ℱ ib(τ_{β}^{α} ∗ (ω^{d} ∗ ρ^{′o}))^{−1}
X′ ∗ T_{F_{d}}^{ρ′} = ((T_{d′}^{ρ})^{−1} ∗ T_{F_{d′}}^{ρ} ∗ ℱ ib(τ_{β}^{ℱ ib(α)} )^{−1}) ⊙ (ℱ ib(α ∗ ρ^{o}) ∗ ℱ ib(τ_{β}^{ω^{d}}))
= (ℱ ib(ω^{d′} ∗ ρ^{o}) ∗ ℱ ib(τ_{β}^{ℱ ib(α)} )^{−1}) ⊙ (ℱ ib(α ∗ ρ^{o}) ∗ ℱ ib(τ_{β}^{ω^{d}}))

hence we are further reduced to showing that :

((ω^{d′} ∗ ρ^{o}) ∗ τ_{β}^{ℱ ib(α)} ) ⊙ (τ_{β}^{ω^{d′}} ∗ (c^{ℱ ib(α)} ∗ ρ^{′o})) = ((α ∗ ρ^{o}) ∗ τ_{β}^{ω^{d}}) ⊙ (τ_{β}^{α} ∗ (ω^{d} ∗ ρ^{′o}))

but this is clear, since both sides agree with τ_{β}^{α⊙ω^{d}}, where τ^{α⊙ω^{d}} denotes as usual the coherence constraint of α ⊙ ω^{d}. □

3.2. Split fibrations and cartesian sections. As usual, we fix a universe U, but some of the constructions in this section shall produce fibrations whose fibre categories will be equivalent, though not necessarily isomorphic to a U-small category.

Definition 3.2.1. Let ℬ be any category.

(i) A split fibration over ℬ is a pair (φ, λ) where φ : 𝒜 → ℬ is a fibration and λ a cleavage for φ whose associated pseudo-functor is strict, in which case we also say that λ is split.

(ii) For i = 1, 2, let (φ_{i} : 𝒜_{i} → ℬ, λ_{i}) be two split fibrations, and F : 𝒜_{1} → 𝒜_{2} a cartesian functor. Let also c_{i} be the pseudo-functor associated with λ_{i}, for i = 1, 2. For every B ∈ Ob(ℬ), let F_{B} : φ_{1}^{−1}B → φ_{2}^{−1}B denote the restriction of F. We say that F is a split cartesian functor (φ_{1}, c_{1}) → (φ_{2}, c_{2}) if the rule : B ↦ F_{B} defines a strict pseudo-natural transformation c_{1} ⇒ c_{2}.

Remark 3.2.2. (i) With the notation of definition 3.2.1(i), it is easily seen that λ is split if and only if for every B ∈ Ob(ℬ) we have c_{1_{B}} = 1_{c_{B}}, and for every pair of morphisms B′′ ⟶^{h} B′ ⟶^{g} B of ℬ we have c_{g∘h} = c_{g}c_{h} and the following diagram commutes :

\xymatrix{c_{g∘h}A \ar[rr]^{λ(c_{g}A,h)} \ar[dr]_{λ(A,g∘h)} & & c_{g}A \ar[dl]^{λ(A,g)} \\ & A} for every A ∈ Ob(φ^{−1}B).

(ii) Likewise, with the notation of definition 3.2.1(ii), and in light of remark 3.1.27(iii), we see that the ℬ-cartesian functor F : 𝒜_{1} → 𝒜_{2} is split if and only if for every morphism g : B′ → B of ℬ and every A ∈ Ob(φ_{1}^{−1}B) we have

c′_{g}(F A) = F (c_{g}A) and F (λ(A, g)) = λ′(F A, g).

(details left to the reader).

Example 3.2.3. The fibration s : ℬ/F 𝒞 → ℬ associated with any functor F : 𝒞 → ℬ (see example 3.1.3(ii)) is split. Indeed, we have a natural cleavage, defined as follows. An object of ℬ/s(ℬ/F 𝒞) is the datum of objects B ∈ Ob(ℬ) and (f : B′ → F C′) ∈ Ob(ℬ/F 𝒞), and a morphism g : B → B′ in ℬ; to such a datum we assign the commutative diagram

\xymatrix{B \ar[r]^{f∘g} \ar[d]_{g} & F C′ \ar[d]^{F 1_{C′}} \\ B′ \ar[r]^{f} & F C′}

which is an object λ(B, f, g) of Morph(ℬ/F 𝒞). Since this object is a ℬ-cartesian morphism of ℬ/F 𝒞, the rule (B, f, g) ↦ λ(B, f, g) extends uniquely to a cleavage for s. By a simple inspection, we find that the corresponding pseudo-functor c associates with every morphism g : B → B′ of ℬ the functor g/F 𝒞 : B′/F 𝒞 → B/F 𝒞 (notation of (1.1.27)). Then it is clear that c is strict.

3.2.4. Cartesian sections of a fibration. Let ℬ be a category, V a universe such that U ⊂ V and ℬ is V-small. The V-small split fibrations over ℬ with essentially U-small fibres are the objects of a 2-category

(U, V)-Split(ℬ)

whose 1-cells are given by the split cartesian functors, and whose 2-cells are the natural ℬtransformations. As usual, we also often write U-Split(ℬ) or just Split(ℬ) for this 2-category. Furthermore, we have an obvious forgetful strict pseudo-functor :

F : (U, V)-Split(ℬ) → (U, V)-Fib(ℬ) (φ, λ) ↦ φ.

On the other hand, if ℬ is U-small, we have as well a natural strict pseudo-functor

C : (U, V)-Fib(ℬ) → (U, V)-Split(ℬ).

Namely, for a fibration φ : 𝒜 → ℬ with essentially U-small fibres, we consider the strict pseudo-functor

𝒜 (−) : ℬ^{o} → (U, V)-Cat B ↦ 𝒜 (B) := Cart_{ℬ} (ℬ/B, 𝒜)

that assigns to every morphism f : B → B′ of ℬ the functor Cart(f_{∗}, 𝒜) for (recall that ℬ/B is fibred over ℬ, by example 3.1.3(i); also f_{∗} : ℬ/B → ℬ/B′ is the ℬ-cartesian functor as in (1.1.25)). The objects of 𝒜 (B) are called the cartesian sections of 𝒜 over B, and the morphisms of 𝒜 (B) are also called morphisms of cartesian sections. Then we set

C(𝒜) := ℱ ib(𝒜 (−))

and we let C(φ) : C(𝒜) → ℬ be the resulting fibration; recall that the fibre category C(φ)^{−1}B is naturally isomorphic to 𝒜 (B) for every B ∈ Ob(ℬ), and C(𝒜) carries a distinguished cleavage whose associated pseudo-functor corresponds – under this identification – to the strict pseudo-functor 𝒜 (−) (remark 3.1.27(i)).

Every ℬ-cartesian functor F : 𝒜_{1} → 𝒜_{2} induces a strict pseudo-natural transformation

Cart_{ℬ} (ℬ/−, F) : 𝒜_{1}(−) ⇒ 𝒜_{2}(−) B ↦ Cart_{ℬ} (ℬ/B, F)

whence a ℬ-cartesian functor

C(F) := ℱ ib(Cart_{ℬ} (ℬ/−, F)) : C(𝒜_{1}) → C(𝒜_{2})

and every natural ℬ-transformation α : F_{1} ⇒ F_{2} of ℬ-cartesian functors F_{1}, F_{2} : 𝒜_{1} → 𝒜_{2} induces a modification

Cart_{ℬ} (ℬ/−, α) : Cart_{ℬ} (ℬ/−, F_{1}) ⇝ Cart_{ℬ} (ℬ/−, F_{2}) B ↦ Cart_{ℬ} (1_{ℬ/B} , α)

whence a natural ℬ-transformation

C(α) := ℱ ib(Cart_{ℬ} (ℬ/−, α)) : C(F_{1}) ⇒ C(F_{2}).

Taking into account the discussion of (3.1.18)-(3.1.21) we easily conclude that the foregoing rules yield a well defined strict pseudo-functor C as sought.

3.2.5. The evaluation functor of a fibration. Let ℬ be a category, and φ : 𝒜 → ℬ a fibration; we claim that there exists a ℬ-cartesian functor

ev^{𝒜} : C(𝒜) → 𝒜

that assigns to every object (B, G : ℬ/B → 𝒜) of C(𝒜) its evaluation G1_{B} ∈ Ob(𝒜_{B}), and to any morphism (f, α : G ⇒ G′ ∘ f_{∗}) : (B, G) → (B′, G′), the composition

G1_{B} ⟶^{α_{1_{B}}} G′f ⟶^{G′(f/B′)} G′1_{B′}

where f/B′ : f → 1_{B′} is the morphism of ℬ/B′ determined by f. In order to check that these rules define a functor, let (g, β : G′ ⇒ G′′ ∘ g_{∗}) : (B′, G′) → (B′′, G′′) be any other morphism; then (g, β) ∘ (f, α) = (g ∘ f, (β ∗ f_{∗}) ⊙ α), and the assertion comes down to the identity

G′′(g/B′′) ∘ β_{1_{B′}} ∘ G′(f/B′) ∘ α_{1_{B}} = G′′(g ∘ f/B′′) ∘ ((β ∗ f_{∗}) ⊙ α)_{1_{B}}.

But we have

β_{1_{B′}} ∘ G′(f/B′) = G′′(g_{∗}(f/B′)) ∘ β_{f} and G′′(g/B′′) ∘ G′′(g_{∗}(f/B′)) = G′′(g ∘ f/B′′)

whence the contention. Moreover, a simple inspection shows that every cartesian functor F : 𝒜_{1} → 𝒜_{2} of fibred ℬ-categories yields a commutative diagram of ℬ-categories :

\xymatrix{C(𝒜_{1}) \ar[r]^{C(F)} \ar[d]_{_{ev_{𝒜_{1}}}} & C(𝒜_{2}) \ar[d]^{_{ev_{𝒜_{2}}}} \\ 𝒜_{1} \ar[r]^{F} & 𝒜_{2}}

and for every natural ℬ-transformation α : F_{1} ⇒ F_{2} of 𝒞 -cartesian functors F_{1}, F_{2} : 𝒜_{1} → 𝒜_{2} we have the identity :

ev^{𝒜_{2}} ∗ C(α) = α ∗ ev^{𝒜_{1}} .

Hence, if B is U-small, the rule 𝒜 ↦ ev^{𝒜} defines a strict pseudo-natural transformation

ev^{•} : F ∘ C ⇒ 1_{Fib(ℬ)} .

Remark 3.2.6. With the notation of (3.2.5), let G, G′ ∈ 𝒜 (B) be two cartesian sections over an object B ∈ Ob(ℬ), and α : G ⇒ G′ a natural ℬ-transformation. Then it is easily seen that α is uniquely determined by its evaluation ev^{𝒜} (1_{B}, α) = α_{1_{B}} : G1_{B} → G′1_{B}. Indeed, every object (f : B′ → B) ∈ Ob(ℬ/B) yields a morphism f/B : f → 1_{B} in ℬ/B, whence a commutative diagram

\xymatrix{Gf \ar[r]^{α_{f}} \ar[d]_{G(f/B)} & G′f \ar[d]^{G′(f/B)} \\ G1_{B} \ar[r]^{α_{1_{B}}} & G′1_{B}}

which determines α_{f} uniquely, since G′(f/B) is cartesian. By the same token, given any morphism g : G1_{B} → G′1_{B} in φ^{−1}B, let us define α_{f}^{g} : Gf → G′f as the unique morphism in φ^{−1}B′ such that G′(f/B) ∘ α_{f}^{g} = g ∘ G(f/B). Then it is easily seen that the rule f ↦ α_{f}^{g} defines a natural ℬ-transformation α^{g} : G ⇒ G′ such that α_{1_{B}}^{g} = g. If G′′ : ℬ/B → 𝒜 is another cartesian functor, and g′ : G′1_{B} → G′′1_{B} is any morphism in φ^{−1}B, clearly we have :

α^{g′} ⊙ α^{g} = α^{g′∘g}.

Theorem 3.2.7. The pseudo-functor C is a strong right 2-adjoint for F, and is fully faithful.

Proof. According to proposition 1.1.15(i) and theorem 2.4.24(i), in order to prove the first assertion it suffices to exhibit a unit and a counit fulfilling the triangular identities (1.1.13). Our candidate counit shall be the pseudo-natural transformation ev^{•}. Taking into account remark 3.1.5(v) and corollary 2.4.29, in order to see that ev^{𝒜} is ℬ-cartesian for every fibration φ : 𝒜 → ℬ, and to prove the second assertion of the theorem, it then suffices to notice :

Claim 3.2.8. The functor ev^{𝒜} is a ℬ-equivalence.

Proof of the claim. It suffices to show that ev^{𝒜} is a fibrewise equivalence (corollary 3.1.28(i)). However, from remark 3.2.6we see already that for every B ∈ Ob(ℬ) the restriction

ev_{|B}^{𝒜} : 𝒜 (B) → 𝒜_{B}

of ev^{𝒜} is fully faithful. To show that ev_{|B}^{𝒜} is essentially surjective, let λ be a unital cleavage for φ, and c its associated unital pseudo-functor. For every A ∈ Ob(𝒜_{B}) consider the functor

i_{A} : ℬ/B → ℬ/φ𝒜 (g : B′ → B) ↦ (A, B, g)

that assigns to every morphism (h/B) : g → g′ of ℬ/B the morphism (A, B, (h/A)) : (A, B, g) → (A, B, g′) of ℬ/φ𝒜. Let also s : Morph(𝒜) → 𝒜 be the source functor (notation of (1.1.30)); it suffices to notice that

β_{B,A}^{λ} := s ∘ λ ∘ i_{A} : ℬ/B → 𝒜 .

is a cartesian section with β_{B,A}^{λ}(1_{B}) = A. ♢

Next, let (φ : 𝒜 → ℬ, λ) be any split fibration with small fibres, and c the strict pseudofunctor associated with λ; denote also by λ^{∗} the distinguished cleavage of C(𝒜), whose associated pseudo-functor is 𝒜 (−) (see (3.2.4)). To define a unit for our adjunction, we need to exhibit a natural split cartesian functor :

(φ, λ) → (C(𝒜), λ^{∗}).

Now, for every B ∈ Ob(ℬ) any A ∈ Ob(𝒜_{B}) define β_{B,A}^{λ} ∈ 𝒜 (B) as in the proof of claim 3.2.8; by virtue of remark 3.2.6, for every morphism f : A′ → A in 𝒜_{B} there exists a unique natural 𝒞 -transformation β_{B,f}^{λ} : β_{B,A′}^{λ} ⇒ β_{B,A}^{λ} such that (β_{B,f}^{λ})_{1_{B}} = f, and the rules A ↦ β_{B,A}^{λ}, f ↦ β_{B,f}^{λ} yield a well defined functor

β_{B}^{λ} : 𝒜_{B} → 𝒜 (B).

According to theorem 3.1.24, it then suffices to show that the rule : B ↦ β_{B}^{λ} defines a strict pseudo-natural transformation

β^{λ} : c ⇒ 𝒜 (−).

The assertion amounts to checking the identities

β_{B,A}^{λ} ∘ g_{∗} = β_{B′,c_{g}A}^{λ} for every (B′ ⟶^{g} B) ∈ Ob(ℬ/B) and every A ∈ Ob(𝒜_{B}) .

However, we have :

β_{B′,c_{g}A}^{λ}(g′) = c_{g′}(c_{g}A) = c_{g∘g′}A = β_{B,A}^{λ}(g ∘ g′) for every object g′ : B′′ → B′ of ℬ/B′

which shows that β_{B,A}^{λ} ∘ g_{∗} and β_{B′,c_{g}A}^{λ} agree on objects; to see that they also agree on morphisms, we consider two objects f : C → B′ and f′ : C′ → B′ of ℬ/B′ and a morphism h/B′ : f → f′ of ℬ/B′ (notation as in the proof of claim 3.2.8), and we notice that, due to remark 3.2.2(i), the three subdiagrams of the following diagram commute

\xymatrix{c_{gf}A \ar[rr]^{β_{B′,cgA}^{λ}(h/B′)} \ar[dr]_{λ(c_{g}A,f)} \ar[ddr]_{λ(A,gf)} &  & c_{gf′}A \ar[ddl]^{λ(A,gf′)} \ar[dl]^{λ(c_{g}A,f′)} \\  & c_{g}A \ar[d]^{λ(A,g)} \\  & A}

The assertion is an immediate consequence. We need to verify that the rule (φ, λ) ↦ β^{λ} yields a strict pseudo-natural transformation

β^{•} : 1_{Split(ℬ)} ⇒ C ∘ F.

To this aim, let G : (φ : 𝒜 → ℬ, λ) → (φ′ : 𝒜′ → ℬ, λ′) be any split ℬ-cartesian functor, and let c and c′ be the pseudo-functors associated with λ and respectively λ′; we come down to showing the commutativity of the diagram

\xymatrix{𝒜_{B} \ar[r]^{β_{B}^{λ}} \ar[d]_{G_{B}} & 𝒜 (B) \ar[d]^{Cart_{ℬ} (ℬ/B,G)} \\ 𝒜′_{B} \ar[r]^{β_{B}^{λ′}} & 𝒜′(B)} for every B ∈ Ob(ℬ) .

However, the composition Cart_{ℬ} (ℬ/B, G) ∘ β_{B}^{λ} assigns to every A ∈ Ob(𝒜_{B}) the cartesian section ℬ/B → 𝒜′ given by the rule : (B′ ⟶^{g} B) ↦ G(c_{g}A), whereas β_{B}^{λ′} ∘ G assigns to the same object the cartesian section given by the rule : (B′ ⟶^{g} B) ↦ c′_{g}(GA); moreover,

if B′′ ⟶^{g′} B is any other object of ℬ/B and h/B : g → g′ is any morphism, we get two commutative diagrams :

\xymatrix{G(c_{g}A) \ar[rr]^{G(β_{B,A}^{λ}(h/B))} \ar[dr]_{G(λ(A,g))} &  & G(c_{g′}A) \ar[dl]^{G(λ(A,g′))} & c′_{g}(GA) \ar[rr]^{β_{B,GA}^{λ′}(h/B)} \ar@{-->}[dr]_{λ′(GA,g)} &  & c′_{g′}(GA) \ar@{-->}[dl]^{λ′(GA,g′)} \\  & GA &  &  & GA.}

Since G is split, remark 3.2.2(ii) says that these two diagrams coincide, i.e. the functors Cart_{ℬ} (ℬ/B, G) ∘ β_{B}^{λ} and β_{B}^{λ′} ∘ G agree on all objects of φ^{−1}B. To see that they agree also on morphisms, let f : A → A′ be any morphism of φ^{−1}B; we have to show that G ∗ β_{B,f}^{λ} = β_{B,Gf}^{λ′}. By remark 3.2.6, it suffices then to compare the evaluations at 1_{B} of both of these natural ℬtransformation; but by definition we have (G ∗ β_{B,f}^{λ})_{1_{B}} = Gf = (β_{B,Gf}^{λ′})_{1_{B}}, as required.

Lastly, let G, G′ : (φ : 𝒜 → ℬ, λ) → (φ′ : 𝒜′ → ℬ, λ′) be two split ℬ-cartesian functors, and γ : G ⇒ G′ a natural ℬ-transformation; we need to show :

Cart_{ℬ} (ℬ/B, γ) ∗ β_{B}^{λ} = β_{B}^{λ′} ∗ (γ_{|𝒜_{B}} ) for every B ∈ Ob(ℬ)

which translates as the identity : γ_{c_{g}A} = c′_{g}(γ_{A}) for every g ∈ Ob(ℬ/B) and A ∈ Ob(𝒜_{B}). However, again by virtue of remark 3.2.2(ii), the following commutative diagrams coincide :

\xymatrix{G(c_{g}A) \ar[r]^{γ_{cgA}} \ar[d]_{G(λ(A,g))} & G′(c_{g}A) \ar[d]^{G′(λ(A,g))} & c′_{g}(GA) \ar[r]^{c′_{g}(γ_{A})} \ar[d]_{λ′(GA,g)} & c′_{g}(G′A) \ar[d]^{λ′(G′A,g)} \\ GA \ar[r]^{γ_{A}} & G′A & GA \ar[r]^{γ_{A}} & G′A}

whence the contention.

It remains to show that the unit and counit thus defined fulfill the triangular identities. Now, let φ : 𝒜 → ℬ be any fibration, (φ′ : 𝒜′ → ℬ, λ′) any split fibration, and c′ the pseudofunctor associated with λ′. Denote by λ^{∗} the distinguished cleavage of C(𝒜) as in (3.2.4), whose associated pseudo-functor is (naturally identified with) 𝒜 (−); then, in view of theorem 3.1.24, the triangular identities come down to the commutativity of the diagrams

[figure: 𝒜 (B); β_{B}^{λ^{∗}}; C(𝒜)(B); Cart_{ℬ} (ℬ/B,ev^{𝒜} ); c′_{B}; β_{B}^{λ′}; 𝒜′(B); _{ev{𝒜′ \atop |B}}; 𝒜 (B); c′_{B}]

for every B ∈ Ob(ℬ). The latter follows by a direct inspection. □

Remark 3.2.9. (i) Let φ : 𝒜 → ℬ be a fibration, and c the pseudo-functor associated with a given cleavage λ for φ. The proof of theorem 3.2.7furnishes a pseudo-natural equivalence

ev_{|•}^{𝒜} : 𝒜 (−) ⇒ c.

In case λ is split, it also furnishes a quasi-inverse for ev_{|•}^{𝒜} which is a strict pseudo-natural equivalence β^{λ} : c ⇒ 𝒜 (−). However, we cannot deduce from this that the unit of the 2adjunction of theorem 3.2.7 is an equivalence, since we cannot ensure the existence of a strict quasi-inverse for β^{λ}. (Especially, ev_{|•}^{𝒜} is not strict, in general.)

(ii) On the other hand, denote by spFib(ℬ) the full 2-subcategory of Fib(ℬ) whose objects are the fibrations that admit a split cleavage. Then claim 3.2.8implies that the inclusion functor

spFib(ℬ) → Fib(ℬ)

is a 2-equivalence which admits a pseudo-inverse given by the rule : (φ : 𝒜 → ℬ) ↦ C(𝒜) for every fibration φ.

3.2.10. For every pair of universes U, V with U ⊂ V, and every 𝒞 ∈ Ob((U, V)-Cat), the set of connected components π_{0}(𝒞) is essentially U-small (see remark 1.2.21(ii)); hence, after replacing π_{0}(𝒞) by a U-small set of the same cardinality, we get a well defined functor

π_{0} : (U, V)-Cat → Set

that is left adjoint to the fully faithful imbedding Set → (U, V)-Cat which assigns to every U-small set the associated discrete category (see example 1.1.6(ii)). Moreover, π_{0} is also a well defined strict pseudo-functor, for the natural 2-category structure on (U, V)-Cat (and here we regard Set as a 2-category whose only 2-cells are identities : see example 2.2.4(i)).

Now, if ℬ is any V-small category, and c : ℬ^{o} → (U, V)-Cat any pseudo-functor, it follows that the composition π_{0}∘c : ℬ^{o} → Set is a strict pseudo-functor, i.e. it is a presheaf on ℬ. Also, every pseudo-natural transformation ω : c ⇒ d of pseudo-functors c, d : ℬ^{o} → (U, V)-Cat induces a morphism of presheaves π_{0} ∗ ω : π_{0} ∘ c → π_{0} ∘ d, and notice that if Ξ : ω ⇝ ω′ is any modification of such pseudo-natural transformations, then π_{0} ∗ ω = π_{0} ∗ ω′ and the induced modification π_{0} ∘ Ξ : π_{0} ∗ ω ⇝ π_{0} ∗ ω is just the identity. In view of theorem 3.1.24, it then follows that the rule c ↦ π_{0} ∘ c yields a well defined strict pseudo-functor

π_{0}^{ℬ} : (U, V)-Fib(ℬ) → ℬ_{U}^{∧}

Explicitly, for every fibration φ : 𝒜 → ℬ, the presheaf π_{0}^{ℬ} (𝒜) is given by the rule : X ↦ π_{0}(φ^{−1}X) for every X ∈ Ob(ℬ). It is then easily seen that π_{0}^{ℬ} is a strong left 2-adjoint to the strict pseudo-functor ℱ ib of (3.1.17).

Remark 3.2.11. The functor ℱ ib of (3.1.17) also admits a right adjoint. Indeed, notice that ℱ ib factors uniquely as the composition of a strict pseudo-functor

spℱ ib : ℬ^{∧} → Split(ℬ)

and the forgetful strict pseudo-functor F (notation of (3.2.4)). In light of theorem 3.2.7, it then suffices to find a right adjoint for spℱ ib. Now, if (φ : 𝒜 → ℬ, λ) is any split fibration, and c : ℬ^{o} → Cat the associated pseudo-functor, notice that the composition

Ob ∘ c : ℬ^{o} → Cat → Set B ↦ Ob(𝒜_{B})

is a presheaf. It is easily seen that the resulting functor

Ob^{ℬ} : Split(ℬ) → ℬ^{o} (φ : 𝒜 → ℬ, λ) ↦ Ob ∘ c

is a right adjoint for spℱ ib. However, we do not get a right 2-adjoint by this rule.

3.3. 2-Fibrations. We wish now to consider the generalization of (3.1.18) where ℬ is an arbitrary V-small 2-category. In this case, for every pseudo-functor c : ℬ^{o} → (U, V)-Cat we get a natural 2-category structure on ℱ ib(c), such that the resulting functor π^{c} : ℱ ib(c) → ℬ is a strict pseudo-functor. Indeed, for every two objects (B, X), (B′, Y) of ℱ ib(c), and any two morphisms (φ_{1}, f_{1}), (φ_{2}, f_{2}) : (B, X) → (B′, Y) of ℱ ib(c), we declare that the 2-cells (φ_{1}, f_{1}) ⇒ (φ_{2}, f_{2}) are the 2-cells α : φ_{1} ⇒ φ_{2} that make commute the diagram

\xymatrix{ & X \ar[dl]_{f_{1}} \ar[dr]^{f_{2}} \\ c_{φ_{1}}Y \ar[rr]^{c_{α,Y}} &  & c_{φ_{2}}Y.}

If (φ_{3}, f_{3}) : (B, X) → (B′, Y) is another morphism and β : (φ_{2}, f_{2}) ⇒ (φ_{3}, f_{3}) another 2cell, the composition β ⊙ α : (φ_{1}, f_{1}) ⇒ (φ_{3}, f_{3}) is given by the composition law for 2-cells of ℬ(B, B′). Since c_{β⊙α,Y} = c_{β,Y} ∘c_{α,Y}, it is easily seen that this composition rule is well defined. Likewise, if (B′′, Z) is any other object of ℱ ib(c), (ψ_{1}, g_{1}), (ψ_{2}, g_{2}) : (B′, Y) → (B′′, Z) any two morphisms and α′ : (ψ_{1}, g_{1}) ⇒ (ψ_{2}, g_{2}) any 2-cell, we let α′ ∗ α : (ψ_{1}, g_{1}) ∘ (φ_{1}, f_{1}) ⇒ (ψ_{2}, g_{2}) ∘ (φ_{2}, f_{2}) be defined by the corresponding composition law for 2-cells in ℬ. In order to check that this rule is well defined, it suffices to show the commutativity of the diagram :

\xymatrix{X \ar[r]^{f_{1}} \ar[dr]_{f_{2}} & c_{φ_{1}}Y \ar[r]^{c_{φ_{1}}(g_{1})} \ar[d]^{c_{α,Y}} & c_{φ_{1}}c_{ψ_{1}}Z \ar[r]^{γ_{(φ_{1},ψ_{1}),Z}^{c}} \ar[d]^{c_{α,c_{ψ_{1}} Z}} & c_{ψ_{1}∘φ_{1}}Z \ar[dd]^{c_{α′∗α,Z}} \\  & c_{φ_{2}}Y \ar[r]^{c_{φ_{2}}(g_{1})} \ar[dr]_{c_{φ_{2}}(g_{2})} & c_{φ_{2}}c_{ψ_{1}}Z \ar[d]^{c_{φ_{2}}(c_{α′,Z})} \\  &  & c_{φ_{2}}c_{ψ_{2}}Z \ar[r]_{γ_{(φ_{2},ψ_{2}),Z}^{c}} & c_{ψ_{2}∘φ_{2}}Z}

(where γ^{c} is the coherence constraint of c); the latter follows directly from remark 2.1.17(ii) and the identity : c_{φ_{2}}(c_{α′,Z}) ∘ c_{α,c_{ψ_{1}}Z} = (c_{α′} ∗ c_{α})_{Z}. It is then obvious that ℱ ib(c) is a 2-category with these composition laws (and with the composition law for 1-cells given by (3.1.18)), since the required associativity and unit axioms hold already in ℬ.

In analogy with the case of usual categories, it is natural to make the following :

Definition 3.3.1. Let 𝒜, ℬ be two V-small 2-categories, π : 𝒜 → ℬ a strict pseudo-functor.

(i) We say that a 1-cell f : A′ → A of 𝒜 is ℬ-cartesian if for every X ∈ Ob(𝒜) the following diagram is cartesian in the category V-Cat :

\xymatrix{𝒜 (X, A′) \ar[r]^{f_{∗}} \ar[d]_{π} & 𝒜 (X, A) \ar[d]^{π} \\ ℬ(πX, πA′) \ar[r]^{(πf)_{∗}} & ℬ(πX, πA).}

(ii) We say that π (or 𝒜) is a 2-fibration over ℬ, if for every 1-cell φ : B′ → B in ℬ and every A ∈ π^{−1}B there exists a ℬ-cartesian 1-cell f : A′ → A in 𝒜 such that π(f) = φ.

(iii) For every B ∈ Ob(ℬ), the fibre π^{−1}B is the 2-subcategory of 𝒜 whose underlying category is the fibre over B of the functor underlying π, and whose 2-cells are the 2-cells α of 𝒜 such that π(α) = i_{B}. We say that π has essentially U-small fibres if π^{−1}B is an essentially U-small 2-category for every B ∈ Ob(ℬ).

Remark 3.3.2. (i) With the notation of definition 3.3.1, notice that a ℬ-cartesian 1-cell of 𝒜 is also a cartesian morphism of the underlying category, since the forgetful functor V-Cat → V-Set commutes with fibre products.

(ii) It is easily seen that a composition of cartesian 1-cells is cartesian, and every invertible 1-cell is cartesian (details left to the reader).

(iii) It follows from (i) that if π is a 2-fibration, then it is also a (usual) fibration on the underlying categories. Moreover, in this case a 1-cell of 𝒜 is cartesian if and only if it is a cartesian morphism of the underlying category : indeed, the necessity has already been remarked in (i). Conversely, let f : A′ → A be a 1-cell which is a cartesian morphism in the underlying category, and pick any 1-cell g : A′′ → A of 𝒜 which is cartesian in the sense of definition 3.3.1(i), and with π(g) = π(f); then there exists an isomorphism h : A′ →^{∼} A′′ in the underlying category of 𝒜 such that f = g ∘ h. By (ii), we deduce that f is a cartesian 1-cell.

(iv) Notice that our definition of 2-fibrations (which follows [81, pp.35–36]) does not include any condition on the existence of cartesian liftings of 2-cells of ℬ. This is somewhat inadequate, and thus our definition should be regarded as provisional. A much more comprehensive treatment of 2-fibrations (including the relevant condition of cartesian liftings of 2-cells) can be found in the paper [44].

Lemma 3.3.3. Let ℬ be a 2-category, and c : ℬ^{o} → (U, V)-Cat a pseudo-functor. We have : (i) The 2-category ℱ ib(c) is a 2-fibration over ℬ. (ii) A 1-cell (φ, f) : (B′, X) → (B, Y) of ℱ ib(c) is cartesian if and only if f : X → c_{φ}Y

is an isomorphism in c_{B′}.

Proof. It suffices to prove (ii). Now, notice that the condition of (ii) is necessary, due to lemma 3.1.20(i). Conversely, let (φ, f) be such a 1-cell with f an isomorphism; consider any object (B′′, Z) of ℱ ib(c), any two 1-cells ψ_{1}, ψ_{2} : B′′ → B′ in ℬ, a 2-cell α : ψ_{1} ⇒ ψ_{2} in ℬ, and any 2-cell β : (φ ∘ ψ_{1}, g_{1}) ⇒ (φ ∘ ψ_{2}, g_{2}) in ℱ ib(c) such that β = φ_{∗} ∗ α. We know already by lemma 3.1.20(i) that for i = 1, 2 there exists a unique 1-cell (ψ_{i}, g′_{i}) : (B′′, Z) → (B′, X) of ℱ ib(c) such that (φ, f) ∘ (ψ_{i}, g′_{i}) = (φ ∘ ψ_{i}, g_{i}), and it remains to check that α yields a 2-cell (ψ_{1}, g′_{1}) ⇒ (ψ_{2}, g′_{2}). To this aim, we consider the diagram :

\xymatrix{Z \ar[r]^{g′_{1}} \ar[dr]_{g′_{2}} & c_{ψ_{1}}X \ar[d]^{c_{α,X}} \ar[r]^{c_{ψ_{1}}(f)} & c_{ψ_{1}}c_{φ}Y \ar[d]^{c_{α,cφY}} \ar[r]^{γ_{(ψ_{1},φ),Y}^{c}} & c_{φ∘ψ_{1}}Y \ar[d]^{c_{φ∗α,Y}} \\  & c_{ψ_{2}}X \ar[r]^{c_{ψ_{2}}(f)} & c_{ψ_{2}}c_{φ}Y \ar[r]^{γ_{(ψ_{2},φ),Y}^{c}} & c_{φ∘ψ_{2}}Y}

(where γ^{c} is the coherence constraint of c). By assumption, the subdiagram obtained after removing the two middle vertical arrows commutes, and we need to check that the triangular subdiagram on the left commutes as well. However, the square subdiagram on the right commutes by remark 2.1.17(ii), and the central square subdiagram commutes by naturality oc c_{α}. Moreover, since f is an isomorphism, the two bottom horizontal arrows are both isomorphisms. The contention follows straightforwardly. □

Example 3.3.4. Let ℬ be any 2-category.

(i) Denote by 𝟙 a chosen final object of Cat, i.e. a category with only one object and one morphism, and consider the constant pseudo-functor F_{𝟙} : ℬ^{o} → Cat with value 𝟙 (notation of (2.5)). It is easily seen that the induced projection :

ℱ ib(F_{𝟙}) → ℬ

is an isomorphism of 2-categories.

(ii) If ℬ has small Hom-categories, fix B ∈ ℬ and consider the strict pseudo-functor

h_{B} : ℬ^{o} → Cat B′ ↦ ℬ(B′, B)

that assigns to every 1-cell f : B′ → B′′ the functor f^{∗} : ℬ(B′′, B) → ℬ(B′, B), and to every 2-cell β : f ⇒ f′ between 1-cells f, f′ : B′ → B′′ the natural transformation β^{∗} : f^{∗} ⇒ f^{′∗} (see example 2.2.8(i)). Then we have a natural isomorphism of 2-categories

ℱ ib(h_{B}) →^{∼} ℬ/B

(notation of example 2.1.11(ii)) which identifies the natural projection ℱ ib(h_{B}) → ℬ with the source strict pseudo-functor s_{B} : ℬ/B → ℬ.

3.3.5. For any pair of 2-fibrations 𝒜, 𝒜′ over ℬ we denote by

2-Cart_{ℬ} (𝒜 , 𝒜′)

the category whose objects are the ℬ-cartesian strict pseudo-functors, i.e. the strict pseudofunctors 𝒜 → 𝒜′ that are ℬ-cartesian functors on the underlying categories; the morphisms are the strict pseudo-natural transformations that are natural ℬ-transformations on the underlying categories. Notice that 2-Cart_{ℬ} (𝒜 , ℱ ib(c)) is a full subcategory of Cart_{ℬ} (𝒜 , ℱ ib(c)), for every pseudo-functor c : ℬ^{o} → Cat and every 2-fibration 𝒜.

Just as in definition 3.1.4(ii), every pair of ℬ-cartesian strict pseudo-functors H : 𝒞 → 𝒜 and K : 𝒜′ → 𝒞′ induces a functor

2-Cart_{ℬ} (H, K) : 2-Cart_{ℬ} (𝒜 , 𝒜′) → 2-Cart_{ℬ} (𝒞 , 𝒞′) G ↦ K ∘ G ∘ H

which assigns to any morphism α : G ⇒ G′ in 2-Cart_{ℬ} (𝒜 , 𝒜′), the natural transformation K ∗ α ∗ H : K ∘ G ∘ H ⇒ K ∘ G′ ∘ H. As usual, in case H = 1_{𝒜} (resp. K = 1_{𝒜′} ) we also denote this functor by 2-Cart_{ℬ} (𝒜 , K) (resp. by 2-Cart_{ℬ} (H, 𝒜′)).

Likewise, if H, H′ : 𝒞 → 𝒜 and K, K′ : 𝒜′ → 𝒞′ are four ℬ-cartesian strict pseudofunctors, every pair of strict pseudo-natural ℬ-transformations β : H ⇒ H′ and γ : K ⇒ K′ induces a natural transformation :

2-Cart_{ℬ} (β, γ) : 2-Cart_{ℬ} (H, K) ⇒ 2-Cart_{ℬ} (H′, K′) G ↦ γ ∗ G ∗ β

and again, if β = 1_{H} (resp. γ = 1_{K}) we denote this natural transformation by 2-Cart_{ℬ} (H, γ) (resp. 2-Cart_{ℬ} (α, K)).

3.3.6. In the situation of (3.3), let d : ℬ^{o} → (U, V)-Cat be another pseudo-functor, and ω : c ⇒ d any pseudo-natural transformation. We claim that the functor ℱ ib(ω) : ℱ ib(c) → ℱ ib(d) defined as in (3.1.19) is a ℬ-cartesian strict pseudo-functor. The assertion amounts to saying that any 2-cell α : (φ_{1}, f_{1}) ⇒ (φ_{2}, f_{2}) as in (3.3) induces a 2-cell

α : ℱ ib(ω)(φ_{1}, f_{1}) ⇒ ℱ ib(ω)(φ_{2}, f_{2}).

In turn, the latter boils down to the commutativity of the diagram :

\xymatrix{ω_{B} ∘ c_{φ_{1}}Y \ar[d]_{(τ_{φ_{1},Y}^{ω} )^{−1}} & ω_{B}X \ar[l]_{ω_{B}f_{1}} \ar[r]^{ω_{B}f_{2}} & ω_{B} ∘ c_{φ_{2}}Y \ar[d]^{(τ_{φ_{2},Y}^{ω} )^{−1}} \\ d_{φ_{1}} ∘ ω_{B′}Y \ar[rr]^{d_{α,ω_{B′} Y}} &  & d_{φ_{2}} ∘ ω_{B′}Y.}

However, since α is a 2-cell in ℱ ib(c), we have the identity :

ω_{B}f_{2} = ω_{B}(c_{α,Y}) ∘ ω_{B}f_{1}

and on the other hand, by remark 2.2.5(i), the naturality of τ^{ω} implies the identity

(τ_{φ_{2},Y}^{ω})^{−1} ∘ ω_{B}(c_{α,Y}) = d_{α,ω_{B′}Y} ∘ (τ_{φ_{1},Y}^{ω})^{−1}

whence the assertion. Lastly, for any two pseudo-natural transformations ω, ω′ : c ⇒ d and every modification Ξ : ω ⇝ ω′, a simple inspection shows that the natural transformation ℱ ib(Ξ) : ℱ ib(ω) ⇒ ℱ ib(ω′) of (3.1.21) is also a strict pseudo-natural transformation.

Lemma 3.3.7. (i) For every two pseudo-functors c, d : ℬ^{o} → (U, V)-Cat, the induced functor:

ℱ ib : PsNat(c, d) → 2-Cart_{ℬ^{o}} (ℱ ib(c), ℱ ib(d))

is an isomorphism of categories.

(ii) (2-Yoneda’s Lemma) For every pseudo-functor c : ℬ^{o} → (U, V)Cat and every X ∈ Ob(ℬ), we have an equivalence of categories pseudo-natural in both arguments :

PsNat(h_{X}, c) →^{∼} c_{X}.

Proof. (i): Notice that a ℬ-cartesian strict pseudo-functor ℱ ib(c) → ℱ ib(d) is completely determined by its underlying cartesian functor of (usual) ℬ-fibrations. Then the assertion is already known from the proof of theorem 3.1.24.

(ii) follows from (i), example 3.3.4(ii) and claim 3.2.8. □

3.3.8. Clearly the 2-fibrations with V-small and essentially U-small fibres over a given V-small 2-category ℬ form a 2-category

(U, V)-2-Fib(ℬ)

whose Hom-categories are given by the categories 2-Cart_{ℬ} (−, −) as in (3.3.5). As usual, this 2-category shall be often just denoted 2-Fib(ℬ). We have a strict and strongly faithful pseudofunctor

ℱ ib_{ℬ} : PsFun(ℬ^{o}, Cat) → 2-Fib(ℬ) F ↦ ℱ ib(F)

but unlike for usual fibrations, ℱ ib_{ℬ} is not a 2-equivalence. As an application of these constructions, we deduce :

Theorem 3.3.9. The 2-category (U, V)-Cat is strongly 2-complete and strongly 2-cocomplete.

Proof. Let F : ℬ → (U, V)Cat be any pseudo-functor from a U-small 2-category ℬ; we set :

ℒ_{F} := 2-Cart_{ℬ^{o}} (ℬ^{o}, ℱ ib(F))

and we define a pseudo-cone π : F_{ℒ_{F}} ⇒ F as follows. First, in light of lemma 3.3.7(i) and example 3.3.4, we have a natural isomorphism of categories

ω : ℒ_{F} →^{∼} PsNat(F_{1}, F)

where F_{1} : ℬ → Cat is the constant pseudo-functor with value 1. Now, let β : F_{1} ⇒ F be any pseudo-cone; for every B ∈ Ob(ℬ) we have then the functor β_{B} : 1 → F B, which is identified with the object β_{B}(∅) ∈ Ob(F B), under the isomorphism of categories :

Fun(1, F B) →^{∼} F B  G ↦ G(∅)  (α : G ⇒ G′) ↦ α_{∅} : G(∅) → G′(∅)

(here Ob(1) = {∅}). Clearly, the rule β ↦ β_{B} yields a functor

π_{B} : PsNat(F_{1}, F) → F B

that assigns to every modification Ξ : β ⇝ β′ the natural transformation Ξ_{B} : β_{B} ⇒ β′_{B}, identified with the morphism Ξ_{B,∅} : β_{B}(∅) → β′_{B}(∅) in F B. Lastly, it is easily seen that the rule B ↦ π_{B} ∘ ω yields a pseudo-cone as sought, with coherence constraint

τ_{f}^{π} : F f ∘ π_{B} ⇒ π_{B′} such that τ_{f,β}^{π} := τ_{f,∅}^{β}

for every pseudo-cone β : F_{1} → F with coherence constraint τ^{β}.

Explicitly, for every B ∈ Ob(ℬ) the functor π_{B} : ℒ_{F} → F B is determined by the identity :

φ(B) = (B, π_{B}(φ)) for every cartesian strict pseudo-functor φ : ℬ^{o} → ℱ ib(F)

and for every pair of cartesian strict pseudo-functors φ, φ′ : ℬ^{o} → ℱ ib(F) and every strict pseudo-natural ℬ^{o}-transformation β : φ ⇒ φ′, we have β_{B} = (1_{B}, π_{B}(β)).

We claim that (ℒ_{F} , π) is a strong 2-limit of F, i.e. for every small category 𝒞, the functor

(3.3.10)  Fun(𝒞 , ℒ_{F}) → PsNat(F_{𝒞} , F)  (φ : 𝒞 → ℒ_{F}) ↦ π ⊙ F_{φ}

is an isomorphism of categories (see definition 2.5.1(i)). Indeed, let α : F_{𝒞} ⇒ F be any pseudo-cone with vertex 𝒞 and basis F; there follows a ℬ^{o}-cartesian strict pseudo-functor ℱ ib(α) : ℱ ib(F_{𝒞} ) → ℱ ib(F), whence a functor

α^{†} := 2-Cart_{ℬ^{o}} (ℬ^{o}, ℱ ib(α)) : 2-Cart_{ℬ^{o}} (ℬ^{o}, ℱ ib(F_{𝒞} )) → ℒ_{F} .

If Θ : α ⇝ β is any modification, we set as well Θ^{†} := 2-Cart_{ℬ^{o}} (ℬ^{o}, ℱ ib(Θ)) : α^{†} → β^{†}. However, a simple inspection shows that ℱ ib(F_{𝒞} ) = ℬ^{o} × 𝒞, the product in the category of 2-categories (where 𝒞 is regarded as a 2-category with trivial 2-cells), fibred over ℬ^{o} via the natural projection p : ℬ^{o} × 𝒞 → ℬ^{o}. There is an obvious functor

c^{𝒞} : 𝒞 → 2-Cart_{ℬ^{o}} (ℬ^{o}, ℱ ib(F_{𝒞} )) X ↦ c_{X}

that assigns to every object X of 𝒞 the constant functor c_{X} : ℬ^{o} → 𝒞 with value X, naturally identified with a section ℬ^{o} → ℬ^{o} × 𝒞 of the projection p. Then, a simple inspection shows that the functor

PsNat(F_{𝒞} , F) → Fun(𝒞 , ℒ_{F})  α ↦ α^{†} ∘ c^{𝒞}  Θ ↦ Θ^{†} ∗ c^{𝒞}

is inverse to the functor (3.3.10), whence the contention.

Next we consider the category \overline{ℱ ib} (F) such that Ob(\overline{ℱ ib} (F)) := Ob(ℱ ib(F)) and

Hom_{\overline{ℱ ib} (F)}(X, X′) := π_{0}(ℱ ib(F)(X, X′)) for every X, X′ ∈ Ob(ℱ ib(F))

(notation of (3.2.10)). The composition law for morphisms in \overline{ℱ ib} (F) is induced in the obvious way by that of 1-cells of ℱ ib(F). For every 1-cell f of ℱ ib(F) we denote by [f] the class of f in Morph(\overline{ℱ ib} (F)) and we let Σ := \{[f] ∈ Morph(\overline{ℱ ib} (F)) | f is ℬ-cartesian \}. We set

ℒ′_{F} := \overline{ℱ ib} (F))[Σ^{−1}]

(notation of theorem 1.6.9). We have a natural functor

ℱ ib(F) → ℒ′_{F}

namely, the composition of the obvious projection ℱ ib(F) → \overline{ℱ ib} (F) with the localization \overline{ℱ ib} (F) → ℒ′_{F}. Recall that ℱ ib(F) carries a distinguished cleavage λ^{∗}, whose associated pseudo-functor is naturally identified with F (remark 3.1.27(i)). Then, the pseudo-cocone ι : F → F_{ℱ ib(F)} associated with λ^{∗} induces a a pseudo-cocone ι′ : F ⇒ F_{ℒ′_{F}} . We claim that (ℒ′_{F} , ι′) is a strong 2-colimit of F, i.e. for every small category 𝒞, the induced functor

(3.3.11)  Fun(ℒ′_{F} , 𝒞) → PsNat(F, F_{𝒞} )  (φ : ℒ′_{F} → 𝒞) ↦ F_{φ} ⊙ ι′

is an isomorphism of categories. Indeed, let α : F ⇒ F_{𝒞} be any pseudo-cocone; there follows a ℬ^{o}-cartesian strict pseudo-functor ℱ ib(α) : ℱ ib(F) → ℱ ib(F_{𝒞} ) →^{∼} ℬ^{o} × 𝒞. After composing with the projection p : ℬ^{o} × 𝒞 → 𝒞 we get a functor

α^{∗} : ℱ ib(F) → 𝒞 .

Since 𝒞 has trivial 2-cells, α^{∗} factors through a unique functor \overline{α}^{∗} : \overline{ℱ ib} (F) → 𝒞. By lemma 3.3.3(ii), a 1-cell (f, g) : (B, X) → (B′, X′) of ℱ ib(F_{𝒞} ) is cartesian if and only if g : X → X′ is an isomorphism of 𝒞. It follows that \overline{α}^{∗} in turn factors uniquely through a functor

α^{‡} : ℒ′_{F} → 𝒞 .

On the other hand, notice that a functor ℱ ib(F) → 𝒞 factors through \overline{ℱ ib} (F) if and only if it induces a strict pseudo-functor ℱ ib(F) → ℬ^{o} × 𝒞, and the latter is ℬ^{o}-cartesian if and only if the resulting functor \overline{ℱ ib} (F) → 𝒞 factors through ℒ′_{F}.

If Θ : α ⇝ β is any modification, we get a natural ℬ-transformation Θ^{∗} := p ∗ ℱ ib(Θ) : α^{∗} ⇒ β^{∗}, which in turn induces a natural transformation

Θ^{‡} : α^{‡} ⇒ β^{‡}.

By a direct inspection, we see that the functor

PsNat(F, F_{𝒞} ) → Fun(ℒ′_{F} , 𝒞)  α ↦ α^{‡}  Θ ↦ Θ^{‡}

is inverse to (3.3.11), whence the contention. □

Example 3.3.12. (i) For instance, let 𝒞_{•} := (𝒞_{i} | i ∈ I) be any small family of small categories. By inspecting the proof of theorem 3.3.9, we see that the 2-product of 𝒞_{•} is represented by the product 𝒞 := ∏_{i∈I} 𝒞_{i}, as constructed in example 1.2.25(i), and the universal cone for 𝒞 is also a universal pseudo-cone for the 2-product of 𝒞_{•}.

(ii) Let F : 𝒞 → ℬ and F′ : 𝒞′ → ℬ be two functors between small categories; by inspecting the proof of theorem 3.3.9, we see that the 2-fibre product of F and F′ (remark 2.5.2(v)) is represented by the category whose objects are all data of the form X := (c, c′, ξ), where c ∈ Ob(𝒞), c′ ∈ Ob(𝒞′), and ξ : F (c) →^{∼} F′(c′) is an isomorphism in ℬ. If X′ := (d, d′, ζ) is another such datum, the morphisms X → X′ are the pairs (φ, φ′), where φ : c → d (resp. φ′ : c′ → d′) is a morphism in 𝒞 (resp. in 𝒞′), and ζ ∘ F (φ) = F′(φ′) ∘ ξ.

Example 3.3.13. (i) Consider a small category ℬ and a pseudo-functor F : ℬ → (U, V)-Cat. In this case, obviously \overline{ℱ ib} (F) = ℱ ib(F), so the strong 2-colimit of F is represented by ℱ ib(F)[Σ^{−1}], where Σ ⊂ Morph(ℱ ib(F)) is the set of cartesian morphisms.

(ii) Suppose next that ℬ is filtered. Then we claim that Σ admits a right calculus of fractions. Indeed, it is clear that Σ satisfies conditions (CF1) and (CF2) of definition 1.6.14(i). Next, let (φ, f) : (B, X) → (B′, Y) and (ψ, g) : (B′′, Z) → (B′, Y) be two morphisms of ℱ ib(F), with (ψ, g) ∈ Σ; hence φ : B′ → B and ψ : B′ → B′′ are morphisms of ℬ, so there exist morphisms φ′ : B → B′′′ and ψ′ : B′′ → B′′′ in ℬ with φ′ ∘ φ = ψ′ ∘ ψ. It follows that (φ′, 1_{F_{φ′}X}) : (B′′′, F_{φ′}X) → (B, X) lies in Σ, and since (ψ, g) is cartesian there exists a unique morphism (ψ′, h) : (B′′′, F_{φ′}X) → (B′′, Y) such that (ψ, g)∘(ψ′, h) = (φ, f)∘(φ′, 1_{F_{φ′}X}). This proves that condition (CF3) holds as well for Σ. Lastly, let (φ, f), (φ′, f′) : (B, X) → (B′, Y) and (ψ, g) : (B′, Y) → (B′′, Z) be three morphisms of ℱ ib(F), with (ψ, g) cartesian, and suppose that (ψ, g) ∘ (φ, f) = (ψ, g) ∘ (φ′, f′); hence φ, φ′ : B′ → B are two morphisms of ℬ, and by assumption there exists a morphism μ : B → B′′′ such that μ ∘ φ = μ ∘ φ′. Then we have the cartesian morphism (μ, 1_{F_{μ}X}) : (B′′′, F_{μ}X) → (B, X), and obviously (ψ, g) ∘ (φ, f) ∘ (μ, 1_{F_{μ}X}) = (ψ, g) ∘ (φ′, f′) ∘ (μ, 1_{F_{μ}X}). Since (ψ, g) is cartesian, it follows that (φ, f) ∘ (μ, 1_{F_{μ}X}) = (φ′, f′) ∘ (μ, 1_{F_{μ}X}). This shows that condition (CF4) holds for Σ.

(iii) In particular, in the situation of (ii), the morphisms in ℱ ib(F)[Σ^{−1}] can be expressed as fractions with denominators in Σ, as detailed by proposition 1.6.16(and remark 1.6.19).

(iv) Let I be a small filtered category, and F : I → (U, V)-Cat any functor; let ℒ be the colimit of F, as explicitly given by example 1.5.10. In particular, the objects of ℒ are equivalence classes [i, X] of pairs (i, X) ∈ Ob(ℱ ib(F)), and the morphisms [f] : [i, X] → [i′, X′] are equivalence classes of morphisms f : F_{φ}X → F_{φ′}X′, for all pairs of morphism (i ⟶^{φ} j ⟵^{φ_{2}} i′) of I. By direct inspection of the construction of ℒ, we get a well defined functor

ℱ ib(F) → ℒ  (i, X) ↦ [i, X]  ((i′ → i, f) : (i, X) → (i′, X′)) ↦ [f].

By the universal property of the localization, this functor extends uniquely to a well defined functor ℱ ib(F)[Σ^{−1}] → ℒ, where Σ is as in (ii), and in view of (iii) and lemma 3.1.20(i), it is easily seen that the latter functor is an equivalence. In other words, the colimit of F represents as well the 2-colimit of the same functor, and more precisely, every universal cocone F ⇒ c_{ℒ} is also a (strict) universal pseudo-cocone.

3.3.14. Let 𝒞 a U-small category, ℬ a V-small category with U-small Hom-sets, and ρ : 𝒞 → ℬ a functor; theorem 3.3.9 enables us to construct both right and left 2-adjoints for the strict pseudo-functor V-Fib(ρ)^{∗} of remark 3.1.5(i). Indeed, combining with theorem 2.6.39and remark 2.6.18(ii) we see first that the strong left 2-Kan extension along ρ^{o} is a strict left 2adjoint L for P := PsFun(ρ^{o}, (U, V)-Cat). On the other hand, by remark 2.4.31, the strict and strong 2-equivalence ℱ ib_{ℬ} of theorem 3.1.24admits a strict and strong pseudo-inverse

Q_{ℬ} : (U, V)-Fib(ℬ) →^{∼} PsFun(ℬ^{o}, (U, V)-Cat).

It follows that the strict pseudo-functor

(U, V)-Fib(ρ)_{!} := ℱ ib_{ℬ} ∘ L ∘ Q_{𝒞} : (U, V)-Fib(𝒞) → (U, V)-Fib(ℬ)

is a left 2-adjoint for ℱ ib_{𝒞} ∘ P ∘ Q_{ℬ} . On the other hand, remark 3.1.27(ii) yields a pseudonatural isomorphism ℱ ib_{𝒞} ∘P →^{∼} (U, V)-Fib(ρ)^{∗} ∘ℱ ib_{ℬ} , so (U, V)-Fib(ρ)_{!} is also left 2-adjoint to (U, V)-Fib(ρ)^{∗}. Likewise, from the strong right 2-Kan extension R along ρ^{o} we get a strict right 2-adjoint for (U, V)-Fib(ρ)^{∗}, namely the pseudo-functor

(U, V)-Fib(ρ)_{∗} := ℱ ib_{ℬ} ∘ R ∘ Q_{𝒞} : (U, V)-Fib(𝒞) → (U, V)-Fib(ℬ).

As usual, we will omit mentioning V, when no ambiguities are likely to arise from the omission.

Remark 3.3.15. (i) Like all 2-adjoints, the pseudo-functors Fib(ρ)_{!} and Fib(ρ)_{∗} are well defined up to pseudo-natural equivalence. However, our construction yields more canonical representatives, that are well defined up to pseudo-natural isomorphisms, since it relies on the strong 2-Kan extensions along ρ^{o}. Moreover, both these 2-adjoints factor naturally through pseudo-functors

Fib(𝒞) → Split(ℬ)

again because this is a feature of strong 2-Kan extensions : see remark 2.6.18(ii).

(ii) In the situation of (3.3.14), we can describe more explicitly the pseudo-functor Fib(ρ)_{∗} as follows. Let 𝒜 → 𝒞 be any fibration with essentially small fibres; then C(Fib(ρ)_{∗}(𝒜)) is the fibration associated with the strict pseudo-functor Fib(ρ)_{∗}(𝒜)(−) : ℬ^{o} → (U, V)-Cat, and by example 3.1.16(ii), for every B ∈ Ob(ℬ) we have an equivalence of categories :

Fib(ρ)_{∗}(𝒜)(B) →^{∼} Cart_{𝒞} (ρ𝒞 /B, 𝒜)

that is pseudo-natural with respect to morphisms B′ → B in ℬ. Thus, Fib(ρ)_{∗}(𝒜) is naturally equivalent to the fibration associated with the strict pseudo-functor

Cart_{𝒞} (ρ𝒞 /−, 𝒜) : ℬ^{o} → (U, V)-Cat B ↦ Cart_{𝒞} (ρ𝒞 /B, 𝒜)

that assigns to every morphism f : B′ → B of ℬ the functor Cart_{𝒞} (ρ𝒞 /f, 𝒜).

(iii) Moreover, we have a pseudo-commutative diagram of 2-categories :

\xymatrix{𝒞^{∧} \ar[r]^{ℱ ib_{𝒞}} \ar[d]_{ρ_{∗}} & Fib(𝒞) \ar[d]^{Fib(ρ)_{∗}} \\ ℬ^{∧} \ar[r]^{ℱ ib_{ℬ}} & Fib(ℬ)}

(see remark 2.4.10). Indeed, let j : Set → Cat be the inclusion functor (that assigns to every set the associated discrete category); by inspecting the constructions of Fib(ρ)_{∗} and ρ_{∗} (see remark 1.3.6(i)), we come down to exhibiting an equivalence :

2-∫_{ρ^{o}} j ∘ F →^{∼} j ∘∫_{ρ^{o}} F for every presheaf F : 𝒞^{o} → Set

pseudo-natural with respect to F. Now, for every B ∈ Ob(ℬ) the category 2-∫_{ρ^{o}} j ∘ F (B) is the 2-limit of the functor j ∘ F ∘ t_{B^{o}} : B^{o}/φ^{o}𝒞^{o} → Cat. But since j is fully faithful and admits a left 2-adjoint (see (3.2.10)), proposition 2.5.11 says that such 2-limit is the discrete category associated with the set representing the 2-limit of F ∘ t_{B^{o}} : B^{o}/φ^{o}𝒞^{o} → Set. By remark 2.5.2(vii), the latter is represented by the (usual) limit of the same functor. But this in turn is none else than the definition of ∫_{φ} F (B^{o}), whence the contention.

(iv) In view of corollary 2.6.42, we also see that if ρ is a fully faithful functor, then Fib(ρ)_{∗} and Fib(ρ)_{!} are both fully faithful pseudo-functors.

(v) For every other universe V′ with V ⊂ V′, we get a diagram of pseudo-functors :

\xymatrix{(U, V)-Fib(𝒞) \ar[r]^{(U,V)-Fib(ρ)_{!}} \ar[d] & (U, V)-Fib(ℬ) \ar[d] & (U, V)-Fib(𝒞) \ar[r]^{(U,V)-Fib(ρ)_{∗}} \ar[d] & (U, V)-Fib(ℬ) \ar[d] \\ (U, V′)-Fib(𝒞) \ar[r]^{(U,V′)-Fib(ρ)_{!}} & (U, V′)-Fib(ℬ) & (U, V′)-Fib(𝒞) \ar[r]^{(U,V′)-Fib(ρ)_{∗}} & (U, V′)-Fib(ℬ)}

whose vertical arrows are the inclusions. A direct inspection of the constructions easily shows that both diagrams are commutative : the details shall be left to the reader.

(vi) Let 𝒞 and 𝒞′ be two small categories, u : 𝒞 → 𝒞′ and v : 𝒞′ → 𝒞 two functors, such that v is left adjoint to u. In light of (1.1.32), we get for every fibration 𝒜 → 𝒞′ and every X ∈ Ob(𝒞) a natural isomorphism of categories :

ω_{X} : Cart_{𝒞′}(v𝒞′/X, 𝒜) →^{∼} 𝒜 (uX)

and for every morphism f : Y → X in 𝒞, a commutative diagram of categories :

\xymatrix{Cart_{𝒞′}(v𝒞′/X, 𝒜) \ar[r]^{ω_{X}} \ar[d]_{Cart_{𝒞′} (v𝒞′/f,𝒜)} & 𝒜 (uX) \ar[d]^{𝒜 (uf)} \\ Cart_{𝒞′}(v𝒞′/Y, 𝒜) \ar[r]^{ω_{Y}} & 𝒜 (uY).}

Combining with (i) and claim 3.2.8, we deduce a natural equivalence of fibrations over 𝒞 :

Fib(v)_{∗}(𝒜) →^{∼} Fib(u)^{∗}(𝒜)

and it is easily seen that the system of such equivalences amounts to a pseudo-natural equivalence of pseudo-functors :

Fib(v)_{∗} →^{∼} Fib(u)^{∗}.

3.3.16. For every small category 𝒞 we have a natural functor

𝒞 /− : 𝒞 → Fib(𝒞) X ↦ (s_{X} : 𝒞 /X → 𝒞)

that assigns to every morphism f : X → Y of 𝒞 the (cartesian) functor f_{∗} : 𝒞 /X → 𝒞 /Y as in (1.1.25). The following proposition upgrades remark 1.3.6(ii) from presheaves to fibrations :

Proposition 3.3.17. Let u : 𝒞 → ℬ be any functor between small categories. With the notation of (3.3.16), we have a pseudo-commutative diagram of 2-categories :

\xymatrix{𝒞 \ar[r]^{u} \ar[d]_{𝒞 /−} & ℬ \ar[d]^{ℬ/−} \\ Fib(𝒞) \ar[r]^{Fib(u)_{!}} & Fib(ℬ).}

Proof. Set Fib(u)_{!}(𝒞 /−) := Fib(u)_{!} ∘ (𝒞 /−) : 𝒞 → Fib(ℬ) and let ℬ/u− : 𝒞 → Fib(ℬ) be the composition of u with the pseudo-funtor ℬ/−. To ease notation, we set :

F := Cart_{ℬ} (Fib(u)_{!}(𝒞 /−), −) : 𝒞^{o} × Fib(ℬ) → (U, V)-Cat
G := Cart_{ℬ} (ℬ/u−, −) : 𝒞^{o} × Fib(ℬ) → (U, V)-Cat

(notation of example 2.2.7(ii)). Let ϑ be the 2-adjunction for the pair (Fib(u)_{!}, Fib(u)^{∗}); we deduce a pseudo-natural equivalence

ω := ϑ ∗ ((𝒞 /−)^{o} × 1_{Fib(ℬ)} ) : F ⇒ Cart_{𝒞} (𝒞 /−, Fib(u)^{∗}).

On the other hand, we have a strict pseudo-natural equivalence :

λ : G ⇒ Cart_{𝒞} (𝒞 /−, Fib(u)^{∗}).

Namely, for every X ∈ Ob(𝒞) and every ℬ-fibration 𝒜, the functor

λ_{X,𝒜} : 𝒜 (uX) → Fib(u)^{∗}𝒜 (X)

assigns to every cartesian section φ ∈ 𝒜 (uX) the unique cartesian section φ^{∗} ∈ 𝒜 ×_{ℬ} 𝒞 (X) whose composition with the projection π : 𝒜 ×_{ℬ} 𝒞 → 𝒜 equals φ ∘ u_{|X}. To every natural ℬ-transformation α : φ ⇒ φ′ between such functors, λ_{X,𝒜} assigns the unique natural 𝒞 - transformation α^{∗} such that π ∗ α^{∗} = α ∗ u_{|X}. We may then pick a quasi-inverse μ for λ, and consider the pseudo-natural equivalence

γ := μ ⊙ ω : F →^{∼} G.

For every X ∈ Ob(𝒞), let i_{X} : Fib(ℬ) → 𝒞^{o} × Fib(ℬ) be the unique strict pseudo-functor whose composition with the projection 𝒞^{o}×Fib(ℬ) → Fib(ℬ) equals 1_{Fib(ℬ)} , and whose composition with the projection 𝒞^{o} × Fib(ℬ) → 𝒞^{o} is the constant pseudo-functor F_{X^{o}} (notation of (2.5)); we deduce the pseudo-natural equivalence

γ ∗ i_{X} : Cart_{ℬ} (Fib(u)_{!}(𝒞 /X), −) →^{∼} Cart_{ℬ} (ℬ/uX, −).

We show more generally :

Claim 3.3.18. Let ℱ and 𝒢 be two ℬ-fibrations, and β_{•} : Cart_{ℬ} (ℱ , −) →^{∼} Cart_{ℬ} (𝒢 , −) a pseudo-natural equivalence. Then β^{∗} := β_{ℱ} (1_{ℱ} ) is a ℬ-equivalence of categories 𝒢 →^{∼} ℱ.

Proof of the claim. Let α_{•} : Cart_{ℬ} (𝒢 , −) → Cart_{ℬ} (ℱ , −) be a quasi-inverse of β_{•}, and set α^{∗} := α_{𝒢} (1_{𝒢}) : ℱ → 𝒢. There follow essentially commutative diagrams :

[figure: Cart_{ℬ} (ℱ , ℱ); Cart_{ℬ} (ℱ ,α^{∗}); β_{ℱ}; Cart_{ℬ} (ℱ , 𝒢); β_{𝒢}; Cart_{ℬ} (𝒢 , 𝒢); Cart_{ℬ} (𝒢 ,β^{∗}); α_{𝒢}; Cart_{ℬ} (𝒢 , ℱ); α_{ℱ}; Cart_{ℬ} (𝒢 , ℱ); Cart_{ℬ} (𝒢 ,α^{∗}); Cart_{ℬ} (𝒢 , 𝒢); Cart_{ℬ} (ℱ , 𝒢); Cart_{ℬ} (ℱ ,β^{∗}); Cart_{ℬ} (ℱ , ℱ).]

From the left diagram we get isomorphisms : α^{∗}∘β^{∗} →^{∼} β_{𝒢} (α^{∗}) = β_{𝒢} ∘α(1_{𝒢}) →^{∼} 1_{𝒢}. Likewise, from the right diagram we get an isomorphism β^{∗} ∘ α^{∗} →^{∼} 1_{ℱ} , whence the contention. ♢

To ease notation, for every X ∈ Ob(𝒞) and every morphism f : X → Y of 𝒞 set

[X] := Fib(u)_{!}(𝒞 /X) and [f] := Fib(u)_{!}(f_{∗}) : [X] → [Y]

According to claim 3.3.18, for every X ∈ Ob(𝒞) we get an equivalence of ℬ-categories :

Γ_{X} := γ_{X,[X]}(1_{[X]}) : ℬ/uX →^{∼} [X].

To conclude, it remains to show that the rule : X ↦ Γ_{X} yields a pseudo-natural equivalence as sought. To this aim, we need to exhibit a coherence constraint for Γ. Now, notice that, after possibly replacing it by a pseudo-naturally isomorphic pseudo-functor, we may assume that Fib(u)_{!} is unital (proposition 2.4.3); then, for every morphism f : X → Y in 𝒞 consider the diagram of oriented squares :

[figure: F (Y, [Y]); γ_{Y,[Y ]}; τ_{f,1_{[Y ]}}^{γ}; F (f,1_{[Y ]}); G(Y, [Y]); G(f,1_{[Y ]}); F (X, [Y]); γ_{X,[Y ]}; G(X, [Y]); F (X, [X]); γ_{X,[X]}; F (1_{X},[f]); G(X, [X]).; G(1_{X},[f]); τ_{1_{X} ,[f]}^{γ}]

where τ_{•,•}^{γ} is the coherence constraint of γ. We deduce isomorphisms of ℬ-cartesian functors :

[f] ∘ Γ_{X} ⟶^{(τ_{1_{X} ,[f]}^{γ})_{1_{[X]}}} γ_{X,[Y]}([f]) ⟵^{(τ_{f,1_{[Y ]}}^{γ})_{1_{[Y ]}}} Γ_{Y} ∘ (uf)_{∗}

so our candidate coherence constraint is :

τ_{f}^{Γ} := (τ_{f,1_{[Y ]}}^{γ})_{1_{[Y ]}}^{−1} ⊙ (τ_{1_{X},[f]}^{γ})_{1_{[X]}} : [f] ∘ Γ_{X} →^{∼} Γ_{Y} ∘ (uf)_{∗}.

With this definition, it is already clear that τ_{1_{X}}^{Γ} = 1_{Γ_{X}} for every X ∈ Ob(𝒞). Next, let f : X → Y and g : Y → Z be two morphisms of 𝒞; we need to check that :

(τ_{g}^{Γ} ∗ (uf)_{∗}) ⊙ ([g] ∗ τ_{f}^{Γ}) = τ_{g∘f}^{Γ} ⊙ (γ_{f_{∗},g_{∗}}^{Fib(u)_{!}} ∗ Γ_{X})

where γ^{Fib(u)_{!}} denotes the coherence constraint of Fib(u)_{!} But the coherence condition for τ^{γ}, relative to the composition of 1-cells (1_{X^{o}}, [g]) ∘ (f^{o}, 1_{[Y]}) = (f^{o}, [g]) = (f^{o}, 1_{[Z]}) ∘ (1_{Y^{o}}, [g]) yields the identities :

(τ_{1_{X},[g]}^{γ})_{[f]} ⊙ ([g] ∗ (τ_{f,1_{[Y ]}}^{γ})) = (τ_{f,[g]}^{γ})_{1_{[Y ]}} = (τ_{f,1_{[Z]}}^{γ})_{[g]} ⊙ (τ_{1_{Y} ,[g]}^{γ} ∗ (uf)_{∗})

By the same token, from the identity : (f^{o}, 1_{[Z]}) ∘ (g^{o}, 1_{[Z]}) = ((g ∘ f)^{o}, 1_{[Z]}) we get :

γ_{X,[Z]}(γ_{f_{∗},g_{∗}}^{Fib(u)_{!}}) ⊙ (τ_{f,1_{[Z]}}^{γ})_{[g]} ⊙ ((τ_{g,1_{[Z]}}^{γ})_{1_{[Z]}} ∗ (uf)_{∗}) = (τ_{g∘f,1_{[Z]}}^{γ})_{1_{[Z]}}

whence :

\begin{aligned}(τ_{g}^{Γ} ∗ (uf)_{∗})⊙([g] ∗ τ_{f}^{Γ}) & =((τ_{g,1_{[Z]}}^{γ})_{1_{[Z]}}^{−1} ∗ (uf)_{∗})⊙(τ_{f,1_{[Z]}}^{γ})_{[g]}^{−1}⊙(τ_{1_{X},[g]}^{γ})_{[f]}⊙([g]∗(τ_{1_{X},[f]}^{γ})_{1_{[X]}}) \\ & =(τ_{g∘f,1_{[Z]}}^{γ})_{1_{[Z]}}^{−1} ⊙ γ_{X,[Z]}(γ_{f_{∗},g_{∗}}^{Fib(u)_{!}})⊙(τ_{1_{X},[g]}^{γ})_{[f]}⊙([g]∗(τ_{1_{X},[f]}^{γ})_{1_{[X]}}). \end{aligned}

Thus, we are reduced to checking that :

(τ_{1_{X},[g∘f]}^{γ})_{1_{[X]}} ⊙ (γ_{f_{∗},g_{∗}}^{Fib(u)_{!}} ∗ Γ_{X}) = γ_{X,[Z]}(γ_{f_{∗},g_{∗}}^{Fib(u)_{!}}) ⊙ (τ_{1_{X},[g]}^{γ})_{[f]} ⊙ ([g] ∗ (τ_{1_{X},[f]}^{γ})_{1_{[X]}}).

But the latter follows once again from the coherence condition for τ^{γ}, applied to the identity : (1_{X}, [g]) ∘ (1_{X}, [f]) = (1_{X}, [g ∘ f]). □

In the same vein, we point out the following :

Proposition 3.3.19. Let 𝒞, 𝒞′ be two small categories, u : 𝒞 → 𝒞′ a functor, and V a universe such that 𝒞^{∧} and 𝒞^{′∧} are V-small. We have a pseudo-commutative diagram of 2-categories :

\xymatrix{(U, V)-Fib(𝒞_{U}^{′∧}) \ar[r]^{(U,V)-Fib(u_{U}^{∧})_{!}} \ar[d]_{(U,V)-Fib(h_{𝒞′} )^{∗}} & (U, V)-Fib(𝒞_{U}^{∧}) \ar[d]^{(U,V)-Fib(h_{𝒞} )^{∗}} \\ (U, V)-Fib(𝒞′) \ar[r]^{(U,V)-Fib(u)^{∗}} & (U, V)-Fib(𝒞).}

Proof. Let ℰ′ be any fibration with V-small fibres over 𝒞^{′∧}; pick a unital cleavage for ℰ′ and let c′ be its associated pseudo-functor. Set ℰ := (U, V)-Fib(u_{U}^{∧})_{!}ℰ′; by inspecting (3.3.14) we see that for every F ∈ Ob(𝒞^{∧}) the fibre category ℰ_{F} represents the strong 2-colimit of

c′ ∘ t_{F}^{o} : (F/u^{∧}𝒞^{′∧}) → (U, V)-Cat

where t_{F} : (F/u^{∧}𝒞^{′∧}) → 𝒞^{′∧} denotes the target functor. Fix a universal pseudo-cocone

e_{•}^{F} : c′ ∘ t_{F}^{o} ⇒ F_{ℰ_{F}} (F ⟶^{φ} u^{∧}G) ↦ (ℰ′_{G} ⟶^{e_{φ}^{F}} ℰ_{F})

and let τ_{•}^{F} be the coherence constraint of e_{•}^{F}. Then ℰ admits a split cleavage d, that assigns to every morphism ψ : F′ → F of 𝒞^{∧} the unique functor ℰ_{ψ} : ℰ_{F} → ℰ_{F′} such that

F_{ℰ_{ψ}} ⊙ e^{F} = e^{F′} ∗ ψ^{∗o}

where ψ^{∗} : F/u^{∧}𝒞^{′∧} → F′/u^{∧}𝒞^{′∧} is defined as in (1.1.25). Hence, V-Fib(h_{𝒞} )^{∗} admits the split cleavage d ∘ h_{𝒞}^{o} . For every X ∈ Ob(𝒞), let η_{X} : h_{X} → u^{∧}h_{uX} be the morphism given by the rule : φ ↦ u(φ) ∈ h_{uX}(uY) for every Y ∈ Ob(𝒞) and every φ ∈ h_{X}(Y). We notice :

Claim 3.3.20. For every X ∈ Ob(𝒞), the morphism η_{ℰ} is an initial object of h_{X}/u^{∧}𝒞^{′∧}.

Proof of the claim. Indeed, In light of remark 1.3.6(ii,iii), we may regard η_{X} as the unit of the natural adjunction for the adjoint pair (u_{!}, u^{∧}). Hence, for every G ∈ Ob(𝒞^{′∧}) and every morphism of presheaves β : h_{X} → u^{∧}G there exists a unique morphism of presheaves β′ : h_{uX} → G on 𝒞′ such that β = u^{∧}(β′) ∘ η_{X} (explicitly, under the canonical bijection provided by Yoneda’s lemma, β corresponds to a unique section s_{β} ∈ (u^{∧}G)X, and β′ corresponds then to the same s_{β}, regarded as an element of G(uX)). The claim is an immediate consequence. ♢

From claim 3.3.20and proposition 2.5.16, we get an equivalence of categories

ε_{X} := e_{η_{X}}^{h_{X}} : ℰ′_{h_{uX}} → ℰ_{h_{X}} for every X ∈ Ob(𝒞).

Claim 3.3.21. The rule : X ↦ ε_{X} yields a pseudo-natural equivalence ε : c′∘(h_{𝒞′} ∘u)^{o} →^{∼} d∘h_{𝒞}^{o} .

Proof of the claim. For every morphism f : Y → X in 𝒞 we have a natural isomorphism :

τ_{f}^{ε} : d_{h_{f}} ∘ ε_{X} = e_{η_{X}∘h_{f}}^{h_{Y}} = e_{u^{∧}(h_{uf})∘η_{Y}}^{h_{Y}} ⟶^{τ_{h_{uf}}^{h_{Y}}} ε_{Y} ∘ c′_{h_{uf}}

and we check that the system (τ_{f}^{ε} | f ∈ Morph(𝒞)) yields a coherence constraint for ε. First, we get τ_{1_{X}}^{ε} = 1_{ε_{X}} for every X ∈ Ob(𝒞), by remark 2.4.2(ii). Next, notice that for every morphism ψ : F′ → F in 𝒞^{∧} we have :

ℰ_{ψ} ∗ τ_{μ}^{F} = τ_{μ}^{F′} for every morphism F′/μ : (F′ → u^{∧}G_{1}) → (F′ → u^{∧}G_{2}) in F′/u^{∧}𝒞^{′∧} .

Thus, let f : X → Y and g : Y → Z be two morphisms in 𝒞; we deduce :

(ε_{Z} ∗ γ_{h_{uf} ,h_{ug}}^{c′}) ⊙ (τ_{g}^{ε} ∗ c′_{uf}) ⊙ (d_{h_{g}} ∗ τ_{f}^{ε}) = (ε_{Z} ∗ γ_{h_{uf} ,h_{ug}}^{c′}) ⊙ (τ_{h_{ug}}^{h_{Z}} ∗ c′_{uf}) ⊙ τ_{h_{uf}}^{h_{Z}}
= (ε_{Z} ∗ γ_{h_{uf} ,h_{ug}}^{c′}) ⊙ τ_{h_{u(g∘f)}}^{h_{Z}}
= τ_{g∘f}^{ε}

whence the contention. ♢ From claim 3.3.21we deduce an equivalence of fibrations over 𝒞 :

Ω_{ℰ′} : Fib(h_{𝒞′} ∘ u)^{∗}ℰ′ →^{∼} Fib(h_{𝒞} )^{∗} ∘ Fib(u_{U}^{∧})_{!}ℰ′.

Next, let φ′ : ℰ′_{1} → ℰ′_{2} be a cartesian functor of fibrations over 𝒞^{′∧}; let ℰ_{i} := Fib(u^{∧})_{!}ℰ′_{i} for i = 1, 2, and set φ := Fib(u_{U}^{∧})_{!}φ′ : ℰ_{1} → ℰ_{2}. For i = 1, 2, pick also unital a cleavage c′_{i} for ℰ′, so that φ′ corresponds to a pseudo-natural transformation ω′ : c′_{1} ⇒ c′_{2}. For every F ∈ Ob(𝒞^{∧}) we pick universal cocones e_{1,•}^{F} : c′_{i} ∘ t_{F}^{o} ⇒ F_{ℰ_{i,F}}, and denote by d_{1} and d_{2} the natural split cleavages for ℰ_{1} and ℰ_{2} described in the foregoing. Then φ corresponds to the strict pseudo-natural transformation ω : d_{1} ⇒ d_{2} that assigns to every F ∈ Ob(𝒞^{∧}) the unique functor ω_{F} : ℰ_{1,F} → ℰ_{2,F} such that

(3.3.22)  e_{2,•}^{F} ⊙ (ω′ ∗ t_{F}^{o}) = F_{ω_{F}} ⊙ e_{1,•}^{F}.

Let ε_{i} : c′_{i} ∘ (h_{𝒞′} ∘ u)^{o} →^{∼} d_{i} ∘ h_{𝒞}^{o} be the pseudo-natural equivalence of claim 3.3.21, for i = 1, 2.

Claim 3.3.23. (ω ∗ h_{𝒞}^{o} ) ⊙ ε_{1} = ε_{2} ⊙ (ω′ ∗ (h_{𝒞′} ∘ u)^{o}).

Proof of the claim. Directly from (3.3.22) we see that the two sides of the identity of the claim agree on every X ∈ Ob(𝒞). It remains to check that the respective coherence constraints agree as well. However, for every F ∈ Ob(𝒞^{∧}) denote by τ_{i,•}^{F} the coherence constraint of e_{i,•}^{F}, for i = 1, 2; the coherence constraint of (ω ∗ h_{𝒞}^{o} ) ⊙ ε_{1} assigns to every morphism f : X → Y of 𝒞 the natural isomorphism of functors ω_{h_{X}} ∗ τ_{1,h_{uf}}^{h_{Y}}, whereas the coherence constraint of ε_{2} ⊙(ω′ ∗(h_{𝒞′} ∘u)^{o}) assigns to f the natural isomorphism τ_{2,h_{uf}}^{h_{Y}} ∗ω′_{h_{uY}}. Then again the required identity follows directly from (3.3.22). ♢

From claim 3.3.23we deduce a commutative diagram of cartesian functors :

\xymatrix{Fib(h_{𝒞′} ∘ u)^{∗}ℰ′_{1} \ar[d]_{Fib(h_{𝒞′} ∘u)^{∗}φ′} \ar[r]^{Ω_{ℰ′_{1}}} & Fib(h_{𝒞} )^{∗}ℰ_{1} \ar[d]^{Fib(h_{𝒞} )^{∗}φ} \\ Fib(h_{𝒞′} ∘ u)^{∗}ℰ′_{2} \ar[r]^{Ω_{ℰ′_{2}}} & Fib(h_{𝒞} )^{∗}ℰ_{2}.}

Lastly, for ℰ′_{1}, ℰ′_{2} as in the foregoing, let φ′_{1}, φ′_{2} : ℰ′ → ℰ′_{2} be two cartesian functors, β′ : φ′_{1} ⇒ φ′_{2} a natural 𝒞^{′∧}-transformation, and set φ_{i} := Fib(u_{U}^{∧})_{!}φ′_{i} for i = 1, 2 and β := Fib(u_{U}^{∧})_{!}β′ : φ_{1} ⇒ φ_{2}. Say that φ′_{1} and φ′_{2} correspond to pseudo-natural transformations ω′_{1}, ω′_{2} : c′_{1} ⇒ c′_{2}, and β′ corresponds to a modification Ξ′ : ω′_{1} ⇝ ω′_{2}; then φ_{1} and φ_{2} correspond to strict pseudonatural transformations ω_{1}, ω_{2} : d_{1} ⇒ d_{2} described as in the foregoing, and β corresponds to the modification Ξ : ω_{1} ⇝ ω_{2} assigning to every F ∈ Ob(𝒞^{∧}) the unique natural transformation Ξ_{F} : ω_{1,F} ⇒ ω_{2,F} with :

e_{2,•}^{F} ∗ (Ξ′ ∘ t_{F}^{o}) = F_{Ξ_{F}} ∗ e_{1,•}^{F}

which specializes to the identity : Ξ_{h_{X}} ⊙ e_{1,η_{X}}^{h_{X}} = e_{2,η_{X}}^{h_{X}} ⊙ Ξ′_{h_{uX}} for every X ∈ Ob(𝒞). The latter in turns yields the identity :

(Ξ ∘ h_{𝒞}^{o} ) ∗ ε_{1} = ε_{2} ∗ (Ξ′ ∘ (h_{𝒞′} ∘ u)^{o})

which finally shows that :

Fib(h_{𝒞} )^{∗}(β) ∗ Ω_{ℰ′_{1}} = Ω_{ℰ′_{2}} ∗ Fib(h_{𝒞′} ∘ u)^{∗}(β′).

Summing up, this shows that the system of equivalences Ω_{•} yields a strict pseudo-natural equivalence of strict pseudo-functors Fib(h_{𝒞′} ∘ u)^{∗} →^{∼} Fib(h_{𝒞} )^{∗} ∘ Fib(u_{U}^{∧})_{!}, as stated. □

3.3.24. We conclude this section by showing that the 2-limits and 2-colimits are computed fibrewise in the 2-category (U, V)-Fib(ℬ), a result that extends the corresponding assertion for the category of presheaves (corollary 1.4.3(ii)). To begin with, let ℬ be any V-small category; with every B ∈ Ob(ℬ) we associate a strict pseudo-functor

fib_{B} : (U, V)-Fib(ℬ) → (U, V)-Cat 𝒜 ↦ 𝒜_{B}.

If 𝒜 and 𝒜′ are two ℬ-fibrations, ψ, ψ′ : 𝒜 → 𝒜′ two ℬ-cartesian functors, and β : ψ ⇒ ψ′ a natural ℬ-transformation, then fib_{B}(ψ) : 𝒜_{B} → 𝒜′_{B} is the restriction of ψ, and fib_{B}(β) : fib_{B}(ψ) ⇒ fib_{B}(ψ′) is the restriction of β. We may then state :

Theorem 3.3.25. With the notation of (3.3.24), we have :

(i) The 2-category (U, V)-Fib(ℬ) is strongly 2-complete and strongly 2-cocomplete. (ii) For every B ∈ Ob(ℬ) and any small 2-category I, the pseudo-functor fib_{B} commutes with the strong 2-limit and strong 2-colimit of any pseudo-functor I → (U, V)-Fib(ℬ).

Proof. We prove the 2-completeness of Fib(ℬ); the 2-cocompleteness follows by the same argument, considering the opposite 2-categories : the details shall be left to the reader. In view of theorem 3.1.24, it suffices to show the corresponding assertions for the 2-category PsFun(ℬ^{o}, Cat) and the similar strict pseudo-functors that we denote as well by

fib_{B} : PsFun(ℬ^{o}, Cat) → Cat c ↦ c_{B}.

Notice that every morphism g : B′ → B of ℬ induces a pseudo-natural transformation

fib_{g} : fib_{B} ⇒ fib_{B′} c ↦ (c_{g} : c_{B} → c_{B′})

with coherence constraint given by the rule :

τ_{β}^{fib_{g}} := (τ_{g}^{β})^{−1}

for every pair of pseudo-natural functors c, d : ℬ^{o} → Cat and every pseudo-natural transformation β : c ⇒ d with coherence constraint τ^{β}. Moreover, if h : B′′ → B′ is any other morphism of ℬ, we get the invertible modification

Γ_{h,g} : fib_{h} ⊙ fib_{g} ⇝ fib_{g∘h} c ↦ γ_{h,g}^{c}

where (δ^{c}, γ^{c}) denotes the coherence constraint of c. Indeed, it is easily seen that the compatibility condition for (Γ_{h,g})_{β} corresponding to a pseudo-natural transformation β : c ⇒ d is equivalent to the coherence axiom for τ^{β} : details left to the reader. Likewise, for every third morphism k : B′′′ → B′′ of ℬ, the composition axiom for pseudo-functors translates as :

(3.3.26)  Γ_{k,g∘h} ⊙ (fib_{k} ∗ Γ_{h,g}) = Γ_{h∘k,g} ⊙ (Γ_{k,h} ∗ fib_{g}).

Let now F : I → PsFun(ℬ^{o}, Cat) be any pseudo-functor; we need to exhibit a 2-limit for F, and by virtue of proposition 2.4.3, we may assume that F i : ℬ^{o} → Cat is unital for every i ∈ Ob(I). Then, for every B ∈ Ob(ℬ) choose a strong 2-limit (L(B), π^{B}) of the pseudofunctor fib_{B} ∘ F (theorem 3.3.9), and let τ^{B} be the coherence constraint of the pseudo-cone π^{B} : F_{L(B)} ⇒ fib_{B} ∘ F. Let g : B′ → B be any morphism of ℬ; by the (strong) universality of π^{B′}, we find a unique functor L(g) : L(B) → L(B′) such that

(3.3.27)  π^{B′} ⊙ F_{L(g)} = (fib_{g} ∗ F) ⊙ π^{B}.

Notice that fib_{1_{B}} ∗ F = 1_{fib_{B}∘F}, since F i is unital for every i ∈ Ob(I); therefore :

L(1_{B}) = 1_{L(B)} for every B ∈ Ob(ℬ) .

Moreover, if h : B′′ → B′ is another morphism of ℬ, we have identities :

π^{B′′} ⊙ F_{L(h)∘L(g)} = (fib_{h} ∗ F) ⊙ (fib_{g} ∗ F) ⊙ π^{B} = ((fib_{h} ⊙ fib_{g}) ∗ F) ⊙ π^{B}
π^{B′′} ⊙ F_{L(h∘g)} = (fib_{h∘g}) ⊙ π^{B}

so there exists a unique natural transformation γ_{h,g}^{L} : L(h) ∘ L(g) ⇒ L(g ∘ h) such that

π^{B′′} ∗ F_{γ_{h,g}^{L}} = (Γ_{h,g} ∘ F) ∗ π^{B}.

Claim 3.3.28. The rules B ↦ L(B) and (g : B′ → B) ↦ L(g) for every B ∈ Ob(ℬ) and every morphism g of ℬ define a unital pseudo-functor L : ℬ^{o} → Cat with coherence constraint given by the system of natural transformations γ_{•,•}^{L}.

Proof of the claim. The unit axiom for γ^{L} is clear from the construction. To check the composition axiom, let g and h be as in the foregoing, and consider as well a third morphism k : B′′′ → B′′ of ℬ. Since π^{B′′′} is a universal pseudo-cone, it suffices to show that

X := π^{B′′′} ∗ (F_{γ_{k,g∘h}^{L}} ⊙ F_{L(k)∗γ_{h,g}^{L}}) = Y := π^{B′′′} ∗ (F_{γ_{h∘k,g}^{L}} ⊙ F_{γ_{k,h}^{L}∗L(g)}).

However, by unwinding the definitions we find :

X = ((Γ_{k,g∘h} ∘ F) ∗ π^{B}) ⊙ (π^{B′′′} ∗ F_{L(k)} ∗ F_{γ_{h,g}^{L}})
= ((Γ_{k,g∘h} ∘ F) ∗ π^{B}) ⊙ ((fib_{k} ∗ F) ∗ π^{B′′} ∗ F_{γ_{h,g}^{L}})
= ((Γ_{k,g∘h} ∘ F) ∗ π^{B}) ⊙ ((fib_{k} ∗ F) ∗ (Γ_{h,g} ∘ F) ∗ π^{B})
= ((Γ_{k,g∘h} ∘ F) ∗ π^{B}) ⊙ (((fib_{k} ∗ Γ_{h,g}) ∘ F) ∗ π^{B})
Y = ((Γ_{h∘k,g} ∘ F) ∗ π^{B}) ⊙ (π^{B′′′} ∗ F_{γ_{k,h}^{L}} ∗ F_{L(g)})
= ((Γ_{h∘k,g} ∘ F) ∗ π^{B}) ⊙ ((Γ_{k,h} ∘ F) ∗ π^{B′} ∗ F_{L(g)})
= ((Γ_{h∘k,g} ∘ F) ∗ π^{B}) ⊙ ((Γ_{k,h} ∘ F) ∗ (fib_{g} ∗ F) ∗ π^{B})
= ((Γ_{h∘k,g} ∘ F) ∗ π^{B}) ⊙ ((Γ_{k,h} ∗ fib_{g}) ∘ F) ⊙ π^{B}

so it suffices to show:

(Γ_{h∘k,g} ∘ F) ⊙ ((Γ_{k,h} ∗ fib_{g}) ∘ F) = (Γ_{k,g∘h} ∘ F) ⊙ ((fib_{k} ∗ Γ_{h,g}) ∘ F)

which follows straightforwardly from (3.3.26). ♢

Next, we notice that for every i ∈ Ob(I) the rule : B ↦ (π_{i}^{B} : L(B) → F i(B)) defines a strict pseudo-natural transformation

π_{i} : L ⇒ F i.

Indeed, the first coherence axiom for π_{i} is easily checked, recalling that both L and F i are unital, and the second one follows directly from the definition of γ^{L} : details left to the reader.

Claim 3.3.29. (i) For every 1-cell φ : i → j in I, the rule :

B ↦ τ_{φ}^{B}

defines an invertible modification Ξ^{φ} : F (φ) ⊙ π_{i} ⇝ π_{j}.

(ii) The system (π_{i} | i ∈ Ob(I)) defines a pseudo-cone

π : F_{L} ⇒ F

with coherence constraint given by the invertible modifications Ξ^{•}.

Proof of the claim. (i): Let τ^{F φ} and τ^{fib_{g}∗F} be the coherence constraints of F φ and respectively fib_{g} ∗ F. We need to verify the compatibility condition :

F j(g) ∗ τ_{φ}^{B} = (τ_{j}^{B′} ∗ L(g)) ⊙ (τ_{g}^{F φ} ∗ π_{i}^{B})

for every morphism g : B′ → B of ℬ. However, comparing the coherence constraints of the two sides of (3.3.27) we get :

(F j(g) ∗ τ_{φ}^{B}) ⊙ (τ_{φ}^{fib_{g}∗F} ∗ π_{i}^{B}) = τ_{φ}^{B′} ∗ L(g).

Since τ_{φ}^{fib_{g}∗F} = (τ_{g}^{F φ})^{−1}, the assertion follows.

(ii): Denote by (δ^{F} , γ^{F}) the coherence constraint of F; we need to check the identities

Ξ^{1_{i}} ⊙ (δ_{i}^{F} ∗ π_{i}) = 1_{π_{i}} Ξ^{ψ} ⊙ (F ψ ∗ Ξ^{φ}) = Ξ^{ψ∘φ} ⊙ (γ_{φ,ψ}^{F} ∗ π_{i})

for every i, j, k ∈ Ob(I) and every pair of 1-cells φ : i → j and ψ : j → k of I. However, the latter are none else than a rephrasing of the coherence axioms for the pseudo-cones π^{B}, for B ranging over all the objects of ℬ. ♢

It remains to check that the pseudo-cone π : F_{L} ⇒ F of claim 3.3.29(ii) is universal. To this aim, let X : ℬ^{o} → Cat be any pseudo-functor, and β : F_{X} ⇒ F any pseudo-cone; we need to exhibit a pseudo-natural trasformation t : X ⇒ L such that π ⊙ F_{t} = β. Let ω : X →^{∼} X^{u} be the isomorphism furnished by proposition 2.4.3, with X^{u} a unital pseudo-functor; we get a commutative diagram of categories :

\xymatrix{PsNat(X^{u}, L) \ar[r] \ar[d]_{PsNat(ω,L)} & PsNat(F_{X^{u}}, F) \ar[d]^{PsNat(F_{ω},F)} \\ PsNat(X, L) \ar[r] & PsNat(F_{X}, F)}

whose vertical arrows are isomorphisms. We need to check that the bottom horizontal arrow is an isomorphism, so it suffices to show that the same holds for the top horizontal arrow; we may thus assume that X is unital. Now, fib_{B} ∗ β is a pseudo-cone with vertex X(B) and basis fib_{B} ∘ F : I → Cat; then there exists a unique functor t_{B} : X(B) → L(B) such that

π^{B} ⊙ F_{t_{B}} = fib_{B} ∗ β for every B ∈ Ob(ℬ) .

Denote by τ^{β} the coherence constraint of β, and by τ^{β_{i}} the coherence constraint of the pseudonatural transformation β_{i} : X ⇒ F i, for every i ∈ Ob(I); we remark :

Claim 3.3.30. For every morphism g : B′ → B of ℬ, the rule :

i ↦ τ_{g}^{β_{i}} for every i ∈ Ob(I)

defines an invertible modification Λ^{g} : (fib_{g} ∗ F) ⊙ (fib_{B} ∗ β) ⇝ (fib_{B′} ∗ β) ⊙ F_{X(g)}.

Proof of the claim. The coherence constraint of (fib_{g} ∗ F) ⊙ (fib_{B} ∗ β) assigns to every 1-cell φ : i → j of I the composition of oriented squares :

[figure: X(B); β_{i,B}; τ_{φ,B}^{β}; F i(B); (τ_{g}^{F φ})^{−1}; F i(g); F φ(B); F i(B′); F φ(B′); X(B); β_{j,B}; F j(B); F j(g); F j(B′)]

whereas the coherence constraint of (fib_{B′} ∗ β) ⊙ F_{X(g)} assigns to φ the natural transformation

τ_{φ,B′}^{β} ∗ X(g) : F φ(B′) ∘ β_{i,B′} ∘ X(g) ⇒ β_{j,B′} ∘ X(g).

Thus, for every such φ we need to show the identity :

τ_{g}^{β_{j}} ⊙ (F j(g) ∗ τ_{φ,B}^{β}) ⊙ ((τ_{g}^{F φ})^{−1} ∗ β_{i,B}) = τ_{φ,B′}^{β} ⊙ (F φ(B′) ∗ τ_{g}^{β_{i}}).

But the latter is equivalent to the compatibility condition for the invertible modification τ_{φ}^{β} : F φ ⊙ β_{i} ⇝ β_{j}, corresponding to the morphism g. ♢

Notice now that (fib_{g}∗F)⊙(fib_{B}∗β) = π^{B′}⊙F_{L(g)∘t_{B}} and (fib_{B′}∗β)⊙F_{X(g)} = π^{B′}⊙F_{t_{B′}∘X(g)} for every morphism g : B′ → B of ℬ. In view of claim 3.3.30, and since π^{B′} is universal, there exists therefore a unique natural isomorphism of functors

τ_{g}^{t} : L(g) ∘ t_{B} ⇒ t_{B′} ∘ X(g) such that π^{B′} ∗ F_{τ_{g}^{t}} = Λ^{g}.

Notice that, by virtue of remark 2.4.2(ii), we have Λ^{1_{B}} = 1_{fib_{B}∗β} for every B ∈ Ob(ℬ), and therefore τ_{1_{B}}^{t} = 1_{t_{B}} for every such B. Let us check that the system (t_{B} | B ∈ Ob(ℬ)) yields the sought pseudo-natural transformation, with coherence constraint given by the system of isomorphisms of functors τ_{•}^{t}. Indeed, the first coherence axiom follows easily from the foregoing; it remains therefore only to show that for every pair of morphisms g : B′ → B and h : B′′ → B′ of ℬ we have :

(t_{B′′} ∗ γ_{h,g}^{X}) ⊙ (τ_{h}^{t} ∗ X(g)) ⊙ (L(h) ∗ τ_{g}^{t}) = τ_{g∘h}^{t} ⊙ (γ_{g,h}^{L} ∗ t_{B})

where γ^{X} denotes the coherence constraint of X. Using the universality of π^{B′′}, we are then reduced to showing the identity :

U := (π^{B′′} ∗F_{t_{B′′}} ∗F_{γ_{h,g}^{X}})⊙(Λ^{h} ∗F_{X(g)})⊙(π^{B′′} ∗F_{L(h)} ∗F_{τ_{g}^{t}}) = V := Λ^{g∘h} ⊙(π^{B′′} ∗F_{γ_{g,h}^{L}} ∗F_{t_{B}}).

However, we have :

U = (π^{B′′} ∗ F_{t_{B′′}} ∗ F_{γ_{h,g}^{X}}) ⊙ (Λ^{h} ∗ F_{X(g)}) ⊙ ((fib_{h} ∗ F) ∗ π^{B′} ∗ F_{τ_{g}^{t}})
= ((fib_{B′′} ∗ β) ∗ F_{γ_{h,g}^{X}}) ⊙ (Λ^{h} ∗ F_{X(g)}) ⊙ ((fib_{h} ∗ F) ∗ π^{B′} ∗ F_{τ_{g}^{t}})
= ((fib_{B′′} ∗ β) ∗ F_{γ_{h,g}^{X}}) ⊙ (Λ^{h} ∗ F_{X(g)}) ⊙ ((fib_{h} ∗ F) ∗ Λ^{g})
V = Λ^{g∘h} ⊙ ((Γ_{h,g} ∘ F) ∗ π^{B} ∗ F_{t_{B}}).

Thus, it suffices to show :

((fib_{B′′} ∗ β) ∗ F_{γ_{h,g}^{X}}) ⊙ (Λ^{h} ∗ F_{X(g)}) ⊙ ((fib_{h} ∗ F) ∗ Λ^{g}) = Λ^{g∘h} ⊙ ((Γ_{h,g} ∘ F) ∗ (fib_{B} ∗ β)).

But a simple inspection shows that the latter identity translates the coherence axiom for β_{i}, relative to the pair of 1-cells (h, g), and for i ranging over all the objects of I.

Claim 3.3.31. (i) π_{i} ⊙ t = β_{i} for every i ∈ Ob(I).

(ii) π ⊙ F_{t} = β.

Proof of the claim. We need to check that the coherence constraint τ^{π_{i}⊙t} of π_{i}⊙t equals τ^{β_{i}}. But for every morphism g : B′ → B of ℬ we have τ_{g}^{π_{i}⊙t} = π_{i}^{B′} ∗ τ_{g}^{t} = Λ_{i}^{g}, whence the contention.

(ii): In view of (i), it suffices to check that β and π ⊙ F_{t} have the same coherence constraints. But the coherence constraint of π ⊙ F_{t} assigns to every 1-cell φ : i → j in I the invertible modification Ξ^{φ} ∗ t, so the assertion follows by a direct inspection of the definitions. ♢

Claim 3.3.31(ii) shows that the functor

(3.3.32)  PsNat(X, L) → PsNat(F_{X}, F)  t ↦ π ⊙ F_{t}

is surjective on objects. To check injectivity on objects, consider two pseudo-natural transformations t, t′ : X ⇒ L such that π ⊙ F_{t} = π ⊙ F_{t′}. Then we have

π^{B} ⊙ F_{t_{B}} = fib_{B} ∗ (π ⊙ F_{t}) = fib_{B} ∗ (π ⊙ F_{t′}) = π^{B} ⊙ F_{t′_{B}} for every B ∈ Ob(ℬ) .

By strong universality of π^{B}, we deduce that t_{B} = t′_{B} for every such B. Moreover, recall that we have attached to π ⊙ F_{t} and every morphism g : B′ → B an invertible modification Λ^{g} : π^{B′} ⊙ F_{L(g)∘t_{B}} ⇝ π^{B′} ⊙ F_{t_{B′}∘X(g)}. Let τ^{t} and τ^{t′} be the coherence constraint of t and t′; by inspecting the definition, we find that

Λ_{i}^{g} = (τ_{g}^{B} ∗ t_{B}) ⊙ (π_{i}^{B′} ∗ τ_{g}^{t}) for every i ∈ Ob(I).

Since π ⊙ F_{t} = π ⊙ F_{t′} and t_{B} = t′_{B}, we get π_{i}^{B′} ∗ τ_{g}^{t} = π_{i}^{B′} ∗ τ_{g}^{t′} for every i ∈ Ob(I), i.e. π^{B′} ∗ F_{τ_{g}^{t}} = π^{B′} ∗ F_{τ_{g}^{t′}}, whence τ_{g}^{t} = τ_{g}^{t′}, by the universality of π^{B′}. Thus, t = t′ as required.

To check that (3.3.32) is faithful, consider pseudo-natural transformations t, t′ : X ⇒ L and modifications Δ, Δ′ : t ⇝ t′ with π ∗ F_{Δ} = π ∗ F_{Δ′}; we deduce:

π^{B}∗(fib_{B}∘F_{Δ}) = fib_{B}∘(π∗F_{Δ}) = fib_{B}∘(π∗F_{Δ′}) = π^{B}∗(fib_{B}∘F_{Δ′}) for every B ∈ Ob(ℬ)

and clearly fib_{B} ∘F_{Δ} and fib_{B} ∘F_{Δ′} are the constant modifications F_{t_{B}} ⇝ F_{t′_{B}} associated with Δ and Δ′. By the universality of π^{B}, it follows that fib_{B} ∘ F_{Δ} = fib_{B} ∘ F_{Δ′} for every B ∈ Ob(ℬ), whence Δ = Δ′, as required.

Lastly, let Δ : π ⊙ F_{t} ⇝ π ⊙ F_{t′} be any modification; for every B ∈ Ob(ℬ) we obtain the modification fib_{B} ∘ Δ : π^{B} ⊙ F_{t_{B}} ⇝ π^{B} ⊙ F_{t′_{B}}, and since π^{B} is universal, there exists a unique natural transformation δ^{B} : t_{B} ⇒ t′_{B} such that fib_{B} ∘ Δ = π^{B} ∗ F_{δ^{B}}. It remains to check that the rule : B ↦ δ^{B} yields a modification δ : t ⇝ t′, since in this case we get Δ = π ∗ F_{δ}, which will prove that (3.3.32) is also full, thus concluding the proof of the theorem. However, Δ is the datum of a modification Δ_{i} : π_{i} ⊙ t ⇝ π_{i} ⊙ t′ for every i ∈ Ob(I); the compatibility condition for Δ_{i} asserts that the following two compositions of oriented squares coincide for every morphism g : B′ → B of ℬ :

[figure: X(B); t_{B}; δ^{B}; L(B); π_{i}^{B}; 1_{π^{B}_{i}}; F i(B); X(B); t_{B}; X(g); τ_{g}^{t}; L(B); L(g); π_{i}^{B}; τ_{g}^{π_{i}}; F i(B); F i(g); X(B); t′_{B}; X(g); τ_{g}^{t′}; L(B); π_{i}^{B}; L(g); τ_{g}^{π_{i}}; F i(B); F i(g); X(B′); t_{B′}; δ^{B′}; L(B′); π_{i}^{B′}; 1_{π{B′ \atop i}}; F i(B′); X(B′); t′_{B′}; L(B′); π_{i}^{B′}; F i(B′); X(B′); t′_{B′}; L(B′); π_{i}^{B′}; F i(B′)]

where τ_{g}^{t}, τ_{g}^{t′} and τ_{g}^{π_{i}} are the coherent constraints of respectively t, t′ and π_{i}. We deduce that

π_{i}^{B′} ∗ (τ_{g}^{t′} ⊙ (L(g) ∗ δ^{B})) = π_{i}^{B′} ∗ ((δ^{B′} ∗ X(g)) ⊙ τ_{g}^{t}) for every i ∈ Ob(I)

and invoking again the universality of π^{B′} we conclude that τ_{g}^{t′}⊙(L(g)∗δ^{B}) = (δ^{B′}∗X(g))⊙τ_{g}^{t}, which is the required compatibility condition for δ. □

3.4. Fibrations in groupoids. Recall that a groupoid is a category all whose morphisms are invertible. To every category ℬ, we may attach the groupoid ℬ^{×} such that Ob(ℬ^{×}) = Ob(ℬ), and whose morphisms are the isomorphisms of ℬ; the composition law for morphisms in ℬ^{×} is the same as that of ℬ, so ℬ^{×} is a subcategory of ℬ. Then clearly every functor G : 𝒜 → ℬ restricts to a functor G^{×} : 𝒜^{×} → ℬ^{×}, and for every universe U, the rules : 𝒞 ↦ 𝒞^{×} and (F : 𝒜 → ℬ) ↦ F^{×} yield a right adjoint (−)^{×} : U-Cat → U-Gpd for the inclusion functor

U-Gpd → U-Cat

of the full subcategory U-Gpd of U-Cat whose objects are the U-small groupoids.

Remark 3.4.1. (i) Notice that every isomorphism of functors α : G →^{∼} H restricts to an isomorphism of functors α^{×} : G^{×} →^{∼} H^{×}.

(ii) Let 𝒜 and ℬ be any two categories, and G : 𝒜 → ℬ any functor. Then it is easily seen that G is essentially surjective if and only if the same holds for G^{×}. Moreover, if G is i-faithful for some i ∈ {0, 1, 2}, then the same holds for G^{×}. The verification in case i = 0 shall be left to the reader. Next, let f : GX → GY be a morphism in ℬ^{×}; if G is fully faithful, there exists g : X → Y in 𝒜 such that Gg = f. But f is invertible, so by the same token there exists h : Y → X in 𝒜 such that Gh = f^{−1}; since G(f ∘ g) = G1_{Y} and G(g ∘ f) = G1_{X}, we deduce f ∘ g = 1_{Y} and g ∘ f = 1_{X}, since G is faithful. This shows that g is a morphism in 𝒜^{×}, and proves the assertion for i = 1. For i = 2, the functor G is an equivalence, so we know already that G^{×} is fully faithful; but we have also already noticed that G^{×} is essentially surjective, whence the assertion for i = 2.

(iii) Since (−)^{×} is a right adjoint, it commutes with all limits of V-Cat (see proposition 1.3.25(iii)). Moreover, by inspecting the construction of example 1.5.10, it is easily seen that (−)^{×} also commutes with all filtered (small) colimits : details left to the reader.

3.4.2. We wish now to upgrade the associated groupoid construction to the 2-category of 𝒞 - fibrations, for any given base category 𝒞. Indeed, let F : 𝒜 → 𝒞 be any fibration; we associate with 𝒜 the category

𝒜^{×}

with Ob(𝒜^{×}) = 𝒜, and whose morphisms are the cartesian morphisms of 𝒜. The composition law for morphisms in 𝒜^{×} is the restriction of that of 𝒜, hence 𝒜^{×} is a subcategory of 𝒜, and we denote by F^{×} : 𝒜^{×} → 𝒞 the restriction of F. Then it is easily seen that F^{×} is also a fibration, and all morphisms of 𝒜^{×} are cartesian for this fibration; moreover, for every X ∈ Ob(𝒞) the fibre category (F^{×})^{−1}X is a groupoid. The latter assertion can be easily checked directly, and it follows also straightforwardly from lemma 3.1.20(i). We call F^{×} the fibration in groupoids associated with F. We say that F is a fibration in groupoids if 𝒜 = 𝒜^{×}. Clearly every cartesian functor G : 𝒜 → ℬ of 𝒞 -fibrations restricts to a (cartesian) functor

G^{×} : 𝒜^{×} → ℬ^{×}.

On the other hand, a given natural 𝒞 -transformation β : G ⇒ H between 𝒞 -cartesian functors G, H : 𝒜 → ℬ induces a natural 𝒞 -transformation β^{×} : G^{×} ⇒ H^{×} if and only if β is an isomorphism of functors, i.e. if and only if β is a morphism of the groupoid Cart_{𝒞} (𝒜 , ℬ)^{×} associated with Cart_{𝒞} (𝒜 , ℬ) as in (3.4). Thus, we get a natural functor

(3.4.3)  Cart_{𝒞} (𝒜 , ℬ)^{×} → Cart_{𝒞} (𝒜^{×}, ℬ^{×})  G ↦ G^{×}  (β : G ⇒ H) ↦ β^{×}

3.4.4. Let U, V be universes such that U ⊂ V, and 𝒞 a V-small category; we denote by

(U, V)-Fib^{×}(𝒞) and (U, V)-Gpd(𝒞)

respectively : the sub-2-category of (U, V)-Fib(𝒞) whose objects are the same as those of (U, V)-Fib(𝒞), and whose Hom-category is Cart_{𝒞} (𝒜 , ℬ)^{×} for every pair (𝒜 , ℬ) of such 𝒞 - fibrations, and : the strong sub-2-category of (U, V)-Fib(𝒞) whose objects are the fibrations in groupoids. With this notation, clearly we get as well a strict pseudo-functor

(−)_{𝒞}^{×} : (U, V)-Fib^{×}(𝒞) → (U, V)-Gpd(𝒞) 𝒜 ↦ 𝒜^{×}

which is given on Hom-categories by the foregoing system of functors (3.4.3). As usual, we may omit mentioning the universes in the above notation, if no ambiguities are likely to arise.

3.4.5. Moreover, every functor u : 𝒞 → 𝒞′ induces strict pseudo-functors

(U, V)-Fib^{×}(u)^{∗} : Fib^{×}(𝒞′) → Fib^{×}(𝒞) (U, V)-Gpd(u)^{∗} : Gpd(𝒞′) → Gpd(𝒞)

defined as the restrictions of Fib(u)^{∗} : Fib(𝒞′) → Fib(𝒞). As usual, these functors will be sometimes just denoted Fib^{×}(u)^{∗} and respectively Gpd(u)^{∗}. By simple inspection we get a commutative diagram of 2-categories :

\xymatrix{Fib^{×}(𝒞′) \ar[r]^{Fib^{×}(u)^{∗}} \ar[d]_{(−)_{𝒞′}^{×}} & Fib^{×}(𝒞) \ar[d]^{(−)_{𝒞}^{×}} \\ Gpd(𝒞′) \ar[r]^{Gpd(u)^{∗}} & Gpd(𝒞).}

3.4.6. Let u : 𝒞 → 𝒞′ be a functor from a U-small category 𝒞 to a V-small category 𝒞′ with U-small Hom-sets. Recall that the source functor s_{X} : u𝒞 /X → 𝒞 is a fibration in groupoids for every X ∈ Ob(𝒞′) (example 3.1.3(i)); taking into account remark 3.3.15(ii) we deduce for every 𝒞 -fibration 𝒜 with V-small fibres and every such X, natural equivalences of categories :

(Fib(u)_{∗}𝒜 (X))^{×} →^{∼} Cart_{𝒞} (u𝒞 /X, 𝒜)^{×} →^{∼} Cart_{𝒞} (u𝒞 /X, 𝒜^{×}) →^{∼} (Fib(u)_{∗}(𝒜^{×}))(X).

There follows a pseudo-commutative diagram of 2-categories :

\xymatrix{Fib^{×}(𝒞) \ar[r]^{(U,V)-Fib^{×}(u)_{∗}} \ar[d]_{(−)_{𝒞}^{×}} & Fib^{×}(𝒞′) \ar[d]^{(−)_{𝒞′}^{×}} \\ Gpd(𝒞) \ar[r]^{(U,V)-Gpd(u)_{∗}} & Gpd(𝒞′)}

where (U, V)-Fib^{×}(u)_{∗} and (U, V)-Gpd(u)_{∗} are the restrictions of (U, V)-Fib(u)_{∗}. Suppose moreover that the category X/u𝒞 is cofiltered for every X ∈ Ob(𝒞′); then, in view of remarks 3.4.1(iii) and 5.5.14(iii), for every such X we have as well a natural equivalence of categories :

(Fib(u)_{!}𝒜 (X))^{×} →^{∼} colim_{X→uY} 𝒜 (Y)^{×} →^{∼} Fib(u)_{!}(𝒜^{×})(X)

where the colimit ranges over the small filtered category (X/u𝒞)^{o}. We get therefore a pseudocommutative diagram of 2-categories :

(3.4.7)  \xymatrix{Fib^{×}(𝒞) \ar[r]^{(U,V)-Fib^{×}(u)_{!}} \ar[d]_{(−)_{𝒞}^{×}} & Fib^{×}(𝒞′) \ar[d]^{(−)_{𝒞′}^{×}} \\ Gpd(𝒞) \ar[r]^{(U,V)-Gpd(u)_{!}} & Gpd(𝒞)}

where (U, V)-Fib^{×}(u)_{!} and (U, V)-Gpd(u)_{!} denote the restrictions of (U, V)-Fib(u)_{!}.

Remark 3.4.8. Let 𝒞 , U and V be as in (3.4.4).

(i) It is easily seen that the pseudo-functor (−)_{𝒞}^{×} is a strong right 2-adjoint for the inclusion pseudo-functor Gpd(𝒞) → Fib^{×}(𝒞), and the counit of the resulting 2-adjunction assigns to every fibration 𝒜 → 𝒞 the inclusion functor 𝒜^{×} → 𝒜 : details left to the reader.

(ii) Notice that a fibration π : 𝒜 → 𝒞 is a fibration in groupoids if and only if π^{−1}X is a groupoid for every X ∈ Ob(𝒞). Indeed, the condition is obviously necessary. Conversely, recall that every morphism f of 𝒜 is the composition of a cartesian morphism and a morphism g such that π(g) = 1_{X} for some X ∈ Ob(𝒞); but if π^{−1}X is a groupoid, then g is an isomorphism in 𝒜, in particular it is cartesian, so the same holds for f, which shows that 𝒜 = 𝒜^{×}.

(iii) By the same token, to every pseudo-functor F : 𝒞^{o} → Cat we attach the pseudo-functor

F^{×} : 𝒞^{o} → Cat X ↦ (F X)^{×}

(whose coherence constraints are the same as those of F). Then we easily see that :

ℱ ib(F)^{×} = ℱ ib(F^{×}).

3.4.9. With the notation of (3.4.4), notice that for every presheaf F ∈ Ob(𝒞_{U}^{∧}), the fibration ℱ ib_{𝒞} (F) is a fibration in groupoid, so we get a well defined strict pseudo-functor

ℱ ib_{𝒞} : 𝒞_{U}^{∧} → (U, V)-Gpd(𝒞)

and by restriction, the pseudo-functor π_{0}^{𝒞} (see (3.2.10)) yields a strong left 2-adjoint

π_{0}^{𝒞} : (U, V)-Gpd(𝒞) → 𝒞_{U}^{∧}.

Moreover, for every functor u : 𝒞 → 𝒞′, a simple inspection yields an essentially commutative diagram

\xymatrix{Gpd(𝒞′) \ar[r]^{Gpd(u)^{∗}} \ar[d]_{π_{0}^{𝒞′}} & Gpd(𝒞) \ar[d]^{π_{0}^{𝒞}} \\ 𝒞_{U}^{′∧} \ar[r]^{u_{U}^{∧}} & 𝒞_{U}^{∧}.}

Furthermore, if 𝒞 is U-small and 𝒞′ is V-small with U-small Hom-sets, we also get the essentially commutative diagram :

(3.4.10)  \xymatrix{Gpd(𝒞) \ar[r]^{Gpd(u)_{!}} \ar[d]_{π_{0}^{𝒞}} & Gpd(𝒞′) \ar[d]^{π_{0}^{𝒞′}} \\ 𝒞_{U}^{∧} \ar[r]^{u_{U!}} & 𝒞_{U}^{′∧}.}

Indeed, for every X ∈ Ob(𝒞′) and every fibration 𝒜 with essentially U-small fibres we have natural equivalences of categories :

(π_{0}^{𝒞′} Gpd(u)_{!}𝒜)(X) →^{∼} π_{0}((Fib(u)_{!}𝒜)(X)) →^{∼} π_{0}(2-colim_{X→uY} 𝒜 (Y)) →^{∼} 2-colim_{X→uY} π_{0}(𝒜 (Y))

since π_{0} is a 2-left adjoint pseudo-functor (proposition 2.5.9(ii)). However, remark 2.5.2(vii) says that the 2-colimit of the (strict) pseudo-functor given by the rule (X → uY) ↦ π_{0}(𝒜 (Y)) also represents the colimit of the same functor, in the category of U-small sets; on the other hand, the latter colimit is represented by u_{U!}(π_{0}^{𝒞} 𝒜)(X), whence the contention.

3.5. Sieves and descent theory. This section develops the basics of descent theory, in the general framework of fibred categories.

Definition 3.5.1. Let ℬ and 𝒞 be two categories, F : ℬ → 𝒞 a functor.

(i) A sieve of 𝒞 is a full subcategory 𝒮 of 𝒞 such that the following holds. If A ∈ Ob(𝒮), and B → A is any morphism in 𝒞, then B ∈ Ob(𝒮).

(ii) If S ⊂ Ob(𝒞) is any subset, there is a smallest sieve 𝒮_{S} of 𝒞 such that S ⊂ Ob(𝒮_{S}); we call 𝒮_{S} the sieve generated by S. If S′ ⊂ Ob(𝒞) is another subset and 𝒮_{S′} ⊂ 𝒮_{S}, we say that S′ is a refinement of S.

(iii) If 𝒮 is a sieve of 𝒞, the inverse image of 𝒮 under F is the full subcategory F^{−1}𝒮 of ℬ with Ob(F^{−1}𝒮) = {B ∈ Ob(ℬ) | F B ∈ Ob(𝒮)} (notice that F^{−1}𝒮 is a sieve).

(iv) If f : X → Y is any morphism in 𝒞, and 𝒮 is any sieve of 𝒞/Y, we shall write 𝒮 ×_{Y} f for the inverse image of 𝒮 under the functor f_{∗} (notation of (1.1.25)).

Remark 3.5.2. Let 𝒞 be a category with small Hom-sets and X an object of 𝒞.

(i) Every sieve 𝒯 of 𝒞 yields a subobject F_{𝒯} of the final object 𝟙_{𝒞} of 𝒞^{∧}, by declaring that F_{𝒯} (Y) ≠ ∅ if and only if Y ∈ Ob(𝒯 ); hence F_{𝒯} (Y) is a set with one element for every Y ∈ Ob(𝒯 ), and is empty for Y ∈ Ob(𝒞) \ Ob(𝒯 ). It is easily seen that the rule 𝒯 ↦ F_{𝒯} establishes a bijection between the set of sieves of 𝒞 and the set of subobjects of 𝟙_{𝒞} . The inverse mapping is given by the rule :

(F ⊂ 𝟙_{𝒞} ) ↦ (ℱ ib(F) ⊂ ℱ ib(𝟙_{𝒞} ) = 𝒞).

(ii) For every sieve 𝒮 of the category 𝒞 /X we may then also consider the presheaf on 𝒞

h_{𝒮} := s_{X!}(F_{𝒮} )

where F_{𝒮} is the presheaf on 𝒞 /X defined as in (i), and s_{X!} : (𝒞 /X)^{∧} → 𝒞^{∧} is the left adjoint to the functor s_{X}^{∧} induced by the source functor s_{X} : 𝒞 /X → 𝒞 (proposition 1.4.13(vi.a)). By proposition 1.4.13(vi.b,c), the presheaf h_{𝒮} is a subobject of h_{X} (notation of (1.2.4)), and by inspecting (1.4.11) we see that

h_{𝒮} (Y) = {f ∈ Hom_{𝒞} (Y, X) | (Y, f) ∈ Ob(𝒮)} for every Y ∈ Ob(𝒞).

For a given morphism f : Y′ → Y in 𝒞, the map h_{𝒮} (f) is just the restriction of Hom_{𝒞} (f, X). By the same token, it also follows that the rule 𝒮 ↦ h_{𝒮} sets up a natural bijection between the subobjects of h_{X} in 𝒞^{∧} and the sieves of 𝒞 /X. The inverse mapping is given by the rule :

(F ⊂ h_{X}) ↦ (ℱ ib(F) ⊂ ℱ ib(h_{X}) = 𝒞 /X)

(see example 3.1.16(i)). Especially, the restriction 𝒮 → 𝒞 of the source functor s_{X} : 𝒞 /X → 𝒞 is a fibration, and example 3.1.16(i) generalizes to a natural isomorphism of 𝒞 -fibrations :

ℱ ib(h_{𝒮} ) →^{∼} 𝒮 .

Combining with lemma 1.4.8, we also deduce a natural isomorphism in 𝒞^{∧} :

(3.5.3)  colim_{𝒮}h_{𝒞} ∘ s_{𝒮} →^{∼} h_{𝒮}

where s_{𝒮} : 𝒮 → 𝒞 is the restriction of the source functor s_{X} of (1.1.24).

(iii) Let S := {X_{i} → X | i ∈ I} be any family of morphisms of 𝒞. Then S generates a given sieve 𝒮 of 𝒞 /X if and only if :

h_{𝒮} = ⋃_{i∈I} Im(h_{X_{i}} → h_{X}).

(Notice that the above union is well defined even in case I is not small.)

(iv) Moreover, if S as in (iii) generates 𝒮 and f : Y → X is any morphism such that the fibre product Y_{i} := Y ×_{X} X_{i} is representable in 𝒞 for every i ∈ I, then 𝒮 ×_{X} f is the sieve generated by the family of induced projections {Y_{i} → Y | i ∈ I} ⊂ Ob(𝒞/Y).

(v) Furthermore, let f : Y → X be any morphism of 𝒞; it is easily seen that the correspondence of (ii) induces a natural identification of subobjects of h_{Y}:

h_{𝒮 ×_{X}f} = h_{𝒮} ×_{h_{X}} h_{Y} for every sieve 𝒮 of 𝒞 /X .

Since the Yoneda embedding is fully faithful, we may sometimes abuse notation, to identify X with the corresponding representable presheaf h_{X}; then we may write h_{𝒮 ×_{X}f} = h_{𝒮} ×_{X} Y. In view of (ii) we deduce an isomorphism of fibred 𝒞 -categories ℱ ib(h_{𝒮} ×_{X} Y) →^{∼} 𝒮 ×_{X} f. On the other hand, let π : h_{𝒮} ×_{X} Y → h_{𝒮} be the natural projection; a direct inspection shows that the foregoing isomorphisms fit into the commutative diagram :

\xymatrix{ℱ ib(h_{𝒮} ×_{X} Y) \ar[r]^{∼} \ar[d]_{ℱ ib(π)} & 𝒮 ×_{X} f \ar[d]^{f_{∗|𝒮}} \\ ℱ ib(h_{𝒮} ) \ar[r]^{∼} & 𝒮}

where the right vertical arrow is the restriction of the functor f_{∗} : 𝒞 /Y → 𝒞 /X (details left to the reader). (Especially, f_{∗|𝒮} is a 𝒞 -cartesian functor, but this assertion is trivial, since every morphism of 𝒞 /X is 𝒞 -cartesian).

3.5.4. Let 𝒞 be a small category, and 𝒮 the sieve of 𝒞 generated by a subset S ⊂ Ob(𝒞). Say that S = {S_{i} | i ∈ I} for a small set I; for every i ∈ I there is a faithful embedding ε_{i} : 𝒞/S_{i} → 𝒮, and for every pair (i, j) ∈ I × I, we define

𝒞/S_{ij} := 𝒞/S_{i} ×_{(ε_{i},ε_{j})} 𝒞/S_{j}

(notation of example 1.2.25(i)). Hence, the objects of 𝒞/S_{ij} are all the triples (X, g_{i}, g_{j}), where X ∈ Ob(𝒞) and g_{l} ∈ Hom_{𝒞} (X, S_{l}) for l = i, j. The natural projections :

π_{ij∗}^{1} : 𝒞/S_{ij} → 𝒞/S_{i} π_{ij∗}^{0} : 𝒞/S_{ij} → 𝒞/S_{j}

are faithful embeddings. We deduce a natural diagram of categories :

∐_{(i,j)∈I×I} [figure: 𝒞/S_{ij}; ∂_{0}; ∂_{1}] ∐_{i∈I} 𝒞/S_{i} ⟶^{ε} 𝒮

where :

∂_{0} := ∐_{(i,j)∈I×I}π_{ij∗}^{0}  ∂_{1} := ∐_{(i,j)∈I×I}π_{ij∗}^{1}  ε := ∐_{i∈I} ε_{i}.

Remark 3.5.5. Notice that, under the current assumptions, the product S_{ij} := S_{i} × S_{j} is not necessarily representable in 𝒞. In case it is, we may consider another category, also denoted 𝒞/S_{ij}, namely the category of S_{ij}-objects of 𝒞 (as in (1.1.24)). The latter is naturally isomorphic to the category with the same name introduced in (3.5.4). Moreover, under this natural isomorphism, the projections π_{ij∗}^{0} and π_{ij∗}^{1} are identified with the functors induced by the natural morphisms π_{ij}^{0} : S_{ij} → S_{j} and respectively π_{ij}^{1} : S_{ij} → S_{i}. Hence, in this case, the notation of (3.5.4) is compatible with (1.1.25).

Lemma 3.5.6. With the notation of (3.5.4), the functor ε induces an isomorphism between 𝒮 and the coequalizer (in the category Cat) of the pair of functors (∂_{0}, ∂_{1}).

Proof. Let 𝒜 be any other object of Cat, and F : ∐_{i∈I} 𝒞/S_{i} → 𝒜 a functor such that F ∘∂_{0} = F ∘ ∂_{1}. We have to show that F factors uniquely through ε. To this aim, we construct explicitly a functor G : 𝒮 → 𝒜 such that G ∘ ε = F. First of all, by the universal property of the coproduct, F is the same as a family of functors (F_{i} : 𝒞/S_{i} → 𝒜 | i ∈ I), and the assumption on F amounts to the system of identities :

(3.5.7)  F_{i} ∘ π_{ij∗}^{1} = F_{j} ∘ π_{ij∗}^{0}  for every i, j ∈ I.

Hence, let X ∈ Ob(𝒮); by assumption there exist i ∈ I and a morphism f : X → S_{i} in 𝒞, so we may set GX := F_{i}f. In case g : X → S_{j} is another morphism in 𝒞, we deduce an object h := (X, f, g) ∈ Ob(𝒞/S_{ij}), so f = π_{ij∗}^{1}h and g = π_{ij∗}^{0}h; then (3.5.7) shows that F_{i}f = F_{j}g, i.e. GX is well-defined.

Next, let φ : X → Y be any morphism in 𝒮; choose i ∈ I and a morphism f_{Y} : Y → S_{i}, and set f_{X} := f_{Y} ∘ φ. We let Gφ := F_{i}(φ : f_{X} → f_{Y}). Arguing as in the foregoing, one verifies easily that Gφ is independent of all the choices, and then it follows easily that G(ψ ∘ φ) = Gψ ∘ Gφ for every other morphism ψ : Y → Z in 𝒮. It is also clear that G1_{X} = 1_{GX}, whence the contention. □

3.5.8. In the situation of (3.5.4), suppose that the set of generators S is the whole of Ob(𝒮); in this case, the augmentation ε can also be used to produce the following 2-categorical presentation of 𝒮, which upgrades the isomorphism (3.5.3). Consider the strict pseudo-functor

G_{𝒮} : 𝒮 → Fib(𝒞) : Y ↦ 𝒞/Y (Z ⟶^{f} Y) ↦ (𝒞 /Z ⟶^{f_{∗}} 𝒞 /Y).

We have a natural strict pseudo-cocone

ε̂: G_{𝒮} ⇒ F_{𝒮}

where F_{𝒮} is the constant pseudo-functor 𝒮 → Fib(𝒞) with value 𝒮, and ε̂_{X} : 𝒞 /X → 𝒮 is the faithful embedding as in (3.5.4), for every X ∈ Ob(𝒮). We may then state :

Lemma 3.5.9. The pseudo-cocone ε̂induces an equivalence of fibrations over 𝒞 :

2-colim_{𝒮}G_{𝒮} →^{∼} 𝒮 .

Proof. By theorem 3.3.25(ii), the 2-colimits are computed fibrewise in the 2-category Fib(𝒞), so we are reduced to checking that ε̂induces an equivalence of categories

2-colim_{𝒮}G_{𝒮 ,X} →^{∼} 𝒮_{X} for every X ∈ Ob(𝒞)

where 𝒮_{X} is the fibre category over X of the fibration 𝒮 → 𝒞, and G_{𝒮 ,X} : 𝒮 → Cat is the strict pseudo-functor that assigns to every Y ∈ Ob(𝒮) the fibre category (𝒞 /Y)_{X} of the fibration 𝒞 /Y → 𝒞 (for every morphism f : Z → Y in 𝒮, the corresponding functor G_{𝒮 ,X} (f) is the restriction of G_{𝒮} (f) = f_{∗}). However, if X ∉ Ob(𝒮), the category 𝒮_{X} is empty, and G_{𝒮 ,X} is the constant pseudo-functor with value equal to the empty category, so the assertion is clear in this case. Suppose then that X ∈ Ob(𝒮), in which case 𝒮_{X} is the category whose unique object is X and whose unique morphism is 1_{X}. Also, for every Y ∈ Ob(𝒮), the category (𝒞 /Y)_{X} is discrete, with set of objects given by Hom_{𝒞} (X, Y). According to example 3.3.13(i), the strong 2-colimit of G_{𝒮 ,X} is represented by the category ℱ ib(G_{𝒮 ,X} )[Σ^{−1}], where Σ is the set of cartesian morphisms of the fibration ℱ ib(G_{𝒮 ,X} ) → 𝒮^{o}. Now, it is easily seen that ℱ ib(G_{𝒮 ,X} ) = (X/𝒮)^{o} = 𝒮^{o}/X^{o}, with structure functor given by the usual source functor 𝒮^{o}/X^{o} → 𝒮^{o}. Hence, Σ is the set of all morphisms of 𝒞^{o}/X^{o}, and since 𝒞^{o}/X^{o} has the final object 1_{X^{o}}, the assertion follows easily from example 1.6.12. □

Definition 3.5.10. Let φ : 𝒜 → ℬ be a fibration, B ∈ Ob(ℬ) an object, 𝒮 ⊂ ℬ/B a sieve, i ∈ {0, 1, 2}, and denote by ι_{𝒮} : 𝒮 → ℬ/B the fully faithful embedding.

(i) We say that 𝒮 is a sieve of φ-i-descent, if the restriction functor :

Cart_{ℬ} (ι_{𝒮} , 𝒜) : 𝒜 (B) → Cart_{ℬ} (𝒮 , 𝒜)

is i-faithful (see remark 1.1.5). (ii) We say that 𝒮 is a sieve of universal φ-i-descent if the sieve 𝒮 ×_{B} f is of φ-i-descent

for every morphism f : B′ → B of ℬ (notation of definition 3.5.1(iv)). (iii) Let f : B′ → B be a morphism in ℬ. We say that f is a morphism of φ-i-descent (resp.

a morphism of universal φ-i-descent), if the sieve generated by {f} is of φ-i-descent (resp. of universal φ-i-descent).

Remark 3.5.11. In the situation of definition 3.5.10, suppose that 𝒮 is the sieve generated by a set of objects {S_{i} → B | i ∈ I} ⊂ Ob(ℬ/B). There follows a natural diagram of categories (notation of (3.5.4)) :

Cart_{ℬ} (𝒮 , 𝒜) ⟶^{ε^{∗}} ∏_{i∈I} [figure: 𝒜 (S_{i}); ∂_{0}^{∗}; ∂_{1}^{∗}] ∏_{(i,j)∈I×I}Cart_{ℬ} (ℬ/S_{ij}, 𝒜)

where ε^{∗} := Cart_{ℬ} (ε, 𝒜) and ∂_{i}^{∗} := Cart_{ℬ} (∂_{i}, 𝒜), for i = 0, 1. With this notation, lemma 3.5.6 easily implies that ε^{∗} induces an isomorphism between Cart_{ℬ} (𝒮 , 𝒜) and the equalizer (in the category Cat) of the pair of functors (∂_{0}^{∗}, ∂_{1}^{∗}).

3.5.12. We would like to exploit the presentation of Cart_{ℬ} (𝒮 , 𝒜) in remark 3.5.11, in order to translate definition 3.5.10 in terms of the fibre categories 𝒜_{S_{i}} and 𝒜_{S_{ij}}. The problem is that such a translation must be carried out via a pseudo-natural equivalence (namely ev), and such equivalences do not respect a presentation as above in terms of equalizers in the category Cat. What we need is to upgrade our presentation of 𝒮 to a new one, which is preserved by pseudonatural transformations. This is achieved as follows. Resume the general situation of (3.5.4). For every i, j, k ∈ I, set 𝒞 /S_{ijk} := 𝒞 /S_{ij} ×_{𝒞} 𝒞 /S_{k}. We have a natural diagram of categories :

(3.5.13)  ∐_{(i,j,k)∈I^{3}} [figure: 𝒞 /S_{ijk}; ∂′_{0}; ∂′_{2}; ∂′_{1}] ∐_{(i,j)∈I^{2}}[figure: 𝒞 /S_{ij}; ∂_{0}; ∂_{1}] ∐_{i∈I} 𝒞 /S_{i} ⟶^{ε} 𝒮

where ∂′_{0} is the coproduct of the natural projections π_{ijk∗}^{0} : 𝒞 /S_{ijk} → 𝒞 /S_{jk} for every i, j, k ∈ I, and likewise ∂′_{1} (resp. ∂′_{2}) is the coproduct of the projections π_{ijk∗}^{1} : 𝒞 /S_{ijk} → 𝒞 /S_{ik} (resp. π_{ijk∗}^{2} : 𝒞 /S_{ijk} → 𝒞 /S_{ij}). We can view (3.5.13) as an augmented 2-truncated semi-simplicial object in Fib(𝒞), i.e. a functor :

F_{𝒮} : Σ_{2}^{+o} → Fib(𝒞)

from the opposite of the category Σ_{2}^{+} whose objects are the ordered sets [−1], [0], [1] and [2], and whose morphisms are the non-decreasing injective maps (this is a subcategory of the category Δ_{2}^{∧} of definition 7.4.1(iii)).

Remark 3.5.14. Suppose that finite products are representable in ℬ, and for every i, j, k ∈ I, set S_{ij} := S_{i} × S_{j}, and S_{ijk} := S_{ij} × S_{k}. Just as in remark 3.5.5, the category 𝒞 /S_{ijk} of S_{ijk}- objects of 𝒞 is naturally isomorphic to the category with the same name introduced in (3.5.12), and under this isomorphism, the functors π_{ijk∗}^{0} are identified with the functors arising from the natural projections π_{ijk}^{0} : S_{ijk} → S_{jk} (and likewise for π_{ijk∗}^{1} and π_{ijk∗}^{2}).

With this notation, denote by Σ_{2} the full subcategory of Σ_{2}^{+} whose objects are the non-empty sets; we have the following 2-category analogue of lemma 3.5.6:

Proposition 3.5.15. The augmentation ε induces an equivalence of categories :

2-colim_{Σ_{2}^{o}} F_{𝒮} →^{∼} 𝒮 .

Proof. By theorem 3.3.25(ii), the 2-colimits are computed fibrewise in the 2-category Fib(𝒞), so we are reduced to checking that ε induces an equivalence of categories

2-colim_{Σ_{2}^{o}} F_{𝒮 ,X} →^{∼} 𝒮_{X} for every X ∈ Ob(𝒞)

where 𝒮_{X} is the fibre category over X of the fibration 𝒮 → 𝒞, and F_{𝒮 ,X} := fib_{X} ∘ F_{𝒮} , where fib_{X} : Fib(𝒞) → Cat is defined as in (3.3.24). Now, if X ∉ Ob(𝒮), the category 𝒮_{X} is empty, and fib_{X} ∘ F_{𝒮} is the constant pseudo-functor with value equal to the empty category, so the assertion trivially holds in this case. Suppose then that X ∈ Ob(𝒮), in which case 𝒮_{X} is the category whose unique object is X and whose unique morphism is 1_{X}. Set

T := ∐_{i∈I} Hom_{𝒞} (X, S_{i})

and notice that F_{𝒮 ,J} [0], F_{𝒮 ,J} [1] and F_{𝒮 ,J} [2] are the discrete categories whose sets of objects are respectively T, T × T and T × T × T. Moreover, the strict pseudo-functor F_{𝒮 ,X} corresponds to the 2-truncated semi-simplicial diagram of sets :

(3.5.16)  \xymatrix{T × T × T \ar[r]^{∂′_{0}} \ar[r]_{∂′_{2}} \ar[r]|{∂′_{1}} & T × T \ar[r]^{∂_{0}} \ar[r]_{∂_{1}} & T}

whose maps ∂_{i} and ∂′_{i} are the natural projections. To conclude, it therefore suffices to show :

Claim 3.5.17. For every set T ≠ ∅, the 2-colimit in the 2-category Cat of the system of discrete categories (3.5.16) is represented by the discrete category with one object.

Proof of the claim. The diagram (3.5.16) can be regarded as a presheaf T_{•} on the category Σ_{2}, and according to example 3.3.13(i), the strong 2-colimit of T_{•} is represented by the category 𝒯 := ℱ ib(T_{•})[Λ^{−1}], where Λ is the set of cartesian morphisms of the fibration ℱ ib(T_{•}) → Σ_{2}; but every morphism of ℱ ib(T_{•}) is cartesian (see (3.1.15)). Denote by 𝒯_{0} the category with Ob(𝒯_{0}) = T, and such that Hom_{𝒯_{0}} (t, t′) contains a unique element τ_{t,t′}, for every t, t′ ∈ T. We define a functor q : 𝒯_{0} → 𝒯 as follows. Recall that the objects of ℱ ib(T_{•}) are the pairs (j, \underline{t}), where j ∈ \{0, 1, 2\} and \underline{t} ∈ T^{j+1}. For every \underline{t} := (t_{0}, t_{1}) ∈ T × T we have morphisms

(0, t_{1}) ⟵^{δ_{0,\underline{t}}} (1, \underline{t}) ⟶^{δ_{1,\underline{t}}} (0, t_{0}) in ℱ ib(T_{•}), and we set

q(t_{0}) := (0, t_{0}) and q(τ_{t_{0},t_{1}}) := [δ_{0,(t_{0},t_{1})}] ∘ [δ_{1,(t_{0},t_{1})}]^{−1} for every t_{0}, t_{1} ∈ T

where we denote by [f] the image in 𝒯 of every morphism f of ℱ ib(T_{•}). Indeed, for every \underline{t} := (t_{0}, t_{1}, t_{2}) ∈ T^{3} and j = 0, 1, 2, we have as well morphisms δ′_{j,\underline{t}} : (2, \underline{t}) → (1, ∂′_{j}(\underline{t})) in ℱ ib(T_{•}), and we may compute :

q(τ_{t_{1},t_{2}}) ∘ q(τ_{t_{0},t_{1}}) = [δ_{0,∂′_{0}\underline{t}}] ∘ [δ_{1,∂′_{0}\underline{t}}]^{−1} ∘ [δ_{0,∂′_{2}\underline{t}}] ∘ [δ_{1,∂′_{2}\underline{t}}]^{−1}

= [δ_{0,∂′_{0}\underline{t}}] ∘ [δ′_{0,\underline{t}}] ∘ [δ′_{0,\underline{t}}]^{−1} ∘ [δ_{1,∂′_{0}\underline{t}}]^{−1} ∘ [δ_{0,∂′_{2}\underline{t}}] ∘ [δ′_{2,\underline{t}}] ∘ [δ′_{2,\underline{t}}]^{−1} ∘ [δ_{1,∂′_{2}\underline{t}}]^{−1}

= [δ_{0,∂′_{0}\underline{t}}] ∘ [δ′_{0,\underline{t}}] ∘ [δ′_{0,\underline{t}}]^{−1} ∘ [δ_{1,∂′_{0}\underline{t}}]^{−1} ∘ [δ_{1,∂′_{0}\underline{t}}] ∘ [δ′_{0,\underline{t}}] ∘ [δ′_{1,\underline{t}}]^{−1} ∘ [δ_{1,∂′_{1}\underline{t}}]^{−1}

= [δ_{0,∂′_{0}\underline{t}}] ∘ [δ′_{0,\underline{t}}] ∘ [δ′_{1,\underline{t}}]^{−1} ∘ [δ_{1,∂′_{1}\underline{t}}]^{−1}
= [δ_{0,∂′_{1}\underline{t}}] ∘ [δ′_{1,\underline{t}}] ∘ [δ′_{1,\underline{t}}]^{−1} ∘ [δ_{1,∂′_{1}\underline{t}}]^{−1}
= q(τ_{t_{0},t_{2}})

as required. Next, let also q′ : 𝒯 → 𝒯_{0} be the unique functor given by the rules :

(0, t_{0}) ↦ t_{0}  (1, (t_{0}, t_{1})) ↦ t_{0}  (2, (t_{0}, t_{1}, t_{2})) ↦ t_{0}  for every t_{0}, t_{1}, t_{2} ∈ T.

Obviously q′ ∘ q = 1_{𝒯_{0}} . Let L : ℱ ib(T_{•}) → 𝒯 be the localization; to conclude, it suffices to exhibit an isomorphism of functors 1_{𝒯} →^{∼} q ∘ q′, and corollary 1.6.11 further reduces to exhibiting an isomorphism ω : L →^{∼} q ∘ q′ ∘ L. We define ω by the rules :

(0, t_{0}) ↦ 1_{(0,t_{0})}  (1, (t_{0}, t_{1})) ↦ [δ_{1,(t_{0},t_{1})}]  (2, (t_{0}, t_{1}, t_{2})) ↦ [δ_{1,(t_{0},t_{2})} ∘ δ′_{2,(t_{0},t_{1},t_{2})}]

for every t_{0}, t_{1}, t_{2} ∈ T. The naturality of ω follows by a straightforward verification. □

3.5.18. Resume the situation of remark 3.5.11, and notice that all the categories appearing in (3.5.13) are fibred over ℬ : indeed, every morphism in each of these categories is cartesian, hence all the functors appearing in (3.5.13) are cartesian. Let us consider now the functor :

Cart_{ℬ} (−, 𝒜) : (Cat/ℬ)^{o} → Cat

that assigns to every ℬ-category 𝒞 the category Cart_{ℬ} (𝒞 , 𝒜). With the notation of remark 3.5.11, we deduce a functor :

Cart_{ℬ} (F_{𝒮} , 𝒜) : Σ_{2}^{+} → Cat

and in light of the foregoing observations, proposition 3.5.15 easily implies that Cart_{ℬ} (ε, 𝒜) induces an equivalence of categories :

(3.5.19)  2-lim_{Σ_{2}} Cart_{ℬ} (F_{𝒮} , 𝒜) →^{∼} Cart_{ℬ} (𝒮 , 𝒜).

Next, suppose that the fibre products S_{ij} := S_{i} ×S_{j} and S_{ijk} := S_{ij} ×S_{k} are representable in ℬ (see remark 3.5.14); in this case, we may compose with the pseudo-equivalence ev_{|•}^{𝒜} of remark 3.2.9(i) : combining with lemma 2.5.3 we finally obtain an equivalence between the category Cart_{ℬ} (𝒮 , 𝒜), and the 2-limit of the pseudo-functor d := ev ∘ Cart_{ℬ} (F_{𝒮} , 𝒜) : Σ_{2} → Cat :

∏_{i∈I} [figure: 𝒜_{S_{i}}; ∂^{0}; ∂^{1}] ∏_{(i,j)∈I^{2}}[figure: 𝒜_{S_{ij}}; ∂^{0}; ∂^{2}; ∂^{1}] ∏_{(i,j,k)∈I^{3}}𝒜_{S_{ijk}}.

Of course, the coface operators ∂^{s} on ∏_{i∈I} 𝒜_{S_{i}} decompose as products of pull-back functors:

π_{ij}^{0∗} : 𝒜_{S_{j}} → 𝒜_{S_{ij}} π_{ij}^{1∗} : 𝒜_{S_{i}} → 𝒜_{S_{ij}}

attached – via the chosen cleavage c of φ – to the projections π_{ij}^{0} : S_{ij} → S_{j} and π_{ij}^{1} : S_{ij} → S_{i} (and likewise for the components π_{ijk}^{t∗} of the other coface operators).

3.5.20. By inspecting the proof of theorem 3.3.9, we may give the following explicit description of this 2-limit. Namely, it is the category whose objects are the data

\underline{X} := (X_{i}, X_{ij}, X_{ijk}, ξ_{i}^{u}, ξ_{ij}^{s}, ξ_{ijk}^{t} | i, j, k ∈ I; s ∈ \{0, 1\}; u, t ∈ \{0, 1, 2\})

where :

X_{i} ∈ Ob(𝒜_{S_{i}}) X_{ij} ∈ Ob(𝒜_{S_{ij}}) X_{ijk} ∈ Ob(𝒜_{S_{ijk}}) for every i, j, k ∈ I

and for every i, j, k ∈ I :

ξ_{i}^{0} : (π_{ik}^{1}π_{ijk}^{1})^{∗}X_{i} →^{∼} X_{ijk}  ξ_{ij}^{0} : π_{ij}^{0∗}X_{j} →^{∼} X_{ij}  ξ_{ijk}^{0} : π_{ijk}^{0∗}X_{jk} →^{∼} X_{ijk}
ξ_{j}^{1} : (π_{jk}^{1}π_{ijk}^{0})^{∗}X_{j} →^{∼} X_{ijk}  ξ_{ij}^{1} : π_{ij}^{1∗}X_{i} →^{∼} X_{ij}  ξ_{ijk}^{1} : π_{ijk}^{1∗}X_{ik} →^{∼} X_{ijk}
ξ_{k}^{2} : (π_{jk}^{0}π_{ijk}^{0})^{∗}X_{k} →^{∼} X_{ijk}  ξ_{ijk}^{2} : π_{ijk}^{2∗}X_{ij} →^{∼} X_{ijk}

are isomorphisms related by the cosimplicial identities :

\begin{aligned}ξ_{ijk}^{0} ∘ π_{ijk}^{0∗}ξ_{jk}^{0} & = ξ_{k}^{2} ∘ γ_{X}^{00} & ξ_{ijk}^{1} ∘ π_{ijk}^{1∗}ξ_{ik}^{0} & = ξ_{k}^{2} ∘ γ_{X}^{01} \\ ξ_{ijk}^{2} ∘ π_{ijk}^{2∗}ξ_{ij}^{0} & = ξ_{j}^{1} ∘ γ_{X}^{02} & ξ_{ijk}^{0} ∘ π_{ijk}^{0∗}ξ_{jk}^{1} & = ξ_{j}^{1} ∘ γ_{X}^{10} \\ ξ_{ijk}^{2} ∘ π_{ijk}^{2∗}ξ_{ij}^{1} & = ξ_{i}^{0} ∘ γ_{X}^{12} & ξ_{ijk}^{1} ∘ π_{ijk}^{1∗}ξ_{ik}^{1} & = ξ_{i}^{0} ∘ γ_{X}^{11} \end{aligned}

where γ^{st} := γ_{d(∂^{s}),d(∂^{t})} : d(∂^{s}) ∘ d(∂^{t}) ⇒ d(∂^{s} ∘ ∂^{t}) denotes the coherence constraint of the cleavage c, for any pair of arrows (∂^{s}, ∂^{t}) in the category Σ_{2}. The morphisms \underline{X} → \underline{Y} in this category are the systems of morphisms :

(X_{i} → Y_{i}, X_{ij} → Y_{ij}, X_{ijk} → Y_{ijk} | i, j, k ∈ I)

that are compatible in the obvious way with the various isomorphisms. However, one may argue as in the proof of proposition 3.5.15, to replace this category by an equivalent one which admits a handier description : given a datum \underline{X}, one can make up an isomorphic datum \underline{X}^{∗} := (X_{i}, X_{ij}^{∗}, X_{ijk}^{∗}, η_{i}, η_{ij}, η_{ijk}), by the rule :

X_{ij}^{∗} := π_{ij}^{1∗}X_{i} X_{ijk}^{∗} := (π_{ik}^{1} ∘ π_{ijk}^{1})^{∗}X_{i}
η_{i}^{0} := 1  η_{ij}^{1} := 1  η_{ijk}^{0} := η_{j}^{1} ∘ γ_{X}^{10}
η_{j}^{1} := (ξ_{i}^{0})^{−1} ∘ ξ_{j}^{1}  η_{ij}^{0} := (ξ_{ij}^{1})^{−1} ∘ ξ_{ij}^{0}  η_{ijk}^{1} := γ_{X}^{11}
η_{k}^{2} := (ξ_{i}^{0})^{−1} ∘ ξ_{k}^{2}  η_{ijk}^{2} := γ_{X}^{12}.

The cosimplicial identities for this new object are subsumed into a single cocycle identity for ω_{ij} := η_{ij}^{0}. Summing up, we arrive at the following description of our 2-limit :

• The objects are all the systems \underline{X} := (X_{i}, ω_{ij}^{X} | i, j ∈ I) where X_{i} ∈ Ob(𝒜_{S_{i}}) for

every i ∈ I, and

ω_{ij}^{X} : π_{ij}^{0∗}X_{j} →^{∼} π_{ij}^{1∗}X_{i}

is an isomorphism in 𝒜_{S_{ij}}, for every i, j ∈ I, fulfilling the cocycle identity :

p_{ijk}^{2∗}ω_{ij}^{X} ∘ p_{ijk}^{0∗}ω_{jk}^{X} = p_{ijk}^{1∗}ω_{ik}^{X} for every i, j, k ∈ I

where, for every i, j, k ∈ I, and t = 0, 1, 2 we have set :

p_{ijk}^{t∗}ω_{••}^{X} := γ_{X}^{1t} ∘ π_{ijk}^{t∗}ω_{••}^{X} ∘ (γ_{X}^{0t})^{−1}.

• The morphisms \underline{X} → \underline{Y} are the systems of morphisms (f_{i} : X_{i} → Y_{i} | i ∈ I) with :

(3.5.21)  ω_{ij}^{Y} ∘ π_{ij}^{0∗}f_{j} = π_{ij}^{1∗}f_{i} ∘ ω_{ij}^{X}  for every i, j ∈ I.

3.5.22. We shall call any pair (X_{•}, ω_{•}) of the above form, a descent datum for the fibration φ, relative to the family \underline{S} := (π_{i} : S_{i} → B | i ∈ I) and the cleavage c. The category of such descent data shall be denoted:

Desc(φ, \underline{S}, c).

Sometimes we may also denote it by Desc(𝒜 , \underline{S}, c), if the notation is not ambiguous. Of course, two different choices of cleavage lead to equivalent categories of descent data, so usually we omit mentioning explicitly c, and write simply Desc(φ, \underline{S}) or Desc(𝒜 , \underline{S}). The foregoing discussion can be summarized, by saying that there is a commutative diagram of categories :

(3.5.23)  \xymatrix{𝒜 (B) \ar[r]^{Cart_{ℬ} (ι_{𝒮} ,𝒜)} \ar[d]_{_{ev_{B}}} & Cart_{ℬ} (𝒮 , 𝒜) \ar[d]^{δ\underline{S}} \\ 𝒜_{B} \ar[r]^{ρ\underline{S}} & Desc(φ, \underline{S}, c)}

whose vertical arrows are equivalences, and where ρ\underline{S} is determined by c. Explicitly, ρ\underline{S} assigns to every C ∈ Ob(𝒜_{B}) the pair (C_{•}, ω_{•}^{C}) where C_{i} := π_{i}^{∗}C, and ω_{ij}^{C} is the composition :

π_{ij}^{0∗} ∘ π_{j}^{∗}C ⟶_{∼}^{γ_{(π{0 \atop ij} ,π_{j})}} (π_{j} ∘ π_{ij}^{0})^{∗}C = (π_{i} ∘ π_{ij}^{1})^{∗}C ⟶_{∼}^{γ_{(π{1 \atop ij} ,π_{i})}^{−1}} π_{ij}^{1∗} ∘ π_{i}^{∗}C

where γ_{(π_{ij}^{1},π_{i})} and γ_{(π_{ij}^{0},π_{j})} are the coherence constraints for the cleavage c (see (3.1.6)). The descent datum (X_{•}, ω_{•}) is said to be effective, if it lies in the essential image of ρ\underline{S}.

We also have an obvious functor :

p\underline{S} : Desc(φ, \underline{S}) → ∏_{i∈I} 𝒜_{S_{i}} (X_{i}, ω_{ij} | i, j ∈ I) ↦ (X_{i} | i ∈ I)

such that :

p\underline{S} ∘ ρ\underline{S} = ∏_{i∈I} π_{i}^{∗} : 𝒜_{B} → ∏_{i∈I} 𝒜_{S_{i}}.

3.5.24. Furthermore, for every morphism f : B′ → B in ℬ, set

\underline{S} ×_{B} f := (π_{i} ×_{B} B′ : S_{i} ×_{B} B′ → B′ | i ∈ I)

which is a generating family for the sieve 𝒮 ×_{B} f (notation of (1.1.24)); then we deduce a pseudo-natural transformation of pseudo-functors (ℬ/B)^{o} → Cat :

ρ : c ∘ i_{B}^{o} ⇒ Desc(φ, \underline{S} ×_{B} −, c) (f : B′ → B) ↦ ρ_{\underline{S}×_{B}f}

(where s_{B} : ℬ/B → ℬ is the source functor of (1.1.24)) fitting into a commutative diagram :

\xymatrix{𝒜 (−) ∘ s_{B}^{o} \ar@{=>}[r]^{Cart_{ℬ} (ι_{𝒮 ×_{B} −} ,𝒜)} \ar@{=>}[d]_{ev∗s_{B}^{o}} & Cart_{ℬ} (𝒮 ×_{B} −, 𝒜) \ar@{=>}[d]^{δ_{\underline{S}×_{B} −}} \\ c ∘ s_{B}^{o} \ar@{=>}[r]^{ρ} & Desc(φ, \underline{S} ×_{B} −, c)}

using which, one can figure out the pseudo-functoriality of the rule : f ↦ Desc(φ, \underline{S} ×_{B} f, c). Namely, every pair of objects f : C → B and f′ : C′ → B, and any morphism h : C′ → C in ℬ/B, yield a commutative diagram :

\xymatrix{S_{j} ×_{B} C′ \ar[d]_{h_{j}:=S_{j}×_{B}h} & S_{ij} ×_{B} C′ \ar[l]_{π_{ij}^{0}} \ar[r]^{π_{ij}^{1}} \ar[d]_{h_{ij}:=S_{ij}×_{B}h} & S_{i} ×_{B} C′ \ar[r]^{π̃_{i}} \ar[d]^{h_{i}:=S_{i}×_{B}h} & C′ \ar[d]^{h} \\ S_{j} ×_{B} C & S_{ij} ×_{B} C \ar[l]^{π_{ij}^{0}} \ar[r]_{π_{ij}^{1}} & S_{i} ×_{B} C \ar[r]_{π_{i}} & C.}

Hence one obtains a functor :

Desc(φ, h, c) : Desc(φ, \underline{S} ×_{B} f, c) → Desc(φ, \underline{S} ×_{B} f′, c)

by the rule :

(X_{i}, ω_{ij}^{X} | i, j ∈ I) ↦ (h_{i}^{∗}X_{i}, ω̃_{ij}^{X} | i, j ∈ I)

where ω̃_{ij}^{X} is the isomorphism that makes commute the diagram :

\xymatrix{h_{ij}^{∗}π_{ij}^{0∗}X_{j} \ar[r]^{γ_{(h_{ij} ,π{0 \atop ij})}} \ar[d]_{h_{ij}^{∗}ω_{ij}} & (π_{ij}^{0} ∘ h_{ij})^{∗}X_{j} & π̃_{ij}^{0∗} ∘ h_{j}^{∗}X_{j} \ar[l]_{γ_{(π{0 \atop ij} ,h_{j})}} \ar[d]^{ω_{ij}^{X}} \\ h_{ij}^{∗}π_{ij}^{1∗}X_{i} \ar[r]^{γ_{(h_{ij} ,π{1 \atop ij})}} & (π_{ij}^{1} ∘ h_{ij})^{∗}X_{i} & π̃_{ij}^{1∗} ∘ h_{i}^{∗}X_{i} \ar[l]_{γ_{(π{1 \atop ij} ,h_{i})}}}

and if f′′ : C′′ → B is a third object, with a morphism g : C′′ → C′, we have a natural isomorphism of functors :

Desc(φ, g, c) ∘ Desc(φ, h, c) ⇒ Desc(φ, h ∘ g, c)

which is induced by the cleavage c, in the obvious fashion.

Theorem 3.5.25. For i = 1, 2, let φ_{i} : 𝒜_{i} → ℬ be two fibrations, F : 𝒜_{1} → 𝒜_{2} a cartesian functor of ℬ-categories, B an object of ℬ and 𝒮 a sieve of ℬ/B generated by the family (S_{i} → B | i ∈ I). We assume that S_{ij} and S_{ijk} are representable in ℬ, for every i, j, k ∈ I (see remark 3.5.14); then we have :

(i) For n ∈ {0, 1, 2} and every i, j, k ∈ I, suppose that (a) 𝒮 is a sieve both of φ_{1}-n-descent and of φ_{2}-(n − 1)-descent.

(b) The restriction F_{i} : φ_{1}^{−1}S_{i} → φ_{2}^{−1}S_{i} of F is n-faithful.

(c) The restriction F_{ij} : φ_{1}^{−1}S_{ij} → φ_{2}^{−1}S_{ij} of F is (n − 1)-faithful.

(d) The restriction F_{ijk} : φ_{1}^{−1}S_{ijk} → φ_{2}^{−1}S_{ijk} of F is (n − 2)-faithful. Then the restriction F_{B} : φ_{1}^{−1}B → φ_{2}^{−1}B of F is n-faithful.

(ii) Suppose that the functors F_{ij} are fully faithful, and the functors F_{ijk} are faithful, for every i, j, k ∈ I. Then the natural commutative diagram

\xymatrix{Cart_{ℬ} (𝒮 , 𝒜_{1}) \ar[r]^{Cart_{ℬ} (𝒮 ,F)} \ar[d] & Cart_{ℬ} (𝒮 , 𝒜_{2}) \ar[d] \\ ∏_{i∈I} φ_{1}^{−1}S_{i} \ar[r]^{∏_{i∈I} F_{i}} & ∏_{i∈I} φ_{2}^{−1}S_{i}}

is 2-cartesian.

Proof. (i): In view of theorem 3.2.7, we may assume that both 𝒜_{1} and 𝒜_{2} are split fibrations (with a suitable choice of cleavages), and F is a split cartesian functor. Recall that the latter condition means the following. For every morphism g : X → Y in ℬ, the induced diagram

\xymatrix{φ_{1}^{−1}Y \ar[r]^{g^{∗}} \ar[d]_{F} & φ_{1}^{−1}X \ar[d]^{F} \\ φ_{2}^{−1}Y \ar[r]^{g^{∗}} & φ_{2}^{−1}X}

commutes (where the horizontal arrows are the pull-back functor given by the chosen cleavages). In this situation, we have a commutative diagram

(3.5.26)  \xymatrix{φ_{1}^{−1}B \ar[r]^{ρ\underline{S}} \ar[d]_{F_{B}} & Desc(φ_{1}, \underline{S}) \ar[d]^{\underline{F}} \\ φ_{2}^{−1}B \ar[r]^{ρ\underline{S}} & Desc(φ_{2}, \underline{S})}

whose right vertical arrow is the functor given by the rule :

\underline{X} := (X_{i}, ω_{ij} | i, j ∈ I) ↦ \underline{F} (\underline{X}) := (F_{i}X_{i}, F_{ij}ω_{ij} | i, j ∈ I).

for every object \underline{X} of Desc(φ_{1}, \underline{S}). By assumption, the top horizontal arrow of (3.5.26) is nfaithful, and the bottom horizontal arrow is (n − 1)-faithful. We need to prove that the left vertical arrow is n-faithful, and it is easily seen that this will follow, once we have shown that the same holds for the right vertical arrow.

Suppose first that n = 0; we have to show that \underline{F} is faithful. However, let \underline{X} and \underline{Y} be two objects of Desc(φ_{1}, \underline{S}), and \underline{h}_{1}, \underline{h}_{2} : \underline{X} → \underline{Y} two morphisms. By definition, \underline{h}_{t} (for t = 1, 2) is a compatible system (h_{t,i} : X_{i} → Y_{i} | i ∈ I), where each h_{t,i} is a morphism in φ_{1}^{−1}S_{i}. Then, \underline{F} (\underline{h}_{t}) is the compatible system (F_{i}h_{t,i} | i ∈ I). Thus, the condition \underline{F} (\underline{h}_{1}) = \underline{F} (\underline{h}_{2}) translates the system of identities F_{i}h_{1,i} = F_{i}h_{2,i} for every i ∈ I. By assumption, each F_{i} is faithful, therefore \underline{h}_{1} = \underline{h}_{2}, as stated.

For n = 1, assumption (d) is empty, (b) means that F_{i} is fully faithful, and (c) means that F_{ij} is faithful for every i, j ∈ I. In light of the previous case, we have only to show that \underline{F} is full. Hence, let \underline{X}, \underline{Y} be as in the foregoing, and (h_{i} : F_{i}X_{i} → F_{i}Y_{i} | i ∈ I) a morphism \underline{F} (\underline{X}) → \underline{F} (\underline{Y}) in Desc(φ_{2}, \underline{S}). By assumption, for every i ∈ I we may find a unique morphism f_{i} : X_{i} → Y_{i} such that F_{i}f_{i} = h_{i}. It remains only to check that the system (f_{i} | i ∈ I) fulfills condition (3.5.21), and since the functors F_{ij} are faithful, it suffices to verify that F_{ij}(3.5.21) holds. However, since F is split cartesian, we have :

F_{ij} ∘ π_{ij}^{0∗}f_{j} = π_{ij}^{0∗} ∘ F_{j}f_{j} F_{ij} ∘ π_{ij}^{1∗}f_{i} = π_{ij}^{1∗} ∘ F_{i}f_{i}

hence we reduce to showing that (F_{ij}ω_{ij}^{Y})∘π_{ij}^{0∗}h_{j} = π_{ij}^{1∗}h_{i}∘(F_{ij}ω_{ij}^{X}), which holds by assumption.

Next, we consider assertion (ii) : the contention is that the functors \underline{F} and :

p_{1,\underline{S}} : Desc(φ_{1}, \underline{S}) → ∏_{i∈I} φ_{1}^{−1}S_{i}

as in (3.5.22), induce an equivalence (p_{1,\underline{S}}, \underline{F}) between Desc(φ_{1}, \underline{S}) and the category 𝒞 consisting of all data of the form \underline{G} := (G_{i}, H_{i}, α_{i}, ω_{ij}^{H} | i, j ∈ I), where G_{i} ∈ Ob(φ_{1}^{−1}S_{i}),

H_{i} ∈ Ob(φ_{2}^{−1}S_{i}), α_{i} : F_{i}G_{i} →^{∼} H_{i} are isomorphisms in φ_{2}^{−1}S_{i}, and \underline{H} := (H_{i}, ω_{ij}^{H} | i, j ∈ I) is

an object of Desc(φ_{2}, \underline{S}). However, given an object as above, set :

ω_{ij}^{H′} := (π_{ij}^{1∗}α_{i}^{−1}) ∘ ω_{ij}^{H} ∘ (π_{ij}^{0∗}α_{j}).

Since φ_{2} is a split fibration, one verifies easily that the datum \underline{H}′ := (H′_{i} := F_{i}G_{i}, ω_{ij}^{H′} | i, j ∈ I) is an object of Desc(φ_{2}, \underline{S}) isomorphic to \underline{H}, and the new datum (F_{i}, H′_{i}, 1_{H′_{i}}, ω_{ij}^{H′} | i, j ∈ I) is isomorphic to \underline{G}; hence 𝒞 is equivalent to the category 𝒞′ whose objects are all data of the form (G_{i}, ω_{ij} | i, j ∈ I) where G_{i} ∈ Ob(φ_{1}^{−1}S_{i}), and (F_{i}G_{i}, ω_{ij} | i, j ∈ I) is an object of Desc(φ_{2}, \underline{S}). By assumption F_{ij} is fully faithful, and F is a split cartesian functor; hence we may find unique isomorphisms ω̃_{ij}^{G} : π_{ij}^{0∗}G_{j} →^{∼} π_{ij}^{1∗}G_{i} such that ω̃_{ij} = F_{ij}ω̃_{ij}^{G}. We claim that the datum (G_{i}, ω_{ij}^{G} | i, j ∈ I) is an object of Desc(φ_{1}, \underline{S}), i.e. the isomorphisms ω_{ij}^{G} satisfy the cocycle condition

(3.5.27)  π_{ijk}^{2∗}ω_{ij}^{G} ∘ π_{ijk}^{0∗}ω_{jk}^{G} = π_{ijk}^{1∗}ω_{ik}^{G}  for every i, j, k ∈ I.

To check this identity, since by assumption the functors F_{ijk} are faithful, it suffices to see that F_{ijk}(3.5.27) holds, which is clear, since the cocycle condition holds for the isomorphisms ω_{ij} (and since F is split cartesian). This shows that (p_{1,\underline{S}}, \underline{F}) is essentially surjective. Next, since the functor p_{1,\underline{S}} is faithful, the same holds for (p_{1,\underline{S}}, \underline{F}). Finally, let

\underline{G} := (G_{i}, ω_{ij} | i, j ∈ I) \underline{G}′ := (G′_{i}, ω′_{ij} | i, j ∈ I)

be two objects of 𝒞′. A morphism \underline{G} → \underline{G}′ consists of a system (α_{i} : G_{i} → G′_{i} | i ∈ I) of morphisms such that (F_{i}α_{i} | i ∈ I) is a morphism

(F_{i}G_{i}, ω_{ij} | i, j ∈ I) → (F_{i}G′_{i}, ω′_{ij} | i, j ∈ I)

in Desc(φ_{2}, \underline{S}). To show that (p_{1\underline{S}}, \underline{F}) is full, and since we know already that this functor is essentially surjective, we may assume that there exist (G_{i}, ω_{ij}^{G} | i, j ∈ I), (G′_{i}, ω_{ij}^{G′} | i, j ∈ I) in Desc(φ_{1}, \underline{S}) such that ω_{ij} = F_{ij}ω_{ij}^{G} and ω′_{ij} = F_{ij}ω_{ij}^{G′} for every i, j ∈ I; in this case, it suffices to verify the identity

(3.5.28)  ω_{ij}^{G′} ∘ π_{ij}^{0∗}α_{j} = π_{ij}^{1∗}α_{i} ∘ ω_{ij}^{G}  for every i, j ∈ I.

Again, the faithfulness of F_{ij} reduces to checking that F_{ij}(3.5.28) holds, which is clear, since F is split cartesian.

Lastly, notice that the case n = 2 of assertion (i) is a formal consequence of (ii). □

3.5.29. In the situation of (3.5.22), let 𝒮 be the sieve generated by the family \underline{S}, and g : B′ → B any morphism in ℬ. We let :

B′_{i} := B′ ×_{B} S_{i} B′_{ij} := B′ ×_{B} S_{ij} B′_{ijk} := B′ ×_{B} S_{ijk} for every i, j, k ∈ I

and denote g_{i} : B′_{i} → S_{i}, g_{ij} : B′_{ij} → S_{ij} and g_{ijk} : B′_{ijk} → S_{ijk} the induced projections.

Corollary 3.5.30. With the notation of (3.5.29), let n ∈ {0, 1, 2}. The following holds : (i) 𝒮 is a sieve of φ-n-descent, if and only if ρ\underline{S} is n-faithful (see (3.5.23)). (ii) Suppose that :

(a) 𝒮 is a sieve of universal φ-n-descent.

(b) The pull-back functors g_{i}^{∗} : φ^{−1}S_{i} → φ^{−1}B′_{i} are n-faithful.

(c) The pull-back functors g_{ij}^{∗} : φ^{−1}S_{ij} → φ^{−1}B′_{ij} are (n − 1)-faithful.

(d) The pull-back functors g_{ijk}^{∗} : φ^{−1}S_{ijk} → φ^{−1}B′_{ijk} are (n − 2)-faithful.

Then the pull-back functor g^{∗} is n-faithful. (iii) Suppose that the functors g_{ij}^{∗} are fully faithful, and the functors g_{ijk}^{∗} are faithful, for

every i, j, k ∈ I. Then the natural essentially commutative diagram :

\xymatrix{Desc(φ, \underline{S}) \ar[r]^{Desc(φ,g)} \ar[d]_{p\underline{S}} & Desc(φ, \underline{S} ×_{B} B′) \ar[d]^{p_{\underline{S}×_{B} B′}} \\ ∏_{i∈I} φ^{−1}S_{i} \ar[r]^{∏_{i∈I} g_{i}^{∗}} & ∏_{i∈I} φ^{−1}B′_{i}}

is 2-cartesian (see remark 2.5.2(v) and example 3.3.12(ii)).

Proof. (i) follows by inspecting (3.5.23).

(ii): Thanks to theorem 3.2.7, we may assume that φ is a split fibration. Now, set

𝒞 := Morph(ℬ)  𝒜_{1} := 𝒞 ×_{(t,φ)} 𝒜  𝒜_{2} := 𝒞 ×_{(s,φ)} 𝒜

where s, t : 𝒞 → ℬ are the source and target functors (see (1.1.30)). The natural projections φ_{i} : 𝒜_{i} → 𝒞 (for i = 1, 2) are two fibrations (see remark 3.1.5(i)). Moreover, t induces a functor t_{|g} : 𝒞/g → ℬ/B (notation of (1.1.26)) and we let 𝒮/g := t_{|g}^{−1}𝒮, which is the sieve of 𝒞/g generated by the cartesian diagrams

D_{i} : \xymatrix{B′_{i} \ar[r]^{g_{i}} \ar[d] & S_{i} \ar[d] \\ B′ \ar[r]^{g} & B} for every i ∈ I.

Notice that the products D_{ij} := D_{i} × D_{j} are represented in 𝒞/g by the diagrams

\xymatrix{B′_{ij} \ar[r]^{g_{ij}} \ar[d] & S_{ij} \ar[d] \\ B′ \ar[r]^{g} & B}  for every i, j ∈ I

and likewise one may represent the triple products D_{ijk} := D_{ij} × D_{k}.

By definition, the objects of 𝒜_{1} (resp. 𝒜_{2}) are the pairs (h : X → Y, a), where h is a morphism in ℬ and a ∈ Ob(φ^{−1}Y) (resp. a ∈ Ob(φ^{−1}X)). A morphism of 𝒜_{1} (resp. of 𝒜_{2})

(h : X → Y, a) → (h′ : X′ → Y′, a′)

is a datum (f_{1}, f_{2}, t), where f_{1} : X → X′ and f_{2} : Y → Y′ are morphisms in ℬ with f_{2} ∘ h = h′ ∘ f_{1}, and t : a → f_{2}^{∗}a′ (resp. t : a → f_{1}^{∗}a′) is a morphism in φ^{−1}Y (resp. in φ^{−1}X). Now, we define a functor F : 𝒜_{1} → 𝒜_{2} of 𝒞 -categories, by the rule :

• (h, a) ↦ (h, h^{∗}a) for every (h, a) ∈ Ob(𝒜_{1}).

• (f_{1}, f_{2}, t) ↦ (f_{1}, f_{2}, h^{∗}t) for every morphism (f_{1}, f_{2}, t) of 𝒜_{1} as above. Notice that if t : a → f_{2}^{∗}a′ is a morphism in φ^{−1}Y, then h^{∗}t : h^{∗}a → h^{∗}f_{2}^{∗}a′ = f_{1}^{∗}h′^{∗}a′ is a morphism of φ^{−1}Y′, since φ is a split fibration.

Notice that a morphism (f_{1}, f_{2}, t) of either 𝒜_{1} or 𝒜_{2} is cartesian if and only if t is an isomorphism; especially, it is clear that F is a cartesian functor. Moreover, for every object h : X → Y of 𝒞, the restriction φ_{1}^{−1}h → φ_{2}^{−1}h of F is isomorphic to the pull-back functor h^{∗} : φ^{−1}Y → φ^{−1}h. Especially, conditions (b)–(d) say that the restriction F_{i} : φ_{1}^{−1}g_{i} → φ_{2}^{−1}g_{i} (resp. F_{ij} : φ_{1}^{−1}g_{ij} → φ_{2}^{−1}g_{ij}, resp. F_{ijk} : φ_{1}^{−1}g_{ijk} → φ_{2}^{−1}g_{ijk}) are n-faithful (resp. (n − 1)- faithful, resp. (n − 2)-faithful). In light of theorem 3.5.25(i), we are then reduced to showing Claim 3.5.31. 𝒮 /g is a sieve both of φ_{1}-n-descent and of φ_{2}-n-descent.

Proof of the claim. Let 𝒟 be any (small) category; we remark first that a functor 𝒟 → 𝒜_{1} is the same as a pair of functors (H : 𝒟 → 𝒜 , K : 𝒟 → 𝒞) such that φ ∘ H = t ∘ K, and likewise one can describe the functors 𝒟 → 𝒜_{2}. Then, it is easily seen that the functors

𝒜 (B) → Cart_{𝒞} (𝒞 /g, 𝒜_{1})  G ↦ (G ∘ t, t)
𝒜 (B′) → Cart_{𝒞} (𝒞 /g, 𝒜_{2})  G ↦ (G ∘ s, s)

are equivalences, and induce equivalences Cart_{ℬ} (𝒮 , 𝒜) → Cart_{𝒞} (𝒮 /g, 𝒜_{1})

Cart_{ℬ} (𝒮 ×_{B} g, 𝒜) → Cart_{𝒞} (𝒮 /g, 𝒜_{2})

(details left to the reader). The claim follows immediately. □

3.5.32. Quite generally, if φ : 𝒜 → ℬ is a fibration over a category ℬ that admits fibre products, the descent data for φ (relative to a fixed cleavage c) also form a fibration :

Dφ : φ-Desc → Morph(ℬ).

Namely, for every morphism f : T′ → T of ℬ, the fibre over f is the category Desc(φ, f) of all descent data (f, A, ξ) relative to the family {f}, and the cleavage c, so A is an object of φ^{−1}T′ and ξ : p_{1}^{∗}A →^{∼} p_{2}^{∗}A is an isomorphism in the category φ^{−1}(T′ ×_{T} T′) satisfying the usual cocycle condition (here p_{1}, p_{2} : T′ ×_{T} T′ → T′ denote the two natural morphisms). Given two objects \underline{A} := (f : T′ → T, A, ξ), \underline{A}′ := (g : W′ → W, A′, ζ) of φ-Desc, the morphisms \underline{A} → \underline{A}′ are the data (h, α) consisting of a commutative diagram :

\xymatrix{W′ \ar[r]^{h} \ar[d]_{g} & T′ \ar[d]^{f} \\ W \ar[r] & T}

and a morphism α : A → A′ such that φ(α) = h and p_{1}^{∗}(α) ∘ ξ = ζ ∘ p_{2}^{∗}(α).

We have a natural cartesian functor of fibrations :

[figure: 𝒜 ×_{(φ,t)} Morph(ℬ); d; p; φ-Desc; Dφ; Morph(ℬ)]

where t : Morph(ℬ) → ℬ is the target functor (see (1.1.30) and example 1.2.25(i)). Namely, to any pair (T, f : S → φT) with T ∈ Ob(𝒜) and f ∈ Ob(Morph(ℬ)), one assigns the canonical descent datum d(T, f) := ρ_{{f}}(T) in Desc(φ, f) associated with the pair as in (3.5.23).

Corollary 3.5.33. In the situation of (3.5.32), let f : B′ → B be a morphism of ℬ, and 𝒮 a sieve of ℬ/B, generated by a family (S_{i} → B | i ∈ I). Let n ∈ {0, 1, 2}, and suppose that :

(a) 𝒮 is a sieve of universal φ-n-descent.

(b) For every i ∈ I, the morphism S_{i} ×_{B} f is of φ-n-descent.

(c) For every i, j ∈ I, the morphism S_{ij} ×_{B} f is of φ-(n − 1)-descent.

(d) For every i, j, k ∈ I, the morphism S_{ijk} ×_{B} f is of φ-(n − 2)-descent. Then f is a morphism of φ-n-descent.

Proof. In view of corollary 3.5.30(i), it is easily seen that a morphism g : T′ → T in ℬ is of φn-descent if and only if the restriction φ^{−1}T → Dφ^{−1}g of d is n-faithful. Set 𝒞 := Morph(ℬ); as in the proof of corollary 3.5.30(ii), the functor t induces a functor t_{|f} : 𝒞 /f → ℬ/B, and we let 𝒮/f := t_{|f}^{−1}𝒮. With this notation, theorem 3.5.25(i) reduces to showing :

Claim 3.5.34. The sieve 𝒮/f is both of p-n-descent and of Dφ-n-descent.

Proof of the claim. By claim 3.5.31, it is already known that 𝒮 /f is of p-n-descent. To show that 𝒮 /f is of Dφ-n-descent, we consider the commutative diagram

\xymatrix{𝒜 (B′) \ar[r] \ar[d] & Cart_{𝒞} (𝒞 /f, φ-Desc) \ar[d] \\ Cart_{ℬ} (𝒮 ×_{B} f, 𝒜) \ar[r] & Cart_{𝒞} (𝒮 /f, φ-Desc)}

whose left (resp. right) vertical arrow is induced by the inclusion 𝒮 ×_{B} f → ℬ/B′ (resp. 𝒮 /f → 𝒮 ×_{B} f) and whose top horizontal arrow is defined as follows. Given a cartesian functor G : ℬ/B′ → 𝒜, we let DG : 𝒞 /f → φ-Desc be the unique cartesian functor determined on the objects of 𝒞/f by the rule :

( \xymatrix{T′ \ar[r]^{g} \ar[d]_{h} & T \ar[d] \\ B′ \ar[r]^{f} & B}) ↦ d(G(h), g).

We leave to the reader the verification the rule G ↦ DG extends to a well defined functor, and then there exists a unique (similarly defined) bottom horizontal arrow that makes commute the foregoing diagram. Moreover, both horizontal arrow thus obtained are equivalences of categories. The claim follows. □

Lemma 3.5.35. Let i ≤ 2 be an integer, F : ℰ → 𝒞 a fibration, X ∈ Ob(𝒞), and 𝒯 ⊂ 𝒮 two sieves of 𝒞 /X. The following holds :

(i) If 𝒯 ×_{X} f is of F -i-descent for every (Y ⟶^{f} X) ∈ 𝒮, then the induced functor

j : Cart_{𝒞} (𝒮 , ℰ) → Cart_{𝒞} (𝒯 , ℰ)

is i-faithful. (ii) If 𝒯 is of universal F -i-descent, the same holds for 𝒮.

Proof. (i): The assertion is trivial for i < 0. We consider first the case where i = 0; thus, let φ, ψ : 𝒮 → ℰ be two 𝒞 -cartesian functors, and α, β : φ ⇒ ψ two natural 𝒞 -transformations such that α_{|𝒯} = β_{|𝒯} . Let also (Y ⟶^{f} X) ∈ Ob(𝒮) be any object and f_{∗} : 𝒞 /Y → 𝒮 the induced functor; then 𝒯 ′ := 𝒯 ×_{X} f ⊂ 𝒞 /Y is a sieve of F -0-descent and

(α ∗ f_{∗})_{|𝒯 ′} = (β ∗ f_{∗})_{|𝒯 ′} .

We deduce that α ∗ f_{∗} = β ∗ f_{∗}; especially α_{f} = β_{f}, which shows that j is faithful. Next, suppose that i = 1, and for φ and ψ as in the foregoing, let α : φ_{|𝒯} ⇒ ψ_{|𝒯} be a given

natural 𝒞 -transformation, and (Y ⟶^{f} X) ∈ Ob(𝒮); then the sieve 𝒯 ′ := 𝒯 ×_{X} f ⊂ 𝒞 /Y is of F -1-descent, and f_{∗} restricts to a functor f_{∗|𝒯 ′} : 𝒯 ′ → 𝒯 , so the natural 𝒞 -transformation

α ∗ (f_{∗|𝒯 ′}) : (φ ∘ f_{∗})_{|𝒯 ′} ⇒ (ψ ∘ f_{∗})_{|𝒯 ′}

extends to a unique natural 𝒞 -transformation α^{(f)} : φ ∘ f_{∗} ⇒ ψ ∘ f_{∗}. Now, let us define

α̃_{f} := α_{1_{Y}}^{(f)} for every (Y ⟶^{f} X) ∈ Ob(𝒮) .

Clearly, α̃_{f} = α_{f} whenever f ∈ Ob(𝒯 ), and it remains to check that α̃ is a natural 𝒞 -

transformation φ ⇒ ψ. Thus, let h/X : (Y′ ⟶^{f′} X) → (Y ⟶^{f} X) be a morphism of 𝒮; we need to show that

ψ(h/X) ∘ α̃_{f} = α̃_{f′} ∘ φ(h/X).

Considering the morphism h/Y : (Y′ ⟶^{h} Y) → 1_{Y} of 𝒞 /Y, and noticing that φ(h/X) = (φ ∘ f_{∗})(h/Y) and ψ(h/X) = (ψ ∘ f_{∗})(h/Y), we come down to checking that α_{h}^{(f)} = α_{1_{Y′}}^{(f′)}. We show more precisely that α^{(f)} ∗ h_{∗} = α^{(f′)}. Indeed, notice that

(α^{(f)} ∗ h_{∗})_{|𝒯 ×_{X}f′} = α ∗ (f′_{∗|𝒯 ×_{X}f′}) = α_{|𝒯 ×_{X}f′}^{(f′)}

and since 𝒯 ×_{X} f′ is of F -0-descent, the assertion follows.

Lastly, for i = 2, we know already that j is fully faithful, so it remains only to show that every cartesian functor φ : 𝒯 → ℰ is isomorphic to the restriction of a cartesian functor 𝒮 → ℰ. Now, for every (f : Y → X) ∈ Ob(𝒮) set 𝒯 ^{(f)} := 𝒯 ×_{X} f; by assumption there exists a 𝒞 -cartesian functor φ^{(f)} : 𝒞 /Y → ℰ with an isomorphism of functors :

ω_{f} : φ_{|𝒯 ^{(f)}}^{(f)} →^{∼} φ ∘ (f_{∗|𝒯 ^{(f)}} ).

Notice that if f ∈ Ob(𝒯 ), we have 𝒯 ^{(f)} = 𝒞 /Y, and in this case we take φ^{(f)} := φ ∘ f_{∗} and ω_{f} := 1_{φ^{(f)}}. Then, for every morphism h/X : (Y′ ⟶^{f′} X) → (Y ⟶^{f′} X) in 𝒮 we get the isomorphism :

φ_{|𝒯 ^{(f′)}}^{(f′)} ⟶^{ω_{f′}} φ ∘ f′_{∗|𝒯 ^{(f′)}} ⟶^{(ω_{f} ∗h_{∗})_{|𝒯 ^{(f′)}}^{−1}} (φ^{(f)} ∘ h_{∗})_{∗|𝒯 ^{(f′)}}

which, as 𝒯 ^{(f′)} is of F -1-descent, extends uniquely to an isomorphism of functors 𝒞 /Y′ → ℰ

τ_{h/X} : φ^{(f′)} →^{∼} φ^{(f)} ∘ h_{∗}.

Claim 3.5.36. The rule : (f : Y → X) ↦ (φ^{(f)} : 𝒞 /Y → ℰ) for every f ∈ Ob(𝒮) defines a pseudo-cocone

φ^{(•)} : G_{𝒮} ⇒ F_{ℰ}

whose coherence constraint is given by the system of isomorphisms τ_{•/X}, and where G_{𝒮} : 𝒮 → Cat is the functor associated with 𝒮 as in (3.5.8).

Proof of the claim. Clearly (τ_{1_{Y} /Y})_{|𝒯 ^{(f)}} = 1_{φ_{|𝒯 ^{(f)}}^{(f)}} for every (f : Y → X) ∈ Ob(𝒮), whence

τ_{1_{Y} /Y} = 1_{φ^{(f)}} by the uniqueness of τ_{1_{Y} /Y}. Next, let (Y′′ ⟶^{f′′} X) ⟶^{h′/X} (Y′ ⟶^{f′} X) ⟶^{h/X} (Y ⟶^{f}

X) be two morphism of 𝒮; the coherence axiom for τ_{•/X} comes down to the identity :

(τ_{h/X} ∗ h′_{∗}) ∘ τ_{h′/X} = τ_{(h∘h′)/X}.

But the latter can be checked after restriction to the sieve 𝒯 ^{(f′′)}, where it follows by a simple inspection : the details are left to the reader. ♢

From claim 3.5.36it follows that there exists a functor

φ̃: 𝒮 → ℰ and an invertible modification Ξ : Fφ̃ ⊙ ε̂ →^{∼} φ^{(•)}

where ε̂: G_{𝒮} ⇒ F_{ℰ} is the universal pseudo-cocone provided by lemma 3.5.9. Explicitly, Ξ is a system of isomorphisms of functors

Ξ_{f} : φ̃ ∘ f_{∗} →^{∼} φ^{(f)} for every (f : Y → X) ∈ Ob(𝒮)

from which it follows easily that φ̃ is 𝒞 -cartesian. The compatibility conditions for Ξ amount to the identities :

(3.5.37)  Ξ_{f} ∗ h_{∗} = τ_{h} ⊙ Ξ_{f′}  for every morphism h/X : f → f′ in 𝒮 .

To conclude, it suffices to exhibit an isomorphism of functors ξ : φ̃_{|𝒯} →^{∼} φ. To this aim, we set

ξ_{g} := (Ξ_{g})_{1_{Y}} : φ̃(g) →^{∼} φ^{(g)}(1_{Y}) = φ(g) for every (g : Y → X) ∈ Ob(𝒯 ) .

To check the naturality of ξ, let h/X : (Y′ ⟶^{g′} X) → (Y ⟶^{g} X) be any morphism of 𝒯 ; notice that τ_{h/X} = 1_{φ^{(g′)}}, whence ξ_{g′} = (Ξ_{g})_{h}, due to (3.5.37). On the other hand, the naturality of Ξ_{g} yields the commutative diagram

\xymatrix{φ̃(g′) = φ̃ ∘ g_{∗}(h) \ar[r]^{(Ξ_{g})_{h}} \ar[d]_{φ̃∘g_{∗}(h/Y)} & φ^{(g)}(h) = φ(g′) \ar[d]^{φ^{(g)}(h/Y)} \\ φ̃(g) = φ̃ ∘ g_{∗}(1_{Y}) \ar[r]^{ξ_{g}} & φ^{(g)}(1_{Y}) = φ(g)}

where φ̃ ∘ g_{∗}(h/Y) = φ̃(h/x) and φ^{(g)}(h/Y) = φ(h/X), whence the contention. (ii): We consider the induced functors

ℰ (X) ⟶^{j′} Cart_{𝒞} (𝒮 , ℰ) ⟶^{j′′} Cart_{𝒞} (𝒯 , ℰ)

and notice that j′′ is i-faithful, by virtue of (i); the same holds for j′′ ∘ j′, by assumption. Then it follows easily that j′ is i-faithful as well, whence the assertion. □

Proposition 3.5.38. Let i ≤ 2 be an integer, F : ℰ → 𝒞 a fibration, X ∈ Ob(𝒞), and 𝒯 , 𝒮 two sieves of 𝒞 /X, such that :

(a) 𝒮 is of universal F -i-descent.

(b) 𝒯 ×_{X} f is of universal F -i-descent for every (Y ⟶^{f} X) ∈ 𝒮. Then 𝒯 is of universal F -i-descent.

Proof. According to lemma 3.5.35(ii), the sieve 𝒮′ := 𝒮 ⋃ 𝒯 is also of universal F -i-descent. It is also clear that 𝒯 ×_{X} f is of F -i-descent for every (Y ⟶^{f} X) ∈ 𝒮′. Thus, we may replace 𝒮 by 𝒮′ and assume that 𝒯 ⊂ 𝒮. Next, let f : Y → X be any morphism of 𝒞; we need to check that 𝒯 ×_{X} f is a sieve of F -i-descent, and by assumption the same holds for 𝒮 ×_{X} f, so it suffices to show that the induced functor Cart_{𝒞} (𝒮 ×_{X} f, ℰ) → Cart_{𝒞} (𝒯 ×_{X} f, ℰ) is i-faithful. In view of lemma 3.5.35(i), we are then reduced to checking that (𝒯 ×_{X} f) ×_{Y} g is a sieve of F -i-descent for every g ∈ Ob(𝒮 ×_{X} f). But we have (𝒯 ×_{X} f)×_{Y} g = 𝒯 ×_{X} (f ∘g), and f ∘ g ∈ Ob(𝒮), so the assertion follows from (b). □

3.6. Profinite groups and Galois categories. Quite generally, for any profinite group P, let P -Set denote the category of discrete finite sets, endowed with a continuous left action of P (the morphisms in P -Set are the P -equivariant maps). Any continuous group homomorphism ω : P → Q of profinite groups induces a restriction functor

Res(ω) : Q-Set → P -Set

in the obvious way. In case the notation is not ambiguous, one writes also Res_{Q}^{P} for this functor. For any two profinite groups P and Q, we denote by

Hom_{cont}(P, Q)

the set of all continuous group homomorphisms P → Q. If φ_{1}, φ_{2} are two such group homomorphisms, we say that φ_{1} is conjugate to φ_{2}, and we write φ_{1} ∼ φ_{2}, if there exists an inner automorphism ω of G, such that φ_{2} = ω ∘ φ_{1}. Clearly the trivial map π → G (whose image is the neutral element of G), is the unique element of a distinguished conjugacy class.

3.6.1. Let P be any profinite group; for any (discrete) finite group G, consider the pointed set Hom_{cont}(P, G)/∼ of conjugacy classes of continuous group homomorphisms P → G. This is also denoted

H_{cont}^{1}(P, G)

and called the first non-abelian continuous cohomology group of P with coefficients in G (so G is regarded as a P -module with trivial P -action). Clearly the formation of H^{1}(P, G) is covariant on the argument G, and controvariant for continuous homomorphisms of profinite groups.

Lemma 3.6.2. Let φ : P → P′ be a continuous homomorphism of profinite groups, and suppose that the induced map of pointed sets :

H_{cont}^{1}(P′, G) → H_{cont}^{1}(P, G) : f ↦ f ∘ φ

is bijective, for every finite group G. Then φ is an isomorphism of topological groups.

Proof. First we show that φ is injective. Indeed, let x ∈ P be any element; we may find an open normal subgroup H ⊂ P such that x ∉ H; taking G := P/H, we deduce that the projection P → P/H factors through φ and a group homomorphism f : P′ → P/H, hence x ∉ Ker φ, as claimed. Moreover, let H′ := Ker f; clearly H′ ∩ P = H, so the topology of P is induced from that of P′. It remains only to show that φ is surjective; to this aim, we consider any continuous surjection f′ : P′ → G′ onto a finite group, and it suffices to show that the restriction of f′ to φP is still surjective. Indeed, let G be the image of φP in G′, denote by i : G → G′ the inclusion map, and let f : P → G be the unique continuous map such that i ∘ f = f′ ∘ φ; by assumption, there exists a continuous group homomorphism g : P′ → G such that f = g ∘ φ. On the other hand, (i ∘ g) ∘ φ = f′ ∘ φ, hence the conjugacy class of i ∘ g equals the conjugacy class of f′, especially i ∘ g is surjective, hence the same holds for i, as required. □

3.6.3. Let G be a profinite group, and H ⊂ G an open subgroup. It is easily seen that H is also a profinite group, with the topology induced from G. Moreover, the restriction functor

Res_{G}^{H} : G-Set → H-Set

admits a left adjoint

Ind_{H}^{G} : H-Set → G-Set.

Namely, to any finite set Σ with a continuous left action of H, one assigns the set Ind_{H}^{G}Σ := G × Σ/∼, where ∼ is the equivalence relation such that

(gh, σ) ∼ (g, hσ) for every g ∈ G, h ∈ H and σ ∈ Σ.

The left G-action on Ind_{H}^{G}Σ is given by the rule : (g′, (g, σ)) ↦ (g′g, σ) for every g, g′ ∈ G and σ ∈ Σ. It is easily seen that this action is continuous, and the reader may check that the functor Ind_{H}^{G} is indeed left adjoint to Res_{G}^{H}.

3.6.4. Moreover, let 1 denote the final object of H-Set; notice that Ind_{H}^{G}1 = G/H, the set of orbits of G under its right translation action by H. Hence, for any finite set Σ with continuous H-action, the unique map t_{Σ} : Σ → 1 yields a G-equivariant map

Ind_{H}^{G}t_{Σ} : Ind_{H}^{G} → G/H

and therefore Ind_{H}^{G} factors through a functor

(3.6.5)  H-Set → G-Set/(G/H).

It is easily seen that (3.6.5) is an equivalence. Indeed, one obtains a natural quasi-inverse, by the rule : (f : Σ → G/H) ↦ f^{−1}(H). The detailed verification shall be left to the reader.

Definition 3.6.6. ([82, Exp.V, Def.5.1]) Let 𝒞 be a category, and F : 𝒞 → Set a functor.

(i) We say that 𝒞 is a Galois category, if 𝒞 is equivalent to P -Set, for some profinite group P. We denote Galois the category whose objects are all the Galois categories, and whose morphisms are the exact functors between Galois categories.

(ii) We say that F is a fibre functor, if F is exact and conservative, and F (X) is a finite set for every X ∈ Ob(𝒞).

(iii) We denote fibre.Fun the 2-category of fibre functors, defined as follows :

(a) The objects are all the pairs (𝒞 , F) consisting of a Galois category 𝒞 and a fibre functor F for 𝒞.

(b) The 1-cells (𝒞_{1}, F_{1}) → (𝒞_{2}, F_{2}) are all the pairs (G, β) consisting of an exact functor G : 𝒞_{1} → 𝒞_{2} and an isomorphism of functors β : F_{1} →^{∼} F_{2} ∘ G.

(c) And for every pair of 1-cells (G′, β′), (G, β) : (𝒞_{1}, F_{1}) → (𝒞_{2}, F_{2}), the 2-cells (G′, β′) → (G, β) are the isomorphisms γ : G′ →^{∼} G such that (F_{2} ∗ γ) ⊙ β′ = β. Composition of 1-cells and 2-cells is defined in the obvious way. We shall also denote simply by G a 1-cell (G, β) as in (b), such that F_{1} = F_{2} ∘ G and β is the identity automorphism of F_{1}.

3.6.7. Notice that any Galois category 𝒞 admits a fibre functor : indeed, if P is any profinite group, the forgetful functor

f_{P} : P -Set → Set

fulfills the conditions of definition 3.6.6(ii), therefore the same holds for the functor f_{P} ∘ β, if β : 𝒞 → P -Set is any equivalence. For any Galois category 𝒞 and any fibre functor F : 𝒞 → Set, we denote

π_{1}(𝒞 , F)

the group of automorphisms of F, and we call it the fundamental group of 𝒞 pointed at F. By definition, for every X ∈ Ob(𝒞), the finite set F (X) is endowed with a natural left action of π_{1}(𝒞 , F). For every X ∈ Ob(𝒞) and every ξ ∈ F (X), the stabilizer H_{X,ξ} ⊂ π_{1}(𝒞 , F) is a subgroup of finite index, and we endow π_{1}(𝒞 , F) with the coarsest group topology for which all such H_{X,ξ} are open subgroups. The resulting topological group π_{1}(𝒞 , F) is profinite, and its natural left action on every F (X) is continuous. Thus, F upgrades to a functor denoted

F^{†} : 𝒞 →^{∼} π_{1}(𝒞 , F)-Set.

A basic result states that F^{†} is an equivalence ([82, Exp.V, Th.4.1]).

Example 3.6.8. Let P be any profinite group. Then there is an obvious injective map

P → π_{1}(P -Set, f_{P})

and [82, Exp.V, Th.4.1] implies that this map is an isomorphism of profinite groups. In other words, the group P can be recovered, up to unique isomorphism, from the category P -Set together with its forgetful functor f_{P}.

Remark 3.6.9. Let 𝒞, 𝒞′ be two Galois categories, and F : 𝒞 → Set a fibre functor. (i) Any exact functor G : 𝒞′ → 𝒞 induces a continuous group homomorphism :

π_{1}(G) : π_{1}(𝒞 , F) → π_{1}(𝒞′, F ∘ G) ω ↦ ω ∗ G.

(ii) Furthermore, any isomorphism β : F′ →^{∼} F of fibre functors of 𝒞 induces an isomorphism of profinite groups :

π_{1}(β) : π_{1}(𝒞′, F) →^{∼} π_{1}(𝒞′, F) ω ↦ β^{−1} ⊙ ω ⊙ β

(see [82, Exp.V, §4] for all these generalities).

(iii) Let now (G, β) : (𝒞_{1}, F_{1}) → (𝒞_{2}, F_{2}) be a 1-cell of fibre.Fun. Combining (i) and (ii), we deduce a natural continuous group homomorphism

π_{1}(G, β) : π_{1}(𝒞_{2}, F_{2}) ⟶^{π_{1}(G)} π_{1}(𝒞_{1}, F_{2} ∘ G) ⟶^{π_{1}(β)} π_{1}(𝒞_{1}, F_{1}).

Proposition 3.6.10. With the notation of remark 3.6.9, the rule that assigns :

• To any object (𝒞 , F) of fibre.Fun, the profinite group π_{1}(𝒞 , F)

• To any 1-cell (G, β) of fibre.Fun, the continuous map π_{1}(G, β) defines a pseudo-functor

π_{1} : fibre.Fun → pf.Grp^{o}

from the 2-category of fibre functors, to the opposite of the category of profinite groups (and continuous group homomorphisms).

Proof. (Here we regard pf.Grp as a 2-category with trivial 2-cells : see example 2.2.4(i)). Let

(𝒞_{1}, F_{1}) ⟶^{(G,β_{G})} (𝒞_{2}, F_{2}) ⟶^{(H,β_{H})} (𝒞_{3}, F_{3})

be any pair of (composable) 1-cells; functoriality on 1-cells amounts to the identity :

π_{1}(G, β_{G}) ∘ π_{1}(H, β_{H}) = π_{1}(H ∘ G, (β_{H} ∗ G) ∘ β_{G})

whose detailed verification we leave to the reader. Next, let γ : (G′, β′) → (G, β) be a 2cell between 1-cells (G′, β′), (G, β) : (𝒞_{1}, F_{1}) → (𝒞_{2}, F_{2}); we have to check that π_{1}(G, β) = π_{1}(G′, β′). This identity boils down to the commutativity of the diagram :

\xymatrix{π_{1}(𝒞_{2}, F_{2}) \ar[r]^{π_{1}(G′)} \ar[d]_{π_{1}(G)} & π_{1}(𝒞_{1}, F_{2} ∘ G′) \ar[d]^{π_{1}(β′)} \\ π_{1}(𝒞_{2}, F_{2} ∘ G) \ar[r]^{π_{1}(β)} \ar@{-->}[ur]^{π_{1}(F_{2}∗γ)} & π_{1}(𝒞_{1}, F_{1}).}

However, the commutativity of the lower triangular subdiagram is clear, hence we are reduced to checking the commutativity of the upper triangular subdiagram; the latter is a special case of the following more general :

Claim 3.6.11. Let 𝒞 and 𝒞′ be two Galois categories, G, G′ : 𝒞′ → 𝒞 two exact functors, β : G′ →^{∼} G an isomorphism, and F : 𝒞 → Set a fibre functor. Then the induced diagram of profinite groups

\xymatrix{ & π_{1}(𝒞 , F) \ar@{-->}[dl]_{π_{1}(G)} \ar@{-->}[dr]^{π_{1}(G′)} \\ π_{1}(𝒞′, F ∘ G) \ar[rr]^{π_{1}(F ∗β)} &  & π_{1}(𝒞′, F ∘ G′)}

commutes.

Proof of the claim. Left to the reader. □

Example 3.6.12. Let ω : P → Q be a continuous group homomorphism between profinite groups. Clearly f_{P} ∘ Res(ω) = f_{Q}, and it is easily seen that the resulting diagram

\xymatrix{P \ar[r]^{ω} \ar[d] & Q \ar[d] \\ π_{1}(P -Set, f_{P}) \ar[r]^{π_{1}(Res(ω))} & π_{1}(Q-Set, f_{Q})}

commutes, where the vertical arrows are the natural identifications given by example 3.6.8: the verification is left as an exercise to the reader.

3.6.13. Let \underline{P} := (P_{i} | i ∈ I) be a cofiltered system of profinite groups, with continuous transition maps, and denote by P the limit of this system, in the category of groups. Then P is naturally a closed subgroup of Q := ∏_{i∈I} P_{i}, and the topology 𝒯 induced by the inclusion map P → Q makes it into a compact and complete topological group. Moreover, since the topology of Q is profinite, the same holds for the topology 𝒯 (details left to the reader). It is then easily seen that the resulting topological group (P, 𝒯 ) is the limit of the system \underline{P} in the category of profinite groups.

Proposition 3.6.14. In the situation of (3.6.13), the natural functor

2-colim_{i∈I} P_{i}-Set → P -Set

is an equivalence.

Proof. The functor is obviously faithful; let us show that it is also full. Indeed, let j ∈ I be any index, Σ, Σ′ two objects of P_{j}-Set, and φ : Σ → Σ′ a P -equivariant map; we need to show that φ is already P_{i}-equivariant, for some index i ∈ I. To this aim, we may as usual replace I by I/j, and assume that j is the final element of I. We may also find an open normal subgroup H_{j} ⊂ P_{j} that acts trivially on both Σ and Σ′. Then, for every i ∈ I, we let H_{i} ⊂ P_{i} be the preimage of P_{j}, and we set \overline{P}_{i} := P_{i}/H_{i}. Let also \overline{P} := P/H, where H ⊂ P is the preimage of H_{j}; by construction, φ is \overline{P} -equivariant. Clearly, we may find i ∈ I such that the image of \overline{P} in the finite group \overline{P}_{j} equals the image of \overline{P}_{i}, and for such index i, the induced map \overline{P} → \overline{P}_{i} is an isomorphism. Especially, φ is P_{i}-equivariant, as sought.

Lastly, we show essential surjectivity. Indeed, let Σ be any object of P -Set; we have to show that the P -action on Σ is the restriction of a continuous P_{i}-action, for a suitable i ∈ I. However, we may find a normal open subgroup H ⊂ P that acts trivially on Σ. Then there exists a normal open subgroup L ⊂ Q (notation of (3.6.13)) such that P ∩ L ⊂ H. We may also assume that there exist a finite subset J ⊂ I and for every i ∈ J an open normal subgroup L_{i} ⊂ P_{i} such that L = ∏_{i∈J} L_{i} × ∏_{i∈I\J} P_{i}. Pick an index j ∈ I that admits morphisms f_{i} : j → i in I, for every i ∈ J, and let L′_{i} ⊂ P_{j} denote the preimage of L_{i} under the corresponding map P_{j} → P_{i}. Finally, set H_{j} := ⋂_{i∈J} L′_{i}. By construction, H contains the preimage of H_{j} in P, and we may therefore assume that H is this preimage. We may replace as usual I by I/j, and assume that j is the final element of I. Then, for every i ∈ I, we let H_{i} denote the preimage of H_{j} in P_{i}, and we set \overline{P}_{i} := P_{i}/H_{i}. Set as well \overline{P} := P/H. Clearly, there exists i ∈ I such that the image of the induced map \overline{P} → \overline{P}_{j} equals the image of \overline{P}_{i}; for such index i, the induced map \overline{P} → \overline{P}_{i} is an isomorphism. Thus, Σ the restriction of an object of P_{i}-Set, as wished. □

3.6.15. We consider now a situation that generalizes slightly that of (3.6.13). Namely, let I be a small filtered category, and

(𝒞_{•}, F_{•}) : I → fibre.Fun i ↦ (𝒞_{i}, F_{i})

a pseudo-functor. By proposition 3.6.10, the composition of π_{1} and (𝒞_{•}, F_{•}) is a functor

π_{1}(𝒞_{•}, F_{•}) : I^{o} → pf.Grp i ↦ P_{i} := π_{1}(𝒞_{i}, F_{i}).

Let P denote the limit (in pf.Grp) of the cofiltered system P_{•}, and set

𝒞 := 2-colim_{I} 𝒞_{•}

where the 2-colimit is formed in the 2-category of small categories. We may then state :

Corollary 3.6.16. In the situation of (3.6.15), there exists a natural equivalence :

𝒞 →^{∼} P -Set.

Proof. Recall that (𝒞_{•}, F_{•}) is the datum of isomorphisms

β_{φ} : F_{j} →^{∼} F_{i} ∘ 𝒞_{φ} for every morphism φ : j → i in I

and 2-cells :

τ_{ψ,φ} : (𝒞_{ψ∘φ}, β_{ψ∘φ}) →^{∼} (𝒞_{ψ}, β_{ψ}) ∘ (𝒞_{φ}, β_{φ}) for every composition j ⟶^{φ} i ⟶^{ψ} k

that – by definition – satisfy the identities :

(3.6.17) (β_{ψ} ∗ 𝒞_{φ}) ⊙ β_{φ} = (F_{k} ∗ τ_{ψ,φ}) ⊙ β_{ψ∘φ}  for every composition j ⟶^{φ} i ⟶^{ψ} k

as well as the composition identities :

((𝒞_{μ}, β_{μ})∗τ_{ψ,φ})⊙τ_{μ,ψ∘φ} = (τ_{μ,ψ} ∗(𝒞_{φ}, β_{φ}))⊙τ_{μ∘ψ,φ} for compositions j ⟶^{φ} i ⟶^{ψ} k ⟶^{μ} l.

Let P_{•}-Set : I → Cat denote the functor given by the rule : i ↦ P_{i}-Set for every i ∈ Ob(I), and φ ↦ Res(P_{φ}), where P_{φ} := π_{1}(𝒞_{φ}, β_{φ}) for every morphism φ of I. In view of proposition 3.6.14and lemma 2.5.3, it suffices to show that the rule : i ↦ F_{i}^{†} for every i ∈ Ob(I) (notation of (3.6.7)), extends to a pseudo-natural isomorphism

F_{•}^{†} : 𝒞_{•} →^{∼} P_{•}-Set.

Indeed, let φ : j → i be any morphism of I, and X any object of 𝒞_{j}; we remark that the bijection

β_{φ}(X) : Res(P_{φ})(F_{j}^{†}X) →^{∼} F_{i}^{†} ∘ 𝒞_{φ}(X)

is P_{i}-equivariant; the proof amounts to unwinding the definitions, and shall be left to the reader. Hence we get an isomorphism of functors β_{φ}^{†} : Res(P_{φ}) ∘ F_{j}^{†} →^{∼} F_{i}^{†} ∘ 𝒞_{φ}, and from (3.6.17) we deduce the identities :

(β_{ψ}^{†} ∗ 𝒞_{φ}) ⊙ (Res(P_{φ}) ∗ β_{φ}^{†}) = (F_{k}^{†} ∗ τ_{ψ,φ}) ⊙ β_{ψ∘φ}^{†} for every composition j ⟶^{φ} i ⟶^{ψ} k.

The latter show that the system β_{•}^{†} fulfills the coherence axiom for a pseudo-natural transformation, as required. □

Remark 3.6.18. (i) Keep the situation of (3.6.15), and let a_{•} : 𝒞_{•} ⇒ 𝒞 be the universal pseudo-cocone induced by the pseudo-functor 𝒞_{•}. We may regard the pseudo-functor (𝒞_{•}, F_{•}) as a pseudo-cocone F_{•} : 𝒞_{•} ⇒ Set whose vertex is the category Set. Then, by the universal property of colimits, we get a functor F : 𝒞 → Set and an isomorphism

σ_{•} : F ∗ a_{•} →^{∼} F_{•}.

On the other hand, let r_{•} : P_{•}-Set ⇒ P -Set be the natural cocone (so r_{i} is the restriction functor corresponding to the natural map P → P_{i}, for every i ∈ Ob(I)); the equivalence G of corollary 3.6.16 is deduced from the pseudo-cocone r_{•} ∘ F_{•}^{†} : 𝒞_{•} → P -Set (where F_{•}^{†} is as in the proof of corollary 3.6.16), so we have an isomorphism of pseudo-functors

t_{•} : G ∗ a_{•} →^{∼} r_{•} ∘ F_{•}^{†}

whence an isomorphism

f_{P} ∗ t_{•} : (f_{P} ∘ G) ∗ a_{•} →^{∼} f_{P} ∗ (r_{•} ∘ F_{•}^{†}) = F_{•}

(notation of (3.6.7)). There follows an isomorphism F ∗ a_{•} →^{∼} (f_{P} ∘ G) ∗ a_{•}; by the universal property of the 2-colimit, the latter must come from a unique isomorphism ϑ : F →^{∼} f_{P} ∘ G. Especially, we see that F is also a fibre functor, and we get a pseudo-cocone

(a_{•}, σ_{•}) : (𝒞_{•}, F_{•}) ⇒ (𝒞 , F).

It is now immediate that (𝒞 , F) is the 2-colimit (in the 2-category fibre.Fun) of the pseudofunctor (𝒞_{•}, F_{•}), and (a_{•}, σ_{•}) is the corresponding universal pseudo-cocone.

(ii) Likewise, r_{•} may be regarded as a universal pseudo-cocone

r_{•} : (P_{•}-Set, f_{P_{•}}) ⇒ (P -Set, f_{P})

(with trivial coherence constraint), and the coherence constraint β_{•}^{†} as in the proof of corollary 3.6.16yields a pseudo-natural equivalence

F_{•}^{†} : (𝒞_{•}, F_{•}) →^{∼} (P_{•}-Set, f_{P_{•}})

as well as an isomorphism

(G, ϑ) ∗ (a_{•}, σ_{•}) →^{∼} r_{•} ⊙ F_{•}^{†}.

Thus, for every i ∈ Ob(I) we get a 2-cell of fibre.Fun :

(G, ϑ) ∘ (a_{i}, σ_{i}) → r_{i} ∘ F_{i}^{†}

whence – by proposition 3.6.10– a commutative diagram of profinite groups :

\xymatrix{P \ar[r] \ar[d]_{π_{1}(G,ϑ)} & P_{i} \ar[d]^{π_{1}(F_{i}^{†})} \\ π_{1}(𝒞 , F) \ar[r]^{π_{1}(a_{i},σ_{i})} & π_{1}(𝒞_{i}, F_{i}).}

3.6.19. Let (𝒞 , F) be a fibre functor, and X a connected object of 𝒞 (example 1.2.16(iv)); pick any ξ ∈ F (X), and let H_{ξ} ⊂ π_{1}(𝒞 , F) be the stabilizer of ξ for the natural left action of π_{1}(𝒞 , F) on F (X). For every object f : Y → X of 𝒞/X, we set

F_{ξ}(f) := F (f)^{−1}(ξ) ⊂ F (Y).

It is clear that the rule f ↦ F_{ξ}(f) yields a functor F_{ξ}^{†} : 𝒞/X → H_{ξ}-Set, which we call the subfunctor of F_{|X} selected by ξ.

Proposition 3.6.20. In the situation of (3.6.19), we have :

(i) 𝒞/X is also a Galois category, and F_{ξ} := f_{H_{ξ}} ∘ F_{ξ}^{†} is a fibre functor for 𝒞/X. (ii) The functor F_{ξ}^{†} induces a natural isomorphism of profinite groups :

π_{1}(F_{ξ}^{†}) : H_{ξ} →^{∼} π_{1}(𝒞/X, F_{ξ}).

Proof. The fibre functor F induces an equivalence of categories

𝒞/X →^{∼} π_{1}(𝒞 , F)-Set/F (X).

On the other hand, since X is connected, there exists a unique isomorphism ω : F (X) →^{∼} G/H_{ξ} of G-sets such that ω(ξ) = H_{ξ}, and then the discussion of (3.6.4) yields an equivalence

π_{1}(𝒞 , F)-Set/F (X) →^{∼} H_{ξ}-Set.

A simple inspection shows that the resulting equivalence 𝒞/X →^{∼} H_{ξ}-Set is none else than the functor F_{ξ}^{†}, so the assertion follows from remark 3.6.9(i) and example 3.6.8. □

3.6.21. Let (𝒞 , F) be a fibre functor, and let us now fix a cleavage c : 𝒞^{o} → Cat for the fibred category t : Morph(𝒞) → 𝒞 (see example 3.1.3(iii)). Also, let I be a small cofiltered category, and X_{•} : I → 𝒞 a functor such that X_{i} is a connected object of 𝒞, for every i ∈ Ob(I); we pick an element

ξ_{•} ∈ lim_{I} F ∘ X_{•}.

In other words, ξ_{•} := (ξ_{i} ∈ F (X_{i}) | i ∈ I) is a compatible system of elements such that

F (φ)(ξ_{j}) = ξ_{i} for every morphism φ : j → i in I.

For every i ∈ I, we denote by H_{i} ⊂ π_{1}(𝒞 , F) the stabilizer of ξ_{i} for the left action of π_{1}(𝒞 , F) on F (X_{i}). Clearly, any morphism j → i induces an inclusion H_{j} ⊂ H_{i}. Furthermore, let

F_{i}^{†} : 𝒞/X_{i} → H_{i}-Set for every i ∈ I

be the subfunctor selected by ξ_{i}, and set F_{i} := f_{H_{i}} ∘ F_{i}^{†}. Let φ : j → i be a morphism of I; to the corresponding morphism X_{φ} : X_{j} → X_{i}, the cleavage c attaches a pull-back functor

X_{φ}^{∗} : 𝒞/X_{i} → 𝒞/X_{j}.

Especially, for any object Y ∈ Ob(𝒞 /X_{i}) we have the cartesian diagram in 𝒞 :

\xymatrix{X_{φ}^{∗}(Y) \ar[r] \ar[d] & Y \ar[d] \\ X_{j} \ar[r]^{X_{φ}} & X_{i}}

whence, since F is exact, a natural bijection :

F (X_{φ}^{∗}(Y)) →^{∼} F (Y) ×_{F (X_{i})} F (X_{j})

which in turns yields a bijection :

F_{j}^{†}(X_{φ}^{∗}(Y)) →^{∼} F (Y) ×_{F (X_{i})} {ξ_{j}} = F_{i}^{†}(Y) × {ξ_{j}}.

That is, we have a natural isomorphism of functors :

(3.6.22)  α_{φ}^{†} : F_{j}^{†} ∘ X_{φ}^{∗} →^{∼} Res_{H_{i}}^{H_{j}} ∘ F_{i}^{†}

and since the X_{φ}^{∗} are exact functors, it is easily seen that the isomorphisms α_{φ} := f_{j} ∗ α_{φ}^{†} yield a pseudo-functor

(3.6.23)  I → fibre.Fun  i ↦ (𝒞/X_{i}, F_{i})  φ ↦ (X_{φ}^{∗}, α_{φ}).

Moreover, α_{φ}^{†} can be seen as a 2-cell of fibre.Fun : F_{j}^{†} ∘ (X_{φ}^{∗}, α_{φ}) →^{∼} Res_{H_{j}}^{H_{i}} ∘ F_{i}^{†}, whence a commutative diagram of profinite groups :

\xymatrix{H_{j} \ar[r] \ar[d]_{π_{1}(F_{j}^{†})} & H_{i} \ar[d]^{π_{1}(F_{i}^{†})} \\ π_{1}(𝒞/X_{j}, F_{j}) \ar[r]^{π_{1}(X_{φ}^{∗},α_{φ})} & π_{1}(𝒞/X_{i}, F_{i})}

whose top horizontal arrow is the inclusion map. Especially, notice that the map π_{1}(X_{φ}^{∗}, α_{φ}) does not depend on the chosen cleavage; this can also be seen by remarking that any two cleavages c, c′ are related by a pseudo-natural isomorphism c →^{∼} c′ (details left to the reader).

3.6.24. Let (𝒞/X, F_{ξ}) be the 2-colimit of the pseudo-functor (3.6.23), as in remark 3.6.18(i), and fix a corresponding universal pseudo-cocone (a_{•}, σ_{•}) : (𝒞/X_{•}, F_{•}) ⇒ (𝒞/X, F_{ξ}). We may then state :

Corollary 3.6.25. In the situation of (3.6.24), there exists a natural isomorphism of profinite groups :

H := ⋂_{i∈Ob(I)} H_{i} →^{∼} π_{1}(𝒞/X, F_{ξ})

which fits into a commutative diagram :

(3.6.26)  \xymatrix{H \ar[r] \ar[d] & H_{i} \ar[d]^{π_{1}(F_{i}^{†})} \\ π_{1}(𝒞/X, F_{ξ}) \ar[r]^{π_{1}(a_{i},σ_{i})} & π_{1}(𝒞/X_{i}, F_{i})}  for every i ∈ Ob(I)

whose top horizontal arrow is the inclusion map.

Proof. By corollary 3.6.16, we have an isomorphism of π_{1}(𝒞/X, F_{ξ}) with the limit of the cofiltered system (π_{1}(𝒞/X_{,}F_{i}) | i ∈ Ob(I)); on the other hand, the discussion of (3.6.21) shows that the latter system is naturally isomorphic to the system (H_{i} | i ∈ Ob(I)). Lastly, the commutativity of (3.6.26) follows from remark 3.6.18(ii). □

3.7. Tensor categories and abelian categories. In this section we assemble some basic definitions and results that pertain to abelian categories and other related classes of categories with extra structure.

Definition 3.7.1. A tensor category is a datum \underline{𝒞} := (𝒞 , ⊗, Φ, Ψ) consisting of a category 𝒞, a functor

⊗ : 𝒞 × 𝒞 → 𝒞 : (X, Y) ↦ X ⊗ Y

and natural isomorphisms :

Φ_{X,Y,Z} : X ⊗ (Y ⊗ Z) →^{∼} (X ⊗ Y) ⊗ Z Ψ_{X,Y} : X ⊗ Y →^{∼} Y ⊗ X

for every X, Y, Z ∈ Ob(𝒞), called respectively the associativity and commutativity constraints of \underline{𝒞} , that satisfy the following axioms.

(a) Coherence axiom : the diagram

\xymatrix{X ⊗ (Y ⊗ (Z ⊗ T)) \ar[r]^{Φ_{X,Y,Z⊗T}} \ar[d]_{X⊗Φ_{Y,Z,T}} & (X ⊗ Y) ⊗ (Z ⊗ T) \ar[r]^{Φ_{X⊗Y,Z,T}} & ((X ⊗ Y) ⊗ Z) ⊗ T \\ X ⊗ ((Y ⊗ Z) ⊗ T) \ar[rr]^{Φ_{X,Y ⊗Z,T}} &  & (X ⊗ (Y ⊗ Z)) ⊗ T \ar[u]_{Φ_{X,Y,Z}⊗T}}

commutes, for every X, Y, Z, T ∈ Ob(𝒞).

(b) Compatibility axiom : the diagram

\xymatrix{X ⊗ (Y ⊗ Z) \ar[r]^{Φ_{X,Y,Z}} \ar[d]_{X⊗Ψ_{Y,Z}} & (X ⊗ Y) ⊗ Z \ar[r]^{Ψ_{X⊗Y,Z}} & Z ⊗ (X ⊗ Y) \ar[d]^{Φ_{Z,X,Y}} \\ X ⊗ (Z ⊗ Y) \ar[r]^{Φ_{X,Z,Y}} & (X ⊗ Z) ⊗ Y \ar[r]^{Ψ_{X,Z}⊗Y} & (Z ⊗ X) ⊗ Y}

commutes, for every X, Y, Z ∈ Ob(𝒞).

(c) Commutation axiom : we have Ψ_{Y,X} ∘ Ψ_{X,Y} = 1_{X⊗Y} for every X, Y ∈ Ob(𝒞)

(d) Unit axiom : there exist an object U ∈ Ob(𝒞) and an isomorphism u : U →^{∼} U ⊗ U such that the functor

(3.7.2)  𝒞 → 𝒞 : X ↦ U ⊗ X

is an equivalence. One says that (U, u) is a unit object of \underline{𝒞} .

Lemma 3.7.3. Let \underline{𝒞} := (𝒞 , ⊗, Φ, Ψ) be any tensor category. The diagram

\xymatrix{X ⊗ (Y ⊗ Z) \ar[r]^{Φ_{X,Y,z}} \ar[d]_{X⊗Ψ_{Y,Z}} & (X ⊗ Y) ⊗ Z \ar[r]^{Ψ_{X,Y} ⊗Z} & (Y ⊗ X) ⊗ Z \\ X ⊗ (Z ⊗ Y) \ar[r]^{Φ_{X,Z,Y}} & (X ⊗ Z) ⊗ Y \ar[r]^{Ψ_{X⊗Z,Y}} & Y ⊗ (X ⊗ Z) \ar[u]_{Φ_{Y,X,Z}}}

commutes, for every X, Y, Z ∈ Ob(𝒞).

Proof. To ease notation, we shall omit the tensor symbol ⊗ between objects, and we shall drop the subscript from Φ and Ψ when we display a diagram. It suffices to consider the diagram

\xymatrix{ & Y (XZ) \ar[rrr]^{Φ} \ar[d]^{Y ⊗Ψ} &  &  & (Y X)Z \ar[d]_{Ψ} \\  & Y (ZX) \ar[r]^{Φ} & (Y Z)X \ar[r]^{Ψ⊗X} & (ZY)X & Z(Y X) \ar[l]_{Φ} \\ (XZ)Y \ar[r]^{Ψ⊗Y} \ar[uur]^{Ψ} & (ZX)Y \ar[u]_{Ψ} &  &  & Z(XY) \ar[u]^{Z⊗Ψ} \ar[lll]_{Φ} & (XY)Z \ar[l]_{Ψ} \ar[uul]_{Ψ⊗Z} \\ X(ZY) \ar[u]^{Φ} &  &  &  &  & X(Y Z) \ar[lllll]_{X⊗Ψ} \ar[u]_{Φ}}

whose two triangular subdiagrams commute by naturality of Ψ, and whose three rectangular subdiagrams commute by compatibility. □

Definition 3.7.4. Let \underline{𝒞} := (𝒞 , ⊗, Φ, Ψ) and \underline{𝒞} ′ := (𝒞′, ⊗′, Φ′, Ψ′) be two tensor categories. A tensor functor \underline{𝒞} → \underline{𝒞} ′ is a pair (F, c) consisting of a functor F : 𝒞 → 𝒞′ and a natural isomorphism

c_{X,Y} : F X ⊗ F Y →^{∼} F (X ⊗ Y) for all X, Y ∈ Ob(𝒞)

such that the following holds.

(a) For every objects X, Y, Z of 𝒞, the diagram

\xymatrix{F X ⊗ (F Y ⊗ F Z) \ar[r]^{F X⊗c_{Y Z}} \ar[d]_{Φ′_{F X,F Y,F Z}} & F X ⊗ F (Y ⊗ Z) \ar[r]^{c_{X,Y ⊗Z}} & F (X ⊗ (Y ⊗ Z)) \ar[d]^{F (Φ_{X,Y,Z})} \\ (F X ⊗ F Y) ⊗ F Z \ar[r]^{c_{X,Y} ⊗F Z} & F (X ⊗ Y) ⊗ F Z \ar[r]^{c_{X⊗Y,Z}} & F ((X ⊗ Y) ⊗ Z)}

commutes.

(b) For all objects X, Y of 𝒞, the diagram

\xymatrix{F X ⊗ F Y \ar[r]^{c_{X,Y}} \ar[d]_{Ψ_{F X,F Y}} & F (X ⊗ Y) \ar[d]^{F (Ψ_{X,Y} )} \\ F Y ⊗ F X \ar[r]^{c_{Y,X}} & F (Y ⊗ X)}

commutes.

(c) If (U, u) is a unit object of \underline{𝒞} , then (F U, c_{U,U}^{−1} ∘ F u) is a unit object of \underline{𝒞} ′.

Remark 3.7.5. (i) Lemma 3.7.3illustrates a general principle valid in every tensor categor \underline{𝒞} : namely, say that X_{1}, …, X_{n} is a sequence of distinct objects of 𝒞, and X′ and X′′ are obtained from these two sequences by taking tensor products several times, and in any order, in which case we say that X′ and X′′ have no repetitions. Now, there will be usually various ways to combine the associativity and commutativity constraints, in order to exhibit some isomorphism X′ →^{∼} X′′. However, the resulting isomorphism shall be independent of the way in which it is expressed as such a combination. This follows from a theorem of Mac Lane. To formalize this result, one could observe that, for any set Σ, there exists a universal tensor category T_{Σ} “generated by Σ”, i.e. such that – for any other tensor category \underline{𝒞} – any mapping Σ → Ob(𝒞) extends uniquely, up to isomorphism, to a tensor functor T_{Σ} → 𝒞. Then Mac Lane’s theorem says that, for every set Σ, and every object X ∈ Ob(T_{Σ}) that has no repetitions, the group Aut_{T_{Σ}}(X) is trivial. Instead of relying on such a general result, we shall make ad hoc verifications, as in the proof of lemma 3.7.3 and of the forthcoming proposition 3.7.6. However, in view of this principle, in the following we shall often omit a detailed description of the isomorphism that we choose to connect two given objects that are thus related : the reader will be able in any case to produce one isomorphism, and the principle says that these choices cannot be source of ambiguities.

(ii) Let 𝒟 be any category, and \underline{𝒞} a tensor category. Then notice that Fun(𝒟, 𝒞) inherits from \underline{𝒞} a natural tensor category structure : we leave to the reader the task of spelling out the details. Moreover notice that if (F, c) : \underline{𝒞} _{1} → \underline{𝒞} _{2} and (F′, c′) : \underline{𝒞} _{2} → \underline{𝒞} _{3} are any two tensor functors between tensor categories, then the composition

(F′ ∘ F, (F′ ∗ c) ⊙ (c′ ∗ (F × F))) : \underline{𝒞} _{1} → \underline{𝒞} _{3}

is again a tensor functor.

Proposition 3.7.6. Let (U, u) be a unit object of a tensor category \underline{𝒞} . Then there exists a unique natural isomorphism

u_{X} : X →^{∼} U ⊗ X for every X ∈ Ob(𝒞)

such that u_{U} = u, and such that the diagrams

\xymatrix{X ⊗ Y \ar[r]^{u_{X⊗Y}} \ar[dr]_{u_{X}⊗Y} & U ⊗ (X ⊗ Y) \ar[d]^{Φ_{U,X,Y}} & X ⊗ Y \ar[r]^{u_{X}⊗Y} \ar[d]_{X⊗u_{Y}} & (U ⊗ X) ⊗ Y \ar[d]^{Ψ_{U,X}⊗Y} \\  & (U ⊗ X) ⊗ Y & X ⊗ (U ⊗ Y) \ar[r]^{Φ_{X,U,Y}} & (X ⊗ U) ⊗ Y}

commute for every X, Y ∈ Ob(𝒞).

Proof. Since (3.7.2) is an equivalence, there exists a unique isomorphism u_{X} fitting into the commutative diagram

\xymatrix{UX \ar[r]^{u⊗X} \ar[dr]_{U⊗u_{X}} & (UU)X \\  & U(UX). \ar[u]_{Φ}}

With this definition, the naturality of the rule : X ↦ u_{X} is clear. In order to check the commutativity of the first diagram, it suffices to show that

(3.7.7)  (U ⊗ Φ_{U,X,Y}) ∘ (U ⊗ u_{XY}) = U ⊗ (u_{X} ⊗ Y).

However, set Θ := (Φ_{U,U,X} ⊗ Y) ∘ Φ_{U,UX,Y}; we have :

Θ ∘ (U ⊗ (u_{X} ⊗ Y)) = (Φ_{U,U,X} ⊗ Y) ∘ ((U ⊗ u_{X}) ⊗ Y) ∘ Φ_{U,X,Y}
= ((u ⊗ X) ⊗ Y) ∘ Φ_{U,X,Y}
= Φ_{UU,X,Y} ∘ (u ⊗ (XY))
= Φ_{UU,X,Y} ∘ Φ_{U,U,XY} ∘ (U ⊗ u_{XY})
= Θ ∘ (U ⊗ Φ_{U,X,Y}) ∘ (U ⊗ u_{XY})

where the first and third identities hold by naturality of Φ, the second and fourth by the definition of u_{X} and respectively u_{X⊗Y}, and the fifth by coherence. Since Θ is an isomorphism, we get (3.7.7). Next, in light of the foregoing, the second diagram commutes if and only if the diagram

(3.7.8)  \xymatrix{XY \ar[r]^{u_{XY}} \ar[dr]_{X⊗u_{Y}} & U(XY) \ar[r]^{Φ} & (UX)Y \ar[d]^{Ψ⊗Y} \\  & X(UY) \ar[r]^{Φ} & (XU)Y}

commutes, and it suffices to check that U ⊗ (3.7.8) commutes. To this aim, we consider the diagram

\xymatrix{U(XY) \ar[rr]^{U⊗(X⊗u_{Y} )} \ar[dd]_{u⊗(XY)} \ar@{-->}[dr]^{U⊗u_{XY}} &  & U(X(UY)) \ar[r]^{U⊗Φ} & U((XU)Y) \ar[d]^{U⊗(Ψ⊗Y)} \\  & U(U(XY)) \ar[rr]^{U⊗Φ} \ar@{-->}[dl]_{Φ} &  & U((UX)Y) \ar[d]^{Φ} \\ (UU)(XY) \ar[rr]^{Φ} &  & ((UU)X)Y & (U(UX))Y. \ar[l]_{Φ⊗Y}}

whose lower subdiagram commutes by the coherence axiom for (U, U, X, Y), whose upper subdiagram is equivalent to (3.7.8), since Ψ_{U,X}^{−1} = Ψ_{X,U}, and whose triangular subdiagram commutes by definition of u_{XY}. Hence, we are reduced to showing that the outer rectangular subdiagram of the above diagram commutes. However, we have a commutative diagram :

\xymatrix{U(X(UY)) \ar[d]_{Φ} & U(XY) \ar[l]_{U⊗(X⊗u_{Y} )} \ar[r]^{u⊗(XY)} \ar[d]_{Φ} & (UU)(XY) \ar[d]^{Φ} \\ (UX)(UY) \ar[d]_{Ψ⊗(UY)} & (UX)Y \ar[l]_{(UX)⊗u_{Y}} \ar[r]^{(u⊗X)⊗Y} \ar[d]_{Ψ⊗Y} & ((UU)X)Y \ar[d]^{Ψ⊗Y} \\ (XU)(UY) & (XU)Y \ar[r]^{(X⊗u)⊗Y} \ar[l]_{(XU)⊗u_{Y}} & (X(UU))Y \\  & X(UY) \ar[dl]_{X⊗(U⊗u_{Y} )} \ar[dr]^{X⊗(u⊗Y)} \ar[u]^{Φ} \\ X(U(UY)) \ar[rr]^{X⊗Φ} \ar[uu]^{Φ} &  & X((UU)Y) \ar[uu]_{Φ}}

so we are further reduced to checking the commutativity of the diagram :

\xymatrix{ & U(X(UY)) \ar[r]^{U⊗Φ} \ar@{-->}[dl]_{Φ} & U((XU)Y) \ar[r]^{U⊗(Ψ⊗Y)} & U((UX)Y) \ar@{-->}[dr]^{Φ} \\ (UX)(UY) \ar[d]_{Ψ⊗(UY)} & U((UY)X) \ar[u]^{U⊗Ψ} \ar[d]_{Φ} & U(U(Y X)) \ar[l]_{U⊗Φ} \ar[r]^{U⊗(U⊗Ψ)} \ar[d]_{Φ} & U(U(XY)) \ar[u]^{U⊗Φ} \ar[d]_{Φ} & (U(UX))Y \ar[d]^{Φ⊗Y} \\ (XU)(UY) & (U(UY))X \ar[d]_{Ψ} \ar[dr]_{Φ⊗X} & (UU)(Y X) \ar[r]^{(UU)⊗Ψ} \ar[d]_{Φ} & (UU)(XY) \ar[r]^{Φ} & ((UU)X)Y \ar[dl]^{Ψ⊗Y} \\  & X(U(UY)) \ar@{-->}[ul]^{Φ} \ar[dr]_{X⊗Φ} & ((UU)Y)X \ar[d]_{Ψ} & (X(UU))Y \\  &  & X((UU)Y). \ar[ur]_{Φ}}

However, the leftmost and the lower triangular subdiagrams commute by compatibility, and the upper rightmost subdiagram commutes by coherence. The lower leftmost subdiagram commutes by naturality of Ψ, and the central square sudiagram commutes by naturality of Φ. The remaining central subdiagram commutes by coherence, and the top rectangular subdiagram is of the form U ⊗ D, where D is a diagram that commutes by virtue of lemma 3.7.3.

The uniqueness of u_{X} is clear by inspecting the second diagram, with Y = U. □

Remark 3.7.9. (i) Keep the notation of proposition 3.7.6. As a consequence of the naturality of u_{X}, we get a commutative diagram

\xymatrix{X \ar[r]^{u_{X}} \ar[d]_{u_{X}} & U ⊗ X \ar[d]^{u⊗u_{X}} \\ U ⊗ X \ar[r]^{u_{U⊗X}} & U ⊗ (U ⊗ X)}

for every object X of 𝒞. In other words :

u_{U⊗X} = U ⊗ u_{X} for every X ∈ Ob(𝒞).

(ii) Let X = Y = U in the second diagram of proposition 3.7.6; we obtain a commutative diagram

\xymatrix{U ⊗ U \ar[r]^{u⊗U} \ar[d]_{U⊗u} & (U ⊗ U) ⊗ U \ar[d]^{Ψ_{U,U} ⊗U} \\ U ⊗ (U ⊗ U) \ar[r]^{Φ_{U,U,U}} & (U ⊗ U) ⊗ U}

Since u = u_{U}, we may combine with (i), to deduce that Ψ_{U,U} ⊗ U = 1_{(U⊗U)⊗U} = 1_{U⊗U} ⊗ U. By naturality of Ψ, it follows that U ⊗ Ψ_{U,U} = U ⊗ 1_{U⊗U}, and since (3.7.2) is an equivalence, we conclude that

Ψ_{U,U} = 1_{U⊗U}.

Example 3.7.10. Let 𝒞 be any category with small Hom-sets, in which finite products are representable. For every pair (X, Y) of objects of 𝒞, pick an object X ⊗ Y representing their product, and fix also two projections p_{X,Y} : X ⊗ Y → X, q_{X,Y} : X ⊗ Y → Y inducing an isomorphism of functors

h_{X⊗Y} →^{∼} h_{X}×h_{Y} : φ ↦ (p_{X,Y} ∘φ, q_{X,Y} ∘φ) for every Z ∈ Ob(𝒞) and φ ∈ h_{X×Y} (Z)

(notation of (1.2.4)). If (X′, Y′) is another such pair, and (g, h) : (X, Y) → (X′, Y′) any morphism in 𝒞 × 𝒞, then there exists a unique morphism f : X ⊗ Y → X′ ⊗ Y′ such that

(p_{X′,Y′} ∘ f, q_{X′,Y′} ∘ f) = (g ∘ p_{X,Y} , h ∘ p_{X,Y})

and we set g ⊗ h := f. These rules define a functor ⊗ : 𝒞 × 𝒞 → 𝒞. For every three objects X, Y, Z there is a natural isomorphism X ⊗(Y ⊗Z) →^{∼} (X ⊗Y)⊗Z that yields an associativity constraint for ⊗; namely, we let

Φ_{X,Y,Z} := (p_{X,Y ⊗Z} ⊗ (p_{Y,Z} ⊗ q_{X,Y ⊗Z})) ⊗ (q_{Y,Z} ∘ q_{X,Y ⊗Z}).

Likewise, we get a commutativity constraint by setting

Ψ_{X,Y} := q_{X,Y} ⊗ p_{X,Y} for every X, Y ∈ Ob(𝒞).

The verifications of the axioms of definition 3.7.1 are lengthy but straightforward, and shall be left to the reader. If U is any final object of 𝒞 (example 1.2.16(iv)), then there exists a unique morphism u : U → U ⊗ U which is easily seen to be an isomorphism, and the pair (U, u) yields a unit for ⊗. In this way, any category with finite products and small Hom-sets is naturally endowed with a structure of tensor category. Notice that, for this tensor structure on 𝒞 (and indeed, for most of the tensor categories that are found in applications), the existence of functorial isomorphisms u_{X} fulfilling the conditions of proposition 3.7.6, is self-evident.

Definition 3.7.11. Let (𝒞 , ⊗, Φ, Ψ) be a tensor category, X ∈ Ob(𝒞) any object, and suppose that the functor

− ⊗ X : 𝒞 → 𝒞 Y ↦ Y ⊗ X

admits a right adjoint :

ℋ om(X, −) : 𝒞 → 𝒞 Y ↦ ℋ om(X, Y).

Then, we call ℋ om(X, −) the internal Hom functor for the object X.

Remark 3.7.12. (i) As usual, the internal Hom functor is determined up to unique isomorphism, if it exists. The counit of adjunction is a morphism of 𝒞

ev_{X,Y} : ℋ om(X, Y) ⊗ X → Y

called the evaluation morphism.

(ii) Suppose that every object of 𝒞 admits an internal Hom functor; then we say briefly that 𝒞 admits an internal Hom functor, and clearly we get a functor

𝒞^{o} × 𝒞 → 𝒞 : (X, Y) ↦ ℋ om(X, Y) for every X, Y ∈ Ob(𝒞).

Moreover, for every X, Y, Z ∈ Ob(𝒞) the composition

[figure: (ℋ om(X, Y) ⊗ ℋ om(Y, Z)) ⊗ X; ∼; (ℋ om(X, Y) ⊗ X) ⊗ ℋ om(Y, Z); ev_{X,Y} ⊗ℋom(Y,Z); Z ⟵^{ev_{Y,Z}} ℋ om(Y, Z) ⊗ Y; Y ⊗ ℋ om(Y, Z); ∼]

corresponds, by adjunction, to a unique composition morphism

ℋ om(X, Y) ⊗ ℋ om(Y, Z) → ℋ om(X, Z).

(iii) In the situation of (ii), notice that the functor 𝒞 → 𝒞 given by the rule : Z ↦ ℋ om(X, ℋ om(Y, Z)), for every Z ∈ Ob(𝒞), is right adjoint to the functor given by the rule : Z ↦ (Z ⊗ X) ⊗ Y →^{∼} Z ⊗ (X ⊗ Y). There follows a natural isomorphism

ℋ om(X, ℋ om(Y, Z)) →^{∼} ℋ om(X ⊗ Y, Z) for every X, Y, Z ∈ Ob(𝒞).

(iv) Moreover, for any unit (U, u) of \underline{𝒞} , we get natural bijections :

Hom_{𝒞} (U, ℋ om(X, Y)) →^{∼} Hom_{𝒞} (U ⊗ X, Y) →^{∼} Hom_{𝒞} (X, Y) for every X, Y ∈ Ob(𝒞).

Also, for every object Y of 𝒞, denote by u_{Y} : Y →^{∼} U⊗Y the isomorphism given by proposition 3.7.6; for every X ∈ Ob(X), it induces natural bijections

Hom_{𝒞} (Y, ℋ om(U, X)) →^{∼} Hom_{𝒞} (Y ⊗ U, X) ⟶^{Hom_{𝒞} (u_{Y} ∘Ψ_{U,Y} ,X)} Hom_{𝒞} (Y, X)

which correspond, via the Yoneda embedding, to a natural isomorphism

ℋ om(U, X) →^{∼} X for every X ∈ Ob(𝒞).

(v) Let X, Y, Z be any three objects of 𝒞; the natural transformation

Hom_{𝒞} (W ⊗ X, Y) → Hom_{𝒞} (W ⊗ (X ⊗ Z), Y ⊗ Z) φ ↦ (φ ⊗ Z) ∘ Φ_{W,X,Z}

corresponds, via the Yoneda embedding, to a unique morphism

t_{X,Y,Z} : ℋ om(X, Y) → ℋ om(X ⊗ Z, Y ⊗ Z).

The reader can check that t_{X,Y,Z} also corresponds, by adjunction, to the morphism

(ev_{X,Y} ⊗ Z) ∘ Φ_{ℋom(X,Y),X,Z} : ℋ om(X, Y) ⊗ (X ⊗ Z) → Y ⊗ Z.

(vi) In the situation of remark 3.7.5(ii), suppose that 𝒞 admits an internal Hom functor; then it is easily seen that the resulting tensor category Fun(𝒟, 𝒞) inherits as well an internal Hom functor, in the obvious way.

The formalism of tensor categories provides the language to deal uniformly with the notions of algebras and their modules that occur in various concrete settings.

Definition 3.7.13. Let (𝒞 , ⊗, Φ, Ψ) be a tensor category, A and B any two objects of 𝒞.

(i) A left A-module (resp. a right B-module) is a datum (X, μ_{X}), consisting of an object X of 𝒞, and a morphism in 𝒞 :

μ_{X} : A ⊗ X → X (resp. μ_{X} : X ⊗ B → X)

called the scalar multiplication of X. (ii) A morphism of left A-modules (X, μ_{X}) → (X′, μ_{X′}) is a morphism f : X → X′ in 𝒞

which makes commute the diagram :

\xymatrix{A ⊗ X \ar[r]^{μ_{X}} \ar[d]_{1_{A}⊗f} & X \ar[d]^{f} \\ A ⊗ X′ \ar[r]^{μ_{X′}} & X′.}

One defines likewise morphisms of right B-modules. (iii) An (A, B)-bimodule is a datum (X, μ_{X}^{l}, μ_{X}^{r}) such that (X, μ_{X}^{l}) is a left A-module,

(X, μ_{X}^{r}) is a right B-module, and the scalar multiplications commute, i.e. the diagram

\xymatrix{A ⊗ (X ⊗ B) \ar[rr]^{1_{A}⊗μ_{X}^{r}} \ar[d]_{Φ_{A,X,B}} &  & A ⊗ X \ar[d]^{μ_{X}^{l}} \\ (A ⊗ X) ⊗ B \ar[r]^{μ_{X}^{l}⊗1_{B}} & X ⊗ B \ar[r]^{μ_{X}^{r}} & X}

commutes. Of course, a morphism of (A, B)-bimodules must be compatible with both left and right multiplication.

We denote by A-Mod_{l} (resp. B-Mod_{r}, resp. (A, B)-Mod) the category of left A-modules (resp. right B-modules, resp. (A, B)-bimodules). For any two left A-modules (resp. right B-modules, resp. (A, B)-bimodules) X and X′, we shall write

Hom_{A_{l}}(X, X′) ( resp. Hom_{B_{r}}(X, X′) ) ( resp. Hom_{(A,B)}(X, X′) )

for the set of morphisms of left A-modules (resp. of right B-modules, resp. of (A, B)- bimodules) X → X′.

3.7.14. In the situation of definition 3.7.13, notice that

Hom_{B_{r}}(X, X′) = Equal( \xymatrix{Hom_{𝒞} (X, X′) \ar[r]^{α} \ar[r]_{β} & Hom_{𝒞} (X ⊗ B, X′)} )

where α (resp. β) is given by the rule :

f ↦ f ∘ μ_{X} ( resp. f ↦ μ_{X′} ∘ (f ⊗ B) ) for every f ∈ Hom_{𝒞} (X, X′)

and similarly for left A-modules. Now, suppose that all equalizers in 𝒞 are representable, and that 𝒞 admits an internal Hom functor; then we may define

ℋ om_{B_{r}}(X, X′) := Equal( \xymatrix{ℋ om(X, X′) \ar[r]^{α} \ar[r]_{β} & ℋ om(X ⊗ B, X′)} )

where α := ℋ om(μ_{X}, X′) and β := ℋ om(X ⊗ B, μ_{X′}) ∘ t_{X,X′,B} (notation of remark 3.7.12(v)). Then, it is easily seen that the bijections of remark 3.7.12(iv) induce natural identifications

Hom_{𝒞} (U, ℋ om_{B_{r}}(X, X′)) →^{∼} Hom_{B_{r}}(X, X′) for every X, X′ ∈ Ob(B_{r}-Mod).

Likewise we may represent in 𝒞 the set of morphisms between two left A-modules, and two (A, B)-modules (details left to the reader).

3.7.15. Let \underline{𝒞} be a tensor category as in (3.7.14) and A, B, C any three objects of 𝒞; suppose that (X, μ_{X}^{l}, μ_{X}^{r}) is an (A, B)-bimodule, and (X′, μ_{X′}^{l}, μ_{X′}^{r}) a (C, B)-bimodule. Then we claim that ℋ := ℋ om_{B_{r}}((X, μ_{X}^{r}), (X′, μ_{X′}^{r})) is naturally a (C, A)-bimodule. For this, we have to exhibit natural morphisms

C ⊗ ℋ ⟶^{μ_{l}} ℋ ⟵^{μ_{r}} ℋ ⊗ A

fulfilling the condition of definition 3.7.13(iii). However, by adjunction, the datum of μ_{l} is the same as that of a morphism C → ℋ om(ℋ , ℋ ), and since the functor ℋ om(X, −) is left exact, the latter is the same as a morphism

C → Equal( [figure: ℋ om(ℋ , ℋ om(X, X′)); ℋom(ℋ ,α); ℋom(ℋ ,β); ℋ om(ℋ , ℋ om(X ⊗ B, X′))]

which in turn – by remark 3.7.12(iii) – corresponds to a morphism

C → Equal( [figure: ℋ om(ℋ ⊗ X, X′); ℋom(ℋ ⊗α,X′); ℋom(ℋ ⊗β,X′); ℋ om(ℋ ⊗ (X ⊗ B), X′)]

and again, the latter is the same as an element of

Equal( \xymatrix{Hom_{𝒞} (C ⊗ (ℋ ⊗ X), X′) \ar[r]^{Hom_{𝒞} (C⊗(ℋ ⊗α),X′)} \ar[r]_{Hom_{𝒞} (C⊗(ℋ ⊗β),X′)} & Hom_{𝒞} (C ⊗ (ℋ ⊗ (X ⊗ B)), X′).}

By unwinding the definition, it is easily seen that the composition

\overline{μ}_{l} : C ⊗ (ℋ ⊗ X) ⟶^{ev_{X,X′}} C ⊗ X′ ⟶^{μ_{X′}^{l}} X′

lies in the above equalizer, and it provides a left C-module structure for ℋ . Likewise, μ_{r} shall be the morphism corresponding to the composition

\overline{μ}_{r} : (ℋ ⊗ X) ⊗ A →^{∼} ℋ ⊗ (A ⊗ X) ⟶^{ℋ ⊗μ_{X}^{l}} ℋ ⊗ X ⟶^{ev_{X,X′}} X′.

Then, the condition that (ℋ , μ_{l}, μ_{r}) is a bimodule, comes down to the commutativity of the diagram

\xymatrix{C ⊗ ((ℋ ⊗ X) ⊗ A) \ar[rr]^{1_{C}⊗\overline{μ}_{r}} \ar[d]_{C⊗Φ_{ℋ ,X,A}} &  & C ⊗ X′ \ar[d]^{μ_{X′}^{l}} \\ C ⊗ (ℋ ⊗ (X ⊗ A)) \ar[r]^{C⊗(ℋ ⊗(μ_{X}^{l}∘Ψ_{X,A}))} & C ⊗ (ℋ ⊗ X) \ar[r]^{_{\overline{μ}_{l}}} & X′}

which is immediate (details left to the reader). We have thus obtained a bifunctor :

(3.7.16)  ℋ om_{B_{r}}(−, −) : (A, B)-Mod^{o} × (C, B)-Mod → (C, A)-Mod.

Likewise, we may define a bifunctor :

ℋ om_{A_{l}}(−, −) : (A, B)-Mod^{o} × (A, C)-Mod → (B, C)-Mod.

3.7.17. Keep the situation of (3.7.15), and suppose moreover that all coequalizers in 𝒞 are representable. Fix an (A, B)-bimodule (X, μ_{X}^{l}, μ_{X}^{r}); the functor

(C, B)-Mod → (C, A)-Mod : X′ ↦ ℋ om_{B_{r}}(X, X′)

admits a left adjoint, the tensor product

(C, A)-Mod → (C, B)-Mod : (X′, μ_{X′}^{l}, μ_{X′}^{l}) ↦ (X′, μ_{X′}^{l}, μ_{X′}^{r}) ⊗_{A} (X, μ_{X}^{l}, μ_{X}^{r})

given by the coequalizer (in 𝒞) of the morphisms :

\xymatrix{X′ ⊗ (A ⊗ X) \ar[r]^{1_{X′}⊗μ_{X}^{l}} \ar[r]_{(μ_{X′}^{r}⊗1_{X})∘Φ_{X′,A,X}} & X′ ⊗ X}

with scalar multiplications induced by μ_{X}^{r} and μ_{X′}^{l}. Likewise, we have a functor :

(A, C)-Mod → (B, C)-Mod : (X′, μ_{X′}^{l}, μ_{X′}^{l}) ↦ (X, μ_{X}^{l}, μ_{X}^{r}) ⊗_{A} (X′, μ_{X′}^{l}, μ_{X′}^{l})

which admits a similar description, and is left adjoint to the functor X′ ↦ ℋ om_{A_{l}}(X, X′) (verifications left to the reader).

3.7.18. Let (U, u) be unit object for \underline{𝒞} . Notice that, for any object A of 𝒞, the rule (Y, μ_{Y}) ↦ (Y, u_{Y}^{−1}, μ_{Y}) (where u_{Y} : Y → U ⊗ Y is the natural isomorphism supplied by proposition 3.7.6), induces a faithful functor A-Mod_{r} → (U, A)-Mod. Letting C := U in (3.7.17), we see that any (A, B)-bimodule X also determines a functor :

A-Mod_{r} → B-Mod_{r} : Y ↦ Y ⊗_{A} X

and likewise for left A-modules.

Example 3.7.19. If 𝒞 = Set is the category of sets (regarded as a tensor category as in example 3.7.21), then a left A-module is just a set X′ with a left action of A, i.e. a map of sets

A × X′ → X′ : (a, x) ↦ a · x.

An (A, A)-bimodule X is a set with both left and right actions of A, such that (a · x) · a′ = a · (x · a′) for every a, a′ ∈ A and every x ∈ X. With this notation, the tensor product X′ ⊗_{A} X is the quotient (X′ × X)/∼, where ∼ is the smallest equivalence relation such that (x′a, x) ∼ (x′, ax) for every x ∈ X, x′ ∈ X′ and a ∈ A.

Definition 3.7.20. Let \underline{𝒞} := (𝒞 , ⊗, Φ, Ψ) be a tensor category, and (U, u) a unit for \underline{𝒞} .

(i) A \underline{𝒞} -semigroup is a datum (M, μ_{M}) consisting of an object M of 𝒞 and a morphism μ_{M} : M ⊗M → M, the multiplication law of \underline{M}, such that (M, μ_{M}, μ_{M}) is a (M, M)- bimodule. A morphism of \underline{𝒞} -semigroups is a morphism φ : M → M′ in 𝒞, such that

μ_{M′} ∘ (φ ⊗ φ) = φ ∘ μ_{M}.

(ii) A \underline{𝒞} -monoid is a datum \underline{M} := (M, μ_{M}, 1_{M}), where (M, μ_{M}) is a semigroup, and 1_{M} : U → M is a morphism in 𝒞, called the unit of \underline{M}, such that

μ_{M} ∘ (1_{M} ⊗ 1_{M}) ∘ u_{M} = 1_{M} = μ_{M} ∘ (1_{M} ⊗ 1_{M}) ∘ u_{M}

where u_{M} : M →^{∼} U ⊗ M is the natural isomorphism provided by proposition 3.7.6. We say that \underline{M} is commutative, if

μ_{M} = μ_{M} ∘ Ψ_{M,M}.

A morphism of monoids \underline{M} → \underline{M}′ is a morphism of semigroups φ : M → M′ such that φ ∘ 1_{M} = 1_{M′}.

Example 3.7.21. (i) Let 𝒞 be any category with small Hom-sets, in which finite products are representable, endow 𝒞 with the tensor category structure described in example 3.7.10, and pick a final object 1_{𝒞} of 𝒞. Then, a \underline{𝒞} -monoid is a datum \underline{M} := (M, μ_{M}, 1_{M}), where μ_{M} : M × M → M and 1_{M} : 1_{𝒞} → M are morphisms of 𝒞, and the axioms for μ_{M} and 1_{M} can be rephrased as requiring that, for every object X of 𝒞, the set M(X) := Hom_{𝒞} (X, M), endowed with the composition law :

M(X) × M(X) →^{∼} Hom_{𝒞} (X, M × M) ⟶^{Hom_{𝒞} (X,μ_{M})} M(X) (m, m′) ↦ m · m′

is a (usual) monoid, with unit Im 1_{M}(X) ∈ M(X). Of course, \underline{M} is commutative, if and only if m · m′ = m′ · m for all objects X of 𝒞 and every m, m′ ∈ M(X).

(ii) In the situation of (i), a \underline{𝒞} -monoid shall also be called simply a 𝒞 -monoid. The category of 𝒞 -monoids admits an initial object which is also a final object, namely \underline{1}_{𝒞} := (1_{𝒞} , μ_{1}, 1_{1}), where μ_{1} is the (unique) morphism 1_{𝒞} × 1_{𝒞} → 1_{𝒞} . (Most of the above can be repeated with the theory of semigroups replaced by any ”algebraic theory” in the sense of [29, Def.3.3.1] : e.g. in this way one can define 𝒞 -groups, 𝒞 -rings, and so on.)

3.7.22. Let \underline{𝒞} and U be as in definition 3.7.20, and \underline{M} := (M, μ_{M}, 1_{M}) a \underline{𝒞} -monoid; of course, we are especially interested in the M-modules which are compatible with the unit and multiplication law of M. Hence we define a left \underline{M}-module as a left M-module (S, μ_{S}) such that the following diagrams commute :

[figure: U ⊗ S; 1_{M} ⊗1_{S}; U ⊗ S; M ⊗ (M ⊗ S); 1_{M} ⊗μ_{S}; Φ_{M,M,S}; M ⊗ S; μ_{S}; M ⊗ S; μ_{S}; S; u_{S}; (M ⊗ M) ⊗ S; μ_{M} ⊗1_{S}; M ⊗ S; μ_{S}; S]

where u_{S} is the isomorphism given by proposition 3.7.6. Likewise we define right \underline{M}-modules, and (\underline{M}, \underline{N})-bimodules, if \underline{N} is a second \underline{𝒞} -monoid; especially, (\underline{M}, \underline{M})-bimodules shall also be called simply \underline{M}-bimodules.

A morphism of left \underline{M}-modules (S, μ_{S}) → (S′, μ_{S′}) is just a morphism of left M-modules, and likewise for right modules and bimodules. For instance, \underline{M} is a \underline{M}-bimodule in a natural way, and an ideal of M is a sub-\underline{M}-bimodule I of \underline{M}. We denote by \underline{M}-Mod_{l} (resp. \underline{M}-Mod_{r}, resp. \underline{M}-Mod) the category of left (resp. right, resp. bi-) \underline{M}-modules; more generally, if \underline{N} is a second \underline{𝒞} -monoid, we have the category (\underline{M}, \underline{N})-Mod of the corresponding bimodules.

Example 3.7.23. Take 𝒞 := Set, regarded as a tensor category, as in example 3.7.10. Then a 𝒞 -monoid is just a usual monoid M, and a left M-module is a datum (S, μ_{S}) consisting of a set S and a scalar multiplication M × S → S : (m, s) ↦ m · s such that

1 · s = s and x · (y · s) = (x · y) · s for every x, y ∈ M and every s ∈ S.

A morphism φ : (S, μ_{S}) → (T, μ_{T}) of M-modules is then a map of sets S → T such that

x · φ(s) = φ(x · s) for every x ∈ M and every s ∈ S

Likewise, an ideal of M is a subset I ⊂ M such that a · x, x · a ∈ I whenever a ∈ I and x ∈ M.

Remark 3.7.24. Let 𝒞 be a complete and cocomplete category, whose colimits are universal (see example 1.4.17), and \underline{M} a 𝒞 -monoid (see example 3.7.21(ii)).

(i) The categories \underline{M}-Mod_{l}, \underline{M}-Mod_{r} and (\underline{M}, \underline{N})-Mod are complete and cocomplete, and the forgetful functor \underline{M}-Mod_{l} → 𝒞 (resp. the same for right modules and bimodules) commutes with all limits and colimits.

(ii) Notice also that the forgetful functor \underline{M}-Mod_{l} → 𝒞 is conservative. Together with (i) and propositions 1.3.18 and 1.3.21(i), this implies that a morphism of left \underline{M}-modules is a monomorphism (resp. an epimorphism) if and only if the same holds for the underlying morphism in 𝒞 (and likewise for right modules and bimodules).

(iii) For each of these categories, the initial object is just the initial object ∅_{𝒞} of 𝒞, endowed with the trivial scalar multiplication. Likewise, the final object is the final object 1_{𝒞} of 𝒞, with scalar multiplication given by the unique morphism M × 1_{𝒞} → 1_{𝒞} . Moreover, the forgetful functor \underline{M}-Mod_{l} → 𝒞 admits a left adjoint, that assigns to any Σ ∈ Ob(𝒞) the free \underline{M}-module \underline{M}^{(Σ)} generated by Σ; as an object of 𝒞, the latter is just M × Σ, and the scalar multiplication is derived from the composition law of \underline{M}, in the obvious way.

For instance, for any n ∈ ℕ, and any left (or right or bi-) \underline{M}-module S, we denote as usual by S^{⊕n} the coproduct of n copies of S.

Remark 3.7.25. Let \underline{𝒞} be a tensor category, \underline{M}, \underline{N}, \underline{P}, \underline{Q} four \underline{𝒞} -monoids.

(i) Let S be a (\underline{M}, \underline{N})-bimodule, S′ a (\underline{P} , \underline{N})-bimodule and S′′ a (\underline{P} , \underline{M})-bimodule. Then it is easily seen that the (P, M)-bimodule (resp. the (P, N)-bimodule) ℋ om_{N_{r}}(S, S′) (resp. S′′ ⊗_{M} S) is actually a (\underline{P} , \underline{M})-bimodule (resp. a (\underline{P} , \underline{N})-bimodule) and the adjunction of (3.7.17) restricts to an adjunction between the corresponding categories of bimodules : the details shall be left to the reader.

(ii) We have as well the analogue of the usual associativity constraints. Namely, for every (\underline{M}, \underline{N})-bimodule S, every (\underline{N}, \underline{P})-bimodule S′ and every (\underline{P} , \underline{Q})-bimodule S′′, there is a natural isomorphism

(S ⊗_{N} S′) ⊗_{P} S′′ →^{∼} S ⊗_{N} (S′ ⊗_{P} S′′) in (\underline{M}, \underline{Q})-Mod

and natural isomorphisms M ⊗_{M} S →^{∼} S →^{∼} S ⊗_{N} N in (\underline{M}, \underline{N})-Mod.

(iii) Also, if \underline{M} is commutative, every left (or right) \underline{M}-module is naturally a (\underline{M}, \underline{M})- bimodule, and we have a commutative constraint

S ⊗_{M} S′ →^{∼} S′ ⊗_{M} S for all left (or right) \underline{M}-modules.

And taking into account (ii), it is easily seen that (\underline{M}-Mod_{l}, ⊗_{M}) is a tensor category.

3.7.26. Let φ : \underline{M}_{1} → \underline{M}_{2} be a morphism of \underline{𝒞} -monoids; we have the (\underline{M}_{1}, \underline{M}_{2})-bimodule :

M_{1,2} := (M_{2}, μ_{M_{2}} ∘ (φ ⊗ 1_{M_{2}}), μ_{M_{2}}).

Letting X := M_{1,2} in (3.7.18), we obtain a base change functor for right modules :

\underline{M}_{1}-Mod_{r} → \underline{M}_{2}-Mod_{r} : X ↦ X ⊗_{M_{1}} M_{2} := X ⊗_{M_{1}} M_{1,2}.

The base change is left adjoint to the restrictions of scalars associated with φ, i.e. the functor :

\underline{M}_{2}-Mod_{r} → \underline{M}_{1}-Mod_{l} : (X, μ_{X}) ↦ (X, μ_{X})_{(φ)} := (X, μ_{X} ∘ (1_{X} ⊗ φ))

(verifications left to the reader). The same can be repeated, as usual, for left modules; for bimodules, one must take the tensor product on both sides : X ↦ M_{2} ⊗_{M_{1}} X ⊗_{M_{1}} M_{2}.

Example 3.7.27. Take 𝒞 = Set, and let M be any monoid, Σ any set, and M^{(Σ)} the free M-module generated by Σ. From the isomorphism

M^{(Σ)} ⊗_{M} {1} →^{∼} {1}^{(Σ)} = Σ

we see that the cardinality of Σ is an invariant, called the rank of the free M-module M^{(Σ)}, and which we denote rk_{M}M^{(Σ)}.

Definition 3.7.28. (i) A pre-additive category is the datum of a category 𝒜 and of an abelian group structure on Hom_{𝒜} (A, B) for every A, B ∈ Ob(𝒜) (especially, the Hom-sets of 𝒜 are not empty), such that the following holds. For every A, B, C ∈ Ob(𝒜), the composition law

Hom_{𝒜} (A, B) × Hom_{𝒜} (B, C) → Hom_{𝒜} (A, C)

is a bilinear pairing.

(ii) A functor F : 𝒜 → ℬ between pre-additive categories is additive if it induces group homomorphisms

Hom_{𝒜} (X, Y) → Hom_{ℬ} (F X, F Y) : φ ↦ F φ for every X, Y ∈ Ob(𝒜) .

Remark 3.7.29. Let 𝒜 be any pre-additive category, and choose a universe U such that 𝒜 is U-small, so that all the finite limits and finite colimits of 𝒜 are well defined as U-presheaves.

(i) If A ∈ Ob(𝒜) is any object, denote by 0_{A} the neutral element of the abelian group End_{𝒜} (A). Suppose that the equalizer of the pair of morphisms 1_{A}, 0_{A} : A → A is representable by an object 0 of 𝒜 (see example 1.2.16(ii)). Then, the datum of a morphism B → 0 is the same as that of a morphism φ : B → A such that φ = 0_{A} ∘ φ. By the bilinearity of the Hom-pairing, the latter condition holds if and only if φ is the neutral element of Hom_{𝒜} (B, A). We conclude that 0 is a final object in 𝒜. Dually, if the coequalizer of the pair (1_{A}, 0_{A}) is representable by some object 0′ of 𝒜, then 0′ is initial in 𝒜. Moreover, if 𝒜 admits a final object 0, then it is easily seen that the unique morphism A → 0 is also the coequalizer of the pair (1_{A}, 0_{A}), so 0 is also an initial object. Conversely, if 𝒜 admits an initial object, then this object is also final in 𝒜, and for any two objects A, B of 𝒜, the neutral element 0_{A,B} of Hom_{𝒜} (A, B) is the unique morphism that factors through 0. We say that 0 is a zero object for 𝒜.

(ii) Suppose that the product A_{1} × A_{2} is representable in 𝒜 for given A_{1}, A_{2} ∈ Ob(𝒜). Denote by p_{i} : A_{1} × A_{2} → A_{i} (i = 1, 2) the projections; then, there are unique morphisms e_{i} : A_{i} → A_{1} × A_{2} (i = 1, 2) such that

(3.7.30)  p_{i} ∘ e_{i} = 1_{A_{i}}  for i = 1, 2 and p_{i} ∘ e_{j} = 0_{A_{j},A_{i}}  for i ≠ j.

Notice that

(3.7.31)  e_{1} ∘ p_{1} + e_{2} ∘ p_{2} = 1_{A_{1}×A_{1}}.

Indeed, we have

p_{i} ∘ (e_{1} ∘ p_{1} + e_{2} ∘ p_{2}) = (p_{i} ∘ e_{1} ∘ p_{1}) + (p_{i} ∘ e_{2} ∘ p_{2}) = p_{i} i = 1, 2

by bilinearity of the Hom pairing, and 1_{A_{1}×A_{2}} is the unique endomorphism φ of A_{1} × A_{2} such that p_{i} ∘ φ = p_{i} for i = 1, 2. It follows that A_{1} × A_{2} also represents the coproduct A_{1} ⨿ A_{2}. Indeed, say that f_{i} : A_{i} → B, for i = 1, 2, are two morphisms to another object B of 𝒜, and set f := f_{1} ∘ p_{1} + f_{2} ∘ p_{2} : A_{1} × A_{2} → B; it is easily seen that f ∘ e_{i} = f_{i} for i = 1, 2, and by virtue of (3.7.31), the morphism f is the unique one that satisfies these identities. Conversely, if the coproduct of A_{1} and A_{2} is representable, a similar argument shows that also A_{1} × A_{2} is representable. We say that A_{1} × A_{2} is a biproduct of A_{1} and A_{2}, and denote it by A_{1} ⊕ A_{2}.

(iii) Notice that the morphisms (p_{i}, e_{i} | i = 1, 2) with the identities (3.7.30) and (3.7.31) characterize A_{1} ⊕ A_{2} up to unique isomorphism. Namely, say that B is another object of 𝒜, for which exist morphisms p′_{i} : B → A_{i} and e′_{i} : A_{i} → B (i = 1, 2) such that p′_{i} ∘ e′_{i} = 1_{A_{i}} for i = 1, 2, and p′_{i} ∘ e′_{j} = 0_{A_{j}A_{i}} for i ≠ j, and moreover e′_{1} ∘ p′_{1} + e′_{2} ∘ p′_{2} = 1_{B}. Then the pair (e′_{1}, e′_{2}) (resp. (p′_{1}, p′_{2})) induces a morphism e′ : A_{1} ⊕ A_{2} → B (resp. p′ : B → A_{1} ⊕ A_{2}) with

p_{j} ∘ p′ ∘ e′ ∘ e_{i} = p′_{j} ∘ e′_{i} = p_{j} ∘ e_{i} for i, j = 1, 2

which – by virtue of the universal properties of the biproduct – implies that p′ ∘ e′ = 1_{A_{1}⊕A_{2}}. Likewise, we may compute

\begin{matrix}e′ ∘ p′ = & (e′ ∘ e_{1} ∘ p_{1} + e′ ∘ e_{2} ∘ p_{2}) ∘ (e_{1} ∘ p_{1} ∘ p′ + e_{2} ∘ p_{2} ∘ p′) \\ = & e′ ∘ e_{1} ∘ p_{1} ∘ p′ + e′ ∘ e_{2} ∘ p_{2} ∘ p′ \\ = & e′_{1} ∘ p′_{1} + e′_{2} ∘ p′_{2} = 1_{B} \end{matrix}

whence the contention.

(iv) Suppose that B_{1} and B_{2} are any other two objects of 𝒜 such that B_{1} ⊕ B_{2} is also representable; given two morphisms f_{1} : A_{1} → B_{1} and f_{2} : A_{2} → B_{2}, we denote by f_{1} ⊕ f_{2} : A_{1} ⊕ A_{2} → B_{1} ⊕ B_{2} the unique morphism such that

p_{B,i} ∘ (f_{1} ⊕ f_{2}) ∘ e_{A,i} = f_{i} for i = 1, 2, and p_{B,i} ∘ (f_{1} ⊕ f_{2}) ∘ e_{A,j} = 0_{A_{j},B_{i}} for i ≠ j.

Notice that

(3.7.32)  f_{1} ⊕ f_{2} = (f_{1} ⊕ 0_{A_{2},B_{2}}) + (0_{A_{1},B_{1}} ⊕ f_{2})

(where the sum is taken in the abelian group Hom_{𝒜} (A_{1} ⊕ A_{2}, B_{1} ⊕ B_{2})); indeed, by bilinearity of the Hom pairing, it is easily seen that the right-hand side of (3.7.32) also satisfies the same identities above that define f_{1} ⊕ f_{2}.

(v) If f : A → B is any morphism of 𝒜, then we define the kernel (resp. cokernel) of f as the equalizer (resp. coequalizer)

Ker f := Equal(f, 0_{A,B}) Coker f := Coequal(f, 0_{A,B}).

Suppose that all kernels and cokernels of 𝒜 are representable in 𝒜, and denote by

ι_{f} : Ker f → A π_{f} : B → Coker f

the natural morphisms in 𝒜. Notice that ι_{f} is a monomorphism, and π_{f} an epimorphism. Notice also that f factors uniquely in 𝒜 as a composition

(3.7.33)  A ⟶^{π_{ι_{f}}} Coker ι_{f} ⟶^{β_{f}} Ker π_{f} ⟶^{ι_{π_{f}}} B.

(vi) Suppose that 𝒜 admits a zero object 0, and let f : A → B be any morphism; by definition Ker f is the presheaf such that

Ker f(C) = {g : B → C | g ∘ f = g ∘ 0_{A,B} = 0_{A,C}}.

If f is a monomorphism, the identity g ∘ f = 0_{A,C} = 0_{B,C} ∘ f implies that g = 0_{B,C}, so Ker f is represented by 0. Dually, if f is an epimorphism, then Coker f is represented by 0.

Remark 3.7.34. Let 𝒜, ℬ be any two pre-additive categories that admit a zero object, and F : 𝒜 → ℬ a functor.

(i) If F is additive, remark 3.7.29(iii) immediately implies that F transforms representable biproducts into representable biproducts. The latter assertion still holds in case F is not necessarily additive, but is either left or right exact. Indeed, suppose that F is left exact, let A_{1} ⊕ A_{2} be any biproduct, and let p_{i}, e_{i} be the morphisms described in remark 3.7.29(ii); by left exactness, F (A_{1} ⊕ A_{2}) represents the product of F A_{1} and F A_{2}, whence an isomorphism F A_{1} ⊕ F A_{2} →^{∼} F (A_{1} ⊕ A_{2}) which identifies F p_{1} and F p_{2} with the natural projections. Moreover, F transforms the final object of 𝒜 into the final object of ℬ (see example 1.2.16(iv)); then, by inspecting the argument in remark 3.7.29(ii), it is easily seen that F identifies as well F e_{i} with the natural injections F A_{i} → F A_{1} ⊕ F A_{2}, for i = 1, 2, so the assertion follows from remark 3.7.29(iii). A similar argument works in case F is right exact.

(ii) Suppose moreover, that all biproducts of 𝒜 are representable. Then we claim that the abelian group structure on Hom_{𝒜} (A, B) is determined by the category 𝒜, i.e. if ℬ is any other pre-additive category, and F : 𝒜 → ℬ is any equivalence of categories, then F induces group isomorphisms (and not just bijections) on Hom sets. Indeed, let A and B be any two objects of 𝒜, and denote by Δ_{A} : A → A ⊕ A (resp. μ_{B} : B ⊕ B → B) the unique morphism such that p_{i} ∘ Δ_{A} = 1_{A} (resp. μ_{B} ∘ e_{i} = 1_{B}) for i = 1, 2. Then we have

f_{1} + f_{2} = μ_{B} ∘ (f_{1} ⊕ f_{2}) ∘ Δ_{A} for every f_{1}, f_{2} : A → B

where f_{1} + f_{2} denotes the sum in the abelian group Hom_{𝒜} (A, B). Indeed, since clearly 0_{A,B} ⊕ 0_{A,B} = 0_{A⊕A,B⊕B}, identity (3.7.32) reduces to checking that f_{1} = μ_{B} ∘ (f_{1} ⊕ 0_{A,B}) ∘ Δ_{A} (and likewise for f_{2}), which follows easily from (3.7.31) : details left to the reader.

(iii) Combining (i) and (ii) we see that if all biproducts of 𝒜 are representable, and F is either left or right exact, then F is additive. More generally, we see that for F to be additive, it suffices that F sends the zero object of 𝒜 to the zero object of ℬ, and that F commutes with the biproducts of the form A ⊕ A, for every A ∈ Ob(𝒜).

Definition 3.7.35. (i) An additive category is a pre-additive category which admits a zero object, and whose biproducts are representable.

(ii) An abelian category is an additive category 𝒜 such that the following holds: (a) All the kernels and cokernels of 𝒜 are representable.

(b) For every morphism f of 𝒜, the morphism β_{f} of (3.7.33) is an isomorphism.

Example 3.7.36. Let 𝒜 be an additive category, A an object of 𝒜, and e : A → A an endomorphism of A such that :

(a) e = e ∘ e, i.e. e is an idempotent element of the ring End_{𝒜} (A).

(b) Ker (e) and Ker (1_{A} − e) are representable in 𝒜. Then the morphisms ι_{e} : Ker (e) → A and ι_{1_{A}−e} : Ker (1_{A} − e) → A induce an isomorphism

ω : Ker (e) ⊕ Ker (1_{A} − e) →^{∼} A.

Indeed, it is easily seen that the morphism ε : A → A⊕A defined by the pair (1_{A} −e, e) factors as a composition

A ⟶^{\overline{ε}} Ker (e) ⊕ Ker (1_{A} − e) ⟶^{ι_{e}⊕ι_{1_{A}−e}} A ⊕ A

for a unique morphism \overline{ε} of 𝒜. Then a direct computation shows that ω ∘ \overline{ε} = 1_{A} and \overline{ε} ∘ ω = 1_{Ker (e)⊕Ker (1_{A}−e)}, whence the assertion (details left to the reader).

Remark 3.7.37. Let 𝒜 be any pre-additive category.

(i) The category 𝒜^{o} is naturally pre-additive : namely, if A, B ∈ Ob(𝒜^{o}) are any two objects, one endows Hom_{𝒜^{o}} (A, B) with the group structure of Hom_{𝒜} (B, A). Likewise, if F : 𝒜 → ℬ is any additive functor between the pre-additive categories 𝒜, ℬ, then also F^{o} : 𝒜^{o} → ℬ^{o} is additive. Moreover, if 𝒜 is additive the same holds for 𝒜^{o}. Indeed, the zero object of 𝒜 is obviously a zero object also in 𝒜^{o}, and biproducts are representable in 𝒜^{o}, since one can take

A^{o} ⊕ B^{o} := (A ⊕ B)^{o} for every A, B ∈ Ob(𝒜) .

Furthermore, if 𝒜 is abelian, the same holds for 𝒜^{o}. Indeed, we can take

Ker f^{o} := (Coker f)^{o}  Coker f^{o} := (Ker f)^{o}  for every morphism f in 𝒜

and with these choices we have :

ι_{f^{o}} = (π_{f})^{o}  π_{f^{o}} = (ι_{f})^{o}  β_{f^{o}} = (β_{f})^{o}

so β_{f^{o}} is an isomorphism whenever the same holds for β_{f}.

(ii) Notice that for every category 𝒞, the category Fun(𝒞 , 𝒜) is pre-additive; indeed, if τ, σ : F ⇒ G are two natural transformations between functors F, G : 𝒞 → 𝒜, then we obtain a natural transformation τ + σ from F to G, by the rule : (τ + σ)_{X} := τ_{X} + σ_{X} for every X ∈ Ob(𝒞) (where the sum denotes the addition law of Hom_{𝒜} (F X, GX)). Clearly, this rule yields an abelian group structure, and the composition of natural transformation defines a bilinear pairing (τ, τ′) ↦ τ ⊙ τ′ on the resulting groups of natural transformations (verification left to the reader).

Moreover, if 𝒜 is an additive (resp. abelian) category, then the same holds for Fun(𝒞 , 𝒜), for every category 𝒞. Indeed, if 0 denotes a zero object of 𝒜, it follows easily from remark 1.4.2(iii) that the constant functor 0_{𝒞} : 𝒞 → 𝒜 with 0_{𝒞} (C) := 0 for every C ∈ Ob(𝒞) is a zero object for Fun(𝒞 , 𝒜). By the same token, if F, G : 𝒞 → 𝒜 are any two functors, then the biproduct F ⊕ G is represented by the functor given by the rule : C ↦ F C ⊕ GC for every C ∈ Ob(𝒞) (where F C ⊕ GC denotes any fixed choice of an object of 𝒜 representing the biproduct of F C and GC); then (F ⊕G)(φ) is the induced morphism F φ⊕Gφ : F C ⊕GC → F C′⊕GC′ for every morphism φ : C → C′ in 𝒞 (details left to the reader). Likewise, for every natural transformation τ : F ⇒ G, the kernel and cokernel of τ are computed argumentwise, and the same holds for the natural morphism β_{τ} of (3.7.33), so if all kernels and cokernels are representable in 𝒜, the same holds for the kernels and cokernels in Fun(𝒞 , 𝒜), and if β_{τ_{C}} is an isomorphism for every C ∈ Ob(𝒞), then the same holds for β_{τ}, whence the contention.

(iii) For any other pre-additive category ℬ, let us denote by Add(ℬ, 𝒜) the full subcategory of Fun(ℬ, 𝒜) whose objects are the additive functors. It follows from (ii) that Add(ℬ, 𝒜) is a pre-additive category.

(iv) Suppose that 𝒜 has small Hom-sets. By definition, for every A, B ∈ Ob(𝒜), the set h_{A}(B) := Hom_{𝒜} (B, A) carries an abelian group structure, such that the presheaf h_{A} factors through an additive functor h_{A}^{†} : 𝒜^{o} → ℤ-Mod from 𝒜^{o} to the category of (small) abelian groups, and the forgetful functor ℤ-Mod → Set. Hence, the Yoneda embedding factors through a fully faithful group-valued Yoneda embedding

h^{†} : 𝒜 → Add(𝒜^{o}, ℤ-Mod).

In view of (iii), we conclude that every pre-additive category is a full subcategory of an abelian category. Moreover, Yoneda’s lemma extends verbatim to the group-valued case : namely, by inspecting the proof of proposition 1.2.6, we see that, for every A ∈ Ob(𝒜) and every additive functor F : 𝒜^{o} → ℤ-Mod there are natural isomorphisms of abelian groups

(3.7.38)  F A →^{∼} Hom_{Add(𝒜^{o},ℤ-Mod)} (h_{A}^{†}, F).

(v) Let f : 𝒜 → ℬ be any functor between pre-additive categories; then the arguments of remark 1.3.6(i) extend verbatim to the present situation : after choosing a universe U large enough so that 𝒜 and ℬ are U-small, f induces functor

f^{∗} : Fun(ℬ^{o}, ℤ-Mod) → Fun(𝒜^{o}, ℤ-Mod)

that admits both left and right adjoints, denoted respectively f_{!} and f_{∗}, and we have :

Proposition 3.7.39. In the situation of remark 3.7.37(v), suppose that f is additive and both 𝒜 and ℬ are additive categories. Then :

(i) Both f^{∗}, f_{!} and f_{∗} are additive functors, and restrict to functors

\xymatrix{Add(ℬ^{o}, ℤ-Mod) \ar[r]^{f^{∗}} & Add(𝒜^{o}, ℤ-Mod). \ar[l]^{f_{!} f_{∗}}}

(ii) The resulting diagram of functors

\xymatrix{𝒜 \ar[r]^{h^{†}} \ar[d]_{f} & Add(𝒜^{o}, ℤ-Mod) \ar[d]^{f_{!}} \\ ℬ \ar[r]^{h^{†}} & Add(ℬ^{o}, ℤ-Mod)}

is essentially commutative.

Proof. (i): Since every left (resp. right) adjoint functor is right (resp. left) exact, remark 3.7.34(iii) says that f^{∗}, f_{∗} and f_{!} are additive. Next, a simple inspection shows that f^{∗} transforms additive functors into additive functors. Let now F : 𝒜^{o} → ℤ-Mod be an additive functor, B ∈ Ob(ℬ) any object, and G := f_{!}F; from the proof of theorem 1.3.4, we see that

GB = colim_{ψ:B→fA} F A

where the colimit ranges over the small category f𝒜^{o}/B of all pairs (A, ψ) consisting of an object A of 𝒜, and a morphism ψ : B → fA in ℬ. Denote by 0_{𝒜} and 0_{ℬ} the zero objects of 𝒜 and ℬ; we wish to show that G is additive, and according to remark 3.7.34(iii), it suffices to check that G(0_{ℬ} ) = 0, and that the natural morphism G(B ⊕ B) → GB ⊕ GB (deduced from the projections p_{i} : B ⊕ B → B) is an isomorphism, for every B ∈ Ob(ℬ).

However, notice that the functor s_{0_{ℬ}} : f𝒜^{o}/0_{ℬ} → 𝒜^{o} is an isomorphism of categories (notation of (1.1.27)); whence a natural isomorphism

G(0_{ℬ} ) →^{∼} colim_{𝒜^{o}}F →^{∼} F (0_{𝒜} ) = 0

where the last identity holds, since F is additive. Next, for any B_{1}, B_{2} ∈ Ob(ℬ) consider the functor

Φ : (f𝒜^{o}/B_{1})×(f𝒜^{o}/B_{2}) → f𝒜^{o}/B_{1} ⊕B_{2} ((A_{1}, ψ_{1}), (A_{2}, ψ_{2})) ↦ (A_{1} ⊕A_{2}, ψ_{1} ⊕ψ_{2}).

Claim 3.7.40. The functor Φ is cofinal.

Proof of the claim. We apply the criterion of proposition 1.5.2. Indeed, let

i := (A, ψ : B_{1} ⊕ B_{2} → fA)

be any object of f𝒜^{o}/B_{1} ⊕ B_{2} and

\begin{matrix}j := & ((A_{1}, φ_{1} : B_{1} → fA_{1}), (A_{2}, φ_{2} : B_{2} → fA_{2})) \\ j′ := & ((A′_{1}, φ′_{1} : B_{1} → fA′_{1}), (A′_{2}, φ′_{2} : B_{2} → fA′_{2})) \end{matrix}

any two objects of (f𝒜^{o}/B_{1}) × (f𝒜^{o}/B_{2}), and suppose that β : i → Φj and β′ : i → Φj′ are morphisms in f𝒜^{o}/B_{1} ⊕ B_{2}, given by morphisms β : A_{1} ⊕ A_{2} → A and β′ : A′_{1} ⊕ A′_{2} → A in 𝒜 such that fβ ∘ (φ_{1} ⊕ φ_{2}) = ψ = fβ′ ∘ (φ′_{1} ⊕ φ′_{2}). We let

j′′ := ((A_{1} ⊕ A′_{1}, δ_{1} : B_{1} → f(A_{1} ⊕ A′_{1})), (A_{2} ⊕ A′_{2}, δ_{2} : B_{2} → f(A_{2} ⊕ A′_{2}))

where δ_{1} := (φ_{1} ⊕ φ′_{1}) ∘ Δ_{B_{1}}, and likewise for δ_{2}. Let also

A_{1} ⊕ A_{2} ⟵^{p} A_{1} ⊕ A′_{1} ⊕ A_{2} ⊕ A′_{2} ⟶^{p′} A′_{1} ⊕ A′_{2}

be the natural projections, which define morphisms

Φp : Φj → Φj′′  Φp′ : Φj′ → Φj′′  in f𝒜^{o}/B_{1} ⊕ B_{2}

A simple inspection shows that Φ(p) ∘ β = Φ(p′) ∘ β′, and therefore i/Φ(f𝒜^{o}/B_{1} ⊕ B_{2}) is connected, whence the claim. ♢

In light of claim 3.7.40, we are reduced to checking that the natural morphism

colim_{(f𝒜^{o}/B_{1})×(f𝒜^{o}/B_{2})} F ∘ s_{B_{1}⊕B_{2}} ∘ Φ → GB_{1} ⊕ GB_{2}

is an isomorphism, for any B_{1}, B_{2} ∈ Ob(ℬ). The latter assertion follows easily by inspecting the definitions, since F is additive. Lastly, a similar argument shows that f_{∗}F is additive, whenever the same holds for F : the reader can spell out the proof as an exercise.

(ii): One may argue as in remark 1.3.6(ii) : in view of (3.7.38), we see that, for every object A of 𝒜, both h_{fA}^{†} and f_{!}h_{A}^{†} represent the same functor : details left to the reader. □

Lemma 3.7.41. For any abelian category 𝒜, the following holds :

(i) 𝒜 is finitely complete and finitely cocomplete.

(ii) The image of any morphism f : A → B of 𝒜 exists in 𝒜, and we have

Coker π_{f} = Im f in Sub(B)

(notation of (1.1.2), example 1.2.26and remark 3.7.29(v)).

Proof. (i): By proposition 1.2.22(i,ii), it suffices to check that all equalizers and coequalizers are representable in 𝒜. However, let g, g′ : X → Y be any pair of morphisms in 𝒜 with same soruces and targets; it easily seen that Ker (g−g′) (resp. Coker (g−g′)) represents the equalizer (resp. the coequalizer) of g and g′, whence the assertion.

(ii): Suppose that f factors through a subobject g : C → B of B; there follows a natural morphism

Coker f → Coker g

which in turns induces a morphism Ker π_{f} → Ker π_{g} of subobjects of B. But since 𝒜 is abelian, Ker π_{g} is also the cokernel of ι_{g} : Ker g → C (notation of remark 3.7.29(v)). However, Ker g = 0 by remark 3.7.29(vi), hence Coker ι_{g} = C, whence the contention. □

Definition 3.7.42. Let 𝒜 be any abelian category with zero object 0 ∈ Ob(𝒜), and

0 ⟶^{0} A ⟶^{f} B ⟶^{g} C ⟶^{0} 0

a sequence of morphisms in 𝒜.

(i) We say that the sequence (f, g) is exact if Ker g = Im f.

(ii) We say that the sequence (0, f, g) is left exact if both (0, f) and (f, g) are exact.

(iii) We say that the sequence (f, g, 0) is right exact if both (f, g) and (g, 0) are exact.

(iv) We say that the sequence (0, f, g, 0) is short exact if (0, f, g) is left exact and (f, g, 0) is right exact.

In other words, the sequence (0, f, g) is left exact if and only if f is a monomorphism, and A represents Ker g. Likewise, (f, g, 0) is right exact if and only if g is an epimorphism and the image of f represents Ker g. The terminology is explained by the following :

Lemma 3.7.43. Let F : 𝒜 → ℬ be any additive functor between abelian categories. Then :

(i) F is left exact if and only if it transforms left exact sequences of 𝒜 into left exact sequences of ℬ.

(ii) F is right exact if and only if it transforms right exact sequences of 𝒜 into right exact

sequences of ℬ. (iii) F is exact if and only if it transforms short exact sequences of 𝒜 into short exact

sequences of ℬ.

Proof. (i): Any left exact functor transforms kernels into kernels and monomorphisms into monomorphisms (proposition 1.3.18(i)), so the condition is necessary. Conversely, suppose that F fulfills this condition; since F is additive, it commutes with finite products, so it suffices to check that it also commutes with equalizers (proposition 1.3.22(i)). Arguing as in the proof of lemma 3.7.41(i), we reduce to checking that F commutes with kernels. But this follows by considering the left exact sequence

0 → Ker f → A ⟶^{f} B for any morphism f of 𝒜

and its image under F. Assertion (ii) admits the dual proof, and (iii) follows by considering the similar short exact sequences

0 → Ker f → A → Im f → 0 0 → Im f → B → Coker f → 0

for every f as in the foregoing, and arguing as in the proof of (i). □

Definition 3.7.44. An abelian tensor category is a tensor category (𝒞 , ⊗, Φ, Ψ) such that 𝒞 is an abelian category, and the functor ⊗ induces bilinear pairings

Hom_{𝒞} (A, B) × Hom_{𝒞} (A′, B′) → Hom_{𝒞} (A ⊗ A′, B ⊗ B′) : (f, g) ↦ f ⊗ g

for every A, A′, B, B′ ∈ Ob(𝒞).

Remark 3.7.45. Let (𝒜 , ⊗, Φ, Ψ) be a tensor category, such that 𝒜 is abelian. If 𝒜 admits an internal Hom functor ℋ om, the functor −⊗A is right exact, and the functor ℋ om(A, −) is left exact for every A ∈ Ob(𝒜), so both are additive, by virtue of remark 3.7.34(iii). Especially, 𝒜 is an abelian tensor category, in this case.

Lemma 3.7.46. Let 𝒜 be any additive category with small Hom-sets, and Σ ⊂ Ob(𝒜) a small subset. We have :

(i) If 𝒜 is abelian, there exists a small full abelian subcategory ℬ of 𝒜 such that Σ ⊂ Ob(ℬ).

(ii) If 𝒜 is small, there exist a complete and cocomplete abelian tensor category (𝒞 , ⊗) with internal Hom functor, and a fully faithful additive functor 𝒜 → 𝒞.

Proof. (i): Let ℬ_{0} be the full subcategory of 𝒜 such that Ob(ℬ_{0}) = Σ; clearly ℬ_{0} is small. Next, for any subcategory 𝒟 of 𝒜, denote by 𝒟′ a subcategory of 𝒜 obtained as follows. For every morphism φ of 𝒟, we pick objects in 𝒜 representing the kernel and cokernel of φ, and for any two objects of 𝒟, we pick an object in 𝒜 representing their product; let Σ′ ⊂ Ob(𝒜) be the resulting subset. Then 𝒟′ is the full subcategory of 𝒜 such that Ob(𝒟′) = Ob(𝒟) ∪ Σ′. It is easily seen that 𝒟′ is small, whenever the same holds for 𝒟. Then we set inductively ℬ_{i+1} := ℬ′_{i} for every i ∈ ℕ. The full subcategory ℬ of 𝒜 with Ob(ℬ) = ⋃_{i∈ℕ} Ob(ℬ_{i}) is still small, and it is abelian, by construction.

(ii): We let 𝒞 := Fun(𝒜 , ℤ-Mod). Then 𝒞 is an abelian category, by virtue of remark 3.7.37(ii), and since ℤ-Mod is complete and cocomplete, the same holds for 𝒞; moreover, the standard tensor product of abelian groups defines a tensor category structure with internal Hom functor on ℤ-Mod, and the latter is inherited by 𝒞 (remarks 3.7.5(ii) and 3.7.12(vi)). It is clear that these two structures amount to an abelian tensor category, and the group-valued Yoneda embedding is the sought fully faithful functor. □

3.7.47. Let 𝒜 be an additive category with small Hom-sets, and

h^{†} : 𝒜^{o} → 𝒜^{†} := Fun(𝒜 , ℤ-Mod)

the fully faithful group-valued Yoneda embedding. For every abelian group G, denote by G_{𝒜} : 𝒜 → ℤ-Mod the constant functor with value G : so, G_{𝒜} (A) := G for every A ∈ Ob(𝒜), and G_{𝒜} (φ) := 1_{G} for every morphism φ in 𝒜. Since 𝒜^{†} is an abelian tensor category with an internal Hom functor ℋ om (see the proof of lemma 3.7.46(ii)), we may define

G ⊗_{ℤ} A := ℋ om(G_{𝒜} , h_{A^{o}}^{†}) for every A ∈ Ob(𝒜) .

If G is free of finite rank, G ⊗_{ℤ} A is a finite direct sum of copies of A, and therefore lies in the essential image of h^{†}; the same holds for a finitely generated G, provided 𝒜 is an abelian category : indeed, we may write G as a cokernel of a map L_{1} → L_{2} of free abelian groups of finite rank, and since the functor ℋ om(−, h_{A}^{†}) is right exact, we see that G ⊗_{ℤ} A is the kernel of a morphism of 𝒜^{o} (i.e. the cokernel of a morphism of 𝒜), so it is represented by an object of 𝒜. Moreover, if φ : G → H is any morphism of abelian groups, we have an obvious induced morphism φ_{𝒜} : G_{𝒜} → H_{𝒜} , whence a morphism φ ⊗_{ℤ} A := ℋ om(φ_{𝒜} , h_{A}^{†}).

Thus, if 𝒜 is abelian, after replacing G ⊗_{ℤ} A by an isomorphic object, we obtain a well defined functor

(3.7.48)  ℤ-Mod_{fg} × 𝒜 → 𝒜  (G, A) ↦ G ⊗_{ℤ} A

where ℤ-Mod_{fg} is the full subcategory of ℤ-Mod whose objects are the finitely generated abelian groups. This functor is not unique, but any two such functors are naturally isomorphic. If 𝒜 is only additive, we can still define such a tensor product functor on the category ℤ-Mod_{fft} × 𝒜 → 𝒜, where ℤ-Mod_{fft} is the full subcategory of ℤ-Mod_{fg} whose objects are the free abelian groups of finite rank.

Remark 3.7.49. Keep the notation of (3.7.47); we have :

(i) From the construction, it is clear that (3.7.48) is a biadditive functor, i.e., for every abelian group G, and every A ∈ Ob(𝒜), the restrictions G ⊗ − and − ⊗ A of (3.7.48) are additive.

(ii) Suppose that 𝒜 is cocomplete; since the tensor product is right exact, it follows easily that (3.7.48) extends to the whole of ℤ-Mod : details left to the reader.

4. Sites and topoi

In this chapter, we assemble some generalities concerning sites and topoi. The main reference for this material is [8].

4.1. Topologies and sites. As in the previous chapter, we fix a universe U such that ℕ ∈ U, and small means U-small throughout. Especially, a presheaf on any category takes its values in U, unless explicitly stated otherwise.

Definition 4.1.1. Let 𝒞 be a category.

(i) A topology on 𝒞 is the datum, for every X ∈ Ob(𝒞), of a set J(X) of sieves of 𝒞/X, fulfilling the following conditions :

(a) (Stability under base change) For every morphism f : Y → X of 𝒞, and every 𝒮 ∈ J(X), the sieve 𝒮 ×_{X} f lies in J(Y).

(b) (Local character) Let X be any object of 𝒞, and 𝒮, 𝒮′ two sieves of 𝒞/X, with 𝒮 ∈ J(X). Suppose that, for every object f : Y → X of 𝒮, the sieve 𝒮′ ×_{X} f lies in J(Y). Then 𝒮′ ∈ J(X).

(c) For every X ∈ Ob(𝒞), we have 𝒞/X ∈ J(X). (ii) In the situation of (i), the elements of J(X) shall be called the sieves covering X. Moreover, say that 𝒮 is the sieve of 𝒞/X generated by a family (f_{i} : X_{i} → X | i ∈ I) of morphisms. If 𝒮 is a sieve covering X, we say that the family (f_{i} | i ∈ I) covers X, or that it is a covering family of X.

(iii) The datum (𝒞 , J) of a category 𝒞 and a topology J := (J(X) | X ∈ Ob(𝒞)) on 𝒞 is called a site, and then 𝒞 is also called the category underlying the site (𝒞 , J). We say that (𝒞 , J) is a small site, if 𝒞 is a small category.

(iv) The set of all topologies on 𝒞 is partially ordered by inclusion: given two topologies J_{1} and J_{2} on 𝒞, we say that J_{1} is finer than J_{2}, if J_{2}(X) ⊂ J_{1}(X) for every X ∈ Ob(𝒞).

Example 4.1.2. Let T be any topological space, and 𝒯 the set of open subsets of T, which is partially ordered by inclusion, and can thus be regarded naturally as a category, as in example 1.1.6(iii). Then the category 𝒯 carries a natural topology J_{T} in the sense of definition 4.1.1 : namely, for every U ∈ 𝒯 we declare that the elements of J_{T} (U) are the sieves 𝒮 ⊂ 𝒯 /U such that ⋃_{(f:U′→U)∈𝒮} U′ = U. In other words, a family (U′_{i} → U | i ∈ I) of morphisms of 𝒯 covers U for the topology J_{T} if and only if ⋃_{i∈I} U_{i} = U.

Remark 4.1.3. Let (𝒞 , J) be any site.

(i) Any finite intersection of sieves covering an object X, again covers X. Indeed, say that 𝒮_{1} and 𝒮_{2} are two sieves covering X; set 𝒮 := 𝒮_{1} ∩ 𝒮_{2} and let f : Y → X be any object of 𝒮_{1}. Then 𝒮 ×_{X} f = 𝒮_{2} ×_{X} f ∈ J(Y), so the assertion follows from the local character of J.

(ii) Also, any sieve of 𝒞/X containing a covering sieve is again a covering sieve. Indeed, if 𝒮 ⊂ 𝒮′, then 𝒮′ ×_{X} f = 𝒞 /Y for every object f : Y → X of 𝒮.

(iii) Let f_{•} := (f_{i} : Y_{i} → X | i ∈ I) be a family of objects of 𝒞/X that generates a sieve 𝒮 covering X, and for every i ∈ I, let (g_{ij} : Z_{ij} → Y_{i} | i ∈ J_{i}) be a family of objects of 𝒞 /Y_{i} that generates a sieve 𝒮_{i} covering Y_{i}. Then the family (f_{i} ∘ g_{ij} : Z_{ij} → X | i ∈ I, j ∈ J_{i}) is a refinement of f_{•} (see definition 3.5.1(ii)) and generates a sieve 𝒮′ covering X. Indeed, say that f : Y → X lies in 𝒮, and pick i ∈ I such that f factors through f_{i} and a morphism g : Y → Y_{i}; then it is easily seen that 𝒮_{i} ×_{Y_{i}} g ⊂ 𝒮′ ×_{X} f.

4.1.4. Suppose that 𝒞 has small Hom-sets. Then, in view of remark 3.5.2(ii), a topology can also be defined by assigning, to any object X of 𝒞, a family J′(X) of subobjects of h_{X}, called the subobjects covering X, such that :

(a) For every X ∈ Ob(𝒞), every R ∈ J′(X), and every morphism Y → X in 𝒞, the fibre product R ×_{X} Y lies in J′(Y).

(b) Say that X ∈ Ob(𝒞), and let R, R′ be two subobjects of h_{X}, such that R ∈ J′(X). Suppose that, for every Y ∈ Ob(𝒞), and every morphism f : h_{Y} → R, we have R′ ×_{X} Y ∈ J′(Y). Then R′ ∈ J′(X).

(c) h_{X} ∈ J′(X) for every X ∈ Ob(𝒞). This viewpoint is adopted in the following :

Definition 4.1.5. Let V be a universe, C := (𝒞 , J) a site, and F ∈ Ob(𝒞_{V}^{∧}).

(i) We say that F is a separated V-presheaf (resp. a V-sheaf) on C, if for every X ∈ Ob(𝒞), every subobject R covering X, and every universe V′ containing V and such that the category 𝒞 has V′-small Hom-sets, the induced morphism :

F (X) = Hom_{𝒞_{V′}^{∧}} (h_{X}, F) → Hom_{𝒞_{V′}^{∧}} (R, F)

is injective (resp. is bijective). For V = U, we shall just say separated presheaf instead of separated U-presheaf, and likewise for sheaves.

(ii) We let C_{V}^{∼} (resp. C_{V}^{sep}) be the full subcategory of 𝒞_{V}^{∧} whose objects are the V-sheaves (resp. the separated V-presheaves) on C. For V = U, this category will be usually denoted simply C^{∼} (resp. C^{sep}).

4.1.6. In the situation of definition 4.1.5(i), say that R = h_{𝒮} for some sieve 𝒮 covering X. In light of (3.5.3), (1.4.5) and proposition 1.2.6(ii), we get a natural identification :

Hom_{𝒞_{V′}^{∧}} (R, F) →^{∼} Hom_{𝒞_{V′}^{∧}} (colim_{𝒮}h_{𝒞} ∘ s_{𝒮} , F) →^{∼} lim_{𝒮^{o}}F ∘ s_{𝒮}^{o} .

Let also (S_{i} → X | i ∈ I) be a generating family for 𝒮. Combining lemma 3.5.6, example 3.1.16(i) and (3.1.15), we get a natural isomorphism :

Hom_{𝒞_{V′}^{∧}} (R, F) →^{∼} Equal(∏_{i∈I} [figure: Cart_{𝒞} (𝒞/S_{i}, ℱ ib(F))] ∏_{(i,j)∈I×I}Cart_{𝒞} (𝒞/S_{ij}, ℱ ib(F)))

which – again by (3.1.15) – we may rewrite more simply as :

Hom_{𝒞_{V′}^{∧}} (R, F) →^{∼} Equal(∏_{i∈I} [figure: F (S_{i})] ∏_{(i,j)∈I×I}Hom_{𝒞_{V′}^{∧}} (h_{S_{i}} ×_{h_{X}} h_{S_{j}}, F)).

The above equalizer can be described explicitly as follows. It consists of all the systems

(a_{i} | i ∈ I) with a_{i} ∈ F (S_{i}) for every i ∈ I

such that, for every i, j ∈ I and every object Y → X of 𝒞/X, we have :

(4.1.7) F (g_{i})(a_{i}) = F (g_{j})(a_{j})  for every pair (S_{i} ⟵^{g_{i}} Y ⟶^{g_{j}} S_{j}) of morphisms in 𝒞/X .

If all S_{ij} := S_{i} ×_{X} S_{j} are representable in 𝒞, the latter expression takes the more familiar form:

Hom_{𝒞_{V′}^{∧}} (R, F) →^{∼} Equal(∏_{i∈I} [figure: F (S_{i})] ∏_{(i,j)∈I×I}F (S_{ij})).

Remark 4.1.8. Let C := (𝒞 , J) be a site such that 𝒞 is a category with small Hom-sets.

(i) The arguments in (4.1.6) yield also the following. A presheaf F on 𝒞 is separated (resp. is a sheaf) on C, if and only if every covering sieve of C is a sieve of 1-descent (resp. of 2-descent) for the fibration s_{F} : ℱ ib(F) → 𝒞 of (3.1.15).

(ii) Let F : 𝒜 → ℬ be a fibration between two categories, and i ≤ 2 an integer. For every X ∈ Ob(ℬ), let J_{F}^{i} (X) denote the set of all sieves 𝒮 ⊂ ℬ/X of universal F -i-descent. Then we claim that J_{F}^{i} is a topology on ℬ. Indeed, it is clear that J_{F}^{i} fulfills conditions (a) and (c) of definition 4.1.1(i), and condition (b) follows from proposition 3.5.38.

(iii) Let F be a presheaf on 𝒞. We deduce from (i) and (ii) that the topology J^{F} := J_{s_{F}}^{2} is the finest on 𝒞 for which F is a sheaf. A subobject R ⊂ h_{X} (for any X ∈ Ob(𝒞)) lies in J^{F} (X) if and only if the natural map F (X′) → Hom_{𝒞^{∧}}(R ×_{X} X′, F) is bijective for every morphism X′ → X in 𝒞.

(iv) More generally, if (F_{i} | i ∈ I) is any family of presheaves on 𝒞, there exists a finest topology for which each F_{i} is a sheaf : namely, the intersection of the topologies J^{F_{i}} as in (iii).

(v) As an important special case, we deduce the existence of a finest topology J on 𝒞 such that all representable presheaves are sheaves on (𝒞 , J). This topology Can_{𝒞} is called the canonical topology on 𝒞. We thus associate to every category 𝒞 a canonical site

Can(𝒞) := (𝒞 , Can_{𝒞} ).

(vi) Another interesting case of (iv) is obtained by taking the family of all presheaves on 𝒞. The corresponding topology J on 𝒞 can be easily described explicitly : namely, one takes J(X) := {𝒞 /X} for every X ∈ Ob(𝒞).

Example 4.1.9. Let 𝒞 be a category, X ∈ Ob(𝒞) any object, and 𝒮 ⊂ 𝒞 /X a sieve.

(i) We say that 𝒮 is an epimorphic sieve (resp. a strict epimorphic sieve) if for every Y ∈ Ob(𝒞) the natural map

h_{Y} (X) → Hom_{𝒞^{∧}}(h_{𝒮} , h_{Y})

is injective (resp. bijective). We say that 𝒮 is a universal epimorphic sieve (resp. a universal strict epimorphic sieve) if 𝒮 ×_{X} h is epimorphic (resp. strict epimorphic) for every morphism h : Y → X of 𝒞. From remark 4.1.8(iii,v) we see that 𝒮 is universal strict epimorphic if and only if it covers X in the canonical topology of 𝒞.

(ii) We say that a family of morphisms f_{•} := (f_{i} : X_{i} → X | i ∈ I) of 𝒞 is epimorphic (resp. strict epimorphic) if it generates an epimorphic (resp. strictly epimorphic) sieve. This is the same as saying for every Y ∈ Ob(𝒞), the natural map

(4.1.10)  Hom_{𝒞} (X, Y) → ∏_{i∈I} Hom_{𝒞} (X_{i}, Y)  g ↦ (g ∘ f_{i} | i ∈ I)

is injective (resp. and its image consists of all the systems of morphisms (g_{i} : X_{i} → Y | i ∈ I)

such that, for every Z ∈ Ob(𝒞), every i, j ∈ I, and every pair of morphisms X_{j} ⟵^{h_{j}} Z ⟶^{h_{i}} X_{i} with f_{i} ∘ h_{i} = f_{j} ∘ h_{j}, we have g_{i} ∘ h_{i} = g_{j} ∘ h_{j}).

(iii) We say that a family f_{•} as in (ii) is effective epimorphic, if it is strict epimorphic, and moreover the fibre products X_{i} ×_{X} X_{j} are representable in 𝒞 for every i, j ∈ I. This is the same as saying that for every Y ∈ Ob(𝒞) the map (4.1.10) identifies Hom_{𝒞} (X, Y) with the equalizer of the two natural maps

∏_{i∈I} [figure: Hom_{𝒞} (X_{i}, Y)] ∏_{(i,j)∈I×I}Hom_{𝒞} (X_{i} ×_{X} X_{j}, Y).

(iv) A family f_{•} as in (ii) is called universal epimorphic (resp. universal effective epimorphic) if (a) the fibre products Y_{i} := X_{i} ×_{X} Y are representable in 𝒞, for every i ∈ I and every morphism Y → X in 𝒞, and (b) all the resulting families (Y_{i} → Y | i ∈ I) are still epimorphic (resp. effective epimorphic). Then clearly such a family generates a universal epimorphic sieve (resp. a universal strict epimorphic sieve).

(v) We say that a morphism f : X′ → X of 𝒞 is an effective epimorphism (resp. a universal epimorphism, resp. a universal effective epimorphism) if the family {f} has the corresponding property. Hence, f is an effective epimorphism if the fibre product X′ ×_{X} X′ is representable in 𝒞, and the induced diagram

\xymatrix{X′ ×_{X} X′ \ar[r]^{p_{1}} \ar[r]_{p_{2}} & X′ \ar[r]^{f} & X}

identifies X with the coequalizer of the two projections p_{1} and p_{2}. (Notice that this condition implies that f is an epimorphism). Likewise, f is a universal epimorphism (resp. a universal effective epimorphism) if for every morphism Y → X, the fibre product X′×_{X} Y is representable in 𝒞 and f ×_{X} Y : X′ ×_{X} Y → Y is an epimorphism (resp. an effective epimorphism).

(vi) Suppose that all fibre products are representable in 𝒞; in this case, proposition 1.2.18(ii) implies that a morphism X′ → X of 𝒞 is a universal epimorphism if and only if it is an epimorphism and the coproduct X ⨿_{X′} X is a universal colimit, in the sense of example 1.4.17(ii).

(vii) Combining (vi) with example 1.4.17(iv), we conclude that all epimorphisms of Set are universal. Also, it is easily seen that all epimorphisms of Set are effective, and hence universal effective. More generally, If 𝒞 is a small category, example 1.2.24(i) and corollary 1.4.3(ii) imply that all epimorphisms in 𝒞^{∧} are universal effective.

4.1.11. Let C := (𝒞 , J) be a small site; directly from definition 4.1.5 (and from (1.4.6)), we see that the category C^{∼} is complete, and the fully faithful inclusion C^{∼} → 𝒞^{∧} commutes with all limits. Moreover, given a presheaf F on 𝒞, it is possible to construct a solution set for F relative to this functor, and therefore one may apply theorem 1.5.13 to produce a left adjoint. However, a more direct and explicit construction of the left adjoint can be given; the latter also provides some additional information which is hard to extract from the former method. Namely, for every X ∈ Ob(𝒞), endow J(X) with the partial ordering induced by inclusion of sieves. Notice that J(X) is small and cofiltered, for every such X, hence the opposite ordered set J(X)^{o} is filtered. For a given presheaf F on 𝒞, set

F^{+}(X) := colim_{𝒮 ∈J(X)^{o}} Hom_{𝒞^{∧}}(h_{𝒮} , F)

where the transition map Hom_{𝒞^{∧}}(h_{𝒮} , F) → Hom_{𝒞^{∧}}(h_{𝒮′} , F) is induced by the monomorphism h_{𝒮′} → h_{𝒮} of subobjects of h_{X}, for every inclusion 𝒮′ ⊂ 𝒮 of sieves covering X. Let (τ_{𝒮} : Hom_{𝒞^{∧}}(h_{𝒮} , F) → F^{+}(X) | 𝒮 ∈ J(X)) be the universal cocone. For a morphism f : Y → X in 𝒞 and a sieve 𝒮 ∈ J(X), the natural projection h_{𝒮} ×_{X} Y → h_{𝒮} induces a map

Hom_{𝒞^{∧}}(h_{𝒮} , F) → Hom_{𝒞^{∧}}(h_{𝒮} ×_{X} Y, F) ⟶^{τ_{𝒮 ×_{X} f}} F^{+}(Y)

whence a map F^{+}f : F^{+}(X) → F^{+}(Y), after taking colimits. Likewise, every morphism F → F′ of presheaves on 𝒞 induces a map Hom_{𝒞^{∧}}(h_{𝒮} , F) → Hom_{𝒞^{∧}}(h_{𝒮} , F′), for every X ∈ Ob(𝒞) and every 𝒮 ∈ J(X), whence a map F^{+}(X) → F^{′+}(X), after taking colimits. Thus, we have a functor :

(4.1.12)  𝒞^{∧} → 𝒞^{∧}  F ↦ F^{+}.

with a natural transformation F (X) → F^{+}(X), since 𝒞/X ∈ J(X) for every X ∈ Ob(𝒞).

Theorem 4.1.13. In the situation of (4.1.11), the following holds :

(i) The inclusion functor i : C^{∼} → 𝒞^{∧} admits the left adjoint

(4.1.14)  𝒞^{∧} → C^{∼}  F ↦ F^{a} := (F^{+})^{+}.

For every presheaf F, we call F^{a} the sheaf associated with F. (ii) Moreover, the functor (4.1.14) is exact.

Proof. We begin with the following :

Claim 4.1.15. Let F be a separated presheaf on 𝒞, and 𝒮_{1} ⊂ 𝒮_{2} two sieves covering some X ∈ Ob(𝒞). Then the natural map Hom_{𝒞^{∧}}(h_{𝒮_{2}} , F) → Hom_{𝒞^{∧}}(h_{𝒮_{1}} , F) is injective.

Proof of the claim. We may find a generating family (f_{i} : S_{i} → X | i ∈ I_{2}) for 𝒮_{2}, and a subset I_{1} ⊂ I_{2}, such that (f_{i} | i ∈ I_{1}) generates 𝒮_{1}. Let s, s′ : h_{𝒮_{2}} → F, whose images agree in Hom_{𝒞^{∧}}(h_{𝒮_{1}} , F). By (4.1.6), s and s′ correspond to families (s_{i} | i ∈ I_{2}), (s′_{i} | i ∈ I_{2}) with s_{i}, s′_{i} ∈ F (S_{i}) for every i ∈ I_{2}, fulfilling the system of identities (4.1.7), and the foregoing condition means that s_{i} = s′_{i} for every i ∈ I_{1}. We need to show that s_{i} = s′_{i} for every i ∈ I_{2}. Hence, let i ∈ I_{2} be any element; by assumption, the natural map

F (S_{i}) → Hom_{𝒞^{∧}}(h_{𝒮_{1}} ×_{X} S_{i}, F)

is injective. However, the objects of 𝒮_{1} ×_{X} f_{i} are all the morphisms g_{i} : Y → S_{i} in 𝒞 such that f_{i} ∘ g_{i} = f_{j} ∘ g_{j} for some j ∈ I_{1} and some g_{j} : Y → S_{j} in 𝒞. If we apply the identities (4.1.7) to these maps g_{i}, g_{j}, we deduce that :

F (g_{i})(s_{i}) = F (g_{j})(s_{j}) = F (g_{j})(s′_{j}) = F (g_{i})(s′_{i}).

In other words, s_{i} and s′_{i} have the same image in Hom_{𝒞^{∧}}(h_{𝒮_{1}} ×_{X} S_{i}, F), hence they agree, as claimed. ♢

Claim 4.1.16. (i) The functor (4.1.12) is left exact.

(ii) For every F ∈ Ob(𝒞^{∧}), the presheaf F^{+} is separated.

(iii) If F is a separated presheaf on 𝒞, then F^{+} is a sheaf on C.

Proof of the claim. (i) is clear, since J(X)^{o} is filtered for every X ∈ Ob(𝒞).

(ii): Let s, s′ ∈ F^{+}(X), and suppose that the images of s and s′ agree in Hom_{𝒞^{∧}}(h_{𝒮} , F^{+}), for some 𝒮 ∈ J(X). We may find a sieve 𝒯 ∈ J(X) such that s and s′ come from elements \overline{s}, \overline{s}′ ∈ Hom_{𝒞^{∧}}(h_{𝒯} , F). Let (g_{i} : S_{i} → X | i ∈ I) be a family of generators for 𝒮; in view of (4.1.6), the images of s and s′ agree in F^{+}(S_{i}) for every i ∈ I. The latter means that, for every i ∈ I, there exists 𝒮_{i} ∈ J(S_{i}), refining 𝒯 ×_{X} g_{i}, such that the images of \overline{s} and \overline{s}′ agree in Hom_{𝒞^{∧}}(h_{𝒮_{i}} , F). For every i ∈ I, let (g_{iλ} : T_{iλ} → S_{i} | λ ∈ Λ_{i}) be a family of generators for 𝒮_{i}, and consider the sieve 𝒯 ′ of 𝒞/X generated by (g_{i} ∘ g_{iλ} : T_{iλ} → X | i ∈ I, λ ∈ Λ_{i}). Then 𝒯 ′ covers X (remark 4.1.3(iii)) and refines 𝒯 , and the images of \overline{s} and \overline{s}′ agree in Hom_{𝒞^{∧}}(h_{𝒯 ′} , F) (as one sees easily, again by virtue of (4.1.6)). This shows that s = s′, whence the contention.

(iii): In view of (ii), it suffices to show that the natural map F^{+}(X) → Hom_{𝒞^{∧}}(h_{𝒮} , F^{+}) is surjective for every 𝒮 ∈ J(X). Hence, say that s ∈ Hom_{𝒞^{∧}}(h_{𝒮} , F^{+}), and let (S_{i} | i ∈ I) be a generating family for 𝒮. By (4.1.6), s corresponds to a system (s_{i} ∈ F^{+}(S_{i}) | i ∈ I) such that the following holds. For every i, j ∈ I, and every pair of morphisms u_{i} : Y → S_{i} and u_{j} : Y → S_{j} in 𝒞/X, we have

(4.1.17)  F^{+}(u_{i})(s_{i}) = F^{+}(u_{j})(s_{j}).

For every i ∈ I, let 𝒮_{i} ∈ J(S_{i}) such that s_{i} is the image of some \overline{s}_{i} ∈ Hom_{𝒞^{∧}}(h_{𝒮_{i}} , F). For every u_{i}, u_{j} as above, set 𝒮_{ij} := (𝒮_{i} ×_{S_{i}} u_{i}) ∩ (𝒮_{j} ×_{S_{j}} u_{j}); since F is separated, (4.1.17) and claim 4.1.15imply that the images of \overline{s}_{i} and \overline{s}_{j} agree in Hom_{𝒞^{∧}}(h_{𝒮_{ij}} , F), for every i, j ∈ I.

However, for every i ∈ I, let (g_{iλ} : T_{iλ} → S_{i} | λ ∈ Λ_{i}) be a generating family for 𝒮_{i}; then \overline{s}_{i} corresponds to a compatible system of sections \overline{s}_{iλ} ∈ F (T_{iλ}), and 𝒮_{ij} is the sieve of all morphisms f : Z → Y such that

u_{i} ∘ f = g_{iλ} ∘ f′_{i} and u_{j} ∘ f = g_{jμ} ∘ f′_{j}

for some λ ∈ Λ_{i}, μ ∈ Λ_{j} and some f′_{i} : Z → T_{iλ}, f′_{j} : Z → T_{jμ}, so by construction we have

(4.1.18)  F (f′_{i})(\overline{s}_{iλ}) = F (f′_{j})(\overline{s}_{jμ})  for every i, j ∈ I and λ ∈ Λ_{i}, μ ∈ Λ_{μ}.

Lastly, let 𝒯 be the sieve of 𝒞/X generated by (g_{i} ∘ g_{iλ} : T_{iλ} → X | i ∈ I, λ ∈ Λ_{i}); then 𝒯 covers X (remark 4.1.3(iii)), and (4.1.18) shows that the system (F (g_{iλ})(\overline{s}_{iλ}) | i ∈ I, λ ∈ Λ_{i}) defines an element of Hom_{𝒞^{∧}}(h_{𝒯} , F), whose image in F^{+}(X) agrees with s. ♢

From claim 4.1.16 we see that the rule : F ↦ F^{a} := (F^{+})^{+} defines a left exact functor 𝒞^{∧} → C^{∼}, with natural transformations η_{F} : F ⇒ i(F^{a}) for every F ∈ Ob(𝒞^{∧}) and ε_{G} : (iG)^{a} ⇒ G for every G ∈ Ob(C^{∼}) fulfilling the triangular identities of (1.1.13). The theorem follows. □

Remark 4.1.19. Let C := (𝒞 , J) be a small site.

(i) It has already been remarked that C^{∼} is complete, and from theorem 4.1.13 we also deduce that C^{∼} is cocomplete, and the functor (4.1.14) (resp. the inclusion functor i : C^{∼} → 𝒞^{∧}) commutes with all colimits (resp. with all limits); more precisely, if F : I → C^{∼} is any functor from a small category I, we have a natural isomorphism in C^{∼} (resp. in 𝒞^{∧}) :

colim_{I} F →^{∼} (colim_{I} i ∘ F)^{a} (resp. i(lim_{I} F) →^{∼} lim_{I} i ∘ F)

(proposition 1.3.25(iii,iv)); especially, limits in C^{∼} are computed argumentwise. Also, all colimits and all epimorphisms are universal in C^{∼} (see examples 1.4.17(ii,iv) and 4.1.9(v,vii)), and filtered colimits in C^{∼} commute with finite limits, since the same holds in 𝒞^{∧}.

(ii) Furthermore, C^{∼} is well-powered and co-well-powered, since the same holds for 𝒞^{∧}. Especially, every morphism f : F → G in C^{∼} admits a well defined image (see example 1.2.26(i)) that can be constructed explicitly as the subobject (Im i(f))^{a} (details left to the reader).

(iii) By composing with the Yoneda embedding, we obtain a functor

h_{C}^{a} : 𝒞 → C^{∼} : X ↦ (h_{X})^{a} for every X ∈ Ob(𝒞)

and since the functor (−)^{a} : 𝒞^{∧} → C^{∼} commutes with small colimits, lemma 1.4.8 yields a natural isomorphism :

colim_{ℱ ib(F)}h_{C}^{a} ∘ s_{F} →^{∼} F for every sheaf F on C.

(iv) From the proof of claim 4.1.16it is clear that the functor

𝒞^{∧} → C^{sep} F ↦ F^{sep} := Im(F → F^{+})

is left adjoint to the inclusion C^{sep} → 𝒞^{∧}. Moreover, we have a natural identification :

F^{a} →^{∼} (F^{sep})^{+} for every F ∈ Ob(𝒞^{∧}).

(v) Let V be a universe such that U ⊂ V; from the definitions, it is clear that the fully faithful inclusion 𝒞_{U}^{∧} ⊂ 𝒞_{V}^{∧} restricts to a fully faithful inclusion

C_{U}^{∼} ⊂ C_{V}^{∼}.

Moreover, by inspecting the proof of theorem 4.1.13, we deduce an essentially commutative diagram of categories :

\xymatrix{𝒞_{U}^{∧} \ar[r] \ar[d] & C_{U}^{∼} \ar[d] \\ 𝒞_{V}^{∧} \ar[r] & C_{V}^{∼}}

whose vertical arrows are the inclusions, and whose horizontal arrows are the functors F ↦ F^{a}.

In practice, one often encounters sites that are not small, but which share many of the properties of small sites. These more general situations are encompassed by the following :

Definition 4.1.20. Let C := (𝒞 , J) be a site.

(i) A topologically generating family for C is a subset G ⊂ Ob(𝒞), such that, for every X ∈ Ob(𝒞), the family

G/X := ⋃_{Y ∈G}{Y } × Hom_{𝒞} (Y, X) ⊂ Ob(𝒞/X)

generates a sieve covering X. (ii) We say that C is a U-site, if 𝒞 has small Hom-sets, and C admits an essentially small

topologically generating family.

4.1.21. Let C := (𝒞 , J) be a U-site, and G an essentially small topologically generating family for C. For every X ∈ Ob(𝒞), denote by J_{G}(X) ⊂ J(X) the set of all sieves covering X which are generated by a subset of G/X (notation of definition 4.1.20(i)).

Lemma 4.1.22. With the notation of (4.1.21), for every X ∈ Ob(𝒞) the following holds : (i) J_{G}(X) is an essentially small set. (ii) J_{G}(X) is a coinitial subset of J(X) (for the partial order given by inclusion of sieves).

Proof. (i) is left to the reader.

(ii): Let 𝒮 ∈ J(X), and say that 𝒮 is generated by a family (f_{i} : S_{i} → X | i ∈ I) of objects of 𝒞 /X (indexed by some not necessarily small set I). Let 𝒮′ be the sieve generated by

⋃_{i∈I}{f_{i} ∘ g | g ∈ G/S_{i}}.

It is easily seen that 𝒮′ ⊂ 𝒮 and 𝒮′ ∈ J_{G}(X). □

Remark 4.1.23. (i) In the situation of (4.1.21), let V be a universe with U ⊂ V, and such that C is a V-small site, so that we have a well defined functor (−)^{+} : 𝒞_{V}^{∧} → 𝒞_{V}^{∧}, as in (4.1.11). Lemma 4.1.22implies that the natural map :

colim_{𝒮 ∈J_{G}(X)^{o}} Hom_{𝒞^{∧}}(h_{𝒮} , F) → F^{+}(X)

is bijective. Therefore, if F is a U-presheaf, F^{+}(X) is essentially U-small, and then the same holds for F^{a}(X). In other words, the restriction to 𝒞_{U}^{∧} of the functor 𝒞_{V}^{∧} → C_{V}^{∼} : F ↦ F^{a} is isomorphic to a functor that factors through C^{∼}.

(ii) We deduce that theorem 4.1.13 holds, more generally, when C is an arbitrary U-site. Likewise, a simple inspection shows that remark 4.1.19(i,iv,v) holds when C is only assumed to be a U-site.

Proposition 4.1.24. Let C := (𝒞 , J) be a U-site. The following holds :

(i) A morphism in C^{∼} is an isomorphism if and only if it is both a monomorphism and an epimorphism.

(ii) All epimorphisms in C^{∼} are universal effective.

Proof. (i): Let φ : F → G be a monomorphism in C^{∼}; then the morphism of presheaves i(φ) : iF → iG is also a monomorphism, and it is easily seen that the cocartesian diagram

𝒟 : \xymatrix{iF \ar[r]^{i(φ)} \ar[d]_{i(φ)} & iG \ar[d]^{α} \\ iG \ar[r] & iG ⨿_{iF} iG}

is also cartesian, hence the same holds for the induced diagram of sheaves 𝒟^{a}. If moreover, φ is an epimorphism, then α^{a} is an isomorphism, hence the same holds for φ = (i(φ))^{a}.

(ii): Let f : F → G be an epimorphism in C^{∼}; in view of remarks 4.1.19(i) and 4.1.23(ii), it suffices to show that f is effective. However, set G′ := Im(i(f)), and let p_{i} : F ×_{G} F → F (for i = 1, 2) be the two projections. Suppose φ : F → X is a morphism in C^{∼} such that φ∘p_{1} = φ∘p_{2}; since i(F ×_{G} F) = iF ×_{iG} iF = iF ×_{G′} iF, the morphism i(φ) factors through a (unique) morphism ψ : G′ → X. On the other hand, it is easily seen that (G′)^{a} = G, hence φ factors through the morphism ψ^{a} : G → X. □

Remark 4.1.25. Proposition 4.1.24(i) implies that every morphism φ : X → Y in C^{∼} factors uniquely (up to unique isomorphism) as the composition of an epimorphism followed by a monomorphism. Indeed; such a factorization is provided by the natural morphisms X → Im(φ) and Im(φ) → Y (see example 1.2.26(i)). If X ⟶^{φ′} Z → Y is another such factorization, then by definition φ′ factors through a unique monomorphism ψ : Im(φ) → Z. However, ψ is an epimorphism, since the same holds for φ′. Hence ψ is an isomorphism.

Proposition 4.1.26. Let (𝒞 , J) be a U-site, X ∈ Ob(𝒞), and R any subobject of h_{X}. The following conditions are equivalent :

(a) The inclusion map i : R → h_{X} induces an isomorphism on associated sheaves

i^{a} : R^{a} →^{∼} h_{X}^{a}.

(b) R covers X.

Proof. (b)⇒(a) : By definition, the natural map R^{a}(X) → Hom_{𝒞^{∧}}(R, R^{a}) is bijective, hence there exists a morphism f : h_{X} → R^{a} in 𝒞^{∧} whose composition with i is the unit of adjunction R → R^{a}. Therefore, f^{a} : h_{X}^{a} → R^{a} is a left inverse for i^{a}. On the other hand, we have a commutative diagram :

\xymatrix{Hom_{𝒞^{∧}}(h_{X}^{a}, h_{X}^{a}) \ar[r] \ar[d] & Hom_{𝒞^{∧}}(R^{a}, h_{X}^{a}) \ar[d] \\ h_{X}^{a}(X) \ar[r] & Hom_{𝒞^{∧}}(R, h_{X}^{a})}

whose bottom and vertical arrows are bijective, so that the same holds also for the top arrow. Set g := i^{a} ∘ f^{a}, and notice that g ∘ i^{a} = i^{a}, therefore g must be the identity of h_{X}^{a}, whence the contention.

(a)⇒(b): Let η_{X} : h_{X} → h_{X}^{a} be the unit of adjunction, and set j := (i^{a})^{−1} ∘ η_{X} : h_{X} → R^{a}. By remarks 4.1.19(iv) and 4.1.23(ii), we may find a covering subobject i_{1} : R_{1} → h_{X}, and a morphism j_{1} : R_{1} → R^{sep} whose image in Hom_{𝒞^{∧}}(R_{1}, R^{a}) equals j ∘i_{1}. Denote by η′_{X} : h_{X} → h_{X}^{sep} the unit of adjunction; by construction, the two morphisms

i^{sep} ∘ j_{1}, η′_{X} ∘ i_{1} : R_{1} → h_{X}^{sep}

have the same image in Hom_{𝒞^{∧}}(R_{1}, h_{X}^{a}). This means that there exists a subobject i_{2} : R_{2} → R_{1} covering X, such that i^{sep} ∘ j_{1} ∘ i_{2} = η′_{X} ∘ i_{1} ∘ i_{2}.

Next, let Y_{•} := (Y_{λ} → X | λ ∈ Λ) be a generating family for the sieve of 𝒞 /X corresponding to R_{2}. There follows, for every λ ∈ Λ, a commutative diagram :

(4.1.27)  \xymatrix{h_{Y_{λ}} \ar[r]^{j_{λ}} \ar[d]_{i_{λ}} & R^{sep} \ar[d]^{i^{sep}} \\ h_{X} \ar[r]^{η′_{X}} & h_{X}^{sep}.}

Then, for every λ ∈ Λ there exists a covering subobject s_{λ} : R_{λ} → h_{Y_{λ}} such that j_{λ} lifts to some t_{λ} : R_{λ} → R, and we pick a generating family (Z_{λμ} → Y_{λ} | μ ∈ Λ_{λ}) for the sieve of 𝒞 /Y_{λ} corresponding to R_{λ}; after replacing Y_{•} by the resulting family (Z_{λμ} → X | λ ∈ Λ, μ ∈ Λ_{λ}) (which still covers X, by virtue of remark 4.1.3(iii)), we may assume that (4.1.27) lifts to a commutative diagram

\xymatrix{h_{Y_{λ}} \ar[r]^{t_{λ}} \ar[d]_{i_{λ}} & R \ar[d] \\ h_{X} \ar[r]^{η′_{X}} & h_{X}^{sep}.}

for every λ ∈ Λ. Then there exists a covering subobject s′_{λ} : R′_{λ} → h_{Y_{λ}} such that i ∘ t_{λ} ∘ s′_{λ} = i_{λ} ∘ s′_{λ} in Hom_{𝒞^{∧}}(R′_{λ}, h_{X}). Lastly, set

R′ := ⋃_{λ∈Λ} Im(i_{λ} ∘ s′_{λ} : R′_{λ} → h_{X})

(notice that R′ ∈ Ob(𝒞_{U}^{∧}) even in case Λ is not a small set). It is easily seen that R′ is a covering subobject of X, and the inclusion map R′ → h_{X} factors through R, so R covers X as well. □

Definition 4.1.28. Let (𝒞 , J) be a site such that 𝒞 has small Hom-sets, and let φ : F → G be a morphism in 𝒞^{∧}.

(i) We say that φ is a covering morphism if, for every X ∈ Ob(𝒞) and every morphism h_{X} → G in 𝒞^{∧}, the image of the induced morphism F ×_{G} h_{X} → h_{X} is a covering subobject of h_{X}.

(ii) We say that φ a bicovering morphism if both φ and the diagonal morphism F → F ×_{G} F induced by φ, are covering morphisms.

Remark 4.1.29. (i) In the situation of definition 4.1.28, let S := {X_{i} → X | i ∈ I} be a family of morphisms in 𝒞, and pick a universe V containing U, such that I is V-small. Using remark 3.5.2(iii), it is easily seen that S covers X if and only if the induced morphism in 𝒞_{V}^{∧}

∐_{i∈I} h_{X_{i}} → h_{X}

is a covering morphism.

(ii) With the notation of definition 4.1.28, recall that, by Yoneda’s lemma, the set of morphisms h_{X} → G in 𝒞^{∧} is naturally identified with GX. Fix s ∈ GX, and let ψ_{s} : h_{X} → G be the corresponding morphism; under this identification, for every Y ∈ Ob(𝒞) the image of the induced map (F ×_{(φ,ψ_{s})} h_{X})(Y) → h_{X}(Y) is then the set of all morphisms f : Y → X such that (Gf)(s) lies in the image of the map φ(f) : F Y → GY. Thus φ is a covering morphism if and only if for every X ∈ Ob(𝒞) and every s ∈ GX, the sieve of 𝒞 /X generated by all morphisms f : Y → X in 𝒞 such that (Gf)(s) ∈ Im(φ_{Y} : F Y → GY) covers X. In other words, φ is a covering morphism if and only if for every X ∈ Ob(𝒞) and every s ∈ GX there exists a covering family (f_{i} : Y_{i} → X | i ∈ I) such that (Gf_{i})(s) ∈ Im(φ_{Y_{i}}) for every i ∈ I.

(iii) Likewise, the diagonal morphism F → F ×_{G} F induced by φ is a covering morphism if and only if for every X ∈ Ob(𝒞) and every two sections s, s′ ∈ F X such that φ_{X}(s) = φ_{X}(s′), the set of all morphisms f : Y → X in 𝒞 such that (F f)(s) = (F f)(s′) generates a sieve covering X. This is the same as saying that for every such X and every such pair (s, s′) there exists a covering family (f_{i} : Y_{i} → X | i ∈ I) such that (F f_{i})(s) = (F f_{i})(s′) for every i ∈ I.

Corollary 4.1.30. Let C := (𝒞 , J) be a U-site, and φ : F → G a morphism in 𝒞^{∧}. We have : (i) φ is a covering morphism if and only if φ^{a} : F^{a} → G^{a} is an epimorphism in C^{∼}. (ii) The diagonal morphism F → F ×_{G} F induced by φ is a covering morphism if and only

if φ^{a} is a monomorphism in C^{∼}. (iii) φ is a bicovering morphism if and only if φ^{a} : F^{a} → G^{a} is an isomorphism in C^{∼}.

Proof. Let V be a universe with U ⊂ V, and such that 𝒞 is V-small. Clearly φ is a covering (resp. bicovering) morphism in 𝒞_{U}^{∧} if and only if the same holds for the image of φ under the fully faithful inclusion 𝒞_{U}^{∧} ⊂ 𝒞_{V}^{∧}. So we may replace U by V, and assume that C is a small site.

(i): Suppose that φ^{a} is an epimorphism; let X be any object of 𝒞, and h_{X} → G a morphism. Since the epimorphisms of C^{∼} are universal (remark 4.1.19(i)), the induced morphism

(φ ×_{G} X)^{a} : (F ×_{G} h_{X})^{a} → h_{X}^{a}

is an epimorphism. Let R ⊂ h_{X} be the image of φ×_{G}h_{X}; then the induced morphism R^{a} → h_{X}^{a} is both a monomorphism and an epimorphism, so it is an isomorphism, by proposition 4.1.24(i). Hence R is a covering subobject, according to proposition 4.1.26.

Conversely, suppose that φ is a covering morphism. By remark 4.1.19(iii), G is the colimit of a family (h_{X_{i}}^{a} | i ∈ I) for certain X_{i} ∈ Ob(𝒞). By definition, the image R_{i} of the induced morphism φ ×_{G} X_{i} : F ×_{G} h_{X_{i}} → h_{X_{i}} covers X_{i}, for every i ∈ I. Now, the induced morphism F ×_{G} h_{X_{i}} → R_{i} is an epimorphism, and the morphism R_{i}^{a} → h_{X_{i}}^{a} is an isomorphism (proposition 4.1.26), hence (φ ×_{G} X_{i})^{a} is an epimorphism, and then the same holds for

colim_{i∈I}(φ ×_{G} X_{i})^{a} : colim_{i∈I} F^{a} ×_{G^{a}} h_{X_{i}}^{a} → colim_{i∈I} h_{X_{i}}^{a} = G^{a}

which is isomorphic to φ^{a}, since the colimits of C^{∼} are universal (remark 4.1.19(i)); so φ^{a} is an epimorphism.

(ii): If δ : F → F ×_{G} F is a covering morphism, the foregoing shows that the induced morphism δ^{a} : F^{a} → F^{a} ×_{G^{a}} F^{a} is both an epimorphism and a monomorphism, hence it is an isomorphism (proposition 4.1.24(i)), so φ^{a} is a monomorphism (proposition 1.2.18(i)). Conversely, if φ^{a} is a monomorphism, δ^{a} is an isomorphism, hence δ is a covering morphism, by (i).

(iii) follows from (i), (ii) and proposition 4.1.24(i). □

Remark 4.1.31. (i) Let C := (𝒞 , J) be a site such that 𝒞 has small Hom-sets, and V a universe containing U, such that 𝒞^{∧} is V-small. For every presheaf F on 𝒞, let J^{∧}(F) be the set of all sieves 𝒮 ⊂ 𝒞^{∧}/F such that the natural morphism

∐_{(f:G→F)∈Ob(𝒮)} G → F

is a covering morphism in 𝒞_{V}^{∧}. Notice that 𝒮 ∈ J^{∧}(F) if and only if there exists a generating family (f_{i} : F_{i} → F | i ∈ I) for 𝒮, indexed by a V-small set I, such that the natural morphism ⨿_{i∈I}F_{i} → F is a covering morphism. Since the functor (−)^{a} : 𝒞_{V}^{∧} → C_{V}^{∼} commutes with arbitrary V-small colimits, corollary 4.1.30(i) says that the latter condition holds if and only if the induced morphism ⨿_{i∈I}F_{i}^{a} → F^{a} is an epimorphism in C_{V}^{∼}.

(ii) We claim that the system (J^{∧}(F) | F ∈ Ob(𝒞^{∧})) is a topology on 𝒞^{∧}. Indeed, it is clear from the definition that if 𝒮 ∈ J^{∧}(F), then 𝒮 ×_{F} f lies in J^{∧}(F′) for every morphism f : F′ → F in 𝒞^{∧}. Next, let 𝒮 , 𝒮′ ⊂ 𝒞^{∧}/F be two sieves, with 𝒮 ∈ J^{∧}(F) and such that 𝒮′ ×_{F} f ∈ J^{∧}(F′) for every (f : F′ → F) ∈ Ob(𝒮). By the foregoing, this means that the induced morphism ⨿_{(g:G→F)∈Ob(𝒮′)} G^{a} ×_{F^{a}} F^{′a} → F^{′a} is an epimorphism in C_{V}^{∼} for every such f. Since (in every category) an arbitrary coproduct of epimorphisms is an epimorphism, it follows that both of the following morphisms are epimorphisms :

∐_{(f:F′→F)∈Ob(𝒮)} ∐_{(g:G→F)∈Ob(𝒮′)} G^{a} ×_{F^{a}} F^{′a} → ∐_{(f:F′→F)∈Ob(𝒮)} F^{′a} → F^{a}

and then so is their composition. But f ∘ (g ×_{F} F′) : G ×_{F} F′ → F lies in 𝒮′ for every f ∈ Ob(𝒮) and every g ∈ Ob(𝒮′), so 𝒮′ ∈ J^{∧}(F), as required. We denote the resulting site

C^{∧} := (𝒞^{∧}, J^{∧}).

In light of remark 4.1.29(ii) it is easily seen that C^{∧} is independent of the choice of V.

4.2. Continuous and cocontinuous functors. We now begin the study of those functors that are compatible with given topologies on their domain and codomain of definition; as explained hereafter, such compatibility can manifest itself in two distinct fashions :

Definition 4.2.1. Let C = (𝒞 , J) and C′ = (𝒞′, J′) be two sites, and g : 𝒞 → 𝒞′ a functor.

(i) We say that g is continuous for the topologies J and J′, if the following holds. For every universe V and every V-sheaf F on C′, the V-presheaf g_{V}^{∧}F is a V-sheaf on C (notation of (1.2.2)). In this case, g_{V}^{∧} clearly induces by restriction a functor

g̃_{V∗} : C_{V}^{′∼} → C_{V}^{∼}.

(ii) We say that g is cocontinuous for the topologies J and J′ if the following holds. For every X ∈ Ob(𝒞), and every covering sieve 𝒮′ ∈ J′(gX), the sieve g_{|X}^{−1}𝒮′ covers X in C (notation of definition 3.5.1(iii) and (1.1.26)).

Example 4.2.2. Let T, T′ be two topological spaces, and f : T → T′ a continuous map; denote by 𝒯 (resp. 𝒯 ′) the category of open subsets of T (resp. of 𝒯 ′), endowed with its natural topology as in example 4.1.2. Then f induces a functor f^{−1} : 𝒯 ′ → 𝒯 : U ↦ f^{−1}U, and it is easily seen that f^{−1} is a continuous functor, according to definition 4.2.1(i).

Lemma 4.2.3. Let C = (𝒞 , J) and C′ = (𝒞′, J′) be two sites, g : 𝒞 → 𝒞′ a functor, and V a universe such that 𝒞 is V-small and 𝒞′ has small Hom-sets. The following conditions are equivalent :

(a) For every V-sheaf F on C′, the presheaf g_{V}^{∧}F is a V-sheaf on C.

(b) For every morphism φ : F → G of V-presheaves on 𝒞 that is bicovering for the topology J, the morphism g_{V!}(φ) : g_{V!}F → g_{V!}G is bicovering for the topology J′.

(c) For every X ∈ Ob(𝒞) and every subobject R ⊂ h_{X} covering X for the topology J, the induced morphism of presheaves g_{V!}R → g_{V!}h_{X} is bicovering for the topology J′.

(d) g is continuous for the topologies J and J′.

Proof. (a)⇒(b): By virtue of remark 1.3.6(iv) we may replace V by a larger universe, and assume that 𝒞 and 𝒞′ are V-small. Then, by corollary 4.1.30(iii), the morphism φ^{a} : F^{a} → G^{a} is an isomorphism, and we need to check that g_{V!}(φ)^{a} is an isomorphism. To this aim, let H be any V-sheaf on C′; we get a commutative diagram :

\xymatrix{Hom_{𝒞^{′∧}}(g_{V!}(G)^{a}, H) \ar[r] \ar[d] & Hom_{𝒞^{′∧}}(g_{V!}(G), H) \ar[r] \ar[d] & Hom_{𝒞^{∧}}(G, g_{V}^{∧}H) \ar[d] & Hom_{𝒞^{∧}}(G^{a}, g_{V}^{∧}H) \ar[l] \ar[d] \\ Hom_{𝒞^{′∧}}(g_{V!}(F)^{a}, H) \ar[r] & Hom_{𝒞^{′∧}}(g_{V!}(F), H) \ar[r] & Hom_{𝒞^{∧}}(F, g_{V}^{∧}H) & Hom_{𝒞^{∧}}(F^{a}, g_{V}^{∧}H) \ar[l]}

whose horizontal arrows are bijections, due to (a). Since φ^{a} is bijective, the right-most vertical arrow is bijective, hence the same holds for the left-most vertical arrow, whence the contention.

(b)⇒(c) and (d)⇒(a) are trivial.

(c)⇒(d): Let V′ be a universe, and F a V′-sheaf on C′; we need to check that g_{V′}^{∧}F is a sheaf on C. To this aim, we may replace V′ by a larger universe, and assume that V ⊂ V′, and that 𝒞 and 𝒞′ are V′-small; then by remark 1.3.6(iv) and corollary 4.1.30(iii) we see that for every X ∈ Ob(𝒞) and every covering subobject R ⊂ h_{X} the induced morphism g_{V′!}(R)^{a} → g_{V′!}(h_{X})^{a} is an isomorphism. Now, we have a commutative diagram :

\xymatrix{Hom_{𝒞^{∧}}(h_{X}, g_{V′}^{∧}F) \ar[r] \ar[d] & Hom_{𝒞^{′∧}}(g_{V′!}h_{X}, F) \ar[d] & Hom_{𝒞^{′∧}}(g_{V′!}(h_{X})^{a}, F) \ar[l] \ar[d] \\ Hom_{𝒞^{∧}}(R, g_{V′}^{∧}F) \ar[r] & Hom_{𝒞^{′∧}}(g_{V′!}R, F) & Hom_{𝒞^{′∧}}(g_{V′!}(R)^{a}, F) \ar[l]}

whose horizontal arrows are bijections; moreover, the right-most vertical arrow is bijective as well, hence the same holds for the left-most vertical arrow, which is the contention. □

Lemma 4.2.4. In the situation of definition 4.2.1, consider the following conditions : (a) g is continuous.

(b) For every covering family (X_{i} → X | i ∈ I) in C, the family (gX_{i} → gX | i ∈ I) covers gX in C′.

(c) For every small covering family (X_{i} → X | i ∈ I) in C, the family (gX_{i} → gX | i ∈ I) covers gX in C′.

(d) For every universe V, and every V-presheaf F on 𝒞′ that is separated for the topology J′, the presheaf g_{V}^{∧}F is separated for the topology J, so that g_{V}^{∧} restricts to a functor

g_{V∗}^{sep} : C_{V}^{′sep} → C_{V}^{sep}.

Then (a)⇒(b)⇔(d)⇒(c). Moreover, if all fibre products are representable in 𝒞, and g commutes with fibre products, then (b)⇒(a). Furthermore, if C is a U-site, then (c)⇒(b).

Proof. Obviously (b)⇒(c).

(a)⇒(b): After replacing U by a larger universe, we may assume that I is a small set and both 𝒞 and 𝒞′ are small. Let F be any sheaf on C′; by assumption, g^{∧}F is a sheaf on C, hence the natural map :

F (gX) = g^{∧}F (X) → ∏_{i∈I} g^{∧}F (X_{i}) = ∏_{i∈I} F (gX_{i})

is injective (by (4.1.6)). This means that the induced morphism

(4.2.5)  ∐_{i∈I} h_{gX_{i}}^{a} → h_{gX}^{a}

is an epimorphism in C^{∼}. Then the assertion follows from corollary 4.1.30and remark 4.1.29(i).

(d)⇒(b) is similar : we may assume that I, 𝒞 and 𝒞′ are small. If F is any separated presheaf on C′, then by assumption g^{∧}F is separated on C, and arguing as in the foregoing, we deduce that the induced morphism ⨿_{i∈I}h_{gX_{i}}^{sep} → h_{gX}^{sep} is an epimorphism in C^{sep}, and then (4.2.5) is an epimorphism in C^{∼}, so we conclude, again by corollary 4.1.30and remark 4.1.29(i).

(b)⇒(d): let F be a separated V-presheaf on C′, and (X_{i} → X | i ∈ I) any covering family in C; in view of (4.1.6), it suffices to show that the induced map

F (gX) = g_{V}^{∧}F (X) → ∏_{i∈I} g_{V}^{∧}F (X_{i}) = ∏_{i∈I} F (gX_{i})

is injective. But this is clear, since (gX_{i} → gX | i ∈ I) is a covering family in C′.

Next, suppose that (b) holds, the fibre products in 𝒞 are representable, and g commutes with all fibre products. For every i, j ∈ I, set X_{ij} := X_{i} ×_{X} X_{j}. To show that (a) holds, it suffices – in view of (4.1.6) – to prove :

Claim 4.2.6. The natural map

g^{∧}F (X) → Hom_{𝒞^{∧}}(Coequal( ∐_{i,j∈I} [figure: h_{X_{ij}}] ∐_{i∈I}h_{X_{i}}), g^{∧}F )

is bijective.

Proof of the claim. Since g_{!} is right exact, and due to (1.3.7), this is the same as the natural map

F (gX) → Hom_{𝒞^{∧}}(Coequal( ∐_{i,j∈I} [figure: h_{gX_{ij}}] ∐_{i∈I}h_{gX_{i}}), F ).

However, by assumption gX_{ij} = gX_{i} ×_{gX} gX_{j}, and then the claim follows by applying (4.1.6) to the covering family (gX_{i} → gX | i ∈ I). ♢

Lastly, suppose that C is a U-site; in order to show that (c)⇒(b), we remark more precisely :

Claim 4.2.7. Let C be a U-site, ℱ := (φ_{i} : X_{i} → X | i ∈ I) any covering family. Then there exists a small set J ⊂ I such that the subfamily (φ_{i} | i ∈ J) covers X.

Proof of the claim. Let 𝒮 ⊂ 𝒞/X be the sieve generated by 𝒮. By lemma 4.1.22, we may find a small covering family ℱ′ := (ψ_{i} : X′_{i} → X | i ∈ I′) (i.e. such that I′ is small), that generates a sieve 𝒮′ ⊂ 𝒮. Then, for every i ∈ I′ we may find γ(i) ∈ I such that ψ_{i} factors through φ_{γ(i)}. The subset J := γI′ will do. ♢

Let ℱ and J be as in claim 4.2.7; then the sieve g𝒮 generated by (g(φ_{i}) | i ∈ I) contains the sieve g𝒮′ generated by (g(φ_{i}) | i ∈ J). Especially, if g𝒮′ is a covering sieve, the same holds for g𝒮, whence the contention. □

4.2.8. In the situation of definition 4.2.1, let V be a universe with U ⊂ V, such that C is a V-site, and 𝒞′ has V-small Hom-sets. Then we may define a functor

ğ_{V}^{∗} : C_{V}^{′∼} → C_{V}^{∼} F ↦ (g_{V}^{∧} ∘ i_{C′}F)^{a}

(where i_{C′} : C_{V}^{′∼} → 𝒞_{V}^{∧} is the forgetful functor). On the other hand, if C is a V-small site and C′ is a V-site, we can define the functors

g_{V!}^{a} : 𝒞_{V}^{∧} → C_{V}^{′∼}  F ↦ (g_{V!}F)^{a}  and g̃_{V}^{∗} := g_{V!}^{a} ∘ i_{C} : C_{V}^{∼} → C_{V}^{′∼}.

As usual, if V = U we often omit the subscript U. Later we shall generalize these constructions to the case where both C and C′ are V-sites : see corollary 4.3.19. For now we remark :

Lemma 4.2.9. In the situation of definition 4.2.1, let V be a universe with U ⊂ V, such that 𝒞 is V-small, and 𝒞′ has V-small Hom-sets. Then the following conditions are equivalent :

(a) g is a cocontinuous functor.

(b) For every covering morphism φ : F → G in 𝒞^{′∧}, the morphism g^{∧}(φ) : g^{∧}F → g^{∧}G is covering.

(c) For every bicovering morphism φ : F → G in 𝒞^{′∧}, the morphism g^{∧}(φ) : g^{∧}F → g^{∧}G is bicovering.

(d) For every X ∈ Ob(𝒞′) and every covering subobject R of h_{X}, the induced morphism g^{∧}R → g^{∧}h_{X} is bicovering.

(e) For every F ∈ Ob(C_{V}^{∼}), the V-presheaf g_{V∗}F is a V-sheaf on C′ (see remark 1.3.6(i)). When these conditions hold, the restriction of g_{V∗} is a right adjoint to ğ_{V}^{∗} denoted

ğ_{V∗} : C_{V}^{∼} → C_{V}^{′∼}

Proof. After replacing V by U, we may assume that 𝒞 is small, and 𝒞′ has small Hom-sets. Recall that the unit of the natural adjunction for the pair (g_{!}, g^{∧}) assigns to every X ∈ Ob(ℬ) the morphism η_{X} : h_{X} → g^{∧}(g_{!}h_{X}) →^{∼} g^{∧}(h_{gX}) such that η_{X}(s) := F s for every X′ ∈ Ob(𝒞) and every (s : X′ → X) ∈ h_{X}(X′) : see remark 1.3.6(v). We notice :

Claim 4.2.10. Let X ∈ Ob(𝒞), and 𝒯 ′ ⊂ 𝒞′/gX a covering sieve; set 𝒯 := g_{|X}^{−1}𝒯 ′. Then :

h_{𝒯} = g^{∧}(h_{𝒯 ′}) ×_{g^{∧}(h_{gX})} h_{X}.

Proof of the claim. For every X′ ∈ Ob(𝒞), the set (g^{∧}(h_{𝒯 ′}) ×_{g^{∧}(h_{gX})} h_{X})(X′) consists of the pairs (t, s) where t : gX′ → gX is an object of 𝒯 ′ and s : X′ → X is a morphism in 𝒞, such that g(s) = t. In other words, this is the set of all s ∈ h_{X}(X′) such that g(s) ∈ Ob(𝒯 ′). This is precisely the definition of h_{𝒯} (X′). ♢

(a)⇒(b): Let X ∈ Ob(𝒞) and ψ : h_{X} → g^{∧}G any morphism in 𝒞^{∧}. By adjunction, ψ corresponds to a morphism ψ′ : h_{gX} → G in 𝒞^{′∧}, such that ψ = g^{∧}(ψ′) ∘ η_{X}. By assumption, ψ′ induces a covering morphism F ×_{G} h_{gX} → h_{gX}, whose image is therefore of the form h_{𝒮′} for some covering sieve 𝒮′ ∈ J′(gX). Since g^{∧} is exact, the epimorphism π : F ×_{G}h_{gX} → h_{𝒮′} induces an epimorphism g^{∧}(π) : g^{∧}(F ×_{G} h_{gX} → h_{gX}) → g^{∧}(h_{𝒮′} ), and notice the following commutative diagram whose three square subdiagrams are cartesian :

\xymatrix{g^{∧}(F) ×_{g^{∧}(G)} h_{X} \ar[r] \ar[d] & g^{∧}(h_{𝒮′} ) ×_{g^{∧}(h_{gX})} h_{X} \ar[r] \ar[d] & h_{X} \ar[d]^{η_{X}} \\ g^{∧}(F ×_{G} h_{gX}) \ar[r]^{g^{∧}(π)} \ar[d] & g^{∧}(h_{𝒮′} ) \ar[r] & g^{∧}(h_{gX}) \ar[d]^{g^{∧}(ψ′)} \\ g^{∧}(F) \ar[rr] &  & g^{∧}(G)}

In view of claim 4.2.10, we deduce that the image of the morphism g^{∧}(F) ×_{g^{∧}(G)} h_{X} → h_{X} induced by g^{∧}(φ) is h_{𝒮} , where 𝒮 := g_{|X}^{−1}𝒮′; but 𝒮 is a covering sieve of X, since g is cocontinuous. This shows that (b) holds.

(b)⇒(c): If φ is a bicovering morphism, (b) implies that both g^{∧}(φ) and the morphism δ : g^{∧}(F) → g^{∧}(F ×_{G} F) induced by φ are covering morphisms. However, g^{∧} is exact, so δ is naturally identified with the diagonal morphism g^{∧}(F) → g^{∧}(F) ×_{g^{∧}(G)} g^{∧}(F) induced by g^{∧}(φ); thus, g^{∧}(φ) is bicovering.

(c)⇒(d) is obvious.

(d)⇒(e): Let F be any sheaf on C; we have to show that the natural map

g_{∗}F (Y) → Hom_{𝒞^{′∧}}(R, g_{∗}F)

is bijective, for every Y ∈ Ob(𝒞′), and every covering subobject of h_{Y}. By adjunction (and by corollary 4.1.30), this is the same as saying that the monomorphism g^{∧}(R) → g^{∧}(h_{Y}) is a covering morphism, which holds by assumption (d).

(e)⇒(a): The assumption implies that, for every sheaf F on C, every X ∈ Ob(𝒞), and every covering sieve 𝒮′ ⊂ 𝒞′/gX, the natural map

g_{∗}F (gX) → Hom_{𝒞^{′∧}}(h_{𝒮′} , g_{∗}F)

is bijective. By adjunction, the same then holds for the natural map

Hom_{𝒞^{∧}}(g^{∧}(h_{gX}), F) → Hom_{𝒞^{∧}}(g^{∧}(h_{𝒮′} ), F)

so the induced morphism g^{∧}(h_{𝒮′} ) → g^{∧}(h_{gX}) is bicovering (proposition 4.1.26). Also, this morphism is a monomorphism (since g^{∧} commutes with all limits); therefore, after base change along the unit of adjunction η_{X} : h_{X} → g^{∧}(h_{gX}), we deduce a covering monomorphism

g^{∧}(h_{𝒮′} ) ×_{g^{∧}(h_{gX})} h_{X} → h_{X}.

Then the contention follows from claim 4.2.10.

Lastly, the assertion concerning the left adjoint ğ_{U}^{∗} is immediate from the definitions. □

Lemma 4.2.11. Let C′ := (𝒞′, J′) be a U-site, C := (𝒞 , J) a small site, and g : 𝒞 → 𝒞′ a continuous functor. Then the following holds :

(i) The functor g̃_{U}^{∗} : C_{U}^{∼} → C_{U}^{′∼} of (4.2.8) is left adjoint to g̃_{U∗}.

(ii) The natural diagrams of functors :

\xymatrix{𝒞 \ar[r]^{g} \ar[d]_{h_{𝒞}^{a}} & 𝒞′ \ar[d]^{h_{𝒞′}^{a}} & 𝒞_{U}^{∧} \ar[rr]^{(−)^{a}} \ar[dr]_{g_{U!}^{a}} &  & C_{U}^{∼} \ar[dl]^{_{g̃_{∗}_{U}}} \\ C_{U}^{∼} \ar[r]^{_{g̃_{∗}_{U}}} & C_{U}^{′∼} &  & C_{U}^{′∼}}

are essentially commutative. (Notation of theorem 4.1.13.)

Proof. The first assertion is straightforward, and the commutativity of the square diagram in (ii) is reduced to the corresponding assertion for (1.3.7), which has already been remarked. To show the commutativity for the triangular diagram in (ii), it suffices to check that for every presheaf F on 𝒞 and every sheaf G of C′ the natural map η_{F} : F → i_{C}(F^{a}) induces a bijection

Hom_{C^{′∼}}(g̃_{U}^{∗}(F^{a}), G) →^{∼} Hom_{C^{′∼}}(g_{U!}^{a}(F), G).

However, by adjunction, the latter is naturally identified with the map

(4.2.12)  Hom_{𝒞^{∧}}(F^{a}, g_{U}^{∧}G) → Hom_{𝒞^{∧}}(F, g_{U}^{∧}G)

induced by η_{F}; but g_{U}^{∧}G is a sheaf, since g is continuous, so (4.2.12) is indeed bijective. □

4.2.13. Let (𝒞 , J) be a small site, (𝒞′, J′) a U-site, u : 𝒞 → 𝒞′ a continuous (resp. cocontinuous) functor. Let also V be a universe such that U ⊂ V. Then it follows easily from remark (1.3.6)(iv) and lemma 4.2.11(i) (resp. from lemma 4.2.9), and from remark 4.1.23(ii) that we have essentially commutative diagrams of categories :

[figure: C_{U}^{∼}; ^{ũ^{∗}^{U}}; C_{U}^{′∼}; _{ũ_{U∗}}; C_{V}^{∼}; _{ũ_{∗}_{V}}; C_{V}^{′∼}; _{ũ_{V∗}}; (resp.; C_{U}^{∼}; ^{ŭ^{U∗}}; C_{U}^{′∼}; _{ŭ_{∗}_{U}}; C_{V}^{∼}; _{ŭ_{V∗}}; C_{V}^{′∼}; _{ŭ_{∗}_{V}}; )]

whose vertical arrows are the inclusion functors. More generally, the diagram for ŭ_{U∗} is well defined and essentially commutative, whenever C is a U-site, and 𝒞′ has small Hom-sets.

Lemma 4.2.14. (i) Let C := (𝒞 , J) and C′ := (𝒞′, J′) be two sites, and v : 𝒞 → 𝒞′, u : 𝒞′ → 𝒞 two functors, such that v is left adjoint to u. The following conditions are equivalent:

(a) u is continuous.

(b) v is cocontinuous. (ii) Moreover, when these conditions hold, then for every universe V such that 𝒞 and 𝒞′ are

V-small, we have natural isomorphisms of functors :

ũ_{V∗} →^{∼} v̆_{V∗}  ũ_{V}^{∗} →^{∼} v̆_{V}^{∗}.

Proof. In view of lemma 4.2.3, we may replace U by a larger universe, after which we may assume that C and C′ are small sites. In this case, the lemma follows from lemma 4.2.9 and proposition 1.3.25(i). □

Lemma 4.2.15. Let (𝒞 , J) be a small site, (𝒞′, J′) a U-site, u : 𝒞 → 𝒞′ a continuous and cocontinuous functor. Then we have :

(i) ũ_{∗} = ŭ^{∗} and this functor admits the left adjoint ũ^{∗} and the right adjoint ŭ_{∗}.

(ii) We have a natural isomorphism of functors

(−)^{a} ∘ u^{∧} →^{∼} ũ_{∗} ∘ (−)^{a} : 𝒞^{′∧} → C^{∼}.

(iii) ũ^{∗} is fully faithful if and only if the same holds for ŭ_{∗}.

(iv) If u is fully faithful, then the same holds for ũ^{∗}. The converse holds, provided the topologies J and J′ are coarser than the canonical topologies.

Proof. (i) is clear by inspecting the definitions.

(ii): For every presheaf G on 𝒞′, let i_{G} : G → G^{a} be the natural morphism; due to lemma 4.2.9the induced morphism u^{∧}(i_{G}) : u^{∧}G → u^{∧}(G^{a}) is bicovering, and notice that u^{∧}(G^{a}) is a sheaf, since u is continuous. Thus, u^{∧}(i_{G}) factors through the natural morphism i_{u^{∧}G} : u^{∧}G → (u^{∧}G)^{a} and a unique morphism in C^{∼}

ω_{G} : (u^{∧}G)^{a} → u^{∧}(G^{a}).

Since i_{u^{∧}G} is bicovering, the same holds for ω_{G}, i.e. the latter is an isomorphism of on C^{∼}. It is then easily seen that the rule G ↦ ω_{G} yields the sought isomorphism.

Assertion (iii) follows from (i) and proposition 1.1.20(iv). Next, suppose that u is fully faithful; then the same holds for ŭ_{∗} (corollary 1.5.19(ii)), so the claim follows from (iii). Lastly, suppose that ũ^{∗} is fully faithful, and both J and J′ are coarser than the canonical topologies on 𝒞 and 𝒞′. In such case, the Yoneda embedding for 𝒞 (resp. 𝒞′) realizes 𝒞 (resp. 𝒞′) as a full subcategory of C^{∼} (resp. of C^{′∼}). Then, from (1.3.7) and the explicit expression of ũ^{∗} provided by lemma 4.2.11(i), we deduce that u is fully faithful. □

Definition 4.2.16. Let C := (𝒞 , J) be a site, ℬ any category, and g : ℬ → 𝒞 a functor. Pick a universe V such that ℬ is V-small and C is a V-site. According to remark 4.1.8(iv), there is a finest topology J_{g} on ℬ such that, for every V-sheaf F on C, the V-presheaf g^{∧}F is a V-sheaf on (ℬ, J_{g}). By lemma 4.2.3, the topology J_{g} is independent of the chosen universe V. We call J_{g} the topology induced via g by J on ℬ. Clearly g is continuous for the sites (ℬ, J_{g}) and C.

Lemma 4.2.17. In the situation of definition 4.2.16, let X be any object of ℬ, and R ⊂ h_{X} any subobject in ℬ^{∧}. We have :

(i) R ∈ J_{g}(X) if and only if for every morphism Y → X in ℬ, the induced morphism g_{V!}(R ×_{X} Y) → h_{g(Y)} is a bicovering morphism in 𝒞_{V}^{∧} (for the topology J on 𝒞).

(ii) Suppose moreover, that either one of the following condition holds: (a) The functor g_{V!}^{a} : ℬ_{V}^{∧} → C_{V}^{∼} commutes with all fibre products.

(b) All fibre products are representable in ℬ, and g commutes with fibre products. Then a family (f_{i} : B_{i} → B | i ∈ I) of morphisms of ℬ generates a covering sieve of J_{g} if and only if (g(f_{i}) : gB_{i} → gB | i ∈ I) generates a covering sieve of J.

Proof. (i): According to remark 4.1.8(iii), we have R ∈ J_{g}(X) if and only if for every morphism Y → X in 𝒞 the natural morphism φ_{Y} : R ×_{X} Y → h_{Y} induces a bijection

(g_{V}^{∧}G)(Y) →^{∼} Hom_{ℬ^{∧}} (R ×_{X} Y, g_{V}^{∧}G) for every G ∈ Ob(C_{V}^{∼}).

By adjunction, this is equivalent to saying that φ_{Y} induces a bijection

G(gY) →^{∼} Hom_{𝒞^{∧}}(g_{V!}(R ×_{X} Y), G) for every G ∈ Ob(C_{V}^{∼}).

The latter is in turn naturally identified with the map

Hom_{C_{V}^{∼}}(g_{V!}(φ_{Y})^{a}, G) : Hom_{C_{V}^{∼}}(h_{gY}^{a} , G) → Hom_{C_{V}^{∼}}(g_{V!}(R ×_{X} Y)^{a}, G).

By Yoneda’s lemma, it follows that R ∈ J_{g}(R) if and only if g_{V!}(φ_{Y})^{a} is an isomorphism, and this is equivalent to the stated condition, by corollary 4.1.30.

(ii): Suppose first that condition (a) holds. Notice that since ℬ is V-small, we may assume that the indexing set I is V-small as well. Let R ⊂ h_{B} (resp. R′ ⊂ h_{gB}) be the subobject generated by the family (f_{i} | i ∈ I) (resp. (g(f_{i}) | i ∈ I)); if R is a covering subobject of h_{B} for J_{g}, then R′ is a covering subobject of h_{gB} for J, by lemma 4.2.4. Conversely, suppose that R′ is a covering subobject of h_{gB} for J; according to (i) and corollary 4.1.30, we need to check that the inclusion j : R → h_{B} induces an isomorphism g_{V!}^{a}(R ×_{B} B′) → h_{gB′}^{a} for every morphism B′ → B in ℬ. Since g_{V!}^{a} commutes with fibre products, we are then easily reduced to checking that j induces an isomorphism g_{V!}^{a}(j) : g_{V!}^{a}R → h_{B}^{a}. However, g_{V!}^{a}(j) is a monomorphism, since g_{V!}^{a} commutes with fibre products (proposition 1.3.18(i)). To show that g_{V!}^{a}(j) is an epimorphism, let S := ∐_{i∈I} h_{B_{i}} (which is an object of ℬ_{V}^{∧}, since I is V-small) and denote by j′ : S → h_{B} the morphism induced by the morphisms f_{i}. The image of g_{V!}^{a}(j) contains the image of g_{V!}^{a}(j′); on the other hand, g_{V!}^{a} commutes with coproducts, since it is a left adjoint, and we have natural identifications : g_{V!}^{a}(h_{B_{i}}) = h_{gB_{i}}^{a} for every i ∈ I. Thus, the image of g_{V!}^{a}(j′) equals that of the morphism φ : ∐_{i∈I} h_{gB_{i}}^{a} → h_{B}^{a} induced by the morphisms g(f_{i}). But since R′ is a covering subobject, φ is an epimorphism, so the same holds for g_{V!}^{a}(j).

Next, suppose that condition (b) holds, and let f_{•} := (f_{i} : B_{i} → B | i ∈ I) be a family of morphisms of ℬ such that (g(f_{i}) | i ∈ I) covers gB in the topology J. Set B_{ij} := B_{i} ×_{B} B_{j} for every i, j ∈ I; the subobject R ⊂ h_{B} generated by f_{•} is the coequalizer in ℬ^{∧} of the natural projections

\xymatrix{∐_{i,j∈I} h_{B_{ij}} \ar[r] \ar[r] & ∐_{i∈I} h_{B_{i}}}

and according to (i) we need to check that for every morphism B′ → B the induced morphism g_{V!}(R ×_{B} B′) → h_{gB′} is a bicovering morphism. However, set as well B′_{i} := B_{i} ×_{B} B′ and B′_{ij} := B_{ij} ×_{B} B′ for every i, j ∈ I; then R ×_{B} B′ is the coequalizer of the induced projections

\xymatrix{∐_{i,j∈I} h_{B′_{ij}} \ar[r] \ar[r] & ∐_{i∈I} h_{B′_{i}}.}

On the other hand, g_{V!} commutes with coequalizers and coproducts, since it is a left adjoint; in view of remark 1.3.6(ii) we deduce that g_{V!}(R ×_{B} B′) is the coequalizer of the projections

\xymatrix{∐_{i,j∈I} h_{gB′_{ij}} \ar[r] \ar[r] & ∐_{i∈I} h_{gB′_{i}}.}

But by assumption we have as well gB′_{i} = gB_{i} ×_{gB} gB′, whence h_{gB_{i}} = h_{gB′_{i}} ×_{h_{gB}} h_{B′}, and likewise for h_{gB′_{ij}}, for every i, j ∈ I (corollary 1.4.3(vi)). We conclude that g_{V!}(R ×_{B} B′) is the subobject of h_{B′} generated by the family (g(f_{i}) ×_{gB} gB′ → gB′ | i ∈ I), and it is therefore a covering subobject, as required. □

Remark 4.2.18. (i) In the situation of lemma 4.2.17, let V′ be another universe with V ⊂ V′, and F a V′-presheaf on ℬ. According to lemma 1.4.8, F is naturally isomorphic to the colimit of the functor h_{ℬ} ∘s_{F} : ℱ ib(F) → ℬ_{V}^{∧} ⊂ ℬ_{V′}^{∧}, where s_{F} : ℱ ib(F) → ℬ is the source functor. For every finite subset S ⊂ Ob(ℱ ib(F)), let F_{S} ⊂ F be the subobject generated by the union of the images of all the morphisms h_{(X,s)} : h_{B} → F with (B, s) ∈ S (notation of (1.4.7)), and notice that F_{S} is isomorphic to an object of ℬ_{V}^{∧}. After replacing each F_{S} by an isomorphic object of ℬ_{V}^{∧}, we obtain therefore F as the filtered colimit of a system (F_{S} | S ∈ Σ_{F}) of objects of ℬ_{V}^{∧}, indexed by the set Σ_{F} of all finite subsets of Ob(ℱ ib(F)).

(ii) Now, let F_{1} ⟶^{φ_{1}} F_{0} ⟵^{φ_{2}} F_{2} be two morphisms in ℬ_{V}^{∧}. Consider the set Σ(φ_{1}, φ_{2}) of all triples (S_{0}, S_{1}, S_{2}) where S_{i} is a finite subset of Ob(ℱ ib(F_{i})) for i = 0, 1, 2, and (B, φ_{j} ∘ s) ∈ S_{0} for j = 1, 2 and every (B, s) ∈ S_{j}. We endow Σ(φ_{1}, φ_{2}) with the partial order such that (S_{0}, S_{1}, S_{2}) ≥ (S′_{0}, S′_{1}, S′_{2}) if and only if S′_{i} ⊂ S_{i} for i = 0, 1, 2. We have a natural functor

τ : Σ(φ_{1}, φ_{2}) → ℬ_{V}^{∧} (S_{0}, S_{1}, S_{2}) ↦ F_{S_{1}} ×_{F_{S_{0}}} F_{S_{2}}

and by a simple inspection we see that the colimit of τ is naturally isomorphic to F_{1} ×_{F_{0}} F_{2}.

(iii) We deduce that G := g_{V′!}^{a}(F_{1} ×_{F_{0}} F_{2}) is naturally isomorphic to the colimit of g_{V′!}^{a} ∘ τ, since g_{V′!}^{a} is a left adjoint; but g_{V′!}^{a} ∘ τ is in turn naturally isomorphic to the functor g_{V!}^{a} ∘ τ, by lemma 4.2.11(ii). Suppose now that g_{V!}^{a} commutes with all fibre products of ℬ_{V}^{∧}. Then G is also naturally isomorphic to the filtered colimit of the system

(g_{V!}^{a}F_{1,S_{1}} ×_{g_{V!}^{a}F_{0,S_{0}}} g_{V!}^{a}F_{2,S_{2}} | (S_{0}, S_{1}, S_{2}) ∈ Σ(φ_{1}, φ_{2})).

But since filtered colimits in ℬ_{V′}^{∧} commute with fibre products, the latter is naturally isomorphic to the fibre product G_{1} ×_{G_{0}} G_{2}, where G_{i} is the filtered colimit of the system (g_{V′!}^{a}F_{S_{i}} | S_{i} ∈ Σ_{F_{i}}) (notice that the projections Σ(φ_{1}, φ_{2}) → Σ_{F_{i}} : (S_{0}, S_{1}, S_{2}) → S_{i} are obviously coinitial functors). But again, since g_{V′!}^{a} commutes with filtered colimits, G_{i} is naturally isomorphic to g_{V′!}^{a}(F_{i}). We conclude that g_{V′!}^{a} commutes with fibre products as well. In particular, condition (ii.a) of lemma 4.2.17is independent of the choice of universe V.

4.3. Morphisms of sites. We observed in example 4.2.2 that every continuous map of topological spaces f : T → T′ induces a continuous functor u := f^{−1} : 𝒯 ′ → 𝒯 between the corresponding categories of open subsets. It is well known that the induced functor on categories of sheaves ũ^{∗} : 𝒯 ^{∼} → 𝒯 ^{′∼} is moreover exact, whereas for a general continuous functor g between arbitrary sites, only the right exactness of g̃^{∗} is always assured. The exactness of g̃^{∗} therefore singles out an interesting class of continuous functors, which will play an important role in our discussion of topoi, and – even more crucially – for the study of the functorial properties of the categories of stacks, in section 5.4. In this section we carry out a preliminary investigation of this class of functors, and prove a useful characterization. Let us begin with :

Definition 4.3.1. (i) Let C = (𝒞 , J) and C′ = (𝒞′, J′) be two sites, and V a universe such that C and C′ are V-small. A morphism of sites C′ → C is the datum of a continuous functor g : 𝒞 → 𝒞′, such that the left adjoint g̃_{V}^{∗} of g̃_{V∗} is exact (notation of definition 4.2.1(i)).

(ii) Let U′ be a universe with U ∈ U′; the U′-small U-sites are the objects of a 2-category

(U, U′)-Site

whose 1-cells are the morphisms of sites, and whose 2-cells β : g ⇒ g′ are the natural transformations g′ ⇒ g between the underlying functors of such morphisms, with the obvious composition laws for 1-cells and 2-cells.

(iii) We shall also be interested in the 2-category of U-lex-sites

(U, U′)-lex.Site

defined as the 2-subcategory of (U, U′)-Site whose objects are the finitely complete U′-small U-sites and whose 1-cells are the morphisms of sites g as in (i) such that the underlying functor g : 𝒞 → 𝒞′ is left exact. For any two such morphisms g, g′, the 2-cells g ⇒ g′ are the same as in (U, U′)-Site.

Remark 4.3.2. (i) With the notation of definition 4.3.1(ii), two different choices of the auxiliary universe U′ do not necessarily yield 2-equivalent 2-categories; nevertheless, such choices are usually ininfluent in the proofs of our results, so we mostly omit mentioning them explicitly, and write U-Site and U-lex.Site for these 2-categories. When the choice of U is clear from the context, we shall likewise drop the mention of U, and write just Site and lex.Site.

(ii) On the other hand, the following proposition 4.3.9 will show that the definition of morphism of sites g : (𝒞′, J′) → (𝒞 , J) depends only on the underlying functor g : 𝒞 → 𝒞′, and not on the choice of a universe V such that C and C′ are V-small.

(iii) Let C := (𝒞 , J) and C′ := (𝒞′, J′) be two sites, u : C′ → C a morphism of sites, and S a set. The constant presheaf on 𝒞 with value S is the constant functor c_{S} : 𝒞^{o} → Set associated with S (see (1.2.9)), and the constant sheaf on C with value S is the associated sheaf c_{S}^{a}. Clearly we have c_{S} = ⨿_{s∈S}c_{{s}}, and each presheaf c_{{s}} is a final object of the category 𝒞^{∧}. It follows that c_{S}^{a} = ⨿_{s∈S}c_{{s}}^{a}, and c_{{s}}^{a} is a final object of C^{∼} (example 1.2.16(iv)). Since ũ^{∗} commutes with all colimits and is left exact by assumption, we deduce that ũ^{∗}(c_{S}^{a}) is naturally isomorphic to the constant sheaf with value S on C′. This assertion may fail for general continuous functors.

For our characterization of morphisms of sites, we shall need the following :

Definition 4.3.3. Let (ℬ, J) be a site, F : 𝒞 → ℬ a fibration, and fix a cleavage λ for F, so that for every X ∈ Ob(𝒞) and every morphism f : B → F X we have the cartesian morphism λ(X, f) : f^{∗}X → X. We say that F is locally cofiltered (relative to the topology J) if the following holds for every B ∈ Ob(ℬ) and every X, X′ ∈ Ob(𝒞_{B}) :

(a) There exists a covering family (B_{i} → B | i ∈ I) in the topology J with Ob(𝒞_{B_{i}}) ≠ ∅ for every i ∈ I.

(b) There exist a covering family (f_{i} : B_{i} → B | i ∈ I) in the topology J, and for every i ∈ I an object Y_{i} ∈ Ob(𝒞_{B_{i}}) with two morphisms f_{i}^{∗}X ← Y_{i} → f_{i}^{∗}X′ in 𝒞_{B_{i}}.

(c) For every pair of morphisms g, g′ : X′ → X in 𝒞_{B} there exist a covering family (f_{i} : B_{i} → B | i ∈ I) in the topology J, and for every i ∈ I a morphism h_{i} : Y_{i} → f_{i}^{∗}X′ in 𝒞_{B_{i}} such that f_{i}^{∗}(g) ∘ h_{i} = f_{i}^{∗}(g′) ∘ h_{i}.

4.3.4. In the situation of definition 4.3.3, suppose that (ℬ, J) and 𝒞 are small; we consider the functor

∫_{a}^{λ} : 𝒞^{∧} → (ℬ, J)^{∼} G ↦ (∫ ^{λ}G)^{a}

where ∫^{λ} : 𝒞^{∧} → ℬ^{∧} is defined as in example 3.1.14(iv). Recall that the sections of ∫^{λ}G(B) are the equivalence classes [X, t] of pairs (X, t) consisting of an object X of F^{−1}B and a section t ∈ GX, for the equivalence relations on the set of such pairs explicited in example 1.2.23(i).

Lemma 4.3.5. With the notation of (4.3.4), let L be any presheaf on 𝒞, and M := ∫^{λ}L. Let also B ∈ Ob(ℬ) and (X, t), (X′, t′) two pairs consisting of objects X, X′ ∈ F^{−1}B and sections t ∈ LX, t′ ∈ LX′. If [X, t] = [X′, t′] in M(B), there exist a covering family (f_{j} : B_{j} → B | j ∈ I) for the topology J, and for each j ∈ I morphisms g_{j} : X_{i} → X, g′_{j} : X_{i} → X′ in 𝒞 such that F g_{j} = F g′_{j} = f_{j} and (Lg_{j})(t) = (Lg′_{j})(t′).

Proof. As in the proof of proposition 1.5.21(ii) we shall say that a diagram of F^{−1}B is a pair of finite sets (A, A′) with A ⊂ Ob(F^{−1}B), A′ ⊂ Morph(F^{−1}B), such that the source and target of every element of A′ lie in A. We notice :

Claim 4.3.6. Let (A, A′) be any diagram of F^{−1}B. There exist a covering family (f_{j} : B_{j} → B | j ∈ I) and for every j ∈ I an object Y_{j} of F^{−1}B_{j} with a system of morphisms (g_{jX} : Y_{j} → X | X ∈ A) such that the following holds. For every j ∈ I, every X, X′, X′′ ∈ A and every h : X → X′′, h′ : X′ → X′′ in A′ we have h ∘ g_{jX} = h′ ∘ g_{jX′}.

Proof of the claim. Let X_{1}, …, X_{n} be the elements of A. Since F is locally cofiltered, by a simple induction on n we find a covering family f′_{•} := (f′_{j′} : B_{j′} → B | j′ ∈ I′) and morphisms g′_{j′i} : Y′_{j′} → X_{i} such that F g′_{j′i} = f′_{j′} for every j′ ∈ I′ and i = 1, …, n. Next, let T ⊂ A′ × A′ be the subset of all pairs (h, h′) such that h and h′ have the same target; we show that for every T′ ⊂ T, and every j′ ∈ I′ there exist a covering family f′′_{••} := (f′′_{j′λ} : B_{j′λ} → B_{j′} | λ ∈ Λ_{j′}) and morphisms g′′_{j′λ} : Y_{j′λ} → Y′_{j′} with F g′′_{j′λ} = f′′_{j′λ} and

h ∘ g′_{j′i} ∘ g′′_{j′λ} = h′ ∘ g′_{j′i′} ∘ g′′_{j′λ}

for every λ ∈ Λ_{j′}, every i, i′, k = 1, …, n and every (h : X_{i} → X_{k}, h′ : X_{i′} → X_{k}) in T′. The claim will follow for T′ = T, by letting f_{•} be the covering family (f′_{j} ∘ f′′_{j′λ} : B_{j′λ} → B | j′ ∈ I, λ ∈ Λ_{j′}), and letting g_{••} be the system of morphisms g′_{j′i} ∘ g′′_{j′λ}.

Now, if T′ = ∅, there is nothing to show. Thus, let (h_{0}, h_{1}) ∈ T′; by induction on the cardinality of T′, we may assume that the assertion is known for the subset T′′ := T′\{(h_{0}, h_{1})}. After replacing f′_{•} by the covering family (f′_{j} ∘ f′′_{j′λ} | j′ ∈ I, λ ∈ Λ_{j′}) and the system g′_{••} with the system of morphisms g′_{j′i} ∘ g′′_{j′λ}, we may then assume that h ∘ g′_{j′i} = h′ ∘ g′_{j′i′} for every i, i′ = 1, …, n, every j ∈ I′ and every (h, h′) ∈ T′′ such that the sources of h and h′ are i and i′ respectively. We may assume that the common target of h_{0} and h_{1} is X_{1}, and say that their sources are X_{i_{0}} and X_{i_{1}} respectively. Since F is locally cofiltered, there exist a covering family f′′_{••} and a system of morphisms g′′_{••} as in the foregoing, such that h_{0} ∘g′_{j′i_{0}} ∘g′′_{j′λ} = h_{1} ∘g′_{j′i_{1}} ∘g′′_{j′λ} for every j′ ∈ I and every λ ∈ Λ_{j′}. Clearly with these choices for f′′_{••} and g′′_{••} the sought identities hold whenever (h, h′) ∈ T′, as required. ♢

Now, the condition [X, t] = [X′, t′] means that for some k ∈ ℕ there exist objects of F^{−1}B

X_{0}, Y_{0}, X_{1}, Y_{1}, …, Y_{k}, X_{k+1}

and sections t_{i} ∈ LX_{i} for i = 0, …, k + 1, with X_{0} = X, X_{k+1} = X′, t_{0} = t and t_{k+1} = t′, as well as morphisms

X_{i} ⟵^{q_{i}} Y_{i} ⟶^{q′_{i}} X_{i+1} such that (Lq_{i})(t_{i}) = (Lq′_{i})(t_{i+1}) for i = 0, …, k.

We apply claim 4.3.6 to the diagram formed by all the objects X_{i}, Y_{i} and all the morphisms q_{i}, q′_{i}, to find a covering family (f_{j} : B_{j} → B | j ∈ I) and morphisms h_{j} : Z_{j} → Y_{i} with F h_{j} = f_{j}, such that g_{ij} := q_{i} ∘ h_{j} = q′_{i} ∘ h_{j} for every j ∈ I and i = 0, …, k. It follows that (Lg_{0j})(t_{0}) = (Lg_{1j})(t_{1}) = ⋯ = (Lg_{k+1,j})(t_{k+1}) for every j ∈ I. Then the assertion holds with g_{j} := g_{0j} and g′_{j} := g_{k+1,j} for every j ∈ I. □

Proposition 4.3.7. With the notation of (4.3.4), we have :

(i) The functor ∫^{λ} commutes with all colimits. (ii) If F is a locally cofiltered fibration, ∫_{a}^{λ} is exact.

Proof. (i): Since the colimits in ℬ^{∧} are computed argumentwise (corollary 1.4.3(ii)), we are reduced to checking that for every B ∈ Ob(ℬ) the functor

𝒞^{∧} → Set G ↦ colim_{(F^{−1}B)^{o}} G ∘ ι_{B}^{o}

commutes with colimits. However, the functor ι_{B}^{∧} : 𝒞^{∧} → (F^{−1}B)^{∧} induced by the inclusion functor ι_{B} : F^{−1}B → 𝒞 commutes with colimits (again, by corollary 1.4.3(ii)), so we come down to checking that the functor Colim_{(F^{−1}B)^{o}} : (F^{−1}B)^{∧} → Set of remark 1.3.3(ii) commutes with colimits, which is clear, as the latter is a left adjoint.

(ii): In light of (i), and since the functor G ↦ G^{a} commutes with colimits, it suffices to check that ∫_{a}^{λ} is left exact, and by proposition 1.3.22(i) we are reduced to showing that ∫_{a}^{λ} commutes with equalizers and finite non-empty products, and that it preserves final objects.

Thus, let E be the presheaf on 𝒞 such that E(X) = {∅} for every X ∈ Ob(𝒞), and set E′ := ∫^{λ}E. For every B ∈ Ob(ℬ) and every s ∈ E^{′a}(B) there is a covering family B_{•} := (B_{i} → B | i ∈ I) in the topology J, and for every i ∈ I an object X_{i} of F^{−1}B_{i} such that s is represented by the system of section ([X_{i}, ∅]) ∈ E′(B_{i}). Conversely, every such system defines a section of E^{′a}(B). Now, since F is locally cofiltered there exists such a covering family B_{•} of B with the property that Ob(F^{−1}B_{i}) is not empty for every i ∈ I; by picking arbitrary X_{i} ∈ Ob(F^{−1}B_{i}) for each i we obtain therefore a section of E^{′a}(B); this shows that E^{′a}(B) ≠ ∅ for every B ∈ Ob(ℬ). Next, let s := ([X_{i}, ∅] | i ∈ I) and s′ := ([X′_{i}, ∅] | i ∈ I) be two sections of E^{′a}(B), with F X_{i} = F X′_{i} = B_{i} for every i ∈ I. Since F is locally cofiltered, we may find for every i ∈ I a covering family (f_{iλ} : B_{iλ} → B_{i} | λ ∈ Λ_{i}) and morphisms g_{iλ} : Y_{iλ} → X_{i}, g′_{iλ} : Y_{iλ} → X′_{i} in 𝒞 with F g_{iλ} = F g′_{iλ} = f_{iλ} for every λ ∈ Λ_{i}. Then both s and s′ are represented by the system ([Y_{iλ}, ∅] | i ∈ I, λ ∈ Λ_{i}). This proves that E^{′a}(B) contains a unique section for every B ∈ Ob(ℬ), i.e. E^{′a} is the final object of (ℬ, J)^{∼}, as required.

To check that ∫_{a}^{λ} commutes with finite products, let G_{1}, G_{2} be two presheaves on 𝒞, and set

G := G_{1} × G_{2}  H := ∫ ^{λ}G  H_{i} := ∫ ^{λ}G_{i}  for i = 1, 2.

We need to check that the projections G → G_{i} induce an isomorphism ω : H^{a} →^{∼} H_{1}^{a} × H_{2}^{a}. To this aim, let as well B ∈ Ob(ℬ) and s ∈ (H_{1}^{a} × H_{2}^{a})(B). Hence s is represented by the datum of a covering family (f_{j} : B_{j} → B | j ∈ I) and sections s_{ij} ∈ H_{i}(B_{j}) for i = 1, 2 and every j ∈ I, fulfilling the following compatibility condition. For every B′ ∈ Ob(ℬ), every j, j′ ∈ I and every pair of morphisms g : B′ → B_{j}, g′ : B′ → B_{j′} such that f_{j} ∘ g_{j} = f_{j′} ∘ g′, we have H_{i}(g)(s_{ij}) = H_{i}(g′)(s_{ij′}) for i = 1, 2. By example 1.2.23(i), each s_{ij} is the class [X_{ij}, t_{ij}] of a pair consisting of an object X_{ij} of F^{−1}B_{j} and a section t_{ij} ∈ G_{i}X_{ij}. Since F is locally cofiltered, we may then find for every j ∈ I a covering family (f_{jλ} : B_{jλ} → B_{j} | λ ∈ Λ_{j}) and for every j ∈ I and every λ ∈ Λ_{j} an object Y_{jλ} of F^{−1}B_{jλ} and morphisms g_{ijλ} : Y_{jλ} → X_{ij} in 𝒞 with F g_{ijλ} = f_{jλ} for i = 1, 2. With this notation, set t_{ijλ} := (G_{i}g_{ijλ})(t_{ij}); we have

[Y_{jλ}, t_{ijλ}] = H_{i}(f_{jλ})(s_{ij}) for i = 1, 2, every j ∈ I and every λ ∈ Λ_{j}.

Thus, the class τ_{jλ} := [Y_{jλ}, (t_{1jλ}, t_{2jλ})] is a section of H(Y_{jλ}) for every j ∈ I and λ ∈ Λ_{j}. Lastly, the family (f_{j} ∘ f_{jλ} : B_{jλ} → B | j ∈ I, λ ∈ Λ_{j}) covers B, and the system of sections τ_{••} defines a section of H^{a}(B) whose image under ω_{B} agrees with s. This shows that ω is an epimorphism. In order to check that ω is a monomorphism, consider t, t′ ∈ H^{a}(B) whose images agree in (H_{1}^{a}×H_{2}^{a})(B); we may find a covering family f_{•} := (f_{j} : B_{j} → B | j ∈ I) such that t and t′ can be represented by compatible systems of sections t_{•} := ([X_{j}, (t_{1j}, t_{2j})] | j ∈ I), t′_{•} := ([X′_{j}, (t′_{1j}, t′_{2j})] | j ∈ I) with X_{j}, X′_{j} ∈ F^{−1}B_{j} and t_{ij}, t′_{ij} ∈ G_{i}X_{j} for i = 1, 2 and every j ∈ I. By assumption, the resulting systems ([X_{j}, t_{ij}] | j ∈ I) and ([X′_{j}, t′_{ij}] | j ∈ I) represent the same sections of H_{i}^{a}(B) for i = 1, 2. This means that there exists for every j ∈ I a covering family (f_{jλ} : B_{jλ} → B_{j} | λ ∈ Λ_{j}) such that if we set t_{ijλ} := G_{i}(λ(f_{jλ}, X_{j}))(t_{ij}) and t′_{ijλ} := G_{i}(λ(f_{jλ}, X′_{j}))(t′_{ij}), we have

[f_{jλ}^{∗}X_{j}, t_{ijλ}] = [f_{jλ}^{∗}X′_{j}, t′_{ijλ}] in H_{i}(B_{jλ}) for i = 1, 2, every j ∈ I and every λ ∈ Λ_{j}.

After replacing f_{•} by the covering family (f_{j} ∘ f_{jλ} : B_{jλ} → B | j ∈ I, λ ∈ Λ_{j}) and the systems t_{•}, t′_{•} by ([f_{jλ}^{∗}X_{j}, (t_{1jλ}, t_{2jλ})] | j ∈ I, λ ∈ Λ) and ([f_{jλ}^{∗}X′_{j}, (t′_{1jλ}, t′_{2jλ})] | j ∈ I, λ ∈ Λ), we may therefore assume from start that

[X_{j}, t_{ij}] = [X′_{j}, t′_{ij}] in H_{i}(B_{j}) for i = 1, 2 and every j ∈ I.

By lemma 4.3.5, we may then find for i = 1, 2 and every j ∈ I a covering family (f_{ijλ} : B_{ijλ} → B_{j} | λ ∈ Λ_{ij}) and morphisms g_{ijλ} : Y_{ijλ} → X_{j}, g′_{ijλ} : Y_{ijλ} → X′_{j} in 𝒞 such that F g_{ijλ} = F g′_{ijλ} = f_{ijλ} and t_{ijλ} := (G_{i}g_{ijλ})(t_{ij}) = (G_{i}g′_{ijλ})(t′_{ij}) for every λ ∈ Λ_{ij}.

For i = 1, 2 and every j ∈ I, let 𝒮_{ij} ⊂ ℬ/B_{j} be the sieve generated by the family f_{ij•}, and pick a generating family (f′_{σ} : B_{σ} → B_{j} | σ ∈ Σ_{j}) for the sieve 𝒮_{1j} ∩ 𝒮_{2j}, which covers B_{j} as well in the topology J. Set Σ := ⋃_{j∈I}{j} × Σ_{j}. By definition, this means that for i = 1, 2 and every (j, σ) ∈ Σ there exist λ ∈ Λ_{1j}, λ′ ∈ Λ_{2j} and a commutative diagram in ℬ :

\xymatrix{B_{σ} \ar[r]^{h_{σ}} \ar[d]_{h′_{σ}} \ar[dr]^{f′_{σ}} & B_{1jλ} \ar[d]^{f_{1jλ}} \\ B_{2jλ′} \ar[r]_{f_{2jλ′}} & B_{j}.}

With this notation, we set Z_{1jσ} := h_{σ}^{∗}Y_{1jλ}, Z_{2jσ} := h_{σ}^{′∗}Y_{2jλ′} and :

s_{1jσ} := G_{1}(λ(h_{σ}, Y_{1jλ}))(t_{1jλ}) s_{2jσ} := G_{2}(λ(h′_{σ}, Y_{2jλ}))(t_{2jλ}) for every (j, σ) ∈ Σ.

Then, both compatible systems ([X_{j}, t_{1j}] | j ∈ I) and ([X′_{j}, t′_{1j}] | j ∈ I) agree with the compatible system ([Z_{1σ}, s_{1σ}] | σ ∈ Σ) in H_{1}^{a}(B), and both compatible systems ([X_{j}, t_{2j}] | j ∈ I) and ([X′_{j}, t′_{2j}] | j ∈ I) agree with the compatible system ([Z_{2σ}, s_{2σ}] | σ ∈ Σ) in H_{2}^{a}(B).

Since F is locally cofiltered, we may find for every σ ∈ Σ a covering family (f_{σλ} : B_{σλ} → B_{σ} | λ ∈ Λ_{σ}) and morphisms g_{iσλ} : Z_{σλ} → Z_{iσ}, with F g_{iσλ} = f_{σλ} for i = 1, 2. Set s_{iσλ} := (G_{i}g_{iσλ})(s_{iσ}) for i = 1, 2, every σ ∈ Σ and every λ ∈ Λ_{σ}. Hence both t and t′ are represented by the compatible system of sections ([Z_{σλ}, (s_{1σλ}, s_{2σλ})] | σ ∈ Σ, λ ∈ Λ_{σ}), so t = t′.

Lastly, let φ_{1}, φ_{2} : G_{1} → G_{2} be two morphisms of presheaves on 𝒞, and E ⊂ G_{1} the equalizer of φ_{1} and φ_{2}; set also H_{i} := ∫^{λ}G_{i}, ψ_{i} := ∫^{λ}φ_{i} for i = 1, 2, and denote by E′ the equalizer of ψ_{1}^{a}, ψ_{2}^{a} : H_{1}^{a} → H_{a}^{2}. We need to check that the induced morphism E^{a} → E′ is an isomorphism. Thus, let B ∈ Ob(ℬ) and s ∈ E′(B); by remark 4.1.19(i), the set E′(B) is the equalizer of the induced maps ψ_{1}^{a}(B), ψ_{2}^{a}(B) : H_{1}^{a}(B) → H_{2}^{a}(B), so we may find a covering family (f_{j} : B_{j} → B | j ∈ I) such that s is represented by a compatible system t_{•} := ([X_{j}, t_{j}] | j ∈ I) with X_{j} ∈ Ob(F^{−1}B_{j}) and t_{j} ∈ G_{1}(B_{j}) for every j ∈ I, such that the systems ([X_{j}, (φ_{1X_{j}})(t_{j})] | j ∈ I) and ([X_{j}, (φ_{2X_{j}})(t_{j})] | j ∈ I) represent the same section of H_{2}^{a}(B). This means that for every j ∈ I there exists a covering family f_{•} := (f_{jλ} : B_{jλ} → B_{j} | λ ∈ Λ_{j}) such that if we set t_{jλ} := G_{1}(λ(f_{jλ}, X_{j}))(t_{j}) we get the following identity in H_{2}(B_{jλ}) :

[f_{jλ}^{∗}X_{j}, φ_{1X_{j}}(t_{jλ})] = [f_{jλ}^{∗}X_{j}, φ_{2X_{j}}(t_{jλ})] for i = 1, 2, every j ∈ I and every λ ∈ Λ_{j}.

After replacing f_{•} by the covering family (f_{jλ} ∘ f_{j} : B_{jλ} → B | j ∈ I, λ ∈ Λ_{j}) and t_{•} by the compatible system ([f_{jλ}^{∗}X_{j}, t_{jλ}] | j ∈ I, λ ∈ Λ_{j}) we may then assume that

[X_{j}, (φ_{1X_{j}})(t_{j})] = [X_{j}, (φ_{2X_{j}})(t_{j})] in H_{2}(B_{j}) for every j ∈ I.

By lemma 4.3.5we may then find for every j ∈ I a covering family (f_{jλ} : B_{jλ} → B_{j} | λ ∈ Λ_{j}) and morphisms g_{jλ}, g′_{jλ} : Y_{jλ} → X_{j} in 𝒞 such that F g_{jλ} = F g′_{jλ} = f_{jλ} and

G_{2}(g_{jλ})((φ_{1X_{j}})(t_{j})) = G_{2}(g′_{jλ})((φ_{2X_{j}})(t_{j})) for every λ ∈ Λ_{j}.

Set Σ := ⋃_{j∈I}{j} × Λ_{j}; since F is locally cofiltered, for every σ ∈ Σ we may then find a covering family (f′_{σλ′} : B′_{σλ′} → B_{σ} | λ′ ∈ Λ′_{σ}) and a morphism h_{σλ′} : Y_{σλ′} → Y_{σ} in 𝒞 such that F h_{σλ′} = f′_{σλ′} and g_{σλ′} := g_{σ} ∘ h_{σλ′} = g′_{σ} ∘ h_{σλ′} for every λ′ ∈ Λ′_{σ}. Set t_{jλλ′} := G_{1}(g_{jλλ′})(t_{j}) for every (j, λ) ∈ Σ and every λ′ ∈ Λ′_{(j,λ)}. The compatible system ([Y_{σλ′}, t_{σλ′}] | σ ∈ Σ, λ′ ∈ Λ′_{σ}) represents again the section s, and by construction we have φ_{1Y_{σλ′}}(t_{σλ′}) = φ_{2Y_{σλ′}}(t_{σλ′}) for every σ ∈ Σ and every λ′ ∈ Λ′_{σ}. This proves that s lies in the image of the map E^{a}(B) → E′(B), so the natural morphism E^{a} → E′ is an epimorphism. To see that it is also a monomorphism, consider two sections s, s′ of E^{a}(B) whose images agree in E′(B); we may find a covering family (f_{j} : B_{j} → B | i ∈ I) and compatible systems ([X_{j}, t_{j}] | j ∈ I), ([X′_{j}, t′_{j}] | j ∈ I) with F X_{j} = F X′_{j} = B_{j} and t_{j} ∈ G_{1}X_{j}, t′_{j} ∈ G_{1}X′_{j} for every j ∈ I. Arguing as in the foregoing, we may then assume that [X_{j}, t_{j}] = [X′_{j}, t′_{j}] in H_{1}(B_{j}) for every j ∈ I. Then by lemma 4.3.5there exist for every j ∈ I a covering family (f_{jλ} : b_{jλ} → B_{j} | j ∈ Λ_{j}) and morphims g_{jλ} : Y_{jλ} → X_{j}, g′_{jλ} : Y_{jλ} → X′_{j} in 𝒞 such that F g_{jλ} = F g′_{jλ} = f_{jλ} and t_{jλ} := (G_{1}g_{jλ})(t_{j}) = (G_{1}g′_{jλ})(t′_{j}) for every λ ∈ Λ_{j}. Clearly φ_{1Y_{jλ}}(t_{jλ}) = φ_{2Y_{jλ}}(t_{jλ}) for every j ∈ I and every λ ∈ Λ_{j}; thus, both s and s′ are represented by the compatible system ([Y_{jλ}, t_{jλ}] | j ∈ I, λ ∈ Λ_{j}), and the proof is concluded. □

4.3.8. Let C := (𝒞 , J) and C′ := (𝒞′, J′) be two sites, g : 𝒞 → 𝒞′ a given functor, and V a universe such that C and C′ are V-small. We shall apply proposition 4.3.7 to the split fibration s : 𝒞′/g𝒞 → 𝒞′ associated with g, with its canonical cleavage λ (see example 3.2.3). Let also t : 𝒞′/g𝒞 → 𝒞 be the target functor (see (1.1.28)); with this notation, notice that

g_{V!} = ∫ ^{λ}∘ t^{∧}.

In light of proposition 4.3.7, it follows already that if s is locally cofiltered, then the functors g_{V!}^{a} and g̃_{V}^{∗} : C_{V}^{∼} → C_{V}^{′∼} are exact. In fact, we have :

Proposition 4.3.9. With the notation of (4.3.8), the following conditions are equivalent : (a) g is a morphism of sites C′ → C.

(b) The fibration s is locally cofiltered, and for every covering family (f_{i} : X_{i} → X | i ∈ I) for the topology J, the family (g(f_{i}) : gX_{i} → gX | i ∈ I) covers gX relative to J′.

Proof. (a)⇒(b) : Since g is continuous, we know already that it transforms covering families for J into covering families for J′ (lemma 4.2.4). Notice as well that g_{V!}^{a} = g̃_{V}^{∗} ∘ (−)^{a} (lemma 4.2.11(ii)); since g̃_{V}^{∗} is exact, the same holds then for g_{V!}^{a}.

Let us check next that condition (a) of definition 4.3.3 holds for s. To this aim, consider the final object E of C^{∼} such that E(X) = {∅} for every X ∈ Ob(𝒞). By assumption, g̃_{V}^{∗}(E) is the final object of C_{V}^{′∼}, i.e. (g̃_{V}^{∗}E)(X′) is a set with one element for every X′ ∈ Ob(𝒞′). However, every section of (g̃_{V}^{∗}E)(X′) is represented by the datum of a covering family (X′_{j} → X′ | j ∈ I) in the topology J′, and a system of sections ([X′_{j} → gX_{j}, ∅] | j ∈ I); especially, Ob(s^{−1}X′_{j}) ≠ ∅ for every j ∈ I, as required.

Next, let l_{i} : X′ → gX_{i} for i = 1, 2 be two objects of s^{−1}X′; in order to check condition (b) of definition 4.3.3 we need to exhibit a covering family (f_{j} : X′_{j} → X′ | j ∈ I) and for every j ∈ I an object h_{j} : X′_{j} → gY_{j} of s^{−1}X′_{j} and morphisms k_{ij} : Y_{j} → X_{i} in 𝒞 that make commute the diagram

(4.3.10)  \xymatrix{X′ \ar[dr]_{l_{i}} & X′_{j} \ar[l]_{f_{j}} \ar[r]^{h_{j}} & gY_{j} \ar[dl]^{g(k_{ij})} \\  & gX_{i}}  for i = 1, 2.

To this aim, set H_{i} := g_{V!}(h_{X_{i}}) = h_{gX_{i}} for i = 1, 2, and H := g_{V!}(h_{X_{1}} × h_{X_{2}}). The pair (l_{1}, l_{2}) yields a section l_{•} of (H_{1} × H_{2})(X′); on the other hand, since g_{V!}^{a} is exact, the natural morphism ω : H → H_{1} ×H_{2} induces an isomorphism on associated sheaves in C^{′∼}. Especially, there exists a covering family (f_{j} | j ∈ I) as sought, such that for every j ∈ I the section (H_{1} × H_{2})(f_{j})(l_{•}) = (l_{1} ∘ f_{j}, l_{2} ∘ f_{j}) agrees with the image under ω of a section [h_{j} : X′_{j} → gY_{j}, (k_{1j}, k_{2j})] of H(X′_{j}). Unwinding the definition, we obtain precisely a diagram (4.3.10).

Lastly, let l_{i} : X′ → gX_{i} for i = 1, 2 be two objects of s^{−1}X′, and t_{1}, t_{2} : X_{1} → X_{2} two morphisms in 𝒞 such that g(t_{1})∘l_{1} = l_{2} = g(t_{2})∘l_{1}; in order to verify condition (c) of definition 4.3.3we need to exhibit a covering family (f_{j} : X′_{j} → X′ | j ∈ I) and for every j ∈ I an object h_{j} : X′_{j} → gY_{j} of s^{−1}X′_{j} with a morphism k_{j} : Y_{j} → X_{1} such that

(4.3.11)  l_{1} ∘ f_{j} = g(k_{j}) ∘ h_{j}  and t_{1} ∘ k_{j} = t_{2} ∘ k_{j}.

To this aim, let t_{i∗} : h_{X_{1}} → h_{X_{2}} be the morphism of presheaves on 𝒞 induced by t_{i}, for i = 1, 2; we set H_{i} := g_{V!}(h_{X_{i}}) = h_{gX_{i}} for i = 1, 2, and denote by E the equalizer of t_{1∗} and t_{2∗}, and by E′ the equalizer of g_{V!}(t_{1∗}), g_{V!}(t_{2∗}) : H_{1} → H_{2}. Since g_{V!}^{a} is exact, the natural morphism ω : g_{V!}E → E′ induces an isomorphism on associated sheaves in C^{∼}. On the other hand, l_{1} defines a section of E′(X′); it follows that there exists a covering family (f_{j} : X′_{j} → X′ | j ∈ I) such that for every j ∈ I the section (E′f_{j})(l_{1}) agrees with the image under ω of a section [h_{j} : X′_{j} → gY_{j}, k_{j}] of (g_{V!}E′)(X′_{j}). Unwinding the definitions, we obtain the identities (4.3.11).

(b)⇒(a): We have already noticed that if s is locally cofiltered, the functor g̃_{V}^{∗} is exact. In order to check that g is continuous, let us consider any sheaf F on 𝒞′, any X ∈ Ob(𝒞), and any covering subobject R ⊂ h_{X} for the topology J. We need to show that the natural map

F (gX) ≃ Hom_{𝒞_{V}^{∧}}(h_{X}, g_{V}^{∗}F) → Hom_{𝒞_{V}^{∧}}(R, g_{V}^{∗}F)

is bijective. By adjunction, the latter is naturally identified with the induced map

Hom_{𝒞_{V}^{′∧}}(g_{V!}(h_{X}), F) → Hom_{𝒞_{V}^{′∧}}(g_{V!}(R), F)

so we come down to checking that the morphism φ : (g_{V!}R)^{a} → (g_{V!}h_{X})^{a} = h_{gX}^{a} induced by the inclusion R → h_{X} is an isomorphism. However, we know already that the functor g_{V!}^{a} is exact, so φ is a monomorphism. Next, let (f_{j} : X_{j} → X | j ∈ I) be a V-small family of generators for the covering sieve of X corresponding to R; there follows a covering morphism

S := ∐_{j∈I} h_{X_{j}} → R → h_{X} in 𝒞^{∧}

whose image under g_{V!} is still a covering morphism : indeed, g_{V!} commutes with coproducts since it is a left adjoint, so g_{V!}(S) is the coproduct of the family (h_{gX_{j}} | j ∈ I), and by assumption g transforms covering families for the topology J into covering families for the topology J′. Thus, the morphism g_{V!}R → g_{V!}h_{X} induces an epimorphism as well on associated sheaves, whence the contention. □

Example 4.3.12. In the situation of (4.3.8), suppose that C is a lex-site and g is left exact. Then g is a morphism of sites if and only if for every covering family (f_{i} : X_{i} → X | i ∈ I) for the topology J, the family (g(f_{i}) : gX_{i} → gX | i ∈ I) covers gX relative to J′. Indeed, under this assumption, the category X/g𝒞 is cofiltered for every X ∈ Ob(𝒞′) (example 1.3.16(i)), so the source fibration s : 𝒞′/g𝒞 → 𝒞′ is trivially locally cofiltered, and the assertion follows from proposition 4.3.9.

Theorem 4.3.13. Let (𝒞 , J) be a U-site, 𝒞′ a category, g : 𝒞′ → 𝒞 a functor, and endow 𝒞′ with the topology J′ induced by g. Suppose that the following conditions hold :

(a) For every X ∈ Ob(𝒞) there exists a family (gY_{i} → X | i ∈ I) of objects of g𝒞′/X that covers X in the topology J.

(b) For every Y, Y′ ∈ Ob(𝒞′) and every morphism φ : gY → gY′ in 𝒞 there exist :

(b.i) a family (ψ_{i} : Z_{i} → Y | i ∈ I) of objects of 𝒞′/Y such that the family (g(ψ_{i}) :

gZ_{i} → gY | i ∈ I) covers gY in the topology J (b.ii) for each i ∈ I a morphism ν_{i} : Z_{i} → Y′ such that φ ∘ g(ψ_{i}) = g(ν_{i}).

(c) For every pair of morphisms φ, φ′ : Y → Y′ in 𝒞′ such that g(φ) = g(φ′) there exists a family (ψ_{i} : Z_{i} → Y | i ∈ I) of objects of 𝒞′/Y such that the family (g(ψ_{i}) : gZ_{i} → gY | i ∈ I) covers gY in the topology J, and φ ∘ ψ_{i} = φ′ ∘ ψ_{i} for every i ∈ I.

Then we have :

(i) g is cocontinuous for the topologies J, J′ and is a morphism of sites (𝒞 , J) → (𝒞′, J′). (ii) g induces an equivalence g̃_{∗} : (𝒞 , J)^{∼} →^{∼} (𝒞′, J′)^{∼}. (iii) A family (ψ_{i} : Y_{i} → Y | i ∈ I) of morphisms of 𝒞′ generates a covering sieve of J′ if

and only if (g(ψ_{i}) : gY_{i} → gY | i ∈ I) generates a covering sieve of J.

Proof. Suppose first that both 𝒞 and 𝒞′ are small. Let s : 𝒞 /g𝒞′ → 𝒞 be the fibration associated with g, as in (4.3.8); we remark :

Claim 4.3.14. The fibration s is locally cofiltered.

Proof of the claim. To check condition (a) of definition 4.3.3, let X ∈ Ob(𝒞); we need to find a covering family (X_{j} → X | j ∈ I) for the topology J, such that Ob(s^{−1}X_{j}) ≠ ∅ for every j ∈ I. However, by condition (a) of the theorem we have a covering family (gY_{j} → X | j ∈ I) for X; we may then choose X_{j} := gY_{j}, since then 1_{gY_{j}} ∈ Ob(s^{−1}X_{j}) for every j ∈ I.

Next, we check condition (b) of definition 4.3.3 : consider any X ∈ Ob(𝒞) and morphisms φ_{i} : X → gY_{i} in 𝒞′ for i = 1, 2; we need to find a covering family (f_{j} : X_{j} → X | j ∈ I) such that for every j ∈ I there exist a morphism ψ_{j} : X_{j} → gZ_{j} in 𝒞 and morphisms ν_{ij} : Z_{j} → Y_{i} with g(ν_{ij})∘ψ_{j} = φ_{i}∘f_{j} for i = 1, 2. To this aim, we use first condition (a) of the theorem to find a covering family f′_{•} := (f′_{j} : gY′_{j} → X | j ∈ I′); then, by condition (b) of the theorem we find for every j ∈ I′ a family (f′_{jλ} : Y′_{jλ} → Y′_{j} | λ ∈ Λ_{j}) such that (g(f′_{jλ}) | λ ∈ Λ_{j}) covers gY′_{j}, and such that for every λ ∈ Λ_{j} there exists a morphism h_{jλ} : Y′_{jλ} → Y_{1} with g(h_{jλ}) = φ_{1}∘f′_{j}∘g(f′_{jλ}). We may then replace f′_{•} by the family (f′_{j} ∘ g(f′_{jλ}) : gY′_{jλ} → X | j ∈ I′, λ ∈ Λ_{j}), and assume from start that for every j ∈ I′ there exists a morphism h_{j} : Y′_{j} → Y_{1} such that g(h_{j}) = φ_{1} ∘ f′_{j}. Next, we apply again condition (b) of the theorem to find for every j ∈ I′ a family (f′′_{jλ′} : Y′′_{jλ′} → Y′_{j} | λ′ ∈ Λ′_{j}) such that (g(f′′_{jλ′}) | λ′ ∈ Λ′_{j}) covers gY′_{j}, and such that for every λ′ ∈ Λ′_{j} there exists a morphism h′_{jλ′} : Y′′_{jλ′} → Y_{2} with g(h′_{jλ′}) = φ_{2} ∘ f′_{j} ∘ g(f′′_{jλ′}). We may then further replace f′_{•} by the family (f′_{j} ∘ g(f′′_{jλ′}) : gY′′_{jλ′} → X | j ∈ I′, λ′ ∈ Λ′_{j}), and the system (h_{j} | j ∈ I′) by the system (h_{j} ∘ f′′_{jλ′} : Y′′_{jλ′} → Y_{1} | j ∈ I′, λ′ ∈ Λ′_{j}), and assume from start that there exists as well for every j ∈ I′ a morphism h′_{j} : Y′_{j} → Y_{2} such that g(h′_{j}) = φ_{2} ∘ f′_{j}. Then we set X_{j} := gY′_{j}, Z_{j} := Y′_{j}, ψ_{j} := 1_{X_{j}}, ν_{1j} := h_{j} and ν_{2j} := h′_{j} for every j ∈ I′; clearly the resulting family f_{•} and systems of morphisms ψ_{•}, ν_{••} will do.

Lastly, we check condition (c) of definition 4.3.3 : let X ∈ Ob(𝒞) and consider morphisms φ : X → gY_{1} in 𝒞 and ψ, ψ′ : Y_{1} → Y_{2} in 𝒞′ such that g(ψ) ∘ φ = g(ψ′) ∘ φ; we need to find a covering family (f_{j} : X_{j} → X | j ∈ I) and for every j ∈ I morphisms h_{j} : X_{j} → gZ_{j} in 𝒞 and ν_{j} : Z_{j} → Y_{1} in 𝒞′ such that φ ∘ f_{j} = g(ν_{j}) ∘ h_{j} and ψ ∘ ν_{j} = ψ′ ∘ ν_{j}. To this aim, we use conditions (a) and (b) of the theorem to get first a covering family f′_{•} := (f′_{j} : gY_{j} → X | j ∈ I), and then for every j ∈ I a family (f′′_{jλ} : Y_{jλ} → Y_{j} | λ ∈ Λ_{j}) such that (g(f′′_{jλ}) | λ ∈ Λ_{j}) covers gY_{j}, and such that for every λ ∈ Λ_{j} there exists a morphism ν′_{jλ} : Y_{jλ} → Y_{1} with g(ν′_{jλ}) = φ ∘ f′_{j} ∘ g(f′′_{jλ}). After replacing f′_{•} by the family (f′_{j} ∘ g(f′′_{jλ}) | j ∈ I, λ ∈ Λ_{j}), we may assume that for every j ∈ I there exists a morphism ν′_{j} : Y_{j} → Y_{1} such that g(ν′_{j}) = φ ∘ f′_{j}. Especially, notice that g(ψ ∘ ν′_{j}) = g(ψ′ ∘ ν′_{j}) for every j ∈ I. By condition (c) of the theorem, there exists therefore for every j ∈ I a family (ν′_{jλ′} : Y′_{jλ′} → Y_{j} | λ′ ∈ Λ′_{j}) such that (g(ν′_{jλ′}) | λ′ ∈ Λ′_{j}) covers gY_{j} and ψ ∘ ν′_{j} ∘ ν′_{jλ′} = ψ′ ∘ ν′_{j} ∘ ν_{jλ} for every λ′ ∈ Λ_{j}. Set I := ⋃_{j∈I}{j} × Λ_{j}; for every (j, λ′) ∈ I we let X_{jλ′} := gY_{jλ′}, f_{jλ′} := f′_{j} ∘ g(ν′_{jλ′}), h_{jλ′} := 1_{X_{jλ′}} and ν_{jλ′} := ν′_{j} ∘ ν′_{jλ′}. Clearly the resulting covering family f_{••} and systems of morphisms h_{••}, ν_{••} will do. ♢

(iii) follows from claim 4.3.14, lemmata 4.2.17(ii) and 4.2.4, and proposition 4.3.9.

(i): The continuity of g holds by definition of J′; then g is a morphism of sites, by claim 4.3.14and proposition 4.3.9. To check the cocontinuity, let Y ∈ Ob(𝒞′) and f_{•} := (f_{j} : X_{j} → gY | j ∈ I) a family of morphisms in 𝒞 generating a covering sieve 𝒮 of gY; we need to show that g_{|Y}^{−1}𝒮 is a covering sieve of Y for the topology J′. However, by (a) we may find for every j ∈ I a family (f_{jλ} : gY_{jλ} → X_{j} | λ ∈ Λ_{j}) covering X_{j} in the topology J; let 𝒮′ ⊂ 𝒮 be the sieve generated by the system (f′_{jλ} := f_{j} ∘ f_{jλ} : gY_{jλ} → gY | j ∈ I, λ ∈ Λ_{j}). Clearly 𝒮′ covers gY in the topology J, and by remark 4.1.3(ii) it suffices to check that g_{|Y}^{−1}𝒮′ is a covering sieve for Y, so we may replace f_{•} by the family f′_{••}, and assume from start that for every j ∈ I we have X_{j} = gY_{j} for some Y_{j} ∈ Ob(𝒞′). By condition (b), we may then find for every j ∈ I a family (h_{jλ} : Y_{jλ} → Y_{j} | λ ∈ Λ_{j}) of morphisms in 𝒞′ such that the family (g(h_{jλ}) | λ ∈ Λ_{j}) covers gY_{j} in the topology J, and for every j ∈ I and λ ∈ Λ_{j} a morphism k_{jλ} : Y_{jλ} → Y in 𝒞′ such that f′′_{jλ} := f_{j} ∘g(h_{jλ}) = g(k_{jλ}). Let 𝒮′′ ⊂ 𝒮 be the sieve generated by the system (f′′_{jλ} : gY_{jλ} → gY | j ∈ I, λ ∈ Λ_{j}). Then 𝒮′′ covers gY in the topology J, and it suffices to check that 𝒯 := g_{|Y}^{−1}𝒮′′ covers Y in the topology J′. However, 𝒯 contains the family (k_{jλ} | j ∈ I, λ ∈ Λ_{j}), which covers Y, by virtue of (iii). The assertion follows.

Claim 4.3.15. (i) Let φ : G → G′ be a morphism of presheaves on 𝒞 such that g^{∧}(φ) : g^{∧}G → g^{∧}G′ is an isomorphism in 𝒞^{′∧}. Then φ is a bicovering morphism.

(ii) Let (η′, ε′) be a unit and counit for the adjoint pair (g̃^{∗}, g̃_{∗}). In order to prove assertion (ii) of the theorem, it suffices to check that η′ : 1_{(𝒞′,J′)^{∼}} ⇒ g̃_{∗}g̃^{∗} is an isomorphism of functors.

Proof of the claim. (i): Indeed, let X ∈ Ob(𝒞) and s ∈ G′X; by condition (a) there exists a covering family (f_{i} : gY_{i} → X | i ∈ I), and by assumption (G′f_{i})(s) ∈ Im(φ_{gY_{i}}) for every i ∈ I, so φ is a covering morphism, by remark 4.1.29(ii). Likewise, let s, s′ ∈ GX such that t := φ_{X}(s) = φ_{X}(s′); it follows that φ_{gY_{i}}((Gf_{i})(s)) = φ_{gY_{i}}((Gf_{i})(s′)) = (G′f_{i})(t), whence (Gf_{i})(s) = (Gf_{i})(s′) for every i ∈ I. In light of remark 4.1.29(iii), the assertion follows.

(ii): Indeed, if η′ is an isomorphism, the triangular identities of (1.1.13) show that g^{∧}(ε′_{G}) will also be an isomorphism for every sheaf G on (𝒞 , J), hence ε′_{G} will be an isomorphism, by (i), and to conclude it will suffice to invoke proposition 1.1.20(i,iii). ♢ (ii): From (i) and lemma 4.2.15(ii) we get an isomorphism of functors 𝒞^{∧} → (𝒞′, J′)^{∼}

ω : (−)^{a} ∘ g^{∧} →^{∼} g̃_{∗} ∘ (−)^{a}.

Let now η : 1_{𝒞^{′∧}} ⇒ g^{∧}∘g_{!} and ε : g_{!}∘g^{∧} ⇒ 1_{𝒞^{∧}} be the unit and counit of the natural adjunction for the pair of functors (g_{!}, g^{∧}), as in remark 1.3.6(iii). We consider the natural transformations of functors on (𝒞′, J′)^{∼} and respectively on (𝒞 , J)^{∼}

η′_{F} : F ⟶^{(η_{F})^{a}} (g^{∧}g_{!}F)^{a} ⟶^{ω_{g_{!}F}} (g^{∧} ∘ g_{!}^{a})(F) ε′_{G} : (g_{!}^{a} ∘ g^{∧})(G) ⟶^{(ε_{F})^{a}} G.

A little diagram chase shows that the pair (η′, ε′) fulfills the triangular identities of (1.1.13), so these are the unit and counit of an adjunction for the pair (g̃^{∗}, g̃_{∗}). By claim 4.3.15(ii), we are thus reduced to showing that η_{F} is a bicovering morphisms, for every presheaf F on 𝒞′.

We check first that η_{F} is a covering morphism : thus, let X ∈ Ob(𝒞^{′∧}), and s ∈ (g^{∧}g_{!}F)(X); then s is a class [φ : gX → gY, σ], where φ is a morphism in 𝒞, and σ ∈ F Y. By condition (b) there exists a family (f_{j} : X_{j} → X | j ∈ I) of morphisms of 𝒞′ such that (φ(f_{j}) | j ∈ I) is a covering family for the topology J, and such that for every j ∈ I there exists a morphism h_{j} : X_{i} → Y with g(h_{j}) = φ∘g(f_{j}). Set σ_{j} := (F h_{j})(σ) ∈ F X_{i} for every j ∈ I; then η_{F,X_{j}}(σ_{j}) is the class [1_{gX_{j}}, σ_{j}] in (g^{∧}g_{!}F)(X_{j}) for every j ∈ I. On the other hand, (g^{∧}g_{!}F)(f_{j})(s) = [φ ∘ g(f_{j}) : gX_{j} → gY, σ] for every j ∈ I. According to remark 4.1.29(ii), it suffices then to show that [1_{gX_{j}}, σ_{j}] = [φ ∘ g(f_{j}), σ] in (g^{∧}g_{!}F)(X_{j}), for every j ∈ I. But h_{j} yields a morphism gX_{j}/h_{j} : (X_{j}, 1_{gX_{j}}) → (Y, φ ∘ g(f_{j})) in gX_{j}/g𝒞′ whence the assertion.

Next, let t, t′ ∈ F X be two sections such that η_{F,X}(t) = η_{F,X}(t′), i.e. [1_{gX}, t] = [1_{gX}, t′] in (g_{!}F)(gX). According to lemma 4.3.5, there exist a covering family (f_{j} : gX_{j} → gX | j ∈ I) for the topology J, and morphisms φ_{j} : gX_{j} → gY_{j} in 𝒞 and h_{j}h′_{j} : Y_{j} → X in 𝒞′ such that

g(h_{j}) ∘ φ_{j} = g(h′_{j}) ∘ φ_{j} = f_{j}  and (F h_{j})(t) = (F h′_{j})(t′)  for each j ∈ I.

Then, by condition (b), we may find for every j ∈ I a family (f′_{jλ} : X_{jλ} → X_{j} | λ ∈ Λ_{j}) such that (g(f′_{jλ}) | λ ∈ Λ_{j}) is a covering family for the topology J, and such that for every λ ∈ Λ_{j} there exists a morphism ν_{jλ} : X_{jλ} → Y_{j} with g(ν_{jλ}) = φ_{j} ∘ g(f′_{jλ}). After replacing (f_{j} | j ∈ Λ) by the covering family (f_{j} ∘ g(f′_{jλ}) : X_{jλ} → X | j ∈ I, λ ∈ Λ), and (φ_{j} | j ∈ J) by the system of morphisms (φ_{j} ∘ g(f′_{jλ}) | j ∈ I, λ ∈ Λ_{j}), we may then assume that for every j ∈ I there exists a morphism ν_{j} : X_{j} → Y_{j} such that φ_{j} = g(ν_{j}), in which case we have g(h_{j} ∘ ν_{j}) = g(h′_{j} ∘ ν_{j}) = f_{j} for every j ∈ I. Then, by condition (c) there exists for every j ∈ I a family (f′′_{jλ′} : X′_{jλ′} → X_{j} | λ′ ∈ Λ′_{j}) of morphisms in 𝒞′ such that (g(f′′_{jλ′}) | λ′ ∈ Λ′_{j}) is a covering family for the topology J, and such that h_{j} ∘ ν_{j} ∘ f′′_{jλ′} = h′_{j} ∘ ν_{j} ∘ f′′_{j}. After replacing (f_{j} | j ∈ Λ) by the covering family (f_{j} ∘ g(f′′_{jλ′}) : X_{jλ′} → X | j ∈ I, λ′ ∈ Λ′_{j}) and (ν_{j} | j ∈ I) by the system (ν_{j} ∘ f′′_{jλ′} | j ∈ I, λ′ ∈ Λ′_{j}), we may then assume that μ_{j} := h_{j} ∘ ν_{j} = h′_{j} ∘ ν_{j} for every j ∈ I. It follows that (F μ_{j})(t) = (F μ_{j})(t′) for every j ∈ I; lastly, the family (μ_{j} : X_{j} → X | j ∈ I) covers X in the topology J′, by virtue of (iii). In view of remark 4.1.29(iii), we conclude that η_{F} is a bicovering morphism, as stated.

This completes the proof of the theorem in case 𝒞 and 𝒞′ are small. Lastly, consider the case where (𝒞 , J) is a U-site and 𝒞′ is a general category, and pick a universe V such that 𝒞 and 𝒞′ are V-small. The theorem then applies to the functor g̃_{V∗}, so the latter is an equivalence, and we also get assertions (i) and (iii), which are independent of the universe U. It remains therefore only to check that g̃_{U∗} is an equivalence, and taking into account the commutative diagram

\xymatrix{(𝒞 , J)_{U}^{∼} \ar[r]^{g̃_{U∗}} \ar[d] & (𝒞′, J′)_{U}^{∼} \ar[d] \\ (𝒞 , J)_{V}^{∼} \ar[r]^{g̃_{V∗}} & (𝒞′, J′)_{V}^{∼}}

(whose vertical arrows are the fully faithful inclusion functors), we already see that g̃_{U∗} is fully faithful. We notice :

Claim 4.3.16. For every covering family (f_{i} : X_{i} → X | i ∈ I) in the topology J there exists an essentially small subset I′ ⊂ I such that (f_{i} : X_{i} → X | i ∈ I′) is still a covering family.

Proof of the claim. Pick an essentially small topologically generating family G ⊂ Ob(𝒞) for the site (𝒞 , J); then for every i ∈ I there exists a covering family (f_{iλ} : X_{iλ} → X_{i} | λ ∈ Λ_{i}) with X_{iλ} ∈ G for every λ ∈ Λ_{i}. Set Λ := ⋃_{i∈I}{i} × Λ_{i}; the family (f′_{iλ} := f_{i} ∘ f_{iλ} : X_{iλ} → X | (i, λ) ∈ Λ) covers X in the topology J. Since Hom_{𝒞} (X_{iλ}, X) is a small set for every (i, λ) ∈ Λ, we may then find an essentially small subset Λ′ ⊂ Λ such that (f′_{iλ} | (i, λ) ∈ Λ′) is still covering; then we can take for I′ ⊂ I the image of Λ′ under the projection Λ → I. ♢

Now, let F be any U-sheaf on (𝒞′, J′); we know already that there exists a V-sheaf G on (𝒞 , J) such that g̃_{∗V}G is isomorphic to F, and we need to check that GX is essentially small for every X ∈ Ob(𝒞). But by condition (a) we may find a covering family (gY_{i} → X | i ∈ I), and by claim 4.3.16we may assume that I is small. Then the induced map GX → ∏_{i∈I} G(gY_{i}) is injective, and G(gY_{i}) →^{∼} F Y_{i} is essentially small for every i ∈ I, whence the contention. □

Remark 4.3.17. (i) Notice that if g is full (resp. faithful), then condition (b) (resp. (c)) of theorem 4.3.13is trivially satisfied.

(ii) It follows easily that theorem 4.3.13 generalizes [17, §2.1, Prop.], which in turn generalizes the implication (i)⇒(ii) in [8, Exp.III, Th.4.1]. One can also show that conditions (a),

(b) and (c) of the theorem are implied by conditions (ℒ 0), (ℒ 1) and (ℒ 2) of [9, Exp.V, Déf.8.1.1], hence theorem 4.3.13generalizes as well [9, Exp.V, Prop.8.1.12].

(iii) A necessary and sufficient condition for a morphism of sites to induce an equivalence of topoi is given in [46, Th.4.7].

(iv) In the situation of theorem 4.3.13, one can show that (𝒞′, J′) does not necessarily have an essentially small topological generating family, and does not necessarily have essentially small Hom-sets, but every essentially small subcategory ℬ ⊂ 𝒞′ is contained in an essentially small subcategory ℬ′ ⊂ 𝒞′ such that the restriction ℬ′ → 𝒞 of g still satisfies conditions (a)–(c) of theorem 4.3.13.

We may now generalize lemma 4.2.11to any continuous functor between U-sites. Indeed, we notice the following further application of theorem 4.3.13, which appears in [8, Exp.III, Th.4.1].

Proposition 4.3.18. Let C := (𝒞 , J) be a U-site, G a topologically generating family for C. Denote by 𝒢 the full subcategory of 𝒞 with Ob(𝒢) = G, and endow 𝒢 with the topology J_{𝒢} induced by J via the inclusion functor u : 𝒢 → 𝒞. Then :

(i) u is cocontinuous for the topologies J, J_{𝒢} and is a morphism of sites C → (𝒢 , J_{𝒥} ). (ii) The induced functor ũ_{∗} : C^{∼} → (𝒢 , J_{𝒢})^{∼} is an equivalence. (iii) A family (ψ_{i} : Y_{i} → Y | i ∈ I) of morphisms of 𝒢 generates a covering sieve of J_{𝒢} if

and only if it generates a covering sieve of J. (iv) (𝒢 , J_{𝒢}) is a U-site.

Proof. (i),(ii),(iii): We apply the criterion of theorem 4.3.13, and taking into account remark 4.3.17(i) we are reduced to checking condition (a) of the theorem. The latter holds by definition of topologically generating family.

(iv): Pick an essentially small topologically generating family H ⊂ Ob(𝒞) for the site C. With claim 4.3.16, we get an essentially small subset G′ ⊂ G such that every X ∈ H has a covering family (X_{i} → X | i ∈ I) in the topology J, with X_{i} ∈ G′ for every i ∈ I. By (iii) and remark 4.1.3(iii), G′ is then a topologically generating family for both C and (𝒢 , J_{𝒢}). □

Corollary 4.3.19. Let C := (𝒞 , J) and C′ := (𝒞′, J′) be two U-sites, g : 𝒞 → 𝒞′ a functor, and V, V′ a pair of universes with U ⊂ V ⊂ V′. We have :

(i) If g is continuous, the following holds : (a) The functor g̃_{V∗} : C_{V}^{′∼} → C_{V}^{∼} admits a left adjoint g̃_{V}^{∗} : C_{V}^{∼} → C_{V}^{′∼}.

(b) There are essentially commutative diagrams of categories :

\xymatrix{C_{V}^{∼} \ar[r]^{^{g̃^{∗}^{V}}} \ar[d]_{i} & C_{V}^{′∼} \ar[d]^{i′} & 𝒞 \ar[r]^{g} \ar[d]_{h_{𝒞}^{a}} & 𝒞′ \ar[d]^{h_{𝒞′}^{a}} \\ C_{V′}^{∼} \ar[r]^{_{g̃{∗ \atop V′}}} & C_{V′}^{′∼} & C_{U}^{∼} \ar[r]^{_{g̃_{∗}_{U}}} & C_{U}^{′∼}}

where i and i′ are the inclusion functors.

(c) Suppose moreover that 𝒞 is finitely complete and g is left exact. Then g̃_{V}^{∗} is exact. (ii) If g is cocontinuous, the following holds :

(a) The functor ğ_{V}^{∗} : C_{V}^{′∼} → C_{V}^{∼} admits a right adjoint ğ_{V∗} : C_{V}^{∼} → C_{V}^{′∼}.

(b) There is an essentially commutative diagram of categories :

\xymatrix{C_{V}^{∼} \ar[r]^{ğ_{V∗}} \ar[d] & C_{V}^{′∼} \ar[d] \\ C_{V′}^{∼} \ar[r]^{ğ_{V′∗}} & C_{V′}^{′∼}}

whose vertical arrows are the inclusion functors. (iii) If g is continuous and cocontinuous, we have natural isomorphisms of functors :

g̃_{V∗} →^{∼} ğ_{V}^{∗} : C_{V}^{′∼} → C_{V}^{∼} (−)^{a} ∘ g_{V}^{∧} →^{∼} g̃_{V∗} ∘ (−)^{a} : 𝒞_{V}^{′∧} → C_{V}^{∼}.

Proof. (i.a): We choose an essentially small topologically generating family G for C, and define the site (𝒢 , J_{𝒢}) and the continuous functor u : (𝒢 , J_{𝒢}) → C as in proposition 4.3.18. By applying lemma 4.2.11(i) to the continuous functor v := g ∘ u, we deduce that ṽ_{V∗} = ũ_{V∗} ∘ g̃_{V∗} admits a left adjoint. Then the assertion follows from proposition 4.3.18(ii).

(i.b): More precisely, g̃_{V}^{∗} = ṽ_{V}^{∗} ∘ ũ_{V∗}. Thus, the essential commutativity of the left diagram follows from (4.2.13). Likewise, the essential commutativity of the right diagram follows from that of the left diagram (with V := U and V′ large enough so that 𝒞 is V′-small) together with lemma 4.2.11(ii).

(i.c) holds by example 4.3.12.

(ii.a): Let u and h be as in the foregoing; from proposition 4.3.18(i) we deduce that h is cocontinuous, hence h̆_{V}^{∗} = ŭ_{V}^{∗} ∘ ğ_{V}^{∗} admits a right adjoint; however ŭ_{V}^{∗} = ũ_{V∗} (lemma 4.2.15(i)), and the latter is an equivalence (proposition 4.3.18(ii)), whence the contention.

(ii.b): More precisely, ğ_{V∗} = h̆_{V∗} ∘ ũ_{V}^{∗}, hence the assertion follows from (4.2.13).

(iii): In view of (i.b) and (ii.b), in order to prove the assertion, we may assume that 𝒞 and 𝒞′ are V-small, and this case is already covered by lemma 4.2.15(i,ii). □

The last result of this section will show the representability of the 2-limit of any (small) cofiltered system of lex-sites. The proof shall use the following :

Lemma 4.3.20. Let I be a small filtered category, U′ a universe, and

𝒞_{•} : I → U′-Cat  i ↦ 𝒞_{i}  (φ : i → j) ↦ (𝒞_{φ} : 𝒞_{i} → 𝒞_{j})

any pseudo-functor such that 𝒞_{i} is finitely complete for every i ∈ Ob(I) and 𝒞_{φ} is a left exact functor for every morphism φ of I. Let also (𝒞 , π_{•}) be a 2-colimit of 𝒞_{•}. Then 𝒞 is finitely complete and π_{i} : 𝒞_{i} → 𝒞 is a left exact functor for every i ∈ Ob(I).

Proof. Notice that the assertions depend only on the equivalence class of the category 𝒞; hence, we may suppose that (𝒞 , π_{•}) is the strong 2-colimit of 𝒞_{•} described explicitly by example 3.3.13(i), i.e. 𝒞 = ℱ ib(𝒞_{•})[Σ^{−1}], where Σ is the set of cartesian morphisms of ℱ ib(𝒞_{•}); then π_{i} is induced by the inclusion functor of the fibre category F^{−1}(i) = 𝒞_{i} into ℱ ib(𝒞_{•}), where F : ℱ ib(𝒞_{•}) → I^{o} is the natural projection. By proposition 1.2.22(i)), in order to check that 𝒞 is finitely complete it suffices to show the following two claims :

Claim 4.3.21. All finite products are representable in 𝒞_{•}.

Proof of the claim. By a simple induction we are easily reduced to showing that the product of two objects (i_{1}^{o}, X_{1}), (i_{2}^{o}, X_{2}) is representable (here we have i_{1}^{o}, i_{2}^{o} ∈ Ob(I^{o}) and X_{t} ∈ Ob(𝒞_{i_{t}}) for t = 1, 2 : see (3.1.18)). Since I^{o} is cofiltered, we may find i ∈ Ob(I) and morphisms φ_{t}^{o} : i^{o} → i_{t}^{o} for t = 1, 2; then (i_{t}^{o}, X_{t}) is isomorphic to (i, 𝒞_{φ_{t}}X_{t}) in 𝒞 for t = 1, 2, so we may assume that i = i_{1} = i_{2}. Since 𝒞_{i} is finitely complete, the product X_{1} × X_{2} is representable in 𝒞_{i}, say by an object P, and let (p_{t} : P → X_{t} | t = 1, 2) be a universal cone. Let us check that

(q_{t} := (1_{i^{o}}, p_{t}) : (i^{o}, P) → (i^{o}, X_{t}) | t = 1, 2)

is a universal cone in 𝒞. Indeed, let (j^{o}, Y) be any object of 𝒞 and (r_{t} : (j^{o}, Y) → (i^{o}, X_{t}) | t = 1, 2) a pair of morphisms of 𝒞; we need to show that there exists a unique morphism

(4.3.22)  r : (j^{o}, Y) → (i^{o}, P)  in 𝒞 such that  q_{t} ∘ r = r_{t}  for t = 1, 2.

By example 3.3.13(ii) we can write r_{t} = g_{t} ∘ s_{t}^{−1}, where s_{t} : (j_{t}^{o}, Y_{t}) → (j^{o}, Y) and g_{t} : (j_{t}^{o}, Y_{t}) → (i^{o}, X_{t}) are morphisms of ℱ ib(𝒞_{•}), and s_{t} is cartesian, for t = 1, 2. By (CF3) of definition 1.6.14(i) we may then find an object (j^{′o}, Y′) of 𝒞 and morphisms s′_{t} : (j^{′o}, Y′) → (j_{t}^{o}, Y_{t}) of ℱ ib(𝒞_{•}) such that s′′ := s_{1} ∘ s′_{1} = s_{2} ∘ s′_{2} and such that s′_{1} is cartesian; therefore s′′ is an isomorphism in 𝒞, so it suffices to check that there exists a unique morphism r′ : (j^{′o}, Y′) → (i^{o}, P) in 𝒞 with q_{t} ∘ r′ = r_{t} ∘ s′′ for t = 1, 2. Thus, we may assume that r_{t} is the class of a morphism (ρ_{t}^{o}, f_{t}) : (j^{o}, Y) → (i^{o}, X_{t}) for t = 1, 2 (notation of (3.1.18), so here ρ_{t} : i → j is a morphism of I and f_{t} : Y → 𝒞_{ρ_{t}}X_{t} is a morphism of 𝒞_{j} for t = 1, 2). Since I is filtered, we may then find a morphism ρ′ : j → j′ in I such that ρ′ ∘ ρ_{1} = ρ′ ∘ ρ_{2}, and since (ρ^{′o}, 1_{𝒞_{ρ}}Y) : (j^{′o}, 𝒞_{ρ}Y) → (j^{o}, Y) is an isomorphism in 𝒞, we may further replace (ρ_{t}^{o}, f_{t}) by its composition with (ρ^{′o}, 1_{𝒞_{ρ}}Y) for t = 1, 2, and assume as well that ρ := ρ_{1} = ρ_{2}. In this situation set P′ = 𝒞_{ρ}P and X′_{t} := 𝒞_{ρ}X_{t}, p′_{t} := 𝒞_{ρ}(p_{t}) for t = 1, 2; we notice that r_{t} factors through a morphism r′_{t} := (1_{j^{o}}, f′_{t}) : (j^{o}, Y) → (j^{o}, X′_{t}) and the cartesian morphism (ρ, 1_{X′_{t}}) : (j^{o}, X′_{t}) → (i^{o}, X_{t}), and moreover

(ρ, 1_{X′_{t}}) ∘ (1_{j^{o}}, p′_{t}) = (1_{i^{o}}, p_{t}) ∘ (ρ, 1_{P′}) in ℱ ib(𝒞_{•}) for t = 1, 2 .

Since 𝒞_{ρ} is left exact, the cone (p′_{t} : P′ → X′_{t} | t = 1, 2) is universal in 𝒞_{j}, so there exists a unique morphism g′ : Y → P′ in 𝒞_{j} such that p′_{t} ∘ g′ = f′_{t} for t = 1, 2. It follows that r := (ρ, g′) : (j^{o}, Y) → (i^{o}, P) fulfills the condition of (4.3.22). It remains to check the uniqueness of r. Thus, let r′ : (j^{o}, Y) → (i^{o}, P) be another morphism of 𝒞 that verifies the identities of (4.3.22). We have r′ = g′ ∘ s^{′−1} for some morphisms g′ and s′ of ℱ ib(𝒞_{•}) with s′ cartesian, and we are reduced to checking that g′ = r ∘ s′. We may then assume that r′ is the class of a morphism (ρ^{′o}, g′′) of ℱ ib(𝒞_{•}). Moreover, the identities (4.3.22) for r′ mean that there exist cartesian morphisms v_{t} : (k_{t}^{o}, Y′_{t}) → (j^{o}, Y) such that q_{t} ∘ r′ ∘ v_{t} = r_{t} ∘ v_{t} in ℱ ib(𝒞_{•}) for t = 1, 2. By (CF3) we may find an object (k^{o}, Y′′) of ℱ ib(𝒞_{•}) and morphisms v′_{t} : (k^{o}, Y′′) → (k_{t}^{o}, Y′_{t}) in ℱ ib(𝒞_{•}) with v′_{1} cartesian, and such that v′′ := v_{1} ∘ v′_{1} = v_{2} ∘ v′_{2}. Since v′′ is an isomorphism in 𝒞, it suffices to check that r ∘ v′′ = r′ ∘ v′′, and we may therefore assume that the identities (4.3.22) hold already in the category ℱ ib(𝒞_{•}), for both r and r′.

We may next find a morphism ν : j′ → j in I such that ν ∘ ρ = ν ∘ ρ′, and it suffices to check that (ν^{o}, 1_{𝒞_{ν}Y} ) ∘ r = (ν^{o}, 1_{𝒞_{ν}Y} ) ∘ r′. We may therefore assume that ρ = ρ′. Then both r and r′ factor uniquely through morphisms (1_{j^{o}}, u), (1_{j^{o}}, u′) : (j^{o}, Y) → (j^{o}, P′) and the cartesian morphism (ρ^{o}, 1_{P′}) : (j^{o}, P′) → (i^{o}, P), and we have

(1_{j^{o}}, p′_{t}) ∘ (1_{j^{o}}, u) = r′_{t} = (1_{j^{o}}, p′_{t}) ∘ (1_{j^{o}}, u′) in ℱ ib(𝒞_{•}) for t = 1, 2 .

Since P′ represents X′_{1} × X′_{2} in 𝒞_{j}, we conclude that u = u′, whence r = r′ as required. ♢

Claim 4.3.23. All equalizers are representable in 𝒞.

Proof of the claim. Let (i_{1}^{o}, X_{1}), (i_{2}^{o}, X_{2}) be objects of 𝒞 and f_{1}, f_{2} : (i_{1}^{o}, X_{1}) → (i_{2}^{o}, X_{2}) morphisms of 𝒞. We need to represent the equalizer of f_{1} and f_{2}, and arguing as in the foregoing we may assume that i := i_{1} = i_{2}, and also that f_{t} is the class of a morphism (φ_{t}, g_{t}) of ℱ ib(𝒞_{•}) for t = 1, 2. Then we may further reduce to the case where φ := φ_{1} = φ_{2}. After replacing (i_{2}^{o}, X_{2}) by (i_{1}^{o}, 𝒞_{φ}X_{2}), we may then assume that f_{t} = (1_{i^{o}}, g_{t}) : (i^{o}, X_{1}) → (i^{o}, X_{2}) for t = 1, 2. By assumption, the equalizer of g_{1} and g_{2} in 𝒞_{i} is representable by some E ∈ Ob(𝒞_{i}), and the universal cone for this equalizer amounts to a morphism u : E → X_{1} in 𝒞_{i} such that g_{1}∘u = g_{2}∘u. We claim that (i^{o}, E) represents the equalizer of f_{1} and f_{2}, and that the morphism (1_{i^{o}}, u) : (i^{o}, E) → (i^{o}, X_{1}) yields a universal cone for this equalizer. Indeed, let (j^{o}, Y) be an object of 𝒞 and h : (j^{o}, Y) → (i^{o}, X_{1}) a morphism in 𝒞 such that f_{1} ∘ h = f_{2} ∘ h; we need to show that h factors uniquely through u and a morphism h′ : (j^{o}, Y) → (i^{o}, E). We reduce easily to the case where h is the class of a morphism (ψ^{o}, k) : (j^{o}, Y) → (i^{o}, X_{1}) of ℱ ib(𝒞_{•}) such that (1_{i^{o}}, g_{1}) ∘ (ψ^{o}, k) = (1_{i^{o}}, g_{2}) ∘ (ψ^{o}, k) in ℱ ib(𝒞_{•}). In this situation, set E′ := 𝒞_{ψ}E, u′ := 𝒞_{ψ}u, X′_{t} := 𝒞_{ψ}X_{t} and g′_{t} := 𝒞_{ψ}g_{t} for t = 1, 2. Then (ψ^{o}, k) factors uniquely through the morphism (1_{j^{o}}, k) : (j^{o}, Y) → (j^{o}, X′_{1}) and the cartesian morphism (ψ^{o}, 1_{X′_{1}}) : (j^{o}, X′_{1}) → (i^{o}, X_{1}), and we have (1_{j^{o}}, g′_{1}) ∘ (1_{j^{o}}, k) = (1_{j^{o}}, g′_{2}) ∘ (1_{j^{o}}, k). Since 𝒞_{ψ} is left exact, E′ represents the equalizer of g′_{1} and g′_{2} in 𝒞_{j}, so there exists a unique morphism k′ : Y → E′ in 𝒞_{j} such that u′ ∘ k′ = k. Then h := (ψ, u′) is the sought morphism. The verification of the uniqueness of h is analogous to that of the corresponding assertion in the proof of claim 4.3.21 : the details shall be left to the reader. ♢

Lastly, the constructions in the proof of claims 4.3.21 and 4.3.23 show that the functors π_{i} commute with finite products and equalizers, so they are left exact, by proposition 1.3.22(i). □

4.3.24. Let now I be a small cofiltered category and C_{•} : I → (U, U′)-lex.Site any pseudofunctor. For every i ∈ Ob(I), say that C_{i} = (𝒞_{i}, J_{i}), and denote by 𝒞_{•} : I^{o} → U′-Cat the composition of C_{•}^{o} with the forgetful functor ^{o}(U, U′)-lex.Site^{o} → U′-Cat. We let (𝒞 , π_{•}) be a strong 2-colimit of the pseudo-functor 𝒞_{•} (theorem 3.3.9), and we endow 𝒞 with the coarsest topology J such that all the functors π_{i} : 𝒞_{i} → 𝒞 are continuous for the topologies J_{i} and J. In light of example 4.3.12and lemma 4.3.20, we see that C := (𝒞 , J) is a lex-site, and we obtain a well defined pseudo-cone π_{•}^{o} : F_{C} ⇒ C_{•}.

Proposition 4.3.25. In the situation of (4.3.24), let F ∈ Ob(𝒞_{U}^{∧}) be any presheaf. We have : (i) The pair (C, π_{•}^{o}) is a strong 2-limit of C_{•}. (ii) F is a sheaf on C if and only if π_{i}^{∧}(F) is a sheaf on C_{i} for every i ∈ Ob(I).

Proof. (i): Let D := (𝒟, J_{𝒟} ) be any finitely complete U-site, and φ_{•} : F_{D} ⇒ C_{•} any pseudocone. Hence for each i ∈ Ob(I) the functor φ_{i} : 𝒞_{i} → 𝒟 is left exact and continuous for the topologies J_{i} and J_{𝒟} . There results a pseudo-cocone φ_{•}^{a} : 𝒞_{•} ⇒ F_{𝒟} , whence a unique functor φ : 𝒞 → 𝒟 such that F_{φ} ⊙ π_{•}^{o} = φ^{a}. The assertion then follows immediately from :

Claim 4.3.26. (i) The functor φ is left exact.

(ii) The functor φ is a morphism of sites C → D.

(iii) C is a U-site.

Proof of the claim. (i): It suffices to check that φ commutes with all finite products and all equalizers of 𝒞 (proposition 1.3.22(i)). Thus, let X_{1}, X_{2} be two objects of 𝒞, and P another object that represents X_{1} × X_{2}; pick also a universal cone for this product, given by a pair of morphisms p_{•} := (p_{t} : P → X_{t} | t = 1, 2). By inspecting the proof of claim 4.3.21, we see that there exist i ∈ Ob(I) and objects P′, X′_{1}, X′_{2} ∈ Ob(𝒞_{i}) such that P′ represents X′_{1} × X′_{2} and the cone p_{•} is isomorphic to 𝒞_{i} ∗ p′_{•}, where p′_{•} := (p′_{t} : P′ → X′_{t} | t = 1, 2) is a universal cone in 𝒞_{i}. Then φ ∗ p_{•} is isomorphic to φ_{i} ∗ p′_{•}, and the latter is universal, since φ_{i} is left exact. This shows that φ commutes with binary products, and hence with all finite products. Similarly, by inspecting the proof of claim 4.3.23 we see that the universal cone of every equalizer in 𝒞 is isomorphic to 𝒞_{i} ∗ τ, for some i ∈ Ob(I) and with τ the universal cone of some equalizer in 𝒞_{i}; then again we deduce easily that φ commutes with equalizers.

(ii): In view of (i) and example 4.3.12, it suffices to show that φ is continuous for the topologies J and J_{𝒟} . To this aim, let J′ be the topology on 𝒞 induced by J_{𝒟} via φ; it suffices to check that J ⊂ J′. However, from lemma 4.2.4 we see that J is the coarsest topology on 𝒞 such that for every i ∈ Ob(I), every X ∈ Ob(𝒞_{i}) and every covering family (g_{λ} : X_{λ} → X | λ ∈ Λ) for the topology J_{i}, the sieve generated by the family (π_{i}(g_{λ}) | λ ∈ Λ) lies in J(π_{i}(X)). Thus, consider any such covering family (g_{λ} | λ ∈ Λ); according to lemma 4.2.17(ii.b) we are further reduced to checking that the family (φ ∘ π_{i}(g_{λ}) | λ ∈ Λ) covers φ ∘ π_{i}(X) = φ_{i}(X) for the topology 𝒟. Since φ ∘ π_{i} is continuous and 𝒞_{i} is finitely complete, the latter follows again from lemma 4.2.4.

(iii): For every i ∈ Ob(I) pick an essentially small topologically generating family G_{i} ⊂ Ob(𝒞_{i}) for the site C_{i}. It suffices to show that G := ⋃_{i∈Ob(I)} π_{i}(G_{i}) is a topologically generating family for the site C. Thus, let X ∈ Ob(𝒞) be any object; by inspecting the construction of 𝒞 we find i ∈ Ob(I) and X_{i} ∈ Ob(𝒞_{i}) such that X = π_{i}(X_{i}), and by assumption the family ℱ := ⋃_{Y ∈G_{i}} Hom_{𝒞_{i}}(Y, X_{i}) covers X_{i} for the topology J_{i}. Hence (π_{i}(g) | g ∈ ℱ) ⊂ ⋃_{Z∈G} Hom_{𝒞} (Z, X) covers X for the topology J, whence the contention. ♢

(ii): The condition is obviously necessary. For the converse, we consider the finest topology J′′ on 𝒞 such that F is a sheaf on the site (𝒞 , J′′) (see remark 4.1.8(iii)); it then suffices to show that J ⊂ J′′. In view of remark 4.1.8(iii), we come down to checking the following. For every i ∈ Ob(I), every X ∈ Ob(𝒞_{i}), every covering family (g_{λ} : X_{λ} → X | λ ∈ Λ) for the topology J_{i} and every morphism f : Y → π_{i}X in 𝒞, the family

(Y ×_{π_{i}X} π_{i}(g_{λ}) : Y ×_{π_{i}X} π_{i}X_{λ} → Y | λ ∈ Λ)

generates a sieve of 2-descent for the fibration ℱ ib(F) → 𝒞 of (3.1.15). But recall that 𝒞 = ℱ ib(𝒞_{•})[Σ^{−1}], where Σ denotes the set of cartesian morphisms of ℱ ib(𝒞_{•}). With this notation, we have π_{i}X = (i^{o}, X) and Y = (j^{o}, Y′) = π_{j}Y′ for some j ∈ Ob(I) and Y′ ∈ Ob(𝒞_{j}). We are then easily reduced to the case where f is the class of a morphism (φ^{o}, t) : (j^{o}, Y′) → (i^{o}, X) of ℱ ib(𝒞_{•}), for some morphism φ : i → j in I and t : Y′ → 𝒞_{φ}X in 𝒞_{j}. Notice now the commutative diagrams :

\xymatrix{π_{j}(𝒞_{φ}X_{λ}) \ar[r]^{π_{j}(𝒞_{φ}g_{λ})} \ar[d]_{(φ,1_{𝒞φX_{λ}} )} & π_{j}(𝒞_{φ}X) \ar[d]^{(φ,1_{𝒞φX} )} \\ π_{i}(X_{λ}) \ar[r]^{π_{i}(g_{λ})} & π_{i}(X)}  for every λ ∈ Λ

whose vertical arrows are cartesian morphisms of ℱ ib(𝒞_{•}), so they are isomorphisms in 𝒞. Moreover, the continuity of 𝒞_{φ} implies that the family (𝒞_{φ}(g_{λ}) | λ ∈ Λ) covers 𝒞_{φ}X for the topology J_{j}. Furthermore, we have (φ^{o}, t) = (1_{j^{o}}, t) ∘ (φ, 1_{𝒞_{φ}X}). Thus, we may replace (g_{λ} | λ ∈ Λ) by (𝒞_{φ}(g_{λ}) | λ ∈ Λ) and assume that i = j and φ = 1_{i}. In this situation, since π_{i} is left exact, we have natural identifications for every λ, μ ∈ Λ :

Y_{λ} := Y ×_{π_{i}X} π_{i}X_{λ} →^{∼} π_{i}(Y′ ×_{X} X_{λ}) Y_{λ} ×_{Y} Y_{μ} →^{∼} π_{i}(Y′ ×_{X} X_{λ} ×_{X} X_{μ}).

So finally we are reduced to checking that the natural map from F (π_{i}Y′) to the equalizers of the restriction maps

∏_{λ∈Λ} [figure: F ∘ π_{i}(Y′ ×_{X} X_{λ})] ∏_{λ,μ∈Λ} F ∘ π_{i}(Y′ ×_{X} X_{λ} ×_{X} X_{μ})

is a bijection. But this is clear, since by assumption F ∘ π_{i} is a sheaf on C_{i}. □

Remark 4.3.27. In the situation of (4.3.24), we also claim that C^{∼} represents the 2-limit of the induced pseudo-functor

C_{•}^{∼} : I → U′-Cat  i ↦ C_{i,U}^{∼}  (φ : i → j) ↦ (C_{φ}^{∼} : C_{i,U}^{∼} → C_{j,U}^{∼}).

Indeed, on the one hand, the natural functor (see definition 2.5.1(ii))

PsFun(𝒞 , Set^{o}) → PsNat(𝒞_{•}, F_{Set}^{o}) (G : 𝒞 → Set^{o}) ↦ F_{G} ⊙ π_{•}

is an equivalence; but we have PsFun(𝒞 , Set^{o}) = 𝒞^{∧}, and a pseudo-cocone G_{•} : 𝒞_{•} ⇒ F_{Set}^{o} is the datum of a system of presheaves G_{i} on 𝒞_{i} for every i ∈ Ob(I), with isomorphisms

τ_{φ}^{G} : G_{i} →^{∼} 𝒞_{φ}^{∧}G_{j} in 𝒞_{i}^{∧} for every morphism φ : i → j in I

fulfilling the usual coherence axioms. Under these identifications, a presheaf G on 𝒞 corresponds to the datum (G_{•}, τ_{•}^{G}) with G_{i} := π_{i}^{∧}G for every i ∈ Ob(I), and with

τ_{φ}^{G} := (τ_{φ}^{π})_{G}^{∧} : π_{i}^{∧}G →^{∼} 𝒞_{φ}^{∧} ∘ π_{j}^{∧}G

where (τ_{φ}^{π})^{∧} : π_{i}^{∧} →^{∼} 𝒞_{φ}^{∧}∘π_{j}^{∧} is the isomorphism of functors induced by the coherence constraint τ_{φ}^{π} : π_{j} →^{∼} 𝒞_{φ} ∘ π_{i} of the universal pseudo-cone π_{•} : F_{𝒞} ⇒ 𝒞_{•}, for every morphism φ : i → j in I. On the other hand, by proposition 4.3.25(ii), the presheaf G is a sheaf on C if and only if G_{i} is a presheaf on C_{i} for every i ∈ Ob(I). Summing up, we obtain a natural equivalence between C^{∼} and the category of pairs (G_{•}, τ_{•}^{G}) as above, such that G_{i} is a sheaf on C_{i} for every i ∈ Ob(I). But the latter is just a description of the category

Cart_{I^{o}}(I^{o}, ℱ ib(C_{•}^{∼}))

which represents the 2-limit of C_{•}^{∼}, by the proof of theorem 3.3.9.

4.4. Topoi. Let U, U′ be a pair of universes with U ∈ U′. A (U, U′)-topos is a U′-small category with U-small Hom-sets which is equivalent to the category of sheaves on a U-small site. We shall often omit the explicit mention of U′, and call “U-topos” such a category, or even just “topos” unless this may give rise to ambiguities.

Remark 4.4.1. Let T be any U-topos.

(i) Remark 4.1.19(i,ii) can be summarized by saying that T is a complete and cocomplete, well-powered and co-well-powered category. Moreover, every epimorphism in T is universal effective, every colimit is universal, and all filtered colimits in T commute with finite limits.

(ii) It follows from (i) that a small family (F_{i} → F | i ∈ I) of morphisms of T is a covering family for the canonical topology if and only if the induced morphism ⨿_{i∈I}F_{i} → F is an epimorphism in T.

(iii) Say that T = C^{∼}, for a small site C := (𝒞 , J); then the set {h_{X}^{a} | X ∈ Ob(𝒞)} is a small topologically generating family for the canonical site Can(T) (see remark 4.1.8(v)). Indeed, by virtue of remark 4.1.19(iii) we may find for every F ∈ Ob(T) a family (X_{i} | i ∈ I) of objects of 𝒞 with an epimorphism ⨿_{i∈I}h_{X_{i}}^{a} → F in the category 𝒞^{∼}, so the resulting family (h_{X_{i}}^{a} → F | i ∈ I) covers F for the canonical topology, by (ii). Especially, Can(T) is a U-site.

(iv) On the other hand, by proposition 4.3.18(ii), if C is a U-site, then C^{∼} is isomorphic to a U-topos.

Theorem 4.4.2. Let C := (𝒞 , J) be a site such that 𝒞 has small Hom-sets. We have :

(i) The Yoneda embedding 𝒞 → 𝒞^{∧} is a morphism of sites h_{𝒞} : C^{∧} → C (notation of remark 4.1.31(ii)) and is cocontinuous for the topologies J and J^{∧}. Moreover the topology J is induced by J^{∧} via the functor h_{𝒞} .

(ii) Suppose that C is a U-site. Then also the forgetful functor i : C^{∼} → 𝒞^{∧}, its left adjoint (−)^{a} : 𝒞^{∧} → C^{∼} and the Yoneda embedding h_{C}^{a} : 𝒞 → C^{∼} are morphisms of sites C^{∧} → Can(C^{∼}), Can(C^{∼}) → C^{∧} and Can(C^{∼}) → C respectively.

(iii) Moreover, in the situation of (ii) the functors (−)^{a} and h_{C}^{a} are cocontinuous for the topologies J, J^{∧} and Can_{C^{∼}}, and the topologies J and J^{∧} are induced by the canonical topology of C^{∼}, via the functors h_{C}^{a} : 𝒞 → C^{∼} and (−)^{a} respectively. Furthermore, for every universe V containing U, the functors i, h_{𝒞} , h_{C}^{a}, (−)^{a} induce equivalences :

\xymatrix{Can(C^{∼})_{V}^{∼} \ar[rr]^{((−)^{a})_{∗}^{∼}} \ar[dr]_{_{h̃{a \atop C∗}}} &  & (C^{∧})_{V}^{∼} \ar[ll]^{ĩ_{∗}} \ar[dl]^{h̃_{𝒞 ∗}} \\  & C_{V}^{∼}.}

(iv) For every topos T, the Yoneda embedding induces an equivalence

h_{T} : T → Can(T)^{∼} F ↦ h_{F} .

Proof. (i): Pick a universe V containing U, and such that C^{∧} is a V-site; denote by J′ the topology on 𝒞 induced by J^{∧} on 𝒞 via h_{𝒞} . Notice that condition (a) of theorem 4.3.13holds for the functor h_{𝒞} , by virtue of remark 4.4.1(iii), and conditions (b) and (c) hold as well, since h_{𝒞} is fully faithful (remark 4.3.17(i)). By theorem 4.3.13(iii) it follows that a family of morphisms (Y_{i} → Y | i ∈ I) in 𝒞 covers Y in the topology J′ if and only if the family (h_{Y_{i}} → h_{Y} | i ∈ I) covers h_{Y} in the topology J^{∧}. Combining with remark 4.1.29(i) we deduce that J′ = J. Moreover, by theorem 4.3.13(ii), the functor (h_{𝒞} )_{V∗}^{∼} : (C^{∧})_{V}^{∼} → C_{V}^{∼} is an equivalence, so the same holds for its left adjoint (h_{𝒞}^{∗})_{V}^{∼}, and especially, h_{𝒞} is a morphism of sites, as stated.

(ii): Let us remark :

Claim 4.4.3. The functor (−)^{a} : 𝒞^{∧} → C^{∼} satisfies conditions (a),(b),(c) of theorem 4.3.13for the topology Can_{C^{∼}}.

Proof of the claim. Indeed, to check condition (a) it suffices to remark that (iF)^{a} is isomorphic to F, for every F ∈ Ob(C^{∼}). Next, let F, G ∈ Ob(𝒞^{∧}) and φ : F^{a} → G^{a} a morphism in C^{∼}; we may find a family (f_{j} : h_{X_{j}} → F | j ∈ I) of morphisms in 𝒞^{∧} such that the induced morphism μ : ⨿_{j∈I}h_{X_{j}} → F is an epimorphism (lemma 1.4.8). The composition

φ_{j} : h_{X_{j}} ⟶^{f_{j}} F → F^{a} ⟶^{φ} G^{a} for every j ∈ J

corresponds to a section σ_{j} ∈ G^{a}(X_{j}). Then we may find for every j ∈ I a covering family (f_{jk} : X_{jk} → X_{j} | k ∈ I_{j}) for the topology J, and a section σ_{jk} : G(X_{jk}) for every k ∈ I_{j}, whose image in G^{a}(X_{jk}) agrees with G^{a}(f_{jk})(σ_{j}). The section σ_{jk} corresponds to a morphism ψ_{jk} : h_{X_{jk}} → G in 𝒞^{∧}, whose composition with the natural morphism G → G^{a} agrees with φ_{j} ∘ h_{f_{jk}}. Moreover, for every j ∈ J the resulting morphism ν_{j} : ⨿_{k∈I_{j}}h_{X_{jk}} → h_{X_{j}} of 𝒞^{∧} induces an epimorphism ν^{a} in C^{∼} (remark 4.1.29(i) and corollary 4.1.30(i)); likewise μ^{a} is an epimorphism in C^{∼}, by virtue of proposition 1.3.25(v). Thus, the family (f_{j}^{a} ∘ h_{f_{jk}}^{a} : h_{X_{jk}}^{a} → F^{a} | j ∈ I, k ∈ I_{j}) covers F^{a} for the topology Can_{C^{∼}} (remark 4.4.1(ii)), and by construction we have φ ∘ (f_{j} ∘ h_{f_{jk}})^{a} = ψ_{jk}^{a} for every j ∈ I and every k ∈ I_{j}. This shows that condition (b) holds. Lastly, let φ, φ′ : F → G be two morphisms of 𝒞^{∧} such that φ^{a} = φ^{′a}, and choose again a family (f_{j} : h_{X_{j}} → F | j ∈ I) as in the foregoing; each f_{j} corresponds to a section σ_{j} ∈ F X_{j}, and by assumption the images of φ_{X_{j}}(σ_{j}) and φ′_{X_{j}}(σ_{j}) agree in G^{a}(X_{j}). Then we may find for every j ∈ I a covering family (f_{jk} : X_{jk} → X_{j} | k ∈ I_{j}) such that

φ_{X_{jk}} ∘ (F f_{jk})(σ_{j}) = (Gf_{jk}) ∘ φ_{X_{j}}(σ_{j}) = (Gf_{jk}) ∘ φ′_{X_{j}}(σ_{j}) = φ′_{X_{jk}} ∘ (F f_{jk})(σ_{j})

for every k ∈ I_{j}. Again, the family (f_{j}^{a} ∘ h_{f_{jk}}^{a} : h_{X_{jk}}^{a} → F^{a} | j ∈ I, k ∈ I_{j}) covers F^{a} for the topology Can_{C^{∼}}, and φ ∘ f_{j} ∘ h_{f_{jk}} = φ′ ∘ f_{j} ∘ h_{f_{jk}} for every j ∈ I and every k ∈ I_{j}. This shows that also condition (c) holds. ♢

Let J′ be the topology on 𝒞^{∧} induced by Can_{C^{∼}} via the functor (−)^{a}; since C^{∼} is isomorphic to a U-site (remark 4.4.1(iv)), theorem 4.3.13 and claim 4.4.3 imply that (−)^{a} : (𝒞^{∧}, J′) → Can(C^{∼}) is continuous and cocontinuous, and a family (F_{j} → F | j ∈ I) of morphisms in 𝒞^{∧} generates a covering sieve for the topology J′ if and only if the family (F_{j}^{a} → F^{a} | j ∈ I) generates a covering sieve for the topology Can_{C^{∼}}. But the latter condition holds if and only if the family (F_{j} → F | j ∈ I) generates a covering sieve for the topology J^{∧}, by virtue of remark 4.1.31(i). In other words, J′ = J^{∧}, and thus theorem 4.3.13also says that (−)^{a} induces an equivalence ((−)^{a})_{V∗}^{∼} : Can(C^{∼})_{V}^{∼} →^{∼} (𝒞^{∧}, J′)_{V}^{∼} for every universe V containing U. Then also the left adjoint of ((−)^{a})_{V∗}^{∼} is an equivalence, and especially (−)^{a} : Can(C^{∼}) → C^{∧} is a morphism of sites. Lastly, since (−)^{a} ∘ i = 1_{C^{∼}}, we see that ĩ_{V∗} : (C^{∧})_{V}^{∼} → Can(C^{∼})_{V}^{∼} is an equivalence as well, and so is its left adjoint (which is ((−)^{a})_{V∗}^{∼}); especially, i : C^{∧} → Can(C^{∼}) is a morphism of sites.

Claim 4.4.4. The Yoneda embedding h_{C}^{a} : 𝒞 → C^{∼} fulfills conditions (a),(b),(c) of theorem 4.3.13, for the canonical topology on C^{∼}.

Proof of the claim. Indeed, condition (a) has already been noticed in remark 4.4.1(iii). Next, let X, Y ∈ Ob(𝒞) and φ : h_{X}^{a} → h_{Y}^{a} any morphism of C^{∼}; then φ corresponds to a section s ∈ h_{Y}^{a} (X), which in turn is represented by the datum of a covering family (f_{i} : X_{i} → X | i ∈ I) for the topology J and a compatible system of morphisms (s_{i} : X_{i} → Y | i ∈ I) in 𝒞. With this notation, h_{f_{i}}^{a}(s) is the image of s in h_{Y}^{a} (X_{i}), so we get h_{s_{i}}^{a} = φ ∘ h_{f_{i}}^{a} for every i ∈ I. Lastly, the family (h_{f_{i}}^{a} : h_{X_{i}}^{a} → h_{X}^{a} | i ∈ I) covers h_{X_{i}}^{a} in the canonical topology, by remarks 4.4.1(ii) and 4.1.29(i) and corollary 4.1.30(i).

To check condition (c), consider morphisms φ, ψ : X → Y in 𝒞 such that h_{φ}^{a} = h_{ψ}^{a}. The latter means that the images \overline{φ} and \overline{ψ} of φ and ψ agree in h_{Y}^{a} (X), so there exists a covering family (f_{i} : X_{i} → X | i ∈ I) such that h_{Y}^{a} (f_{i})(\overline{φ}) = h_{Y}^{a} (f_{i})(\overline{ψ}) for every i ∈ I. This in turn translates as the identity φ ∘ f_{i} = ψ ∘ f_{i} for every i ∈ I. To conclude, we observe as in the foregoing that the family (h_{f_{i}}^{a} : h_{X_{i}}^{a} → h_{X}^{a} | i ∈ I) covers h_{X}^{a} in the topology Can_{C^{∼}}. ♢

Let J′ be the topology on 𝒞 induced by Can_{C^{∼}} via h_{𝒞}^{a}; since we have already noticed that C^{∼} is isomorphic to a U-site, claim 4.4.4and theorem 4.3.13imply that h_{𝒞}^{a} induces an equivalence H_{V} := (h_{C}^{a})_{V∗}^{∼} : Can(C^{∼})_{V}^{∼} →^{∼} (𝒞 , J′)_{V}^{∼} for every universe V containing U. Moreover, arguing as in the foregoing we deduce that J′ = J. Thus, we get an equivalence H : Can(C^{∼})^{∼} →^{∼} C^{∼}, which as usual implies that h_{C}^{a} : C^{∼} → C is a morphism of sites, concluding the proof of (ii) and (iii). Lastly, by a simple inspection we see that H ∘ h_{C^{∼}} is naturally isomorphic to 1_{C^{∼}}, whence (iv). □

Definition 4.4.5. (i) Let S and T be two topoi. A morphism of topoi f : T → S is a datum

(f^{∗}, f_{∗}, η)  where f_{∗} : T → S  f^{∗} : S → T

are two functors such that f^{∗} is left exact and left adjoint to f_{∗}, and η : 1_{S} ⇒ f_{∗}f^{∗} is a unit of an adjunction ϑ for the pair (f^{∗}, f_{∗}) (these are sometimes called geometric morphisms : see [30, Def.2.12.1]). We shall say that η and ϑ are respectively the unit and the adjunction of f.

(ii) Let f := (f^{∗}, f_{∗}, η_{f}) : T′′ → T′ and g := (g^{∗}, g_{∗}, η_{g}) : T′ → T be two morphisms of topoi. The composition g ∘ f is the morphism

(f^{∗} ∘ g^{∗}, g_{∗} ∘ f_{∗}, (g_{∗} ∗ η_{f} ∗ g^{∗}) ⊙ η_{g}) : T′′ → T

(see remark 1.1.17(i)). To check the associativity of this composition law, it suffices to notice that (g_{∗} ∗ η_{f} ∗ g^{∗}) ⊙ η_{g} is the unit of the composition of the adjunctions determined by η_{f} and η_{g}, as defined in remark 1.1.17(i). For every topos T, let i_{T} : 1_{T} →^{∼} 1_{T} be the identity automorphism of 1_{T}; the datum

(1_{T} , 1_{T} , , i_{T})

yields a morphism of topoi T → T which is neutral for left and right compositions relative to the foregoing composition law.

(iii) Let f, g : T → S be two morphisms of topoi. A natural transformation τ : f ⇒ g is just a natural transformation of functors τ_{∗} : f_{∗} ⇒ g_{∗}. Notice that, in view of remark 1.1.17(ii), the datum of τ_{∗} is the same as the datum of a natural transformation τ^{∗} : g^{∗} ⇒ f^{∗}.

(iv) Let U′ be a universe with U ∈ U′; the (U, U′)-topoi are the objects of the 2-category

(U, U′)-Topos

whose 1-cells are the morphisms of topoi, and whose 2-cells are the natural transformations between such morphisms. Obviously this category depends on the choice of U′, but two different choices are related by a 2-equivalence of 2-categories, so we shall usually omit mentioning an explicit choice of U′, and write U-Topos for this 2-category. Moreover, when the choice of U is clear from the context, we shall likewise drop the mention of U and write just Topos.

Example 4.4.6. Let T_{•} := (T_{i} | i ∈ I) be any U-small family of (U, U′)-topoi.

(i) The product category T := ∏_{i∈I} T_{i} is a topos (see example 1.2.25(i)). Indeed, pick for each i ∈ I a small site C_{i} := (𝒞_{i}, J_{i}) with an equivalence C_{i}^{∼} →^{∼} T_{i}; according to example 1.2.25(i), the coproduct of the family of categories (𝒞_{i} | i ∈ I), is represented by a category 𝒞 whose set of objects is the disjoint union ⨿_{i∈I}Ob(𝒞_{i}). We endow 𝒞 with the topology J such that J(i, X) := J_{i}(X) for every (i, X) ∈ Ob(𝒞). Then it is easily seen that there is a natural isomorphism of categories

(𝒞 , J)^{∼} →^{∼} ∏_{i∈I} C_{i}^{∼}.

However, the product of the categories C_{i}^{∼} also represents the 2-product, in the 2-category U′-Cat, of the same family of categories, and the standard universal cone is also a universal pseudo-cone (see example 3.3.12); there follows an equivalence of categories

(𝒞 , J)^{∼} →^{∼} T

whence the contention.

(ii) Moreover, T represents the 2-coproduct of the family T_{•} in the 2-category Topos. Indeed, for every i ∈ I let π_{i} : T → T_{i} be the projection functor. We define a right adjoint e_{i} : T_{i} → T for π_{i} by the rule : X ↦ (X_{j} | j ∈ I) for every X ∈ Ob(T_{i}), where X_{i} := X and X_{j} := 1_{T_{j}} is a fixed choice of a final object in T_{j}, for every j ∈ I \ {i}. It is easily seen that π_{i} is left exact, and there is a natural choice of adjunction for the pair (π_{i}, e_{i}), whence a well defined morphism of topoi

φ_{i} : T_{i} → T such that φ_{i}^{∗} = π_{i} and φ_{i∗} := e_{i}.

Next, let S be another topos, and (ψ_{i} : T_{i} → S | i ∈ I) a given family of morphisms of topoi. By the universal property of the product, there exists a unique functor λ^{∗} : S → T such that

(4.4.7)  φ_{i}^{∗} ∘ λ^{∗} = ψ_{i}^{∗}  for every i ∈ I

and it is easily seen that λ^{∗} is left exact. We construct a right adjoint λ_{∗} : T → S for λ^{∗} by the following rule. Given any object X_{•} := (X_{i} | i ∈ I) of T, let λ_{∗}(X_{•}) be any object of S representing the product ∏_{i∈I} ψ_{i∗}X_{i}, and fix as well a universal cone (p_{i}^{X} : λ_{∗}(X_{•}) → ψ_{i∗}X_{i} | i ∈ I). Then, for every morphism f_{•} : X_{•} → Y_{•} in T there is a unique morphism λ_{∗}(f_{•}) : λ_{∗}(X_{•}) → λ_{∗}(Y_{•}) that makes commute the diagram

\xymatrix{λ_{∗}(X_{•}) \ar[r]^{λ_{∗}(f_{•})} \ar[d]_{p_{i}^{X}} & λ_{∗}(Y_{•}) \ar[d]^{p_{i}^{Y}} \\ ψ_{i∗}X_{i} \ar[r]^{ψ_{i∗}f_{i}} & ψ_{i∗}Y_{i}} for every i ∈ I.

Thus, the rules X_{•} ↦ λ_{∗}(X_{•}) and f_{•} ↦ λ_{∗}(f_{•}) yield a well defined functor as sought, unique up to unique isomorphism. Then it is easily seen that λ^{∗} is left adjoint to λ_{∗}, and there is even a natural choice of adjunction for the pair (λ^{∗}, λ_{∗}) (details left to the reader), so the latter yields a well defined morphism of topoi λ : T → S. Lastly, (4.4.7) implies that λ_{∗} ∘ φ_{∗} is right adjoint to ψ_{i}^{∗}, whence an isomorphism of functors

λ_{∗} ∘ φ_{∗} →^{∼} ψ_{i∗} for every i ∈ I

which shows that the system (φ_{i} | i ∈ I) is a universal pseudo-cocone, whence the contention.

Proposition 4.4.8. Let T, T′ be two topoi, f : T → T′ a left exact functor, φ a morphism of T.

(i) If f commutes with all small colimits in T, the following holds : (a) f is continuous for the canonical topologies on T and T′.

(b) There exists a morphism of topoi F : T′ → T, unique up to unique isomorphism,

such that F^{∗} = f. (ii) f is conservative if and only if it reflects epimorphisms. (iii) If f is exact, the natural morphism f(Im φ) → Im(fφ) is an isomorphism.

Proof. (i.a): Let S := {g_{i} : X_{i} → X | i ∈ I} be a small covering family of morphisms in Can_{T}; by lemma 4.2.4, it suffices to show that fS := {f(g_{i}) | i ∈ I} is a covering family in Can_{T′}. However, our assumption implies that the induced morphism ⨿_{i∈I}X_{i} → X is an epimorphism, hence the same holds for the induced morphism ⨿_{i∈I}fX_{i} → fX in T′ (by proposition 1.3.18(ii)). But all epimorphisms are universal effective in T′ (remark 4.4.1(i)), hence (f(g_{i}) | i ∈ I) is a universal effective family, as required.

(i.b): By corollary 4.3.19(i.a,i.c), the functor f gives rise to a morphism of topoi

(f̃^{∗}, f̃_{∗}, η) : Can(T′)^{∼} → Can(T)^{∼}

(where η is any choice of unit of adjunction), and it suffices to show that f̃^{∗} is isomorphic to f, under the natural identifications T →^{∼} Can(T)^{∼}, T′ →^{∼} Can(T′)^{∼} given by the Yoneda embeddings. The latter follows from corollary 4.3.19(i.b).

(iii) follows easily from remark 4.1.25.

(ii) follows from propositions 4.1.24(i) and 1.3.21. □

Remark 4.4.9. (i) Let us consider the 2-category (U, U′)-Topos^{∗} whose objects are the same as those of the 2-category (U, U′)-Topos, and such that for every two U-topoi T and S, the 1-cells f : T → S are the left exact functors f : S → T that commute with all small colimits. For any pair of such functors f, f′ : S → T, the 2-cells β : f ⇒ f′ of (U, U′)-Topos^{∗} are the natural transformations β : f′ ⇒ f. As usual, we shall often write U-Topos^{∗} or just Topos^{∗} for this 2-category. We have a natural strict pseudo-functor

(−)^{∗} : Topos → Topos^{∗}

that is the identity on objects, and assigns to every morphism f := (f^{∗}, f_{∗}, η_{f}) : T → S of topoi the functor f^{∗} : S → T, and to every natural transformation β : (f^{∗}, f_{∗}, η_{f}) ⇒ (g^{∗}, g_{∗}, η_{g}) the adjoint transformation (β, η_{f}, η_{g})^{†} : g^{∗} ⇒ f^{∗}. Then proposition 4.4.8(i.b) can be restated by saying that (−)^{∗} is a strict 2-equivalence of 2-categories.

(ii) Notice also the strict pseudo-functors

U′-Cat^{o} ⟵^{(−)^{∗}} Topos ⟶^{(−)_{∗}} U′-Cat

defined as follows. The pseudo-functor (−)_{∗} is the identity map on objects : T ↦ T, and is given on morphisms by the rule : (F_{∗}, F^{∗}, η_{F}) ↦ F_{∗} (notation of definition 4.4.5(i)). On natural transformations it is also the identity : (τ : (F^{∗}, F_{∗}, η_{F}) ⇒ (G^{∗}, G_{∗}, η_{G})) ↦ (τ : F_{∗} ⇒ G_{∗}).

The pseudo-functor (−)^{∗} is given on objects by the rule : T ↦ T^{o}, and on morphisms by the rule : (F^{∗}, F_{∗}, η_{F}) ↦ (F^{∗})^{o}. On natural transformations it is defined by the rule :

(τ : (F^{∗}, F_{∗}, η_{F}) ⇒ (G^{∗}, G_{∗}, η_{G})) ↦ ((τ, ϑ_{F} , ϑ_{G})^{†o} : F^{∗o} ⇒ G^{∗o})

where ϑ^{F} and ϑ^{G} are the adjunctions for the pairs (F^{∗}, F_{∗}) and respectively (G^{∗}, G_{∗}) whose units are η_{F} and respectively η_{G}, and (τ, ϑ_{F} , ϑ_{G})^{†} is the adjoint to τ, as in remark 1.1.17(ii). The pseudo-functoriality of (−)^{∗} follows straightforwardly from remark 1.1.17(iv,v) : the details shall be left to the reader.

Example 4.4.10. Let T be a topos.

(i) Say that T = (𝒞 , J)^{∼} for a small site (𝒞 , J). According to remark 4.1.8(vi), the category 𝒞^{∧} is also a topos, and since the functor F ↦ F^{a} of theorem 4.1.13is left exact, it determines a morphism of topoi T → 𝒞^{∧}. (On the other hand, the category T^{∧} is too large to be a U-topos.)

(ii) Let f := (f^{∗}, f_{∗}, η) : T → S be any morphism of topoi; we have an essentially commutative diagram whose horizontal arrows are the Yoneda embeddings :

\xymatrix{T \ar[r]^{h_{T}} \ar[d]_{f_{∗}} & Can(T)^{∼} \ar[d]^{(f^{∗})_{∗}^{∼}} \\ S \ar[r]^{h_{S}} & Can(S)^{∼}.}

Indeed, an isomorphism (f^{∗})_{∗}^{∼} ∘ h_{T} →^{∼} h_{S} ∘ f_{∗} is given explicitly as follows. Let ϑ be the adjunction for the pair (f^{∗}, f_{∗}) resulting from the unit η. So, ϑ consists of a system of bijections:

h_{X} ∘ f^{∗}(Y) = Hom_{T} (f^{∗}Y, X) ⟶^{ϑ_{Y,X}} Hom_{S}(Y, f_{∗}X) = h_{f_{∗}X}(Y)

natural in X ∈ Ob(T) and Y ∈ Ob(S). The naturality with respect to morphisms Y′ → Y in S then implies that the rule Y ↦ ϑ_{Y,X} is an isomorphism of sheaves ϑ_{•,X} : h_{X} ∘ f^{∗} →^{∼} h_{f_{∗}X} for every X ∈ Ob(T). Lastly, the naturality with respect to morphisms X → X′ in T says that the rule X ↦ ϑ_{•,X} yields an isomorphism of functors as sought : the details are left to the reader.

4.4.11. Global section functors. In view of theorem 4.4.2(iv), the objects of any topos T may be thought of as sheaves on Can(T), and one uses often the suggestive notation

X(S) := Hom_{T} (S, X) for every X, S ∈ Ob(T).

The elements of X(S) are also called the S-sections of X. If 1_{T} is any final object of T, one defines the global sections functor Γ : T → Set by the rule :

U ↦ Γ(T, U) := U(1_{T}).

Moreover Γ admits a left adjoint :

(−)_{T} : Set → T : S ↦ S_{T} := S × 1_{T}

(the coproduct of S copies of 1_{T}) and one calls S_{T} the constant sheaf with value S.

Proposition 4.4.12. With the notation of (4.4.11), we have :

(i) The functor (−)_{T} is exact, hence the pair ((−)_{T} , Γ) yields a morphism of topoi

Γ_{T} : T → Set.

(ii) For every pair of morphisms of topoi u, v : T → Set there exists a unique natural transformation β : u_{∗} ⇒ v_{∗}, and β is an isomorphism of functors.

(iii) For every morphism f : T → S of topoi, there exists a unique isomorphism :

Γ_{S} ∘ f →^{∼} Γ_{T} .

Proof. (See [8, Exp.IV, §4.3]). For the proof of (i), we may assume that T = C^{∼} for a small site C := (𝒞 , J). Then, for every (small) set S, let c_{S} : 𝒞^{o} → Set be the constant presheaf on 𝒞 with value S; since the colimits in 𝒞^{o} are computed argumentwise (corollary 1.4.3(ii)), we have c_{S} = ∐_{s∈S} c_{{s}}, whence S_{T} = c_{S}^{a}, since the functor (−)^{a} commutes with small colimits. Since (−)^{a} is an exact functor (theorem 4.1.13(ii)), it suffices to check that the same holds for the functor Set → 𝒞^{o} : S ↦ c_{S}. But more precisely, the latter commutes with all small limits and all small colimits, again because the latter are computed argumentwise.

(ii): Since u^{∗} is exact, u^{∗}({∅}) is a final object of T, and we get natural bijections :

u_{∗}X →^{∼} Hom_{Set}({∅}, u_{∗}X) →^{∼} Hom_{T} (u^{∗}({∅}), X) for every X ∈ Ob(T)

i.e. the presheaf u_{∗} : T → Set on the category T^{o} is represented by the initial object u^{∗}({∅})^{o}. The same applies to v_{∗}, and then Yoneda’s lemma yields natural bijections

Hom_{(T^{o})^{∧}}(u_{∗}, v_{∗}) →^{∼} Hom_{T} (v^{∗}({∅}), u^{∗}({∅})) →^{∼} {∅}

whence the contention.

(iii) is an immediate consequence of (ii). □

Remark 4.4.13. (i) If T is a U-topos, C := (𝒞 , J) is a U-lex-site, and g : Can(T) → C is a morphism of sites, then the underlying functor g : 𝒞 → T is left exact. Indeed, by corollary 4.3.19(i.b) we have an essentially commutative diagram of functors whose vertical arrows are the Yoneda embeddings :

(4.4.14)  \xymatrix{𝒞 \ar[r]^{g} \ar[d]_{h_{𝒞}^{a}} & T \ar[d]^{h_{T}} \\ C^{∼} \ar[r]^{_{g̃_{∗}}} & Can(T)^{∼}.}

Then, h_{T} is an equivalence (theorem 4.4.2(iv)) and g̃^{∗} is left exact, since g is a morphism of sites; it suffices therefore to show that h_{𝒞}^{a} is left exact. But the functor (−)^{a} : 𝒞^{∧} → C^{∼} is left exact by theorem 4.1.13(ii), so the assertion follows from corollary 1.4.3(vi).

(ii) Every morphism of U-sites u : C′ → C induces a morphism of topoi

ũ := (ũ^{∗}, ũ_{∗}, η̃^{u}) : C^{′∼} → C^{∼}

after choosing a unit η̃^{u} for the adjoint pair (ũ^{∗}, ũ_{∗}) (corollary 4.3.19(i)).

(iii) Conversely, a morphism (f^{∗}, f_{∗}, η) : T → S of topoi determines a morphism of sites

f^{∗} : Can(T) → Can(S).

Indeed, g := f^{∗} : S → T is continuous for the canonical topologies, by virtue of proposition 4.4.8(i), and by corollary 4.3.19(i.b) we have an essentially commutative diagram of functors as in (4.4.14), with 𝒞 := S and C = Can(S). In this case, both vertical arrows of (4.4.14) are equivalences (theorem 4.4.2(iv)). Since g is left exact, it then follows that the same holds for g̃^{∗}, whence the assertion.

(iv) Likewise, for every pair of U-sites C := (𝒞 , J), C′ := (𝒞′, J′), for every cocontinuous functor v : 𝒞 → 𝒞′, it is clear that v̆^{∗} is an exact functor, so v induces a morphism of topoi

v̆ := (v̆^{∗}, v̆_{∗}, η̆^{v}) : C^{∼} → C^{′∼}

(corollary 4.3.19(ii)), after choosing a unit η̆^{v} for the adjoint pair (v̆^{∗}, v̆_{∗}).

(v) In the situation of (iv), suppose that v admits a right adjoint u. Then u is continuous for the topologies of the sites C and C′, and ũ^{∗} is isomorphic to v̆^{∗} (lemma 4.2.14). Especially, ũ^{∗} is exact, so u is a morphism of sites, and we have an isomorphism of morphisms of topoi :

ũ →^{∼} v̆.

4.4.15. We define as follows a pair of pseudo-functors

\xymatrix{Site \ar[r]^{T} & Topos. \ar[l]^{Can}}

Let C be a U-site; by remark 4.4.1(iv) we may find an isomorphism of topoi

ω_{C} : C^{∼} →^{∼} T(C)

with a U-topos T(C), and it is easily seen that we may take this topos to be also U′-small. Then the rule : C ↦ T(C) defines the pseudo-functor T on objects. Next, let g : C → C′ be any morphism of U-sites; by remark 4.4.13(ii), any choice of a unit η^{g} for the adjoint pair (g̃^{∗}, g̃_{∗}) yields a morphism of topoi g̃: C^{∼} → C^{′∼}. There follows easily a unique morphism of U-topoi

T(g) := (T(g)^{∗}, T(g)_{∗}, η^{T(g)}) : T(C) → T(C′)

that is identified with (g̃^{∗}, g̃_{∗}, η^{g}) via the isomorphisms ω_{C} and ω_{C′} (details left to the reader). Lastly, let β : g ⇒ g′ be a natural transformation between morphisms of sites g, g′ : C → C′; there follows a natural transformation β^{∧} : g^{′∧} ⇒ g^{∧} (notation of (1.2.2)) that induces by restriction a natural transformation

β_{∗}^{∼} : g̃′_{∗} ⇒ g̃_{∗}

which is in turn identified via ω_{C} and ω_{C′} with a unique natural transformation

T(β) : T(g′)_{∗} ⇒ T(g)_{∗}.

If g′′ : C → C′ is a third morphism of sites and β′ : g′ ⇒ g′′ is another natural transformation, it is clear that

T(β′ ⊙ β) = T(β) ⊙ T(β′).

Likewise, if h, h′ : C′ → C′′ is another pair of morphisms of sites and α : h ⇒ h′ is another natural transformation, we deduce easily from (1.2.2) the identity :

T(α ∗ β) = T(β) ∗ T(α).

For a composable pair C ⟶^{g} C′ ⟶^{g′} C′′ of morphisms of U-sites, it is easily seen that T(g′)_{∗} ∘ T(g)_{∗} = T(g′ ∘ g)_{∗}, and the coherence constraint γ_{g,g′}^{T} of T is given by 1_{T(g′∘g)_{∗}}. Likewise, T(1_{C})_{∗} = 1_{T(C)} for every U-site C, and the coherence constraint δ_{C}^{T} is given by the identity automorphism of 1_{T(C)}. Then the required coherence axioms are trivially satified; however, notice that T is not a strict pseudo-functor, since we do not have necessarily T(g)^{∗} ∘ T(g′)^{∗} = T(g′ ∘ g)^{∗}. This completes the construction of T.

The pseudo-functor Can assigns to every U-topos T the U-site Can(T) and to every morphism of U-topoi f : T → S the morphism of sites f^{∗} : Can(T) → Can(S) (see remark 4.4.13(iii)). Lastly, let (f^{∗}, f_{∗}, η^{f}), (g^{∗}, g_{∗}, η^{g}) : T → S be two morphisms of topoi and β : f_{∗} ⇒ g_{∗} a natural transformation; then we let Can(β) := (β, η^{f}, η^{g})^{†} : g^{∗} ⇒ f^{∗}, the adjoint transformation of β, as defined in remark 1.1.17(ii). Taking into account remark 1.1.17(iv,v,vi), we easily see that these rules define a strict pseudo-functor Can as sought.

Remark 4.4.16. Notice that the pseudo-functor Can factors through the inclusion strict pseudofunctor lex.Site → Site. We then get as well a pair of pseudo-functors :

\xymatrix{lex.Site \ar[r]^{lex.T} & Topos. \ar[l]^{lex.Can}}

where lex.T is the restriction of T.

Theorem 4.4.17. (i) The pseudo-functor T is right 2-adjoint to the pseudo-functor Can. (ii) The pseudo-functor lex.T is right 2-adjoint to the pseudo-functor lex.Can. (iii) The pseudo-functors Can and lex.Can are fully faithful.

Proof. (i): We construct as follows a pseudo-natural equivalence h_{•} : 1_{Topos} ⇒ T ∘ Can. For every U-topos T, we choose a quasi-inverse h_{T}^{∗} : T ∘ Can(T) → T for the composition h_{T ∗} : T → T ∘ Can(T) of the Yoneda embedding h_{T} : T → Can(T)^{∼} with the isomorphism ω_{Can(T)} of (4.4.15). We also fix an isomorphism of functors η^{h_{T}} : 1_{T∘Can(T)} →^{∼} h_{T ∗} ∘ h_{T}^{∗}, and we consider the morphism of U-topoi :

h_{T} := (h_{T}^{∗} , h_{T ∗}, η^{h_{T}}) : T → T ∘ Can(T).

In order to show that the rule T ↦ h_{T} defines the sought pseudo-natural equivalence, we need to explicit its coherence constraint; the latter amounts to an isomorphism of functors

τ_{f}^{h} : T(f^{∗})_{∗} ∘ h_{T ∗} →^{∼} h_{S∗} ∘ f_{∗}

for every morphism of U-topoi (f^{∗}, f_{∗}, η^{f}) : T → S. However, example 4.4.10(ii) yields an isomorphism of functors τ_{f}^{h} : (f^{∗})_{∗}^{∼} ∘ h_{T} →^{∼} h_{S} ∘ f_{∗}, so we can take :

τ_{f}^{h} := ω_{(S,Can_{S})} ∗ τ_{f}^{h}.

To check the coherence axioms for τ^{h}, notice first that T and Can are unital pseudo-functors, so that the first coherence axiom amounts to the identity : τ_{1_{T}}^{h} = 1_{h_{T ∗}} for every U-topos T. The latter follows from the identity : τ_{1_{T}}^{h} = 1_{h_{T}}, which is clear from the construction of τ^{h}.

Next, for two morphisms of U-topoi f := (f^{∗}, f_{∗}, η^{f}) : T → T′, g := (g^{∗}, g_{∗}, η^{g}) : T′ → T′′, we come down to checking that

(4.4.18)  (η_{g}^{h} ∗ f_{∗}) ⊙ ((g^{∗})_{∗}^{∼} ∗ η_{f}^{h}) = η_{g∘f}^{h}.

Now, recall that η_{f}^{h} is induced by the unique adjunction ϑ^{f} for the pair (f^{∗}, f_{∗}) whose unit is η^{f}, and likewise for η_{g}^{h} and η_{g∘f}^{h}; on the other hand, η_{g∘f}^{h} is precisely the unit of the adjunction ϑ^{g} ∘ϑ^{f} (notation of remark 1.1.17(i)). From this, the identity (4.4.18) follows straightforwardly.

Likewise, let f, f′ : T → S be two morphisms of topoi, and β : f ⇒ f′ a natural transformation; in order to check the naturality of τ^{h} we come down to showing that :

τ_{f′}^{h} ⊙ (β^{†})_{∗}^{∼} = (h_{S} ∗ β) ⊙ τ_{f}^{h}.

The latter follows by inspecting the characterization of β^{†} furnished by remark 1.1.17(iii), and this completes the construction of h_{•}. From the pseudo-natural equivalence h_{•} we construct the sought 2-adjunction as the composition of the pseudo-natural transformation

H_{T} ∗ (Can^{o} × 1_{Site}) : Hom_{Site}(Can, −) ⇒ Hom_{Topos}(T ∘ Can, T)

with the pseudo-natural transformation

H_{Topos} ∗ (h_{•}^{o} × 1_{Topos}) : Hom_{Topos}(T ∘ Can, T) ⇒ Hom_{Topos}(1_{Topos}, T).

This composition assigns to every U-topos T and every U-site C a functor

Φ_{T,C} : Hom_{Site}(Can(T), C) → Hom_{Topos}(T, T(C))

as required, and it remains to check that this functor is an equivalence for every such T and C.

Explicitly, Φ_{T,C} assigns to every morphism of sites g : Can(T) → C the morphism of topoi T(g) ∘ h_{T} : T → T(C), and to every 2-cell β : g ⇒ g′ of Site the natural transformation T(β) ∗ h_{T} : T(g) ∘ h_{T} ⇒ T(g′) ∘ h_{T}. Let us show the essential surjectivity : thus, say that C = (𝒞 , J), and let f := (f^{∗}, f_{∗}, η^{f}) : T → T(C) be a morphism of topoi; from theorem 4.4.2(ii) we see that g := f^{∗} ∘ ω_{C} ∘ h_{𝒞}^{a} : 𝒞 → T is a morphism of sites g : Can(T) → C, and we have an isomorphism of functors h_{T} ∘ g →^{∼} g̃^{∗} ∘ h_{𝒞}^{a} (corollary 4.3.19(i.b)). We deduce an isomorphism :

(4.4.19)  h_{T ∗} ∘ f^{∗} ∘ ω_{C} ∘ h_{𝒞}^{a} →^{∼} ω_{Can(T)} ∘ g̃^{∗} ∘ h_{𝒞}^{a} .

Claim 4.4.20. Let D be any category, k, k′ : C^{∼} → D two functors, λ : k ∘ h_{𝒞}^{a} →^{∼} k′ ∘ h_{𝒞}^{a} an isomorphism of functors, and suppose that k commutes with all small colimits. Then there exists an isomorphism of functors λ′ : k →^{∼} k′.

Proof of the claim. Let G ⊂ Ob(𝒞) be an essentially small topologically generating subset, 𝒢 ⊂ 𝒞 the full subcategory with Ob(𝒢) = G; endow 𝒢 with the topology J_{𝒢} induced by J via the inclusion functor u : 𝒢 → 𝒞, and set G := (𝒢 , J_{𝒢}). Then ũ_{∗} : C^{∼} → G^{∼} is an equivalence (proposition 4.3.18), hence the same holds for its left adjoint ũ^{∗} : G^{∼} → C^{∼}. According to corollary 4.3.19(i.b), we have an isomorphism of functors μ : ũ^{∗} ∘ h_{𝒢}^{a} →^{∼} h_{𝒞}^{a} ∘ i. There follows a unique isomorphism λ′ : k ∘ ĩ^{∗} ∘ h_{𝒢}^{a} →^{∼} k′ ∘ ĩ^{∗} ∘ h_{𝒢}^{a} such that

(k′ ∗ μ) ⊙ λ′ = (λ ∗ i) ⊙ (k ∗ ω).

Now, if k ∘ĩ^{∗} and k′ ∘ĩ^{∗} are isomorphic, the same follows for k and k′, since ĩ^{∗} is an equivalence (details left to the reader). Thus, we may replace C by G and λ by λ′, and assume from start that C is a small site. Next, let F ∈ Ob(C^{∼}); we have a universal cocone τ^{F} given by the rule :

(X, s) ↦ (τ_{(X,s)}^{F} : h_{X}^{a} → F) for every (X, s) ∈ Ob(ℱ ib(F))

where τ_{(X,s)}^{F} is characterized as the morphism of sheaves whose composition with the natural morphism of presheaves h_{X} → h_{X}^{a} is the unique morphism of presheaves t_{(X,s)}^{F} : h_{X} → F such that (t_{(X,s)}^{F})_{X}(1_{X}) = s (remark 4.1.19(iii)). By assumption, the cocone k ∗ τ^{F} is still universal, hence there exists a unique isomorphism λ′_{F} : kF →^{∼} k′F that makes commute the diagram :

\xymatrix{k(h_{X}^{a}) \ar[r]^{k(τ_{(X,s)}^{F})} \ar[d]_{λ_{X}} & kF \ar[d]^{λ′_{F}} \\ k′(h_{X}^{a}) \ar[r]^{k′(τ_{(X,s)}^{F})} & k′F}  for every (X, s) ∈ Ob(ℱ ib(F))

and we are reduced to checking that the rule : F ↦ λ′_{F} is a natural transformation k ⇒ k′. Thus, let φ : F → G be any morphism of sheaves on C; recall that φ induces a functor

ℱ ib(φ) : ℱ ib(F) → ℱ ib(G) (X, s) ↦ (X, φ_{X}(s))

and the foregoing characterizations of τ^{F} and τ^{G} easily imply that :

(4.4.21)  φ ∘ τ_{(X,s)}^{F} = τ_{(X,φ_{X}(s))}^{G}  for every (X, s) ∈ Ob(ℱ ib(F)) .

Consider now for every (X, s) ∈ Ob(ℱ ib(F)) the diagram :

[figure: k′(h_{X}^{a}); k′(τ_{(X,s)}^{F}); k′F; k′(φ); k(h_{X}^{a}); k(τ_{(X,s)}^{F}); λ_{X}; kF; k(φ); λ′_{F}; k(h_{X}^{a}); k(τ_{(X,φ(s))}^{G}); λ_{X}; kG; λ′_{G}; k′(h_{X}^{a}); k′(τ_{(X,φ(s))}^{G}); k′G]

We have to prove the commutativity of the right trapezoidal subdiagram, and by (4.4.21), the inner and outer square subdiagrams both commute; moreover, also the upper and lower trapezoidal subdiagrams commute, by construction. Now, since by assumption the cocone k ∗ τ^{F} is still universal, it suffices to check that k′(φ) ∘ λ′_{F} ∘ k(τ_{(X,s)}^{F}) = λ′_{G} ∘ k(φ) ∘ k(τ_{(X,s)}^{F}). The latter follows from a straightforward diagram chase, left to the reader. ♢

Since g̃^{∗} commutes with small colimits, claim 4.4.20and (4.4.19) yield an isomorphism :

h_{T ∗} ∘ f^{∗} ∘ ω_{C} →^{∼} ω_{Can(T)} ∘ g̃^{∗}.

From this, we further deduce an isomorphism of functors :

f^{∗} →^{∼} h_{T}^{∗} ∘ T(g)^{∗}

whence, finally, an isomorphism of their respective right adjoint functors : T(g)_{∗} ∘ h_{T ∗} →^{∼} f_{∗}.

Next, we check that Φ_{T,C} is faithful : let g, g′ : Can(T) → C be two morphisms of sites, and β, β′ : g ⇒ g′ two 2-cells of Site such that T(β)_{∗} ∗ h_{T} = T(β′)_{∗} ∗ h_{T}; we need to show that β = β′. However, the condition means that β̃_{∗} ∘ h_{T} = β̃′_{∗} ∘ h_{T}; i.e. for every X ∈ Ob(T), every A ∈ Ob(𝒞) and every morphism φ : gA → X in T, we have φ ∘ β_{A} = φ ∘ β′_{A}. Letting X := gA and φ := 1_{gA}, we get β_{A} = β′_{A}, for every A ∈ Ob(𝒞), as required.

Lastly, in order to check that Φ_{T,C} is full, let α : T(g) ∘ h_{T} ⇒ T(g′) ∘ h_{T} be any natural transformation, with g and g′ as in the foregoing. This is the same as a natural transformation α′ : g̃_{∗} ∘ h_{T} ⇒ g̃′_{∗} ∘ h_{T}. The latter assigns to every X ∈ Ob(T) a morphism of sheaves α′_{X} : h_{X} ∘ g^{o} → h_{X} ∘ g^{′o}; in particular, for every A ∈ Ob(𝒞), we deduce a map

(α′_{gA})_{A} : Hom_{T} (gA, gA) → Hom_{T} (g′A, gA) and we set : β_{A} := (α′_{gA})_{A}(1_{gA}).

Let us check that the rule : A ↦ β_{A} yields a natural transformation β : g′ → g. Indeed, let φ : A → B be any morphism of 𝒞; we need to show that g(φ) ∘ β_{A} = β_{B} ∘ g′(φ). However, the naturality of the rule X ↦ α′_{X} for every X ∈ Ob(T) implies the identity :

g(φ) ∘ (α′_{gA})_{A}(1_{gA}) = (α′_{gB})_{A}(g(φ))

and the naturality of the rule : Y ↦ (α′_{gB})_{Y} for every Y ∈ Ob(𝒞) yields the identity :

(α′_{gB})_{A}(g(φ)) = (α′_{gB})_{B}(1_{gB}) ∘ g′(φ)

whence the contention. It remains to check that T(β)_{∗} ∗ h_{T} = α, or equivalently, that β̃_{∗} ∗ h_{T} = α′. The latter comes down to the identity : (α′_{X})_{A}(φ) = φ ∘ β_{A} for every X ∈ Ob(T), every A ∈ Ob(𝒞) and every morphism φ : gA → X in T. This identity in turn follows easily from the naturality of the rule : X ↦ α′_{X} (details left to the reader).

(ii): Notice that – by virtue of example 4.3.12– the same rules defining h_{•} also yield a pseudonatural transformation 1_{Topos} ⇒ lex.T ∘ lex.Can. We use this pseudo-natural transformation as in the foregoing to construct the sought 2-adjunction for the pair (lex.Can, lex.T); the details shall be left to the reader.

(iii): Since h_{•} is a pseudo-natural equivalence, the assertion for Can follows from corollary 2.4.29. The same arguments applies to lex.Can. □

Remark 4.4.22. As an application of theorem 4.4.17 we deduce the representability of the 2limit of any cofiltered system of topoi. Indeed, let I be a small cofiltered category, and T_{•} : I → (U, U′)-Topos any pseudo-functor. By proposition 4.3.25, the pseudo-functor lex.Can ∘ T_{•} : I → lex.Site admits a 2-limit (C, π_{•}). By theorem 4.4.17(ii) and proposition 2.5.9(i), the pair (lex.T(C), lex.T∗π_{•}) is a 2-limit of the pseudo-functor lex.T∘lex.Can∘T_{•} : I → (U, U′)-Topos. But by theorem 4.4.17(iii) and corollary 2.4.29, we have a pseudo-natural equivalence ω : lex.T ∘ lex.Can →^{∼} 1_{Topos}, so (lex.T(C), (ω ∗ T_{•}) ⊙ (lex.T ∗ π_{•})) is a 2-limit for T_{•} (lemma 2.5.3). Combining with remark 4.3.27, we see that the 2-limit of T_{•} is also a 2-limit of the induced pseudo-functor

I → U′-Cat  i ↦ T_{i}  (φ : i → j) ↦ (T_{φ∗} : T_{i} → T_{j}).

4.5. Fibred sites. The formalism developed in this section and the next one shall allow us to deal with families of sites or of topoi, indexed by arbitrary categories. Especially, we will explain how to combine the members of such a family into a single total site or total topos.

Definition 4.5.1. (i) Let 𝒞 be a category and V a universe. A fibred site with V-small fibres over 𝒞 is a datum (𝒜 , p, J_{•}) consisting of a fibration with V-small fibre categories

p : 𝒜 → 𝒞

and of a topology J_{X} on the fibre category 𝒜_{X}, for every X ∈ Ob(𝒞), such that the following holds. For every cleavage λ of p and every morphism φ : X → Y in 𝒞, the functor c_{φ} : 𝒜_{Y} → 𝒜_{X} is a morphism of sites (𝒜_{X}, J_{X}) → (𝒜_{Y} , J_{Y}), where c : 𝒞^{o} → V-Cat is the pseudo-functor associated with λ.

(ii) A datum (𝒜 , p, J_{•}) as in (i) is a fibred lex-site with V-small fibres if (𝒜_{X}, J_{X}) is a lex-site for every X ∈ Ob(𝒞) and the functors c_{φ} are left exact for every morphism φ of 𝒞 and every cleavage λ with associated pseudo-functor c. As usual, we shall mostly omit mentioning the universe V, unless the omission would be source of ambiguities.

(iii) Let (𝒜 , p, J_{•}) be a fibred site over the category 𝒞. We endow 𝒜 with the coarsest topology J_{𝒜} such that the inclusion functor 𝒜_{X} → 𝒜 is continuous for the topologies J_{X} and J_{𝒜} , for every X ∈ Ob(𝒞). The total site of (𝒜 , p, J_{•}) is the resulting site

(𝒜 , J_{𝒜} ).

(iv) Let (𝒜 , p, J_{•}) and (𝒜′, p′, J′_{•}) be two fibred sites over 𝒞. A morphism of fibred sites

φ : (𝒜 , p, J_{•}) → (𝒜′, p′, J′_{•})

is a 𝒞 -cartesian functor φ : 𝒜′ → 𝒜 whose restriction φ_{X} : 𝒜′_{X} → 𝒜_{X} is a morphism of sites (𝒜_{X}, J_{X}) → (𝒜′_{X}, J′_{X}), for every X ∈ Ob(𝒞). If (𝒜 , p, J_{•}) and (𝒜′, p′, J′_{•}) are fibred lex-sites, we say that φ is a morphism of fibred lex-sites, if it is 𝒞 -cartesian, and the restriction φ_{X} : (𝒜_{X}, J_{X}) → (𝒜′_{X}, J′_{X}) is a morphism of lex-sites, for every X ∈ Ob(𝒞).

Remark 4.5.2. (i) With the notation of definition 4.5.1, it is clear that if there exists a cleavage of p with associated pseudo-functor c, such that c_{φ} is a morphism of sites (𝒜_{X}, J_{X}) → (𝒜_{Y} , J_{Y}) for every morphism φ : X → Y, then the same holds for every cleavage of p (and in this case, p is a fibred site). Likewise, if c_{φ} is also left exact for every such φ, then the same holds for every cleavage of p (and then p is a fibred lex-site).

(ii) Let (𝒜 , p, J_{•}) and (𝒜 , J_{𝒜} ) be as in definition 4.5.1(iii). By lemma 4.2.4, we have 𝒮 ∈ J_{𝒜} (A) for every A ∈ Ob(𝒜) and every sieve 𝒮 ⊂ 𝒜 /A such that

(4.5.3)  𝒮 ∩ (𝒜_{pA}/A) ∈ J_{pA}(A).

(iii) In the situation of (ii), suppose moreover that 𝒞 is small and (𝒜_{X}, J_{X}) is a U-site for every X ∈ Ob(𝒞); under these assumptions, the fibre category 𝒜_{X} has small Hom-sets for every such X, but the same does not necessarily hold for the category 𝒜. However, for every A, A′ ∈ Ob(𝒜) the set Hom_{𝒜} (A, A′) is essentially small : indeed, let λ be a cleavage for 𝒜 and c its associated pseudo-functor; since every morphism f : A′ → A in 𝒜 factors as a morphism A′ → c_{p(f)}A of 𝒜_{pA′} and the cartesian morphism λ(A, p(f)) : c_{p(f)}A → A, the assertion follows easily. Thus, after replacing 𝒜 by an isomorphic category, we may assume that 𝒜 has small Hom-sets, and in this case we claim that (𝒜 , J_{𝒜} ) is a U-site. Indeed, choose for every X ∈ Ob(𝒞) an essentially small topologically generating family G_{X} ⊂ Ob(p^{−1}X) for (𝒜_{X}, J_{X}); then it is clear from (ii) that ⋃_{X∈Ob(𝒞)} G_{X} is an essentially small topologically generating family for (𝒜 , J_{𝒜} ).

(iv) Clearly the fibred sites (resp. the fibred lex-sites) over 𝒞 with V-small fibres are the objects of a 2-category :

V-fib.Site(𝒞) (resp. V-fib.lex.Site)

whose 1-cells are the morphisms of fibred sites (resp. of fibred lex-sites), and whose 2-cells g ⇒ g′ are the natural 𝒞 -transformations g′ ⇒ g between the underlying functor of such morphisms, with the obvious composition laws for 1-cells and 2-cells. We have an obvious forgetful strict pseudo-functor :

V-fib.Site(𝒞) → ^{o}(V-Fib(𝒞))^{o} (𝒜 , p, J_{•}) ↦ (𝒜 , p).

(v) Let 𝒞 be a category with V-small Hom-sets and whose set of objects is essentially V-small. Taking into account theorem 3.1.24, it is easily seen that the pseudo-functor ℱ ib_{𝒞} induces strict and strong 2-equivalences of 2-categories :

\underline{ℱ ib} _{𝒞} : PsFun(𝒞 , V-Site) → V-fib.Site(𝒞)
lex.\underline{ℱ ib} _{𝒞} : PsFun(𝒞 , V-lex.Site) → V-fib.lex.Site(𝒞).

Namely, we have an obvious forgetful strict pseudo-functor

Φ : V-Site → ^{o}(V-Cat)^{o} (𝒜 , J) ↦ 𝒜

whence a pseudo-functor PsFun(𝒞 , Φ) : PsFun(𝒞 , V-Site) → PsFun(𝒞 ,^{o}(V-Cat)^{o}) (see remark 2.2.17(i)). Then, since^{o}𝒞^{o} = 𝒞^{o}, we have a strict isomorphism of 2-categories :

PsFun(𝒞 ,^{o}(V-Cat)^{o}) →^{∼} ^{o}PsFun(𝒞^{o}, V-Cat)^{o}

(see (2.2.13)) and a strict and strong 2-equivalence

oℱ ib_{𝒞}^{o} : ^{o}PsFun(𝒞^{o}, V-Cat)^{o} →^{∼} ^{o}(V-Fib(𝒞))^{o}.

The composition PsFun(𝒞 , V-Site) → ^{o}(V-Fib(𝒞))^{o} of these pseudo-functors assigns to every pseudo-functor

\underline{𝒜} : 𝒞 → V-Site X ↦ (𝒜_{X}, J_{X})

the fibration 𝒜′ := ℱ ib(^{o}(Φ ∘ \underline{𝒜} )^{o}) whose fibre category 𝒜′_{X} over every X ∈ Ob(𝒞) is naturally identified with 𝒜_{X}. Then \underline{ℱ ib} (\underline{𝒜} ) is the fibred site (𝒜′, J′_{•}) where J′_{X} is the topology on 𝒜′_{X} corresponding to J_{X} under this identification 𝒜′_{X} →^{∼} 𝒜_{X}, for every X ∈ Ob(𝒞).

(vi) By remark 2.4.31, the pseudo-functor \underline{ℱ ib} _{𝒞} admits a strict and strong pseudo-inverse

\underline{c}^{•} : V-fib.Site → PsFun(𝒞 , V-Site) \underline{𝒜} := (𝒜 , p, J_{•}) ↦ \underline{c}\underline{𝒜}

which, to every fibred site (𝒜 , p, J_{•}) over 𝒞, assigns the pseudo-functor c^{𝒜} : 𝒞^{o} → V-Cat associated with a cleavage of 𝒜, and for every X ∈ Ob(𝒞), endows the category c_{X}^{𝒜} = 𝒜_{X} with the topology J_{X}. Obviously \underline{c}^{•} restricts to a strict and strong pseudo-inverse for lex.\underline{ℱ ib} _{𝒞} .

4.5.4. Let 𝒞 be a category, (𝒜 , p, J_{•}) a fibred lex-site over 𝒞, and (𝒜 , J_{𝒜} ) its total site. For every X ∈ Ob(𝒞) denote by i_{X} : 𝒜_{X} → 𝒜 the inclusion functor. We have :

Proposition 4.5.5. In the situation of (4.5.4), the following holds :

(i) The functor i_{X} commutes with fibre products and equalizers for every X ∈ Ob(𝒞). (ii) For every A ∈ Ob(𝒜) and every sieve 𝒮 ⊂ 𝒜 /A, we have 𝒮 ∈ J_{𝒜} (A) if and only if

𝒮 satisfies condition (4.5.3). (iii) The functor i_{X} is both continuous and cocontinuous for the topologies J_{X} and J_{𝒜} , for

every X ∈ Ob(𝒞). (iv) A presheaf F on 𝒜 is a sheaf for the topology J_{𝒜} if and only if i_{X}^{∧}F is a sheaf on

(𝒜_{X}, J_{X}) for every X ∈ Ob(𝒞).

(v) Suppose moreover that 𝒞 is small, and (𝒜_{X}, J_{X}) is a U-site for every X ∈ Ob(𝒞). Then we have an essentially commutative diagram of categories :

\xymatrix{𝒜^{∧} \ar[r]^{i_{X}^{∧}} \ar[d]_{(−)^{a}} & 𝒜_{X}^{∧} \ar[d]^{(−)^{a}} \\ (𝒜 , J_{𝒜} )^{∼} \ar[r]^{i_{X∗}^{∼}} & (𝒜_{X}, J_{X})^{∼}} for every X ∈ Ob(𝒞).

Proof. Choose a cleavage λ for p and let c : 𝒞^{o} → V-Cat be the associated pseudo-functor (for some universe V such that 𝒜_{X} is a V-small category for every X ∈ Ob(𝒞)).

(i): Let A′ ⟶^{f′} A ⟵^{f′′} A′′ be two morphisms in 𝒜_{X}; pick P ∈ Ob(𝒜_{X}) representing A′ ×_{A} A′′ in 𝒜_{X}, and let A′ ⟵^{q′} P ⟶^{q′′} A′′ be a universal cone. Let then B ∈ Ob(𝒜) be any object and A′ ⟵^{g′} B ⟶^{g′′} A′′ two morphisms in 𝒜 such that f′ ∘ g′ = f′′ ∘ g′′. It follows easily that φ := p(g′) = p(g′′), so g′ (resp. g′′) factors through a morphism h′ : B → c_{φ}A′ (resp. h′′ : B → c_{φ}A′′) in 𝒜_{pB} and the cartesian morphism λ(A′, φ) : c_{φ}A′ → A′ (resp. λ(A′′, φ) : c_{φ}A′′ → A′′), and since c_{φ} is left exact, we get a unique morphism h : B → c_{φ}P in 𝒜_{pB} such that c_{φ}(q′) ∘ h = h′ and c_{φ}(q′′) ∘ h = h′′. Then k := λ(φ, P) ∘ h : B → P is the unique morphism in 𝒜 such that q′ ∘ k = g′ and q′′ ∘ k = g′′. This shows that P represents the fibre product of A′ and A′′ over A in 𝒜. Similarly one shows the assertion for equalizers : the details shall be left to the reader.

(ii): In view of (i) and lemma 4.2.4, it suffices to check that the family 𝒥 of sieves verifying condition (4.5.3) yields a topology on 𝒜. Now, condition (c) of definition 4.1.1(i) obviously holds for 𝒥. Next, let 𝒮 ⊂ 𝒜 /A be a sieve fulfilling condition (4.5.3), and f : A′ → A a morphism of 𝒜; we set φ := p(f) and factor f through a morphism f′ : A′ → A′′ := c_{φ}A in p^{−1}A′ and the cartesian morphism f′′ := λ(A, φ) : A′′ → A. Then let 𝒮′ := 𝒮 ×_{A} f′′, and notice that if (g_{λ} : B_{λ} → A | λ ∈ Λ) generates 𝒮_{|A} := 𝒮 ∩ (𝒜_{pA}/A), then the family (c_{φ}(g_{λ}) : c_{φ}B_{λ} → A′′ | λ ∈ Λ) lies in 𝒮′, and the latter family covers A′′ for the topology J_{pA′′}, since c_{φ} is continuous for the topologies J_{pA} and J_{pA′′}, and since 𝒮_{|A} covers A for the topology J_{pA}. Then 𝒮 ×_{A} f = 𝒮′ ×_{A′′} f′ contains the family (A′ ×_{A′′} c_{φ}B_{λ} → A′ | λ ∈ Λ), which covers A′ for the topology J_{pA′}. This shows that condition (a) of definition 4.1.1(i) holds for 𝒥. Lastly, let 𝒮 ⊂ 𝒜 /A be an element of 𝒥, and consider another sieve 𝒯 ⊂ 𝒜 /A such that for every (f : A′ → A) ∈ Ob(𝒮) the sieve 𝒯 ×_{A} f lies in 𝒥. Let 𝒮_{|A} ⊂ 𝒜_{pA}/A be as in the foregoing, and define likewise 𝒯_{|A}; then 𝒮_{|A} ∈ J_{pA}(A) and it is easily seen that

𝒯_{|A} ×_{A} f = (𝒯 ×_{A} f) ∩ (𝒜_{pA′}/A′) for every (f : A′ → A) ∈ Ob(𝒮_{|A}) .

Hence 𝒯_{|A} ∈ J_{pA}(A), so 𝒯 ∈ 𝒥; this shows that 𝒥 fulfills condition (b) of definition 4.1.1(i). (iii) follows immediately from (ii). (v): Pick a universe V containing U, and such that 𝒜 is V-small; in view of remarks 4.1.23(i)

and 4.5.2(iii) we are easily reduced to checking the essential commutativity of the diagram :

\xymatrix{𝒜_{V}^{∧} \ar[r]^{i_{X,V}^{∧}} \ar[d]_{(−)^{+}} & 𝒜_{X,V}^{∧} \ar[d]^{(−)^{+}} \\ 𝒜_{V}^{∧} \ar[r]^{i_{X}^{∧}} & 𝒜_{X,V}^{∧}}  for every X ∈ Ob(𝒞)

where the functors (−)^{+} are defined as in (4.1.11). We may then replace U by V and assume that 𝒜 is a small category. For every X ∈ Ob(𝒞) and every A ∈ Ob(𝒜_{X}) we endow J_{X}(A) and J_{𝒜} (A) with the ordering induced by inclusion of sieves, and consider the map

J_{X}(A) → J_{𝒜} (A) 𝒮 ↦ 𝒮^{∗}

that assigns to every 𝒮 ∈ J_{X}(A) the sieve 𝒮^{∗} ∈ J_{𝒜} (A) generated by Ob(𝒮) ⊂ Ob(𝒜 /A). This map is obviously injective, and according to (ii), its image is a cofinal subset in the opposite ordered set J_{𝒜} (A)^{o}. Moreover, for every morphism f : A′ → A in 𝒜_{X}, clearly we have

(𝒮 ×_{A} f)^{∗} = 𝒮^{∗} ×_{A} f for every 𝒮 ∈ J_{X}(A) .

Furthermore, for every such 𝒮 and every presheaf F on 𝒜_{X}, we have a natural identification of F (𝒮) := Hom_{𝒜_{X}^{∧}} (h_{𝒮} , F) with the equalizer of the two natural maps (see (4.1.6))

(4.5.6)  Equal( ∏_{(B→A)∈Ob(𝒮)}[figure: F (B)] ∏_{(B→A←B′)∈Ob(𝒮)×Ob(𝒮)} F (B ×_{A} B′))

and for every inclusion 𝒮′ ⊂ 𝒮 of sieves covering A, the map F (𝒮) → F (𝒮′) corresponds, under this identification, to the restriction of the projection

(4.5.7)  ∏_{(B→A)∈Ob(𝒮)}F (B) → ∏_{(B→A)∈Ob(𝒮′)}F (B).

Recall that

F^{+}(A) := colim_{𝒮 ∈J_{X}(A)^{o}} F (𝒮)

and for every morphism f : A′ → A in 𝒜_{X}, the map F^{+}(f) : F^{+}(A) → F^{+}(A′) is the colimit of the maps F (𝒮) → F (𝒮 ×_{A} f) → F^{+}(A′) induced by the projection 𝒮 ×_{A} f → 𝒮, for every 𝒮 ∈ J_{X}(A)^{o}. Now, if F = i_{X}^{∧}G for a presheaf G on 𝒜, it follows from (4.1.6) that the equalizer (4.5.6) is also naturally identified with G(𝒮^{∗}), and likewise, for every inclusion of sieves 𝒮′ ⊂ 𝒮 in J_{X}(A), the map G(𝒮^{∗}) → G(𝒮^{′∗}) corresponds to the restriction of (4.5.7), under this same identification. After taking colimits, we thus get an isomorphism

(i_{X}^{∧}G)^{+}(A) →^{∼} G^{+}(A)

natural in both the presheaf G and the object A ∈ Ob(𝒜_{X}). The assertion follows.

(iv): The condition is obviously necessary. For the converse, let F be a presheaf on 𝒜 such that i_{X}^{∧}(F) is a sheaf on (𝒜_{X}, J_{X}) for every X ∈ Ob(𝒞); denote by F^{a} the sheaf on (𝒜 , J_{𝒜} ) associated with F, and by j : F → F^{a} the natural map of presheaves. For every X ∈ Ob(𝒞) we get a commutative diagram of presheaves on 𝒜_{X} :

\xymatrix{i_{X}^{∧}(F) \ar[r]^{i_{X}^{∧}(j)} \ar[d] & i_{X}^{∧}(F^{a}) \ar[d] \\ (i_{X}^{∧}F)^{a} \ar[r]^{(i_{X}^{∧}(j))^{a}} & (i_{X}^{∧}F^{a})^{a}}

whose vertical arrows are isomorphisms, by assumption. On the other hand, in view of (v), the morphism (i_{X}^{∧}(j))^{a} is naturally identified with i_{X}^{∧}(j^{a}) : i_{X}^{∧}(F^{a}) → i_{X}^{∧}((F^{a})^{a}); now, j^{a} is an isomorphism, hence the same holds for i_{X}^{∧}(j^{a}), and then also for i_{X}^{∧}(j), for every X ∈ Ob(𝒞). But this means that j is already an isomorphism, i.e. F is a sheaf on (𝒜 , J_{𝒜} ). □

Example 4.5.8. In the situation of (4.5.4), suppose that 𝒞 admits a final object X_{0}. Then the functor i_{X_{0}} is a morphism of sites (𝒜 , J_{𝒜} ) → (𝒜_{X_{0}}, J_{X}). Indeed, let A_{0} be any final object of 𝒜_{X_{0}}; it is easily seen that A_{0} is also a final object of 𝒜, and taking into account propositions 1.3.22(i) and 4.5.5(i) we deduce that i_{X_{0}} is left exact. Then the assertion follows from proposition 4.5.5(iii) and example 4.3.12.

Proposition 4.5.9. Let u : (𝒜′, p′, J′_{•}) → (𝒜 , p, J_{•}) be a morphism of fibred lex-sites over a category 𝒞, and denote by (𝒜 , J_{𝒜} ), (𝒜′, J_{𝒜′} ) the corresponding total sites. Then u is a morphism of sites (𝒜′, J_{𝒜′} ) → (𝒜 , J_{𝒜} ).

Proof. Proposition 4.5.5(i,ii) easily implies that u is continuous for the topologies J_{𝒜} and J_{𝒜′} , so it remains only to check that the fibration s : 𝒜′/u𝒜 → 𝒜′ is locally cofiltered for the topology J_{𝒜′} (proposition 4.3.9). Thus, let X ∈ Ob(𝒞) and A′ ∈ Ob(𝒜′_{X}); the fibration s_{(X)} : 𝒜′_{X}/u_{X}𝒜_{X} → 𝒜′_{X} is locally cofiltered, by proposition 4.3.9, hence there exists a covering family g_{•} := (g_{λ} : A′_{λ} → A′ | λ ∈ Λ) relative to the topology J′_{X} such that Ob(s_{(X)}^{−1}A′_{λ}) ≠ ∅ for every λ ∈ Λ; but the family g_{•} covers A′ also relative to the topology J_{𝒜′} (proposition 4.5.5(ii)), so s fulfills condition (a) of definition 4.3.3. To check condition (b), consider morphisms f_{i} : A′ → uA_{i} in 𝒜′ for i = 1, 2, and let φ_{i} : X → Y_{i} be the image of f_{i} in 𝒞 for i = 1, 2. Pick any cleavage for 𝒜, and let c be its associated pseudo-functor; then f_{i} = u(f′′_{i}) ∘ f′_{i} where f′_{i} : A′ → u(c_{φ_{i}}A_{i}) is a morphism in 𝒜′_{X} and f′′_{i} : c_{φ_{i}}A_{i} → A_{i} is a cartesian morphism of 𝒜, for i = 1, 2. Since s_{(X)} is locally cofiltered, we may then find a covering family g_{•} as in the foregoing, and for every λ ∈ Λ a morphism f_{λ} : A′_{λ} → uA_{λ} in 𝒜′_{X} and morphisms h_{i,λ} : A_{λ} → c_{φ_{1}}A_{1} in 𝒜_{X} for i = 1, 2, such that u(h_{i,λ}) ∘ f_{λ} = f′_{i} ∘ g_{λ}. Then we get morphisms in 𝒜 /u𝒜′

(A′_{λ}/f′′_{i} ∘ h_{i,λ}) : (A′_{λ}, f_{λ}) → (A′_{λ}, f_{i} ∘ g_{λ}) for i = 1, 2 and every λ ∈ Λ

whence the contention. Lastly, let (f_{i} : A′ → uA_{i} | i = 1, 2) be a pair of morphisms in 𝒜′, and h_{1}, h_{2} : A_{1} → A_{2} two morphisms of 𝒜 with u(h_{i}) ∘ f_{1} = f_{2} for i = 1, 2. Let φ_{i} : X → Y_{i} be the image of f_{i} in 𝒞 for i = 1, 2; there exist morphisms h′_{1}, h′_{2} : c_{φ_{1}}A_{1} → c_{φ_{2}}A_{2} in 𝒜_{X} and cartesian morphisms f′_{i} : c_{φ_{i}}A_{i} → A_{i} in 𝒜 such that h_{i} ∘ f′_{1} = f′_{2} ∘ h′_{i} for i = 1, 2. Since s_{(X)} is locally cofiltered, we may then find g_{•} as in the foregoing, and for every λ ∈ Λ a morphism f_{λ} : A′_{λ} → uA_{λ} in 𝒜′_{X} and a morphism h_{λ} : A_{λ} → c_{φ_{1}}A_{1} in A_{X} such that h′_{1} ∘ h_{λ} = h′_{2} ∘ h_{λ}. It follows that h_{1} ∘ f′_{1} ∘ h_{λ} = h_{2} ∘ f′_{1} ∘ h_{λ}, which shows that condition (c) holds as well for s. □

Remark 4.5.10. Let V be any universe. Proposition 4.5.9says that the rule that assigns to every fibred lex-site over 𝒞 with V-small fibres its total site defines a strict pseudo-functor

totSite : V-fib.lex.Site(𝒞) → V-Site

that assigns to every morphism of lex-sites the underlying functor, and is the identity on 2-cells.

4.5.11. Let (𝒜 , p, J_{•}) be a fibred site over a category 𝒞, and u : 𝒞′ → 𝒞 any functor. Set 𝒜′ := Fib(u)^{∗}𝒜, and recall that for every X ∈ Ob(𝒞′) the natural projection π : 𝒜′ → 𝒜 restricts to an isomorphism of categories π_{X} : 𝒜′_{X} →^{∼} 𝒜_{uX}. Then we get a fibred site denoted

𝒞′ ×_{(𝒞 ,u)} (𝒜 , p, J_{•}) or more simply 𝒞′ ×_{𝒞} (𝒜 , p, J_{•})

whose underlying fibration is p′ : 𝒜′ → 𝒞′ and where the topology J_{X} on 𝒜′_{X} is induced by J_{X} via the isomorphism π_{X}, for every X ∈ Ob(𝒞′). Obviously, if (𝒜 , p, J_{•}) is a fibred lex-site, the same holds for 𝒞′ ×_{𝒞} (𝒜 , p, J_{•}), and in this case, proposition 4.5.5(ii,iv) easily implies that π is a continuous and cocontinuous functor for the respective total sites (𝒜 , J_{𝒜} ), (𝒜′, J_{𝒜′} ).

4.5.12. In the situation of (4.5.11), let v : 𝒞′ → 𝒞 be another functor, and β : u ⇒ v any natural transformation. After choosing a cleavage λ for the fibration 𝒜, we attach to β a 𝒞′- cartesian functor Fib(β)_{𝒜}^{∗} : Fib(v)^{∗}𝒜 → Fib(u)^{∗}𝒜, as in (3.1.29). Explicitly, let c be the pseudo-functor associated with λ; then for every X ∈ Ob(𝒞′) the restriction 𝒜_{vX} → 𝒜_{uX} of Fib(β)_{𝒜}^{∗} is given by the functor c_{β_{X}}. Hence, Fib(β)_{𝒜}^{∗} is a morphism of fibred sites denoted :

β ×_{𝒞} (𝒜 , p, J_{•}) : 𝒞′ ×_{(𝒞 ,u)} (𝒜 , p, J_{•}) → 𝒞′ ×_{(𝒞 ,v)} (𝒜 , p, J_{•}).

It is easily seen that β ×_{𝒞} (𝒜 , p, J_{•}) is independent, up to isomorphisms of morphisms of fibred sites, of the choice of cleavage λ : the details shall be left to the reader.

4.5.13. Let 𝒞 , 𝒟 be two small categories, t : 𝒟 → 𝒞 a functor, and u : (𝒜_{0}, p_{0}, J_{0,•}) → (𝒜_{1}, p_{1}, J_{1,•}) a morphism of fibred lex sites over 𝒞. Set

(ℬ_{i}, q_{i}, J′_{i,•}) := 𝒟 ×_{𝒞} (𝒜_{i}, p_{i}, J_{i,•}) for i = 0, 1

(notation of (4.5.11)). Clearly u induces a morphism of fibred lex sites

v := 𝒟 ×_{𝒞} u : (ℬ_{0}, q_{0}, J′_{0,•}) → (ℬ_{1}, q_{1}, J′_{1,•}).

Then u and v are also morphisms of the respective total sites u : (𝒜_{0}, J_{0}) → (𝒜_{1}, J_{1}) and v : (ℬ_{0}, J′_{0}) → (ℬ_{1}, J′_{1}) (proposition 4.5.9). For i = 0, 1, let π_{i} : ℬ_{i} → 𝒜_{i} be the projection. We deduce a commutative diagram of categories :

ℰ  :  \xymatrix{𝒜_{0}^{∼} \ar[r]^{ũ_{∗}} \ar[d]_{π̃_{0∗}} & 𝒜_{1}^{∼} \ar[d]^{π̃_{1∗}} \\ ℬ_{0}^{∼} \ar[r]^{ṽ_{∗}} & ℬ_{1}^{∼}}

and notice that each functor of the diagram ℰ admits a left adjoint; after choosing an adjunction for each of the resulting adjoint pairs of functors, we can view ℰ as a square of links, oriented by the identity 1_{π̃_{1∗}∘ũ_{∗}} (see (2.3.8)).

Proposition 4.5.14. In the situation of (4.5.13), suppose that (𝒜_{i,X}, J_{i,X}) is a U-site for every X ∈ Ob(𝒞) and i = 0, 1. Then the base change transformation

Υ(ℰ) : ṽ^{∗} ∘ π̃_{1∗} → π̃_{0∗} ∘ ũ^{∗}

is an isomorphism of functors.

Proof. We consider first the oriented diagram of categories :

ℰ′  :  [figure: 𝒜_{0}^{∧}; u^{∧}; π_{0}^{∧}; 𝒜_{1}^{∧}; π_{1}^{∧}; 𝒜_{0}^{∼}; ũ_{∗}; π̃_{0∗}; i_{𝒜_{0}}; 𝒜_{1}^{∼}; π̃_{1∗}; i_{𝒜_{1}}; ℬ_{0}^{∼}; ṽ_{∗}; i_{ℬ_{0}}; ℬ_{1}^{∼}; i_{ℬ_{1}}; ℬ_{0}^{∧}; v^{∧}; ℬ_{1}^{∧}]

whose diagonal arrows are the inclusion functors, and all whose orientations are identities. Moreover, each functor in ℰ′ still admits a left adjoint, so as usual ℰ′ can be regarded as an oriented diagram of links. By adding a further identity orientation for the “front face” we obtain a cubical diagram as in (2.3.21), which obviously commutes on 2-cells, except that the orientations do not agree with those of the corresponding diagram in (2.3.21), nor with those of its variant considered in remark 2.3.22(iii). However, the induced diagram^{†}ℰ′ of adjoint squares as in (2.3.8) still commutes on 2-cells, due to proposition 2.3.4, and moreover its orientations agree with those of the variant from remark 2.3.22(iii). More precisely, the left and right faces of^{†}ℰ′ are oriented by^{o}(1_{i_{ℬ_{1}} ∘ṽ_{∗}})^{†} and respectively^{o}(1_{i_{𝒜_{1}} ∘ũ_{∗}})^{†}, and the top and bottom faces are oriented by^{o}(1_{i_{ℬ_{1}} ∘π̃_{1∗}})^{†} and respectively^{o}(1_{i_{ℬ_{0}} ∘π̃_{0∗}})^{†}.

Claim 4.5.15. The base change transformations

Υ(1_{i_{ℬ_{1}} ∘π̃_{1∗}}) : (−)_{ℬ_{1}}^{a} ∘ π_{1}^{∧} → π̃_{1∗} ∘ (−)_{𝒜_{1}}^{a} and Υ(1_{i_{ℬ_{0}} ∘π̃_{0∗}}) : (−)_{ℬ_{0}}^{a} ∘ π_{0}^{∧} → π̃_{0∗} ∘ (−)_{𝒜_{0}}^{a}

are isomorphisms of functors.

Proof of the claim. We check the assertion for Υ(1_{i_{ℬ_{0}} ∘π̃_{0∗}}); the same argument shall apply also to Υ(1_{i_{ℬ_{1}} ∘π̃_{1∗}}). Since i_{ℬ_{0}} is a fully faithful functor, the counit ε^{ℬ_{0}} of the chosen adjunction for the pair ((−)_{ℬ_{0}}^{a} , i_{ℬ_{0}} ) is an isomorphism (proposition 1.1.20(iii)), so we are reduced to checking that ((−)_{ℬ_{0}}^{a} ∘ π_{0}^{∧}) ∗ η^{𝒜_{0}} is an isomorphism, where η^{𝒜_{0}} and ε^{𝒜_{0}} are the unit and counit of an adjunction for the pair ((−)_{𝒜_{0}}^{a} , i_{𝒜_{0}} ). But recall that there exists an isomorphism of functors : (−)_{ℬ_{0}}^{a} ∘ π_{0}^{∧} →^{∼} π̃_{0∗} ∘ (−)_{𝒜_{0}}^{a} (corollary 4.3.19(iii)), so it suffices to show that (−)_{𝒜_{0}}^{a} ∗ η^{𝒜_{0}} is an isomorphism. However, ε^{𝒜_{0}} is an isomorphism, since i_{𝒜_{0}} is fully faithful, hence the contention follows from the triangular identities of (1.1.13). ♢

In light of claim 4.5.15, proposition 2.3.10, and remark 2.3.22(i,iii), we are then reduced to showing that the base change transformation

Υ(1_{π_{1}^{∧}∘u^{∧}}) : v_{!} ∘ π_{1}^{∧} → π_{0}^{∧} ∘ u_{!}

is an isomorphism of functors. To this aim, recall that for every A_{0} ∈ Ob(𝒜_{0}) and every presheaf F on 𝒜_{1}, the set u_{!}F (A_{0}) represents the colimit of the functor

F ∘ t_{A_{0}}^{o} : (A_{0}/u𝒜_{1})^{o} → Set

where t_{A_{0}} denotes the usual target functor as in (1.1.27). Thus, u_{!}F (A_{0}) is the set of equivalence classes [s, φ]_{u} of pairs (s, φ), where φ : A_{0} → uA_{1} is a morphism of 𝒜_{0}, and s ∈ F A_{1}. For every morphism ψ : A_{0} → A′_{0} in 𝒜_{0}, the map u_{!}F (ψ) is given by the rule : [s, φ]_{u} ↦ [s, φ ∘ ψ]_{u} for every such [s, φ]_{u}. Moreover, for every morphism f : F → F′ of presheaves on 𝒜_{0}, the morphism u_{!}f : u_{!}F → u_{!}F′ is given by the rule : [s, φ]_{u} ↦ [f_{A_{1}}(s), φ]_{u} for every such [s, φ]_{u}. We have a standard adjunction for the pair (u_{!}, u^{∧}) whose unit and counit (η^{u}, ε^{u}) are given by the following rules, for every presheaf F on 𝒜_{1} and F′ on 𝒜_{0} :

η_{F,A_{1}}^{u}(s) := [s, 1_{uA_{1}}]_{u} ∈ u^{∧}u_{!}F (A_{1})  for every A_{1} ∈ Ob(𝒜_{1}) and every s ∈ F A_{1}
ε_{F′,A_{0}}^{u}[s, A_{0} ⟶^{φ} uA′_{1}]_{u} := F′φ(s) ∈ F′A_{0} for every s ∈ F (uA′_{1}).

A similar description applies to the functor v_{!} and the unit and counit of its standard adjunction. Thus, for every such F, the morphism (v_{!}π_{1}^{∧} ∗ η^{u})_{F} : v_{!}π_{1}^{∧}F → v_{!}π_{1}^{∧}u^{∧}u_{!}F is given by the rule:

[s, B_{0} ⟶^{φ} vB_{1}]_{v} ↦ [[s, 1_{uπ_{1}B_{1}}]_{u}, φ]_{v} for every s ∈ F π_{1}B_{1}

and the morphism (ε^{v} ∗ π_{0}^{∧}u_{!})_{F} : v_{!}v^{∧}π_{0}^{∧}u_{!}F → π_{0}^{∧}u_{!}F is given by the rule :

[[t, π_{0}vA′_{1} ⟶^{ψ} uA′′_{1}]_{u}, A_{0} ⟶^{φ} vA′_{1}]_{v} ↦ π_{0}^{∧}u_{!}F (φ)[t, π_{0}vA′_{1} ⟶^{ψ} uA′′_{1}]_{u} = [t, ψ ∘ π_{0}(φ)]_{u}

for every t ∈ F A′′_{1}. Summing up, Υ(1_{π_{1}^{∧}∘u^{∧}}) is then given by the rule :

[s, B_{0} ⟶^{φ} vB_{1}]_{v} ↦ [s, π_{0}(φ)]_{u} for every s ∈ F π_{1}B_{1}.

Consider then the functor

ρ^{B_{0}} : (B_{0}/vℬ_{1})^{o} → (π_{0}B_{0}/u𝒜_{1})^{o} (B_{0} ⟶^{φ} vB_{1}) ↦ (π_{0}B_{0} ⟶^{π_{0}(φ)} π_{0}vB_{1} = uπ_{1}B_{1})

and notice that t_{π_{0}B_{0}}^{o} ∘ ρ^{B_{0}} = π_{1}^{o} ∘ t_{B_{0}}^{o}, so that ρ^{B_{0}} induces a morphism of presheaves on the category Set^{o}, as in (the dual of) remark 1.2.11(i) :

colim_{(π_{0}B_{0}/u𝒜_{1})^{o}} F ∘ t_{π_{0}B_{0}}^{o} → colim_{(ℬ_{0}/vℬ_{1})^{o}} F ∘ π_{1}^{o} ∘ t_{B_{0}}^{o}

which corresponds to a map ω : v_{!}π_{1}^{∧}F (B_{0}) → π_{0}^{∧}u_{!}F (B_{0}), and by a direct inspection it is easily seen that ω = Υ(1_{π_{1}^{∧}∘u^{∧}})_{F,B_{0}}. In view of proposition 1.5.2, we are then reduced to showing :

Claim 4.5.16. For every X ∈ Ob(ℬ_{0}) we have :

(i) The categories X/vℬ_{1} and π_{0}X/u𝒜_{1} are cofiltered.

(ii) The functor ρ^{X} is cofinal.

Proof of the claim. (i): We show that X/vℬ_{1} is cofiltered; the same argument shall apply to π_{0}X/u𝒜_{1} as well. Indeed, let us check that X/vℬ_{1} is not the empty category : to this aim, let v_{q_{0}X} : ℬ_{1,q_{0}X} → ℬ_{0,q_{0}X} be the restriction of v to the fibre categories over q_{0}X, so it suffices to check that X/v_{q_{0}X}ℬ_{1,q_{0}X} is not empty; but ℬ_{1,q_{0}X} is finitely complete, hence it admits a final object Y_{0}, and vY_{0} is a final object of ℬ_{0,q_{0}X}, since v is left exact. The unique morphism X → vY_{0} is then an object of X/v_{q_{0}X}ℬ_{1,q_{0}X}.

Next, consider two objects (Y_{1}, φ_{1}), (Y_{2}, φ_{2}) of X/vℬ_{1}; for i = 1, 2 we may find cartesian morphisms β_{i} : Y′_{i} → Y_{i} in ℬ_{1} such that q_{1}(β_{i}) = q_{0}(φ_{i}), and then there exists a unique morphism φ′_{i} : X → vY′_{i} in the fibre category ℬ_{0,q_{0}X} such that v(β_{i}) ∘ φ′_{i} = φ_{i}. The pair (Y′_{i}, φ′_{i}) is an object of X/vℬ_{1}, and β_{i} is a morphism (Y′_{i}, φ′_{i}) → (Y_{i}, φ_{i}) for i = 1, 2. Since ℬ_{1,q_{0}X} is finitely complete, the product Z := Y′_{1} × Y′_{2} is representable, and since v_{q_{0}X} is left exact, vZ represents vY′_{1}×vY′_{2}, whence a morphism ψ′ : X → vZ in ℬ_{0,q_{0}X} whose composition with the projection v(p_{i}) : vZ → vY′_{i} agrees with φ′_{i}, for i = 1, 2. Thus, (Z, ψ′) ∈ Ob(X/vℬ_{1})

and we have morphisms (Z, ψ′) ⟶^{X/p_{i}} (Y′_{i}, φ′_{i}) ⟶^{X/β_{i}} (Y_{i}, φ_{i}) for i = 1, 2.

Lastly, let X/α_{1}, X/α_{2} : (Y_{1}, φ_{1}) → (Y_{2}, φ_{2}) be two morphisms of X/vℬ_{1}. We find cartesian morphisms β_{i} : Y′_{i} → Y_{i} in ℬ_{1} such that q_{1}(β_{i}) = q_{0}(φ_{i}) for i = 1, 2, and notice that q_{1}(α_{1} ∘β_{1}) = q_{0}(φ_{2}) = q_{1}(α_{2} ∘β_{1}). Then there exist unique morphisms α′_{i} : Y′_{1} → Y′_{2} in ℬ_{1,q_{0}X} such that β_{2} ∘ α′_{i} = α_{i} ∘ β_{1} for i = 1, 2. Since ℬ_{1,q_{0}X} is finitely complete, the equalizer of the pair (α′_{1}, α′_{2}) is representable in ℬ_{1,q_{0}X} by a morphism λ′ : E → Y′_{1} in ℬ_{1,q_{0}X}. Set λ := β_{1} ∘ λ′; it follows that α_{1} ∘ λ = α_{2} ∘ λ. Moreover, for i = 1, 2 we may find a morphism φ′_{i} : X → vY′_{i} in ℬ_{0,q_{0}X} such that v(β_{i}) ∘ φ′_{i} = φ_{i}. We compute :

v(β_{2}∘α′_{1})∘φ′_{1} = v(α_{1}∘β_{1})∘φ′_{1} = v(α_{1})∘φ_{1} = φ_{2} = v(α_{2})∘φ_{1} = v(α_{2}∘β_{1})∘φ′_{1} = v(β_{2}∘α′_{2})∘φ′_{1}

whence v(α′_{1}) ∘ φ′_{1} = v(α′_{2}) ∘ φ′_{1}, since v(β_{2}) is cartesian. However, v(λ) is the equalizer of the pair (v(α′_{1}), v(α′_{2})), since v_{q_{0}X} is left exact; so finally φ′_{1} factors through a unique morphism φ′′_{1} : X → vE and v(λ′) : vE → vY′_{1}. Thus, we get a morphism X/λ : (E, φ′′_{1}) → (Y_{1}, φ_{1}) in X/vℬ_{1} such that (X/α_{1}) ∘ (X/λ) = (X/α_{2}) ∘ (X/λ), whence the contention.

(ii): In view of (i), it suffices to show that conditions (a) and (b) of lemma 1.5.7(i) hold for ρ^{X}. To this aim, let u_{q_{0}X} : 𝒜_{1,q_{0}X} → 𝒜_{0,q_{0}X} be the restriction of u to the fibre categories, and (Y, φ) any object of (π_{0}X/u𝒜_{1})^{o}. Arguing as in the proof of (i), we find (Y′, φ′) ∈ Ob((π_{0}X/u_{q_{0}X}𝒜_{1,q_{0}X})^{o}) and a morphism (Y, φ) → (Y′, φ′) in (π_{0}X/u𝒜_{1})^{o}. But notice that ρ^{X} restricts to an isomorphism of categories (X/v_{q_{0}X}ℬ_{1,q_{0}X})^{o} →^{∼} (π_{0}X/u_{q_{0}X}𝒜_{1,q_{0}X})^{o}. Condition (a) is an immediate consequence. Lastly, let (Y, φ) ∈ Ob(X/vℬ_{1}), (Z, φ′) ∈ Ob(π_{0}X/u𝒜_{1}), and a pair of morphisms in π_{0}X/u𝒜_{1} :

π_{0}X/β_{i} : ρ^{X}(Y, φ) = (π_{1}Y, π_{0}(φ)) → (Z, φ′) i = 1, 2.

Hence, β_{1}, β_{2} : π_{1}Y → Z are morphisms of ℬ_{1} with u(β_{1}) ∘ π_{0}(φ) = u(β_{2}) ∘ π_{0}(φ). Arguing as in the proof of (i) we find a cartesian morphism β : Y′ → Y in ℬ_{1} and a morphism ψ : X → vY′ in ℬ_{0,q_{0}X} with v(β) ∘ ψ = φ. Notice that p_{1}(β_{1} ∘ π_{1}(β)) = p_{0}(φ′) = p_{1}(β_{2} ∘ π_{1}(β)). Then we may find Z′′ ∈ Ob(𝒜_{1,q_{0}X}) and a cartesian morphism γ : Z′′ → Z in 𝒜_{1} such that p_{1}(γ) = p_{0}(φ′), and for i = 1, 2, a morphism β′′_{i} : π_{1}Y′ → Z′ in 𝒜_{1,q_{0}X} such that β_{1} ∘ π_{1}(β) = γ ∘ β′′_{i}. Since π_{1} restricts to an isomorphism ℬ_{1,q_{0}X} →^{∼} 𝒜_{1,q_{0}X}, there exist Z′ ∈ Ob(ℬ_{1,q_{0}X}) with π_{1}Z′ = Z′′, and unique morphisms β′_{i} : Y′ → Z′ in ℬ_{1,q_{0}X} such that π_{1}(β′_{i}) = β′′_{i} for i = 1, 2. We compute :

u(γ) ∘ uπ_{1}(β′_{1}) ∘ π_{0}(ψ) = u(β_{1} ∘ π_{1}(β)) ∘ π_{0}(ψ) = u(β_{1}) ∘ π_{0}(φ)
= u(β_{2}) ∘ π_{0}(φ)
= u(β_{2} ∘ π_{1}(β)) ∘ π_{0}(ψ)
= u(γ) ∘ uπ_{1}(β′_{2}) ∘ π_{0}(ψ).

Since u(γ) is cartesian and π_{0} restricts to an isomorphism ℬ_{0,q_{0}X} →^{∼} 𝒜_{0,q_{0}X} , we conclude that v(β′_{1}) ∘ ψ = v(β′_{2}) ∘ ψ. Let λ : E → Y′ be the equalizer of the pair (β′_{1}, β′_{2}) in ℬ_{1,q_{0}X}; since v_{q_{0}X} is left exact, v(λ) : vE → vY′ represents the equalizer of (v(β′_{1}), v(β′_{2})) in 𝒜_{1,q_{0}X}, so finally ψ factor through v(λ) and a unique morphism μ : X → vE in 𝒜_{1,q_{0}X}. We have (E, μ) ∈ Ob(X/vℬ_{1}), and a morphism X/(β ∘ λ) : (E, μ) → (Y, φ) in X/vℬ_{1} such that ρ^{X}(X/(β ∘ λ)) equalizes π_{0}X/β_{1} and π_{0}X/β_{2} in (π_{0}X/u𝒜_{1})^{o}. This concludes the proof of condition (b). □

4.6. Fibred topoi. For every small category I, we consider next the 2-category

PsFun(I, Topos)

whose objects T : I → Topos shall be called the fibred topoi over I. The 1-cells, that is the pseudo-natural transformations ω : T ⇒ S shall be called morphisms of fibred topoi over I, and denoted by a simple arrow T → S. The 2-cells, i.e. the modifications ω ⇝ ω′ in PsFun(I, Topos) shall be called also transformations of morphisms of fibred topoi over I.

Remark 4.6.1. (i) Recall that the definition of the 2-category Topos involves the choice of two universes U, U′ with U ∈ U′, so that every object of Topos is both a U-topos and a small U′-category (see definition 4.4.5(iv)). Then, consider any pseudo-functor

F : I → U′-Cat

such that (a): F_{i} is an U-topos for every i ∈ Ob(I), and (b): for every morphism φ : i → j of I there exists a morphism of topoi f_{φ} : F_{i} → F_{j} with F_{φ} = f_{φ∗}. Then we get a fibred topos T over I such that T_{i} := F_{i} for every i ∈ Ob(I) and T_{φ} := f_{φ} for every morphism φ of I. This fibred topos depends on the choice of the morphisms of topoi f_{φ}, but any two such choices yield isomorphic fibred topoi over I.

(ii) In particular, any functor F : I → U′-Cat fulfilling (a) and (b) can be regarded as a fibred topos over I; in this case, the coherence constraint (δ^{T} , γ^{T}) of the resulting T shall be given by identity automorphisms of functors :

δ_{i}^{T} = 1_{T_{i}}  γ_{φ,ψ}^{T} = 1_{T_{φ}∘T_{ψ}}  in U′-Cat

which are however not necessarily identity 2-cells of the 2-category Topos, since one does not necessarily have f_{ψ}^{∗} ∘ f_{φ}^{∗} = f_{ψ∘φ}^{∗}, nor f_{1_{i}}^{∗} = 1_{T_{i}}. Hence, even when it is given by an actual functor, T will only in general be a (non-strict) pseudo-functor, when regarded as a fibred topos.

(iii) From the pseudo-functors of remark 4.4.9(ii) we deduce two strict pseudo-functors :

PsFun(I^{o}, U′-Cat)^{o} ⟵^{[−]^{∗}} PsFun(I, Topos) ⟶^{[−]_{∗}} PsFun(I, U′-Cat).

• Namely, [−]_{∗} assigns to every fibred topos T : I → Topos over I the composition

T_{∗} := (−)_{∗} ∘ T : I → U′-Cat  i ↦ T_{i}  (i ⟶^{φ} j) ↦ (T_{φ∗} : T_{i} → T_{j})

and to every morphism of fibred topoi ω : T → S the pseudo-natural transformation ω_{∗} := (−)_{∗} ∗ ω : T_{∗} ⇒ S_{∗}. To every transformation Ξ : ω ⇝ ω′ it assigns the modification

Ξ_{∗} := (−)_{∗} ∘ Ξ : ω_{∗} ⇝ ω′_{∗}.

• Likewise, [−]^{∗} assigns to every fibred topos T over I the pseudo-functor

T^{∗} := ((−)^{∗} ∘ T)^{o} : I^{o} → U′-Cat  i^{o} ↦ T_{i}^{o}  (j^{o} ⟶^{φ^{o}} i^{o}) ↦ (T_{φ}^{∗o} : T_{j}^{o} → T_{i}^{o})

and to every morphism of fibred topoi ω : T → S the pseudo-natural transformation ω^{∗} := ((−)^{∗} ∗ ω)^{o} : S^{∗} ⇒ T^{∗}. To every transformation Ξ : ω ⇝ ω′ it assigns the modification

Ξ^{∗} := ((−)^{∗} ∘ Ξ)^{o} : ω^{∗} ⇝ ω^{′∗}.

4.6.2. There are several useful ways of attaching a fibration to any fibred topos T over I : namely, with the notation of (3.1.18) and remark 4.6.1(iii), we can consider the fibrations

π : ℱ ib(T_{∗}) → I^{o} and π′ : ℱ ib(T^{∗}) → I.

More precisely, we have strict pseudo-functors :

U′-Fib(I)^{o} ⟵^{ℱ ib_{I}^{o}∘[−]^{∗}} PsFun(I, Topos) ⟶^{ℱ ib_{I^{o}}∘[−]_{∗}} U′-Fib(I^{o}).

• We have also a natural pseudo-functor from fibred topoi over I to fibred sites over I : namely, the composition

\underline{Can} : PsFun(I, Topos) ⟶^{PsFun(I,Can)} PsFun(I, U′-Site) ⟶^{\underline{ℱ ib} _{I}} U′-fib.Site(I)

where Can : Topos → Site is the pseudo-functor of (4.4.15), and \underline{ℱ ib} _{I} is the 2-equivalence of remark 4.5.2(v). Explicitly, \underline{Can}(T) is given as follows. Since I is a usual category, we have oI^{o} = I^{o}, so T^{∗} induces a pseudo-functor

I^{o} ⟶^{^{o}T^{∗}} ^{o}(U′-Cat) ⟶^{(−)^{o}} U′-Cat  i^{o} ↦ T_{i}  (j^{o} ⟶^{φ} i^{o}) ↦ (T_{φ}^{∗} : T_{j} → T_{i})

where (−)^{o} denotes the strict isomorphism of example 2.1.24. The fibration

p : \underline{Can}(T) := ℱ ib((−)^{o} ∘^{o}T^{∗}) → I

carries a natural structure of fibred lex-site : namely, the category p^{−1}(i) is naturally identified with T_{i} for every i ∈ Ob(I), and inherits the latter’s canonical topology; also, p admits a natural cleavage whose associated pseudo-functor c is identified with (−)^{o}∘^{o}T^{∗}, so for every morphism φ : i → j of I the functor c_{φ} : p^{−1}(j) → p^{−1}(i) is in turn identified with the left exact functor T_{φ}^{∗} : T_{j} → T_{i}, which is continuous for the canonical topologies (remark 4.4.13(iii)).

• Conversely, to every fibred site we may attach a fibred topos, via the pseudo-functors

\underline{T} : U′-fib.Site(I) ⟶^{\underline{c}^{•}} PsFun(I, U′-Site) ⟶^{PsFun(I,T)} PsFun(I, Topos)
\underline{lex.T} : U′-fib.lex.Site(I) ⟶^{lex.\underline{c}^{•}} PsFun(I, U′-lex.Site) ⟶^{PsFun(I,lex.T)} PsFun(I, Topos)

where T : Site → Topos and lex.T : lex.Site → Topos are as in (4.4.15), and \underline{c}^{•} (resp. lex.\underline{c}^{•}) is a strict and strong pseudo-inverse for \underline{ℱ ib} _{I} (resp. for lex.\underline{ℱ ib} _{I} : see remark 4.5.2(vi)).

Lemma 4.6.3. With the notation of (4.6.2), there exists a natural isomorphism of I-categories

ℱ ib(T^{∗}) →^{∼} ℱ ib(T_{∗})^{o}.

Proof. For every morphism φ : i → j of I, let (η_{φ}, ε_{φ}) be the unit and counit for the adjunction for the pair (T_{φ}^{∗}, T_{φ∗}) that defines the morphism of topoi T_{φ} : T_{i} → T_{j}. This adjunction assigns to every morphism f : T_{φ}^{∗}Y → X (for any X ∈ Ob(T_{i}) and Y ∈ Ob(T_{j})) an adjoint morphism

f^{†} := T_{φ∗}(f) ∘ η_{φ,Y} : Y → T_{φ∗}X.

Notice that ℱ ib(T^{∗}) and ℱ ib(T_{∗}) have the same set of objects, and the morphisms of ℱ ib(T^{∗}) (resp. ℱ ib(T_{∗})) are the pairs (φ, f) where φ : i → j is a morphism of I and f : T_{φ}^{∗}Y → X (resp. f : Y → T_{φ∗}X) is a morphism of T_{i} (resp. of T_{j}). We claim that the rules

(i, X) ↦ (i, X) (φ, f) ↦ (φ, f^{†})

yield the sought isomorphism of categories. Indeed, denote also by (δ, γ) the coherence constraint of the pseudo-functor T^{∗}; hence

γ_{(φ,ψ)} : T_{ψφ}^{∗} →^{∼} T_{φ}^{∗}T_{ψ}^{∗} and δ_{i} : T_{1_{i}}^{∗} →^{∼} 1_{T_{i}}

are isomorphisms of functors, for every i ∈ Ob(I) and every pair of morphisms φ : i → j and ψ : j → k in I, and according to remark 1.1.17(ii), there follow adjoint isomorphisms

γ_{(φ,ψ)}^{†} : T_{ψ∗}T_{φ∗} →^{∼} T_{ψ∘φ∗} and δ_{i}^{†} : 1_{T_{i}} →^{∼} T_{1_{i}∗}.

With this notation, we need to check the identity

(f ∘ T_{φ}^{∗}g ∘ γ_{(φ,ψ),Z})^{†} = γ_{(φ,ψ),X}^{†} ∘ T_{ψ∗}(f^{†}) ∘ g^{†}

for every pair of morphisms (φ, f : T_{φ}^{∗}Y → X) and (ψ, g : T_{ψ}^{∗}Z → Y) of ℱ ib(T^{∗}). However, on the one hand we have

(f ∘ T_{φ}^{∗}(g) ∘ γ_{(φ,ψ),Z})^{†} = T_{ψφ∗}(f ∘ T_{φ}^{∗}g ∘ γ_{(φ,ψ),Z}) ∘ η_{ψφ,Z}

and on the other hand :

\begin{matrix}γ_{(φ,ψ),X}^{†} ∘ T_{ψ∗}(f^{†}) ∘ g^{†} = & γ_{(φ,ψ),X}^{†} ∘ T_{ψ∗}T_{φ∗}(f) ∘ T_{ψ∗}(η_{φ,Y}) ∘ T_{ψ∗}(g) ∘ η_{ψ,Z} \\ = & T_{ψφ∗}(f) ∘ γ_{(φ,ψ),T_{φ}^{∗}Y}^{†} ∘ T_{ψ∗}T_{φ∗}T_{φ}^{∗}(g) ∘ T_{ψ∗}(η_{φ,T_{ψ}^{∗}Z}) ∘ η_{ψ,Z} \\ = & T_{ψφ∗}(f) ∘ γ_{(φ,ψ),T_{φ}^{∗}Y}^{†} ∘ T_{ψ∗}T_{φ∗}T_{φ}^{∗}(g) ∘ η_{(φ,ψ),Z} \end{matrix}

where (η_{(φ,ψ)}, ε_{(φ,ψ)}) denotes the unit and counit of the adjunction that defines the composition T_{ψ} ∘ T_{φ} : T_{i} → T_{k}, by virtue of remark 1.1.17(i). Thus, we are reduced to checking that :

T_{ψφ∗}(T_{φ}^{∗}g ∘ γ_{(φ,ψ),Z}) ∘ η_{ψφ,Z} = γ_{(φ,ψ),T_{φ}^{∗}Y}^{†} ∘ T_{ψ∗}T_{φ∗}T_{φ}^{∗}(g) ∘ η_{(φ,ψ),Z}.

But we have γ_{(φ,ψ),T_{φ}^{∗}Y}^{†} ∘ T_{ψ∗}T_{φ∗}T_{φ}^{∗}(g) = T_{ψφ∗}T_{φ}^{∗}(g) ∘ γ_{(φ,ψ),T_{φ}^{∗}T_{ψ}^{∗}Z}^{†}, so we are further reduced to showing the identity :

T_{ψφ∗}(γ_{(φ,ψ),Z}) ∘ η_{ψφ,Z} = γ_{(φ,ψ),T_{φ}^{∗}T_{ψ}^{∗}Z}^{†} ∘ η_{(φ,ψ),Z}.

But using the explicit expressions of remark 1.1.17(ii) we see that :

γ_{(φ,ψ)}^{†} ∗ T_{φ}^{∗}T_{ψ}^{∗} = (T_{ψφ∗} ∗ ε_{(φ,ψ)} ∗ T_{φ}^{∗}T_{ψ}^{∗})⊙(T_{ψφ∗} ∗ γ_{(φ,ψ)} ∗ T_{ψ∗}T_{φ∗}T_{φ}^{∗}T_{ψ}^{∗})⊙(η_{ψφ} ∗ T_{ψ∗}T_{φ∗}T_{φ}^{∗}T_{ψ}^{∗})

and on the other hand

(η_{ψφ} ∗ T_{ψ∗}T_{φ∗}T_{φ}^{∗}T_{ψ}^{∗}) ⊙ η_{(φ,ψ)} = (T_{ψφ∗}T_{ψφ}^{∗} ∗ η_{(φ,ψ)}) ⊙ η_{ψφ}

so it suffices to check that :

γ_{(φ,ψ)} = (ε_{(φ,ψ)} ∗ T_{φ}^{∗}T_{ψ}^{∗}) ⊙ (γ_{(φ,ψ)} ∗ T_{ψ∗}T_{φ∗}T_{φ}^{∗}T_{ψ}^{∗}) ⊙ (T_{ψφ}^{∗} ∗ η_{(φ,ψ)}).

The latter follows from the identity :

(γ_{(φ,ψ)} ∗ T_{ψ∗}T_{φ∗}T_{φ}^{∗}T_{ψ}^{∗}) ⊙ (T_{ψφ}^{∗} ∗ η_{(φ,ψ)}) = (T_{φ}^{∗}T_{ψ}^{∗} ∗ η_{(φ,ψ)}) ⊙ γ_{(φ,ψ)}

together with the triangular identities for (η_{(φ,ψ)}, ε_{(φ,ψ)}). □

Definition 4.6.4. (i) For any two categories 𝒜 , ℬ and any functor F : 𝒜 → ℬ we denote

Σ(𝒜 /ℬ)

the category of sections of F, which is the subcategory of Fun(ℬ, 𝒜) whose objects are the functors G : ℬ → 𝒜 such that F ∘ G = 1_{ℬ} and whose morphisms are the natural transformations β : G ⇒ G′ with F ∗ β = i_{ℬ} (where i_{ℬ} is the identity automorphism of 1_{ℬ} ).

(ii) Let T be a fibred topos over I, and π, π′ its associated fibrations as in (4.6.2); we let :

Top(T)_{∗} := Σ(ℱ ib(T_{∗})/I^{o}) and Top(T) := Σ(ℱ ib(T^{∗})/I)^{o}.

By virtue of lemma 4.6.3and remark 1.1.19(i) we have a natural isomorphism of categories :

(4.6.5)  Top(T) →^{∼} Top(T)_{∗}.

4.6.6. Let ω : T → S be any morphism of fibred topoi over I. From the pseudo-natural transformations ω^{∗} : S^{∗} ⇒ T^{∗} and ω_{∗} : T_{∗} ⇒ S_{∗} we get the functors

\begin{aligned}ℱ ib(ω^{∗}) & : ℱ ib(S^{∗}) → ℱ ib(T^{∗}) \\ ℱ ib(ω_{∗}) & : ℱ ib(T_{∗}) → ℱ ib(S_{∗}) \end{aligned}

(notation of (3.1.19)); whence functors

Top(ω)^{∗} : Top(S) → Top(T)  E ↦ ℱ ib(ω^{∗}) ∘ E
Top(ω)_{∗} : Top(T)_{∗} → Top(S)_{∗}  E ↦ ℱ ib(ω_{∗}) ∘ E.

By virtue of (4.6.5) we may then identify Top(ω)_{∗} with a functor which we shall also denote

Top(ω)_{∗} : Top(T) → Top(S).

Moreover, ω also induces a morphism of fibred sites

\underline{Can}(ω) := ℱ ib((−)^{o} ∗^{o}ω^{∗}) : \underline{Can}(T) → \underline{Can}(S).

Let ω : T → S and ν : S → U be two morphisms of fibred topoi over I. A simple inspection of the definition shows that

(4.6.7)

Top(ω)^{∗} ∘ Top(ν)^{∗} = Top(ν ∘ ω)^{∗}
Top(ν)_{∗} ∘ Top(ω)_{∗} = Top(ν ∘ ω)_{∗}
\underline{Can}(ν) ∘ \underline{Can}(ω) = \underline{Can}(ν ∘ ω).

4.6.8. Let ω, ω′ : T → S be two morphisms of fibred topoi over I, and Ξ : ω ⇝ ω′ a transformation. Then Ξ induces natural transformations

ℱ ib(Ξ_{∗}) : ℱ ib(ω_{∗}) ⇒ ℱ ib(ω′_{∗}) ℱ ib(Ξ^{∗}) : ℱ ib(ω^{∗}) ⇒ ℱ ib(ω^{′∗})

whence natural transformations :

Top(Ξ)^{∗} : Top(ω′)^{∗} ⇒ Top(ω)^{∗}  E ↦ ℱ ib(Ξ^{∗}) ∗ E
Top(Ξ)_{∗} : Top(ω)_{∗} ⇒ Top(ω′)_{∗}  E ↦ ℱ ib(Ξ_{∗}) ∗ E.

Moreover, Ξ induces a natural transformation of morphisms of fibred sites :

\underline{Can}(Ξ) := ℱ ib((−)^{o} ∘^{o}Ξ^{∗}) : \underline{Can}(ω′) → \underline{Can}(ω).

Remark 4.6.9. (i) Let T be a fibred topos over I with coherence constraint (δ^{T} , γ^{T}), and T^{u} : I → Topos the unital pseudo-functor associated with T (proposition 2.4.3); by (4.6.7) and remark 2.2.14(i), the pseudo-natural isomorphism α^{T} : T →^{∼} T^{u} induces an isomorphism of categories Top(α^{T})^{∗} : Top(T^{u}) →^{∼} Top(T). Then the following description of Top(T^{u}) – obtained by direct inspection – also applies to Top(T), up to natural isomorphism of categories:

• the objects of Top(T^{u}) are the systems E_{•} := ((E_{i}, E_{φ}) | i ∈ Ob(I), φ ∈ Morph(I)) with E_{i} ∈ Ob(T_{i}) for every i ∈ Ob(I) and where

E_{φ} : T_{φ}^{∗}E_{j} → E_{i} for every φ : i → j in I

is a morphism of T_{i}, such that the following diagram commutes :

\xymatrix{T_{ψ}^{∗}T_{φ}^{∗}E_{k} \ar[r]^{T_{ψ}^{∗}E_{φ}} & T_{ψ}^{∗}E_{j} \ar[d]^{E_{φ}} \\ T_{ψ∘φ}^{∗}E_{k} \ar[r]^{E_{ψ∘φ}} \ar[u]^{(γ_{(φ,ψ)}^{T^{u}})_{E_{k}}^{†}} & E_{i}}

for every pair of morphisms i ⟶^{φ} j ⟶^{ψ} k of I, and E_{1_{i}} = 1_{E_{i}} for every i ∈ Ob(I).

• the morphisms f_{•} : E_{•} → F_{•} are the systems (f_{i} | i ∈ Ob(I)) where f_{i} : E_{i} → F_{i} is a morphism of T_{i} for every i ∈ Ob(I), such that the following diagram commutes :

\xymatrix{T_{φ}^{∗}E_{j} \ar[r]^{T_{φ}^{∗}f_{j}} \ar[d]_{E_{φ}} & T_{φ}^{∗}F_{j} \ar[d]^{F_{φ}} \\ E_{i} \ar[r]^{f_{i}} & F_{i}} for every morphism φ : i → j of I.

• the composition of morphisms f_{•} : E_{•} → F_{•} and g_{•} : F_{•} → G_{•} is defined by the

obvious rule : (g_{•} ∘ f_{•})_{i} := g_{i} ∘ f_{i} for every i ∈ Ob(I). (ii) Likewise, notice that for every unital fibred topos T over I, the objects of Top(T)_{∗} are

the systems F_{•} := (F_{i}, F_{φ} | i ∈ Ob(I), φ ∈ Morph(I)) with F_{i} ∈ Ob(T_{i}) for every i ∈ Ob(I) and F_{φ} : F_{j} → T_{φ∗}F_{i} a morphism of T_{j}, for every morphism φ : i → j of I. This datum is required to satisfy the identity

γ_{(φ,ψ),F_{i}}^{T} ∘ (T_{φ∗}F_{φ}) ∘ F_{ψ} = F_{ψ∘φ}

for every pair i ⟶^{φ} j ⟶^{ψ} k of morphisms of I, and F_{1_{i}} = 1_{F_{i}} for every i ∈ Ob(I), where, as usual, γ^{T} denotes the coherence constraint of T. The morphisms f_{•} : F_{•} → F′_{•} are the systems (f_{i} : F_{i} → F′_{i} | i ∈ Ob(I)) where f_{i} is a morphism of T_{i} for every i ∈ Ob(I) and

F′_{φ} ∘ f_{j} = (T_{φ∗}f_{i}) ∘ F_{φ} for every morphism φ : i → j of I.

(iii) To every morphism ω : T → S of fibred topoi over I there corresponds a morphism of unital fibred topoi ω^{u} : T^{u} → S^{u} (see remark 2.4.2(i)); thus, in order to describe Top(ω)^{∗}, we may assume without loss of generality that T and S are unital. In this case, we get the following explicit description. By definition, the coherence constraint of ω assigns to every morphism φ : i → j of I a natural transformation

[figure: S_{j}; ω_{j}^{∗}; S_{φ}^{∗}; τ_{φ}^{ω†}; T_{j}; T_{φ}^{∗}; S_{i}; ω_{i}^{∗}; T_{i}.]

Then Top(ω)^{∗} is given by the rule : E_{•} ↦ ((ω_{i}^{∗}(E_{i}), E_{φ}^{ω}) | i ∈ Ob(I), φ ∈ Morph(I)), with

E_{φ}^{ω} : T_{φ}^{∗} ∘ ω_{j}^{∗}(E_{j}) ⟶^{τ_{φ,E_{j}}^{ω†}} ω_{i}^{∗} ∘ S_{φ}^{∗}(E_{j}) ⟶^{ω_{i}^{∗}(E_{φ})} ω_{i}^{∗}(E_{i})

and by assigning to every morphism g_{•} : E_{•} → E′_{•} of Top(S) the morphism

(ω_{i}^{∗}(g_{i}) : ω_{i}^{∗}(E_{i}) → ω_{i}^{∗}(E′_{i}) | i ∈ Ob(I)).

Proposition 4.6.10. The functor Top(ω)^{∗} is left adjoint to Top(ω)_{∗}.

Proof. Arguing as in remark 4.6.9(i), we reduce easily to the case where T and S are unital. In light of the identification (4.6.5), it suffices to exhibit an adjunction for the pair of functors

\xymatrix{Top(S)_{∗} \ar[r]^{Ω^{∗}} & Top(T) \ar[l]^{Ω_{∗}}}

resulting from the pair (Top(ω)^{∗}, Top(ω)_{∗}) via these identifications. Explicitly, Ω^{∗} assigns to every object F_{•} of Top(S)_{∗} (see remark 4.6.9(ii)) the following object of Top(T) :

(ω_{i}^{∗}(F_{i}), F_{φ}^{Ω} := ω_{i}^{∗}(F_{φ}^{†}) ∘ (τ_{φ}^{ω})_{F_{j}}^{†} | i ∈ Ob(I), φ ∈ Morph(I))

where F_{φ}^{†} : T_{φ}^{∗}F_{j} → F_{i} is the morphism of T_{i} corresponding to F_{φ} under the adjunction of T_{φ}, i.e. F_{φ}^{†} = ε_{E_{i}}^{T_{φ}} ∘ (T_{φ}^{∗}F_{φ}), where ε^{T_{φ}} is the counit of the adjunction of T_{φ}. For every morphism f_{•} : F_{•} → F′_{•} of Top(S)_{∗} we have Ω^{∗}(f_{•}) = (ω_{i}^{∗}f_{i} | i ∈ Ob(I)).

Likewise, Ω_{∗} assigns to every object E_{•} of Top(T) the following object of Top(S)_{∗} :

(ω_{i∗}(E_{i}), E_{φ}^{Ω} := (τ_{φ}^{ω})_{E_{i}}^{−1} ∘ ω_{j∗}(E_{φ}^{†}) | i ∈ Ob(I), φ ∈ Morph(I))

where E_{φ}^{†} : E_{j} → T_{φ∗}E_{i} is the morphism of T_{j} corresponding to E_{φ} under the adjunction of T_{φ}, i.e. E_{φ}^{†} = (T_{φ∗}E_{φ}) ∘ η_{E_{j}}^{T_{φ}}, where η^{T_{φ}} is the unit of T_{φ}. For every morphism g_{•} : E_{•} → E′_{•} of Top(T) we have Ω_{∗}(g_{•}) = (ω_{i∗}g_{i} | i ∈ Ob(I)).

Now, let β : Ω^{∗}(F_{•}) → E_{•} be a morphism of Top(T); thus, β is a system of morphisms (β_{i} : ω_{i}^{∗}(F_{i}) → E_{i} | i ∈ Ob(I)) such that the following diagram commutes :

(4.6.11)  \xymatrix{T_{φ}^{∗}ω_{j}^{∗}F_{j} \ar[r]^{T_{φ}^{∗}β_{j}} \ar[d]_{F_{φ}^{Ω}} & T_{φ}^{∗}E_{j} \ar[d]^{E_{φ}} \\ ω_{i}^{∗}F_{i} \ar[r]^{β_{i}} & E_{i}}  for every morphism i ⟶^{φ} j of I.

Our candidate adjunction assigns to β the system of morphisms

β^{†} := (β_{i}^{†} := ω_{i∗}(β_{i}) ∘ η_{F_{i}}^{ω_{i}} : F_{i} → ω_{i∗}E_{i} | i ∈ Ob(I))

where η^{ω_{i}} is the unit of ω_{i} : T_{i} → S_{i}, so β_{i}^{†} is the morphism corresponding to β_{i} under the adjunction of ω_{i}. Thus, we need to show the commutativity of the diagram :

\xymatrix{F_{j} \ar[r]^{β_{j}^{†}} \ar[d]_{F_{φ}} & ω_{j∗}E_{j} \ar[d]^{E_{φ}^{Ω}} \\ S_{φ∗}F_{i} \ar[r]^{S_{φ∗}(β_{i}^{†})} & S_{φ∗}ω_{i∗}E_{i}} for every morphism i ⟶^{φ} j of I.

Set R := ω_{j} ∘ T_{φ}, and let η^{R} and ε^{R} be the unit and counit of R; we consider the diagram :

\xymatrix{F_{j} \ar[r]^{η_{F_{j}}^{R}} \ar[d]_{β_{j}^{†}} & R_{∗}R^{∗}F_{j} \ar[r]^{R_{∗}(F_{φ}^{Ω})} \ar[d]_{R_{∗}T_{φ}^{∗}(β_{j})} & R_{∗}ω_{i}^{∗}F_{i} \ar[r]^{(τ_{φ}^{ω})_{ω^{∗}_{i}F_{i}}^{−1}} \ar[d]^{R_{∗}(β_{i})} & S_{φ∗}ω_{i∗}ω_{i}^{∗}F_{i} \ar[d]^{S_{φ∗}ω_{i∗}(β_{i})} \\ ω_{j∗}E_{j} \ar[r]_{ω_{j∗}(η_{E_{j}}^{Tφ})} & R_{∗}T_{φ}^{∗}E_{j} \ar[r]_{R_{∗}(E_{φ})} & R_{∗}E_{i} \ar[r]_{(τ_{φ}^{ω})_{E_{i}}^{−1}} & S_{φ∗}ω_{i∗}E_{i}.}

Recalling that η_{F_{j}}^{R} = ω_{j∗}(η_{ω_{j}^{∗}F_{j}}^{T_{φ}}) ∘ η_{F_{j}}^{ω_{j}}, we see that the left square commutes, by naturality of η^{T_{φ}}. The commutativity of the central square follows trivially from that of (4.6.11), and that of the right square follows from the naturality of τ_{φ}^{ω}. Notice now that the composition of the three bottom horizontal arrows equals E_{φ}^{Ω}. We are therefore reduced to checking the identity :

(τ_{φ}^{ω})_{ω_{i}^{∗}F_{i}}^{−1} ∘ R_{∗}(F_{φ}^{Ω}) ∘ η_{F_{j}}^{R} = S_{φ∗}(η_{F_{i}}^{ω_{i}}) ∘ F_{φ}.

We compute :

R_{∗}(F_{φ}^{Ω}) ∘ η_{F_{j}}^{R} = R_{∗}(ω_{i}^{∗}(F_{φ}^{†}) ∘ τ_{φ,F_{j}}^{ω†}) ∘ η_{F_{j}}^{R}
= R_{∗}ω_{i}^{∗}(F_{φ}^{†}) ∘ R_{∗}(ε_{ω_{i}^{∗}S_{φ}^{∗}F_{j}}^{R}) ∘ R_{∗}R^{∗}(τ_{φ,ω_{i}^{∗}S_{φ}^{∗}F_{j}}^{ω} ∘ η_{F_{j}}^{S_{φ}∘ω_{i}}) ∘ η_{F_{j}}^{R}
= R_{∗}ω_{i}^{∗}(F_{φ}^{†}) ∘ R_{∗}(ε_{ω_{i}^{∗}S_{φ}^{∗}F_{j}}^{R}) ∘ η_{R_{∗}ω_{i}^{∗}S_{φ}^{∗}F_{j}}^{R} ∘ τ_{φ,ω_{i}^{∗}S_{φ}^{∗}F_{j}}^{ω} ∘ η_{F_{j}}^{S_{φ}∘ω_{i}}
= R_{∗}ω_{i}^{∗}(F_{φ}^{†}) ∘ τ_{φ,ω_{i}^{∗}S_{φ}^{∗}F_{j}}^{ω} ∘ η_{F_{j}}^{S_{φ}∘ω_{i}}
= τ_{φ,ω_{i}^{∗}F_{i}}^{ω} ∘ S_{φ∗}ω_{i∗}ω_{i}^{∗}(F_{φ}^{†}) ∘ η_{F_{j}}^{S_{φ}∘ω_{i}}
= τ_{φ,ω_{i}^{∗}F_{i}}^{ω} ∘ S_{φ∗}ω_{i∗}ω_{i}^{∗}(ε_{E_{i}}^{S_{φ}}) ∘ S_{φ∗}ω_{i∗}ω_{i}^{∗}S_{φ}^{∗}(F_{φ}) ∘ η_{F_{j}}^{S_{φ}∘ω_{i}}
= τ_{φ,ω_{i}^{∗}F_{i}}^{ω} ∘ S_{φ∗}ω_{i∗}ω_{i}^{∗}(ε_{E_{i}}^{S_{φ}}) ∘ η_{S_{φ∗}F_{i}}^{S_{φ}∘ω_{i}} ∘ F_{φ}.

So we are further reduced to showing that :

S_{φ∗}ω_{i∗}ω_{i}^{∗}(ε_{E_{i}}^{S_{φ}}) ∘ η_{S_{φ∗}F_{i}}^{S_{φ}∘ω_{i}} = S_{φ∗}(η_{F_{i}}^{ω_{i}}).

But we have

S_{φ∗}ω_{i∗}ω_{i}^{∗}(ε_{E_{i}}^{S_{φ}}) ∘ η_{T_{φ∗}F_{i}}^{S_{φ}∘ω_{i}} = S_{φ∗}ω_{i∗}ω_{i}^{∗}(ε_{E_{i}}^{S_{φ}}) ∘ S_{φ∗}(η_{S_{φ}^{∗}S_{φ∗}F_{i}}^{ω_{i}}) ∘ η_{S_{φ∗}F_{i}}^{S_{φ}}
= S_{φ∗}(ω_{i∗}ω_{i}^{∗}(ε_{E_{i}}^{S_{φ}}) ∘ η_{S_{φ}^{∗}S_{φ∗}F_{i}}^{ω_{i}}) ∘ η_{S_{φ∗}F_{i}}^{S_{φ}}
= S_{φ∗}(η_{F_{i}}^{ω_{i}}) ∘ S_{φ∗}(ε_{F_{i}}^{S_{φ}}) ∘ η_{S_{φ∗}F_{i}}^{S_{φ}}

so it suffices to invoke the triangular identities for the pair (η^{S_{φ}}, ε^{S_{φ}}) to conclude. Likewise one shows that for every morphism λ : F_{•} → Ω_{∗}E_{•} in Top(S)_{∗}, the system

(λ_{i}^{†} := ε_{E_{i}}^{ω_{i}} ∘ ω_{i}^{∗}(λ_{i}) : ω_{i}^{∗}F_{i} → E_{i} | i ∈ Ob(I))

is a morphism Ω^{∗}F_{•} → E_{•} in Top(T) : the verification shall be left to the reader. In view of remark 1.1.17(ii), it is then clear that these two rules yield mutually inverse bijections between Hom_{Top(T)}(Ω^{∗}(F_{•}), E_{•}) and Hom_{Top(S)_{∗}}(F_{•}, Ω_{∗}(E_{•})). Lastly, the naturality of the rule (β : Ω^{∗}(F_{•}) → E_{•}) ↦ β^{†} with respect to both F_{•} and E_{•} follows directly from the same property for the adjunction of each morphism ω_{i} (details left to the reader). □

Remark 4.6.12. In the situation of proposition 4.6.10, denote by λ_{S} : Top(S) →^{∼} Top(S)_{∗} the natural isomorphism of (4.6.5). Then we have Top(ω)_{∗} ∘ Top(ω)^{∗} = λ_{S}^{−1} ∘ Ω_{∗} ∘ Ω^{∗} ∘ λ_{S}, and Top(ω)^{∗} ∘ Top(ω)_{∗} = Ω^{∗} ∘ Ω_{∗}. It follows that from the proof of proposition 4.6.10 we can extract an adjunction for the pair (Top(ω)^{∗}, Top(ω)_{∗}), whose unit η^{Top(ω)} and counit ε^{Top(ω)} are given explicitly by the rules :

F_{•} ↦ (η_{F_{i}}^{ω_{i}} | i ∈ Ob(I)) E_{•} ↦ (ε_{E_{i}}^{ω_{i}} | i ∈ Ob(I))

where (η^{ω_{i}}, ε^{ω_{i}}) are the unit and counit for the adjoint pair (ω_{i}^{∗}, ω_{i∗}), for every i ∈ Ob(I).

4.6.13. Let π : 𝒜 → I and ρ : J → I be any two functors; we have an obvious functor

Σ(𝒜 /ρ)^{∗} : Σ(𝒜 /I) → Σ(J ×_{I} 𝒜 /J) (G : I → 𝒜) ↦ J ×_{I} G

that assigns to every morphism α : G ⇒ G′ in Σ(𝒜 /I) the natural transformation J ×_{I} α : J ×_{I} G ⇒ J ×_{I} G′. Especially, if I and J are small categories, and T is any fibred topos over I, recalling the natural identification ℱ ib((T ∘ ρ)^{∗}) →^{∼} J ×_{I} ℱ ib(T^{∗}) of remark 3.1.5(ii), we deduce a functor

Top(T/ρ)^{∗} := Σ(ℱ ib(T^{∗})/ρ)^{∗o} : Top(T) → Top(T ∘ ρ).

Explicitly, Top(T/ρ)^{∗} assigns to every E_{•} as in remark 4.6.9(i) the datum (E_{ρ(j)}, E_{ρ(φ)} | j ∈ Ob(J), φ ∈ Morph(J)), and to every morphism f_{•} : E_{•} → F_{•} of Top(T), the morphism of Top(T ∘ ρ) given by the system (f_{ρ(j)} : E_{ρ(j)} → F_{ρ(j)} | j ∈ Ob(J)). We wish next to exhibit left and right adjoints for Top(T/ρ)^{∗}. To this aim, we observe more generally :

Proposition 4.6.14. Let ρ : J → I be a functor between small categories, U′ a universe containing U, and c : I^{o} → U′-Cat any pseudo-functor such that

(a) The category c_{i} is complete for every i ∈ Ob(I).

(b) The functor c_{f} : c_{i} → c_{i′} commutes with small limits, for every morphism i′ ⟶^{f} i in I. Then Σ(ℱ ib(c)/ρ)^{∗} : Σ(ℱ ib(c)/I) → Σ(ℱ ib(c ∘ ρ^{o})/J) admits a right adjoint.

Proof. By proposition 2.4.3we may assume that c is unital. Now, set 𝒜 := ℱ ib(c), denote by π : 𝒜 → I the projection, and let F_{•} : J → J ×_{I} 𝒜 be any object of Σ(J ×_{I} 𝒜 /J); hence F_{•} assigns to every j ∈ Ob(J) an object F_{j} ∈ c_{ρ(j)}, and to every morphism ψ : j → j′ in J a morphism F_{ψ} : F_{j} → c_{ρ(ψ)}F_{j′} of c_{ρ(j)}. We set

Λ_{i}(j, φ) := c_{φ}F_{j} for every i ∈ Ob(I) and (j, φ : i → ρ(j)) ∈ Ob(i/ρJ).

For every morphism (j, φ) ⟶^{i/ψ} (j′, φ′) of i/ρJ we define Λ_{i}(i/ψ) : Λ_{i}(j, φ) → Λ_{i}(j′, φ′) as the composition

c_{φ}F_{j} ⟶^{c_{φ}F_{ψ}} c_{φ}c_{ρ(ψ)}F_{j′} ⟶^{(γ_{φ,ρ(ψ)}^{c})_{F_{j′}}} c_{φ′}F_{j′}

where γ^{c} is the coherence constraint of c. Also, for every morphism f : i′ → i in I, let us set

Λ_{f}(j, φ) := (γ_{f,φ}^{c})_{F_{j}}^{−1} : Λ_{i′}(j, φ ∘ f) → c_{f}(Λ_{i}(j, φ)) for every (j, φ) ∈ Ob(i/ρJ).

Claim 4.6.15. (i) The rules : (j, φ) ↦ Λ_{i}(j, φ) and i/ψ ↦ Λ(i/ψ) for every object (j, φ) and morphism i/ψ of i/ρJ define a functor

Λ_{i}^{F_{•}} : i/ρJ → π^{−1}(i) for every i ∈ Ob(I).

(ii) Let the functor f/ρJ : i/ρJ → i′/ρJ be as in (1.1.27). The rule : (j, φ) ↦ Λ_{f}(j, φ) defines a natural transformation

Λ_{f}^{F_{•}} : Λ_{i′}^{F_{•}} ∘ (f/ρJ) ⇒ c_{f} ∘ Λ_{i}^{F_{•}}.

Proof of the claim. (i): Since c is unital, a simple inspection shows that Λ_{i}(1_{(j,φ)}) = 1_{Λ(j,φ)} for

every (j, φ) ∈ Ob(ρJ/i). Next consider a pair of morphisms (j, φ) ⟶^{i/ψ} (j′, φ′) ⟶^{i/ψ′} (j′′, φ′′) of i/ρJ. We have

Λ_{i}(i/ψ′) ∘ Λ_{i}(i/ψ) = (γ_{φ′,ρ(ψ′)}^{c})_{F_{j′′}} ∘ c_{φ′}F_{ψ′} ∘ (γ_{φ,ρ(ψ)}^{c})_{F_{j′}} ∘ c_{φ}F_{ψ}
= (γ_{φ′,ρ(ψ′)}^{c})_{F_{j′′}} ∘ (γ_{φ,ρ(ψ)}^{c} ∗ c_{ρ(ψ′)})_{F_{j′}} ∘ c_{φ}c_{ρ(ψ)}(F_{ψ′}) ∘ c_{φ}F_{ψ}
= (γ_{φ′,ρ(ψ′)}^{c})_{F_{j′′}} ∘ (γ_{φ,ρ(ψ)}^{c} ∗ c_{ρ(ψ′)})_{F_{j′}} ∘ c_{φ}((γ_{ρ(ψ),ρ(ψ′)}^{c −1})_{F_{j′′}} ∘ F_{ψ′∘ψ})
= (γ_{φ,ρ(ψ′∘ψ)}^{c})_{F_{j′′}} ∘ c_{φ}F_{ψ′∘ψ}
= Λ_{i}(i/ψ′ ∘ ψ)

whence the contention.

(ii): It suffices to remark the commutativity of the diagram :

\xymatrix{c_{φ∘f}F_{j} \ar[r]^{c_{φ∘f} F_{ψ}} \ar[d]_{(γ_{f,φ}^{c})_{F_{j}}^{−1}} & c_{φ∘f}c_{ρ(ψ)}F_{j′} \ar[d]^{(γ_{f,φ}^{c})_{c_{ρ(ψ)}F_{j′}}^{−1}} \ar[r]^{(γ_{φ∘f,ρ(ψ)}^{c})_{F_{j′}}} & c_{ρ(ψ)∘φ∘f}F_{j′} \ar[d]^{(γ_{f,φ′}^{c})_{F_{j′}}^{−1}} \\ c_{f}c_{φ}F_{j} \ar[r]_{c_{f} c_{φ}F_{ψ}} & c_{f}c_{φ}c_{ρ(ψ)}F_{j′} \ar[r]_{c_{f} (γ_{φ,ρ(ψ)}^{c})_{F_{j′}}} & c_{f}c_{ρ(ψ)∘φ}F_{j′}}

for every morphism (i/ψ) : (j, φ) → (j′, φ′) of i/ρJ. ♢

In view of claim 4.6.15(i), for every i ∈ Ob(I) we choose λ(F_{•}, i) ∈ Ob(c_{i}) representing the limit of Λ_{i}^{F_{•}}, and a universal cone τ^{F_{•},i} : c_{λ(F_{•},i)} ⇒ Λ_{i}^{F_{•}}. In view of claim 4.6.15(ii) there follows a cone

Υ_{f}^{F_{•}} := Λ_{f}^{F_{•}} ⊙ (τ^{F_{•},i′} ∗ (f/ρJ)) : c_{λ(F_{•},i′)} ⇒ c_{f} ∘ Λ_{i}^{F_{•}}.

On the other hand, notice that the cone c_{f} ∗ τ^{F_{•},i} : c_{c_{f} λ(F_{•},i)} ⇒ c_{f} ∘ Λ_{i}^{F_{•}} is still universal, since c_{f} commutes with small limits; hence there exists a unique morphism in c_{i′}

λ(F_{•}, f) : λ(F_{•}, i′) → c_{f}λ(F_{•}, i) such that (c_{f} ∗ τ^{F_{•},i}) ⊙ c_{λ(F_{•},f)} = Υ_{f}^{F_{•}}.

Claim 4.6.16. The rules : i ↦ λ(F_{•}, i) and f ↦ λ(F_{•}, f) for every i ∈ Ob(I) and every morphism f of I define a functor

λ(F_{•}) : I → 𝒜 .

Proof of the claim. Since c is unital, a simple inspection shows that Λ_{1_{i}}^{F_{•}} = 1_{Λ_{i}^{F•}} for every

i ∈ Ob(I), whence λ(F_{•}, 1_{i}) = 1_{λ(F_{•},i)}. Next, let i′′ ⟶^{f′} i′ ⟶^{f} i be two morphisms of I; we need to check that λ(F_{•}, f) ∘ λ(F_{•}, f′) = λ(F_{•}, f ∘ f′), and by the universality of c_{f∘f′} ∗ τ^{F_{•},i} it suffices to show that X := (c_{f∘f′} ∗ τ^{F_{•},i}) ⊙ c_{λ(F_{•},f)∘λ(F_{•},f′)} = (c_{f∘f′} ∗ τ^{F_{•},i}) ⊙ c_{λ(F_{•},f∘f′)}. We compute :

X = (c_{f∘f′} ∗ τ^{F_{•},i}) ⊙ c_{(γ_{f′,f}^{T})_{λ(F•,i)}} ⊙ c_{c_{f′}λ(F_{•},f)} ⊙ c_{λ(F_{•},f′)}
= (γ_{f,f′}^{c} ∗ Λ_{i}^{F_{•}}) ⊙ (c_{f′}c_{f} ∗ τ^{F_{•},i}) ⊙ c_{c_{f′}λ(F_{•},f)} ⊙ c_{λ(F_{•},f′)}
= (γ_{f,f′}^{c} ∗ Λ_{i}^{F_{•}}) ⊙ c_{f′} ∗ (Λ_{f}^{F_{•}} ⊙ (τ^{F_{•},i′} ∗ (f/ρJ))) ⊙ c_{λ(F_{•},f′)}
= (γ_{f,f′}^{c} ∗ Λ_{i}^{F_{•}}) ⊙ (c_{f′} ∗ Λ_{f}^{F_{•}}) ⊙ ((Λ_{f′}^{F_{•}} ⊙ (τ^{F_{•},i′′} ∗ (f′/ρJ))) ∗ (f/ρJ))
= (γ_{f,f′}^{c} ∗ Λ_{i}^{F_{•}}) ⊙ (c_{f′} ∗ Λ_{f}^{F_{•}}) ⊙ (Λ_{f′}^{F_{•}} ∗ (f/ρJ)) ⊙ (τ^{F_{•},i′′} ∗ (f ∘ f′/ρJ)).

So we are reduced to checking that

(γ_{f,f′}^{c} ∗ Λ_{i}^{F_{•}}) ⊙ (c_{f′} ∗ Λ_{f}^{F_{•}}) ⊙ (Λ_{f′}^{F_{•}} ∗ (f/ρJ)) = Λ_{f∘f′}^{F_{•}}.

But the latter follows directly from the coherence axioms for γ^{c}. ♢

It is clear that the functor λ(F_{•}) of claim 4.6.16 is a section of the projection 𝒜 → I. Next, let β_{•} : F_{•} → F′_{•} be any morphism of Σ(J ×_{I} 𝒜 /J); we deduce easily a natural transformation

Λ_{i}^{β_{•}} : Λ_{i}^{F_{•}} ⇒ Λ_{i}^{F′_{•}}  (j, φ) ↦ (c_{φ}β_{j} : c_{φ}F_{j} → c_{φ}F′_{j})  for every i ∈ Ob(I)

such that

(4.6.17) (c_{f}Λ_{i}^{β_{•}}) ⊙ Λ_{f}^{F_{•}} = Λ_{f}^{F′_{•}} ⊙ (Λ_{i′}^{β_{•}} ∗ (f/ρJ))  for every morphism f : i′ → i of I

whence a unique morphism in c_{i}

λ(β_{•}, i) : λ(F_{•}, i) → λ(F′_{•}, i) such that τ^{F′_{•},i} ⊙ c_{λ(β_{•},i)} = Λ_{i}^{β_{•}} ⊙ τ^{F_{•},i}.

Claim 4.6.18. The rule i ↦ λ(β_{•}, i) defines a natural transformation

λ(β_{•}) : λ(F_{•}) ⇒ λ(F′_{•}).

Proof of the claim. Let f : i′ → i be any morphism of I; as usual, we reduce to checking the identity : X := (c_{f} ∗ τ^{F′_{•},i}) ⊙ c_{λ(F′_{•},f)} ⊙ c_{λ(β_{•},i′)} = Y := (c_{f} ∗ τ^{F′_{•},i}) ⊙ (c_{f} ∗ c_{λ(β_{•},i)}) ⊙ c_{λ(F_{•},f)}. We compute :

X = Υ_{f}^{F′_{•}} ⊙ c_{λ(β_{•},i′)} = Λ_{f}^{F′_{•}} ⊙ (Λ_{i′}^{β_{•}} ⊙ τ^{F_{•},i′}) ∗ (f/ρJ)
Y = c_{f}(Λ_{i}^{β_{•}} ⊙ τ^{F_{•},i}) ⊙ c_{λ(F_{•},f)} = c_{f}Λ_{i}^{β_{•}} ⊙ Υ_{f}^{F_{•}}

so the assertion follows from (4.6.17). ♢

Let β_{•} : F_{•} → F′_{•} and β′_{•} : F′_{•} → F′′_{•} be two morphisms of Σ(J ×_{I} 𝒜 /J); by a simple inspection we see that Λ_{i}^{β′_{•}∘β_{•}} = Λ_{i}^{β′_{•}} ⊙ Λ_{i}^{β_{•}} for every i ∈ Ob(I), whence λ(β′_{•} ∘ β_{•}) = λ(β′_{•}) ⊙ λ(β_{•}). It is also easily seen that λ(1_{F_{•}}) = 1_{λ(F_{•})} for every object F_{•} of Σ(J ×_{I} 𝒜 /J), so we have finally obtained a functor

Σ(𝒜 /ρ)_{∗} : Σ(J ×_{I} 𝒜 /J) → Σ(𝒜 /I)  F_{•} ↦ λ(F_{•})  (β_{•} : F_{•} → F′_{•}) ↦ λ(β_{•}).

To see that Σ(𝒜 /ρ)_{∗} is the sought right adjoint, consider any objects E_{•} of Σ(𝒜 /I) and F_{•} of Σ(J ×_{I} 𝒜 /J), and a morphism β_{•} : E_{•} → λ(F_{•}) in Σ(𝒜 /I). It is easily seen that the rule :

j ↦ β_{j}^{†} : E_{ρ(j)} ⟶^{β_{ρ(j)}} λ(F_{•}, ρ(j)) ⟶^{τ_{(j,1_{ρ(j)})}^{F•,ρ(j)}} F_{j} for every j ∈ Ob(J)

defines a morphism β_{•}^{†} : J ×_{I} E_{•} → F_{•} in Σ(J ×_{I} 𝒜 /J) (details left to the reader). Conversely, let α_{•} : J ×_{I} E_{•} → F_{•} be a morphism in Σ(J ×_{I} 𝒜 /J); for every i ∈ Ob(I) and (j, φ) ∈ Ob(i/ρJ) we let τ_{(j,φ)}^{α_{•},i} := c_{φ}α_{j} ∘ E_{φ} : E_{i} → c_{φ}F_{j}. It is easily seen that the rule : (j, φ) ↦ τ_{(j,φ)}^{α_{•},i} defines a cone τ^{α_{•},i} : c_{E_{i}} ⇒ Λ_{i}^{F_{•}} (details left to the reader); there follows a unique morphism

α_{i}^{†} : E_{i} → λ(F_{•}, i) in c_{i} such that τ^{F_{•},i} ⊙ c_{α_{i}^{†}} = τ^{α_{•},i}.

Claim 4.6.19. The rule i ↦ α_{i}^{†} defines a morphism α_{•}^{†} : E_{•} → λ(F_{•}) in Σ(𝒜 /I).

Proof of the claim. As usual we reduce to checking that

X := (c_{f} ∗ τ^{F_{•},i}) ⊙ c_{c_{f} α_{i}^{†}} ⊙ c_{E_{f}} = Y := (c_{f} ∗ τ^{F_{•},i}) ⊙ c_{λ(F_{•},f)} ⊙ c_{α_{i′}^{†}}

for every morphism f : i′ → i in I. However, we have :

X = (c_{f} ∗ τ^{α,i}) ⊙ c_{E_{f}} Y = Υ_{f}^{F_{•}} ⊙ c_{α_{i′}^{†}} = Λ_{f}^{F_{•}} ⊙ (τ^{α,i′} ∗ (f/ρJ)).

So we come down to checking that

(c_{f}c_{φ}α_{j}) ∘ (c_{f}E_{φ}) ∘ E_{f} = (γ_{f,φ}^{c})_{F_{j}}^{−1} ∘ c_{φ∘f}α_{j} ∘ E_{φ∘f} for every (j, φ) ∈ Ob(i/ρJ).

To this aim, it suffices to notice that we have : (γ_{f,φ}^{c})_{F_{j}} ∘ (c_{f}c_{φ}α_{j}) = c_{φ∘f}α_{j} ∘ (γ_{f,φ}^{c})_{E_{ρ(j)}} and (γ_{f,φ}^{c})_{E_{ρ(j)}} ∘ (c_{f}E_{φ}) ∘ E_{f} = E_{φ∘f}. ♢

Claim 4.6.20. For every morphism α_{•} : J ×_{I} E_{•} → F_{•} in Σ(J ×_{I} 𝒜 /J) and β_{•} : E_{•} → λ(F_{•}) in Σ(𝒜 /I) we have (α_{•}^{†})^{†} = α_{•} and (β_{•}^{†})^{†} = β_{•}.

Proof of the claim. The assertion concerning α_{•} follows by direct inspection. Next, let us consider for every i ∈ Ob(I) and every (j, φ) ∈ Ob(i/ρJ) the diagram of morphisms in c_{i}

𝒟_{(j,φ)}  :  \xymatrix{E_{i} \ar[r]^{E_{φ}} \ar[dr]_{β_{i}} & c_{φ}E_{ρ(j)} \ar[r]^{c_{φ}β_{ρ(j)}} & c_{φ}λ(F_{•}, ρ(j)) \ar[d]^{c_{φ}τ_{(j,1_{ρ(j)})}^{F•,ρ(j)}} \\  & λ(F_{•}, i) \ar[r]_{τ_{(j,φ)}^{F•,i}} \ar[ur]^{λ(F_{•},φ)} & c_{φ}F_{j}.}

The assertion is equivalent to the commutativity of 𝒟_{(j,φ)} for every such i and (j, φ). However, notice that (Λ_{φ}^{F_{•}})_{(j,1_{ρ(j)})} = 1_{c_{φ}F_{j}}, since c is unital; this implies that the lower triangular subdiagram commutes. The commutativity of the upper triangular subdiagram is clear. ♢

Lastly, it is easily seen that the rule : β_{•} ↦ β_{•}^{†} is natural in both E_{•} and F_{•} (details left to the reader); in view of claim 4.6.20, this rule then establishes the sought adjunction between Σ(𝒜 /ρ)^{∗} and Σ(𝒜 /ρ)_{∗}. □

Corollary 4.6.21. In the situation of (4.6.13), the functor Top(T/ρ)^{∗} admits both a left and a right adjoint, denoted respectively :

Top(T/ρ)_{!} : Top(T ∘ ρ) → Top(T)  and  Top(T/ρ)_{∗} : Top(T ∘ ρ) → Top(T).

Proof. It is easily seen that the isomorphism of categories (4.6.5) (and the corresponding one for T ∘ ρ) identifies the functor Top(T/ρ)^{∗} with Σ(ℱ ib(T_{∗})/ρ^{o})^{∗}, hence the existence of the right adjoint Top(T/ρ)_{∗} follows from proposition 4.6.14. On the other hand, by applying the same proposition to Σ(ℱ ib(T^{∗})/ρ)^{∗} : Σ(ℱ ib(T^{∗})/I) → Σ(ℱ ib(T ∘ ρ)^{∗}/J) we see that the latter admits as well a right adjoint, hence its opposite functor Top(T/ρ)^{∗} admits a left adjoint. □

Remark 4.6.22. (i) Let ρ : J → I be any functor between small categories and T any fibred topos over I. Corollary 4.6.21 implies that the functor Top(T/ρ)^{∗} commutes with all small limits and all small colimits (proposition 1.3.25(iii,iv)).

(ii) Let 𝟙 be a final object of the category Cat (i.e. a category with one object and one morphism); for every t ∈ Ob(I) we have a unique functor ρ^{t} : 𝟙 → I that sends the unique object of 𝟙 to t. Clearly Top(T ∘ ρ^{t}) = T_{t} and the induced functor

Top(T/ρ^{t})^{∗} : Top(T) → T_{t}

assigns to every datum E_{•} as in remark 4.6.9(i) the object E_{t} and to every morphism f_{•} : E_{•} → F_{•} of Top(T) the morphism f_{t} : E_{t} → F_{t} of T_{t}. The assertion that Top(T/ρ^{t})^{∗} commutes with limits and colimits for every t ∈ Ob(I) then means that the small limits and colimits in the category Top(T) are computed fibrewise.

(iii) Moreover, the observation of (ii) determines the limits and colimits in Top(T) up to unique isomorphism. Indeed, let F : Λ → Top(T) be any functor from a small category Λ, and let (L_{•}, τ) be a pair consisting of an object L_{•} of Top(T) representing the colimit of F, and a universal cocone τ : F ⇒ c_{L_{•}}. By (ii) we know that L_{t} ∈ Ob(T_{t}) represents the colimit of F^{t} := Top(T/ρ^{t})^{∗} ∘ F : Λ → T_{t} and τ^{t} := Top(T/ρ^{t})^{∗} ∗ τ : F^{t} ⇒ c_{L_{t}} is a universal cocone for every t ∈ Ob(I). Now, let φ : s → t be any morphism in I; we get a natural transformation

F^{φ} : T_{φ}^{∗} ∘ F^{t} ⇒ F^{s} λ ↦ ((F λ)_{φ} : T_{φ}^{∗}(F λ)_{t} → (F λ)_{s}).

Since T_{φ}^{∗} admits a right adjoint, the cocone T_{φ}^{∗} ∗ τ^{t} : T_{φ}^{∗} ∘ F^{t} ⇒ c_{T_{φ}^{∗}L_{t}} is still universal (proposition 1.3.25(iv)), hence there exists a unique morphism

L′_{φ} : T_{φ}^{∗}L_{t} → L_{s} in T_{s} such that τ^{s} ⊙ F^{φ} = c_{L′_{φ}} ⊙ (T_{φ}^{∗} ∗ τ^{t}).

But then we must have L′_{φ} = L_{φ} necessarily, so L_{•} is completely determined by a choice of universal cocones (τ^{t} | t ∈ Ob(I)). Likewise we see that the limit of F is completely determined by the choice of a system of pairs (M_{t}, μ^{t} : F^{t} ⇒ c_{M_{t}} | t ∈ Ob(I) with M_{t} ∈ Ob(T_{t}) representing the limit of F^{t}, and μ_{t} a universal cone, for every t ∈ Ob(I).

(iv) By direct inspection, we see that the functor

Top(T/ρ^{t})_{!} : T_{t} → Top(T)

assigns to every E ∈ Ob(T_{t}) the datum (E_{i}, E_{φ} | i ∈ Ob(I), φ ∈ Morph(I)) such that :

E_{i} := ∐_{f:i→t} T_{f}^{∗}E for every i ∈ Ob(I)

where the f ranges over all the morphisms i → t in I. For every morphism φ : i → j of I, the morphism E_{φ} : T_{φ}^{∗}E_{j} → E_{i} is the unique one fitting in the commutative diagram :

\xymatrix{T_{φ}^{∗}T_{f}^{∗}E \ar[r]^{γ_{φ,f}^{T^{∗}}} \ar[d] & T_{f∘φ}^{∗}E \ar[d] \\ E_{j} \ar[r]^{E_{φ}} & E_{i}}  for every morphism f : j → t in I

where γ^{T^{∗}} is the coherence constraint of T^{∗}, and where the vertical arrows are induced by the chosen universal cocones for the representatives E_{i} and E_{j} of the foregoing coproducts. For every morphism h in T_{t}, the morphism Top(T/ρ^{t})_{!}(h) is the natural transformation that assigns to every i ∈ Ob(I) the coproduct of morphisms ∐_{f:i→t} T_{f}^{∗}(h) : details left to the reader.

4.6.23. Let T be a fibred topos over I; we consider the associated fibred lex-site π : \underline{Can}(T) → I as in (4.6.2), and its total site (\underline{Can}(T), J). Also, for every t ∈ Ob(I), we denote by i_{t} : T_{t} → \underline{Can}(T) the inclusion functor.

Theorem 4.6.24. (i) With the notation of (4.6.23), we have a natural equivalence of categories:

a_{T} : Top(T) →^{∼} (\underline{Can}(T), J)_{U}^{∼}

and a morphism of sites :

b_{T} : Can(Top(T)) → (\underline{Can}(T), J)

fitting into an essentially commutative diagram (notation of remark 4.6.22(ii)) :

\xymatrix{Can(Top(T))_{U}^{∼} \ar[dr]_{b_{T ∗}^{∼}} & Top(T) \ar[r]^{Top(T/ρ^{t})^{∗}} \ar[d]_{a_{T}} \ar[l]_{h_{Top(T)}} & T_{t} \ar[d]^{h_{T_{t}}} \\ & (\underline{Can}(T), J)_{U}^{∼} \ar[r]^{(i_{t})_{U∗}^{∼}} & Can(T_{t})_{U}^{∼}} for every t ∈ Ob(I).

(ii) The category Top(T) is an U-topos, which we call the total topos of the fibred topos T.

Proof. Let us remark more generally :

Claim 4.6.25. For every pseudo-functor c : I → lex.Site there exists a natural isomorphism of categories (notation of (4.4.15) and remarks 4.5.10and 4.5.2(v)) :

α_{c} : Top(T ∘ c) →^{∼} T(totSite ∘ \underline{ℱ ib} _{I}(c)).

Proof of the claim. By proposition 2.4.3, we may assume that c is unital, and we let γ^{c} be the coherence constraint of c. By proposition 4.5.5(iv), a presheaf F on the total site of \underline{ℱ ib} _{I}(c) is a sheaf if and only if its restriction F_{i} to each fibre site c_{i} is a sheaf. Hence, such a sheaf F is the datum F_{•} := (F_{i}, F_{φ} : F_{j} → (c_{φ})_{∗}^{∼}F_{i} | i ∈ Ob(I), (φ : i → j) ∈ Morph(I)) of a system of sheaves and morphisms of sheaves that make commute the diagrams

\xymatrix{F_{k} \ar[d]_{F_{ψ}} \ar[r]^{F_{ψ∘φ}} & (c_{ψ∘φ})_{∗}^{∼}F_{i} \\ (c_{ψ})_{∗}^{∼}F_{j} \ar[r]^{(c_{ψ})_{∗}^{∼}F_{φ}} & (c_{ψ})_{∗}^{∼} ∘ (c_{φ})_{∗}^{∼}F_{i} \ar[u]_{(γ_{φ,ψ}^{c})_{∗,F_{i}}^{∼}}}

for every pair of morphisms i ⟶^{φ} j ⟶^{ψ} k of I. Namely, F_{φ,X} := F (φ, 1_{c_{φ}X}) : F_{j}X → F_{i}(c_{φ}X) for every such φ, and every X ∈ Ob(T_{j}). Recall now that, for every site (𝒞 , J), the topos T(𝒞 , J) is isomorphic to (𝒞 , J)^{∼}; especially, T(c_{i}) is isomorphic to the category of sheaves on the site c_{i}, for every i ∈ Ob(I). Under this identification, we then see that a datum F_{•} as in the foregoing corresponds precisely to an object of the category Top(T ∘ c)_{∗}, and likewise it is easily seen that the morphisms of Top(T∘c)_{∗} correspond naturally to the morphisms of sheaves on the total site of \underline{ℱ ib} _{I}(c). Then the sought isomorphism is the composition of this natural identification with the isomorphism Top(T ∘ c)_{∗} →^{∼} Top(T ∘ c)_{∗} of lemma 4.6.3. ♢

Recall now that the unit η : 1_{Topos} → T ∘ Can of the 2-adjoint pair (Can, T) is a pseudonatural equivalence (theorem 4.4.17(iii)); there follows a pseudo-natural equivalence η_{T} : T →^{∼} T ∘ Can ∘ T. Then the sought equivalence a_{T} is the composition of Top(η_{T}) : Top(T) → Top(T ∘ Can ∘ T) with the isomorphism α_{Can∘T} of claim 4.6.25. The essential commutativity of the square subdiagram of the diagram of (i) follows by direct inspection.

Now, (\underline{Can}(T), J) is isomorphic to an U-site (remark 4.5.2(iii)), hence (\underline{Can}(T), J)_{U}^{∼} is isomorphic to an U-topos (remark 4.4.1(iv)), whence (ii). Especially, for every pseudo-functor c : I → lex.Site the category Top(T ∘ c) is an U-topos, and then the 2-adjunction of theorem 4.4.17assigns to the isomorphism α_{c} of claim 4.6.25a morphism of sites :

β_{c} : Can(Top(T ∘ c)) → totSite ∘ \underline{ℱ ib} _{I}(c)

so that we may let b_{T} := β_{Can∘T} ∘ Can(Top(η_{T})). The essential commutativity of the triangular subdiagram of (i) follows from the explicit construction of the unit of this 2-adjunction, provided by the proof of theorem 4.4.17. □

Remark 4.6.26. (i) By unwinding the constructions in the proof of theorem 4.6.24, we see that the equivalence a_{T} assigns to every object E := ((E_{i}, E_{φ}) | i ∈ Ob(I), φ ∈ Morph(I)) of Top(T) the datum (h_{E_{i}}, g_{E_{φ}} | i ∈ Ob(I), φ ∈ Morph(I)), consisting of the sheaf h_{E_{i}} on T_{i} represented by E_{i} for every i ∈ Ob(I), and the morphism

g_{E_{φ}} : h_{E_{j}} ⟶^{h_{E^{†}_{φ}}} h_{T_{φ∗}E_{i}} →^{∼} (T_{φ}^{∗})_{∗}^{∼}h_{E_{i}} for every (i ⟶^{φ} j) ∈ Morph(I).

Here E_{φ}^{†} is the adjoint to the morphism E_{φ}, relative to the adjunction for the pair (T_{φ}^{∗}, T_{φ∗}) that defines the morphism of topoi T_{φ} : T_{i} → T_{j}. The natural identification h_{T_{φ∗}E_{i}} →^{∼} (T_{φ}^{∗})_{∗}^{∼}h_{E_{i}} is deduced as well from the same adjunction, so the morphism of sheaves g_{E_{φ}} turns out be given by the rule : (f : X → E_{j}) ↦ (E_{φ} ∘ T_{φ}^{∗}f) for every X ∈ Ob(S_{j}) and every f ∈ h_{E_{j}}(X).

(ii) On the other hand, the morphism of sites b_{T} of theorem 4.6.24is a composition

\underline{Can}(T) ⟶^{h_{\underline{Can}(T)}^{a}} (\underline{Can}(T), J)^{∼} ⟶^{a_{T}^{−1}} Top(T)

where we have denoted by a_{T}^{−1} any choice of a quasi-inverse of a_{T}. To describe b_{T} more explicitly, consider any (i, X) ∈ Ob(\underline{Can}(T)); hence i ∈ Ob(I) and X ∈ Ob(T_{i}); the proof of theorem 4.6.24 assigns to (i, X) the system (F (i, X)_{j}, F (i, X)_{φ} | j ∈ Ob(I), φ ∈ Morph(I)) where F (i, X)_{j} is the restriction to Can(T_{j}) of the sheaf h_{(i,X)}^{a} on (\underline{Can}(T), J). Now, let G(i, X)_{j} be the restriction of h_{(i,X)} to T_{j}; for every Y ∈ Ob(T_{j}) we have

G(i, X)_{j}(Y) = {(φ, f) | φ ∈ Hom_{I}(j, i), f ∈ Hom_{T_{j}}(Y, T_{φ}^{∗}X)}.

For every morphism h : Y′ → Y in T_{j}, the map G(i, X)_{j}(h) is given by the rule : (φ, f) ↦ (φ, f ∘ h). Then set

[i, X, j] := ∐_{φ:j→i} T_{φ}^{∗}X ∈ Ob(T_{j})

and fix a universal cocone (e_{X,φ} : T_{φ}^{∗}X → [i, X, j] | j ⟶^{φ} i). Notice the morphism of presheaves

(4.6.27)  G(i, X)_{j} → h_{[i,X,j]}  (φ, f : Y → T_{φ}^{∗}X) ↦ (e_{X,φ} ∘ f : Y → [i, X, j]).

Let us check that (4.6.27) is bicovering. Indeed, (4.6.27) is clearly a monomorphism, hence it suffices to show that it is a covering morphism. Thus, let f : Y → [i, X, j] be any morphism on T_{j} and for every φ ∈ Hom_{I}(j, i) set Y_{φ} := Y ×_{[i,X,j]} T_{φ}^{∗}. The induced cocone (e_{φ} : Y_{φ} → Y | j ⟶^{φ} i) is still universal (remark 4.4.1(i)) hence it defines a covering family for Y in the canonical topology of T_{j} (remark 4.4.1(ii)). But clearly f ∘ e_{φ} lies in the image of the map G(i, X)_{j}(Y_{φ}) → h_{[i,X,j]}(Y_{φ}) for every φ : j → i, whence the contention. Hence, (4.6.27)induces an isomorphism G(i, X)_{j}^{a} →^{∼} h_{[i,X,j]}, from the sheaf G(i, X)_{j}^{a} on Can(T_{j}) associated with the presheaf G(i, X)_{j}; the latter is naturally isomorphic to F (i, X)_{j}, by proposition 4.5.5(v). Summing up, we have exhibited a natural isomorphism of sheaves on Can(T_{j}):

F (i, X)_{j} →^{∼} h_{[i,X,j]} for every j ∈ Ob(I).

(iii) Next, according to the proof of theorem 4.6.24, for every morphism ψ : j → k in I, the morphism of sheaves F (i, X)_{ψ} : F (i, X)_{k} → (T_{ψ}^{∗})_{∗}^{∼}F (i, X)_{j} assigns to every Y ∈ Ob(T_{k}) the map h_{(i,X)}^{a}(ψ, 1_{T_{ψ}^{∗}Y}) : h_{(i,X)}^{a}(k, Y) → h_{(i,X)}^{a}(j, T_{ψ}^{∗}Y). Then, for every such φ and Y let

G(i, X)_{ψ,Y} := h_{(i,X)}(ψ, 1_{T_{ψ}^{∗}Y}) : G(i, X)_{k}(Y) → (T_{ψ}^{∗})^{∧}G(i, X)_{j}(Y).

Explicitly, if γ^{T^{∗}} denotes the coherence constraint of the pseudo-functor T^{∗}, we get :

G(i, X)_{ψ,Y} (φ, f) = (φ ∘ ψ, γ_{(ψ,φ),X}^{T^{∗}} ∘ T_{φ}^{∗}f) for every (φ, f) ∈ G(i, X)_{k}(Y).

The rule : Y ↦ G(i, X)_{ψ,Y} defines a morphism G(i, X)_{ψ} : G(i, X)_{k}(Y) → (T_{ψ}^{∗})^{∧}G(i, X)_{j} of presheaves, and G(i, X)_{ψ}^{a} = F (i, X)_{ψ} for every morphism ψ of I. Now, denote by [i, X, ψ] : T_{ψ}^{∗}[i, X, k] → [i, X, j] the unique morphism in T_{j} fitting into the commutative diagrams :

\xymatrix{T_{ψ}^{∗}T_{φ}^{∗}X \ar[r]^{γ_{(ψ,φ),X}^{T^{∗}}} \ar[d]_{T_{ψ}^{∗}(e_{X,φ})} & T_{φ∘ψ}^{∗}X \ar[d]^{e_{X,φ∘ψ}} \\ T_{ψ}^{∗}[i, X, k] \ar[r]^{[i,X,ψ]} & [i, X, j]} for every φ : k → i.

The foregoing discussion easily implies that we have a commutative diagram of sheaves :

\xymatrix{F (i, X)_{k} \ar[r]^{∼} \ar[d]_{F (i,X)_{ψ}} & h_{[i,X,k]} \ar[d]^{h_{[i,X,ψ]}^{†}} \\ (T_{ψ}^{∗})_{∗}^{∼}F (i, X)_{j} \ar[r]^{∼} & (T_{ψ}^{∗})_{∗}^{∼}h_{[i,X,j]}}

whose horizontal arrows are the isomorphisms constructed in (ii), and the adjoint h_{[i,X,ψ]}^{†} of h_{[i,X,ψ]} is described explicitly as in (i). Summing up, we find that (up to natural isomorphism) the functor b_{T} is given by the rule :

(i, X) ↦ ([i, X, j], [i, X, ψ] | j ∈ Ob(I), ψ ∈ Morph(I)) for every (i, X) ∈ Ob(\underline{Can}(T)).

Corollary 4.6.28. Let I, J be two small categories, and T, S two fibred topoi over I. We have :

(i) For every morphism ω : T → S of fibred topoi over I, let η^{Top(ω)} be the unit of the adjunction for the pair of functors (Top(ω)^{∗}, Top(ω)_{∗}) provided by remark 4.6.12. Then (Top(ω)^{∗}, Top(ω)_{∗}, η^{Top(ω)}) is a morphism of topoi :

Top(ω) : Top(T) → Top(S).

(ii) For every functor ρ : J → I, let η^{Top(T/ρ)} be the unit of any adjunction for the pair (Top(T/ρ)^{∗}, Top(T/ρ)_{∗}). Then (Top(T/ρ)^{∗}, Top(T/ρ)_{∗}, η^{Top(T/ρ)}) is a morphism of topoi :

Top(T/ρ) : Top(T ∘ ρ) → Top(T).

(iii) For every pair ω : T → S, ν : S → U of morphisms of fibred topoi over I, we have :

Top(ν ∘ ω) = Top(ν) ∘ Top(ω).

Proof. Assertion (ii) is clear from remark 4.6.22(i).

(i): In view of proposition 4.6.10 and theorem 4.6.24, there remains only to check that Top(ω)^{∗} is left exact. Arguing as in remark 4.6.9(i,iii) we may assume that T and S are unital; then a functor E_{•} : J → Top(S) assigns to every j ∈ Ob(J) a datum E_{•,j} := (E_{ij}, E_{φ,j} | i ∈ Ob(I), φ ∈ Morph(I)) as in remark 4.6.9(i), and to every morphism f : j → j′ a system of morphisms (E_{i,f} : E_{ij} → E_{ij′} | i ∈ Ob(I)) such that

E_{φ,j′} ∘ S_{φ}^{∗}E_{i′,f} = E_{i,f} ∘ E_{φ,j} for every morphism φ : i → i′ of I.

The limit of E_{•} in the category Top(S) is the same as the colimit of E_{•}^{o} in the category Σ(ℱ ib(S^{∗})/I), and we construct the latter following remark 4.6.22(iii). Thus, for every i ∈ Ob(I) we consider the functor (E_{•}^{o})^{i} : J → ℱ ib(S^{∗}) given by the rules : j ↦ E_{ij} for every j ∈ Ob(J) and f ↦ E_{i,f}^{o} for every morphism f of J. Then (E_{•}^{o})^{i} factors through the inclusion functor S_{i}^{o} → ℱ ib(S^{∗}), and the colimit of the resulting functor (\overline{E}_{•}^{o})^{i} : J → S_{i}^{o} in S_{i}^{o} is the limit of the opposite functor \overline{E}_{•}^{i} : J^{o} → S_{i}, so we pick L(i) ∈ Ob(S_{i}) representing the latter limit and a universal cone τ^{i} : c_{L(i)} ⇒ E_{•}^{i}. Next, every morphism φ : i → i′ in I induces a natural transformation

\overline{E}_{•}^{φ} : S_{φ}^{∗} ∘ \overline{E}_{•}^{i′} ⇒ \overline{E}_{•}^{i} j ↦ E_{φ,j} : S_{φ}^{∗}E_{i′j} → E_{ij}.

Then there exists a unique morphism L(φ) : S_{φ}^{∗}L(i′) → L(i) of S_{i} fitting into the commutative diagram :

𝒟  :  \xymatrix{S_{φ}^{∗} ∘ c_{L(i′)} \ar@{=>}[r]^{c_{L(φ)}} \ar@{=>}[d]_{S_{φ}^{∗}∗τ^{i′}} & c_{L(i)} \ar@{=>}[d]^{τ^{i}} \\ S_{φ}^{∗} ∘ \overline{E}_{•}^{i′} \ar@{=>}[r]^{_{\overline{E}_{φ}_{•}}} & _{\overline{E}i•}}

and remark 4.6.22(iii) shows that L_{•} := (L(i), L(φ) | i ∈ Ob(I) φ ∈ Morph(I)) is an object of Top(S) representing the limit of E_{•}, and the system of cones (τ^{i} | i ∈ Ob(I)) adds up to a universal cone τ : c_{L_{•}} ⇒ E_{•}. Now, set

F_{•} := Top(ω)^{∗} ∘ E_{•} : J → Top(T).

We define likewise the functors \overline{F}_{•}^{i} : J → T_{i} for every i ∈ Ob(I) and the natural transformations \overline{F}_{•}^{φ} : T_{φ}^{∗} ∘ \overline{F}_{•}^{i′} ⇒ \overline{F}_{•}^{i} for every morphism φ : i → i′ of I; then by a simple inspection we deduce from 𝒟 the commutative diagram

𝒟^{ω}  :  \xymatrix{T_{φ}^{∗} ∘ c_{ω_{i′}^{∗}L(i′)} \ar@{=>}[r]^{c_{L(φ)^{ω}}} \ar@{=>}[d]_{T_{φ}^{∗}∗ω_{i′}^{∗}∗τ^{i′}} & c_{ω_{i}^{∗}L(i)} \ar@{=>}[d]^{ω_{i}^{∗}∗τ^{i}} \\ T_{φ}^{∗} ∘ \overline{F}_{•}^{i′} \ar@{=>}[r]^{_{\overline{F}_{φ}_{•}}} & _{\overline{F} i •}}

with (ω_{i}^{∗}L(i), L(φ)^{ω} | i ∈ Ob(I), φ ∈ Morph(I)) := Top(ω)^{∗}L_{•} as in remark 4.6.9(ii). Lastly, if J is a finite category, then by assumption ω_{i}^{∗} ∗ τ^{i} is still a universal cone for every i ∈ Ob(I), and therefore the commutativity of 𝒟^{ω} determines again L(φ)^{ω} uniquely. In this case, remark 4.6.22(iii) shows that Top(ω)^{∗}L_{•} represents the limit of F_{•} and Top(ω)^{∗} ∗ τ is a universal cone.

(iii) follows easily from the explicit description of η^{Top(ω)}, η^{Top(ν)} and η^{Top(ν∘ω)} given by remark 4.6.12. □

Remark 4.6.29. Let I be any small category.

(i) From corollary 4.6.28(i,iii) it follows easily that the rules : T ↦ Top(T), ω ↦ Top(ω) and Ξ ↦ Top(Ξ)_{∗} for every fibred topos T over I, every morphism ω of such fibred topoi, and every transformation Ξ of such morphisms, define a strict pseudo-functor

Top : PsFun(I, Topos) → Topos.

(ii) It is also easily seen that the isomorphisms of claim 4.6.25 yield a pseudo-natural isomorphism of pseudo-functors :

α_{•} : Top ∘ PsFun(I, lex.T) →^{∼} T ∘ totSite ∘ lex.\underline{ℱ ib} _{I}.

Moreover, let \underline{c}^{•} : fib.lex.Site(I) → PsFun(I, lex.Site) be a strict and strong pseudo-inverse for the 2-equivalence lex.\underline{ℱ ib} _{I} (see remark 4.5.2(v)); then from α_{•} ∗ \underline{c}^{•} we deduce a pseudonatural isomorphism of pseudo-functors (notation of (4.6.2)) :

a_{•} : Top ∘ \underline{lex.T} →^{∼} T ∘ totSite.

By 2-adjunction, from α_{•} and a_{•} we then deduce as well pseudo-natural transformations

β_{•} : Can ∘ Top ∘ PsFun(I, lex.T) → totSite ∘ lex.\underline{ℱ ib} _{I}  c ↦ β_{c}
b_{•} : Can ∘ Top ∘ \underline{lex.T} → totSite

where β_{c} is defined as in the proof of theorem 4.6.24: the details are left to the reader. (iii) Lastly, the rule T ↦ a_{T} of theorem 4.6.24(i) defines the pseudo-natural equivalence :

a_{•} := PsFun(I, η) ⊙ (a_{•} ∗ \underline{Can}) : Top →^{∼} T ∘ totSite ∘ \underline{Can}

where η : 1_{Topos} → T ∘ Can is the unit of the 2-adjoint pair (Can, T). Again by 2-adjunction, the rule : T ↦ b_{T} yields likewise a pseudo-natural transformation

b_{•} : Can ∘ Top → totSite ∘ \underline{Can}.

4.7. Localization and points of a topos. Let 𝒞 be a category, and f : Y → X any morphism of 𝒞. We consider the source functor s_{X} : 𝒞/X → 𝒞 as in (1.1.24), and the induced functor

(s_{X})_{|f} : (𝒞 /X)/f → 𝒞 /(s_{X}f) = 𝒞 /Y

as in (1.1.26), which is obviously an isomorphism of categories, hence it induces a bijection :

{sieves of 𝒞 /Y } →^{∼} {sieves of (𝒞 /X)/f } 𝒮 ↦ (s_{X})_{|f}^{−1}𝒮

(notation of definition 3.5.1(iii)). Fix a universe V with U ⊂ V and such that 𝒞 is V-small, so that the functor (s_{X})_{V!} : (𝒞 /X)_{V}^{∧} → 𝒞_{V}^{∧} is well defined. We notice that for every subobject R of the presheaf h_{f} on 𝒞 /X, the presheaf (s_{X})_{V!}R is a subobject of h_{Y} : more precisely, in light of (1.4.11) we see that for a sieve 𝒮 of (𝒞 /X)/f and a sieve 𝒯 of 𝒞 /Y, we have :

(4.7.1)  h_{𝒯} = (s_{X})_{V!}h_{𝒮} ⇔ 𝒮 = (s_{X})_{|f}^{−1}𝒯 .

Let now J be a topology on 𝒞; endow 𝒞 /X with the topology J_{X} induced by J via s_{X}, and set

C := (𝒞 , J) C/X := (𝒞 /X, J_{X}).

Since (s_{X})_{V!} commutes with fibre products (proposition 1.4.13(vi.c)), lemma 4.2.17(ii.a) says that a subobject R ⊂ h_{f} covers f for the topology J_{X} if and only if the induced morphism (s_{X})_{V!}R → (s_{X})_{V!}(h_{f}) = h_{Y} covers Y for the topology J. In other words, a family of morphisms (h_{λ}/X : (f_{λ} : Y_{λ} → Y) → f | λ ∈ Λ) covers f for the topology J_{X} if and only if the family (h_{λ} : Y_{λ} → Y | λ ∈ Λ) covers Y for the topology J. In view of (4.7.1), it follows easily that s_{X} is both continuous and cocontinuous for the topologies J and J_{X}.

4.7.2. Suppose now that C is a U-site; then the same holds for C/X as well : indeed, if G ⊂ Ob(𝒞) is an essentially small topologically generating family for C, then G/X ⊂ Ob(𝒞 /X) is an essentially small topologically generating family for C/X (notation of definition 4.1.20(i)). By virtue of corollary 4.3.19, the functor s̃_{X∗} ≃ s̆_{X}^{∗} : C^{∼} → (C/X)^{∼} admits therefore both a left adjoint s̃_{X}^{∗} and a right adjoint s̆_{X∗}. We introduce a special notation and terminology for these functors :

• The functor s̃_{X∗} shall be also denoted j_{X}^{∗}, and called the functor of restriction to X.

• The functor s̆_{X∗} shall be denoted j_{X∗}, and called the direct image functor.

• The functor s̃_{X}^{∗} shall be denoted j_{X!}, and called the functor of extension by empty. Thus, j_{X}^{∗} is right adjoint to j_{X!}, and left adjoint to j_{X∗}. However, a left adjoint for j_{X}^{∗} can be exhibited alternatively by a more explicit construction, which will allow us to extract some useful additional properties of the category (C/X)^{∼}. Indeed, even though 𝒞 /X is not necessarily Usmall, we can invoke proposition 1.4.13(vi.a) and remark 4.1.23(ii), in order to define a functor by the same rule as in (4.2.8), namely

j_{X!} : (C/X)_{U}^{∼} → C_{U}^{∼} F ↦ (s_{X!} ∘ i_{C/X}F)^{a}

where i_{C/X} : (C/X)^{∼} → (𝒞 /X)^{∧} is the forgetful functor. Moreover, remark 4.1.19(v) (and again remark 4.1.23(ii)) implies that this functor is (isomorphic to) the restriction of (s̃_{X})_{V}^{∗}, and since the inclusion functor C_{U}^{∼} → C_{V}^{∼} is fully faithful, we deduce that it is also a left adjoint to j_{X}^{∗}. Furthermore, recall that s_{X!} factors through the source functor s_{h_{X}} : 𝒞^{∧}/h_{X} → 𝒞^{∧} and an equivalence e_{X} : (𝒞 /X)^{∧} →^{∼} 𝒞^{∧}/h_{X} (proposition 1.4.13(vi.b)); consequently, j_{X!} factors through the source functor s_{h_{X}^{a}} : C^{∼}/h_{X}^{a} → C^{∼} and the composition

ẽ_{X} : (C/X)^{∼} ⟶^{i_{C/X}} (𝒞 /X)^{∧} ⟶^{e_{X}} 𝒞^{∧}/h_{X} ⟶^{(−)_{|h_{X}}^{a}} C^{∼}/h_{X}^{a}

where (−)_{|h_{X}}^{a} is induced by (−)^{a} : 𝒞^{∧} → C^{∼} and the object h_{X} ∈ Ob(𝒞^{∧}), as in (1.1.26).

Remark 4.7.3. (i) Let g : Y → Z be any morphism in 𝒞; then we have the sites C/Y and C/Z as in (4.7), as well as the functor g_{∗} : 𝒞 /Y → 𝒞 /Z of (1.1.25). The isomorphism of categories (s_{Z})_{|g} : (𝒞 /Z)/g →^{∼} 𝒞 /Y identifies g_{∗} with the source functor s_{g} : (𝒞 /Z)/g → 𝒞 /Z. Hence, the discussion of (4.7.2) applies to g_{∗} as well, and since s_{Z} ∘ g_{∗} = s_{Y}, we see that J_{Y} is the topology induced by J_{Z} via g_{∗} on 𝒞 /Y; so, g_{∗} is continuous and cocontinuous for the topologies J_{Y} and J_{Z}. Moreover, if C is a U-site, then (g̃_{∗})_{∗} : (C/Z)^{∼} → (C/Y)^{∼} admits a left adjoint (g̃_{∗})^{∗} and a right adjoint (ğ_{∗})_{∗}. In agreement with the foregoing, we let

j_{g}^{∗} := (g̃_{∗})_{∗}  j_{g∗} := (ğ_{∗})_{∗}  j_{g!} := (g̃_{∗})^{∗}.

Clearly j_{g}^{∗}∘j_{Z}^{∗} = j_{Y}^{∗}, and we have isomorphisms of functors : j_{Z∗}∘j_{g∗} →^{∼} j_{Y ∗} and j_{Z!}∘j_{g!} →^{∼} j_{Y!}.

(ii) By propositions 4.4.8(i) and 1.3.25(iii,iv), we also see that the pairs (j_{X}^{∗}, j_{X∗}) and (j_{g}^{∗}, j_{g∗}) determine morphisms of topoi, unique up to unique isomorphism :

j_{X} : (C/X)^{∼} → C^{∼} j_{g} : (C/Y)^{∼} → (C/Z)^{∼}.

(iii) Suppose that all finite products of 𝒞 are representable. Then for every X ∈ Ob(𝒞), the source functor s_{X} admits a right adjoint

p_{X} : 𝒞 → 𝒞 /X Y ↦ (p_{X,Y} : X × Y → X)

where X × Y is any choice of a representative for the product of X and Y, and p_{X,Y} is the corresponding natural projection : see proposition 1.4.13(iii). Since s_{X} is cocontinuous for the topologies J and J_{X}, the functor p_{X} is a morphism of sites C/X → C (remark 4.4.13(v)), and we have an isomorphism of morphisms of topoi :

j_{X} →^{∼} p̃_{X}.

Proposition 4.7.4. With the notation of (4.7.2), the following holds :

(i) The functor ẽ_{X} is an equivalence.

(ii) We have essentially commutative diagrams :

\xymatrix{(C/X)^{∼} \ar[r]^{i_{C/X}} \ar[d]_{ẽ_{X}} & (𝒞 /X)^{∧} \ar[d]^{e_{X}} & (𝒞 /X)^{∧} \ar[d]_{e_{X}} \ar[r]^{(−)^{a}} & (C/X)^{∼} \ar[d]^{ẽ_{X}} \\ C^{∼}/h_{X}^{a} \ar[r]^{(i_{C})_{|h_{X}}} & 𝒞^{∧}/h_{X} & 𝒞^{∧}/h_{X} \ar[r]^{(−)_{|h_{X}}^{a}} & C^{∼}/h_{X}^{a}}

where (i_{C})_{|h_{X}} and (−)_{|h_{X}}^{a} are the functors attached to the adjoint pair ((−)^{a}, i_{C}) and the object h_{X} ∈ Ob(𝒞^{∧}), as in example 1.2.28(i).

Proof. Let η_{F} : h_{X} → h_{X}^{a} be the natural morphism in 𝒞^{∧}. We remark :

Claim 4.7.5. For every presheaf F on 𝒞 /X there exists a cartesian diagram in 𝒞^{∧} :

\xymatrix{s_{X!}(F^{a}) \ar[r] \ar[d]_{e_{X}(F^{a})} & (s_{X!}F)^{a} \ar[d]^{e_{X}(F)^{a}} \\ h_{X} \ar[r]^{η_{X}} & h_{X}^{a}.}

Proof of the claim. As already pointed out in the proof of proposition 1.4.13(vi.b), the presheaf 𝟙_{𝒞 /X} := h_{1_{X}} is a final object of (𝒞 /X)^{∧}, and the equivalence e_{X} assigns to every presheaf F on 𝒞 /X the h_{X}-presheaf e_{X}(F) : s_{X!}F → s_{X!}(𝟙_{𝒞 /X}) →^{∼} h_{X} deduced from the unique morphism F → 𝟙_{𝒞 /X}. Moreover, e_{X} admits the quasi-inverse

e′_{X} : 𝒞^{∧}/h_{X} → (𝒞 /X)^{∧} (φ : G → h_{X}) ↦ s_{X}^{∧}(G) ×_{s_{X}^{∧}(h_{X})} 𝟙_{𝒞 /X}

where the fibre product is defined via the unit of adjunction η_{𝟙} : 𝟙_{𝒞 /X} → s_{X}^{∧}(s_{X!}𝟙_{𝒞 /X}) →^{∼} s_{X}^{∧}(h_{X}) and the morphism s_{X}^{∧}(φ). Thus, we have a natural isomorphism

F →^{∼} s_{X}^{∧}(s_{X!}F) ×_{s_{X}^{∧}(h_{X})} 𝟙_{𝒞 /X} in (𝒞 /X)^{∧} .

On the other hand, the functor s_{X}^{∧} commutes with the functor G ↦ G^{a} (corollary 4.3.19(iii)), so there follows an isomorphism

F^{a} →^{∼} s_{X}^{∧}(s_{X!}F)^{a} ×_{s_{X}^{∧}(h_{X}^{a})} 𝟙_{𝒞 /X} in (C/X)^{∼}.

But it is easily seen that the morphism η_{𝟙}^{a} : 𝟙_{𝒞 /X} → s_{X}^{∧}(h_{X}^{a}) is the composition of η_{𝟙} and the natural morphism s_{X}^{∧}(h_{X}) → s_{X}^{∧}(h_{X}^{a}), so we get a commutative diagram with cartesian squares:

\xymatrix{F^{a} \ar[r]^{α} \ar[d] & s_{X}^{∧}((s_{X!}F)^{a} ×_{h_{X}^{a}} h_{X}) \ar[r] \ar[d] & s_{X}^{∧}(s_{X!}F)^{a} \ar[d] \\ 𝟙_{𝒞 /X} \ar[r] & s_{X}^{∧}(h_{X}) \ar[r] & s_{X}^{∧}(h_{X}^{a}).}

Especially, α induces an isomorphism F^{a} →^{∼} e′_{X}((s_{X!}F)^{a} ×_{h_{X}^{a}} h_{X}), whence an isomorphism

e_{X}(F^{a}) →^{∼} ((s_{X!}F)^{a} ×_{h_{X}^{a}} h_{X} → h_{X})

and the claim follows. ♢

(i): Let φ : G → h_{X}^{a} be any object of C^{∼}/h_{X}^{a}; set G′ := G ×_{h_{X}^{a}} h_{X}, and let φ′ : G′ → h_{X} be the induced projection. Let also F := e′_{X}(φ′), which is a presheaf on 𝒞 /X. We claim that F is a sheaf for the topology J_{X}. Indeed, by claim 4.7.5we have an isomorphism in 𝒞^{∧}/h_{X} :

s_{X!}(F^{a}) →^{∼} (s_{X!}F)^{a} ×_{h_{X}^{a}} h_{X} →^{∼} G^{′a} ×_{h_{X}^{a}} h_{X} →^{∼} G′

and since e′_{X} is quasi-inverse to e_{X}, we have an isomorphism of h_{X}-presheaves s_{X!}(F) →^{∼} G′. Thus, F →^{∼} F^{a} in (𝒞 /X)^{∧}, as required. Next we claim that the resulting functor

ẽ′_{X} : C^{∼}/h_{X}^{a} → (C/X)^{∼} (φ : G ⟶^{φ} h_{X}^{a}) ↦ e′_{X}(G ×_{h_{X}^{a}} h_{X} ⟶^{φ×_{h^{a}_{X}} h_{X}} h_{X})

is a quasi-inverse for ẽ_{X}. Indeed, for every φ as in the foregoing we have an isomorphism

ẽ_{X} ∘ ẽ′_{X}(φ) →^{∼} (φ ×_{h_{X}^{a}} h_{X})^{a} →^{∼} φ in C^{∼}/h_{X}^{a}

which is natural with respect to φ. Lastly, for every sheaf F on C/X we have ẽ′_{X} ∘ ẽ_{X}(F) = e′_{X}((e_{X}(F)^{a} ×_{h_{X}^{a}} h_{X}), which is isomorphic to e′_{X}(e_{X}(F)) ≃ F, by claim 4.7.5.

(ii): For every sheaf F on C/X, claim 4.7.5yields an isomorphism :

e_{X}(i_{C/X}F) →^{∼} (ẽ_{X}(F) ×_{h_{X}^{a}} h_{X} → h_{X}).

The essential commutativity of the left square diagram in (ii) is an immediate consequence; since e_{X} and ẽ_{X} are both equivalences, and since (−)_{|h_{X}}^{a} is left adjoint to (i_{C})_{|h_{X}}, we deduce the essential commutativity also for the right square diagram. □

Remark 4.7.6. (i) Under the equivalence ẽ_{X} of proposition 4.7.4, the functor j_{X}^{∗} is identified with the functor :

C^{∼} → C^{∼}/h_{X}^{a} F ↦ (F × h_{X}^{a} → h_{X}^{a})

and j_{X!} is identified with the source functor s_{h_{X}^{a}} : C^{∼}/h_{X}^{a} → C^{∼} of (1.1.24).

(ii) Likewise, the functor j_{g}^{∗} of remark 4.7.3(i) is identified with the functor

C^{∼}/h_{Z}^{a} → C^{∼}/h_{Y}^{a} (F → h_{Z}^{a}) ↦ (F ×_{h_{Z}^{a}} h_{Y}^{a} ↦ h_{Y}^{a})

and j_{g!} is identified with the functor (h_{g}^{a})_{∗} : C^{∼}/h_{Y}^{a} → C^{∼}/h_{Z}^{a} induced by h_{g}^{a} : h_{Y}^{a} → h_{Z}^{a}.

Proposition 4.7.7. Let C := (𝒞 , J) and C′ := (𝒞′; J′) be two sites, u : 𝒞 → 𝒞′ a continuous functor, X any object of 𝒞. Then we have :

(i) The functor u_{|X} : 𝒞 /X → 𝒞′/uX is continuous for the sites C/X and C′/uX.

(ii) We have essentially commutative diagrams :

\xymatrix{(C′/uX)^{∼} \ar[r]^{^{ũ_{|X}^{∗}}} \ar[d]_{ẽ_{uX}} & (C/X)^{∼} \ar[d]^{ẽ_{X}} & (C/X)^{∼} \ar[d]_{ẽ_{X}} \ar[r]^{^{ũ_{|X}^{∗}}} & (C′/uX)^{∼} \ar[d]^{ẽ_{uX}} \\ C^{′∼}/h_{uX}^{a} \ar[r]^{(ũ_{∗})_{|h^{a}_{X}}} & C^{∼}/h_{X}^{a} & C^{∼}/h_{X}^{a} \ar[r]^{(u^{∗})_{|h^{a}_{X}}} & C^{′∼}/h_{uX}^{a}.}

(iii) If u is a morphism of sites C′ → C, then u_{|X} is a morphism of sites C′/uX → C/X.

Proof. (i): Combining propositions 4.7.4(ii) and 1.4.19(i) with example 1.2.28(iii) we get an essentially commutative diagram, whose vertical arrows are equivalences :

\xymatrix{(C′/uX)^{∼} \ar[r]^{(u_{|X})^{∧}∘i_{C′/X}} \ar[d]_{ẽ_{uX}} & (𝒞 /X)^{∧} \ar[d]^{e_{X}} \\ C^{′∼}/h_{uX}^{a} \ar[r]^{(u^{∧}∘i_{C′})_{|h_{X}}} & 𝒞^{∧}/h_{X}.}

Now, let F be a sheaf on C′/uX, and denote by φ : G → h_{uX}^{a} the h_{uX}^{a}-sheaf ẽ_{uX}(F). By proposition 4.7.4(ii), it suffices to show that (u^{∧} ∘ i_{C′})_{|h_{X}}(φ) ≃ (u^{∧}G ×_{u^{∧}h_{uX}^{a}} h_{X} → h_{X}) lies in the essential image of (i_{C})_{|h_{X}}. But the morphism h_{X} → u^{∧}h_{uX}^{a} is the composition of

the natural morphisms h_{X} → h_{X}^{a} and h_{X}^{a} → u^{∧}h_{uX}^{a}, hence u^{∧}G ×_{u^{∧}h_{uX}^{a}} h_{X} →^{∼} G′ ×_{h_{X}^{a}} h_{X},

with G′ := u^{∧}G ×_{u^{∧}h_{uX}^{a}} h_{X}^{a}, and the latter is a sheaf on C, since u is continuous, whence the contention.

(ii): Recall that we have a natural isomorphism ũ^{∗}(h_{X}^{a}) →^{∼} h_{uX}^{a} in C^{′∼} (lemma 4.2.11(ii)). By example 1.2.28(i), we deduce an adjoint pair of functors :

(ũ^{∗})_{|h_{X}^{a}} : C^{∼}/h_{X}^{a} → C^{′∼}/h_{uX}^{a} (ũ_{∗})_{|h_{X}^{a}} : C^{′∼}/h_{uX}^{a} → C^{∼}/h_{X}^{a}.

Now, the proof of (i) yields a natural isomorphism of functors :

ẽ_{X} ∘ (u_{|X})_{∗}^{∼} →^{∼} (−)_{|h_{X}}^{a} ∘ e_{X} ∘ i_{C/X} ∘ (u_{|X})_{∗}^{∼}

→^{∼} (−)_{|h_{X}}^{a} ∘ e_{X} ∘ (u_{|X})^{∧} ∘ i_{C′/uX}

→^{∼} (−)_{|h_{X}}^{a} ∘ (u^{∧} ∘ i_{C′})_{|h_{X}} ∘ ẽ_{uX}

and a direct inspection of the constructions shows that (−)_{|h_{X}}^{a} ∘ (u^{∧} ∘ i_{C′})_{|h_{X}} is isomorphic to the functor (ũ_{∗})_{|h_{X}^{a}}, so that the left square diagram of (ii) is essentially commutative. Since ẽ_{X} and ẽ_{uX} are equivalences, we deduce that the same holds also for the right square diagram.

(iii): By assumption, the functor ũ^{∗} is left exact, so the same holds for (ũ^{∗})_{|h_{X}^{a}} (proposition 1.4.19(ii)), and by (ii) the same follows for ũ_{|X}^{∗}, whence the contention. □

Example 4.7.8. (i) Let T be a topos, and U ∈ Ob(T) any object. By proposition 4.7.4(i) and remark 4.7.3(iii), we have a natural equivalence

(Can(T)/U)^{∼} →^{∼} Can(T)^{∼}/h_{U}

identifying the functor j_{U}^{∗} : Can(T)^{∼} → (Can(T)/U)^{∼} with p_{h_{U}} : Can(T)^{∼} → Can(T)^{∼}/h_{U}. The source functor s_{h_{U}} : Can(T)^{∼}/h_{U} → Can(T)^{∼} is a left adjoint for p_{h_{U}}, but the latter admits also a right adjoint, which is naturally identified with j_{U∗} : (Can(T)/U)^{∼} → Can(T)^{∼}. On the other hand, the Yoneda equivalence T →^{∼} Can(T)^{∼} induces an equivalence T/U →^{∼} Can(T)^{∼}/h_{U}, which shows that T/U is a topos, and identifies p_{h_{U}} in turn with the functor

p_{U} : T → T/U X ↦ X_{|U} := (X × U → U).

Hence, the latter is a morphisms of U-sites Can(T) → Can(T/U), and we denote again by

j_{U} := (j_{U}^{∗}, j_{U∗}, η^{U}) : T/U → T

the corresponding morphism of topoi. Moreover, the source functor s_{U} is a left adjoint for j_{U}^{∗}, and shall be denoted also by j_{U!} : T/U → T. Likewise, any morphism f : U′ → U in T determines, up to unique isomorphism, a morphism of topoi

j_{f} := (j_{f}^{∗}, j_{f∗}, η^{f}) : T/U′ → T/U

with an isomorphism j_{U} ∘ j_{f} →^{∼} j_{U′} of morphisms of topoi.

(ii) Recall that the target functor t : Morph(T) → T is a fibration (example 3.1.3(iii)); pick a cleavage for t, and let c : T^{o} → V-Cat be its associated pseudo-functor (for a universe V such that T is V-small). We may choose the cleavage so that for every morphism f : U′ → U, the functor c_{f} : T/U → T/U′ agrees with j_{f}^{∗} (the details are left to the reader). Notice then that c^{o} : T → V-Cat^{o} factors through the forgetful strict pseudo-functor

Link(V-Cat^{o}) → V-Cat^{o}  𝒜 ↦ 𝒜  (F, G, η, ε) ↦ F

and a pseudo-functor

T → Link(V-Cat^{o})  U ↦ T/U  (U′ ⟶^{f} U) ↦ (j_{f}^{∗}, j_{f∗}, η^{f}, ε^{f})

where ε^{f} is the counit for the adjunction of the pair (j_{f}^{∗}, j_{f∗}), whose unit is η^{f}. We compose the latter with the strict isomorphism of 2-categories Link(V-Cat^{o}) →^{∼} ^{o}Link(V-Cat) provided by proposition 2.3.4, and notice that the resulting pseudo-functor maps T to the strong sub-2-category^{o}Topos of^{o}Link(V-Cat). Thus, we obtain a well defined fibred topos over T :

T/− : T = ^{o}T → Topos  U ↦ T/U  (U′ ⟶^{f} U) ↦ j_{f}.

(iii) In case U is a subobject of the final object 1_{T} of T, the morphism j_{U} of (i) is called an open subtopos of T. In this case we denote by CU the full subcategory of T such that

Ob(CU) = {X ∈ Ob(T) | j_{U}^{∗}X = 1_{T/U}}.

Then CU is a topos, called the complement of U in T, and the inclusion functor i_{∗} : CU → T admits a left adjoint i^{∗} : T → CU, namely, the functor which assigns to every X ∈ Ob(T) the push-out X_{|CU} in the cocartesian diagram :

\xymatrix{X × U \ar[r]^{p_{X}} \ar[d]_{p_{U}} & X \ar[d] \\ U \ar[r] & X_{|CU}}

where p_{X} and p_{U} are the natural projections. Moreover, i^{∗} is an exact functor, hence the adjoint pair (i^{∗}, i_{∗}) defines a morphism of topoi CU → T, unique up to unique isomorphism. (See [8, Exp.IV, Prop.9.3.4].)

Remark 4.7.9. (i) Let f : T′ → T be a morphism of topoi, U ∈ Ob(T), and let us fix final objects 1_{T} of T and 1_{T′} of T′. For every Y ∈ Ob(T′), let also u_{Y} : Y → 1_{T′} be the unique morphism in T′. Suppose that s : T′ → T/U is a morphism of topoi with an isomorphism ω : f →^{∼} j_{U} ∘ s. Hence, we have a functorial isomorphism (notation of remark 1.1.17(ii))

ω_{X}^{†} : s^{∗}(j_{U}^{∗}X) →^{∼} f^{∗}X for every X ∈ Ob(T).

So, s^{∗}1_{U} →^{∼} f^{∗}1_{T} is a final object of T′, hence u_{s^{∗}1_{U}} : s^{∗}1_{U} → 1_{T′} is an isomorphism. Let

Δ_{U} : 1_{U} → (U × U ⟶^{j_{U}^{∗} (U)} U) be the diagonal morphism; then

σ(s, ω) := ω_{U}^{†} ∘ s^{∗}(Δ_{U}) ∘ u_{s^{∗}1_{U}}^{−1} : 1_{T′} → f^{∗}U

is an element of Γ(T′, f^{∗}U) (notation (4.4.11)). Moreover, for every object φ : X → U of T/U, we have a cartesian diagram in T/U :

𝒟 : \xymatrix{X \ar[rr]^{Γ_{φ}} \ar[dd]_{φ/U} \ar[dr]^{φ} & & X × U \ar[dd]^{j_{U}^{∗} φ} \ar[dl]_{j_{U}^{∗} X} \\ & U \\ U \ar[ur]_{1_{U}} \ar[rr]_{Δ_{U}} & & U × U \ar[ul]^{j_{U}^{∗} U}}

where Γ_{φ} is the graph of φ, and s^{∗}𝒟 is isomorphic to the cartesian diagram (in T′) :

ℰ : \xymatrix{s^{∗}φ \ar[r]^{ω_{X}^{†}∘s^{∗}Γ_{φ}} \ar[d]_{u_{s^{∗}1_{U}} ∘s^{∗}(φ/U)} & f^{∗}X \ar[d]^{f^{∗}φ} \\ 1_{T′} \ar[r]^{σ(s,ω)} & f^{∗}U.}

This shows that s^{∗} – and therefore also s – is determined, up to isomorphism, by σ(s, ω).

(ii) Conversely, if σ ∈ Γ(T′, f^{∗}U) is any global section, we may define a functor s^{∗} : T/U → T′ by means of the cartesian diagram ℰ : this amounts to choosing a representative in T′ for the fibre product 1_{T′} ×_{(σ,f^{∗}φ)} f^{∗}X, for every object φ : X → U of T/U. We claim that there exists an isomorphism of functors

s^{∗} ∘ j_{U}^{∗} →^{∼} f^{∗}.

Indeed, recall that j_{U}^{∗}X = (q_{X} : X × U → U), where X × U is a representative of the product of X and U in T, and q_{X} is the natural projection; let also p_{X} : X × U → X be the other projection. Likewise, pick a representative f^{∗}X × f^{∗}U for the fibre product of f^{∗}X and f^{∗}U in T′, and let p_{f^{∗}X} : f^{∗}X × f^{∗}U → f^{∗}X and q_{f^{∗}X} : f^{∗}X × f^{∗}U → f^{∗}U be the projections.

Since f^{∗} is left exact, there exists a unique isomorphism λ_{X} : f^{∗}X × f^{∗}U →^{∼} f^{∗}(X × U) in T′ such that f^{∗}(p_{X}) ∘ λ_{X} = p_{f^{∗}X} and f^{∗}(q_{X}) ∘ λ_{X} = q_{f^{∗}X}. We deduce a commutative diagram :

\xymatrix{f^{∗}X \ar[r]^{1_{f^{∗}X}×(σ∘u_{f^{∗}X})} \ar[d]_{u_{f^{∗}X}} & f^{∗}X × f^{∗}U \ar[d]^{q_{f^{∗}X}} \ar[r]^{λ} & f^{∗}(X × U) \ar[dl]^{f^{∗}j_{U}^{∗} X} \\ 1_{T′} \ar[r]^{σ} & f^{∗}U}

whose square subdiagram is cartesian (the details are left to the reader). The assertion follows straightforwardly. Let us also define a functor t : T′ → T′/f^{∗}U, by the rule :

tY := σ ∘ u_{Y} for every Y ∈ Ob(T′).

By inspecting the diagram ℰ, we deduce natural bijections :

Hom_{T′}(Y, s^{∗}φ) →^{∼} Hom_{T′/f^{∗}U}(tY, f^{∗}φ) for every Y ∈ Ob(T′) and φ ∈ Ob(T/U).

Since f^{∗} is exact, it follows easily that s^{∗} is left exact (indeed, s^{∗} commutes with all the limits with which f^{∗} commutes). Moreover, since all colimits are universal in T (see (4.1.19)(i)), it is easily seen that s^{∗} commutes with all colimits. Then, by proposition 4.4.8(i.b), the functor s^{∗} determines a morphism of topoi s : T′ → T/U, unique up to unique isomorphism, with an isomorphism f →^{∼} j_{U} ∘ s.

(iii) Next, let s, s′ : T′ → T/U be two morphisms of topoi, ω : f →^{∼} j_{U}∘s and ω′ : f →^{∼} j_{U}∘s′ two isomorphisms, and β : s ⇒ s′ any natural transformation such that

(4.7.10)  (j_{U} ∗ β) ⊙ ω = ω′.

By remark 1.1.17(iv,vi), it follows that ω^{′†} = ω^{†} ⊙ (β^{†} ∗ j_{U}^{∗}). We may then compute :

\begin{matrix}σ(s′, ω′) = ω_{U}^{′†} ∘ s^{′∗}(Δ_{U}) ∘ u_{s^{′∗}1_{U}}^{−1} = & ω_{U}^{†} ∘ β_{j_{U}^{∗} U}^{†} ∘ s^{′∗}(Δ_{U}) ∘ u_{s^{′∗}1_{U}}^{−1} \\ = & ω_{U}^{†} ∘ s^{∗}(Δ_{U}) ∘ β_{1_{U}}^{†} ∘ u_{s^{′∗}1_{U}}^{−1} \\ = & σ(s, ω). \end{matrix}

By considering the cartesian diagram ℰ of (i), we then deduce that for every object φ : X → U of T/U there exists a unique isomorphism

λ_{φ} : s^{′∗}φ →^{∼} s^{∗}φ such that ω_{X}^{†} ∘ s^{∗}Γ_{φ} ∘ λ_{φ} = ω_{X}^{′†} ∘ s^{′∗}Γ_{φ}.

Since ω_{X}^{†} is an isomorphism, the latter identity yields :

s^{∗}Γ_{φ} ∘ λ_{φ} = s^{∗}Γ_{φ} ∘ β_{φ}^{†}.

However, Γ_{φ} is a monomorphism, hence the same holds for s^{∗}Γ_{φ} (proposition 1.3.18(i)), whence λ_{φ} = β_{φ}^{†}. Hence, β is the unique isomorphism of functors s →^{∼} s′ verifying (4.7.10).

(iv) Lastly, consider a morphism ψ : U → V of T, and σ ∈ Γ(T′, f^{∗}U). Let s^{U} : T/U → T′ be the morphism of topoi deduced from σ as in (ii), with its associated isomorphism of functors ω^{U} : s^{U∗} ∘ j_{U}^{∗} →^{∼} f^{∗}. Likewise, let s^{V} : T/V → T′ be the morphism of topoi and ω^{V} : s^{V ∗} ∘ j_{V}^{∗} →^{∼} f^{∗} the isomorphism deduced, again as in (ii), from f^{∗}(ψ) ∘ σ ∈ Γ(T′, f^{∗}V). A direct inspection of the constructions yields a natural isomorphism of functors

β : s^{V ∗} →^{∼} s^{U∗} ∘ j_{ψ}^{∗} such that ω^{V} = ω^{U} ⊙ (s^{U∗} ∗ γ_{ψ,V}^{† −1}) ⊙ (β ∗ j_{V}^{∗})

where γ_{ψ,V} : j_{V} ∘ j_{ψ} →^{∼} j_{U} is the coherence constraint of the fibred topos T/− of example 4.7.8(ii). In view of (iii), it follows easily that for every pair (s, ω) as in (i) we have :

σ(j_{ψ} ∘ s, (γ_{ψ,V}^{−1} ∗ s) ⊙ ω) = f^{∗}(ψ) ∘ σ(s, ω)

Definition 4.7.11. Let T be a topos.

(i) A point of T (or a T -point) is a morphism of topoi Set → T. If ξ = (ξ^{∗}, ξ_{∗}) is a point, and F is any object of T, the set ξ^{∗}F is usually denoted by F_{ξ}.

(ii) If ξ is a T -point, and f : T → S is any morphism of topoi, we denote by f(ξ) the S-point f ∘ ξ. If β : ξ → ξ′ is any morphism of T -points, we let likewise f(β) := f_{∗} ∗ β.

(iii) A neighborhood of ξ is a pair (U, a), where U ∈ Ob(T), and a ∈ U_{ξ}. A morphism of neighborhoods (U, a) → (U′, a′) is a morphism f : U → U′ in T such that f_{ξ}(a) = a′. The category of all neighborhoods of ξ shall be denoted Nbd(ξ).

(iv) A family (f_{λ} : T_{λ} → T | λ ∈ Λ) of morphisms of topoi is conservative, if the functor

T → ∏_{λ∈Λ} T_{λ} F ↦ (f_{λ}^{∗}F | λ ∈ Λ)

is conservative (definition 1.1.4(ii)) (the product ∏_{λ∈Λ} T_{λ} is formed in a sufficiently large universe containing U). We say that T has enough points, if T has a conservative set of points.

(v) Consider a graph Γ (see definition 1.6.1(i)) and a system (C_{S} | S ∈ Ob(Topos)), where C_{S} is a given set of morphisms of graphs Γ → S, for every topos S, such that f^{∗}φ ∈ C_{S′}, for every φ ∈ C_{S} and every morphism f : S′ → S of topoi. Let T_{•} := (T_{λ} ⟶^{f_{λ}} T | λ ∈ Λ) be a conservative family of morphisms of topoi, and P(S, φ) a property defined on elements φ ∈ C_{S}, for every topos S; we say that P can be checked on T_{•}, if for every φ ∈ C_{T} we have :

• P(T, φ) holds if and only if P(T_{λ}, f_{λ}^{∗} ∘ φ) holds for every λ ∈ Λ. Moreover, we say that P can be checked on stalks, if for every topos S and every conservative set Ω of S-points, the property P can be checked on Ω.

Example 4.7.12. Let T be a topos, and (U_{λ} → 1_{T} | λ ∈ Λ) any family of morphisms of T covering the final object 1_{T} (for the canonical topology of T); then the induced family of morphisms of topoi (j_{U_{λ}} : T/U_{λ} → T | λ ∈ Λ) is conservative. For the proof, we easily reduce to the case where Λ is a small set, by lemma 4.1.22; then it suffices to recall that, by remark 4.4.1(i), every X ∈ Ob(T) is the coequalizer of the induced pair of morphisms of T

\xymatrix{∐_{λ,μ∈Λ} U_{λ} × U_{μ} × X \ar[r] \ar[r] & ∐_{λ∈Λ} U_{λ} × X}

(the details are left to the reader).

Remark 4.7.13. (i) With the notation of definition 4.7.11, notice that

Nbd(ξ)^{o} = ℱ ib(ξ^{∗})

where the fibration π : ℱ ib(ξ^{∗}) → T^{o} is defined as in (3.1.15).

(ii) Moreover, the category Nbd(ξ) is finitely complete. Indeed, let Λ be any finite category, and F : Λ → Nbd(ξ) any functor; say that F λ = (U_{λ}, a_{λ}) for every λ ∈ Ob(Λ). Then it is easily seen that the limit of F is represented by (U, a), where U ∈ Ob(T) represents the limit of the functor π^{o} ∘ F : Λ → T, and a ∈ U_{ξ} = lim_{Λ} ξ^{∗} ∘ π^{o} ∘ F is the unique element whose image under the induced projection U_{ξ} → U_{λ,ξ} agrees with a_{λ}, for every λ ∈ Ob(Λ).

(iii) Let ξ be a point of the topos T; from (ii) it follows that the category Nbd(ξ) is cofiltered. (iv) In view of (i) and (1.4.7), for every F ∈ Ob(T) there is a natural cocone :

[figure: Nbd(ξ)^{o}; Hom_{T} (π,F); c_{F_{ξ}}; τ_{ξ,F}; Set]

assigning to every neighborhood (U, a) of ξ, the map τ_{ξ,F} (U, a) : Hom_{T} (U, F) → F_{ξ} such that s ↦ s_{ξ}(a) for every s : U → F. Then lemma 1.4.8says that τ_{ξ,F} induces a natural bijection

(4.7.14)  colim_{Nbd(ξ)^{o}} Hom_{T} (π, F) →^{∼} F_{ξ}  for every F ∈ Ob(T).

(v) It follows from remark 4.7.9(iii), that a neighborhood (U, a) of ξ is the same as the datum of an isomorphism class of a point ξ_{U} of the topos T/U which lifts ξ, i.e. such that ξ ≃ j_{U}(ξ_{U}). Moreover, say that ξ_{U′} is another lifting of ξ, corresponding to a neighborhood (U′, a′) of ξ; then, by inspecting the constructions of remark 4.7.9(i,ii) we see that, under this identification, a morphism (U, a) → (U′, a′) of neighborhoods of ξ corresponds to the datum of :

a morphism φ : U → U′ in T and an isomorphism j_{φ}(ξ_{U}) →^{∼} ξ_{U′} of T -points

where j_{φ} is the morphism of topoi T/U → T/U′ induced by φ.

Proposition 4.7.15. Let T be a topos with enough points; we have :

(i) For every object U of T, the topos T/U has enough points.

(ii) More precisely, let Ω be a conservative set of T -points, and denote by j_{U}^{−1}Ω the set of all T/U-points ξ_{U} such that there exists ξ ∈ Ω with an isomorphism ξ →^{∼} j_{U}(ξ_{U}); then j_{U}^{−1}Ω is a conservative set of T/U-points.

Proof. Let φ : X → Y be a morphism in T/U such that φ_{ξ} is an epimorphism for every ξ ∈ j_{U}^{−1}Ω; by proposition 1.3.21(ii), it suffices to show that φ is an epimorphism, and the latter will follow, if we show that the same holds for j_{U!}φ. Thus suppose by way of contradiction, that j_{U!}φ is not an epimorphism; then there exist ξ ∈ Ω and y ∈ (j_{U!}Y)_{ξ}, such that y does not lie in the image of (j_{U!}φ)_{ξ}. Let a := (j_{U!}π_{Y})_{ξ}(y) ∈ U_{ξ}, where π_{Y} : Y → 1_{T/U} = 1_{U} is the unique morphism in T/U. By remark 4.7.13(v), we may find a lifting (ξ_{U} : Set → T/U, ω_{U}) of ξ such that

(4.7.16)  σ(ξ_{U}, ω_{U}) = a

where σ(ξ_{U}, ω_{U}) is defined as in remark 4.7.9(i). After replacing ξ by j_{U}(ξ_{U}), we may assume that ω_{U} = 1_{ξ}, in which case (4.7.16) means that a = ξ_{U}^{∗}Δ_{U}, where Δ_{U} : 1_{T/U} → j_{U}^{∗}j_{U!}1_{T/U} is the unit of adjunction (i.e the diagonal U → U × U). We have a commutative diagram of sets :

\xymatrix{ξ_{U}^{∗}X \ar[r]^{ξ_{U}^{∗} φ} \ar[d]_{ξ_{U}^{∗} Δ_{X}} & ξ_{U}^{∗}Y \ar[d]^{ξ_{U}^{∗} Δ_{Y}} \\ (j_{U!}X)_{ξ} \ar[r]^{(j_{U!}φ)_{ξ}} & (j_{U!}Y)_{ξ} = ξ_{U}^{∗}(j_{U}^{∗}j_{U!}Y)}

where Δ_{X} : X → j_{U}^{∗}j_{U!}X is the unit of adjunction, and likewise for Δ_{Y}. More plainly, Δ_{Y} : Y × 1_{T/U} → Y × (j^{∗}j_{U!}1_{T/U}) is the product 1_{Y} × Δ_{U}, and under this identification, ξ_{U}^{∗}Δ_{Y} is the mapping ξ_{U}^{∗}Y → ξ_{U}^{∗}Y × (j^{∗}j_{U!}1_{T/U}) given by the rule : z ↦ (z, a) for every z ∈ ξ_{U}^{∗}Y. Especially, we see that y lies in the image of ξ_{U}^{∗}Δ_{Y}. But by assumption, the map ξ_{U}^{∗}φ is surjective, hence y lies in the image of (j_{U!}ξ)_{ξ}, a contradiction. □

4.7.17. Remark 4.7.13(v) prompts the following construction. Let us consider the fibred topos

T/− : T → Topos  U ↦ T/U  (U′ ⟶^{φ} U) ↦ (T/U′ ⟶^{j_{φ}} T/U)

of example 4.7.8(ii). Denote also by 1_{T} a fixed final object of T, and for every U ∈ Ob(T) let φ_{U} : U → 1_{T} be the unique morphism in T; the source functor yields a natural isomorphism T/1_{T} →^{∼} T that identifies j_{φ_{U}} : T/U → T/1_{T} with j_{U} : T/U → T. Moreover, the rule : U ↦ φ_{U} clearly defines a natural transformation

φ_{•} : 1_{T} ⇒ c_{1_{T}}

where c_{1_{T}} : T → T is the constant functor with value 1_{T}. For a sufficiently large universe V, example 2.2.7(i) yields a strict pseudo-functor H_{Topos} : Topos^{o} × Topos → V-Cat which restricts to a strict pseudo-functor

Pt : Topos → V-Cat S ↦ Pt(S) := Hom_{Topos}(Set, S)

assigning to every topos S the category of S-points. We consider the composition

Pt_{T} := Pt ∘ (T/−) : T → V-Cat U ↦ Pt(T/U)

as well as the pseudo-natural transformation

Pt_{T} ∗ φ_{•} : Pt_{T} ⇒ F_{Pt(T)}

from Pt_{T} to the constant pseudo-functor F_{Pt(T)} with value Pt(T). There follows a cartesian functor of fibrations over T^{o} :

ℱ ib(Pt_{T} ∗ φ_{•}) : ℱ ib(Pt_{T}) → T^{o} × Pt(T).

Let also σ_{ξ} : T^{o} → T^{o} × Pt(T) be the cartesian section such that σ_{ξ}(U) := (U, ξ) for every U ∈ Ob(T). Lastly, we consider the 2-cartesian diagram in the category Fib(T^{o}) :

[figure: ℒ (ξ) := ℱ ib(Pt_{T}) ×_{T^{o}×Pt(T)}^{2} T^{o}; ℱ ib(Pt_{T}); ℱ ib(Pt_{T} ∗φ_{•}); T^{o}; σ_{ξ}; T^{o} × Pt(T)]

and we set

Lift(ξ) := ℒ (ξ)^{o}.

Taking into account theorem 3.3.25(ii), we see that for every U ∈ Ob(T), the fibre ℒ (ξ)_{U} of the fibration ℒ (ξ) → T^{o} is the category whose objects are the pairs (ξ_{U}, ω_{U}), where ξ_{U} is a T/U-point, and ω_{U} : ξ →^{∼} j_{U}(ξ_{U}) is an isomorphism of T -points. The morphisms (ξ_{U}, ω_{U}) → (ξ′_{U}, ω′_{U}) are the morphisms of T/U-points

β : ξ_{U} → ξ′_{U} such that ω_{U′} = j_{U}(β) ⊙ ω_{U}

(with the obvious composition law). Hence, the objects of Lift(ξ) are the triples (U, ξ_{U}, ω_{U}) with U ∈ Ob(T) and (ξ_{U}, ω_{U}) ∈ Ob(ℒ (ξ)_{U}). The morphisms (U, ξ_{U}, ω_{U}) → (V, ξ_{V} , ω_{V}) are the pairs (φ, β), where φ : U → V is a morphism in T, and β : ξ_{V} → j_{φ}(ξ_{U}) is a morphism of T/V -points, such that

(γ_{φ,V} ∗ ξ_{U}) ⊙ j_{V} (β) ⊙ ω_{V} = ω_{U}

where γ_{φ,V} : j_{V} ∘ j_{φ} →^{∼} j_{U} is the coherence constraint of T/−. Then, remark 4.7.9 shows that ℒ (ξ) is a fibration in groupoids, and we have an equivalence of categories :

(4.7.18)  N_{ξ} : Lift(ξ) →^{∼} Nbd(ξ)  (U, ξ_{U}, ω_{U}) ↦ (U, σ(ξ_{U}, ω_{U}))

where σ(ξ_{U}, ω_{U}) ∈ U_{ξ} is defined as in remark 4.7.9(i).

4.7.19. Notice next that, since ℒ (ξ) is a fibration in groupoids, the natural projection ℒ (ξ) → ℱ ib(Pt_{T}) factors through the inclusion pseudo-functor ℱ ib(Pt_{T}^{×}) = ℱ ib(Pt_{T})^{×} → ℱ ib(Pt_{T}) (notation of remark 3.4.8(iii)). To every small set X, we wish now to attach a functor

Ξ^{X} : ℱ ib(Pt^{×}) → Set (ζ : Set → S) ↦ ζ^{∗}ζ_{∗}X.

To this aim, according to remark 3.1.23(i), it will suffice to exhibit a lax-natural transformation

α^{X} : ^{o}Pt^{×} ⇒ ^{o}F_{Set}

where F_{Set} is the constant pseudo-functor Topos → V-Cat with value Set. We need therefore to give for every S ∈ Ob(Topos) a functor α_{S}^{X} : Pt(S)^{×} → Set, together with a coherence constraint for the resulting system (α_{S}^{X} | S ∈ Ob(Topos)). Now, let us set

α_{S}^{X}(ξ) := ζ^{∗}ζ_{∗}X for every ζ ∈ Ob(Pt(S)).

Next, let β : ζ →^{∼} ζ′ be any morphism in Pt(S)^{×}; i.e. β : ζ_{∗} →^{∼} ζ′_{∗} is an isomorphism of functors. Then we define α_{S}^{X}(β) : ζ^{∗}ζ_{∗}X → ζ^{′∗}ζ′_{∗}X as the composition

ζ^{∗}ζ_{∗}X ⟶^{(ζ^{∗}∗β)_{X}} ζ^{∗}ζ′_{∗}X ⟶^{(β^{†}∗ζ′_{∗})_{X}^{−1}} ζ^{′∗}ζ′_{∗}X

where β^{†} : ζ^{′∗} →^{∼} ζ^{∗} is the transpose of the natural transformation β (see remark 1.1.17(ii)). From remark 1.1.17(vi) we see already that α_{S}^{X}(1_{ζ}) = 1_{ζ^{∗}ζ_{∗}X}. Next, let β′ : ζ′ → ζ′′ be another morphism of Pt(S)^{×}; we compute :

α_{S}^{X}(β′) ∘ α_{S}^{X}(β) = ((β^{′†} ∗ ζ′′_{∗})^{−1} ⊙ (ζ^{′∗} ∗ β′) ⊙ (β^{†} ∗ ζ′_{∗})^{−1} ⊙ (ζ^{∗} ∗ β))_{X}
= ((β^{′†} ∗ ζ′′_{∗})^{−1} ⊙ (β^{†} ∗ ζ′′_{∗})^{−1} ⊙ (ζ^{∗} ∗ β′) ⊙ (ζ^{∗} ∗ β))_{X}
= (((β′ ⊙ β)^{†} ∗ ζ′′_{∗})^{−1} ⊙ (ζ^{∗} ∗ (β′ ⊙ β)))_{X}
= α_{S}^{X}(β′ ⊙ β)

where the second equality follows from (1.1.11), and the third follows from remark 1.1.17(vi). Next, let f : S → S′ be any morphism of topoi, and ε^{f} : f^{∗} ∘ f_{∗} ⇒ 1_{S} the counit of the adjoint pair (f^{∗}, f_{∗}); our candidate coherence constraint for α_{•}^{X} assigns to every S-point ζ the map

τ_{f,ζ}^{X} : f(ζ)^{∗}f(ζ)_{∗}X = ζ^{∗} ∘ f^{∗} ∘ f_{∗} ∘ ζ_{∗}X ⟶^{(ζ^{∗}∗ε^{f} ∗ζ_{∗})_{X}} ζ^{∗}ζ_{∗}X

(see example 2.2.6(ii)). We need to check that the rule ζ ↦ τ_{f,ζ}^{X} yields a natural transformation

τ_{f}^{X} : α_{S′}^{X} ∘ Pt(f)^{×} ⇒ α_{S}^{X}.

Thus, let β : ζ_{∗} →^{∼} ζ′_{∗} be any morphism of Pt(S)^{×}; the assertion comes down to showing the commutativity of the diagram :

\xymatrix{ζ^{∗}f^{∗}f_{∗}ζ_{∗}X \ar[r]^{(ζ^{∗}f^{∗}f_{∗}∗β)_{X}} \ar[d]_{(ζ^{∗}∗ε^{f} ∗ζ_{∗})_{X}} & ζ^{∗}f^{∗}f_{∗}ζ′_{∗}X \ar[r]^{((f_{∗}∗β)^{†}∗f_{∗}ζ′_{∗})_{X}^{−1}} \ar[d]_{(ζ^{∗}∗ε^{f} ∗ζ′_{∗})_{X}} & ζ^{′∗}f^{∗}f_{∗}ζ′_{∗}X \ar[d]^{(ζ^{′∗}∗ε^{f} ∗ζ′_{∗})_{X}} \\ ζ^{∗}ζ_{∗}X \ar[r]^{(ζ^{∗}∗β)_{X}} & ζ^{∗}ζ′_{∗}X \ar[r]^{(β^{†}∗ζ′_{∗})_{X}^{−1}} & ζ^{′∗}ζ′_{∗}X}

However, the commutativity of the left square subdiagram follows from the naturality of ζ^{∗} ∗ε^{f}. Next, recall that (f_{∗} ∗ β)^{†} = β^{†} ∗ f^{∗} (remark 1.1.17(vi)); then the commutativity of the right square subdiagram follows by applying (1.1.11) to the natural transformations β^{†} and ε^{f} ∗ ζ′_{∗}.

Lastly, we have to check the coherence axioms for τ_{•}^{X}. Since ^{o}Pt^{×} and ^{o}F_{Set} are both strict pseudo-functors, we come down to showing that

τ_{1_{S}}^{X} = 1_{α_{S}^{X}} and τ_{g}^{X} ⊟ τ_{f}^{X} = τ_{g∘f}^{X}

for every topos S and every pair of morphisms of topoi S ⟶^{f} S′ ⟶^{g} S′′ (see remark 2.2.5(i)). These follow easily by direct inspection. This completes the construction of α^{X}, and thus also of Ξ^{X} := ℱ ib(α^{X}). After composing with the projection ℒ (ξ) → ℱ ib(Pt_{T}^{×}) → ℱ ib(Pt^{×}), we deduce a functor

Ξ^{X,ξ} : Lift(ξ)^{o} → Set (U, ξ_{U}, ω_{U}) ↦ ξ_{U}^{∗}ξ_{U∗}X.

Lemma 4.7.20. (i) For every (S, ζ) ∈ Ob(ℱ ib(Pt^{×})) and every set X, let ε_{X}^{(S,ζ)} : ζ^{∗}ζ_{∗}X → X be the counit of adjunction. Then the rule : (S, ζ) ↦ ε_{X}^{(S,ζ)} defines a natural cocone

Ξ^{X} ⇒ c_{X}.

(ii) The cocone of (i) induces a natural bijection :

colim_{Lift(ξ)^{o}} Ξ^{X,ξ} →^{∼} X for every (small) set X.

Proof. (i): Let ν^{X} : ^{o}Pt^{×} ⇒ ^{o}F_{Set} be the strict pseudo-natural transformation such that

ν_{S}^{X} : Pt(S)^{×} → Set

is the constant functor with value X, for every topos S (i.e. ν_{S}^{X}(ζ) := X for every S-point ζ, and ν_{S}^{X}(β) = 1_{X} for every invertible morphism β of S-points). For every such S, let also

ε_{X}^{(S,•)} : Ob(Pt(S)) → Morph(Set)

be the map that associates with every S-point ζ, the map of sets ε_{X}^{(S,ζ)}. We notice :

Claim 4.7.21. The rule : S ↦ ε_{X}^{(S,•)} for every topos S, defines a modification :

ε_{X}^{(•,•)} : ν^{X} ⇝ α^{X}.

Proof of the claim. First, we need to check that ε_{X}^{(S,•)} is a natural transformation α_{S}^{X} ⇒ ν_{S}^{X}, for every topos S. The latter assertion amounts to the commutativity of the diagram :

[figure: ζ^{∗}ζ_{∗}X; ε_{X}^{(S,ζ)}; α_{S}^{X}(β); X; ζ^{′∗}ζ′_{∗}X; ε_{X}^{(S,ζ′)}; X]

for every isomorphism β : ζ →^{∼} ζ′ of S-points. Since (β^{†} ∗ ζ′_{∗})_{X}^{−1} = ((β^{−1})^{†} ∗ ζ′_{∗})_{X} (remark 1.1.17(iv)), this is in turn the same as the identity :

ε_{X}^{(S,ζ′)} ∘ ((β^{−1})^{†} ∗ ζ′_{∗})_{X} = ε_{X}^{(S,ζ)} ∘ (ζ^{∗} ∗ β^{−1})_{X}

which follows from remark 1.1.17(iii). Next, we need to check the compatibility condition for ε_{X}^{(S,•)}, which comes down to the identity :

ε_{X}^{(S,ζ)} ⊙ τ_{f}^{X} = ε_{X}^{(S,f(ζ))}

for every morphism of topoi f : S → S′ and every S-point ζ, where τ_{f}^{X} denotes the coherence constraint of α^{X}. The latter follows by direct inspection. ♢

By remark 3.1.23(ii), the sought cocone is then ℱ ib(ε_{X}^{(•,•)}) : Ξ^{X} ⇒ ℱ ib(ν^{X}) = c_{X}.

(ii): Let π : ℱ ib(Pt^{×}) → Topos^{o} be the fibration arising from the strict pseudo-functor Pt^{×}. We let 𝒞 be the fibre product in the cartesian square of categories :

\xymatrix{Morph(ℱ ib(Pt^{×})) \ar[r]^{p} \ar[dr]_{Morph(π)} & 𝒞 \ar[r]^{s′} \ar[d]_{π′} & ℱ ib(Pt^{×}) \ar[d]^{π} \\  & Morph(Topos^{o}) \ar[r]^{s} & _{Topos_{o}}}

where s and s′ ∘ p are the source functors. By remark 3.1.27(ii), we have a natural identification

𝒞 →^{∼} ℱ ib(Pt^{×} ∘ s^{o})

and recall that Morph(π) is a fibration as well, by example 3.1.9. To every morphism of topoi f : S′ → S, we attach the functor

Σ_{f}^{X} : π^{′−1}(f) = Pt(S)^{×} → Set ζ ↦ Γ(S′, f^{∗}ζ_{∗}X)

where Γ(S′, −) : S′ → Set is defined as in (4.4.11), after fixing a final object 1_{S′} of S′. To every isomorphism α : ζ_{1} →^{∼} ζ_{2} of S-points, the functor Σ_{f}^{X} assigns the map

Σ_{f}^{X}(α) := Γ(S′, f^{∗}α_{X}) : Γ(S′, f^{∗}ζ_{1∗}X) → Γ(S′, f^{∗}ζ_{2∗}X).

Next, notice that if (S′_{1} ⟶^{f_{1}} S_{1})^{o} and (S′_{2} ⟶^{f_{2}} S_{2})^{o} are two objects of Morph(Topos^{o}), then a morphism f_{1}^{o} → f_{2}^{o} in this category is a commutative diagram of morphisms of topoi :

𝒟_{g,g′}  :  \xymatrix{S′_{2} \ar[r]^{f_{2}} \ar[d]_{g′} & S_{2} \ar[d]^{g} \\ S′_{1} \ar[r]^{f_{1}} & S_{1}.}

The (split) cleavage Pt^{×} ∘ s^{o} of 𝒞 associates with 𝒟_{g,g′} the functor

Pt(g)^{×} : Pt(S_{2})^{×} → Pt(S_{1})^{×} ζ ↦ g(ζ)

that assigns to every morphism α of Pt(S_{2})^{×} the morphism g(α) of Pt(S_{1})^{×}. To the diagram 𝒟_{g,g′} and every object ζ of Pt(S_{2})^{×} we attach as well the map :

τ_{ζ}^{g,g′} : Γ(S′_{1}, f_{1}^{∗}g_{∗}ζ_{∗}X) ⟶^{Γ(S′_{1},Υ(𝒟_{g,g′})∗ζ_{∗})_{X}} Γ(S′_{1}, g′_{∗}f_{2}^{∗}ζ_{∗}X) ⟶^{ω_{f^{∗}_{2}ζ∗X}^{g′}} Γ(S′_{2}, f_{2}^{∗}ζ_{∗}X)

where Υ(𝒟_{g,g′}) is the base change transformation associated with the diagram 𝒟_{g,g′}, viewed as an oriented square diagram of links, whose orientation is the identity (see (2.3.8)). The second map in this composition is induced by the unique isomorphism of morphisms of topoi ω^{g′} : Γ_{S′_{1}} ∘ g′_{∗} →^{∼} Γ_{S′_{2}} (proposition 4.4.12(ii)).

Claim 4.7.22. (i) For every diagram 𝒟_{g,g′}, the rule : ζ ↦ τ_{ζ}^{g,g′} defines a natural transformation

[figure: Pt(S_{2})^{×}; Σ_{f_{2}}^{X}; Pt(g)^{×}; τ^{g,g′}; Set; Pt(S_{1})^{×}; Σ_{f_{1}}^{X}; Set.]

(ii) The rule : 𝒟_{g,g′} ↦ τ^{g,g′} provides the coherence constraint for a lax-natural transformation

Σ_{•}^{X} : ^{o}(Pt^{×} ∘ s^{o}) ⇒ ^{o}F_{Set}.

Proof of the claim. (i): For every object (S′ ⟶^{f} S)^{o} of Morph(Topos^{o}), consider the functor

σ_{f} : Pt(S)^{×} → S ζ ↦ ζ_{∗}X

that assigns to every isomorphism α as in the foregoing, the morphism α_{X} of S. With this notation, τ^{g,g′} is the composition of the following oriented squares :

[figure: Pt(S_{2})^{×}; σ_{f_{2}}; Pt(g)^{×}; 1_{g∗∘σ_{f_{2}}}; S_{2}; f_{2}^{∗}; g_{∗}; Υ(𝒟_{g,g′}); S′_{2}; Γ_{S′_{2}}; g′_{∗}; ω^{g′}; Set; Pt(S_{1})^{×}; σ_{f_{1}}; S_{1}; f_{1}^{∗}; S′_{2}; Γ_{S′_{1}}; Set.]

(ii): Clearly τ^{1_{S},1_{S′}} = 1_{Σ_{f}^{X}}, for every morphism of topoi f : S′ → S. According to remark 2.2.5(i), it remains to check the identity

τ^{g,g′} ⊟ τ^{h,h′} = τ^{g∘h,g′∘h′}

for every composable pair of diagrams 𝒟_{g,g′} : f_{1}^{o} → f_{2}^{o} and 𝒟_{h,h′} : f_{2}^{o} → f_{3}^{o}. However, by proposition 2.1.9we have :

τ^{g,g′} ⊟ τ^{h,h′} = (1_{g_{∗}∘σ_{f_{2}}} ⊟ 1_{h_{∗}∘σ_{f_{3}}}) □ (Υ(𝒟_{g,g′}) ⊟ Υ(𝒟_{h,h′})) □ (ω^{g′} ⊟ ω^{h′}).

Clearly 1_{g_{∗}∘σ_{f_{2}}} ⊟1_{h_{∗}∘σ_{f_{3}}} = 1_{(g∘h)_{∗}∘σ_{f_{3}}}, and we have ω^{g′} ⊟ω^{h′} = ω^{g′∘h′}, by the uniqueness properties of the isomorphisms ω^{•}. Lastly, Υ(𝒟_{g,g′}) ⊟ Υ(𝒟_{h,h′}) = Υ(𝒟_{g,g′} □ 𝒟_{h,h′}), by proposition 2.3.12(i), whence the claim. ♢

From claim 4.7.22(ii) and remark 3.1.23(i) we deduce a functor ℱ ib(Σ_{•}^{X}) : 𝒞 → Set.

Next, notice that, for every object (S′ ⟶^{f} S)^{o} of Morph(Topos^{o}), the objects of the fibre category Morph(π)^{−1}(f) are the triples (ζ, ζ′, β), where ζ is an S-point, ζ′ is an S′-point, and β : ζ →^{∼} f(ζ′) is an isomorphism of S-points. The morphisms (ζ_{1}, ζ′_{1}, β_{1}) → (ζ_{2}, ζ′_{2}, β_{2}) are the pairs (α, α′) where α : ζ_{1} →^{∼} ζ_{2} and α′ : ζ′_{1} →^{∼} ζ′_{2} are isomorphisms of S-points and respectively S′-points, such that

(4.7.23)  (f_{∗} ∗ α′) ⊙ β_{1} = β_{2} ⊙ α.

If (γ, γ′) : (ζ_{2}, ζ′_{2}, β_{2}) → (ζ_{3}, ζ′_{3}, β_{3}) is another such morphism, we have

(γ, γ′) ∘ (α, α′) = (γ ⊙ α, γ′ ⊙ α′) in Morph(π)^{−1}(f).

The (split) cleavage c of Morph(ℱ ib(Pt^{×})) attaches to the diagram 𝒟_{g,g′}, the functor

c_{g,g′} : Morph(π)^{−1}(f_{2}) → Morph(π)^{−1}(f_{1}) (ζ, ζ′, β) ↦ (g(ζ), g′(ζ′), g(β))

that assigns to every morphism (α, α′) of Morph(π)^{−1}(f_{2}) the morphism (g(α), g′(α′)). With this notation, p is the (cartesian) functor associated with the strict pseudo-natural transformation

t : c ⇒ Pt^{×} ∘ s^{o}

that assigns to every morphims of topoi f : S′ → S the functor

Morph(π)^{−1}(f) → Pt(S)  (ζ, ζ′, β) ↦ ζ  (α, α′) ↦ α.

Now, to every morphism of topoi f : S′ → S, and every (ζ, ζ′, β) ∈ Ob(Morph(π)^{−1}(f)), let us attach the map b_{f,(ζ,ζ′,β)} : Γ(S′, f^{∗}ζ_{∗}X) → ζ^{∗}ζ_{∗}X defined as the composition :

Γ(S′, f^{∗}ζ_{∗}X) ⟶^{Γ(S′,(η^{ζ′}∗f^{∗}ζ_{∗})_{X})} Γ(S′, ζ′_{∗}ζ^{′∗}f^{∗}ζ_{∗}X) ⟶^{ω_{ζ^{′∗}f^{∗}ζ∗X}^{ζ′}} ζ^{′∗}f^{∗}ζ_{∗}X ⟶^{(β^{†}∗ζ_{∗})_{X}} ζ^{∗}ζ_{∗}X

where η^{ζ′} : 1_{S′} ⇒ ζ′_{∗}ζ^{′∗} is the unit of adjunction, and ω^{ζ′} : Γ_{S′} ∘ ζ′ →^{∼} 1_{Set} is the unique isomorphism of morphisms of topoi provided by proposition 4.4.12(ii).

Claim 4.7.24. (i) For every morphism of topoi f : S′ → S, the rule : (ζ, ζ′, β) ↦ b_{f,(ζ,ζ′,β)} defines a natural transformation

b_{f} : Σ_{f}^{X} ∘ t_{f} ⇒ α_{S}^{X} ∘ t_{f}.

(ii) The rule : f ↦ b_{f} defines a modification

b_{•} : (α^{X} ∗ s^{o}) ⊙^{o}t ⇝ Σ_{•}^{X} ⊙^{o}t.

Proof of the claim. (i): Let (α, α′) : (ζ_{1}, ζ′_{1}, β_{1}) → (ζ_{2}, ζ′_{2}, β_{2}) be a morphism in Morph(π)^{−1}(f), and set U := (ζ′_{2∗} ∗ α^{′†} ∗ f^{∗}ζ_{2∗})_{X}^{−1} ∘ (ζ′_{2∗} ∗ α′ ∗ f^{∗}ζ_{2∗})_{X}. We notice that :

\begin{aligned}(η^{ζ′_{2}} ∗ f^{∗}ζ_{2∗})_{X} ∘ (f^{∗}(α_{X}) & = U ∘ (η^{ζ′_{1}} ∗ f^{∗}ζ_{2∗})_{X} ∘ (f^{∗}(α_{X}) \\ & = U ∘ (ζ′_{1∗}ζ_{1}^{′∗}f^{∗} ∗ α)_{X} ∘ (η^{ζ′_{1}} ∗ f^{∗}ζ_{1∗})_{X} \end{aligned}

where the first equality holds by remark 1.1.17(iii), and the second one follows from (1.1.11). We are then reduced to checking the commutativity of the diagram :

[figure: Γ(S′, ζ′_{1∗}ζ_{1}^{′∗}f^{∗}ζ_{1∗}X); ω_{ζ{′∗ \atop 1}f^{∗}ζ_{1∗}X}^{ζ′_{1}}; Γ(S′,ζ′_{1∗}ζ_{1}^{′∗}f^{∗}α_{X}); ζ_{1}^{′∗}f^{∗}ζ_{1∗}X; (β_{1}^{†}∗ζ_{1∗})_{X}; ζ_{1}^{′∗}f^{∗}α_{X}; ζ_{1}^{∗}ζ_{1∗}X; ζ_{1}^{∗}α_{X}; Γ(S′, ζ′_{1∗}ζ_{1}^{′∗}f^{∗}ζ_{2∗}X); ω_{ζ{′∗ \atop 1}f^{∗}ζ_{2∗}X}^{ζ′_{1}}; Γ(S′,(α′∗ζ_{1}^{′∗}f^{∗}ζ_{2∗})_{X}); ζ_{1}^{′∗}f^{∗}ζ_{2∗}X; (β_{1}^{†}∗ζ_{2∗})_{X}; ζ_{1}^{∗}ζ_{2∗}X; Γ(S′, ζ′_{2∗}ζ_{1}^{′∗}f^{∗}ζ_{2∗}X); Γ(S′,(ζ′_{2∗}∗α^{′†}∗f^{∗}ζ_{2∗})_{X}^{−1}); ω_{ζ{′∗ \atop 1}f^{∗}ζ_{2∗}X}^{ζ′_{2}}; ζ_{1}^{′∗}f^{∗}ζ_{2∗}X; (α^{′†}∗f^{∗}ζ_{2∗})_{X}^{−1}; (β_{1}^{†}∗ζ_{2∗})_{X}; ζ_{1}^{∗}ζ_{2∗}X; (α^{†}∗ζ_{2∗})_{X}^{−1}; Γ(S′, ζ′_{2∗}ζ_{2}^{′∗}f^{∗}ζ_{2∗}X); ω_{ζ{′∗ \atop 2}f^{∗}ζ_{2∗}X}^{ζ′_{2}}; ζ_{2}^{′∗}f^{∗}ζ_{2∗}X; (β_{2}^{†}∗ζ_{2∗})_{X}; ζ_{2}^{∗}ζ_{2∗}X.]

However, the commutativity of the top two squares and of the bottom square on the left is clear; the commutativity of the bottom square on the right follows from remark 1.1.17(iv,v) and (4.7.23). Lastly, the commutativity of the left square on the central row follows immediately from the uniqueness properties of the isomorphisms ω^{ζ′_{1}} and ω^{ζ′_{2}} (proposition 4.4.12(ii)), whence the contention.

(ii): Consider a diagram 𝒟_{g,g′} as in the foregoing, and an object (ζ, ζ′, β) of Morph(π)^{−1}(f_{2}). We have to check the compatibility condition :

b_{f_{2},(ζ,ζ′,β)} ∘ τ_{ζ}^{g,g′} = τ_{g,ζ}^{X} ∘ b_{f_{1},(g(ζ),g(ζ′),g(β))}.

To this aim, notice that ω^{g′(ζ′)} = ω^{ζ′} ⊙ (ω^{g′} ∗ ζ′_{∗}), due to proposition 4.4.12(ii); then it suffices to show the commutativity of the diagram :

[figure: Γ(S′_{1}, f_{1}^{∗}g_{∗}ζ_{∗}X); Γ(S′_{1},Υ(𝒟_{g,g′})∗ζ_{∗})_{X}; Γ(S′_{1},(η^{g′(ζ′)}∗f_{1}^{∗}g_{∗}ζ_{∗})_{X}); Γ(S′_{1}, g′_{∗}f_{2}^{∗}ζ_{∗}X); ω_{f^{∗}_{2}ζ∗X}^{g′}; Γ(S′_{2}, f_{2}^{∗}ζ_{∗}X); Γ(S′_{2},(η^{ζ′}∗f_{2}^{∗}ζ_{∗})_{X}); Γ(S′_{1}, g′(ζ′)_{∗}g′(ζ′)^{∗}f_{1}^{∗}g_{∗}ζ_{∗}X); ω_{ζ′_{∗}ζ^{′∗}g^{′∗}f^{∗}_{1}g∗ζ∗X}^{g′}; Γ(S′_{2}, ζ′_{∗}g′(ζ′)^{∗}f_{1}^{∗}g_{∗}ζ_{∗}X); ω_{ζ^{′∗}g^{′∗}f^{∗}_{1}g∗ζ∗X}^{ζ′}; Γ(S′_{2},(ζ^{′∗}f_{2}^{∗}∗ε^{g}∗ζ_{∗})_{X}); Γ(S′_{2}, ζ′_{∗}ζ^{′∗}f_{2}^{∗}ζ_{∗}X); ω_{ζ^{′∗}f^{∗}_{2}ζ∗X}^{ζ′}; ζ^{′∗}g^{′∗}f_{1}^{∗}g_{∗}ζ_{∗}X; (g(β)^{†}∗g(ζ)_{∗})_{X}; ζ^{′∗}f_{2}^{∗}g^{∗}g_{∗}ζ_{∗}X; (ζ^{′∗}f_{2}^{∗}∗ε^{g}∗ζ_{∗})_{X}; ζ^{′∗}f_{2}^{∗}ζ_{∗}X; (β^{†}∗ζ_{∗})_{X}; g(ζ)^{∗}g(ζ)_{∗}X; (ζ^{∗}∗ε^{g}∗ζ_{∗})_{X}; ζ^{∗}ζ_{∗}X.]

However, the commutativity of the bottom square follows from (1.1.11) and remark 1.1.17(vi), and that of the central square is clear, by the naturality of ω^{ζ′}. In order to show the commutativity of the top square, set A := ζ_{∗}X, and recall that

Υ(𝒟_{g,g′}) = (g′_{∗}f_{2}^{∗} ∗ ε^{g}) ⊙ (η^{g′} ∗ f_{1}^{∗}g_{∗}) and η^{g′(ζ′)} = (g′_{∗} ∗ η^{ζ′} ∗ g^{′∗}) ⊙ η^{g′}

(proposition 2.3.10). We are then reduced to checking the commutativity of the diagram :

\xymatrix{ & Γ(S′_{1}, g′_{∗}g^{′∗}f_{1}^{∗}g_{∗}A) \ar[dl]_{Γ(S′_{1},(g′_{∗}∗η^{ζ′}∗g^{′∗}f_{1}^{∗})_{A})} \ar[d]|{ω_{g^{′∗}f^{∗}_{1}g∗A}^{g′}} \ar[r]^{Γ(S′_{1},(g′_{∗}f_{2}^{∗}∗ε^{g})_{A})} & Γ(S′_{1}, g′_{∗}f_{2}^{∗}A) \ar[d]|{ω_{f^{∗}_{2}A}^{g′}} \\ Γ(S′_{1}, g′(ζ′)_{∗}g′(ζ′)^{∗}f_{1}^{∗}g_{∗}A) \ar[dr]_{ω_{ζ′_{∗}g′(ζ′)^{∗}f^{∗}_{1}g∗A}^{g′}} & Γ(S′_{2}, g^{′∗}f_{1}^{∗}g_{∗}A) \ar[d]|{Γ(S′_{2},(η^{ζ′}∗g^{′∗}f_{1}^{∗}g_{∗})_{A})} \ar[r]^{Γ(S′_{2},(f_{2}^{∗}∗ε^{g}∗)_{A})} & Γ(S′_{2}, f_{2}^{∗}A) \ar[d]|{Γ(S′_{2},(η^{ζ′}∗f_{2}^{∗})_{A})} \\  & Γ(S′_{2}, ζ′_{∗}g′(ζ′)^{∗}f_{1}^{∗}g_{∗}A) \ar[r]^{Γ(S′_{2},(ζ′_{∗}f_{2}^{∗}∗ε^{g})_{A})} & Γ(S′_{2}, ζ′_{∗}ζ^{′∗}f_{2}^{∗}A).}

However, it is easily seen that the two square subdiagrams and the triangular subdiagram commute (details left to the reader), whence the contention. ♢

From claim 4.7.24(ii) and remark 3.1.23(ii) we get the natural transformation

ℱ ib(b_{•}) : ℱ ib(Σ_{•}^{X}) ∘ p ⇒ Ξ^{X} ∘ s′ ∘ p.

Lastly, let q : ℒ (ξ) → ℱ ib(Pt^{×}) be the natural projection, s_{ℒ (ξ)} : Morph(ℒ (ξ)) → ℒ (ξ) the source functor, and set G := ℱ ib(Σ_{•}^{X}) ∘ p ∘ Morph(q); we deduce the natural transformation

β : ℱ ib(b_{•}) ∗ Morph(q) : G ⇒ Ξ^{X,ξ} ∘ s_{ℒ (ξ)} .

By theorem 1.3.4, the latter induces a natural transformation

(4.7.25)  ∫ ^{s_{ℒ (ξ)}} G ⇒ Ξ^{X,ξ}.

Notice that, under the natural identification Morph(ℒ (ξ)) →^{∼} Morph(Lift(ξ))^{o}, the functor s_{ℒ (ξ)} correspond to the functor t_{Lift(ξ)}^{o}, where t_{Lift(ξ)} : Morph(Lift(ξ)) → Lift(ξ) is the target functor. Moreover, t_{Lift(ξ)} is a fibration, by virtue of remark 4.7.13(ii) and example 3.1.3(iii). Furthermore, for every (U, ξ_{U}, ω_{U}) ∈ Ob(Lift(ξ)), the fibre category t^{−1}(U, ξ_{U}, ω_{U}) is

Lift(ξ)/(U, ξ_{U}, ω_{U}) →^{∼} Lift(ξ_{U}).

Hence, the restriction of β to the fibre category t^{−1}(U, ξ_{U}, ω_{U}) is a cocone

(4.7.26)  [figure: Lift(ξ_{U})^{o}; G; c_{ξ^{∗}_{U} ξ_{U∗}X}; Set.]

By a direct inspection of the construction, we see that (4.7.26) is the cocone τ_{ξ,ξ_{U∗}X} ∗N_{ξ_{U}}^{o}, where N_{ξ_{U}} : Lift(ξ_{U}) →^{∼} Nbd(ξ_{U}) is the equivalence (4.7.18), and τ_{ξ,ξ_{U∗}X} is the universal cocone of remark 4.7.13(iv). Hence, (4.7.26) is a universal cocone, for every (U, ξ_{U}, ω_{U}) ∈ Ob(Lift(ξ)); taking into account example 3.1.14(iv), we deduce that (4.7.25) is an isomorphism of functors, and we get a natural isomorphism :

colim_{Morph(Lift(ξ))^{o}} G →^{∼} colim_{Lift(ξ)^{o}} Ξ^{X,ξ}.

Now, notice that the functor

(4.7.27)  Lift(ξ)^{o} → Morph(Lift(ξ))^{o}  (U, ξ_{U}, ω_{U}) ↦ 1_{(U,ξ_{U} ,ω_{U})}

is cofinal. A simple inspection reveals that the composition of G with (4.7.27) is the constant functor with value X, whence the contention. □

4.7.28. Suppose now that T = C^{∼} for some small site C := (𝒞 , J). For a point ξ of T, we may define another category Nbd(ξ, C), whose objects are the pairs (U, a) where U ∈ Ob(𝒞) and a ∈ (h_{U}^{a})_{ξ}; the morphisms (U, a) → (U′, a′) are the morphisms f : U → U′ in 𝒞 such that (h_{f}^{a})_{ξ}(a) = a′. (Notation of remark 4.1.19(iii).) The rule (U, a) ↦ (h_{U}^{a}, a) defines a functor

(4.7.29)  Nbd(ξ, C) → Nbd(ξ).

Proposition 4.7.30. In the situation of (4.7.28), we have :

(i) The category Nbd(ξ, C) is cofiltered.

(ii) The functor (4.7.29) is cofinal.

Proof. To begin with, since ξ^{∗} commutes with all colimits, remark 4.1.19(iii) implies easily that, for every object (U, a) of Nbd(ξ) there exist an object (V, b) of Nbd(ξ, C) and a morphism (h_{V}^{a} , b) → (U, a) in Nbd(ξ). Hence, it suffices to show (i).

Thus, let (U_{1}, a_{1}) and (U_{2}, a_{2}) be two objects of Nbd(ξ, C); since Nbd(ξ) is cofiltered, we may find an object (F, b) of Nbd(ξ) and morphisms φ_{i} : (F, b) → (h_{U_{i}}^{a}, a_{i}) (for i = 1, 2). By the foregoing, we may also assume that (F, b) = (h_{V}^{a} , b) for some object (V, b) of Nbd(ξ, C), in which case φ_{i} ∈ h_{U_{i}}^{a}(V) for i = 1, 2. By remark 4.1.19(iv), we may find a sieve 𝒮 covering V such that φ_{i} ∈ Hom_{𝒞^{∧}}(h_{𝒮} , h_{U_{i}}^{sep}) for both i = 1, 2.

On the other hand, (3.5.3) and proposition 4.1.26yield a natural isomorphism :

colim_{𝒮}h^{a} ∘ s →^{∼} h_{V}^{a} .

Therefore, since ξ^{∗} commutes with all colimits, we may find (f : S → V) ∈ Ob(𝒮) and c ∈ (h_{S}^{a})_{ξ} such that h_{f}^{a} : (h_{S}^{a}, c) → (h_{V}^{a} , b) is a morphism of neighborhoods of ξ.

For i = 1, 2, denote by \overline{φ}_{S,i} ∈ h_{U}^{sep}(S) the image of φ_{i} (under the map induced by the natural morphism h_{S} → h_{𝒮} coming from (3.5.3)). Pick any φ_{S,i} ∈ Hom_{𝒞} (S, V) in the preimage of \overline{φ}_{S,i}; then φ_{S,i} defines a morphism (S, c) → (U_{i}, a_{i}) in Nbd(ξ, C).

Next, suppose that φ_{1}, φ_{2} : (U, a) → (U′, a′) are two morphisms in Nbd(ξ, C); arguing as in the foregoing, we may find an object (V, b) of Nbd(ξ, C), and ψ ∈ h_{U}^{sep}(V) = Hom_{𝒞^{∧}}(h_{V}^{sep}, h_{U}^{sep}) whose image in h_{U}^{a}(V) yields a morphism ψ^{a} : (h_{V}^{a} , b) → (h_{U}^{a}, a) in Nbd(ξ), and such that φ_{1}^{sep} ∘ ψ = φ_{2}^{sep} ∘ ψ in h_{U′}^{sep}(V). We may then find a covering subobject i : R → h_{V}, such that φ_{1} ∘(ψ ∘i) = φ_{2} ∘(ψ ∘i) in Hom_{𝒞^{∧}}(R, h′_{U}). Again, by combining (3.5.3) and proposition 4.1.26, we deduce that there exists a morphism β : (V′, b′) → (V, b) in Nbd(ξ, C), such that φ_{1} ∘ (ψ ∘ β) = φ_{2} ∘ (ψ ∘ β). This completes the proof of (i). □

As a corollary of proposition 4.7.30and of remark (4.7.13)(iv), we deduce, for every sheaf F on C, a natural isomorphism :

colim_{Nbd(ξ,C)^{o}} F ∘ ι_{C}^{o} →^{∼} F_{ξ}

where ι_{C} : Nbd(ξ, C) → 𝒞 is the functor such that (U, a) ↦ U for every object (U, a).

4.8. Algebra on a topos. Let T be any topos, and endow T with the structure of tensor category as explained in example 3.7.10 (so, the tensor functor is given by fixed choices of products for every pairs of objects of T, and any final object 1_{T} can be taken for unit object of (T, ⊗)). We notice that (T, ⊗) admits an internal Hom functor (see remark 3.7.12(ii)). Indeed, let X and X′ be any two objects of T. It is easily seen that the presheaf on T :

U ↦ Hom_{T/U}(X′_{|U}, X_{|U}) = Hom_{T} (X′×U, X)

is actually a sheaf on (T, C_{T}) (notation of example 4.7.8(i)), so it is an object of T, denoted :

ℋ om_{T} (X′, X).

The functor :

T → T : X ↦ ℋ om_{T} (X′, X)

is right adjoint to the functor T → T : Y ↦ Y × X′, so it is an internal Hom functor for X′. If f : T → S is a morphism of topoi, and Y ∈ Ob(S), we have a natural isomorphism in S :

(4.8.1)  ℋ om_{S}(Y, f_{∗}X) →^{∼} f_{∗}ℋ om_{T} (f^{∗}Y, X)

which, on every U ∈ Ob(S), induces the natural bijection :

Hom_{S}(Y ×U, f_{∗}X) →^{∼} Hom_{T} (f^{∗}Y ×f^{∗}U, X)

given by the adjunction (f^{∗}, f_{∗}). By general nonsense, from (4.8.1) we derive a natural morphism in S :

f_{∗}ℋ om_{T} (X′, X) → ℋ om_{S}(f_{∗}X′, f_{∗}X)

and in T :

f^{∗}ℋ om_{S}(Y′, Y) ⟶^{ϑ_{f}} ℋ om_{T} (f^{∗}Y′, f^{∗}Y) for any Y, Y′ ∈ Ob(S).

Moreover, if g : U → T is another morphism of topoi, the diagram :

(4.8.2) \xymatrix{g^{∗}f^{∗}ℋ om_{S}(Y′, Y) \ar[rr]^{g^{∗}ϑ_{f}} \ar@{-->}[dr]_{ϑ_{f∘g}} &  & g^{∗}ℋ om_{T} (f^{∗}Y′, f^{∗}Y) \ar[dl]^{ϑ_{g}} \\  & ℋ om_{U}(g^{∗}f^{∗}Y′, g^{∗}f^{∗}Y)}

commutes, up to a natural isomorphism.

4.8.3. Let A be any object of T, and (X, μ_{X}) a left A-module for the tensor category structure on T as in (4.8); for every object U of T we obtain a left A_{|U}-module (on the topos T/U : see example 4.7.8(i)), by the rule :

(X, μ_{X})_{|U} := (X_{|U}, μ_{X} × 1_{U}).

If (X′, μ_{X′}) is another left A-module, it is easily seen that the presheaf on T :

U ↦ Hom_{A_{|U} -Mod_{l}}((X, μ_{X})_{|U}, (X′, μ_{X′})_{|U})

is actually a sheaf for the canonical topology, so it is an object of T, denoted :

ℋ om_{A_{l}}((X, μ_{X}), (X′, μ_{X′}))

(or just ℋ om_{A_{l}}(X, X′), if the notation is not ambiguous). The same considerations can be repeated for the sets of morphisms of right B-modules, and of (A, B)-bimodules, so one gets objects ℋ om_{B_{r}}(X, X′) and ℋ om_{(A,B)}(X, X′). By a simple inspection, we see that these objects are naturally isomorphic to the objects denoted in the same way in (3.7.14), so the notation is not in conflict with loc.cit.; it also follows that ℋ om_{A_{l}}(X, X′) is the equalizer of two morphisms in T :

\xymatrix{ℋ om_{T} (X, X′) \ar[r] \ar[r] & ℋ om_{T} (A × X, X′)} .

In the same vein, let A, B, C ∈ Ob(T) be any three objects, S an (A, B)-bimodule, S′ a (C, B)-bimodule, and S′′ a (C, A)-bimodule. Then the (C, B)-bimodule ℋ om_{B_{r}}(S, S′) and the (C, B)-bimodule S′′ ⊗_{A} S (see (3.7.17)) are the sheaves on (T, C_{T}) associated with the presheaves given by the rules : U ↦ Hom_{B_{|U,r}}(S_{|U}, S′_{|U}), and respectively : U ↦ S′(U) ⊗_{M(U)} S(U) for every object U of T. Furthermore, the general theory of monoids, their modules and their tensor products, developed in section 3.7 is available in the present situation, so we have a well defined notion of T -monoid (see example 3.7.21(ii) and remark 3.7.24). Via the equivalence of theorem (4.4.2), a T -monoid \underline{M} is also the same as a sheaf of monoids M on the site (T, C_{T}), and a left (resp. right, resp. bi-) \underline{M}-module is the same as the datum of a sheaf S in (T, C_{T}), such that S(U) is a left (resp. right, resp. bi-) M(U)-module, for every object U of T.

4.8.4. Let f : T_{1} → T_{2} be a morphism of topoi, A_{1} an object of T_{1}, and (X, μ_{X}) a left A_{1}- module. Since f_{∗} is left exact, we have a natural isomorphism : f_{∗}(A_{1} × X) →^{∼} f_{∗}A_{1} × f_{∗}X, so we obtain a left f_{∗}A_{1}-module :

f_{∗}(X, μ_{X}) := (f_{∗}X, f_{∗}μ_{X})

which we denote just f_{∗}X, unless the notation is ambiguous. Likewise, since f^{∗} is left exact, from any object A_{2} of T_{2}, and any left A_{2}-module (Y, μ_{Y}), we obtain a left f^{∗}A_{2}-module :

f^{∗}(Y, μ_{Y}) := (f^{∗}Y, f^{∗}μ_{Y}).

The same considerations apply of course, also to right modules and to bimodules. Furthermore, let A_{i}, B_{i}, C_{i} be three objects of T_{i} (for i = 1, 2); since f_{∗} is left exact, for any (A_{1}, B_{1})- bimodule X and any (C_{1}, A_{1})-bimodule X′ we have a natural morphism of (f_{∗}C_{1}, f_{∗}B_{1})- bimodules :

f_{∗}X′ ⊗_{f_{∗}A_{1}} f_{∗}X → f_{∗}(X′ ⊗_{A_{1}} X)

and since f^{∗} is exact, for any (A_{2}, B_{2})-bimodule Y and any (C_{2}, A_{2})-bimodule Y′ we have a natural isomorphism :

f^{∗}Y′ ⊗_{f^{∗}A_{2}} f^{∗}Y →^{∼} f^{∗}(Y′ ⊗_{A_{2}} Y)

of (f^{∗}C_{2}, f^{∗}B_{2})-bimodules.

4.8.5. The constructions of the previous paragraphs also apply to presheaves on T : this can be seen, e.g. as follows. Pick a universe V such that T is V-small; then T_{V}^{∧} is a V-topos. Hence, if A is any V-presheaf on T, and X, X′ ∈ Ob(T_{V}^{∧}) two left A-modules, we may construct ℋ om_{A_{l}}(X, X′) as an object in T_{V}^{∧}. Now, if A, X, X′ lie in the full subcategory T_{U}^{∧} of T_{V}^{∧}, it is easily seen that also ℋ om_{A_{l}}(X, X′) lies in T_{U}^{∧}. Likewise, if A, B, C are two U-presheaves on T, we may define X′ ⊗_{A} X in T_{U}^{∧}, for any (A, B)-bimodule X and (C, A)-bimodule X′, and this tensor product will still be left adjoint to the ℋ om-functor for presheaves on T.

Moreover, we have a natural morphism of topoi i_{V} : (T, C_{T})_{V}^{∼} → T_{V}^{∧}, given by the forgetful functor and its left adjoint i_{V}^{∗}, which is given by the rule : F ↦ F^{a} : see example 4.4.10(i). The restriction of i_{V}^{∗} to the full subcategory T_{U}^{∧} is isomorphic to a functor that factors through the Yoneda embedding T → (T, C_{T})_{V}^{∼}, therefore the discussion of (4.8.4) specializes to show that, for every U-presheaf A on T, and every A-module X ∈ Ob(T_{U}^{∧}), the object X^{a} ∈ Ob(T) is naturally an A^{a}-module. Also, for any (A, B)-bimodule X and (C, A)-bimodule X′, such that A, B, C, X, X′ are U-small, we have a natural isomorphism of (C^{a}, B^{a})-bimodules :

X^{′a} ⊗_{A^{a}} X^{a} →^{∼} (X′ ⊗_{A} X)^{a}.

The following definition gathers some further notions – specific to monoids over a topos – which shall be used in this work.

Definition 4.8.6. Let T be a topos, \underline{M} a T -monoid, S a left (resp. right, resp. bi-) \underline{M}-module.

(i) S is said to be of finite type, if there exist a covering family (U_{λ} → 1_{T} | λ ∈ Λ) of the final object of T, and for every λ ∈ Λ an integer n_{λ} ∈ ℕ and an epimorphism of left (resp. right, resp. bi-) \underline{M}_{|U_{λ}}-modules : \underline{M}_{|U_{λ}}^{⊕n_{λ}} → S_{|U_{λ}}.

(ii) S is finitely presented, if there exist a covering family (U_{λ} → 1_{T} | λ ∈ Λ) of the final object of T, and for every λ ∈ Λ integers m_{λ}, n_{λ} ∈ ℕ and morphisms f_{λ}, g_{λ} : \underline{M}_{|U_{λ}}^{⊕m_{λ}} → \underline{M}_{|U_{λ}}^{⊕n_{λ}} whose coequalizer – in the category of left (resp. right, resp. bi-) \underline{M}_{|U_{λ}}-modules – is isomorphic to S_{|U_{λ}}.

(iii) S is said to be coherent, if it is of finite type, and for every object U in T, every

submodule of finite type of S_{|U} is finitely presented. (iv) S is said to be invertible, if there exist a covering family (U_{λ} → 1_{T} | λ ∈ Λ), and

for every λ ∈ Λ, an isomorphism \underline{M}_{|U_{λ}} →^{∼} S_{|U_{λ}} of left (resp. right, resp. bi-) \underline{M}_{|U}- modules. (Thus, every invertible module is finitely presented.)

(v) An ideal I ⊂ \underline{M} is said to be invertible, (resp. of finite type, resp. finitely presented, resp. coherent) if it is such, when regarded as an \underline{M}-bimodule.

Example 4.8.7. (i) Take again T = Set. Then an M-module S is of finite type if and only if there exists a finite subset Σ ⊂ S, such that S = M · Σ, with obvious notation. In this case, we say that Σ is a finite system of generators of S (and we also say that S is finitely generated; likewise, an ideal of finite type is also called finitely generated). We say that S is cyclic, if S = M · s for some s ∈ S.

(ii) If S and S′ are two M-modules, the coproduct S ⊕ S′ is the disjoint union of S and S′, with scalar multiplication given by the disjoint union of the laws μ_{S} and μ_{S′}. The product S × S′ is the cartesian product of the underlying sets, with scalar multiplication given by the rule : x · (s, s′) := (x · s, x · s′) for every x ∈ M, s, ∈ S, s′ ∈ S′.

For future use, let us also make the : Definition 4.8.8. Let T be any topos, \underline{P} a T -monoid, and (N, +, 0) any monoid.

(i) We say that \underline{P} is N-graded, if it admits a morphism of monoids π : \underline{P} → N_{T}, where N_{T} is the constant sheaf of monoids arising from N (the coproduct of copies of the final object 1_{T} indexed by N). For every n ∈ N we let \underline{P}_{n} := π^{−1}(n_{T}), the preimage of the global section corresponding to n. Then

\underline{P} = ∐_{n∈N} \underline{P}_{n}

the coproduct of the objects \underline{P}_{n}, and the multiplication law of \underline{P} restricts to a map \underline{P}_{n} × \underline{P}_{m} → \underline{P}_{n+m}, for every n, m ∈ N. Especially, each \underline{P}_{n} is a \underline{P}_{0}-module, and \underline{P} is also the direct sum of the \underline{P}_{n}, in the category of \underline{P}_{0}-modules. The morphism π is called the grading of \underline{P}.

(ii) In the situation of (i), let S be a left (resp. right, resp. bi-) \underline{P} -module. We say that S is N-graded, if it admits a morphism of \underline{P} -modules π_{S} : S → N_{T}, where N_{T} is regarded as a \underline{P} -bimodule via the grading π of \underline{P}. Then S is the coproduct S = ∐_{n∈N} S_{n}, where S_{n} := π_{S}^{−1}(n_{T}), and the scalar multiplication of S restricts to morphisms \underline{P}_{n} × S_{m} → S_{n+m}, for every n, m ∈ N. The morphism π_{S} is called the grading of S.

(iii) A morphism \underline{P} → \underline{Q} of N-graded T -monoids is a morphism of monoids that respects the gradings, with obvious meaning. Likewise one defines morphisms of N-graded \underline{P} -modules.

Example 4.8.9. Take T = Set, and let M be any commutative monoid. Then we claim that the only invertible object in the tensor category M-Mod_{l} is M; i.e. if S and S′ are any two (M, M)-bimodules, then S ⊗_{M} S′ ≃ M if and only if S and S′ are both isomorphic to M.

Indeed, let φ : S ⊗_{M} S′ →^{∼} M be an isomorphism, and choose s_{0} ∈ S, s′_{0} ∈ S′ such that φ(s_{0} ⊗ s′_{0}) = 1. Consider the morphisms of left M-modules :

M ⟶^{α} S ⟶^{β} M M ⟶^{α′} S′ ⟶^{β′} M

such that :

α(m) = m · s_{0} β(s) = φ(s ⊗ s′_{0}) α′(m) = m · s′_{0} β′(s′) = φ(s_{0} ⊗ s′)

for every m ∈ M, s ∈ S, s′ ∈ S′; we notice that β ∘ α = 1_{M} = β′ ∘ α′. There follows natural morphisms :

S′ ⟶^{γ} M ⊗_{M} S′ ⟶^{α⊗_{M} S′} S ⊗_{M} S′ ⟶^{β⊗_{M} S′} M ⊗_{M} S′ ⟶^{γ^{−1}} S′

whose composition is the identity 1_{S′}. However, it is easily seen that φ ∘ (α ⊗_{M} S′) ∘ γ = β′ and γ^{−1} ∘ (β ⊗_{M} S′) ∘ φ^{−1} = α′, thus α′ ∘ β′ = 1_{S′}, hence both α′ and β′ are isomorphisms, and the same holds for α and β.

Example 4.8.10. Let \underline{M} be a T -monoid, and ℒ a \underline{M}-bimodule. For every n ∈ ℕ, let ℒ^{⊗n} := ℒ ⊗\underline{M} ⋯ ⊗\underline{M} ℒ, the n-fold tensor power of ℒ. The ℕ-graded \underline{M}-bimodule

Tens• ; \underline{M}ℒ := ∐_{n∈ℕ} ℒ^{⊗n}

is naturally a ℕ-graded T -monoid, with composition law induced by the natural morphisms ℒ^{⊗n} ⊗\underline{M} ℒ^{⊗m} →^{∼} ℒ^{⊗n+m}, for every n, m ∈ ℕ. (Here we set ℒ^{⊗0} := \underline{M}.) If ℒ is invertible, Tens• ; \underline{M}ℒ is a commutative ℕ-graded T -monoid, which we also denote Sym• ; \underline{M}ℒ.

Remark 4.8.11. (i) Let f : T → S be a morphism of topoi, \underline{M} := (M, μ_{M}) a T -semigroup, and \underline{N} := (N, μ_{N}) a S-semigroup. Then clearly f_{∗}\underline{M} := (f_{∗}M, f_{∗}μ_{M}) is a S-semigroup, and f^{∗}\underline{N} := (f^{∗}N, f^{∗}μ_{N}) is a T -semigroup.

(ii) Furthermore, if 1_{M} : 1_{T} → M (resp. 1_{N} : 1_{S} → N) is a unit for M (resp. for N), then notice that f_{∗}1_{T} = 1_{S} (resp. f^{∗}1_{S} = 1_{T}), since the final object is the empty product; it follows that f_{∗}1_{M} (resp. f^{∗}1_{S}) is a unit for f_{∗}\underline{M} (resp. for f^{∗}\underline{N}).

(iii) Obviously, if \underline{M} (resp. \underline{N}) is commutative, the same holds for f_{∗}\underline{M} (resp. f^{∗}\underline{N}).

(iv) If X is a left \underline{M}-module, then f_{∗}X is a left f_{∗}\underline{M}-module, and if Y is a left \underline{N}-module, then f^{∗}Y is a left f^{∗}\underline{N}-module. The same holds for right modules and bimodules.

(v) Moreover, let ε_{M} : f^{∗}f_{∗}\underline{M} → \underline{M} (resp. η_{N} : \underline{N} → f_{∗}f^{∗}\underline{N}) be the counit (resp. unit) of adjunction. Then the counit (resp. unit) :

ε_{X} : f^{∗}f_{∗}X → X_{(ε_{M})} (resp. η_{Y} : Y → f_{∗}f^{∗}Y_{(η_{N})})

is a morphism of f^{∗}f_{∗}\underline{M}-modules (resp. of \underline{N}-modules) (notation of (3.7.26)). (Details left to the reader.)

(vi) Let φ : f^{∗}\underline{N} → \underline{M} be a morphism of T -monoids. Then the functor

\underline{N}-Mod_{l} → \underline{M}-Mod_{l} : Y ↦ M ⊗_{f^{∗}N} f^{∗}Y

is left adjoint to the functor :

\underline{M}-Mod_{l} → \underline{N}-Mod_{l} : X ↦ f_{∗}X_{(η_{N})}.

(And likewise for right modules and bimodules : details left to the reader.)

(vii) The considerations of (4.8.5) also apply to monoids : we get that, for any presheaf of monoids \underline{M} := (M, μ_{M}, 1_{M}) on T, the datum \underline{M}^{a} := (M^{a}, μ_{M}^{a}, 1_{M}^{a}) is a T -monoid, and we have a well defined functor :

\underline{M}-Mod_{l} → \underline{M}^{a}-Mod_{l} X ↦ X^{a}.

(And as usual, the same applies to right modules and bimodules.)

4.8.12. Let T be a topos, U any object of T, and \underline{M} a T -monoid. As a special case of remark 4.8.11(i), we have the T/U-monoid j_{U}^{∗} \underline{M} = \underline{M}_{|U}, and if we take φ := 1_{j_{U}^{∗} \underline{M}} in remark 4.8.11(vi), we deduce that the functor

j_{U}^{∗} : \underline{M}-Mod_{l} → \underline{M}_{|U}-Mod_{l} Y ↦ Y_{|U}

admits the right adjoint j_{U∗}. Now, suppose that X → U is any left \underline{M}_{|U}-module. The scalar multiplication of X is a U-morphism μ_{X} : M × X → X and j_{U!}μ_{X} is the same morphism, seen as a morphism in T (notation of example 4.7.8(i)). In other words, j_{U!} induces a faithful functor on left modules, also denoted :

j_{U!} : \underline{M}_{|U}-Mod_{l} → \underline{M}-Mod_{l}.

It is easily seen that this functor is left adjoint to the foregoing functor j_{U}^{∗}. Especially, this functor is right exact; it is not generally left exact, since it does not preserve the final object (unless U = 1_{T}). However, it does commute with fibre products, and therefore transforms monomorphisms into monomorphisms. All this holds also for right modules and bimodules.

4.8.13. Let T be any category as in example 3.7.21(i), denote by 1_{T} a final object of T, and by \underline{M} any T -monoid. A pointed left \underline{M}-module is a datum

(S, 0_{S})

consisting of a left \underline{M}-module S and a morphism of \underline{M}-modules 0_{S} : 1_{T} → S, where 0 is the final object of M-Mod_{l}. Often we shall write S instead of (S, 0_{S}), unless this may give rise to ambiguities. As usual, a morphism φ : S → T of pointed modules is a morphism of M-modules, such that 0_{T} = φ ∘ 0_{S}. In other words, the resulting category is just 0/M-Mod_{l}, and shall be denoted \underline{M}-Mod_{l∘}.

Likewise one may define the category \underline{M}-Mod_{r∘} of right \underline{M}-modules, and (\underline{M}, \underline{N})-Mod_{∘} of pointed bimodules, for given T -monoids \underline{M} and \underline{N}.

Remark 4.8.14. Let T be a category as in remark 3.7.24.

(i) For reasons that will become readily apparent, for many purposes the categories of pointed modules are more useful than the non-pointed variant of (3.7.22). In any case, we have a faithful functor :

(4.8.15)  \underline{M}-Mod_{l} → \underline{M}-Mod_{l∘}  S ↦ S_{∘} := (S ⊕ 0, 0_{S})

where 0_{S} : 0 → S ⊕ 0 is the obvious inclusion map. Thus, we may – and often will, without further comment – regard any \underline{M}-module as a pointed module, in a natural way. (The same can of course be repeated for right modules and bimodules.)

(ii) In turn, when dealing with pointed M-modules, things often work out nicer if \underline{M} itself is a pointed T -monoid. The latter is the datum (\underline{M}, 0_{M}) of a T -monoid \underline{M} and a morphism of \underline{M}-modules 0_{M} : 0 → M. A morphism of pointed T -monoids is of course just a morphism f : \underline{M} → \underline{M}′ of T -monoids, such that f ∘ 0_{M} = 0_{M′}. As customary, we shall often just write \underline{M} instead of (\underline{M}, 0_{M}), unless we wish to stress that \underline{M} is pointed.

(iii) Let (\underline{M}, 0_{M}) be a pointed T -monoid; a pointed left (\underline{M}, 0_{M})-module is a pointed left \underline{M}-module S, such that 0 · s = 0 for every s ∈ S. A morphism of pointed left (\underline{M}, 0_{M})- modules is just a morphism of pointed left \underline{M}-modules. As usual, these gadgets form a category (\underline{M}, 0_{M})-Mod_{l∘}. Similarly we have the right and bi-module variant of this definition.

(iv) The forgetful functor from the category of pointed T -monoids to the category of T - monoids, admits a left adjoint :

\underline{M} ↦ (\underline{M}_{∘}, 0_{M_{∘}}).

Namely, M_{∘} is the \underline{M}-module M ⊕ 0, the zero map 0_{M_{∘}} : 0 → M ⊕ 0 is the obvious inclusion, and the scalar multiplication M ×M_{∘} → M_{∘} is extended to a multiplication law μ : M_{∘}×M_{∘} → M_{∘} in the unique way for which (\underline{M}_{∘}, μ, 0_{M_{0}}) is a pointed monoid. The unit of adjunction \underline{M} → \underline{M}_{∘} is the obvious inclusion map.

(v) If \underline{M} is a (non-pointed) monoid, the restriction of scalars

(\underline{M}_{∘}, 0_{M_{∘}})-Mod_{l∘} → \underline{M}-Mod_{l∘}

is an isomorphism of categories. Namely, any pointed left \underline{M}-module S is naturally a pointed left \underline{M}_{∘}-module : the given scalar multiplication M × S → S extends to a scalar multiplication M_{∘} × S → S whose restriction 0 × S → S factors through the zero section 0_{S} (and likewise for right modules and bimodules).

(vi) Let T be a topos. The notions introduced thus far for non-pointed T -monoids, also admit pointed variants. Thus, a pointed module (S, 0_{S}) is said to be of finite type if the same holds for S, and S is finitely presented if, locally on T, it is the coequalizer of two morphisms between free \underline{M}-modules of finite type.

Example 4.8.16. Take T := Set, and let M be any monoid; then a pointed left M-module is just a left M-module S endowed with a distinguished zero element 0 ∈ S, such that m · 0 = 0 for every m ∈ M. A morphism φ : S → S′ of pointed left M-modules is just a morphism of left M-modules such that φ(0) = 0 (and similarly for right modules and bimodules.)

Likewise, a pointed monoid is endowed with a distinguished zero element, denoted 0 as usual, such that 0 · x = 0 for every x ∈ M.

Remark 4.8.17. Let T be a category as in remark 3.7.24, and \underline{M} a T -monoid.

(i) Regardless of whether \underline{M} is pointed or not, the category \underline{M}-Mod_{l∘} is also complete and cocomplete; for instance, if (S, 0_{S}) and (S′, 0_{S′}) are two pointed modules, the coproduct (S′′, 0_{S′′}) := (S, 0_{S}) ⊕ (S′, 0_{S′}) is defined by the push-out (in the category \underline{M}-Mod_{l}) of the cocartesian diagram :

\xymatrix{0 ⊕ 0 \ar[r]^{0_{S}⊕0_{S′}} \ar[d] & S ⊕ S′ \ar[d] \\ 0 \ar[r]^{0_{S′′}} & S′′.}

Likewise, if φ′ : S′ → S and φ′′ : S′′ → S are two morphisms in \underline{M}-Mod_{l∘}, the fibre product S′ ×_{S} S′′ in the category \underline{M}-Mod_{l} is naturally pointed, and represents the fibre product in the category of pointed modules. All this holds also for right modules and bimodules.

(ii) The forgetful functor \underline{M}-Mod_{l∘} → T_{∘} := 1_{T} /T to the category of pointed objects of T, commutes with all limits, since it is a right adjoint; it also commutes with all colimits. This forgetful functor admits a left adjoint, that assigns to any Σ ∈ Ob(T) the free pointed \underline{M}-module \underline{M}^{(Σ)_{∘}}. If \underline{M} is pointed, the latter is defined as the push-out in the cocartesian diagram

\xymatrix{1_{T} × Σ \ar[r] \ar[d] & M × Σ \ar[d] \\ 1_{T} \ar[r] & _{\underline{M}_{(Σ)_{∘}}}}

and if \underline{M} is not pointed, one defines it via the equivalence of remark 4.8.14(v) : by a simple inspection we find that in this case \underline{M}^{(Σ)_{∘}} = (\underline{M}^{(Σ)})_{∘}, where \underline{M}^{(Σ)} is the free (unpointed) \underline{M}-module, as in remark 3.7.24(iii).

Notice as well that the forgetful functors T_{∘} → T and \underline{M}-Mod_{l∘} → \underline{M}-Mod_{l} both commute with all connected colimits, hence the same also holds for the forgetful functor \underline{M}-Mod_{l∘} → T. (See definition 1.2.19(vii).) The same can be repeated for right modules and bimodules.

(iii) Moreover, if φ : S → S′ is any morphism in \underline{M}-Mod_{l∘}, we may define Ker φ and Coker φ (in the category \underline{M}-Mod_{l∘}); namely, the kernel is the limit of the diagram S ⟶^{φ} S′ ← 0 and the cokernel is the colimit of the diagram 0 ← S ⟶^{φ} S′. Especially, if S is a submodule of S′, we have a well defined quotient S′/S of pointed left \underline{M}-modules. Furthermore, we say that a sequence of morphisms of pointed left \underline{M}-modules :

0 → S′ ⟶^{φ} S ⟶^{ψ} S′′ → 0

is right exact, if ψ induces an isomorphism Coker φ →^{∼} S′′; we say that it is left exact, if φ induces an isomorphism S′ →^{∼} Ker ψ, and it is short exact if it is both left and right exact. (Again, all this can be repeated also for right modules and bimodules.)

Example 4.8.18. Take T = Set, and let M be a pointed or not-pointed monoid. Then the argument from example 3.7.27can be repeated for the free pointed M-modules : if Σ is any set, we have

M^{(Σ)_{∘}} ⊗_{M} {1} →^{∼} {1}^{(Σ)_{∘}} = Σ_{∘}

where Σ_{∘} is the disjoint union of Σ and the final object of Set (a set with one element). Hence, the cardinality of Σ is an invariant, called the rank of the free pointed M-module M^{(Σ)_{∘}}, and denoted rk_{M}^{∘}M^{(Σ)_{∘}}.

4.8.19. Let T be a topos, (\underline{M}, 0_{M}), (\underline{N}, 0_{N}) and (\underline{P} , 0_{P}) three pointed T -monoids, S, (resp. S′) a pointed (\underline{M}, \underline{N})-bimodule (resp. (\underline{P} , \underline{N})-bimodule); we denote

Hom_{(\underline{N},0_{N})_{r}}(S, S′)

the set of all morphisms of pointed right \underline{N}-modules S → S′. As usual, the presheaf

ℋ om_{(\underline{N},0_{N})_{r}}(S, S′) : U ↦ Hom_{(\underline{N},0_{N})_{r|U}} (S_{|U}, S′_{|U})

(with obvious notation) is a sheaf on (T, C_{T}), hence it is represented by an object of T. Indeed, this object is also the fibre product in the cartesian diagram :

\xymatrix{ℋ om_{(\underline{N},0_{N})_{r}}(S, S′) \ar[r] \ar[d] & ℋ om_{N_{r}}(S, S′) \ar[d]^{0_{S}^{∗}} \\ ℋ om_{N_{r}}(0, 0) \ar[r]^{0_{S′∗}} & ℋ om_{N_{r}}(0, S′).}

Especially, ℋ om_{(\underline{N},0_{N})_{r}}(S, S′) is naturally a (\underline{P} , \underline{M})-bimodule, and moreover, it is pointed : its zero section represents the unique morphism S → S′ which factors through 0.

Notice also that, for every pointed (\underline{P} , \underline{M})-bimodule S′′, the tensor product S′′ ⊗_{M} S is naturally pointed, and as in the non-pointed case, the functor

(4.8.20)  (\underline{P} , \underline{M})-Mod_{∘} → (\underline{P} , \underline{N})-Mod_{∘} : S′′ ↦ S′′ ⊗_{M} S

is left adjoint to the functor

(\underline{P} , \underline{N})-Mod_{∘} → (\underline{P} , \underline{M})-Mod_{∘} : S′ ↦ ℋ om_{(\underline{N},0_{N})_{r}}(S, S′).

By general nonsense, the functor (4.8.20) is right exact; especially, for any right exact sequence T′ → T → T′′ → 0 of pointed (\underline{P} , \underline{M})-bimodules, the induced sequence

T′ ⊗_{M} S → T ⊗_{M} S → T′′ ⊗_{M} S → 0

is again right exact.

Remark 4.8.21. Suppose \underline{M}, \underline{N} and \underline{P} are non-pointed T -monoids, S is a (\underline{M}, \underline{N})-bimodule and S′′ a (\underline{P} , \underline{M})-bimodule.

(i) If S and S′′ are pointed, one may define a tensor product S′′ ⊗_{M} S in the category (\underline{P} , \underline{N})-Mod_{∘}, if one regards S as a pointed (\underline{M}_{∘}, \underline{N}_{∘})-bimodule, and S′′ as a (\underline{P}_{∘}, \underline{M}_{∘})- bimodule as in remark 4.8.14(v); then one sets simply S′′ ⊗_{M} S := S′′ ⊗_{M_{∘}} S, which is then viewed as a pointed (\underline{P} , \underline{N})-bimodule. In this way one obtains a left adjoint to the corresponding internal Hom-functor ℋ om_{N} from pointed (\underline{P} , \underline{N})-bimodules to pointed (\underline{P} , \underline{M})-bimodules (details left to the reader).

(ii) Lastly, if neither S nor S′′ is pointed, notice the natural isomorphism :

(S′′ ⊗_{M} S)_{∘} →^{∼} S′′_{∘} ⊗_{M_{∘}} S_{∘} in the category (\underline{P} , \underline{N})-Mod_{∘}.

Definition 4.8.22. In the situation of (4.8.19), let P = N := (1_{T})_{∘}, and notice that – with these choices of P and N – a pointed (\underline{P} , \underline{M})-bimodule (resp. a pointed (\underline{M}, \underline{N})-bimodule) is just a right pointed \underline{M}-module (resp. a left pointed \underline{M}-module), and a pointed (\underline{P} , \underline{N})-module is just a pointed object of T.

(i) We say that S is a flat pointed left \underline{M}-module (or briefly, that S is \underline{M}-flat), if the functor (4.8.20) transforms short exact sequences of right pointed \underline{M}-modules, into short exact sequences of pointed T -objects. Likewise, we define flat pointed right \underline{M}-modules. (ii) Let φ : \underline{M} → \underline{M}′ be a morphism of pointed T -monoids. We say that φ is flat, if \underline{M}′ is

a flat left \underline{M}-module, for the module structure induced by φ.

Remark 4.8.23. (i) In the situation of remark 4.8.11(i), suppose that \underline{M} := (M, 0_{M}) is a pointed T -monoid and \underline{N} := (N, 0_{N}) a pointed S-monoid. By arguing as in remark 4.8.11(ii), we see that f^{∗}\underline{N} := (f^{∗}N, f^{∗}0_{N}) is a pointed T -monoid, and f_{∗}\underline{M} := (f_{∗}M, f_{∗}0_{M}) is a pointed S-monoid.

(ii) Likewise, if (X, 0_{X}) is a pointed left \underline{M}-module, and (Y, 0_{Y}) a pointed left \underline{N}-module, the f_{∗}(X, 0_{X}) := (f_{∗}X, f_{∗}0_{X}) is a pointed f_{∗}\underline{M}-module, and f^{∗}(Y, 0_{Y}) := (f^{∗}Y, f^{∗}0_{Y}) is a pointed f^{∗}\underline{N}-module (and likewise for right modules and bimodules).

(iii) Also, just as in remark 4.8.11(vii), the associated sheaf functor F ↦ F^{a} transforms a presheaf \underline{M} of pointed monoids on T, into a pointed T -monoid \underline{M}^{a}, and sends pointed left (resp. right, resp. bi-) \underline{M}-modules to pointed left (resp. right, resp. bi-) \underline{M}^{a}-modules.

(iv) Moreover, if φ : f^{∗}\underline{N} → \underline{M} is a morphism of pointed T -monoids, then – in view of the discussion of (4.8.19) – the adjunction of remark 4.8.11(vi) extends to pointed modules : we leave the details to the reader.

(v) Furthermore, in the situation of (4.8.12), we may also define a functor

j_{U!} : \underline{M}_{|U}-Mod_{l∘} → \underline{M}-Mod_{l∘}

which will be a left adjoint to j_{U}^{∗}. Indeed, let (X, 0_{X}) be a left pointed \underline{M}_{|U}-module; the functor from (4.8.12) yields a morphism j_{U!}0_{X} of (non-pointed) \underline{M}-modules, and we define j_{U!}(X, 0_{X}) to be the push-out (in the category \underline{M}-Mod_{l}) of the diagram 0 ← j_{U!}0_{X} ⟶^{j_{U!}0_{X}} j_{U!}X. The latter is endowed with a natural morphism 0 → j_{U!}(X, 0_{X}), so we have a well defined pointed left \underline{M}-module. We leave to the reader the verification that the resulting functor, called extension by zero, is indeed left adjoint to the restriction functor.

(vi) It is convenient to extend definition 4.8.22 to non-pointed modules and monoids : namely, if S is a non-pointed left \underline{M}-module, we shall say that S is flat, if the same holds for the pointed left \underline{M}_{∘}-module S_{∘}. Likewise, we say that a morphism φ : \underline{M} → \underline{N} of non-pointed T -monoids is flat, if the same holds for φ_{∘}.

Lemma 4.8.24. Let T be a topos, U any object of T, and denote by i_{∗} : CU → T the inclusion functor of the complement of U in T (see example 4.7.8(iii)). Let also \underline{M}, \underline{N}, \underline{P} be three pointed T -monoids. Then the following holds :

(i) The functor j_{U!} of extension by zero is faithful, and transforms exact sequences of pointed left \underline{M}_{|U}-modules, into exact sequences of pointed left \underline{M}-modules (and likewise for right modules and bimodules).

(ii) For every pointed (\underline{M}, \underline{N})-bimodule S and every pointed (\underline{P}_{|U}, \underline{M}_{|U})-bimodule S′, the natural morphism of pointed (\underline{P} , \underline{N})-modules

j_{U!}(S′ ⊗_{M_{|U}} S_{|U}) → j_{U!}S′ ⊗_{M} S

is an isomorphism. (iii) If S is flat pointed left \underline{M}_{|U}-module, then j_{U!}S is a flat pointed left \underline{M}-module (and

likewise for right modules). (iv) For every pointed (\underline{M}, \underline{N})-bimodule S, and every pointed (i^{∗}\underline{P} , i^{∗}\underline{M})-bimodule S′, the

natural morphism of pointed (\underline{P} , \underline{N})-bimodules

i_{∗}S′ ⊗_{M} S → i_{∗}(S′ ⊗_{i^{∗}M} i^{∗}S)

is an isomorphism.

(v) If S is a flat pointed left i^{∗}\underline{M}-module, then i_{∗}S is a flat left pointed \underline{M}-module (and likewise for right modules).

(vi) If S is a flat pointed left \underline{M}-module, then S_{|U} is a flat left pointed \underline{M}_{|U}-module.

Proof. (i): Let us show first that j_{U!} is faithful. Indeed, suppose that φ, ψ : S → S′ are two morphisms of left pointed \underline{M}_{|U}-modules, such that j_{U!}φ = j_{U!}ψ. We need to show that φ = ψ.

Let p : S′ → S′′ be the coequalizer of φ and ψ; then j_{U!}p is the coequalizer of j_{U!}φ and j_{U!}ψ (since j_{U!} is right exact); hence we are reduced to showing that a morphism p : S′ → S′′ is an isomorphism if and only if the same holds for j_{U!}p. This follows from remark 3.7.24(ii) and the following more general :

Claim 4.8.25. Let φ : X → X′, A → X, A → B be three morphisms in T. Then φ is a monomorphism (resp. an epimorphism) if and only if the same holds for the induced morphism φ ⨿_{A} B : X ⨿_{A} B → X′ ⨿_{A} B.

Proof of the claim. We may assume that T = C^{∼} for some small site C := (𝒞 , J). Then φ ⨿_{A} B = (iφ ⨿_{iA} iB)^{a}, where i : C^{∼} → 𝒞^{∧} is the forgetful functor. Since the functor F ↦ F^{a} is exact, we are reduced to the case where T = 𝒞^{∧}, and in this case the assertion can be checked argumentwise, i.e. we may assume that T = Set, where the claim is obvious. ♢

Next, we already know that j_{U!} transforms right exact sequences into right exact sequences. To conclude, it suffices then to check that j_{U!} transforms monomorphisms into monomorphisms.To this aim, we apply again remark 3.7.24(ii) and claim 4.8.25.

(ii) is proved by general nonsense, and (iii) is an immediate consequence of (i) and (ii) : we leave the details to the reader.

(iv): By (4.8.4), we have j_{U}^{∗}(i_{∗}S′ ⊗_{M} S) ≃ 0 ⊗_{j_{U}^{∗} M} j_{U}^{∗}S ≃ 1_{T/U}, hence i_{∗}S′ ⊗_{M} S ∈ Ob(CU). Notice now that, for every object X of CU, the counit of adjunction i^{∗}i_{∗}X → X is an isomorphism (proposition 1.1.20(iii)); by the triangular identities of (1.1.13), it follows that the same holds for the unit of adjunction i_{∗}X → i_{∗}i^{∗}i_{∗}X. Especially, the natural morphism :

i_{∗}S′ ⊗_{M} S → i_{∗}i^{∗}(i_{∗}S′ ⊗_{M} S) →^{∼} i_{∗}(i^{∗}i_{∗}S′ ⊗_{i^{∗}M} i^{∗}S) →^{∼} i_{∗}(S′ ⊗_{i^{∗}M} i^{∗}S).

is an isomorphism. The latter is the morphism of assertion (iv).

(v) follows easily from (iv) and its proof.

(vi): In view of (i), it suffices to show that the functor S′ ↦ j_{U!}(S′ ⊗_{M_{|U}} S_{|U}) transforms exact sequences into exact sequences. The latter follows easily from (ii). □

Proposition 4.8.26. Let P(T, \underline{M}, S) be the property : “S is a flat pointed left \underline{M}-module” (for a monoid \underline{M} on a topos T). Then P can be checked on stalks. (See definition 4.7.11(v).)

Proof. Suppose first that S_{ξ} is a flat left \underline{M}_{ξ}-module for every ξ in a conservative set of T -points; let φ : X → X′ be a monomorphism of pointed right \underline{M}-modules; by (4.8.4) we have a natural isomorphism

(φ ⊗_{M} S)_{ξ} →^{∼} φ_{ξ} ⊗_{M_{ξ}} S_{ξ}

in the category of pointed sets, and our assumption implies that these morphisms are monomorphisms. Since an arbitrary product of monomorphisms is a monomorphism, remark 1.2.21(ii) shows that φ ⊗_{M} S is also a monomorphism, whence the contention.

Next, suppose that S is a flat pointed left \underline{M}-module. We have to show that the functor

(4.8.27)  S′ ↦ S′ ⊗_{M_{ξ}} S_{ξ}

from pointed right \underline{M}_{ξ}-modules to pointed sets, preserves monomorphisms.

However, let (U, ξ_{U}, ω_{U}) be any lifting of ξ (see (4.7.17)); in view of (4.8.4), we have

P_{U}(S′) := (ξ_{U}^{∗}ξ_{U∗}S′) ⊗_{A_{ξ}} S_{ξ} ≃ (ξ_{U}^{∗}ξ_{U∗}S′) ⊗_{ξ_{U}^{∗} A_{|U}} ξ_{U}^{∗}S_{|U} ≃ ξ_{U}^{∗}(ξ_{U∗}S′ ⊗_{A_{|U}} S_{|U})

and then lemma 4.8.24(vi) implies that the functor S′ → P_{U}(S′) preserves monomorphisms. By lemma 4.7.20, remark 4.7.13(ii) and (4.7.18), the functor (4.8.27) is a filtered colimit of such functors P_{U}, hence it preserves monomorphisms as well. □

4.8.28. We wish now to introduce a few notions that pertain to the special class of commutative T -monoids. When T = Set, these notions are well known, and we wish to explain quickly that they generalize without problems, to arbitrary topoi.

To begin with, for every category T as in example 3.7.21(i), we denote by Mnd_{T} (resp. Mnd_{T ∘}) the category of commutative non-pointed (resp. pointed) T -monoids; in case T = Set, we shall usually drop the subcript, and write just Mnd (resp. Mnd_{∘}). Notice that if \underline{M} is any (pointed or not pointed) commutative T -monoid, every left or right \underline{M}-module is a \underline{M}- bimodule in a natural way, hence we shall denote indifferently by \underline{M}-Mod (resp. \underline{M}-Mod_{∘}) the category of non-pointed (resp. pointed) left or right \underline{M}-modules.

The following lemma is a special case of a result that holds more generally, for every ”algebraic theory” in the sense of [29, Def.3.3.1] (see [29, Prop.3.4.1, Prop.3.4.2]).

Lemma 4.8.29. Let T be a topos. We have :

(i) The category Mnd_{T} admits arbitrary limits and colimits.

(ii) In the category Mnd_{T}, filtered colimits commute with all finite limits.

(iii) The forgetful functor ι : Mnd_{T} → T that assigns to a monoid its underlying object of T, commutes with all limits, and with all filtered colimits.

Proof. (iii): Commutation with limits holds because ι admits a left adjoint : namely, to an object Σ of T one assigns the free monoid ℕ_{T}^{(Σ)} generated by Σ, defined as the sheaf associated with the presheaf of monoids

U ↦ ℕ^{(Σ(U))} for every U ∈ Ob(T)

where ℕ is the additive monoid of natural numbers (see remark 4.8.11(vii)). One verifies easily that this T -monoid represents the functor

\underline{M} ↦ Hom_{T} (Σ, \underline{M}) Mnd_{T} → Set.

Moreover, if I is any small category, and F : I → Mnd_{T} any functor, one checks easily that the limit of ι ∘ F can be endowed with a unique composition law (indeed, the limit of the composition laws of the monoids F_{i}), such that the resulting monoid represents the limit of F.

A similar argument also shows that Mnd_{T} admits arbitrary filtered colimits, and that ι commutes with filtered colimits. It is likewise easy to show that the product of two T -monoids \underline{M} and \underline{N} is also the coproduct of \underline{M} and \underline{N}. To complete the proof of (i), it suffices therefore to show that any two maps f, g : \underline{M} → \underline{N} admit a coequalizer; the latter is obtained as the coequalizer \underline{N}′ (in the category T) of the two morphisms :

\xymatrix{\underline{M} × \underline{N} \ar[r]_{μ_{N} ∘(g×1_{N})} \ar[r]^{μ_{N} ∘(f×1_{N})} & \underline{N}.}

We leave to the reader the verification that the composition law of N descends to a (necessarily unique) composition law on \underline{N}′.

(ii) follows from (iii) and the fact that the same assertion holds in T (remark 4.4.1(i)). □

Example 4.8.30. (i) For instance, if T = Set, the product M_{1} × M_{2} of any two commutative monoids is representable in Mnd; its underlying set is the cartesian product of M_{1} and M_{2}, and the composition law is the obvious one.

(ii) As usual, the kernel Ker φ (resp. cokernel Coker φ) of a map of T -monoids φ : \underline{M} → \underline{N} is defined as the fibre product (resp. push-out) of the diagram of T -monoids

\underline{M} ⟶^{φ} \underline{N} ← \underline{1}_{T} (resp. \underline{1}_{T} ← \underline{M} ⟶^{φ} \underline{N}).

Especially, if \underline{M} ⊂ \underline{N}, one defines in this way the quotient \underline{N}/\underline{M}.

(iii) Also, if T = Set, and φ_{1} : M → M_{1}, φ_{2} : M → M_{2} are two maps in Mnd, the push-out M_{1} ⨿_{M} M_{2} can be described as follows. As a set, it is the quotient (M_{1} × M_{2})/∼, where ∼ denotes the minimal equivalence relation such that

(m_{1}, m_{2} · φ_{2}(m)) ∼ (m_{1} · φ_{1}(m), m_{2}) for every m ∈ M, m_{1} ∈ M_{1}, m_{2} ∈ M_{2}

and the composition law is the unique one such that the projection M_{1} × M_{2} → M_{1} ⨿_{M} M_{2} is a map of monoids. We deduce the following :

Lemma 4.8.31. Let G be an abelian group. The following holds :

(i) If φ : M → N and ψ : M → G are two morphisms of monoids (in the topos T = Set), G ⨿_{M} N is the quotient (G × N)/≈, where ≈ is the equivalence relation such that :

(g, n) ≈ (g′, n′) ⇔ (ψ(a) · g, φ(b) · n) = (ψ(b) · g′, φ(a) · n′) for some a, b ∈ M.

(ii) If φ : G → M and ψ : G → N are two morphisms of monoids, the set underlying M ⨿_{G}N is the set-theoretic quotient (M ×N)/G for the G-action defined via (φ, ψ^{−1}).

(iii) Especially, if M is a monoid and G is a submonoid of M, then the set underlying M/G is the set-theoretic quotient of M by the translation action of G.

Proof. (i): One checks easily that the relation ≈ thus defined is transitive. Let ∼ be the equivlence relation defined as in example 4.8.30(iii). Clearly :

(g, n · ψ(m)) ≈ (g · φ(m), n) for every g ∈ G, n ∈ N and m ∈ M

hence (g, n) ∼ (g′, n′) implies (g, n) ≈ (g′, n′). Conversely, suppose that (ψ(a) · g, φ(b) · n) = (ψ(b) · g′, φ(a) · n′) for some g ∈ G, n ∈ N and a, b ∈ M. Then :

(g, n) = (g, φ(a) · φ(a)^{−1} · n) ∼ (ψ(a) · g, φ(a)^{−1} · n) = (ψ(b) · g′, φ(a)^{−1} · n)

as well as : (g′, n′) = (g′, φ(b) · φ(a)^{−1} · n) ∼ (ψ(b) · g′, φ(a)^{−1} · n). Hence (g, n) ∼ (g′, n′) and the claim follows.

(ii) follows directly from example 4.8.30(iii), and (iii) is a special case of (ii). □

4.8.32. Let T be a topos. For any T -ring \underline{R}, we let \underline{R}-Mod be the category of \underline{R}-modules (defined in the usual way); especially, we may consider the T -ring ℤ_{T} (the constant sheaf with value ℤ : see (4.4.11)). Then ℤ_{T} -Mod is the category of abelian T -groups. The forgetful functor ℤ_{T} -Mod → Mnd_{T} admits a right adjoint :

Mnd_{T} → ℤ_{T} -Mod : \underline{M} ↦ \underline{M}^{×}.

The latter can be defined as the fibre product in the cartesian diagram :

\xymatrix{\underline{M}^{×} \ar[r] \ar[d] & \underline{M} × \underline{M} \ar[d]^{μ_{M}} \\ 1_{T} \ar[r]^{1_{M}} & M.}

For i = 1, 2, let p_{i} : \underline{M} × \underline{M} → \underline{M} be the projections, and p′_{i} : \underline{M}^{×} → \underline{M} the restriction of p_{i}; for every U ∈ Ob(T), the image of p′_{i}(U) : \underline{M}^{×}(U) → \underline{M}(U) consists of all sections x which are invertible, i.e. for which there exists y ∈ \underline{M}(U) such that μ_{M}(x, y) = 1_{M}. It is easily seen that such inverse is unique, hence p′_{i} is a monomorphism, p′_{1} and p′_{2} define the same subobject of \underline{M}, and this subobject \underline{M}^{×} is the largest abelian T -group contained in \underline{M}. We say that \underline{M} is sharp, if \underline{M}^{×} = 1_{T}. The inclusion functor, from the full subcategory of sharp T -monoids, to Mnd_{T}, admits a left adjoint

\underline{M} ↦ \underline{M}^{♯} := \underline{M}/\underline{M}^{×}.

We call \underline{M}^{♯} the sharpening of \underline{M}.

4.8.33. Let \underline{S} be a submonoid of a commutative T -monoid \underline{M}, and F_{S} : Mnd_{T} → Set the functor that assigns to any commutative T -monoid \underline{N} the set of all morphisms f : \underline{M} → \underline{N} such that f(\underline{S}) ⊂ \underline{N}^{×}. We claim that F_{S} is representable by a T -monoid \underline{S}^{−1}\underline{M}.

In case T = Set, one may realize \underline{S}^{−1}\underline{M} as the quotient (\underline{S} × \underline{M})/∼ for the equivalence relation such that (s_{1}, x_{1}) ∼ (s_{2}, x_{2}) if and only if there exists t ∈ \underline{S} such that ts_{1}x_{2} = ts_{2}x_{1}. The composition law of \underline{S}^{−1}\underline{M} is the obvious one; then the class of a pair (s, x) is denoted naturally by s^{−1}x. This construction can be repeated on a general topos : letting X := \underline{S} × \underline{M}, the foregoing equivalence relation can be encoded as the equalizer R of two maps X ×X ×S → \underline{M}, and the quotient under this equivalence relation shall be represented by the coequalizer of two other maps R → X; the reader may spell out the details, if he wishes. Equivalently, \underline{S}^{−1}\underline{M} can be realized as the sheaf on (T, C_{T}) associated with the presheaf :

T → Mnd : U ↦ \underline{S}(U)^{−1}\underline{M}(U)

(see remark 4.8.11(vii)). The natural morphism \underline{M} → \underline{S}^{−1}\underline{M} is called the localization map. For T = Set, and f ∈ M any element, we shall also use the standard notation :

M_{f} := S_{f}^{−1}M where S_{f} := {f^{n} | n ∈ ℕ}.

Lemma 4.8.34. Let f_{1} : \underline{M} → \underline{N}_{1} and f_{2} : \underline{M} → \underline{N}_{2} be morphisms of T -monoids, \underline{S} ⊂ \underline{M}, \underline{S}_{i} ⊂ \underline{N}_{i} (i = 1, 2) three submonoids, such that f_{i}(\underline{S}) ⊂ \underline{S}_{i} for i = 1, 2. Then the natural morphism :

(\underline{S}_{1} · \underline{S}_{2})^{−1}(\underline{N}_{1} ⨿\underline{M} \underline{N}_{2}) → \underline{S}_{1}^{−1}\underline{N}_{1} ⨿\underline{S}^{−1}\underline{M} \underline{S}_{2}^{−1}\underline{N}_{2}

is an isomorphism.

Proof. One checks easily that both these T -monoids represent the functor Mnd_{T} → Set that assigns to any T -monoid \underline{P} the pairs of morphisms (g_{1}, g_{2}) where g_{i} : \underline{N}_{i} → \underline{P} satisfies g_{i}(\underline{S}_{i}) ⊂ \underline{P}^{×}, for i = 1, 2, and g_{1} ∘ f_{1} = g_{2} ∘ f_{2}. The details are left to the reader. □

4.8.35. The forgetful functor ℤ-Mod_{T} → Mnd_{T} from abelian T -groups to commutative T -monoids, admits a left adjoint

\underline{M} ↦ \underline{M}^{gp} := \underline{M}^{−1}\underline{M}.

A commutative T -monoid \underline{M} is said to be integral if the unit of adjunction \underline{M} → \underline{M}^{gp} is a monomorphism. The functor \underline{M} ↦ \underline{M}^{gp} commutes with all colimits, since all left adjoints do; it does not commute with arbitrary limits (see example 4.8.36(v)).

We denote by Int.Mnd_{T} the full subcategory of Mnd_{T} consisting of all integral monoids; when T = Set, we omit the subscript, and write just Int.Mnd. The natural inclusion ι : Int.Mnd_{T} → Mnd_{T} admits a left adjoint :

Mnd_{T} → Int.Mnd_{T} : \underline{M} ↦ \underline{M}^{int}.

Namely, \underline{M}^{int} is the image (in the category T) of the unit of adjunction \underline{M} → \underline{M}^{gp}. It follows easily that the category Int.Mnd_{T} is cocomplete, since the colimit of a family (\underline{M}_{λ} | λ ∈ Λ) of integral monoids is represented by

(colim_{λ∈Λ} ι\underline{M}_{λ})^{int}.

Likewise, Int.Mnd_{T} is complete, and limits commute with the forgetful functor to T; to check this, it suffices to show that

L := lim_{λ∈Λ} ι(\underline{M}_{λ})

is integral. However, by lemma 4.8.29(iii) we have L ⊂ ∏_{λ∈Λ} \underline{M}_{λ} ⊂ ∏_{λ∈Λ} \underline{M}_{λ}^{gp}, whence the claim.

Example 4.8.36. (i) Take T = Set; if M is any monoid, and a ∈ M is any element, we say that a is regular, if the map M → M given by the rule : x ↦ a · x is injective. It is easily seen that M is integral if and only if every element of M is regular.

(ii) For an arbitrary topos T, notice that the T -monoid \underline{G}^{a} associated with a presheaf of groups \underline{G} on T, is a T -group : indeed, the condition \underline{G}^{×} = \underline{G} implies \underline{(}G^{a})^{×} = \underline{G}^{a}, since the functor F ↦ F^{a} is exact. More precisely, for every presheaf \underline{M} of monoids on T, we have a natural isomorphism :

(\underline{M}^{gp})^{a} →^{∼} (\underline{M}^{a})^{gp} for every T -monoid \underline{M}

since both functors are left adjoint to the forgetful functor from T -groups to presheaves of monoids on T.

(iii) It follows from (ii) that a T -monoid \underline{M} is integral if and only if \underline{M}(U) is an integral monoid, for every U ∈ Ob(T). Indeed, if \underline{M} is integral, then \underline{M}(U) ⊂ \underline{M}^{gp}(U) for every such U, so \underline{M}(U) is integral. Conversely, by definition \underline{M}^{gp} is the sheaf associated with the presheaf U ↦ \underline{M}(U)^{gp}; now, if \underline{M}(U) is integral, we have \underline{M}(U) ⊂ \underline{M}(U)^{gp}, and consequently \underline{M} ⊂ \underline{M}^{gp}, since the functor F ↦ F^{a} is exact.

(iv) We also deduce from (ii) that the functor \underline{M} ↦ \underline{M}^{a} sends presheaves of integral monoids, to integral T -monoids. Therefore we have a natural isomorphism :

(4.8.37)  (\underline{M}^{int})^{a} →^{∼} (\underline{M}^{a})^{int}

as both functors are left adjoint to the forgetful functor from integral T -monoids, to presheaves of monoids on T. In the same vein, it is easily seen that the forgetful functor Int.Mnd_{T} → T commutes with filtered colimits : indeed, (4.8.37) and lemma 4.8.29(iii) reduce the assertion to showing that the colimit of a filtered system of presheaves of integral monoids is integral, which can be verified directly.

(v) Take T = Set, and let φ : M → N be an injective map of monoids; if N (hence M) is integral, one sees easily that the induced map φ^{gp} : M^{gp} → N^{gp} is also injective. This may fail, when N is not integral : for instance, if M is any integral monoid, and N := M_{∘} is the pointed monoid associated with M as in remark 4.8.14(iv), then for the natural inclusion i : M → M_{∘} we have i^{gp} = 0, since (M_{∘})^{gp} = {1}.

Lemma 4.8.38. Let T be a topos with enough points, \underline{M} an integral T -monoid, and \underline{N} ⊂ \underline{M} a T -submonoid. Then \underline{M}/\underline{N} is an integral T -monoid.

Proof. In light of example 4.8.36(iv), we are reduced to the case where T = Set. Moreover, since the natural morphism \underline{M}/\underline{N} → \underline{N}^{−1}M/\underline{N}^{gp} is an isomorphism, we may assume that \underline{N} is an abelian group. Now, notice that (\underline{M}/\underline{N})^{gp} = \underline{M}^{gp}/\underline{N} since the functor P ↦ P^{gp} commutes with colimits. On the other hand, \underline{M}/\underline{N} is the set-theoretic quotient of \underline{M} by the translation action of \underline{N} (lemma 4.8.31(iii)). This shows that the unit of adjunction \underline{M}/\underline{N} → (\underline{M}/\underline{N})^{gp} is injective, as required. □

4.8.39. Let M be an integral monoid. Classically, one says that M is saturated, if we have :

M = {a ∈ M^{gp} | a^{n} ∈ M for some integer n > 0}.

In order to globalize the class of saturated monoid to arbitrary topoi, we make the following :

Definition 4.8.40. Let T be a topos, φ : \underline{M} → \underline{N} a morphism of integral T -monoids. (i) We say that φ is exact if the diagram of commutative T -monoids

𝒟_{φ} : \xymatrix{\underline{M} \ar[r]^{φ} \ar[d] & \underline{N} \ar[d] \\ _{\underline{M}_{gp}} \ar[r]^{φ^{gp}} & _{\underline{N}_{gp}}}

is cartesian (where the vertical arrows are the natural morphisms). (ii) For any integer k > 0, the k-Frobenius map of \underline{M} is the endomorphism k_{M} of \underline{M} given by the rule : x ↦ x^{k} for every U ∈ Ob(T) and every x ∈ \underline{M}(U). We say that \underline{M} is k-saturated, if k_{M} is an exact morphism.

(iii) We say that \underline{M} is saturated, if \underline{M} is integral and k-saturated for every integer k > 0.

We denote by Sat.Mnd_{T} the full subcategory of Int.Mnd_{T} whose objects are the saturated T -monoids. As usual, when T = Set, we shall drop the subscript, and just write Sat.Mnd for this category. The above definition (and several of the related results in section 6.2) is borrowed from [159].

Remark 4.8.41. (i) Clearly, when T = Set, definition 4.8.40(iii) recovers the classical notion of saturated monoid. Again, for usual monoids, it is easily seen that the forgetful functor Sat.Mnd → Int.Mnd admits a left adjoint, that assigns to any integral monoid M its saturation M^{sat}. The latter is the monoid consisting of all elements x ∈ M^{gp} such that x^{k} ∈ M for some integer k > 0; especially, the torsion subgroup of M^{gp} is always contained in M^{sat}. The easy verification is left to the reader. Clearly, M is saturated if and only if M = M^{sat}. More generally, the unit of adjunction M → M^{sat} is just the inclusion map.

(ii) For a general topos T, and a morphism φ as in definition 4.8.40(i), notice that φ is exact if and only if the induced map of monoids φ(U) : \underline{M}(U) → \underline{N}(U) is exact for every U ∈ Ob(T). Indeed, if 𝒟_{φ} is cartesian, then the same holds for the induced diagram 𝒟_{φ}(U) of monoids; since the natural map \underline{M}(U)^{gp} → \underline{M}^{gp}(U) is injective (and likewise for \underline{N}), it follows easily that the diagram of monoids 𝒟_{φ(U)} is cartesian, i.e. φ(U) is exact. For the converse, notice that 𝒟_{φ} is of the form (h𝒟_{φ})^{a}, where h : T → T^{∧} is the Yoneda embedding, and F ↦ F^{a} denotes the associated sheaf functor T^{∧} → (T, C_{T})^{∼} = T; the assumption means that h𝒟 is a cartesian diagram in T^{∧}, hence 𝒟 is exact in T, since the associated sheaf functor is exact.

(iii) Example 4.8.36(iii) and (ii) imply that a T -monoid \underline{M} is saturated, if and only if \underline{M}(U) is a saturated monoid, for every U ∈ Ob(T). We also remark that, in view of example 4.8.36(ii), the functor F ↦ F^{a} takes presheaves of k-saturated (resp. saturated) monoids, to k-saturated (resp. saturated) T -monoids : indeed, if η : \underline{M} → \underline{M}^{gp} is the unit of adjunction for a presheaf of monoids \underline{M}, then η^{a} : \underline{M}^{a} → (\underline{M}^{gp})^{a} = (\underline{M}^{a})^{gp} is the unit of adjunction for the associated T -monoid, hence it is clear the functor F ↦ F^{a} preserves exact morphisms.

(iv) It follows easily that the inclusion functor Sat.Mnd_{T} → Int.Mnd_{T} admits a left adjoint, namely the functor

Int.Mnd_{T} → Sat.Mnd_{T} : \underline{M} ↦ \underline{M}^{sat}

that assigns to \underline{M} the sheaf associated with the presheaf U → \underline{M}′(U) := \underline{M}(U)^{sat} on T (notice that the functor \underline{M} ↦ \underline{M}′ from presheaves of integral monoids, to presheaves of saturated monoids, is left adjoint to the inclusion functor). Just as in example 4.8.36(iv), we deduce a natural isomorphism

(4.8.42)  (\underline{M}^{sat})^{a} →^{∼} (\underline{M}^{a})^{sat}  for every T -monoid \underline{M}

since both functors are left adjoint to the forgetful functor from Sat.Mnd_{T}, to presheaves of integral monoids on T.

(v) By the usual general nonsense, the saturation functor commutes with all colimits. Moreover, the considerations of (4.8.35) can be repeated for saturated monoids : first, the category Sat.Mnd_{T} is cocomplete, and arguing as in example 4.8.36(iv), one checks that filtered colimits commute with the forgetful functor Sat.Mnd_{T} → T; next, if F : Λ → Sat.Mnd_{T} is a functor from a small category Λ, then for each integer k > 0, the induced diagram of integral monoids

lim_{Λ} 𝒟_{k_{F}} : \xymatrix{lim_{Λ} F \ar[r]^{^{lim^{Λ} k^{F}}} \ar[d] & lim_{Λ} F \ar[d] \\ lim_{Λ} F^{gp} \ar[r]^{_{lim_{Λ} k{gp \atop F}}} & lim_{Λ} F^{gp}}

is cartesian; since the natural morphism

(lim_{Λ} F)^{gp} → lim_{Λ} F^{gp}

is a monomorphism, it follows easily that the limit of F is saturated, hence Sat.Mnd_{T} is complete, and furthermore all limits commute with the forgetful functor to T.

4.8.43. In view of remark 4.8.11(i,ii,iii), a morphism of topoi f : T → S induces functors :

(4.8.44)  f_{∗} : Mnd_{T} → Mnd_{S}  f^{∗} : Mnd_{S} → Mnd_{T}

and one verifies easily that (4.8.44) is an adjoint pair of functors.

Lemma 4.8.45. Let f : T → S be a morphism of topoi, \underline{M} an S-monoid. We have : (i) If \underline{M} is integral (resp. saturated), f^{∗}\underline{M} is an integral (resp. saturated) T -monoid. (ii) More precisely, there is a natural isomorphism :

f^{∗}(\underline{M}^{int}) →^{∼} (f^{∗}\underline{M})^{int} (resp. f^{∗}(\underline{M}^{sat}) →^{∼} (f^{∗}\underline{M})^{sat}, if \underline{M} is integral).

(iii) If φ is an exact morphism of integral S-monoids, then f^{∗}φ is an exact morphism of integral T -monoids.

Proof. To begin with, notice that the adjoint pair (f^{∗}, f_{∗}) of (4.8.44) restricts to a corresponding adjoint pair of functors between the categories of abelian T -groups and abelian S-groups (since the condition \underline{G} = \underline{G}^{×} for monoids, is preserved by any left exact functor).

There follows a natural isomorphism :

(f^{∗}\underline{M})^{gp} →^{∼} f^{∗}(\underline{M}^{gp}) for every S-monoid \underline{M}

since both functors are left adjoint to the functor f_{∗} from abelian T -groups to S-monoids. Now, if \underline{M} is an integral S-module, and η : \underline{M} → \underline{M}^{gp} is the unit of adjunction, it is easily seen that f^{∗}η : f^{∗}\underline{M} → (f^{∗}\underline{M})^{gp} is also the unit of adjunction. From this and proposition 4.4.8(iii), we deduce the assertion concerning f^{∗}(\underline{M}^{int}).

By the same token, we get assertion (iii) of the lemma. Especially, if \underline{M} is saturated, then the same holds for f^{∗}\underline{M}. The assertion concerning f^{∗}(\underline{M}^{sat}) follows by the usual argument. □

Lemma 4.8.46. (i) The functor f^{∗} of (4.8.44) commutes with all finite limits and all colimits.

(ii) Let P(T, M) be the property “M is an integral (resp. saturated) T -monoid”. Then P can be checked on every conservative set of morphisms of topoi (see definition 4.7.11(v).)

Proof. (i): Concerning finite limits, in light of lemma 4.8.29(iii) we are reduced to the assertion that f^{∗} : S → T is left exact, which holds by definition. Next f^{∗} commutes with colimits, because it is a left adjoint.

(ii): A T -monoid \underline{M} is integral if and only if the unit of adjunction η : \underline{M} → \underline{M}^{int} is an isomorphism. However, f^{∗}(\underline{M}^{int}) →^{∼} (f^{∗}\underline{M})^{int}, by lemma 4.8.45(ii), and f^{∗}η : \underline{M} → (f^{∗}\underline{M})^{int} is the unit of adjunction. The assertion is an immediate consequence. The same argument applies as well to saturated T -monoids. □

Example 4.8.47. (i) For instance, the unique morphism of topoi Γ : T → Set (see proposition 4.4.12(i,ii)) induces a pair of adjoint functors :

(4.8.48) Mnd_{T} → Mnd : M ↦ Γ(T, M)  and Mnd → Mnd_{T} : P ↦ P_{T}

where P_{T} is the constant sheaf of monoids on (T, C_{T}) with value P.

(ii) Specializing lemma 4.8.45(ii) to this adjoint pair, we obtain natural isomorphisms :

(4.8.49)  (M_{T})^{int} →^{∼} (M^{int})_{T}  (M_{T})^{sat} →^{∼} (M^{sat})_{T}

of functors Mnd → Int.Mnd_{T} and Int.Mnd → Sat.Mnd_{T}. Especially, if M is an integral (resp. saturated) monoid, then the constant T -monoid M_{T} is integral (resp. saturated).

(iii) If ξ is any T -point, notice also that the stalk M_{T,ξ} is isomorphic to M, since ξ is a section of Γ : T → Set.

4.8.50. Let T be a topos, \underline{R} a T -ring. We have a forgetful functor \underline{R}-Alg → Mnd_{T} that assigns to a (unital, commutative) \underline{R}-algebra (\underline{A}, +, ·, 1_{A}) its multiplicative T -monoid (\underline{A}, ·). If T = Set, this functor admits a left adjoint Mnd → R-Alg : M ↦ R[M]. Explicitly, R[M] = ⨁_{x∈M} xR, and the multiplication law is uniquely determined by the rule :

xa · yb := (x · y)ab for every x, y ∈ M and a, b ∈ R.

For a general topos T, the above construction globalizes to give a left adjoint

(4.8.51)  Mnd_{T} → \underline{R}-Alg : \underline{M} ↦ \underline{R}[\underline{M}].

The latter is the sheaf on (T, C_{T}) associated with the presheaf U ↦ \underline{R}(U)[\underline{M}(U)], for every U ∈ Ob(T). The functor (4.8.51) commutes with arbitrary colimits (since it is a left adjoint); especially, if \underline{M} → \underline{M}_{1} and \underline{M} → \underline{M}_{2} are two morphisms of monoids, we have a natural identification :

(4.8.52)  \underline{R}[\underline{M}_{1} ⨿\underline{M} \underline{M}_{2}] →^{∼} \underline{R}[\underline{M}_{1}] ⊗_{\underline{R}[\underline{M}]} \underline{R}[\underline{M}_{2}].

By inspecting the universal properties, we also get a natural isomorphism :

(4.8.53)  \underline{S}^{−1}\underline{R}[\underline{M}] →^{∼} \underline{R}[\underline{S}^{−1}\underline{M}]

for every monoid \underline{M} and every submonoid \underline{S} ⊂ \underline{M}.

4.8.54. Likewise, if \underline{M} is any T -monoid, let \underline{R}[\underline{M}]-Mod denote as usual the category of modules over the T -ring \underline{R}[\underline{M}]; we have a forgetful functor \underline{R}[\underline{M}]-Mod → \underline{M}-Mod. When T = Set, this functor admits a left adjoint M-Mod → R[M]-Mod : S ↦ R[S]. Explicitly, R[S] is the free R-module with basis given by S, and the R[M]-module structure on R[S] is determined by the rule:

xa · sb := μ_{S}(x, s)ab for every x ∈ M, s ∈ S and a, b ∈ R.

For a general topos T, this construction globalizes to give a left adjoint

\underline{M}-Mod → \underline{R}[\underline{M}]-Mod : (S, μ_{S}) ↦ \underline{R}[S]

which is defined as the sheaf associated with the presheaf U ↦ \underline{R}(U)[S(U)] in T^{∧}.

Example 4.8.55. (i) Take T = Set, let (M, μ_{M}, 1_{M}) be any monoid, and Λ any set. Then the coproduct M^{(Λ)} of Λ copies of M is representable in the category of monoids, as the subset of M^{Λ} consisting of all sequences a_{•} := (a_{λ} | λ ∈ Λ) such that {λ ∈ Λ | a_{λ} ≠ 0} is a finite set; the composition law on M^{(Λ)} is the obvious one :

a_{•} · b_{•} := (μ_{M}(a_{λ}, b_{λ}) | λ ∈ Λ) for every a_{•}, b_{•} ∈ M^{(Λ)}.

(ii) Let also R be any ring, and (ℕ, +, 0) the monoid of non-negative integers; then the ring

R[ℕ^{(Λ)}]

is the free polynomial R-algebra R[T_{λ} | λ ∈ Λ] on the set of variables (T_{λ} | λ ∈ Λ).

(iii) We shall later encounter also the following variant. Let p be a prime integer, and (ℕ[1/p], +, 0) the additive monoid of all non-negative rationals of the form a/p^{n}, for every a, n ∈ ℕ. Notice that ℕ[1/p] is the colimit of the direct system of monoids

ℕ → ℕ → ℕ ⋯

with transition maps given by multiplication by p. Then, for every set Λ, the ring

R[T_{λ}^{1/p^{∞}} | λ ∈ Λ] := R[ℕ[1/p]^{(Λ)}]

is the colimit of the system of R-algebras R[ℕ^{(Λ)}] → R[ℕ^{(Λ)}] → R[ℕ^{(Λ)}] → ⋯ with transition maps : R[T_{λ} | λ ∈ Λ] → R[T_{λ} | λ ∈ Λ] such that T_{λ} ↦ T_{λ}^{p} for all λ ∈ Λ. Thus, the system (T_{λ}^{1/p^{n}} | λ ∈ Λ, n ∈ ℕ) is a family of generators for this R-algebras, and the free polynomial R-algebra R[T_{λ} | λ ∈ Λ] is naturally an R-subalgebra of R[T_{λ}^{1/p^{∞}} | λ ∈ Λ].

4.9. Torsors on a topos. In this section we discuss torsors over (not necessarily abelian) group objects of a topos. Then we explain some basic notions concerning the points of the étale and Zariski topoi of a scheme, and we conclude with the proof of Hilbert’s theorem 90 (lemma 4.9.27(iv)).

Definition 4.9.1. Let T be a topos, and G a T -group.

(i) A left G-torsor is a left G-module (X, μ_{X}), inducing an isomorphism

(μ_{X}, p_{X}) : G × X → X × X

(where p_{X} : G×X → X is the natural projection) and such that there exists a covering morphism U → 1_{T} in T for which X(U) ≠ ∅. This is the same as saying that the unique morphism X → 1_{T} is an epimorphism.

(ii) A morphism of left G-torsors is just a morphism of the underlying G-modules. Likewise, we define right G-torsors, G-bitorsors, and morphisms between them. We let:

H^{1}(T, G)

be the set of isomorphism classes of right G-torsors. (iii) A (left or right or bi-) G-torsor (X, μ_{X}) is said to be trivial, if Γ(T, X) ≠ ∅.

Remark 4.9.2. (i) In the situation of definition 4.9.1, notice that H^{1}(T, G) always contains a distinguished element, namely the class of the trivial G-torsor (G, μ_{G}).

(ii) Conversely, suppose that (X, μ_{X}) is a trivial left G-torsor, and say that σ ∈ Γ(T, X); then we have a cartesian diagram :

\xymatrix{G \ar[r]^{μ_{σ}} \ar[d] & X \ar[d]^{1_{X}×σ} \\ G × X \ar[r]^{(μ_{X},p_{X})} & X × X}

which shows that (X, μ_{X}) is isomorphic to (G, μ_{G}) (and likewise for right G-torsors).

(iii) Notice that every morphism f : (X, μ_{X}) → (X′, μ_{X′}) of G-torsors is an isomorphism. Indeed, the assertion can be checked locally on T (i.e., after pull-back by a covering morphism U → 1_{T}). Then we may assume that X admits a global section σ ∈ Γ(T, X), in which case σ′ := σ ∘ f ∈ Γ(T, X′). Then, arguing as in (ii), we get a commutative diagram :

\xymatrix{ & G \ar[dl]_{μ_{σ}} \ar[dr]^{μ_{σ′}} \\ X \ar[rr]^{f} &  & X′}

where both μ_{σ} and μ_{σ′} are isomorphisms, and then the same holds for f.

(iv) The tensor product of a G-bitorsor and a left G-bitorsor is a left G-torsor. Indeed the assertion can be checked locally on T, so we are reduced to checking that the tensor product of a trivial G-bitorsors and a trivial left G-torsor is the trivial left G-torsor, which is obvious.

(v) Likewise, if G_{1} → G_{2} is any morphism of T -groups, and X is a left G_{1}-torsor, it is easily seen that the base change G_{2} ⊗_{G_{1}} X yields a left G_{2}-torsor (and the same holds for right torsors and bitorsors). Hence the rule G ↦ H^{1}(T, G) is a functor from the category of T -groups, to the category of pointed sets. One can check that H^{1}(T, G) is an essentially small set (see [78, Chap.III, §3.6.6.1]).

(vi) Let f : T → S be a morphism of topoi; if X is a left G-torsor, the f_{∗}G-module f_{∗}X is not necessarily a f_{∗}G-torsor, since we may not be able to find a covering morphism U → 1_{S} such that f_{∗}X(U) ≠ ∅. On the other hand, if H a S-group and Y a left H-torsor, then it is easily seen that f^{∗}Y is a left f^{∗}H-torsor.

4.9.3. Let f : T′ → T be a morphism of topoi, and G a T′-monoid; we define a U-presheaf R^{1}f_{∗}^{∧}G on T, by the rule :

U ↦ H^{1}(T′/f^{∗}U, G_{|f^{∗}U}).

(More precisely, since this set is only essentially small, we replace it by an isomorphic small set). If φ : U → V is any morphism in T, and X is any G_{|f^{∗}V} -torsor, then X ×_{f^{∗}V} f^{∗}U is a G_{|f^{∗}U}-torsor, whose isomorphism class depends only on the isomorphism class of X; this defines the map R^{1}f_{∗}^{∧}G(φ), and it is clear that R^{1}f_{∗}^{∧}G(φ ∘ ψ) = R^{1}f_{∗}^{∧}G(ψ) ∘ R^{1}f_{∗}^{∧}G(φ), for any other morphism ψ : W → U in T. Lastly, we denote by :

R^{1}f_{∗}G

the sheaf on (T, C_{T}) associated with the presheaf R^{1}f_{∗}^{∧}G. Notice that the object R^{1}f_{∗}G is pointed, i.e. it is endowed with a natural global section :

τ_{f,G} : 1_{T} → R^{1}f_{∗}G

namely, the morphism associated with the morphism of presheaves 1_{T} → R^{1}f_{∗}^{∧}G which, for every U ∈ Ob(T), singles out the isomorphism class τ_{f,G}(U) ∈ R^{1}f_{∗}^{∧}G(U) of the trivial G_{|f^{∗}U}-torsor.

4.9.4. Let g : T′′ → T′ be another morphism of topoi, and G a T′′-group. Notice that :

f^{∗∧}R^{1}g_{∗}^{∧}G = R^{1}(f ∘ g)_{∗}^{∧}G

hence the natural morphism (in T^{′∧}) R^{1}f_{∗}^{∧}G → R^{1}f_{∗}G induces a morphism R^{1}(f ∘ g)_{∗}^{∧}G → f^{∗∧}R^{1}g_{∗}G in T^{∧}, which yields, after taking associated sheaves, a morphism in T :

(4.9.5)  R^{1}(f ∘ g)_{∗}G → f_{∗}R^{1}g_{∗}G.

One sees easily that this is a morphism of pointed objects of T, i.e. the image of the global section τ_{f∘g,G} under this map, is the global section f_{∗}τ_{g,G}.

Next, suppose that U ∈ Ob(T) and X is any g_{∗}G_{|f^{∗}U}-torsor (on T′/f^{∗}U); we may form the g^{∗}g_{∗}G_{|g^{∗}f^{∗}U}-torsor g^{∗}X, and then base change along the natural morphism g^{∗}g_{∗}G → G, to obtain the G_{|g^{∗}f^{∗}U}-torsor G ⊗_{g^{∗}g_{∗}G} g^{∗}X. This rule yields a map R^{1}f_{∗}^{∧}(g_{∗}G) → R^{1}(f ∘ g)_{∗}^{∧}G, and after taking associated sheaves, a natural morphism of pointed objects :

(4.9.6)  R^{1}f_{∗}(g_{∗}G) → R^{1}(f ∘ g)_{∗}G.

Remark 4.9.7. As a special case, let h : S′ → S be a morphism of topoi, H a S′-group. If we take T′′ := S′, T′ := S, g := h and f : S → Set the (essentially) unique morphism of topoi, (4.9.6) and (4.9.5) boil down to maps of pointed sets :

(4.9.8)  H^{1}(S, h_{∗}H) → H^{1}(S′, H) → Γ(S, R^{1}h_{∗}H).

These considerations are summarized in the following :

Theorem 4.9.9. In the situation of (4.9.4), there exists a natural exact sequence of pointed objects of T :

1_{T} → R^{1}f_{∗}(g_{∗}G) → R^{1}(f ∘ g)_{∗}G → f_{∗}R^{1}g_{∗}G.

Proof. The assertion means that (4.9.6) identifies R^{1}f_{∗}(g_{∗}G) with the subobject :

R^{1}(f ∘ g)_{∗}G ×_{f_{∗}R^{1}g_{∗}G} f_{∗}τ_{g,G}

(briefly : the preimage of the trivial global section). We begin with the following :

Claim 4.9.10. In the situation of remark 4.9.7, the sequence of maps (4.9.8) identifies the pointed set H^{1}(S, h_{∗}H) with the preimage of the trivial global section τ_{h,H} of R^{1}h_{∗}H.

Proof of the claim. Notice first that a global section Y of R^{1}h_{∗}^{∧}H maps to the trivial section τ_{h,H} of R^{1}h_{∗}H if and only if there exists a covering morphism U → 1_{S} in (S, C_{S}), such that Y (h^{∗}U) ≠ ∅. Thus, let X be a right h_{∗}H-torsor; the image in H^{1}(S′, H) of its isomorphism class is the class of the H-torsor Y := h^{∗}X ⊗_{h^{∗}h_{∗}H} H. The latter defines a global section of R^{1}h_{∗}H. However, by definition there exists a covering morphism U → 1_{S} such that X(U) ≠ ∅, hence also h^{∗}X(h^{∗}U) ≠ ∅, and therefore Y (h^{∗}U) ≠ ∅. This shows that the image of H^{1}(S, h_{∗}H) lies in the preimage of τ_{h,H}.

Moreover, notice that h_{∗}Y (U) ≠ ∅, hence h_{∗}Y is a h_{∗}H-torsor. Now, let ε_{H} : h^{∗}h_{∗}H → H (resp. η_{h_{∗}H} : h_{∗}H → h_{∗}h^{∗}h_{∗}H) be the counit (resp. unit of adjunction); we have a natural morphism α : h^{∗}X → Y_{(ε_{H})} of h^{∗}h_{∗}H-modules, whence a morphism :

h_{∗}α : h_{∗}h^{∗}X → h_{∗}Y_{(h_{∗}ε_{H})}

of h_{∗}h^{∗}h_{∗}H-modules. On the other hand, the unit of adjunction η_{X} : X → h_{∗}h^{∗}X_{(η_{h∗H})} is a morphism of h_{∗}H-modules (remark 4.8.11(v)). Since h_{∗}ε_{H} ∘ η_{h_{∗}H} = 1_{h_{∗}H} (see (1.1.13)), the composition h_{∗}α ∘ η_{X} is a morphism of h_{∗}H-modules, hence it is an isomorphism, by remark 4.9.2(iii). This implies that the first map of (4.9.8) is injective.

Conversely, suppose that the class of a H-torsor X′ gets mapped to τ_{h,H}; we need to show that the class of X′ lies in the image of H^{1}(S, h_{∗}H). However, the assumption means that there exists a covering morphism U → 1_{S} such that X′(h^{∗}U) ≠ ∅; by adjunction we deduce that h_{∗}X′(U) ≠ ∅, hence h_{∗}X′ is a h_{∗}H-torsor. In order to conclude, it suffices to show that the image in H^{1}(S′, H) of the class of h_{∗}X′ is the class of X.

Now, the counit of adjunction h^{∗}h_{∗}X′ → X′ is a morphism of h^{∗}h_{∗}H-modules (remark 4.8.11(v)); by adjunction it induces a map h^{∗}h_{∗}X′ ⊗_{h^{∗}h_{∗}H} H → H of H-torsors, which must be an isomorphism, according to remark 4.9.2(iii). ♢

If we apply claim 4.9.10 with S := T′/f^{∗}U, S′ := T′′/(g^{∗}f^{∗}U) and h := g/(g^{∗}f^{∗}U), for U ranging over the objects of T, we deduce an exact sequence of presheaves of pointed sets :

1_{T} → R^{1}f_{∗}^{∧}(g_{∗}H) → R^{1}(f ∘ g)_{∗}^{∧}G → f^{∗∧}R^{1}g_{∗}G

from which the theorem follows, after taking associated sheaves. □

4.9.11. Let f : T′ → T be a morphism of topoi, U an object of T, G a T′-group, p : X → U a right G_{|U}-torsor. Then, for every object V of T we have an induced sequence of maps of sets

X(f^{∗}V) ⟶^{p_{∗}} U(f^{∗}V) ⟶^{∂} R^{1}f_{∗}^{∧}G(V)

where p_{∗} is deduced from p, and for every σ ∈ U(f^{∗}V), we let ∂(σ) be the isomorphism class of the right G_{|f^{∗}V} -torsor (X ×_{U} f^{∗}V → f^{∗}V). Clearly the image of p_{∗} is precisely the preimage of (the isomorphism class of) the trivial G_{|f^{∗}V} -torsor. After taking associated sheaves, we deduce a natural sequence of morphisms in T :

(4.9.12)  f_{∗}X ⟶^{p} f_{∗}U ⟶^{∂} R^{1}f_{∗}G

such that the preimage of the global section τ_{f,G} is precisely the image in Γ(T′, U) of the set of global sections of X.

4.9.13. A ringed topos is a pair (T, 𝒪_{T}) consisting of a topos T and a (unital, associative) T -ring 𝒪_{T}, called the structure ring of T. A morphism f : (T, 𝒪_{T}) → (S, 𝒪_{S}) of ringed topoi is the datum of a morphism of topoi f : T → S and a morphism of T -rings :

f^{♮} : f^{∗}𝒪_{S} → 𝒪_{T} .

We denote, as usual, by 𝒪_{T}^{×} ⊂ 𝒪_{T} the subobject representing the invertible sections of 𝒪_{T}. For every object U of T, and every s ∈ 𝒪_{T} (U), let D(s) ⊂ U be the subobject such that :

Hom_{T} (V, D(s)) := {φ ∈ U(V) | φ^{∗}s ∈ 𝒪_{T}^{×}(V)}.

We say that (T, 𝒪_{T}) is locally ringed, if D(0) = ∅_{T} (the initial object of T), and moreover

D(s) ∪ D(1 − s) = U for every U ∈ Ob(T), and every s ∈ 𝒪_{T} (f).

A morphism of locally ringed topoi f : (T, 𝒪_{T}) → (S, 𝒪_{S}) is a morphism of ringed topoi such that

f^{∗}D(s) = D(f^{♮}(U)(f^{∗}s)) for every U ∈ Ob(S) and every s ∈ 𝒪_{S}(U).

If T has enough points, then (T, 𝒪_{T}) is locally ringed if and only if the stalks 𝒪_{T,ξ} of the structure ring at all the points ξ of T are local rings. Likewise, a morphism f : (T, 𝒪_{T}) → (S, 𝒪_{S}) of ringed topoi is locally ringed if and only if, for every T -point ξ, the induced map 𝒪_{S,f(ξ)} → 𝒪_{T,ξ} is a local ring homomorphism.

4.9.14. In the rest of this section we present a few results concerning the special case of topologies on a scheme. Hence, for any scheme X, we shall denote by X_{ét} (resp. by X_{Zar}) the small étale (resp. the small Zariski) site on X. It is clear that X_{Zar} is a small site, and it is not hard to show that X_{ét} is a U-site ([9, Exp.VII, §1.7]). The inclusion of underlying categories :

u_{X} : X_{Zar} → X_{ét}

is a continuous functor (see definition 4.2.1(i)) commuting with finite limits, whence a morphism of topoi :

ũ_{X} := (ũ_{X}^{∗}, ũ_{X∗}) : X_{ét}^{∼} → X_{Zar}^{∼}.

such that the diagram of functors :

\xymatrix{X_{Zar} \ar[r]^{u_{X}} \ar[d] & X_{ét} \ar[d] \\ X_{Zar}^{∼} \ar[r]^{_{ũ_{∗}_{X}}} & X_{ét}^{∼}}

commutes, where the vertical arrows are the Yoneda embeddings (lemma 4.2.11).

The topoi X_{Zar}^{∼} and X_{ét}^{∼} are locally ringed in a natural way, and by faithfully flat descent, we see easily that ũ_{X∗}𝒪_{X_{ét}} = 𝒪_{X_{Zar}}. By inspection, ũ_{X} is a morphism of locally ringed topoi.

4.9.15. For any ring R (of our fixed universe U), denote by Sch/R the category of R-schemes, and by Sch/R_{Zar} (resp. Sch/R_{ét}) the big Zariski (resp. étale) site on Sch/R. For R = ℤ, we shall usually just write Sch_{Zar} and Sch_{ét} for these sites. The morphisms u_{X} of (4.9.13) are actually restrictions of a single morphism of sites :

u : Sch_{Zar} → Sch_{ét}

which, for every universe V such that U ∈ V, induces a morphism of V-topoi :

ũ_{V} : (Sch_{ét})_{V}^{∼} → (Sch_{Zar})_{V}^{∼}.

4.9.16. Let X be scheme; a geometric point of X is a morphism of schemes ξ : Spec κ → X, where κ is an arbitrary separably closed field. Notice that both the Zariski and étale topoi of Spec κ are equivalent to the category Set, so ξ induces a topos-theoretic point ξ_{ét}^{∼} : (Spec κ)_{ét}^{∼} → X_{ét}^{∼} of X_{ét}^{∼} (and likewise for X_{Zar}^{∼}). A basic feature of both the Zariski and étale topologies, is that every point of X_{Zar}^{∼} and X_{ét}^{∼} arise in this way.

More precisely, we say that two geometric points ξ and ξ′ of X are equivalent, if there exists a third such point ξ′′ which factors through both ξ and ξ′. It is easily seen that this is an equivalence relation on the set of geometric points of X, and two topos-theoretic points ξ_{ét}^{∼} and ξ_{ét}^{′∼} are isomorphic if and only if the same holds for the points ξ_{Zar}^{∼} and ξ_{Zar}^{′∼}, if and only if ξ is equivalent to ξ′.

Definition 4.9.17. Let X be a scheme, x a point of X, and \overline{x} : Spec κ → X a geometric point. (i) We let κ(x) be the residue field of the local ring 𝒪_{X,x}, and set

|x| := Spec κ(x)  κ(\overline{x}) := κ  |\overline{x}| := Spec κ(\overline{x})

If \{x\} ⊂ X is the image of \overline{x}, we say that \overline{x} is localized at x, and that x is the support of \overline{x}. We say that \overline{x} is strict if κ(\overline{x}) is a separable closure of κ(x).

(ii) We associate with \overline{x} a strict geometric point \overline{x}^{st}, as follows. Let κ(\overline{x}^{st}) be the separable closure of κ(x) inside κ(\overline{x}); the inclusion κ(\overline{x}^{st}) ⊂ κ(\overline{x}) defines a morphism of schemes :

(4.9.18)  |\overline{x}| → |\overline{x}^{st}| := Spec κ(\overline{x}^{st})

and \overline{x} is the composition of (4.9.18) and a unique strict geometric point localized at x

\overline{x}^{st} : |\overline{x}^{st}| → X.

(iii) The localization of X at x is the local scheme

X(x) := Spec 𝒪_{X,x}.

The strict henselization of X at \overline{x} is the strictly local scheme

X(\overline{x}) := Spec 𝒪_{X,\overline{x}}

where 𝒪_{X,\overline{x}} denotes the strict henselization of 𝒪_{X,x} relative to the geometric point \overline{x} ([66, Ch.IV, Déf.18.8.7]) (recall that a local ring is called strictly local, if it is henselian with separably closed residue field; a scheme is called strictly local, if it is the spectrum of a strictly local ring: see [66, Ch.IV, Déf.18.8.2]). By definition, the geometric point \overline{x} lifts to a unique geometric point of X(\overline{x}), which shall be denoted again by \overline{x}.

(iv) Moreover, we shall denote by

i_{x} : X(x) → X i\overline{x} : X(\overline{x}) → X(x)

the natural morphisms of schemes, and if ℱ is any sheaf on X_{Zar} (resp. X_{ét}), we let

ℱ (x) := i_{x}^{∗}ℱ ℱ (\overline{x}) := i∗ ; \overline{x}ℱ (x)

so ℱ (x) is a sheaf on X(x)_{Zar} (resp. X(x)_{ét}) and ℱ (\overline{x}) is a sheaf on X(\overline{x})_{Zar} (resp. X(\overline{x})_{ét}). (v) If f : Y → X is any morphism of schemes, we let

f^{−1}(x) := Y ×_{X} |x| f^{−1}(\overline{x}) := Y ×_{X} |\overline{x}| Y (x) := Y ×_{X} X(x) Y (\overline{x}) := Y ×_{X} X(\overline{x}).

Also, if \overline{y} is any geometric point of Y, we define f(\overline{y}) as the geometric point f ∘ \overline{y} of X, and we call f(\overline{y})^{st} the strict image of \overline{y} in X. Notice that the natural identification |\overline{y}| = |f(\overline{y})| induces a morphism of schemes :

|\overline{y}^{st}| → |f(\overline{y})^{st}|.

4.9.19. Many discussions concerning the Zariski or étale site of a scheme, only make appeal to general properties of these two topologies, and therefore apply indifferently to either of them, with only minor verbal changes. For this reason, to avoid tiresome repetitions, the following notational device is often useful. Namely, instead of referring each time to X_{Zar} and X_{ét} in the course of an argument, we shall write just X_{τ}, with the convention that τ ∈ {Zar, ét} has been chosen arbitrarily at the beginning of the discussion. In the same manner, a τ-point of X will mean a point of the topos X_{τ}^{∼}, and a τ-open subset of X will be any object of the site X_{τ}. With this convention, a Zar-point is a usual point of X, whereas an ét-point shall be a geometric point. Likewise, if ξ is a given τ-point of X, the localization X(ξ) makes sense for both topologies : if τ = ét, then X(ξ) is the strict henselization as in definition 4.9.17(ii); if τ = Zar, then X(ξ) is the usual localization of X at the (Zariski) point ξ. If τ = ét, the support of ξ is given by definition 4.9.17(i); if τ = Zar, then the support of ξ is just ξ itself (and correspondingly, in this case ξ is localized at ξ). Furthermore, 𝒪_{X,ξ} is a local ring if τ = Zar, and it is a strictly local ring, in case τ = ét.

4.9.20. Let f : X → Y be a morphism of schemes, \overline{x} a geometric point of X, and set \overline{y} := f(\overline{x}). The natural morphism f_{x} : X(x) → Y (y) induces a unique local morphism of strictly local schemes

f\overline{x} : X(\overline{x}) → Y (\overline{y})

([66, Ch.IV, Prop.18.8.8(ii)]) that fits in a commutative diagram :

[figure: |\overline{x}|; \overline{x}; X(\overline{x}); f\overline{x}; i\overline{x}; X(x); f_{x}; i_{x}; X; f; |\overline{y}|; \overline{y}; Y (\overline{y}); i\overline{y}; Y (y); i_{y}; Y.]

Let now ℱ be any sheaf on Y_{ét}; there follows a natural isomorphism :

f ∗ ; \overline{x}ℱ (\overline{y}) →^{∼} (f^{∗}ℱ)(\overline{x}).

Notice also the natural bijections :

(4.9.21)  \begin{aligned}ℱ\overline{y} & →^{∼} Γ(Y (\overline{y}), ℱ (\overline{y})) →^{∼} Γ(|\overline{y}|, \overline{y}^{∗}ℱ (\overline{y})) \\ f^{∗}ℱ\overline{x} & →^{∼} Γ(X(\overline{x}), f^{∗}ℱ (\overline{x})) →^{∼} Γ(|\overline{x}|, \overline{x}^{∗}f^{∗}ℱ (\overline{x})) \end{aligned}

which induce a natural identification :

(4.9.22)  ℱ\overline{y} →^{∼} f^{∗}ℱ\overline{x} : σ ↦ f ∗ ; \overline{x}(σ).

4.9.23. Let X be a scheme, x, x′ ∈ X any two points, such that x is a specialization of x′. Choose a geometric point \overline{x} localized at x. The localization map 𝒪_{X,x} → 𝒪_{X,x′} induces a natural specialization morphism of X-schemes :

X(x′) → X(x).

Set W := X(\overline{x}) ×_{X(x)} X(x′). The natural map g : W → X(x′) is faithfully flat, and is the limit of a cofiltered system of étale morphisms; hence we may find w ∈ W lying over the closed point of X(x′), and the induced map κ(x′) → κ(w) is algebraic and separable. Choose also a geometric point \overline{w} of W localized at w, and set \overline{x}′ := g(\overline{w}). Then g induces an isomorphism g\overline{w} : W (\overline{w}) →^{∼} X(\overline{x}′), whence a unique morphism

(4.9.24)  X(\overline{x}′) → X(\overline{x})

which makes commute the diagram :

\xymatrix{W (\overline{w}) \ar[r]^{g\overline{w}} \ar[d]_{i\overline{w}} & X(\overline{x}′) \ar[d] \ar[r]^{i_{\overline{x}′}} & X(x′) \ar[d] \\ W (w) \ar[r] & X(\overline{x}) \ar[r]^{i\overline{x}} & X(x)}

where the left bottom arrow is the natural projection, and the right-most vertical arrow is the specialization map. In this situation, we say that \overline{x} is a specialization of \overline{x}′ (and that \overline{x}′ is a generization of \overline{x}), and we call (4.9.24) a strict specialization morphism. Combining with (4.9.21), we obtain the strict specialization map induced by (4.9.24)

(4.9.25)  𝒢\overline{x} → 𝒢\overline{x}′

for every sheaf 𝒢 on X_{ét}.

Remark 4.9.26. (i) In the situation of (4.9.20), suppose that 𝒢 = f^{∗}ℱ for a sheaf ℱ on Y_{ét}. Then (4.9.25) is a map ℱ_{f(\overline{x})} → ℱ_{f(\overline{x}′)}. By inspecting the definition, it is easily seen that the latter agrees with the strict specialization map for ℱ induced by a unique strict specialization morphism Y (f(\overline{x}′)) → Y (f(\overline{x})).

(ii) Notice that (4.9.24) and (4.9.25) depend not only on the choice of w (which may not be unique, when X(x) is not unibranch) but also on the geometric point \overline{w}. Indeed, the group of automorphisms of the X(x′)-scheme X(\overline{x}′) is naturally isomorphic to the Galois group Gal(κ(x′)^{s}/κ(x′)) ([66, Ch.IV, (18.8.8.1)]).

Lemma 4.9.27. Let X be a scheme, ℱ a sheaf on X_{ét}. We have :

(i) The counit of the adjunction ε_{ℱ} : ũ_{X}^{∗} ∘ ũ_{X∗}ℱ → ℱ is a monomorphism.

(ii) Suppose there exist a sheaf 𝒢 on X_{Zar}, and an epimorphism f : ũ^{∗}𝒢 → ℱ (resp. a

monomorphism f : ℱ → ũ^{∗}𝒢). Then ε_{ℱ} is an isomorphism. (iii) The functor ũ_{X}^{∗} is fully faithful. (iv) (Hilbert 90) R^{1}ũ_{X∗}𝒪_{X_{ét}}^{×} = 1_{X_{Zar}}.

Proof. (i): The assertion can be checked on the stalks. Hence, let ξ be any geometric point of X; we have to show that the natural map (ũ_{X∗}ℱ)_{ξ} → ℱ_{ξ} is injective. To this aim, say that s, s′ ∈ (ũ_{X∗}ℱ)_{ξ}, and suppose that the image of s in ℱ_{ξ} agrees with the image of s′; we may find an open neighborhood U of ξ in X_{Zar}, such that s and s′ lie in the image of ℱ (U), and by assumption, there exists an étale morphism f : V → U such that the images of s and s′ coincide in ℱ (V). However, f(V) ⊂ U is an open subset ([64, Ch.IV, Th.2.4.6]), and the induced map V → f(V) is a covering morphism in X_{ét}; it follows that the images of s and s′ agree already in ℱ (f(V)), therefore also in (ũ_{X∗}ℱ)_{ξ}.

(iii): According to proposition 1.1.20(iii), it suffices to show that the unit of the adjunction η_{𝒢} : 𝒢 → ũ_{X∗} ∘ ũ_{X}^{∗}𝒢 is an isomorphism, for every 𝒢 ∈ Ob(X_{Zar}^{∼}). However, we have morphisms :

ũ_{X}^{∗}𝒢 ⟶^{ũ_{X}^{∗}(η_{𝒢} )} ũ_{X}^{∗} ∘ ũ_{X∗} ∘ ũ_{X}^{∗}𝒢 ⟶^{ε_{ũ^{∗}_{X} 𝒢}} ũ_{X}^{∗}𝒢 .

whose composition is the identity of ũ_{X}^{∗}𝒢 (see (1.1.13)); also, (i) says that ε_{ũ_{X}^{∗}𝒢} is a monomorphism, and then it follows formally that it is actually an isomorphism (e.g. from the dual of [28, Prop.1.9.3]). Hence the same holds for ũ_{X}^{∗}(η_{𝒢}), and by considering the stalks of the latter, we conclude that also η_{𝒢} is an isomorphism, as required.

(ii): Suppose first that f : ũ^{∗}𝒢 → ℱ is an epimorphism. We have just seen that η_{𝒢} is an isomorphism, therefore we have a morphism ũ^{∗} ∘ ũ_{∗}f : ũ^{∗}𝒢 → ũ^{∗} ∘ ũ_{∗}ℱ whose composition with ε_{ℱ} is f; especially, ε_{ℱ} is an epimorphism, so the assertion follows from (i).

In the case of a monomorphism f : ℱ → ũ^{∗}𝒢, set ℋ := ũ^{∗}𝒢 ⨿_{ℱ} ũ^{∗}𝒢; we may represent f as the equalizer of the two natural maps j_{1}, j_{2} : ũ^{∗}𝒢 → ℋ . However, the natural morphism ũ^{∗}(𝒢 ⨿ 𝒢) → ℋ is an epimorphism, hence the counit ε_{ℋ} is an isomorphism, by the previous case. Then (iii) implies that j_{i} = ũ^{∗}j′_{i} for morphisms j′_{i} : 𝒢 → ũ_{∗}ℋ (i = 1, 2). Let ℱ′ be the equalizer of j′_{1} and j′_{2}; then ũ^{∗}ℱ′ ≃ ℱ, and since we have already seen that the unit of adjunction is an isomorphism, the assertion follows from the triangular identitities of (1.1.13).

(iv): The assertion can be checked on the stalks. To ease notation, set ℱ := R^{1}ũ_{X∗}𝒪_{X_{ét}}^{×}. Let ξ be any geometric point of X, and say that s ∈ ℱ_{ξ}; pick a (Zariski) open neighborhood U ⊂ X of ξ such that s lies in the image of ℱ (U). We may then find a Zariski open covering (U_{λ} → U | λ ∈ Λ) of U, such that the image of s in ℱ (U_{i}) is represented by a 𝒪_{U_{λ},ét}^{×}-torsor on U_{λ,ét}, for every λ ∈ Λ. After replacing U by any U_{λ} containing the support of ξ, we may assume that s is the image of the isomorphism class of some 𝒪_{U_{ét}}^{×}-torsor X_{ét} on U_{ét}. By faithfully flat descent, there exist a 𝒪_{U_{Zar}}^{×}-torsor X on U_{Zar}, and an isomorphism of 𝒪_{U_{ét}}^{×}-torsors :

X_{ét} →^{∼} 𝒪_{U_{ét}}^{×} ⊗_{ũ_{U}^{∗} 𝒪_{U_{Zar}}^{×}} ũ_{U}^{∗}X.

However, after replacing U by a smaller open neighborhood of ξ, we may suppose that X(U) ≠ ∅, therefore X_{ét}(U) ≠ ∅ as well, i.e. s is the image of the trivial section of ℱ (U). □

5. Stacks

Let (𝒞 , J) be any site, and consider the rule that assigns to every U ∈ Ob(𝒞) the category (U, J_{U})^{∼} of sheaves on 𝒞 /U, for the topology J_{U} induced by J via the source functor 𝒞 /U → 𝒞. Every morphism g : U′ → U of 𝒞 induces a pull-back functor g̃^{∗} : (U, J_{U})^{∼} → (U′, J_{U′})^{∼}, but if g′ : U′′ → U′ is another such morphism, the composition g̃′^{∗} ∘ g̃^{∗} is not usually equal, but only isomorphic to the pull-back functor induced by g ∘ g′. Moreover, given a covering family (U_{i} → U | i ∈ I), we can describe the category (U, J_{U})^{∼} only up to equivalence in terms of the corresponding categories on the U_{i} and their intersections; this “gluing up to equivalence” is a 2-categorical manipulation of descent data relative to the given covering, so it involves not only double intersections U_{i} × U_{j}, but also triple intersections. Summing up, the rule U ↦ (U, J_{U})^{∼} does not quite define a sheaf of categories on C, but rather a kind of 2-categorical analogue of the latter : it is the first example of a structure that is often encountered in algebraic geometry, and whose systematic study is the subject of the theory of stacks developed in this chapter.

5.1. Prestacks and stacks on a site. As usual, U will denote a chosen universe, and small will be synonymous with U-small, throughout this section.

Definition 5.1.1. Let C := (𝒞 , J) be a site, φ : 𝒜 → 𝒞 a fibration, and i ≤ 2 an integer. We say that φ is an i-separated prestack if for every X ∈ Ob(𝒞), every 𝒮 ∈ J(X) is a sieve of φ-i-descent (see definition 3.5.10). A 2-separated prestack is called a stack. For every universe V, we denote by

V-i-PreStack(C) (resp. V-Stack(C) )

the full 2-subcategory of V-Fib(𝒞) whose objects are the i-separated prestacks (resp. the stacks) on C. Hence V-(−1)-PreStack(C) = V-Fib(𝒞) and V-2-PreStack(C) = V-Stack(C). As usual, if V = U we often drop the mention of the universe from this notation.

Example 5.1.2. Let (𝒞 , J) be any site, F any presheaf on 𝒞, and s_{F} : ℱ ib(F) → 𝒞 the fibration attached to F, as in (3.1.15). Let also X ∈ Ob(𝒞) be any object, and 𝒮 any sieve of 𝒞 /X; by (3.1.15) and remark 3.5.2(ii) we have a commutative diagram

\xymatrix{F X = Hom_{𝒞^{∧}}(h_{X}, F) \ar[r] \ar[d] & Hom_{𝒞^{∧}}(h_{𝒮} , F) \ar[d] \\ ℱ ib(F)(X) \ar[r] & Cart_{𝒞} (𝒮 , ℱ ib(F))}

whose vertical arrows are bijections, and whose top (resp. bottom) horizontal arrow is induced by the inclusion h_{𝒮} → h_{X} (resp. 𝒮 ⊂ 𝒞 /X). We conclude that ℱ ib(F) is a 0-separated prestack; by the same token, ℱ ib(F) is a 1-separated prestack (resp. a stack) if and only if F is a separated presheaf (resp. a sheaf). Thus, we get a commutative diagram of functors

(5.1.3)  \xymatrix{(𝒞 , J)^{∼} \ar[r] \ar[d] & Stack(𝒞 , J) \ar[d] \\ 𝒞^{∧} \ar[r]^{ℱ ib} & 0-PreStack(𝒞 , J)}

(whose vertical arrows are the inclusion functors) which allows us to regard (𝒞 , J)^{∼} as a full subcategory of Stack(𝒞 , J).

5.1.4. We can characterize 0-separated and 1-separated prestacks on a small site C := (𝒞 , J) by means of the following construction. Let 𝒜 → 𝒞 be any fibration with small fibres. To every X ∈ Ob(𝒞) and every pair of cartesian sections σ, σ′ ∈ 𝒜 (X) we attach the presheaf

𝒞art(σ, σ′) : (𝒞 /X)^{o} → Set (Y ⟶^{f} X) ↦ Cart_{𝒞} (σ ∘ f_{∗}, σ′ ∘ f_{∗})

which assigns to every morphism (Z ⟶^{g} X) ⟶^{h/X} (Y ⟶^{f} X) of 𝒞 /X the map

𝒞art(σ, σ′)(f) → 𝒞art(σ, σ′)(g) β ↦ β ∗ h_{∗}

(with f_{∗} : 𝒞 /Y → 𝒞 /X and h_{∗} : 𝒞 /Z → 𝒞 /Y as in (1.1.25)). Notice that :

(5.1.5) (f_{∗})^{∧}𝒞art(σ, σ′) = 𝒞art(σ ∘ f_{∗}, σ′ ∘ f_{∗})  for every (Y ⟶^{f} X) ∈ Ob(𝒞 /X).

Moreover, for every σ, σ′, σ′′ ∈ 𝒜 (X), the system of composition maps

Cart_{𝒞} (σ ∘ f_{∗}, σ′ ∘ f_{∗}) × Cart_{𝒞} (σ′ ∘ f_{∗}, σ′′ ∘ f_{∗}) → Cart_{𝒞} (σ ∘ f_{∗}, σ′′ ∘ f_{∗}) (β, β′) ↦ β′ ⊙ β

clearly defines a morphism of presheaves on 𝒞 /X :

(5.1.6)  𝒞art(σ, σ′) × 𝒞art(σ′, σ′′) → 𝒞art(σ, σ′′).

Furthermore, every pair of isomorphisms of cartesian sections : μ : σ →^{∼} τ, μ′ : σ′ →^{∼} τ′ induces an isomorphism of presheaves

(5.1.7) 𝒞art(σ, σ′) →^{∼} 𝒞art(τ, τ′)  (β : σ ∘ f_{∗} ⇒ σ′ ∘ f_{∗}) ↦ (μ′ ∗ f_{∗}) ⊙ β ⊙ (μ^{−1} ∗ f_{∗}).

Lastly, every cartesian functor F : 𝒜 → 𝒜′ of 𝒞 -fibrations yields a morphism of presheaves :

(5.1.8)  𝒞art(σ, σ′) → 𝒞art(F ∘ σ, F ∘ σ′)  β ↦ F ∗ β.

Lemma 5.1.9. With the notation of (5.1.4), the following conditions are equivalent : (a) The fibration 𝒜 → 𝒞 is 0-separated (resp. 1-separated).

(b) For every X ∈ Ob(𝒞) and every pair of cartesian sections σ, σ′ ∈ 𝒜 (X), the presheaf 𝒞art(σ, σ′) is separated (resp. is a sheaf) on the site C/X (notation of (4.7)).

Proof. Let X ∈ Ob(𝒞) be any object, and 𝒮 ⊂ 𝒞 /X any covering sieve; choose a generating family (f_{i} : Y_{i} → X | i ∈ I) for 𝒮, and for every i, j ∈ I set 𝒞 /Y_{ij} := 𝒞 /Y_{i} ×_{𝒞 /X} 𝒞 /Y_{j}. There follows a commutative diagram :

\xymatrix{𝒜 (X) \ar[r]^{Cart_{𝒞} (j,𝒜)} \ar[dr]_{ρ} & Cart_{𝒞} (𝒮 , 𝒜) \ar[d]^{ε^{∗}} \\  & ∏_{i∈I} 𝒜 (Y_{i}) \ar[r]^{∂_{0}^{∗}} \ar[r]_{∂_{1}^{∗}} & ∏_{i,j∈I} Cart_{𝒞} (𝒞 /Y_{ij}, 𝒜)}

where j : 𝒮 → 𝒞 /X is the inclusion functor, ρ is the unique functor whose composition with the projection onto the factor 𝒜 (Y_{i}) agrees with Cart_{𝒞} (f_{i∗}, 𝒜), for every i ∈ I, and the functors ε^{∗}, ∂_{0}^{∗} and ∂_{1}^{∗} are as in remark 3.5.11. Clearly ε^{∗} is faithful; it follows that ρ is faithful if and only if the same holds for Cart_{𝒞} (j, 𝒜). On the other hand, the family ((f_{i}/X) : f_{i} → 1_{X} | i ∈ I) of morphisms of 𝒞 /X covers 1_{X}, according to (4.7). Taking into account (5.1.5), we conclude easily that 𝒜 is 0-separated if and only if all the presheaves 𝒞art(σ, σ′) are separated. Next, by remark 3.5.11the functor ε^{∗} induces an isomorphism of Cart_{𝒞} (𝒮 , 𝒜) onto the equalizer E of ∂_{0}^{∗} and ∂_{1}^{∗}, and clearly ρ factors through a unique functor ρ′ : 𝒜 (X) → E. It follows that ρ′ is fully faithful if and only if the same holds for Cart_{𝒞} (j, 𝒜). Again, this means that 𝒜 is 1-separated if and only if all the presheaves 𝒞art(σ, σ′) are sheaves. □

Theorem 5.1.10. Let C := (𝒞 , J) be any small site. The inclusion strict pseudo-functor 1-PreStack(C) → Fib(𝒞) admits a strict and strong left 2-adjoint pseudo-functor :

Fib(𝒞) → 1-PreStack(C) 𝒜 ↦ 𝒜^{sep}.

Proof. Let 𝒜 → 𝒞 be any fibration with small fibre categories; pick a unital cleavage λ for 𝒜, and let c be its associated unital pseudo-functor. For every X ∈ Ob(𝒞) consider the functor

β_{X}^{λ} : 𝒜_{X} → 𝒜 (X) A ↦ β_{X,A}^{λ}

defined as in the proof of theorem 3.2.7. For every A, A′ ∈ Ob(𝒜_{X}), consider as well the map

ω_{A,A′} : Hom_{𝒜_{X}} (A, A′) → 𝒞art(β_{X,A}^{λ}, β_{X,A′}^{λ})(1_{X}) (f : A → A′) ↦ β_{X,f}^{λ}.

Let ℋ_{A,A′} be the sheaf on C/X associated with the presheaf 𝒞art(β_{X,A}^{λ}, β_{X,A′}^{λ}), and denote

ω̃_{A,A′} : Hom_{𝒜_{X}} (A, A′) → H_{A,A′} := ℋ_{A,A′}(1_{X})

the composition of ω_{A,A′} with the natural map 𝒞art(β_{X,A}^{λ}, β_{X,A′}^{λ})(1_{X}) → H_{A,A′}. For every A, A′, A′′ ∈ Ob(𝒜_{X}), we have morphisms of presheaves as in (5.1.6)

𝒞art(β_{X,A}^{λ}, β_{X,A′}^{λ}) × 𝒞art(β_{X,A′}^{λ}, β_{X,A′′}^{λ}) → 𝒞art(β_{X,A}^{λ}, β_{X,A′′}^{λ})

whence a morphism of sheaves ℋ_{A,A′} × ℋ_{A′,A′′} → ℋ_{A,A′′}, which induces a composition map :

H_{A,A′} × H_{A′,A′′} → H_{A,A′′}.

It is easily seen that this composition law is associative, for every A, A′, A′′, A′′′ ∈ Ob(𝒜_{X}), and ω̃_{A,A}(1_{A}) is neutral for left and right composition. Hence, we have a category 𝒜_{X}^{sep} whose set of objects is Ob(𝒜_{X}), and such that Hom_{𝒜_{X}^{sep}} (A, A′) := H_{A,A′} for every A, A′ ∈ Ob(𝒜_{X}), with the foregoing composition law. Furthermore, the system of maps ω̃_{••} yields a functor

ω̃^{X} : 𝒜_{X} → 𝒜_{X}^{sep}

which is the identity map on objects. Next, notice that for every morphism g : Y → X in 𝒞 and every A ∈ Ob(𝒜_{X}) we have β_{X,A}^{λ}(g) = β_{Y,c_{g}A}^{λ}(1_{Y}) = c_{g}A; it follows that there exists a unique isomorphism of cartesian sections

(5.1.11)  β_{X,A}^{λ} ∘ g_{∗} →^{∼} β_{Y,c_{g}A}^{λ}  such that 1_{Y} ↦ 1_{c_{g}A}.

By virtue of (5.1.5)and (5.1.7), the isomorphisms (5.1.11)induce an isomorphism of presheaves

μ_{(A,A′)}^{g} : g_{∗}^{∧}𝒞art(β_{X,A}^{λ}, β_{X,A′}^{λ}) →^{∼} 𝒞art(β_{Y,c_{g}A}^{λ}, β_{Y,c_{g}A′}^{λ}) for every A, A′ ∈ Ob(𝒜_{X}) .

Since g_{∗} is continuous and cocontinuous for the topologies of C/X and C/Y (remark 4.7.3(i)), combining with lemma 4.2.15(ii) we deduce a natural isomorphism of sheaves on C/Y :

μ̃_{(A,A′)}^{g} : j_{g}^{∗}ℋ_{A,A′} →^{∼} ℋ_{c_{g}A,c_{g}A′}

whence a map d_{A,A′}^{g} : H_{A,A′} ⟶^{ℋ_{A,A′}(g/X)} ℋ_{A,A′}(g) \underrightarrow{μ_{(A,A′),1_{Y}}^{g}} H_{c_{g}A,c_{g}A′} and it is then easily

seen that the rules : A ↦ c_{g}A for every A ∈ Ob(𝒜_{X}) and f ↦ d_{A,A′}^{g}(f) for every morphism f : A → A′ of 𝒜_{X}^{sep} define a functor

d_{g} : 𝒜_{X}^{sep} → 𝒜_{Y}^{sep}.

Furthermore, a direct inspection yields a commutative diagram :

\xymatrix{Hom_{𝒜_{X}} (A, A′) \ar[r]^{ω̃_{A,A′}} \ar[d]_{c_{g,(A,A′)}} & H_{A,A′} \ar[d]^{d_{A,A′}^{g}} \\ Hom_{𝒜_{Y}} (c_{g}A, c_{g}A′) \ar[r]^{ω̃_{cgA,cgA′}} & H_{c_{g}A,c_{g}A′}}

where c_{g,(A,A′)} is the map given by the functor c_{g} : 𝒜_{X} → 𝒜_{Y}. In other words :

d_{g} ∘ ω̃^{X} = ω̃^{Y} ∘ c_{g}.

Next, let Y′ ⟶^{g′} Y ⟶^{g} X be two morphisms of 𝒞 /X; for every A ∈ Ob(𝒜_{X}) there exists a unique isomorphism of cartesian sections

(5.1.12)  β_{Y′,c_{g′}c_{g}A}^{λ} →^{∼} β_{Y′,c_{gg′}A}^{λ}  such that 1_{Y′} ↦ (γ_{(g,g′),A}^{c} : c_{g′}c_{g}A →^{∼} c_{gg′}A)

where γ_{(•,•)}^{c} denotes the coherence constraint of c. Denote by

γ_{(g,g′),A}^{d} ∈ Hom_{𝒜_{Y′}^{sep}} (c_{g}c_{g′}A, c_{gg′}A)

the image of (5.1.12), where the latter is seen as an element of 𝒞art(β_{Y′,c_{g′}c_{g}A}^{λ}, β_{Y′,c_{gg′}A}^{λ})(1_{Y′}). A direct inspection of the construction yields a commutative diagram :

\xymatrix{(g ∘ g′)_{∗}^{∧}𝒞art(β_{X,A}^{λ}, β_{X,A′}^{λ}) \ar[r]^{g_{∗}^{′∧}(μ_{(A,A′)}^{g})} \ar[d]_{μ_{(A,A′)}^{g∘g′}} & g_{∗}^{′∧}𝒞art(β_{Y,c_{g}A}^{λ}, β_{Y,c_{g}A′}^{λ}) \ar[d]^{μ_{(cgA,cgA′)}^{g′}} \\ 𝒞art(β_{Y′,c_{gg′}A}^{λ}, β_{Y′,c_{gg′}A′}^{λ}) \ar[r]^{δ_{(A,A′)}^{g,g′}} & 𝒞art(β_{Y′,c_{g′}c_{g}A}^{λ}, β_{Y′,c_{g′}c_{g}A′}^{λ})}

where δ_{(A,A′)}^{g,g′} is the isomorphism of presheaves (5.1.7) induced by the isomorphisms (5.1.12). Let (δ_{(A,A′)}^{g,g′})^{a} be the isomorphism of sheaves on C/X associated with δ_{(A,A′)}^{g,g′}; there follows a commutative diagram :

\xymatrix{H_{A,A′} \ar[r]^{d_{A,A′}^{g}} \ar[d]_{d_{A,A′}^{gg′}} & H_{c_{g}A,c_{g}A′} \ar[d]^{d_{cgA,cgA′}^{g′}} \\ H_{c_{gg′}A,c_{gg′}A} \ar[r]^{d_{(A,A′)}^{g,g′}} & H_{c_{g′}c_{g}A,c_{g′}c_{g}A′}}

with d_{(A,A′)}^{g,g′} := (δ_{(A,A′)}^{g,g′})_{1_{Y′}}^{a}. This translates as the commutativity of the diagram :

\xymatrix{c_{g′}c_{g}A \ar[r]^{d_{g′}∘d_{g}(f)} \ar[d]_{γ_{(g,g′),A}^{d}} & c_{g′}c_{g}A′ \ar[d]^{γ_{(g,g′),A′}^{d}} \\ c_{gg′}A \ar[r]^{d_{gg′}(f)} & c_{gg′}A′}  for every f ∈ Hom_{𝒜_{X}^{sep}} (A, A′)

which means that the rule : A ↦ γ_{(g,g′),A}^{d} yields an isomorphism of functors d_{g′} ∘ d_{g} →^{∼} d_{gg′}. Then it is easily seen that the rules : X ↦ 𝒜_{X}^{sep} and g ↦ d_{g} for every X ∈ Ob(𝒞) and every morphism g of 𝒞 yield a unital pseudo-functor whose coherence constraint is the system of isomorphisms γ_{••}^{d} : the detailed verification shall be left to the reader. Likewise, it follows easily that the rule X ↦ ω̃^{X} defines a strict pseudo-natural transformation ω̃^{•} : c ⇒ d. Set

𝒜^{sep} := ℱ ib(d)

and let λ^{∗} be the distinguished cleavage of 𝒜^{sep}, whose associated pseudo-functor is naturally identified with d (see remark 3.1.27(i)). Then we have a unique cartesian functor

ω̃_{𝒜} : 𝒜 → 𝒜^{sep}

restricting to ω̃^{X} : 𝒜_{X} → 𝒜_{X}^{sep} for every X ∈ Ob(𝒞). A direct inspection then shows that

β_{X,A}^{λ^{∗}} = ω̃_{𝒜} ∘ β_{X,A}^{λ} for every X ∈ Ob(𝒞) and A ∈ Ob(𝒜_{X})

and we have a natural isomorphism of presheaves :

ℋ_{A,A′} →^{∼} 𝒞art(β_{X,A}^{λ^{∗}}, β_{X,A′}^{λ^{∗}}) for every A, A′ ∈ Ob(𝒜_{X})

which identifies the morphism of presheaves induced by ω̃_{𝒜} as in (5.1.8) :

𝒞art(β_{X,A}^{λ}, β_{X,A′}^{λ}) → 𝒞art(ω̃_{𝒜} ∘ β_{X,A}^{λ}, ω̃_{𝒜} ∘ β_{X,A′}^{λ}) = 𝒞art(β_{X,A}^{λ^{∗}}, β_{X,A′}^{λ^{∗}})

with the natural bicovering morphism of presheaves 𝒞art(β_{X,A}^{λ}, β_{X,A′}^{λ}) → ℋ_{A,A′}. Especially, 𝒞art(β_{X,A}^{λ^{∗}}, β_{X,A′}^{λ^{∗}}) is a sheaf on C/X for every such X, A, A′, so 𝒜^{sep} is 1-separated over C, by lemma 5.1.9. Consider now any 1-separated fibration 𝒜′ over C, and any cartesian functor F : 𝒜 → 𝒜′. For every X ∈ Ob(𝒞) and A, A′ ∈ 𝒜_{X}, the morphism of presheaves

(5.1.13)  𝒞art(β_{A}^{λ}, β_{A′}^{λ}) → 𝒞art(F ∘ β_{A}^{λ}, F ∘ β_{A′}^{λ})

as in (5.1.8) factors uniquely through ℋ_{A,A′}, since 𝒞art(F ∘ β_{A}^{λ}, F ∘ β_{A′}^{λ}) is a sheaf on C/X (lemma 5.1.9). There follows a map F_{A,A′}^{sep} : H_{A,A′} → Hom_{𝒜′} (F A, F A′), whose composition with ω̃_{A,A′} equals F_{A,A′} : Hom_{𝒜} (A, A′) → Hom_{𝒜′} (F A, F A′). It is easily seen that the rules : A ↦ F^{sep}A := F A and (A, A′) ↦ F_{A,A′}^{sep} yield a well defined functor F_{|X}^{sep} : 𝒜_{X}^{sep} → 𝒜′_{X} such that F_{|X}^{sep} ∘ ω̃^{X} agrees with the restriction 𝒜_{X} → 𝒜′_{X} of F.

Let λ′ be a unital cleavage for 𝒜′, and c′ its associated pseudo-functor; then F corresponds to a pseudo-natural transformation α : c ⇒ c′, and denote by τ_{•}^{α} the coherence constraint of α. For every X ∈ Ob(𝒞) and A ∈ Ob(𝒜_{X}) we have also a unique isomorphism

(5.1.14)  β_{F A}^{λ′} →^{∼} F ∘ β_{A}^{λ}  such that 1_{X} ↦ 1_{F A}.

Explicitly, (5.1.14) assigns to every (g : Y → X) ∈ Ob(𝒞 /X) the isomorphism τ_{g,A}^{α} : c′_{g}F A →^{∼} F c_{g}A of 𝒜′_{Y}. On the other hand, we have a morphism of presheaves

(5.1.15)  𝒞art(β_{A}^{λ}, β_{A}^{λ}) → 𝒞art(β_{F A}^{λ′}, β_{F A}^{λ′})

described explicitly as follows. For every (g : Y → X) ∈ Ob(𝒞 /X), every morphism of cartesian sections t_{•} : β_{A}^{λ} ∘ g_{∗} ⇒ β_{A′}^{λ} ∘ g_{∗} is determined by t_{1_{Y}} : namely, t_{h} = c_{h}(t_{1_{Y}}) for every h ∈ Ob(𝒞 /Y); then (5.1.15) maps every such t_{•} to the morphism t′_{•} : β_{F A}^{λ′} ∘ g_{∗} ⇒ β_{F A′}^{λ′} ∘ g_{∗} with t′_{h} := c′_{h}(F t) for every h ∈ Ob(𝒞 /Y). With this notation, the condition that τ_{g}^{α} is a natural isomorphism c′_{g} ∘ F →^{∼} F ∘ c_{g} for every g ∈ Ob(𝒞 /X) translates as the commutativity of the diagram of presheaves on 𝒞 /X :

\xymatrix{𝒞art(β_{A}^{λ}, β_{A}^{λ}) \ar[r] \ar[d] & 𝒞art(β_{F A}^{λ′}, β_{F A′}^{λ′}) \ar[d] \\ 𝒞art(F ∘ β_{A}^{λ}, F ∘ β_{A}^{λ}) \ar[r] & 𝒞art(F ∘ β_{A}^{λ}, β_{F A}^{λ′})}

whose top horizontal (resp. left vertical) arrow is (5.1.15) (resp. (5.1.13)) and whose other two arrows are the morphisms (5.1.7) induced by the isomorphisms (5.1.14) (and by the identities of β_{F A}^{λ′} and of F ∘ β_{A}^{λ}). After taking associated sheaves, we obtain a similar diagram, in which 𝒞art(β_{A}^{λ}, β_{A}^{λ}) is replaced by ℋ_{A,A′}. Unwinding the definitions, we find that the commutativity of the latter diagram yields the commutativity of the following diagram of morphisms of 𝒜_{Y} :

\xymatrix{c′_{g}F A \ar[r]^{c′_{g}F^{sep}f} \ar[d]_{τ_{g,A}^{α}} & c′_{g}F A′ \ar[d]^{τ_{g,A′}^{α}} \\ F c_{g}A \ar[r]^{F^{sep}d_{g}f} & F c_{g}A′} for every f ∈ H_{A,A′}.

Thus, τ_{g}^{α} extends to a natural isomorphism F_{|Y}^{sep} ∘ c_{g} →^{∼} c′_{g} ∘ F_{|X}^{sep}, and then it is clear that the rule X ↦ F_{|X}^{sep} yields a pseudo-natural transformation α′ : d ⇒ c′ whose coherence constraint is again τ_{•}^{α}. Finally, α′ corresponds to a unique cartesian functor F^{sep} : 𝒜^{sep} → 𝒜′ whose restriction 𝒜_{X}^{sep} → 𝒜′_{X} agrees with F_{|X}^{sep} for every X ∈ Ob(𝒞). It is then clear that F^{sep} is the unique such cartesian functor whose composition with ω̃_{𝒜} equals F. This universal property then easily implies that the rules : 𝒜 ↦ 𝒜^{sep} and F ↦ F^{sep} yield a left adjoint (−)^{sep} for the inclusion functor 1-PreStack(C)Fib(𝒞) : the details shall be left to the reader.

Lastly, let μ : F ⇒ G be any natural 𝒞 -transformation of cartesian functors F, G : 𝒜 → 𝒜′ between 𝒞 -fibrations. Then for every X ∈ Ob(𝒞) and every A ∈ 𝒜_{X}, the morphism μ_{A} : F A → GA determines a unique natural transformation μ_{A}^{∗} : F ∘ β_{A}^{λ} ⇒ G ∘ β_{A}^{λ} such that 1_{X} ↦ μ_{A}. There follows for every A, A′ ∈ Ob(𝒜) a commutative diagram of presheaves :

\xymatrix{𝒞art(β_{A}^{λ}, β_{A′}^{λ}) \ar[r] \ar[d] & 𝒞art(F ∘ β_{A}^{λ}, F β_{A′}^{λ}) \ar[d] \\ 𝒞art(G ∘ β_{A}^{λ}, G ∘ β_{A′}^{λ}) \ar[r] & 𝒞art(F ∘ β_{A}^{λ}, G ∘ β_{A′}^{λ})}

whose top horizontal and left vertical arrows are the morphisms (5.1.13), and whose other two arrows are the morphisms (5.1.7) induced by μ_{A}^{∗} and μ_{A′}^{∗}. After taking associated sheaves on C/X, we deduce a commutative diagram

\xymatrix{Hom_{𝒜^{sep}} (A, A′) \ar[r] \ar[d] & Hom_{ℬ^{sep}} (F A, F B) \ar[d] \\ Hom_{𝒜^{sep}} (GA, GA′) \ar[r] & Hom_{𝒜^{sep}} (F A, GA′)}

which shows that μ is also a natural transformation F^{sep} ⇒ G^{sep}. Clearly the rule μ ↦ μ^{sep} is inverse to the rule (ν : F^{sep} ⇒ G^{sep}) ↦ ν ∗ ω̃_{𝒜} , so (−)^{sep} is the sought strong left 2-adjoint of the inclusion pseudo-functor. □

5.1.16. Let (𝒞 , J) be a small site. Our next task is to construct the analogue for stacks of the functor F ↦ F^{+} on presheaves (see (4.1.11)). To this aim, recall that for every X ∈ Ob(𝒞) the set J(X) of sieves covering X is cofiltered, and if 𝒮′ ⊂ 𝒮 are two such sieves and F : ℰ → 𝒞 any fibration, the inclusion functor i : 𝒮′ → 𝒮 induces a functor

Cart_{𝒞} (i, ℰ) : Cart_{𝒞} (𝒮 , ℰ) → Cart_{𝒞} (𝒮′, ℰ)

whence a well defined filtered system of small categories C_{X} := (Cart_{𝒞} (𝒮 , ℰ) | 𝒮 ∈ J(X)^{o}) associated with every object X of 𝒞. If ℰ has small fibres, we may then consider the functor

c_{F}^{+} : 𝒞^{o} → Cat X ↦ colim_{𝒮 ∈J(X)^{o}} Cart_{𝒞} (𝒮 , ℰ)

where the colimit is explicitly given by the construction detailed in example 1.5.10. For any morphism f : Y → X in 𝒞, the corresponding functor c_{F}^{+}(f) : c_{F}^{+}(X) → c_{F}^{+}(Y) is obtained as follows. For every 𝒮 ∈ J(X), the functor f_{∗|𝒮} : 𝒮 ×_{X} f → 𝒮 induces a functor

Cart_{𝒞} (f_{∗|𝒮} , ℰ) : Cart_{𝒞} (𝒮 , ℰ) → Cart(𝒮 ×_{X} f, ℰ) → c_{F}^{+}(Y)

(see remark 3.5.2(v)) and the system of such functors obviously forms a cocone of vertex c_{F}^{+}(Y) and basis C_{X} : J(X)^{o} → Cat. There follows a unique functor c_{F}^{+}(f) : c_{F}^{+}(X) → c_{F}^{+}(Y) whose composition with the natural functor Cart_{𝒞} (𝒮 , ℰ) → c_{F}^{+}(X) agrees with Cart_{𝒞} (f_{∗|𝒮} , ℰ) for every 𝒮 ∈ J(X). It is then easily seen that

c_{F}^{+}(f ∘ g) = c_{F}^{+}(g) ∘ c_{F}^{+}(f) for every pair of morphisms Z ⟶^{g} Y ⟶^{f} X in 𝒞 .

With the notation of (3.1.18), we set

ℰ^{+} := ℱ ib(c_{F}^{+})

and let F^{+} : ℰ^{+} → 𝒞 be the structure functor of ℰ^{+}. Recall also that ℰ^{+} is endowed with a distinguished split cleavage λ_{F}^{+} whose associated pseudo-functor is naturally identified with c_{F}^{+} (see remark 3.1.27(i)). Thus, the objects of ℰ^{+} are represented by the pairs

[X, ψ : 𝒮 → ℰ]

with X ∈ Ob(𝒞) and ψ is a 𝒞 -cartesian functor defined on some covering sieve 𝒮 ∈ J(X). A morphism [f, σ] : [X′, ψ′ : 𝒮′ → ℰ] → [X, ψ : 𝒮 → ℰ] is represented by the datum of a morphism f : X′ → X in 𝒞, and a natural 𝒞 -transformation σ : ψ′_{|𝒮′′} ⇒ (ψ ∘ f_{∗})_{|𝒮′′} , where 𝒮′′ ⊂ 𝒮′ ∩ (𝒮 ×_{X} f) is a sieve covering X′.

5.1.17. Next, let F′ : ℰ′ → 𝒞 be another fibration with small fibres, and G : ℰ → ℰ′ any 𝒞 -cartesian functor; the system of induced functors

Cart_{𝒞} (𝒮 , G) : Cart_{𝒞} (𝒮 , ℰ) → Cart_{𝒞} (𝒮 , ℰ′) for every X ∈ Ob(𝒞) and 𝒮 ∈ J(X)

yields, after taking colimits, a strict pseudo-natural transformation

c_{G}^{+} : c_{F}^{+} ⇒ c_{F′}^{+}  and then a 𝒞 -cartesian functor :  G^{+} := ℱ ib(c_{G}^{+}) : ℰ^{+} → ℰ^{′+}.

Moreover, if ω : G ⇒ G′ is any natural 𝒞 -transformation of 𝒞 -cartesian functors G, G′ : ℰ → ℰ′, we get a system of natural transformations

Cart_{𝒞} (𝒮 , ω) : Cart_{𝒞} (𝒮 , G) ⇒ Cart_{𝒞} (𝒮 , G′) for every X ∈ Ob(𝒞) and 𝒮 ∈ J(X)

whence again, after taking colimits, a modification

c_{ω}^{+} : c_{G}^{+} ⇝ c_{G′}^{+} whence a natural transformation : ω^{+} := Fib(c_{ω}^{+}) : G^{+} ⇒ G^{′+}.

It is then clear that the rules : (F : ℰ → 𝒞) ↦ F^{+}, (G : ℰ → ℰ′) ↦ G^{+} and (ω : G ⇒ G′) ↦ ω^{+} yield a well defined strict pseudo-functor

(−)^{+} : Fib(𝒞) → Fib(𝒞).

Moreover, since 𝒞 /X ∈ J(X) for every X ∈ Ob(𝒞), we have an obvious strict pseudo-natural transformation of strict pseudo-functors :

j_{ℰ} : Cart_{𝒞} (𝒞 /−, ℰ) ⇒ c_{ℰ}^{+}

(notation of (3.2.4)), whence a 𝒞 -cartesian functor

j_{ℰ} := Fib(j_{F}) : C(ℰ) → ℰ^{+}

and it is clear that the rule (F : ℰ → 𝒞) ↦ j_{ℰ} yields a strict pseudo-natural transformation of strict pseudo-functors.

Remark 5.1.18. Let F : ℰ → 𝒞 be any fibration with small fibres, X ∈ Ob(𝒞) any object, and 𝒮 any sieve of 𝒞 /X covering X. By unwinding the definitions, and taking into account lemma 3.1.20(i), we see that a 𝒞 -cartesian functor 𝒮 → ℰ^{+} is the datum of :

• for every object Y ⟶^{f} X of 𝒮, a sieve 𝒮^{(f)} covering Y and a 𝒞 -cartesian functor

σ^{(f)} : 𝒮^{(f)} → ℰ

• for every morphism h/X : (Y′ ⟶^{f′} X) → (Y ⟶^{f} X) of 𝒮, a sieve 𝒮^{(h/X)} ⊂ 𝒮^{(f′)} ∩ (𝒮^{(f)} ×_{Y} h) covering Y′ and a natural isomorphism of 𝒞 -functors

σ^{(h/X)} : σ_{|𝒮^{(h/X)}}^{(f′)} →^{∼} (c_{F}^{+}(h)(σ^{(f)}))_{|𝒮^{(h/X)}} = (σ^{(f)} ∘ h_{∗})_{|𝒮^{(h/X)}}

where h_{∗} : 𝒮^{(f)} ×_{Y} h → 𝒮^{(f)} is the functor (Z ⟶^{g} Y′) ↦ (Z ⟶^{h∘g} Y)

• for every h/X as in the foregoing and every other morphism h′/X : (Y′′ ⟶^{f′′} X) →

(Y′ ⟶^{f} X) of 𝒮_{σ}, a covering sieve 𝒮^{(h,h′/X)} ⊂ 𝒮^{(h′/X)} ∩ (𝒮^{(h/X)} ×_{Y′} h′) such that

(5.1.19)  (σ^{(h/X)} ∗ h′_{∗})_{|𝒮^{(h,h′/X)}} ⊙ σ_{|𝒮^{(h,h′/X)}}^{(h′/X)} = σ_{|𝒮^{(h,h′/X)}}^{(h∘h′/X)} .

Lemma 5.1.20. Let i ≤ 2 be an integer, F : ℰ → 𝒞 an i-separated prestack, X ∈ Ob(𝒞). Then every inclusion 𝒯 ⊂ 𝒮 of sieves of 𝒞 /X covering X induces an i-faithful functor

j : Cart_{𝒞} (𝒮 , ℰ) → Cart_{𝒞} (𝒯 , ℰ).

Proof. It is an immediate consequence of lemma 3.5.35(i). □

Lemma 5.1.21. Let 𝒞 be a small category, F : ℰ → 𝒞 a fibration with small fibres. We have : (i) The prestack F^{+} : ℰ^{+} → 𝒞 is 0-separated. (ii) If F is a 0-separated prestack, then F^{+} is a 1-separated prestack. (iii) If F is a 1-separated prestack, then F^{+} is a stack. (iv) If F is a stack, the functor j_{ℰ} : C(ℰ) → ℰ^{+} is an equivalence.

(v) There exists an essentially commutative diagram of strict pseudo-functors (notation of definition 4.1.5(ii)) :

\xymatrix{𝒞^{∧} \ar[r]^{ℱ ib} \ar[d]_{(−)^{+}} & 0-PreStack(𝒞 , J) \ar[d]^{(−)^{+}} \\ (𝒞 , J)^{sep} \ar[r]^{ℱ ib} & 1-PreStack(𝒞 , J).}

Proof. Let X ∈ Ob(𝒞) be any object, and 𝒯 any covering sieve of 𝒞 /X; let also [φ], [ψ] ∈ (F^{+})^{−1}X, and 𝒮 a sieve covering X such that [φ] and [ψ] are represented by 𝒞 -cartesian functors φ, ψ : 𝒮 → ℰ.

(i): Let [α], [β] : [φ] → [ψ] be morphisms of (F^{+})^{−1}X whose images agree under the functor

j : (F^{+})^{−1}X → Cart_{𝒞} (𝒯 , ℰ^{+}).

deduced from the split cleavage c_{F}^{+}. We need to check that α = β. To this aim, may assume that [α] and [β] are represented by natural 𝒞 -transformations α, β : φ ⇒ ψ; then, for every (Y ⟶^{f} X) ∈ Ob(𝒯 ) there exists a sieve 𝒯 ^{(f)} ⊂ 𝒮 ×_{X} f covering Y, such that

(α ∗ f_{∗})_{|𝒯 ^{(f)}} = (β ∗ f_{∗})_{|𝒯 ^{(f)}} .

Now, let 𝒮′ ⊂ 𝒮 be the sieve generated by ⋃_{f∈Ob(𝒯 )} f_{∗}(Ob(𝒯 ^{(f)})); by remark 4.1.3(iii), the sieve 𝒮′ covers X, and clearly α_{|𝒮′} = β_{|𝒮′} , whence the contention.

(ii): Let α : j([φ]) ⇒ j([ψ]) be a given natural 𝒞 -transformation; we need to check that α = j(α′) for some sieve 𝒮′ ⊂ 𝒮 covering X and some natural 𝒞 -transformation α′ : φ_{|𝒮′} ⇒ ψ_{|𝒮′} . However, α is described by a datum as follows :

• For every (Y ⟶^{f} X) ∈ Ob(𝒯 ), a sieve 𝒯 ^{(f)} ⊂ 𝒮 ×_{X} f covering Y and a natural

𝒞 -transformation α^{(f)} : (φ ∘ f_{∗})_{|𝒯 ^{(f)}} ⇒ (ψ ∘ f_{∗})_{|𝒯 ^{(f)}}

• for every morphism (h/X) : (Y′ ⟶^{f′} X) → (Y ⟶^{f} X) in 𝒯 , a sieve 𝒯 ^{(h/X)} covering

Y′ and contained in 𝒯 ^{(f′)} ∩ (𝒯 ^{(f)} ×_{Y} h) such that
α_{|𝒯 ^{(h/X)}}^{(f′)} = (α^{(f)} ∗ h_{∗})_{|𝒯 ^{(h/X)}} .

But then, in view of lemma 5.1.20, we may take already 𝒯 ^{(h/X)} := 𝒯 ^{(f′)} ∩ (𝒯 ^{(f)} ×_{Y} h). With this notation, let 𝒮′ be the sieve generated by ⋃_{f∈Ob(𝒯 )} f_{∗}(Ob(𝒯 ^{(f)})); then 𝒮′ covers X, by remark 4.1.3(iii). For every (Y ⟶^{g} X) ∈ Ob(𝒮′), pick any f ∈ Ob(𝒯 ) such that g = g′ ∘ f for some g′ ∈ Ob(𝒯 ^{(f)}), and set

α′_{g} := α_{g′}^{(f)} : φ(g) → ψ(g).

Let us check that this definition is independent of the choices of f and g′. Indeed, suppose that g = g′′ ∘ f′ for some other f′ ∈ Ob(𝒯 ) and some g′′ ∈ Ob(𝒯 ^{(f′)}); then we have

(α^{(f)} ∗g′_{∗})_{|𝒯 ′} = (α^{(g)})_{|𝒯 ′} = (α^{(f′)} ∗g′′_{∗})_{|𝒯 ′} with 𝒯 ′ := 𝒯 ^{(g)} ∩ (𝒯 ^{(f)} ×_{Y} g′) ∩ (𝒯 ^{(f′)} ×_{Y} g′′)

and by invoking again lemma 5.1.20, we deduce that

(α^{(f)} ∗ g′_{∗})_{|𝒯 ′′} = (α^{(f′)} ∗ g′′_{∗})_{|𝒯 ′′} with 𝒯 ′′ := (𝒯 ^{(f)} ×_{Y} g′) ∩ (𝒯 ^{(f′)} ×_{Y} g′′) .

Especially, we get : α_{g′}^{(f)} = (α^{(f)} ∗ g′_{∗})_{1_{Y}} = (α^{(f′)} ∗ g′′_{∗})_{1_{Y}} = α_{g′′}^{(f′)}. It is then clear that α′ is a well defined natural 𝒞 -transformation as sought.

(v) follows now from (ii) and a direct inspection of the construction of the functors (−)^{+} for presheaves and for prestacks.

(iii): Let σ : 𝒯 → ℰ^{+} be a 𝒞 -cartesian functor; hence, this is a datum

((𝒮^{(f)}, σ^{(f)} | f ∈Ob(𝒯 )), (𝒮^{(h/X)}, σ^{(h/X)} | f′ ⟶^{h} f in 𝒯 ), (𝒮^{(h,h′/X)} | f′′ ⟶^{h′} f′ ⟶^{h} f in 𝒯 ))

as in remark 5.1.18. But then, lemma 5.1.20says that we may even take :

𝒮^{(h/X)} := 𝒮^{(f′)} ∩ (𝒮^{(f)} ×_{Y} h) 𝒮^{(h,h′/X)} := 𝒮^{(f′′)} ∩ (𝒮^{(f′)} ×_{Y′} h′) ∩ (𝒮^{(f)} ×_{Y} (h ∘ h′))

for every f ∈ Ob(𝒯 ), every morphism f′ ⟶^{h} f in 𝒯 , and every pair f′′ ⟶^{h′} f′ ⟶^{h} f of morphisms in 𝒯 . Let 𝒯 ′ ⊂ 𝒞 /X be the sieve generated by ⋃_{f∈Ob(𝒯 )} f_{∗}(Ob(𝒮^{(f)})), which covers X, by remark 4.1.3(iii). We consider the functor

G_{𝒯 ′} : 𝒯 ′ → Cat/(𝒞 /X) (Y ⟶^{g} X) ↦ (𝒞 /X)/g = 𝒞 /Y

defined as in (3.5.8), and we construct a pseudo-cocone

ψ_{•} : G_{𝒯 ′} ⇒ F_{ℰ}

as follows. For every (Y ⟶^{g} X) ∈ Ob(𝒯 ′), choose g′ ∈ Ob(𝒯 ) and g′′ ∈ Ob(𝒮^{(g′)}) with

g = g′ ∘ g′′ and set : ψ_{g} := σ^{(g′)} ∘ g′′_{∗} : 𝒞 /Y → ℰ .

Next, let g_{1}, g_{2} ∈ Ob(𝒯 ′) and h/X : (Z_{1} ⟶^{g_{1}} X) → (Z_{2} ⟶^{g_{2}} X) a morphism of 𝒯 ′; with this notation, we have the 𝒞 -cartesian functors

σ^{(g′_{1})} ∘ g′′_{1∗}, σ^{(g′_{2})} ∘ (g′′_{2} ∘ h)_{∗} : 𝒞 /Z_{1} → ℰ

and two isomorphisms of 𝒞 -functors

(σ^{(g′_{1})} ∘ g′′_{1∗})_{|𝒮^{(g_{1})}} ⟸^{σ^{(g{′′ \atop 1}/X)}} σ^{(g_{1})} ⟹^{σ^{(g{′′ \atop 2}∘h/X)}} (σ^{(g′_{2})} ∘ (g′′_{2} ∘ h)_{∗})_{|𝒮^{(g_{1})}} .

Since F is 1-separated, it follows that the composition σ_{|𝒮^{(g_{1})}}^{(g′′_{2}∘h/X)} ⊙ (σ^{(g′′_{1}/X)})_{|𝒮^{(g_{1})}}^{−1} extends uniquely to an isomorphism of 𝒞 -functors

τ_{h/X}^{ψ} : ψ_{g_{1}} →^{∼} ψ_{g_{2}} ∘ h_{∗}.

We claim that the rule g ↦ ψ_{g} yields a pseudo-cocone as sought, with coherence constraint given by the system of isomorphisms τ_{•}^{ψ}. Indeed, let h′/X : (Z_{0} ⟶^{g_{0}} X) → (Z_{1} ⟶^{g_{1}} X) be another morphism of 𝒯 ′; then we have as well the isomorphisms of 𝒞 -functors

\xymatrix{(σ^{(g′_{0})} ∘ g′′_{0∗})_{|𝒮^{(g_{0})}} & σ^{(g_{0})} \ar@{=>}[l]_{σ^{(g{′′ \atop 0}/X)}} \ar@{=>}[r]^{σ^{(g{′′ \atop 1}∘h′/X)}} \ar@{=>}[d]^{σ^{(g{′′ \atop 2}∘h′∘h/X)}} & (σ^{(g′_{1})} ∘ (g′′_{1} ∘ h′)_{∗})_{|𝒮^{(g_{0})}} \\  & (σ^{(g′_{2})} ∘ (g′′_{2} ∘ h ∘ h′)_{∗})_{|𝒮^{(g_{0})}}}

and with this notation, we have :

(τ_{h′/X}^{ψ})_{|𝒮^{(g_{0})}} = σ^{(g′′_{1}∘h′/X)} ⊙ (σ^{(g′′_{0}/X)})^{−1} (τ_{h∘h′/X}^{ψ})_{|𝒮^{(g_{0})}} = (σ^{(g′′_{2}∘h∘h′/X)}) ⊙ (σ^{(g′′_{0}/X)})^{−1}.

Set 𝒮 := 𝒮^{(g′′_{2}∘h,h′/X)} ∩ 𝒮^{(g′′_{1},h′/X)} ⊂ 𝒮^{(g_{0})}. We claim that

(τ_{h∘h′/X}^{ψ})_{|𝒮} = (τ_{h/X}^{ψ} ∗ h′_{∗})_{|𝒮} ⊙ (τ_{h′/X}^{ψ})_{|𝒮} .

Indeed, the latter identity follows by combining the following ones :

\begin{matrix}σ_{|𝒮}^{(g′′_{2}∘h∘h′/X)} = & (σ^{(g′′_{2}∘h/X)} ∗ h′_{∗})_{|𝒮} ⊙ σ_{|𝒮}^{(h′/X)} \\ σ_{|𝒮}^{(g′′_{1}∘h′/X)} = & (σ^{(g′′_{1}/X)} ∗ h′_{∗})_{|𝒮} ⊙ σ_{|𝒮}^{(h′/X)} \end{matrix}

that are provided by remark 5.1.18. Since 𝒮 covers Z_{0} and F is 1-separated, it follows that

τ_{h∘h′/X}^{ψ} = (τ_{h/X}^{ψ} ∗ h′_{∗}) ⊙ τ_{h′/X}^{ψ}

which shows that τ^{ψ} satisfies the required coherence axioms. By lemma 3.5.9, we deduce that there exist a 𝒞 -functor ψ : 𝒯 ′ → ℰ and an invertible modification Ξ : F_{ψ} ⊙ ε̂_{𝒯 ′} ⇝ ψ_{•}, where ε̂_{𝒯 ′} : G_{𝒯 ′} ⇒ F_{𝒯 ′} is the universal pseudo-cocone defined as in (3.5.8); explicitly, this amounts to a system of oriented squares with invertible 2-cells :

[figure: 𝒞 /Y; g′′_{∗}; g_{∗}; Ξ_{g}; 𝒯 ′; ψ; 𝒮^{(g′)}; σ^{(g′)}; ℰ]  for every g ∈ Ob(𝒯 ′)

from which it follows easily that ψ is 𝒞 -cartesian (here g = g′ ∘ g′′ is the factorisation used to define ψ_{g}). The compatibility condition for Ξ amounts to the identity :

(5.1.22) Ξ_{g_{2}} ∗ h_{∗} = τ_{h/X}^{ψ} ⊙ Ξ_{g_{1}}  for every morphism (Z_{1} ⟶^{g_{1}} X) ⟶^{h/X} (Z_{2} ⟶^{g_{2}} X) of 𝒯 ′ .

Thus, the functor ψ represents an object [ψ] of the fibre category ℰ^{+}(X), and to conclude, we need to check that j([ψ]) is isomorphic to σ. However, since we know already from (ii) that ℰ^{+} is a 1-separated prestack, by lemma 5.1.20 it suffices to find an isomorphism between the images of j([ψ]) and σ in Cart_{𝒞} (𝒯 ′, ℰ^{+}). This comes down to exhibiting isomorphisms of functors

ϑ_{g} : ψ ∘ g_{∗} →^{∼} σ^{(g)} for every g ∈ Ob(𝒯 ′)

(where g_{∗} : 𝒮^{(f)} → 𝒯 ′ is the usual functor with t ↦ g ∘ t for every t ∈ Ob(𝒮^{(f)})) such that

(5.1.23) σ^{(h/X)} ⊙ ϑ_{g_{1}|𝒮^{(h/X)}} = (ϑ_{g_{2}} ∗ h_{∗})_{|𝒮^{(h/X)}}  for every morphism g_{1} ⟶^{h/X} g_{2} of 𝒯 ′ .

To this aim, we set :

ϑ_{g} := (σ^{(g′′/X)})^{−1} ⊙ (Ξ_{g})_{|𝒮^{(g)}} .

Since ℰ is 1-separated, the identities (5.1.23) can be checked after restriction to any covering sieve contained in 𝒮^{(h/X)}; but on such a suitable sieve we may apply the identities (5.1.19), and combining with (5.1.22) we conclude easily : the details shall be left to the reader.

(iv): It suffices to check that j_{ℰ} is a fibrewise equivalence when ℰ is a stack (corollary 3.1.28(i)), and this follows by direct inspection of the definitions. □

Theorem 5.1.24. For every small site (𝒞 , J), the inclusion strict pseudo-functor

F : Stack(𝒞 , J) → Fib(𝒞)

admits a left 2-adjoint

Fib(𝒞) → Stack(𝒞 , J) (F : ℰ → 𝒞) ↦ (F^{a} : ℰ^{a} → 𝒞)

which assigns to every fibration over 𝒞 its associated stack. Moreover, we have a pseudocommutative diagram of pseudo-functors (see remark 2.4.10) :

(5.1.25)  \xymatrix{𝒞^{∧} \ar[r]^{ℱ ib} \ar[d]_{(−)^{a}} & Fib(𝒞) \ar[d]^{(−)^{a}} \\ (𝒞 , J)^{∼} \ar[r]^{ℱ ib} & Stack(𝒞 , J)}

whose left vertical arrow is the usual left adjoint to the inclusion functor for presheaves.

Proof. Recall first that the counit of the 2-adjoint pair (F, C) of (3.2.4) is a 𝒞 -equivalence ev^{ℰ} : C(ℰ) → ℰ for every fibration F : ℰ → 𝒞 (theorem 3.2.7). Moreover, the unit of the same 2-adjoint pair assigns to every split fibration (F, λ) a 𝒞 -equivalence η_{ℰ} : ℰ → C(ℰ), deduced from the strict pseudo-natural equivalence β^{λ} defined in the proof of theorem 3.2.7(see remark 3.2.9(i)). There follows a 𝒞 -equivalence :

ω_{ℰ} :=: C(ℰ)^{+} ⟶^{(ev^{ℰ} )^{+}} ℰ^{+} ⟶^{η_{ℰ^{+}}} C(ℰ^{+}).

By inspecting the definitions, we see that ω_{ℰ} assigns to every object [X, φ : 𝒮 → C(ℰ)] of C(ℰ)^{+} the cartesian section ω_{ℰ} ([X, φ]) ∈ ℰ^{+}(X) given by the rule :

(f : Y → X) ↦ (𝒮 ×_{X} f ⟶^{f_{∗}} 𝒮 ⟶^{ev^{ℰ} ∘φ} ℰ)

and notice that ev^{ℰ} ∘ φ ∘ f_{∗}(h) = φ_{f∘h}(1_{Z}) for every (h : Z → Y) ∈ Ob(𝒮 ×_{X} f). For every morphism g/X : (f′ : Y′ → X) → (f : Y → X) in 𝒞 /X, the induced natural transformation

ω_{ℰ} ([X, φ])(g) : ω_{ℰ} ([X, φ])(f′) ⇒ ω_{ℰ} ([X, φ])(f) is the identity ev^{ℰ} ∘ φ ∘ f′_{∗} →^{∼} ev^{ℰ} ∘ φ ∘ f_{∗} ∘ g_{∗}.

Claim 5.1.26. (i) For every two 𝒞 -fibrations 𝒜 ⟶^{φ} 𝒞 ⟵^{φ′} 𝒜′, the functor

C_{𝒜 ,𝒜′} : Cart_{𝒞} (𝒜 , 𝒜′) → Cart_{𝒞} (C(𝒜), C(𝒜′))  F ↦ C(F)  (α : F ⇒ F′) ↦ C(α)

is an equivalence.

(ii) For every 𝒞 -fibration 𝒜 → 𝒞 the following diagram of functors commutes:

\xymatrix{Cart_{𝒞} (C(ℰ), 𝒜) \ar[r]^{(−)^{+}} \ar[d]_{C_{C(ℰ),𝒜}} & Cart_{𝒞} (C(ℰ)^{+}, 𝒜^{+}) \ar[d]^{Cart_{𝒞} (j_{C(ℰ)} ,𝒜^{+})} \\ Cart_{𝒞} (C(C(ℰ)), C(𝒜)) \ar[r]^{Cart_{𝒞} (C(C(ℰ)),j_{𝒜} )} & Cart_{𝒞} (C(C(ℰ)), 𝒜^{+}).}

Proof of the claim. Assertion (ii) follows by direct inspection. For (i) let us notice the commutative diagram :

\xymatrix{Cart_{𝒞} (𝒜 , 𝒜′) \ar@{-->}[dr]_{Cart_{𝒞} (ev^{𝒜} ,𝒜′)} \ar[rr]^{C_{𝒜 ,𝒜′}} &  & Cart_{𝒞} (C(𝒜), C(𝒜′)) \ar[dl]^{Cart_{𝒞} (𝒜 ,ev^{𝒜′} )} \\  & Cart_{𝒞} (C(𝒜), 𝒜′)}

where all arrows except possibly C_{𝒜 ,𝒜} are equivalences; then the same must hold for C_{𝒜 ,𝒜′} . ♢

Claim 5.1.27. For every fibration 𝒜 → 𝒞 the diagram of functors :

\xymatrix{Cart_{𝒞} (ℰ^{+}, 𝒜) \ar[r]^{Cart_{𝒞} (j_{ℰ} ,𝒜)} \ar[d]_{C_{ℰ^{+},𝒜}} & Cart_{𝒞} (C(ℰ), 𝒜) \ar[d]^{(−)^{+}} \\ Cart_{𝒞} (C(ℰ^{+}), C(𝒜)) \ar[r]^{Cart_{𝒞} (ω_{ℰ} ,j_{𝒜} )} & Cart_{𝒞} (C(ℰ)^{+}, 𝒜^{+})}

is essentially commutative (notation of claim 5.1.26(i)).

Proof of the claim. Let G : ℰ^{+} → 𝒜 be any 𝒞 -cartesian functor; the 𝒞 -cartesian functor G′ := (G ∘ j_{ℰ})^{+} : C(ℰ)^{+} → 𝒜^{+} can be described as follows. Let [X, ψ : 𝒮 → C(ℰ)] be any object of C(ℰ)^{+}; then G′([X, ψ]) = [X, G ∘ j_{ℰ} ∘ ψ : 𝒮 → 𝒜], and notice that j_{ℰ} ∘ ψ is the 𝒞 -cartesian functor given by the rule :

(f : Y → X) ↦ [Y, ψ_{f} : 𝒞 /Y → ℰ] for every f ∈ Ob(𝒮) .

On the other hand, the 𝒞 -cartesian functor G′′ : j_{𝒜} ∘ C(G) ∘ ω_{ℰ} : C(ℰ)^{+} → 𝒜^{+} is described as follows. For any [X, ψ] as in the foregoing, G′′([X, ψ]) = [X, G ∘ ψ′ : 𝒞 /X → 𝒜], where ψ′ : 𝒮 → ℰ^{+} is the 𝒞 -cartesian functor given by the rule :

(f : Y → X) ↦ [Y, ev^{ℰ} ∘ ψ ∘ f_{∗}] for every f ∈ Ob(𝒞 /X)

and notice that ev^{ℰ} ∘ ψ ∘ f_{∗} : 𝒮 ×_{X} f → ℰ is in turn given by the rule :

(g : Z → Y) ↦ ψ_{f∘g}(1_{Z}) for every g ∈ Ob(𝒮 ×_{X} f) .

Let ψ′′ : 𝒮 → 𝒜 be the restriction of ψ′; then [X, ψ′] = [X, ψ′′] in ℰ^{+}, and 𝒮 ×_{X} f = 𝒞 /Y for every (f : Y → X) ∈ Ob(𝒮). For such f, we need to compare the cartesian functors ψ_{f}, ev^{ℰ} ∘ ψ ∘ f_{∗} : 𝒞 /Y → ℰ. However, ψ_{f}(1_{Y}) = ev^{ℰ} ∘ ψ ∘ f_{∗}(1_{Y}), so there exists a unique isomorphism of functors

τ_{ψ,f} : ev^{ℰ} ∘ ψ ∘ f_{∗} →^{∼} ψ_{f} such that (τ_{ψ,f})_{1_{Y}} = 1_{ψ_{f} (1_{Y} )}.

Explicitly, every object (g : Z → Y) of 𝒞 /Y yields a morphism g/Y : g → 1_{Y} in 𝒞 /Y, and therefore (τ_{ψ,f})_{g} is determined by the commutativity of the diagram :

[figure: ψ_{f∘g}(1_{Z}); (τ_{ψ,f})_{g}; (ev^{ℰ} ∘ψ∗f_{∗})_{g/Y}; ψ_{f}(g); ψ_{f} (g/Y); ψ_{f}(1_{Y}); ψ_{f}(1_{Y})]

However, (ev^{ℰ} ∘ ψ ∘ f_{∗})_{g/Y} = ψ_{f}(g/Y) ∘ ψ_{g/X,1_{Z}}, so that

(τ_{ψ,f})_{g} = ψ_{g/X,1_{Z}} for every (g : Z → Y) ∈ Ob(𝒞 /Y).

We claim that we get an isomorphism of 𝒞 -cartesian functors

G([τ_{ψ}]) : G′′([X, ψ]) →^{∼} G′([X, ψ]) f ↦ G([1_{Y} , τ_{ψ,f}]) for every (f : Y → X) ∈ Ob(𝒮) .

Indeed, let g/X : (f′ : Y′ → X) → (f : Y → X) be any morphism in 𝒮; the assertion comes down to the commutativity of the diagram :

[figure: G([Y′, ev^{ℰ} ∘ ψ ∘ f′_{∗}]); G([1_{Y′},τ_{ψ,f′}]); G([Y′, ev^{ℰ} ∘ ψ ∘ f_{∗} ∘ g_{∗}]); G([1_{Y′},τ_{ψ,f} ∗g_{∗}]); G([Y′, ψ_{f′}]); G([1_{Y′},ψ_{g/X}]); G([Y′, ψ_{f} ∘ g_{∗}]).]

The latter in turn follows by applying G to the system of identities :

ψ_{g/X,h} ∘ ψ_{h/X,1_{Z}} = ψ_{g∘h/X,1_{Z}} for every (h : Z → Y′) ∈ Ob(𝒞 /Y′)

which hold by functoriality of ψ, applied to the morphisms f′ ∘ h ⟶^{h/X} f′ ⟶^{g/X} f of 𝒮.

Next, we check the naturality of the rule ψ ↦ G([τ_{ψ}]). Hence, let [X_{i}, ψ_{i} : 𝒮_{i} → C(ℰ)] for i = 1, 2 be two objects of C(ℰ)^{+}, and [t, σ] : [X_{2}, ψ_{2}] → [X_{1}, ψ_{1}] a morphism; i.e. t : X_{2} → X_{1} is a morphism of 𝒞 and σ : ψ_{2|𝒮_{3}} ⇒ (ψ_{1} ∘ t_{∗})_{|𝒮_{3}} is a natural 𝒞 -transformation defined on a sieve 𝒮_{3} ⊂ 𝒮_{2} ∩ (𝒮_{1} ×_{X_{1}} t). We have

G′([t, σ]) = [t, (G ∘ j_{ℰ}) ∗ σ : G ∘ j_{ℰ} ∘ ψ_{2|𝒮_{3}} ⇒ G ∘ j_{ℰ} ∘ (ψ_{1} ∘ t_{∗})_{|𝒮_{3}} ].

On the other hand, notice that ω_{ℰ} ([t, σ]) : ω_{ℰ} ([X_{2}, ψ_{2}]) → ω_{ℰ} ([X_{1}, ψ_{1}]) is the natural transformation σ′ : ψ′_{2} ⇒ ψ′_{1} ∗ t_{∗}, where ψ′_{i} is the functor such that (f : Y → X_{i}) ↦ [Y, ev^{ℰ} ∘ ψ_{i} ∘ f_{∗}] for every f ∈ Ob(𝒞 /X_{i}), and σ′_{f} := ev^{ℰ} ∗ σ ∗ f_{∗}, with f_{∗} : 𝒮_{3} ×_{X_{2}} f → 𝒮_{3} for every f ∈ Ob(𝒞 /X_{2}). Explicitly, σ′_{f} assigns to every (g : Z → Y) ∈ Ob(𝒮_{3} ×_{X_{2}} f) the morphism

σ_{f∘g,1_{Z}} : ψ_{2,f∘g}(1_{Z}) → ψ_{1,t∘f∘g}(1_{Z}).

Thus, we need to check the commutativity of the diagram :

\xymatrix{[X_{2}, G ∘ ψ′_{2}] \ar[r]^{G([τ_{ψ_{2}}])} \ar[d]_{[t,G∗σ′]} & [X_{2}, G ∘ j_{ℰ} ∘ ψ_{2}] \ar[d]^{[t,(G∘j_{ℰ} )∗σ]} \\ [X_{1}, G ∘ ψ′_{1}] \ar[r]^{G([τ_{ψ_{1}}])} & [X_{1}, G ∘ j_{ℰ} ∘ ψ_{1}].}

which in turn follows from the commutativity of the diagram :

\xymatrix{ψ_{2,f∘g}(1_{Z}) \ar[r]^{σ_{f∘g,1_{Z}}} \ar[d]_{ψ_{2,g/X_{2},1_{Z}}} & ψ_{1,t∘f∘g}(1_{Z}) \ar[d]^{ψ_{1,g/X_{1},1_{Z}}} \\ ψ_{2,f}(g) \ar[r]^{σ_{f,g}} & ψ_{1,t∘f}(g)}

for every f ∈ Ob(𝒞 /X_{2}) and every g ∈ Ob(𝒮_{3} ×_{X_{2}} f). Since g/X_{1} = t_{∗}(g/X_{2}) in 𝒮_{1}, the assertion holds by naturality of σ. Lastly, the naturality with respect to natural 𝒞 -transformations G ⇒ G′ is immediate from the definitions. ♢

Now, let 𝒜 → 𝒞 be a stack; then j_{𝒜} is an equivalence, by lemma 5.1.21(iv), and it follows easily that Cart_{𝒞} (ω_{ℰ} , j_{𝒜} ) is an equivalence as well. The same holds for C_{ℰ^{+},𝒜} and C_{C(ℰ),𝒜} , by claim 5.1.26(i). We deduce that the functor (−)^{+} appearing in the diagrams of claims 5.1.27 and 5.1.26(ii) admits both left and right quasi-inverse functors, so it is an equivalence. By claim 5.1.27, we finally conclude that Cart_{𝒞} (j_{ℰ} , 𝒜) is an equivalence as well. Now, set

ℰ^{a} := ℰ^{+++}.

The composition Cart_{𝒞} (j_{ℰ} , 𝒜) ∘ Cart_{𝒞} (j_{ℰ^{+}}, 𝒜) ∘ Cart_{𝒞} (j_{ℰ^{++}}, 𝒜) is an equivalence

λ_{ℰ ,𝒜} : Cart_{𝒞} (ℰ^{a}, 𝒜) →^{∼} Cart_{𝒞} (C(C(C(ℰ))), 𝒜).

The rules (ℰ , 𝒜) ↦ Cart_{𝒞} (ℰ^{a}, 𝒜) and (ℰ , 𝒜) ↦ Cart_{𝒞} (C(C(C(ℰ))), 𝒜) yield strict pseudofunctors

Cart_{𝒞} ((−)^{a}, −), Cart_{𝒞} (C ∘ C ∘ C, −) : Fib(𝒞)^{o} × Stack(𝒞 , J) → Cat

and clearly the rule (ℰ , 𝒜) ↦ λ_{ℰ ,𝒜} defines a strict pseudo-natural equivalence of functors

λ : Cart_{𝒞} ((−)^{a}, −) ⇒ Cart_{𝒞} (C ∘ C ∘ C, −).

On the other hand, by composing evaluation functors we get as well a strict pseudo-natural equivalence C ∘ C ∘ C ⇒ 1_{Fib(𝒞)}, whence an induced strict pseudo-natural equivalence

λ′ : Cart_{𝒞} (−, F) →^{∼} Cart_{𝒞} (C ∘ C ∘ C, −)

where F : Stack(𝒞 , J) → Fib(𝒞) is the (forgetful) inclusion strict pseudo-functor. After choosing a quasi-inverse for λ (see theorem 2.4.12) and composing with λ′, we obtain the required (non-strict) pseudo-natural equivalence

Cart_{𝒞} (−, F) →^{∼} Cart_{𝒞} ((−)^{a}, −).

Lastly, the essential commutativity of (5.1.25) follows directly from lemma 5.1.21(v). □

5.2. Covering morphisms of prestacks. In this section we wish to characterize the i-faithful functors of prestacks (for i = 0, 1, 2), by means of certain conditions that are analogous to those given in definition 4.1.28for presheaves; to this aim, we make the following :

Definition 5.2.1. Let (𝒞 , J) be a site, F : ℰ → 𝒞 and F′ : ℰ′ → 𝒞 two fibrations, and φ : ℰ′ → ℰ a 𝒞 -cartesian functor.

(i) φ is a 0-covering if for every Y ∈ Ob(ℰ) there exist a covering family (f_{i} : X_{i} → F Y | i ∈ I) and for every i ∈ I an object Y′_{i} ∈ Ob(ℰ′_{X_{i}}) and a 𝒞 -cartesian morphism h_{i} : φY′_{i} → Y in ℰ such that F (h_{i}) = f_{i}.

(ii) φ is a 1-covering if for every X ∈ Ob(𝒞), every Y_{1}, Y_{2} ∈ Ob(ℰ′_{X}) and every morphism g : φY_{1} → φY_{2} in ℰ_{X} there exist a covering family (f_{i} : X_{i} → X | i ∈ I), and morphisms Y_{2} ⟵^{g_{i}} Z_{i} ⟶^{h_{i}} Y_{1} in ℰ′, where h_{i} is 𝒞 -cartesian for every i ∈ I, such that

φ(g_{i}) = g ∘ φ(h_{i}) and F′(h_{i}) = f_{i}.

(iii) φ is a 2-covering if for every X ∈ Ob(𝒞), every Y_{1}, Y_{2} ∈ Ob(ℰ′_{X}), and every pair g_{1}, g_{2} : Y_{1} → Y_{2} of morphisms in ℰ′_{X} with φ(g_{1}) = φ(g_{2}) there exist a covering family (f_{i} : X_{i} → X | i ∈ I), and for every i ∈ I a 𝒞 -cartesian morphism

h_{i} : Z_{i} → Y_{1} in ℰ′ such that g_{1} ∘ h_{i} = g_{2} ∘ h_{i} and F′(h_{i}) = f_{i}.

Remark 5.2.2. Suppose that (ℰ ⟶^{F} 𝒞 , λ) and (ℰ′ ⟶^{F′} 𝒞 , λ′) are split fibrations over 𝒞, and φ : (ℰ′, λ′) → (ℰ , λ) is a split cartesian functor. Let c and c′ be the strict pseudo-functors associated with λ and λ′. Then we can rephrase the conditions of definition 5.2.1as follows :

(i) φ is 0-covering if and only if for every Y ∈ Ob(ℰ) there exist a covering family (f_{i} : X_{i} → F Y | i ∈ I) and for every i ∈ I an object Y′_{i} ∈ Ob(ℰ′_{X_{i}}) and an isomorphism

φY′_{i} →^{∼} c_{f_{i}}Y in ℰ_{X_{i}}.

(ii) φ is 1-covering if and only if for every X ∈ Ob(𝒞), every Y_{1}, Y_{2} ∈ Ob(ℰ′_{X}) and every morphism g : φY_{1} → φY_{2} in ℰ_{X} we have a covering family (f_{i} : X_{i} → X | i ∈ I), and for every i ∈ I a morphism g_{i} : c′_{f_{i}}Y_{1} → c′_{f_{i}}Y_{2} in ℰ′_{X_{i}} such that φ(g_{i}) = c_{f_{i}}(g).

(iii) φ is a 2-covering if for every X ∈ Ob(𝒞), every Y_{1}, Y_{2} ∈ Ob(ℰ′_{X}), and every pair g_{1}, g_{2} : Y_{1} → Y_{2} of morphisms in ℰ′_{X} with φ(g_{1}) = φ(g_{2}) there exists a covering family (f_{i} : X_{i} → X | i ∈ I) such that c′_{f_{i}}(g_{1}) = c′_{f_{i}}(g_{2}).

Lemma 5.2.3. Consider a site (𝒞 , J), and an essentially commutative diagram of the 2-cate-gory Fib(𝒞), whose vertical arrows are equivalences of categories :

\xymatrix{ℰ′ \ar[r]^{φ} \ar[d]_{ω′} & ℰ \ar[d]^{ω} \\ ℱ′ \ar[r]^{ψ} & ℱ .}

Let also i ∈ {0, 1, 2}. Then φ is i-covering if and only if the same holds for ψ.

Proof. Denote ℰ ⟶^{F} 𝒞 ⟵^{F′} ℰ′ and ℱ ⟶^{G} 𝒞 ⟵^{G′} ℱ′ the respective fibrations. The assumption means that there exists an isomorphism of functors β : ψ ∘ ω′ →^{∼} ω ∘ φ such that G ∗ β = 1_{F′}.

Suppose that φ is 0-covering, let Y ∈ Ob(ℱ) be any object, and set X := GY; by assumption there exists Z ∈ Ob(ℰ_{X}) with an isomorphism t : ωZ →^{∼} Y in ℱ_{X} (corollary 3.1.28(i)). Then there exist a covering family (f_{j} : X_{j} → X | j ∈ I) and for every j ∈ I a cartesian morphism h_{j} : φZ′_{j} → Z with F h_{j} = f_{j}. It follows that

h′_{j} := t ∘ ω(h_{j}) ∘ β_{Z′_{j}} : ψ(ω′Z′_{j}) → Y

is cartesian and Gh′_{j} = f_{j} for every j ∈ I, so ψ is 0-covering.

Conversely, suppose that ψ is 0-covering; let Y ∈ Ob(ℰ) and X := F Y. By assumption there exist a covering family (f_{j} : X_{j} → X | j ∈ I) and for every j ∈ I a cartesian morphism h_{j} : ψZ_{j} → ωY with Gh_{j} = f_{j}. Then we find for every j ∈ I an isomorphism t_{j} : ω′Z′_{j} →^{∼} Z_{j} in ℱ′_{X_{j}}; it follows that h_{j} := h ∘ ψ(t_{j}) ∘ β_{Z′_{j}}^{−1} : ω(φZ′_{j}) → ωY is cartesian, and since ω is an equivalence, there exists a unique cartesian morphism h′_{j} : φZ′_{j} → Y in ℰ with ω(h′_{j}) = h_{j}, for every j ∈ I (details left to the reader); by construction, F h′_{j} = f_{j}, so φ is 0-covering.

Next, if φ is 1-covering, let X ∈ Ob(𝒞) and g : ψY_{1} → ψY_{2} any morphism in ℱ_{X}. We find isomorphisms t_{i} : ω′Z_{i} →^{∼} Y_{i} in ℱ′_{X} (i = 1, 2), and a morphism g′ : ψ(ω′Z_{1}) → ψ(ω′Z_{2}) such that ψ(t_{2}) ∘ g′ = g ∘ ψ(t_{1}); then β_{Z_{2}} ∘ g′ = ω(g′′) ∘ β_{Z_{1}} for a morphism g′′ : φZ_{1} → φZ_{2} in ℰ_{X}. By assumption, there exist a covering family (f_{j} : X_{j} → X | j ∈ I) and morphisms

Z_{2} ⟵^{g_{j}} W_{j} ⟶^{h_{j}} Z_{1} in ℰ′ for every j ∈ I, such that h_{j} is cartesian with F′h_{j} = f_{j}, and φ(g_{j}) = g′′ ∘ φ(h_{j}). Set h′_{j} := t_{1} ∘ ω′(h_{j}) and g′_{j} := t_{2} ∘ ω′(g_{j}); then h′_{j} is cartesian with G′(h′_{j}) = f_{j}, and ψ(g′_{j}) = g ∘ ψ(h′_{j}) for every j ∈ I. This shows that ψ is 1-covering.

Conversely, suppose that ψ is 1-covering, and let g : φY_{1} → φY_{2} be a morphism in ℰ_{X}. Then there exist a covering family (f_{j} : X_{j} → X | j ∈ I) and morphisms ω′Y_{2} ⟵^{g_{j}} Z_{j} ⟶^{h_{j}} ω′Y_{1} in ℱ′ such that h_{j} is cartesian with G′h_{j} = f_{j} and β_{Y_{2}} ∘ ψ(g_{j}) = ω(g) ∘ β_{Y_{1}} ∘ ψ(h_{j}) for every j ∈ I. We pick also an isomorphism t_{j} : ω′Z′_{j} →^{∼} Z_{j} in ℱ′_{X_{j}} for every j ∈ I; then there exist a unique cartesian morphism h′_{j} : Z′_{j} → Y_{1} with ω′(h′_{j}) = h_{j} ∘ t_{j} and a unique morphism g′_{j} : Z′_{j} → Y_{2} with ω′(g′_{j}) = g_{j} ∘ t_{j}. A direct computation shows that ω(g ∘ φ(h′_{j})) = ω(φ(g′_{j})), whence g ∘ φ(h′_{j}) = φ(g′_{j}) for every j ∈ I. Thus, φ is 1-cartesian.

Lastly, suppose that φ is 2-covering, and let g_{1}, g_{2} : Y_{1} → Y_{2} be two morphisms in ℱ′_{X} such that ψ(g_{1}) = ψ(g_{2}). Pick isomorphisms t_{i} : ω′Z_{i} →^{∼} Y_{i} for i = 1, 2; then there exists for i = 1, 2 a unique morphism g′_{i} : Z_{1} → Z_{2} such that t_{2} ∘ ω′(g′_{i}) = g_{i} ∘ t_{1}. It follows that

ψ(t_{2}) ∘ β_{Z_{2}}^{−1} ∘ ω(φ(g′_{1})) = ψ(t_{2}) ∘ ψ(ω′g_{1}) ∘ β_{Z_{1}}^{−1}
= ψ(g_{1} ∘ t_{1}) ∘ β_{Z_{1}}^{−1}
= ψ(g_{2} ∘ t_{1}) ∘ β_{Z_{1}}^{−1}
= ψ(t_{2}) ∘ β_{Z_{2}}^{−1} ∘ ω(φ(g′_{2}))

whence φ(g′_{1}) = φ(g′_{2}). Then we have a covering family (f_{j} : X_{j} → X | j ∈ I) and for every j ∈ I a cartesian morphism h_{j} : W_{j} → Z_{1} such that g′_{1} ∘ h_{j} = g′_{2} ∘ h_{j} and F h_{j} = f_{j}. Hence h′_{j} := t_{1} ∘ ω′(h_{j}) : ω′W_{j} → Y_{1} is cartesian with G′h′_{j} = f_{j} and g_{1} ∘ h′_{j} = g_{2} ∘ h′_{j}. This proves that ψ is 2-covering. Conversely, suppose that ψ is 2-covering, and let g_{1}, g_{2} : Y_{1} → Y_{2} be two morphisms in ℰ′_{X} such that φ(g_{1}) = φ(g_{2}). Set g′_{i} := ω′(g_{i}) for i = 1, 2; then ψ(g′_{1}) = ψ(g′_{2}), so there exist a covering family (f_{j} : X_{j} → X | j ∈ I) and for every j ∈ I a cartesian morphism h_{j} : Z_{j} → ω′Y_{1} such that g′_{1} ∘h_{j} = g′_{2} ∘h_{j} and G′h_{j} = f_{j}. For every j ∈ I pick an isomorphism t_{j} : ω′Z′_{j} →^{∼} Z_{j} in G^{′−1}X; there exists a unique cartesian morphism h′_{j} : Z′_{j} → Y_{1} such that ω′(h′_{j}) = h_{j} ∘ t_{j}. It is easily seen that ω′(g_{1} ∘ h′_{j}) = ω′(g_{2} ∘ h′_{j}), whence g_{1} ∘ h′_{j} = g_{2} ∘ h′_{j}, and F′h′_{j} = f_{j} for every j ∈ I. Thus, φ is 2-covering. □

Lemma 5.2.4. Let (𝒞 , J) be a site, F : ℰ → 𝒞 , F′ : ℰ′ → 𝒞 , F′′ : ℰ′′ → 𝒞 three fibrations, φ : ℰ′′ → ℰ′ and ψ : ℰ′ → ℰ two cartesian functors, and i ∈ {0, 1, 2}. We have:

(i) If φ and ψ are i-covering, the same holds for ψ ∘ φ.

(ii) If ψ ∘ φ is i-covering, and φ is j-covering for every j < i, then ψ is i-covering.

(iii) If ψ ∘ φ is i-covering and ψ is j-covering for every j > i, then φ is i-covering.

Proof. In view of lemma 5.2.3and claim 3.2.8we may replace φ and ψ by C(φ) and C(ψ), and assume from start that φ and ψ are split cartesian functors. We then denote by c, c′ and c′′ the strict pseudo-functors associated with the split cleavages of ℰ, ℰ′ and ℰ′′ respectively.

(i): Suppose first that i = 0, let Y ∈ Ob(ℰ) be any object, and set X := F Y; by assumption there exist a covering family (f_{j} : X_{j} → X | j ∈ I) and for every j ∈ I an isomorphism h_{j} : ψY′_{j} →^{∼} c_{f_{j}}Y in ℰ_{X_{j}}. Likewise, for every j ∈ I there exist a covering family (f_{jλ} : X_{jλ} → X_{j} | λ ∈ Λ_{j}) and for every λ ∈ Λ_{j} an isomorphism h_{jλ} : φY′′_{jλ} →^{∼} c′_{f_{jλ}}Y′_{j} in ℰ′_{X_{jλ}}. Then the family (f′_{jλ} := f_{j} ∘ f_{jλ} | j ∈ I, λ ∈ Λ_{j}) covers X and c_{f_{jλ}}(h_{j}) ∘ ψ(h_{jλ}) : ψ ∘ φ(Y′′_{jλ}) → c_{f′_{jλ}}Y is an isomorphism in ℰ_{X_{jλ}} for every j ∈ I and λ ∈ Λ_{j}, whence the contention, in this case.

Next, suppose that i = 1. Let X ∈ Ob(𝒞), Y′′_{1}, Y′′_{2} ∈ Ob(ℰ′′_{X}), and g : ψ ∘ φY′′_{1} → ψ ∘ φY′′_{2} any morphism in ℰ_{X}. By assumption there exist a covering family (f_{j} : X_{j} → X | j ∈ I) and for every j ∈ I a morphism

g_{j} : φ(c′′_{f_{j}}Y′′_{1}) = c′_{f_{j}}(φY′′_{1}) → c′_{f_{j}}(φY′′_{2}) = φ(c′′_{f_{j}}Y′′_{2}) in ℰ′_{X_{j}} such that ψ(g_{j}) = c_{f_{j}}(g).

Since φ is 1-covering, we find for every j ∈ I a covering family (f_{jλ} : X_{jλ} → X_{j} | λ ∈ Λ_{j}) and for every λ ∈ Λ_{j} a morphism

g_{jλ} : c′′_{f_{jλ}}(c′′_{f_{j}}Y′′_{1}) → c′′_{f_{jλ}}(c′′_{f_{j}}Y′′_{2})  in ℰ′_{X_{j}} such that  φ(g_{jλ}) = c′_{f_{jλ}}(g_{j}).

Then the family (f′_{jλ} := f_{j} ∘ f_{jλ} | j ∈ I, λ ∈ Λ_{j}) covers X, and ψ ∘ φ(g_{jλ}) = c_{f′_{jλ}}(g) for every j ∈ I and every λ ∈ Λ_{j}. It follows that ψ ∘ φ is 1-covering.

Lastly, let i = 2; consider morphisms g_{1}, g_{2} : Y′′_{1} → Y′′_{2} in ℰ′′_{X} with ψ ∘ φ(g_{1}) = ψ ∘ φ(g_{2}). By assumption, there exists a covering family (f_{j} : X_{j} → X | j ∈ I) such that

φ(c′′_{f_{j}}(g_{1})) = c′_{f_{j}}(φ(g_{1})) = c′_{f_{j}}(φ(g_{2})) = φ(c′′_{f_{j}}(g_{2})) for every j ∈ I.

We may then find for every j ∈ I a covering family (f_{jλ} : X_{jλ} → X_{j} | λ ∈ Λ_{j}) such that c′′_{f_{jλ}} ∘ c′′_{f_{j}}(g_{1}) = c′′_{f_{jλ}} ∘ c′′_{f_{j}}(g_{2}) for every λ ∈ Λ_{j}. We conclude that ψ ∘ φ is 2-covering.

(ii): Again, we suppose first that i = 0, in which case the assumption means that ψ ∘ φ is 0covering (and there is no condition on φ), and we need to check that ψ is 0-covering. However, the assertion follows straightforwardly from the definitions (details left to the reader).

Next, suppose that i = 1, in which case ψ ∘ φ is 1-covering and φ is 0-covering. Let g : ψY′_{1} → ψY′_{2} be a morphism in ℰ_{X}, for some X ∈ Ob(𝒞); by assumption there exist for t = 1, 2 a covering family (f_{tj} : X_{tj} → X | j ∈ I_{t}) and isomorphisms h_{tj} : φY′′_{tj} →^{∼} c′_{f_{j}}Y′_{t} in F^{′−1}X_{j} for every j ∈ I_{t}. Notice then that for every morphism f′ : X′ → X_{ti} in 𝒞 we get an isomorphism c′_{f′}(h_{tj}) : φ(c′′_{f′}Y′′_{tj}) →^{∼} c′_{f_{j}∘f′}Y′_{t} in ℰ′_{X′}. We deduce that, after replacing the covering families f_{1•} and f_{2•} by a common refinement, we may assume that I := I_{1} = I_{2} and f_{j} := f_{1j} = f_{2j} for every j ∈ I. In this situation, for every j ∈ I there exists a morphism

g_{j} : ψ ∘ φ(Y′′_{1j}) → ψ ∘ φ(Y′′_{2j})  in ℰ_{X_{j}} such that  ψ(h_{2j}) ∘ g_{j} = c_{f_{j}}(g) ∘ ψ(h_{1j}).

By our assumption on ψ ∘ φ, for every j ∈ I we then find a covering family (f_{jλ} : X_{jλ} → X_{j} | λ ∈ Λ_{j}) and for every λ ∈ Λ_{j} a morphism

g_{jλ} : Y′′_{1j} → Y′′_{1j}  in ℰ′′_{X_{jλ}}  such that ψ ∘ φ(g_{jλ}) = c_{f_{jλ}}(g_{j}).

It follows that ψ(c′_{f_{jλ}}h_{2j}) ∘ ψ(φ(g_{jλ})) = c_{f_{j}∘f_{jλ}}(g) ∘ ψ(c′_{f_{j}}h_{1j}) for every j ∈ I and λ ∈ Λ_{j}. Since the family (f_{i} ∘ f_{iλ} | i ∈ I, λ ∈ Λ_{i}) covers X, we conclude that ψ is 1-covering.

Next, suppose that i = 2, so ψ ∘ φ is 2-covering and φ is j-covering for j = 0, 1. Let X ∈ Ob(𝒞) and g_{1}, g_{2} : Y′_{1} → Y′_{2} two morphisms of ℰ′_{X} such that ψ(g_{1}) = ψ(g_{2}). Arguing as in the previous case, we find a covering family (f_{j} : X_{j} → X | j ∈ I) and isomorphisms h_{tj} : φY′′_{tj} →^{∼} c′_{f_{j}}Y′_{t} in ℰ′_{X_{j}} for t = 1, 2 and every j ∈ I. In this situation, for every j ∈ I and t = 1, 2 there exists a unique morphism g_{tj} : φY′′_{1i} → φY′′_{2i} in ℰ′_{X_{j}} such that

(5.2.5)  h_{2j} ∘ g_{tj} = c_{f_{j}}(g_{t}) ∘ h_{1j}.

Notice that ψ(h_{2j}) ∘ ψ(g_{1j}) = c′_{f_{j}}(ψ(g_{1})) = c′_{f_{j}}(ψ(g_{2})) = ψ(h_{2j}) ∘ ψ(g_{2j}), whence

ψ(g_{1j}) = ψ(g_{2j}) for every j ∈ I.

Then, since φ is 1-covering, for every j ∈ I and t = 1, 2 we find a covering family f_{tj•} := (f_{tjλ} : X_{tjλ} → X_{j} | λ ∈ Λ_{tj}) and morphisms

g_{tjλ} : c′′_{f_{tjλ}}Y′′_{1j} → c′′_{f_{tjλ}}Y′′_{2j}  in ℰ′′_{X_{tjλ}}  such that φ(g_{tjλ}) = c′_{f_{tjλ}}(g_{tj})

and arguing as in the foregoing, we may replace f_{1j•} and f_{2j•} by a common refinement, and assume that Λ_{j} := Λ_{1j} = Λ_{2j} for every j ∈ I, and f_{jλ} := f_{1jλ} = f_{2jλ}, for every j ∈ I and every λ ∈ Λ_{j}. We deduce that

ψ ∘ φ(g_{1jλ}) = c′′_{f_{jλ}}(ψ(g_{1j})) = c′′_{f_{jλ}}(ψ(g_{2j})) = ψ ∘ φ(g_{2jλ}) for every j ∈ I and Λ ∈ Λ_{j}.

Since ψ ∘ φ is 2-covering, there exists therefore a covering family (f_{jλλ′} : X_{jλλ′} → X_{jλ} | λ′ ∈ Λ_{jλ}) such that c′′_{f_{jλλ′}}(g_{1jλ}) = c′′_{f_{jλλ′}}(g_{2jλ}) for every λ′ ∈ Λ_{jλ}. Therefore c′_{f_{jλλ′}} ∘ c′_{f_{jλ}}(g_{1j}) = φ(c′′_{f_{jλλ′}}(g_{1jλ})) = φ(c′′_{f_{jλλ′}}(g_{2jλ})) = c′_{f_{jλλ′}} ∘ c′_{f_{jλ}}(g_{2j}) for every j ∈ I, λ ∈ Λ_{j} and λ′ ∈ Λ_{jλ}. Set f′_{jλλ′} := f_{j} ∘ f_{jλ} ∘ f_{jλλ′} for every such j, λ, λ′; combining with (5.2.5), we deduce easily that c_{f′_{jλλ′}}(g_{1}) = c_{f′_{jλλ′}}(g_{2}), and we conclude that ψ is 2-covering, as sought.

(iii): The case where i = 2 is immediate from the definitions.

Say that i = 1, so ψ ∘ φ is 1-covering and ψ is 2-covering; let X ∈ Ob(𝒞) and g : φY′′_{1} → φY′′_{2} a morphism in ℰ′_{X}. By assumption, there exist a covering family (f_{j} : X_{j} → | j ∈ I) and a morphism g_{j} : c_{f_{j}}Y′′_{1} → c_{f_{j}}Y′′_{2} in ℰ′′_{X_{j}} for every j ∈ I such that

ψ ∘ φ(g_{j}) = c_{f_{j}}(ψ(g)) = ψ(c′_{f_{j}}(g)).

Then, since ψ is 2-covering, we find for every j ∈ I a covering family (f_{jλ} : X_{jλ} → X_{j} | λ ∈ Λ_{j}) such that c′_{f_{jλ}}(φ(g_{j})) = c′_{f_{jλ}} ∘ c′_{f_{j}}(g) for every λ ∈ Λ_{j}, i.e. φ(c′′_{f_{jλ}}(g_{j})) = c′_{f_{j}∘f_{jλ}}(g). This shows that φ is 1-covering.

In order to deal with the case where i = 0, let us remark :

Claim 5.2.6. Suppose ψ is j-covering for j = 1, 2. Let X ∈ Ob(𝒞) and Z, Z′ ∈ Ob(ℰ′_{X}) such that ψZ and ψZ′ are isomorphic in ℰ_{X}. Then there exists a covering family (f_{j} : X_{j} → X | j ∈ I) such that c′_{f_{j}}Z and c′_{f_{j}}Z′ are isomorphic in ℰ′_{X_{j}} for every j ∈ I.

Proof of the claim. Let l : ψZ →^{∼} ψZ′ be an isomorphism in ℰ_{X}; since ψ is 1-covering, there exist a covering family (f′_{j} : X′_{j} → X | j ∈ I) and for every j ∈ I a morphism l_{j} : c′_{f′_{j}}Z → c′_{f′_{j}}Z′ such that ψ(l_{j}) = c_{f′_{j}}(l). Likewise, there exist a covering family (f′′_{j} : X′_{j} → X | j ∈ I′) and for every j ∈ I′ a morphism l′_{j} : c′_{f′′_{j}}Z′ → c′_{f′′_{j}}Z such that ψ(l′_{i}) = c_{f′′_{j}}(l^{−1}). After replacing (f′_{j} | j ∈ I) and (f′′_{j} | j ∈ I′) by a common refinement, we may then assume that I = I′ and f_{j} := f′_{j} = f′′_{j} for every j ∈ I (details left to the reader). It follows that φ(k_{j} ∘ l_{j}) = c_{f_{j}}(1_{Z}) = φ(1_{c′_{f_{j}} Z}) and φ(l_{j} ∘ k_{j}) = c_{f_{j}}(1_{Z′}) = φ(1_{c′_{f_{j}} Z′}) for every j ∈ I. Then, since ψ is 2-covering, for every j ∈ I there exists a covering family (f_{jλ} : X_{jλ} → X_{j} | λ ∈ Λ_{j}) such that c′_{f_{jλ}}(k_{j} ∘ l_{j}) = c′_{f_{jλ}}(1_{c′_{f_{j}} Z}); likewise, for every j ∈ I there exists a covering family (f′_{jλ} : X′_{jλ} → X_{j} | λ ∈ Λ′_{j}) such that c′_{f′_{jλ}}(l_{j} ∘ k_{j}) = c′_{f′_{jλ}}(1_{c′_{f_{j}} Z′}). After replacing these families by a common refinement, we may assume that Λ_{j} = Λ′_{j} for every j ∈ I, and f_{jλ} = f′_{jλ} for every j ∈ I and every λ ∈ Λ_{j}. Then, set h_{jλ} := f_{j} ∘ f_{jλ} for every j ∈ I and every λ ∈ Λ_{j}; we deduce easily that c′_{f′_{jλ}}(l_{j}) is an isomorphism c′_{h_{jλ}}Z →^{∼} c′_{h_{jλ}}Z′, whence the claim. ♢

Now, suppose that ψ ∘ φ is 0-covering and ψ is j-covering for j = 1, 2, and let X ∈ Ob(𝒞), Y′ ∈ Ob(ℰ′_{X}); by assumption there exist a covering family (f_{j} : X_{j} → X | j ∈ I) and for every j ∈ I an isomorphism ψ ∘ φY′′_{j} →^{∼} c_{f_{j}}ψY′ = ψ(c′_{f_{j}}Y′) in ℰ_{X}. By claim 5.2.6, there exists for every j ∈ I a covering family (f_{jλ} : X_{jλ} → X_{j} | j ∈ I) such that c′_{f_{jλ}}(φY′′_{j}) = φ(c′′_{f_{jλ}}Y′′_{j}) is isomorphic to c′_{f_{j}∘f_{jλ}}Y′ in ℰ′_{X_{i}} for every λ ∈ Λ_{j}. This shows that φ is 0-covering, and concludes the proof of the lemma. □

Lemma 5.2.7. Let C := (𝒞 , J) be a site, ℰ ⟶^{F} 𝒞 ⟵^{F′} ℰ′ two fibrations, and φ : ℰ → ℰ′ a 𝒞 -cartesian functor. The following conditions are equivalent :

(i) φ is 1-covering (resp. 2-covering).

(ii) For every X ∈ Ob(𝒞) and every σ, σ′ ∈ ℰ (X), the induced morphism :

(5.2.8)  𝒞art(σ, σ′)^{a} → 𝒞art(φ ∘ σ, φ ∘ σ′)^{a}

of sheaves on C/X is an epimorphism (resp. a monomorphism).

Proof. Recall that – as described in (5.1.8) – for every X, σ, σ′ as in (ii) we have a morphism of presheaves φ_{σ,σ′}^{∗} : 𝒞art(σ, σ′) → 𝒞art(φ ∘ σ, φ ∘ σ′) on 𝒞 /X, and (5.2.8) is the morphism of associated sheaves (φ_{σ,σ′}^{∗})^{a} on the site C/X. Then the assertion follows easily by inspecting the definitions, and taking into account corollary 4.1.30(i,ii) and remark 4.1.29(ii,iii). □

Proposition 5.2.9. Let (𝒞 , J) be a site, ℰ ⟶^{F} 𝒞 ⟵^{F′} ℰ′ two fibrations, φ : ℰ → ℰ′ a 𝒞 - cartesian functor, and i ∈ {0, 1, 2} such that ℰ is i-separated and ℰ′ is (i − 1)-separated. Then φ is i-faithful if and only if it is j-covering for every j ≥ 2 − i.

Proof. We consider first the case where i = 0, so by assumption ℰ is a 0-separated prestack (and no conditions on ℰ′); the assertion is that φ is faithful if and only if it is 2-covering. However, if φ is faithful, it is obviously 2-covering. Thus, suppose that φ is 2-covering, and let X ∈ Ob(𝒞) and g_{1}, g_{2} : Y → Z two morphisms in ℰ_{X} such that φ(g_{1}) = φ(g_{2}); by assumption there exist a covering family (f_{j} : X_{j} → X | j ∈ I) and for every j ∈ I a cartesian morphism h_{j} : Y_{j} → Y in ℰ such that F h_{j} = f_{j} and g_{1} ∘ h_{j} = g_{2} ∘ h_{j}. We may then find two cartesian sections ψ_{Y} , ψ_{Z} ∈ ℰ (X) such that ψ_{Y} (1_{X}) = Y and ψ_{Z}(1_{X}) = Z, and for i = 1, 2 there exists a unique natural 𝒞 -transformation σ_{i} : ψ_{Y} ⇒ ψ_{Z} such that σ_{i,1_{X}} = g_{i}. It follows that (σ_{1})_{|𝒮} = (σ_{2})_{|𝒮} , so σ_{1} = σ_{2}, since ℰ is 0-separated; hence g_{1} = g_{2}, so φ is faithful.

In case i = 1, the prestack ℰ is 1-separated and ℰ′ is 0-separated, and we need to check that φ is fully faithful if and only if it is j-covering for j = 1, 2. Again, if φ is fully faithful, obviously it is j-covering for j = 1, 2. Conversely, if the latter condition holds, by the foregoing case we know already that φ is faithful, so it remains only to check that φ is full. Thus, let X ∈ Ob(𝒞) and g : φY_{1} → φY_{2} any morphism in ℰ′_{X}; by assumption, there exist a covering

family (f_{j} : X_{j} → X | j ∈ I) and morphisms Y_{2} ⟵^{g_{j}} Z_{j} ⟶^{h_{j}} Y_{1} with h_{j} cartesian, such that F h_{j} = f_{j} and φ(g_{j}) = g ∘ φ(h_{j}) for every j ∈ I. As in the foregoing, we pick for i = 1, 2 a cartesian section ψ_{i} ∈ ℰ (X) such that ψ_{i}(1_{X}) = Y_{i}, and then there exists a unique natural 𝒞 -transformation σ : φ ∘ ψ_{1} ⇒ φ ∘ ψ_{2} such that σ_{1_{X}} = g. Since ψ_{1}(f_{j}/X) : ψ_{1}(f_{j}) → Y_{1} is cartesian, there exists moreover a unique isomorphism in F^{−1}X_{j}

h′_{j} : ψ_{1}(f_{j}) →^{∼} Z_{j} such that h_{j} ∘ h′_{j} = ψ_{1}(f_{j}/X).

Likewise, since ψ_{2}(f_{j}/X) : ψ_{2}(f_{j}) → Y_{2} is cartesian, there exists a unique morphism in F^{−1}X_{j}

g′_{j} : ψ_{1}(f_{j}) → ψ_{2}(f_{j}) such that g_{j} ∘ h′_{j} = ψ_{2}(f_{j}/X) ∘ g′_{j}.

We get therefore for every j ∈ I a unique natural 𝒞 -transformation

σ_{j} : ψ_{1} ∘ f_{j∗} ⇒ ψ_{2} ∘ f_{j∗} such that σ_{j,1_{X_{j}}} = g′_{j}.

Claim 5.2.10. φ ∗ σ_{j} = σ ∗ f_{j} for every j ∈ I.

Proof of the claim. It suffices to check that φ(σ_{j,1_{X_{j}}}) = (σ ∗ f_{j})_{1_{X_{j}}}, i.e. φ(g′_{j}) = σ_{f_{j}} for every j ∈ I. However : φ(ψ_{2}(f_{j}/X))∘φ(g′) = φ(g_{j})∘φ(h′_{j}) = g∘φ(h_{j})∘φ(h′_{j}) = g∘φ(ψ_{1}(f_{j}/X)) = φ(ψ_{2}(f_{j}/X)) ∘ σ_{f_{j}}, whence the assertion, since φ(ψ_{2}(f_{j}/X)) is cartesian. ♢

Recall now that the proof of theorem 3.2.7 yields a natural equivalence η_{Cℰ)} : C(ℰ) →^{∼} C(C(ℰ)), induced by the strict pseudo-natural equivalence β^{λ} associated with the natural split cleavage λ of C(ℰ). We set ψ_{i}^{∗} := η_{C(ℰ)}(ψ_{i}) for i = 1, 2. Explicitly, ψ_{i}^{∗}(f) = ψ_{i} ∘ f_{∗} for every f ∈ Ob(𝒞 /X), and ψ_{i,g/X}^{∗} = 1_{ψ_{i}^{∗}(f′)} for every morphism g/X : f′ → f in 𝒞 /X. Let 𝒮 ⊂ 𝒞 /X be the sieve generated by (f_{j} | j ∈ I); we construct as follows a natural 𝒞 -transformation

σ^{†} : ψ_{|𝒮}^{∗} ⇒ ψ_{|𝒮}^{∗} .

For every (f : X′ → X) ∈ Ob(𝒮), pick j ∈ J such that there exists a factorisation f = f_{j} ∘ f′ for some morphism f′ : X → X_{j} of 𝒞, and set σ_{f}^{†} := σ_{j}∗f′_{∗}. Let us check that σ_{f}^{†} is independent of the choice of factorisation : indeed, if f = f′′ ∘ f_{k} for some other k ∈ I and some morphism f′′ : X′ → X_{k}, we have φ ∗ (σ_{j} ∗ f′) = σ ∗ f_{j} ∗ f′ = σ ∗ f_{k} ∗ f′′ = φ ∗ (σ_{k} ∗ f′′) by claim 5.2.10, whence the contention, since φ is faithful. Likewise, the naturality of the rule f ↦ σ_{f}^{†} can be checked after composition with φ, where it follows easily (details left to the reader).

Now, since ℰ is 1-separated, the same holds for C(ℰ), hence σ^{†} extends to a natural 𝒞 - transformation σ^{∗} : ψ_{1}^{∗} ⇒ ψ_{2}^{∗}. Then there exists a unique natural 𝒞 -transformation σ′ : ψ_{1} ⇒ ψ_{2} such that η_{C(ℰ)}(σ′) = σ^{∗}. We claim that

φ ∗ σ′ = σ.

Indeed, it suffices to show that η_{C(ℰ)}(φ ∗ σ′) = η_{C(ℰ)}(σ), i.e. that C(φ) ∗ σ^{∗} = η_{C(ℰ)}(σ), and since ℰ′ is 0-separated, we are reduced to checking that (C(φ) ∗ σ^{∗})_{|𝒮} = η_{C(ℰ)}(σ)_{|𝒮} , i.e. that C(φ) ∗ σ^{†} = η_{C(ℰ)}(σ)_{|𝒮} . Then we are further reduced to showing that φ ∗ σ_{f_{j}}^{†} = σ ∗ f_{j∗} for every j ∈ I. But the latter identities hold by claim 5.2.10. Lastly, set g′ := σ′_{1_{X}}; we deduce that φ(g′) = g, and this shows that φ is full, as required.

In case i = 2, the prestack ℰ′ is 1-separated and ℰ is a stack, and we need to check that φ is an equivalence if and only if it is j-covering for j = 0, 1, 2. Again, if φ is an equivalence, clearly it is j-covering for all j; conversely, if this condition holds, by the foregoing we know already that φ is fully faithful, so it remains only to check that φ is essentially surjective. To this aim, let X ∈ Ob(𝒞) and Y′ ∈ Ob(ℰ′_{X}); by assumption, there exist a covering family (f_{j} : X_{j} → X | j ∈ I) and for every j ∈ I a cartesian morphism h_{j} : φY_{j} → Y′ with F′h_{j} = f_{j}. Pick, for every j ∈ I a cartesian section ψ_{j} ∈ ℰ (X_{j}) with ψ_{j}(1_{X_{j}}) = Y_{j}. Now, for every j, k ∈ I consider the category 𝒞 /X_{jk} := 𝒞 /X_{j} ×_{(f_{j∗},f_{k∗})} 𝒞 /X_{k} (see (3.5.4)). The objects

of 𝒞 /X_{jk} are the pairs of morphisms X_{j} ⟵^{f′_{j}} X′ ⟶^{f′_{k}} X_{k} such that f′ := f_{j} ∘ f′_{j} = f_{k} ∘ f′_{k}. We

have two natural projections 𝒞 /X_{j} ⟵^{π_{jk}^{0}} 𝒞 /X_{jk} ⟶^{π_{jk}^{1}} 𝒞 /X_{k} that send every such pair (f′_{j}, f′_{k}) to f′_{j} and respectively f′_{k}. For every such pair (f′_{j}, f′_{k}) and s = j, k we get cartesian morphisms ψ_{s}(f′_{s}/X_{s}) : ψ_{s}(f′_{s}) → Y_{s} such that F ψ_{s}(f′_{s}/X_{s}) = f′_{s}. There follows a unique isomorphism

τ : φ ∘ ψ_{j}(f′_{j}) →^{∼} φ ∘ ψ_{k}(f′_{k}) such that h_{k} ∘ φ(ψ_{k}(f′_{k}/X_{k})) ∘ τ = h_{j} ∘ φ(ψ_{j}(f′_{j}/X_{j}))

and since φ is fully faithful, we get a unique isomorphism

ω_{(f′_{j},f′_{k})}^{ij} : ψ_{j}(f′_{j}) →^{∼} ψ_{k}(f′_{k}) such that φ(ω_{(f′_{j},f′_{k})}^{ij}) = τ

and we claim that the rule (f′_{j}, f′_{k}) ↦ ω_{(f′_{j},f′_{k})}^{ij} yields an isomorphism of functors

ω^{ij} : ψ_{j} ∘ π_{jk}^{0}(f′_{j}, f′_{k}) →^{∼} ψ_{k} ∘ π_{jk}^{1}(f′_{j}, f′_{k}).

Indeed, let (X_{j} ⟵^{f′′_{j}} X′′ ⟶^{f′′_{k}} X_{k}) be another object of 𝒞 /X_{jk}; a morphism (f′′_{j}, f′′_{k}) → (f′_{j}, f′_{k}) is a morphism t : X′′ → X′ in 𝒞 such that f′_{j} ∘ t = f′′_{j} and f′_{k} ∘ t = f′′_{k}, and the assertion amounts to the identity :

A := ψ_{k}(t/X_{k}) ∘ ω_{(f′′_{j},f′′_{k})}^{jk} = B := ω_{(f′_{j},f′_{k})}^{jk} ∘ ψ_{j}(t/X_{j}).

However, we have :

\begin{matrix}h_{k} ∘ (φ ∘ ψ_{k}(f′_{k}/X_{k})) ∘ φ(A) = & h_{k} ∘ φ(ψ_{k}(f′′_{k}/X_{k}) ∘ ω_{(f′′_{j},f′′_{k})}^{jk}) \\ = & h_{j} ∘ φ(ψ_{j}(f′′_{j}/X_{j})) \\ = & h_{j} ∘ φ(ψ_{j}(f′_{j}/X_{j})) ∘ φ(ψ_{j}(t/X_{j})) \\ = & h_{k} ∘ (φ ∘ ψ_{k}(f′_{k}/X_{k})) ∘ φ(B) \end{matrix}

whence the contention, since h_{k} ∘ (φ ∘ ψ_{k}(f′_{k}/X_{k})) is cartesian and φ is faithful. Next, for every j, k, l ∈ I set 𝒞 /X_{jkl} := 𝒞 /X_{jk} ×_{𝒞 /X} 𝒞 /X_{l}; the objects of this category are the triples (f′_{j} : X′ → X_{j}, f′_{k} : X′ → X_{k}, f′_{l} : X′ → X_{l}) such that f_{j} ∘ f′_{j} = f_{k} ∘ f′_{k} = f_{l} ∘ f′_{l}. We have obvious projection functors

π_{jkl}^{0} : 𝒞 /X_{jkl} → 𝒞 /X_{kl}  π_{jkl}^{1} : 𝒞 /X_{jkl} → 𝒞 /X_{jl}  π_{jk}^{2} : 𝒞 /X_{jkl} → 𝒞 /X_{jk}.

With this notation, the uniqueness properties of ω^{ij} easily imply the cocycle identity :

(ω^{jk} ∗ π_{jkl}^{2}) ⊙ (ω^{kl} ∗ π_{jkl}^{0}) = ω^{jl} ∗ π_{jkl}^{1} for every j, k, l ∈ I

(details left to the reader). Let 𝒮 ⊂ 𝒞 /X be the sieve generated by the family (f_{j} | j ∈ I); in view of proposition 3.5.15, we deduce that there exist a cartesian functor ψ : 𝒮 → ℰ and a system of isomorphisms (η_{j} : ψ_{j} →^{∼} ψ ∗ f_{j∗} | j ∈ I) fulfilling the compatibility condition

(η_{k} ∗ π_{jk}^{1}) ⊙ ω^{jk} = η_{j} ∗ π_{jk}^{0} for every j, k ∈ I.

Since ℰ is stack, we may then find a cartesian section ψ′ ∈ ℰ (X) with an isomorphism ψ′_{|𝒮} →^{∼} ψ. Set Y := ψ′(1_{X}); to conclude the proof, it will suffice to exhibit an isomorphism Y′ →^{∼} φY in ℰ′. To this aim, pick as well a cartesian section ψ̃ ∈ ℰ′(X) such that ψ̃(1_{X}) = Y′; we set ψ̃_{j} := ψ̃ ∘ f_{j∗} : 𝒞 /X_{j} → ℰ′ for every j ∈ I. Notice that ψ̃_{j} ∘ π_{jk}^{0} = ψ̃_{k} ∘ π_{jk}^{1} for every j, k ∈ I.

Since h_{j} is cartesian, there exists a unique isomorphism h′_{j} : ψ̃(f_{j}) →^{∼} φY_{j} in F^{′−1}X_{j} such that h_{j} ∘ h′_{j} = ψ̃(f_{j}/X). There follows an isomorphism of cartesian functors

η̃_{j} : ψ̃_{j} →^{∼} φ ∘ ψ_{j}  such that η̃_{j,1_{X_{j}}} = h′_{j}  for every j ∈ I.

Explicitly, for every (f′ : X′ → X_{j}) ∈ Ob(𝒞 /X_{j}), the isomorphism η̃_{j,f′} : ψ̃_{j}(f′) →^{∼} φ∘ψ_{j}(f′) is characterized by the identity :

φ(ψ_{j}(f′/X_{j})) ∘ η̃_{j,f′} = h′_{j} ∘ ψ̃_{j}(f′/X_{j}) in ℰ′.

We claim that the following diagram commutes for every j, k ∈ I :

[figure: ψ̃_{j} ∘ π_{jk}^{0}; η̃_{j}∗π_{jk}^{0}; ψ̃_{k} ∘ π_{jk}^{1}; η̃_{k}∗π_{jk}^{1}; φ ∘ ψ_{j} ∘ π_{jk}^{0}; φ∗ω^{ij}; φ ∘ ψ_{k} ∘ π_{jk}^{1}.]

Indeed, let (f′_{j}, f′_{k}) be any object of 𝒞 /X_{jk}; we compute :

\begin{matrix}h_{k} ∘ φ(ψ_{k}(f′_{k}/X_{k})) ∘ (η̃_{k} ∗ π_{jk}^{1})_{(f′_{j},f′_{k})} = & h_{k} ∘ h′_{k} ∘ ψ̃_{k}(f′_{k}/X_{k}) \\ = & ψ̃(f_{k}/X) ∘ ψ̃(f′_{k}/X) \\ = & ψ̃(f_{k} ∘ f′_{k}/X) \\ = & ψ̃(f_{j} ∘ f′_{j}/X) \\ = & h_{j} ∘ h′_{j} ∘ ψ̃_{j}(f′_{j}/X_{j}) \\ = & h_{j} ∘ φ(ψ_{j}(f′_{j}/X_{j})) ∘ η̃_{j,f′} \\ = & h_{k} ∘ φ(ψ_{k}(f′_{k}/X_{k})) ∘ φ(ω_{(f′_{j},f′_{k})}^{ij}) ∘ η̃_{j,f′} \end{matrix}

whence the claim, as usual, since h_{k} ∘φ(ψ_{k}(f′_{k}/X_{k})) is cartesian. In view of proposition 3.5.15, we deduce that there exists a unique isomorphism of cartesian functors

ψ̃_{|𝒮} →^{∼} φ ∘ ψ = (φ ∘ ψ′)_{|𝒮} .

But since ℰ′ is 1-separated, the latter comes from a unique isomorphism ψ̃ →^{∼} φ ∘ ψ′, whence the sought isomorphism Y′ →^{∼} φY. □

Proposition 5.2.11. Let (𝒞 , J) be a small site, F : ℰ → 𝒞 a fibration, and j ∈ {0, 1, 2}. Then: (i) The unit of adjunction η_{ℰ} : ℰ → ℰ^{a} is i-covering for i = 0, 1, 2. (ii) ℰ is j-separated if and only if η_{ℰ} is j-faithful. (iii) A cartesian functor φ : ℰ → ℰ′ of 𝒞 -fibrations is j-covering if and only if the same

holds for the induced functor φ^{a} : ℰ^{a} → ℰ^{′a}.

Proof. (i): Quite generally, let G : 𝒜 → ℬ be a 𝒞 -cartesian functor of fibrations, and H : 𝒜′ →^{∼} 𝒜 a 𝒞 -equivalence of categories; then it is easily seen that G is i-covering for some i ∈ {0, 1, 2} if and only if the same holds for G ∘ H. Taking into account lemma 5.2.4(i), we are therefore reduced to checking :

Claim 5.2.12. The functor j_{ℰ} : C(ℰ) → ℰ^{+} is i-covering for i = 0, 1, 2.

Proof of the claim. Let [X, ψ : 𝒮 → ℰ] be any object of ℰ^{+}, and pick a generating family (f_{j} : X_{j} → X | j ∈ I) for 𝒮; then (X_{j}, ψ ∘ f_{j} : 𝒞 /X_{j} → ℰ) is an object of C(ℰ) and f_{j} induces a cartesian morphism j_{ℰ} (X_{j}, ψ ∘ f_{j∗}) → [X, ψ] for every j ∈ I. Thus, j_{ℰ} is 0-covering.

Next, let [t, σ] : [X′, ψ′ : 𝒞 /X′ → ℰ] → [X, ψ : 𝒞 /X → ℰ] be a morphism in ℰ^{+}. By definition, t : X′ → X is a morphism of 𝒞, and σ : ψ′_{|𝒮} ⇒ (ψ ∘ t_{∗})_{|𝒮} is a natural 𝒞 - transformation defined on a covering sieve 𝒮 ⊂ 𝒞 /X′. Again, we pick a generating family (f_{j} : X′_{i} → X′ | j ∈ I) for 𝒮, and denote by (f_{j}, h_{j}) : (X′_{j}, ψ′ ∘ f_{j∗}) → (X′, ψ′) the cartesian morphism in C(ℰ) induced by f_{j}. Then σ ∗ f_{j∗} : ψ′ ∘ f_{j∗} ⇒ ψ ∘ (t ∘ f_{j})_{∗} defines a morphism (t ∘ f_{j}, g_{j}) : (X′_{j}, ψ′ ∘ f_{j∗}) → (X, ψ) in C(ℰ) such that j_{ℰ} (t ∘ f_{j}, g_{j}) = [t, σ] ∘ j_{ℰ} (f_{j}, h_{j}) for every j ∈ I. This proves that j_{ℰ} is 1-covering.

Lastly, let (t_{1}, σ_{1}), (t_{2}, σ_{2}) : (X′, ψ′ : 𝒞 /X′ → ℰ) → (X, ψ : 𝒞 /X → ℰ) be two morphisms in C(ℰ) such that j_{ℰ} (t_{1}, σ_{1}) = j_{ℰ} (t_{2}, σ_{2}). Especially, this means that t := t_{1} = t_{2}, and there exists a covering sieve 𝒮 ⊂ 𝒞 /X′ such that (σ_{1})_{|𝒮} = (σ_{2})_{|𝒮} . We pick a generating family (f_{j} : X′_{j} → X′ | j ∈ I) for 𝒮, and let (f_{j}, h_{j}) : (X′_{j}, ψ′ ∘ f_{j∗}) → (X′, ψ′) be the cartesian morphism induced by f_{j}; then (t, σ_{1}) ∘ (f_{j}, h_{j}) = (t, σ_{2}) ∘ (f_{j}, h_{j}) for every j ∈ I, so j_{ℰ} is 2-covering. ♢

(ii): From (i) and proposition 5.2.9 we see already that if ℰ is j-separated, then η_{ℰ} is jfaithful. Suppose then that η_{ℰ} is j-faithful; let X ∈ Ob(𝒞) be any object, and 𝒮 ⊂ 𝒞 /X any covering sieve; we consider the induced commutative diagram of categories :

\xymatrix{ℰ (X) \ar[r] \ar[d]_{Cart_{𝒞} (𝒞 /X,η_{ℰ} )} & Cart_{𝒞} (𝒮 , ℰ) \ar[d]^{Cart_{𝒞} (𝒮 ,η_{ℰ} )} \\ ℰ^{a}(X) \ar[r] & Cart_{𝒞} (𝒮 , ℰ^{a})}

whose bottom horizontal arrow is an equivalence, since ℰ^{a} is a stack, and whose vertical arrows are j-faithful, by virtue of corollary 3.1.28(ii.a). It follows that the top horizontal arrow is jfaithful as well, which means that ℰ is j-separated.

(iii): We have an essentially commutative diagram of cartesian functors

\xymatrix{ℰ \ar[r]^{φ} \ar[d] & ℰ′ \ar[d] \\ ℰ^{a} \ar[r]^{φ^{a}} & ℰ^{′a}}

whose vertical arrows are i-covering for i = 0, 1, 2, by (i). Then the assertion follows immediately from lemma 5.2.4(i,ii). □

5.3. Local calculus of fractions. Let C := (𝒞 , J) be a site, 𝒜 → 𝒞 a fibration; choose a unital cleavage λ for 𝒜, and let c : 𝒞^{o} → Cat be the associated unital pseudo-functor. To ease notation, for every morphism f : X′ → X in 𝒞 we let f^{∗} := c_{f} : 𝒜_{X} → 𝒜_{X′}, and denote by γ_{•,•}^{c} the coherence constraint of c. Consider a system of sets Σ_{•} := (Σ_{X} | X ∈ Ob(𝒞)) with :

• Σ_{X} ⊂ Morph(𝒜_{X}) for every X ∈ Ob(𝒞)

• f^{∗}(Σ_{X}) ⊂ Σ_{X′} for every morphism f : X′ → X of 𝒞.

Proposition 5.3.1. In the situation of (5.3), there exist a 1-separated prestack 𝒜 {Σ_{•}^{−1}} over C, and a cartesian functor

L_{{Σ}} : 𝒜 → 𝒜 {Σ_{•}^{−1}}

with the following properties :

(i) L_{{Σ}}f is an isomorphism in 𝒜 {Σ_{•}^{−1}}, for every f ∈ Σ := ⋃_{f∈Ob(𝒞)} Σ_{X}.

(ii) For every 1-separated prestack ℰ on C, and every cartesian functor F : 𝒜 → ℰ such that F f is invertible in ℰ for every f ∈ Σ, there exists a unique cartesian functor F′ : 𝒜 {Σ_{•}^{−1}} → ℬ such that F = F′ ∘ L_{{Σ}}.

Proof. According to theorem 1.6.9, for every morphism f : X′ → X of 𝒞 the functor f^{∗} extends uniquely to a functor f_{Σ}^{∗} : 𝒜_{X}[Σ_{X}^{−1}] → 𝒜_{X′}[Σ_{X′}^{−1}], and γ_{f,g}^{c} : f^{∗} ∘ g^{∗} →^{∼} (g ∘ f)^{∗} extends uniquely to an isomorphism of functors γ_{f,g}^{Σ} : f_{Σ}^{∗} ∘ g_{Σ}^{∗} →^{∼} (g ∘ f)_{Σ}^{∗} for every composable pair (f, g) of morphisms of 𝒞 (corollary 1.6.11). It is then easily seen that the rules : X ↦ 𝒜_{X}[Σ_{X}^{−1}] for every X ∈ Ob(𝒞) and f ↦ f_{Σ}^{∗} for every morphism f of 𝒞 yield a unital pseudo-functor c[Σ_{•}^{−1}] : 𝒞^{o} → Cat with coherence constraint γ_{••}^{Σ}, and we set

𝒜 [Σ_{•}^{−1}] := ℱ ib(c[Σ_{•}^{−1}]).

From theorem 1.6.9 it follows easily that the system of localization functors 𝒜_{X} → 𝒜_{X}[Σ_{X}^{−1}] yields a unique cartesian functor L′ : 𝒜 → 𝒜 [Σ_{•}^{−1}] such that the following holds. For every fibration ℱ over 𝒞, and every cartesian functor F : 𝒜 → ℱ such that F f is invertible in ℱ for every f ∈ Σ, there exists a unique cartesian functor F′ : 𝒜 [Σ_{•}^{−1}] → ℱ such that F = F′ ∘ L′. By theorem 5.1.10 we conclude that the composition of L′ with the natural cartesian functor 𝒜 [Σ_{•}^{−1}] → 𝒜 {Σ_{•}^{−1}} := 𝒜 [Σ_{•}^{−1}]^{sep} fulfills the stated conditions. □

Definition 5.3.2. In the situation of (5.3), we shall say that the system Σ_{•} admits a right local calculus of fractions if the following conditions hold for every X ∈ Ob(𝒞) : (LCF1) The set Σ_{X} contains the isomorphisms of 𝒜_{X}. (LCF2) The set Σ_{X} fulfills condition (CF2) of definition 1.6.14. (LCF3) For every morphism f : A → B in 𝒜_{X} and every s : C → B in Σ_{X}, there exist a

covering family (φ_{λ} : X_{λ} → X | λ ∈ Λ) in the site C, and for every λ ∈ Λ a morphism g_{λ} : D_{λ} → φ_{λ}^{∗}C in 𝒜_{X_{λ}} and t_{λ} : D_{λ} → φ_{λ}^{∗}A in Σ_{X_{λ}} such that φ_{λ}^{∗}(f) ∘ t_{λ} = φ_{λ}^{∗}(s) ∘ g_{λ}.

(LCF4) If f, g : A → B are any two morphisms in 𝒜_{X} such that s ∘ f = s ∘ g for some s : B → C in Σ_{X}, then there exist a covering family (φ_{λ} : X_{λ} → X | λ ∈ Λ) in the site C and for every λ ∈ Λ an element t_{λ} : D_{λ} → φ_{λ}^{∗}A in Σ_{X_{λ}} such that φ^{∗}(f) ∘ t_{λ} = φ_{λ}^{∗}(g) ∘ t_{λ}.

5.3.3. In the situation of (5.3), for every X ∈ Ob(𝒞) and A ∈ Ob(𝒜_{X}) let Σ_{X/A} be the full subcategory of 𝒜_{X}/A whose objects are the elements of Σ_{X} with target equal to A. For every

morphism (Y′ ⟶^{φ′} X) ⟶^{ψ/X} (Y ⟶^{φ} X) of 𝒞 /X notice that ψ^{∗} : 𝒜_{Y} → 𝒜_{Y′} induces a functor

(ψ/X)^{∗} : 𝒜_{Y} /φ^{∗}A → 𝒜_{Y′}/φ^{′∗}A (I ⟶^{f} φ^{∗}A) ↦ (ψ^{∗}I ⟶^{γ_{(ψ,φ),A}^{c}∘ψ^{∗}(f)} φ^{′∗}A)

Suppose now that Σ_{•} fulfills condition (LCF1) of definition 5.3.2; then (ψ/X)^{∗} restricts to

(ψ/X)^{∗} : Σ_{Y/φ^{∗}A} → Σ_{Y′/φ^{′∗}A}.

Moreover, if (ψ′/X) : (Y′′ ⟶^{φ′′} X) → (Y′ ⟶^{φ′} X) is another morphism of 𝒞 /X, we have an isomorphism in Σ_{Y′′/φ^{′′∗}A} for every (s : I → φ^{∗}A) ∈ Ob(Σ_{Y/φ^{∗}A})

(γ_{(ψ′,ψ),I}^{c}/X) : (ψ′/X)^{∗} ∘ (ψ/X)^{∗}(s) →^{∼} (ψ ∘ ψ′/X)^{∗}(s).

Clearly the rule s ↦ (γ_{(ψ′,ψ),I}^{c}/X) yields an isomorphism of functors γ_{ψ′,ψ}^{Σ} : (ψ′/X)^{∗} ∘ (ψ/X)^{∗} →^{∼} (ψ ∘ ψ′/X)^{∗}, and then it easily seen that the rules : φ ↦ Σ_{Y/φ^{∗}A} for every (φ : Y → X) ∈ Ob(𝒞 /X) and ψ/X ↦ (ψ/X)^{∗} for every morphism ψ/X of 𝒞 /X define a unital pseudo-functor Σ_{•/A} : (𝒞 /X)^{o} → Cat, whose coherence constraint is given by the system of isomorphisms γ_{ψ′,ψ}^{Σ}. There follows a fibration

S_{A} := ℱ ib(Σ_{•/A}) → 𝒞 /X

and we denote by λ_{A} its natural unital cleavage. Next, for A, B ∈ Ob(𝒜_{X}) consider the functor

ℋ_{A,B} : S_{A}^{o} → Set (Y ⟶^{φ} X, I ⟶^{s} φ^{∗}A) ↦ {(s, f) | f ∈ Hom_{𝒜_{Y}} (I, φ^{∗}B)}

which assigns to every morphism (ψ/X, h/φ^{∗}A) : (φ, s) → (φ′, s′) of S_{A} the map

ℋ_{A,B}(ψ/X, h/φ^{∗}A) : ℋ_{A,B}(φ′, s′) → ℋ_{A,B}(φ, s) (s′, f) ↦ (s, ((ψ/X)^{∗}f) ∘ h).

Lemma 5.3.4. With the notation of (5.3.3), suppose that the system Σ_{•} admits a right local calculus of fraction. Then the fibration S_{A} is locally cofiltered over the site C/X.

Proof. Condition (LCF1) implies immediately condition (a) of definition 4.3.3. Next, let (φ : Y → X) ∈ Ob(𝒞 /X), and (φ, s : I → φ^{∗}A) and (φ, s′ : I′ → φ^{∗}A) be two objects of Σ_{Y/φ^{∗}A}. By (LCF3) we may find a covering (ψ_{λ} : Y_{λ} → Y | λ ∈ Λ) for the topology J, and for every λ ∈ Λ a morphism f_{λ} : J → ψ_{λ}^{∗}I in 𝒜_{Y_{λ}} and an element t_{λ} : J → ψ_{λ}^{∗}I′ of Σ_{Y_{λ}} such that s′′_{λ} := ψ_{λ}^{∗}(s) ∘ f_{λ} = ψ_{λ}^{∗}(s′) ∘ t_{λ}. By (LCF1) and (LCF2), we have u_{λ} := γ_{(φ,ψ_{λ}),A} ∘ s′′_{λ} ∈ Σ_{Y_{λ}/(φ∘ψ_{λ})^{∗}A} for every λ ∈ Λ, and it follows that t_{λ} and f_{λ} yield morphisms u_{λ} → (ψ/X)^{∗}(s′) and respectively u_{λ} → (ψ/X)^{∗}(s). Hence S_{A} fulfills condition (b) of definition 4.3.3.

Lastly, consider two morphisms (g/φ^{∗}A), (g′/φ^{∗}) : (s : I → φ^{∗}A) → (s′ : I → φ^{∗}A) in Σ_{Y/φ^{∗}A}. In other words, g, g′ : I → I′ are morphisms in 𝒜_{Y} with s′∘g = s′∘g′ = s. By (LCF3) there exist then a covering family (ψ_{λ} : Y_{λ} → Y | λ ∈ Λ) and for every λ ∈ Λ an element h_{λ} : J → ψ_{λ}^{∗}I of Σ_{𝒜_{Y_{λ}}} with t_{λ} := ψ_{λ}^{∗}(g)∘h_{λ} = ψ_{λ}^{∗}(g′)∘h_{λ}. Notice that u_{λ} := ψ_{λ}^{∗}(s′)∘t_{λ} = ψ_{λ}^{∗}(s)∘h_{λ} lies in Σ_{Y_{λ},(ψ_{λ})^{∗}φ^{∗}A}. Hence h_{λ} yields a morphism (h/(φ ∘ ψ_{λ})^{∗}A) : γ_{(φ,ψ_{λ}),A} ∘ u_{λ} → (ψ/X)^{∗}(s) in Σ_{Y_{λ},(φ∘ψ_{λ})^{∗}A} with (ψ_{λ}/X)^{∗}(g) ∘ (h/(φ ∘ ψ_{λ})^{∗}A) = (ψ_{λ}/X)^{∗}(g′) ∘ (h/(φ ∘ ψ_{λ})^{∗}A). This shows that condition (c) holds as well. □

5.3.5. With the notation of (4.3.4), consider the presheaf on 𝒞 /X and the sheaf on C/X :

M_{A,B} := ∫ ^{λ^{A}} ℋ_{A,B} and H_{A,B} := ∫_{a}^{λ_{A}} ℋ_{A,B}.

Recall that for every (φ : Y → X) ∈ Ob(𝒞 /X), the set M_{AB}(φ) consists of the equivalence classes of [s, f] of pairs with s ∈ Ob(Σ_{Y/φ^{∗}A}) and (s, f) ∈ ℋ_{AB}(φ, s). We shall denote by {s, f} ∈ H_{A,B}(φ) the image of such a class [s, f]. Taking into account lemmata 4.3.5and 5.3.4 we see that if Σ_{•} admits a right local calculus of fraction, the following holds :

(a) For every σ ∈ H_{AB}(φ) we have a covering family (φ_{λ} : Y_{λ} → Y | λ ∈ Λ) in the site C, and for every λ ∈ Λ a morphism g_{λ} : C_{λ} → φ_{λ}^{∗}B in 𝒜_{Y_{λ}} and an element s_{λ} : C_{λ} → φ_{λ}^{∗}A of Σ_{Y_{λ}} such that H_{AB}(φ_{λ}/X)(σ) is the image of [s_{λ}, g_{λ}] in H_{AB}(φ ∘ φ_{λ}/X).

(b) For every two pairs (s : I → φ^{∗}A, f), (s′ : I′ → φ^{∗}A, f′) ∈ ℋ_{AB}(φ, s) with [s, f] = [s′, f′] in M_{AB}(φ, s) there exist a covering family (φ_{λ} : Y_{λ} → Y | λ ∈ Λ) in the site C, and for every λ ∈ Λ two morphisms in 𝒜_{Y_{λ}}

φ_{λ}^{∗}I′ ⟵^{t′_{λ}} J_{λ} ⟶^{t_{λ}} φ_{λ}^{∗}I with (φ_{λ}/X)^{∗}(f) ∘ t_{λ} = (φ_{λ}/X)^{∗}(f′) ∘ t′_{λ}

and such that (φ_{λ}/X)^{∗}(s) ∘ t_{λ} = (φ_{λ}/X)^{∗}(s′) ∘ t′_{λ} lies in Σ_{Y_{λ},(φ∘φ_{λ})^{∗}A}. Let λ_{Σ} and λ_{{Σ}} be the cleavages of 𝒜 [Σ_{•}^{−1}] and respectively 𝒜 {Σ_{•}^{−1}} deduced from the cleavage λ of 𝒜, as in the proofs of proposition 5.3.1 and theorem 5.1.10. For every A ∈ Ob(𝒜 {Σ_{•}^{−1}}_{X}) = Ob(𝒜_{X}[Σ_{•}^{−1}]) = Ob(𝒜_{X}) define the cartesian sections β_{A}^{λ_{Σ}} : 𝒞 /X → 𝒜 [Σ_{•}^{−1}] and β_{X,A}^{λ_{{Σ}}} : 𝒞 /X → 𝒜 {Σ_{•}^{−1}} as in the proof of claim 3.2.8. Let also L_{Σ} : 𝒜 → 𝒜 [Σ_{•}^{−1}] be the localization functor; we have a natural morphism of presheaves on 𝒞 /X :

M_{A,B} → 𝒞art(β_{A}^{λ_{Σ}}, β_{B}^{λ_{Σ}}) for every A, B ∈ Ob(𝒜_{X})

which assigns to every [s, f] ∈ M_{A,B}(φ) the unique natural 𝒞 -transformation β_{A}^{λ_{Σ}} ∘ φ_{∗} → β_{B}^{λ_{Σ}} ∘ φ_{∗} such that 1_{Y} ↦ L_{Σ}(f) ∘ L_{Σ}(s)^{−1}, for every φ ∈ Ob(𝒞 /X). We deduce a morphism H_{A,B} → 𝒞art(β_{A}^{λ_{Σ}}, β_{B}^{λ_{Σ}})^{a} of associated sheaves on C/X. But recall that the proof of theorem 5.1.10yields as well a natural identification

𝒞art(β_{A}^{λ_{Σ}}, β_{B}^{λ_{Σ}})^{a} →^{∼} 𝒞art(β_{A}^{λ_{{Σ}}}, β_{B}^{λ_{{Σ}}}).

The composition of the latter two morphisms is a morphism of sheaves on C/X :

(5.3.6)  H_{AB} → 𝒞art(β_{X,A}^{λ_{{Σ}}}, β_{X,B}^{λ_{{Σ}}})  {s, f} ↦ L_{{Σ}}(f) ∘ L_{{Σ}}(s)^{−1}.

Proposition 5.3.7. Suppose that the system Σ_{•} admits a right local calculus of fraction. Then (5.3.6) is an isomorphism of sheaves for every A, B ∈ Ob(𝒜_{X}).

Proof. We define for every X ∈ Ob(𝒞) a category

ℒ_{X}

whose set of objects is Ob(𝒜_{X}), and such that Hom_{ℒ_{X}} (A, B) := H_{AB}(1_{X}) for every A, B ∈ Ob(𝒜_{X}). For every A, B, C ∈ Ob(𝒜_{X}), the composition law

(5.3.8)  H_{AB}(1_{X}) × H_{BC}(1_{X}) → H_{AC}(1_{X})

is obtained as follows. Let (φ : Y → X) ∈ Ob(𝒞 /X) and [s : I → φ^{∗}A, f] ∈ M_{AB}(φ) and [s′ : I′ → φ^{∗}B, f′] ∈ M_{BC}(φ) any two sections; by virtue of (LCF3) we may find a covering family (φ_{λ} : Y_{λ} → Y | λ ∈ Λ) and for every λ ∈ Λ, morphisms in 𝒜_{Y_{λ}} :

φ_{λ}^{∗}I ⟵^{s_{λ}} J_{λ} ⟶^{f_{λ}} φ_{λ}^{∗}I′ such that (φ_{λ}/X)^{∗}(f) ∘ s_{λ} = (φ_{λ}/X)^{∗}(s′) ∘ f_{λ} and s_{λ} ∈ Σ_{Y_{λ}}.

Set t_{λ} := (φ_{λ}/X)^{∗}(s)∘s_{λ} : J_{λ} → (φ∘φ_{λ})^{∗}A and g_{λ} := (φ_{λ}/X)^{∗}(f′)∘f_{λ} : J_{λ} → (φ∘φ_{λ})^{∗}B for every λ. For λ, μ ∈ Λ, consider morphisms Y_{λ} ⟵^{ψ_{λ}} Z ⟶^{ψ_{μ}} Y_{μ} such that ψ := φ_{λ} ∘ ψ_{λ} = φ_{μ} ∘ ψ_{μ}.

Claim 5.3.9. H_{AB}(ψ_{λ}/X){t_{λ}, g_{λ}} = H_{AB}(ψ_{μ}/X){t_{μ}, g_{μ}}.

Proof of the claim. We need to show that {(ψ_{λ}/X)^{∗}t_{λ}, (ψ_{λ}/X)^{∗}g_{λ}} = {(ψ_{μ}/X)^{∗}t_{μ}, (ψ_{μ}/X)^{∗}g_{μ}}. However, by (LCF3) we may find a covering family (ρ_{ν} : Z_{ν} → Z | ν ∈ Λ′) and for every

ν ∈ Λ′, morphisms ρ_{ν}^{∗}ψ_{μ}^{∗}J_{μ} ⟵^{s_{ν}} J_{ν} ⟶^{s′_{ν}} ρ_{ν}^{∗}ψ_{λ}^{∗}J_{λ} in 𝒜_{Z_{ν}} with s_{ν} ∈ Σ_{Z_{ν}}, and such that

(ρ_{ν}/Y)^{∗}(ψ_{μ}/Y)^{∗}(s_{μ}) ∘ s_{ν} = (ρ_{ν}/Y)^{∗}(ψ_{λ}/Y)^{∗}(s_{λ}) ∘ s′_{ν}.

We compute :

(ρ_{ν}/Y)^{∗}((ψ/X)^{∗}(s′) ∘ (ψ_{μ}/Y)^{∗}(f_{μ})) ∘ s_{ν} = (ρ_{ν}/Y)^{∗}((ψ_{μ}/X)^{∗}((φ_{μ}/X)^{∗}(s′) ∘ f_{μ})) ∘ s_{ν}
= (ρ_{ν}/Y)^{∗}((ψ_{μ}/X)^{∗}((φ_{μ}/X)^{∗}(f) ∘ s_{λ})) ∘ s_{ν}
= (ρ_{ν}/Y)^{∗}((ψ/X)^{∗}(f) ∘ (ψ_{μ}/Y)^{∗}(s_{μ})) ∘ s_{ν}
= (ρ_{ν}/Y)^{∗}((ψ/X)^{∗}(f) ∘ (ψ_{λ}/Y)^{∗}(s_{λ})) ∘ s′_{ν}
= (ρ_{ν}/Y)^{∗}((ψ/X)^{∗}(s′) ∘ (ψ_{λ}/Y)^{∗}(f_{λ})) ∘ s′_{ν}.

By (LCF4) it follows that for every ν ∈ Λ′ there exist a covering family (ρ_{νν′} : Z_{νν′} → Z_{ν} | ν′ ∈ Λ′_{ν}) and for every ν′ ∈ Λ′_{ν} an element t_{νν′} : J_{νν′} → ρ_{νν}^{∗}J_{ν} of Σ_{Z_{νν′}} such that

(ρ_{νν′}/Y)^{∗}(ρ_{ν}/Y)^{∗}((ψ_{μ}/Y)^{∗}(f_{μ}) ∘ s_{ν}) ∘ t_{νν′} = (ρ_{νν′}/Y)^{∗}(ρ_{ν}/Y)^{∗}((ψ_{λ}/Y)^{∗}(f_{λ}) ∘ s′_{ν}) ∘ t_{νν′}.

Set ρ′_{νν′} := ρ_{ν} ∘ ρ_{νν′} for every ν′ ∈ Λ′_{ν}, and

T_{λνν′} := (ψ_{λ} ∘ ρ′_{νν′}/X)^{∗}t_{λ}  G_{λνν′} := (ψ_{λ} ∘ ρ′_{νν′}/X)^{∗}g_{λ}  γ_{λνν′} := γ_{(ρ′_{νν′},ψ_{λ}),J_{λ}}^{c}

and define likewise T_{μνν′}, G_{μνν′} and γ_{μνν′}. We compute :

[T_{λνν′}, G_{λνν′}] = [T_{λνν′} ∘ γ_{λνν′} ∘ (ρ_{νν′}/Z)^{∗}(s′_{ν}) ∘ t_{νν′}, G_{λνν′} ∘ γ_{λνν′} ∘ (ρ′_{νν′}/Z)^{∗}(s′_{ν}) ∘ t_{νν′}]

= [T_{μνν′} ∘ γ_{μνν′} ∘ (ρ_{νν′}/Z)^{∗}(s_{ν}) ∘ t_{νν′}, G_{μνν′} ∘ γ_{μνν′} ∘ (ρ_{νν′}/Z)^{∗}(s_{ν}) ∘ t_{νν′}]

= [T_{μνν′}, G_{μνν′}]

whence the contention. ♢

Claim 5.3.9 implies that the system ({t_{λ}, g_{λ}} | λ ∈ Λ) defines a unique section of H_{AC}(φ) that we denote {s′, f′} ∘ {s, f}. Notice that the argument proves as well that the construction of {s′, f′} ∘ {s, f} is independent of all auxiliary choices. A direct inspection shows then that we obtain a well defined morphism of presheaves

M_{AB} × M_{BC} → H_{AC} ([s, f], [s′, f′]) ↦ {s′, f′} ∘ {s, f}

which therefore induces a unique morphism of sheaves on C/X :

H_{AB} × H_{BC} → H_{AC} such that ({s, f}, {s′, f′}) ↦ {s′, f′} ∘ {s, f}.

Then the sought map (5.3.8) is the map of sets obtained by evaluating the foregoing morphism on the object 1_{X} ∈ Ob(𝒞 /X). Let us check that {1_{φ^{∗}B}, 1_{φ^{∗}B}} ∘ σ = σ = σ ∘ {1_{φ^{∗}A}, 1_{φ^{∗}A}} for every A, B ∈ Ob(𝒜_{X}) and every φ ∈ Ob(𝒞 /X) : indeed, it suffices to check these identities for σ = {s, f}, for any (s, f) ∈ ℋ_{A,B}(φ), and the latter follow by simple inspection. Lastly, in order to show the associativity property for our composition law, consider A, B, C, D ∈ Ob(𝒜_{X}) and (s : I → φ^{∗}A, f), (s′ : I′ → φ^{∗}B, f′), (s′′ : I′′ → φ^{∗}C, f′′) ∈ ℋ_{AB}(φ), for some (φ : Y → X) ∈ Ob(𝒞 /X). By (LCF3) we may find a covering family (φ_{λ} : Y_{λ} → Y | λ ∈ Λ) and for every λ ∈ Λ, morphisms in 𝒜_{Y_{λ}} :

φ_{λ}^{∗}I ⟵^{t_{λ}} J_{λ} ⟶^{g_{λ}} φ_{λ}^{∗}I′ ⟵^{t′_{λ}} J′_{λ} ⟶^{g′_{λ}} φ_{λ}^{∗}I′′ with t_{λ}, t′_{λ} ∈ Σ_{Y_{λ}}

such that (φ_{λ}/X)^{∗}(f) ∘ t_{λ} = (φ_{λ}/X)^{∗}s′ ∘ g_{λ} and (φ_{λ}/X)^{∗}(f′) ∘ t′_{λ} = (φ_{λ}/X)^{∗}(s′′) ∘ g′_{λ}. Then we may also find for every λ ∈ Λ a covering family (φ_{λμ} : Y_{λμ} → Y_{λ} | μ ∈ Λ_{λ}) and for every μ ∈ Λ_{λ}, morphisms in 𝒜_{Y_{λμ}} :

φ_{λμ}^{∗}J_{λ} ⟵^{t_{λμ}} J_{λμ} ⟶^{g_{λμ}} φ_{λμ}^{∗}J′_{λ} such that (φ_{λμ}/Y)^{∗}(g_{λ}) ∘ t_{λμ} = (φ_{λμ}/Y)^{∗}(t_{λ}) ∘ g_{λμ}

and with t_{λμ} ∈ Σ_{Y_{λμ}}. For every such λ and μ set ψ_{λμ} := φ_{λ} ∘ φ_{λμ} and

(t′_{λμ}, g′_{λμ}) := ((ψ_{λμ}/X)^{∗}(s) ∘ (φ_{λμ}/Y)^{∗}(t_{λ}) ∘ t_{λμ}, (ψ_{λμ}/X)^{∗}(f′′) ∘ (φ_{λμ}/Y)^{∗}(g_{λ}) ∘ g_{λμ}).

It then follows easily that the system ({t′_{λμ}, g′_{λμ}} | λ ∈ Λ, μ ∈ Λ_{λ}) represents both {s′′, f′′} ∘ ({s′, f′} ∘ {s, f}) and ({s′′, f′′} ∘ {s′, f′}) ∘ {s, f}. The sought associativity property is a straightforward consequence. This concludes the construction of ℒ_{X}.

Next, notice that for every morphism φ : Y → X in 𝒞 and every A, B ∈ Ob(𝒜_{X}) we have a natural isomorphism of presheaves on 𝒞 /Y :

M_{φ^{∗}A,φ^{∗}B} →^{∼} (φ_{∗})^{∧}M_{A,B}

Namely, for every (ψ : Z → Y) ∈ Ob(𝒞 /Y), the isomorphism is given by the map :

M_{φ^{∗}A,φ^{∗}B}(ψ) →^{∼} M_{A,B}(φ ∘ ψ) [s, f] ↦ [γ_{(ψ,φ),A}^{c} ∘ s, γ_{(ψ,φ),B}^{c} ∘ f]

Since φ_{∗} is continuous and cocontinuous for the sites C/X and C/Y (remark 4.7.3(i)), combining with lemma 4.2.15(ii) there follows a natural isomorphism of sheaves on C/Y :

(5.3.10)  H_{φ^{∗}A,φ^{∗}B} →^{∼} j_{φ}^{∗}H_{A,B}.

We deduce a map

d_{φ,(A,B)} : Hom_{ℒ_{X}} (A, B) → H_{A,B}(φ) →^{∼} Hom_{ℒ_{Y}} (φ^{∗}A, φ^{∗}B)

namely the composition of the restriction map H_{A,B}(φ/X) : H_{A,B}(1_{X}) → H_{A,B}(φ) with the evaluation at 1_{Y} ∈ Ob(𝒞 /Y) of the inverse of the foregoing isomorphism of sheaves. A simple inspection shows that the rules : A ↦ φ^{∗}A and σ ↦ d_{φ,(A,B)}(σ) for every A, B ∈ 𝒜_{X} and every σ ∈ Hom_{ℒ_{X}} (A, B) yield a well defined functor

d_{φ} : ℒ_{X} → ℒ_{Y} .

Furthermore, for every pair Z ⟶^{ψ} Y ⟶^{φ} X of morphisms of 𝒞, the natural isomorphism γ_{ψ,φ}^{c} : ψ^{∗} ∘ φ^{∗} →^{∼} (φ ∘ ψ)^{∗} induces a natural isomorphism

γ_{ψ,φ}^{d} : d_{ψ} ∘ d_{φ} →^{∼} d_{φ∘ψ} A ↦ {1_{ψ^{∗}φ^{∗}A}, γ_{(ψ,φ),A}^{c}}.

Clearly the rules : X ↦ ℒ_{X} and φ ↦ d_{φ} for every X ∈ Ob(𝒞) and every morphism φ of 𝒞 define a unital pseudo-functor d : 𝒞^{o} → Cat, with coherence constraint γ_{•,•}^{d}. We set :

ℒ := ℱ ib(d)

and we denote by λ_{ℒ} the natural cleavage of ℒ. Then, for every X ∈ Ob(𝒞) and every A ∈ Ob(ℒ_{X}) let β_{X,A}^{λ_{ℒ}} : 𝒞 /X → ℒ be the cartesian section defined as in the proof of claim 3.2.8; by inspecting the constructions, we see that for every A, B ∈ Ob(ℒ_{X}), the system of isomorphisms (5.3.10) induces a natural isomorphism of presheaves on 𝒞 /X :

H_{AB} →^{∼} 𝒞art(β_{X,A}^{λ_{ℒ}}, β_{X,B}^{λ_{ℒ}}).

Especially, 𝒞art(σ, σ′) is a sheaf on C/X for every pair of cartesian sections σ, σ′ ∈ ℒ (X), and therefore ℒ is a 1-separated prestack (lemma 5.1.9). We have a natural functor

F_{X} : 𝒜_{X} → ℒ_{X} for every X ∈ Ob(𝒞)

that is the identity on objects and is given by the rule : (f : A → B) ↦ {1_{A}, f} on morphisms f of 𝒜_{X}. Clearly the rule X ↦ F_{X} defines a strict pseudo-natural transformation F_{•} : c ⇒ d, whence a cartesian functor

L := ℱ ib(F_{•}) : 𝒜 → ℒ .

By construction, L(s) is an isomorphism in ℒ, for every s ∈ ⋃_{X∈Ob(𝒞)} Σ_{X}, hence L factors uniquely through a cartesian functor L′ : 𝒜 {Σ_{•}^{−1}} → ℒ and the universal cartesian functor L_{{Σ}} of proposition 5.3.1. To construct conversely a functor ℒ → 𝒜 {Σ_{•}^{−1}}, we begin by evaluating at 1_{X} the morphism of sheaves (5.3.6), to get a map

L′′_{X,A,B} : Hom_{ℒ_{X}} (A, B) → Hom_{𝒜 {Σ_{•}^{−1}}_{X}} (A, B).

Then, a direct inspection of the constructions yields a functor L′′_{X} : ℒ_{X} → 𝒜 {Σ_{•}^{−1}}_{X} that is the identity on objects, and is given on morphisms by the rule : σ ↦ L′′_{X,A,B}(σ) for every A, B ∈ Ob(ℒ_{X}), and every σ ∈ Hom_{ℒ_{X}} (A, B) : the details shall be left to the reader.

Lastly, if c_{{Σ}} denotes the pseudo-functor associated with the cleavage λ_{{Σ}}, we easily see that the rule : X ↦ ℒ′′_{X} for every X ∈ Ob(𝒞) defines a strict pseudo-natural 𝒞 -transformation L′′_{•} : d ⇒ c_{{Σ}}, whence a cartesian functor

L′′ := ℱ ib(L′′_{•}) : ℒ → 𝒜 {Σ_{•}^{−1}}.

A direct inspection shows that L′∘L′′ = 1_{ℒ} . In order to show that L′′∘L′ = 1_{𝒜 {Σ_{•}^{−1}}}, it suffices to check that L′′ ∘ L′ ∘ L_{{Σ}} = L_{{Σ}}, due to the universal property of L_{{Σ}}. Thus, we come down to the proving that L′′ ∘ L = L_{{Σ}}, which again holds by direct inspection. Summing up, we have proven that L′′ is an isomorphism of categories; the proposition follows at once. □

5.4. Functorial properties of the categories of stacks. In section 4.2 we have attached to every continuous or cocontinuous functor between sites certain natural functors on the corresponding categories of sheaves. Hereafter we shall likewise associate with such functors certain natural pseudo-functors on the corresponding 2-categories of stacks.

Proposition 5.4.1. Let (𝒞 , J) and (𝒞′, J′) be two sites, u : 𝒞 → 𝒞′ a cocontinuous functor, and i ∈ {0, 1, 2}. The following holds :

(i) For every i-covering cartesian functor ℰ_{1} ⟶^{φ} ℰ_{2} of 𝒞′-fibrations ℰ_{1} ⟶^{F_{1}} 𝒞′ ⟵^{F_{2}} ℰ_{2}, the functor Fib(u)^{∗}(φ) : 𝒞 ×_{𝒞′} ℰ_{1} → 𝒞 ×_{𝒞′} ℰ_{2} is i-covering (notation of remark 3.1.5(i)).

(ii) For every i-separated fibration ℱ → 𝒞, the fibration Fib(u)_{∗}(ℱ) is i-separated.

Proof. (i): Recall that the objects of 𝒞 ×_{𝒞′} ℰ_{j} for j = 1, 2 are the pairs (X, Y) with X ∈ Ob(𝒞), Y ∈ Ob(ℰ_{j}), such that uX = F_{j}Y. The morphisms (X, Y) → (X′, Y′) are the pairs (f, g) where f : X → X′ is a morphism in 𝒞, g : Y → Y′ is a morphism in ℰ_{j}, and u(f) = F_{j}(g).

Suppose first that i = 0, and let (X, Y) ∈ Ob(𝒞 ×_{𝒞′} ℰ_{2}) be any object. By assumption, there exist a covering family (f′_{j} : X′_{j} → uX | j ∈ I) and for every j ∈ I a cartesian morphism h_{j} : φY_{j} → Y in ℰ_{2} with F_{2}h_{j} = f′_{j}. Let 𝒮 ⊂ 𝒞′/uX be the sieve generated by (f′_{j} | j ∈ I); since u is cocontinuous, there exists a covering family (f_{k} : X_{k} → X | k ∈ I′) such that u(f_{k}) ∈ Ob(𝒮) for every k ∈ I′. Then, for every k ∈ I′ pick j(k) ∈ I such that u(f_{k}) = f′_{j(k)} ∘ f′′_{k} for some morphism f′′_{k} : uX_{k} → X′_{j(k)} in 𝒞′, and choose a cartesian morphism h′_{k} : Z_{k} → Y_{j(k)} in ℰ_{1} such that F_{1}h′_{k} = f′′_{k}. Then h′′_{k} := h_{j(k)} ∘ φ(h′_{k}) : φZ_{k} → Y is cartesian and F_{2}h′′_{k} = u(f_{k}); therefore (f_{k}, h′′_{k}) : (X_{k}, φZ_{k}) → (X, Y) is a cartesian morphism in 𝒞 ×_{𝒞′} ℰ_{2} for every j ∈ I. This proves that Fib(u)^{∗}(φ) is 0-covering.

Next, suppose that i = 1, and let (1_{X}, g) : (X, φY_{1}) → (X, φY_{2}) be a morphism in 𝒞 ×_{𝒞′} ℰ_{2}. By assumption there exist a covering family (f′_{j} : X′_{j} → uX | j ∈ I) in 𝒞′ and morphisms

Y_{2} ⟵^{g_{j}} Z_{j} ⟶^{h_{j}} Y_{1} in ℰ_{1} with h_{j} cartesian, such that φ(g_{j}) = g ∘ φ(h_{j}) and F_{1}h_{j} = f′_{j} for every j ∈ I. Then, choose a covering family (f_{k} : X_{k} → X | k ∈ I′) as in the foregoing, so that for every k ∈ I′ there exist j(k) ∈ I and a factorization u(f_{k}) = f′_{j(k)} ∘ f′′_{k} for some morphism f′′_{k} : uX_{k} → X′_{j(k)} in 𝒞′. Choose also for every k ∈ I′ a cartesian morphism h′_{k} : Z′_{k} → Z_{j(k)} with F_{1}h′_{k} = f′′_{k}. Then h′′_{k} := h_{j(k)} ∘ h′_{k} : Z′_{k} → Y_{1} is cartesian and F_{1}h′′_{k} = u(f_{k}), so we get a cartesian morphism (f_{k}, h′′_{k}) : (X_{k}, Z′_{k}) → (X, Y_{1}) such that (1_{X}, g) ∘ (f_{k}, φ(h′′_{k})) = (f_{k}, φ(g_{j(k)} ∘ h′_{k})) for every k ∈ I′. This shows that Fib(u)^{∗}(φ) is 1-covering.

Lastly, say that i = 2, and let (1_{X}, g_{1}), (1_{X}, g_{2}) : (X, Y_{1}) → (X, Y_{2}) be two morphisms in 𝒞 ×_{𝒞′} ℰ_{1} with φ(g_{1}) = φ(g_{2}). By assumption there exist a covering family (f′_{j} : X′_{j} → uX | j ∈ I) and a cartesian morphism h_{j} : Z_{j} → Y_{1} in ℰ_{1} such that g_{1} ∘ h_{j} = g_{2} ∘ h_{j} and F_{1}h_{j} = f′_{j} for every j ∈ I. We pick again a covering family (f_{k} : X_{k} → X | k ∈ I′) in 𝒞 such that for every k ∈ I′ there exist j(k) ∈ I and a factorization u(f_{k}) = f′_{j(k)}∘f′′_{k}, and we choose for every k ∈ I′ a cartesian morphism h′_{k} : Z′_{k} → Z_{j(k)} such that F_{1}h′_{k} = f′′_{k}. Then h′′_{k} := h_{j(k)} ∘ h′_{k} : Z′_{k} → Y_{1} is cartesian and F_{1}h′′_{k} = u(f_{k}); therefore (f_{k}, h′′_{k}) : (X_{k}, Z′_{k}) → (X, Y_{1}) is cartesian morphism in 𝒞 ×_{𝒞′} ℰ_{1} such that (1_{X}, g_{1}) ∘ (f_{k}, h′′_{k}) = (1_{X}, g_{2}) ∘ (f_{k}, h′′_{k}) for every k ∈ I′. This proves that Fib(u)^{∗}(φ) is 2-covering.

(ii): See (3.3.14) for the definition of the pseudo-functor Fib(u)_{∗}. Let X ∈ Ob(𝒞) be any object, and 𝒮 ⊂ 𝒞 /X a covering sieve; we consider the essentially commutative diagram :

[figure: Fib(u)_{∗}ℱ (X); Cart_{𝒞} (Fib(u)^{∗}(𝒞 /X), ℱ); Cart_{𝒞} ((Fib(u)^{∗}(𝒞 /X))^{a}, ℱ^{a}); Cart_{𝒞} (𝒮 , Fib(u)_{∗}ℱ); Cart_{𝒞} (Fib(u)^{∗}(𝒮), ℱ); Cart_{𝒞} ((Fib(u)^{∗}(𝒮))^{a}, ℱ^{a}).]

We need to show that the left vertical arrow is i-faithful, and we know that the horizontal arrows of the left square subdiagram are equivalences. We are thus reduced to checking that the central vertical arrow is i-faithful. But since ℱ is i-separated, the unit of adjunction ℱ → ℱ^{a} is i-faithful (proposition 5.2.11(ii)), so the same holds for the horizontal arrows of the right square subdiagram (corollary 3.1.28(ii.a)), and we are further reduced to checking that the right vertical arrow is an equivalence. To this aim, it suffices to check that the inclusion functor t : 𝒮 → 𝒞 /X induces an equivalence (Fib(u)^{∗}(𝒮))^{a} →^{∼} (Fib(u)^{∗}(𝒞 /X))^{a} (corollary 3.1.28(ii.b)). Now, recall that 𝒞 /X is isomorphic to ℱ ib(h_{X}), where h_{X} is the presheaf on 𝒞 represented by X; likewise, 𝒮 is isomorphic to ℱ ib(h_{𝒮} ), for a covering subobject s : h_{𝒮} → h_{X}. We deduce natural isomorphisms :

(Fib(u)^{∗}(𝒮))^{a} →^{∼} ℱ ib(u^{∧}h_{𝒮} )^{a} →^{∼} ℱ ib((u^{∧}h_{𝒮} )^{a})

and likewise (Fib(u)^{∗}(𝒞 /X))^{a} →^{∼} ℱ ib((u^{∧}h_{X})^{a}), by theorem 5.1.24 and remark 3.2.9(ii). Clearly, these isomorphisms identify s with ℱ ib((u^{∧}s)^{a}); the latter is an isomorphism, according to lemma 4.2.9, whence the contention. □

Definition 5.4.2. Let C := (𝒞 , J) and C′ := (𝒞′, J′) be two sites, and u : 𝒞 → 𝒞′ a functor. We say that u : C′ → C is a weak morphism of sites if for every universe V and every V-stack ℰ on C′, the fibration V-Fib(u)^{∗}ℰ is a V-stack on C. Then, for every such V, the pseudo-functor V-Fib(u)^{∗} induces by restriction a (strict) pseudo-functor

V-St(u)_{∗} : V-Stack(C′) → V-Stack(C).

Remark 5.4.3. (i) From example 5.1.2, it follows that every weak morphism of sites C′ → C is a continuous functor for the topologies of C and C′.

(ii) We shall see hereafter that every morphism of sites is a weak morphism of sites.

Proposition 5.4.4. Let C := (𝒞 , J) and C′ := (𝒞′, J′) be two sites, u : 𝒞 → 𝒞′ a functor, and V a universe such that 𝒞 is V-small and 𝒞′ has V-small Hom-sets. The following conditions are equivalent :

(a) For every V-stack ℰ on C′, the fibration V-Fib(u)^{∗}ℰ is a V-stack on C.

(b) For every morphism φ : ℰ → ℱ in V-Fib(𝒞) that is i-covering for i = 0, 1, 2, the morphism V-Fib(u)_{!}φ : V-Fib(u)_{!}ℰ → V-Fib(u)_{!}ℱ is i-covering for i = 0, 1, 2.

(c) For every X ∈ Ob(𝒞) and every covering sieve 𝒮 ⊂ 𝒞 /X, the induced morphism of 𝒞′-fibrations V-Fib(u)_{!}𝒮 → V-Fib(u)_{!}(𝒞 /X) is i-covering for i = 0, 1, 2.

(d) u : C′ → C is a weak morphism of sites.

Proof. (a)⇒(b): In view of remark 3.3.15(v), we may replace V by a larger universe, and assume that both 𝒞 and 𝒞′ are V-small. Then, according to propositions 5.2.9 and 5.2.11(iii), the functor φ^{a} : ℰ^{a} → ℱ^{a} is an equivalence, and we need to check that (V-Fib(u)_{!}φ)^{a} is an equivalence. To this aim, for every V-stack 𝒜 on C′, we consider the diagram :

\xymatrix{Cart_{𝒞′}(Fib(u)_{!}(ℱ)^{a}, 𝒜) \ar[r] \ar[d]|{Cart_{𝒞′} (Fib(u)_{!}(φ)^{a},𝒜)} & Cart_{𝒞′}(Fib(u)_{!}(ℱ), 𝒜) \ar[r] \ar[d]|{Cart_{𝒞′} (Fib(u)_{!}(φ),𝒜)} & Cart_{𝒞} (ℱ , Fib(u)^{∗}𝒜) \ar[d]|{Cart_{𝒞} (φ,Fib(u)^{∗}𝒜)} \\ Cart_{𝒞′}(Fib(u)_{!}(ℰ)^{a}, 𝒜) \ar[r] & Cart_{𝒞′}(Fib(u)_{!}(ℰ), 𝒜) \ar[r] & Cart_{𝒞} (ℰ , Fib(u)^{∗}𝒜)}

whose left square subdiagram is induced by the natural cartesian functors

Fib(u)_{!}(ℱ) → Fib(u)_{!}(ℱ)^{a} and Fib(u)_{!}(ℰ) → Fib(u)_{!}(ℰ)^{a}

and whose right square subdiagram is given by the coherence constraint of the 2-adjunction for the pair (V-Fib(u)_{!}, V-Fib(u)^{∗}). Thus, both subdiagrams are essentially commutative, and the same then holds for their composition. Moreover, all horizontal arrows are equivalences, so the left-most vertical arrow is an equivalence if and only if the same holds for the right-most one, and in light of lemma 2.1.13(iii) we are reduced to checking that Cart_{𝒞′}(φ, Fib(u)^{∗}𝒜) is an equivalence for every such 𝒜. But by assumption, V-Fib(u)^{∗}𝒜 is a stack on C, so we have as well the pseudo-commutative diagram

\xymatrix{Cart_{𝒞} (ℱ^{a}, Fib(u)^{∗}𝒜) \ar[d]_{Cart_{𝒞} (φ^{a},Fib(u)^{∗}𝒜)} \ar[r] & Cart_{𝒞} (ℱ , Fib(u)^{∗}𝒜) \ar[d]^{Cart_{𝒞} (φ,Fib(u)^{∗}𝒜)} \\ Cart_{𝒞} (ℰ^{a}, Fib(u)^{∗}𝒜) \ar[r] & Cart_{𝒞} (ℰ , Fib(u)^{∗}𝒜)}

whose horizontal arrows are equivalences. Hence it suffices to show that Cart_{𝒞} (φ^{a}, Fib(u)^{∗}𝒜) is an equivalence for every V-stack 𝒜 on C′, which follows by invoking again lemma 2.1.13(iii).

(b)⇒(c): This is clear from propositions 5.2.9 and 5.2.11(iii), after inspecting the pseudocommutative diagram 5.1.25.

(c)⇒(d): Let V′ be any universe; we need to check that V′-Fib(u)^{∗}ℰ is a V′-stack on C, for every V′-stack ℰ on C′. To this aim, we can replace V′ by a larger universe, and assume that V ⊂ V′, and both 𝒞 and 𝒞′ are V′-small. Moreover, in light of remark 3.3.15(v) and our assumption (c), we see that for every X ∈ Ob(𝒞) and every covering sieve 𝒮 ⊂ 𝒞 /X, the induced morphism of 𝒞′-fibrations V′-Fib(u)_{!}𝒮 → V′-Fib(u)_{!}(𝒞 /X) is i-covering for i = 0, 1, 2. Now, the coherence constraint for the 2-adjoint pair (V′-Fib(u)_{!}, V′-Fib(u)^{∗}) yields an essentially commutative diagram :

\xymatrix{𝒞art_{𝒞} (𝒞 /X, V′-Fib(u)^{∗}ℰ) \ar[r] \ar[d] & 𝒞art_{𝒞} (𝒮 , V′-Fib(u)^{∗}ℰ) \ar[d] \\ 𝒞art_{𝒞′}(V′-Fib(u)_{!}(𝒞 /X), ℰ) \ar[r] & 𝒞art_{𝒞′}(V′-Fib(u)_{!}𝒮 , ℰ)}

whose vertical arrow are equivalences. Thus, we are reduced to checking that the bottom horizontal arrow is an equivalence. But since ℰ is a stack, we have as well the following pseudocommutative commutative diagram, whose vertical arrows are again equivalences :

\xymatrix{𝒞art_{𝒞′}(V′-Fib(u)_{!}(𝒞 /X)^{a}, ℰ) \ar[r] \ar[d] & 𝒞art_{𝒞′}(V′-Fib(u)_{!}(𝒮)^{a}, ℰ) \ar[d] \\ 𝒞art_{𝒞′}(V′-Fib(u)_{!}(𝒞 /X), ℰ) \ar[r] & 𝒞art_{𝒞′}(V′-Fib(u)_{!}𝒮 , ℰ)}

Hence, it suffices to check that the top horizontal arrow of this latter diagram is an equivalence. But the functor V′-Fib(u)_{!}(𝒮)^{a} → V′-Fib(u)_{!}(𝒞 /X)^{a} is an equivalence, by propositions 5.2.9 and 5.2.11(iii); the contention is an immediate consequence.

(d)⇒(a) is trivial. □

5.4.5. Let C := (𝒞 , J) and C′ := (𝒞′, J′) be two sites; we wish to give some useful criteria for a functor u : 𝒞 → 𝒞′ to be a weak morphism of sites C′ → C. To this aim, we consider the following conditions :

(C0) For every X ∈ Ob(𝒞) and every covering family (X_{i} → X | i ∈ I) for the topology

J, the family (uX_{i} → uX | i ∈ I) covers uX for the topology J′. (C1) u is continuous for the topologies J and J′. (C2) Condition (C0) holds, and for every X ∈ Ob(𝒞), every covering family (X_{i} → X | i ∈

I) for the topology J admits a refinement (X′_{i} → X | i ∈ I′) that still covers X, such that the fibre products X′_{i} ×_{X} X′_{j} and X′_{i} ×_{X} X′_{j} ×_{X} X′_{k} are representable in 𝒞 for every i, j, k ∈ I and u commutes with these fibre products.

(C3) Condition (C0) holds, and there exists a universe V containing U such that 𝒞 and 𝒞′

are V-small and the functor u_{V!}^{a} : 𝒞_{V}^{∧} → C_{V}^{′∼} commutes with fibre products. (C4) u : C′ → C is a morphism of sites.

Remark 5.4.6. (i) Recall that (C1) means that u^{∧} transforms sheaves on C′ into sheaves on C. Likewise, according to lemma 4.2.4, condition (C0) holds if and only if u^{∧} transforms separated presheaves on C′ into separated presheaves on C.

(ii) We have (C4)⇒(C3)⇒(C1)⇒(C0), by lemma 4.2.4. By inspecting the proof of lemma 4.2.4we also easily see that (C2)⇒(C1).

(iii) In the situation of (5.4.5), suppose that C is a lex-site and u is left exact. Then example 4.3.12says that u fulfills condition (C4) if and only if it fullfils condition (C0).

5.4.7. In the situation of (5.4.5), let 𝒜′ → 𝒞′ be a fibration, and X ∈ Ob(𝒞). Set 𝒜 := Fib(u)^{∗}𝒜′ → 𝒞 and let π : 𝒜 → 𝒜′ be the natural projection. For every cartesian section σ ∈ 𝒜′(uX) there exists a unique cartesian section

u_{|X}^{∗}(σ) ∈ 𝒜 (X) such that π ∘ u_{|X}^{∗}(σ) = σ ∘ u_{|X}.

Also, for every pair of cartesian sections σ, τ ∈ 𝒜′(uX) and every natural 𝒞 -transformation β : σ ⇒ τ there exists a unique natural 𝒞 -transformation

u_{|X}^{∗}(β) : u_{|X}^{∗}(σ) ⇒ u_{|X}^{∗}(τ) such that π ∗ u_{|X}^{∗}(β) = β ∗ u_{|X}.

This characterization easily implies that the rules : σ ↦ u_{|X}^{∗} and β ↦ u_{|X}^{∗}(β) for every such σ and β, define a functor u_{|X}^{∗} : 𝒜′(uX) → 𝒜 (X) fitting into the commutative diagram :

\xymatrix{𝒜′(uX) \ar[r]^{u_{|X}^{∗}} \ar[d]_{_{ev{𝒜′ \atop uX}}} & 𝒜 (X) \ar[d]^{_{ev_{𝒜}_{X}}} \\ 𝒜′_{uX} \ar[r] & 𝒜_{X}}

whose bottom horizontal arrow is the isomorphism of categories induced by π, and whose vertical arrows are the evaluation functors of (3.2.5). Since the latter are equivalences, it follows that the same holds for u_{|X}^{∗}. Notice that for every (f : Y → X) ∈ Ob(𝒞) and every cartesian section σ ∈ 𝒜′(uX) we have :

u_{|Y}^{∗} (σ ∘ (uf)_{∗}) = u_{|X}^{∗}(σ) ∘ f_{∗}

(detail left to the reader). Hence, for every pair of cartesian sections σ, τ ∈ 𝒜′(uX) and every (f : Y → X) ∈ Ob(𝒞 /X) we deduce a bijection

Cart(σ, τ)(uf) →^{∼} Cart(u_{|X}^{∗}σ, u_{|X}^{∗}τ)(f) β ↦ u_{|Y}^{∗} β.

Finally, it is easily seen that this system of bijections amounts to an isomorphism of presheaves

u_{|X}^{∧}𝒞art(σ, τ) →^{∼} 𝒞art(u_{|X}^{∗}σ, u_{|X}^{∗}τ).

Proposition 5.4.8. In the situation of (5.4.5), let F : ℰ → 𝒞′ be any fibration. We have : (i) If ℰ is 0-separated, and (C0) holds, then Fib(u)^{∗}(ℰ) is 0-separated. (ii) If ℰ is 1-separated, and (C1) holds, then Fib(u)^{∗}(ℰ) is 1-separated. (iii) If (C2) holds, then u is a weak morphism of sites C′ → C.

Proof. To ease notation, set ℱ := Fib(u)^{∗}(ℰ), and let π : ℱ → ℰ be the natural projection.

(i): Let X ∈ Ob(𝒞) be any object, and σ, τ ∈ ℱ (X) two cartesian sections; according to lemma 5.1.9 it suffices to show that the presheaf 𝒞art(σ, τ) is separated on the site C/X. However, the discussion of (5.4.7) yields cartesian sections σ′, τ′ ∈ ℰ (uX) with isomorphisms u_{|X}^{∗}σ′ →^{∼} σ and u_{|X}^{∗}τ′ →^{∼} τ, whence isomorphisms of presheaves

𝒞art(σ, τ) →^{∼} 𝒞art(u_{|X}^{∗}σ′, u_{|X}^{∗}τ′) →^{∼} u_{|X}^{∧}𝒞art(σ′, τ′).

Since ℰ is 0-separated, lemma 5.1.9 says that 𝒞art(σ′, τ′) is a separated presheaf on the site C′/uX. On the other hand, the explicit description of the covering sieves of the sites C/X and C′/uX furnished by (4.7) easily implies that the functor u_{|X} also fulfills condition (C0), relative to these sites (details left to the reader). Then the assertion follows from remark 5.4.6(i).

(ii) is similar : arguing as in the foregoing, we are reduced to checking that u_{|X}^{∧}(𝒞art(σ′, τ′)) is a sheaf on C/X. However, 𝒞art(σ′, τ′) is a sheaf on C′/uX, by lemma 5.1.9, and u_{|X} is continuous on the sites C/X and C′/uX, by proposition 4.7.7, whence the assertion.

(iii): Suppose that (C2) holds, and that ℰ is a stack on C′; let X ∈ Ob(𝒞) be any object, 𝒮 ⊂ 𝒞 /X a sieve covering X, and 𝒮′ ⊂ 𝒮 a refinement covering X, generated by a family f_{•} := (f_{i} : X′_{i} → X | i ∈ I) with the properties of condition (C2). We get restriction functors :

ℱ (X) ⟶^{ρ} Cart_{𝒞} (𝒮 , ℱ) ⟶^{ρ′} Cart_{𝒞} (𝒮′, ℱ)

and we need to prove that ρ is an equivalence. We check first that ρ is fully faithful. To this aim, notice that ℱ is 0-separated by virtue of (i), hence ρ′ is faithful (lemma 5.1.20), and we are then easily reduced to showing that ρ′ ∘ ρ is fully faithful. Now, choose a cleavage c for ℰ, so that c ∘ u^{o} is a cleavage for ℱ. Then Cart_{𝒞} (𝒮′, ℱ) is equivalent to the category of descent data Desc(ℱ , f_{•}, c ∘ u^{o}) as in (3.5.22). However, π induces an isomorphism of categories ℱ_{X} →^{∼} ℰ_{uX}, and since u commutes with the fibre products X_{i} ×_{X} X_{j} and X_{i} ×_{X} X_{j} ×_{X} X_{k}, a simple inspection shows that the natural functor

ℱ_{X} → Desc(ℱ , f_{•}, c ∘ u^{o})

is naturally identified with the corresponding functor

ℰ_{uX} → Desc(ℰ , u(f_{•}), c).

The latter is an equivalence, since ℰ is a stack. Thus ρ′ ∘ ρ is even an equivalence, and therefore ρ is fully faithful, as stated. Since X and 𝒮 are arbitrary, we have thus proved that ℱ is 1separated; but then ρ′ is fully faithful, again by lemma 5.1.20, and it follows easily that ρ is an equivalence if and only if the same holds for ρ′ ∘ ρ. For the latter, the assertion has just been shown, so finally ℱ is a stack, as required. □

Lemma 5.4.9. In the situation of (5.4.5), let φ : ℰ_{1} → ℰ_{2} be a 𝒞′-cartesian functor between two fibrations over 𝒞′. Suppose that (C0) holds, and moreover that for every X ∈ Ob(𝒞′) there exists a covering family (uY_{j} → X | j ∈ I) for the topology J′. Let also i ∈ {0, 1, 2} such that Fib(u)^{∗}(φ) is i-covering for the topology J. Then φ is i-covering for the topology J′.

Proof. In view of lemma 5.2.3, we may replace φ by C(φ), and reduce to the case where ℰ_{1} and ℰ_{2} are split 𝒞′-fibrations, and φ is a split cartesian functor. For t = 1, 2 let c_{t} : 𝒞′ → Cat be the strict pseudo-functor associated with the split cleavage of ℰ_{t}. Suppose first that i = 0, and let X ∈ Ob(𝒞′) and E ∈ Ob(ℰ_{2,X}). By assumption, we may find a covering family (f_{j} : uY_{j} → X | j ∈ I) and we set E_{j} := c_{2,f_{j}}E ∈ Ob(ℰ_{2,uY_{j}}) for every j ∈ I. Since Fib(u)^{∗}(φ) is 0-covering, we may find for every j ∈ I a covering family (f_{jj′} : Y_{jj′} → Y_{j} | j′ ∈ Λ_{j}) and for every j′ ∈ Λ_{j} an object E′_{jj′} ∈ Ob(ℰ_{1,uY_{jj′}}) with an isomorphism φE′_{jj′} →^{∼} c_{2,u(f_{jj′})}E_{j}. Set h_{jj′} := f_{j} ∘ u(f_{jj′}) : uY_{jj′} → X for every j ∈ I and j′ ∈ Λ_{j}; since (C0) holds for u, the family

(h_{jj′} | j ∈ I, j′ ∈ Λ_{j}) covers X, and φE′_{jj′} →^{∼} c_{2,h_{jj′}}E for every j ∈ I and j′ ∈ Λ_{j}. This shows

that φ is 0-covering.

Next, suppose that i = 1, and let X ∈ Ob(𝒞′) and g : φE_{1} → φE_{2} a morphism in ℰ_{2,X}. We pick a covering family (f_{j} : uY_{j} → X | j ∈ I), and set E_{tj} := c_{1,f_{j}}E_{t} for t = 1, 2 and every j ∈ I; also, let g_{j} := c_{2,f_{j}}(g) : φE_{1j} → φE_{2j} for every j ∈ J. Since Fib(u)^{∗}(φ) is 1-covering, for every j ∈ I there exist a covering family (f_{jj′} : Y_{jj′} → Y_{j} | j′ ∈ Λ_{j}) and for every j′ ∈ Λ_{j} a morphism g_{jj′} : c_{1,u(f_{jj′})}E_{1j} → c_{1,u(f_{jj′})}E_{2j} with φ(g_{jj′}) = c_{2,u(f_{jj′})}(g_{j}) = c_{2,u(f_{jj′})∘f_{j}}(g). Arguing as in the foregoing, we easily deduce that φ is 1-covering (details left to the reader).

Lastly, suppose that i = 2, and let X ∈ Ob(𝒞′) and g_{1}, g_{2} : E_{1} → E_{2} two morphisms in ℰ_{1,X} such that φ(g_{1}) = φ(g_{2}). Pick again a covering family (f_{j} : uY_{j} → X | j ∈ I) and set g_{tj} := c_{1,f_{j}}(g_{t}) for t = 1, 2 and every j ∈ I. Then φ(g_{1j}) = φ(g_{2j}) for every j ∈ I, and since Fib(u)^{∗}(φ) is 2-covering, there exists for every such j a covering family (f_{jj′} : Y_{jj′} → Y_{j} | j′ ∈ Λ_{j}) such that c_{1,u(f_{jj′})}(g_{1j}) = c_{1,u(f_{jj′})}(g_{2j}) for every j′ ∈ Λ_{j}. Arguing as in the foregoing, it follows easily that φ is 2-covering. □

5.4.10. Let C := (𝒞 , J) be a small site and define the site C_{U}^{∧} := (𝒞_{U}^{∧}, J^{∧}) as in remark 4.1.31(ii). By theorem 4.4.2(i), the Yoneda embedding is a cocontinuous morphism of sites h_{𝒞} : C_{U}^{∧} → C. Thus, for every universe V containing U and every i-separated V-prestack φ : ℰ → 𝒞 on 𝒞, the fibration

V-Fib(h_{𝒞} )_{∗}(ℰ) → 𝒞_{U}^{∧}

is an i-separated prestack for the topology J^{∧} (proposition 5.4.1(ii)). Moreover, since h_{𝒞} is fully faithful, remark 3.3.15(iv) and corollary 2.4.29imply that the counit of 2-adjunction

V-Fib(h_{𝒞} )^{∗} ∘ V-Fib(h_{𝒞} )_{∗}(ℰ) → ℰ

is an equivalence of categories. Thus, V-Fib(h_{𝒞} )_{∗}(ℰ) is a natural extension of ℰ to the site C^{∧} which contains C as a full subcategory with the topology induced from J^{∧}.

Remark 5.4.11. (i) In the situation of (5.4.10), we can describe more explicitly the fibration V-Fib(h_{𝒞} )_{∗}(ℰ), as follows. We consider the functor

c_{φ}^{▽} : (𝒞_{U}^{∧})^{o} → V-Cat F ↦ Cart_{𝒞} (ℱ ib(F), ℰ)

which assigns to every morphism of presheaves β : F → F′, the functor Cart_{𝒞} (ℱ ib(β), ℰ) : c_{φ}^{▽}(F′) → c_{φ}^{▽}(F). Then, combining remark 3.3.15(ii) and example 3.1.16(iii) we get a natural equivalence of 𝒞_{U}^{∧}-categories :

V-Fib(h_{𝒞} )_{∗}(ℰ) →^{∼} ℱ ib(c_{φ}^{▽}).

(ii) Let i : C_{U}^{∼} → 𝒞_{U}^{∧} be the inclusion functor, and (−)^{a} : 𝒞_{U}^{∧} → C_{U}^{∼} its left adjoint. We have a pseudo-natural equivalence of pseudo-functors :

V-Fib((−)^{a})_{∗} →^{∼} V-Fib(i)^{∗}.

Indeed, let ℰ be any fibration on 𝒞_{U}^{∧}; in view of remark 3.3.15(ii), the fibration V-Fib((−)^{a})_{∗}(ℰ) is naturally equivalent to the fibration associated with the strict pseudo-functor

F ↦ Cart_{𝒞_{U}^{∧}}((−)^{a}𝒞_{U}^{∧}/F, ℰ) for every F ∈ Ob(C_{U}^{∼}).

On the other hand, C(Fib(i)^{∗}(ℰ)) is the fibration associated with the strict pseudo-functor

F ↦ Cart_{𝒞_{U}^{∧}}(𝒞_{U}^{∧}/iF, ℰ) for every F ∈ Ob(C_{U}^{∼}).

It suffices then to notice that (−)^{a}𝒞_{U}^{∧}/F = 𝒞_{U}^{∧}/iF for every sheaf F on the site C.

Theorem 5.4.12. Let C := (𝒞 , J) be a small site, h_{𝒞} : 𝒞 → 𝒞_{U}^{∧} the Yoneda embedding and i : C_{U}^{∼} → 𝒞_{U}^{∧} the inclusion functor. For every universe V containing U we have :

(i) The strict pseudo-functor V-Fib(h_{𝒞} )^{∗} restricts to a 2-equivalence ;

V-Stack(C_{U}^{∧}) →^{∼} V-Stack(C).

(ii) Likewise, i and its left adjoint (−)^{a} : 𝒞_{U}^{∧} → C_{U}^{∼} induce 2-equivalences :

\xymatrix{V-Stack(C_{U}^{∧}) \ar[r]^{V-Fib(i)^{∗}} & V-Stack(Can(C_{U}^{∼})). \ar[l]^{V-Fib((−)^{a})^{∗}}}

Proof. (i): Let η and ε be the unit and counit of a 2-adjunction for the pair of pseudo-functors (V-Fib(h_{𝒞} )^{∗}, V-Fib(h_{𝒞} )_{∗}). Since h_{𝒞} is a cocontinuous morphism of sites C^{∧} → C (theorem 4.4.2(i)), we know already that V-Fib(h_{𝒞} )_{∗} sends V-stacks on C to V-stacks on C^{∧} (proposition 5.4.1(ii)); according to corollary 2.4.32(i), it then suffices to show that for every V-stack ℰ on C^{∧} and ℰ′ on C, the fibration V-Fib(h_{𝒞} )^{∗}(ℰ) is a V-stack on C, and η_{ℰ} and ε_{ℰ′} are equivalences. The assertion for ε_{ℰ′} has already been noticed in (5.4.10). Next, to ease notation, let us drop the prefix V, and set ℱ := Fib(h_{𝒞} )^{∗}(ℰ). Let also i_{ℱ} : ℱ → ℱ^{a} be the natural morphism of fibrations on 𝒞, and denote by

β_{ℰ} : ℰ → Fib(h_{𝒞} )_{∗}(ℱ^{a})

the composition of η_{ℰ} and Fib(h_{𝒞} )_{∗}(i_{ℱ} ). Then ε_{ℱ^{a}} ∘ (Fib(h_{𝒞} )^{∗}β_{ℰ}) is isomorphic to

ε_{ℱ^{a}} ∘ (Fib(h_{𝒞} )^{∗} ∘ Fib(h_{𝒞} )_{∗}(i_{ℱ} )) ∘ (Fib(h_{𝒞} )^{∗}η_{ℰ})

which is in turn isomorphic to i_{ℱ} ∘ε_{ℱ} ∘(Fib(h_{𝒞} )^{∗}η_{ℰ}), and the latter is isomorphic to i_{ℱ} , by virtue of the triangular modifications associated with the pair (η, ε) (theorem 2.4.24(i)). Since ε_{ℱ^{a}} is an equivalence, and i_{ℱ} is j-covering for j = 0, 1, 2, it follows that Fib(h_{𝒞} )^{∗}(β_{ℰ}) is j-covering for j = 0, 1, 2. Since the functor h_{𝒞} verifies condition (C0) of (5.4.5), lemma 5.4.9implies that β_{ℰ} is also j-covering for j = 0, 1, 2. But Fib(h_{𝒞} )_{∗}(ℰ) is a stack (proposition 5.4.1(ii)), hence β_{ℰ} is an equivalence (proposition 5.2.9). Therefore, Fib(h_{𝒞} )^{∗}(β) is an equivalence as well, and by the foregoing, the same then holds for i_{ℱ} . The latter means that Fib(h_{𝒞} )^{∗}(ℰ) is a stack, and it also follows that η_{ℰ} is an equivalence, as sought.

(ii): By theorem 4.4.2(iii), the functor (−)^{a} is cocontinuous for the topologies J^{∧} and Can_{C^{∼}}; taking into account remark 5.4.11(ii) and proposition 5.4.1(ii), it follows that i is a weak morphism of sites C_{U}^{∧} → Can(C_{U}^{∼}). Also, (−)^{a} is a weak morphism of sites Can(C_{U}^{∼}) → C_{U}^{∧}, due to proposition 5.4.8(iii). Moreover, since (−)^{a} ∘ i is isomorphic to 1_{C^{∼}}, we have a pseudonatural isomorphism of pseudo-functors:

V-Fib(i)^{∗} ∘ V-Fib((−)^{a})^{∗} →^{∼} 1_{V-Stack(Can(C^{∼}))}.

Next, let ℰ be any stack on C^{∧} with V-small fibres; recall that ℰ is naturally equivalent to the split fibration associated with the strict pseudo-functor Cart_{𝒞^{∧}}(𝒞^{∧}/−, ℰ). Likewise, V-Fib((−)^{a})^{∗} ∘ V-Fib(i)^{∗}(ℰ) is naturally equivalent to the split fibration associated with the strict pseudo-functor

𝒞^{∧} → V-Cat F ↦ Cart_{𝒞^{∧}}(𝒞^{∧}/F^{a}, ℰ).

By virtue of corollary 2.4.32(i), we are thus reduced to checking that for every F ∈ Ob(𝒞^{∧}), the natural morphism j_{F} : F → F^{a} induces an equivalence of categories Cart_{𝒞^{∧}}(𝒞^{∧}/F^{a}, ℰ) →^{∼} Cart_{𝒞^{∧}}(𝒞^{∧}/F, ℰ). To this aim, we notice :

Claim 5.4.13. For every stack ℰ on C^{∧} and every bicovering morphism f : G → G′ of (𝒞^{∧})_{V}^{∧} (for the topology J^{∧} of 𝒞^{∧}), the functor Cart_{𝒞^{∧}}(ℱ ib(f), ℰ) is an equivalence.

Proof of the claim. Indeed, according to theorem 5.1.24 we have an essentially commutative diagram :

\xymatrix{Cart_{𝒞} (ℱ ib(G^{′a}), ℰ) \ar[r]^{Cart_{𝒞} (ℱ ib(f^{a}),ℰ)} \ar[d] & Cart_{𝒞} (ℱ ib(G^{a}), ℰ) \ar[d] \\ Cart_{𝒞} (ℱ ib(G′), ℰ) \ar[r]^{Cart_{𝒞} (ℱ ib(f),ℰ)} & Cart_{𝒞} (ℱ ib(G), ℰ)}

whose vertical arrows are equivalences. By assumption, f^{a} : G^{a} → G^{′a} is an isomorphism of sheaves on (C^{∧})_{V}^{∼}, hence the top horizontal arrow is an isomorphism of categories, and therefore the bottom horizontal arrow is an equivalence, as claimed. ♢

In view of claim 5.4.13, and recalling the natural identifications

𝒞^{∧}/F →^{∼} ℱ ib(h_{F}) 𝒞^{∧}/F^{a} →^{∼} ℱ ib(h_{F^{a}})

(example 3.1.16(i)), we are then further reduced to showing :

Claim 5.4.14. j_{F} induces a bicovering morphism h_{j_{F}} : h_{F} → h_{F^{a}} of (𝒞^{∧})_{V}^{∧}.

Proof of the claim. Consider any V-presheaf G on 𝒞^{∧}, denote by G^{a} ∈ (C^{∧})_{V}^{∼} the sheaf associated with G, and let j_{G} : G → G^{a} be the natural morphism; since h_{𝒞} is continuous and cocontinuous for the topologies J and J^{∧} (theorem 4.4.2(i)), lemma 4.2.15(ii) implies that h_{𝒞}^{∧}(G^{a}) is a V-sheaf on C, and the morphism of V-presheaves h_{𝒞}^{∧}(G) → h_{𝒞}^{∧}(G^{a}) on 𝒞 is bicovering. Taking G := h_{F}, we get h_{𝒞}^{∧}(h_{F}) = F, whence an isomorphism t : h_{𝒞}^{∧}(h_{F}^{a}) →^{∼} F^{a} that identifies h_{𝒞}^{∧}(j_{G}) with j_{F}. Likewise, we have h_{𝒞}^{∧}(h_{F^{a}}) = F^{a}, and since h_{𝒞}^{∧} restricts to an equivalence (C^{∧})_{V}^{∼} →^{∼} C_{V}^{∼} (theorem 4.4.2(iii)), there follows a unique isomorphism s : h_{F}^{a} →^{∼} h_{F^{a}} such that h_{𝒞}^{∧}(s) = t. Summing up, we deduce that h_{𝒞}^{∧}(s ∘ j_{G}) = j_{F} = h_{𝒞}^{∧}(h_{j_{F}}); since the functor h_{𝒞^{∧}}^{∧} : 𝒞^{∧} → (𝒞^{∧})_{V}^{∧} is fully faithful, we get s∘j_{G} = h_{j_{F}}, and since s is an isomorphism, the claim follows. □

Corollary 5.4.15. In the situation of (5.4.5), suppose that condition (C3) holds for some universe V. Then u is a weak morphism of sites C′ → C.

Proof. By remark 4.2.18(iii) we may assume that also 𝒞^{∧} and 𝒞^{′∧} are V-small. We consider the essentially commutative diagram :

\xymatrix{𝒞 \ar[r]^{u} \ar[d]_{h_{𝒞}} & 𝒞′ \ar[d]^{h_{C′}^{a}} \\ 𝒞_{V}^{∧} \ar[r]^{u_{V!}^{a}} & C_{V}^{′∼}}

and we endow 𝒞_{V}^{∧} with the topology J^{∧} as in remark 4.1.31, and C_{V}^{′∼} with its canonical topology. We notice that, with these topologies, the functor u_{V!}^{a} satisfies condition (C2) : indeed, condition (C0) holds for u_{V!}^{a} by definition of the topology J^{∧}; also, all fibre products of 𝒞_{V}^{∧} are representable, and by assumption u_{V!}^{a} commutes with fibre products. Let V′ be a universe containing V and such that C_{V}^{∧} and C_{V}^{′∼} are V′-small; by the foregoing case, we deduce that V′-Fib(u_{V!}^{a})^{∗} sends stacks on Can(C_{V}^{′∼}) to stacks on (𝒞_{V}^{∧}, J^{∧}). On the other hand, theorem 5.4.12implies that V′-Fib(h_{C′}^{a})^{∗} restricts to a 2-equivalence from the 2-category of V′-stacks on Can(C_{V}^{′∼}) to that of V′-stacks on (𝒞′, J′), so we may assume that ℰ = V′-Fib(h_{C′}^{a})^{∗}(ℰ′) for some V′-stack on Can(C_{V}^{′∼}), and we set ℰ′′ := V′-Fib(u_{V!}^{a})^{∗}(ℰ′). Then Fib(u)^{∗}(ℰ) is equivalent to V′-Fib(h_{𝒞} )^{∗}(ℰ′′), and the latter is a stack, by theorem 5.4.12(i). □

Example 5.4.16. Let V be a universe, C := (𝒞 , J) a V-small site, and X ∈ Ob(𝒞). Define the site C/X as in (4.7), and recall that the source functor s_{X} : 𝒞 /X → 𝒞 is continuous and cocontinuous for the topologies of C and C/X. Moreover, the functor s_{V!} commutes with fibre products, by proposition 1.4.13(vi.c), hence condition (C3) holds for s_{X}, and by corollary 5.4.15we conclude that s_{X} is a weak morphism of sites C → C/X.

Proposition 5.4.17. Let C := (𝒞 , J) and C′ := (𝒞′, J′) be two U-sites, and u : C → C′ a morphism of sites such that :

(a) u induces an equivalence of categories ũ_{∗} : C^{∼} →^{∼} C^{′∼}.

(b) The subset u(Ob(𝒞′)) ⊂ Ob(𝒞) is a topologically generating family for the site C. Then, for every universe V such that C is a V-site, the following holds :

(i) u induces an equivalence of categories ũ_{V∗} : C_{V}^{∼} →^{∼} C_{V}^{′∼}.

(ii) u induces a 2-equivalence of 2-categories V-Fib(u)^{∗} : V-Stack(C) →^{∼} V-Stack(C′).

Proof. (i): To ease notation, set v := ũ_{∗}. Recall that Can(C^{∼}) and Can(C^{′∼}) are isomorphic to U-sites (remark 4.4.1(iv)). By corollary 4.3.19(i.b) and theorem 4.4.2(iii) there follows, for every universe V containing U, an essentially commutative diagram

\xymatrix{Can(C^{∼})_{V}^{∼} \ar[r]^{ṽ_{V∗}} \ar[d] & Can(C^{′∼})_{V}^{∼} \ar[d] \\ C_{V}^{∼} \ar[r]^{ũ_{V∗}} & C_{V}^{′∼}}

whose vertical arrows are equivalences. Since v is an equivalence, the same holds for ṽ_{∗}, and then also for ũ_{∗}. Next, let V be any universe such that C is a V-site, and pick another universe V′ containing V and U. We deduce a commutative diagram

\xymatrix{C_{V}^{∼} \ar[r]^{ũ_{V∗}} \ar[d] & C_{V}^{′∼} \ar[d] \\ C_{V′}^{∼} \ar[r]^{ũ_{V′∗}} & C_{V′}^{′∼}}

whose vertical arrows are the fully faithful inclusion functors and whose bottom horizontal arrow is an equivalence, by the foregoing case. It follows already that ũ_{V∗} is fully faithful. We may now argue as in the proof of theorem 4.3.13: let F be any V-sheaf on C′; we know already that there exists a V′-sheaf G on C such that ũ_{∗V′}G is isomorphic to F, and we need to check that GX is essentially V-small for every X ∈ Ob(𝒞). But by condition (b) we may find a covering family (uY_{i} → X | i ∈ I), and by claim 4.3.16 we may assume that I is V-small. Then the induced map GX → ∏_{i∈I} G(uY_{i}) is injective, and by assumption G(uY_{i}) ≃ F Y_{i} is essentially V-small for every i ∈ I, whence the contention.

(ii): Suppose first that V contains U and both 𝒞^{∧} and 𝒞^{′∧} are V-small; notice that u fulfills condition (C3) of (5.4.5) (remark 5.4.6(ii)), and therefore V-Fib(u)^{∗} sends V-stacks on C to Vstacks on C′ (corollary 5.4.15). Likewise, V-Fib(v)^{∗} sends V-stacks on Can(C^{∼}) to V-stacks on Can(C^{′∼}). By corollary 4.3.19(i.b) and theorem 5.4.12(ii) we then get a pseudo-commutative diagram

\xymatrix{V-Stack(Can(C^{∼})) \ar[r]^{V-Fib(v)^{∗}} \ar[d] & V-Stack(Can(C^{′∼})) \ar[d] \\ V-Stack(C) \ar[r]^{V-Fib(u)^{∗}} & V-Stack(C′)}

whose vertical arrows are 2-equivalences. Since v is an equivalence, V-Fib(v)^{∗} is also a 2equivalence, and then the same holds for V-Fib(u)^{∗}. Lastly, let V be any universe such that C is a V-small site, and pick a universe V′ containing V and U, and such that 𝒞^{∧} and 𝒞^{′∧} are V′-small; we deduce a commutative diagram

\xymatrix{V-Stack(C) \ar[r]^{V-Fib(u)^{∗}} \ar[d] & V-Stack(C′) \ar[d] \\ V′-Stack(C) \ar[r]^{V′-Fib(u)^{∗}} & V′-Stack(C′)}

whose vertical arrows are the fully faithful inclusion strict pseudo-functors, and whose bottom horizontal arrow is a 2-equivalence, by the foregoing case. It follows already that V-Fib(u)^{∗} is fully faithful. We remark :

Claim 5.4.18. Let T ⊂ Ob(𝒞) be an essentially V-small topologically generating family of the site C, and F : ℰ → 𝒞 a V′-stack on C such that F^{−1}X is an essentially V-small category for every X ∈ T. Then ℰ is equivalent to a V-stack on C.

Proof of the claim. Denote by 𝒯 ⊂ 𝒞 the full subcategory with Ob(𝒯 ) = T, let j : 𝒯 → 𝒞 be the inclusion functor, and endow 𝒯 with the topology J_{𝒯} induced by J via the functor j. Let also η_{ℰ} : ℰ → ℱ := V′-Fib(j)_{∗} ∘ V′-Fib(j)^{∗}(ℰ) be the unit of adjunction. By proposition 4.3.18(ii), the functor j is a morphism of V′-sites C → (𝒯 , J_{𝒯} ), hence ℰ′ := V′-Fib(j)^{∗}(ℰ) is a V′-stack on (𝒯 , J_{𝒯} ) (remark 5.4.6(ii) and corollary 5.4.15), and by assumption its fibres are V-small, i.e. it is a V-stack on (𝒯 , J_{𝒯} ). Moreover, since j is cocontinuous for the topologies J and J_{𝒯} (proposition 4.3.18(i)), we also see that ℱ is a V′-stack on C (proposition 5.4.1(ii)). Furthermore, since j is fully faithful, the same holds for V′-Fib(j)_{∗} (remark 3.3.15(iv)), so the counit of adjunction ε_{ℰ′} : V′-Fib(j)^{∗}(ℱ) → ℰ′ is an equivalence. By virtue of the triangular modifications for the pair (η, ε) (theorem 2.4.24(i)), it follows that V′-Fib(j)^{∗}(η_{ℰ}) is an equivalence as well, and then it is i-covering for i = 0, 1, 2 (proposition 5.2.9). By invoking lemma 5.4.9 we conclude finally that η_{ℰ} is i-covering for i = 0, 1, 2, so it is an equivalence, again by proposition 5.2.9. Summing up, we are reduced to checking that if ℰ′ is a V-stack on (𝒯 , J_{𝒯} ), then ℱ′ := V′-Fib(j)_{∗}(ℰ′) is a V-stack on C. To this aim, pick any cleavage c : 𝒯 ^{o} → V-Cat for ℰ′; by construction, the fibre of ℱ′ over any X ∈ Ob(𝒞) is the 2-limit of the pseudofunctor c ∘ t_{X^{o}} : X^{o}/j^{o}𝒯 ^{o} → V-Cat, and it suffices to remark that X^{o}/j^{o}𝒯 ^{o} is a V-small category. ♢

Now, by assumption 𝒞 admits a V-small topologically generating family T, and by assumption (b), for every X ∈ T we may find a covering family (uY_{i} → X | i ∈ I_{X}). By claim 4.3.16, for every X ∈ T there exists then a V-small subset I′_{X} ⊂ I_{X} such that (uY_{i} → X | i ∈ I′_{X}) is still a covering family; set T′ := {uY_{i} | X ∈ T, i ∈ I_{X}}. Clearly T′ is a V-small topologically generating family for C. Lastly, let ℰ′ be any V-stack on C′; we know already that there exists a V′-stack ℰ on C such that V′-Fib(u)^{∗}(ℰ) is equivalent to ℰ′. But the latter condition implies that the fibre of ℰ over each X ∈ T′ is a V-small category; by claim 5.4.18we deduce that ℰ is equivalent to a V-stack on C, and this shows that V-Fib(u)^{∗} is a 2-equivalence, as stated. □

Corollary 5.4.19. For every U-site C := (𝒞 , J) the inclusion pseudo-functor Stack(C) → Fib(𝒞) admits a left 2-adjoint

Fib(𝒞) → Stack(C) ℰ ↦ ℰ^{a}.

Proof. Let T ⊂ Ob(𝒞) be an essentially small topologically generating family, 𝒯 ⊂ 𝒞 the full subcategory with Ob(𝒯 ) = T, and j : 𝒯 → 𝒞 the inclusion functor. Endow 𝒯 with the topology J_{𝒯} induced by J via j. Let also ℰ → 𝒞 be any fibration with essentially small fibres, and set ℰ′ := Fib(j)^{∗}(ℰ). Let ℰ^{′a} be the stack associated with ℰ′ (theorem 5.1.24), and denote by i : ℰ′ → ℰ^{′a} the natural morphism of fibrations over 𝒯 . Choose any universe V containing U and such that 𝒞 is V-small, and consider the composition

β_{ℰ} : ℰ ⟶^{η_{ℰ}} V-Fib(j)_{∗}(ℰ′) ⟶^{V-Fib(j)_{∗}(i)} ℰ^{∗} := V-Fib(j)_{∗}(ℰ^{′a})

where η_{ℰ} is the unit of adjunction. By arguing as in the proof of claim 5.4.18, we see that ℰ^{∗} is a stack with essentially small fibres on C and that V-Fib(j)^{∗}(η_{ℰ}) is an equivalence of categories. Moreover, since j is fully faithful, the counit of adjunction V-Fib(j)^{∗} ∘ V-Fib(j)_{∗} ⇒ 1_{V-Fib(𝒯 )} is a pseudo-natural equivalence, so we have an essentially commutative diagram of functors

\xymatrix{V-Fib(j)^{∗} ∘ V-Fib(j)_{∗}(ℰ′) \ar[r]^{V-Fib(j)^{∗}∘V-Fib(j)_{∗}(i)} \ar[d] & V-Fib(j)^{∗}(ℰ^{∗}) \ar[d] \\ ℰ′ \ar[r]^{i} & ℰ^{′a}}

whose vertical arrows are equivalences of categories. Since i is l-covering for l = 0, 1, 2 (proposition 5.2.11(i)), we conclude that the same holds for V-Fib(j)^{∗}(β_{ℰ}), and then also for β_{ℰ}, by virtue of lemma 5.4.9. Let us denote by ℰ_{V}^{a} the V-stack associated with ℰ (a priori, its fibres are only essentially V-small), and i_{V} : ℰ → ℰ_{V}^{a} the unit of adjunction; there exists a morphism of fibrations ω_{ℰ} : ℰ_{V}^{a} → ℰ^{∗}, pseudo-natural in ℰ, such that the functor ω_{ℰ} ∘ i_{V} is isomorphic to β_{ℰ}. Since i_{V} is l-covering for l = 0, 1, 2 (proposition 5.2.11(i)), it follows that the same holds for ω_{ℰ} (lemma 5.2.4(ii)), and then the latter is an equivalence of categories (proposition 5.2.9). Summing up, the rule : ℰ ↦ ω_{ℰ} yields a pseudo-natural equivalence of pseudo-functors, so the pseudo-functor given by the rule : ℰ ↦ ℰ^{∗} is the sought left 2-adjoint. □

5.4.20. Let now C := (𝒞 , J) and C′ := (𝒞′, J′) be two U-sites, u : C → C′ a weak morphism of sites (definition 5.4.2); then, for every universe V with U ⊂ V, the pseudo-functor V-St(u)_{∗} admits a left 2-adjoint

V-St(u)^{∗} : V-Stack(C′) → V-Stack(C)

defined as follows. We choose an essentially U-small topologically generating subset G ⊂ Ob(𝒞′) and let 𝒢 ⊂ 𝒞′ be the full subcategory with Ob(𝒢) = G. We endow 𝒢 with the topology J_{𝒢} induced by J′ via the inclusion functor j : 𝒢 → 𝒞′. Recall that j is a morphism of sites C′ → (𝒢 , J_{𝒢}) and j̃_{∗} : C^{′∼} →^{∼} (𝒢 , J_{𝒢})^{∼} is an equivalence (proposition 4.3.18(ii)). Then V-St(j)_{∗} is well defined (remark 5.4.6(ii)) and it is a 2-equivalence (proposition 5.4.17(ii)), and V-St(u)^{∗} is given by the composition :

V-Stack(C′) ⟶^{V-St(j)_{∗}} V-Stack(𝒢 , J_{𝒢}) ⟶^{V-Fib(u∘j)_{!}} V-Fib(𝒞) ⟶^{(−)^{a}} V-Stack(C)

where (−)^{a} is the pseudo-functor of corollary 5.4.19. It is then easily seen that the resulting pseudo-functor is a left 2-adjoint for V-St(u)_{∗}; especially, it is independent, up to pseudo-natural equivalence, of the choice of G (remark 2.4.28(iii)); the details shall be left to the reader. As usual, we shall often omit mentioning V, in case V = U.

Example 5.4.21. By remark 5.4.6(ii), every morphism of sites u : C := (𝒞 , J′) → C′ := (𝒞′, J) fulfills (C3), hence it is a weak morphism of sites (corollary 5.4.15), and so it induces a pseudo-functor V-St(u)_{∗}. If C and C′ are U-sites, we have also the pseudo-functor V-St(u)^{∗}.

Proposition 5.4.22. Let C := (𝒞 , J) and C′ := (𝒞′, J′) be two small sites, u : C → C′ a morphism of sites, i ∈ {0, 1, 2}, and f : ℰ → ℱ an i-covering cartesian functor of 𝒞′- fibrations. We have :

(i) The functor Fib(u)_{!}(f) : Fib(u)_{!}(ℰ) → Fib(u)_{!}(ℱ) is i-covering.

(ii) If ℰ and ℱ are stacks on C′, then St(u)^{∗}(f) : St(u)^{∗}(ℰ) → St(u)^{∗}(ℱ) is i-covering.

Proof. We show first the following special case :

Claim 5.4.23. The proposition holds if u is a morphism of lex-sites.

Proof of the claim. By proposition 5.2.11(iii), it suffices to show assertion (i) of the proposition. To this aim, we consider the essentially commutative diagram

\xymatrix{Fib(u)_{!}(C(ℰ)) \ar[r]^{Fib(u)_{!}(C(f))} \ar[d]_{Fib(u)_{!}(ev^{ℰ} )} & Fib(u)_{!}(C(ℱ)) \ar[d]^{Fib(u)_{!}(ev^{ℱ} )} \\ Fib(u)_{!}(ℰ) \ar[r]^{Fib(u)_{!}(f)} & Fib(u)_{!}(ℱ)}

where ev^{•} is the pseudo-natural equivalence described in (3.2.5). Then both vertical arrows of the diagram are equivalences (lemma 2.4.11(ii)), so Fib(u)_{!}(f) is i-covering if and only if the same holds for Fib(u)_{!}(C(f)) (lemma 5.2.3), and by the same token, f is i-covering if and only if the same holds for C(f). Thus, we may replace f by C(f), and assume from start that f is a split cartesian functor (ℰ , λ^{ℰ}) → (ℱ , λ^{ℱ} ). In this case, let c^{ℰ} and c^{ℱ} be the strict pseudofunctors associated with the cleavages λ^{ℰ} and λ^{ℱ} ; then Fib(u)_{!}(ℰ) is equivalent to the fibration associated with the pseudo-functor

d^{ℰ} : 𝒞^{o} → Cat X ↦ 2-colim_{(X/u𝒞′)^{o}}c^{ℰ} ∘ t_{X}^{o}

where t_{X} : X/u𝒞′ → 𝒞′ is the target functor. Hence, d_{X}^{ℰ} = ℱ ib(c^{ℰ} ∘t_{X}^{o})[Σ_{ℰ ,X}^{−1}], where Σ_{ℰ ,X} is the set of cartesian morphisms of ℱ ib(c^{ℰ} ∘ t_{X}^{o}) (see the proof of theorem 3.3.9). The universal pseudo-cocone π^{ℰ ,X} : c^{ℰ} ∘ t_{X}^{o} ⇒ F_{d_{X}^{ℰ}} is the strict pseudo-natural transformation that assigns to every object (X′, ψ : X → uX′) of (X/u𝒞′)^{o} the functor

π_{(X′,ψ)}^{ℰ ,X} : c_{X′}^{ℰ} → d_{X}^{ℰ} E ↦ ((X′, ψ), E).

To every morphism φ : X → Y of 𝒞 we then attach the functor

d_{φ}^{ℰ} : d_{Y}^{ℰ} → d_{X}^{ℰ} such that π^{ℰ ,X} ∗ φ^{∗o} = F_{d_{φ}^{ℰ}} ⊙ π^{ℰ ,Y}

where φ^{∗} : Y/u𝒞′ → X/u𝒞′ is the functor induced by φ (see example 2.2.8(ii)); unwinding the definitions, we find that d_{φ}^{ℰ} is given by the rule :

((X′, ψ : Y → uX′), E) ↦ ((X′, ψ ∘ φ), E) for every ((X′, ψ), E) ∈ Ob(d_{Y}^{ℰ}).

Likewise we describe Fib(u)_{!}(ℱ) up to equivalence; next, the split cartesian functor f corresponds to a strict pseudo-natural transformation c^{f} : c^{ℰ} ⇒ c^{ℱ} , and under the foregoing identifications, the cartesian functor Fib(u)_{!}(f) corresponds to the strict pseudo-natural transformation d^{f} : d^{ℰ} ⇒ d^{ℱ} assigning to every X ∈ Ob(𝒞) the functor :

d_{X}^{f} : d_{X}^{ℰ} → d_{X}^{ℱ} ((X′, ψ), E) ↦ ((X′, ψ), c_{X′}^{f}E).

The objects of ℱ ib(d^{ℰ}) are then the data (X, X′, ψ, E) with X ∈ Ob(𝒞) and ((X′, ψ), E) ∈ Ob(d_{X}^{ℰ}), and likewise for the objects of ℱ ib(d^{ℱ} ). After this preparation, suppose that f is 0-covering, and let us check that the same holds for Fib(u)_{!}(f), where the latter is naturally identified with ℱ ib(d^{f}). Thus, consider any object (X, X′, ψ : X → uX′, F) of ℱ ib(d^{ℱ} ); according to remark 5.2.2, we need to find a covering family (φ_{λ} : Y_{λ} → X | λ ∈ Λ) and for every λ ∈ Λ an object \underline{E}_{λ} := ((Y′_{λ}, ψ_{λ}), E) ∈ d_{Y_{λ}}^{ℰ} with an isomorphism

((Y′_{λ}, ψ_{λ}), c_{Y′_{λ}}^{f}E_{λ}) = d_{Y_{λ}}^{f}((Y′_{λ}, ψ_{λ}), E_{λ}) →^{∼} d_{φ_{λ}}^{ℱ}((X′, ψ), F) = ((X′, ψ ∘ φ_{λ}), F) in d_{Y_{λ}}^{ℱ}.

But since f is 0-covering, there exist a covering (τ_{λ} : Y′_{λ} → X′ | λ ∈ Λ) for the topology J′ and an object E_{λ} ∈ Ob(c_{Y′_{λ}}^{ℰ}) with an isomorphism ω_{λ} : c_{Y′_{λ}}^{f}E_{λ} →^{∼} c_{τ_{λ}}^{ℱ}F in c_{Y′_{λ}}^{ℱ} for every λ ∈ Λ. Thus, let us set Y_{λ} := X ×_{uX′} uY′_{λ}, and denote by φ_{λ} : Y_{λ} → X the induced projection, for every λ ∈ Λ. Since u is continuous, the family (u(τ_{λ}) | λ ∈ Λ) covers uX′ for the topology J, and therefore (φ_{λ} | λ ∈ Λ) covers X. Lastly, the sought isomorphism shall be the composition of the isomorphism ((Y′_{λ}, ψ_{λ}), c_{Y′_{λ}}^{f}E_{λ}) →^{∼} ((Y′_{λ}, ψ_{λ}), c_{τ_{λ}}^{ℱ}F) of ℱ ib(c^{ℱ} ∘ t_{Y_{λ}}^{o}) induced by ω_{λ}, and the cartesian morphism ((Y′_{λ}, ψ_{λ}), c_{τ_{λ}}^{ℱ}F) → ((X′, ψ ∘ φ_{λ}), F) of ℱ ib(c^{ℱ} ∘ t_{Y_{λ}}^{o}).

Next, suppose that f is 1-covering, and let ((X′_{i}, ψ_{i} : X → uX′_{i}), E_{i}) for i = 1, 2 be two objects of d_{X}^{ℰ} with a morphism g : ((X′_{1}, ψ_{1}), c_{X′_{1}}^{f}E_{1}) → ((X′_{2}, ψ_{2}), c_{X′_{2}}^{f}E_{2}) in d_{X}^{ℱ}. Notice that (Z/u𝒞′)^{o} is a filtered category for every Z ∈ Ob(𝒞) (example 1.3.16(i)), hence Σ_{ℱ ,Z} admits a right calculus of fraction (example 3.3.13(ii)); moreover, every cartesian morphism (σ, s) : ((Z′′, ψ′), F′) → ((Z′, ψ), F) in Σ_{ℱ ,Z} admits the factorization :

(5.4.24)  (σ, s) = λ^{ℱ} (F, σ) ∘ (1_{Z′′}, s)  in ℱ ib(c^{ℱ} ∘ t_{Z}^{o})

where (1_{X′′}, s) is an isomorphism of c_{X′′}^{ℱ} (lemma 3.1.20(i)), so g can be represented by a pair

((X′_{1}, ψ_{1}), c_{X′_{1}}^{f}E_{1}) ⟵^{λ^{ℱ} (c_{X′_{1}}^{f}E_{1},ρ)} ((X′′_{1}, ψ′_{1}), c_{ρ}^{ℱ} ∘ c_{X′_{1}}^{f}E_{1}) ⟶^{g′} ((X′_{2}, ψ_{2}), c_{X′_{2}}^{f}E_{2})

where ρ : X′′_{1} → X′_{1} is a morphism of 𝒞′ and ψ′_{1} : X → uX′′_{1} is a morphism of 𝒞 such that u(ρ) ∘ ψ′_{1} = ψ_{1}. Also, g′ is a morphism in ℱ ib(c^{ℱ} ∘ t_{X}^{o}). Then in turn, g′ is a pair (X/μ, t), where X/μ is a morphism (X′′_{1}, ψ′_{1}) → (X′_{2}, ψ_{2}) in X/u𝒞′, i.e. a morphism μ : X′′_{1} → X′_{2} in 𝒞′ with u(μ) ∘ ψ_{1} = ψ_{2}, and g′ is a morphism in c_{X′′_{1}}^{ℱ}

g′ : c_{X′′_{1}}^{f} ∘ c_{ρ}^{ℰ} E_{1} = c_{ρ}^{ℱ} ∘ c_{X′_{1}}^{f}E_{1} → c_{μ}^{ℱ} ∘ c_{X′_{2}}^{f}E_{2} = c_{X′′_{1}}^{f} ∘ c_{μ}^{ℰ}E_{2}.

According to remark 5.2.2, we need to exhibit a covering family (φ_{λ} : Y_{λ} → X | λ ∈ Λ), and for every λ ∈ Λ a morphism g_{λ} : ((X′_{1}, ψ_{1} ∘ φ_{λ}), E_{1}) → ((X′_{2}, ψ_{2} ∘ φ_{λ}), E_{2}) in d_{Y_{λ}}^{ℰ} such that d_{X′_{1}}^{f}(g_{λ}) = d_{φ_{λ}}^{ℱ}(g). Now, since f is 1-covering, there exist a covering family (τ_{λ} : Y′′_{λ} → X′′_{1} | λ ∈ Λ) and for every λ ∈ Λ a morphism

g′_{λ} : c_{τ_{λ}}^{ℰ} ∘ c_{ρ}^{ℰ} E_{1} → c_{τ_{λ}}^{ℰ} ∘ c_{μ}^{ℰ}E_{2} in c_{Y′′_{λ}}^{ℰ} such that c_{Y′′_{λ}}^{f}(g′_{λ}) = c_{τ_{λ}}^{ℱ}(g′).

We set Y_{λ} := X ×_{uX′′_{1}} uY′′_{λ}, and we let φ_{λ} : Y_{λ} → X and ψ′′_{λ} : Y_{λ} → uY′′_{λ} be the induced projections for every λ ∈ Λ; arguing as in the foregoing it is easily seen that the family (g_{λ} | λ ∈ Λ) covers X, and we define g_{λ} as the morphism of d_{Y_{λ}}^{ℰ} represented by the pair of morphisms :

((X′_{λ}, ψ_{1} ∘ φ_{λ}), E_{1}) ⟵^{λ^{ℰ} (E_{1},ρ∘τ_{λ})} ((Y′′_{λ}, ψ′′_{λ}), c_{ρ∘τ_{λ}}^{ℰ}E_{1}) ⟶^{(Y_{λ}/μ∘τ_{λ},g′_{λ})} ((X′_{2}, ψ_{2} ∘ φ_{λ}), E_{2}).

In view of remark 3.2.2(ii) we see that d_{Y_{λ}}^{f}(g′_{λ}) is represented by the pair

((X′_{λ}, ψ_{1}φ_{λ}), c_{X′_{1}}^{f}E_{1}) ⟵^{λ^{ℱ} (c_{X′_{1}}^{f}E_{1},ρτ_{λ})} ((Y′′_{λ}, ψ′′_{λ}), c_{Y′′_{λ}}^{f}c_{ρτ_{λ}}^{ℰ}E_{1}) ⟶^{(Y_{λ}/μτ_{λ},c_{τ_{λ}}^{ℱ}(g′))} ((X′_{2}, ψ_{2}φ_{λ}), c_{X′_{2}}^{f}E_{2})

and by remark 3.2.2(i) we have :

λ^{ℱ} (c_{X′_{1}}^{f}E_{1}, ρτ_{λ}) = λ^{ℱ} (c_{X′_{1}}^{f}E_{1}, ρ) ∘ λ^{ℱ} (c_{ρ}^{ℱ} c_{X′_{1}}^{f}E_{1}, τ_{λ}).

On the other hand, (Y_{λ}/μ ∘ τ_{λ}, c_{τ_{λ}}^{ℱ}(g′)) is the composition :

((Y′′_{λ},ψ′′_{λ}), c_{Y′′_{λ}}^{f}c_{ρτ_{λ}}^{ℰ}E_{1}) ⟶^{λ^{ℱ} (c_{ρ}^{ℱ} c_{X′_{1}}^{f}E_{1},τ_{λ})} ((X′′_{λ}, ψ′_{1}φ_{λ}), c_{ρ}^{ℱ} c_{X′_{1}}^{f}E_{1}) ⟶^{d_{φ_{λ}}^{ℱ}(X/μ,g′)} ((X′_{2}, ψ_{2}φ_{λ}), c_{X′_{2}}^{f}E_{1})

so d_{Y_{λ}}^{f}(g′_{λ}) is also represented by the pair

((X′_{λ}, ψ_{1}φ_{λ}), c_{X′_{1}}^{f}E_{1}) ⟵^{λ^{ℱ} (c_{X′_{1}}^{f}E_{1},ρ)} ((X′′_{λ}, ψ′_{1}φ_{λ}), c_{ρ}^{ℱ} c_{X′_{1}}^{f}E_{1}) ⟶^{d_{φ_{λ}}^{ℱ}(X/μ,g′)} ((X′_{2}, ψ_{2}φ_{λ}), c_{X′_{2}}^{f}E_{1})

which represents d_{φ_{λ}}^{ℱ}(g) as well, as required.

Lastly, suppose that f is 2-covering, and consider objects ((X′_{j}, ψ_{j}), E_{j}) of d_{X}^{ℰ} for j = 1, 2, and two morphisms g_{1}, g_{2} : ((X′_{1}, ψ_{1}), E_{1}) → ((X′_{2}, ψ_{2}), E_{2}) in d_{X}^{ℰ} such that

(5.4.25)  d_{X}^{f}(g_{1}) = d_{X}^{f}(g_{2}).

We need to exhibit a covering family (φ_{λ} : Y_{λ} → X | λ ∈ Λ) such that d_{φ_{λ}}^{ℰ}(g_{1}) = d_{φ_{λ}}^{ℰ}(g_{2}) for every λ ∈ Λ. We reduce easily to the case where g_{1} and g_{2} are two classes of morphisms of ℱ ib(c^{ℰ} ∘ t_{X}^{o}). In this case, (5.4.25) means that there exists a cartesian morphism h : ((X′′_{1}, ψ′_{1}), F) → ((X′_{1}, ψ_{1}), c_{X′_{1}}^{f}E_{1}) in ℱ ib(c^{ℱ} ∘t_{X}^{o}) such that ℱ ib(c^{f} ∘t_{X}^{o})(g_{1})∘h = ℱ ib(c^{f} ∘ t_{X}^{o})(g_{2}) ∘ h. However, we have a factorization h = λ^{ℱ} (c_{X′_{1}}^{f}E_{1}, μ) ∘ (1_{X′′_{1}}, h′) as in (5.4.24), where (1_{X′′_{1}}, h′) is an isomorphism of c_{X′′_{1}}^{ℱ}, so we may assume that h = λ^{ℱ} (c_{X′_{1}}^{f}E_{1}, μ) = ℱ ib(c^{f} ∘ t_{X}^{o})(λ^{ℰ} (E_{1}, μ)). Then we may as well replace g_{i} by g_{i} ∘ λ^{ℰ} (E_{1}, μ) for i = 1, 2, and assume from start that

ℱ ib(c^{f} ∘ t_{X}^{o})(g_{1}) = ℱ ib(c^{f} ∘ t_{X}^{o})(g_{2}).

Especially, there exist a morphism ρ : X′_{1} → X′_{2} in 𝒞′ such that u(ρ)∘ψ_{1} = ψ_{2}, and morphisms g′_{i} : E_{1} → c_{ρ}^{ℰ} E_{2} such that g_{i} = (ρ, g′_{i}) for i = 1, 2, and c_{X′_{1}}^{f}(g′_{1}) = c_{X′_{1}}^{f}(g′_{2}). Then, since f is 2-covering, we find a covering family (τ_{λ} : Y′_{λ} → X′_{1} | λ ∈ Λ) such that

(5.4.26)  c_{τ_{λ}}^{ℰ}(g′_{1}) = c_{τ_{λ}}^{ℰ}(g′_{2})  for every λ ∈ Λ.

We set again Y_{λ} := X ×_{uX′_{1}} uY′_{λ}, and we let φ_{λ} : Y_{λ} → X and ψ_{λ} : Y_{λ} → Y′_{λ} be the induced projections, for every λ ∈ Λ. We claim that the resulting covering (φ_{λ} | λ ∈ Λ) of X will do.

Indeed, d_{φ_{λ}}^{ℰ}(g_{i}) : ((X′_{1}, ψ_{1} ∘ φ_{λ}), E_{1}) → ((X′_{2}, ψ_{2} ∘ φ_{λ}), E_{2}) is represented by the morphism (Y_{λ}/ρ, g′_{i}) of ℱ ib(c^{ℰ} ∘ t_{Y_{λ}}^{o}) for i = 1, 2, and (5.4.26) implies easily that

(Y_{λ}/ρ, g′_{1}) ∘ λ^{ℰ} (E_{1}, τ_{λ}) = (Y_{λ}/ρ, g′_{2}) ∘ λ^{ℰ} (E_{1}, τ_{λ})

where λ^{ℰ} (E_{1}, τ_{λ}) : ((Y′_{λ}, ψ_{λ}), c_{τ_{λ}}^{ℰ}E_{1}) → ((X′_{1}, ψ_{1} ∘ φ_{λ}), E_{1}) is a cartesian morphism of the fibration ℱ ib(c^{ℰ} ∘ t_{Y_{λ}}^{o}), whence the contention. ♢

We consider now the essentially commutative diagram

\xymatrix{Fib(u)_{!}(ℰ) \ar[r]^{Fib(u)_{!}(f)} \ar[d]_{Fib(u)_{!}(η_{ℰ} )} & Fib(u)_{!}(ℱ) \ar[d]^{Fib(u)_{!}(η_{ℱ} )} \\ Fib(u)_{!}(ℰ^{a}) \ar[r]^{Fib(u)_{!}(f^{a})} & Fib(u)_{!}(ℱ^{a})}

where η_{ℰ} : ℰ → ℰ^{a} and η_{ℱ} : ℱ → ℱ^{a} are the natural morphisms, and notice that Fib(u)_{!}(η_{ℰ})^{a} and Fib(u)_{!}(η_{ℱ} )^{a} are equivalences, by virtue of proposition 5.4.4. Then Fib(u)_{!}(f)^{a} is icovering if and only if the same holds for Fib(u)_{!}(f^{a})^{a} (lemma 5.2.3), and finally Fib(u)_{!}(f) is i-covering if and only if the same holds for Fib(u)_{!}(f^{a}) (proposition 5.2.11(iii)). Summing up, we see that, in order to prove the proposition, we may assume that ℰ and ℱ are stacks on C′, and moreover, in this case it suffices to show assertion (ii).

Now, by lemma 4.2.11(ii) we have the essentially commutative diagram :

\xymatrix{𝒞′ \ar[r]^{u} \ar[d]_{h_{C′}^{a}} & 𝒞 \ar[d]^{h_{C}^{a}} \\ C_{U}^{′∼} \ar[r]^{_{ũ_{∗}}} & C_{U}^{∼}}

whose vertical arrows are the Yoneda embeddings. Especially, for every universe V containing U and such that 𝒞_{U}^{∧} is V-small, V-St(h_{C′}^{a})_{∗} is a 2-equivalence from the 2-category of stacks on Can(C_{U}^{′∼}) to that of stacks on C (theorem 5.4.12), and therefore there exists a cartesian functor f′ : ℰ′ → ℱ′ of V-stacks on Can(C_{U}^{′∼}) with an essentially commutative diagram :

𝒟 : \xymatrix{ℰ \ar[r]^{f} \ar[d] & ℱ \ar[d] \\ V-St(h_{C′}^{a})_{∗}(ℰ′) \ar[r]^{V-St(h_{C′}^{a})_{∗}(f′)} & V-St(h_{C′}^{a})_{∗}(ℱ′)}

whose vertical arrows are equivalences. We notice :

Claim 5.4.27. Let C := (𝒞 , J) be a small site, h_{C}^{a} : 𝒞 → C^{∼} the Yoneda embedding, φ : 𝒜 → ℬ a morphism of V-Fib(C^{∼}), and j ∈ {0, 1, 2}. Then φ is j-covering for the canonical topology of C^{∼} if and only if V-Fib(h_{C}^{a})^{∗}(φ) is j-covering for the topology J.

Proof of the claim. Suppose that φ is j-covering; since h_{C}^{a} is cocontinuous for the topologies J (theorem 4.4.2(iii)) and Can_{C^{∼}}, proposition 5.4.1(i) says that V-Fib(h_{C}^{a})^{∗}(φ) is j-covering. The converse follows from lemma 5.4.9. ♢

Since f is i-covering, the same holds for V-St(h_{C′}^{a})_{∗}(f′), by lemma 5.2.3, and then also for f′, by claim 5.4.27. Moreover, considering the essentially commutative diagram V-St(u)^{∗}(𝒟) and applying again lemma 5.2.3, we are reduced to checking that V-St(u)^{∗} ∘ V-St(h_{C′}^{a})_{∗}(f′) is i-covering. To ease notation, set S := V-St(h_{C}^{a})_{∗} ∘ V-St(h_{C}^{a})^{∗} ∘ V-St(u)^{∗} ∘ V-St(h_{C′}^{a})_{∗}; recalling that V-St(h_{C}^{a})^{∗} is a pseudo-inverse for the 2-equivalence V-St(h_{C}^{a})_{∗}, we get an essentially commutative diagram

\xymatrix{V-St(u)^{∗} ∘ V-St(h_{C′}^{a})_{∗}(ℰ′) \ar[r]^{V-St(u)^{∗}∘V-St(h_{C′}^{a})_{∗}(f′)} \ar[d] & V-St(u)^{∗} ∘ V-St(h_{C′}^{a})_{∗}(ℱ′) \ar[d] \\ S(ℰ) \ar[r]^{S(f′)} & S(ℱ)}

whose vertical arrows are equivalences; invoking once again lemma 5.2.3, we are then reduced to checking that S(f) is i-covering. However, we have pseudo-natural equivalences :

S →^{∼} V-St(h_{C}^{a})_{∗} ∘ V-St(ũ^{∗})^{∗} ∘ V-St(h_{C′}^{a})^{∗} ∘ V-St(h_{C′}^{a})_{∗} →^{∼} V-St(h_{C}^{a})_{∗} ∘ V-St(ũ^{∗})^{∗}

which, after applying once more lemma 5.2.3, further reduces to checking that the functor V-St(h_{C}^{a})_{∗} ∘ V-St(ũ^{∗})^{∗}(f′) is i-covering. The latter follows from claims 5.4.23and 5.4.27. □

5.4.28. Let C := (𝒞 , J) and C′ := (𝒞′, J′) be two small sites, and u : 𝒞 → 𝒞′ a cocontinuous functor; by proposition 5.4.1(ii), the pseudo-functor Fib(u)_{∗} restricts to a pseudo-functor

St(ŭ)_{∗} : Stack(C) → Stack(C′)

and it is easily seen that the latter admits a left 2-adjoint

St(ŭ)^{∗} : Stack(C′) ⟶^{i} Fib(𝒞′) ⟶^{Fib(u)^{∗}} Fib(𝒞) ⟶^{(−)_{C}^{a}} Stack(C)

with i the inclusion pseudo-functor. On the other hand, by remark 4.4.13(iii,iv), the functor u induces a morphism of sites ŭ^{∗} : Can(C^{∼}) → Can(C^{′∼}), whence a 2-adjoint pair of pseudofunctors (St(ŭ^{∗})^{∗}, St(ŭ^{∗})_{∗}). Define also h_{C}^{a} : 𝒞 → C^{∼}, h_{C′}^{a} : 𝒞′ → C^{′∼} as in remark 4.1.19(iii).

Proposition 5.4.29. (i) In the situation of (5.4.28), we have pseudo-commutative diagrams :

\xymatrix{Stack(Can C^{′∼}) \ar[r]^{St(ŭ^{∗})^{∗}} \ar[d]_{St(h_{C′}^{a})_{∗}} & Stack(Can C^{∼}) \ar[d]^{St(h_{C}^{a})_{∗}} & Stack(Can C^{∼}) \ar[r]^{St(ŭ^{∗})_{∗}} \ar[d]_{St(h_{C}^{a})_{∗}} & Stack(Can C^{′∼}) \ar[d]^{St(h_{C′}^{a})_{∗}} \\ Stack(C′) \ar[r]^{St(ŭ)^{∗}} & Stack(C) & Stack(C) \ar[r]^{St(ŭ)_{∗}} & Stack(C′).}

(ii) If u is also a weak morphism of sites C′ → C, we have a pseudo-natural equivalence

St(ŭ)^{∗} →^{∼} St(u)_{∗}.

(iii) If u admits a right adjoint v : 𝒞′ → 𝒞 that is a weak morphism of sites C → C′, we have pseudo-natural equivalences :

St(v)_{∗} →^{∼} St(ŭ)_{∗} St(v)^{∗} →^{∼} St(ŭ)^{∗}.

Proof. (i): Since St(h_{C}^{a})_{∗} and St(h_{C′}^{a})_{∗} are 2-equivalences (theorem 5.4.12(i,ii)), it suffices to show the pseudo-commutativity of the left square diagram. To this aim, consider the sites C^{∧} and C^{′∧} on the categories 𝒞^{∧} and 𝒞^{′∧} defined in remark 4.1.31(ii), and notice that since u^{∧} : 𝒞^{∧} → 𝒞^{′∧} commutes with all limits and all colimits, it induces a morphism of topoi C^{′∧} → C^{∧} (proposition 4.4.8(i)), whence a morphism of sites u^{∧} : C^{′∧}) → C^{∧}) (remark 4.4.13(iii)); combining with lemma 4.2.9 and theorem 4.4.2(ii) we deduce an essentially commutative diagram of sites:

\xymatrix{Can(C^{′∼}) \ar[r]^{^{ŭ^{∗}}} \ar[d]_{(−)_{C′}^{a}} & Can(C^{∼}) \ar[d]^{(−)_{C}^{a}} \\ C^{′∧} \ar[r]^{u^{∧}} & C^{∧}.}

Notice as well that St((−)_{C}^{a})_{∗} : Stack(Can(C^{∼})) → Stack(C^{∧}) is a 2-equivalence (proposition 5.4.17(ii) and theorem 4.4.2(iii)), and likewise for St((−)_{C′}^{a})_{∗}. There follows a pseudocommutative diagram (details left to the reader) :

\xymatrix{Stack(Can(C^{′∼})) \ar[r]^{St(ŭ^{∗})^{∗}} \ar[d]_{St((−)_{C′}^{a})_{∗}} & Stack(Can(C^{∼})) \ar[d]^{St((−)_{C}^{a})_{∗}} \\ Stack(C^{′∧}) \ar[r]^{St(u^{∧})^{∗}} & Stack(C^{∧}).}

Thus, we are reduced to checking that the following diagram pseudo-commutes :

\xymatrix{Fib(𝒞^{′∧}) \ar[r]^{Fib(u^{∧})_{!}} \ar[d]_{Fib(h_{𝒞′} )^{∗}} & Fib(𝒞^{∧}) \ar[r]^{(−)_{C^{∧}}^{a}} \ar[d]_{Fib(h_{𝒞} )^{∗}} & Stack(C^{∧}) \ar[d]^{St(h_{𝒞} )_{∗}} \\ Fib(𝒞′) \ar[r]^{Fib(u)^{∗}} & Fib(𝒞) \ar[r]^{(−)_{C}^{a}} & Stack(C).}

However, the pseudo-commutativity of the left square subdiagram follows from proposition 3.3.19. For the right subdiagram, it suffices to observe that h_{𝒞} is cocontinuous for the sites C and C^{∧} (theorem 4.4.2(i)) and invoke proposition 5.4.1(i).

(ii) follows by inspecting the definitions, and the first pseudo-natural equivalence of (iii) follows from remark 3.3.15(vi); then the second pseudo-natural equivalence of (iii) follows from the first one and from remark 2.4.28(iii). □

5.4.30. Let C := (𝒞 , J) and C′ := (𝒞′, J′) be two U-sites, and u : C′ → C a weak morphism of sites. We have a commutative diagram of 2-categories :

𝒟_{u}  :  \xymatrix{Stack(C′) \ar[r]^{i_{C′}} \ar[d]_{St(u)_{∗}} & Fib(𝒞′) \ar[d]^{Fib(u)^{∗}} \\ Stack(C) \ar[r]^{i_{C}} & Fib(𝒞)}

where the inclusion pseudo-functors i_{C} and i_{C′} admit left 2-adjoints (−)_{C}^{a} : Fib(𝒞) → Stack(C) and (−)_{C′}^{a} : Fib(𝒞′) → Stack(C′) (corollary 5.4.19). After fixing an adjunction for these two pairs of 2-adjoint pseudo-functors, we may then regard 𝒟_{u} as a square of weak links as in (2.3.19), oriented by the identity pseudo-natural transformation. If we similarly fix adjunctions also for the pairs (Fib(u)_{!}, Fib(u)^{∗}) and (St(u)^{∗}, St(u)_{∗}), the diagram 𝒟_{u} becomes even an oriented square of links.

Corollary 5.4.31. In the situation of (5.4.30), suppose that the functor u : 𝒞 → 𝒞′ is also cocontinuous for the topologies J and J′. Then the base change transformation

Υ(𝒟_{u}) : (−)_{C}^{a} ∘ Fib(u)^{∗} → St(u)_{∗} ∘ (−)_{C′}^{a}

is a pseudo-natural equivalence.

Proof. Let V be a universe with U ⊂ V, and such that 𝒞 and 𝒞′ are V-small; we get a commutative diagram of 2-categories :

\xymatrix{V-Stack(C′) \ar[rrr]^{V-i_{C′}} \ar[ddd]_{V-St(u)_{∗}} &  &  & V-Fib(𝒞′) \ar[ddd]^{V-Fib(u)^{∗}} \\  & Stack(C′) \ar[r]^{i_{C′}} \ar[d]_{St(u)_{∗}} \ar@{-->}[ul]_{j_{C′}} & Fib(𝒞′) \ar[d]^{Fib(u)^{∗}} \ar@{-->}[ur]^{j_{𝒞′}} \\  & Stack(C) \ar[r]^{i_{C}} \ar[dl]_{j_{C}} & Fib(𝒞) \ar[dr]^{j_{𝒞}} \\ V-Stack(C) \ar[rrr]^{V-i_{C}} &  &  & V-Fib(𝒞)}

whose horizontal arrows are the inclusion pseudo-functors; since the latter all admit left 2adjoints, we may regard this diagram as an oriented cubical diagram of weak links as in 2.3.21, whose orientations are given by identities. Especially, the lower and upper trapezoidal subdiagrams are oriented respectively by 1_{j_{𝒞} ∘i_{C}} and 1_{j_{𝒞′} ∘i_{C′}}, and we remark :

Claim 5.4.32. Υ(1_{j_{𝒞} ∘i_{C}}) and Υ(1_{j_{𝒞′} ∘i_{C′}}) are pseudo-natural equivalences.

Proof of the claim. We show the assertion for Υ(1_{j_{𝒞} ∘i_{C}}) : the same argument shall apply to Υ(1_{j_{𝒞′} ∘i_{C′}}) as well. Let (η, ε) (resp. (V-η, V-ε)) be the unit and counit of a 2-adjunction for the 2-adjoint pair ((−)_{C}^{a}, i_{C}) (resp. for the 2-adjoint pair (V-(−)_{C}^{a}, V-i_{C})); since V-i_{C} is fully faithful, V-ε is a pseudo-natural equivalence, so it suffices to check that the same holds for (V-(−)_{C}^{a} ∘ j_{𝒞} ) ∗ η. But in view of the pseudo-natural equivalence V-(−)_{C}^{a} ∘ j_{𝒞} →^{∼} j_{C} ∘ (−)_{C}^{a}, we are further reduced to checking that (−)_{C}^{a} ∗ η is a pseudo-natural equivalence; the latter follows from the triangular identities for the pair (η, ε), since ε is a pseudo-natural equivalence. ♢

Denote by V-𝒟_{u} the “front face” of the foregoing diagram (with its identity orientation); from claim 5.4.32 and remark 2.3.22(i) we see that if Υ(V-𝒟_{u}) is a pseudo-natural equivalence, the same holds for Υ(𝒟_{u}). Thus, we may replace U by V, and assume from start that 𝒞 and 𝒞′ are small. Let now ℱ be any fibration on 𝒞′; the functor Υ(𝒟_{u})_{ℱ} : (Fib(u)^{∗}ℱ)^{a} → St(u)_{∗}(ℱ^{a}) is obtained explicitly as follows. First, let η_{ℱ} : ℱ → ℱ^{a} be the unit of the chosen 2-adjunction for the pair ((−)_{C′}^{a}, i_{C′}). Then Υ(𝒟_{u})_{ℱ} is the composition of the induced functor

(Fib(u)^{∗}η_{ℱ} )^{a} : (Fib(u)^{∗}ℱ)^{a} → (St(u)_{∗}(ℱ^{a}))^{a}

with the counit of adjunction ε_{ℱ^{a}} : (St(u)_{∗}(ℱ^{a}))^{a} → St(u)_{∗}(ℱ^{a}). We need to check that Υ(𝒟_{u})_{ℱ} is an equivalence; however, ε_{ℱ^{a}} is an equivalence, so it suffices to check that the same holds for (Fib(u)^{∗}η_{ℱ} )^{a}. By propositions 5.2.9 and 5.2.11(iii) we are reduced to showing that Fib(u)^{∗}(η_{ℱ} ) : Fib(u)^{∗}ℱ → St(u)_{∗}(ℱ^{a}) is i-covering for i = 0, 1, 2. Since u is cocontinuous, the latter assertion follows from propositions 5.4.1and 5.2.11(i). □

5.5. Sheaves of categories. We wish next to elucidate the relationship between stacks and the related notion of sheaf of categories on a given site. We begin by recalling the following :

Definition 5.5.1. Let C := (𝒞 , J) be any site, and 𝒜 any other category.

(i) A presheaf on 𝒞 with values in 𝒜 is any object of the category (notation of (1.1.10))

(𝒞 , 𝒜)^{∧} := Fun(𝒞^{o}, 𝒜).

(ii) Let V be a universe such that 𝒜 has V-small Hom-sets; following [59, Ch.0, §3.1], we say that such a presheaf ℱ is a sheaf on C with values in 𝒜 if for every T ∈ Ob(𝒜), the rule :

U ↦ ℱ_{T} (U) := Hom_{𝒜} (T, ℱ (U)) for every U ∈ Ob(𝒞)

defines a V-sheaf (of sets) ℱ_{T} on C. This condition is obviously independent of the choice of V. We denote by

(C, 𝒜)^{∼}

the full subcategory of (𝒞 , 𝒜)^{∧} whose objects are the sheaves on C with values in 𝒜.

Remark 5.5.2. (i) Let C := (𝒞 , J) be a site and ℱ a presheaf on 𝒞 with values in a category 𝒜. According to (4.1.6), the presheaf ℱ is a sheaf on C with values in 𝒜 if and only if, for every T ∈ Ob(𝒜), every U ∈ Ob(𝒞), and every sieve 𝒮 ∈ J(U) the natural map

ℱ_{T} (U) → lim_{𝒮^{o}}ℱ_{T} ∘ s_{𝒮}^{o}

is bijective (where the limit is taken in a sufficiently large universe V). The latter in turns is equivalent to the following. For every U and 𝒮 as in the foregoing, the induced cone :

(ℱ (f) : ℱ (U) → ℱ (U′) | (f : U′ → U) ∈ Ob(𝒮))

is universal in 𝒜. Especially, a sheaf on C with values in Set is just a usual sheaf (of sets).

In case 𝒞 is small, 𝒜 is complete, and all the fibre products in 𝒞 are representable, arguing similarly we see that ℱ is a sheaf on C with values in 𝒜 if and only if, for every U ∈ Ob(𝒞) and every small family (U_{i} → U | i ∈ I) of objects of 𝒞 /U that generate a sieve covering U, the restriction morphisms ℱ (U) → ℱ (U_{i}) induce an isomorphism

ℱ (U) →^{∼} Equal(∏_{i∈I} [figure: ℱ (U_{i})] ∏_{(i,j)∈I×I}ℱ (U_{i} ×_{U} U_{j})) in 𝒜 .

(ii) Let 𝒜′ be another category, and F : 𝒜 → 𝒜′ a functor that commutes with limits; it follows from (i) that (𝒞 , F)^{∧} := Fun(𝒞^{o}, F) : (𝒞 , 𝒜)^{∧} → (𝒞 , 𝒜′)^{∧} restricts to a functor

(C, F)^{∼} : (C, 𝒜)^{∼} → (C, 𝒜′)^{∼}.

(iii) Suppose that 𝒜 is complete and V-small for a universe V such that U ⊂ V. Consider a functor

ℱ_{•} : Λ → (C, 𝒜)^{∼} λ ↦ ℱ_{λ}

from a small category Λ. Recall that the limit L of ℱ_{•} in (𝒞 , 𝒜)^{∧} is computed argumentwise, i.e. L(U) represents the limit of the induced functor

ℱ_{•}(U) : Λ → 𝒜 λ ↦ ℱ_{λ}(U)

for every U ∈ Ob(𝒞) (corollary 1.4.1(ii)). We claim that L is a sheaf on C with values in 𝒜, and therefore it represents the limit of ℱ_{•} in (C, 𝒜)^{∼}; especially, the latter category is complete. Indeed, for any T ∈ Ob(𝒜) we have natural identifications

Hom_{𝒜} (T, L(U)) →^{∼} lim_{λ∈Ob(Λ)} Hom_{𝒜} (T, ℱ_{λ}(U)) = lim_{λ∈Ob(Λ)} ℱ_{λ,T} (U) →^{∼} ( lim_{λ∈Ob(Λ)} ℱ_{λ,T} )(U)

and the limit of the functor ℱ_{•,T} from Λ to the category of V-presheaves (of sets) on 𝒞 is a V-sheaf, since by assumption every ℱ_{λ,T} is a V-sheaf (remark 4.1.19(i)). This shows that L_{T} is a V-sheaf on C for every T ∈ Ob(𝒜), as required.

(iv) Let C′ := (𝒞′, J′) be another site, and u : 𝒞 → 𝒞′ a continuous functor for the topologies J and J′. Then u induces a functor

(u, 𝒜)^{∧} := Fun(u^{o}, 𝒜) : (𝒞′, 𝒜)^{∧} → (𝒞 , 𝒜)^{∧}

and notice that for every presheaf ℱ′ on 𝒞′ with values in 𝒜, and every T ∈ Ob(𝒜) we have

(5.5.3)  u^{∧}(ℱ′_{T}) = ((u, 𝒜)^{∧}ℱ′)_{T} .

It follows that (u, 𝒜)^{∧} restricts to a functor

(ũ, 𝒜)_{∗} : (C′, 𝒜)^{∼} → (C, 𝒜)^{∼}.

Likewise, every natural transformation α : u ⇒ v between such continuous functors u, v : 𝒞 → 𝒞′ induces a natural transformation

(α, 𝒜)^{∧} := Fun(α^{o}, 𝒜) : (v, 𝒜)^{∧} ⇒ (u, 𝒜)^{∧}

which yields by restriction a natural transformation

(α̃, 𝒜)_{∗} : (v, 𝒜)_{∗}^{∼} ⇒ (u, 𝒜)_{∗}^{∼}.

(v) For every U ∈ Ob(𝒞), the topology J induces a topology J_{U} on 𝒞 /U (see (4.7.2)), and we let C/U := (𝒞 /U, J_{U}). The source functor s_{U} : 𝒞 /U → 𝒞 is continuous for the topologies J and J_{U}, hence (iv) yields a well defined functor

(s̃_{U}, 𝒜)_{∗} : (C, 𝒞)^{∼} → (C/U, 𝒞)^{∼} ℱ ↦ ℱ_{|U}.

Likewise, if g : U → V is any morphism of 𝒞, the corresponding functor g_{∗} : 𝒞 /U → 𝒞 /V is continuous for the topologies J_{U} and J_{V}, so it restricts to a functor

(g̃, 𝒜)_{∗} : (C/V, 𝒜)^{∼} → (C/U, 𝒜)^{∼}

generalizing remark 4.7.3(i). We may then consider the category

(C/•, 𝒜)^{∼}

whose objects are the pairs (U, ℱ) consisting of objects U ∈ Ob(𝒞) and ℱ ∈ Ob(C/U, 𝒜)^{∼}. A morphism (U, ℱ) → (V, 𝒢) is the datum of a morphism g : U → V of 𝒞 and a morphism ℱ → (g̃, 𝒜)_{∗}𝒢 in (C/U, 𝒜)^{∼}. It follows easily from (5.5.3) that the resulting functor

(C/•, 𝒜)^{∼} → 𝒞 (U, ℱ) ↦ U

is a fibration. Moreover, for every U ∈ Ob(𝒞), every covering family (U_{i} → U | i ∈ I) for J_{U} generates a sieve of 1-descent for this fibration, which is even of 2-descent, if 𝒜 is complete and 𝒞 is small.

5.5.4. In this section we are mainly interested in presheaves and sheaves with values in the category Cat of small categories, but the following auxiliary construction shall also be useful. For every category ℬ whose fibred products are representable we consider the 2-category

Cat^{∗}(ℬ)

of category objects of ℬ, whose objects are the data 𝒞^{∗} := (O, M, s, t, 𝟙, c) such that O, M ∈ Ob(ℬ), and

(5.5.5)  \xymatrix{M \ar[r]^{s} \ar[r]_{t} & O}  𝟙 : O → M  M ×_{(t,s)} M ⟶^{c} M

are morphisms in ℬ called respectively the source, target, identity and composition laws of 𝒞^{∗}, fulfilling the identities :

s ∘ 𝟙 = 1_{O} = t ∘ 𝟙

and making commute the diagrams of morphisms of ℬ :

\xymatrix{M ×_{(t,s)} M ×_{(t,s)} M \ar[r]^{1_{M} ×c} \ar[d]_{c×1_{M}} & M ×_{(t,s)} M \ar[d]^{c} & M \ar[r]^{(1_{M} ,𝟙∘t)} \ar[d]_{(𝟙∘s,1_{M})} \ar[dr]^{1_{M}} & M ×_{(t,s)} M \ar[d]^{c} \\ M ×_{(t,s)} M \ar[r]^{c} & M & M ×_{(t,s)} M \ar[r]^{c} & M.}

Notice that the datum of c includes the choice of a representative for the fibre product M×_{(t,s)}M, and we (implicitly) fix as well a universal cone M ← M ×_{(t,s)} M → M for this fibre product.

The 1-cells (F_{1}, F_{2}) : 𝒞^{∗} := (O, M, s, t, 𝟙, c) → 𝒞^{′∗} := (O′, M′, s′, t′, 𝟙′, c′) in Cat^{∗}(ℬ) are the pairs of morphisms (F_{1} : O → O′, F_{2} : M → M′) of ℬ such that

s′ ∘ F_{2} = F_{1} ∘ s  t′ ∘ F_{2} = F_{1} ∘ t  𝟙′ ∘ F_{1} = F_{2} ∘ 𝟙  c′ ∘ (F_{2} ×_{(t,s)} F_{2}) = F_{2} ∘ c.

The composition law of 1-cells is given by the obvious rule :

(F′_{1}, F′_{2}) ∘ (F_{1}, F_{2}) := (F′_{1} ∘ F_{1}, F′_{2} ∘ F_{2}) for every pair of 1-cells 𝒞^{∗} ⟶^{(F_{1},F_{2})} 𝒞^{′∗} ⟶^{(F′_{1},F′_{2})} 𝒞^{′′∗} .

For every pair of category objects 𝒞^{∗} and 𝒞^{′∗} of ℬ, and every pair of 1-cells (F_{1}, F_{2}), (G_{1}, G_{2}) : 𝒞^{∗} → 𝒞^{′∗}, the 2-cells β : (F_{1}, F_{2}) ⇒ (G_{1}, G_{2}) are the morphisms β : O → M′ of ℬ such that

(5.5.6)  s′ ∘ β = F_{1}  t′ ∘ β = G_{1}  c′ ∘ (β ∘ s, G_{2}) = c′ ∘ (G_{1}, β ∘ t)

(where (β ∘ s, G_{2}) and (G_{1}, β ∘ t) are morphisms M → M′ ×_{(t′,s′)} M′).

Remark 5.5.7. (i) We have a natural strict and strong 2-equivalence of 2-categories :

(5.5.8)  V-Cat →^{∼} Cat^{∗}(V-Set)  for every universe V.

Namely, to each V-small category 𝒞 we assign the object [𝒞] := (Ob(𝒞), Morph(𝒞), s, t, 𝟙, c) where s, t, 𝟙, c encode the source, target, identity and composition laws for 𝒞 in the obvious way. Then any functor F : 𝒞 → 𝒞′ induces an obvious 1-cell [F] : [𝒞] → [𝒞′], and every natural transformation β : F ⇒ F′ induces a 2-cell [β] : [F_{1}] ⇒ [F_{2}]. The quasi-inverse assigns to every datum 𝒞^{∗} := (O, M, s, t, 𝟙, c) the category [𝒞^{∗}] with Ob([𝒞^{∗}]) := O and Hom_{[𝒞^{∗}]}(X, Y) := s^{−1}(X) ∩ t^{−1}(Y) for every X, Y ∈ O, with the composition law induced by c in the obvious way, and with 1_{X} := 𝟙(X) for every X ∈ O. Then every 1-cell (F_{1}, F_{2}) : 𝒞^{∗} → 𝒞^{′∗} induces a functor [F_{1}, F_{2}] : [𝒞^{∗}] → [𝒞^{′∗}] and every 2-cell β : (F_{1}, F_{2}) ⇒ (G_{1}, G_{2}) induces a natural transformation [β] : [F_{1}, F_{2}] ⇒ [G_{1}, G_{2}], in the obvious fashion.

(ii) We have obvious objects and morphisms functors :

\xymatrix{V-Cat \ar[r]^{Ob} \ar[r]_{Morph} & V-Set}  for every universe V

which – in terms of the 2-equivalence (5.5.8) – translate as the functors Cat^{∗}(V-Set) → V-Set that extract from each datum (O, M, s, t, 𝟙, c) the set O and respectively the set M, and that assign to every morphism (F_{1}, F_{2}) of Cat^{∗}(V-Set) the map F_{1} and respectively the map F_{2}.

(iii) Notice that both Ob and Morph are representable functors. Indeed, let 𝟙 be the category with one object and one morphism, and let 𝟚 be the category with exactly two objects a and b, and a single morphism from a to b (and no morphisms from b to a). Then it is easily seen that 𝟙 (resp. 𝟚) represents the presheaf Ob (resp. Morph) on V-Cat^{o} : the details shall be left to the reader. Especially, Ob and Morph commute with all the limits of V-Cat (example 1.3.15(ii)).

Moreover, obviously a functor F : 𝒜 → 𝒜′ is an isomorphism in V-Cat if and only if both Ob(F) : Ob(𝒜) → Ob(𝒜′) and Morph(F) : Morph(𝒜) → Morph(𝒜′) are bijections.

(iv) Taking into account (iii) and remark 5.5.2(ii), it follows immediately that a presheaf of V-small categories F on 𝒞 is a sheaf of categories on C if and only if Ob(F) := Ob ∘ F and Morph(F) := Morph ∘ F are sheaves (of sets) on C.

(v) For every site C := (𝒞 , J), the categories (𝒞 , V-Cat)^{∧} and (C, V-Cat)^{∼} inherit from V-Cat a natural structure of 2-category. Namely, (𝒞 , V-Cat)^{∧} can be described as the sub-2-category of of PsFun(𝒞 , V-Cat) whose objects are the strict pseudo-functors, whose 1-cells are all the strict pseudo-natural transformations (where 𝒞 is regarded as usual, as a 2-category with trivial 2-cells), and whose 2-cells are all the modifications β ⇝ β′, for every pair of 1-cells β, β′ : F ⇒ F′ of (𝒞 , V-Cat)^{∧}. Similarly we describe (C, V-Cat)^{∼} as the strong sub-2-category of (𝒞 , V-Cat)^{∧} whose objects are the sheaves of V-small categories on the site C (see definition 2.4.9(iii)). Then (5.5.8) generalizes to strict and strong 2-equivalences

(5.5.9)  (𝒞 , V-Cat)^{∧} →^{∼} Cat^{∗}(𝒞_{V}^{∧})  (C, V-Cat)^{∼} →^{∼} Cat^{∗}(C_{V}^{∼}).

(vi) Let ℬ and ℬ′ be any two categories whose fibre products are representable, and u : ℬ → ℬ′ any functor that commutes with fibre products. Then u induces a strict pseudo-functor

Cat^{∗}(u) : Cat^{∗}(ℬ) → Cat^{∗}(ℬ′) (O, M, s, t, 𝟙, c) ↦ (u(0), u(M), u(s), u(𝟙), u(c)).

Indeed, we can regard u(c) as a morphism u(M) ×_{(u(t),u(s))} u(M) → u(M) (since u commutes with fibre products), and then it is clear that the rule defining Cat^{∗}(u) takes objects of Cat^{∗}(ℬ) to objects of Cat^{∗}(ℬ′). To every 1-cell (F_{1}, F_{2}) and every 2-cell β, the pseudo-functor Cat^{∗}(u) assigns likewise respectively the 1-cell (u(F_{1}), u(F_{2})) and the 2-cell u(β) of Cat^{∗}(ℬ′).

If v : ℬ → ℬ′ is another functor commuting with fibre products, and α : u ⇒ v is any natural transformation, we get a strict pseudo-natural transformation

Cat^{∗}(α) : Cat^{∗}(u) ⇒ Cat^{∗}(v) (O, M, s, t, 𝟙, c) ↦ (α_{O}, α_{M}).

Indeed, notice that u(M ×_{(t,s)} M) represents u(M) ×_{(u(t),u(s))} u(M) since u commutes with fibre products, and likewise for v(M ×_{(t,s)} M), and under these identifications, the morphism α_{M×_{(t,s)}M} corresponds to α_{M} ×_{(u(t),u(s))} α_{M}, whence it is easily seen that (α_{O}, α_{M}) is a 1-cell of Cat^{∗}(ℬ′), and the strict pseudo-functoriality of Cat^{∗}(α) is then immediate from the definition.

(vii) Clearly the rules u ↦ Cat^{∗}(u) and α ↦ Cat^{∗}(α) define a strict pseudo-functor from the sub-2-category of Cat whose objects are all the small categories whose fibre products are representable, whose 1-cells are the functor commuting with fibre products and whose 2cells are the natural transformations, to the 2-category whose objects are the small 2-categories, whose 1-cells are the strict pseudo-functors, and whose 2-cells are the strict pseudo-natural transformations.

5.5.10. Let now C := (𝒞 , J) be a U-site; then we deduce that the strongly faithful inclusion pseudo-functor (C, V-Cat)^{∼} → (𝒞 , V-Cat)^{∧} admits a strong left 2-adjoint :

(5.5.11)  (𝒞 , V-Cat)^{∧} → (C, V-Cat)^{∼}  F ↦ F^{a}

for every universe V with U ⊂ V. Namely, by remark 5.5.7(vi,vii), the exact left adjoint (−)^{a} : 𝒞_{V}^{∧} → C_{V}^{∼} provided by remark 4.1.23(ii) induces first a strong left 2-adjoint Cat^{∗}((−)^{a}) : Cat^{∗}(𝒞_{V}^{∧}) → Cat^{∗}(C_{V}^{∼}) to the inclusion of 2-categories Cat^{∗}(C_{V}^{∼}) → Cat^{∗}(C_{V}^{∧}). Then, combining with the strong and strict 2-equivalences (5.5.9) we get the sought strong left 2adjoint. It is also easily seen that (5.5.11) is an exact functor on the underlying categories.

5.5.12. Let C := (𝒞 , J) and C′ := (𝒞′, J′) be two U-sites, u : 𝒞 → 𝒞′ a functor, U′ a universe with U ⊂ U′ and such that 𝒞 and 𝒞′ are U′-small and the functor u_{U′!}^{a} : 𝒞_{U′}^{∧} → C_{U′}^{′∼} commutes with fibre products. Arguing as in (5.5.10), and in light of remark 4.2.18(iii), we deduce that for every universe V with U ⊂ V, the pseudo-functor u_{V!}^{a} induces a pseudo-functor

(u, V-Cat)_{!}^{a} : (𝒞 , V-Cat)^{∧} → (C′, V-Cat)^{∼}.

In case u is continuous for the topologies J and J′, corollary 4.3.19(i) and remark 5.5.7(vi,vii) easily imply that (u, V-Cat)_{!}^{a} restricts to a strong and strict left 2-adjoint

(ũ, V-Cat)^{∗} : (C, V-Cat)^{∼} → (C′, V-Cat)^{∼}

for the strict pseudo-functor (ũ, V-Cat)_{∗} of remark 5.5.2(iv).

5.5.13. Let C := (𝒞 , J) and C′ := (𝒞′, J′) be two U-sites, u : 𝒞 → 𝒞′ a cocontinuous functor, and V a universe with U ⊂ V. Arguing as in (5.5.10) we see that the adjoint pair of left exact functors (ŭ_{V}^{∗}, ŭ_{V∗}) provided by corollary 4.3.19(ii) induces a strict and strong 2-adjoint pair of strict pseudo-functors

\xymatrix{(C′, V-Cat)^{∼} \ar[r]^{(ŭ,V-Cat)^{∗}} & (C, V-Cat)^{∼}. \ar[l]^{(ŭ,V-Cat)_{∗}}}

Remark 5.5.14. (i) In the situation of (5.5.12), suppose that u is a continuous functor and ũ_{V∗} is an equivalence, so that ũ_{V}^{∗} is its quasi-inverse. From the explicit description of the functors (ũ, V-Cat)_{∗} and (ũ, V-Cat)^{∗}, we then deduce straightforwardly that the latter are mutually quasi-inverse equivalences and strong 2-equivalences as well.

(ii) Combining with theorem 4.4.2(iii), we deduce especially that for every U-site C := (𝒞 , J), the Yoneda embedding h_{𝒞}^{a} : 𝒞 → C_{U}^{∼} induces a strong 2-equivalence of 2-categories :

(h̃_{𝒞}^{a} , V-Cat)_{∗} : (Can(C_{U}^{∼}), V-Cat)^{∼} →^{∼} (C, V-Cat)^{∼}.

(iii) Let C := (𝒞 , J) and C′ := (𝒞′, J′) be two sites such that 𝒞 is small and 𝒞′ has small Hom-sets, and let u : 𝒞 → 𝒞′ be any continuous functor for the topologies J and J′. Recall that for every universe V with U ⊂ V, the functor (u, V-Cat)^{∧} admits a left adjoint

(u, V-Cat)_{!} : (𝒞 , V-Cat)^{∧} → (𝒞′, V-Cat)^{∧}

computed by left Kan extensions : see theorem 1.3.4. If C′ is a U-site, we can compose this functor with the functor (5.5.11), to get a functor (u, V-Cat)_{!}^{a} : (𝒞 , V-Cat)^{∧} → (C′, V-Cat)^{∼}, whose restriction to (𝒞 , V-Cat)^{∼} is a left adjoint for (ũ, V-Cat)_{∗}. If C is no longer a small site, but only a U-site, we can pick as usual an essentially small topologically generating full subcategory 𝒢 ⊂ 𝒞, endowed with the topology J_{𝒢} induced by J via the inclusion functor j : 𝒢 → 𝒞, so that (j̃, V-Cat)_{∗} : (C, V-Cat)^{∼} → ((𝒢 , J_{𝒢}), V-Cat)^{∼} is an equivalence, by (i); then (ũ, V-Cat)_{∗} admits the left adjoint (ũ ∘ ȷ, V-Cat)^{∗} ∘ (j̃, V-Cat)_{∗}. This yields another construction for the functor underlying the strict pseudo-functor (ũ, V-Cat)^{∗} of (5.5.12), which is available under weaker assumptions, but is not obviously pseudo-functorial for the 2-category structures of (C, V-Cat)^{∼} and (C′, V-Cat)^{∼}.

(iv) On the other hand, in the situation of (iii), recall that the strict pseudo-functor

PsFun(u^{o}, V-Cat) : PsFun(𝒞^{′o}, V-Cat) → PsFun(𝒞^{o}, V-Cat)

admits a left 2-adjoint that can be computed by the strong left 2-Kan extension 2-∫^{u^{o}} along u^{o} (remark 2.6.18(ii) and theorem 3.3.9). The latter restricts to a strict pseudo-functor

(u, V-Cat)_{!} : (𝒞 , V-Cat)^{∧} → (𝒞′, V-Cat)^{∧}.

Moreover, the 2-adjunction for the pair (2-∫^{u^{o}}, PsFun(u^{o}, V-Cat)) restricts to a natural functor

(−)_{F,G}^{†} : Hom_{(𝒞 ,V-Cat)^{∧}}(G ∘ u^{o}, F) → Hom_{(𝒞 ,V-Cat)^{∧}}(G, 2-∫ ^{u^{o}} F )

for every presheaf of categories F on 𝒞 and G on 𝒞′ (remark 2.6.24). But in this generality, though such functors are fully faithful, it is not clear whether they are equivalences of categories.

(v) However, suppose now additionally that the category X/u𝒞 is cofiltered for every X ∈ Ob(𝒞′) (this condition is fulfilled, notably, in case C and C′ are two lex-sites, and u is a morphism of lex-sites C′ → C, by virtue of example 1.3.16(i)). Recall that for every presheaf of V-small categories F on 𝒞 and every X ∈ Ob(𝒞′), the value 2-∫^{u^{o}} F (X) represents the 2-colimit of the strict pseudo-functor F ∘ t_{X}^{o} : (X/u𝒞)^{o} → V-Cat; but then such 2-colimit is represented by the colimit of the same functor, and every universal cocone for such colimit is also a universal pseudo-cocone (example 3.3.13(iv)). This means that the restriction to (𝒞 , V-Cat)^{∧} of the strong left 2-Kan extension 2-∫^{u^{o}} agrees with the left Kan extension ∫^{u^{o}} of theorem 1.3.4; moreover, a direct inspection shows that the functor (−)_{F,G}^{†} agrees on objects with the standard adjunction for the pair (∫^{u^{o}}, (u, V-Cat)^{∧}), provided by the proof of theorem 1.3.4. I.e., under these assumptions, (u, V-Cat)_{!} is indeed a strong and strict left 2-adjoint for (u, V-Cat)^{∧}, and we get an essentially commutative diagram of 2-categories :

\xymatrix{(𝒞 , V-Cat)^{∧} \ar[r]^{(u,V-Cat)_{!}} \ar[d] & (𝒞′, V-Cat)^{∧} \ar[d] \\ PsFun(𝒞^{o}, V-Cat) \ar[r]^{2-∫^{u^{o}}} & PsFun(𝒞^{′o}, V-Cat)}

whose vertical arrows are the inclusion strict pseudo-functors (with a different choice of left 2-Kan extension, this diagram is then still at least pseudo-commutative). Lastly, under these same assumptions, we obtain as well a pseudo-commutative diagram :

\xymatrix{(𝒞 , V-Cat)^{∧} \ar[r]^{(u,V-Cat)_{!}} \ar[d]_{ℱ ib_{𝒞}} & (𝒞′, V-Cat)^{∧} \ar[d]^{ℱ ib_{𝒞′}} \\ V-Fib(𝒞) \ar[r]^{V-Fib(u)_{!}} & V-Fib(𝒞′).}

Arguing as in (iii), we get, again under the current assumptions, an alternative construction of the strict pseudo-functor (ũ, V-Cat)^{∗}, as the restriction to (𝒞 , V-Cat)^{∼} of the composition of the foregoing pseudo-functor (u, V-Cat)_{!} with the pseudo-functor (5.5.11).

Lemma 5.5.15. Let C := (𝒞 , J) be any site, and consider a presheaf of categories on 𝒞

𝒜_{•} : 𝒞^{o} → Cat  X ↦ 𝒜_{X}  (f : Y → X) ↦ (𝒜_{f} : 𝒜_{X} → 𝒜_{Y}).

The following holds :

(i) If 𝒜_{•} is a sheaf on C, the associated prestack ℱ ib(𝒜_{•}) on C is 1-separated.

(ii) Suppose that C is a U-site, and that ℱ ib(𝒜_{•}) is a stack on C, and let j : 𝒜_{•} → 𝒜_{•}^{a} be the unit of adjunction (notation of (5.5.10)). Then the induced morphism of prestacks ℱ ib(j) : ℱ ib(𝒜_{•}) → ℱ ib(𝒜_{•}^{a}) is an equivalence of categories.

Proof. (i): By lemma 5.1.9, it suffices to show that for every X ∈ Ob(𝒞) and every pair of cartesian sections σ, σ′ : 𝒞 /X → ℱ ib(𝒜_{•}), the presheaf 𝒞art(σ, σ′) is a sheaf on the site C/X. Set A := σ(1_{X}) and A′ := σ′(1_{X}); it is easily seen that 𝒞art(σ, σ′) is isomorphic to the presheaf

H_{AA′} : (𝒞 /X)^{o} → Set (f : Y → X) ↦ Hom_{𝒜_{Y}} (𝒜_{f}(A), 𝒜_{f}(A′))

that assigns to every morphism (h/X) : (Y ⟶^{f} X) → (Y′ ⟶^{f′} X) of 𝒞 /X the induced map

Hom_{𝒜′_{Y}} (𝒜_{f′}(A), 𝒜_{f′}(A′)) → Hom_{𝒜_{Y}} (𝒜_{f}(A), 𝒜_{f}(A′)) g ↦ 𝒜_{h}(g).

Now, let 𝟙_{𝒞 /X} be a final object of (𝒞 /X)^{∧} (see example 1.2.16(v)); it is easily seen that 𝟙_{𝒞 /X} is a sheaf on C/X. Recall that the source functor s_{X} : 𝒞 /X → 𝒞 is continuous for the topologies of C and C/X. Denote also by (O_{𝒜} , M_{𝒜} , s_{𝒜} , t_{𝒜} , 𝟙_{𝒜} ) the object of (𝒞 , Cat^{∗})^{∧} corresponding to 𝒜_{•}; by remark 5.5.7(iv), both O_{𝒜} and M_{𝒜} are sheaves on C, hence we get sheaves s̃_{X∗}(O_{𝒜} ) and s̃_{X∗}(M_{𝒜} ) on C/X, as well as source and target morphisms of sheaves s̃_{X∗}(s_{𝒜} ), s̃_{X∗}(t_{𝒜} ) : s̃_{X∗}(M_{𝒜} ) → s̃_{X∗}(O_{𝒜} ). There follows a cartesian diagram of presheaves :

\xymatrix{H_{AA′} \ar[r] \ar[d] & s̃_{X∗}(M_{𝒜} ) \ar[d]^{(s̃_{X∗}(s_{𝒜} ),s̃_{X∗}(t_{𝒜} ))} \\ 𝟙_{𝒞 /X} \ar[r]^{τ} & s̃_{X∗}(O_{𝒜} × O_{𝒜} )}  on 𝒞 /X

where τ_{f} : {𝟙} = 𝟙_{𝒞 /X} (X) → Ob(𝒜_{Y} × 𝒜_{Y}) is the map such that 𝟙 ↦ (𝒜_{f}(A), 𝒜_{f}(A′)), for every morphism f : Y → X of 𝒞. But then H_{AA′} is a sheaf on C/X, as required.

(ii): By assumption, ℱ ib(𝒜_{•}) is 2-separated, and ℱ ib(𝒜_{•}^{a}) is 1-separated, by (i); in view of proposition 5.2.9, it then suffices to check that ℱ ib(j) is i-covering for i = 0, 1, 2. The latter follows easily from remark 5.2.2, taking into account the explicit description of 𝒜_{•}^{a} from (5.5.10). □

5.5.16. Fix a universe V with U⊂V, and a U-site C := (𝒞 , J_{𝒞} ). Consider the pseudo-functor

ℱ ib_{C}^{a} : (C, V-Cat)^{∼} → V-Stack(C) 𝒜_{•} ↦ ℱ ib(𝒜_{•})^{a}

composition of the restriction of the pseudo-functor ℱ ib_{𝒞} of theorem 3.1.24with (−)^{a} of corollary 5.4.19(where (C, V-Cat)^{∼} is a 2-category as in remark 5.5.7(v)). Let also C′ := (𝒞′, J_{𝒞′}) be another U-site and u : C′ → C a weak morphism of sites (definition 5.4.2); we wish to attach an orientation to the induced diagram :

(5.5.17)  \xymatrix{(C′, V-Cat)^{∼} \ar[r]^{(ũ,V-Cat)_{∗}} \ar[d]_{ℱ ib_{C′}^{a}} & (C, V-Cat)^{∼} \ar[d]^{ℱ ib_{C}^{a}} \\ V-Stack(C′) \ar[r]^{V-St(u)_{∗}} & V-Stack(C)}

i.e. a pseudo-natural transformation

Δ^{u} : ℱ ib_{C}^{a} ∘ (ũ, V-Cat)_{∗} ⇒ V-St(u)_{∗} ∘ ℱ ib_{C′}^{a}.

To this aim, notice that remark 3.1.27(ii) yields an oriented diagram of 2-categories :

(5.5.18)  [figure: (C′, V-Cat)^{∼}; (ũ,V-Cat)_{∗}; ℱ ib_{𝒞′}; ⊥^{u}; (C, V-Cat)^{∼}; ℱ ib_{𝒞}; V-Fib(𝒞′); V-Fib(u)^{∗}; V-Fib(𝒞)]

whose orientation ⊥^{u} is a strict pseudo-natural isomorphism of strict pseudo-functors. On the other hand, the square of links 𝒟_{u} of (5.4.30), oriented by the identity pseudo-natural transformation, yields a base change square :

(5.5.19)  [figure: V-Fib(𝒞′); V-Fib(u)^{∗}; (−)_{C′}^{a}; Υ(𝒟_{u}); V-Fib(𝒞); (−)_{C}^{a}; V-Stack(C′); V-St(u)_{∗}; V-Stack(C)]

and then we take Δ^{u} :=⊥^{u} □ Υ(𝒟_{u}). In case we wish to emphasize the dependence on V, we shall also write V-Δ^{u} for this orientation of (5.5.17).

Remark 5.5.20. In the situation of (5.5.16), suppose that u is a morphism of sites; in this case, the pseudo-functors (ũ, V-Cat)_{∗} and V-St(u)_{∗} admit left 2-adjoints (ũ, V-Cat)^{∗} and V-St(u)^{∗}, and after choosing units and counits for these 2-adjoint pairs, and for the 2-adjoint pair (V-Fib(u)_{!}, V-Fib(u)^{∗}) we then get well defined links

(ũ, V-Cat) := ((ũ, V-Cat)^{∗}, (ũ, V-Cat)_{∗}, η^{(ũ,V-Cat)}, ε^{(ũ,V-Cat)}):(C′, V-Cat)^{∼}→(C, V-Cat)^{∼}
V-St(u) := (V-St(u)^{∗}, V-St(u)_{∗}, η^{V-St(u)}, ε^{V-St(u)}) : V-Stack(C′) → V-Stack(C)
V-Fib(u) :=(V-Fib(u)_{!}, V-Fib(u)^{∗}, η^{V-Fib(u)}, ε^{V-Fib(u)}) : V-Fib(𝒞′) → V-Fib(𝒞)

of the 2-category V-\overline{2-Cat}, and the data

((−)_{C′}^{a}, (−)_{C}^{a}, Υ(𝒟_{u})) and (ℱ ib_{𝒞′}, ℱ ib_{𝒞} , ⊥^{u})

can be regarded as 1-cells V-Fib(u) → V-St(u) and (ũ, V-Cat) → V-Fib(u) in the 2-category wLink(V-\overline{2-Cat}) (see (2.3.19)) whose composition is the 1-cell

(ℱ ib_{C′}^{a}, ℱ ib_{C}^{a}, Δ^{u}) : (ũ, V-Cat) → V-St(u).

5.5.21. Let V be a universe with U ∈ V; we shall denote by

(U, V)-wSite

the sub-2-category of (U, V)-Site (see definition 4.3.1(ii)) whose objects are all the V-small Usites, whose 1-cells are the weak morphisms of sites, and whose 2-cells g ⇒ g′ are the natural transformations of functors g′ ⇒ g. As in remark 4.3.2(i), we shall usually drop the mention of U and V, and write simply wSite for this 2-category. We have a strict pseudo-functor

(−, V-Cat)^{∼} : wSite → V-\overline{2-Cat} C ↦ (C, V-Cat)^{∼}

(notation of remark 2.2.16) that assigns to every morphism of U-sites u the strict pseudofunctor (ũ, V-Cat)_{∗}, and to every natural transformation β : u ⇒ v the 2-cell (β̃, V-Cat)_{∗} : (ṽ, V-Cat)_{∗} ⇒ (ũ, V-Cat)_{∗}. Likewise, we get a pseudo-functor

V-Stack : wSite → V-\overline{2-Cat}  C ↦ V-Stack(C)  (u : C′ → C) ↦ V-St(u)_{∗}.

To define the coherence constraints (δ^{V-Stack}, γ^{V-Stack}) of V-Stack, and the action of V-Stack on natural transformations β : u ⇒ v between weak morphisms of sites u, v : C′ → C, notice that V-Stack(𝒞 , J) ⊂ V-Fib(𝒞) for every U-site (𝒞 , J), and V-St(u)_{∗} is the restriction of V-Fib(u)^{∗}, for every morphism of sites u; we may then simply define (δ^{V-Stack}, γ^{V-Stack}) as the restriction of the coherence constraint (δ^{V-Fib}, γ^{V-Fib}) of the pseudo-functor V-Fib of (3.1.29), and for any β as above, let V-St(β)_{∗} : V-St(v) ⇒ V-St(u)_{∗} be the restriction of V-Fib(β)^{∗}.

With this notation, it would be tempting to state that the rule : C ↦ ℱ ib_{C}^{a} of (5.5.16) defines a pseudo-natural transformation ℱ ib_{•}^{a} : (−, V-Cat)^{∼} ⇒ V-Stack whose coherence constraint is given by the orientations Δ^{u}. However, notice first that Δ^{u} points in the direction opposite to the one which is required for such a coherence constraint. This can be fixed, by stating instead that ℱ ib_{•}^{a} should be a pseudo-natural transformation^{o}(−, V-Cat)^{∼} ⇒ ^{o}V-Stack. But since the orientations Δ^{u} are not, in general, pseudo-natural equivalences, we have rather :

Proposition 5.5.22. There exists a lax-natural transformation

oℱ ib_{•}^{a} : ^{o}(−, V-Cat)^{∼} ⇒ ^{o}V-Stack  C ↦ ^{o}ℱ ib_{C}^{a}  for every U-site C

whose coherence constraint attaches to every weak morphism of U-sites u : C′ → C the square o(5.5.17), with its orientation^{o}Δ^{u}.

Proof. Consider the forgetful strict pseudo-functor

Φ : wSite → ^{o}V-Cat^{o} (𝒞 , J) ↦ 𝒞

that assigns to every weak morphism of sites u : (𝒞′, J′) → (C, J) the functor u : 𝒞 → 𝒞′, and to every 2-cell β : u ⇒ v of wSite the natural transformation β : v ⇒ u. From corollary 3.1.36we deduce a pseudo-natural transformation of pseudo-functors :

oℱ ib_{•} : ^{o}(−, V-Cat)^{∼} ⇒ ^{o}V-Fib ∘^{o}Φ  (𝒞 , J) ↦ ^{o}ℱ ib_{𝒞}  for every U-site (𝒞 , J)

whose coherence constraint assigns to every morphism of sites u the oriented square diagram (5.5.18). It then suffices to exhibit a lax-natural transformation

o(−)_{•}^{a} : ^{o}V-Fib ∘^{o}Φ ⇒ ^{o}V-Stack  C ↦ ^{o}(−)_{C}^{a}  for every U-site C

whose coherence constraint assigns to every weak morphism of U-sites u : C′ → C the oriented square Υ(𝒟_{u}). Indeed, we will then define ^{o}ℱ ib_{•}^{a} := ^{o}(−)_{•}^{a} ⊙ ^{o}ℱ ib_{•}. Now, notice that the pseudo-functors V-Stack and V-Fib factor through well-defined pseudo-functors

V-LStack : wSite → Link(V-\overline{2-Cat}) V-LFib : ^{o}V-Cat^{o} → Link(V-\overline{2-Cat}).

Namely, V-LStack assigns to every site C the 2-category V-Stack(C) and to every weak morphism of sites u : C′ → C the link V-St(u) defined as in remark 5.5.20; likewise, V-LFib assigns to every category 𝒞 the 2-category V-Fib(𝒞), and to every functor u : 𝒞 → 𝒞′ the link V-Fib(u). Next, let i_{C} : V-Stack(C) → V-Fib(𝒞) be the inclusion strict pseudo-functor; notice that the rule : C ↦ i_{C} for every site C yields a pseudo-natural transformation

i_{•} : V-LStack ⇒ V-LFib ∘ Φ

whose coherence constraints are given by the oriented diagrams 𝒟_{u} (notice that i_{•} is not strict, even though 𝒟_{u} is oriented by the identity pseudo-natural transformation). According to remark 2.2.5(ii), the pseudo-natural transformation i_{•} corresponds to a pseudo-functor ĩ_{•} : wSite → ℳ := 2-Morph(Link(V-\overline{2-Cat})), and we consider the composition :

Ψ : wSite \underrightarrow{ĩ_{•}} ℳ ⟶^{Υ_{V-\overline{2-Cat}}^{o}} (2-Morph(V-\overline{2-Cat}^{o}))^{o} →^{∼} ^{o}2-Morph(^{o}V-\overline{2-Cat})

with the pseudo-functor Υ_{V-\overline{2-Cat}}^{o} of theorem 2.3.17 and the strict isomorphism of 2-categories given by example 2.1.23(i). Let s, t : 2-Morph(^{o}V-\overline{2-Cat}) → ^{o}V-\overline{2-Cat} be the source and target strict pseudo-functors; by definition, ^{o}Ψ is a lax-natural transformation s ∘ ^{o}Ψ ⇒ t ∘ ^{o}Ψ ⇒. Lastly, a direct inspection shows that s ∘^{o}Ψ = LStack and t ∘^{o}Ψ ⇒= LFib ∘ Φ, so we may let o(−)_{•}^{a} := ^{o}Ψ. □

5.5.23. We consider now an oriented square :

[figure: D′ := (𝒟′, J_{𝒟′} ); v′; u′; β; C′ := (𝒞′, J_{𝒞′}); u; D := (𝒟, J_{𝒟} ); v; C := (𝒞 , J_{𝒞} )]

where v and v′ are morphisms of U-sites, and u and u′ are weak morphisms of U-sites. In case also u and u′ are morphisms of sites, we get an induced oriented diagram of topoi :

(5.5.24)  [figure: D̃′; v′; u′; β̃_{∗}; C̃′; ũ; D̃; ṽ; C̃.]

Even when u and u′ are only weak morphisms of sites, both ũ_{∗} and ũ′_{∗} admit left adjoint functors, hence (5.5.24) may still be regarded as an oriented square of links in the 2-category V-Cat. On the other hand, we deduce a diagram of 2-categories :

[figure: V-Stack(D′); St(v′)_{∗}; St(u′)_{∗}; Δ^{v′}; V-Stack(C′); St(u)_{∗}; (D′, V-Cat)^{∼}; Δ^{u′}; (v′,V-Cat)_{∗}; (u′,V-Cat)_{∗}; (β̃,V-Cat)_{∗}; (C′, Cat)^{∼}; (ũ,V-Cat)_{∗}; Δ^{u}; (D, V-Cat)^{∼}; (ṽ,V-Cat)_{∗}; Δ^{v}; (C, V-Cat)^{∼}; V-Stack(D); St(v)_{∗}; V-Stack(C)]

whose diagonal arrows are the pseudo-functors ℱ ib^{a}. We complete it by adding the orientation

St(β)_{∗}^{γ} := (γ_{u′,v}^{Stack})^{−1} ⊙ St(β)_{∗} ⊙ γ_{v′,u}^{Stack} : St(u)_{∗} ∘ St(v′)_{∗} ⇒ St(v)_{∗}St(u′)_{∗}

for the external square subdiagram. In light of proposition 5.5.22, we then see that the resulting cubical diagram commutes on 2-cells, in the sense of remark 2.3.22(iii).

5.5.25. In the situation of (5.5.23), we associate with v the links (ṽ, V-Cat), V-St(v) and V-Fib(v) of the 2-category V-\overline{2-Cat}, as in remark 5.5.20, as well as 1-cells

((−)_{D}^{a}, (−)_{C}^{a}, Υ(𝒟^{v})) : V-Fib(v) → V-St(v) (ℱ ib_{𝒟} , ℱ ib_{𝒞} , ⊥^{v}) : (ṽ, V-Cat) → V-Fib(v)

in the 2-category wLink(V-\overline{2-Cat}), whose composition is the 1-cell :

(ℱ ib_{D}^{a}, ℱ ib_{C}^{a}, Δ^{v}) : (ṽ, V-Cat) → V-St(v)

(and likewise for v′). Additionally, we get two more 1-cells of wLink(V-\overline{2-Cat}) :

((ũ′, Cat)_{∗}, (ũ, Cat)_{∗}, (β̃, Cat)_{∗}) : (ṽ′, Cat) → (ṽ, Cat)
(St(u′)_{∗}, St(u)_{∗}, St(β)_{∗}^{γ}) : St(v′) → St(v).

Corollary 5.5.26. With the notation of (5.5.25), let 𝒜_{•} ∈ Ob((C′, Cat)^{∼}). We have :

(i) If ℱ ib_{𝒞′}(𝒜_{•}) is a stack on C′, then Δ_{𝒜_{•}}^{u} is an equivalence of categories. (ii) If u is also cocontinuous, then Δ^{u} is a pseudo-natural equivalence. (iii) Υ(ℱ ib_{D}^{a}, ℱ ib_{C}^{a}, Δ^{v}) is a pseudo-natural equivalence. (iv) In the situation of (5.5.23), suppose that both ℱ ib_{𝒞′}(𝒜_{•}) and ℱ ib_{𝒟′} ((ṽ′, Cat)^{∗}𝒜_{•})

are stacks. Then ℱ ib_{D}^{a}(Υ((β̃, Cat)_{∗})_{𝒜_{•}} ) is an equivalence if and only if the same holds for Υ(St(β)_{∗}^{γ})_{ℱ ib(𝒜_{•})} .

Proof. (i): It suffices to show that Υ(𝒟^{u})_{ℰ} is an equivalence if ℰ is a stack on C′. Thus, let (η^{C}, ε^{C}) be the unit and counit for the 2-adjoint pair (()_{C}^{a}, i_{C}) (notation of (5.4.30)) and define likewise (η^{C′}, ε^{C′}). Since i_{C} is fully faithful, ε^{C} is a pseudo-natural equivalence, and likewise for ε^{C′}. From the triangular identities of theorem 2.4.24(i) we deduce that also η^{C} ∗ i_{C} and η^{C′} ∗ i_{C′} are pseudo-natural equivalences; the assertion follows directly.

(ii): Again, it suffices to check that Υ(𝒟^{u}) is a pseudo-natural equivalence if u is cocontinuous; this is known by corollary 5.4.31.

Claim 5.5.27. In order to prove (iii), we may assume that v is a morphism of small lex-sites.

Proof of the claim. Let T (resp. T′) be the site whose underlying category is C_{U}^{∼} (resp D_{U}^{∼}), with its canonical topology. Pick a universe V′ with V ⊂ V′, such that T and T′ are V′-small. By direct inspection, we see that V-Δ^{v} is the restriction of V′-Δ^{v}; hence if Υ(V′-Δ^{v}) is a pseudonatural equivalence, the same holds for Υ(V-Δ^{v}). We may then replace V by V′, and suppose that C, D, T, T′ are V-small; notice that w := ũ^{∗} : T′ → T is a morphism of lex-sites (remark 4.4.13(ii)). We consider the induced essentially commutative diagram of morphisms of sites :

[figure: T′; w; h_{𝒟}^{a}; α; T; h_{𝒞}^{a}; D; v; C]

(lemma 4.2.11(ii)) which, according to (5.5.23), induces a diagram :

[figure: (D, V-Cat)^{∼}; (ṽ,V-Cat)_{∗}; ℱ ib_{D}^{a}; (α̃,Cat)_{∗}; (C, V-Cat)^{∼}; ℱ ib_{C}^{a}; (T′, V-Cat)^{∼}; Δ^{h^{a}_{𝒟}}; (w̃,V-Cat)_{∗}; ℱ ib_{T′}^{a}; Δ^{w}; (h̃_{𝒟}^{a} ,Cat)_{∗}; (T, V-Cat)^{∼}; ℱ ib_{T}^{a}; (h̃_{𝒞}^{a} ,Cat)_{∗}; Δ^{h^{a}_{𝒞}}; V-Stack(T′); V-St(w)_{∗}; V-St(h_{𝒟}^{a} )_{∗}; St(α)_{∗}^{γ}; V-Stack(T); V-St(h_{𝒞}^{a} )_{∗}; V-Stack(D); V-St(v)_{∗}; V-Stack(C)]

completed by adding the orientation Δ^{v} for the external square subdiagram. Indeed, this diagram is obtained by rotating suitably the corresponding diagram of (5.5.23), and notice that, after such rotation, the diagram still commutes on 2-cells, but now in the sense of (2.3.21). The four diagonal arrows of the diagrams are 2-equivalences, by theorem 4.4.2(iii), proposition 5.4.17(ii), and remark 5.5.14(ii). Then Υ((α̃, Cat)_{∗}) and Υ(St(α)_{∗}^{γ}) are pseudo-natural equivalences, by remark 2.3.11. Combining with (ii), theorem 4.4.2(iii) and remark 2.3.22(i), we conclude that if Υ(Δ^{w}) is a pseudo-natural equivalence, the same holds for Υ(Δ^{v}), whence the claim. ♢

(iii): By claim 5.5.27, we shall henceforth suppose that v is a morphism of small lex-sites. By remark 2.3.20we have Υ(Υ(𝒟^{v})) = 1_{i_{C}∘V-St(v)_{∗}}^{†}, and since

1_{i_{C}∘V-St(v)_{∗}} : Fib(v) ∘ ((−)_{C′}^{a}, i_{C′}, η^{C}, ε^{C}) ⇒ ((−)_{C}^{a}, i_{C}, η^{C}, ε^{C}) ∘ St(v)

is an invertible 2-cell in the category Link(V-\overline{2-Cat}), the strict isomorphisms of proposition 2.3.4 show that 1_{i_{C}∘V-St(v)_{∗}}^{†} is invertible as well. On the other hand, Υ(⊥^{v}) is not necessarily a pseudo-natural equivalence, but in order to conclude the proof of (iii) it will suffice to show that (−)_{D}^{a} ∗ Υ(⊥^{v}) is a pseudo-natural equivalence. To this aim, notice that (5.5.18) (for the morphism v) can be further decomposed as a chain of three oriented squares :

[figure: (D, V-Cat)^{∼}; (ṽ,V-Cat)_{∗}; j_{D}; ⊥_{1}^{v}; (C, V-Cat)^{∼}; j_{C}; (𝒟, V-Cat)^{∧}; (v,V-Cat)^{∧}; j_{𝒟}; ⊥_{2}^{v}; (𝒞 , V-Cat)^{∧}; j_{𝒞}; PsFun(𝒟^{o}, V-Cat); PsFun(v^{o},V-Cat); ℱ ib_{𝒟}; ⊥_{3}^{v}; PsFun(𝒞^{o}, V-Cat); ℱ ib_{𝒞}; V-Fib(𝒟); V-Fib(v)^{∗}; V-Fib(𝒞)]

where j_{C}, j_{𝒞} , j_{D} and j_{𝒟} are the inclusion pseudo-functors, and where now ℱ ib_{𝒞} and ℱ ib_{𝒟} are strong 2-equivalences (theorem 3.1.24). The orientation ⊥_{3}^{v} is still a pseudo-natural equivalence, and both orientations ⊥_{1}^{v} and ⊥_{2}^{v} are identities. After choosing as usual a unit and counit (η, ε) for the 2-adjoint pair ((v, V-Cat)_{!}, (v, V-Cat)^{∧}) and a unit and counit (η′, ε′) for the 2-adjoint pair (2-∫^{v^{o}}, PsFun(v^{o}, V-Cat)), we may regard the three squares as oriented squares of weak links. Then Υ(⊥_{3}^{v}) is well defined, and is a pseudo-natural equivalence (remark 2.3.11). Next, let 𝒜_{•} be any presheaf of categories on 𝒞, and set ℬ_{•} := (v, V-Cat)^{∧}𝒜; by definition we have

Υ(⊥_{2}^{v})_{𝒜_{•}} = ε′_{ℬ_{•}} ⊙ 2-∫ ^{v^{o}}η_{𝒜_{•}} .

But the discussion of remark 5.5.14(v) shows that

ε′_{ℬ_{•}} = ε_{ℬ_{•}} and 2-∫ ^{v^{o}}η_{𝒜_{•}} = (v, V-Cat)_{!}(η_{𝒜_{•}} ).

Hence, the triangular identities of (1.1.13) yield Υ(⊥_{2}^{v})_{𝒜_{•}} = 1_{ℬ_{•}} . Thus, we are further reduced to checking that ℱ ib_{D}^{a} ∗ Υ(⊥_{1}^{v}) is a pseudo-natural equivalence, or equivalently, that the same holds for ^{o}(ℱ ib_{D}^{a}) ∗^{o}Υ(⊥_{1}^{v}). However, let [−]_{C}^{a} (resp. [−]_{D}^{a}) be the left 2-adjoint of j_{C} (resp. of j_{D}), and (η^{C}, ε^{C}) (resp. (η^{D}, ε^{D})) the unit and counit of the 2-adjoint pair ([−]_{C}^{a}, j_{C}) (resp. ([−]_{D}^{a}, j_{D})); by proposition 2.3.10we have

oΥ(⊥_{1}^{v}) = (Υ(^{o}(⊥_{1}^{v})^{†}) = ^{o}((j_{D}∘(ṽ, V-Cat)^{∗})∗ε^{C})⊙(j_{D}∗ ⊥_{1}^{v†} ∗j_{C})⊙(η^{D}∗((v, V-Cat)_{!}∘j_{C})))

and notice that ε^{C} is a pseudo-natural equivalence, since j_{C} is fully faithful (corollary 2.4.29). Also ⊥_{1}^{v†} is a pseudo-natural equivalence, by virtue of proposition 2.3.4. Therefore, let ℬ_{•} be any sheaf of categories on C, and set ℬ′_{•} := (v, V-Cat)_{!}ℬ_{•}; we are reduced to checking that the cartesian functor ℱ ib_{D}^{a}(η_{ℬ′_{•}}^{C} ) is an equivalence of categories. But remark 5.2.2 easily implies that ℱ ib_{𝒟} (η_{ℬ′_{•}}^{C} ) is t-covering for t = 0, 1, 2, so the assertion follows from propositions 5.2.11(iii) and 5.2.9.

(iv): Under the stated assumptions, (i) says that both Δ_{𝒜_{•}}^{u} and Δ_{(ṽ′,Cat)^{∗}𝒜}^{u′} are equivalences, and both Υ(Δ^{v}) and Υ(Δ^{v′}) are pseudo-natural equivalences, by virtue of (iii). Then the assertion follows by arguing as in remark 2.3.22(i). □

Example 5.5.28. (i) Let C := (𝒞 , J) be a U-site, and A any small category. The constant presheaf of categories on 𝒞 with value A is the presheaf A_{𝒞} such that A_{𝒞} (X) := A for every X ∈ Ob(𝒞), and A_{𝒞} (f) := 1_{A} for every morphism f of 𝒞. The associated sheaf A_{𝒞}^{a} is then the constant sheaf of categories on C with value A. Likewise, we say that ℱ ib(A_{𝒞} ) is the constant fibration on 𝒞 with value A, and ℱ ib(A_{𝒞} )^{a} is the constant stack on C with value A.

(ii) Now, let C′ := (C′, J′) be another U-site, and u : C′ → C a morphism of sites; we notice that (ũ, Cat)^{∗}A_{𝒞} is isomorphic to the constant sheaf of categories on C′ with value A. Indeed, for the proof we are easily reduced to the corresponding assertion for constant sheaves of sets, which is known by remark 4.3.2(iii).

(iii) In light of (i) and lemma 5.5.15(iii), we deduce that the stack St(u)^{∗}(ℱ ib(A_{𝒞} )^{a}) is equivalent to the constant stack on C′ with value A.

5.5.29. In section (5.7) we will prove two base change theorems, asserting that, for suitable oriented diagrams of sites as in (5.5.23), the base change map Υ(St(β)_{∗}^{γ}) is a pseudo-natural equivalence. To this aim we shall apply corollary 5.5.26(iv), thereby reducing the assertion to the corresponding one for Υ((β̃, Cat)_{∗}), i.e. we shall deduce a base change theorem for stacks from one for sheaves of categories. The latter in turn can be reduced to the corresponding assertion for Υ(β_{∗}^{∼}), as we explain hereafter. Indeed, by remark 5.5.7(v,vi,vii) we deduce from (5.5.23) a diagram of 2-categories whose four diagonal arrows are 2-equivalences :

[figure: (D′, Cat)^{∼}; (v′,Cat)_{∗}; (u′,Cat)_{∗}; (C′, Cat)^{∼}; (ũ,Cat)_{∗}; Cat^{∗}(D^{′∼}); Cat^{∗}(ṽ′_{∗}); Cat^{∗}(ũ′_{∗}); Cat^{∗}(β̃_{∗}); Cat^{∗}(C^{′∼}); Cat^{∗}(ũ_{∗}); Cat^{∗}(D^{∼}); Cat^{∗}(ṽ_{∗}); Cat^{∗}(C^{∼}); (D, Cat)^{∼}; (ṽ,Cat)_{∗}; (C, Cat)^{∼}]

whose four external trapezoidal subdiagrams are strictly commutative, and can therefore be oriented by adding in the respective identity pseudo-natural transformations. We further complete the diagram by inserting the orientation

(β̃, Cat)_{∗} : (ũ, Cat)_{∗} ∘ (ṽ′, Cat)_{∗} ⇒ (ṽ, Cat)_{∗} ∘ (ũ′, Cat)_{∗}

for the “front face” of the resulting cubical diagram 𝒟. We regard 𝒟 as a diagram of 1-cells and 2-cells in V-\overline{2-Cat} (for a suitable universe V); then it is clear that 𝒟 commutes on 2-cells in the sense of (2.3.21). Moreover, all the horizontal arrows in 𝒟 admit left 2-adjoint pseudofunctors, which we regard as left adjoint 1-cells, in the 2-category V-\overline{2-Cat}; after fixing such a system of left adjoints, we may regard 𝒟 as a diagram oriented squares of links in V-\overline{2-Cat}, and especially, the 2-cells Υ((β̃, Cat)_{∗}) and Υ(Cat^{∗}(β̃_{∗})) are then well defined. Invoking again remarks 2.3.11), and 2.3.22(i) we deduce that Υ((β̃, Cat)_{∗}) is invertible if and only if the same holds for Υ(Cat^{∗}(β̃_{∗})) (recall that the invertible 2-cells of V-\overline{2-Cat} are the equivalence classes of pseudo-natural equivalences of pseudo-functors). According to proposition 2.3.24, we have Υ(Cat^{∗}(β̃_{∗})) = Cat^{∗}(Υ(β_{∗}^{∼})), where Υ(β_{∗}^{∼}) is the base change transformation associated with the oriented square of links (5.5.24); so, if Υ(β_{∗}^{∼}) is invertible, the same holds for Υ((β̃, Cat)_{∗}).

5.5.30. Here is a first illustration of the method explained in (5.5.29). Consider a morphism of sites u : C′ := (𝒞′, J′) → C := (𝒞 , J); we assume that all finite products are representable in 𝒞 and 𝒞′, and that u commutes with such products. For every X, Y ∈ Ob(𝒞) we choose a representative X × Y ∈ Ob(𝒞) for the product of X and Y, and let X ⟵^{q_{X,Y}} X × Y ⟶^{p_{X,Y}} Y be the universal projections. For every X ∈ Ob(𝒞) we get a morphism of sites p_{X} : C/X → C that assigns to every Y ∈ Ob(𝒞) the projection p_{X,Y} : see remark 4.7.3(iii). Likewise we define the morphism of sites p_{uX} : C′/uX → C′. For every Y ∈ Ob(𝒞), there is by assumption a unique isomorphism β_{Y} : u(X × Y) →^{∼} uX × uY in 𝒞′ such that p_{uX,uY} ∘ β_{Y} = u(p_{X,Y}), and

q_{uX,uY} ∘ β_{Y} = u(q_{X,Y}), where uX ⟵^{q_{uX,uY}} uX × uY ⟶^{p_{uX,uY}} uY are likewise the universal

projections. Especially, β_{Y} /X : u_{|X}(p_{X,Y}) →^{∼} p_{uX,uY} is an isomorphism in 𝒞′/uX (notation of (1.1.26)). Thus, we get an oriented square diagram of sites :

[figure: C′/uX; p_{uX}; u_{|X}; β; C′; u; C/X; p_{X}; C.]

Proposition 5.5.31. In the situation of (5.5.30), the base change transformation Υ(St(β)_{∗}^{γ}) is a pseudo-natural equivalence.

Proof. To begin with, we remark :

Claim 5.5.32. Let 𝒜_{•} ∈ Ob((C′, Cat)^{∼}) such that ℰ := ℱ ib_{𝒞′}(𝒜_{•}) is a stack on C′. Then the fibration ℱ ib_{𝒞′/uX} (p̃_{uX}, Cat)^{∗}𝒜_{•} is a stack on C′/uX.

Proof of the claim. The isomorphism of functors p̃_{uX}^{∗} ≃ s̃_{uX∗} of remark 4.7.3(iii) induces an isomorphism of pseudo-functors (p̃_{uX}, Cat)^{∗} →^{∼} (s̃_{uX}, Cat)_{∗}, hence it suffices to show that ℱ := ℱ ib_{𝒞′/uX} (s̃_{uX}, Cat)_{∗}𝒜_{•} = ℱ ib_{𝒞′/uX} (𝒜_{•} ∘ s_{uX}^{o}) ≃ 𝒞′/uX ×_{𝒞′} ℰ is a stack. Thus, let (Z ⟶^{h} uX) ∈ Ob(𝒞′/uX) and 𝒮′ ⊂ (𝒞′/uX)/h a sieve covering h for the site C′/uX; under the natural identification of categories

(𝒞′/uX)/h →^{∼} 𝒞′/Z

the sieve 𝒮′ corresponds to a sieve 𝒮 ⊂ 𝒞′/Z covering Z for the site C (see (4.7)). Then the restriction functor

ℱ (h) := Cart_{𝒞′/uX} ((𝒞′/uX)/h, ℱ) → Cart_{𝒞′/uX} (𝒮′, ℱ)

is naturally identified with the restriction functor

ℰ (Z) := Cart_{𝒞′}(𝒞′/Z, ℰ) → Cart_{𝒞′}(𝒮 , ℰ)

whence the contention. ♢

By claim 5.5.32, lemma 5.5.15(ii), corollary 5.5.26(iv) and the discussion of (5.5.29), we are reduced to showing that the base change transformation Υ(β_{∗}^{∼}) : p̃_{X}^{∗} ∘ ũ_{∗} → ũ_{|X∗} ∘ p̃_{uX}^{∗} is an isomorphism of functors. Now, recall that the source functor s_{X} : 𝒞 /X → 𝒞 is continuous for the sites C and C/X, and is left adjoint to p_{X}; more precisely, we have an explicit adjunction :

ϑ_{h,Z} : Hom_{𝒞} (Y, Z) →^{∼} Hom_{𝒞 /X} (h, p_{X,Z}) for every (Y ⟶^{h} X) ∈ Ob(𝒞 /X) and Z ∈ Ob(𝒞)

that assigns to every morphism f : Y → Z of 𝒞 the unique morphism f^{∗}/X : h → p_{X,Z} of 𝒞 /X such that q_{X,Z} ∘ f^{∗} = f. The adjunction ϑ_{h,Z} induces a corresponding adjunction for the pair (s̃_{X∗}, p̃_{X∗}), as described in remark 1.1.19(iii,iv). Explicitly, the unit of this induced adjunction assigns to every sheaf ℱ on C the morphism of sheaves :

η_{ℱ}^{X} : ℱ → ℱ ∘ s_{X}^{o} ∘ p_{X}^{o} Y ↦ (ℱ (q_{X,Y}) : ℱ (Y) → ℱ (X × Y))

and the counit assigns to every sheaf 𝒢 on C/X the morphism of sheaves :

ε_{𝒢}^{X} : 𝒢 ∘ p_{X}^{o} ∘ s_{X}^{o} → 𝒢 (Y ⟶^{h} X) ↦ (𝒢 (h^{∗}/X) : 𝒢 (p_{X,Y}) → 𝒢 (h)).

The same description applies to the adjunction for the pair (s̃_{uX∗}, p̃_{uX∗}), and we get therefore isomorphisms of links

\begin{matrix}ℒ_{X} := & (s̃_{X∗}, p̃_{X∗}, η^{X}, ε^{X}) →^{∼} p̃_{X} := (p̃_{X}^{∗}, p̃_{X∗}, η^{p_{X}}, ε^{p_{X}}) \\ ℒ_{uX} := & (s̃_{uX∗}, p̃_{uX∗}, η^{uX}, ε^{uX}) →^{∼} p̃_{uX} := (p̃_{uX}^{∗}, p̃_{uX∗}, η^{p_{uX}}, ε^{p_{uX}}). \end{matrix}

Hence, it suffices to check that the base change transformation for the oriented diagram of links:

𝒟  :  [figure: (C′/uX)^{∼}; ℒ_{uX}; ũ_{|X}; β̃_{∗}; C^{′∼}; ũ; (C/X)^{∼}; ℒ_{X}; C^{∼}]

is an isomorphism of functors Υ(𝒟) : s̃_{X∗} ∘ ũ_{∗} →^{∼} ũ_{|X∗} ∘ s̃_{uX∗}. However, a direct computation that we leave to the reader easily shows that Υ(𝒟) is the identity automorphism of s̃_{X∗} ∘ ũ_{∗} = ũ_{|X∗} ∘ s̃_{uX∗} □ 5.6. Stacks in groupoids and ind-finite stacks. Let C := (𝒞 , J) be any site; for every universe V denote by

V-Stack^{×}(C) and V-StGpd(C)

respectively the strong 2-subcategory of V-Fib^{×}(𝒞) whose objects are the V-stacks on C and the strong 2-subcategory of V-Stack(C) whose objects are the V-stacks in groupoids on C, i.e. the V-stacks on C that are fibrations in groupoids. For every 𝒞 -fibration F : ℰ → 𝒞, every X ∈ Ob(𝒞), and every sieve 𝒮 ⊂ 𝒞 /X, the inclusion ℰ^{×} → ℰ yields a commutative diagram

\xymatrix{ℰ (X)^{×} \ar[r] \ar[d] & ℰ^{×}(X) \ar[d] \\ Cart_{𝒞} (𝒮 , ℰ)^{×} \ar[r] & Cart_{𝒞} (𝒮 , ℰ^{×})}

whose horizontal arrows are isomorphisms of categories, due to remark 3.4.8(i); notice that the source functor s : 𝒮 → 𝒞 is a fibration in groupoids, by remark 3.5.2(ii). Suppose now that ℰ is i-separated on C for some i ∈ {0, 1, 2}; from remark 3.4.1(ii) we deduce that the left vertical arrow is an i-faithful functor whenever 𝒮 ∈ J(X), and then the same holds for the right vertical arrow, so finally ℰ^{×} is also i-separated on C. Especially, the pseudo-functor (−)_{𝒞}^{×} restricts to a pseudo-functor

(−)_{C}^{×} : V-Stack^{×}(C) → V-StGpd(C).

Next, if C is a small site, taking into account remark 3.4.1(iii) we get a natural identification:

(ℰ_{X}^{+})^{×} →^{∼} colim_{𝒮 ∈J(X)} Cart_{𝒞} (𝒮 , ℰ)^{×} →^{∼} colim_{𝒮 ∈J(X)} Cart_{𝒞} (𝒮 , ℰ^{×}) = ((ℰ^{×})^{+})_{X}

whence natural isomorphisms of 𝒞 -fibrations :

(5.6.1)  (ℰ^{+})^{×} →^{∼} (ℰ^{×})^{+}  (ℰ^{a})^{×} →^{∼} (ℰ^{×})^{a}.

If C′ := (𝒞′, J′) is another site, and u : C → C′ a weak morphism of sites, it is also clear that V-St(u)_{∗} restricts to pseudo-functors

V-St^{×}(u)_{∗} : V-Stack^{×}(C) → V-Stack^{×}(C′) V-StGpd(u)_{∗} : V-StGpd(C) → V-StGpd(C′)

that make commute the diagram of 2-categories :

(5.6.2)  \xymatrix{V-Stack^{×}(C) \ar[d]_{(−)_{C}^{×}} \ar[r]^{V-St^{×}(u)_{∗}} & V-Stack^{×}(C′) \ar[d]^{(−)_{C′}^{×}} \\ V-StGpd(C) \ar[r]^{V-StGpd(u)_{∗}} & V-StGpd(C′).}

5.6.3. Consider any U-site C := (𝒞 , J), and let j : 𝒯 → 𝒞 be the inclusion functor of an essentially small subcategory whose set of objects is a topologically generating family for C; we endow as usual 𝒯 with the topology J_{𝒯} induced by J via j, so that j is a continuous functor for J and J_{𝒯} . Say that 𝒞 is U′-small for some universe U′, and recall that the rule : 𝒜 ↦ 𝒜^{a} := U′-Fib(j)_{∗}(U′-Fib(j)^{∗}(𝒜)^{a}) for every fibration 𝒜 with small fibres on 𝒞 yields a left 2-adjoint for the inclusion pseudo–functor Stack(C) → Fib(𝒞) (see the proof of corollary 5.4.19). Combining with (3.4.5) and (3.4.6) we deduce a pseudo-commutative diagram :

\xymatrix{V-Fib^{×}(𝒞) \ar[r]^{(−)^{a}} \ar[d]_{(−)_{𝒞}^{×}} & V-Stack^{×}(C) \ar[d]^{(−)_{C}^{×}} \\ V-Gpd(𝒞) \ar[r]^{(−)^{a}} & V-StGpd(C)}

for every universe V containing U.

Proposition 5.6.4. For every morphism of U-sites u : C′ := (𝒞′, J′) → C := (𝒞 , J), and every universe V containing U, the pseudo-functor V-St(u)^{∗} restricts to pseudo-functors

V-St^{×}(u)^{∗} : V-Stack^{×}(C) → V-Stack^{×}(C′) V-StGpd(u)^{∗} : V-StGpd(C) → V-StGpd(C′)

and we have a pseudo-commutative diagram :

\xymatrix{V-Stack^{×}(C) \ar[r]^{V-St^{×}(u)^{∗}} \ar[d]_{(−)_{C}^{×}} & V-Stack^{×}(C′) \ar[d]^{(−)_{C′}^{×}} \\ V-StGpd(C) \ar[r]^{V-StGpd(u)^{∗}} & V-StGpd(C′).}

Proof. Endow the topoi T := C_{U}^{∼} and T′ := C_{U}^{′∼} with their canonical topologies; we get an essentially commutative diagram of morphisms of sites :

\xymatrix{T′ \ar[r]^{u′} \ar[d]_{h_{C′}^{a}} & T \ar[d]^{h_{C}^{a}} \\ C′ \ar[r]^{u} & C}  with u′ := ũ^{∗}

(lemma 4.2.11(ii)). After replacing U by a larger universe, we may assume that T and T′ are small; notice also that u′ is a morphism of lex-sites (remark 4.4.13(ii)). Especially, the proposition is already known for u′, by virtue of (3.4.7) and (5.6.1).

Claim 5.6.5. The proposition holds for h_{C}^{a} and h_{C′}^{a}.

Proof of the claim. It suffices to check the assertion for h_{C}^{a}, since the same argument will work for h_{C′}^{a}. Now, let 𝒜 be a stack in groupoids on C; we need to check that St(h_{C}^{a})^{∗}(𝒜) is a stack of groupoids on T. However, since St(h_{C}^{a})_{∗} is a 2-equivalence (proposition 5.4.17(ii)), we may assume that 𝒜 = St(h_{C}^{a})_{∗}ℬ for a stack ℬ on T, and we may moreover assume that ℬ is a stack in groupoids, by (5.6.1). But since St(h_{C}^{a})^{∗}(𝒜) is equivalent to ℬ, the assertion follows. This shows that the pseudo-functor StGpd(h_{C}^{a})^{∗} is well defined, and then it is clearly a pseudo-inverse for StGpd(h_{C}^{a})_{∗}. But then the required pseudo-commutativity of the resulting diagram follows easily from the commutativity of (5.6.2) : details left to the reader. ♢

Now, let 𝒜 be a stack in groupoids on C; we need to check that ℬ := St(u)^{∗}(𝒜) is a stack in groupoids on C′, and by claim 5.6.5it suffices to show that St(h_{C′}^{a})^{∗}(ℬ) is a stack in groupoids on T′; but the latter is equivalent to St(u′)^{∗} ∘ St(h_{C}^{a})^{∗}(𝒜), and the proposition is already known for both u′ and h_{C}^{a}. This shows that StGpd(u)^{∗} is well defined.

Lastly, we consider the diagram of 2-categories :

\xymatrix{StGpd(T) \ar[rrr]^{StGpd(u′)^{∗}} \ar[ddd]|{StGpd(h_{C}^{a})^{∗}} &  &  & StGpd(T′) \ar[ddd]|{StGpd(h_{C′}^{a})^{∗}} \\  & Stack^{×}(T) \ar[r]^{St^{×}(u′)^{∗}} \ar@{-->}[ul]^{(−)_{T}^{×}} \ar[d]_{St^{×}(h_{C}^{a})^{∗}} & Stack^{×}(T′) \ar[d]^{St^{×}(h_{C′}^{a})^{∗}} \ar@{-->}[ur]_{(−)_{T′}^{×}} \\  & Stack^{×}(C) \ar[r]^{St^{×}(u)^{∗}} \ar@{-->}[dl]_{(−)_{C}^{×}} & Stack^{×}(C′) \ar@{-->}[dr]^{(−)_{C′}^{×}} \\ StGpd(C) \ar[rrr]^{StGpd(u)^{∗}} &  &  & StGpd(C′)}

whose inner and outer square subdiagrams are pseudo-commutative. Moreover, by the foregoing, we know that also the left, the right, and the top trapezoidal subdiagrams pseudo-commute.

Since St^{×}(h_{C}^{a}) is a 2-equivalence, a little diagram chase shows easily that the bottom trapezoidal subdiagram pseudo-commutes as well, whence the proposition. □

5.6.6. Let C := (𝒞 , J) be any U-site; according to example 5.1.2, for every universe V containing U, the pseudo-functor ℱ ib_{𝒞} of (3.4.9) restricts to a (strict) pseudo-functor :

ℱ ib_{C} : C_{V}^{∼} → V-StGpd(C)

and then it is clear that π_{0}^{𝒞} induces a left 2-adjoint pseudo-functor

π_{0}^{C} : V-StGpd(C) → C_{V}^{∼} ℰ ↦ (π_{0}^{𝒞} (ℰ))^{a}.

Lemma 5.6.7. In the situation of (5.6.6), let φ : ℰ → ℱ be a cartesian functor of fibrations in groupoids over 𝒞. The following holds :

(i) φ is 0-covering (for the topology of C) if and only if π_{0}^{𝒞} (φ)^{a} : π_{0}^{𝒞} (ℰ)^{a} → π_{0}^{𝒞} (ℱ)^{a} is an epimorphism of sheaves on C.

(ii) If φ is i-covering for i = 0, 1, then π_{0}^{𝒞} (φ)^{a} is an isomorphism.

Proof. Both assertions follow by direct inspection of the definitions, taking into account corollary 4.1.30and remark 4.1.29(ii,iii). □

5.6.8. Let u : C′ := (𝒞′, J′) → C := (𝒞 , J) be a morphism of U-sites; then for every universe V containing U the induced diagram of 2-categories

\xymatrix{V-StGpd(C) \ar[r]^{V-StGpd(u)^{∗}} \ar[d]_{π_{0}^{C}} & V-StGpd(C′) \ar[d]^{π_{0}^{C′}} \\ C^{∼} \ar[r]^{_{ũ_{∗}}} & C^{′∼}}

is essentially commutative. Indeed, lemma 5.6.7(ii) and proposition 5.2.11(i) imply that for every fibration ℰ over 𝒞′, the unit of adjunction ℰ → ℰ^{a} induces an isomorphism π_{0}^{𝒞′} (ℰ)^{a} →^{∼} π_{0}^{C′}(ℰ^{a}) of sheaves on C′. The assertion follows easily, taking into account the essential commutativity of (3.4.10) : the details are left to the reader.

5.6.9. Ind-finite stacks. It is easy to say when a sheaf of groups G on a topological space 𝒯 is ind-finite, when 𝒯 admits a basis of quasi-compact open subsets : in which case, one simply asks that the group of U-sections G(U) is a filtered union of finite groups, for every quasi-compact open subset U of 𝒯 . This definition is however not suitable for more general topological spaces, nor of course for arbitrary sites. Before we explain a definition that is appropriate for the general case, let us introduce a notion of quasi-compactness for such context:

Definition 5.6.10. Let (𝒞 , J) be any site, and X ∈ Ob(𝒞). We say that X is quasi-compact for the topology J, if for every covering sieve 𝒮 ∈ J(X) there exists a finite subset S ⊂ 𝒮 that generates a sieve covering X.

Obviously this definition recovers the standard one, in the case of the site of open subsets of any topological space.

Definition 5.6.11. Let C := (𝒞 , J) be a site, and G a sheaf of groups on C. We say that G is ind-finite if the following holds. For every X ∈ Ob(𝒞) and every finite subset Σ ⊂ GX, there exists a covering family (f_{i} : X_{i} → X | i ∈ I) for the topology J such that the set Gf_{i}(Σ) := {Gf_{i}(σ) | σ ∈ Σ} generates a finite subgroup of GX_{i}, for every i ∈ I.

Remark 5.6.12. The first observation is that if G is ind-finite on the site (𝒞 , J), then for every quasi-compact X ∈ Ob(𝒞) the group GX is ind-finite, i.e. it is a filtered union of finite groups. Indeed, let Σ ⊂ GX be any finite subset; by assumption we have a covering family (X_{i} → X | i ∈ I) such that Gf_{i}(Σ) generates a finite group H_{i} for every i ∈ I, and since X is quasi-compact, we may assume that I is a finite set. But the natural map GX → ∏_{i∈I} GX_{i} is injective, and maps the subgroup H generated by Σ into the finite group ∏_{i∈I} H_{i}, so H is finite, whence the claim.

5.6.13. For every group G and every subset Σ ⊂ G, let us write ⟨Σ⟩ ⊂ G for the subgroup generated by Σ. Definition 5.6.11 prompts the following construction. Let C := (𝒞 , J) be any site, and H any presheaf of groups on 𝒞. For every n ∈ ℕ and every X ∈ Ob(𝒞) we set :

(H^{n})_{f}(X) := {(σ_{1}, …, σ_{n}) ∈ H^{n}(X) | ⟨σ_{1}, …, σ_{n}⟩ is a finite group}.

Clearly the rule : X ↦ (H^{n})_{f}(X) yields a sub-presheaf H_{f}^{n} of the presheaf of sets H^{n}, for every n ∈ ℕ. If H is a sheaf on C, (H^{n})_{f} is not necessarily a sheaf, hence we consider the sheaf

(H^{n})_{lf}

defined as the smallest subsheaf of sets of H^{n} containing (H^{n})_{f}. The latter is also (naturally isomorphic to) the sheaf of sets (H^{n})_{f}^{a} on C associated with the presheaf (H^{n})_{f}. Clearly every morphism φ : H → K of presheaves of groups on 𝒞 induces a morphism of presheaves of sets

(φ^{n})_{f} : (H^{n})_{f} → (K^{n})_{f} for every n ∈ ℕ.

If H and K are sheaves of groups on C, we get also an induced morphism of sheaves of sets

(φ^{n})_{lf} : (H^{n})_{lf} → (K^{n})_{lf} for every n ∈ ℕ.

Thus, in the notation of definition 5.5.1, we get by these rules two well defined functors

(−)_{f}^{n} : (𝒞 , V-Grp)^{∧} → 𝒞_{V}^{∧}  (−)_{lf}^{n} : (C, V-Grp)^{∼} → C_{V}^{∼}  for every n ∈ ℕ

for every universe V.

Remark 5.6.14. Let C := (𝒞 , J) be a site, and H a sheaf of groups on C.

(i) Notice that (H^{n})_{f} is a separated presheaf for every n ∈ ℕ, since it is a sub-presheaf of the sheaf H^{n}; hence (H^{n})_{lf} = (H^{n})_{f}^{+} (claim 4.1.16(ii)). This means that for every X ∈ Ob(𝒞), the set (H^{n})_{lf}(X) consists of all (σ_{1}, …, σ_{n}) ∈ H^{n}(X) for which there exists a covering family (f_{i} : X_{i} → X | i ∈ I) for the topology J such that ⟨Gf_{i}(σ_{1}), …, Gf_{i}(σ_{n})⟩ is a finite subgroup of GX_{i}, for every i ∈ I.

(ii) We deduce easily from (i) that H is ind-finite if and only if H^{n} = (H^{n})_{lf} for every n ∈ ℕ. (iii) It also follows from (i) that (H^{n})_{f}(X) = (H^{n})_{lf}(X) for every quasi-compact object

X of 𝒞. Indeed, let \underline{σ} := (σ_{1}, …, σ_{n}) ∈ (H^{n})_{lf}(X), and pick a covering family (f_{i} : X_{i} → X | i ∈ I) verifying the condition of (i) relative to \underline{σ}; since X is quasi-compact, we may assume that I is a finite set, and then arguing as in remark 5.6.12we deduce that \underline{σ} ∈ (H^{n})_{f}(X).

Lemma 5.6.15. Let (𝒞 , J) be a site, and φ : H → K a bicovering morphism of presheaves of groups on 𝒞. Then (φ^{n})_{f} : (H^{n})_{f} → (K^{n})_{f} is bicovering for every n ∈ ℕ.

Proof. Let X ∈ Ob(𝒞) and \underline{σ} := (σ_{1}, …, σ_{n}) ∈ (K^{n})_{f}(X); according to remark 4.1.29(ii) we have to exhibit a covering family (f_{i} : X_{i} → X | i ∈ I) such that (Kf_{i}(σ_{1}), …, Kf_{i}(σ_{n})) lies in the image of the map (φ^{n})_{f,X_{i}} : (H^{n})_{f}(X_{i}) → (H^{n})_{f}(X_{i}) for every i ∈ I. But by assumption, for every j = 1, …, n there exists a covering sieve 𝒮_{j} ⊂ 𝒞 /X such that Kf(σ_{j}) lies in the image of the map φ_{Y} : HY → KY for every f : Y → X in 𝒮_{j}. Then the sieve 𝒮 := 𝒮_{1} ∩⋯∩𝒮_{n} still covers X (remark 4.1.3(i)), and for every g : Y → X in 𝒮 we see that K^{n}g(\underline{σ}) = φ_{Y}^{n} (\underline{τ}) for some \underline{τ} := (τ_{1}, …, τ_{n}) ∈ H^{n}(Y). To conclude, it then suffices to exhibit for every such g and τ a covering sieve 𝒮′ ⊂ 𝒮 /Y such that H^{n}g′(\underline{τ}) ∈ (H^{n})_{f}(Y′) for every g′ : Y′ → Y in 𝒮′. However, by construction we have φ_{Y}^{n} (\underline{τ}) ∈ (K^{n})_{f}(Y); the latter means that the subgroup ⟨φ_{Y} (τ_{1}), …, φ_{Y} (τ_{n})⟩ ⊂ KY is finite. This in turn means that there exists an integer N ≥ 2, such that for every \underline{j} ∈ \{1, …, n\}^{N} we may find an integer M(\underline{j}) < N and

\underline{j}′ ∈ \{1, …, n\}^{M(\underline{j})} with ∏_{k=1}^{N} φ_{Y} (τ_{j_{k}}) = ∏_{k=1}^{M(\underline{j})} φ_{Y} (τ_{j′_{k}}). By remark 4.1.29(iii), this implies that there exists a covering sieve 𝒮 ′ ; \underline{j} ⊂ 𝒞 /Y such that ∏_{k=1}^{N} Hf(τ_{j_{k}}) = ∏_{k=1}^{M(\underline{j})} Hf(τ_{j′_{k}}) for every f : Y′ → Y in 𝒮 ′ ; \underline{j}. Then the sieve 𝒮′ := ⋂_{\underline{j}∈\{1,...,n\}^{N}} 𝒮 ′ ; \underline{j} covers Y, and we deduce that the sequence H^{n}f(\underline{τ}) generates a finite subgroup of H^{n}Y′, for every f : Y′ → Y in 𝒮′; i.e. H^{n}f(\underline{τ}) ∈ (H^{n})_{f}(Y′), as required.

Lastly, let \underline{σ}, \underline{σ}′ ∈ (H^{n})_{f}(X) be two sections such that φ_{X}^{n}(\underline{σ}) = φ_{X}^{n}(\underline{σ}′); according to remark 4.1.29(iii) we need to exhibit a covering sieve 𝒮 ⊂ 𝒞 /X such that H^{n}f(\underline{σ}) = H^{n}f(\underline{σ}′) for every f : Y → X in 𝒮. But by assumption, for i = 1, …, n there exists a sieve 𝒮_{i} such that Hf(σ_{i}) = Hf(σ′_{i}) for every f in 𝒮_{i}; clearly 𝒮 := 𝒮_{1} ∩ ⋯ ∩ 𝒮_{n} will do. □

5.6.16. Let now C := (𝒞 , J) and C′ := (𝒞′, J′) be two sites, u : 𝒞′ → 𝒞 a functor, and H a presheaf of groups on 𝒞. Clearly u^{∧}H is a presheaf of groups on 𝒞′, and we have :

u^{∧}((H^{n})_{f}) = ((u^{∧}H)^{n})_{f} for every n ∈ ℕ.

Proposition 5.6.17. In the situation of (5.6.16), suppose that u is cocontinuous for the topologies J and J′, and H is a sheaf of groups on C. Then the following holds :

(i) ŭ^{∗}H is a sheaf of groups on C′, and ŭ^{∗}((H^{n})_{lf}) = ((ŭ^{∗}H)^{n})_{lf} for every n ∈ ℕ.

(ii) If H is ind-finite, the same holds for ŭ^{∗}H.

Proof. (i): Since the functor (−)^{a} is exact, it is clear that the group law K × K → K of every presheaf of groups K on 𝒞′ yields a group law K^{a} × K^{a} → K^{a} for the sheaf K^{a} : details left to the reader; the first assertion of the proposition is an immediate consequence.

To check the sought identity, recall that by definition ŭ^{∗}((H^{n})_{lf}) = (u^{∧}((H^{n})_{f}^{a}))^{a}, and

(u^{∧}((H^{n})_{f}^{a}))^{a} = (u^{∧}((H^{n})_{f}))^{a}

by lemma 4.2.9 and corollary 4.1.30(iii). Next, (u^{∧}((H^{n})_{f}))^{a} = (((u^{∧}H)^{n})_{f})^{a}, by (5.6.16). Lastly, (((u^{∧}H)^{n})_{f})^{a} = ((((u^{∧}H)^{a})^{n})_{f})^{a}, by lemma 5.6.15, whence the contention.

(ii) follows immediately from (i) and remark 5.6.14(ii). □

Lemma 5.6.18. In the situation of (5.6.16), suppose that the category X/u𝒞′ is cofiltered for every X ∈ Ob(𝒞), and let K be a presheaf of groups on 𝒞′. Then u_{!}K is a presheaf of groups on 𝒞, and we have a natural isomorphism of presheaves :

u_{!}((K^{n})_{f}) →^{∼} ((u_{!}K)^{n})_{f} for every n ∈ ℕ.

Proof. Under the stated assumption, the functor u_{!} is exact (cp. the proof of corollary 1.5.19(i)), so we easily deduce that u_{!}K is a presheaf of groups, as in the proof of proposition 5.6.17(i).

Next, we get a natural morphism of presheaves ω : u_{!}((K^{n})_{f}) → ((u_{!}K)^{n})_{f} as follows. First, the unit of adjunction is a morphism η_{K} : K → u^{∧}u_{!}K, which induces a morphism (η_{K}^{n})_{f} : (K^{n})_{f} → ((u^{∧}u_{!}K)^{n})_{f}. Then, by (5.6.16) we have ((u^{∧}u_{!}K)^{n})_{f} = u^{∧}(((u_{!}K)^{n})_{f}), and the resulting morphism (K^{n})_{f} → u^{∧}(((u_{!}K)^{n})_{f}) yields by adjunction the sought morphism.

Explicitly, for every X ∈ Ob(𝒞), every element of u_{!}((K^{n})_{f})(X) is the class [\underline{σ}] of some \underline{σ} := (σ_{1}, …, σ_{n}) ∈ K^{n}(Y), for an object f : X → uY of X/u𝒞′, and ω_{X}([\underline{σ}]) is the section ([σ_{1}], …, [σ_{n}]) ∈ ((u_{!}K)^{n})_{f}(X), where [σ_{i}] ∈ u_{!}K(X) is the class of σ_{i}, for i = 1, …, n.

To show the injectivity of ω_{X}, let [\underline{σ}], [\underline{τ}] ∈ u_{!}((K^{n})_{f})(X) such that ω_{X}([\underline{σ}]) = ω_{X}([\underline{τ}]); we may assume that \underline{σ}, \underline{τ} ∈ K^{n}Y for some morphism f : X → uY of 𝒞′, and the assumption means that for i = 1, …, n there exists a morphism X/h_{i} : (g_{i} : X → uY_{i}) → f in X/u𝒞′ such that Kh_{i}(σ_{i}) = Kh_{i}(τ_{i}) in KY_{i}. Since X/u𝒞′ is cofiltered, we may then find an object g′ : X → uY′ of X/u𝒞′ and morphisms X/h′_{i} : g′ → g_{i} of X/u𝒞′ with (X/h_{i}) ∘ (X/h′_{i}) = (X/h_{j}) ∘ (X/h′_{j}) for every i, j = 1, …, n. Recall that h_{i} : Y_{i} → Y and h′_{i} : Y′ → Y_{i} are morphisms in 𝒞′, and the foregoing identity means that h := h_{i} ∘ h′_{i} = h_{j} ∘ h′_{j} for every such i, j. We conclude that Kh(σ_{i}) = Kh(τ_{i}) for i = 1, …, n, whence [\underline{σ}] = [\underline{τ}], as required.

To check the surjectivity of ω_{X}, let ([σ_{1}], …, [σ_{n}]) ∈ ((u_{!}K)^{n})_{f}(X); hence, for i = 1, …, n there exists an object f_{i} : X → uY_{i} of X/u𝒞′ such that σ_{i} ∈ KY_{i}. Arguing as in the foregoing, we easily reduce to the case where Y := Y_{1} = ⋯ = Y_{n} and f := f_{1} = ⋯ = f_{n}. For every morphism X/h : (g : X → uY′) → f in X/u𝒞′, let G_{h} := ⟨Kh(σ_{1}), …, Kh(σ_{n})⟩ ⊂ KY′, and set G := ⟨[σ_{1}], …, [σ_{n}]⟩ ⊂ u_{!}K(X). The source morphism (X/u𝒞′)/f → X/u𝒞′ is coinitial (example 1.5.9(i)), and a direct inspection shows that the group G is the colimit of the functor

G_{•} : ((X/u𝒞′)/f)^{o} → Grp h ↦ G_{h}.

To every pair of objects X/h : (g : X → uY′) → f and X/h : (g : X → uY′′) → f of (X/u𝒞′)/f, and every morphism (X/l)/f : (X/h′) → (X/h) in (X/u𝒞′)/f, the functor G_{•} assigns the group homomorphism G_{h} → G_{h′} given by the restriction of Kh : KY′ → KY′′. By assumption, G is a finite group; let us then remark more generally :

Claim 5.6.19. Let Γ_{•} : I → Grp be a functor from a filtered category I, such that : (a) Γ_{i} is a finitely generated group for every i ∈ Ob(I)

(b) Γ_{φ} is a surjective group homomorphism for every morphism φ of I

(c) The colimit G of Γ_{•} is a finite group. Let also τ_{•} : Γ_{•} ⇒ c_{G} be a universal cocone. Then there exists a cofinal functor ψ : J → I such that τ ∗ ψ is an isomorphism of functors Γ_{•} ∘ ψ →^{∼} c_{G}.

Proof of the claim. By virtue of proposition 1.5.21(ii) we may assume that (I, ≤) is a filtered partially ordered set. Since I is filtered, the colimit of Γ_{•} commutes with the forgetful functor Φ : Grp → Set. Especially, for every x ∈ G we may find i(x) ∈ Ob(I) and x′ ∈ Γ_{i(x)} such that τ_{i(x)}(x′) = x. Since I is filtered and G is finite, we may then find i ∈ Ob(I) with i ≥ i(x) for every x ∈ G. Hence the rule : x ↦ Γ_{i(x),i}(x′) defines a set-theoretic section σ_{i} : G → Γ_{i} of τ_{i}. Then for every j ≥ i, the composition σ_{j} := Γ_{ij} ∘ σ_{i} is a set-theoretic section of τ_{j}. After replacing I by the cofinal subset {j ∈ I | j ≥ i} we may therefore assume that there exists a cone σ_{•} : c_{G} ⇒ Φ ∘ Γ_{•} such that τ_{i} ∘ σ_{i} = 1_{G} for every i ∈ I. Next, for every x, y ∈ G we may find i(x, y) ∈ I such that

σ_{i(x,y)}(xy) = σ_{i(x,y)}(x) · σ_{i(x,y)}(y)

(details left to the reader). Again, we may then pick i_{0} ∈ I such that i_{0} ≥ i(x, y) for every x, y ∈ G, in which case it is easily seen that σ_{i_{0}}(xy) = σ_{i_{0}}(x) · σ_{i_{0}}(y) for every x, y ∈ G, i.e. σ_{i_{0}} is a group homomorphism. Then clearly σ_{j} is a group homomorphism for every j ≥ i_{0}, so after replacing I by the cofinal subset of elements ≥ i_{0}, we may assume that σ_{•} is even a cone c_{G} ⇒ Γ_{•}, and that I admits an initial element i_{0}. Now, let g_{1}, …, g_{k} be a finite system of generators for Γ_{i_{0}}. For t = 1, …, k and every l ∈ I we have τ_{l} ∘ σ_{l} ∘ τ_{l}(g_{t}) = τ_{l} ∘ Γ_{i_{0},l}(g_{t}); hence we may find j_{t} ∈ I such that

σ_{j_{t}} ∘ τ_{j_{t}}(g_{t}) = Γ_{i_{0},j_{t}}(g_{t}).

Then pick again j ∈ I such that j ≥ j_{t} for every t = 1, …, k, and set g′_{t} := Γ_{i_{0},j}(g_{t}) for every such t. It follows that for every t = 1, …, k we have :

σ_{j} ∘ τ_{j}(g′_{t}) = σ_{j} ∘ τ_{j} ∘ Γ_{i_{0},j}(g_{t})
= σ_{j} ∘ τ_{j_{t}} ∘ Γ_{i_{0},j_{t}}(g_{t})
= Γ_{j_{t},j} ∘ σ_{j_{t}} ∘ τ_{j_{t}} ∘ Γ_{i_{0},j_{t}}(g_{t})
= Γ_{j_{t},j} ∘ Γ_{i_{0},j_{t}}(g_{t})
= g′_{t}.

Notice that g′_{1}, …, g′_{k} is a generating system for Γ_{j}, since Γ_{i_{0},j} is a surjective map. We conclude that σ_{j} ∘ τ_{j} = 1_{Γ_{j}}, so τ_{j} and σ_{j} are mutually inverse group isomorphisms. After replacing I by the cofinal subset {l ∈ I | l ≥ j}, we may then assume that j = i_{0} is the initial element of I. Then, for every i ∈ I the map σ_{i} = Γ_{i_{0},i} ∘ σ_{i_{0}} is surjective; but σ_{i} is also injective by construction, so σ_{i} is an isomorphism for every i ∈ I, and finally, the same follows for τ_{i}. ♢

From claim 5.6.19we deduce that there exists a morphism X/h : (X → uY′) → f in X/u𝒞′ such that G_{h} is a finite group, and therefore \underline{σ}′ := K^{n}h(σ_{1}, …, σ_{n}) ∈ (K^{n})_{f}(Y′). Then the class [\underline{σ}′] ∈ u_{!}(K^{n})(X) lies in u_{!}((K^{n})_{f})(X) and ω_{X}([\underline{σ}′]) = ([σ_{1}], …, [σ_{n}]) as required. □

Theorem 5.6.20. In the situation of (5.6.16), suppose that u : C → C′ is a morphism of sites, and let K be a sheaf of groups K on C′. Then ũ^{∗}K is a sheaf of groups on C, and we have a natural isomorphism of sheaves :

ω_{u,K}^{(n)} : ũ^{∗}((K^{n})_{lf}) →^{∼} ((ũ^{∗}K)^{n})_{lf} for every n ∈ ℕ.

Proof. By assumption, for every universe V such that C and C′ are V-small, the functor ũ^{∗} : C^{′∼} → C^{∼} is exact; as in the proof of proposition 5.6.17(i), we deduce that ũ^{∗}K is a sheaf of groups. We get therefore for every such V a well defined functor

(ũ, V-Grp)^{∗} : (C′, Grp)^{∼} → (C, V-Grp)^{∼} K ↦ ũ^{∗}K.

Next, the unit of adjunction η_{K} : K → ũ_{∗}ũ^{∗}K induces a morphism of presheaves (η_{K}^{n})_{f} : (K^{n})_{f} → ((ũ_{∗}ũ^{∗}K)^{n})_{f}. By (5.6.16), we have ((ũ_{∗}ũ^{∗}K)^{n})_{f} = u^{∧}(((ũ^{∗}K)^{n})_{f}), so by adjunction (η_{K}^{n})_{f} corresponds to a morphism of presheaves u_{!}((K^{n})_{f}) → ((ũ^{∗}K)^{n})_{f}. After taking associated sheaves, we get a morphism of sheaves ω : (u_{!}((K^{n})_{f}))^{a} → ((ũ^{∗}K)^{n})_{lf}. Lastly, the inclusion of presheaves (K^{n})_{f} → (K^{n})_{lf} induces an isomorphism (u_{!}((K^{n})_{f}))^{a} →^{∼} ũ^{∗}((K^{n})_{lf}), by lemma 4.2.11(ii), and ω_{u,K}^{(n)} is the composition of ω with the inverse of this isomorphism. With the notation of (5.6.13), we have therefore a natural transformation :

ω_{u,•}^{(n)} : ũ^{∗} ∘ (−)_{lf}^{n} ⇒ (−)_{lf}^{n} ∘ (ũ, Grp)^{∗} K ↦ ω_{u,K}^{(n)}

and it remains to check that ω_{u,K}^{(n)} is an isomorphism. We begin with the following more explicit description of the map (ω_{u,K}^{(n)})_{X} : ũ^{∗}((K^{n})_{lf})(X) → ((ũ^{∗}K)^{n})_{lf}(X), for every X ∈ Ob(𝒞). First, by definition every element of ũ^{∗}((K^{n})_{lf})(X) is the equivalence classes [\underline{σ}]_{f} of a sequence \underline{σ} := (σ_{1}, …, σ_{n}) ∈ K^{n}Y, for a morphism f : X → uY in 𝒞, such that for some covering sieve 𝒮 ⊂ 𝒞′/Y, the subgroup ⟨Kg(σ_{1}), …, Kg(σ_{n})⟩ ⊂ KY′ is finite for every (g : Y′ → Y) ∈ Ob(𝒮). Likewise, every element of ((ũ^{∗}K)^{n})_{lf}(X) is a sequence ([τ_{1}]_{f_{1}}, …, [τ_{n}]_{f_{n}}) such that [τ_{i}]_{f_{i}} ∈ ũ^{∗}K(X) for every i = 1, …, n is the equivalence class of a section τ_{i} ∈ KY_{i}, for a morphism f_{i} : X → uY_{i} in 𝒞, and there exists a covering sieve 𝒯 ⊂ 𝒞 /X such that the subgroup ⟨[τ_{1}]_{f_{1}∘h}, ⋯, [τ_{n}]_{f_{n}∘h}⟩ ⊂ (ũ^{∗}K)(X′) is finite for every (h : X′ → X) ∈ Ob(𝒯 ).

Now, given such class [\underline{σ}]_{f} and covering sieve 𝒮, the sieve u(𝒮) ⊂ 𝒞 /uY generated by {u(g) | g ∈ Ob(𝒮)} covers uY (lemma 4.2.4), and 𝒯 := f ×_{uY} u(𝒮) covers X for the topology J. For every (h : X′ → X) ∈ Ob(𝒯 ) there exist (g : Y′ → Y) ∈ Ob(𝒮) and a morphism f′ : X′ → uY′ in 𝒞 with f ∘ h = u(g) ∘ f′. It follows that

[σ_{i}]_{f∘h} = [σ_{i}]_{u(g)∘f′} = [Kg(σ_{i})]_{f′} for i = 1, …, n

and therefore ⟨[σ_{1}]_{f∘h}, …, [σ_{n}]_{f∘h}⟩ is a finite subgroup of (ũ^{∗}K)(X′). Then, by unwinding the definition, we find that (ω_{u,K}^{(n)})_{X} is given by the rule :

[\underline{σ}]_{f} ↦ ([σ_{1}]_{f}, …, [σ_{n}]_{f}).

Next, endow T := C^{∼} and T′ := C^{′∼} with their canonical topologies, and consider the essentially commutative diagram of sites provided by lemma 4.2.11(ii) :

[figure: T; u′; v; β; T′; v′; C; u; C′] with u′ := ũ^{∗}, v′ := h_{C′}^{a} and v := h_{C}^{a}.

After replacing U by a larger universe, we may assume that T and T′ are small; we notice :

Claim 5.6.21. The theorem holds for u′ and every sheaf H on T′.

Proof of the claim. Recall that u′ is a morphism of lex-sites (remark 4.4.13(ii)), and in particular u′ fulfills the condition of lemma 5.6.18, so we get an isomorphism of presheaves ω : u′_{!}((H^{n})_{f}) →^{∼} ((u′_{!}H)^{n})_{f}. Taking into account lemma 4.2.11(ii), we deduce an isomorphism of sheaves ω^{a} : ũ^{′∗}((H^{n})_{lf}) →^{∼} ((ũ^{′∗}H)^{n})_{lf}. But a direct inspection of the construction shows that the latter agrees with ω_{u′,H}^{(n)}. ♢

Claim 5.6.22. The theorem holds as well for v and v′.

Proof of the claim. It suffices to show the claim for v, since the same argument will work for v′ as well. Recall now that v̆^{∗} = ṽ_{∗} : C^{∼} → T^{∼} is an equivalence (theorem 4.4.2(iii)), and ṽ^{∗} is its quasi-inverse. It follows that (ṽ, Grp)_{∗} : (C, Grp)^{∼} → (T, Grp)^{∼} is also an equivalence, and its inverse is (ṽ, Grp)^{∗}. Then, let H be any sheaf of groups on C; in order to check that ω_{v,H} is an isomorphism, we may assume that H = ṽ_{∗}L for a sheaf of groups L on T, and it suffices to check the commutativity of the diagram :

[figure: ṽ^{∗}((ṽ_{∗}L^{n})_{lf}); ω_{v,ṽ∗L}^{(n)}; ((ṽ^{∗}ṽ_{∗}L)^{n})_{lf}; ((ε_{L})^{n})_{lf}; ṽ^{∗}ṽ_{∗}((L^{n})_{lf}); ε_{(L^{n})_{lf}}; (L^{n})_{lf}]

where ε_{•} : ṽ^{∗}ṽ_{∗} →^{∼} 1_{T^{∼}} is the counit for the adjoint (ṽ^{∗}, ṽ_{∗}). However, for every sheaf F on T, and every X ∈ Ob(T) the elements of ṽ^{∗}ṽ_{∗}F (X) are the equivalence classes [σ]_{f}, where f : X → vY is a morphism in T, and σ ∈ ṽ_{∗}F (Y) = F (vY); with this notation, the map ε_{F,X} : ṽ^{∗}ṽ_{∗}F (X) →^{∼} F X is given by the rule : [σ]_{f} ↦ F f(σ). Then the assertion follows after simple inspection, by combining with the foregoing explicit description of ω_{v,ṽ_{∗}L}^{(n)}. ♢

We deduce an oriented diagram of categories :

[figure: (T′, Grp)^{∼}; (−)_{lf}^{n}; (ũ′,Grp)^{∗}; ω_{v′,•}^{(n)}; T^{′∼}; _{ũ_{′∗}}; (C′, Grp)^{∼}; φ; (−)_{lf}^{n}; ω_{u,•}^{(n)}; (v′,Grp)^{∗}; (ũ,Grp)^{∗}; C^{′∼}; _{ũ_{∗}}; _{ṽ_{′∗}}; ψ; (C, Grp)^{∼}; (−)_{lf}^{n}; ω_{v,•}^{(n)}; (ṽ,Grp)^{∗}; C^{∼}; v^{∗}; (T, Grp)^{∼}; (−)_{lf}^{n}; T^{∼}.]

which we complete by adding the orientation ω_{u′,•}^{(n)} for the “front face”. Here φ and ψ denote the identifications induced by β :

(ũ′, Grp)^{∗} ∘ (ṽ′, Grp)^{∗} →^{∼} (ṽ, Grp)^{∗} ∘ (ũ, Grp)^{∗} and ũ^{′∗} ∘ ṽ^{′∗} →^{∼} ṽ^{∗} ∘ ũ^{∗}.

Explicitly, for every sheaf F′ on C′ and every X ∈ Ob(T), the elements of ũ^{′∗} ∘ ṽ^{′∗}F (X) are the equivalences classes [[τ]_{g}]_{f}, where f : X → u′Y is a morphism of T, g : Y → v′Z is a morphism of T′, and τ ∈ F Z; likewise one describes the elements of ṽ^{∗} ∘ ũ^{∗}F (X), and notice

that β_{Z} ∘ u′(g) ∘ f is a morphism X → vuZ in T. Then ψ_{F,X} : ũ^{′∗} ∘ ṽ^{′∗}F (X) →^{∼} ṽ^{∗} ∘ ũ^{∗}F (X)

is given by the rule :

[[τ]_{g}]_{f} ↦ [[τ]_{β_{Z}∘u′(g)∘f}]_{1_{uZ}}.

If F is a sheaf of groups on T, the same rule defines the group isomorphism φ_{F,X}. Now, a simple inspection shows that this oriented diagram commutes on 2-cells, in the sense of (2.3.21). By claims 5.6.21and 5.6.22we know that ω_{u′,•}^{(n)}, ω_{v,•}^{(n)} and ω_{v′,•}^{(n)} are isomorphisms; we conclude that the same holds for ω_{u,•}^{(n)}, as stated. □

Corollary 5.6.23. In the situation of theorem 5.6.20, the following holds :

(i) If K is ind-finite, the same holds for ũ^{∗}K.

(ii) If ũ^{∗} is a conservative functor and ũ^{∗}K is ind-finite, then K is ind-finite.

Proof. The assertions are immediate consequences of theorem 5.6.20and remark 5.6.14(ii). □

Proposition 5.6.24. In the situation of (5.6.16), suppose that u is continuous and there exists a topologically generating family G ⊂ Ob(𝒞′) such that uY is quasi-compact for the topology J, for every Y ∈ G. Then, if H is an ind-finite sheaf on C, the sheaf ũ_{∗}H is ind-finite on C′.

Proof. Let X ∈ Ob(𝒞′) be any object, and Σ ⊂ ũ_{∗}H(X) = H(uX) any finite set. Pick a covering family (f_{i} : Y_{i} → X | i ∈ I) with Y_{i} ∈ G for every i ∈ I; by remark 5.6.12, the group ũ_{∗}H(Y_{i}) = H(uY_{i}) is ind-finite, hence Hf_{i}(Σ) generates a finite subgroup of H(uY_{i}) for every i ∈ I, whence the assertion. □

5.6.25. C := (𝒞 , J) be a site, and 𝒜 → 𝒞 a fibration; for every X ∈ Ob(𝒞) and every cartesian section σ ∈ Ob(𝒜 (X)) we define the presheaf on 𝒞 /X :

𝒞art^{×}(σ) := 𝒞art(σ^{×}, σ^{×})

(notation of (3.4.2)). Clearly, for every f ∈ Ob(𝒞 /X), the set 𝒞art^{×}(σ)(f) carries a natural group structure, given by composition of automorphisms of the cartesian section (σ ∘ f_{∗})^{×} of 𝒜^{×}, so 𝒞art^{×}(σ) is a presheaf of groups on 𝒞 /X.

Definition 5.6.26. With the notation of (5.6.25), we say that 𝒜 is an ind-finite prestack on C, if 𝒞art^{×}(σ) an ind-finite sheaf of groups on C/X, for every X ∈ Ob(𝒞), and every σ ∈ Ob(𝒜 (X)) (notation of (4.7)). We say that 𝒜 is an ind-finite stack on C, if it is both an ind-finite prestack and a stack on C.

5.6.27. Let u : C := (𝒞 , J) → C′ := (𝒞′, J′) be a morphism of small sites, ℰ a stack on C′; let also X ∈ Ob(𝒞′), and σ, τ ∈ Ob(ℰ (X)). Set ℰ′ := St(u)^{∗}ℰ, and denote

η_{ℰ} : ℰ → St(u)_{∗}ℰ′

the unit of the 2-adjunction for the 2-adjoint pair (St(u)^{∗}, St(u)_{∗}). According to (5.4.7), the natural projection π : St(u)_{∗}ℰ′ → ℰ′ induces an equivalence of categories :

u_{|X}^{∗} : ℰ′(uX) →^{∼} St(u)_{∗}ℰ′(X).

We may then find σ′, τ′ ∈ Ob(ℰ′(uX)) and isomorphisms

ω_{1} : η_{ℰ} ∘ σ →^{∼} u_{|X}^{∗}(σ′) ω_{2} : η_{ℰ} ∘ τ →^{∼} u_{|X}^{∗}(τ′).

Taking into account lemma 5.1.9we deduce isomorphisms of sheaves on C′/X :

𝒞art(η_{ℰ} ∘ σ, η_{ℰ} ∘ τ) →^{∼} 𝒞art(u_{|X}^{∗}σ′, u_{|X}^{∗}τ′) →^{∼} ũ_{|X∗}𝒞art(σ′, τ′)

namely the composition of the isomorphism (5.1.7)induced by ω_{1} and ω_{2}, with the isomorphism of presheaves given in (5.4.7). Then (5.1.8) yields as well a morphism of sheaves :

𝒞art(σ, τ) → 𝒞art(η_{ℰ} ∘ σ, η_{ℰ} ∘ τ) β ↦ η_{ℰ} ∗ β.

By adjunction, the composition of these morphisms induces a morphism of sheaves on C/uX :

(5.6.28)  ũ_{|X}^{∗}𝒞art(σ, τ) → 𝒞art(σ′, τ′).

Remark 5.6.29. (i) Let (f : Y → uX) ∈ Ob(𝒞 /uX). The evaluation at f of the morphism (5.6.28) can be described explicitly as follows. For every element [α] ∈ ũ_{|X}^{∗}𝒞art(σ, τ)(f) we may find a covering family (k_{λ} : Y_{λ} → Y | λ ∈ Λ) and for every λ ∈ Λ a morphism h_{λ} : Y_{λ} → uZ_{λ} in 𝒞 and a morphism g_{λ} : Z_{λ} → X in 𝒞′ with f ∘ k_{λ} = u(g_{λ}) ∘ h_{λ}, such that [α] is represented by a system of morphisms of cartesian sections (α_{λ} : σ ∘ g_{λ∗} ⇒ τ ∘ g_{λ∗}). Set

α_{λ}^{∗} := π(ω_{2,g_{λ}} ∘ η_{ℰ} (α_{λ,1_{Z_{λ}}}) ∘ ω_{1,g_{λ}}^{−1}) : σ′(ug_{λ}) → τ′(ug_{λ}) for every λ ∈ Λ.

Then for every λ ∈ Λ there exists a unique morphism α_{λ}^{∗∗} : σ′(f ∘ k_{λ}) → τ′(f ∘ k_{λ}) in ℰ′_{Y_{λ}} that makes commute the following diagram in ℰ′ :

\xymatrix{σ′(f ∘ k_{λ}) \ar[r]^{α_{λ}^{∗∗}} \ar[d]_{σ′(h_{λ}/uX)} & τ′(f ∘ k_{λ}) \ar[d]^{τ′(h_{λ}/uX)} \\ σ′(ug_{λ}) \ar[r]^{α_{λ}^{∗}} & τ′(ug_{λ})}

and there exists a unique morphism of cartesian sections α^{∗∗} : σ′ ∘ f_{∗} ⇒ τ′ ∘ f∗ such that α_{k_{λ}}^{∗∗} = α_{λ}^{∗∗} for every λ ∈ Λ. Then the map (5.6.28) is given by the rule : [α] ↦ α^{∗∗}.

(ii) The morphism (5.6.28) depends obviously on the choice of σ′, τ′, ω_{1} and ω_{2}. However, say that σ′′, τ′′ ∈ Ob(ℰ′(uX)) are two other cartesian sections with isomorphisms ω′_{1} : η_{ℰ} ∘σ →^{∼} u_{|X}^{∗}(σ′′) and ω′_{2} : η_{ℰ} ∘ τ →^{∼} u_{|X}^{∗}(τ′′). Then there exists unique isomorphisms ρ_{1} : σ′ →^{∼} σ′′ and

ρ_{2} : τ′ →^{∼} τ′′ in ℰ′(uX) such that

ω′_{i} = u_{|X}^{∗}(ρ_{i}) ∘ ω_{i} for i = 1, 2

and in light of (i) it is easily seen that we get a commutative diagram :

\xymatrix{ & 𝒞art(σ, τ) \ar[dl] \ar@{-->}[dr] \\ 𝒞art(σ′, τ′) \ar[rr]^{∼} &  & 𝒞art(σ′′, τ′′)}

whose downward arrows are the morphisms (5.6.28) relative to the choices (ω_{1}, ω_{2}) and respectively (ω′_{1}, ω′_{2}), and whose horizontal arrow is the isomorphism (5.1.7) induced by (ρ_{1}, ρ_{2}).

Proposition 5.6.30. The morphism (5.6.28) is an isomorphism.

Proof. Let us check that (5.6.28) is a monomorphism. Thus, let (f : Y → uX) ∈ Ob(𝒞 /uX), and [α], [β] ∈ ũ_{|X}^{∗}𝒞art(σ, τ)(f) whose images agree in 𝒞art(σ′, τ′)(f); we need to show that [α] = [β]. We may find a covering family (k_{λ} : Y_{λ} → Y | λ ∈ Λ) such that the restrictions [α_{λ}], [β_{λ}] ∈ ũ_{|X}^{∗}𝒞art(σ, τ)(f ∘ k_{λ}) of [α] and [β] are in the image of the natural map u_{|X!}𝒞art(σ, τ)(f ∘ k_{λ}) → ũ_{|X}^{∗}𝒞art(σ, τ)(f ∘ k_{λ}), for every λ ∈ Λ. Then the images of [α_{λ}] and [β_{λ}] agree in 𝒞art(σ′, τ′)(f ∘ k_{λ}), and it suffices to check that [α_{λ}] = [β_{λ}] for every such λ. Thus, we may assume from start that [α], [β] are in the image of the natural map u_{|X!}𝒞art(σ, τ)(f) → ũ_{|X}^{∗}𝒞art(σ, τ)(f). This means that there exists a commutative diagram

\xymatrix{uZ′ \ar[dr]_{u(g′)} & Y \ar[d]^{f} \ar[r]^{h} \ar[l]_{h′} & uZ \ar[dl]^{u(g)} \\  & uX}

in 𝒞 such that [α] and [β] are represented by natural 𝒞 -transformations

α : σ ∘ g_{∗} ⇒ τ ∘ g_{∗} β : σ ∘ g′_{∗} ⇒ τ ∘ g′_{∗}.

By proposition 4.3.9, the fibration s : 𝒞 /u𝒞′ → 𝒞 is locally cofiltered relative to the topology J of C, hence there exist a covering family (k_{λ} : Y_{λ} → Y | λ ∈ Λ) and for every λ ∈ Λ a morphism h′′_{λ} : Y_{λ} → uZ_{λ} in 𝒞, and two morphisms l_{λ} : Z_{λ} → Z, l′_{λ} : Z_{λ} → Z′ such that

h ∘ k_{λ} = u(l_{λ}) ∘ h′′_{λ}  and h′ ∘ k_{λ} = u(l′_{λ}) ∘ h′′_{λ}  for every λ ∈ Λ.

It suffices to check that the images of [β] and [α] in 𝒞art(σ, τ)(f ∘k_{λ}) coincide for every λ ∈ Λ; but these images are the classes of [α∗l_{λ}] and respectively [β ∗l′_{λ}]. Hence, we may assume from start that Z = Z′, h = h′ and g = g′. With the notation of (5.4.7), the condition on [α] and [β] amounts then to the identity :

(5.6.31)  u_{|Z}^{∗}(η_{ℰ} ∗ α) ∗ h_{∗} = u_{|Z}^{∗}(η_{ℰ} ∗ β) ∗ h_{∗}  in ℰ′(Y).

Recall that the evaluation functor ev_{Z}^{ℰ} : ℰ (Z) → ℰ_{Z} is an equivalence of categories for every Z ∈ Ob(𝒞), natural with respect to morphism in 𝒞 (claim 3.2.8); this equivalence assigns to the cartesian sections σ ∘ g_{∗} and τ ∘ g_{∗} their evaluations σ_{g} := σ(g) and τ_{g} := τ(g) in c_{X}, and to α and β the morphisms α_{0} := α_{1_{Z}} : σ_{g} → τ_{g} and β_{0} := β_{1_{Z}} : σ_{g} → τ_{g} in ℰ_{Z}. We wish next to similarly interpret the identity (5.6.31) as an equality of morphisms in the fibre category ℰ′_{Y}. To this aim, let us pick a unital cleavage for ℰ, and let c be its associated unital pseudo-functor; denote also by γ^{c} the coherence constraint of c. Recall that for every Y ∈ Ob(𝒞), the fibre category Fib(u)_{!}ℰ_{Y} of the 𝒞 -fibration Fib(u)_{!}ℰ represents the 2-colimit of the pseudo-functor

F_{Y} := c ∘ t_{Y}^{o} : (Y/u𝒞′)^{o} → Cat (g : Y → uZ) ↦ c_{Z} = ℰ_{Z}.

The latter is realized by the localization ℱ ib(F_{Y})[Σ_{Y}^{−1}], where Σ_{Y} denotes the set of cartesian morphisms of ℱ ib(F_{Y}). Recall that the objects of ℱ ib(F_{Y})[Σ_{Y}^{−1}] are the data

(Z, h : Y → uZ, T) with (Z, h) ∈ Ob(Y/u𝒞′) and T ∈ Ob(ℰ_{Z}) .

The morphisms are the pairs [Y/g, t] : (Z, h, T) → (Z′, h′, T′) where Y/g : h → h′ is a morphism in Y/u𝒞′ and t : T → c_{g}T′ is a morphism in ℰ_{Z}. Moreover, every morphism k : Y → Y′ in 𝒞 induces a 𝒞 -cartesian functor

k^{∗} : ℱ ib(F_{Y′}) → ℱ ib(F_{Y})  (Z, h, T) ↦ (Z, h ∘ k, T)  [Y′/g, t] ↦ [Y/g, t]

which therefore extends uniquely to a functor

ℱ ib(F_{Y′})[Σ_{Y′}^{−1}] → ℱ ib(F_{Y})[Σ_{Y}^{−1}]

and the system of such functors provides a natural split cleavage for Fib(u)_{!}ℰ. Notice then the rules : Y ↦ ℱ ib(F_{Y}) and k ↦ k^{∗} for every Y ∈ Ob(𝒞) and every morphism k of 𝒞, define a pseudo-functor d : 𝒞^{o} → Cat, and the foregoing description amounts to saying that Fib(u)_{!}ℰ is represented by the fibration which in the proof of proposition 5.3.1is denoted

ℱ ib(d)[Σ_{•}^{−1}] → 𝒞 .

By proposition 5.3.1, the natural functor j : Fib(u)_{!}ℰ → ℰ′ factors then as a composition

Fib(u)_{!}ℰ ⟶^{L_{{Σ}}} ℱ ib(d){Σ_{•}^{−1}} ⟶^{j′} ℰ′.

Furthermore, for every Z ∈ Ob(𝒞′) we have a natural identification u_{|uZ}^{∗} : St(u)_{∗}ℰ′_{Z} →^{∼} ℰ′_{uZ}. Hence, η_{ℰ} restricts to a functor of fibre categories η_{ℰ ,Z} : ℰ_{Z} → ℰ′_{uZ} for every such Z. However, ℰ_{Z} is also the fibre category over 1_{uZ} ∈ Ob(uZ/u𝒞′) of the (uZ/u𝒞′)-fibration ℱ ib(F_{uZ}); in terms of the foregoing realization of Fib(u)_{!}ℰ_{uZ}, the functor η_{ℰ ,Z} is then just given by the rule :

ℰ_{Z} → ℱ ib(F_{uZ})[Σ_{uZ}^{−1}] ⟶^{j_{uZ}} ℰ′_{uZ} T ↦ j_{uZ}(Z, 1_{uZ}, T)

where j_{uZ} is the restriction of j. Summing up, we find that the objects η_{ℰ} (σ_{g}) and η_{ℰ} (τ_{g}) of ℰ′_{uZ} are respectively j_{uZ}(Z, 1_{uZ}, σ_{g}) and j_{uZ}(Z, 1_{uZ}, τ_{g}), and (5.6.31) translates as the identity:

j_{uZ}[Y/1_{Z}, α_{0}] = j_{uZ}[Y/1_{Z}, β_{0}].

But since j is 2-covering (proposition 5.2.11(i)), we deduce that there exists a covering family (k_{λ} : Y_{λ} → Y | λ ∈ Λ) in 𝒞 such that [Y_{λ}/1_{Z}, α_{0}] = [Y_{λ}/1_{Z}, β_{0}] in ℱ ib(F_{Y_{λ}})[Σ_{Y_{λ}}^{−1}] for every λ ∈ Λ. Since it suffices to check that the images of [α] and [β] coincide in 𝒞art(σ, τ)(f ∘ k_{λ}) for every λ, we may then assume from start that [Y/1_{Z}, α_{0}] = [Y/1_{Z}, β_{0}] in ℱ ib(F_{Y})[Σ_{Y}^{−1}].

Claim 5.6.32. The system Σ_{•} := (Σ_{Y} | Y ∈ Ob(𝒞)) admits a right local calculus of fractions.

Proof of the claim. Conditions (LCF1) and (LCF2) of definition 5.3.2 obviously hold for Σ_{•}. Next, let (Y/k′, l) : (h′, T′) → (h, T) be a morphism in ℱ ib(F_{Y}) and (Y/k′′, t) : (h′′, T′′) → (h, T) an element of Σ_{Y}; i.e. we have a commutative diagram of morphisms of 𝒞 :

\xymatrix{uZ′ \ar[dr]_{u(k′)} & Y \ar[d]^{h} \ar[r]^{h′′} \ar[l]_{h′} & uZ′′ \ar[dl]^{u(k′′)} \\  & uZ}

together with a morphism l : T′ → c_{k′}T and an isomorphism t : T′′ →^{∼} c_{k′′}T in 𝒜_{Y}. By proposition 4.3.9, the fibration s : 𝒞 /u𝒞′ → 𝒞 is locally cofiltered relative to the topology J of C, hence there exist a covering family (g_{λ} : Y_{λ} → Y | λ ∈ Λ) and for every λ ∈ Λ a morphism h_{λ} : Y_{λ} → uZ_{λ} in 𝒞, and two morphisms k′_{λ} : Z_{λ} → Z′, k′′_{λ} : Z_{λ} → Z′′ such that

h′ ∘ g_{λ} = u(k′_{λ}) ∘ h_{λ}  h′′ ∘ g_{λ} = u(k′′_{λ}) ∘ h_{λ}  k′ ∘ k′_{λ} = k′′ ∘ k′′_{λ}  for every λ ∈ Λ

whence a well defined morphism in 𝒜_{Y_{λ}}, for every λ ∈ Λ :

l_{λ} : c_{k′_{λ}}T′ ⟶^{c_{k′_{λ}}l} c_{k′_{λ}}c_{k′}T ⟶^{γ_{(k′_{λ},k′),T}^{c}} c_{k′∘k′_{λ}}T ⟶^{γ_{(k{′′ \atop λ},k′′),T}^{c −1}} c_{k′′_{λ}}c_{k′′}T ⟶^{(c_{k{′′ \atop λ}}t)^{−1}} c_{k′′_{λ}}T′′

which then yield a commutative diagram in ℱ ib(F_{Y_{λ}}) for every such λ :

\xymatrix{(h_{λ}, c_{k′_{λ}}T′) \ar[r]^{(Y_{λ}/k′_{λ},t_{λ})} \ar[d]_{(Y_{λ}/k′′_{λ},l_{λ})} & g_{λ}^{∗}(h′, T′) \ar[d]^{g_{λ}^{∗}(Y/k′,l)} \\ g_{λ}^{∗}(h′′, T′′) \ar[r]^{g_{λ}^{∗}(Y/k′′,t)} & g_{λ}^{∗}(Y/h, T)}

where we take for t_{λ} the identity of c_{k′_{λ}}T′. This shows that condition (LCF3) holds for Σ_{•}. Lastly, suppose that we have a commutative diagram in ℱ ib(𝒜_{Y}) :

\xymatrix{(h, T) \ar[r]^{(Y/k,l)} \ar[r]_{(Y/k′,l′)} & (h′, T′) \ar[r]^{(Y/k′′,t)} & (h′′, T′′)}

with (Y/k′′, t) ∈ Σ_{Y}. This means that we have a commutative diagram in 𝒞 :

\xymatrix{ & Y \ar[dl]_{h} \ar[d]^{h′} \ar[dr]^{h′′} \\ uZ \ar[r]^{u(k)} \ar[r]_{u(k′)} & uZ′ \ar[r]^{u(k′′)} & uZ′′}

and morphisms l : T → c_{k}T′, l′ : T → c_{k′}T′ and an isomorphism t : T′ →^{∼} c_{k′′}T′′ in 𝒜_{Y} with

k′′ ∘ k = k′′ ∘ k′ and γ_{(k,k′′),T′′}^{c} ∘ c_{k}(t) ∘ l = γ_{(k′,k′′),T′′}^{c} ∘ c_{k′}(t) ∘ l′.

Since the fibration s : 𝒞 /u𝒞′ → 𝒞 is locally cofiltered, there exist a covering family (g_{λ} : Y_{λ} → Y | λ ∈ Λ) and for every λ ∈ Λ, morphisms h_{λ} : Y_{λ} → uZ_{λ} in 𝒞 and k_{λ} : Z_{λ} → Z in 𝒞′ such that h ∘ g_{λ} = u(k_{λ}) ∘ h_{λ} in 𝒞 and k ∘ k_{λ} = k′ ∘ k_{λ} in 𝒞′. We compute :

γ_{(k∘k_{λ},k′′),T′′}^{c} ∘ c_{k∘k_{λ}}(t) ∘ γ_{(k,k_{λ}),T′}^{c} ∘ c_{k_{λ}}(l) = γ_{(k∘k_{λ},k′′),T′′}^{c} ∘ γ_{(k,k_{λ}),c_{k′′}T′′}^{c} ∘ c_{k_{λ}}c_{k}(t) ∘ c_{k_{λ}}(l)

= γ_{(k_{λ},k′′∘k),T′′}^{c} ∘ c_{k_{λ}}(γ_{(k,k′′),T′′}^{c}) ∘ c_{k_{λ}}c_{k}(t) ∘ c_{k_{λ}}(l)

= γ_{(k_{λ},k′′∘k′),T′′}^{c} ∘ c_{k_{λ}}(γ_{(k,k′′),T′′}^{c} ∘ c_{k}(t) ∘ l)
= γ_{(k_{λ},k′′∘k′),T′′}^{c} ∘ c_{k_{λ}}(γ_{(k′,k′′),T′′}^{c} ∘ c_{k′}(t) ∘ l′)

= γ_{(k_{λ},k′′∘k′),T′′}^{c} ∘c_{k_{λ}}(γ_{(k′,k′′),T′′}^{c})∘c_{k_{λ}}c_{k′}(t)∘c_{k_{λ}}(l′)

= γ_{(k′∘k_{λ},k′′),T′′}^{c} ∘ γ_{(k′,k_{λ}),c_{k′′}T′′}^{c} ∘ c_{k_{λ}}c_{k′}(t) ∘ c_{k_{λ}}(l′)

= γ_{(k′∘k_{λ},k′′),T′′}^{c} ∘ c_{k′∘k_{λ}}(t) ∘ γ_{(k′,k_{λ}),T′}^{c} ∘ c_{k_{λ}}(l′)

whence γ_{(k,k_{λ}),T′}^{c} ∘ c_{k_{λ}}(l) = γ_{(k′,k_{λ}),T′}^{c} ∘ c_{k_{λ}}(l′). We deduce a commutative diagram in ℱ ib(F_{Y_{λ}})

\xymatrix{(h_{λ}, c_{k_{λ}}T) \ar[r]^{(Y/k_{λ},t_{λ})} & g_{λ}^{∗}(h, T) \ar[r]^{g_{λ}^{∗}(Y/k,l)} \ar[r]_{g_{λ}^{∗}(Y/k′,l′)} & g_{λ}^{∗}(h′, T′)}  for every λ ∈ Λ

where t_{λ} is the identity of c_{k_{λ}}T. This shows that (LFC4) holds for Σ_{•}. ♢

To ease notation, set A := L_{{Σ}}(Z, h, σ_{g}) and B := L_{{Σ}}(Z, h, τ_{g}), and define the presheaf M_{A,B} on 𝒞 /uZ and the sheaf H_{A,B} on C/uZ as in (5.3.5). Then

L_{{Σ}}[Y/1_{Z}, α] = L_{{Σ}}[Y/1_{Z}, β] ∈ H_{A,B}(h)

so there exists a covering family (k_{λ} : Y_{λ} → Y | λ ∈ Λ) such that the images of [Y_{λ}/1_{Z}, α] and [Y_{λ}/1_{Z}, β] agree in M_{A,B}(h ∘ g_{λ}). Hence as usual we may assume from start that the images of [Y/1_{Z}, α_{0}] and [Y/1_{Z}, β_{0}] agree in M_{A,B}(h). Then, in view of claim 5.6.32, condition (b) of (5.3.5) tells us that there exist a covering family (k_{λ} : Y_{λ} → Y | λ ∈ Λ), and for every λ ∈ Λ a morphism h_{λ} : Y_{λ} → uZ_{λ} in 𝒞 and a morphism l_{λ} : Z_{λ} → Z in 𝒞′ such that h ∘ k_{λ} = u(l_{λ}) ∘ h_{λ} and with c_{l_{λ}}(α_{0}) = c_{l_{λ}}(β_{0}). This means that the images of [α] and [β] agree in ũ_{|X}^{∗}𝒞art(σ, τ)(f ∘ k_{λ}) for every λ ∈ Λ, so [α] = [β], as required.

Next, let us check that (5.6.28) is an epimorphism. To this aim, set ℰ′′ := Fib(u)_{!}ℰ, and let η′_{ℰ} : ℰ → Fib(u)^{∗}ℰ′′ be the unit of the 2-adjunction for the 2-adjoint pair (Fib(u)_{!}, Fib(u)^{∗}); as in(5.6.27), the projection π′ : Fib(u)^{∗}ℰ′′ → ℰ′′ induces an equivalence of categories

u_{|X}^{∗} : ℰ′′(uX) →^{∼} Fib(u)^{∗}ℰ′′(X)

so we may find σ′′, τ′′ ∈ Ob(ℰ′′(uX)) with isomorphisms :

η′_{ℰ} ∘ σ →^{∼} u_{|X}^{∗}(σ′′) η′_{ℰ} ∘ τ →^{∼} u_{|X}^{∗}(τ′′).

Since Fib(u)^{∗}(j) ∘ η′_{ℰ} : ℰ → St(u)_{∗}ℰ′ is isomorphic to η_{ℰ}, we deduce isomorphisms

σ′ →^{∼} j ∘ σ′′ τ′ →^{∼} j ∘ τ′′

and it suffices to show that the corresponding morphism of sheaves on C/uX

ũ_{|X}^{∗}𝒞art(σ, τ) → 𝒞art(j ∘ σ′′, j ∘ τ′′)

is an epimorphism. Thus, let (f : Y → X) ∈ Ob(𝒞 /X) and μ′ ∈ 𝒞art(j ∘ σ′′, j ∘ τ′′)(f); we need to show that μ′ : j ∘ σ′′ ∘ f_{∗} ⇒ j ∘ τ′′ ∘ f_{∗} is the image of some [μ] ∈ ũ_{|X}^{∗}𝒞art(σ, τ)(f). Set σ′′_{0} := σ′′(f), τ′′_{0} := τ′′(f), and recall that μ′ is determined by its evaluation

μ′_{0} := μ′_{1_{Y}} : j(σ′′_{0}) → j(τ′′_{0}).

Since j is 1-covering (proposition 5.2.11(i)), there exist a covering family (k_{λ} : Y_{λ} → Y | λ ∈ Λ), and for every λ ∈ Λ a morphism μ′′_{λ} : σ′′(f ∘k_{λ}) → τ′′(f ∘k_{λ}) in ℰ′′_{Y_{λ}} such that μ′_{k_{λ}} = j(μ′′_{λ}). Now, suppose that for every λ ∈ Λ there exists μ_{λ} ∈ ũ_{|X}^{∗}𝒞art(σ, τ)(f ∘ k_{λ}) whose image in 𝒞art(σ′, τ′)(f ∘ k_{λ}) equals μ′ ∗ k_{λ}. Then, for every λ, λ′ ∈ Λ, and every pair of morphisms

Y_{λ′} ⟵^{k′′_{λλ′}} Y_{λλ′} ⟶^{k′_{λλ′}} Y_{λ} in 𝒞 such that k_{λλ′} := k_{λ} ∘ k′_{λλ′} = k_{λ′} ∘ k′′_{λλ′}, let μ_{λλ′} and μ_{λ′λ} be

the images of μ_{λ} and respectively μ_{λ′} in ũ_{|X}^{∗}𝒞art(σ, τ)(k_{λλ′} ∘ f); it follows that μ_{λλ′} and μ_{λ′λ} are both mapped to μ′ ∗ k_{λλ′} by the morphism (5.6.28). Since we already know that the latter is a monomorphism, we deduce that μ_{λλ′} = μ_{λ′λ} for every such pair of morphisms k′_{λλ′}, k′′_{λλ′}. Then the system (μ_{λ} | λ ∈ Λ) determines a unique μ ∈ ũ_{|X}^{∗}𝒞art(σ, τ)(f) whose image in 𝒞art(σ′, τ′)(f) equals μ′, as sought. Thus, we may replace μ by μ ∗ k_{λ} for every λ ∈ Λ, and assume from start that there exists a morphism

μ′′_{0} : L_{{Σ}}σ′′_{0} → L_{{Σ}}τ′′_{0}  in ℱ ib(d){Σ_{•}^{−1}}_{Y} such that  μ′_{0} = j′(μ′′_{0}).

Set σ′′′ := L_{{Σ}}σ′′ and τ′′′ := L_{{Σ}}τ′′; then μ′′_{0} corresponds to a unique morphism of cartesian sections μ′′ : σ′′′ ∘ f_{∗} ⇒ τ′′′ ∘ f_{∗} with j′ ∗ μ′′ = μ′. Recall that σ′′_{0} (resp. τ′′_{0}) is a datum (Z, h, T) (resp. (Z′, h′, T′)). Now, according to condition (a) of (5.3.5), we may find a covering family (k_{λ} : Y_{λ} → Y | λ ∈ Λ), and for every λ ∈ Λ, an object (Z_{λ}, h_{λ} : Y_{λ} → uZ_{λ}, T_{λ}) of ℱ ib(F_{Y_{λ}}),

and morphisms (Z, h ∘ k_{λ}, T) ⟵^{(Y_{λ}/l_{λ},s_{λ})} (Z_{λ}, h_{λ}, T_{λ}) ⟶^{(Y_{λ}/l′_{λ},t_{λ})} (Z′, h′ ∘ k_{λ}, T′) in ℱ ib(F_{Y_{λ}}),

where s_{λ} : T_{λ} → c_{l_{λ}}T is an isomorphism in ℰ_{Z_{λ}}, and such that we have a commutative diagram in ℱ ib(d){Σ_{•}^{−1}}_{Y_{λ}} :

\xymatrix{(Z, h ∘ k_{λ}, T) \ar[d] \ar[r]^{L_{\{Σ\}}(Y_{λ}/l′_{λ},t_{λ})∘L_{\{Σ\}}(Y_{λ}/l_{λ},s_{λ})^{−1}} & (Z′, h′ ∘ k_{λ}, T) \ar[d] \\ σ′′′(f ∘ k_{λ}) \ar[r]^{μ′′_{k_{λ}}} & τ′′′(f ∘ k_{λ})}

whose vertical arrows are isomorphisms. The latter is equivalent to a commutative diagram :

\xymatrix{(Z_{λ}, h_{λ}, c_{l_{λ}}T) \ar[d] \ar[r]^{L_{\{Σ\}}(Y_{λ}/1_{Z_{λ}},t_{λ}∘s_{λ}^{−1})} & (Z_{λ}, h_{λ}, c_{l′_{λ}}T′) \ar[d] \\ σ′′′(f ∘ k_{λ}) \ar[r]^{μ′′_{k_{λ}}} & τ′′′(f ∘ k_{λ})}

whose vertical arrows are again isomorphisms. After replacing σ′′′ and τ′′′ by isomorphic cartesian sections, we may therefore assume that μ′′_{k_{λ}} = L_{{Σ}}(Y_{λ}/1_{Z_{λ}}, t_{λ} ∘ s_{λ}^{−1}) for every λ ∈ Λ.

Arguing as in the foregoing, we may then assume that there exist a morphism h : Y → uZ in 𝒞 and a morphism t : T → T′ in ℰ_{Z} such that μ′_{0} = j(Y/1_{Z}, t) : j(Z, h, T) → j(Z, h, T′).

Next, since the fibration s : 𝒞 /u𝒞′ → 𝒞 is locally cofiltered over the site C, we may find a covering family (k_{λ} : Y_{λ} → Y | λ ∈ Λ), and for every λ ∈ Λ a morphism h_{λ} : Y_{λ} → uZ_{λ}

in 𝒞 and two morphisms X ⟵^{g_{λ}} Z_{λ} ⟶^{g′_{λ}} Z such that f ∘ k_{λ} = g_{λ} ∘ h_{λ} and h′′ ∘ k_{λ} = g′_{λ} ∘ h_{λ}.

Then again, we may replace σ′ and τ′ by isomorphic cartesian sections, and assume that μ′_{k_{λ}} = j(Y_{λ}/1_{Z_{λ}}, c_{g′_{λ}}t) : (Z_{λ}, h_{λ}, c_{g′_{λ}}T) → (Z_{λ}, h_{λ}, c_{g′_{λ}}T′). Arguing once again as in the foregoing, we may finally assume that we have still μ′_{0} = j(Y/1_{Z}, t) : j(Z, h, T) → j(Z, h, T′), and moreover there exists a morphism g : Z → X in 𝒞 such that u(g) ∘ h = f. But then, t corresponds to a morphism of cartesian sections μ : σ ∘ f_{∗} → τ ∘ f_{∗}, and the pair (Y/g : f → h, μ) yields the sought section [μ]. □

Theorem 5.6.33. In the situation of (5.6.27), the following holds :

(i) If ℰ is ind-finite, then the same holds for ℰ′.

(ii) If ũ^{∗} is a conservative functor and ℰ′ is ind-finite, then ℰ is ind-finite.

Proof. (i): In view of proposition 5.6.4, we may replace ℰ by ℰ^{×}, and assume from start that ℰ is a stack in groupoids, in which case we need to show that for every Y ∈ Ob(𝒞) and every σ′ ∈ Ob(ℰ′(Y)), the sheaf of groups 𝒞art(σ′, σ′) is ind-finite on C/Y.

Since the natural functor j : ℰ′′ := Fib(u)_{!}ℰ → ℰ′ is 0-covering (proposition 5.2.11(i)), there exist a covering family (φ_{λ} : Y_{λ} → Y | λ ∈ Λ), and for every λ ∈ Λ a cartesian section σ′′_{λ} ∈ Ob(ℰ′′(Y_{λ})) with an isomorphism j ∘ σ′′_{λ} →^{∼} σ′ ∘ φ_{λ∗}. Clearly it suffices to show that 𝒞art(σ′ ∘ φ_{λ∗}, σ′ ∘ φ_{λ∗}) is ind-finite for every λ ∈ Λ. We may thus replace Y by Y_{λ} for every such λ, and assume from start that σ′ = j ∘ σ′′ for some σ′′ ∈ Ob(ℰ′′(Y)). Recall that σ′′(1_{Y}) ∈ Ob(ℰ′′_{Y}) is a datum (X, f : Y → uX, T) where (X, f) ∈ Ob(Y/u𝒞′) and T ∈ Ob(ℰ_{X}). Let then τ : 𝒞′/X → ℰ and τ′ : 𝒞 /uX → ℰ′′ be cartesian sections with τ_{1_{X}} = T and τ′(1_{uX}) := (X, 1_{uX}, T). As explained in the proof of of proposition 5.6.30, we have an isomorphism :

η′_{ℰ} ∘ τ →^{∼} u_{|X}^{∗}(τ′)

where η′_{ℰ} : ℰ → Fib(u)^{∗}ℰ′′ is the unit of a 2-adjunction for the 2-adjoint pair (Fib(u)_{!}, Fib(u)^{∗}). Since Fib(u)^{∗}(j) ∘ η′_{ℰ} : ℰ → St(u)_{∗}ℰ′ is isomorphic to the unit η_{ℰ} : ℰ → St(u)_{∗}ℰ′ of a 2adjunction for the 2-adjoint pair (Fib(u)_{!}, Fib(u)^{∗}), there follows an isomorphism

η_{ℰ} ∘ τ →^{∼} u_{|X}^{∗}(j ∘ τ′).

From proposition 5.6.30and corollary 5.6.23(i) we conclude that 𝒞art(j ∘τ′, j ∘τ′) is ind-finite on C/uX. On the other hand, we have τ′ ∘ f_{∗} ≃ σ′′, whence j ∘ τ′ ∘ f_{∗} ≃ σ′. Combining with (5.1.5), there follows an isomorphism of sheaves 𝒞art(σ′, σ′) →^{∼} (f_{∗})_{∗}^{∼}𝒞art(j ∘ τ′, j ∘ τ′). But recall that f_{∗} is both continuous and cocontinuous for the topologies of the sites C/Y and C/uX (remark 4.7.3(i)); then the assertion follows from proposition 5.6.17(ii) and lemma 4.2.15(i).

(ii): We may reduce as in (i) to the case where ℰ is a stack in groupoids. Next notice that if ũ^{∗} is conservative, ũ_{|X}^{∗} is conservative for every X ∈ Ob(𝒞′), by virtue of proposition 4.7.7(ii). Then the assertion follows immediately from proposition 5.6.30and corollary 5.6.23(ii). □

Proposition 5.6.34. In the situation of (5.6.16), suppose that u is a weak morphism of sites, and moreover there exists a topologically generating family G ⊂ Ob(𝒞′) such that uY is quasicompact for the topology J, for every Y ∈ G. Then, for every ind-finite stack ℰ on C, the stack St(u)_{∗}ℰ is ind-finite on C′.

Proof. In view of the pseudo-commutative diagram (5.6.2) we may assume that ℰ is a stack in groupoids, in which case the same holds for ℰ′ := St(u)_{∗}ℰ. Now, let X ∈ Ob(𝒞′) and σ, τ ∈ ℰ′(X). Recall that u induces a continuous functor u_{|X} : 𝒞′/X → 𝒞 /uX (proposition 4.7.7(i)), and moreover G/X ⊂ Ob(𝒞′/X) is a topologically generating family, according to

(4.7.2); then it is easily seen that for every (Y ⟶^{g} X) ∈ Ob(G/X) the object uY ⟶^{u(g)} uX of 𝒞 /uX is quasi-compact. On the other hand, arguing as in the proof of proposition 5.4.8(i), we find σ′, τ′ ∈ ℰ (uX) and isomorphisms of sheaves

𝒞art(σ, τ) →^{∼} ũ_{|X∗}𝒞art(σ′, τ′).

The assertion then follows from proposition 5.6.24. □

5.6.35. Stalk of a stack over a T -point. Let T be a topos, and ξ := (ξ^{∗}, ξ_{∗}) : Set → T a T -point (see definition 4.7.11(i)). For every stack ℰ on the canonical site Can(T) we set

ℰ_{ξ} := St(ξ^{∗})^{∗}ℰ .

Likewise, for every morphism φ : ℰ → ℱ of stacks on Can(T), we let

φ_{ξ} := St(ξ^{∗})^{∗}(φ) : ℰ_{ξ} → ℱ_{ξ}.

Remark 5.6.36. (i) Recall that Set is equivalent to (𝟙, J)^{∼}, where 𝟙 is the category with one object and one morphism, and J is the unique topology on 𝟙. It follows easily that a stack on the canonical site Can(Set) of Set amounts to the datum of a category, and a morphism of stacks on Can(Set) is just an arbitrary functor.

(ii) Likewise, a functor F : 𝒜 → ℬ between categories is 0-covering (resp. 1-covering, resp. 2-covering) when regarded as a morphism of stacks on Can(Set), if and only if it is essentially surjective (resp. full, resp. faithful) : the details shall be left to the reader.

Proposition 5.6.37. With the notation of (5.6.35), let Ω be a conservative set of T -points (see definition 4.7.11(iv)). Then for every i = 0, 1, 2 the following conditions are equivalent :

(a) φ is i-covering.

(b) The morphism φ_{ξ} is i-covering for every ξ ∈ Ω.

Proof. (a)⇒(b) follows from proposition 5.4.22(ii).

(b)⇒(a): For i = 1 and i = 2, the assertion follows easily from lemma 5.2.7and proposition 5.6.30. Suppose then that φ_{ξ} is 0-covering for every ξ ∈ Ω; by lemma 5.6.7(i), it follows that π_{0}(φ_{ξ}) : π_{0}(ℰ_{ξ}) → π_{0}(ℱ_{ξ}) is a surjection. But according to (5.6.8), π_{0}(ℰ_{ξ}) is naturally identified with the stalk π_{0}^{T} (ℰ)_{ξ} of the sheaf π_{0}^{T} (ℰ) on Can(T), and likewise for π_{0}(ℱ_{ξ}). Thus, the morphism of sheaves π_{0}^{T} (φ) : π_{0}^{T} (ℰ) → π_{0}^{T} (ℱ) is an epimorphism, and therefore φ is 0-covering, again by lemma 5.6.7(i). □

5.7. Stacks on fibred sites and fibred topoi. Let 𝒞 and 𝒞′ be two categories, u : 𝒞′ → 𝒞 any functor, (𝒜 , p, J_{•}) a fibred site over 𝒞, and (𝒜′, p′_{•}, J′_{•}) := 𝒞′ ×_{𝒞} (𝒜 , p, J_{•}) the induced fibred site as in (4.5.11); denote also by (𝒜 , J_{𝒜} ) and (𝒜′, J_{𝒜′} ) the respective fibred sites. Then it follows easily from proposition 4.5.5(ii,iv) that the projection π : 𝒜′ → 𝒜 fulfills condition (C2) of (5.4.5), relative to the topologies J_{𝒜} and J_{𝒜′} , so it is a weak morphism of sites π : (𝒜 , J_{𝒜} ) → (𝒜′, J_{𝒜′} ), and we get a well defined pseudo-functor

St(π)_{∗} : Stack(𝒜 , J_{𝒜} ) → Stack(𝒜′, J_{𝒜′} ).

If 𝒞 and 𝒞′ are small and (𝒜_{X}, J_{X}) is a U-site for every X ∈ Ob(𝒞), then (𝒜 , J_{𝒜} ) and (𝒜′, J_{𝒜′} ) are also U-sites (remark 4.5.2(iii)), and in this case also the left 2-adjoint St(π)^{∗} of St(π)_{∗} is well defined; moreover, π is cocontinuous (see (4.5.11)), so we have a pseudo-natural equivalence St(π)_{∗} →^{∼} St(π̆)^{∗} (proposition 5.4.29(ii)).

As a special case, for every X ∈ Ob(𝒞) the inclusion functor i_{X} : 𝒜_{X} → 𝒜 is a weak morphism of sites (𝒜 , J_{𝒜} ) → (𝒜_{X}, J_{X}), and we get a well defined pseudo-functor

St(i_{X})_{∗} : Stack(𝒜 , J_{𝒜} ) → Stack(𝒜_{X}, J_{X}).

If 𝒞 is small and (𝒜_{X}, J_{X}) is a U-site for every X ∈ Ob(𝒞), also St(i_{X})^{∗} is well defined.

Proposition 5.7.1. In the situation of (5.7), we have :

(i) Let φ : ℰ → ℰ′ be a cartesian functor of fibrations over 𝒜, and i ∈ {0, 1, 2}. The following conditions are equivalent :

(a) φ is i-covering for the topology J_{𝒜} .

(b) Fib(i_{X})^{∗}(φ) is i-covering for the topology J_{X}, for every X ∈ Ob(𝒞).

(ii) Suppose that 𝒞 is small, and (𝒜_{X}, J_{X}) is a U-site for every X ∈ Ob(𝒞). Then the following diagram of 2-categories is pseudo-commutative :

\xymatrix{Fib(𝒜) \ar[r]^{Fib(i_{X})^{∗}} \ar[d]_{(−)^{a}} & Fib(𝒜_{X}) \ar[d]^{(−)^{a}} \\ Stack(𝒜 , J_{𝒜} ) \ar[r]^{St(i_{X})_{∗}} & Stack(𝒜_{X}, J_{X})} for every X ∈ Ob(𝒞).

Proof. (i.a)⇒(i.b): This follows directly from propositions 5.4.1(i) and 4.5.5(iii).

(i.b)⇒(i.a): Suppose first that i = 0, and let X ∈ Ob(𝒜) and E′ ∈ Ob(ℰ′_{X}); pick cleavages for ℰ and ℰ′, and let c and c′ be the corresponding associated pseudo-functors. By assumption, there exist a covering family f_{•} : (f_{i} : X_{i} → X | i ∈ I) for the topology J_{pX}, and for every i ∈ I an object E_{i} ∈ Ob(ℰ_{X_{i}}) such that φE_{i} is isomorphic to c′_{f_{i}}E′ in ℰ′_{X_{i}}. Since f_{•} is also a covering family for the topology J_{𝒜} , this shows that φ is 0-covering, as required. The cases where i = 1, 2 are similar : the details shall be left to the reader.

(ii): The assertion is a special case of corollary 5.4.31. □

Corollary 5.7.2. Let j ≤ 2 be any integer. The following holds :

(i) In the situation of (4.3.24), let ℰ be a fibration on 𝒞. Then ℰ is a j-separated prestack on C if and only if Fib(π_{i})^{∗}(ℰ) is a j-separated prestack on C_{i} for every i ∈ Ob(I). (ii) In the situation of (5.7), let ℰ be a fibration on 𝒜. Then ℰ is a j-separated prestack on (𝒜 , J_{𝒜} ) if and only if Fib(i_{X})^{∗}(ℰ) is j-separated on (𝒜_{X}, J_{X}) for every X ∈ Ob(𝒞).

Proof. (i): The condition is necessary, by virtue of corollary 5.4.15 and proposition 5.4.8(i,ii). For the converse, we argue as in the proof of proposition 4.3.25(ii). Indeed, suppose that Fib(π_{i})^{∗}(ℰ) is a j-separated prestack on C_{i} for every i ∈ Ob(I); according to remark 4.1.8(ii) there exists a topology J′ on 𝒞 such that, for every X ∈ Ob(𝒞), the set J′(X) consists of the sieves 𝒮 ⊂ 𝒞 /X of universal ℰ -j-descent. It then suffices to prove that J ⊂ J′. We come down to checking the following. For every i ∈ Ob(I), every X ∈ Ob(𝒞_{i}), every covering family g_{•} := (g_{λ} : X_{λ} → X | λ ∈ Λ) for the topology J_{i}, and every morphism f : Y → π_{i}X in 𝒞, the family (Y ×_{π_{i}X} π_{i}(g_{λ}) : Y ×_{π_{i}X} π_{i}X_{λ} → Y | λ ∈ Λ) generates a sieve of ℰ -j-descent. Arguing as in the proof of proposition 4.3.25(ii), we are first reduced to the case where Y = π_{i}Y′ for some Y′ ∈ Ob(𝒞_{i}) and f = π_{i}(t) for some morphism t : Y′ → X of 𝒞_{i}, and then we may even replace the family g_{•} by (Y′ ×_{X} g_{λ} | λ ∈ Λ) and reduce further to the case where t = 1_{X}. Finally, pick a cleavage c for ℰ; we need to show that the natural functor

ℰ (π_{i}X) → Desc(ℰ , π_{i}(g_{•}), c)

is j-faithful (notation of (3.5.22)). But since π_{i} is left exact, this functor is naturally identified with the corresponding functor for ℰ_{i} := Fib(π_{i})^{∗}(ℰ) :

ℰ_{i}(X) → Desc(ℰ_{i}, g_{•}, c ∘ π_{i}^{o})

and the latter is j-faithful, since ℰ_{i} is a j-separated prestack on C_{i}.

(ii): Again, the condition is necessary, by corollary 5.4.15and proposition 5.4.8(i,ii). For the converse we argue as in the proof of proposition 4.5.5. Namely, we let J′ be the topology on 𝒜 such that for every A ∈ Ob(𝒜) the set J′(A) consists of the sieves 𝒮 ⊂ 𝒜 /A of universal ℰ -j-descent, and we check that J_{𝒜} ⊂ J′, assuming that ℰ_{X} := Fib(i_{X})^{∗}(ℰ) is a j-separated prestack on (𝒜_{X}, J_{X}) for every X ∈ Ob(𝒞). To this aim, we consider any morphism f : A′ → A in 𝒜 and any family (g_{λ} : B_{λ} → A | λ ∈ Λ) of morphisms in 𝒜_{pA} covering A for the topology J_{pA}; after choosing a cleavage for 𝒜, we then construct the family g′_{•} := (g′_{λ} : B′_{λ} → A′ | λ ∈ Λ) of morphisms of 𝒜_{pA′} as in the proof of proposition 4.5.5. Let i_{pA′} : 𝒜_{pA′} → 𝒜 be the inclusion functor; taking into account lemma 3.5.35(ii), we reduce to checking that the sieve of 𝒜 /A′ generated by the family i_{pA′}(g′_{•}) := (i_{pA′}(g_{λ}) | λ ∈ Λ) is of ℰ -j-descent. Now, choose a cleavage c for ℰ, and set

B′_{λμ} := B′_{λ} ×_{A′} B′_{μ}  and B′_{λμν} := B′_{λμ} ×_{A′} B′_{ν}  for every λ, μ, ν ∈ Λ

and notice that i_{pA′}B′_{λμ} represents the fibre product i_{pA′}B′_{λ} ×_{i_{pA′}A′} B′_{μ}, and likewise i_{pA′}B′_{λμν} represents i_{pA′}B′_{λμ} ×_{i_{pA′}A′} i_{pA′}B′_{ν} (remark 4.5.2(ii)), so that the category Desc(ℰ , i_{pA′}(g′_{•}), c) is well defined, and we are reduced to checking that the induced map

ℰ (i_{pA′}A′) → Desc(ℰ , i_{pA′}(g′_{•}), c)

is j-faithful. But the latter is naturally identified with the analogous map

ℰ_{pA′}(A′) → Desc(ℰ_{pA′}, g′_{•}, c ∘ i_{pA′}^{o})

which is j-faithful by assumption. □

Proposition 5.7.3. In the situation of (4.5.13), suppose that (𝒜_{i,X}, J_{i,X}) is a U-site for every X ∈ Ob(𝒞) and i = 0, 1. Then the induced base change transformation :

Υ(St(1_{u∘π_{1}})_{∗}^{γ}) : St(v)^{∗} ∘ St(π_{1})_{∗} → St(π_{0})_{∗} ∘ St(u)^{∗}

is a pseudo-natural equivalence.

Proof. Recall that the projections π_{0} and π_{1} satisfy condition (C2) from (5.4.5) : see (5.7); in particular, they are weak morphisms of sites, and moreover they are cocontinuous (see (4.5.11)). We are then in the situation contemplated in (5.5.23), and by corollary 5.5.26(ii), both Δ^{π_{0}} and Δ^{π_{1}} are pseudo-natural equivalences, and the same holds for Υ(Δ^{u}) and Υ(Δ^{v}), by virtue of corollary 5.5.26(iii). Hence, arguing as in remark 2.3.22(i), we are reduced to checking that Υ((1_{u∘π_{1}}^{∼}, Cat)_{∗}) is a pseudo-natural equivalence. The latter assertion follows from proposition 4.5.14and the discussion of (5.5.29). □

Definition 5.7.4. Let (𝒜 , p, J_{•}) and (𝒜′, p′, J′_{•}) be two fibred sites over a category 𝒞. A weak morphism of fibred sites φ : (𝒜 , p, J_{•}) → (𝒜′, p′, J′_{•}) is a functor φ : 𝒜′ → 𝒜 with p∘φ = p′, whose restriction φ_{X} : 𝒜′_{X} → 𝒜_{X} is a weak morphism of sites (𝒜_{X}, J_{X}) → (𝒜′_{X}, J′_{X}), for every X ∈ Ob(𝒞).

Proposition 5.7.5. Let (𝒜 , p, J_{•}) and (𝒜′, p′, J′_{•}) be two fibred lex-sites over a category 𝒞, and φ : (𝒜 , p, J_{•}) → (𝒜′, p′, J′_{•}) a weak morphism of fibred sites. Then φ is a weak morphism of the respective total sites (𝒜 , J_{𝒜} ) → (𝒜′, J_{𝒜′} ).

Proof. For every X ∈ Ob(𝒞), let i_{x} : 𝒜_{X} → 𝒜 and i′_{X} : 𝒜′_{X} → 𝒜′ be the inclusion functors, and φ_{X} : 𝒜′_{X} → 𝒜_{X} the restriction of φ; in light of corollary 5.7.2(ii), it suffices to check that for every stack ℰ over (𝒜 , J_{𝒜} ), the fibration ℱ := Fib(i′_{X})^{∗} ∘ Fib(φ)^{∗}(ℰ) is a stack on (𝒜′_{X}, J′_{X}). But we have a natural identification ℱ →^{∼} Fib(φ_{X})^{∗} ∘Fib(i_{X})^{∗}(ℰ), and Fib(i_{X})^{∗}(ℰ) is a stack on (𝒜_{X}, J_{X}), again by corollary 5.7.2(ii), whence the contention. □

5.7.6. Let (𝒜 , p, J_{•}) be a small fibred lex-site over a small category 𝒞; pick a unital cleavage λ for p, and let c : 𝒞^{o} → Cat be the associated pseudo-functor. Clearly c^{o} factors through a pseudo-functor

c̃ : 𝒞 → lex.Site X ↦ (𝒜_{X}, J_{X})

and the forgetful functor lex.Site → Cat^{o}. Let Σ_{𝒜} be the set of cartesian morphisms of 𝒜. By remark 3.3.13(i), we know that 𝒜 [Σ_{𝒜}^{−1}] represents the strong 2-colimit of c :

2-colim_{𝒞^{o}}c →^{∼} 𝒜 [Σ_{𝒜}^{−1}].

Suppose now that 𝒞 is cofiltered. Then, the construction of (4.3.24) endows 𝒜 [Σ_{𝒜}^{−1}] with a natural topology J_{𝒜}^{∗} such that (𝒜 [Σ_{𝒜}^{−1}], J_{𝒜}^{∗} ) is a strong 2-limit of c̃, and again by direct inspection we see that the localization functor L_{𝒜} : 𝒜 → 𝒜 [Σ_{𝒜}^{−1}] is continuous for the topology J_{𝒜}^{∗} and the topology J_{𝒜} of the total site (𝒜 , J_{𝒜} ). Moreover, for every X ∈ Ob(X), the composition 𝒜_{X} → 𝒜 [Σ_{𝒜}^{−1}] of the inclusion functor i_{X} : 𝒜_{X} → 𝒜 and F_{𝒜} , is a morphism of sites

L_{𝒜} ∘ i_{X} : (𝒜 [Σ_{𝒜}^{−1}], J_{𝒜}^{∗} ) → (𝒜_{X}, J_{X})

and the rule : X ↦ L_{𝒜} ∘ i_{X} defines both a universal pseudo-cone τ̃ : F_{(𝒜 [Σ_{𝒜}^{−1}],J_{𝒜}^{∗} )} ⇒ c̃ and a universal pseudo-cocone τ : c ⇒ F_{𝒜 [Σ_{𝒜}^{−1}]} . Recall as well that Σ_{𝒜} admits a right calculus of fraction (example 3.3.13(ii)), so that the category X/L_{𝒜} 𝒜 is cofiltered for every X ∈ Ob(𝒜 [Σ_{𝒜}^{−1}]) (proposition 1.6.25(i)). Then the source fibration s : 𝒜 [Σ_{𝒜}^{−1}]/L_{𝒜} 𝒜 → 𝒜 [Σ_{𝒜}^{−1}] is trivially locally cofiltered, and from proposition 4.3.9we see that L_{𝒜} is a morphism of sites

L_{𝒜} : (𝒜 [Σ_{𝒜}^{−1}], J_{𝒜}^{∗} ) → (𝒜 , J_{𝒜} ).

Proposition 5.7.7. In the situation of (5.7.6), the following holds :

(i) The functor L_{𝒜 ∗}^{∼} : (𝒜 [Σ_{𝒜}^{−1}], J_{𝒜}^{∗} )^{∼} → (𝒜 , J_{𝒜} )^{∼} is fully faithful.

(ii) Suppose that every object of 𝒜 is quasi-compact for the topology J_{𝒜} (see definition 5.6.10), and let F and ℰ be respectively a sheaf and a stack on (𝒜 , J_{𝒜} ). Then:

(a) L_{𝒜!} F is a sheaf on (𝒜 [Σ_{𝒜}^{−1}], J_{𝒜}^{∗} ).

(b) Fib(L_{𝒜} )_{!}ℰ is a stack on (𝒜 [Σ_{𝒜}^{−1}], J_{𝒜}^{∗} ).

Proof. (i) follows easily from corollary 1.6.11.

Next, let F ∈ Ob(𝒜^{∧}) and A ∈ Ob(𝒜 [Σ_{𝒜}^{−1}]); recall that L_{𝒜!} F (A) represents the colimit of

F ∘ t_{A}^{o} : (A/L_{𝒜} 𝒜)^{o} → Set (A → L_{𝒜} B) ↦ F B.

We define as follows a functor Φ_{A} : 𝒞 /pA → A/L_{𝒜} 𝒜. For every φ ∈ Ob(𝒞 /pA) we let

Φ_{A}(φ) := (c_{φ}A, (L_{𝒜} λ(A, φ))^{−1} : A → L_{𝒜} (c_{φ}A))

(see (3.1.6)). For every morphism ψ/pA : φ → φ′ of 𝒞 /pA we have φ = φ′ ∘ ψ, and we set

Φ_{A}(ψ/pA) := A/L_{𝒜} (λ(1_{A}, ψ)) : Φ_{A}(φ) → Φ_{A}(φ′).

Claim 5.7.8. The functor Φ_{A} is coinitial.

Proof of the claim. Since 𝒞 is cofiltered, the same holds for 𝒞 /A (example 1.5.9(i)), hence we may apply the criterion of (the dual of) lemma 1.5.7(i). First, for every object (B, f : A → L_{𝒜} B) of A/L_{𝒜} 𝒜 we may find a morphism φ : pB → pA of 𝒞 and a morphism f′ : c_{φ}A → B in p^{−1}(pB) such that f = L_{𝒜} (f′) ∘ L_{𝒜} (λ(A, φ))^{−1} (details left to the reader), so that f′ yields a morphism Φ_{A}(φ) → (B, f) in A/L_{𝒜} 𝒜. Next, let φ : X → pA be any object of 𝒞 /pA and A/g, A/h : Φ_{A}(φ) → (B, f : A → L_{𝒜} B) two morphisms of A/L_{𝒜} 𝒜; by definition, we have g = f ∘ L_{𝒜} (λ(A, φ)) = h, so condition (b) of lemma 1.5.7(i) is trivially verified. ♢

By claim 5.7.8, the set L_{𝒜!} F (A) also represents the colimit of the functor

F ∘ (t_{A} ∘ Φ_{A})^{o} : (𝒞 /pA)^{o} → Set (φ : X → pA) ↦ F (c_{φ}A).

(ii.a): Notice first that, under the stated assumptions, for every X ∈ Ob(𝒞), every B ∈ Ob(𝒜_{X}) is quasi-compact for the topology J_{X} : indeed, if 𝒮 ∈ J_{X}(B), let 𝒮′ be the sieve of 𝒜 /B generated by 𝒮; we have 𝒮′ ∈ J_{𝒜} (B), so by assumption, there exists a finite subset {f_{i} : B_{i} → B | i ∈ I} ⊂ 𝒮′ that generates a sieve 𝒮′′ covering B in (𝒜 , J_{𝒜} ). But for every i ∈ I there exist (f′_{i} : C_{i} → B) ∈ 𝒮 and a morphism g_{i} : B_{i} → C_{i} in 𝒜 such that f_{i} = f′_{i} ∘ g_{i}; the family (f′_{i} | i ∈ I) still generates a covering sieve for J_{𝒜} , and then the same family covers B also in the site (𝒜_{X}, J_{X}). Next we remark :

Claim 5.7.9. Let A ∈ Ob(𝒜 [Σ_{𝒜}^{−1}]) = Ob(𝒜), and 𝒮 a sieve of 𝒜 [Σ_{𝒜}^{−1}]/A. Then 𝒮 ∈ J_{𝒜}^{∗} (A) if and only if there exist a cartesian morphism h : A′ → A in 𝒜 and a finite covering family (f_{i} : A′_{i} → A′) in (𝒜_{pA′}, J_{pA′}) with L_{𝒜} (h ∘ f_{i}) ∈ 𝒮 for every i ∈ I.

Proof of the claim. Since L_{𝒜} is continuous and L_{𝒜} h is an isomorphism in 𝒜 [Σ_{𝒜}^{−1}], the stated condition implies that 𝒮 ∈ J_{𝒜}^{∗} (A). For the converse, recall that J_{𝒜}^{∗} is the coarsest topology such that L_{𝒜} ∘ i_{X} is a continuous functor for the topology J_{X} and J_{𝒜}^{∗} , for every X ∈ Ob(𝒞). Thus, for every B ∈ Ob(𝒜), denote by J_{𝒜}^{∗∗}(B) the set of all sieves of 𝒜 [Σ_{𝒜}^{−1}]/B fulfilling the condition of the claim; it suffices then to check that J_{𝒜}^{∗∗} := (J_{𝒜}^{∗∗}(B) | B ∈ Ob(A)) is a topology on 𝒜 [Σ_{𝒜}^{−1}], and that L_{𝒜} ∘ i_{X} is a continuous functor for the topologies J_{X} and J_{𝒜}^{∗∗}, for every X ∈ Ob(𝒞). To check stability under base change for J_{𝒜}^{∗∗} (see definition 4.1.1(i)), consider any morphism g : B → A in 𝒜 [Σ_{𝒜}^{−1}]; we need to show that 𝒮 ×_{A} g ∈ J_{𝒜}^{∗∗}(B). However, since Σ_{𝒜} admits a right calculus of fraction, there exist a cartesian morphism h′ : B′ → B in 𝒜 and a morphism k : B′ → A in 𝒜 such that g = L_{𝒜} (k) ∘ L_{𝒜} (h′)^{−1}. Suppose now that 𝒮′ := 𝒮 ×_{A} L_{𝒜} (k) ∈ J_{B′}^{∗∗}; this means that there exist a cartesian morphism h′′ : B′′ → B′ and a finite covering family (f′_{i} : B′′_{i} → B′) in (𝒜_{pB′}, J_{pB′}) such that L_{𝒜} (h′′ ∘ f′_{i}) ∈ 𝒮′ for every i ∈ I. Therefore L_{𝒜} (h′ ∘ h′′ ∘ f′_{i}) ∈ 𝒮′ ×_{B′} L_{𝒜} (h′)^{−1} = 𝒮 ×_{A} g, and since h′ ∘ h′′ is cartesian, we get 𝒮 ×_{A} g ∈ J_{𝒜}^{∗∗}. Thus, we may replace B by B′, and assume from start that g = L_{𝒜} (k) for some morphism k : B → A in 𝒜. In this case, since 𝒞 is cofiltered, we can find X ∈ Ob(𝒞) with morphisms φ : X → pB and ψ : X → pA′ such that p(g)∘φ = p(h)∘ψ; then we may also find a morphism k′ : B′ → A′′ in p^{−1}X, and cartesian morphisms h′ : B′ → B and h′′ : A′′ → A′ in 𝒜 such that p(h′) = φ, p(h′′) = ψ, and k ∘ h′ = h ∘ h′′ ∘ k′ in 𝒜. For every i ∈ I there exist a morphism f′_{i} : A′′_{i} → A′′ in p^{−1}X and a cartesian morphism h′′_{i} : A′′_{i} → A′_{i} such that f_{i} ∘ h′′_{i} = h′′ ∘ f′_{i}. Since c_{ψ} is continuous for the topologies J_{pA′} and J_{X}, it follows easily that (f′_{i} | i ∈ I) is a covering family in (𝒜_{X}, J_{X}) (details left to the reader), and by construction L_{𝒜} (h ∘ h′′ ∘ f′_{i}) ∈ 𝒮 for every i ∈ I. Since h ∘ h′′ is cartesian, we may therefore replace A′ by A′′, h by h ∘ h′′ and the family (f_{i} | i ∈ I) by (f′_{i} | i ∈ I), and assume from start that there exist a morphism k′ : B′ → A′ in 𝒜_{pA′} and a cartesian morphism h′ : B′ → B in 𝒜 such that h ∘ k′ = k ∘ h′. Let (f′_{i} : A′_{i} ×_{A′} B′ → B′ | i ∈ I) be the induced finite covering family of B′ in 𝒜_{pA′}, and for every i ∈ I let also k′_{i} : A′_{i} ×_{A′} B′′ → A′_{i} be the second projection. It suffices to check that L_{𝒜} (h′ ∘ f′_{i}) ∈ 𝒮 ×_{A} g for every i ∈ I, i.e. that L_{𝒜} (k∘h′∘f′_{i}) ∈ 𝒮 for every such i. However, L_{𝒜} (k∘h′∘f′_{i}) = L_{𝒜} (h∘k′∘f′_{i}) = L_{𝒜} (h∘f_{i}∘k′_{i}), whence the contention. To check the local character of J_{𝒜}^{∗∗}, suppose that 𝒯 is another sieve of 𝒜 [Σ_{𝒜}^{−1}]/A such that for every (t : A′ → A) ∈ 𝒮 we have 𝒯 ×_{A} t ∈ J_{𝒜}^{∗∗}(A′); we need to show that 𝒯 ∈ J^{∗∗}(A). In particular, for every i ∈ I there exist a cartesian morphism h_{i} : B_{i} → A′_{i} in 𝒜 and a finite covering family (g_{iλ} : B_{iλ} → B_{i} | λ ∈ Λ_{i}) in (𝒜_{pB_{i}}, J_{pB_{i}}) such that L_{𝒜} (h_{i} ∘ g_{iλ}) ∈ 𝒯 ×_{A} L_{𝒜} (h ∘ f_{i}) for every λ ∈ Λ_{i}, i.e. L_{𝒜} (h ∘ f_{i} ∘ h_{i} ∘ g_{iλ}) ∈ 𝒯 for every such λ. Now, since 𝒞 is cofiltered, we may find a morphism φ : Y → pA′ in 𝒞 such that for every i ∈ I there exists a morphism φ_{i} : Y → pB_{i} with φ = p(h_{i}) ∘ φ_{i}. Pick then any cartesian morphism k : C → A′ with p(k) = φ, and a cartesian morphism k_{i} : C_{i} → B_{i} such that p(k_{i}) = φ_{i}, for every i ∈ I. With this notation, for every i ∈ I there exists a unique morphism f′_{i} : C_{i} → C in 𝒜_{Y} such that k ∘ f′_{i} = f_{i} ∘ h_{i} ∘ k_{i}. Since c_{φ} is continuous for the topologies J_{Y} and J_{pA′}, it follows easily that (f′_{i} : C_{i} → C | i ∈ I) is a finite covering family in (𝒜_{Y} , J_{Y}) (details left to the reader). Lastly, for every i ∈ I and λ ∈ Λ_{i} pick a cartesian morphism k_{iλ} : C_{iλ} → B_{iλ} with p(k_{iλ}) = φ_{i}. Then for every such i and λ there exists a unique morphism g′_{iλ} : C_{iλ} → C_{i} in 𝒜_{Y} such that g_{iλ} ∘ k_{iλ} = k_{i} ∘ g′_{iλ}, and arguing as in the foregoing, we easily see that (g′_{iλ} | λ ∈ Λ_{i}) is a finite covering family in (𝒜_{Y} , J_{Y}), for every i ∈ I. We conclude that (f′_{iλ} := f′_{i} ∘ g′_{i,λ} : C_{iλ} → C | i ∈ I, λ ∈ Λ_{i}) is a finite covering family in (𝒜_{Y} , J_{Y}) as well. But we have k ∘ f′_{iλ} = f_{i} ∘ h_{i} ∘ g_{iλ} ∘ k_{iλ} for every i ∈ I and λ ∈ Λ_{i}; since h ∘ k is a cartesian morphism, the assertion follows. Thus, J_{𝒜}^{∗∗} is indeed a topology on 𝒜 [Σ_{𝒜}^{−1}]. In order to prove that L_{𝒜} ∘ i_{X} is continuous for J_{𝒜}^{∗∗} and J_{X}, it suffices now to check that for every covering family (f_{λ} : B_{λ} → B | λ ∈ Λ) in 𝒜_{X}, the family (L_{𝒜} (f_{λ}) | λ ∈ Λ) covers B for the topology J_{𝒜}^{∗∗} (lemma 4.2.4). Since, as we have already observed, B is quasi-compact for the topology J_{X}, we are easily reduced to the case where Λ is a finite set (details left to the reader); but then the assertion follows immediately from the definition of J_{𝒜}^{∗∗}(B). ♢

We shall show that the natural morphism of presheaves L_{𝒜!} F → (L_{𝒜!} F)^{+} is an isomorphism (notation of (4.1.11)); the lemma will follow immediately. To this aim, claim 5.7.9 reduces to checking the following assertion. For every A ∈ Ob(𝒜) and every finite covering family (f_{i} : A_{i} → A | i ∈ I) in (𝒜_{pA}, J_{pA}), the natural map :

L_{𝒜!} F A → E := Equal(∏_{i∈I} [figure: L_{𝒜!} F (A_{i})] ∏_{(i,j)∈I×I}L_{𝒜!} F (A_{i} ×_{A} A_{j}))

is bijective. Thus, let ℐ be the finite category whose set of objects is the disjoint union of I and I×I, and with two morphisms i ← (i, j) → j for every (i, j) ∈ I×I (and of course, the identity morphism of each object); the covering family (f_{i} | i ∈ I) induces a functor A_{•} : ℐ^{o} → 𝒜_{pA} given by the rule : i ↦ A_{i} and (i, j) ↦ A_{ij} := A_{i} ×_{A} A_{j} for every i, j ∈ I, and which assigns to every morphism l : (i, j) → k of ℐ the projection π_{l} : A_{ij} → A_{k}. We consider the functor

Ψ : ℐ × (𝒞 /pA) → 𝒜

that assigns to every pair (t, φ/pA) ∈ Ob(ℐ) × Ob(𝒞 /pA) the object c_{φ}(A_{t}), and to every morphism (l, ψ) : (t, φ/pA) → (t′, φ′/pA) the composition :

c_{φ}(A_{t}) ⟶^{λ(1_{A_{t}},ψ)} c_{φ′}(A_{t}) ⟶^{c_{φ′}(π_{l})} c_{φ′}(A_{t′}).

For every (t, φ) ∈ Ob(ℐ × (𝒞 /pA)), let also [t] (resp. [φ]) be the subcategory of ℐ (resp. of 𝒞 /pA) with Ob([t]) := {t} (resp. with Ob([φ]) = {φ}), and denote by Ψ_{t} : 𝒞 /pA → 𝒜 (resp Ψ^{φ} : ℐ → 𝒜) the restriction of Ψ to [t] × (𝒞 /pA) →^{∼} 𝒞 /pA (resp. to ℐ × [φ] →^{∼} ℐ). With this notation, L_{𝒜!} F (A_{t}) represents the colimit of F ∘ Ψ_{t}^{o}, for every t ∈ Ob(ℐ), and since ℐ is finite and (𝒞 /pA)^{o} is filtered, we get natural identifications :

E = lim_{t∈Ob(ℐ)} colim_{(𝒞 /pA)^{o}} F ∘ Ψ_{t}^{o} →^{∼} colim_{φ∈Ob(𝒞 /pA)} lim_{ℐ}F ∘ (Ψ^{φ})^{o}.

But since c_{φ} is left exact and continuous for the topologies J_{X} and J_{Y}, for every morphism φ : X → Y of 𝒞, and since the restriction of F to 𝒜_{X} is a sheaf for the topology J_{X}, for every X ∈ Ob(𝒞), we see that F (c_{φ}A) represents the limit of the functor F ∘ (Ψ^{φ})^{o}, for every such φ, so the functor (𝒞 /pA)^{o} → Set given by the rule φ ↦ lim_{ℐ} F ∘ (Ψ^{φ})^{o} is naturally identified with F ∘ (t_{A} ∘ Φ_{A})^{o}. Summing up, we conclude that E represents the colimit of F ∘ (t_{A} ∘ Φ_{A})^{o}, whence (i).

(ii.b): By claim 3.2.8, we may assume that ℰ = ℱ ib(E_{•}) for a presheaf of categories E_{•} on 𝒜. Then recall that ℱ := Fib(L_{𝒜} )_{!}ℰ is the fibration associated with the strict pseudo-functor computed by the strong left 2-Kan extension :

F_{•} := 2-∫ ^{L_{𝒜}^{o}} E_{•} A ↦ F_{A} := 2-colim_{(A/L_{𝒜} 𝒜)^{o}} E_{•} ∘ t_{A}^{o}.

However, since (A/L_{𝒜} 𝒜)^{o} is filtered, the 2-colimit of E_{•} ∘ t_{A}^{o} is represented by the colimit of the same functor, for every A ∈ Ob(𝒜) (example 3.3.13(iv)), and in view of claim 5.7.8, this is also the colimit of the functor E_{•} ∘ (t_{A} ∘ Φ_{A})^{o} : (𝒞 /pA)^{o} → Cat, for every such A.

It suffices to show that the natural functor C(ℱ) → ℱ^{+} (see (5.1.17)) is an equivalence of categories, and in view of claim 5.7.9 (and of corollary 3.1.28(i)) we are reduced to checking the following assertion. For every A ∈ Ob(𝒜) and every finite covering family f_{•} := (f_{i} : A_{i} → A | i ∈ I) in (𝒜_{pA}, J_{pA}), the natural functor

F_{A} →^{∼} ℱ (A) → Desc(ℱ , L_{𝒜} (f_{•}))

is an equivalence. Now, every morphism ψ/A : φ → φ′ of 𝒞 /pA induces a functor

δ_{ψ/A} : Desc(ℰ , c_{φ′}(f_{•})) → Desc(ℰ , c_{φ}(f_{•}))

as follows. Notice that, since c_{φ} is left exact, a descent datum for ℰ relative to the covering c_{φ}(f_{•}) := (c_{φ}(f_{i}) | i ∈ I) is a pair (e_{•}, ω_{••}), where e_{•} := (e_{i} | i ∈ I) is a system of objects e_{i} ∈ Ob(E_{c_{φ}A}) for every i ∈ I, and ω_{••} is a system of isomorphisms ω_{ij} : E_{c_{φ}(π_{ij})}(e_{i}) →^{∼} E_{c_{φ}(π_{ji})}(e_{j}) fulfilling the usual cocycle condition (here we denote by π_{ij} : A_{ij} → A_{i} and π_{ji} : A_{ij} → A_{j} the projections). A morphism h_{•} : (e_{•}, ω_{••}) → (e_{•}, ω_{••}) of such descent data is a system of morphisms (h_{i} : e_{i} → e′_{i} | i ∈ I) compatible in the obvious fashion with ω_{ij} and ω′_{ij}. Then δ_{ψ/A} is given by the rules :

\begin{matrix}(e_{•}, ω_{••}) & ↦ ((E_{λ(1_{A_{i}},ψ)}(e_{i}) | i ∈ I), (E_{λ(1_{A_{ij}} ,ψ)}(ω_{ij}) | i, j ∈ I)) \\ h_{•} & ↦ ((E_{λ(1_{A_{i}},ψ)}(h_{i}) | i ∈ I) \end{matrix}

for every object (e_{•}, ω_{••}) and every morphism h_{•} of Desc(ℰ , c_{φ′}(f_{•})). We thus obtain a functor δ : (𝒞 /pA)^{o} → Cat that assigns to every φ ∈ Ob(𝒞 /pA) the category Desc(ℰ , c_{φ}(f_{•})), and to every morphism ψ/A the functor δ_{ψ/A}, and a direct inspection of the definitions yields a natural identification

Desc(ℱ , L_{𝒜} (f_{•})) →^{∼} colim_{(𝒞 /pA)^{o}} δ.

Lastly, since ℰ is a stack on (𝒜 [Σ_{𝒜}^{−1}], J_{𝒜}^{∗} ), the natural functor E_{c_{φ}A} → Desc(ℰ , c_{φ}(f_{•})) is an equivalence for every φ ∈ Ob(𝒞 /pA), and we get an induced equivalence

2-colim_{(𝒞 /pA)^{o}}E_{•} ∘ (t ∘ Φ_{A})^{o} →^{∼} 2-colim_{(𝒞 /pA)^{o}}δ.

But again, these 2-colimits are represented by the colimits of the same functors, whence the contention. □

5.7.10. Keep the notation of (5.7.6), and let now φ : (𝒜 , p, J_{•}) → (𝒜′, p′, J′_{•}) be a morphism of small fibred lex-sites over the small category 𝒞. We have φ(Σ_{𝒜′} ) ⊂ Σ_{𝒜} , whence a commutative diagram of categories, for every X ∈ Ob(𝒞) :

\xymatrix{𝒜′_{X} \ar[r]^{i′_{X}} \ar[d]_{φ_{X}} & 𝒜′ \ar[r]^{L_{𝒜′}} \ar[d]_{φ} & 𝒜′[Σ_{𝒜′}^{−1}] \ar[d]^{φ_{Σ}} \\ 𝒜_{X} \ar[r]^{i_{X}} & 𝒜 \ar[r]^{L_{𝒜}} & 𝒜 [Σ_{𝒜}^{−1}]}

where i_{X} and i′_{X} are the inclusion functors. Let as well λ′ be a unital cleavage for p′, and c′ its associated pseudo-functor, which factors likewise through a pseudo-functor c̃′ : 𝒞 → lex.Site; then φ corresponds to a unique pseudo-natural transformation ω : c′ ⇒ c, and the latter in turn can also be regarded as a 2-cell ω̃ : c̃ ⇒ c̃′ of lex.Site. Now, since 𝒜 [Σ_{𝒜}^{−1}] is a strong 2-colimit for c, there exists a unique functor ψ : 𝒜′[Σ_{𝒜′}^{−1}] → 𝒜 [Σ_{𝒜}^{−1}] such that τ ⊙ F_{ψ} = ω; clearly we must then have ψ = φ_{Σ}. On the other hand, since 𝒜 [Σ_{𝒜}^{−1}] is a strong 2-limit for c̃, there exists a unique morphism of sites ψ̃: (𝒜 [Σ_{𝒜}^{−1}], J_{𝒜}^{∗} ) → (𝒜′[Σ_{𝒜′}^{−1}], J_{𝒜′}^{∗} ) such that Fψ̃ ⊙ τ̃ = ω̃; thus, we must have ψ̃ = ψ, and this shows that φ_{Σ} is a morphism of sites as well, so we get a commutative diagram of morphisms of sites :

\xymatrix{(𝒜 [Σ_{𝒜}^{−1}], J_{𝒜}^{∗} ) \ar[r]^{L_{𝒜}} \ar[d]_{φ_{Σ}} & (𝒜 , J_{𝒜} ) \ar[d]^{φ} \\ (𝒜′[Σ_{𝒜′}^{−1}], J_{𝒜′}^{∗} ) \ar[r]^{L_{𝒜′}} & (𝒜′, J_{𝒜′} )}

which we orient by the identity transformation 1_{φ∘L_{𝒜}} . Following (5.5.23), the resulting oriented diagram then induces an oriented square of 2-categories :

[figure: Stack(𝒜 [Σ_{𝒜}^{−1}], J_{𝒜}^{∗} ); St(1_{φ∘L_{𝒜}} )_{∗}^{γ}; St(L_{𝒜} )_{∗}; St(φ_{Σ})_{∗}; Stack(𝒜 , J_{𝒜} ); St(φ)_{∗}; Stack(𝒜′[Σ_{𝒜′}^{−1}], J_{𝒜′}^{∗} ); St(L_{𝒜′} )_{∗}; Stack(𝒜′, J_{𝒜′} )]

which in turn can be regarded as an oriented square of links of the 2-category V-\overline{2-Cat}, for a suitable universe V : see (5.5.25).

Proposition 5.7.11. In the situation of (5.7.10), suppose that every object of 𝒜 is quasicompact for the topology J_{𝒜} . Then Υ(St(1_{φ∘L_{𝒜}} )_{∗}^{γ}) is a pseudo-natural equivalence.

Proof. Let ℰ be any stack on (𝒜 , J_{𝒜} ); as usual, we may assume that ℰ = ℱ ib(E_{•}) for a presheaf of categories E_{•} on 𝒜, and by virtue of lemma 5.5.15(ii) we may also assume that E_{•} is a sheaf of categories on (𝒜 , J_{𝒜} ). According to corollary 5.5.26(iv), and the discussion of (5.5.29), we are then reduced to checking :

Claim 5.7.12. (i) ℱ ib((L_{𝒜}^{∼}, Cat)^{∗}E_{•}) is a stack on (𝒜 [Σ_{𝒜}^{−1}], J_{𝒜}^{∗} ).

(ii) Υ((1_{φ∘L_{𝒜}} )_{∗}^{∼}) is an isomorphism of functors.

Proof of the claim. (i): On one hand the natural morphism (L_{𝒜} , Cat)_{!}E_{•} → (L_{𝒜}^{∼}, Cat)^{∗}E_{•} is an isomorphism, by proposition 5.7.7(ii.a); on the other hand, from claim 5.7.8 and lemma 1.5.7(ii) we see that the category A/L_{𝒜} 𝒜 is cofiltered for every A ∈ Ob(𝒜), hence remark 5.5.14(v) yields an equivalence of categories ℱ ib((L_{𝒜} , Cat)_{!}E_{•}) →^{∼} Fib(L_{𝒜} )_{!}(ℰ). Lastly, Fib(L_{𝒜} )_{!}(ℰ) is a stack, by proposition 5.7.7(ii.b), whence the assertion.

(ii): By definition, we have :

Υ((1_{φ∘L_{𝒜}} )_{∗}^{∼}) = (ε^{L_{𝒜′}} ∗ φ̃_{Σ∗}L̃_{𝒜}^{∗} ) ⊙ (L̃_{𝒜′}^{∗} φ̃_{∗} ∗ η^{L_{𝒜}} )

where η^{L_{𝒜}} (resp. ε^{L_{𝒜′}} ) is the unit (resp. the counit) of an adjunction for the pair (L̃_{𝒜}^{∗} , L̃_{𝒜 ∗}) (resp. (L̃_{𝒜′}^{∗} , L̃_{𝒜′∗})). However, L̃_{𝒜′∗} is fully faithful (proposition 5.7.7(i)), hence ε^{L_{𝒜′}} is an isomorphism (proposition 1.1.20(iii)). Thus, we are reduced to checking that L̃_{𝒜′}^{∗} φ̃_{∗} ∗ η^{L_{𝒜}} is an isomorphism. However, since also ε^{L_{𝒜}} is an isomorphism, we know already that L̃_{𝒜}^{∗} ∗ η^{L_{𝒜}} is an isomorphism, due to the triangular identities of (1.1.13). Thus, we are further reduced to exhibiting an arbitrary isomorphism of functors :

L̃_{𝒜′}^{∗} ∘ φ̃_{∗} →^{∼} φ̃_{Σ∗} ∘ L̃_{𝒜}^{∗} .

Hence, let F be any sheaf on (𝒜 , J_{𝒜} ), and A′ ∈ Ob(𝒜′); from the proof of proposition 5.7.7(i) we see that L̃_{𝒜′}^{∗} ∘ φ̃_{∗}F (A′) represents the colimit of the functor

Ψ : (𝒞 /p′A′)^{o} → Set  (X ⟶^{f} p′A′) ↦ F φ(c′_{f}A′)  (f ⟶^{g/p′A′} f′) ↦ F φ(λ′(1_{A′}, g))

whereas φ̃_{Σ∗} ∘ L̃_{𝒜}^{∗} F (A′) represents the colimit of the functor

Ψ′ : (𝒞 /p′A′)^{o} → Set  (X ⟶^{f} p′A′) ↦ F c_{f}(φA′)  (f ⟶^{g/p′A′} f′) ↦ F (λ′(1_{φA′}, g)).

Now, let τ^{ω} denote the coherence constraint of the pseudo-natural transformation ω associated with φ as in (5.7.10), so that ω_{X} : 𝒜_{X} → 𝒜′_{X} is the restriction of φ, for every X ∈ Ob(𝒞); we claim that the following system of maps yields an isomorphism of functors Ψ →^{∼} Ψ′ :

(5.7.13)  (F (τ_{f,A′}^{ω}) : F φ(c′_{f}A′) →^{∼} F c_{f}(φA′) | f ∈ Ob(𝒞 /p′A′)).

More precisely, we show that for every two objects f : X → p′A′ and f′ : X′ → p′A′ and every morphism g/p′A′ : f → f′ of 𝒞 /p′A′ the following diagram commutes :

(5.7.14)  \xymatrix{c_{f}(φA′) \ar[r]^{τ_{f,A′}^{ω}} \ar[d]_{λ(1_{φA′},g/p′A′)} & φ(c′_{f}A′) \ar[d]^{φ(λ′(1_{A′},g/p′A′))} \\ c_{f′}(φA′) \ar[r]^{τ_{f′,A′}^{ω}} & φ(c′_{f′}A′).}

To this aim, denote by γ^{c′} the coherence constraint of c′, and recall that, under the natural identification 𝒜′ →^{∼} ℱ ib(c′), the morphism λ′(1_{A′}, g/p′A′) of 𝒜′ corresponds to the morphism

(g, γ_{(g,f′),A′}^{c′ −1} : c′_{f}A′ → c′_{g}c′_{f′}A′) : (X, c′_{f}A′) → (X′, c′_{f′}A′).

Likewise we describe λ(1_{φA′}, g) in terms of the coherence constraint γ^{c} of c; since λ and λ′ are unital, the assertion comes down to the commutativity of the diagram :

\xymatrix{c_{g}c_{f′}(φA′) \ar[rr]^{γ_{(g,f′),φA′}^{c}} \ar[d]_{c_{g}(τ_{f′,A′}^{ω})} &  & c_{f}(φA′) \ar[d]^{τ_{f,A′}^{ω}} \\ c_{g} ∘ φ(c′_{f′}A′) \ar[r]^{τ_{g,c{′ \atop f′} A′}^{ω}} & φ(c′_{g}c′_{f′}A′) \ar[r]^{ω_{X}(γ_{(g,f′),A′}^{c′})} & φ(c′_{f}A′)}

which in turn follows from the coherence axiom for τ^{ω}. So finally, the colimit of the system (5.7.13) yields a bijection t_{F,A′} : L̃_{𝒜′}^{∗} ∘ φ̃_{∗}F (A′) →^{∼} φ̃_{Σ∗} ∘ L̃_{𝒜}^{∗} F (A′) for every A′ ∈ Ob(𝒜′), and we need to check that the rule : A′ ↦ t_{F,A′} defines a morphism of sheaves t_{F}. Thus, let

(β_{F,f}^{A} : F (c_{f}A) → L̃_{𝒜}^{∗} F (A) | f ∈ Ob(𝒞 /pA)) for every A ∈ Ob(𝒜)

be a fixed system of universal cocones, and h : B → A any morphism in 𝒜 [Σ_{𝒜}^{−1}]; for a given object f : X → pA of 𝒞 /pA, we may find a morphism g : Y → pB in 𝒞 and a morphism h_{g,f} : c_{g}B → c_{f}A in 𝒜 such that L_{𝒜} (λ(A, f))^{−1} ∘ h = L_{𝒜} (h_{g,f}) ∘ L_{𝒜} (λ(B, g))^{−1}, and by unraveling the definitions, it is easily seen that :

L̃_{𝒜}^{∗} F (h) ∘ β_{F,f}^{A} = β_{F,g}^{B} ∘ F h_{g,f}.

Likewise, fix a system of universal cocones :

(β_{F,f}^{′A′} : F φ(c′_{f}A′) → L̃_{𝒜′}^{∗} φ̃_{∗}F (A′) | f ∈ Ob(𝒞 /p′A′)) for every A′ ∈ Ob(𝒜′) .

If h′ : B′ → A′ is any morphism of 𝒜′[Σ_{𝒜′}^{−1}], and f : X → p′A′ any object of 𝒞 /p′A′, we may find a morphism g : Y → p′B′ in 𝒞 and a morphism h′_{g,f} : c′_{g}B′ → c′_{f}A′ of 𝒜′ such that L_{𝒜′} (λ′(A, f))^{−1} ∘ h′ = L_{𝒜′} (h′_{g,f}) ∘ L_{𝒜′} (λ′(B, g))^{−1}, and again it is easily seen that :

L̃_{𝒜′}^{∗} φ̃_{∗}F (h′) ∘ β_{F,f}^{′A′} = β_{F,g}^{′B′} ∘ F φ(h′_{g,f}).

Now, for every such h′ we need to check the identity :

(5.7.15)  t_{F,B′} ∘ L̃_{𝒜′}^{∗} φ̃_{∗}F (h′) = φ̃_{Σ∗}L̃_{𝒜}^{∗} F (h′) ∘ t_{F,A′}

and it suffices to check that the sides of (5.7.15) agree after composition with β_{F,f}^{′A′}, for every f ∈ Ob(𝒞 /p′A′). Notice that if g : Y → p′B′ is a suitable choice for constructing h′_{g,f}, then the same choice can be used for constructing (φ_{Σ}h′)_{g,f} (details left to the reader). We compute:

\begin{aligned}t_{F,B′} ∘ L̃_{𝒜′}^{∗} φ̃_{∗}F (h′) ∘ β_{F,f}^{′A′} & = t_{F,B′} ∘ β_{F,g}^{′B′} ∘ F φ(h′_{g,f}) = β_{F,g}^{φB′} ∘ F (τ_{g,B′}^{ω}) ∘ F φ(h′_{g,f}) \\ φ̃_{Σ∗}L̃_{𝒜}^{∗} F (h′) ∘ t_{F,A′} ∘ β_{F,f}^{′A′} & = φ̃_{Σ∗}L̃_{𝒜}^{∗} F (h′) ∘ β_{F,f}^{φA′} ∘ F (τ_{f,A′}^{ω}) = β_{F,g}^{φB′} ∘ F (φ_{Σ}h′)_{g,f} ∘ F (τ_{f,A′}^{ω}). \end{aligned}

Thus, we are reduced to checking that we may choose (φ_{Σ}h′)_{g,f} such that :

φ(h′_{g,f}) ∘ τ_{g,B′}^{ω} = τ_{f,A′}^{ω} ∘ (φ_{Σ}h′)_{g,f}.

In other words, we have to check the identity :

(5.7.16) L_{𝒜} ((τ_{f,A′}^{ω})^{−1} ∘ φ(h′_{g,f}) ∘ τ_{g,B′}^{ω}) ∘ L_{𝒜} (λ(φB′, g))^{−1} = L_{𝒜} (λ(φA′, f))^{−1} ∘ (φ_{Σ}h′).

However, if we regard g as a morphism g/p′B′ : g → 1_{p′B′} of 𝒞 /p′B′, we get

λ′(1_{B′}, g/p′B′) = λ′(B′, g) and λ(1_{φB′}, g/p′B′) = λ(φB′, g).

Likewise, we have corresponding identities for λ′(A′, g) and λ(φA′, g), by regarding f as a morphism f/p′A′ : f → 1_{p′A′} of 𝒞 /p′A′. Then (5.7.14) yields the identities

φ(λ′(A′, f)) ∘ τ_{f,A′}^{ω} = λ(φA′, f) φ(λ′(B′, g)) ∘ τ_{g,B′}^{ω} = λ(φB′, g).

(notice that τ_{1_{A′},A′}^{ω} = 1_{φA′} and τ_{1_{B′},B′}^{ω} = 1_{φB′}, since c and c′ are unital : see remark 2.4.2(ii)). Summing up, we deduce that (5.7.16) is equivalent to the identity :

L_{𝒜} (φ(λ′(A′, f)) ∘ φ(h′_{g,f})) ∘ L_{𝒜} (φ(λ′(B′, g)))^{−1} = φ_{Σ}h′

which follows immediately from the definition of h′_{g,f} and the identity φ_{Σ} ∘ L_{𝒜′} = L_{𝒜} ∘ φ.

This conclude the construction of the morphism of sheaves t_{F} : L̃_{𝒜′}^{∗} ∘ φ̃_{∗}F →^{∼} φ̃_{Σ∗} ∘ L̃_{𝒜}^{∗} F. To conclude the proof, it remains to check that the rule F ↦ t_{F} yields the sought isomorphism of functors. Thus, let ν : F → G be a morphism of sheaves on (𝒜 , J_{𝒜} ); we come down to checking the identity :

(5.7.17)  φ̃_{Σ∗}L̃_{𝒜}^{∗} (ν)_{A′} ∘ t_{F,A′} = t_{G,A′} ∘ L̃_{𝒜}^{∗} φ̃_{∗}(ν)  for every A′ ∈ Ob(𝒜′)

and again, it suffices to check that the two sides of (5.7.17) agree after composition with β_{F,f}^{′A′}, for every f ∈ Ob(𝒞 /p′A′). We compute :

\begin{aligned}φ̃_{Σ∗}L̃_{𝒜}^{∗} (ν)_{A′} ∘ t_{F,A′} ∘ β_{F,f}^{′A′} & = φ̃_{Σ∗}L̃_{𝒜}^{∗} (ν)_{A′} ∘ β_{F,f}^{φA′} ∘ F (τ_{f,A′}^{ω}) = β_{G,f}^{φA′} ∘ ν_{c_{f} (φA′)} ∘ F (τ_{f,A′}^{ω}) \\ t_{G,A′} ∘ L̃_{𝒜}^{∗} φ̃_{∗}(ν) ∘ β_{F,f}^{′A′} & = t_{G,A′} ∘ β_{G,f}^{′A′} ∘ ν_{φ(c′_{f} A′)} = β_{G,f}^{φA′} ∘ G(τ_{f,A′}^{ω}) ∘ ν_{φ(c′_{f} A′)}. \end{aligned}

So we come down to showing that ν_{c_{f} (φA′)} ∘ F (τ_{f,A′}^{ω}) = G(τ_{f,A′}^{ω}) ∘ ν_{φ(c′_{f} A′)}. The latter is clear, since ν is a morphism of sheaves. □

5.7.18. Stacks on fibred topoi. Recall that the pseudo-functor T of (4.4.15) assigns to every site C a topos T(C) with an isomorphism ω_{C} : C^{∼} →^{∼} T(C), and the unit of the 2-adjoint pair (Can, T) exhibited in the proof of theorem 4.4.17 assigns to every topos T an equivalence h_{T ∗} :→^{∼} T ∘ Can(T) whose composition with ω_{Can(T)}^{−1} equals the Yoneda imbedding h_{T} : T →^{∼} T^{∼}. Also, for every natural transformation β : u ⇒ v of morphisms of topoi u, v : T′ → T, the isomorphisms ω_{Can(T)} and ω_{Can(T′)} identify T ∘ Can(β) with (β^{†})_{∗}^{∼}, where β^{†} is the adjoint transformation to β. Now, consider any oriented square of morphisms of topoi :

[figure: T′; u; f′; β; T; f; S′; v; S]

(i.e. β is a natural transformation f_{∗} ∘ u_{∗} ⇒ v_{∗} ∘ f′_{∗}). There follows a diagram of oriented squares of topoi :

[figure: (Can T′)^{∼}; (u^{∗})^{∼}; (f^{′∗})^{∼}; (Can T)^{∼}; (f^{∗})^{∼}; T′; u; f′; h_{T′}; β; T; f; h_{T}; S′; v; h_{S′}; S; h_{S}; (Can S′)^{∼}; (v^{∗})^{∼}; (Can S)^{∼}]

whose unmarked orientations are identified, via the isomorphisms ω_{•}, with the coherence constraints of the pseudo-natural transformation h_{•∗}. We complete it by inserting the orientation (β^{†})_{∗}^{∼} for the external square. Then the resulting cubical diagram commutes on 2-cells, in the sense of (2.3.21). Since the four diagonal arrows are equivalences (theorem 4.4.2(iv)) it follows that Υ(β) is an isomorphism of functors if and only if the same holds for Υ((β^{†})_{∗}^{∼}) (remark 2.3.22(ii)).

5.7.19. Let ω : T → S be a morphism of fibred topoi over a small category I; let also t ∈ Ob(I) be any object, and define the functor ρ^{t} : 𝟙 → I as in remark 4.6.22(ii). By direct inspection, we see that :

Ψ_{t}^{∗} := Top(T/ρ^{t})^{∗} ∘ Top(ω)^{∗} = ω_{t}^{∗} ∘ Top(S/ρ^{t})^{∗}.

Indeed, both functors attach to every object E_{•} : I → ℱ ib(S^{∗}) of Top(S) the object ω_{t}^{∗}(E_{t}) ∈ Ob(T_{t}), and to every morphism β_{•} : E_{•} → E′_{•}, the morphism ω_{t}^{∗}(β_{t}). Similarly, we get :

Φ_{t} := Top(S/ρ^{t})^{∗} ∘ Top(ω)_{∗} = ω_{t∗} ∘ Top(T/ρ^{t})^{∗}.

The adjoint of 1_{Ψ_{t}^{∗}} yields therefore an orientation for the following diagram of topoi :

[figure: T_{t}; Top(T/ρ^{t}); ω_{t}; 1_{Ψ^{∗}_{t}}^{†}; Top(T); Top(ω); S_{t}; Top(S/ρ^{t}); Top(S)]

which we regard as an oriented square of links in the 2-category of categories (see (2.3.8)).

Lemma 5.7.20. With the notation of (5.7.19), we have : Υ(1_{Ψ_{t}^{∗}}^{†}) = 1_{Φ_{t}}.

Proof. We may as usual assume that T and S are unital; then we may replace Top(S) by the isomorphic category Top(S)_{∗}, and Top(ω) by the adjunction Ω := (Ω^{∗}, Ω_{∗}, η^{Ω}) described in the proof of proposition 4.6.10. Then Top(S/ρ^{t})^{∗} is replaced by Σ(ℱ ib(S_{∗})/ρ^{to})^{∗} (see the proof of corollary 4.6.21), and we come down to considering the oriented square of links :

[figure: T_{t}; Top(T/ρ^{t}); ω_{t}; 1_{Λ^{∗}_{t}}^{†}; Top(T); Ω; S_{t}; Σ(ℱ ib(S_{∗})/ρ^{to}); Top(S)_{∗}]  where Λ_{t}^{∗} := Top(T/ρ^{t})^{∗} ∘ Ω^{∗}

whose bottom horizontal arrow is given by Σ(ℱ ib(S_{∗})/ρ^{to})^{∗}, its right adjoint, and a unit for this adjoint pair. Set Φ′_{t} := Σ(ℱ ib(S_{∗})/ρ^{to})^{∗} ∘ Ω_{∗}; by proposition 2.3.10, it suffices to show that Υ(^{o}1_{Λ_{t}^{∗}}) = ^{o}1_{Φ′_{t}}. However, let (η^{ω_{i}}, ε^{ω_{i}}) be the unit and counit of the adjunction for the pair (ω_{i}^{∗}, ω_{i∗}) defining the morphism of topoi ω_{i}, for every i ∈ Ob(I); by definition we have :

Υ(^{o}1_{Λ_{t}^{∗}}) = (ε^{^{o}ω_{t}} ∗^{o}Σ(ℱ ib(S_{∗})/ρ^{to})^{∗} ∗^{o}Ω_{∗}) ⊙ (^{o}ω_{t∗} ∗^{o}Top(T/ρ^{t})^{∗} ∗ η^{^{o}Ω})

and recall that ε^{^{o}ω_{t}} = ^{o}η^{ω_{t}}; likewise, η^{^{o}Ω} = ^{o}ε^{Ω}, where ε^{Ω} denotes the counit for the adjoint pair Ω. Thus, we get :

oΥ(^{o}1_{Λ_{t}^{∗}}) = (ω_{t∗} ∗ Top(T/ρ^{t})^{∗} ∗ ε^{Ω}) ⊙ (η^{ω_{t}} ∗ Σ(ℱ ib(S_{∗})/ρ^{to})^{∗} ∗ Ω_{∗}) = (Φ_{t} ∗ ε^{Ω}) ⊙ (η^{ω_{t}} ∗ Φ_{t}).

But by inspecting the proof of proposition 4.6.10 we see that ε^{Ω} is the natural transformation that assigns to every object E_{•} : I → ℱ ib(T^{∗}) of Top(T) the system of morphisms (ε^{ω_{i}} : ω_{i}^{∗} ∘ ω_{i∗}E_{i} → E_{i} | i ∈ Ob(I)). Thus, finally, the natural transformation (Φ_{t} ∗ ε^{Ω}) ⊙ (η^{ω_{t}} ∗ Φ_{t}) attaches to every such E_{•} the morphism (ω_{t∗} ∗ ε^{ω_{t}}) ⊙ (η^{ω_{t}} ∗ ω_{t∗}), which is indeed 1_{ω_{t∗}E_{t}}, by the triangular identities of (1.1.13). □

5.7.21. Keep the situation of (5.7.19), and to ease notation set

v_{t} := Top(S/ρ^{t})^{∗}  v′_{t} := Top(T/ρ^{t})^{∗}  S := Can ∘ Top : PsFun(I, Topos) → Site.

Arguing as in (5.5.23), we deduce an essentially commutative diagram :

[figure: Stack(Can T_{t}); St(ω_{t}^{∗})_{∗}; St(v′_{t})_{∗}; St(1_{Ψ^{∗}_{t}})_{∗}^{γ}; Stack(S(T)); St(Top(ω)^{∗})_{∗}; Stack(Can S_{t}); St(v_{t})_{∗}; Stack(S(S))]

which we regard as an oriented square in Link(V-\overline{2-Cat}) (see (5.5.25)); then we may state :

Proposition 5.7.22. Υ(St(1_{Ψ_{t}^{∗}})_{∗}^{γ}) is a pseudo-natural equivalence.

Proof. Let ℰ be any stack on the canonical site of Top(T); we need to check that Υ(St(1_{Ψ_{t}^{∗}})_{∗}^{γ})_{ℰ} is an equivalence of categories. To this aim, in view of claim 3.2.8 we may replace ℰ by the split fibration C(ℰ), so that ℰ = ℱ ib(𝒜_{•}) for a presheaf of categories 𝒜_{•} on Top(T); then by lemma 5.5.15(ii) we may as well replace 𝒜_{•} by 𝒜_{•}^{a}, and assume that 𝒜_{•} is a sheaf of categories on Top(T).

Claim 5.7.23. Υ((1_{Ψ_{t}^{∗}}^{∼}, Cat)_{∗}) is a pseudo-natural equivalence.

Proof of the claim. As explained in (5.5.29), it suffices to check that Υ((1_{Ψ_{t}^{∗}})_{∗}^{∼}) is an isomorphism of functors. By (5.7.18), the latter holds if and only if Υ(1_{Ψ_{t}^{∗}}^{†}) is an isomorphism of functors. This in turn is clear from lemma 5.7.20. ♢

In light of claim 5.7.23 and corollary 5.5.26(iv), it suffices to check that ℱ ib((ṽ′_{t}, Cat)^{∗}𝒜_{•}) is a stack on the canonical site of T_{t}. For every site C, let

ω_{C} : Cat^{∗}(C^{∼}) →^{∼} (C, Cat)^{∼}

be the strict and strong 2-equivalence of (5.5.9). We may assume that 𝒜_{•} = ω_{Top(T)}(𝒜^{∗}) for a category object 𝒜^{∗} of Can(Top(T))^{∼} and then we need to check that ℱ ib(ω_{T_{t}}∘Cat^{∗}(ṽ_{t}^{′∗})(𝒜^{∗})) is a stack on Can(T_{t}). Since h_{Top(T)} : Top(T) → Can(Top(T))^{∼} is an equivalence, we may assume that 𝒜^{∗} = Cat^{∗}(h_{Top(T)})(ℬ^{∗}) for some ℬ^{∗} ∈ Ob(Cat^{∗}(Top(T))); by corollary 4.3.19(i.b), it then suffices to check that ℱ ib(ω_{T_{t}} ∘ Cat^{∗}(h_{T_{t}}) ∘ Cat^{∗}(v′_{t})(ℬ^{∗})) is a stack. However, theorem 4.6.24(i) yields an equivalence of categories a : Top(T) →^{∼} (Can(T), J)^{∼} and a morphism of sites b : Can(Top(T)) → (Can(T), J) with isomorphisms of functors :

h_{T_{t}} ∘ v′_{t} →^{∼} ĩ_{t∗} ∘ a a →^{∼} b̃_{∗} ∘ h_{Top(T)}

where i_{t} : T_{t} → Can(T) is the inclusion functor. There follow isomorphisms of fibrations :

ℱ ib(ω_{T_{t}} ∘ Cat^{∗}(h_{T_{t}}) ∘ Cat^{∗}(v′_{t})ℬ^{∗}) →^{∼} ℱ ib(ω_{T_{t}} ∘ Cat^{∗}(ĩ_{t∗}) ∘ Cat^{∗}(a)ℬ^{∗})
→^{∼} ℱ ib((ĩ_{t}, Cat)_{∗} ∘ ω_{(Can(T),J)} ∘ Cat^{∗}(a)ℬ^{∗})
→^{∼} Fib(i_{t})^{∗}(ℱ ib(ω_{(Can(T),J)} ∘ Cat^{∗}(a)ℬ^{∗}))

and by corollary 5.7.2, it then suffices to check that ℱ ib(ω_{(Can(T),J)} ∘ Cat^{∗}(a)ℬ^{∗}) is a stack on (Can(T), J). We are then further reduced to showing that ℱ ib((b̃, Cat)_{∗}𝒜_{•}) is a stack on (Can(T), J), or equivalently, that the same holds for Fib(b)^{∗}(ℱ ib(𝒜_{•})) = Fib(b)^{∗}(ℰ). The latter holds, by corollary 5.4.15. □

5.7.24. Consider now an oriented diagram of fibred topoi over a small category I :

[figure: T′; μ; ω′; Ξ; T; ω; S′; ν; S]

i.e. Ξ : ω ∘ μ ⇝ ν ∘ ω′ is a modification. For every t ∈ Ob(I) we get an oriented diagram :

[figure: S_{t}; ν_{t}^{∗}; ω_{t}^{∗}; S′_{t}; ω_{t}^{′∗}; Top(S); Top(ν)^{∗}; Top(S/ρ^{t})^{∗}; Top(Ξ)^{∗}; Top(ω)^{∗}; Top(S′); Top(ω′)^{∗}; Top(S′/ρ^{t})^{∗}; Top(T); Top(μ)^{∗}; Top(T/ρ^{t})^{∗}; Top(T′); Top(T′/ρ^{t})^{∗}; T_{t}; μ_{t}^{∗}; T′_{t}.]

whose four unmarked 2-cells are identities. We complete it by adding the orientation

Ξ_{t}^{†} := (Ξ_{t}, η^{(ω∘μ)_{t}}, η^{(ν∘ω′)_{t}})^{†} : (ν ∘ ω′)_{t}^{∗} ⇒ (ω ∘ μ)_{t}^{∗}

for the front face (notation of remark 1.1.17(ii)). Then a direct inspection shows that the resulting cubical diagram commutes both on 1-cells and 2-cells. As in (5.7.21), there follows an essentially commutative oriented diagram of 2-categories :

[figure: Stack(Can T′_{t}); St(μ_{t}^{∗})_{∗}; St(ω_{t}^{′∗})_{∗}; Stack(Can T_{t}); St(ω_{t}^{∗})_{∗}; Stack(S(T′)); St(Top(μ)^{∗})_{∗}; St(Top(ω′)^{∗})_{∗}; St(Top(Ξ)^{∗})_{∗}^{γ}; Stack(S(T)); St(Top(ω)^{∗})_{∗}; Stack(S(S′)); St(Top(ν)^{∗})_{∗}; Stack(S(S)); Stack(Can S′_{t}); St(ν_{t}^{∗})_{∗}; Stack(Can S_{t})]

whose four unmarked 2-cells are defined as in the diagram of (5.7.21), and which we complete to a cubical diagram that commutes on 2-cells, by adding the orientation for the front face

St(Ξ_{t}^{†})_{∗}^{γ} : St(ω_{t}^{∗})_{∗} ∘ St(μ_{t}^{∗})_{∗} ⇒ St(ν_{t}^{∗})_{∗} ∘ St(ω_{t}^{′∗})_{∗}.

Thus, let ℰ be any stack on the site Can(Top(T)), and set ℰ_{t} := St(Top(T/ρ^{t})^{∗})^{∗}ℰ; in light of proposition 5.7.22and remark 2.3.22(i), we conclude that :

St(Top(S′/ρ^{t})^{∗})^{∗} ∗ Υ(St(Top(Ξ)^{∗})_{∗}^{γ})_{ℰ} is an equivalence ⇔ Υ(St(Ξ_{t}^{†})_{∗}^{γ})_{ℰ_{t}} is an equivalence.

5.7.25. Next, consider an oriented diagram of fibred lex-sites over I :

𝒟 : [figure: \underline{𝒜}; φ′; ψ′; β; \underline{𝒜} ′; ψ; \underline{ℬ}; φ; \underline{ℬ} ′.]

For every t ∈ Ob(I) the restriction 𝒟_{t} of the diagram 𝒟 to the fibre categories over t yields another oriented square of 2-categories

St(𝒟_{t})_{∗} : [figure: Stack(𝒜_{t}); St(φ′_{t})_{∗}; St(ψ′_{t})_{∗}; St(β_{t})_{∗}^{γ}; Stack(𝒜′_{t}); St(ψ_{t})_{∗}; Stack(ℬ_{t}); St(φ_{t})_{∗}; Stack(ℬ′_{t}).]

For every fibred site \underline{𝒜} := (𝒜 , p, J_{•}) over I, let also i_{𝒜 ,t} : 𝒜_{t} → 𝒜 be the inclusion functor.

Corollary 5.7.26. In the situation of (5.7.25), let ℰ be any stack on the total site of \underline{𝒜} ′, and set ℰ_{t} := St(i_{𝒜′,t})_{∗}ℰ for every t ∈ Ob(I). The following conditions are equivalent :

(a) Υ(St(β)_{∗}^{γ})_{ℰ} is an equivalence.

(b) Υ(St(β_{t})_{∗}^{γ})_{ℰ_{t}} is an equivalence for every t ∈ Ob(I).

Proof. By invoking the pseudo-natural transformation b_{•} : S ∘ \underline{lex.T} → totSite of remark 4.6.29(ii), we get an oriented cubical diagram of fibred sites :

[figure: S(\underline{T} \underline{𝒜} ); S(\underline{T} φ′); S(\underline{T} ψ′); b\underline{𝒜}; S(\underline{T} \underline{𝒜} ′); S(\underline{T} ψ); b_{\underline{𝒜} ′}; τ_{1}; totSite(\underline{𝒜} ); φ′; ψ′; β; totSite(\underline{𝒜} ′); τ_{2}; ψ; totSite(\underline{ℬ} ); φ; totSite(\underline{ℬ} ′); τ_{3}; S(\underline{T} \underline{ℬ} ); S(\underline{T} φ); b\underline{ℬ}; τ_{4}; S(\underline{T} \underline{ℬ} ′); b_{\underline{ℬ} ′}]

whose front face is oriented by S(\underline{T}β) = Top(\underline{T}β)^{∗}, and where the orientations τ_{1}, …, τ_{4} are given by the coherence constraints of b_{•}; again, this cubical diagram commutes on 2-cells. After applying termwise as usual the pseudo-functor St(−)_{∗}, we deduce a similar cubical diagram whose front and back faces are oriented by St(Top(\underline{T}β)^{∗})_{∗}^{γ} and respectively St(β)_{∗}^{γ}, and whose other four faces are oriented by St(τ_{i})_{∗}^{γ} for i = 1, …, 4.

Claim 5.7.27. St(b\underline{𝒜} )_{∗} is a 2-equivalence for every fibred lex-site \underline{𝒜} over I.

Proof of the claim. By construction, b\underline{𝒜} is the composition of an isomorphism of canonical sites S(\underline{T} \underline{𝒜} ) →^{∼} Can ∘ T(totSite \underline{𝒜} ) with the counit ε_{totSite(\underline{𝒜} )} : Can ∘ T(totSite \underline{𝒜} ) → totSite(\underline{𝒜} ). Thus, it suffices to check that St(ε_{totSite(\underline{𝒜} )})_{∗} is a 2-equivalence. But the natural isomorphism T(totSite \underline{𝒜} ) →^{∼} totSite(\underline{𝒜} )^{∼} identifies ε_{totSite(\underline{𝒜} )} with the Yoneda morphism h_{totSite(\underline{𝒜} )}^{a} : totSite(\underline{𝒜} )^{∼} → totSite(\underline{𝒜} ), so the assertion follows from theorem 5.4.12. ♢

Claim 5.7.28. For every fibred lex-site \underline{𝒜} over I we have pseudo-natural equivalences :

St(ω_{𝒜_{t}}^{∗} )^{∗} ∘ St(Top(\underline{T}(\underline{𝒜} )/ρ^{t})^{∗})^{∗} ∘ St(b\underline{𝒜} )^{∗} →^{∼} St(h_{𝒜_{t}}^{a} )^{∗} ∘ St(i_{𝒜 ,t})_{∗} for every t ∈ Ob(I)

where ω_{𝒜_{t}} : 𝒜_{t}^{∼} →^{∼} T(𝒜_{t}) is the isomorphism of topoi as in (4.4.15).

Proof of the claim. Recall that St(h_{𝒜_{t}}^{a} )^{∗} and its 2-adjoint St(h_{𝒜_{t}}^{a} )_{∗} are 2-equivalences (theorem 5.4.12); then, from proposition 5.4.29(i,ii) we get pseudo-natural equivalences :

St(h_{𝒜_{t}}^{a} )^{∗} ∘ St(i_{𝒜 ,t})_{∗} →^{∼} St(h_{𝒜_{t}}^{a} )^{∗} ∘ St(ĭ_{𝒜 ,t})^{∗} →^{∼} St(ĭ_{𝒜 ,t}^{∗})^{∗} ∘ St(h_{totSite(\underline{𝒜} )}^{a})^{∗}.

We are thus reduced to checking the essential commutativity of the following diagram of sites :

\xymatrix{Can(𝒜_{t}^{∼}) \ar[r]^{ĭ_{\underline{𝒜} ,t}^{∗}} \ar[d]_{Top(\underline{T}(\underline{𝒜} )/ρ^{t})^{∗}∘ω_{𝒜_{t}}^{∗}} & Can(totSite \underline{𝒜} )^{∼} \ar[d]^{h_{totSite(\underline{𝒜} )}^{a}} \\ Can ∘ Top(\underline{T} \underline{𝒜} ) \ar[r]^{b\underline{𝒜}} & totSite(\underline{𝒜} ).}

The latter follows by direct inspection. ♢

Now, on the one hand, it is clear that condition (a) holds if and only if St(i_{ℬ,t} )_{∗} ∗ Υ(St(β)_{∗}^{γ})_{ℰ} is an equivalence for every t ∈ Ob(I). By claim 5.7.28, the latter holds if and only if :

(c) (St(Top(\underline{T}(\underline{ℬ} )/ρ^{t})^{∗})^{∗} ∘ St(b\underline{ℬ} )^{∗}) ∗ Υ(St(β)_{∗}^{γ})_{ℰ} is an equivalence for every t ∈ Ob(I). Set ℰ′ := St(b\underline{𝒜} ′)^{∗}(ℰ). By claim 5.7.27and remarks 2.3.11and 2.3.22(i), condition (c) in turn holds if and only if :

(d) St(Top(\underline{T}(\underline{ℬ} )/ρ^{t})^{∗})^{∗} ∗ Υ(St(Top(\underline{T}β)^{∗})_{∗}^{γ})_{ℰ′} is an equivalence for every t ∈ Ob(I). Then, the discussion of (5.7.24) shows that (d) holds if and only if :

(e) Υ(St((Tβ_{t})^{†})_{∗}^{γ})_{St(Top(\underline{T}(\underline{𝒜} ′)/ρ^{t})^{∗})^{∗}ℰ′} is an equivalence for every t ∈ Ob(I). By invoking again claim 5.7.28, we see that (e) in turn holds if and only if :

(f) Υ(St((β_{t}^{∼})^{†})_{∗}^{γ})_{St(h_{𝒜′_{t}}^{a} )^{∗}ℰ_{t}} is an equivalence for every t ∈ Ob(I).

However, from the counit ε : Can ∘ T ⇒ 1_{Site} of the 2-adjoint pair (Can, T) and the system of isomorphisms ω_{C} : C^{∼} →^{∼} T(C) of (4.4.15) we get as usual an oriented cubical diagram

[figure: Stack(Can 𝒜_{t}^{∼}); St(φ_{t}^{′∗})_{∗}; St(ψ̃_{t}^{′∗})_{∗}; St(h_{𝒜_{t}}^{a} )_{∗}; Stack(Can 𝒜_{t}^{′∼}); St(ψ̃_{t}^{∗})_{∗}; St(h_{𝒜′_{t}}^{a} )_{∗}; Stack(𝒜_{t}); St(φ′_{t})_{∗}; St(ψ′_{t})_{∗}; St(β_{t})_{∗}^{γ}; Stack(𝒜′_{t}); St(ψ_{t})_{∗}; Stack(ℬ_{t}); St(φ_{t})_{∗}; Stack(ℬ′_{t}); Stack(Can ℬ_{t}^{∼}); St(φ_{t}^{∗})_{∗}; St(h_{ℬ_{t}}^{a} )_{∗}; Stack(Can ℬ_{t}^{′∼}).; St(h_{ℬ′_{t}}^{a} )_{∗}]

commuting on 2-cells, whose four unmarked orientations are pseudo-natural equivalences, and whose front face is oriented by St((β_{t}^{∼})^{†})_{∗}^{γ}. Combining with remarks 2.3.11 and 2.3.22(i) we finally conclude that (f)⇔(b). □

Example 5.7.29. Let u : C′ := (𝒞′, J′) → C := (𝒞 , J) be a morphism of lex-sites.

(i) Recall that the target functor t_{𝒞} : Morph(𝒞) → 𝒞 is a fibration whose fibre category t_{𝒞}^{−1}X is naturally identified with 𝒞 /X, for every X ∈ Ob(𝒞) (example 3.1.3(iii)). Fix a unital cleavage for this fibration, and let c be the associated pseudo-functor; it is easily seen that for every morphism φ : X → Y in 𝒞, the functor c_{φ} : 𝒞 /Y → 𝒞 /X is right adjoint to φ_{∗} : 𝒞 /X → 𝒞 /Y, hence it is left exact : explicitly, c_{φ} assigns to every object Z ⟶^{h} Y of 𝒞 /Y the induced projection (X ×_{Y} Z → X) ∈ Ob(𝒞 /X), where X ×_{Y} Z is a choice of representative for the fibre product of X and Z over Y (detail left to the reader). This description implies easily that c_{φ} is a morphism of lex-sites C/X → C/Y, for every such φ (notation of (4.7)). The collection of sites (C/X | X ∈ Ob(𝒞)) then endows the fibration t_{𝒞} with a well defined structure of fibred lex-site (Morph(𝒞), t_{𝒞} , J_{•}^{𝒞} ).

(ii) The functor u : 𝒞 → 𝒞′ induces a commutative diagram of categories :

\xymatrix{Morph(𝒞) \ar[r]^{Morph(u)} \ar[d]_{t_{𝒞}} & Morph(𝒞′) \ar[d]^{t_{𝒞′}} \\ 𝒞 \ar[r]^{u} & 𝒞′.}

Let now ℬ be any category, and F : ℬ → 𝒞 a given functor; we set

𝒜 := Fib(F)^{∗}(Morph(𝒞)) 𝒜′ := Fib(F ∘ u)^{∗}(Morph(𝒞′))

and let

Morph(𝒞) ⟵^{π} 𝒜 ⟶^{φ} ℬ ⟵^{φ′} 𝒜′ ⟶^{π′} Morph(𝒞′)

be the natural projections. There is a unique functor

g : 𝒜 → 𝒜′ such that φ′ ∘ g = φ and π′ ∘ g = Morph(u) ∘ π

and since u is left exact, it is easily seen that g is cartesian (details left to the reader). Moreover, the restriction φ^{−1}B → φ^{′−1}B of g is naturally identified with u_{|F B} : 𝒞 /F B → 𝒞′/uF B, which is a morphism of lex-sites C′/uF B → C/F B. Thus, g is a morphism of fibred lex-sites

ℬ ×_{𝒞′} (Morph(𝒞′), t_{𝒞′}, J_{•}^{𝒞′}) → ℬ ×_{𝒞} (Morph(𝒞), t_{𝒞} , J_{•}^{𝒞} )

and therefore, g is also a morphism of the induced total sites (proposition 4.5.9)

g : (𝒜′, J_{𝒜′} ) → (𝒜 , J_{𝒜} ).

(iii) Suppose moreover that ℬ is cofiltered, in which case the pseudo-functor c^{o} factors through a pseudo-functor c̃ : ℬ → lex.Site as in (5.7.6), whose strong 2-limit is represented by the localization 𝒜 [Σ_{𝒜}^{−1}], endowed with a certain topology J_{𝒜}^{∗} , where Σ_{𝒜} denotes the set of cartesian morphisms of 𝒜. Likewise, the corresponding localization 𝒜′[Σ_{𝒜′}^{−1}] of 𝒜′ carries a natural topology J_{𝒜′}^{∗} , and for every B ∈ Ob(ℬ) the composition of the inclusion functor i_{F B} : 𝒞 /F B → 𝒜 (resp. i_{uF B} : 𝒞′/uF B → 𝒜′) with the localization functor L_{𝒜} : 𝒜 → 𝒜 [Σ_{𝒜}^{−1}] (resp. L_{𝒜′} : 𝒜′ → 𝒜′[Σ_{𝒜′}^{−1}]) is a morphism of sites

l_{F B} : (𝒜 [Σ_{𝒜}^{−1}], J_{𝒜}^{∗} ) → C/F B (resp. l_{uF B} : (𝒜′[Σ_{𝒜′}^{−1}], J_{𝒜′}^{∗} ) → C′/uF B ) .

Hence, for every B ∈ Ob(ℬ) we get an oriented square of sites :

[figure: (𝒜′[Σ_{𝒜′}^{−1}], J_{𝒜′}^{∗} ); l_{uF B}; g_{Σ}; C′/uF B; u_{|F B}; (𝒜 [Σ_{𝒜}^{−1}], J_{𝒜}^{∗} ); l_{F B}; C/F B]

whose orientation is the identity 1_{l_{F B}∘g_{Σ}}, and where g_{Σ} is the localization of g (see (5.7.10)). With this notation, we have the following :

Corollary 5.7.30. In the situation of example 5.7.29(iii), suppose furthermore that ℬ admits a final object B_{0}, and that every object of 𝒞′ is quasi-compact for the topology J′. Then the base change transformation

Υ(St(1_{l_{F B_{0}}∘g_{Σ}})_{∗}^{γ}) : St(l_{F B_{0}})^{∗} ∘ St(u_{|F B_{0}})_{∗} → St(g_{Σ})_{∗} ∘ St(l_{uF B_{0}})^{∗}

is a pseudo-natural equivalence.

Proof. We regard C/F B_{0} as a fibred lex-site over the category 𝟙 with one object and one morphism; then we may define the fibred lex-site ℬ ×_{𝟙} C/F B_{0} over ℬ, as in (4.5.11) : its underlying category is ℬ × 𝒞 /F B_{0}, and its fibre categories are all naturally identified with 𝒞 /F B_{0}, and endowed with the topology of C/F B_{0}. Likewise we define the fibred lex-site ℬ ×_{𝟙} C′/uF B_{0}, and we endow as well the underlying categories with their respective total site topologies. According to example 4.5.8, the inclusion functors i_{F B_{0}} : 𝒞 /F B_{0} → ℬ × 𝒞 /F B_{0} and i_{uF B_{0}} : 𝒞′/uF B_{0} → ℬ × 𝒞′/uF B_{0} are then morphisms of sites :

i_{F B_{0}} : ℬ ×_{𝟙} C/F B_{0} → C/F B_{0} i_{uF B_{0}} : ℬ ×_{𝟙} C′/uF B_{0} → C′/uF B_{0}.

For every B ∈ Ob(ℬ), let t_{B} : B → B_{0} be the unique morphism in ℬ; we have a natural isomorphism of categories

(𝒞 /F B_{0})/F t_{B} →^{∼} 𝒞 /F B

that identifies the source functor s_{F t_{B}} : (𝒞 /F B_{0})/F t_{B} → 𝒞 /F B_{0} with the functor (F t_{B})_{∗} : 𝒞 /F B → 𝒞 /F B_{0}, and likewise for the category (𝒞′/uF B_{0})/uF t_{B}. According to (5.5.30), for every such B we have then an oriented square of sites :

[figure: C′/uF B; p_{uF t_{B}}; u_{|F B}; β_{B}; C′/uF B_{0}; u_{|F B_{0}}; C/F B; p_{F t_{b}}; C/F B_{0}.]

It is easily seen that the system of orientations (β_{B} | B ∈ Ob(ℬ)) amounts to an orientation β for the central square subdiagram in the following diagram of oriented squares of sites :

[figure: (𝒜′[Σ_{𝒜′}^{−1}], J_{𝒜′}^{∗} ); L_{𝒜′}; g_{Σ}; (𝒜′, J_{𝒜′} ); g; p_{𝒜′}; β; ℬ ×_{𝟙} C′/uF B_{0}; ℬ×u_{|F B_{0}}; i_{uF B_{0}}; C′/uF B_{0}; u_{|F B_{0}}; (𝒜 [Σ_{𝒜}^{−1}], J_{𝒜}^{∗} ); L_{𝒜}; (𝒜 , J_{𝒜} ); p_{𝒜}; ℬ ×_{𝟙} C/F B_{0}; i_{F B_{0}}; C/F B_{0}.]

Here p_{𝒜} is the cartesian functor whose restriction to fibre categories 𝒞 /F B_{0} → 𝒞 /F B agrees with p_{F t_{B}}, for every B ∈ Ob(ℬ), and the orientations of the left and right square are identities. Under our assumptions, it is easily seen that every object of 𝒜 is quasi-compact for the topology J_{𝒜′} ; then the base change transformation associated with the left square subdiagram is a pseudo-natural equivalence, by virtue of proposition 5.7.11. The same holds for the base change transformation associated with the central square, by proposition 5.5.31 and corollary 5.7.26. A simple inspection shows that the composition of the three top (resp. bottom) horizontal arrows equals l_{uF B_{0}} (resp. l_{F B_{0}}). It remains therefore only to check that the base change transformation

St(i_{F B_{0}})^{∗}St(u_{|F B_{0}})_{∗} → St(ℬ × u_{|F B_{0}})_{∗}St(i_{uF B_{0}})^{∗}

associated with the right square is a pseudo-natural equivalence. To this aim, we remark :

Claim 5.7.31. Let ℱ_{•} be any sheaf of categories on C′/uF B_{0} such that ℱ ib(ℱ_{•}) is a stack on C′/uF B_{0}. Then ℱ ib((ĩ_{uF B_{0}}^{∗}, Cat)^{∗}ℱ_{•}) is a stack on ℬ ×_{𝟙} C′/uF B_{0}.

Proof of the claim. notice first that, since B_{0} is a final object in ℬ, the projection q_{uF B_{0}} : ℬ × 𝒞′/uF B_{0} → 𝒞′/uF B_{0} is a left adjoint for the functor i_{uF B_{0}}. On the other hand, proposition 4.5.5(iv) easily implies that q_{uF B_{0}} is continuous for the sites C′/uF B_{0} and ℬ ×_{𝟙} C′/uF B_{0}. Combining with lemma 4.2.14(ii), we deduce a natural isomorphism of functors :

(5.7.32)  ĩ_{uF B_{0}}^{∗} →^{∼} q̃_{uF B_{0}∗}.

Thus, it suffices to check that ℱ ib((q̃_{uF B_{0}∗}, Cat)_{∗}ℱ_{•}) is a stack. The latter follows easily from corollary 5.7.2(ii). ♢

In light of claim 5.7.31, corollary 5.5.26(iv), and the discussion of (5.5.29), we are then reduced to checking that the base change transformation

ĩ_{F B_{0}}^{∗} ∘ (u_{|F B_{0}})_{∗}^{∼} → (ℬ × u_{|F B_{0}})_{∗}^{∼} ∘ ĩ_{uF B_{0}}^{∗}

is an isomorphism of functors. But notice that i_{uF B_{0}} is fully faithful, and it is both continuous and cocontinuous (proposition 4.5.5(iii)); it follows that ĩ_{uF B_{0}}^{∗} is fully faithful (lemma 4.2.15(iv)), hence the unit of the adjoint pair (ĩ_{uF B_{0}}^{∗}, ĩ_{uF B_{0}∗}) is an isomorphism. Thus, let η and ε be the unit and counit for the adjoint pair (ĩ_{F B_{0}}^{∗}, ĩ_{F B_{0}∗}); we are reduced to checking that ε ∗ ((ℬ × u_{|F B_{0}})_{∗}^{∼} ∘ ĩ_{uF B_{0}}^{∗}) is an isomorphism. However, let q_{F B_{0}} : ℬ × 𝒞 /F B_{0} → 𝒞 /F B_{0} be the projection; arguing as in the proof of claim 5.7.31we get an isomorphism of functors

(5.7.33)  ĩ_{F B_{0}}^{∗} →^{∼} q̃_{F B_{0}∗}.

From (5.7.32) and (5.7.33) there follow isomorphisms of functors :

(ℬ × u_{|F B_{0}})_{∗}^{∼} ∘ ĩ_{uF B_{0}}^{∗} →^{∼} (ℬ × u_{|F B_{0}})_{∗}^{∼} ∘ q̃_{uF B_{0}∗} →^{∼} q̃_{F B_{0}∗} ∘ (u_{|F B_{0}})_{∗}^{∼} →^{∼} ĩ_{F B_{0}}^{∗} ∘ (u_{|F B_{0}})_{∗}^{∼}.

So it suffices to check that ε ∗ (ĩ_{F B_{0}}^{∗} ∘ (u_{|F B_{0}})_{∗}^{∼}) is an isomorphism; but again, arguing as in the foregoing we see that ĩ_{F B_{0}}^{∗} is fully faithful, so η is an isomorphism, and finally the assertion follows, taking into account the triangular identities of (1.1.13). □

Remark 5.7.34. In the situation of example 5.7.29(ii), the functors φ : 𝒜 → ℬ and s_{𝒞} ∘ π : 𝒜 → 𝒞 induce a functor

s : 𝒜 → ℬ × 𝒞 (B, f : X → Y) ↦ (B, X)

(namely, s is the unique functor whose composition with the projections ℬ ← ℬ × 𝒞 → 𝒞 equals respectively φ and s_{𝒞} ∘ π). The functor s admits a right adjoint :

q : ℬ × 𝒞 → 𝒜

that assigns to every (B, X) ∈ Ob(ℬ × 𝒞) the object (B, p_{X,B} : X × F B → F B) ∈ Ob(𝒜), where p_{X,B} is the natural projection. To every morphism (β, f) : (B, X) → (B′, X′) of ℬ ×𝒞, the functor q assigns the morphism q(β, f) := (β, 𝒟) of 𝒜, with 𝒟 the commutative square :

\xymatrix{X × F B \ar[r]^{p_{X,B}} \ar[d]_{f×F β} & F B \ar[d]^{F β} \\ X′ × F B′ \ar[r]^{p_{X′,B′}} & F B′.}

Let us regard (𝒞 , J) as a fibred site over the category 𝟙 with one object and one morphism; then ℬ ×_{𝟙} (𝒞 , J) is a fibred category over ℬ, and taking into account remark 4.7.3(iii), it is easily seen that q is a morphism of fibred sites

q : ℬ ×_{𝒞} (Morph(𝒞), t_{𝒞} , J_{•}^{𝒞} ) → ℬ ×_{𝟙} (𝒞 , J)

and therefore it is as well a morphism of the respective total sites

q : (𝒜 , J_{𝒜} ) → (ℬ × 𝒞 , J_{ℬ×𝒞} ).

It follows that s is cocontinuous for the topologies J_{𝒜} and J_{ℬ×𝒞} (lemma 4.2.14(i)); moreover, we easily deduce from example 5.4.16that s is a weak morphism of fibred sites

s : ℬ ×_{𝟙} (𝒞 , J) → ℬ ×_{𝒞} (Morph(𝒞), t_{𝒞} , J_{•}^{𝒞} )

(see definition 5.7.4); therefore it is as well a weak morphism of the respective total sites

s : (ℬ × 𝒞 , J_{ℬ×𝒞} ) → (𝒜 , J_{𝒜} )

(proposition 5.7.5). Combining with proposition 5.4.29(ii,iii), we deduce a pseudo-natural equivalence of pseudo-functors :

St(s)_{∗} →^{∼} St(q)^{∗}.

6. Monoids and polyhedra

Unless explicitly stated otherwise, every monoid encountered in this chapter shall be commutative. For this reason, we shall usually economize adjectives, and write just “monoid” when referring to commutative monoids.

6.1. Monoids. If M is any monoid, we shall usually denote the composition law of M by multiplicative notation: (x, y) ↦ x · y (so 1 is the neutral element). However, sometimes it is convenient to be able to switch to an additive notation; to allow for that, we shall denote by (log M, +) the monoid with additive composition law, whose underlying set is the same as for the given monoid (M, ·), and such that the identity map is an isomorphism of monoids (then, the neutral element of log M is denoted by 0). For emphasis, we may sometimes denote by log : M →^{∼} log M the identity map, so that one has the tautological identities :

log 1 = 0  and  log(x · y) = log x + log y for every x, y ∈ M.

Conversely, if (M, +) is a given monoid with additive composition law, we may switch to a multiplicative notation by writing (exp M, ·), in the same way.

6.1.1. For any monoid M, and any two subsets S, S′ ⊂ M, we let :

S · S′ := {s · s′ | s ∈ S, s′ ∈ S′}

and S^{a} is defined recursively for every a ∈ ℕ, by the rule :

S^{0} := {1} and S^{a} := S · S^{a−1} if a > 0.

Notice that the pair (𝒫(M), ·) consisting of the set of all subsets of M, together with the composition law just defined, is itself a monoid : the neutral element is the subset {1}. In the same vein, the exponential notation for subsets of M becomes a multiplicative notation in the monoid (log 𝒫(M), +) = (𝒫(log M), +), i.e. we have the tautological identity : log S^{a} = a · log S, for every S ∈ 𝒫(M) and every a ∈ ℕ.

Furthermore, for any two monoids M and N, the set Hom_{Mnd}(M, N) is naturally a monoid. The composition law assigns to any two morphisms φ, ψ : M → N their product φ · ψ, given by the rule : φ · ψ(m) := φ(m) · ψ(m) for every m ∈ M.

Basic examples of monoids are the set (ℕ, +) of natural numbers, and the non-negative real (resp. rational) numbers (ℝ_{+}, +) (resp. (ℚ_{+}, +)), with their standard addition laws.

6.1.2. Given a surjection X → Y of monoids, it may not be possible to express Y as a quotient of X – a problem relevant to the construction of presentations for given monoids, in terms of free monoids. For instance, consider the monoid (ℤ, ⊙) consisting of the set ℤ with the composition law ⊙ such that :

x ⊙ y := \{ \begin{matrix}x + y & if either x, y ≥ 0 or x, y ≤ 0 \\ max(x, y) & otherwise \end{matrix}

for every x, y ∈ ℤ. Define a surjective map φ : ℕ^{⊕2} → (ℤ, ⊙) by the rule (n, m) ↦ n ⊙ −m, for every n ∈ ℕ. Then one verifies easily that Ker φ = {0}, and nevertheless φ is not an isomorphism. The right way to proceed is indicated by the following :

Lemma 6.1.3. Every surjective map of monoids is an effective epimorphism (in the category Mnd).

Proof. (See example 4.1.9(v) for the notion of effective epimorphism.) Let π : M → N be a surjection of monoids. For every monoid X, we have a natural diagram of sets :

\xymatrix{Hom_{Mnd}(N, X) \ar[r]^{j} & Hom_{Mnd}(M, X) \ar[r]_{p_{2}^{∗}} \ar[r]^{p_{1}^{∗}} & Hom_{Mnd}(M ×_{N} M, X)}

where p_{1}, p_{2} : M ×_{N} M → M are the two natural projections, and we have to show that the map j identifies Hom_{Mnd}(N, X) with the equalizer of p_{1}^{∗} and p_{2}^{∗}. First of all, the surjectivity of π easily implies that j is injective. Hence, let φ : M → X be any map such that φ∘p_{1} = φ∘p_{2}; we have to show that φ factors through π. To this aim, it suffices to show that the map of sets underlying φ factors as a composition φ′ ∘ π, for some map of sets φ′ : N → X, since φ′ will then be necessarily a morphism of monoids. However, the forgetful functor F : Mnd → Set commutes with fibre products (lemma 4.8.29(iii)), and F (π) is an effective epimorphism, since in the category Set all surjections are effective epimorphisms. The assertion follows. □

6.1.4. Lemma 6.1.3 allows to construct presentations for an arbitrary monoid M, as follows. First, we choose a surjective map of monoids F := ℕ^{(S)} → M, for some set S. Then we choose another set T and a surjection of monoids ℕ^{(T)} → F ×_{M} F. Composing with the natural projections p_{1}, p_{2} : F ×_{M} F → F, we obtain a diagram :

(6.1.5)  \xymatrix{ℕ^{(T)} \ar[r]^{q_{1}} \ar[r]_{q_{2}} & ℕ^{(S)} \ar[r] & M}

which, in view of lemma 6.1.3, identifies M to the coequalizer of q_{1} and q_{2}.

Definition 6.1.6. Let M be a monoid, Σ ⊂ M a subset.

(i) Let (e_{σ} | σ ∈ Σ) be the natural basis of the free monoid ℕ^{(Σ)}. We say that Σ is a system of generators for M, if the map of monoids ℕ^{(Σ)} → M such that e_{σ} ↦ σ for every σ ∈ Σ, is a surjection.

(ii) M is said to be finitely generated if it admits a finite system of generators.

(iii) M is said to be fine if it is integral and finitely generated.

(iv) A finite presentation for M is a diagram such as (6.1.5) that identifies M to the co-

equalizer of q_{1} and q_{2}, and such that, moreover, S and T are finite sets.

(v) We say that a morphism of monoids φ : M → N is finite, if N is a finitely generated M-module, for the M-module structure induced by φ.

Lemma 6.1.7. (i) Every finitely generated monoid admits a finite presentation.

(ii) Let M be a finitely generated monoid, and (N_{i} | i ∈ I) a filtered family of monoids. Then the natural map :

colim_{i∈I} Hom_{Mnd}(M, N_{i}) → Hom_{Mnd}(M, colim_{i∈I} N_{i})

is a bijection.

Proof. (i): Let M be a finitely generated monoid, and choose a surjection π : ℕ^{(S)} → M with S a finite set. We have seen that M is the coequalizer of the two projections p_{1}, p_{2} : P := ℕ^{(S)}×_{M} ℕ^{(S)} → ℕ^{(S)}. For every finitely generated submonoid N ⊂ P, let p_{1,N}, p_{2,N} : N → ℕ^{(S)} be the restrictions of p_{1} and p_{2}, and denote by C_{N} the coequalizer of p_{1,N} and p_{2,N}. By the universal property of C_{N}, the map π factors through a map of monoids π_{N} : C_{N} → M, and since π is surjective, the same holds for π_{N}. It remains to show that π_{N} is an isomorphism, for N large enough. We apply the functor M ↦ ℤ[M] of (4.8.50), and we derive that ℤ[M] is the coequalizer of the two maps ℤ[p_{1}], ℤ[p_{2}] : ℤ[P] → ℤ[S], i.e. ℤ[M] ≃ ℤ[S]/I, where I is the ideal generated by Im(ℤ[p_{1}] − ℤ[p_{2}]). Clearly I is the colimit of the filtered system of analogous ideals I_{N} generated by Im(ℤ[p_{1,N}] − ℤ[p_{2,N}]), for N ranging over the filtered family ℱ of finitely generated submonoids of P. By noetherianity, there exists N ∈ ℱ such that I = I_{N}, therefore ℤ[M] is the coequalizer of ℤ[p_{1,N}] and ℤ[p_{2,N}]. But the latter coequalizer is also the same as ℤ[C_{N}], whence the contention.

(ii): This is a standard consequence of (i). Indeed, say that f_{1}, f_{2} : M → N_{i} are two morphisms whose compositions with the natural map N_{i} → N := colim_{i∈I} N_{i} agree, and pick a finite set of generators x_{1}, …, x_{n} for M. For any morphism φ : i → j in the filtered category I, denote by g_{φ} : N_{i} → N_{j} the corresponding morphism; then we may find such a morphism φ, so that g_{φ} ∘ f_{1}(x_{k}) = g_{φ} ∘ f_{2}(x_{k}) for every k ≤ n, whence the injectivity of the map in (ii). Next, let f : M → N be a given morphism, and pick a finite presentation (6.1.5); we deduce a morphism g : ℕ^{(S)} → N, and since S is finite, it is clear that g factors through a morphism g_{i} : ℕ^{(S)} → N_{i} for some i ∈ I. Set g′_{i} := g_{i} ∘ q_{1} and g′′_{i} := g_{i} ∘ q_{2}; by assumption, after composing g′_{i} and of g′′_{i} with the natural map N_{i} → N, we obtain the same map, so by the foregoing there exists a morphism φ : i → j in I such that g_{φ} ∘ g′_{i} = g_{φ} ∘ g′′_{i}. It follows that g_{φ} ∘ g_{i} factors through M, whence the surjectivity of the map in (ii). □

Definition 6.1.8. Let M be a monoid, I ⊂ M an ideal.

(i) We say that I is principal, if it is cyclic, when regarded as an M-module.

(ii) The radical of I is the ideal rad(I) consisting of all x ∈ M such that x^{n} ∈ I for every

sufficiently large n ∈ ℕ. If I = rad(I), we also say that I is a radical ideal. (iii) A face of M is a submonoid F ⊂ M with the following property. If x, y ∈ M are any

two elements, and xy ∈ F, then x, y ∈ F. (iv) Notice that the complement of a face is always an ideal. We say that I is a prime ideal

of M, if M \I is a face of M.

Proposition 6.1.9. Let M be a finitely generated monoid, and S a finitely generated M-module. Then we have :

(i) Every submodule of S is finitely generated. (ii) Especially, every ideal of M is finitely generated.

Proof. Of course, (ii) is a special case of (i). To show (i), let S′ ⊂ S be an M-submodule, Σ ⊂ S′ any system of generators. Let 𝒫′(Σ) be the set of all finite subsets of Σ, and for every A ∈ 𝒫′(Σ), denote by S′_{A} ⊂ S′ the submodule generated by A; clearly S′ is the filtered union of the family (S′_{A} | A ∈ 𝒫′(Σ)), hence ℤ[S′] is the filtered union of the family of ℤ[M]-submodules (ℤ[S′] | S ∈ 𝒫′(Σ)). Since ℤ[M] is noetherian and ℤ[S] is a finitely generated ℤ[M]-module, it follows that ℤ[S′_{A}] = ℤ[S′] for some finite subset A ⊂ Σ, whence the contention. □

Definition 6.1.10. (i) Let M be a monoid. The spectrum of M is the set :

Spec M

consisting of all prime ideals of M. It has a natural partial ordering, given by inclusion of prime ideals; the minimal element of Spec M is the empty ideal ∅ ⊂ M, and the maximal element is:

𝔪_{M} := M \M^{×}.

(ii) The dimension of M, denoted dim M ∈ ℕ ∪ {+∞}, is defined as the supremum of all r ∈ ℕ such that there exists a chain of strict inclusions of prime ideals of M :

𝔭_{0} ⊂ 𝔭_{1} ⊂ ⋯ ⊂ 𝔭_{r}.

(iii) The height of a prime ideal 𝔭 ∈ Spec M is defined as ht 𝔭 := dim M_{𝔭}.

(iv) A facet of M is the complement of a prime ideal of M of height one.

(v) Any morphism φ : M → N of monoids induces a natural map :

φ^{∗} : Spec N → Spec M 𝔭 ↦ φ^{−1}𝔭

of partially ordered sets. We say that φ is local, if φ(𝔪_{M}) ⊂ 𝔪_{N}.

Remark 6.1.11. Let M be any monoid.

(i) It is clear from definition 6.1.10(i) that dim M = 0 if and only if M = M^{×}.

(ii) Moreover, if (I_{λ} | λ ∈ Λ) is any collection of ideals of M then it is easily seen that both ⋃_{λ∈Λ} I_{λ} and ⋂_{λ∈Λ} I_{λ} are ideals of M. If (𝔭_{λ} | λ ∈ Λ) is any family of prime ideals of M, then ⋃_{λ∈Λ} 𝔭_{λ} is a prime ideal of M.

Corollary 6.1.12. Let M be any fine and sharp monoid. The set 𝔪_{M} \𝔪_{M}^{2} is finite, and is the unique minimal system of generators of M.

Proof. It is easily seen that any system of generators of M must contain Σ := 𝔪_{M} \𝔪_{M}^{2}, hence the latter must be a finite set. On the other hand, suppose that there exists an element x_{0} ∈ M which is not contained in the submonoid M′ generated by Σ. Then we may write x_{0} = x_{1}y_{1} for some x_{1}, y_{1} ∈ 𝔪_{M}, with x_{1} ∉ M′, so x_{1} admits a similar decomposition. Proceeding in this way, we obtain a sequence of elements (x_{n} | n ∈ ℕ) with the property that Mx_{n} ⊂ Mx_{n+1} for every n ∈ ℕ. We claim that Mx_{n} ≠ Mx_{n+1} for every n ∈ ℕ. Indeed, if the inequality fails for some n ∈ ℕ, we may write x_{n+1} = ax_{n} for some a ∈ M, and on the other hand, we have by construction x_{n} = yx_{n+1} for some y ∈ 𝔪_{M}; summing up, we get x_{n} = yax_{n}, whence ya = 1, since M is integral, therefore y ∈ M^{×}, a contradiction.

Thus, from the given x_{0}, we have produced an infinite strictly ascending chain of ideals of M, which is ruled out by virtue of proposition 6.1.9(ii). This means that x_{0} cannot exist, and the corollary follows. □

Lemma 6.1.13. Let f_{1} : M → N_{1} and f_{2} : M → N_{2} be two local morphisms of monoids. If N_{1} and N_{2} are sharp, then N_{1} ⨿_{M} N_{2} is sharp.

Proof. Let (a, b) ∈ N_{1} × N_{2}, and suppose there exist c ∈ M, a′ ∈ N_{1}, b′ ∈ N_{2} such that (a, b) = (a′f_{1}(c), b′) and (1, 1) = (a′, f_{2}(c)b′); since N_{2} is sharp, we deduce f_{2}(c) = b′ = 1, so b = 1. Then, since f_{2} is local, we get c ∈ M^{×}, hence f_{1}(c) = 1 and a = a′ = 1. One argues symmetrically in case (1, 1) = (a′f_{1}(c), b′) and (a, b) = (a′, f_{2}(c)b′). We conclude that (a, b) represents the unit class in N_{1} ⨿_{M} N_{2} if and only if a = b = 1. Now, suppose that the class of (a, b) is invertible in N_{1} ⨿_{M} N_{2}; it follows that there exists (c, d) such that ac = 1 and bd = 1, which implies that a = 1 and b = 1, whence the contention. □

Lemma 6.1.14. Let S ⊂ M be any submonoid. The localization j : M → S^{−1}M induces an injective map j^{∗} : Spec S^{−1}M → Spec M which identifies Spec S^{−1}M with the subset of Spec M consisting of all prime ideals 𝔭 such that 𝔭 ∩ S = ∅.

Proof. For every 𝔭 ∈ Spec M, denote by S^{−1}𝔭 the ideal of S^{−1}M generated by the image of 𝔭. We claim that 𝔭 = j^{∗}(S^{−1}𝔭) for every 𝔭 ∈ Spec M such that 𝔭 ∩ S = ∅. Indeed, clearly 𝔭 ⊂ j^{∗}(S^{−1}𝔭); next, if f ∈ j^{∗}(S^{−1}𝔭), there exist s ∈ S and g ∈ 𝔭 such that s^{−1}g = f in S^{−1}M; therefore there exists t ∈ S such that tg = tsf in M, especially tsf ∈ 𝔭, hence f ∈ 𝔭, since t, s ∉ 𝔭. Likewise, one checks easily that S^{−1}𝔭 is a prime ideal if 𝔭 ∩ S = ∅, and 𝔮 = S^{−1}(j^{∗}𝔮) for every 𝔮 ∈ Spec S^{−1}M, whence the contention. □

Remark 6.1.15. (i) If we take S_{𝔭} := M\𝔭, the complement of a prime ideal 𝔭 of M, we obtain the monoid

M_{𝔭} := S_{𝔭}^{−1}M

and Spec M_{𝔭} ⊂ Spec M is the subset consisting of all prime ideals 𝔮 of M contained in 𝔭.

(ii) Likewise, if 𝔭 ⊂ M is any prime ideal, the spectrum Spec(M \𝔭) is naturally identified with the subset of Spec M consisting of all prime ideals 𝔮 containing 𝔭 (details left to the reader).

(iii) Let S ⊂ M be any submonoid. Then there exists a smallest face F of M containing S (namely, the intersection of all the faces that contain S). It is easily seen that F is the subset of all x ∈ M such that xM ∩ S ≠ ∅. From this characterization, it is clear that S^{−1}M = F^{−1}M. In other words, every localization of M is of the type M_{𝔭} for some 𝔭 ∈ Spec M.

Lemma 6.1.16. Let M be a monoid, and I ⊂ M any ideal. Then rad(I) is the intersection of all the prime ideals of M containing I.

Proof. It is easily seen that a prime ideal containing I also contains rad(I). Conversely, say that f ∈ M \ rad(I); let φ : M → M_{f} be the localization map. Denote by 𝔪 the maximal ideal of M_{f}. We claim that I ⊂ φ^{−1}𝔪, Indeed, otherwise there exist g ∈ I, h ∈ M and n ∈ ℕ such that g^{−1} = f^{−n}h in M_{f}; this means that there exists m ∈ ℕ such that f^{m+n} = f^{m}gh in M, hence f^{m+n} ∈ I, which contradicts the assumption on f. On the other hand, obviously f ∉ φ^{−1}𝔪. □

Lemma 6.1.17. (i) Let M be a monoid, and G ⊂ M^{×} a subgroup.

(a) The map given by the rule : I ↦ I/G establishes a natural bijection from the set of ideals of M onto the set of ideals of M/G.

(b) Especially, the natural projection π : M → M/G induces a bijection :

π^{∗} : Spec M/G → Spec M

(ii) Let (M_{i} | i ∈ I) be any finite family of monoids, and for each j ∈ I, denote by π_{j} : ∏_{i∈I} M_{i} → M_{j} the natural projection. The induced map

∏_{i∈I} Spec M_{i} → Spec ∏_{i∈I} M_{i} : (𝔭_{i} | i ∈ I) ↦ ⋃_{i∈I} π_{i}^{∗}𝔭_{i}

is a bijection. (iii) Let (M_{i} | i ∈ I) be any filtered system of monoids. The natural map

Spec colim_{i∈I} M_{i} → lim_{i∈I} Spec M_{i}

is a bijection.

Proof. (i): By lemma 4.8.31(iii), M/G is the set-theoretic quotient of M by the translation action of G. By definition, any ideal I of M is stable under the G-action, hence the quotient I/G is well defined, and one checks easily that it is an ideal of M/G. Moreover, if 𝔭 ⊂ M is a prime ideal, it is easily seen that 𝔭/G is a prime ideal of M/G. Assertions (a) and (b) are straightforward consequences.

(ii): The assertion can be rephrased by saying that every face F of ∏_{i∈I} M_{i} is a product of faces F_{i} ⊂ M_{i}. However, if \underline{m} := (m_{i} | i ∈ I) ∈ F, then, for each i ∈ I we can write \underline{m} = \underline{m}(i) · \underline{n}(i), where, for each j ∈ I, the j-th-component of \underline{m}(i) (resp. of \underline{n}(i)) equals 1 (resp. m_{j}), unless j = i, in which case it equals m_{i} (resp. 1). Thus, \underline{m}(i) ∈ F for every i ∈ I, and the contention follows easily.

(iii): Denote by M the colimit of the system (M_{i} | i ∈ I), and φ_{i} : M_{i} → M the natural morphisms of monoids, as well as φ_{f} : M_{i} → M_{j} the transition maps, for every morphism f : i → j in I. Recall that the set underlying M is the colimit of the system of sets (M_{i} | i ∈ I) (lemma 4.8.29(iii)). Let now 𝔭_{•} := (𝔭_{i} | ∈ I) be a compatible system of prime ideals, i.e. such that 𝔭_{i} ∈ Spec M_{i} for every i ∈ I, and φ_{f}^{−1}𝔭_{j} = 𝔭_{i} for every f : i → j. We let β(𝔭_{•}) := ⋃_{i∈I} φ_{i}(𝔭_{i}). We claim that β(𝔭_{•}) is a prime ideal of M. Indeed, suppose that x, y ∈ M and xy ∈ β(𝔭_{•}); since I is filtered, we may find i ∈ I, x_{i}, y_{i} ∈ M_{i}, and z_{i} ∈ 𝔭_{i}, such that x = φ_{i}(x_{i}), y = φ_{i}(y_{i}), and xy = φ_{i}(z_{i}). Especially, φ_{i}(z_{i}) = φ_{i}(x_{i}y_{i}), so there exists a morphism f : i → j such that φ_{f}(z_{i}) = φ_{f}(x_{i}y_{i}). But φ_{f}(z_{i}) ∈ 𝔭_{j}, so either φ_{f}(x_{i}) ∈ 𝔭_{j} or φ_{f}(y_{i}) ∈ 𝔭_{j}, and finally either x ∈ 𝔭 or y ∈ 𝔭, as required.

Let 𝔭 ⊂ M be any prime ideal; it is easily seen that β(φ_{i}^{−1}𝔭 | i ∈ I) = 𝔭. To conclude, it suffices to show that 𝔭_{i} = φ_{i}^{−1}β(𝔭_{•}), for every compatible system 𝔭_{•} as above, and every i ∈ I. Hence, fix i ∈ I and pick x_{i} ∈ M_{i} such that φ_{i}(x_{i}) ∈ β(𝔭_{•}); then there exist j ∈ I and x_{j} ∈ 𝔭_{j} such that φ_{i}(x_{i}) = φ_{j}(x_{j}). Since I is filtered, we may find k ∈ I and morphisms f : i → k and g : j → k such that φ_{f}(x_{i}) = φ_{g}(x_{j}), so φ_{f}(x_{i}) ∈ 𝔭_{k}, and finally x_{i} ∈ 𝔭_{i}, as sought. □

Remark 6.1.18. In case (M_{i} | i ∈ I) is an infinite family of monoids, the natural map of lemma 6.1.17(ii) is still injective, but it is not necessarily surjective. For instance, let I be any infinite set, and let 𝒰 ⊂ 𝒫(I) be a non-principal ultrafilter; denote by ^{∗}ℕ the quotient of ℕ^{I} under the equivalence relation ∼_{𝒰} such that (a_{i} | i ∈ I) ∼_{𝒰} (b_{i} | i ∈ I) if and only if there exists U ∈ 𝒰 such that a_{i} = b_{i} for every i ∈ U. It is clear that the composition law on ℕ^{I} descends to^{∗}ℕ; the resulting structure (^{∗}ℕ, +) is called the monoid of hypernatural numbers. Denote by π : ℕ^{I} → ^{∗}ℕ the projection, and let \underline{0} ∈ ℕ^{I} be the unit; then it is easily seen that \{π(0)\} is a face of^{∗}ℕ, but π^{−1}(π(0)) is not a product of faces F_{i} ⊂ ℕ.

Remark 6.1.19. (i) Notice that not all epimorphisms in Mnd are surjections on the underlying sets; for instance, every localization map M → S^{−1}M is an epimorphism.

(ii) If φ : N → M is a map of fine monoids (see definition 6.1.6(vi)), then it will follow from corollary 6.4.2that N ×_{M} N is also finitely generated. If M is not integral, then this fails in general : a counter-example is provided by the morphism φ constructed in (6.1.2).

(iii) Let Σ be any set; it is easily seen that a free monoid M ≃ ℕ^{(Σ)} admits a unique minimal system of generators, in natural bijection with Σ. Especially, the cardinality of Σ is determined by the isomorphism class of M; this invariant is called the rank of the free monoid M. This is the same as the rank of M as an ℕ-module (see example 3.7.27).

(iv) A submonoid of a finitely generated monoid is not necessarily finitely generated. For instance, consider the submonoid M ⊂ ℕ^{⊕2}, with M := {(0, 0)} ∪ {(a, b) | a > 0}. However, the following result shows that a face of a finitely generated monoid is again finitely generated.

Lemma 6.1.20. Let f : M → N be a map of monoids, F ⊂ N a face of N, and Σ ⊂ N a system of generators for N. Then :

(i) N^{×} is a face of N, and f^{−1}F is a face of M.

(ii) Σ ∩ F is a system of generators for F.

(iii) If N is finitely generated, Spec N is a finite set, and dim N is finite.

(iv) If N is finitely generated (resp. fine) then the same holds for F^{−1}N.

Proof. (i) and (ii) are left to the reader, and (iii) is an immediate consequence of (ii). To show (iv), notice that – in view of (ii) – the set Σ ∪ {f^{−1} | f ∈ F ∩ Σ} is a system of generators of F^{−1}N. □

Definition 6.1.21. (i) If M is a pointed or not pointed monoid (see remark 4.8.14(ii)), and S is a pointed M-module, we say that S is integral, if for every x, y ∈ M and every s ∈ S such that xs = ys ≠ 0, we have x = y. The annihilator ideal of S is the ideal

Ann_{M}(S) := {m ∈ M | ms = 0 for every s ∈ S}.

If s ∈ S is any element, we also write Ass_{M}(s) := Ass_{M}(Ms). The support of S is the subset :

Supp S := {𝔭 ∈ Spec M | S_{𝔭} ≠ 0}.

(ii) A pointed monoid (M, 0_{M}) is called integral, if it is integral when regarded as a pointed M-module; it is called fine, if it is finitely generated and integral in the above sense.

(iii) The forgetful functor Mnd_{∘} → Set (notation of (4.8.28)) admits a left adjoint, which assigns to any set Σ the free pointed monoid ℕ_{∘}^{(Σ)} := (ℕ^{(Σ)})_{∘}.

(iv) A morphism φ : M → N of pointed monoids is local, if both M, N ≠ 0, and φ is local when regarded as a morphism of non-pointed monoids.

Remark 6.1.22. (i) Quite generally, a (non-pointed) monoid M is finitely generated (resp. free, resp. integral, resp. fine) if and only if M_{∘} has the corresponding property for pointed monoids. However, there exist integral pointed monoids which are not of the form M_{∘} for any non-pointed monoid M.

(ii) Let (M, 0_{M}) be a pointed monoid. An ideal of (M, 0_{M}) is a pointed submodule I ⊂ M. Just as for non-pointed monoids, we say that I is a prime ideal, if M \ I is a (non-pointed) submonoid of M, and a non-pointed submonoid which is the complement of a prime ideal, is called a face of M. Hence the smallest ideal is {0}. Notice though, that {0} is not necessarily a prime ideal, hence the spectrum Spec (M, 0_{M}) does not always admit a least element. However, if M = N_{∘} for some non-pointed monoid N, the natural morphism of monoids N → N_{∘} induces a bijection :

Spec (N_{∘}, 0_{N_{∘}}) → Spec N.

(iii) Let I ⊂ M be any ideal; the inclusion map I → M can be regarded as a morphism of pointed M-modules (if M is not pointed, this is achieved via the faithful embedding (4.8.15)), whence a pointed M-module M/I, with a natural morphism M → M/I. The latter map is also a morphism of monoids, for the obvious monoid structure on M/I. One checks easily that if M is integral, M/I is an integral pointed monoid.

(iv) Let M be a (pointed or not pointed) monoid, 𝔭 ⊂ M a prime ideal. Then the natural morphism of monoids M → M/𝔭 induces a bijection :

Spec M/𝔭 →^{∼} {𝔮 ∈ Spec M | 𝔭 ⊂ 𝔮} = Spec M \𝔭.

(v) Let M be a (pointed or not pointed) monoid, and S ≠ 0 a pointed M-module. Then the support of S contains at least the maximal ideal of M. This trivial observation shows that a pointed M-module is 0 if and only if its support is empty.

(vi) Let M be a pointed monoid, and Σ ⊂ M a non-pointed submonoid. The localization Σ^{−1}M (defined in the category of monoids, as in (4.8.33)) is actually a pointed monoid : its zero element 0_{Σ^{−1}M} is the image of 0_{M}.

(vii) If M is a (pointed or not pointed) monoid, Σ ⊂ M any non-pointed submonoid, and S a pointed M-module, we let as usual Σ^{−1}S := Σ^{−1}M ⊗_{M} S (see remark 4.8.21(i), if M is not pointed). The resulting functor M-Mod_{∘} → Σ^{−1}M-Mod_{∘} is exact. Indeed, it is right exact, since it is left adjoint to the restriction of scalars arising from the localization map M → Σ^{−1}M (see (3.7.26)), and one verifies directly that it commutes with finite limits. Also, it is clear that

Supp Σ^{−1}S = Supp S ∩ Spec Σ^{−1}M.

(viii) Let M be a pointed monoid, and N ⊂ M a pointed submonoid. Since the final object 1 of the category of pointed monoids is not isomorphic to the initial object 1_{∘}, the push-out of the diagram 1 ← N → M is not an interesting object (it is always isomorphic to 1). Even if we form the quotient M/N in the category of non-pointed monoids, we still get always 1, since 0_{M} ∈ N, and therefore in the quotient M/N the images of 0_{M} and of the unit of M coincide.

The only case that may give rise to a non-trivial quotient, is when N is non-pointed; in this situation we may form M/N in the category of non-pointed monoids, and then remark that the image of 0_{M} yields a zero element 0_{M/N} for M/N, so the latter is a pointed monoid.

Example 6.1.23. (i) Let M be a (pointed or not pointed) monoid, G ⊂ M^{×} a subgroup, and S a pointed M-module. Then M/G ⊗_{M} S = S/G is the set of orbits of S under the induced G-action.

(ii) In the situation of (i), notice that the functor

M-Mod → M/G-Mod : S ↦ S/G

is exact, hence M/G is a flat M-module. (See definition 4.8.22(i).)

(iii) Likewise, if Σ ⊂ M is a non-pointed submonoid, then the localization Σ^{−1}M is a flat M-module, due to remark 6.1.22(vii).

(iv) Suppose that S is an integral pointed M-module (with M either pointed or not pointed), and let Σ ⊂ M be a non-pointed submonoid. Then Σ^{−1}S is also an integral pointed Σ^{−1}Mmodule. Indeed, suppose that the identity

(6.1.24)  (s^{−1}a) · (s′^{−1}b) = (s^{−1}a) · (s′′^{−1}c) ≠ 0

holds for some a, b, c ∈ S and s, s′, s′′ ∈ Σ; we need to check that s′^{−1}b = s′′^{−1}c, or equivalently, that s′′b = s′c in Σ^{−1}S. However, (6.1.24) is equivalent to s′′ab = s′ac in Σ^{−1}S, and the latter holds if and only if there exists t ∈ Σ such that ts′′ab = ts′ac in S. The two sides in the latter identity are ≠ 0, as the same holds for the two sides of the identity (6.1.24); therefore s′′b = s′c holds already in S, and the contention follows.

(v) Likewise, in the situation of (iv), S/Σ := S⊗_{M} M/Σ is an integral pointed M/Σ-module. Indeed, notice the natural identification S/Σ = Σ^{−1}S ⊗_{Σ^{−1}M} (Σ^{−1}M)/Σ^{gp} which – in view of (iv) – reduces the proof to the case where Σ is a subgroup of M^{×}. Then the assertion is easily verified, taking into account (i).

Especially, if M is an integral pointed monoid, and Σ ⊂ M is any non-pointed submonoid, then both Σ^{−1}M and M/Σ are integral pointed monoids (this generalizes lemma 4.8.38).

(vi) Let G be any abelian group, φ : M → G a morphism of non-pointed monoids. Then G_{∘} is a flat M_{∘}-module. For the proof, we may – in light of (iii) – replace M by M^{gp}, thereby reducing to the case where M is a group. Next, by (ii), we may assume that φ is injective, in which case G is a free M-module with basis G/M.

Remark 6.1.25. (i) Let M → N and M → N′ be morphisms of pointed monoids; N and N′ can be regarded as pointed M-modules in an obvious way, hence we may form the tensor product N′′ := N ⊗_{M} N′; the latter is endowed with a unique monoid structure such that the maps e : N → N′′ and e′ : N′ → N′′ given by the rule n ↦ n ⊗ 1 for all n ∈ N (resp. n′ ↦ 1 ⊗ n′ for all n′ ∈ N′) are morphisms of monoids. Just as for usual ring homomorphisms, the monoid N′′ is a coproduct of N and N′ over M, i.e. there is a unique isomorphism of pointed monoids:

(6.1.26)  N ⊗_{M} N′ →^{∼} N ⨿_{M} N′

that identifies e and e′ to the natural morphisms N → N ⨿_{M} N′ and N′ → N ⨿_{M} N′. As usual all this extends to non-pointed monoids. (Details left to the reader.)

(ii) Especially, if we take M = {1}_{∘}, the initial object in Mnd_{∘}, we obtain an explicit description of the pointed monoid N ⊕ N′ : it is the quotient (N × N′)/∼, where ∼ denotes the minimal equivalence relation such that (x, 0) ∼ (0, x′) for every x ∈ N, x′ ∈ N′. From this, a direct calculation shows that a direct sum of pointed integral monoids is again a pointed integral monoid.

Remark 6.1.27. (i) Clearly every pointed M-module S is the colimit of the filtered family of its finitely generated submodules. Moreover, S is the colimit of a filtered family of finitely presented pointed M-modules. Recall the standard argument : pick a countable set I, and let 𝒞 be the (small) full subcategory of the category M-Mod_{∘} whose objects are the coequalizers of every pair of maps of pointed M-modules p, q : M^{(I_{1})} → M^{(I_{2})}, for every finite sets I_{1}, I_{2} ⊂ I (this means that, for every such pair p, q we pick one representative for this coequalizer). Then there is a natural isomorphism of pointed M-modules :

colim_{i𝒞/S} ι_{S} →^{∼} S

where i : 𝒞 → M-Mod_{∘} is the inclusion functor, and ι_{S} is the functor as in (1.1.27).

(ii) If S is finitely generated, we may find a finite filtration of S by submodules 0 = S_{0} ⊂ S_{1} ⊂ ⋯ ⊂ S_{n} = S such that S_{i+1}/S_{i} is a cyclic M-module, for every i = 1, …, n.

(iii) Notice that if S is integral and S′ ⊂ S is any submodule, then S/S′ is again integral. Moreover, if S is integral and cyclic, we have a natural isomorphism of M-modules :

S →^{∼} M/Ann_{M}(S).

(Details left to the reader.)

(iv) Suppose furthermore, that M^{♯} is finitely generated, and S is any pointed M-module. Lemma 6.1.17(i.a) and proposition 6.1.9(ii) easily imply that every ascending chain

I_{0} ⊂ I_{1} ⊂ I_{2} ⊂ ⋯

of ideals of M is stationary; especially, the set {Ann_{M}(s) | s ∈ S \ {0}} admits maximal elements. Let I be a maximal element in this set; a standard argument as in commutative algebra shows that I is a prime ideal : indeed, say that xy ∈ I = Ann_{M}(s) and x ∉ I; then xs ≠ 0, hence y ∈ Ann_{M}(xs) = I, by the maximality of I. Now, if S is also finitely generated, it follows that we may find a finite filtration of S as in (ii) such that, additionally, each quotient S_{i+1}/S_{i} is of the form M/𝔭, for some prime ideal 𝔭 ⊂ M.

These properties make the class of integral pointed modules especially well behaved : essentially, the full subcategory M-Int.Mod_{∘} of M-Mod_{o} consisting of these modules mimics closely a category of A-modules for a ring A, familiar from standard linear algebra. This shall be amply demonstrated henceforth. For instance, we point out the following combinatorial version of Nakayama’s lemma :

Proposition 6.1.28. Let M be a (pointed or not pointed) monoid, S a finitely generated integral pointed M-module, and S′ ⊂ S a pointed submodule, such that

S = S′ ∪ 𝔪_{M} · S.

Then S = S′.

Proof. After replacing S by S/S′, we may assume that 𝔪_{M}S = S, in which case we have to check that S = 0. Suppose then, that S ≠ 0; from remark 6.1.27(ii,iii) it follows that S admits a (pointed) submodule T ⊂ S such that S/T ≃ M/𝔪_{M}. Especially, 𝔪_{M} · (S/T) = 0, i.e. 𝔪_{M}S ⊂ T, and therefore S = T, which contradicts the choice of T. The contention follows. □

Remark 6.1.29. (i) The integrality assumption cannot be omitted in proposition 6.1.28. Indeed, take M := ℕ and S := 0_{∘}, where 0 denotes the final ℕ-module. Then S ≠ 0, but 𝔪_{M}S = S.

(ii) Let us say that an element of the M-module S is primitive, if it does not lie in 𝔪_{M}S. We deduce from proposition 6.1.28, the following :

Corollary 6.1.30. Let M be a sharp (pointed or not pointed) monoid, S a finitely generated integral pointed monoid. Then the set S \ 𝔪_{M}S of primitive elements of S is finite, and is the unique minimal system of generators of S.

Proof. Indeed, it is easily seen that every system of generators of S must contain all the primitive elements, so S \𝔪_{M}S must be finite. On the other hand, let S′ ⊂ S be the submodule generated by the primitive elements; clearly S′ ∪ 𝔪_{M}S = S, hence S′ = S, by proposition 6.1.28. □

6.1.31. Let R be any ring, M a non-pointed monoid. Notice that the M-module underlying any R[M]-module is naturally pointed, whence a forgetful functor R[M]-Mod → M-Mod_{∘}. The latter admits a left adjoint

M-Mod_{∘} → R[M]-Mod : (S, 0_{S}) ↦ R⟨S⟩ := Coker R[0_{S}].

Likewise, the monoid (A, ·) underlying any (commutative unital) R-algebra A is naturally pointed, whence a forgetful functor R-Alg → Mnd_{∘}, which again admits a left adjoint

Mnd_{∘} → R-Alg : (M, 0_{M}) ↦ R⟨M⟩ := R[M]/(0_{M})

where (0_{M}) ⊂ R[M] denotes the ideal generated by the image of 0_{M}.

If (M, 0_{M}) is a pointed monoid, and S is a pointed (M, 0_{M})-module, then notice that R⟨S⟩ is actually a R⟨M⟩-module, so we have as well a natural functor

(M, 0_{M})-Mod_{∘} → R⟨M⟩-Mod S ↦ R⟨S⟩

which is again left adjoint to the forgetful functor.

For instance, let I ⊂ M be an ideal; from the foregoing, it follows that R⟨M/I⟩ is naturally an R[M]-algebra, and we have a natural isomorphism :

R⟨M/I⟩ →^{∼} R[M]/IR[M].

Explicitly, for any x ∈ F := M \I, let \overline{x} ∈ R⟨M/I⟩ be the image of x; then R⟨M/I⟩ is a free R-module, with basis (\overline{x} | x ∈ F). The multiplication law of R⟨M/I⟩ is determined as follows. Given x, y ∈ F, then \overline{x} · \overline{y} = \overline{xy} if xy ∈ F, and otherwise it equals zero.

Notice that if I_{1} and I_{2} are two ideals of M, we have a natural identification :

R⟨M/(I_{1} ∩ I_{2})⟩ →^{∼} R⟨M/I_{1}⟩ ×_{R⟨M/(I_{1}∪I_{2})⟩} R⟨M/I_{2}⟩.

These algebras will play an important role in section 12.5. As a special case, suppose that 𝔭 ⊂ M is a prime ideal; then the inclusion M \𝔭 ⊂ M induces an isomorphism of R-algebras :

R[M \𝔭] →^{∼} R⟨M/𝔭⟩.

Furthermore, general nonsense yields a natural isomorphism of R-modules :

(6.1.32) R⟨S ⊗_{M} S′⟩ →^{∼} R⟨S⟩ ⊗_{R⟨M⟩} R⟨S′⟩  for all pointed M-modules S and S′.

6.1.33. Let A be a commutative ring with unit, and f : M → (A, ·) a morphism of pointed monoids. Then f induces (forgetful) functors :

A-Mod → M-Mod_{∘} A-Alg → M/Mnd_{∘}

(notation of (1.1.24)) which admit left adjoints :

M-Mod_{∘} → A-Mod  S ↦ S ⊗_{M} A := ℤ⟨S⟩ ⊗_{ℤ⟨M⟩} A
M/Mnd_{∘} → A-Alg  N ↦ N ⊗_{M} A := ℤ⟨N⟩ ⊗_{ℤ⟨M⟩} A.

Sometimes we may also use the notation :

S ⊗_{M} N := ℤ⟨S⟩ ⊗_{ℤ⟨M⟩} N and S ⊗^{L}_{M} K_{•} := ℤ⟨S⟩ ⊗^{L}_{ℤ⟨M⟩} K_{•}

for any pointed M-module S, any A-module N and any object K_{•} of D^{−}(A-Mod). The latter derived tensor product is obtained by tensoring with a flat ℤ⟨M⟩-flat resolution of ℤ⟨S⟩. (Such resolutions can be constructed combinatorially, starting from a simplicial resolution of S.) All the verifications are standard, and shall be left to the reader.

Moreover, for any ideal I ⊂ M and any A-module N, we shall write f(I)N, or sometimes just IN, for the A-submodule of N generated by the system (f(x) · y | x ∈ I, y ∈ N).

Definition 6.1.34. Let M be a pointed monoid, A a commutative ring with unit, φ : M → (A, ·) a morphism of pointed monoids, N an A-module. We say that N is φ-flat (or just M-flat, if no ambiguity is likely to arise), if the functor

M-Int.Mod_{∘} → A-Mod : S ↦ S ⊗_{M} N

is exact, in the sense that it sends exact sequences of pointed integral M-modules, to exact sequences of A-modules. We say that N is faithfully φ-flat if this functor is exact in the above sense, and we have S ⊗_{M} N = 0 if and only if S = 0.

Remark 6.1.35. (i) Notice that the functor S ↦ S ⊗_{M} N of definition 6.1.34 is right exact in the categorical sense (i.e. it commutes with finite colimits), since it is a right adjoint. However, even when N is faithfully flat, this functor is not always left exact in the categorical sense : it does not commute with finite products, nor with equalizers, in general.

(ii) Let M and S be as in definition 6.1.34, and let R be any non-zero commutative unital ring. Denote by φ : M → (R⟨M⟩, ·) the natural morphism of pointed monoids. In light of (6.1.32), it is clear that if R⟨S⟩ is a flat R⟨M⟩-module, then S is a flat pointed M-module, and the latter condition implies that R⟨S⟩ is φ-flat.

(iii) Let P be a pointed monoid, A a ring, φ : P → (A, ·) a morphism of monoids, f : A → B a ring homomorphism. If A is φ-flat and f is flat, then B is (f ∘ φ)-flat. Conversely, if B is (f ∘ φ)-flat and f is faithfully flat, then A is φ-flat : the verifications are standard, and shall be left to the reader.

Lemma 6.1.36. Let M be a monoid, A a ring, φ : M → (A, ·) a morphism of monoids, and assume that φ is local and A is φ-flat. Then A is faithfully φ-flat.

Proof. Let S be an integral pointed M-module, and suppose that S ⊗_{M} A = {0}; we have to show that S = {0}. Say that s ∈ S, and let Ms ⊂ S be the M-submodule generated by s. Since A is φ-flat, it follows easily that Ms ⊗_{M} A = {0}, hence we are reduced to the case where S is cyclic. By remark 6.1.27(iii), we may then assume that S = M/I for some ideal I ⊂ M. It follows that S ⊗_{M} A = A/φ(I)A, so that φ(I) generates A. Since φ is local, this implies that I = M, whence the contention. □

Lemma 6.1.37. Let M be an integral pointed monoid, I, J ⊂ M two ideals, A a ring, α : M → (A, ·) a morphism of monoids, N an α-flat A-module, and S a flat M-module. Then :

\begin{aligned}IS ∩ JS & = (I ∩ J)S \\ α(I)N ∩ α(J)N & = α(I ∩ J)N. \end{aligned}

Proof. We consider the commutative ladder of pointed M-modules, with exact rows and injective vertical arrows :

(6.1.38)  \xymatrix{0 \ar[r] & I ∩ J \ar[r] \ar[d] & I \ar[r] \ar[d] & I/(I ∩ J) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & J \ar[r] & M \ar[r] & M/(I ∪ J) \ar[r] & 0}

By assumption, the ladder of A-modules (6.1.38)⊗_{M} N has still exact rows and injective vertical arrows. Then, the snake lemma gives the following short exact sequence involving the cokernels of the vertical arrows :

0 → JN/(I ∩ J)N → N/IN ⟶^{p} N/(IN + JN) → 0

(where we have written JN instead of α(J)N, and likewise for the other terms). However Ker p = JN/(IN ∩ JN), whence the second stated identity. The first stated identity can be deduced from the second, by virtue of remark 6.1.35(ii). □

Remark 6.1.39. By inspection of the proof, we see that the first identity of lemma 6.1.37holds, more generally, whenever ℤ⟨S⟩ is φ-flat, where φ : M → ℤ⟨M⟩ is the natural morphism of pointed monoids.

Proposition 6.1.40. Let M be a pointed integral monoid, A a ring, φ : M → (A, ·) a morphism of monoids, N an A-module. Then we have :

(i) The following conditions are equivalent : (a) N is φ-flat.

(b) Tor_{i}^{ℤ⟨M⟩}(ℤ⟨T ⟩, N) = 0 for every i > 0 and every pointed integral M-module T.

(c) Tor_{1}^{ℤ⟨M⟩}(ℤ⟨M/I⟩, N) = 0 for every ideal I ⊂ M.

(d) The natural map I ⊗_{M} N → N is injective for every ideal I ⊂ M. (ii) If moreover, M^{♯} is finitely generated, then the conditions (a)-(d) of (i) are equivalent to

either of the following two conditions :

(e) Tor_{1}^{ℤ⟨M⟩}(ℤ⟨M/𝔭⟩, N) = 0 for every prime ideal 𝔭 ⊂ M.

(f) The natural map 𝔭 ⊗_{M} N → N is injective for every prime ideal 𝔭 ⊂ M.

Proof. Clearly (a)⇒(b)⇒(c)⇒(e). Next, by considering the short exact sequence of pointed integral M-modules 0 → I → M → M/I → 0 we easily see that (c)⇔(d) and (e)⇔(f).

(c)⇒(a) : Let Σ := (0 → S′ → S → S′′ → 0) be a short exact sequence of pointed integral M-modules; we need to show that the induced map S′ ⊗_{M} N → S ⊗_{M} N is injective. Since the sequence ℤ⟨Σ⟩ is still exact, the long Tor-exact sequence reduces to showing that Tor_{1}^{ℤ⟨M⟩}(ℤ⟨T ⟩, N) = 0 for every pointed integral M-module T. In view of remark 6.1.27(i), we are easily reduced to the case where T is finitely generated; next, using remark 6.1.27(ii,iii), the long exact Tor-sequence, and an easy induction on the number of generators of T, we may assume that T = M/I, whence the contention.

Lastly, if M^{♯} is finitely generated, then remark 6.1.27(iv) shows that, in the foregoing argument, we may further reduce to the case where T = M/𝔭 for a prime ideal 𝔭 ⊂ M; this shows that (e)⇒(a). □

Lemma 6.1.41. Let M be a pointed monoid, S a pointed M-module, and suppose that the following conditions hold for S :

(F1) If s ∈ S and a ∈ Ann_{M}(s), then there exists b ∈ Ann_{M}(a) such that s ∈ bS.

(F2) If a_{1}, a_{2} ∈ M and s_{1}, s_{2} ∈ S satisfy the identity a_{1}s_{1} = a_{2}s_{2} ≠ 0, then there exist

b_{1}, b_{2} ∈ M and t ∈ S such that s_{i} = b_{i}t for i = 1, 2 and a_{1}b_{1} = a_{2}b_{2}. Then the natural map I ⊗_{M} S → S is injective for every ideal I ⊂ M.

Proof. Let I be an ideal, and suppose that two elements a_{1}⊗s_{1} and a_{2}⊗s_{2} of I⊗_{M} S are mapped to the same element of S. If a_{i}s_{i} = 0 for i = 1, 2, then (F1) says that there exist b_{1}, b_{2} ∈ M and t_{1}, t_{2} ∈ S such that a_{i}b_{i} = 0 and s_{i} = b_{i}t_{i} for i = 1, 2; thus a_{i} ⊗ s_{i} = a_{i} ⊗ b_{i}t_{i} = a_{i}b_{i} ⊗ s_{i} = 0 in I ⊗_{M} S. In case a_{i}s_{i} ≠ 0, pick b_{1}, b_{2} ∈ M and t ∈ S as in (F2); we conclude that a_{1} ⊗ s_{1} = a_{1} ⊗ b_{1}t = a_{1}b_{1} ⊗ t = a_{2}b_{2} ⊗ t = a_{2} ⊗ s_{2} in I ⊗_{M} S, whence the contention. □

Theorem 6.1.42. Let M be an integral pointed monoid, S a pointed M-module. The following conditions are equivalent :

(a) S is M-flat.

(b) For every morphism M → P of pointed monoids, P ⊗_{M} S is P -flat.

(c) For every short exact sequence Σ of integral pointed M-modules, the sequence Σ⊗_{M} S is again short exact.

(d) Conditions (F1) and (F2) of lemma 6.1.41hold for S.

Proof. Clearly (b)⇒(a)⇒(c).

(c)⇒(d): To show (F1), set I := Ma, and denote by i : I → M the inclusion; (c) implies that the induced map i ⊗_{M} S : I ⊗_{M} S → S is injective. However, we have a natural isomorphism I →^{∼} M/Ann_{M}(a) of M-modules (remark 6.1.27(iii)), whence an isomorphism : I ⊗_{M} S →^{∼} S/Ann_{M}(a)S, and under this identification, i ⊗_{M} S is induced by the map S → S : s ↦ as. Thus, multiplication by a maps the subset S \Ann_{M}(a)S injectively into itself, which is the claim.

For (F2), notice that ℤ⟨S⟩ is φ-flat under condition (c), for φ : M → ℤ⟨M⟩ the natural morphism. Now, say that a_{1}s_{1} = a_{2}s_{2} ≠ 0 in S; set I := Ma_{1}, J := Ma_{2}; the assumption means that a_{1}s_{1} ∈ IS ∩ JS, in which case remark 6.1.39 shows that there exist t ∈ S and b_{1}, b_{2} ∈ M such that a_{1}b_{1} = a_{2}b_{2}, and a_{1}s_{1} = a_{1}b_{1}t, hence a_{2}s_{2} = a_{2}b_{2}t. Since we have seen that multiplication by a_{1} maps S \Ann_{M}(a_{1})S injectively into itself, we deduce that s_{1} = b_{1}t, and likewise we get s_{2} = b_{2}t.

To prove that (d)⇒(b), we observe :

Claim 6.1.43. Let P be a pointed monoid, Λ a small locally directed category (see definition 1.2.19(iv)), and S_{•} : Λ → P -Mod_{∘} a functor, such that S_{λ} fulfills conditions (F1) and (F2), for every λ ∈ Ob(Λ). Then the colimit of S_{•} also fulfills conditions (F1) and (F2).

Proof of the claim. In light of remark 1.2.21(ii), we may assume that Λ is either discrete or connected. Suppose first that Λ is connected; then remark 4.8.17(ii) allows to check directly that conditions (F1) and (F2) hold for the colimit of S_{•}, since they hold for every S_{λ}. If Λ is discrete, the assertion is that conditions (F1) and (F2) are preserved by arbitrary (small) direct sums, which we leave as an exercise for the reader. ♢

To a given pointed M-module S, we attach the small category S^{∗}, such that :

Ob(S^{∗}) = S\{0} and Hom_{S^{∗}}(s′, s) = {a ∈ M | as = s′}.

The composition of morphisms is induced by the composition law of M, in the obvious way. Notice that S^{∗} is locally directed if and only if S satisfies condition (F2).

We define a functor F : S^{∗} → M-Mod_{∘} as follows. For every s ∈ Ob(S^{∗}) we let F (s) := M, and for every morphism a : s′ → s we let F (a) := a·1_{M}. We have a natural transformation τ : F ⇒ c_{S}, where c_{S} : S^{∗} → M-Mod_{∘} is the constant functor associated with S; namely, for every s ∈ Ob(S^{∗}), we let τ_{s} : M → S be the map given by the rule a ↦ as for all a ∈ M. There follows a morphism of pointed M-modules :

(6.1.44)  colim_{S^{∗}} F → S

Claim 6.1.45. If S fulfills conditions (F1), the map (6.1.44) is an isomorphism.

Proof of the claim. Indeed, we have a natural decomposition of S^{∗} as coproduct of a family (S_{i}^{∗} | i ∈ I) of connected subcategories (for some small set I : see remark 1.2.21(ii)); especially we have Hom_{S^{∗}}(s, s′) = ∅ if s ∈ Ob(S_{i}^{∗}) and s′ ∈ Ob(S_{j}^{∗}) for some i ≠ j in I.

For each i ∈ I, let F_{i} : S_{i}^{∗} → M-Mod_{∘} be the restriction of F. There follows a natural isomorphism :

⨁_{i∈I} colim_{S_{i}^{∗}} F_{i} →^{∼} colim_{S^{∗}} F.

Since the colimit of F_{i} commutes with the forgetful functor to sets, an inspection of the definitions yields the following explicit description of the colimit T_{i} of F_{i}. Every element of T_{i} is represented by some pair (s, a) where s ∈ Ob(S_{i}^{∗}) ⊂ S\{0} and a ∈ M; such pair is mapped to as by (6.1.44), and two such pairs (s, a), (s′, a′) are identified in T_{i} if there exists b ∈ M such that bs = s′ and ba′ = a.

Hence, denote by S_{i} the image under (6.1.44) of T_{i}; we deduce first, that S_{i} ∩ S_{j} = {0} if i ≠ j. Indeed, say that t ∈ S_{i} ∩ S_{j}; by the foregoing, there exist s_{i} ∈ Ob(S_{i}^{∗}), s_{j} ∈ Ob(S_{j}^{∗}), and a_{i}, a_{j} ∈ M, such that a_{i}s_{i} = t = a_{j}s_{j}. If t ≠ 0, we get morphisms a_{i} : t → s_{i} and a_{j} : t → s_{j} in S^{∗}; say that t ∈ S_{k}^{∗} for some k ∈ I; it then follows that S_{i}^{∗} = S_{k}^{∗} = S_{j}^{∗}, a contradiction. Next, it is clear that (6.1.44) is surjective. It remains therefore only to show that each T_{i} maps injectively onto S_{i}. Hence, say that (s_{1}, a_{1}) and (s_{2}, a_{2}) represent two elements of T_{i} with t := a_{1}s_{1} = a_{2}s_{2}. If t ≠ 0, we get, as before, morphisms a_{1} : t → s_{1} and a_{2} : t → s_{2} in S_{i}^{∗}, and the two pairs are identified in T_{i} to the pair (t, 1). Lastly, if t = 0, condition (F1) yields b ∈ M and s′ ∈ S such that a_{1}b = 0 and s_{1} = bs′, whence a morphism b : s_{1} → s′ in S_{i}^{∗}, and F_{i}(b)(a_{1}, s_{1}) = (0, s′) which represents the zero element of T_{i}. The same argument applies as well to (a_{2}, s_{2}), and the claim follows. ♢

Claim 6.1.46. Let M → P be any morphism of pointed monoids, and S a pointed M-module fulfilling conditions (F1) and (F2). Then the natural map I ⊗_{M} S → P ⊗_{M} S is injective, for every ideal I ⊂ P. Proof of the claim. From claim 6.1.45we deduce that P ⊗_{M} S is the locally directed colimit of the functor P ⊗_{M} F, and notice that the pointed P -module P fulfills conditions (F1) and (F2); by claim 6.1.43 we deduce that P ⊗_{M} S also fulfills the same conditions, so the claim follows from lemma 6.1.41. ♢

After these preliminaries, suppose that conditions (F1) and (F2) hold for S, and let Σ := (0 → T′ → T → T′′ → 0) be a short exact sequence of pointed M-modules. We wish to show that Σ ⊗_{M} S is still short exact. However, if U′′ ⊂ T′′ is any M-submodule, let U ⊂ T be the preimage of U′′, and notice that the induced sequence 0 → T′ → U → U′′ → 0 is still short exact. Since a filtered colimit of short exact sequences is short exact, remark 6.1.27(i) allows to reduce to the case where T′′ is finitely generated.

We shall argue by induction on the number n of generators of T′′. Hence, suppose first that T′′ is cyclic, and let t ∈ T be any element whose image in T′′ is a generator. Set C := Mt ⊂ T, and let C′ ⊂ T′ be the preimage of C. We obtain a cocartesian (and cartesian) diagram of pointed M-modules :

𝒟 : \xymatrix{C′ \ar[r] \ar[d] & C \ar[d] \\ T′ \ar[r] & T.}

The induced diagram 𝒟 ⊗_{M} S is still cocartesian, hence the same holds for the diagram of sets underlying 𝒟 ⊗_{M} S (remark 4.8.17(ii)). Especially, if the induced map C′ ⊗_{M} S → C ⊗_{M} S is injective, the same will hold for the map T′ ⊗_{M} S → T ⊗_{M} S. We may thus replace T′ and T by respectively C′ and C, which allows to assume that also T is cyclic. In this case, pick a generator u ∈ T; we claim that there exists a unique multiplication law μ_{T} on T, such that the surjection p : M → T : a ↦ au is a morphism of pointed monoids. Indeed, for every t, t′ ∈ T, write t = au for some a ∈ M, and set μ_{T} (t, t′) := at′. Using the linearity of p we easily check that μ_{T} (t, t′) does not depend on the choice of a, and the resulting composition law μ_{T} is commutative and associative. Then T′ is an ideal of T, so claim 6.1.46 tells us that the map T′ ⊗_{M} S → T ⊗_{M} S is injective, as required.

Lastly, suppose that n > 1, and the assertion is already known whenever T′′ is generated by at most n − 1 elements. Let U′′ ⊂ T′′ be a pointed M-submodule, such that U′′ is generated by at most n − 1 elements, and T′′/U′′ is cyclic. Denote by U ⊂ T the preimage of U′′; we deduce short exact sequences Σ′ := (0 → T′ → U → U′′ → 0) and Σ′′ := (0 → U → T → T′′/U′′ → 0), and by inductive assumption, both Σ′ ⊗_{M} S and Σ′′ ⊗_{M} S are short exact. Therefore the natural map T′ ⊗_{M} S → T ⊗_{M} S is the composition of two injective maps, hence it is injective, as stated. □

Remark 6.1.47. In the situation of remark 6.1.35(ii), suppose that M is pointed integral. Then theorem 6.1.42implies that S is a flat pointed M-module if and only if R⟨S⟩ is φ-flat.

Corollary 6.1.48. Let M be an integral pointed monoid, S a pointed M-module. Then

(i) The following conditions are equivalent :

(a) S is M-flat.

(b) For every ideal I ⊂ M, the induced map I ⊗_{M} S → S is injective. (ii) If moreover M^{♯} is finitely generated, then these conditions are equivalent to : (c) For every prime ideal 𝔭 ⊂ M, the induced map 𝔭 ⊗_{M} S → S is injective.

Proof. (i): Indeed, by remark 6.1.47and proposition 6.1.40(i) (together with (6.1.32)), both (a) and (b) hold if and only if ℤ⟨S⟩ is φ-flat, for φ : M → ℤ⟨M⟩ the natural morphism.

(ii): This follows likewise from proposition 6.1.40(ii). □

Corollary 6.1.49. Let φ : M → N be a morphism of pointed monoids, G ⊂ M^{×}, H ⊂ N^{×} two subgroups such that φ(G) ⊂ H, and denote by \overline{φ} : M/G → N/H the induced morphism. Let also S be any N-module. We have :

(i) If S_{(φ)} is a flat M-module, then S/H_{(\overline{φ})} is a flat M/G-module (notation of (3.7.26)). (ii) If moreover, M is a pointed integral monoid and S is a pointed integral H-module,

then also the converse of (i) holds.

Proof. (i): We have a natural isomorphism

(S/H)_{(\overline{φ})} →^{∼} N/H ⊗_{N/φG} S_{(φ)}/φG.

However, by example 6.1.23(ii), N/H is a flat N/φG-module, and S_{(φ)}/φG is a flat M/Gmodule, whence the contention.

(ii): By theorem 6.1.42, it suffices to check that conditions (F1) and (F2) of lemma 6.1.41 hold in S_{(φ)}, and notice that, by the same token, both conditions hold for the M/G-module S/H_{(\overline{φ})} = S_{(φ)}/G, since M/G is pointed integral (example 6.1.23(v)).

Hence, say that φ(a)s = 0 for some a ∈ M and s ∈ S; we may then find b ∈ M, t′ ∈ S and g ∈ G such that φ(gb)t = s and ab = 0. Setting t := φ(g)t′, we deduce that (F1) holds.

Next, say that φ(a_{1})s_{1} = φ(a_{2})s_{2} ≠ 0 for some a_{1}, a_{2} ∈ M and s_{1}, s_{2} ∈ S; then we may find g ∈ G, h_{1}, h_{2} ∈ H, b_{1}, b_{2} ∈ M and t′ ∈ S such that ga_{1}b_{1} = a_{2}b_{2} and

(6.1.50)  φ(b_{i})h_{i}t′ = s_{i}  for i = 1, 2.

After replacing b_{1} by gb_{1} and h_{1} by h_{1}φ(g^{−1}), we reduce to the case where a_{1}b_{1} = a_{2}b_{2}. From (6.1.50) we deduce that

φ(a_{1}b_{1})h_{1}t′ = φ(a_{1})s_{1} = φ(a_{2})s_{2} = φ(a_{2}b_{2})h_{2}t′ = φ(a_{1}b_{1})h_{2}t′

whence h_{1} = h_{2}, since S is a pointed integral H-module. Setting t := h_{1}t′, we see that (F2) holds. □

Corollary 6.1.51. Let φ : M → N be a flat morphism of pointed monoids, with M pointed integral, and let 𝔭 ⊂ N be any prime ideal. Then the morphism M/φ^{−1}𝔭 → N/𝔭 induced by φ is also flat.

Proof. Let F := N \𝔭; by theorem 6.1.42, it suffices to check that conditions (F1) and (F2) of lemma 6.1.41 hold for the φ^{−1}F -module F. However, since 0 ∉ F, condition (F1) holds trivially. Moreover, by assumption these two conditions hold for the M-module N; hence, say that φ(a_{1}) · s_{1} = φ(a_{2}) · s_{2} in F, for some a_{1}, a_{2} ∈ φ^{−1}F and s_{1}, s_{2} ∈ F. It follows that there exist b_{1}, b_{2} ∈ M and t ∈ N such that a_{1}b_{1} = a_{2}b_{2} in M, and φ(b_{i}) · t = s_{i} for i = 1, 2. Since F is a face, this implies that φ(b_{1}), φ(b_{2}), t ∈ F, so (F2) holds for F, as stated. □

Another corollary is the following analogue of a well known criterion due to Lazard.

Proposition 6.1.52. Let M be an integral pointed monoid, S an integral pointed M-module, R a non-zero commutative ring with unit. The following conditions are equivalent :

(a) S is M-flat.

(b) S is the colimit of a filtered system of free pointed M-modules (see remark 4.8.17(ii)).

(c) R⟨S⟩ is a flat R⟨M⟩-module.

Proof. Obviously (b)⇒(a) and (b)⇒(c).

(c)⇒(a) has already been observed in remark 6.1.35(ii).

(a)⇒(b): It suffices to prove that if S is flat, the category S^{∗} attached to S as in the proof of theorem 6.1.42, is pseudo-filtered. However, a simple inspection of the construction shows that S^{∗} is pseudo-filtered if and only if S satisfies both condition (F2) of lemma 6.1.41, and the following further condition. For every a, b ∈ M and s ∈ S such that as = bs ≠ 0, there exist t ∈ S and c ∈ M such that ac = bc and ct = s. This condition is satisfied by every integral pointed M-module, whence the contention. □

6.1.53. Consider now, a cartesian diagram of integral pointed monoids :

𝒟(P_{0}, I, P_{1}) : \xymatrix{P_{0} \ar[r] \ar[d] & P_{1} \ar[d] \\ P_{2} \ar[r] & P_{3}}

where P_{2} (resp. P_{3}) is a quotient P_{0}/I (resp. P_{1}/IP_{1}) for some ideal I ⊂ P_{0}, and the vertical arrows of 𝒟(P_{0}, I, P_{1}) are the natural surjections. In this situation, it is easily seen that the induced map I → IP_{1} is bijective; especially, I is both a P_{0}-module and a P_{1}-module. Let φ_{i} : P_{i} → ℤ⟨P_{i}⟩ be the units of adjunction, for i = 0, 1, 2. Let also M be any ℤ⟨P_{0}⟩-module.

Lemma 6.1.54. In the situation of (6.1.53), we have :

(i) Let J ⊂ P_{0} be any ideal. Then S := P_{0}/J admits a three-step filtration

0 ⊂ Fil_{0}S ⊂ Fil_{1}S ⊂ Fil_{2}S = S

such that Fil_{0}S and gr_{2}S are P_{2}-modules, and gr_{1}S is a P_{1}-module. (ii) The following conditions are equivalent :

(a) M is φ_{0}-flat.

(b) M ⊗_{P_{0}} P_{i} is φ_{i}-flat and Tor_{1}^{ℤ⟨P_{0}⟩}(M, ℤ⟨P_{i}⟩) = 0, for i = 1, 2. (iii) The following conditions are equivalent :

(a) Tor_{1}^{ℤ⟨P_{0}⟩}(M, ℤ⟨P_{i}⟩) = 0 for i = 1, 2, 3.

(b) Tor_{1}^{ℤ⟨P_{i}⟩}(M ⊗_{P_{0}} P_{i}, ℤ⟨P_{3}⟩) = 0 for i = 1, 2. (iv) Suppose moreover, that P_{3} ≠ 0 is a free pointed P_{2}-module. Then the ℤ⟨P_{0}⟩-module

M is φ_{0}-flat if and only if the ℤ⟨P_{1}⟩-module M ⊗_{P_{0}} P_{1} is φ_{1}-flat.

Proof. (i): Define Fil_{1}S := Ker(S → P_{0}/(I ∪ J)) = (I ∪ J)/J. Then it is already clear that S/Fil_{1}S is a P_{2}-module. Next, let Fil_{0}S := Ker(Fil_{1}S → (I ∪ JP_{1})/JP_{1}) = JP_{1}/J. Since IP_{1} = I, we see that Fil_{0}S is a P_{2}-module, and (I ∪ JP_{1})/JP_{1} is a P_{1}-module.

(ii): Clearly (a)⇒(b), hence suppose that (b) holds, and let us prove (a). By proposition 6.1.40(i), it suffices to show that Tor_{1}^{ℤ⟨P_{0}⟩}(M, ℤ⟨P_{0}/I⟩) = 0 for every ideal J ⊂ P_{0}. In view of (i), we are then reduced to showing that Tor_{1}^{ℤ⟨P_{0}⟩}(M, ℤ⟨S⟩) = 0, whenever S is a P_{i}-module, for i = 1, 2. However, for such P_{i}-module S, we have a change of rings spectral sequence

E_{pq}^{2} : Tor_{p}^{ℤ⟨P_{i}⟩}(Tor_{q}^{ℤ⟨P_{0}⟩}(M, ℤ⟨P_{i}⟩), ℤ⟨S⟩) ⇒ Tor_{p+q}^{ℤ⟨P_{0}⟩}(M, ℤ⟨S⟩).

Under assumption (b), we deduce : Tor_{1}^{ℤ⟨P_{0}⟩}(M, ℤ⟨S⟩) = Tor_{1}^{ℤ⟨P_{i}⟩}(M ⊗_{P_{0}} P_{i}, ℤ⟨S⟩) = 0.

(iii): Notice that the induced diagram of rings ℤ⟨𝒟(P_{0}, I, P_{1})⟩ is still cartesian. Then, this is a special case of [75, Lemma 3.4.15].

(iv): Suppose that M ⊗_{P_{0}} P_{1} is φ_{1}-flat, and P_{3} is a free pointed P_{2}-module, say P_{3} ≃ P_{2}^{(Λ)_{∘}}, for some set Λ ≠ ∅; then the ℤ⟨P_{2}⟩-module ℤ⟨P_{3}⟩ is isomorphic to ℤ⟨P_{2}⟩^{(Λ)}, especially it is faithfully flat, and we deduce that Tor_{1}^{ℤ⟨P_{0}⟩}(M, ℤ⟨P_{i}⟩) = 0 for i = 1, 2, by (iii). On the other hand, M ⊗_{P_{0}} P_{3} is φ_{3}-flat, so it also follows that M ⊗_{P_{0}} P_{2} is φ_{2}-flat, by proposition 6.1.52. Summing up, this shows that M fulfills condition (ii.b), hence also (ii.a), as sought. □

6.1.55. Let now P → Q be an injective morphism of integral pointed monoids, and suppose that P^{♯} and Q^{♯} are finitely generated monoids, and Q is a finitely generated P -module. Denote φ_{P} : P → ℤ⟨P ⟩ and φ_{Q} : Q → ℤ⟨Q⟩ the usual units of adjunction.

Theorem 6.1.56. In the situation of (6.1.55), let M be a ℤ⟨P ⟩-module, and suppose that the ℤ⟨Q⟩-module M ⊗_{P} Q is φ_{Q}-flat. Then M is φ_{P} -flat.

Proof. Using lemma 6.1.54(iv), and an easy induction, it suffices to show that there exists a finite chain

(6.1.57)  P = Q_{0} ⊂ Q_{1} ⊂ ⋯ ⊂ Q_{n} = Q

of inclusions of integral pointed monoids, and for every j = 0, …, n − 1 an ideal I_{j} ⊂ Q_{j}, and a cartesian diagram of integral pointed monoids 𝒟(Q_{j}, I_{j}, Q_{j+1}) as in (6.1.53), and such that Q_{j+1}/I_{j} ≠ 0 is a free pointed Q_{j}/I_{j}-module. (Notice that each Q_{i}^{♯} is a quotient of Q_{i}/P^{×}, and the latter is a submodule of Q/P^{×}, hence Q_{i}^{♯} is still a finitely generated monoid, by proposition 6.1.9(i).) If P = Q, there is nothing to prove; so we may assume that P is strictly contained in Q, and – invoking again proposition 6.1.9(i) – we further reduce to showing that there exist a monoid Q_{1} ⊂ Q strictly containing P, and an ideal I ⊂ P, such that the diagram 𝒟(P, I, Q_{1}) fulfills the above conditions.

Suppose first that the support of Q/P contains only the maximal ideal 𝔪_{P} of P, and let x_{1}, …, x_{r} be a finite system of generators for 𝔪_{P} (proposition 6.1.9(ii)). For each i = 1, …, r, the localization (Q/P)_{x_{i}} is a P_{x_{i}}-module with empty support, hence (Q/P)_{x_{i}} = 0 (remark 6.1.22(v)). It follows that every element of Q/P is annihilated by some power of x_{i}, and since Q/P is finitely generated, we may find N ∈ ℕ large enough, such that x_{i}^{N}Q/P = 0 for i = 1, …, r. After replacing N by some possibly larger integer, we get 𝔪_{P}^{N} · Q/P = 0, and we may assume that N is the least integer with this property. If N = 0, there is nothing to prove; hence suppose that N > 0, and set Q_{1} := P ∪ 𝔪_{P}^{N−1}Q. Notice that Q_{1} is a monoid, and Q_{1}/P ≠ 0 is annihilated by 𝔪_{P}. Especially, 𝔪_{P} Q_{1} = 𝔪_{P}. Moreover, the induced map P/𝔪_{P} → Q_{1}/𝔪_{P} is injective and Q_{1} is an integral pointed module, therefore the group P^{×} acts freely on Q_{1}\𝔪_{P}, i.e. Q_{1}/𝔪_{P} is a free pointed P/𝔪_{P} -module. It follows easily that if we take I := 𝔪_{P}, we do obtain a diagram 𝒟(P, 𝔪_{P} , Q_{1}) with the sought properties, in this case.

For the general case, let 𝔭 ⊂ P be a minimal element of Supp Q/P (for the ordering given by inclusion). Then the induced morphism P_{𝔭} → Q_{𝔭} still satisfies the conditions of (6.1.55) (lemma 6.1.20(iv)). Moreover, Supp Q_{𝔭}/P_{𝔭} = {𝔭P_{𝔭}} by remark 6.1.22(vii). By the previous case, we deduce that there exists a chain of inclusions of integral pointed monoids P_{𝔭} ⊂ Q′_{1} ⊂ Q_{𝔭}, such that the resulting diagram 𝒟(P_{𝔭}, 𝔭P_{𝔭}, Q′_{1}) is cartesian, and Q′_{1}/𝔭P_{𝔭} ≠ 0 is a free pointed P_{𝔭}/𝔭P_{𝔭}-module. Let \overline{e}_{1}, …, \overline{e}_{d} be a basis of the latter P_{𝔭}/𝔭P_{𝔭}-module, with \overline{e}_{1} = 1. Hence, \overline{e}_{i} ∈ Q_{𝔭} \ 𝔭P_{𝔭} for every i = 1, …, d, and after multiplying \overline{e}_{2}, …, \overline{e}_{d} by a suitable element of P \ 𝔭, we may assume that each \overline{e}_{i} is the image in Q_{𝔭} of an element e_{i} ∈ Q. Moreover, for every i, j ≤ d, either \overline{e}_{i}\overline{e}_{j} = 0, or else there exist a_{ij} ∈ P_{𝔭} and k(i, j) ≤ d such that \overline{e}_{i}\overline{e}_{j} = a_{ij}\overline{e}_{k(i,j)}. Furthermore, fix a system of generators x_{1}, …, x_{r} for 𝔭; then, for every i ≤ r and every j ≤ d we have x_{i}\overline{e}_{j} ∈ 𝔭P_{𝔭}. Again, after multiplying \overline{e}_{2}, …, \overline{e}_{d} by some c ∈ P \𝔭, we may assume that a_{ij} ∈ P for every i, j ≤ d, and moreover that x_{i}\overline{e}_{j} lies in the image of 𝔭 for every i ≤ r and j ≤ d.

And if we multiply yet again e_{2}, …, e_{d} by a suitable element of P \𝔭, we may finally reach a system of elements e_{1}, …, e_{d} ∈ Q such that e_{1} = 1 and :

• For every i, j ≤ d, we have either e_{i}e_{j} = 0 or else e_{i}e_{j} = a_{ij}e_{k(i,j)}.

• x_{i}e_{j} ∈ 𝔭 for every i ≤ r and j ≤ d.

Clearly these elements span a P -module Q_{1} which is a monoid containing P and contained in Q; moreover, by construction we have 𝔭Q_{1} = 𝔭, hence the resulting diagram 𝒟(P, 𝔭, Q_{1}) is cartesian. Notice also that (Q_{1}/𝔭)_{𝔭} ≃ Q′_{1}/𝔭P_{𝔭}, and that P/𝔭 = P′_{∘}, where P′ := P \𝔭 is an integral (non-pointed) monoid. To conclude it suffices now to apply the following

Claim 6.1.58. Let P′ be an integral non-pointed monoid, S a pointed P′_{∘}-module, and \underline{e} := (e_{1}, …, e_{d}) a system of generators for S. Suppose that S ⊗_{P′_{∘}} P_{∘}^{′gp} is a free pointed P_{∘}^{′gp}- module, and the image of \underline{e} is a basis for this module. Then S is a free pointed P′_{∘}-module, with basis \underline{e}.

Proof of the claim. If \underline{e} is not a basis, we have a relation in S of the type a_{1}e_{1} = a_{2}e_{2}, for some a_{1}, a_{2} ∈ P′. This relation must persist in S ⊗_{P′_{∘}} P_{∘}^{′gp}, and implies that a_{1} = a_{2} = 0 in P_{∘}^{′gp}. However, under the stated assumptions the localization map P′_{∘} → P_{∘}^{′gp} is injective, a contradiction. □

6.2. Integral monoids. We begin presently the study of a special class of monoids, the integral non-pointed monoids, and the subclass of saturated monoids (see definition 4.8.40(iii)). Later, we shall complement this section with further results on fine monoids (see sections 6.4 and 12.5). Throughout this section, all the monoids under consideration shall be non-pointed.

Definition 6.2.1. Let φ : M → N be a morphism of monoids.

(i) φ is said to be integral if, for any integral monoid M′, and any morphism M → M′, the push-out N ⊗_{M} M′ is integral.

(ii) φ is said to be strongly flat (resp. strongly faithfully flat) if the induced morphism ℤ[φ] : ℤ[M] → ℤ[N] is flat (resp. faithfully flat).

Lemma 6.2.2. Let f : M → N and g : N → P be two morphisms of monoids.

(i) If f and g are integral (resp. strongly flat), the same holds for g ∘ f.

(ii) If f is integral (resp. strongly flat), and M → M′ is any other morphism, then the

morphism 1_{M′} ⊗_{M} f : M′ → M′ ⊗_{M} N is integral (resp. strongly flat). (iii) If f is integral, and S ⊂ M and T ⊂ N are any two submonoids such that f(S) ⊂ T,

then the induced morphism S^{−1}M → T^{−1}N is integral. (iv) If S ⊂ M is any submonoid, the natural map M → S^{−1}M is strongly flat.

(v) If f is integral, then the same holds for f^{int}, and the natural map M^{int} ⊗_{M} N → N^{int} is an isomorphism.

(vi) The unit of adjunction M → M^{int} is an integral morphism.

Proof. (i) is obvious. Assertion (ii) for integral maps is likewise clear, and (ii) for strongly flat morphisms follows from (4.8.52). Assertion (iv) follows immediately from (4.8.53).

(iii): Let S^{−1}M → M′ be any map of integral monoids; in view of lemma 4.8.34, we have P := M′ ⊗_{S^{−1}M} T^{−1}N ≃ T^{−1}(M′ ⊗_{M} N), hence P is integral, which is the claim.

(v): The second assertion follows easily by comparing the universal properties (details left to the reader); using this and and (ii), we deduce that f^{int} is integral.

(vi) is left to the reader. □

Theorem 6.2.3. Let φ : M → N be a morphism of integral monoids. Consider the following conditions :

(a) φ is integral.

(b) φ is flat (see remark 4.8.23(vi)).

(c) φ is flat and injective.

(d) φ is strongly flat.

(e) For every field k, the induced map k[φ] : k[M] → k[N] is flat. Then : (e) ⇔ (d) ⇔ (c) ⇒ (b) ⇔ (a).

Proof. This result appears in [111, Prop.4.1(1)], with a different proof.

(a)⇒(b): Let I ⊂ M be any ideal; we consider the M-module :

R(M, I) := ⨁_{n∈ℕ} I^{n}

where I^{n} denotes, for each n ∈ ℕ, the n-th power of I in the monoid (𝒫(M), ·) of (6.1.1). Then there exists an obvious multiplication law on R(M, I), such that the latter is a ℕ-graded integral monoid, and the inclusion M → R(M, I) in degree zero is a morphism of monoids. We call R(M, I) the Rees monoid associated with M and I.

Denote also by j : R(M, I) → M ×ℕ the natural inclusion map. By assumption, the monoid R(M, I) ⊗_{M} N is integral. However, the natural map R(M, I) ⊗_{M} N → (R(M, I) ⊗_{M} N)^{gp} factors through j⊗_{M} N. The latter means that, for every n ∈ ℕ, the induced map I^{n}⊗_{M} N → N is injective. Then the assertion follows from proposition 6.1.40and remark 4.8.21(ii).

(b)⇒(a): Let M → M′ be any morphism of integral monoids; we need to show that the natural map M′ ⊗_{M} N → M^{′gp} ⊗_{M^{gp}} N^{gp} is injective (see remark 6.1.25(i)). The latter factors through the morphism M′ ⊗_{M} N → M^{′gp} ⊗_{M} N, which is injective by theorem 6.1.42 and remark 4.8.21(ii). We are thus reduced to proving the injectivity of the natural map :

M^{′gp} ⊗_{M^{gp}} (M^{gp} ⊗_{M} N) → M^{′gp} ⊗_{M^{gp}} N^{gp}.

By comparing the respective universal properties, it is easily seen that M^{gp} ⊗_{M} N is the localization φ(M)^{−1}N, which of course injects into N^{gp}. Then the contention follows from the following general :

Claim 6.2.4. Let G be a group, T → T′ an injective morphism of monoids, G → P a morphism of monoids. Then the natural map P ⊗_{G} T → P ⊗_{G} T′ is injective.

Proof of the claim. This follows easily from remark 6.1.25(i) and lemma 4.8.31(ii). ♢ (d)⇒(e) and (c)⇒(b) are trivial. (e)⇒(c): The flatness of φ has already been noticed in remark 6.1.35(ii). To show that φ is

injective, let a_{1}, a_{2} ∈ M and let k be any field. Under assumption (e), the image in k[N] of the annihilator Ann_{k[M]}(a_{1} − a_{2}) generates the ideal Ann_{k[N]}(φ(a_{1}) − φ(a_{2})). Set b := a_{1}a_{2}^{−1}; it follows that Ann_{k[M^{gp}]}(1 − b) generates Ann_{k[N^{gp}]}(1 − φ(b)). However, one checks easily that the annihilator of 1 − b in k[M^{gp}] is either 0 if b is not a torsion element of M^{gp}, or else is generated (as an ideal) by 1 + b + ⋯ + b^{n−1}, where n is the order of b in the group M^{gp}. Now, if φ(a_{1}) = φ(a_{2}), we have φ(b) = 1, hence the annihilator of 1 − b cannot be 0, and in fact k[N^{gp}] = Ann_{k[N^{gp}]}(1 − φ(b)) = nk[N^{gp}]. Since k is arbitrary, it follows that n = 1, i.e. a_{1} = a_{2}.

(c)⇒(d): since φ is injective, N_{∘} is an integral pointed M_{∘}-module, so the assertion is a special case of proposition 6.1.52. □

Remark 6.2.5. (i) Let G be a group; then every morphism of monoids M → G is integral. Indeed, lemma 6.2.2(i,vi) reduces the assertion to the case where M is integral, in which case it is an immediate consequence of theorem 6.2.3and example 6.1.23(vi).

(ii) Let M be an integral monoid, and S a flat pointed M_{∘}-module. From theorem 6.1.42we see that T := S\{0} is an M-submodule of S, hence S = T_{∘}.

(iii) Let φ : M → N be a morphism of integral monoids, and set Γ := Coker φ^{gp}; then the natural map π : N → Γ defines a grading on N (see definition 4.8.8(i)), which we call the φ-grading. As usual, we shall write N_{γ} instead of π^{−1}(γ), for every γ ∈ Γ. We shall use the additive notation for the composition law of Γ; especially, the neutral element shall be denoted by 0. Clearly φ factors through a morphism of monoids M → N_{0}, and each graded summand N_{γ} is naturally a M-module.

(iv) With the notation of (iii), we claim that the induced morphism \overline{φ} : φ(M) → N is flat (hence strongly flat, by theorem 6.2.3), if and only if N_{γ} is a filtered union of cyclic M-modules, for every γ ∈ Γ. Indeed, notice that a cyclic M-submodule of N_{γ} is a free φ(M)-module of rank one (since N is integral), hence the condition implies that N_{γ} is a flat φ(M)-module, hence \overline{φ} flat. Conversely, suppose that \overline{φ} is flat, and let n_{1}, n_{2} ∈ N_{γ} (for some γ ∈ Γ); this means that there exist a_{1}, a_{2} ∈ M such that φ(a_{1})n_{1} = φ(a_{2})n_{2} in N. Then, condition (F2) of theorem 6.1.42says that there exist n′ ∈ N and b_{1}, b_{2} ∈ M such that n_{i} = φ(b_{i})n′ for i = 1, 2; especially, n′ ∈ N_{γ}, which shows that N_{γ} is a filtered union of cyclic M-modules.

(v) With the notation of (iii), notice as well, that a morphism φ : M → N of integral monoids is exact (see definition 4.8.40(i)) if and only if Ker φ^{gp} ⊂ M and φ induces an isomorphism M/Ker φ^{gp} →^{∼} N_{0}. (Details left to the reader.)

Theorem 6.2.6. Let M → N be a finite, injective morphism of integral monoids, and S a pointed M_{∘}-module. Then S is M_{∘}-flat if and only if N_{∘} ⊗_{M_{∘}} S is N_{∘}-flat.

Proof. In light of theorem 6.1.42, we may assume that N_{∘} ⊗_{M_{∘}} S is N_{∘}-flat, and we shall show that S is M_{∘}-flat. To this aim, let 1 denote the trivial monoid (the initial and final object in the category of monoids); any pointed M_{∘}-module X is a pointed 1_{∘}-module by restriction of scalars, and if X and Y are any two pointed M_{∘}-modules, we define a M_{∘}-module structure on X ⊗_{1_{∘}} Y by the rule

a · (x ⊗ y) := ax ⊗ ay for every a ∈ M_{∘}, x ∈ X and y ∈ Y.

With this notation, we remark :

Claim 6.2.7. Let φ : M → G be a morphism of monoids, where G is a group and M is integral. Then a pointed M_{∘}-module S is M_{∘}-flat if and only if the same holds for the M_{∘}- module S ⊗_{1_{∘}} G_{∘}.

Proof of the claim. Suppose that S is M_{∘}-flat; then S = T_{∘} for some M-module T (remark 6.2.5(ii)), and it is easily seen that S ⊗_{1_{∘}} G_{∘} = (T ×G)_{∘}. By theorem 6.1.42, it suffices to check that conditions (F1) and (F2) of lemma 6.1.41hold for (T × G)_{∘}.

Hence, suppose that a ∈ M_{∘} and h ∈ (T × G)_{∘} satisfy the identity ah = 0; in this case, a simple inspection shows that either a = 0 or h = 0; condition (F1) follows straightforwardly. To check (F2), say that a_{1} · (s_{1}, g) = a_{2} · (s_{2}, g_{2}) ≠ 0, for some a_{i} ∈ M, s_{i} ∈ S, g_{i} ∈ G (i = 1, 2); since (F2) holds for S, we may find b_{1}, b_{2} ∈ M and t ∈ S such that a_{1}b_{1} = a_{2}b_{2} and b_{i}t = s_{i} (i = 1, 2). Notice that g := φ(b_{1})^{−1}g_{1} = φ(b_{2})^{−1}g_{2} in G, therefore b_{i} · (t, g) = (s_{i}, g_{i}) for i = 1, 2, whence the contention.

Conversely, suppose that S ⊗_{1_{∘}} G_{∘} is M_{∘}-flat; we wish to show that (F1) and (F2) hold for S. However, suppose that as = 0 for some a ∈ M_{∘} and s ∈ S with s ≠ 0; then a · (s ⊗ e) = 0 (where e ∈ G is the neutral element); since (F1) holds for S ⊗_{1_{∘}} G_{∘}, we deduce that there exist b ∈ M_{∘} and t ⊗ g ∈ S ⊗_{1_{∘}} G_{∘} such that ba = 0 and s ⊗ e = b · (t ⊗ g) = bt ⊗ φ(b)g; this implies that bt = s, so (F1) holds for S, as sought.

Lastly, suppose that a_{1}s_{1} = a_{2}s_{2} ≠ 0 for some a_{i} ∈ M and s_{i} ∈ S (i = 1, 2). It follows that a_{1} · (s_{1} ⊗ φ(a_{2})) = a_{2} · (s_{2} ⊗ φ(a_{1})). By applying condition (F2) to this identity in S ⊗_{1_{∘}} G_{∘}, we deduce that the same condition holds also for S. ♢

Next, we observe that there is a natural isomorphism of N_{∘}-modules :

(6.2.8) N_{∘} ⊗_{M_{∘}} (S ⊗_{1_{∘}} N_{∘}^{gp}) →^{∼} (N_{∘} ⊗_{M_{∘}} S) ⊗_{1_{∘}} N_{∘}^{gp}  n ⊗ (s ⊗ g) ↦ (n ⊗ s) ⊗ φ(n)g

whose inverse is given by the rule : (n ⊗ s) ⊗ g ↦ n ⊗ (s ⊗ φ(n)^{−1}g) for every n ∈ N, s ∈ S, g ∈ N_{∘}^{gp}. We leave to the reader the verification that these maps are well defined, and they are inverse to each other. In view of (6.2.8) and claim 6.2.7, we may then replace S by S ⊗_{1_{∘}} N_{∘}^{gp}, which allows to assume that S is an integral pointed M_{∘}-module and N_{∘} ⊗_{M_{∘}} S is an integral pointed N_{∘}-module. In this case, in view of proposition 6.1.52we know that ℤ⟨N_{∘} ⊗_{M_{∘}} S⟩ is a flat ℤ⟨N_{∘}⟩-module, and it suffices to show that ℤ⟨S⟩ is a flat ℤ⟨M_{∘}⟩-module.

However, under our assumptions, the ring homomorphism ℤ⟨M_{∘}⟩ → ℤ⟨N_{∘}⟩ is finite and injective, so the assertion follows from [86, Part II, Th.1.2.4] and (6.1.32). □

Lemma 6.2.9. Let M be an integral monoid, and S ⊂ M a submonoid. We have : (i) The natural map S^{−1}M^{sat} → (S^{−1}M)^{sat} is an isomorphism. (ii) If M is saturated, then M/S is saturated, and if S is a group, also the converse holds. (iii) The inclusion map M → M^{sat} is a local morphism. (iv) The inclusion M ⊂ M^{sat} induces a natural bijection :

Spec M^{sat} →^{∼} Spec M.

Proof. (i) and (iii) are left to the reader. For (ii), the natural isomorphism S^{−1}M/S^{gp} →^{∼} M/S, together with (i), reduces to the case where S is a group, in which case it suffices to remark that (M/S)^{sat} = M^{sat}/S. Lastly, to show (iv) it suffices to prove that, for any face F ⊂ M, the submonoid F^{sat} ⊂ M^{sat} is a face, and F^{sat} ∩ M = F, and that every face of M^{sat} is of this form. These assertions are easy exercises, which we leave as well to the reader. □

Lemma 6.2.10. Let M be a saturated monoid such that M^{♯} is fine. Then there exists an isomorphism of monoids :

M →^{∼} M^{♯} × M^{×}

and if M is fine, M^{×} is a finitely generated abelian group. Moreover, the projection M → M^{♯} induces a bijection :

Spec M^{♯} →^{∼} Spec M : 𝔭 ↦ 𝔭 × M^{×}.

Proof. Under the stated assumptions, G := M^{gp}/M^{×} is a free abelian group of finite rank, hence the projection M^{gp} → G admits a splitting σ : G → M^{gp}. Set M_{0} := M ∩ σ(G); it is easily seen that M = M_{0} × M^{×}, whence M_{0} ≃ M^{♯}. If M is fine, M^{gp} is a finitely generated abelian group, hence the same holds for its direct factor M^{×}. The last assertion can be proven directly, or can be regarded as a special case of lemma 6.1.17(i.b). □

Definition 6.2.11. Let φ : M → N be a morphism of integral monoids.

(i) We say that φ is k-saturated (for some integer k > 0), if the push-out P ⊗_{M} N is integral and k-saturated, for every morphism M → P with P integral and k-saturated. (ii) We say that φ is saturated, if the following holds. For every morphism of monoids M → P such that P is integral and saturated, the monoid P ⊗_{M} N is also integral and saturated.

Clearly, if φ is k-saturated for every integer k > 0, then φ is saturated.

Lemma 6.2.12. Let φ : M → N be a morphism of integral monoids, and S ⊂ M, T ⊂ N two submonoids such that φ(S) ⊂ T. The following holds :

(i) The localization map M → S^{−1}M is saturated.

(ii) If φ is saturated, the same holds for the morphisms S^{−1}M → T^{−1}N and M/S → N/T induced by φ.

(iii) If S and T are two groups, then φ is saturated if and only if the same holds for the

induced morphism M/S → N/T. (iv) If φ is saturated, then the natural map N ⊗_{M} M^{sat} → N^{sat} is an isomorphism.

Proof. (i) follows from the standard isomorphism : S^{−1}M ⊗_{M} N →^{∼} φ(S)^{−1}N, together with lemma 6.2.9(i). Next, let M/S → P and S^{−1}M → Q be morphisms of monoids. Then

P ⊗_{M/S} N/T ≃ (P ⊗_{M} N)/T and Q ⊗_{S^{−1}M} T^{−1}N ≃ T^{−1}(Q ⊗_{M} N)

(lemma 4.8.34) so assertions (ii) and (iii) follow from lemma 6.2.9(i,ii).

(iv) follows by comparing the universal properties. □

Example 6.2.13. (i) Let M be an integral monoid, I ⊂ M an ideal, and consider again the Rees monoid R(M, I) of the proof of theorem 6.2.3. Clearly R(M, I) is an integral monoid. However, easy examples show that, even when M is saturated, R(M, I) is not generally saturated. More precisely, the following holds. For every ideal J ⊂ M, set

J^{sat} := {a ∈ M^{gp} | a^{n} ∈ J^{n} for some integer n > 0}

where J^{n} denotes the n-th power of J in the monoid (𝒫, ·) of (6.1.1). Then J^{sat} is an ideal of M^{sat}. With this notation, we have the identity :

R(M, I)^{sat} = ⨁_{n∈ℕ}(I^{n})^{sat}.

(Verification left to the reader.)

(ii) For instance, take M := ℚ_{+}^{⊕2}, and let I ⊂ M be the ideal consisting of all pairs (x, y) such that x + y > 1. Then I^{n} = {(x, y) ∈ ℚ_{+}^{⊕2} | x + y > n}, and it is easily seen that I^{n} = (I^{n})^{sat} for every n ∈ ℕ, hence in this case R(M, I) is saturated, in view of (ii). It is easily seen that R(M, I) does not fulfill condition (F2) of lemma 6.1.41, hence the natural inclusion map i : M → R(M, I) is not flat (theorem 6.1.42), hence it is not an integral morphism, according to theorem 6.2.3. On the other hand, we have :

Lemma 6.2.14. With the notation of example 6.2.13(ii), the morphism i is saturated.

Proof. We prefer to work with the multiplicative notation, so we shall argue with the monoid (exp ℚ_{+}^{⊕2}, ·) (see (6.1)). Indeed, let φ : M → P be any morphism of monoids, with P saturated. Clearly R(M, I) is the direct sum of the P -modules P ⊗_{M} I^{n}, for all n ∈ ℕ.

Claim 6.2.15. The natural map P ⊗_{M} I^{n} → I^{n}P is an isomorphism, for every n ∈ ℕ

Proof of the claim. Indeed, the map is obviously surjective. Hence, suppose that a_{1}x_{1} = a_{2}x_{2} for some a_{1}, a_{2} ∈ I^{n}, x_{1}, x_{2} ∈ P such that x_{1} ⊗a_{1} = x_{2} ⊗a_{2}. For every ϑ ∈ ℚ with 0 ≤ ϑ ≤ 1, set a_{ϑ} := a_{1}^{ϑ} · a_{2}^{1−ϑ} and notice that

x_{ϑ} := a_{1}a_{ϑ}^{−1}x_{1} = a_{2}a_{ϑ}^{−1}x_{2} ∈ P^{gp}

and if N ∈ ℕ is large enough, so that Nϑ ∈ ℕ, then x_{ϑ}^{N} = x_{1}^{Nϑ}x_{2}^{Nϑ} ∈ P, hence x_{ϑ} ∈ P.

Next, for any (a, b), (a′, b′) ∈ ℚ_{+}^{⊕2}, set (a, b) ∨ (a′, b′) := (min(a, a′), min(b, b′)). Choose an increasing sequence 0 := ϑ_{0} < ϑ_{1} < ⋯ < ϑ_{n} := 1 of rational numbers, such that

b_{i} := a_{ϑ_{i}} ∨ a_{ϑ_{i+1}} ∈ I^{n} for every i = 0, …, n − 1.

Then there exist c_{i}, d_{i} ∈ ℚ_{+}^{⊕2} such that a_{ϑ_{i}} = b_{i}c_{i} and a_{ϑ_{i+1}} = b_{i}d_{i} for i = 0, …, n − 1. We may then compute in I^{n} ⊗_{M} P :

a_{ϑ_{i}} ⊗ x_{ϑ_{i}} = b_{i}c_{i} ⊗ x_{ϑ_{i}} = b_{i} ⊗ c_{i}x_{ϑ_{i}} and likewise a_{ϑ_{i+1}} ⊗ x_{ϑ_{i+1}} = b_{i} ⊗ d_{i}x_{ϑ_{i+1}}.

By construction, we have c_{i}x_{ϑ_{i}} = d_{i}x_{ϑ_{i+1}} for i = 0, …, n − 1, whence the contention. ♢

In view of claim 6.2.15, we are reduced to showing that R(P, IP) is saturated, and by example 6.2.13, this comes down to proving that (I^{n}P)^{sat} = I^{n}P for every n ∈ ℕ. However, say that x ∈ P^{gp}, and x^{n} = a_{1}x_{1} ⋯ a_{n}x_{n} for some a_{i} ∈ I and x_{i} ∈ P; set a := (a_{1} ⋯ a_{n})^{1/n}, and notice that a ∈ I. Then x^{n} = a^{n} · x_{1} ⋯ x_{n}, so that xa^{−1} ∈ P, and finally x ∈ IP, as required. □

Lemma 6.2.14shows that a saturated morphism is not necessarily integral. Notwithstanding, we shall see later that integrality holds for an important class of saturated morphisms (corollary 6.4.6). Now we wish to globalize the class of saturated morphisms, to an arbitrary topos. Of course, we could define the notion of saturated morphism of T -monoids, just by repeating word by word definition 6.2.11(ii). However, it is not clear that the resulting condition would be of a type which can be checked on stalks, in the sense of definition 4.7.11(v). For this reason, we prefer to proceed as in Tsuji’s work [159].

Lemma 6.2.16. Let T be a topos, φ : \underline{M} → \underline{N} and ψ : \underline{N} → \underline{P} two morphisms of integral T -monoids. We have:

(i) If φ and ψ are exact, the same holds for ψ ∘ φ.

(ii) If ψ ∘ φ is exact, the same holds for φ.

(iii) Consider a commutative diagram of integral T -monoids :

(6.2.17)  \xymatrix{\underline{M} \ar[r]^{φ} \ar[d]_{ψ} & \underline{N} \ar[d]^{ψ′} \\ _{\underline{M}_{′}} \ar[r]^{φ′} & \underline{N}′.}

Then the following holds :

(a) If (6.2.17) is a cartesian diagram and φ′ is exact, then φ is exact.

(b) If (6.2.17) is cocartesian (in the category Int.Mnd_{T}) and φ is exact, then φ′ is exact.

Proof. For all these assertions, remark 4.8.41(ii) easily reduces to the case where T = Set, which therefore we assume from start. Now, (i) and (ii) are left to the reader.

(iii.a): Let x ∈ M^{gp} such that φ^{gp}(x) ∈ N; hence (φ′ ∘ ψ)^{gp}(x) = ψ′(φ^{gp}(x)) ∈ N′, and therefore ψ^{gp}(x) ∈ M′, since φ′ is exact. It follows that x ∈ M, so φ is exact.

(iii.b): Let x ∈ (M′)^{gp} such that (φ′)^{gp}(x) ∈ N′; then we may write (φ′)^{gp}(x) = φ′(y)·ψ′(z) for some y ∈ M′ and z ∈ N. Therefore, (φ′)^{gp}(xy^{−1}) = ψ′(z); since the functor P ↦ P^{gp} commutes with colimits, it follows that we may find w ∈ M^{gp} such that ψ^{gp}(w) = xy^{−1} and φ^{gp}(w) = z. Since φ is exact, we deduce that w ∈ M, therefore xy^{−1} ∈ M′, and finally x ∈ M′, whence the contention. □

6.2.18. Let T be a topos; for two morphisms \underline{P} ← \underline{M} → \underline{N} of integral T -monoids, we set

\underline{N} ⊗^{int}_{M} \underline{P} := (\underline{N} ⊗_{M} \underline{P})^{int}

which is the push-out of these morphisms, in the category Int.Mnd_{T}. Notice that, for every morphism f : T′ → T of topoi, the natural morphism of T′-monoids

(6.2.19)  f^{∗}(\underline{N} ⊗^{int}_{M} \underline{P}) → f^{∗}\underline{N} ⊗^{int}_{f^{∗}M} f^{∗}\underline{P}

is an isomorphism (by lemmata 4.8.45(i) and 4.8.46(i)).

Let now φ : \underline{M} → \underline{N} be a morphism of integral T -monoids. For any integer k > 0, let k_{M} and k_{N} be the k-Frobenius maps of M and N (definition 4.8.40(ii)), and consider the cocartesian diagram :

(6.2.20)  \xymatrix{\underline{M} \ar[r]^{^{k^{M}}} \ar[d]_{φ} & \underline{M} \ar[d]^{φ′} \\ \underline{N} \ar[r]^{k_{M} ⊗^{int}_{M} 1_{N}} & \underline{P} .}

The endomorphism k_{N} factors through k_{M} ⊗^{int}_{M} 1_{N} and a unique morphism β : \underline{P} → \underline{N} such that β ∘ φ′ = φ. A simple inspection shows that k_{P} = (k_{M} ⊗_{M} 1_{N}) ∘ β. Now, if k_{M} is exact, then the same holds for k_{M} ⊗^{int}_{M} 1_{N}, in view of lemma 6.2.16(iii.b). Hence, if \underline{P} is k-saturated, then β is exact (lemma 6.2.16(ii)), and if \underline{M} is k-saturated, also the converse holds. Likewise, if \underline{M} is k-saturated and β is exact, then \underline{N} is k-quasi-saturated.

These considerations motivate the following :

Definition 6.2.21. Let T be a topos, φ : \underline{M} → \underline{N} a morphism of integral T -monoids. (i) A commutative diagram of integral T -monoids :

\xymatrix{\underline{M} \ar[r]^{φ} \ar[d] & \underline{N} \ar[d] \\ _{\underline{M}_{′}} \ar[r]^{φ′} & _{\underline{N}_{′}}}

is called an exact square, if the induced morphism \underline{M}′ ⊗^{int}_{M} \underline{N} → \underline{N}′ is exact. (ii) φ is said to be k-quasi-saturated if the commutative diagram :

(6.2.22)  \xymatrix{\underline{M} \ar[r]^{φ} \ar[d]_{_{k_{M}}} & \underline{N} \ar[d]^{_{k_{N}}} \\ \underline{M} \ar[r]^{φ} & \underline{N}}

is an exact square (the vertical arrows are the k-Frobenius maps). (iii) φ is said to be quasi-saturated if it is k-quasi-saturated for every integer k > 0.

Proposition 6.2.23. Let T be a topos.

(i) If (6.2.17) is an exact square, and \underline{M} → \underline{P} is any morphism of integral T -monoids, the square \underline{P} ⊗^{int}_{M} (6.2.17) is exact.

(ii) Consider a commutative diagram of integral T -monoids :

(6.2.24)  \xymatrix{\underline{M} \ar[r]^{φ_{1}} \ar[d]_{ψ} & \underline{N} \ar[d] \ar[r]^{φ_{2}} & \underline{P} \ar[d]^{ψ′} \\ _{\underline{M}_{′}} \ar[r]^{φ′_{1}} & _{\underline{N}_{′}} \ar[r]^{φ′_{2}} & _{\underline{P}_{′}}}

and suppose that the left and right square subdiagrams of (6.2.24) are exact. Then the same holds for the square diagram :

\xymatrix{\underline{M} \ar[r]^{φ_{2}∘φ_{1}} \ar[d]_{ψ} & \underline{P} \ar[d]^{ψ′} \\ _{\underline{M}_{′}} \ar[r]^{φ′_{2}∘φ′_{1}} & \underline{P}′.}

Proof. (i): We have a commutative diagram :

\xymatrix{(\underline{P} ⊗^{int}_{M} \underline{M}′) ⊗^{int}_{M} \underline{N} \ar[r]^{σ} \ar[d]_{ω} & \underline{P} ⊗^{int}_{M} (\underline{M}′ ⊗^{int}_{M} \underline{N}) \ar[d]^{1_{P} ⊗^{int}_{M} α} \\ (\underline{P} ⊗^{int}_{M} \underline{M}′) ⊗^{int}_{P} (\underline{P} ⊗^{int}_{M} \underline{N}) \ar[r]^{β} & _{\underline{P} ⊗^{int}_{M} \underline{N}_{′}}}

where ω, σ are the natural isomorphisms, and β and α : \underline{M}′ ⊗^{int}_{M} \underline{N} → \underline{N}′ are the natural maps. By assumption, α is exact, hence the same holds for 1_{P} ⊗^{int}_{M} α, by lemma 6.2.16(iii.b). So β is exact, which is the claim.

(ii): Let α : \underline{M}′ ⊗^{int}_{M} \underline{N} → \underline{N}′ and β : \underline{N}′ ⊗^{int}_{N} \underline{P} → \underline{P}′ be the natural maps; by assumptions, these are exact morphisms. However, we have a natural commutative diagram :

\xymatrix{(\underline{M}′ ⊗^{int}_{M} \underline{N}) ⊗^{int}_{N} \underline{P} \ar[r]^{ω} \ar[d]_{α ⊗^{int}_{N} 1_{P}} & \underline{M}′ ⊗^{int}_{M} \underline{P} \ar[d]^{γ} \\ _{\underline{N}_{′} ⊗^{int}_{N} \underline{P}} \ar[r]^{β} & _{\underline{P}_{′}}}

where ω is the natural isomorphism, and γ is the map deduced from ψ′ and φ′_{2} ∘ φ′_{1}. Then the assertion follows from lemma 6.2.16(i,iii.b). □

Corollary 6.2.25. Let T be a topos, φ : \underline{M} → \underline{N}, ψ : \underline{N} → \underline{P} two morphisms of integral T -monoids, and h, k > 0 any two integers. The following holds :

(i) If φ is both h-quasi-saturated and k-quasi-saturated, then φ is hk-quasi-saturated. (ii) If φ and ψ are k-quasi-saturated, the same holds for ψ ∘ φ. (iii) If \underline{M} → \underline{P} is any map of integral T -monoids, and φ is k-quasi-saturated (resp. quasi-

saturated), then the same holds for φ ⊗^{int}_{M} 1_{P} : \underline{P} → \underline{N} ⊗^{int}_{M} \underline{P}. (iv) Let \underline{S} ⊂ φ^{−1}(\underline{N}^{×}) be a T -submonoid. Then φ is k-quasi-saturated (resp. quasi-

saturated) if and only if the same holds for the induced map φ_{S} : \underline{S}^{−1}\underline{M} → \underline{N}.

(v) Let \underline{G} ⊂ Ker φ be a subgroup. Then φ is k-quasi-saturated (resp. quasi-saturated) if and only if the same holds for the induced map \overline{φ} : \underline{M}/\underline{G} → \underline{N}.

(vi) φ is quasi-saturated if and only if it is p-quasi-saturated for every prime number p. (vii) \underline{M} is k-saturated (resp. saturated) if and only if the unique morphism of T -monoids

\{1\} → \underline{M} is k-quasi-saturated (resp. quasi-saturated). (viii) If φ is k-quasi-saturated (resp. quasi-saturated), and \underline{M} is k-saturated (resp. satu-

rated), then \underline{N} is k-saturated (resp. saturated). (ix) If \underline{M} is integral, and \underline{N} is a T -group, φ is quasi-saturated.

Proof. (i) and (ii) are straightforward consequences of proposition 6.2.23(ii).

To show (iii), set \underline{P}′ := \underline{N} ⊗^{int}_{M} \underline{P} and let us remark that we have a commutative diagram

\xymatrix{\underline{P} \ar[d]_{φ ⊗^{int}_{M} 1_{P}} \ar[r] & \underline{P}_{1} \ar[r] \ar[d] & \underline{P} \ar[d]^{φ ⊗^{int}_{M} 1_{P}} \\ _{\underline{P}_{′}} \ar[r] & _{\underline{P}_{2}} \ar[r] & _{\underline{P}_{′}}}

such that :

• the composition of the top (resp. bottom) arrows is the k-Frobenius map

• the left square subdiagram is (6.2.22) ⊗^{int}_{M} \underline{P}

• the right square subdiagram is cocartesian (hence exact). then the assertion follows from proposition 6.2.23.

(iv): Suppose that φ is k-quasi-saturated. Then the same holds for \underline{S}^{−1}φ : \underline{S}^{−1}M → \underline{S}^{−1}M ⊗^{int}_{M} \underline{N}, according to (iii). However, we have a natural isomorphism \underline{S}^{−1}\underline{M} ⊗^{int}_{M} \underline{N} →^{∼} φ(\underline{S})^{−1}\underline{N} = N, so φ_{S} is k-quasi-saturated.

Conversely, suppose that φ_{S} is k-quasi-saturated. By (ii), in order to prove that the same holds for φ, it suffices to show that the localisation map \underline{M} → \underline{S}^{−1}\underline{M} is k-quasi-saturated. But this is clear, since (6.2.22) becomes cocartesian if we take for φ the localisation map.

(v): To begin with, \underline{M}/\underline{G} is an integral monoid, by lemma 4.8.38. Suppose that φ is k-quasi-saturated. Arguing as in the proof of (iv) we see that the natural map \underline{N} → (\underline{M}/\underline{G}) ⊗^{int}_{M} \underline{N} is an isomorphism, hence \overline{φ} is k-quasi-saturated, by (iii).

Conversely, suppose that \overline{φ} is k-quasi-saturated. By (ii), in order to prove that φ is saturated, it suffices to show that the same holds for the projection \underline{M} → \underline{M}/\underline{G}. By (iii), we are further reduced to showing that the unique map \underline{G} → \{1\} is k-quasi-saturated, which is trivial.

(vi) is a straightforward consequence of (i). Assertion (vii) can be verified easily on the definitions. Next, suppose that φ is quasi-saturated and \underline{M} is saturated. By (ii) and (vii) it follows that the unique morphism \{1\} → \underline{N} is quasi-saturated, hence (vii) implies that \underline{N} is saturated, so (viii) holds.

(ix): In view of (iv), we are reduced to the case where \underline{M} is also a group, in which case the assertion is obvious. □

Proposition 6.2.26. Let φ : M → N be an integral morphism of integral monoids, k > 0 an integer. The following conditions are equivalent :

(a) φ is k-quasi-saturated.

(b) φ is k-saturated.

(c) The push-out P of the cocartesian diagram (6.2.20), is k-saturated.

Proof. (a)⇒(b) by virtue of corollary 6.2.25(iii,viii), and trivially (b)⇒(c). Lastly, the implication (c)⇒(a) was already remarked in (6.2.18). □

Corollary 6.2.27. Let φ : M → N be an integral morphism of integral monoids. Then φ is quasi-saturated if and only if it is saturated. □

Proposition 6.2.26motivates the following :

Definition 6.2.28. Let T be a topos, φ : \underline{M} → \underline{N} a morphism of integral T -monoids, k > 0 an integer. We say that φ is k-saturated (resp. saturated) if φ is integral and k-quasi-saturated (resp. and quasi-saturated).

In case T = Set, we see that an integral morphism of integral monoids is saturated in the sense of definition 6.2.28, if and only if it is saturated in the sense of the previous definition 6.2.11. We may now state :

Corollary 6.2.29. Let P(T, φ) be the property “φ is an integral (resp. exact, resp. k-saturated, resp. saturated) morphism of integral T -monoids” (for a topos T). Then P can be checked on stalks. (See definition 4.7.11(v).)

Proof. The fact that integrality of a T -monoid can be checked on stalks, has already been established in lemma 4.8.46(ii). For the property “φ is an integral morphism” (between integral T -monoids), it suffices to apply theorem 6.2.3and proposition 4.8.26.

For the property “φ is exact” one applies lemma 4.8.46(iii). From this, and from (6.2.19) one deduces that also the properties of k-saturation and saturation can be checked on stalks. □

Lemma 6.2.30. Let f : M → N be a morphism of integral monoids. We have :

(i) If f is exact, then f is local.

(ii) Conversely, if f is an integral and local morphism, then f is exact.

Proof. (i) is left to the reader as an exercise.

(ii): Suppose x ∈ M^{gp} is an element such that b := f_{S}(x) ∈ N. Write x = z^{−1}y for certain y, z ∈ M; therefore b · f(z) = f(y) holds in N, and theorems 6.2.3, 6.1.42 imply that there exist c ∈ N and a_{1}, a_{2} ∈ M, such that 1 = c · f(a_{1}), b = c · f(a_{2}) and ya_{1} = za_{2}. Since f is local, we deduce that a_{1} ∈ M^{×}, hence x = a_{1}^{−1}a_{2} lies in M. □

Proposition 6.2.31. Let f : M → N be an integral map of integral monoids, k > 0 an integer, and N = ⨁_{γ∈Γ} N_{γ} the f-grading of N (see remark 6.2.5(iii)). Then the following conditions are equivalent :

(a) f is k-quasi-saturated.

(b) N_{kγ} = N_{γ}^{k} for every γ ∈ Γ. (Here the k-th power of subsets of N is taken in the monoid 𝒫(N), as in (6.1.1).)

Proof. (a)⇒(b): Let π : N^{gp} → Γ be the projection, suppose that y ∈ N_{kγ} for some γ ∈ Γ, and pick x ∈ N^{gp} such that π(x) = γ. This means that y = x^{k} · f^{gp}(z) for some z ∈ M^{gp}. By (a), it follows that we may find a pair (a, b) ∈ M × N and an element w ∈ M^{gp}, such that (aw^{−k}, bw) = (z, x) in M^{gp} × N^{gp}. Especially, b, b · f(a) ∈ N_{γ}, and consequently y = b^{k−1} · (b · f(a)) ∈ N_{kγ}, as stated.

(b)⇒(a): Let S := f^{−1}(N^{×}); by lemmata 6.2.30(ii) and 6.2.2(iii), the induced map f_{S} : S^{−1}M → N is exact and integral. Moreover, by corollary 6.2.25(iv), f is k-quasi-saturated if and only if the same holds for f_{S}, and clearly the f-grading of N agrees with the f_{S}-grading. Hence we may replace f by f_{S} and assume from start that f is exact. In this case, G := Ker f^{gp} ⊂ M, and corollary 6.2.25(v) says that f is k-quasi-saturated if and only if the same holds for the induced map \overline{f} : M/G → N; moreover \overline{f} is still integral, since it is deduced from f by push-out along the map M → M/G. Hence we may replace f by \overline{f}, thereby reducing to the case where f is injective. Also, f is flat, by theorem 6.2.3. The assertion boils down to the following. Suppose that (x, y) ∈ M^{gp} × N^{gp} is a pair such that a := f^{gp}(x) · y^{k} ∈ N; we have to show that there exists a pair (m, n) ∈ M × N whose class in the push-out P of the diagram

N ⟵^{f} M ⟶^{k_{M}} M

agrees with the image of (x, y) in P^{gp}. However, set γ := π(y), and notice that a ∈ N_{kγ}, hence we may write a = n_{1} ⋯ n_{k} for certain n_{1}, …, n_{k} ∈ N_{γ}. Then, according to remark 6.2.5(iv), we may find n ∈ N_{γ} and x_{1}, …, x_{k} ∈ M such that n_{i} = n · f(x_{i}) for every i = 1, …, k. It follows that y · n^{−1} ∈ f^{gp}(M^{gp}), say y = n · f(z) for some z ∈ M^{gp}. Set m := x_{1} ⋯ x_{k}; then (x, y) = (x, n · f(z)) represents the same class as (x · z^{k}, n) in P^{gp}. Especially, f(x · z^{k}) · n^{k} = a = f(m) · n^{k}, hence m = x · n^{k}, since f is injective. The claim follows. □

Corollary 6.2.32. Let f : M → N be an integral and local morphism of integral and sharp monoids. Then :

(i) f is exact and injective.

(ii) If furthermore, f is saturated, then Coker f^{gp} is a torsion-free abelian group.

Proof. (i): It follows from lemma 6.2.31 that f is exact. Next, suppose that f(a_{1}) = f(a_{2}) for some a_{1}, a_{2} ∈ M. By theorem 6.2.3 and 6.1.42, it follows that there exist b_{1}, b_{2} ∈ M and t ∈ N such that 1 = f(b_{1})t = f(b_{2})t and a_{1}b_{1} = a_{2}b_{2}. Since N is sharp, we deduce that f(b_{1}) = f(b_{2}) = 1, and since f is local and M is sharp, we get b_{1} = b_{2} = 1, thus a_{1} = a_{2}, whence x = 1, which is the contention.

(ii): Let us endow N with its f-grading. Suppose now that g ∈ G is a torsion element, and say that g^{k} = 1 in G for some integer k > 0; by propositions 6.2.31 and 6.2.26, we then have N_{1} = N_{g}^{k}. Especially, there exist a_{1}, …, a_{k} ∈ N_{g}, such that a_{1} ⋯ a_{k} = 1. Since N is sharp, we must have a_{i} = 1 for i = 1, …, k, hence g = 1. □

Corollary 6.2.33. Let φ : M → N be a morphism of integral monoids, F ⊂ N any face, and φ_{F} : φ^{−1}F → F the restriction of φ.

(i) If φ is flat, then the same holds for φ_{F}, and the induced map Coker φ_{F}^{gp} → Coker φ^{gp} is injective.

(ii) If φ is saturated, the same holds for φ_{F}.

Proof. (i): The fact that φ_{F} is flat, is a special case of corollary 6.1.51. The assertion about cokernels boils down to showing that the induced diagram of abelian groups :

\xymatrix{(φ^{−1}F)^{gp} \ar[r]^{φ_{F}^{gp}} \ar[d] & F^{gp} \ar[d] \\ M^{gp} \ar[r]^{φ^{gp}} & N^{gp}}

is cartesian. However, say that φ^{gp}(a_{1}a_{2}^{−1}) = f_{1}f_{2}^{−1} for some a_{1}, a_{2} ∈ M and f_{1}, f_{2} ∈ F. This means that φ(a_{1})f_{2} = φ(a_{2})f_{1} in N; by condition (F2) of theorem 6.1.42, we deduce that there exist b_{1}, b_{2} ∈ M and t ∈ N such that b_{2}a_{1} = b_{1}a_{2} and f_{i} = t · φ(b_{i}) (i = 1, 2). Since F is a face, it follows that b_{1}, b_{2} ∈ φ^{−1}F, hence a_{1}a_{2}^{−1} = b_{1}b_{2}^{−1} ∈ (φ^{−1}F)^{gp}, as required.

(ii): By (i), the morphism φ_{F} is integral, hence it suffices to show that φ_{F} quasi-saturated (proposition 6.2.26), and to this aim we shall apply the criterion of proposition 6.2.31. Indeed, let N = ⨁_{γ∈Γ} N_{γ} (resp. F = ⨁_{γ∈Γ_{F}} F_{γ}) be the φ-grading (resp. the φ_{F} -grading); according to (i), the induced map j : Γ_{F} → Γ is injective. This means that F_{γ} = F ∩ N_{j(γ)} for every γ ∈ Γ_{F}. Hence, for every integer k > 0 we may compute

F_{kγ} = F ∩ N_{j(γ)}^{k} = (F ∩ N_{j(γ)})^{k} = F_{γ}^{k}

where the first identity follows by applying proposition 6.2.31 (and proposition 6.2.26) to the saturated map φ, and the second identity holds because F is a face of N. □

6.3. Polyhedral cones. Fine monoids can be studied by means of certain combinatorial objects, which we wish to describe. Part of the material that follows is borrowed from [73]. Again, all the monoids in this section are non-pointed.

6.3.1. Quite generally, a convex cone is a pair (V, σ), where V is a finite dimensional ℝ-vector space, and σ ⊂ V is a non-empty subset such that :

ℝ_{+} · σ = σ = σ + σ

where the addition is formed in the monoid (𝒫(V), +) as in (6.1.1), and scalar multiplication by the set ℝ_{+} is given by the rule :

ℝ_{+} · S := {r · s | r ∈ ℝ_{+}, s ∈ S} for every S ∈ 𝒫(V) .

A subset S ⊂ σ is called a ray of σ, if it is of the form ℝ_{+} · {s}, for some s ∈ σ\{0}.

We say that (V, σ) is a closed convex cone if σ is closed as a subset of V (of course, V is here regarded as a topological space via any choice of isomorphism V ≃ ℝ^{n}). We denote by ⟨σ⟩ ⊂ V the ℝ-vector space generated by σ. To a convex cone (V, σ) one assigns the dual cone (V^{∨}, σ^{∨}), where V^{∨} := Hom_{ℝ}(V, ℝ), the dual of V, and :

σ^{∨} := {u ∈ V^{∨} | u(v) ≥ 0 for every v ∈ σ}.

Also, the opposite cone of σ is the cone

−σ := {−v ∈ V | v ∈ σ}.

Notice that σ^{∨} and −σ are always closed cones. Notice as well that the restriction of the addition law of V determines a monoid structure (σ, +) on the set σ. A map of cones

φ : (W, σ_{W}) → (V, σ_{V})

is an ℝ-linear map φ : W → V such that φ(σ_{W}) ⊂ σ_{V}. Clearly, the restriction of φ yields a map of monoids (σ_{W} , +) → (σ_{V} , +). If S ⊂ V is any subset, we set :

S^{⊥} := {u ∈ V^{∨} | u(s) = 0 for every s ∈ S} ⊂ V^{∨}.

Lemma 6.3.2. Let (V, σ) be a closed convex cone, Then, under the natural identification V →^{∼} (V^{∨})^{∨}, we have (σ^{∨})^{∨} = σ.

Proof. This follows from [40, Ch.II, §5, n.3, Cor.5]. □

6.3.3. A convex polyhedral cone is a cone (V, σ) such that σ is of the form :

σ := {r_{1}v_{1} + ⋯ + r_{s}v_{s} ∈ V | r_{i} ≥ 0 for every i ≤ s}

for a given set of vectors v_{1}, …, v_{s} ∈ V, called generators for the cone σ. One also says that {v_{1}, …, v_{s}} is a generating set for σ, and that ℝ_{+} · v_{1}, …, ℝ_{+} · v_{s} are generating rays for σ. We say that σ is a simplicial cone, if it is generated by a system of linearly independent vectors.

Lemma 6.3.4. Let (V, σ) be a convex polyhedral cone, S a finite generating set for σ. Then :

(i) For every v ∈ σ there is a subset T ⊂ S consisting of linearly independent vectors, such that v is contained in the convex polyhedral cone generated by T.

(ii) (V, σ) is a closed convex cone.

Proof. (i): Let T ⊂ S be a subset such that v is contained in the cone generated by T; up to replacing T by a subset, we may assume that T is minimal, i.e. no proper subset of T generates a cone containing v. We claim that T consists of linearly independent vectors. Otherwise, we may find a linear relation of the form ∑_{w∈T} a_{w} · w = 0, for certain a_{w} ∈ ℝ, at least one of which is non-zero; we may then assume that :

(6.3.5)  a_{w} > 0  for at least one vector w ∈ T.

Say also that v = ∑_{w∈T} b_{w} · w with b_{w} ∈ ℝ_{+}; by the minimality assumption on T, we must actually have b_{w} > 0 for every w ∈ T. We deduce the identity : v = ∑_{w∈T} (b_{w} − ta_{w}) · w for every t ∈ ℝ; let t_{0} be the supremum of the set of positive real numbers t such that b_{w} − ta_{w} ≥ 0 for every w ∈ T. In view of (6.3.5) we have t_{0} ∈ ℝ_{+}; moreover b_{w} −t_{0}a_{w} ≥ 0 for every w ∈ T, and b_{w} − t_{0}a_{w} = 0 for at least one vector w, which contradicts the minimality of T.

(ii): In view of (i), we are reduced to the case where σ is generated by finitely many linearly independent vectors, and for such cones the assertion is clear (details left to the reader). □

6.3.6. A face of a convex cone σ is a subset of the form σ ∩ Ker u, for some u ∈ σ^{∨}. The dimension (resp. codimension) of a face τ of σ is the dimension of the ℝ-vector space ⟨τ⟩ (resp. of the ℝ-vector space ⟨σ⟩/⟨τ⟩). A facet of σ is a face of codimension one. Notice that if σ is a polyhedral cone, and τ is a face of σ, then (V, τ) is also a convex polyhedral cone; indeed if S ⊂ V is a generating set for σ, then S ∩ τ is a generating set for τ.

Lemma 6.3.7. Let (V, σ) be a convex polyhedral cone. Then the faces of (V, σ) are the same as the faces of the monoid (σ, +).

Proof. We may assume that σ ≠ {0}, and let v_{1}, …, v_{n} be a system of generators for (V, σ). Let F be a face of the polyhedral cone σ, and pick u ∈ σ^{∨} such that F = σ ∩ Ker u. Then σ \ F = {x ∈ σ | u(x) > 0}, and this is an ideal of the monoid (σ, +); so F is a face of (σ, +).

Conversely, suppose that F is a face of (σ, +). First, let us show that F is a cone in V. Indeed, let f ∈ F, and r > 0 any real number; we have to prove that r · f ∈ F. To this aim, it suffices to show that (r/N) · f ∈ F for some integer N > 0, so that, after replacing r by r/N for N large enough, we may assume that 0 < r < 1. In this case, (1 − r) · f ∈ σ, and we have f = r · f + (1 − r) · f, so that r · f ∈ F, since F is a face of (σ, +). Notice now that S := ℝ_{+}v_{1} ∪ ⋯ ∪ ℝ_{+}v_{n} is a system of generators for (σ, +); therefore S ∩ F is a system of generators for F (lemma 6.1.20(ii)), and it is a union of generating rays, since F is a cone. Thus, F is the polyhedral cone spanned by {v_{1}, …, v_{n}} ∩ F.

Let N := σ + (−F); it is a convex polyhedral cone, since the same holds for σ and F. Now, let u_{1}, …, u_{k} ∈ N^{∨} be a system of generators of the ℝ-subspace of V^{∨} spanned by N^{∨}, and set u := u_{1}+⋯+u_{k}. Notice that F ⊂ Ker u, since F ∪(−F) ⊂ N. Conversely, let x ∈ σ∩Ker u; then −x ∈ Ker u_{i} for every i = 1, …, k, and therefore −x ∈ N^{∨∨} = N (lemmata 6.3.2 and 6.3.4(ii)). Hence, we may write −x = m − f for some m ∈ σ and f ∈ F, or equivalently, f = m+x, which shows that x ∈ F. Summing up, F = σ∩Ker u, i.e. F is a face of (V, σ). □

Proposition 6.3.8. Let (V, σ) be a convex polyhedral cone. The following holds : (i) Any intersection of faces of σ is still a face of σ. (ii) If τ is a face of σ, and γ is a face of (V, τ), then γ is a face of σ. (iii) Every proper face of σ is the intersection of the facets that contain it.

Proof. (i): Say that τ_{i} = σ ∩ Ker u_{i}, where u_{1}, …, u_{n} ∈ σ^{∨}. Then ⋂_{i=1}^{n} τ_{i} = σ ∩ Ker ∑_{i=1}^{n} u_{i}.

(ii): Say that τ = σ ∩ Ker u and γ = τ ∩ Ker v, where u ∈ σ^{∨} and v ∈ τ^{∨}. Then, for large r ∈ ℝ_{+}, the linear form v′ := v + ru is non-negative on any given finite generating set of σ, hence it lies in σ^{∨}, and γ = σ ∩ Ker v′.

(iii): To begin with, we prove the following :

Claim 6.3.9. (i) Every proper face of σ is contained in a facet.

(ii) Every face of σ of codimension 2 is the intersection of exactly two facets.

Proof of the claim. We may assume that ⟨σ⟩ = V. Let τ be a face of σ of codimension at least two, and denote by \overline{σ} be the image of σ in the quotient \overline{V} := V/⟨τ⟩; clearly (\overline{V} , \overline{σ}) is again a polyhedral cone. Moreover, choose u ∈ σ^{∨} such that τ = σ ∩ Ker u; the linear form u descends to \overline{u} ∈ \overline{σ}^{∨}, therefore \overline{σ} ∩ Ker \overline{u} = \{0\} is a face of \overline{σ}.

(ii): If τ has codimension two, \overline{V} has dimension two. Suppose that the assertion is known for (\overline{V} , \overline{σ}); then we find exactly two facets \overline{γ}_{1}, \overline{γ}_{2} of \overline{σ} whose intersection is \{0\}. Their preimages in V intersect σ in facets γ_{1}, γ_{2} that satisfy γ_{1} ∩ γ_{2} = τ. Hence, we may assume from start that τ = {0} and V has dimension two, in which case the verification is easy, and shall be left to the reader.

(i): Arguing by induction on the codimension, it suffices to show that τ is contained in a proper face spanning a larger subspace. To this aim, suppose that the claim is known for \overline{σ}; since \{0\} is a face of \overline{σ} of codimension at least two, it is contained in a proper face \overline{γ}; the preimage γ of the latter intersects σ in a proper face containing τ. Thus again, we are reduced to the case where τ = {0}. Pick u_{0} ∈ σ^{∨} such that σ ∩ Ker u_{0} = {0}; choose also any other u_{1} ∈ V^{∨} such that σ ∩ Ker u_{1} ≠ {0}. Since dim_{ℝ} V^{∨} ≥ 2, we may find a continuous map f : [0, 1] → V^{∨} \{0} with f(0) = u_{0} and f(1) = u_{1}. Let ℙ_{+}(V) be the topological space of rays of V (i.e. the topological quotient V \ {0}/ ∼ by the equivalence relation such that v ∼ v′ if and only if v and v′ generate the same ray), and define likewise ℙ_{+}(V^{∨}); let also Z ⊂ P := ℙ_{+}(V) × ℙ_{+}(V^{∨}) be the incidence correspondence, i.e. the subset of all pairs (\overline{v}, \overline{u}) such that u(v) = 0, for any representative u of the class \overline{u} and v of the class \overline{v}. Finally, let ℙ_{+}(σ) ⊂ ℙ_{+}(V) be the image of σ \{0}. Then Z (resp. ℙ_{+}(σ)) is a closed subset of P (resp. of ℙ_{+}(V)), hence Y := Z ∩ (ℙ_{+}(σ) × ℙ_{+}(V^{∨})) is a closed subset of P. Since the projection π : P → ℙ_{+}(V^{∨}) is proper, π(Y) is closed in ℙ_{+}(V^{∨}). Let \overline{f} : [0, 1] → ℙ_{+}(V^{∨}) be the composition of f and the natural projection V^{∨} \{0} → ℙ_{+}(V^{∨}); then f^{−1}(π(Y)) is a closed subset of [0, 1], hence it admits a smallest element, say a (notice that a > 0). Moreover, u_{a} ∈ σ^{∨}; indeed, otherwise we may find v ∈ σ\{0} such that u_{a}(v) < 0, and since u_{0}(v) > 0, we would have u_{b}(v) = 0 for some b ∈ (0, a). The latter means that f(b) ∈ π(Y), which contradicts the definition of a. Since by construction, σ ∩ Ker u_{a} ≠ {0}, the claim follows. ♢

Let τ be any face of σ; to show that (iii) holds for τ, we argue by induction on the codimension of τ. The case of codimension 2 is covered by claim 6.3.9(ii). If τ has codimension > 2, we apply claim 6.3.9(i) to find a proper face γ containing τ; by induction, τ is the intersection of facets of γ, and each of these is the intersection of two facets in σ (again by claim 6.3.9(ii)), whence the contention. □

6.3.10. Suppose σ spans V (i.e. ⟨σ⟩ = V) and let τ be a facet of σ; by definition there exists an element u_{τ} ∈ σ^{∨} such that τ = σ ∩ Ker u_{τ}, and one sees easily that the ray R_{τ} := ℝ_{+} · u_{τ} ⊂ σ^{∨} is well-defined, independently of the choice of u_{τ}. Hence the half-space :

H_{τ} := {v ∈ V | u_{τ}(v) ≥ 0}

depends only on τ. Recall that the interior (resp. the topological closure) of a subset E ⊂ V is the largest open subset (resp. the smallest closed subset) of V contained in E (resp. containing E). The topological boundary of E is the intersection of the topological closures of E and of its complement V \E.

Proposition 6.3.11. Let (V, σ) be a convex polyhedral cone, such that σ spans V. We have: (i) The topological boundary of σ is the union of its facets. (ii) If σ ≠ V, then σ = ⋂_{τ⊂σ} H_{τ}, where τ ranges over the facets of σ. (iii) The rays R_{τ}, where τ ranges over the facets of σ, generate the cone σ^{∨}.

Proof. (i): Notice that σ spans V if and only if the interior of σ is not empty. A proper face τ is the intersection of σ with a hyperplane Ker u ⊂ V with u ∈ σ^{∨} \{0}; therefore, every neighborhood U ⊂ V of any point v ∈ τ intersects V \ σ. This shows that τ lies in the topological boundary of σ.

Conversely, if v is in the boundary of σ, choose a sequence (v_{i} | i ∈ ℕ) of points of V \σ, converging to the point v; by lemma 6.3.2, for every i ∈ ℕ there exists u_{i} ∈ σ^{∨} such that u_{i}(v_{i}) < 0. Up to multiplication by scalars, we may assume that the vectors u_{i} lie on some sphere in V^{∨} (choose any norm on V^{∨}); hence we may find a convergent subsequence, and we may then assume that the sequence (u_{i} | i ∈ ℕ) converges to an element u ∈ V^{∨}. Necessarily u ∈ σ^{∨}, therefore v lies on the face σ ∩ Ker u, and the assertion follows from proposition 6.3.8(iii).

(ii): Suppose, by way of contradiction, that v lies in every half-space H_{τ}, but v ∉ σ. Choose any point v′ in the interior of σ, and let t ∈ [0, 1] be the largest value such that w := tv + (1 − t)v′ ∈ σ. Clearly w lies on the boundary of σ, hence on some facet τ, by (i). Say that τ = σ ∩ Ker u; then u(v′) > 0 and u(w) = 0, so u(v) < 0, a contradiction.

(iii): When σ = V, there is nothing to prove, hence we may assume that σ ≠ V. In this case, suppose that u ∈ σ^{∨}, and u is not in the cone C generated by the rays R_{τ}. Applying lemma 6.3.2 to the cone (V^{∨}, C), we deduce that there exists a vector v ∈ V with v ∈ H_{τ} for all the facets τ of σ, and u(v) < 0, which contradicts (ii). □

Corollary 6.3.12. Let (V, σ) and (V, σ′) be two convex polyhedral cones. Then :

(i) (Farkas’ theorem) The dual (V^{∨}, σ^{∨}) is also a convex polyhedral cone.

(ii) If τ is a face of σ, then τ^{∗} := σ^{∨}∩τ^{⊥} is a face of σ^{∨} such that ⟨τ^{∗}⟩ = ⟨τ⟩^{⊥}. Especially:

(6.3.13)  dim_{ℝ}⟨τ⟩ + dim_{ℝ}⟨τ^{∗}⟩ = dim_{ℝ} V.

The rule τ ↦ τ^{∗} is a bijection from the set of faces of σ to those of σ^{∨}. The smallest face of σ is σ ∩ (−σ).

(iii) (V, σ ∩ σ′) is a convex polyhedral cone, and every face of σ ∩ σ′ is of the form τ ∩ τ′, for some faces τ of σ and τ′ of σ′.

Proof. (i): Set W := ⟨σ⟩ ⊂ V, and pick a basis u_{1}, …, u_{k} of W^{⊥}; by proposition 6.3.11(iii), the assertion holds for the dual (W^{∨}, σ^{∨}) of the cone (W, σ). However, W^{∨} ≃ V^{∨}/W^{⊥}, hence the dual cone (V^{∨}, σ^{∨}) is generated by lifts of generators of (W^{∨}, σ^{∨}), together with the vectors u_{i} and −u_{i}, for i = 1, …, k.

(ii): Notice first that the faces of σ^{∨} are exactly the cones σ^{∨} ∩ {u}^{⊥}, for u ∈ σ = (σ^{∨})^{∨}. For a given v ∈ σ, let τ be the smallest face of σ such that v ∈ τ; this means that τ^{∨} ∩ {v}^{⊥} = τ^{⊥} (where (V^{∨}, τ^{∨}) is the dual of (V, τ)). Hence σ^{∨}∩{v}^{⊥} = σ^{∨}∩(τ^{∨}∩{v}^{⊥}) = τ^{∗}, so every face of σ^{∨} has the asserted form. The rule τ ↦ τ^{∗} is clearly order-reversing, and from the obvious inclusion τ ⊂ (τ^{∗})^{∗} it follows that τ^{∗} = ((τ^{∗})^{∗})^{∗}, hence this map is a bijection. It follows from this, that the smallest face of σ is (σ^{∨})^{∗} = σ ∩ (σ^{∨})^{⊥} = (σ^{∨})^{⊥} = σ ∩ (−σ). In particular, we see that (σ ∩ (−σ))^{∗} = σ^{∨}, and furthermore, (6.3.13) holds for τ := σ ∩ (−σ). Identity (6.3.13) for a general face τ can be deduced by inserting τ in a maximal chain of faces of σ, and comparing with the dual chain of faces of σ^{∨} (details left to the reader). Finally, it is clear that ⟨τ⟩ ⊂ ⟨τ^{∗}⟩^{⊥}; since these spaces have the same dimension, we deduce that ⟨τ⟩^{⊥} = ⟨τ^{∗}⟩.

(iii): Indeed, lemma 6.3.2 implies that σ^{∨} + σ^{′∨} is the dual of σ ∩ σ′, hence (i) implies that σ ∩ σ′ is polyhedral. It also follows that every face τ of σ ∩ σ′ is the intersection of σ ∩ σ′ with the kernel of a linear form u + u′, for some u ∈ σ^{∨} and u′ ∈ σ^{′∨}. Consequently, τ = (σ ∩ Ker u) ∩ (σ′ ∩ Ker u′). □

Corollary 6.3.14. For a convex polyhedral cone (V, σ), the following conditions are equivalent: (a) σ ∩ (−σ) = {0}.

(b) σ contains no non-zero linear subspaces.

(c) There exists u ∈ σ^{∨} such that σ ∩ Ker u = {0}.

(d) σ^{∨} spans V^{∨}.

Proof. (a) ⇔ (b) since σ ∩ (−σ) is the largest subspace contained in σ. Next, (a) ⇔ (c) since σ ∩ (−σ) is the smallest face of σ. Finally, (a) ⇔ (d) since dim_{ℝ}⟨σ ∩ (−σ)⟩ + dim_{ℝ}⟨σ^{∨}⟩ = n (corollary 6.3.12(ii)). □

6.3.15. A convex polyhedral cone fulfilling the equivalent conditions of corollary 6.3.14 is said to be strictly convex. Suppose that (V, σ) is strictly convex; then proposition 6.3.11(iii) says that σ is generated by the rays R_{τ}, where τ ranges over the facets of σ^{∨}. The rays R_{τ} are uniquely determined by σ, and are called the extremal rays of σ. Moreover, these R_{τ} form the unique minimal set of generating rays for σ. Indeed, concerning the minimality : for each facet τ of σ^{∨}, pick v_{τ} ∈ σ with ℝ_{+} · v_{τ} = R_{τ}; suppose that v_{τ_{0}} = ∑_{τ∈S} t_{τ} · v_{τ} for some subset S of the set of facets of σ^{∨}, and appropriate t_{τ} > 0, for every τ ∈ S. It follows easily that u(v_{τ}) = 0 for every u ∈ τ_{0}, and every τ ∈ S. But by definition of R_{τ}, this implies that S = {τ_{0}}, which is the claim. Concerning uniqueness : suppose that Σ is another system of generating rays; especially, for any facet τ ⊂ σ^{∨}, the ray R_{τ} is in the convex cone generated by Σ; it follows easily that there exists ρ ∈ Σ such that u(ρ) = 0 for every u ∈ τ, in which case ρ = R_{τ}. This shows that Σ must contain all the extremal rays.

Example 6.3.16. (i) Suppose that dim_{ℝ} V = 2, and (V, σ) is a strictly convex polyhedral cone, and assume that σ generates V. Then the only face of codimension two of σ is {0}, so it follows easily from claim 6.3.9(ii) that σ admits exactly two facets, and these are also the extremal rays of σ, especially σ is simplicial. Of course, these assertions are rather obvious; in dimension > 2, the general situation is much more complicated.

(ii) Let (V, σ) be a convex polyhedral cone, and suppose that σ spans V. Let τ be a face of σ. Notice that (σ, +)^{gp} = (V, +), and (τ, +) is a face of the monoid (σ, +), by lemma 6.3.7. Hence we may view the localization τ^{−1}σ naturally as a submonoid of (V, +), and it is easily seen that τ^{−1}σ is a convex cone. By proposition 6.3.11(iii), the polyhedral cone σ^{∨} is generated by its extremal rays ℝl_{1}, …, ℝl_{n}, and by proposition 6.3.8(iii), we may assume that τ = σ ∩ Ker (l_{1} + ⋯ + l_{k}) for some k ≤ n. Clearly l_{i}(v) ≥ 0 for every v ∈ τ^{−1}σ and every i ≤ k. Conversely, if l ∈ (τ^{−1}σ)^{∨}, we must have τ ⊂ Ker l and l ∈ σ^{∨}; if we write l = ∑_{i=1}^{n} a_{i}l_{i} for some a_{i} ≥ 0, it follows easily that a_{i} = 0 for every i > k. On the other hand, suppose that v ∈ V satisfies the inequalities l_{i}(v) ≥ 0 for i = 1, …, k; then, for every i = k + 1, …, n we may find u_{i} ∈ τ such that l_{i}(v + u_{i}) ≥ 0, hence v + u_{k+1} + ⋯ + u_{n} ∈ σ, and therefore v ∈ τ^{−1}σ. This shows that τ^{−1}σ is a closed convex cone, and its dual (τ^{−1}σ)^{∨} is the convex cone generated by l_{1}, …, l_{k}; especially, it is a convex polyhedral cone, and then the same holds for τ^{−1}σ, by virtue of lemma 6.3.2and corollary 6.3.12(i).

(iii) In the situation of (ii), let v ∈ τ be any element that lies in the relative interior of τ, i.e. in the complement of the union of the facets of τ. Denote by S_{v} ⊂ τ the submonoid generated by v. Then we claim that

τ^{−1}σ = S_{v}^{−1}σ.

Indeed, let s ∈ σ and t ∈ τ be any two element; in view of proposition 6.3.11(i), it is easily seen that there exists an integer N > 0 such that v − N^{−1}t ∈ τ, hence t′ := Nv − t ∈ τ. Therefore s − t = (s + t′) − Nv ∈ S^{−1}σ, and the assertion follows.

Lemma 6.3.17. Let f : V → W be a linear map of finite dimensional ℝ-vector spaces, (V, σ) a convex polyhedral cone. The following holds :

(i) (W, f(σ)) is a convex polyhedral cone.

(ii) Suppose moreover, that σ ∩ Ker f does not contain non-zero linear subspaces of V. Then, for every face τ of f(σ) there exists a face τ′ of σ such that f(τ′) ⊂ τ, and f restricts to an isomorphism : ⟨τ′⟩ →^{∼} ⟨τ⟩.

Proof. (i) is obvious. To show (ii) we argue by induction on n := dim_{ℝ} Ker f. The assertion is obvious when n = 0, hence suppose that n > 0 and that the claim is already known whenever Ker f has dimension < n. Let τ be a face of f(σ); then f^{−1}τ is a face of f^{−1}f(σ), hence σ ∩ f^{−1}τ is a face of σ = σ ∩ f^{−1}f(σ). In view of proposition 6.3.8(ii), we may then replace σ by σ ∩ f^{−1}τ, and therefore assume from start that τ = f(σ). We may as well assume that V = ⟨σ⟩ and W = ⟨τ⟩. The assumption on σ implies especially that σ ≠ V, hence σ is the intersection of the half-spaces H_{γ} corresponding to its facets γ (proposition 6.3.11(ii)). For each facet γ of σ, let u_{γ} be a chosen generator of the ray R_{γ} (notation of (6.3.10)). Since σ ∩ Ker f does not contain non-zero linear subspaces, we may find a facet γ such that V′ := Ker u_{γ} does not contain Ker f. Then, the inductive assumption applies to the restriction f_{|V′} : V′ → W and the convex polyhedral cone σ ∩ V′, and yields a face τ′ of the latter, such that f induces an isomorphism ⟨τ′⟩ →^{∼} ⟨f(σ ∩ V′)⟩. Finally, ⟨f(σ ∩ V′)⟩ = W, since V′ does not contain Ker f. □

Lemma 6.3.18. Let f : V → W be a linear map of finite dimensional ℝ-vector spaces, f^{∨} : W^{∨} → V^{∨} the transpose of f, and (W, σ) a convex polyhedral cone. Then :

(i) (V, f^{−1}σ) is a convex polyhedral cone and (f^{−1}σ)^{∨} = f^{∨}(σ^{∨}).

(ii) For every face δ of f^{−1}σ, there exists a face τ of σ such that δ = f^{−1}τ. If furthermore, ⟨σ⟩+f(V) = W, we may find such a τ so that additionally, f induces an isomorphism:

V/⟨δ⟩ →^{∼} W/⟨τ⟩.

(iii) Conversely, for every face τ of σ, the cone f^{−1}τ is a face of f^{−1}σ, and (f^{−1}τ)^{∗} is the smallest face of (f^{−1}σ)^{∨} containing f^{∨}(τ^{∗}) (notation of corollary 6.3.12(ii)).

Proof. (i): By corollary 6.3.12(i), (W^{∨}, σ^{∨}) is a convex polyhedral cone, hence we may find u_{1}, …, u_{s} ∈ σ^{∨} such that σ = ⋂_{i=1}^{s} u_{i}^{−1}(ℝ_{+}). Therefore f^{−1}σ = ⋂_{i=1}^{s}(u_{i} ∘ f)^{−1}(ℝ_{+}). Let γ ⊂ V^{∨} be the cone generated by the set {u_{1} ∘f, …, u_{s} ∘f}; then f^{−1}σ = γ^{∨}, and the assertion results from lemma 6.3.2and a second application of (i).

(iii): For every u ∈ σ^{∨}, we have : f^{−1}(σ ∩ Ker u) = (f^{−1}σ) ∩ Ker u ∘ f. Since we already know that (f^{−1}σ)^{∨} = f^{∨}(σ^{∨}), we see that the faces of f^{−1}σ are exactly the subsets of the form f^{−1}τ, where τ ranges over the faces of σ. Next, for any such τ, the set f^{∨}(τ^{∗}) consists of all u ∈ V^{∨} of the form u = w ∘ f for some w ∈ σ^{∨} such that w(τ) = 0. From this description it is clear that f^{∨}(τ^{∗}) ⊂ (f^{−1}τ)^{∗}. To show that (f^{−1}τ)^{∗} is the smallest face containing f^{∨}(τ^{∗}), it then suffices to prove that f^{∨}(τ^{∗})^{⊥} ∩ f^{−1}σ ⊂ f^{−1}τ. However, let v ∈ f^{∨}(τ^{∗})^{⊥} ∩ f^{−1}σ; then w ∘ f(v) = 0 for every w ∈ τ^{∗}, i.e. f(v) ∈ (τ^{∗})^{∗} = τ, whence the contention.

(ii): The first assertion has already been shown; hence, suppose that ⟨σ⟩ + f(V) = W. We deduce that σ^{∨} ∩ Ker f^{∨} does not contain non-zero linear subspaces of W^{∨}; indeed, if u ∈ W^{∨}, and both u and −u lie in σ^{∨}, then u vanishes on ⟨σ⟩, and if u ∈ Ker f^{∨}, then u vanishes as well on f(V), hence u = 0. We may then apply lemma 6.3.17(ii) to find a face γ of σ^{∨} such that f^{∨}(γ) ⊂ δ^{∗} and f^{∨} restricts to an isomorphism : ⟨γ⟩ →^{∼} ⟨δ^{∗}⟩. Especially, δ^{∗} is the smallest face of (f^{−1}σ)^{∨} containing f^{∨}(γ), hence δ^{∗} = (f^{−1}γ^{∗})^{∗} by (iii), i.e. δ = f^{−1}γ^{∗}. We also deduce that f induces an isomorphism : V/⟨δ^{∗}⟩^{⊥} →^{∼} W/⟨γ⟩^{⊥}. Since ⟨δ^{∗}⟩^{⊥} = ⟨δ⟩ and ⟨γ⟩^{⊥} = ⟨γ^{∗}⟩, the second assertion holds with τ := γ^{∗}. □

Lemma 6.3.19. Let (V, σ) and (V′, σ′) be two convex polyhedral cones. Then :

(i) (V ⊕ V′, σ × σ′) is a convex polyhedral cone.

(ii) Every face of σ × σ′ is of the form τ × τ′, for some faces τ of σ and τ′ of σ′.

Proof. Indeed, σ × σ′ = (p_{1}^{−1}σ) ∩ (p_{2}^{−1}σ′), where p_{1} and p_{2} are the natural projections of V ⊕ V′ onto V and V′. Hence, assertions (i) and (ii) follow from corollary 6.3.12(iii) and lemma 6.3.18(i). □

6.3.20. Let (L, +) be a free abelian group of finite rank, σ ⊂ L_{ℝ} := L ⊗_{ℤ} ℝ a convex polyhedral cone. We say that σ is L-rational (or briefly : rational, when there is no danger of ambiguity) if it admits a generating set consisting of elements of L. Then it is clear that every face of a rational convex polyhedral cone is again rational (see (6.3.6)). On the other hand, let

(M, +) ⊂ (L, +)

be a submonoid of L; we shall denote by (L_{ℝ}, M_{ℝ}) the convex cone generated by M (i.e. the smallest convex cone in L_{ℝ} containing the image of M). If M is fine, M_{ℝ} is a convex polyhedral cone. Later we shall also find useful to consider the subset :

M_{ℚ} := {m ⊗ q | m ∈ M, q ∈ ℚ_{+}} ⊂ L_{ℚ} := L ⊗_{ℤ} ℚ

which is a submonoid of L_{ℚ}.

Proposition 6.3.21. Let (L, +) be a free abelian group of finite rank, with dual

L^{∨} := Hom_{ℤ}(L, ℤ).

Let also (L_{ℝ}, σ) and (L_{ℝ}, σ′) be two L-rational convex polyhedral cones. We have : (i) The dual (L_{ℝ}^{∨}, σ^{∨}) is an L^{∨}-rational convex polyhedral cone. (ii) (L_{ℝ}, σ ∩ σ′) is also an L-rational convex polyhedral cone. (iii) Let g : L′ → L (resp. h : L → L′) be a map of free abelian groups of finite rank,

and denote by g_{ℝ} : L′_{ℝ} → L_{ℝ} (resp. h_{ℝ} : L_{ℝ} → L′_{ℝ}) the induced ℝ-linear map. Then, (L′_{ℝ}, g_{ℝ}^{−1}σ) and (L′_{ℝ}, h_{ℝ}(σ)) are L′-rational.

(iv) Let L′ be another free abelian group of finite rank, and (L′_{ℝ}, σ′) an L′-rational convex polyhedral cone. Then (L_{ℝ} ⊕ L′_{ℝ}, σ × σ′) is L ⊕ L′-rational.

Proof. (i) and (ii) follow easily, by inspecting the proof of corollary 6.3.12(i),(iii) : the details shall be left to the reader.

(iii): The assertion concerning h_{ℝ}(σ) is obvious. To show the assertion for g_{ℝ}^{−1}σ, one argues as in the proof of lemma 6.3.18(i) : by (i), we may find u_{1}, …, u_{s} ∈ L^{∨} such that σ = ⋂_{i=1}^{s} u_{i,ℝ}^{−1}(ℝ_{+}). Therefore g_{ℝ}^{−1}σ = ⋂_{i=1}^{s}(u_{i} ∘ g)_{ℝ}^{−1}(ℝ_{+}). Let γ ⊂ V^{∨} be the cone generated by the set {(u_{1} ∘ g)_{ℝ}, …, (u_{s} ∘ g)_{ℝ}}; then g_{ℝ}^{−1}σ = γ^{∨}, and the assertion results from lemma 6.3.2and a second application of (i).

Lastly, arguing as in the proof of lemma 6.3.19(i), one derives (iii) from (ii) and (iii). □

Parts (i) and (iii) of the following proposition provide the bridge connecting convex polyhedral cones to fine monoids.

Proposition 6.3.22. Let (L, +) be a free abelian group of finite rank, (L_{ℝ}, σ) an L-rational convex polyhedral cone, and set σ_{L} := L ∩ σ. Then :

(i) (Gordan’s lemma) σ_{L} is a fine and saturated submonoid of L, and L ∩ ⟨σ⟩ = σ_{L}^{gp}. (ii) For every v ∈ L_{ℝ}, the subset L ∩ (σ − v) is a finitely generated σ_{L}-module. (iii) For any submonoid M ⊂ L, we have : M_{ℚ} = M_{ℝ} ∩ L_{ℚ} and M^{sat} = M_{ℝ} ∩ L.

Proof. (Here σ − v ⊂ L_{ℝ} denotes the translate of σ by the vector −v, i.e. the subset of all w ∈ L_{ℝ} such that w + v ∈ σ.) Choose v_{1}, …, v_{s} ∈ L that generate σ, and set

C_{ε} := {∑_{i=1}^{s} t_{i}v_{i} | t_{i} ∈ [0, ε] for i = 1, …, s} for every ε > 0.

(i): Clearly L ∩ σ is saturated. Since C_{1} is compact and L is discrete, C_{1} ∩ L is a finite set. We claim that C_{1} ∩ L generates the monoid σ_{L}. Indeed, if v ∈ σ_{L}, write v = ∑_{i=1}^{s} r_{i}v_{i}, with r_{i} ≥ 0 for every i = 1, …, s; hence r_{i} = m_{i} + t_{i} for some m_{i} ∈ ℕ and t_{i} ∈ [0, 1[, and therefore v = v′ + ∑_{i=1}^{s} m_{i}v_{i}, where v′, v_{1}, …, v_{s} ∈ C_{1} ∩ L. Next, it is clear that σ_{L}^{gp} ⊂ L ∩ ⟨σ⟩; for the converse, say that w ∈ L ∩ ⟨σ⟩, and write w = w_{1} − w_{2}, for some w_{1}, w_{2} ∈ σ. Then w_{1} = ∑_{i=1}^{s} t_{i}v_{i} for some t_{i} ≥ 0; we pick t′_{i} ∈ ℕ such that t′_{i} ≥ t_{i} for every i ≤ s, and we set w′_{i} := ∑_{i=1}^{s} t′_{i}v_{i}. It follows that w = w′_{1} − w′_{2}, where w′_{2} := w_{2} + (w′_{1} − w_{1})), and notice that w′_{1} ∈ σ_{L} and w′_{2} ∈ σ; then we must have w′_{2} ∈ σ_{L} as well, and therefore w ∈ σ_{L}^{gp}.

(ii) is similar : from the compactness of C_{1} one sees that L ∩ (C_{1} − v) is a finite set; on the other hand, arguing as in the proof of (i), one checks easily that the latter set generates the σ_{L}-module L ∩ (σ − v).

(iii): Let x ∈ M_{ℝ} ∩ L_{ℚ}; then we may write

(6.3.23)  x = ∑_{i=1}^{n} r_{i} ⊗ m_{i}  where m_{i} ∈ M, r_{i} > 0 for every i ≤ n.

Claim 6.3.24. In the situation of proposition 6.3.22(iii), let x ∈ M_{ℝ} ∩ L_{ℚ}, and write x as in (6.3.23). Then, for every ε > 0 there exist q_{1}, …, q_{n} ∈ ℚ_{+} with |r_{i} − q_{i}| < ε for every i = 1, …, n, and such that x = ∑_{i=1}^{n} q_{i} ⊗ m_{i}.

Proof of the claim. Up to a reordering, we may assume that m_{1}, …, m_{k} form a basis of the ℚ-vector space generated by m_{1}, …, m_{n}, therefore m_{k+i} = ∑_{j=1}^{k} q_{ij}m_{j} for a matrix

A := (q_{ij} | i = 1, …, n − k; j = 1, …, k)

with entries in ℚ. Let \underline{r} := (r_{1}, …, r_{k}) and \underline{r}′ := (r_{k+1}, …, r_{n}); since x ∈ L_{ℚ}, we deduce that \underline{b} := \underline{r} + \underline{r}′ · A ∈ ℚ^{⊕k}. Moreover, if \underline{s} := (s_{1}, …, s_{k}) ∈ ℚ^{⊕k} and \underline{s}′ := (s_{k+1}, …, s_{n}) ∈ ℚ^{⊕n−k} satisfy the identity \underline{b} = \underline{s} + \underline{s}′ · A, then ∑_{i=1}^{n} s_{i} ⊗ m_{i} = x. If we choose \underline{s}′ very close to \underline{r}′, then \underline{s} shall be very close to \underline{r}; especially, we can achieve that both \underline{s} and \underline{s}′ are vectors with positive coordinates. ♢

Claim 6.3.24shows that x ∈ M_{ℚ}, whence the first stated identity; for the second identity, we are reduced to showing that M^{sat} = M_{ℚ} ∩ L, which is immediate. □

For various algebraic and geometric applications of the theory of polyhedral cones, one is led to study subdivisions of a given cone, in the sense of the following definition 6.3.25. Later we shall see a more abstract notion of subdivision, in the context of general fans, which however finds its roots and motivation in the intuitive manipulations of polyhedra that we formalize hereafter.

Definition 6.3.25. Let V be a finite dimensional ℝ-vector space.

(i) A fan in V is a finite set Δ consisting of convex polyhedral cones of V, such that : • for every σ ∈ Δ, and every face τ of σ, also τ ∈ Δ;

• for every σ, τ ∈ Δ, the intersection σ ∩ τ is also an element of Δ, and is a face of

both σ and τ. (ii) We say that Δ is a simplicial fan if all the elements of Δ are simplicial cones. (iii) Suppose that V = L ⊗_{ℤ} ℝ for some free abelian group L; then we say that Δ is

L-rational if the same holds for every τ ∈ Δ. (iv) A refinement of the fan Δ is a fan Δ′ in V with ⋃_{σ∈Δ} σ = ⋃_{τ∈Δ′} τ, and such that every

τ ∈ Δ is the union of the γ ∈ Δ′ contained in τ.

(v) A subdivision of a convex polyhedral cone (V, σ) is a refinement of the fan Δ_{σ} consisting of σ and its faces.

Lemma 6.3.26. Let (V, σ) be any convex polyhedral cone, Δ a subdivision of (V, σ). We let

Δ^{s} := {τ ∈ Δ | ⟨τ⟩ = ⟨σ⟩}.

Then ⋃_{τ∈Δ^{s}} τ = σ.

Proof. Let τ_{0} ∈ Δ be any element. Then σ′ := ⋃_{τ≠τ_{0}} τ is a closed subset of σ. If ⟨τ_{0}⟩ ≠ ⟨σ⟩, then σ \ τ_{0} is a dense open subset of σ contained in σ′; it follows that σ′ = σ in this case. Especially, τ_{0} = ⋃_{τ≠τ_{0}}(τ ∩ τ_{0}); since each τ ∩ τ_{0} is a face of τ_{0}, we see that there must exist τ ≠ τ_{0} such that τ_{0} is a face of τ. The lemma follows immediately. □

Example 6.3.27. (i) Certain useful subdivisions of a polyhedral cone σ are produced by means of auxiliary real-valued functions defined on σ. Namely, let us say that a continuous function f : σ → ℝ is a roof, if the following holds. There exists a non-empty finite set of ℝ-linear forms {l_{1}, …, l_{n}} on V, such that f(v) = min(l_{1}(v), …, l_{n}(v)) for every v ∈ σ. The concept of roof shall be reintroduced in section 6.6, in a more abstract and general guise; however, in order to grasp the latter, it is useful to keep in mind its more concrete polyhedral incarnation. We attach to f a subdivision of σ, as follows. For every i, j = 1, …, n define l_{ij} := l_{i} − l_{j}, and let τ_{i} ⊂ V^{∨} be the polyhedral cone σ^{∨} + ℝl_{i1} + ⋯ + ℝl_{in}. From the identity l_{ik} = l_{ij} + l_{jk} we easily deduce that τ_{i}^{∨} ∩ τ_{j}^{∨} is a face of both τ_{i}^{∨} and τ_{j}^{∨}, for every i, j = 1, …, n. Denote by Θ the smallest subdivision of σ containing all the τ_{i}^{∨}; it is easily seen that

σ = ⋃_{i=1}^{n} τ_{i}^{∨}

and the restriction of f to each τ_{i}^{∨} agrees with l_{i}.

(ii) Conversely, let f : σ → ℝ a continuous function; suppose there exist a subdivision Θ of σ, and a system (l_{τ} | τ ∈ Θ) of ℝ-linear forms on V such that

• f(v) = l_{τ}(v) for every τ ∈ Θ and every v ∈ τ.

• f(u + v) ≥ f(u) + f(v) for every u, v ∈ σ.

Then we claim that f is a roof on σ. Indeed, let Θ^{s} ⊂ Θ be the subset of all τ that span ⟨σ⟩. Notice fist that the system (l_{τ} | τ ∈ Θ^{s}) already determines f uniquely, by virtue of lemma 6.3.26. Next, let τ, τ′ ∈ Θ^{s} be any two elements, and pick an element v of the interior of τ. For any u ∈ τ′ and any ε > 0 we have, by assumption : f(v + εu) ≥ f(v) + f(εu). If ε is small enough, we have as well v + εu ∈ τ, in which case the foregoing inequality can be written as :

l_{τ}(v) + ε · l_{τ}(u) = l_{τ}(v + εu) ≥ l_{τ}(v) + ε · l_{τ′}(u)

whence l_{τ}(u) ≥ l_{τ′}(u) = f(u) and the assertion follows.

Proposition 6.3.28. Let f : V → W be a linear map of finite dimensional ℝ-vector spaces, (V, σ) a convex polyhedral cone, and h : σ → f(σ) the restriction of f. Then :

(i) There exists a subdivision Δ of (W, f(σ)) such that :

h^{−1}(a + b) = h^{−1}(a) + h^{−1}(b) for every τ ∈ Δ and every a, b ∈ τ.

(ii) Suppose moreover that V = L ⊗_{ℤ} ℝ, W = L′ ⊗_{ℤ} ℝ and f = g ⊗_{ℤ} 1_{ℝ} for a map g : L → L′ of free abelian groups. If σ is L-rational, then we may find an L-rational subdivision Δ such that (i) holds.

Proof. Let V_{0} be the largest linear subspace contained in σ ∩ Ker f. Notice that, under the assumptions of (ii), we have : V_{0} = ℝ ⊗_{ℤ} Ker g. One verifies easily that the proposition holds for the given map f and the cone (V, σ), if and only if it holds for the induced map \overline{f} : V/V_{0} → W/f(V_{0}) and the cone (V/V_{0}, \overline{σ}) (where \overline{σ} is the image of σ in V/V_{0}). Hence, we may replace f by \overline{f}, and assume from start that σ∩Ker f contains no non-zero linear subspaces. Moreover, we may assume that σ spans V and f(σ) spans W.

(i): Let S be the set of faces τ of σ such that f restricts to an isomorphism ⟨τ⟩ →^{∼} W.

Claim 6.3.29. Let λ ⊂ f(σ) be any ray. Then :

(i) λ′ := h^{−1}λ is a strictly convex polyhedral cone. Especially, λ′ is generated by its extremal rays (see (6.3.15)).

(ii) For every extremal ray ρ of λ′ with ρ ⊄ Ker f, there exists τ ∈ S such that ρ = τ ∩ f^{−1}(λ).

Proof of the claim. λ′ is a convex polyhedral cone by lemma 6.3.18(i) and corollary 6.3.12(iii). To see that λ′ is strictly convex, notice that any subspace L ⊂ f^{−1}(λ) lies already in Ker f, and if L ⊂ σ, we must have L = {0} by assumption. Let ρ be an extremal ray of λ′ which is not contained in Ker f; notice that λ′ is the intersection of the polyhedral cones λ_{1} := h^{−1}⟨λ⟩ and λ_{2} := f^{−1}λ, hence we can find faces δ_{i} of λ_{i} (i = 1, 2) such that ρ = δ_{1}∩δ_{2} (corollary 6.3.12(iii) and lemma 6.3.18(i)). However, the only proper face of λ_{2} is Ker f (lemma 6.3.18(ii)), hence δ_{2} = λ_{2}. Likewise, f^{−1}⟨λ⟩ has no proper faces, hence δ_{1} = γ ∩ f^{−1}⟨λ⟩ for some face γ of σ (again by corollary 6.3.12(iii)). Since λ_{2} is a half-space in f^{−1}⟨λ⟩, we deduce easily that either δ_{1} = ρ or δ_{1} = ⟨ρ⟩. Especially, dim_{ℝ}(f^{−1}⟨λ⟩)/⟨δ_{1}⟩ = dim_{ℝ} Ker f. We may then apply lemma 6.3.18(ii) to the embedding f^{−1}⟨λ⟩ ⊂ V, to find a face τ of σ such that :

τ ∩ f^{−1}⟨λ⟩ = δ_{1}  ⟨τ⟩ ∩ f^{−1}⟨λ⟩ = ⟨ρ⟩  dim_{ℝ} V/⟨τ⟩ = dim_{ℝ} Ker f.

It follows that ⟨τ⟩ ∩ Ker f = {0}, and therefore τ ∈ S, as required. ♢

We construct as follows a subdivision of (W, f(σ)). For every τ ∈ S, let F (τ) be the set consisting of the facets of the polyhedral cone f(τ); set also F := ⋃_{τ∈S} F (τ). Notice that, for every γ ∈ F, the subspace ⟨γ⟩ is a hyperplane of W; we let :

U := f(σ)\ ⋃_{γ∈F}⟨γ⟩.

Then U is an open subset of f(σ), and the topological closure \overline{C} of every connected component C of U is a convex polyhedral cone. Moreover, if C and D are any two such connected components, the intersection \overline{C} ∩ \overline{D} is a face of both \overline{C} and \overline{D}. We let Δ be the subdivision of f(σ) consisting of the cones \overline{C} – where C ranges over all the connected components of U – together with all their faces.

Claim 6.3.30. For every δ ∈ Δ and every τ ∈ S, the intersection δ ∩ f(τ) is a face of δ.

Proof of the claim. Due to proposition 6.3.8(iii), we may assume that δ is the topological closure of a connected component C of U. We may also assume that f(τ) ≠ W, otherwise there is nothing to prove; in that case, we have f(τ) = ⋂_{γ∈F (τ)} H_{γ}, where, for each γ ∈ F (τ), the half-space H_{γ} is the unique one that contains both f(τ) and γ (proposition 6.3.11(ii)). It then suffices to show that δ ∩ H_{γ} is a face of δ for each such H_{γ}. We may assume that δ ⊄ H_{γ}. Since C is connected and C ⊂ W \⟨γ⟩, it follows that δ ⊂ −H_{γ}, the topological closure of the complement of H_{γ}. Hence (−H_{γ})^{∨} ⊂ δ^{∨} (where (−H_{γ})^{∨} is the dual of the polyhedral cone (W, −H_{γ})), and therefore δ ∩ H_{γ} = δ ∩ H_{γ} ∩ (−H_{γ}) = δ ∩ ⟨γ⟩ is indeed a face of δ. ♢

Next, for every w ∈ f(σ), let I(w) := {τ ∈ S | w ∈ f(τ)}.

Claim 6.3.31. Let δ ∈ Δ, and w_{1}, w_{2} ∈ δ. Then I(w_{1} + w_{2}) ⊂ I(w_{1}) ∩ I(w_{2}).

Proof of the claim. Suppose first that w_{1}+w_{2} is contained in a face δ′ of δ; say that δ′ = δ∩Ker u, for some u ∈ δ^{∨}. This means that u(w_{1} + w_{2}) = 0, hence u(w_{1}) = u(w_{2}) = 0, i.e. w_{1}, w_{2} ∈ δ′. Hence, we may replace δ by δ′, and assume that δ is the smallest element of Δ containing w_{1} + w_{2}. Thus, suppose that τ ∈ I(w_{1} + w_{2}); therefore w_{1} + w_{2} ∈ f(τ) ∩ δ. From claim 6.3.30 we deduce that δ ⊂ f(τ), hence τ ∈ I(w_{1}) ∩ I(w_{2}), as claimed. ♢

Finally, we are ready to prove assertion (i). Hence, let a, b ∈ f(σ) be any two vectors that lie in the same element of Δ. Clearly :

h^{−1}(a) + h^{−1}(b) ⊂ h^{−1}(a + b)

hence it suffices to show the converse inclusion. However, directly from claim 6.3.29(ii) we derive the identity :

h^{−1}(ℝ_{+} · w) = (σ ∩ Ker f) + ∑_{τ∈I(w)}(τ ∩ f^{−1}(ℝ_{+} · w)) for every w ∈ f(σ).

Taking into account claim 6.3.31, we are then reduced to showing that :

τ ∩ f^{−1}(ℝ_{+} · (a + b)) ⊂ (τ ∩ f^{−1}(ℝ_{+} · a)) + (τ ∩ f^{−1}(ℝ_{+} · b)) for every τ ∈ I(a + b).

The latter assertion is obvious, since f restricts to an isomorphism ⟨τ⟩ →^{∼} W.

(ii): By inspecting the construction, one verifies easily that the subdivision Δ thus exhibited shall be L-rational, whenever σ is. □

6.3.32. Later we shall also be interested in rational variants of the identities of proposition 6.3.28(i). Namely, consider the following situation. Let g : L → L′ be a map of free abelian groups of finite rank, g_{ℝ} : L_{ℝ} → L′_{ℝ} the induced ℝ-linear map, and (L_{ℝ}, σ) an L-rational convex polyhedral cone; set τ := g_{ℝ}(σ), and denote by h_{ℝ} : σ → τ (resp. h_{ℚ} : σ ∩ L_{ℚ} → τ ∩ L′_{ℚ}) the restriction of g_{ℝ}. We point out, for later reference, the following observation :

Lemma 6.3.33. In the situation of (6.3.32), suppose that :

h_{ℝ}^{−1}(x_{1}) + h_{ℝ}^{−1}(x_{2}) = h_{ℝ}^{−1}(x_{1} + x_{2}) for every x_{1}, x_{2} ∈ τ

(where the sum is taken in the monoid (𝒫(σ), +)). Then we have as well :

h_{ℚ}^{−1}(x_{1}) + h_{ℚ}^{−1}(x_{2}) = h_{ℚ}^{−1}(x_{1} + x_{2}) for every x_{1}, x_{2} ∈ τ ∩ L′_{ℚ}.

Proof. Let x_{1}, x_{2} ∈ τ ∩ L′_{ℚ} be any two elements, and v ∈ h_{ℚ}^{−1}(x_{1} + x_{2}), so we may write v = v_{1} + v_{2} for some v_{i} ∈ h_{ℝ}^{−1}(x_{i}) (i = 1, 2). Let also u_{1}, …, u_{k} be a finite system of generators for σ^{∨}, and set

J_{i} := {j ≤ k | u_{j}(v_{i}) = 0} E_{i} := g_{ℝ}^{−1}(x_{i}) ∩ ⋂_{j∈J_{i}} Ker u_{j} (i = 1, 2).

Clearly L_{ℚ} ∩ E_{i} is a dense subset of E_{i} for i = 1, 2, hence, in any neighborhood of (x_{1}, x_{2}) in L_{ℝ}^{⊕2} we may find a solution (y_{1}, y_{2}) ∈ L_{ℚ}^{⊕2} for the system of equations

g_{ℝ}(y_{i}) = x_{i}  u_{j}(y_{i}) = 0  for i = 1, 2 and every j ∈ J_{i}.

Since u_{j}(x_{i}) > 0 for every j ∉ J_{i}, we will also have u_{j}(y_{i}) > 0 for every j ∉ J_{i}, provided y_{i} is sufficiently close to x_{i}. The lemma follows. □

6.3.34. We conclude this section with some considerations that shall be useful later, in our discussion of normalized lengths for model algebras (see (17.1.37)). Keep the notation of proposition 6.3.22, and for every subset U ⊂ L_{ℝ}, let

𝒮_{L,σ}(U) := {L ∩ (σ − v) | v ∈ U} and set 𝒮_{L,σ} := 𝒮_{L,σ}(L_{ℝ}).

There is a natural L-module structure on 𝒮_{L,σ}; namely, notice that

(L ∩ (σ − v)) + l = L ∩ (σ − (v − l)) for every v ∈ L_{ℝ} and l ∈ L

hence the rule τ_{l} : S ↦ S + l defines a bijection of 𝒮_{L,σ} onto itself, for every l ∈ L, and clearly τ_{l} ∘ τ_{l′} = τ_{l+l′} for every l, l′ ∈ L. Also, for every S ∈ 𝒮_{L,σ} define

Ω(σ, S) := {v ∈ L_{ℝ} | L ∩ (σ − v) = S}

and denote by \overline{Ω}(σ, S) the topological closure of Ω(σ, S) in L_{ℝ}. For given u ∈ L_{ℝ}^{∨} and r ∈ ℝ, set H_{u,r} := {v ∈ L_{ℝ} | u(v) ≥ r}. We shall say that a subset of L_{ℝ} is ℚ-linearly constructible, if it lies in the boolean subalgebra of 𝒫(L_{ℝ}) generated by the subsets H_{u⊗_{ℚ}1_{ℝ},r}, for u ranging over all the ℚ-linear forms L_{ℚ} → ℚ, and r ranging over all rational numbers.

Proposition 6.3.35. With the notation of (6.3.34), the following holds :

(i) 𝒮_{L,σ}(U) is a finite set, for every bounded subset U ⊂ L_{ℝ}.

(ii) 𝒮_{L,σ} is a finitely generated L-module.

(iii) For every non-empty S ∈ 𝒮_{L,σ}, the subset Ω(σ, S) is ℚ-linearly constructible.

(iv) Suppose moreover, that σ spans L_{ℝ}. Then, for every S ∈ 𝒮_{L,σ}, the subset Ω(σ, S) is

contained in the topological closure of its interior (see (6.3.10)).

(v) For every S ∈ 𝒮_{L,σ}, and every v ∈ \overline{Ω}(σ, S), we have S ⊂ L ∩ (σ − v).

Proof. (i): Define C_{ε} as in the proof of proposition 6.3.22; since U is bounded, it is contained in the union of finitely many subsets of L_{ℝ} of the form C_{1} + l, for l ranging over a finite subset of L. On the other hand, τ_{l} induces a bijection

𝒮_{L,σ}(C_{1}) →^{∼} ℒ_{L,σ}(C_{1} − l) for every l ∈ L.

Hence, it suffices to check the assertion for U = C_{1}. However, the proof of proposition 6.3.22(ii) shows that L ∩ (σ − v) is generated by L ∩ (C_{1} − v); if v ∈ C_{1}, the latter subset is contained in C′ := C_{1} ∪ (−C_{1}), which is a compact subset of L_{ℝ}. Therefore L ∩ C′ is a finite set, and the claim follows.

(ii): We have already observed that the L-module 𝒮_{L,σ} is generated by 𝒮_{L,σ}(C_{1}), and this is a finite set, by (i).

(iii): Fix a minimal system S_{1}, …, S_{n} of generators of the L-module 𝒮_{L,σ} (i.e. the S_{i} are chosen representatives for the orbits of the L-action on 𝒮_{L,σ}). After replacing S_{i} by some translates S_{i} + l (for an appropriate l ∈ L) we may also assume that either S_{i} = ∅, or else 0 ∈ S_{i}, and notice that this implies :

(6.3.36)  S_{i} ⊂ ⟨σ⟩ ∩ L = σ_{L}^{gp}  for every i = 1, …, n

(proposition 6.3.22(i)). Set

A_{ij} := {l ∈ L | S_{i} ⊂ S_{j} − a} for every i, j ≤ n

and notice that A_{ij} is a σ_{L}-module, for every i, j ≤ n.

Claim 6.3.37. If S_{i}, S_{j} ≠ ∅, the σ_{L}-module A_{ij} is finitely generated.

Proof of the claim. Fix l ∈ L such that σ_{L} + l ⊂ S_{i}. Next, say that x_{1}, …, x_{t} is a finite system of generators for the σ_{L}-module S_{j} (proposition 6.3.22(ii)); by virtue of (6.3.36), for every s = 1, …, t, we may write x_{s} = a_{s} −b_{s} for certain a_{s}, b_{s} ∈ σ_{L}. Set l′ := b_{1} +⋯+b_{t}, and notice that S_{j} ⊂ σ_{L} − l′. Now, if S_{i} ⊂ S_{j} − a, we deduce that σ_{L} + l ⊂ σ_{L} − a − l′, especially l ∈ σ_{L} − a − l′, i.e. a ∈ σ_{L} − (l + l′). This shows that A_{ij} is isomorphic to an ideal of σ_{L}, and then the claim follows from proposition 6.1.9(ii). ♢

Now, let i, j ≤ n such that S_{i}, S_{j} ≠ ∅. Suppose first that i ≠ j, and let A′_{ij} ⊂ A_{ij} be any finite generating system for the σ_{L}-module A_{ij}. From the construction, it is clear that every element of LS_{j} that contains S_{i}, must contain S_{j} − l, for some l ∈ A′_{ij}. To deal with the case where i = j, we remark, more generally :

Claim 6.3.38. Let P be any fine and saturated monoid, M ⊂ P^{gp} a non-empty finitely generated P -submodule, and a ∈ P^{gp} an element such that aM ⊂ M. Then a ∈ P.

Proof of the claim. Pick any m ∈ M, and denote by M′ ⊂ M the submodule generated by (a^{k}m | k ∈ ℕ). According to proposition 6.1.9(i), there exists N ≥ 0 such that M′ is generated by the finite system (a^{k}m | k = 0, …, N). Especially, a^{N+1}m ∈ M′, and therefore there exist x ∈ P and i ≤ N such that a^{N+1}m = a^{i}mx in M; it follows that a^{N+1−i} ∈ P, and finally a ∈ P, since P is saturated. ♢

From (6.3.36) we see that A_{ii} ⊂ σ_{L}^{gp}, if S_{i} ≠ ∅; combining with claim 6.3.38, we deduce that A_{ii} = σ_{L}. Moreover, notice as well that if S_{i} = S_{i} − a for some a ∈ σ_{L}^{gp}, then both a and −a ∈ A_{ii}, so that a ∈ σ_{L}^{×}. Thus, let A′_{ii} be any set of representatives of 𝔪_{σ} \𝔪_{σ}^{2}, where 𝔪_{σ} denotes the maximal ideal of σ_{L}^{♯}. If a ∈ L, and S_{i} − a contains strictly S_{i}, then a is a non-invertible element of σ_{L}, and taking into account corollary 6.1.12, we see that A′_{ii} is finite, and there exists l ∈ A′_{ii} such that S_{i} − l ⊂ S_{i} − a. Next, for every i ≤ n such that S_{i} ≠ ∅, set

𝒮^{i} := ⋃_{j}{S_{j} + l | l ∈ A′_{ij}}

where j ≤ n runs over the indices such that S_{j} ≠ ∅. Summing up, we conclude that 𝒮^{i} is a finite set for every i ≤ n with S_{i} ≠ ∅, and if an element of 𝒮_{L,σ} contains strictly S_{i}, then it contains some element of 𝒮^{i}. Lastly, in order to prove assertion (iii), we may assume that S = S_{i} for some i ≤ n, and notice that :

(6.3.39)  Ω(σ, S_{i}) = {v ∈ L_{ℝ} | S ⊂ σ − v} \ ⋃_{S′∈𝒮^{i}} {v ∈ L_{ℝ} | S′ ⊂ σ − v}.

Since S is finitely generated (proposition 6.3.22(ii)), we reduce to showing that, for every a ∈ L, the subset Ω(σ, a) := {v ∈ L_{ℝ} | a ∈ σ − v} is ℚ-linearly constructible, which follows easily from proposition 6.3.21(i) and lemma 6.3.2.

(iv): We remark :

Claim 6.3.40. Let C_{ε} be as in the proof of proposition 6.3.22. Then, for every S ∈ 𝒮_{L,σ} and every a ∈ Ω(σ, S) there exists ε > 0 such that a + C_{ε} ⊂ Ω(σ, S).

Proof of the claim. Since σ is closed in L_{ℝ}, for every a ∈ L_{ℝ} and every b ∈ L_{ℝ} \ Ω(σ, a) there exists ε > 0 such that (b + C_{ε}) ∩ Ω(σ, a) = ∅. Taking into account (6.3.39), the claim follows easily. ♢

If σ span L_{ℝ}, the subset C_{ε} has non-empty interior U_{ε}, for every ε > 0, and the topological closure of U_{ε} equals C_{ε}. The assertion is then an immediate consequence of claim 6.3.40.

(v): The assertion follows easily from proposition 6.3.22(ii) : the details shall be left to the reader. □

6.3.41. Let L be as in (6.3.34), and for all integers n, m > 0 set

\frac{1}{m}L := \{v ∈ L_{ℚ} | mv ∈ L\} \frac{1}{m}L[1/n] := ⋃_{k≥0} \frac{1}{n^{k}m}L.

For future reference, let us also point out :

Lemma 6.3.42. With the notation of (6.3.41), let Ω ⊂ L_{ℝ} be a ℚ-linearly constructible subset. Then we have :

(i) The topological closure of Ω in L_{ℝ} is again ℚ-linearly constructible.

(ii) There exists an integer m > 0 such that \frac{1}{m}L[1/n] ∩ Ω is dense in Ω, for every n > 1.

Proof. (i): Ω is a finite union of non-empty subsets of the form H_{1} ∩ ⋯ ∩ H_{k}, where each H_{i} is either of the form H_{u⊗1_{ℝ},r} for some non-zero ℚ-linear form u of L_{ℚ} and some r ∈ ℚ (and this is a closed subset of L_{ℝ}), or else is the complement in L_{ℝ} of a subset of this type (and then its closure is a half-space H_{−u⊗1_{ℝ},r}). One verifies that the closure of H_{1} ∩ ⋯ ∩ H_{k} is the intersection of the closures of H_{1}, …, H_{k}, whence the assertion.

(ii): We may assume that Ω = Ω_{1} ∩ Ω_{2}, where Ω_{1} is a finite intersection of rational hyplerplanes, and Ω_{2} is a finite intersection of open half-spaces (i.e. of complements of closed half-spaces). Suppose that \frac{1}{m}L[1/n] ∩ Ω_{1} is dense in Ω_{1}; then clearly \frac{1}{m}L[1/n] ∩ Ω is dense in Ω. Hence, we may further assume that Ω is a non-empty intersection of rational hyperplanes. In this case, Ω is of the form V_{ℝ} + v_{0}, where v_{0} ∈ L_{ℚ}, and V_{ℝ} = V ⊗_{ℤ} ℝ for some subgroup V ⊂ L. Notice that L[1/n] ∩ V_{ℝ} is dense in V_{ℝ} for every integer n > 1. Then, any integer m > 0 such that v_{0} ∈ \frac{1}{m}L will do. □

6.4. Fine and saturated monoids. This section presents the further developments of the theory of fine and saturated monoids. Again, all the monoids in this section are non-pointed. We begin with a few corollaries of proposition 6.3.22(i,iii).

Corollary 6.4.1. Let M be an integral monoid, such that M^{♯} is fine. We have :

(i) The inclusion map M → M^{sat} is a finite morphism of monoids. (ii) Especially, if M is fine, any monoid N with M ⊂ N ⊂ M^{sat}, is fine.

Proof. (i): From lemma 6.2.9(ii) we deduce that M^{sat} is a finitely generated M-module if and only if (M^{♯})^{sat} is a finitely generated M^{♯}-module. Hence, we may replace M by M^{♯}, and assume that M is fine. Pick a surjective group homomorphism φ : ℤ^{⊕n} → M^{gp}; it is easily seen that :

φ^{−1}(M^{sat}) = (φ^{−1}M)^{sat}

and clearly it suffices to show that φ^{−1}N is finitely generated, hence we may replace M by φ^{−1}M, and assume throughout that M^{gp} is a free abelian group of finite rank. In this case, proposition 6.3.22(i,iv) already implies that M^{sat} is finitely generated. Let a_{1}, …, a_{k} ∈ M^{sat} be a finite system of generators, and pick integers n_{1}, …, n_{k} > 0 such that a_{i}^{n_{i}} ∈ M for i = 1, …, k. For every i = 1, …, k let Σ_{i} := {a_{i}^{j} | j = 0, …, n_{i} − 1}; it is easily seen Σ_{1} ⋯ Σ_{k} ⊂ M^{sat} is a system of generators for the M-module M^{sat} (where the product of the sets Σ_{i} is formed in the monoid 𝒫(M^{sat}) of (6.1.1)).

(ii) follows from (i), in view of proposition 6.1.9(i). □

Corollary 6.4.2. Let f : M_{1} → M and g : M_{2} → M be two morphisms of monoids, such that M_{1} and M_{2} are finitely generated, and M is integral. Then the fibre product M_{1} ×_{M} M_{2} is a finitely generated monoid, and if M_{1} and M_{2} are fine, the same holds for M_{1} ×_{M} M_{2}.

Proof. If the monoids M, M_{1} and M_{2} are integral, M_{1} ×_{M} M_{2} injects in M_{1}^{gp} ×_{M^{gp}} M_{2}^{gp} (lemma 4.8.29(iii)), hence it is integral. To show that the fibre product is finitely generated, choose surjective morphisms ℕ^{⊕a} → M_{1} and ℕ^{⊕b} → M_{2}, for some a, b ∈ ℕ; by composition we get maps of monoids φ : ℕ^{⊕a} → M, ψ : ℕ^{⊕b} → M, such that the induced morphism P := ℕ^{⊕a} ×_{M} ℕ^{⊕b} → M_{1} ×_{M} M_{2} is surjective. Hence it suffices to show that P is finitely generated. To this aim, let L := Ker(φ^{gp} − ψ^{gp} : ℤ^{⊕a+b} → M^{gp}); for every i = 1, …, a + b, denote also by π_{i} : ℤ^{⊕a+b} → ℤ the projection onto the i-th direct summand. The system {π_{i} | i = 1, …, a+b} generates a rational convex polyhedral cone σ ⊂ L^{∨}⊗_{ℤ}ℝ, and one verifies easily that P = L ∩ σ^{∨}, so the assertion follows from propositions 6.3.21(i) and 6.3.22(i). □

Corollary 6.4.3. (i) Let f : M → N be morphisms of monoids, with M finitely generated. Then Ker f is a finitely generated monoid.

(ii) Let f, f′ : M → N be two morphisms of monoids, with M finitely generated, and N integral. Then Equal(f, f′) is finitely generated.

Proof. (i): By assumption, there exists r ∈ ℕ and a surjective morphism of monoids g : ℕ^{⊕r} → M. Clearly Ker f = g(Ker f ∘g), hence it suffices to check that Ker g∘f is finitely generated, so we may assume that M is a fine monoid. In that case, it is easily seen that Ker f = (Ker f)^{gp} ∩ M = (Ker f)^{gp} ×_{M^{gp}} M, so the assertion follows from corollary 6.4.2.

(ii): Let i : M → M^{gp} be the unit of adjunction; it is easily seen that Equal(f, f′) = Ker((f^{gp} − f^{′gp}) ∘ i), so by (i), the monoid Equal(f, f′) is finitely generated. □

Remark 6.4.4. Both corollaries 6.4.2and 6.4.3can fail for non-integral monoids. For instance, let I := {0, 1}, which is a monoid with the composition law such that 0 + 0 = 0 and 1 + 1 = 1 = 0 + 1. Consider the surjective morphism of monoids f : ℕ → I such that f(0) = 0 and f(n) = 1 for every n > 0. For i = 1, 2, let also π_{i} : N^{⊕2} → ℕ be the projection; then ℕ ×_{I} ℕ = Equal(f ∘ π_{1}, f ∘ π_{2}) is not finitely generated.

Corollary 6.4.5. Let (Γ, +, 0) be an integral monoid, M a finitely generated Γ-graded monoid. Then M_{0} is a finitely generated monoid, and M_{γ} is a finitely generated M_{0}-module, for every γ ∈ Γ.

Proof. We have M_{0} = M ×_{Γ} {0}, hence M_{0} is finitely generated, by corollary 6.4.2. The given element γ ∈ Γ determines a unique morphism of monoids ℕ → Γ such that 1 ↦ γ. Let p_{1} : M′ := M ×_{Γ} ℕ → ℕ and p_{2} : M′ → M be the two natural projections; by lemma 4.8.29(iii), we have M_{γ} = p_{2}(p_{1}^{−1}(1)). In light of corollary 6.4.2, M′ is still finitely generated, hence we are reduced to the case where Γ = ℕ and γ = 1. In this case, pick a finite set of generators S for M. One checks easily that M_{1} ∩ S generates the M_{0}-module M_{1}. □

Corollary 6.4.6. Let M be an integral monoid, such that M^{♯} is fine, and φ : M → N a saturated morphism of monoids. Then φ is flat.

Proof. In view of corollary 6.4.1(i) and theorem 6.2.6, it suffices to show that the M^{sat}-module M^{sat} ⊗_{M} N is flat. Hence we may replace M by M^{sat}, and assume that M is saturated. Let I ⊂ M be any ideal, and define R(M, I) as in the proof of theorem 6.2.3; by assumption, R(M, I)^{sat} ⊗_{M} N is a saturated – especially, integral – monoid, i.e. the natural map

R(M, I) ⊗_{M} N → R(M, I)^{gp} ⊗_{M^{gp}} N^{gp}

is injective. The latter factors through the morphism j⊗_{M} N, where j : R(M, I) → M ×ℕ is the obvious inclusion. In light of example 6.2.13(i), we deduce that the induced map I^{sat} ⊗_{M} N → N is injective. Now, if I is a prime ideal, then I^{sat} = I, hence the contention follows from corollary 6.1.48(ii). □

The following corollary generalizes lemma 6.2.10.

Corollary 6.4.7. Let f : M → N be a local, flat and saturated morphism of fine monoids, with M sharp. Then there exists an isomorphism of monoids

g : N →^{∼} N^{♯} × N^{×}

that fits into a commutative diagram

\xymatrix{M \ar[r]^{f^{♯}} \ar[d]_{f} & N^{♯} \ar[d] \\ N \ar[r]^{g} & N^{♯} × N^{×}}

whose right vertical arrow is the natural inclusion map.

Proof. From lemma 6.2.30(ii), we know that f is exact, and since M is sharp, we easily deduce that f(M)^{gp} ∩ N^{×} = {1}. Hence, the induced group homomorphism M^{gp} ⊕ N^{×} → N^{gp} is injective. On the other hand, since f is flat, local and saturated, the same holds for f^{♯} : M → N^{♯} (lemma 6.2.12(iii) and corollary 6.4.6); then corollary 6.2.32(ii) says that the cokernel of the induced group homomorphism M^{gp} → (N^{♯})^{gp} = N^{gp}/N^{×} is a free abelian group G (of finite rank). Summing up, we obtain an isomorphism of abelian groups :

h : M^{gp} ⊕ N^{×} ⊕ G →^{∼} N^{gp}

extending the map f^{gp}. Set N_{0} := N ∩ h(M^{gp} ⊕ G); it follows easily that the natural map N_{0} × N^{×} → N is an isomorphism; especially, the projection N → N^{♯} maps N_{0} isomorphically onto N^{♯}, and the contention follows. □

6.4.8. Let (M, ·) be a fine (non-pointed) monoid, so that M^{gp} is a finitely generated abelian group. We set M_{ℝ}^{gp} := log M^{gp} ⊗_{ℤ} ℝ, and we let M_{ℝ} be the convex polyhedral cone generated by the image of log M. Then (M_{ℝ}, +) is a monoid, and we have a natural morphism of monoids

φ : log M → (M_{ℝ}, +).

Proposition 6.4.9. With the notation of (6.4.8), we have :

(i) The rule : F ↦ F_{ℝ} establishes a bijection between the set of faces of M and the set of faces of M_{ℝ}.

(ii) The map φ induces a bijection

φ^{∗} : Spec M_{ℝ} → Spec M.

Proof. Clearly, we may assume that M ≠ {1}. Let 𝔭 ⊂ M be a prime ideal; we denote by 𝔭_{ℝ} the ideal of (M_{ℝ}, +) generated by all elements of the form r · φ(x), where r is any strictly positive real number, and x is any element of 𝔭. We also denote by (M \𝔭)_{ℝ} the convex cone of M_{ℝ}^{gp} generated by the image of M \𝔭.

Claim 6.4.10. M_{ℝ} is the disjoint union of (M \𝔭)_{ℝ} and 𝔭_{ℝ}.

Proof of the claim. To begin with, we show that M_{ℝ} = (M\𝔭)_{ℝ} ∪ 𝔭_{ℝ}. Indeed, let x ∈ M_{ℝ}; then we may write x = ∑_{i=1}^{h} m_{i} ⊗ a_{i} for certain a_{1}, …, a_{h} ∈ ℝ_{+} and m_{1}, …, m_{h} ∈ M. We may assume that a_{1}, …, a_{k} ∈ 𝔭 and a_{k+1}, …, a_{h} ∈ M \𝔭. Now, if k = 0 we have x ∈ (M \𝔭)_{ℝ}, and otherwise x ∈ 𝔭_{ℝ}, which shows the assertion.

It remains to show that (M \𝔭)_{ℝ} ∩ 𝔭_{ℝ} = ∅. To this aim, suppose by way of contradiction, that this intersection contains an element x; this means that we have finite subsets S_{0} ⊂ M \𝔭 and S_{1} ⊂ M such that S_{1} ∩ 𝔭 ≠ ∅, and an identity of the form :

(6.4.11)  x = ∑_{σ∈S_{0}} σ ⊗ a_{σ} = ∑_{σ∈S_{1}} σ ⊗ b_{σ}

where a_{σ} > 0 for every σ ∈ S_{0} and b_{σ} > 0 for every σ ∈ S_{1}. For every σ ∈ S_{0}, choose a rational number a′_{σ} ≥ a_{σ}; after adding the summand ∑_{σ∈S_{0}} σ ⊗ (a′_{σ} − a_{σ}) to both sides of (6.4.11), we may assume that a_{σ} ∈ ℚ_{+} for every σ ∈ S_{0}. Let N ⊂ M be the submonoid generated by S_{1}; it follows that x ∈ N_{ℝ} ∩ M_{ℚ} = N_{ℚ} (proposition 6.3.22(iii)), hence we may assume that all the coefficients a_{σ} and b_{σ} are rational and strictly positive (see remark 6.3.24). We may further multiply both sides of (6.4.11) by a large integer, to obtain that these coefficients are actually integers. Then, up to further multiplication by some integer, the identity of (6.4.11) lifts to an identity between elements of log M, of the form : ∑_{σ∈S_{0}} a_{σ} · σ = ∑_{σ∈S_{1}} b_{σ} · σ. The latter is absurd, since S_{1} ∩ 𝔭 ≠ ∅ and S_{0} ∩ 𝔭 = ∅. ♢

Claim 6.4.10 implies that 𝔭_{ℝ} is a prime ideal of M_{ℝ}, and clearly 𝔭 ⊂ φ^{∗}(𝔭_{ℝ}). Since we have as well M \𝔭 ⊂ φ^{−1}(M \𝔭)_{ℝ}, we deduce that 𝔭 = φ^{∗}(𝔭_{ℝ}). Hence the rule 𝔭 ↦ 𝔭_{ℝ} yields a right inverse φ_{∗} : Spec M → Spec M_{ℝ} for the natural map φ^{∗}. To show that φ_{∗} is also a left inverse, let 𝔮 ⊂ M_{ℝ} be a prime ideal; by lemma 6.3.7and proposition 6.3.21(i), the face M_{ℝ}\𝔮 is of the form M_{ℝ} ∩ Ker u, for some u ∈ M_{ℝ}^{∨} ∩ (log M^{gp})^{∨}. Then it is easily seen that M_{ℝ}\𝔮 is the convex cone generated by φ(M) ∩ Ker u, in other words, M_{ℝ} \ 𝔮 = φ^{−1}(Ker u)_{ℝ}. Again by claim 6.4.10, it follows that 𝔮 = (M \φ^{−1}(Ker u))_{ℝ} = (φ^{∗}𝔮)_{ℝ}, as stated. The argument also shows that every face of M_{ℝ} is of the from (M \𝔭)_{ℝ} for a unique prime ideal 𝔭, which settles assertion (i). □

Corollary 6.4.12. Let M be a fine monoid. We have :

(i) dim M = rk_{ℤ}(M^{gp}/M^{×}).

(ii) dim(M \𝔭) + ht 𝔭 = dim M for every 𝔭 ∈ Spec M.

(iii) If M ≠ {1} is sharp (see (4.8.32)), there exists a local morphism M → ℕ.

(iv) If M is sharp and M^{gp} is a torsion-free abelian group of rank r, there exists an injective morphism of monoids M → ℕ^{⊕r}.

Proof. (i): By proposition 6.4.9, the dimension of M can be computed as the length of the longest chain F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{d} of strict inclusions of faces of M_{ℝ}. On the other hand, given such a maximal chain, denote by r_{i} the dimension of the ℝ-vector space spanned by F_{i}; in view proposition 6.3.8(ii),(iii), it is easily seen that r_{i+1} − r_{i} = 1 for every i = 0, …, d − 1. Since M_{ℝ} ∩ (−M_{ℝ}) is the minimal face of M_{ℝ}, we deduce that

dim M = dim_{ℝ} M_{ℝ}^{gp} − dim_{ℝ} M_{ℝ} ∩ (−M_{ℝ}).

Clearly dim_{ℝ} M_{ℝ}^{gp} = rk_{ℤ}M^{gp}; moreover, by proposition 6.4.9, the face M_{ℝ}∩(−M_{ℝ}) is spanned by the image of the face M^{×} of M. whence the assertion.

(ii) is similar : again proposition 6.3.8(ii),(iii) implies that, every face F of M_{ℝ} fits into a maximal strictly ascending chain of faces of M_{ℝ}, and the length of any such maximal chain is dim M, by (i).

(iii): Notice that rk_{ℤ}M^{gp} > 0, by (i). By proposition 6.4.9(i), M_{ℝ} is strictly convex, therefore, by proposition 6.3.21(i), we may find a non-zero linear map φ : M^{gp} ⊗_{ℤ} ℚ → ℚ, such that M_{ℝ} ∩ Ker φ ⊗_{ℚ} ℝ = {0} and φ(M) ⊂ ℚ_{+}. A suitable positive integer of φ will do.

(iv): Under the stated assumption, we may regard M as a submonoid of M_{ℝ}, and the latter contains no non-zero linear subspaces. By corollary 6.3.14 and proposition 6.3.21(i), we may then find r linearly independent forms u_{1}, …, u_{r} : M^{gp} ⊗_{ℤ} ℚ → ℚ which are positive on M. It follows that u_{1} ⊗_{ℚ} ℝ, …, u_{r} ⊗_{ℚ} ℝ generate a polyhedral cone σ^{∨} ⊂ M_{ℝ}^{∨}, so its dual cone σ ⊂ M_{ℝ}^{gp} contains M_{ℝ}. By construction, σ admits precisely r extremal rays, say the rays generated by the vectors v_{1}, …, v_{r}, which we can pick in M_{ℚ}^{gp}, in which case they form a basis of the latter ℚ-vector space. Now, every x ∈ M_{ℝ} can be written uniquely in the form x = ∑_{i=1}^{r} a_{i}v_{i} for certain a_{1}, …, a_{r} ∈ ℚ_{+}; since M is finitely generated, we may find an integer N > 0 independent of x, such that Na_{i} ∈ ℕ for every i = 1, …, r. In other words, M is contained in the monoid generated by N^{−1}v_{1}, …, N^{−1}v_{r}; the latter is isomorphic to ℕ^{⊕r}. □

6.4.13. For any monoid M, the dual of M is the monoid

M^{∨} := Hom_{Mnd}(M, ℕ)

(see (6.1.1)). As usual, there is a natural morphism

M → M^{∨∨} : m ↦ (φ ↦ φ(m)) for every m ∈ M and φ ∈ M^{∨}.

We say that M is reflexive, if this morphism is an isomorphism.

Proposition 6.4.14. Let M be a monoid. We have :

(i) M^{∨} is integral, saturated and sharp.

(ii) If M is finitely generated, M^{∨} is fine, and we have a natural identification :

(M^{∨})_{ℝ} →^{∼} (M_{ℝ})^{∨}

(where (M_{ℝ})^{∨} is defined as in (6.3.1)). Moreover, dim M = dim M^{∨}. (iii) If M is finitely generated and sharp, we have a natural identification :

(M^{∨})^{gp} →^{∼} (M^{gp})^{∨}.

(iv) If M is fine, sharp and saturated, then M is reflexive.

Proof. (i): It is easily seen that the natural group homomorphism

(6.4.15)  (M^{∨})^{gp} → (M^{gp})^{∨} := Hom_{ℤ}(M^{gp}, ℤ)

is injective. Now, say that φ ∈ (M^{∨})^{gp} and Nφ ∈ M^{∨} for some N ∈ ℕ; we may view φ as group homomorphism φ : M^{gp} → ℤ, and the assumption implies that φ(M) ⊂ ℤ ∩ ℚ_{+} = ℕ, whence the contention.

(ii): Indeed, let x_{1}, …, x_{n} be a system of generators of M. Define a group homomorphism f : (M^{gp})^{∨} → ℤ^{⊕n} by the rule : φ ↦ (φ(x_{1}), …, φ(x_{n})) for every f : M^{gp} → ℤ. Then M^{∨} = φ^{−1}(ℕ^{⊕n}), and since (M^{gp})^{∨} is fine, corollary 6.4.2 implies that M^{∨} is fine as well. Next, the injectivity of (6.4.15) implies especially that (M^{∨})^{gp} is torsion-free, hence (6.4.15)⊗_{ℤ} ℝ is still injective; its restriction to (M^{∨})_{ℝ} factors therefore through an injective map f : (M^{∨})_{ℝ} → (M_{ℝ})^{∨}. The latter map is determined by the image of M^{∨}, and by inspecting the definitions, we see that f(φ) := φ^{gp} ⊗ 1 for every φ ∈ M^{∨}. To prove that f is an isomorphism, it suffices to show that it has dense image. However, say that φ ∈ (M_{ℝ})^{∨}; then φ : M^{gp} → ℝ is a group homomorphism such that φ(M) ⊂ ℝ_{+}. Since M is finitely generated, in any neighborhood of φ in M_{ℝ}^{gp} we may find some φ′ : M^{gp} → ℚ_{+}, and then Nφ′ ∈ M^{∨} for some integer N ∈ ℕ large enough. It follows that φ′ is in the image of f, whence the contention.

The stated equality follows from the chain of identities :

dim M = dim M_{ℝ} = dim(M_{ℝ})^{∨} = dim(M^{∨})_{ℝ} = dim M^{∨}

where the first and the last follow from proposition 6.4.9(ii), and the second follows from corollary 6.3.12(ii).

(iii): Let us show first that, under these assumptions, (6.4.15) ⊗_{ℤ} ℝ is an isomorphism. Indeed, if M is sharp, (M_{ℝ})^{∨} spans (M_{ℝ}^{gp})^{∨} (corollary 6.3.14 and proposition 6.4.9(i)); then the assertion follows from (ii). We deduce that (M^{∨})^{gp} and (M^{gp})^{∨} are free abelian groups of the same rank, hence we may find a basis φ_{1}, …, φ_{r} of (M^{∨})^{gp} (resp. ψ_{1}, …, ψ_{r} of (M^{gp})^{∨}), and positive integers N_{1}, …, N_{r} such that (6.4.15) is given by the rule : φ_{i} ↦ N_{i}ψ_{i} for every i = 1, …, r. But then necessarily we have N_{i} = 1 for every i ≤ r, and (iii) follows.

(iv): It is easily seen that M^{∨} = (M_{ℝ})^{∨} ∩ (M^{gp})^{∨} (notation of (6.4.8)). After dualizing again we find : M^{∨∨} = ((M^{∨})_{ℝ})^{∨} ∩ (M^{∨gp})^{∨}. From (ii) we deduce that ((M^{∨})_{ℝ})^{∨} = (M_{ℝ})^{∨∨} = M_{ℝ} (lemma 6.3.2), and from (iii) we get : (M^{∨gp})^{∨} = (M^{gp})^{∨∨} = M^{gp}. Hence M^{∨∨} = M_{ℝ} ∩ M^{gp} = M (proposition 6.3.22(iii)). □

Remark 6.4.16. (i) Let M be a sharp and fine monoid. Proposition 6.4.14(iii) implies that the natural map

Hom_{Mnd}(M, ℚ_{+})^{gp} → Hom_{Mnd}(M, ℚ)

is an isomorphism. Indeed, it is easily seen that this map is injective. For the surjectivity, one uses the identification Hom_{Mnd}(M, ℚ) →^{∼} (M^{∨})^{gp} ⊗_{ℤ} ℚ, which follows from loc.cit. (Details left to the reader.)

(ii) For i = 1, 2, let N_{i} → N be two morphisms of monoids. By general nonsense, we have a natural isomorphism :

(N_{1} ⊗_{N} N_{2})^{∨} →^{∼} N_{1}^{∨} ×_{N^{∨}} N_{2}^{∨}.

(iii) If f_{i} : M_{i} → M (i = 1, 2) are morphisms of fine, saturated and sharp monoids, there exists a natural surjection :

(6.4.17)  M_{1}^{∨} ⊗_{M^{∨}} M_{2}^{∨} → (M_{1} ×_{M} M_{2})^{∨}

whose kernel is the subgroup of invertible elements. Indeed, set P := M_{1}^{∨} ⊗_{M^{∨}} M_{2}^{∨}; in view of (ii) and proposition 6.4.14(iv), we have a natural identification P^{∨∨} →^{∼} (M_{1} ×_{M} M_{2})^{∨}, and the sought map is its composition with the double duality map P → P^{∨∨}. Moreover, clearly P is finitely generated, and it is also integral and saturated, since saturation commutes with colimits. Hence – again by proposition 6.4.14(iv) – the double duality map induces an isomorphism

P/P^{×} →^{∼} P^{∨∨}.

(iv) In the situation of (iii), if f_{i} : M_{i} → M (i = 1, 2) are epimorphisms, then (6.4.17) is an isomorphism. Indeed, in this case the dual morphisms f_{i}^{∨} : M^{∨} → M_{i}^{∨} are injective, so that P is sharp (lemma 6.1.13), whence the claim.

Theorem 6.4.18. Let M be a saturated monoid, such that M^{♯} is fine. We have :

(i) M = ⋂_{ht 𝔭=1} M_{𝔭} (where the intersection runs over the prime ideals of M of height one).

(ii) If moreover, dim M = 1, then there is an isomorphism of monoids :

M^{×} × ℕ →^{∼} M.

(iii) Suppose that M^{gp} is a torsion-free abelian group, and let R be any normal domain. Then the group algebra R[M] is a normal domain as well.

Proof. (i): Pick a decomposition M = M^{♯} ×M^{×} as in lemma 6.2.10, and notice that M^{♯} is fine, sharp and saturated. The prime ideals of M are of the form 𝔭 = 𝔭_{0} × M^{×}, where 𝔭_{0} is a prime ideal of M^{♯}. Then it is easily seen that M_{𝔭} = M_{𝔭_{0}}^{♯} × M^{×}. Therefore, the sought assertion holds for M if and only if it holds for M^{♯}, and therefore we may replace M by M^{♯}, which reduces to the case where M is sharp, hence the natural morphism φ : log M → M_{ℝ} is injective. In such situation, we have M = M_{ℝ} ∩ M^{gp} and M_{𝔭} = M_{𝔭,ℝ} ∩ M^{gp} for every prime ideal 𝔭 ⊂ M (proposition 6.3.22(iii) and lemma 6.2.9(i)). Thus, we are reduced to showing that

M_{ℝ} = ⋂_{ht 𝔭=1} M_{𝔭,ℝ}.

However, set τ := (M \𝔭)_{ℝ}; by inspecting the definitions, one sees that M_{𝔭,ℝ} = M_{ℝ} + (−τ), and proposition 6.4.9shows that τ is a facet of M_{ℝ}, hence M_{𝔭,ℝ} is the half-space denoted H_{τ} in (6.3.10). Then the assertion is a rephrasing of proposition 6.3.11(ii).

(ii): Arguing as in the proof of (i), we may reduce again to the case where M is sharp, in which case M = M_{ℝ} ∩ M^{gp}. The foregoing shows that, in case dim M = 1, the cone M_{ℝ} is a half-space, whose boundary hyperplane is the only non-trivial face σ of M_{ℝ}. However, σ is generated by the image of the unique non-trivial face of M, i.e. by M^{×} = {1} (proposition 6.4.9(i)), hence σ = {0}, so M_{ℝ} is a half-line. Now, let u : M_{ℝ}^{gp} → ℝ be a non-zero linear form, such that u(M) ≥ 0, and x_{1}, …, x_{n} a system of non-zero generators for M; say that u(x_{1}) is the least of the values u(x_{i}), for i = 1, …, n. Since M is saturated, it follows easily that every value u(x_{i}) is an integer multiple of u(x_{1}) (proposition 6.3.22(iii)), and then x_{1} is a generator for M, so M ≃ ℕ.

(iii): To begin with, R[M] ⊂ R[M^{gp}], and since M^{gp} is torsion-free, it is clear that [M^{gp}] is a domain, hence the same holds for R[M]. Furthermore, from (i) we derive : R[M] = ⋂_{ht𝔭=1} R[M_{𝔭}], hence it suffices to show that R[M_{𝔭}] is normal whenever 𝔭 has height one. However, we have R[M_{𝔭}] ≃ R[M_{𝔭}^{×}] ⊗_{R} R[ℕ] in light of (ii), and since M_{𝔭}^{×} is torsion free, it is a filtered colimit of a family of free abelian groups of finite rank, so everything is clear. □

Example 6.4.19. Let M be a fine, sharp and saturated monoid of dimension 2.

(i) By corollary 6.4.12(i) and example 6.3.16, we see that M admits exactly two facets, which are fine saturated monoids of dimension one; by theorem 6.4.18(ii) each of these facets is generated by an element, say e_{i} (for i = 1, 2). From proposition 6.3.22(iii) it follows that ℚ_{+}e_{1} ⊕ ℚ_{+}e_{2} = M_{ℚ}. Especially, we may find an integer N > 0 large enough, such that :

ℕe_{1} ⊕ ℕe_{2} ⊂ M ⊂ ℕ\frac{e_{1}}{N} ⊕ ℕ\frac{e_{2}}{N} .

(ii) Moreover, clearly e_{1} and e_{2} are unimodular elements of M^{gp} (i.e. they generate direct summands of the latter free abelian group of rank 2). We may then find a basis f_{1}, f_{2} of M^{gp} with e_{1} = f_{1}, and e_{2} = af_{1} + bf_{2}, where a, b ∈ ℤ and (a, b) = 1. After replacing f_{2} by some element of the form cf_{2} + df_{1} with c ∈ {1, −1} and d ∈ ℤ, we may assume that b > 0 and 0 ≤ a < b. Clearly, such a normalized pair (a, b) determines the isomorphism class of M, since M_{ℝ} is the strictly convex cone of M_{ℝ}^{gp} whose extremal rays are generated by e_{1} and e_{2}, and M = M^{gp} ∩ M_{ℝ}.

(iii) More precisely, suppose that M′ is another fine, sharp and saturated monoid of dimension 2, and φ : M → M′ an isomorphism. Pick a basis f′_{1}, f′_{2} of M′^{gp} and a normalized pair (a′, b′) as in (ii), such that e′_{1} := f_{1} and e′_{2} := a′f′_{1} + b′f′_{2} generate the two facets of M′. Clearly, φ must send a facet of M onto a facet of M′; we distinguish two possibilities :

• either φ(e_{1}) = e′_{1} and φ(e_{2}) = e′_{2}, in which case we get φ(f_{2}) = b^{−1}(a′−a)f′_{1}+b^{−1}b′f′_{2}; especially, b′, a − a′ ∈ bℤ. By considering φ^{−1}, we get symmetrically that b ∈ b′ℤ, so b = b′ and therefore (a′, b) = 1 = (a, b) and 0 ≤ a, a′ < b, whence a = a′

• or else φ(e_{1}) = e′_{2} and φ(e_{2}) = e′_{1}, in which case we get φ(f_{2}) = b^{−1}(1 − aa′)f_{1} + b^{−1}b′af_{2}. It follows again that b′ ∈ bℤ, so b = b′, arguing as in the previous case. Moreover, 0 ≤ a′ < b, and 1 − aa′ ∈ bℤ. In other words, the class of a′ in the group (ℤ/bℤ)^{×} is the inverse of the class of a.

Conversely, it is easily seen that if M′ is as above, f′_{1}, f′_{2} is a basis of M′^{gp}, and the two facets of M′ are generated by f′_{1} and a′f′_{1} + b′f′_{2}, for a pair (a′, b′) normalized as in (ii), and such that aa′ ≡ 1 (mod b), then there exists an isomorphism M →^{∼} M′ of monoids (details left to the reader). Hence, set (ℤ/bℤ)^{†} := (ℤ/bℤ)^{×}/∼, where ∼ denotes the smallest equivalence relation such that [a] ∼ [a]^{−1} for every [a] ∈ (ℤ/bℤ)^{×}. We conclude that there exists a natural bijection between the set of isomorphism classes of fine, sharp and saturated monoids of dimension 2, and the set of pairs (b, [a]), where b > 0 is an integer, and [a] ∈ (ℤ/bℤ)^{†}.

6.4.20. Let A be a ring, P a monoid. Recall that 𝔪_{P} is the maximal (prime) ideal of P (see definition 6.1.10(i)). The 𝔪_{P} -adic filtration of P is the descending sequence of ideals :

⋯ ⊂ 𝔪_{P}^{3} ⊂ 𝔪_{P}^{2} ⊂ 𝔪_{P}

where 𝔪_{P}^{n} is the n-th power of 𝔪 in the monoid 𝒫(P) (see (6.1.1)). It induces a 𝔪_{P} -adic filtration Fil_{•}M on any P -module M and any A[P]-algebra B, defined by letting Fil_{n}M := 𝔪_{P}^{n} M and Fil_{n}B := A[𝔪_{P}^{n}] · B, for every n ∈ ℕ.

Lemma 6.4.21. Suppose that P is fine. We have :

(i) The 𝔪_{P} -adic filtration is separated on P.

(ii) If P is sharp, P \𝔪_{P}^{n} is a finite set, for every n ∈ ℕ.

Proof. (i): Indeed, choose A to be a non-zero noetherian ring, set J := ⋂_{n≥0} A[𝔪_{P}^{n}] and notice that J is generated by 𝔪_{P}^{∞} := ⋂_{n∈ℕ} 𝔪_{P}^{n}. On the other hand, J is annihilated by an element of 1+A[𝔪_{P}] ([126, Th.8.9]). Thus, suppose x ∈ 𝔪_{P}^{∞}, and pick y ∈ A[𝔪_{P}] such that (1−y)x = 0; we may write y = a_{1}t_{1} + ⋯ + a_{r}t_{r} for certain a_{1}, …, a_{r} ∈ A and t_{1}, …, t_{r} ∈ 𝔪_{P}. Therefore x = a_{1}xt_{1} + ⋯ + a_{r}xt_{r} in A[P], which is absurd, since P is integral.

Assertion (ii) is immediate from the definition. □

6.4.22. Keep the assumptions of lemma 6.4.21. It turns out that P can actually be made into a graded monoid, albeit in a non-canonical manner. We proceed as follows. Let ε : P → P^{sat} the inclusion map, T ⊂ P^{sat} the torsion subgroup, set Q := P^{sat}/T, and let π : P^{sat} → Q be the natural surjection. We may regard log Q as a submonoid of the polyhedral cone Q_{ℝ}, lying in the vector space Q_{ℝ}^{gp}, as in (6.4.8). Since Q_{ℝ} is a rational polyhedral cone, the same holds for Q_{ℝ}^{∨}, hence we may find a ℚ-linear form γ : log Q^{gp} ⊗_{ℤ} ℚ → ℚ, which is non-negative on log Q, and such that Q_{ℝ} ∩ Ker γ ⊗_{ℚ} ℝ is the minimal face of Q_{ℝ}, i.e. the ℝ-vector space spanned by the image of Q^{×}. If we multiply γ by some large positive integer, we may achieve that γ(log P) ⊂ ℕ. We set :

gr_{n}^{γ}P := (γ ∘ π ∘ ε)^{−1}(n) for every n ∈ ℕ.

It is clear that gr_{n}^{γ}P · gr_{m}^{γ}P ⊂ gr_{n+m}^{γ}P, hence

P = ∐_{n∈ℕ} gr_{n}^{γ}P

is a ℕ-graded monoid, and consequently :

A[P] = ⨁_{n∈ℕ} A[gr_{n}^{γ}P]

is a graded A-algebra. Moreover, it is easily seen that gr_{0}^{γ}P = P^{×}. More generally, for x ∈ P, let μ(x) be the maximal n ∈ ℕ such that x ∈ 𝔪_{P}^{n}; then there exists a constant C ≥ 1 such that :

γ(x) ≥ μ(x) ≥ C^{−1}γ(x) for every x ∈ P

so that the 𝔪_{P} -adic filtration and the filtration defined by gr_{•}^{γ}P, induce the same topology on P and on A[P]. As a corollary of these considerations, we may state the following “regularity criterion” for fine monoids :

Proposition 6.4.23. Let P be an integral monoid such that P^{♯} is fine. Then we have

rk_{P^{×}}^{∘}𝔪_{P} /𝔪_{P}^{2} ≥ dim P

(notation of example 4.8.18) and the equality holds if and only if P^{♯} is a free monoid.

Proof. (Notice that 𝔪_{P} /𝔪_{P}^{2} is a free pointed P^{×}-module, since P^{×} obviously acts freely on 𝔪_{P} \𝔪_{P}^{2} .) Since 𝔪_{P^{♯}}/𝔪_{P^{♯}}^{2} = (𝔪_{P} /𝔪_{P}^{2} )⊗_{P} P^{♯}, we may replace P by P^{♯}, and assume from start that P is sharp and fine. Then, the rank of 𝔪_{P} /𝔪_{P}^{2} equals the cardinality of the set Σ := 𝔪_{P} \ 𝔪_{P}^{2} , which is finite, by lemma 6.4.21. We have a surjective morphism of monoids φ : ℕ^{(Σ)} → P, that sends the basis of ℕ^{(Σ)} bijectively onto Σ ⊂ P (corollary 6.1.12). The sought inequality follows immediately, and it is also clear that we have equality, in case P is free. Conversely, if equality holds, φ^{gp} must be a surjective group homomorphism between free abelian group of the same finite rank (corollary 6.4.12(i)), so it is an isomorphism, and then the same holds for φ. □

Proposition 6.4.24. Let P be a fine and sharp monoid, A a noetherian local ring. Set S_{P} := 1 + A[𝔪_{P}]; then we have :

dim S_{P}^{−1}A[P] = dim A + dim P.

Proof. To begin with, the assumption that P is sharp implies that S_{P}^{−1}A[P] is local. Next, notice that A[P] is a free A-module, hence the natural map A → S_{P}^{−1}A[P] is a flat and local ring homomorphism. Let k be the residue field of A; in view of [126, Th.15.1(ii)], we deduce :

dim S_{P}^{−1}A[P] = dim A + dim S_{P}^{−1}k[P].

Hence we are reduced to showing the stated identity for A = k. However, clearly S_{P}^{−1}k[P] = k[P]_{𝔪}, where 𝔪 is the maximal ideal generated by the image of 𝔪_{P}, hence it suffices to apply claim 11.6.37(ii) and corollary 6.4.12(i), to conclude. □

6.4.25. Let A be a ring, and P a fine and sharp monoid. We define :

A[[P]] := lim_{n∈ℕ} A⟨P/𝔪_{P}^{n} ⟩.

Alternatively, this is the completion of A[P] for its A[𝔪_{P}]-adic topology. In view of the finiteness properties of the 𝔪_{P} -adic filtration (lemma 6.4.21(ii)), one may present A[[P]] as the ring of formal infinite sums ∑_{σ∈P} a_{σ} ·σ, with arbitrary coefficients a_{σ} ∈ A, where the multiplication and addition are defined in the obvious way. Moreover, we may use a morphism of monoids γ : log P → ℕ as in (6.4.22), to see that :

(6.4.26)  A[[P]] = ∏_{n∈ℕ} A[gr_{n}^{γ}P]

where A[gr_{n}^{γ}P] · A[gr_{m}^{γ}P] ⊂ A[gr_{n+m}^{γ}P] for every m, n ∈ ℕ. So any element x ∈ A[[P]] can be decomposed as an infinite sum

x = ∑_{n∈ℕ} gr_{n}^{γ}x.

The term gr_{0}^{γ}x ∈ gr_{0}^{γ}A = A does not depend on the chosen γ : it is the constant term of x, i.e. the image of x under the natural projection A[[P]] → A.

Corollary 6.4.27. Let P be a fine and sharp monoid, A a noetherian local ring. Then : (i) For any local morphism P → A (see definition 6.1.10(v)), we have the inequality:

dim A ≤ dim A/𝔪_{P} A + dim P.

(ii) dim A[P] = dim A[[P]] = dim A + dim P.

Proof. (i): Set A_{0} := A/𝔪_{P} A, and B := S_{P}^{−1}A_{0}[P], where S_{P} ⊂ A_{0}[P] is the multiplicative subset 1 + A_{0}[𝔪_{P}]; if we denote by gr_{•}A (resp. gr_{•}B) the graded A_{0}-algebra associated to the 𝔪_{P} -adic filtration on A (resp. on B), we have a natural surjective homomorphism of graded A_{0}-algebras :

gr_{•}B → gr_{•}A.

Hence dim A = dim gr_{•}A ≤ dim gr_{•}B = dim B, by [126, Th.15.7]. Then the assertion follows from proposition 6.4.24.

(ii): Set B := S_{P}^{−1}A[P] and let 𝔪_{A} (resp 𝔪_{B}) be the maximal ideal of A (resp. of B); notice that A[[P]] is the (𝔪_{P} B)-adic completion of the local noetherian ring B, so A[[P]] is a local noetherian ring as well, with maximal ideal 𝔫 := 𝔪_{A}[[P]] + A[[𝔪_{P}]], and the 𝔫-adic completion of A[[P]] is naturally isomorphic to the 𝔪_{B}-adic completion B^{∧} of B. Hence we get

dim A[[P]] = dim B^{∧} = dim B

so the second identity follows from proposition 6.4.24. Next, clearly we have dim A ≥ dim B. On the other hand, let 𝔮 ⊂ A[P] be any prime ideal, set 𝔭 := 𝔮 ∩ A and denote by κ the residue field of the local ring A_{𝔭}; with this notation, [126, Th.15.1(ii)] says that

dim A[P]_{𝔮} = dim A_{𝔭} + dim A[P]_{𝔮} ⊗_{A} κ = dim A_{𝔭} + dim κ[P]_{𝔮} ≤ dim A + dim κ[P]

so the assertion follows from 11.6.37(ii) and corollary 6.4.12(i). □

6.4.28. Now we wish to state and prove the combinatorial versions of the Artin-Rees lemma, and of the local flatness criterion (see [126, Th.22.3]). Namely, let P be a pointed monoid, such that P^{♯} is finitely generated; let also (A, 𝔪_{A}) be a local noetherian ring, N a finitely generated A-module, and consider a morphism of pointed monoids :

α : P → (A, ·)

The following is our version of the Artin-Rees lemma :

Lemma 6.4.29. In the situation of (6.4.28), let J ⊂ P be an ideal, M a finitely generated P -module, M_{0} ⊂ M a submodule. Then there exists c ∈ ℕ such that :

(6.4.30)  J^{n}M ∩ M_{0} = J^{n−c}(J^{c}M ∩ M_{0})  for every n > c.

Proof. Set \overline{M} := M/P^{×}, \overline{M}_{0} := M_{0}/P^{×} and \overline{J} := J/P^{×}, the set-theoretic quotients for the respective natural P^{×}-actions. Notice that \overline{J} is an ideal of P^{♯} and \overline{M}_{0} ⊂ \overline{M} is an inclusion of \overline{P} -modules. Moreover, any set of generators of the ideal \overline{J} (resp. of the P^{♯}-module \overline{M}_{0}) lifts to a set of generators for J (resp. for the P -module M_{0}). Furthermore, it is easily seen that (6.4.30) is equivalent to the identity \overline{J}^{n}\overline{M} ∩ \overline{M}_{0} = \overline{J}^{n−c}(\overline{J}^{c}\overline{M} ∩ \overline{M}_{0}). Hence, we are reduced to the case where P = P^{♯} is a finitely generated monoid. Then ℤ[P] is noetherian, ℤ[M] is a ℤ[P]-module of finite type, and we notice that :

ℤ[J^{n}M ∩ M_{0}] = J^{n}ℤ[M] ∩ ℤ[M_{0}] ℤ[J^{n−c}(J^{c}M ∩ M_{0})] = J^{n−c}(J^{c}ℤ[M] ∩ ℤ[M_{0}]).

Thus, the assertion follows from the standard Artin-Rees lemma [126, Th.8.5]. □

Proposition 6.4.31. In the situation of (6.4.28), suppose moreover that P^{♯} is fine (see remark 4.8.14(vi)), and let 𝔪_{α} := α^{−1}𝔪_{A}. Then the following conditions are equivalent :

(a) N is α-flat (see definition 6.1.34).

(b) Tor_{i}^{ℤ⟨P ⟩}(N, ℤ⟨M⟩) = 0 for every i > 0 and every integral pointed P -module M.

(c) Tor_{1}^{ℤ⟨P ⟩}(N, ℤ⟨P/𝔪_{α}⟩) = 0.

(d) The natural map :

(𝔪_{α}^{n}/𝔪_{α}^{n+1}) ⊗_{P} N → 𝔪_{α}^{n}N/𝔪_{α}^{n+1}N

is an isomorphism of A-modules, for every n ∈ ℕ (notation of (6.1.33)).

Proof. The assertion (a)⇔(b) is just a restatement of proposition 6.1.40(i), and holds in greater generality, without any assumption on either A or the pointed integral monoid P. Also, obviously (b)⇒(c), As for the remaining assertions, let S := α^{−1}(A^{×}); since the localization P → S^{−1}P is flat, the natural maps :

Tor_{i}^{ℤ⟨P ⟩}(N, ℤ⟨M⟩) → Tor_{i}^{ℤ⟨S^{−1}P ⟩}(N, ℤ⟨S^{−1}M⟩)

are isomorphisms, for every i ∈ ℕ and every P -module M. Also, notice that the two P -modules appearing in (d) are actually S^{−1}P -modules (and the natural map is ℤ⟨S^{−1}P ⟩-linear). Hence, we can replace everywhere P by S^{−1}P, which allows to assume that α is local, i.e. 𝔪_{α} = 𝔪_{P}.

(c)⇒(d): For every n ∈ ℕ, we have a short exact sequence of pointed P -modules :

0 → 𝔪_{P}^{n} /𝔪_{P}^{n+1} → P/𝔪_{P}^{n+1} → P/𝔪_{P}^{n} → 0.

It is easily seen that 𝔪_{P}^{n} /𝔪_{P}^{n+1} is a free P/𝔪_{P} -module (in the category of pointed modules), so the assumption implies that Tor_{1}^{ℤ⟨P ⟩}(N, 𝔪_{P}^{n} /𝔪_{P}^{n+1}) = 0 for every n ∈ ℕ. By looking at the induced long Tor-sequences, we deduce that the natural map

Tor_{1}^{ℤ⟨P ⟩}(N, P/𝔪_{P}^{n+1}) → Tor_{1}^{ℤ⟨P ⟩}(N, P/𝔪_{P}^{n})

is injective for every n ∈ ℕ. Then, a simple induction shows that, under assumption (c), all these modules vanish. The latter means that the natural map :

𝔪_{P}^{n} ⊗_{P} N → 𝔪_{P}^{n} N

is an isomorphism, for every n ∈ ℕ. We consider the commutative ladder with exact rows :

\xymatrix{𝔪_{P}^{n+1} ⊗_{P} N \ar[r] \ar[d] & 𝔪_{P}^{n} ⊗_{P} N \ar[r] \ar[d] & (𝔪_{P}^{n} /𝔪_{P}^{n+1}) ⊗_{P} N \ar[r] \ar[d] & 0 \\ 𝔪_{P}^{n+1}N \ar[r] & 𝔪_{P}^{n} N \ar[r] & 𝔪_{P}^{n} N/𝔪_{P}^{n+1}N \ar[r] & 0.}

By the foregoing, the two left-most vertical arrows are isomorphisms, hence the same holds for the right-most, whence (c).

(d)⇒(c): We have to show that the natural map u : 𝔪_{P} ⊗_{P} N → 𝔪_{P} N is an isomorphism. To this aim, we consider the 𝔪_{P} -adic filtrations on these two modules; for the associated graded modules one gets :

gr_{n}(𝔪_{P} N) = 𝔪_{P}^{n} N/𝔪_{P}^{n+1}N gr_{n}(𝔪_{P} ⊗_{P} N) = (𝔪_{P}^{n} /𝔪_{P}^{n+1}) ⊗_{P} N

for every n ∈ ℕ; whence maps of A-modules :

gr_{n}(𝔪_{P} ⊗_{P} N) ⟶^{gr_{n}u} gr_{n}(𝔪_{P} N).

which are isomorphism by assumption. To conclude, it suffices to show :

Claim 6.4.32. For every ideal I ⊂ P, the the A-module I ⊗_{P} N is 𝔪_{P} A-adically separated.

Proof of the claim. Indeed, notice that the ideal I/P^{×} ⊂ P^{♯} is finitely generated (proposition 6.1.9(ii)), hence the same holds for I, so I ⊗_{P} N is a finitely generated A-module. Then the contention follows from [126, Th.8.10]. ♢

(c)⇒(b): We argue by induction on i. For i = 1, suppose first that M = P/I for some ideal I ⊂ P (notice that any such quotient is an integral pointed P -module); in this case, the assertion to prove is that the natural map v : I ⊗_{P} N → IN is an isomorphism. However, consider the 𝔪_{P} -adic filtration on P/I; for the associated graded module we have :

gr_{n}(P/I) = (𝔪_{P}^{n} ∪ I)/(𝔪^{n+1} ∪ I) for every n ∈ ℕ

and it is easily seen that this is a free pointed P^{×}-module, for every n ∈ ℕ. Hence, by inspecting the long exact Tor-sequences associated to the exact sequences

0 → gr_{n}(P/I) → P/(𝔪_{P}^{n+1} ∪ I) → P/(𝔪_{P}^{n} ∪ I) → 0

our assumption (c), together with a simple induction yields :

(6.4.33)  Tor_{1}^{ℤ⟨P ⟩}(N, ℤ⟨P/(𝔪_{P}^{n} ∪ I)⟩)  for every n ∈ ℕ.

Now, fix n ∈ ℕ; in light of lemma 6.4.29, there exists k ≥ n such that 𝔪_{P}^{k} ∩ I ⊂ 𝔪_{P}^{n} I. We deduce surjective maps of A-modules :

I ⊗_{P} N → \frac{I}{𝔪_{P}^{k} ∩ I} ⊗_{P} N → \frac{I}{𝔪_{P}^{n} I} ⊗_{P} N →^{∼} \frac{I ⊗_{P} N}{𝔪_{P}^{n} (I ⊗_{P} N)}.

On the other hand, (6.4.33) says that the natural map (𝔪_{P}^{n} ∪ I) ⊗_{P} N → (𝔪_{P}^{n} ∪ I)N is an isomorphism, so the same holds for the induced composed map :

\frac{I}{𝔪_{P}^{k} ∩ I} ⊗_{P} N →^{∼} \frac{𝔪_{P}^{k} ∪ I}{𝔪_{P}^{k}} ⊗_{P} N → \frac{(𝔪_{P}^{k} ∪ I)N}{𝔪_{P}^{k} N} .

Consequently, the kernel of v is contained in 𝔪_{P}^{n} (I ⊗_{P} N); since n is arbitrary, we are reduced to showing that the 𝔪_{P} -adic filtration is separated on I ⊗_{P} N, which is claim 6.4.32.

Next, again for i = 1, let M be an arbitrary integral P -module. In view of remark 6.1.27(i), we may assume that M is finitely generated; moreover, remark 6.1.27(ii), together with an easy induction further reduces to the case where M is cyclic, in which case, according to remark 6.1.27(iii), M is of the form P/I for some ideal I, so the proof is complete in this case.

Finally, suppose i > 1 and assume that the assertion is already known for i − 1. We may similarly reduce to the case where M = P/I for some ideal I as in the foregoing; to conclude, we observe that :

Tor_{i}^{ℤ⟨P ⟩}(N, ℤ⟨P/I⟩) ≃ Tor_{i−1}^{ℤ⟨P ⟩}(N, ℤ⟨I⟩).

Since obviously I is an integral P -module, the contention follows. □

As a corollary, we have the following combinatorial going-down theorem, which is proved in the same way as its commutative algebra counterpart.

Corollary 6.4.34. In the situation of (6.4.28), assume that P^{♯} is fine, and that A is α-flat. Let 𝔭 ⊂ 𝔮 be two prime ideals of P, and 𝔮′ ⊂ A a prime ideal such that 𝔮 = α^{−1}𝔮′. Then there exists a prime ideal 𝔭′ ⊂ 𝔮′ such that 𝔭 = α^{−1}𝔭′.

Proof. Let β : P_{𝔮} → A_{𝔮′} be the morphism induced by α; it is easily seen that A_{𝔮′} is β-flat, and moreover (P_{𝔭})^{♯} = (P^{♯})_{𝔭}^{♯} is still fine (lemma 6.1.20(iv)). Hence we may replace α by β, which (in view of remark 6.1.15(i)) allows to assume that 𝔮 (resp. 𝔮′) is the maximal ideal of P (resp. of A). Next, let P_{0} := P/𝔭, A_{0} := A/𝔭A and denote by α_{0} : P_{0} → A_{0} the morphism induced from α; it is easily seen that A_{0} is α_{0}-flat : for instance, the natural map (𝔪_{P}^{n} /𝔪_{P}^{n+1})⊗_{P} A_{0} → 𝔪_{P}^{n} A_{0}/𝔪_{P}^{n+1}A_{0} is of the type f ⊗_{A} A_{0}, where f is the map in proposition 6.4.31(c), thus if the latter is bijective, so is the former. Moreover P_{0}^{♯} is a quotient of P^{♯}, hence it is again fine. Therefore we may replace P by P_{0} and A by A_{0}, which allows to further assume that 𝔭 = {0}, and it suffices to show that there exists a prime ideal 𝔮′ ⊂ A, such that α^{−1}𝔮′ = {0}. Set Σ := P \ {0}; it is easily seen that the natural morphism P → Σ^{−1}P is injective; moreover, its cokernel C (in the category of pointed P -modules) is integral, so that Tor_{1}^{ℤ⟨P ⟩}(A, ℤ⟨C⟩) = 0 by assumption. It follows that the localization map A → Σ^{−1}A is injective, especially Σ^{−1}A ≠ {0}, and therefore it contains a prime ideal 𝔮′′. The prime ideal 𝔮′ := 𝔮′′ ∩ A will do. □

6.4.35. Fractional ideals. Let P be an integral monoid. A fractional ideal of P is a P - submodule I ⊂ P^{gp} such that I ≠ ∅ and x · I ⊂ P for some x ∈ P. Clearly the union and the intersection of finitely many fractional ideals, are again fractional ideals. We may also define the product of two fractional ideals I_{1}, I_{2} ⊂ P^{gp} : namely, the subset

I_{1}I_{2} := {xy | x ∈ I_{1}, y ∈ I_{2}} ⊂ P^{gp}

which is again a fractional ideal, by an easy inspection. If I is a fractional ideal of P, we say that I is finitely generated, if it is such, when regarded as a P -module. For any two fractional ideals I_{1}, I_{2}, we let

(I_{1} : I_{2}) := {x ∈ P^{gp} | x · I_{2} ⊂ I_{1}}.

It is easily seen that (I_{1} : I_{2}) is a fractional ideal of P (if x ∈ I_{2} and yI_{1} ⊂ P, then clearly xy(I_{1} : I_{2}) ⊂ P). We set

I^{−1} := (P : I)  and I^{∗} := (I^{−1})^{−1}  for every fractional ideal I ⊂ P^{gp}.

Clearly J^{−1} ⊂ I^{−1}, whenever I ⊂ J, and I ⊂ I^{∗} for all fractional ideals I, J. We say that I is reflexive if I = I^{∗}. We remark that I^{−1} is reflexive, for every fractional ideal I ⊂ P^{gp}. Indeed, we have I^{−1} ⊂ (I^{−1})^{∗}, and on the other hand (I^{−1})^{∗} = (I^{∗})^{−1} ⊂ I^{−1}. It follows that I^{∗} is reflexive, for every fractional ideal I. Moreover, I^{∗} ⊂ J^{∗}, whenever I ⊂ J; especially, I^{∗} is the smallest reflexive fractional ideal containing I. Notice furthermore, that aI^{−1} = (a^{−1}I)^{−1} for every a ∈ P^{gp}; therefore, aI^{∗} = (aI)^{∗}, for every fractional ideal I and a ∈ P^{gp}.

Lemma 6.4.36. Let P be any integral monoid, I, J ⊂ P^{gp} two fractional ideals. Then : (i) (IJ)^{∗} = (I^{∗}J^{∗})^{∗}. (ii) I^{∗} is the intersection of the invertible fractional ideals of P that contain I (see definition

4.8.6(iv)).

Proof. (i): Since IJ ⊂ I^{∗}J^{∗}, we have (IJ)^{∗} ⊂ (I^{∗}J^{∗})^{∗}. To show the converse inclusion, it suffices to check that I^{∗}J^{∗} ⊂ (IJ)^{∗}, since (IJ)^{∗} is reflexive, and (I^{∗}J^{∗})^{∗} is the smallest reflexive fractional ideal containing I^{∗}J^{∗}. Now, let a ∈ I be eny element; we get aJ^{∗} = (aJ)^{∗} ⊂ (IJ)^{∗}, so IJ^{∗} ⊂ (IJ)^{∗}, and therefore (IJ^{∗})^{∗} ⊂ (IJ)^{∗}. Lastly, let b ∈ J^{∗} be any element; we get bI^{∗} = (bI)^{∗} ⊂ (IJ^{∗})^{∗}, so I^{∗}J^{∗} ⊂ (IJ^{∗})^{∗}, whence the lemma.

(ii): It suffices to unwind the definitions. Indeed, a ∈ P^{gp} lies in I^{∗} if and only if aI^{−1} ⊂ P, if and only if ab ∈ P, for every b ∈ P^{gp} such that bI ⊂ P. In other words, a ∈ I^{∗} if and only if a ∈ b^{−1}P for every b ∈ P^{gp} such that I ⊂ b^{−1}P, which is the contention. □

6.4.37. Let P be any integral monoid. We denote by Div(P) the set of all reflexive fractional ideals of P. We define a composition law on Div(P) by the rule :

I ⊙ J := (IJ)^{∗} for every I, J ∈ Div(P).

It follows easily from lemma 6.4.36(i) that ⊙ is an associative law; indeed we may compute :

(I ⊙ J) ⊙ K = ((IJ)^{∗}K)^{∗} = (IJK)^{∗} = (I(JK)^{∗})^{∗} = I ⊙ (J ⊙ K)

for every I, J, K ∈ Div(P). Clearly I ⊙ J = J ⊙ I and P ⊙ I = I, for every I, J ∈ Div(P), so (Div(P), ⊙) is a commutative monoid. Notice as well that if I ⊂ P, then also I^{∗} ⊂ P (lemma 6.4.36(ii)), so the subset of all reflexive fractional ideals contained in P is a submonoid Div_{+}(P) ⊂ Div(P).

Example 6.4.38. Let A be an integral domain, and K the field of fractions of A. Classically, one defines the notion of fractional ideal : see e.g. [126, p.80] and [34, Ch.VII, §1, no.1], but notice that the definitions in these two references differ slightly, as the zero ideal is a fractional ideal according to the latter, but not according to the former. Additionally, one has a notion of reflexive fractional ideal of A, which are also called divisorial fractional ideals in [34, Ch.VII, §1, no.1]. In our terminology, these are understood as follows. Set A′ := A∩K^{×}, and notice that the monoid (A, ·) is naturally isomorphic to the integral pointed monoid A′_{∘}. Then a fractional ideal of A is an A′_{∘}-submodule of K_{∘}^{×} = K of the form I_{∘}, where I ⊂ K^{×} is a fractional ideal of A′. Likewise one may define the reflexive fractional ideals of A. The set Div(A) of all reflexive fractional ideals of A is then endowed with the unique monoid structure, such that the map Div(A′) → Div(A) given by the rule I ↦ I_{∘} is an isomorphism of monoids.

Lemma 6.4.39. Let P be an integral monoid, and G ⊂ P^{×} a subgroup. We have :

(i) The rule I ↦ I/G induces a bijection from the set of fractional ideals of P to the set of fractional ideals of P/G.

(ii) A fractional ideal I of P is reflexive if and only if the same holds for I/G.

(iii) The rule I ↦ I/G defines an isomorphism of monoids

Div(P) →^{∼} Div(P/G).

(iv) If P^{♯} is fine, every fractional ideal of P is finitely generated.

Proof. The first assertion is left to the reader. Next, we remark that I^{−1}/G = (I/G)^{−1} and (IJ)/G = (I/G) · (J/G), for every fractional ideals I, J of P, which imply immediately assertions (ii) and (iii). Lastly, suppose that P^{♯} is finitely generated, and let I be any fractional ideal of P; pick x ∈ I^{−1}; since P is integral, I is finitely generated if and only if the same holds for xI. Hence, in order to show (iv), we may assume that I ⊂ P, in which case the assertion follows from proposition 6.1.9(ii) and lemma 6.1.17(i.a). □

In order to characterize the monoids P such that Div(P) is a group, we make the following :

Definition 6.4.40. Let P be an integral monoid, and a ∈ P^{gp} any element.

(a) We say that a is power-bounded, if there exists b ∈ P such that a^{n}b ∈ P for all n ∈ ℕ.

(b) We say that P is completely saturated, if all power-bounded elements of P^{gp} lie in P.

Example 6.4.41. Let (Γ, ≤) be an ordered abelian group, and set Γ^{+} := {γ ∈ Γ | γ ≤ 1}. Then Γ^{+} is always a saturated monoid, but it is completely saturated if and only if the convex rank of Γ is ≤ 1 (see [75, Def.6.1.20]). The proof shall be left as an exercise for the reader.

Proposition 6.4.42. Let P be an integral monoid. We have :

(i) (Div(P), ⊙) is an abelian group if and only if P is completely saturated.

(ii) If P is fine and saturated, then P is completely saturated.

(iii) Let A be a Krull domain, and set A′ := A\{0}. Then (A′, ·) is a completely saturated monoid.

Proof. (i): Suppose that I ∈ Div(I) admits an inverse J in the monoid (Div(P), ⊙), and notice that I ⊙ I^{−1} ⊂ P; it follows easily that I ⊙ (J ∪ I^{−1})^{∗} = P, hence I^{−1} ⊂ J, by the uniqueness of the inverse. On the other hand, if J strictly contains I^{−1}, then IJ strictly contains P, which is absurd. Thus, we see that Div(P) is a group if and only if I ⊙I^{−1} = P for every I ∈ Div(P). Now, suppose first that P is completely saturated. In view of lemma 6.4.36(ii), we are reduced to showing that P is contained in every invertible fractional ideal containing I^{−1}I. Hence, say that I^{−1}I ⊂ aP for some a ∈ P^{gp}; equivalently, we have a^{−1}I^{−1}I ⊂ P, i.e. a^{−1}I^{−1} ⊂ I^{−1}, and then a^{−k}I^{−1} ⊂ I^{−1} for every integer k ∈ ℕ. Say that b ∈ I^{−1} and c ∈ I^{∗}; we conclude that a^{−k}bc ∈ P for every k ∈ ℕ, so a^{−1} ∈ P, by assumption, and finally P ⊂ aP, as required.

Conversely, suppose that Div(P) is a group, and let a ∈ P^{gp} be any power-bounded element. By definition, this means that the P -submodule I of P^{gp} generated by (a^{k} | k ∈ ℕ) is a fractional ideal of P. Then I^{−1} is a reflexive fractional ideal, and by assumption I^{−1} admits an inverse, which must be I^{∗}, by the foregoing. On the other hand, by construction we have aI ⊂ I, hence aI^{∗} = (aI)^{∗} ⊂ I^{∗}. We deduce that aP = a(I^{∗} ⊙ I^{−1}) = aI^{∗} ⊙ I^{−1} ⊂ I^{∗} ⊙ I^{−1} = P, i.e. a ∈ P, as stated.

(ii) is a special case of the following :

Claim 6.4.43. Let P be any fine and saturated monoid, M ⊂ P^{gp} a non-empty finitely generated P -submodule, and a ∈ P^{gp} an element such that aM ⊂ M. Then a ∈ P.

Proof of the claim. Pick any m ∈ M, and denote by M′ ⊂ M the submodule generated by (a^{k}m | k ∈ ℕ). According to proposition 6.1.9(i), there exists N ≥ 0 such that M′ is generated by the finite system (a^{k}m | k = 0, …, N). Especially, a^{N+1}m ∈ M′, and therefore there exist x ∈ P and i ≤ N such that a^{N+1}m = a^{i}mx in M; it follows that a^{N+1−i} ∈ P, and finally a ∈ P, since P is saturated. ♢

(iii): See [126, §12] for the basic generalities on Krull domains. One is immediately reduced to the case where A is a valuation ring whose value group Γ has rank ≤ 1. Taking into account (i) and lemma 6.4.39, it then suffices to show that the monoid A′/A^{×} is completely sataurated. However, the latter is isomorphic to the submonoid Γ^{+} of elements ≤ 1 in Γ, so the assertion follows from example 6.4.41. □

6.4.44. Let φ : P → Q be a morphism of integral monoids, and I any fractional ideal of P; notice that IQ := φ^{gp}(I)Q ⊂ Q^{gp} is a fractional ideal of Q. Moreover, the identities

(I_{1} ∪ I_{2})Q = I_{1}Q ∪ I_{2}Q  (I_{1}I_{2})Q = (I_{1}Q) · (I_{2}Q)  for all fractional ideals I_{1}, I_{2} ⊂ P^{gp}

are immediate from the definitions. Likewise, if A an integral domain and α : P → (A\{0}, ·) a morphism of monoids, then the A-submodule IA := α^{gp}(I)A of the field of fractions of A is a fractional ideal of the ring A (in the usual commutative algebraic meaning : see example 6.4.38), and we have corresponding identities :

(I_{1} ∪ I_{2})A = I_{1}A + I_{2}A  (I_{1}I_{2})A = (I_{1}A) · (I_{2}A)  for all fractional ideals I_{1}, I_{2} ⊂ P^{gp}.

Lemma 6.4.45. In the situation of (6.4.44), suppose that φ is flat and A is α-flat, and let I, J, J′ ⊂ P^{gp} be three fractional ideals, with I finitely generated. Then we have :

(i) (J : I)Q = (JQ : IQ) and (J : I)A = (J : I)A.

(ii) Especially, if I is reflexive, the same holds for IQ and IA (see example 6.4.38).

(iii) Suppose furthermore that A is local, and α is a local morphism. Then JA = J′A if

and only if J = J′. (iv) If P is fine, the rule I ↦ IQ and I ↦ IA define morphisms of monoids

Div(φ) : Div(P) → Div(Q) Div(α) : Div(P) → Div(A)

(where Div(A) is defined as in example 6.4.38), and Div(α) is injective, if α is local and A is a local domain.

Proof. (i): Say that I = a_{1}P ∪ ⋯ ∪ a_{n}P for elements a_{1}, …, a_{n} ∈ P^{gp}. Then

(J : I) = a_{1}^{−1}J ∩ ⋯ ∩ a_{n}^{−1}J and (JQ : IQ) = a_{1}^{−1}JQ ∩ ⋯ ∩ a_{n}^{−1}JQ

and likewise for (J : I)A, hence the assertion follows from an easy induction, and the following

Claim 6.4.46. For any two fractional ideals J_{1}, J_{2} ⊂ P, we have (J_{1} ∩ J_{2})Q = J_{1}Q ∩ J_{2}Q and (J_{1} ∩ J_{2})A = J_{1}A ∩ J_{2}A.

Proof of the claim. Pick any x ∈ P such that xJ_{1}, xJ_{2} ⊂ P; since P is an integral monoid, and A is an integral domain, it suffices to show that x(J_{1} ∩ J_{2})Q = xJ_{1}Q ∩ xJ_{2}Q and likewise for x(J_{1} ∩ J_{2})A, and notice that x(J_{1} ∩ J_{2}) = xJ_{1} ∩ xJ_{2}. We may thus assume that J_{1} and J_{2} are ideals of P, in which case the assertion is lemma 6.1.37. ♢

(ii): Suppose that I is reflexive; from (i) we deduce that ((IA)^{−1})^{−1} = IA. The assertion is an immediate consequence, once one remarks that, for any fractional ideal J ⊂ A, there is a natural isomorphism of A-modules : J^{−1} →^{∼} J^{∨} := Hom_{A}(J, A). Indeed, the isomorphism assigns to any x ∈ J^{−1} the map μ_{x} : J → A : a ↦ xa for every a ∈ J (details left to the reader).

(iii): We may assume that JA = J′A, and we prove that J = J′, and by replacing J′ by J ∪ J′, we may assume that J ⊂ J′. Then the contention follows easily from lemma 6.1.36.

(iv): This is immediate from (i) and (iii). □

Remark 6.4.47. In the situation of lemma 6.4.45(iv), obviously Div(φ) restricts to a morphism of submonoids :

Div_{+}(φ) : Div_{+}(P) → Div_{+}(Q).

6.4.48. Next, suppose that P is fine and saturated, and let I ⊂ P^{gp} be any fractional ideal. Then theorem 6.4.18(i) easily implies that :

I^{−1} = ⋂_{ht 𝔭=1}(I_{𝔭})^{−1}

where the intersection – running over the prime ideals of P of height one – is taken within Hom_{P} (I, P^{gp}), which naturally contains all the (I_{𝔭})^{−1}. The structure of the fractional ideals of P_{𝔭} when ht 𝔭 = 1 is very simple : quite generally, theorem 6.4.18(ii) easily implies that if dim P = 1, then all fractional ideals are cyclic, and then clearly they are reflexive. On the other hand, I is finitely generated, by lemma 6.4.39(iv). We deduce that I is reflexive if and only if :

(6.4.49)  I = ⋂_{ht 𝔭=1} I_{𝔭}.

Indeed, suppose that (6.4.49) holds; then we have I^{∗} = ⋂_{ht 𝔭=1}(I_{𝔭}^{−1})^{−1} = ⋂_{ht 𝔭=1} I_{𝔭}, since we have just seen that I_{𝔭} is a reflexive fractional ideal of P_{𝔭}, for every prime ideal 𝔭 of height one.

Proposition 6.4.50. Let P be a fine and saturated monoid, and denote by D ⊂ Spec P the subset of all prime ideals of height one. Then the mapping :

(6.4.51)  ℤ^{⊕D} → Div(P) : ∑_{ht 𝔭=1} n_{𝔭}[𝔭] ↦ ⋂_{ht 𝔭=1} 𝔪_{P_{𝔭}}^{n_{𝔭}}

is an isomorphism of abelian groups.

Proof. Here 𝔪_{P_{𝔭}} ⊂ P_{𝔭} is the maximal ideal, and for n ≥ 0, the notation 𝔪_{P_{𝔭}}^{n} means the usual n-th power operation in the monoid 𝒫(P^{gp}), which we extend to all integers n, by letting 𝔪_{P_{𝔭}}^{n} := 𝔪_{P_{𝔭}}^{−n} whenever n < 0.

In order to show that (6.4.51) is well defined, set I := ⋂_{ht 𝔭=1} 𝔪_{𝔭}^{n_{𝔭}}. Pick, for every 𝔭 such that n_{𝔭} < 0, an element x_{𝔭} ∈ 𝔭, and set y_{𝔭} := x_{𝔭}^{−n_{𝔭}}; if n_{𝔭} ≥ 0, set y_{𝔭} := 1. Then it is easy to check (using theorem 6.4.18(i)) that ∏_{ht 𝔭=1} y_{𝔭} lies in I^{−1}, hence I is a fractional ideal. Next, for given 𝔭, 𝔭′ ∈ D, notice that (P_{𝔭})_{𝔭′} = P^{gp}; it follows that

(6.4.52)  I_{𝔭} = 𝔪_{𝔭}^{n_{𝔭}}  for every 𝔭 ∈ D

therefore I is reflexive. Furthermore, it is easily seen (from theorem 6.4.18(ii)), that every reflexive ideal of P_{𝔭} is of the form 𝔪_{P}^{n} for some integer n, and moreover 𝔪_{P}^{n} = 𝔪_{P}^{m} if and only if n = m. Then (6.4.49) implies that the mapping (6.4.51) is surjective, and the injectivity follows from (6.4.52). It remains to check that (6.4.51) is a group homomorphism, and to this aim we may assume – in view of lemma 6.4.45(iv) – that dim P = 1, in which case the assertion is immediate. □

6.4.53. A morphism φ : I → J of fractional ideals of P is, by definition, a morphism of P -modules. Let x, y ∈ I be any two elements; we may find a, b ∈ P such that ax = by in I, and therefore aφ(x) = φ(ax) = φ(by) = bφ(y); thus, φ(y) = (b^{−1}a) · φ(x) = (x^{−1}y) · φ(x). This shows that, for every morphism φ : I → J of fractional ideals, there exists c ∈ P^{gp} such that φ(x) = cx for every x ∈ I. Especially, I ≃ J if and only if there exists a ∈ P^{gp} such that I = aJ. Likewise one may characterize the morphisms and isomorphisms of fractional ideals of an integral domain. We denote

\overline{Div}(P)

the set of isomorphism classes of reflexive fractional ideals of P. From the foregoing, it is clear that if I ≃ I′, we have I ⊙ J ≃ I′ ⊙ J for every J ∈ Div(P); therefore the composition law of Div(P) descends to a composition law for \overline{Div}(P), which makes it into a (commutative) monoid, and if P is completely saturated, then \overline{Div}(P) is an abelian group. We also deduce an exact sequence of monoids

(6.4.54)  1 → P^{×} → P^{gp} ⟶^{j_{P}} Div(P) → \overline{Div}(P) → 1

where j_{P} is given by the rule A: a ↦ aP for every a ∈ P; especially, j_{P} restricts to a morphism of monoids

j_{P}^{+} : P → Div_{+}(P).

Likewise, we define \overline{Div}(A), for any integral domain A : see example 6.4.38. Moreover, in the situation of (6.4.44), we see from lemma 6.4.45(iv) that if α is local, P is fine, A is α-flat, and φ is flat, then Div(φ) and Div(α) descend to well defined morphisms of monoids

\overline{Div}(φ) : \overline{Div}(P) → \overline{Div}(Q) \overline{Div}(α) : \overline{Div}(P) → \overline{Div}(A).

Proposition 6.4.55. Let P be a fine and saturated monoid, I, J ⊂ P^{gp} two fractional ideals, A a local integral domain, and α : P → (A, ·) a local morphism of monoids. We have :

(i) (I : I) = P.

(ii) Suppose that A is α-flat. Then IA ≃ JA if and only if I ≃ J. Especially, in this case \overline{Div}(α) is an injective map.

Proof. (i): Clearly it suffices to show that (I : I) ⊂ P. Hence, say that x ∈ (I : I), and pick any a ∈ I; it follows that x^{n}a ∈ P for every n > 0; in the additive group log P^{gp} we have therefore the identity n · log(x) + log(a) ∈ log P, so log(x) + n^{−1} log(a) ∈ (log P)_{ℝ} for every n > 0. Since (log P)_{ℝ} is a convex polyhedral cone in (log P^{gp})_{ℝ}, we deduce that x ∈ (log P)_{ℝ} ∩ (log P^{gp}) = log P (proposition 6.3.22(iii)), as claimed.

(ii): We may assume that IA is isomorphic to JA, and we show that I is isomorphic to J. Indeed, the assumption means that a(IA) = JA for some x ∈ Frac(A); therefore, a ∈ (JA : IA) and a^{−1} ∈ (IA : JA), so

A = (IA : JA) · (JA : IA) = ((I : J) · (J : I))A

by virtue of lemma 6.4.45(i). Since A is local, it follows that there exist a ∈ (I : J) and b ∈ (J : I) such that α(ab) ∈ A^{×}, whence ab ∈ P^{×}, since α is local. It follows easily that I = aJ, as asserted. □

Example 6.4.56. (i) Let P be a fine and saturated monoid, and D ⊂ Spec P the subset of all prime ideals of height one; for every 𝔭 ∈ D, denote

v_{𝔭} : P → P_{𝔭}^{♯} →^{∼} ℕ

the composition of the localization map, and the natural isomorphism resulting from theorem 6.4.18(ii). A simple inspection shows that the isomorphism (6.4.51) identifies the map j_{P} of (6.4.54) with the morphism of monoids

v_{P} : P^{gp} → ℤ^{⊕D} x ↦ (v_{𝔭}^{gp}(x) | 𝔭 ∈ D).

With this notation, the isomorphism (6.4.51) is the map given by the rule :

k_{•} ↦ v_{P}^{−1}(k_{•} + ℕ^{⊕D}) for every k_{•} ∈ ℤ^{⊕D}.

(ii) Suppose now that P is sharp and dim P = 2, in which case D = {𝔭_{1}, 𝔭_{2}} contains exactly two elements. According to example 6.4.19(ii), we may find a basis f_{1}, f_{2} of P^{gp}, such that the two facets P \ 𝔭_{1} and P \ 𝔭_{2} of P are generated respectively by e_{1} := f_{1} and e_{2} := af_{1} + bf_{2}, for some a, b ∈ ℕ, with a < b and (a, b) = 1. It follows easily that P is a submonoid of the free monoid

Q := ℕe′_{1} ⊕ ℕe′_{2} where e′_{1} := b^{−1}e_{1} and e′_{2} := b^{−1}e_{2}

and Q^{gp}/P^{gp} ≃ ℤ/bℤ (details left to the reader). The induced map Spec Q → Spec P is a homeomorphism; especially Q admits two prime ideals 𝔮_{1}, 𝔮_{2} of height one, so that 𝔮_{i} ∩ P = 𝔭_{i} for i = 1, 2, whence – by proposition 6.4.50– a natural isomorphism

s^{∗} : Div(Q) →^{∼} Div(P)

and notice that j_{Q} : Q^{gp} → Div(Q) is the isomorphism given by the rule : e′_{i} ↦ 𝔮_{i} for i = 1, 2. Moreover, we have commutative diagrams of monoids :

\xymatrix{P \ar[r]^{s} \ar[d]_{v_{𝔭_{i}}} & Q \ar[d]^{v_{𝔮_{i}}} \\ ℕ \ar[r]^{t_{i}} & ℕ} (i = 1, 2).

Clearly, Q \ 𝔮_{i} is the facet generated by e′_{i}, so v_{𝔮_{i}} is none else than the projection onto the direct factor ℕe′_{3−i}, for i = 1, 2. In order to compute v_{𝔭_{i}}, it then suffices to determine t_{i}, or equivalently t_{i}^{gp}. However, set τ_{i} := v_{𝔮_{i}}^{gp} ∘ s^{gp}; clearly τ_{1}(f_{2}) = τ_{1}(e′_{2} − ae′_{1}) = 1, so τ_{1} is surjective. Also, τ_{2}(f_{1}) = b and τ_{2}(f_{2}) = −a, so τ_{2} is surjective as well; therefore both t_{1} and t_{2} are the identity endomorphism of ℕ. Summing up, we find that

j_{P} = s^{∗} ∘ j_{Q} ∘ s^{gp}

and the morphism v_{P} is naturally identified with s^{gp} : P^{gp} → Q^{gp}. Especially, we have obtained a natural isomorphism

\overline{Div}(P) →^{∼} ℤ/bℤ.

We may then rephrase in more intrinsic terms the classification of example 6.4.19(iii) : namely the isomorphism class of P is completely determined by the datum of \overline{Div}(P) and the equivalence class of the height one prime ideals of P in the quotient set \overline{Div}(P)^{†} defined as in loc.cit.

(iii) In the situation of (ii), a simple inspection yields the following explicit description of all reflexive fractional ideals of P. Recall that such ideals are of the form

I_{k_{1},k_{2}} := 𝔪_{1}^{k_{1}} ∩ 𝔪_{2}^{k_{2}} = {x ∈ P^{gp} | v_{𝔭_{1}}(x) ≥ k_{1}, v_{𝔭_{2}}(x) ≥ k_{2}}

where 𝔪_{i} is the maximal ideal of P_{𝔭_{i}}, and k_{i} ∈ ℤ, for i = 1, 2. Then

I_{k_{1},k_{2}} = {x_{1}e_{1} + x_{2}e_{2} | x_{1}, x_{2} ∈ b^{−1}ℤ, x_{1} ≥ b^{−1}k_{2}, x_{2} ≥ b^{−1}k_{1}} ∩ P^{gp} for all k_{1}, k_{2} ∈ ℤ.

With this notation, the cyclic reflexive ideals are then those of the form

(x_{1}f_{1} + x_{2}f_{2})P = I_{x_{2}b,x_{1}b−x_{2}a} with x_{1}, x_{2} ∈ ℤ.

Especially, we see that the classes of 𝔭_{1} = I_{0,1} and 𝔭_{2} = I_{1,0} are both of order b in \overline{Div}(P).

Lemma 6.4.57. Let P be a fine and saturated monoid of dimension 2, and denote by b the order of the finite cyclic group \overline{Div}(P). We have :

𝔪_{P}^{[b/2]} ⊂ I · I^{−1} for every I ∈ Div(P)

(where [b/2] denotes the largest integer ≤ b/2).

Proof. Notice first that the assertion holds for a given I ∈ Div(P), if and only if it holds for xI, for any x ∈ P^{gp}. If b = 1, then P = ℕ^{⊕2}, in which case \overline{Div}(P) = 0, so every reflexive fractional ideal of P is isomorphic to P, and the assertion is clear. Hence, assume that b > 1; let 𝔭_{1}, 𝔭_{2} be the two prime ideals of height one of P, and define Q, e_{1}, e_{2} and I_{k_{1},k_{2}} for every k_{1}, k_{2} ∈ ℤ, as in example 6.4.56(ii,iii). With this notation, notice that

𝔪_{P} \ 𝔪_{P}^{2} = {e_{1}, e_{2}} ∪ Σ where Σ ⊂ {x_{1}e_{1} + x_{2}e_{2} | x_{1}, x_{2} ∈ b^{−1}ℤ, 0 ≤ x_{1}, x_{2} < 1}.

It follows easily that, for every i ∈ ℕ, every element of 𝔪_{P}^{i} is of the form x_{1}e_{1} + x_{2}e_{2} with x_{1}, x_{2} ∈ b^{−1}ℕ and max(x_{1}, x_{2}) ≥ b^{−1}i. Hence, let I ∈ Div(P) and x := x_{1}e_{1} + x_{2}e_{2} ∈ 𝔪_{P}^{[b/2]}, and say that bx_{1} ≥ [b/2]. According to example 6.4.56(iii), we may assume that I = 𝔭_{2}^{j} = I_{j,0} for some j ∈ {0, …, b − 1}. Moreover, notice that the assertion holds for I if and only if it holds for I^{−1}, whose class in \overline{Div}(P) agrees with the class of 𝔭_{2}^{b−j}. Clearly, either j ≤ [b/2] or b − j ≤ [b/2]; hence, we may assume that j ∈ {0, …, [b/2]}. Thus, P ⊂ I^{−1}, and I ⊂ I · I^{−1}, and clearly x ∈ I, so we are done in this case. The case where bx_{2} ≥ [b/2] is dealt with in the same way, by writing I = 𝔭_{1}^{j} for some non-negative j ≤ [b/2] : the details are left to the reader. □

If f : P → Q is a general morphism of integral monoids, and I a fractional ideal of Q, the P -module f^{gp−1}(I) is not necessarily a fractional ideal of P (for instance, consider the natural map P → P^{gp}). One may obtain some positive results, by restricting to the class of morphisms introduced by the following :

Definition 6.4.58. Let f : P → Q be a morphism of monoids. We say that f is of Kummer type, if f is injective, and the induced map f_{ℚ} : P_{ℚ} → Q_{ℚ} is surjective (notation of (6.3.20)).

Lemma 6.4.59. Let f : P → Q be a morphism of monoids of Kummer type, S_{Q} ⊂ Q a submonoid, and set S_{P} := f^{−1}S_{Q}. We have :

(i) The map Spec f : Spec Q → Spec P is bijective; especially dim P = dim Q.

(ii) If Q^{×} is a torsion-free abelian group, P is the trivial monoid (resp. is sharp) if and

only if the same holds for Q. (iii) The induced morphism S_{P}^{−1}P → S_{Q}^{−1}Q is of Kummer type. (iv) If P is integral, the unit of adjunction P → P^{sat} is of Kummer type.

(v) Suppose that P is integral and saturated. Then f^{♯} : P^{♯} → Q^{♯} is of Kummer type.

(vi) If both P and Q are integral, and P is saturated, then f is exact.

Proof. (ii) and (iv) are trivial, and (iii) is an exercise for the reader.

(i): Let F, F′ ⊂ Q be two faces such that f^{−1}F = f^{−1}F′, and say that x ∈ F. Then x^{n} ∈ f(P) for some n > 0, so x^{n} ∈ f(f^{−1}F′), whence x ∈ F′, which implies that Spec f is injective. Next, for a given face F of P, let F′ ⊂ Q be the subset of all x ∈ Q such that there exists n > 0 with x^{n} ∈ f(F). It is easily seen that F′ is a face of Q, and moreover f^{−1}F′ = F, which shows that Spec f is also surjective.

(v): Clearly the map (P^{♯})_{ℚ} → (Q^{♯})_{ℚ} is surjective. Now, let x, y ∈ P such that the images of f(x) and f(y) agree in Q^{♯}, i.e. there exists u ∈ Q^{×} with u · f(x) = f(y); we may find n > 0 such that u^{n}, u^{−n} ∈ f(P). Say that u^{n} = f(v), u^{−n} = f(w); since f(vw) = 1, we have vw = 1, and moreover f(vx^{n}) = f(y^{n}), so vx^{n} = y^{n}. Therefore x^{n}y^{−n}, x^{−n}y^{n} ∈ P, and since P is saturated we deduce that xy^{−1}, x^{−1}y ∈ P, so the images of x and y agree in P^{♯}.

(vi): Notice first that f^{gp} is injective, since the same holds for f. Suppose x ∈ P^{gp} and f^{gp}(x) ∈ Q; we may then find an integer k > 0 and y ∈ P such that f(y) = f(x)^{k}. Since P is saturated, it follows that x ∈ P, so f is exact. □

6.4.60. Suppose that φ : P → Q is a morphism of integral monoids of Kummer type, with P saturated, and let I ⊂ Q^{gp} be a fractional ideal. Then φ^{∗}I := φ^{gp−1}(I) is a fractional ideal of P. Indeed, by assumption there exists a ∈ Q such that aI ⊂ Q; we may find k > 0 and b ∈ P such that a^{k} = φ(b), so φ(bx) ∈ φ(P)^{gp} ∩ Q = φ(P) for every x ∈ φ^{∗}I, since φ is exact (lemma 6.4.59(v)); therefore b · φ^{∗}(I) ⊂ P.

6.4.61. In the situation of (6.4.60), suppose that both P and Q are fine and saturated, and let gr_{•}Q^{gp} be the φ-grading of Q, indexed by (Γ, +) := Q^{gp}/P^{gp} (see remark 6.2.5(iii)); for every x ∈ Q^{gp}, denote \overline{x} ∈ Γ the image of x. Let also I ⊂ Q^{gp} be any fractional ideal, and denote by gr_{•}I the Γ-grading on I deduced from the φ-grading of Q^{gp}; arguing as in (6.4.60), it is easily seen that, more generally, φ^{∗}(x^{−1}gr\overline{x}I) is a fractional ideal of P, for every x ∈ Q (details left to the reader). For every prime ideal 𝔮 of height one in Q, we have a commutative diagram of monoids :

(6.4.62)  \xymatrix{P \ar[r]^{φ} \ar[d]_{v_{φ^{−1}𝔮}} & Q \ar[d]^{v_{𝔮}} \\ ℕ \ar[r]^{e_{𝔮}} & ℕ}

where v_{𝔮} and v_{φ^{−1}𝔮} are defined as in example 6.4.56(i), and e_{𝔮} is the multiplication by a non-zero (positive) integer, which we call the ramification index of φ at 𝔮, and we denote also e_{𝔮}.

Lemma 6.4.63. In the situation of (6.4.61), suppose that I is a reflexive fractional ideal. Then φ^{∗}(a^{−1} · gr\overline{a}I) is a reflexive fractional ideal of P, for every a ∈ Q^{gp}.

Proof. Clearly, we may replace I by a^{−1}I, and reduce to the case where a = 1, in which case we have to check that φ^{∗}I is a reflexive fractional ideal. However, according to example 6.4.56(i), we may write I = v_{Q}^{−1}(k_{•} + ℕ^{⊕D}), where D ⊂ Spec Q is the subset of the height one prime ideals, and k_{•} ∈ ℤ^{⊕D}. Set

k′_{•} := ([e_{𝔮}^{−1}k_{𝔮}] | 𝔮 ∈ D)

(where, for a real number x, we let [x] be the smallest integer ≥ x). Since Spec φ is bijective (lemma 6.4.59(i)), the commutative diagrams (6.4.62)imply that φ^{∗}I = v_{P}^{−1}(k′_{•}+ℕ^{⊕D}), whence the contention. □

Example 6.4.64. Let P be as in example 6.4.56(ii), set Q := Div_{+}(P), and take φ := j_{P}^{+} : P → Q (notation of (6.4.53)). The discussion of loc.cit. shows that φ is a morphism of Kummer type, and notice that the φ-grading of Q is indexed by Q^{gp}/P^{gp} = \overline{Div}(P). Now, pick any x ∈ Q, and let \overline{x} ∈ \overline{Div}(P) be the equivalence class of x; by lemma 6.4.63, the P - module gr\overline{x}Q is isomorphic to a reflexive fractional ideal of P. We claim that the isomorphism class of gr\overline{x}Q is precisely \overline{x}^{−1} (where the inverse is formed in the commutative group \overline{Div}(P)). Indeed, let a ∈ P^{gp} be any element; by definition, we have a ∈ φ^{∗}(x^{−1}gr\overline{x}Q) if and only if φ^{gp}(a) ∈ x^{−1}gr\overline{x}Q, if and only if ax ∈ Q, if and only if a ∈ x^{−1}, whence the claim. Thus, the family

(gr_{γ}Div_{+}(P) | γ ∈ \overline{Div}(P))

is a complete system of representatives for the isomorphism classes of the reflexive fractional ideals of a fine, sharp and saturated monoid P of dimension 2.

Remark 6.4.65. Further results on reflexive fractional ideals for monoids, and their divisor class groups can be found in [48].

6.5. Fans. According to Kato ([112, §9]), a fan is to a monoid what a scheme is to a ring. More prosaically, the theory of fans is a reformulation of the older theory of rational polyhedral decompositions, developed in [116].

Definition 6.5.1. (i) A monoidal space is a datum (T, 𝒪_{T}) consisting of a topological space T and a sheaf of monoids 𝒪_{T} on T.

(ii) A morphism of monoidal spaces is a datum

(f, log f) : (T, 𝒪_{T}) → (S, 𝒪_{S})

consisting of a continuous map f : T → S, and a morphism log f : f^{∗}𝒪_{S} → 𝒪_{T} of sheaves of monoids that is local, i.e. whose stalk (log f)_{t} : 𝒪_{S,f(t)} → 𝒪_{T,t} is a local morphism, for every t ∈ T. The strict locus of (f, log f) is the subset

Str(f, log f) ⊂ T

consisting of all t ∈ T such that (log f)_{t} is an isomorphism. (iii) We say that a monoidal space (T, 𝒪_{T}) is sharp, (resp. integral, resp. saturated) if 𝒪_{T}

is a sheaf of sharp (resp. integral, resp. integral and saturated) monoids. (iv) For any monoidal space (resp. integral monoidal space) (T, 𝒪_{T}), the sharpening (resp.

the saturation) of (T, 𝒪_{T}) is the sharp monoidal space (T, 𝒪_{T})^{♯} := (T, 𝒪_{T}^{♯}) (resp. (T, 𝒪_{T})^{sat} := (T, 𝒪_{T}^{sat})).

It is easily seen that the rule (T, 𝒪_{T}) ↦ (T, 𝒪_{T})^{♯} extends to a functor from the category of monoidal spaces to the full subcategory of sharp monoidal spaces. This functor is right adjoint to the corresponding fully faithful embedding of categories.

Likewise, the functor (T, 𝒪_{T}) ↦ (T, 𝒪_{T})^{sat} is right adjoint to the fully faithful embedding of the category of saturated monoidal spaces, into the category of integral monoidal spaces.

6.5.2. Let P be any monoid; for every f ∈ P, let us set

D(f) := {𝔭 ∈ Spec P | f ∉ 𝔭}.

Notice that D(f) ∩ D(g) = D(fg) for every f, g ∈ P. We endow Spec P with the topology having a basis consisting of the subsets D(f), for every f ∈ P. Notice that 𝔪_{P} is the only closed point of Spec P (especially, Spec P is trivially quasi-compact).

By lemma 6.1.14, the localization map j_{f} : P → P_{f} induces an identification j_{f}^{∗} : Spec P_{f} →^{∼} D(f). It is easily seen that (j_{f}^{∗})^{−1}D(fg) = D(j(g)) ⊂ Spec P_{f}; in other words, the topology of Spec P_{f} agrees with the topology induced from Spec P, via j^{∗}.

Next, for every f ∈ P we set :

𝒪_{Spec P} (D(f)) := P_{f}.

We claim that 𝒪_{Spec P} (D(f)) depends only on the open subset D(f), up to natural isomorphism (and not on the choice of f). More precisely, say that D(f) ⊂ D(g) for two given elements f, g ∈ P; it follows that the image of g in P_{f} lies outside the maximal ideal 𝔪_{P_{f}}, hence g ∈ P_{f}^{×}, and therefore the localization map j_{f} : P → P_{f} factors uniquely through a morphism of monoids :

j_{f,g} : P_{g} → P_{f}.

Likewise, if D(g) ⊂ D(f) as well, the localization j_{g} : P → P_{g} factors through a unique map j_{g,f} : P_{f} → P_{g}, whence the identities :

j_{f,g} ∘ j_{g,f} ∘ j_{f} = j_{f} j_{g,f} ∘ j_{f,g} ∘ j_{g} = j_{g}

and since j_{f} and j_{g} are epimorphisms, we see that j_{f,g} and j_{g,f} are mutually inverse isomorphisms.

6.5.3. Say that D(f) ⊂ D(g) ⊂ D(h) for some f, g, h ∈ P; by direct inspection, it is clear that j_{f,g} ∘ j_{g,h} = j_{f,h}, so the rule D(f) ↦ P_{f} yields a well defined presheaf of monoids on the site 𝒞_{P} of open subsets of Spec P of the form D(f) for some f ∈ P. Then 𝒪_{Spec P} is trivially a sheaf on 𝒞_{P} (notice that if D(f) = ⋃_{i∈I} D(g_{i}) is an open covering of D(f), then D(g_{i}) = D(f) for some i ∈ I). According to [59, Ch.0, §3.2.2] it follows that 𝒪_{Spec P} extends uniquely to a well defined sheaf of monoids on Spec P, whence a monoidal space (Spec P, 𝒪_{Spec P}). By inspecting the construction, we find natural identifications :

(6.5.4)  (𝒪_{Spec P})_{𝔭} →^{∼} P_{𝔭}  for every 𝔭 ∈ Spec P

and moreover :

P →^{∼} Γ(Spec P, 𝒪_{Spec P}).

It is also clear that the rule

(6.5.5)  P ↦ (Spec P, 𝒪_{Spec P})

defines a functor from the category Mnd^{o} to the category of monoidal spaces.

Proposition 6.5.6. The functor (6.5.5) is right adjoint to the functor :

(T, 𝒪_{T}) ↦ Γ(T, 𝒪_{T})

from the category of monoidal spaces, to the category Mnd^{o}.

Proof. Let f : P → Γ(T, 𝒪_{T}) be a map of monoids. We define a morphism

φ_{f} := (φ_{f}, log φ_{f}) : (T, 𝒪_{T}) → (Spec P, 𝒪_{Spec P})

as follows. Given t ∈ T, let f_{t} : P → 𝒪_{T,t} be the morphism deduced from f, and denote by 𝔪_{t} ⊂ 𝒪_{T,t} the maximal ideal. We set φ_{f}(t) := f_{t}^{−1}(𝔪_{t}). In order to show that φ_{f} is continuous, it suffices to prove that U_{s} := φ_{f}^{−1}(D(s)) is open in M_{T}, for every s ∈ P. However, U_{s} = {t ∈ T | f_{t}(s) ∈ 𝒪_{T,t}^{×}}, and it is easily seen that this condition defines an open subset (details left to the reader). Next, we define log φ_{f} on the basic open subsets D(s). Indeed, let j_{s} : Γ(T, 𝒪_{T}) → 𝒪_{T} (U_{s}) be the natural map; by construction, j_{s} ∘ f(s) is invertible in 𝒪_{T} (U_{s}), hence j_{s} ∘ f extends to a unique map of monoids :

P_{s} = 𝒪_{Spec P} (D(s)) → φ_{f∗}𝒪_{T} (D(s)).

By [59, Ch.0, §3.2.5], the above rule extends to a unique morphism 𝒪_{Spec P} → φ_{f∗}𝒪_{T} of sheaves of monoids, whence – by adjunction – a well defined morphism log φ_{f} : φ_{f}^{∗}𝒪_{Spec P} → 𝒪_{T}. In order to show that (φ_{f}, log φ_{f}) is the sought morphism of monoidal spaces, it remains to check that (log φ_{f})_{t} : P_{φ_{f} (t)} → 𝒪_{T,t} is a local morphism, for every t ∈ T. However, let i_{t} : P → P_{φ_{f} (t)} be the localization map; by construction, we have (log φ_{f})_{t} ∘ i_{t} = f_{t}, and the contention is a straightforward consequence.

Conversely, say that (φ, log φ) : (T, 𝒪_{T}) → (Spec P, 𝒪_{Spec P}) is a morphism of monoidal spaces; then log φ corresponds to a unique morphism ψ : 𝒪_{Spec P} → φ_{∗}𝒪_{T}, and we set

f_{φ} := Γ(Spec P, ψ) : P → Γ(T, 𝒪_{T}).

By inspecting the definitions, it is easily seen that f_{φ_{f}} = f for every morphism of monoids f as above. To conclude, it remains only to show that the rule (φ, log φ) ↦ f_{φ} is injective. However, for a given morphism of monoidal spaces (φ, log φ) as above, and every point t ∈ T, we have a commutative diagram of monoids :

\xymatrix{P \ar[r]^{f_{φ}} \ar[d] & Γ(T, 𝒪_{T}) \ar[d]^{j_{t}} \\ P_{φ(t)} \ar[r]^{log φ_{t}} & 𝒪_{T,t}} .

Since log φ_{t} is local, it follows that φ(t) = (f_{φ} ∘ j_{t})^{−1}𝔪_{t}, especially f_{φ} determines φ : T → Spec P. Finally, since the map P → P_{φ(t)} is an epimorphism, we see that log φ_{t} is determined by f_{φ} as well, and the proposition follows. □

Definition 6.5.7. Let (T, 𝒪_{T}) be a sharp monoidal space.

(i) We say that (T, 𝒪_{T}) is an affine fan, if there exist a monoid P and an isomorphism of sharp monoidal spaces (Spec P, 𝒪_{Spec P})^{♯} →^{∼} (T, 𝒪_{T}).

(ii) In the situation of (i), if P can be chosen to be finitely generated (resp. fine), we say

that (T, 𝒪_{T}) is a finite (resp. fine) affine fan. (iii) We say that (T, 𝒪_{T}) is a fan, if there exists an open covering T = ⋃_{i∈I} U_{i}, such that the

induced sharp monoidal space (U_{i}, 𝒪_{T |U_{i}}) is an affine fan, for every i ∈ I. We denote by Fan the full subcategory of the category of monoidal spaces, whose objects are the fans.

(iv) In the situation of (iii), if the covering (U_{i} | i ∈ I) can be chosen, so that (U_{i}, 𝒪_{T |U_{i}}) is a finite (resp. fine) affine fan for every i ∈ I, we say that (T, 𝒪_{T}) is locally finite (resp. locally fine).

(v) We say that the fan (T, 𝒪_{T}) is finite (resp. fine) if it is locally finite (resp. locally fine) and quasi-compact.

(vi) Let (T, 𝒪_{T}) be a fan. The simplicial locus T_{sim} ⊂ T is the subset of all t ∈ T such that 𝒪_{T,t} is a free monoid of finite rank.

Remark 6.5.8. (i) For every monoid P, let T_{P} denote the affine fan (Spec P)^{♯}. In light of proposition 6.5.6, it is easily seen that the functor P ↦ T_{P} is an equivalence from the opposite of the full subcategory of sharp monoids, to the category of affine fans.

(ii) Since the saturation functor commutes with localizations (lemma 6.2.9(i)), it is easily seen that the saturation of a fan is a fan, and more precisely, the saturation of an affine fan T_{P}, is naturally isomorphic to T_{P^{sat}}.

(iii) Let Q_{1} and Q_{2} be two monoids; since the product P × Q is also the coproduct of P and Q in the category Mnd (see example 4.8.30(i)), we have a natural isomorphism in the category of fans :

T_{P ×Q} →^{∼} T_{P} × T_{Q}.

More generally, suppose that P → Q_{i}, for i = 1, 2, are two morphisms of monoids. Then we have a natural isomorphism of fans :

T_{Q_{1}⊗_{P} Q_{2}} →^{∼} T_{Q_{1}} ×_{T_{P}} T_{Q_{2}}.

From this, a standard argument shows that fibre products are representable in the category of fans.

(iv) Furthermore, lemma 6.1.17(ii) implies that the natural map :

π : Spec (P × Q) → Spec P × Spec Q

is a homeomorphism (where the product of Spec P and Spec Q is taken in the category of topological spaces and continuous maps).

(v) Moreover, we have natural isomorphisms of monoids :

𝒪_{T_{P ×Q},π^{−1}(s,t)} →^{∼} 𝒪_{T_{P} ,s} × 𝒪_{T_{Q},t} for every s ∈ Spec P and t ∈ Spec Q.

(vi) For any fan T := (T, 𝒪_{T}), and any monoid M, we shall use the standard notation :

T (M) := Hom_{Fan}((Spec M)^{♯}, T).

Especially, if T is an affine fan, say T = (Spec P)^{♯}, then T (M) = Hom_{Mnd}(P, M^{♯}); for instance, if T is affine, T (ℕ) is a monoid, and T (ℚ_{+})^{gp} is a ℚ-vector space. Furthermore, by standard general nonsense we have natural identifications of sets :

(T_{1} ×_{T} T_{2})(M) →^{∼} T_{1}(M) ×_{T (M)} T_{2}(M)

for any pair of T -fans T_{1} and T_{2}, and every monoid M. If T, T_{1} and T_{2} are affine, this identification is also an isomorphism of monoids.

Example 6.5.9. (i) The topological space underlying the affine fan (Spec ℕ, 𝒪_{Spec ℕ})^{♯} consists of two points : Spec ℕ = {∅, 𝔪}, where 𝔪 := ℕ\{0} is the closed point. The structure sheaf 𝒪 := 𝒪_{Spec ℕ} is determined as follows. The two stalks are 𝒪_{∅} = {1} (the trivial monoid) and 𝒪_{𝔪} = ℕ; the global sections are Γ(Spec ℕ, 𝒪) = ℕ.

(ii) Let (T, 𝒪_{T}) be any fan, P any monoid, with maximal ideal 𝔪_{P}, and φ : T_{P} := (Spec P, 𝒪_{Spec P})^{♯} → (T, 𝒪_{T}) a morphism of fans. Say that φ(𝔪_{P}) ∈ U for some affine open subset U ⊂ F; then φ(Spec P) ⊂ U, hence φ factors through a morphism of fans T_{P} → (U, 𝒪_{T |U}). In view of proposition 6.5.6, such a morphism corresponds to a unique morphism of monoids φ^{♮} : 𝒪_{T} (U) → P^{♯}, and then φ(𝔪_{P}) = φ^{♮−1}(𝔪_{P}) ∈ Spec 𝒪_{T} (U) = U. The map on stalks determined by φ is the local morphism

𝒪_{T,φ(𝔪_{P})} →^{∼} 𝒪_{T} (U)_{φ(𝔪_{P})}/𝒪_{T} (U)_{φ(𝔪_{P})}^{×} → P^{♯}

obtained from φ^{♮} after localization at the prime ideal φ^{♮−1}(𝔪_{P}).

(iii) For any two monoids M and N, denote by loc.Hom_{Mnd}(M, N) the set of local morphisms of monoids M → N. The discussion in (ii) leads to a natural identification :

T (P) →^{∼} ∐_{t∈T} loc.Hom_{Mnd}(𝒪_{T,t}, P^{♯})

For any monoid P. The support of a P -point φ ∈ T (P) is the unique point t ∈ T such that φ corresponds to a local morphism 𝒪_{T,t} → P^{♯}.

Example 6.5.10. (i) Let P be any monoid, k > 0 any integer, set T_{P} := (Spec P)^{♯}, and let k_{P} : P → P be the k-Frobenius map of P (definition 4.8.40(ii)). Then

k_{T_{P}} := Spec k_{P} : T_{P} → T_{P}

is a well defined endomorphism inducing the identity on the underlying topological space.

(ii) More generally, let F be any fan; for every integer k > 0 we have the k-Frobenius endomorphism

k_{F} : F → F

which induces the identity on the underlying topological space, and whose restriction to any affine open subfan U ⊂ F is the endomorphism k_{U} defined as in (i).

6.5.11. Let P be a monoid, M a P -module, and set T_{P} := (Spec P)^{♯}. We define a presheaf M^{∼} on the site of basic affine open subsets D(f) ⊂ Spec P (for all f ∈ P), by the rule :

U ↦ M^{∼}(U) := M ⊗_{P} 𝒪_{T_{P}} (U)

(and for an inclusion U′ ⊂ U of basic open subsets, the corresponding morphism M^{∼}(U) → M^{∼}(U′) is deduced from the restriction map 𝒪_{T_{P}} (U) → 𝒪_{T_{P}} (U′)). It is easily seen that M^{∼} is a sheaf, hence it extends to a well defined sheaf of 𝒪_{T_{P}} -modules on T_{P} ([59, Ch.0, §3.2.5]). Clearly Γ(T_{P} , M^{∼}) = M, and the rule M ↦ M^{∼} yields a well defined functor P -Mod → 𝒪_{T_{P}} -Mod, which is left adjoint to the global section functor on 𝒪_{T_{P}} -modules : ℳ ↦ Γ(T_{P} , ℳ) (verification left to the reader).

Definition 6.5.12. Let (T, 𝒪_{T}) be a fan, ℳ a 𝒪_{T} -module. We say that ℳ is quasi-coherent, if there exist an open covering T = ⋃_{i∈I} U_{i} of T by affine open subsets, and for each i ∈ I an 𝒪_{T} (U_{i})-module M_{i} with an isomorphism of 𝒪_{T |U_{i}}-modules ℳ_{|U_{i}} →^{∼} M_{i}^{∼}.

Remark 6.5.13. (i) Let (T, 𝒪_{T}) be a fan, ℳ a quasi-coherent 𝒪_{T} -module, and U ⊂ T be any open subset, such that (U, 𝒪_{T |U}) is an affine fan (briefly : an affine open subfan of T). Then, since U admits a unique closed point t ∈ U, it is easily seen that ℳ_{|U} is naturally isomorphic to ℳ_{t}^{∼} as a 𝒪_{T |U}-module.

(ii) In the same vein, if ℳ is an invertible 𝒪_{T} -module (see definition 4.8.6(iv)), then the restriction ℳ_{|U} of ℳ to any affine open subset, is isomorphic to 𝒪_{T |U}.

(iii) For any fan (T, 𝒪_{T}), the sheaf of abelian groups 𝒪_{T}^{gp} is quasi-coherent (exercise for the reader). Suppose that T is integral; then an 𝒪_{T} -submodule ℐ ⊂ 𝒪_{T}^{gp} is called a fractional ideal (resp. a reflexive fractional ideal) of 𝒪_{T} if ℐ is quasi-coherent, and ℐ (U) is a fractional ideal (resp. a reflexive fractional ideal) of 𝒪_{T} (U), for every affine open subset U ⊂ T.

(iv) Let P be any integral monoid, I ⊂ P^{gp} a fractional ideal of P, and set T_{P} := (Spec P)^{♯}. It follows easily from lemma 6.4.39(i) that I^{∼} ⊂ 𝒪_{T_{P}}^{gp} is a fractional ideal of 𝒪_{T_{P}}, and I^{∼} is reflexive if and only if I is a reflexive fractional ideal of P (lemma 6.4.39(ii)).

(v) Suppose that T is locally finite; in this case, it follows easily from proposition 6.1.9(ii) that every quasi-coherent ideal of 𝒪_{T} is coherent. Likewise, if T is also integral, and ℐ ⊂ 𝒪_{T}^{gp} is a (quasi-coherent) fractional ideal of 𝒪_{T}, then ℐ is coherent, provided the stalks ℐ_{t} are finitely generated 𝒪_{T,t}-modules, for every t ∈ T. (Details left to the reader.)

Remark 6.5.14. (i) Let T be any integral fan. We define a sheaf 𝒟iv_{T} on T, by letting 𝒟iv_{T} (U) be the set of all reflexive fraction ideals of 𝒪_{U}, for every open subset U ⊂ T.

(ii) Now, suppose that T is locally fine; in this case, we can endow 𝒟iv_{T} with a natural structure of T -monoid, as follows. First, we define a presheaf of monoids on the site 𝒞_{T} of affine open subsets of T by the rule :

U ↦ 𝒟iv_{T} (U) := (Div(𝒪_{T} (U)), ⊙)

(notation of (6.4.37)) and for an inclusion U′ ⊂ U of affine open subset, the corresponding morphism of monoids 𝒟iv_{T} (U) → 𝒟iv_{T} (U′) is deduced from the flat map 𝒪_{T} (U) → 𝒪_{T} (U′), by virtue of lemma 6.4.45(iv). Arguing as in (6.5.3), we see that 𝒟iv_{T} is a sheaf on 𝒞_{T}, and then [59, Ch.0, §3.2.2] implies that 𝒟iv_{T} extends uniquely to a sheaf of monoids on T. It is then clear that the sheaf of sets underlying this T -monoid is (naturally isomorphic to) the sheaf defined in (i).

(iii) In the situation of (ii), we have likewise a T -submonoid 𝒟iv_{T}^{+} ⊂ 𝒟iv_{T} (remark 6.4.47), and we may also define a T -monoid \overline{𝒟iv} _{T} (see (6.4.53)). Moreover, we have the global version of (6.4.54) : namely, the sequence of T -monoids

1 → 𝒪_{T}^{gp} ⟶^{j_{T}} 𝒟iv_{T} → \overline{𝒟iv} _{T} → 1

is exact (recall that 𝒪_{T}^{×} = 1_{T}, the initial T -monoid), and j_{T} restricts to a map of T -monoids

𝒪_{T} → 𝒟iv_{T}^{+}.

Indeed, the assertion can be checked on the stalks over each t ∈ T, where it reduces to the exact sequence (6.4.54) for P := 𝒪_{T,t}. Lastly, we remark that if T is locally fine and saturated, then 𝒟iv_{T} and \overline{𝒟iv} _{T} are abelian T -groups (proposition 6.4.42(i,ii)).

6.5.15. If f : T′ → T is a morphism of fans, and ℳ is any 𝒪_{T} -module, then we define as usual the 𝒪_{T′}-module :

f^{∗}ℳ := f^{−1}ℳ ⊗_{f^{−1}𝒪_{T}} 𝒪_{T′}

where f^{−1}ℳ denotes the usual sheaf-theoretic inverse image of ℳ (so f^{−1}𝒪_{T} means here what was denoted f^{∗}𝒪_{T} in definition 6.5.1(ii)). The rule ℳ ↦ f^{∗}ℳ yields a left adjoint to the functor

𝒪_{T′}-Mod → 𝒪_{T} -Mod 𝒩 ↦ f_{∗}𝒩

(verification left to the reader). Notice that if ℳ is quasi-coherent, then f^{∗}ℳ is a quasi-coherent 𝒪_{T′}-module. Indeed, the assertion is local on T′, hence we are reduced to the case where T′ = (Spec P′) and T = (Spec P) for some monoids P and P′. In this case, the functor M ↦ f^{∗}(M^{∼}) : P -Mod → 𝒪_{T′}-Mod is left adjoint to the functor ℳ ↦ Γ(T′, ℳ) on 𝒪_{T′}- modules. The latter functor also admits the left adjoint given by the rule : M ↦ (M ⊗_{P} P′)^{∼}, whence a natural isomorphism of 𝒪_{T′}-modules :

f^{∗}(M^{∼}) →^{∼} (M ⊗_{P} P′)^{∼}.

6.5.16. Let T := (T, 𝒪_{T}) be a fan, t ∈ T any point. The height of t is :

ht_{T} (t) := dim 𝒪_{T,t} ∈ ℕ ∪ {+∞}

(see definition 6.1.10(ii)) and the dimension of T is dim T := sup(ht_{T} (t) | t ∈ T). (If T = ∅ is the empty fan, we let dim T := −∞.)

Suppose that T is locally finite; then it follows from (6.5.4) and lemma 6.1.20(iii),(iv) that the height of any point of T is an integer. Moreover, let U(t) ⊂ T denote the subset of all points x ∈ T which specialize to t (i.e. such that the topological closure of {x} in T contains t); clearly U(t) is the intersection of all the open neighborhoods of t in T, and we have a natural homeomorphism :

(6.5.17)  Spec 𝒪_{T,t} →^{∼} U(t).

Especially, if T is locally finite, U(t) is a finite set, and moreover U(t) is an open subset : indeed, if U ⊂ T is any finite affine open neighborhood of t, we have U(t) ⊂ U, hence U(t) can be realized as the intersection of the finitely many open neighborhoods of t in U. In this case, (6.5.17) induces an isomorphism of fans :

(6.5.18)  (Spec 𝒪_{T,t})^{♯} →^{∼} (U(t), 𝒪_{T |U(t)}).

Therefore, for every h ∈ ℕ, let T_{h} ⊂ T be the subset of all points of T of height ≤ h; clearly U(t) ⊂ T_{h} whenever t ∈ T_{h}, hence the foregoing shows that – if T is locally finite – T_{h} is an open subset of T for every h ∈ ℕ, and T = ⋃_{h∈ℕ} T_{h}.

Notice also that the simplicial locus of a fan T is closed under generizations. Therefore, T_{sim} is an open subset of T, whenever T is locally finite.

6.5.19. In the situation of remark 6.5.8(v), suppose additionally that P and Q are finitely generated. The natural projection P × Q → P induces a morphism j : T_{P} → T_{P ×Q} of affine fans, and it is easily seen that j(t) = π^{−1}(t, ∅) for every t ∈ T_{P}. It follows that the restriction of j is a homeomorphism U(t) →^{∼} U(j(t)) for every t ∈ T_{P}, and moreover log j : j^{∗}𝒪_{T_{P ×Q}} → 𝒪_{T_{P}} is an isomorphism. We conclude that j is an open immersion.

6.5.20. Let P be a fine, sharp and saturated monoid, and set Q := P^{∨} (notation of (6.4.13)). By proposition 6.4.14(iv), we have a natural identification P →^{∼} Q^{∨}. By proposition 6.4.9(ii) and corollary 6.3.12(ii), the rule

(6.5.21)  F ↦ F^{∗} := F_{ℝ}^{∗} ∩ P

establishes a natural bijection from the faces of Q to those of P. For every face F of Q, set 𝔭_{F} := P \F^{∗}; there follows a natural bijection F ↦ 𝔭_{F} between the set of all faces of Q and Spec P, such that

F ⊂ F′ ⇔ 𝔭_{F} ⊂ 𝔭_{F′}.

Moreover, set T_{P} := (Spec P)^{♯}; we have natural identifications :

F^{∨} →^{∼} 𝒪_{T_{P} ,𝔭_{F}} for every face F of Q

under which, the specialization maps 𝒪_{T_{P} ,𝔭_{F′}} → 𝒪_{T_{P} ,𝔭_{F}} correspond to the restriction maps (F′)^{∨} → F^{∨} : φ ↦ φ_{|F}.

Definition 6.5.22. Let T := (T, 𝒪_{T}) be a fan.

(i) An integral (resp. a rational) partial subdivision of T is a morphism f : (T′, 𝒪_{T′}) → T of fans such that, for every t ∈ T′, the group homomorphism

(log f)_{t}^{gp} : 𝒪_{T,f(t)}^{gp} → 𝒪_{T′,t}^{gp} (resp. the ℚ-linear map (log f)_{t}^{gp} ⊗_{ℤ} 1_{ℚ})

is surjective. (ii) If f : T′ → T is an integral (resp. rational) partial subdivision, and the induced map

T′(ℕ) → T (ℕ) (resp. T′(ℚ_{+}) → T (ℚ_{+})) : φ ↦ f ∘ φ

is bijective, we say that f is an integral (resp. a rational) subdivision of T. (iii) A morphism of fans f : T′ → T is finite (resp. proper), if the fibre f^{−1}(t) is a finite

(resp. and non-empty) set, for every t ∈ T.

(iv) A subdivision T′ → T of T is simplicial, if T′_{sim} = T′.

Remark 6.5.23. (i) Let T be any integral fan. Then the counit of adjunction

T^{sat} → T

is an integral subdivision. This morphism is also a homeomorphism on the underlying topological spaces, in light of 6.2.9(iv).

(ii) Let f : T′ → T be any integral subdivision. Then f restricts to a bijection T′_{0} →^{∼} T_{0} on the sets of points of height zero. Indeed, notice that if t′ ∈ T′ is a point of height zero, then f(t′) ∈ T_{0} since the map (log f)_{t′}^{gp} must be local; moreover loc.Hom_{Mnd}(𝒪_{T,t′}, ℕ) consists of precisely one element, namely the unique map σ_{t} : ℕ → {1}, and if t′_{1}, t′_{2} ∈ T′_{0} have the same image in T_{0}, the sections σ_{t′_{1}} and σ_{t′_{2}} have the same image in T (ℕ), hence they must coincide, so that t′_{1} = t′_{2}, as claimed.

(iii) Let f : T′ → T be an integral subdivision of locally fine and saturated fans. In general, the image of a point t′ ∈ T′ of height one may have height strictly greater than one. On the other hand, for any t ∈ T of height one, and any t′ ∈ f^{−1}(t), the map ℤ →^{∼} 𝒪_{T,t}^{gp} → 𝒪_{T′,t′}^{gp} must be surjective (theorem 6.4.18(ii)), therefore 𝒪_{T′,t′}^{gp} is a cyclic group; however 𝒪_{T′,t′} is also sharp and saturated, so it must be either the trivial monoid {1} or ℕ. The first case is excluded by (ii), so ht(t′) = 1, and moreover (log f)_{t′} is an isomorphism (and there exists a unique such isomorphism). Since the induced map T′(ℕ) → T (ℕ) is bijective, it follows easily that f^{−1}(t) consists of exactly one point, and therefore f restricts to an isomorphism f^{−1}(T_{1}) →^{∼} T_{1}.

Proposition 6.5.24. Let f : T′ → T be a morphism of fans, with T′ locally finite, and consider the following conditions :

(a) The induced map T′(ℕ) → T (ℕ) is injective.

(b) For every integral saturated monoid P, the induced map T′(P) → T (P) is injective.

(c) f is a partial rational subdivision. Then we have : (a) ⇔ (b) ⇒ (c).

Proof. Obviously (b) ⇒ (a). Conversely, assume that (a) holds, let P be a saturated monoid, and suppose we have two sections in T′(P) whose images in T (P) agree. In light of example 6.5.9(iii), this means that we may find two points t′_{1}, t′_{2} ∈ T′, such that f(t′_{1}) = f(t′_{2}) = t, and two local morphisms of monoids σ_{i} : 𝒪_{T′,t′_{1}} → P/P^{×} whose compositions with log f_{t′_{i}} (i = 1, 2) yield the same morphism 𝒪_{T,t} → P/P^{×}, and we have to show that these maps are equal. In view of lemma 6.2.9(ii), we may then replace P by P/P^{×}, and assume that P is sharp. Since the stalks of 𝒪_{T′} are finitely generated, the morphisms σ_{i} factor through a finitely generated submonoid M ⊂ P. We may then replace P by its submonoid M^{sat}, which allows to assume additionally that P is finitely generated (corollary 6.4.1(ii)). In this case, we may find an injective map j : P → ℕ^{⊕r} (corollary 6.4.12(iv); notice that j is trivially a local morphism), hence we may replace σ_{i} by j ∘ σ_{i} (for i = 1, 2), after which we may assume that P = ℕ^{⊕r} for some r ∈ ℕ. Let δ : P → ℕ be the local morphism given by the rule : (x_{1}, …, x_{r}) ↦ x_{1} + ⋯ + x_{r} for every x_{1}, …, x_{r} ∈ ℕ; the compositions δ ∘ σ_{i} (for i = 1, 2) are two elements of T′(ℕ) whose images agree in T (ℕ), hence they must coincide by assumption. This implies already that t′_{1} = t′_{2}. Next, let π_{k} : P → ℕ (for k = 1, …, r) be the natural projections, and fix k ≤ r; the morphisms π_{k} ∘ σ_{i} for i = 1, 2 are not necessarily local, but they determine elements of T′(ℕ) whose images agree again in T (ℕ), hence they must coincide. Since k is arbitrary, we deduce that σ_{1} = σ_{2}, as stated.

Next, we suppose that (b) holds, and we wish to show assertion (c); the latter is local on F′, hence we may assume that both F and F′ are affine, say F = (Spec Q)^{♯} and F′ = (Spec Q′)^{♯}, with Q′ finitely generated and sharp, and then we are reduced to checking that the map Q^{gp} ⊗_{ℤ} ℚ → Q^{′gp} ⊗_{ℤ} ℚ induced by f is surjective, or equivalently, that the dual map :

Hom_{Mnd}(Q′, ℚ) → Hom_{Mnd}(Q, ℚ)

is injective. To this aim, we may further assume that Q′ is integral, in which case, by remark 6.4.16(i) we have Hom_{Mnd}(Q′, ℚ) = Hom_{Mnd}(Q′, ℚ_{+})^{gp}; the contention is an easy consequence. □

6.5.25. In light of proposition 6.5.24, we may ask whether the surjectivity of the map on P - points induced by a morphism f of fans can be similarly characterized. This turns out to be the case, but more assumptions must be made on the morphism f, and also some additional restrictions must be imposed on the type of monoid P. Namely, we shall consider monoids of the form Γ_{+}, where (Γ, ≤) is any totally ordered abelian group, and Γ_{+} ⊂ Γ is the subgroup of all elements ≤ 1 (where 1 ∈ Γ denotes the neutral element). With this notation, we have the following :

Proposition 6.5.26. Let f : T′ → T be a finite partial integral subdivision, with T locally finite. The following conditions are equivalent :

(a) The induced map T′(ℕ) → T (ℕ) is surjective.

(b) For every totally ordered abelian group (Γ, ≥), the induced map T′(Γ_{+}) → T (Γ_{+}) is surjective.

Proof. Obviously, we need only to show that (a) ⇒ (b). Thus suppose, by way of contradiction, that (a) holds, but nevertheless there exist a totally ordered abelian group (Γ, ≤), and an element of T (Γ_{+}) which is not in the image of T′(Γ_{+}). Such element corresponds to a local morphism of monoids φ : 𝒪_{T,t} → Γ_{+}, for some t ∈ T, and the assumption means that φ does not factor through the monoid 𝒪_{T′,s}, for any s ∈ f^{−1}(t). Set

P := 𝒪_{T,t}^{int} Q_{s} := P^{gp} ×_{𝒪_{T′,s}^{gp}} 𝒪_{T′,s}^{int} for every s ∈ f^{−1}(t)

Notice that, since the map P^{gp} → 𝒪_{T′,s}^{gp} is surjective, we have

Q_{s}/Q_{s}^{×} ≃ 𝒪_{T′,s}^{int}/(𝒪_{T′,s}^{int})^{×}

and there is a natural injective morphism of monoids g_{s} : P → Q_{s}, determined by the pair (i, (log f)_{s}^{int}), where i : P^{int} → P^{gp} is the natural morphism; moreover, P^{gp} = Q_{s}^{gp} for every s ∈ f^{−1}(t). Clearly φ factors through a morphism \overline{φ} : P → Γ_{+}; since the unit of adjunction 𝒪_{T,t} → P is surjective, it follows that P is sharp and \overline{φ} is local. Moreover, the group homomorphism \overline{φ}^{gp} : P^{gp} → Γ factors uniquely through each Q_{s}^{gp}. Our assumption then states that we may find, for each s ∈ f^{−1}(t), an element x_{s} ∈ Q_{s} whose image in Γ lies in the complement of Γ_{+}, i.e. the image of x_{s}^{−1} lies in the maximal ideal 𝔪 ⊂ Γ_{+}. Let P′ ⊂ P^{gp} be the submonoid generated by P and by (x_{s}^{−1} | s ∈ f^{−1}(t)). By construction, P′ is finitely generated, and the morphism \overline{φ} extends uniquely to a morphism P′ → Γ_{+}, which maps each x_{s}^{−1} into 𝔪. It follows that all the x_{s}^{−1} lie in the maximal ideal of P′. Let us now pick any local morphism ψ′ : P′ → ℕ (corollary 6.4.12(iii)); by restriction, ψ′ induces a local morphism ψ : P → ℕ, which – according to (a) – must factor through a local morphism ψ_{s} : Q_{s} → ℕ, for at least one s ∈ f^{−1}(t). However, on the one hand we have ψ^{′gp} = ψ^{gp} = ψ_{s}^{gp}; on the other hand ψ′(x_{s}^{−1}) ≠ 0, hence ψ_{s}(x_{s}) = ψ^{′gp}(x_{s}) ∉ ℕ, a contradiction. □

Example 6.5.27. Let T be a locally fine fan, φ : F → T an integral subdivision, with F locally fine and saturated, and k > 0 an integer. Suppose we have a commutative diagram of fans :

(6.5.28)  \xymatrix{F \ar[r]^{φ} \ar[d]_{g} & T \ar[d]^{_{k_{T}}} \\ F \ar[r]^{φ} & T.}

where k_{T} is the k-Frobenius endomorphism (example 6.5.10(ii)). Then we claim that necessarily g = k_{F}. Indeed, suppose that this fails; then we may find a point t ∈ F such that the composition of g and the open immersion j_{t} : U(t) → F is not equal to j_{t} ∘ k_{U(t)}. Set P := 𝒪_{F,t}; then g ∘ j_{t} ≠ j_{t} ∘ k_{U(t)} in F (P). However, an easy computation shows that φ(g ∘ j_{t}) = φ(j_{t} ∘ k_{U(t)}), which contradicts proposition 6.5.24.

6.5.29. Let P := ∐_{n∈ℕ} P_{n} be a ℕ-graded monoid (see definition 4.8.8); then P_{0} is a submonoid of P, every P_{n} is a P_{0}-module, and P_{+} := ∐_{n>0} P_{n} is an ideal of P. For every a ∈ P, the localization P_{a} is ℤ-graded in an obvious way, and we denote by P_{(a)} ⊂ P_{a} the submonoid of elements of degree 0. Notice that there is a natural identification of P_{0}-monoids

(6.5.30)  P_{(a^{n})} →^{∼} P_{(a)}  for every integer n > 0.

Set as well :

D_{+}(a) := (Spec P_{(a)})^{♯}

and notice that the natural map P → P_{(a)} induces a morphism of fans π_{a} : D_{+}(a) → T_{P} := (Spec P_{0})^{♯}. If b ∈ P is any other element, in order to determine the fibre product D_{+}(a) ×_{T_{P}} D_{+}(b) we may assume – in light of (6.5.30) – that a, b ∈ P_{n} for the same integer n, in which case we have natural isomorphisms

P_{(a)} ⊗_{P_{0}} P_{(b)} →^{∼} P_{(ab)} ⟵^{∼} P_{(a)}[b^{−1}a]

(see remark 6.1.25(i)) onto the localization of P_{(a)} obtained by inverting its element a^{−1}b; this is of course the same as P_{(b)}[a^{−1}b]. In other words D_{+}(a) ×_{T_{P}} D_{+}(b) is naturally isomorphic to D_{+}(ab), identified to an open subfan in both D_{+}(a) and D_{+}(b). We may then glue the fans D_{+}(a) for a ranging over all the elements of P, to obtain a new fan, denoted :

Proj P

called the projective fan associated with P. By inspecting the construction, we see that the morphisms π_{a} assemble to a well defined morphism of fans π_{P} : Proj P → T_{P}. Each element a ∈ P yields an open immersion j_{a} : D_{+}(a) → Proj P, and if b ∈ P is any other element, j_{ab} factors through an open immersion D_{+}(ab) → D_{+}(a).

6.5.31. Let φ : P → P′ be a morphism of ℕ-graded monoids (so φP_{n} ⊂ P′_{n} for every n ∈ ℕ). Set :

G(φ) := ⋃_{a∈P} D_{+}(φ(a)) ⊂ Proj P′.

Notice that, for every a ∈ P, φ induces a morphism φ_{(a)} : P_{(a)} → P′_{(a)}, whence a morphism of affine fans (Proj φ)_{a} : D_{+}(φ(a)) → D_{+}(a) ⊂ Proj P. Moreover, if b ∈ P is any other element, it is easily seen that (Proj φ)_{a} and (Proj φ)_{b} agree on D_{+}(φ(a)) ∩ D_{+}(φ(b)). Therefore, the morphisms (Proj φ)_{a} glue to a well defined morphism :

Proj φ : G(φ) → Proj P.

Notice that G(φ) = Proj P′, whenever φP generates the ideal P′_{+}. Moreover, we have

(6.5.32)  (Proj φ)^{−1}D_{+}(a) = D_{+}(φ(a))  for every a ∈ P.

Indeed, say that D_{+}(b) ⊂ G(φ) for some b ∈ P′, and (Proj φ)D_{+}(b) ⊂ D_{+}(a). In order to show that D_{+}(b) ⊂ D(φ(a)), it suffices to check that D_{+}(bφ(c)) ⊂ D_{+}(φ(a)) for every c ∈ P. However, the assumption means that the natural map

P_{(c)} → P′_{(φ(c))} → P′_{(bφ(c))} → P′_{(bφ(c))}/P_{(bφ(c))}^{′×}

factors through the localization P_{(c)} → P_{(ac)}. This is equivalent to saying that φ(c^{−1}a) is invertible in P′_{(bφ(c))}, in which case the localization P′_{(φ(c))} → P′_{(bφ(c))} factors through the localization P′_{(φ(c))} → P′_{(φ(ac))}. The latter means that the open immersion D_{+}(bφ(c)) ⊂ D_{+}(φ(c)) factors through the open immersion D_{+}(φ(ac)) ⊂ D_{+}(φ(c)), as claimed.

6.5.33. In the situation of (6.5.29), set Y := Proj P to ease notation. Let M be a ℤ-graded P -module; for every a ∈ P, let M_{(a)} ⊂ M_{a} := M ⊗_{P} P_{a} be the P_{(a)}-submodule of degree zero elements (for the natural grading on M_{a}). We deduce a quasi-coherent 𝒪_{D_{+}(a)}-module M_{(a)}^{∼} (see definition 6.5.12). Moreover, if b ∈ P is any other element, we have a natural identification

ω̃_{a,b} : M_{(a)|D_{+}(a)∩D_{+}(b)}^{∼} →^{∼} M_{(b)|D_{+}(a)∩D_{+}(b)}^{∼}.

This can be verified as follows. First, in view of (6.5.30), we may assume that a, b ∈ P_{n}, for some n ∈ ℕ, in which case we consider the P -linear morphism :

M_{(a)} → M_{(b)} ⊗_{P_{(b)}} P_{(b)}[a^{−1}b] : \frac{x}{a^{m}} ↦ \frac{x}{b^{m}} ⊗ \frac{b^{m}}{a^{m}} for every x ∈ M_{nm}.

It is easily seen that this map is actually P_{(a)}-linear, hence it extends to a 𝒪_{T} (D_{+}(a) ∩ D_{+}(b))- linear morphism :

ω_{a,b} : M_{(a)} ⊗_{P_{(a)}} P_{(a)}[b^{−1}a] →^{∼} M_{(b)} ⊗_{P_{(b)}} P_{(b)}[a^{−1}b].

Moreover, ω_{a,b} ∘ ω_{b,a} is the identity map, hence ω_{a,b} induces the sought isomorphism ω̃_{a,b}. Furthermore, for any a, b, c ∈ P, set D_{+}(a, b, c) := D_{+}(a) ∩ D_{+}(b) ∩ D_{+}(c); we have the identity :

ω̃_{a,c|D_{+}(a,b,c)} = ω̃_{b,c|D_{+}(a,b,c)} ∘ ω̃_{a,b|D_{+}(a,b,c)}

which shows that the locally defined sheaves M_{(a)}^{∼} glue to a well defined 𝒪_{Y} -module, which we shall denote M^{∼}. Especially, for every n ∈ ℤ, let P (n) be the ℤ-graded P -module such that P (n)_{k} := P_{n+k} for every k ∈ ℤ (with the convention that P_{n} := ∅ if n < 0); we set :

𝒪_{Y} (n) := P (n)^{∼}.

Every element a ∈ P_{n} induces a natural isomorphism :

𝒪_{Y} (n)_{|D_{+}(a)} →^{∼} 𝒪_{D_{+}(a)} : x ↦ f^{−k}x for every local section x.

Hence on the open subset :

U_{n}(P) := ⋃_{a∈P_{n}} D_{+}(a)

the sheaf 𝒪_{Y} (n) restricts to an invertible 𝒪_{U_{n}(P)}-module (see definition 4.8.6(iv)). Especially, if P_{1} generates P_{+}, the 𝒪_{Y} -modules 𝒪_{Y} (n) are invertible, for every n ∈ ℤ.

6.5.34. In the situation of (6.5.31), let M be a ℤ-graded P -module. Then M′ := M ⊗_{P} P′ is a ℤ-graded P′-module, with the grading defined by the rule :

(6.5.35)  M′_{n} := ⋃_{j+k=n} Im(M_{j} ⊗_{P_{0}} P′_{k} → M′).

There follows a P_{(a)}-linear morphism :

(6.5.36)  M_{(a)} → M′_{(φ(a))} : \frac{x}{a^{k}} ↦ \frac{x ⊗ 1}{φ(a)^{k}}  for every a ∈ P

and since both localization and tensor product commute with arbitrary colimits, it is easily seen that (6.5.36) extends an injective P′_{(φ(a))}-linear map

M_{(a)} ⊗_{P_{(a)}} P′_{(φ(a))} → M′_{(φ(a))}

whence a map of 𝒪_{D_{+}(φ(a))}-modules (Proj φ)^{∗}M_{|D_{+}(φ(a))}^{∼} → (M′)_{|_{D_{+}(φ(a))}}^{∼}, and the system of such maps, for a ranging over the elements of P, is compatible with all open immersions D_{+}(φ(ab)) ⊂ D_{+}(a), whence a well defined monomorphism of 𝒪_{G(φ)}-modules

(6.5.37)  (Proj φ)^{∗}M^{∼} → (M′)_{|G(φ)}^{∼}.

Moreover, if a ∈ P_{1}, then for every m ∈ M_{j} and x ∈ P′_{k}, we may write

\frac{m ⊗ x}{φ(a)^{j+k}} = \frac{m}{a^{j}} ⊗ \frac{x}{φ(a)^{j}}

so the above map is an isomorphism on D_{+}(a). Thus, (6.5.37) restricts to an isomorphism on the open subset

G_{1}(φ) := ⋃_{a∈P_{1}} D_{+}(φ(a)).

Especially (6.5.37) is an isomorphism whenever P_{1} generates P_{+}. Notice as well that G_{1}(φ) ⊂ U_{1}(P′) ∩ G(φ), and actually G_{1}(φ) = U_{1}(P′) if φ(P) generates P′_{+}.

6.5.38. Let P be as in (6.5.29), and f : P_{0} → Q a given morphism of monoids. Then P′ := P ⊗_{P_{0}} Q is naturally ℕ-graded, so that the natural map f_{P} : P → P′ is a morphism of graded monoids. Every element of P′ is of the form a ⊗ b = (a ⊗ 1) · (1 ⊗ b), where a ∈ P and b ∈ Q. Then lemma 4.8.34yields a natural isomorphism of Q-monoids :

P_{(a)} ⊗_{P_{0}} Q[b^{−1}] →^{∼} P′_{(a⊗b)}

whence an isomorphism of affine fans :

β_{a⊗b} : D_{+}(a ⊗ b) →^{∼} D_{+}(a) ×_{T_{P}} D(b)

such that (π_{a} ×_{T_{P}} j_{b}^{∗}) ∘ β_{a⊗b} = π_{a⊗b}, where j_{b}^{∗} : D(b) → T_{Q} := (Spec Q)^{♯} is the natural open immersion. Especially, it is easily seen that the isomorphisms β_{a⊗1} assemble to a well defined isomorphism of fans :

(Proj f_{P} , π_{P′}) : Proj P′ →^{∼} Proj P ×_{T_{P}} T_{Q}

such that (π_{P} ×_{T_{P}} 1_{T_{Q}}) ∘ (Proj f_{P} , π_{P′}) = π_{P′}. Lastly, if g : Q → R is another morphism of monoids, T_{R} := (Spec R)^{♯}, and P′′ := P′ ⊗_{Q} R, then we have the identity :

(6.5.39)  ((Proj f_{P} , π_{P′}) ×_{T_{Q}} 1_{T_{R}}) ∘ (Proj g_{P′}, π_{P′′}) = (Proj (g ∘ f)_{P} , π_{P′′}).

Moreover, for every ℤ-graded module M, the map (Proj f_{P})^{∗}M^{∼} → (M ⊗_{P_{0}} Q)_{|G(f_{P})}^{∼} of (6.5.37) is an isomorphism, regardless of whether or not P_{1} generates P_{+} (verification left to the reader). Especially, we get a natural identification :

(Proj f_{P})^{∗}𝒪_{Y} (n) →^{∼} 𝒪_{Y′}(n) for every n ∈ ℤ

where Y := Proj P and Y′ := Proj P′.

6.5.40. In the situation of (6.5.38), let φ : R → P be a morphism of ℕ-graded monoids. There follow morphisms of fans :

Proj φ : G(φ) → Proj R Proj(f_{P} ∘ φ) : G(f_{P} ∘ φ) → Proj R

and in view of (6.5.32), it is easily seen that :

(6.5.41)  G(f_{P} ∘ φ) = (Proj f_{P})^{−1}(G(φ)).

6.5.42. Let now (T, 𝒪_{T}) be any fan. A ℕ-graded 𝒪_{T} -monoid is a ℕ-graded T -monoid 𝒫, with a morphism 𝒪_{T} → 𝒫 of T -monoids. We say that such a 𝒪_{T} -monoid is quasi-coherent, if it is such, when regarded as a 𝒪_{T} -module. To a quasi-coherent ℕ-graded 𝒪_{T} -monoid 𝒫, we attach a morphism of fans :

π_{𝒫} : Proj 𝒫 → T

constructed as follows. First, for every affine open subfan U ⊂ T, the monoid 𝒫(U) is ℕgraded, so we have the projective fan Proj 𝒫(U), and the morphism of monoids 𝒪_{T} (U) → 𝒫(U) induces a morphism of fans Proj 𝒫(U) → U. Next, say that U_{1}, U_{2} ⊂ T are two affine open subsets; for any affine open subset V ⊂ U_{1} ∩ U_{2} we have restriction maps ρ_{V,i} : 𝒫(U_{i}) → 𝒫(V) inducing isomorphisms of graded 𝒪_{T} (V)-monoids :

𝒫(U_{i}) ⊗_{𝒪_{T} (U_{i})} 𝒪_{T} (V) →^{∼} 𝒫(V).

whence isomorphisms of V -fans :

Proj 𝒫(V) →^{∼} Proj 𝒫(U_{i}) ⊗_{𝒪_{T} (U_{i})} 𝒪_{T} (V) ⟶^{Proj ρ_{V,i}} Proj 𝒫(U_{i}) ×_{U_{i}} V

which in turn yield natural identifications :

ϑ_{V} : Proj 𝒫(U_{1}) ×_{U_{1}} V →^{∼} Proj 𝒫(U_{2}) ×_{U_{2}} V.

If W ⊂ V is a smaller affine open subset, (6.5.39) implies that ϑ_{V} ×_{V} 1_{W} = ϑ_{W}, and therefore the isomorphisms ϑ_{V} glue to a single isomorphism of U_{1} ∩ U_{2}-fans :

Proj 𝒫(U_{1}) ×_{U_{1}} (U_{1} ∩ U_{2}) →^{∼} Proj 𝒫(U_{2}) ×_{U_{2}} (U_{1} ∩ U_{2})

which is furthermore compatible with base change to any triple intersection U_{1}∩U_{2}∩U_{3} of affine open subsets (details left to the reader). In such situation, we may glue the fans Proj 𝒫(U) – with U ⊂ T ranging over all the open affine subsets – along the above isomorphisms, to obtain the sought fan Proj 𝒫; the construction also comes with a well defined morphism to T, as required. Then, for every such open affine U, the induced morphism Proj 𝒫(U) → Proj 𝒫 is an open immersion; finally a direct inspection shows that, for every smaller affine open subset V ⊂ U we have :

U_{n}(𝒫(U)) ∩ Proj 𝒫(V) = U_{n}(𝒫(V)) for every n ∈ ℕ

(where the intersection is taken in Proj 𝒫). Hence the union of all the open subsets U_{n}(𝒫(U)) is an open subset U_{n}(𝒫) ⊂ Proj𝒫, intersecting each Proj 𝒫(U) in its subset U_{n}(𝒫(U)).

6.5.43. To ease notation, set Y := Proj 𝒫, and let π : Y → T be the projection. Let ℳ be a ℤ-graded 𝒫-module, quasi-coherent as a 𝒪_{T} -module; for every affine open subset U ⊂ T, the graded 𝒫(U)-module ℳ (U) yields a quasi-coherent 𝒪_{π^{−1}U}-module ℳ_{U}^{∼}, and every inclusion of affine open subset U′ ⊂ U induces a natural isomorphism ℳ_{U|U′}^{∼} →^{∼} ℳ_{U′}^{∼} of 𝒪_{π^{−1}U′}-modules. Therefore the modules ℳ_{U}^{∼} glue to a well defined 𝒪_{Y} -module ℳ^{∼}.

For every n ∈ ℤ, denote by ℳ (n) the ℤ-graded 𝒫-module such that ℳ (n)_{k} := ℳ_{n+k} for every k ∈ ℤ (especially, with the convention that 𝒫_{k} := 0 whenever k < 0, we obtain in this way the 𝒫-module 𝒫(n)). We set :

𝒪_{Y} (n) := 𝒫(n)^{∼} and ℳ^{∼}(n) := ℳ^{∼} ⊗_{𝒪_{Y}} 𝒪_{Y} (n).

Clearly the restriction of 𝒪_{Y} (n) to U_{n}(𝒫) is invertible, for every n ∈ ℤ.

Moreover, for every n ∈ ℤ, the scalar multiplication 𝒫(n) ⊗_{𝒪_{T}} ℳ → ℳ (n) determines a well defined morphism of 𝒪_{Y} -modules :

ℳ^{∼}(n) → ℳ (n)^{∼}

and arguing as in (6.5.34) we see that the restriction of this map is an isomorphism on U_{1}(𝒫). Especially, we have natural morphisms of 𝒪_{Y} -modules :

𝒪_{Y} (n) ⊗_{𝒪_{Y}} 𝒪_{Y} (m) → 𝒪_{Y} (n + m) for every n, m ∈ ℤ

whose restrictions to U_{1}(𝒫) are isomorphisms.

Example 6.5.44. Let T be a fan, ℒ an invertible 𝒪_{T} -module, and set 𝒫(ℒ) := Sym_{𝒪_{T}}^{•} ℒ (see example 4.8.10). Then the morphism

π_{𝒫(ℒ)} : ℙ(ℒ) := Proj 𝒫(ℒ) → T

is an isomorphism. Indeed, the assertion can be checked locally on every affine open subset U ⊂ T, hence say that U = (Spec P)^{♯} for some monoid P, and ℒ ≃ 𝒪_{T |U}, in which case the P -monoid 𝒫(ℒ)(U) is isomorphic to P × ℕ (with its natural morphism P → P × ℕ : x ↦ (x, 0) for every x ∈ P), and the sought isomorphism corresponds to the natural identification :

(6.5.45)  P = (P × ℕ)_{(1,1)}

where (1, 1) ∈ P × {1} = (P × ℕ)_{1}. Likewise, 𝒪_{ℙ(ℒ)} (n) is the 𝒪_{ℙ(ℒ)} -module associated with the graded (P × ℕ)-module P × ℕ(n) = ℒ^{⊗n}(U) ⊗_{P} (P × ℕ), so (6.5.45) induces a natural isomorphism

π_{𝒫(ℒ)}^{∗} ℒ^{⊗n} →^{∼} 𝒪_{ℙ(ℒ)} (n) for every n ∈ ℕ.

6.5.46. In the situation of (6.5.42), let φ : 𝒫 → 𝒫′ be a morphism of quasi-coherent ℕgraded 𝒪_{T} -monoids (defined in the obvious way). By the foregoing, for every affine open subset U ⊂ T, we have an induced morphism Proj φ(U) : G(φ(U)) → Proj𝒫(U) of U-fans, where G(φ(U)) ⊂ Proj 𝒫(U) is an open subset of Proj 𝒫′. Let V ⊂ U be a smaller affine open subset; in light of (6.5.41), we have

G(φ(V)) = G(φ(U)) ∩ Proj 𝒫(V).

It follows that the union of all the open subsets G(φ(U)) is an open subset G(φ) such that

G(φ) ∩ Proj 𝒫(U) = G(φ(U)) for every affine open subset U ⊂ T

and the morphisms Proj φ(U) assemble to a well defined morphism

Proj φ : G(φ) → Proj 𝒫.

Moreover, if ℳ is a ℤ-graded quasi-coherent 𝒫-module, the morphisms (6.5.37) assemble to a well defined morphism of 𝒪_{G(φ)}-modules :

(6.5.47)  (Proj φ)^{∗}ℳ^{∼} → (ℳ′)_{|G(φ)}^{∼}

where the grading of ℳ′ := ℳ ⊗_{𝒫} 𝒫′ is defined as in (6.5.35). Likewise, the union of all open subsets G_{1}(φ(U)) is an open subset G_{1}(φ) ⊂ U_{1}(𝒫) ∩ G(φ), such that the restriction of (6.5.47) to G_{1}(φ) is an isomorphism. Especially, set Y := Proj 𝒫 and Y′ := Proj 𝒫′; we have a natural morphism :

(Proj φ)^{∗}𝒪_{Y} (n) →^{∼} 𝒪_{Y′}(n)_{|G(φ)}

which is an isomorphism, if 𝒫_{1} generates 𝒫_{+} := ∐_{n>0} 𝒫_{n} locally on T.

6.5.48. On the other hand, let f : T′ → T be a morphism of fans. The discussion in (6.5.38) implies that f induces a natural isomorphism of T′-fans :

(6.5.49)  Proj f^{∗}𝒫 →^{∼} Proj 𝒫 ×_{T} T′.

Moreover, set Y := Proj 𝒫, Y′ := Proj f^{∗}𝒫, and let π_{Y} : Y′ → Y be the projection deduced from (6.5.49); then there follows a natural identification :

𝒪_{Y′}(n) →^{∼} π_{Y}^{∗} 𝒪_{Y} (n) for every n ∈ ℤ.

6.5.50. Keep the notation of (6.5.42), and to ease notation, set Y := Proj 𝒫. Let 𝒞 be the category whose objects are all the pairs (ψ : X → T, ℒ), where ψ is a morphism of fans, and ℒ is an invertible 𝒪_{X}-module; the morphisms (ψ : X → T, ℒ) → (ψ′ : X′ → T, ℒ′) are the pairs (β, h), where β : X → X′ is a morphism of T -fans, and h : β^{∗}ℒ′ →^{∼} ℒ is an isomorphism of 𝒪_{X′}-modules (with composition of morphisms defined in the obvious way). Consider the functor

F_{𝒫} : 𝒞^{o} → Set

which assigns to any object (ψ, ℒ) of 𝒞, the set consisting of all morphisms of graded 𝒪_{X}- monoids

g : ψ^{∗}𝒫 → Sym_{𝒪_{X}}^{•} ℒ

which are epimorphisms on the underlying 𝒪_{X}-modules (notation of example 4.8.10). On a morphism (β, h) as in the foregoing, and an element g′ ∈ F_{𝒫} (ψ′, ℒ′), the functor acts by the rule :

F_{𝒫} (β, h) := (Sym_{𝒪_{X}}^{•} h) ∘ β^{∗}g′.

Lemma 6.5.51. In the situation of (6.5.50), the following holds :

(i) The object (π_{𝒫|U_{1}(𝒫)} : U_{1}(𝒫) → T, 𝒪_{Y} (1)_{|U_{1}(𝒫)} ) represents the functor F_{𝒫} .

(ii) If 𝒫 is an integral T -monoid, the 𝒪_{T} -monoid 𝒫^{sat} admits a unique grading such that the unit of adjunction 𝒫 → 𝒫^{sat} is a ℕ-graded morphism, and there is a natural isomorphism of Proj 𝒫-fans :

Proj (𝒫^{sat}) →^{∼} (Proj 𝒫)^{sat}.

Proof. (i): The proof is mutatis mutandis, the same as that of lemma 10.6.33(with some minor simplifications). We leave it as an exercise for the reader.

(ii): The first assertion shall be left to the reader. The second assertion is local on Proj 𝒫, hence we may assume that T = (Spec P_{0}), and 𝒫 = P^{∼} for some ℕ-graded integral P_{0}- monoid P. Let a ∈ P^{sat} be any element; by definition we have a^{n} ∈ P for some n > 0, and we know that the open subsets D_{+}(a) et D_{+}(a^{n}) coincide in Proj(𝒫^{sat}); hence we come down to showing that (P_{(a)})^{sat} = (P^{sat})_{(a)} for every a ∈ P, which can be left to reader. □

Definition 6.5.52. Let (T, 𝒪_{T}) be a fan (resp. an integral fan), ℐ ⊂ 𝒪_{T} an ideal (resp. a fractional ideal) of 𝒪_{T}.

(i) Let f : X → T be a morphism of fans (resp. of integral fans); then f^{−1}ℐ is an ideal (resp. a fractional ideal) of f^{−1}𝒪_{T}, and we let :

ℐ𝒪_{X} := log f(f^{−1}ℐ) · 𝒪_{X}

which is the smallest ideal (res. fractional ideal) 𝒪_{X} containing the image of f^{−1}ℐ. (ii) A blow up of the ideal ℐ is a morphism of fans (resp. of integral fans) φ : T′ → T which enjoys the following universal property. The ideal (resp. fractional ideal) ℐ 𝒪_{T′} is invertible, and every morphism of fans (resp. of integral fans) X → T such that ℐ 𝒪_{X} is invertible, factors uniquely through φ.

6.5.53. Let T be a fan (resp. an integral fan), ℐ ⊂ 𝒪_{T} a quasi-coherent ideal (resp. fractional ideal), and consider the ℕ-graded 𝒪_{T} -monoid :

ℬ(𝒥) := ∐_{n∈ℕ} ℐ^{n}

where ℐ^{n} ⊂ 𝒪_{T} is the ideal (resp. fractional ideal) associated with the presheaf U ↦ ℐ (U)^{n} for every open subset U ⊂ T (notation of (6.1.1), with the convention that ℐ^{0} := 𝒪_{T}) and the multiplication law of ℬ(ℐ) is defined in the obvious way.

Proposition 6.5.54. The natural projection

Proj ℬ(ℐ) → T

is a blow up of the ideal ℐ.

Proof. We shall consider the case where T is not necessarily integral, and ℐ ⊂ 𝒪_{T}; the case of a fractional ideal of an integral fan is proven in the same way. Set Y := Proj ℬ(ℐ); to begin with, let us show that ℐ𝒪_{Y} is invertible. The assertion is local on T, hence we may assume that T = (Spec P)^{♯}, and ℐ = I^{∼} for some ideal I ⊂ P, so Y = Proj B(I), where B(I) = ∐_{n∈ℕ} I^{n}. Let a ∈ B(I)_{1} = I be any element; then the restriction of ℐ𝒪_{Y} to D_{+}(a) is generated by 1 = a/a ∈ B(I)_{(a)}, so clearly ℐ_{|D_{+}(a)} ≃ 𝒪_{Y}; since U_{1}(B(I)) = Y (notation of (6.5.33)), the contention follows.

Next, let φ : X → T be a morphism of fans, such that ℐ𝒪_{X} is an invertible ideal. It follows easily that ℐ^{n}𝒪_{X} is invertible for every n ∈ ℕ, so the natural map of ℕ-graded 𝒪_{X}-monoids

φ^{∗}Sym_{𝒪_{T}}^{•} (ℐ) →^{∼} Sym_{𝒪_{X}}^{•} (ℐ𝒪_{X}) → ℬ(ℐ𝒪_{X})

is an isomorphism. On the other hand, the projection Proj ℬ(ℐ𝒪_{X}) → X is an isomorphism (example (6.5.44)), whence – in view of (6.5.49) – a natural morphism of T -fans :

(6.5.55)  X → Proj ℬ(ℐ).

To conclude, it remains to show that (6.5.55) is the only morphism of T -fans from X to Proj ℬ(ℐ). The latter assertion can be checked again locally on T, so we are reduced as above to the case where T is the spectrum of P, and ℐ is associated with I. We may also assume that X = (Spec Q)^{♯}, and φ is given by a morphism of monoids f : P → Q. Then the hypothesis means that the ideal f(I)Q is isomorphic to Q (see remark 6.5.13(ii)), hence it is generated by an element of the form f(a), for some a ∈ I, and the endomorphism x ↦ f(a)x of f(I)Q, is an isomorphism. In such situation, it is clear that f factors uniquely through a morphism of monoids P → B(I)_{(a)}; namely, one defines g : B(I)_{(a)} → Q by the rule : a^{−k}x ↦ f(a)^{−k}f(x) (for every x ∈ I^{k}), which is well defined, by the foregoing observations. The morphism (Spec g)^{♯} : X → D_{+}(a) must then agree with (6.5.55). □

Example 6.5.56. (i) Let P be a monoid, I ⊂ P any finitely generated ideal, {a_{1}, …, a_{n}} a finite system of generators of I; set T := (Spec P)^{♯}, and let φ : T′ → T be the blow up of the ideal I^{∼} ⊂ 𝒪_{T}. Then T′ admits an open covering consisting of the affine fans D_{+}(f_{i}). The latter are the spectra of the monoids Q_{i} consisting of all fractions of the form a · f_{i}^{−t}, for every a ∈ I^{n}; we have a · f_{i}^{−t} = b · f_{i}^{−s} in Q_{i} if and only if there exists k ∈ ℕ such that a · f_{i}^{s+k} = b · f_{i}^{t+k}, if and only if the two fractions are equal in P_{f_{i}}, in other word, Q_{i} is the submonoid of P_{f_{i}} generated by P and {f_{j} · f_{i}^{−1} | j ≤ n}, for every i = 1, …, n.

(ii) Consider the special case where P is fine, and the ideal I ⊂ P is generated by two elements f, g ∈ P. Let t ∈ T′ be any point; up to swapping f and g, we may assume that t corresponds to a prime ideal 𝔭 ⊂ P [f/g], hence φ(t) corresponds to 𝔮 := j^{−1}𝔭 ⊂ P, where j : P → P [f/g] is the natural map. We have the following two possibilities :

• Either f/g ∈ 𝔭, in which case let y ∈ F′ := P [f/g] \ 𝔭 be any element; writing x = y·(f/g)^{n} for some n ≥ 0 and y ∈ P, we deduce that n = 0, so x = y ∈ F := P \𝔮, therefore F′ = j(F). Notice as well that in this case f/g is not invertible in P [f/g], hence P [f/g]^{×} = P^{×}, whence dim P [f/g] = dim P, and , by corollary 6.4.12(i).

• Or else f/g ∉ 𝔭, in which case the same argument yields F′ = j(F)[f/g]. In this case, f/g is invertible in P [f/g] if and only if it is invertible in the face F′, hence ht 𝔭 = dim P − rk_{ℤ}F′ ≥ dim P − dim F − 1, by corollary 6.4.12(i),(ii).

In either event, corollary 6.4.12(i),(ii) implies the inequality :

1 ≥ ht(φ(t)) − ht(t) ≥ 0 for every t ∈ T′.

6.6. Special subdivisions. In this section we explain how to construct – either by geometrical or combinatorial means – useful subdivisions of given fans.

6.6.1. Let T be any locally fine and saturated fan, and t ∈ T any point. By reflexivity (proposition 6.4.14(iv)), the elements s ∈ 𝒪_{T,t}^{gp} ⊗_{ℤ} ℚ correspond bijectively to ℚ-linear forms ρ_{s} : U(t)(ℚ_{+})^{gp} → ℚ, and s ∈ 𝒪_{T,t}^{gp} if and only if ρ_{s} restricts to a morphism of monoids U(t)(ℕ) → ℤ. Moreover, this bijection is compatible with specialization maps : if t′ is a generization of t in T, then the form U(t′)(ℚ_{+})^{gp} → ℚ induced by the image of s in 𝒪_{T,t′}^{gp} ⊗_{ℤ} ℚ is the restriction of ρ_{s} (see (6.5.20)).

Hence, any global section λ ∈ Γ(T, 𝒪_{T}^{gp}) yields a well defined function

ρ_{λ} : T (ℕ) → ℤ

whose restriction to U(t)(ℕ) is the restriction of a ℤ-linear form on U(t)(ℕ)^{gp}, for every t ∈ T; conversely, any such function arises from a unique global section of 𝒪_{T}^{gp}. Likewise, we have a natural isomorphism between the ℚ-vector space of global sections λ of 𝒪_{T}^{gp} ⊗_{ℤ} ℚ, and the space of functions ρ_{λ} : T (ℚ_{+}) → ℚ with a corresponding linearity property.

Let now ρ : T (ℚ_{+}) → ℚ be any function; we may attach to ρ a sheaf of fractional ideals of 𝒪_{T,ℚ} (notation of (6.3.20)), by the rule :

ℐ_{ρ,ℚ}(U) := {s ∈ 𝒪_{T,ℚ}(U) | ρ_{s} ≥ ρ_{|U}} for every open subset U ⊂ T

In this generality, not much can be said concerning ℐ_{ρ,ℚ}; to advance, we restrict our attention to a special class of functions, singled out by the following :

Definition 6.6.2. Let T be a locally fine and saturated fan.

(a) A roof on T is a function :

ρ : T (ℚ_{+}) → ℚ

such that, for every t ∈ T, there exist k := k(t) ∈ ℕ \ {0} and ℚ-linear forms

λ_{1}, …, λ_{k} : U(t)(ℚ_{+})^{gp} → ℚ

with ρ(s) = min(λ_{i}(s) | i = 1, …, k) for every s ∈ U(t)(ℚ_{+}).

(b) An integral roof on T is a roof ρ on T such that ρ(s) ∈ ℤ for every s ∈ T (ℕ).

6.6.3. The interest of the notion of roof on a fan T, is that it encodes in a geometrical way, an integral subdivision of T, together with a coherent sheaf of fractional ideals of 𝒪_{T} (see definition 4.8.6(iii)). This shall be seen in several steps. To begin with, let T and ρ be as in definition 6.6.2(a). For any t ∈ T, pick a system \underline{λ} := \{λ_{1}, …, λ_{k}\} of ℚ-linear forms fulfilling condition (b) of the definition; then for every i = 1, …, k let us set :

U(t, i)(ℕ) := {x ∈ U(t)(ℕ) | ρ(x) = λ_{i}(x)}.

Notice that U(t)(ℕ) = 𝒪_{T,t}, by proposition 6.4.14(iv). Moreover, say that \underline{λ} is irredundant for t if no proper subsystem of \underline{λ} fulfills condition (b) of definition 6.6.2relative to U(t).

Lemma 6.6.4. With the notation of (6.6.3), the following holds :

(i) U(t, i)(ℕ) is a saturated fine monoid for every i ≤ k.

(ii) There is a unique system of ℚ-linear forms which is irredundant for t.

(iii) If \underline{λ} is irredundant for t, then dim U(t, i)(ℕ) = ht_{T} (t) for every i = 1, …, k.

Proof. (i): We leave to the reader the verification that U(t, i) is a saturated monoid. Next, let σ_{i} ⊂ U(t)(ℝ_{+})^{gp∨} be the convex polyhedral cone spanned by the linear forms

((λ_{j} − λ_{i}) ⊗_{ℚ} ℝ | j = 1, …, k).

Then σ_{i} is 𝒪_{T,t}^{gp∨}-rational, so σ_{i}^{∨} is 𝒪_{T,t}^{gp}-rational, and σ_{i}^{∨} ∩ 𝒪_{T,t}^{gp} is a fine monoid (propositions 6.3.21(i), and 6.3.22(i)), therefore the same holds for U(t, i)(ℕ) = σ_{i}^{∨} ∩ 𝒪_{T,t} (corollary 6.4.2).

(iii): Notice that ht_{T} (t) = dim U(t)(ℕ), by proposition 6.4.14(ii) and (6.5.18). In view of corollary 6.4.12(i), it follows already that dim U(t, i) ≤ ht_{T} (t). Now, let λ_{1}, …, λ_{k} be an irredundant system, and suppose, by contradiction, that dim U(t, i)(ℕ) < dim U(t)(ℕ) for some i ≤ k. Especially, σ_{i}^{∨} ∩ U(t)(ℝ_{+}) does not span the ℝ-vector space U(t)(ℝ_{+})^{gp}, and therefore the dual σ_{i} + U(t)(ℝ)^{∨} is not strictly convex (corollary 6.3.14). After relabeling, we may assume that i = 1. Hence there exist a_{j}, b_{j} ∈ ℝ_{+} and φ, φ′ ∈ U(t)(ℝ)^{∨}, and an identity :

∑_{j=2}^{k} a_{j}(λ_{j} − λ_{i}) + φ = − ∑_{j=2}^{k} b_{j}(λ_{j} − λ_{i}) − φ′.

Moreover, ∑_{j=2}^{k}(a_{j} + b_{j}) > 0. It follows that there exist ψ ∈ U(t)(ℝ)^{∨}, and non-negative real numbers (c_{j} | j = 2, … k) such that

λ_{i} = ∑_{j=2}^{k} c_{j}λ_{j} + ψ and ∑_{j=2}^{k} c_{j} = 1.

On the other hand, the irredundancy condition means that there exists x ∈ U(t, 1)(ℕ) such that λ_{j}(x) > λ_{1}(x) for every j > 1. Since ψ(x) ≥ 0, we get a contradiction.

(ii): The assertion is clear, if ht_{T} (t) ≤ 1. Hence suppose that the height of t is ≥ 2, and let \underline{λ} := \{λ_{1}, …, λ_{k}\} and \underline{μ} := \{μ_{1}, …, μ_{r}\} be two irredundant systems for t. Fix i ≤ k, and pick x ∈ U(t, i)(ℕ) which does not lie on any proper face of U(t, i)(ℕ) (the existence of x is ensured by (iii) and proposition 6.4.9(i)); say that μ_{1}(x) = ρ(x). Since i is arbitrary, the assertion shall follow, once we have shown that μ_{1} agrees with λ_{i} on the whole of U(t, i)(ℕ).

However, by definition we have μ_{1}(y) ≥ λ_{i}(y) for every y ∈ U(t, i)(ℕ), and then it is easily seen that Ker(μ_{1} − λ_{i}) ∩ U(t, i)(ℕ) is a face of U(t, i)(ℕ); since x ∈ Ker(μ_{1} − λ_{i}), we deduce that μ_{1} − λ_{i} vanishes identically on U(t, i)(ℕ). □

6.6.5. Henceforth, we denote by \underline{λ}(t) := \{λ_{1}, …, λ_{k}\} the irredundant system of ℚ-linear forms for t. Let 1 ≤ i, j ≤ k; then we claim that U(t, i, j)(ℕ) := U(t, i)(ℕ) ∩ U(t, j)(ℕ) is a face of both U(t, i)(ℕ) and U(t, j)(ℕ). Indeed say that x, x′ ∈ U(t, i)(ℕ) and x + x′ ∈ U(t, i, j)(ℕ); these conditions translate the identities :

λ_{i}(x) + λ_{i}(x′) = λ_{j}(x) + λ_{j}(x′)  λ_{i}(x) ≤ λ_{j}(x)  λ_{i}(x′) ≤ λ_{j}(x′)

whence x, x′ ∈ U(t, i, j)(ℕ). Define :

U(t, i) := (Spec U(t, i)(ℕ)^{∨})^{♯}  U(t, i, j) := (Spec U(t, i, j)(ℕ)^{∨})^{♯}  for every i, j ≤ k.

According to (6.5.20) and (6.5.18), the inclusion maps U(t, i, j)(ℕ) → U(t, l)(ℕ) (for l = i, j) are dual to open immersions

(6.6.6)  U(t, i) ← U(t, i, j) → U(t, j).

We may then attach to t and ρ_{|U(t)} the fan U(t, ρ) obtained by gluing the affine fans U(t, i) along their common intersections U(t, i, j). The duals of the inclusions U(t, i)(ℕ) → U(t)(ℕ) determine a well defined morphism of locally fine and saturated fans :

(6.6.7)  U(t, ρ) → U(t)

which, by construction, induces a bijection on ℕ-points : U(t, ρ)(ℕ) →^{∼} U(t)(ℕ), so it is a rational subdivision, according to proposition 6.5.24.

6.6.8. Let now t′ ∈ T be a generization of t; clearly the system \underline{λ}′ := \{λ′_{1}, …, λ′_{k}\} consisting of the restrictions λ′_{i} of the linear forms λ_{i} to the ℚ-vector subspace U(t′)(ℚ_{+})^{gp}, fulfills condition (b) of definition 6.6.2, relative to t′. However, \underline{λ}′ may fail to be irredundant; after relabeling, we may assume that the subsystem {λ′_{1}, …, λ′_{l}} is irredundant for t′, for some l ≤ k. With the foregoing notation, we have obvious identities :

U(t′, i)(ℕ) = U(t, i)(ℕ) ∩ U(t′)(ℕ) U(t′, i, j)(ℕ) = U(t, i, j)(ℕ) ∩ U(t′)(ℕ)

for every i, j ≤ l; whence, in light of remark 6.4.16(iii), a commutative diagram of fans :

\xymatrix{U(t′, i) \ar[d] & U(t′, i, j) \ar[l] \ar[r] \ar[d] & U(t′, j) \ar[d] \\ U(t, i) ×_{U(t)} U(t′) & U(t, i, j) ×_{U(t)} U(t′) \ar[l] \ar[r] & U(t, i) ×_{U(t)} U(t′)}

whose top horizontal arrows are the open immersions (6.6.6) (with t replaced by t′), whose bottom horizontal arrows are the open immersions (6.6.6)×_{U(t)}U(t′), and whose vertical arrows are natural isomorphisms. Since U(t′) is an open subset of U(t), we deduce an open immersion

j_{t,t′} : U(t′, ρ) → U(t, ρ).

If t′′ is a generization of t′, it is clear that j_{t′,t′′} ∘ j_{t,t′} = j_{t,t′′}, hence we may glue the fans U(t, ρ) along these open immersions, to obtain a locally fine and saturated fan T (ρ). Furthermore, the morphisms (6.6.7) glue to a single rational subdivision :

(6.6.9)  T (ρ) → T.

Remark 6.6.10. (i) In the language of definition 6.3.25the foregoing lengthy procedure translates as the following simple geometric operation. Given a fan Δ (consisting of a collection of convex polyhedral cones of a ℝ-vector space V), a roof on Δ is a piecewise linear function F := ⋃_{σ∈Δ} σ → ℝ, which is concave on each σ ∈ Δ (and hence it is a roof on each such σ, in the sense of example 6.3.27). Then, such a roof determines a natural refinement Δ′ of Δ; namely, Δ′ is the coarsest refinement such that, for each σ′ ∈ Δ′, the function ρ_{|σ′} is the restriction of a ℝ-linear form on V. This refinement Δ′ corresponds to the present T (ρ).

(ii) Moreover, let P be a fine, sharp and saturated monoid of dimension d, set T_{P} := (Spec P)^{♯}, and suppose that f : T → T_{P} is any integral, fine, proper and saturated subdivision. Then f corresponds to a geometrical subdivision Δ of the strictly convex polyhedral cone T_{P} (ℝ_{+}) = P_{ℝ}^{∨}, and we claim that Δ can be refined by the subdivision associated with a roof on T_{P}. Namely, let Δ_{d−1} be the subset of Δ consisting of all σ of dimension d − 1; every σ ∈ Δ_{d−1} is the intersection of a d-dimensional element of Δ and a hyperplane H_{σ} ⊂ P_{ℝ}^{gp∨}; such hyperplane is the kernel of a linear form λ_{σ} on P_{ℝ}^{gp∨}. Let us define

ρ(x) := ∑_{σ∈Δ_{d−1}} min(0, λ_{σ}(x)) for every x ∈ T_{P} (ℝ_{+}).

Then it is easily seen that the subdivision of T_{P} (ℝ_{+}) associated with the roof ρ as in (i), refines the subdivision Δ. In the language of fans, this construction translates as follows. For every point σ ∈ T of height d − 1, let H_{σ} ⊂ P^{gp} be the kernel of the surjection P^{gp} → 𝒪_{T,σ}^{gp} induced by log f; notice that H_{σ} is a free abelian group of rank one, and pick a generator s_{σ} of H_{σ}, which – as in (6.6.1) – corresponds to a function λ_{σ} : T_{P} (ℕ) → ℤ, so we may again consider the integral roof ρ on T_{P} defined as in the foregoing. Then it is easily seen that the rational subdivision T (ρ) → T_{P} associated with ρ, factors as the composition of f and a (necessarily unique) integral subdivision g : T (ρ) → T. More precisely, for every mapping ε : {σ ∈ T | ht(σ) = d − 1} → {0, 1}, let us set

λ_{ε} := ∑_{ht(σ)=d−1} ε(σ) · λ_{σ} and U(ε)(ℕ) := {x ∈ T_{P} (ℕ) | ρ(x) = λ_{ε}(x)}.

Whenever U(ε)(ℕ) has dimension d, let t_{ε} ∈ T (ρ) be the unique point such that U(ε)(ℕ)^{∨} = 𝒪_{T (ρ),t_{ε}}. As the reader may check, there exists a unique closed point τ ∈ T, such that ε(σ) · λ_{σ}(x) ≤ 0 for every x ∈ U(τ)(ℕ) and every σ ∈ U(τ) of height d−1. Then we have g(t_{ε}) = τ, and the restriction U(t_{ε}) → U(τ) of g is deduced from the inclusion U(ε)(ℕ) ⊂ U(τ)(ℕ) of submonoids of T_{P} (ℕ).

(iii) Furthermore, in the situation of (ii), the roof ρ on T_{P} can also be viewed as a roof on T, and then it is clear from the construction that the morphism g : T (ρ) → T is also the subdivision of T attached to the roof ρ.

We wish now to establish some basic properties of the sheaf of fractional ideals

ℐ_{ρ} := ℐ_{ρ,ℚ} ∩ 𝒪_{T}^{gp}

attached to a given roof on T. First we remark :

Lemma 6.6.11. Keep the notation of (6.6.5), and let s ∈ 𝒪_{T,t}^{gp} ⊗_{ℤ} ℚ be any element such that ρ_{s} ≥ ρ_{|U(t)}. Then we have :

(i) There exist φ ∈ (𝒪_{T,t})_{ℚ} (notation of (6.3.20)) and c_{1}, …, c_{k} ∈ ℝ_{+} such that :

ρ_{s} = ρ_{φ} + ∑_{i=1}^{k} c_{i}λ_{i}  ∑_{i=1}^{k} c_{i} = 1.

(ii) The stalk ℐ_{ρ,t} is a finitely generated 𝒪_{T,t}-module.

Proof. (i): Let σ ⊂ U(t)(ℝ_{+})^{gp} be the convex polyhedral cone spanned by the linear forms ((λ_{i} − ρ_{s}) ⊗_{ℚ} ℝ | i = 1, …, k). Then the assumption on s means that

U(t)(ℝ_{+}) ∩ σ^{∨} = U(t)(ℝ_{+}) ∩ Ker ρ_{s} ⊗_{ℚ} ℝ.

Especially σ^{∨} ∩ U(t)(ℝ_{+}) does not span U(t)(ℝ_{+})^{gp}. Then one can repeat the proof of lemma 6.6.4(iii) to derive the assertion.

(ii): By remark 6.4.16(i), we may write λ_{i} = ρ_{s_{i}} − ρ_{s′_{i}}, where s_{i}, s′_{i} ∈ (𝒪_{T,t})_{ℚ} for each i ≤ k; pick N ∈ ℕ large enough, so that Ns′_{i} ∈ 𝒪_{T,t} for every i ≤ k, and set τ := N ∑_{i=1}^{k} s′_{i}. Then τ + ℐ_{ρ,t} ⊂ 𝒪_{T,t}^{gp} ∩ (𝒪_{T,t})_{ℚ} = 𝒪_{T,t}. By proposition 6.1.9(ii), we deduce that τ + ℐ_{ρ,t} is a finitely generated ideal, whence the contention. □

Proposition 6.6.12. Let T be a locally fine and saturated fan, ρ a roof on T. Then the associated fractional ideal ℐ_{ρ} of 𝒪_{T} is coherent.

Proof. In view of lemma 6.6.11(ii) and remark 6.5.13(v), it suffices to show that ℐ_{ρ} is quasicoherent, i.e. for every generization t′ of t, the image of ℐ_{ρ,t} in 𝒪_{T,t′} generates the 𝒪_{T,t′}- module ℐ_{ρ,t′}. Fix such t′; by propositions 6.3.21(i) and 6.4.9(i), there exists λ ∈ 𝒪_{T,t} = U(t)(ℕ)^{∨} such that U(t′)(ℕ) = Ker λ; especially, we see that U(t′)(ℕ)^{gp} is a direct summand of U(t)(ℕ)^{gp}. Now, let s′ ∈ ℐ_{ρ,t′} be any local section; it follows that we may find s ∈ 𝒪_{T,t}^{gp} such that ρ_{s} : U(t)(ℕ)^{gp} → ℤ is a ℤ-linear extension of the corresponding ℤ-linear form ρ_{s′} : U(t′)(ℕ)^{gp} → ℤ. Let also {λ_{1}, …, λ_{k}} be the irredundant system of ℚ-linear forms for t (relative to the roof ρ). For every i ≤ k we have the following situation :

λ_{ℝ} := λ ⊗_{ℚ} ℝ ∈ U(t, i)(ℝ_{+})^{∨} (ρ_{s} − λ_{i}) ⊗_{ℚ} ℝ ∈ U(t′, i)(ℝ_{+})^{∨}.

However, U(t′, i)(ℝ_{+}) = U(t, i)(ℝ_{+}) ∩ Ker λ_{ℝ}, hence U(t′, i)(ℝ_{+})^{∨} = U(t, i)(ℝ_{+})^{∨} + ℝλ_{ℝ}. Especially, there exist r_{i} ∈ ℝ_{+} and φ ∈ U(t)(ℝ_{+})^{∨} such that (ρ_{s} − λ_{i}) ⊗_{ℚ} ℝ = φ − r_{i}λ_{ℝ}. Let N be an integer greater than max(r_{1}, …, r_{k}); it follows that s + Nλ ∈ ℐ_{ρ,t} and its image in ℐ_{ρ,t′} equals s′. □

Proposition 6.6.13. In the situation of (6.6.3) suppose that ρ is an integral roof on T. Then the morphism (6.6.9) is the saturation of a blow up of the fractional ideal ℐ_{ρ}.

Proof. We have to exhibit an isomorphism f : T (ρ) → X := Proj ℬ(ℐ)^{sat} of T -fans. We begin with :

Claim 6.6.14. (i) The fractional ideal ℐ_{ρ}𝒪_{T (ρ)} is invertible.

(ii) For every n ∈ ℕ, denote by nρ : T (ℚ_{+}) → ℚ_{+} the function given by the rule x ↦ n · ρ(x) for every x ∈ T (ℚ_{+}). Then :

ℬ(ℐ)^{sat} = ℬ′ := ∐_{n∈ℕ} ℐ_{nρ}

Proof of the claim. (ii): The assertion is local on T, hence we may assume that T = U(t) for some t ∈ T, in which case, denote by \underline{λ} := \{λ_{1}, …, λ_{k}\} the irredundant system of ℚlinear forms for t. Let n ∈ ℕ, and s ∈ ℐ_{nρ}(U(t)); it is easily seen that the T -monoid ℬ′ is saturated, hence it suffices to show that there exists an integer k > 0 such that kρ_{s} = ρ_{s′} for some s′ ∈ ℐ^{k}(U(t)) (notation of (6.6.1)). However, lemma 6.6.11(ii) implies more precisely that we may find such k, so that the corresponding s′ lies in the ideal generated by \underline{λ}.

(i): The assertion is local on T (ρ), hence we consider again t ∈ T and the corresponding \underline{λ} as in the foregoing. It suffices to show that 𝒥 := ℐ_{ρ}𝒪_{U(t,i)} is invertible for every i = 1, …, k (notation of (6.6.5)). However, by inspecting the constructions it is easily seen that 𝒥 (U(t, i)) consists of all s ∈ (U(t, i)(ℕ)^{∨})^{gp} such that ρ_{s}(x) ≥ ρ(x) for every x ∈ U(t, i)(ℚ_{+}), i.e. ρ_{s}(x) ≥ λ_{i}(x) for every x ∈ U(t, i)(ℚ_{+}). However, since ρ is integral, we have λ_{i} ∈ 𝒪_{T,t}^{gp}; if we apply lemma 6.6.11(i) with T replaced by U(t, i), we conclude that 𝒥 (U(t, i)) is the fractional ideal generated by λ_{i}, whence the contention. ♢

In view of claim 6.6.14(i) we see that there exists a unique morphism f of T -fans from T (ρ) to X. It remains to check that f is an isomorphism; the latter assertion is local on X, hence we may assume that T = U(t) for some t ∈ T, and then we let again \underline{λ} be the irredundant system for t. A direct inspection yields a natural identification of 𝒪_{T,t}-monoids :

ℬ′(U(t, i))_{(λ_{i})} →^{∼} U(t, i)(ℕ)^{∨} for every i = 1, …, k

whence an isomorphism U(t, i) →^{∼} D_{+}(λ_{i}) ⊂ X, which – by uniqueness – must coincide with the restriction of f. On the other hand, the proof of claim 6.6.14(ii) also shows that X = D_{+}(λ_{1}) ∪ ⋯ ∪ D_{+}(λ_{k}), and the proposition follows. □

Example 6.6.15. Let P be a fine, sharp and saturated monoid.

(i) The simplest non-trivial roofs on T_{P} := (Spec P)^{♯} are the functions ρ_{λ} such that

ρ_{λ}(x) := min(0, λ(x)) for every x ∈ T_{P} (ℚ_{+}).

where λ ≠ 0 is a given element of Hom_{ℚ}(T_{P} (ℚ_{+})^{gp}, ℚ) ≃ P^{gp} ⊗_{ℤ} ℚ. Such a ρ_{λ} is an integral roof, provided λ ∈ P^{gp}. In the latter case, we may write λ = ρ_{s_{1}} − ρ_{s_{2}}, for some s_{1}, s_{2} ∈ P. Set ρ′ := ρ_{λ} + ρ_{s_{2}}, i.e. ρ′ = min(ρ_{s_{1}}, ρ_{s_{2}}); clearly T (ρ_{λ}) = T (ρ′), and on the other hand lemma 6.6.11(i) implies that the ideal ℐ_{ρ′} is the saturation of the ideal generated by s_{1} and s_{2}.

(ii) More generally, any system λ_{1}, …, λ_{n} ∈ Hom_{ℚ}(T_{P} (ℚ_{+})^{gp}, ℚ) of ℚ-linear forms yields a roof ρ on T_{P}, such that ρ(x) := ∑_{i=1}^{n} min(0, λ_{i}(x)) for every x ∈ T_{P} (ℚ_{+}). A simple inspection shows that the corresponding subdivision T (ρ) → T can be factored as the composition of n subdivisions g_{i} : T_{i} → T_{i−1}, where T_{0} := T_{P}, T_{n} := T (ρ), and each g_{i} (for i ≤ n) is the subdivision of T_{i−1} corresponding to the roof ρ_{λ_{i}} as defined in (i).

(iii) These subdivisions of T_{P} ”by hyperplanes” are precisely the ones that occur in remark 6.6.10(ii),(iii). Summing up, we conclude that every proper integral and saturated subdivision g : T → T_{P} of T_{P} can be dominated by another subdivision f : T (ρ) → T_{P} of the type considered in (ii), so that f factors as the composition of g and a subdivision h : T (ρ) → T which is also of the type (ii). Especially, both f and h can be realized as the composition of finitely many saturated blow up of ideals generated by at most two elements of P.

6.6.16. Let P be a fine, sharp and saturated monoid. A proper, integral, fine and saturated subdivision of

T_{P} := (Spec P)^{♯}

is essentially equivalent to a (P^{gp})^{∨}-rational subdivision of the polyhedral cone σ := P_{ℝ}^{∨} (see (6.4.8) and definition 6.3.25). A standard way to subdivide a polyhedron σ consists in choosing a point x_{0} ∈ σ \{0}, and forming all the polyhedra x_{0} ∗ F, where F is any proper face of σ, and x_{0} ∗ F denotes the convex span of x_{0} and F. We wish to describe the same operation in terms of the topological language of affine fans.

Namely, pick any non-zero φ ∈ T_{P} (ℚ_{+}) (φ corresponds to the point x_{0} in the foregoing). Let U(φ) ⊂ Spec P be the set of all prime ideals 𝔭 such that φ(P \𝔭) ≠ {0}; in other words, the complement of U(φ) is the topological closure of the support of φ in T_{P}, especially, U(φ) is an open subset of T_{P}. Denote by j : P = Γ(T_{P} , 𝒪_{T_{P}}) → Γ(U(φ), 𝒪_{T_{P}}) the restriction map. The morphism of monoids :

P → Γ(U(φ), 𝒪_{T_{P}}) × ℚ_{+} x ↦ (j(x), φ(x))

determines a cocartesian diagram of fans

\xymatrix{U(φ) × (Spec ℚ_{+})^{♯} \ar[d]_{ψ′} \ar[r]^{β′} & U(φ) × (Spec ℕ)^{♯} \ar[d]^{ψ} \\ T_{P} \ar[r]^{β} & T_{φ^{−1}ℕ} := (Spec φ^{−1}ℕ)^{♯}}

Lemma 6.6.17. With the notation of (6.6.16), the morphisms ψ and ψ′ are proper rational subdivisions, which we call the subdivisions centered at φ.

Proof. Notice that both β and β′ are homeomorphisms on the underlying topological spaces, and moreover both log β^{gp} ⊗_{ℤ} 1_{ℚ} and log β^{′gp} ⊗_{ℤ} 1_{ℚ} are isomorphisms. Thus, it suffices to show that ψ is a proper rational subdivision, hence we may replace P by φ^{−1}ℕ, which allows to assume that φ is a morphism of monoids P → ℕ, and ψ is a morphism of fans

T′ := U(φ) × (Spec ℕ)^{♯} → T_{P} .

In this situation, by inspecting the construction, we find that ψ restricts to an isomorphism :

ψ^{−1}U(φ) →^{∼} U(φ)

and the preimage of the closed subset T_{P} \ U(φ) is the preimage of the closed point 𝔪 ∈ (Spec ℕ)^{♯} under the natural projection T′ → (Spec ℕ)^{♯}; in view of the discussion of (6.5.19), this is naturally identified with U(φ)×{𝔪}. Moreover, the restriction U(φ)×{𝔪} → T_{P} \U(φ) of ψ is the map (t, 𝔪) ↦ t ∪ φ^{−1}𝔪 (recall that t ⊂ P is a prime ideal which does not contain φ^{−1}𝔪). Thus, the assertion will follow from :

Claim 6.6.18. The ℚ-linear map

log ψ_{(t,𝔪)}^{gp} ⊗_{ℤ} 1_{ℚ} : P^{gp} ⊗_{ℤ} ℚ → (𝒪_{T_{P} ,t}^{gp} × ℤ) ⊗_{ℤ} ℚ

is surjective for every t ∈ U(φ), and the induced map :

(6.6.19)  T′(ℚ_{+}) → T_{φ^{−1}ℕ}(ℚ_{+}) = T_{P} (ℚ_{+})

is bijective.

Proof of the claim. Indeed, let F_{t} ⊂ P^{∨} be the face of P^{∨} corresponding to the point t ∈ U(φ), under the bijection (6.5.21); then φ ∉ F_{t}, whence a natural isomorphism of monoids :

(F_{t} + ℕφ)^{∨} →^{∼} 𝒪_{T_{P} ,t} × ℕ : λ ↦ (λ_{|F} , λ(φ))

whose inverse, composed with log ψ_{(t,𝔪)}, yields the restriction map P → (F_{t} + ℕφ)^{∨}. This interpretation makes evident the surjectivity of log ψ_{(t,𝔪)}^{gp} ⊗_{ℤ} 1_{ℚ}. In view of example 6.5.9(iii), the bijectivity of (6.6.19) is also clear, if one remarks that :

P_{ℝ}^{∨} = ⋃_{t∈U(φ)}(F_{t} + ℕφ)_{ℝ}.

The latter identity is obvious from the geometric interpretation in terms of polyhedral cones. A formal argument runs as follows. Let φ′ ∈ P_{ℝ}^{∨}; since P spans P_{ℝ}^{gp}, the cone (P_{ℝ})^{∨} is strictly convex (corollary 6.3.14), hence the line φ′ + ℝφ ⊂ P_{ℝ}^{gp} is not contained in P_{ℝ}^{∨}, therefore there exists a largest r ∈ ℝ such that φ′ − rφ ∈ P_{ℝ}^{∨}, and necessarily r ≥ 0. If φ′ − rφ = 0, the assertion is clear; otherwise, let F be the minimal face of P^{∨} such that φ′ − rφ ∈ F_{ℝ}, so that φ′ = (φ′ − rφ) + rφ ∈ (F + ℕφ)_{ℝ}. Thus, we are reduced to showing that φ ∉ F. But notice that φ′ − rφ lies in the relative interior of F; therefore, if φ ∈ F, we may find ε > 0 such that φ′ − (r + ε)φ still lies in F_{ℝ}, contradicting the definition of r. □

6.6.20. Let P be a fine, sharp and saturated monoid with d := dim P > 0. Lemma 6.6.17 is frequently used to construct subdivisions centered at an interior point of T_{P}, i.e. a point φ ∈ T_{P} (ℕ) which does not lie on any proper face of T_{P} (ℕ) (equivalently, the support of φ is the closed point 𝔪_{P} of T_{P}). In this case U(φ) = T_{P} \{𝔪_{P} } = (T_{P})_{d−1}. By lemma 6.6.17, the ℕ-point φ lifts to a unique ℚ_{+}-point φ̃ of (T_{P})_{d−1} × (Spec ℕ)^{♯}, and by inspecting the definitions, it is easily seen that – under the identification of remark 6.5.8(iii) – the support of φ̃ is the point (∅, 𝔪_{ℕ}), where ∅ ∈ T_{P} is the generic point. More precisely, we may identify (Spec ℕ)^{♯}(ℚ_{+}) with ℚ_{+}, and (T_{P})_{d−1}(ℚ_{+}) with a cone in the ℚ-vector space T_{P} (ℚ_{+})^{gp}, and then φ̃ corresponds to the point (0, 1) ∈ T_{P} (ℚ_{+})^{gp} × ℚ_{+}.

Suppose now that we have an integral roof

ρ : (T_{P})_{d−1}(ℚ_{+}) → ℚ

and let π : T (ρ) → (T_{P})_{d−1} be the associated subdivision. Let also ψ : (T_{P})_{d−1} × (Spec ℕ)^{♯} → T_{P} be the subdivision centered at φ; we deduce a new subdivision :

(6.6.21)  T^{∗} := T (ρ) × (Spec ℕ)^{♯} ⟶^{π×1_{(Spec ℕ)^{♯}}} (T_{P})_{d−1} × (Spec ℕ)^{♯} ⟶^{ψ} T_{P}

whose restriction to the preimage of (T_{P})_{d−1} is T_{P} -isomorphic to π.

Lemma 6.6.22. In the situation of (6.6.20), there exist an integral roof

ρ̃: T_{P} (ℚ_{+}) → ℚ

whose restriction to (T_{P})_{d−1}(ℚ_{+}) agrees with ρ, and a morphism T^{∗} → T (ρ̃) of T_{P} -fans, whose underlying continuous map is a homeomorphism.

Proof. According to remark 6.5.8(vi), we have a natural identification :

T_{P} (ℚ_{+}) = T (ρ)(ℚ_{+}) × (Spec ℕ)^{♯}(ℚ_{+}) = T (ρ)(ℚ_{+}) × ℚ_{+} ⊂ T_{P} (ℚ_{+})^{gp} × ℚ.

mapping the point φ to (0, 1). For a given c ∈ ℝ, denote by ρ_{c} : T_{P} (ℚ_{+}) → ℚ the function given by the rule : (x, y) ↦ ρ(x) + cy for every x ∈ T (ρ)(ℚ_{+}) and every y ∈ ℚ_{+}.

Let t ∈ T (ρ) be any point of height d − 1; by assumption, there exists a ℚ-linear form λ_{t} : U(π(t))(ℚ_{+})^{gp} → ℚ whose restriction to U(t)(ℚ_{+}) agrees with the restriction of ρ. Therefore, the restriction of ρ_{c} to U(t, 𝔪_{ℕ})(ℚ_{+}) = U(t)(ℚ_{+}) × ℚ_{+} agrees with the restriction of the ℚ-linear form

λ_{t,c} : U(π(t))(ℚ_{+})^{gp} × ℚ = T_{P} (ℚ_{+})^{gp} → ℚ (x, y) ↦ λ_{t}(x) + cy.

For any two points t, t′ ∈ T (ρ) of height d − 1, with π(t) = π(t′), pick a finite system of generators {x_{1}, …, x_{n}} for U(t′)(ℕ), and let y_{1}, …, y_{n} ∈ U(t)(ℚ)^{gp}, a_{1}, …, a_{n} ∈ ℚ such that

x_{i} = y_{i} + a_{i}φ in the vector space U(π(t))(ℚ_{+})^{gp}.

In case x_{i} ∈ U(t)(ℕ), we shall have a_{i} = 0, and otherwise we remark that a_{i} > 0. Indeed, if a_{i} < 0 we would have x_{i} − a_{i}φ ∈ U(t)(ℚ_{+})^{gp} ∩ T_{P} (ℚ_{+}) ⊂ U(π(t))(ℚ_{+}); however, U(π(t))(ℚ_{+}) is a proper face of T_{P} (ℚ_{+}), hence φ ∈ U(π(t))(ℚ_{+}), a contradiction.

We may then find c > 0 large enough, so that λ_{t′}(x_{i}) < λ_{t,c}(x_{i}) for every x_{i} ∉ U(t)(ℕ). It follows easily that

(6.6.23)  λ_{t′,c}(x) < λ_{t,c}(x)  for all x ∈ U(t′, 𝔪_{ℕ})(ℚ_{+})\U(t, 𝔪_{ℕ})(ℚ_{+}).

Clearly we may choose c large enough, so that (6.6.23) holds for every pair t, t′ as above, and then it is clear that ρ_{c} will be a roof on T_{P}. Notice that the points of height d of T^{∗} are precisely those of the form (t, 𝔪_{ℕ}), for t ∈ T_{P} of height d − 1, so the points of T (ρ_{c}) of height d are in natural bijection with those of T^{∗}, and if τ ∈ T (ρ_{c}) corresponds to τ^{∗} ∈ T^{∗} under this bijection, we have an injective map U(τ^{∗})(ℕ) → U(τ)(ℕ), commuting with the induced projections to T_{P} (ℕ). There follows a morphism of T_{P} -fans U(τ^{∗}) → U(τ) inducing a bijection U(τ^{∗})(ℚ_{+}) →^{∼} U(τ)(ℚ_{+}). Since T (ρ_{c}) (resp. T^{∗}) is the union of all such U(τ) (resp. U(τ^{∗})), we deduce a morphism T^{∗} → T (ρ_{c}) inducing a homeomorphism on underlying topological spaces. Lastly, say that τ^{∗} = (t, 𝔪); then U(τ^{∗})(ℕ)^{gp} = U(t)(ℕ)^{gp} ⊕ ℤφ, and on the other hand U(t)(ℕ)^{gp} is a direct factor of U(τ)(ℕ)^{gp} (since the specialization map 𝒪_{T (ρ_{c}),τ}^{gp} → 𝒪_{T,t}^{gp} is surjective). It follows easily that we may choose for c a suitable positive integer, in such a way that the resulting roof ρ̃:= ρ_{c} will also be integral. □

6.6.24. Let P be a fine, sharp and saturated monoid, and set as usual T_{P} := (Spec P)^{♯}. There is a canonical choice of a point in T_{P} (ℕ) which does not lie on any proper face of T_{P} (ℕ); namely, one may take the ℕ-point φ_{P} defined as the sum of the generators of the onedimensional faces of T_{P} (ℕ) (such faces are isomorphic to ℕ, by theorem 6.4.18(ii)).

Set d := dim P; if ρ is a given integral roof for (T_{P})_{d−1}, and c ∈ ℕ is a sufficiently large, we may then attach to the datum (φ_{P} , ρ, c) an integral roof ρ̃ of T_{P} extending ρ as in lemma 6.6.22, and such that ρ̃(φ_{P}) = c. More generally, let T be a fine and saturated fan of dimension d, and suppose we have a given integral roof ρ_{d−1} on T_{d−1}; for every point t ∈ T of height d we have the corresponding canonical point φ_{t} in the “interior” of U(t)(ℕ), and we may then pick an integer c_{d} large enough, so that ρ_{d−1} extends to an integral roof ρ_{d} : T (ℚ_{+}) → ℚ with ρ_{d}(φ_{t}) = c_{d} for every t ∈ T of height d, and such that the associated subdivision is T_{P} -homeomorphic to T (ρ_{d−1}) × (Spec ℕ).

This is the basis for the inductive construction of an integral roof on T_{P} which is canonical in a certain restricted sense. Indeed, fix an increasing sequence of positive integers \underline{c} := (c_{2}, …, c_{d}); first we define ρ_{1} : T_{1}(ℚ_{+}) → ℚ_{+} to be the identically zero map. This roof is extended recursively to a function ρ_{h} on T_{h}, for each h = 2, …, d, by the rule given above, in such a way that ρ_{h}(φ_{t}) = c_{i} for every point t of height i ≤ h. By the foregoing we see that the sequence \underline{c} can be chosen so that ρ_{d} shall again be an integral roof.

6.6.25. More generally, let 𝒮 := {P_{1}, …, P_{k}} be any finite set of fine, sharp and saturated monoids. We let 𝒮 -Fan be the full subcategory of Fan whose objects are the fans T such that, for every t ∈ T, there exist P ∈ 𝒮 and an open immersion U(t) ⊂ (Spec P)^{♯}. Then the foregoing shows that we may find a sequence of integers \underline{c}(𝒮) := (c_{2}, …, c_{d}), with d := max(dim P_{i} | i = 1, …, k) such that the following holds. Every object T of 𝒮 -Fan is endowed with an integral roof ρ_{T} : T (ℚ_{+}) → ℚ_{+} such that :

• ρ_{T} (φ_{t}) = c_{i} whenever ht(t) = i ≥ 2, and ρ_{T} vanishes on T_{1}(ℚ_{+}).

• Every open immersion g : T′ → T in 𝒮 -Fan determines an open immersion g̃ : T′(ρ_{T′}) → T (ρ_{T}) such that the diagram

\xymatrix{T′(ρ_{T′}) \ar[r]^{g̃} \ar[d]_{π_{T′}} & T (ρ_{T}) \ar[d]^{π_{T}} \\ T′ \ar[r]^{g} & T}

commutes (where π_{T} and π_{T′} are the subdivisions associated to ρ_{T} and ρ_{T′}).

• If dim T = d, there exists a natural rational subdivision

(6.6.26)  T_{d−1}(ρ_{T_{d−1}}) × (Spec ℕ)^{♯} → T (ρ_{T})

which is a homeomorphism on the underlying topological spaces.

6.6.27. Notice as well that, by construction, T_{1}(ρ_{T_{1}}) = T_{1}; hence, by composing the morphisms (6.6.26), we obtain a rational subdivision of T (ρ_{T}) :

T_{1}×(Spec ℕ^{⊕d−1})^{♯} → ⋯ → T_{d−2}(ρ_{T_{d−2}})×(Spec ℕ^{⊕2})^{♯} → T_{d−1}(ρ_{T_{d−1}})×(Spec ℕ)^{♯} → T (ρ_{T}).

In view of theorem 6.4.18(ii), it is easily seen that every affine open subsect of T_{1} is isomorphic to (Spec ℕ)^{♯}, and any two such open subsets have either empty intersection, or else intersect in their generic points. In any case, we deduce a natural epimorphism

(ℚ_{+})_{T_{1}} → 𝒪_{T_{1},ℚ}

(notation of (6.3.20)) from the constant T_{1}-monoid arising from ℚ_{+}; whence an epimorphism :

ϑ_{T} : (ℚ_{+}^{⊕d})_{T (ρ_{T})} → 𝒪_{T (ρ_{T}),ℚ}

which is compatible with open immersions g : T′ → T in 𝒮 -Fan, in the following sense. Set d′ := dim T′, and notice that d′ ≤ d; denote by π_{dd′} : ℚ_{+}^{⊕d} → ℚ_{+}^{⊕d′} the projection on the first d′ direct summands; then the diagram of T′-monoids :

(6.6.28)  \xymatrix{(ℚ_{+}^{⊕d})_{T′(ρ_{T′})} \ar[r]^{g^{∗}ϑ_{T}} \ar[d]_{(π_{dd′})_{T′(ρ_{T′})}} & g̃^{∗}𝒪_{T (ρ_{T}),ℚ} \ar[d]^{(log g̃)_{ℚ}} \\ (ℚ_{+}^{⊕d′})_{T′(ρ_{T′})} \ar[r]^{ϑ_{T′}} & 𝒪_{T′(ρ_{T′}),ℚ}}

commutes.

6.6.29. Let f : T′ → T be a proper morphism, with T′ locally fine, such that the induced map T′(ℚ_{+}) → T (ℚ_{+}) is injective, and let s ∈ T be any element. For every t ∈ f^{−1}(s), we set

G_{t} := U(t)(ℚ_{+})^{gp} ∩ U(s)(ℕ)^{gp}  H_{t} := U(t)(ℕ)^{gp}  δ_{t} := (G_{t} : H_{t})

and define δ(f, s) := max(δ_{t} | t ∈ f^{−1}(s)).

Lemma 6.6.30. For every s ∈ T we have :

(i) δ(f, s) ∈ ℕ \ {0}.

(ii) If t, t′ ∈ f^{−1}(s), and t is a specialization of t′ in T′, then δ_{t′} ≤ δ_{t}.

(iii) If f is a rational subdivision, the following conditions are equivalent : (a) δ(f, s) = 1.

(b) U(s)(ℕ) = ⋃_{t∈f^{−1}(s)} U(t)(ℕ).

Proof. (i): Since f^{−1}(s) is a finite non-empty set, the assertion means that δ_{t} < +∞ for every t ∈ f^{−1}(s). However, for such a t, let P := 𝒪_{T,s} and Q := 𝒪_{T′,t}; then

U(t)(ℚ_{+})^{gp} = Hom_{ℤ}(Q^{gp}, ℚ) and U(t)(ℕ)^{gp} = Hom_{ℤ}(Q^{gp}, ℤ)

(remark 6.4.16(i) and proposition 6.4.14(iii)). By proposition 6.5.24 we know that the map (log f)_{t}^{gp} ⊗_{ℤ} ℚ : P^{gp} ⊗_{ℤ} ℚ → Q^{gp} ⊗_{ℤ} ℚ is surjective. Since Q is finitely generated, it follows that the image Q′ of P^{gp} in Q^{gp} is a subgroup of finite index, and then it is easily seen that δ_{t} = (Q^{gp} : Q′).

(ii): Under the stated assumptions we have :

G_{t′} = G_{t} ∩ U(t′)(ℚ_{+})^{gp} H_{t′} = H_{t} ∩ U(t′)(ℚ_{+})^{gp}

whence the contention.

(iii): Assume that (b) holds, and let φ ∈ U(t)(ℚ_{+})^{gp} ∩ U(s)(ℕ)^{gp} for some t ∈ f^{−1}(s). Pick any element φ_{0} ∈ U(t)(ℕ) which does not lie on any proper face of 𝒪_{T′,t}^{∨}; we may then find an integer a > 0 large enough, so that φ + aφ_{0} ∈ U(t)(ℚ_{+}). Set φ_{1} := φ + aφ_{0} and φ_{2} := aφ_{0}; then φ_{1}, φ_{2} ∈ U(t)(ℚ_{+}) ∩ U(s)(ℕ) = U(t)(ℕ), hence φ = φ_{1} − φ_{2} ∈ U(t)(ℕ)^{gp}. Since φ is arbitrary, we see that δ_{t} = 1, whence (a).

Conversely, suppose that (a) holds, and let φ ∈ U(s)(ℕ); then there exists t ∈ f^{−1}(s) such that φ ∈ U(t)(ℚ_{+}). Thus, φ ∈ U(t)(ℕ)^{gp} ∩ U(t)(ℚ_{+}) = U(t)(ℕ), whence (b). □

Theorem 6.6.31. Every locally fine and saturated fan T admits an integral, proper, simplicial subdivision f : T′ → T, whose restriction f^{−1}T_{sim} → T_{sim} is an isomorphism of fans.

Proof. Let T be such a fan. By induction on h ∈ ℕ, we shall construct a system of integral, proper simplicial subdivisions f_{h} : S(h) → T_{h} of the open subsets T_{h} (notation of (6.5.16)), such that, for every h ∈ ℕ, the restriction f_{h+1}^{−1}(T_{h}) → T_{h} of f_{h+1} is isomorphic to f_{h}, and such that f_{h}^{−1}T_{h,sim} → T_{h,sim} is an isomorphism. Then, the colimit of the morphisms f_{h} will be the sought subdivision of T.

For h ≤ 1, we may take S(h) := T_{h}.

Next, suppose that h > 1, and that f_{h−1} : S(h − 1) → T_{h−1} has already been given; as a first step, we shall exhibit a rational, proper simplicial subdivision of T_{h}. Indeed, for every t ∈ T_{h} \ T_{h−1}, choose a ℕ-point φ_{t} ∈ U(t)(ℕ) in the following way. If t ∈ T_{sim}, then let φ_{t} be the (unique) generator of an arbitrarily chosen one-dimensional face of U(t)(ℕ); and otherwise take any point φ_{t} which does not lie on any proper face of U(t)(ℕ).

With these choices, notice that U(φ_{t}) × (Spec ℕ)^{♯} = U(t) in case t ∈ T_{sim}, and otherwise U(φ_{t}) = U(t)_{h−1} (notation of (6.6.16)). By lemma 6.6.17, we obtain corresponding rational subdivisions U(φ_{t}) × (Spec ℕ)^{♯} → U(t) of U(t) centered at φ_{t}. Notice that if t lies in the simplicial locus, this subdivision is an isomorphism, and in any case, it restricts to an isomorphism on the preimage of U(t)_{h−1}.

By composing with the restriction of f_{h−1} × (Spec ℕ)^{♯}, we get a rational subdivision:

g_{t} : T′_{t} := f_{h−1}^{−1}U(φ_{t}) × (Spec ℕ)^{♯} → U(t)

whose restriction to the preimage of U(t)_{h−1} is an isomorphism. Moreover, g_{t} is an isomorphism if t lies in T_{sim}. Also notice that g_{t} is simplicial, since the same holds for f_{h−1}.

If t, t′ are any two distinct points of T of height h, we deduce an isomorphism

g_{t}^{−1}(U(t) ∩ U(t′)) →^{∼} g_{t′}^{−1}(U(t) ∩ U(t′))

hence we may glue the fans T′_{t} and the morphisms g_{t} along these isomorphism, to obtain the sought simplicial rational subdivision g : T′ → T_{h}.

For the next step, we shall refine g locally at every point s of height h; i.e. for such s, we shall find an integral, proper simplicial subdivision f_{s} : T′′_{s} → U(s), whose restriction to U(s)_{h−1} agrees with g, hence with f_{h−1}. Once this is accomplished, we shall be able to build the sought subdivision f_{h} by gluing the morphisms f_{s} and f_{h−1} along the open subsets U(s)_{h−1}.

Of course, if s lies in the simplicial locus of T, we will just take for f_{s} the restriction of g, which by construction is already an isomorphism.

Henceforth, we may assume that T = U(s) is an affine fan of dimension h with s ∉ T_{sim}, and g : T′ → T is a given proper rational simplicial subdivision, whose restriction to g^{−1}T_{h−1} is an integral subdivision. We wish to apply the criterion of lemma 6.6.30(iii), which shows that g is an integral subdivision if and only if δ(g, s) = 1. Thus, let t_{1}, …, t_{k} be the points of T′ such that δ_{t_{i}} = δ(g, s) for every i = 1, …, k. Since δ(g, s) is anyway a strictly positive integer (lemma 6.6.30(i)), a simple descending induction reduces to the following :

Claim 6.6.32. Given g as above, we may find a proper rational simplicial subdivision g′ : T′′ → T such that the following holds :

(i) The restriction of g′ to g^{′−1}T_{h−1} is isomorphic to the restriction of g.

(ii) Let t′_{1}, …, t′_{k′} ∈ T′′ be the points such that δ_{t′_{i}} = δ(g′, s) for every i = 1, …, k′. We have δ(g′, s) ≤ δ(g, s), and if δ(g′, s) = δ(g, s) then k′ < k.

Proof of the claim. Set t := t_{1}; by definition, there exists φ′ ∈ U(s)(ℕ) which lies in U(t)(ℚ_{+}) \ U(t)(ℕ); this means that there exists a morphism φ fitting into a commutative diagram :

\xymatrix{𝒪_{T,s} \ar[r]^{(log g)_{t}} \ar[d]_{φ′} & 𝒪_{T′,t} \ar[d]^{φ} \\ ℕ \ar[r] & ℚ_{+}.}

Say that 𝒪_{T′,t} ≃ ℕ^{⊕r}, and let (π_{1}, …, π_{r}) be the (essentially unique) basis of 𝒪_{T′(h),t}^{∨}; then φ = a_{1}π_{1} + ⋯ a_{r}π_{r}, for some a_{1}, …, a_{r} ≥ 0, and after subtracting some positive integer multiple of π, we may assume that 0 ≤ a_{i} < 1 for every i = 1, …, r. Moreover, the coefficients are all strictly positive if and only if φ is a local morphism; more generally, we let t′ be the unique generization of t such that φ factors through a local morphism 𝒪_{T′,t′} → ℕ. Set e := ht(t′), and denote by π_{1}, …, π_{e} the basis of 𝒪_{T′,t′}^{∨}, so that :

(6.6.33)  φ = b_{1}π_{1} + ⋯ + b_{e}π_{e}

for unique rational coefficients b_{1}, …, b_{e} such that 0 < b_{i} < 1 for every i = 1, …, e.

Denote by Z ⊂ T′ the topological closure of {t′}; for every u ∈ Z ∩ T′, the morphism φ factors through a morphism φ_{u} : 𝒪_{T,u} → ℚ_{+}, and we may therefore consider the subdivision of U(u) centered at φ_{u} as in (6.6.16), which fits into a commutative diagram of fans :

\xymatrix{(U(u)\Z) × (Spec ℚ_{+})^{♯} \ar[r] \ar[d] & (U(u)\Z) × (Spec ℕ)^{♯} \ar[d]^{ψ_{u}} \\ U(u) \ar[r] & U′(u) := (Spec φ_{u}^{−1}ℕ)^{♯}.}

We complete the family (ψ_{u} | u ∈ Z), by letting U′(u) := U(u) and ψ_{u} := 1_{U(u)} for every u ∈ T′ \ Z. Notice then, that the topological spaces underlying U(u) and U′(u) agree for every u ∈ T′, and for every u_{1}, u_{2} ∈ T′, the restrictions of ψ_{u_{1}} and ψ_{u_{2}} :

ψ_{u_{i}}^{−1}(U(u_{1}) ∩ U(u_{2})) → U(u_{1}) ∩ U(u_{2}) (i = 1, 2)

are isomorphic. Furthermore, by construction, each restriction U(u) → T of g factors uniquely through a morphism β_{u} : U′(u) → T, hence the family (β_{u} ∘ ψ_{u} | u ∈ T′) glues to a well defined morphism of fans g′ : T′′ → T. By a direct inspection, it is easily seen that g′ is a proper rational simplicial subdivision which fulfills condition (i) of the claim.

Moreover, the map of topological spaces underlying g′ factors naturally through a continuous map p : T′′ → T′, so that g′ = g ∘ p. The restriction V := p^{−1}(T′ \Z) → T of g′ is isomorphic to the restriction T′ \Z → T of g, hence :

δ_{t} = δ_{p(t)} for every t ∈ p^{−1}(T′ \Z).

It follows that if k > 1 and T′ \Z contains at least one of the points t_{2}, …, t_{k}, then δ(g′, s) ≥ δ(g, s). On the other hand, p^{−1}(T′ \Z) contains at most k − 1 points u of T′′ such that δ_{u} = δ(g, s), and for the remaining points u′ ∈ p^{−1}(T′ \Z) we have δ_{u′} < δ(g, s). Since obviously

δ(g′, s) = max(δ(g′_{|p^{−1}(T′\Z)}, s), δ(g′_{|p^{−1}Z}, s))

we see that condition (ii) holds provided we show :

(6.6.34)  δ(g′_{|p^{−1}Z}, s) = max(b_{1}, …, b_{e}) · δ(g, s).

Hence, let us fix u ∈ Z, and let (v, x) ∈ (U(u)\Z) × (Spec ℕ)^{♯} be any point (see (6.5.19)); if x = ∅, then (v, x) ∉ p^{−1}Z, so it suffices to consider the points of the form (v, 𝔪) (where 𝔪 ⊂ ℕ is the maximal ideal). Moreover, say that ht(u) = d; in view of lemma 6.6.30(ii), it suffices to consider the points (v, 𝔪) such that ht(v) = d − 1. There are exactly e such points, namely the prime ideals v_{i} := (π_{i} ∘ σ)^{−1}𝔪, where π_{1}, …, π_{e} are as in (6.6.33), and σ : 𝒪_{T′,u} → 𝒪_{T′,t′} is the specialization map.

In order to estimate δ_{(v,𝔪)} for some v := v_{i}, we look at the transpose of the map

(log g′)_{(v,𝔪)}^{gp} : 𝒪_{T,s}^{gp} → 𝒪_{T′,v}^{gp} × ℤ : z ↦ ((log g)_{v}^{gp}(z), φ^{gp}(z)).

Let (p_{1}, …, p_{d}) be the basis of 𝒪_{T,u}^{∨}, ordered in such a way that p_{i} = π_{i}∘σ for every i = 1, …, e. By a little abuse of notation, we may then denote (p_{j} | j ≠ i) the basis of 𝒪_{T′,v}^{∨}, so that the dual group (𝒪_{T′,v}^{gp} × ℤ)^{∨} admits the basis {p_{j}^{gp} | j ≠ i} ∪ {q}, where q is the natural projection onto ℤ. Set as well p′_{i} := p_{i} ∘(log g)_{u}^{gp} for every i = 1, …, d. With this notation, the above transpose is the group homomorphism given by the rule :

p_{j} ↦ p′_{i} for j ≠ i and q ↦ b_{1}p′_{1} + ⋯ + b_{e}p′_{e}

from which we deduce easily that δ_{(v,𝔪)} = b_{i} · δ_{u}, whence (6.6.34). □

Proposition 6.6.35. Let (Γ, +, 0) be a fine monoid, M a fine Γ-graded monoid. Then :

(i) There exists a finite set of generators C := {γ_{1}, …, γ_{k}} of Γ, with the following property. For every γ ∈ Γ, we may find a_{1}, …, a_{k} ∈ ℕ such that:

γ = a_{1}γ_{1} + ⋯ + a_{k}γ_{k} and M_{γ} = M_{γ_{1}}^{a_{1}} ⋯ M_{γ_{k}}^{a_{k}}.

(ii) There exists a subgroup H ⊂ Γ^{gp} of finite index, such that :

M_{aγ} = M_{γ}^{a} for every γ ∈ H ∩ Γ and every integer a > 0.

(In (i) and (ii) we use the multiplication law of 𝒫(M), as in (6.1.1)).

Proof. Obviously we may assume that M maps surjectively onto Γ, in which case G := Γ^{gp} is finitely generated, and its image G′ into G_{ℝ} := G ⊗_{ℤ} ℝ is a free abelian group of finite rank. The same holds as well for the image L of log M^{gp} in M_{ℝ}^{gp} := log M^{gp} ⊗_{ℤ} ℝ. Let p : G → G′ be the natural projection. According to proposition 6.3.22(iii), we have :

(6.6.36)  M_{ℚ} = M_{ℝ} ∩ M_{ℚ}^{gp}

(notation of (6.3.20)). Let f_{ℝ}^{gp} : M_{ℝ}^{gp} → G_{ℝ} be the induced ℝ-linear map, and denote by f_{ℝ} : M_{ℝ} → G_{ℝ} the restriction of f_{ℝ}^{gp}. By proposition 6.3.28(ii), we may find a G′-rational subdivision Δ of f_{ℝ}(M_{ℝ}) such that for every σ ∈ Δ and every a, b ∈ σ we have :

(6.6.37)  f_{ℝ}^{−1}(a + b) = f_{ℝ}^{−1}(a) + f_{ℝ}^{−1}(b)

After choosing a refinement, we may assume that Δ is a simplicial fan (theorem 6.6.31). Let τ ∈ Δ be any cone; by proposition 6.3.22(i), the monoid N := τ ∩ G′ is finitely generated, and then the same holds for M ×_{G′} N, by corollary 6.4.2. However, the latter is just M′ := ⨁_{γ∈p^{−1}N} M_{γ} (lemma 4.8.29(iii)). Set Γ′ := Γ ∩ p^{−1}N.

Claim 6.6.38. p(Γ′) generates τ.

Proof of the claim. Clearly the Γ-grading of M induces a surjection M_{ℚ} → Γ_{ℚ} (notation of (6.3.20)). By proposition 6.3.22(iii), we have Γ_{ℚ} = f_{ℝ}(M_{ℝ}) ∩ G_{ℚ}, hence N ⊂ τ ∩ G_{ℚ} ⊂ Γ_{ℚ}. Thus, for every n ∈ N we may find a ∈ ℕ such that a · n ∈ p(Γ), hence a · n ∈ p(Γ′). On the other hand, N generates τ, since the latter is G′-rational. The claim follows. ♢

In view of claim 6.6.38, we may replace M by M′, and Γ by Γ′, which allows to assume that f_{ℝ}(M_{ℝ}) is a simplicial cone, and (6.6.37) holds for every a, b ∈ f_{ℝ}(M_{ℝ}). Next, let S := {e_{1}, …, e_{n}} ⊂ p(Γ) be a set of generators of the cone f_{ℝ}(M_{ℝ}); from the discussion in (6.3.15) we see that, up to replacing S by a subset, the rays ℝ_{+} · e_{i} (with i = 1, …, n) are precisely the extremal rays of f_{ℝ}(M_{ℝ}), especially, the vectors e_{1}, …, e_{n} are ℝ-linearly independent. Choose g_{1}, …, g_{n} ∈ Γ such that p(g_{i}) = e_{i} for every i = 1, …, n. According to corollary 6.4.5, there exist finite subsets Σ_{1} …, Σ_{n} ⊂ M, such that :

M_{g_{i}} = M_{0} · Σ_{i} for every i = 1, …, n.

Claim 6.6.39. (M_{0})_{ℚ} = f_{ℝ}^{−1}(0) ∩ M_{ℚ}^{gp}.

Proof of the claim. To begin with, we may write f_{ℝ}^{−1}(0) = (f_{ℝ}^{gp})^{−1}(0) ∩ M_{ℝ}, hence f_{ℝ}^{−1}(0) ∩ M_{ℚ}^{gp} = (f_{ℚ}^{gp})^{−1}(0) ∩ M_{ℚ}, by (6.6.36). Now, suppose x ∈ (f_{ℚ}^{gp})^{−1}(0) ∩ M_{ℚ}; then we may find an integer a > 0 such that ax = m ⊗ 1 for some m ∈ log M. Say that m ∈ log M_{γ}; then f_{ℚ}^{gp}(γ) = 0, therefore γ is a torsion element of G′, and consequently bm ∈ log M_{0} for an integer b > 0 large enough. We conclude that x = (bm) ⊗ (ba)^{−1} ∈ (M_{0})_{ℚ}, as claimed. ♢

On the other hand, since ℝ_{+}e_{i} is a G′-rational polyhedral cone, f_{ℝ}^{−1}(ℝ_{+}e_{i}) is an L-rational polyhedral cone (proposition 6.3.21(ii,iii)), hence it admits a finite set of generators S_{i} ⊂ M_{ℚ}. Up to replacing the elements of S_{i} by some positive rational multiples, we may assume that f_{ℝ}(s) is either 0 or e_{i}, for every s ∈ S_{i}. Then, set S′_{i} := {s ∈ S_{i} | f_{ℝ}(s) = e_{i}}; we easily get :

(6.6.40) f_{ℝ}^{−1}(e_{i}) = f_{ℝ}^{−1}(0) + T_{i}  where : T_{i} := {∑_{s∈S′_{i}} t_{s} · s | t_{s} ∈ ℝ_{+}, ∑_{s∈S′_{i}} t_{s} = 1}

(so T_{i} is the convex hull of S′_{i}, and as usual, the addition of sets in (6.6.40) refers to the addition law of 𝒫(M_{ℝ}^{gp}), see (6.1.1)). Next, we may find an integer a > 0 such that a · s lies in the image of log M, for every s ∈ S_{i}. After replacing e_{i} by a · e_{i} and S_{i} by {a · s | s ∈ S_{i}}, we may then achieve that (6.6.40) holds, and furthermore S′_{i} lies in the image of log M, therefore in the image of log M_{g_{i}}. It follows easily that (6.6.40) still holds with S′_{i} replaced by the set Σ_{i} ⊗ 1 := {m ⊗ 1 | m ∈ Σ_{i}}. Let Σ_{0} ⊂ log M be a finite set of generators for the monoid M_{0}; claim 6.6.39 implies that Σ_{0} is also a set of generators for the L-rational polyhedral cone f_{ℝ}^{−1}(0). Let P ⊂ M be the submonoid generated by Σ := Σ_{0} ∪ Σ_{1} ∪ ⋯ ∪ Σ_{n}, and Γ′′ ⊂ Γ the submonoid generated by g_{1}, …, g_{n}; clearly the Γ-grading of M restricts to a Γ′′-grading on P. Notice that g_{1} ⊗ 1, …, g_{n} ⊗ 1 are linearly independent in G_{ℚ}, since the same holds for e_{1}, …, e_{n}; especially, Γ′′ ≃ ℕ^{⊕n}.

Claim 6.6.41. (i) The set Σ ⊗ 1 := {s ⊗ 1 | s ∈ Σ} generates the cone M_{ℝ}.

(ii) P_{a+b} = P_{a} · P_{b} for every a, b ∈ Γ′′.

(iii) There exists a finite set A ⊂ M such that M = A · P.

Proof of the claim. By (6.6.40)and the foregoing discussion, we know that (Σ_{0}∪Σ_{i})⊗1 generate f_{ℝ}^{−1}(ℝ_{+}e_{i}), for every i = 1, …, n. Since the additivity property (6.6.37) holds for every a, b ∈ f_{ℝ}(M_{ℝ}), assertion (i) follows. (ii) is a straightforward consequence of the definitions. Next, from (i) and proposition 6.3.22(iii), we deduce that M_{ℚ} = P_{ℚ}. Thus, let m_{1}, …, m_{r} be a system of generators for the monoid M; it follows that there are integers k_{1}, …, k_{r} > 0, such that m_{1}^{k_{1}}, …, m_{r}^{k_{r}} ∈ P, and therefore the subset A := {∏_{i=1}^{r} m_{i}^{t_{i}} | 0 ≤ t_{i} < k_{i} for every i ≤ r} fulfills the condition of (iii). ♢

We introduce a partial ordering on G, by declaring that a ≤ b for two elements a, b ∈ G, if and only if b − a ∈ Γ′′. Now, let a ∈ G be any element; we set

G(a) := {g ∈ G | g ≤ a}.

The subset G(a) inherits a partial ordering from G, and a is the maximum of the elements of G(a); moreover, notice that every non-empty finite subset S ⊂ G(a) admits a supremum sup S ∈ G(a). Indeed, it suffices to show the assertion for a set of two elements S = {b_{1}, b_{2}}; then a − b_{i} = ∑_{j=1}^{n} k_{ij}g_{j} for certain k_{ij} ∈ ℕ, so that sup(b_{1}, b_{2}) = a − ∑_{j=1}^{n} min(k_{1j}, k_{2j}) · g_{j}.

Let A be as in claim 6.6.41(iii), and denote by B ⊂ Γ the image of A; for every a ∈ G, let also B(a) := B ∩ G(a); when B(a) ≠ ∅, invoking several times claim 6.6.41(ii,iii), we get :

M_{a} = ⋃_{b∈B(a)} M_{b} · P_{a−b} = ⋃_{b∈B(a)} M_{b} · P_{sup B(a)−b} · P_{a−sup B(a)} ⊂ M_{sup B(a)} · P_{a−sup B(a)}.

Finally, say that a−sup B(a) = ∑_{i=1}^{n} t_{i}g_{i} for certain t_{1}, …, t_{n} ∈ ℕ; applying once more claim 6.6.41(ii), we conclude that :

M_{a} ⊂ M_{sup B(a)} · ∏_{i=1}^{n} M_{g_{i}}^{t_{i}}.

The converse inclusion is clear, and therefore condition (i) of the proposition is fulfilled with :

C := {g_{1}, …, g_{n}} ∪ {sup B(a) | a ∈ G, B(a) ≠ ∅}.

(ii): For h ∈ Γ^{′′gp}, say h = ∑_{i=1}^{n} a_{i}g_{i}, with integers a_{1}, …, a_{n}, we let |h| := ∑_{i=1}^{n} |a_{i}|g_{i} ∈ Γ′′. Choose any positive integer α such that :

(6.6.42)  |b| ≤ α · ∑_{i=1}^{n} g_{i}  for every b ∈ B ∩ Γ^{′′gp}

and let H ⊂ Γ^{′′gp} be the subgroup generated by αg_{1}, …, αg_{n}.

Claim 6.6.43. B(h) = B(kh) for every h ∈ H and every integer k > 0.

Proof of the claim. Let h := ∑_{i=1}^{n} α_{i}g_{i} ∈ H, and suppose that b ∈ B(kh) for some k > 0; therefore kh − b ∈ Γ′′, hence b ∈ Γ^{′′gp}, and we can write b = ∑_{i=1}^{n} β_{i}g_{i} for integers β_{1}, …, β_{n}, such that kα_{i} − β_{i} ≥ 0 for every i = 1, …, n. In this case, (6.6.42) implies that α_{i} ≥ 0 for every i ≤ n, and α_{i} ≥ α ≥ β_{i} whenever β_{i} > 0. It follows easily that k′h − b ∈ Δ for every integer k′ > 0, whence the claim. ♢

Using claims 6.6.41(ii) and 6.6.43, and arguing as in the foregoing, we may compute :

M_{ah} = M_{sup B(ah)} · P_{ah−sup B(ah)}
= M_{sup B(h)} · P_{ah−sup B(h)}
= M_{sup B(h)} · P_{h−sup B(h)} · P_{h}^{a−1}
⊂ M_{h}^{a}.

The converse inclusion is clear, so (ii) holds. □

6.6.44. Let M be an integral monoid, and w ∈ log M^{gp} any element. For ε ∈ {1, −1} we have a natural inclusion

j_{ε} : log M → M(ε) := log M + εℕw

(i.e. M(ε) is the submonoid of M^{gp} generated by M and w^{ε}). Let us write w := b^{−1}a for some a, b ∈ M; then the induced morphisms of affine schemes ι_{ε} := Spec ℤ[j_{ε}] have a natural geometric interpretation. Namely, let f : X → Spec ℤ[M] be the blow up of the ideal I ⊂ ℤ[M] generated by a and b; we have X = U_{1} ∪ U_{−1}, where U_{ε}, for ε = ±1, is the largest open subscheme of X such that w^{ε} ∈ 𝒪_{X}(U_{ε}). Then ι_{ε} is naturally identified with the restriction U_{ε} → Spec ℤ[M] of the blow up f. More generally, by adding to M any finite number of elements of M^{gp}, we may construct in a combinatorial fashion, the standard affine charts of a blow up of an ideal of ℤ[M] generated by finitely many elements of M. These considerations explain the significance of the following flattening theorem :

Theorem 6.6.45. Let j : M → N be an inclusion of fine monoids. Then there exist a finite set Σ ⊂ log M^{gp}, and an integer k > 0 such that the following holds :

(i) For every mapping ε : Σ → {±1}, the induced inclusion :

M(ε) := log M + ∑_{σ∈Σ} ε(σ)ℕσ → N(ε) := log N + ∑_{σ∈Σ} ε(σ)ℕσ

is a flat morphism of fine monoids. (ii) Suppose that j is a flat morphism, and let ι : M → M^{sat} be the natural inclusion. Denote by Q the push-out of the diagram N ⟵^{j} M ⟶^{ι∘k_{M}} M^{sat} (where k_{M} is the k-Frobenius). Then the natural map M^{sat} → Q^{sat} is flat and saturated.

Proof. (i): (Notice that log M, log N and P (ε) := ∑_{σ∈Σ} ε(σ)ℕσ may be regarded as submonoids of log N^{gp}, and then the above sum is taken in the monoid (𝒫(log N^{gp}), +) defined as in (6.1.1).) Set G := N^{gp}/M^{gp}, and let N^{gp} = ⨁_{γ∈G} N_{γ}^{gp} be the j-grading of N^{gp} (remark 6.2.5(iii)); notice that this grading restricts to j-gradings for N and N(ε). Let also Γ := {γ ∈ G | N_{γ} ≠ ∅}, and choose a finite generating set {γ_{1}, …, γ_{r}} of Γ with the properties of proposition 6.6.35(i). According to corollary 6.4.5, for every i ≤ r there exists a finite subset Σ_{i} := {t_{i1}, …, t_{in_{i}}} ⊂ M such that N_{γ_{i}} = N_{0} · Σ_{i}. Moreover, N_{0} is a finitely generated monoid, by corollary 6.4.2. Let Σ_{0} be a finite system of generators for N_{0}, and for every i ≤ r define Σ′_{i} := {t_{ij} − t_{il} | 1 ≤ j < l ≤ n_{i}}. We claim that the subset Σ := Σ_{0} ∪ Σ′_{1} ∪ ⋯ ∪ Σ′_{r} will do. Indeed, first of all notice that Σ ⊂ log M^{gp}. We shall apply the flatness criterion of remark 6.2.5(iv). Thus, we have to show that, for every g ∈ Γ, the M(ε)-module N(ε)_{g} is a filtered union of cosets {m} + M(ε) (for certain m ∈ N_{g}). Hence, let x_{1}, x_{2} ∈ N(ε)_{g}; we may write x_{i} = m_{i} + p_{i} for some m_{i} ∈ log N and p_{i} ∈ P (ε) (i = 1, 2); since {x_{i}} + M(ε) ⊂ {m_{i}} + M(ε), we may then assume that p_{1} = p_{2} = 0, hence x_{1}, x_{2} ∈ N. Thus, it suffices to show that N_{g} is contained in a filtered union of cosets as above. However, by assumption there exist a_{1}, …, a_{r} ∈ ℕ such that g = ∑_{i=1}^{r} a_{i}γ_{i} and N_{g} = N_{γ_{1}}^{a_{1}} ⋯ N_{γ_{r}}^{a_{r}}; we are then easily reduced to the case where g = γ_{i} for some i ≤ r. Therefore, x_{j} = σ_{j} + b_{j}, where σ_{j} ∈ Σ_{i} and b_{j} ∈ N_{0}, for j = 1, 2. Notice that P (ε) contains either σ_{1} − σ_{2}, or σ_{2} − σ_{1} (or both); in the first occurrence, set σ := σ_{2}, and otherwise, let σ := σ_{1}. Likewise, say that Σ_{0} = {y_{1}, …, y_{n}}, so that b_{j} = ∑_{s=1}^{n} a_{js}y_{s} for certain a_{js} ∈ ℕ (j = 1, 2); for every s ≤ n, we set a_{s}^{∗} := min(a_{1s}, a_{2s}) if y_{s} ∈ P (ε), and otherwise we set a_{s}^{∗} := max(a_{1s}, a_{2s}). One sees easily that x_{1}, x_{2} ∈ {σ + ∑_{s=1}^{n} a_{s}^{∗}y_{s}} + M(ε), whence the contention.

(ii): By proposition 6.6.35(ii), there exists a subgroup H ⊂ G of finite index, such that :

π^{−1}(h^{n}) = π^{−1}(h)^{n}

for every integer n > 0 and every h ∈ H. Let k := (G : H), and define N′ as the fibre product in the cartesian diagram :

(6.6.46)  \xymatrix{N′ \ar[r]^{π′} \ar[d]_{μ} & G \ar[d]^{_{k_{G}}} \\ N \ar[r]^{π} & G}

The trivial morphism 0_{M} : M → G (i.e. the unique one that factors through {1}) and the inclusion j satisfy the identity : k_{G} ∘ 0_{M} = π ∘ j, hence they determine a well-defined map φ : M → N′.

Claim 6.6.47. φ is flat and saturated.

Proof of the claim. For the flatness, we shall apply the criterion of remark 6.2.5(iv). First, φ is injective, since μ ∘ φ = j. Next, notice that the sequence of abelian groups :

0 → M^{gp} ⟶^{φ^{gp}} (N′)^{gp} ⟶^{(π′)^{gp}} G → 0

is short exact; indeed, this is none else than the pullback ℰ ∗ k_{G}^{gp} along the morphism k_{G}, of the short exact sequence

(6.6.48)  ℰ := (0 → M^{gp} ⟶^{j^{gp}} N^{gp} ⟶^{π^{gp}} G → 0).

It follows that φ is flat if and only if, for every x ∈ π′(N′), the preimage (π′)^{−1}(x) is a filtered union of cosets of the form {n} · φ(M′). However, the induced map (π′)^{−1}(g) → π^{−1}(g^{k}) is a bijection for every g ∈ G, and the flatness of j implies that π^{−1}(x^{k}) is a filtered union of cosets, whence the contention. Notice also that Im k_{G} ⊂ H; hence, by the same token, we derive that (π′)^{−1}(g^{n}) = (π′)^{−1}(g)^{n} for every g ∈ G, therefore φ is quasi-saturated, by proposition 6.2.31. Since we know already that j is integral (theorem 6.2.3), the claim follows. ♢

Next, we wish to consider the commutative diagram of monoids :

(6.6.49)  \xymatrix{M \ar[r]^{j} \ar[d]_{_{k_{M}}} & N \ar[d]^{ψ} \\ M \ar[r]^{φ} & N′}

such that ψ is the map determined by the pair of morphisms (f, k_{N}). Let P be the push-out of the maps j and k_{M}; the maps φ and ψ determine a morphism τ : P → N′.

Claim 6.6.50. (i) The diagram (6.6.46)^{gp} of associated abelian groups, is cartesian. (ii) The diagram of abelian groups (6.6.49)^{gp} is cocartesian (i.e. τ^{gp} is an isomorphism). (iii) There exists a morphism λ : N′ → P such that λ ∘ τ = k_{P} and τ ∘ λ = k_{N′}.

Proof of the claim. (i): Suppose that x ∈ (N′)^{gp} is any element such that (π′)^{gp}(x) = 1 and μ^{gp}(x) = 1; we may write x = b^{−1}a for some a, b ∈ N′, and it follows that π′(a) = π′(b) and μ(a) = μ(b), hence a = b in N′, since the forgetful functor Mnd → Set commutes with fibre products. Thus x = 1. On the other hand, suppose that π^{gp}(b^{−1}a) = x^{k} for some a, b ∈ N and x ∈ G; therefore, π(b^{k−1}a) = (bx)^{k}, so there exists c ∈ N′ such that μ(c) = b^{k−1}a and π′(c) = bx. Likewise, there exists d ∈ N′ with μ(d) = b^{k} and π′(d) = b. Consequently, (π′)^{gp}(d^{−1}c) = x and μ(d^{−1}c) = b^{−1}a. The assertion follows.

(ii): Quite generally, let E := (0 → A → B → C → 0) be a short exact sequence of objects in an abelian category 𝒞, and for every object X of 𝒞, let k_{X} := k · 1_{X} : X → X. Then one has a natural map of complexes α_{E} : E ∗ k_{C} → E (resp. β_{E} : E → k_{A} ∗ E) from E to the pullback of E along k_{C} (resp. from the push-out of E along k_{A} to E), and a natural isomorphism ω_{E} : k_{A} ∗E →^{∼} E ∗k_{C} in the category Ext_{𝒞} (C, A) of extensions of C by A (this is the category whose objects are all short exact sequences in 𝒞 of the form 0 → A → X → C → 0, and whose morphisms are the maps of complexes which are the identity on A and C). These maps are related by the identities :

(6.6.51)  β_{E} ∘ α_{E} ∘ ω_{E} = k · 1_{k_{A}∗E}  ω_{E} ∘ β_{E} ∘ α_{E} = k · 1_{E∗k_{C}}.

We leave to the reader the construction of ω_{E}. In the case at hand, we obtain a natural isomorphism ω_{ℰ} : k_{M}^{gp} ∗ ℰ →^{∼} ℰ ∗ k_{G}, where ℰ is the short exact sequence of (6.6.48). An inspection of the construction shows that the map τ^{gp} is precisely the isomorphism defined by ω_{ℰ}.

(iii): Let μ′ : N → P be the natural map, and set λ := μ′∘μ. By inspecting the constructions, one checks easily that λ^{gp} is the map defined by β_{ℰ} ∘ α_{ℰ}. Then the assertion follows from (6.6.51). ♢

Let P′ be the push-out of the diagram N′ ⟵^{φ} M ⟶^{ι} M^{sat}; from claim 6.6.47, lemma 6.2.2(i) and corollary 6.2.25(iii), we deduce that the natural map M^{sat} → P′ is flat and saturated, hence P′ is saturated. On the other hand, directly from the definitions we get a cocartesian diagram

(6.6.52)  \xymatrix{P \ar[r] \ar[d]_{τ} & Q \ar[d]^{τ′} \\ N′ \ar[r] & P′.}

The induced diagram (6.6.52)^{sat} of saturated monoids is still cocartesian (remark 4.8.41(v)); however, claim 6.6.50(iii) implies easily that τ^{sat} is an isomorphism, therefore the same holds for (τ′)^{sat}, and assertion (ii) follows. □

7. Homological algebra

This chapter lays the foundations of homological and homotopical algebra that shall be needed in the rest of the treatise. For some standard facts that we do not repeat here, we use the book [163] as a quick reference. Other two useful references are the books [110] and [162].

7.1. Complexes in an additive category. Let 𝒜 be any additive category. We denote by C(𝒜) the category of (cochain) complexes of objects of 𝒜. Hence, an object of C(𝒜) is a pair

K^{•} := (K^{•}, d_{K}^{•})

consisting of a system of objects (K^{n} | n ∈ ℤ) and morphisms (d_{K}^{n} : K^{n} → K^{n+1} | n ∈ ℤ) of 𝒜, called the differentials of the complex K^{•}, such that

d_{K}^{n+1} ∘ d_{K}^{n} = 0 for every n ∈ ℤ.

The morphisms φ^{•} : K^{•} → L^{•} in C(𝒜) are the systems of morphisms (φ^{n} : K^{n} → L^{n} | n ∈ ℤ) of 𝒜 such that

φ^{n+1} ∘ d_{K}^{n} = d_{L}^{n+1} ∘ φ^{n} for every n ∈ ℤ.

We will usually omit the subscript when referring to the differentials of a complex, unless there is a danger of confusion. Also, let I ⊂ ℤ be any (bounded or unbounded) interval, i.e. I is either of the form ℤ ∩ [a, +∞[, or ℤ∩] − ∞, b] (for some a, b ∈ ℤ), or the intersection of any of these two. We shall denote by

C^{I}(𝒜)

the full subcategory of C(𝒜) whose objects are the complexes K^{•} such that K^{i} = 0 whenever i ∉ I (where we fix a zero object 0 of 𝒜). For instance, if I = ℤ ∩ [a, +∞[, then C^{I}(𝒜) is also denoted C^{≥a}(𝒜), and likewise for the case of an upper bounded interval. Moreover, we set

C^{−}(𝒜) := ⋃_{n∈ℤ} C^{≤n}(𝒜)  C^{+}(𝒜) := ⋃_{n∈ℤ} C^{≥n}(𝒜)  C^{b}(𝒜) := ⋃_{n∈ℤ} C^{[−n,n]}(𝒜)

so C^{−}(𝒜) (resp. C^{+}(𝒜), resp. C^{b}(𝒜)) is the full subcategory of C(𝒜) whose objects are the bounded above (resp. bounded below, resp. bounded) complexes of 𝒜.

7.1.1. For every a ∈ ℤ, the inclusion functor

(7.1.2)  C^{≥a}(𝒜) → C(𝒜)  (resp. C^{≤a}(𝒜) → C(𝒜) )

admits a right (resp. left) adjoint called the brutal truncation functor, and denoted

t^{≥a} : C(𝒜) → C^{≥a}(𝒜) (resp. t^{≤a} : C(𝒜) → C^{≤a}(𝒜) ) .

Namely, for any complex K^{•}, we let t^{≥a}(K^{•}) be the unique object of C^{≥a}(𝒜) that agrees with K^{•} in all degrees ≥ a, and with the same differentials as K^{•}, in this range of degrees (and likewise for t^{≤a}(K^{•})). If φ^{•} : K^{•} → L^{•} is any morphism in C(𝒜), then t^{≥a}φ^{•} is the unique morphism t^{≥a}K^{•} → t^{≥a}L^{•} which agrees with φ^{•} in all degrees ≥ a (and likewise for t^{≤a}φ^{•}).

Moreover, if 𝒜 is an abelian category, (7.1.2) also admits a left (resp. a right) adjoint

τ^{≥a} : C(𝒜) → C^{≥a}(𝒜) (resp. τ^{≤a} : C(𝒜) → C^{≤a}(𝒜) )

called the normalized truncation functor (or just the truncation functor). Namely, we fix representatives for all kernels and cokernels of 𝒜, and for any complex K^{•}, we let τ^{≥a}(K^{•}) be the unique object of C^{≥a}(𝒜) whose term in degree a equals Coker d_{K}^{a−1}, and which agrees with K^{•} in all degrees > a, with the same differentials as K^{•}, in this range of degrees; the differential in degree a is the unique morphism Coker d_{K}^{a−1} → K^{a+1} whose composition with the natural map K^{a} → Coker d_{K}^{a−1} equals d_{K}^{a−1}. If φ^{•} : K^{•} → L^{•} is any morphism in C(𝒜), then τ^{≥a}φ^{•} is the unique morphism τ^{≥a}K^{•} → τ^{≥a}L^{•} which agrees with φ^{•} in all degrees > a, and which, in degree a, is the morphism Coker d_{K}^{a} → Coker d_{L}^{a} induced by φ^{a}.

Likewise, τ^{≤a}(K^{•}) is the unique complex of C^{≤a}(𝒜) whose term in degree a equals Ker d_{K}^{a}, and which agrees with K^{•} in all degrees < a, with the same differentials as K^{•}, in all degrees < a − 1; the differential in degree a − 1 is the restriction of d_{K}^{a−1}. Again, for φ^{•} as above, τ^{≤a}φ^{•} agrees with φ^{•} in degrees < a, and is the restriction Ker d_{K}^{a} → Ker d_{L}^{a} of φ^{a} in degree a.

7.1.3. There is an obvious functor

•[0] : 𝒜 → C(𝒜) : A ↦ A[0]

that sends any object A of 𝒜 to the complex with A placed in degree zero, i.e. such that A[0]^{i} equals A if i = 0, and equals 0 otherwise (clearly, there is a unique such complex). On the other hand, the shift operator is the functor

C(𝒜) → C(𝒜) : K^{•} → K^{•}[1]

given by the rule :

K^{•}[1]^{n} := K^{n+1}  d_{K[1]}^{n} := −d_{K}^{n+1}  for every n ∈ ℤ.

Clearly the shift operator is an automorphism of C(𝒜), and one defines the operator K^{•} ↦ K^{•}[n], for every n ∈ ℤ, as the n-th power of the shift operator (in the automorphism group of C(𝒜)). Then, we can combine the two previous operators, to define the complex

A[n] := (A[0])[n] for every n ∈ ℤ and every A ∈ Ob(𝒜) .

Furthermore, notice that the shift operator restricts to functors

C^{+}(𝒜) → C^{+}(𝒜)  C^{−}(𝒜) → C^{−}(𝒜)  C^{b}(𝒜) → C^{b}(𝒜).

Definition 7.1.4. Let 𝒜 be an additive category, φ^{•}, ψ^{•} : K^{•} → L^{•} two morphisms in C(𝒜).

(i) A (chain) homotopy from ψ^{•} to φ^{•} is the datum s^{•} := (s^{n} : K^{n} → L^{n−1} | n ∈ ℤ) of a system of morphisms in 𝒜 such that

φ^{n} − ψ^{n} = s^{n+1} ∘ d_{K}^{n} + d_{L}^{n−1} ∘ s^{n} for every n ∈ ℤ.

(ii) We say that φ^{•} and ψ^{•} as in (i) are chain homotopic if there is a chain homotopy between them. It is easily seen that this defines an equivalence relation ∼ on the set Hom_{C(𝒜)} (K^{•}, L^{•}), which is preserved by composition of morphisms : if φ^{•} ∼ ψ^{•} and α^{•} : K′^{•} → K^{•}, β^{•} : L^{•} → L′^{•} are any two morphisms, then φ^{•} ∘ α^{•} ∼ ψ^{•} ∘ α^{•} and β^{•} ∘ φ^{•} ∼ β^{•} ∘ ψ^{•}. It follows that, for every interval I ⊂ ℤ there exists a well defined homotopy category

Hot^{I}(𝒜)

whose objects are the same as those of C^{I}(𝒜), and whose morphisms are the homotopy classes of morphisms of complexes, and a natural functor

(7.1.5)  C^{I}(𝒜) → Hot^{I}(𝒜)

which is the identity on objects, and the quotient map on Hom-sets. If I = ℤ, we write Hot(𝒜) instead of Hot^{ℤ}(𝒜), and we denote

Hot_{𝒜} (K^{•}, L^{•})

the set of morphisms K^{•} → L^{•} in Hot(𝒜). We may also define the subcategories Hot^{+}(𝒜), Hot^{−}(𝒜) and Hot^{b}(𝒜) as in (7.1).

(iii) We also say that a morphism φ : K^{•} → L^{•} is a homotopy equivalence, if the class of φ^{•} is an isomorphism in Hot(𝒜), i.e. if there exists a morphism ψ^{•} : L^{•} → K^{•} such that ψ^{•} ∘ φ^{•} ∼ 1_{K^{•}} and φ^{•} ∘ ψ^{•} ∼ 1_{L^{•}}. We say that φ^{•} is null-homotopic if it is chain homotopic to the zero morphism K^{•} → L^{•}. We say that a complex K^{•} is homotopically trivial, if the zero endomorphism 0 · 1_{K^{•}} is a homotopy equivalence.

Remark 7.1.6. Let 𝒜 and 𝒜′ be any two additive categories.

(i) If F : 𝒜 → 𝒜′ is any additive functor, we get induced functors

C(F) : C(𝒜) → C(𝒜′)  Hot(F) : Hot(𝒜) → Hot(𝒜′)  K^{•} ↦ F K^{•}

by the rule :

(F K^{•})^{n} := F (K^{n}) and d_{F K^{•}}^{n} := F (d_{K}^{n}) for every n ∈ ℤ and every K^{•} ∈ Ob(C(𝒜)) .

More generally, we get induced functors C^{I}(F), Hot^{I}(F) for any interval I ⊂ ℤ, as well as C^{+}(F), C^{−}(F) and C^{b}(F) (and the corresponding homotopy category variants), in the obvious fashion. Also, notice that the shift operators descend to a functor on the homotopy category :

Hot(𝒜) → Hot(𝒜)  A^{•} ↦ A^{•}[n]  for every n ∈ ℤ

and the latter restrict to endofunctors of the subcategories Hot^{+}(𝒜), Hot^{−}(𝒜) and Hot^{b}(𝒜).

(ii) The category C(𝒜) is additive. Namely, if K^{•}, L^{•} are any two complexes of 𝒜, and φ^{•}, ψ^{•} : K^{•} → L^{•} any two morphisms, we define

(φ + ψ)^{i} := φ^{i} + ψ^{i} for every i ∈ ℤ

and it is clear that this rule yields a natural abelian group structure on Hom_{𝒜} (K^{•}, L^{•}), such that composition of morphisms is a bilinear operation. The zero object is the (unique) complex 0 which has the zero object of 𝒜 in each degree, and biproducts in C(𝒜) admit natural representatives, defined by the rule :

(A ⊕ B)^{i} := A^{i} ⊕ B^{i} for every A^{•}, B^{•} ∈ Ob(C(𝒜)) and every i ∈ ℤ .

Moreover, it is clear that if φ_{1}^{•}, φ_{2}^{•} : K^{•} → L^{•} are any two morphisms in C(𝒜) such that φ_{1}^{•} ∼ φ_{2}^{•}, we have φ_{1}^{•} + ψ^{•} ∼ φ_{2}^{•} + ψ^{•} for every other morphism ψ^{•} : K^{•} → L^{•}; hence there is a unique abelian group structure on Hot_{𝒜} (K^{•}, L^{•}) for every K^{•}, L^{•} ∈ Ob(C(𝒜)), that makes Hot(𝒜) into an additive category, such that (7.1.5) is an additive functor.

Furthermore, if 𝒜 is abelian, the same holds for C(𝒜). Indeed, we can take

(Ker φ)^{i} := Ker φ^{i}  (Coker φ)^{i} := Coker φ^{i}  for every morphism φ^{•} of C(𝒜)

and it is clear that the resulting morphism β_{φ^{•}} as in (3.7.33) is an isomorphism. It is then also clear that, for any additive functor F as in (i), the induced functors C(F) and Hot(F) are additive, and the same holds as well for the shift operators on C(𝒜) and Hot(𝒜).

All these considerations extend more generally to the full subcategories C^{I}(𝒜) (for any interval I ⊂ ℤ), C^{−}(𝒜), C^{+}(𝒜) and C^{b}(𝒜), as well as their homotopy category variants : the detailed verifications shall be left to the reader.

(iii) Taking into account remark 3.7.37(i), we get a natural identification

C(𝒜)^{o} →^{∼} C(𝒜^{o}) K^{•} ↦ (K^{o})^{•}

where

(K^{o})^{i} := (K^{−i})^{o} and d_{K^{o}}^{i} := (−1)^{i+1} · (d_{K}^{−i−1})^{o}

for every i ∈ ℤ and every K^{•} ∈ Ob(C(𝒜)). If φ^{•} : K^{•} → L^{•} is any morphism in C(𝒜), then (φ^{o})^{i} := φ^{−i} for every i ∈ ℤ. Also, if s^{•} is a homotopy between morphisms f^{•}, g^{•} : K^{•} → L^{•}, then the system

(s^{o})^{•} := ((−1)^{i} · (s^{1−i})^{o} | i ∈ ℤ)

is a homotopy between (f^{o})^{•}, (g^{o})^{•} : (L^{o})^{•} → (K^{o})^{•}, whence an isomorphism of categories :

Hot(𝒜)^{o} →^{∼} Hot(𝒜^{o}).

One may wonder about the reason for inserting apparently artificial signs in the above formulas; the point is that this sign convention makes the foregoing natural identifications behave slightly better in connection with additive contravariant functors (and their extensions to complexes : see e.g. the construction of the Hom^{•} functor in example 7.1.7). On the other hand, there is a drawback as well : indeed, notice that by applying twice our natural isomorphism, we do not get the identity automorphism of C(𝒜)^{oo} = C(𝒜); rather

d_{K^{oo}}^{•} = −d_{K}^{•} and (s^{oo})^{•} = −s^{•}

for every object K^{•} and every homotopy s^{•} of C(𝒜) (details left to the reader).

(iv) The indexing notation that makes use of superscript to denote the degrees in a complex, is known traditionally as cohomological degree notation. Sometimes it is more natural to switch to the homological degree notation, that makes use of subscript indexing; namely, one associates with any cochain complex K^{•}, the chain complex K_{•} given by the rule :

K_{n} := K^{−n}  and d_{n} := d^{−n} : K_{n} → K_{n−1}  for every n ∈ ℤ.

(v) If 𝒜 is complete (resp. cocomplete) then the same holds for C^{I}(𝒜), for any interval I ⊂ ℤ. Indeed, we may regard these categories as full subcategories of the category Fun(ℤ, 𝒜), where the ordered set ℤ (endowed with its standard ordering) is viewed as a category as explained in example 1.1.6(iii), and therefore the assertion follows from corollary 1.4.1(ii,iv), which shows more precisely that the limits and colimits in C^{I}(𝒜) are formed degree-wise.

(vi) If 𝒜 is complete (resp. cocomplete), then all products (resp. coproducts) of Hot^{I}(𝒜) are representable, for every interval I ⊂ ℤ. Indeed, let K_{•}^{•} := (K_{j}^{•} | j ∈ J) be any family of objects of Hot^{I}(𝒜) indexed by a small set J; we claim that the product K^{•} in the category C^{I}(𝒜) of the family K_{•}^{•} also represents the product of the same family in Hot^{I}(𝒜). Indeed, let M^{•} be any other object of Hot^{I}(𝒜), and π_{j}^{•} : K^{•} → K_{j}^{•} the canonical projection, for every j ∈ J; from any morphism φ^{•} : M^{•} → K^{•} we obtain a system (φ_{j}^{•} := π_{j}^{•} ∘ φ^{•} : M^{•} → K_{j}^{•} | j ∈ J) of morphisms in Hot^{I}(𝒜). Conversely, given such a system of morphisms, for every j ∈ J pick arbitrarily a morphism of complexes φ̃_{j}^{•} : M^{•} → K_{j}^{•} in the class of φ_{j}^{•}; the system (φ̃_{j}^{•} | j ∈ J) yields by (v) a unique morphism of complexes φ̃^{•} : M^{•} → K^{•} and we let φ^{•} be the homotopy class of φ̃^{•}. We need to show that φ^{•} does not depend on the choices of representatives φ̃_{j}^{•}; to this aim, we come down to checking that if each φ̃_{j}^{•} is null-homotopic, then the same holds for φ̃^{•}. Hence, pick for every j ∈ J a homotopy s_{j}^{•} from φ̃_{j}^{•} to the zero morphism. Since the products are computed degree-wise in C^{I}(𝒜), the system (s_{j}^{n} : M^{n} → K_{j}^{n−1} | j ∈ J) corresponds, for every n ∈ I, to a unique morphism s^{n} : M^{n} → K^{n−1} of 𝒜, and it is easily seen that the system (s^{n} | n ∈ I) yields a well defined homotopy from φ̃^{•} to the zero morphism M^{•} → K^{•}. By applying the same argument to the opposite category 𝒜^{o} and invoking (iii) we get the assertion for coproducts.

Example 7.1.7. Denote by ℤ-Mod_{fft} the additive category of free abelian groups of finite rank. We have a natural isomorphism of categories :

ℤ-Mod_{fft}^{o} →^{∼} ℤ-Mod_{fft} : P ↦ P^{∨} := Hom_{ℤ}(P, ℤ)

that sends any ℤ-linear map f : P → Q to its transpose f^{∨} : Q^{∨} → P^{∨}. In view of remark 7.1.6(iii) there follow natural isomorphisms of categories

C(ℤ-Mod_{fft})^{o} →^{∼} C(ℤ-Mod_{fft})  Hot(ℤ-Mod_{fft})^{o} →^{∼} Hot(ℤ-Mod_{fft})  K^{•} ↦ K^{∨•}.

Definition 7.1.8. Let 𝒜 be an abelian category, K^{•} any object of C(𝒜), and i ∈ ℤ any integer. (i) The cohomology of K^{•} in degree i is the object of 𝒜 :

H^{i}K^{•} := Ker d^{i}/Im d^{i−1}.

(ii) We say that K^{•} is acyclic in degree i (resp. acyclic) if H^{i}K^{•} = 0 (resp. if H^{j}K^{•} = 0 for every j ∈ ℤ).

Remark 7.1.9. (i) Clearly, for every i ∈ ℤ, the rule K^{•} ↦ H^{i}K^{•} extends to a functor

H^{i} : C(𝒜) → 𝒜 .

(ii) Moreover, if φ^{•}, ψ^{•} : K^{•} → L^{•} are chain homotopic morphisms in C(𝒜), then it is easily seen that, for every i ∈ ℤ, the induced morphisms in cohomology

H^{i}φ^{•}, H^{i}ψ^{•} : H^{i}K^{•} → H^{i}L^{•}

coincide. Hence, the cohomology functor H^{i} factors (uniquely) through a functor

H^{i} : Hot(𝒜) → 𝒜 for every i ∈ ℤ.

(iii) Notice as well that the cohomology is a self-dual operation : namely, we have an essentially commutative diagram of functors

\xymatrix{C(𝒜)^{o} \ar[rr]^{∼} \ar@{-->}[dr]_{(H^{i})^{o}} &  & C(𝒜^{o}) \ar[dl]^{H^{−i}} \\  & 𝒜^{o}}  for every i ∈ ℤ

whose top horizontal arrow is the isomorphism of remark 7.1.6(iii). Indeed, if K^{•} is any complex of 𝒜, the morphism d_{K}^{i−1} factors uniquely through a morphism e_{K}^{i−1} : K^{i−1} → Ker d_{K}^{i}, and we have a natural isomorphism H^{i}K^{•} →^{∼} Coker e_{K}^{i−1}. On the other hand, the morphism d_{K}^{i} factors uniquely through a morphism f_{K}^{i} : Coker d_{K}^{i−1} → K^{i} and we have also the natural isomorphism H^{i}K^{•} →^{∼} Ker f_{K}^{i}, so the assertion follows from remark 3.7.37(i).

(iv) Just like in remark 7.1.6(iv), it is sometimes convenient to switch to a subscript notation; thus, for any chain complex K_{•} one sets

H_{n}K_{•} := H^{−n}K_{•} for every n ∈ ℤ

and calls this object of 𝒜 the homology of K_{•} in degree n.

7.1.10. A double complex of 𝒜 is an object of

C_{2}(𝒜) := C(C(𝒜))

(and likewise for a morphism of double complexes); i.e. a double complex is a triple

K^{••} := (K^{••}, d_{h}^{••}, d_{v}^{••})

consisting of a system (K^{pq} | p, q ∈ ℤ) of objects of 𝒜, and morphisms d_{h}^{pq}, d_{v}^{pq} called respectively the horizontal and vertical differentials, fitting into a commutative diagram

\xymatrix{K^{pq} \ar[r]^{d_{h}^{pq}} \ar[d]_{d_{v}^{pq}} & K^{p+1,q} \ar[d]^{d_{v}^{p+1,q}} \\ K^{p,q+1} \ar[r]^{d_{h}^{pq}} & K^{p+1,q+1}}  for every p, q ∈ ℤ

and such that

d_{h}^{p+1,q} ∘ d_{h}^{pq} = 0  d_{v}^{p,q+1} ∘ d_{v}^{pq} = 0  for every p, q ∈ ℤ.

7.1.11. There are natural functors

C_{2}(𝒜) → C_{2}(𝒜) : K^{••} ↦ fl_{𝒜} (K^{••}) C_{2}(𝒜) → C(𝒜) : K^{••} ↦ (K^{••})^{Δ}

where :

• The flip fl_{𝒜} (K^{••}) of K^{••} is the double complex F^{••} such that F^{pq} := K^{qp} for every p, q ∈ ℤ, with differentials deduced from those of K^{••}, in the obvious way.

• The diagonal (K^{••})^{Δ} is the complex D^{•} such that D^{p} := K^{pp} for every p ∈ ℤ and with differentials

d_{v}^{p+1,q} ∘ d_{h}^{pq} : D^{p} → D^{p+1} for every p ∈ ℤ.

Suppose that all coproducts (resp. all products) are representable in 𝒜. Then there are two other natural functors

Tot_{𝒜}^{⊕} : C_{2}(𝒜) → C(𝒜) ( resp. Tot_{𝒜}^{Π} : C_{2}(𝒜) → C(𝒜) )

defined as follows. The total complex Tot_{𝒜}^{⊕}(K^{••}) (resp. Tot_{𝒜}^{Π}(K^{••})) is the complex T^{•} with

T^{n} := ⨁_{p+q=n} K^{pq} ( resp. T^{n} := ∏_{p+q=n} K^{pq} )

and with differentials T^{n} → T^{n+1} given by the sum (resp. the product) of the morphisms

d_{h}^{pq} + (−1)^{p} · d_{v}^{pq} : K^{pq} → K^{p,q+1} ⊕ K^{p+1,q} for all p, q ∈ ℤ such that p + q = n.

We usually drop the subscript 𝒜, unless the omission would be a source of ambiguities, and we often omit as well the superscript ⊕ when dealing with the total complex functor; to avoid confusion, we stipulate that the notation Tot shall always refer to the functor Tot^{⊕}, so if we need to use the other total complex functor, we shall denote it explicitly by Tot^{Π}. Notice that we have natural isomorphisms

Tot(K^{••}) →^{∼} Tot(fl_{𝒜} (K^{••})) (resp. Tot^{Π}(K^{••}) →^{∼} Tot^{Π}(fl_{𝒜} (K^{••})) )

given, in each degree n ∈ ℤ, by the direct sum (resp. the direct product) of the morphisms (−1)^{pq} · 1_{K^{pq}}, for every p, q ∈ ℤ such that p + q = n.

7.1.12. The category of triple complexes of 𝒜 can be realized in two equivalent ways : namely, we have natural identifications :

(7.1.13)  C_{2}(C(𝒜)) →^{∼} C(C_{2}(𝒜))

and we denote either of these categories by C_{3}(𝒜). In other words, a triple complex is a system

(K^{ijk}, d_{1}^{ijk}, d_{2}^{ijk}, d_{3}^{ijk} | i, j, k ∈ ℤ)

such that, for every fixed p ∈ ℤ, the subsystems

(K^{p,••}, d_{2}^{p,••}, d_{3}^{p,••})  (K^{•,p,•}, d_{1}^{•,p,•}, d_{3}^{•,p,•})  (K^{•,•,p}, d_{1}^{•,•,p}, d_{2}^{•,•,p})

are double complexes of 𝒜. Corresponding to the two interpretations of C_{3}(𝒜), we have two different way of forming total complexes out of triple complexes; namely, if 𝒜 is cocomplete we have the functors

Tot_{𝒜} ∘ Tot_{C(𝒜)} , Tot_{𝒜} ∘ C(Tot_{𝒜} ) : C_{3}(𝒜) → C(𝒜)

and likewise for the Tot^{Π} variant, if 𝒜 is complete. However, a simple computation shows that these two functors agree under the natural identification (7.1.13) : namely, both functors yield a complex whose differential is the direct sum (resp. direct product) of the morphisms

d_{1}^{ijk} + (−1)^{i} · d_{2}^{ijk} + (−1)^{i+j} · d_{3}^{ijk} : K^{ijk} → K^{i+1,j,k} ⊕ K^{i,j+1,k} ⊕ K^{i,j,k+1}.

We shall therefore denote these functors indifferently by Tot (resp. by Tot^{Π}), just as for double complexes.

7.1.14. Let 𝒜 , 𝒜′, 𝒜′′ be three additive categories, and

B : 𝒜 × 𝒜′ → 𝒜′′

a biadditive functor, i.e. such that, for every A ∈ Ob(𝒜) and every A′ ∈ Ob(𝒜′), both functors

B(A, −) : 𝒜′ → 𝒜′′ B(−, A′) : 𝒜′ → 𝒜′′

are additive. The functor B induces a functor

B^{••} : C(𝒜) × C(𝒜′) → C_{2}(𝒜′′)

by the rule

B^{ij}(K^{•}, L^{•}) := B(K^{i}, L^{j}) for every K^{•} ∈ Ob(C(𝒜)) and L^{•} ∈ Ob(C(𝒜′)) .

The differentials of B^{••}(K^{•}, L^{•}) in degree (p, q) are respectively B(d_{K}^{p}, 1_{L^{q}}) and B(1_{K^{p}}, d_{L}^{q}). Suppose that 𝒜′′ is cocomplete (resp. complete); then, by composing with the functor Tot^{⊕} (resp. with Tot^{Π}), we deduce a biadditive functor

B_{⊕}^{•} : C(𝒜) × C(𝒜′) → C(𝒜′′) (resp. B_{Π}^{•} : C(𝒜) × C(𝒜′) → C(𝒜′′)) .

For a general additive category 𝒜′′, the foregoing functors are still well defined at least on bounded above or bounded below complexes, i.e. we have functors

B_{−}^{•} : C^{−}(𝒜) × C^{−}(𝒜′) → C^{−}(𝒜′′) B_{+}^{•} : C^{+}(𝒜) × C^{+}(𝒜′) → C^{+}(𝒜′′).

Example 7.1.15. (i) If 𝒜 is any additive category, the construction of (7.1.14) extends the biadditive functor

Hom_{𝒜} (−, −) : 𝒜^{o} × 𝒜 → ℤ-Mod

to a functor Hom_{𝒜 ,Π}^{•} : C(𝒜) × C(𝒜^{o}) → C(ℤ-Mod) (notice the swapping of the two factors, that is required in order to conform with established sign conventions) and combining with the natural identification of remark 7.1.6(iii), we obtain a biadditive functor

Hom_{𝒜}^{•} (−, −) : C(𝒜) × C(𝒜)^{o} → C(ℤ-Mod) : (L^{•}, K^{•}) ↦ Hom_{𝒜 ,Π}^{•} ((K^{o})^{•}, L^{•}).

We shall denote by d_{K,L}^{•} the differential of the complex Hom_{𝒜}^{•} (K^{•}, L^{•}), for every K^{•}, L^{•} ∈ Ob(C(𝒜)). Let n ∈ ℤ be any integer; with this notation, a simple inspection shows that :

• d_{K,L}^{n} is the product, for every j ∈ ℤ, of the maps

Hom_{𝒜} (K^{j−n}, L^{j}) → Hom_{𝒜} (K^{j−n}, L^{j+1}) ⊕ Hom_{𝒜} (K^{j−n−1}, L^{j})

given by the rule : (φ : K^{j−n} → L^{j}) ↦ d_{L}^{j} ∘ φ − (−1)^{n} · φ ∘ d_{K}^{j−n−1}.

• Hom_{C(𝒜)} (K^{•}, L^{•}[n]) = Ker d_{K,L}^{n}.

• Let φ^{•}, ψ^{•} : K^{•} → L^{•}[n] be any two morphisms; then the set of homotopies from φ^{•} to ψ^{•} is naturally identified with the subset

{s^{•} ∈ Hom_{𝒜}^{−1−n}(K^{•}, L^{•}) | d_{K,L}^{−1−n}(s^{•}) = ψ^{•} − φ^{•}}.

• Consequently, we have a natural identification :

Hot_{𝒜} (K^{•}, L^{•}[n]) →^{∼} H^{n}Hom_{𝒜}^{•} (K^{•}, L^{•}).

(ii) The constructions of (i) can be used to endow C(𝒜) with a natural 2-category structure, whose 2-cells are given by homotopies of complexes. Indeed, let us write

s^{•} : φ^{•} ⇒ ψ^{•}

if s^{•} := (s^{n} | n ∈ ℕ) is a homotopy from φ^{•} to ψ^{•}. Then, if λ^{•} : K^{•} → L^{•} is another morphism, and t^{•} : ψ^{•} ⇒ λ^{•} another homotopy, we define a composition law by setting

s^{•} ⊙ t^{•} := (s^{n} + t^{n} | n ∈ ℕ)

and it is immediate that s^{•} ⊙ t^{•} is a homotopy φ^{•} ⇒ λ^{•}. Moreover, if β^{•}, γ^{•} : L^{•} → P^{•} are two other morphisms in C(𝒜), and u : β^{•} ⇒ γ^{•} another homotopy, we have a Godement composition law by the rule

u^{•} ∗ s^{•} := (β^{n+1} ∘ s^{n} + u^{n} ∘ ψ^{n} | n ∈ ℕ) : β^{•} ∘ φ^{•} ⇒ γ^{•} ∘ ψ^{•}.

The associativity of the laws ∗ and ⊙ thus defined are easily checked by direct computation.

Now, suppose that δ^{•} : L^{•} → P^{•} is yet another morphism, and v^{•} : γ^{•} ⇒ δ^{•} another homotopy. A direct calculation yields the identity

(u^{•} ∗ s^{•}) ⊙ (v^{•} ∗ t^{•}) = (u^{•} ⊙ v^{•}) ∗ (s^{•} ⊙ t^{•}) + c^{•}

where c^{n} := d_{P}^{n−2} ∘ u^{n−1} ∘ t^{n} − u^{n} ∘ t^{n+1} ∘ d_{K}^{n} for every n ∈ ℤ, which can be rewritten as

c^{•} = d_{K,L}^{−2}((u ∘ t)^{•}) where (u ∘ t)^{n} := u^{n−1} ∘ t^{n} for every n ∈ ℤ.

Summing up, we conclude that C(𝒜) carries a 2-category structure, whose 2-cells φ ⇒ ψ (for any two 1-cells φ, ψ : K^{•} → L^{•}) are the classes

\overline{s} ∈ Hom_{𝒜}^{−1}(K^{•}, L^{•})/Im(d_{K,L}^{−2}) such that d_{K,L}^{−1}(\overline{s}) = ψ^{•} − φ^{•}.

(iii) Moreover, if F is any additive functor as in remark 7.1.6(i), the induced functor C(F) extends to a pseudo-functor for the 2-category structures given by (ii); indeed, if s^{•} : K^{•} ⇒ L^{•} is a homotopy, obviously the system (F s^{n} | n ∈ ℕ) is a homotopy F s^{•} : F (K^{•}) ⇒ F (L^{•}).

Example 7.1.16. (i) Suppose that (𝒜 , ⊗, Φ, Ψ) is a tensor abelian category, and set

B(A, A′) := A ⊗ A′ for every A, A′ ∈ Ob(𝒜) .

Following (7.1.14), we get a corresponding functor B_{⊕}^{••}, for which we use the notation

− ⊠ − : C(𝒜) × C(𝒜) → C_{2}(𝒜).

If all coproducts are representables in 𝒜, we get as well the functor B_{⊕}^{•}, which we denote :

K^{•} ⊗ L^{•} := Tot(K^{•} ⊠ L^{•}).

If 𝒜 is not cocomplete, K^{•} ⊗ L^{•} is still well defined, provided K^{•} and L^{•} are both bounded below or if they are both bounded above. The commutativity constraints for (𝒜 , ⊗) yield natural isomorphisms K^{•} ⊠ L^{•} →^{∼} fl_{𝒜} (L^{•} ⊠ K^{•}), as well as

(7.1.17)  Ψ_{K,L}^{•} : K^{•} ⊗ L^{•} →^{∼} L^{•} ⊗ K^{•}.

Namely, one takes the direct sum of the maps (−1)^{pq} · Ψ_{K^{p},L^{q}}, for every p, q ∈ ℤ.

(ii) Likewise, if P^{•} is another complex of 𝒜, the associativity constraints of 𝒜 assemble to an isomorphism of triple complexes

(Φ_{K^{i},L^{j},P^{k}} | i, j, k) : (K^{•} ⊠ L^{•}) ⊠ P^{•} →^{∼} K^{•} ⊠ (L^{•} ⊠ P^{•})

whence, taking into account the discussion of (7.1.12), a natural isomorphism in C(𝒜) :

Φ_{K,L,P}^{•} : K^{•} ⊗ (L^{•} ⊗ P^{•}) →^{∼} (K^{•} ⊗ L^{•}) ⊗ P^{•}.

With these natural isomorphisms, C(𝒜) is then a tensor abelian category as well.

(iii) In the situation of (i), notice that the natural morphism K^{i} ⊗L^{j} → (K^{•} ⊗L^{•})^{i+j} induces morphisms

Ker (d_{K}^{i}) ⊗ Ker (d_{L}^{j}) → Ker (d_{K⊗L}^{i+j})
(Ker (d_{K}^{i}) ⊗ Im (d_{L}^{j−1})) ⊕ (Im (d_{K}^{i−1}) ⊗ Ker (d_{L}^{j})) → Im (d_{K⊗L}^{i+j})

so, the induced map Ker (d_{K}^{i}) ⊗ Ker (d_{L}^{j}) → H^{i+j}(K^{•} ⊗ L^{•}) factors through a natural pairing :

H^{i}(K^{•}) ⊗ H^{j}(L^{•}) → H^{i+j}(K^{•} ⊗ L^{•}) for every i, j ∈ ℤ.

(iv) Moreover, if P^{•} is a third complex, in view of (ii) we get a commutative diagram

\xymatrix{H^{i}K^{•} ⊗ (H^{j}L^{•} ⊗ H^{k}P^{•}) \ar[d]_{γ} \ar[r]^{α} & (H^{i}K^{•} ⊗ H^{j}L^{•}) ⊗ H^{k}P^{•} \ar[r]^{β} & H^{i+j}(K^{•} ⊗ L^{•}) ⊗ H^{k}P^{•} \ar[d]^{δ} \\ H^{i}K^{•} ⊗ H^{j+k}(L^{•} ⊗ P^{•}) \ar[r]^{λ} & H^{i+j+k}(K^{•} ⊗ (L^{•} ⊗ P^{•})) \ar[r]^{σ} & H^{i+j+k}((K^{•} ⊗ L^{•}) ⊗ P^{•})}

where α is the associativity constraint, β, γ, δ and λ are given by the above pairing, and σ is deduced from Φ_{K,L,P}^{•}.

(v) If 𝒜 also admits an internal Hom functor, we may define as well a functor

ℋ om^{••} : C(𝒜) × C(𝒜)^{o} → C_{2}(𝒜)

following the trace of example 7.1.15(i); namely, we set

ℋ om^{p,q}(K^{•}, L^{•}) := ℋ om(K^{−p}, L^{q}) for every p, q ∈ ℤ

with differentials

d_{h}^{p,q} := ℋ om(1_{K^{−p}}, d_{L}^{q})  d_{v}^{p,q} := (−1)^{p+1} · ℋ om(d_{K}^{−p−1}1_{L^{q}})  for every p, q ∈ ℤ.

If all products are representable in 𝒜, we may then define

ℋ om^{•}(K^{•}, L^{•}) := Tot^{Π}ℋ om^{••}(K^{•}, L^{•}).

(vi) Suppose that 𝒜 is both complete and cocomplete, and P^{•} is any other complex; to ease notation, set H^{•} := ℋ om^{•}(K^{•}, L^{•}) and N^{•} := P^{•} ⊗ K^{•}. We have natural isomorphisms

Hom_{C(𝒜)} (P^{•}, H^{•}) →^{∼} \xymatrix{Equal(Hom_{𝒜}^{0} (P^{•}, H^{•}) \ar[r]^{d_{v}} \ar[r]_{d_{h}} & Hom_{𝒜}^{1} (P^{•}, H^{•}))}

where

d_{h} := ∏_{n∈ℤ} Hom_{𝒜} (P^{n}, d_{H}^{n}) and d_{v} := ∏_{n∈ℤ} Hom_{𝒜} (d_{P}^{n} , H^{n+1})

(see example 7.1.15(i)). However, notice that

Hom_{𝒜}^{a} (P^{•}, H^{•}) = ∏_{n∈ℤ} Hom_{𝒜} (P^{n}, ∏_{p+q=n+a} ℋ om(K^{−p}, L^{q}))
= ∏_{n∈ℤ} ∏_{p+q=n+a} Hom_{𝒜} (P^{n}, ℋ om(K^{−p}, L^{q}))
= ∏_{n∈ℤ} ∏_{p+q=n+a} Hom_{𝒜} (P^{n} ⊗ K^{−p}, L^{q})
= ∏_{q∈ℤ} Hom_{𝒜} ( ⨁_{n−p=q−a} P^{n} ⊗ K^{−p}, L^{q})

for every n, a ∈ ℤ, whence

Hom_{C(𝒜)} (P^{•}, H^{•}) →^{∼} \xymatrix{Equal(Hom_{𝒜}^{0} (N^{•}, L^{•}) \ar[r]^{d′_{v}} \ar[r]_{d′_{h}} & Hom_{𝒜}^{1} (N^{•}, L^{•}))}

where

d′_{h} := ∏_{n∈ℤ} Hom_{𝒜} (d_{N}^{q}, L^{q}) and d′_{v} := ∏_{n∈ℤ} Hom_{𝒜} (N^{q}, d_{L}^{q})

so finally :

Hom_{C(𝒜)} (P^{•}, ℋ om^{•}(K^{•}, L^{•})) →^{∼} Hom_{C(𝒜)} (P^{•} ⊗ K^{•}, L^{•})

which says that ℋ om^{•} is an internal Hom functor for C(𝒜).

(vii) In the situation of (vi), set

ℋ om_{C(𝒜)} (K^{•}, L^{•}) := Ker (d^{0} : H^{0} → H^{1})

and take P^{•} := Z[0], where Z is any object of 𝒜; it is easily seen that the natural map

Hom_{𝒜} (Z, ℋ om_{C(𝒜)} (K^{•}, L^{•})) → Hom_{C(𝒜)} (P^{•}, H^{•}) → Hom_{C(𝒜)} (Z[0] ⊗ K^{•}, L^{•})

is an isomorphism : details left to the reader.

7.1.18. Suppose that 𝒜 is an additive category with small Hom-sets, and let (G^{•}, d_{G}^{•}) be any complex of free abelian groups of finite rank; with the notation of (3.7.47), we obtain an object (G_{𝒜}^{•} , d_{𝒜}^{•} ) of C(𝒜^{†}); on the other hand, if (K^{•}, d_{K}^{•}) is any object of C(𝒜), we may also consider the object h_{K}^{†} := (h_{K^{n}}^{†}, h_{d^{n}}^{†} | n ∈ ℕ) of C(𝒜^{†}), and since 𝒜^{†} is an abelian tensor category, we may form the tensor product

G^{•} ⊠_{ℤ} K^{•} := ℋ om^{••}(G_{𝒜}^{•} , h_{K}^{†})

according to example 7.1.16(v). Arguing as in (3.7.47), we see that this object of C_{2}(𝒜^{†}) is isomorphic to an object of C_{2}(𝒜), and after choosing representing objects, we get a functor

C(ℤ-Mod_{fft}) × C(𝒜) → C_{2}(𝒜) (G^{•}, K^{•}) ↦ G^{•} ⊠_{ℤ} K^{•}

which is additive in both arguments. If all direct sums are representable in 𝒜, or else, if G^{•} is a bounded complex, it is then natural to define

G^{•} ⊗_{ℤ} K^{•} := Tot G^{•} ⊠_{ℤ} K^{•} for every G^{•} and K^{•} as above.

Likewise, we set K^{•} ⊠_{ℤ} G^{•} := fl_{𝒜} (G^{•} ⊠_{ℤ} K^{•}) and K^{•} ⊗_{ℤ} G^{•} := Tot(K^{•} ⊠_{ℤ} G^{•}).

Remark 7.1.19. (i) With the notation of (7.1.18), notice the natural isomorphism

K^{•}[1] →^{∼} ℤ[1] ⊗_{ℤ} K^{•} for every K^{•} ∈ Ob(C(𝒜))

which explains the sign convention in the definition of the shift operator in (7.1.3).

(ii) Moreover, denote by 𝕂⟨1⟩^{•} ∈ Ob(C^{[−1,0]}(ℤ-Mod)) the object such that

𝕂⟨1⟩^{−1} := ℤ 𝕂⟨1⟩^{0} := ℤ ⊕ ℤ

and with differential d^{−1} given by the rule : n ↦ (n, −n) for every n ∈ ℤ. Let e_{0} := (0, 1) and e_{1} := (1, 0) be the canonical basis of 𝕂⟨1⟩^{0}; we have two morphisms

ι_{i}^{•} : ℤ[0] → 𝕂⟨1⟩^{•} for i = 0, 1

given, in degree zero, by the rule : n ↦ n · e_{i} for every n ∈ ℕ. For any pair of morphisms φ^{•}, ψ^{•} : L^{•} → M^{•} in C(𝒜), and any homotopy s^{•} := (s^{n} | n ∈ ℤ) from φ^{•} to ψ^{•}, we obtain a morphism of complexes

σ^{•} : 𝕂⟨1⟩^{•} ⊗_{ℤ} L^{•} → M^{•}

as follows. For every n ∈ ℤ, the morphism σ^{n} : L^{n} ⊕ L^{n} ⊕ L^{n+1} → M^{n} restricts to φ^{n} (resp. ψ^{n}, resp. s^{n}) on the first (resp. second, resp. third) summand. Clearly

(7.1.20)  φ^{•} = σ^{•} ∘ (ι_{0}^{•} ⊗_{ℤ} L^{•})  ψ^{•} = σ^{•} ∘ (ι_{1}^{•} ⊗_{ℤ} L^{•}).

Conversely, the datum of a morphism σ^{•} : 𝕂⟨1⟩^{•} ⊗_{ℤ} L^{•} → M^{•} yields a homotopy from σ^{•} ∘ (ι_{1}^{•} ⊗_{ℤ} L^{•}) to σ^{•} ∘ (ι_{0}^{•} ⊗_{ℤ} L^{•}).

(iii) If 𝒜 is abelian, the functor −⊠_{ℤ} − extends to the category C(ℤ-Mod_{fg})×C(𝒜), and if 𝒜 is also cocomplete, we can even extend it to the whole of C(ℤ-Mod) × C(𝒜) and we have as well a corresponding functor − ⊗_{ℤ} − on this category (see (3.7.47) and remark 3.7.49(ii)).

(iv) Let M^{•} and N^{•} be any two complexes of abelian groups, and f_{0}^{•}, f_{1}^{•} : M^{•} → N^{•} two homotopy equivalent morphisms. If K^{•} is any object of C(𝒜) and the tensor product M^{•}⊗_{ℤ}K^{•}, N^{•} ⊗_{ℤ} K^{•} are both representable in C(𝒜), then the induced morphisms f_{0}^{•} ⊗_{ℤ} K^{•} and f_{1}^{•} ⊗_{ℤ} K^{•} are homotopy equivalent. Indeed, by (ii), we have a morphism τ^{•} : 𝕂⟨1⟩^{•} ⊗_{ℤ} M^{•} → N^{•} such that τ^{•} ∘ (ι_{i}^{•} ⊗_{ℤ} 1_{M}) = f_{i}^{•} for i = 0, 1; by example 7.1.16(ii) the tensor product τ^{•} ⊗_{ℤ} K^{•} is naturally identified with a morphism τ_{K}^{•} : 𝕂⟨1⟩^{•} ⊗_{ℤ} (M^{•} ⊗_{ℤ} K^{•}) → N^{•} ⊗_{ℤ} K^{•} such that τ_{K}^{•} ∘ (ι_{i}^{•} ⊗ 1_{M⊗_{ℤ}K}) = f_{i}^{•} ⊗_{ℤ} K^{•} for i = 0, 1, whence the claim, again by (ii).

(v) Let P^{•} be any bounded complex of free abelian groups of finite rank. Notice the natural identification

(P^{•} ⊗_{ℤ} K^{•})^{o} →^{∼} P^{∨•} ⊗_{ℤ} (K^{o})^{•} in C(𝒜^{o})

deduced from example 7.1.7and remark 7.1.6(iii).

Proposition 7.1.21. Let 𝒜 be any additive category, and F : C(𝒜) → ℬ any functor. The following conditions are equivalent :

(a) F factors uniquely through the natural functor C(𝒜) → Hot(𝒜).

(b) F φ^{•} is an isomorphism, for every homotopy equivalence φ^{•} in C(𝒜).

Proof. Clearly (a)⇒(b), hence, suppose that (b) holds, consider two chain homotopic morphisms φ^{•}, ψ^{•} : L^{•} → M^{•} in C(𝒜), and define σ^{•} as in remark 7.1.19(ii); in view of (7.1.20), assertion (a) will follow, once we know that

F (ι_{0}^{•} ⊗_{ℤ} L^{•}) = F (ι_{1}^{•} ⊗_{ℤ} L^{•}).

To show the latter identity, consider the morphism

p^{•} : 𝕂⟨1⟩^{•} → ℤ[0]

such that p^{0} : ℤ ⊕ ℤ → ℤ is the addition map : (a, b) ↦ a + b for every a, b ∈ ℤ. Clearly

p^{•} ∘ ι_{0}^{•} = 1_{ℤ[0]} = p^{•} ∘ ι_{1}^{•}

so we are easily reduced to showing that F (p^{•} ⊗_{ℤ} L^{•}) is an isomorphism in ℬ. By assumption (b), the latter in turn will follow, once we have shown that p^{•} ⊗_{ℤ} L^{•} is a homotopy equivalence. Thus, it suffices to exhibit a morphism j^{•} : ℤ[0] → 𝕂⟨1⟩^{•} such that

(7.1.22)  p^{•} ∘ j^{•} = 1_{ℤ[0]}

and (j^{•} ∘ p^{•}) ⊗_{ℤ} L^{•} is homotopy equivalent to 1_{𝕂⟨1⟩⊗_{ℤ}L}. Set 𝕂⟨1, 1⟩^{•} := 𝕂⟨1⟩^{•} ⊗_{ℤ} 𝕂⟨1⟩^{•}; by remark 7.1.19(ii), the datum of a homotopy from (j^{•} ∘ p^{•}) ⊗_{ℤ} L^{•} to 1_{𝕂⟨1⟩⊗_{ℤ}L} is equivalent to that of a morphism

τ^{•} : 𝕂⟨1, 1⟩^{•} ⊗_{ℤ} L^{•} → 𝕂⟨1⟩^{•} ⊗_{ℤ} L^{•}

such that

τ^{•} ∘ (ι_{0}^{•} ⊗_{ℤ} 𝕂⟨1⟩^{•} ⊗_{ℤ} L^{•}) = (j^{•} ∘ p^{•}) ⊗_{ℤ} L^{•} and τ^{•} ∘ (ι_{1}^{•} ⊗_{ℤ} 𝕂⟨1⟩^{•} ⊗_{ℤ} L^{•}) = 1_{𝕂⟨1⟩⊗_{ℤ}L}.

Notice that (7.1.22) is fulfilled with j^{•} := ι_{0}^{•}; we are then further reduced to showing the existence of a morphism

t^{•} : 𝕂⟨1, 1⟩^{•} → 𝕂⟨1⟩^{•}

such that

t^{•} ∘ (ι_{0}^{•} ⊗_{ℤ} 𝕂⟨1⟩^{•}) = ι_{0}^{•} ∘ p^{•} and t^{•} ∘ (ι_{1}^{•} ⊗_{ℤ} 𝕂⟨1⟩^{•}) = 1_{𝕂⟨1⟩}.

To this aim, let e_{0}, e_{1} be the canonical basis of 𝕂⟨1⟩^{0}, so that d^{−1}(1) = e_{0} − e_{1}. Then :

𝕂⟨1, 1⟩^{k} is spanned by \{ \begin{matrix}1 ⊗ 1 & for k = −2 \\ (e ⊗ e_{i}, e_{i} ⊗ e | i = 0, 1) & for k = −1 \\ (e_{i} ⊗ e_{j} | i, j = 0, 1) & for k = 0 \end{matrix}

with differentials given by the rules :

\begin{matrix}d^{−2}(1 ⊗ 1) := & (e_{0} − e_{1}) ⊗ 1 − 1 ⊗ (e_{0} − e_{1}) \\ d^{−1}(1 ⊗ e_{i}) := & (e_{0} − e_{1}) ⊗ e_{i} \\ d^{−1}(e_{i} ⊗ 1) := & e_{i} ⊗ (e_{0} − e_{1}). \end{matrix}

We define a morphism t^{•} as sought, by the rules :

t^{−2}(1 ⊗ 1) := 0
t^{−1}(1 ⊗ e_{0}) = t^{−1}(e_{0} ⊗ 1) := 0
t^{−1}(e_{1} ⊗ 1) = t^{−1}(1 ⊗ e_{1}) := 1
t^{0}(e_{i} ⊗ e_{0}) := e_{0} (i = 0, 1)
t^{0}(e_{i} ⊗ e_{1}) := e_{i} (i = 0, 1).

A direct inspection shows that t^{•} satisfies the required identities, and concludes the proof. □

7.1.23. In the situation of (7.1.14), let A ∈ Ob(𝒜), A′ ∈ Ob(𝒜′) and let P be any free abelian group of finite rank. With the notation of (3.7.47), it is easily seen that there exist natural isomorphisms in 𝒜′′

(7.1.24)  B(P ⊗_{ℤ} A, A′) →^{∼} P ⊗_{ℤ} B(A, A′) →^{∼} B(A, P ⊗_{ℤ} A′).

Now, let “?” be either + or −, and K^{•} (resp. L^{•}) any object of C^{?}(𝒜) (resp. of C^{?}(𝒜′)); let also P^{•} be any bounded complex of free abelian groups of finite rank. The maps (7.1.24) assemble to a natural isomorphism of triple complexes

ω^{ijk} : B(P^{i} ⊗_{ℤ} K^{j}, L^{k}) →^{∼} P^{i} ⊗_{ℤ} B(K^{j}, L^{k}) i, j, k ∈ ℤ

whence, taking into account the discussion of (7.1.12), an isomorphism of total complexes

ω_{K,L,P}^{•} : B_{?}^{•}(P^{•} ⊗_{ℤ} K^{•}, L^{•}) →^{∼} P^{•} ⊗_{ℤ} B_{?}^{•}(K^{•}, L^{•})

(notation of (7.1.18)). Likewise, we get a natural isomorphism

σ_{P,K,L}^{•} : B_{?}^{•}(K^{•}, P^{•} ⊗_{ℤ} L^{•}) →^{∼} P^{•} ⊗_{ℤ} B_{?}^{•}(K^{•}, L^{•}).

(Notice that σ_{P,K,L}^{•} involves composition with the flip operator C(fl_{𝒜} ) : C_{3}(𝒜) →^{∼} C_{3}(𝒜), hence, in each degree p ∈ ℤ, it is the direct sum of morphisms (−1)^{ij}σ^{ijk}, for all i, j, k ∈ ℤ such that i + j + k = p, and where σ^{ijk} is a natural isomorphism as in (7.1.24)).

Especially, take P^{•} := 𝕂⟨1⟩^{•}; it is easily seen that ω_{K,L,𝕂⟨1⟩}^{•} identifies the morphism

B_{?}^{•}(ι_{i}^{•} ⊗_{ℤ} 1_{K}, L^{•}) : B_{?}^{•}(K^{•}, L^{•}) → B_{?}^{•}(𝕂⟨1⟩^{•} ⊗_{ℤ} K^{•}, L^{•})

with the morphism

ι_{i}^{•} ⊗_{ℤ} B_{?}^{•}(K^{•}, L^{•}) : B_{?}^{•}(K^{•}, L^{•}) → 𝕂⟨1⟩^{•} ⊗_{ℤ} B_{?}^{•}(K^{•}, L^{•}).

Hence, suppose that φ_{0}^{•}, φ_{1}^{•} : L^{•} → M^{•} are two homotopy equivalent morphisms in C^{?}(𝒜′); taking into account remark 7.1.19(ii), we deduce that the induced morphisms B_{?}^{•}(K^{•}, φ_{i}^{•}) (for i = 0, 1) are also homotopy equivalent. A similar argument shows that the functor

C^{?}(𝒜) → C^{?}(𝒜′′) : K^{•} ↦ B_{?}^{•}(K^{•}, L^{•})

likewise preserves homotopy equivalences. We conclude that the functor B_{?}^{•} descends to the homotopy category, to give a functor

B_{?}^{•} : Hot^{?}(𝒜) × Hot^{?}(𝒜′) → Hot^{?}(𝒜′′)

which, in case 𝒜′′ is cocomplete (resp. complete), can even be extended to a functor

B_{⊕}^{•} : Hot(𝒜) × Hot(𝒜′) → Hot(𝒜′′) (resp. B_{Π}^{•} : Hot(𝒜) × Hot(𝒜′) → Hot(𝒜′′)) .

Example 7.1.25. (i) Consider any abelian tensor category (𝒜 , ⊗, Φ, Ψ). Following (7.1.23), the tensor product of example 7.1.16(i) descends to a biadditive functor

⊗ : Hot^{?}(𝒜) × Hot^{?}(𝒜) → Hot^{?}(𝒜)

for “?” equal to either + or −. If 𝒜 is cocomplete, this functor is even defined on the whole of Hot(𝒜).

(ii) The constructions of (7.1.23) can also be applied to the functor Hom_{𝒜}^{•} of example 7.1.15(i) : taking into account remark 7.1.6(iii), we deduce that the functor Hom_{𝒜}^{•} descends to a biadditive functor :

Hot_{𝒜}^{•} : Hot(𝒜) × Hot(𝒜)^{o} → Hot(ℤ-Mod) : (L^{•}, K^{•}) ↦ Hom_{𝒜}^{•} (K^{•}, L^{•}).

7.2. Filtered complexes and spectral sequences. Let 𝒜 be any abelian category; a filtered object of 𝒜 is a datum

(A, Fil^{•}A)

consisting of an object A of 𝒜 and a system of subobjects (Fil^{p}A | p ∈ ℤ) of A, with

Fil^{p+1}A ⊂ Fil^{p}A for every p ∈ ℤ.

If (B, Fil^{•}B) is any other filtered object of 𝒜, a morphism of filtered objects

(7.2.1)  (A, Fil^{•}A) → (B, Fil^{•}B)

is just a morphism u : A → B in 𝒜 which restricts to a morphism

Fil^{p}u : Fil^{p}A → Fil^{p}B for every p ∈ ℤ.

With these morphisms, clearly the filtered objects of 𝒜 form a category

Fil.𝒜 .

Moreover, any functor F : 𝒜 → ℬ induces a functor

Fil.F : Fil.𝒜 → Fil.ℬ

that assigns to any filtered object (A, Fil^{•}A) of 𝒜 the filtered object (F A, Fil^{•}F A) such that Fil^{k}F A := Im(F (Fil^{k}A) → F A) for every k ∈ ℤ.

7.2.2. To any object (A, Fil^{•}A) of Fil.𝒜 we attach its associated graded object, which is the system of objects of 𝒜 :

(gr^{n}A | n ∈ ℤ) where gr^{n}A := Fil^{n}A/Fil^{n+1}A for every n ∈ ℤ.

Notice that any morphism u(A, Fil^{•}) → (B, Fil^{•}) of filtered objects of 𝒜 induces a system of morphisms in 𝒜 :

gr^{p}u : gr^{p}A → gr^{p}B for every p ∈ ℤ.

We say that the filtration Fil^{•}A of a filtered object (A, Fil^{•}A) is bounded above (resp. bounded below) if there exists N ∈ ℤ such that Fil^{N}A = A (resp. such that Fil^{N}A = 0). We say that the filtration Fil^{•} is finite if it is bounded above and below. Clearly, if Fil^{•}A is bounded above (resp. below), then gr^{n}A = 0 whenever −n (resp. n) is sufficiently large.

Definition 7.2.3. Let 𝒜 be an abelian category, and r_{0} ∈ ℕ any integer.

(i) A (cohomological) r_{0}-spectral sequence in 𝒜 is a datum

((E_{r}^{pq}, d_{r}^{pq}, β_{r}^{pq}) | p, q, r ∈ ℤ, r ≥ r_{0})

consisting of :

• objects E_{r}^{pq} of 𝒜 and morphisms d_{r}^{pq} : E_{r}^{pq} → E_{r}^{p+r,q−r+1} in 𝒜, such that :

(7.2.4)  d_{r}^{p+r,q−r+1} ∘ d_{r}^{pq} = 0  for every p, q, r ∈ ℤ with r ≥ r_{0}.

• isomorphisms

β_{r}^{pq} : Ker d_{r}^{pq}/Im d_{r}^{p−r,q+r−1} →^{∼} E_{r+1}^{pq} for every p, q, r ∈ ℤ with r ≥ r_{0}.

(ii) Let (E_{•}^{••}, d_{E,•}^{••}, β_{E,•}^{pq}) and (F_{•}^{••}, , d_{F,•}^{••}, β_{F,•}^{pq}) be two r_{0}-spectral sequences in 𝒜. A morphism of r_{0}-spectral sequences

u_{•}^{••} : E_{•}^{••} → F_{•}^{••}

is a system (u_{r}^{pq} : E_{r}^{pq} → F_{r}^{pq} | p, q, r ∈ ℤ, r ≥ r_{0}) of morphisms of 𝒜, such that the diagrams

\xymatrix{E_{r}^{pq} \ar[r]^{d_{E,r}^{pq}} \ar[d]_{u_{r}^{pq}} & E^{p+r,q−r+1} \ar[d]^{u_{r}^{p+r,q−r+1}} & E_{r}^{pq} \ar[d]^{u_{r}^{pq}} & Ker d_{E,r}^{pq} \ar[r] \ar[l] & Ker d_{E,r}^{pq}/Im d_{E,r}^{p−r,q+r−1} \ar[r]^{β_{E,r}^{pq}} & E_{r+1}^{pq} \ar[d]_{u_{r+1}^{pq}} \\ F_{r}^{pq} \ar[r]^{d_{F,r}^{pq}} & F^{p+r,q−r+1} & F_{r}^{pq} & Ker d_{F,r}^{pq} \ar[r] \ar[l] & Ker d_{E,r}^{pq}/Im d_{E,r}^{p−r,q+r−1} \ar[r]^{β_{F,r}^{pq}} & F_{r+1}^{pq}}

commute for every p, q, r ∈ ℤ with r ≥ r_{0} (where the unmarked arrows are the natural monomorphisms and epimorphisms). Obviously, the r_{0}-spectral sequences of 𝒜 and their morphisms form a category, which we denote by

Sp.Seq_{r_{0}}(𝒜)

7.2.5. Let r ∈ ℕ be any integer. Clearly, for every integer s ≥ r there is a natural functor

Sp.Seq_{r}(𝒜) → Sp.Seq_{s}(𝒜)

that simply forgets the terms E_{k}^{pq} with r ≤ k < s. One may also define a natural functor form s-spectral sequences to r-spectral sequences. Namely, let E_{•}^{••} be any (r + 1)-spectral sequence of 𝒜. The décalage of E^{••} is the r-spectral sequence (Déc (E)^{••}, d_{Déc (E)}^{••}) given by the rule

Déc (E)_{s}^{p,q−p} := E_{s+1}^{p+q,−p}  d_{Déc (E),s}^{p,q−p} := d_{E,s+1}^{p+q,−p}  for every p, q, s ∈ ℤ with s ≥ r

and whose isomorphisms β_{Déc (E)}^{••} are deduced from the corresponding isomorphisms for E^{••}, in the obvious way. Clearly, this rule extends to a natural functor

Déc : Sp.Seq_{r+1}(𝒜) → Sp.Seq_{r}(𝒜).

7.2.6. Let E_{•}^{••} be any r_{0}-spectral sequence; with the notation of definition 7.2.3we define, by induction on k − r, two systems of subobjects

(Z_{k}(E_{r}^{pq}), B_{k}(E_{r}^{pq}) ⊂ E_{r}^{pq} | p, q, r, k ∈ ℤ, k > r ≥ r_{0})

as follows. First, we set

Z_{r+1}(E_{r}^{pq}) := Ker d_{r}^{pq}  B_{r+1}(E_{r}^{pq}) := Im d_{r}^{p−r,q+r−1}  for every p, q, r ∈ ℤ with r ≥ r_{0}.

Next, say that i ≥ 2 is any integer, and suppose that both Z_{k}(E_{r}^{pq}) and B_{k}(E_{r}^{pq}) have already been defined, for every p, q, r, k ∈ ℤ with k > r ≥ r_{0} and k − r < i; we let

Z_{r+i}(E_{r}^{pq}) := π_{pqr}^{−1}Z_{r+i}(E_{r+1}^{pq}) B_{r+i}(E_{r}^{pq}) := π_{pqr}^{−1}B_{r+i}(E_{r+1}^{pq})

where π_{pqr} is the composition

Z_{r+1}(E_{r}^{pq}) → Z_{r+1}(E_{r}^{pq})/B_{r+1}(E_{r}^{pq}) ⟶^{β_{r}^{pq}} E_{r+1}^{pq}.

Then it is clear that the system (β_{r}^{pq} | p, q, r ∈ ℤ, r ≥ r_{0}) induces isomorphisms

β_{rk}^{pq} : Z_{k}(E_{r}^{pq})/B_{k}(E_{r}^{pq}) →^{∼} E_{k}^{pq} for every p, q, r, k ∈ ℤ with k > r ≥ r_{0}

and for every p, q, r, k, n ∈ ℤ with k, n > r ≥ r_{0} we have inclusions

B_{k}(E_{r}^{pq}) ⊂ B_{k+1}(E_{r}^{pq}) ⊂ Z_{n+1}(E_{r}^{pq}) ⊂ Z_{n}(E_{r}^{pq}) ⊂ E_{r}^{pq}.

Definition 7.2.7. In the situation of (7.2.6), we define :

(i) an abutment for the spectral sequence E_{•}^{••} to be a datum consisting of • a system of subobjects (B_{∞}(E_{r_{0}}^{pq}), Z_{∞}(E_{r_{0}}^{pq}) ⊂ E_{r_{0}}^{pq} | p, q ∈ ℤ) such that

B_{k}(E_{r_{0}}^{pq}) ⊂ B_{∞}(E_{r_{0}}^{pq}) ⊂ Z_{∞}(E_{r_{0}}^{pq}) ⊂ Z_{k}(E_{r_{0}}^{pq}) for every p, q, k ∈ ℤ with k > r_{0}

• a system of filtered objects ((E_{∞}^{n}, Fil^{•}E_{∞}^{n}) | n ∈ ℤ) of 𝒜

• a system of isomorphisms in 𝒜

β_{∞}^{pq} : E_{∞}^{pq} := Z_{∞}(E_{r_{0}}^{pq})/B_{∞}(E_{r_{0}}^{pq}) →^{∼} gr^{p}E_{∞}^{p+q} for every p, q ∈ ℤ.

We summarize these conditions via the traditional notation :

E_{r}^{pq} ⇒ E_{∞}^{p+q}.

For any n ∈ ℤ, we say that E_{•}^{••} is convergent in degree n, if for every p, q ∈ ℤ with p + q = n there exists k ∈ ℕ such that B_{k}(E_{r_{0}}^{pq}) = B_{∞}(E_{r_{0}}^{pq}) and Z_{k}(E_{r_{0}}^{pq}) = Z_{∞}(E_{r_{0}}^{pq}). We say that E_{•}^{••} is convergent, if it is convergent in all degrees.

(iii) Let (E_{•}^{••}, B_{∞}(E_{r_{0}}^{••}), Z_{∞}(E_{r_{0}}^{••}), E_{∞}^{•}, β_{E,∞}^{pq}) and (F_{•}^{••}, B_{∞}(F_{r_{0}}^{••}), Z_{∞}(F_{r_{0}}^{••}), F_{∞}^{•}, β_{F,∞}^{pq}) be two r_{0}-spectral sequences in 𝒜 with abutments. A morphism of r_{0}-spectral sequences with abutments is a pair

u_{•}^{••} : E_{•}^{••} → F_{•}^{••} (v^{n} : (E_{∞}^{n}, Fil^{•}E_{∞}^{n}) → (F_{∞}^{n}, Fil^{•}F_{∞}^{n}) | n ∈ ℤ)

such that :

• u_{•}^{••} is a morphism of r_{0}-spectral sequences.

• v^{n} is a morphism of filtered objects of 𝒜, for every n ∈ ℤ.

• u_{r_{0}}^{pq}(B_{∞}(E_{r_{0}}^{pq})) ⊂ B_{∞}(F_{r_{0}}^{pq}) and u_{r_{0}}^{pq}(Z_{∞}(E_{r_{0}}^{pq})) ⊂ Z_{∞}(F_{r_{0}}^{pq}) for every p, q ∈ ℤ.

• The following diagrams commute, for every p, q ∈ ℤ :

\xymatrix{E_{r_{0}}^{pq} \ar[d]_{u_{r_{0}}^{pq}} & Z_{∞}(E_{r_{0}}^{pq}) \ar[l] \ar[r] & E_{∞}^{pq} \ar[r]^{β_{E,∞}^{pq}} & gr^{p}E_{∞}^{p+q} \ar[d]^{gr^{p}v^{p+q}} \\ F_{r_{0}}^{pq} & Z_{∞}(F_{r_{0}}^{pq}) \ar[l] \ar[r] & E_{∞}^{pq} \ar[r]^{β_{F,∞}^{pq}} & gr^{p}F_{∞}^{p+q}}

(where the unmarked arrows are the natural epimorphisms and monomorphisms).

7.2.8. For every r ∈ ℕ, we denote by

Sp.Seq_{r}^{∞}(𝒜)

the category of r-spectral sequences with abutments (and their morphisms, as in definition 7.2.7(ii)). For every integer s > r there is an obvious functor

Sp.Seq_{r}^{∞}(𝒜) → Sp.Seq_{s}^{∞}(𝒜)

that simply forgets the terms E_{k}^{pq} with r ≤ k < s, and replaces the terms B_{∞}(E_{r}^{pq}), Z_{∞}(E_{r}^{pq}) with their images β_{rs}^{pq}(B_{∞}(E_{r}^{pq})), β_{rs}^{pq}(Z_{∞}(E_{r}^{pq}) in E_{s}^{pq}.

Remark 7.2.9. Let r_{0} ∈ ℕ, n ∈ ℤ and E_{•}^{••} any r_{0}-spectral sequence of 𝒜.

(i) Notice that if E_{r}^{pq} = 0 for some p, q ∈ ℤ and r ≥ r_{0}, then E_{s}^{pq} = 0 for every s ≥ r.

(ii) We say E_{•}^{••} is bounded above (resp. bounded below) in degree n, if there exist r, N ∈ ℕ with r ≥ r_{0}, such that

E_{r}^{pq} = 0 for all p, q ∈ ℤ such that p + q = n and q > N (resp. p > N).

If E_{•}^{••} is bounded above in degree n + 1 and bounded below in degree n − 1, notice that for every p, q ∈ ℤ with p + q = n there exists an integer r ≥ r_{0} such that both d_{s}^{pq} and d_{s}^{p−s,q+s−1} are zero morphisms for every s ≥ r, in which case β_{s}^{pq} is an isomorphism E_{s+1}^{pq} →^{∼} E_{s}^{pq}, for every s ≥ r, and for every p, q, s, k ∈ ℤ with k > s ≥ r and p + q = n we have :

Z_{k}(E_{s}^{pq}) = Z_{s+1}(E_{s}^{pq}) B_{k}(E_{s}^{pq}) = B_{s+1}(E_{s}^{pq}).

Definition 7.2.10. Let 𝒜 be any abelian category, and n ∈ ℤ.

(i) A filtered complex (K^{•}, Fil^{•}K^{•}) of 𝒜 is just an object of the category Fil.C(𝒜).

(ii) We say that Fil^{•}K^{•} is bounded above (resp. bounded below, resp. finite) in degree n, if the filtration Fil^{•}K^{n} on the object K^{n} of 𝒜 is bounded above (resp. bounded below, resp. finite). We say that Fil^{•}K^{•} is bounded, if it is finite in all degrees (see (7.2.2)).

Remark 7.2.11. Likewise, we may define a filtered double complex as an object of the category Fil.C_{2}(𝒜). Then, the flip and diagonal functors of (7.1.11) can be upgraded to functors

Fil.C_{2}(𝒜) → Fil.C_{2}(𝒜) Fil.C_{2}(𝒜) → Fil.C(𝒜).

If all coproducts (resp. all products) are representable in 𝒜, the total complex functor Tot^{⊕} (resp. Tot^{Π}) induces a corresponding functor

Fil.Tot^{⊕} : Fil.C_{2}(𝒜) → Fil.C(𝒜) (resp. Fil.Tot^{Π} : Fil.C_{2}(𝒜) → Fil.C(𝒜)

(see (7.2)) which we shall often denote just by Tot (resp. by Tot^{Π}).

7.2.12. Any filtered complex of 𝒜 determines a spectral sequence of 𝒜, whose terms are defined as follows. For every p, q ∈ ℤ and r ∈ ℕ, set

Z(K)_{r}^{pq} := Ker (\overline{d} : (Fil^{p}K^{•})^{p+q} → (Fil^{p}K^{•})^{p+q+1}/(Fil^{p+r}K^{•})^{p+q+1})
D(K)_{r}^{pq} := (Fil^{p+1}K^{•})^{p+q} + d_{K}^{p+q−1}((Fil^{p−r+1}K^{•})^{p+q−1}) ⊂ (Fil^{p−r+1}K^{•})^{p+q}
B(K)_{r}^{pq} := D(K)_{r}^{pq} ∩ Z(K)_{r}^{pq}
E(K)_{r}^{pq} := Z(K)_{r}^{pq}/B(K)_{r}^{pq}

where \overline{d} is the morphism in 𝒜 induced by the differential

d_{Fil^{p}K}^{p+q} : (Fil^{p}K^{•})^{p+q} → (Fil^{p}K^{•})^{p+q+1}

of the complex Fil^{p}K^{•}. Notice that

(Fil^{p+r}K^{•})^{p+q} ⊂ Z(K)_{r}^{pq} ⊂ (Fil^{p}K^{•})^{p+q} for every p, q ∈ ℤ and r ∈ ℕ.

ALMOST RINGS AND PERFECTOID SPACES 545

Lemma 7.2.13. With the notation of (7.2.12), we have :

B(K)_{r}^{pq} = Z(K)_{r}^{p+1,q−1} + d_{K}^{p+q−1}(Z(K)_{r−1}^{p−r+1,q+r−2}) for every p, q ∈ ℤ and r ≥ 1. Proof. Notice first that

D(K)_{r}^{pq} ∩ (Fil^{p}K^{•})^{p+q} = (Fil^{p+1}K^{•})^{p+q} + (Im d_{Fil^{p−r+1}K}^{p+q−1} ∩ (Fil^{p}K^{•})^{p+q})

and

Im d_{Fil^{p−r+1}K}^{p+q−1} ∩ (Fil^{p}K^{•})^{p+q} = d_{K}^{p+q−1}(Z(K)_{r−1}^{p−r+1,q+r−2}).

On the other hand, we have : d_{K}^{p+q}(D(K)_{r}^{pq}) ∩ (Fil^{p+r}K^{•})^{p+q+1} = Im d_{Fil^{p+1}K}^{p+q} ∩ (Fil^{p+r}K^{•})^{p+q+1} = d_{K}^{p+q}(Z(K)_{r}^{p+1,q−1}). The lemma follows easily, by comparing these identities. □ 7.2.14. It is easily seen that d_{Fil^{p}K}^{p+q} restricts to a morphism

(7.2.15)  D(K)_{r}^{pq} + Z(K)_{r}^{pq} → Z(K)_{r}^{p+r,q−r+1}

which sends D(K)_{r}^{pq} into B(K)_{r}^{p+r,q−r+1}, and therefore induces a morphism

d(K)_{r}^{pq} : E(K)_{r}^{pq} → E(K)_{r}^{p+r,q−r+1} for every p, q ∈ ℤ and r ∈ ℕ.

Clearly d(K)_{r}^{p+r,q−r+1} ∘ d(K)_{r}^{pq} = 0. Furthermore, a simple inspection shows that Ker d(K)_{r}^{pq} = Z(K)_{r+1}^{pq}/(Z(K)_{r+1}^{pq} ∩ D(K)_{r}^{pq}) Im d(K)_{r}^{p−r,q+r−1} = D(K)_{r+1}^{pq}/D(K)_{r}^{pq} whence a natural isomorphism

E(K)_{r+1}^{pq} →^{∼} Ker d(K)_{r}^{pq}/Im d(K)_{r}^{p−r,q+r−1} for every p, q ∈ ℤ and r ∈ ℕ

so the system (E(K)_{•}^{••}, d(K)_{•}^{••}) is indeed a spectral sequence. Remark 7.2.16. (i) A simple inspection shows that

E(K)_{0}^{pq} = gr^{p}K^{p+q} and d(K)_{0}^{pq} = d_{gr^{p}K}^{p+q} for every p, q ∈ ℤ

so the resulting complex (E(K)_{0}^{p,•}, d(K)^{p,•}) is just gr^{p}K^{•}[q], for every p, q ∈ ℤ.

(ii) In view of (i), we have a natural isomorphism

E(K)_{1}^{pq} →^{∼} H^{p+q}(gr^{p}K^{•}) for every p, q ∈ ℤ

which identifies d(K)_{1}^{pq} with a morphism

H^{p+q}(gr^{p}K^{•}) → H^{p+q+1}(gr^{p+1}K^{•}).

A direct inspection shows that the latter is the boundary morphism in degree p + q arising (by the snake lemma) from the short exact of complexes

0 → gr^{p+1}K^{•} → Fil^{p}K^{•}/Fil^{p+2}K^{•} → gr^{p}K^{•} → 0.

7.2.17. With the notation of (7.2.6) and (7.2.12), it is easily seen that

\begin{matrix}Z_{k}(E(K)_{r}^{pq}) = & Im (Z(K)_{k}^{pq} → E(K)_{r}^{pq}) \\ B_{k}(E(K)_{r}^{pq}) = & Im (B(K)_{k}^{pq} → E(K)_{r}^{pq}) \end{matrix}

for every p, q, r, k ∈ ℤ with k > r ≥ 0. We may then define a natural abutment for E(K)_{•}^{••}, as follows. For every p, q ∈ ℤ we set

Z_{∞}(E(K)_{0}^{pq}) := Im(Ker d_{Fil^{p}K}^{p+q} → E(K)_{0}^{pq})
B_{∞}(E(K)_{0}^{pq}) := Im((Fil^{p}K^{p+q} ∩ Im d_{K}^{p+q−1}) → E(K)_{0}^{pq})

as well as

E(K)_{∞}^{n} := H^{n}K^{•}  Fil^{p}E(K)_{∞}^{n} := Im(H^{n}(Fil^{p}K^{•}) → H^{n}(K^{•}))  for every p, n ∈ ℤ.

By inspecting the definitions we find natural isomorphisms

β(K)_{∞}^{pq} : E(K)_{∞}^{pq} := Z_{∞}(E(K)_{0}^{pq})/B_{∞}(E(K)_{0}^{pq}) →^{∼} gr^{p}E(K)_{∞}^{p+q}

and the datum A(K)_{∞}^{••} := (B_{∞}(E(K)_{0}^{pq}), Z_{∞}(E(K)_{0}^{pq}), β(K)_{∞}^{pq}) is the sought abutment. Summing up, we get a well defined functor

Fil.C(𝒜) → Sp.Seq_{0}^{∞}(𝒜) (K^{•}, Fil^{•}K^{•}) ↦ (E(K)_{•}^{••}, A(K)_{∞}^{••}).

Clearly, if Fil^{•}K^{•} is bounded in degree n, then Fil^{•}E(K)_{∞}^{n} is a finite filtration.

Proposition 7.2.18. (i) Let (K^{•}, Fil^{•}K^{•}) be a filtered complex in 𝒜, and N ∈ ℕ such that :

H^{n−1}(K^{•}/Fil^{−p}K^{•}) = 0  and  H^{n+1}(Fil^{p}K^{•}) = 0  for every p ≥ N.

Then the 1-spectral sequence with abutment (E(K)_{•}^{••}, A(K)_{∞}^{••}) is convergent in degree n.

(ii) Especially, the conclusion of (i) holds if Fil^{•}K^{•} is bounded above in degree n − 1 and bounded below in degree n + 1.

Proof. Obviously (i)⇒(ii). Hence, let us assume that the condition of (i) holds, and consider, for every r ∈ ℕ and every p, q ∈ ℤ with p + q = n, the composition

H^{n}(gr^{p}K^{•}) ⟶^{δ} H^{n+1}(Fil^{p+1}K^{•}) ⟶^{H^{n+1}(π_{r})} H^{n+1}(Fil^{p+1}K^{•}/Fil^{p+r}K^{•})

where δ is the boundary morphism attached by the snake lemma to the short exact sequence of complexes 0 → Fil^{p+1}K^{•} → Fil^{p}K^{•} → gr^{p}K^{•} → 0, and π_{r} : Fil^{p+1}K^{•} → Fil^{p+1}K^{•}/Fil^{p+r}K^{•} is the projection. By direct inspection, we see that :

Ker (H^{n+1}(π_{r}) ∘ δ) = Z_{r}(E_{1}^{pq}) and Ker δ = Z_{∞}(E_{1}^{pq}).

On the other hand, the long exact cohomology sequence attached to the short exact sequence of complexes 0 → Fil^{p+r}K^{•} → Fil^{p+1}K^{•} → Fil^{p+1}K^{•}/Fil^{p+r}K^{•} → 0 shows that Ker H^{n+1}(π_{r}) is a quotient of H^{n+1}(Fil^{p+r}K^{•}). Thus, H^{n+1}(π_{r}) is a monomorphism for every r ∈ ℕ such that p + r ≥ N, and in that case we get Z_{r}(E_{1}^{pq}) = Z_{∞}(E_{1}^{pq}). Likewise, consider the composition :

H^{n−1}(Fil^{p−r+1}K^{•}/Fil^{p}K^{•}) ⟶^{H^{n−1}(j_{r})} H^{n−1}(K^{•}/Fil^{p}K^{•}) ⟶^{δ′} H^{n}(gr^{p}K^{•})

where δ′ is the boundary morphism attached by the snake lemma to the short exact sequence 0 → Fil^{p}K^{•} → K^{•} → K^{•}/Fil^{p}K^{•} → 0, and j_{r} : Fil^{p−r+1}K^{•}/Fil^{p}K^{•} → K^{•}/Fil^{p}K^{•} is the natural inclusion. By direct inspection we see that

Im(δ′ ∘ H^{n−1}(j_{r})) = B_{r}(E_{1}^{pq}) and Im δ′ = B_{∞}(E_{1}^{pq}).

On the other hand, the long exact cohomology sequence attached to the short exact sequence 0 → Fil^{p−r+1}K^{•}/Fil^{p}K^{•} → K^{•}/Fil^{p}K^{•} → K^{•}/Fil^{p−r+1}K^{•} → 0 shows that Coker H^{n−1}(j_{r}) is a subobject of H^{n−1}(K^{•}/Fil^{p−r+1}K^{•}). Especially, H^{n−1}(j_{r}) is an epimorphism whenever p − r + 1 ≤ −N, and in that case we get B_{r}(E_{1}^{pq}) = B_{∞}(E_{1}^{pq}). □

7.2.19. The constructions of examples 7.1.15(i), 7.1.16(i) admit filtered counterparts. Namely, suppose that (𝒜 , ⊗, Φ, Ψ) is a tensor abelian category, and let (A, Fil^{•}A) and (B, Fil^{•}B) be any two objects of Fil.𝒜. Suppose that :

• at least one of the filtrations Fil^{•}A and Fil^{•}B is bounded

• or else, 𝒜 is cocomplete. In this situation we define a filtration Fil^{•}(A ⊗ B) on A ⊗ B, by ruling that

Fil^{p}(A ⊗ B) := ∑_{k∈ℤ} Im (Fil^{k}A ⊗ Fil^{p−k}B → A ⊗ B) for every p ∈ ℤ.

In case 𝒜 is cocomplete, clearly, this rule yields a functor

Fil.𝒜 × Fil.𝒜 → Fil.𝒜 .

Next, suppose that 𝒜 is cocomplete, and let (K^{•}, Fil^{•}K^{•}) and (L^{•}, Fil^{•}L^{•}) be any two filtered complexes of 𝒜; we get an induced filtration Fil^{•}K^{•} ⊠L^{•} on the double complex K^{•} ⊠L^{•}, and applying the functor Tot of remark 7.2.11we obtain a filtration on K^{•} ⊗ L^{•}, whence a functor

Fil.C(𝒜) × Fil.C(𝒜) → Fil.C(𝒜) (Fil^{•}K, Fil^{•}L) ↦ Fil^{•}K^{•} ⊗ L^{•}.

Notice that if 𝒜 is not cocomplete, the filtered double complex Fil^{•}K^{•}⊠L^{•} is still well defined, provided at least one of the filtrations Fil^{•}K^{•} and Fil^{•}L^{•} is bounded. If moreover both K^{•} and L^{•} lie in C^{−}(𝒜) or C^{+}(𝒜), then also Fil^{•}K^{•} ⊗ L^{•} is well defined.

7.2.20. Let (A, Fil^{•}A) and (B, Fil^{•}B) be any two objects of Fil.𝒜. We define a filtration Fil^{•}Hom_{𝒜} (A, B) on the abelian group Hom_{𝒜} (A, B), by ruling that

Fil^{p}Hom_{𝒜} (A, B) := {f : A → B | f(Fil^{k}A) ⊂ Fil^{p+k}B for every k ∈ ℤ}

for every p ∈ ℤ. Clearly this rule yields a functor

(7.2.21)  Fil.𝒜 × (Fil.𝒜)^{o} → Fil.ℤ-Mod.

Let now (K^{•}, Fil^{•}K^{•}) and (L^{•}, Fil^{•}L^{•}) be any two filtered complexes of 𝒜; we define a filtration Fil^{•}Hom_{𝒜}^{••}(K^{•}, L^{•}) on the double complex of abelian groups Hom_{𝒜}^{••}(K^{•}, L^{•}) (see example 7.1.15(i)), by ruling that

Fil^{k}Hom_{𝒜}^{p,q}(K^{•}, L^{•}) := Fil^{k}Hom_{𝒜} (K^{−p}, L^{q}) for every p, q, k ∈ ℤ

(where the right-hand side is the filtered abelian group obtained by applying the functor (7.2.21) to the filtered objects (K^{−p}, Fil^{•}K^{−p}) and (L^{q}, Fil^{•}L^{q}) of 𝒜). After applying the functor Tot^{Π} of remark 7.2.11, we deduce a filtration on Hom_{𝒜}^{•} (K^{•}, L^{•}), whence a functor

Fil.C(𝒜) × Fil.C(𝒜)^{o} → Fil.C(ℤ-Mod) (Fil^{•}L^{•}, Fil^{•}K^{•}) ↦ Fil^{•}Hom_{𝒜}^{•} (K^{•}, L^{•}).

Taking into account remark 7.1.15(i), we also see that

Hom_{Fil.C(𝒜)} (K^{•}, L^{•}) = Fil^{0}Hom^{•}(K^{•}, L^{•}) ∩ ker d_{K,L}^{0}.

Definition 7.2.22. Let 𝒜 be any abelian category, k ∈ ℤ any integer, φ^{•}, ψ^{•} : (K^{•}, Fil^{•}K^{•}) → (L^{•}, Fil^{•}L^{•}) any two morphisms of filtered complexes of 𝒜.

(i) A homotopy of order k from φ^{•} to ψ^{•} is an element s^{•} ∈ Fil^{k}Hom_{𝒜}^{−1}(K^{•}, L^{•}) such that d_{K,L}^{−1}(s^{•}) = ψ^{•} − φ^{•} (notation of example 7.1.15(i)).

(ii) We write φ^{•} ∼_{k} ψ^{•} if there exists a homotopy of order k from ψ^{•} to φ^{•}. It is easily seen that if this is the case, and α^{•} : K′^{•} → K^{•}, β^{•} : L^{•} → L′^{•} are any two morphisms, then φ^{•} ∘ α^{•} ∼_{k} ψ^{•} ∘ α^{•} and β^{•} ∘ φ^{•} ∼_{k} β^{•} ∘ ψ^{•}. Moreover, ∼_{k} is an equivalence relation on Hom_{Fil.C(𝒜)} (K^{•}, L^{•}). It follows that there exists a well defined filtered homotopy category of order k

Fil.Hot(𝒜 , k)

whose objects are the same as those of Fil.C(𝒜), and whose morphisms are the order k homotopy classes of morphisms of complexes. Furthermore, we have a natural functor

Fil.C(𝒜) → Fil.Hot(𝒜 , k)

which is the identity on objects, and the quotient map on Hom-sets.

7.2.23. With the notation of definition 7.2.22, suppose that k ≥ 0, and consider any element s^{•} ∈ Fil^{k}Hom_{𝒜}^{•} (K^{•}, L^{•}); then σ := d_{K,L}^{−1}(s^{•}) : (K^{•}, Fil^{•}K^{•}) → (L^{•}, Fil^{•}L^{•}) is a morphism of filtered complexes, and a simple inspection shows that

σ^{p+q}(Z(K)_{r}^{pq}) ⊂ Ker d_{Fil^{p}L}^{p+q} for every p, q, r ∈ ℤ with r ≥ k

so the induced map E(σ)_{r}^{pq} : E(K)_{r}^{pq} → E(L)_{r}^{pq} vanishes for every p, q, r ∈ ℤ with r ≥ k. Consequently, we obtain a commutative diagram of functors

\xymatrix{Fil.C(𝒜) \ar[r] \ar[d] & Sp.Seq_{0}^{∞}(𝒜) \ar[d] \\ Fil.Hot(𝒜 , k) \ar[r] & Sp.Seq_{k}^{∞}(𝒜)}  for every k ≥ 0

whose top horizontal arrow is the functor of (7.2.17), and whoses left (resp. right) vertical arrow is the functor of definition 7.2.22(ii) (resp. the forgetful functor of (7.2.8)).

7.2.24. We show next, how to lift the décalage functor of (7.2.5), to a functor on complexes

Déc : Fil.C(𝒜) → Fil.C(𝒜).

This construction is borrowed from [54]. Namely, given (K^{•}, Fil^{•}K^{•}) as in (7.2.12), we set

D^{•} := K^{•} and Fil^{p}D^{n} := Z(K)_{1}^{p+n,−p} for every p, n ∈ ℤ.

It follows straightforwardly from (7.2.15), that the differential d_{K}^{•} of K^{•} restricts to a morphism Fil^{p}D^{n} → Fil^{p}D^{n+1} for every p, n ∈ ℤ, so (D^{•}, Fil^{•}D^{•}) is an object of Fil.C(𝒜), and we let

Déc (K^{•}, Fil^{•}K^{•}) := (D^{•}, Fil^{•}D^{•}).

Every morphism (7.2.1) induces in the obvious fashion a morphism

Déc (K^{•}, Fil^{•}K^{•}) → Déc (L^{•}, Fil^{•}L^{•})

so Déc is a functor as sought. Moreover, it is easily seen that if Fil^{•}K^{•} is a bounded filtration, the same holds for the filtration of Déc (K^{•}, Fil^{•}K^{•}). Notice that this definition is not self-dual, i.e. it is not preserved after switching to the opposite category 𝒜^{o}.

7.2.25. We wish next to compare the spectral sequences

E(K)_{•}^{••} and E(Déc K)_{•}^{••}

attached, as in (7.2.12), to (K^{•}, Fil^{•}) and respectively to Déc (K^{•}, Fil^{•}K). To this aim, notice that

Z(Déc K)_{0}^{p,n−p} = Z(K)_{1}^{p+n,−p} D(Déc K)_{0}^{p,n−p} = Z(K)_{1}^{p+1+n,−p−1}

and we have

Z(K)_{1}^{p+1+n,−p−1} ⊂ (Fil^{p+1+n}K^{•})^{n} ⊂ D(K)_{1}^{p+n,−p} ⊂ Z(K)_{1}^{p+n,−p}

whence a natural epimorphism

u^{p,n−p} : E(Déc K)_{0}^{p,n−p} → Déc (E(K))_{0}^{p,n−p} for every p, n ∈ ℤ.

With this notation, we have :

Proposition 7.2.26. With the notation of (7.2.24), the following holds :

(i) The system (u^{p,n−p} | p, n ∈ ℤ) extends to an epimorphism in Sp.Seq_{0}(𝒜)

u_{•}^{••} : E(Déc K)_{•}^{••} → Déc (E(K))_{•}^{••}.

(ii) The morphism u^{•} induces isomorphisms in Sp.Seq_{1}(𝒜)

(E(Déc K)_{r}^{••} | r ≥ 1) →^{∼} (Déc (E(K))_{r}^{••} | r ≥ 1).

Proof. For any filtered complex (L^{•}, Fil^{•}L^{•}) of 𝒜, let

L′^{•} := L^{•}  (Fil^{p}L′^{•})^{n} := (Fil^{p−n}L^{•})^{n}  for every p, n ∈ ℤ.

It is easily seen that

(L^{•}, Fil^{•}L^{•})′ := (L′^{•}, Fil^{•}L′^{•})

is a filtered complex of 𝒜, and a direct inspection of the definitions yields the identities

E(L)_{r}^{p,n−p} = E(L′)_{r+1}^{p+n,−p}  d(L)_{r}^{p,n−p} = d(L′)_{r+1}^{p+n,−p}  for every p, n ∈ ℤ and r ∈ ℕ

which add up to an identity in Sp.Seq_{0}(𝒜) :

(7.2.27)  (E(L)_{r}^{••} | r ≥ 0) = Déc (E(L′)_{r}^{••} | r ≥ 1).

Especially, set (M^{•}, Fil^{•}M^{•}) := (Déc (K^{•}, Fil^{•}K^{•}))′. Explicitly, we have

M^{•} = K^{•}  (Fil^{p}M^{•})^{n} = Z(K)_{1}^{p,n−p} ⊂ (Fil^{p}K^{•})^{n}  for every p, n ∈ ℤ

so that the identity morphism 1_{K} : M^{•} → K^{•} is a morphism

(M^{•}, Fil^{•}M^{•}) → (K^{•}, Fil^{•}K^{•}) in Fil.C(𝒜)

which in turns yields a morphism in Sp.Seq_{0}(𝒜) :

E(M)_{•}^{••} → E(K)_{•}^{••}.

Combining with (7.2.27), we get a natural morphism

v_{•}^{••} : E(Déc K)_{•}^{••} → Déc (E(K))_{•}^{••}

and a direct inspection shows that u_{•}^{••} = v_{•}^{••}, whence (i). To check (ii), we remark that

Z(M)_{r}^{pq} = Ker (\overline{d} : Z(K)_{1}^{pq} → Z(K)_{1}^{p,q+1}/Z(K)_{1}^{p+r,q−r+1}) = Z(K)_{r}^{pq} for every r ≥ 1

whence :

B(M)_{r}^{pq} = B(K)_{r}^{pq} for every r ≥ 1

due to lemma (7.2.13). □

7.3. Derived categories and derived functors. Let 𝒜 be any additive category. With the notation of remark 7.1.19(ii), set

ℂ^{•} := Coker (ι_{1}^{•} : ℤ[0] → 𝕂⟨1⟩^{•}).

Explicitly, ℂ^{•} is the object of C^{[−1,0]}(ℤ-Mod) such that ℂ^{−1} = ℂ^{0} = ℤ and with d_{ℂ}^{−1} = 1_{ℤ}. The morphism ι_{0}^{•} induces a short exact sequence of complexes of abelian groups :

0 → ℤ[0] ⟶^{ι^{•}} ℂ^{•} ⟶^{π^{•}} ℤ[1] → 0.

Now, let φ^{•} : K^{•} → L^{•} be any morphism in C(𝒜). We define the cone of φ^{•} as the push-out in the cocartesian diagram of C(𝒜) :

\xymatrix{K^{•} \ar[r]^{ι^{•}⊗_{ℤ}1_{K}} \ar[d]_{φ^{•}} & ℂ^{•} ⊗_{ℤ} K^{•} \ar[d]^{β^{•}} \\ L^{•} \ar[r]^{ψ^{•}} & (Cone φ)^{•}.}

Let also γ^{•} : 𝕂⟨1⟩^{•} → ℂ^{•} be the natural projection; by remark 7.1.19(ii), the composition

β^{•} ∘ (γ^{•} ⊗_{ℤ} K^{•}) : 𝕂⟨1⟩^{•} ⊗_{ℤ} K^{•} → (Cone φ)^{•}

corresponds to a homotopy from ψ^{•} ∘ φ^{•} to β^{•} ∘ (γ^{•} ∘ ι_{1}^{•}) ⊗_{ℤ} K^{•}, and the latter is obviously the zero endomorphism of K^{•}. Moreover, we get a natural identification

(7.3.1)  Coker ψ^{•} →^{∼} Coker (ι^{•} ⊗_{ℤ} K^{•}) →^{∼} K^{•}[1]

and we let ∂^{•} be the unique morphism of complexes which fits in the commutative diagram

\xymatrix{ & (Cone φ)^{•} \ar[dl]_{−π^{•}} \ar[dr]^{∂^{•}} \\ Coker ψ^{•} \ar[rr] &  & K^{•}[1]}

whose horizontal arrow is the identification (7.3.1), and where π^{•} is the natural projection. Summing up, we have attached to φ^{•} a natural sequence of morphisms of complexes

Θ(φ^{•}) : K^{•} ⟶^{φ^{•}} L^{•} ⟶^{ψ^{•}} (Cone φ)^{•} ⟶^{∂^{•}} K^{•}[1]

such that ψ^{•} ∘ φ^{•} is homotopically trivial, and the induced sequence of morphisms in 𝒜

0 → L^{i} ⟶^{ψ^{i}} (Cone φ)^{i} ⟶^{∂^{i}} K^{i+1} → 0

is split exact for every i ∈ ℤ, i.e. such that (Cone φ)^{i} = L^{i} ⊕ K^{i+1}, and ψ_{i} (resp. −∂^{i}) is the natural monomorphism (resp. epimorphism) induced by this direct sum decomposition. A direct inspection shows that the differential

d_{Cone φ}^{i−1} : L^{i−1} ⊕ K^{i} → L^{i} ⊕ K^{i+1}

is given by the matrix

[ \begin{matrix}d_{L}^{i−1} & φ^{i} \\ 0 & −d_{K}^{i} \end{matrix} ] .

Moreover, it is easily seen that every morphism β^{•} : φ_{1}^{•} → φ_{2}^{•} in Morph(C(𝒜)) induces a natural commutative diagram in C(𝒜) :

(7.3.2) \xymatrix{Θ(φ_{1}^{•}) \ar[d]_{Θ(β^{•})} & K_{1}^{•} \ar[r]^{φ_{1}^{•}} \ar[d]_{β_{1}^{•}} & L_{1}^{•} \ar[r]^{ψ_{1}^{•}} \ar[d]_{β_{2}^{•}} & (Cone φ_{1})^{•} \ar[d]_{Cone(β_{1},β_{2})^{•}} \ar[r]^{∂_{1}^{•}} & K_{1}^{•}[1] \ar[d]^{β_{1}^{•}[1]} \\ Θ(φ_{2}^{•}) & K_{2}^{•} \ar[r]^{φ_{2}^{•}} & L_{2}^{•} \ar[r]^{ψ_{2}^{•}} & (Cone φ_{2})^{•} \ar[r]^{∂_{2}^{•}} & K_{2}^{•}[1]}

where ψ_{1}^{•} and ψ_{2}^{•} are the natural morphisms, as in the foregoing (details left to the reader).

Definition 7.3.3. Let 𝒜 be any additive category, and denote by T(𝒜) either of the categories C(𝒜) or Hot(𝒜). With the notation of (7.3) :

(i) We call Θ(φ^{•}) the true triangle associated to the morphism φ^{•}, and ∂^{•} is the boundary morphism of Θ(φ^{•}).

(ii) A triangle of T(𝒜) is any sequence of morphisms of T(𝒜) :

(7.3.4)  A^{•} ⟶^{α^{•}} B^{•} ⟶^{β^{•}} C^{•} ⟶^{γ^{•}} A^{•}[1].

(iii) Let Θ_{i} := (A_{i}^{•} ⟶^{α_{i}^{•}} B_{i}^{•} ⟶^{β_{i}^{•}} C_{i}^{•} ⟶^{γ_{i}^{•}} A_{i}^{•}[1]) for i = 1, 2 be two triangles of T(𝒜). A morphism of triangles Θ_{1} → Θ_{2} is a commutative diagram of morphisms of T(𝒜) :

\xymatrix{A_{1}^{•} \ar[r]^{α_{1}^{•}} \ar[d]_{τ^{•}} & B_{1}^{•} \ar[r]^{β_{1}^{•}} \ar[d] & C_{1}^{•} \ar[r]^{γ_{1}^{•}} \ar[d] & A_{1}^{•}[1] \ar[d]^{τ^{•}[1]} \\ A_{2}^{•} \ar[r]^{α_{2}^{•}} & B_{2}^{•} \ar[r]^{β_{2}^{•}} & C_{2}^{•} \ar[r]^{γ_{2}^{•}} & A_{2}^{•}[1].}

Morphisms of triangles can be composed in the obvious fashion, and clearly the system of all triangles of T(𝒜) and their morphisms, forms a category.

(iv) A distinguished triangle of T(𝒜) is a triangle of T(𝒜) that is isomorphic to a true

triangle. We denote by

Θ.T(𝒜)

the full subcategory of the category of triangles of T(𝒜) whose objects are the distinguished triangles.

(v) Let ℬ be any other additive category. A triangulated functor from T(𝒜) to T(ℬ) is a pair (F, τ) consisting of a functor F : T(𝒜) → T(ℬ) and a natural isomorphism

τ_{K}^{•} : F (K^{•}[1]) →^{∼} (F K^{•})[1] for every K^{•} ∈ Ob(T(𝒜))

such that, for every distinguished triangle (7.3.4) of T(𝒜), the triangle

F A^{•} ⟶^{F α^{•}} F B^{•} ⟶^{F β^{•}} F C^{•} ⟶^{τ_{A}^{•}∘F γ^{•}} F A^{•}[1]

is distinguished in T(ℬ). We also say that F is triangulated if there exists a natural isomorphism τ as above, such that (F, τ) is a triangulated functor.

Remark 7.3.5. (i) With the terminology of definition 7.3.3, we may say that diagram (7.3.2) is a morphism of distinguished triangles, and clearly the rules : φ^{•} ↦ Θ(φ^{•}) for any morphism φ^{•} in C(𝒜) and β^{•} ↦ Θ(β^{•}) for any morphism β^{•} in Morph(C(𝒜)) amount to a functor

(7.3.6)  Morph(C(𝒜)) → Θ.C(𝒜).

Moreover, if φ^{•} is a morphism of C^{+}(𝒜) (resp. C^{−}(𝒜), resp. C^{b}(𝒜)), then clearly (Cone φ)^{•} lies in the same subcategory of C(𝒜), so (7.3.6) restricts to functors

Morph(C^{?}(𝒜)) → Θ.C^{?}(𝒜) with “?” equal to either +, −, or b

with obvious notation.

(ii) Let F : 𝒜 → ℬ be any additive functor between additive categories. Directly from the definitions, it is clear that the induced functors C(F) and Hot(F) are both triangulated.

(iii) Suppose now that 𝒜 is an abelian category, so that the same holds for C(𝒜), and the discussion of (7.3) yields a short exact sequence

0 → L^{•} ⟶^{ψ^{•}} (Cone φ^{•}) ⟶^{∂^{•}} K^{•}[1] → 0 in C(𝒜)

such that the boundary map in degree i of the induced long exact cohomology sequence is none else than φ^{i+1}, so we get a natural acyclic complex

⋯ → H^{i−1}L^{•} ⟶^{ψ^{i}} H^{i−1}(Cone φ)^{•} ⟶^{∂^{i−1}} H^{i}K^{•} ⟶^{φ^{i}} H^{i}L^{•} → ⋯

for every morphism φ^{•} in C(𝒜). However, one of the subtleties of distinguished triangles, is that they provide a language for expressing certain exactness and cohomological assertions, that remains available even for general additive but not necessarily abelian categories. For instance, let us show the following : Proposition 7.3.7. Let 𝒜 be any additive category. For any complex L^{•} of 𝒜, the functors

C(𝒜) → C(ℤ-Mod)  K^{•} ↦ Hom_{𝒜}^{•} (L^{•}, K^{•})
Hot(𝒜) → Hot(ℤ-Mod)  K^{•} ↦ Hot_{𝒜}^{•} (L^{•}, K^{•})

are triangulated (notation of example 7.1.25). Proof. Let φ^{•} : K_{1}^{•} → K_{2}^{•} be any morphism of complexes of 𝒜. The universal properties of the push-out and of the cokernel yield a unique commutative diagram

[figure: Hom_{𝒜}^{•} (L^{•}, Θ(φ^{•})); Hom_{𝒜}^{•} (L^{•}, K_{2}^{•}); Hom_{𝒜}^{•} (L^{•}, (Cone φ^{•})); β^{•}; Hom_{𝒜}^{•} (L^{•}, K_{1}^{•}[1]); γ^{•}; Θ(Hom_{𝒜}^{•} (L^{•}, φ^{•})); Hom_{𝒜}^{•} (L^{•}, K_{2}^{•}); Cone Hom_{𝒜}^{•} (L^{•}, φ^{•}); Hom_{𝒜}^{•} (L^{•}, K_{1}^{•})[1].]

Now, since the sequence 0 → K_{2}^{i} → (Cone φ)^{i} → K_{1}^{i+1} → 0 is split exact in each degree i ∈ ℤ, it is easily seen that the same holds for the induced sequence

0 → Hom_{𝒜}^{i} (L^{•}, K_{2}^{•}) → Hom_{𝒜}^{i} (L^{•}, (Cone φ)^{•}) → Hom_{𝒜}^{i} (L^{•}, K_{1}^{•}[1]) → 0.

It follows that γ^{•} is the obvious natural identification and β^{•} is the direct product of the natural identifications

Hom_{𝒜} (L^{i}, φ^{j} ⊕ φ^{j+1}) →^{∼} Hom_{𝒜} (L^{i}, φ^{j}) ⊕ Hom_{𝒜} (L^{i}, φ^{j+1}) for every i, j ∈ ℤ

whence the assertion for the functor Hom_{𝒜}^{•} . The assertion for Hot_{𝒜}^{•} follows immediately. □

Remark 7.3.8. Resume the situation of (7.1.23). It is easily seen that, mutatis mutandis, the proof of proposition 7.3.7shows also the following. For every K^{•} ∈ Ob(Hot(𝒜)) the functor

Hot^{−}(𝒜′) → Hot^{−}(𝒜′′) L^{•} ↦ B_{−}^{•}(K^{•}, L^{•})

is triangulated. Likewise, the same holds for the functor

Hot^{−}(𝒜) → Hot^{−}(𝒜′′) K^{•} ↦ B_{−}^{•}(K^{•}, L^{•})

for every L^{•} ∈ Ob(Hot(𝒜′)) : details left to the reader.

Lemma 7.3.9. With the notation of (7.3), let φ_{1}^{•}, φ_{2}^{•} : K^{•} → L^{•} be any two morphisms in C(𝒜). The following holds :

(i) Any homotopy s^{•} from φ_{1}^{•} to φ_{2}^{•} induces a natural isomorphism

γ_{s}^{•} : (Cone φ_{1})^{•} →^{∼} (Cone φ_{2})^{•} in C(𝒜)

fitting into a commutative diagram

[figure: K^{•}; φ_{1}^{•}; L^{•}; ψ_{1}^{•}; (Cone φ_{1})^{•}; ∂_{φ_{1}}^{•}; γ_{s}^{•}; K^{•}[1]; K^{•}; φ_{2}^{•}; L^{•}; ψ_{2}^{•}; (Cone φ_{2})^{•}; ∂_{φ_{2}}^{•}; K^{•}[1]]

whose top (resp. bottom) row is Θ(φ_{1}^{•}) (resp. Θ(φ_{2}^{•})). (ii) There is a natural isomorphism

ω^{•} : (Cone ψ_{1})^{•} →^{∼} K^{•}[1] ⊕ (ℂ^{•} ⊗_{ℤ} L^{•}) in C(𝒜)

fitting into a commutative diagram

[figure: L^{•}; ψ_{1}^{•}; (Cone φ_{1})^{•}; τ^{•}; ∂_{φ_{1}}^{•}; (Cone ψ_{1})^{•}; ω^{•}; ∂_{ψ_{1}}^{•}; L^{•}[1]; K^{•}[1]; K^{•}[1] ⊕ (ℂ^{•} ⊗_{ℤ} L^{•}); −p_{1}^{•}; p_{2}^{•}; L^{•}[1]]

whose top row is Θ(ψ_{1}^{•}), and where p_{1}^{•} is the natural projection, and p_{2}^{•} is the morphism given by the matrix

[ \begin{matrix}0 \\ φ_{1}^{i+1} \\ −1_{L^{i+1}} \end{matrix} ] : L^{i} ⊕ K^{i+1} ⊕ L^{i+1} → L^{i+1} for every i ∈ ℤ.

Proof. (i): By assumption, φ_{1}^{i} = φ_{2}^{i} + d^{i−1}s^{i} + s^{i+1}d^{i} for every i ∈ ℤ. We let γ_{s}^{i} be the automorphism of L^{i} ⊕ K^{i+1} given by the matrix

[ \begin{matrix}1_{L^{i}} & s^{i+1} \\ 0 & 1_{K^{i+1}} \end{matrix} ] for every i ∈ ℤ.

A direct computation shows that the system (γ_{s}^{i} | i ∈ ℤ) is the sought isomorphism, and the commutativity of the resulting diagram as in (i) follows by a simple inspection.

(ii): Set σ^{•} := (ι^{•} ⊗_{ℤ} 1_{L}) ∘ φ_{1}^{•}. By definition, we have a natural identification

(Cone σ)^{•} = (Cone ψ_{1})^{•}

as well as a commutative diagram

[figure: K^{•}; φ_{1}^{•}; ι^{•}⊗_{ℤ}1_{K}; L^{•}; ι^{•}⊗_{ℤ}1_{L}; ψ_{1}^{•}; ℂ^{•} ⊗_{ℤ} L^{•}; ℂ^{•} ⊗_{ℤ} K^{•}; (Cone φ_{1})^{•}; τ^{•}; ∂_{φ_{1}}^{•}; (Cone ψ_{1})^{•}; ∂_{σ}^{•}; K^{•}[1]; −1_{K[1]}; K^{•}[1]; K^{•}[1]]

whose two square subdiagrams on the top ladder are cocartesian. Notice that ℂ^{•} is homotopically trivial (details left to the reader); by remark 7.1.19(iv), the same then holds for ℂ^{•} ⊗_{ℤ} L^{•}. Thus, σ^{•} is null-homotopic. In light of (i) and (7.3.1), we deduce an isomorphism as sought, fitting into a commutative diagram

[figure: K^{•}; ℂ^{•} ⊗_{ℤ} L^{•}; (Cone ψ_{1})^{•}; ω^{•}; ∂_{σ}^{•}; K^{•}[1]; K^{•}; 0; ℂ^{•} ⊗_{ℤ} L^{•}; K^{•}[1] ⊕ (ℂ^{•} ⊗_{ℤ} L^{•}); ∂_{0}^{•}; K^{•}[1]]

whose bottom row is the true triangle associated to the zero morphism K^{•} → ℂ^{•} ⊗_{ℤ} L^{•}. Explicitly, the differentials

d_{(Cone ψ_{1})}^{i}, d_{K[1]⊕(ℂ⊗_{ℤ}L)}^{i} : L^{i} ⊕ K^{i+1} ⊕ L^{i+1} → L^{i+1} ⊕ K^{i+2} ⊕ L^{i+2}

are given by the matrices

[ \begin{matrix}d_{L}^{i} & φ_{1}^{i+1} & 1_{L^{i+1}} \\ 0 & −d_{K}^{i+1} & 0 \\ 0 & 0 & −d_{L}^{i+1} \end{matrix} ]  [ \begin{matrix}d_{L}^{i} & 0 & 1_{L^{i+1}} \\ 0 & −d_{K}^{i+1} & 0 \\ 0 & 0 & −d_{L}^{i+1} \end{matrix} ]  for every i ∈ ℤ

from which we see that we can take for ω^{i} the automorphism of L^{i} ⊕ K^{i+1} ⊕ L^{i+1} given by the matrix

[ \begin{matrix}1_{L^{i}} & 0 & 0 \\ 0 & 1_{K^{i+1}} & 0 \\ 0 & φ_{1}^{i+1} & 1_{L^{i+1}} \end{matrix} ]  for every i ∈ ℤ

and then the commutativity of the right square subdiagram of (ii) is immediate. Lastly, it is easily seen that ∂_{0}^{•} = −p_{1}^{•}, therefore

−p_{1}^{•} ∘ ω^{•} ∘ τ^{•} = ∂_{σ}^{•} ∘ τ^{•} = ∂_{φ_{1}}^{•}

which shows the commutativity of the left square subdiagram of (ii). □

Remark 7.3.10. (i) In the situation of lemma 7.3.9, notice that the diagram

[figure: K^{•}; φ_{1}^{•}; L^{•}; ψ_{1}^{•}; (Cone φ_{1})^{•}; ∂_{φ_{1}}^{•}; γ_{s}^{•}; K^{•}[1]; K^{•}; φ_{2}^{•}; L^{•}; ψ_{2}^{•}; (Cone φ_{2})^{•}; ∂_{φ_{2}}^{•}; K^{•}[1]]

commutes in Hot(𝒜), and amounts to a natural isomorphism

Θ(φ_{1}^{•}) →^{∼} Θ(φ_{2}^{•}) in Θ.Hot(𝒜) .

(ii) Let Θ := (A^{•} ⟶^{α^{•}} B^{•} ⟶^{β^{•}} C^{•} ⟶^{γ^{•}} A^{•}[1]) be any triangle of C(𝒜) or Hot(𝒜). We obtain a new triangle by setting

Θ[1] := (B^{•} ⟶^{β^{•}} C^{•} ⟶^{γ^{•}} A^{•}[1] ⟶^{−α^{•}[1]} B^{•}[1])

and clearly, the rule Θ ↦ Θ[1] yields an endofunctor of the category of triangles. The change of sign on the last arrow is needed to ensure that this operator restricts to an endofunctor of the subcategory of distinguished triangles, at least up to homotopy. Indeed, we may state :

Proposition 7.3.11. With the notation of remark 7.3.10(ii), suppose that Θ is a distinguished triangle of Hot(𝒜). Then the same holds for Θ[1], so we have an automorphism

Θ.Hot(𝒜) →^{∼} Θ.Hot(𝒜) Θ ↦ Θ[1].

Proof. We may assume that Θ = Θ(α^{•}) for some morphism α^{•} : A^{•} → B^{•} of complexes of 𝒜, in which case C^{•} = (Cone α)^{•}, and it suffices to show the more precise :

Claim 7.3.12. There is a natural isomorphism

Θ(β^{•}) →^{∼} Θ(α^{•})[1] in Θ.Hot(𝒜) .

Proof of the claim. Indeed, notice that the natural projection p_{1}^{•} : A^{•}[1] ⊕ (ℂ^{•} ⊗_{ℤ} B^{•}) → A^{•}[1] represents an isomorphism in Hot(𝒜), whose inverse is the class of the natural monomorphism e_{1}^{•} : A^{•}[1] → A^{•}[1] ⊕ (ℂ^{•} ⊗_{ℤ} B^{•}). Taking into account lemma 7.3.9(ii), we then get the commutative diagram in Hot(𝒜) :

[figure: B^{•}; β^{•}; (Cone α)^{•}; (Cone β)^{•}; p_{1}^{•}∘ω^{•}; ∂_{β}^{•}; B^{•}[1]; B^{•}; β^{•}; (Cone α)^{•}; ∂_{α}^{•}; A^{•}[1]; −α^{•}; B^{•}[1]]

where ω^{•} : (Cone β)^{•} →^{∼} A^{•}[1] ⊕ (ℂ^{•} ⊗_{ℤ} B^{•}) is the isomorphism provided by lemma 7.3.9(ii). The claim follows. □

Definition 7.3.13. Let 𝒜 be any abelian category.

(i) A quasi-isomorphism is a morphism φ^{•} : K^{•} → L^{•} in C(𝒜) such that H^{i}φ^{•} is an isomorphism for every i ∈ ℤ.

(ii) With the notation of (7.3), we let Σ be the subset of Morph(C(𝒜)) consisting of all quasi-isomorphisms. We set

D(𝒜) := C(𝒜)[Σ^{−1}]

(see remark 1.6.10(i)) and we call this category the derived category of 𝒜.

(iii) More generally, if I ⊂ ℤ is any interval as in (7.1), and a ∈ ℤ is any integer, we denote

D^{I}(𝒜)  D^{≥a}(𝒜)  D^{≤a}(𝒜)  D^{+}(𝒜)  D^{−}(𝒜)  D^{b}(𝒜)

the essential image in D(𝒜) of the categories C^{I}(𝒜), respectively C^{≥a}(𝒜), C^{≤a}(𝒜), C^{+}(𝒜), C^{−}(𝒜), C^{b}(𝒜).

Remark 7.3.14. (i) Notice that, if φ^{•} is a quasi-isomorphism in C(𝒜), and ψ^{•} is any other morphism that is homotopy equivalent to φ^{•}, then ψ^{•} is a quasi-isomorphism as well (remark 7.1.9(ii)). We may then say that a morphism φ^{•} in Hot(𝒜) is a quasi-isomorphism, if it admits a representative in C(𝒜) which is a quasi-isomorphism; by the foregoing, this property can be checked on any representative for the homotopy class φ^{•}.

(ii) In the same vein, notice that, by virtue of proposition 7.1.21, the localization functor C^{?}(𝒜) → D^{?}(𝒜) factors uniquely through a functor

(7.3.15)  Hot^{?}(𝒜) → D^{?}(𝒜)

whenever “?” equals either +, −, b, or any interval I ⊂ ℤ.

(iii) In light of remark 7.1.9(iii), we see that the isomorphism of remark 7.1.6(iii) identifies the subset Σ^{o} ⊂ Ob(C(𝒜)^{o}) with the system of quasi-isomorphisms of C(𝒜^{o}). Combining with remark 1.6.10(ii) we get natural isomorphisms of categories :

D^{I}(𝒜)^{o} →^{∼} D^{−I}(𝒜^{o})  D^{+}(𝒜)^{o} →^{∼} D^{−}(𝒜^{o})  D^{b}(𝒜)^{o} →^{∼} D^{b}(𝒜^{o})

where I ⊂ ℤ is any interval, and we set −I := {−a | a ∈ I}.

(iv) Notice that the brutal truncation functors of (7.1.1) do not transform quasi-isomorphism into quasi-isomorphisms, hence they do not descend to the derived category. On the other hand, the normalized truncation functors do preserve quasi-isomorphisms, so they yield functors

τ^{≥a} : D(𝒜) → D^{≥a}(𝒜)  and τ^{≤a} : D(𝒜) → D^{≤a}(𝒜).  for every a ∈ ℤ

Moreover, let i^{≥a} : C^{≥a}(𝒜) → C(𝒜) and j^{≥a} : D^{≥a}(𝒜) → D(𝒜) be the inclusion functors; so the counit of the adjoint pair (τ^{≥a}, i^{≥a}) is just the identity endofunctor 1_{C^{≥a}(𝒜)} , and it is easily seen that the unit of adjunction 1_{C(𝒜)} ⇒ i^{≥a} ∘ τ^{≥a} induces a natural transformation η : 1_{D(𝒜)} ⇒ j^{≥a} ∘ τ^{≥a}. Then, the triangular identities for the adjunction (τ^{≥a}, i^{≥a}) imply corresponding triangular identities for η and the identity endofunctor 1_{D^{≥a}(𝒜)} ; by proposition 1.1.15(i), we conclude that τ^{≥a} is left adjoint to j^{≥a}. Likewise, τ^{≤a} is right adjoint to the inclusion functor D^{≤a}(𝒜) → D(𝒜).

(v) Directly from the definition (and the universal property of localization), we see that, for every i ∈ ℤ, the cohomology functor in degree i factors uniquely through a functor

H^{i} : D(𝒜) → 𝒜 .

The set of such functors is not faithful : i.e. two morphisms φ^{•}, ψ^{•} : K^{•} → L^{•} such that H^{i}(φ^{•}) = H^{i}(ψ^{•}) for every i ∈ ℤ do not necessarily coincide.

(vi) Moreover, the shift operators clearly descend to functors on the derived category :

D(𝒜) → D(𝒜)  A^{•} ↦ A^{•}[n]  for every n ∈ ℕ.

Hence, we have a category of distinguished triangles Θ.D^{?}(𝒜) as in definition 7.3.3(iv), and a notion of triangulated functor D^{?}(𝒜) → D^{?}(ℬ) as in definition 7.3.3(v), for “?” equal to +, −, b or ℤ, and for any pair of abelian categories 𝒜, ℬ.

Theorem 7.3.16. Let 𝒜 be any abelian category, and denote by

Σ_{?} ⊂ Morph(Hot^{?}(𝒜)) with “?” equal to either +, −, b, or any interval I ⊂ ℤ

the subset of all quasi-isomorphisms. We have :

(i) The functor (7.3.15) factors uniquely through an equivalence :

Hot^{?}(𝒜)[Σ_{?}^{−1}] →^{∼} D^{?}(𝒜).

(ii) The set Σ_{?} admits both a right and a left calculus of fractions.

Proof. To begin with, we point out :

Claim 7.3.17. For any abelian category 𝒜, the following holds :

(iii) Let K^{•}, L^{•} be any two complexes of 𝒜, with L^{•} ∈ Ob(C^{≤a}(𝒜)). If there exists a quasi-isomorphism K^{•} → L^{•}, then the counit of adjunction τ^{≤a}K^{•} → K^{•} is a quasiisomorphism.

(iv) In order to prove the theorem, it suffices to show that Σ admits a right calculus of fractions.

Proof of the claim. (iii) follows immediately from remark 7.3.14(iv) (details left to the reader).

(iv): Indeed, suppose that Σ admits a right calculus of fraction; in view of remark 7.3.14(iii) it follows that Σ admits as well a left calculus of fraction. Next, in case “?” equals ℤ, assertion (i) says that the induced functor Hot(𝒜)[Σ^{−1}] →^{∼} D(𝒜) is an equivalence; the latter follows immediately by comparing the respective universal properties : details left to the reader. So, the theorem is completely proven, for “?” equal to ℤ.

Next, from (iii), the case “?”= ℤ of the theorem, and proposition 1.6.21 we get assertion (i) of the theorem for “?” equal to ]−∞, a], and we also deduce that Σ_{]−∞,a]} admits a right calculus of fractions. We check directly that Σ_{]−∞,a]} admits also a left calculus of fractions. Indeed, (the duals of) axioms (CF1) and (CF2) are obviously fulfilled. For (CF3), let us consider any pair of morphisms f^{•} : B^{•} → A^{•} and s^{•} : B^{•} → C^{•} in Hot^{]−∞,a]}(𝒜), with s^{•} a quasi-isomorphism; since (the dual of) (CF3) is already known to hold for Σ, we may find morphisms t : A → D and g : C → D in Hot(𝒜), such that t ∘ f = g ∘ s, and where t is a quasi-isomorphism. But then, by remark 7.3.14(iv), we may replace D (resp. t, resp. g) by τ^{≤a}D (resp. by τ^{≤a}t, resp. by τ^{≤a}g), after which we may assume that t ∈ Σ_{]−∞,a]}, whence (CF3). The proof of (CF4) is similar, and shall be left to the reader.

Dualizing the foregoing case, and taking into account remark 7.3.14(iii), we then get both assertions (i) and (ii) of the theorem also for “?” equal to [a, +∞[, so the proof of theorem is complete for the case when “?” equals any unbounded interval I ⊂ ℤ. One argues likewise to show the theorem in case “?” equals either + or − (details left to the reader). Lastly, for “?” equal to either b or an interval [a, b], it suffices to apply proposition 1.6.21 and (iii) to the fully faithful inclusion functors

Hot^{[a,b]}(𝒜) → Hot^{≥a}(𝒜) Hot^{b}(𝒜) → Hot^{+}(𝒜)

and argue as in the foregoing cases to complete the proof of the claim. ♢

By claim 7.3.17(ii), it remains only to show that Σ admits a right calculus of fractions. To this aim, we check the conditions of definition 1.6.14. (CF1) and (CF2) are obvious. Next, consider two morphisms φ^{•} : A^{•} → B^{•} and ψ^{•} : C^{•} → B^{•} in C^{?}(𝒜), with φ^{•} a quasi-isomorphism. Let β^{•} : B^{•} → (Cone ψ)^{•} be the natural morphism; with the notation of (7.3.2), we have an induced commutative diagram

[figure: A^{•}; β^{•}∘φ^{•}; φ^{•}; (Cone ψ)^{•}; (Cone β ∘ φ)^{•}; γ^{•}; A^{•}[1]; φ^{•}[1]; B^{•}; β^{•}; (Cone ψ)^{•}; (Cone β)^{•}; B^{•}[1]]

where γ^{•} := Cone (φ^{•}, 1)^{•}, and by the 5-lemma, it follows that γ^{•} is a quasi-isomorphism. However, lemma 7.3.9(ii) identifies γ^{•} with a morphism (Cone β ∘ φ)^{•} → C^{•}[1] ⊕ (ℂ^{•} ⊗_{ℤ} B^{•}), whence a commutative diagram

\xymatrix{(Cone β ∘ φ)^{•} \ar[d]_{π^{•}∘γ^{•}} \ar[r] & A^{•}[1] \ar[d]^{φ^{•}[1]} \\ C^{•}[1] \ar[r]^{ψ^{•}[1]} & B^{•}[1]}

where π^{•} : C^{•}[1] ⊕ (ℂ^{•} ⊗_{ℤ} B^{•}) → C^{•}[1] is the natural projection. Since ℂ^{•} ⊗_{ℤ} B^{•} is homotopically trivial (see the proof of lemma 7.3.9(ii)), we see that π^{•} is also a quasi-isomorphism, which shows (CF3). Lastly, let f^{•}, g^{•} : X^{•} → Y^{•} be any two morphisms in Hot^{?}(𝒜), and s^{•} : Y^{•} → Z^{•} a quasi-isomorphism such that s^{•} ∘ f^{•} = s^{•} ∘ g^{•} in Hot^{?}(𝒜). We rewrite the latter condition as s^{•} ∘ (f^{•} − g^{•}) = 0, and we notice the exact sequence

H^{−1}Hot_{𝒜}^{•} (X^{•}, (Cone s)^{•}) → H^{0}Hot_{𝒜}^{•} (X^{•}, Y^{•}) → H^{0}Hot_{𝒜}^{•} (X^{•}, Z^{•})

induced, via remark 7.3.5(i), by the distinguished triangle Hot_{𝒜}^{•} (X^{•}, Θ(s^{•})) provided by proposition 7.3.7. Taking into account example 7.1.15(i), we deduce that there exists a morphism

h^{•} : X^{•} → (Cone s^{•})[−1] such that ∂_{s}^{•}[−1] ∘ h^{•} = f^{•} − g^{•} in Hot^{?}(𝒜)

where ∂_{s}^{•} : (Cone s)^{•} → Y^{•}[1] is the boundary morphism of the true triangle Θ(s^{•}). Let now ∂_{h}^{•} : (Cone h)^{•} → X^{•}[1] be the boundary morphism of Θ(h^{•}); from proposition 7.3.11 it follows easily that

h^{•} ∘ ∂_{h}^{•}[−1] = 0 in Hot^{?}(𝒜)

whence (f^{•} − g^{•}) ∘ ∂_{h}^{•}[−1] = 0, and to deduce (CF4), it suffices to remark :

Claim 7.3.18. ∂_{h}^{•} is a quasi-isomorphism.

Proof of the claim. By assumption, s^{•} is a quasi-isomorphism; in light of the long exact sequence of remark 7.3.5(i), it follows easily that H^{i}(Cone s)^{•} = 0 for every i ∈ ℤ. By the same token, we see that H^{i}∂_{h}^{•} is an isomorphism for every i ∈ ℤ, which is the claim. □

Corollary 7.3.19. For every abelian category 𝒜, and every interval I ⊂ ℤ, we have : (i) D^{?}(𝒜) is an additive category for “?” equal to either +, −, b or I. (ii) If 𝒜 is complete, and its products are exact, i.e. the functor Lim_{S} : Fun(S, 𝒜) → 𝒜

is exact for every small discrete category S, then the products are representable in D^{I}(𝒜), and the localization functor j^{I} : C^{I}(A) → D^{I}(A) commutes with direct products.

(iii) Dually, if 𝒜 is cocomplete, and its direct sums are exact, i.e. the functor Colim_{S} : Fun(S, 𝒜) → 𝒜 is exact for every small discrete category S, then the direct sums are representable in D^{I}(𝒜), and the localization functor j^{I} commutes with direct sums.

Proof. (ii): In light of remark 7.1.6(vi), it suffices to check that the localization functor

Hot^{I}(𝒜) → D^{I}(𝒜)

commutes with direct products. Hence, consider any family (K_{j}^{•} | j ∈ J) of objects of Hot^{I}(𝒜) indexed by a small set J, set K^{•} := ∏_{j∈J} K_{j}^{•}, the direct product in the category C^{I}(𝒜), and denote by π_{j}^{•} : K^{•} → K_{j}^{•} the canonical projection, for every j ∈ J; we know that K^{•} represents also the direct product of the family K_{•}^{•} in Hot^{I}(𝒜). Hence, let L^{•} be any other object of Hot^{I}(𝒜), and φ^{•}, φ^{′•} : L^{•} → K^{•} any two morphisms in D^{I}(𝒜) such that π_{j}^{•} ∘ φ^{•} = π_{j}^{•} ∘ φ^{′•} for every j ∈ J; we need to check that φ^{•} = φ^{′•}. After composing with a quasi-isomorphism L^{′•} → L^{•} of Hot^{I}(𝒜), we may assume that φ^{•} and φ^{′•} are two morphisms in Hot^{I}(𝒜) (theorem 7.3.16(ii)), and we may find for every j ∈ J a quasi-isomorphism ψ_{J}^{•} : K_{j}^{•} → K_{j}^{′•} in Hot^{I}(𝒜) such that ψ_{j}^{•} ∘ π_{j}^{•} ∘ φ^{•} = ψ_{j}^{•} ∘ π_{j}^{•} ∘ φ^{′•}. However, we have

(7.3.20)  H^{r}K^{•} = ∏_{j∈J} H^{r}K_{j}^{•}  for every r ∈ ℤ

and likewise for K^{′•} := ∏_{j∈J} K_{j}^{′•}, hence ψ^{•} := ∏_{j∈J} ψ_{j}^{•} : K^{•} → K^{′•} is a quasi-isomorphism with ψ^{•} ∘ φ^{•} = ψ^{•} ∘ φ^{′•} in Hot^{I}(𝒜), whence the assertion. Conversely, given a system of morphisms (φ_{j}^{•} : L^{•} → K_{j}^{•} | j ∈ J) in D^{I}(𝒜), we know by theorem 7.3.16(ii) that there exist for every j ∈ J a quasi-isomorphism ψ_{j}^{•} : K_{j}^{•} → K_{j}^{′•} and a morphism φ̃_{j}^{•} : L^{•} → K^{′•} in C^{I}(𝒜) whose class in Hot^{I}(𝒜) represents ψ_{j}^{•} ∘ φ_{j}^{•} in D^{I}(A). The systems (φ̃_{j}^{•} | j ∈ J) and (ψ_{j}^{•} | j ∈ J) yield morphisms φ̃^{•} : L^{•} → K^{′•} := ∏_{j∈J} K_{j}^{′•} and ψ^{•} : K^{•} → K^{′•}, and we come down to checking that ψ^{•} is a quasi-isomorphism. However, under the identification (7.3.20) and the corresponding one for H^{r}K^{′•}, the map H^{r}ψ^{•} equals ∏_{j∈J} H^{r}ψ_{j}^{•}, whence the contention.

(iii) follows from (ii), by duality.

(i): The proof of (ii) already shows that all finite products and coproducts are representable in D^{?}(𝒜), and obviously the zero complex is a zero object for D^{?}(𝒜). It remains to check that D(𝒜) is pre-additive. More precisely, there exists a unique pre-additive structure on D^{?}(𝒜) such that the localization functor Hot^{?}(𝒜) → D^{?}(𝒜) is additive. Indeed, given any pair of morphisms φ^{•}, φ^{′•} : K^{•} → L^{•} in D^{?}(𝒜), we may find a quasi-isomorphism ψ^{•} : L^{•} → L^{′•} in Hot^{?}(𝒜) such that ψ^{•} ∘ φ^{•} and ψ^{•} ∘ φ^{′•} are represented by morphisms β^{•}, β^{′•} : K^{•} → L^{′•} of Hot^{?}(𝒜) (theorem 7.3.16(ii)), and we set φ^{•} + φ^{′•} := (ψ^{′•})^{−1} ∘ (β^{•} + β^{′•}). It is easily seen that this definition does not depend on the choice of ψ^{•}, and that it provides the required group law on Hom_{D^{?}(𝒜)} (K^{•}, L^{•}) : the details are left to the reader. □

7.3.21. Let now 𝒜 be an abelian category, ℬ an additive subcategory of 𝒜, and set

Σ_{ℬ,+} := Σ ∩ Morph(Hot^{+}(ℬ)) Σ_{ℬ,−} := Σ ∩ Morph(Hot^{−}(ℬ))

where Σ ⊂ Morph(Hot(𝒜)) is the set of all quasi-isomorphisms. We may define the categories

D_{ℬ}^{+}(𝒜) := Hot^{+}(ℬ)[Σ_{ℬ,+}^{−1} ] D_{ℬ}^{−}(𝒜) := Hot^{−}(ℬ)[Σ_{ℬ,−}^{−1} ]

and clearly the inclusion functor ℬ → 𝒜 induces natural functors

i^{+} : D_{ℬ}^{+}(𝒜) → D^{+}(𝒜) i^{−} : D_{ℬ}^{−}(𝒜) → D^{−}(𝒜).

Proposition 7.3.22. In the situation of (7.3.21), suppose additionally that the following holds : (a) ℬ is a full subcategory of 𝒜.

(b) For every A ∈ Ob(𝒜), there exists B ∈ Ob(ℬ) with a monomorphism A → B in 𝒜. Then the functor i^{+} is an equivalence, and Σ_{ℬ,+} admits a left calculus of fractions.

Proof. Set Σ_{+} := Σ ∩ Morph(Hot^{+}(𝒜)). By theorem 7.3.16(i), it suffices to show that the induced functor

Hot^{+}(ℬ)[Σ_{ℬ,+}^{−1} ] → Hot^{+}(𝒜)[Σ_{+}^{−1}]

is an equivalence. In light of remark 1.6.24 and theorem 7.3.16(ii), we are then reduced to checking :

Claim 7.3.23. Suppose that conditions (a) and (b) of the proposition hold. Then, for every K^{•} ∈ C^{+}(𝒜) there exists L^{•} ∈ C^{+}(ℬ) with a quasi-isomorphism K^{•} → L^{•}.

Proof of the claim. Say that K^{•} ∈ C^{≥a}(𝒜) for some a ∈ ℤ; we shall construct first a double complex A^{••} ∈ C^{≥a}(C^{≥−1}(𝒜)) such that

• A^{p,−1} = K^{p} and d_{h}^{p−1,−1} = d_{K}^{p−1} for every p ∈ ℤ.

• A^{pq} ∈ Ob(ℬ) for every p ∈ ℤ and every q ≥ 0.

• For every p ∈ ℤ, the cohomology of the complex (A^{p•}, d_{v}^{p•}) vanishes in every degree. To this aim, we proceed by induction on p. Hence, we let A^{pq} = 0 for every p < a and every q ∈ ℤ. Suppose that p ≥ a, and that A^{ij} is already defined for every i < a and every j ∈ ℤ, as well as all the differentials d_{v}^{ij} and d_{h}^{i−1,j} for every i < a and every j ∈ ℤ. Then we construct A^{pj} and the differentials d_{v}^{p,j−1}, d_{h}^{p−1,j} by induction on j ∈ ℤ. Namely, for j < −1 we set A^{pj} = 0, and for j = −1 we set A^{p,−1} = K^{p} and d_{h}^{p−1,−1} = d_{K}^{p−1}. Thus, suppose that j ≥ 0, and A^{p,j−1} is already given, as well as d_{h}^{p−1,j−1} and d_{h}^{p,j−2}; we define the object B^{pj} of 𝒜 as the coproduct in the push-out diagram

\xymatrix{A^{p−1,j} \ar[r]^{g} & B^{pj} \\ Coker d_{v}^{p−1,j−2} \ar[u]^{_{\overline{d}_{v}}} \ar[r]^{_{\overline{d}_{h}}} & Coker d_{v}^{p,j−2} \ar[u]_{f}}

where \overline{d}_{v} and \overline{d}_{h} are induced by d_{v}^{p−1,j−1} and respectively d_{h}^{p−1,j−1}. Notice that, by inductive assumption, \overline{d}_{v} is a monomorphism, so the same holds for f (details left to the reader). By assumption (b), we may find a monomorphism h : B^{pq} → C for some C ∈ Ob(ℬ), and we set A^{pq} := C, d_{h}^{p−1,j} := h ∘ g and d_{v}^{p,j−1} := h ∘ f ∘ p, where p : A^{p,j−1} → Coker d_{v}^{p,j−2} is the natural projection. Then it is easily seen that d_{h}^{p−1,j} ∘ d_{v}^{p−1,j−1} = d_{v}^{p,j−1} ∘ d_{h}^{p−1,j−1}, and moreover

Ker d_{v}^{p,j−1}/Im d_{v}^{p,j−2} = 0

as required. Now, let t^{≥0} : C_{2}(𝒜) → C(C^{≥0}(𝒜)) be the brutal truncation functor, and set

B^{••} := t^{≥0}A^{••}  C^{••} := K^{•}[0]  L^{•} := Tot B^{••}.

(Hence, C^{••} is the double complex whose rows C^{•q} are the zero complex, except for q = 0, where it agrees with K^{•}). The differentials d_{v}^{•,−1} of A^{••} induce a morphism f^{•} : K^{•} → L^{•}, and it remains only to check that f^{•} is a quasi-isomorphism. To this aim, notice that the differentials d_{v}^{•,−1} also induce a morphism of double complexes

g^{••} : C^{••} → B^{••}

such that Tot g^{••} = f^{•}. Moreover, we have standard convergent spectral sequences

E_{2}^{pq} := H^{q}(C^{p,•}) ⇒ H^{p+q}Tot C^{••} F_{2}^{pq} := H^{q}(B^{p,•}) ⇒ H^{p+q}Tot B^{••}

and by construction, g^{••} induces an isomorphism of spectral sequences E_{2}^{••} →^{∼} F_{2}^{••}, whence the claim. □

Remark 7.3.24. (i) In view of remark 7.3.14(iii), we see that the dual of proposition 7.3.22 also holds. Namely, in the situation of (7.3.21), suppose that

(a) ℬ is a full subcategory of 𝒜.

(b) For every A ∈ Ob(𝒜), there exists B ∈ Ob(ℬ) with an epimorphism B → A in 𝒜. Then the functor i^{−} is an equivalence, and Σ_{ℬ,−} admits a right calculus of fractions.

(ii) Two special cases of proposition 7.3.22 (and its dual) are especially important. Namely, let ℐ (resp. 𝒫) be the full subcategory of 𝒜 whose objects are the injective (resp. projective) objects of 𝒜. We say that 𝒜 has enough injectives (resp. enough projectives) if ℐ (resp. 𝒫) satisfies condition (b) of proposition 7.3.22 (resp. condition (b) of (i)). When this holds, proposition 7.3.22(and its dual) can be further sharpened; namely, we have :

Theorem 7.3.25. Let 𝒜 be any abelian category, and define the subcategories ℐ and 𝒫 of 𝒜 as in remark 7.3.24(ii). Then the following holds :

(i) Let I^{•} be any object of C^{+}(ℐ). Let also L^{•} be any object of C(𝒜), and α^{•} : I^{•} → L^{•} any quasi-isomorphism. Then there exists a morphism β : L^{•} → I^{•} in C(𝒜) such that

β^{•} ∘ α^{•} = 1_{I^{•}} in Hot(𝒜) .

(ii) If 𝒜 has enough injectives, the inclusion functor ℐ → 𝒜 induces an equivalence

Hot^{+}(ℐ) →^{∼} D^{+}(𝒜).

(iii) Dually, if 𝒜 has enough projectives, the inclusion 𝒫 → 𝒜 induces an equivalence

Hot^{−}(𝒫) →^{∼} D^{−}(𝒜).

Proof. (i): To begin with, we remark :

Claim 7.3.26. In the situation of (i), let also K^{•} be any acyclic complex of 𝒜, and φ^{•} : K^{•} → I^{•} any morphism in C(𝒜). Then φ^{•} is null-homotopic.

Proof of the claim. We need to exhibit a system of morphisms (s^{n} : K^{n} → I^{n−1} | n ∈ ℤ) with

(7.3.27)  φ^{n} = d_{I}^{n−1} ∘ s^{n} + s^{n+1} ∘ d_{K}^{n}

for every n ∈ ℤ. To this aim, say that I^{•} ∈ Ob(C^{≥a}(𝒜)) for some a ∈ ℤ; then we let s^{j} be the zero morphism for every j ≤ a. Next, suppose that j ≥ a, and that s^{j} has already been given, so that (7.3.27) holds with n = j − 1. We compute :

\begin{matrix}(φ^{j} − d_{I}^{j−1} ∘ s^{j}) ∘ d_{K}^{j−1} = & d_{I}^{j−1} ∘ φ^{j−1} − d_{I}^{j−1} ∘ s^{j} ∘ d_{K}^{j−1} \\ = & d_{I}^{j−1} ∘ (φ^{j−1} − s^{j} ∘ d_{K}^{j−1}) \\ = & d_{I}^{j−1} ∘ d_{I}^{j−2} ∘ s^{j−1} \\ = & 0. \end{matrix}

Since Im d_{K}^{j−1} = Ker d_{K}^{j}, it follows that φ^{j} −d_{I}^{j−1} ∘s^{j} factors through a morphism Im d_{K}^{j} → I^{j}, and since I^{j} is injective, the latter can be extended to a morphism s^{j+1} : K^{j+1} → I^{j}. By construction, (7.3.27) holds for n = j, with this choice of s^{j+1}, whence the claim. ♢

Now, under the condition of (i), the complex Cone α^{•} is acyclic (remark 7.3.5(iii)), so the boundary morphism ∂^{•} : Cone α^{•} → I^{•}[1] is null-homotopic, by claim 7.3.26. Let us then fix a homotopy s^{•} from the zero morphism to ∂^{•}; in each degree j ∈ ℤ, the morphism s^{j} is the sum of a morphism s_{L}^{j} : L^{j} → I^{j} and a morphism s_{I}^{j} : I^{j+1} → I^{j}, and the relation

−∂^{j} = d_{I[1]}^{j−1} ∘ s^{j} + s^{j+1} ∘ d_{Cone φ}^{j} for every j ∈ ℤ

translates as the pair of identities :

\begin{matrix}−d_{I}^{j} ∘ s_{L}^{j} + s_{L}^{j+1} ∘ d_{L}^{j} = & 0 \\ −d_{I}^{j} ∘ s_{I}^{j} + s_{L}^{j+1} ∘ α^{i+1} + s_{I}^{j+1} ∘ d_{I}^{i+1} = & 1_{I^{j+1}} \end{matrix}

the first of which says that the system (s_{L}^{j} | j ∈ ℤ) amounts to a morphism of complexes s_{L}^{•} : L^{•} → I^{•}, and the second says that the system (s_{I}^{j} | j ∈ ℤ) yields a homotopy from 1_{I} to s_{L}^{•} ∘ α^{•}. In other words, the morphism β^{•} := s_{L}^{•} will do.

(ii): By proposition 7.3.22, it suffices to check that every quasi-isomorphism in Hot^{+}(ℐ) is an isomorphism. But this follows immediately from (i).

(iii): Clearly the subcategory of injective objects of 𝒜^{o} is 𝒫^{o}, so the assertion follows from (ii) and remark 7.3.14(iii). □

Corollary 7.3.28. With the notation of theorem 7.3.25, the natural map

Hot_{𝒜} (K^{•}, I^{•}) → Hom_{D(𝒜)} (K^{•}, I^{•})

is an isomorphism, for every K^{•} ∈ Ob(C(𝒜)) and every I^{•} ∈ Ob(C^{+}(ℐ)).

Proof. By virtue of theorem 7.3.16, every morphism g^{•} : K^{•} → I^{•} in D(𝒜) is represented by a fraction (f^{•}, α^{•}), where f^{•} : K^{•} → L^{•} is any morphism in Hot(𝒜), and α^{•} : I^{•} → L^{•} is a quasi-isomorphism. Then, by theorem 7.3.25(i) we may find a morphism β^{•} : L^{•} → I^{•} in Hot(𝒜) such that β^{•} ∘ α^{•} = 1_{I^{•}}; clearly β^{•} is also a quasi-isomorphism, and g^{•} = β^{•} ∘ f^{•} in D(𝒜), so the map of the corollary is surjective. For the injectivity, suppose that f_{1}^{•}, f_{2}^{•} : K^{•} → I^{•} are any two morphisms in Hot(𝒜) whose images agree in D(𝒜); invoking again theorem 7.3.16, we then find a quasi-isomorphism α^{•} : I^{•} → L^{•} in Hot(𝒜) such that α^{•} ∘ f_{1}^{•} = α^{•} ∘ f_{2}^{•} in Hot(𝒜). Then pick again β^{•} as in the foregoing; we deduce that

f_{1}^{•} = β^{•} ∘ α^{•} ∘ f_{1}^{•} = β^{•} ∘ α^{•} ∘ f_{2}^{•} = f_{2}^{•} in Hot(𝒜)

whence the assertion. □

Definition 7.3.29. Let 𝒜 and ℬ be two abelian categories, and F : C^{+}(𝒜) → D(ℬ) a functor. Denote ω_{𝒜}^{+} : C^{+}(𝒜) → D^{+}(𝒜) and ω_{𝒜}^{−} : C^{−}(𝒜) → D^{−}(𝒜) the localization functors.

(i) A right derived functor of F is a pair (RF, μ) consisting of a functor

RF : D^{+}(𝒜) → D(ℬ)

and a natural transformation

μ : F ⇒ RF ∘ ω_{𝒜}^{+}

which satisfies the following universal property. For every other pair (G, ζ) consisting of a functor G : D^{+}(𝒜) → D(ℬ) and a natural transformation ζ : F ⇒ G ∘ ω_{𝒜}^{+}, there exists a unique natural transformation ξ : RF ⇒ G such that

(7.3.30)  ζ = (ξ ∗ ω_{𝒜}^{+}) ∘ μ.

(ii) Dually, if F : C^{−}(𝒜) → D(ℬ) is any functor, a left derived functor of F is a pair (LF, μ) consisting of a functor

LF : D^{−}(𝒜) → D(ℬ)

and a natural transformation

μ : F ⇒ LF ∘ ω_{𝒜}^{−}

such that ((LF)^{o}, μ^{o}) is a right derived functor of F^{o}, under the natural identifications of remark 7.3.14(iii).

(iii) Especially, let φ : 𝒜 → ℬ be any additive functor, and for “?” equal to + or −, let ω_{ℬ}^{?} : C^{?}(ℬ) → D^{?}(ℬ) be the localization functor. Then a right (resp. left) derived functor of ω_{ℬ}^{+} ∘ C^{+}(φ) (resp. of ω_{ℬ}^{−} ∘ C^{−}(φ)) shall just be called a right (resp. left) derived functor of φ, and we shall just write Rφ (resp. Lφ) to denote this functor.

Remark 7.3.31. Keep the situation of definition 7.3.29.

(i) As usual, if the right derived functor of F exists, it is unique up to isomorphism. More precisely, let (RF, μ) be any pair as in definition 7.3.29(i); if (G, ζ) is any other such pair fulfilling the same universal condition, there exists a unique isomorphism of functors ξ : RF →^{∼} G such that (7.3.30) holds. Likewise one characterizes left functors up to isomorphism.

(ii) For F as in definition 7.3.29(i) (resp. definition 7.3.29(ii)), we shall use, for every p ∈ ℤ and every object K^{•} of D^{+}(𝒜) (resp. of D^{−}(𝒜)), the standard notation :

R^{p}F K^{•} := H^{p}(RF K^{•}) (resp. L^{p}F K^{•} := H^{p}(LF K^{•})).

Also, if φ is as in definition 7.3.29(iii), and the right (resp. left) derived functor of φ exists, then we have a natural transformation

φA → R^{0}φA[0] (resp. L^{0}φA[0] → φA) for every A ∈ Ob(𝒜)

and it is easily seen that this transformation is an isomorphism of functors if and only φ is left exact (resp. right exact : details left to the reader). We say that an object A of 𝒜 is φ-acyclic if R^{p}φA[0] = 0 (resp. L^{p}φA[0] = 0) for every p ≠ 0.

(iii) Suppose that 𝒜 has enough injectives (see remark 7.3.24(ii)), and F : C^{+}(𝒜) → D(ℬ) is a functor that factors through Hot^{+}(𝒜) (via the natural functor (7.1.5)). Then the right derived functor of F exists, and can be constructed as follows. First, say that f^{•} : I^{•} → J^{•} is a quasi-isomorphism, with I^{•}, J^{•} ∈ Ob(C^{+}(ℐ)) (where ℐ is as in remark 7.3.24(ii)). By theorem 7.3.25(i), it follows that f^{•} is a homotopy equivalence, so F f^{•} is an isomorphism by assumption, and therefore F induces a functor

RF_{ℐ} : D_{ℐ}^{+}(𝒜) → D^{+}(ℬ)

(notation of (7.3.21)). However, according to proposition 7.3.22, the natural functor i^{+} : D_{ℐ}^{+}(𝒜) → D^{+}(𝒜) admits a quasi-inverse j^{+} : D^{+}(𝒜) → D_{ℐ}^{+}(𝒜); in view of claim 7.3.23, the functor j^{+} can be described by choosing, for every object K^{•} of C^{+}(𝒜), a quasi-isomorphism

τ_{K}^{•} : K^{•} → I_{K}^{•} with I_{K}^{•} ∈ Ob(C^{+}(ℐ)) .

With this notation, we let

RF := RF_{ℐ} ∘ j^{+}

and we define a natural transformation μ : F ⇒ RF ∘ ω_{𝒜}^{+}, by the rule :

K^{•} ↦ F (τ_{K}^{•}) for every K^{•} ∈ Ob(C^{+}(𝒜)) .

Now, suppose that G : D^{+}(𝒜) → D(ℬ) is another functor, with a natural transformation ζ : F ⇒ G ∘ ω_{𝒜}^{+}. There follows a commutative diagram in D(ℬ)

\xymatrix{F K^{•} \ar[r]^{ζ_{K}^{•}} \ar[d]_{F τ_{K}^{•}} & GK^{•} \ar[d]^{Gτ_{K}^{•}} \\ F I_{K}^{•} \ar[r]^{ζ_{I_{K}}^{•}} & GI_{K}^{•}}

in which Gτ_{K}^{•} is an isomorphism. It follows that (7.3.30)holds if and only if ξ_{K} = (Gτ_{K}^{•})^{−1}∘ζ_{I_{K}}^{•} for every K^{•} ∈ Ob(C^{+}(𝒜)), and therefore (RF, μ) is a derived functor of F, as stated. This is essentially the original construction of the right derived functor proposed by Grothendieck in [84], which predates the invention of derived categories. Dually, one obtains likewise a left derived functor of F, in case 𝒜 has enough projectives : in this case, one starts by fixing, for every bounded above complex K^{•} of 𝒜, a quasi-isomorphism P_{K}^{•} → K^{•} with P^{•} in C^{+}(𝒫).

(iv) A slight generalization of [84, Lemme 3.3.1] yields a construction which works only for additive functors F : 𝒜 → ℬ, but which requires only that 𝒜 has enough F -injective objects. Namely, let us suppose that 𝒜 contains a full additive subcategory ℳ such that :

(a) For every A ∈ Ob(𝒜), there exists a monomorphism A → M, with M ∈ Ob(ℳ).

(b) For every acyclic bounded below complex K^{•} ∈ C^{+}(ℳ), the complex F K^{•} is acyclic. Then we say that ℳ is an F -injective subcategory of 𝒜, and we obtain a right derived functor of F as follows. Let f^{•} : M^{•} → N^{•} be any quasi-isomorphism in C^{+}(ℳ); then Cone f^{•} is acyclic, so the same holds for F (Cone f^{•}), by (b), and since C^{+}(F) is a triangulated functor (remark 7.3.5(ii)), we see that F f^{•} is still a quasi-isomorphism. Thus, C^{+}(F) induces a functor

RF_{ℳ} : D_{ℳ}^{+} (𝒜) → D^{+}(ℬ).

On the other hand, proposition 7.3.22 says that the natural functor i^{+} : D_{ℳ}^{+} (𝒜) → D^{+}(𝒜) admits a quasi-inverse j^{+} : D^{+}(𝒜) → D_{ℳ}^{+} (𝒜), so we may set again

RF := RF_{ℳ} ∘ j^{+}

and one obtains as in (iii) a natural transformation μ, such that the pair (RF, μ) fulfills the required universal property : details left to the reader. This derived functor RF is triangulated (see remark 7.3.14(vi)); moreover, the construction implies that every object of ℳ is F -acyclic.

(v) Dually, we say that 𝒜 has enough F -projective objects (for F as in (iv)) if the category 𝒜^{o} has enough F^{o}-objects, i.e. there exists a full additive subcategory ℳ such that :

(a) For every A ∈ Ob(𝒜), there exists an epimorphism M → A, with M ∈ Ob(ℳ).

(b) For every acyclic bounded above complex K^{•} ∈ C^{−}(ℳ), the complex F K^{•} is acyclic. In this case we say that ℳ is an F -projective subcategory of 𝒜, and by dualizing the constructions of (iv) one obtains a triangulated left derived functor LF; also, every object of ℳ is F -acyclic. This terminology agrees with that of [110, Def.13.3.4] (to check the equivalence of our definition with that of loc.cit. one can argue as in the proof of [110, Prop.13.3.5(ii)]).

(vi) The above quoted [84, Lemme 3.3.1] deals with the special case of (iv), where the additive full subcategory ℳ fulfills condition (a), but (b) is replaced by the more restrictive :

(c) For every M, M′ ∈ Ob(ℳ) and every short exact sequence in 𝒜

Σ  :  0 → M′ → M → M′′ → 0

the object M′′ is also in ℳ, and Σ induces a short exact sequence

F (Σ)  :  0 → F (M′) → F (M) → F (M′′) → 0.

To see that (c)⇒(b), let (M^{•}, d_{M}^{•}) be any acyclic object of C^{+}(ℳ); an easy induction shows that the induced sequences

Σ^{i} : 0 → Im d^{i} → M^{i+1} → Im d^{i+1} → 0

are short exact and Im d^{i} ∈ Ob(ℳ) for every i ∈ ℤ. By assumption, the sequences F (Σ^{i}) are then also short exact, so F (Im d^{i}) = Im(F d^{i}) for every i ∈ ℤ, and F M^{•} is acyclic. Notice also that any short exact sequence 0 → I′ → I → I′′ → 0 in 𝒜 with I′ injective, is split, and I′′ is injective if and only if the same holds for I. Hence, if 𝒜 has enough injectives, the category ℳ := ℐ fulfills conditions (a) and (c) for every additive functor F. Conversely, suppose that ℳ fulfills condition (a), and moreover, that every A ∈ Ob(𝒞) isomorphic to a direct factor of an object of ℳ, is an object of ℳ. Then ℳ contains every injective object of 𝒜. Indeed, if I is an injective object of 𝒞, then by (a) we can find a monomorphism I → M with M in ℳ; hence I is a direct summand of M, so it is in ℳ, by our assumption.

Example 7.3.32. Let 𝒜 be any abelian category with enough injectives; recall that the functor

Hom_{𝒜}^{•} : C(𝒜) × C(𝒜)^{o} → C(ℤ-Mod)

factors through Hot(𝒜) × Hot(𝒜)^{o} (see example 7.1.25); hence, after fixing an equivalence j^{+} : D^{+}(𝒜) → D_{ℐ}^{+}(𝒜) and arguing as in remark 7.3.31(iii), we deduce a functor

RHom_{𝒜}^{•} : D^{+}(𝒜) × Hot(𝒜)^{o} → D(ℤ-Mod)

such that, for every K^{•} ∈ Ob(Hot(𝒜)), the restriction

(7.3.33)  D^{+}(𝒜) → D(ℤ-Mod) : L^{•} ↦ RHom_{𝒜}^{•} (K^{•}, L^{•})

is the derived functor of the functor Hom_{𝒜}^{•} (K^{•}, −) : C^{+}(𝒜) → D(ℤ-Mod). Moreover, (7.3.33) is triangulated, by proposition 7.3.7. Furthermore, let φ^{•} : K_{1}^{•} → K_{2}^{•} be any quasiisomorphism in Hot(𝒜); we claim that RHom_{𝒜}^{•} (φ^{•}, L^{•}) is a quasi-isomorphism, for every L^{•} ∈ Ob(D^{+}(𝒜)). Indeed, set I_{L}^{•} := j^{+}L^{•}; by example 7.1.15(i) and corollary 7.3.28we have natural isomorphisms of abelian groups

H^{n}RHom_{𝒜}^{•} (φ^{•}, L^{•}) →^{∼} Hot_{𝒜} (φ^{•}, I_{L}^{•}[n]) →^{∼} Hom_{D(𝒜)} (φ^{•}, I_{L}^{•}[n])

whence the contention. It follows that RHom_{𝒜}^{•} descends to an additive functor

RHom_{𝒜}^{•} : D^{+}(𝒜) × D(𝒜)^{o} → D(ℤ-Mod)

with a natural isomorphism of abelian groups, for every K^{•} in D(𝒜) and every L^{•} in D^{+}(𝒜) :

(7.3.34)  R^{n}Hom_{𝒜}^{•} (K^{•}, L^{•}) →^{∼} Hom_{D(𝒜)} (K^{•}, L^{•}[n]).

Remark 7.3.35. In the situation of example 7.3.32, our methods do not allows us to extend the functor RHom_{𝒜}^{•} to unbounded complexes in both arguments, but still we can show that every L^{•} ∈ D(𝒜) and every distinguished triangle of D(𝒜)

Θ : K^{•} → K^{′•} → K^{′′•} → K^{•}[1]

induce a long exact Ext-sequence :

Hom_{D(𝒜)} (L^{•}, K^{•})→Hom_{D(𝒜)} (L^{•}, K^{′•})→Hom_{D(𝒜)} (L^{•}, K^{′′•})→Hom_{D(𝒜)} (L^{•}, K^{•}[1])→⋯ Indeed, let L^{′•} → L^{•} be any quasi-isomorphism; we may assume that Θ is already a distinguished triangle of Hot(𝒜), and due to proposition 7.3.7and remark 7.3.5(iii), we get the long exact sequence

Hot_{𝒜} (L^{′•}, K^{•}) → Hot_{𝒜} (L^{′•}, K^{′•}) → Hot_{𝒜} (L^{′•}, K^{′′•}) → Hot_{𝒜} (L^{′•}, K^{•}[1]) → ⋯

and on the other hand, theorem 7.3.16(ii) implies that Hom_{D(𝒜)} (L^{•}, K^{•}) is isomorphic to the colimit of the system (Hot_{𝒜} (L^{′•}, K^{•}) | L^{′•} → L^{•}) indexed by the filtered set of all such quasiisomorphisms (proposition 1.6.16(i)). Since all filtered colimits are exact in the category of abelian groups, the assertion follows. Likewise, from Θ and any L^{•} ∈ Ob(D(𝒜)) we get as well the long exact Ext-sequence :

Hom_{D(𝒜)} (K^{•}[1], L^{•})→Hom_{D(𝒜)} (K^{′′•}, L^{•})→Hom_{D(𝒜)} (K^{′•}, L^{•})→Hom_{D(𝒜)} (K^{•}, L^{•})→⋯

Example 7.3.36. (i) Remark 7.3.31(iv) can be adapted to biadditive functors. Namely, in the situation of (7.1.23), let us say that an object A of 𝒜 (resp. A′ of 𝒜′) is left B-flat (resp. right B-flat) if the additive functor B(A, −) (resp. B(−, A′)) is exact. We assume that :

• 𝒜 has enough left B-flat objects and 𝒜′ has enough right B-flat objects; i.e. for every M ∈ Ob(𝒜) (resp. M′ ∈ Ob(𝒜′)) there exists an epimorphism A → M (resp. A′ → M′), where A (resp. A′) is a left (resp. right) B-flat object of 𝒜 (resp. of 𝒜′).

Let us denote by ℱ (resp. ℱ′) the full subcategory of 𝒜 (resp. of 𝒜′) whose objects are the left B-flat (resp. right B-flat) objects of 𝒜 (of 𝒜′). By remark 7.3.24(i), the natural functors

(7.3.37)  D_{ℱ}^{−}(𝒜) → D^{−}(𝒜)  D_{ℱ′}^{−} (𝒜′) → D^{−}(𝒜′)

are equivalences. On the other hand, let P^{•} be any object of C^{−}(ℱ), and f^{•} : K^{•} → L^{•} any quasi-isomorphism in C^{−}(𝒜′); we wish to show that the induced morphism B_{−}^{•}(P^{•}, f^{•}) is a quasi-isomorphism as well. By remark 7.3.8, it suffices to check that B_{−}^{•}(P^{•}, Cone f^{•}) is acyclic. However, the double complex B^{••}(P^{•}, Cone f^{•}) yields a convergent spectral sequence

E_{1}^{ij} := H^{i}B(P^{j}, Cone f^{•}) ⇒ H^{i+j}B_{−}^{•}(P^{•}, Cone f^{•})

and since P^{j} is left B-flat and Cone f^{•} is acyclic, we get E_{1}^{ij} = 0 for every i, j ∈ ℤ, whence the claim. Likewise, we see that B^{•}(g^{•}, Q^{•}) is a quasi-isomorphism, whenever Q^{•} ∈ Ob(C^{−}(ℱ′)) and g^{•} is any quasi-isomorphism in C^{−}(𝒜). Then, as in remark 7.3.31(iv), after fixing quasiinverse functors for the equivalences (7.3.37), we obtain a functor

LB_{−}^{•} : D^{−}(𝒜) × D^{−}(𝒜′) → D^{−}(𝒜′′)

with a natural transformation

μ : LB_{−}^{•} ∘ (ω_{𝒜}^{−} × ω_{𝒜′}^{−} ) → ω_{𝒜′′}^{−} ∘ B_{−}^{•}

such that, for every K^{•} ∈ Ob(C^{−}(𝒜)), the induced functor

D^{−}(𝒜′) → D^{−}(𝒜′′) : X^{•} ↦ LB^{•}(K^{•}, X^{•})

together with the corresponding restriction of μ, is a left derived functor of B_{−}^{•}(K^{•}, −). Symmetrically, for every K^{•} ∈ Ob(C^{−}(𝒜′)), the functor LB_{−}^{•}(−, K^{•}) provides a left derived functor of B_{−}^{•}(−, K^{•}) : details left to the reader.

(ii) By remark 7.3.31(ii), the following conditions are equivalent : (a) The induced morphism

B^{∗}(A, A′) := L^{0}B^{•}(A[0], A′[0]) → B(A, A′)

is an isomorphism for every A ∈ Ob(𝒜) and every A′ ∈ Ob(𝒜′).

(b) The functor B(A, −) : 𝒜′ → 𝒜′′ is right exact for every A ∈ Ob(𝒜).

(c) The functor B(−, A′) : 𝒜 → 𝒜′′ is right exact for every A′ ∈ Ob(𝒜).

Moreover, B^{∗} is obviously another biadditive functor 𝒜 × 𝒜′ → 𝒜′′, and every right (resp. left) B-flat object of 𝒜 (of 𝒜′) is also a right (resp. left) B^{∗}-flat object. Thus, 𝒜 has enough right B^{∗}-flat objects and 𝒜′ has enough left B^{∗}-flat objects, so we may define as well the left defived functor LB^{∗} of B^{∗}. Furthermore, a simple inspection shows that the functors B^{∗}(A, −) and B^{∗}(−, A′) are right exact, for every A ∈ Ob(𝒜) and A′ ∈ Ob(𝒜′), and the induced morphism

LB^{∗} → LB

is an isomorphism of functors. Lastly, an object P of 𝒜 (resp. P′ of 𝒜′) is right (resp. left) B^{∗}-flat if and only if L^{1}B(P, −) (resp. L^{1}B(−, P′)) is the (constant) zero functor. The detailed verifications of all these assertions shall be left as exercises for the reader.

7.3.38. Let (𝒜 , ⊗, Φ, Ψ) be any abelian tensor category; example 7.3.36 applies especially to the functor ⊗. In this case, it is clear that an object of 𝒜 is left ⊗-flat if and only if it is right ⊗-flat, and such objects shall therefore be called simply ⊗-flat. Thus, suppose that 𝒜 has enough ⊗-flat objects; we conclude that the induced tensor functor for complexes of 𝒜 (see example 7.1.16) admits a left derived functor

− ⊗^{L} − : D^{−}(𝒜) × D^{−}(𝒜) → D^{−}(𝒜)

called the derived tensor product. Explicitly, this is defined as follows : for every bounded above complex K^{•} we fix a quasi-isomorphism ρ_{K}^{•} : P_{K}^{•} → K^{•} with P_{K}^{•} a bounded above complex of ⊗-flat objects, and we set

K^{•} ⊗^{L} L^{•} := P_{K}^{•} ⊗ P_{L}^{•}.

By inspecting the construction, we also see that K^{•} ⊗^{L} L^{•} is naturally isomorphic – via ρ_{K}^{•} and ρ_{L}^{•} – to both K^{•} ⊗ P_{L}^{•} and P_{K}^{•} ⊗ L^{•}. Moreover, if A is any object of 𝒜, clearly the functor A ⊗ − : 𝒜 → 𝒜 is right exact if and only if the same holds for the functor − ⊗ A : 𝒜 → 𝒜, and remark 7.3.31(ii) implies that the latter condition holds if and only if the induced morphism

H^{0}(A[0] ⊗^{L} B[0]) → A ⊗ B

is an isomorphism for every B ∈ Ob(𝒜). We also let

Tor_{i}^{𝒜} (K^{•}, L^{•}) := H_{i}(K^{•} ⊗^{L} L^{•}) for every i ∈ ℤ.

In case 𝒜 = A-Mod for some ring A, it is customary to denote this functor by − ⊗^{L}_{A} −, and then one also writes Tor_{i}^{A} instead of Tor_{i}^{A-Mod}.

Remark 7.3.39. (i) Using the commutativity and associativity constraints for the tensor product in 𝒜, we deduce – in light of example 7.1.16(i,ii) – natural associativity isomorphisms

K^{•} ⊗^{L} (L^{•} ⊗^{L} Q^{•}) →^{∼} (K^{•} ⊗^{L} L^{•}) ⊗^{L} Q^{•} in D^{−}(𝒜)

as well as commutativity isomorphisms

K^{•} ⊗^{L} L^{•} →^{∼} L^{•} ⊗^{L} K^{•} in D^{−}(𝒜)

for any bounded above complexes K^{•}, L^{•}, and Q^{•}.

(ii) In the situation of (7.3.38), take 𝒜 = A-Mod for some ring A, and suppose furthermore that φ : A → B is a ring homomorphism, K^{•} a bounded above complex of A-modules, and L^{•} a complex of B-modules. Notice that P_{K}^{•} ⊗_{A} L^{•} is naturally a complex of B-modules; also, if L^{•} → Q^{•} is any morphism of complexes of B-modules, then the induced map P_{K}^{•} ⊗_{A} L^{•} → P_{K}^{•} ⊗_{A} Q^{•} is B-linear. It follows easily that the derived tensor product yields a functor

D^{−}(A-Mod) × D^{−}(B-Mod) → D^{−}(B-Mod) (K^{•}, L^{•}) ↦ K^{•} ⊗^{L}_{A} L^{•}

such that, denoting φ^{∗} : D^{−}(B-Mod) → D^{−}(A-Mod) the “forgetful” functor, we have a natural isomorphism

K^{•} ⊗^{L}_{A} φ^{∗}L^{•} →^{∼} φ^{∗}(K^{•} ⊗^{L}_{A} L^{•}) in D^{−}(A-Mod).

7.3.40. Let now M_{1}^{•}, M_{2}^{•}, N_{1}^{•}, N_{2}^{•} be any four objects of C^{−}(𝒜), and to ease notation, set

M_{12}^{•} := M_{1}^{•} ⊗ M_{2}^{•}  N_{12}^{•} := N_{1}^{•} ⊗ N_{2}^{•}  P_{12}^{•} := P_{M_{12}}^{•}  ρ_{12}^{•} := ρ_{M_{12}}^{•}

as well as P_{i}^{•} := P_{M_{i}}^{•} and ρ_{i}^{•} := ρ_{M_{i}}^{•} for i = 1, 2. There is a commutative diagram in D^{−}(𝒜)

\xymatrix{P_{1}^{•} ⊗ P_{2}^{•} \ar[rr]^{φ_{12}^{•}} \ar[dr]_{ρ_{1}^{•}⊗ρ_{2}^{•}} &  & P_{12}^{•} \ar[dl]^{ρ_{12}^{•}} \\  & M_{12}^{•}}

where ρ_{12}^{•} is an isomorphism, so φ_{12}^{•} is uniquely determined, whence a map

(M_{1}^{•} ⊗^{L} N_{1}^{•}) ⊗ (M_{2}^{•} ⊗^{L} N_{2}^{•}) →^{∼} (P_{1}^{•} ⊗ P_{2}^{•}) ⊗ N_{12}^{•} ⟶^{φ_{12}^{•}⊗N_{12}^{•}} P_{12}^{•} ⊗ N_{12}^{•} →^{∼} M_{12}^{•} ⊗^{L} N_{12}^{•}.

Taking into account example 7.1.16(iii) and remark 7.3.14(v), we deduce a bilinear pairing

Tor_{i}^{𝒜} (M_{1}^{•}, N_{1}^{•}) ⊗ Tor_{j}^{𝒜} (M_{2}^{•}, N_{2}^{•}) → Tor_{i+j}^{𝒜}(M_{12}^{•}, N_{12}^{•}) for every i, j ∈ ℤ.

Moreover, suppose that M_{3}^{•} and N_{3}^{•} are two other bounded above complexes of 𝒜; by inspecting the constructions, we find a commutative diagram

\xymatrix{P_{1}^{•} ⊗ (P_{2}^{•} ⊗ P_{3}^{•}) \ar[r]^{P_{1}^{•}⊗φ_{23}^{•}} \ar[dr]_{ρ_{1}^{•}⊗(ρ_{2}^{•}⊗ρ_{3}^{•})} \ar[ddd]_{Φ_{P}^{•}} & P_{1}^{•} ⊗ P_{23}^{•} \ar[r]^{φ_{1,23}^{•}} \ar[d]^{ρ_{1}^{•}⊗ρ_{23}^{•}} & P_{1,23}^{•} \ar[dl]^{ρ_{1,23}^{•}} \ar[ddd]^{P_{Φ_{M}}^{•}} \\  & M_{1,23}^{•} \ar[d]^{Φ_{M}^{•}} \\  & M_{12,3}^{•} \\ (P_{1}^{•} ⊗ P_{2}^{•}) ⊗ P_{3}^{•} \ar[r]^{φ_{12}^{•}⊗P_{3}^{•}} \ar[ur]^{(ρ_{1}^{•}⊗ρ_{2}^{•})⊗ρ_{3}^{•}} & P_{12}^{•} ⊗ P_{3}^{•} \ar[r]^{φ_{12,3}^{•}} \ar[u]_{ρ_{12}^{•}⊗ρ_{3}^{•}} & P_{12,3}^{•} \ar[ul]_{ρ_{12,3}^{•}}}

where M_{1,23}^{•} := M_{12}^{•} ⊗ M_{3}^{•}, M_{23}^{•} := M_{2}^{•} ⊗ M_{3}^{•}, M_{1,23}^{•} := M_{1}^{•} ⊗ M_{23}^{•}, and likewise for P_{1,23}^{•}, P_{23}^{•}, and P_{12,3}^{•} and the morphism φ_{23}^{•}, φ_{1,23}^{•}, φ_{12,3}^{•}, ρ_{23}^{•}, ρ_{1,23}^{•}, ρ_{12,3}^{•}. Here Φ_{M}^{•} and Φ_{P}^{•} are the associativity constraints.

Therefore, set T_{i}^{j} := Tor_{i}^{𝒜} (M_{j}^{•}, N_{j}^{•}) for every i ∈ ℤ and j = 1, 2, 3, and also

T_{i}^{jk} := Tor_{i}^{𝒜} (M_{jk}^{•}, N_{jk}^{•})  T_{i}^{1,23} := Tor_{i}^{𝒜} (M_{1,23}^{•}, N_{1,23}^{•})  T_{i}^{12,3} := Tor_{i}^{𝒜} (M_{12,3}^{•}, N_{12,3}^{•})

for every i ∈ ℤ, with j = 1, 2 and k = j + 1; in light of example 7.1.16(iv), we deduce a commutative diagram in 𝒜 :

(7.3.41)  \xymatrix{T_{i}^{1} ⊗ (T_{j}^{2} ⊗ T_{k}^{3}) \ar[r] \ar[d] & T_{i}^{1} ⊗ T_{j+k}^{23} \ar[r] & T_{i+j+k}^{1,23} \ar[d] \\ (T_{i}^{1} ⊗ T_{j}^{2}) ⊗ T_{k}^{3} \ar[r] & T_{i+j}^{12} ⊗ T_{k}^{3} \ar[r] & T_{i+j+k}^{12,3}}

whose horizontal arrows are given by the above bilinear pairing, and whose left (resp. right) vertical arrow is the associativity constraint (resp. is induced by the associativity constraint Φ_{M}^{•}).

Lemma 7.3.42. Let 𝒜 be any abelian category and a, b ∈ ℤ any two integers. We have :

(i) Let K^{•} ∈ Ob(D^{≤b}(𝒜)) and L^{•} ∈ Ob(D^{≥a}(𝒜)) be any two complexes, and suppose that 𝒜 has enough injectives. Then the complex RHom_{𝒜}^{•} (K^{•}, L^{•}) lies in D^{≥a−b}(𝒜). (ii) Suppose that (𝒜 , ⊗, Φ, Ψ) is an abelian tensor category with enough ⊗-flat objects,

and let K^{•} ∈ Ob(D^{≤a}(𝒜)) and L^{•} ∈ Ob(D^{≤b}(𝒜)). then K^{•} ⊗^{L} L^{•} ∈ Ob(D^{≤a+b}(𝒜)).

Proof. (i) follows easily from (7.3.34) and remark 7.3.14(iv) : details left to the reader. Assertion (ii) follows immediately from the construction of the derived tensor product. □

7.3.43. Complexes of modules over a ring. We conclude this section with a discussion of a few special features of the category of complexes of modules over a ring and of its derived category.

7.3.44. Let A be any ring; to ease notation we set

C(A) := C(A-Mod)  Hot(A) := Hot(A-Mod)  D(A) := D(A-Mod)

and likewise we define C^{I}(A), Hot^{I}(A) and D^{I}(A) for any interval I ⊂ ℤ.

Proposition 7.3.45. With the notation of (7.3.44), the following holds :

(i) Let (K_{n}^{•}, φ_{n}^{•} : K_{n}^{•} → K_{n+1}^{•} | n ∈ ℕ) be any direct system of objects of C^{I}(A), and L^{•} its colimit in C^{I}(A). Then L^{•} represents also the colimit of (j^{I}(K_{n}^{•}), j^{I}(φ_{n}^{•}) | n ∈ ℕ) in D^{I}(A).

(ii) Let (K_{n}^{•}, φ_{n}^{•} : K_{n+1}^{•} → K_{n}^{•} | n ∈ ℕ) be any inverse system of complexes of Amodules, and L^{•} its limit in the category C(A). Suppose that for every i ∈ ℤ the inverse system (K_{n}^{i}, φ_{n}^{i} | n ∈ ℕ) satisfies the Mittag-Leffler condition. Then for every object C^{•} of D(A) we have a short exact sequence

0 → lim_{n∈ℕ}^{1} Hom_{D(A)}(C^{•}, K_{n}^{•}[−1]) → Hom_{D(A)}(C^{•}, L^{•}) → lim_{n∈ℕ} Hom_{D(A)}(C^{•}, K_{n}^{•}) → 0.

Proof. (i): We have commutative diagrams in C(A) :

\xymatrix{ & K_{i}^{•} \ar[r]^{ψ_{i}^{•}} \ar[d] & K_{i}^{•} ⊕ K_{i+1}^{•} \ar[d] \\ 0 \ar[r] & ⨁_{n∈ℕ} K_{n}^{•} \ar[r]^{ψ^{•}} & ⨁_{n∈ℕ} K_{n}^{•} \ar[r] & L^{•} \ar[r] & 0}  for every i ∈ ℕ

whose bottom rows are the same short exact sequence for every i ∈ ℕ, and with ψ_{i}^{r}(x) := (x, −φ_{i}^{r}(x)) for every i ∈ ℕ, every r ∈ ℤ and every x ∈ K_{i}^{r}. For every object C^{•} of D(A) set

P (k) := ∏_{n∈ℕ} Hom_{D(A)}(K_{n}^{•}, C^{•}[k])  ψ̃(k) := Hom_{D(A)}(ψ^{•}, C^{•}[k])  for every k ∈ ℤ.

By corollary 7.3.19(ii) and remark 7.3.35, there follows a long exact sequence :

Hom_{D(A)}(L^{•}, C^{•}[k]) → P (k) ⟶^{ψ̃(k)} P (k) → Hom_{D(A)}(L^{•}, C^{•}[k + 1]) → ⋯

and commutative diagrams for every i ∈ ℕ and k ∈ ℤ :

(7.3.46)  \xymatrix{P (k) \ar[r]^{ψ̃(k)} \ar[d] & P (k) \ar[d] \\ Hom_{D(A)}(K_{i}^{•} ⊕ K_{i+1}^{•}, C^{•}[k]) \ar[r] & Hom_{D(A)}(K_{i}^{•}, C^{•}[k])}

whose bottom rows are induced by ψ_{i}^{•}. Thus, Ker ψ̃(0) consists of the systems of morphisms (β_{n}^{•} : K_{n}^{•} → C^{•} | n ∈ ℕ) such that β_{n+1}^{•} ∘ φ_{n}^{•} = β_{n}^{•} for every n ∈ ℕ. In other words, Ker ψ̃(0) is naturally identified with the set of cocones with basis (K_{n}^{•} | n ∈ ℕ) and vertex C^{•} in the category D(A). Moreover, considering (7.3.46) with k = −1, a simple induction on i shows that ψ̃(−1) is surjective (details left to the reader); summing up, we have a natural identification

Hom_{D(A)}(L^{•}, C^{•}) →^{∼} Hom_{D(A)}(colim_{n∈ℕ} j^{I}(K_{n}^{•}), C^{•})

whence the assertion.

(ii): We argue similarly, considering the commutative diagram in C(A) :

\xymatrix{0 \ar[r] & L^{′•} \ar[r] & ∏_{n∈ℕ} K_{n}^{•} \ar[r]^{μ^{•}} \ar[d] & ∏_{n∈ℕ} K_{n}^{•} \ar[r] \ar[d] & 0 \\  &  & K_{i}^{•} ⊕ K_{i+1}^{•} \ar[r]^{μ_{i}^{•}} & K_{i}^{•}}  for every i ∈ ℕ

whose top row is independent of i and with μ_{i}^{r}(x, y) := x − φ_{i}^{r}(y) for every i ∈ ℕ, every r ∈ ℤ and every (x, y) ∈ K_{i}^{r} ⊕ K_{i+1}^{r}. Notice that the top row is a short exact sequence in C(A), since the system (K_{n}^{r} | n ∈ ℕ) satisfies the Mittag-Leffler condition for every r ∈ ℤ. For every k ∈ ℤ and any C^{•} ∈ Ob(D(A)) set

Q(k) := ∏_{n∈ℕ} Hom_{D(A)}(C^{•}, K_{n}^{•}[k])  μ̃(k) := Hom_{D(A)}(C^{•}, μ^{•}[k]).

By corollary 7.3.19(ii) and remark 7.3.35, there follows a long exact sequence

Hom_{D(A)}(C^{•}, L^{′•}[k]) → Q(k) ⟶^{μ̃(k)} Q(k) → Hom_{D(A)}(C^{•}, L^{′•}[k + 1]) →

as well as commutative diagrams as in (7.3.46). Then, a simple inspection yields a natural identification of Ker μ̃(0) with the set of cones in D(A) with basis (K_{n}^{•} | n ∈ ℕ) and vertex C^{•}. Lastly, Coker μ̃(−1) is naturally identified with lim_{n∈ℕ}^{1} Hom_{D(A)}(C^{•}, K_{n}^{•}[−1]), whence the contention. □

7.3.47. Minimal resolutions. Let A be a local ring, k its residue field, and :

⋯ ⟶^{d_{3}} L_{2} ⟶^{d_{2}} L_{1} ⟶^{d_{1}} L_{0} ⟶^{ε} M

a resolution of an A-module M of finite type. We say that (L_{•}, d_{•}, ε) is a finite-free resolution if each L_{i} is a free A-module of finite rank. We say that (L_{•}, d_{•}, ε) is a minimal free resolution of M if it is a finite-free resolution, and moreover the induced maps k ⊗_{A} L_{i} → k ⊗_{A} Im d_{i} are isomorphisms for all i ∈ ℕ (where we let d_{0} := ε). One verifies easily that if A is a coherent ring, then every finitely presented A-module admits a minimal resolution.

7.3.48. Let \underline{L} := (L_{•}, d_{•}, ε) and \underline{L}′ := (L′_{•}, d′_{•}, ε′) be two free resolutions of M. A morphism of resolutions \underline{L} → \underline{L}′ is a map of complexes φ_{•} : (L_{•}, d_{•}) → (L′_{•}, d′_{•}) that extends to a commutative diagram

[figure: L_{•}; ε; φ_{•}; M; L′_{•}; ε′; M]

Lemma 7.3.49. Let \underline{L} := (L_{•}, d_{•}, ε) be a minimal free resolution of an A-module M of finite type, \underline{L}′ := (L′_{•}, d′_{•}, ε′) any other finite-free resolution, φ_{•} : \underline{L}′ → \underline{L} a morphism of resolutions. Then φ_{•} is an epimorphism (in the category of complexes of A-modules), Ker φ_{•} is a null homotopic complex of free A-modules, and there is an isomorphism of complexes :

L′_{•} →^{∼} L_{•} ⊕ Ker φ_{•}.

Proof. Suppose first that \underline{L} = \underline{L}′. We set d_{0} := ε, L_{−1} := M, φ_{−1} := 1_{M} and we show by induction on n that φ_{n} is an isomorphism. Indeed, this holds for n = −1 by definition. Suppose that n ≥ 0 and that the assertion is known for all j < n; by a little diagram chasing (or the five lemma) we deduce that φ_{n−1} induces an automorphism Im d_{n} →^{∼} Im d_{n}, therefore φ_{n} ⊗_{A} 1_{k} : k ⊗_{A} L_{n} → k ⊗_{A} L_{n} is an automorphism (by minimality of \underline{L}), so the same holds for φ_{n} (e.g. by looking at the determinant of φ_{n}).

For the general case, by standard arguments we construct a morphism of resolutions: ψ_{•} : \underline{L} → \underline{L}′. By the foregoing case, φ_{•} ∘ ψ_{•} is an automorphism of \underline{L}, so φ_{•} is necessarily an epimorphism, and \underline{L′} decomposes as claimed. Finally, it is also clear that Ker φ_{•} is an acyclic bounded above complex of free A-modules, hence it is null homotopic. □

Remark 7.3.50. (i) Suppose that A is a coherent local ring, and let \underline{L} := (L_{•}, d_{•}, ε) and \underline{L}′ := (L′_{•}, d′_{•}, ε′) be two minimal resolutions of the finitely presented A-module M. It follows easily from lemma 7.3.49that \underline{L} and \underline{L}′ are isomorphic as resolutions of M.

(ii) Moreover, any two isomorphisms \underline{L} → \underline{L}′ are homotopic, hence the rule: M ↦ L_{•} extends to a functor

A-Mod_{coh} → Hot(A-Mod)

from the category of finitely presented A-modules to the homotopy category of complexes of A-modules.

(iii) The sequence of A-modules (Syz_{A}^{i}M := Im d_{i} | i > 0) is determined uniquely by M (up to non-unique isomorphism). The graded module Syz_{A}^{•}M is sometimes called the syzygy of the module M. Moreover, if \underline{L}′′ is any other finite free resolution of M, then we can choose a morphism of resolutions \underline{L}′′ → \underline{L}, which will be a split epimorphism by lemma 7.3.49, and the submodule d_{•}(L′′_{•}) ⊂ L_{•} decomposes as a direct sum of Syz_{A}^{•}M and a free A-module of finite rank.

Lemma 7.3.51. Let A → B a faithfully flat homomorphism of coherent local rings, M a finitely presented A-module. Then there exists an isomorphism of graded B-modules :

B ⊗_{A} Syz_{A}^{•}M → Syz_{B}^{•}(B ⊗_{A} M).

Proof. Left to the reader. □

Definition 7.3.52. A ring homomorphism A → B is said to be essentially of finite type (resp. essentially of finite presentation) if B is a localization of a finite type (resp. finitely presented) A-algebra; in this case one also says that B is an A-algebra of essentially finite type (resp. an essentially finitely presented A-algebra).

Proposition 7.3.53. Let A → B be a flat and essentially finitely presented local ring homomorphism of local rings, M an A-flat finitely presented B-module. Then the B-module M admits a minimal free resolution

Σ_{•} : ⋯ ⟶^{d_{3}} L_{2} ⟶^{d_{2}} L_{1} ⟶^{d_{1}} L_{0} ⟶^{d_{−1}} L_{−1} := M.

Moreover, Σ is universally A-exact, i.e. for every A-module N, the complex Σ_{•} ⊗_{A} N is still exact.

Proof. To start out, let us notice :

Claim 7.3.54. In order to prove the proposition, it suffices to show that, for every A-flat finitely presented B-module N, and every B-linear surjective map d : L → N, from a free B-module L of finite rank, Ker d is also an A-flat finitely presented B-module.

Proof of the claim. Indeed, in that case, we can build inductively a minimal resolution Σ_{•} of M, such that N_{i} := Ker(d_{i} : L_{i} → L_{i−1}) is an A-flat finitely presented B-module for every i ∈ ℕ. Namely, suppose that a complex Σ_{•}^{(i)} with these properties has already been constructed, up to degree i, and let κ_{B} be the residue field of B; by Nakayama’s lemma, we may find a surjection d_{i+1} : L_{i+1} → N_{i}, where L_{i+1} is a free B-module of rank dim_{κ_{B}}(N_{i} ⊗_{B} κ_{B}). Under the assumption of the claim, the resulting complex Σ_{•}^{(i+1)} : (L_{i+1} → L_{i} → L_{i−1} → ⋯ → M) fulfills the sought conditions, up to degree i + 1.

It is easily seen that the complex Σ_{•} thus obtained shall be universally A-exact. ♢ Let us write A as the union of the filtered family (A_{λ} | λ ∈ Λ) of its noetherian subalgebras.

Say that B = C_{𝔭}, for some finitely presented A-algebra C, and a prime ideal 𝔭 ⊂ C, and M = N_{𝔭} for some finitely presented C-module N. We may find λ ∈ Λ, a finitely generated A_{λ}-algebra C_{λ} and a C_{λ}-module N_{λ} such that C = C_{λ} ⊗_{A_{λ}} A, and N = N_{λ} ⊗_{A_{λ}} A; for every μ ≥ λ, let C_{μ} := A_{μ} ⊗_{A_{λ}} C_{λ} and N_{μ} := A_{μ} ⊗_{A_{λ}} N_{λ}; also, denote by 𝔭_{μ} the preimage of 𝔭 in C_{μ}, and set B_{μ} := (C_{μ})_{𝔭_{μ}}, M_{μ} := (N_{μ})_{𝔭_{μ}}. According to [65, Ch.IV, Cor.11.2.6.1(ii)], we may assume that M_{μ} is a flat A_{μ}-module, for every μ ≥ λ. Moreover, suppose that d : L → M is a B-linear surjection from a free B-module L of rank r; then we may find μ ∈ Λ such that d descends to a B_{μ}-linear surjection d_{μ} : L_{μ} → M_{μ} from a free B_{μ}-module L_{μ} of rank r. It follows easily that K_{μ} := Ker d_{μ} is a flat A_{μ}-module, for every μ ≥ λ, and the induced map K_{μ} ⊗_{B_{μ}} B → K := Ker d is a surjection, whose kernel is a quotient of Tor_{1}^{A_{μ}}(A, M_{μ})_{𝔭}; the latter vanishes, since M_{μ} is A_{μ}-flat. Hence, K is A-flat; furthermore, K_{μ} is clearly a finitely generated B_{μ}-module, hence K is a finitely presented B-module. Then the proposition follows from claim 7.3.54. □

7.4. Simplicial objects. In this section, we introduce the simplicial formalism, which provides the language for the homotopical algebra of section 7.10.

Definition 7.4.1. Let 𝒞 be any category, and k ∈ ℕ any integer.

(i) We denote by Δ the simplicial category, whose objects are the finite ordered sets :

[n] := {0 < 1 < ⋯ < n} for every n ∈ ℕ

and whose morphisms are the non-decreasing functions. (ii) Δ is a full subcategory of the augmented simplicial category Δ^{∧}, whose set of objects is Ob(Δ) ∪ {∅}, with Hom_{Δ^{∧}}(∅, [n]) consisting of the unique mapping of sets ∅ → [n], for every n ∈ ℕ. It is convenient to set [−1] := ∅.

(iii) The augmented k-truncated simplicial category Δ_{k}^{∧}, is the full subcategory of Δ^{∧} whose objects are the elements of Ob(Δ^{∧}) of cardinality ≤ k+1. The k-truncated simplicial category is the full subcategory Δ_{k} of Δ_{k}^{∧} whose set of objects is Ob(Δ_{k}^{∧})\{∅}.

(iv) A simplicial object (resp. an augmented simplicial object, resp. a k-truncated simplicial object, resp. a k-truncated augmented simplicial object) of 𝒞 is a functor Δ^{o} → 𝒞 (resp. (Δ^{∧})^{o} → 𝒞, resp. Δ_{k}^{o} → 𝒞, resp. (Δ_{k}^{∧})^{o}). The morphisms of simplicial objects of 𝒞 are just the natural transformations (and likewise for the truncated or augmented variants). Clearly, these objects form a category, and we use the notation

s.𝒞 := Fun(Δ^{o}, 𝒞)  ŝ.𝒞 := Fun((Δ^{∧})^{o}, 𝒞)
s_{k}.𝒞 := Fun(Δ_{k}^{o}, 𝒞)  ŝ_{k}.𝒞 := Fun((Δ_{k}^{∧})^{o}, 𝒞).

(v) Dually, a cosimplicial object F^{•} of 𝒞 is a functor F : Δ → 𝒞, or – which is the same – a simplicial object in 𝒞^{o}. Likewise one defines the truncated or augmented cosimplicial variants, and we set

c.𝒞 := Fun(Δ, 𝒞)  ĉ.𝒞 := Fun(Δ^{∧}, 𝒞)
c_{k}.𝒞 := Fun(Δ_{k}, 𝒞)  ĉ_{k}.𝒞 := Fun(Δ_{k}^{∧}, 𝒞).

7.4.2. Notice the front-to-back involution :

(7.4.3)  Δ → Δ :  (α : [n] → [m]) ↦ (α^{∨} : [n] → [m]) for every n, m ∈ ℕ

defined as the endofunctor which induces the identity on Ob(Δ), and such that :

α^{∨}(i) := m − α(n − i) for every α ∈ Hom_{Δ}([n], [m]) and every i ∈ [n].

Another construction of interest is the endofunctor

γ : Δ → Δ

given by the rule : [n] ↦ [n + 1] for every n ∈ ℕ, and which takes any morphism α : [n] → [m] of Δ, to the morphism γ(α) : [n + 1] → [m + 1] which is the unique extension of α such that γ(α)(n + 1) := m + 1. Notice that γ restricts to functors γ_{k} : Δ_{k+1} → Δ_{k} for every k ∈ ℕ.

• Given a simplicial object F of 𝒞, one gets a cosimplicial object F^{o} of 𝒞, by the (obvious) rule : (F^{o})[n] := F [n] for every n ∈ ℕ, and F^{o}(α) := F (α)^{o} for every morphism α in Δ.

• Moreover, by composing a simplicial (resp. cosimplicial) object F (resp. G) with the involution (7.4.3), one obtains a simplicial (resp. cosimplicial) object F^{∨} (resp. G^{∨}). Likewise, given a morphism α : F_{1} → F_{2}, the Godement product α^{∨} := α ∗ (7.4.3) is a morphism F_{1}^{∨} → F_{2}^{∨}.

• For F and G as above, we may also consider the simplicial (resp. cosimplicial) object γF := F ∘ γ^{o} (resp. γG := G ∘ γ), and this definition extends again to morphisms, by taking Godement products. The object γF (resp. γG) is called the path space of F (resp. of G). If F is a (k + 1)-truncated simplicial object, then we can consider γ_{k}F := F ∘ γ_{k}^{o}, which is a k-truncated simplicial object (and likewise for truncated cosimplicial objects).

7.4.4. There is an obvious fully faithful functor :

𝒞 → s.𝒞 : A ↦ s.A ( resp. 𝒞 → s_{k}.𝒞 : A ↦ s_{k}.A )

that assigns to each object A of 𝒞 the constant simplicial object s.A (resp. constant truncated simplicial object s_{k}.A) such that s.A[n] := A for every n ∈ ℕ (resp. for every n ≤ k), and s.A(α) := 1_{A} for every morphism α of Δ (resp. of Δ_{k}). Of course, we have as well augmented variants ŝ.A and ŝ_{k}.A, and cosimplicial versions c.A, c_{k}.A, ĉ.A, ĉ_{k}.A.

Moreover, we have, for every integer k ∈ ℕ, the k-truncation functor

s.trunc_{k} : s.𝒞 → s_{k}.𝒞 ( resp. ŝ.trunc_{k} : ŝ.𝒞 → ŝ_{k}.𝒞 )

that assigns to any simplicial (resp. augmented simplicial) object F : Δ^{o} → 𝒞 (resp. F : (Δ^{∧})^{o} → 𝒞) its composition with the inclusion functor Δ_{k}^{o} → Δ^{o} (resp. (Δ_{k}^{∧})^{o} → (Δ^{∧})^{o}).

Again, we have as well the corresponding cosimplicial versions c.trunc_{k} and ĉ.trunc_{k}. Also, for every n ∈ ℕ, we have the functor

•[n] : s.𝒜 → 𝒜 A ↦ A[n].

Lastly, any functor φ : ℬ → 𝒞 induces functors

s.φ : s.ℬ → s.𝒞 s_{k}.φ : s_{k}.ℬ → s_{k}.𝒞 : F ↦ φ ∘ F

and there are of course augmented variants ŝ.φ and ŝ_{k}.φ, as well as the corresponding cosimplicial versions.

Example 7.4.5. (i) For every integer k ≥ −1, we denote by Δ_{k} the simplicial set represented by [k]. Explicitly, Δ_{k} is given by the rule

Δ_{k}[n] := Hom_{Δ}([n], [k]) and Δ_{k}[φ] := Hom_{Δ}(φ, [k])

for every n ∈ ℕ and every morphism φ in Δ. For instance, Δ_{−1} (resp. Δ_{0}) is the constant simplicial set associated to the empty set (resp. to the set with one element); this is also the initial (resp. final) object for s.Set.

(ii) Any morphism φ : [k] → [n] in Δ^{∧} induces a morphism which we shall denote

Δ_{φ} := Hom_{Δ^{∧}}(−, φ) : Δ_{k} → Δ_{n}

given by left composition with φ on Δ_{k}[i], for every i ∈ ℕ.

7.4.6. For given n ∈ ℕ, and every i = 0, …, n, let

ε_{i} : [n − 1] → [n] (resp. η_{i} : [n + 1] → [n])

be the unique injective map in Δ^{∧} whose image misses i (resp. the unique surjective map in Δ with two elements mapping to i). The morphisms ε_{i} (resp. η_{i}) are called face maps (resp. degeneracy maps). By direct inspection, one checks that they fulfill the identities :

ε_{j} ∘ ε_{i} = ε_{i} ∘ ε_{j−1}  if i < j
η_{j} ∘ η_{i} = η_{i} ∘ η_{j+1}  if i ≤ j
η_{j} ∘ ε_{i} = \{ \begin{matrix}ε_{i} ∘ η_{j−1} & if i < j \\ 1 & if i = j or i = j + 1 \\ ε_{i−1} ∘ η_{j} & if i > j + 1. \end{matrix}

Example 7.4.7. (i) For instance, notice the identities :

ε_{i}^{∨} = ε_{n−i}  η_{i}^{∨} = η_{n−i}  for every n ∈ ℕ and every i = 0, …, n.

(ii) For every r, s, i ∈ ℕ with i ≤ r, we set

ε_{r,i}^{s} := ε_{i} ∘ ⋯ ∘ ε_{i} : [r] → [r + s].

This is the injective mapping whose image is {0, …, i − 1, s + i, …, r + s}; for instance, the front-to-back dual

ε_{r,0}^{s∨} := ε_{r+s} ∘ ⋯ ∘ ε_{r+1} : [r] → [r + s]

is just the natural inclusion map. We shall also use the notation

ε_{−1,0}^{s} : [−1] → [s] for every s ∈ ℕ

for the unique morphism ∅ → [s] in Δ^{∧}; of course, we have ε_{−1,0}^{s∨} = ε_{−1,0}^{s} for every s ∈ ℕ.

7.4.8. It is easily seen that every morphism α : [n] → [m] in Δ admits a unique factorization α = ε ∘ η, where the monomorphism ε is uniquely a composition of faces :

ε = ε_{i_{1}} ∘ ⋯ ∘ ε_{i_{s}} with 0 ≤ i_{s} ≤ ⋯ ≤ i_{1} ≤ m

and the epimorphism η is uniquely a composition of degeneracy maps :

η = η_{j_{1}} ∘ ⋯ ∘ η_{j_{t}} with 0 ≤ j_{1} < ⋯ < j_{t} ≤ m

(see [163, Lemma 8.1.2]). It follows that, to give a simplicial object A[•] of a category 𝒞, it suffices to give a sequence of objects (A[n] | n ∈ ℕ) of 𝒞, together with face operators

∂_{i} := A[ε_{i}] : A[n] → A[n − 1] i = 0, …, n

for every integer n > 0 and degeneracy operators

σ_{i} := A[η_{i}] : A[n] → A[n + 1] i = 0, …, n

for every n ∈ ℕ, satisfying the following simplicial identities :

(7.4.9)

∂_{i} ∘ ∂_{j} = ∂_{j−1} ∘ ∂_{i}  if i < j
σ_{i} ∘ σ_{j} = σ_{j+1} ∘ σ_{i}  if i ≤ j
∂_{i} ∘ σ_{j} = \{ \begin{matrix}σ_{j−1} ∘ ∂_{i} & if i < j \\ 1 & if i = j or i = j + 1 \\ σ_{j} ∘ ∂_{i−1} & if i > j + 1. \end{matrix}

Under this correspondence we have ∂_{i} = A(ε_{i}) and σ_{i} = A(η_{i}) ([163, Prop.8.1.3]). Likewise, a k-truncated simplicial object of 𝒞 is the same as the datum of a sequence (A[n] | n = 0, …, k) of objects of 𝒞, and of a system of face and degeneracy operators restricted to this sequence of objects, and fulfilling the same identities (7.4.9).

Dually, a cosimplicial object A[•] of 𝒞 is the same as the datum of a sequence (A[n] | n ∈ ℕ) of objects of 𝒞, together with coface operators

∂^{i} : A[n − 1] → A[n] i = 0, …, n

and codegeneracy operators

σ^{i} : A[n + 1] → A[n] i = 0, …, n

which satisfy the cosimplicial identities :

(7.4.10)

∂^{j} ∘ ∂^{i} = ∂^{i} ∘ ∂^{j−1}  if i < j
σ^{j} ∘ σ^{i} = σ^{i} ∘ σ^{j+1}  if i ≤ j
σ^{j} ∘ ∂^{i} = \{ \begin{matrix}∂^{i} ∘ σ^{j−1} & if i < j \\ 1 & if i = j or i = j + 1 \\ ∂^{i−1} ∘ σ^{j} & if i > j + 1 \end{matrix}

and likewise for k-truncated cosimplicial objects.

7.4.11. An augmented simplicial object of a category 𝒞 can be viewed as the datum of a simplicial object A[•] of 𝒞, together with an object A[−1] ∈ Ob(𝒞), and a morphism ε : A[0] → A[−1], which is an augmentation, i.e. such that :

ε ∘ ∂_{0} = ε ∘ ∂_{1}.

Dually, an augmented cosimplicial object of 𝒞 can be viewed as a cosimplicial object A[•], together with a morphism η : A[−1] → A[0] in 𝒞, such that η^{o} is an augmentation for A^{o}[•]. We say that η is an augmentation for A[•].

Remark 7.4.12. Let A be a simplicial object of the category 𝒞.

(i) For every n ∈ ℕ we have γA[n] := A[n + 1] (notation of (7.4.2)), and the face operators γA[ε_{i}] : γA[n + 1] → γA[n] for i ≤ n + 1 (resp. degeneracy operators γA[η_{i}] : γA[n] → γA[n + 1] for i ≤ n) of γA are ∂_{i} : A[n + 2] → A[n + 1] (resp. σ_{i} : A[n + 1] → A[n + 2]); i.e. we drop ∂_{n+2} and σ_{n+1}. Likewise for the truncated variants.

(ii) The discarded faces ∂_{n+2} and degeneracies σ_{n+1} can be used to produce natural morphisms

s.A[0] ⟶^{f_{A}} γA ⟶^{g_{A}} A.

Namely, we set

f_{A}[n] := σ_{n} ∘ ⋯ ∘ σ_{1}  g_{A}[n] := ∂_{n+1}  for every n ∈ ℕ.

For every k ∈ ℕ, the same operation on an object A ∈ Ob(s_{k+1}.𝒞) yields natural morphisms

s.trunc_{k}A ⟶^{f_{A}} γ_{k}A ⟶^{g_{A}} s.trunc_{k}A.

(iii) Since there is a unique morphism σ_{n,0} : [n] → [0] in Δ for every n ∈ ℕ, it is easily seen that the system (A[σ_{n,0}] : A[0] → A[n]) defines a natural morphism

s.A[0] → A in s.𝒞 .

(iv) Likewise, suppose ε : A[0] → A[−1] is an augmentation for A; since there is exactly one morphism ε_{−1,0}^{n+1} : ∅ → [n] in Δ^{∧} for every n ∈ ℕ, we see that the system (A[ε_{−1,0}^{n+1}] | n ∈ ℕ) defines a natural morphism

A → s.A[−1] in s.𝒞 .

Definition 7.4.13. Let 𝒞 be any category. Denote by

e_{i} : Δ^{o} → [1]/Δ^{o} i = 0, 1

the functor that assigns to each [n] ∈ Ob(Δ) the unique morphism [n] → [1] of Δ whose image is {i}. Let also t : [1]/Δ^{o} → Δ^{o} be the target functor (see (1.1.24)). Let A and B be two simplicial objects of 𝒞, and f, g : A → B two morphisms.

(i) A homotopy from f to g is the datum of a natural transformation

u : A ∘ t ⇒ B ∘ t such that u ∗ e_{0} = f and u ∗ e_{1} = g.

(ii) Dually, if C and D are cosimplicial objects of 𝒞, and p, q : C → D any two morphisms in c.𝒞, then a homotopy from p to q is a homotopy from p^{o} to q^{o} in s.𝒞^{o}.

Remark 7.4.14. (i) In the situation of definition 7.4.13, suppose that p : A′ → A and q : B → B′ are any two morphisms in s.𝒞; then the natural transformation

(q ∗ t) ∘ u ∘ (p ∗ t) : A′ ∘ t ⇒ B′ ∘ t

is a homotopy from q ∘ f ∘ p to q ∘ g ∘ p. Moreover, if F : 𝒞 → 𝒟 is any functor, then

F ∗ u : F A ∗ t ⇒ F B ∗ t

is a homotopy from F f to F g.

(ii) However, unlike the case for chain homotopies, simplicial homotopies cannot be composed in this generality; hence, the simplicial category s.𝒞 cannot be made into a 2-category, by taking the homotopies as 2-cells.

(iii) In the same vein, for a general category 𝒞, the relation :

R(f, g) ⇔ there exists a homotopy from f to g

on the set Morph(s.𝒞) of morphisms of s.𝒞 is neither symmetric nor transitive (though it will follow from theorem 7.4.60 that this is an equivalence relation, in case 𝒞 is abelian). Since t ∘ e_{i} = 1_{Δ^{o}} for i = 0, 1, the natural transformation f ∗ t : A ∘ t ⇒ B ∘ t is a homotopy from f to f, for every morphism f : A → B of s.C, so the relation R is at least reflexive. Let R^{∗} be the minimal equivalence relation on Morph(s.𝒞) such that R(f, g) ⇒ R^{∗}(f, g) for every f, g ∈ Morph(f, g). By virtue of (i), we have R^{∗}(f, g) ⇒ R^{∗}(q ∘ f ∘ p, q ∘ g ∘ p) for every pair of morphisms f, g : A → B and every pair of morphisms p : A′ → A and q : B → B′ in s.𝒞. Hence, there exists a well defined simplicial homotopy category

s.Hot(𝒞)

with Ob(s.Hot(𝒞)) = Ob(s.𝒞), and with morphisms given by the sets :

s.Hot_{𝒞} (A, B) := Hom_{s.𝒞} (A, B)/R^{∗} for every A, B ∈ Ob(s.𝒞)

of R^{∗}-equivalence classes of morphisms A → B in s.𝒞. The composition law for s.Hot(𝒞) is given by the unique system of maps s.Hot_{𝒞} (A, B) × s.Hot_{𝒞} (B, C) → s.Hot_{𝒞} (A, C) such that we get a projection functor :

s.𝒞 → s.Hot(𝒞) A ↦ A

by the rule : f ↦ [f], the class of f in s.Hot_{𝒞} (A, B), for every A, B ∈ Ob(s.𝒞) and every morphism f : A → B. Likewise, by virtue of (i), every functor F : 𝒞 → 𝒟 induces a functor

s.Hot(F) : s.Hot(𝒞) → s.Hot(𝒟)  A ↦ F A  [f] ↦ [F f].

Definition 7.4.15. (i) Let 𝒞 be any category, and A[•] any augmented simplicial object of 𝒞, with augmentation given by a morphism φ : A → s.A[−1] in 𝒞. Then we say that A is homotopically trivial if the class of φ is an isomorphism in s.Hot(𝒞).

(ii) Dually, we say that an augmented cosimplicial object B[•] of 𝒞 is homotopically trivial, if B[•]^{o} is a homotopically trivial object of s.𝒞^{o}.

7.4.16. Notice that there are exactly n + 1 morphisms φ : [n] → [1] for every [n] ∈ Ob(Δ), and they can be labeled by the cardinality of φ^{−1}(0) : for every n ∈ ℕ and every k ≤ n + 1, we shall write φ_{n,k} : [n] → [1] for the unique morphism such that φ_{n,k}^{−1}(0) has cardinality k. With this notation, notice that

φ_{n,k} ∘ ε_{i} = \{ \begin{matrix}φ_{n−1,k} & if i ≥ k \\ φ_{n−1,k−1} & if i < k \end{matrix}  and φ_{n,k} ∘ η_{i} = \{ \begin{matrix}φ_{n+1,k} & if i ≥ k \\ φ_{n+1,k+1} & if i < k. \end{matrix}

Hence, a homotopy u from f to g as in definition 7.4.13, is the same as a system of morphisms

u_{n,k} : A[n] → B[n] for every n ∈ ℕ and every k ≤ n + 1

such that u_{n,n+1} = f[n] and u_{n,0} = g[n] for every n ∈ ℕ, and the diagrams

\xymatrix{A[n] \ar[r]^{u_{n,k}} \ar[d]_{∂_{i}} & B[n] \ar[d]^{∂_{i}} & A[n] \ar[r]^{u_{n,k}} \ar[d]_{σ_{i}} & B[n] \ar[d]^{σ_{i}} \\ A[n − 1] \ar[r]^{u_{n−1,k−a}} & B[n − 1] & A[n + 1] \ar[r]^{u_{n+1,k+a}} & B[n + 1]}

commute for every n ∈ ℕ and every k ≤ n + 1, where a := 0 if i ≥ k, and a := 1 if i < k.

7.4.17. A bisimplicial object in a category 𝒞 is an object of the category s.(s.𝒞). The latter can also be regarded as the category of all functors Δ^{o} × Δ^{o} → 𝒞; it follows that a bisimplicial object of 𝒞 is the same as a system (A[p, q] | (p, q) ∈ ℕ × ℕ) of objects of 𝒞, together with morphisms

A[α, β] : A[p, q] → A[p′, q′] for all morphisms α : [p′] → [p], β : [q′] → [q] of Δ

compatible with compositions of morphisms in Δ, in the obvious way. More generally, we may define inductively the category of n-simplicial objects s^{n}.𝒜, for every n ∈ ℕ, by letting s^{0}.𝒜 := 𝒜, and s^{n}.𝒜 := s.(s^{n−1}.𝒜), for every n > 0. The diagonal functor

Δ → Δ × Δ  [n] ↦ ([n], [n])  α ↦ (α, α)  for all n ∈ ℕ and all morphisms α of Δ

induces a functor

Δ_{𝒞} : s^{2}.𝒞 → s.𝒞 A ↦ A^{Δ}.

Especially, we have A^{Δ}[n] := A[n, n] for every n ∈ ℕ, and the face operators ∂_{i} on A^{Δ}[n] are of the form A[ε_{i}, ε_{i}], for every i = 0, …, n (and likewise for the degeneracies). Also, the flip functor

Δ × Δ → Δ × Δ ([m], [n]) ↦ ([n], [m])

induces an endofunctor

fl : s^{2}.𝒜 → s^{2}.𝒜

in the obvious way. Furthermore, the endofunctors

Δ × Δ ⟶^{γ×1_{Δ}} Δ × Δ ⟵^{1_{Δ}×γ} Δ × Δ

induce functors

s^{2}.𝒜 ⟵^{γ_{1}} s^{2}.𝒜 ⟶^{γ_{2}} s^{2}.𝒜

that admit descriptions as in remark 7.4.12(i). Correspondingly, we get natural morphisms

g_{A}^{(i)} : γ_{i}A → A for i = 1, 2 and every A ∈ Ob(s^{2}.𝒜)

as in remark 7.4.12(ii).

Remark 7.4.18. (i) Let 𝒞 be a category whose finite coproducts are representable. Let also f.Set be the category of finite sets. To every object S of s.f.Set and every X ∈ Ob(s.𝒞), we attach a bisimplicial object S ⊠ X of 𝒞 as follows. For every n, m ∈ ℕ, we let S ⊠ X[n, m] be the coproduct of finitely many copies of X[m], indexed by the elements of S[n]; hence, for every a ∈ S[n] we have a natural morphism i_{a} : X[m] → S ⊠ X[n, m]. If φ : [n] → [n′] and ψ : [m] → [m′] are any two morphisms in Δ^{o}, we let S⊠X[φ, ψ] : S⊠X[n, m] → S⊠X[n′, m′] be the unique morphism such that S ⊠X[φ, ψ]∘i_{a} = i_{S[φ](a)} ∘X[ψ] for every a ∈ S[n]. Clearly, this rules extends to a well defined functor

s.f.Set × s.𝒞 → s^{2}.𝒞 (S, X) ↦ S ⊠ X.

Likewise, we define X ⊠ S := fl(S ⊠ X) (notation of (7.4.17)). If all coproducts of 𝒞 are representable, we may even extend the above construction to arbitrary simplicial sets.

(ii) In the same vein, let 𝒜 be any abelian category, and M any object of s.ℤ-Mod_{fg} (notation of (3.7.47)). For any A ∈ Ob(s.𝒜), we may define a bisimplicial object M ⊠_{ℤ} A of 𝒜, by the rule : [n, m] ↦ M[n] ⊗_{ℤ} A[n] for every n, m ∈ ℕ and [φ, ψ] ↦ M[φ] ⊗_{ℤ} A[ψ] for all morphisms φ, ψ of Δ (where these mixed tensor products are as defined in (3.7.47)). Clearly these rules yield a well defined functor

s.ℤ-Mod_{fg} × s.𝒜 → s^{2}.𝒜 (M, A) ↦ M ⊠_{ℤ} A.

Likewise, we set A ⊠_{ℤ} M := fl(M ⊠_{ℤ} A).

(iii) Furthermore, if (𝒞 , ⊗) is any tensor category, and X, Y any two simplicial objects of 𝒞, we may define a bisimplicial object X ⊠ Y, by the same rule as in (ii). This yields a functor

s.𝒞 × s.𝒞 → s^{2}.𝒞 (X, Y) ↦ X ⊠ Y.

In this situation (resp. in the situation of (i), resp. of (ii)), we shall let also

X ⊗ Y := (X ⊠ Y)^{Δ} (resp. S ⊗ X := (S ⊠ X)^{Δ}, resp. M ⊗_{ℤ} A := (M ⊠_{ℤ} A)^{Δ} ) which we shall call the tensor product of X and Y (resp. of S and X, resp. of M and A). Notice the natural identifications

S ⊠ A →^{∼} (S ⊗ s.ℤ) ⊠_{ℤ} A for every S ∈ Ob(s.f.Set) and every A ∈ Ob(s.𝒜) .

Likewise, if U is any unit object for (𝒞 , ⊗), we get natural identifications

S ⊠ X →^{∼} (S ⊗ s.U) ⊠ X for every S ∈ Ob(s.f.Set) and every X ∈ Ob(s.𝒞)

via the isomorphisms u_{X} : X →^{∼} U ⊗ X provided by proposition 3.7.6. In the same vein, we have a natural identification Δ_{0}⊗X →^{∼} X for every X ∈ Ob(𝒞) (notation of example 7.4.5(i)).

(iv) Notice also that if f, g : S → T are any two morphisms of simplicial finite sets, then – in light of remark 7.4.14(i) – any homotopy u from f to g induces a homotopy u ⊗ X from f ⊗ X to g ⊗ X.

(v) Let f, g : A → B be as in definition 7.4.13. Notice that the datum of a homotopy u from f to g is the same as that of a morphism

ũ : Δ_{1} ⊗ A → B such that ũ ∘ (Δ_{ε_{0}} ⊗ 1_{A}) = f and ũ ∘ (Δ_{ε_{1}} ⊗ 1_{A}) = g

(where ε_{0}, ε_{1} : [0] → [1] are the face maps : notation of example 7.4.5(ii)). Indeed, given u, we construct ũ as follows. For every n ∈ ℕ and every φ ∈ Δ_{1}[n], let ũ[n] be the unique morphism such that ũ[n] ∘ i_{φ} = u_{φ}. The naturality of u easily implies that this rule amounts to a morphism ũ as sought. Conversely, given ũ, we can construct a natural transformation u, by reversing the foregoing rule.

Remark 7.4.19. (i) Let (𝒞 , ⊗) be a tensor category with internal Hom functor ℋ om, and unit object U. To every two objects X, Y of s.𝒞, we may attach the object of 𝒞

ℋ om_{s.𝒞} (X, Y) := \xymatrix{Equal(∏_{n∈ℕ} ℋ om(X[n], Y [n]) \ar[r]^{d_{1}} \ar[r]_{d_{0}} & ∏_{φ:[n]→[m]} ℋ om(X[m], Y [n]))}

where the second product ranges over the morphisms φ of Δ, and where

d_{0} := ∏_{φ:[n]→[m]} ℋ om(X[φ], Y [n]) and d_{1} := ∏_{φ:[n]→[m]} ℋ om(X[m], Y [φ]).

Arguing as in example 7.1.16(vi), it is easily seen that there are natural isomorphisms

Hom_{𝒞} (Z, ℋ om_{s.𝒞} (X, Y)) →^{∼} Hom_{s.𝒞} (s.Z ⊗ X, Y)

for every Z ∈ Ob(𝒞) and every X, Y ∈ Ob(s.𝒞).

(ii) Suppose additionally, that all finite coproducts of 𝒞 are representable. For every Z ∈ Ob(𝒞), and every X ∈ Ob(s.𝒞), consider the simplicial set

Hom_{𝒞} (Z, X)

such that Hom_{𝒞} (Z, X)[n] := Hom_{𝒞} (Z, X[n]) for every n ∈ ℕ, with face and degeneracies deduced from those of X, in the obvious way. For any k ∈ ℕ, let also Δ_{k} be the simplicial set defined in example 7.4.5(i); we have natural isomorphisms

Hom_{𝒞} (Z, ℋ om_{s.𝒞} (Δ_{k} ⊗ s.U, X)) →^{∼} Hom_{s.𝒞} (Δ_{k} ⊗ s.Z, X)
→^{∼} Hom_{s.Set}(Δ_{k}, Hom_{𝒞} (Z, X))
→^{∼} Hom_{𝒞} (Z, X)[k]

where the last isomorphism follows from Yoneda’s lemma (proposition 1.2.6(ii) : details left to the reader). Applying again Yoneda’s lemma, we deduce a natural isomorphism

ℋ om_{s.𝒞} (Δ_{k} ⊗ s.U, X) →^{∼} X[k] for every k ∈ ℕ.

(iii) Notice that Δ_{ψ}∘Δ_{φ} = Δ_{ψ∘φ} for every two morphisms φ : [k] → [k′] and ψ : [k′] → [k′′] of Δ^{∧} (notation of example 7.4.5(ii)). Hence, the system (Δ_{i} | i ∈ ℕ) amounts to an object of c.s.Set. Moreover, since the Yoneda isomorphisms are natural in both X and Δ_{k}, we get a commutative diagram

\xymatrix{ℋ om_{s.𝒞} (Δ_{k′} ⊗ s.U, X) \ar[r]^{∼} \ar[d]_{ℋom_{s.𝒞} (Δ_{φ}⊗s.U,X)} & X[k′] \ar[d]^{X[φ]} \\ ℋ om_{s.𝒞} (Δ_{k} ⊗ s.U, X) \ar[r]^{∼} & X[k]}

for every morphism φ as above.

(iv) Notice as well that the considerations of (ii) and (iii) can be repeated, mutatis mutandis, for truncated simplicial objects : if X and Y are objects of s_{n}.𝒞 and Z is an object of 𝒞, then ℋ om_{s_{n}.𝒞} (X, Y) (resp. Hom_{𝒞} (Z, X)) shall be an object of 𝒞 (resp. of s_{k}.Set), and we shall have natural isomorphisms

ℋ om_{s_{n}.𝒞} (s.trunc_{n}(Δ_{k} ⊗ s.U), X) →^{∼} X[k] for every k ≤ n

and similarly for the commutative diagrams of (iii) (details left to the reader).

7.4.20. Let 𝒞 be a finitely cocomplete category. Theorem 1.3.4 say that, for every integer k ∈ ℕ, the k-truncation functor on s.𝒞 admits a left adjoint

sk_{k} : s_{k}.𝒞 → s.𝒞

which is called the k-th skeleton functor. By inspecting the proof of loc.cit. we see that, for every k-truncated simplicial object F, this adjoint is calculated by the rule :

(7.4.21)  sk_{k}A[n] := colim_{φ:[i]→[n]} F [i]

where i ranges over all the integers ≤ k, and φ over all the morphisms [i] → [n] in Δ^{o}, and the transition maps F [i] → F [j] in the colimit are the morphisms F [ψ] given by all commutative triangles

(7.4.22)  \xymatrix{[i] \ar[rr]^{ψ} \ar[dr]_{φ} &  & [j] \ar[dl]^{φ} \\  & [n]}  in Δ^{o}.

A morphism α : [n] → [m] in Δ^{o} induces a morphism sk_{k}F [α] : sk_{k}A[n] → sk_{k}A[m]; namely, for every φ : [i] → [m] one has a natural morphism j_{φ} : F [i] → sk_{k}F [m], and sk_{k}F [α] is the colimit of the system of morphisms

j_{α∘ψ} : F [i] → sk_{k}A[m] for all ψ : [i] → [n].

It is clear that, for every n, m ≤ k and every α : [n] → [m], the colimit (7.4.21) is realized by F [n], and under this identification, sk_{k}A[α] agrees with F [α], so the unit of adjunction

F → s.trunc_{k} ∘ sk_{k}F

is an isomorphism. Dually, if all finite limits are representable in 𝒞, the truncation functor admits a right adjoint

cosk_{k} : s_{k}.𝒞 → s.𝒞

called the k-th coskeleton functor, and a simple inspection of the proof of loc.cit. yields the rule:

cosk_{k}F [n] := lim_{φ:[n]→[i]} F [i]

where i ranges over the integers ≤ k, and φ : [n] → [i] over the morphisms in Δ^{o}, and the transition maps are as in the foregoing (except that the downwards arrows in the commutative triangles (7.4.22) are reversed). Especially, we easily deduce that the counit of adjunction

s.trunc_{k} ∘ cosk_{k}F → F

is an isomorphism. Moreover, if ℬ is another category with small Hom-sets, whose finite colimits (resp. finite limits) are all representable, and φ : ℬ → 𝒞 is any functor, then for any F ∈ Ob(s_{k}.ℬ) there is a natural transformation

sk_{k}(s_{k}.φF) → s.φ(sk_{k}F) ( resp. s.φ(cosk_{k}F) → cosk_{k}(s_{k}.φF) )

which is an isomorphism, if φ is right exact (resp. if φ is left exact).

7.4.23. Let 𝒜 be an abelian category, and A any object of s.𝒜. For every n > 0, set

d_{n} := ∑_{i=0}^{n}(−1)^{i} · ∂_{i} A[n] → A[n − 1].

Directly from the simplicial identities (7.4.9) we may compute

\begin{matrix}d_{n} ∘ d_{n+1} = & ∑_{i=0}^{n} ∑_{j=0}^{n+1}(−1)^{i+j} · ∂_{i} ∘ ∂_{j} \\ = & ∑_{i=0}^{n} ∑_{j>i}^{n+1}(−1)^{i+j} · ∂_{j−1} ∘ ∂_{i} + ∑_{i=0}^{n} ∂_{i} ∘ ∂_{i} + ∑_{i=0}^{n} ∑_{j<i}^{n}(−1)^{i+j} · ∂_{i} ∘ ∂_{j} \\ = & ∑_{i=0}^{n} ∑_{j−1>i}^{n+1}(−1)^{i+j} · ∂_{j−1} ∘ ∂_{i} + ∑_{i=0}^{n} ∑_{j<i}^{n}(−1)^{i+j} · ∂_{i} ∘ ∂_{j} \\ = & 0 \end{matrix}

for every n ∈ ℕ, so we are led to the following :

Definition 7.4.24. Let 𝒜 be an abelian category, and k ∈ ℕ any integer.

(i) If A[•] is any simplicial object of 𝒜, with face operators ∂_{i}, the unnormalized complex associated to A[•] is the complex (A_{•}, d_{•}) ∈ Ob(C^{≤0}(𝒜)) such that A_{n} := A[n] for every n ∈ ℕ, and d_{n} is defined as in (7.4.23), for every n > 0.

(ii) If A[•] is a k-truncated simplicial object of 𝒜, the unnormalized complex associated to A[•] as the complex (A_{•}, d_{•}) where A_{n} and d_{n} are defined as in (i) for every n ≤ k, and A_{n} := 0, d_{n} := 0 for every n > k.

(iii) If A[•] is as in (i) (resp. as in (ii)), the normalized complex associated to A[•] is the subcomplex N_{•}A of A_{•} such that

N_{0}A := A[0] and N_{n}A := ⋂_{i=1}^{n} Ker ∂_{i} for every n > 0 (resp. for 0 < n ≤ k).

So, the differential N_{n}A → N_{n−1}A equals ∂_{0}, for every n ∈ ℕ (resp. for every n ≤ k). (iv) If A[•] is as in (i) or (ii), the homology of A in degree n is

H_{n}A := H_{n}A_{•} for every n ∈ ℕ.

(v) Let ε : A → A_{−1} be an augmentation for A[•]. One says that the augmented simplicial object (A, ε) is aspherical, if H_{n}A = 0 for every n > 0, and ε induces an isomorphism

H_{0}A →^{∼} A_{−1}.

7.4.25. Let 𝒜 be an abelian category, and recall that s^{n}.𝒜 and s_{k}.𝒜 are both abelian categories as well, for every n, k ∈ ℕ (remark 3.7.37(ii)); also, clearly the rule of definition 7.4.24(iii) yields natural additive functors

N_{𝒜} : s.𝒜 → C^{≤0}(𝒜)  N_{𝒜 ,k} : s_{k}.𝒜 → C^{[−k,0]}(𝒜)  A[•] ↦ N_{•}A  for every k ∈ ℕ

and the rules of definition 7.4.24(i,ii) yield additive functors

U_{𝒜} : s.𝒜 → C^{≤0}(𝒜)  U_{𝒜 ,k} : s_{k}.𝒜 → C^{[−k,0]}(𝒜)  A[•] ↦ A_{•}  for every k ∈ ℕ.

Remark 7.4.26. Let 𝒜 and A be as in (7.4.23); directly from the simplicial identities (7.4.9) we may compute

d_{n} ∘ σ_{k} = ∑_{i=0}^{k−1}(−1)^{i} · σ_{k−1} ∘ ∂_{i} + ∑_{i=k+2}^{n}(−1)^{i} · σ_{k} ∘ ∂_{i−1}.

Especially, if we let

D_{0} := 0  and D_{n}A := ∑_{i=0}^{n−1} Im (σ_{i} : A[n − 1] → A[n])  for every n > 0

we see that d_{n} restricts to a morphism d′_{n} : D_{n}A → D_{n−1}A for every n > 0, hence (D_{•}A, d′_{•}) is a subcomplex of A_{•}, called the degenerate subcomplex, and clearly we obtain an additive functor

D_{𝒜} : s.𝒜 → C(𝒜) A ↦ D_{•}A.

Proposition 7.4.27. The natural injections induce a decomposition

A_{•} = N_{•}A ⊕ D_{•}A in C^{≤0}(𝒜) .

Proof. First, we notice that N_{n}A ∩ D_{n}A = 0 for every n ∈ ℕ. Indeed, it suffices to check that N_{n}A ∩ Im(σ_{i}) = 0 for every i = 0, …, n − 1, but the latter follows easily from the identity ∂_{i+1} ∘ σ_{i} = 1_{A[n−1]}. To conclude the proof, it suffices to show that N_{n}A + D_{n}A = A_{n} for every n ∈ ℕ. Indeed, set K_{0} := A_{n} and define inductively K_{i} := K_{i−1} ∩ Ker ∂_{i} for every i = 1, …, n; to prove the latter assertion, it suffices to check that

(7.4.28)  (1_{A_{n}} − σ_{i} · ∂_{i})(K_{i}) ⊂ K_{i+1}  for every i = 0, …, n.

However, we have :

∂_{j} ∘ (1_{A_{n}} − σ_{i} · ∂_{i}) = ∂_{j} − σ_{i−1} ∘ ∂_{i−1} ∘ ∂_{j} whenever j < i

and ∂_{i} ∘ (1_{A_{n}} − σ_{i} · ∂_{i}) = ∂_{i} − ∂_{i} = 0, whence (7.4.28). □

Example 7.4.29. (i) Let S be any simplicial set, and set ℤ^{S} := S ⊗ s.ℤ (notation of remark 7.4.18(iii)); we wish to give an explicit description of the complex N_{•}ℤ^{S} . To begin with, notice that ℤ^{S}[n] is the free abelian group with basis indexed by S[n], for every n ∈ ℕ, and for every x ∈ S[n] denote by e_{x} ∈ ℤ^{S}[n] the corresponding basis element. Next, set

D_{n}S := ⋃_{i=0}^{n−1} Im (σ_{i} : S[n − 1] → S[n]) and N_{n}S := S[n] \ D_{n}S for every n ∈ ℕ.

From proposition 7.4.27, we see that N_{n}ℤ^{S} can be naturally identified with the direct summand of ℤ^{S}[n] generated by the system (e_{x} | x ∈ N_{n}S), for every n ∈ ℕ. In order to describe the differential d_{n}, let us define \overline{e}_{x} := e_{x} if x ∈ N_{n}S, and \overline{e}_{x} := 0 if x ∈ D_{n}S. Then we may write

d_{n}(\overline{e}_{x}) = ∑_{i=0}^{n}(−1)^{i} · \overline{e}_{∂_{i}x} for every n ∈ ℕ and every x ∈ N_{n}S.

The verifications are straightforward, and shall be left to the reader.

(ii) For instance, for any i ∈ ℕ consider the simplicial set Δ_{i} of example 7.4.5(i), and set

𝕂⟨i⟩_{•} := N_{•}ℤ^{Δ_{i}}

where ℤ^{Δ_{i}} is defined as in (i). Notice that this notation agrees with that of remark 7.1.19(ii). It is easily seen that a morphism [n] → [i] of Δ lies in D_{n}Δ_{i} if and only if it is not injective; hence N_{n}Δ_{i} is the set of all injective maps φ : [n] → [i] of Δ. We deduce a natural isomorphism

𝕂⟨i⟩_{n} →^{∼} Λ_{ℤ}^{n+1}ℤ^{⊕i+1} for every i, n ∈ ℕ.

Namely, to any map φ as above, we assign the exterior product e_{φ(0)} ∧ ⋯ ∧ e_{φ(n)}, where e_{0}, …, e_{i} denotes the canonical basis of ℤ^{⊕i+1}. Under this isomorphism, the differential of 𝕂⟨i⟩_{•} gets identified with the differential of the Koszul complex attached to the sequence 1_{i+1} := (1, …, 1) ∈ ℤ^{⊕i+1}, that will be introduced in (7.8). Summing up, we obtain natural short exact sequences

0 → 𝕂⟨i⟩_{•}[1] → K_{•}(1_{i+1}) → ℤ[0] → 0 for every i ∈ ℕ

where [1] denotes the shift operator, and ℤ[0] is the complex with ℤ in degree zero : see (7.1.3).

(iii) Let 𝒜 be any abelian category, A ∈ Ob(s.𝒜) and Z ∈ Ob(𝒜). Notice that the simplicial set Hom_{𝒜} (Z, A) of remark 7.4.19(ii) is actually a simplicial abelian group, and a direct inspection of the arguments of loc.cit. yields a natural isomorphism of abelian groups

Hom_{s.ℤ-Mod}(ℤ^{Δ_{i}}, Hom_{𝒜} (Z, A)) →^{∼} Hom_{𝒜} (Z, A[i]) for every i ∈ ℕ.

7.4.30. Keep the notation of remark 7.4.26, and let

j_{•}^{A} : N_{•}A → A_{•} q_{•}^{A} : A_{•} → N_{•}A

be respectively the injection and the projection with kernel D_{•}A. Then the rules : A ↦ j_{•}^{A} and A ↦ q_{•}^{A} define natural transformations

j_{•} : N_{𝒜} ⇒ U_{𝒜} q_{•} : U_{𝒜} ⇒ N_{𝒜} .

With this notation, we may state :

Theorem 7.4.31. With the notation of remark (7.4.30), we have :

(i) The injection j_{•}^{A} is a homotopy equivalence, and D_{•}A is homotopically trivial.

(ii) More precisely, there exist natural modifications

j_{•} ∘ q_{•} ⇝ 1_{U_{𝒜}} q_{•} ∘ j_{•} ⇝ 1_{N_{𝒜}}

on the 2-category C(𝒜) as in example 7.1.15(ii) (see definition 2.2.9). (iii) Especially, the natural map

H_{i}N_{•}A → H_{i}A

is an isomorphism, for every i ∈ ℕ.

Proof. Clearly (ii)⇒(i), and (i)⇒(iii), by virtue of remark 7.1.9(ii). To show (ii), set φ_{0}^{A} := 1_{A_{0}} and φ_{n}^{A} := 1_{A[n]} − σ_{n−1} ∘ ∂_{n} : A_{n} → A_{n} for every n > 0. We notice :

Claim 7.4.32. The system (φ_{n}^{A} | n ∈ ℕ) defines an endomorphism of A_{•}, which is homotopically equivalent to 1_{A_{•}}.

Proof of the claim. Indeed, let s_{n} := (−1)^{n} · σ_{n} : A_{n} → A_{n+1} for every n ∈ ℕ, and define s_{n} := 0 for every n < 0. Using the simplicial identities (7.4.9) we compute :

\begin{matrix}s_{n−1} ∘ d_{n} + d_{n+1} ∘ s_{n} = & (−1)^{n−1} · ∑_{j=0}^{n}(−1)^{j} · (σ_{n−1} ∘ ∂_{j} − ∂_{j} ∘ σ_{n}) \\ = & φ_{n}^{A} − 1_{A[n]} + (−1)^{n−1} · ∑_{j=0}^{n−1}(−1)^{j} · (σ_{n−1} ∘ ∂_{j} − σ_{n−1} ∘ ∂_{j}) \\ = & φ_{n}^{A} − 1_{A[n]} \end{matrix}

for every n > 0, and for n = 0 we have as well s_{−1} ∘ d_{0} + d_{1} ∘ s_{0} = 0, whence the claim. ♢

Notice that the homotopy exhibited in the proof of claim 7.4.32 is natural in A, so it already yields the first sought modification. Next, for every simplicial object B of 𝒜, let γB and g_{B} : γB → B be as in remark 7.4.12(i,ii), and define inductively

β^{0}A := A  and β^{n+1}A := Ker g_{β^{n}A}  for every n ∈ ℕ.

A simple inspection shows that

N_{n+i}A ⊂ β^{n}A[i]  and β^{n}A[0] = N_{n}A  for every i, n ∈ ℕ.

Hence, we may define a subcomplex B_{•}^{(n)} of A_{•} for every n ∈ ℕ, as follows. For i = 0, …, n we let B_{i}^{(n)} := N_{i}A, and for i > n we let B_{i}^{(n)} := (β^{n}A)[n + i] (notation of (7.1.3)). The differential of B_{•}^{(n)} is of course just the restriction of that of A_{•}. By construction, N_{•}A is a subcomplex of B_{•}^{(n)}, for all n ∈ ℕ. Next, we define an endomorphism f_{•}^{(n)} of B_{•}^{(n)}, by setting

f_{i}^{(n)} := 1_{B_{i}^{(n)}}  for every i ≤ n, and f_{i}^{(n)} := φ_{i}^{β^{n}A}  for i > n.

Notice that the restriction of f_{•}^{(n)} to the subcomplex N_{•}A is just the inclusion map N_{•}A → B_{•}^{(n)}. Clearly B^{(n+1)} ⊂ B^{(n)} for every n ∈ ℕ, and we remark that f_{•}^{(n)} factors through the inclusion map B^{(n+1)} ⊂ B^{(n)}. Indeed, since B_{i}^{(n)} = B_{i}^{(n+1)} for every i ≤ n, the assertion is obvious for this range of degrees; so we have only to check that φ_{•}^{β^{n}A} factors through β^{n+1}A_{•} for every n ∈ ℕ, and an easy induction reduces to checking that φ_{•}^{A} factors through B_{•}^{(1)}. But the latter assertion comes down to the identity ∂_{i} ∘ φ_{i}^{A} = 0 for every i > 0, which follows easily from the simplicial identities (7.4.9).

If (s_{i} | i ∈ ℕ) is the homotopy between 1_{β^{n}A_{•}} and φ_{•}^{β^{n}A} supplied by claim 7.4.32, then we obtain a homotopy (t_{i}^{(n)} | i ∈ ℕ) between 1_{B_{•}^{(n)}} and f_{•}^{(n)}, by setting

t_{i}^{(n)} := 0  for every i < n, and t_{i}^{(n)} := s_{i−n}  for i ≥ n.

Notice that Im t_{i}^{(n)} ⊂ D_{i+1}A for every i, n ∈ ℕ. Thus, for every n ∈ ℕ, let h_{•}^{(n)} : B_{•}^{(n)} → B_{•}^{(n+1)} be the morphism of complexes deduced from f_{•}^{(n)} and j_{•}^{(n)} : B^{(n)} → A_{•} the inclusion map, and set q_{•}^{(n)} := h_{•}^{(n)} ∘ ⋯ ∘ h_{•}^{(0)}; it follows easily that the composition

p_{•}^{(n)} := j_{•}^{(n+1)} ∘ q_{•}^{(n)} : A_{•} → A_{•}

is homotopically equivalent to 1_{A_{•}}, for every n ∈ ℕ. More precisely, a direct inspection shows that the system of morphisms

τ_{i}^{(n)} := ∑_{k=0}^{n−1} j_{•}^{(k+1)} ∘ t_{•}^{(k+1)} ∘ q_{•}^{(k)} for every i ∈ ℕ

provides a homotopy between 1_{A_{•}} and p_{•}^{(n)}. Furthermore, it is clear that

p_{i}^{(n)} = p_{i}^{(m)}  and τ_{i}^{(n)} = τ_{i}^{(m)}  for every m ≥ n and every i ≤ n

so, we finally get an endomorphism p_{•} : A_{•} → A_{•} by setting p_{i} := p_{i}^{(i)} for every i ∈ ℕ, and a homotopy τ_{•} between p_{•} and 1_{A_{•}}, with τ_{i} := τ_{i}^{(i)} for every i ∈ ℕ. By construction, p_{•} factors through j_{•}^{A}, and moreover, the restriction of p_{•} to the subcomplex N_{•}A is just j_{•}^{A}, so j_{•}^{A} is a homotopy equivalence, via natural homotopies, as stated.

Lastly, notice that Im τ_{i} ⊂ D_{i+1}A for every i ∈ ℕ, from which it follows easily that p_{i}(D_{i}A) ⊂ D_{i}A for every i ∈ ℕ, and then the foregoing implies that Ker p_{•} = D_{•}A. We conclude that D_{•}A is homotopically trivial, as stated. □

7.4.33. By iterating U, we get a functor from bisimplicial objects to double complexes

U_{𝒜}^{2} : s^{2}.𝒜 ⟶^{s.U_{𝒜}} s.C(𝒜) ⟶^{U_{C(𝒜)}} C(C(𝒜)) A[•, •] ↦ A_{••}

and notice that this functor is naturally isomorphic to the functor

s^{2}.𝒜 ⟶^{U_{s.𝒜}} C(s.𝒜) ⟶^{C(U_{𝒜} )} C(C(𝒜)).

In the same vein, theorem 7.4.31yields natural decompositions of additive functors :

(7.4.34)  U_{𝒜}^{2} = (C(N_{𝒜} ) ⊕ C(D_{𝒜} )) ∘ (N_{s.𝒜} ⊕ D_{s.𝒜} )

and if we let

N_{𝒜}^{2} := C(N_{𝒜} ) ∘ N_{s.𝒜} : s^{2}.𝒜 → C(C(𝒜)) A ↦ N_{••}A

the natural morphism

i_{•}^{A} : Tot N_{••}A → Tot A_{••}

is a homotopy equivalence.

7.4.35. We wish next to exhibit two natural transformations

A_{•}^{Δ} ⟶^{AW_{•}^{A}} TotA_{••} ⟶^{Sh_{•}^{A}} A_{•}^{Δ} for every A ∈ Ob(s^{2}.𝒜) .

Namely, for every n ∈ ℕ, define :

• Sh_{n}^{A} as the sum, for all p, q ∈ ℕ such that p + q = n, of the shuffle maps

Sh_{p,q}^{A} := ∑_{μν} ε_{μν} · A[η_{ν_{1}} ∘ ⋯ ∘ η_{ν_{q}}, η_{μ_{1}} ∘ ⋯ ∘ η_{μ_{p}}] : A_{p,q} → A_{n,n}

where the sum ranges over the shuffle permutations (μ, ν) of type (p, q) of the set {0, …, p + q − 1} (these are the permutations described in [75, §4.3.15]), and ε_{μν} is the sign of the permutation (μ, ν)

• AW_{n}^{A} as the sum, for all p, q ∈ ℕ such that p + q = n, of the Alexander-Whitney maps

AW_{p,q}^{A} := A[ε_{p,0}^{q∨}, ε_{q,0}^{p}] : A_{n,n} → A_{p,q}

(notation of example 7.4.7(ii)). Clearly, for every p, q ∈ ℕ the rule A ↦ A_{p,q} defines a functor

•[p, q] : s^{2}.𝒜 → 𝒜

and the maps Sh_{p,q}^{A} and AW_{p,q}^{A} yield natural transformations

Sh_{p,q} : •[p, q] ⇒ •[p + q, p + q] AW_{p,q} : •[p + q, p + q] ⇒ •[p, q].

Proposition 7.4.36. With the notation of (7.4.25), the sequence (Sh_{n}^{A} | n ∈ ℕ) defines a morphism of chain complexes

Sh_{•}^{A} : TotA_{••} → A_{•}^{Δ} for every A ∈ Ob(s^{2}.𝒜) .

Proof. For any bisimplicial object A of 𝒜, set A_{−1,q} = A_{p,−1} := 0 for every p, q ∈ ℤ, and let

A[ε_{0}, 1_{[q]}] : A_{0,q} → A_{−1,q}  A[1_{[p]}, ε_{0}] : A_{p,0} → A_{p,−1}  A[ε_{0}, ε_{0}] : A_{0,0} → A_{−1,−1}

be the zero maps; likewise, let Sh_{−1,q}^{A} : A_{−1,q} → A_{q−1,q−1} and Sh_{p,−1}^{A} : A_{p,−1} → A_{p−1,p−1} be the zero maps, and notice that also Sh_{0,0}^{A} is the zero map. We define

d_{p,q}^{A,h} := ∑_{i=0}^{p}(−1)^{i} ·A[ε_{i}, 1_{[q]}] : A_{p,q} → A_{p−1,q} d_{p,q}^{A,v} := ∑_{i=0}^{q}(−1)^{j} ·A[1_{[p]}, ε_{i}] : A_{p,q} → A_{p,q−1}

so, for every n ∈ ℕ the differential d_{n} in degree n of TotA_{••} is the sum of the maps

d_{p,q}^{A,h} + (−1)^{p} · d_{p,q}^{A,v} : A_{p,q} → A_{p−1,q} ⊕ A_{p,q−1} for all p, q ∈ ℕ such that p + q = n

whereas the differential of A_{•}^{Δ} is the morphism

d_{n}^{A} := ∑_{i=1}^{n} A[ε_{i}, ε_{i}] : A_{n,n} → A_{n−1,n−1}

and we have to check the identity

(7.4.37)  d_{p+q}^{A} ∘ Sh_{p,q}^{A} = Sh_{p−1,q}^{A} ∘ d_{p,q}^{A,h} + (−1)^{p} · Sh_{p,q−1}^{A} ∘ d_{p,q}^{A,v}  for all p, q ∈ ℕ.

Now, we set B := γ_{1}A, C := γ_{2}A, D := γ_{2}B (notation of (7.4.17)), but for the purpose of this proof, we shall modify the differentials of the double complexes B_{••} and C_{••} in certain low degrees : namely, we define

d_{0,q}^{B,h} := A[ε_{0}, 1_{[q]}]  d_{p,0}^{C,v} := A[1_{[p]}, ε_{0}]  for every p, q ∈ ℕ.

Claim 7.4.38. With the foregoing notation, the following holds :

(i) Sh_{p,q}^{A} = (−1)^{q} · Sh_{p−1,q}^{D} ∘ A[1_{[p]}, η_{q}] + Sh_{p,q−1}^{D} ∘ A[η_{p}, 1_{[q]}] for every p, q ∈ ℕ. (ii) A[ε_{p+q−1}, ε_{p+q−1}] ∘ Sh_{p−1,q−1}^{D} = Sh_{p−1,q−1}^{A} ∘ A[ε_{p}, ε_{q}] for every p, q > 0. (iii) A[ε_{p+q}, ε_{p+q}] ∘ Sh_{p,q}^{A} = (−1)^{q} · Sh_{p−1,q}^{A} ∘ A[ε_{p}, 1_{[q]}] + Sh_{p,q−1}^{A} ∘ A[1_{[p]}, ε_{q}] for every

p, q ∈ ℕ.

Proof of the claim. (i): First, we notice the identities :

(7.4.39)  A[1_{[p+q]}, η_{p+q}] ∘ Sh_{p,q}^{A} = Sh_{p,q}^{C} ∘ A[1_{[p]}, η_{q}] A[η_{p+q}, 1_{[p+q]}] ∘ Sh_{p,q}^{A} = Sh_{p,q}^{B} ∘ A[η_{p}, 1_{[q]}].  for every p, q ∈ ℕ

that are deduced from the identities

\begin{matrix}η_{μ_{1}} ∘ ⋯ ∘ η_{μ_{p}} ∘ η_{p+q} = & η_{q} ∘ η_{μ_{1}} ∘ ⋯ ∘ η_{μ_{p}} \\ η_{ν_{1}} ∘ ⋯ ∘ η_{ν_{q}} ∘ η_{p+q} = & η_{p} ∘ η_{ν_{1}} ∘ ⋯ ∘ η_{ν_{q}} \end{matrix}

which in turn follow from the simplicial identities for degeneracy maps. By applying the first (resp. second) identity (7.4.39) with A replaced by B (resp. by C) and p replaced by p − 1 (resp. with q replaced by q − 1) we get :

(7.4.40)  A[1_{[p+q]}, η_{p+q−1}] ∘ Sh_{p−1,q}^{B} = Sh_{p−1,q}^{D} ∘ A[1_{[p]}, η_{q}] A[η_{p+q−1}, 1_{[p+q]}] ∘ Sh_{p,q−1}^{C} = Sh_{p,q−1}^{D} ∘ A[η_{p}, 1_{[q]}].  for every p, q ∈ ℕ.

Now, suppose first that p, q > 0, and let (μ, ν) be any (p, q)-shuffle of {0, …, p + q − 1}; then either μ_{p} = p + q − 1 or ν_{q} = p + q − 1. In the first (resp. second) case, after removing μ_{p} (resp. ν_{q}) we get a (p − 1, q)-shuffle (resp. a (p, q − 1)-shuffle) (\overline{μ}, \overline{ν}) with

ε_{\overline{μ}\overline{ν}} = (−1)^{q} · ε_{μν} (resp. ε_{\overline{μ}\overline{ν}} = ε_{μν}).

However, the first (resp. second) left-hand side of (7.4.40) contains precisely all the terms of the first (resp. second) type occurring in the definition of Sh_{p,q}^{A}, so we get (i) in this case. The cases where either p = 0 or q = 0 can be dealt with by a similar, but simpler, argument.

(ii) follows by naturality of Sh_{p−1,q−1}, applied to the morphism g_{A}^{(1)} ∘ g_{B}^{(2)} : D → A from (7.4.17).

(iii): The case where p = q = 0 is obvious, and the other cases follow by composing both sides of (i) with A[ε_{p+q}, ε_{p+q}], applying (ii), and recalling that η_{q} ∘ ε_{q} = 1_{[q]} : details left to the reader. ♢

Now, a simple inspection shows that (7.4.37) follows from claim 7.4.38(iii) and the following

Claim 7.4.41. d_{p+q−1}^{D} ∘ Sh_{p,q}^{A} = Sh_{p−1,q}^{A} ∘ d_{p−1,q}^{B,h} + (−1)^{p} · Sh_{p,q−1}^{A} ∘ d_{p,q−1}^{C,v} for every p, q ∈ ℕ.

Proof of the claim. Consider first the case where p = 0. If q ≤ 1, it is easily seen that both sides of the stated identity vanish; if q ≥ 2, the left-hand side is

d_{q−1}^{D} ∘ A[η_{0} ∘ ⋯ ∘ η_{q−1}, 1_{[q]}] = ∑_{i=0}^{q−1}(−1)^{i} · A[η_{0} ∘ ⋯ ∘ η_{q−1} ∘ ε_{i}, ε_{i}]

and the right-hand side is

A[η_{0} ∘ ⋯ ∘ η_{q−2}, 1_{[q]}] ∘ d_{0,q−1}^{C,v} = ∑_{i=0}^{q−1}(−1)^{i} · A[η_{0} ∘ ⋯ ∘ η_{q−2}, ε_{i}]

so in this case the assertion comes down to the obvious identity

η_{0} ∘ ⋯ ∘ η_{q−1} ∘ ε_{i} = η_{0} ∘ ⋯ ∘ η_{q−2} : [q − 2] → [0] for every i = 0, …, q − 1.

Likewise we deal with the case where q = 0. It follows already that (7.4.37) holds for these values of (p, q), and for every bisimplicial object A of 𝒜; especially, we get

(7.4.42)  d_{q}^{D} ∘ Sh_{0,q}^{D} = Sh_{0,q−1}^{D} ∘ d_{0,q}^{D,v}  for every q ∈ ℕ.

Next, for p = q = 1, a direct computation shows that both sides equal

A[ε_{0} ∘ η_{0}, 1_{[1]}] − A[1_{[1]}, ε_{0} ∘ η_{0}] : A_{1,1} → A_{1,1}.

We prove now, by induction on q, that the assertion holds for p = 1. This is already known for q ≤ 1, so suppose that r > 1, and that the assertion holds for p = 1 and every q < r. The latter implies that also (7.4.37) holds for these values of (p, q), and for every bisimplicial object A of 𝒜; especially, we get

(7.4.43)  d_{r}^{D} ∘ Sh_{1,r−1}^{D} = Sh_{0,r−1}^{D} ∘ d_{1,r−1}^{D,h} − Sh_{1,r−2}^{D} ∘ d_{1,r−1}^{D,v}.

On the other hand, claim 7.4.38(i) says that

d_{r}^{D} ∘ Sh_{1,r}^{A} = (−1)^{r} · d_{r}^{D} ∘ Sh_{0,r}^{D} ∘ A[1_{[1]}, η_{r}] + d_{r}^{D} ∘ Sh_{1,r−1}^{D} ∘ A[η_{1}, 1_{[r]}]

Combining with (7.4.42) and (7.4.43) we obtain

d_{r}^{D} ∘Sh_{1,r}^{A} = (−1)^{r} ·Sh_{0,r−1}^{D} ∘d_{0,r}^{D,v} ∘A[1_{[1]}, η_{r}]+(Sh_{0,r−1}^{D} ∘d_{1,r−1}^{D,h} −Sh_{1,r−2}^{D} ∘d_{1,r−1}^{D,v})∘A[η_{1}, 1_{[r]}].

However, we have

d_{1,r−1}^{D,h} ∘ A[η_{1}, 1_{[r]}] = A[η_{1} ∘ ε_{0}, 1_{[r]}] − A[1_{[1]}, 1_{[r]}] = A[ε_{0} ∘ η_{0}, 1_{[r]}] − 1_{A[1,r]}

so we can rewrite d_{r}^{D} ∘ Sh_{1,r}^{A} = φ_{1} + φ_{2} − φ_{3}, where

φ_{1} := Sh_{0,r−1}^{D} ∘ ((−1)^{r} · d_{0,r}^{D,v} ∘ A[1_{[1]}, η_{r}] − 1_{A[1,r]}) = (−1)^{r} · Sh_{0,r−1}^{D} ∘ A[1_{[1]}, η_{r−1}] ∘ d_{1,r−1}^{C,v}
φ_{2} := Sh_{0,r−1}^{D} ∘ A[ε_{0} ∘ η_{0}, 1_{[r]}]
φ_{3} := Sh_{1,r−2}^{D} ∘ d_{1,r−1}^{D,v} ∘ A[η_{1}, 1_{[r]}].

On the other hand, using claim 7.4.38(i), we can compute

Sh_{0,r}^{A} ∘ d_{0,r−1}^{B,h} = Sh_{0,r−1}^{D} ∘ A[η_{0}, 1_{[r]}] ∘ d_{0,r−1}^{B,h} = φ_{2}
Sh_{1,r−1}^{A} ∘ d_{1,r−1}^{C,v} = ((−1)^{r−1} · Sh_{0,r−1}^{D} ∘ A[1_{[1]}, η_{r−1}] + Sh_{1,r−2}^{D} ∘ A[η_{1}, 1_{[r−1]}]) ∘ d_{1,r−1}^{C,v}
= φ_{3} − φ_{1}

which shows that the claim holds for p = 1 and q = r, and concludes the induction.

Lastly, we prove the claim for every p, q ∈ ℕ, by induction on p+q; notice that the foregoing already shows that the assertion holds whenever p + q ≤ 2. Thus, let r > 2, and suppose that the claim is already known for every pair (p, q) such that p + q < r; then also (7.4.37) holds for such values of p and q, and especially we get

(7.4.44) d_{p+q}^{D} ∘ Sh_{p,q}^{D} = Sh_{p−1,q}^{D} ∘ d_{p,q}^{D,h} + (−1)^{p} · Sh_{p,q−1}^{D} ∘ d_{p,q}^{D,v}  whenever p + q < r.

Let (p′, q′) be a pair such that p′ + q′ = r; combining (7.4.44) with claim 7.4.38(i), we get

d_{p′+q′−1}^{D} ∘ Sh_{p′,q′}^{A} = (−1)^{q′} · (Sh_{p′−2,q′}^{D} ∘ d_{p′−1,q′}^{D,h} − (−1)^{p′} · Sh_{p′−1,q′−1}^{D} ∘ d_{p′−1,q′}^{D,v}) ∘ A[1_{[p′]}, η_{q′}]
+ (Sh_{p′−1,q′−1}^{D} ∘ d_{p′,q′−1}^{D,h} + (−1)^{p′} · Sh_{p′,q′−2}^{D} ∘ d_{p′,q′−1}^{D,v}) ∘ A[η_{p′}, 1_{[q′]}].

On the other hand, after noticing that

\begin{matrix}d_{p′,q′−1}^{D,h} ∘ A[η_{p′}, 1_{[q′]}] − (−1)^{p′} · 1_{A[p′,q′]} = & A[η_{p′−1}, 1_{[q′]}] ∘ d_{p′−1,q′}^{B,h} \\ d_{p′−1,q′}^{D,v} ∘ A[1_{[p′]}, η_{q′}] − (−1)^{q′} · 1_{A[p′,q′]} = & A[1_{[p′]}, η_{q′−1}] ∘ d_{p′,q′−1}^{C,v} \end{matrix}

we may apply claim 7.4.38(i), to compute

Sh_{p′−1,q′}^{A} ∘ d_{p′−1,q′}^{B,h} = ((−1)^{q′} · Sh_{p′−2,q′}^{D} ∘ A[1_{[p′−1]}, η_{q′}] + Sh_{p′−1,q′−1}^{D} ∘ A[η_{p′−1}, 1_{[q′]}]) ∘ d_{p′−1,q′}^{B,h}
= (−1)^{q′} · Sh_{p′−2,q′}^{D} ∘ d_{p′−1,q′}^{D,h} ∘ A[1_{[p′]}, η_{q′}] − (−1)^{p′} · Sh_{p′−1,q′−1}^{D}
+ Sh_{p′−1,q′−1}^{D} ∘ d_{p′,q′−1}^{D,h} ∘ A[η_{p′}, 1_{[q′]}]

and

Sh_{p′,q′−1}^{A} ∘ d_{p′,q′−1}^{C,v} = ((−1)^{q′−1} · Sh_{p′−1,q′−1}^{D} ∘ A[1_{[p′]}, η_{q′−1}]
+ Sh_{p′,q′−2}^{D} ∘ A[η_{p′}, 1_{[q′−1]}]) ∘ d_{p′,q′−1}^{C,v}
= (−1)^{q′−1} · Sh_{p′−1,q′−1}^{D} ∘ d_{p′−1,q′}^{D,v} ∘ A[1_{[p′]}, η_{q′}] + Sh_{p′−1,q′−1}^{D}
+ Sh_{p′,q′−2}^{D} ∘ d_{p′,q′−1}^{D,v} ∘ A[η_{p′}, 1_{[q′]}].

Finally, the sought identity for the pair (p′, q′) follows by comparing the last two identities with the previous one for d_{p′+q′−1}^{D} ∘ Sh_{p′,q′}^{A}, and this concludes the proof of the inductive step. □

Proposition 7.4.45. With the notation of (7.4.25), the sequence (AW_{n}^{A} | n ∈ ℕ) defines a morphism of chain complexes

AW_{•}^{A} : A_{•}^{Δ} → TotA_{••} for every A ∈ Ob(s^{2}.𝒜) .

Proof. Denote by d_{•}^{T} the differential of TotA_{••}, and keep the notation of the proof of proposition 7.4.36; we have to check the identity

(7.4.46)  AW_{n−1} ∘ d_{n}^{A} = d_{n}^{T} ∘ AW_{n}  for every n ∈ ℕ.

However, say that p, q ∈ ℕ and p + q = n; the projection of the right-hand side of (7.4.46) on the direct summand A_{p−1,q} of (TotA_{••})_{n−1} equals

(7.4.47)  d_{p,q}^{A,h} ∘ AW_{p,q} + (−1)^{p−1} · d_{p−1,q+1}^{A,v} ∘ AW_{p−1,q+1}.

Unwinding the definition, (7.4.47) is found to be

∑_{i=0}^{p}(−1)^{i} · A[ε_{p,0}^{q∨} ∘ ε_{i}, ε_{q,0}^{p}] − (−1)^{p} · ∑_{i=0}^{q+1}(−1)^{i} · A[ε_{p−1,0}^{q+1∨}, ε_{q+1,0}^{p−1} ∘ ε_{i}]

which we may rewrite as

(7.4.48)  ∑_{i=0}^{p−1}(−1)^{i} · A[ε_{p,0}^{q∨} ∘ ε_{i}, ε_{q,0}^{p}] + ∑_{i=p}^{p+q}(−1)^{i} · A[ε_{p−1,0}^{q+1∨}, ε_{q+1,0}^{p−1} ∘ ε_{i−p+1}].

On the other hand, the projection of the left-hand side of (7.4.46) onto A_{p−1,q} equals

∑_{i=0}^{n}(−1)^{i} · A[ε_{i} ∘ ε_{p−1,0}^{q∨}, ε_{i} ∘ ε_{q,0}^{p−1}].

To compare the latter with (7.4.48), it suffices to remark that

ε_{i} ∘ ε_{p−1,0}^{q∨} = \{ \begin{matrix}ε_{p,0}^{q∨} ∘ ε_{i} & if i < p \\ ε_{p−1,0}^{q+1∨} & if i ≥ p \end{matrix} and ε_{i} ∘ ε_{q,0}^{p−1} = \{ \begin{matrix}ε_{q,0}^{p} & if i < p \\ ε_{q+1,0}^{p−1} ∘ ε_{i} & if i ≥ p \end{matrix}

which are all deduced from the simplicial identities for the ε_{i}. The proposition follows. □

7.4.49. Propositions 7.4.36and 7.4.45yield natural transformations of functors

\xymatrix{Tot U_{𝒜}^{2} \ar[r]^{^{Sh^{•}}} & U_{𝒜} ∘ Δ_{𝒜} . \ar[l]^{_{AW_{•}}}}

Now, denote by h_{𝒜} : C(𝒜) → Hot(𝒜) the natural functor (this induces the identity on the objects, and the projection on the group of morphisms); there follow natural transformations

\xymatrix{h_{𝒜} ∘ Tot U_{𝒜}^{2} \ar[r]^{h_{𝒜} ∗Sh_{•}} & h_{𝒜} ∘ U_{𝒜} ∘ Δ_{𝒜} . \ar[l]^{h_{𝒜} ∗AW_{•}}}  between functors s^{2}.𝒜 → Hot(𝒜).

Theorem 7.4.50 (Eilenberg-Zilber-Cartier). With the notation of (7.4.49), we have : (i) h_{𝒜} ∗ AW_{•} and h_{𝒜} ∗ Sh_{•} are mutually inverse isomorphisms of functors. (ii) More precisely, there exist natural modifications (see definition 2.2.9)

AW_{•} ∘ Sh_{•} ⇝ 1_{Tot U_{𝒜}^{2}} Sh_{•} ∘ AW_{•} ⇝ 1_{U_{𝒜} ∘Δ_{𝒜}} .

Proof. Let p_{•}^{A} : Tot A_{••} → Tot N_{••}A be the projection whose kernel is the sum of the remaining three direct summands in the decomposition Tot (7.4.34); explicitly, in each degree n ∈ ℕ, this kernel is the sum of the subobjects Im A[η_{i}, 1_{[q]}] and Im A[1_{[p]}, η_{j}] of A_{p,q}, for every i = 0, …, p − 1, j = 0, …, q − 1, and every p, q ∈ ℕ such that p + q = n. Likewise, let j_{•}^{A} : N_{•}A^{Δ} → A_{•}^{Δ} and q_{•}^{A} : A_{•}^{Δ} → N_{•}A^{Δ} be as in (7.4.30); theorem 7.4.31 and the discussion of (7.4.33) show that the pairs (p_{•}^{A}, i_{•}^{A}) and (j_{•}^{A}, q_{•}^{A}) induce mutually inverse isomorphisms in Hot(𝒜), so it suffices to check that the same holds for the compositions

\overline{Sh}_{•}^{A} := q_{•}^{A} ∘ Sh_{•}^{A} ∘ i_{•}^{A} : Tot N_{••}A → N_{•}A^{Δ} \overline{AW}_{•}^{A} := p_{•}^{A} ∘ AW_{•}^{A} ∘ j_{•}^{A} : N_{•}A^{Δ} → Tot N_{••}A

for every bisimplicial object A of 𝒜. However, we have

Claim 7.4.51. p_{•}^{A} ∘ AW_{•}^{A} ∘ Sh_{•}^{A} ∘ i_{•}^{A} = 1_{Tot N_{••}A}.

Proof of the claim. More precisely, say that p + q = n, consider the composition

f_{p,q} : A_{p,q} ⟶^{Sh_{p,q}^{A}} A_{p+q,p+q} ⟶^{AW_{n}^{A}} (Tot A_{••})_{n}

and let g_{p,q} : A_{p,q} → (Tot A_{••})_{n} be the inclusion map; we shall show that f_{p,q} − g_{p,q} ⊂ Ker p_{n}^{A}. Indeed, we have

f_{p,q} = ∑_{i=0}^{n} ∑_{(μ,ν)} ε_{μν} · A[η_{ν_{1}} ∘ ⋯ ∘ η_{ν_{q}} ∘ ε_{i,0}^{n−i∨}, η_{μ_{1}} ∘ ⋯ ∘ η_{μ_{p}} ∘ ε_{n−i,0}^{i}]

(notation of (7.4.35), and ε_{n−i,0}^{i} is defined as in the proof of proposition 7.4.45). However, if i < p, then η_{μ_{1}} ∘ ⋯ ∘ η_{μ_{p}} ∘ ε_{n−i,0}^{i} is of the form τ ∘ η_{μ_{p}} for some τ : [p + q − i − 1] → [q], so the corresponding term does lie in Ker p_{n}^{A}. If i > p, then η_{ν_{1}} ∘ ⋯ ∘ η_{ν_{q}} ∘ ε_{i,0}^{n−i∨} is of the form τ ∘ η_{k} for some k ≤ ν_{q} and some τ : [i − 1] → [p], so this term likewise lies in Ker p_{n}^{A}. It remains to consider the terms with i = p; however, a simple inspection shows that η_{ν_{1}} ∘ ⋯ ∘ η_{ν_{q}} ∘ ε_{p,0}^{n−p∨} is of the same form as above, unless ν_{1} is either p − 1 or p (details left to the reader); furthermore, if ν_{1} = p − 1, then μ_{p} ≥ p, in which case η_{μ_{1}} ∘ ⋯ ∘ η_{μ_{p}} ∘ ε_{n−i,0}^{i} is of the form described above. In all these cases, the corresponding term again lies in Ker p_{n}^{A}. So, it remains only to consider the single case where (μ, ν) is the identity permutation, whose sign equals 1; in this case, the corresponding term is none else than A[1_{[p]}, 1_{[q]}], whence the claim. ♢

Claim 7.4.51 and theorem 7.4.31(ii) already yield the existence of the sought modification AW_{•} ∘ Sh_{•} ⇝ 1_{Tot U_{𝒜}^{2}} . Next we define, for every n ∈ ℕ, a natural transformation

s_{n} : •[n] ∘ Δ_{𝒜} ⇒ •[n + 1] ∘ Δ_{𝒜}

(notation of (7.4.4); so s_{n}^{A} is a morphism A[n, n] → A[n+1, n+1] for every object A of s^{2}.𝒜). The construction is by induction on n : for n = 0 we let s_{0}^{A} : A[0, 0] → A[1, 1] be the zero morphism; for n > 0 we set

s_{n}^{A} := Sh_{n}^{A′} ∘ AW_{n}^{A′} ∘ A[η_{0}, η_{0}] − s_{n−1}^{A′} where A′ := (γ_{2} ∘ γ_{1}(A^{∨}))^{∨}

(notation of (7.4.17) and (7.4.2) : explicitly, we have A′[p, q] := A[p + 1, q + 1] for every p, q ∈ ℕ, and A′[ε_{i}, ε_{j}] := A[ε_{i+1}, ε_{j+1}], and likewise for A′[ε_{i}, η_{j}], A′[η_{i}, ε_{j}] and A′[η_{i}, η_{j}], for every face and degeneracy map of Δ). We have

Claim 7.4.52. The system (q_{n+1}^{A} ∘s_{n}^{A} ∘j_{n}^{A} | n ∈ ℕ) is a homotopy q_{•}^{A} ∘Sh_{•}^{A} ∘AW_{•}^{A} ∘j_{•}^{A} ⇒ 1_{N_{•}A^{Δ}}.

Proof of the claim. Denote by d_{•}^{A} the differential of A_{•}^{Δ}, and let d_{0}^{A} : A_{0}^{Δ} → 0 and s_{−1}^{A} : 0 → A_{0}^{Δ} be the zero morphisms. We check, more precisely, that

q_{n}^{A} ∘ (d_{n+1}^{A} ∘ s_{n}^{A} + s_{n−1}^{A} ∘ d_{n}^{A}) = q_{n}^{A} ∘ Sh_{n}^{A} ∘ AW_{n}^{A} − q_{n}^{A} for every n ∈ ℕ.

We argue by induction on n, and the assertion is clear for n = 0. For n = 1, notice that

Sh_{1}^{A} ∘ AW_{1}^{A} = A[ε_{1} ∘ η_{0}, 1_{[1]}] + A[1_{[1]}, ε_{0} ∘ η_{0}] : A_{1,1} → A_{1,1}.

It follows that

s_{1}^{A} = Sh_{1}^{A′} ∘ AW_{1}^{A′} ∘ A[η_{0}, η_{0}] = (A[ε_{1} ∘ η_{0} ∘ η_{0}, η_{0}] + A[η_{0}, η_{1}])

whence

d_{2}^{A} ∘ s_{1}^{A} = Sh_{1}^{A} ∘ AW_{1}^{A} − 1_{A[1,1]} + A[ε_{1} ∘ ε_{1} ∘ η_{0}, ε_{1} ∘ η_{0}]

(details left to the reader). Since Im A[η_{0}, η_{0}] ⊂ Ker q_{2}^{A}, the assertion follows in this case. Next, suppose that r > 1, and that the sought identity is already known for every n < r, and every bisimplicial object A; especially, it holds for A′, so if we let d_{•}^{A′} be the differential of A′_{•}, we get

(7.4.53)  q_{r−1}^{A′} ∘ (d_{r}^{A′} ∘ s_{r−1}^{A′} + s_{r−2}^{A′} ∘ d_{r−1}^{A′}) = q_{r−1}^{A′} ∘ Sh_{r−1}^{A′} ∘ AW_{r−1}^{A′} − q_{r−1}^{A′}.

On the other hand, after noticing that

d_{r}^{A′} ∘ A[η_{0}, η_{0}] = 1_{A[r,r]} − A[η_{0}, η_{0}] ∘ d_{r−1}^{A′}

we may compute

d_{r}^{A′} ∘ s_{r}^{A} = d_{r}^{A′} ∘ Sh_{r}^{A′} ∘ AW_{r}^{A′} ∘ A[η_{0}, η_{0}] − d_{r}^{A′} ∘ s_{r−1}^{A′}
= Sh_{r}^{A′} ∘ AW_{r}^{A′} ∘ d_{r}^{A′} ∘ A[η_{0}, η_{0}] − d_{r}^{A′} ∘ s_{r−1}^{A′}
= Sh_{r}^{A′} ∘ AW_{r}^{A′} − Sh_{r}^{A′} ∘ AW_{r}^{A′} ∘ A[η_{0}, η_{0}] ∘ d_{r−1}^{A′} − d_{r}^{A′} ∘ s_{r−1}^{A′}

which, combined with (7.4.53), implies

(7.4.54)  q_{r−1}^{A′} ∘ (d_{r}^{A′} ∘ s_{r}^{A} + s_{r−1}^{A} ∘ d_{r−1}^{A′}) = −q_{r−1}^{A′}.

Furthermore, notice the natural morphism in s^{2}.𝒜

(g_{A^{∨}}^{(1)} ∘ g_{γ_{1}A^{∨}}^{(2)})^{∨} : A′ → A

supplied by (7.4.17); explicitly, for every p, q ∈ ℕ, this morphism is given by the discarded face operator A[ε_{0}, ε_{0}] : A[p + 1, q + 1] → A[p, q]. Then, the naturality of s_{•}, Sh_{•} and AW_{•} implies

A[ε_{0}, ε_{0}] ∘ s_{r}^{A} = A[ε_{0}, ε_{0}] ∘ Sh_{r}^{A′} ∘ AW_{r}^{A′} ∘ A[η_{0}, η_{0}] − A[ε_{0}, ε_{0}] ∘ s_{r−1}^{A′}
= Sh_{r}^{A} ∘ AW_{r}^{A} ∘ A[η_{0} ∘ ε_{0}, η_{0} ∘ ε_{0}] − s_{r−1}^{A} ∘ A[ε_{0}, ε_{0}]
= Sh_{r}^{A} ∘ AW_{r}^{A} − s_{r−1}^{A} ∘ A[ε_{0}, ε_{0}].

Lastly, recalling that d_{r}^{A} = A[ε_{0}, ε_{0}]−d_{r−1}^{A′}, the latter identity can be added to (7.4.54), to deduce

q_{r−1}^{A′} ∘ (d_{r+1}^{A} ∘ s_{r}^{A} + s_{r−1}^{A} ∘ d_{r}^{A}) = q_{r−1}^{A′} ∘ Sh_{r}^{A} ∘ AW_{r}^{A} − q_{r−1}^{A′}.

Now, to prove the assertion in degree r, it suffices to observe that q_{r}^{A} factors through q_{r−1}^{A′}. ♢

Claim 7.4.52and theorem 7.4.31(ii) supply the second sought modification, and conclude the proof of the theorem. □

7.4.55. Let (𝒜 , ⊗) be an abelian tensor category, A[•] and B[•] two objects of s.𝒜, and define the bisimplicial objects A ⊠ B and B ⊠ A, as well as the simplicial objects A ⊗ B and B ⊗ A of 𝒜 as in remark 7.4.18(iii). Notice that the system of commutativity constraints (Ψ_{A[n],B[n]} | n ∈ ℕ) amounts to an isomorphism

Ψ_{A⊗B} : A ⊗ B →^{∼} B ⊗ A in s.𝒜

whence an isomorphism Ψ_{(A⊗B)_{•}} : (A ⊗ B)_{•} →^{∼} (B ⊗ A)_{•} on the respective unnormalized complexes.

Proposition 7.4.56. With the notation of (7.4.55), the diagram of chain complexes

\xymatrix{Tot(A ⊠ B)_{••} \ar[r]^{Ψ_{A•,B•}^{•}} \ar[d]_{_{Sh{A⊠B \atop •}}} & Tot(B ⊠ A)_{••} \ar[d]^{_{Sh{B⊠A \atop •}}} \\ (A ⊗ B)_{•} \ar[r]^{Ψ_{(A⊗B)•}} & (B ⊗ A)_{•}}

commutes, where Ψ_{A_{•},B_{•}}^{•} is the commutativity constraint for the unnormalized chain complexes A_{•} and B_{•}, as in (7.1.17).

Proof. The assertion boils down to the identity

(7.4.57)  (−1)^{pq} · Sh_{q,p}^{B⊠A} ∘ Ψ_{A[p],B[q]} = Ψ_{A[n],B[n]} ∘ Sh_{p,q}^{A⊠B}

for every p, q ∈ ℕ with p + q = n. For the latter, suppose first that p = 0, in which case

Sh_{p,q}^{A⊠B} = A[η_{0} ∘ ⋯ ∘ η_{q−1}] ⊗ 1_{B[q]} and Sh_{q,p}^{B⊠A} = 1_{B[q]} ⊗ A[η_{0} ∘ ⋯ ∘ η_{q−1}]

from which we derive (7.4.57), using the naturality of Ψ (details left to the reader). Likewise we argue for the case where q = 0. For the general case, we proceed by induction on n. The cases n = 0, 1 have already been dealt with, so suppose r ≥ 2, and that the sought identity is already known for every pair of integers whose sum is < n, and every objects A, B of s.𝒜. By the foregoing, we may also assume that both p, q > 0, and then claim 7.4.38(i) implies that

Sh_{p,q}^{A⊠B} = (−1)^{q} · Sh_{p−1,q}^{γA⊠γB} ∘ (1_{A[p]} ⊗ B[η_{q}]) + Sh_{p,q−1}^{γA⊠γB} ∘ (A[η_{p}] ⊗ 1_{B[q]}).

However, it follows easily from remark 3.7.34(iii) that Ψ is an additive functor in both of its arguments; combining with the inductive assumption, we deduce that

Ψ_{A[n],B[n]} ∘ Sh_{p,q}^{A⊠B} = (−1)^{pq} · Sh_{q,p−1}^{γB⊠γA} ∘ Ψ_{A[p],B[q+1]} ∘ (1_{A[p]} ⊗ B[η_{q}])
+ (−1)^{p(q−1)} · Sh_{q−1,p}^{γB⊠γA} ∘ Ψ_{A[p+1],B[q]} ∘ (A[η_{p}] ⊗ 1_{B[q]})
= (−1)^{pq} · Sh_{q,p−1}^{γB⊠γA} ∘ (B[η_{q}] ⊗ 1_{A[p]}) ∘ Ψ_{A[p],B[q]}
+ (−1)^{p(q−1)} · Sh_{q−1,p}^{γB⊠γA} ∘ (1_{B[q]} ⊗ A[η_{p}]) ∘ Ψ_{A[p],B[q]}
= (−1)^{pq} · Sh_{q,p}^{B⊠A} ∘ Ψ_{A[p],B[q]}

where the last equality follows again from claim 7.4.38(i) and the additivity of Ψ. □

7.4.58. The shuffle map is also associative, in the following sense. Let

A := (A[p, q, r] | p, q, r ∈ ℕ)

be any triple simplicial object of the abelian category 𝒜, and denote by A^{(1,2)} (resp. A^{(2,3)}) the diagonal bisimplicial object of 𝒜 extracted from A by the rule

A^{(1,2)}[p, q] := A[p, p, q] ( resp. A^{(2,3)}[p, q] := A[p, q, q] ) for every p, q ∈ ℕ.

Let also A_{•••} be the triple chain complex associated to A, and A_{•••}^{Δ} the diagonal chain complex extracted from A_{•••}. Moreover, denote by A′ (resp. A′′) the bisimplicial object of s.𝒜 given by the rule :

[p, q] ↦ A[p, q, •] ( resp. [p, q] ↦ A[•, p, q] ) for every p, q ∈ ℕ.

Proposition 7.4.59. With the notation of (7.4.58), we have a commutative diagram in C(𝒜)

\xymatrix{Tot(A_{•••}) \ar[r] \ar[d] & Tot(A_{••}^{(1,2)}) \ar[d]^{_{Sh{A^{(1,2)} \atop •}}} \\ Tot(A_{••}^{(2,3)}) \ar[r]^{_{Sh{A^{(2,3)} \atop •}}} & A_{•••}^{Δ}}

whose top horizontal (resp. left vertical) arrow is obtained as the composition of Sh_{•}^{A′} (resp. Sh_{•}^{A′′}) with the functor Tot ∘ C(U_{𝒜} ) : C(s.𝒜) → C(C(𝒜)) → C(𝒜).

Proof. For every p, q, r ∈ ℕ, denote by

\begin{matrix}Sh_{p,q}^{A′}[r] : & A[p, q, r] → A[p + q, p + q, r] = A^{(1,2)}[p + q, r] \\ Sh_{q,r}^{A′′}[p] : & A[p, q, r] → A^{(2,3)}[p, q + r] \end{matrix}

respectively the [r]-component of Sh_{p,q}^{A′} and the [p]-component of Sh_{q,r}^{A′′}. The assertion boils down to the identity :

Sh_{p+q,r}^{A^{(1,2)}} ∘ Sh_{p,q}^{A′}[r] = Sh_{p,q+r}^{A^{(2,3)}} ∘ Sh_{q,r}^{A′′}[p] for every p, q, r ∈ ℕ.

To check the latter, set

D′ := γ_{2} ∘ γ_{1}A′ D^{(1,2)} := γ_{2} ∘ γ_{1}(A^{(1,2)})

and define likewise D′′ and D^{(2,3)} (notation of (7.4.17)). Also, let A[p, q, r] := 0 whenever one of the indices p, q, r is < 0, and define Sh_{p,q} to be the zero map, when either p or q is strictly negative (cp. the proof of proposition 7.4.36). We argue by induction on n := p + q + r; the case n = 0 is trivial, so suppose that n > 0, and that the assertion is already known for all indices whose sum is < n, and all triple simplicial objects of 𝒜. Notice as well that the assertion trivially holds as well if any of the indices p, q, r is strictly negative, since in this case both sides are the zero map. Hence, we may assume that p, q, r ∈ ℕ; in this case, by applying claim 7.4.38(i), first to A^{(1,2)} and then to A′, we compute

Sh_{p+q,r}^{A^{(1,2)}} ∘ Sh_{p,q}^{A′}[r] = (−1)^{r} · Sh_{p+q−1,r}^{D^{(1,2)}} ∘ A[1_{[p+q]}, 1_{[p+q]}, η_{r}] ∘ Sh_{p,q}^{A′}[r]
+ Sh_{p+q,r−1}^{D^{(1,2)}} ∘ A[η_{p+q}, η_{p+q}, 1_{[r]}] ∘ Sh_{p,q}^{A′}[r]
= (−1)^{r} · Sh_{p+q−1,r}^{D^{(1,2)}} ∘ Sh_{p,q}^{A′}[r + 1] ∘ A[1_{[p]}, 1_{[q]}, η_{r}]
+ Sh_{p+q,r−1}^{D^{(1,2)}} ∘ Sh_{p,q}^{D′}[r] ∘ A[η_{p}, η_{q}, 1_{[r]}]  (by (7.4.39))
= (−1)^{q+r} · Sh_{p+q−1,r}^{D^{(1,2)}} ∘ Sh_{p−1,q}^{D′}[r + 1] ∘ A[1_{[p]}, η_{q}, η_{r}]
+ (−1)^{r} · Sh_{p+q−1,r}^{D^{(1,2)}} ∘ Sh_{p,q−1}^{D′}[r + 1] ∘ A[η_{p}, 1_{[q]}, η_{r}]
+ Sh_{p+q,r−1}^{D^{(1,2)}} ∘ Sh_{p,q}^{D′}[r] ∘ A[η_{p}, η_{q}, 1_{[r]}]
= (−1)^{q+r} · Sh_{p−1,q+r}^{D^{(2,3)}} ∘ Sh_{q,r}^{D′′}[p] ∘ A[1_{[p]}, η_{q}, η_{r}]
+ (−1)^{r} · Sh_{p,q+r−1}^{D^{(2,3)}} ∘ Sh_{q−1,r}^{D′′}[p + 1] ∘ A[η_{p}, 1_{[q]}, η_{r}]
+ Sh_{p,q+r−1}^{D^{(2,3)}} ∘ Sh_{q,r−1}^{D′′}[p + 1] ∘ A[η_{p}, η_{q}, 1_{[r]}]

where the last identity holds by inductive assumption. On the other hand, by applying claim 7.4.38(i) to A^{(2,3)} we get

Sh_{p,q+r}^{A^{(2,3)}} ∘ Sh_{q,r}^{A′′}[p] = (−1)^{q+r} · Sh_{p−1,q+r}^{D^{(2,3)}} ∘ A[1_{[p]}, η_{q+r}, η_{q+r}] ∘ Sh_{q,r}^{A′′}[p]
+ Sh_{p,q+r−1}^{D^{(2,3)}} ∘ A[η_{p}, 1_{[q+r]}, 1_{[q+r]}] ∘ Sh_{q,r}^{A′′}[p]
= (−1)^{q+r} · Sh_{p−1,q+r}^{D^{(2,3)}} ∘ Sh_{q,r}^{D′′}[p] ∘ A[1_{[p]}, η_{q}, η_{r}]
+ Sh_{p,q+r−1}^{D^{(2,3)}} ∘ Sh_{q,r}^{A′′}[p + 1] ∘ A[η_{p}, 1_{[q]}, 1_{[r]}]  (by (7.4.39))

and after applying again claim 7.4.38(i) to A′′ and comparing with the foregoing expression for Sh_{p+q,r}^{A^{(1,2)}} ∘ Sh_{p,q}^{A′}[r], we obtain the sought identity. □

Theorem 7.4.60 (Dold-Puppe-Kan). For any abelian category 𝒜, and any k ∈ ℕ, we have : (i) The functors N_{𝒜} and N_{𝒜 ,k} are equivalences. (ii) If f, g are any two morphisms in s.𝒜, then there exists a simplicial homotopy from f

to g if and only if there exists a chain homotopy from N_{•}f to N_{•}g.

Proof. We easily reduce to the case where 𝒜 is small, and then there exists a fully faithful embedding 𝒜 → ℬ, where ℬ is a complete and cocomplete abelian tensor category, with internal Hom functor (lemma 3.7.46).

(i): We first construct an explicit quasi-inverse for the functors N_{ℬ} and N_{ℬ,k} , as follows. For every i ∈ ℕ, consider the cochain complex 𝕂⟨i⟩_{•} defined in example 7.4.29(ii); notice that every morphism φ : [i] → [i′] in Δ induces a morphism

𝕂⟨φ⟩_{•} := N_{•}ℤ^{Δ_{φ}} : 𝕂⟨i⟩_{•} → 𝕂⟨i′⟩_{•}

(notation of remark 7.4.19(iii)). Hence, the system (𝕂⟨i⟩_{•} | i ∈ ℕ) amounts to a cosimplicial object of C^{≤0}(ℤ-Mod). Now, let U be a unit of the tensor category ℬ; for any object C_{•} of C^{≤0}(ℬ) (resp. of C^{[−k,0]}(ℬ)), we set

K_{C}[i] := ℋ om_{C(ℬ)} (𝕂⟨i⟩_{•} ⊗_{ℤ} U[0], C_{•}) for every i ∈ ℕ (resp. for every i ≤ k)

where ℋ om_{C(ℬ)} is the functor constructed in example 7.1.16(vii), and the mixed tensor product is defined as in (7.1.18). By the foregoing, it is clear that the system (K_{C}[i] | i ∈ ℕ) amounts to an object of s.ℬ (resp. of s_{k}.ℬ). In order to compute N_{•}K_{C}, let us set

ℤ_{+}^{Δ_{i}} := ∑_{n=1}^{i} Im (ℤ^{Δ_{εn}} : ℤ^{Δ_{i−1}} → ℤ^{Δ_{i}})  ℤ_{0}^{Δ_{i}} := ℤ^{Δ_{i}}/ℤ_{+}^{Δ_{i}}  \overline{𝕂}⟨i⟩_{•} := N_{•}ℤ_{0}^{Δ_{i}}

for every i > 0, as well as ℤ_{0}^{Δ_{0}} := ℤ^{Δ_{0}} and \overline{𝕂}⟨0⟩_{•} := 𝕂⟨0⟩_{•}; thus, \overline{𝕂}⟨i⟩_{•} is also the quotient of 𝕂⟨i⟩_{•} by the sum of the images of the morphisms 𝕂⟨ε_{n}⟩, for n = 1, …, i. With this notation, a simple inspection of the definition shows that

N_{i}K_{C} = ℋ om_{C(ℬ)} (\overline{𝕂}⟨i⟩_{•} ⊗_{ℤ} U[0], C_{•}) for every i ∈ ℕ (resp. for every i ≤ k).

On the other hand, using the explicit description of example 7.4.29(ii), it is easily seen that

\overline{𝕂}⟨i⟩_{n} →^{∼} \{ \begin{matrix}ℤ & if n = i or n = i − 1 ≥ 0 \\ 0 & otherwise. \end{matrix}

More precisely, \overline{𝕂}⟨i⟩_{i} (resp. \overline{𝕂}⟨i⟩_{i−1}) is generated by the basis element e_{1} of 𝕂⟨i⟩_{i} (resp. e_{ε_{0}} of 𝕂⟨i⟩_{i−1}) corresponding to the identity map 1_{[i]} (resp. corresponding to ε_{0} : [i − 1] → [i]). Furthermore, example 7.4.29(i) shows that the differential 𝕂⟨i⟩_{i} → 𝕂⟨i⟩_{i−1} maps e_{1} to e_{ε_{0}}, so it corresponds to the identity map ℤ → ℤ, under the foregoing identification. Taking into account remark 3.7.12(iv), we deduce that N_{i}K_{C} is the kernel of the morphism

(7.4.61)  C_{i} ⊕ C_{i−1} → C_{i−1} ⊕ C_{i−2}

given by the matrix

( \begin{matrix}(−1)^{i} · d_{i}^{C} & 1_{C_{i−1}} \\ 0 & (−1)^{i+1} · d_{i−1}^{C} \end{matrix} )

and since d_{i−1}^{C} ∘ d_{i}^{C} = 0, the latter is just C_{i}; more precisely, C_{i} is identified with this kernel, via the monomorphism

(7.4.62)  (1_{C_{i}}, (−1)^{i+1} · d_{i}^{C}) : C_{i} → C_{i} ⊕ C_{i−1}.

This identification N_{i}K_{C} →^{∼} C_{i} can also be described as follows. For every Z ∈ Ob(ℬ), let

Hom_{ℬ} (Z, C_{•})

be the cochain complex such that Hom_{ℬ} (Z, C_{•})_{n} := Hom_{ℬ} (Z, C_{n}) for every n ∈ ℤ, with differentials induced by those of C_{•}, in the obvious way; then we have natural identifications

Hom_{C(ℤ-Mod)}(\overline{𝕂}⟨i⟩_{•}, Hom_{ℬ} (Z, C_{•})) →^{∼} Hom_{C(ℬ)} (\overline{𝕂}⟨i⟩_{•} ⊗_{ℤ} Z[0], C_{•}) →^{∼} Hom_{ℬ} (Z, N_{i}K_{C})

(see example 7.1.16(vii)) whose composition with the induced isomorphism

Hom_{ℬ} (Z, N_{i}K_{C}) →^{∼} Hom_{ℬ} (Z, C_{i})

is given by the rule :

(7.4.63)  (φ_{•} : \overline{𝕂}⟨i⟩_{•} → Hom_{ℬ} (Z, C_{•})) ↦ (φ_{i}(ε_{1}) : Z → C_{i}).

It remains to determine the differential d_{i+1}^{N} : N_{i+1}K_{C} → N_{i}K_{C}; by definition, the latter is induced by the morphism \overline{𝕂}⟨ε_{0}⟩_{•} : \overline{𝕂}⟨i⟩_{•} → \overline{𝕂}⟨i + 1⟩_{•}. In turn, the foregoing description says that \overline{𝕂}⟨ε_{0}⟩_{•} is naturally identified with the morphism of C(ℤ-Mod)

\xymatrix{\overline{𝕂}⟨i⟩_{•} \ar[d] & 0 \ar[r] & 0 \ar[r] \ar[d] & ℤ \ar[d]_{1_{ℤ}} \ar[r] & ℤ \ar[d] \ar[r] & 0 \\ \overline{𝕂}⟨i + 1⟩_{•} & 0 \ar[r] & ℤ \ar[r] & ℤ \ar[r] & 0 \ar[r] & 0}

(where the two copies of ℤ on the top horizontal row are placed in homological degrees i and i − 1), so d_{i+1}^{N} is deduced from the morphism in C(ℬ)

\xymatrix{0 \ar[r] & C_{i+1} ⊕ C_{i} \ar[r] \ar[d] & C_{i} ⊕ C_{i−1} \ar[d]_{1} \ar[r] & 0 \ar[r] \ar[d] & 0 \\ 0 \ar[r] & 0 \ar[r] & C_{i} ⊕ C_{i−1} \ar[r] & C_{i−1} ⊕ C_{i−2} \ar[r] & 0}

whose rows are given by the morphisms (7.4.61). Therefore, via the identification (7.4.62), the morphism d_{i+1}^{N} becomes none else than (−1)^{i+1} · d_{i+1}^{C}, i.e. we have obtained a natural isomorphism

ω_{•}^{C} : N_{•}K_{C} →^{∼} C_{•}

for every complex in C^{≤0}(ℬ) (resp. in C^{[−k,0]}(ℬ)).

Conversely, let B[•] be any object of s.ℬ, and Z any object of ℬ; clearly

N_{•}(Δ_{k} ⊗ s.Z) = 𝕂⟨i⟩_{•} ⊗_{ℤ} Z[0] for every i ∈ ℕ.

In view of example 7.1.16(vii) and remark 7.4.19(ii), we deduce a natural transformation

(7.4.64)  \xymatrix{Hom_{ℬ} (Z, B[i]) \ar[r]^{a} & Hom_{s.ℬ} (Δ_{i} ⊗ s.Z, B) \ar[d]^{b} \\ Hom_{ℬ} (Z, K_{N_{•}B}[i]) & Hom_{C(ℬ)} (𝕂⟨i⟩_{•} ⊗_{ℤ} Z[0], N_{•}B) \ar[l]_{c}}

which, by Yoneda’s lemma, comes from a unique morphism in ℬ

ψ_{i}^{B} : B[i] → K_{N_{•}B}[i] for every i ∈ ℕ.

The same construction applies, in case B is an object of s_{k}.ℬ : we need only replace the group Hom_{s.ℬ} (Δ_{i} ⊗ s.Z, B) by Hom_{s_{k}.ℬ} (s.trunc_{k}(Δ_{i} ⊗ s.Z), B) in (7.4.64) : see remark 7.4.19(iv). Moreover, remark 7.4.19(iii) implies that the system ψ^{B} := (ψ_{i}^{B} | i ∈ ℕ) amounts to a morphism B → K_{N_{•}B} in s.ℬ (resp. in s_{k}.ℬ). By the same token, the ℤ-linear isomorphism a maps the abelian subgroup Hom_{ℬ} (Z, N_{i}B) isomorphically onto the subgroup

Hom_{s.ℬ} (ℤ_{0}^{Δ_{i}} ⊗_{ℤ} s.Z, B) →^{∼} Hom_{s.ℤ-Mod}(ℤ_{0}^{Δ_{i}}, Hom_{ℬ} (Z, B))

for every Z ∈ Ob(ℬ) and every i ∈ ℕ, where Hom_{ℬ} (Z, B) is the simplicial abelian group as in example 7.4.29(iii). Explicitly, if β : Z → N_{i}B is any morphism, then a(β) is the unique morphism ℤ_{0}^{Δ_{i}} → Hom_{ℬ} (Z, B) of simplicial abelian groups such that a(β)(e_{1}) = β, where e_{1} ∈ \overline{𝕂}⟨i⟩_{i} ⊂ ℤ_{0}^{Δ_{i}}[i] is the basis element described in the foregoing. Likewise, we have natural identifications

Hom_{C(ℤ-Mod)}(𝕂⟨i⟩_{•}, Hom_{ℬ} (Z, N_{•}B)) →^{∼} Hom_{C(ℬ)} (𝕂⟨i⟩_{•} ⊗_{ℤ} Z[0], N_{•}B)

and b restricts to a map

Hom_{s.ℤ-Mod}(ℤ_{0}^{Δ_{i}}, Hom_{ℬ} (Z, B)) → Hom_{C(ℤ-Mod)}(\overline{𝕂}⟨i⟩_{•}, Hom_{ℬ} (Z, N_{•}B)) φ ↦ N_{•}φ.

Again, the same applies to an object of s_{k}.ℬ, by taking suitable truncated variants of the above constructions. Taking into account (7.4.63), we conclude that ψ^{B} induces an isomorphism

N_{•}ψ^{B} : N_{•}B →^{∼} N_{•}K_{N_{•}B}

which is inverse to ω^{N_{•}B}. To finish the proof of assertion (i) for the category ℬ, it then suffices to remark :

Claim 7.4.65. The functors N_{ℬ} and N_{ℬ,k} are conservative.

Proof of the claim. We show, by induction on n, that if a morphism h in s.ℬ or s_{k}.ℬ (for any k ∈ ℕ) induces an isomorphism N_{•}h, then h[n] is an isomorphism for every n ∈ ℕ (resp. for every n ≤ k). The assertion is obvious for n = 0, hence suppose that n > 0, and that h[n−1] is known to be an isomorphism whenever h is a morphism in s.ℬ or in s_{k}.ℬ (for arbitrary k ∈ ℕ) such that N_{•}h is an isomorphism. Let h : A → B be any such morphism in s.ℬ or in s_{k}.ℬ. If h is a morphism in s_{0}.ℬ, we are done, hence we may suppose that either h is a morphism in s.ℬ or k > 0. Set

A′ := Ker(g_{A} : γA → A) ( resp. A′ := Ker(g_{A} : γ_{k−1}A → s.trunc_{k−1}A) )

as well as B′ := Ker(g_{B}) (notation of remark 7.4.12(ii)). Notice that g_{A} is an epimorphism, since ∂_{n+2} admits the section σ_{n+1}, for every n ∈ ℕ (resp. for every n ≤ k). Therefore, we have a commutative diagram in s.ℬ with exact rows

\xymatrix{0 \ar[r] & A′ \ar[r] \ar[d]_{h′} & γA \ar[r] \ar[d]_{γh} & A \ar[r] \ar[d]^{h} & 0 \\ 0 \ar[r] & A′ \ar[r] & γA \ar[r] & A \ar[r] & 0}

(resp. a corresponding diagram in s_{k−1}.ℬ). By inspecting the definitions, it is easily seen that N_{•}A′ = (N_{•}A)[−1], N_{•}A′ = (N_{•}A)[−1], and N_{•}h′ = (N_{•}h)[−1]; especially, N_{•}h′ is an isomorphism, so h′[n − 1] is an isomorphism, by inductive assumption. The same holds also for h[n − 1], and we conclude that γh[n] is an isomorphism. But γh[n] = h[n + 1], so we are done. ♢

Lastly, in order to prove assertion (i) for the original category 𝒜, it suffices to notice :

Claim 7.4.66. Let C_{•} be any object of C(𝒜) (resp. of C^{[−k,0]}(𝒜)), and regard C_{•} as an object of C(ℬ) (resp. of C^{[−k,0]}(ℬ)), via the fully faithful embedding 𝒜 → ℬ. Then K_{C} is isomorphic to an object of s.𝒜 (resp. s_{k}.𝒜), regarded as a full subcategory of s.ℬ (resp. of s_{k}.ℬ), via the same embedding.

Proof of the claim. This follows easily, by remarking that 𝕂⟨i⟩_{•} lies in C^{[−i,0]}(ℤ-Mod) for every i ∈ ℕ, and 𝕂⟨i⟩_{j} is a finitely generated abelian group for every i, j ∈ ℕ : details left to the reader. ♢

(ii): First, let f, g : A → B be two morphisms in s.𝒜, and u : Δ_{1} ⊗ A → B a homotopy from f to g (see remark 7.4.18(v)); especially,

u ∘ (Δ_{ε_{1}} ⊗ A) = f and u ∘ (Δ_{ε_{0}} ⊗ A) = g.

Notice that

ℤ_{•}^{Δ_{1}} ⊗_{ℤ} A_{•} = Tot(Δ_{1} ⊠ A)_{••}

(notation of remark 7.4.18(iii) and example 7.4.29(i)); there follows a morphism in C(𝒜)

ũ_{•} : ℤ_{•}^{Δ_{1}} ⊗_{ℤ} A_{•} ⟶^{Sh_{•}} (Δ_{1} ⊗ A)_{•} ⟶^{q_{•}} N_{•}(Δ_{1} ⊗ A) ⟶^{N_{•}u} N_{•}B

where Sh_{•} denotes the shuffle map for the bisimplicial object A ⊠ Δ_{1}, and q_{•} is the projection defined in (7.4.30). Moreover, the maps Δ_{ε_{i}}[0] : Δ_{0}[0] → Δ_{1}[0] (i = 0, 1) induce morphisms

ẽ_{i,n} := ℤ^{Δ_{ε_{i}}[0]} ⊗_{ℤ} A_{n} : A_{n} → ℤ^{Δ_{1}[0]} ⊗_{ℤ} A_{n} ⊂ (ℤ_{•}^{Δ_{1}} ⊗_{ℤ} A_{•})_{n}

that amount to morphisms of cochain complexes ẽ_{i,•} : A_{•} → ℤ_{•}^{Δ_{1}} ⊗_{ℤ} A_{•} (i = 0, 1), and a simple inspection of the definitions shows that

Sh_{•} ∘ ẽ_{i,•} = (Δ_{ε_{i}} ⊗ A)_{•} for i = 0, 1

whence

ũ_{•} ∘ ẽ_{1,•} = f_{•} and ũ_{•} ∘ ẽ_{0,•} = g_{•}.

The construction makes it clear that ẽ_{i} restricts to a morphism N_{•}A → 𝕂⟨1⟩_{•} ⊗_{ℤ} N_{•}A, and the latter is none else than the map ι_{i} ⊗_{ℤ} N_{•}A, with the notation of remark 7.1.19(ii). We conclude that the morphism

\overline{u}_{•} : 𝕂⟨1⟩_{•} ⊗_{ℤ} N_{•}A ↪ ℤ_{•}^{Δ_{1}} ⊗_{ℤ} A_{•} \underrightarrow{ũ_{•}} N_{•}B

is a homotopy from N_{•}f to N_{•}g (notation of (7.4.30)).

Conversely, suppose that \overline{v}_{•} : 𝕂⟨1⟩_{•} ⊗_{ℤ} N_{•}A → N_{•}B is a homotopy from N_{•}f to N_{•}g. Since 𝕂⟨1⟩_{•} is a direct summand of ℤ_{•}^{Δ_{1}} and N_{•}A is a direct summand of A_{•}, we may extend \overline{v}_{•} to a morphism

ṽ_{•} : ℤ_{•}^{Δ_{1}} ⊗_{ℤ} A_{•} → N_{•}B

such that ṽ_{•} is the zero morphism on the direct summands other than 𝕂⟨1⟩_{•} ⊗_{ℤ} N_{•}A. Next, consider the composition

v_{•} : N_{•}(Δ_{1} ⊗ A) ⟶^{j_{•}} (Δ_{1} ⊗ A)_{•} ⟶^{AW_{•}} A_{•} ⊗_{ℤ} ℤ_{•}^{Δ_{1}} ⟶^{Ψ_{•}} ℤ_{•}^{Δ_{1}} ⊗_{ℤ} A_{•} \underrightarrow{ṽ_{•}} N_{•}B

where j_{•} is the natural injection (see (7.4.30)), AW_{•} is the Alexander-Whitney map for the bisimplicial object A ⊠ Δ_{1} (notice that Δ_{1} ⊗ A = (A ⊠ Δ_{1})^{Δ}), and Ψ_{•} is the commutativity constraint (see example 7.1.16(i)). By (i), the morphism v_{•} comes from a unique morphism

v : Δ_{1} ⊗ A → B in s.𝒜 .

On the other hand, since N_{p+q}A is contained in the kernel of A[ε_{p,0}^{q∨}] for every p, q ∈ ℕ, it is easily seen that the diagram

\xymatrix{N_{•}A \ar[r]^{ι_{i}⊗_{ℤ}N_{•}A} \ar[d]_{N_{•}(Δ_{ε_{i}}⊗A)} & 𝕂⟨1⟩_{•} ⊗_{ℤ} N_{•}A \ar[d] \\ N_{•}(Δ_{1} ⊗ A) \ar[r]^{Ψ_{•}∘AW_{•}∘j_{•}} & ℤ_{•}^{Δ_{1}} ⊗_{ℤ} A_{•}}

commutes for i = 0, 1. We conclude that v is a homotopy from f to g. □

Corollary 7.4.67. Let 𝒜 be any abelian category. We have :

(i) If k ∈ ℕ is any integer, and A any k-truncated simplicial object of 𝒜, then

H_{i}(cosk_{k}A) = 0 for every i ≥ k.

(ii) Every homotopically trivial augmented simplicial object of 𝒜 is aspherical.

Proof. (i): Denote by t_{≤k} : C^{≤0}(𝒜) → C^{[−k,0]}(𝒜) the brutal truncation functor (see (7.1.1)). In light of theorem 7.4.60, we see that t_{≤k} admits a right adjoint v_{k} : C^{[−k,0]}(𝒜) → C^{≤0}(𝒜), and clearly there are natural isomorphisms

N_{•}cosk_{k}A →^{∼} v_{k}N_{•}A for every A ∈ Ob(s_{k}.𝒜) .

Taking into account theorem 7.4.31(iii), we are then reduced to showing

Claim 7.4.68. H_{i}(v_{k}K_{•}) = 0 for every (K_{•}, d_{•}) ∈ Ob(C^{[−k,0]}) and every i ≥ k.

Proof of the claim. Indeed it is easily seen that :

(v_{k}K_{•})_{i} = \{ \begin{matrix}K_{i} & for ≤ k \\ Ker d_{k} & for i = k + 1 \\ 0 & for i > k + 1 \end{matrix}

and the differential of v_{k}K_{•} in degree ≤ k agrees with that of K_{•}, whereas in degree k + 1 it is the natural inclusion map (details left to the reader). The claim follows immediately. ♢

(ii) follows directly from theorems 7.4.60(ii) and 7.4.31(iii), and remark 7.1.9(ii). □

7.5. Simplicial sets. This section, which complements the previous one, collects some classical material pertaining to the homotopy theory of the category of simplicial sets. The presentation is borrowed from [80] and [108], where much more may be found.

7.5.1. To begin with, notice that the category s.Set of simplicial sets is none else than the category of presheaves on the category Δ, so s.Set is complete and cocomplete, and all limits and colimits in s.Set are computed argumentwise (see (4.1)); also, all colimits and monomorphisms are universal, and all epimorphisms are universal effective.

Example 7.5.2. (i) We have already introduced in example 7.4.5the simplicial set Δ_{k} (for any k ∈ ℕ), which is just the image of [k] under the Yoneda embedding

h : Δ → s.Set.

To ease notation, for any morphism φ : [n] → [m] in Δ, we shall usually write φ : Δ_{n} → Δ_{m} instead of Δ_{φ}. By Yoneda’s lemma (proposition 1.2.6), we have natural identifications

(7.5.3)  A[n] →^{∼} Hom_{s.Set}(Δ_{n}, A)  for every A ∈ Ob(s.Set) and every n ∈ ℕ.

(ii) More generally, the category of simplicial sets contains all the products Δ_{n} × Δ_{m} (for every m, n ∈ ℕ). Furthermore, if S is any set and A any simplicial set, we may form the product s.S × A (notation of (7.4.4)), which we shall denote simply by S × A. Explicitly, we have

(S × A)[n] := S × A[n] for every n ∈ ℕ

and the faces and degeneracies of S × A are derived from those of A, in the obvious fashion. (iii) Moreover, lemma 1.4.8yields a natural presentation

A →^{∼} colim_{hΔ/A} h_{s.Set} ∘ ι_{A}

for every simplicial set A. By inspecting the definitions, we see that the latter amounts to the following description of A. Notice the natural identification Δ_{j}[i] →^{∼} Hom_{s.Set}(Δ_{i}, Δ_{j}) provided by (7.5.3); we have a natural diagram in s.Set

(7.5.4)  ∐_{i,j∈ℕ}(A[i] × Δ_{i}[j]) × Δ_{j} [figure: p_{•}; c_{•}]∐_{n∈ℕ} A[n] × Δ_{n} ⟶^{t_{•}} A

where p_{•} are c_{•} are the (unique) morphisms which restrict respectively to morphisms

{a} × Δ_{j} ⟵^{φ} {(a, φ)} × Δ_{i} ⟶^{1_{Δ_{i}}} {A[φ](a)} × Δ_{i}

for every i, j ∈ ℕ and every (a, φ) ∈ A[i]×Δ_{j}[i], and where t_{•} is the (unique) morphism whose restriction to {b}×Δ_{n} → A is the morphism corresponding to b ∈ A[n] under the identification (7.5.3), for every b ∈ A[n]. Then t_{•} is an epimorphism, and (7.5.4) naturally identifies A with the coequalizer of p_{•} and c_{•}.

(iv) Another useful simplicial set is the boundary of Δ_{k}, denoted

∂Δ_{k} for every integer k > 0

which is defined as the smallest subobject of Δ_{k} containing the images of all the face morphisms ε_{i} : Δ_{k−1} → Δ_{k}, for i = 0, …, k (see (7.4.6)). Thus, we have a natural epimorphism in s.Set

(7.5.5)  [k] × Δ_{k−1} → ∂Δ_{k}  for every k > 0

whose restriction to the subobject {i} × Δ_{k−1} agrees with ε_{i}, for every i = 0, …, k.

Notice that the simplicial identities yield a cartesian diagram in Δ^{∧}

\xymatrix{[k − 2] \ar[r]^{ε_{j−1}} \ar[d]_{ε_{i}} & [k − 1] \ar[d]^{ε_{i}} \\ [k − 1] \ar[r]^{ε_{j}} & [k]} whenever 0 ≤ i < j ≤ k.

Since the Yoneda embedding commutes with representable limits (corollary 1.4.3(vi)), there follows a cartesian diagram in s.Set

\xymatrix{Δ_{k−2} \ar[r]^{ε_{j−1}} \ar[d]_{ε_{i}} & Δ_{k−1} \ar[d]^{ε_{i}} \\ Δ_{k−1} \ar[r]^{ε_{j}} & Δ_{k}}  for 0 ≤ i < j ≤ k

(details left to the reader). Now, denote by S_{k} ⊂ [k]×[k] the subset of all pairs (i, j) with i < j; since (7.5.5) is effective (by (7.5.1)), we deduce a commutative diagram

\xymatrix{S_{k} × Δ_{k−2} \ar[r]^{ε′_{•}} \ar[r]_{ε′′_{•}} & [k] × Δ_{k−1} \ar[r] & ∂Δ_{k}}  in s.Set

which presents ∂Δ_{k} as the coequalizer of ε′_{•} and ε′′_{•}, where ε′_{•} (resp. ε′′_{•}) is the morphism whose restriction to {(i, j)} × Δ_{k−2} agrees with ε_{i} : {(i, j)} × Δ_{k−2} → {j} × Δ_{k−1} (resp. with ε_{j−1} : {(i, j)} × Δ_{k−2} → {i} × Δ_{k−1}) for every (i, j) ∈ S_{k}. As an immediate corollary, we see that if A is any simplicial set, a morphism ∂Δ_{k} → A is the same as the datum of an ordered sequence x_{0}, …, x_{k} of k + 1 elements of A[k − 1], such that

∂_{i}x_{j} = ∂_{j−1}x_{i} whenever 0 ≤ i < j ≤ k.

Moreover, if x ∈ A[n] is any element, we may regard x as a morphism \overline{x} : Δ_{n} → A via the natural identification (7.5.3), and then we shall often denote by

∂x : ∂Δ_{n} → A

the restriction of \overline{x} to the suboboject ∂Δ_{n}. Lastly, we set ∂Δ_{0} := s.∅.

(v) Another important simplicial set is Λ_{k}^{n}, which is defined for every k, n ∈ ℕ such that k > 0 and n = 0, …, k; namely, it is the smallest subobject of Δ_{k} that contains the images of all the face morphisms ε_{i} : Δ_{k−1} → Δ_{k}, except for the face ε_{n}. Thus, for every such k and n we have a natural monomorphism

ι_{k}^{n} : Λ_{k}^{n} → Δ_{k}.

The same argument as in (iv) yields a presentation of Λ_{k}^{n} as the coequalizer of two morphisms :

\xymatrix{S_{k}^{n} × Δ_{k−2} \ar[r]^{ε′_{•}} \ar[r]_{ε′′_{•}} & ([k] \ \{n\}) × Δ_{k−1} \ar[r] & Λ_{k}^{n}}

where S_{k}^{n} ⊂ S_{k} is the subset of all pairs (i, j) with i, j ≠ n, and ε′_{•}, ε′′_{•} are the restrictions of the morphisms with the same name appearing in (iii).

(vi) Let ψ : X → Y be any morphism of simplicial sets, y ∈ Y [0] any element, and j_{y} : Δ_{0} → Y the corresponding morphism of simplicial sets; the resulting fibre product

ψ^{−1}(y) := Δ_{0} ×_{Y} X

is called the fibre of ψ over y. Explicitly, set {y_{n}} := Im j_{y}[n] for every n ∈ ℕ. Then

ψ^{−1}(y)[n] = ψ[n]^{−1}(y_{n}) for every n ∈ ℕ

and the faces and degeneracies of ψ^{−1}(y) are the restrictions of the corresponding maps for X.

Remark 7.5.6. (i) From the simplicial identities of (7.4.6) we also get a commutative diagram

\xymatrix{Δ_{n+1} \ar[r]^{η_{i}} \ar[d]_{η_{j+1}} & Δ_{n} \ar[d]^{η_{j}} \\ Δ_{n} \ar[r]^{η_{i}} & Δ_{n−1}}

for every integer n > 0 and every i, j ≤ n − 1 with i ≤ j. We claim that this diagram is cocartesian in s.Set. Indeed, let X be any simplicial set, and f, g : Δ_{n} → X any two morphisms such that f ∘ η_{i} = g ∘ η_{j+1}. We set h := f ∘ ε_{j+1} : Δ_{n−1} → X, and we notice that

h = f ∘ η_{i} ∘ ε_{i} ∘ ε_{j+1} = g ∘ η_{j+1} ∘ ε_{i} ∘ ε_{j+1} = g ∘ η_{j+1} ∘ ε_{j+2} ∘ ε_{i} = g ∘ ε_{i}.

Therefore :

\begin{matrix}h ∘ η_{j} = & g ∘ ε_{i} ∘ η_{j} = g ∘ η_{j+1} ∘ ε_{i} = f ∘ η_{i} ∘ ε_{i} = f \\ h ∘ η_{i} = & f ∘ ε_{j+1} ∘ η_{i} = f ∘ η_{i} ∘ ε_{j+2} = g ∘ η_{j+1} ∘ ε_{j+2} = g. \end{matrix}

Lastly, since both η_{i} and η_{j} are epimorphisms, h is uniquely determined by either of the identities h ∘ η_{j} = f and h ∘ η_{i} = g, whence the assertion.

(ii) We point out that also the diagram

\xymatrix{Δ_{n+1} \ar[r]^{η_{i}} \ar[d]_{η_{i}} & Δ_{n} \ar[d]^{1_{Δn}} \\ Δ_{n} \ar[r]^{1_{Δn}} & Δ_{n}}

is trivially cocartesian for every n ∈ ℕ and every i ≤ n, since η_{i} is an epimorphism.

(iii) Let k ∈ ℕ be any integer. The categories Set of all small sets and f.Set of finite sets, both satisfy the conditions of (7.4.20), so we get left and right adjoints

sk_{k}, cosk_{k} : s_{k}.Set → s.Set sk_{k}, cosk_{k} : s_{k}.f.Set → s.f.Set

for the respective k-truncation functors.

(iv) For any simplicial set X and any integer r > 0, let us say that an element x ∈ X[r] is a degenerate simplex, if x = σ_{i}y for some i = 0, …, r − 1 and some y ∈ X[r − 1]. We claim that, for any k-truncated simplicial set Y, and every r > k, every element of sk_{k}Y [r] is a degenerate simplex. Indeed, let α : [k] → [r] be any morphism in Δ^{o}; notice first that, under the identification Y [k] →^{∼} sk_{k}Y [k] given by the unit of adjunction, the map sk_{k}Y [α] : sk_{k}Y [k] → sk_{k}Y [r] corresponds to the natural map j_{α} : F [k] → sk_{k}Y [r]. But sk_{k}Y [r] is the union of the images of all such maps (see example 1.2.23(i)), and on the other hand, any such map factors through some degeneracy map σ_{i}, whence the assertion.

(v) For any simplicial set X and every k, n ∈ ℕ with n ≥ k, the counits of adjunction give a natural commutative diagram

\xymatrix{sk_{k}(s.trunc_{k}X) \ar[rr]^{ε_{X}^{(k,n)}} \ar[dr]_{ε_{X}^{(k)}} &  & sk_{n}(s.trunc_{n}X) \ar[dl]^{ε_{X}^{(n)}} \\  & X}

amounting to a cocone with vertex X, and it follows easily from the discussion of (7.4.20) that s.trunc_{k}(ε_{X}^{(k,n)}) is an isomorphism for every such k, n ∈ ℕ, and the resulting morphism

colim_{n∈ℕ} sk_{n}(s.trunc_{n}X) → X

is an isomorphism. We define the n-th skeleton of X as the simplicial subset

Sk_{n}X := Im ε_{X}^{(n)} for every n ∈ ℕ.

Lemma 7.5.7. Let X be any simplicial set, r > 0 any integer, and x, y ∈ X[r] be any two degenerate simplices (see remark 7.5.6(iii)). The following holds :

(i) If ∂x = ∂y, then x = y.

(ii) There exists a unique pair (p, z) where p : [r] → [n] is an epimorphism and z ∈ X[n]

is a non-degenerate simplex such that x = X[p](z). (iii) The natural morphism sk_{n}(s.trunc_{n}X) → Sk_{n}X is an isomorphism for every n ∈ ℕ.

Proof. (i): Say that x = σ_{m}z and y = σ_{n}w for some z, w ∈ X[r − 1]. If m = n, we have

z = ∂_{m}σ_{m}z = ∂_{m}x = ∂_{m}y = ∂_{m}σ_{m}w = w

whence the assertion. Hence, we may assume that m < n, in which case we get :

z = ∂_{m}x = ∂_{m}σ_{n}w = σ_{n−1}∂_{m}w

so that x = σ_{m}σ_{n−1}∂_{m}w = σ_{n}σ_{m}∂_{m}w. Thus, we may replace z by σ_{m}∂_{m}w and m by n, and reduce to the foregoing case, so the proof is complete.

(ii): The existence of such a pair (p, z) is obvious. Next, suppose that (p′ : [r] → [n′], z′) is another such pair; from remark 7.5.6, it follows easily that there exists a cocartesian diagram of the form

\xymatrix{Δ_{r} \ar[r]^{Δ_{p}} \ar[d]_{Δ_{p′}} & Δ_{n} \ar[d]^{Δ_{q′}} \\ Δ_{n′} \ar[r]^{Δ_{q}} & Δ_{k}}

for a suitable k ≤ n, n′ and epimorphisms q : [n] → [k] and q′ : [n′] → [k] (details left to the reader). Regarding z and z′ as morphisms x : Δ_{n} → X and respectively x′ : [n′] → X, our assumption gives the identity :

z ∘ Δ_{p} = z′ ∘ Δ_{p′}

whence a unique w ∈ X[k] such that X[q′](w) = z and X[q](w) = z′. Since z and z′ are non-degenerate, it follows that n = k = n′ and q = q′ = 1_{Δ_{k}}, whence z = z′, as stated.

(iii): Set Y := sk_{n}(s.trunc_{n}X) for every n ∈ ℕ; the natural map f[k] : Y [k] → Sk_{n}[k] is surjective for every k ∈ ℕ, and is bijective for every k ≤ n, so it suffices to check that f[k] is injective for every k > n. Thus, fix k > N, and let y, y′ ∈ Y [k] be any two simplices such that f[k](y) = f[k](y′). Remark 7.5.6(iv) tells us that y and y′ are degenerate, say y = Y [p](z) and y′ = Y [p′](z′) for epimorphisms p : [k] → [m] and p : [k] → [m′], and non-degenerate simplices z ∈ Y [m], z′ ∈ Y [m′]. We must then have m, m′ ≤ n, and therefore x := f[m](z) and x′ := f[m′](z′) are two non-degenerate simplices of Sk_{n}X, such that

Sk_{n}[p](x) = f[k](y) = f[k](y′) = Sk_{n}[p′](x′).

By (ii), we deduce that p = p′ and x = x′, whence y = y′, as required. □

Remark 7.5.8. Let X be any simplicial set and n ∈ ℕ any integer.

(i) Clearly, Sk_{n}X ⊂ Sk_{n+1}X, and we have

(7.5.9)  X = ⋃_{r∈ℕ} Sk_{r}X.

We say that X has dimension ≤ n, if X = Sk_{n}X. Let s.Set_{≤n} be the full subcategory of s.Set whose objects are the simplicial set of dimension ≤ n; it follows easily from lemma 7.5.7(iii) that the adjunction (sk_{n}, trunc_{n}) establishes an equivalence

s.Set_{≤n} →^{∼} s_{k}.Set.

(ii) Moreover, Sk_{n}X can be obtained by “attaching n-cells” to Sk_{n−1}X, if n > 0. Namely, let E_{X}^{n} ⊂ X[n] denote the set of non-degenerate n-simplices of X; we get a commutative diagram

𝒟 : \xymatrix{E_{X}^{n} × ∂Δ_{n} \ar[r]^{E_{X}^{n}×i_{n}} \ar[d] & E_{X}^{n} × Δ_{n} \ar[d]^{c} \\ Sk_{n−1}X \ar[r] & Sk_{n}X}

where i_{n} : ∂Δ_{n} → Δ_{n} and the bottom horizontal arrow are the natural inclusion maps, and c is the unique morphism whose restriction to the factor \{x\} × Δ_{n} is the morphism \overline{x} : Δ_{n} → X corresponding to x, for every x ∈ E_{X}^{n}. We claim that 𝒟 is a cocartesian diagram. For the proof, notice that all the simplicial sets in 𝒟 have dimension ≤ n; by (i), it then suffices to check that trunc_{n}𝒟 is cocartesian, i.e. that the diagram of sets 𝒟[k] is cocartesian, for every k ≤ n (see (7.5.1)). The latter assertion is clear for k < n, since in this case both horizontal arrows of 𝒟[k] are isomorphisms. For k = n, notice that the complement S of E_{X}^{n} × ∂Δ_{n}[n] in E_{X}^{n} × Δ_{n}[n] is naturally identified with E_{X}^{n}, which is also the complement of Sk_{n−1}X[n] in Sk_{n}X[n]; it then suffices to remark that, under these natural bijections, the restriction of c[n] to S is identified with the identity map E_{X}^{n} →^{∼} E_{X}^{n} (details left to the reader).

Definition 7.5.10. Let φ : A → B and ψ : X → Y be two morphisms of simplicial sets.

(i) We say that ψ has the right lifting property with respect to φ, if for every commutative diagram

\xymatrix{A \ar[r]^{α} \ar[d]_{φ} & X \ar[d]^{ψ} \\ B \ar[r]^{β} & Y}  in s.Set

there exists a morphism γ : B → X such that γ ∘ φ = α and ψ ∘ γ = β. (ii) We say that ψ : X → Y is a Kan fibration (or briefly, that ψ is a fibration), if ψ has the right lifting property with respect to all the monomorphisms ι_{k}^{n} : Λ_{k}^{n} → Δ_{k}, for every k, n ∈ ℕ with k > 0 and n ≤ k (notation of example 7.5.2(v)).

(iii) We say that φ is a retract of ψ if there exists a commutative diagram of simplicial sets

\xymatrix{A \ar[r] \ar[d]_{φ} \ar[rr]^{1_{A}} & X \ar[d]^{ψ} \ar[r] & A \ar[d]^{φ} \\ B \ar[r] \ar[rr]_{1_{B}} & Y \ar[r] & B.}

(iv) We say that a simplicial set A is fibrant if the (unique) morphism A → Δ_{0} is a fibration.

Remark 7.5.11. In light of example 7.5.2(iv,v), it is clear that a simplicial set A is fibrant if and only if the following Kan extension condition holds. For every n, k ∈ ℕ with 0 ≤ k ≤ n + 1, and every sequence x_{0}, …, x_{k−1}, x_{k+1}, …, x_{n+1} of elements of A[n] such that

∂_{i}x_{j} = ∂_{j−1}x_{i} whenever 0 ≤ i < j ≤ n + 1 and i, j ≠ k

there exists an element x ∈ A[n + 1] such that

∂_{i}x = x_{i} for every i = 0, …, k − 1, k + 1, …, n + 1.

Example 7.5.12. Let A be any simplicial group, i.e. an object of s.Grp, where Grp denotes the category of groups. Then the simplicial set underlying A is fibrant. Indeed, suppose that n, k and x_{0}, …, x_{n+1} are as in remark 7.5.11. We construct by induction on i = −1, …, n + 1 an element y_{i} ∈ A[n + 1] such that ∂_{j}y_{i} = x_{j} whenever 0 ≤ j ≤ i, j ≠ k. Indeed, the condition is fulfilled trivially for i = −1, by setting y_{−1} := 1, the neutral element of the group A[n + 1]. Suppose that i ≥ 0, and y_{i−1} has already been exhibited as required; if i = k, we set y_{i} := y_{i−1}, which again trivially fulfills the stated condition. Otherwise, consider the element u := x_{i}^{−1} · (∂_{i}y_{i−1}), and notice that

∂_{j}u = (∂_{j}x_{i}^{−1}) · (∂_{j}∂_{i}y_{i−1}) = (∂_{j}x_{i}^{−1}) · (∂_{i−1}∂_{j}y_{i−1}) = (∂_{j}x_{i}^{−1}) · (∂_{i−1}x_{j}) = 1

for every j = 0, …, i − 1 with j ≠ k. Hence, set y_{i} := y_{i−1} · (σ_{i}u)^{−1}; we have

∂_{j}y_{i} = (∂_{j}y_{i−1}) · (∂_{j}σ_{i}u)^{−1} = x_{i} · (σ_{i−1}∂_{j}u) = x_{i} whenever 0 ≤ j < i.

Likewise :

∂_{i}y_{i} = (∂_{i}y_{i−1}) · (∂_{i}σ_{i}u)^{−1} = (∂_{i}y_{i−1}) · (∂_{i}σ_{i}∂_{i}y_{i−1})^{−1} · (∂_{i}σ_{i}x_{i}) = x_{i}

as needed. Thus, the element x := y_{n+1} fulfills the condition of remark 7.5.11.

For any given morphism ψ of s.Set, the set of all morphisms φ such that ψ has the right lifting property with respect to φ is closed under certain elementary operations, that are singled out in the following :

Definition 7.5.13. Let Σ be a set of monomorphisms of s.Set.

(i) We say that Σ is saturated if the following conditions hold :

(a) All isomorphisms of s.Set lie in Σ.

(b) For every countable system of morphisms in s.Set

A_{0} ⟶^{φ_{0}} A_{1} ⟶^{φ_{1}} A_{2} → ⋯

such that φ_{n} ∈ Σ for every n ∈ ℕ, also the induced morphism

A_{0} → colim_{n∈ℕ} A_{n}

lies in Σ.

(c) For any cocartesian diagram of simplicial sets :

\xymatrix{A \ar[r] \ar[d]_{φ} & A′ \ar[d]^{φ′} \\ B \ar[r] & B ⨿_{A} A′}

such that φ ∈ Σ, we have φ′ ∈ Σ.

(d) If (φ_{i} : A_{i} → B_{i} | i ∈ I) is an arbitrary family of elements of Σ, then also

∐_{i∈I} φ_{i} : ∐_{i∈I} A_{i} → ∐_{i∈I} B_{i}

lies in Σ.

(e) Any retract of any element of Σ lies also in Σ (see definition 7.5.10(iii)). (ii) The saturation of Σ is the intersection of all saturated sets of monomorphisms of s.Set

containing Σ.

7.5.14. With this terminology, for any morphism ψ of s.Set, let Σ(ψ) be the set of monomorphisms φ of s.Set such that ψ has the right lifting property with respect to φ. We point out the following simple observation :

Lemma 7.5.15. With the notation of (7.5.14), the following holds :

(i) Σ(ψ) is saturated, for any morphism ψ of s.Set.

(ii) For any cartesian square of simplicial sets

\xymatrix{X′ \ar[r] \ar[d]_{ψ′} & X \ar[d]^{ψ} \\ Y′ \ar[r] & Y}

we have Σ(ψ) ⊂ Σ(ψ′). (iii) If ψ′ is any retract of ψ, then Σ(ψ) ⊂ Σ(ψ′) (see definition 7.5.10(iii)). (iv) Especially, in both (ii) and (iii), if ψ is a fibration, the same holds for ψ′, and for any

y ∈ Y [0], the fibre ψ^{−1}(y) is a fibrant simplicial set (see example 7.5.2(vi)).

Proof. Left to the reader. □

7.5.16. The Kan extension condition isolates a class of simplicial sets on which the standard constructions of homotopy theory can be carried out. Many of these constructions come down to exhibiting certain morphisms from a product of the type Δ_{k} × Δ_{1} (for variable k ∈ ℕ) to given simplicial sets. To manipulate with ease such basic products, it will be useful to have at our disposal presentations for them, in the vein of example 7.5.2(iv,v).

To this aim, notice that Δ is a full subcategory of the category POSet of partially ordered sets (notation of example 1.1.6(iii)). We may then extend the Yoneda embedding of Δ into s.Set to a well defined functor

(7.5.17)  POSet → s.Set  (P, ≤) ↦ Δ_{P} .

Namely, for every partially ordered set (P, ≤), let h_{P} : POSet^{o} → Set be the image of P under the Yoneda embedding of the category POSet; then Δ_{P} is the restriction of h_{P} to the subcategory Δ^{o}. We notice :

Lemma 7.5.18. The functor (7.5.17) is fully faithful and commutes with all limits.

Proof. The Yoneda embedding commutes with representable limits (corollary 1.4.3(vi)), and the same holds for the restriction functor POSet^{∧} → s.Set, since limits are computed argumentwise in both of these categories (corollary 1.4.3(ii)). To see that (7.5.17) is faithful, it suffices to remark that there is a natural isomorphism

(7.5.19)  Δ_{P} [0] →^{∼} P  for every partially ordered set (P, ≤)

which induces a natural identification : Δ_{φ} = φ, for every morphism φ : (P, ≤) → (Q, ≤) of partially ordered sets. To check that (7.5.17) is full, let (P, ≤) and (Q, ≤) be any two objects of POSet, and f : Δ_{P} → Δ_{Q} a morphism of simplicial sets. The natural identification (7.5.19) yields a map φ := f[0] : P → Q, and it suffices to check that φ is a map of ordered sets such that Δ_{φ} = f. Thus, let x_{0}, x_{1} ∈ P be any two elements with x_{0} < x_{1}; there follows a unique morphism ψ : Δ_{1} → P of partially ordered sets such that ψ(i) = x_{i} for i = 0, 1. Then ψ ∈ Δ_{P} [1], and we may set

ψ′ := f[1](ψ) ∈ Δ_{Q}[1]  y_{i} := Δ_{Q}[ε_{i−1}](ψ′)  for i = 0, 1

(notation of (7.4.8)). Tracing back the definitions, we see that φ(x_{i}) = y_{i} for i = 0, 1, and the existence of ψ′ tells us that y_{0} ≤ y_{1}, i.e. φ is a morphism in POSet, as required. It remains only to show that Δ_{φ}[n] = f[n] for every n ∈ ℕ. However, let x : [n] → P be any element of Δ_{P} [n] and set y := f[n](x), so the image of x (resp. of y) is an increasing sequence (x_{0}, …, x_{n}) of elements of P (resp. (y_{0}, …, y_{n}) of Q); we have to check that φ(x_{k}) = y_{k} for every k = 0, …, n. Now, for any such k ≤ n, let β_{k} : [0] → [n] be the unique map such that β_{k}(0) = k; the correspondence (7.5.19) identifies x_{k} with Δ_{P} [β_{k}](x) and y_{k} with Δ_{Q}[β_{k}](y), whence the contention (details left to the reader). □

7.5.20. The category of partially ordered sets is complete; especially, all finite products

[k_{0}] × ⋯ × [k_{n}]

are representable in POSet : explicitly, the partial ordering on such a product is defined by declaring that

\underline{a} := (a_{0}, …, a_{n}) ≤ \underline{b} := (b_{0}, …, b_{n}) if and only if a_{i} ≤ b_{i} for every i = 0, …, n

for every pair of elements \underline{a}, \underline{b} ∈ [k_{0}] × ⋯ × [k_{n}]. Now, for any k, r ∈ ℕ with r ≤ k, consider the morphism in POSet

φ_{k,r} : [k + 1] → [k] × [1] such that φ_{k,r}(i) = \{ \begin{matrix}(0, i) & for i = 0, …, r \\ (1, i − 1) & for i = r + 1, …, k + 1. \end{matrix}

It is easily seen that every morphism [t] → [k] × [1] in POSet (for any t ∈ ℕ) factors through some φ_{k,r}, so we get an epimorphism of simplicial sets

φ_{k} : [k] × Δ_{k+1} → Δ_{k} × Δ_{1}

whose restriction to each subobject {r} × Δ_{k+1} is the morphism Δ_{φ_{k,r}} (notation of (7.5.17)). Now, let r, s be any two integers such that 0 ≤ r < s ≤ k; obviously, the intersection of the images of φ_{k,r} and φ_{k,s} is the subset

{(0, 0), …, (0, r), (1, s), …, (1, k)}

so we get a cartesian diagram in POSet

\xymatrix{[t] \ar[r]^{ε_{t,r+1}^{s−r}} \ar[d]_{ε_{t,r+1}^{s−r}} & [k + 1] \ar[d]^{φ_{k,r}} \\ [k + 1] \ar[r]^{φ_{k,s}} & [k] × [1]}  with t := k + 1 − s + r

(notation of example (7.4.7)(ii)). Since (7.5.17) commutes with fibre products, and φ_{k} is an effective epimorphism (by (7.5.1)), we conclude that the diagram

(7.5.21)  ⋃_{0≤r<s≤k}{(r, s)} × Δ_{k+1−s+r} [figure: ε′_{•}; ε′′_{•}; [k] × Δ_{k+1}; φ_{k}; Δ_{k} × Δ_{1}]

identifies Δ_{k} ×Δ_{1} with the coequalizer of ε′_{•} and ε′′_{•}, where ε′_{•} (resp. ε′′_{•}) is the morphism whose restrictions to each subobject {(r, s)} × Δ_{t} agrees with ε_{t,r+1}^{s−r} : {(r, s)} × Δ_{t} → {r} × Δ_{k+1} (resp. with ε_{t,r+1}^{s−r} : {(r, s)} × Δ_{t} → {s} × Δ_{k+1}).

However, (7.5.21) can be simplified as follows. Notice that if i_{1}, …, i_{r} is any sequence of non-negative integers such that 1 ≥ i_{n+1} − i_{n} ≥ 0 for every n = 0, …, r − 1, then

ε_{i_{r}} ∘ ⋯ ∘ ε_{i_{1}} = ε_{t,i_{1}}^{r} for every t ∈ ℕ.

From this observation, it is easily seen that the terms {(r, s)} × Δ_{k+1−s−r} in (7.5.21) with s > r + 1 are redundant; so we arrive at the presentation

(7.5.22)  [k − 1] × Δ_{k} [figure: ε′_{•}; ε′′_{•}; [k] × Δ_{k+1}; φ_{k}; Δ_{k} × Δ_{1}]

where each term {r} × Δ_{k} corresponds to the term {(r, r + 1)} × Δ_{k} of (7.5.21), and the morphisms ε′_{•}, ε′′_{•} are redefined accordingly.

7.5.23. We consider now the following sets of monomorphisms of s.Set :

• Σ_{1} is the set of all morphisms ι_{k}^{n} : Λ_{k}^{n} → Δ_{k}, for every k, n ∈ ℕ with k ≥ n.

• Σ_{2} is the set of all morphisms

i_{k}^{j} : (∂Δ_{k} × Δ_{1}) ∪ (Δ_{k} × Λ_{1}^{j}) → Δ_{k} × Δ_{1} for every k ∈ ℕ and j = 0, 1.

• Σ_{3} is the set of all monomorphisms of the form

i_{K,L}^{j} : (K × Δ_{1}) ∪ (L × Λ_{1}^{j}) → L × Δ_{1}

where K → L is an arbitrary monomorphism of s.Set, and j = 0, 1. With this notation, we may now state :

Proposition 7.5.24. The sets Σ_{1}, Σ_{2} and Σ_{3} have the same saturation.

Proof. In order to check that the saturation of Σ_{1} contains Σ_{2}, fix k ∈ ℕ; if j = 0 (resp. if j = 1), for every r = 0, …, k + 1, denote by A_{r} ⊂ Δ_{k} × Δ_{1} the image of the restriction of φ_{k} to the subobject {0, …, r} × Δ_{k+1} (resp. the subobject {k − r + 1, …, k + 1} × Δ_{k+1}), and define B_{r} as the fibre product in the cartesian diagram

\xymatrix{B_{r} \ar[r] \ar[d] & (∂Δ_{k} × Δ_{1}) ∪ (Δ_{k} × Λ_{1}^{j}) \ar[d]^{i_{k}^{j}} \\ A_{r} \ar[r] & Δ_{k} × Δ_{1}.}

Set also A_{−1} := s.∅ (the initial object of s.Set). Now, a straightforward induction reduces to checking that the natural morphisms

B_{k} ⨿_{B_{r}} A_{r} → B_{k} ⨿_{B_{r+1}} A_{r+1} for r = −1, …, k − 1

lie in the saturation of Σ_{1}. To this aim, it suffices to show that the same holds for the natural morphisms

τ_{r} : B_{r+1} ⨿_{B_{r}} A_{r} → A_{r+1} for r = −1, …, k − 1.

This, in turns, follows immediately from :

Claim 7.5.25. (i) The natural isomorphism {0} × Δ_{k+1} →^{∼} A_{0} identifies B_{0} with {0} × Λ_{k+1}^{1}. (ii) For every r = 0, …, k − 1 there is a commutative diagram in s.Set

\xymatrix{Δ_{k} \ar[r]^{ε_{r+1}} \ar[d]_{ε_{r+1}} & Λ_{k+1}^{r+2} \ar[r]^{ι_{k+1}^{r+2}} \ar[d] & Δ_{k+1} \ar[d] \\ A_{r} \ar[r] & B_{r+1} ⨿_{B_{r}} A_{r} \ar[r]^{τ_{r}} & A_{r+1}}

whose two square subdiagrams are cocartesian.

Proof of the claim. To ease notation, we shall identify {r} × Δ_{k} with its image under φ_{k}, which is a subobject of Δ_{k} × Δ_{1}, for every r = 0, …, k. Notice that B_{r+1} ⨿_{B_{r}} A_{r} is the smallest subobject of Δ_{k} × Δ_{1} that contains A_{r} and Z_{r+1} := ({r + 1} × Δ_{k+1}) ∩ B_{k}. However, a morphism [t] → B_{k} is the same as an increasing sequence of elements of [k] × [1]

(7.5.26)  (a_{n}, b_{n})  n = 0, …, t

such that :

(a) either the cardinality of the set {a_{0}, …, a_{t}} is strictly less than k + 1

(b) or else b_{n} ≠ j for every n = 0, …, t.

On the other hand, recall that the inclusion {r + 1} × Δ_{k+1} ⊂ Δ_{k} × Δ_{1} corresponds to the inclusion map of partially ordered sets φ_{k,r+1} (notation of (7.5.20)). It follows that an injective morphism Δ_{t} → Z_{r+1} corresponds to a strictly increasing sequence (7.5.26) contained in

{(0, 0), …, (0, r + 1), (1, r + 1), …, (1, k)}

and fulfilling either of the foregoing conditions (a) or (b). Taking r = −1, we already see that (i) holds. If r ≥ 0, take t := k, and notice that no such sequence can have a_{r} ≠ r + 1, so we see that of all the k + 2 monomorphisms

1_{{r+1}} × ε_{j} : {r + 1} × Δ_{k} → {r + 1} × Δ_{k+1}

only those with j ≠ r + 1, r + 2 factor through Z_{r+1}. On the other hand, from the presentation (7.5.22) we see that the intersection of A_{r} and {r + 1} × Δ_{k+1} is the image of {r} × Δ_{k}, which maps to both of them via the morphism ε_{r+1}. From this, we conclude already that the left square subdiagram of (ii) is indeed cocartesian, if we let the central vertical arrow to be the morphism that naturally identifies Λ_{k+1}^{r+2} with the subobject {r+1}×Λ_{k+1}^{r+2} of {r+1}×Δ_{k+1}. By the same token, it is clear the right square subdiagram is likewise cocartesian, provided we let the rightmost vertical arrow to be the morphism that naturally identifies Δ_{k+1} with {r + 1} × Δ_{k+1}. ♢

Next, let us check that the saturation of Σ_{2} contains Σ_{3}. Indeed, let μ : K → L be any monomorphism of s.Set, and fix j ∈ {0, 1}; according to example 7.5.2(iii), there exist :

• A countable system of monomorphisms

K_{−1} := K ⟶^{μ_{−1}} K_{0} → ⋯ → K_{n} ⟶^{μ_{n}} K_{n+1} → ⋯

such that μ is isomorphic to the induced morphism

K → colim_{n∈ℕ} K_{n}.

• For every n ∈ ℕ, a set I_{n} and an epimorphism K_{n−1} ⨿ (I_{n} × Δ_{n}) → K_{n} whose

restriction to K_{n−1} agrees with μ_{n−1}.

There follows, for every integer n ∈ ℕ, a cocartesian diagram

\xymatrix{(K_{n−1} × Δ_{1}) ∪ (K_{n} × Λ_{1}^{j}) \ar[r] \ar[d]_{i_{K_{n−1},Kn}^{j}} & (K_{n−1} × Δ_{1}) ∪ (L × Λ_{1}^{j}) \ar[d]^{φ_{n}} \\ K_{n} × Δ_{1} \ar[r] & (K_{n} × Δ_{1}) ∪ (L × Λ_{1}^{j})}

and the system (φ_{n} | n ∈ ℕ) induces a morphism

(K × Δ_{1}) ∪ (L × Λ_{1}^{j}) → colim_{n∈ℕ}(K_{n} × Δ_{1}) ∪ (L × Λ_{1}^{j}) = L × Δ_{1}

which is isomorphic to i_{K,L}^{j}. Thus, we may assume that there exist a set I and an epimorphism K ⨿ (I × Δ_{n}) → L for some n ∈ ℕ, and we shall argue by induction on n. Set Q := (I × Δ_{n}) ×_{L} K and notice that Q is a subobject of I × Δ_{n}. Also, since all epimorphisms are effective, the induced commutative diagram

\xymatrix{Q \ar[r] \ar[d] & K \ar[d] \\ I × Δ_{n} \ar[r] & L}

is cocartesian. Furthermore, since all colimits are universal in s.Set, we deduce that

Q = ∐_{i∈I_{n}} Q_{i} where Q_{i} := ({i} × Δ_{n}) ∩ Q for every i ∈ I.

We are then reduced to the case where L = Δ_{n}. If n = 0, then clearly K is either Δ_{0} or Δ_{−1}, and in either case we have i_{K,L}^{j} ∈ Σ_{2}.

Next, suppose that n > 0, and that the assertion is already known for every integer k < n. If K = Δ_{n}, we are done; otherwise, it is easily seen that the inclusion K → Δ_{n} factors through ∂Δ_{n}, and arguing as in the foregoing, we are reduced to considering the morphisms i_{K,∂Δ_{n}}^{j} and i_{∂Δ_{n},Δ_{n}}^{j}. The latter lies in Σ_{2}, and from example 7.5.2(iv) it is clear that there is an epimorphism K ⨿ ([n] × Δ_{n−1}) → ∂Δ_{n}, so i_{K,∂Δ_{n}}^{j} lies in the saturation of Σ_{2}, by inductive assumption.

Lastly, we show that the saturation of Σ_{3} contains Σ_{1}. To this aim, it suffices to construct a commutative diagram

\xymatrix{Λ_{k}^{r} \ar[r] \ar[d]_{ι_{k}^{r}} & (Λ_{k}^{r} × Δ_{1}) ∪ (Δ_{k} × Λ_{1}^{j}) \ar[r] \ar[d]^{i_{Λ^{r}_{k},Δ_{k}}^{j}} & Λ_{k}^{r} \ar[d]^{ι_{k}^{r}} \\ Δ_{k} \ar[r]^{i} & Δ_{k} × Δ_{1} \ar[r]^{p} & Δ_{k}}

for every n, r ∈ ℕ with r ≤ k, such that p ∘ i = 1_{Δ_{k}}. However, notice that a morphism [t] → (Λ_{k}^{r} × Δ_{1}) ∪ (Δ_{k} × Λ_{1}^{j}) (for any t ∈ ℕ) is the same as an increasing sequence (7.5.26) of elements of [k] × [1] such that

(a) either b_{n} ≠ j for every n = 0, …, t

(b) or else, the cardinality of of the set {a_{0}, …, a_{t}} ∪ {r} is strictly less than k + 1. Now, any such p and i shall be the images, under the functor (7.5.17), of corresponding maps of ordered sets

[k] ⟶^{i^{∗}} [k] × [1] ⟶^{p^{∗}} [k] such that p^{∗} ∘ i^{∗} = 1_{[k]}.

We obtain a suitable j ∈ {0, 1} and suitable p^{∗}, i^{∗} by the following rule. If r < k, we let j := 1, and define p^{∗} and i^{∗} as the maps such that

i^{∗}(a) = (a, 1) for every a = 0, …, k p^{∗}(a, b) = \{ \begin{matrix}a & if b = 1 or a ≤ r \\ r & otherwise. \end{matrix}

If r = k, we let j := 0, and set

i^{∗}(a) := (a, 0) for every a = 0, …, k p^{∗}(a, b) = \{ \begin{matrix}a & if b = 0 \\ r & otherwise. \end{matrix}

The reader may easily check that the resulting maps i and p will do. □

Definition 7.5.27. A monomorphism in s.Set is an anodyne extension, if it lies in the saturation of the set Σ_{1}.

In view of lemma 7.5.15(i), we see that a fibration has the right lifting property with respect to every anodyne extension. We also notice :

Corollary 7.5.28. Let K → L be any anodyne extension, and A → B any monomorphism. Then the induced monomorphism

(K × B) ∪ (L × A) → L × B

is an anodyne extension.

Proof. Let Σ be the set of monomorphisms K′ → L′ such that the induced morphism

(K′ × B) ∪ (L′ × A) → L′ × B

is an anodyne extension. It is easily seen that Σ is saturated (details left to the reader; cp. the proof of proposition 7.5.24); taking into account proposition 7.5.24, it then suffices to check that Σ contains the morphisms i_{X,Y}^{j}, where f : X → Y is an arbitrary monomorphism (notation of (7.5.23)). However, for any such f we have a commutative diagram

\xymatrix{(((X × Δ_{1}) ∪ (Y × Λ_{1}^{j})) × B) ∪ ((Y × Δ_{1}) × A) \ar[r] \ar[d] & ((Y × Δ_{1}) × B \ar[d] \\ (((X × B) ∪ (Y × A)) × Δ_{1}) ∪ ((Y × B) × Λ_{1}^{j}) \ar[r]^{i_{(X×B)∪(Y ×A),Y ×B}^{j}} & (Y × B) × Δ_{1}}

whose vertical arrows are isomorphisms. Then the contention follows by appealing again to proposition 7.5.24. □

Theorem 7.5.29. For every morphism f : X → Y of simplicial sets there exists a commutative diagram

\xymatrix{X \ar[rr]^{i_{f}} \ar[dr]_{f} &  & E_{f} \ar[dl]^{p_{f}} \\  & Y}  in s.Set

such that i_{f} is an anodyne extension, and p_{f} is a fibration.

Proof. For every k, n ∈ ℕ with k ≥ min(1, n), consider the set ℒ_{k}^{n} of all commutative diagrams

\xymatrix{Λ_{k}^{n} \ar[r] \ar[d]_{ι_{k}^{n}} & X \ar[d]^{f} \\ Δ_{k} \ar[r] & Y}

and set

Λ_{f} := ∐_{n∈ℕ} ∐_{k≥min(1,n)}ℒ_{k}^{n} × Λ_{k}^{n} L_{f} := ∐_{n∈ℕ} ∐_{k≥min(1,n)}ℒ_{k}^{n} × Δ_{k}.

There follows a natural morphism

\xymatrix{Λ_{f} \ar[r]^{g} \ar[d]_{ι} & X \ar[d]^{f} \\ L_{f} \ar[r] & Y}

with ι an anodyne extension. Define E_{f}^{0} as the push-out in the induced cocartesian diagram

\xymatrix{Λ_{f} \ar[r]^{g} \ar[d]_{ι} & X \ar[d]^{i_{f}^{0}} \\ L_{f} \ar[r] & E_{f}^{0}}

so that i_{f}^{0} is anodyne as well, and we have a commutative diagram

\xymatrix{X \ar[rr]^{i_{(X,f)}^{0}} \ar[dr]_{f} &  & E_{f}^{0} \ar[dl]^{p_{f}^{0}} \\  & Y.}

Clearly, the rule (X, f) ↦ (E_{f}^{0}, p_{f}^{0}) extends to a well defined functor

\underline{E}^{0} : s.Set/Y → s.Set/Y

and the rule (X, f) ↦ i_{(X,f)}^{0} yields a natural transformation

i^{0} : 1_{s.Set/Y} ⇒ \underline{E}^{0}.

Thus, we may define inductively :

\underline{E}^{n}(X, f) := \underline{E}^{0}(\underline{E}^{n−1}(X, f)) for every integer n ≥ 1

so \underline{E}^{n}(X, f) is a pair (E_{f}^{n}, p_{f}^{n}), and we get as well as a natural transformation

i_{(X,f)}^{n} := i_{\underline{E}^{n−1}(X,f)}^{0} : E_{f}^{n−1} → E_{f}^{n} for every integer n ≥ 1

which is again an anodyne extension. We set

E_{f} := colim_{n∈ℕ} E_{f}^{n}

where the transition maps in the colimit are given by the system of morphisms (i_{f}^{n} | n ∈ ℕ). The colimit of the system of morphism (p_{f}^{n} | n ∈ ℕ) is then a morphism p_{f} : E_{f} → Y; moreover, the induced morphism i_{f} : X → E_{f} is anodyne, and obviously p_{f} ∘ i_{f} = f. To conclude the proof, it suffices to show that p_{f} is a fibration. Thus, consider any commutative diagram

\xymatrix{Λ_{k}^{n} \ar[r]^{g} \ar[d]_{ι_{k}^{n}} & E_{f} \ar[d]^{p_{f}} \\ Δ_{k} \ar[r]^{h} & Y.}

It follows easily from example 7.5.2(v) that – for r ∈ ℕ large enough – g factors through a morphism g_{r} : Λ_{k}^{n} → E_{f}^{r} and the natural morphism E_{f}^{r} → E_{f}, whence a commutative diagram

[figure: Λ_{k}^{n}; g_{r}; ι_{k}^{n}; E_{f}^{r}; p_{f}^{r}; i_{f}^{r+1}; E_{f}^{r+1}; p_{f}^{r+1}; Δ_{k}; h; Y; Y.]

But then, by construction of E_{f}^{n+1}, we see that h lifts to a morphism h′ : Δ_{k} → E_{f}^{n+1} such that p_{f}^{r+1} ∘ h′ = h and h′ ∘ ι_{k}^{n} = i_{f}^{r+1} ∘ g_{r}. The composition of h′ with the natural morphism E_{f}^{r+1} → E_{f} is a morphism h′′ : Δ_{k} → E_{f} such that p_{f} ∘h′′ = h and h′′∘ι_{k}^{n} = g, as required. □

A useful way to produce new fibrant simplicial sets out of old ones is provided by the general construction introduced in the following :

Definition 7.5.30. Let A and B be any two simplicial sets; the function complex

s.ℋ om(A, B)

associated with A and B is the simplicial set given by the rule :

n ↦ Hom_{s.Set}(Δ_{n} × A, B) φ ↦ Hom_{s.Set}(Δ_{φ} × 1_{A}, B)

for every n ∈ ℕ and every morphism φ of Δ.

Remark 7.5.31. (i) Notice that the system of isomorphisms A[n] →^{∼} Hom_{s.Set}(Δ_{n}, A) given by Yoneda’s lemma for every n ∈ ℕ, amounts to a natural identification

ι_{A} : A →^{∼} s.ℋ om(Δ_{0}, A) for every A ∈ Ob(s.Set).

(ii) Also, if A, B and C are any three simplicial sets, there is a natural transformation

τ_{A,B,C} : s.ℋ om(A, B) → s.ℋ om(A × C, B × C)

which, to any n ∈ ℕ, assigns the map

Hom_{s.Set}(Δ_{n} × A, B) → Hom_{s.Set}(Δ_{n} × A × C, B × C) φ ↦ φ × 1_{C}.

(iii) There is a natural evaluation morphism

ev_{A,B} : A × s.ℋ om(A, B) → B for every A, B ∈ Ob(s.Set)

which, to every n ∈ ℕ, assigns the map

A[n] × Hom_{s.Set}(Δ_{n} × A, B) → B[n] (a, f) ↦ f[n](1_{Δ_{n}}, a).

Indeed, for any morphism φ : [k] → [n] in Δ, and (a, f) any element of (A × ℋ om(A, B))[n] we may compute

B[φ] ∘ ev_{A,B}[n](a, f) = B[φ] ∘ f[n](1_{Δ_{n}}, a)
= f[k] ∘ (Δ_{n}[φ] × A[φ])(1_{Δ_{n}}, a)
= f[k](φ, A[φ](a))
= f[k] ∘ (Δ_{φ} × 1_{A})[k](1_{Δ_{k}}, A[φ](a))
= (f ∘ (Δ_{φ} × 1_{A}))[k](1_{Δ_{k}}, A[φ](a))
= ev_{A,B}[k](A[φ](a), f ∘ (Δ_{φ} × 1_{A}))
= ev_{A,B}[k] ∘ (A × ℋ om(A, B))[φ](a, f)

which shows that ev_{A,B} is a morphism of simplicial sets.

(iv) Clearly s.Set is a tensor category, with tensor product given by the product of simplicial sets, and with Δ_{0} as unit object. We claim that the functor

s.ℋ om : (s.Set)^{o} × s.Set → s.Set

is an internal Hom functor for (s.Set, ×, Δ_{0}). Indeed, given A, B, C ∈ Ob(s.Set) and any morphism f : A × B → C, we obtain a morphism

A ⟶^{ι_{A}} s.ℋ om(Δ_{0}, A) ⟶^{τ_{A,B,C}} s.ℋ om(B, A × B) ⟶^{ℋom(B,f)} s.ℋ om(B, C).

Conversely, given a morphism g : A → ℋ om(B, C), we get a morphism

A × B ⟶^{g×1_{B}} s.ℋ om(B, C) × B →^{∼} B × s.ℋ om(B, C) ⟶^{ev_{B,C}} C.

It is easily seen that these two rules yield mutually inverse natural bijections, as required (details left to the reader).

7.5.32. Let f : A → B and g : C → D be morphisms of simplicial sets. We have a commutative diagram

\xymatrix{s.ℋ om(D, A) \ar[r]^{s.ℋom(D,f)} \ar[d]_{s.ℋom(g,A)} & s.ℋ om(D, B) \ar[d]^{s.ℋom(g,B)} \\ s.ℋ om(C, A) \ar[r]^{s.ℋom(C,f)} & s.ℋ om(C, B)}

whence an induced morphism in s.Set

(7.5.33)  s.ℋ om(D, A) → s.ℋ om(D, B) ×_{s.ℋom(C,B)} s.ℋ om(C, A).

Proposition 7.5.34. In the situation of (7.5.32), suppose that f is a fibration and g a monomorphism. Then (7.5.33) is a fibration as well.

Proof. The datum of a commutative diagram

\xymatrix{Λ_{k}^{i} \ar[r] \ar[d]_{ι_{k}^{i}} & s.ℋ om(D, A) \ar[d] \\ Δ_{k} \ar[r] & s.ℋ om(D, B) ×_{s.ℋom(C,B)} s.ℋ om(C, A)}

is equivalent, by remark 7.5.31(iv), to that of a commutative diagram

\xymatrix{(Λ_{k}^{i} × D) ∪ (Δ_{k} × C) \ar[r] \ar[d]_{g} & A \ar[d]^{f} \\ Δ_{k} × D \ar[r] & B.}

On the other hand, f has the right lifting property with respect to g (corollary 7.5.28); it follows easily that (7.5.33) has the right lifting property with respect to ι_{k}^{i}, as stated. □ Corollary 7.5.35. Let A be any fibrant simplicial set. We have :

(i) s.ℋ om(D, A) is fibrant, for any simplicial set D.

(ii) Any monomorphism D → D′ of simplicial sets induces a fibrant morphism

sℋ om(D′, A) → s.ℋ om(D, A).

Proof. (i): Take B := Δ_{0}, C := Δ_{−1}, and let f : A → B, g : C → D be the unique morphisms of simplicial sets. In view of proposition 7.5.34, it suffices to notice the natural identifications

s.ℋ om(D, Δ_{0}) →^{∼} Δ_{0} →^{∼} s.ℋ om(Δ_{−1}, D)

for every simplicial set A. The proof of (ii) is similar : details left to the reader. □

7.5.36. Let f, g : X → Y be two morphisms of simplicial sets. Recall (see remark 7.4.18(v)) that a homotopy u from f to g is the datum of a commutative diagram of simplicial sets :

[figure: X × Δ_{0}; 1_{X}×ε_{0}; X × Δ_{1}; u; X × Δ_{0}; 1_{X}×ε_{1}; X; f; Y; X.; g]

Definition 7.5.37. In the situation of (7.5.36), suppose that i : X′ → X is a monomorphism of simplicial sets, such that f ∘ i = g ∘ i. We say that u is a homotopy from f to g relative to X′, if u is a homotopy from f to g, and the following diagram commutes :

\xymatrix{X′ × Δ_{1} \ar[d]_{i×1_{Δ_{1}}} \ar[r] & X′ \ar[d]^{f∘i} \\ X × Δ_{1} \ar[r]^{u} & Y}

where the top horizontal arrow is the natural projection. Remark 7.5.38. Let f, g : X → Y be any two morphisms of simplicial sets.

(i) Of course, a homotopy from f to g can be viewed as a homotopy relative to the sub-object Δ_{−1} ⊂ X, so definition 7.5.37 generalizes the previous notion. Notice also that to f one may attach a constant homotopy

X × Δ_{1} ⟶^{1_{X}×η_{0}} X ⟶^{f} Y.

(ii) Let u be any homotopy from f to g relative to a subobject X′ of X, and h : Y → Z any morphism of simplicial sets. Then clearly h ∘ u is a homotopy from h ∘ f to h ∘ g relative to X′. Likewise, if t : Z → X is any other morphism, then u ∘ (t × Δ_{1}) is a homotopy from f ∘ t to g ∘ t relative to Z ×_{X} X′.

7.5.39. Let A be any simplicial set; on the set Hom_{s.Set}(Δ_{0}, A) ≃ A[0] we consider the binary relation ∼ such that

a ∼ b  ⇔  there exists a homotopy from a to b.

Lemma 7.5.40. With the notation of (7.5.39), suppose that A is fibrant. Then ∼ is an equivalence relation on A[0].

Proof. Clearly a ∼ b if and only if there exists a morphism u : Δ_{1} → X such that u ∘ ε_{0} = a and u ∘ ε_{1} = b. Taking u := a ∘ σ_{0}, where σ_{0} : Δ_{1} → Δ_{0} is the unique morphism, and a ∈ A[0] any element, we obtain the reflexivity of ∼. Next, suppose that a ∼ b and b ∼ c for some a, b, c ∈ A[0], and pick u, v : Δ_{1} → A that give a homotopy from a to b, and respectively from b to c. Then it is easily seen that the datum of u and v determines a unique morphism t : Λ_{2}^{1} → A such that t ∘ ε_{0} = u and t ∘ ε_{2} = v. By assumption, t extends to a morphism s : Δ_{2} → A, and then s ∘ ε_{1} is a homotopy from a to c, whence the transitivity of ∼.

Lastly, let u : Δ_{1} → A be a homotopy from a to b, for some a, b ∈ A[0]; we let t : Λ_{2}^{0} → A be the unique morphism such that t ∘ ε_{2} = u and t ∘ ε_{1} = b ∘ σ_{0}. By assumption t extends to a morphism s : Δ_{2} → A, and it is easily seen that s ∘ ε_{0} is a homotopy from b to a, which shows that ∼ is also symmetric. □

7.5.41. More generally, if i : K → X is any monomorphism in s.Set, and Y is any simplicial set, we consider on Hom_{s.Set}(X, Y) the binary relation

f ∼_{K} g  ⇔  there exists a homotopy from f to g relative to K.

In case K = s.∅, we simply write f ∼ g instead of f ∼_{s.∅} g.

Theorem 7.5.42. With the notation of (7.5.41), suppose that Y is fibrant. Then ∼_{K} is an equivalence relation.

Proof. The fibres of the map of sets

(7.5.43)  Hom_{s.Set}(X, Y) → Hom_{s.Set}(K, Y)  f ↦ f ∘ i

give a partition of Hom_{s.Set}(X, Y), and if f ∼_{K} g, then f and g lie in the same fibre of (7.5.43). Hence, we may restrict attention to a single fibre of (7.5.43), say the fibre over the morphism h : K → Y. Now, h corresponds to a morphism h^{∗} : Δ_{0} → s.ℋ om(K, Y) in s.Set, and likewise, the datum of a pair of morphisms f, g : X → Y such that f ∘i = h = g∘i corresponds to that of a pair of morphisms f^{∗}, g^{∗} : Δ_{0} → s.ℋ om(X, Y) with i^{∗} ∘ f^{∗} = h^{∗} = i^{∗} ∘ g^{∗}, where

i^{∗} : s.ℋ om(X, Y) → s.ℋ om(K, Y)

is the morphism deduced from i. Furthermore, a homotopy from f to g relative to K is the same as a morphism t : Δ_{1} → s.ℋ om(X, Y) such that i^{∗} ∘ t factors through h^{∗} and such that

t ∘ ε_{0} = f^{∗} t ∘ ε_{1} = g^{∗}.

Set A := i^{∗−1}(h^{∗}) (notation of example 7.5.2(vi)). We conclude that t corresponds to a homotopy between the elements of A[0] induced by f^{∗} and g^{∗}. However, A is fibrant by corollary 7.5.35(ii) and lemma 7.5.15(iv), so the assertion follows from lemma 7.5.40. □

7.5.44. Let A be any simplicial set, ξ ∈ A[0] any element, and denote also by ξ : Δ_{0} → A the corresponding morphism of simplicial sets. We shall frequently abuse notation, and denote indifferently by ξ the unique element of A[n] that lies in the image of the map ξ[n]. We call the pair (A, ξ) a pointed simplicial set. With this terminology, we have :

Definition 7.5.45. Let (A, ξ) be a pointed simplicial set, such that A is fibrant.

(i) For every integer n ∈ ℕ, let A(ξ, n) be the set of all a ∈ A[n] such that ∂a factors through ξ (notation of example 7.5.2(iv)). We set

π_{n}(A, ξ) := A(ξ, n)/∼_{∂Δ_{n}}

(where ∼_{∂Δ_{n}} is the homotopy equivalence relation defined in (7.5.41)), and for every n > 0 we call π_{n}(A, ξ) the n-th homotopy group of A.

(ii) Notice that π_{0}(A, ξ) is actually independent of ξ, so we shall usually just denote it π_{0}(A). We say that A is connected if the cardinality of π_{0}(A) equals one.

Lemma 7.5.46. In the situation of definition 7.5.45, let n ∈ ℕ be any integer and a ∈ A(ξ, n) any simplex. The following conditions are equivalent :

(a) a ∼_{∂Δ_{n}} ξ.

(b) There exists b ∈ A[n + 1] such that ∂_{i}b = ξ for every i = 1, …, n + 1 and ∂_{0}b = a.

Proof. (a)⇒(b): Let h : Δ_{n} × Δ_{1} → A be any morphism in s.Set. From the presentation (7.5.22) we see that h is the same as a system x_{0}, …, x_{n} of elements of A[n + 1] such that

∂_{i}x_{i} = ∂_{i}x_{i−1} for i = 1, …, n.

Namely, x_{i} := h ∘ Δ_{φ_{n,i}} for every i = 0, …, n, where φ_{n,i} : [n + 1] → [n] × [1] is defined as in (7.5.20). Moreover, recall that the fully faithful functor (7.5.17) associates with any map of partially ordered sets f : [n] → [n] × [1] a morphism Δ_{f} : Δ_{n} → Δ_{n} × Δ_{1}, and it is easily seen that Δ_{f} factors through the subobject (∂Δ_{n} × Δ_{1}) ∪ (Δ_{n} × ∂Δ_{1}) if and only if f fulfills the following condition. Set f(j) := (r_{j}, s_{j}) for every j ∈ [n]; then, for every j = 0, …, n − 1 we have either r_{j} = r_{j+1} or s_{j} = s_{j+1} (details left to the reader). It follows easily that h is a homotopy from ξ to a relative to ∂Δ_{n} if and only if we have

∂_{0}x_{0} = a  ∂_{n+1}x_{n} = ξ  and ∂_{j}x_{i} = ξ for 0 < i ≤ n and j ≠ i, i + 1.

Consider then the system y_{•} := (y_{1}, …, y_{n+2}) of elements of A[n + 1] such that y_{n+2} := ξ and y_{i} = x_{i−1} for every i = 1, …, n + 1. A direct calculation shows that

∂_{i}y_{j} = ∂_{j−1}y_{i} whenever 1 ≤ i < j ≤ n + 2

so y_{•} corresponds to a unique morphism φ : Λ_{n+2}^{0} → A whose restriction with the i-th face of Λ_{n+2}^{n+2} agrees with y_{i}, for every i = 1, …, n+2. Since A is fibrant, we may then find c ∈ A[n+2] such that ∂_{i}c = y_{i} for every i = 1, …, n + 2. Set b := ∂_{0}c; then

∂_{i}b = ∂_{i}∂_{0}c = ∂_{0}∂_{i+1}c = ∂_{0}x_{i} for every i = 0, …, n + 1

whence the assertion.

(b)⇒(a): Consider the map f : [n] × [1] → [n + 1] in the category POSet given by the rule :

(i, 0) ↦ i and (i, 1) ↦ n + 1 for every i ∈ [n]

and let \overline{b} : Δ_{n+1} → A be the morphism in s.Set corresponding to b; then it is easily seen that \overline{b} ∘ Δ_{f} : Δ_{n} × Δ_{1} → A is a homotopy from a to ξ relative to ∂Δ_{n} (details left to the reader). □

7.5.47. Let (A, ξ) be any pointed simplicial set, such that A is fibrant. To justify the name of π_{n}(A, ξ), we shall endow it with a natural group structure, for every n > 0. To this aim, say that α, β ∈ π_{n}(A, ξ) are any two classes, and pick representatives x, y ∈ A(ξ, n) for α and respectively β. Then, obviously the sequence

ξ, …, ξ, x, y

of n + 1 elements of A[n] satisfies the compatibility condition of remark 7.5.11, with k := n. We may therefore find z ∈ A[n + 1] such that ∂_{n−1}z = x, ∂_{n+1}z = y, and ∂_{i}z = ξ for i = 0, …, n − 2. The first observation is :

Lemma 7.5.48. With the notation of (7.5.47), the class of ∂_{n}z in π_{n}(A, ξ) depends only on α and β (and not on z, nor the choice of representatives x and y).

Proof. Let x′ and y′ be two other elements of A(ξ, n), and h (resp. h′) a homotopy from x to x′ (resp. from y to y′) relative to ∂Δ_{n}. Let also z (resp. z′) be a morphism Δ_{n+1} → A such that

z ∘ ε_{i} = ξ = z′ ∘ ε_{i} for i = 0, …, n − 2

and

z ∘ ε_{n−1} = x  z′ ∘ ε_{n−1} = x′  z ∘ ε_{n+1} = y  z′ ∘ ε_{n+1} = y′.

Then there exists a unique morphism ψ : (Δ_{n+1} × ∂Δ_{1}) ∪ (Λ_{n+1}^{n} × Δ_{1}) → A such that • ψ ∘ (1_{Δ_{n+1}} × ε_{0}) = z and ψ ∘ (1_{Δ_{n+1}} × ε_{1}) = z′.

• ψ ∘ (ε_{i} × 1_{Δ_{1}}) factors through ξ for i = 0, …, n − 2.
• ψ ∘ (ε_{n−1} × 1_{Δ_{1}}) = h and ψ ∘ (ε_{n+1} × 1_{Δ_{1}}) = h′

and it is easily seen that ψ ∘ (ε_{n} × 1_{Δ_{1}}) is a homotopy from ∂_{n}z to ∂_{n}z′. □

7.5.49. Keep the notation of (7.5.47); lemma 7.5.48says that the rule : (α, β) ↦ ∂_{n}z yields a well defined pairing

(7.5.50) π_{n}(A, ξ) × π_{n}(A, ξ) → π_{n}(A, ξ) (α, β) ↦ α · β  for every integer n > 0.

For any x ∈ A(ξ, n), we shall write [x] for the class of x in π_{n}(A, ξ).

Theorem 7.5.51. For every integer n > 0, the pairing (7.5.50) defines a group law on π_{n}(A, ξ), whose neutral element is the class [ξ].

Proof. Fix n > 0 and any α ∈ π_{n}(A, ξ); to see that [ξ] · α = α · [ξ] = α, pick any representative x : Δ_{n} → A for α, and set z := x ∘ η_{n}, z′ := x ∘ η_{n+1}. From the simplicial identities (7.4.6) it is easily seen that ∂_{i}z = ∂_{i}z′ = ξ for i = 0, …, n − 2, and ∂_{n−1}z = ∂_{n+1}z′ = x, so ∂_{n}z and ∂_{n}z′ represent respectively [ξ] · α and α · [ξ]. But we have ∂_{n}z = ∂_{n}z′ = x, whence the claim.

Next, we check that the map

π_{n}(A, ξ) → π_{n}(A, ξ) : β ↦ α · β

is surjective. Indeed, given any y ∈ A(ξ, n), there exists a morphism φ : Λ_{n+1}^{n+1} → A such that φ ∘ ε_{i} = ξ for i = 0, …, n − 2, φ ∘ ε_{n−1} = x and φ ∘ ε_{n} = y. We can then extend φ to a morphism z : Δ_{n} → A, and if we denote by β (resp. γ) the class of y (resp. of ∂_{n+1}z) in π_{n}(A, ξ), we see that α · γ = β.

Likewise, one shows that right multiplication by α defines a surjection on π_{n}(A, ξ) (details left to the reader). It remains only to check the associativity of (7.5.50), and to this aim, let x, y, z ∈ A(ξ, n) be any three elements, representing respectively α, β, γ ∈ π_{n}(A, ξ). Then there exist u_{n−1}, u_{n+1}, u_{n+2} ∈ A[n + 1] such that

∂_{i}u_{n−1} = ∂_{i}u_{n+1} = ∂_{i}u_{n+2} = ξ for every i = 0, …, n − 2

and

\begin{matrix}∂_{n−1}u_{n−1} = & x & ∂_{n+1}u_{n−1} = & y \\ ∂_{n−1}u_{n+1} = & ∂_{n}u_{n−1} & ∂_{n+1}u_{n+1} = & z \\ ∂_{n−1}u_{n+2} = & y & ∂_{n+1}u_{n+2} = & z. \end{matrix}

By remark 7.5.11, we may then find some v ∈ A[n+2] such that ∂_{i}v equals ξ for i = 0, …, n− 2, and equals u_{i} for i = n − 1, n + 1, n + 2. It follows that

∂_{n−1}∂_{n}v = x ∂_{n+1}∂_{n}v = ∂_{n}u_{n+2} and ∂_{i}∂_{n}v = ξ for i = 0, …, n − 2

We may then compute :

\begin{matrix}(α · β) · γ = & [∂_{n}u_{n−1}] · γ \\ = & [∂_{n}u_{n+1}] \\ = & [∂_{n}∂_{n+1}v] \\ = & [∂_{n}∂_{n}v] \\ = & α · (β · γ) \end{matrix}

as required. □

Remark 7.5.52. (i) Let us denote

s.Set_{∘}^{f}

the category of fibrant pointed simplicial sets, whose objects are all the pairs (A, ξ) consisting of a simplicial set A and an element ξ ∈ A[0]. A morphism f : (A, ξ) → (A′, ξ′) in s.Set_{∘}^{f} is a morphism f : A → A′ of simplicial sets such that f[0](ξ) = ξ′. It follows easily from remark 7.5.38(ii) that, for every integer n > 0 (resp. for n = 0), the rule (A, ξ) ↦ π_{n}(A, ξ) (resp. (A, ξ) ↦ π_{0}(A)) yields a functor

π_{n} : s.Set_{∘}^{f} → Grp (resp. π_{0} : s.Set_{∘}^{f} → Set_{∘})

with values in the category of groups (resp. of pointed sets). Namely, if f is a morphism in s.Set_{∘}^{f} as in the foregoing, and x ∈ A(ξ, n) (resp. x ∈ A[0]) is any element, we let π_{n}(f, ξ)[x] (resp. π_{0}(f)[x]) be the class of f[n](x) in π_{n}(A′, ξ′) (resp. in π_{0}(A′)). A simple inspection shows that this rule is well defined and does yield a group homomorphism π_{n}(A, ξ) → π_{n}(A′, ξ′), whenever n > 0.

(ii) Let X and Y be two fibrant simplicial sets, ξ ∈ X[0] any element, and denote also by ξ the image of the corresponding morphism Δ_{0} → X, which is therefore a simplicial subset of X. Let f, g : X → Y be two morphisms of simplicial sets such that f[0](ξ) = g[0](ξ) and f ∼_{ξ} g. Then we claim that

π_{0}(f) = π_{0}(g) and π_{n}(f, ξ) = π_{n}(g, ξ) for every n > 0.

Indeed, let u be a homotopy from f to g relative to ξ; if x ∈ X(ξ, n), denote also by x : Δ_{n} → X the corresponding morphism, and notice that u ∘ (x × 1_{Δ_{1}}) : Δ_{n} × Δ_{1} → Y is a homotopy from f[n](x) to g[n](x) relative to ∂Δ_{n}, whence the contention.

7.5.53. Let p : X → Y be a fibration between fibrant simplicial sets X and Y. Let also ξ ∈ X[0] be any vertex, and set ξ′ := p[0](ξ) and F := p^{−1}(ξ′) (notation of example 7.5.2(vi)), so that F is fibrant as well, by lemma 7.5.15(iv). For any n ∈ ℕ and any a ∈ Y (ξ′, n), we get a commutative diagram

\xymatrix{Λ_{n}^{0} \ar[r]^{c} \ar[d] & X \ar[d]^{p} \\ Δ_{n} \ar[r]^{a} & Y}

where c is the unique morphism in s.Set that factors through ξ : Δ_{0} → X. We may then find b ∈ X[n] such that p[n](b) = a and ∂_{i}b = ξ for every i = 1, …, n.

Lemma 7.5.54. In the situation of (7.5.53), we have ∂_{0}b ∈ F (ξ, n − 1), and the class [∂_{0}b] ∈ π_{n−1}(F, ξ) depends only on [a] ∈ π_{n}(Y, ξ′) (and is independent of the representative a for [a] and of the simplex b).

Proof. We have ∂_{i}∂_{0}b = ∂_{0}∂_{i+1}b = ξ for every i = 0, …, n − 1, whence the first assertion. Next, suppose that a′ ∈ Y (ξ′, n) is another simplex such that a ∼_{∂Δ_{n}} a′, and pick any homotopy h : Δ_{n} × Δ_{1} → Y from a to a′ relative to ∂Δ_{n}. Let also b′ ∈ X[n] be any simplex such that p(b′) = a′ and ∂_{i}b′ = ξ for i = 1, …, n. We obtain a commutative diagram

\xymatrix{(Δ_{n} × ∂Δ_{1}) ∪ (Λ_{n}^{0} × Δ_{1}) \ar[d] \ar[r]^{c′} & X \ar[d]^{p} \\ Δ_{n} × Δ_{1} \ar[r]^{h} & Y}

where c′ is the unique morphism whose restriction to Δ_{n} × Λ_{1}^{0} (resp. to Δ_{n} × Λ_{1}^{1}) agrees with b (resp. with b′) and whose restriction to Λ_{n}^{0} × Δ_{1} is the unique morphism that factors through ξ : Δ_{0} → X. Since p is a fibration, we may then find a morphism h′ : Δ_{n} × Δ_{1} → X which extends c′, and such that p ∘ h′ = h (corollary 7.5.28). Set h′′ := h′ ∘ (ε_{0} × 1_{Δ_{1}}) : Δ_{n−1} × Δ_{1} → X. It is easily seen that h′′ is a homotopy from ∂_{0}b to ∂_{0}b′ relative ∂Δ_{n−1}, whence the lemma. □

7.5.55. Keep the situation of (7.5.53); it follows from lemma 7.5.54 that the rule [a] ↦ [∂_{0}b] yields a well defined map

δ_{n} : π_{n}(Y, ξ′) → π_{n−1}(F, ξ) for every n > 0

and it is easily seen that δ_{n}[ξ] = [ξ′], i.e. δ_{n} is a map of pointed sets. Let also i : F → X be the natural monomorphism; there follows, for every n ∈ ℕ, a natural sequence of pointed sets

π_{n+1}(X, ξ) ⟶^{π_{n+1}(p,ξ)} π_{n+1}(Y, ξ′) ⟶^{δ_{n+1}} π_{n}(F, ξ) ⟶^{π_{n}(i,ξ)} π_{n}(X, ξ) ⟶^{π_{n}(p,ξ)} π_{n}(Y, ξ′)

called the homotopy sequence in degree n attached to the fibration p (and the vertex ξ). Let \underline{1} denote the monoid with one element (the initial object in the category of monoids); noticed that the category Set_{∘} can also be regarded naturally as the category of pointed (left) \underline{1}-modules (see (4.8.13)), and therefore the homotopy sequence in every degree is naturally a sequence of pointed \underline{1}-modules.

Theorem 7.5.56. With the notation of (7.5.55), the following holds :

(i) The homotopy sequence attached to p is exact in every degree n ∈ ℕ, i.e. we have

Im π_{n+1}(p, ξ) = Ker δ_{n+1}  Im δ_{n+1} = Ker π_{n}(i, ξ)  Im π_{n}(i, ξ) = Ker π_{n}(p, ξ).

(ii) δ_{n} is a group homomorphism for every n > 1.

Proof. (See remark 4.8.17(iii) for the notion of kernel of a morphism of pointed \underline{1}-modules.)

(ii): Let a_{n−1}, a_{n}, a_{n+1} ∈ Y (ξ′, n) be three elements, b_{n−1}, b_{n}, b_{n+1} ∈ X[n] and φ ∈ Y [n + 1] four simplices such that

• p(b_{i}) = a_{i} and ∂_{j}b_{i} = ξ for every i = n − 1, n, n + 1 and j = 1, …, n

• ∂_{i}φ = ξ′ for i = 0, …, n − 2 and ∂_{i}φ = a_{i} for i = n − 1, n, n + 1.

We deduce a commutative diagram

\xymatrix{Λ_{n+1}^{0} \ar[r]^{c} \ar[d] & X \ar[d]^{p} \\ Δ_{n+1} \ar[r]^{φ} & Y}

where c is the unique morphism whose restriction to the i-th face of Λ_{0}^{n+1} factors through ξ if i = 1, …, n − 2, and equals b_{i} for i = n − 1, n, n + 1. Since p is a fibration, we may then find ψ : Δ_{n+1} → X whose restriction to Λ_{n+1}^{0} agrees with c, and such that p ∘ ψ = φ. Set t := ∂_{0}ψ ∈ X[n]; then p[n](t) = ∂_{0}(p[n + 1](ψ)) = ∂_{0}φ = ξ, so that t ∈ F [n]. Moreover, ∂_{i}t = ∂_{0}∂_{i+1}ψ for every i = 0, …, n, hence ∂_{i}t = ξ for i < n − 2 and ∂_{i}t = ∂_{0}b_{i+1} for i = n − 2, n − 1, n. This means that

[∂_{0}b_{n}] = [∂_{0}b_{n−1}] · [∂_{0}b_{n+1}] in π_{n−1}(F, ξ)

and on the other hand we have [a_{n}] = [a_{n−1}] · [a_{n+1}] in π_{n}(Y, ξ′), whence the assertion.

(i): The inclusion Im π_{n}(i, ξ) ⊂ Ker π_{n}(p, ξ) is immediate, and the identity Im δ_{n+1} = Ker π_{n}(i, ξ) follows easily from lemma 7.5.46, by inspection of the definition of δ_{n+1}. The inclusion Im π_{n+1}(p, ξ) ⊂ Ker δ_{n+1} is likewise immediate, from the definition of δ_{n+1} (details left to the reader).

• Next, let us show that Ker δ_{n+1} ⊂ Im π_{n+1}(p, ξ). Indeed, let a ∈ Y (ξ′, n + 1), pick b ∈ X[n + 1] such that p[n + 1](b) = a and ∂_{i}b = ξ for every i = 1, …, n + 1, and suppose that ∂_{0}b ∼_{∂Δ_{n}} ξ. Let h : Δ_{n} × Δ_{1} → F be a homotopy from ∂_{0}b to ξ relative to ∂Δ_{n}. Then we get a unique morphism

t : (Δ_{n+1} × Λ_{1}^{1}) ∪ (∂Δ_{n+1} × Δ_{1}) → X

whose restriction to Δ_{n+1} × Λ_{1}^{1} agrees with b, whose restriction to Λ_{n+1}^{0} × Δ_{1} factors through ξ, and whose restriction to (Im ε_{0}) × Δ_{1} ⊂ ∂Δ_{n+1} × Δ_{1} agrees with h. Since X is fibrant, t extends to a morphism t′ : Δ_{n+1} × Δ_{1} → X, and we denote by x ∈ X[n + 1] the restriction of t′ to Δ_{n+1} × Λ_{1}^{0}. Then it is easily seen that x ∈ X(ξ, n + 1), and p ∘ t′ is a homotopy relative to ∂Δ_{n+1} from a to p(x), whence the assertion.

• Lastly, let b ∈ X(ξ, n) be any simplex such that [p(b)] = [ξ′] in π_{n}(Y, ξ′), and pick any homotopy h : Δ_{n} × Δ_{1} → Y from p(b) to ξ′ relative to ∂Δ_{n}. We get a commutative diagram

\xymatrix{(Δ_{n} × Λ_{1}^{0}) ∪ (∂Δ_{n} × Δ_{1}) \ar[r]^{φ} \ar[d] & X \ar[d]^{p} \\ Δ_{n} × Δ_{1} \ar[r]^{h} & Y}

where φ is the unique morphism whose restriction to ∂Δ_{n} × Δ_{1} factors through ξ, and whose restriction to Δ_{n} × Λ_{1}^{0} agrees with b. Since p is a fibration, we may then find a morphism h′ : Δ_{n} × Δ_{1} → X extending φ and such that p ∘ h′ = h. Set b′ := h′ ∘ (1_{Δ_{n}} × ε_{1}), so that h′ is a homotopy from b to b′ relative to ∂Δ_{n}; however, notice that p(b′) = ξ′, so b′ ∈ F (ξ, n), therefore [b] ∈ π_{n}(i, ξ), which concludes the proof of the theorem. □

7.5.57. The following device shall be useful for explaining the functorial behaviour of the homotopy groups under change of base points. Let A be any fibrant simplicial set; we set

π(A) := A[1]/∼_{∂Δ_{1}} .

Given a ∈ A[1], we denote by [a] ∈ π(A) the class of a. Evidently, the vertices ∂_{i}a ∈ A[0] (i = 0, 1) depend only on [a], so we shall also denote them by ∂_{i}[a]. We use the notation

[a] : x → y

to signal that x = ∂_{0}[a] and y = ∂_{1}[a]. If [b] : y → z is another class, we define a composition [b] ∘ [a] : x → z as follows. Let φ : Λ_{2}^{1} → A be the unique morphism such that φ ∘ ε_{0} = a and φ ∘ ε_{2} = b; since A is fibrant, φ extends to a morphism c : Δ_{2} → A, and we let

[b] ∘ [a] := [∂_{1}c].

In order to show that this rule is well defined, suppose that a′, b′ ∈ A[1] and c′ ∈ A[2] are any three other simplices such that [a] = [a′], [b] = [b′], ∂_{0}c′ = a′ and ∂_{2}c′ = b′. Pick a homotopy h (resp. h′) from a to a′ (resp. from b to b′) relative to ∂Δ_{1}; there follows a morphism

t : (Δ_{2} × ∂Δ_{1}) ∪ (Λ_{2}^{1} × Δ_{1}) → A

whose restriction to Δ_{2} × Λ_{1}^{0} (resp. to Δ_{2} × Λ_{1}^{1}) agrees with c (resp. with c′) and whose restriction to (Im ε_{0}) × Δ_{1} (resp. to (Im ε_{2}) × Δ_{1}) agrees with h (resp. with h′). Since A is fibrant, t extends to a morphism Δ_{2} × Δ_{1} → A, whose restriction to (Im ε_{1}) × Δ_{1} is a homotopy from ∂_{1}c to ∂_{1}c′ relative to ∂Δ_{1}, whence the claim. Moreover, we claim that this composition law is associative : indeed, the proof is the same as that of the associativity of the group law on π_{1}(A, ξ) (details left to the reader). Furthermore, for every vertex x ∈ A[0], let 1_{x} : Δ_{1} → A be the unique morphism which factors through x; then [a] ∘ [1_{x}] = [a] for every y ∈ A[0] and every [a] : x → y, since

∂_{0}(a ∘ η_{1}) = 1_{x}  ∂_{1}(a ∘ η_{1}) = a  ∂_{2}(a ∘ η_{1}) = a.

Likewise, we see that [1_{x}] ∘ [b] = [b] for every [b] : y → x, by considering ∂(b ∘ η_{0}) = b (details left to the reader). Lastly, for any [a] : x → y consider the unique morphism t : Λ_{2}^{0} → A such that t ∘ ε_{1} = 1_{x} and t ∘ ε_{0} = a; since A is fibrant, t extends to a morphism c : Δ_{2} → A, and it follows that [∂_{2}c] ∘ [a] = 1_{x}. Similarly, we see easily that [a] admits a right inverse. We conclude that the datum

\underline{π}(A) := (A[0], π(A), ∘)

is a groupoid, i.e. a category all whose morphisms are isomorphisms. It shall be called the fundamental groupoid of A.

Remark 7.5.58. Let f : X → Y be any morphism of fibrant simplicial sets. A simple inspection of the definitions shows that f induces a functor

\underline{π}(f) : \underline{π}(X) → \underline{π}(Y).

Explicitly, \underline{π}(f) is the functor whose map on objects X[0] → Y [0] is given by f[0], and whose map on morphisms is induced by f[1]. It is then clear that the rule f ↦ \underline{π}(f) defines a functor

\underline{π} : s.Set → Gpd

with values in the full subcategory of Cat whose objects are all (small) groupoids.

7.5.59. Now, let n ∈ ℕ be any integer, x ∈ A[0] any vertex, b : Δ_{n} → A any element of A(x, n), and [α] : x → y any element of π(A). We have a unique morphism

t : (∂Δ_{n} × Δ_{1}) ∪ (Δ_{n} × Λ_{1}^{1}) → A

such that :

• the restriction of t to ∂Δ_{n} ×Δ_{1} equals α∘p, where p : ∂Δ_{n} ×Δ_{1} → Δ_{1} is the natural projection

• the restriction of t to Δ_{n} × Λ_{1}^{1} agrees with b. Since A is fibrant, t extends to a morphism h : Δ_{n} × Δ_{1} → A, and we denote by ∂_{1}h ∈ A[n] the restriction of h to Δ_{n} × Λ_{1}^{0}. Notice that ∂_{1}h ∈ A(y, n).

Lemma 7.5.60. With the notation of (7.5.59), the class [∂_{1}h] ∈ π_{n}(A, y) depends only on [α] and [b] ∈ π_{n}(A, x) (and is independent of h and of the representatives α and b for these classes).

Proof. Say that [β] = [α] in π(A), so we have a homotopy γ : Δ_{1} × Δ_{1} → A from α to β relative to ∂Δ_{1}. Suppose also that [b] = [b′] in π_{n}(A, x), and let H : Δ_{n} × Δ_{1} → A be a given homotopy from b to b′ relative to ∂Δ_{n}. Pick also a morphism h′ : Δ_{n} × Δ_{1} → A whose restriction to ∂Δ_{n} × Δ_{1} (resp. to Δ_{n} × Λ_{1}^{1}) equals β ∘ p (resp. b′). To ease notation, set

K := (Λ_{1}^{1} × Δ_{1}) ∪ (Δ_{1} × ∂Δ_{1}) and L := Δ_{1} × Δ_{1}

and notice that the monomorphism K → L is anodyne. We then have a unique morphism

u : (∂Δ_{n} × L) ∪ (Δ_{n} × K) → A

such that • the restriction of u to ∂Δ_{n} × L equals q ∘ γ, where q : ∂Δ_{n} × L → L is the natural

projection

• the restriction of u to Δ_{n} × (Λ_{1}^{1} × Δ_{1}) equals H

• the restriction of u to Δ_{n} × (Λ_{1}^{1} × Δ_{1}) (resp. to Δ_{n} × (Λ_{1}^{0} × Δ_{1})) equals h (resp. h′). By corollary 7.5.28, the morphism u extends to a morphism h′′ : Δ_{n} × L → A, and it is easily seen that the restriction of h′′ to Δ_{n} × (Λ_{1}^{0} × Δ_{1}) is a homotopy from ∂_{1}h to ∂_{1}h′ relative to ∂Δ_{n}, whence the assertion. □

7.5.61. In light of lemma 7.5.60, the rule (α, b) ↦ [∂_{1}h] yields a well defined mapping

π_{n}(A, [α]) : π_{n}(A, x) → π_{n}(A, y)

for every n ∈ ℕ, every x, y ∈ A[0] and every [α] : x → y. For n = 0, a simple inspection shows that the resulting map π_{0}(A, [α]) : π_{0}(A) → π_{0}(A) is the identity. Thus, in the following we assume that n > 0, in which case we have :

Proposition 7.5.62. With the notation of (7.5.61), the following holds for every n > 0 : (i) π_{n}(A, [α]) is a group homomorphism, and π_{n}(A, [1_{x}]) = 1_{π_{n}(A,x)} for every x ∈ A[0]. (ii) If [β] : y → z is any other element of π(A), we have

π_{n}(A, [β]) ∘ π_{n}(A, [α]) = π_{n}(A, [β] ∘ [α]).

(iii) Therefore, the rule

x ↦ π_{n}(A, x)  [α] ↦ π_{n}(A, [α])  for every x ∈ A[0] and every [α] ∈ π(A)

yields a well defined functor

π_{n} : \underline{π}(A) → Grp.

Proof. (i): Let b, b′ ∈ A(x, n) be any two elements, and pick a morphism c : Δ_{n+1} → A such that ∂_{i}c = x for i = 0, …, n−2, ∂_{n−1}c = b and ∂_{n+1}c = b′. Let also [α] : x → y be any element of π(A), and choose morphisms h, h′ : Δ_{n} × Δ_{1} → A whose restrictions to ∂Δ_{n} × Δ_{1} equal p∘α (where p is as in (7.5.59)) and whose restrictions to Δ_{n} ×Λ_{1}^{1} agree with b and respectively b′. Then there exists a unique morphism T : (∂Δ_{n+1} × Δ_{1}) ∪ (Δ_{n+1} × Λ_{1}^{1}) → A such that

• for i = 0, …, n − 2, the restriction of T to (Im ε_{i}) × Δ_{1} equals p′ ∘ α, where p′ :

(Im ε_{i}) × Δ_{1} → Δ_{1} is the projection

• the restriction of T to (Im ε_{n−1}) × Δ_{1} (resp. to (Im ε_{n+1}) × Δ_{1}) agrees with h (resp.

with h′)
• the restriction of T to Δ_{n+1} × Λ_{1}^{1} agrees with c.

Since A is fibrant, T extends to a morphism H : Δ_{n+1} × Δ_{1} → A, and we denote by U : Δ_{n}×Δ_{1} → A the restriction of H to (Im ε_{n})×Δ_{1}. By inspecting the definition, it is easily seen that ∂_{1}U is a representative for both π_{n}(A, [α])([b] · [b′]) and π_{n}(A, [α])([b]) · π_{n}(A, [α])([b′]), so π_{n}(A, [α]) is a group homomorphism. A simple inspection shows that π_{n}(A, [1_{x}]) is the identity map of π_{n}(A, x).

(ii): Pick any morphism φ : Δ_{2} → A such that ∂_{0}φ = α and ∂_{2}φ = β. Let also b and h as in (7.5.59), and choose similarly a morphism h′ : Δ_{n} × Δ_{1} → A whose restriction to Δ_{n} × Λ_{1}^{1} equals ∂_{1}h, and whose restriction to ∂Δ_{n} × Δ_{1} equals p ∘ β. Then there exists a unique morphism u : (∂Δ_{n} × Δ_{2}) ∪ (Δ_{n} × Λ_{2}^{1}) → A such that

• the restriction of u to ∂Δ_{n} × Δ_{2} equals q ∘ φ, where q : ∂Δ_{n} × Δ_{2} → Δ_{2} is the projection

• the restriction of u to Δ_{n} × (Im ε_{0}) (resp. to Δ_{n} × (Im ε_{2})) equals h (resp. h′).

Since A is fibrant, u extends to a morphism w : Δ_{n} × Δ_{2} → A, and we denote by w_{1} : Δ_{n} × Δ_{1} → A the restriction of w to Δ_{n} × (Im ε_{1}). Hence, the restriction of w_{1} to ∂Δ_{n} × Δ_{1} equals p ∘ (∂_{1}φ), and its restriction ∂_{1}w_{1} to Δ_{n} × Λ_{1}^{0} (resp. to Δ_{n} × Λ_{1}^{1}) equals b (resp. ∂_{1}h′). Unwinding the definitions, we find that

\begin{matrix}π_{n}(A, [β]) ∘ π_{n}(A, [α])([b]) = & π_{n}(A, [β])([∂_{1}h]) \\ = & [∂_{1}w_{1}] \\ = & π_{n}(A, [∂_{1}φ])([b]) \\ = & π_{n}(A, [β] ∘ [α])([b]) \end{matrix}

whence the contention. Of course, (iii) results by combining (i) and (ii). □

Remark 7.5.63. Let f : X → Y be any morphism of fibrant simplicial sets, [α] : x → x′ a morphism in \underline{π}(X), and set y := f[0](x), y′ := f[0](y′) and β := f[1](α), so that [β] = \underline{π}(f)[α] in \underline{π}(Y) (notation of remark 7.5.58). A direct inspection shows that the resulting diagram

\xymatrix{π_{n}(X, x) \ar[r]^{π_{n}(X,[α])} \ar[d]_{π_{n}(f,x)} & π_{n}(X, x′) \ar[d]^{π_{n}(f,x′)} \\ π_{n}(Y, y) \ar[r]^{π_{n}(Y,[β])} & π_{n}(Y, y′)}

commutes (details left to the reader).

Definition 7.5.64. Consider any commutative diagram of simplicial sets :

\xymatrix{X′ \ar[rr]^{f} \ar[rr]_{g} \ar[dr]_{p′} &  & X \ar[dl]^{p} \\  & Y.}

(i) A p-fibrewise homotopy from f to g is a homotopy h from f to g such that p ∘ h is a constant homotopy (see remark 7.5.38(i)).

(ii) Let also K ⊂ X′ be any simplicial subset. A p-fibrewise homotopy from f to g relative to K is a p-fibrewise homotopy from f to g which is also a homotopy relative to K (see definition 7.5.37). We write

f ∼^{p} g (resp. f ∼^{p}_{K} g)

if there exists a p-fibrewise homotopy from f to g (resp. relative to K). (iii) Recall that, for every n ∈ ℕ, we may regard any x ∈ X[n] as a morphism x : Δ_{n} → X.

Then, we say that p is a minimal fibration, if p is a fibration, and

a = b ⇔ a ∼^{p}_{∂Δ_{n}} b

for every n ∈ ℕ and every a, b ∈ X[n] such that p[n](a) = p[n](b). (iv) A strong deformation retract of p is a datum (i, r, h) consisting of a monomorphism

i : X′ → X, an epimorphism r : X → X′ such that

p ∘ i ∘ r = p and r ∘ i = 1_{X′}

and a p-fibrewise homotopy h : X × Δ_{1} → X from 1_{X} to i ∘ r relative to X′.

(v) We say that f is a fibrewise homotopy equivalence from p′ to p, if there exists a morphism of simplicial sets f′ : X → X′ with

p′ ∘ f′ = p  f′ ∘ f ∼^{p′} 1_{X′}  f ∘ f′ ∼^{p} 1_{X}.

(vi) For any simplicial set A, let p_{A} : A → Δ_{0} be the unique morphism of simplicial sets. If A is fibrant, we say that A is minimal, if p_{A} is a minimal fibration. We say that a morphism of simplicial sets f : A → A′ is a homotopy equivalence, if f is a fibrewise homotopy equivalence from p_{A} to p_{A′}. We say that a monomorphism i : A′ → A is a strong deformation retract of A, if there exist a morphism r : A → A′ and a homotopy h from 1_{A} to i ∘ r, such that (i, r, h) is a strong deformation retract of p_{A}.

Remark 7.5.65. (i) By inspecting the definitions, it is easily seen that a composition of fibrewise homotopy equivalences is a fibrewise homotopy equivalence. Especially, a composition of homotopy equivalences is a homotopy equivalence.

(ii) Let (i, r, h) be any strong deformation retract; then, clearly, both i and r are fibrewise homotopy equivalences.

Lemma 7.5.66. Let p : X → Y be any fibration, p′ : X′ → Y any object of s.Set/Y, and i : K → X′ any monomorphism. Then :

(i) ∼^{p}_{K} is an equivalence relation on Hom_{s.Set/Y} (p′, p).

(ii) More precisely, let f, f′, f′′ ∈ Hom_{s.Set/Y} (p′, p) be any three elements such that

f ∼^{p} f′ f′ ∼^{p}_{K} f′′

and h a p-fibrewise homotopy from f to f′. Then there exists a p-fibrewise homotopy h′ from f to f′′ such that

h′ ∘ (i × 1_{Δ_{1}}) = h ∘ (i × 1_{Δ_{1}}).

(iii) Let f, f′ be as in (ii), and suppose that f is a fibrewise homotopy equivalence. Then the same holds for f′.

Proof. (i): It is a relative variant of the proof of theorem 7.5.42. Namely, consider first the case of ∼^{p}_{K}; on the one hand, by proposition 7.5.34, the morphism

t : s.ℋ om(X′, X) → P := s.ℋ om(X′, Y) ×_{s.ℋom(K,Y)} s.ℋ om(K, X)

induced by p and i is a fibration. On the other hand, let f, f′ ∈ Hom_{s.Set/Y} (p′, p) be any two elements; the datum of a homotopy h from f to f′ relative to K is the same as that of a morphism

h^{∗} : Δ_{1} → s.ℋ om(X′, X)

such that s.ℋ om(i, X) ∘ h^{∗} factors through a morphism a : Δ_{0} → s.ℋ om(K, X) and the unique morphism π : Δ_{1} → Δ_{0}. Then, h is a p-fibrewise homotopy if and only if

\begin{matrix}s.ℋ om(K, p) ∘ a = & s.ℋ om(i, Y) ∘ b \\ s.ℋ om(X′, p) ∘ h^{∗} = & b ∘ π \end{matrix}

for the unique morphism b : Δ_{0} → s.ℋ om(X′, Y) corresponding to p′. Now, the pair (a, b) determines a unique morphism (a, b) : Δ_{0} → P, and the foregoing conditions tell us that t∘h^{∗} = (a, b)∘π. Thus, set Q := t^{−1}(a, b) (notation of example 7.5.2(vi)); by lemma 7.5.15(iv), the simplicial set Q is fibrant, and we have a natural bijection

Q[0] →^{∼} {f ∈ Hom_{s.Set/Y} (p′, p) | f ∘ i = a}

which identifies the restriction of the relation ∼^{p}_{K} with the relation ∼ on Q[0]. The latter is an equivalence relation, by lemma 7.5.40, whence the contention.

(ii): Let again b : Δ_{0} → s.ℋ om(X′, Y) be the morphism corresponding to p′, and set

P_{b} := q^{−1}(b) Q_{b} := P_{b} ×_{P} s.ℋ om(X′, X)

where q : P → s.ℋ om(X′, Y) is the natural projection. So, the morphism t restricts to a fibration t_{b} : Q_{b} → P_{b}. By assumption, f′ and f′′ can be regarded as morphisms Δ_{0} → s.ℋ om(X′, X) such that a := s.ℋ om(i, X) ∘ f′ = s.ℋ om(i, X) ∘ f′′, and there exists a morphism g : Δ_{1} → s.ℋ om(X′, X) such that s.ℋ om(i, X) ∘ g factors through a and s.ℋ om(X′, p) ∘ g factors through b, and moreover g ∘ ε_{1} = f′, g ∘ ε_{0} = f′′. Likewise, the homotopy h can be regarded as a morphism Δ_{1} → s.ℋ om(X′, X) such that s.ℋ om(X′, p)∘h factors through b, and moreover h ∘ ε_{1} = f, h ∘ ε_{0} = f′. The pair (h, g) then amounts to a morphism (h, g) : Λ_{2}^{1} → Q_{b} fitting into a commutative diagram

\xymatrix{Λ_{2}^{1} \ar[d]_{ι_{2}^{1}} \ar[rr]^{(h,g)} &  & Q_{b} \ar[d]^{t_{b}} \\ Δ_{2} \ar[r]^{η_{0}} & Δ_{1} \ar[r] & P_{b}}

(namely, (h, g) is the unique morphism such that (h, g) ∘ ε_{0} = h and (h, g) ∘ ε_{2} = g). Thus, (h, g) extends to a morphism u : Δ_{2} → Q_{b} such that t_{b} ∘u factors as well through η_{0}; it is easily seen that the morphism u ∘ ε_{1} yields the sought homotopy h′.

(iii): By assumption, there exists a morphism g : X → X′ of s.Set/Y such that g ∘ f ∼^{p} 1_{X′} and f ∘ g ∼^{p′} 1_{X}. On the other hand, since f ∼^{p} f′, remark 7.5.38(ii) implies that g ∘ f ∼^{p′} g ∘ f′ and f ∘ g ∼^{p} f′ ∘ g. Then the assertion follows from (i). □

7.5.67. Consider a fibration p : X → Y of simplicial sets, and two morphisms f_{0}, f_{1} : A → Y, and for i = 0, 1 define the simplicial set X_{i} as the fibre product in the cartesian diagram

\xymatrix{X_{i} \ar[r] \ar[d]_{p_{i}} & X \ar[d]^{p} \\ A \ar[r]^{f_{i}} & Y.}

Proposition 7.5.68. In the situation of (7.5.67), suppose that f_{0} ∼ f_{1} (notation of (7.5.41)). Then there exists a fibrewise homotopy equivalence from p_{0} to p_{1}. Proof. Let h : A × Δ_{1} → Y be a homotopy from f_{0} to f_{1}, and define the simplicial set H as the fibre product in the resulting diagram whose two square subdiagrams are cartesian

\xymatrix{X_{i} \ar[r]^{j_{i}} \ar[d]_{p_{i}} & H \ar[r] \ar[d]_{p_{H}} & X \ar[d]^{p} \\ A \ar[r]^{A×ε_{i}} & A × Δ_{1} \ar[r]^{h} & Y}  for i = 0, 1.

Then, p_{H} is a fibration (lemma 7.5.15(iv)), hence there exist commutative diagrams

\xymatrix{X_{i} \ar[r]^{j_{i}} \ar[d]_{X_{i}×ε_{i}} & H \ar[d]^{p_{H}} \\ X_{i} × Δ_{1} \ar[r]^{p_{i}×Δ_{1}} \ar[ur]^{ϑ_{i}} & A × Δ_{1}}  for i = 0, 1

(proposition 7.5.24) whence two commutative diagrams with cartesian square subdiagrams

\xymatrix{X_{i} \ar[r]^{ω_{i}} \ar[d]_{X_{i}×ε_{1−i}} \ar[rr]^{p_{i}} & X_{1−i} \ar[r]^{p_{1−i}} \ar[d]^{j_{1−i}} & A \ar[d]^{A×ε_{1−i}} \\ X_{i} × Δ_{1} \ar[r]^{ϑ_{i}} \ar[rr]_{p_{i}×Δ_{1}} & H \ar[r]^{p_{H}} & A × Δ_{1}} for i = 0, 1.

Notice now that

ϑ_{0} ∘ (X_{0} × ε_{1}) = j_{1} ∘ ω_{0} = ϑ_{1} ∘ (X_{1} × ε_{1}) ∘ ω_{0} = ϑ_{1} ∘ (ω_{0} × Δ_{1}) ∘ (X_{0} × ε_{1})

from which we deduce that there exists a commutative diagram

\xymatrix{X_{0} × Λ_{2}^{0} \ar[r]^{β} \ar[d]_{X_{0}×ι_{2}^{0}} & H \ar[d]^{p_{H}} \\ X_{0} × Δ_{2} \ar[r]^{p_{0}×η_{1}} \ar[ur]^{γ} & A × Δ_{1}}

where β is the unique morphism such that

β ∘ (X_{0} × ε_{1}) = ϑ_{0} and β ∘ (X_{0} × ε_{2}) = ϑ_{1} ∘ (ω_{0} × Δ_{1}).

Set γ_{0} := γ ∘ (X_{0} × ε_{0}) : X_{0} × Δ_{1} → H, and notice that η_{1} ∘ ε_{0} : Δ_{1} → Δ_{1} factors through ε_{0} : Δ_{0} → Δ_{1}; it follows that γ_{0} factors through j_{0} : X_{0} → H and a morphism \overline{γ}_{0} : X_{0} × Δ_{1} → X_{0}. Lastly, notice that

\begin{matrix}j_{0} ∘ \overline{γ}_{0} ∘ (X_{0} × ε_{0}) = & ϑ_{0} ∘ (X_{0} × ε_{0}) = j_{0} \\ j_{0} ∘ \overline{γ}_{0} ∘ (X_{0} × ε_{1}) = & ϑ_{1} ∘ (ω_{0} × Δ_{1}) ∘ (X_{0} × Δ_{0}) = ϑ_{1} ∘ (X_{1} × ε_{0}) ∘ ω_{0} = j_{0} ∘ ω_{1} ∘ ω_{0} \end{matrix}

whence

\overline{γ}_{0} ∘ (X_{0} × ε_{0}) = 1_{X_{0}} and \overline{γ}_{0} ∘ (X_{0} × ε_{1}) = ω_{1} ∘ ω_{0}

so \overline{γ}_{0} is a homotopy from 1_{X_{0}} to ω_{1} ∘ ω_{0}. A similar construction yields likewise a homotopy from 1_{X_{1}} to ω_{0} ∘ ω_{1}, whence the proposition (details left to the reader). □

Proposition 7.5.69. Every fibration p : X → Y admits a strong deformation retract (j, r, h) such that p ∘ j is a minimal fibration.

Proof. For every k, n ∈ ℕ with n ≥ k, denote by

Sk_{k}X ⟶^{ε_{X}^{(k,n)}} Sk_{n}X ⟶^{η_{X}^{(n)}} X

the inclusion maps (notation of remark 7.5.6(v)) and set p^{(n)} := p ∘ η_{X}^{(n)} for every n ∈ ℕ.

Claim 7.5.70. For every n ∈ ℕ there exist morphisms

j^{(n)} : Z^{(n)} → Sk_{n}X  r^{(n)} : Sk_{n}X → Z^{(n)}  h^{(n)} : Sk_{n}X × Δ_{1} → X

such that :

(a) r^{(n)} ∘ j^{(n)} = 1_{Z^{(n)}} and p^{(n)} ∘ j^{(n)} ∘ r^{(n)} = p^{(n)}.

(b) h^{(n)} is a homotopy from η_{X}^{(n)} to η_{X}^{(n)} ∘ j^{(n)} ∘ r^{(n)}.

(c) Both p∘h^{(n)} and the restriction h_{Z}^{(n)} : Z^{(n)} ×Δ_{1} → X of h^{(n)} are constant homotopies.

(d) ε_{X}^{(k,n)} restricts to a morphism e^{(k,n)} : Z^{(k)} → Z^{(n)} for every k ≤ n.

(e) s.trunc_{k}e^{(k,n)} : s.trunc_{k}Z^{(n)} → s.trunc_{k}Z^{(k)} is an isomorphism, and s.trunc_{k}h^{(n)} =

s.trunc_{k}h^{(k)} for every k ≤ n.
(f) The induced map Z^{(n)}[n] → X[n]/∼^{p}_{∂Δ_{n}} is injective.

Proof of the claim. We construct inductively the sought simplicial sets and morphisms, as follows. First, set Z^{(−1)} := s.∅. Next, let n ∈ ℕ be any integer such that either n = 0 or else Z^{(n−1)} ⊂ Sk_{n−1}X, r^{(n−1)} and h^{(n−1)} have already been defined. Since Z^{(n−1)}[n] consists entirely of degenerate simplices (see remark 7.5.6(iv)), lemma 7.5.7(i) implies that the natural projection Z^{(n−1)}[n] → X[n]/∼^{p}_{∂Δ_{n}} is injective.

Now, for n = 0, set C_{0} := X[0]/ ∼^{p}_{∂Δ_{n}}, and if n > 0, let C_{n} ⊂ X[n]/ ∼^{p}_{∂Δ_{n}} be the subset of all equivalence classes of elements x ∈ X[n] such that ∂_{i}x ∈ Z^{(n−1)}[n − 1] for every i = 0, …, n (notice that ∂_{i}x depends only on the class of x in X[n]/ ∼^{p}_{∂Δ_{n}}). Pick a representative x ∈ X[n] for every equivalence class \overline{x} ∈ C_{n} that does not lie in the image of Z^{(n−1)}[n], and let R_{n} ⊂ X[n] be the subset of all such chosen representatives; we denote by T^{(n)} ⊂ s.trunc_{n}X the n-truncated simplicial set such that

T^{(n)}[n] = Z^{(n−1)}[n] ∪ R_{n}  and T^{(n)}[k] = Z^{(n−1)}[k]  for every k < n.

The inclusion map T^{(n)} → s.trunc_{n}X induces a morphism sk_{n}T^{(n)} → Sk_{n}X, so we may set

Z^{(n)} := Im (sk_{n}T^{(n)} → Sk_{n}X)

and we let j^{(n)} : Z^{(n)} → Sk_{n}X be the inclusion map. With this definition, we already see that conditions (d) and (f) are fulfilled, as well the part of condition (e) concerning Z^{(n)}. Next, we define the morphism r^{(n)} : Sk_{n}X → Z^{(n)}. Namely, we set r^{(n)}[k] := r^{(n−1)}[k] for every k < n, and the map r^{(n)}[n] is constructed as follows. If n = 0, we let r^{(0)}[0] : X[0] → Z^{(0)}[0] = R_{0} be the map that sends any x ∈ X[0] to the unique representative of the equivalence class of x in X[0]/∼^{p}_{∂Δ_{n}} that lies in R_{0}. In case n > 0, let x ∈ Sk_{n}X[n] = X[n] be any element; if x ∈ Z^{(n)}[n], we let r^{(n)}[n](x) := x, and if x ∉ Z^{(n)}[n] we let h_{∂x} : ∂Δ_{n} × Δ_{1} → X be the composition

∂Δ_{n} × Δ_{1} ⟶^{(∂x)×1_{Δ_{1}}} Sk_{n−1}X × Δ_{1} ⟶^{h^{(n−1)}} X.

The pair (h_{∂x}, x) amounts to a morphism γ_{x} fitting into a commutative diagram

\xymatrix{(Δ_{n} × Λ_{j}^{1}) ∪ ((∂Δ_{n}) × Δ_{1}) \ar[rr]^{γ_{x}} \ar[d] &  & X \ar[d]^{p} \\ Δ_{n} × Δ_{1} \ar[r] & Δ_{n} \ar[r]^{p∘x} & Y}

whose horizontal (resp. vertical) unmarked arrow is the natural projection (resp. the natural inclusion map). Since p is a fibration, proposition 7.5.24 and lemma 7.5.15(i) imply that γ_{x} extends to a morphism h_{x} : Δ_{n} × Δ_{1} → X which, by inspection, is a p-fibrewise homotopy from x to

x′ := h_{x} ∘ (1_{Δ_{n}} × ε_{1}) : Δ_{n} → X

(notation of example 7.5.2(iv)). Moreover, ∂_{i}x′ ∈ Z^{(n−1)}[n − 1] for every i = 0, …, n, so there exists a unique x′′ ∈ T^{(n)}[n] such that x′ ∼^{p}_{∂Δ_{n}} x′′, and we set

r^{(n)}[n](x) := x′′.

We have to check that the resulting system r^{(n)}[•] := (r^{(n)}[k] | k = 0, …, n) is a morphism

s.trunc_{n}X → T^{(n)}

of n-truncated simplicial sets. If n = 0, there is nothing to show. If n > 0, since the subsystem (r^{(n)}[k] | k = 0, …, n − 1) is already, by assumption, a morphism s.trunc_{n−1}X → s.trunc_{n−1}Z^{(n−1)}, we have only to show that

∂(r^{(n)}[n](x)) = r^{(n)}[n − 1] ∘ ∂x for every x ∈ X[n].

If x ∈ T^{(n)}[n], the identity is obvious, since the face and degeneracies of T^{(n)} are the restrictions of those of X. Hence, suppose x ∉ T^{(n)}[n]; then, by construction, ∂(r^{(n)}[n](x)) is the morphism h_{∂x} ∘ (1_{∂Δ_{n}} × ε_{1}) : ∂Δ_{n} → X. The latter is the same as

(η_{X}^{(n−1)} ∘ j^{(n−1)} ∘ r^{(n−1)})[n − 1] ∘ ∂x

(because h^{(n−1)} is a homotopy from η_{X}^{(n−1)} to η_{X}^{(n−1)} ∘ j^{(n−1)} ∘ r^{(n−1)}) whence the contention. By adjunction, the morphism r^{(n)}[•] yields the required morphism

r^{(n)} : Sk_{n}X → Z^{(n)}.

Also, again by adjunction, the identities of (a) can be checked on the respective n-truncations, where they are clear (details left to the reader).

It remains to exhibit the homotopy h^{(n)} : Sk_{n}X × Δ_{1} → X. However, to any given x ∈ Sk_{n}X[n] \ Z^{(n)}[n], we have already attached a p-fibrewise homotopy h_{x} from x to x′, and we have x′ ∼^{p}_{∂Δ_{n}} r^{(n)}[n](x); by lemma 7.5.66(ii), it follows that there exists a p-fibrewise homotopy h′_{x} : Δ_{n} × Δ_{1} → X from x to r^{(n)}[n](x) such that

(7.5.71)  h′_{x} ∘ (i_{n} × 1_{Δ_{1}}) = h_{x} ∘ (i_{n} × 1_{Δ_{1}}) = h_{∂x}

where i_{n} : ∂Δ_{n} → Δ_{n} is the inclusion map. If x ∈ Z^{(n)}[n], we have x = r^{(n)}[n](x), so we may take h′_{x} to be a constant homotopy. There results a well defined map

μ : X[n] → s.ℋ om(Δ_{1}, X)[n] x ↦ h′_{x}.

On the other hand, by remark 7.5.31(iv), the homotopy h^{(n−1)} corresponds to a morphism of simplicial sets

ν^{(n−1)} : Sk_{n−1}X → s.ℋ om(Δ_{1}, X).

We claim that the datum of μ and the maps (ν^{(n−1)}[k] | k = 0, …, n−1) amounts to a morphism of n-truncated simplicial sets

(7.5.72)  s.trunc_{n}X → s.trunc_{n}(s.ℋ om(Δ_{1}, X)).

The assertion comes down to checking the identity

(7.5.73) ∂_{i} ∘ μ(x) = ν^{(n−1)}[n − 1] ∘ ∂_{i}(x)  for every x ∈ X[n] and every i = 0, …, n

and recall that the i-th face operator on s.ℋ om(Δ_{1}, X)[n]

∂_{i} : Hom_{s.Set}(Δ_{n} × Δ_{1}, X) → Hom_{s.Set}(Δ_{n−1} × Δ_{1}, X)

is given by the rule : φ ↦ φ ∘ (ε_{i} × 1_{Δ_{1}}) for every morphism φ : Δ_{n} × Δ_{1} → X.

Hence, if x ∈ Z^{(n)}[n], the morphism ∂_{i}(h′_{x}) is the constant homotopy from ∂_{i}x to itself; since h^{(n−1)} fulfills condition (c), this is the same as ν^{(n−1)}[n − 1](∂_{i}x), so in this case (7.5.73) holds. In case x ∉ Z^{(n)}[n], identity (7.5.71) says that ∂(h′_{x}) = h_{∂x}. But a simple inspection shows that h_{∂x} = ν^{(n−1)}[n − 1] ∘ ∂x, so (7.5.73) holds also in this case, and we get the sought morphism (7.5.72); by adjunction, the latter corresponds to a morphism

h^{(n)} : Sk_{n}X × Δ_{1} → X

as required, and clearly condition (e) for h^{(n)} is fulfilled by construction. Moreover, a simple inspection shows that (7.5.72) fits into a commutative diagram

\xymatrix{ & s.trunc_{n}X \\ s.trunc_{n}X \ar[ur]^{1_{s.truncnX}} \ar[dr]_{s.trunc_{n}(j^{(n)}∘r^{(n)})} \ar[r] & s.trunc_{n}(s.ℋ om(Δ_{1}, X)) \ar[u]_{s.trunc_{n}(s.ℋom(ε_{0},X))} \ar[d]^{s.trunc_{n}(s.ℋom(ε_{1},X))} \\  & s.trunc_{n}X}

where we have used the natural identification X →^{∼} s.ℋ om(Δ_{0}, X) for the target of the morphisms s.ℋ om(ε_{i}, X) (i = 0, 1). By adjunction, this diagram translates as our condition (b) for h^{(n)}. To conclude the proof of the claim, it remains to check condition (c). However, in order to prove that p ∘ h^{(n)} is a constant homotopy it suffices – by adjunction – to show that the morphism ν^{(n)} : Sk_{n}X → s.ℋ om(Δ_{1}, X) corresponding to h^{(n)} fits into a commutative diagram

𝒟 : \xymatrix{Sk_{n}X \ar[r]^{ν^{(n)}} \ar[d]_{p^{(n)}} & s.ℋ om(Δ_{1}, X) \ar[d]^{s.ℋom(Δ_{1},p)} \\ Y = s.ℋ om(Δ_{0}, Y) \ar[r]^{s.ℋom(π,Y)} & s.ℋ om(Δ_{1}, Y)}

where π is the unique morphism Δ_{1} → Δ_{0}. Again by adjunction, it then suffices to check that s.trunc_{n}𝒟 commutes; the latter assertion follows easily by inspecting the construction of the morphism (7.5.72). We argue similarly to show that h_{Z}^{(n)} is a constant homotopy : the assertion comes down to checking the commutativity of the diagram

𝒟_{Z} : [figure: Z^{(n)}; ν^{(n)}∘j^{(n)}; s.ℋ om(Δ_{1}, X); Z^{(n)} = s.ℋ om(Δ_{0}, Z^{(n)}); s.ℋom(π,Z^{(n)}); s.ℋ om(Δ_{1}, Z^{(n)}); s.ℋom(Δ_{1},η_{X}^{(n)}∘j^{(n)})]

and since the counit of adjunction sk_{n}T^{(n)} → Z^{(n)} is an epimorphism, we are reduced to showing that s.trunc_{n}𝒟_{Z} commutes, which again follows by inspecting the construction of (7.5.72) : details left to the reader. ♢

Now, if the system (Z^{(n)}, j^{(n)}, r^{(n)}, h^{(n)} | n ∈ ℕ) fulfills conditions (a)–(f) of claim 7.5.70, we let

Z := colim_{n∈ℕ} Z^{(n)}

where the transition maps Z^{(k)} → Z^{(n)} are the morphisms e^{(k,n)}, for every k, n ∈ ℕ with k ≤ n. In view of (7.5.9), the colimit of the system of monomorphisms (j^{(n)} | n ∈ ℕ) (resp. epimorphisms (r^{(n)} | n ∈ ℕ)) is therefore a monomorphism

j : Z → X (resp. an epimorphism r : X → Z)

such that r ∘ j = 1_{Z} and p ∘ j ∘ r = p. It is also clear that there exists a unique homotopy h : X × Δ_{1} → X from 1_{X} to j ∘ r, such that

s.trunc_{n}(h) = s.trunc_{n}(h^{(n)}) for every n ∈ ℕ

and then the resulting datum (j, r, h) is the sought strong deformation retract. It remains to check that p ∘ j is a minimal fibration. First, it follows from lemma 7.5.15(iv) that p ∘ j is a fibration. Lastly, fix n ∈ ℕ, and suppose that z, z′ ∈ Z[n] are any two elements such that z ∼^{p∘j}_{∂Δ_{n}} z′; then j(z) ∼^{p}_{∂Δ_{n}} j(z′), so z = z′, by condition (f) of claim 7.5.70. □

Proposition 7.5.74. Consider a commutative diagram of simplicial sets

\xymatrix{X′ \ar[rr]^{f} \ar[dr]_{p′} &  & X \ar[dl]^{p} \\  & Y}

such that p and p′ are minimal fibrations, and f is a fibrewise homotopy equivalence. Then f is an isomorphism of simplicial sets.

Proof. The proposition is an immediate consequence of the following more precise :

Claim 7.5.75. In the situation of the proposition, suppose that :

(a) f′ : X′ →^{∼} X is an isomorphism such that p ∘ f′ = p′ and f ∼^{p} f′.

(b) p is a minimal fibration. Then f is an isomorphism as well.

Proof of the claim. We show first that f[n] is injective for every n ∈ ℕ. We argue by induction on n ∈ ℕ. For n = 0, notice that ∂Δ_{0} = s.∅; hence, if a, b ∈ X′[0] and f[0](a) = f[0](b), assumption (a) yields :

f′[0](a) ∼^{p}_{∂Δ_{0}} f[0](a) f′[0](b) ∼^{p}_{∂Δ_{0}} f[0](b)

whence f′[0](a) ∼^{p}_{∂Δ_{0}} f′[0](b) by virtue of lemma 7.5.66(i); then assumption (b) implies that f′[0](a) = f′[0](b), and finally a = b, as stated. Next, suppose that n > 0 and f[n − 1] is already known to be injective, and let a, b ∈ X′[n] be any two elements with f[n](a) = f[n](b); condition (a) says that there exists a p-fibrewise homotopy h_{a} (resp. h_{b}) : Δ_{n} × Δ_{1} → X from f[n](a) to f′[n](a) (resp. from f[n](b) to f′[n](b)), and on the other hand, the inductive assumption implies that ∂_{i}a = ∂_{i}b, for every i = 0, …, n, so the restrictions of h_{a} and h_{b} are both equal to the same morphism w : ∂Δ_{n} × Δ_{1} → X. There follows a commutative diagram

\xymatrix{(Δ_{n} × Λ_{2}^{2}) ∪ (∂Δ_{n} × Δ_{2}) \ar[r]^{u} \ar[d] & X \ar[d]^{p} \\ Δ_{n} × Δ_{2} \ar[r]^{v} & Y}

such that

u ∘ (1_{Δ_{n}} × ε_{1}) = h_{a} u ∘ (1_{Δ_{n}} × ε_{0}) = h_{b}

and the restriction of u to ∂Δ_{n} × Δ_{2} equals w ∘ (1_{∂Δ_{n}} × η_{0}). Also, v is the composition of the projection Δ_{n} × Δ_{2} → Δ_{n} and the morphism (p ∘ f)[n](a) : Δ_{n} → Y. Then there exists a morphism t : Δ_{n} × Δ_{2} → X that extends u and lifts v; it is easily seen that t ∘ (Δ_{n} × ε_{2}) : Δ_{n} × Δ_{1} → X is a p-fibrewise homotopy from f′[n](a) to f′[n](b) relative to ∂Δ_{n}, and since p is minimal, we conclude that f′[n](a) = f′[n](b), so finally a = b, as required.

Next, we show by induction on n ∈ ℕ that f[k] is surjective, for every k < n. If n = 0, there is nothing to show, hence suppose that n > 0, and let x ∈ X[n] be any element; by inductive assumption, for every i = 0, …, n there exists a_{i} ∈ X′[n−1] such that f[n−1](a_{i}) = ∂_{i}x. Since the injectivity of f has already been established, we see easily that ∂_{i}a_{j} = ∂_{j−1}a_{i} whenever 0 ≤ i < j ≤ n, so the system (a_{0}, …, a_{n}) determines a unique morphism

w′ : ∂Δ_{n} → X′

such that f ∘ w′ is the restriction of x : Δ_{n} → X to ∂Δ_{n} (see example 7.5.2(iv)). Now, let h : X′ × Δ_{1} → X be a p-fibrewise homotopy from f to f′; we deduce a commutative diagram

\xymatrix{(∂Δ_{n} × Δ_{1}) ∪ (Δ_{n} × Λ_{1}^{0}) \ar[r]^{u′} \ar[d] & X \ar[d]^{p} \\ Δ_{n} × Δ_{1} \ar[r]^{v′} & Y}

such that the restriction of u′ to ∂Δ_{n} × Δ_{1} (resp. to Δ_{n} × Λ_{1}^{0}) agrees with h ∘ (w′ × 1_{Δ_{1}}) (resp. with the composition of the projection Δ_{n} × Λ_{1}^{0} →^{∼} Δ_{n} and the morphism x). Also, v′ is a constant homotopy. There follows a morphism t′ : Δ_{n} × Δ_{1} → X that extends u′ and lifts v′. Let

x′ : Δ_{n} → X′

be the unique morphism such that f′ ∘ x′ = t′ ∘ (1_{Δ_{n}} × ε_{1}). By construction, we have

f′ ∘ x′ ∘ i_{n} = f′ ∘ w′

(where i_{n} : ∂Δ_{n} → Δ_{n} is the natural inclusion); whence x′ ∘ i_{n} = w′, and we obtain yet another commutative diagram

\xymatrix{(Δ_{n} × Λ_{2}^{0}) ∪ (∂Δ_{n} × Δ_{2}) \ar[r]^{u′′} \ar[d] & X \ar[d]^{p} \\ Δ_{n} × Δ_{2} \ar[r]^{v′′} & Y}

such that

u′′ ∘ (1_{Δ_{n}} × ε_{1}) = t′ u′′ ∘ (1_{Δ_{n}} × ε_{2}) = h ∘ (x′ × 1_{Δ_{1}})

and where v′′ factors through the projection Δ_{n} × Δ_{2} → Δ_{n}. As usual, we find a morphism t′′ : Δ_{n} × Δ_{2} → X that lifts v′′ and extends u′′, and it is easily seen that t′′ ∘ (Δ_{n} × ε_{0}) is a p-fibrewise homotopy from x to f[n](x′) relative to ∂Δ_{n}. By the minimality of p, it follows that x = f[n](x′), whence the sought surjectivity of f[n]. □

Definition 7.5.76. Let f : X → Y be any morphism of fibrant simplicial sets. We say that f is a weak equivalence, if the induced map π_{n}(f, ξ) is an isomorphism, for every ξ ∈ X[0] and every n ∈ ℕ.

Proposition 7.5.77. Let f : X → Y and g : Y → Z be any two morphisms of fibrant simplicial sets. The following holds :

(i) If f has the right lifting property with respect to the natural monomorphism i_{n} : ∂Δ_{n} → Δ_{n} for every n ∈ ℕ, then f is both a fibration and a weak equivalence.

(ii) If f is a homotopy equivalence, then it is a weak equivalence.

(iii) If any two of the three morphisms f, g, g ∘ f is a weak equivalence, then so is the third.

Proof. (i): It follows easily from lemma 7.5.15(i) and remark 7.5.8 that f has the right lifting property with respect to every monomorphism of simplicial sets (details left to the reader). Especially, f is a fibration. It also follows easily that f[0] is surjective. Moreover, suppose that a, b ∈ X[0] and [f(a)] = [f(b)] in π_{0}(Y), and pick a homotopy h : Δ_{1} → Y from f(a) to f(b); there follows a commutative diagram

\xymatrix{∂Δ_{1} \ar[d]_{i_{1}} \ar[r]^{c} & X \ar[d]^{f} \\ Δ_{1} \ar[r]^{h} & Y}

such that the restriction of c to Λ_{1}^{0} (resp. to Λ_{1}^{1}) agrees with a (resp. with b). By assumption, c extends to a morphism h′ : Δ_{1} → X such that f ∘ h′ = h. We conclude that π_{0}(f) is bijective. Next, let ξ ∈ X[0] be any vertex, and set F := f^{−1}(f(ξ)). By lemma 7.5.15(ii), the unique morphism F → Δ_{0} has the right lifting properties with respect to i_{n}, for every n ∈ ℕ. By the foregoing, we know already that π_{0}(F) is a set of cardinality one; moreover, we have :

Claim 7.5.78. π_{n}(F, ξ) = 0 for every n ≥ 1.

Proof of the claim. Let a ∈ F (ξ, n) be any element; then there exists a unique morphism b : ∂Δ_{n+1} → F whose restriction to Λ_{n+1}^{0} factors through ξ, and whose composition with ε_{0} : Δ_{n} → ∂Δ_{n+1} agrees with a. By assumption, b extends to a morphism c : Δ_{n+1} → X, and then the claim follows from lemma 7.5.46. ♢

The proposition now follows easily from theorem 7.5.56and claim 7.5.78.

(ii): Let f′ : Y → X be a morphism and h : X × Δ_{1} → X a homotopy from γ_{0} := 1_{X} to γ_{1} := f′ ∘ f. By adjunction, h corresponds to a morphism β : X → s.ℋ om(Δ_{1}, X) fitting into the commutative diagrams

\xymatrix{X \ar[r]^{β} \ar[dr]_{γ_{i}} & s.ℋ om(Δ_{1}, X) \ar[d]^{s.ℋom(ε_{i},X)} \\ & s.ℋ om(Λ_{1}^{i}, X) = X} i = 0, 1.

Recall that s.ℋ om(Δ_{1}, X) is fibrant (corollary 7.5.35(i)).

Claim 7.5.79. β and γ_{1} are weak equivalences.

Proof of the claim. As for β, since γ_{0} is an isomorphism, it suffices to show that p_{0} := s.ℋ om(ε_{0}, X) is a weak equivalence. To this aim, by (i), we are reduced to checking that p_{0} has the right lifting property with respect to the monomorphisms i_{n}, for every n ∈ ℕ. Thus, let a : ∂Δ_{n} → s.ℋ om(Δ_{1}, X) be a morphism such that p_{0}∘a extends to a morphism b : Δ_{n} → X; by adjunction, a and b determine a unique morphism (Δ_{n} × Λ_{1}^{0}) ∪ (∂Δ_{n} × Δ_{1}) → X which, since X is fibrant, extends to a morphism Δ_{n}×Δ_{1} → X. The latter corresponds, by adjunction, to a unique morphism Δ_{n} → s.ℋ om(Δ_{1}, X) which is the sought extension of a.

The same argument shows as well that s.ℋ om(ε_{1}, X) is a weak equivalence, and so the same follows for γ_{1}, as stated. ♢

Now, if f is a homotopy equivalence, we have also a homotopy h′ : Y × Δ_{1} → Y from 1_{Y} to f ∘ f′. Taking into account claim 7.5.79, we deduce that both

π_{n}(g, f(ξ)) ∘ π_{n}(f, ξ) and π_{n}(f, f′(ξ′)) ∘ π_{n}(f′, ξ′)

are bijections, for every n ∈ ℕ, every ξ ∈ X[0] and every ξ′ ∈ Y [0]. The assertion follows.

(iii): The only non-trivial assertion is that g is a weak equivalence if both f and g ∘ f are. However, in this case it is clear that π_{0}(g) is an isomorphism, so it remains only to check that π_{n}(g, y) is an isomorphism for every y ∈ Y [0]. Thus, fix such vertex y; since π_{0}(f) is a bijection, there exist x ∈ X[0] and a morphism [α] : y → y′ := f[0](x) in \underline{π}(Y) (notation of (7.5.57)). Set z := g[0](y), z′ := g[0](z′) and β := g[1](α); there follows a commutative diagram (see remark 7.5.63)

\xymatrix{π_{n}(Y, y) \ar[r]^{π_{n}(Y,[α])} \ar[d]_{π_{n}(g,y)} & π_{n}(Y, y′) \ar[d]^{π_{n}(g,y′)} & π_{n}(X, x) \ar[l]_{π_{n}(f,x)} \ar[dl]^{π_{n}(g∘f,x)} \\ π_{n}(Z, z) \ar[r]^{π_{n}(Z,[β])} & π_{n}(Z, z′)}

from which we see first that π_{n}(g, y′) is an isomorphism, and then the assertion follows. □

The following is the main result of this section :

Theorem 7.5.80 (Whitehead). Every weak equivalence of fibrant simplicial sets is a homotopy equivalence.

Proof. We begin with the following special case :

Claim 7.5.81. Let f : X → Y be any minimal fibration of fibrant simplicial sets, and suppose that f is a weak equivalence. Then f is an isomorphism.

Proof of the claim. We consider first the case where Y = Δ_{0} (notice that Δ_{0} is trivially a fibrant simplicial set), so that f is the unique morphism X → Δ_{0}. We show by induction that X[n] contains exactly one element, for every n ∈ ℕ. Say first that n = 0; since π_{0}(Δ_{0}) is non-empty, we may find a vertex ξ ∈ X[0], and we set ξ′ := f[0](ξ). Let a ∈ X[0] be any other vertex; since f[0](a) = ξ′, by assumption we may find a homotopy from a to ξ, and therefore a = ξ, since X is minimal. Next, suppose that n > 0, and the assertion is already known for every integer < n; let b ∈ X[n] be any simplex, and notice that ∂b must factor through ξ, by inductive assumption, so b ∈ X(ξ, n). However, obviously π_{n}(Δ_{0}, ξ′) = 0, hence b ∼_{∂Δ_{n}} ξ, and finally b = ξ, again by the minimality of X.

Next, consider an arbitrary minimal fibration f, fix ξ′ ∈ Y [0] and pick any ξ ∈ f[0]^{−1}(ξ′) (notice that Y [0] and f[0]^{−1}(ξ′) are non-empty, since Y is fibrant and f is a fibration). Set F := f^{−1}(ξ′); theorem 7.5.56(i) implies that π_{n}(F, ξ) has cardinality equal to one, for every n ∈ ℕ. On the other hand, since f is minimal, it is easily seen that the same holds for F (details left to the reader) therefore F ≃ Δ_{0}, by the foregoing case. Since ξ′ is arbitrary, the claim follows. ♢

Now, let f : X → Y be any weak equivalence, with X and Y fibrant simplicial sets. By theorem 7.5.29, there exist an anodyne extension i : X → E and a fibration p : E → Y such that p ∘ i = f. Since Y is fibrant, the same holds for E.

Claim 7.5.82. The morphism i is a strong deformation retract of E.

Proof of the claim. Since i is anodyne and X is fibrant, the identity 1_{X} extends to a morphism r : E → X. Next, consider the unique morphism

t : (X × Δ_{1}) ∪ (E × ∂Δ_{1}) → E

whose restriction to X × Δ_{1} is the composition of i with the projection onto X, and whose restriction with E ×Λ_{1}^{0} (resp. with E ×Λ_{1}^{1}) agrees with 1_{E} (resp. with i∘r). Since E is fibrant, t extends to a morphism h : E × Δ_{1} → E (corollary 7.5.28), and it is easily seen that (i, r, h) is a strong deformation retract for the unique morphism E → Δ_{0}. ♢

By claim 7.5.82 and remark 7.5.65(ii) we see that i is a homotopy equivalence, hence it suffices to check that the same holds for p. Moreover, i is a weak equivalence (proposition 7.5.77(ii)), hence the same holds for p (proposition 7.5.77(iii)). Hence, we may replace X by E and f by p, and assume from start that f is also a fibration. We may then find a strong deformation retract (j : F → X, r, h) of f such that q := f ∘ j is a minimal fibration. Arguing as in the foregoing, we see that j is a weak equivalence, and therefore the same holds for q (again, by proposition 7.5.77(iii)); then claim 7.5.81 says that q is an isomorphism. Lastly, r is a homotopy equivalence (remark 7.5.65(ii)), so the same holds for f = q ∘ r. □

7.6. Graded rings.

Definition 7.6.1. Let (Γ, +, 0) be a commutative monoid, R a ring.

(i) A Γ-graded (associative, unital) R-algebra is a pair \underline{B} := (B, gr_{•}B) consisting of an associative unital R-algebra B and a Γ-grading B = ⨁_{γ∈Γ} gr_{γ}B of the R-module B, such that

1 ∈ gr_{0}B  and gr_{γ}B · gr_{γ′}B ⊂ gr_{γ+γ′}B  for every γ, γ′ ∈ Γ.

A morphism of Γ-graded R-algebras is a map of R-algebras which is compatible with the gradings, in the obvious way.

(ii) Let \underline{B} := (B, gr_{•}B) be a Γ-graded R-algebra. A Γ-graded (left) \underline{B}-module is a datum \underline{M} := (M, gr_{•}M) consisting of a B-module M and a Γ-grading M = ⨁_{γ∈Γ} gr_{γ}M of the R-module underlying M, such that

gr_{γ}B · gr_{γ′}M ⊂ gr_{γ+γ′}M for every γ, γ′ ∈ Γ.

A morphism of Γ-graded \underline{B}-module is a map f : M → N of B-modules, such that f(gr_{γ}M) ⊂ gr_{γ}N, for every γ ∈ Γ. Likewise we define Γ-graded right \underline{B}-modules and Γ-graded \underline{B}- bimodules, and their respective morphisms.

(iii) A graded ideal of \underline{B} is a Γ-graded sub-\underline{B}-bimodule (I, gr_{•}I) of \underline{B}. Obviously, in this case we must have gr_{γ}I = I ∩ gr_{γ}B for every γ ∈ Γ.

(iv) If f : Γ′ → Γ is any morphism of commutative monoids, and \underline{M} is any Γ-graded R-module, we define the Γ′-graded R-module Γ′ ×_{Γ} \underline{M} by setting

gr_{γ}(Γ′ ×_{Γ} M) := gr_{f(γ)}M for every γ ∈ Γ′.

Notice that if \underline{B} := (B, gr_{•}B) is a Γ-graded R-algebra, then Γ′ ×_{Γ} \underline{B} is a Γ′-graded R-algebra, with multiplication and addition laws induced by those of B, in the obvious way. Likewise, if (M, gr_{•}M) is a Γ-graded \underline{B}-module, then Γ′ ×_{Γ} M is naturally a Γ′-graded Γ′ ×_{Γ} \underline{B}-module.

(v) Furthermore, if N is any Γ′-graded R-module, we define the Γ-graded R-module N_{/Γ} whose underlying R-module is the same as N, and whose grading is given by the rule

gr_{γ}(N_{/Γ}) := ⨁_{γ′∈f^{−1}(γ)} gr_{γ}N.

Just as in (iii), if \underline{C} := (C, gr_{•}C) is a Γ′-graded R-algebra, then we get a Γ-graded R-algebra \underline{C}_{/Γ}, whose underlying R-algebra is the same as C. Lastly, if (N, gr_{•}N) is a Γ′-graded \underline{C}- module, then N_{/Γ} is a Γ-graded \underline{C}_{/Γ}-module.

In this section we will consider only commutative graded algebras, but in the next section (7.7) we will encounter certain special classes of non-commutative graded algebras as well.

Remark 7.6.2. (i) For instance, the R-algebra R[Γ] is naturally a Γ-graded R-algebra, when endowed with the Γ-grading such that gr_{γ}R[Γ] := γR for every γ ∈ Γ.

(ii) Let R be a ring; consider a cartesian diagram of monoids

\xymatrix{Γ_{3} \ar[r] \ar[d] & Γ_{1} \ar[d] \\ Γ_{2} \ar[r] & Γ_{0}}

and let B be any Γ_{1}-graded R-algebra. A simple inspection of the definitions yields an identity of Γ_{2}-graded R-algebras :

Γ_{2} ×_{Γ_{0}} B_{/Γ_{0}} = (Γ_{3} ×_{Γ_{1}} B)_{/Γ_{2}}.

Definition 7.6.3. Let A be a ring, S ⊂ A the multiplicative subset of regular elements of A. (i) For every ring homomorphism A → B, we denote the integral closure of A in B by :

i.c.(A, B).

(ii) The total ring of fractions of A is the ring :

Frac(A) := S^{−1}A.

(iii) The normalization A is A^{ν} := i.c.(A, Frac(A)). We say that A is normal if A = A^{ν}. (iv) We say that a A is nice if either A = 0 or Frac(A) has Krull dimension zero.

Remark 7.6.4. Let A be a ring.

(i) Notice that the localization map A → Frac(A) is injective, and every regular element of Frac(A) is invertible : the details are left to the reader.

(ii) If A has Krull dimension zero, then every regular element of A is invertible : indeed, if a ∈ A is regular and 𝔭 ∈ Spec A, the image of a in A_{𝔭} is still regular, hence a ∉ 𝔭A_{𝔭}, because 𝔭A_{𝔭} is the nilradical of A_{𝔭}. Thus, a doesn’t lie in any prime ideal of A, whence the contention.

(iii) If A is reduced and has finitely many minimal prime ideals, then A is nice. Indeed, in this case Frac(A) has also finitely many minimal prime ideals 𝔭_{1}, …, 𝔭_{k}; now, if x ∈ Frac(A)\ (𝔭_{1} ∪ ⋯ 𝔭_{k}), and x is not invertible, then x is not regular, by (i). Say that xy = 0 for some y ∈ Frac(A)\{0}; it follows that y ∈ 𝔭_{1} ∩⋯∩𝔭_{k} = 0, a contradiction. So every non-invertible element of Frac(A) lies in 𝔭_{1} ∪ ⋯ ∪ 𝔭_{k}, whence the contention.

(iv) If A is noetherian and has no embedded prime ideals, then A is nice. Indeed, in this case Frac(A) fulfills the same conditions; then, take again x ∈ Frac(A) in the complement of the finitely many minimal prime ideals. If x is not invertible, it is not regular according to (i), hence it lies in some associated prime ideal of A ([126, Th.6.1(ii)]), a contradiction.

(v) Every flat ring homomorphism f : A → B maps the regular elements of A to the regular elements of B, hence it extends uniquely to a flat ring homomorphism

Frac(f) : Frac(A) → Frac(B).

Moreover, if f is injective, the same holds for Frac(f), since the localization map B → Frac(B) is injective by (i) : details left to the reader.

(vi) Let f : A → B be a flat ring homomorphism, such that A is nice and Spec κ(𝔭) ⊗_{A} B is either empty or has Krull dimension zero for every prime ideal 𝔭 of A (where κ(𝔭) denotes the residue field of 𝔭). Then B is nice and Frac(f) induces an isomorphism of B-algebras :

f′ : Frac(A) ⊗_{A} B →^{∼} Frac(B).

Indeed, let b ∈ B be any regular element; since the induced map g : B → Frac(A) ⊗_{A} B is flat, g(b) is regular in Frac(A) ⊗_{A} B. On the other hand, by assumption Spec Frac(A) and the fibres of Spec(f) are either empty or have Krull dimension zero, hence Frac(A) ⊗_{A} B is either 0 or has Krull dimension zero, so g(b) is invertible, by (ii). This shows that g extends to a ring homomorphism Frac(B) → Frac(A) ⊗_{A} B which is the inverse of f′, whence the contention.

(vii) If A is nice, (A_{𝔭})^{ν} = (A^{ν})_{𝔭} for every 𝔭 ∈ Spec A. Indeed, in light of (vi) we have :

(A_{𝔭})^{ν} = i.c.(A_{𝔭}, Frac(A_{𝔭})) = i.c.(A_{𝔭}, Frac(A)_{𝔭}) = (A^{ν})_{𝔭}.

(viii) Especially, if A is a nice and reduced ring, then A is normal if and only if A is locally normal, i.e. if and only if A_{𝔭} is a reduced local normal domain for every 𝔭 ∈ Spec A. Indeed, the condition is sufficent, by virtue of (vii). Conversely, suppose that A is nice, reduced and normal. In view of (vii), it suffices to check that A_{𝔭} is a domain for every 𝔭 ∈ Spec A, and since A_{𝔭} is reduced, this amounts to showing that A_{𝔭} has a unique minimal prime ideal. However, A_{𝔭} is nice due to (vi), hence X := Spec Frac(A_{𝔭}) has Krull dimension zero, and especially it is a totally disconnected topological space. Moreover, it follows from (i) that the inclusion map X → Spec A_{𝔭} has dense image, so it must contain every minimal prime ideal of Spec A_{𝔭}, and then X is precisely the set of minimal prime ideals of A_{𝔭}. Now, if X contains two distinct points, we may find non-empty open and closed subsets U, U′ ⊂ X with U ∩ U′ = ∅ and U ∪ U′ = X; then there is an idempotent e ∈ Frac(A_{𝔭}) which vanishes exactly on U. Since A_{𝔭} is normal by (vii), it follows that e ∈ A_{𝔭}, but then Spec A_{𝔭} is the disjoint union of two non-empty open and closed subsets, which is absurd, since it has a unique closed point.

Definition 7.6.5. Let (Γ, +, 0) be an integral monoid, \underline{A} := (A, gr_{•}A) a Γ-graded ℤ-algebra. Set gr_{γ}^{∗}A := gr_{γ}A \ {0} for every γ ∈ Γ, and S^{∗} := ⋃_{γ∈Γ} gr_{γ}^{∗}A.

(i) We say that \underline{A} is a ∗-domain if we have gr_{γ}^{∗}A · gr_{γ′}^{∗}A ⊂ gr_{γ+γ′}^{∗}A for every γ, γ′ ∈ Γ.

(ii) We say that \underline{A} is a ∗-field if A ≠ 0 and every element of S^{∗} is invertible in A.

(iii) Notice that if \underline{A} is a ∗-domain, then every element of S^{∗} is regular in A, and obviously S^{∗} is a multiplicative subset of A, so we have a unique Γ^{gp}-grading on

Frac^{∗}(\underline{A}) := S^{∗−1}A

such that the localization map A → Frac^{∗}(\underline{A}) is a morphism of Γ^{gp}-graded ℤ-algebras.

Lemma 7.6.6. Let \underline{A} be a Γ-graded ∗-field, and \underline{M} := (M, gr_{•}M) a Γ-graded \underline{A}-module. Then M is a free A-module, and it admits a basis consisting of homogeneous elements of M.

Proof. Indeed, notice first that gr_{0}A is a field, and Γ_{0} := {γ ∈ Γ | gr^{∗}A ≠ ∅} is an abelian group (with the addition law of Γ). Now, let Λ ⊂ Γ be a system of representatives for Γ/Γ^{0} (say, with 0 ∈ Λ), and for every λ ∈ Λ, pick a basis (m_{λ,i} | i ∈ I_{λ}) for the gr_{0}A-vector space gr_{λ}M. Let us show that Σ := (m_{λ,i} | λ ∈ Λ, i ∈ I_{λ}) is a basis for the A-module M. To see that Σ is a generating family for M, consider any γ ∈ Γ and any m ∈ gr_{γ}M; then there exist λ ∈ Λ and γ′ ∈ Γ_{0} such that γ = λ+γ′, and for any a ∈ gr_{γ′}A we have a gr_{0}A-linear combination a^{−1}m = ∑_{i∈J} a_{i}m_{λ,i} for a finite subset J ⊂ I_{λ}. Thus m = ∑_{i∈J} aa_{i}m_{λ,i}, whence the contention. Next, say that we have an A-linear relation of the form : ∑_{λ∈Λ} ∑_{i∈I_{λ}} a_{λ,i}m_{λ,i} = 0 (where a_{λ,i} = 0 for all but finitely many indices (λ, i)); then it is easily seen that we have already ∑_{i∈I_{λ}} a_{λ,i}m_{λ,i} = 0 for every λ ∈ Λ. Then it also follows straightforwardly that ∑_{i∈I_{λ}} gr_{γ}(a_{λ,i})m_{λ,i} = 0 for every γ ∈ Γ_{0}, where gr_{γ}(a_{λ,i}) denotes the image of a_{λ,i} under the projection A ↦ gr_{γ}A. For every such γ, pick b_{γ} ∈ gr_{γ}^{∗}A; then we get the gr_{0}A-linear relation ∑_{i∈I_{λ}} b_{γ}^{−1}gr_{γ}(a_{λ,i})m_{λ,i} = 0 in gr_{λ}M, and therefore gr_{γ}(a_{λ,i}) = 0 for every λ ∈ Λ, every γ ∈ Γ_{0} and every i ∈ I_{λ}, i.e. a_{λ,i} = 0 for all such λ and i, as required. □

Remark 7.6.7. (i) For every Γ-graded ∗-domain \underline{A}, the Γ^{gp}-graded ℤ-algebra Frac^{∗}(\underline{A}) is a ∗- field, and the natural maps A → Frac^{∗}(\underline{A}) → Frac(A) are injective (details left to the reader).

(ii) Let \underline{A} and \underline{B} be two Γ-graded ∗-domains, and f : \underline{A} → \underline{B} an injective morphism of Γgraded ℤ-algebras. Then f extends to a unique morphism of Γ^{gp}-graded ℤ-algebras Frac(f)^{∗} : Frac^{∗}(\underline{A}) → Frac^{∗}(\underline{B}), and taking into account (i) and lemma 7.6.6, we see that Frac^{∗}(f) is a flat and injective ring homomorphism, hence it extends in turn uniquely to a flat and injective homomorphism of total rings of fractions Frac(f) : Frac(A) → Frac(B), by remark 7.6.4(v).

(iii) Especially, let (\underline{A}_{λ} := (A_{λ}, gr_{•}^{λ}A) | λ ∈ Λ) be a filtered system of Γ-graded ℤ-algebras, with injective transition maps, such that \underline{A}_{λ} is a ∗-domain for every λ ∈ Λ. It follows easily that the colimit A of the filtered system of underlying rings (A_{λ} | λ ∈ Λ) carries a unique Γ-grading gr_{•} such that the natural ring homomorphism j_{λ} : A_{λ} → A is a morphism of Γgraded ℤ-algebras \underline{A}_{λ} → \underline{A} := (A, gr_{•}); moreover, \underline{A} is also a ∗-domain, and the resulting cocone (\underline{A}_{λ} → \underline{A} | λ ∈ Λ) is universal in the category of Γ-graded ℤ-algebras (details left to the reader). Furthermore, according to (ii), for every λ, μ ∈ Λ with μ ≥ λ, the transition map A_{λ} → A_{μ} extends uniquely to a flat and injective ring homomorphism Frac(A_{λ}) → Frac(A_{μ}); similarly we get a unique flat and injective extension Frac(j_{λ}) : Frac(A_{λ}) → Frac(A) of j_{λ}, for every λ ∈ Λ. Then it is easily seen that the resulting cocone (Frac(j_{λ}) | λ ∈ Λ) is universal.

(iv) In the situation of (iii), suppose moreover that A_{λ} is nice for every λ ∈ Λ. Then the same holds for A. Indeed, from (iii) we deduce a natural identification of Spec Frac(A) with the inverse limit of the cofiltered system of topological spaces (Spec Frac(A_{λ}) | λ ∈ Λ). But it is easily seen that the limit of a cofiltered system of spectral spaces of dimension zero has dimension zero (details left to the reader), whence the contention.

Proposition 7.6.8. Let (Γ, +, 0) be an abelian group, \underline{A} := (A, gr_{•}A) a Γ-graded ∗-field, p ≥ 1 the characteristic exponent of the field gr_{0}A, and suppose that Γ\{0} has no p-torsion elements. Then A is a nice reduced normal ring.

Proof. Clearly \underline{A} is the filtered colimit of its Γ-graded ℤ-subalgebras Δ ×_{Γ} \underline{A}, where Δ ranges over the finitely generated subgroups of Γ. Moreover, A^{ν} is the filtered union of its subrings (Δ×_{Γ} \underline{A})^{ν}. Taking into account remark 7.6.7(iv), we are thus easily reduced to the case where Γ is finitely generated, say Γ = Δ_{1}⊕⋯⊕Δ_{k}, for cyclic abelian groups Δ_{1}, …, Δ_{k}. Furthermore, after replacing Γ by a subgroup, we may assume that gr_{γ}^{∗} ≠ ∅ for every γ ∈ Γ, in which case it is easily seen that gr_{γ}A is a one-dimensional gr_{0}A-vector space, for every γ ∈ Γ (details left to the reader), and we deduce an isomorphism of gr_{0}A-algebras :

(Δ_{1} ×_{Γ} \underline{A}) ⊗_{gr_{0}A} ⋯ ⊗_{gr_{0}A} (Δ_{k} ×_{Γ} \underline{A}) →^{∼} A.

In view of remark 7.6.4(iii) we are then reduced to checking :

Claim 7.6.9. If Γ is a cyclic abelian group, A is a smooth gr_{0}A-algebra.

Proof of the claim. Let γ be a generator of Γ; recall that gr_{γ}^{∗}A ≠ ∅, and pick any a ∈ gr_{γ}^{∗}A. If Γ →^{∼} ℤ, then it is easily seen that the map : P (T) ↦ P (a) yields an isomorphism of gr_{0}Aalgebras gr_{0}A[T, T^{−1}] →^{∼} A. Lastly, say that Γ = ℤ/nℤ for some n ∈ ℕ \ {0}; by assumption n is invertible in gr_{0}A, and b := a^{n} ∈ gr_{0}^{∗}A. Then the same rule yields an isomorphism of gr_{0}A-algebras gr_{0}A[T]/(T^{n} − b) →^{∼} A. □

Corollary 7.6.10. Let Γ be a monoid, \underline{A} := (A, gr_{•}A) a Γ-graded ∗-domain and suppose that Γ^{gp} is a torsion-free abelian group. Then A is a domain.

Proof. In view of remark 7.6.7(i) we may replace \underline{A} by the Γ^{gp}-graded ℤ-algebra Frac^{∗}(A), and assume from start that Γ is a torsion-free abelian group, and \underline{A} is a ∗-field. Then \underline{A} is the filtered colimit of the system of Γ-graded ℤ-subalgebras Δ ×_{Γ} \underline{A}, where Δ ranges over the finitely generated subgroups of Γ, and it suffices to prove the assertion for such subalgebras. Thus, we may assume Γ = ℤ^{⊕r} for some r ∈ ℕ, and after replacing Γ by a subgroup, we may even assume that gr_{γ}^{∗}A ≠ ∅ for every γ ∈ Γ. In this case, the proof of proposition 7.6.8 shows that A is isomorphic to gr_{0}A[T_{1}^{±1}, …, T_{r}^{±1}], whence the contention. □

Proposition 7.6.11. Suppose that (Γ′, +) → (Γ, +) is a morphism of fine monoids, B a finitely generated (resp. finitely presented) Γ-graded R-algebra, and M a finitely generated (resp. finitely presented) Γ-graded B-module. We have :

(i) Γ′ ×_{Γ} B is a finitely generated (resp. finitely presented) R-algebra.

(ii) Γ′ ×_{Γ} M is a finitely generated (resp. finitely presented) Γ′ ×_{Γ} B-module.

(iii) gr_{γ}M is a finitely generated (resp. finitely presented) gr_{0}B-module, for every γ ∈ Γ.

Proof. Let B_{M} denote the direct sum B ⊕ M, endowed with the R-algebra structure given by the rule :

(b_{1}, m_{1}) · (b_{2}, m_{2}) = (b_{1}b_{2}, b_{1}m_{2} + b_{2}m_{1}) for every b_{1}, b_{2} ∈ B and m_{1}, m_{2} ∈ M.

Notice that B_{M} is characterized as the unique R-algebra structure for which M is an ideal with M^{2} = 0, the natural projection B_{M} → B is a map of R-algebras, and the B-module structure on M induced via π agrees with the given B-module structure on M.

Claim 7.6.12. The following conditions are equivalent :

(a) The R-algebra B_{M} is finitely generated (resp. finitely presented).

(b) B is a finitely generated (resp. finitely presented) R-algebra and M is a finitely generated (resp. finitely presented) B-module.

Proof of the claim. (b)⇒(a): Suppose first that B is a finitely generated R-algebra, and M is a finitely generated B-module. Pick a system of generators Σ_{B} := {b_{1}, …, b_{s}} for B and Σ_{M} := {m_{1}, …, m_{k}} for M. Then it is easily seen that Σ_{B} ∪ Σ_{M} generates the R-algebra B_{M}.

For the finitely presented case, pick a surjection of R-algebras φ : R[T_{1}, …, T_{s}] → B and of B-module ψ : B^{⊕k} → M. Let Σ′_{B} be a finite system of generators of the ideal Ker φ. Pick also a finite system b_{1}, …, b_{r} of generators of the B-module Ker ψ; we may write b_{i} = ∑_{j=1}^{k} b_{ij}e_{j} for certain b_{ij} ∈ B (where e_{1}, …, e_{k} is the standard basis of B^{⊕k}). For every i ≤ r and j ≤ k, pick P_{ij} ∈ φ^{−1}(b_{ij}). It is easily seen that B_{M} is isomorphic to the R-algebra R[T_{1}, …, T_{s+k}]/I, where I is generated by Σ′_{B} ∪ {∑_{j=1}^{k} P_{ij}T_{j+s} | i = 1, …, r} ∪ {T_{i+s}T_{j+s} | 0 ≤ i, j ≤ k}.

(a)⇒(b): Suppose that B_{M} is a finitely generated R-algebra, and let c_{1}, …, c_{n} be a system of generators. For every i = 1, …, n, we may write c_{i} = b_{i} + m_{i} for unique b_{i} ∈ B and m_{i} ∈ M. Since M^{2} = 0, it is easily seen that m_{1}, …, m_{n} is a system of generators for the B-module M, and clearly b_{1}, …, b_{n} is a system of generators for the R-algebra B.

Next, suppose that B_{M} is finitely presented over R. We may find a system of generators b_{1}, …, b_{s}, m_{1}, …, m_{k} of B_{M} with b_{i} ∈ B and m_{j} ∈ M for every i ≤ s and j ≤ k. We deduce a surjection of R-algebras

φ : R[T_{1}, …, T_{s+k}]/(T_{s+i}T_{s+j} | 0 ≤ i, j ≤ k) → B_{M}

such that T_{i} ↦ b_{i} for every i ≤ s and T_{j+s} ↦ m_{j} for every j ≤ k. It is easily seen that Ker φ is generated by the classes of finitely many polynomials P_{1}, …, P_{r}, where

P_{i} = Q_{i}(T_{1}, …, T_{s}) + ∑_{j=1}^{k} T_{s+j}Q_{ij}(T_{1}, …, T_{s}) i = 1, …, r

for certain polynomials Q_{i}, Q_{ij} ∈ R[T_{1}, …, T_{s}]. It follows easily that B = R[T_{1}, …, T_{s}]/I, where I is the ideal generated by Q_{1}, …, Q_{r}, and M is isomorphic to the B-module B^{⊕k}/N, where N is the submodule generated by the system {∑_{j=1}^{k} Q_{ij}(b_{1}, …, b_{s})e_{j} | i = 1, …, r} ♢

Suppose now that B is a finitely generated R-algebra; then B is generated by finitely many homogeneous elements, say b_{1}, …, b_{s} of degrees respectively γ_{1}, …, γ_{s}. Thus, we may define surjections of monoids

(7.6.13)  ℕ^{⊕s} → Γ : e_{i} ↦ γ_{i}  for i = 1, …, s

(where e_{1}, …, e_{s} is the standard basis of ℕ^{⊕s}) and of R-algebras φ : C → B, where C := R[ℕ^{⊕s}] is a free polynomial R-algebra. Notice that C is a ℕ^{⊕s}-graded R-algebra, and via (7.6.13) we may regard φ as a morphism of Γ-graded R-algebras C_{/Γ} → B. Then I := ker φ is a Γ-graded ideal of C, and if we set P := ℕ^{⊕s} ×_{Γ} Γ′ we deduce an isomorphism of Γ′-graded R-algebras

B′ := Γ′ ×_{Γ} B →^{∼} (P ×_{ℕ^{⊕s}} C)_{/Γ′}/(Γ′ ×_{Γ} I)

(see remark 7.6.2(ii)).

Claim 7.6.14. P ×_{ℕ^{⊕s}} C is a finitely presented R-algebra.

Proof of the claim. Indeed, this R-algebra is none else than R[P], hence the assertion follows from corollary 6.4.2and lemma 6.1.7(i). ♢

From claim 7.6.14it follows already that B′ is a finitely generated R-algebra. Now, suppose that M is a finitely generated B-module, and set M′ := Γ′ ×_{Γ} M; notice that

(7.6.15)  Γ′ ×_{Γ} (B_{M}) = B′_{M′}.

In view of claim 7.6.12, we deduce that M′ is a finitely generated B′-module. Next, in case B is a finitely presented R-algebra, I is a finitely generated ideal of C_{/Γ}; as we have just seen, this implies that Γ′ ×_{Γ} I is a finitely generated (Γ′ ×_{Γ} C_{/Γ})-module, and then claim 7.6.14 shows that B′ is a finitely presented R-algebra. This concludes the proof of (i).

Lastly, if moreover M is a finitely presented B-module, assertion (i), together with (7.6.15) and claim 7.6.12say that M′ is a finitely presented B′-module; thus, also assertion (ii) is proven.

(iii): For any given γ ∈ Γ, let us consider the morphism f : ℕ → Γ such that 1 ↦ γ, and set B′ := ℕ ×_{Γ} B. By (i), the R-algebra B′ is finitely generated (resp. finitely presented), and the B′-module M′ := ℕ ×_{Γ} M is finitely generated (resp. finitely presented). After replacing B by B′ and M by M′, we may then assume from start that Γ = ℕ, and we are reduced to showing that gr_{1}M is a finitely generated (resp. finitely presented) gr_{0}B-module.

Let m_{1}, …, m_{t} be a system of generators of M consisting of homogeneous elements of degrees respectively j_{1}, …, j_{t}. We endow B^{⊕t} with the ℕ-grading such that

gr_{k}B^{⊕t} := ⨁_{i=1}^{t} gr_{k−j_{i}}Be_{i}

(where e_{1}, …, e_{t} is the standard basis of B^{⊕t}); then the B-linear map B^{⊕t} → M given by the rule e_{i} ↦ m_{i} for every i = 1, …, t is a morphism of ℕ-graded B-modules, and if M is finitely presented, its kernel is generated by finitely many homogeneous elements b_{1}, …, b_{s}. In the latter case, endow again B^{⊕s} with the unique ℕ-grading such that the B-linear map φ : B^{⊕s} → B^{⊕t} given by the rule e_{i} ↦ b_{i} for every i = 1, …, s is a morphism of ℕ-graded B-modules. Now, in order to check that gr_{1}M is a finitely generated gr_{0}B-module, it suffices to show that the same holds for gr_{1}B^{⊕t}. The latter is a direct sum of gr_{0}B-modules isomorphic to either gr_{0}B or gr_{1}B. Likewise, if M is finitely presented, gr_{1}M = Coker gr_{1}φ, and again, gr_{1}B^{⊕s} is a direct sum of gr_{0}B-modules isomorphic to either gr_{0}B or gr_{1}B; hence in order to check that gr_{1}M is a finitely presented gr_{0}B-module, it suffices to show that gr_{1}B is a finitely presented gr_{0}B-module. In either event, we are reduced to the case where Γ = ℕ and M = B.

However, from (ii) we deduce especially that gr_{0}B = {0} ×_{ℕ} B is a finitely generated (resp. finitely presented) R-algebra, hence B is a finitely generated (resp. finitely presented) B_{0}- algebra as well; we may then assume that R = gr_{0}B. Let Σ be a system of homogeneous generators for the R-algebra B; we may then also assume that

(7.6.16)  Σ ∩ gr_{0}B = ∅.

Then it is easily seen that the R-module gr_{1}B is generated by Σ ∩ gr_{1}B. Lastly, if B is a finitely presented R-algebra, we consider the natural surjection ψ : R[Σ] → B from the free polynomial R-algebra generated by the set Σ, and endow R[Σ] with the unique grading for which ψ is a map of ℕ-graded R-algebras; then I := Ker ψ is a finitely generated ℕ-graded ideal with gr_{0}I = 0. As usual, we pick a finite system Σ′ of homogeneous generators for I; clearly B_{1} is isomorphic to gr_{1}B_{0}[Σ]/gr_{1}I. On the other hand, (7.6.16) easily implies that gr_{1}R[Σ] is a free R-module of finite rank, and moreover gr_{1}I is generated by Σ′ ∩ gr_{1}I; especially, gr_{1}B is a finitely presented R-module in this case, and the proof is complete. □

7.6.17. Let (Γ, +) be a monoid, R a ring, \underline{B} := (B, gr_{•}B) a Γ-graded R-algebra, and M a Γ-graded \underline{B}-module. We denote by M[γ] the Γ-graded \underline{B}-module whose underlying B-module is M, and whose grading is given by the rule :

gr_{β}M[−γ] := ⨁_{δ+γ=β} gr_{δ}M for every γ ∈ Γ.

Remark 7.6.18. (i) In the situation of (7.6.17), pick any system x := (x_{i} | i ∈ I) of homogeneous generators of M, and say that x_{i} ∈ gr_{γ_{i}}M for every i ∈ I. Then we may define a surjective map of Γ-graded \underline{B}-modules

L := ⨁_{i∈I} B[−γ_{i}] → M : e_{i} ↦ x_{i} for every i ∈ I

where (e_{i} | i ∈ I) denotes the canonical basis of the free B-module L (notice that e_{i} ∈ gr_{γ_{i}}L for every i ∈ I).

(ii) In case M is a finitely generated B-module, we may pick a finite system x as above, and then L shall be a free B-module of finite rank.

(iii) Especially, suppose that B is a coherent ring and M is finitely presented as a B-module; then, in the situation of (ii), the kernel of the surjection L → M shall be again a finitely presented Γ-graded \underline{B}-module, so we can repeat the above construction, and find inductively a resolution

Σ : ⋯ → L_{n} ⟶^{d_{n}} L_{n−1} → ⋯ → L_{0} ⟶^{d_{0}} M

such that L_{n} is a free B-module of finite rank, and the map d_{n} is a morphism of Γ-graded \underline{B}-modules, for every n ∈ ℕ.

(iv) In the situation of (iii), suppose furthermore that B is a flat R-algebra, in which case gr_{γ}B is a flat R-module, for every γ ∈ Γ. Then it is clear that the resolution Σ yields, in each degree γ ∈ Γ a flat resolution Σ_{γ} of the R-module gr_{γ}M.

7.6.19. Graded rings and diagonalizable groups. We conclude this section we a brief review of the well known correspondance between Γ-gradings on a module M (for a given commutative group Γ), and actions of the diagonalizable group D(Γ) on M. Let S be a scheme; on the category Sch/S of S-schemes we have the presheaf of rings :

𝒪_{Sch/S} : Sch/S → ℤ-Alg (X → S) ↦ Γ(X, 𝒪_{X}).

Also, let M be an 𝒪_{S}-module; following [55, Exp.I, Déf.4.6.1], we attach to M the 𝒪_{Sch/S}- module 𝒲_{M} given by the rule : (f : X → S) ↦ Γ(X, f^{∗}M). If M and N are two quasicoherent 𝒪_{S}-modules, and f : S′ → S an affine morphism, it is easily seen that

(7.6.20)  Γ(S′, ℋ om_{𝒪_{Sch/S}} (𝒲_{M}, 𝒲_{N})) = Hom_{𝒪_{S}} (M, f_{∗}𝒪_{S′} ⊗_{𝒪_{S}} N)

(see [55, Exp.I, Prop.4.6.4] for the details).

Let f : G → S be a group S-scheme, i.e. a group object in the category Sch/S. If f is affine, we say that G is an affine group S-scheme; in that case, the mutiplication law G ×_{S} G → G and the unit section S → G correspond respectively to morphisms of 𝒪_{S}-algebras

Δ_{G} : f_{∗}𝒪_{G} → f_{∗}𝒪_{G} ⊗_{𝒪_{S}} f_{∗}𝒪_{G} ε_{G} : 𝒪_{S} → f_{∗}𝒪_{G}

which make commute the diagram :

\xymatrix{f_{∗}𝒪_{G} \ar[r]^{Δ_{G}} \ar[d]_{Δ_{G}} & f_{∗}𝒪_{G} ⊗_{𝒪_{S}} f_{∗}𝒪_{G} \ar[d]^{1_{f∗𝒪_{S}} ⊗Δ_{G}} \\ f_{∗}𝒪_{G} ⊗_{𝒪_{S}} f_{∗}𝒪_{G} \ar[r]^{Δ_{G}⊗1_{f∗𝒪_{S}}} & f_{∗}𝒪_{G} ⊗_{𝒪_{S}} f_{∗}𝒪_{G} ⊗_{𝒪_{S}} f_{∗}𝒪_{G}}

as well as a similar diagram, which expresses the unit property of ε_{G} : see [55, Exp.I, §4.2].

Example 7.6.21. Let G be any commutative group. The presheaf of groups

D_{S}(G) : Sch/S → ℤ-Mod (X → S) ↦ Hom_{ℤ-Mod}(G, 𝒪_{X}^{×}(X))

is representable by an affine group S-scheme D_{S}(G), called the diagonalizable group scheme attached to G. Explicitly, if S = Spec R is an affine scheme, the underlying S-scheme of D_{S}(G) is Spec R[G], and the group law is given by the map of R-algebras

Δ_{G} : R[G] → R[G] ⊗_{R} R[G] →^{∼} R[G × G] g ↦ (g, g) for every g ∈ G

with unit ε_{G} : R[G] → R given by the standard augmentation (see [55, Exp.I, §4.4]). For a general scheme S, we have D_{S}(G) = D_{Spec ℤ}(G) ×_{Spec ℤ} S (with the induced group law and unit section).

Definition 7.6.22. Let M be an 𝒪_{S}-module, G a group S-scheme. A G-module structure on M is the datum of a morphism of presheaves of groups on Sch/S :

h_{G} → 𝒜ut _{𝒪_{Sch/S}} (𝒲_{M})

(where h_{G} denotes the Yoneda embedding : see (1.2.4)).

7.6.23. Suppose now that f : G → S is an affine group S-scheme, and M a quasi-coherent 𝒪_{S}-module; in view of (7.6.20), a G-module structure on M is then the same as a map of 𝒪_{S}-modules

μ_{M} : M → f_{∗}𝒪_{G} ⊗_{𝒪_{S}} M

which makes commute the diagrams :

\xymatrix{M \ar[r]^{μ_{M}} \ar[d]_{μ_{M}} & f_{∗}𝒪_{G} ⊗_{𝒪_{S}} M \ar[d]^{Δ_{G}⊗1_{M}} & M \ar[r]^{μ_{M}} \ar[dr]_{1_{M}} & f_{∗}𝒪_{G} ⊗_{𝒪_{S}} M \ar[d]^{ε_{G}⊗1_{M}} \\ f_{∗}𝒪_{G} ⊗_{𝒪_{S}} M \ar[r]^{1_{f∗𝒪_{G}} ⊗μ_{M}} & f_{∗}𝒪_{G} ⊗_{𝒪_{S}} f_{∗}𝒪_{G} ⊗_{𝒪_{S}} M &  & M.}

Example 7.6.24. Let Γ be a commutative group; a D_{S}(Γ)-module structure on a quasi-coherent 𝒪_{S}-module M is the datum of a morphism of 𝒪_{S}-modules

μ_{M} : M → 𝒪_{S}[Γ] ⊗_{𝒪_{S}} M = ℤ[Γ] ⊗_{ℤ} M

which makes commute the diagrams of (7.6.23). If S = Spec R is affine, M is associated with an R-module which we denote also by M; in this case, μ_{M} is the same as a system (μ_{M}^{(γ)} | γ ∈ Γ) of 𝒪_{S}-linear endomorphisms of M, such that :

• for every x ∈ M, the subset {γ ∈ Γ | μ_{M}^{(γ)}(x) ≠ 0} is finite.

• μ_{M}^{(γ)} ∘ μ_{M}^{(τ)} = δ_{γ,τ} · 1_{M} and ∑_{γ∈Γ} μ_{M} = 1_{M}.

In other words, the μ_{M}^{(γ)} form an orthogonal system of projectors of M, summing up to the identity 1_{M}. This is the same as the datum of a Γ-grading on M : namely, for a given D_{S}(Γ)- module structure μ_{M}, one lets

gr_{γ}M := μ_{M}^{(γ)}(M) for every γ ∈ Γ

and conversely, given a Γ-grading gr_{•}M on M, one defines μ_{M} as the R-linear map given by the rule : x ↦ γ ⊗ x for every γ ∈ Γ and every x ∈ gr_{γ}M.

7.6.25. Suppose now that g : X → S is an affine S-scheme, and f : G → S an affine group S-scheme. A G-action on X is a morphism of presheaves of groups :

h_{G} → 𝒜ut_{Sch/S^{∧}}(h_{X}).

(notation of (1.2.4)); the latter is the same as a morphism of S-schemes

(7.6.26)  G ×_{S} X → X

inducing a G-module structure on g_{∗}𝒪_{X} :

g_{∗}𝒪_{X} → f_{∗}𝒪_{G} ⊗_{𝒪_{S}} g_{∗}𝒪_{X}

which is also a morphism of 𝒪_{S}-algebras. For instance, if S = Spec R is affine, and G = D_{S}(Γ) for an abelian group Γ, we may write X = Spec A for some R-algebra B, and in view of example 7.6.24, the G-action on X is the same as the datum of a Γ-graded R-algebra structure on B, in the sense of definition 7.6.1(i).

Remark 7.6.27. (i) Suppose that Γ is an integral monoid. Then, to a Γ-graded R-algebra B, the correspondance described in (7.6.25) attaches a D_{S}(Γ^{gp})-action on Spec B (where S := Spec R), given by the map of R-algebras

ϑ_{B} : B → B[Γ] ⊂ B[Γ^{gp}] : b ↦ b · γ for every γ ∈ Γ, and every b ∈ gr_{γ}B.

(ii) If Γ is a finite abelian group whose order is invertible in 𝒪_{S}, then D_{S}(Γ) is an étale S-scheme. Indeed, in light of (4.8.52), the assertion is reduced to the case where Γ = ℤ/nℤ for some integer n > 0 which is invertible in 𝒪_{S}. However, R[ℤ/nℤ] ≃ R[T]/(T^{n} − 1), which is an étale R-algebra, if n ∈ R^{×}.

(iii) More generally, suppose that Γ is a finitely generated abelian group, such that the order of its torsion subgroup is invertible in 𝒪_{S}. Then we may write Γ = L ⊕ Γ_{tor}, where L is a free abelian group of finite rank, and Γ_{tor} is a finite abelian group as in (ii). In view of (4.8.52) and (ii), we conclude that D_{S}(Γ) is a smooth S-scheme in this case.

(iv) Let Γ be as in (iii), and suppose that X is an S-scheme with an action of G := D_{S}(Γ). Then the corresponding morphism (7.6.26)and the projection p_{G} : G×_{S}X → G induce an automorphism of the G-scheme G×_{S} X, whose composition with the projection p_{X} : G×_{S} X → X equals (7.6.26). We then deduce that both (7.6.26) and p_{X} are smooth morphisms. This observation, together with the above correspondance between Γ-graded algebras and D_{S}(Γ)-actions, is the basis for a general method that allows to prove properties of graded rings, provided they can be translated as properties of the corresponding schemes which are well behaved under smooth base changes. We shall present hereafter a few applications of this method.

Proposition 7.6.28. Let Γ be an integral monoid, B a Γ-graded (commutative, unital) ring, and suppose that the order of any torsion element of Γ^{gp} is invertible in B. Then :

(i) nil(B[Γ]) = nil(B) · B[Γ]. (ii) nil(B) is a Γ-graded ideal of B.

Proof. (i): Clearly nil(B) · B[Γ] ⊂ nil(B[Γ]). To show the converse inclusion, it suffices to prove that nil(B[Γ]) ⊂ 𝔭B[Γ] for every prime ideal 𝔭 ⊂ B, or equivalently that B/𝔭[Γ] is a reduced ring for every such 𝔭. Since the natural map Γ → Γ^{gp} is injective, we may further replace Γ by Γ^{gp}, and assume that Γ is an abelian group. In this case, B/𝔭[Γ] is the filtered union of the rings B/𝔭[H], where H runs over the finitely generated subgroups of Γ; it suffices therefore to prove that each B/𝔭[H] is reduced, so we may assume that Γ is finitely generated, and the order of its torsion subgroup is invertible in B. In this case, B/𝔭[Γ] is a smooth B/𝔭algebra (remark 7.6.27(iii)), and the assertion follows from [66, Ch.IV, Prop.17.5.7].

(ii): The assertion to prove is that nil(B) = ⨁_{γ∈Γ} nil(B) ∩ gr_{γ}B. However, let ϑ_{B} : B → B[Γ] be the map defined as in remark 7.6.27(i); now, (i) implies that ϑ_{B} restricts to a map nil(B) → nil(B) · B[Γ], whence the contention. □

Proposition 7.6.29. Let (Γ, +, 0) be an integral monoid, f : A → B a morphism of Γ-graded rings, and suppose that the order of any torsion element of Γ^{gp} is invertible in A. We have :

(i) (i.c.(A, B))[Γ^{gp}] = i.c.(A[Γ^{gp}], B[Γ^{gp}]).

(ii) The grading of B restricts to a Γ-grading on the subring i.c.(A, B).

(iii) Suppose moreover that B is reduced. Then for every γ ∈ Γ such that gr_{γ}i.c.(A, B) ≠ 0 there exists an integer m > 0 with gr_{mγ}A ≠ 0.

Proof. (i): We easily reduce to the case where Γ is finitely generated, in which case A[Γ^{gp}] is a smooth A-algebra (remark 7.6.27(iii)), and the assertion follows from [64, Ch.IV, Prop.6.14.4].

(ii): Define ϑ_{A} and ϑ_{B} as in remark 7.6.27(i), and consider the commutative diagram :

\xymatrix{A \ar[r]^{ϑ_{A}} \ar[d]_{f} & A[Γ] \ar[d]^{f[Γ]} \\ B \ar[r]^{ϑ_{B}} & B[Γ].}

Say that b ∈ i.c.(A, B); then ϑ_{B}(b) ∈ i.c.(A[Γ], B[Γ]). In light of (i), we deduce that ϑ_{B}(b) ∈ (i.c.(A, B))[Γ^{gp}] ∩ B[Γ] = (i.c.(A, B))[Γ]. The claim follows easily.

(iii): Let x be any non-zero element of gr_{γ}i.c.(A, B). By definition there exist an integer n > 0 and a_{1}, …, a_{n} ∈ A such that x^{n} + a_{1}x^{n−1} + ⋯ + a_{n} = 0 in B. For i = 1, …, n let a′_{i} be the image of a_{i} under the projection A → gr_{iγ}A; it is easily seen that x^{n}+a′_{1}x^{n−1}+⋯+a′_{n} = 0. If a′_{i} = 0 for every i = 1, …, n, we deduce that x^{n} = 0, which is absurd, since B is reduced. Thus, we have a′_{m} ≠ 0 for at least one index m ≤ n, so gr_{mγ}A ≠ 0. □

Corollary 7.6.30. Let Γ be an integral monoid, \underline{A} := (A, gr_{•}A) a Γ-graded ∗-domain, Γ^{ν} ⊂ Γ^{gp} the saturation of Γ, and suppose that the order of any torsion element of Γ^{gp} is invertible in A. Then there exists a unique Γ^{ν}-grading gr_{•}A^{ν} on A^{ν} (notation of definition 7.6.3(iii)) such that (A^{ν}, gr_{•}A^{ν}) is a Γ^{ν}-graded nice reduced ∗-domain, and the inclusion map A → A^{ν} is a morphism of Γ^{ν}-graded rings.

Proof. From proposition 7.6.8 and remark 7.6.7(i) we see that A^{ν} = i.c.(A, Frac^{∗}(A)). The corollary then follows straightforwardly from proposition 7.6.29. □

7.7. Differential graded algebras. The material of this paragraph shall be applied in section 7.10, in order to study certain strictly anti-commutative graded algebras constructed via homotopical algebra. Especially, the graded algebras appearing in this paragraph are usually not commutative, unless it is explicitly said otherwise.

Definition 7.7.1. (i) Let A be any ring, and \underline{B} := (B, gr_{•}B) a ℤ-graded (associative, unital) A-algebra. We say that \underline{B} is strictly anti-commutative (or alternating), if we have

(a) a · b = (−1)^{pq} · b · a for every p, q ∈ ℤ, every a ∈ gr_{p}B, and every b ∈ gr_{q}B

(b) a · a = 0 for every p ∈ ℤ and every a ∈ gr_{2p+1}B.

If only condition (a) holds, we say that \underline{B} is anti-commutative.

(ii) Let \underline{M} := (M, gr_{•}M) be any ℤ-graded \underline{B}-bimodule. An A-linear graded derivation from \underline{B} to \underline{M} is a morphism of graded A-modules ∂ : \underline{B} → \underline{M} such that

∂(xy) = ∂(x) · y + x · ∂(y) for every x, y ∈ B.

Remark 7.7.2. (i) Let A-Alt be the full subcategory of the category of ℤ-graded A-algebras, whose objects are the strictly anti-commutative graded A-algebras. Define also the category A-Alg.Mod as in [75, Def.2.5.22]. We have an obvious functor

gr_{0,1} : A-Alt → A-Alg.Mod (B, gr_{•}B) ↦ (gr_{0}B, gr_{1}B)

and it is easily seen that the functor gr_{0,1} is right adjoint to the functor

Λ_{•}^{•} : A-Alg.Mod → A-Alt (B, M) ↦ Λ_{B}^{•}M

that assigns to every object (B, M) the graded A-algebra of the B-linear exterior powers of M; the details shall be left to the reader.

(ii) For every pair of graded A-algebras (B, gr_{•}B), (B′, gr_{•}B′) we define a tensor product

(B′′, gr_{•}B′′) := (B, gr_{•}B) ⊗_{A} (B′, gr_{•}B′)

which is the graded A-algebra with B′′ := B ⊗_{A} B′, and gr_{n}B′′ := ⨁_{i+j=n} gr_{i}B ⊗_{A} gr_{j}B′ for every n ∈ ℤ. The multiplication law of B′′ is the direct sum of the maps

(gr_{i}B ⊗_{A}gr_{j}B′)×(gr_{k}B ⊗_{A}gr_{l}B′) → gr_{i+k}B ⊗_{A}gr_{j+l}B′ (b⊗b′, c⊗c′) ↦ (−1)^{jk}bc⊗b′c′.

It is easily seen that if (B, gr_{•}B) and (B′, gr_{•}B′) are anti-commutative (resp. alternating), the same holds for their tensor product. Moreover, if (B, gr_{•}B) and (B′, gr_{•}B′) are alternating, then (B′′, gr_{•}B′′) represents the coproduct of (B, gr_{•}B) and (B′, gr_{•}B′) in the category A-Alt. Again, we leave the details to the reader.

(iii) Likewise, all finite coproducts are representable in A-Alg.Mod : namely, the coproduct of any two objects (B, M), (B′, M′) is represented by :

(B′′, M′′) := (B ⊗_{A} B′, (M ⊗_{A} B′) ⊕ (B ⊗_{A} M′))

with the obvious universal cocone (B, M) → (B′′, M′′) ← (B′, M′). By (i), (ii), and proposition 1.3.25(iv), we deduce a natural isomorphism of alternating graded A-algebras :

Λ_{B′′}^{•}M′′ →^{∼} Λ_{B}^{•}M ⊗_{A} Λ_{B′}^{•}M′.

Definition 7.7.3. Let A be any ring.

(i) A differential graded A-algebra is the datum of • a complex (B^{•}, d_{B}^{•}) of A-modules

• an A-linear map μ^{pq} : B^{p} ⊗_{A} B^{q} → B^{p+q} for every p, q ∈ ℤ such that the following holds :

(a) Set B := ⨁_{p∈ℤ} B^{p}; then the system of maps μ^{••} adds up to a map μ : B ⊗_{A} B → B, and one requires that the resulting pair (B, μ) is an associative unital ℤ-graded Aalgebra. Then, one sets a · b := μ(a ⊗ b), for every a, b ∈ B.

(b) We have the identities

d_{B}^{p+q}(a · b) = d_{B}^{p}(a) · b + (−1)^{p} · a · d_{B}^{q}(b) for every p, q ∈ ℤ and every a ∈ B^{p}, b ∈ B^{q}.

We call B the graded A-algebra associated to B^{•}. Then we also say that B^{•} is anti-commutative (resp. strictly anti-commutative) if the same holds for B (definition 7.7.1).

(ii) A morphism B^{•} → C^{•} of differential graded A-algebras is a map of complexes of Amodules such that the induced map of associated graded A-modules is a map of A-algebras. We denote the resulting category of differential graded A-algebras by :

A-dgAlg.

We shall also consider the full subcategory of A-dgAlg denoted

A-dgAlt

whose objects are the strictly anti-commutative differential graded A-algebras.

(iii) Let (B^{•}, d_{B}^{•}) be a differential graded A-algebra, B its associated graded A-algebra, and (M^{•}, d_{M}^{•}) a complex of A-modules.

(a) We say that M^{•} is a left B^{•}-module if the A-module M := ⨁_{p∈ℤ} M^{p} is a graded left B-module (for the natural ℤ-grading on M), and we have

d_{M}^{p+q}(b · m) = (d_{B}^{p}b) · m + (−1)^{p} · b · d_{M}^{q}(m)

for every p, q ∈ ℤ, every b ∈ B^{p}, and every m ∈ M^{q}.

(b) We say that M^{•} is a right B^{•}-module if M is a graded right B-module, and we have

d_{M}^{p+q}(m · a) = d_{M}^{q}(m) · a + (−1)^{q} · m · d_{M}^{p}(a)

for every p, q ∈ ℤ, every b ∈ B^{p}, and every m ∈ M^{q}.

(c) We say that M^{•} is a B-bimodule if it is both a left and right B^{•}-module, and with these B^{•}-modules structures, the A-module M becomes a B-bimodule (i.e. the left multiplication commutes with the right multiplication).

We call M the graded B-module associated to M^{•}. A morphism M^{•} → N^{•} of left (resp. right, resp. bi-) B^{•}-modules is a map of complexes of A-modules, such that the induced map of associated graded A-modules is a map of left (resp. right, resp. bi-) B-modules.

Remark 7.7.4. Let B^{•} be any differential graded A-algebra.

(i) Notice that condition (b) of definition 7.7.3(i) is the same as saying that μ^{••} induces a map of complexes

μ^{•} : B^{•} ⊗_{A} B^{•} → B^{•}.

Moreover, in light of example 7.1.16(i), we see that B^{•} is anti-commutative if and only if the diagram

\xymatrix{B^{•} ⊗_{A} B^{•} \ar[dr]_{μ^{•}} \ar[rr]^{∼} &  & B^{•} ⊗_{A} B^{•} \ar[dl]^{μ^{•}} \\  & B^{•}}

commutes, where the horizontal arrow is the isomorphism (7.1.17) that swaps the two tensor factors. Hence, in some sense this is actually a commutativity condition.

(ii) Likewise, if M^{•} is a complex of A-modules with a graded left (right) B-module structure on the associated graded A-module M, then M^{•} is a left (resp. right) B^{•}-module if and only if the scalar multiplication of the B-module M induces a map of complexes

B^{•} ⊗_{A} M^{•} → M^{•} ( resp. M^{•} ⊗_{A} B^{•} → M^{•} ).

(iii) It is easily seen that the multiplication maps μ^{pq} induce A-linear maps

(H^{p}B^{•}) ⊗_{A} (H^{q}B^{•}) → H^{p+q}B^{•} for every p, q ∈ ℤ

anf if we let H^{•}B^{•} := ⨁_{p∈ℤ} H^{p}B^{•}, then the resulting map

(H^{•}B^{•}) ⊗_{A} (H^{•}B^{•}) → H^{•}B^{•}

endows H^{•}B^{•} with a structure of ℤ-graded associative unital A-algebra, which shall be strictly anti-commutative whenever the same holds for B^{•}. Likewise, if M^{•} is any left (resp. right, resp. bi) B^{•}-module, then H^{•}M^{•} is naturally a ℤ-graded left (resp. right, resp. bi) H^{•}B^{•}-module.

(iv) Let M^{•} be a left B^{•}-module, and denote by μ_{M}^{pq} : B^{p} ⊗_{A} M^{q} → M^{p+q} the (p, q)-graded component of the scalar multiplication of M^{•}, for every p, q ∈ ℤ. Then M^{•}[1] is also naturally a left B^{•}-module, with scalar multiplication μ_{M[1]}^{••} given by the rule :

μ_{M[1]}^{pq} := (−1)^{p} · μ_{M}^{p,q+1} for every p, q ∈ ℤ.

Likewise, if N^{•} is a right B^{•}-module, with scalar multiplication μ_{N}^{••}, then N^{•}[1] is naturally a right B^{•}-module, with multiplication μ_{N[1]}^{••} given by the rule :

μ_{N[1]}^{pq} := μ_{N}^{p,q+1} for every p, q ∈ ℤ.

Lastly, if P^{•} is a B^{•}-bimodule, then the left and right B^{•}-module structure defined above on P^{•}[1], amount to a natural B-bimodule structure on P^{•}[1].

(v) Let C^{•} be any ℤ-graded A-algebra, N^{•} any ℤ-graded left (resp. right, resp. bi-) Bmodule. We let N[1]^{•} be the graded A-module given by the rule N[1]^{p} := N^{p+1} for every p ∈ ℤ. We shall always view N[1]^{•} as a left (resp. right, resp. bi-) B-module, via the scalar multiplications obtained from those of N^{•}, following the rules spelled out in (iv). This ensures that the functor from B^{•}-modules to H^{•}B^{•}-modules that assigns to any B^{•}-module M^{•} its homology H^{•}M^{•}, is compatible with shift operators.

7.7.5. Let A be ring, (B^{•}, d_{B}^{•}) any differential graded A-algebra, and I^{•} ⊂ B^{•} a (graded) two-sided ideal of B^{•} (i.e. a bi-submodule of the B^{•}-bimodule B^{•}). Let

∂ : H^{•}(B^{•}/I^{•}) → H^{•}I^{•}[1]

denote the natural map induced by the short exact sequence of complexes

0 → I^{•} → B^{•} → B^{•}/I^{•} → 0.

We have :

Lemma 7.7.6. In the situation of (7.7.5), the map ∂ is an A-linear graded derivation of the graded A-algebra H^{•}(B^{•}/I^{•}).

Proof. Indeed, let a and b be any two cycles of the complex B^{•}/I^{•} in degree p and q, and \overline{a}, \overline{b} the respective classes; lift a and b to some elements ã ∈ B^{p} and b̃ ∈ B^{q}, so that ∂(\overline{a} · \overline{b}) is the class in H^{p+q+1}I^{•} of d_{B}(ã · b̃) = d_{B}(ã) · b̃ + (−1)^{p} · ã · d_{B}(b̃). Since d_{B}(ã) (resp. d_{B}(b̃)) represents the class of ∂(\overline{a}) (resp. of ∂(\overline{b})), the assertion follows from the explicit description of the bimodule structure on I^{•}[1] provided by remark 7.7.4(iv). □

Remark 7.7.7. Let B^{•} and C^{•} be two differential graded A-algebras, and denote by B and C the respective associated graded A-algebras. As in remark 7.7.2(ii), we may endow B⊗_{A}C with a natural structure of graded A-algebra. In terms of morphisms of complexes, the multiplication law of B ⊗_{A} C then corresponds to the composition

(B^{•} ⊗_{A} C^{•}) ⊗_{A} (B^{•} ⊗_{A} C^{•}) →^{∼} (B^{•} ⊗_{A} B^{•}) ⊗_{A} (C^{•} ⊗_{A} C^{•}) ⟶^{μ_{B}^{•}⊗_{A}μ_{C}^{•}} B^{•} ⊗_{A} C^{•}

where the first isomorphism is obtained by composing the associativity isomorphisms of example 7.1.16(ii) and the isomorphisms (7.1.17) that swap the tensor factors. Here μ_{B}^{•} and μ_{C}^{•} are the multiplication maps of B^{•} and C^{•}. Thus, we obtain on the complex B^{•} ⊗_{A} C^{•} a natural structure of differential graded A-algebra, and taking into account remark 7.7.2(ii) it is easily seen that if B^{•} and C^{•} are strictly anti-commutative, then B^{•} ⊗_{A} C^{•} represents the coproduct of B^{•} and C^{•} in A-dgAlt.

7.7.8. Suppose now that both B^{•} and C^{•} are bounded above complexes; presumably, in this case there is a canonical way to define a differential graded algebra structure as well on (suitable representatives for)

D^{•} := B^{•} ⊗^{L}_{A} C^{•}

in such a way that this structure is well defined as an object of the derived category of A-dgAlg (the latter should be, as usual, the localization of A-dgAlg, by the multiplicative system of quasiisomorphisms). More modestly, we shall endow the graded A-module H_{•}D^{•} with a natural structure of associative graded A-algebra, in such a way that the natural map

(7.7.9)  H_{•}D^{•} := ⨁_{i∈ℤ} Tor_{i}^{A}(B^{•}, C^{•}) → H_{•}(B^{•} ⊗_{A} C^{•})

is a morphism of graded A-algebras. The multiplication of H^{•}D^{•} is defined as the composition

H_{i}D^{•} ⊗_{A} H_{j}D^{•} ⟶^{α} Tor_{i+j}^{A}(B^{•} ⊗_{A} B^{•}, C^{•} ⊗_{A} C^{•}) ⟶^{μ} H_{i+j}D^{•} for every i, j ∈ ℤ

where α is the bilinear pairing provided by (7.3.40), and with μ := Tor_{i+j}^{A}(μ_{B}^{•}, μ_{C}^{•}).

Let us check first that the foregoing rule does define an associative multiplication on H_{•}D^{•}. To this aim, set B_{2}^{•} := B^{•} ⊗_{A} B^{•}, B_{3}^{•} := B^{•} ⊗_{A} B_{2}^{•} and define likewise C_{2}^{•} and C_{3}^{•}; a little diagram chase, together with (7.3.41), reduces to verifying the commutativity of the diagram

\xymatrix{Tor_{i}^{A}(B_{2}^{•}, C_{2}^{•}) ⊗_{A} H_{j}D^{•} \ar[r]^{μ⊗_{A}1_{H_{i}D^{•}}} \ar[d] & H_{i}D^{•} ⊗_{A} H_{j}D^{•} \ar[d] & H_{i}D^{•} ⊗_{A} Tor_{j}^{A}(B_{2}^{•}, C_{2}^{•}) \ar[l]_{1_{H_{j} D^{•}}⊗_{A}μ} \ar[d] \\ Tor_{i+j}^{A}(B_{3}^{•}, C_{3}^{•}) \ar[r]^{γ} & Tor_{i+j}^{A}(B_{2}^{•}, C_{2}^{•}) & Tor_{i+j}^{A}(B_{3}^{•}, C_{3}^{•}) \ar[l]_{δ}}

whose vertical arrows are the bilinear pairings of (7.3.40), and with

γ := Tor_{i+j}^{A}(μ_{B}^{•} ⊗_{A} 1_{B^{•}}, μ_{C}^{•} ⊗_{A} 1_{C^{•}}) δ := Tor_{i+j}^{A}(1_{B^{•}} ⊗_{A} μ_{B}^{•}, 1_{C^{•}} ⊗_{A} μ_{C}^{•}).

We show the commutativity of the left subdiagram; the same argument applies to the right one. Unwinding the definitions, we come down to checking the commutativity, in D(A-Mod), of the diagram of complexes

\xymatrix{(P_{B_{2}}^{•} ⊗_{A} C_{2}^{•}) ⊗_{A} (P_{B}^{•} ⊗_{A} C^{•}) \ar[r]^{∼} \ar[d]_{ϑ^{•}} & (P_{B_{2}}^{•} ⊗_{A} P_{B}^{•}) ⊗_{A} C_{3}^{•} \ar[r]^{φ_{12,3}^{•}⊗_{A}1_{C^{•}_{3}}} & P_{B_{3}}^{•} ⊗_{A} C_{3}^{•} \ar[d]^{η^{•}} \\ (P_{B}^{•} ⊗_{A} C^{•}) ⊗_{A} (P_{B}^{•} ⊗_{A} C^{•}) \ar[r]^{∼} & (P_{B}^{•} ⊗_{A} P_{B}^{•}) ⊗_{A} C_{2}^{•} \ar[r]^{φ_{1,23}^{•}⊗_{A}1_{C^{•}_{3}}} & P_{B_{2}}^{•} ⊗_{A} C_{2}^{•}}

(notation of (7.3.38)), with ϑ^{•} := (P_{μ_{B}}^{•} ⊗_{A}μ_{C}^{•})⊗_{A}1_{P_{B}^{•}⊗_{A}C^{•}} and η^{•} := P_{μ_{B}⊗_{A}1_{B}}^{•} ⊗_{A}(μ_{C}^{•} ⊗_{A}1_{C^{•}}), and where φ_{12,3}^{•} and φ_{1,23}^{•} are as in (7.3.40) (and with the isomorphisms given by compositions of associativity and swapping isomorphisms). A direct inspection shows that this diagram commutes up to homotopy, as required.

Next, we check that (7.7.9) is a map of graded A-algebras. Unwinding the definitions, we see that – with the notation of (7.3.40) – the multiplication of H_{•}D^{•} is the map on homology induced by the composition

(P_{B}^{•} ⊗_{A} C^{•}) ⊗_{A} (P_{B}^{•} ⊗_{A} C^{•}) →^{∼} (P_{B}^{•} ⊗_{A} P_{B}^{•}) ⊗_{A} C_{2}^{•} → P_{B_{2}}^{•} ⊗_{A} C^{•} → P_{B}^{•} ⊗_{A} C

where the first isomorphism is again a composition of associativity isomorphisms and isomorphisms that swap the factors; the second map is φ_{12}^{•} ⊗_{A} μ_{C}^{•}, where φ_{12}^{•} : P_{B}^{•} ⊗_{A} P_{B}^{•} → P_{B_{2}}^{•} is defined as in (7.3.40). The last map is P_{μ_{B}}^{•} ⊗_{A} 1_{C^{•}}, where P_{μ_{B}}^{•} is given by (7.3.38). Since the associativity and swapping isomorphisms are obviously natural in all their arguments, we come down to checking the commutativity, in D(A-Mod), of the diagram of complexes

\xymatrix{(P_{B}^{•} ⊗_{A} P_{B}^{•}) ⊗_{A} C_{2}^{•} \ar[rr]^{φ_{12}^{•}⊗_{A}μ_{C}^{•}} \ar[d]_{(ρ_{B}^{•}⊗_{A}ρ_{B}^{•})⊗_{A}1_{C^{•}_{2}}} &  & P_{B_{2}}^{•} ⊗_{A} C^{•} \ar[d]^{P_{μ_{B}}^{•} ⊗_{A}1_{C^{•}}} \\ B_{2}^{•} ⊗_{A} C_{2}^{•} \ar[r]^{μ_{B}^{•}⊗_{A}μ_{C}^{•}} & B^{•} ⊗_{A} C^{•} & P_{B}^{•} ⊗_{A} C^{•}. \ar[l]_{ρ_{B}^{•}⊗_{A}1_{C^{•}}}}

The latter is further reduced to the commutativity of

\xymatrix{P_{B}^{•} ⊗_{A} P_{B}^{•} \ar[rr]^{φ_{12}^{•}} \ar[d]_{ρ_{B}^{•}⊗_{A}ρ_{B}^{•}} &  & P_{B_{2}}^{•} \ar[d]^{P_{μ_{B}}^{•}} \\ B_{2}^{•} \ar[r]^{μ_{B}^{•}} & B^{•} & P_{B}^{•}. \ar[l]_{ρ_{B}^{•}}}

But a simple inspection shows that this diagram indeed commutes up to homotopy.

Lastly, we claim that if B^{•} and C^{•} are both anti-commutative, then the same holds for H_{•}D^{•}. Indeed, consider the diagram

\xymatrix{(P_{B}^{•} ⊗_{A} P_{B}^{•}) ⊗_{A} C_{2}^{•} \ar[rrrr]^{∼} \ar[dr]_{(ρ_{B}^{•}⊗_{A}ρ_{B}^{•})⊗_{A}1_{C^{•}_{2}}} \ar[ddd]^{φ_{12}^{•}⊗_{A}μ_{C}^{•}} &  &  &  & (P_{B}^{•} ⊗_{A} P_{B}^{•}) ⊗_{A} C_{2}^{•} \ar[dl]^{(ρ_{B}^{•}⊗_{A}ρ_{B}^{•})⊗_{A}1_{C^{•}_{2}}} \ar[ddd]_{φ_{12}^{•}⊗_{A}μ_{C}^{•}} \\  & B_{2}^{•} ⊗_{A} C_{2}^{•} \ar[rr]^{∼} \ar[dr]_{μ_{B}^{•}⊗_{A}μ_{C}^{•}} &  & B_{2}^{•} ⊗_{A} C_{2}^{•} \ar[dl]^{μ_{B}^{•}⊗_{A}μ_{C}^{•}} \\  &  & B^{•} ⊗_{A} C^{•} \\ P_{B_{2}}^{•} ⊗_{A} C^{•} \ar[rr]^{P_{μ_{B}}^{•} ⊗_{A}1_{C^{•}}} &  & P_{B}^{•} ⊗_{A} C^{•} \ar[u]_{ρ_{B}^{•}⊗_{A}1_{C^{•}}} &  & P_{B_{2}}^{•} ⊗_{A} C^{•}. \ar[ll]_{P_{μ_{B}}^{•} ⊗_{A}1_{C^{•}}}}

(The upper isomorphism is obtained from the automorphism of P_{B}^{•} ⊗_{A} P_{B}^{•} that swaps the two factors, and from the automorphism of C_{2}^{•} of the same type; likewise for the lower isomorphism.) The assumption on B^{•} and C^{•} says that the inner triangular subdiagram commutes, and the same holds – up to homotopy – for the upper and the left and right subdiagrams, by a simple inspection. Then also the outer rectangular subdiagram commutes up to homotopy, and the contention follows easily.

Remark 7.7.10. (i) Simplicial A-algebras are an important source of differential graded algebras, thanks to the following construction. Let R be any simplicial A-algebra, R_{•} the associated chain complex of A-modules, and denote by μ_{R} : R ⊗_{A} R → R the multiplication map of R. By considering the shuffle map for the bisimplicial A-module R ⊠_{A} R (notation as in (7.4.55)), we deduce a natural map of complexes

μ_{R_{•}} : R_{•} ⊗_{A} R_{•} ⟶^{Sh_{•}^{R⊠_{A}R}} (R ⊗_{A} R)_{•} ⟶^{(μ_{R})_{•}} R_{•}

and taking into account propositions 7.4.56 and 7.4.59, it is easily seen that (R_{•}, μ_{R_{•}}) is a strictly anti-commutative differential graded algebra. By remark 7.7.4(iii), we deduce that the graded A-module H_{•}R := ⨁_{p∈ℕ} H_{p}R is naturally an ℕ-graded associative unital and strictly anti-commutative A-algebra.

(ii) Likewise, if M is any R-module, then we obtain on the associated chain complex M_{•} a natural structure of R_{•}-bimodule, so that H_{•}M is naturally a graded H_{•}R-bimodule.

(iii) Clearly, a morphism φ : R → S of simplicial A-algebras induces a morphism φ_{•} : R_{•} → S_{•} of differential graded A-algebras, and a morphism f : M → N of R-modules induces a morphism φ_{•} : M_{•} → N_{•} of R_{•}-bimodules.

7.8. Koszul algebras and regular sequences. For any ring A, let us consider the category

A-dAlg.Mod

whose objects are the triples (B, M, ∂), where B is an A-algebra, M is a B-module, and ∂ : M → B is an A-linear map. The morphisms (B, M, ∂) → (B′, M′, ∂′) in A-dAlg.Mod are the pairs (f, g) where f : B → B′ is a map of A-algebras, and g : B′ ⊗_{B} M → M′ is a B′-linear map, such that

f ∘ ∂(m) = ∂′ ∘ g(1 ⊗ m) for every m ∈ M

with the obvious composition law for such morphisms. We have a natural functor :

A-dgAlt → A-dAlg.Mod (B^{•}, d_{B}^{•}) ↦ (Ker d_{B}^{0}, B^{−1}, d_{B}^{−1})

which admits a left adjoint

K_{•} : A-dAlg.Mod → A-dgAlt (B, M, ∂) ↦ K_{•}(B, M, ∂)

that associates to every object (B, M, ∂) its Koszul algebra K_{•}(B, M, ∂), whose underlying strictly anti-commutative A-algebra is the exterior algebra Λ_{B}^{•}M, as in remark 7.7.2(i). The differentials of the complex K_{•}(B, M, ∂) are given by the maps

d_{n+1} : Λ_{B}^{n+1}M → Λ_{B}^{n}M x_{0} ∧ ⋯ ∧ x_{n} ↦ ∑_{i=0}^{n}(−1)^{i}∂(x_{i}) · x_{0} ∧ ⋯ ∧ x_{i−1} ∧ x_{i+1} ∧ ⋯ ∧ x_{n}

for every n ∈ ℕ. The detailed verifications shall be left to the reader.

Remark 7.8.1. (i) In light of remark 7.7.2(iii), it is easily seen that the coproduct of any two objects (B, M, ∂) and (B′, M′, ∂′) of A-dAlg.Mod is representable by the object

(B′′ := B ⊗_{A} B′, M′′ := (M ⊗_{A} B′) ⊕ (B ⊗_{A} M′), ∂′′)

where ∂′′ : M′′ → B′′ is the A-linear map such that

∂′′(m ⊗ b′, b ⊗ m′) := ∂(m) ⊗ b′ + b ⊗ ∂′(m′) for every m ∈ M, m′ ∈ M′, b ∈ B, b′ ∈ B′.

Combining with remark 7.7.7and proposition 1.3.25(iv), we deduce a natural isomorphism :

K_{•}(B′′, M′′, ∂′′) →^{∼} K_{•}(B, M, ∂) ⊗_{A} K_{•}(B′, M′, ∂′) in A-dgAlt.

(ii) Let f := (f_{1}, …, f_{r}) be a finite sequence of elements of A, and I ⊂ A the ideal generated by f; let also ∂_{f} : A^{⊕r} → A be the A-linear form such that ∂_{f}(a_{1}, …, a_{r}) := ∑_{i=1}^{r} f_{i}a_{i} for every a_{1}, …, a_{r} ∈ A. The Koszul complex of the sequence f is the complex :

K_{•}(f) := K_{•}(A, A^{⊕r}, ∂_{f})

(see [61, Ch.III, §1.1]). Thus, for r = 1, so that f = (f) for a single element f ∈ A, we have

K_{•}(f) = (0 → A ⟶^{f} A → 0)

concentrated in homological degrees 0 and 1. In the general case, combining with (i) we get a natural isomorphism :

K_{•}(f) →^{∼} K_{•}(f_{1}) ⊗_{A} ⋯ ⊗_{A} K_{•}(f_{r}) in A-dgAlt.

(iii) For every complex of A-modules M^{•}, we also use the customary notation :

K_{•}(f, M^{•}) := M^{•} ⊗_{A} K_{•}(f) K^{•}(f, M^{•}) := Tot^{•}(Hom_{A}^{•}(K_{•}(f), M^{•})) and denote by H_{•}(f, M^{•}) (resp. H^{•}(f, M^{•})) the homology of K_{•}(f, M^{•}) (resp. the cohomology of K^{•}(f, M^{•})). Especially, if M is any A-module :

H_{0}(f, M) = M/IM H^{0}(f, M) = Hom_{A}(A/I, M) = Ann_{M}(I)

(where, as usual, we regard M as a complex placed in degree 0).

Lemma 7.8.2. With the notation of remark 7.8.1(iii), let g : A → B be a ring homomorphism, and M^{•} (resp. N^{•}) a complex of A-modules (resp. B-modules). The following holds :

(i) For every x ∈ I, scalar multiplication by x induces the zero endomorphism on the objects K_{•}(f, M^{•}) and K^{•}(f, M^{•}) of Hot(A-Mod).

(ii) Especially, H_{i}(f, M^{•}) and H^{i}(f, M^{•}) are A/I-modules, for every i ∈ ℤ.

(iii) If I = A, the complexes K_{•}(f, M^{•}) and K^{•}(f, M^{•}) are homotopically trivial.

(iv) Denote by g(f) the image in B of the sequence f. Then we have natural identifications

K_{•}(f, N^{•}) →^{∼} K_{•}(g(f), N^{•})  K^{•}(f, N^{•}) →^{∼} K^{•}(g(f), N^{•})  in C(B-Mod).

(v) We have a natural isomorphism

K^{•}(f, M^{•}) →^{∼} K_{•}(f, M^{•})[−r] in C(A-Mod).

Proof. (i): It suffices to notice that scalar multiplication by f induces the zero endomorphism of K_{•}(f), for every f ∈ A : indeed, a homotopy from f · 1_{K_{•}(f)} to the zero endomorphism is given by the system of maps (s_{n} | n ∈ ℕ) with s_{n} := 0 for n ≠ 0, and s_{0} := 1_{A}.

(ii) and (iii) are immediate consequences of (i), and (iv) follows directly from the definitions. (v): Recall that for every free A-module L of finite rank and every A-module M we have a

natural isomorphism of A-modules

Hom_{A}(L, M) →^{∼} L^{∨} ⊗_{A} M where L^{∨} := Hom_{A}(L, A).

There follows, for every complex L_{•} of free A-modules of finite rank, a natural isomorphism

Hom_{A}^{•}(L_{•}, M^{•}) →^{∼} L_{•}^{∨} ⊠_{A} M^{•} in C_{2}(A-Mod)

(notation of example 7.1.16(i)). Moreover, for any two free A-modules of finite rank L_{1} and L_{2} we have as well a natural isomorphism

(L_{1} ⊗_{A} L_{2})^{∨} →^{∼} L_{1}^{∨} ⊗_{A} L_{2}^{∨}

whence, for any complexes L_{1,•} and L_{2,•} of free A-modules of finite rank, a natural isomorphism

(L_{1,•} ⊠_{A} L_{2,•})^{∨} →^{∼} L_{1,•}^{∨} ⊠_{A} L_{2,•}^{∨} in C_{2}(A-Mod).

Summing up, it then suffices to observe that there is a natural isomorphism

K_{•}(f) →^{∼} K_{•}(f)^{∨}[1] in C(A-Mod)

for every f ∈ A. □

7.8.3. Let A be a ring, f := (f_{1}, …, f_{r}) a sequence of elements of A, and f′ := (f_{1}, …, f_{r−1}); let also I ⊂ A (resp. I′ ⊂ A) be the ideal generated by the sequence f (resp. by f′). We have a short exact sequence of complexes : 0 → A[0] → K_{•}(f_{r}) → A[1] → 0, and after tensoring with K_{•}(f′), in view of remark 7.8.1(ii) we get a distinguished triangle :

K_{•}(f′) → K_{•}(f) → K_{•}(f′)[1] ⟶^{∂} K_{•}(f′)[1].

By inspecting the definitions one checks easily that the boundary map ∂ is induced by multiplication by f_{r}. There follow exact sequences :

(7.8.4)  0 → H_{0}(f_{r}, H_{p}(f′, M)) → H_{p}(f, M) → H^{0}(f_{r}, H_{p−1}(f′, M)) → 0

for every A-module M and for every p ∈ ℕ, and together with remark 7.8.1(iii), we deduce :

Lemma 7.8.5. With the notation of (7.8.3), the following conditions are equivalent : (a) H_{i}(f, M) = 0 for every i > 0.

(b) The scalar multiplication by f_{r} is a bijection on H_{i}(f′, M) for every i > 0, and is an injection on M/I′M. □

Definition 7.8.6. We say that the sequence f := (f_{1}, …, f_{r}) of elements of A is completely secant on the A-module M, if we have H_{i}(f, M) = 0 for every i > 0.

The interest of definition 7.8.6 is due to its relation to the notion of regular sequence of elements of A (see e.g. [37, §9, n.6]). Namely, we have the following criterion :

Proposition 7.8.7. With the notation of (7.8.3), the following conditions are equivalent : (a) The sequence f is M-regular.

(b) For every j ≤ r, the sequence (f_{1}, …, f_{j}) is completely secant on M.

Proof. Lemma 7.8.5 shows that (b) implies (a). Conversely, suppose that (a) holds; we show that (b) holds, by induction on r. If r = 0, there is nothing to prove. Assume that the assertion is already known for all j < r. Since f is M-regular by assumption, the same holds for the subsequence f′ := (f_{1}, …, f_{r−1}), and f_{r} is regular on M/(f′)M. Hence H_{p}(f′, M) = 0 for every p > 0, by inductive assumption. Then lemma 7.8.5 shows that H_{p}(f, M) = 0 for every p > 0, as claimed. □

Notice that any permutation of a completely secant sequence is again completely secant, whereas a permutation of a regular sequence is not always regular. As an application of the foregoing, we point out the following :

Corollary 7.8.8. With the notation of (7.8.3), the following holds :

(i) If a sequence (f, g) of elements of A is M-regular, and M is f-adically separated, then (g, f) is M-regular.

(ii) If (n_{1}, …, n_{r}) is any sequence of strictly positive integers, then f is completely secant on M (resp. M-regular) if and only if the same holds for the sequence (f_{1}^{n_{1}}, …, f_{r}^{n_{r}}).

Proof. (i): According to proposition 7.8.7, we only need to show that the sequence (g) is completely secant, i.e. that g is regular on M. Hence, suppose that gm = 0 for some m ∈ M; it suffices to show that m ∈ f^{n}M for every n ∈ ℕ. We argue by induction on n. By assumption g is regular on M/fM, hence m ∈ fM, which shows the claim for n = 1. Let n > 1, and suppose we already know that m = f^{n−1}m′ for some m′ ∈ M. Hence 0 = gf^{n−1}m′, so gm′ = 0 and the foregoing case shows that m′ = fm′′ for some m′′ ∈ M, thus m = f^{n}m′′, as required.

(ii): We deal first with the assertion for completely secant sequences. First, notice that, since every permutation of a completely secant sequence is still completely secant, it suffices to show the assertion for the sequence of integers (1, 1, …, 1, n_{r}). However, set f′ := (f_{1}, …, f_{r−1}); by lemma 7.8.5, f is completely secant if and only if scalar multiplication by f_{r} is a bijection on H_{i}(f′, M) for every i > 0, and an injection on M/(f′)M. But f_{r} fulfills the latter conditions if and only if f_{r}^{n_{r}} does, whence the contention. The assertion for M-regular sequences follows from the foregoing, in view of proposition 7.8.7. □

7.8.9. Let A be a ring, f := (f_{1}, …, f_{r}) a completely secant sequence of elements of A (see definition 7.8.6); denote by J the ideal generated by f, and set A_{0} := A/J. We may regard the complex A_{0}[0] as an (especially simple) differential graded A-algebra, with multiplication \overline{μ}^{•} deduced from that of A_{0}, in the obvious way. We may then state :

Proposition 7.8.10. In the situation of (7.8.9), the A_{0}-module J/J^{2} is free of rank r, and there exists a unique isomorphism of strictly anti-commutative graded A-algebras

(7.8.11)  Λ_{A_{0}}^{•}(J/J^{2}) →^{∼} H_{•}(A_{0}[0] ⊗^{L}_{A} A_{0}[0])

which restricts, in degree 1, to the natural identification

Λ_{A_{0}}^{1}(J/J^{2}) = J/J^{2} →^{∼} Tor_{1}^{A}(A_{0}, A_{0}).

Proof. By assumption, the Koszul complex, with its natural augmentation, yields a resolution

ε^{•} : K_{•}(f) → A_{0}[0]

by free A-modules. Hence, there is a unique isomorphism ω^{•} : P_{A_{0}}^{•} →^{∼} K_{•}(f) in D(A-Mod), whose composition with ε^{•} agrees with ρ_{A_{0}}^{•} (notation of (7.3.38)). Then ε^{•} is a map of differential graded A-algebras, and we easily deduce a commutative diagram in D(A-Mod)

\xymatrix{P_{A_{0}}^{•} ⊗_{A} P_{A_{0}}^{•} \ar[r]^{P • ; \overline{μ}} \ar[d]_{ω^{•}⊗_{A}ω^{•}} & P_{A_{0}}^{•} \ar[d]^{ω^{•}} \\ K_{•}(f) ⊗_{A} K_{•}(f) \ar[r] & K_{•}(f)}

whose bottom horizontal arrow is the multiplication map of K_{•}(f), and where P • ; \overline{μ} is defined as in (7.3.38). We conclude that ω^{•} induces an isomorphism of anti-commutative graded A-algebras

(7.8.12)  H_{•}(A_{0}[0] ⊗^{L}_{A} A_{0}[0]) →^{∼} K_{•}(f, A_{0}[0]) →^{∼} H_{•}((Λ_{A}^{•}(A^{⊕r}), d_{f,•}) ⊗_{A} A_{0}).

By simple inspection, we see that d_{f,i} ⊗_{A} 1_{A_{0}} = 0 for every i ∈ ℤ, whence an isomorphism of strictly anti-commutative A-graded algebras :

H_{•}((Λ_{A}^{•}(A^{⊕r}), d_{f,•}) ⊗_{A} A_{0}) →^{∼} Λ_{A_{0}}^{•}(A_{0}^{⊕r}).

Combining with (7.8.12), we obtain in degree one a natural isomorphism

Λ_{A_{0}}^{1}(A_{0}^{⊕r}) = A_{0}^{⊕r} →^{∼} Tor_{1}^{A}(A_{0}, A_{0}) →^{∼} J/J^{2}

which, finally, delivers the sought isomorphism of differential graded A-algebras. The uniqueness of (7.8.11) is clear, since the exterior algebra is generated by its degree one summand. □

Definition 7.8.13. Let A be a ring, f := (f_{1}, …, f_{r}) a finite sequence of elements of A, and M an A-module. Set also R := ℤ[T_{1}, …, T_{r}], let I ⊂ R (resp. J ⊂ A) be the ideal generated by T := (T_{1}, …, T_{r}) (resp. by f), and denote by gr_{•}R the graded ring associated with the I-adic filtration of R; we associate with f the R-module structure on M such that T_{i}x := f_{i}x for every x ∈ M and every i = 1, …, r, and denote likewise by gr_{•}M the graded gr_{•}R-module associated with the J-adic filtration on M. We say that f is M-quasi-regular, if the natural map

gr_{•}R ⊗_{gr_{0}R} gr_{0}M → gr_{•}M

is an isomorphism.

Lemma 7.8.14. (i) With the notation of definition 7.8.13, we have a natural isomorphism :

R/I ⊗^{L}_{R} M →^{∼} K_{•}(f, M) in D(R-Mod).

(ii) If moreover f is completely secant on A, we have a natural isomorphism :

A/J ⊗^{L}_{A} M →^{∼} K_{•}(f, M) in D(A-Mod).

Proof. (i): Clearly T is a regular sequence on R, hence H_{i}(T, R) = 0 for every i > 0 (proposition 7.8.7), i.e. K_{•}(T, R) is a resolution of R/I by free R-modules (remark 7.8.1(iii)). On the other hand, let g : R → A be the unique ring homomorphism such that g(T_{i}) := f_{i} for i = 1, …, r; clearly the R-module structure of M is obtained by restriction of scalars along g from its A-module structure, so we have a natural identification K_{•}(T, M) →^{∼} K_{•}(f, M) (lemma 7.8.2(iv)). The assertion follows immediately.

(ii): If f is completely secant on A, then K_{•}(f) is a resolution of A/J by free A-modules (remark 7.8.1(iii)), whence the assertion. □

The following result summarizes and completes the list of interdependencies found thus far between the properties of finite sequences of elements in a ring that have been introduced at various stages in the text.

Proposition 7.8.15. In the situation of definition 7.8.13, consider the following conditions : (a) The sequence f is M-regular.

(b) The sequence f is completely secant on M.

(c) Tor_{1}^{R}(R/I, M) = 0, or equivalently, H_{1}(f, M) = 0.

(d) The sequence f is M-quasi-regular.

(e) T_{1} is gr_{•}M-regular, and f_{1}M ∩ J^{n+1}M = f_{1}J^{n}M for every n ∈ ℕ.

(f) M is J-adically complete and separated.

(g) For every k = 0, …, r, the A-module M/ ∑_{i=1}^{k} f_{i}M is J-adically separated. Then (a)⇒(b)⇒(c)⇒(d)⇒(e), and if (d) holds, then (f)⇒(g)⇒(a).

Proof. (a)⇒(b) is already known, by proposition 7.8.7, and (b)⇒(c), by lemma 7.8.14(i).

(c)⇒(d): The assumption means that the natural map I ⊗_{R} M → IM is bijective. On the other hand, notice that gr_{0} := R/I is isomorphic to ℤ, and gr_{n}R := I^{n}/I^{n+1} is a free ℤ-module for every n ∈ ℕ; consider then for every n ∈ ℕ the commutative diagram :

\xymatrix{0 \ar[r] & I^{n+1} ⊗_{R} M \ar[r] \ar[d] & I^{n} ⊗_{R} M \ar[r] \ar[d] & gr_{n}R ⊗_{R} M \ar[d] \ar[r] & 0 \\ 0 \ar[r] & I^{n+1}M \ar[r] & I^{n}M \ar[r] & gr_{n}M \ar[r] & 0}

all of whose vertical arrows are surjections, and whose horizontal rows are short exact sequences, as Tor_{1}^{R}(gr_{n}R, M) = 0. Then, a simple induction shows that all three vertical arrows are isomorphisms for every n ∈ ℕ, whence the assertion.

(d)⇒(e): It is easily seen that I^{n+1}/(T_{1}I^{n} + I^{n+2}) is a free ℤ-module for every n ∈ ℕ, hence we get a short exact sequence

0 → gr_{n}R ⊗_{gr_{0}R} gr_{0}M ⟶^{φ} gr_{n+1}R ⊗_{gr_{0}R} gr_{0}M → (I^{n+1}/(T_{1}I^{n} + I^{n+2})) ⊗_{gr_{0}R} gr_{0}M → 0

for every n ∈ ℕ where φ is induced by scalar multiplication by T_{1} on R, which maps I^{n} into I^{n+1} for every such n. Then (d) implies that T_{1} is regular on the gr_{•}R-module gr_{•}M; moreover

{x ∈ I^{n}M | f_{1}x ∈ I^{n+2}M} = I^{n+1}M for every n ∈ ℕ

whence f_{1}I^{n}M ∩ I^{n+2}M = f_{1}I^{n+1}M for every n ∈ ℕ. By a simple induction on k, we deduce that f_{1}I^{n−k}M ∩ I^{n+2}M = f_{1}I^{n+1}M for k = 0, …, n, whence the sought identity.

(d) and (f)⇒(g): We shall argue by induction on the length r of f. For r = 0, there is nothing to prove. Next, suppose that r ≥ 1, and the assertion is already known for every M-quasi-regular sequence of length r − 1; by the foregoing, (e) holds for M, so that scalar multiplication by f_{1} on the A-module M induces an injective map gr_{n}M → gr_{n+1}M for every n ∈ ℕ. Since the J-adic topology is separated on M, we deduce already that f_{1} is M-regular and the J-adic topology on f_{1}M agrees with the topology induced by the J-adic topology of M; since f_{1}M is isomorphic to M, we conclude that f_{1}M is complete for the topology induced by M, and by virtue of proposition 8.2.13(i,v), condition (f) holds therefore as well for \overline{M} := M/f_{1}M.

Now, let R′ := ℤ[T_{2}, …, T_{r}] ⊂ R, set I′ := I ∩ R′, J′ := I′A, and denote by gr_{•}R′ the graded ring associated with the I′-adic filtration of R′. The projection R → R/(T_{1}) ←^{∼} R′ induces isomorphisms of rings and respectively graded rings :

gr_{0}R →^{∼} gr_{0}R′ (gr_{•}R)/(\overline{T}_{1}) →^{∼} gr_{•}R′

where \overline{T}_{1} ∈ gr_{1}R denotes the class of T_{1}. Let us endow \overline{M} with the R′-module structure obtained by restriction of scalars along the inclusion map R′ → R, and let gr_{•}\overline{M} be the graded gr_{•}R′-module associated with the J′-adic filtration on \overline{M}. Then clearly the J′-adic topology of \overline{M} coincides with its J-adic topology; moreover, notice that

gr_{•}R′ ⊗_{gr_{•}R} gr_{•}M = ⨁_{n∈ℕ} \frac{I^{n}M}{f_{1}I^{n−1}M + I^{n+1}M} gr_{•}\overline{M} = ⨁_{n∈ℕ} \frac{I^{n}M}{(f_{1}M ∩ I^{n}M) + I^{n+1}M}

(here we set I^{−1}M := M). Hence, (e) also implies that the projection gr_{•}M → gr_{•}\overline{M} induces isomorphisms of gr_{0}R-modules and respectively graded gr_{•}R′-modules :

gr_{0}M →^{∼} gr_{0}\overline{M} gr_{•}R′ ⊗_{gr_{•}R} gr_{•}M →^{∼} gr_{•}\overline{M}

Therefore, the natural map

gr_{•}R′ ⊗_{gr_{0}R′} gr_{0}\overline{M} → gr_{•}\overline{M}

is bijective, i.e. the sequence f′ := (f_{2}, …, f_{n}) is \overline{M}-quasi-regular; by inductive assumption, \overline{M}/ ∑_{i=2}^{k} f_{i}\overline{M} = M/ ∑_{i=1}^{k} f_{i}M is then J-adically separated for k = 1, …, n, as required.

(d) and (g)⇒(a): Again, we argue by induction on r. If r = 0, there is nothing to show; suppose then that r ≥ 1, and the assertion is already known for every A-module M fulfilling condition (g), relative to any M-quasi-regular sequence of length r − 1. As in the foregoing, we see that f_{1} is M-regular, and f′ is \overline{M}-quasi-regular; then by inductive assumption, f′ is \overline{M}-regular, so f is M-regular. □

Remark 7.8.16. (i) Condition (g) of proposition 7.8.15holds in either of the following cases :

(a) A is noetherian, M is an A-module of finite type and I lies in the Jacobson radical of A ([126, Th.8.1(i)]).

(b) For a given fine monoid P, the ring A is P -graded, M is a P -graded A-module, and each of f_{1}, …, f_{r} are homogeneous elements whose degrees lie in the maximal ideal 𝔪 of P : indeed, recall that the 𝔪-adic filtration is separated on P (lemma 6.4.21(i)).

(ii) In the situation of definition 7.8.13, notice that the sequence f is quasi-regular on A if and only if the following two conditions hold :

(a) J/J^{2} is a free A/J-module of rank r

(b) the natural map of graded A/J-algebras

Sym_{A/J}^{•}(J/J^{2}) → ⨁_{n∈ℕ} J^{n}/J^{n+1}

is an isomorphism : the details shall be left to the reader. See also corollary 7.9.22(iv).

For future reference, we point out :

Lemma 7.8.17. (Cp. [20, Exp.VII, Prop.1.8(ii)]]) Let A be a ring, n ≥ 1 an integer, and f := (f_{0}, …, f_{n}) a completely secant sequence of elements of A. The following holds :

(i) The sequences g := (f_{0}, f_{1} − f_{0}T_{1}, …, f_{n} − f_{0}T_{n}) and g′ := (f_{1} − f_{0}T_{1}, …, f_{n} − f_{0}T_{n}) are completely secant in the polynomial A-algebra A[T_{1}, …, T_{n}].

(ii) Let A[f_{1}/f_{0}, …, f_{n}/f_{0}] ⊂ A[1/f_{0}] be the A-subalgebra generated by f_{1}/f_{0}, …, f_{n}/f_{0}. Then the kernel of the surjective map of A-algebras

A[T_{1}, …, T_{n}] → A[f_{1}/f_{0}, …, f_{n}/f_{0}]  T_{i} ↦ f_{i}/f_{0}  for i = 1, …, n

is the ideal I ⊂ A[T_{1}, …, T_{n}] generated by the sequence g′.

Proof. (i): Since the inclusion map A → B := A[T_{1}, …, T_{n}] is flat, it is clear that the sequence f is completely secant in B. Define the B-linear forms ∂_{f}, ∂_{g} : B^{⊕n+1} → B as in remark 7.8.1(ii), so that K_{•}(f) = K_{•}(B, B^{⊕n+1}, ∂_{f}), and likewise for K_{•}(g). We have an isomorphism

(1_{B}, φ) : (B, B^{⊕n+1}, ∂_{g}) →^{∼} (B, B^{⊕n+1}, ∂_{f}) in A-dAlg.Mod

with φ : B^{⊕n+1} →^{∼} B^{⊕n+1} the B-linear automorphism such that φ(e_{0}) := e_{0} and φ(e_{i}) := e_{i} − T_{i}e_{0} for i = 1, …, n. The assertion for g is an immediate consequence.

Combining with lemma 7.8.5, we see that the scalar multiplication by f_{0} is injective on H_{i}(g′, B), for every i ∈ ℕ. Hence, the natural map

H_{i}(g′, B) → H_{i}(g′, B[f_{0}^{−1}])

is injective for every i ∈ ℕ, and in order to show that g′ is completely secant in B, we may then assume that f_{0} ∈ A^{×}. To this aim, we consider also the sequence T := (T_{1}, …, T_{n}); we have an isomorphism

(ψ, ψ^{⊕n}) : (B, B^{⊕n}, ∂_{T}) →^{∼} (B, B^{⊕n}, ∂_{g′}) in A-dAlg.Mod

where ψ : B →^{∼} B is the ring isomorphism such that ψ(T_{i}) := f_{i} − f_{0}T_{i} for i = 1, …, n. Since the sequence T is obviously B-regular, the assertion then follows from proposition 7.8.7.

(ii) We have a commutative diagram of rings :

\xymatrix{B/IB \ar[r] \ar[d] & A[f_{1}/f_{0}, …, f_{n}/f_{0}] \ar[d] \\ B/IB[1/f_{0}] \ar[r] & A[1/f_{0}]}

whose vertical arrows are the localizations, and it is easily seen that the bottom horizontal arrow is an isomorphism. On the other hand, from (i) and lemma 7.8.5 we see that the scalar multiplication by f_{0} is injective on B/I; hence the top horizontal arrow is injective, whence the contention. □

Definition 7.8.18. Let 𝒜 be any additive category, and A_{•} any object of Fun(ℕ^{o}, 𝒜) (where the partially ordered set (ℕ, ≤) is regarded as category, as in example 1.1.6(iii)). In other words, A_{•} is a datum (A_{•}, φ_{••}) consisting of a family (A_{p} | p ∈ ℕ) of objects of 𝒜 and a system of morphisms φ_{q,p} : A_{q} → A_{p} of 𝒜 for every integers q ≥ p ≥ 0, such that

φ_{q,p} ∘ φ_{r,q} = φ_{r,p} for every r ≥ q ≥ p ≥ 0.

The objects of Fun(ℕ^{o}, 𝒜) are called also inverse systems of 𝒜 indexed by ℕ. Then :

(i) We say that A_{•} is essentially zero, if for every p ∈ ℕ there exists q ≥ p such that φ_{q,p} is the zero morphism.

(ii) We say that A_{•} is uniformly essentially zero, if there exists c ∈ ℕ such that φ_{p+c,p} is the zero morphism for every p ∈ ℕ. In this case, the smallest of such c is called the step of A_{•}.

Recall that if Fun(ℕ^{o}, 𝒜) is an additive category, and it is even abelian, if the same holds for 𝒜; moreover, the kernels and cokernels in Fun(ℕ^{o}, 𝒜) are computed argumentwise : see remark 3.7.37(ii). We notice :

Lemma 7.8.19. Let 𝒜 be any abelian category, and consider a short exact sequence

0 → A′_{•} → A_{•} → A′′_{•} → 0 in Fun(ℕ^{o}, 𝒜) .

Then, A_{•} is essentially zero (resp. uniformly essentially zero) if and only if the same holds for both A′_{•} and A′′_{•}.

Proof. Indeed, it is clear that if A_{n} → A_{m} is the zero morphism for some n ≥ m, then the same holds for the morphisms A′_{n} → A′_{m} and A′′_{n} → A′′_{m}. Conversely, suppose that A′_{n} → A′_{m} and A′′_{p} → A′′_{n} are the zero morphisms, for some p ≥ n ≥ m; then it follows easily that the same holds for the morphism A_{p} → A_{m}. □

7.8.20. Let f := (f_{1}, …, f_{r}) and g := (g_{i} | i = 1, …, r) be two finite sequences of elements of a ring A; we set fg := (f_{i}g_{i} | i = 1, …, r) and define as follows a map of Koszul complexes

φ_{g} : K_{•}(fg) → K_{•}(f)

(see remark 7.8.1(ii)). First, suppose that r = 1; then f = (f), g = (g) and the sought map φ_{g} is the commutative diagram :

[figure: 0; A; fg; g; A; 0; 0; A; f; A; 0.]

For the general case we let :

φ_{g} := φ_{g_{1}} ⊗_{A} ⋯ ⊗_{A} φ_{g_{r}}.

Especially, for every m, n ≥ 0 we have maps φ_{f^{n}} : K_{•}(f^{n+m}) → K_{•}(f^{m}), whence maps

φ_{f^{n}}^{•} : K^{•}(f^{m}, M) → K^{•}(f^{m+n}, M)

and clearly φ_{f^{p+q}}^{•} = φ_{f^{p}}^{•} ∘ φ_{f^{q}}^{•} for every m, p, q ≥ 0.

7.8.21. In the situation of (7.8.20), set A_{r} := ℤ[T_{1}, …, T_{r}], and denote by β_{f} : A_{r} → A the unique ring homomorphism such that β_{f}(T_{i}) := f_{i} for every i = 1, …, r. Let also I_{r} ⊂ A_{r} be the ideal generated by (T_{1}, …, T_{r}), and set I := I_{r}A. We consider the following conditions :

(a)_{f} The inverse system (Tor_{i}^{A_{r}}(A_{r}/I_{r}^{n}, A) | n ∈ ℕ) is essentially zero for every i > 0. (b)_{f} The inverse system (Tor_{i}^{A}(A/I^{n}, A/I) | n ∈ ℕ) is essentially zero for every i > 0. (c)_{f} The inverse system (H_{i}(f, I^{n}) | n ∈ ℕ) is essentially zero for every i > 0. (d)_{f} The inverse system (H_{i}K_{•}(f^{n}) | n ∈ ℕ) is essentially zero for every i > 0. (e)_{f} The inverse system (Tor_{i+1}^{A}(A/I^{n}, M) | n ∈ ℕ) is essentially zero, for every i, k ∈ ℕ

and every A/I^{k}-module M. (f)_{f} For every i ∈ ℤ and every bounded above complex C_{•} of flat A-modules such that

for every j ∈ ℤ the annihilator ideal of H_{j}C_{•} = 0 contains a power of I, the inverse system (H_{i}(I^{n}C_{•}) | n ∈ ℕ) is essentially zero.

Moreover, we shall also consider the conditions (a)_{f}^{un}, …, (f)_{f}^{un} obtained from (a)_{f}, …, (f)_{f} after replacing “essentially zero” by “uniformly essentially zero”.

Remark 7.8.22. Condition (d)_{f} seems to have been first considered in [83, Lemma 2.4], and in particular, our remark 7.8.45 was already observed in [83, Lemma 2.5 and th.2.8]. The same condition reappears in [85, Exp.II, Lemme 9], as well as in in [1, Lemma 3.1.1], and it is baptised “weak proregularity” in an errata^{1} for that paper. A non-commutative extension is proposed in [160]. For motivation, we point out the following lemma, which says that completely secant sequences f trivially satisfy condition (a)_{f}^{un} :

Lemma 7.8.23. With the notation of (7.8.21), let k ∈ ℕ such that H_{k}(f, A) = 0. Then we have

Tor_{k}^{A_{r}}(A_{r}/I_{r}^{n}, A) = 0 for every n ∈ ℕ.

1see http://www.math.purdue.edu/~lipman/papers/homologyfix.pdf

Proof. Notice first that A_{r}/I_{r} = ℤ, and I_{r}^{n}/I_{r}^{n+1} is a free ℤ-module, for every n ∈ ℕ; using the long Tor-exact sequences arising from the short exact sequences

0 → I_{r}^{n}/I_{r}^{n+1} → A_{r}/I_{r}^{n+1} → A_{r}/I_{r}^{n} → 0

a simple induction reduces to the case where n = 1 (details left to the reader). In this case, the assertion follows from lemma 7.8.14(i). □

Lemma 7.8.24. With the notation of (7.8.21), for every i, p ∈ ℕ with i > 0, and every A_{r}/I_{r}^{p}- module N, the natural morphism Tor_{i}^{A_{r}}(A_{r}/I_{r}^{n+p}, N) → Tor_{i}^{A_{r}}(A_{r}/I_{r}^{n}, N) is the zero map.

Proof. Let Fil^{•}N be the I_{r}-adic filtration of N, and gr^{•}N the associated graded A_{r}/I_{r}-module; the long exact Tor-sequences arising from the short exact sequences 0 → Fil^{q+1}N → Fil^{q}N → gr^{q}N → 0 (for every q ∈ ℕ), and an easy induction argument, reduce to the case where p = 1 (details left to the reader), so N is an A_{r}/I_{r}-module, i.e. a ℤ-module; we may moreover reduce to the case where N is a finitely generated ℤ-module, and then we may even assume that N = ℤ/aℤ for some a ∈ ℕ (details left to the reader). In this situation, set B := A_{r}/aA_{r}, J := I_{r}B_{r}, consider the standard 2-spectral sequence ([163, Th.5.6.6])

E_{pq}^{2} := Tor_{p}^{B}(Tor_{q}^{A_{r}}(A_{r}/I_{r}^{n}, B), ℤ/aℤ) ⇒ Tor_{p+q}^{A_{r}}(A_{r}/I_{r}^{n}, ℤ/aℤ)

and notice that, on the one hand, the scalar multiplication by a is injective on A_{r}/I_{r}^{n}, and on the other hand, the complex 0 → A_{r} → A_{r} → 0 with differential given by a · 1_{A_{r}} is a free resolution of B; we deduce that E_{pq}^{2} = 0 whenever q > 0, whence natural isomorphisms :

Tor_{i}^{B}(B/J^{n}, ℤ/aℤ) →^{∼} Tor_{i}^{A_{r}}(A_{r}/I_{r}^{n}, ℤ/aℤ) for every i, n ∈ ℕ

and we are reduced to checking that the natural map

Tor_{i}^{B}(B/J^{n+1}, ℤ/aℤ) → Tor_{i}^{B}(B/J^{n}, ℤ/aℤ)

vanishes for every i > 0. To this aim, notice that the Koszul complex K_{•} := K_{•}(T_{•}, B) associated with the sequence T_{•} := (T_{1}, …, T_{k}) of elements of B, is a resolution of Z/aℤ consisting of free B-modules (proposition 7.8.7), so we have a natural isomorphism

B/J^{n} ⊗_{B} K_{•} →^{∼} B/J^{n} ⊗^{L}_{B} ℤ/aℤ in D(B-Mod).

Now, denote by gr_{•}B the standard graded ℤ/aℤ-algebra structure on B such that gr_{1}B is generated by T_{1}, …, T_{k}. For every i ∈ ℕ we define a grading on K_{i} as follows. Let (e_{1}, …, e_{k}) be the canonical basis of B^{⊕k}, and ℱ_{i} the set of all strictly increasing maps {1, …, i} → {1, …, r}; for every φ ∈ ℱ_{i} and every r ∈ ℕ set

e_{φ} := e_{φ(1)} ∧ ⋯ ∧ e_{φ(i)} and gr_{r}K_{i} := ∑_{φ∈ℱ_{i}} e_{φ} · gr_{r−i}B.

A simple inspection then shows that the differential of the Koszul complex is, in each degree, a map of graded ℤ/aℤ-modules, therefore we get a natural decomposition

K_{•} →^{∼} ⨁_{r∈ℕ} gr_{r}K_{•} in C(ℤ/aℤ-Mod)

and since J^{n} = ⨁_{r≥n} gr_{r}B, the complex K_{•} ⊗_{B} B/J^{n} admits a corresponding decomposition, and we have

(gr_{r}K_{•} ⊗_{B} B/J^{n})_{i} = \{ \begin{matrix}gr_{r}K_{i} & if r < i + n \\ 0 & otherwise. \end{matrix}

Since H_{i}K_{•} = 0 for every i > 0, it follows easily that

gr_{r}H_{i}(K_{•} ⊗_{B} B/J^{n}) = 0 for every i, r, n ∈ ℕ such that r ≠ i − 1 + n and i > 0.

Thus, fix i, n ∈ ℕ with i > 0, and consider any x ∈ Tor_{i}^{B}(B/J^{n+1}, ℤ/aℤ); by the foregoing, we have x ∈ H_{i}(gr_{i+n}K_{•} ⊗_{B} B/J^{n+1}), and H_{i}(gr_{i+n}K_{•} ⊗_{B} B/J^{n}) = 0, so the image of x vanishes in Tor_{i}^{B}(B/J^{n}, ℤ/aℤ), as required. □

Proposition 7.8.25. With the notation of (7.8.21), fix i ∈ ℕ, and let also g := (g_{i} | i = 1, …, s) be another sequence of elements of A that generates an ideal J. We have :

(i) (a)_{f} ⇔ (b)_{f} ⇔ (c)_{f} ⇔ (d)_{f} ⇔ (e)_{f} ⇔ (f)_{f}. (ii) (a)_{f}^{un} ⇒ (b)_{f}^{un} ⇔ (c)_{f}^{un} ⇔ (e)_{f}^{un} ⇔ (f)_{f}^{un}. (iii) If the radical of I equals the radical of J, then (a)_{f} ⇔ (a)_{g}. (iv) If I = J, then (a)_{f}^{un} ⇔ (a)_{g}^{un}.

Proof. Let us check first that (a)_{f} ⇒ (b)_{f} and (a)_{f}^{un} ⇒ (b)_{f}^{un}. To this aim, we consider the change of rings spectral sequence

E(n)_{ij}^{2} := Tor_{i}^{A}(Tor_{j}^{A_{r}}(A_{r}/I_{r}^{n}, A), A/I) ⇒ Tor_{i+j}^{A_{r}}(A_{r}/I_{r}^{n}, A/I).

Notice that the projection A_{r}/I_{r}^{m} → A_{r}/I_{r}^{n} induces a morphism of spectral sequences

(7.8.26)  E(m)_{••}^{•} → E(n)_{••}^{•}  for every m ≥ n.

Then the assertion is the case p = 0 of the following :

Claim 7.8.27. If condition (a)_{f} (resp. (a)_{f}^{un}) holds, then the inverse system (E(n)_{i+1,0}^{p+2} | n ∈ ℕ) is essentially zero (resp. uniformly essentially zero) for every i, p ∈ ℕ.

Proof of the claim. We fix i ∈ ℕ, and we argue by descending induction on p. Notice first that

E(n)_{i,0}^{i+1} = E(n)_{i,0}^{∞} for every i, n ∈ ℕ.

Then, lemma 7.8.24 trivially implies that the assertion holds for every i, p ∈ ℕ with p ≥ i. Now, let 0 ≤ q ≤ i − 1, and suppose that the assertion is already known for every p > q (with our fixed i). We get an exact sequence of inverse systems

0 → (E(n)_{i+1,0}^{q+3} | n ∈ ℕ) → (E(n)_{i+1,0}^{q+2} | n ∈ ℕ) → (E(n)_{i−q−1,q+1}^{q+2} | n ∈ ℕ)

and by inductive assumption the system (E(n)_{i+1,0}^{q+3} | n ∈ ℕ) is essentially zero (resp. uniformly essentially zero). Moreover, (a)_{f} (resp. (a)_{f}^{un}) implies that the same holds for the system (E(n)_{i−q−1,q+1}^{q+2} | n ∈ ℕ). Then the contention follows from lemma 7.8.19. ♢

• In order to show that (b)_{f} ⇒ (a)_{f} we remark :

Claim 7.8.28. Let j ∈ ℕ be any integer. Then :

(i) If (E(n)_{0,j}^{2} | n ∈ ℕ) is an essentially zero inverse system, the same holds for the inverse system (Tor_{j}^{A_{r}}(A_{r}/I_{r}^{n}, A) | n ∈ ℕ).

(ii) If (Tor_{j}^{A_{r}}(A_{r}/I_{r}^{n}, A) | n ∈ ℕ) is an essentially zero inverse system, the same holds for

the inverse system (E_{i,j}^{p+2} | n ∈ ℕ), for every i, p ∈ ℕ. (iii) The inverse system (E(n)_{0,j}^{j+2} | n ∈ ℕ) is essentially zero. (iv) (b)_{f} implies that (E(n)_{i,0}^{j+2} | n ∈ ℕ) is an essentially zero system for every i ∈ ℕ.

Proof of the claim. (i): Denote by φ_{n,c} : Tor_{j}^{A_{r}}(A_{r}/I_{r}^{n+c}, A) → Tor_{j}^{A_{r}}(A_{r}/I_{r}^{n}, A) the natural map. The hypothesis means that for every n ∈ ℕ there exists c ∈ ℕ such that φ_{n,c} ⊗_{A} A/I = 0, i.e. Im φ_{n,c} ⊂ I · Tor_{j}^{A_{r}}(A_{r}/I_{r}^{n}, A). By a simple induction we deduce that for every n, k ∈ ℕ there exists d ∈ ℕ such that Im φ_{n,c} ⊂ I^{k} · Tor_{j}^{A_{r}}(A_{r}/I_{r}^{n}, A), and letting k = n, the assertion follows.

(ii): Obviously, the assumption implies that (E_{i,j}^{2} | n ∈ ℕ) is an essentially zero inverse system; the assertion is an immediate consequence.

(iv) is clear, and notice that E(n)_{0,j}^{j+2} = E(n)_{0,j}^{∞} for every n ∈ ℕ. In light of lemma 7.8.24, assertion (iii) follows easily as well. ♢

In light of claim 7.8.28(i) it suffices to show that (E(n)_{0,j}^{2}) is essentially zero for every j > 0. We argue by induction on j; for j = 1 we consider the exact sequence of inverse systems :

(E(n)_{2,0}^{2} | n ∈ ℕ) → (E(n)_{0,1}^{2} | n ∈ ℕ) → (E(n)_{0,1}^{3}) → 0.

By claim 7.8.28(iii,iv), the first and third terms are essentially zero, and therefore the same holds for the middle term, by lemma 7.8.42.

Next, let j > 1, and suppose that the inverse systems (E(n)_{0,k}^{2} | n ∈ ℕ) are essentially zero for 0 < k < j. By claim 7.8.28(i,ii) we deduce that (E(n)_{i,k}^{p+2} | n ∈ ℕ) is essentially zero for every p, i ∈ ℕ and whenever 0 < k < j. We show, by descending induction on p, that (E(n)_{0,j}^{p+2} | n ∈ ℕ) is essentially zero for every p ∈ ℕ. First, the assertion holds for p ≥ j, by virtue of claim 7.8.28(iii). Suppose then that 0 ≤ k < j, and that the assertion is already known for every p > k; we consider the exact sequence of inverse systems

(E(n)_{k,j−k−1}^{k+2} | n ∈ ℕ) → (E(n)_{0,j}^{k+2} | n ∈ ℕ) → (E(n)_{0,j}^{k+3}) → 0.

Our inductive assumtions imply that the first and third terms are essentially zero; then the same holds for the middle term (lemma 7.8.42), and the proof is concluded.

• Next, we show that (b)_{f} ⇒ (e)_{f} and (b)_{f}^{un} ⇒ (e)_{f}^{un}. For any A/I^{k}-module M, we need to check that the inverse system (Tor_{i+1}^{A}(A/I^{n}, M) | n ∈ ℕ) is essentially zero (resp. uniformly essentially zero); to this aim, we consider the I-adic filtration Fil^{•}M on M, and the associated graded A/I-module gr_{•}M. Applying the long exact Tor-sequences arising from the short exact sequences 0 → Fil^{q+1}M → Fil^{q}M → gr^{q}M → 0, together with lemma 7.8.19, we reduce to checking that (Tor_{i+1}^{A}(A/I^{n}, gr^{•}M) | n ∈ ℕ) is an essentially zero (resp. uniformly essentially zero) inverse system. Thus, we may suppose that IM = 0, in which case we show more precisely :

Claim 7.8.29. Let p > 0 be any integer, and suppose that (Tor_{i}^{A}(A/I^{n}, A/I) | n ∈ ℕ) is an essentially zero (resp. uniformly essentially zero) inverse system for every i = 1, …, p. Then, for every A/I-module M the inverse system (Tor_{p}^{A}(A/I^{n}, M) | n ∈ ℕ) is essentially zero (resp. uniformly essentially zero).

Proof of the claim. We consider the spectral sequence

E(n)_{i,j}^{2} := Tor_{i}^{A/I}(Tor_{j}^{A}(A/I^{n}, A/I), M) ⇒ Tor_{i+j}^{A}(A/I^{n}, M) for every n ∈ ℕ

and the corresponding morphisms of spectral sequences as in (7.8.26), induced by the projections A/I^{m} → A/I^{n}. Notice that E(n)_{i,0}^{2} = 0, and therefore E(n)_{i,0}^{∞} = 0 for every i > 0. On the other hand, our assumption implies that the inverse system (E(n)_{i,j}^{2} | n ∈ ℕ) is essentially zero (resp. uniformly essentially zero) for every i, j ∈ ℕ such that j > 0 and i + j = q. Summing up, we deduce that the inverse system (E(n)_{i,j}^{∞} | n ∈ ℕ) is essentially zero (resp. uniformly essentially zero) for every i, j ∈ ℕ such that i + j = p. Taking into account lemma 7.8.19, the contention now follows by a simple induction. ♢

• We check next that (e)_{f} ⇒ (f)_{f} and (e)_{f}^{un} ⇒ (f)_{f}^{un}. To this aim, notice that H_{i}(I^{n}C_{•}) =

H_{i}(I^{n}[0]⊗^{L}_{A} C_{•}) for every i, n ∈ ℤ, since C_{•} is a complex of flat A-modules. On the other hand, we have a standard spectral sequence

E(n)_{p,q}^{2} := Tor_{p}^{A}(I^{n}, H_{q}C_{•}) ⇒ H_{p+q}(I^{n}[0] ⊗^{L}_{A} C_{•}).

Since Tor_{p}^{A}(I^{n}, H_{q}C_{•}) →^{∼} Tor_{p+1}^{A}(A/I^{n}, H_{q}C_{•}) for every p, n ∈ ℕ, condition (e)_{f} (resp. (e)_{f}^{un}) implies that (E(n)_{p,q}^{2} | n ∈ ℕ) is an essentially zero (resp. uniformly essentially zero) inverse system for every p ∈ ℕ and every q ∈ ℤ, hence the same holds for the system (E(n)_{p,q}^{∞} | n ∈ ℕ).

Since C_{•} is bounded above, lemma 7.8.19easily implies that H_{i}(I^{n}[0] ⊗^{L}_{A} C_{•}) is an essentially zero (resp. uniformly essentially zero) system for every i ∈ ℤ (details left to the reader), whence the contention.

Clearly (f)_{f} ⇒ (c)_{f} and (f)_{f}^{un} ⇒ (c)_{f}^{un}, since the Koszul complex is a bounded complex of flat A-modules whose cohomology is an A/I-module in each degree (lemma 7.8.2(ii)).

• To show that (c)_{f} ⇒ (b)_{f} and (c)_{f}^{un} ⇒ (b)_{f}^{un} we consider the spectral sequence

E(n)_{i,j}^{2} := Tor_{i}^{A}(I^{n}, H_{j}K_{•}(f)) ⇒ H_{i+j}(f, I^{n}) for every n ∈ ℕ

and the corresponding system of morphisms of spectral sequences as in (7.8.26). Recalling that H_{0}K_{•}(f_{•}) = A/I, as well as the natural isomorphism

(7.8.30)  Tor_{i}^{A}(I^{n}, A/I) →^{∼} Tor_{i+1}^{A}(A/I^{n}, A/I)  for every i, n ∈ ℕ

the assertion will follow from the case j = p = 0 of the following more general :

Claim 7.8.31. If (c)_{f} (resp. (c)_{f}^{un}) holds, the inverse system (E(n)_{i,j}^{p+2} | n ∈ ℕ) is essentially zero (resp. uniformly essentially zero) for every i, j, p ∈ ℕ.

Proof of the claim. We argue by induction on i. For i = 0, notice that E(n)_{0,0}^{2} = E(n)_{0,0}^{∞} for every n ∈ ℕ; condition (c)_{f} (resp. (c)_{f}^{un}) then yields the claim for the system (E(n)_{0,0}^{2} | n ∈ ℕ). In light of (7.8.30) and claim 7.8.29, we deduce that (E(n)_{0,j}^{2} | n ∈ ℕ) is essentially zero (resp. uniformly essentially zero) for every j ∈ ℕ, and then the same follows for (E(n)_{0,j}^{p+2} | n ∈ ℕ), for every j, p ∈ ℕ.

Next, suppose that k > 0, and the claim is already known for every i < k and every j, p ∈ ℕ. We show, by descending induction on q, that (E(n)_{k,0}^{q} | n ∈ ℕ) is an essentially zero (resp. uniformy essentially zero) system for every q ≥ 2. Indeed, notice first that E(n)_{k,0}^{k+1} = E(n)_{k,0}^{∞}; in light of (c)_{f} (resp. (c)_{f}^{un}), the assertion follows already for every q ≥ k + 1. Next, let p ≤ k and suppose that the assertion is already known for q = p + 1. Notice the exact sequence of inverse systems :

0 → (E(n)_{k,0}^{p+1} | n ∈ ℕ) → (E(n)_{k,0}^{p} | n ∈ ℕ) → (E(n)_{k−p,p−1}^{p} | n ∈ ℕ).

By inductive assumption, both the first and third terms are essentially zero (resp. uniformly essentially zero), so the same holds for the middle term. We know now that (E(n)_{k,0}^{2} | n ∈ ℕ) is essentially zero (resp. uniformly essentially zero); from (7.8.30) and claim 7.8.29we deduce that the same holds for (E(n)_{k,j}^{2} | n ∈ ℕ), for every j ∈ ℕ, and finally the same follows for (E(n)_{k,j}^{p+2} | n ∈ ℕ), for every p, j ∈ ℕ, whence the claim. ♢

• Lastly, let us check that (a)_{f} ⇔ (d)_{f}. To this aim, set T_{•} := (T_{1}, …, T_{r}), and notice that H_{i}(T_{•}^{n}, A_{r}) = 0 for every i, n > 0 (proposition 7.8.7), whence a natural isomorphism

Tor_{i}^{A_{r}}(A_{r}/T_{•}^{n}A_{r}, A) →^{∼} H_{i}(K_{•}(T_{•}^{n}) ⊗_{A_{r}} A) →^{∼} H_{i}(f^{n}, A) for every i, n ∈ ℕ.

On the other hand, it is clear that for every n ∈ ℕ there exists m ∈ ℕ such that

I_{r}^{m} ⊂ T_{•}^{n}A_{r} ⊂ I_{r}^{n}

whence the contention.

(iv): Set B_{r} := A_{r} ⊗_{ℤ} A, and notice that β_{f} factors as a composition

A_{r} ⟶^{β′_{f}} B_{r} ⟶^{γ_{f}} A_{r}

where β′_{f} : A_{r} → B_{r} is the ring homomorphism such that β′_{f}(T_{i}) := 1 ⊗ f_{i} for i = 1, …, r, and γ_{f} is the map of ℤ-algebras induced by the pair (β_{f}, 1_{A}). Let us endow B_{r} with the A_{r}-module structure given by β′_{f} and A with the B_{r}-module structure given by γ_{f}; there follows, for every n ∈ ℕ a change of rings spectral sequence

E(n)_{i,j}^{2} := Tor_{i}^{B_{r}}(Tor_{j}^{A_{r}}(A_{r}/I_{r}^{n}, B_{r}), A) ⇒ Tor_{i+j}^{A_{r}}(A_{r}/I_{r}^{n}, A).

Claim 7.8.32. Tor_{j}^{A_{r}}(A_{r}/I_{r}^{n}, B_{r}) = 0 for every j > 0.

Proof of the claim. We have a change of rings spectral sequence

E_{p,q}^{2} := Tor_{p}^{A_{r}}(Tor_{q}^{ℤ}(A, A_{r}), A_{r}/I_{r}^{n}) ⇒ Tor_{p+q}^{ℤ}(A, A_{r}/I_{r}^{n})

and clearly E_{p,q}^{2} = 0 for every q > 0, so there follows a natural isomorphism

Tor_{p}^{A_{r}}(B_{r}, A_{r}/I_{r}^{n}) →^{∼} Tor_{p}^{ℤ}(A, A_{r}/I_{r}^{n}) for every p ∈ ℕ.

However, A_{r}/I_{r}^{n} is a free ℤ-module, whence the claim. ♢ In view of claim 7.8.32we have E_{i,j}^{2} = 0 for every j > 0, whence a natural isomorphism

Tor_{i}^{B_{r}}(B_{r}/I_{r}^{n}B_{r}, A) →^{∼} Tor_{i}^{A_{r}}(A_{r}/I_{r}^{n}, A) for every i ∈ ℕ.

Especially, (a)_{f} is equivalent to :

(g)_{f} The inverse system (Tor_{i}^{B_{r}}(B_{r}/I_{r}^{n}B_{r}, A) | n ∈ ℕ) is essentially zero for every i > 0 and (a)_{f}^{un} is equivalent to the corresponding condition (g)_{f}^{un}.

Now, set (f, g) := (f_{1}, …, f_{r}, g_{1}, …, g_{s}). In light of the foregoing we see that in order to prove the equivalence (a)_{f} ⇔ (a)_{g} it suffices to check that

(g)_{f} ⇔ (g)_{(f,g)}  and (g)_{g} ⇔ (g)_{(f,g)}

and likewise for the uniformly essentially zero variants, so we may assume from start that s > r and f_{i} = g_{i} for i = 1, …, r. In this case, an easy induction on s − r further reduces to the case where s = r + 1. Now, let us say that g_{s} = ∑_{i=1}^{r} a_{i}f_{i} for some a_{1}, …, a_{r} ∈ A, and consider the automorphism of A-algebras ω : B_{r+1} →^{∼} B_{r+1} such that ω(T_{i}) := T_{i} for i = 1, …, r and ω(T_{r+1}) := T_{r+1} − ∑_{i=1}^{r} a_{i}T_{i}. Clearly ω(I_{r+1}^{n}B_{r+1}) = I_{r+1}^{n}B_{r+1} for every n ∈ ℕ, and γ_{g} ∘ ω(g) = (f, 0) := (f_{1}, …, f_{r}, 0). It follows easily that (g)_{g} ⇔ (g)_{(f,0)} and (g)_{g}^{un} ⇔ (g)_{(f,0)}^{un}, so we may further assume that g = (f, 0).

Let π : B_{r+1} → B_{r} be the map of A-algebras such that π(T_{i}) := T_{i} for i = 1, …, r and π(T_{r+1}) := 0, and notice that γ_{(f,0)} = γ_{f} ∘ π; if we endow B_{r} with the B_{r+1}-module structure given by π, there follows, for every n ∈ ℕ, a change of rings spectral sequence

E(n)_{i,j}^{2} := Tor_{i}^{B_{r}}(Tor_{j}^{B_{r+1}}(B_{r+1}/I_{r+1}^{n}B_{r+1}, B_{r}), A) ⇒ Tor_{i+j}^{B_{r+1}}(B_{r+1}/I_{r+1}^{n}B_{r+1}, A)

as well as a system of morphisms of spectral sequences as in (7.8.26).

Claim 7.8.33. (i) Tor_{j}^{B_{r+1}}(B_{r+1}/I_{r+1}^{n+1}B_{r+1}, B_{r}) = 0 for every j > 1.

(ii) The projection B_{r+1}/I_{r+1}^{n+1} → B_{r+1}/I_{r+1}^{n} induces the zero map

Tor_{1}^{B_{r+1}}(B_{r+1}/I_{r+1}^{n+1}B_{r+1}, B_{r}) → Tor_{1}^{B_{r+1}}(B_{r+1}/I_{r+1}^{n}B_{r+1}, B_{r}) for every n ∈ ℕ.

Proof of the claim. The B_{r+1}-module B_{r} admits the free resolution B_{r+1} → B_{r+1} ⟶^{π} B_{r} given by scalar multiplication by T_{r+1}. Assertion (i) follows immediately, and we also deduce that

Tor_{1}^{B_{r+1}}(B_{r+1}/I_{r+1}^{n+1}B_{r+1}, B_{r}) →^{∼} Ker(B_{r+1}/I_{r+1}^{n+1}B_{r+1} ⟶^{T_{r+1}} B_{r+1}/I_{r+1}^{n+1}B_{r+1}) ⊂ I_{r+1}^{n}/I_{r+1}^{n+1}.

Moreover, the transition maps in (ii) are induced by the inclusions I_{r+1}^{n} ⊂ I_{r+1}^{n−1} for every n > 0, whence (ii). ♢

From claim 7.8.33 we see that E(n)_{i,j}^{2} = 0 for every j > 1, and (E(n)_{i,1}^{2} | n ∈ ℕ) is a uniformly essentially zero inverse system. Hence, E(n)_{i,j}^{∞} is uniformly essentially zero for every j > 0, and E(n)_{i,0}^{∞} = E(n)_{i,0}^{2} for every n, i ∈ ℕ. Taking into account lemma 7.8.19 we conclude that (g)_{(f,0)} (resp. (g)_{(f,0)}^{un}) holds if and only if (E(n)_{i,0}^{2} | n ∈ ℕ) is essentially zero (resp. uniformly essentially zero) for every i > 0. But E(n)_{i,0}^{2} = Tor_{i}^{B_{r}}(B_{r}/I_{r}^{n}B_{r}, A), so the latter condition is precisely (g)_{f} (resp. (g)_{f}^{un}).

(iii): By assumption, there exists an integer k > 0 such that f_{i}^{k} ∈ J and g_{j}^{k} ∈ I for every i = 1, …, r and every j = 1, …, s. By the foregoing proof of (iv), we deduce that

(a)_{f} ⇔ (a)_{(f,g^{k})} and (a)_{g} ⇔ (a)_{(f^{k},g)}.

Hence, we are reduced to checking that (a)_{(f^{k},g)} ⇔ (a)_{(f,g)} ⇔ (a)_{(f,g^{k})}. We show the first equivalence; obviously the same argument will yield the second one. To this aim, for every integers n ≥ m ≥ 0 let φ_{•}^{(n,m)} : K_{•}(f^{n}, g^{n}) → K_{•}(f^{m}, g^{m}) and ψ_{•}^{(n,m)} : K_{•}(f^{n}, g^{kn}) → K_{•}(f^{m}, g^{km}) be the transition morphisms of the inverse systems (K_{•}(f^{n}, g^{n}) | n ∈ ℕ) and (K_{•}(f^{n}, g^{kn}) | n ∈ ℕ). Then, for every n, k ∈ ℕ we have morphisms

K_{•}(f^{kn}, g^{k^{2}n}) ⟶^{β_{•}} K_{•}(f^{kn}, g^{kn}) ⟶^{β′_{•}} K_{•}(f^{n}, g^{kn}) ⟶^{β′′_{•}} K_{•}(f^{n}, g^{n})

such that β′_{•} ∘ β_{•} = ψ_{•}^{(kn,n)} and β′′_{•} ∘ β′_{•} = φ_{•}^{(kn,n)} (see (7.8.20)), whence the contention. □

Remark 7.8.34. (i) By carefully tracking the estimates arising in the proof, it is possible to refine proposition 7.8.25(ii) as follows. For every r ∈ ℕ and every sequence of integers (c(i) | i ∈ ℕ) there exists a sequence of integers (d(i) | i ∈ ℕ), such that the following holds : for A and f of length r as in (7.8.21), suppose that the inverse system (Tor_{i+1}^{A_{r}}(A_{r}/I_{r}^{n}, A) | n ∈ ℕ) is uniformly essentially zero with step ≤ c(i), for every i ∈ ℕ. Then the inverse system (H_{i+1}(f, I^{n}) | n ∈ ℕ) is uniformly essentially zero of step ≤ d(i), for every i ∈ ℕ.

(ii) A shorter alternative proof for the refinement of (i) can be proven as follows. Fix such a sequence c_{•}, and suppose that the assertion fails; then we may find a system of pairs ((R_{λ}, f_{λ}) | λ ∈ ℕ) consisting of a ring R_{λ} and a sequence f_{λ} of r elements of R_{λ}, for every λ, such that the following holds. For every λ ∈ ℕ, denote by β_{λ} : A_{r} → R_{λ} the ring homomorphism associated with the sequence f_{λ}, as in (7.8.21); then :

(a) the inverse system (Tor_{i+1}^{A_{r}}(A_{r}/I_{r}^{n}, R_{λ}) | n ∈ ℕ) is uniformly essentially zero with step ≤ c(i), for every i, λ ∈ ℕ

(b) the inverse system (H_{i+1}(f, I^{n}) | n ∈ ℕ) is uniformly essentially zero of step d(i, λ), for every i ∈ ℕ

(c) the sequence of integers (d(i_{0}, λ) | λ ∈ ℕ) is strictly increasing, for some i_{0} ∈ ℕ. Now, set J := ⊕_{λ∈ℕ}R_{λ}, B := A_{r} ⊕ J, and endow B with the unique ring structure such that (a, b) · (a′, b′) := (aa′, ba′ + a′b) for every a, a′ ∈ A_{r} and b, b′ ∈ J. Then J is an ideal of B with J^{2} = 0, the natural projection induces a ring isomorphism ω : B/J →^{∼} A_{r}, and the inclusion map β : A_{r} → B is a ring homomorphism; moreover, the A_{r}-module structure induced via ω on J agrees with the A_{r}-module structure induced by the system of maps (β_{λ} | λ ∈ ℕ). We consider the sequence t := (T_{1}, …, T_{r}) of A_{r}, and its image f := β(t) in B, and we notice the natural isomorphisms of A_{r}-modules

Tor_{i+1}^{A_{r}}(A_{r}/I_{r}^{n}, B) →^{∼} ⨁_{λ∈ℕ} Tor_{i+1}^{A_{r}}(A_{r}/I_{r}^{n}, A_{r}) for every i, n ∈ ℕ.

In light of condition (a), we deduce that B satisfies condition (a)_{f}^{un}, so it also satisfies (c)_{f}^{un}, by virtue of proposition 7.8.25(ii); on the other hand, clearly we have as well natural isomorphisms

H_{i}(f, I_{r}^{n}B) →^{∼} H_{i}(t, I_{r}^{n}) ⊕ ⨁_{λ∈ℕ} H_{i}(f_{λ}, I_{r}^{n}R_{λ}) for every i, n ∈ ℕ

which, in view of our assumption (c), imply that the inverse system (H_{i_{0}}(f, I_{r}^{n}B) | n ∈ ℕ) cannot be uniformly essentially zero, a contradiction.

7.8.35. Let f := (f_{1}, …, f_{r}) be a finite sequence of elements of a ring A, and set

I := f_{1}A + ⋯ + f_{r}A and J := ⋃_{n∈ℕ} Ann_{A}(I^{n}).

Denote by \overline{f} := (\overline{f}_{1}, …, \overline{f}_{r}) the image of f in \overline{A} := A/J.

Proposition 7.8.36. With the notation of (7.8.35), the following conditions are equivalent : (a) The ring A satisfies condition (a)_{f} (resp. (a)_{f}^{un}, resp. (c)_{f}^{un}) of (7.8.21).

(b) \overline{A} satisfies condition (a)\overline{f} (resp. (a)un \overline{f}, resp (c)un \overline{f}) and there exists k ∈ ℕ with I^{k}J = 0.

Proof. (a)⇒(b): by proposition 7.8.25(i) the ring A satisfies condition (c)_{f}, so there exists k ∈ ℕ such that the natural map H_{r}(f, I^{k}) → H_{r}(f, A) is the zero morphism. In other words

I^{k} ∩ Ann_{A}(I) = 0.

Therefore we have as well I^{k} ∩ J = 0, and it follows easily that J = Ann_{A}(I^{k}). If A satisfies (c)un \overline{f}, notice that for every n ≥ k the natural map I^{n} → I^{n}\overline{A} is then bijective, and thus the same holds for the induced map H_{i}(f, I^{n}) → H_{i}(f, I^{n}\overline{A}), for every i ∈ ℕ, so \overline{A} fulfills (c)un \overline{f}.

If A satisfies (a)\overline{f} (resp. (a)un \overline{f}), define A_{r} and β_{f} as in (7.8.21), and endow A and J with the A_{r}-module structure induced by β_{f}; from lemma 7.8.24 we deduce that the inverse system T (J, i)_{•} := (Tor_{i}^{A_{r}}(A_{r}/I_{r}^{n}, J) | n ∈ ℕ) is uniformly essentially zero for every i > 0. By assumption, the inverse system T (A, i) := (Tor_{i}^{A_{r}}(A_{r}/I_{r}^{n}, A) | n ∈ ℕ) is essentially zero (resp. uniformly essentially zero); moreover, the short exact sequence of A_{r}-modules 0 → J → A → \overline{A} → 0 yields exact sequences of inverse systems :

(7.8.37) T (J, i)_{•} → T (A, i)_{•} → T (\overline{A}, i)_{•} := (Tor_{i}^{A_{r}}(A_{r}/I_{r}^{n}, \overline{A}) | n ∈ ℕ) → T (J, i − 1)_{•}

for every i > 0. By virtue of lemma 7.8.19, it follows already that T (\overline{A}, i)_{•} is essentially zero (resp. uniformly essentially zero) for every i > 1. By the same token, the case i = 1 yields the exact sequence of inverse systems

T (A, 1)_{•} → T (\overline{A}, 1) → (J/I^{n}J | n ∈ ℕ) ⟶^{φ_{•}} (A/I^{n} | n ∈ ℕ).

But the foregoing shows that φ_{n} is injective for every n ≥ k, hence Ker φ_{n} is uniformly essentially zero, and finally T (\overline{A}, 1)_{•} is essentially zero (resp. uniformly essentially zero), again by lemma 7.8.19. Summing up, this shows that \overline{A} fulfills condition (a)\overline{f} (resp. (a)un \overline{f}), as stated.

(b)⇒(a): Suppose first that \overline{A} satisfies (a)_{f} (resp. (a)_{f}^{un}); by assumption, J is an A_{r}/I_{r}^{k}- module, so the inverse system T (J, i)_{•} is uniformly essentially zero for every i > 0, by lemma 7.8.24; the inverse system T (\overline{A}, i)_{•} is essentially zero (resp. uniformly essentially zero), and thus the same holds also for T (A, i)_{•}, in view of (7.8.37) and lemma 7.8.19. Lastly, if \overline{A} satisfies (c)un \overline{f}, then it also satisfies condition (a)_{f}, by proposition 7.8.25(i), hence A satisfies (a)_{f}, by the foregoing case; but as we have seen, this implies that the natural map H_{i}(f, I^{n}) → H_{i}(f, I^{n}\overline{A}) is bijective for every i ∈ ℕ and sufficiently large n, so A satisfies (c)un \overline{f}. □

7.8.38. Let A, I and f be as in (7.8.35), and for every n > 0 denote by I^{(n)} ⊂ A the ideal generated by f^{n} := (f_{i}^{n} | i = 1, …, r). For all m ≥ n > 0 we deduce natural commutative diagrams of complexes :

\xymatrix{K_{•}(f^{m}) \ar[r] \ar[d]_{_{φ_{f^{m−n}}}} & A/I^{(m)}[0] \ar[d]^{π_{mn}} \\ K_{•}(f^{n}) \ar[r] & A/I^{(n)}[0]}

(notation of (7.8.20)) where π_{mn} is the natural surjection, whence a compatible system of maps:

(7.8.39)  Hom_{D(A-Mod)}(A/I^{(n)}[0], M^{•}) → Hom_{D(A-Mod)}(K_{•}(f^{n}), M^{•}).

for every n ≥ 0 and every complex M^{•} in D^{+}(A-Mod). Since K_{•}(f^{n}) is a complex of free A-modules, (7.8.39) translates as a direct system of maps :

(7.8.40)  R^{i}Hom_{A}(A/I^{(n)}, M^{•}) → H^{i}(f^{n}, M^{•})  for all n ∈ ℕ and every i ∈ ℤ.

Notice that I^{nr−r+1} ⊂ I^{(n)} ⊂ I^{n} for every n > 0, hence the colimit of the system (7.8.40) is equivalent to a natural map :

(7.8.41)  colim_{n∈ℕ} R^{i}Hom_{A}^{•}(A/I^{n}, M^{•}) → colim_{n∈ℕ} H^{i}(f^{n}, M^{•})  for every i ∈ ℤ.

Lemma 7.8.42. With the notation of (7.8.38), the following conditions are equivalent : (a) The map (7.8.41) is an isomorphism for every M^{•} ∈ Ob(D^{+}(A-Mod)) and all i ∈ ℤ.

(b) colim_{n∈ℕ} H^{i}(f^{n}, J) = 0 for every i > 0 and every injective A-module J.

(c) The ring A satisfies condition (d)_{f} of (7.8.21).

Proof. (a) ⇒ (b) is obvious. Next, if J is an injective A-module, we have natural isomorphisms

(7.8.43)  H^{i}(f^{n}, J) ≃ Hom_{A}(H_{i}K_{•}(f^{n}), J)  for all n ∈ ℕ.

which easily implies that (c) ⇒ (b).

(b) ⇒ (c) : Indeed, for any p ∈ ℕ let us choose an injection φ : H_{i}K_{•}(f^{n}) → J into an injective A-module J. By (7.8.43) we can regard φ as an element of H^{i}(f^{n}, J); by (b) the image of φ in Hom_{A}(H_{i}K_{•}(f^{q}), J) must vanish if q > p is large enough. This can happen only if H_{i}K_{•}(f^{q}) → H_{i}K_{•}(f^{p}) is the zero map.

(b) ⇒ (a) : Let M^{•} → J^{•} be an injective resolution of the complex M^{•}. The double complex colim_{n∈ℕ} Hom_{A}^{•}(K_{•}(f^{n}), J^{•}) determines two spectral sequences :

E_{1}^{pq} := colim_{n∈ℕ} Hom_{A}(K_{p}(f^{n}), H^{q}J^{•}) ⇒ colim_{n∈ℕ} R^{p+q}Hom_{A}^{•}(K_{•}(f^{n}), M^{•})
F_{1}^{pq} := colim_{n∈ℕ} H^{p}(f^{n}, J^{q}) ≃ colim_{n∈ℕ} Hom_{A}(H_{p}K_{•}(f^{n}), J^{q}) ⇒ colim_{n∈ℕ} R^{p+q}Hom_{A}^{•}(K_{•}(f^{n}), M^{•}).

Clearly E_{1}^{pq} = 0 whenever q > 0, and (b) says that F_{1}^{pq} = 0 for p > 0. Hence these two spectral sequences degenerate and we deduce natural isomorphisms :

colim_{n∈ℕ} R^{q}Hom_{A}^{•}(A/I^{(n)}, M^{•}) ≃ F_{2}^{0q} →^{∼} E_{2}^{q0} ≃ colim_{n∈ℕ} H^{q}(f^{n}, M^{•}).

By inspection, one sees easily that these isomorphisms are the same as the maps (7.8.41). □

Lemma 7.8.44. In the situation of (7.8.38), suppose that the following holds. For every finitely presented quotient B of A and every i = 1, …, r, there exists p ∈ ℕ such that

Ann_{B}(f_{i}^{q}) = Ann_{B}(f_{i}^{p}) for every q ≥ p.

Then the ring A satisfies condition (d)_{f} of (7.8.21).

Proof. We shall argue by induction on r. If r = 1, then f = (f) for a single element f ∈ A. In this case, our assumption ensures that there exists p ∈ ℕ such that Ann_{A}(f^{q}) = Ann_{A}(f^{p}) for every q ≥ p. It follows easily that H_{1}(φ_{f^{p}}) : H_{1}K_{•}(f^{p+k}) → H_{1}K_{•}(f^{k}) is the zero map for every k ≥ 0 (notation of (7.8.20)), whence the claim.

Next, suppose that r > 1 and that the claim is known for all sequences of less than r elements. Set g := (f_{1}, …, f_{r−1}) and f := f_{r}. Specializing (7.8.4) to our current situation, we derive short exact sequences :

0 → H_{0}(f^{n}, H_{p}K_{•}(g^{n})) → H_{p}K_{•}(f^{n}) → H^{0}(f^{n}, H_{p−1}K_{•}(g^{n})) → 0

for every p > 0 and n ≥ 0; for a fixed p, this is an inverse system of exact sequences. By induction, the inverse system (H_{i}K_{•}(g^{n}) | n ∈ ℕ) is essentially zero for i > 0; in light of lemma 7.8.19, we deduce already that the inverse system (H_{i}K_{•}(f^{n}) | n ∈ ℕ) is essentially zero for all i > 1. To conclude, we are thus reduced to showing that the inverse system (T_{n} := H^{0}(f^{n}, H_{0}K_{•}(g^{n})) | n ∈ ℕ) is essentially zero. However A_{n} := H_{0}K_{•}(g^{n}) = A/(g_{1}^{n}, …, g_{r−1}^{n}) is a finitely presented quotient of A for any fixed n ∈ ℕ, hence the foregoing case r = 1 shows that the inverse system (T_{mn} := Ann_{A_{n}}(f^{m}) | m ∈ ℕ) is essentially zero. Let m ≥ n be chosen so that T_{mn} → T_{nn} is the zero map; then the composition T_{m} = T_{mm} → T_{mn} → T_{nn} = T_{n} is zero as well. □

Remark 7.8.45. Notice that the condition of lemma 7.8.44 holds when A is noetherian. Combining with lemma 7.8.42, we conclude that if A is noetherian, (7.8.41) is an isomorphism, for every finite system f of elements of A, and every i ∈ ℕ. See also corollary 7.9.22(iii).

7.9. Filtered rings and Rees algebras. Some of the following material is borrowed from [27, Appendix III], where much more can be found.

Definition 7.9.1. Let R be a ring, A an R-algebra.

(i) An R-algebra filtration on A is an increasing exhaustive filtration Fil_{•}A indexed by ℤ and consisting of R-submodules of A, such that :

1 ∈ Fil_{0}A  and Fil_{i}A · Fil_{j}A ⊂ Fil_{i+j}A  for every i, j ∈ ℤ.

The pair \underline{A} := (A, Fil_{•}A) is called a filtered R-algebra. (ii) Let M be an A-module. An \underline{A}-filtration on M is an increasing exhaustive filtration

Fil_{•}M consisting of R-submodules, and such that :

Fil_{i}A · Fil_{j}M ⊂ Fil_{i+j}M for every i, j ∈ ℤ.

The pair \underline{M} := (M, Fil_{•}M) is called a filtered \underline{A}-module. (iii) Let U be an indeterminate. The Rees algebra of \underline{A} is the ℤ-graded subring of A[U, U^{−1}]

R(\underline{A})_{•} := ⨁_{i∈ℤ} U^{i} · Fil_{i}A.

We occasionally consider also the two ℕ-graded subrings of R(\underline{A})_{•} denoted :

R^{−}(\underline{A})_{•} := A[U] ∩ R(\underline{A})_{•} and R^{+}(\underline{A})_{•} := A[U^{−1}] ∩ R(\underline{A})_{•}

(iv) Let \underline{M} := (M, Fil_{•}M) be a filtered \underline{A}-module. The Rees module of \underline{M} is the graded

R(\underline{A})_{•}-module :

R(\underline{M})_{•} := ⨁_{i∈ℤ} U^{i} · Fil_{i}M.

We have similarly the R^{−}(\underline{A})-submodule R^{−}(\underline{M})_{•} := ⨁_{i∈ℕ} U^{i} ·Fil_{i}M and the R^{+}(\underline{A})-

submodule R^{+}(\underline{M})_{•} := ⨁_{i∈ℕ} U^{−i} · Fil_{−i}M

Lemma 7.9.2. Let R be a ring, \underline{A} := (A, Fil_{•}A) a filtered R-algebra, gr_{•}\underline{A} the associated graded R-algebra, R(\underline{A})_{•} ⊂ A[U, U^{−1}] the Rees algebra of \underline{A}. Then there are natural isomorphisms of graded R-algebras :

R(\underline{A})_{•}/UR(\underline{A})_{•} ≃ gr_{•}\underline{A} R(\underline{A})_{•}[U^{−1}] ≃ A[U, U^{−1}]

and of R-algebras :

R(\underline{A})_{•}/(1 − U)R(\underline{A})_{•} ≃ A.

Proof. The isomorphisms with gr_{•}\underline{A} and with A[U, U^{−1}] follow directly from the definitions. For the third isomorphism, it suffices to remark that A[U, U^{−1}]/(1 − U) ≃ A. □

Example 7.9.3. Let A be any ring, I ⊂ A any ideal, and endow A with its I-adic filtration Fil_{•}A, so that Fil_{−i}A := I^{i} for every i ∈ ℕ and Fil_{i}A = A for every i > 0. The pair \underline{A} := (A, Fil_{•}) is obviously a filtered A-algebra, and we denote by R(A, I)_{•} its Rees algebra.

Definition 7.9.4. Let R be a ring, \underline{A} := (A, Fil_{•}A) a filtered R-algebra.

(i) Suppose that A is of finite type over R, let x := (x_{1}, …, x_{n}) be a finite set of generators for A as an R-algebra, and k := (k_{1}, …, k_{n}) a sequence of n integers; the good filtration Fil_{•}A attached to the pair (x, k) is the R-algebra filtration such that Fil_{i}A is the R-submodule generated by all the elements of the form

∏_{j=1}^{n} x_{j}^{a_{j}}  where :  ∑_{j=1}^{n} a_{j}k_{j} ≤ i and a_{1}, …, a_{n} ≥ 0

for every i ∈ ℤ. A filtration Fil_{•}A on A is said to be good if it is the good filtration attached to some system of generators x and some sequence of integers k.

(ii) The filtration Fil_{•}A is said to be positive if it is the good filtration associated with a

pair (x, k) as in (i), such that moreover k_{i} > 0 for every i = 1, …, n. (iii) Let M be a finitely generated A-module. An \underline{A}-filtration Fil_{•}M is called a good filtra-

tion if R(M, Fil_{•}M)_{•} is a finitely generated R(\underline{A})_{•}-module.

Example 7.9.5. Let A := R[t_{1}, …, t_{n}] be the free R-algebra in n indeterminates. Choose any sequence k := (k_{1}, …, k_{n}) of integers, and denote by Fil_{•}A the good filtration associated with t := (t_{1}, …, t_{n}) and k. Then R(A, Fil_{•}A)_{•} is isomorphic, as a graded R-algebra, to the free polynomial algebra A[U] = R[U, t_{1}, …, t_{n}], endowed with the grading such that U ∈ gr_{1}A[U] and t_{j} ∈ gr_{k_{j}}A[U] for every j ≤ n. Indeed, a graded isomorphism can be defined by the rule : U ↦ U and t_{j} ↦ U^{k_{j}} · t_{j} for every j ≤ n. The easy verification shall be left to the reader.

7.9.6. Let \underline{A} and M be as in definition 7.9.4(iii); suppose that m := (m_{1}, …, m_{n}) is a finite system of generators of M, and let k := (k_{1}, …, k_{n}) be an arbitrary sequence of n integers. To the pair (m, k) we attach a filtered \underline{A}-module \underline{M} := (M, Fil_{•}M), by declaring that :

(7.9.7)  Fil_{i}M := m_{1} · Fil_{i−k_{1}}A + ⋯ + m_{n} · Fil_{i−k_{n}}A  for every i ∈ ℤ.

Notice that the homogeneous elements m_{1} ·U^{k_{1}}, …, m_{n} ·U^{k_{n}} generate the graded Rees module R(\underline{M})_{•}, hence Fil_{•}M is a good \underline{A}-filtration. Conversely :

Lemma 7.9.8. For every good filtration Fil_{•}M on M, there exist a sequence m of generators of M and a sequence of integers k, such that Fil_{•}M is of the form (7.9.7).

Proof. Suppose that Fil_{•}M is a good filtration; then R(\underline{M})_{•} is generated by finitely many homogeneous elements m_{1} · U^{k_{1}}, …, m_{n} · U^{k_{n}}. Thus,

R(\underline{M})_{i} := U^{i} · Fil_{i}M = m_{1} · U^{k_{1}} · A_{i−k_{1}} + ⋯ + m_{n} · U^{k_{n}} · A_{i−k_{n}} for every i ∈ ℤ

which means that the sequences m := (m_{1}, …, m_{n}) and k := (k_{1}, …, k_{n}) will do. □

7.9.9. Let A → B and A → C be maps of noetherian rings, and suppose that either B or C is an A-algebra of finite type; let also M be a finitely generated B-module, N a finitely generated C-module, and I ⊂ B any ideal. Then the A-modules

T_{i} := Tor_{i}^{A}(M, N)

inherit natural B ⊗_{A} C-module structures, and we remark :

Lemma 7.9.10. In the situation of (7.9.9), the B ⊗_{A} C-module T_{i} is finitely generated for every i ∈ ℕ.

Proof. Say that B is an A-algebra of finite type; the same argument will apply also to the case where C is an A-algebra of finite type. Then, we have B = D/J for some free polynomial A-algebra D of finite type, and some ideal J ⊂ D. Hence, M is also a D-module of finite type; then D is a noetherian ring, so we may find a resolution L_{•} → M consisting of free Dmodules of finite type, and since D is a free A-module, we get natural A-linear isomorphisms ω_{i} : T_{i} →^{∼} H_{i}(L_{•} ⊗_{A} N) for every i ∈ ℕ. Moreover, L_{i} ⊗_{A} N is naturally a complex of D ⊗_{A} Cmodules of finite type, so T_{i} inherits via ω_{i} a structure of D ⊗_{A} C-module of finite type. Lastly, a simple inspection shows that the latter D ⊗_{A} C-module structure on T_{i} agrees with the one induced from the natural B ⊗_{A} C-module structure, via the surjection D ⊗_{A} C → B ⊗_{A} C, whence the contention. □

We endow B (resp. B ⊗_{A} C) with its I-adic (resp. I · (B ⊗_{A} C)-adic) filtration as in example 7.9.3, and denote by \underline{B} (resp. \underline{B} ⊗_{A} C) the resulting filtered ring. Also, set Fil_{n}M := Fil_{n}B ·M for every n ∈ ℤ; we define a \underline{B} ⊗_{A} C-filtration on T_{i}, by the rule :

Fil_{n}T_{i} := Im (Tor_{i}^{A}(Fil_{n}M, N) → T_{i}) for every i ∈ ℕ and n ∈ ℤ.

Proposition 7.9.11. The \underline{B} ⊗_{A} C-filtration Fil_{•}T_{i} is good, for every i ∈ ℕ.

Proof. Pick a finite set of generators f_{1}, …, f_{r} for I, and consider the surjective map of graded B-algebras

(7.9.12) B[U, t_{1}, …, t_{r}] → R(\underline{B})_{•} : U ↦ U ∈ R(\underline{B})_{1} t_{i} ↦ f_{i}  for i = 1, …, r

(for the grading of B[U, t_{1}, …, t_{r}] that places the indeterminates t_{1}, …, t_{r} in degree −1, and U in degree 1). The B ⊗_{A} C-module

(7.9.13)  P_{i,•} := ⨁_{n∈ℤ} Tor_{i}^{A}(Fil_{n}M, N)

carries a natural structure of graded R(\underline{B})_{•}⊗_{A}C-module, whence a graded B⊗_{A}C[U, t_{1}, …, t_{r}]- module structure as well, via (7.9.12), and it suffices to show :

Claim 7.9.14. P_{i,•} is a finitely generated B ⊗_{A} C[U, t_{1}, …, t_{r}]-module, for every i ∈ ℕ.

Proof of the claim. Notice that R(M, Fil_{•}M)_{•} is a finitely generated R(\underline{B})_{•}-module; a fortiori, it is a finitely generated graded B[U, t_{1}, …, t_{r}]-module. By remark 7.6.18(iv), we may then find a resolution

⋯ → L_{n} ⟶^{d_{n}} L_{n−1} → ⋯ → L_{0} ⟶^{d_{0}} R(M, Fil_{•}M)_{•}

where each L_{n} is a free B[U, t_{1}, …, t_{r}]-module of finite rank, each d_{n} is a morphism of graded B[U, t_{1}, …, t_{r}]-modules, and the restriction of the resolution to the summands of degree i is a flat resolution of the B-module I^{i}M, for every i ∈ ℕ. There follows a natural isomorphism of graded B ⊗_{A} C-modules

(7.9.15)  P_{i,•} →^{∼} H_{i}(L_{•} ⊗_{A} N)  for every i ∈ ℕ

and a simple inspection shows that the B ⊗_{A} C[U, t_{1}, …, t_{r}]-module structure on P_{i,•} deduced via (7.9.15) agrees with the foregoing one, so the assertion follows. □

7.9.16. In the situation of (7.9.9), set B_{n} := B/I^{n} for every n ∈ ℕ. We deduce, for every i ∈ ℕ, a morphism of projective systems of B ⊗_{A} C-modules

X_{•}^{i} := (B_{n} ⊗_{B} Tor_{i}^{A}(M, N) | n ∈ ℕ) ⟶^{φ_{i,•}} Y_{•}^{i} := (Tor_{i}^{A}(B_{n} ⊗_{B} M, N) | n ∈ ℕ)

where the transition maps of X_{•}^{i} and Y_{•}^{i} are induced by the projections B_{n+1} → B_{n}, for every n ∈ ℕ, and the morphisms φ_{i,n} are induced by the projections M → B_{n} ⊗_{B} M.

Corollary 7.9.17. With the notation of (7.9.16), we have :

(i) The morphism φ_{i,•} is an isomorphism of pro-B ⊗_{A} C-modules, for every i ∈ ℕ.

(ii) The morphism φ_{i,•} induces an isomorphism of B ⊗_{A} C-modules :

lim_{n∈ℕ} B_{n} ⊗_{B} Tor_{i}^{A}(M, N) →^{∼} lim_{n∈ℕ} Tor_{i}^{A}(B_{n} ⊗_{B} M, N) for every i ∈ ℕ.

Proof. (i): The assertion means that the systems (Ker φ_{i,n} | n ∈ ℕ) and (Coker φ_{i,n} | n ∈ ℕ) are essentially zero, for every i ∈ ℕ. We shall prove the more precise :

Claim 7.9.18. For every i ∈ ℕ the systems (Ker φ_{i,n} | n ∈ ℕ) and (Coker φ_{i,n} | n ∈ ℕ) are uniformly essentially zero (see definition 7.8.18(ii)).

Proof of the claim. The long exact Tor_{•}^{A}(−, N)-sequence arising from the short exact sequence

0 → I^{n}M → M → M/I^{n}M → 0

yields a natural identification

Coker φ_{i,n} = Ker (U^{n} : P_{i−1,−n} → P_{i−1,0})

where P_{i−1,•} is defined as in (7.9.13), and U^{n} denotes the scalar multiplication by the same element of R(\underline{B})_{•}, for the natural R(\underline{B})_{•}-module structure of P_{i−1,•}. Under this identification, the system (Coker φ_{i,n} | n ∈ ℕ) becomes a direct summand of the system of B ⊗_{A} C-modules

(7.9.19)  (Ker (U^{n} : P_{i−1,•} → P_{i−1,•}) | n ∈ ℕ)

whose transition maps are given by multiplication by U. By the same token, the inverse system (Ker φ_{i,n} | n ∈ ℕ) is naturally identified with the inverse system (Fil_{−n}T_{i}/I^{n}T_{i} | n ∈ ℕ), for every i ∈ ℕ. Now, it follows easily from proposition 7.9.11 and lemma 7.9.8 that, for every i ∈ ℕ, there exists c ∈ ℕ such that

Fil_{−n−c}T_{i} ⊂ I^{n}T_{i} for every n ∈ ℕ

whence the sought vanishing for the maps between kernels. Lastly, since B ⊗_{A} C[U, t_{1}, …, t_{r}] is noetherian, claim 7.9.14 implies that there exists c ∈ ℕ such that Ker U^{c+n} = Ker U^{c} for every n ∈ ℕ. Therefore the transition map Ker U^{c+n} → Ker U^{n} of (7.9.19) vanishes for every n ∈ ℕ, and the proof of the claim is complete. ♢

(ii) is a standard consequence of (i) : see [163, Prop.3.5.7]. □

7.9.20. Let f := (f_{1}, …, f_{r}) be a finite sequence of elements of a ring A, and I ⊂ A the ideal generated by f. Denote by Fil_{•}A the I-adic filtration of A, and set B_{•} := R^{+}(A, Fil_{•}A) (notation of definition 7.9.1(iii)); let moreover f_{0} be the image in B_{0} of the sequence f. The inclusion map I = B_{1} ⊂ B_{•} extends uniquely to a map of A-algebras

(7.9.21)  Sym_{A}^{•}(I) → B_{•}.

Part (iv) of the following corollary is included in [37, §9, Th.1]; an earlier reference is [128, Ch.I, Th.1] where the same result is proved under the assumption that f is a regular sequence.

Corollary 7.9.22. In the situation of (7.9.20), the following holds :

(i) B_{•} satisfies condition (c)_{f_{0}} if and only if B_{•} satisfies condition (c)_{f_{0}}^{un}, if and only if A satisfies condition (c)_{f}^{un} (notation of (7.8.21)).

(ii) B_{•} satisfies condition (a)_{f_{0}}^{un} if and only if A satisfies condition (a)_{f}^{un}.

(iii) If A is noetherian, then A satisfies condition (a)_{f}^{un}.

(iv) If H_{1}(f, A) = 0, then (7.9.21) is an isomorphism.

Proof. (iii): It suffices to apply claim 7.9.18 with A (resp. I, resp. B, resp. C) replaced by A_{r} (resp. by I_{r}, resp. by A_{r}, resp. by A).

(i): Indeed, if condition (c)_{f_{0}} holds, for every i ∈ ℕ there exists k ∈ ℕ such that the natural map H_{i}(f_{0}, I^{k}B_{•}) → H_{i}(f_{0}, B) is the zero map. However, clearly the latter is the direct sum of the natural maps

H_{i}(f, I^{n+k}) → H_{i}(f, I^{n}) for every n ∈ ℕ

whence (c)_{f}^{un}. By the same token, we immediately see that (c)_{f}^{un} ⇒ (c)_{f_{0}}^{un}.

(ii): Define A_{r}, I_{r}, β_{f} : A_{r} → A and β_{f_{0}} : A_{r} → B_{•} as in (7.8.21), endow A_{r} with its I_{r}-adic filtration Fil_{•}A_{r}, and set

B_{r,•} := R(A_{r}, Fil_{•}A_{r}) C_{•} := B_{r,•} ⊗_{A_{r}} A.

Let also t be the sequence (T_{1}, …, T_{r}) of homogenous elements of degree zero in B_{r,•}, and g the image of t under the natural map of graded rings B_{r,•} → C_{•}; since the ring B_{r,•} is noetherian, (iii) tells us that it satisfies condition (a)_{t}^{un}, which means that we have uniformly essentially zero inverse systems

K_{r,•} := (Ker (I_{r}^{n} ⊗_{A_{r}} B_{r,•} → I_{r}^{n}B_{r,•}) | n ∈ ℕ) and T_{r,•}^{i} := (Tor_{i}^{A_{r}}(I_{r}^{n}, B_{r,•}) | n ∈ ℕ)

for every i > 0. Suppose now that (a)_{f}^{un} holds; this means that also the inverse systems

K_{•} := (Ker (I_{r}^{n} ⊗_{A_{r}} A → I^{n}) | n ∈ ℕ) and T_{•}^{i} := (Tor_{i}^{A_{r}}(I_{r}^{n}, A) | n ∈ ℕ) for every i > 0

are uniformly essentially zero; in this case, we notice :

Claim 7.9.23. For every i > 0 there exists c ∈ ℕ such that I^{c} · T_{n}^{i} = 0 for every n ∈ ℕ.

Proof of the claim. By assumption, for every i ∈ ℕ there exists c ∈ ℕ such that the transition map φ_{n+c,n} : T_{n+c}^{i} → T_{n}^{i} is the zero morphism for every n ∈ ℕ. Now, if a ∈ I^{c}, the scalar multiplication by a on I^{n} factors as the composition of an A-linear map I^{n} → I^{n+c} and the inclusion map I^{n+c} → I^{n}. It follows that scalar multiplication by a on T_{n}^{i} factors as the composition of an A-linear map T_{n}^{i} → T_{n+c}^{i} and φ_{n+c,n}, whence the claim. ♢

Claim 7.9.24. If C_{•} satisfies condition (a)_{g}^{un}, then B_{•} satisfies condition (a)_{f_{0}}^{un}.

Proof of the claim. We have a natural surjective homomorphism f_{•} : C_{•} → B_{•} of graded rings, such that Ker f_{•} = ⊕_{n∈ℕ}K_{n}, and arguing as in the proof of claim 7.9.23, we see that there exists c ∈ ℕ such that I^{c} · K_{n} = 0 for every n ∈ ℕ. Set

J_{C} := ⋃_{n∈ℕ} Ann_{C_{•}}(I^{n}) J_{B} := ⋃_{n∈ℕ} Ann_{B_{•}}(I^{n})

and \overline{C} := C_{•}/J_{C}; it follows that the projection C_{•} → \overline{C}_{•} factors through a surjective ring homomorphism B_{•} → \overline{C} whose kernel equals J_{B}, so that f_{•}(J_{C}) = J_{B}. Suppose now that C_{•} satisfies condition (a)_{g}^{un}, and let \overline{g} be the image of g in \overline{C}; then there exists k ∈ ℕ such that I^{k}J_{C} = 0, and \overline{C} satisfies condition (a)un \overline{g} (proposition 7.8.36). Hence I^{k}J_{B} = 0, and the assertion follows, again by proposition 7.8.36. ♢

In view of claim 7.9.24, we are reduced to checking that for every i > 0 the inverse systems

K′_{•} := (Ker (I_{r}^{n} ⊗_{A_{r}} C_{•} → I_{r}^{n}C_{•}) | n ∈ ℕ) and T_{•}^{′i} := (Tor_{i}^{A_{r}}(I_{r}^{n}, C_{•}) | n ∈ ℕ)

are uniformly essentially zero. However, notice that the natural map

K_{r,•} ⊗_{A_{r}} A → K′_{•}

is an epimorphism of inverse systems, so we get already the assertion for K′_{•}. Next, we remark:

Claim 7.9.25. The inverse system (H_{i}(A ⊗^{L}_{A_{r}} B_{r,•} ⊗^{L}_{A_{r}} I_{r}^{n}) | n ∈ ℕ) is uniformly essentially zero for every i > 0.

Proof of the claim. We have a spectral sequence :

E(n)_{pq}^{2} := Tor_{p}^{A_{r}}(A, Tor_{q}^{A_{r}}(B_{r,•}, I_{r}^{n})) ⇒ H_{p+q}(A ⊗^{L}_{A_{r}} B_{r,•} ⊗^{L}_{A_{r}} I_{r}^{n}) for every n ∈ ℕ

hence it suffices to check that the inverse system (E(n)_{pq}^{2} | n ∈ ℕ) is uniformly essentially zero for every p, q ∈ ℕ such that p + q > 0. The latter assertion is already known for every q > 0, since the system T_{r,•}^{q} is uniformly essentially zero. For q = 0, we consider the short exact sequence of inverse systems

0 → K_{r,•} → (B_{r,•} ⊗_{A_{r}} I_{r}^{n} | n ∈ ℕ) → (I_{r}^{n}B_{r,•} | n ∈ ℕ) → 0

whence, for every p ∈ ℕ an exact sequence of inverse systems

X_{•}^{p} := Tor_{p}^{A_{r}}(A, K_{r,n} | n ∈ ℕ) → (E(n)_{2}^{p0} | n ∈ ℕ) → Y_{•}^{p} := (Tor_{p}^{A_{r}}(A, I_{r}^{n}B_{r,•}) | n ∈ ℕ).

Now, the system X_{•}^{p} is uniformly essentially zero for every p ∈ ℕ, since the same holds for K_{r,•}; moreover, Y_{•}^{p} is uniformly essentially zero for p > 0, since the same holds for T_{•}^{p}. Hence, (E(n)_{2}^{p0} | n ∈ ℕ) is uniformly essentially zero for p > 0, and the proof is concluded. ♢

Lastly, for every n ∈ ℕ let us consider the spectral sequence

E(n)_{pq}^{2} := Tor_{p}^{A_{r}}(Tor_{q}^{A_{r}}(A, B_{r,•}), I_{r}^{n}) ⇒ H_{p+q}(A ⊗^{L}_{A_{r}} B_{r,•} ⊗^{L}_{A_{r}} I_{r}^{n})

and notice that (E(n)_{p0}^{2} | n ∈ ℕ) = T_{•}^{′p} for every p ∈ ℕ. To conclude, it suffices therefore to show :

Claim 7.9.26. The inverse system (E(n)_{p0}^{k} | n ∈ ℕ) is uniformly essentially zero, for every k ≥ 2 and every p > 0.

Proof of the claim. We fix p > 0 and argue by descending induction on k. Notice that E(n)_{p0}^{p+1} = E(n)_{p0}^{∞} for every n ∈ ℕ, so claim 7.9.25 implies that the assertion holds whenever k ≥ p + 1. Next, let 2 ≤ i ≤ p and suppose that the assertion is known for every k > i; we have an exact sequence of inverse systems

(E(n)_{p0}^{i+1} | n ∈ ℕ) → (E(n)_{p0}^{i} | n ∈ ℕ) → (E(n)_{p−i,i−1}^{i} | n ∈ ℕ).

In view of our inductive assumption (and of lemma 7.8.19), we are therefore reduced to showing that the inverse system (E(n)_{p−i,i−1}^{i} | n ∈ ℕ) is uniformly essentially zero. However, by claim 7.9.23 there exists c ∈ ℕ such that I_{r}^{c} · Tor_{i−1}^{A_{r}}(A, B_{r,•}) = 0, and then the contention follows from lemma 7.8.24. ♢

Conversely, if (a)_{\overline{f}_{0}}^{un} holds for B_{•}, then we get immediately condition (a)un \overline{f} for A, since the latter is a direct summand of the A_{r}-module B_{•}.

(iv): We consider first the case where A = A_{r} and f = (T_{1}, …, T_{r}), so I = I_{r} (notation of (7.8.21)). We have natural A_{r}-linear surjections :

⊗_{A_{r}}^{n}I_{r} → Sym_{A_{r}}^{n}I_{r} ⟶^{β_{n}} I_{r}^{n} for every n ∈ ℕ

and we need to check that β_{n} is an isomorphism for every n ∈ ℕ. The assertion is clear for n = 0, 1. Let then n ≥ 1, and notice that ⊗_{A_{r}}^{n}I_{r}^{n} can be presented as the quotient L_{n}/R_{n} where:

• L_{n} is the free ℤ-module with basis ℬ_{n} given by all sequences (m_{1}, …, m_{n}) consisting of unital monomials m_{1}, …, m_{n} of total degree > 0 in the variables T_{1}, …, T_{r}

• R_{n} ⊂ L_{n} is the subgroup generated by all elements of the form :

(m_{1}, …, m_{i−1}, m′m_{i}, m_{i+1}, …, m_{n}) − (m_{1}, …, m_{j−1}, m′m_{j}, m_{j+1}, …, m_{n})

for every (m_{1}, …, m_{n}) ∈ ℬ_{n}, every m′ ∈ ℬ_{1}, and every 1 ≤ i < j ≤ n (details left to the reader). Likewise, Sym_{A_{r}}^{n}I_{r} can be presented as the quotient L_{n}/R′_{n}, where R′_{n} ⊂ L_{n} is the subgroup generated by R_{n} and the additional system of elements of the form :

(m_{1}, …, m_{n}) − (m_{σ(1)}, …, m_{σ(n)})

for every (m_{1}, …, m_{n}) ∈ ℬ_{n} and every permutation σ of {1, …, n}. Moreover, for every n ∈ ℕ and every λ ∈ {1, …, r} we have a ℤ-linear map

φ_{λ,n} : L_{n} → L_{n+1}  m_{•} := (m_{1}, …, m_{n}) ↦ (T_{λ}, m_{1}, …, m_{n})  for every m_{•} ∈ ℬ_{n}

and clearly φ_{λ,n}(R_{n}) ⊂ R_{n+1} and φ_{λ,n}(R′_{n}) ⊂ R′_{n+1} for every such n and λ. With this notation, we come down to checking that for every pair (m_{1}, …, m_{n}), (m′_{1}, …, m′_{n}) ∈ ℬ_{n} such that d(m_{•}) := ∏_{i=1}^{n} m_{n} = ∏_{i=1}^{n} m′_{n}, we have δ := (m_{1}, …, m_{n}) − (m′_{1}, …, m′_{n}) ∈ R′_{n}. We argue by induction on n. The case n = 1 is trivial. If n = d(m_{•}), then m_{i}, m′_{i} ∈ {T_{1}, …, T_{r}} for i = 1, …, n, so clearly there exists a permutation σ with m′_{i} = m_{σ(i)} for every i = 1, …, n. Let then d(m_{•}) > n ≥ 2, and suppose that the assertion is known for every p_{•}, p′_{•} ∈ ℬ_{n−1}. Pick λ ∈ {1, …, r} such that T_{λ} divides some m_{i} (and hence, some m′_{j}); it suffices to prove that there exists p_{•}, p′_{•} ∈ ℬ_{n−1} such that

m_{•} − φ_{λ,n−1}(p_{•}), m′_{•} − φ_{λ,n−1}(p′_{•}) ∈ R′_{n}  and  ∏_{i=1}^{n} m_{i} = T_{λ} · ∏_{i=2}^{n} p_{i} = T_{λ} · ∏_{i=2}^{n} p′_{i}.

Indeed, in that case, by inductive assumption we have p_{•} −p′_{•} ∈ R′_{n−1}, so that φ_{λ,n−1}(p_{•} −p′_{•}) ∈ R′_{n}, whence the assertion. It suffices to exhibit p_{•}, as the same argument will apply to give p′_{•}. Now, say that T_{λ} divides m_{i}; if m_{i} = T_{λ}, there exists j ∈ {1, …, n} such that m_{j} has degree > 1, so m_{j} = ab for two unital monomials a, b of degrees > 0. Then let q_{•} ∈ ℬ_{n} be the basis element such that q_{i} := am_{i}, q_{j} := b, and q_{l} := m_{l} for every l ≠ i, j. Hence m_{•} − q_{•} ∈ R_{n}, and after replacing m_{•} by q_{•} we may assume that the degree of m_{i} is > 1. In that case, m_{i} = cT_{λ} for a monomial c of degree > 0, and we define q′_{•} ∈ ℬ_{n} by setting q′_{1} := T_{λ}m_{1}, q′_{i} := c, and q′_{l} := m_{l} for every l ≠ i, 1. Again, m_{•} − q′_{•} ∈ R_{n}, so after replacing m_{•} by q′_{•} we may moreover assume that i = 1. Then, say that m_{1} = dT_{λ} for some monomial d of degree > 0; the sought p_{•} ∈ ℬ_{n−1} is obtained by setting p_{1} := m_{2}d and p_{i} := m_{i+1} for every i = 2, … n − 1.

Next, let A be a general ring; since H_{1}(f, A) = 0, we have Tor_{1}^{A_{r}}(A_{r}/I_{r}^{n}, A) = 0 for every n ∈ ℕ (lemma 7.8.23); i.e. the natural A ⊗_{A_{r}} I_{r}^{n} → I^{n} is bijective for every n ∈ ℕ. Composing with the natural isomorphism Sym_{A}^{•}(A ⊗_{A_{r}} I_{r}) →^{∼} A ⊗_{A_{r}} Sym_{A_{r}}^{•}(I_{r}), the assertion for A and f follows from that for A_{r} and (T_{1}, …, T_{r}). □

Remark 7.9.27. By carefully tracking the estimates in the proof, it is easily seen that for every sequence of integers (c(i) | i ∈ ℕ) there exists a sequence of integers (d(i) | i ∈ ℕ) such that the following refinement of corollary 7.9.22(ii) holds. Suppose that for every i ∈ ℕ the inverse system (Tor_{i}^{A_{r}}(A_{r}/I_{r}^{n}, A) | n ∈ ℕ) is uniformly essentially zero with step ≤ c(i); then the inverse system (Tor_{i}^{A_{r}}(A_{r}/I_{r}^{n}, B_{•}) | n ∈ ℕ) is uniformly essentially zero with step ≤ d(i).

7.9.28. Let (A, Fil_{•}A) be a filtered ring, i.e. a filtered ℤ-algebra, in the sense of definition 7.9.1(i), and denote by gr_{•}A the associated graded ring. Suppose that the filtration Fil_{•}A is exhaustive and separated, that is

⋃_{n∈ℤ} Fil_{n}A = A  and  ⋂_{n∈ℤ} Fil_{n}A = 0.

The filtration Fil_{•}A defines a linear topology 𝒯 on the ℤ-module A, and we suppose moreover that 𝒯 is complete (and notice that 𝒯 is obviously separated). Furthermore, let f := (f_{1}, …, f_{r}) be a finite sequence of elements of A; for every i = 1, …, r, pick n_{i} ∈ ℤ such that f_{i} ∈ Fil_{n_{i}}A, denote by \overline{f}_{i} ∈ gr_{n_{i}}A the image of f_{i}, and let \overline{f} := (\overline{f}_{1}, …, \overline{f}_{r}) be the resulting sequence of elements of gr_{•}A.

Proposition 7.9.29. In the situation of (7.9.28), suppose that the sequence \overline{f} is completely secant (see definition 7.8.6). Then the same holds for the sequence f.

Proof. We define as follows a filtration on the Koszul complex K_{•}(f) = K_{•}(f, A), i.e. a compatible system of filtrations on the exterior powers Λ_{A}^{i}(A^{⊕r}) for i = 1, …, r. To this aim, for every i ≤ r denote by S_{i} the set of all strictly increasing maps {1, …, i} → {1, …, r}; for every φ ∈ S_{i} set

e_{φ} := e_{φ(1)} ∧ ⋯ ∧ e_{φ(i)} and n_{φ} := n_{φ(1)} + ⋯ + n_{φ(i)}.

We let Fil_{•}Λ_{A}^{i}(A^{⊕r}) be the good filtration associated, as in (7.9.6), with the basis (e_{φ} | φ ∈ S_{i}) and the sequence of integers (n_{φ} | φ ∈ S_{i}). The explicit description in (7.8) shows that the differentials of the Koszul complex are morphisms of filtered abelian groups, for the resulting filtrations on the terms K_{i}(f). Thus, we have the sought filtration Fil_{•}K_{•}(f, A), and we let gr_{•}K_{•}(f, A) be the associated complex of graded abelian groups; with this notation, a simple inspection shows that we have a natural isomorphism of complexes of abelian groups

(7.9.30)  gr_{•}K_{•}(f, A) →^{∼} K_{•}(\overline{f}, gr_{•}A).

For every j, k ∈ ℤ with j ≥ k we set

K_{•}^{(j,k)}(f, A) := Fil_{j}K_{•}(f, A)/Fil_{k−1}K_{•}(f, A).

We endow K_{•}^{(j,k)}(f, A) with the filtration induced by Fil_{•}K_{•}(f, A), and we let gr_{•}K_{•}^{(j,k)}(f, A) be the complex of graded abelian groups associated with this filtered subquotient of K_{•}(f, A). According to remark 7.2.16(i), we have a 1-spectral sequence

E_{1}^{pq} := H^{p+q}(gr_{−p}K_{•}^{(j,k)}(f, A)) ⇒ H^{p+q}K_{•}^{(j,k)}(f, A).

However, (7.9.30) and the assumption on \overline{f} imply that E_{1}^{pq} = 0 whenever p + q < 0, and then proposition 7.2.18 implies that H^{n}(K_{•}^{(j,k)}(f, A)) = 0 for every j, k, n with j ≥ k and n < 0. Set as well K_{•}^{(k)}(f, A) := K_{•}(f, A)/Fil_{k−1}K_{•}(f, A) for every k ∈ ℤ; it follows easily that

H^{n}K_{•}^{(k)}(f, A) = 0 for every n < 0 and every k ∈ ℤ.

Next, since K_{n}(f, A) is a free A-module of finite rank for every n ∈ ℤ, and since the filtration Fil_{•}A is separated and defines a complete topology, we get a natural isomorphism of abelian groups

K_{n}(f, A) →^{∼} lim_{k∈ℤ} K_{n}^{(k)}(f, A) for every n ∈ ℤ.

Lastly, in view of [163, Prop.3.5.7, Th.3.5.8] we conclude that H^{n}K_{•}(f, A) = 0 for every n < 0, i.e. the sequence f is completely secant, as stated. □

7.10. Some homotopical algebra. The methods of homotopical algebra allow to construct derived functors of non-additive functors, in a variety of situations. In this section, we present the basics of this theory, beginning with its cornerstone, the standard resolution associated with a triple, as explained in the following paragraph.

7.10.1. A triple (⊤, η, μ) on a category 𝒞 is the datum of a functor ⊤ : 𝒞 → 𝒞 together with natural transformations :

η : 1_{𝒞} ⇒ ⊤ μ : ⊤ ∘ ⊤ ⇒ ⊤

such that the following diagrams commute :

[figure: (⊤ ∘ ⊤) ∘ ⊤; μ∗⊤; ⊤ ∘ (⊤ ∘ ⊤); ⊤∗μ; ⊤ ∘ ⊤; μ; ⊤; ⊤∗η; 1_{⊤}; ⊤ ∘ ⊤; μ; ⊤; η∗⊤; 1_{⊤}; ⊤ ∘ ⊤; μ; ⊤; ⊤.]

Dually, a cotriple (⊥, ε, δ) on a category 𝒞 is a functor ⊥: 𝒞 → 𝒞 together with natural transformations :

ε :⊥⇒ 1_{𝒞} δ :⊥⇒⊥ ∘ ⊥

such that the following diagrams commute :

[figure: ⊥; δ; δ; ⊥ ∘ ⊥; δ∗⊥; ⊥; 1_{⊥}; 1_{⊥}; δ; ⊥ ∘ ⊥; ⊥∗δ; ⊥ ∘(⊥ ∘ ⊥); (⊥ ∘ ⊥)∘ ⊥; ⊥; ⊥ ∘ ⊥; ⊥∗ε; ε∗⊥; ⊥ .]

7.10.2. A cotriple (⊥, ε, δ) on 𝒞 and an object A of 𝒞 yield a simplicial object ⊥A[•] in 𝒞; namely, for every n ∈ ℕ set ⊥ A[n] :=⊥^{n+1} A, and define face and degeneracy operators :

\begin{aligned}∂_{i} := (⊥^{i} ∗ε∗ ⊥^{n−i})_{A} & : ⊥ A[n] →⊥ A[n − 1] \\ σ_{i} := (⊥^{i} ∗δ∗ ⊥^{n−i})_{A} & : ⊥ A[n] →⊥ A[n + 1]. \end{aligned}

Using the foregoing commutative diagrams, one verifies easily that the simplicial identities (7.4.9) hold. Moreover, the morphism ε_{A} :⊥A → A defines an augmentation ⊥A[•] → A.

Dually, a triple (⊤, η, μ) and an object A of 𝒞 determine a cosimplicial object ⊤A[•] := ⊤^{•+1}A, such that

\begin{aligned}∂^{i} := (⊤^{i} ∗ η ∗ ⊤^{n−i})_{A} & : ⊤^{n}A → ⊤^{n+1}A \\ σ^{i} := (⊤^{i} ∗ μ ∗ ⊤^{n−i})_{A} & : ⊤^{n+2}A → ⊤^{n+1}A \end{aligned}

which is augmented by the morphism η_{A} : A → ⊤A. Clearly, the rule : A ↦⊥A[•] (resp. A ↦ ⊤A[•]) defines a functor

𝒞 → ŝ.𝒞 (resp. 𝒞 → ĉ.𝒞) .

7.10.3. An adjoint pair of functors (G : ℬ → 𝒜 , F : 𝒜 → ℬ) with its unit η and counit ε (see (1.1.13)), determines a triple (⊤, η, μ) on ℬ, where :

⊤ := F ∘ G : ℬ → ℬ μ := F ∗ ε ∗ G : ⊤ ∘ ⊤ ⇒ ⊤

as well as a cotriple (⊥, ε, δ) on 𝒜, where :

⊥:= G ∘ F : 𝒜 → 𝒜 δ := G ∗ η ∗ F : ⊥ ⇒ ⊥ ∘ ⊥ .

Indeed, the naturality of ε and η easily implies the commutativity of the diagrams of (7.10.1).

Remark 7.10.4. (i) Notice that a cotriple on 𝒞 is the same as a triple on 𝒞^{o} : more precisely, to every triple (⊤, η, μ) on 𝒞 there corresponds a dual cotriple on 𝒞^{o}, given by the datum

(⊤, η, μ)^{o} := (⊤^{o}, η^{o}, μ^{o}).

(ii) Also, clearly the dual of the cosimplicial set ⊤A[•] is the simplicial set in 𝒞^{o} attached to the dual cotriple (⊤, η, μ)^{o} and the object A (seen as an object of 𝒞^{o}).

(iii) Recall that every adjoint pair (G : ℬ → 𝒜 , F : 𝒜 → ℬ) yields by duality an adjoint pair (F^{o} : 𝒜^{o} → ℬ^{o}, G^{o} : ℬ^{o} → 𝒜^{o}), and if η (resp. ε) is the unit (resp. the counit) of a given adjunction for the pair (G, F), then ε^{o} (resp. η^{o}) is a unit (resp. a counit) of an adjunction for the dual pair (F^{o}, G^{o}) (see remark 1.1.19(iv)). Then, by direct inspection we easily see that the cotriple (⊥, ε, δ) attached to the adjoint pair (G, F) and to (η, ε) is the dual of the triple attached to the adjoint pair (F^{o}, G^{o}) and to (ε^{o}, η^{o}).

The following proposition explains how to use triples and cotriples to construct functorial resolutions.

Proposition 7.10.5. In the situation of (7.10.3), the following holds :

(i) For every A ∈ Ob(𝒜) and B ∈ Ob(ℬ), the augmented simplicial objects :

⊥GB[•] ⟶^{ε_{GB}} GB F ⊥A[•] ⟶^{F ∗ε_{A}} F A

are homotopically trivial (notation of (7.10.2)).

(ii) Dually, the same holds for the augmented cosimplicial objects :

GB ⟶^{G∗η_{B}} G⊤B[•] F A ⟶^{η_{F A}} ⊤F A[•].

Proof. Notice that

F ⊥A[n] = ⊤F A[n] for every [n] ∈ Ob(Δ^{∧}).

Therefore, for every morphism φ : [n] → [m] in Δ^{∧}, we have two morphisms

F ⊥A[φ] : F ⊥A[m] → F ⊥A[n] ⊤F A[φ] : F ⊥A[n] → F ⊥A[m].

Now, recall that the augmentation ε_{A} defines a natural morphism

⊥A[ε_{−1,0}^{•+1}] :⊥A[•] → s.A [n] ↦⊥A[ε_{−1,0}^{n+1}]

(notation of remark 7.4.12(iv)), and one sees that the system (⊤F A[ε_{−1,0}^{n+1}] | [n] ∈ Ob(Δ)) defines a morphism

⊤F A[ε_{−1,0}^{•+1}] : s.F A → F ⊥A[•]

which is right inverse to F ⊥A[ε_{−1,0}^{•+1}]. For every n, k ∈ ℕ such that k ≤ n + 1, set

u_{n,k} := (⊤F A[ε_{n−k,0}^{k}]) ∘ (F ⊥A[ε_{n−k,0}^{k}])

(notation of example 7.4.7(ii)).

Claim 7.10.6. The system u_{••} defines a homotopy from 1_{F⊥A[•]} to (⊤F A[ε_{−1,0}^{•+1}])∘(F ⊥A[ε_{−1,0}^{•+1}]) (see (7.4.16)).

Proof of the claim. By unwinding the definitions, we see that F ⊥A[ε_{n−k,0}^{k}] is the composition

F (ε_{⊥^{n+1−k}A} ∘ ⋯ ∘ ε_{⊥^{n−1}A} ∘ ε_{⊥^{n}A}) : F ⊥^{n+1}A → F ⊥^{n+1−k}A for k > 0

and ⊤F A[ε_{n−k,0}^{k}] is the composition

η_{F ⊥^{n}A} ∘ η_{F ⊥^{n−1}A} ∘ ⋯ ∘ η_{F ⊥^{n+1−k}A} : F ⊥^{n+1−k}A → F ⊥^{n+1}A for k > 0

and both equal 1_{F ⊥^{n+1}A} for k = 0. Now, since the face operator ∂_{i} of F ⊥A[n] is F ⊥^{i} (ε_{⊥^{n−i}A}) for every i ≤ n, the naturality of η yields a commutative diagram

\xymatrix{F ⊥^{n}A \ar[r]^{∂_{i−1}} \ar[d]_{_{η_{F⊥^{n}A}}} & F ⊥^{n−1}A \ar[d]^{_{η_{F⊥^{n−1}A}}} \\ F ⊥^{n+1}A \ar[r]^{∂_{i}} & F ⊥^{n}A}  for every i > 0 and every n ∈ ℕ

whereas ∂_{0} ∘ η_{F⊥^{n}A} = 1_{F⊥^{n}A} for every n ∈ ℕ; whence the identities :

∂_{i} ∘ ⊤F A[ε_{n−k,0}^{k}] = \{ \begin{matrix}⊤F A[ε_{n−k,0}^{k−1}] & for i < k \\ ⊤F A[ε_{n−k−1,0}^{k}] ∘ ∂_{i−k} & for i ≥ k. \end{matrix}

On the other hand, we have the simplicial identities :

ε_{i} ∘ ε_{n−k,0}^{k} = \{ \begin{matrix}ε_{n−k}^{k+1} & for i < k \\ ε_{n+1−k}^{k} ∘ ε_{i−k} & for i ≥ k. \end{matrix}

Summing up, we conclude that

∂_{i} ∘ u_{n,k} = \{ \begin{matrix}u_{n−1,k−1} ∘ ∂_{i} & for i < k \\ u_{n−1,k} ∘ ∂_{i} & for i ≥ k. \end{matrix}

Arguing likewise, one deduces as well the corresponding commutation rule – spelled out in (7.4.16) – for the degeneracies σ_{i}, and the claim follows. ♢

Likewise, notice that

G⊤B[n] =⊥GB[n] for every [n] ∈ Δ^{∧}.

Therefore, for every morphism φ : [n] → [m] in Δ^{∧}, we have as well two morphisms

⊥GB[φ] :⊥GB[m] →⊥GB[n] G⊤B[φ] :⊥GB[n] →⊥GB[m]

and the system (G⊤B[ε_{−1,0}^{n+1}] | [n] ∈ Ob(Δ^{∧})) defines a morphism

G⊤B[ε_{−1,0}^{•+1}] : s.GB →⊥GB[•]

which is right inverse to the morphism ⊥GB[ε_{−1,0}^{•+1}] deduced from the augmentation. Arguing as in the proof of claim 7.10.6, one checks that the rule

v_{n,k} := (G⊤B[ε_{n−k,0}^{k∨}]) ∘ (⊥GB[ε_{n−k,0}^{k∨}]) for every n, k ∈ ℕ such that k ≤ n + 1

yields a homotopy v from 1_{⊥GB[•]} to (G⊤B[ε_{−1,0}^{•+1}]) ∘ (⊥GB[ε_{−1,0}^{•+1}]).

The dual statements admit the dual proof. □

7.10.7. We explain first an application of the foregoing constructions to the construction of the derived functors of certain additive functors. Indeed, let now 𝒜 be an abelian category, and (⊥, ε, δ) a cotriple on 𝒜. In view of (7.10.2) and (7.4.25) we get a functor

△_{0} := U∘⊥^{•}: 𝒜 → C^{≤0}(𝒜)

that assigns to any object A of 𝒜 the unnormalized chain complex associated to the simplicial object ⊥ A[•]. Also, the augmentation ε_{A} :⊥ A[•] → A induces a natural transformation U ∗ ε : △_{0} ⇒ •[0] (notation of (7.1.3)). Then we can consider the composition :

△ : C^{−}(𝒜) ⟶^{C^{−}(△_{0})} C^{−}(C^{≤0}(𝒜)) ⟶^{Tot} C^{−}(𝒜)

and notice that Tot∘C^{−}(•[0]) is the identity automorphism of C^{−}(𝒜), so U∗ε induces a natural transformation :

ω_{K^{•}} : △K^{•} → K^{•} for every K^{•} ∈ C^{−}(𝒜) .

Proposition 7.10.8. Let 𝒜 , ℬ and 𝒜′ be three abelian categories, φ : 𝒜 → 𝒜′ an additive functor, and (G : ℬ → 𝒜 , F : 𝒜 → ℬ) an adjoint pair of functors. Denote by Gℬ the full subcategory of 𝒜 with Ob(Gℬ) := {GB | B ∈ Ob(ℬ)}, and suppose that :

(a) G and φ ∘ G are both exact.

(b) F is exact and faithful. Then we have :

(i) ω_{K^{•}} is a quasi-isomorphism for every K^{•} ∈ Ob(C^{−}(𝒜)).

(ii) Gℬ is a φ-projective subcategory of 𝒜 (see remark 7.3.31(v)).

Proof. Let (⊥, ε, δ) be the cotriple associated with the adjoint pair (G, F), as in in (7.10.3). By proposition 7.10.5(i), the counit ε_{A} induces a quasi-isomorphism F (△_{0}A) →^{∼} F A[0] for every A ∈ Ob(𝒜), where △_{0} is defined as in (7.10.7); then the natural transformation △_{0}A → A[0] is already a quasi-isomorphism for every such A, since F is exact and faithful. The assertion then follows from [163, Lemma 2.7.3].

(ii): For every i ∈ ℕ, consider the functor

φ_{i} := H_{i} ∘ C(φ) ∘ △_{0} : 𝒜 → ℬ A ↦ H_{i}(φ(△_{0}A)).

Again by proposition 7.10.5(i), ε_{GB} induces a quasi-isomorphism △_{0}(GB) →^{∼} GB[0] for every B ∈ Ob(ℬ), so φ_{i}(GB) = 0 for every i > 0, and we get a natural transformation φ_{0} ⇒ φ whose restriction to Gℬ is an isomorphism of functors. Since G, F and φ are exact, the same holds for ⊥ and φ∘⊥; hence, every short exact sequence Σ := (0 → A′ → A → A′′ → 0) of 𝒜 induces a short exact sequence

φ(△_{0}Σ) : 0 → φ(△_{0}A′) → φ(△_{0}A) → φ(△_{0}A′′) → 0 in C(𝒜′) .

Especially, if φ_{i}(A) = φ_{i}(A′′) = 0 for every i > 0, we have as well φ_{i}(A′) = 0 for every i > 0, by the long exact cohomology sequence associated with φ(T Σ). Let then X_{•} := (X_{n}, d_{n} | n ∈ ℕ) be any acyclic object of C^{−}(Gℬ); by considering the short exact sequences Σ_{n} := (0 → Im(d_{n+1}) → X_{n} → Im(d_{i}) → 0), we deduce inductively that φ_{i}(Im d_{n}) = 0 for every i > 0 and every n ∈ ℕ, so the sequence φ_{0}(Σ_{n}) is still short exact for every such n. Hence, the complex φ_{0}(X_{•}) is still acyclic, and then the same holds for the complex φ(X_{•}). Lastly, since F is faithful, the counit ε_{A} : GF A → A is an epimorphism (proposition 1.1.20(ii)), whence the assertion. □

Remark 7.10.9. (i) Proposition 7.10.8(i) implies that the functor △ of (7.10.7) takes quasiisomorphisms to quasi-isomorphisms, so it descends to a well defined functor

△ : D^{−}(𝒜) → D_{Gℬ}^{−} (𝒜)

(notation of (7.3.21)) and taking into account remark 7.3.24(i) we see that △ is a quasi-inverse for the functor i^{−} : D_{Gℬ}^{−} (𝒜) → D^{−}(𝒜) induced by the inclusion functor i : Gℬ → 𝒜. By virtue of proposition 7.10.8(ii) and remark 7.3.31(v), we then conclude that the left derived functor of φ is given by :

Lφ : D^{−}(𝒜) → D^{−}(𝒜′) K^{•} ↦ φ(△K^{•}).

(ii) In view of remark 7.10.4, the constructions of (7.10.7) can be dualized : every triple (⊤, η, μ) on 𝒜 induces first a functor ▽_{0} : 𝒜 → C^{≥0}(𝒜) that attaches to every A ∈ Ob(𝒜) the unnormalized cochain complex of the cosimplicial object ⊤A[•] (the latter is obtained by regarding ⊤A[•] as a simplicial object of 𝒜^{o}, and applying the corresponding functor U : s.𝒜^{o} → C^{≤0}(𝒜^{o}) = C^{≥0}(𝒜)^{o} : see remark 7.1.6(iii)), which in turns induces a functor

▽ : C^{≥0}(𝒜) → C^{≥0}(𝒜)

together with a natural transformation τ_{K^{•}} : K^{•} → ▽K^{•} for every K^{•} ∈ Ob(C^{≥0}(𝒜)).

(iii) Likewise, proposition 7.10.8 dualizes as follows. Let 𝒜 , ℬ and 𝒜′ be three abelian categories, φ : 𝒜 → 𝒜′ an additive functor, and (F : 𝒜 → ℬ, G : ℬ → 𝒜) an adjoint pair of functors (so, now F is left adjoint to G). Define the subcategory Gℬ as in proposition 7.10.8, and suppose that conditions (a) and (b) of loc.cit. hold. Then the morphism τ_{K^{•}} of (ii) is a quasi-isomorphism for every K^{•} ∈ D^{+}(𝒜), and Gℬ is a φ-injective subcategory of 𝒜. Invoking remark 7.3.31(iv), we deduce that, in this situation, the right derived functor of φ is given by :

Rφ : D^{+}(𝒜) → D^{+}(𝒜′) K^{•} ↦ φ(▽K^{•}).

Example 7.10.10. (i) The following application is already suggested in [77, App.2, Rem.3.7]. Let I, J be two small categories, 𝒜 a complete abelian category, and f : I → J a functor. By theorem 1.3.4, the right Kan extension ∫_{f} along f is right adjoint to Fun(f, 𝒜) : Fun(J, 𝒜) → Fun(I, 𝒜). Moreover, the limit functor Lim_{J} : Fun(J, 𝒜) → 𝒜 is right adjoint to the constant functor c_{J} : 𝒜 → Fun(J, 𝒜). It is easily seen that the functor Fun(f, 𝒜) is exact; suppose then that :

(a) f is essentially surjective

(b) Both Lim_{I} and Lim_{j/fI} for every j ∈ Ob(J) are exact. It is easily seen that (a) implies that Fun(f, 𝒜) is faithful, and (b) implies that ∫_{f} is exact; taking into account remark 7.10.9(iii), we deduce that the right derived functor of Lim_{J} is given by :

R Lim_{J} : D^{+}(Fun(J, 𝒜)) → D^{+}(𝒜) K^{•} ↦ Lim_{J}(▽K^{•})

where ▽ : C^{+}(Fun(J, 𝒜)) → C^{+}(Fun(J, 𝒜)) is induced by the triple associated with the adjoint pair (Fun(f, 𝒜), ∫_{f}).

(ii) With the notation of (i), take for I a discrete subcategory of J, such that every object of J is isomorphic to some element of Ob(I), and let f : I → J be the inclusion functor. Then f is essentially surjective, and the category j/fI is discrete for every j ∈ Ob(J). Suppose then that the products are exact in the complete abelian category 𝒜, i.e. that the functor Lim_{S} : Fun(S, 𝒜) → 𝒜 is exact for every small discrete category S. In this situation, conditions (a) and (b) of (i) are verified, so we get a construction of R Lim_{J} via the standard resolutions of proposition 7.10.5(ii). Let us explicit this construction, for any given functor F : J → 𝒜 : j ↦ F_{j}. First, for every n ∈ ℕ and j ∈ Ob(J), let S(n, j) be the set of all functors β : [n] → J with β(0) = j and β(i) ∈ Ob(I) for every i = 1, …, n (recall that [n] is the set of integers {0, …, n}, with its standard ordering). Then ▽_{0}F is the complex of Fun(J, 𝒜) :

j ↦ (▽_{0}F)_{j}^{•} := (0 → ∏_{β∈S(1,j)} F_{β(1)} ⟶^{d_{j}^{0}} ∏_{β∈S(2,j)} F_{β(2)} ⟶^{d_{j}^{1}} ⋯)

where, in view of the sign conventions of remark 7.1.6(iii), we set d_{j}^{n} = ∑_{i=0}^{n+1}(−1)^{i+n+1} · ∂_{j}^{i+1}, and ∂_{j}^{i} is the unique morphism that makes commute the diagram :

\xymatrix{∏_{β∈S(n+1,j)} F_{β(n+1)} \ar[r]^{∂_{j}^{i}} \ar[d]_{p_{γ∘ε_{i}}} & ∏_{β∈S(n+2,j)} F_{β(n+2)} \ar[d]^{p_{γ}} \\ F_{γ∘ε_{i}(n+1)} \ar[r]^{F_{γ(ε_{i}(n+1)→n+2)}} & F_{γ(n+2)}}  for every γ ∈ S(n + 2, j)

where ε_{i} : [n + 1] → [n + 2] is the i-th face map, and p_{γ} and p_{γ∘ε_{i}} are the natural projections. To every morphism φ : j → k of J, there corresponds the morphism (▽_{0}F)_{φ}^{•} : (▽_{0}F)_{j}^{•} → (▽_{0}F)_{k}^{•} of C^{≥0}(𝒜) that makes commute the diagram :

[figure: ∏_{β∈S(n,j)} F_{β(n)}; (▽_{0}F)_{φ}^{n−1}; p_{φ^{∗}(γ)}; ∏_{β∈S(n,k)} F_{β(n)}; p_{γ}; F_{γ(n)}; F_{γ(n)}]  for every n ∈ ℕ \ {0} and every γ ∈ S(n, k)

where φ^{∗} : S(n, k) → S(n, j) is the map given by the rule :

(k ⟶^{ψ_{0}} β(1) ⟶^{ψ_{1}} ⋯ ⟶^{ψ_{n−1}} β(n)) ↦ (j ⟶^{ψ_{0}∘φ} β(1) ⟶^{ψ_{1}} ⋯ ⟶^{ψ_{n−1}} β(n)).

Lastly, the augmentation τ_{F} : F → ▽_{0}F is the unique map that makes commute the diagram :

\xymatrix{F_{j} \ar[r]^{τ_{F,j}} \ar[dr]_{F_{γ(0→1)}} & ∏_{β∈S(1,j)} F_{β(1)} \ar[d]^{p_{γ}} \\ & F_{γ(1)}} for every γ ∈ S(1, j) and every j ∈ Ob(J).

(iii) We deduce from (ii) the following explicit expression for the complex lim_{J}(▽_{0}F). Notice that for every X ∈ Ob(𝒜) and every n ∈ ℕ, the datum of a morphism X → lim_{J}(▽_{0}F)_{•}^{n} is equivalent to that of a system f_{•} := (f_{β} | β ∈ Fun([n + 1], J)), where f_{β} : X → F_{β(n+1)} is a morphism of 𝒜 for every β ∈ Fun([n + 1], J), fulfilling the compatibility condition :

f_{φ^{∗}(β)} = f_{β} for every φ ∈ Morph(J) and every β ∈ Fun([n + 1, J]).

In turn, such a system f_{•} is equivalent to that of a system g_{•} := (g_{γ} | γ ∈ Fun([n], J)), where g_{γ} : X → F_{γ(n)} is also a morphism of 𝒜 (and without any compatibility condition); namely, to the system g_{•} there corresponds the system f_{•} with f_{β} := g_{β∘ε_{0}}, where ε_{0} : [n] → [n + 1] is the 0-th face map, so that ε_{0}(i) := i + 1 for i = 0, …, n. Hence, lim_{J}(▽_{0}F)_{•}^{n} is represented by ∏_{γ∈Fun([n],J)} F_{γ(n)}, for every n ∈ ℕ. Under these identifications, it is easily seen that d^{n} := lim_{J} d_{•}^{n} : lim_{J}(▽_{0}F)_{•}^{n} → lim_{J}(▽_{0}F)_{•}^{n+1} corresponds to the map ∑_{i=0}^{n+1}(−1)^{i+n+1}∂^{i} : ∏_{γ∈Fun([n],J)} F_{γ(n)} → ∏_{γ∈Fun([n+1],J)} F_{γ(n)}, where ∂^{i} is the unique morphism of 𝒜 that makes commute the diagram :

\xymatrix{∏_{γ∈Fun([n],J)} F_{γ(n)} \ar[r]^{∂^{i}} \ar[d]_{p_{β∘ε_{i}}} & ∏_{γ∈Fun([n+1],J)} F_{γ(n+1)} \ar[d]^{p_{β}} \\ F_{β∘ε_{i}(n)} \ar[r]^{F_{γ(ε_{i}(n)→n+1)}} & F_{β(n+1)}}  for every β ∈ Fun([n + 1], J)

where again, ε_{i} : [n] → [n + 1] is the i-th face map, and p_{β} and p_{β∘ε_{i}} are the natural projections. Summing up, lim_{J}(▽_{0}F) is (up to natural isomorphisms of C(𝒜)) the complex :

0 → ∏_{β∈Fun([0],J)} F_{β(0)} ⟶^{d^{0}} ∏_{β∈Fun([1],J)} F_{β(1)} ⟶^{d^{1}} ⋯

with the differentials d^{n} given above.

(iv) Let G : J′ → J be a functor; by remark 1.2.11(i), we have a natural transformation

(lim_{G} 1_{𝒜} )_{F} : Lim_{J} F → Lim_{J′}(F ∘ G).

Since the functor Fun(G, 1_{𝒜} ) : Fun(J, 𝒜) → Fun(J′, 𝒜) is exact, we deduce a natural transformation

(R lim_{G} 1_{𝒜} )_{F^{•}} : R lim_{J} F^{•} → R lim_{J′}(F^{•} ∘ G) for every F^{•} ∈ Ob(D^{+}(Fun(J, 𝒜))) .

We may upgrade this transformation to a natural map of cochain complexes

λ_{F} : lim_{J} ▽_{0}F → lim_{J′} ▽_{0}(F ∘ G)

as follows. It suffices to exhibit a natural map μ_{F} that makes commute the diagram

\xymatrix{ & F ∘ G \ar[dl]_{τ_{F} ∗G} \ar[dr]^{τ_{F ∘G}} \\ (▽_{0}F) ∘ G \ar[rr]^{μ_{F}} &  & ▽_{0}(F ∘ G)}

where τ_{F} and τ_{F ∘G} are the respective augmentations, as in (ii). For every n ∈ ℕ, we let μ_{F}^{n} be the unique morphism that makes commute the diagram :

\xymatrix{∏_{β∈S(n,Gj′)} F_{β(n)} \ar[r]^{(μ_{F}^{n})_{j′}} \ar@{-->}[dr]_{p_{G∗(γ)}} & ∏_{β∈S(n,j′)} F_{Gβ(n)} \ar[d]^{p_{γ}} \\  & F_{Gγ(n)}} for every j′ ∈ Ob(J′) and every γ ∈ S(n, j′)

where G_{∗} : S(n, j′) → S(n, Gj′) is the map such that (β : [n] → J′) ↦ (G ∘ β : [n] → J).

In terms of the natural identifications of (iii), it follows that λ_{F} corresponds to the unique morphism of complexes that makes commute the diagram :

\xymatrix{∏_{γ∈Fun([n],J)} F_{γ(n)} \ar[r]^{λ_{F}^{n}} \ar@{-->}[dr]_{p_{G∗(β)}} & ∏_{γ′∈Fun([n],J′)} F_{Gγ′(n)} \ar[d]^{p_{β}} \\ & F_{Gβ(n)}} for every n ∈ ℕ and every β ∈ Fun([n], J′).

(v) Let I, J and the inclusion functor f : I → J be as in (ii), and suppose now that 𝒜 is cocomplete and that the direct sums are exact in 𝒜, i.e. that the functor Colim_{S} : Fun(S, 𝒜) → 𝒜 is exact for every small discrete category S. Then we can follow remark 7.10.9(i) to obtain a construction of the left derived functor L Colim_{J} of the functor Colim_{J}, which shall be dual to (ii). Let us explicit the resulting complex colim_{J}(△_{0}F) for any given functor F : J → 𝒜. Proceeding as in (iii), we find that it is the complex of C^{−}(𝒜) :

⋯ ⟶^{d_{2}} ⨁_{β∈Fun([1],J)} F_{β(0)} ⟶^{d_{1}} ⨁_{β∈Fun([0],J)} F_{β(0)} → 0

with d_{n} = ∑_{i=0}^{n}(−1)^{i} · ∂_{n−i}, and ∂_{i} is the morphism that makes commute the diagram :

\xymatrix{F_{γ(0)} \ar[r]^{F_{γ(0→ε_{i}(0))}} \ar[d]_{e_{γ}} & F_{γ∘ε_{i}(0)} \ar[d]^{e_{γ∘ε_{i}}} \\ ⨁_{β∈Fun([n],J)} F_{β(0)} \ar[r]^{∂_{i}} & ⨁_{β∈Fun([n−1],J)} F_{β(0)}}  for every γ ∈ Fun([n], J)

where e_{γ} and e_{γ∘ε_{i}} are the natural monomorphisms.

7.10.11. Let 𝒜 be a complete abelian category with exact products, J a small category, I ⊂ J a discrete subcategory such that the inclusion functor f : I → J is essentially surjective. Then the functor ▽ : C^{+}(Fun(J, 𝒜)) → C^{+}(Fun(J, 𝒜)) is well defined, as in example 7.10.10(ii).

We consider also the special case where we take 𝒜 = ℤ, and we denote the corresponding functor by ▽′ : C^{+}(Fun(J, ℤ-Mod)) → C^{+}(Fun(J, ℤ-Mod)).

Proposition 7.10.12. In the situation of (7.10.11), we have a natural isomorphism in C(𝒜) :

Hom_{𝒜}^{•} (C^{•}, lim_{J}(▽D^{•})) →^{∼} lim_{J}(▽′Hom_{𝒜}^{•} (C^{•}, D^{•}))

for every C^{•} ∈ Ob(C^{−}(𝒜)) and every D^{•} ∈ Ob(C^{+}(Fun(J, 𝒜))).

Proof. (Here the functor Hom_{𝒜}^{•} (−, −) is defined as in example 7.1.15(i).) We have two functors

Hom_{𝒜} (−, Lim_{J}), Lim_{J} Hom_{𝒜} (−, −) : 𝒜^{o} × Fun(J, 𝒜) → ℤ-Mod

and an isomorphism ω : Hom_{𝒜} (−, Lim_{J}) →^{∼} Lim_{J} Hom_{𝒜} (−, −). Then, for every K^{•} ∈ Ob(C(𝒜)) and every L^{•} ∈ Ob(C(Fun(J, 𝒜))), the system (ω_{K^{q},L^{p}} | p, q ∈ ℤ) yields an isomorphism : Hom_{𝒜}^{•} (K^{•}, lim_{J} L^{•}) →^{∼} lim_{J} Hom_{𝒜}^{•} (K^{•}, L^{•}) in C(ℤ-Mod). Especially, we get a natural isomorphism:

(7.10.13)  Hom_{𝒜}^{•} (C^{•}, Lim_{J}(▽D^{•})) →^{∼} Lim_{J} Hom_{𝒜}^{•} (C^{•}, ▽D^{•}).  in C(ℤ-Mod)

and recall that ▽D^{•} = Tot(▽_{0}D^{•}) (see remark 7.10.9(ii)); also, Hom_{𝒜}^{•} is the composition

C(𝒜) × C(𝒜)^{o} ⟶^{1_{C(𝒜)} ×Φ} C(𝒜) × C(𝒜^{o}) ⟶^{Hom_{𝒜}^{••}} C_{2}(𝒜) ⟶^{Tot_{𝒜}^{Π}} C(𝒜)

where Φ : C(𝒜)^{o} →^{∼} C(𝒜^{o}) is the isomorphism of categories of remark 7.1.6(iii), and Hom_{𝒜}^{••} is deduced from the biadditive functor Hom_{𝒜} : 𝒜 × 𝒜^{o} → 𝒜, as prescribed in (7.1.14). Consider the functor H : C(𝒜) × 𝒜^{o} → C(𝒜) such that H(L^{•}, K)^{p} := Hom_{𝒜} (K, L^{p}) for every ((L^{•}, d_{L}^{•}), K) ∈ Ob(C(𝒜)×𝒜^{o}) and every p ∈ ℤ, with differential d^{p} := Hom_{𝒜} (K, d_{L}^{p}) for every such p. The functor H is biadditive, so by applying to H the construction of (7.1.14) we get another functor :

Hom_{𝒜}^{•••} := H^{••} : C_{2}(𝒜) × C(𝒜^{o}) → C_{3}(𝒜) (L^{••}, K^{•}) ↦ Hom_{𝒜}^{•••}(K^{•}, L^{••}).

With this notation, for every K^{•} ∈ Ob(C(𝒜)) and every L^{••} ∈ Ob(C_{2}(𝒜)) we have :

Hom_{𝒜}^{••}(K^{•}, Tot^{Π}(L^{••})) = C(Tot_{𝒜}^{Π})(Hom_{𝒜}^{•••}(K^{•}, L^{••}))

(recall that C(Tot_{𝒜}^{Π})(M^{•••})^{nr} = Π_{p+q=n}M^{pqr}, and Hom_{𝒜}^{•••}(K^{•}, L^{••})^{pqr} = Hom_{𝒜} (K^{r}, L^{pq}) for every p, q, r ∈ ℤ). Summing up, we get a natural isomorphism :

(7.10.14)  Hom_{𝒜}^{•} (C^{•}, Tot(▽_{0}D^{•})) →^{∼} Tot_{𝒜}^{Π} ∘ C(Tot_{𝒜}^{Π})(Hom_{𝒜}^{•••}(Φ(C^{•}), ▽_{0}D^{•}))

and recall that (▽_{0}D^{•})^{pq} = ⊤^{p+1}(D^{q}) for every p, q ∈ ℕ. Next, by a direct inspection of the constructions in (ii), we get an isomorphism of functors 𝒜^{o} × 𝒜 → C^{≥0}(ℤ-Mod) :

Hom_{𝒜} (−, ▽_{0}(−)) →^{∼} ▽′_{0}Hom_{𝒜} (−, −)

whence a natural isomorphism :

Hom_{𝒜}^{•••}(Φ(C^{•}), ▽_{0}D^{•}) →^{∼} ▽′_{0}Hom_{𝒜}^{••}(Φ(C^{•}), D^{•}) in C_{3}(ℤ-Mod).

Lastly, recall that Tot_{𝒜}^{Π} ∘ C(Tot_{𝒜}^{Π}) = Tot_{𝒜}^{Π} ∘ Tot_{C(𝒜)}^{Π} , and we have Tot_{C(𝒜)}^{Π} (M^{•••})^{pn} = ∏_{q+r=n} M^{pqr} for every p, n ∈ ℤ (see (7.1.12)), and notice that

Tot_{C(𝒜)}^{Π} (▽′_{0}Hom_{𝒜}^{••}(Φ(C^{•}), D^{•})) = ▽′_{0}(Tot_{𝒜}^{Π}Hom_{𝒜}^{••}(Φ(C^{•}), D^{•})) = ▽′_{0}(Hom_{𝒜}^{•} (C^{•}, D^{•})).

Combining with (7.10.14) and (7.10.13), the assertion follows. □

Remark 7.10.15. (i) In the situation of proposition 7.10.12, suppose moreover that 𝒜 has enough projective objets. Then we may choose a quasi-isomorphism P^{•} → C^{•} in C^{−}(𝒜), with P^{•} a complex of projectives; from the double complexes

Hom_{𝒜} (P^{•}, lim_{J}(▽D^{•}))  and  lim_{J}(▽′Hom_{𝒜} (P^{•}, D^{•}))

and taking into account example 7.10.10(i), we then obtain two spectral sequences

E_{2}^{pq} := Ext_{𝒜}^{p} (C^{•}, R^{q} lim_{J} D^{•}) F_{2}^{pq} := R^{p} lim_{J} Ext_{𝒜}^{q} (C^{•}, D^{•})

that converge both to Hom_{D(𝒜)} (C^{•}, R lim_{J} D^{•}[p + q]) (and with finite induced filtrations on the abutment : see [163, §5.6]). These spectral sequences already appear in [106, Th.4.2, Th.4.3] and in [144, Th.1].

(ii) On the other hand, by applying proposition 7.10.12 to the opposite category 𝒜^{o}, and taking into account example 7.10.10(v), we deduce a natural isomorphism in C(𝒜) :

Hom_{𝒜}^{•} (colim_{J}(△D^{•}), C^{•}) →^{∼} lim_{J}(▽′Hom_{𝒜}^{•} (D^{•}, C^{•}))

for every C^{•} ∈ Ob(C^{+}(𝒜)) and every D^{•} ∈ C^{−}(Fun(J, 𝒜)), where △ : C^{−}(Fun(J, 𝒜)) → C^{−}(Fun(J, 𝒜)) is the functor associated with the adjoint pair (∫^{f}, Fun(f, 𝒜)). If moreover, 𝒜 has enough injective objects, we may choose a quasi-isomorphism C^{•} → Q^{•} in C^{+}(𝒜) with a complex Q^{•} of injectives, and arguing as in (i) we deduce two spectral sequences :

E_{2}^{pq} := Ext_{𝒜}^{p} (L^{q} colim_{J} D^{•}, C^{•}) F_{2}^{pq} := R^{p} lim_{J} Ext_{𝒜}^{q} (D^{•}, C^{•})

that converge both to Hom_{D(𝒜)} (L colim_{J} D^{•}, C^{•}[p + q]).

(iii) Even in case 𝒜 does not have enough projective objects, we can still deduce from proposition 7.10.12a spectral sequence

E_{2}^{pq} := R^{p} lim_{J} Hom_{D(𝒜)} (C^{•}, D^{•}[q]) ⇒ Hom_{D(𝒜)} (C^{•}, R lim_{J} D^{•}[p + q]).

To this aim, we notice first that proposition 7.10.12yields a spectral sequence

F (C^{•})_{2}^{pq} := R^{p} lim_{J} Hot_{𝒜} (C^{•}, D^{•}[q]) ⇒ Hot_{𝒜} (C^{•}, R lim_{J} D^{•}[p + q]).

Next, let Σ be the full subcategory of Hot^{−}(𝒜)/C^{•} whose objects are the quasi-isomorphisms C^{′•} → C^{•}. By virtue of theorem 7.3.16, we deduce a spectral sequence :

colim_{C^{′•}∈Ob(Σ)} F (C^{′•})_{2}^{pq} ⇒ Hom_{D(𝒜)} (C^{•}, R lim_{J} D^{•}[p + q]).

On the other hand, set G(C^{′•})_{j}^{pq} := ∏_{β∈S(p,j)} Hot_{𝒜} (C^{′•}, D_{β(p)}^{•}) for every p, q ∈ ℕ every j ∈ Ob(J), and every object C^{′•} → C^{•} of Σ (notation of example 7.10.10(ii)). Then F (C^{′•})_{2}^{pq} is the kernel of a map ∏_{φ∈Morph(J)} G(C^{′•})_{t(φ)}^{pq} → ∏_{j∈Ob(J)} G(C^{′•})_{j}^{pq}. Since Σ is filtered, we are then reduced to checking that for any family (L_{λ}^{•} | λ ∈ Λ) of objects of C^{+}(𝒜) we have

colim_{C^{′•}∈Ob(Σ)} ∏_{λ∈Λ} Hot_{𝒜} (C^{′•}, D_{λ}^{•}) = ∏_{λ∈Λ} Hom_{D(𝒜)} (C^{•}, D_{λ}^{•}).

The latter follows easily from corollary 7.3.19(ii), remark 7.1.6(vi), and theorem 7.3.16.

7.10.16. Now, denote by Set_{∘} the category of pointed sets, whose objects are all the pairs (S, s) where S is a small set, and s ∈ S is any element; the morphisms f : (S, s) → (S′, s′) are the mappings f : S → S′ such that f(s) = s′. Next, let A be any ring, and consider the pointed forgetful functor

F : A-Mod → Set_{∘} M ↦ (M, 0_{M})

(where 0_{M} ∈ M is the zero element of M). The functor F admits the left adjoint

L : Set_{∘} → A-Mod (S, s) ↦ A^{(S)}/As

(where A^{(S)} denotes the free A-module with basis given by S, so As ⊂ A^{(S)} is the direct summand generated by the basis element s ∈ S). According to proposition 7.10.5, the adjoint pair (L, F) yields a functor

⊥_{•}^{A}: A-Mod → ŝ.A-Mod

into the category of augmented simplicial A-modules (notation of (7.4.11)), such that

F ⊥_{•}^{A} M → FM

is a homotopically trivial augmented pointed simplicial set, for every A-module M. This functor is obtained by iterating the functor ⊥^{A}:= L ∘ F, so in each degree it consists of a free A-module. We call ⊥_{•}^{A} M the standard free resolution of M.

Remark 7.10.17. Notice that ⊥_{•}^{A} 0 = s.0, the constant simplicial A-module associated with the trivial A-module (see (7.4.4)). This is the reason why we prefer the adjoint pair (L, F), rather than the analogous pair considered in [103, I.1.5.5], arising from the forgetful functor from A-modules to non-pointed sets.

7.10.18. Let R be a simplicial A-algebra, and define the category R-Mod of R-modules, as in [75, §8.1]. Notice that any morphism S → R of simplicial rings induces a base change functor

S-Mod → R-Mod (M[n] | n ∈ ℕ) ↦ (R[n] ⊗_{S[n]} M[n] | n ∈ ℕ)

which is left adjoint to the forgetful functor (details left to the reader).

Now, by applying the functors ⊥_{•}^{R[n]} to the terms M[n] of a given R-module M, we obtain an augmented simplicial R-module

(7.10.19)  ⊥_{•}^{R} M → M

which amounts to a bisimplicial A-module, whose columns ⊥_{•}^{R[n]} M[n] → M[n] are augmented simplicial R[n]-modules, for every n ∈ ℕ. Likewise, the row ⊥_{n}^{R} M is an R-module, for every n ∈ ℕ.

Lemma 7.10.20. In the situation of (7.10.18), let φ : M → N be any quasi-isomorphism of R-modules. Then the induced morphism

⊥_{n}^{R} φ :⊥_{n}^{R} M →⊥_{n}^{R} N

is a homotopy equivalence, for every n ∈ ℕ.

Proof. Set ⊥_{−1}^{R} M := M, and notice the natural isomorphisms

(7.10.21)  ⊥_{n}^{R} M →^{∼} R⊗_{s.ℤ} ⊥^{s.ℤ} ∘ ⊥_{n−1}^{R} M  for every n ∈ ℕ

(where s.ℤ is the constant simplicial ring arising from ℤ (see (7.4.4)), which is the initial object in the category of simplicial rings). The assumption means that ⊥_{−1}^{R} φ is a quasi-isomorphism. However, (7.10.21) and Whitehead’s theorem ([103, I.2.2.3]) imply that if – for a given n ∈ ℕ – the map ⊥_{n−1}^{R} φ is a quasi-isomorphism, then ⊥_{n}^{R} φ is a homotopy equivalence. We may thus conclude by a simple induction on n. □

Example 7.10.22. Let M and N two R-modules. We may define two functors

M ⊗^{ℓ}_{R} − ( resp. − ⊗^{ℓ}_{R} N ) : R-Mod → R-Mod

by the rules :

P ↦ M ⊗_{R} (⊥_{•}^{R} P)^{Δ} ( resp. P ↦ (⊥_{•}^{R} P)^{Δ} ⊗_{R} N ) for every R-module P

where Δ is the diagonal functor, from 𝒞 -bisimplicial to 𝒞 -simplicial objects (see (7.4.17)). (i) We claim that these functors descend to the derived category. That is, suppose that

φ : P → P′ is a quasi-isomorphism; then M ⊗^{ℓ}_{R} φ and φ ⊗^{ℓ}_{R} N are both quasi-isomorphisms. Indeed, lemma 7.10.20implies that the induced morphisms

M⊗_{R} ⊥_{n}^{R} φ : M⊗_{R} ⊥_{n}^{R} P → M⊗_{R} ⊥_{n}^{R} P′

are quasi-isomorphisms, for every n ∈ ℕ, and likewise for φ⊗_{R} ⊥_{n}^{R} N, so the assertion follows from Eilenberg-Zilber’s theorem 7.4.50(i).

(ii) Also, we claim that the notation M ⊗^{ℓ}_{R} N is unambiguous. That is, we may compute this

object by applying the functor M ⊗^{ℓ}_{R} − to N, or by applying the functor − ⊗^{ℓ}_{R} N to M, and the resulting two R-modules are naturally isomorphic in D(R-Mod). Indeed, it suffices to check that the natural morphisms

M ⊗_{R} (⊥_{•}^{R} N)^{Δ} ⟵^{α} (⊥_{•}^{R} M)^{Δ} ⊗_{R} (⊥_{•}^{R} N)^{Δ} ⟶^{β} (⊥_{•}^{R} M)^{Δ} ⊗_{R} N

induced by the augmentation (7.10.19), are both quasi-isomorphisms. We check this for α; the same argument shall apply to β. To ease notation, set L := (⊥_{•}^{R} N)^{Δ}. Then α is deduced by extracting the diagonal from the augmented simplicial R-module

(7.10.23)  ⊥_{•}^{R} M ⊗_{R} L → M ⊗_{R} L

(so, the n-th column of s.L is a constant and flat simplicial R[n]-module, for every n ∈ ℕ). Thus, we are reduced to checking that the columns of (7.10.23) are aspherical. However, for every n ∈ ℕ, the n-th column is the augmented simplicial R[n]-module (⊥_{•}^{R[n]} M[n])⊗_{R[n]}L[n]; since L[n] is flat, it then suffices to recall that the standard free resolution is aspherical.

(iii) The foregoing discussion yields a well defined functor

− ⊗^{ℓ}_{R} − : D(R-Mod) × D(R-Mod) → D(R-Mod)

called the left derived tensor product. The same method shall be applied hereafter to construct derived functors of certain non-additive functors.

Remark 7.10.24. Let M and N be as in example 7.10.22and set P := (⊥_{•}^{R} M)⊗_{R} s.N (this is a simplicial R-module; especially, it is a bisimplicial A-module); by Eilenberg-Zilber’s theorem

7.4.50, the cochain complex associated with M ⊗^{ℓ}_{R} N is naturally isomorphic, in D(A-Mod) to the complex Tot(P^{••}), where P^{••} denotes the double complex associated with P. There follows a spectral sequence :

E_{pq}^{1} := Tor_{q}^{R[p]}(M[p], N[p]) ⇒ H_{p+q}(M ⊗^{ℓ}_{R} N)

whose differentials d_{pq}^{1} : E_{pq}^{1} → E_{p−1,q}^{1} are obtained as follows. For every integer q ∈ ℕ, let T_{q} be the R-module given by the rule : T_{q}[p] := E_{pq}^{1}, with face and degeneracy maps deduced from those of M, N and R, in the obvious way. Then d_{pq}^{1} is the differential in degree p of the chain complex T_{q•} associated with T_{q} (details left to the reader).

Example 7.10.25. (i) Another useful cotriple arises from the forgetful functor A-Alg → Set and its left adjoint, that attaches to any set S the free A-algebra A[S]. There results, for every A-algebra B, a standard simplicial resolution by free A-algebras

F_{A}(B) → B.

Now, if R is again a simplicial A-algebra, and S any R-algebra, we can proceed as in the foregoing, to obtain a bisimplicial resolution F_{R}(S) → S whose columns F_{R[n]}(S[n]) → S[n] are augmented simplicial R[n]-algebras, for every n ∈ ℕ. A simple inspection shows that the proof of lemma 7.10.20carries over – mutatis mutandis – to the present situation, hence a quasiisomorphism S → S′ of R-algebra induces a morphism F_{R}(S)[n] → F_{R}(S′)[n] of R-algebras, which is a homotopy equivalence on the underlying R-modules, for every n ∈ ℕ.

(ii) We may then define a derived tensor product for R-algebras, proceeding as in example 7.10.22. Namely, if S and S′ are any two R-algebras, we define two functors

S ⊗^{ℓ}_{R} − ( resp. − ⊗^{ℓ}_{R} S′ ) : R-Alg → R-Alg

by the rules :

T ↦ S ⊗_{R} F_{R}(T)^{Δ} ( resp. T ↦ F_{R}(T)^{Δ} ⊗_{R} N ) for every R-algebra T.

Arguing like in loc.cit. we see that both functors transform quasi-isomorphisms into quasiisomorphisms, hence they descend to the derived category D(R-Alg), and moreover, the two

functors are naturally isomorphic, so the notation S ⊗^{ℓ}_{R} S′ is unambiguous : the detailed verification is left as an exercise for the reader.

7.10.26. Let A-Alg.Mod be the category of all pairs (B, M), where B is any A-algebra, and M any B-module. The morphisms (B, M) → (B′, M′) are the pairs (f, φ), where f : B → B′ is a morphism of A-algebras, and φ : M → f^{∗}M′ a B-linear map (here f^{∗}M′ denotes the Bmodule obtained from the B′-module M′, by restriction of scalars along the map f). Suppose now that we have a commutative diagram of categories :

\xymatrix{A-Alg.Mod \ar[r]^{T} \ar[d]_{F} & 𝒞 \ar[d]^{f} \\ A-Alg \ar[r]^{g} & ℬ}

such that :

• For every X ∈ Ob(B), the fibre f^{−1}X is an abelian category whose filtered colimits are representable and exact.

• F is given by the rule (B, M) ↦ B for every (B, M) ∈ Ob(A-Alg.Mod).

• For every A-algebra B, the restriction

T_{B} : F^{−1}B → f^{−1}(gB)

of T commutes with filtered colimits, i.e., if ((B, M_{i}) | i ∈ I) is any filtered system of objects of A-Alg.Mod (over the same A-algebra B), then the induced morphism

colim_{i∈I} T_{B}M_{i} → T_{B}(colim_{i∈I} M_{i})

is an isomorphism. The functors f and g extend to functors s.f : s.𝒞 → s.ℬ, respectively s.g : s.A-Alg → s.ℬ, and notice that – for R any simplicial A-algebra – T induces a functor

T_{R} : R-Mod → s.𝒞_{/R} := s.f^{−1}(s.gR) (M[n] | n ∈ ℕ) ↦ (T_{R[n]}M[n] | n ∈ ℕ).

For every R-module M, we set

LT_{R}(M) := T_{R}(⊥_{•}^{R} M)^{Δ}.

The augmentation T_{R}(⊥_{•}^{R} M) → T_{R}M can be regarded as a morphism of bisimplicial objects

T_{R}(⊥_{•}^{R} M) → s.T_{R}M

(the columns of s.T_{R}M are constant 𝒞 -simplicial objects), whence a morphism in s.𝒞_{/R} :

(7.10.27)  LT_{R}(M) → s.T_{R}M^{Δ} = T_{R}M  for every R-module M.

In many applications, s.𝒞_{/R} will be also an abelian category, but anyhow we can state the following :

Proposition 7.10.28. With the notation of (7.10.26), suppose that 𝒞 is an abelian category. Then the following holds :

(i) If M is a flat R-module, then (7.10.27) is a quasi-isomorphism.

(ii) If φ : M → N is a quasi-isomorphism of R-modules, then the induced map

LT_{R}(M) → LT_{R}(N)

is a quasi-isomorphism.

Proof. (i): The assumption means that M[n] is a flat R[n]-module for every n ∈ ℕ. Denote by C the unnormalized double complex associated with T_{R}(⊥_{•}^{R} M), and by C^{Δ} (resp. by D) the unnormalized complex associated with LT_{R}(M) (resp. to T_{R}M). By Eilenberg-Zilber’s theorem 7.4.50, we have a natural quasi-isomorphism

(7.10.29)  C^{Δ} → Tot(C)

given by the Alexander-Whitney map of (7.4.35), and a simple inspection shows that the map C^{Δ} → D induced by (7.10.27) factors through (7.10.29). Hence, it suffices to show that the map

Tot(C) → D

induced by the augmentation of T_{R}(⊥_{•}^{R} M), is a quasi-isomorphism, under the stated condition. To this aim, it suffices to check that the augmented 𝒞 -simplicial object

(7.10.30)  T_{R}(⊥_{•}^{R[n]} M[n]) → T_{R}M[n]

is aspherical for every n ∈ ℕ. However, since M[n] is a flat R[n]-module, it can be written as a filtered colimit of a system of free R[n]-modules ([120, Ch.I, Th.1.2]); on the other hand, the functor ⊥_{•}^{R[n]} commutes with all filtered colimits, and the same holds for T_{R}, by assumption. Since the filtered colimits of the fibres of the functor f are exact, we are then reduced to checking the assertion in case M[n] is a free R[n]-module; but in this case, (7.10.30) is even homotopically trivial, by virtue of proposition 7.10.5.

(ii): In light of Eilenberg-Zilber’s theorem 7.4.50, it suffices to show that the induced map

T_{R}(⊥_{n}^{R} M) → T_{R}(⊥_{n}^{R} N)

is a quasi-isomorphism for every n ∈ ℕ. In turns, this follows readily from lemma 7.10.20. □

Remark 7.10.31. (i) The proof of proposition 7.10.28(i) applies as well to the derived tensor product; namely, for any two R-modules M and N there is a natural morphism of R-modules

M ⊗^{ℓ}_{R} N → M ⊗_{R} N

which is a quasi-isomorphism, if either M or N is a flat R-module (details left to the reader).

(ii) Especially, let S and S′ be any two R-algebras. Then, since the standard free resolution F_{R}(S) of example 7.10.25is a flat simplicial R-module, (i) implies that the R-module underly-

ing the R-algebra S ⊗^{ℓ}_{R}S′, is naturally isomorphic, in D(R-Mod), to the derived tensor product

of the R-modules underlying S and S′. In other words, the notation − ⊗^{ℓ}_{R} − is unambiguous, whether one refers to derived tensor products of algebras, or of their underlying modules.

(iii) Denote by σ, ω : R-Mod → R-Mod respectively the suspension and loop functors ([103, I.3.2.1]), and recall that σ is left adjoint to ω. A simple inspection of the definitions yields natural identifications

(σM) ⊗_{R} N →^{∼} σ(M ⊗_{R} N) (ωM) ⊗_{R} N →^{∼} ω(M ⊗_{R} N)

for every R-modules M and N. By the same token, it is clear that σ and ω transform flat R-modules into flat R-modules. In view of (i), we deduce natural isomorphisms

(σM) ⊗^{ℓ}_{R} N →^{∼} σ(M ⊗^{ℓ}_{R} N)  (ωM) ⊗^{ℓ}_{R} N →^{∼} ω(M ⊗^{ℓ}_{R} N)  in D(R-Mod)

for every R-modules M and N.

(iv) Let f : N → N′ be any morphism of R-modules; in the same vein, we get a natural identification :

Cone(M ⊗_{R} f) →^{∼} M ⊗_{R} Cone f for every R-module M

(see [103, I.3.2.2] for the definition of the cone of a morphism of R-modules). It follows

immediately that the functor M ⊗^{ℓ}_{R} − : D(R-Mod) → D(R-Mod) transforms distinguished triangles into distinguished triangles (see [103, I.3.2.2.4] for the definition of distinguished triangle in D(R-Mod)). For future reference, let us also point out :

Lemma 7.10.32. Let R be a simplicial A-algebra, X and Y two R-modules, n, m ∈ ℕ two integers, and suppose that H_{i}X = 0 = H_{j}Y for every i < n and j < m. Then

H_{i}(X ⊗^{ℓ}_{R} Y) = 0 for every i < n + m.

Proof. Set σ^{0} := 1_{R-Mod}, and define inductively σ^{k} : R-Mod → R-Mod by the rule : σ^{k} := σ ∘ σ^{k−1} for every k > 0. Define likewise ω^{k}, and notice that σ^{k} is left adjoint to ω^{k}, for every k ∈ ℕ (remark 7.10.31(iii)). Let f : σ^{n} ∘ ω^{n}X → X denote the counit of adjunction, and notice that Cone f = 0 in D(R-Mod). In view of remark 7.10.31(iv,v), we deduce that the natural morphisms

σ^{n}(ω^{n}X ⊗^{ℓ}_{R} Y) → (σ^{n} ∘ ω^{n}X) ⊗^{ℓ}_{R} Y → X ⊗^{ℓ}_{R} Y

are isomorphisms in D(R-Mod). Likewise, the natural morphism

σ^{m}(ω^{n}X ⊗^{ℓ}_{R} ω^{m}Y) → ω^{n}X ⊗^{ℓ}_{R} Y

is an isomorphism in D(R-Mod), so finally the same holds for the morphism

σ^{n+m}(ω^{n}X ⊗^{ℓ}_{R} ω^{m}Y) → X ⊗^{ℓ}_{R} Y

whence the claim. □

7.10.33. In view of proposition 7.10.28, it is clear that if 𝒞 is an abelian category, LT_{R} yields a well defined functor on derived categories

LT_{R} : D(R-Mod) → D(s.𝒞_{/R}) M ↦ LT_{R}(M)

(where D(s.𝒞_{/R}) is the localization of s.𝒞_{/R} with respect to the class of quasi-isomorphisms, and likewise for D(R-Mod) : see [75, Def.8.1.3]). More generally, suppose that 𝒞 is endowed with a functor

Φ : 𝒞 → 𝒜

to an abelian category 𝒜 (usually, this will be a forgetful functor of some sort), and denote by D_{Φ}(s.𝒞_{/R}) the localization of s.𝒞_{/R} with respect to the system of its morphisms whose image under s.Φ are quasi-isomorphisms in s.𝒜. Then, since clearly

s.Φ ∘ LT_{R} = L(s.Φ ∘ T_{R})

we see that LT_{R} descends to a well defined functor

LT_{R} : D(R-Mod) → D_{Φ}(s.𝒞_{/R}).

We will need also the following slight refinement :

Corollary 7.10.34. In the situation of proposition 7.10.28, the following holds :

(i) Let φ : M → N be a morphism of R-modules, and n ∈ ℕ an integer such that

H_{i}φ : H_{i}M → H_{i}N

is an isomorphism, for every i < n. Then the same holds for the induced map

H_{i}(LT_{R}φ) : H_{i}(LT_{R}M) → H_{i}(LT_{R}N).

(ii) Suppose that T (B, 0) = 0_{gB} (the initial and final object of f^{−1}(gB)) for every Aalgebra B. Let n ∈ ℕ be an integer, and M an R-module such that H_{i}M = 0 for every i < n. Then H_{i}(LT_{R}M) = 0_{gR_{i}} for every i < n.

Proof. (i): Denote by

φ_{X} : X → cosk_{n}X for every X ∈ Ob(R-Mod)

the unit of adjunction (see (7.4.20)). Taking into account corollary 7.4.67(i), we have the following properties :

• X[i] = cosk_{n}X[i] and φ[i] is the identity map of X[i], for every i ≤ n.

• H_{i}(cosk_{n}X) = 0 for every i ≥ n.

In view of proposition 7.10.28, we are then easily reduced to checking the assertion for the special where N := cosk_{n}M, and φ := φ_{M}. However, since in this case N[i] = M[i] for every i ≤ n, it is clear that ⊥_{•}^{R[i]} φ[i] is the identity automorphism of ⊥_{•}^{R[i]} M[i], for every i ≤ n. After applying the functor T_{R}, and extracting the diagonal, we see that the induced morphism LT_{R}M → LT_{R}N in s.𝒞 is given by the identity automorphism of (LT_{R}M)[i], in every degree i ≤ n, whence the contention.

(ii): In light of (i), we may assume that M = s.0 (the trivial R-module). In this case, the assertion follows easily from remark 7.10.17. □

7.10.35. In the situation of (7.10.26), take 𝒞 := A-Alg.Mod, f := F, and g the identity automorphism of A-Alg. If B → B′ is a map of A-algebras, and (B, M) ∈ Ob(𝒞), we define B′ ⊗_{B} (B, M) := (B′, B′ ⊗_{B} M).

Corollary 7.10.36. In the situation of (7.10.35), suppose moreover that, for every flat morphism B → B′ of A-algebras, the natural map

B′ ⊗_{B} T (B, M) → T (B′ ⊗_{B} (B, M))

is an isomorphism, for every (B, M) ∈ Ob(𝒞). Then, for every flat morphism φ : R → S of simplicial V -algebras, the natural morphism

(7.10.37)  S ⊗_{R} LT_{R}M → LT_{S}(S ⊗_{R} M)

is an isomorphism in D(S-Mod), for every R-module M.

Proof. (Recall that φ is flat if and only if S[n] is a flat R[n]-algebra, for every n ∈ ℕ. The map of the proposition is deduced from the natural morphism ⊥_{•}^{R} M →⊥_{•}^{S} (S ⊗_{R} M), given by functoriality of the standard free resolution.)

Let π :⊥_{•}^{R} M → M be the standard free resolution of M; by the flatness assumption on S, the morphism S ⊗_{R} π : S⊗_{R} ⊥_{•}^{R} M → S ⊗_{R} M is a free resolution of the S-module S ⊗_{R} M, whence a natural isomorphism

LT_{S}(S ⊗_{R} M) →^{∼} T_{S}((S⊗_{R} ⊥_{•}^{R} M)^{Δ}) in D(S-Mod)

by proposition 7.10.28(i) and Eilenberg-Zilber’s theorem 7.4.50. On the other hand, the assumptions on T yield a natural isomorphism

S ⊗_{R} LT_{R}(M) = S ⊗_{R} T_{R}((⊥_{•}^{R} M)^{Δ}) →^{∼} T_{S}((S⊗_{R} ⊥_{•}^{R} M)^{Δ}) in D(S-Mod).

By combining these isomorphisms, we get an isomorphism S ⊗_{R} LT_{R}M → LT_{S}(S ⊗_{R} M), and a simple inspection shows that the latter is realized by the natural map (7.10.37). □

7.10.38. Let now R be any simplicial A-algebra, and I ⊂ R any ideal, and set R_{0} := R/I. The positive Rees algebra of I is the graded R-algebra

R^{+}(R, I)^{•} := ⨁_{p∈ℕ} I^{p}

(cp. definition 7.9.1(iii) and example 7.9.3). Notice the natural isomorphism of graded R_{0}- algebras

R^{+}(R, I) ⊗_{R} R_{0} →^{∼} gr_{I}^{•}R := ⨁_{p∈ℕ} I^{p}/I^{p+1}

as well as the exact sequence of R^{+}(R, I)-modules

0 → gr_{I}^{•+1}R → R^{+}(R, I) ⊗_{R} R/I^{2} → gr_{I}^{•}R → 0

deduced from the natural projection R/I^{2} → R_{0}. Next, pick any quasi-isomorphism of Ralgebras P → R_{0}, with P a flat R-algebra; there follows an exact sequence of P ⊗_{R} R^{+}(R, I)- modules

0 → P ⊗_{R} gr_{I}^{•+1}R → P ⊗_{R} R^{+}(R, I) ⊗_{R} R/I^{2} ⟶^{β} P ⊗_{R} gr_{I}^{•}R → 0

and notice that β is actually a morphism of P ⊗_{R} R^{+}(R, I)-algebras. According to remark 7.7.10(i), after forming the associated cochain complexes, we obtain an epimorphism β^{•} of differential graded (P ⊗_{R} R^{+}(R, I))^{•}-algebras, whose kernel (P ⊗_{R} gr_{I}^{•+1}R)^{•} is a two-sided

ideal of (P ⊗_{R} R^{+}(R, I)⊗_{R} R/I^{2})^{•}. In this situation, we deduce a map of H_{•}(R_{0} ⊗^{ℓ}_{R} R^{+}(P, I))- modules

(7.10.39)  δ : H_{•}(R_{0} ⊗^{ℓ}_{R} gr_{I}^{•}R) → H_{•}(R_{0} ⊗^{ℓ}_{R} gr_{I}^{•+1}R)[1]

which is a graded derivation, according to lemma 7.7.6(recall that here the shift [1] refers to the homological grading; the notation gr^{•+1} denotes also a shift in degrees, which however does not alter the signs of the scalar multiplication in the way prescribed by remark 7.7.4(v)).

7.10.40. Keep the situation of (7.10.38), and suppose furthermore that R is a constant Aalgebra and I a constant ideal, i.e. R = s.B, and I = s.J for some A-algebra B and some ideal J ⊂ B. We set B_{0} := B/J and

G^{•} := R_{0} ⊗^{ℓ}_{R} gr_{I}^{•}R  Λ_{•} := Λ_{B}^{•}(J/J^{2})  S^{•} := Sym_{B}^{•}(J/J^{2})

where the derived tensor product is formed in D(R-Alg), as in example 7.10.25, so G^{•} is represented by P ⊗_{R} gr_{I}^{•}R, and H_{•}G^{•} is a bigraded A-algebra, strictly anti-commutative for the (homological) grading H_{•}, and commutative for the (cohomological) grading deduced from the grading of G^{•} (see remark 7.7.10(i)). Also, Λ_{•} (resp. S^{•}) is a strictly anti-commutative (resp. a commutative) graded B_{0}-algebra, and we use the homological (resp. cohomological) conventions for grading, so Λ_{p} (resp. S^{p}) is placed in degree −p (resp. p). Moreover, a standard calculation yields a natural isomorphism of B_{0}-modules

H_{0}G^{0} →^{∼} B_{0} H_{1}G^{0} →^{∼} Tor_{1}^{B}(B_{0}, B_{0}) →^{∼} J/J^{2}

whence a natural map Λ_{•} → H_{•}G^{0} of strictly anti-commutative graded B_{0}-algebras, restricting to an isomorphism in degrees ≤ 1. On the other hand, we have a surjective map of commutative graded B_{0}-algebras

(7.10.41)  S^{•} → gr_{J}^{•}B := ⨁_{p∈ℕ} J^{p}/J^{p+1}

restricting as well to an isomorphism in degrees ≤ 1; since P ⊗_{R} gr_{I}^{•}R is naturally a simplicial gr_{J}^{•}B-algebra, we obtain a natural map of bigraded S^{•}-algebras

ω_{•}^{•} : Λ_{•} ⊗_{B} S^{•} → H_{•}G^{•}.

Now, H_{•}G^{•} has been endowed with a derivation δ : H_{•}G^{•} → H_{•}G^{•+1}[1] in (7.10.39); on the other hand, there exists on Λ_{•} ⊗_{B} S^{•} a natural S^{•}-linear graded derivation

∂ : Λ_{•} ⊗_{B} S^{•} → Λ_{•} ⊗_{B} S^{•+1}[1]

as explained in [103, I.4.3.1.2]. We may then consider the diagram of S^{•}-linear maps

(7.10.42)  \xymatrix{Λ_{•} ⊗_{B} S^{•} \ar[r]^{ω_{•}^{•}} \ar[d]_{∂} & H_{•}G^{•} \ar[d]^{δ} \\ Λ_{•} ⊗_{B} S^{•+1}[1] \ar[r]^{ω_{•}^{•+1}[1]} & H_{•}G^{•+1}[1].}

Definition 7.10.43. (i) In the situation of (7.10.40), we say that the ideal J is quasi-regular, if the following conditions hold :

• The map ω_{•}^{•} restricts to an isomorphism ω_{•}^{0} : Λ_{•} →^{∼} H_{•}G^{0}.

• The B_{0}-module J/J^{2} is flat. (ii) Let R be any simplicial A-algebra, and I ⊂ R any ideal. We say that I is quasi-regular,

if I[n] is a quasi-regular ideal of R[n], for every n ∈ ℕ.

Remark 7.10.44. (i) Definition 7.10.43 is due to Quillen ([139, Def.8.4]). Notice that a sequence of elements (f_{1}, …, f_{r}) in a ring B is B-quasi-regular, in the sense of definition 7.8.13, if and only if it generates an ideal J ⊂ B such that J/J^{2} is a free B_{0}-module of rank r, and (7.10.41) is an isomorphism (notation of (7.10.40)). The relationship between these two quasiregularity notions is partially elucidated by the following proposition 7.10.45(ii).

(ii) Suppose that \underline{B} := (B_{i} | i ∈ I) is any filtered system of rings, and \underline{J} := (J_{i} ⊂ B_{i} | i ∈ I) a filtered system of ideals, such that J_{i} is quasi-regular in B_{i}, for every i ∈ I. Denote by B (resp. J) the colimit of \underline{B} (resp. of \underline{J}); then it is easily seen that J is quasi-regular in B.

(iii) Let B be any ring, and C := B[X_{i} | i ∈ I] a free B-algebra (for any set I). Then the ideal J := (X_{i} | i ∈ I) ⊂ C is quasi-regular. Indeed, (ii) reduces to the case where I is a finite set, for which the assertion is a special case of proposition 7.10.47below.

Proposition 7.10.45. In the situation of (7.10.40), we have :

(i) (7.10.42) is a commutative diagram.

(ii) Suppose that J is a quasi-regular ideal. Then (a) ω_{•}^{•} is an isomorphism.

(b) There exists a natural isomorphism of B_{0}-modules :

Tor_{i}^{B}(B_{0}, B/J^{n}) →^{∼} Coker(∂_{i+1}^{n−2} : Λ_{i+1} ⊗_{B} S^{n−2} → Λ_{i} ⊗_{B} S^{n−1}) for every n, i > 0.

Proof. (i): Since both derivations are S^{•}-linear, it suffices to check that (7.10.42) commutes in (upper) degree 0. Moreover, since the B-algebra Λ_{•} is generated by Λ_{1}, it suffices to check the commutativity of the diagram

(7.10.46)  \xymatrix{Λ_{1} \ar[r]^{ω_{1}^{0}} \ar[d]_{∂_{1}^{0}} & H_{1}G^{0} \ar[d]^{δ_{1}^{0}} \\ S^{1} \ar[r]^{ω_{0}^{1}} & H_{0}G^{1}.}

However, according to [103, I.4.3.1.2], the map ∂_{1}^{0} is just the identity of J/J^{2}. The map δ_{1}^{0} is the boundary map arising – by the snake lemma – from the exact sequence of complexes

0 → JP^{•}/J^{2}P^{•} → P^{•}/J^{2}P^{•} → P^{•}/JP^{•} → 0

(where P^{•} denotes the cochain complex associated with the flat resolution P → s.B_{0}). The same boundary map is used to define the isomorphism ω_{1}^{0}, and ω_{0}^{1} is the natural isomorphism J/J^{2} → H_{0}(P^{•}/JP^{•}). Summing up, the commutativity of (7.10.46) follows by simple inspection.

(ii.a): We prove, by induction on n ∈ ℕ, that every ω_{•}^{n} is an isomorphism. For n = 0, 1 there is nothing to show, so suppose that n > 1, and the assertion is already known for every degree < n. In order to prove the assertion in degree n, it suffices to check that ω_{0}^{n} : S^{n} → gr_{J}^{n}B is an isomorphism; indeed, in this case gr_{J}^{n}B is a flat B_{0}-module, and then a standard spectral sequence argument allows to conclude that ω_{j}^{n} is also an isomorphism, for every j ∈ ℕ.

Let P → R_{0} be a resolution as in (7.10.38). The I-adic filtration has finite length on P/I^{k}P, for every k ∈ ℕ, and therefore yields a convergent spectral sequence

E_{pq}^{1} ⇒ Tor_{p+q}^{B}(B_{0}, B/J^{k})

with

E_{pq}^{1} := \{ \begin{matrix}Tor_{p+q}^{B}(B_{0}, gr_{J}B^{−p}) & for p > −k \\ 0 & otherwise \end{matrix}

whose differentials E_{pq}^{1} → E_{p−1,q}^{1} agree – by direct inspection – with the derivation δ_{p+q}^{−p}, whenever p > 1 − k. Especially, for k = n + 1, we get a complex

Σ : H_{2}G^{n−2} ⟶^{δ_{2}^{n−2}} H_{1}G^{n−1} ⟶^{δ_{1}^{n−1}} gr_{J}^{n}B → 0.

On the other hand, since J/J^{2} is a flat B_{0}-module, the corresponding sequence

Σ′ : Λ_{2} ⊗_{B} S^{n−2} ⟶^{∂} Λ_{1} ⊗_{B} S^{n−1} ⟶^{∂} S^{n} → 0

is an exact complex (this is a segment of the Koszul complex of [103, I.4.3.1.7]). In light of (i), the maps ω_{2}^{n−2} and ω_{1}^{n−1} yield a morphism of complexes Σ′ → Σ, so we are reduced to checking that Σ is exact, and the inductive assumption already says that Σ is exact at the middle term H_{1}G^{n−1}. Now notice that, since the Koszul complex is exact, the inductive assumption implies that E_{pq}^{2} = 0 for all (p, q) such that p > 1 − n and q > 0; moreover, obviously we have E_{pq}^{1} = 0 for p + q < 0 or p > 0, therefore

E_{00}^{∞} = E_{00}^{1} and E_{−n,n}^{∞} = E_{−n,n}^{2} = Coker δ_{1}^{n−1}.

But a simple inspection shows that the natural map H_{0}P → E_{00}^{∞} is an isomorphism, so E_{−n,n}^{∞} = 0, therefore δ_{1}^{n−1} is surjective, and Σ is exact, as required.

(ii.b): We use the previous spectral sequence, for k = n. Taking into account (ii.a), and the exactness of the Koszul complex ([103, I.4.3.1.7]) we get

E_{pq}^{1} = 0  for p + q < 0 or p > 0 or p ≤ −n
E_{p,q}^{2} = 0  for every (p, q) with p > 1 − n and q > 0
E_{1−n,i+n−1}^{2} →^{∼} Coker ∂_{i+1}^{n−2}  for every i ∈ ℕ.

It follows that E_{1−n+r,i+n−r}^{r} = 0 for every r ≥ 2 and every i ∈ ℕ; indeed, this is clear if 1 − n + r > 0, and if the latter fails, we get 1 − n + r > 1 − n and i + n − r > 0, so the stated vanishing holds nevertheless. We deduce that E_{1−n,i+n−1}^{∞} = E_{1−n,i+n−1}^{2} for every i ∈ ℕ. Next, suppose that i > 0. Clearly we have E_{p,i−p}^{∞} = 0 both for p > 0 and p ≤ −n; and if 0 ≥ p > 1 − n, we get i − p ≥ i > 0, so again E_{p,i−p}^{∞} = 0. In conclusion, E_{p,i−p}^{∞} = 0 except possibly for p = 1 − n, and the contention follows. □

Proposition 7.10.47. Let B be a ring, f_{•} := (f_{1}, …, f_{r}) a finite sequence of elements of B. Denote by J the ideal generated by f_{•}, set B_{0} := B/J, and consider the following conditions :

(a) The sequence f_{•} is B-regular.

(b) The sequence f_{•} is completely secant in B.

(c) The ideal J is quasi-regular and J/J^{2} is a free B_{0}-module of rank r.

(d) The sequence f_{•} is B-quasi-regular. Then (a)⇒(b)⇒(c)⇒(d), and if B is J-adically complete and separated, then (d)⇒(a).

Proof. (a)⇒(b) and (c)⇒(d) have already been pointed out respectively in proposition 7.8.7 and proposition 7.10.45(ii.a), and the same for the implication (d)⇒(a), when B is J-adically complete and separated.

(b)⇒(c): This becomes a paraphrase of proposition 7.8.10, once we have identified the

graded B-algebra H_{•}G^{•} with the graded B-algebra H_{•}(B_{0}[0] ⊗^{L}_{B} B_{0}[0]). However, denote by P^{•} the cochain complex associated with the flat resolution P → s.B_{0}; according to remark 7.7.10(i), P^{•} is naturally a differential graded B-algebra, and the induced map ε^{•} : P^{•} → B_{0}[0] is a morphism of differential graded B-algebras (for the multiplication law \overline{μ}^{•} on B_{0}[0] inherited from that of B_{0}). Then there exists in D(B-Mod) a unique isomorphism ω^{•} : P_{B_{0}}^{•} → P^{•} whose composition with ε^{•} agrees with ρ_{B_{0}}^{•} (notation of (7.3.38)), and moreover we get a commutative diagram in D(B-Mod) :

\xymatrix{P_{B_{0}}^{•} ⊗_{B} P_{B_{0}}^{•} \ar[r]^{P • ; \overline{μ}} \ar[d]_{ω^{•}⊗_{B}ω^{•}} & P_{B_{0}}^{•} \ar[d]^{ω^{•}} \\ P^{•} ⊗_{B} P^{•} \ar[r] & P^{•}}

whose bottom horizontal arrow is the multiplication map of P^{•}, and where P • ; \overline{μ} is defined as in (7.3.38). The assertion follows immediately. □

The following is a well known theorem of Quillen, which is found in his typewritten notes [139, Th.8.8] that have been circulated since the late 60s, but have never been published. A version of this theorem, under more restrictive assumptions, is also found in [3, Ch.XIII, Prop.3].

Theorem 7.10.48. Let R be any simplicial A-algebra, and I ⊂ R a quasi-regular ideal such that H_{0}I = 0. Then we have :

H_{i}I^{n} = 0 for every n ∈ ℕ and every i < n.

Proof. We argue by induction on n ∈ ℕ. For n ≤ 1 there is nothing to prove, so suppose that n > 1, and that the assertion is already known for I^{n−1}. Set R_{k} := R/I^{k+1} for every k ∈ ℕ, and let

Λ_{•} ⊗_{R} S^{•} := Λ_{R}^{•}(I/I^{2}) ⊗_{R} Sym_{R}^{•}(I/I^{2})

which is a bigraded R_{0}-module, endowed with the bigraded derivation ∂_{•}^{•} such that ∂_{•}^{•}[i] is the Koszul derivation of Λ_{•} ⊗_{R} S^{•}[i], defined as in [103, I.4.3.1.2]. Set as well

C_{i} := Coker (∂_{i}^{n−3} : Λ_{i} ⊗_{R} S^{n−3} → Λ_{i−1} ⊗_{R} S^{n−2}) for every i ∈ ℕ.

From the inductive assumption and lemma 7.10.32, we already know that

(7.10.49)  H_{i}(I ⊗^{ℓ}_{R} I^{n−1}) = 0  for every i < n.

However, according to remark 7.10.24, there is a spectral sequence

E_{pq}^{1} := Tor_{q}^{R[p]}(I[p], I[p]^{n−1}) ⇒ H_{p+q}(I ⊗^{ℓ}_{R} I^{n−1})

and proposition 7.10.45(ii.b) yields natural isomorphisms

E_{pq}^{1} →^{∼} Tor_{q+1}^{R[p]}(R_{0}[p], I[p]^{n−1}) →^{∼} Tor_{q+2}^{R[p]}(R_{0}[p], R_{n−2}[p]) →^{∼} C_{q+3}[p] for every q > 0

under which, the differentials E_{pq}^{1} → E_{p−1,q}^{1} are identified with those of the chain complex defining the R_{0}-module C_{q+3}. Now, since I/I^{2} is a flat R_{0}-module, [103, I.4.3.1.7] says that the sequence of R_{0}-modules

Σ^{i} : 0 → Λ_{n+i−3} ⟶^{∂} Λ_{n+i−4} ⊗_{R} S^{1} → ⋯ → Λ_{i} ⊗_{R} S^{n−3} ⟶^{∂} Λ_{i−1} ⊗_{R} S^{n−2} → C_{i} → 0

is an exact complex, for every i > 0. On the other hand, since H_{0}I/I^{2} = 0, (and again, since I/I^{2} is a flat R_{0}-module), combining [103, I.4.3.2.1] and lemma 7.10.32, we see that

H_{i}(Λ_{j} ⊗_{R} S^{q−j}) = 0 for every q ∈ ℕ, every j ≤ q, and every i < q.

We recall the following general

Claim 7.10.50. Let 𝒜 be any abelian category, d ∈ ℕ any integer, and consider an exact sequence

0 → K_{d} ⟶^{f_{d}} K_{d−1} → ⋯ → K_{1} ⟶^{f_{1}} K_{0} → 0

of objects of C(𝒜), such that K_{i} = 0 in D(𝒜), for every i = 1, …, d − 1. Then there is a natural isomorphism

K_{0} →^{∼} K_{d}[d − 1] in D(𝒜) .

Proof of the claim. For every i = 0, …, d, set Z_{i} := Ker f_{i}. We have short exact sequences

0 → Z_{i} → K_{i} → Z_{i−1} → 0 for i = 1, …, d.

Since K_{i} = 0 in D(𝒜), there follows natural isomorphisms Z_{i−1} →^{∼} Z_{i}[1] for every i = 1, …, d − 1. However, Z_{0} = K_{0} and Z_{d−1} = K_{d}, whence the claim. ♢

By applying claim 7.10.50to the exact sequence cosk_{n+i−3}Σ^{i}, we easily deduce that

H_{j}C_{i} = 0 for every i > 0 and every j < n + i − 3

so finally E_{pq}^{2} = 0 for every q > 0 and every p < n + q, and clearly E_{pq}^{1} = 0 for every q < 0. Consequently, for every i < n the term E_{i−q,q}^{∞} = 0 vanishes for q > 0, and equals E_{i,0}^{2} for q = 0. Lastly, a simple calculation shows that

E_{i,0}^{2} = H_{i}(I ⊗_{R} I^{n−1}) for every i ∈ ℕ.

Summing up, and taking into account 7.10.49, we conclude that H_{i}(I ⊗_{R} I^{n−1}) = 0 for every i < n, so we are reduced to checking :

Claim 7.10.51. The natural map H_{i}(I ⊗_{R} I^{n−1}) → H_{i}(I^{n}) is surjective, for every i ≤ n.

Proof of the claim. Indeed, denote by K the kernel of the epimorphism I ⊗_{R} I^{n−1} → I^{n}; as in the foregoing, we get natural isomorphisms

K[i] →^{∼} Tor_{1}^{R[i]}(R_{0}[i], I[i]^{n−1}) →^{∼} Tor_{2}^{R[i]}(R_{0}[i], R[i]/I[i]^{n−1}) →^{∼} C_{3}[i] for every i ∈ ℕ

that amount to an isomorphism K →^{∼} C_{3} of R_{0}-modules. We have already seen that H_{i}C_{3} = 0 for every i < n, whence the claim. □

7.11. Injective modules, flat modules and indecomposable modules.

7.11.1. Indecomposable modules. Recall that a unital (not necessarily commutative) ring R is said to be local if R ≠ 0 and, for every x ∈ R either x or 1−x is invertible. If R is commutative, this definition is equivalent to the usual one.

7.11.2. Let 𝒞 be any abelian category and M an object of 𝒞. One says that M is indecomposable if it is non-zero and cannot be presented in the form M = N_{1} ⊕ N_{2} with non-zero objects N_{1} and N_{2}. If such a decomposition exists, then End_{𝒞} (N_{1}) × End_{𝒞} (N_{2}) ⊂ End_{𝒞} (M), especially the unital ring End_{𝒞} (M) contains an idempotent element e ≠ 1, 0 and therefore it is not a local ring.

However, if M is indecomposable, it does not necessarily follow that End_{𝒞} (M) is a local ring. Nevertheless, one has the following:

Theorem 7.11.3 (Krull-Remak-Schmidt). Let (A_{i})_{i∈I} and (B_{j})_{j∈J} be two finite families of objects of 𝒞, such that:

(a) ⊕_{i∈I}A_{i} ≃ ⊕_{j∈J}B_{j}.

(b) End_{𝒞} (A_{i}) is a local ring for every i ∈ I, and B_{j} ≠ 0 for every j ∈ J.

Then we have :

(i) There is a surjection φ : I → J, and isomorphisms B_{j} →^{∼} ⊕_{i∈φ^{−1}(j)}A_{i}, for every j ∈ J. (ii) Especially, if B_{j} is indecomposable for every j ∈ J, then I and J have the same

cardinality, and φ is a bijection.

Proof. Clearly, (i)⇒(ii). To show (i), let us begin with the following :

Claim 7.11.4. Let M_{1}, M_{2} be two objects of 𝒞, and set M := M_{1} ⊕ M_{2}. Denote by e_{i} : M_{i} → M (resp. p_{i} : M → M_{i}) the natural injection (resp. projection) for i = 1, 2. Suppose that α : P → M is a subobject of M, such that p_{1}α : P → M_{1} is an isomorphism. Then the natural morphism β : P ⊕ M_{2} → M is an isomorphism.

Proof of the claim. Denote by π_{P} : P ⊕M_{2} → P and π_{2} : P ⊕M_{2} → M_{2} the natural projections; then β := απ_{P} + e_{2}π_{2}. Of course, the assertion follows easily by applying the 5-lemma (which holds in any abelian category) to the commutative ladder with exact rows :

[figure: 0; M_{2}; P ⊕ M_{2}; π_{P}; β; P; p_{1}α; 0; 0; M_{2}; e_{2}; M; p_{1}; M_{1}; 0.]

Equivalently, one can argue directly as follows. By definition, Coker β represents the functor

𝒞 → ℤ-Mod : X ↦ Ker Hom_{𝒞} (β, X)

and likewise for Coker p_{1}α; however, it is easily seen that these functors are naturally isomorphic, hence the natural morphism Coker β → Coker p_{1}α is an isomorphism, so β is an epimorphism. Next, let t : Ker β → P ⊕ M_{2} be the natural morphism; since p_{1}α is an isomorphism, π_{P} ∘ t = 0, so t factors through a morphism \overline{t} : Ker β → M_{2}. It follows that e_{2} ∘ \overline{t} = β ∘ t = 0, therefore \overline{t} = 0, and finally Ker β = 0, since t is a monomorphism. ♢

Set A := ⊕_{i∈I}A_{i}, and denote e_{i} : A_{i} → A (resp. p_{i} : A → A_{i}) the natural injection (resp. projection), for every i ∈ I. Also, endow I′ := I ∪ {∞} with any total ordering (I′, <) for which ∞ is the maximal element.

Claim 7.11.5. Let (a_{j} : A → A)_{j∈J} be a finite system of endomorphisms such that ∑_{j∈J} a_{j} = 1_{A}. Then there exists a mapping φ : I → J such that the following holds :

(i) a_{φ(i)}e_{i} : A_{i} → A is a monomorphism, for every i ∈ I, and let P_{i} := Im(a_{φ(i)}e_{i}). (ii) For every l ∈ I′, the natural morphism β_{l} : ⨁_{i<l} P_{i}⊕⨁_{i≥l} A_{i} → A is an isomorphism.

Proof of the claim. We both define φ(l) and prove (ii), by induction on l. If l is the smallest element of I, then β_{l} = 1_{A}, so (ii) is obvious. Hence, let l ∈ I be any element, l′ ∈ I′ the successor of l, and assume that β_{l} is an isomorphism. Set a′_{j} := β_{l}^{−1}a_{j}β_{l} for every j ∈ J, and M_{l} := ⨁_{i<l} P_{i} ⊕ ⨁_{i≥l} A_{i}; also, let e′_{l} : A_{l} → M_{l} and p′_{l} : M_{l} → A_{l} be the natural morphisms. Clearly ∑_{j∈J} a′_{j} = 1_{M_{l}}, so ∑_{j∈J} p′_{l}a′_{j}e′_{l} = 1_{A_{l}}; since End_{𝒞} (A_{l}) is a local ring, it follows that there exists j_{l} ∈ J such that p′_{l}a′_{j_{l}}e′_{l} is an automorphism of A_{l}. However,

(7.11.6)  a′_{j_{l}}e′_{l} = β_{l}^{−1}a_{j_{l}}β_{l}e′_{l} = β_{l}^{−1}a_{j_{l}}e_{l}

so assertion (i) follows for the index l, by setting φ(l) := j_{l}.

Next, set M_{l,1} := A_{l} and M_{l,2} := ⨁_{i<l} P_{i} ⊕ ⨁_{i>l} A_{i}. The foregoing argument provides a subobject α : P′_{l} := Im(a′_{φ(l)}e′_{l}) → M_{l} = M_{l,1} ⊕ M_{l,2} such that p′_{l}α : P′_{l} → M_{l,1} is an isomorphism. By claim 7.11.4, it follows that the natural map β′_{l′} : P′_{l} ⊕ M_{l,2} → M_{l} is an isomorphism. On the other hand, (7.11.6) implies that β_{l} restricts to an isomorphism γ_{l} : P′_{l} →^{∼} P_{l}. We deduce a commutative diagram

\xymatrix{P′_{l} ⊕ M_{l,2} \ar[r]^{β′_{l′}} \ar[d]_{γ_{l}⊕1_{M_{l,2}}} & M_{l} \ar[d]^{β_{l}} \\ M_{l′} \ar[r]^{β_{l′}} & A}

whose vertical arrows and top arrow are isomorphisms. Thus, the bottom arrow is an isomorphism as well, as required. ♢

For every j ∈ J, let e′_{j} : B_{j} → A (resp. p′_{j} : A → B_{j}) be the injection (resp. projection) deduced from a given isomorphism as in (a), and set a_{j} := e′_{j}p′_{j} for every j ∈ J. Then ∑_{j∈J} a_{j} = 1_{A}, so claim 7.11.5 yields a mapping φ : I → J such that the following holds. Set P_{j} := ⊕_{i∈φ^{−1}(j)}Im(a_{j}e_{i}) for every j ∈ J; then the natural morphism β : ⊕_{j∈J}P_{j} →^{∼} A is an isomorphism. However, clearly β(P_{j}) is a subobject of Im(e′_{j}), for every j ∈ J. Hence, φ must be a surjection, and β restricts to isomorphisms P_{j} →^{∼} Im(e′_{j}) for every j ∈ J, whence isomorphisms P_{j} →^{∼} B_{j}, from which (ii) follows immediately. □

Remark 7.11.7. There are variants of theorem 7.11.3, that hold under different sets of assumptions. For instance, in [76, Ch.I, §6, Th.1] it is stated that part (ii) of the theorem still holds when I and J are small (not necessarily finite) sets, provided that 𝒞 admits a small set of generators and that all small filtered colimits are representable and exact in 𝒞. (One can moreover show that the latter result still holds even in the absence of a small set of generators for 𝒞.)

7.11.8. Let us now specialize to the case of the category A-Mod, where A is a (commutative) local ring, say with residue field k. Let M be any finitely generated A-module; arguing by induction on the dimension of M ⊗_{A} k, one shows easily that M admits a finite decomposition M = ⊕_{i=1}^{r}M_{i}, where M_{i} is indecomposable for every i ≤ r.

Lemma 7.11.9. Let A be a henselian local ring, and M an A-module of finite type. Then M is indecomposable if and only if End_{A}(M) is a local ring.

Proof. In view of (7.11.2), we can assume that M is indecomposable, and we wish to prove that End_{A}(M) is local. Thus, let φ ∈ End_{A}(M).

Claim 7.11.10. The subalgebra A[φ] ⊂ End_{A}(M) is integral over A.

Proof of the claim. This is standard: choose a finite system of generators (f_{i})_{1≤i≤r} for M, then we can find a matrix a := (a_{ij})_{1≤i,j≤r} of elements of A, such that φ(f_{i}) = ∑_{j=1}^{r} a_{ij}f_{j} for every i ≤ r. The matrix a yields an endomorphism ψ of the free A-module A^{⊕r}; on the other hand, let e_{1}, …, e_{r} be the standard basis of A^{⊕r} and define an A-linear surjection π : A^{⊕r} → M by the rule: e_{i} ↦ f_{i} for every i ≤ r. Then φ ∘ π = π ∘ ψ; by Cayley-Hamilton, ψ is annihilated by the characteristic polynomial χ(T) of the matrix a, whence χ(φ) = 0. ♢

Since A is henselian, claim 7.11.10implies that A[φ] decomposes as a finite product of local rings. If there were more than one non-zero factor in this decomposition, the ring A[φ] would contain an idempotent e ≠ 1, 0, whence the decomposition M = eM ⊕ (1 − e)M, where both summands would be non-zero, which contradicts the assumption. Thus, A[φ] is a local ring, so that either φ or 1_{M} − φ is invertible. Since φ was chosen arbitrarily in End_{A}(M), the claim follows. □

Corollary 7.11.11. Let A be a henselian local ring. Then:

(i) If (M_{i})_{i∈I} and (N_{j})_{j∈J} are two finite families of indecomposable A-modules of finite type such that ⊕_{i=1}^{r}M_{i} ≃ ⊕_{j=1}^{s}N_{j}, then there is a bijection β : I →^{∼} J such that M_{i} ≃ N_{β(i)} for every i ∈ I.

(ii) If M and N are two finitely generated A-modules such that M^{⊕k} ≃ N^{⊕k} for some

integer k > 0, then M ≃ N. (iii) If M, N and X are three finitely generated A-modules such that X ⊕ M ≃ X ⊕ N,

then M ≃ N.

Proof. It follows easily from theorem 7.11.3and lemma 7.11.9; the details are left to the reader. □

Proposition 7.11.12. Let A → B be a faithfully flat map of local rings, M and N two finitely presented A-modules with a B-linear isomorphism ω : B ⊗_{A} M →^{∼} B ⊗_{A} N. Then M ≃ N.

Proof. Under the standing assumptions, the natural B-linear map

B ⊗_{A} Hom_{A}(M, N) → Hom_{B}(B ⊗_{A} M, B ⊗_{A} N)

is an isomorphism ([75, Lemma 2.4.29(i.a)]). Hence we can write ω = ∑_{i=1}^{r} b_{i} ⊗ φ_{i} for some b_{i} ∈ B and φ_{i} : M → N (1 ≤ i ≤ r). Denote by k and K the residue fields of A and B respectively; we set \overline{φ}_{i} := 1_{k} ⊗_{A} φ_{i} : k ⊗_{A} M → k ⊗_{A} N. From the existence of ω we deduce easily that n := dim_{k} k ⊗_{A} M = dim_{k} k ⊗_{A} N. Hence, after choosing bases, we can view \overline{φ}_{1}, …, \overline{φ}_{r} as endomorphisms of the k-vector space k^{⊕n}. We consider the polynomial p(T_{1}, …, T_{r}) := det(∑_{i=1}^{r} T_{i} · \overline{φ}_{i}) ∈ k[T_{1}, …, T_{r}]. Let \overline{b}_{1}, …, \overline{b}_{r} be the images of b_{1}, …, b_{r} in K; it follows that p(\overline{b}_{1}, …, \overline{b}_{r}) ≠ 0, especially p(T_{1}, …, T_{r}) ≠ 0.

Claim 7.11.13. The proposition holds if k is an infinite field or if k = K.

Proof of the claim. Indeed, in either of these cases we can find \overline{a}_{1}, …, \overline{a}_{r} ∈ k such that p(\overline{a}_{1}, …, \overline{a}_{r}) ≠ 0. For every i ≤ r choose an arbitrary representative a_{i} ∈ A of \overline{a}_{i}, and set φ := ∑_{i=1}^{r} a_{i}φ_{i}. By construction, 1_{k} ⊗_{A} φ : k ⊗_{A} M → k ⊗_{A} N is an isomorphism. By Nakayama’s lemma we deduce that φ is surjective. Exchanging the roles of M and N, the same argument yields an A-linear surjection ψ : N → M. Finally, [126, Ch.1, Th.2.4] shows that both ψ ∘ φ and φ ∘ ψ are isomorphisms, whence the claim. ♢

Let A^{h} be the henselization of A, and A^{sh}, B^{sh} the strict henselizations of A and B respectively. Now, the induced map A^{sh} → B^{sh} is faithfully flat, the residue field of A^{sh} is infinite, and ω induces a B^{sh}-linear isomorphism B^{sh} ⊗_{A} M →^{∼} B^{sh} ⊗_{A} N. Hence, claim 7.11.13 yields an A^{sh}-linear isomorphism β : A^{sh} ⊗_{A} M →^{∼} A^{sh} ⊗_{A} N. However, A^{sh} is the colimit of a filtered family (A_{λ} | λ ∈ Λ) of finite étale A^{h}-algebras, so β descends to an A_{λ}-linear isomorphism β_{λ} : A_{λ} ⊗_{A} M ≃ A_{λ} ⊗_{A} N for some λ ∈ Λ. The A^{h}-module A_{λ} is free, say of finite rank n, hence β_{λ} can be regarded as an A^{h}-linear isomorphism (A^{h} ⊗_{A} M)^{⊕n} →^{∼} (A^{h} ⊗_{A} N)^{⊕n}. Then A^{h} ⊗_{A} M ≃ A^{h} ⊗_{A} N, by corollary 7.11.11(ii). Since the residue field of A^{h} is k, we conclude by another application of claim 7.11.13. □

7.11.14. Injective hulls. The notion of injective hull plays a central role in the theory of local duality : for a noetherian local ring, one constructs a dualizing module as the injective hull of the residue field (see [85, Exp.IV, Th.4.7]), and in section 11.6, injective hulls of the residue fields of a monoid algebra will also enable us to perform a certain computation of local cohomology, which is a crucial step in the proof of Hochster’s theorem. We present here the basic results on injective hulls, in the context of arbitrary abelian categories.

Definition 7.11.15. Let 𝒜 be any abelian category, and f : N → M a monomorphism in 𝒜.

(i) We say that M is an essential extension of N if the following holds. For any subobject P ⊂ M we have either P = 0 or P ∩ Im f ≠ 0 (here 0 denotes the zero object of 𝒜 : see remark 3.7.29(i)).

(ii) We say that M is a proper essential extension of N if it is an essential extension of N,

and f is not an isomorphism. (iii) We say that M is an injective hull of N, if M is both an essential extension of N, and

an injective object of 𝒜.

Lemma 7.11.16. Let 𝒜 be an abelian category, and I an object of 𝒜. Suppose that : (a) 𝒜 is cocomplete.

(b) All colimits of 𝒜 are universal (see example 1.4.17). Then we have :

(i) I is injective if and only if it does not admit any proper essential extensions.

(ii) If N → M is any monomorphism, the set of all essential extensions of N contained in M admits maximal elements.

Proof. (i): Suppose that I is injective, and let f : I → M be any monomorphism which is not an isomorphism. Then f admits a left inverse, so I is a direct summand of M, hence M is not an essential extension of I. Conversely, suppose that I does not admit proper essential extensions; let f : N → M be a monomorphism in 𝒜 and g : N → I any morphism in 𝒜. We consider the cocartesian diagram in 𝒜

\xymatrix{N \ar[r]^{f} \ar[d]_{g} & M \ar[d]^{g′} \\ I \ar[r]^{f′} & P}

and notice that f′ is a monomorphism, since the same holds for f. By Zorn’s lemma – and due to conditions (a) and (b) – we may find a maximal subobject Q ⊂ P such that Q ∩ I = 0, and clearly P/Q is an essential extension of I (via h); therefore P/Q = I, so P = I ⊕ Q, and if we let p : P → I be the resulting projection, we see that p ∘ g′ : M → I is an extension of f. This shows that I is injective.

(ii): By Zorn’s lemma, it suffices to check the following assertion. Suppose that (P_{j} | j ∈ J) is a non-empty totally ordered family of essential extensions of N contained in M (for some totally ordered small indexing set J). Then P := ⋃_{j∈J} P_{j} is again an essential extension of N (notice that this union – i.e. colimit – is a subobject of M, due to condition (b)). However, say that Q ⊂ P and Q ∩ N = 0; due to condition (b), we have Q = ⋃_{j∈J}(Q ∩ P_{j}), and clearly (Q ∩ P_{j}) ∩ N = 0, so Q ∩ P_{j} = 0 for every j ∈ J, and finally Q = 0. □

Proposition 7.11.17. Let 𝒜 be an abelian category fulfilling condition (a) and (b) of lemma 7.11.16, and M any object of 𝒜. We suppose moreover that :

(c) 𝒜 has enough injective objects. Then we have :

(i) M admits an injective hull. More precisely, if M → I is any monomorphism into an injective object, then a maximal essential extension of M in I is an injective hull of M. (ii) Let f : M → E be an injective hull of M, and g : M → I any monomorphism into an

injective object. Then g factors through f and a monomorphism h : E → I. (iii) If f : M → E and g : M → E′ are two injective hulls of M, there exists an isomor-

phism h : E →^{∼} E′ such that h ∘ f = f′.

Proof. (i): Let E ⊂ I be such a maximal essential extension of M (which exists by lemma 7.11.16(ii)); by virtue of lemma 7.11.16(i), it suffices to check that E does not admit proper essential extensions. However, suppose that E → E′ is a proper essential extension; since I is injective, the inclusion morphism E → I extends to a morphism f : E′ → I. By maximality of E, we must then have Ker f ≠ 0; on the other hand, obviously E ∩ Ker f = 0, which is absurd, since E′ is an essential extension of E.

(ii): Since I is injective, g extends to a morphism h : E → I, and clearly M ∩ Ker h = 0. Since E is an essential extension of M, it follows that Ker h = 0.

(iii): By (ii) there exists a monomorphism h : E → E′ such that h ∘ f = f′. Since E is injective, it follows that Im h is a direct summand of E′; but E′ is an essential extension of M, so Im h = E′, i.e. h is also an epimorphism, hence an isomorphism. □

7.11.18. We specialize now to the case where 𝒜 is the category A-Mod of A-modules, with A an arbitrary noetherian ring. Recall that if M is any A-module, then the set Ass M of all associated primes of M consists of the prime ideals 𝔭 ⊂ A such that there exists m ∈ M with Ann_{A}(m) = 𝔭. (This is the correct definition only for noetherian rings : we shall see in definition 10.5.1a more general notion that is well behaved for arbitrary rings.) By proposition 7.11.17(i,iii) the injective hull of M exists and is well defined up to (in general, non-unique) isomorphism, and we shall denote by E_{A}(M) a choice of such hull.

Lemma 7.11.19. Let A be a noetherian ring. The following holds :

(i) If (I_{λ} | λ ∈ Λ) is a (small) family of injective A-modules, then I := ⨁_{λ∈Λ} I_{λ} is an injective A-module.

(ii) If S ⊂ A is any multiplicative subset, and M any injective A-module, then M_{S} is an injective A_{S}-module.

Proof. (i): Let us first recall :

Claim 7.11.20. Let R be any ring, M an R-module. The following conditions are equivalent :

(a) M is an injective R-module.

(b) For every ideal J ⊂ R, every R-linear map J → M extends to an R-linear map R → M.

(c) Ext_{R}^{1}(R/J, M) = 0 for every ideal J ⊂ R.

Proof of the claim. Of course, (a)⇒(b)⇒(c). To check that (c)⇒(a), let N ⊂ P be an inclusion of R-modules, and f : N → M an A-linear map. We let S be the set of all pairs (N′, f′), where N′ ⊂ P is an R-submodule containing N, and f′ : N′ → M an R-linear map extending f. The set S is partially ordered, by declaring that (N′, f′) ≥ (N′′, f′′) if N′′ ⊂ N′ and f′_{|N′′} = f′′, for any two pairs (N′, f′), (N′′, f′′) ∈ S. By Zorn’s lemma, S admits a maximal element (Q, g). Suppose Q ≠ P, and let x ∈ P \ Q. Set Q′ := Q + Rx and J := Ann_{R}(Q′/Q); there follows an exact sequence of R-modules

0 → Q → Q′ → R/J → 0

and then (c) implies that the restriction map Hom_{R}(Q′, M) → Hom_{R}(Q, M) is surjective; especially, g extetnds to an R-linear map Q′ → M, contradicting the maximality of Q. Hence Q = P, which shows (a). ♢

In view of claim 7.11.20, it suffices to show that every A-linear map f : J → I from any ideal J ⊂ A, extends to an A-linear map A → I. However, since J is finitely generated, there exists a finite subset Λ′ ⊂ Λ such that the image of f is contained in I′ := ⨁_{λ∈Λ′} I_{λ}. Clearly I′ is an injective A-module, so f extends to an A-linear map A → I′, and the assertion follows.

(ii): Let J ⊂ A_{S} be any ideal, and write J = I_{S} for some ideal I ⊂ A; since A is noetherian, we have

Ext_{A_{S}}^{1}(A_{S}/J, M_{S}) = A_{S} ⊗_{A} Ext_{A}^{1}(A/I, M)

so the assertion follows from claim 7.11.20. □

We may now state :

Proposition 7.11.21. Let A be any noetherian ring, M any A-module. We have :

(i) For every 𝔭 ∈ Spec A, the A-module E_{A}(A/𝔭) is indecomposable (see (7.11.2)). (ii) Let I be any non-zero injective A-module, and 𝔭 ∈ Ass I any associated prime. Then E_{A}(A/𝔭) is a direct summand of I. Especially, if I is indecomposable, then I is isomorphic to E_{A}(A/𝔭).

(iii) Ass_{A}M = Ass_{A}E_{A}(M).

(iv) If 𝔭, 𝔮 ∈ Spec A are any two prime ideals, then the A-modules E_{A}(A/𝔭) and E_{A}(A/𝔮)

are isomorphic if and only if 𝔭 = 𝔮.

(v) E_{A_{S}}(M_{S}) ≃ A_{S} ⊗_{A} E_{A}(M) for any multiplicative subset S ⊂ A.

Proof. (i): Set B := A/𝔭, and suppose that E_{A}(B) is decomposable; especially, there exist non-zero submodules M_{1}, M_{2} ⊂ E_{A}(B) with M_{1} ∩ M_{2} = 0. Thus, (M_{1} ∩ B) ∩ (M_{2} ∩ B) = 0, and since B is a domain, we deduce that M_{i} ∩ B = 0 for i = 1, 2. But since E_{A}(B) is an essential extension of B, this is absurd.

(ii): By assumption, there exists x ∈ I such that Ann_{A}(x) = 𝔭, so the submodule Ax ⊂ I is isomorphic to A/𝔭; then, by proposition 7.11.17(ii) there exists a monomorphism f : E_{A}(A/𝔭) → I, and since E_{A}(A/𝔭) in injective, the image of f is a direct summand of I.

(iii): Clearly Ass_{A}(M) ⊂ Ass_{A}E_{A}(M). Conversely, suppose 𝔭 ∈ Ass_{A}E_{A}(M); then there exists an A-submodule N ⊂ E_{A}(M) isomorphic to A/𝔭. Since E_{A}(M) is an essential extension of M, we have N ∩ M ≠ 0, so 𝔭 ∈ Ass_{A}(M).

(iv) follows directly from (iii).

(v): We know already that E′ := A_{S} ⊗_{A} E_{A}(M) contains M_{S} and is injective (lemma 7.11.19(iii)), so it remains only to check that E′ is an essential extension of M. Thus, let x ∈ E′ \ M_{S}; we have to check that N := A_{S}x ∩ M_{S} ≠ 0, and clearly we may assume that x ∈ E_{A}(M). Set ℱ := {Ann_{A}(tx) | t ∈ S}; since A is noetherian, ℱ admits maximal elements, and notice that A_{S}x = A_{S}tx for any t ∈ S. Hence, we may replace x by tx for some t ∈ S, after which we may assume that Ann_{A}(x) is maximal in ℱ. We may write Ax∩M = Ix for some ideal I ⊂ A, so N = I_{S}x; let a_{1}, …, a_{k} ∈ A be a system of generators for I, and notice that Ix ≠ 0, since E_{A}(M) is an essential extension of M. Suppose that N = 0; then there exists t ∈ S such that the identity ta_{i}x = 0 holds in A for i = 1, …, k. However, Ann_{A}(x) = Ann_{A}(tx) by construction, so Ix = 0, a contradiction. □

Theorem 7.11.22. Let A be a noetherian ring, I an injective A-module. We have : (i) I decomposes as a direct sum of the form

(7.11.23)  I ≃ ⨁_{𝔭∈Spec A} E_{A}(A/𝔭)^{(R_{𝔭})}

for a system of (small) sets (R_{𝔭} | 𝔭 ∈ Spec A). (ii) Moreover, the cardinality of R_{𝔭} equals dim_{κ(𝔭)} Hom_{A_{𝔭}}(κ(𝔭), I_{𝔭}), for every 𝔭 ∈ Spec A (where κ(𝔭) := A_{𝔭}/𝔭_{𝔭}). Especially, this cardinality is independent of the decomposition (7.11.23).

Proof. (i): Denote by ℱ the set of all indecomposable injective submodules of I, and by 𝒮 the set of all subsets 𝒢 ⊂ ℱ such that the natural map I_{𝒢} := ⨁_{G∈𝒢} G → I is injective. The set 𝒮 is partially ordered by inclusion, and by Zorn’s lemma, 𝒮 admits a maximal element ℳ; in light of proposition 7.11.21(ii), it suffices to check that I_{ℳ} = I. However, I_{ℳ} is injective (lemma 7.11.19(i)), hence I = I_{ℳ} ⊕ J for some A-module J, and it is easily seen that J is injective as well. We are thus reduced to showing that J = 0. Now, if J ≠ 0, let 𝔭 ∈ Ass_{A}J be any associated prime ([126, Th.6.1(i)]); then E_{A}(A/𝔭) is an indecomposable injective direct summand of J (proposition 7.11.21(i,ii)), and therefore I_{ℳ} ⊕ E_{A}(A/𝔭) is a submodule of I, contradicting the maximality of ℳ.

(ii): In light of (i) and proposition 7.11.21(iii,v) we have

Hom_{A_{𝔭}}(κ(𝔭), I_{𝔭}) = Hom_{A_{𝔭}}(κ(𝔭), E_{A}(A/𝔭)_{𝔭}^{(R_{𝔭})}) = κ(𝔭)^{(R_{𝔭})} ⊗_{κ(𝔭)} Hom_{A_{𝔭}}(κ(𝔭), E_{A_{𝔭}}(κ(𝔭)))

so we are reduced to showing the following :

Claim 7.11.24. Let 𝔪 ⊂ A be any maximal ideal, and set κ := A/𝔪; then

d := dim_{κ} Hom_{A}(κ, E_{A}(κ)) = 1.

Proof of the claim. Since κ ⊂ E_{A}(κ), obviously d ≥ 1. On the other hand, we have a natural identification

Hom_{A}(κ, E_{A}(κ)) = F := {x ∈ E_{A}(κ) | 𝔪x = 0}.

If d > 1, we may find x ∈ F such that Ax ∩ κ = 0; but this is absurd, since E_{A}(κ) is an essential extension of κ. □

7.11.25. When dealing with the more general coherent rings that appear in sections 11.4 and 11.5, the injective hull is no longer suitable for the study of local cohomology, but it turns out that one can use instead a “coh-injective hull”, which works just as well.

Definition 7.11.26. Let A be a ring; we denote by A-Mod_{coh} the full subcategory of A-Mod consisting of all coherent A-modules.

(i) An A-module J is said to be coh-injective if the functor

A-Mod_{coh} → A-Mod^{o} : M ↦ Hom_{A}(M, J)

is exact. (ii) An A-module M is said to be ω-coherent if it is countably generated, and every finitely

generated submodule of M is finitely presented.

Lemma 7.11.27. Let A be a ring.

(i) Let M, J be two A-modules, N ⊂ M a submodule. Suppose that J is coh-injective and both M and M/N are ω-coherent. Then the natural map:

Hom_{A}(M, J) → Hom_{A}(N, J)

is surjective. (ii) Let (J_{λ} | λ ∈ Λ) be a filtered family of coh-injective A-modules. Then colim_{λ∈Λ} J_{λ} is

coh-injective. (iii) Assume that A is coherent, let M_{•} be an object of D^{−}(A-Mod_{coh}) and I^{•} a bounded

below complex of coh-injective A-modules. Then the natural map

Hom_{A}^{•}(M_{•}, I^{•}) → RHom_{A}^{•}(M_{•}, I^{•})

is an isomorphism in D(A-Mod).

Proof. (i): Since M is ω-coherent, we may write it as an increasing union ⋃_{n∈ℕ} M_{n} of finitely generated, hence finitely presented submodules. For each n ∈ ℕ, the image of M_{n} in M/N is a finitely generated, hence finitely presented submodule; therefore N_{n} := N ∩ M_{n} is finitely generated, hence it is a coherent A-module, and clearly N = ⋃_{n∈ℕ} N_{n}. Suppose φ : N → J is any A-linear map; for every n ∈ ℕ, set P_{n+1} := M_{n} + N_{n+1} ⊂ M, and denote by φ_{n} : N_{n} → J the restriction of φ. We wish to extend φ to a linear map φ′ : M → J, and to this aim we construct inductively a compatible system of extensions φ′_{n} : P_{n} → J, for every n > 0. For n = 1, one can choose arbitrarily an extension φ′_{1} of φ_{1} to P_{1}. Next, suppose n > 0, and that φ′_{n} is already given; since P_{n} ⊂ M_{n}, we may extend φ′_{n} to a map φ′′_{n} : M_{n} → J. Since M_{n} ∩ N_{n+1} = N_{n}, and since the restrictions of φ′′_{n} and φ_{n+1} agree on N_{n}, there exists a unique A-linear map φ′_{n+1} : P_{n+1} → J that extends both φ′′_{n} and φ_{n+1}.

(ii): Recall that a coherent A-module is finitely presented. However, it is well known that an A-module M is finitely presented if and only if the functor Q ↦ Hom_{A}(M, Q) on Amodules, commutes with filtered colimits (see e.g. [75, Prop.2.3.16(ii)]). The assertion is an easy consequence.

(iii): Since A is coherent, one can find a resolution φ : P_{•} → M_{•} consisting of free Amodules of finite rank (cp. [75, §7.1.20]). One looks at the spectral sequences

E_{1}^{pq} : Hom_{A}(P_{p}, I^{q}) ⇒ R^{p+q}Hom_{A}^{•}(M_{•}, I^{•})
F_{1}^{pq} : Hom_{A}(M_{p}, I^{q}) ⇒ H^{p+q}Hom_{A}^{•}(M_{•}, I^{•}).

The resolution φ induces a morphism of spectral sequences F_{1}^{••} → E_{1}^{••}; on the other hand, since I^{q} is coh-injective, we have : E_{2}^{pq} ≃ Hom_{A}(H_{p}F_{•}, I^{q}) ≃ Hom_{A}(H_{p}M_{•}, I^{q}) ≃ F_{2}^{pq}, so the induced map F_{2}^{pq} → E_{2}^{pq} is an isomorphism for every p, q ∈ ℕ, whence the claim. □

7.11.28. Let A be a coherent ring, Z ⊂ Spec A a constructible closed subset; we denote by

A-Mod_{coh,Z}

the full subcategory of A-Mod_{coh} whose objects are the (coherent) A-modules with support contained in Z. Let I ⊂ A be any finitely generated ideal such that Z = Spec A/I; it is easily seen that

Ob(A-Mod_{coh,Z}) = ⋃_{n∈ℕ} Ob(A/I^{n}-Mod_{coh}).

Now, consider a functor

T : A-Mod_{coh,Z}^{o} → ℤ-Mod.

Notice that T M is naturally an A-module, for every coherent A-module M : indeed, if a ∈ A is any element, we may define the scalar multiplication by a on T M as the ℤ-linear endomorphism T (a · 1_{M}). In other words, T factors through the forgetful functor A-Mod → ℤ-Mod; especially, if we set

H_{T} := colim_{n∈ℕ} T (A/I^{n})

we get a natural A-linear map

M →^{∼} colim_{n∈ℕ} Hom_{A}(A/I^{n}, M) → colim_{n∈ℕ} Hom_{A}(T M, T (A/I^{n})) → Hom_{A}(T M, H_{T})

whence a bilinear pairing

M × T M → H_{T}

which in turns yields a natural transformation

ω_{M} : T M → Hom_{A}(M, H_{T}) for every M ∈ Ob(A-Mod_{coh,Z}).

Lemma 7.11.29. In the situation of (7.11.28), the following conditions are equivalent : (a) ω is an isomorphism of functors T →^{∼} Hom_{A}(−, H_{T}).

(b) T is left exact.

Proof. Clearly (a)⇒(b). For the converse, suppose first that Z = Spec A, in which case, notice that H_{T} = T A and ω_{A} is the natural isomorphism T A →^{∼} Hom_{A}(A, T A); then, if M is any coherent A-module, pick a finite presentation

Σ : A^{⊕n} → A^{⊕m} → M → 0

and apply the 5-lemma to the resulting ladder of A-modules ω_{Σ} with left exact rows, to deduce that ω_{M} is an isomorphism. Next, for a general ideal I and M an A-module with Supp M ⊂ Z, we may find n ∈ ℕ such that M is an A/I^{n}-module; since A/I^{k} is coherent, the foregoing case then shows that the induced map

T M → Hom_{A}(M, T (A/I^{k}))

is an isomorphism, for every k ≥ n. To conclude, it suffices to remark that the natural map

colim_{k∈ℕ} Hom_{A}(M, T (A/I^{k})) → Hom_{A}(M, H_{T})

is an isomorphism, since M is finitely presented ([75, Prop.2.3.16(ii)]). □

We wish next to present a criterion that allows to detect, among the functors T as in (7.11.28), those that are exact. A complete characterization shall be given only for a restricted class of coherent ring; namely, we make the following :

Definition 7.11.30. Let A be a coherent ring. We say that A is an Artin-Rees ring, if the following holds. For every finitely generated ideal I ⊂ A, every coherent A-module M, and every finitely generated A-submodule N ⊂ M, the I-adic topology of M induces the I-adic topology on N.

We can then state :

Proposition 7.11.31. In the situation of (7.11.28) suppose furthermore that A is an Artin-Rees ring. Then the following conditions are equivalent :

(a) H_{T} is a coh-injective A-module.

(b) T is exact.

Proof. Clearly (a)⇒(b). For the converse, let M be any coherent A-module, and N ⊂ M any finitely generated A-submodule; it suffices to check that the induced map

Hom_{A}(M, H_{T}) → Hom_{A}(N, H_{T})

is surjective. However, let f : N → H_{T} be any A-linear map; since N is finitely presented, there exists n ∈ ℕ such that f factors through an A-linear map f_{n} : N → T (A/I^{n}) ([75, Prop.2.3.16(ii)]), and clearly I^{n}N ⊂ Ker f_{n}, so I^{n}N ⊂ Ker f as well. Since the I-adic topology of N agrees with the topology induced by the I-adic topology of M, there exists k ∈ ℕ such that I^{k}M ∩ N ⊂ I^{n}N. Set \overline{N} := N/(I^{k}M ∩ N); then \overline{N} is a finitely generated A-submodule of M/I^{k}M, and especially it is a coherent A-module. By construction, f factors through an A-linear map \overline{f} : \overline{N} → H_{T}; assumption (b) and lemma 7.11.29 imply that \overline{f} extends to an A-linear map M/I^{k}M → H_{T}, and the resulting map M → H_{T} extends f, whence (a). □

Remark 7.11.32. (i) In the terminology of definition 7.11.30, the standard Artin-Rees lemma implies that every noetherian ring is an Artin-Rees ring. We shall see later that if V is any valuation ring, then every essentially finitely presented V -algebra is an Artin-Rees ring (corollary 9.1.28and theorem 11.4.46).

(ii) On the other hand, if A is noetherian, claim 7.11.20 easily implies that an A-module is coh-injective if and only if it is injective.

(iii) Combining (i) and (ii) with proposition 7.11.31, we recover [85, Exp.IV, Prop.2.1].

Example 7.11.33. Let κ_{0} be a field, A a noetherian κ_{0}-algebra, 𝔪 ⊂ A a maximal ideal such that κ := A/𝔪 is a finite extension of κ_{0}, and set Z := {𝔪} ⊂ Spec A. Then every object of A-Mod_{coh,Z} is a finite dimensional κ_{0}-vector space, and therefore the functor

T : A-Mod_{coh,Z} → κ_{0}-Mod M ↦ Hom_{κ_{0}}(M, κ_{0})

is exact. Proposition 7.11.31and remark 7.11.32(ii) then say that

H_{T} := colim_{n∈ℕ} Hom_{κ_{0}}(A/𝔪^{n}, κ_{0})

is an injective A-module. More precisely, notice that Hom_{A}(κ, H_{T}) = Ann_{H_{T}} (𝔪) = T (κ) ≃ κ, therefore H_{T} is the injective hull of the residue field κ.

7.11.34. Flatness criteria. The following generalization of the local flatness criterion answers affirmatively a question raised in [65, Ch.IV, Rem.11.3.12].

Lemma 7.11.35. Let A be a ring, I ⊂ A an ideal, B a finitely presented A-algebra, 𝔭 ⊂ B a prime ideal containing IB, and M a finitely presented B-module. Then the following two conditions are equivalent:

(a) M_{𝔭} is a flat A-module.

(b) M_{𝔭}/IM_{𝔭} is a flat A/I-module and Tor_{1}^{A}(M_{𝔭}, A/I) = 0.

Proof. Clearly, it suffices to show that (b)⇒(a), hence we assume that (b) holds. We write A as the union of a filtered family (A_{λ} | λ ∈ Λ) of its ℤ-subalgebras of finite type, and set A′ := A/I, I_{λ} := I ∩ A_{λ}, A′_{λ} := A_{λ}/I_{λ} for every λ ∈ Λ. Then, for some λ ∈ Λ, the A-algebra B descends to an A_{λ}-algebra B_{λ} of finite type, and M descends to a finitely presented B_{λ}-module M_{λ}. For every μ ≥ λ we set B_{μ} := A_{μ} ⊗_{A_{λ}} B_{λ} and M_{μ} := B_{μ} ⊗_{B_{λ}} M_{λ}. Up to replacing Λ by a cofinal family, we can assume that B_{μ} and M_{μ} are defined for every μ ∈ Λ. Then, for every λ ∈ Λ let g_{λ} : B_{λ} → B be the natural map, and set 𝔭_{λ} := g_{λ}^{−1}𝔭.

Claim 7.11.36. There exists λ ∈ Λ such that M_{λ,𝔭_{λ}}/I_{λ}M_{λ,𝔭_{λ}} is a flat A′_{λ}-module.

Proof of the claim. Set B′_{λ} := B_{λ} ⊗_{A_{λ}} A′_{λ} and M′_{λ} := M_{λ} ⊗_{A_{λ}} A′_{λ} for every λ ∈ Λ; clearly the natural maps

colim_{λ∈Λ} A′_{λ} → A/IA  colim_{λ∈Λ} B′_{λ} → B/IB  colim_{λ∈Λ} M′_{λ} → M/IM

are isomorphisms. Then the claim follows from (b) and [65, Ch.IV, Cor.11.2.6.1(i)]. ♢

In view of claim 7.11.36, we can replace Λ by a cofinal subset, and thereby assume that M_{λ,𝔭_{λ}}/I_{λ}M_{λ,𝔭_{λ}} is a flat A′_{λ}-module for every λ ∈ Λ.

Claim 7.11.37. (i) More generally, let R_{•} := (R_{λ} | λ ∈ Λ) be any system of rings indexed by any filtered set Λ, and N_{•} := (N_{λ} | λ ∈ Λ), N′_{•} := (N′_{λ} | λ ∈ Λ) two systems consisting of R_{λ}-modules N_{λ} and N′_{λ} for every λ ∈ Λ, and R_{λ}-linear transition maps N_{λ} → N_{μ}, N′_{λ} → N′_{μ} for every λ, μ ∈ Λ with λ ≤ μ. Denote by R, N, N′ the colimits of R_{•}, N_{•} and respectively N′_{•}; then the natural maps R_{λ} → R, N_{λ} → N, N′_{λ} → N induce isomorphisms :

colim_{λ∈Λ} Tor_{i}^{R_{λ}}(N_{λ}, N′_{λ}) → Tor_{i}^{R}(N, N′) for every i ∈ ℕ.

(ii) For every λ, μ ∈ Λ with μ ≥ λ, the natural map:

f_{λμ} : B_{μ,𝔭_{μ}} ⊗_{B_{λ}} Tor_{1}^{A_{λ}}(M_{λ,𝔭_{λ}}, A′_{λ}) → Tor_{1}^{A_{μ}}(M_{μ,𝔭_{μ}}, A′_{μ})

is surjective.

Proof of the claim. (i): For every λ ∈ Λ, let L_{•}(N′_{λ}) be the canonical free resolution of the R_{λ}-module N′_{λ} ([37, §3, n.3]). Similarly, denote by L_{•}(N′) the canonical free resolution of the R-module N′. It follows from [38, Ch.II, §6, n.6, Cor.] and the exactness properties of filtered colimits, that the natural map: colim_{λ∈Λ} L_{•}(N′_{λ}) → L_{•}(N′) is an isomorphism. Hence :

colim_{λ∈Λ} H_{•}(N_{λ} ⊗^{L}_{R_{λ}} N′_{λ}) ≃ colim_{λ∈Λ} H_{•}(N_{λ} ⊗_{R_{λ}} L_{•}(N′_{λ}))
≃ H_{•}(N ⊗_{R} colim_{λ∈Λ} L_{•}(N′_{λ}))
≃ H_{•}(N ⊗_{R} L_{•}(N′))
≃ H_{•}(N ⊗^{L}_{R} N′).

(ii): We use the change of rings spectral sequences for Tor ([163, Th.5.6.6])

E_{pq}^{2} : Tor_{p}^{A_{μ}}(Tor_{q}^{A_{λ}}(M_{λ,𝔭_{λ}}, A_{μ}), A′_{μ}) ⇒ Tor_{p+q}^{A_{λ}}(M_{λ,𝔭_{λ}}, A′_{μ})
F_{pq}^{2} : Tor_{p}^{A′_{λ}}(Tor_{q}^{A_{λ}}(M_{λ,𝔭_{λ}}, A′_{λ}), A′_{μ}) ⇒ Tor_{p+q}^{A_{λ}}(M_{λ,𝔭_{λ}}, A′_{μ}).

Since F_{10}^{2} = F_{20}^{2} = 0, the natural map

α : F_{01}^{2} := A′_{μ} ⊗_{A′_{λ}} Tor_{1}^{A_{λ}}(M_{λ,𝔭_{λ}}, A′_{λ}) → Tor_{1}^{A_{λ}}(M_{λ,𝔭_{λ}}, A′_{μ})

is an isomorphism. On the other hand, we have a surjection:

β : Tor_{1}^{A_{λ}}(M_{λ,𝔭_{λ}}, A′_{μ}) → E_{10}^{2} := Tor_{1}^{A_{μ}}(M_{μ,𝔭_{λ}}, A′_{μ})

and f_{λμ} = (β ∘ α)_{𝔭_{μ}}, whence the claim. ♢ We deduce from claim 7.11.37(i) that the natural map

colim_{λ∈Λ} Tor_{1}^{A_{λ}}(M_{λ,𝔭_{λ}}, A′_{λ}) → Tor_{1}^{A}(M_{𝔭}, A′) = 0

is an isomorphism. However, Tor_{1}^{A_{λ}}(M_{λ,𝔭_{λ}}, A′_{λ}) is a finitely generated B_{λ,𝔭_{λ}}-module by [37, §6, n.4, Cor.]. We deduce that f_{λμ,𝔭_{λ}} = 0 for some μ ≥ λ; therefore Tor_{1}^{A_{μ}}(M_{μ,𝔭_{μ}}, A′_{μ}) = 0, in view of claim 7.11.37(ii), and then the local flatness criterion of [61, Ch.0, §10.2.2] says that M_{μ,𝔭_{μ}} is a flat A_{μ}-module, so finally M_{𝔭} is a flat A-module, as stated. □

8. Complements of topology and topological algebra

This chapter is a miscellanea of results of topology and topological algebra that shall be needed in the rest of the treatise.

8.1. Spectral spaces and constructible subsets. The class of spectral topological spaces was first introduced in Hochster’s paper [90], where it was shown to coincide precisely with the class of those spaces that are homeomorphic to (Zariski) spectra of commutative rings. Since then, spectral spaces have reappeared in a variety of unrelated contexts; their ubiquity has all to do with the very agreeable properties of their boolean algebras of constructible subsets; especially, the constructible subsets of a spectral space generate a quasi-compact and separated topology. The many corollaries springing from this basic observation add up to a handy toolkit for dealing in a uniform way with the kind of non-separated spaces that most frequently occur in geometric applications of an algebraic bent.

In this section we introduce spectral spaces and prove their basic properties. We also collect a few results showing the interplay between algebraic and topological properties of schemes and their constructible subsets.

Definition 8.1.1. Let X be a topological space.

(i) We say that X is quasi-separated if the intersection of any two quasi-compact open subsets of X is still quasi-compact.

(ii) We say that X is coherent if it is quasi-compact and it admits a basis consisting of

quasi-compact open subsets, and closed under finite intersections. (iii) We say that X is reducible if there exist non-empty closed subsets Z, Z′ ⊂ X such that

Z ∪ Z′ = X and Z, Z′ ≠ X. We say that X is irreducible if it is non-empty and not reducible.

(iv) We say that X is sober if for every irreducible closed subset Z ⊂ X (with the topology induced from X) there exists a unique point η_{Z} ∈ Z such that Z is the topological closure of {η_{Z}} in X. In this case, we say that η_{Z} is the generic point of Z.

(v) We say that X is spectral if it is sober and coherent.

(vi) We say that X is noetherian, if for every descending chain

⋯ ⊂ Z_{2} ⊂ Z_{1} ⊂ Z_{0}

of closed subsets of X, there exists n ∈ ℕ such that Z_{m} = Z_{n} for every m ≥ n. (vii) We say that X is locally coherent (resp. locally spectral, resp. locally noetherian) if

every point of X admits an open neighborhood which is coherent (resp. spectral, resp. noetherian).

(viii) We say that X is T_{0} if, for every pair of points x, y ∈ X, there exists an open subset

U ⊂ X such that U ∩ {x, y} contains exactly one point. (ix) We say that X is compact if it is quasi-compact and separated.

Remark 8.1.2. (i) Let X be a topological space, (Z_{i} | i ∈ I) a family of irreducible closed subsets of X, totally ordered by inclusion, and denote by Z the topological closure in X of ⋃_{i∈I} Z_{i}. Then Z is also irreducible. Indeed, say that Z = Y_{1} ∪ Y_{2} for two closed subsets Y_{1}, Y_{2} of X. If Z_{i} ⊂ Y_{1} ∩ Y_{2} for every i ∈ I, then Z ⊂ Y_{1}; otherwise, say that Z_{i} ⊄ Y_{1} for some i ∈ I; then Z_{i} ⊂ Y_{2} for every i ∈ I, whence Z ⊂ Z_{2}.

(ii) By Zorn’s lemma, it follows that every irreducible closed subset of X is contained in a maximal one. A maximal irreducible closed subset of X is called an irreducible component of X, and clearly X is the union of its irreducible components.

(iii) Let Z_{1}, Z_{2} ⊂ X be any two closed subsets, and endow Z_{1}, Z_{2} and Z_{3} := Z_{1} ∪ Z_{2} with the topology induced by X. For j ≤ 3, denote by I_{j} the set of all irreducible components of Z_{i}; then I_{3} ⊂ I_{1} ∪ I_{2}. Indeed, let Y ∈ I_{3}; then Y = (Y ∩ Z_{1}) ∪ (Y ∩ Z_{2}), so we must have Y = Y ∩ Z_{i} for some i ∈ {1, 2}.

(iv) Let k ∈ ℕ be any integer, and Z_{•} := (Z_{i} | i = 0, …, k) a strictly descending finite chain of irreducible closed subsets of X. The integer k is the length of Z_{•}. The supremum of the lengths of all such chains is an invariant called the (Krull) dimension of X, and denoted

dim X

(especially, dim ∅ = −∞). We say that X has finite Krull dimension if dim X ∈ ℕ ∪ {−∞}.

(v) Suppose that X is the union of finitely many closed subsets Z_{1}, …, Z_{k}. Then it follows directly from (iii) that dim X = sup(dim Z_{1}, …, dim Z_{k}).

(vi) If X is sober, then X is also T_{0}. Indeed, let x, y ∈ X be any two distinct points, and denote by Z_{y} the topological closure of {y} in X; if X is sober, we may assume – up to swapping the roles of x and y – that x ∉ Z_{y}. In this case, the set X \Z_{y} is an open neighborhood of x that does not contain y, whence the claim.

Lemma 8.1.3. Let X be a topological space, T ⊂ X any subset, \overline{T} the topological closure of T in X, and endow T and \overline{T} with the topologies induced from X. The following holds :

(i) T is irreducible if and only if the same holds for \overline{T}.

(ii) If T is non-empty and open in X, and X is irreducible, then T is irreducible.

(iii) If X is sober, and T is locally closed in X, then T is sober as well.

(iv) Suppose that T = T_{1} ∪ T_{2} for two subsets T_{1} and T_{2}, and endow also T_{1} and T_{2} with

the topologies induced from X. If T_{1}, T_{2} and X are sober, the same holds for T.

(v) Let f : X → Y be any continuous map, and endow f(X) with the topology induced by Y. If X is irreducible, the same holds for f(X).

Proof. For any subset Y ⊂ X, let \overline{Y} denote the topological closure of Y in X.

(i): Suppose that T is irreducible, and say that \overline{T} = V ∪ V′ for two closed subsets V, V′ of \overline{T}; then T = (T ∩ V) ∪ (T ∩ V′), so we may assume that T = T ∩ V, in which case \overline{T} = V. Conversely, suppose that \overline{T} is irreducible, and T = Z_{1} ∪ Z_{2} for some closed subsets Z_{1}, Z_{2} of T; then \overline{T} = \overline{Z}_{1} ∪ \overline{Z}_{2}, so we may assume that \overline{T} = \overline{Z}_{1}, in which case T = T ∩ \overline{Z}_{1} = Z_{1}.

(ii): Indeed, say that T = Z_{1} ∪ Z_{2} for two non-empty closed subsets of T; then X = \overline{Z}_{1} ∪ \overline{Z}_{2} ∪ (X \ T). By assumption, X ⊄ X \ T; as X is irreducible, we may then assume that X = \overline{Z}_{1}, whence T = Z_{1}.

(iii): This is clear if T is closed in X, so we may assume that T is open in X. However, say that Z ⊂ T is an irreducible closed subset; by (i), the topological closure \overline{Z} of Z in X is irreducible, and clearly its generic point η\overline{Z} lies in Z, so η\overline{Z} is the unique point of Z such that the closure of \{η\overline{Z}\} in T equals Z.

(iv): Let Z be any irreducible closed subset of T, and set Z_{i} := Z ∩T_{i} for i = 1, 2. According to (i), \overline{Z} is irreducible; since \overline{Z} = \overline{Z}_{1} ∪ \overline{Z}_{2}, we may then assume that \overline{Z} = \overline{Z}_{1}, especially \overline{Z}_{1} is irreducible, so Z_{1} is irreducible, by (i). Therefore Z_{1} admits a generic point η, since T_{1} is sober. By construction, the topological closure of \{η\} in T equals T ∩ \overline{Z} = Z, and it remains to see that η is the unique point of Z with this property. However, if η′ is any other such point, then clearly the topological closures in X of both \{η\} and \{η′\} equal \overline{Z}, so η = η′, since X is sober.

(v): Indeed, suppose that f(X) = Z_{1} ∪ Z_{2} for two closed subsets Z_{1}, Z_{2} of f(X); then X = f^{−1}(Z_{1}) ∪ f^{−1}(Z_{2}), so we may assume that X = f^{−1}Z_{1}, in which case f(X) = Z_{1}. □

Proposition 8.1.4. Let X be any topological space. Then X is noetherian if and only if the following conditions hold :

(a) Every closed subset of X has a finite number of irreducible components.

(b) For every descending chain (Z_{n} | n ∈ ℕ) of irreducible closed subsets of X, there exists N ∈ ℕ such that Z_{n} = Z_{N} for every n ≥ N.

Proof. (This was first shown in in [136, Prop.1.1], with a different proof.) Suppose first that X is noetherian. Then (b) clearly holds. To show (a), let S denote the set of all closed subsets of X that have infinitely many irreducible components; since X is noetherian, if S is not empty, it admits minimal elements, so say that Z is such a minimal element. Then clearly Z cannot be irreducible, so Z = Z_{1} ∪ Z_{2} for some non-empty closed subsets of X strictly contained in Z. By minimality of Z, we have Z_{1}, Z_{2} ∉ S; but then remark 8.1.2(iii) implies that Z_{1} ∪ Z_{2} ∉ S as well, a contradiction. For the converse, we show more precisely the following :

Claim 8.1.5. Let X be any topological space, Z_{•} := (Z_{n} | n ∈ ℕ) a descending non-stationary chain of closed subsets of X, such that Z_{n} has a finite number of irreducible components, for every n ∈ ℕ. Then there exists a descending, non-stationary chain (Z′_{n} | n ∈ ℕ) of closed subsets of X, such that Z′_{n} is an irreducible component of Z_{n}, for every n ∈ ℕ.

Proof of the claim. For every n ∈ ℕ, let I_{n} be the (finite) set of irreducible components of Z_{n}, and denote by I′_{n} ⊂ I_{n} the subset of all T ∈ I_{n} such that T ∈ I_{m} for every m ≥ n. Notice that, since Z_{•} is descending and non-stationary, the subset I′′_{n} := I_{n} \ I′_{n} is non-empty, for every n ∈ ℕ. Moreover, for every n > 0, every T ∈ I′′_{n} and every T′ ∈ I_{n−1} such that T ⊂ T′, we must have T′ ∈ I′′_{n−1}; indeed, otherwise we would have T′ ∈ I_{n}, hence T = T′, hence T ∈ I′_{n}, a contradiction. Hence, for every n ∈ ℕ, denote by S_{n} the set of all chains (T_{k} | k = 0, …, n) such that

• T_{k} ∈ I′′_{k} for every k = 0, …, n.

• T_{k} ⊂ T_{k−1} for every k = 1, …, n.

The foregoing observations easily imply that S_{n} is a finite non-empty set for every n ∈ ℕ. Moreover, we have an obvious map S_{m} → S_{n} whenever m ≥ n, that assigns to any T_{•} ∈ S_{m} the truncated chain (T_{k} | k = 0, …, n). The limit S of the system (S_{n} | n ∈ ℕ) is then nonempty; say that (T_{•}^{k} | k ∈ ℕ) is any element of S, and set Z′_{n} := T_{n}^{n} for every n ∈ ℕ. We claim that the chain (Z′_{n} | n ∈ ℕ) will do. Indeed, suppose by way of contradiction, that the chain Z′_{•} is stationary, so there exists N ∈ ℕ such that Z′_{n} = Z′_{N} for every n ≥ N; but then we would have Z_{N} ∈ I_{n} for every n ≥ N, hence Z_{N} ∈ I′_{N}, which is absurd. □

Remark 8.1.6. Let X be a topological space, U ⊂ X an open subset, and endow U with the topology induced from X.

(i) If X is quasi-separated, then the same holds for U. If X is coherent, then X is quasiseparated. Indeed, let (U_{i} | i ∈ I) be a basis for X consisting of quasi-compact open subsets and closed under finite intersections, and U_{1}, U_{2} two quasi-compact open subsets. Then there exist finite subsets J_{i} ⊂ I such that U_{i} = ⋃_{i∈J_{i}} U_{i} for i = 1, 2; consequently U_{1} ∩ U_{2} = ⋃_{(j,j′)∈J_{1}×J_{2}}(U_{j} ∩ U_{j′}), and by assumption each U_{j} ∩ U_{j′} is quasi-compact, whence the claim.

(ii) If X is coherent (resp. spectral) and U is quasi-compact, then U is coherent (resp. spectral) for the topology induced from X. Indeed, lemma 8.1.3(iii) says that U is sober whenever the same holds for X, and if (U_{i} | i ∈ I) is a basis of quasi-compact open subsets of X, then (U ∩ U_{i} | i ∈ I) is a basis of quasi-compact open subsets of U.

(iii) It follows easily from (ii) that if X is locally coherent (resp. locally spectral) then the same holds for U.

(iv) If X is noetherian, every open subset of X is quasi-compact, so X is coherent. Also, every locally closed subset of X is noetherian as well.

(v) Furthermore, suppose that T_{1}, …, T_{k} is any finite family of subsets of X, such that X = ⋃_{i=1}^{k} T_{i}, and endow each T_{i} with the topology induced from X; then, if each T_{i} is noetherian, the same holds for X.

8.1.7. Let us denote by Top the category of topological spaces (whose morphisms are the continuous maps), and by Sober the full subcategory of Top whose objects are the sober topological spaces.

Proposition 8.1.8. The inclusion functor

Sober → Top

admits a left adjoint.

Proof. For any topological space X, denote by S_{X} the set of all irreducible closed subsets of X. If Z is any irreducible closed subset of X, we shall use the notation η_{Z} to refer to Z, when we wish to regard the latter as an element of S_{X}. From lemma 8.1.3(i,ii) we see that, for every open subset U ⊂ X, we have a natural bijection

S_{U} →^{∼} \{η_{Z} ∈ S_{X} | Z ∩ U ≠ ∅\} T ↦ η\overline{T}

(where, for any subset T of X, we let \overline{T} be the topological closure of T in X). We may then identify S_{U} with its image in S_{X}, and we notice

Claim 8.1.9. (i) S_{U} ∩ S_{V} = S_{U∩V} for every open subsets U, V ⊂ X.

(ii) For any family (U_{i} | i ∈ I) of open subsets of X, we have S_{⋃_{i∈I} U_{i}} = ⋃_{i∈I} S_{U_{i}}.

Proof of the claim. (i): Clearly S_{U∩V} ⊂ S_{U} ∩ S_{V}. For the converse inclusion, suppose that Z is closed subset of X, such that Z ∩U, Z ∩V ≠ ∅ and Z ∩U ∩V = ∅. Then Z = (Z\U)∪(Z\V), so Z is not irreducible in X. The assertion is an immediate consequence.

(ii) is trivial. ♢ From claim 8.1.9 we see that the family (S_{U} | U is open in X) is a topology on S_{X}, and we

endow S_{X} with this topology. Next, there is a natural map

f_{X} : X → S_{X} x ↦ η\overline{\{x\}}

and it is easily seen that f_{X}^{−1}(S_{U}) = U for every open subset U ⊂ X; especially f_{X} is continuous. It also follows that the rule Z ↦ f_{X}^{−1}Z establishes a bijection from the closed subsets of S_{X} to those of X; since this bijection respects inclusion of closed subsets, we deduce that it restricts to a bijection from the irreducible closed subsets of S_{X} to those of X. Moreover, in light of lemma 8.1.3(v), it is easily seen that the inverse of this bijection assigns to any irreducible closed subset Z of X, the topological closure of f_{X}(Z) in S_{X}.

Claim 8.1.10. S_{X} is a sober topological space.

Proof of the claim. Let Z be any irreducible closed subset of X, and denote by S_{Z} the topological closure of {η_{Z}} in S_{X}; then S_{Z} = S_{X} \S_{U}, where S_{U} is the largest open subset of S_{X} that does not contain η_{Z}, i.e. U is the largest open subset of X that does not intersect Z. So U = X \Z, and therefore f_{X}^{−1}(S_{Z}) = Z. We conclude that the topological closure of f_{X}(Z) in S_{X} equals S_{Z}. Summing up, we see that the rule η_{Z} ↦ S_{Z} establishes a bijection from S_{X} to the set of irreducible closed subsets of S_{X}, whence the claim. ♢

Now, let Y be any sober space, and g : X → Y any continuous map; in light of claim 8.1.10, it suffices to show that there exists a unique continuous map h : S_{X} → Y such that g = h ∘ f_{X}. However, it is easily seen that any continuous map h with this property must assign to every η_{Z} ∈ S_{X} the generic point η_{g(Z)} of the topological closure of g(Z) in S (lemma 8.1.3(v)); conversely, one checks easily that the rule

η_{Z} ↦ η_{g(Z)} for every irreducible closed subset Z of X

defines a map h : S_{X} → Y such that h^{−1}U = S_{g^{−1}U} for every open subset U ⊂ Y; especially, h is continuous : the details shall be left to the reader. □

Remark 8.1.11. Keep the notation of the proof of proposition 8.1.8, and let U ⊂ X be any open subset; it is easily seen that U = ∅ if and only if S_{U} = ∅. It follows that the rule V ↦ f_{X}^{−1}V defines a bijection from the set of open subsets of S_{X} to the set of open subsets of X. Consequently, X is quasi-compact (resp. quasi-separated) if and only if the same holds for S_{X}, and X is coherent if and only if S_{X} is spectral.

Remark 8.1.12. It is clear that the topological space underlying every affine scheme is spectral, and that of any scheme is locally spectral. Conversely, it is shown in [90] that to every spectral space X one can attach naturally a commutative ring A_{X} whose spectrum is isomorphic to X. This rule X ↦ A_{X} is functorial with respect to continuous maps of a certain type, which are introduced in the next definition.

Definition 8.1.13. Let f : X → Y be a continuous map of topological spaces, T ⊂ X any subset, and endow T with the topology induced from X.

(i) We say that f is quasi-compact (resp. quasi-separated) if, for every quasi-compact (resp. quasi-separated) open subset U ⊂ X, the preimage f^{−1}U is quasi-compact (resp. quasi-separated).

(ii) We say that f is spectral if, for every pair of quasi-compact quasi-separated open subsets U ⊂ X, V ⊂ Y with f(U) ⊂ V, the restriction f_{|U} : U → V is quasi-compact.

(iii) We say that T is retro-compact in X if the inclusion map T → X is quasi-compact. (iv) We say that T is globally constructible if T lies in the boolean algebra of subsets of X

generated by all retro-compact open subsets of X.

(v) We say that T is constructible if every point of X admits an open neighborhood U such that T ∩ U is globally constructible in U.

(vi) We say that T is pro-constructible (resp. ind-constructible) if every point of X admits an open neighborhood U such that T ∩ U is the intersection (resp. the union) of constructible subsets of U.

Remark 8.1.14. Let f : X → Y and g : Y → Z be two continuous maps of topological spaces. (i) If f and g are quasi-compact (resp. quasi-separated), then the same holds for g ∘ f. (ii) If f and g are spectral, and both X and Y are locally coherent, then g ∘ f is spectral.

Indeed, say that U ⊂ X and V ⊂ Z are quasi-compact and quasi-separated open subsets such that g ∘ f(U) ⊂ V; for every x ∈ U we may find a quasi-compact and quasi-separated open subset W_{x} ⊂ Y such that g(W_{x}) ⊂ V and f(x) ∈ W_{x}. Since X is locally coherent, we may find a quasi-compact and quasi-separated open neighborhood U_{x} of x in X, such that f(U_{x}) ⊂ W_{x}. Then both f_{|U_{x}} : U_{x} → W_{x} and g_{|W_{x}} : W_{x} → V are quasi-compact, so (i) says that the same holds for their composition (g ∘ f)_{|U_{x}} : U_{x} → V. However, finitely many such U_{x} suffice to cover U, whence the assertion.

(iii) Let us say that f is strongly spectral if the following holds. For every quasi-compact inclusion map U → U′ between open subsets of Y, the induced open immersion f^{−1}U → f^{−1}U′ is also quasi-compact. We claim that if f is strongly spectral, then f is spectral. Indeed, let U ⊂ X, V ⊂ Y be two quasi-compact and quasi-separated open subsets, and V′ ⊂ V another quasi-compact open subset; by assumption, the induced inclusion map i : f^{−1}V′ → f^{−1}V is quasi-compact, so f_{|U}^{−1}V′ = i^{−1}U is quasi-compact, whence the claim.

(iv) Obviously, a composition of strongly spectral maps is strongly spectral. Moreover, it is easily seen that any open immersion i : U → X is strongly spectral. In view of (iii), it follows that every open immersion is spectral.

Lemma 8.1.15. Let X be any topological space. We have :

(i) X is quasi-separated if and only if it admits a covering consisting of retro-compact open quasi-separated subsets.

(ii) X is sober if and only if it admits a covering consisting of sober open subsets.

(iii) The following conditions are equivalent :

(a) X is locally coherent and quasi-separated.

(b) X admits a basis consisting of quasi-compact open subsets, which is closed under

finite intersections. (iv) The following conditions are equivalent :

(a) X is coherent (resp. spectral).

(b) X is quasi-compact and admits a covering consisting of coherent (resp. spectral) retro-compact open subsets.

(c) X is quasi-compact, quasi-separated and locally coherent (resp. locally spectral).

(v) X is noetherian if and only if it is locally noetherian and quasi-compact.

(vi) If X is locally noetherian, then X is locally coherent and quasi-separated.

Proof. (i): The condition is trivially necessary. Suppose then that (U_{i} | i ∈ I) is a covering of X such that U_{i} is quasi-separated, retro-compact and open in X for every i ∈ I; pick any two quasicompact open subsets U, U′. We may find a finite subset J ⊂ I such that U ∪ U′ ⊂ ⋃_{j∈J} U_{j}, and our assumptions imply that, for every j ∈ J, the subset V_{j} := U ∩U′ ∩U_{j} is quasi-compact, so the same holds for ⋃_{j∈J} V_{j} = U ∩ U′.

(ii): Again, necessity is trivial. For the converse, let (U_{i} | i ∈ I) be an open covering of X such that U_{i} is sober for every i ∈ I. Say that Z ⊂ X is any irreducible closed subset, and that Z_{i} := U_{i} ∩ Z ≠ ∅ for some i ∈ I; then Z_{i} is an irreducible subset of U_{i} and the topological closure of Z_{i} in X equals Z (lemma 8.1.3(ii)). Let η be the generic point of Z_{i}; then clearly the topological closure of {η} in X equals Z. It remains to check that η is the unique element of Z with this property. However, say that η′ is another such element; then it is easily seen that η′ ∈ U_{i} as well, and η′ is then also the generic point of Z_{i}, whence the contention.

(iii): It is easily seen that (b)⇒(a). Conversely, suppose that (a) holds, and choose a covering (U_{i} | i ∈ I) of X consisting of coherent open subsets; for each i ∈ I we then have a basis (U_{ij} | j ∈ J_{i}) of the topology of U_{i}, consisting of quasi-compact open subsets. Set ℱ := (U_{ij} | i ∈ I, j ∈ I_{j}), and let ℱ′ be the family of all finite non-empty intersections of elements of ℱ; then ℱ′ is a basis of the topology of X, and since X is quasi-separated, any element of ℱ′ is still a quasi-compact open subset of X, whence (b).

(iv): Obviously (a)⇒(b), and (b)⇒(c) by virtue of (i) and remark 8.1.6(i). Lastly, (c)⇒(a), by (iii) and (ii).

(v): Suppose that X is locally noetherian and quasi-compact, in which case it admits a finite covering (U_{i} | i ∈ I) consisting of open noetherian subsets. Now, let Z_{•} := (Z_{n} | n ∈ ℕ) be a descending chain of closed subsets of X; by assumption, every resulting chain (Z_{n}∩U_{i} | n ∈ ℕ) is stationary, and since I is finite, it follows that Z_{•} is stationary as well.

(vi): X is locally coherent, by virtue of remark 8.1.6(iv); next, if U, V ⊂ X are quasi-compact open subsets, then they are noetherian, by (v), so their intersection is quasi-compact. □

Proposition 8.1.16. Let f : X → Y be a continuous map of topological spaces, (Y_{i} | i ∈ I) an open covering of Y, and f_{i} : f^{−1}Y_{i} → Y_{i} the restriction of f, for every i ∈ I. We have :

(i) If Y is locally coherent, then f is quasi-compact if and only if f_{i} is quasi-compact for every i ∈ I.

(ii) If both X and Y are locally coherent, then f is quasi-separated and spectral if and

only if f_{i} is quasi-separated and spectral for every i ∈ I. (iii) Suppose that both X and Y are locally coherent, and let (X_{i} | i ∈ I) be an open

covering of X such that X_{i} ⊂ f^{−1}Y_{i} for every i ∈ I. Then f is spectral if and only if the restriction X_{i} → Y_{i} of f is spectral for every i ∈ I.

Proof. (i): Clearly, if f is quasi-compact, the same holds for every f_{i}. Hence we may assume that f_{i} is quasi-compact for every i ∈ I, and we have to check that f is quasi-compact. To this aim, let U ⊂ Y be any quasi-compact open subset, and set U_{i} := Y_{i} ∩ U for every i ∈ I; since Y is locally coherent, we may find, for every i ∈ I, a covering (U_{i,λ} | λ ∈ Λ_{i}) of U_{i} consisting of quasi-compact open subsets. Since U is quasi-compact, finitely many of these U_{i,λ} suffice to cover U, hence f^{−1}U is covered by finitely many subsets of the type f_{i}^{−1}U_{i,λ}, and the latter are quasi-compact by assumption, so the same holds for U.

(iii): To begin with, we remark

Claim 8.1.17. Suppose that both X and Y are locally coherent. Then f is spectral if and only if it is strongly spectral (see remark 8.1.14(iii)).

Proof of the claim. In view of remark 8.1.14(iii), we may assume that f is spectral, and we show that f is strongly spectral. Indeed, let U ⊂ U′ be any inclusion of open subset of Y, with U retro-compact in U′. We need to check that f^{−1}U is retro-compact in f^{−1}U′, and by (i), the assertion can be checked locally on f^{−1}U′. However, since X and Y are locally coherent, for every x ∈ f^{−1}U′ we may find quasi-compact and quasi-separated open neighborhoods V′ of x in f^{−1}U′ and W′ of f(x) in U′ such that f(V′) ⊂ W′. By assumption, W′∩U is quasi-compact, so the same holds for V := V′ ∩ f^{−1}U (because f is spectral); since V′ is quasi-separated, the inclusion map V → V′ is then quasi-compact, as required. ♢

Now, in light of claim 8.1.17, for the proof of (iii) we may assume that each restriction X_{i} → Y_{i} is strongly spectral, and it suffices to check that the same holds for f. However, let U_{1} ⊂ U_{2} be any quasi-compact inclusion map of open subsets of Y; by (i), the induced inclusion map U_{1} ∩Y_{i} → U_{2} ∩Y_{i} is also quasi-compact, so by assumption the same holds for the inclusion

X_{i} ∩ f^{−1}U_{1} = X_{i} ∩ f^{−1}(U_{1} ∩ Y_{i}) → X_{i} ∩ f^{−1}(U_{2} ∩ Y_{i}) = X_{i} ∩ f^{−1}U_{2} for every i ∈ I.

Again (i) then implies that the inclusion f^{−1}U_{1} → f^{−1}U_{2} is quasi-compact, as required.

(ii): Again, we may assume that f_{i} is quasi-separated and spectral for every i ∈ I, and we show that the same follows for f. By (iii), we know already that f is spectral, so we let U ⊂ Y be any quasi-separated open subset, and show that f^{−1}U is quasi-separated in X. To this aim, choose a family of quasi-compact open subsets (U_{i,λ} | λ ∈ Λ_{i}) which gives a covering of U ∩Y_{i}, for every i ∈ I; since U is quasi-separated, U_{i,λ} is retro-compact in U for every i ∈ I and every λ ∈ Λ_{i}, hence f^{−1}U_{i,λ} is retro-compact in f^{−1}U, by claim 8.1.17. Now, let V_{1}, V_{2} ⊂ f^{−1}U be any two quasi-compact open subsets; it follows that V_{1} ∩ f^{−1}U_{i,λ} and V_{2} ∩ f^{−1}U_{i,λ} are quasi-compact for every i ∈ I and every λ ∈ Λ_{i}. Since f_{i} is quasi-separated, we deduce that V_{i,λ} := V_{1} ∩ V_{2} ∩ f^{−1}U_{i,λ} is also quasi-compact, for every such i and λ. Since V_{1} is quasicompact, finitely many of these f^{−1}U_{i,λ} suffice to cover V_{1}, and therefore finitely many V_{i,λ} cover V_{1} ∩ V_{2}, so the latter is quasi-compact, as required. □

Example 8.1.18. We return to remark 8.1.12 : with the terminology of definition 8.1.13(ii), we may now say that the continuous map Spec φ : Spec B → Spec A attached to any ring homomorphism φ : A → B is spectral (verification left to the reader). In light of proposition 8.1.16(iii) it follows that the continuous map underlying any morphism of schemes f : X → Y is spectral as well. Moreover, let (Y_{i} | i ∈ I) be any covering of Y consisting of open subschemes; proposition 8.1.16(i,ii) implies that f is quasi-compact (resp. quasi-separated) if and only the same holds for each restriction f_{|Y_{i}} : f^{−1}Y_{i} → Y_{i}.

Lemma 8.1.19. Let X be a topological space, U ⊂ X an open subset, T ⊂ X any subset, and endow U and T with the topology induced from X. The following holds :

(i) Every closed subset of X is retro-compact. The union of any finite number of retrocompact subsets of X is still retro-compact. The intersection of any open retro-compact subset of X with an arbitrary retro-compact subset of X is retro-compact.

(ii) Every globally constructible subset of X is retro-compact, and it is a finite union of subsets of the form V \V′, where V and V′ are retro-compact open subsets of X.

(iii) Suppose that T is retro-compact, and let S ⊂ T be any subset. If S is retro-compact in T, then it is retro-compact in X as well. If T is also open and S is globally constructible in T, then S is globally constructible in X as well.

(iv) T is pro-constructible if and only if X \T is ind-constructible.

(v) If T is retro-compact (resp. globally constructible, resp. pro-constructible, resp. indconstructible) in X, then T ∩ U is retro-compact (resp. globally constructible, resp. constructible, resp. pro-constructible, resp. ind-constructible) in U.

(vi) The constructible subsets of X form a boolean algebra.

(vii) If X is quasi-separated and U is quasi-compact, then U is retro-compact in X.

(viii) Suppose that X is quasi-compact; then we have :

(a) If T is retro-compact in X, then T is quasi-compact.

(b) Every globally constructible subset of X is quasi-compact.

(ix) Suppose that X is quasi-compact and quasi-separated; then we have :

(a) U is retro-compact in X if and only if it is quasi-compact.

(b) The globally constructible subsets of X are precisely the finite unions of subsets of the form V \V′, where V and V′ are arbitrary quasi-compact open subsets of X.

(x) Suppose that X is coherent; then we have :

(a) T is constructible in X if and only if it is globally constructible in X (in which case T is also retro-compact, by (ii)).

(b) T is pro-constructible (resp. ind-constructible) if and only if it is the intersection (resp. the union) of a family of constructible subsets of X.

(c) The constructible open (resp. closed) subsets of X are precisely the quasi-compact open subsets (resp. the complements of quasi-compact open subsets) of X.

(d) If T is retro-compact in X and S is any constructible (resp. pro-constructible, resp. ind-constructible) subset of X, then T ∩ S is constructible (resp. pro-constructible, resp. ind-constructible) in T.

(e) If T is retro-compact in X, then it is a coherent topological space, and the inclusion map i_{T} : T → X is quasi-separated.

(f) If T is constructible in X, and S is any constructible subset of T, then S is constructible in X.

(xi) If X is spectral and T is constructible in X, then T is spectral as well.

Proof. (i) shall be left to the reader, and (ii) follows easily from (i).

(iii): The first assertion is obvious, and the second follows from (ii) and the first assertion. (v): The proof for the case where T is retro-compact or globally constructible is left to the

reader. Suppose that T is constructible in X; then, for any given point x ∈ U, we may find an open neighborhood U′ of x in X, such that T ∩ U′ is globally constructible in U′; then by the foregoing case, we know that U′ ∩ U is an open neighborhood of x in U such that T ∩ U′ ∩ U is globally constructible in U′ ∩ U, so T ∩ U is constructible in U. The assertion for the case where T is pro-constructible or ind-constructible, is an immediate consequence.

(vi): It is easily seen that the complement of a constructible subset is constructible. Next, suppose that T and T′ are two constructible subsets of X, and x ∈ X is any point; then we may find open neighborhoods U and U′ of x such that T ∩ U (resp. T′ ∩ U′) is globally constructible in U (resp. in U′). By (v), it follows that T ∩ U ∩ U′ and T ∩ U ∩ U′ are globally constructible in U ∩ U′, and therefore the same holds for both (T ∩ T′) ∩ U ∩ U′ and (T ∪ T′) ∩ U ∩ U′. We conclude that both T ∩ T′ and T ∪ T′ are constructible, whence the assertion.

(iv) follows easily from from (vi).

(vii) and (viii.a) are obvious, and (viii.b) follows from (viii.a) and (ii).

(ix.a) follows from (vii) and (viii.a), and (ix.b) follows from (ix.a) and (ii).

(x.a): Indeed, by assumption we may find for every x ∈ X an open neighborhood U_{x} of x in X such that T ∩ U_{x} is globally constructible in U_{x}; by (v) we may then assume that U_{x} is quasi-compact, so U_{x} is retro-compact, by (ix.a) and remark 8.1.6(i), and consequently T ∩ U_{x} is globally constructible in X, by (iii). Finitely many of such U_{x} suffice to cover X, whence the assertion.

(x.b): Indeed, say that T is ind-constructible; arguing as in the proof of (x.a) we may find, for every x ∈ X, a quasi-compact open subset U_{x} such that T ∩ U_{x} is the union of a family (T_{i,x} | i ∈ I_{x}) of constructible subsets of U_{x}. By (x.a) and remark 8.1.6(ii), each T_{i,x} is globally constructible in U_{x}, hence also in X, by (viii.a), (iii) and remark 8.1.6(i); since T = ⋃_{x∈X} ⋃_{i∈I_{x}} T_{i,x}, the claim follows. The assertion for pro-constructible subsets is reduced to this case, by taking complements and using (iv).

(x.c): Indeed, every quasi-compact open subset is constructible, by (vii); conversely, (x.a) and (ix.b) show that every constructible subset of X is quasi-compact. The assertion about constructible closed subsets of X then follows immediately, taking into account (vi).

(x.d): In view of (x.b), it suffices to consider the case where S is constructible, and so S is even globally constructible in X, by (x.a). Then, we are further reduced to the case where S is an open retro-compact subset of X, i.e. S is open and quasi-compact in X, by (x.c), and it suffices to check that T ∩ S is retro-compact in T. Thus, let V be any quasi-compact open subset of T; since X is coherent, we may write V = V′ ∩ T, for some quasi-compact open subset V′ of X (details left to the reader), and then V ∩ (T ∩ S) = (V′ ∩ S) ∩ T. Now, V′ ∩ S is open and quasi-compact in X (remark 8.1.6(i)) and therefore V ∩ (T ∩ S) is quasi-compact in T, and therefore also in T ∩ S, since the latter is open in T.

(x.e): T is quasi-compact by (viii.a), and if (U_{i} | i ∈ I) is a basis of quasi-compact open subsets of X which is closed under finite intersections, then (T ∩ U_{i} | i ∈ I) is a basis of quasicompact open subsets for T which is also closed under finite intersections, so T is coherent. Next, let U ⊂ X be any open subset (then U is quasi-separated, by remark 8.1.6(i)), and V_{1}, V_{2} any two quasi-compact open subsets of U ∩ T; then V_{1} and V_{2} are also quasi-compact in T, so the same follows for V_{1} ∩V_{2} (again, by remark 8.1.6(i)), which shows that i_{T} is quasi-separated.

(x.f): In view of (x.e), the topological space T is coherent, so S is globally constructible in T by (x.a), and in light of (ii) and (vi) we may assume that S = V \V′, for two retro-compact open subsets V, V′ of T. Then V and V′ are quasi-compact in T, by (viii.a), and since X is coherent, we may find quasi-compact open subsets W, W′ of X such that V = W ∩ T and V′ = W′ ∩ T, so that S = (W \ W′) ∩ T, and the latter is constructible in X, again by (vi) and (x.c).

(xi): In light of (x.e) and (x.a), it remains to show that T is sober, and taking into account (ii) and lemma 8.1.3(iv), we may assume that T is locally closed in X, in which case it suffices to invoke lemma 8.1.3(iii). □

Remark 8.1.20. Let f : X → Y be any quasi-compact continuous map of topological spaces.

(i) If T ⊂ X is a retro-compact subset of X, then f(T) is a retro-compact subset of Y. Indeed, if U ⊂ Y is any quasi-compact open subset, we have f(T) ∩ U = f(T ∩ f^{−1}U), and f^{−1}U is a quasi-compact open subset of X, by assumption.

(ii) If f is quasi-separated, Y is locally coherent, and T ⊂ Y is any constructible (resp. pro-constructible, resp. ind-constructible) subset in Y, then f^{−1}T enjoys the same property in X. To see this, we may assume that Y is coherent, in which case X is quasi-separated and T is constructible (resp. an intersection, resp. a union of constructible subsets, by lemma 8.1.19(x.b)); then it suffices to consider the case where T is constructible, and we are further reduced to the case where T is open and quasi-compact (lemma 8.1.19(ix.b,x.a)), for which the assertion follows from lemma 8.1.19(vii).

(iii) If f is quasi-separated, then f is spectral. Indeed, say that U ⊂ X and V ⊂ U are quasi-compact and quasi-separated open subsets such that f(U) ⊂ V, and let V′ ⊂ V be any quasi-compact open subset of V; then f^{−1}V is quasi-compact and quasi-separated, and f_{|U}^{−1}V′ = U ∩ f^{−1}V′, which is quasi-compact, under the current assumptions, because U and f^{−1}V′ are quasi-compact open subsets of f^{−1}V.

(iv) Suppose that X is locally coherent, T ⊂ X is any retro-compact subset, and endow T with the topology induced from X. Then the inclusion map i_{T} : T → X is spectral. To see this, first we invoke proposition 8.1.16(i,iii) to reduce to the case where X is coherent. Then the assertion follows from (iii) and lemma 8.1.19(x.e). Together with lemma 8.1.19(i), we see especially that any closed immersion Z → X is quasi-compact, quasi-separated and spectral.

Proposition 8.1.21. Let X be a coherent topological space. Then X is noetherian if and only if every irreducible closed subset is constructible.

Proof. If X is noetherian, then every open subset of X is constructible, by virtue of lemma 8.1.19(x.c), so the condition is necessary. Conversely, suppose that X is coherent, and that every irreducible closed subset of X is constructible. Denote by ℱ the set of closed non-constructible subsets of X, partially ordered by inclusion. If (Z_{λ} | λ ∈ Λ) is a non-empty totally ordered subset of ℱ, let us show that Z := ⋂_{λ∈Λ} Z_{λ} lies in ℱ. Indeed, if Z ∉ ℱ, then X \ Z would be a quasi-compact open subset of X (lemma 8.1.19(x.c)), and since X \ Z = ⋃_{λ∈Λ} X \ Z_{λ}, we would have X \ Z = X \ Z_{λ} for some λ ∈ Λ, i.e. Z = Z_{λ}, which is absurd, since Z_{λ} ∈ ℱ. By Zorn’s lemma, ℱ admits therefore a minimal element Z_{0}, if it is non-empty. Let us check that Z_{0} is irreducible : indeed, otherwise we have Z_{0} = Z′_{0} ∪ Z′′_{0} for two closed subsets Z′_{0}, Z′′_{0} strictly contained in Z_{0}; by minimality of Z_{0}, the subsets Z′_{0} and Z′′_{0} are therefore constructible, and then the same holds for Z_{0}, a contradiction. Now, by assumption every irreducible subset of X is constructible; hence we must have ℱ = ∅, and this easily implies that X is noetherian, again by virtue of lemma 8.1.19(x.c). □

8.1.22. Let X_{•} := (X_{i} | i ∈ Ob(I)) be a system of sober and quasi-compact topological spaces, indexed by a category I, with continuous transition maps p_{u} : X_{i} → X_{j} for every morphism u : i → j in I, and set

X := lim_{i∈I} X_{i}

where the limit is taken in the category of topological spaces. For every i ∈ Ob(I), denote by p_{i} : X → X_{i} the induced projection, and recall that the system of all subsets of the form ⋂_{j∈J} p_{j}^{−1}U_{j}, where J ranges over all finite subsets of Ob(I), and U_{j} ranges over all open subsets of X_{j}, is a basis of the topology of X.

Proposition 8.1.23. In the situation of (8.1.22), the following holds :

(i) X is a sober topological space.

(ii) Suppose furthermore, that the system X_{•} is cofiltered. We have : (a) If X_{i} ≠ ∅ for every i ∈ Ob(I), then X is quasi-compact and non-empty.

(b) If p_{u} is quasi-compact for every morphism u of I, then p_{i} is quasi-compact, for every i ∈ Ob(I).

Proof. (i) follows immediately from corollary 1.3.26(i,ii) and proposition 8.1.8.

(ii.a): By proposition 1.5.21(ii), we may assume that I is a partially ordered set, and to ease notation, for every u : i → j in I (i.e. for every i, j ∈ I such that i ≤ j), we set p_{ij} := p_{u}.

Now, consider the family ℱ consisting of all compatible systems Z_{•} := (Z_{i} | i ∈ I), where : • Z_{i} is a non-empty closed subset of X_{i}, for every i ∈ I

• p_{ij}(Z_{i}) ⊂ Z_{j} for every i, j ∈ I with i ≤ j and endow ℱ with a partial ordering, by declaring that, for any Z_{•}, Z′_{•} ∈ ℱ, we have Z_{•} ≤ Z′_{•} if and only if Z_{i} ⊂ Z′_{i} for every i ∈ I. Suppose now that (Z_{•}^{λ} | λ ∈ Λ) is a totally ordered subset of ℱ, and set C_{i} := ⋂_{λ∈Λ} Z_{i}^{λ} for every i ∈ I.

Claim 8.1.24. The resulting system C_{•} lies in ℱ.

Proof of the claim. Indeed, the assertion comes down to saying that C_{i} ≠ ∅ for every i ∈ I. But notice that, for every finite subset Λ′ ∈ Λ and every i ∈ I, the intersection ⋂_{λ∈Λ′} Z_{i}^{λ} is obviously closed and non-empty in X_{i}. Since X_{i} is quasi-compact for every i ∈ I, the claim follows. ♢

Since (X_{i} | i ∈ I) ∈ ℱ, we have ℱ ≠ ∅. From claim 8.1.24and Zorn’s lemma, we deduce that ℱ admits minimal elements. We notice:

Claim 8.1.25. Let Z_{•} be a minimal element of ℱ. We have :

(i) Z_{i} is irreducible in X_{i}, for every i ∈ I.

(ii) p_{ij}(Z_{i}) is dense in Z_{j}, for every i, j ∈ I with i ≤ j.

Proof of the claim. (i): Suppose, by way of contradiction, that there exist i ∈ I and closed non-empty proper subsets Z^{(1)}, Z^{(2)} of Z_{i}, such that Z_{i} = Z^{(1)} ∪ Z^{(2)}. For t = 1, 2, we consider the compatible system Z_{•}^{(t)} such that

Z_{j}^{(t)} := \{ \begin{matrix}Z_{j} ∩ p_{ji}^{−1}Z^{(t)} & if j ≤ i \\ Z_{j} & otherwise. \end{matrix}

By the minimality of Z_{•}, neither Z_{•}^{(1)} nor Z_{•}^{(2)} lies in ℱ, hence there must exist j, k ∈ I such that Z_{j}^{(1)} = Z_{k}^{(2)} = ∅. Since I is cofiltered, we may then assume that j = k, in which case Z_{k} = Z_{k} ∩ p_{ki}^{−1}Z_{i} = Z_{k}^{(1)} ∪ Z_{k}^{(2)} = ∅, which is absurd.

(ii): Fix any i, j ∈ I with i ≤ j, and denote by T the topological closure of p_{ij}(Z_{i}) in Z_{j}. We consider the compatible system T_{•} such that

T_{k} := \{ \begin{matrix}p_{kj}^{−1}T & if k ≤ j \\ Z_{k} & otherwise. \end{matrix}

Notice that T_{k} = Z_{k} for every k ≤ i; especially T_{k} ≠ ∅ for every k ∈ I, and therefore T_{•} ∈ ℱ. By the minimality of Z_{•}, we must have T_{•} = Z_{•}, and therefore T = Z_{j}, as stated. ♢

Let now Z_{•} be any minimal element of ℱ; according to claim 8.1.25, every Z_{i} is irreducible, say with generic point η_{i}, and p_{ij} maps η_{i} to η_{j}, whenever i, j ∈ I with i ≤ j. The compatible system of points (η_{i} | i ∈ I) is then a point of X, so the latter is not empty.

Next, notice that every open covering of X can be refined to a covering of the form :

(8.1.26)  X = ⋃_{i∈I} p_{i}^{−1}U_{i}  for a family of open subsets U_{i} ∈ X_{i}

hence, in order to prove that X is quasi-compact, it suffices to check that, for a covering as (8.1.26), there exists a finite subset J ⊂ I such that the family (p_{j}^{−1}U_{j} | j ∈ J) already covers X. To this aim, set

Z_{i} := X_{i}\U_{i}  and Z′_{i} := ⋂_{j≥i} p_{ij}^{−1}Z_{j}  for every i ∈ I.

Notice that (8.1.26) means that ⋂_{i∈I} p_{i}^{−1}Z_{i} = ∅, and a fortiory we have

(8.1.27)  ⋂_{i∈I} p_{i}^{−1}Z′_{i} = ∅

as well. However, by construction we have p_{ij}(Z′_{i}) ⊂ Z′_{j} for every i, j ∈ I with i ≤ j, so (Z′_{i} | i ∈ I) is a cofiltered system of sober and quasi-compact topological spaces (lemma 8.1.3(iii)), and (8.1.27) shows that the limit of this system is the empty topological space; in light of the foregoing, we conclude that there exists i ∈ I such that Z′_{i} = ∅. Since X_{i} is quasicompact, this in turns means that there exists a finite subset J ⊂ {j ∈ I | j ≥ i} such that ⋂_{j∈J} p_{ij}^{−1}Z_{j} = ∅. Lastly, this means precisely that the family (p_{j}^{−1}U_{j} | j ∈ J) covers X.

(ii.b): Fix i_{0} ∈ I, and let U_{i_{0}} be any quasi-compact open subset of X_{i_{0}}; the subset I_{0} := {j ∈ I | j ≤ i_{0}} is cofinal in I, so we may replace I by I_{0} and assume that i_{0} is the largest element of I. In this case, for every j ≤ i_{0} we may set U_{j} := p_{ji_{0}}^{−1}U, and (U_{i} | i ∈ I) is a cofiltered system of quasi-compact and sober topological spaces (lemma 8.1.3(iii)), hence its limit U is quasi-compact (and sober), by (i) and (ii.a); on the other hand, the induced map U → X is an open immersion, whence the contention. □

Remark 8.1.28. Proposition 8.1.23fails in general for non-cofiltered systems of quasi-compact and sober spaces. For instance, a fibre product of such spaces shall not be quasi-compact, in general.

Definition 8.1.29. Let (X, 𝒯 ) be a topological space. The set of ind-constructible subsets of X is closed under finite unions and finite intersections, and forms a basis for a topology 𝒞 on X called the constructible topology. We shall denote by X^{c} the topological space (X, 𝒞).

Remark 8.1.30. Let f : X → Y be a continuous map of topological spaces.

(i) The map f is not necessarily continuous with respect to the constructible topologies on X and Y, but remark 8.1.20(ii) says that if f is quasi-separated and quasi-compact, and Y is locally coherent, then the induced map

f^{c} : X^{c} → Y^{c}

(whose underlying map of sets is the same as f) is continuous.

(ii) Let U be any open subset of a topological space X, and endow U with the topology induced from X. Then the topology of U^{c} is finer than the topology on U induced from the topology of X^{c} (lemma 8.1.19(v)). The same holds if X is coherent and U is retro-compact (lemma 8.1.19(x.d)).

(iii) Suppose that X is locally coherent. By lemma 8.1.19(iv, x.b), it follows that the open (resp. closed) subsets of X^{c} are the ind-constructible (resp. pro-constructible) subsets of X.

Proposition 8.1.31. Let f : X → Y be any spectral map between locally coherent topological spaces, T ⊂ Y any subset. The following holds :

(i) f^{c} is continuous.

(ii) If T is constructible (resp. ind-constructible, resp. pro-constructible) in Y, then f^{−1}T is constructible (resp. ind-constructible, resp. pro-constructible) in X.

Proof. Clearly, it suffices to check (ii). However, let T ⊂ Y be any subset as in (ii), and pick a covering (U_{i} | i ∈ I) of Y consisting of open coherent subsets; it suffices to check that f^{−1}(T ∩U_{i}) is constructible (resp. ind-constructible, resp. pro-constructible) in f^{−1}U_{i} for every i ∈ I, so we may replace Y by any U_{i}, and assume that Y is coherent, in which case, lemma 8.1.19(x.b) implies that it suffices to consider the case where T is constructible in Y. Then, T shall be a finite union of subsets of the form T′ \ T′′, where T′′ ⊂ T′ are any two quasi-compact open subsets of Y (lemma 8.1.19(ix.b,x.a)), and we may further reduce to the case where T is an open quasi-compact subset of Y. Now, let U be any coherent open subset of X; it suffices to check that W := U ∩ f^{−1}V is quasi-compact in U (lemma 8.1.19(x.c)), and the latter holds, since f is spectral. □

Lemma 8.1.32. Let X be a locally coherent topological space, T ⊂ X any subset, and endow T with the topology induced from X. We have :

(i) If T is either constructible or open in X, the topology on T induced from X^{c} agrees with the topology of T^{c}.

(ii) The topology of X^{c} is finer than that of X.

Proof. (i): It suffices to show that :

(a) if S is any ind-constructible subset of X, then S ∩ T is ind-constructible in T

(b) if S is any ind-constructible subset of T, then S is ind-constructible in X. However, fix a basis (U_{i} | i ∈ I) of X consisting of coherent open subsets, and set T_{i} := T ∩ U_{i} for every i ∈ I. If S is as in (a), and if S ∩ T_{i} is ind-constructible in T_{i} for every i ∈ I, then clearly S ∩ T is ind-constructible in T. If S is as in (b), and S ∩ U_{i} is ind-constructible in U_{i} for every i ∈ I, then S is ind-constructible in X. Hence, it suffices to prove (a) and (b) with X and T replaced by respectively U_{i} and T_{i}, for every i ∈ I. We may then assume from start that X is coherent. In this case, T is either open or retro-compact in X (lemma 8.1.19(x.a)), hence the topology of T^{c} is finer than the topology 𝒯 on T induced from X^{c} (remark 8.1.30(ii)), whence (a). To show (b), suppose first that T is constructible in X; then, any ind-constructible subset of T is a union of constructible subsets in T (lemma 8.1.19(x.b,e)), and therefore it is also a union of constructible subsets in X (lemma 8.1.19(x.f)), whence (b), in this case. Lastly, if T is open in X, and S is ind-constructible in T, pick a covering (T_{i} | i ∈ I) of T consisting of open coherent subsets; then S_{i} := S ∩ T_{i} is ind-constructible in T_{i} for every i ∈ I, so for every such i we may find a family (S_{i,λ} | λ ∈ Λ_{i}) of globally constructible subsets of T_{i}, whose union is S_{i} (lemma 8.1.19(x.a,b)). However, T_{i} is retro-compact in X, so S_{i,λ} is constructible in X, for every i ∈ I and every λ ∈ Λ_{i} (lemma 8.1.19(iii)); finally, S = ⋃_{i∈I} ⋃_{λ∈Λ_{i}} S_{i,λ} is ind-constructible in X, so (b) holds also in this case.

(ii): We have to check that every open subset U of X is ind-constructible in X; however, U is ind-constructible in U, so the assertion follows from the foregoing. □

8.1.33. We come now to the main theorem of this section. To prepare the proof, we introduce the following terminology. Let ℱ be any family of subsets of a topological space T; we say that ℱ is a closed filter (resp. a constructible filter) if :

• every Z ∈ ℱ is closed (resp. constructible) in T

• ∅ ∉ ℱ and ℱ ≠ ∅

• if Z, Z′ ∈ ℱ, then Z ∩ Z′ ∈ ℱ

• if Z ∈ ℱ and Z′ is any closed (resp. constructible) subset of T containing Z, then

Z′ ∈ ℱ. The center of a filter ℱ is the subset ⋂_{Z∈ℱ} Z. It is easily seen that a topological space T is quasi-compact if and only if every closed filter of T has non-empty center.

Theorem 8.1.34. Let X be a topological space. We have :

(i) If X is spectral, X^{c} is compact.

(ii) If X is locally spectral and quasi-separated, then X^{c} is locally compact and separated.

Proof. (We say that a topological space is locally compact if it admits a covering consisting of compact open subsets.)

(i): We check first that X^{c} is separated. To this aim, let x, y be any two points of X, and denote by \overline{\{x\}} and \overline{\{y\}} the topological closures of \{x\} and \{y\} with respect to 𝒯 . If x ∉ \overline{\{y\}}, it follows from lemma 8.1.32(ii) that we may find a constructible subset T ⊂ X such that x ∈ T and y ∉ T; then the complement S of T in X is also constructible, and T and S separate the points x and y. Likewise one argues in the symmetric case where y ∉ \overline{\{x\}}. Lastly, if neither of these conditions hold, we must have x = y, since X is sober and both x and y are generic points of the same closed subset of X.

To show that X^{c} is quasi-compact, we check that its closed filters have non-empty center. However, let ℱ be any such filter, and denote by ℱ′ the set of all constructible subsets T of X such that there exists F ∈ ℱ with F ⊂ T. Clearly ℱ′ is a constructible filter of X, thus, we come down to :

Claim 8.1.35. (i) ℱ and ℱ′ have the same center.

(ii) Every constructible filter of X has non-empty center.

Proof of the claim. (i): Clearly the center of ℱ′ contains the center of ℱ. For the converse, it suffices to remark that every closed subset of X is pro-constructible in X (lemma 8.1.32(ii) and remark 8.1.30(iii)), and therefore it is the intersection of the constructible subsets that contain it (lemma 8.1.19(x.b)).

(ii): In view of lemma 8.1.19(xi), every such filter can be regarded as a cofiltered system of spectral spaces, with continuous transition maps. Then the assertion is a special case of proposition 8.1.23(ii.a). ♢

(ii): Lemma 8.1.32 and (i) imply that X^{c} is locally compact. To see that X is separated, let x, y ∈ X be any two distinct points. We may then find a quasi-compact open subset U of X containing both x and y, and the induced map U^{c} → X^{c} is an open immersion (lemma 8.1.32), so we may replace X by U, and assume from start that X is is spectral (lemmata 8.1.15(iv) and 8.1.19(vii)), which is the case covered by (i). □

Corollary 8.1.36. Let X be any spectral topological space, U ⊂ X a subset. We have : (i) U is open and quasi-compact in X^{c} if and only if it is constructible in X. (ii) X^{c} is a spectral topological space. (iii) If Y is any locally spectral topological space, the same holds for Y^{c}.

Proof. (i): Suppose first that U is open and quasi-compact in X^{c}; then U is a union of a family (U_{i} | i ∈ I) constructible subsets of X (lemma 8.1.19(x.b)), and there exists a finite subset J ⊂ I such that (U_{i} | i ∈ J) already covers U, so U is constructible.

Conversely, suppose that U is constructible in X, in which case it is both open and closed in X^{c}; since the latter is quasi-compact (theorem 8.1.34(i)), we see that U is quasi-compact in X^{c}.

(ii): Indeed, (i) and lemma 8.1.19(vi) easily imply that X^{c} is coherent, and X^{c} is trivially sober, since in any separated space the irreducible closed subsets are exactly those that contain a unique point.

(iii) follows from (ii) and lemma 8.1.32: details left to the reader. □

Corollary 8.1.37. Let f : X → Y be a continuous map of locally spectral topological spaces. The following holds :

(i) f is spectral if and only if the same holds for f^{c}. (ii) Suppose furthermore that f is spectral. Then :

(a) f is quasi-compact if and only if the same holds for f^{c} : X^{c} → Y^{c}.

(b) f is quasi-separated if and only if the same holds for f^{c}.

Proof. We notice first :

Claim 8.1.38. In order to prove the corollary, we may assume that Y is spectral.

Proof of the claim. Let (Y_{i} | i ∈ I) be a covering of Y consisting of spectral open subsets; then each Y_{i}^{c} is open in Y^{c}, and its topology agrees with the topology induced from Y^{c} (lemma 8.1.32(i,ii)), so (Y_{i}^{c} | i ∈ I) is an open covering of Y^{c}. Then the claim follows immediately from proposition 8.1.16(iii) and corollary 8.1.36(iii). ♢

Thus, we assume henceforth that Y is spectral. Next, we remark, quite generally :

Claim 8.1.39. Let g : T → S be any continuous map between locally spectral topological spaces, with S locally compact. Then g is spectral.

Proof of the claim. By lemma 8.1.16(iii), we may assume that S is compact and that T is quasicompact and quasi-separated. Now, let V ⊂ S be any quasi-compact open subset; since S is separated, V is also closed in S, hence g^{−1}V is closed in T, so it is quasi-compact, since the same holds for T. Thus, g is quasi-compact, and it is also obviously quasi-separated (due to remark 8.1.6(i)), so the claim follows from remark 8.1.20(iii). ♢

(i): In light of claim 8.1.39, proposition 8.1.31(i) and theorem 8.1.34(i), we see that if f is spectral, the same holds for f^{c}, hence we may assume that f^{c} is spectral, and we show that the same holds for f. Thus, let U ⊂ X and V ⊂ Y be quasi-compact and quasi-separated open subsets such that f(U) ⊂ V; it suffices to check that f_{|U} : U → V is quasi-compact, and since U^{c} (resp. V^{c}) is open in X^{c} (resp. in Y^{c}, by lemma 8.1.32(ii)), the restriction f_{|U}^{c} : U^{c} → V^{c} is still spectral, so we may replace X by U and Y by V, and assume that X is quasi-compact and quasi-separated, in which case it is spectral (lemma 8.1.15(iv)), and we need to show that f is quasi-compact. Now, let V ⊂ Y be any quasi-compact open subset; then V is spectral (remark 8.1.6(ii)), and V^{c} is open and quasi-compact in Y^{c} (lemma 8.1.32(ii) and theorem 8.1.34(i)). Hence, V^{c} is closed in Y^{c}, and therefore f^{−1}V^{c} is closed, and therefore quasi-compact, in X^{c}. Since the topology of the latter is finer than that of X, we conclude that f^{−1}V is quasi-compact in X.

(ii.a): If f is quasi-compact, it follows that X admits a finite covering consisting of spectral open subsets, and if U is any such subset, the restriction f_{|U} is quasi-compact, since f is spectral. Clearly it suffices to check that f_{|U}^{c} is quasi-compact for every such U, so we may replace X by U and assume that X is spectral as well, in which case, notice that f is obviously quasiseparated. Now, say that U ⊂ Y^{c} is open and quasi-compact; so, U is constructible in Y (corollary 8.1.36(i)), and then V := f^{−1}U is constructible in X, by remark 8.1.20(ii), so it is quasi-compact in X^{c} (corollary 8.1.36(i)).

Conversely, say that f^{c} is quasi-compact, and U is a quasi-compact open subset of Y. Then U is constructible in Y, so it is open and quasi-compact in Y^{c} (corollary 8.1.36(i)), therefore f^{−1}U is quasi-compact in X^{c}, and since the topology of X^{c} is finer than that of X (lemma 8.1.32(ii)), we conclude that f^{−1}U is quasi-compact in X.

(ii.b): Suppose that f is quasi-separated; then X is quasi-separated, so X^{c} is separated (theorem 8.1.34(ii)), and it follows easily that f^{c} is quasi-separated (details left to the reader). Conversely, if f^{c} is quasi-separated, then X^{c} is quasi-separated, since Y^{c} is separated (by theorem 8.1.34(i)). Then, say that U, V are two quasi-compact open subsets of X; we may cover U (resp. V) by a finite family (U_{i} | i ∈ I) (resp. (V_{j} | j ∈ J)) of spectral open subsets, and the resulting maps U_{i}^{c} → X^{c}, V_{j}^{c} → X^{c} are open immersions (lemma 8.1.32). Moreover, each U_{i}^{c} and V_{j}^{c} is compact, by theorem 8.1.34(i), so U_{i}^{c} ∩ V_{j}^{c} is quasi-compact in X^{c}, for every i ∈ I and every j ∈ J; consequently U ∩ V is quasi-compact in X^{c}, and a fortiori also in X, since the topology of the latter is coarser than that of X^{c} (lemma 8.1.32(ii)). □

Corollary 8.1.40. (i) In the situation of (8.1.22), suppose moreover that :

(a) The topological space X_{i} is spectral for every i ∈ Ob(I).

(b) The transition map p_{u} is quasi-compact, for every morphism u of I. Then X is spectral.

(ii) Suppose furthermore, that the system X_{•} is cofiltered. Then :

(a) For every constructible (resp. open and quasi-compact) subset U ⊂ X there exist i ∈ Ob(I) and a constructible (resp. open and quasi-compact) subset U_{i} ⊂ X_{i} such that U = p_{i}^{−1}U_{i}.

(b) For any i ∈ Ob(I) and any two constructible subsets U, U′ of X_{i} such that p_{i}^{−1}U ⊂ p_{i}^{−1}U′, there exists a morphism u : j → i in I, such that p_{u}^{−1}U ⊂ p_{u}^{−1}U′.

Proof. (i): By proposition 8.1.23(i) we know already that X is sober, so it remains only to check that X is coherent. However, under the stated conditions, the datum (X_{i}^{c} | i ∈ Ob(I)) is a system of quasi-compact and separated topological spaces, with continuous transition maps (theorem 8.1.34 and remark 8.1.30(i)); denote by X^{c} the limit of this system (in the category of topological spaces). Now, the product Y := ∏_{i∈Ob(I)} X_{i}^{c} is likewise quasi-compact and separated, and we remark :

Claim 8.1.41. The natural map X^{c} → Y is a closed immersion.

Proof of the claim. For every i ∈ Ob(I), denote by q_{i} : Y → X_{i}^{c} the natural projection; the image of X^{c} in Y is the intersection of the subsets Y_{u} := {y_{•} ∈ Y | p_{u} ∘ q_{i}(y_{•}) = q_{j}(y_{•})}, for every morphism u : i → j in I, so it suffices to check that Y_{u} is closed in Y, for every such u. This is a special case of the following more general assertion. Let f, g : T → T′ be two continuous maps of topological spaces, with T′ separated; then the subset

S := {t ∈ T | f(t) = g(t)}

is closed in T. To show the latter, denote F : T → T′×T′ the map such that F (t) = (f(t), g(t)) for every t ∈ T; then S = F^{−1}Δ_{T′}, where Δ_{T′} := {(t′, t′) | t′ ∈ T′} is the diagonal of T′ × T′, and F is continuous for the product topology on T′ × T′. We then come down to showing that Δ_{T′} is closed in T′ × T′, which holds if (and only if) T′ is separated. ♢

It follows from claim 8.1.41 that X^{c} is quasi-compact and separated; the topology of X^{c} is obviously finer than that of X, so the latter is quasi-compact as well. It remains to check that X admits a basis of quasi-compact open subsets closed under finite intersections. To this aim, let J ⊂ Ob(I) be any finite subset, pick a quasi-compact open subset U_{j} of X_{j} for each j ∈ J, and set

V := ∏_{i∈I\J} X_{j} × ∏_{j∈J} U_{j} U := X ∩ V.

It suffices to show that U is quasi-compact in X. However, notice that U_{j} is constructible in X_{j}, so it is both open and closed in X_{j}^{c}, and therefore V is closed in Y; since X^{c} is closed in Y by claim 8.1.41, it follows that U is quasi-compact in X^{c}, and a fortiori, also in X.

(ii.a): Suppose first that U is open and quasi-compact; then U is a finite union of open subsets of the form U_{J} := ⋂_{j∈J} p_{j}^{−1}U_{j}, where J ⊂ Ob(I) is a finite subset, and U_{j} is an open quasicompact subset of X_{j}, for every j ∈ J. For such a given U_{J}, since I is cofiltered, there exists i ∈ Ob(I) with morphisms u_{j} : i → j for every j ∈ J, and therefore U_{J} = p_{i}^{−1}V_{i}, where V_{i} := ⋂_{j∈J} u_{j}^{−1}U_{j} is quasi-compact in X_{i}, since each u_{j} is quasi-compact and X_{i} is quasiseparated (remark 8.1.6(ii)). Thus, U = ⋃_{j∈J′} V_{j} for a finite subset J′ ⊂ Ob(I) and quasicompact open subsets V_{j} ⊂ X_{j}; again, we may find i ∈ Ob(I) and a morphism v_{j} : i → j for every j ∈ J′, so that U = p_{i}^{−1}U_{i}, where U_{i} := ⋃_{j∈J′} v_{j}^{−1}V_{j} is open and quasi-compact in X_{i}.

The case where U is constructible is reduced easily to the foregoing case, taking into account (i) and lemma 8.1.19(ix.b,x.a), and arguing as above, using the assumption that X_{•} is cofiltered (details left to the reader).

(ii.b) Let Z := U \U′, set Z_{u} := p_{u}^{−1}Z for every morphism u : j → i in I, and endow Z_{u} with the topology induced from X_{j}^{c}. From theorem 8.1.34(i) and lemma 8.1.19(xi) we see that Z_{•} := (Z_{u} | u ∈ Ob(I/i)) is a cofiltered system of compact topological spaces (see example 1.5.9(i)), and the assumption means that the limit of Z_{•} is empty, so there exists u ∈ Ob(I/i) such that Z_{u} = ∅, whence the assertion. □

Corollary 8.1.42. Let X be any spectral topological space, T ⊂ X a pro-constructible subset, and endow T with the topology induced from X. Then T is spectral and retro-compact in X.

Proof. By lemma 8.1.19(x.b), we may write T as the intersection of a system (T_{i} | i ∈ I) of constructible subsets of X. Let J be the set of all finite subsets of I, and for every S ∈ J, endow T_{S} := ⋂_{i∈S} T_{i} with the topology induced from X. Then T_{S} is a constructible subset of X for every S ∈ J, and clearly the topological space T is the limit of the (cofiltered) inverse system (T_{S} | S ∈ J). By lemma 8.1.19(xi) each such T_{S} is spectral and retro-compact in X (lemma 8.1.19(x.a)); therefore, if S ⊂ S′ are any two elements of J, then T_{S} is constructible in T_{S′} (lemma 8.1.19(x.d)), hence also retro-compact in T_{S′} (lemma 8.1.19(x.a)), i.e. the induced map T_{S} → T_{S′} is quasi-compact, and the assertion follows from corollary 8.1.40(i) and proposition 8.1.23(ii.b). □

Proposition 8.1.43. Let (X, 𝒯 ) be a compact topological space, 𝒰 a family of open and closed subsets of X, and endow X with the coarsest topology 𝒯_{𝒰} containing 𝒰 . Then the following conditions are equivalent :

(a) (X, 𝒯_{𝒰} ) is a T_{0} topological space.

(b) (X, 𝒯_{𝒰} ) is a spectral space. Moreover, if these conditions hold, then (X, 𝒯_{𝒰} )^{c} = (X, 𝒯 ).

Proof. From remark 8.1.2(vi) we get (b)⇒(a).

(a)⇒(b): Since 𝒯 is compact, every U ∈ 𝒰 is compact as well in 𝒯 , and therefore it is quasi-compact in 𝒯_{𝒰} . Thus, (X, 𝒯_{𝒰} ) is coherent, and it remains only to check that X is sober. To this aim, let Z be any irreducible closed subset of (X, 𝒯_{𝒰} ), and set

𝒰_{Z} := {U ∈ 𝒰 | U ∩ Z ≠ ∅} and Z′ := Z ∩ ⋂_{U∈𝒰_{Z}} U.

Since 𝒯_{𝒰} is T_{0}, the subset Z′ contains at most one point of X, and to conclude, it suffices to check that Z′ ≠ ∅. However, notice that Z is closed also in 𝒯 , hence it is compact in this latter topology, so the same holds for every finite intersection U_{1}∩⋯∩U_{k} ∩Z with U_{1}, …, U_{k} ∈ 𝒰_{Z}. We are then reduced to showing that every such finite intersection is non-empty. Suppose that the latter fails, and set V_{i} := Z \ U_{i} for every i = 1, …, k; then Z = V_{1} ∪ ⋯ ∪ V_{k}, and since Z is irreducible in 𝒯_{𝒰} , it follows that Z = V_{i} for some i ≤ k, which is absurd.

Lastly, suppose that (X, 𝒯_{𝒰} ) is spectral; we claim that in this case the identity map of X yields a continuous map i : (X, 𝒯 ) → (X, 𝒯_{𝒰} )^{c}. Indeed, let T ⊂ X be a subset that is globally constructible for the topology 𝒯_{𝒰} ; in view of lemma 8.1.19(x.a,x.b), it suffices to show that T is open in the topology 𝒯 . Then lemma 8.1.19(ix.b) further reduces to the case where T = V \V′ for two quasi-compact open subsets V, V′ of (X, 𝒯_{𝒰} ). However, V and V′ are finite unions of subsets of the form U_{1} ∩ ⋯ ∩ U_{k}, where k ∈ ℕ is arbitrary, and U_{1}, …, U_{k} ∈ 𝒰 ; it follows that V and V′ are open and closed in (X, 𝒯 ), and therefore the same holds for their difference. Lastly, since (X, 𝒯 ) is separated and (X, 𝒯_{𝒰} ) is quasi-compact (theorem 8.1.34(i)), it follows that i is a homeomorphism, whence the proposition. □

Definition 8.1.44. Let X be a topological space, x, x′ ∈ X any two points.

(i) We say that x is a specialization of x′ in X, or equivalently, that x′ is a generization of x in X, if x lies in the topological closure of {x′} in X.

(ii) We say that x is an immediate specialization of x′ in X, or equivalently, that x′ is an immediate generization of x in X, if x ≠ x′, x is a specialization of x′ in X, and any other point of X that is both a specialization of x′ and a generization of x in X is equal to either x or x′.

(iii) We denote by X(x) the set of all generizations of x in X, and we endow X(x) with the

topology induced by X. (iv) Let f : X → Y be any continuous map of topological spaces. We say that f is

specializing (resp. generizing) if the following holds. For every x ∈ X and every specialization (resp. generization) y of f(x) in Y, there exists a specialization (resp. generization) x′ of x in X, such that f(x′) = y.

Remark 8.1.45. Let X be a topological space, and x any point of X.

(i) Clearly X(x) is the intersection of all open neighborhoods of x in X. Especially, if X is coherent, X(x) is a pro-constructible subset of X. If X is T_{0} (i.e. for any two distinct points of X, at least one of them admits an open neighborhood not containing the other), x is the unique closed point of X(x). If X is spectral, the same holds for X(x), and the inclusion map j_{x} : X(x) → X is quasi-compact (corollary 8.1.42); since X(x) is quasi-separated, j_{x} is also quasi-separated, and even spectral, by remark 8.1.20(iii).

(ii) Clearly, the relation

x ≤ x′ ⇔ x is a specialization of x′ in X

defines a preordering on X (see example 1.1.6(iii)), and every continuous map f : X → Y is also a morphism of preordered sets (X, ≤) → (Y, ≤). If X is T_{0}, then (X, ≤) is even a partially ordered set. Suppose that X is T_{0} and non-empty; in general, (X, ≤) admits neither maximal nor minimal points, but if X is sober, it follows easily from remark 8.1.2(ii) that (X, ≤) admits maximal points. If X ≠ ∅ is quasi-compact and T_{0}, then (X, ≤) also admits minimal elements: indeed, if (x_{i} | i ∈ I) is any (non-empty) totally ordered subset of X, let \overline{\{x_{i}\}} be the topological closure of \{x_{i}\} in X, for every i ∈ I; then Z := ⋂_{i∈I} \overline{\{x_{i}\}} ≠ ∅, so any point of Z is smaller than every x_{i}, and then the assertion follows as usual, by Zorn’s lemma.

(iii) Suppose that X is sober. Then, X is noetherian if and only if the following holds : (a) Every closed subset of X has finitely many maximal points.

(b) Every descending sequence (x_{n} | n ∈ ℕ) in (X, ≤) is stationary, i.e. it admits N ∈ ℕ

such that x_{n} = x_{N} for every n ≥ N. Indeed, for a sober space, (a) and (b) are rephrasing of the corresponding conditions of proposition 8.1.4. It is also clear that, in this situation, the dimension of X is the supremum of the lengths of all finite strictly descending sequences in (X, ≤) (see remark 8.1.2(iv)).

(iv) If f : X → Y is a continuous map of T_{0} topological spaces, it is easily seen that f is also a map of partially ordered sets (X, ≤) → (Y, ≤) for the partial orderings induced by specializations, as in (ii).

Definition 8.1.46. (i) Let X be a T_{0} topological space. We denote by

Max X and Min X

the sets of maximal and respecively minimal points of the partially ordered set (X, ≤), where ≤ is the partial ordering induced by specialization, as in remark 8.1.45(ii).

(ii) Let f : X → Y be a continuous map of T_{0} topological spaces. We say that f is maximizing (resp. minimizing) if it restricts to a map

Max(f) : Max X → Max Y (resp. Min(f) : Min X → Min Y).

Proposition 8.1.47. Let X be any locally spectral topological space, T ⊂ X a pro-constructible subset, and U ⊂ X an ind-constructible subset. The following holds :

(i) The topological closure of T in X is the set of all specializations of all points of T. (ii) U is open in X if and only if it contains the generizations of all its points. (iii) T is dense in X if and only if Max X ⊂ T (notation of definition 8.1.46(i)).

Proof. (i): Denote by \overline{T} the topological closure of T in X, and by T^{s} the set of all specializations in X of points of T. Clearly T^{s} ⊂ \overline{T}, so we need only prove the converse inclusion. However, let V be any open subset of X; on the one hand, the topological closure of T ∩ V in V equals V ∩ \overline{T}. On the other hand, say that x ∈ V is a specialization in X of a point y ∈ T; then clearly y ∈ T ∩ V, i.e. x is a specialization in V of y, so V ∩ T^{s} is the set of specializations in V of the points of V ∩ T. Since X admits a covering consisting of spectral open subsets, we may then replace X by any of these subsets, and assume from start that X is spectral.

Now, suppose that there exists x ∈ \overline{T} \T^{s}, denote by ℱ the family of all quasi-compact open neighborhoods of x in X, and notice that ℱ is cofiltered, since X is quasi-separated, and not empty, since X is quasi-compact. Since x ∈ \overline{T}, we must have W ∩ T ≠ ∅ for every W ∈ ℱ; on the other hand, we have

⋂_{W ∈ℱ} (W ∩ T) = ∅

since x ∉ T^{s}. But both T and the elements of ℱ are closed in X^{c}, so (W ∩ T | W ∈ ℱ) is a closed filter of X^{c}, therefore its center cannot be empty (theorem 8.1.34(i) and (8.1.33)), a contradiction.

(ii) follows easily from (i), by considering the pro-constructible subset X \U.

(iii) is an immediate consequence of (i). □

Example 8.1.48. Let X be a spectral space of dimension ≤ 0. It follows easily from proposition 8.1.47(i) that every quasi-compact open subset of X is closed. Then X is separated : indeed, let x, x′ ∈ X be any two distinct points; since X is sober, we may assume that x does not lie in the topological closure of {x′}, in which case there exists a quasi-compact open neighborhood U of x that does not contain x′, and the foregoing implies that X \ U is an open neighborhood of x′, whence the contention. Conversely, every separated and compact topological space whose quasi-compact open subsets form a basis of the topology is a spectral space of dimension ≤ 0; indeed, such a space is coherent, and it is obviously sober, since all its points are closed. A topological space with this properties is sometimes called a compact boolean space : see [115, p.168,169], where it is also explained that, by virtue of the Stone representation theorem, such spaces are characterized as those topological spaces which are homeomorphic to the maximal spectra of (unital) boolean algebras. More precisely, to such any compact boolean space X, one attaches its characteristic ring R_{X}, which is the boolean algebra of all continuous functions X → ℤ/2ℤ, and then X is naturally homeomorphic to Max R_{X}.

Corollary 8.1.49. Let X be a spectral space, and T ⊂ X a subset that contains the generizations of all its points, and endow T with the topology induced by X. The following conditions are equivalent :

(a) T is quasi-compact.

(b) T is a pro-constructible subset of X.

(c) T is spectral.

Proof. We know already that (b)⇒(c), by corollary 8.1.42, and clearly (c)⇒(a).

(a)⇒(b): For every x ∈ X \T, let \overline{\{x\}} be the topological closure of \{x\} in X; by assumption, \overline{\{x\}} ∩ T = ∅, so for every t ∈ T there exists a quasi-compact open subset U_{x,t} such that t ∈ U_{t,x} and \overline{\{x\}} ∩ T = ∅. Since T is quasi-compact, we then have a finite subset S_{x} ⊂ T, such that T ⊂ U_{x} := ⋃_{t∈S_{x}} U_{t,x}, and clearly U_{x} is a constructible open subset of X. We have T = ⋂_{x∈X\T} U_{x}, whence (b). □

Corollary 8.1.50. Let f : X → Y be a continuous map of locally spectral topological spaces, and T ⊂ X (resp. S ⊂ Y) any pro-constructible subset. Then :

(i) If f is quasi-compact, f(T) is a pro-constructible subset of Y.

(ii) If f is spectral and generizing, the topological closure in X of f^{−1}S is the preimage of the topological closure of S in Y.

Proof. (i): The assertion is local on Y, hence we may assume that Y is spectral; then X admits a finite covering consisting of open spectral subsets, and we are easily reduced to the case where X is spectral as well. Now, by theorem 8.1.34(i,ii), the subset T is closed, and therefore quasicompact, in X^{c}. Then f(T) is quasi-compact in Y^{c}; hence it is closed in Y^{c}, and therefore pro-constructible in Y.

(ii): Let \overline{S} denote the topological closure of S in Y; clearly the topological closure of f^{−1}S lies in f^{−1}\overline{S}, so we have only to check the converse inclusion. Thus, say that x ∈ f^{−1}\overline{S}, so s := f(x) ∈ \overline{S}, and therefore s is a specialization of a point s′ ∈ S (proposition 8.1.47(i)); since f is generizing, it follows that there exists a generization x′ of x in X, such that f(x′) = s′. This shows that x lies in the set of specializations of the points of f^{−1}S, and the contention follows. □

Proposition 8.1.51. Let f : Y → X be a surjective, spectral map of locally spectral topological spaces, and suppose that f is either specializing or generizing. Then X is the (topological) quotient of Y, under the equivalence relation induced by f.

Proof. The assertion means that a subset Z ⊂ X is open (resp. closed) in X if and only if Z′ := f^{−1}Z is closed in Y. For any subset Z of X (resp. of Y), we denote by \overline{Z} the topological closure of Z in X (resp. in Y). Of course, we may assume that Z′ is closed in Y, and we check that Z is closed in X. Now, Z′ is pro-constructible in Y (lemma 8.1.32(ii)), so f(Z′) = Z is proconstructible in X (corollary 8.1.50(i)). If f is generizing, it follows that Z′ = \overline{f^{−1}Z} = f^{−1}\overline{Z} (corollary 8.1.50(ii)); since f is surjective, we deduce that Z = \overline{Z}, i.e. Z is closed, as required. Lastly, suppose that f is specializing, and notice that \overline{Z} is the set of all specializations of all points of Z (proposition 8.1.47(i)); thus, let x ∈ Z, x′ ∈ X a specialization of x and y ∈ Z′ such that f(y) = x; since f is specializing, there exists y′ ∈ Y with f(y′) = x′. But since Z′ is closed, we have y′ ∈ Z′, so x′ ∈ Z, as required. □

Corollary 8.1.52. Let X be a spectral topological space such that Min X (resp. Max X) is finite. If Z ⊂ X is a subset closed under specializations and generizations, then Z is open and closed.

Proof. When Max X is finite, any such Z is a finite union of irreducible components, hence it is closed. But the same applies to the complement of Z, whence the contention. When Min X is finite, consider the continuous map

g : Y := ∐_{x∈Min X}X(x) → X

whose restriction to each X(x) is the inclusion map. Now, Y is spectral, and g is surjective, continuous and spectral (remark 8.1.45(i)), and it is obviously also generizing; on the other hand, clearly the assertion holds for Y, hence for X, by proposition 8.1.51. □

Remark 8.1.53. Hochster defines in [90] an involution

X ↦ X^{∗}

of the category of spectral topological spaces and spectral maps, with the following properties. For every spectral space X, the set underlying X^{∗} is the same as that underlying X, and the constructible closed subsets of X form a basis of the topology of X^{∗}. He shows that the partially ordered set (X^{∗}, ≤) is the opposite of (X, ≤) ([90, Prop.8]). Using this construction, it is possible to give an alternative proof of corollary 8.1.52.

8.1.54. In the study of the valuation spectrum to be carried out in section 9.2 we shall encounter certain subsets of a spectral space that are not pro-constructible, but nevertheless are spectral spaces, when endowed with the subspace topology. In the following proposition we describe the general principle underlying these constructions.

Let X be any topological space, S ⊂ X × X a subset of specializations of X : i.e. for every (x, y) ∈ S the point y is a specialization in X of the point x, in which case we shall also say that y is an S-admissible specialization of x, and x is an S-admissible generization of y. We say that a subset T ⊂ X is S-closed if every S-admissible specialization of every point of T lies also in T. Furthermore, we shall say that S is a transitive set of specializations, if the following holds : if x, y, z ∈ X are any three points, and (x, y), (y, z) ∈ S, then (x, z) ∈ S as well. We may then state :

Proposition 8.1.55. Let X be any spectral space, S a given set of specializations of X, set

Y := {x ∈ X | x has no proper S-admissible specializations }

and endow Y with the topology induced by the inclusion map Y → X. Suppose that :

(S1) Every y ∈ Y has a fundamental system of open neighborhoods in X consisting of

constructible S-closed open subsets. (S2) For every x ∈ X there exists an S-admissible specialization y of x that lies in Y.

Then the following holds :

(i) For every x ∈ X there exists a unique point r(x) ∈ Y that is an S-admissible specialization of x in X.

(ii) The topological space Y is spectral, and the resulting retraction

r : X → Y

is spectral. (iii) More precisely, a subset T ⊂ Y is constructible in Y if and only if r^{−1}T is con-

structible in X.

Proof. (i): Suppose that a given x ∈ X has two S-admissible specialization y, y ∈ Y; by remark 8.1.2(vi) we know that X is T_{0}, so we may assume that there exists an open neighborhood U of y that does not contain y′. If V ⊂ U is any open neighborhood of y, then x ∈ V, but V does not contain the S-admissible specialization y′ of x, contradicting (S1).

(ii): Let 𝒫 (resp. 𝒬) be the set of all open subsets T ⊂ Y such that r^{−1}T is constructible (resp. open and constructible) in X.

Claim 8.1.56. 𝒫 is a basis of the topology of Y, and Y is a T_{0} topological space.

Proof of the claim. Since X is T_{0}, condition (S1) implies easily that the same holds for Y. Next, notice that we have U = r^{−1}r(U) for every open and S-closed subset U ⊂ X, and therefore U = r^{−1}(U ∩ Y) for every such U. Thus, U ∩ Y ∈ 𝒫 for every open, constructible and S-closed subset U ⊂ X, and the claim follows from (S1). ♢

Let now 𝒯 be the topology whose basis is 𝒬; clearly every element of 𝒬 is open and closed in the topology 𝒯 . Moreover, since X is quasi-compact, it is easily seen that the same holds for (Y, 𝒯 ). Then proposition 8.1.43 and claim 8.1.56 show that Y is spectral. It is also clear that r is continuous and quasi-compact, so it is spectral, by remark 8.1.20(iii).

(iii): Proposition 8.1.43 also shows that 𝒬 is the set of constructible subsets of Y, whence the assertion. □

Corollary 8.1.57. In the situation of proposition 8.1.55, suppose moreover that the following condition holds :

(S3) For every x ∈ X, any two S-admissible specializations are comparable, i.e. if y, z ∈ X are any two S-admissible specializations of x, then either y is an S-admissible specialization of z or else z is an S-admissible specialization of y.

Then, for every subset T ⊂ X the following conditions are equivalent :

(a) T = r^{−1}r(T).

(b) T is S-closed, and every S-admissible generization of every point of T lies in T.

Proof. It is easily seen that (b)⇒(a). Hence, suppose that (a) holds, let x ∈ T be any point, and y ∈ X any S-admissible generization of x. By (S3), either x is an S-admissible specialization of r(y), or r(y) is an S-admissible specialization of x. If r(y) is an S-admissible specialization of x, then r(x) = r(y), by proposition 8.1.55(i); therefore y ∈ T, by (a). If x is an S-admissible specialization of r(y), we must have x = r(y), since r(y) ∈ Y; then (a) yields again y ∈ T.

Lastly, let y be an S-admissible specialization of x. By (S3), either r(x) is an S-admissible specialization of y, or y is an S-admissible specialization of r(x). If r(x) is an S-admissible specialization of y, then r(x) = r(y), by proposition 8.1.55(i); then (a) yields y ∈ T. If y is an S-admissible specialization of r(x), then y = r(x), since r(x) ∈ Y; then again y ∈ T, by (a). □

The foregoing notions shall be applied also to the study of schemes and their underlying topological spaces, which are always locally spectral (see remark 8.1.12). For future reference, we make the following :

Definition 8.1.58. Let X be any scheme.

(i) We say that X is locally coherent, if 𝒪_{X} is a coherent sheaf of rings.

(ii) We say that X is coherent, if it is locally coherent, quasi-compact and quasi-separated.

Notice that the topological space underlying a coherent scheme is always spectral, by lemma 8.1.15(iv). We conclude this section with a few miscellaneous results concerning the topology of schemes.

Proposition 8.1.59. Let X be any reduced and coherent scheme, and endow the set Max X of maximal points of X with the topology induced from X. Then Max X is quasi-compact.

Proof. Clearly we have Max U = U ∩ Max X for any open subset U ⊂ X; then, if (U_{i} | i ∈ I) is any finite covering of X consisting of affine open subsets, it suffices to show that Max U_{i} is quasi-compact for every i ∈ I. We may therefore assume from start that X is affine, say X = Spec A for some reduced coherent ring A. We notice :

Claim 8.1.60. Let R be any reduced ring and f ∈ R any element; we have :

Max (Spec R_{f}) = Max (Spec R) ∩ Spec R/Ann_{R}(f).

Proof of the claim. Set I := Ann_{R}(f) and let 𝔭 be any minimal prime ideal of R; then 𝔭 ∈ Max (Spec R_{f}) if and only if 𝔭 ∈ Max (Spec R) and f ∉ 𝔭. Thus, it suffices to check that f ∉ 𝔭 if and only if I ⊂ 𝔭. However, if f ∉ 𝔭, clearly I ⊂ 𝔭. For the converse, notice that I_{𝔭} = Ann_{R_{𝔭}}(f), and R_{𝔭} is a field, since R is reduced; hence I_{𝔭} is either R_{𝔭} or 0, depending on whether the image of f is invertible or not in R_{𝔭}, and the assertion follows easily. ♢

Now, pick any covering (U_{i} | i ∈ I) of Max X consisting of open subsets; we have to show that there exists a finite subset J ⊂ I such that (U_{i} | i ∈ J) already covers Max X. To this aim, we may assume without loss of generality that every U_{i} is of the form Max (Spec A_{f_{i}}) for some f_{i} ∈ A. Notice that Ann_{A}(f_{i}) is an ideal of finite type, since A is coherent; in view of claim 8.1.60, we deduce that for every i ∈ I there exists a constructible open subset V_{i} of X such that Max X\U_{i} = V_{i} ∩ Max X. By assumption, Max X ∩ ⋂_{i∈I} V_{i} = ∅; since every V_{i} contains the generizations of all its points, it follows that ⋂_{i∈I} V_{i} = ∅. By corollary 8.1.36(i) and theorem 8.1.34, we deduce that there exists a finite subset J ⊂ I such that ⋂_{i∈J} V_{i} = ∅. Then clearly this subset J will do. □

Lemma 8.1.61. Let A be a ring, I ⊂ A an ideal. Then :

(i) The following are equivalent : (a) The map A → A/I is flat.

(b) The map A → A/I is a localization.

(c) For every prime ideal 𝔭 ⊂ A containing I, we have IA_{𝔭} = 0, especially, V (I) is

closed under generizations in Spec A. (ii) Suppose I fulfills the conditions (a)-(c) of (i). Then the following are equivalent :

(a) I is finitely generated.

(b) I is generated by an idempotent.

(c) V (I) ⊂ Spec A is open.

Proof. (i): Clearly (b)⇒(a). Also (a)⇒(c), since every flat local homomorphism is faithfully flat. Suppose that (c) holds; we show that the natural map B := (1 + I)^{−1}A → A/I is an isomorphism. Indeed, notice that IB is contained in the Jacobson radical of B, hence it vanishes, since it vanishes locally at every maximal ideal of B.

(ii): Clearly (ii.c)⇒(ii.b)⇒(ii.a). If (ii.a) holds, then V (I) is constructible and closed under generizations, by (ii.c), so it is open (proposition 8.1.47(ii)), i.e. (ii.a)⇒(ii.c). □

8.1.62. For every ring A, we let ℐ (A) be the set of all ideals I ⊂ A fulfilling the equivalent conditions (i.a)-(i.c) of lemma 8.1.61, and 𝒵 (A) the set of all closed subset Z ⊂ Spec A that are closed under generizations in Spec A. In light of lemma 8.1.61(i.c), we have a natural mapping:

(8.1.63)  ℐ (A) → 𝒵 (A) : I ↦ V (I).

Lemma 8.1.64. The mapping (8.1.63) is a bijection, whose inverse assigns to any Z ∈ 𝒵 (A) the ideal I[Z] consisting of all the elements f ∈ A such that fA_{𝔭} = 0 for every 𝔭 ∈ Z.

Proof. Notice first that I[V (I)] = I for every I ∈ ℐ (A); indeed, clearly I ⊂ I[V (I)], and if f ∈ I[V (I)], then fA_{𝔭} = 0 for every prime ideal 𝔭 containing I, hence the image of f in A/I vanishes, so f ∈ I. To conclude the proof, it remains to show that if Z is closed and closed under generizations, then V (I[Z]) = Z. However, say that Z = V (J), let 𝔭 ∈ Z be any prime ideal, and suppose that f ∈ J; then Spec A_{𝔭} ⊂ Z, hence f is nilpotent in A_{𝔭}, so there exist n ∈ ℕ and an open neighborhood U ⊂ Spec A of 𝔭 such that f^{n} = 0 in U. Since Z is quasi-compact, finitely many such U suffice to cover Z, hence we may find n ∈ ℕ large enough such that f^{n} ∈ I[Z], whence the contention. □

The following result improves upon [136, Cor.2.6].

Proposition 8.1.65. Let R be a ring, R′ any finitely generated R-algebra, φ : Spec R′ → Spec R the induced morphism of schemes, Σ ⊂ Spec R any subset, and endow Σ (resp. φ^{−1}Σ) with the topology induced from Spec R (resp. from Spec R′). If Σ is a noetherian topological space, then the same holds for φ^{−1}Σ.

Proof. Set X := Spec R and X′ := Spec R′; we may find an integer n ∈ ℕ and a closed immersion X′ → Spec R[T_{1}, …, T_{n}], so it suffices to prove the assertion for R′ = R[T_{1}, …, T_{n}], and by factoring the map R → R′ as a composition R → R[T_{1}, …, T_{n−1}] → R′, an easy induction reduces to the case where R′ = R[T].

Now suppose, by way of contradiction, that the proposition fails, and let ℱ be the family of all closed subsets Z of Σ such that φ^{−1}Z is not noetherian. Since Σ is noetherian, and ℱ is not empty, ℱ admits minimal elements. Let Z be a minimal element of ℱ, and endow Z with the topology induced from Σ; then Z is also noetherian (remark 8.1.6(iv)), so we may replace Σ by Z, and assume that no proper closed subset of Σ lies in ℱ. We claim that in this case, Σ is irreducible. Indeed, let Σ_{1}, …, Σ_{k} be the finitely many irreducible components of Σ (proposition 8.1.4); if k > 1, we have Σ_{i} ∉ ℱ for i = 1, …, k, but then remark 8.1.6(v) easily implies that Σ ∉ ℱ as well, a contradiction. Hence, the topological closure \overline{Σ} of Σ in X is irreducible in X (lemma 8.1.3(i)); we endow \overline{Σ} with the reduced closed subscheme structure induced by the closed immersion \overline{Σ} → X, so \overline{Σ} is isomorphic to the spectrum of some quotient of R. We may then replace X by \overline{Σ}, after which we may assume that R is a domain, and Σ is dense in X. In this case, denote by K the field of fractions of R. We notice :

Claim 8.1.66. Let A be a domain, K the field of fractions of A, I ⊂ A[T] any ideal, and p(T) ∈ I a monic polynomial that generates I · K[T]. Then p(T) generates I.

Proof of the claim. Consider any g(T) ∈ I, and let g(T) = q(T) · p(T) + r(T) be the euclidean division of g by p in K[T]. Since p generates I · K[T], we have r = 0, and since p is monic, it is easily seen that q ∈ A[T], whence the claim. ♢

Now, suppose that (Z_{n} | n ∈ ℕ) is a strictly descending chain of closed subsets of φ^{−1}Σ, and for each n ∈ ℕ, let \overline{Z}_{n} be the topological closure of Z_{n} in Spec R′, and I_{n} ⊂ R′ the largest ideal such Spec R′/I_{n} = \overline{Z}_{n}. There follows an increasing chain of ideals (I_{n} | n ∈ ℕ). The corresponding sequence (I_{n} · K[T] | n ∈ ℕ) is stationary and consists of principal ideals of K[T], so we may find N ∈ ℕ and a monic polynomial p(T) such that p(T) generates I_{n} · K[T] for every n ≥ N. Moreover, we may find some f ∈ R\{0} such that p(T) ∈ R_{f}[T] (where R_{f} := R[f^{−1}]), and we may even assume that p(T) ∈ I_{n} · R_{f}[T] for every n ≥ N, in which case claim 8.1.66 shows that p(T) generates I_{n} · R_{f}[T] for every n ≥ N. Set U := Spec R_{f}; we conclude that the chain of subsets (\overline{Z}_{n} ∩ φ^{−1}U | n ∈ ℕ) is stationary. However, Z′_{n} := Z_{n} ∩ φ^{−1}U = \overline{Z}_{n} ∩ φ^{−1}(U ∩ Σ) for every n ∈ ℕ, so the chain (Z′_{n} | n ∈ ℕ) is stationary as well. On the other hand, by construction U ≠ ∅, so Σ′ := Σ\U is a proper closed subset of Σ, therefore φ^{−1}Σ′ is noetherian, and especially, the chain (Z_{n} ∩ φ^{−1}Σ′ | n ∈ ℕ) is stationary. Summing up, we deduce that the chain (Z_{n} | n ∈ ℕ) is stationary, which is absurd. □

8.2. Topological groups. This section collects a few generalities on topological groups that shall be used in later sections.

Definition 8.2.1. A topological group is a pair (G, 𝒯_{G}) where (G, ·, e_{G}) is a group and 𝒯_{G} a topology on the set underyling G, such that :

• the composition law of G is a continuous map (G, 𝒯_{G}) × (G, 𝒯_{G}) → (G, 𝒯_{G})

• the rule g ↦ g^{−1} for every g ∈ G yields a continuous map (G, 𝒯_{G}) → (G, 𝒯_{G}).

A morphism of topological groups φ : (G, 𝒯_{G}) → (H, 𝒯_{H}) is a group homomorphism φ : G → H that is continuous for the topologies 𝒯_{G} and 𝒯_{H}.

Remark 8.2.2. Let (G, 𝒯 ) be any topological abelian group.

(i) If U ⊂ G is any open neighborhood of the neutral element 0, set −U := {−g | g ∈ U}. Clearly −U is still an open neighborhood of 0 in G, and thus the same holds for U ∩ −U. Especially, G admits a fundamental system of open neighborhoods of 0 consisting of subsets U such that U = −U.

(ii) Let 𝒰 be any fundamental system of open neighborhoods of 0 in G. It is easily seen that a subset V ⊂ G is open if and only if for every g ∈ V there exists U ∈ 𝒰 such that g + U := {g + h | h ∈ U} ⊂ V. Especially, 𝒰 determines completely the topology of G. Notice also that, due to the continuity of the composition law of G, for every U ∈ 𝒰 there exists U′ ∈ 𝒰 such that U′ + U′ := {g + g′ | g, g′ ∈ U′} ⊂ U.

(iii) Conversely, let 𝒰 be any family of subsets of G such that :

(a) 0 ∈ U for every U ∈ 𝒰

(b) U = −U for every U ∈ 𝒰

(c) for every U ∈ 𝒰 there exists U′ ∈ 𝒰 such that U′ + U′ ⊂ U

(d) for every U, U′ ∈ 𝒰 there exists U′′ ∈ 𝒰 such that U′′ ⊂ U ∩ U′.

(e) for every U ∈ 𝒰 and every g ∈ U there exists U′ ∈ 𝒰 such that g + U′ ⊂ U. Then there exists a unique topology 𝒯 on G such that 𝒰 is a fundamental system of neighborhoods of 0 in G relative to the topology 𝒯 , and (G, 𝒯 ) is a topological group. Indeed, the uniqueness is clear from (ii), which also describes 𝒯 explicitly, and it is easily seen that the composition law G × G → G and the map G → G : g ↦ −g are both continuous for this topology : details left to the reader.

(iv) Let φ : G′ → G be a group homomorphism with G′ also abelian, and endow G′ with the topology 𝒯 ′ induced by 𝒯 . Then (G′, 𝒯 ′) is a topological group. Indeed, pick any fundamental system 𝒰 of open neighborhoods of 0 in G fulfilling conditions (a)-(e) of (iii); it is easily seen that the system φ^{−1}𝒰 := (φ^{−1}U | U ∈ 𝒰 ) is a fundamental system of open neighborhoods of the neutral element 0′ of G′ relative to the topology 𝒯 ′, and clearly φ^{−1}𝒰 fulfills the same conditions (a)-(e), whence the assertion.

(v) Likewise, let ψ : G → G′′ be a group homomorphism with G′′ also abelian; for any system 𝒰 of open neighborhoods of 0 as in (iv), set ψ(𝒰 ) := (ψ(U) | U ∈ 𝒰 ); it is easily seen that the system ψ(𝒰 ) fulfills again conditions (a)-(e) of (iii), hence there exists a unique topology 𝒯 ′′ on G′′ for which ψ(𝒰 ) is a fundamental system of open neighborhoods of the neutral element 0′′ ∈ G′′, and such that (G′′, 𝒯 ′′) is a topological abelian group. If ψ is surjective, it is easily seen that 𝒯 ′′ agrees with the topology induced by 𝒯 via ψ. More generally, ψ(G) is an open subgroup of G′′ for the topology 𝒯 ′′, and the topology on ψ(G′′) induced by 𝒯 ′′ agrees with the quotient topology induced by 𝒯 via the surjection ψ : G → ψ(G) : details left to the reader. Especially, 𝒯 ′′ is independent of the choice of 𝒰 , and is the finest topology on G′′ such that ψ : (G, 𝒯 ) → (G′′, 𝒯 ′′) is a morphism of topological abelian groups.

(vi) Let (G′, 𝒯 ′) be another topological abelian group, and φ : G → G′ a group homomorphism. Then φ is continuous if and only if it is continuous at the point 0 ∈ G, i.e. if and only if for every open neighborhood U′ ⊂ G′ of the neutral element 0′ ∈ G′ there exists an open neighborhood U ⊂ G of 0 in G with φ(U) ⊂ U′. Indeed, the condition is obviously necessary. Conversely, suppose this condition holds, and let V′ ⊂ G′ be any open subset; for every g ∈ φ^{−1}V we may pick an open neighborhood U′ of 0′ in G′ such that U′ ⊂ −φ(g) + V, and by assumption there exists an open neighborhood U ⊂ G of 0 such that U ⊂ φ^{−1}(U′), whence g + U ⊂ φ^{−1}V, so φ^{−1}V is open in G, whence the contention.

(vii) Notice that G is separated if and only if the subset {0} is closed in G. Indeed, the condition is obviously necessary. For the converse, suppose that {0} is closed, and let g ∈ G \ {0} be any other point; it suffices to show that there exist open neighborhoods U and U′ respectively of 0 and of g, such that U ∩ U′ = ∅. However, by assumption there exists an open neighborhood V of 0 that does not contain g; then let U be any open neighborhood of 0 such that U = −U and U + U ⊂ V, and set U′ := g + U. Suppose u ∈ U ∩ U′; then there exists u′ ∈ U such that u = g + u′, whence g = u − u′ ∈ U + U ⊂ V, a contradiction.

Lemma 8.2.3. Let G be a topological abelian group, I_{0} ⊂ G_{0} ⊂ G two subgroups, that we endow with the topology induced from G. Also, let 𝒯_{0} and 𝒯 be the quotient topologies of \overline{G}_{0} := G_{0}/I_{0} and respectively \overline{G} := G/I_{0}. The following holds :

(i) 𝒯_{0} agrees with the topology induced by 𝒯 via the inclusion map j : \overline{G}_{0} → \overline{G}.

(ii) Suppose that I_{0} is closed in G_{0}, denote by I ⊂ G the topological closure of I_{0} in G, and endow \overline{G}′ := G/I with the quotient topology 𝒯 ′. Then the induced map

j′ : \overline{G}_{0} → \overline{G}′

is injective, and 𝒯_{0} agrees with the topology induced by 𝒯 ′ via j′.

Proof. (i): We remark, quite generally :

Claim 8.2.4. Let

\xymatrix{X′ \ar[r]^{g′} \ar[d]_{f′} & X \ar[d]^{f} \\ Y′ \ar[r]^{g} & Y}

be a cartesian diagram of topological spaces. If f is an open map, the same holds for f′.

Proof of the claim. Every open subset of X′ is a union of subsets of the form U ×_{Y} V = g^{′−1}U∩f^{′−1}V, for arbitrary open subsets U ⊂ X and V ⊂ Y′. Then f′(U×_{Y} V) = V ∩g^{−1}f(U) is open in Y′, whence the claim. ♢ We consider the diagram

\xymatrix{G_{0} \ar[r] \ar[d]_{π_{0}} & G \ar[d]^{π} \\ _{\overline{G}_{0}} \ar[r]^{j} & \overline{G}}

whose vertical arrows are the projections, and whose top horizontal arrow is the inclusion map. It is easily seen that this diagram is cartesian, provided we endow G_{0} and \overline{G}_{0} with the topologies induced by the inclusions into G and respectively \overline{G}. Then π is an open map, so the same holds for π_{0}, by claim 8.2.4, whence the assertion.

(ii): Since I_{0} is closed in G_{0}, we have I_{0} = G_{0} ∩ I, so j′ is clearly injective. Let 𝒯′_{0} be the topology on \overline{G}_{O} induced by 𝒯 ′ via j′; the identity map is a continuous bijection

(\overline{G}_{0}, 𝒯_{0}) → (\overline{G}_{0}, 𝒯′_{0})

so it suffices to check that it is also a closed map. Now, let π′ : G → \overline{G}′ and π′_{0} : G_{0} → \overline{G}_{0} be the projections, and let Z ⊂ \overline{G}_{0} be a subset that is closed relative to the topology 𝒯_{0}; by definition, this means that π_{0}^{′−1}Z = Z + I_{0} is closed in G_{0}. Let (Z + I_{0})^{c} be the topological closure of Z + I_{0} in G; it is easily seen that (Z + I_{0})^{c} = π^{′−1}π′((Z + I_{0})^{c}), so π′((Z + I_{0})^{c}) is closed in \overline{G}′, and hence \overline{G}_{0} ∩ π′((Z + I_{0})^{c}) is closed in \overline{G}_{0}, relative to the topology 𝒯′_{0}; lastly

\overline{G}_{0} ∩ π′((Z + I_{0})^{c}) = (G_{0} ∩ (Z + I_{0})^{c})/I_{0} = (Z + I_{0})/I_{0} = Z

since Z + I_{0} is closed in G_{0}. The assertion follows. □

Lemma 8.2.5. Let f : G → H and g : H → K be two continuous homomorphisms of topological groups. If g ∘ f is an open map, the same holds for g.

Proof. Set h := g ∘ f, and let U be any open neighborhood of the neutral element e_{H} in H; since h(f^{−1}U) ⊂ g(U), we see that g(U) contains an open neighborhood of the neutral element e_{K} in K. Now, let x ∈ g(U) be any element, and pick y ∈ U such that h(y) = x; let also V be an open neighborhood of e_{H} in H such that y · V ⊂ U; it follows that g(y · V) ⊂ g(U). On the other hand, g(V) contains an open neighborhood W of e_{K}, so x · W is an open neighborhood of x contained in g(U). Since x is arbitrary, the assertion follows. □

Definition 8.2.6. Let G be any abelian topological group.

(i) A Cauchy net on G is a family (g_{λ} | λ ∈ Λ) of elements of G indexed by a filtered partially ordered set Λ, such that the following holds. For every open neighborhood U of the neutral element 0 ∈ G there exists λ ∈ Λ such that g_{μ} − g_{ν} ∈ U for every μ, ν ≥ λ.

(ii) We say that two Cauchy nets g_{•} := (g_{λ} | λ ∈ Λ) and g′_{•} := (g′_{λ′} | λ′ ∈ Λ′) of G are equivalent, and we write g_{•} ∼ g′_{•}, if the following holds. For every open neighborhood U of 0 ∈ G there exist λ ∈ Λ and λ′ ∈ Λ′ such that g_{μ} − g′_{μ′} ∈ U for every μ ≥ λ and every μ′ ≥ λ′.

(iii) We say that an element h ∈ G is a limit point for the Cauchy net g_{•} := (g_{λ} | λ ∈ Λ) if g_{•} is equivalent to the trivial Cauchy net (h_{λ′} := h | λ′ ∈ Λ′) indexed by a partially ordered set Λ′ that has exactly one element. In this case, we also say that g_{•} converges to h in G.

(iv) We say that G is complete if every Cauchy net of G admits a limit point. We denote by

AbTopGrp  and AbTopGrp_{cp.sep.}

respectively the category of abelian topological groups (with morphisms given by the continuous group homomorphisms) and its full subcategory whose objects are the complete and separated topological groups.

Remark 8.2.7. (i) In view of remark 8.2.2(vii), it is easily seen that an abelian topological group G is separated if and only if every Cauchy net of G admits at most one limit point. In this case, the unique limit point of a Cauchy net (g_{λ} | λ ∈ Λ) shall be often denoted

lim_{λ∈Λ} g_{λ}.

(ii) Moreover, it is easily seen that the relation ∼ introduced by definition 8.2.6is indeed an equivalence relation on the set of all Cauchy nets on G. Clearly, if g_{•} ∼ g′_{•}, then an element h ∈ G is a limit point for g_{•} if and only if it is a limit point for g′_{•}.

(iii) Furthermore, if φ : G → G′ is a continuous group homomorphism of topological abelian groups, and (g_{λ} | λ ∈ Λ) a Cauchy net of G, then φ(g_{•}) := (φ(g_{λ}) | λ ∈ Λ) is a Cauchy net of G′, and if g′_{•} is another Cauchy net of G with g_{•} ∼ g′_{•}, then φ(g_{•}) ∼ φ(g′_{•}).

(iv) Suppose that G is complete and separated, and let H ⊂ G be any subgroup, that we endow with the topology induced by G. Then it is easily seen that H is complete if and only if it is topologically closed in G.

Theorem 8.2.8. (i) The inclusion functor AbTopGrp_{cp.sep} → AbTopGrp admits a left adjoint

AbTopGrp → AbTopGrp_{cp.sep.} (G, 𝒯 ) ↦ (G^{∧}, 𝒯 ^{∧})

called the completion functor.

(ii) For every topological abelian group (G, 𝒯 ), the unit of adjunction i_{G} : G → G^{∧} is called the completion map of G. It has dense image, and 𝒯 agrees with the topology induced by 𝒯 ^{∧} via i_{G}.

(iii) Let φ : (G, 𝒯 ) → (G′, 𝒯 ′) be a morphism of topological abelian groups with dense image and with G′ complete and separated, and such that 𝒯 agrees with the topology induced by 𝒯 ′ via φ. By the adjunction of (i), the map φ factors through i_{G} and a unique continuous group homomorphism φ^{∧} : G^{∧} → G′, and φ^{∧} is an isomorphism of topological groups.

Proof. Let (G, +, 0) be any topological abelian group. We let G^{∧} be the set of equivalence classes of Cauchy nets of G. For any such Cauchy net g_{•}, we denote by [g_{•}] the class of g_{•} in G^{∧}. We define a composition law on G^{∧} as follows. Given two Cauchy nets g_{•} := (g_{λ} | λ ∈ Λ) and g′_{•} := (g′_{λ′} | λ′ ∈ Λ′), let Λ × Λ′ be the product of Λ and Λ′ in the category of partially ordered sets (whose underlying set is the cartesian product of Λ and Λ′); it is easily seen that the family g′′_{(•,•)} := (g_{λ} + g′_{λ′} | (λ, λ′) ∈ Λ × Λ′) is also a Cauchy net of G, and we set

[g_{•}] + [g′_{•}] := [g′′_{(•,•)}].

One checks easily that [g′′_{(•,•)}] depends only on the classes of g_{•} and g′_{•}, so we get by this rule a well defined mapping G^{∧} × G^{∧} → G^{∧}. Clearly the neutral element for this addition law is the class [0] of any constant Cauchy net (0_{λ} | λ ∈ Λ) with 0_{λ} := 0 for every λ ∈ Λ. Also, set −g_{•} := (−g_{λ} | λ ∈ Λ); then it is easily seen that [g_{•}] + [−g_{•}] = [0]. The associativity and commutativity of the addition law are likewise immediate, so G^{∧} is indeed an abelian group.

Claim 8.2.9. The set G^{∧} is essentially small.

Proof of the claim. Let g_{•} := (g_{λ} | λ ∈ Λ) be any Cauchy net of G. Fix any fundamental system 𝒰 of open neighborhoods of 0 in G, and endow it with the partial ordering such that U ≤ U′ if and only U′ ⊂ U, for every U, U′ ∈ 𝒰 ; for every U ∈ 𝒰 there exists λ(U) ∈ Λ such that g_{λ} − g_{μ} ∈ U for every λ, μ ≥ λ(U). It is easily seen that the family (g_{λ(U)} | U ∈ 𝒰 ) is a Cauchy net of G equivalent to g_{•}. Thus, G^{∧} has the cardinality of the set of equivalence classes G_{𝒰}^{∧} of all Cauchy nets indexed by 𝒰 ; clearly 𝒰 is small, so the same holds for G_{𝒰}^{∧} , whence the claim. ♢

In light of claim 8.2.9, we may replace G^{∧} by an isomorphic small group, that we shall still denote by G^{∧}. We also have a natural group homomorphism

i_{G} : G → G^{∧} g ↦ [g]

that assigns to every g ∈ G the class of any constant Cauchy net with value g. Next, by remark 8.2.2(i), we may find a fundamental system 𝒰 of open neighborhoods of 0 in G such that −U = U for every U ∈ 𝒰 ; for every U ∈ 𝒰 , we denote by U^{∧} ⊂ G^{∧} the subset consisting of the classes of all Cauchy nets (g_{λ} | λ ∈ Λ) fulfilling the following condition. There exists U′ ∈ 𝒰 such that g_{λ} + U′ ⊂ U for every λ ∈ Λ.

Claim 8.2.10. The system 𝒰^{∧} := (U^{∧} | U ∈ 𝒰 ) fulfills conditions (a)-(e) of remark 8.2.2(iii).

Proof of the claim. Indeed, (a) and (d) are obvious. To see that (b) holds, let (g_{λ} | λ ∈ Λ) be any Cauchy net, and U′, U ∈ 𝒰 such that g_{λ} + U′ ⊂ U for every λ ∈ Λ; then −g_{λ} + (−U′) ⊂ −U, i.e. −g_{λ} +U′ ⊂ U for every λ ∈ Λ, whence the contention. Next, for any U ∈ 𝒰 , pick U′ ∈ 𝒰 such that U′ + U′ ⊂ U; to check (c) it suffices to show that U^{′∧} + U^{′∧} ⊂ U^{∧}. Thus, say that (g_{λ} | λ ∈ Λ) and (g′_{λ′} | λ′ ∈ Λ′) are two Cauchy nets whose classes lie in U^{′∧}, and choose U′′ ∈ 𝒰 such that g_{λ} + U′′, g′_{λ′} + U′′ ⊂ U′ for every λ ∈ Λ and every λ′ ∈ Λ′; it follows that g_{λ} + g′_{λ′} + U′′ ⊂ U′ + U′ ⊂ U for every such λ and λ′, as arrested. To check (e), let (g_{λ} | λ ∈ Λ) be a Cauchy net, U, U′ ∈ 𝒰 such that g_{λ} + U′ ⊂ U for every λ ∈ Λ, and (g′_{λ′} | λ′ ∈ Λ) another Cauchy net such that there exists U′′ ∈ 𝒰 with g′_{λ′} + U′′ ⊂ U′ for every λ′ ∈ Λ′; it follows that g_{λ} + g′_{λ′} + U′′ ⊂ U for every such λ and λ′, whence the contention. ♢

From claim 8.2.10 and remark 8.2.2(iii) we deduce that there exists a unique topology 𝒯 ^{∧} on G^{∧} for which 𝒰^{∧} is a fundamental system of open neighborhoods of 0, and (G^{∧}, 𝒯 ^{∧}) is a topological group. Moreover, we claim that

i_{G}^{−1}(U^{∧}) = U for every U ∈ 𝒰 .

Indeed, the inclusion U ⊂ i_{G}^{−1}(U^{∧}) is obvious. For the converse, suppose that the class of the Cauchy net g_{•} := (g_{λ} | λ ∈ Λ) lies in U^{∧}, and there exists h ∈ G such that i_{G}(h) = [g_{•}]. This means that there exists U′ ∈ 𝒰 such that g_{λ} + U′ ⊂ U for every λ ∈ Λ and λ_{0} ∈ Λ such that h − g_{λ} ∈ U′ for every λ ≥ λ_{0}; thus, h ∈ g_{λ} + U′ ⊂ U, whence the contention. Taking into account remark 8.2.2(vi), it follows that i_{G} is a continuous group homomorphism, and more precisely, the topology of G agrees with the topology induced by 𝒯 ^{∧} via i_{G}. Furthermore, it is clear that the image of i_{G} is dense in G^{∧}. Let us remark, more precisely :

Claim 8.2.11. (i) For every U ∈ 𝒰 and every open subset V ⊂ G^{∧}, we have :

V ⊂ U^{∧}  ⇔  i_{G}^{−1}V ⊂ U.

(ii) For every U ∈ 𝒰 , the subset U^{∧} lies in the topological closure i_{G}(U)^{c} of i_{G}(U) in G^{∧}.

Proof of the claim. (i): By the foregoing, we know already that if V ⊂ U^{∧}, then i_{G}^{−1}V ⊂ U. Conversely, suppose that i_{G}^{−1}V ⊂ U, and let h ∈ V \ U^{∧}; then h is the class of a Cauchy net h_{•} := (h_{λ} | λ ∈ Λ) with the following property. For every U′ ∈ 𝒰 there exist x_{U′} ∈ U′ and λ(U′) ∈ Λ such that g_{U′} := h_{λ(U′)} + x_{U′} ∉ U. It is easily seen that the family g_{•} := (g_{U′} | U′ ∈ 𝒰 ) is a Cauchy net of G equivalent to h_{•}. We may thus replace h_{•} by g_{•}, and assume from start that h_{λ} ∈ G \ U for every λ ∈ Λ. Next, since V is open in G^{∧}, and since we know already that i_{G} is continuous, there exists λ ∈ Λ such that i_{G}(h_{λ}) ∈ V; by assumption, this implies that h_{λ} ∈ U, a contradiction.

(ii): Explicitly, i_{G}(U)^{c} is the set of equivalence classes of all Cauchy nets g_{•} := (g_{λ} | λ ∈ Λ) such that ([g_{•}] + V^{∧}) ∩ i_{G}(U) ≠ ∅ for every V ∈ 𝒰 . Now, suppose that [g_{•}] ∈ U^{∧}, so we may assume that g_{λ} ∈ U for every λ ∈ U, and for any given V ∈ 𝒰 pick V′ ∈ 𝒰 such that V′ + V′ ⊂ V; we may then find λ(V′) ∈ Λ such that g_{λ} − g_{λ′} ∈ V′ for every λ, λ′ ≥ λ(V′). Set Λ′ := {λ ∈ Λ | λ ≥ λ(V′)}, h := g_{λ(V′)} and k_{λ} := h − g_{λ} for every λ ∈ Λ′. Clearly k_{•} := (k_{λ} | λ ∈ Λ′) is a Cauchy net, h ∈ U and [g_{•}] + [k_{•}] = i_{G}(h); lastly, we have k_{λ} + V′ ⊂ V′ + V′ ⊂ V for every λ ∈ Λ′, whence [k_{•}] ∈ V^{∧}. ♢

Claim 8.2.12. (G^{∧}, 𝒯 ^{∧}) is complete and separated.

Proof of the claim. Let (g_{•,i} | i ∈ I) a family of Cauchy nets of G indexed by a partially ordered set I, such that ([g_{•,i}] | i ∈ I) is a Cauchy net in G^{∧}. Hence, for every i ∈ I we have a partially ordered set Λ_{i} such that g_{•,i} = (g_{λ} | λ ∈ Λ_{i}). For every U ∈ 𝒰 we may find i(U) ∈ I such that [g_{•,i(U)}] − [g_{•,j}] ∈ U^{∧} for every j ≥ i(U). The latter means that there exist a Cauchy net (h_{k} | k ∈ K) of G and U′ ∈ 𝒰 such that h_{k} + U′ ⊂ U for every k ∈ K, as well as λ(U) ∈ Λ_{i(U)}, μ(U, j) ∈ Λ_{j} and k_{0} ∈ K such that g_{μ,j} − g_{λ,i(U)} − h_{k} ∈ U′ for every k ≥ k_{0}, every μ ≥ μ(U, j) and every λ ≥ λ(U). Therefore g_{μ,j} ∈ g_{λ,i(U)} + h_{k} + U′ ⊂ g_{λ,i(U)} + U for every such j, μ and λ. Lastly, we may also assume that g_{λ,i(U)} − g_{λ(U),i(U)} ∈ U for every λ ≥ λ(U), whence g_{μ,j} ∈ g_{λ(U),i(U)} + U + U for every j ≥ i(U) and every μ ≥ μ(U, j). Hence, the system l_{•} := (l_{U} := g_{λ(U),i(U)} | U ∈ 𝒰 ) is a Cauchy net in G (for the partial ordering on 𝒰 such that U ≤ U′ if and only if U′ ⊂ U, for every U, U′ ∈ 𝒰 ), and it is easily seen that [l_{•}] is a limit point of ([g_{•,i}] | i ∈ I), which shows that G^{∧} is complete.

Lastly, suppose that g_{•} := (g_{λ} | λ ∈ Λ) is a Cauchy net such that [g_{•}] lies in U^{∧} for every U ∈ 𝒰 . This means that for every such U there exists a Cauchy net h_{•}^{U} := (h_{λ′}^{U} | λ′ ∈ Λ_{U}) such that h_{λ′}^{U} ∈ U for every λ′ ∈ Λ_{U} and such that g_{•} ∼ h_{•}^{U}. The latter condition implies that there exist λ(U) ∈ Λ and λ′(U) ∈ Λ_{U} such that g_{λ} − h_{λ′}^{U} ∈ U for every λ ≥ λ(U) and every λ′ ≥ λ′(U). We conclude that g_{λ} ∈ U + U for every λ ≥ λ(U), so g_{λ} is equivalent to [0], which shows that G^{∧} is separated. ♢

Next, consider a continuous group homomorphism φ : G → H to a complete and separated topological abelian group H. In view of remark 8.2.7(i,iii), it follows easily that φ factors through i_{G} and the group homomorphism φ^{∧} : G^{∧} → H that assigns to every equivalence class [g_{•}] the unique limit point of the Cauchy net φ(g_{•}) in H. In view of remark 8.2.2(vi), we need to check that φ^{∧} is continuous at the point 0. Thus, let V ⊂ H be any open neighborhood of the neutral point 0_{H} of H, and let g_{•} := (g_{λ} | λ ∈ Λ) be a Cauchy net in G with h := φ^{∧}([g_{•}]) ∈ V; we may find an open neighborhood V′ of 0_{H} in H such that V′ = −V′ and h+V′+V′+V′ ⊂ V, and set U := φ^{−1}V′, so that U = −U. Now, let g′_{•} := (g′_{λ′} | λ′ ∈ Λ′) be any Cauchy net in G such that [g′_{•}] ∈ [g_{•}] + U^{∧}. The condition means that there exist a Cauchy net (h_{i} | i ∈ I) in G and an open neighborhood U′ of 0 in G such that h_{i} + U′ ⊂ U for every i ∈ I, and moreover there exist λ_{0} ∈ Λ, λ′_{0} ∈ Λ′, and i_{0} ∈ I such that g′_{λ′} − g_{λ} − h_{i} ∈ U′ for every λ ≥ λ_{0}, every λ′ ≥ λ′_{0} and every i ≥ i_{0}. Thus, g′_{λ′} ∈ g_{λ} + U, and consequently φ(g′_{λ′}) ∈ φ(g_{λ}) + V′ for every λ ≥ λ_{0} and λ′ ≥ λ′_{0}. Set h′ := φ^{∧}([g′_{•}]); we may also assume that φ(g_{λ}) − h, h′ − φ(g′_{λ′}) ∈ V′ for every λ ≥ λ_{0} and every λ′ ≥ λ′_{0}. We deduce that

h′ ∈ φ(g′_{λ′}) + V′ ⊂ φ(g_{λ}) + V′ + V′ ⊂ h + V′ + V′ + V′ ⊂ V

i.e. φ^{∧}([g_{•}] + U^{∧}) ⊂ V, whence the assertion. Lastly, since the image of i_{G} is dense in G^{∧}, the uniqueness of φ^{∧} is clear.

(iii): The existence of the continuous group homomorphism φ^{∧} : G^{∧} → G′ has just been established. Next, since φ has dense image, every g′ ∈ G′ is the unique limit point of a Cauchy net of G′ of the form (φ(g_{λ}) | λ ∈ Λ) for a family g_{•} := (g_{λ} | λ ∈ Λ) of elements of G, and since 𝒯 agrees with the topology induced by 𝒯 ′, it is easily seen that g_{•} is a Cauchy net in G. It then follows that φ^{∧}([g_{•}]) = g′, which shows that φ^{∧} is surjective. Let g_{•} := (g_{λ} | λ ∈ Λ) be a Cauchy net in G whose class [g_{•}] lies in the kernel of φ^{∧}; this means that (φ(g_{λ}) | λ ∈ Λ) converges to 0 in G′, and since 𝒯 is induced by 𝒯 ′, it follows easily that g_{•} converges to 0 in G, which shows that φ^{∧} is also injective. It remains therefore only to check that the topology on G^{∧} induced by 𝒯 ′ via φ^{∧} is finer than 𝒯 ^{∧}. To this aim, consider any U ∈ 𝒰 ; by assumption, there exists an open neighborhood W of 0 in G′ such that φ^{−1}W ⊂ U. Set V := φ^{∧ −1}W; then i_{G}^{−1}V ⊂ U, and therefore V ⊂ U^{∧}, by virtue of claim 8.2.11(i). Since 𝒰^{∧} is a fundamental system of open neighborhoods of 0 in G^{∧}, the assertion follows. □

Proposition 8.2.13. Let (G, 𝒯_{G}) be a topological abelian group, H ⊂ G a subgroup, and endow H (resp. G/H) with the topology 𝒯_{H} (resp. 𝒯_{G/H}) induced by 𝒯_{G} via the inclusion map j : H → G (resp. via the projection π : G → G/H). Let also i_{G} : G → G^{∧} be the completion map. Then we have :

(i) The resulting sequence of separated completions is exact :

0 → H^{∧} ⟶^{j^{∧}} G^{∧} ⟶^{π^{∧}} (G/H)^{∧}.

(ii) The topology 𝒯_{H}^{∧} of H^{∧} agrees with the one induced by the topology 𝒯_{G}^{∧} of G^{∧} via j^{∧}. (iii) The image of j^{∧} is the topological closure in G^{∧} of i_{G}(H). (iv) Endow L := G^{∧}/H^{∧} with the quotient topology 𝒯_{L} induced by G^{∧} via the projection

G^{∧} → L. Also, let k : L → (G/H)^{∧} be the injective map deduced from π^{∧}. Then 𝒯_{L} agrees with the subspace topology induced by (G/H)^{∧} via k.

(v) Suppose that G admits a countable fundamental system of open neighborhoods of 0. Then π^{∧} is surjective.

Proof. (i): Let g_{•} := (g_{λ} | λ ∈ Λ) be a Cauchy net of H; a direct inspection of the definitions shows that 0 is a limit point of g_{•} (relative to the topology 𝒯_{H}) if and only if it is a limit point of j(g_{•}), relative to the topology 𝒯_{G}, whence the injectivity of j^{∧}. Next, by functoriality we have π^{∧} ∘ j^{∧} = (π ∘ j)^{∧} = 0 : H^{∧} → (G/H)^{∧}. To conclude, suppose then that g_{•} := (g_{λ} | λ ∈ Λ) is a Cauchy net of G such that π(g_{•}) := (π(g_{λ}) | λ ∈ Λ) admits 0 as limit point, and pick a fundamental system 𝒰 of open neighborhoods of 0 in G; the assertion means that for every U ∈ 𝒰 there exists λ(U) ∈ Λ such that π(g_{λ}) ∈ π(U) for every λ ≥ λ(U), i.e. g_{λ} ∈ H + U := {h + u | h ∈ H, u ∈ U} for every such λ. Thus, for every U ∈ 𝒰 pick h_{U} ∈ H such that g_{λ(U)} − h_{U} ∈ U; it follows that the families (g_{λ(U)} | U ∈ 𝒰 ) and h_{•} := (h_{U} | U ∈ 𝒰 ) are also Cauchy nets in G, and

g_{•} ∼ (g_{λ(U)} | U ∈ 𝒰 ) ∼ h_{•}

which shows that the class of g_{•} in G^{∧} lies in the image of j^{∧}, as stated.

(ii): Let h_{•} := (h_{λ} | λ ∈ Λ) be a Cauchy net of H whose class [h_{•}] ∈ H^{∧} lies in j^{∧ −1}(U^{∧}), for a given U ∈ 𝒰 (notation of the proof of theorem 8.2.8); this means that there exist a Cauchy net g_{•} := (g_{λ′} | λ′ ∈ Λ′) of G, and U′ ∈ 𝒰 , such that g_{λ′} + U′ ⊂ U for every λ′ ∈ Λ′, and such that h_{•} ∼ g_{•}. Pick any U′′ ∈ 𝒰 such that U′′ + U′′ ⊂ U′; after replacing Λ and Λ′ by some cofinal subsets, we may assume that h_{λ} −g_{λ′} ∈ U′′ for every λ ∈ Λ and every λ′ ∈ Λ′. Whence, h_{λ} + U′′ ∈ g_{λ′} + U′′ + U′′ ⊂ U for every λ ∈ Λ, which shows that H^{∧} ∩ U^{∧} ⊂ (H ∩ U)^{∧}. The converse inclusion is clear, and the assertion follows.

(iii): On the one hand, the image of j^{∧} is a complete subgroup of G^{∧}, so it is topologically closed (remark 8.2.7(iv)); on the other hand, i_{G}(H) is dense in in H^{∧}, whence the contention.

(iv): Denote by (\overline{G}, \overline{𝒯} _{G}) the maximal separated quotient of G (i.e. the quotient G/\{0\}^{c}, where {0}^{c} denotes the topological closure of {0} in G), endowed with the quotient topology induced by G; let also \overline{H} be the topological closure of the image of H in \overline{G}, and endow \overline{H} with the topology \overline{𝒯} _{H} induced by \overline{𝒯} _{G} via the inclusion map \overline{H} → \overline{G}. Endow as well \overline{G}/\overline{H} with the quotient topology \overline{𝒯} _{G/H} induced by \overline{G}; we get a sequence of separated abelian topological groups

0 → \overline{H} → \overline{G} → \overline{G}/\overline{H} → 0

and after taking separated completions, a commutative ladder of continuous group homomorphisms :

\xymatrix{0 \ar[r] & H^{∧} \ar[r] \ar[d] & G^{∧} \ar[r] \ar[d] & (G/H)^{∧} \ar[d] \\ 0 \ar[r] & _{\overline{H}_{∧}} \ar[r] & _{\overline{G}_{∧}} \ar[r] & (\overline{G}/\overline{H})^{∧}.}

On the other hand, it is easily seen that 𝒯_{H} (resp. 𝒯_{G/H}) agrees with the topology induced by \overline{𝒯} _{H} (resp. by \overline{𝒯} _{G/H}) via the natural map H → \overline{H} (resp. G/H → \overline{G}/\overline{H}) : details left to the reader. It follows immediately that the central and right-most vertical arrows are isomorphisms of topological groups. By the 5-lemma, we deduce that the left-most vertical arrow is a group isomorphism; but then it is also a homeomorphism, by virtue of (ii). Summing up, we may replace G by \overline{G} and H by \overline{H}, and assume from start that G is separated and H is topologically closed in G; especially, the completion map G → G^{∧} is also injective. We now consider the commutative diagram of continuous group homomorphisms

\xymatrix{G/H \ar[r]^{f} \ar[dr]_{i_{G/H}} & L \ar[r]^{i_{L}} \ar[d]^{k} & L^{∧} \ar[dl]^{g} \\  & (G/H)^{∧}}

where i_{G/H} and i_{L} are the completion maps. By (iii) and lemma 8.2.3(ii), the map f is injective, and 𝒯_{G/H} agrees with the topology induced by 𝒯_{L} via f. By theorem 8.2.8(ii), the topology 𝒯_{L} in turn is induced via i_{L} by the topology 𝒯_{L}^{∧} of its completion L^{∧}. Thus, 𝒯_{G/H} is induced by 𝒯_{L}^{∧} via i_{L} ∘ f; lastly, i_{L} ∘ f has dense image. Taking into account theorem 8.2.8(iii), we conclude that g is an isomorphism of topological groups, whence the assertion.

(v): Let \overline{g}_{•} := (\overline{g}_{λ} | λ ∈ Λ) be a Cauchy net in G/H, and pick a countable fundamental system 𝒰 := (U_{n} | n ∈ ℕ) of open neighborhoods of 0 in G. A simple induction shows that we may assume :

(8.2.14)  U_{n+1} + U_{n+1} ⊂ U_{n}  for every n ∈ ℕ.

Now, for every n ∈ ℕ there exists λ(n) ∈ Λ such that \overline{g}_{λ} −\overline{g}_{μ} ∈ π(U_{n}) for every λ, μ ≥ λ(n). It follows that the family (\overline{g}_{λ(n)} | n ∈ ℕ) is a Cauchy net in G/H equivalent to g_{•} (for the standard ordering on ℕ); thus, we may assume from start that the class [\overline{g}_{•}] in (G/H)^{∧} is represented by a Cauchy sequence (\overline{g}_{n} | n ∈ ℕ) of G/H, such that \overline{g}_{n+1} − \overline{g}_{n} ∈ π(U_{n}) for every n ∈ ℕ. Pick an arbitrary sequence (g_{n} | n ∈ ℕ) in G with π(g_{n}) = \overline{g}_{n} for every n ∈ ℕ; by construction, for every n ∈ ℕ there exists h_{n} ∈ H such that g_{n+1} + h_{n} − g_{n} ∈ U_{n}. Set g′_{n} := g_{n} + ∑_{i=0}^{n−1} h_{i} for every n ∈ ℕ; we then see that g′_{n+1} − g′_{n} ∈ U_{n} for every n ∈ ℕ. Lastly, using (8.2.14), a simple induction shows more generally that g′_{i} − g′_{j} ∈ U_{n−1} whenever i ≥ j > 0. Therefore g′_{•} := (g′_{n} | n ∈ ℕ) is a Cauchy sequence in G, and clearly π^{∧}([g′_{•}]) = [g_{•}], whence the contention. □

Corollary 8.2.15. Suppose that the topological group G is the product of a (small) family (G_{i} | i ∈ I) of topological abelian groups, and for every i ∈ I denote by G_{i}^{∧} the separated completion of G_{i}. Then the natural map j : G → ∏_{i∈I} G_{i}^{∧} factors uniquely through the completion map G → G^{∧} and an isomorphism of topological groups :

G^{∧} →^{∼} ∏_{i∈I} G_{i}^{∧}.

Proof. It is easily seen that the product P := ∏_{i∈I} G_{i}^{∧} is complete and separated; also, j has dense image, and the topology of G is induced by that of P via j. Then the assertion follows from theorem 8.2.8(iii). □

Corollary 8.2.16. Let G_{•} := (G_{i} | i ∈ Ob(I)) be any system of topological abelian groups, indexed by any small category I. We have :

(i) If G_{i} is complete and separated for every i ∈ Ob(I), then lim_{I} G_{•} is a complete and separated topological abelian group.

(ii) Suppose that I is cofiltered and countable (see example 1.5.26), and that the system G_{•} satisfies the following Mittag-Leffler condition :

(ML) For every i ∈ Ob(I) there exists a morphism φ : j → i in I such that for every morphism ψ : k → j in I we have

Im(G(φ ∘ ψ) : G_{k} → G_{i}) = Im(G(φ) : G_{j} → Gi).

Then the natural map

(lim_{I} G_{•})^{∧} → lim_{I} G_{•}^{∧}

is an isomorphism of topological groups.

Proof. (i): Indeed, the limit L of the system G_{•} is a closed subgroup of the product P := ∏_{i∈Ob(I)} G_{i}, and the topology of L is induced by that of P via this closed inclusion map; then the assertion follows from corollary 8.2.15and remark 8.2.7(iv).

(ii): In light of theorem 8.2.8(ii), it is easily seen that the topology of L is induced by that of L′ := lim_{I} G_{•}^{∧}, via the natural map β : L → L′. Taking into account (i) and theorem 8.2.8(iii), we are then reduced to checking that β has dense image. Moreover, by virtue of example 1.5.26, we may assume that I is the totally ordered set ℕ^{o}. Now, for every i, j ∈ ℕ with j ≥ i let φ_{j,i} : G_{j} → G_{i} be the transition map; for every i ∈ ℕ we set

H_{i} := ⋂_{j≥i} Im φ_{j,i}

and we endow H_{i} with the topology induced by G_{i}. Condition (ML) easily implies that φ_{j,i} restricts to a surjective continuous group homomorphism H_{j} → H_{i} for every i, j ∈ ℕ with j ≥ i, and the natural morphisms

lim_{ℕ^{o}} H_{•} → L  lim_{ℕ^{o}} H_{•}^{∧} → L′

are isomorphisms of topological groups. We may then replace G_{•} by H_{•}, and assume from start that the map φ_{j,i} is a surjection, for every j ≥ i. In this case we come down to checking that the inclusion map H_{i} → H_{i}^{∧} is dense for every i ∈ Ob(I), which we know by theorem 8.2.8(ii). □

Corollary 8.2.17. Let G be any topological abelian group, H ⊂ G an open subgroup, H^{∧} and G^{∧} the separated completions of H and G, and i_{G} : G → G^{∧} the completion map. Then :

(i) H^{∧} is an open subgroup of G^{∧}, and the natural map G/H → G^{∧}/H^{∧} is a group isomorphism. (ii) H = i_{G}^{−1}H^{∧}. (iii) Suppose that G admits a fundamental system (H_{λ} | λ ∈ Λ) of open neighborhoods of

0, consisting of open subgroups. Then (H_{λ}^{∧} | λ ∈ Λ) is a fundamental system of open neighboroods of 0 for G^{∧}.

Proof. Under these assumptions, G induces the discrete topology on the quotient G/H, hence the latter is complete and separated; taking into account proposition 8.2.13(i,ii), we get an exact sequence

0 → H^{∧} → G^{∧} → G/H

from which both (i) and (ii) follow easily. Assertion (iii) follows by inspecting the proof of theorem 8.2.8(i). □

Proposition 8.2.18. Let Λ ≠ ∅ be a finite set, ((H_{λ}, +, 0_{λ}) | λ ∈ Λ) a family of topological abelian groups, φ : ∏_{λ∈Λ} H_{λ} → G a continuous ℤ-multilinear map into a complete and separated topological abelian group (G, +, 0_{G}), and for every λ ∈ Λ denote by i_{λ} : H_{λ} → H_{λ}^{∧} the completion map. Then there exists a unique continuous ℤ-multilinear map

φ^{∧} : ∏_{λ∈Λ} H_{λ}^{∧} → G  such that  φ^{∧} ∘ (∏_{λ∈Λ} i_{λ}) = φ.

Proof. For every subset Δ ⊊ Λ and every h_{•} := (h_{λ} | λ ∈ Δ) ∈ ∏_{λ∈Δ} H_{λ}, let

j_{Δ,h_{•}} : ∏_{λ∈Λ\Δ} H_{λ} → ∏_{λ∈Λ} H_{λ}

be the map such that j_{Δ,h_{•}}(k_{•}) := (l_{λ} | λ ∈ Λ) for every k_{•} := (k_{λ} | λ ∈ Λ \ Δ) ∈ ∏_{λ∈Λ\Δ} H_{λ}, where l_{λ} := h_{λ} for every λ ∈ Δ, and l_{λ} := k_{λ} for every λ ∈ Λ \ Δ. Let us first show :

Claim 8.2.19. Let ((H_{λ}, +, 0_{λ}) | λ ∈ Λ) and G be as in the proposition, and ψ : ∏_{λ∈Λ} H_{λ} → G a ℤ-multilinear map. Then the following conditions are equivalent :

(a) ψ is continuous.

(b) ψ ∘ j_{Δ,h_{•}} is continuous at (0_{λ} | λ ∈ Λ \ Δ) for every Δ ⊊ Λ and every h_{•} ∈ ∏_{λ∈Δ} H_{λ}.

Proof of the claim. Clearly every such map j_{Δ,h_{•}} is continuous, hence (a)⇒(b). Conversely, let h_{•} ∈ ∏_{λ∈Λ} H_{λ} be any element, and for every subset Δ ⊂ Λ set as well j_{Δ} := j_{Δ,k_{•}}, where k_{•} := (h_{λ} | λ ∈ Δ); we have

ψ(h_{•} + u_{•}) = ψ(h_{•}) + ∑_{Δ⊊Λ} ψ ∘ j_{Δ}(u_{λ} | λ ∈ Λ \ Δ) for every u_{•} := (u_{λ} | λ ∈ Λ) ∈ ∏_{λ∈Λ} H_{λ}

which easily implies that (b)⇒(a) : the details are left to the reader. ♢

Now, the uniqueness of φ^{∧} is clear, since the map ∏_{λ_{i}nΛ} i_{λ} has dense image. For the existence of φ^{∧}, we are easily reduced to checking that for every μ ∈ Λ the map φ extends to a continuous ℤ-multilinear map φ_{μ} : H_{μ}^{∧} × ∏_{λ∈Λ\{μ}} H_{λ} → G, so that φ_{μ} ∘ (i_{μ} × ∏_{λ∈Λ\{μ}} 1_{H_{λ}}) = φ.

Thus, let h_{•} ∈ ∏_{λ∈Λ\{μ}} H_{λ} be any element; the composition φ_{h_{•}} := φ ∘ j_{Λ\{μ},h_{•}} : H_{μ} → G is a continuous group homomorphism, which by adjunction factors through i_{μ} and a unique continuous group homomorphism φ_{h_{•}}^{∧} : H_{μ}^{∧} → G, and we set

φ_{μ}(x, h_{•}) := φ_{h_{•}}^{∧}(x) for every x ∈ H_{μ}^{∧} and every h_{•} ∈ ∏_{λ∈Λ\{μ}} H_{λ}.

To check that φ_{μ} is ℤ-multilinear, say that h_{•} ∈ ∏_{λ∈Λ\{μ}} H_{λ}, ν ∈ Λ \ {μ}, and k ∈ H_{ν}; set h′_{•} := (h′_{λ} | λ ∈ Λ \ {μ}) and h′′_{•} := (h′′_{λ} | λ ∈ Λ \ {μ}) with h′′_{λ} = h′_{λ} := h_{λ} if λ ≠ ν, and h′_{ν} := k, h′′_{ν} := h_{ν} + h′_{ν}. Since φ is ℤ-multilinear, φ_{h′′_{•}} = φ_{h_{•}} + φ_{h′_{•}}, whence φ_{h′′_{•}}^{∧} = φ_{h_{•}}^{∧} + φ_{h′_{•}}^{∧}, so that φ_{μ}(x, h′′_{•}) = φ_{μ}(x, h_{•}) + φ_{μ}(x, h′_{•}) for every x ∈ H_{μ}^{∧}, whence the assertion.

Next, for every x ∈ H_{μ}^{∧}, let also φ_{μ,x} : ∏_{λ∈Λ\{μ}} H_{λ} → G be the map such that φ_{μ,x}(h_{•}) := φ_{h_{•}}(x) for every h_{•} ∈ ∏_{λ∈Λ\{μ}} H_{λ}. To show the continuity of φ_{μ}, by claim 8.2.19and a simple induction on the cardinality c of Λ, it suffices to check that :

(a) φ_{μ} is continuous at (0_{λ} | λ ∈ Λ) ∈ H_{μ}^{∧} × ∏_{λ∈Λ\{μ}} H_{λ}

(b) if c ≥ 2, then for every x ∈ H_{μ}^{∧} the map φ_{μ,x} is continuous at (0_{λ} | λ ∈ Λ \ {μ}). To this aim, for every open neighborhood U of 0_{μ} in H_{μ}, define the open neighborhood U^{∧} of 0_{μ} in H_{μ}^{∧}, as in the proof of theorem 8.2.8, so that U^{∧} lies in the topological closure of i_{μ}(U) in H_{μ}^{∧} (claim 8.2.11(ii)). Let V ⊂ G be any open neighborhood of 0_{G}, and W ⊂ G another open neighborhood of 0_{G} with W + W + W ⊂ V, and denote by W^{c} (resp. (W + W)^{c}) the topological closure of W (resp. of W + W) in G; since φ is continuous, there exist a family U_{•} := (U_{λ} | λ ∈ Λ) such that U_{λ} is an open neighborhood of 0_{λ} in H_{λ} for every λ ∈ Λ, and φ(∏_{λ∈Λ} U_{λ}) ⊂ W. It follows that φ_{μ}(U_{μ}^{∧} × ∏_{λ∈Λ\{μ}} U_{λ}) ⊂ W^{c} ⊂ W + W ⊂ V, which shows (a). Lastly, say that x is the limit of a Cauchy net (x_{i} | i ∈ I) in H_{μ}, and keep U_{•} as in the foregoing; hence, there exists i ∈ I such that x_{j}−x_{k} ∈ U_{μ} for every j, k ∈ I with j, k ≥ i. Since φ is continuous, there exist moreover for every λ ∈ Λ \ {μ} an open neighborhood U′_{λ} ⊂ U_{λ} of 0_{λ} such that φ({x_{i}} × ∏_{λ∈Λ\{μ}} U′_{λ}) ⊂ W. Thus, φ({x_{j}} × ∏_{λ∈Λ\{μ}} U′_{λ}) ⊂ W + W for every j ≥ i, so φ_{μ}({x} × ∏_{λ∈Λ\{μ}} U′_{λ}) ⊂ (W + W)^{c} ⊂ W + W + W ⊂ V, whence (b). □

8.3. Topological rings and topological modules. If T is any topological space and S ⊂ T any subset, we shall denote by S^{c} the topological closure of S in T.

Definition 8.3.1. (i) A topological ring is a pair (A, 𝒯_{A}) where A is a (commutative, unital) ring and 𝒯_{A} a topology on the set underlying A, such that :

• the multiplication law of A is a continuous map (A, 𝒯_{A}) × (A, 𝒯_{A}) → (A, 𝒯_{A})

• the pair ((A, +, 0), 𝒯_{A}) is a topological group. (ii) Let (A, 𝒯_{A}) be a topological ring, M an A-module, 𝒯_{M} a topology on the set underlying

M. We say that (M, 𝒯_{M}) is a topological (A, 𝒯_{A})-module (briefly : a topological A-module) if • the scalar multiplication is a continuous map (A, 𝒯_{A}) × (M, 𝒯_{M}) → (M, 𝒯_{M})

• the pair ((M, +, 0), 𝒯_{M}) is a topological abelian group. (iii) Let (A, 𝒯_{A}), (B, 𝒯_{B}) be two topological rings. A morphism of topological rings is a

ring homomorphism A → B that is continuous for the topologies 𝒯_{A} and 𝒯_{B}. Such a morphism (A, 𝒯_{A}) → (B, 𝒯_{B}) is also called a topological (A, 𝒯_{A})-algebra. The topological A-algebras form a category

A-TopAlg

whose morphisms are the continuous A-algebra homomorphisms. Especially, the ring ℤ with its discrete topology is a topological ring, and ℤ-TopAlg is the category of topological rings and continuous ring homomorphisms. Likewise, the topological (A, 𝒯_{A})-modules form a category

A-TopMod

whose morphisms are the continuous A-linear maps.

(iv) Let (A, 𝒯_{A}) be a topological ring, B → A a ring homomorphism, (M, 𝒯_{M}) a topological A-module. We say that the topology 𝒯_{M} is B-linear if there exists a fundamental system of open neighborhoods of 0 ∈ M for the topology 𝒯_{M}, consisting of B-submodules of M.

(v) Especially, in the situation of (iv) we say that the topology 𝒯_{A} is B-linear, if the same holds for the topological A-module (A, 𝒯_{A}). We say also simply that 𝒯_{A} is linear if it is Alinear. We denote by

A-TopAlg_{B-lin}

the full subcategory of A-TopAlg whose objects are the B-linear topological A-algebras.

Remark 8.3.2. (i) Let A be a ring, and 𝒯_{A} a topology on A such that (A, 𝒯_{A}) is a topological group, for the addition law of A. Let 𝒰 be a fundamental system of open neighborhoods of 0 in A; it is easily seen that (A, 𝒯_{A}) is a topological ring if and only if the system 𝒰 fulfills the following conditions :

(a) For every U ∈ 𝒰 there exists U′ ∈ 𝒰 such that U′ · U′ := {a · b | a, b ∈ U′} ⊂ U.

(b) For every a ∈ A and every U ∈ 𝒰 there exists U′ ∈ 𝒰 such that a · U′ ⊂ U. (ii) Especially, if I_{•} := (I_{λ} | λ ∈ Λ) is any system of ideals of A that is cofiltered by

inclusion, then there is a unique group topology 𝒯_{I_{•}} on A for which I_{•} is a fundamental system of open neighborhoods of 0, and (A, 𝒯_{I_{•}}) is a topological ring. We call 𝒯_{I_{•}} the I_{•}-adic topology on A. Likewise, if M is any A-module, the I_{•}-adic topology on M is the unique group topology 𝒯_{M} for which (I_{λ}M | λ ∈ Λ) is a fundamental system of open neighborhoods of 0 in M, and (M, 𝒯_{M}) is a topological (A, 𝒯_{I_{•}})-module.

Remark 8.3.3. Let (A, 𝒯_{A}) be any topological ring and (M, 𝒯_{M}) any topological A-module.

(i) Let (A^{∧}, 𝒯_{A}^{∧}) be the separated completion of the topological group underlying A; from proposition 8.2.18we see that the multiplication law of A induces a continuous ℤ-bilinear map A^{∧} × A^{∧} → A^{∧}, which furnishes A^{∧} with a natural structure of topological ring, such that the completion map A → A^{∧} is a morphism of topological rings. Likewise, the separated completion (M^{∧}, 𝒯_{M}^{∧}) of (M, 𝒯_{M}) is naturally a topological A^{∧}-module.

(ii) Suppose that 𝒯_{A} is the linear topology defined by a cofiltered system I_{•} := (I_{λ} | λ ∈ Λ) of ideals of A, and let I ⊂ A be any open ideal. Also, suppose that 𝒯_{M} is the I_{•}-adic topology on M. Then it follows easily from (i) and corollary 8.2.17(iii), that 𝒯_{M}^{∧} is the A^{∧}-linear topology defined by the cofiltered system ((I_{λ}M)^{∧} | λ ∈ Λ) of submodules of M^{∧}. Moreover, (IM)^{∧} is an open A^{∧}-submodule in M^{∧}, the natural map M/IM → M^{∧}/(IM)^{∧} is an isomorphism, and the completion map M → M^{∧} identifies (IM)^{∧} with the topological closure of the image of IM in M^{∧} (corollary 8.2.17(i)).

(iii) In the situation of (ii), suppose furthermore that for every λ ∈ Λ there exists μ ∈ Λ with I_{μ} ⊂ I_{λ}^{2} (where I_{λ}^{2} is the usual square of the ideal I_{λ}). Then it is easily seen that the I_{•}-adic topology on IM agrees with the topology induced by 𝒯_{M} on IM.

(iv) In the situation of (iii), suppose additionally that Λ is countable and I is finitely generated. Then the image of (IM)^{∧} in M^{∧} equals IM^{∧} : indeed, by assumption we may find k ∈ ℕ and a surjective A-linear map φ : M^{⊕k} → IM, and it is easily seen that φ is an open map; especially, the quotient topology induced on IM via the map φ agrees with 𝒯_{IM}. By proposition 8.2.13(v), the separated completion of φ is a surjective map

φ^{∧} : (M^{⊕k})^{∧} → (IM)^{∧}

and on the other hand we have a natural identification (M^{⊕k})^{∧} →^{∼} M^{∧⊕k}, whence the contention (details left to the reader).

Example 8.3.4. (i) Let A ≠ 0 be a ring, Σ a set, and ℱ a filter of Σ, i.e. a family of subsets of Σ fulfilling the following conditions : (a) for every F, F′ ∈ ℱ we have F ∩ F′ ∈ ℱ; (b) for every F ⊂ F′ ⊂ Σ with F ∈ ℱ, we have F′ ∈ ℱ. Let us endow A with the discrete topology, and M := A^{(Σ)} with the linear topology 𝒯_{M} defined by the system of submodules (A^{(F)} | F ∈ ℱ). Notice that 𝒯_{M} is separated if and only if ⋂_{F ∈ℱ} F = ∅. Notice as well the natural isomorphism of A-modules

M/A^{(F)} →^{∼} A^{(Σ\F)} for every subset F ⊂ Σ.

For F ⊂ F′ ⊂ Σ, these isomorphisms identify the natural projection M/A^{(F)} → M/A^{(F′)} with the A-linear map π_{F,F′} : A^{(Σ\F)} → A^{(Σ\F′)} : (a_{σ} | σ ∈ Σ \ F) ↦ (a_{σ} | σ ∈ Σ \ F′). The completion of M is then isomorphic to the inverse limit of the cofiltered system ((A^{Σ\F} | F ∈ ℱ), π_{••}). If 𝒯_{M} is separated, a direct inspection shows that this limit is represented by the submodule L of A^{Σ} consisting of all sequences (a_{σ} | σ ∈ Σ) with support Λ := {σ ∈ Σ | a_{σ} ≠ 0} satisfying the following condition : Λ ∩ (Σ \ F) is a finite set for every F ∈ ℱ.

(ii) In the situation of (i), suppose moreover that for every finite or countable subset Λ ⊂ Σ there exists F ∈ ℱ with Λ ∩ F = ∅. Then M is complete and separated. Indeed, the condition implies that ⋂_{F ∈ℱ} F = ∅, so M is separated. To check the completeness of M, say that a_{•} := (a_{σ} | σ ∈ Σ) ∈ L; if the support Λ of a_{•} is not finite, we may find a countable subset Λ′ ⊂ Λ, and by assumption there exists F ∈ ℱ such that Λ′∩F = ∅. But then Λ′ ⊂ Λ∩(Σ\Λ′), so Λ ∩ (Σ \ Λ′) is not a finite set, a contradiction. Thus a_{•} ∈ M, whence the assertion.

Example 8.3.5. Let A be any topological ring whose topology is linear, complete and separated, I ⊂ A a closed ideal, and endow A/I with the quotient topology. In case A admits a fundamental system of open neighborhoods of 0 consisting of a countable family of ideals, then A/I is still complete and separated (proposition 8.2.13(v)). But if this latter condition is omitted, A/I will still be separated, but it will not necessarily be complete. For a counterexample, take A := K[T_{λ} | λ ∈ Λ], where K is a non-zero ring, and Λ is the (small) set of all countable ordinal numbers, and let I ⊂ A be the ideal generated by (T_{λ} − T_{μ} | λ, μ ∈ Λ). We endow A with the linear topology defined by the cofiltered system of ideals (J_{λ}^{n} | (λ, n) ∈ Λ × ℕ), where J_{λ} ⊂ A is the ideal generated by (T_{μ} | μ ∈ Λ, μ > λ), for every λ ∈ Λ. Let Σ ⊂ A be the subset of monomials, which is in natural bijection with the set of all maps φ : Λ → ℕ such that φ^{−1}(ℕ \ {0}) is a finite set. For every (λ, n) ∈ Λ × ℕ, set Σ_{λ,n} := J_{λ}^{n} ∩ Σ, and let ℱ be the set of all subsets F ⊂ Σ such that Σ_{λ,n} ⊂ F for some (λ, n) ∈ Λ × ℕ. Then we have a natural identification A →^{∼} K^{(Σ)}, and the topology of A is defined by the cofiltered system of K-submodules (K^{(F)} | F ∈ ℱ). It is easily seen that ℱ fulfills the condition of example 8.3.4, hence A is complete and separated. Now, the quotient A/I is isomorphic to K[T], endowed with its T -adic topology, which is separated but not complete.

Lemma 8.3.6. Let A be a topological ring, B ⊂ A an open subring, C the integral closure of B in A; endow B and C with the topologies induced by A, and denote by A^{∧}, B^{∧} and C^{∧} the respective separated completions. Then C^{∧} is the integral closure of B^{∧} in A^{∧}.

Proof. Notice that C^{∧} = B^{∧} · C, hence C^{∧} lies in the integral closure of B^{∧} in A^{∧}, and we are therefore reduced to checking that B is integrally closed in A if and only if B^{∧} is integrally closed in A^{∧}. Let j : A → A^{∧} be the completion map. Suppose first that B is integrally closed in A, and let a ∈ A^{∧} be any element that is integral over B^{∧}; pick any monic polynomial P (T) ∈ B^{∧}[T] with P (a) = 0. Since B is open in A, we may then find a′ ∈ A and a monic polynomial Q(T) ∈ B[T] such that

j(a′) − a, P (a′), j(Q(a′)) − P (j(a′)) ∈ B^{∧}.

It follows that Q(a′) ∈ j^{−1}B^{∧} = B (corollary 8.2.17(ii)). Set R(T) := Q(T) − Q(a′); then R(T) ∈ B[T] and R(a′) = 0, so a′ ∈ B, by assumption, and finally a ∈ B^{∧}, as required. Conversely, if a ∈ A is integral over B, it follows easily that j(a) is integral over B^{∧}, and hence j(a) ∈ B^{∧}, if B^{∧} is integrally closed in A^{∧}; in this case, a ∈ j^{−1}B^{∧} = B, which shows that B is integrally closed in A. □

8.3.7. For any topological ring A, let 𝒞_{A} denote the full subcategory of A-TopAlg whose objects are the complete and separated topological A-algebras whose topology is linear. Then the finite coproducts of 𝒞_{A} are representable. Indeed, consider more generally any two topological A-modules M, N whose topologies are A-linear; we denote by 𝒯_{M,N}^{⊗} the A-linear topology on M ⊗_{A} N defined by the system of submodules

Im(M′ ⊗_{A} N + M ⊗_{A} N′ → M ⊗_{A} N)

where M′ (resp. N′) ranges over the system of all open A-submodules of M (resp. of N), and following [59, Ch.0,§7.7.2] we denote by

M ˆ⊗_{A}N

the separated completion of (M ⊗_{A} N, 𝒯_{M,N}^{⊗}). Notice that if (M_{λ} | λ ∈ Λ) and (N_{λ′} | λ′ ∈ Λ′) are any two fundamental systems of open submodules of M and respectively N, we have a natural isomorphism of topological A-modules

M ˆ⊗_{A}N →^{∼} lim_{(λ,λ′)∈Λ×Λ′} M/M_{λ} ⊗_{A} N/N_{λ′}

where the tensor products M/M_{λ} ⊗_{A} N/N_{λ′} are endowed with their discrete topologies.

By the same token, if M′ and N′ are two other topological A-modules whose topologies are A-linear, and f : M → M′ and g : N → N′ are any two continuous A-linear maps, then the map f ⊗_{A} g : M ⊗_{A} N → M′ ⊗_{A} N′ is continuous for the topologies 𝒯_{M,N}^{⊗} and 𝒯_{M′,N′}^{⊗}, and therefore its separated completion is a well defined continuous A-linear map

f ˆ⊗_{A}g : M ˆ⊗_{A}N → M′ ˆ⊗_{A}N′.

Now, if B and C are any two topological A-algebras whose topologies are linear, notice that the topology 𝒯_{B,C}^{⊗} is defined by a cofiltered system of ideals of B ⊗_{A} C, hence (B ⊗_{A} C, 𝒯_{B,C}^{⊗}) is a topological A-algebra. We claim that if B and C are complete and separated, then B ˆ⊗_{A}C represents the coproduct of B and C in the category 𝒞_{A}. For the proof, consider any two morphisms f : B → D and g : C → D in the category 𝒞_{A}; there exists a unique morphism of A-algebras h : B ⊗_{A} C → D such that h(b ⊗ c) := f(b) · g(c) for every b ∈ B and c ∈ C, and since D is complete and separated, it remains only to check that h is continuous for the topology 𝒯_{B,C}^{⊗}. However, let I ⊂ D be any open ideal; then J := f^{−1}I (resp. K := g^{−1}I) is an open ideal of B (resp. of C) and clearly h(J ⊗_{A} C + B ⊗_{A} K) ⊂ I, whence the contention.

Definition 8.3.8. Let (A, 𝒯 ) be any topological ring, and S ⊂ A any subset.

(i) S ⊂ A is bounded in A, if for every open neighborhood U of 0 ∈ A there exists an open neighborhood V of 0 ∈ A such that v · s ∈ U for every v ∈ V and s ∈ S.

(ii) For every n ∈ ℕ, let S(n) := {a_{1} ⋯ a_{n} | a_{1}, …, a_{n} ∈ S}. We say that S is power bounded in A if the subset ⋃_{n∈ℕ} S(n) is bounded in A. An element a ∈ A is power bounded (resp. topologically nilpotent) in A, if the subset {a} is power bounded in A (resp. if the sequence (a^{n} | n ∈ ℕ) converges to 0 in the topology 𝒯 ). We denote by

A^{∘} and A^{∘∘}

respectively the subset of all power-bounded elements and the subset of all topologically nilpotent elements of A. An ideal I ⊂ A is topologically nilpotent, if for every open neighborhood U of 0 in A there exists n ∈ ℕ such that I^{n} ⊂ U.

(iii) We say that (A, 𝒯 ) is adic (resp. c-adic) if there exists an ideal I of A such that

(I^{n} | n ∈ ℕ) (resp. ((I^{n})^{c} | n ∈ ℕ) )

is a fundamental system of open neighborhood of zero in A. Then, we call any such I an ideal of adic definition (resp. an ideal of c-adic definition) of (A, 𝒯 ), and we also say that 𝒯 is Iadic (resp. I-c-adic). If M is any A-module, the I-adic topology on M is the (unique) A-linear topology such that (I^{n}M | n ∈ ℕ) is a fundamental system of open neighborhoods of 0.

(iv) We say that (A, 𝒯 ) is f-adic if there exists an open subring A_{0} ⊂ A that is adic for the topology induced by 𝒯 , and such that A_{0} admits a finitely generated ideal of adic definition.

(v) Let (A, 𝒯 ) be any f-adic ring. An open subring A_{0} of A is called a ring of definition of A, if 𝒯 induces a linear topology on A_{0}.

(vi) We say that (A, 𝒯 ) is a Tate ring if it is f-adic and A^{×} ∩ A^{∘∘} ≠ ∅.

Remark 8.3.9. Let (A, 𝒯 ) be any topological ring.

(i) Let S_{1}, S_{2} ⊂ A be two bounded subsets. Clearly S_{1} ∪ S_{2} is bounded. Moreover, also S_{1}S_{2} := {s_{1}s_{2} | s_{1} ∈ S_{1}, s_{2} ∈ S_{2}} is bounded. Indeed, for any given open neighborhood U of 0 in A, let us pick an open neighborhood V of 0 in A such that S_{1}V ⊂ U, and an open neighborhood V′ of 0 in A such that S_{2}V′ ⊂ V; then S_{1}S_{2}V′ ⊂ U, whence the contention.

(ii) Let T_{1}, …, T_{k} ⊂ A be any finite family of subsets. Then T := T_{1} ∪ ⋯ ∪ T_{k} is power bounded in A if and only if every T_{1}, …, T_{k} is power bounded in A. Indeed, set S_{i} := ⋃_{n∈ℕ} T_{i}(n) for i = 1, …, k (notation of definition 8.3.8(ii)); by definition, T is power bounded if and only if the product S_{1} ⋯ S_{k} is bounded, so the assertion follows from (i).

(iii) Let T ⊂ A be any subset. Then T is bounded (resp. power bounded) in A if and only if the same holds for the topological closure T^{c} of T in A. Indeed, suppose that T is bounded, and let U ⊂ A be any open neighborhood of 0 ∈ A; pick an open neighborhood U′ of 0 in A such that U′ + U′ ⊂ U. By assumption, there exists an open neighborhood V of 0 in A such that T · V ⊂ U′, whence T^{c} · V ⊂ U^{′c}; but notice that U^{′c} ⊂ U′ + U′ : indeed, if x ∈ U^{′c}, we have (x − U′) ∩ U′ ≠ ∅, whence x ∈ U′ + U′. Summing up, we get T^{c} · V ⊂ U, which shows that T^{c} is bounded. Next, suppose that T is power bounded, i.e. T′ := ⋃_{n∈ℕ} T (n) is bounded, and notice that ⋃_{n∈ℕ} T^{c}(n) ⊂ T^{′c}; by the foregoing T^{′c} is bounded, so T^{c} is power bounded.

(iv) Let a ∈ A be any element, and n ≥ 1 any integer. Then a is topologically nilpotent if and only if the same holds for a^{n}. Indeed, obviously if a is topologically nilpotent, the same holds for a^{n}. Conversely, suppose that a^{n} is topologically nipotent; since the map A → A : b ↦ a^{r}b is continuous for every r ∈ ℕ, it follows that the sequences (a^{kn+r} | k ∈ ℕ) converge to 0 in A for every r = 0, …, n − 1; the claim is an immediate consequence.

(v) Let a, b ∈ A be any two elements. If a is power bounded and b is topologically nilpotent, ab is topologically nilpotent. Indeed, for any open neighborhood U of 0 in A there exists an open neighborhood V of 0 in A such that {b^{k} | k ∈ ℕ} · V ⊂ U, and on the other hand, there exists n ∈ ℕ such that a^{k} ∈ V for every integer k ≥ n; thus, (ab)^{k} ∈ U for every k ≥ n, whence the claim.

(vi) The class of f-adic rings was introduced by R.Huber in [98]. Such topological rings are now often called Huber rings.

Remark 8.3.10. Let (A, 𝒯 ) be a topological ring whose topology 𝒯 is ℤ-linear.

(i) The subset A^{∘∘} is an additive subgroup of A. Indeed, let a, b ∈ A^{∘∘} be any two elements, U any open additive subgroup in A, and pick an open neighborhood V of 0 in A such that V · V ⊂ U; we may find r ∈ ℕ such that a^{i}, b^{i} ∈ V for every i ≥ r. Then, we may also find an integer s ≥ r such that a^{i}b^{j} ∈ U for every i, j ∈ ℕ such that i + j ≥ s and min(i, j) < r. It follows easily that (a + b)^{n} ∈ U for every n ≥ s, and since U is arbitrary, we conclude that a + b is topologically nilpotent.

(ii) For any subset T ⊂ A, denote by ⟨T ⟩ ⊂ A the additive subgroup generated by T in A. Then T is bounded (resp. power bounded) if and only if the same holds for ⟨T ⟩. Indeed, suppose that T is bounded, and let U ⊂ A be any open additive subgroup; by assumption there exists an open neighborhood V of 0 in A such that T · V ⊂ U. But then clearly ⟨T ⟩ · V ⊂ U, so ⟨T ⟩ is bounded. Likewise, suppose that T is power bounded, and set S := ⋃_{n∈ℕ} T (n) (notation of definition 8.3.8(ii)); since S is bounded, we know already that the same holds for ⟨S⟩. However, clearly ⟨S⟩ = ⋃_{n∈ℕ}⟨T ⟩(n), so ⟨T ⟩ is power bounded.

(iii) Let T ⊂ A be any power bounded subset; in light of (ii), it is easily seen that the ℤsubalgebra ℤ[T] ⊂ A generated by T is bounded in A, and especially, every element of ℤ[T] is power bounded. Combining with remark 8.3.9(ii), we deduce that A^{∘} is the filtered union of all the bounded subrings of A. Especially, A^{∘} is a subring of A.

(iv) Moreover, A^{∘} is integrally closed in A, and A^{∘∘} is a radical ideal of A^{∘}. Indeed, say that x ∈ A is integral over A^{∘}, so that there exist a_{1}, …, a_{n} ∈ A^{∘} (for some n ∈ ℕ) such that x^{n} + a_{1}x^{n−1} + ⋯ + a_{n} = 0, and set B := ℤ[a_{1}, …, a_{n}] ⊂ A; by (iii), the subring B is bounded in A, and it is easily seen that B[x] = ∑_{i=0}^{n−1} Bx^{i}, so B[x] is bounded as well, therefore B[x] ⊂ A^{∘}, which shows the first assertion. Lastly, (i) and remark 8.3.9(v) imply easily that A^{∘∘} is an ideal of A^{∘}, and remark 8.3.9(iv) shows that A^{∘∘} is a radical ideal.

(v) Lastly, if the topology 𝒯 is also complete and separated, A^{∘∘} lies in the Jacobson radical of A^{∘}. Indeed, in this case, for every a ∈ A^{∘∘} the series 1 − a + a^{2} − a^{3} + ⋯ converges in A to a unique element b such that b · (1 + a) = 1. Now, b lies in the topological closure ℤ[a]^{c} of ℤ[a] ⊂ A, and notice that ℤ[a] is the additive subgroup generated by the subset T := {a^{n} | n ∈ ℕ}; since a is power bounded, obviously T is a power bounded subset, and then the same holds for ℤ[a]^{c}, by (ii) and remark 8.3.9(iii). Thus, b ∈ A^{∘}, whence the contention.

Lemma 8.3.11. Let A be a ring, and I ⊂ A an ideal; consider a cartesian diagram of topological A-modules

\xymatrix{M_{0} \ar[r] \ar[d] & M_{1} \ar[d]^{f_{1}} \\ M_{2} \ar[r]^{f_{2}} & M_{3}}

and suppose that the topology of M_{3} is discrete, and the topologies of M_{1} and M_{2} are I-adic. Then the topology of M_{0} is I-adic as well.

Proof. By assumption, we may find c ∈ ℕ such that f_{1}(I^{c}M_{1}) = f_{2}(I^{c}M_{2}) = 0, and it follows easily that I^{n}M_{1} × I^{n}M_{2} ⊂ M_{0} for every n ≥ c. Especially, (I^{n}M_{1} × I^{n}M_{2} | n ≥ c) is a fundamental system of open neighborhoods of 0 in M_{0}. We also see that

I^{n+c}M_{1} × I^{n+c}M_{2} = I^{n}(I^{c}M_{1} × I^{c}M_{2}) ⊂ I^{n}M_{0} ⊂ I^{n}M_{1} × I^{n}M_{2} for every n ∈ ℕ

whence the claim. □

Lemma 8.3.12. Let A be a topological ring, (M, 𝒯_{M}) a topological A-module whose topology is A-linear, complete and separated, and I ⊂ A an ideal such that the I-adic topology on M is finer than 𝒯_{M}. Suppose moreover that either one of the following conditions holds :

(a) I^{n}M is closed in the topology 𝒯_{M}, for every n ∈ ℕ.

(b) I is finitely generated. Then M is separated and complete for the I-adic topology.

Proof. Let (J_{λ} | λ ∈ Λ) be a fundamental system of open submodules in M; by assumption, for every λ ∈ Λ there exists n(λ) ∈ ℕ such that I^{n(λ)}M ⊂ J_{λ}. Suppose first that (a) holds; it follows that the natural map

M/I^{n}M → lim_{λ∈Λ} M/(I^{n}M + J_{λ})

is injective for every n ∈ ℕ. Taking into account example 1.5.15(ii), we deduce that both of the induced maps

M → lim_{n∈ℕ} M/I^{n}M  and  lim_{n∈ℕ} M/I^{n}M → lim_{(λ,n)∈Λ×ℕ} M/(I^{n}M + J_{λ})

are injective. However, notice that the subset Σ ⊂ Λ × ℕ of all (n, λ) such that n ≥ n(λ), is cofinal, hence the composition of these two maps is an isomorphism. We conclude that both these maps are bijective, and the assertion follows.

Next, suppose that (b) holds, and let a_{1}, …, a_{r} be a finite system of generators for I. Clearly M is separated for the I-adic topology. Now, suppose x_{•} := (x_{n} | n ∈ ℕ) is a sequence of elements of M such that x_{n} − x_{m} ∈ I^{n}M for every n, m ∈ ℕ with m ≥ n. The sequence x_{•} converges for the topology 𝒯_{M} to an element x ∈ M, and it remains to check that x is also the limit of x_{•} for the I-adic topology of M. To this aim, for every n ∈ ℕ, let

S_{n} := {(j_{1}, …, j_{r}) ∈ ℕ^{r} | j_{1} + ⋯ + j_{r} = n}

and for every σ := (j_{1}, …, j_{r}) ∈ ℕ^{r} set a^{σ} := a_{1}^{j_{1}} ⋯ a_{r}^{j_{r}}. For every k, l ∈ ℕ we may find inductively a system of elements (y_{l,σ} | σ ∈ S_{k}) of M such that

• x_{k+l} − x_{k} = ∑_{σ∈S_{k}} y_{l,σ}a^{σ}

• y_{l′,σ} − y_{l,σ} ∈ I^{l}M whenever l′ ≥ l and for every σ ∈ ℕ^{r}.

Thus, the sequence (y_{l,σ} | l ∈ ℕ) converges in the topology 𝒯_{M} to some element y_{σ} ∈ M for every σ ∈ ℕ^{r}, and we get x−x_{k} = ∑_{σ∈S_{k}} y_{σ}a^{σ}; so this difference lies in I^{k}M for every k ∈ ℕ, whence the contention. □

Proposition 8.3.13. Let A be a ring, I ⊂ A an ideal, and M a faithful A-module of finite type. The following holds :

(i) If M is I-adically complete and separated, then I lies in the Jacobson radical of A.

(ii) If moreover A is noetherian, then M is I-adically complete and separated if and only if the same holds for A.

Proof. (i): Let A^{∧} be the I-adic completion of A, so that M is naturally an A^{∧}-module. Suppose by contradiction that there exists a maximal ideal 𝔪 ⊂ A that does not contain I. Then there exist a ∈ I and b ∈ 𝔪 with a + b = 1, whence b = 1 − a ∈ (A^{∧})^{×} (with b^{−1} = ∑_{n∈ℕ} a^{n}), so that M = 𝔪M, and therefore M_{𝔪} = 0, by Nakayama’s lemma. On the other hand, since by assumption we have Ann_{A}(M) = 0, we get Ann_{A_{𝔪}}(M_{𝔪}) = 0, so A_{𝔪} = 0, which is absurd.

(ii): If A is I-adically complete and separated, the same holds for M, by [126, Th.8.7]. Conversely, if M is I-adically complete and separated, then I lies in the Jacobson radical of A, by (i). Moreover, since M is faithful and of finite type, there exists k ∈ ℕ and an injective map f : A → M^{⊕k} (details left to the reader). Clearly M^{⊕k} is I-adically complete and separated, and f(A) is a closed subset for the I-adic topology of M^{⊕k} ([126, Th.8.10]), hence A is complete and separated for the topology 𝒯 induced by M^{⊕k}; but the latter agrees with the I-adic topology ([126, Th.8.6]). □

Proposition 8.3.14. Let A be a noetherian ring, I ⊂ J ⊂ A two ideals, and M an A-module; set \overline{A} := A/I, \overline{J} := J/I and \overline{M} := M/IM. Suppose that M is I-adically separated and complete, and that either one of the following conditions holds :

(a) M is of finite type.

(b) M is flat. Then M is J-adically separated and complete if and only if the same holds for \overline{M}.

Proof. For every A-module N, let N^{∧} be the J-adic completion of N. We notice :

Claim 8.3.15. For every flat A-module P, the following holds :

(i) Every A-module of finite type N induces an isomorphism f_{N} : N ⊗_{A} P^{∧} →^{∼} (N ⊗_{A} P)^{∧}. (ii) Moreover, P^{∧} is a flat A^{∧}-module.

Proof of the claim. (i): We consider a short exact sequence of A-modules of finite type

Σ  :  0 → N_{1} → N_{2} → N_{3} → 0

and we first show that the induced sequence of J-adic completions :

0 → (N_{1} ⊗_{A} P)^{∧} → (N_{2} ⊗_{A} P)^{∧} → (N_{3} ⊗_{A} P)^{∧} → 0

is still short exact. To this aim, since Σ ⊗_{A} P is still short exact, it suffices to check that the J-adic topology of N_{1} ⊗_{A} P is induced by that of N_{2} ⊗_{A} P (proposition 8.2.13(i,v)). But by the Artin-Rees lemma there exists c ∈ ℕ such that J^{n+c}N_{2} ∩ N_{1} ⊂ J^{n}N_{1} ⊂ J^{n}N_{2} ∩ N_{1} for every n ∈ ℕ. Since P is a flat A-module, it follows that

J^{n+c}(N_{2} ⊗_{A} P) ∩ (N_{1} ⊗_{A} P) ⊂ J^{n}(N_{1} ⊗_{A} P) ⊂ J^{n}(N_{2} ⊗_{A} P) ∩ (N_{1} ⊗_{A} P)

for every n ∈ ℕ, whence the assertion. Now, let φ : A^{⊕n} → N be an A-linear surjection, and N′ := Ker φ. By the foregoing, we have a commutative ladder :

\xymatrix{ & N′ ⊗_{A} P^{∧} \ar[r] \ar[d]_{f_{N′}} & A^{⊕n} ⊗_{A} P^{∧} \ar[r] \ar[d]^{f_{A^{⊕n}}} & N ⊗_{A} P^{∧} \ar[r] \ar[d]^{f_{N}} & 0 \\ 0 \ar[r] & (N′ ⊗_{A} P)^{∧} \ar[r] & (A^{⊕n} ⊗_{A} P)^{∧} \ar[r] & (N ⊗_{A} P)^{∧} \ar[r] & 0}

whose bottom (resp. top) horizontal row is short exact (resp. right exact). Clearly f_{A^{⊕n}} is an isomorphism, hence f_{N} is surjective for every A-module N of finite type. Especially, f_{N′} is surjective; but the snake lemma yields an exact sequence Ker f_{A^{⊕n}} → Ker f_{N} → Coker f_{N′}, so f_{N} is an isomorphism.

(ii): It is easily seen that the natural map P → A^{∧} ⊗_{A} P induces an isomorphism on J-adic completions; after replacing A by A^{∧}, and P by A^{∧} ⊗_{A} P, we may thus assume from start that A is J-adically complete and separated, and then we need to check that P is a flat A-module. Now, let i : L → A be the inclusion map of an ideal; by (i), the map i ⊗_{A} P^{∧} : L ⊗_{A} P^{∧} → P^{∧} is naturally identified with the J-adic completion (i ⊗_{A} P)^{∧} of i ⊗_{A} P : L ⊗_{A} P → P. Since P is a flat A-module, i ⊗_{A} P is injective; then the foregoing shows that the same holds for (i ⊗_{A} P)^{∧}, hence also for i ⊗_{A} P^{∧}, so P^{∧} is a flat A-module ([126, Th.7.2]). ♢

Now, if (b) holds, claim 8.3.15(ii) implies that the natural map M^{∧}/IM^{∧} → \overline{M}^{∧} is an isomorphism; the latter holds also under condition (a), by [126, Th.8.11]. In either case, we already deduce that if M is J-adically complete and separated, then the same holds for \overline{M}.

Conversely, suppose that \overline{M} is J-adically complete and separated. If (a) holds, let \overline{A}_{0} := \overline{A}/Ann\overline{A}(\overline{M}); then \overline{M} is a faithful \overline{A}_{0}-module of finite type, and therefore \overline{A}_{0} is J-adically complete and separated (proposition 8.3.13(ii)), so that every \overline{A}_{0}-module of finite type is Jadically complete and separated ([126, Th.8.7]). Now, notice that, for every n ∈ ℕ, the \overline{A}- module I^{n}M/I^{n+1}M is a quotient of I^{n}/I^{n+1} ⊗\overline{A} \overline{M}, hence it is an \overline{A}_{0}-module of finite type, so it is J-adically complete and separated. By [126, Th.8.7], this means that the natural map I^{n}M/I^{n+1}M → (I^{n}M/I^{n+1}M) ⊗_{A} A^{∧} is an isomorphism for every n ∈ ℕ. Since A^{∧} is a flat A-algebra, it follows easily that, for every n ∈ ℕ, the natural map

f_{n} : M/I^{n}M → (M/I^{n}M) ⊗_{A} A^{∧} →^{∼} M^{∧}/I^{n}M^{∧}

is an isomorphism. Since M is I-adically complete and separated, the limit of the system (f_{n} | n ∈ ℕ) is an isomorphism from M to the I-adic completion of M^{∧}. But M^{∧} is I-adically complete and separated (lemma 8.3.12), so M is J-adically complete and separated, as stated.

Lastly, if (b) holds, then M^{∧} is a flat A-module (claim 8.3.15(ii)), and therefore, for every n ∈ ℕ, both vertical arrows of the commutative diagram :

\xymatrix{(I^{n}/I^{n+1}) ⊗\overline{A} \overline{M} \ar[r] \ar[d] & (I^{n}/I^{n+1}) ⊗\overline{A} M^{∧}/IM^{∧} \ar[d] \\ I^{n}M/I^{n+1}M \ar[r] & I^{n}M^{∧}/I^{n+1}M^{∧}}

are isomorphisms. The same holds for the top horizontal arrow, by claim 8.3.15(i), so also for the bottom horizontal one. Then we argue as in the foregoing case, to conclude. □

Remark 8.3.16. Claim 8.3.15is a special case of proposition 14.2.43, and for local rings it can already be found in [15, Cor.3.15]. For a non-noetherian variant, see [75, Lemma 7.1.6].

Corollary 8.3.17. Let A be a noetherian ring, I ⊂ J ⊂ A two ideals, M an A-module that is either flat or of finite type, and such that M/IM is J-adically complete and separated. Denote by M_{I}^{∧} (resp. M_{J}^{∧}) the I-adic (resp. J-adic) completion of M. Then the natural map

α : M_{I}^{∧} → M_{J}^{∧}

is an isomorphism.

Proof. Let also A_{I}^{∧} be the I-adic completion of A; if M is a flat (resp. finitely generated) Amodule, then M_{I}^{∧} is a flat (resp. finitely generated) A_{I}^{∧}-module (claim 8.3.15(ii)); moreover, M/IM is trivially I-adically complete and separated. Hence, the natural map M/IM → M_{I}^{∧}/IM_{I}^{∧} is bijective, by claim 8.3.15(i) (resp. by [126, Th.8.11]). Especially, M_{I}^{∧}/IM_{I}^{∧} is Jadically complete and separated, so the same holds for M_{I}^{∧}, by proposition 8.3.14. Thus, there exists a unique A-linear map β : M_{J}^{∧} → M_{I}^{∧} whose composition with the J-adic completion map i_{M,J} : M → M_{J}^{∧} agrees with the I-adic completion map i_{M,I} : M → M_{I}^{∧}. Likewise, by definition we have α ∘ i_{M,I} = i_{M,J}, so that

α ∘ β ∘ i_{M,J} = i_{M,J} and β ∘ α ∘ i_{M,I} = i_{M,I}.

By the universal properties of i_{M,I} and i_{M,J}, it follows that α ∘ β = 1_{M_{J}^{∧}} and β ∘ α = 1_{M_{I}^{∧}}, whence the assertion. □

Proposition 8.3.18. Let (A, 𝒯 ) be any f-adic ring. We have :

(i) If A_{0} is any ring of definition of A, then the topology of A_{0} is adic and admits a finitely generated ideal of adic definition.

(ii) A subring of A is a ring of definition of A if and only if it is open and bounded in A. (iii) The following conditions are equivalent :

(a) The topology 𝒯 is linear.

(b) The topology 𝒯 is adic and admits a finitely generated ideal of adic definition.

Proof. Fix an open subring A_{1} and a finitely generated ideal I of A_{1}, such that 𝒯 induces the I-adic topology on A_{1}.

(i): Since A_{0} is open in A, it follows that there exists an integer n > 0 with I^{n} ⊂ A_{0}; set J := I^{n}A_{0}. If K is any open ideal of A_{0}, we may likewise find m ∈ ℕ such that I^{mn} ⊂ K, so that J^{m} ⊂ K and J^{m} is open in A_{0}. Hence, 𝒯 induces the J-adic topology on A_{0}; however, J is not necessarily finitely generated (unless A_{1} ⊂ A_{0}). Hence, let T ⊂ A_{0} be a finite system of generators of the ideal I^{n} of A_{1}, and set J′ := A_{0}T; then J′ ⊂ J, and on the other hand J^{2} = J′I_{0}^{n} ⊂ J′, so the J-adic topology on A_{0} coincides with the J′-adic topology.

(ii): Clearly, every ring of definition of A is open and bounded. Conversely, suppose that A_{0} is open and bounded. As in the foregoing, we find an integer n > 0 such that I^{n} ⊂ A_{0} and we set J := I^{n}A_{0}. Let now U be any open neighborhood of zero in A_{0}; since A_{1} admits a fundamental system of open neighborhoods of zero consisisting of additive subgroups, we may assume that U is an additive subgroup of A_{0}. Since A_{0} is bounded, there exists m ∈ ℕ such that x · a ∈ U for every x ∈ I^{m} and every a ∈ A_{0}; then clearly J^{m} ⊂ U. Lastly, it is easily seen that J^{k} is open in A_{0} for every k ∈ ℕ, whence (ii).

(iii): If 𝒯 is linear, then A is bounded in itself, so the assertion follows directly from (ii). □

Corollary 8.3.19. Let (A, 𝒯 ) be any f-adic ring. We have :

(i) If A_{0} and A_{1} are rings of definition of A, then the same holds for A_{0} ∩ A_{1} and A_{0} · A_{1}. (ii) Let B (resp. C) be a bounded (resp. open) subring of A, with B ⊂ C. Then there

exists a ring of definition A_{0} of A with B ⊂ A_{0} ⊂ C. (iii) (A, 𝒯 ) is adic if and only if A is bounded in the topology 𝒯 . (iv) A^{∘} is the filtered union of all the subrings of definition of A.

Proof. (i): The assertion for A_{0} ∩ A_{1} is immediate from proposition 8.3.18(ii). The assertion for A_{0} · A_{1} follows likewise, taking into account that A has a fundamental system of open neighborhoods of zero consisting of additive subgroups : details left to the reader.

(ii): Let A_{1} be any ring of definition of A; by proposition 8.3.18(ii), the subring C_{1} := A_{1}∩C is a ring of definition of A, and we may take A_{0} := B · C_{1}.

(iii): If (A, 𝒯 ) is adic, then clearly A is bounded. Conversely, if A is bounded, (ii) implies that A is a subring of definition of (A, 𝒯 ), and then (A, 𝒯 ) is adic, by proposition 8.3.18(i).

(iv): By remark 8.3.10(iii), we know that A^{∘} is the filtered union of the family ℱ of bounded subrings of A; notice that any subring containing an open subring of A is also open in A. Since the family of bounded and open subring of A is not empty (proposition 8.3.18(ii)), we conclude that this family is cofinal in ℱ, whence the contention. □

Corollary 8.3.20. Let (A, 𝒯_{A}) be any topological ring, B ⊂ A an open subring, and endow B with the topology 𝒯_{B} induced from 𝒯_{A}. The following holds :

(i) (A, 𝒯_{A}) is f-adic if and only if the same holds for (B, 𝒯_{B}).

(ii) Suppose that (A, 𝒯_{A}) is a Tate ring, and B is a ring of definition of A. Then we have : (a) B contains an element of A^{×} ∩ A^{∘∘}.

(b) Let s ∈ B ∩ A^{×} ∩ A^{∘∘} be any element. Then sB is an ideal of adic definition for B, and B_{s} = A.

(iii) Conversely, let C be any ring, f ∈ C any element, D the image of C in the localization C_{f}, and endow C_{f} with the unique D-linear topology 𝒯_{f} such that {f^{n}D | n ∈ ℕ} is a fundamental system of neighborhoods of 0 in C_{f}. Then (C_{f}, 𝒯_{f}) is a Tate ring.

Proof. (i): Directly from the definition we see that if (B, 𝒯_{B}) is f-adic, the same holds for (A, 𝒯_{A}). Conversely, if (A, 𝒯_{A}) is f-adic, let A_{0} ⊂ A be any subring of definition; by corollary 8.3.19(ii) we may find a subring of definition A_{1} of A contained in A_{0} ∩ B; especially, A_{1} is an open subring of B whose topology is adic with a finitely generated ideal of adic definition (proposition 8.3.18(i)), whence the assertion.

(ii.a) is clear. Next, for every a ∈ A there exists n ∈ ℕ such that s^{n}a ∈ B, so B_{s} = A. Also, the map a ↦ s^{n}a is an automorphism of (A, 𝒯_{A}), so s^{n}B is open for every n ∈ ℕ. Lastly, since B is bounded in A, for every open neighborhood U of 0 ∈ A there exists n ∈ ℕ such that s^{n}B ⊂ U, whence (ii.b).

(iii): Explicitly, the open subsets of (C_{f}, 𝒯_{f}) are the arbitrary unions of subsets of the form a + f^{m}D, where a ∈ C_{f} is arbitrary and m ∈ ℕ is any integer; indeed, it is easily seen that the intersection of any two subsets of this type is either empty, or equal to one of them, so such arbitrary unions define a topology 𝒯_{f}, and it is easily seen that (C_{f}, 𝒯_{f}) is a topological ring (details left to the reader). A simple inspection then shows that (C_{f}, 𝒯_{f}) is a Tate ring. □

Lemma 8.3.21. Let A be any c-adic, complete and separated topological ring, I, J ⊂ A two ideals, with I finitely generated. We have :

(i) I is open in A if and only if the same holds for I^{c}.

(ii) If I is an ideal of c-adic definition for A, the following holds : (a) A is adic and I is an ideal of adic definition for A.

(b) J is open in A if and only if the same holds for J^{c}.

Proof. Assertion (i) is a special case of [75, Lemma 5.3.5(ii) and 5.3.8(i)], and (ii.a) follows directly from (i). Lastly, suppose that J^{c} is open in A; by (ii.a) there exists n ∈ ℕ such that I^{n} ⊂ J^{c} ⊂ J + I^{n+1}. Letting M := (I^{n} + J)/J, it follows that IM = M, and since A is complete and separated for its I-adic topology, I lies in the Jacobson radical of A (remark 8.3.10(v)); therefore M = 0 by Nakayama’s lemma, i.e. I^{n} ⊂ J, so J is open in A. □

Lemma 8.3.22. Let A be a complete and separated adic topological ring, I ⊂ A an ideal of adic definition, M an A-module whose I-adic topology 𝒯_{M} is separated, N ⊂ M a finitely generated submodule, and N^{c} the topological closure of N in (M, 𝒯_{M}). We have :

(i) If the I-adic topology 𝒯_{N} of N^{c} agrees with the one induced by 𝒯_{M}, then N = N^{c}. (ii) N is open in (M, 𝒯_{M}) if and only if the same holds for N^{c}.

Proof. (i): By assumption, IN^{c} is an open submodule of N^{c} for the topology induced by (M, 𝒯_{M}), hence N^{c} = N + IN^{c}, and the assertion follows from [126, Th.8.4].

(ii): If N is open in M, then N = N^{c} and the assertion is clear. Conversely, if N^{c} is open in M, then I^{n}M ⊂ N^{c} for some n ∈ ℕ, hence 𝒯_{N} agrees with the topology induced from 𝒯_{M}; then we get again N = N^{c}, by (i). □

Definition 8.3.23. Let (A, 𝒯 ) and (A′, 𝒯 ′) be two topological rings, and f : A → A′ a ring homomorphism.

(i) We say that f is c-adic (resp. adic) if 𝒯 and 𝒯 ′ are linear, and the family of ideals

((f(I) · A′)^{c} | I open ideal in A) (resp. (f(I) · A′ | I open ideal in A) )

is a fundamental system of open neighborhoods of zero in A′.

(ii) Suppose that A and A′ are f-adic. Then we say that f is f-adic if there are rings of definition A_{0} of A and A′_{0} of A′ such that f(A_{0}) ⊂ A′_{0} and the restriction f_{|A_{0}} is adic.

(iii) Let Q be any property of ring homomorphisms (e.g. “of finite type”, “flat”, “étale” and so on). We say that f is an adically Q morphism (resp. a c-adically Q morphism) if f is adic (resp. c-adic) and for every open ideal I ⊂ A the induced map A/I → A′/(IA′)^{c} is a Q ring homomorphism.

Lemma 8.3.24. Let f : (A, 𝒯 ) → (A′, 𝒯 ′) be any ring homomorphism of topological rings.

(i) If f is c-adic, the following holds :

(a) f is a continuous ring homomorphism.

(b) If the topology 𝒯 is c-adic, the same holds for 𝒯 ′, and if I is any ideal of c-adic definition for A, then f(I)A′ is an ideal of c-adic definition for A′.

(c) If the topologies 𝒯 and 𝒯 ′ are f-adic, linear and complete, then f is adic. (ii) If the topologies 𝒯 and 𝒯 ′ are adic, the following conditions are equivalent :

(a) f is adic.

(b) There exists an ideal of adic definition I of A such that f(I) · A′ is an ideal of adic definition of A′.

(c) For every ideal I of adic definition of A, the ideal f(I) · A′ is of adic definition for A′. (iii) Suppose that f is an f-adic map of f-adic topological rings. Then the following holds :

(a) f is continuous and bounded (i.e. if S is any bounded subset of A, the subset f(S) is bounded in A′). Especially, f(A^{∘}) ⊂ A^{′∘}.

(b) For every ring of definition A_{0} of A and A′_{0} of A′ such that f(A_{0}) ⊂ A′_{0}, and every ideal of adic definition I of A_{0}, the ideal f(I) · A′_{0} is of adic definition for A′_{0}.

(c) For every ring of definition A_{0} of A and every open subring B ⊂ A′ such that f(A_{0}) ⊂ B, there exists a ring of definition A′_{0} of A′ such that f(A_{0}) ⊂ A′_{0} ⊂ B.

(d) If B ⊂ A, B′ ⊂ A′ are open subrings with f(B) ⊂ B′, then the restriction f_{|B} : B →

B′ is f-adic, for the topologies of B and B′ induced by 𝒯 and 𝒯 ′. (iv) If f is a continuous open map, and the topologies 𝒯 and 𝒯 ′ are f-adic, then f is f-adic.

(v) Suppose that f(A) is open in A′, that 𝒯 is a linear topology and that 𝒯 ′ is adic. Then f is adic if and only if it is a continuous open map.

Proof. (i.a) is obvious, and (i.c) follows easily from proposition 8.3.18(iii) and lemma 8.3.21(i). To show (i.b), let us first remark, quite generally :

Claim 8.3.25. Let f : X → Y be any continuous map between arbitrary topological spaces, and S, T ⊂ X two subsets such that S^{c} = T^{c}. Then (fS)^{c} = (fT)^{c}.

Proof of the claim. It suffices to show that fS ⊂ (fT)^{c}. However, clearly we have T ⊂ f^{−1}((fT)^{c}), therefore S ⊂ T^{c} ⊂ f^{−1}((fT)^{c}), whence the contention. ♢

Claim 8.3.26. Let B be any topological ring, J_{1}, J_{2} ⊂ B two additive subgroups. Denote J_{1}J_{2} the additive subgroup generated by (j_{1}j_{2} | j_{1} ∈ J_{1}; j_{2} ∈ J_{2}), and define likewise J_{1}^{c}J_{2}^{c}. Then

(J_{1}J_{2})^{c} = (J_{1}^{c}J_{2}^{c})^{c} for every n ∈ ℕ.

Proof of the claim. Since J_{1}J_{2} ⊂ J_{1}^{c}J_{2}^{c}, we have (J_{1}J_{2})^{c} ⊂ (J_{1}^{c}J_{2}^{c})^{c}. For the converse, it suffices to show that J_{1}^{c}J_{2}^{c} ⊂ (J_{1}J_{2})^{c}. However, for every k ∈ ℕ, consider the map

μ_{k} : (B^{⊕2})^{⊕k} → B (b_{ij} | i = 1, 2; j = 1, …, k) ↦ ∑_{t=1}^{k} b_{1t}b_{2t}.

Then J_{1}^{c}J_{2}^{c} = ⋃_{k∈ℕ} μ_{k}((J_{1}^{c} ⊕ J_{2}^{c})^{⊕k}). Now, μ_{k} is clearly continuous for the product topology on (B^{⊕2})^{⊕k}, and (J_{1}^{c} ⊕ J_{2}^{c})^{⊕k} is the topological closure of (J_{1} ⊕ J_{2})^{⊕k} in this topology, so μ_{k}((J_{1}^{c} ⊕ J_{2}^{c})^{⊕k}) is contained in the topological closure in B of μ_{k}((J_{1} ⊕ J_{2})^{⊕k}) (claim 8.3.25), which in turns is contained in (J_{1}J_{2})^{c}, whence the contention. ♢

Now, let I be any ideal of c-adic definition of A; by assumption, for every open ideal I′ of A′ there exists n ∈ ℕ such that (f((I^{n})^{c})A′)^{c} ⊂ I. Set J := f(I)A′; then J is open in B, and by claims 8.3.26and 8.3.25we deduce that

(f((I^{n})^{c})A′)^{c} = ((f((I^{n})^{c}))^{c}A′)^{c} = ((f(I^{n}))^{c}A′)^{c} = (f(I^{n})A′)^{c} = (J^{n})^{c}

whence the contention.

(ii): If I_{1}, I_{2} are any two ideals of adic definition of A, then there exist n, m ∈ ℕ such that I_{1}^{n} ⊂ I_{2}^{m} ⊂ I_{1}. The assertion is an easy consequence : details left to the reader.

(iii.a): The continuity of f is obvious. Suppose that S ⊂ A is a bounded subset, and let U′ ⊂ A′ be any open neighborhood of zero; we have to exhibit an open neighborhood of zero V′ in A′ such that f(S) · V′ ⊂ U′. To this aim, let A_{0} and A′_{0} be rings of definition of A and A′, such that f(A_{0}) ⊂ A′_{0}, and I ⊂ A_{0} an ideal of adic definition such that IA′_{0} is an ideal of adic definition of A′_{0}; we may assume that U is an open ideal of A′_{0}, and we may find n, m ∈ ℕ such that f(I^{n}) ⊂ U′ and S · I^{m} ⊂ I^{n}. Thus, f(S) · f(I^{m}) ⊂ U′, so V′ := I^{m}A′_{0} will do.

(iii.b): Let A_{1} and A′_{1} be rings of definitions of A and A′ such that f(A_{1}) ⊂ A′_{1}, and J ⊂ A_{1} an ideal of adic definition such that K := f(J) · A′_{1} is an ideal of adic definition of A′_{1}. Then there exist n, m ∈ ℕ such that J^{n} ⊂ I and K^{m} ⊂ A′_{0}, so that K^{n+m} = f(J^{n}) · K^{m} ⊂ f(I) · A′_{0}; especially, f(I) · A′_{0} is open in A′_{0}. Moreover, since I is topologically nilpotent and finitely generated in A_{0}, it is easily seen that f(I)A′_{0} is topologically nilpotent in A′_{0}; thus f(I)A′_{0} is an ideal of adic definition of A′_{0}, as stated.

(iii.c) follows easily from (iii.a) and corollary 8.3.19(ii).

(iii.d): By corollary 8.3.20(i), both B and B′ are f-adic rings. Let A_{0} ⊂ A and A′_{0} ⊂ A′ be subrings of definition with f(A_{0}) ⊂ A′_{0}; hence B_{0} := A_{0} ∩B and B′_{0} := A′_{0} ∩B′ are subrings of definition of B and B′ respectively, with f(B_{0}) ⊂ B′_{0} (proposition 8.3.18(ii)); then the assertion follows from (iii.b).

(iv): Let A_{0} and A′_{0} be rings of definition of A and A′; after replacing A_{0} by A_{0} ∩ f^{−1}A′_{0} we may assume that f(A_{0}) ⊂ A′_{0} (proposition 8.3.18(ii)); by assumption, f(A_{0}) is open in A′, so we may further replace A′_{0} by f(A′_{0}), and assume from start that f restricts to an open surjective map A_{0} → A′_{0}. In this case, let I ⊂ A_{0} be any ideal of adic definition; it follows easily that f(I) is an ideal of adic definition for A′_{0}, so f is adic.

(v) If f is a continuous open map, then for every pair of open ideals J ⊂ A′ and I ⊂ A with f(I) ⊂ J, we have f(I) · A′ ⊂ J, and clearly f(I) · A′ is open in A′. This shows that f is adic.

Conversely, if f is adic and f(A) is open in A′, let J ⊂ f(A′) be an open ideal of A′. For every open ideal I ⊂ A, the ideal f(I) · A′ is open in A′, hence the same holds for f(I) · J; but f(I) · J ⊂ f(I), so f(I) is open for every such I, and thus f is open. □

Example 8.3.27. Let (A, 𝒯 ) be a topological ring, f : A → A′ a surjective ring homomorphism, and endow A′ with the topology 𝒯 ′ induced by A via f. We have :

(i) It is easily seen that a subset U′ ⊂ A′ is open if and only if U′ = f(U) for some open subset U of A. It follows that (A′, 𝒯 ′) is a topological ring.

(ii) Moreover, if (A, 𝒯 ) is c-adic, the same holds for (A′, 𝒯 ′), and f : (A, 𝒯 ) → (A′, 𝒯 ′) is c-adic. Indeed, let I ⊂ A be an ideal of c-adic definition, and set J := f(I); by (i) we know that f((I^{n})^{c}) is open in A’ for every n ∈ ℕ, and then claim 8.3.25 implies that f((I^{n})^{c}) = (J^{n})^{c}, so the latter is open in A′ for every n ∈ ℕ, again by (i), and the system of such ideals is a fundamental system of open neighborhoods of 0 in A′.

(iii) Likewise, if (A, 𝒯 ) is adic (resp. f-adic), the same holds for (A′, 𝒯 ′), and f is adic (resp f-adic).

Definition 8.3.28. For any f-adic ring A, set

X_{A} := Spec A X_{A}^{∘∘} := Spec A/A^{∘∘}A.

We call X_{A}^{∘∘} the non-analytic locus of X_{A} and its complement X_{A} \ X_{A}^{∘∘} the analytic locus.

Lemma 8.3.29. Let A and B be two f-adic rings, f : A → B a continuous ring homomorphism, and set φ := Spec f. We have :

(i) X_{A}^{∘∘} = {𝔭 ∈ X_{A} | 𝔭 is open in A}. (ii) φ restricts to a map X_{B}^{∘∘} → X_{A}^{∘∘}. (iii) If f is an injective and open map, φ restricts to an isomorphism of schemes :

X_{B} \ X_{B}^{∘∘} →^{∼} X_{A} \ X_{A}^{∘∘}.

(iv) The homomorphism f is f-adic if and only if φ^{−1}(X_{A}^{∘∘}) = X_{B}^{∘∘}.

(v) Let J ⊂ A be any ideal. Then J is open if and only if Spec A/J ⊂ X_{A}^{∘∘}.

Proof. (i) and (ii) are clear.

(v): If J is open, then the same holds for every prime ideal 𝔭 containing J; in light of (i) it then follows that A^{∘∘} ⊂ 𝔭 for every such 𝔭, so the radical of J also contains A^{∘∘}, i.e. Spec A/J ⊂ X_{A}^{∘∘}. Conversely, suppose that the radical of J contains A^{∘∘}, and pick any subring of definition A_{0} ⊂ A, and a finitely generated ideal I ⊂ A_{0} of adic definition; it follows easily that J contains I^{n} for every sufficiently large n ∈ ℕ, so J is open.

Next, let us check that if f is adic, then φ^{−1}(X_{A}^{∘∘}) = X_{B}^{∘∘}. Indeed, pick subrings of definition A_{0} ⊂ A, B_{0} ⊂ B such that f(A_{0}) ⊂ B_{0}, and an ideal of adic definition I of A_{0}; then J := f(I) · B_{0} is an ideal of adic definition of B_{0} (lemma 8.3.24(iii.b)). It is easily seen that X_{A}^{∘∘} = Spec A/IA and X_{B}^{∘∘} = Spec B/JB, whence the stated condition.

(iii): Notice first that f is f-adic in the current situation, so the foregoing already shows that φ^{−1}(X_{A}^{∘∘}) = X_{B}^{∘∘}. The latter means that X_{B} \ X_{B}^{∘∘} = ⋃_{s∈A^{∘∘}} Spec B_{s}. Now, let 𝔭 ∈ X_{A} \ X_{A}^{∘∘} be any non-open prime ideal of A; then there exists s ∈ A^{∘∘} such that s ∉ 𝔭, and it suffices to check that the localization f_{s} : A_{s} → B_{s} is an isomorphism. However, f_{s} is obviously injective; let b ∈ B be any element; since A is open in B, there exists n ∈ ℕ such that s^{n}b ∈ A, so f_{s} is surjective as well.

(iv): By the foregoing, we may assume that φ^{−1}(X_{A}^{∘∘}) = X_{B}^{∘∘}, and we check that f is adic. Indeed, pick a subring of definition B_{0} ⊂ B; since f is continuous, f^{−1}B_{0} is open in A_{0}, so we may find a subring of definition A_{0} of A such that A_{0} ⊂ f^{−1}B_{0} (corollary 8.3.19(ii)). Let f_{0} : A_{0} → B_{0} be the restriction of f, and set φ_{0} := Spec f_{0}; we get a commutative diagram

\xymatrix{X_{B} \ar[r]^{φ} \ar[d] & X_{A} \ar[d] \\ X_{B_{0}} \ar[r]^{φ_{0}} & X_{A_{0}}}

whose vertical arrows are induced by the corresponding inclusion maps. The assumption implies that φ^{−1}(X_{A} \ X_{A}^{∘∘}) = X_{B} \ X_{B}^{∘∘}; together with (iii), we deduce that φ_{0}^{−1}(X_{A_{0}} \ X_{A_{0}}^{∘∘}) = X_{B_{0}} \ X_{B_{0}}^{∘∘}, which is equivalent to φ_{0}^{−1}(X_{A_{0}}^{∘∘}) = X_{B_{0}}^{∘∘}. On the other hand, it suffices to show that f_{0} is an adic ring homomorphism, so we may replace A and B by A_{0} and B_{0}, and assume from start that A and B are adic topological rings. However, our assumption on φ^{−1}(X_{A}^{∘∘}) means that the radical of f(A^{∘∘}) · B equals B^{∘∘}; now, pick any finitely generated ideal of adic definition I of A (resp. J of B); then the radical of f(I) · B also equals B^{∘∘}, and according to (v), the latter condition implies that f(I) · B is open in B, so f is adic. □

Proposition 8.3.30. With the notation of definition 8.3.28, let U ⊂ X_{A} be a quasi-compact open subset containing the analytic locus. We have :

(i) There exists a unique ring topology 𝒯_{U} on A_{U} := 𝒪_{X_{A}}(U) such that (A_{U}, 𝒯_{U}) is f-adic and the restriction map ρ_{U} : A → A_{U} is open.

(ii) Suppose moreover that A admits a complete, separated and noetherian ring of definition. Then (A_{U}, 𝒯_{U}) is complete and separated.

(iii) Let φ : A → B be any morphism of f-adic topological rings, such that the image of Spec φ lies in U. Then the resulting map φ_{U} : (A_{U}, 𝒯_{U}) → B is continuous.

Proof. (i): Clearly, there exists a unique topology 𝒯_{U} on A_{U} such that ρ_{U} is open and (A_{U}, 𝒯_{U}) is a topological group for its additive group structure, and we need to check that (A_{U}, 𝒯_{U}) is a topological ring. Let I be a finitely generated ideal of adic definition for a ring of definition A_{0} of A; the assertion comes down to the following. For every a ∈ A_{U} and every n ∈ ℕ there exists k ∈ ℕ such that

a · ρ_{U}(I^{k}) ⊂ ρ_{U}(I^{n}).

Now, pick a finite system of generators (f_{1}, …, f_{r}) of I, and notice that A^{∘∘} is the radical of the ideal IA^{∘} in the subring A^{∘} of A (remark 8.3.10(iv)), hence

(8.3.31)  X_{A}^{∘∘} = Spec A/IA.

Let j : U → X_{A} be the inclusion map; then j_{∗}𝒪_{U} is a quasi-coherent 𝒪_{X_{A}}-module ([59, Ch.I, Cor.9.2.2]), and on the other hand, from (8.3.31) and our assumption on U we see that Spec A_{f_{i}} ⊂ U. Hence

A_{f_{i}} = 𝒪_{X_{A}}(Spec A_{f_{i}}) = j_{∗}𝒪_{U}(Spec A_{f_{i}}) = j_{∗}𝒪_{U}(X_{A})_{f_{i}} = A_{U}[f_{i}^{−1}] for i = 1, …, r.

Thus, there exists m ∈ ℕ such that f_{i}^{m}a ∈ ρ_{U}(A) for every i = 1, …, r. Then there also exists t ∈ ℕ such that ρ_{U}(I^{t}) · f_{i}^{m}a ⊂ ρ_{U}(I^{n}) for i = 1, …, r, and the assertion follows easily. It is then also clear that the topology 𝒯_{U} is f-adic, and ρ_{U}(A_{0}) is a ring of definition of A_{U}, with ρ_{U}(I) as ideal of adic definition.

(ii): Let A_{0} ⊂ A be a noetherian, complete and separated subring of definition; it suffices to check that A_{0} ∩ Ker ρ_{U} is a closed ideal, but this holds, by [126, Th.8.10(i)].

(iii): Let H ⊂ B be any open additive subgroup; then φ_{U}^{−1}H contains the open subgroup ρ_{U}(φ^{−1}H), so it is open, whence the contention. □

Lemma 8.3.32. Let f : (A, 𝒯_{A}) → (B, 𝒯_{B}) be a c-adic morphism of topological rings. Then:

(i) The completion map (A, 𝒯_{A}) → (A^{∧}, 𝒯_{A}^{∧}) is c-adic, and A is c-adic if and only if the same holds for A^{∧}.

(ii) The separated completion f^{∧} : (A^{∧}, 𝒯_{A}^{∧}) → (B^{∧}, 𝒯_{B}^{∧}) of f is c-adic.

(iii) Let Q be any property of ring homomorphisms. If f is a c-adically Q morphism, the

same holds for f^{∧} (see definition 8.3.23(iii)). (iv) If the topology 𝒯_{A} is I-adic for some finitely generated ideal I ⊂ A, the topology 𝒯_{A}^{∧}

is IA^{∧}-adic.

Proof. (i): Let (I_{λ} | λ ∈ Λ) be any fundamental system of open ideals of A, and set I_{λ}^{∧} := (I_{λ}A^{∧})^{c} for every λ ∈ Λ. From remark 8.3.3(ii) we see that (I_{λ}^{∧} | λ ∈ Λ) is a fundamental system of open neighborhoods of zero in A^{∧}, so 𝒯_{A}^{∧} is a linear topology, and the completion map A → A^{∧} is c-adic. In view of lemma 8.3.24(i.b), it follows that if A is c-adic, the same holds for A^{∧}. For the converse, suppose that A^{∧} is c-adic, and pick an ideal of c-adic definition of the form (IA^{∧})^{c}, where I is an open ideal of A; for every n ∈ ℕ we let (I^{n})^{c} (resp. (I^{n})^{∧}) be the topological closure of I^{n} in A (resp. in A^{∧}); then

(I^{n})^{c} = A ∩ (I^{n})^{∧} for every n ∈ ℕ.

Now, let J ⊂ A be any open ideal; the topological closure J^{∧} of J in A^{∧} is open in A^{∧}, so there exists n ∈ ℕ such that (I^{n})^{∧} ⊂ J^{∧}, and therefore (I^{n})^{c} ⊂ A ∩ J^{∧} = J, which says that A is c-adic with I as ideal of c-adic definition.

(ii): We know already that 𝒯_{A}^{∧} and 𝒯_{B}^{∧} are both linear topologies. Next, by assumption, for every open ideal J ⊂ B there exists λ ∈ Λ such that (I_{λ}B)^{c} ⊂ J; it follows that (I_{λ}^{∧}B^{∧})^{c} = (I_{λ}B^{∧})^{c} ⊂ (JB^{∧})^{c}. Likewise, for every λ ∈ Λ there exists an open ideal J of B such that J ⊂ (I_{λ}B)^{c}, and therefore (JB^{∧})^{c} ⊂ (I_{λ}^{∧}B^{∧})^{c}. Since the family ((JB^{∧})^{c} | J open in B) is a fundamental system of open ideals of B, it follows that the same holds for the family ((I_{λ}^{∧}B^{∧})^{c} | λ ∈ Λ), whence the contention.

(iii): This follows easily from (ii), taking into account the natural isomorphisms

A^{∧}/(IA^{∧})^{c} →^{∼} A/I and B^{∧}/(JB^{∧})^{c} →^{∼} B/J

for every open ideals I of A and J of B, as well as the equality : ((IA^{∧})^{c}B^{∧})^{c} = (IB^{∧})^{c}, which identify the map A^{∧}/(IA^{∧})^{c} → B^{∧}/(IB^{∧})^{c} induced by f^{∧} with the map A/I → B/(IB)^{c} induced by f.

(iv) follows easily from remark 8.3.3(ii,iv). □

Proposition 8.3.33. Let A be an f-adic topological ring, A_{0} an open subring of A, and A_{0}^{∧}, A^{∧} the separated completions of A_{0} and respectively A. We have :

(i) A^{∧} is an f-adic topological ring.

(ii) If A_{0} is a ring of definition of A, then A_{0}^{∧} is a ring of definition for A^{∧}.

(iii) The natural map j : A ⊗_{A_{0}} A_{0}^{∧} → A^{∧} is a ring isomorphism.

Proof. By corollary 8.2.17(i), the natural map A_{0}^{∧} → A^{∧} is an open immersion, so (i) and (ii) follow from lemma 8.3.32(iv).

(iii): Suppose first that A_{0} is a ring of definition of A. Since A_{0} is open in A, the topology induced by A on B := A/A_{0} via the projection A → B is discrete, so the completion map B → B^{∧} is an isomorphism of topological groups, and there follows a commutative ladder :

[figure: 0; A_{0}^{∧}; A ⊗_{A_{0}} A_{0}^{∧}; j; B ⊗_{A_{0}} A_{0}^{∧}; j_{B}; 0; 0; A_{0}^{∧}; A^{∧}; B; 0]

whose bottom row is short exact, by proposition 8.2.13(v). It follows that the top row is exact as well, and to conclude, it suffices to check that j_{B} is an isomorphism. To this aim, let us write B as the colimit of the filtered system (B_{λ} | λ ∈ Λ) of its finitely generated A_{0}-submodules; clearly it suffices to show that the restriction j_{λ} : B_{λ} ⊗_{A_{0}} A_{0}^{∧} → B_{λ} of j_{B} is an isomorphism for every such λ. However, for every λ ∈ Λ there exists an open ideal I_{λ} ⊂ A_{0} such that I_{λ}B_{λ} = 0, so j_{λ} factors as the composition of the isomorphism B_{λ} ⊗_{A_{0}} A_{0}^{∧} →^{∼} B_{λ} ⊗_{A_{0}} A_{0}^{∧}/I_{λ}A_{0}^{∧} and the natural identifications B_{λ} ⊗_{A_{0}} A_{0}^{∧}/I_{λ}A_{0}^{∧} →^{∼} B_{λ} ⊗_{A_{0}} A_{0}/I_{λ} →^{∼} B_{λ}, whence the contention, in this case. Next, if A_{0} is an arbitrary open subring of A, then A_{0} is an f-adic ring (corollary 8.3.20(i)); we let A_{1} be a ring of definition of A_{0}, and we notice that A_{1} is also a ring of definition for A (proposition 8.3.18(ii)). By the foregoing, both natural maps A_{0} ⊗_{A_{1}} A_{1}^{∧} → A_{0}^{∧} and A ⊗_{A_{1}} A_{1}^{∧} → A^{∧} are isomorphisms, so the same follows for j. □

Proposition 8.3.34. Let A, B, B′ be three f-adic topological rings and f : B → A, g : B → B′ two f-adic ring homomorphisms. Then we have :

(i) There exists a unique f-adic ring topology 𝒯_{A′} on A′ := B′⊗_{B} A such that the resulting diagram (with f′ := 1_{B′} ⊗_{B} f and g_{A} := g ⊗_{B} 1_{A})

(8.3.35)  \xymatrix{B \ar[r]^{f} \ar[d]_{g} & A \ar[d]^{g_{A}} \\ B′ \ar[r]^{f′} & (A′, 𝒯_{A′})}

is cocartesian in the category ℤ-TopAlg_{ℤ-lin} (notation of definition 8.3.1(v)). Moreover, both f′ and g_{A} are f-adic ring homomorphisms.

(ii) Suppose additionally that f is open. Then the same holds for f′, and (8.3.35) is cocartesian in the category ℤ-TopAlg.

(iii) In the situation of (ii), suppose moreover that B is a ring of definition of A, and g induces an isomorphism g^{∧} : B^{∧} →^{∼} B^{′∧} on separated completions; then

(a) g_{A} induces an isomorphism g_{A}^{∧} : A^{∧} →^{∼} A^{′∧} on separated completions (b) A^{′∘} (resp. A′^{∘∘}) is the image of B′ ⊗_{B} A^{∘} (resp. of B′ ⊗_{B} A^{∘∘}) in A′.

Proof. (i): In light of corollary 8.3.19(i) and lemma 8.3.24(iii.b) we may find subrings of definitions B_{0} ⊂ B, A_{0} ⊂ A and B′_{0} ⊂ B′ such that f and g restrict to adic ring homomorphisms f_{0} : B_{0} → A_{0} and g_{0} : B_{0} → A′_{0}. Let also I_{0} ⊂ B_{0} be an ideal of adic definition; denote by \overline{A}′_{0} ⊂ A′ the image of the induced map A′_{0} := B′_{0} ⊗_{B_{0}} A′_{0} → A′, and endow \overline{A}′_{0} with its I_{0}\overline{A}′_{0}-adic topology. There exists a unique group topology 𝒯 ′ on A′ such that \overline{A}′_{0} is an open subgroup, and we claim that 𝒯 ′ is a ring topology. This comes down to the following assertion. For every a′ ∈ A′ and every n ∈ ℕ there exists m ∈ ℕ such that

a′ · I_{0}^{m}\overline{A}′_{0} ⊂ I_{0}^{n}\overline{A}′_{0}.

However, we may write a′ as a finite sum of terms of the form b′ ⊗ a, with a ∈ A and b′ ∈ B′, so we are easily reduced to the case where a′ = b′ ⊗ a for such a and b′. But since f and g are f-adic, we may find, for every n ∈ ℕ, integers m_{1}, m_{2} ∈ ℕ such that a · I_{0}^{m_{1}}A_{0} ⊂ I^{n}A_{0}, and b′ · I_{0}^{m_{2}}B′_{0} ⊂ B′_{0}; obviously m := m_{1} + m_{2} will do. It is now clear that f′ and g_{A} are both f-adic ring homomorphisms for this topology 𝒯_{A′} on A′. Lastly, let (C, 𝒯_{C}) be any topological ring whose topology is ℤ-linear, h : B′ → C and k : A → C two continuous ring homomorphisms such that k ∘ f = h ∘ g; there follows a unique ring homomorphism l : A′ → C such that l ∘ i = h and l ∘ g_{A} = k. To conclude, we have to check that l is continuous, and it suffices to show that the restriction \overline{l}_{0} : \overline{A}′_{0} → C is continuous. But the topology of \overline{A}′_{0} agrees with the quotient topology induced from the I_{0}A′_{0}-adic topology 𝒯_{A′_{0}} on A′_{0}, so we reduced to checking that the induced map l_{0} : (A′_{0}, 𝒯_{A′_{0}}) → C is continuous in a neighborhood of 0. Thus, let U be a neighborhood of 0 in C; since 𝒯_{C} is ℤ-linear, we may assume that U is a subgroup of C. Moreover, there exists an open neighborhood U′ of 0 in C such that U′ · U′ ⊂ U. Then we may find n ∈ ℕ such that k(I_{0}^{n}A_{0}), h(I_{0}^{n}B′_{0}) ⊂ U′, so that l_{0}(I_{0}^{2n}A′_{0}) ⊂ U, whence the contention.

(ii): Indeed, if f is open, we may replace A_{0} by f(B_{0}) in the foregoing, in which case the induced map B′_{0} → A′_{0} is surjective and open; since the same holds for the projection A′_{0} → \overline{A}′_{0}, we see that f′ is open. Next, let C be any topological ring, and h, k two continuous ring homomorphisms as in the foregoing. We have to show that the induced map l : A′ → C is continuous, and we are again reduced to showing that the same holds for its restriction \overline{l}_{0} : \overline{A}′_{0} → C. But we have just seen that the topology of \overline{A}′_{0} agrees with the quotient topology induced by the surjective map f′_{0} : B′_{0} → \overline{A}′ (the restriction of f′); thus \overline{l}_{0} is continuous if and only if the same holds for \overline{l}_{0} ∘ f′_{0}, and the latter is none else than the restriction of h.

(iii): Indeed, under these assumptions, we get a cocartesian diagram of rings

\xymatrix{B′ \ar[rr]^{f′} \ar[d]_{j} &  & A′ \ar[d] \\ B^{′∧} \ar[r]^{g^{∧−1}} & B^{∧} \ar[r] & B^{∧} ⊗_{B′} A′ \ar[r]^{∼} & B^{∧} ⊗_{B} A \ar[r]^{∼} & A^{∧}}

where j is the completion map, and the righmost isomorphism on the bottom row is provided by proposition 8.3.33(iii). Since the natural map B^{∧} → A^{∧} is injective, it follows especially that Ker f′ ⊂ Ker j, and therefore f′ induce an isomorphism B′^{∧} →^{∼} \overline{A}_{0}^{′∧} on separated completions.

Invoking again proposition 8.3.33(iii) we deduce a commutative diagram

\xymatrix{B^{∧} ⊗_{B} A \ar[r] \ar[d] & B^{′∧} ⊗_{B′} A′ \ar[r] & \overline{A}_{0}^{′∧} ⊗\overline{A}′_{0} A′ \ar[d] \\ A^{∧} \ar[rr]^{g_{A}^{∧}} &  & A^{′∧}}

whose top horizontal and vertical arrows are isomorphisms, so the same holds for the bottom horizontal arrow. Set \overline{A} := Im g_{A}, and endow \overline{A} with the quotient topology induced by A via the surjection A → \overline{A}. Recall that the topology of A (resp. of A′) agrees with the topology induced by A^{∧} (resp. by A^{′∧}) via the completion map A → A^{∧} (resp. A′ → A^{′∧}); since we already know that g_{A}^{∧} is an isomorphism of topological rings, it follows easily that the topology of \overline{A} agrees with the topology induced by A′ via the inclusion map \overline{A} → A′ (details left to the reader). Moreover, recall that g_{A} is f-adic, so lemma 8.3.24(iii) implies that the image of B′ ⊗_{B} A^{∘} lies in A^{′∘}; for the converse, say that x ∈ A^{′∘}, and write x = ∑_{i=1}^{n} b_{i} ⊗ a_{i} for some n ∈ ℕ and elements a_{i} ∈ A, b_{i} ∈ B′, for i = 1, …, n. Let I ⊂ B be any ideal of definition; we may find m ∈ ℕ such that a_{i}I^{m} ⊂ A^{∘∘} for every i = 1, …, n, and since g^{∧} is an isomorphism, we may also find b′_{i} ∈ B, b′′_{i} ∈ I^{m}B′ such that b_{i} = g(b′_{i}) + b′′_{i} for i = 1, …, n. Clearly b′′_{i} ⊗ a_{i} lies in the image of B′ ⊗_{B} A^{∘∘} for every i = 1, …, n, so y := 1 ⊗ (∑_{i=1}^{n} a_{i}b′_{i}) ∈ \overline{A} ∩ A^{′∘} ⊂ \overline{A}^{∘}, and it remains only to observe that \overline{A}^{∘} = g_{A}(A^{∘}), because the kernel of the surjection A → \overline{A} is contained in the kernel of the completion map A → A^{∧} (details left to the reader). Likewise, if x ∈ A^{′∘∘}, then y ∈ \overline{A} ∩ A^{′∘∘} = \overline{A}^{∘∘} = g(A^{∘∘}), which shows that A′^{∘∘} lies in the image of B′ ⊗_{B} A^{∘∘}, and the converse inclusion is obvious. □

8.3.36. Let A be any topological ring; for any two subsets S, T ⊂ A we denote by S · T the additive subgroup of A generated by {st | s ∈ S, t ∈ T }. We define inductively

T^{0} := {1}  and T^{n+1} := T · T^{n}  for every n ∈ ℕ.

Now, let (T_{λ} | λ ∈ Λ) be any (small) family of subsets of A; then, for every ν ∈ ℕ^{(Λ)} the additive subgoup

T^{ν} := ∏_{λ∈Λ} T_{λ}^{ν(λ)}

is well defined. We consider the polynomial A-algebra

A[X_{•}] := A[X_{λ} | λ ∈ Λ]

and we also set X^{ν} := ∏_{λ∈Λ} X_{λ}^{ν(λ)} for every such ν. Furthermore, let 𝒰_{A} be the set of all open neighborhood U of 0 in A, and set

U[T_{•}X_{•}] := { ∑_{ν∈ℕ^{(Λ)}} a_{ν}X^{ν} ∈ A[X_{•}] | a_{ν} ∈ T^{ν} · U for every ν ∈ ℕ^{(Λ)}} for every U ∈ 𝒰_{A} .

Proposition 8.3.37. In the situation of (8.3.36), suppose that A is an f-adic ring, and T_{λ} · A is an open ideal of A for every λ ∈ Λ, and let A_{0} ⊂ A be a subring of definition. Then we have :

(i) There exists a unique ring topology 𝒯_{A,T_{•}} on A[X_{•}] for which the family

𝒰_{A[X_{•}]} := (U[T_{•}X_{•}] | U ∈ 𝒰_{A})

is a fundamental system of open neighborhoods of 0.

(ii) The topological ring

A[X_{•}]_{T} := (A[X_{•}], 𝒯_{A,T_{•}})

is f-adic, and A_{0}[T_{•}X_{•}] is a subring of definition of A[X_{•}]. Moreover, if I ⊂ A_{0} is an ideal of adic definition, then I[T_{•}X_{•}] is an ideal of adic definition for A_{0}[T_{•}X_{•}].

(iii) The structure map h : A → A[X_{•}] is f-adic for the topology 𝒯_{A,T_{•}}, and the subset {tX_{λ} | λ ∈ Λ, t ∈ T_{λ}} is power bounded in A[X_{•}].

(iv) Moreover, for every f-adic ring B, every family (b_{λ} | λ ∈ Λ) of elements of B, and every continuous ring homomorphism f : A → B such that F := {f(t)·b_{λ} | λ ∈ Λ, t ∈ T_{λ}} is power bounded in B, there exists a unique continuous ring homomorphism g : (A[X_{•}], 𝒯_{A,T_{•}}) → B with g ∘ h = f and g(X_{λ}) = b_{λ} for every λ ∈ Λ.

Proof. (i): Notice that every element of 𝒰_{A[X_{•}]} is an additive subgroup of A[X_{•}]; it follows easily that there exists a unique group topology on A[X_{•}] that admits 𝒰_{A[X_{•}]} as a fundamental system of neighborhoods of 0. Moreover, for U ∈ 𝒰_{A} we may find V ∈ 𝒰_{A} with V · V ⊂ U, whence V [T_{•}X_{•}] · V [T_{•}X_{•}] ⊂ U[T_{•}X_{•}]. Therefore, it remains only to check that for every P ∈ A[X_{•}] and every U ∈ 𝒰_{A} there exists V ∈ 𝒰_{A} such that

P · V [T_{•}X_{•}] ⊂ U[T_{•}X_{•}].

To this aim, notice first that, since A is f-adic and T_{λ} · A is open in A for every λ ∈ Λ, then T^{ν} · U ∈ 𝒰_{A} for every ν ∈ ℕ^{(Λ)} and every U ∈ 𝒰_{A} (details left to the reader). Say that P = ∑_{ν∈ℕ^{(Λ)}} a_{ν}X^{ν}; since a_{ν} = 0 except for finitely many ν ∈ ℕ^{(Λ)}, we may find V ∈ 𝒰_{A} small enough that a_{ν} · V ∈ T^{ν} · U for every ν ∈ ℕ^{(Λ)}. It is easily seen that such V will do.

(ii): From the foregoing, it is clear that A_{0}[T_{•}X_{•}] is an open subring of A[X_{•}] and I[T_{•}X_{•}] is an ideal of A_{0}[T_{•}X_{•}]. Moreover, I^{n}[T_{•}X_{•}] = I[T_{•}X_{•}]^{n} for every n ∈ ℕ, so the topology 𝒯_{A,T_{•}} induces the I[T_{•}X_{•}]-adic topology on A_{0}[T_{•}X_{•}].

(iii): The continuity of h is obvious; moreover, h restricts to a map A_{0} → A_{0}[T_{•}X_{•}], and I · A_{0}[T_{•}X_{•}] = I[T_{•}X_{•}], so h is f-adic. Lastly, we have tX_{λ} ∈ A_{0}[T_{•}X_{•}] for every λ ∈ Λ and t ∈ T_{λ}, whence the assertion.

(iv): Obviously there exists a unique map of A-algebras g : A[X_{•}] → B such that g(X_{λ}) = b_{λ} for every λ ∈ Λ, and it suffices to check that g is continuous. To this aim, for any open additive subgroup U of B pick V ∈ 𝒰_{B} such that F^{n} · V ⊂ U for every n ∈ ℕ. Set also W := f^{−1}V; then g(W [T_{•}X_{•}]) ⊂ U, whence the contention. □

8.3.38. Modules of finite type over topological rings. Let (A, 𝒯_{A}) be a topological ring, and M any A-module of finite type. We choose any surjective A-linear map π : A^{⊕n} → M (for some n ∈ ℕ) and we endow A^{⊕n} := A × ⋯ × A with the product of the topologies of its direct factors, and M with the topology 𝒯_{M}^{A} induced by A^{⊕n} via the map π.

Lemma 8.3.39. With the notation of (8.3.38), the following holds :

(i) The topology 𝒯_{M}^{A} is independent of the choice of π.

(ii) π : (A, 𝒯_{A})^{n} → (M, 𝒯_{M}^{A}) is an open map.

Proof. (i): Let π′ : A^{⊕m} → M be any other A-linear surjection; we define π′′ : A^{⊕n+m} = A^{⊕n} ⊕ A^{⊕m} → M as the unique A-linear map whose restriction to A^{⊕n} ⊕ 0 agrees with π and whose restriction to 0 ⊕ A^{⊕m} agrees with π′. It suffices to check that the topologies on M induced by A^{⊕n} and A^{⊕m} via π and respectively π′ agree with the topology 𝒯 ′′ induced by A^{⊕n+m} via π′′. Thus, we are reduced to comparing 𝒯_{M}^{A} and 𝒯 ′′. Now, let e_{1}, …, e_{n+m} be the standard basis of A^{⊕n+m}, and for each i = 1, …, m choose f_{i} ∈ A^{⊕n} ⊕ 0 such that π′′(f_{i}) = π′′(e_{i+n}); we consider the A-linear automorphism

ω : A^{⊕n+m} →^{∼} A^{⊕n+m} e_{j} ↦ \{ \begin{matrix}e_{j} & for j = 1, …, n \\ e_{j} − f_{j−n} & for j = n + 1, …, n + m. \end{matrix}

Clearly ω : (A, 𝒯_{A})^{n+m} →^{∼} (A, 𝒯_{A})^{n+m} is a homeomorphism, and the restriction of π′′ ∘ ω to A^{⊕n} ⊕ 0 still agrees with π. We may then replace π′′ by π′′ ∘ ω and assume from start that 0 ⊕ A^{⊕m} ⊂ Ker π′′. Now, let U ⊂ M be any subset; we have U ∈ 𝒯_{M}^{A} if and only if π^{−1}U is open in (A, 𝒯_{A})^{n}, and the latter holds if and only if (π^{−1}U) × A^{⊕m} is open in (A, 𝒯_{A})^{n+m}, which in turn is equivalent to U ∈ 𝒯 ′′, whence the contention.

(ii): Let U ⊂ A^{⊕n} be any open subset and set U′ := U+Ker π := {u+v | u ∈ U, v ∈ Ker π}; then U′ is open in A^{⊕n} and π^{−1}(πU) = U′, whence the contention. □

Definition 8.3.40. In the situation of (8.3.38), we call 𝒯_{M}^{A} the canonical topology of the Amodule M of finite type.

Proposition 8.3.41. Let (A, 𝒯_{A}) be a topological ring, M_{1}, …, M_{k} a finite sequence of Amodules of finite type, (N, 𝒯_{N}) a topological A-module and β : M_{1} × ⋯ × M_{k} → N any A-multilinear map. Then we have :

(i) The canonical topology of the A-module M_{1} × ⋯ × M_{k} is the product of the canonical topologies 𝒯_{M_{1}}^{A}, …, 𝒯_{M_{k}}^{A}.

(ii) β is a continuous map (M_{1}, 𝒯_{M_{1}}^{A}) × ⋯ × (M_{k}, 𝒯_{M_{K}}^{A}) → (N, 𝒯_{N}).

Proof. (i): By a simple induction on k, we are easily reduced to the case where k = 2. Let then 𝒯 ′ be the product of the topologies 𝒯_{M_{1}}^{A} and 𝒯_{M_{2}}; pick A-linear surjections π_{i} : A^{⊕n_{i}} → M_{i} for i = 1, 2. It is easily seen that if U ∈ 𝒯 ′, then (π_{1} × π_{2})^{−1}U is open in (A, 𝒯_{A})^{n_{1}+n_{2}}, hence U ∈ 𝒯_{M_{1}×M_{2}}^{A} (details left to the reader). Conversely, if U ∈ 𝒯_{M_{1}×M_{2}}^{A}, then (π_{1} × π_{2})^{−1}U is a union of subsets of the form V_{1} × V_{2}, where V_{i} ⊂ A^{⊕n_{1}} is an arbitrary open subset, for i = 1, 2; therefore (π_{1} × π_{2})(V_{1} × V_{2}) = (π_{1}V_{1}) × (π_{2}V_{2}) ∈ 𝒯 ′′ since π_{1} and π_{2} are open maps (lemma 8.3.39(ii)), and so U ∈ 𝒯 ′′.

(ii): Pick A-linear surjections π_{i} : A^{⊕n_{i}} → M_{i} for i = 1, …, k, and set n := ∑_{i=1}^{k} n_{i} and β′ := β ∘ (π_{1} × ⋯ × π_{k}). It is easily seen that β′ : (A, 𝒯_{A})^{n} → (N, 𝒯_{N}) is a continuous map (details left to the reader). Now, let U ⊂ N be any open subset; we have β^{−1}(U) = (π_{1} × ⋯ × π_{k})(β^{′−1}U), which is open in M_{1} × ⋯ × M_{k}, since π_{1} × ⋯ × π_{k} is an open map (lemma 8.3.39(ii)). □

Corollary 8.3.42. Let (A, 𝒯_{A}) be any topological ring, B → A a ring homomorphism, I ⊂ A an ideal, M and N two A-modules of finite type. The following holds :

(i) (M, 𝒯_{M}^{A}) is a topological A-module.

(ii) Every A-linear surjection M → N is a continuous open map (M, 𝒯_{M}^{A}) → (N, 𝒯_{N}^{A}). (iii) If the topology 𝒯_{A} is B-linear (resp. I-adic), the same holds for 𝒯_{M}^{A}.

Proof. Pick an A-linear surjection π : A^{⊕n} → M.

(i): Let σ_{M} : M ⊕ M → M and μ_{M} : A × M → M be respectively the addition law and the scalar multiplication law of M; we need to check that these maps are continuous for the product topologies (M, 𝒯_{M}^{A}) × (M, 𝒯_{M}^{A}) and (A, 𝒯_{A}) × (M, 𝒯_{M}^{A}). However, we have the commutative diagrams

\xymatrix{A^{⊕n} ⊕ A^{⊕n} \ar[r]^{σ_{A^{n}}} \ar[d]_{π⊕π} & A^{⊕n} \ar[d]^{π} & A × A^{⊕n} \ar[r]^{μ_{A^{n}}} \ar[d]_{1_{A}×π} & A^{⊕n} \ar[d]^{π} \\ M ⊕ M \ar[r]^{σ_{M}} & M & A × M \ar[r]^{μ_{M}} & M}

where σ_{A^{n}} and μ_{A^{n}} are the addition law and the scalar multiplication law of A^{⊕n}, which are clearly continuous for the product topologies. Now, if U ∈ 𝒯_{M}^{A}, then V := π^{−1}U is open in (A, 𝒯_{A})^{n} and W := σ_{A^{n}}^{−1}(V) is open in A^{⊕n} ⊕ A^{⊕n}; hence σ_{M}^{−1}(U) = (π ⊕ π)(W) is open in M ⊕ M, since π ⊕ π is open (lemma 8.3.39(ii)). Likewise one checks the continuity of μ_{M}.

(ii): The continuity of f : M → N follows from (i) and proposition 8.3.41(ii). Next, we easily reduce to checking that the map f ∘ π : (A, 𝒯_{A})^{n} → (N, 𝒯_{N}^{A}) is open; but this map is surjective, so the assertion follows from lemma 8.3.39(i).

(iii): Suppose that (J_{λ} | λ ∈ Λ) is a fundamental system of open neighborhoods of 0 ∈ A consisting of B-submodules; since π is open, the family (π(J_{λ}^{⊕n}) | λ ∈ Λ) is a fundamental system of open neighborhoods of 0 ∈ M (details left to the reader); in view of (i), the assertion follows. Likewise, if 𝒯_{A} is the I-adic topology, then (I^{k}A^{⊕n} | k ∈ ℕ) is a fundamental system of open neighborhoods of 0 ∈ A^{⊕n}, so (I^{k}M | k ∈ ℕ) is a fundamental system of open neighborhoods of 0 ∈ M, i.e. 𝒯_{M}^{A} is the I-adic topology of M. □

Remark 8.3.43. (i) Let (A, 𝒯_{A}) be a topological ring and B any finite A-algebra. In light of proposition 8.3.41(ii) and corollary 8.3.42(i), it is easily seen that (B, 𝒯_{B}^{A}) is a topological Aalgebra; moreover, if 𝒯_{A} is the I-adic topology (for some ideal I ⊂ A), then 𝒯_{B}^{A} is the IB-adic topology (corollary 8.3.42(ii)).

(ii) Let N be any finite B-module. On the one hand, we may endow N with its canonical topology 𝒯_{N}^{B} as B-module of finite type (that is, relative to the topological ring (B, 𝒯_{B}^{A})); on the other hand, by restriction of scalars, N is also an A-module of finite type, so we have as well the canonical topology 𝒯_{N}^{A}. We claim that these topologies coincide. Indeed, pick an A-linear surjection π : A^{⊕n} → B and a B-linear surjection π′ : B^{⊕m} → N; there follows an A-linear surjection π^{⊕m} : A^{⊕nm} → B^{⊕m} and by virtue of proposition 8.3.41(i) the product topology of B^{⊕m} = (B, 𝒯_{B}^{A}) × ⋯ × (B, 𝒯_{B}^{A}) agrees with the topology induced by A^{⊕nm} via π^{⊕m}. Then the topology 𝒯_{N}^{B} is also induced by π′ ∘ π^{⊕m}, whence the contention.

(iii) In the situation of (i), suppose that the topology 𝒯_{A} is f-adic; let A_{0} ⊂ A be a subring of definition, I_{0} ⊂ A_{0} a finitely generated ideal of adic definition and (x_{λ} | λ ∈ Λ) a finite system of generators of I_{0}. Say that B = ∑_{i=1}^{n} Ab_{i} for some elements b_{1}, …, b_{n} ∈ B with b_{1} := 1, and let π : A^{⊕n} → B be the A-linear surjection such that π(a_{1}, …, a_{n}) = ∑_{i=1}^{n} a_{i}b_{i} for every a_{1}, …, a_{n} ∈ A. We have a system (c_{ijk} | i, j, k = 1, …, n) of elements of A such that

b_{i}b_{j} = ∑_{k=1}^{n} c_{ijk}b_{k} for every i, j = 1, …, n.

After replacing I_{0} by I_{0}^{k} for k ∈ ℕ large enough, we may assume that I_{0}c_{ijk} ∈ A_{0} for every i, j, k = 1, …, n. Let B_{0} ⊂ B be the A_{0}-submodule generated by {1} ∪ {b_{i}x_{λ} | λ ∈ Λ, i = 2, …, n}. It is easily seen that B_{0} is a subring of B, and moreover J := π(I_{0}^{⊕n}) ⊂ B_{0}, so B_{0} is open in B (lemma 8.3.39(ii)). Furthermore, a simple calculation shows that J is an ideal of B_{0}, and clearly I_{0}B_{0} ⊂ J. Lastly, the family (π((I_{0}^{k+1})^{⊕n}) | k ∈ ℕ) is a fundamental system of neighborhoods of 0 ∈ B, and we have π((I_{0}^{k+1})^{⊕n}) = I_{0}^{k}J for every k ∈ ℕ, so the topology of B_{0} induced by the inclusion into (B, 𝒯_{B}^{A}) is I_{0}B_{0}-adic. We conclude that the topology 𝒯_{B}^{A} is f-adic, and the structure map A → B is f-adic and restricts to an adic finite ring homomorphism A_{0} → B_{0} between subrings of definition.

8.3.44. Linearly compact modules. Let A be any ring, and M any A-module; recall that a coset of M is a subset of the form x + N, where N ⊂ M is any submodule, and x ∈ M any element. If A is a topological ring, and if M is a topological A-module, we say that a coset x+N is open (resp closed) if it is an open (resp. closed) subset of M, equivalently, if N is an open (resp. closed) submodule. We say that a collection (x_{i} + N_{i} | i ∈ I) of cosets of M has the finite intersection property, if ⋂_{j∈J}(x_{j} + N_{j}) ≠ ∅ for every finite subset J ⊂ I.

Definition 8.3.45. Let A be a topological ring, M a topological A-module whose topology is A-linear and separated. We say that M is linearly compact if we have ⋂_{i∈I}(x_{i} + N_{i}) ≠ ∅ for every system (x_{i} + N_{i} | i ∈ I) of open cosets of M with the finite intersection property.

Remark 8.3.46. Let A be a topological ring, M a topological A-module whose topology is A-linear, and N ⊂ M a submodule that is closed for the topology of M.

(i) Then N is the intersection of the open submodules that contain it. Indeed, for every x ∈ M \ N there exists an open submodule N′ ⊂ M with x + N′ ∩ N = ∅, so that x does not lie in the open submodule N + N′, whence the assertion.

(ii) Suppose that M is linearly compact, and let (x_{i} + N_{i} | i ∈ I) be a family of closed cosets with the finite intersection property. Then it follows easily from (i) that ⋂_{i∈I}(x_{i} + N_{i}) ≠ ∅ : the details are left to the reader.

(iii) Endow N with the topology induced by M. Then, if M is linearly compact, the same holds for N. Indeed, if (x_{i} + N_{i} | i ∈ I) is a family of open cosets of N with the finite intersection property, then each x_{i} + N_{i} is also a closed coset in N, therefore in M as well, and one concludes with (ii).

(iv) Moreover, endow M/N with the topology induced by the projection p : M → M/N; then if M is linearly compact, the same holds for M/N. Indeed, clearly the topology of M/N is A-linear and separated, since N is closed in M, and if (x_{i} + Q_{i} | i ∈ I) is any family of open cosets of M/N with the finite intersection property, then each p^{−1}(x_{i} + Q_{i}) is an open coset of M, so that ⋂_{i∈I}(x_{i} + Q_{i}) = p(⋂_{i∈I} p^{−1}(x_{i} + Q_{i})) ≠ ∅.

(v) Let f : M → M′ be a continuous homomorphism of A-modules, and suppose that M is linearly compact, and that the topology of M′ is A-linear and separated. Then f(M) is a closed subset of M′. Indeed, let x be any element of the topological closure of f(M) in M′, and denote by 𝒰 the system of open submodules of M′; then f(M) ∩ (x + N′) ≠ ∅ for every N′ ∈ 𝒰 , so that the family of cosets (f^{−1}(x + N′) | N′ ∈ 𝒰 ) of M has the finite intersection property. By assumption, there exists y ∈ ⋂_{N′∈𝒰} f^{−1}(x + N′), and then x − f(y) ∈ ⋂_{N′∈𝒰} N′ = {0}, whence the assertion.

(vi) If M is linearly compact, the completion map j : M → M^{∧} is an isomorphism of topological modules. Indeed, it follows easily from corollary 8.2.17(iii) that the topology of M^{∧} is A-linear, hence the image of j is a closed submodule, by (v); but then j is bijective, since its image is dense in M^{∧}, and since M is separated. Then we conclude with theorem 8.2.8(ii).

(vii) If M is an artinian A-module, then M is linearly compact for its discrete topology. Indeed, let (x_{i} + N_{i} | i ∈ I) be any family of cosets of M with the finite intersection property; since M is artinian, there exists a finite subset J ⊂ I such that N_{J} := ⋂_{j∈J} N_{j} = ⋂_{i∈I} N_{i}, and notice that Q_{J} := ⋂_{j∈J}(x_{i} + N_{i}) = y + N_{J} for any y ∈ Q. Now, for every i ∈ I we have N_{J} ⊂ N_{i}, and since Q_{J} ∩ (x_{i} + N_{i}) ≠ ∅, we must have Q_{J} ⊂ x_{i} + N_{i}, so that ⋂_{i∈I}(x_{i} + N_{i}) = Q_{J} ≠ ∅.

Proposition 8.3.47. Let A be a topological ring, and (M_{λ} | λ ∈ Λ) a family of linearly compact topological A-modules. Then the topological product ∏_{λ∈Λ} M_{λ} is linearly compact.

Proof. It is easily seen that the topology of M := ∏_{λ∈Λ} M_{λ} is A-linear and separated. Hence, let 𝒰 be any set of open cosets of M with the finite intersection property; it remains to check that ⋂_{U∈𝒰} U ≠ ∅. To this aim, consider the set ℱ of all such sets; we endow ℱ with the partial ordering such that 𝒰 ≤ 𝒰′ if and only if 𝒰 ⊂ 𝒰′. Then, let also ℱ /𝒰 be the subset of all elements of ℱ larger than 𝒰 . If (𝒰_{i} | i ∈ I) is any totally ordered subset of ℱ /𝒰 , it is easily seen that ⋃_{i∈I} 𝒰_{i} lies also in ℱ /𝒰 ; by Zorn’s lemma, ℱ /𝒰 then admits a maximal element 𝒱 . Clearly it suffices to check that ⋂_{V ∈𝒱} V ≠ ∅, hence we may assume from start that 𝒰 is maximal in ℱ. Next, since the projection p_{λ} : M → M_{λ} is an open map, (p_{λ}(U) | U ∈ 𝒰 ) is a system of open cosets of M_{λ}, for every λ ∈ Λ, and obviously it has the finite intersection property, since the same holds for 𝒰 . Hence, for every λ ∈ Λ pick x_{λ} ∈ ⋂_{U∈𝒰} p_{λ}(U). Notice that p_{λ}(p_{λ}^{−1}(x_{λ} + N) ∩ U) = (x_{λ} + N) ∩ p_{λ}(U) for every λ ∈ Λ, every U ∈ 𝒰 , and every open submodule N ⊂ M_{λ}. Especially, p_{λ}^{−1}(x_{λ} + N) ∩ U ≠ ∅ for every such λ, U and N. Moreover, for every finite subset 𝒰′ ⊂ 𝒰 we have ⋂_{U∈𝒰′} U ∈ 𝒰 , by maximality of 𝒰 . Hence, for every λ, U and N as in the foregoing, the family 𝒰 ∪ {p_{λ}^{−1}(x_{λ} + N)} has the finite intersection property, so that it must coincide with 𝒰 , by maximality of 𝒰 . Especially, for every finite subset Λ′ ⊂ Λ, and every system (N_{λ} | λ ∈ Λ′) such that N_{λ} is an open submodule of M_{λ} for every λ ∈ Λ′, we have U ∩ ⋂_{λ∈Λ′} p_{λ}^{−1}(x_{λ} + N_{λ}) ≠ ∅ for every U ∈ 𝒰 . Since U is a closed subset of M, it follows that (x_{λ} | λ ∈ Λ) ∈ U for every U ∈ 𝒰 , as required. □

Part (ii) of the following corollary is already found in [106, Th.7.1].

Corollary 8.3.48. Let A be a topological ring, J any category, and F : J → A-TopMod any functor such that F j is a linearly compact A-module for every j ∈ Ob(J). We have :

(i) lim_{J} F is a linearly compact A-module.

(ii) Let also φ : A-TopMod → A-Mod be the forgetful functor, and suppose moreover that J is cofiltered. Then R^{p} lim_{J}(φ ∘ F) = 0 for every p > 0.

Proof. (i): Set P := ∏_{j∈Ob(J)} F j and Q := ∏_{φ∈Morph(J)} F ∘ t(φ), where t : Morph(J) → J is the target functor of (1.1.30). According to the proof of proposition 1.2.22(i), the limit L of F is represented by the kernel of a continuous map P → Q; since P and Q are both linearly compact (proposition 8.3.47), it follows that L is a closed submodule of P, and we conclude with remark 8.3.46(iii).

(ii): Set G := φ ∘ F. Let ℱ be the set of finite subsets S ⊂ Morph(J), and endow ℱ with the partial ordering induced by inclusion of finite subsets. For every S ∈ ℱ, let J_{S} be the smallest subcategory of J with J_{S} ⊂ Morph(J_{S}), and denote by G_{S} : J_{S} → A-Mod the restriction of F; we compute R lim_{J} G and R lim_{J_{S}} G_{S} by the canonical resolutions lim_{J} ▽G and lim_{J_{S}} ▽G_{S} exhibited in example 7.10.10(ii,iii) (with the notation of loc.cit., we take f : I → J to be the inclusion functor of the discrete subcategory I with Ob(I) = Ob(J)). Recall that lim_{J} ▽^{p}G = ∏_{β∈Fun([p],J)} G_{β(p)}; especially, lim_{J} ▽^{p}G is linearly compact for the product topology (proposition 8.3.47). Moreover, a simple inspection shows that the differential d^{p} : lim_{J} ▽^{p}G → lim_{J} ▽^{p+1}G is a continuous map, for every p ∈ ℕ. From these explicit descriptions, we also see that lim_{J_{S}} ▽G_{S} is a quotient of the complex lim_{J} ▽G, for every S ∈ ℱ, and we have :

(8.3.49)  lim_{J} ▽G = lim_{S∈ℱ} lim_{J_{S}} ▽G_{S}  in C(A-Mod).

For any finite subset T ⊂ Ob(J), any p > 0, and every j ∈ T let U_{j} ⊂ S(p, j) be a finite subset. Clearly there exists S ∈ ℱ such that every element of ⋃_{j∈T} U_{j} is a functor [p] → J_{S}.

Claim 8.3.50. There exists a category J_{S}^{∗} with an initial object e, and functors J_{S} ⟶^{t} J_{S}^{∗} ⟶^{u} J whose composition is the inclusion functor J_{S} → J.

Proof of the claim. Clearly Ob(J_{S}) is a finite set; then, since J is cofiltered, we can find j_{0} ∈ Ob(J) and a system of morphisms (f_{j} : j_{0} → j | j ∈ Ob(J_{S})) of J such that g∘f_{j} = h∘f_{j} for every j, j′ ∈ Ob(J_{S}) and every g, h ∈ Hom_{J}(j, j′) ∩ S. Let then J_{S}^{∗} be the unique category whose set of objects is the disjoint union Ob(J_{S})⊔{e}, such that e is the unique initial object of J_{S}^{∗}, and such that J_{S} is a fully faithful subcategory of J_{S}^{∗}. We let t be the inclusion functor, and we take u to be the unique functor such that u ∘ t is the inclusion functor, and with u(e) := j_{0}, and u(e → j) := f_{j} for every j ∈ Ob(J_{S}). ♢

With the notation of claim 8.3.50, let G_{S}^{∗} := G ∘ u : J_{S}^{∗} → A-Mod, and denote also by ▽G_{S}^{∗} the canonical resolution provided by example 7.10.10(ii). Then u and t induce maps of complexes

(8.3.51)  lim_{J} ▽G → lim_{J_{S}^{∗}} ▽G_{S}^{∗} → lim_{J_{S}} ▽G_{S}

whose composition is the natural projection. Moreover, lim_{J_{S}^{∗}} is isomorphic to the functor such that X ↦ X(e) for every X ∈ Ob(Fun(J_{S}^{∗}, A-Mod)), and in particular lim_{J_{S}^{∗}} is exact, so that R^{p} lim_{J_{S}^{∗}} = 0 for every p > 0. It follows that the projection lim_{J} ▽G → lim_{J_{S}} ▽G_{S} induces the zero map H^{p}(lim_{J} ▽G) → H^{p}(lim_{J_{S}} ▽G_{S}) for every p > 0. Thus, for every p > 0 and every x ∈ Ker d^{p}, the image of x in lim_{J_{S}} ▽^{p}G_{S} lies in the image of the corresponding differential d_{S}^{p−1} : lim_{J_{S}} ▽^{p−1}G_{S} → lim_{J_{S}} ▽^{p}G_{S}. In view of (8.3.49) and the surjectivity of the composed map (8.3.51), this implies that x lies in the topological closure of the image of d^{p−1}; but the image of d^{p−1} is closed (remark 8.3.46(v)), whence the assertion. □

Example 8.3.52. Let (A, 𝔪) be a local noetherian ring, complete and separated for its 𝔪-adic topology. Then every A-module M of finite type is linearly compact for its 𝔪-adic topology. Indeed, M is 𝔪-adically complete and separated ([126, Th.8.7]), so corollary 8.3.48(i) reduces to checking that M_{n} := M/𝔪^{n}N is linearly compact for every n ∈ ℕ. But M_{n} is an A/𝔪^{n}- module of finite type, and A/𝔪^{n} is artinian, so the assertion follows from remark 8.3.46(vii) and [126, Th.3.1(iii)].

Corollary 8.3.53. Let (A, 𝔪) be a local noetherian ring, complete and separated for its 𝔪-adic topology, J a filtered category, M^{•} ∈ Ob(C^{−}(Fun(J, A-Mod))), and N^{•} ∈ C^{+}(A-Mod). Suppose that M(j)^{p} is an A-module of finite type for every p ∈ ℤ and every j ∈ Ob(J), and that H^{q}(N) is an A-module of finite type for every q ∈ ℤ. Then the natural map

Ext_{A}^{n}(colim_{J} M^{•}, N^{•}) → lim_{J} Ext_{A}^{n}(M^{•}, N^{•})

is an isomorphism.

Proof. Since J is filtered, the functor Colim_{J} : Fun(J, A-Mod) → A-Mod is exact, so the natural transformation L Colim_{J} M^{•} → colim_{J} M^{•} is an isomorphism in D(A-Mod). By example 7.10.15(ii), we have therefore a spectral sequence

E_{2}^{pq} := R^{p} lim_{J} Ext_{A}^{q}(M^{•}, N^{•}) ⇒ Ext_{A}^{p+q}(colim_{J} M^{•}, N^{•})

and we are reduced to checking that E_{2}^{pq} = 0 for every p > 0. In light of example 8.3.52 and corollary 8.3.48(ii), it then suffices to check that Ext_{A}^{q}(M(j)^{•}, N^{•}) is an A-module of finite type, for every q ∈ ℕ and every j ∈ Ob(J). Now, since A is noetherian, we may find a quasiisomorphism P^{•} → M(j)^{•} in C^{−}(A-Mod), such that P^{n} is a free A-module of finite rank for every n ∈ ℤ; the double complex Hom_{A}(P^{•}, N^{•}) yields a spectral sequence

F_{2}^{pq} := Ext_{A}^{p}(P^{•}, H^{q}N^{•}) ⇒ Ext_{A}^{p+q}(M(j)^{•}, N^{•}).

Since the induced filtration on the abutment is finite, we are further reduced to checking that Ext_{A}^{p}(P^{•}, H^{q}N^{•}) is an A-module of finite type for every p, q ∈ ℤ, which is clear, since A is noetherian. □

Remark 8.3.54. With more work, one can see that corollary 8.3.53 still holds for every M^{•} ∈ Ob(C^{−}(Fun(J, A-Mod))) and every N^{•} ∈ C^{+}(A-Mod) (without finiteness assumptions on the terms M(j)^{p}). Indeed, for every j ∈ Ob(J) let ℱ_{j} be the set of all subcomplexes M^{′•} ⊂ M(j)^{•} such that M^{′p} is an A-module of finite type for every p ∈ ℤ. We let I be the category with Ob(I) = ⋃_{j∈Ob(J)} ℱ_{j} × {j}, and such that the morphisms (M^{′•}, j) → (M^{′′•}, k) are the commutative diagrams of C^{−}(A-Mod) :

𝒟(f^{•}, φ)  :  \xymatrix{M^{′•} \ar[r] \ar[d]_{f^{•}} & M(j)^{•} \ar[d]^{M(φ)^{•}} \\ M^{′′•} \ar[r] & M(k)^{•}}

whose horizontal arrows are the inclusion maps, for every (M^{′•}, j), (M^{′′•}, k) ∈ Ob(I) and every morphism φ : j → k of J, and with composition law such that 𝒟(f^{•}, φ) ∘ 𝒟(g^{•}, ψ) := 𝒟(f^{•} ∘ g^{•}, φ ∘ ψ) for any composable pair of morphisms of I. We have a functor π : I → J such that (M^{′•}, j) ↦ j for every (M^{′•}, j) ∈ Ob(I), and 𝒟(f^{•}, φ) ↦ φ for every morphism 𝒟(f^{•}, φ) of I. Moreover, we have a functor F : I → C^{−}(A-Mod) such that F (M^{′•}, j) := M^{′•} for every (M^{′•}, j) ∈ Ob(I) and F (𝒟(f^{•}, φ)) := f^{•} for every morphism 𝒟(f^{•}, φ) of I. It is easily seen that I is filtered, and π^{o} : I^{o} → J^{o} is a fibration (details left to the reader).

Furthermore, for every j ∈ Ob(J) let ι_{j} : π^{−1}(j) → I be the inclusion functor; it is easily seen that colim_{π^{−1}(j)} F ∘ ι_{j} = M(j)^{•}. In view of example 3.1.14(iv), we get a natural identification : ∫^{π} F = M^{•}, so that colim_{I} F = colim_{J} M^{•}. By corollary 8.3.53, we have natural isomorphisms for every n ∈ ℕ :

Ext_{A}^{n}(colim_{J} M^{•}, N^{•}) →^{∼} Ext_{A}^{n}(colim_{I} F, N^{•}) →^{∼} lim_{I} Ext_{A}^{n}(F, N^{•}) →^{∼} lim_{J} ∫_{π} Ext_{A}^{n}(F, N^{•}).

Lastly, notice that the category π^{−1}(j) is also filtered, for every j ∈ Ob(J); invoking again corollary 8.3.53together with example 3.1.14(iii), we deduce a natural isomorphism

∫_{π} Ext_{A}^{n}(F, N^{•}) →^{∼} Ext_{A}^{n}(M^{•}, N) for every n ∈ ℕ

whence the assertion.

8.4. Topologically local and topologically henselian rings. Localization and henselization are basic techniques in the study of commutative algebra; in this section we introduce suitable variants of these constructions for f-adic topological rings.

Definition 8.4.1. (i) Let \underline{A} := A be any f-adic ring.

(i) We say that A is topologically local if A^{∘∘} lies in the Jacobson radical of A^{∘}. (ii) We say that A is topologically henselian if (A^{∘}, A^{∘∘}) is a henselian pair.

Proposition 8.4.2. Let A be any f-adic ring, and B any open subring of A. We have : (i) A is topologically local if and only if the same holds for B. (ii) If A is topologically henselian, then A is also topologically local. (iii) A is topologically henselian if and only if the same holds for B. (iv) Suppose that A is topologically henselian, let C be any finite A-algebra, and endow C

with its canonical A-module topology 𝒯_{C}^{A} (see remark 8.3.43(i)). Then (C, 𝒯_{C}^{A}) is a topologically henselian f-adic ring.

Proof. (i): Indeed, suppose first that B is topologically local, and let a ∈ A^{∘∘} be any element; we may find n ∈ ℕ such that a^{n+1} ∈ B^{∘∘}, therefore 1 − a^{n+1} ∈ B^{∘×} and

(1 − a)^{−1} = (1 + a + ⋯ + a^{n})/(1 − a^{n+1}) ∈ A^{∘}

so a lies in the Jacobson radical of A^{∘}. Conversely, suppose that A is topologically local, and let b ∈ B^{∘∘} be any element; then 1−b ∈ A^{∘×}, and we may find n ∈ ℕ such that b^{n+1}/(1−b) ∈ B^{∘}. Therefore

(1 − b)^{−1} = b^{n+1}/(1 − b) + 1 + b + ⋯ + b^{n} ∈ B^{∘}

so that b lies in the Jacobson radical of B^{∘}.

(ii): Quite generally, if (R, I) is a henselian pair, then I lies in the Jacobson radical of R (details left to the reader).

(iii): Indeed, suppose first that B is topologically henselian; then B is topologically local, by (ii), and therefore the same holds for A, by (i); in view of [75, Rem.5.1.10(ii)] we then see that A is topologically henselian if and only if the same holds for the open subring C := B[A^{∘∘}] (which is f-adic, by corollary 8.3.20(i)). However, it is easily seen that the inclusion map B^{∘} → C^{∘} is integral, and moreover the ideal A^{∘∘} of C^{∘} is the radical of B^{∘∘} · C^{∘} (details left to the reader). Then the assertion follows from [75, Rem.5.1.10(i,v)].

In order to prove the converse, consider – for every scheme X – the set oc(X) of all open and closed subsets of X; clearly any morphism of schemes φ : Y → X induces a mapping

oc(φ) : oc(X) → oc(Y) Z ↦ φ^{−1}Z.

With this notation we have, quite generally :

Claim 8.4.3. Let φ : Y → X be any closed (resp. open) and surjective morphism of schemes, and p_{1}, p_{2} : Y ×_{X} Y → Y the induced projections. Then the diagram of sets

\xymatrix{oc(X) \ar[r]^{oc(φ)} & oc(Y) \ar[r]^{oc(p_{1})} \ar[r]_{oc(p_{2})} & oc(Y ×_{X} Y)}

identifies oc(X) with the equalizer of the mappings oc(p_{1}) and oc(p_{2}).

Proof of the claim. Since φ is surjective, oc(φ) is injective, and obviously its image lies in the equalizer of oc(p_{1}) and oc(p_{2}). Conversely, let Z be an open and closed subset of Y such that p_{1}^{−1}Z = p_{2}^{−1}Z. Since φ is surjective, we have Z ⊂ φ^{−1}φ(Z), and we claim that in fact Z = φ^{−1}φ(Z). Indeed, let y ∈ φ^{−1}φ(Z) be any point; then there exists z ∈ Z such that φ(y) = φ(z), and we may find w ∈ Y ×_{X} Y such that p_{1}(w) = z and p_{2}(w) = y. The condition on Z then implies that y ∈ Z as well, as required. Next, set Z′ := Y \ Z, and notice that Z′ lies as well in the equalizer of oc(p_{1}) and oc(p_{2}), so we have also Z′ = φ^{−1}φ(Z′); it follows that φ(Z) ∩ φ(Z′) = ∅, and since φ is a closed (resp. open) mapping, we conclude that φ(Z) is an open and closed subset of X whose preimage in Y equals Z, whence the claim. ♢

Now, if A is topologically henselian, the foregoing shows that the same holds for C, and taking into account lemma 8.3.29(iii), to conclude the proof of (iii) it suffices to remark :

Claim 8.4.4. Let f : R → S be an integral ring homomorphism, I ⊂ R any ideal. Suppose that the image of the induced morphism of schemes φ : Spec S → Spec R contains Spec R \ Spec R/I. Then we have :

(i) I lies in the Jacobson radical of R if and only if IS lies in the Jacobson radical of S. (ii) The pair (R, I) is henselian if and only if the same holds for the pair (S, IS).

Proof of the claim. Set S′ := S × (R/I) and let π : R → R/I be the projection; the pair (f, π) determines a unique integral ring homomorphism f′ : R → S′, and it is easily seen that the pair (S, IS) is henselian if and only if the same holds for the pair (S′, IS′). Likewise, obviously IS lies in the Jacobson radical of S if and only if IS′ lies in the Jacobson radical of S′. Moreover, Spec f′ is surjective; thus, we may replace S by S′, and assume from start that φ is surjective.

(i): Suppose first that I lies in the Jacobson radical of R, and let 𝔪 ⊂ S be any maximal ideal; then 𝔪′ := φ(𝔪) is a maximal ideal of R ([126, Th.9.4(i)]), therefore I ⊂ 𝔪′, and hence IS ⊂ 𝔪, so IS lies in the Jacobson radical of S. Conversely, suppose that the latter condition holds, and let 𝔪′ be any maximal ideal of R; by assumption, we may find a prime ideal 𝔪 of S with φ(𝔪) = 𝔪′, and then 𝔪 must be a maximal ideal, so IS ⊂ 𝔪, and finally I ⊂ 𝔪′.

(ii): If (R, I) is henselian, the same holds for (S, IS) ([75, Rem.5.1.10(v)]), so we may assume that (S, IS) is henselian, and we show that the same holds for (R, I). Indeed, let R → R′ be any finite ring homomorphism; we have to check that the induced map R′ → R′/IR′ restricts to a bijection oc(Spec R′) →^{∼} oc(Spec R′/IR′). However, set S′ := R′⊗_{R}S, and notice that the pair (S′, IS′) is henselian (again, by [75, Rem.5.1.10(v)]), and moreover the induced map Spec S′ → Spec R′ is surjective; thus, we may replace R by R′, and we reduce to showing that the projection R → R/I induces a bijection oc(Spec R) → oc(Spec R/I). However, φ is a universally closed morphism ([60, Ch.II, Prop.6.1.10]), so claim 8.4.3 further reduces to checking that the induced projections S → S_{0} := S/IS and S ⊗_{R} S → S_{0} ⊗_{R} S_{0} induce bijections

oc(Spec S) → oc(Spec S_{0}) oc(Spec S ⊗_{R} S) → oc(Spec S_{0} ⊗_{R} S_{0}).

In turns, this is clear, since (S, IS) and (S ⊗_{R} S, IS ⊗_{R} S) are both henselian pairs (again, by [75, Rem.5.1.10(v)]). ♢

(iv): Let A_{0} ⊂ A be a subring of definition, and I_{0} ⊂ A_{0} an ideal of adic definition; by remark 8.3.43(iii), we know that the topology 𝒯_{C}^{A} is f-adic, and C admits a subring C_{0} of definition such that the structure map A → C restricts to an adic finite ring homomorphism A_{0} → C_{0}. Then the pair (C_{0}, I_{0}C_{0}) is henselian (([75, Rem.5.1.10(v)]), and therefore (C, 𝒯_{C}^{A}) is topologically henselian, by (iii). □

Remark 8.4.5. If A is a topologically local f-adic ring, we have a homeomorphism

(8.4.6)  1 + A^{∘∘} →^{∼} 1 + A^{∘∘} : x ↦ x^{−1}.

Indeed, say that a ∈ A^{∘∘}; then there exists n ∈ ℕ such that

(8.4.7)  1/(1 − a) − (1 + a + ⋯ + a^{n}) = a^{n+1}/(1 − a) ∈ A^{∘∘}

and since A^{∘∘} is an ideal of A^{∘}, we deduce that 1/(1 − a) ∈ 1 + A^{∘∘}. It remains only to check that (8.4.6) is continuous on A^{∘∘}. However, if a, b ∈ A^{∘∘}, we may write

(1 − a)^{−1} − (1 − a − b)^{−1} = (1 − a)^{−1} · (1 − 1/(1 − (1 − a)^{−1}b))

so we are reduced to checking the continuity of (8.4.6) at the point x = 1, and the latter follows easily from (8.4.7). Notice that the same argument proves more precisely that

(1 + U)^{−1} = 1 + U

for every open additive subgroup U ⊂ A^{∘∘} such that U · U ⊂ U : details left to the reader.

8.4.8. Let A be any f-adic ring, B a ring of definition of A, and I a finitely generated ideal of adic definition of B. We let

B_{loc} := (1 + I)^{−1}B and A_{loc} := A ⊗_{B} B_{loc}

and endow B_{loc} with the unique topology such that the localization map B → B_{loc} is adic, and A_{loc} with the unique f-adic ring topology 𝒯_{loc} such that the inclusion map B_{loc} → A_{loc} is open and the induced map A → A_{loc} is f-adic (proposition 8.3.34(ii)). We claim that A_{loc} is topologically local. Indeed, suppose that a ∈ A_{loc}^{∘∘}; we may find n ∈ ℕ such that a^{n} ∈ IB_{loc}, so a^{n} = b/(1 + t) for some b, t ∈ I. Therefore, 1 − a^{n} = (1 + t − b)/(1 + t) ∈ B_{loc}^{×}, and arguing as in the proof of proposition 8.4.2(i) we deduce that 1 − a ∈ A_{loc}^{×}, whence the claim. It follows especially that A_{loc} is independent of the choice of I : indeed, suppose that J ⊂ I is another ideal of adic definition for B, and let A′ := (1 + J)^{−1}A; then there exist a unique map of A-algebras ρ′ : A′ → A_{loc}, and a natural isomorphism A_{loc} →^{∼} (1+I)^{−1}A′ of A-algebras that identifies ρ′ with the localization map; however, the foregoing shows that the image of 1 + I lies in A^{′×}, so ρ′ is bijective, and a simple inspection then shows that ρ′′ is also an isomorphism of topological rings. Lastly, let f : A → C be any continuous ring homomorphism of fadic rings, with C topologically local, and let C_{0} be a ring of definition of C, and J an ideal of adic definition for C_{0}; by the foregoing, we may replace I by a smaller open ideal, and assume that f(I) ⊂ J. Moreover C_{0} is topologically local, by proposition 8.4.2(i), whence f(1 + I) ⊂ 1 + J ⊂ C_{0}^{×}, so that f factors uniquely through the localization map A → A_{loc} and a ring homomorphism g : A_{loc} → C. Furthermore, g is a continuous map : indeed, g(IB_{loc}) = g((1 + I)^{−1}I) ⊂ (1 + J)^{−1}J ⊂ J, whence the assertion. Thus, the localization map ρ : A → A_{loc} is initial in the category of topologically local topological A-algebras with f-adic topologies; especially, the pair (A_{loc}, ρ) is determined up to unique isomorphism, and we call it the topological localization of the f-adic ring A.

8.4.9. For any topological ring R, denote by

R-TopAlg_{f−adic} and R-TopHens_{f−adic}

the full subcategories of R-TopAlg whose objects are respectively the f-adic topological Ralgebras, and the topologically henselian f-adic topological R-algebras (definition 8.3.1(iii)).

Now, let again A, B and I as in (8.4.8), denote by (B_{I}^{h}, I^{h}) the henselization of the pair (B, I), endow B_{I}^{h} with its I^{h}-adic topology, and set

A^{h} := A ⊗_{B} B_{I}^{h}.

We endow A^{h} with the unique topology 𝒯_{A}^{h} such that (A^{h}, 𝒯_{A}^{h}) is an f-adic topological ring, the natural map B_{I}^{h} → A^{h} is open and the natural map A → A^{h} is f-adic (proposition 8.3.34(ii)). We may then state :

Theorem 8.4.10. With the notation of (8.4.9), the following holds :

(i) (A^{h}, 𝒯_{A}^{h}) is a topologically henselian f-adic ring, independent of the choice of B and I, up to unique isomorphism of topological A-algebras.

(ii) The rule : A ↦ (A^{h}, 𝒯_{A}^{h}) extends to a functor

ℤ-TopAlg_{f−adic} → ℤ-TopHens_{f−adic}

that is left adjoint to the forgetful functor (here ℤ is endowed with its discrete topology).

Proof. Let us consider as well the full subcategories of ℤ-TopAlg denoted

ℤ-TopAlg_{adic} and ℤ-TopHens_{adic}

whose objects are respectively the adic topological rings, and the adic topological rings R such that the pair (R, R^{∘∘}) is henselian (notice that R^{∘∘} is an ideal of R). For every adic ring R, we let (R^{h}, (R^{∘∘})^{h}) the henselization of the pair (R, R^{∘∘}), and we endow R^{h} with the unique ring topology 𝒯_{R}^{h} such that the natural map R → R^{h} is adic. We notice :

Claim 8.4.11. The rule R ↦ (R^{h}, 𝒯_{R}^{h}) extends to a functor

ℤ-TopAlg_{adic} → ℤ-TopHens_{adic}

that is left adjoint to the forgetful functor.

Proof of the claim. More generally, let f : R → S and g : R → R′ be two ring homomorphisms, and I ⊂ R and J ⊂ S two ideals such that :

• f(I) ⊂ J

• J lies in the Jacobson radical ideal of S

• g is unramified, and g ⊗_{R} R/I is an isomorphism. Then we claim that there exists at most one ring homomorphism h : R′ → S such that h∘g = f. Indeed, let h, h′ : R′ → S be any two such maps; according to [66, Ch.IV, Prop.17.4.6], the maximal subscheme U ⊂ Spec S such that (Spec h)_{|U} = (Spec h′)_{|U} is open and closed in Spec S, and on the other hand, U contains Spec S/J, since g ⊗_{R} R/I is an isomorphism. But Spec S/J meets every open and closed subset of Spec S, since J lies in the Jacobson radical of S, whence the assertion. Next, suppose that the pair (S, J) is henselian, and g is étale; in this case, the map g ⊗_{R} S : S → R′ ⊗_{R} S admits a section s : R′ ⊗_{R} S → S ([66, Ch.IV, Prop.18.5.4]), and we set h := s ∘ (R′ ⊗_{R} f) : R′ → S. By the foregoing, h is the unique ring homomorphism such that h ∘ g = f. Now, let (R^{h}, I^{h}) be the henselization of the pair (R, I), and recall that R^{h} is the colimit of a filtered system (g_{λ} : R → R′_{λ} | λ ∈ Λ) of étale ring homomorphisms such that g_{λ} ⊗_{R} R/I is an isomorphism for every λ ∈ Λ. Summing up, we conclude that f factors uniquely through a morphism of R-algebras R^{h} → S. Lastly, if R and S are adic topological rings, then every continuous ring homomorphism g : R → S maps I := R^{∘∘} into J := S^{∘∘}, and if (S, S^{∘∘}) is henselian, the foregoing yields a unique ring homomorphism f^{h} : R^{h} → S^{h} extending f; to conclude, it suffices to check that f^{h} is continuous. However, if J′ is any ideal of adic definition for S, we may find an ideal I′ of adic definition for R such that f(I′) ⊂ J′; by construction, I′R^{h} is an ideal of adic definition of R^{h}, and f^{h}(I′R^{h}) ⊂ J′, whence the contention. ♢

(i): Let us check first that A^{h} is independent of the choice of B and I. Indeed, since B^{∘∘} is the radical of I, there exists a unique isomorphism φ : B_{I}^{h} →^{∼} B^{h} of B-algebras, where B^{h} is as in the foregoing; it is then clear that the I^{h}-adic topology on B^{h} agrees with the topology 𝒯_{B}^{h}, under the isomorphism φ. Let us endow A ⊗_{B} B^{h} with the unique ring topology such that the natural map B^{h} → A ⊗_{B} B^{h} is open (proposition 8.3.34(ii)); it follows that φ ⊗_{B} A : A^{h} →^{∼} A ⊗_{B} B^{h} is an isomorphism of topological A-algebras, which shows that A^{h} does not depend on I. By the same token, we also deduce that A^{h} is topologically henselian (proposition 8.4.2(iii)). Next, let C ⊂ A be another ring of definition, I_{C} ⊂ C an ideal of definition of C, and (C^{h}, I_{C}^{h}) the henselization of the pair (C, I_{C}); endow C^{h} with its I_{C}^{h}-adic topology, set A′ := A ⊗_{C} C^{h}, and endow again A′ with the unique ring topology such that the natural map C^{h} → A′ is open and the induced map A → A′ is f-adic. We need to show :

Claim 8.4.12. There exists a unique isomorphism of topological A-algebras A^{h} →^{∼} A′.

Proof of the claim. In light of corollary 8.3.19(i) we are easily reduced to the case where B ⊂ C, and since we have already established that the construction of C^{h} and B^{h} is independent of the ideals of definition, we may moreover assume that I_{C} = I (details left to the reader). In this situation, set D := B^{h}⊗_{B}C; the induced ring homomorphism B^{h} → D identifies I^{h} = B^{h}⊗_{B}I with the ideal ID of D, and therefore the pair (D, ID) is henselian ([75, Rem.5.1.10(ii)]). Let us endow D with its ID-adic topology; then D^{∘∘} is the radical of ID, and therefore D is topologically henselian ([75, Rem.5.1.10(i)]); moreover, the natural map C → D is continuous, so by claim 8.4.11 it factors uniquely through a continuous map C^{h} → D of C-algebras. We claim that the latter is an isomorphism of topological rings. Indeed, let E be any other object of ℤ-TopHens_{adic}; any continuous ring homomorphism B → E admits a unique continuous extension B^{h} → E, so the set of continuous ring homomorphisms D → E is in natural bijection with the set of continuous ring homomorphisms C → E, and the latter are as well in natural bijection with the set of continuous ring homomorphisms C^{h} → E, whence the claim (details left to the reader). There follows an isomorphism of A-algebras

ψ : A^{h} →^{∼} A ⊗_{C} D →^{∼} A′

and since the structure maps of these A-algebras are f-adic, it is clear that ψ is an isomorphism of topological rings. Furthermore, suppose that ψ′ : A^{h} →^{∼} A′ is another isomorphism of topological A-algebras, and set R := ψ(B^{h})·ψ′(B^{h}); then R is a ring of definition of A′ (corollary 8.3.19(i)), so it is an object of ℤ-TopHens_{adic} (proposition 8.4.2(iii)), and the restrictions B^{h} → R of ψ and ψ′ agree on B. By claim 8.4.11, it follows that ψ and ψ′ agree on B^{h}, and therefore they coincide, whence the contention. ♢

Lastly, let f : A → R be any continuous ring homomorphism from A to a topologically henselian f-adic topological ring, and pick any ring of definition R_{0} of R; after replacing B by B ∩ f^{−1}R_{0} we may assume that f restricts to a map f_{0} : B → R_{0} (proposition 8.3.18(ii)), and since R_{0} is topologically henselian (proposition 8.4.2(iii)), the map f_{0} factors uniquely through a continuous ring homomorphism f_{0}^{h} : B^{h} → E (claim 8.4.11). The datum of f and f_{0}^{h} determines a unique continuous map of A-algebras A^{h} → E (proposition 8.3.34(ii)), so the proof is concluded. □

Definition 8.4.13. Let A be any f-adic topological ring. The topological A-algebra (A^{h}, 𝒯_{A}^{h}) provided by theorem 8.4.10is called the topological henselization of A.

Remark 8.4.14. Our notions of topological henselian f-adic ring and of topological henselization already appear in [101, §3.1], where they are called respectively henselian f-adic ring and henselization of an f-adic ring.

Corollary 8.4.15. Let A be any f-adic topological ring, and denote by A_{loc}, A^{h}, A^{∧} respectively the topologically localization, the topological henselization, and the separated completion of A. The following holds :

(i) The localization map A → A_{loc} and the henselization map A → A^{h} induce isomorphisms of topological rings on separated completions :

A^{h∧} ⟵^{∼} A^{∧} →^{∼} A_{loc}^{∧}.

(ii) The inclusion maps A^{∘} → A and A^{∘∘} → A induce natural identifications :

(A^{∘})_{loc} →^{∼} (A_{loc})^{∘}  (A^{∘})^{h} →^{∼} (A^{h})^{∘}  (A^{∘})^{∧} →^{∼} (A^{∧})^{∘}
A^{∘∘} ⊗_{A^{∘}} A_{loc}^{∘} →^{∼} (A_{loc})^{∘∘}  A^{∘∘} ⊗_{A^{∘}} A^{∘h} →^{∼} (A^{h})^{∘∘}  (A^{∘∘})^{∧} →^{∼} (A^{∧})^{∘∘}.

(iii) Let R be any open subring of A^{∘}, and denote by (R^{h}, R^{∘∘h}) the henselization of the pair (R, R^{∘∘}). Then we have :

(a) There exists a unique ring topology 𝒯_{R}^{h} on R^{h} such that the natural map R → R^{h} is f-adic and (R^{h}, 𝒯_{R}^{h}) is isomorphic to the topological henselization of R.

(b) Endow A ⊗_{R} R^{h} with the unique f-adic topology such that the natural map R^{h} → A ⊗_{R} R^{h} is open (see proposition 8.3.34(ii)). There exists a unique isomorphism of topological A-algebras A^{h} →^{∼} A ⊗_{R} R^{h}.

Proof. (i): Let B be any ring of definition of A; a simple inspection of the construction in (8.4.8) shows that the natural map

A′ := A ⊗_{B} B_{loc} → A_{loc}

is an isomorphism of topological rings, for the f-adic topology on A′ described in proposition 8.3.34(ii). Moreover, the localization map B → B_{loc} clearly induces an isomorphism B^{∧} →^{∼} (B_{loc})^{∧}. Then the assertion for A_{loc}^{∧} follows from proposition 8.3.34(iii.a). Likewise, the natural map B → B^{h} induces an isomorphism on separated completions, so the same argument yields the assertion for A^{h∧}.

(ii): The assertions for (A_{loc})^{∘} and (A_{loc})^{∘∘} follow by the same token, from proposition 8.3.34(iii.b). Likewise, we get the assertions for (A^{h})^{∘} and (A^{h})^{∘∘}. In the case of the completion functor, we know that the natural map A ⊗_{B} B^{∧} → A^{∧} is an isomorphism of topological rings (proposition 8.3.33(iii)), so we may still apply proposition 8.3.34(iii) to deduce that (A^{∧})^{∘} (resp. (A^{∧})^{∘∘}) is the topological closure in A^{∧} of the image of A^{∘} (resp. of A^{∘∘}).

(iii): Suppose first that R = A; especially, A^{∘} = A. Then, according to corollary 8.3.19(iv), the ring A is the filtered union of the system (B_{λ} | λ ∈ Λ) of its subrings of definitions containing B, whence an induced isomorphism of A-algebras

A^{h} →^{∼} colim_{λ∈Λ} B_{λ} ⊗_{B} B^{h}

where B^{h} denotes the topological henselization of B. On the other hand, B is also a subring of definition of B_{λ}, and B_{λ}^{∘∘} = B_{λ} ∩ B^{∘∘} for every λ ∈ Λ. Let (B_{λ}^{h}, J_{λ}) be the henselization of the pair (B_{λ}, B_{λ}^{∘∘}); by theorem 8.4.10(i) we deduce a unique isomorphism

B_{λ}^{h} →^{∼} B_{λ} ⊗_{B} B^{h} for every λ ∈ Λ

of topological B_{λ}-algebras. Lastly, we have a unique isomorphism of A-algebras

colim_{λ∈Λ} B_{λ}^{h} →^{∼} R^{h}

whence the sought isomorphism φ : A^{h} →^{∼} R^{h} of A-algebras. The uniqueness of φ follows from the universal property of the henselization functor. If we may endow R^{h} with the topology induced from A^{h} via φ, clearly both (iii.a) and (iii.b) hold in this case.

Next, if R ⊂ A^{∘} is an arbitrary open subring, choose a ring of definition B of A contained in R; by the foregoing case, R^{h} is naturally isomorphic to the topological henselization of R, so that we have unique isomorphisms

R^{h} →^{∼} R ⊗_{A_{0}} A_{0}^{h} A^{h} →^{∼} A ⊗_{A_{0}} A_{0}^{h}

respectively of topological R-algebras and topological A-algebras, such that both natural maps A_{0}^{h} → R^{h} and A_{0}^{h} → A^{h} are open. There follows an isomorphism of topological A-algebras φ : A^{h} →^{∼} A ⊗_{R} R^{h} as sought. The uniqueness of φ follows again from the universal property of the topological henselization A^{h} of A. □

8.5. Graded structures on topological rings. Let Γ be a monoid, A := ⨁_{γ∈Γ} A_{γ} a Γ-graded ring, and 𝒯 a topology on A. The Γ-graded structure of A is not usually inherited by the completion (A, 𝒯 )^{∧} of (A, 𝒯 ). This observation motivates the following definition, that introduces a more flexible notion of graded structure on topological rings, preserved under completions and related operations.

Definition 8.5.1. Let (A, 𝒯 ) be a separated topological ring, Γ a monoid.

(i) A Γ-pre-graded structure on (A, 𝒯 ) is a datum (A, \underline{B}, Γ) consisting of a subring B ⊂ A with a Γ-graded ℤ-algebra structure \underline{B} on B (definition 7.6.1(i)) such that the following holds :

(a) The topology induced on B by 𝒯 agrees with the linear topology defined by a cofiltered system of graded ideals of B (see definition 7.6.1(iii)).

(b) B is a dense subset of A. (ii) A Γ-graded structure on (A, 𝒯 ) is a Γ-pre-graded structure (A, \underline{B}) such that gr_{γ}B is a

closed subset of A, for every γ ∈ Γ.

(iii) Let (A, \underline{B}, Γ) and (A′, \underline{B}′, Γ′) be two topological rings with pre-graded structures. A morphism of topological rings with pre-graded structures (A, \underline{B}) → (A′, \underline{B}′) is a pair (f, φ) consisting of a continuous ring homomorphism f : A → A′ and a morphism of monoids φ : Γ → Γ′ such that f(B) ⊂ B′, and such that the restriction f_{|B} : B → B′ induces a morphism of Γ-graded ℤ-algebras (notation of definition 7.6.1(iv))

\underline{B} → Γ ×_{Γ′} \underline{B}′.

Clearly, we get therefore a category of topological rings with pre-graded structures :

pre-gr.TopAlg.

We also have the full subcategory of pre-gr.TopAlg denoted

gr.TopAlg

whose objects are the topological rings with graded structures.

Remark 8.5.2. Let (Γ, +, 0) be a monoid, (A, \underline{B}) a Γ-pre-graded structure on the topological ring (A, 𝒯 ). Pick a fundamental system (J_{λ} | λ ∈ Λ) of graded open ideals of B, and endow gr_{γ}B with the topology 𝒯_{γ} induced by 𝒯 , for every γ ∈ Γ.

(i) Since J_{λ} = ⨁_{γ∈Γ} gr_{γ}J_{λ} for every λ ∈ Λ, the topology 𝒯_{γ} agrees with the linear topology defined by the cofiltered system of gr_{0}B-submodules (gr_{γ}J_{λ} | λ ∈ Λ), for every γ ∈ Γ. Moreover, a graded ideal I of B is closed in the topology of B if and only if gr_{γ}I is closed in gr_{γ}B for every γ ∈ Γ. Indeed, since I ∩ gr_{γ}B = gr_{γ}I for every such γ, the condition is clearly necessary. Conversely, notice that the natural projection π_{γ} : B → gr_{γ}B is continuous for every γ ∈ Γ, hence the same holds for the composition ρ_{γ} : B → gr_{γ}B/I of π_{γ} with the quotient map gr_{γ}B → gr_{γ}B/I; now, if gr_{γ}I is closed in gr_{γ}B, the topology of gr_{γ}B/I is separated, hence Ker ρ_{γ} is closed in B, and to conclude it suffices to remark that I is the intersection of such kernels, for γ ranging over all elements of Γ.

(ii) For every γ ∈ Γ, denote by (gr_{γ}B^{∧}, 𝒯_{γ}^{∧}) the completion of (gr_{γ}B, 𝒯_{γ}). Also, let gr_{γ}J_{λ}^{∧} be the topological closure of gr_{γ}J_{λ} in gr_{γ}B^{∧}, for every λ ∈ Λ and γ ∈ Γ, and set

J′_{λ} := ∏_{γ∈Γ} gr_{γ}J_{λ}^{∧}  B′ := ∏_{γ∈Γ} gr_{γ}B^{∧}  J′′_{λ} := ⨁_{γ∈Γ} gr_{γ}J_{λ}^{∧}  B′′ := ⨁_{γ∈Γ} gr_{γ}B^{∧}.

Notice that B′′ (resp. B′) is a ring (resp. a gr_{0}B-module), and J′′_{λ} (resp. J′_{λ}) is an ideal of B′′ (resp. a gr_{0}B-submodule of B′), for every λ ∈ Λ. We endow B′ with the linear topology defined by the cofiltered system of submodules (J′_{λ} | λ ∈ Λ). Notice that B′ is complete and separated, and the induced map

j_{B} : B → B′

is continuous; more precisely, the topology of B agrees with the topology induced by B′ via j_{B}. Moreover, for every γ ∈ Γ, let π′_{γ} : B′ → gr_{γ}B^{∧} be the projection; since J′_{λ} ⊂ π_{γ}^{′−1}(gr_{γ}J_{λ}) for every λ ∈ Λ, we see that π′_{γ} is continuous for the topology 𝒯_{γ}, for every γ ∈ Γ.

(iii) Next, since B is dense in A, the inclusion map B → A extends uniquely to an isomorphism between the completion (A^{∧}, 𝒯 ^{∧}) of (A, 𝒯 ) and the completion of B (theorem 8.2.8(iii)); it follows that j_{B} factors uniquely through a continuous map of topological gr_{0}Bmodules

j_{A} : A^{∧} → B′ a ↦ (a_{γ} | γ ∈ Γ)

and j_{A} is injective, since the topology of B is induced by that of B′ (proposition 8.2.13(i)). For every γ ∈ Γ we call π_{γ}^{∧} := π′_{γ} ∘ j_{A} : A^{∧} → gr_{γ}B^{∧} : a ↦ a_{γ} the canonical γ-projection and set

a_{S} := ∑_{γ∈S} a_{γ} for every a ∈ A^{∧} and every finite subset S ⊂ Γ.

(iv) For every finite subset S ⊂ Γ, let i_{S} : gr_{S}B := ⨁_{γ∈S} gr_{γ}B → A be the inclusion map; since J_{λ} is a graded ideal for every λ ∈ Λ, it is easily seen that the topology on gr_{S}B induced by A via i_{S} is the product topology ∏_{γ∈S} 𝒯_{γ}. Hence, i_{S} extends uniquely to a continuous map

∏_{γ∈S}(gr_{γ}B^{∧}, 𝒯_{γ}^{∧}) = (gr_{S}B)^{∧} → (A^{∧}, 𝒯 ^{∧})

which is still injective (proposition 8.2.13(i)), whence an injective map B′′ → A^{∧}. We endow the Γ-graded ring \underline{B}′′ := (B′′, gr_{•}B′′) with the topology induced from A^{∧} via this map, and set

(A, \underline{B})^{∧} := (A^{∧}, \underline{B}′′).

(v) Notice that, in case Γ ≠ 0, the separation condition on 𝒯 can be deduced from condition (b) in definition 8.5.1: indeed, if γ, γ′ are two distinct elements of Γ, we have

{0}^{c} ⊂ gr_{γ}B ∩ gr_{γ′}B = 0

whence the contention. Thus, the separation condition in definition 8.5.1only serves to rule out the somewhat trivial case of a non-separated topological space A endowed with the {0}-graded ℤ-algebra structure.

Proposition 8.5.3. Let (Γ, +, 0) be a monoid, (A, \underline{B}) a Γ-pre-graded structure on the topological ring (A, 𝒯 ). We have :

(i) The topology 𝒯 is linear.

(ii) In the situation of remark 8.5.2(iii), the system (a_{S} | S ⊂ Γ), with S ranging over the filtered set of finite subsets of Γ, is a Cauchy net in B′′, whose unique limit point is a.

(iii) In the situation of remark 8.5.2(iv), the datum (A, \underline{B})^{∧} is a Γ-graded structure on (A^{∧}, 𝒯 ^{∧}).

(iv) The inclusion functor gr.TopAlg → pre-gr.TopAlg admits a left adjoint

pre-gr.TopAlg → gr.TopAlg (A, \underline{B}, Γ) ↦ (A, \underline{B}, Γ)^{c}

that assigns to each topological ring with pre-graded structure its associated graded structure.

Proof. Fix a fundamental system (J_{λ} | λ ∈ Λ) of graded open ideals of B.

(i): Since A is separated, the completion map (A, 𝒯 ) → (A^{∧}, 𝒯 ^{∧}) is injective, B is naturally identified with a dense subring of A^{∧}, and the inclusion map B → A^{∧} extends to an isomorphism between (A^{∧}, 𝒯 ^{∧}) and the completion B^{∧} of B ([41, Ch.II, §3, n.7, Prop.13]). However, for every λ ∈ Λ, let J_{λ}^{c} be the topological closure of J_{λ} in B^{∧}; then (J_{λ}^{c} | λ ∈ Λ) is a fundamental system of open ideals in B^{∧} (remark 8.3.3(ii)), and lastly, (J_{λ}^{c} ∩ A | λ ∈ Λ) is a fundamental system of open ideals in A, whence the contention.

(iii): By remark 8.2.7(iv), the subset gr_{γ}B′′ is closed in the topology of A^{∧}, for every γ ∈ Γ. Moreover, B ⊂ B′′, so B′′ is dense in A^{∧}. In light of the proof of (i), it remains only to check that J_{λ}^{c} ∩ B′′ = J′′_{λ}, for every λ ∈ Λ. To this aim, say that x ∈ B′′ ∩ J_{λ}^{c} for some such λ; then there exists a finite subset S ⊂ Γ such that x = ∑_{γ∈S} x_{γ} with x_{γ} ∈ gr_{γ}B′′ for every γ ∈ S. It follows that π_{γ}^{∧}(x) = x_{γ} for γ ∈ S and π_{γ}^{∧}(x) = 0 otherwise. On the other hand, from claim 8.3.25we get

π_{γ}^{∧}(x) ∈ π_{γ}^{∧}(J_{λ}^{c}) ⊂ (π_{γ}(J_{λ}))^{c} = gr_{γ}J_{λ}^{∧}

and the contention follows.

(ii): In the light of the proof of (iii), we have to check that for every λ ∈ Λ there exists a finite subset S_{J} ⊂ Γ such that a_{S} − a_{S′} ∈ J′′_{λ} for every finite subsets S, S′ ⊂ Γ containing S_{J}. However, the proof of (i) shows that the topological closure J_{λ}^{c} of J_{λ} in A is an open ideal of A; since B is dense in A, it follows that there exists b ∈ B such that a − b ∈ J_{λ}^{c}. Write b = ∑_{γ∈T} b_{γ} for some finite subset T ⊂ Γ; in view of claim 8.3.25, we deduce that

π_{γ}^{∧}(b − a) = b_{γ} − a_{γ} ∈ π_{γ}^{∧}(J_{λ}^{c}) ⊂ π_{γ}(J)^{c} = gr_{γ}J_{λ}^{∧} for every γ ∈ Γ

(notation of remark 8.5.2(ii)). Especially, a_{γ} ∈ gr_{γ}J_{λ}^{∧} for every γ ∈ Γ \ T, and consequently π_{γ}^{∧}(a_{S} − a_{S′}) ∈ gr_{γ}J_{λ}^{∧} for every γ ∈ Γ and every S, S′ ⊂ Γ containing T; i.e. a_{S} − a_{S′} ∈ J′′_{λ} for every such S, S′, so that S_{J} := T will do. By the same token, a − a_{S} = (a − b) + (b − a_{S}) ∈ J_{λ}^{c} for every finite subset S ⊂ Γ containing T, so a is the unique limit of this Cauchy net.

(iv): For every γ ∈ Γ, let gr_{γ}B^{c} be the topological closure of gr_{γ}B in A, and set \underline{B}^{c} := ⊕_{γ∈Γ}gr_{γ}B^{c}; clearly \underline{B}^{c} ⊂ \underline{B}′′, and in light of (iii), it is clear that the pair (A, \underline{B}^{c}) is a topological ring with graded ring structure. Moreover, if (f, φ) : (A, \underline{B}, Γ) → (X, \underline{Y} , Γ′) is any morphism to a topological ring with graded structure, then claim 8.3.25implies easily that f maps \underline{B}^{c} into \underline{Y}, so we get the sought left adjoint by setting (A, \underline{B}, Γ)^{c} := (A, \underline{B}^{c}, Γ). □

Example 8.5.4. Let Γ be a monoid and (A, \underline{B}) a topological ring with Γ-graded structure. We have the following elementary operations :

(i) If I ⊂ B is any graded ideal, let I^{c} (resp. I_{A}) be the topological closure of I in B (resp. in A), and endow A/I_{A} with the quotient topology arising from the projection A → A/I_{A}. Remark 8.5.2(i) implies that I^{c} is a graded ideal of B : namely gr_{γ}I^{c} is the topological closure of gr_{γ}I in B, for every γ ∈ Γ. It follows that B/I^{c} is a Γ-graded subring of A/I_{A}, and the topology induced by A/I_{A} on B/I^{c} agrees with the quotient topology induced by B (lemma 8.2.3(ii)), so the former is linear, and we get therefore a topological ring with pre-graded structure (A/I_{A}, \underline{B}/I^{c}). We then may form the associated graded structure

(A, \underline{B})/I := (A/I_{A}, \underline{B}/I^{c})^{c}

(notation of proposition 8.5.3(iv)). Explicitly, this is the pair (A/I_{A}, \underline{C}), where

gr_{γ}C := (Im(gr_{γ}B → A/I_{A}))^{c} = (gr_{γ}B/I^{c})^{c} for every γ ∈ Γ.

(ii) Keep the situation of (i), and suppose additionally that A is complete and separated. Then we claim that Im(gr_{γ}B → A/I_{A}) is already closed in A/I_{A}, so the graded subring of A/I_{A} is just B/I^{c} in this case. Indeed, under these assumptions gr_{γ}B is also complete and separated, hence the canonical γ-projection π_{γ}^{∧} : A/I_{A} → gr_{γ}C^{∧} factors through the completion map gr_{γ}(B/I^{c}) → gr_{γ}C^{∧}; on the other hand, π_{γ}^{∧} restricts to an injective map gr_{γ}C → gr_{γ}C^{∧}, so there results an injective map gr_{γ}C → gr_{γ}(B/I^{c}), whose restriction to gr_{γ}(B/I^{c}) is the identity map. Thus, gr_{γ}C = gr_{γ}(B/I^{c}), as stated.

(iii) Lastly, let \underline{C} := (C, gr_{•}C) ⊂ \underline{B} be any closed Γ-graded subring, and we endow C and the topological closure C^{c} of C in A with the topologies induced from A; then gr_{γ}C = C∩gr_{γ}B is closed in gr_{γ}B, and hence also in A, for every γ ∈ Γ, so the pair

(A, \underline{B}) ∩ C := (C^{c}, \underline{C})

is again a topological ring with Γ-graded structure.

8.5.5. Let p ≥ 2 be a prime integer, (A, \underline{B}) any Γ-graded topological ring, a ∈ A^{∧} any element, and (a_{γ} | γ ∈ Γ) the sequence attached to a, as in remark 8.5.2(iii). For every finite subset S ⊂ Γ, let S^{∗} be the set of non-constant mappings {1, …, p} → S; choose also a cyclic subgroup G of order p of the group of permutations of {1, …, p}, and endow S^{∗} with the right G-action given by the rule : g(φ) := φ ∘ g for every g ∈ G and φ ∈ S^{∗}. Clearly, for every φ ∈ S^{∗}, the product a_{φ(i)} ⋯ a_{φ(p)} depends only on the class \overline{φ} of φ in S^{∗}/G, so we denote it a\overline{φ}. Also, notice that every such class \overline{φ} has cardinality p, since p is a prime; with this notation, we may then write

(a^{p})_{S} = c_{S} + p · d_{S} where c_{S} := ∑_{γ∈S}(a_{γ})^{p} and d_{S} := ∑_{\overline{φ}∈S^{∗}/G} a\overline{φ}.

Lemma 8.5.6. With the notation of (8.5.5), we have :

(i) The systems (c_{S} | S ⊂ Γ) and (d_{S} | S ⊂ Γ) are Cauchy nets in B′′.

(ii) Suppose moreover that the topology of A is coarser than the p-adic topology, and the p-Frobenius p_{Γ} of Γ is injective. Then the following conditions are equivalent :

(a) a is topologically nilpotent in A^{∧}.

(b) a_{γ} is topologically nilpotent in B′′ for every γ ∈ Γ.

Proof. (i): According to proposition 8.5.3(ii), for every open graded ideal J ⊂ B′′ there exists a finite subset S_{J} ⊂ Γ such that a_{γ} ∈ gr_{γ}J for every γ ∈ Γ \ S_{J}. Now, if S ⊂ Γ is any finite subset containing S_{J}, let S_{1}^{∗} be the subset of S^{∗} consisting of all mappings {1, …, p} → S whose image is contained in S_{J}, and set S_{2}^{∗} := S^{∗} \ S_{1}^{∗}. Clearly a\overline{φ} ∈ J for every \overline{φ} ∈ S_{2}^{∗}/G (notation of remark 8.5.2(ii)), and S_{1}^{∗} is (finite and) independent of S, whence d_{S} − d_{S′} ∈ J for every finite subsets S, S′ ⊂ Γ containing S_{J}. The assertion for d_{S} follows. Likewise, it is clear that c_{S} − c_{S′} ∈ J^{p} for every S, S′ as in the foregoing, so the assertion for c_{S} follows immediately.

(ii): Let I ⊂ B be any graded open ideal; by assumption there exists n ∈ ℕ such that p^{n} ∈ B. We consider the image \overline{a} of a in the quotient topological ring with Γ-graded structure (A, \underline{B})/(I +pB), as in example 8.5.4, whose underlying topological ring is A/J, where J is the topological closure of I + pB in A. Let also I^{∧} (resp. J^{∧}) be the topological closure of I (resp. of J) in A^{∧}. Then the system of γ-projections (\overline{a}_{γ} | γ ∈ Γ) of \overline{a} is the image of (a_{γ} | γ ∈ Γ). Now, if a is topologically nilpotent, we may find k ∈ ℕ such that \overline{a}^{p^{k}} = 0; since p ∈ J and since p_{Γ} is injective, assertion (i) implies that (\overline{a}_{γ})^{p^{k}} = 0 for every γ ∈ Γ, i.e. (a_{γ})^{p^{k}} ∈ J^{∧}, and therefore (a_{γ})^{p^{kn}} ∈ (J^{∧})^{n} ⊂ I^{∧} (claim 8.3.26). Since I is arbitrary, this shows that a_{γ} is topologically nilpotent for every γ ∈ Γ. Conversely, if the latter condition holds, for every γ ∈ Γ there exists i ≥ n such that \overline{a}_{γ}^{p^{i}} = 0. On the other hand, notice that the topology of A/J is discrete, so A/J = B/I^{c}, where I^{c} denotes the topological closure of I in B; especially, there exists a finite set S ⊂ Γ such that \overline{a}_{γ} = 0 for every γ ∈ Γ \ S. Thus, we may find j ≥ n large enough, such that \overline{a}_{γ}^{p^{j}} = 0 for every γ ∈ Γ; but then (i) implies that \overline{a}^{p^{j}} = 0, and again, since I is arbitrary, this implies that a is topologically nilpotent. □

Example 8.5.7. (i) Let Γ, Δ be two monoids, and (A, \underline{B}) a topological ring with Δ-graded structure. Endow the ring A[Γ] with the unique topology such that the inclusion map A → A[Γ] is adic, and B[Γ] with its natural Δ ⊕ Γ-graded structure. Then we claim that the pair

(A, \underline{B})[Γ] := (A[Γ], B[Γ])

is a topological ring with Δ ⊕ Γ-graded structure. Indeed, let (I_{λ} | λ ∈ Λ) be a fundamental system of Δ-graded open ideals for B; then the family (I_{λ}^{c}[Γ] | λ ∈ Λ) is a fundamental system of open ideals for A[Γ], and I_{λ}^{c}[Γ] ∩ B[Γ] = I_{λ}[Γ] for every λ ∈ Λ. Moreover, it is easily seen that the projection p_{γ} : A[Γ] → A : ∑_{γ′∈Γ} γ · a_{γ} ↦ a_{γ} is a continuous map, for every γ ∈ Γ; now, each direct summand gr_{(δ,γ)}B[Γ] equals p_{γ}^{−1}(gr_{δ}B) ⋂_{γ′≠γ}(Ker p_{γ′}), so it is a closed subset, whence the contention.

(ii) Let φ : Γ → Γ′ be any morphism of monoids, (A, \underline{B}) any topological ring with Γ-graded structure. We obtain a topological ring with Γ′-pre-graded structure (A, \underline{B}_{/Γ′}), and by virtue of proposition 8.5.3(iv), we may then form the associated Γ′-graded structure

(A, \underline{B})_{/Γ′} := (A, \underline{B}_{/Γ′})^{c}.

(iii) In the situation of (ii), notice also that the pair (1_{A}, φ) yields a morphism of topological rings with graded structures :

(A, \underline{B}, Γ) → (A, \underline{B}, Γ)_{/Γ′}.

Moreover, suppose that (A, \underline{D}, Γ′) is another Γ′-graded structure on A, such that (1_{A}, φ) induces a morphism of topological rings with graded structures

(A, \underline{B}, Γ)_{/Γ′} → (A, \underline{D}, Γ′)

and set (A, \underline{C}) := (A, \underline{B}, Γ)_{/Γ′}. Then we claim that \underline{D} = \underline{C}. Indeed, by assumption gr_{γ′}D is closed in A, hence also in gr_{γ′}C, for every γ′ ∈ Γ′; arguing as in remark 8.5.2(i) we easily see that D is a closed subset of C, whence the contention, as D is dense in A, hence also in B.

We deduce that the morphism (1_{A}, φ) enjoys the following universal property. For every morphism (f, ψ) : (A, \underline{B}, Γ) → (A′, \underline{B}′, Γ′) such that ψ factors through φ, there exists a unique morphism of topological rings with Γ′-graded structures (f, 1_{Γ′}) : (A, \underline{B}, Γ)_{/Γ′} → (A′, \underline{B}′, Γ′) such that (f, ψ) = (f, 1_{Γ′}) ∘ (1_{A}, ψ).

8.5.8. To state the following proposition 8.5.11, it is convenient to introduce a special class of monoids; we shall say that a monoid (Γ, +, 0) is weakly integral if we have the following partial cancellation property :

2 · γ + δ = 2 · γ + δ′ ⇒ γ + δ = γ + δ′ for every γ, δ, δ′ ∈ Γ.

Lemma 8.5.9. Let (Γ, +, 0) be any monoid, n ≥ 2 any integer, and suppose that the n-Frobenius endomorphism of Γ is injective. Then Γ is weakly integral.

Proof. Indeed, let γ, δ, δ′ ∈ Γ be three elements such that 2 · γ + δ = 2 · γ + δ′. Then clearly n · γ + δ = n · γ + δ′. A simple induction on k then shows that n · γ + k · δ = n · γ + k · δ′ for every k ∈ ℕ. Especially n · (γ + δ) = n · (γ + δ′), whence γ + δ = γ + δ′, as required. □

Lemma 8.5.10. Let (Γ, +, 0) be a weakly integral monoid, (A, \underline{B}) a topological ring with Γgraded structure, a ∈ A, α ∈ Γ and f ∈ gr_{α}B any three elements. Then we have :

(i) π_{2α+δ}^{∧}(f^{2}a) = f · π_{α+δ}^{∧}(fa) for every δ ∈ Γ.

(ii) π_{γ}^{∧}(fa) = 0 whenever γ ∉ α + Γ.

Proof. Since π_{γ} is a continuous map for every γ ∈ Γ, it suffices to check these identities for a ∈ B, and then there exists a finite subset S ⊂ Γ such that a = a_{S}.

(ii): We have fa = ∑_{δ∈S} f · π_{δ}^{∧}(a), and f · π_{δ}(a) ∈ gr_{α+δ}B for every δ ∈ S, whence the assertion.

(i): Clearly π_{2α+δ}^{∧}(f^{2}a) = ∑_{γ∈T} a · π_{γ}^{∧}(fa), where T := {γ ∈ S | α + γ = 2α + δ}. But due to (ii), we have π_{γ}^{∧}(fa) = 0 unless γ ∈ α + Γ, so we may replace T by the subset of all elements of the form α+δ′ such that 2α+δ′ = 2α+δ. Since Γ is weakly integral, T = {α+δ}, whence the assertion. □

Proposition 8.5.11. Let (Γ, +, 0) be a monoid, (A, \underline{B}) a Γ-graded structure on the topological ring (A, 𝒯 ), and i_{0} : gr_{0}B → A the inclusion map. We have :

(i) Suppose that the topology of A is f-adic, complete and separated. Then :

(a) There exists a finitely generated graded ideal J of B such that JA is an ideal of adic definition for A.

(b) If moreover, J = (gr_{0}J) · B or else Γ is weakly integral, 𝒯 induces on B the

J-adic topology. (ii) Suppose that i_{0} is c-adic. Then A is c-adic if and only if the same holds for gr_{0}B. (iii) Suppose that A is complete and separated, and i_{0} is c-adic. Then A is f-adic if and

only if the same holds for gr_{0}B.

Proof. (i.a): Let I be any finitely generated ideal of adic definition of A; by assumption, we may find a graded ideal I_{B} of B and an integer n ∈ ℕ with I^{n} ∩ B ⊂ I_{B} ⊂ I. For every subset S of A we denote as usual by S^{c} the topological closure of S in A; since I^{n} is an open and closed subset of A, we have

I^{n} = I^{n} ∩ B^{c} = (I^{n} ∩ B)^{c} ⊂ (I^{n} ∩ I_{B})^{c}

so that I^{n} = (I^{n} ∩ I_{B}) + I^{n+1}. We may then find a finitely generated graded subideal J of I_{B} such that I^{n} = (I^{n}∩J)+I^{n+1}. On the other hand, since A is I-adically complete and separated, the ideal I is contained in the Jacobson radical of A (remark 8.3.10(v)), so that I^{n} = (I^{n} ∩J)A, by Nakayama’s lemma, and therefore I^{n} ⊂ JA; by construction, we have as well JA ⊂ I, so J is an ideal with the sought properties.

(i.b): We prove more precisely :

Claim 8.5.12. In the situation of (i), let J ⊂ B be any graded ideal, and suppose that either one of the following conditions holds :

(a) Γ is integral.

(b) Γ is weakly integral and JA is open in A.

(c) J = (gr_{0}J) · B. Then J = B ∩ JA for every n ∈ ℕ.

Proof of the claim. Say that x ∈ B ∩ JA, so we may find a mapping φ : {1, …, k} → Γ and elements a_{i} ∈ gr_{φ(i)}J, y_{i} ∈ A for i = 1, …, k, such that x = ∑_{i=1}^{k} a_{i}y_{i}, as well as a finite subset S ⊂ Γ such that x = x_{S}. Suppose first that J = (gr_{0}J) · B, in which case we may assume that a_{i} ∈ gr_{0}J for i = 1, …, k; we compute

x = ∑_{γ∈S} ∑_{i=1}^{k} a_{i} · π_{γ}^{∧}(y_{i})

whence the claim, in this case. Next, suppose that Γ is integral, so that the natural map Γ → Γ^{gp} is injective; we may then replace Γ by Γ^{gp}, and assume from start that Γ is an abelian group.

We compute

x = ∑_{γ∈S} ∑_{i=1}^{k} a_{i} · π_{γ−φ(i)}^{∧}(y_{i})

so the claim follows also in this case. Lastly, suppose that condition (b) holds; then we may find homogeneous elements f_{1}, …, f_{r} ∈ J such that ∑_{i=1}^{r} f_{i}A is an open ideal of A, and the same holds therefore for the ideal I := ∑_{i=1}^{r} f_{i}^{2}A. Next, we may find y′_{i} ∈ B such that y_{i} −y′_{i} ∈ I for i = 1, …, k, and clearly it suffices to check that x′ := x−∑_{i=1}^{k} y′_{i}a_{i} ∈ J. We may then replace x by x′, and assume as well that x ∈ I; in this case, lemma 8.5.10 says that π_{γ}(x) ∈ ∑_{i=1}^{r} f_{i}B for every γ ∈ S, whence the contention. ♢

(ii): Due to lemma 8.3.32(i) and proposition 8.5.3(iii), we may replace (A, \underline{B}) by (A, \underline{B})^{∧}, after which we may assume that A is complete and separated. We know already that if the topology of gr_{0}B is c-adic, the same holds for 𝒯 (lemma 8.3.24(i.b)). Thus, suppose that 𝒯 is c-adic, and let I be any ideal of c-adic definition of A; by assumption, there exists an open ideal I_{0} ⊂ gr_{0}B with (I_{0}A)^{c} ⊂ I. Now, let J ⊂ gr_{0}B be any other open ideal; we know that there exists m ∈ ℕ such that (I^{m})^{c} ⊂ (JA)^{c}, and taking into account claim 8.3.26 we deduce that (I_{0}^{m}A)^{c} ⊂ (JA)^{c}. Combining with claim 8.3.25we get

π_{0}^{∧}(I_{0}^{m}A)^{c} ⊂ π_{0}^{∧}(JA)^{c}.

But notice that, since A is complete, π_{0}^{∧}(I_{0}^{m}A) = I_{0}^{m} and π_{0}^{∧}(JA) = J, so finally (I_{0}^{m})^{c} ⊂ J, whence the assertion.

(iii): Suppose first that gr_{0}B is f-adic (in which case, it is also adic, by proposition 8.3.18(iii)). Then lemmata 8.3.24(i.b) and 8.3.21(ii.a) imply that the same holds for A. Conversely, suppose that A is f-adic (and adic), and let I ⊂ A be any finitely generated ideal of adic definition. By assumption, there exists an open ideal J ⊂ gr_{0}B such that I^{n} ⊂ (JA)^{c} ⊂ I for some n ∈ ℕ. Since I^{n} is both open and closed in A, we deduce

I^{n} = I^{n} ∩ (JA)^{c} = (I^{n} ∩ JA)^{c} = I^{n+1} + (JA ∩ I^{n}).

It follows that there exists a finitely generated graded subideal K ⊂ J such that I^{n} ⊂ KA + I^{n+1}, and so I^{n} ⊂ KA ⊂ I, by virtue of Nakayama’s lemma and remark 8.3.10(v). Then (K^{r}A | r ∈ ℕ) is a fundamental system of open ideals in A, and therefore (K^{r}A∩gr_{0}B | r ∈ ℕ) is a fundamental system of open ideals in gr_{0}B. But from claims 8.3.26and 8.3.25we get

K^{r} ⊂ K^{r}A ∩ gr_{0}B ⊂ π_{0}^{∧}(K^{r}A) ⊂ π_{0}^{∧}(K^{r}B)^{c} = K^{r}

so K^{r}A ∩ gr_{0}B = K^{r} for every r ∈ ℕ, and finally gr_{0}B is f-adic. □

8.6. Homological algebra for topological modules. Let A be any topological ring. The category A-TopMod of topological A-modules and continuous A-linear maps is additive and with representable kernels and cokernels but, generally, not abelian; indeed, if f : M → N is a continuous map of topological A-modules, Coker(Ker f → M) is not necessarily isomorphic to Ker(N → Coker f), since the quotient topology (induced from M) on Im f may be finer than the subspace topology (induced from N). It is therefore useful to introduce the following :

Definition 8.6.1. Let A be a topological ring, f : M → N a morphism of topological Amodules. We say that f is strict, if the natural map

Coker(Ker f → M) → Ker(N → Coker f)

is an isomorphism of topological A-modules (where these kernels and cokernels are formed in the additive category A-TopMod : see remark 3.7.29(v)).

The category A-TopMod is exact in the sense of [140]; namely, the admissible monomorphisms (resp. epimorphisms) are the continuous injections (resp. surjections) f : M → N that are strict, in the sense of definition 8.6.1, i.e. that induce homeomorphisms M →^{∼} f(M) (resp. M/Ker f →^{∼} N), where f(M) (resp. M/Ker f) is endowed with the subspace (resp. quotient) topology induced from N (resp. from M). An admissible epimorphism is also called a quotient map of topological A-modules. Correspondingly there is a well defined class of admissible short exact sequences of topological A-modules. The following lemma exhibits a useful class of admissible short exact sequences.

Lemma 8.6.2. Consider an inverse system, with surjective transition maps

(\underline{E}_{n} | n ∈ ℕ) : 0 → (M′_{n} | n ∈ ℕ) → (M_{n} | n ∈ ℕ) → (M′′_{n} | n ∈ ℕ) → 0

of short exact sequences of discrete A-modules. Then the induced complex of inverse limits :

lim_{n∈ℕ} \underline{E}_{n} : 0 → M′ := lim_{n∈ℕ} M′_{n} → M := lim_{n∈ℕ} M_{n} → M′′ := lim_{n∈ℕ} M′′_{n} → 0

is an admissible short exact sequence of topological A-modules.

Proof. For every pair of integers i, j ∈ ℕ with i ≤ j, let φ_{ji} : M_{j} → M_{i} be the transition map in the inverse system (M_{n} | n ∈ ℕ), and define likewise φ′_{ji} and φ′′_{ji}; the assumption means that all these maps are onto. Set K_{ji} := Ker φ_{ji} and define likewise K′_{ji}, K′′_{ji}. By the snake lemma we deduce, for every i ∈ ℕ, an inverse system of short exact sequences :

0 → (K′_{ji} | j ≥ i) → (K_{ji} | j ≥ i) → (K′′_{ji} | j ≥ j) → 0

where again, all the transition maps are surjective. However, by definition, the decreasing family of submodules (K_{i} := lim_{j≥i} K_{ji} | i ∈ ℕ) is a fundamental system of open neighborhoods of 0 ∈ M (and likewise one defines the topology on the other two inverse limits). It follows already that the surjection M → M′′ is a quotient map of topological A-modules. To conclude it suffices to remark :

Claim 8.6.3. The natural map K′_{i} := lim_{j≥i} K′_{ji} → K_{i} induces an identification :

K′_{i} = K_{i} ∩ M′ for every i ∈ ℕ.

Proof of the claim. Indeed, for every j ≥ i we have a left exact sequence :

\underline{F}_{j} : (0 → K′_{ji} → M′_{j} ⊕ K_{ji} ⟶^{α} M_{j})

where α(m′, m) = m′ − m for every m′ ∈ M′_{j} and m ∈ K_{ji}. Therefore lim_{j≥i} \underline{F}_{j} is the analogous left exact sequence 0 → K′_{i} → M′ ⊕ K_{i} → M, whence the claim. □

Remark 8.6.4. Let (C^{•}, d^{•}) be a complex of topological A-modules with continuous differentials. Then, for every i ∈ ℤ, the cohomology H^{i}C^{•} inherits a well defined topology : namely, we can take either

Coker(d^{i−1} : C^{i−1} → Ker d^{i}) or Ker(d^{i} : Coker d^{i−1} → C^{i+1})

(where these kernels and cokernels are formed in the additive category A-TopMod) and by lemma 8.2.3(i), these two topologies coincide (regardless of whether the differentials are strict or otherwise). Notice also that for every i ∈ ℤ the following conditions are equivalent :

(a) The differential d^{i} factors through an open map C^{i} → Ker d^{i+1}.

(b) The differential d^{i} is strict and the topology of H^{i+1}C^{•} is discrete.

8.6.5. Consider an admissible short exact sequence of complexes of topological A-modules

0 → (C_{1}^{•}, d_{1}^{•}) ⟶^{f^{•}} (C_{2}^{•}, d_{2}^{•}) ⟶^{g^{•}} (C_{3}^{•}, d_{3}^{•}) → 0

i.e. a double complex of A-TopMod whose rows 0 → C_{1}^{k} → C_{2}^{k} → C_{3}^{k} → 0 are admissible short exact sequences for every k ∈ ℤ. For every i ∈ ℤ and for j = 1, 2, 3, denote by δ_{j}^{i} : C^{i} → Ker d_{j}^{i+1} the continuous map induced by the differential d_{j}^{i} : C_{j}^{i} → C_{j}^{i+1} of C_{j}^{•}.

Proposition 8.6.6. In the situation of (8.6.5), fix also i ∈ ℤ. We have :

(i) If δ_{2}^{i} and δ_{3}^{i−1} are open maps, the same holds for δ_{1}^{i}. (ii) If δ_{1}^{i} and δ_{3}^{i} are open maps, the same holds for δ_{2}^{i}. (iii) If δ_{1}^{i} and δ_{2}^{i−1} are open maps, the same holds for δ_{3}^{i−1}.

Proof. (i): Endow D := g^{i+1}(Ker d_{2}^{i+1}) with the quotient topology induced by the restriction h : Ker d_{2}^{i+1} → D of g^{i+1}, and notice that Im d_{3}^{i} ⊂ D; we claim that d_{3}^{i} induces a continuous and open map C_{3}^{i} → D. Indeed, since g^{i} is an admissible epimorphism, it suffices to check that the composition d_{3}^{i} ∘ g^{i} : C_{2}^{i} → D is continuous and open; however, the latter equals h ∘ δ_{2}^{i}, whence the assertion. Moreover, f^{i+1} restricts to an admissible monomorphism Ker d_{1}^{i+1} → Ker d_{2}^{i+1}. Summing up, we may replace C_{j}^{i+1} with Ker d_{j}^{i+1} for j = 1, 2, and C_{3}^{i+1} with D, and assume from start that C_{j}^{i+2} = 0 for j = 1, 2, 3. Then d_{j}^{i} : C_{j}^{i} → C_{j}^{i+1} is open for j = 2, 3, and it remains to check that d_{1}^{i} is open as well.

Next, set \overline{C}_{1}^{i} := Coker d_{1}^{i−1} and \overline{C}_{2}^{i} := Coker d_{2}^{i−1} ∘ f^{i−1} (endowed with the topologies induced by C_{1}^{i} and respectively C_{2}^{i}). Clearly d_{j}^{i} factors through a continuous map \overline{d}_{j}^{i} : \overline{C}_{j}^{i} → C_{j}^{i+1} for j = 1, 2, and f^{i} (resp. g^{i}) induces an injective (resp. surjective) continuous map \overline{f}^{i} : \overline{C}_{1}^{i} → \overline{C}_{2}^{i} (resp. \overline{g}^{i} : \overline{C}_{2}^{i} → C_{3}^{i}). Moreover, by lemma 8.2.3(i) the topology of \overline{C}_{1}^{i} is induced by the topology of \overline{C}_{2}^{i} via \overline{f}^{i}, i.e. \overline{f}^{i} is an admissible monomorphism. Likewise, \overline{g}^{i} is an admissible epimorphism. We may then replace C_{j}^{i} by \overline{C}_{j}^{i} for j = 1, 2, and assume from start C_{1}^{i−1} = 0, in which case we obtain a commutative diagram of continuous maps

(8.6.7)  \xymatrix{0 \ar[r] & 0 \ar[r] \ar[d] & C_{2}^{i−1} \ar[r]^{g^{i−1}} \ar[d]_{d_{2}^{i−1}} & C_{3}^{i−1} \ar[d]^{d_{3}^{i−1}} \ar[r] & 0 \\ 0 \ar[r] & C_{1}^{i} \ar[r]^{f^{i}} \ar[d]_{d_{1}^{i}} & C_{2}^{i} \ar[r]^{g^{i}} \ar[d]_{d_{2}^{i}} & C_{3}^{i} \ar[r] \ar[d]^{d_{3}^{i}} & 0 \\ 0 \ar[r] & C_{1}^{i+1} \ar[r]^{f^{i+1}} & C_{2}^{i+1} \ar[r]^{g^{i+1}} & C_{3}^{i+1} \ar[r] & 0}

whose rows are admissible short exact sequences, and where d_{2}^{i} and δ_{3}^{i−1} are still open. Consider the diagram of topological A-modules

(8.6.8)  \xymatrix{Ker (d_{3}^{i} ∘ g^{i}) \ar[r]^{^{\overline{g}^{i}}} \ar[d]_{_{\overline{d}_{i}_{2}}} & Ker d_{3}^{i} \ar[d] \\ Ker g^{i+1} \ar[r] & 0}

deduced from the right bottom square of (8.6.7). In view of claim 8.2.4 we deduce that \overline{g}^{i} and \overline{d}_{2}^{i} are open maps. Moreover, f^{i} and f^{i+1} factor through admissible monomorphisms C_{1}^{i} → Ker (d_{3}^{i} ∘ g^{i}) and C_{1}^{i+1} → Ker g^{i+1}. Likewise, d_{2}^{i−1} and d_{3}^{i−1} factor through continuous maps C_{2}^{i−1} → Ker (d_{3}^{i} ∘ g^{i}) and C_{3}^{i−1} → Ker d_{3}^{i}. Thus, we may replace the bottom right square of (8.6.7) with (8.6.8), and assume from start that C_{3}^{i+1} = 0 as well. In this situation, g^{i−1} and f^{i+1} are both isomorphisms of topological A-modules, and d_{3}^{i−1} is an open map, so we are reduced to showing :

Claim 8.6.9. Consider a diagram of topological A-modules

\xymatrix{ &  & C_{4} \ar[d]^{h} \\ 0 \ar[r] & C_{1} \ar[r]^{f} & C_{2} \ar[r]^{g} \ar[d]^{k} & C_{3} \ar[r] & 0 \\  &  & C_{5}}

such that :

(a) The horizontal row is an admissible short exact sequence.

(b) g ∘ h and k are open maps and k ∘ h = 0.

Then k ∘ f is an open map.

Proof of the claim. After pulling back the short exact admissible sequence of (f, g) along the map g ∘ h, we obtain the short exact sequence of topological modules

0 → C_{1} ⟶^{f′} C_{2} ×_{C_{3}} C_{4} ⟶^{g′} C_{4} → 0.

Notice that g′ is still an admissible epimorphism, by claim 8.2.4. Likewise, f′ is an admissible monomorphism, since its composition with the projection π : C_{2} ×_{C_{3}} C_{4} → C_{2} equals f, which is an admissible monomorphism by assumption. Moreover, since g ∘ h is open, the same holds for π, again by claim 8.2.4, and therefore also for k′ := k ∘ π : C_{2} ×_{C_{3}} C_{4} → C_{5}. Furthermore, the pair (h, 1_{C_{4}}) defines a continuous section s : C_{4} → C_{2} ×_{C_{3}} C_{4} for g′, whence a continuous isomorphism of A-modules

ω : C_{1} × C_{4} →^{∼} C_{2} ×_{C_{3}} C_{4} (x, y) ↦ f′(x) + s(y) = (f(x) + h(y), y)

and we notice that ω is an isomorphism of topological A-modules : indeed, let us endow f(C_{1}) with the topology induced by the inclusion into C_{2}, so that f factors through an isomorphism u : C_{1} →^{∼} f(C_{1}) of topological A-modules; then the inverse of ω is the continuous map

C_{2} ×_{C_{3}} C_{4} → C_{1} × C_{4} (a, b) ↦ (u^{−1}(a − h(b)), b)

whence the assertion. Summing up, we deduce that the continuous map ψ := k ∘ π ∘ ω : C_{1} × C_{4} → C_{5} is open, and notice that ψ(x, y) = k ∘ f(x) for every (x, y) ∈ C_{1} × C_{4}, since k ∘ h = 0. Lastly, let U ⊂ C_{1} be any open subset; then U × C_{4} is open in C_{1} × C_{4}, and ψ(U × C_{4}) = k ∘ f(U) is open in C_{5}, whence the claim. ♢

(ii): Define the A-module D and the A-linear map h : Ker d_{2}^{i+1} → D as in (i), and endow D with the topology induced by the inclusion into C_{3}^{i+1}. Then h is continuous, and it is also an open map, since the same holds for h ∘ δ_{2}^{i} = δ_{3}^{i} ∘ g^{i} : C_{2}^{i} → D (lemma 8.2.5). We may then argue as in the proof of (i), to reduce to the case where C_{j}^{i+2} = 0 for j = 1, 2, 3, in which case d_{1}^{i} and d_{3}^{i} are both open maps, and we must show that the same holds for d_{2}^{i}.

Next, we pull back the admissible short exact sequence (f^{i}, g^{i}) (resp. (f^{i+1}, g^{i+1})) along the map g^{i} (resp. along the map d_{3}^{i} ∘ g^{i}) to get the commutative diagram of topological A-modules:

(8.6.10)  [figure: 0; C_{1}^{i}; f^{′i}; d_{1}^{i}; C_{2}^{i} ×_{C_{3}^{i}} C_{2}^{i}; g^{′i}; d_{2}^{′i}; C_{2}^{i}; 0; 0; C_{1}^{i+1}; f^{′i+1}; C_{2}^{i+1} ×_{C_{3}^{i+1}} C_{2}^{i}; g^{′i+1}; π^{i+1}; C_{2}^{i}; d_{3}^{i}∘g^{i}; 0; 0; C_{1}^{i+1}; f^{i+1}; C_{2}^{i+1}; g^{i+1}; C_{3}^{i+1}; 0.]

Arguing as in the proof of claim 8.6.9we easily see that the rows of (8.6.10) are still admissible short exact sequences. Moreover, the pair (1_{C_{2}^{i}}, 1_{C_{2}^{i}}) (resp. (d_{2}^{i}, 1_{C_{2}^{i}})) gives a section s^{i} (resp. s^{i+1}) for g^{′i} (resp. for g^{′i+1}). Furthermore, π^{i+1} is an open map, by claim 8.2.4, and if π^{i} : C_{2}^{i} ×_{C_{3}^{i+1}} C_{2}^{i} → C_{2}^{i} denotes the projection, we have

d_{2}^{i} ∘ π^{i} = π^{i+1} ∘ d_{2}^{′i}.

Thus, in order to show that d_{2}^{i} is open, it suffices to check that the same holds for d_{2}^{′i} (lemma 8.2.5). Summing up, we may replace the exact rows (f^{i}, g^{i}) and (f^{i+1}, g^{i+1}) by (f^{′i}, g^{′i}) and (f^{′i+1}, g^{′i+1}), and assume also from start that C_{3}^{i} = C_{3}^{i+1} and d_{3}^{i} = 1_{C_{3}^{i}}, and moreover, that g^{i} and g^{i+1} admit continuous sections s^{i} and respectively s^{i+1}, such that d_{2}^{i} ∘ s^{i} = s^{i+1}. Arguing as in the proof of claim 8.6.9we deduce isomorphisms of topological A-modules

ω^{k} : C_{1}^{k} × C_{3}^{i} →^{∼} C_{2}^{k}  (x, y) ↦ f^{k}(x) + s^{k}(y)  for k = i, i + 1

fitting into a commutative diagram

\xymatrix{C_{1}^{i} × C_{3}^{i} \ar[r]^{ω^{i}} \ar[d]_{d_{1}^{i}×1_{C^{i}_{3}}} & C_{2}^{i} \ar[d]^{d_{2}^{i}} \\ C_{1}^{i+1} × C_{3}^{i} \ar[r]^{ω^{i+1}} & C_{2}^{i+1}}

and since d_{1}^{i} is an open map, the same holds for d_{1}^{i} × 1_{C_{3}^{i}}, whence the assertion.

(iii): Arguing as in the proof of (i), we reduce to the case where C_{j}^{i+2} = 0 for every j = 1, 2, 3, in which case d_{1}^{i} is an open map. Then, we define \overline{C}_{j}^{i} for j = 1, 2 as in the proof of (i), so that d_{j}^{i} factors through a continuous map \overline{d}_{j}^{i} : \overline{C}_{j}^{i} → C_{j}^{i+1} for j = 1, 2. Notice that \overline{d}_{1}^{i} is an open map, and δ_{2}^{i−1} induces an open map Coker f^{i−1} → Ker \overline{d}_{2}^{i}. We may then replace C_{j}^{i} by \overline{C}_{j}^{i} for j = 1, 2, and assume from start that C_{1}^{i−1} = 0, in which case we obtain a commutative diagram (8.6.7) whose rows are admissible short exact sequences, and where d_{1}^{i} and δ_{2}^{i−1} are open maps. Next, we may argue again as in the proof of (i), to reduce to the case where C_{3}^{i+1} = 0 as well, in which case we have to show that d_{3}^{i−1} is an open map. To this aim, we consider the cartesian diagram

\xymatrix{C_{1}^{i} ×_{C_{2}^{i+1}} C_{2}^{i} \ar[r]^{π_{2}^{i}} \ar[d]_{π_{1}^{i}} & C_{2}^{i} \ar[d]^{d_{2}^{i}} \\ C_{1}^{i} \ar[r]^{f^{i+1}∘d_{1}^{1}} & C_{2}^{i+1}}

and we notice that the pair (1_{C_{1}^{i}}, f^{i}) defines a continuous section s : C_{1}^{i} → C_{1}^{i} ×_{C_{2}^{i+1}} C_{2}^{i} for π_{1}^{i}. Likewise, the pair consisting of the zero map Ker d_{2}^{i} → C_{1}^{i} and the inclusion map Ker d_{2}^{i} → C_{2}^{i} defines a continuous map t : Ker d_{2}^{i} → C_{1}^{i} ×_{C_{2}^{i+1}} C_{2}^{i}, whence an isomorphism of topological A-modules

ω : C_{1}^{i} × Ker d_{2}^{i} →^{∼} C_{1}^{i} ×_{C_{2}^{i+1}} C_{2}^{i} (x, y) ↦ s(x) + t(y) = (x, f^{i}(x) + y).

Furthermore, under the current assumptions, f^{i+1} is an isomorphism of topological A-modules, so f^{i+1} ∘ d_{1}^{i} is an open map, and then the same holds for π_{2}^{i}, according to claim 8.2.4. Lastly, notice the commutative diagram

\xymatrix{C_{1}^{i} × C_{2}^{i−1} \ar[r]^{π_{2}^{i−1}} \ar[d]_{1_{C^{i}_{1}}×δ_{2}^{i−1}} & C_{2}^{i−1} \ar[rr]^{g^{i−1}} &  & C_{3}^{i−1} \ar[d]^{d_{3}^{i−1}} \\ C_{1}^{i} × Ker d_{2}^{i} \ar[r]^{ω} & C_{1}^{i} ×_{C_{3}^{i+1}} C_{2}^{i} \ar[r]^{π_{2}^{i}} & C_{2}^{i} \ar[r]^{g^{i}} & C_{3}^{i}}

where π_{2}^{i−1} is the projection. Since δ_{2}^{i−1} is an open map, the same holds for 1_{C_{1}^{i}} × δ_{2}^{i−1}, and then also for d_{3}^{i−1} ∘ g^{i−1} ∘ π_{2}^{i−1}. By lemma 8.2.5, we conclude that d_{3}^{i−1} is open as well, as required. □

8.6.11. Consider an object C^{••} of C_{2}(A-TopMod), i.e. a double complex of topological Amodules whose horizontal and vertical differentials

d_{h}^{p,q} : C^{p,q} → C^{p+1,q} d_{v}^{p,q} : C^{p,q} → C^{p,q+1}

are continuous maps, and therefore induce continuous maps

δ_{h}^{p,q} : C^{p,q} → Ker d_{h}^{p+1,q}  δ_{v}^{p,q} : C^{p,q} → Ker d_{v}^{p,q+1}  for every p, q ∈ ℤ.

With this notation, we have :

Corollary 8.6.12. In the situation of (8.6.11), suppose moreover that :

(a) C^{p,q} = 0 whenever p < 0 or q < 0.

(b) The maps δ_{v}^{p,q} are open for every p, q ∈ ℤ.
(c) The maps δ_{h}^{p,q} are open for every p ∈ ℤ and every q > 0.

Then the maps δ_{h}^{p,0} are open for every p ∈ ℤ.

Proof. To ease notation, set

B_{v}^{pq} := Im d_{v}^{p,q−1}  Z_{v}^{pq} := Ker d_{v}^{p,q}  H_{v}^{p,q} := Coker δ_{h}^{p,q}  for every p, q ∈ ℤ

and denote by d_{B}^{p,q} : B_{v}^{p,q} → B^{p+1,q} and d_{Z}^{p,q} : Z_{v}^{p,q} → Z^{p+1,q} the restrictions of d_{h}^{p,q}, for every p, q ∈ ℤ. Then (B_{v}^{•,q}, d_{B}^{•,q}) and (Z_{v}^{•,q}, d_{Z}^{•,q}) are complexes of topological A-modules, and we get admissible short exact sequences of complexes

\begin{matrix}0 → (Z_{v}^{•,q}, d_{Z}^{•,q}) → (C^{•,q}, d_{h}^{•,q}) → (B_{v}^{•,q+1}, d_{B}^{•,q+1}) → 0 \\ 0 → (B_{v}^{•,q}, d_{B}^{•,q}) → (Z_{v}^{•,q}, d_{Z}^{•,q}) → H_{v}^{•,q} → 0. \end{matrix}

Let also

δ_{B}^{p,q} : B_{v}^{p,q} → Kerd_{B}^{p+1,q} δ_{Z}^{p,q} : Z_{v}^{p,q} → Kerd_{Z}^{p+1,q}

be the continuous maps induced by d_{B}^{p,q} and respectively d_{Z}^{p,q}, for every p, q ∈ ℤ.

Claim 8.6.13. For every p, q the following holds :

(d) If δ_{Z}^{p,q} is an open map, the same holds for δ_{B}^{p,q}.

(e) If δ_{B}^{p−1,q+1} is an open map, the same holds for δ_{Z}^{p,q}, provided q > 0.

Proof of the claim. Indeed, H_{v}^{•,q} is complex of discrete topological A-modules, for every q ∈ ℤ, due to assumption (b), so its differentials d_{H}^{•,q} trivially induce open maps H_{v}^{p,q} → Ker d_{H}^{p+1,q} for every q; then (d) follows from proposition 8.6.6(i). Likewise, (e) follows from assumption (b) and proposition 8.6.6(i). ♢

We now show that, for every n ∈ ℤ, the maps δ_{B}^{p,q} are open for every p, q ∈ ℤ such that p + q = n and q > 0. We argue by induction on p. For p < −1, assumption (a) says that B_{v}^{p,q} = B_{v}^{p+1,q} = 0, so the assertion trivially holds. Suppose then that p ≥ −1, and that we already know that δ_{B}^{p−1,q+1} is open; from claim 8.6.13we deduce that δ_{B}^{p,q} is also open, as stated. Especially, we see that δ_{B}^{p,1} is an open map, for every p ∈ ℤ. Lastly, notice that (a) and (b) also imply that the topology of Z_{v}^{p,0} is discrete for every p ∈ ℤ; then we apply proposition 8.6.6(ii) to the admissible short exact sequence 0 → Z_{v}^{•,0} → C^{•,0} → B_{v}^{•,1} to conclude. □

8.6.14. Let A be any topological ring, (C^{•}, d^{•}) any complex of topological A-modules, φ^{•} : C^{•} → C^{•} an endomorphism of C^{•} and h^{•} a chain homotopy from 1_{C^{•}} to φ^{•} in the category C(A-TopMod), so that φ^{i} : C^{i} → C^{i} and h^{i} : C^{i} → C^{i−1} are continuous A-linear maps, for every i ∈ ℤ.

Lemma 8.6.15. In the situation of (8.6.14), fix i ∈ ℤ and suppose that φ^{i} = 0. Then we have : (i) d^{i−1} induces an open and surjective map δ^{i} : C^{i−1} → Ker d^{i}. (ii) Ker d^{i} is a direct summand of the topological A-module C^{i} (i.e. the inclusion map

Ker d^{i} → C^{i} admits a continuous left inverse).

Proof. The assumption means that

d^{i−1} ∘ h^{i} + h^{i+1} ∘ d^{i} = 1_{C^{i}}.

Now, set e^{i} := d^{i−1} ∘ h^{i}; we compute

(1_{C^{i}} − e^{i}) ∘ e^{i} = h^{i+1} ∘ d^{i} ∘ d^{i−1} ∘ h^{i} = 0

and therefore e^{i} is an idempotent continuous endomorphism of C^{i}. According to example 3.7.36, it follows that e^{i} and 1_{C^{i}} − e^{i} induce a natural isomorphism of topological A-modules

(8.6.16)  Ker (e^{i}) ⊕ Ker (1_{C^{i}} − e^{i}) →^{∼} C^{i}.

Notice as well that d^{i} ∘ e^{i} = 0, whence d^{i} = d^{i} ∘ (1_{C^{i}} − e^{i}), and we deduce easily that

Ker (1_{C^{i}} − e^{i}) = Ker d^{i}

whence (ii). By the same token, we see that e^{i} restricts to the identity map on Ker d^{i}, i.e. δ^{i} admits a continuous right inverse, given by the restriction Ker d^{i} → C^{i−1} of h^{i}. Taking into account lemma 8.2.5, we then get also (i). □

8.6.17. Extensions of topological rings. Let A be any topological ring whose topology is linear; we shall consider for any A-algebra C, the category

Exal_{A}(C)

whose objects are all short exact sequences of A-modules

(8.6.18)  Σ : 0 → M → E ⟶^{ψ} C → 0

such that E is an A-algebra (with A-module structure given by the structure map A → E), and ψ is a map of A-algebras. The morphisms in Exal_{A}(C) are the commutative ladders of A-modules whose central vertical arrow g is a map of A-algebras :

(8.6.19)  [figure: 0; M′; E′; g; C; 0; 0; M′′; E′′; C; 0.]

For any topological A-algebra C whose topology is linear, we shall consider the category

Exaltop_{A}(C)

whose objects are the short exact sequences of A-modules (8.6.18) such that E is a topological A-algebra whose topology is linear (and again, with A-module structure given by the structure map A → E), and ψ is a continuous and open map of topological A-algebras, whose kernel M is a discrete topological space, for the topology induced from E. The morphisms in Exaltop_{A}(C) are the commutative ladders (8.6.19) such that g is a continuous map of topological A-algebras.

8.6.20. Now, let A be as in (8.6.17), C any topological A-algebra whose topology is linear, φ : A → C the structure morphism, and (I_{λ} | λ ∈ Λ) (resp. (J_{λ′} | λ′ ∈ Λ′)) a collection of open ideals of A (resp. of C), which is a fundamental system of open neighborhoods of 0. Let Λ′′ ⊂ Λ×Λ′ be the subset of all (λ, λ′) such that φ(I_{λ}) ⊂ J_{λ′}; the set Λ′′ is partially ordered, by declaring that (λ, λ′) ≤ (μ, μ′) if and only if I_{μ} ⊂ I_{λ} and J_{μ′} ⊂ J_{λ′}. Set A_{λ} := A/I_{λ} for every λ ∈ Λ (resp. C_{λ′} := C/J_{λ′} for every λ′ ∈ Λ′); for (λ, λ′), (μ, μ′) ∈ Λ′′ with (λ, λ′) ≤ (μ, μ′), the surjection π_{μ′λ′} : C_{μ′} → C_{λ′} induces a functor

Exal_{A_{λ}}(C_{λ′}) → Exal_{A_{μ}}(C_{μ′}) Σ ↦ Σ ∗ π_{μ′λ′}

(see [75, §2.5.5]) and clearly the rule (λ, λ′) ↦ Exal_{A_{λ}}(C_{λ′}) yields a pseudo-functor

E : (Λ′′, ≤) → Cat.

Moreover, we have a pseudo-cocone :

(8.6.21)  E ⇒ Exaltop_{A}(C)

defined as follows. To any (λ, λ′) ∈ Λ′′ and any object Σ_{λ,λ′} : 0 → M → E_{λ′} → C_{λ′} → 0 of Exal_{A_{λ}}(C_{λ′}), one assigns the extension (8.6.18) obtained by pulling back Σ_{λ,λ′} along the projection π_{λ′} : C → C_{λ′}. Let β : E → E_{λ′} be the induced projection; we endow E with the linear topology defined by the fundamental system of all open ideals of the form ψ^{−1}J ∩ β^{−1}J′ where J (resp. J′) ranges over the set of open ideals of C (resp. over the set of all ideals of E_{λ′}). With this topology, it is easily seen that both ψ and the induced structure map A → E are continuous ring homomorphisms. Moreover, if J ⊂ J_{λ′}, then ψ^{−1}J ∩ β^{−1}J′ = J × (J′ ∩ M), from which it follows that ψ is an open map. Furthermore, M ∩ β^{−1}0 = 0, which shows that M is discrete, for the topology induced by the inclusion map M → E. Summing up, we have attached to Σ_{λ,λ′} a well defined object Σ_{λ,λ′} ∗ π_{λ′} of Exaltop_{A}(C), and it is easily seen that the rule Σ_{λ,λ′} ↦ Σ_{λ,λ′} ∗ π_{λ′} is functorial in Σ_{λ,λ′}, and for (λ, λ) ≤ (μ, μ′) in Λ′′, there is a natural isomorphism in Exaltop_{A}(C) :

Σ_{λ,λ′} ∗ π_{λ′} →^{∼} (Σ_{λ,λ′} ∗ π_{μ′λ′}) ∗ π_{μ′}

(details left to the reader).

Lemma 8.6.22. The pseudo-cocone (8.6.21) induces an equivalence of categories :

β : 2-colim_{Λ′′} E →^{∼} Exaltop_{A}(C).

Proof. Let Σ as in (8.6.18) be any object of Exaltop_{A}(C). By assumption, there exists an open ideal J ⊂ E such that M ∩ J = 0. Since ψ is an open map, there exists λ′ ∈ Λ′ such that J_{λ′} ⊂ ψ(J), and after replacing J by J ∩ ψ^{−1}J_{λ′}, we may assume that ψ(J) = J_{λ′}. Likewise, if φ_{E} : A → E is the structure morphism, there exists λ ∈ Λ such that I_{λ} ⊂ φ_{E}^{−1}J, and it follows that the induced extension

Σ_{λ,λ′} : 0 → M → E/J → C_{λ′} → 0

is an object of Exal_{A_{λ}}(C_{λ′}). We notice :

Claim 8.6.23. There exists a natural isomorphism Σ →^{∼} Σ_{λ,λ′} ∗ π_{λ′} in Exaltop_{A}(C).

Proof of the claim. (i): By construction, ψ and the projection π_{J} : E → E/J define a unique morphism γ : E → (E/J)×_{C_{λ′}} C of A-algebras, restricting to the identity map on M (which is an ideal in both of these A-algebras). It is clear that γ is an isomorphism, and therefore it yields a natural isomorphism Σ →^{∼} Σ_{λ,λ′} ∗ π_{λ′} in Exal_{A}(C). It remains to check that γ is continuous and open. For the continuity, it suffices to remark that, for every ideal I ⊂ E/J and every μ′ ∈ Λ′, the ideals γ^{−1}(I ×_{C_{λ′}} C) = π_{J}^{−1}I and γ^{−1}(E/J ×_{C_{λ′}} J_{μ′}) = ψ^{−1}J_{μ′} are open in E, which is obvious, since J is an open ideal and ψ is continuous. Lastly, let I ⊂ E be any open ideal such that I ⊂ J ∩ ψ^{−1}J_{λ′}; since ψ is an open map, it is easily seen that γ(I) = 0 × ψ(I) is an open ideal of (E/J) ×_{C_{λ′}} C, so γ is open. ♢

From claim 8.6.23 we see already that β is essentially surjective. It also follows easily that β is full. Indeed, consider any morphism s : Σ′ → Σ′′ of Exaltop_{A}(C) as in (8.6.19), and pick an open ideal J′′ ⊂ E′′ with J ∩ M′′ = 0; set J′ := g^{−1}J′′, and notice that J′ ∩ M′ = 0. Moreover, if the image of J′′ in C equals J_{λ′} for some λ′ ∈ Λ′, then clearly the same holds for the image of J′ in C. Therefore, in this case the foregoing construction yields objects Σ′_{λ,λ′} and Σ′′_{λ,λ′} of Exal_{A_{λ}}(C_{λ′}) (for a suitable λ ∈ Λ), whose middle terms are respectively E′/J′ and E′′/J′′, and s descends to a morphism s_{λ′} : Σ′_{λ,λ′} → Σ′′_{λ,λ′}, whose middle term is the map g_{λ′} : E′/J′ → E′′/J′′ induced by g. By inspecting the proof of claim 8.6.23, we deduce a commutative diagram

\xymatrix{E′ \ar[r]^{∼} \ar[d]_{g} & (E′/J′) ×_{C_{λ′}} C \ar[d]^{g_{λ′}×_{C_{λ′}} C} \\ E′′ \ar[r]^{∼} & (E′′/J′′) ×_{C_{λ′}} C}

whose horizontal arrows are the maps that define the isomorphisms Σ′ →^{∼} Σ′_{λ,λ′} ∗ π_{λ′} and Σ′′ →^{∼} Σ′′_{λ,λ′} ∗ π_{λ′} in Exaltop_{A}(C). It follows easily that s_{λ} ∗ π_{λ′} = s, whence the assertion. Lastly, the faithfulness of β is immediate, since the projections π_{λ′} are surjective maps. □

8.6.24. Let A be as in 8.6.17, and C any A-algebra (resp. any topological A-algebra); we denote by

nilExal_{A}(C) ( resp. nilExaltop_{A}(C) )

the full subcategory of Exal_{A}(C) (resp. of Exaltop_{A}(C)) whose objects are the nilpotent extensions of C, i.e. those extensions (8.6.18), where M is a nilpotent ideal of E. Moreover, in the situation of (8.6.20), clearly E restricts to a pseudo-functor

nilE : (Λ′′, ≤) → Cat (λ, λ′) ↦ nilExal_{A_{λ}}(C_{λ′}).

Also, (8.6.21) restricts to a pseudo-cocone on nilE, and lemma 8.6.22 immediately implies an equivalence of categories

(8.6.25)  2-colim_{Λ′′} nilE →^{∼} nilExaltop_{A}(C).

Proposition 8.6.26. Let A and C be as in (8.6.20). The following holds :

(i) Suppose that J_{λ′}^{2} is open in C, for every λ′ ∈ Λ′. Then the forgetful functor

(8.6.27)  nilExaltop_{A}(C) → nilExal_{A}(C)

is fully faithful. (ii) Suppose additionally, that :

(a) C is noetherian, and I ⊂ C is an ideal such that the topology of C is I-adic.

(b) I_{λ}^{2} is open in A, for every λ ∈ Λ.

Then the essential image of (8.6.27) is the (full) subcategory of all nilpotent extensions (8.6.18) such that the C-module M/M^{2} is annihilated by a power of I.

Proof. (i): The functor is obviously faithful, and in light of (8.6.25), we come down to the following situation. We have a commutative ladder of extensions of A-algebras

\xymatrix{0 \ar[r] & M \ar[r] \ar[d] & E \ar[r]^{ψ} \ar[d]^{g} & C \ar[r] \ar[d]^{π_{λ′}} & 0 \\ 0 \ar[r] & N \ar[r] & E′ \ar[r] & C_{λ′} \ar[r] & 0}

for some λ′ ∈ Λ′, whose top (resp. bottom) row is an object of nilExaltop_{A}(C) (resp. of nilExal_{A_{λ}}(C_{λ′}), for some λ ∈ Λ), and we need to show that the kernel of g contains an open ideal. To this aim, we remark :

Claim 8.6.28. For any open ideal J ⊂ E, the ideal J^{2} is open as well.

Proof of the claim. Recall that M is a discrete A-module for the topology induced by E; then, after replacing J by a smaller open ideal, we may assume that J ∩ M = 0. We have I := ψ(J^{2}) = ψ(J)^{2}, so the assumption in (i) say that I is open in C, and therefore ψ^{−1}I = J^{2} ⊕ M is open in E, so finally J ∩ ψ^{−1}I = J^{2} is open in E, as stated. ♢

Now, set J := ψ^{−1}J_{λ′}; then J is an open ideal of E, and clearly g(I) ⊂ N. Say that N^{k} = 0; then g(I^{k}) = 0, and I^{k} is an open ideal of E, by claim 8.6.28, whence the contention.

(ii): It is easily seen that, for every extension (8.6.18) in the essential image of (8.6.27), we must have I^{k}(M/M^{2}) = 0 for every sufficiently large k ≥ 0 (details left to the reader). Conversely, consider a nilpotent extension Σ as in (8.6.18), with I^{k}(M/M^{2}) = 0 and M^{t} = 0 for some k, t ∈ ℕ. Pick a finite system f := (f_{1}, …, f_{r}) of elements of E whose images in B form a system of generators for I^{k}; notice that the ideal (f^{n})B is open in B, and (f^{n})M = 0 for every n ≥ t. There follows an inverse system of exact sequences (notation of remark 7.8.1(ii))

H_{1}(f^{n}, B) → M ⟶^{β_{n}} E/(f^{n})E → B/(f^{n})B → 0 for every n ≥ t

with transition maps induced by the morphisms φ_{f} of (7.8.20). As B is noetherian, lemma 7.8.44and remark 7.8.45imply that the system (H_{1}(f^{n}, B) | n ∈ ℕ) is essentially zero. Therefore, the same holds for the inverse system (Ker β_{n} | n > 0). However, the transition map Ker β_{n+1} → Ker β_{n} is obviously injective for every n ≥ t, so we conclude that β_{n} is injective for some sufficiently large integer n. For such n, we obtain a nilpotent extension

(8.6.29)  0 → M → E/(f^{n})E → B/(f^{n})B → 0.

Lastly, let φ : A → B be the structure map, and pick λ ∈ Λ such that I_{λ} ⊂ φ^{−1}I^{k}; it is easily seen that I_{λ}^{t}M = 0 and φ(I_{λ}^{s}) ⊂ (f^{n})B for s ∈ ℕ large enough. Therefore, some sufficiently large power of I_{λ} annihilates E/(f^{n})E; under our assumption (b), such power of I_{λ} contains another open ideal I_{μ}, so (8.6.29) is an object of nilExal_{A_{μ}}(B/(f^{n})B) whose image in nilExaltop_{A}(B) agrees with Σ. □

Proposition 8.6.30. Let A be a topological ring (whose topology is linear), B a noetherian A-algebra, I ⊂ B an ideal, and suppose that :

(a) The structure map A → B is continuous for the I-adic topology on B.

(b) For every open ideal J ⊂ A, the ideal J^{2} is also open. Then the following conditions are equivalent :

(c) B (with its I-adic topology) is a formally smooth A-algebra.

(d) Ω_{B/A}^{1} ⊗_{B} B/I is a projective B/I-module, and H_{1}(𝕃_{B/A} ⊗_{B} B/I) = 0.

Proof. More generally, let A and C be as in (8.6.20), and M a discrete C-module, i.e. a Cmodule annihilated by an open ideal; we denote by Exaltop_{A}(C, M) the C-module of square zero topological A-algebra extensions of C by M. Likewise, for every (λ, λ′) ∈ Λ′′, and every C_{λ′}-module M, let Exal_{A_{λ}}(C_{λ′}, M) be the C_{λ′}-module of isomorphism classes of square zero A_{λ}-algebra extensions of C_{λ′} by M. For (λ, λ′) ≤ (μ, μ′), we get a natural map of C_{μ′}-modules

(8.6.31)  Exal_{A_{λ}}(C_{λ′}, M) → Exal_{A_{μ}}(C_{μ′}, M)  for every C_{λ′}-module M

and (8.6.25) implies a natural isomorphism :

colim_{(λ,λ′)∈Λ′′} Exal_{A_{λ}}(C_{λ′}, M) →^{∼} Exaltop_{A}(C, M)

which is well defined for every discrete C-module M. By inspecting the definitions, and taking into account [63, Ch.0, Prop.19.4.3], it is easily seen that C is a formally smooth A-algebra if and only if Exaltop_{A}(C, M) = 0 for every such discrete C-module M. We also have a natural C-linear map :

(8.6.32)  Exaltop_{A}(C, M) → Exal_{A}(C, M)  for every discrete C-module M

and proposition 8.6.26(i) shows that (8.6.32) is an injective map, provided J_{λ′}^{2} is an open ideal of C, for every λ′ ∈ Λ′. Moreover, for C := B (with its I-adic topology), the map (8.6.32) is an isomorphism, by virtue of proposition 8.6.26(ii).

Now, recall the natural isomorphism of C_{λ}-modules ([103, Ch.III, Th.1.2.3])

Exal_{A}(B, M) →^{∼} Ext_{B}^{1}(𝕃_{B/A}, M) for every B-module M.

Combining with the foregoing, we deduce a natural B-linear isomorphism

Exaltop_{A}(B, M) →^{∼} Ext_{B}^{1}(𝕃_{B/A}, M)

for every B-module M annihilated by some ideal I^{k}. Summing up, B is a formally smooth Aalgebra if and only if Ext_{B}^{1}(𝕃_{B/A}, M) vanishes for every discrete B-module M. Set B_{0} := B/I; by considering the I-adic filtration on a given M, a standard argument shows that the latter condition holds if and only if it holds for every B_{0}-module M, and in turns, this is equivalent to the vanishing of Ext_{B_{0}}^{1}(𝕃_{B/A} ⊗_{B} B_{0}, M) for every B_{0}-module M. This last condition is equivalent to (d), as stated. □

We conclude this section with a list of topological corollaries of the conditions (a)_{f} − (f)_{f} that were introduced in (7.8.21).

8.6.33. Let A be a ring, f := (f_{1}, …, f_{r}) a finite sequence of elements of A that generate an ideal I ⊂ A. Let also (C_{•}, d_{•}) be any bounded above complex of flat A-modules such that, for every j ∈ ℤ the annihilator ideal of H_{j}C_{•} contains a power of I. Denote also by A^{∧} the I-adic completion of A, and by C_{•}^{∧} the complex whose term (resp. differential) in degree j is the I-adic completion of C_{j} (resp. of d_{j}), for every j ∈ ℤ. Lastly, set Q_{•} := Cone(A[0] → A^{∧}[0]), the cone of the morphism given by the completion map, and let f^{∧} be the image in A^{∧} of the sequence f.

Corollary 8.6.34. In the situation of (8.6.33), the following holds :

(i) If A satisfies condition (a)_{f} of (7.8.21), the natural maps

C_{•} → A^{∧} ⊗_{A} C_{•} → C_{•}^{∧}

are quasi-isomorphisms. (ii) The following conditions are equivalent : (a) A satisfies condition (a)_{f}.

(b) A^{∧} satisfies condition (a)_{f^{∧}} and Q_{•} ⊗^{L}_{A} A/I[0] is acyclic.

Proof. (i): According to [163, Th.3.5.8], for every i ∈ ℤ we have a natural short exact sequence

0 → L := lim_{n∈ℕ}^{1}H_{i+1}(C_{•} ⊗_{A} A/I^{n}) → H_{i}(C^{∧}) ⟶^{ω} lim_{n∈ℕ} H_{i}(C_{•} ⊗_{A} A/I^{n}) → 0.

On the other hand, the complex C_{•} fulfills condition (f)_{f} of (7.8.21), so the kernel and cokernel of the natural morphism of inverse systems (H_{i}(C_{•}) | n ∈ ℕ) → (H_{i}(C_{•} ⊗_{A} A/I^{n}) | n ∈ ℕ) are both essentially zero, hence the induced map

H_{i}(C_{•}) → lim_{n∈ℕ} H_{i}(C_{•} ⊗_{A} A/I^{n})

is an isomorphism for every i ∈ ℤ, and L = 0 ([163, Prop.3.5.7]). Hence, ω is an isomorphism, and it is easily seen that the resulting isomorphism H_{i}(C^{∧}) →^{∼} H_{i}(C) is the inverse of the map induced by the completion map C_{•} → C_{•}^{∧}, hence the latter is a quasi-isomorphism. Next we remark :

Claim 8.6.35. Let B be any ring, g a finite sequence of elements of B that generates an ideal J, and K_{•} a bounded above complex of B-modules. The following conditions are equivalent :

(a) K_{•} ⊗^{L}_{B} B/J[0] is acyclic.
(b) K_{•}(g, K_{•}) is acyclic.

Proof of the claim. (a)⇒(b): For every q ∈ ℕ we consider the spectral sequences

E_{pq}^{2} := H_{p}(K_{•} ⊗^{L}_{B} H_{q}K_{•}(g)[0]) ⇒ H_{p+q}(g, K_{•})
F (q)_{ij}^{2} := Tor_{i}^{B/J}(H_{j}(K_{•} ⊗^{L}_{B} B/J[0]), H_{q}K_{•}(g)) ⇒ E_{i+j,q}^{2} for every q ∈ ℤ.

Assumption (a) implies that F (q)_{ij}^{2} = 0 for every i, j, q ∈ ℤ, so that E_{pq}^{2} = 0 for every p ∈ ℤ,

whence (b). Conversely, if (b) holds, we show – by induction on n – that H_{n}(K_{•}⊗^{L}_{B}B/J[0]) = 0 for every n ∈ ℤ. First, since K_{•} is bounded above, we may find n_{0} ∈ ℤ such that the assertion holds for every n ≤ n_{0}. Next, let k > n_{0}, and suppose that the assertion is already known for every n < k. It follows that F (q)_{ij}^{2} = 0 whenever j < k, and clearly the same holds also whenever i < 0; thus F (q)_{0k}^{2} = E_{kq}^{2} for every q ∈ ℤ. By the same token, we deduce that E_{pq}^{2} = 0 for every p < k, and clearly the same holds whenever q < 0. Summing up, it follows

that E_{k0}^{2} = H_{k}(g, K_{•}) = 0; however, F (0)_{0k}^{2} = H_{k}(K_{•} ⊗^{L}_{B}B/J[0]), whence the contention. ♢

Claim 8.6.36. Suppose that A satisfies condition (a)_{f} of (7.8.21), let k ∈ ℕ be any integer, and N any A/I^{k}-module. The completion map A → A^{∧} induces an isomorphism

N →^{∼} A^{∧} ⊗_{A} N  and Tor_{i}^{A}(A^{∧}, N) = 0  for every i > 0.

Proof of the claim. Notice that, since K_{j}(f) is a free A-module of finite rank for every j ∈ ℤ, the natural map A^{∧} ⊗_{A} K_{•}(f) → K_{•}(f)^{∧} is an isomorphism. But we have already shown that the completion map K_{•}(f) → K_{•}(f)^{∧} is a quasi-isomorphism, hence K_{•}(f, Q_{•}) is acyclic, and in

light of claim 8.6.35we see that the same holds for Q_{•} ⊗^{L}_{A} A/I[0]. Let M be any A/I-module; from the change of rings spectral sequence

Tor_{i}^{A/I}(H_{j}(Q_{•} ⊗_{A} A/I[0]), M) ⇒ H_{i+j}(Q_{•} ⊗^{L}_{A} M[0])

we deduce that Q_{•} ⊗^{L}_{A} M[0] is also acyclic. Lastly, by considering, for t = 0, …, k − 1 the distinguished triangles

Q_{•} ⊗^{L}_{A} I^{t}N/I^{t+1}N[0] → Q_{•} ⊗^{L}_{A} N/I^{t+1}N[0] → Q_{•} ⊗^{L}_{A} N/I^{t}N[0] → Q_{•} ⊗^{L}_{A} I^{t}N/I^{t+1}N[1]

a simple induction shows that Q_{•} ⊗^{L}_{A} N[0] is acyclic as well; the latter assertion is equivalent to the claim. ♢

Now, let us consider the spectral sequence

E_{pq}^{2} := Tor_{p}^{A}(A^{∧}, H_{q}C_{•}) ⇒ H_{p+q}(A^{∧}[0] ⊗^{L}_{A} C_{•})

and notice that A^{∧}[0] ⊗^{L}_{A} C_{•} = A^{∧} ⊗_{A} C_{•}, since C_{•} is a complex of flat A-modules. By assumption, for every q ∈ ℤ there exists k ∈ ℕ such that I^{k} · H_{q}C_{•} = 0; taking into account claim 8.6.36we conclude that E_{pq}^{2} = 0 whenever p > 0, and E_{0q}^{2} = H_{q}C_{•} for every q ∈ ℤ; it is then easily seen that the resulting isomorphism H_{q}C_{•} →^{∼} H_{q}(A^{∧} ⊗_{A} C_{•}) is the map induced by the completion map A → A^{∧}.

(ii.a)⇒(ii.b): We have already remarked in the proof of (i) that if A satisfies condition (a)_{f},

then Q_{•} ⊗^{L}_{A} A/I[0] is acyclic. Next, define A_{r} and β_{f} : A_{r} → A as in (7.8.21), and consider the spectral sequence

E(n)_{ij}^{2} := H_{i}(Tor_{j}^{A_{r}}(A_{r}/I_{r}^{n}, A) ⊗^{L}_{A} Q_{•}) ⇒ H_{i+j}(A_{r}/I_{r}^{n} ⊗^{L}_{A_{r}} Q_{•}) for every n ∈ ℕ.

In light of claim 8.6.36we see that E(n)_{ij}^{2} = 0 for every i ∈ ℤ and j, n ∈ ℕ, so A_{r}/I_{r}^{n} ⊗^{L}_{A_{r}} Q_{•} is acyclic, and therefore the completion map A → A^{∧} induces quasi-isomorphisms

(8.6.37)  A_{r}/I_{r}^{n} ⊗_{A_{r}} A →^{∼} A_{r}/I_{r}^{n} ⊗_{A_{r}} A^{∧}  for every n ∈ ℕ

so A^{∧} satisfies (a)_{f}.

(ii.b)⇒(ii.a): Arguing as in the proof of claim 8.6.36we see that N ⊗^{L}_{A} Q_{•} is acyclic for every k ∈ ℕ and every A/I^{k}-module N. Then we find again E(n)_{ij}^{2} = 0 for every i ∈ ℤ and every j, n ∈ ℕ, as well as the quasi-isomorphisms (8.6.37), and the assertion follows. □

Corollary 8.6.38. In the situation of (7.8.35), set Z := Spec A/I, and suppose that U := Spec A \ Z is an affine scheme. Then \overline{A} satisfies condition (c)un \overline{f} of (7.8.21).

Proof. Endow A with its I-adic topology, and A_{U} := 𝒪_{U}(U) with the f-adic topology 𝒯_{U} provided by proposition 8.3.30(i). Clearly, the restriction map ρ_{U} : A → A_{U} factors through \overline{A}, so the latter is an open subring of A_{U}. Now, by definition there is no prime ideal of U that contains all the elements \overline{f}_{1}, …, \overline{f}_{r}; since U is affine by assumption, it follows that there exist g_{1}, …, g_{r} ∈ A_{U} such that ∑_{i=1}^{r} g_{i} · \overline{f}_{i} = 1 in A_{U}. Next, since \overline{A} is a ring of definition of A_{U}, and \overline{I} := I/J an ideal of adic definition, it follows that there exists k ∈ ℕ such that g_{i} · \overline{I}^{k} ⊂ \overline{A} for i = 1, …, r. It follows easily that the scalar multiplication by g_{i} induces an \overline{A}-linear map \overline{I}^{n+k} → \overline{I}^{n} for every n ∈ ℕ, and the latter in turn induces a map of complexes

ψ_{i,n} : K_{•}(\overline{f}, \overline{I}^{n+k}) → K_{•}(\overline{f}, \overline{I}^{n}) for every n ∈ ℕ and i = 1, …, r.

On the other hand, scalar multiplication by \overline{f}_{i} induces a map of complexes

φ_{i,n} : K_{•}(\overline{f}, \overline{I}^{n}) → K_{•}(\overline{f}, \overline{I}^{n}) for every n ∈ ℕ and i = 1, …, r

that equals the zero morphism in the homotopy category (see lemma 7.8.2(i)). Summing up, we conclude that ∑_{i=1}^{r} φ_{i,n} ∘ ψ_{i,n} is the zero morphism K_{•}(\overline{f}, \overline{I}^{n+k}) → K_{•}(\overline{f}, \overline{I}^{n}) in the homotopy category, and on the other hand it coincides with the inclusion map. The assertion follows. □

Remark 8.6.39. (i) Let A be a ring, J ⊂ A an ideal of finite type; endow A with its J-adic topology, denote by A^{∧} the completion of A, and set X_{A} := Spec A, X_{A^{∧}} := Spec A^{∧}. Let also U ⊂ X_{A} be an affine open subset containing the analytic locus of X_{A} (see definition 8.3.28), and set U^{∧} := U ×_{X_{A}} X_{A^{∧}}. Moreover, let

A_{U} := 𝒪_{X_{A}}(U) A_{U^{∧}} := 𝒪_{X_{A^{∧}}}(U^{∧}) τ_{A} := Ker(A → A_{U}) τ_{A^{∧}} := Ker(A^{∧} → A_{U^{∧}})

and denote by (τ_{A}A^{∧})^{c} the topological closure of τ_{A}A^{∧} in A^{∧}. Then we have

τ_{A^{∧}} ⊂ (τ_{A}A^{∧})^{c}.

Indeed, endow A_{U} with the f-adic topology characterized by proposition 8.3.30(i), and let A_{U}^{∧} be the completion of A_{U}; then A/τ_{A} is a ring of definition of A_{U}, hence its completion (A/τ_{A})^{∧} = A^{∧}/(τ_{A}A^{∧})^{c} is a ring of definition of A_{U}^{∧}, and the natural map A^{∧}/(τ_{A}A^{∧})^{c} ⊗_{A} A_{U} → A_{U}^{∧} is an isomorphism (propositions 8.2.13(iii) and 8.3.33(i,ii)). On the other hand, A_{U^{∧}} = A^{∧} ⊗_{A} A_{U}, so we get a commutative diagram of rings :

\xymatrix{A^{∧} \ar[r] \ar[d] & A^{∧}/(τ_{A}A^{∧})^{c} \ar[d] \\ A_{U^{∧}} \ar[r] & A_{U}^{∧}}

whose right vertical arrow is injective, whence the assertion.

(ii) In the situation of (i), say furthermore that X_{A} \ U = Spec A/I, for some ideal I ⊂ A generated by a finite sequence f := (f_{1}, …, f_{r}) of elements of A, and suppose that τ_{A} · I^{k} = 0 for some k ∈ ℕ. Then τ_{A} ∩ I^{n} = 0 for some n ∈ ℕ. Indeed, by virtue of corollary 8.6.38 and proposition 7.8.36, the ring A satisfies condition (c)_{f}^{un} of (7.8.21), and the proof of proposition 7.8.36shows that the sought identity follows from condition (c)_{f}. In particular, τ_{A} is a discrete subset of A, and the natural map τ_{A} → τ_{A}A^{∧} is an isomorphism; then, τ_{A} is (naturally identified with) a discrete subset of A^{∧} as well, and (i) implies that the natural map τ_{A} → τ_{A^{∧}} is an isomorphism as well.

(iii) Consider now the special case where J = bA for some b ∈ A. Then we claim more precisely that Ann_{A^{∧}}(b^{n}) is the topological closure of the image of Ann_{A}(b^{n}) in A^{∧}, for every n ∈ ℕ. Indeed, let x ∈ Ann_{A^{∧}}(b^{n}); for every k ∈ ℕ we may find x_{k} ∈ A and y_{k} ∈ A^{∧} such that x = x_{k} + b^{k}y_{k} in A^{∧}. Then b^{n}x_{k} ∈ (b^{n+k}A^{∧}) ∩ A = b^{n+k}A; say that b^{n}x_{k} = b^{n+k}z_{k} with z_{k} ∈ A. Hence the sequence (x_{k} − b^{k}z_{k} | k ∈ ℕ) lies in Ann_{A}(b^{n}) and converges b-adically to x, whence the contention.

Corollary 8.6.40. In the situation of (8.6.33), let B be a ring, φ : B → A a ring homomorphism, K ⊂ B and J ⊂ I two ideals. Set M := K ⊗_{B} A, and denote by

ψ : M → A

the A-linear map induced by φ. Suppose moreover that A satisfies condition (a)_{f} of (7.8.21), and the image of ψ is open in the I-adic topology of A. Then we have :

(i) There exists n ∈ ℕ such that I^{n}M ∩ Ker ψ = 0.

(ii) A is J-adically complete and separated if and only if the same holds for M.

Proof. (i): Notice first that Ker ψ = Tor_{1}^{B}(B/K, A). Especially, Ker ψ is naturally a B/Kmodule. By assumption, there exists k ∈ ℕ such that I^{k} ⊂ Im ψ = φ(K) · A; we deduce that Ker ψ is also an A/I^{k}-module. Next, we consider the short exact sequences

0 → Ker ψ → M → Im ψ → 0
0 → Im ψ → A → A/Im ψ → 0.

There follows exact sequences of inverse systems

T_{•} :=(Tor_{1}^{A_{r}}(A_{r}/I_{r}^{n}, Im ψ) | n ∈ ℕ)→K_{•} := (A/I^{n} ⊗_{A} Ker ψ | n ∈ ℕ)→(M/I^{n}M | n ∈ ℕ)
T′_{•} :=(Tor_{2}^{A_{r}}(A_{r}/I_{r}^{n}, A/Im ψ) | n ∈ ℕ) → T_{•} → T′′_{•} := (Tor_{1}^{A_{r}}(A_{r}/I_{r}^{n}, A) | n ∈ ℕ).

Now, since A satisfies condition (a)_{f}, the inverse system T′′_{•} is essentially zero, and the same holds for T′_{•}, by virtue of lemma 7.8.24; hence, T_{•} is essentially zero, by lemma 7.8.19. Moreover, K_{n} = Ker ψ for every n ≥ k, and the transition maps K_{n+1} → K_{n} are the identities. Then the assertion follows by a simple diagram chase : we leave the details to the reader.

(ii): Since Im ψ is open in A for the I-adic topology, it is also open in A for the J-adic topology, and the latter agrees with the topology induced by the J-adic topology of A; therefore, A is J-adically complete and separated if and only if the same holds for Im ψ. On the other hand, the J-adic topology on Im ψ also agrees with the quotient topology induced by the J-adic topology of M. Moreover, (ii) implies that the J-adic topology of M induces the discrete topology on Ker ψ. Then, proposition 8.2.13(i,v) yields a short exact sequence of J-adic completions :

0 → Ker ψ → M^{∧} → (Im ψ)^{∧} → 0

which shows that Im ψ is J-adically complete and separated if and only if the same holds for M, whence (ii). □

9. Complements of commutative algebra

This chapter is a miscellanea of results of commutative algebra that shall be needed in the rest of the treatise.

9.1. Valuation theory. This section is a selection of topics in valuation theory, and will be complemented by the section 9.2, devoted to Huber’s theory of the valuation spectrum.

Definition 9.1.1. Recall that an ordered abelian group is a datum (Γ, ·, 1, ≤) consisting of an abelian group (Γ, ·, 1) and a total ordering ≤ on Γ (see example 1.1.6(iii)) such that

γ ≤ γ′ ⇒ γ · γ′′ ≤ γ′ · γ′′ for every γ, γ′, γ′′ ∈ Γ.

(i) We denote by Γ^{+} ⊂ Γ the submonoid of all γ ∈ Γ such that γ ≤ 1.

(ii) A subgroup Δ of the ordered abelian group Γ is said to be convex if it satisfies the following condition. If δ ∈ Δ^{+} := Δ ∩ Γ^{+}, and γ ∈ Γ is any element with 1 ≥ γ ≥ δ, then γ ∈ Δ. The spectrum of Γ is the set of all convex subgroups of Γ, denoted

Spec Γ.

(iii) The convex rank of Γ, denoted

c.rk(Γ) ∈ ℕ ∪ {∞}

is the supremum over the lengths r of the chains 0 ⊂ Δ_{1} ⊊ ⋯ ⊊ Δ_{r} := Γ of convex subgroups of Γ. The rational rank of Γ is

rk(Γ) := dim_{ℚ}(Γ ⊗_{ℤ} ℚ) ∈ ℕ ∪ {∞}.

(iv) A morphism of ordered abelian groups is a group homomorphism that is an orderpreserving map for the underlying totally ordered sets.

Remark 9.1.2. Let (Γ, ·, 1, ≤) be any ordered abelian group.

(i) It is easily seen that Γ is a torsion-free abelian group. Moreover, we have quite generally

c.rk(Γ) ≤ rk(Γ)

(details left to the reader).

(ii) Notice as well that there exists a unique structure of ordered abelian group on Γ ⊗_{ℤ} ℚ such that the natural map Γ → Γ ⊗_{ℤ} ℚ is a morphism of ordered groups. Namely, we have γ ⊗ q ≤ 1 in Γ ⊗_{ℤ} ℚ whenever γ ≤ 1 in Γ and q ≥ 0. Moreover, for every q ∈ ℚ we have a well defined group endomorphism

Γ ⊗_{ℤ} ℚ → Γ ⊗_{ℤ} ℚ (γ ⊗ r) ↦ (γ ⊗ r)^{q} := γ ⊗ qr.

(iii) A subgroup Δ of Γ is convex if and only if there exists an ordered abelian group structure on Γ/Δ such that the projection Γ → Γ/Δ is a morphism of ordered abelian groups. Then the ordered abelian group structure with this property is unique. Moreover, notice that the intersection of an arbitrary family of convex subgroups of Γ is still convex. Especially, for every subgroup Δ of Γ, there exists a minimal convex subgroup of Γ containing Δ, called the convex hull of Δ; explicitly, it is the subgroup of all elements γ ∈ Γ such that there exists δ ∈ Δ^{+} with δ ≤ γ ≤ δ^{−1}.

(iv) There is a natural inclusion-reversing bijection

Spec Γ →^{∼} Spec Γ^{+} Δ ↦ Γ^{+} \ Δ^{+}

(notation of definitions 6.1.10(i) and 9.1.1(ii)). Indeed, it is easily seen that Γ^{+} \ Δ^{+} is a prime ideal of Γ^{+}, for every convex subgroup Δ of Γ. Conversely, for any prime ideal 𝔭 ⊂ Γ^{+}, the subset Δ(𝔭)^{+} := Γ^{+} \ 𝔭 is a submonoid, and we claim that the subgroup Δ(𝔭) generated by Δ(𝔭)^{+} is convex. Indeed, suppose that γ ∈ Δ(𝔭)^{+} and γ′ ∈ Γ^{+} is any other element such that γ ≤ γ′; then β := γ · γ^{′−1} ∈ Γ^{+}, therefore γ′ cannot lie in 𝔭, for otherwise the same would hold for γ = β · γ′. The contention follows.

(v) Let f : Γ → Γ′ be a morphism of ordered abelian groups. Then f induces a mapping

Spec Γ′ → Spec Γ Δ ↦ f^{−1}Δ.

For instance, if Δ ⊂ Γ is any convex subgroup, and π : Γ → Γ/Δ the projection, then Spec π : Spec Γ/Δ → Spec Γ is an injective map, whose image is the subset of all convex subgroups of Γ containing Δ. In the same vein, notice that the inclusion map i : Γ → Γ ⊗_{ℤ} ℚ induces a bijection

Spec i : Spec Γ ⊗_{ℤ} ℚ →^{∼} Spec Γ

(details left to the reader). Moreover, if f as above is injective, then Spec f is surjective : indeed, if Δ ⊂ Γ is any convex subgroup, let Δ′ be the convex hull of f(Δ) in Γ′; then it is easily seen that f^{−1}Δ′ = Δ.

(vi) If c.rk(Γ) = 1, we may find an injective morphism of ordered abelian groups

ρ : (Γ, ·, 1, ≤) → (ℝ, +, 0, ≤).

Indeed, pick an element γ ∈ Γ with γ > 1. For every δ ∈ Γ and every positive integer n, there exists a largest integer k(n) such that γ^{k(n)} < δ^{n}. Then (k(n)/n | n ∈ ℕ) is a Cauchy sequence and we let ρ(γ) := lim_{n→∞} k(n)/n. One verifies easily that ρ is an order-preserving group homomorphism, and since the convex rank of Γ equals one, it follows that ρ is injective.

(vii) To any γ ∈ Γ^{+} we may attach two convex subgroups

o(γ) := ⋂_{n∈ℕ}{δ ∈ Γ | γ ≤ δ^{n} ≤ γ^{−1}} and O(γ) := ⋃_{n∈ℕ}{δ ∈ Γ | γ^{n} ≤ δ ≤ γ^{−n}}.

Clearly o(γ) ⊂ O(γ), and if γ ≠ 1, the quotient Q(γ) := O(γ)/o(γ) has convex rank one (for its natural ordered group structure as in (iii)). Indeed, suppose that δ, μ are two elements of O(γ) with classes \overline{δ}, \overline{μ} ∈ Q(γ)_{+} and \overline{δ} ≠ 1; the assumptions mean that there exist n, k ∈ ℕ such that δ^{k} < γ and γ^{n} ≤ μ ≤ γ^{−n}. Then δ^{kn} ≤ μ; thus, the only convex subgroups of Q(γ) are Q(γ) and {1}, as claimed.

(viii) Standard examples of ordered abelian groups are (ℚ, +, 0, ≤) and (ℝ, +, 0, ≤); the latter is also isomorphic to the ordered group (ℝ_{>0}, ·, 1, ≤) of strictly positive real numbers. We shall also use the notation : ℚ_{>0} := ℚ ∩ ℝ_{>0}, ℝ_{+} := ℝ_{>0} ∪ {0} and ℚ_{+} := ℚ_{>0} ∪ {0}.

Definition 9.1.3. Let A be any ring, (Γ, ·, 1, ≤) any ordered abelian group; we extend the ordering and the composition law of Γ to Γ_{∘} := Γ ∪ {0} (cp. remark 4.8.14(i)), by the rule

0 < γ  and 0 · γ = 0 = γ · 0  for every γ ∈ Γ.

Moreover, for every q ∈ ℚ_{+} we extend to (Γ⊗_{ℤ}ℚ)_{∘} the q-th power operation of remark 9.1.2(ii), by declaring that 0^{0} := 1 and 0^{q} := 0 for every strictly positive q ∈ ℚ.

(i) A Γ-valued semi-norm on A is a mapping

v : A → Γ_{∘}

such that :

• v(0) = 0 and v(1) ≤ 1.

• v(a − b) ≤ max(v(a), v(b)) for every a, b ∈ A.
• v(ab) ≤ v(a) · v(b) for every a, b ∈ A.

The value group of v is the subgroup

Γ_{v}

of Γ generated by Im (v) \ {0}. The rank of v is the convex rank of its value group. (ii) We say that the semi-norm v is power-multiplicative if we have :

v(a^{n}) = v(a)^{n} for every a ∈ A and every integer n > 0.

(iii) We say that the semi-norm v is a valuation if we have :

v(1) = 1  and v(ab) = v(a) · v(b)  for every a, b ∈ A.

(iv) We say that v is a norm if v(a) ≠ 0 for every a ≠ 0 in A.

(v) We say that the semi-norm v is real-valued if its value group is a subgroup of ℝ_{>0}.

(vi) Let Γ and Γ′ be two ordered abelian groups, v (resp. v′) a Γ-valued (resp. Γ′-valued) semi-norm on A. We say that v is equivalent to v′ if there exist an ordered abelian group Γ′′ and a semi-norm v′′ : A → Γ′′, together with injective maps of ordered abelian groups φ : Γ′′ → Γ and φ′ : Γ′′ → Γ′ such that φ_{∘} ∘ v′′ = v and φ′_{∘} ∘ v′′ = v′ (notation of remark 4.8.14(i)).

Remark 9.1.4. Let A and Γ be as in definition 9.1.3, and | · |_{A} any Γ-valued semi-norm on A.

(i) Notice that |1|_{A} = |1 · 1|_{A} ≤ |1|_{A}^{2} ≤ |1|_{A}, whence |1|_{A} = |1|_{A}^{2}, so that either |1|_{A} = 1 or |1|_{A} = 0. In case |1|_{A} = 0, then |a|_{A} = |a · 1|_{A} ≤ |a|_{A} · 0 = 0, and in this case | · |_{A} is the trivial semi-norm with constant value 0.

(ii) We have |−a|_{A} = |a|_{A} for every a ∈ A. Indeed, |−a|_{A} = |0−a|_{A} ≤ max(|0|_{A}, |a|_{A}) = |a|_{A}; after replacing a by −a we get also |a|_{A} ≤ | − a|_{A}, whence the claim.

(iii) Let a, b ∈ A be any two elements such that |a|_{A} > |b|_{A}. Then we have |a + b|_{A} = |a|_{A}. Indeed, suppose that |a + b|_{A} < |a|_{A}; taking into account (ii) we get

|a|_{A} ≤ max(|a + b|_{A}, | − b|_{A}) < |a|_{A}

which is absurd.

(iv) Let S ⊂ A be any multiplicative subset such that |s|_{A} ≠ 0 for every s ∈ S. If | · |_{A} is a valuation, there exists a unique valuation

| · |_{S^{−1}A} : S^{−1}A → Γ_{∘}

whose composition with the localization map A → S^{−1}A agrees with | · |_{A}.

(v) Notice that the subsets

Ker (| · |_{A}) := {a ∈ A | |a|_{A} = 0} ⊂ (A, | · |_{A})^{+} := {a ∈ A | |a|_{A} ≤ 1}

are respectively an ideal and a subring of A. Clearly | · |_{A} is the composition of the projection

A → A/Ker (| · |_{A})

and a unique Γ-valued norm on A/Ker (| · |_{A}). Moreover, if | · |_{A} is a valuation, then Ker (| · |_{A}) is a prime ideal of A called the support of | · |_{A}, and we define the residue field of | · |_{A} as

κ(| · |_{A}) := Frac A/Ker (| · |_{A}).

In light of (iv), in this case | · |_{A} factors uniquely through the projection A → κ(| · |_{A}) and a valuation | · |_{κ} on κ(| · |_{A}), called the residual valuation of | · |_{A}. Conversely, if 𝔭 is any prime ideal of A, and | · |_{κ(𝔭)} is any valuation on the residue field κ(𝔭) of 𝔭, then the composition of | · |_{κ(𝔭)} with the projection A → κ(𝔭) is a valuation on A.

(vi) With the notation of (v), we claim that if | · |_{A} is a power-multiplicative semi-norm, then the subring (A, | · |_{A})^{+} is integrally closed in A. Indeed, let a ∈ A be any element that satisfies an identity of the type

a^{n} + b_{1}a^{n−1} + ⋯ + b_{n} = 0 with b_{1}, …, b_{n} ∈ (A, | · |_{A})^{+}.

It follows that |a|_{A}^{n} = |a^{n}|_{A} ≤ max(|b_{i}|_{A} · |a|_{A}^{n−i} | i = 1, …, n). Hence, say that |a|_{A}^{n} ≤ |b_{i}| · |a|_{A}^{n−i} for some i ≤ n; if |a| = 0, obviously a ∈ (A, | · |_{A})^{+}. Otherwise, we deduce |a|_{A}^{i} ≤ |b_{i}|_{A} ≤ 1, whence |a|_{A} ≤ 1, and the claim follows.

(vii) Let 𝔭 ⊂ A be any prime ideal. Then there exists a unique (up to equivalence) rank zero valuation on A with support 𝔭, defined by the rule :

|x| := \{ \begin{matrix}0 & if x ∈ 𝔭 \\ 1 & otherwise. \end{matrix}

Any valuation of this type is called a trivial valuation on A.

Example 9.1.5. (i) Let A be any ring, P a monoid, (Γ, ·, 1, ≤) an ordered abelian group, and φ : P → Γ_{∘} a morphism of monoids. We deduce a mapping (notation of (4.8.50))

v_{φ} : A[P] → Γ_{∘}  ∑_{x∈P} a_{x} · x ↦ max(φ(x) | x ∈ P, a_{x} ≠ 0)

(where the maximum of the empty set is taken to be 0 ∈ Γ_{∘}, so that w_{φ}(0) = 0). It is easily seen that v_{φ} is a Γ-valued seminorm on A[P]. Also, v_{φ} is a norm if and only if φ(P) ⊂ Γ.

(ii) Suppose moreover that A is a reduced ring, P is integral and P^{gp} is a torsion-free abelian group; then v_{φ} is valuation. Indeed, consider the first the case where A is a domain; then the same holds for A[P], since the latter is a subring of A[P^{gp}], which in turn is the filtered union of its subrings A[G], with G ranging over all free abelian subrings of finite rank contained in P^{gp}. Now, let

a := ∑_{x∈P} a_{x} · x b := ∑_{x∈P} b_{x} · x

be any two non-zero elements of A[P], and set α := v_{φ}(a) and β := v_{φ}; let also a′ := ∑_{φ(x)=α} a_{x} · x, and b′ := ∑_{φ(x)=β} b_{x} · x. It is easily seen that v_{φ}(ab) = v_{φ}(a′b′); on the other hand we have a′b′ ≠ 0 since A[P] is a domain, so we must have v_{φ}(a′b′) = αβ, whence the assertion. If A is any reduced ring, we have a natural inclusion A → ∏_{𝔭} A_{𝔭}, where 𝔭 ranges over all minimal prime ideals of A, and each factor A_{𝔭} is a field. Then we get likewise an inclusion A[P] → ∏_{𝔭} A_{𝔭}[P], and the assertion is easily reduced to the foregoing case : details left to the reader.

9.1.6. Real-valued semi-norms, of course, occur frequently in algebraic and analytic questions. The datum (A, | · |_{A}) of a ring and a real valued semi-norm | · |_{A} is usually called a (real valued) semi-normed ring. If | · |_{A} is power-multiplicative, one also says that (A, | · |_{A}) is a uniform semi-normed ring. A morphism of semi-normed rings f : (A, | · |_{A}) → (B, | · |_{B}) is a ring homomorphism f : A → B for which there exists a real number C ≥ 0 such that

(9.1.7)  |f(a)|_{B} ≤ C · |a|_{A}  for every a ∈ A.

Clearly a composition of morphisms of semi-normed ring is again a morphism of semi-normed rings, hence the semi-normed rings and their morphisms form a category 𝒮, and we let u.𝒮 ⊂ 𝒮 be the full subcategory whose objects are the uniform semi-normed rings. Notice that if f verifies (9.1.7) and (B, | · |_{B}) is a uniform semi-normed ring, then we have

|f(a)|_{B}^{n} = |f(a^{n})|_{B} ≤ C · |a^{n}| ≤ C · |a|^{n} for every n ∈ ℕ and a ∈ A

whence |f(a)|_{B} ≤ C^{1/n} · |a|_{A} for every such n and a; especially, (9.1.7) holds with C ≤ 1.

Lemma 9.1.8. Let (A, | · |) be any (real valued) semi-normed ring. Then the following holds : (i) The sequence (|a^{n}|^{1/n} | n ∈ ℕ) converges in ℝ for every a ∈ A. We set

|a|^{∗} := lim_{n→+∞} |a^{n}|^{1/n}.

(ii) The mapping | · |^{∗} : A → ℝ is a power-multiplicative semi-norm on A.

(iii) |a|^{∗} ≤ |a| for every a ∈ A.

(iv) The rule : (A, |·|) ↦ (A, |·|^{∗}) defines a left adjoint for the inclusion functor u.𝒮 → 𝒮.

Proof. (i): Fix a ∈ A, and set ν_{n} := |a^{n}|^{1/n} for every integer n > 0. Suppose first that ν_{k} = 0 for some k ∈ ℕ; it follows easily that ν_{n} = 0 for every n ≥ k, whence the assertion, in this case. Hence, we may assume that ν_{n} > 0 for every n ∈ ℤ; we notice that

ν_{kn} ≤ ν_{n} for every k, n > 0

and therefore

ν_{kn+r} ≤ (|a^{kn}| · |a^{r}|)^{1/(kn+r)} ≤ ν_{n}^{1−1/(k+1)} · |a^{r}|^{1/(nk+r)}

for every k, n, r ∈ ℕ with 0 ≤ r < n and k > 0. Now, for a given integer n > 0 and a real number ε > 0, pick k ∈ ℕ large enough, so that |a^{r}|^{1/(nk)} > 1 − ε for every r = 0, …, n − 1, and 1/(k + 1) < ε; it follows that

ν_{t} ≤ ν_{n}^{1−ε} · (1 − ε) for every integer t ≥ kn.

Consequently

lim sup_{n→+∞} ν_{n} = lim inf_{n→+∞} ν_{n}

whence the assertion.

(ii): Fix a, b ∈ A and a real number ε > 0; by (i), there exists k_{0} ∈ ℕ such that

|a^{k}|^{1/k} < |a|^{∗} · (1 + ε)  and |b^{k}|^{1/k} < |b|^{∗} · (1 + ε)  for every k > k_{0}.

Pick an integer t ≥ 2k_{0}, and large enough, so that

|a^{i}|^{1/n}, |b^{i}|^{1/n} < 1 + ε for every n ≥ t and every i ≤ k_{0}.

Now, we have :

|(a + b)^{n}|^{1/n} ≤ max{|a^{i}|^{1/n} · |b^{n−i}|^{1/n} | i = 0, …, n}.

On the other hand, we may assume that |a|^{∗} ≥ |b|^{∗}; set also

M := max((|a|^{∗} · (1 + ε))^{1−ε}, |a|^{∗} · (1 + ε)).

Then, for every n ∈ ℕ such that n ≥ t and k_{0}/n ≤ ε, we get

|a^{i}|^{1/n} · |b^{n−i}|^{1/n} ≤ \{ \begin{matrix}|a|^{∗} · (1 + ε) & whenever i, n − i > k_{0} \\ M · (1 + ε) & otherwise. \end{matrix}

We immediately deduce that |a + b|^{∗} ≤ max(|a|^{∗}, |b|^{∗}) for every a, b ∈ A.

The inequality |ab|^{∗} ≤ |a|^{∗} · |b|^{∗} for every a, b ∈ A is an immediate consequence of the corresponding property for | · |, and the power-multiplicative condition follows easily from the definition of | · |^{∗}.

(iii) follows easily by remarking that |a^{n}| ≤ |a|^{n} for every a ∈ A.

(iv): Let (B, | · |_{B}) be a uniform semi-normed ring, and f : (A, | · |) → (B, | · |_{B}) a morphism of semi-normed rings, so that (9.1.7) holds for some C ≥ 0. We get : |f(a)|_{B} = |f(a^{n})|_{B}^{1/n} ≤ C^{1/n} · |a^{n}|^{1/n} for every a ∈ A and n ∈ ℕ. After taking the limit for n → +∞, it follows that |f(a)|_{B} ≤ |a|^{∗} for every a ∈ A, so that f : (A, | · |^{∗}) → (B, | · |_{B}) is a morphism in u.𝒮. The assertion follows immediately. □

Example 9.1.9. (i) Let A be any ring, A_{0} ⊂ A a subring, I ⊂ A_{0} an ideal. We define, for every n ∈ ℕ, the A_{0}-submodule of A

I^{−n} := {a ∈ A | aI^{n} ⊂ A_{0}}

and we consider the order function associated with I

ν_{I} : A → ℤ ∪ {±∞} a ↦ sup{n ∈ ℤ | a ∈ I^{n}}

(where the supremum of the empty set is taken to be −∞). It is easily seen that • ν_{I}(a + b) ≥ min(ν_{I}(a), ν_{I}(b)) for every a, b ∈ A

• ν_{I}(ab) ≥ ν_{I}(a) + ν_{I}(b) for every a, b ∈ A
• ν_{I}(a) ≥ 0 if and only if a ∈ A_{0}.

Let also

A(I) := {a ∈ A | ν_{I}(a) ≠ −∞}.

It is easily seen that A(I) is a subring of A. Moreover, fix any real number ρ ∈]0, 1[, and set

|a|_{I} := ρ^{ν_{I}(a)} ∈ ℝ_{≥0}. for every a ∈ A(I)

(with the convention that ρ^{+∞} := 0). It is clear that | · |_{I} is a semi-norm on A(I). The corresponding power-multiplicative semi-norm | · |_{I}^{∗} provided by lemma 9.1.8 is sometimes called the asymptotic Samuel function of I.

(ii) In the situation of (i), let J ⊂ A_{0} be another ideal, with I ⊂ J. Then it is easily seen that, for every a ∈ A, we have either ν_{J}(a) ≥ ν_{I}(a) ≥ 0, or else 0 > ν_{I}(a) ≥ ν_{J}(a). Therefore

A(J) ⊂ A(I)

and the two foregoing conditions can be unified in the following assertion. For every a ∈ A(J) we have either ν_{J}(a) = 0, in which case ν_{I}(a) = 0 as well, or else

0 ≤ ν_{I}(a)/ν_{J}(a) ≤ 1.

We deduce a corresponding inequality for asymptotic Samuel functions : namely, for every a ∈ A(J) with |a|_{J}^{∗} ≠ 0 we have either |a|_{J}^{∗} = 1, in which case |a|_{I}^{∗} = 1 as well, or else

0 ≤ \frac{log |a|_{I}^{∗}}{log |a|_{J}^{∗}} ≤ 1.

(iii) For instance, for any integer n > 0 we have A(I) = A(I^{n}) and

ν_{I^{n}}(a^{n}) ≥ ν_{I}(a) ≥ n · ν_{I^{n}}(a) for every a ∈ A

from which it follows easily that

|a|_{I^{n}}^{∗} = (|a|_{I}^{∗})^{1/n} for every a ∈ A(I).

(iv) Let I and I′ two ideals of A_{0}, and suppose that the I-adic topology on A_{0} agrees with the I′-adic topology. Combining (ii) and (iii) we deduce that

A(I) = A(I′)

and for every a ∈ A(I) we have |a|_{I}^{∗} = 1 (resp. |a|_{I}^{∗} = 0) if and only if |a|_{I′}^{∗} = 1 (resp. |a|_{I′}^{∗} = 0); also, there exists a real number C ≥ 0 (independent of a) such that

0 ≤ \frac{log |a|_{I}^{∗}}{log |a|_{I′}^{∗}} , \frac{log |a|_{I′}^{∗}}{log |a|_{I}^{∗}} ≤ C whenever |a|_{I}^{∗} ≠ 1, 0.

(v) We notice as well that the asymptotic Samuel function is independent of the subring A_{0}. Namely, in the situation of (i), suppose that A′_{0} is another subring of A such that I is also an ideal of A′_{0}. Then we may set

I^{′n} := I^{n}  for every n > 0, and I^{′−k} := {a ∈ A | aI^{k} ⊂ A′_{0}}  for every k ≥ 0

(where I^{0} := A_{0}). Using the system of powers (I^{n} | n ∈ ℤ) we define the function ν_{I} and the subring A(I) as in (i), and with the system of powers (I^{′n} | n ∈ ℤ) we may define likewise the function ν_{I′} and the subring A(I′). After fixing ρ ∈]0, 1[, we then may define correspondingly the semi-norms |·|_{I} and |·|_{I′}, as well as the associated power multiplicative semi-norms |·|_{I}^{∗} and | · |_{I′}^{∗}. Obviously, for every a ∈ A we have ν_{I}(a) > 0 if and only ν_{I′}(a) > 0, and ν_{I}(a) = ν_{I′}(a) for every a ∈ A such that ν_{I}(a) > 0. Moreover, suppose that ν_{I}(a) = −n for some n ∈ ℕ; this means especially that aI^{n} ⊂ A_{0}, so that aI^{n+1} ⊂ I ⊂ A′_{0}, whence ν_{I}(a) ≥ −n − 1. It follows easily that

A(I) = A(I′) and |a|_{I}^{∗} = |a|_{I′}^{∗} for every a ∈ A(I).

(vi) It follows especially from (v) that the subring (A(I), | · |_{I}^{∗})^{+} of A(I) is independent of A_{0} (notation of remark 9.1.4(v)), and moreover, if B ⊂ A is any subring such that I is also an ideal of B, then B ⊂ (A(I), | · |_{I}^{∗})^{+}. More generally, we have :

Proposition 9.1.10. In the situation of example 9.1.9(i), consider any subring B of A with

I ⊂ B ⊂ (A(I), | · |_{I}^{∗})^{+}.

Then the following holds :

(i) A(I) ⊂ A(IB) and |a|_{I}^{∗} = |a|_{IB}^{∗} for every a ∈ (A(I), | · |_{I}^{∗})^{+}.

(ii) Suppose moreover that I is a finitely generated ideal of A_{0}. Then A(I) = A(IB) and |a|_{I}^{∗} = |a|_{IB}^{∗} for every a ∈ A(I).

Proof. (i): Arguing as in example 9.1.9(v), we easily see that ν_{IB}(a) ≥ ν_{I}(a) for every a ∈ A such that ν_{I}(a) > 0, and ν_{IB}(a) ≥ ν_{I}(a) − 1 if ν(a) ≤ 0. This already implies the stated inclusion of subrings, as well as the inequality

(9.1.11)  |a|_{IB}^{∗} ≤ |a|_{I}^{∗}  for every a ∈ A(I).

Now, suppose that |a|_{IB}^{∗} = ρ^{c} for some real number c > 0. Set n(k) := ν_{IB}(a^{k}) for every k ∈ ℕ. It follows that lim_{k→+∞} n(k)/k = c, and especially n(i) > 0 for every sufficiently large i ∈ ℕ. It suffices to check that |a^{i}|_{I}^{∗} ≤ ρ^{ic}, so we may replace a by a^{i} for such sufficiently large i, and assume from start that ν_{IB}(a) ≥ 0. In this situation, for every k ∈ ℕ we may then write a^{k} = ∑_{j=1}^{r} a_{i}b_{i}, for some a_{1}, …, a_{r} ∈ I^{n(k)} and b_{1}, …, b_{r} ∈ B. Since by assumption |b_{i}|_{I}^{∗} ≤ 1 for every i ≤ r, taking into account lemma 9.1.8(ii,iii) we deduce that

|a^{k}|_{I}^{∗} ≤ sup(|x|_{I}^{∗} | x ∈ I^{n(k)}) ≤ sup(|x|_{I} | x ∈ I^{n(k)}) ≤ ρ^{n(k)}

whence |a|_{I}^{∗} ≤ ρ^{n(k)/k} for every k ∈ ℕ, and therefore |a|_{I}^{∗} ≤ |a|_{IB}^{∗}.

(ii): Indeed, we know already that A(I) ⊂ A(IB), and to show the converse inclusion, pick any finite system of generators (t_{1}, …, t_{r}) for I, and let a ∈ A(IB) be any element. This means that a · (IB)^{n} ⊂ B ⊂ A(I) for some n ∈ ℕ. Hence we may find m ∈ ℕ such that

a · t_{1}^{k_{1}} ⋯ t_{r}^{k_{r}} · I^{m} ⊂ A_{0} for every (k_{1}, …, k_{r}) ∈ ℕ^{r} with k_{1} + ⋯ + k_{r} = n

whence a · I^{m+n} ⊂ A_{0}, so a ∈ A(I), as stated. Next, to conclude, in light of (9.1.11), it suffices to show that |a|_{I}^{∗} ≤ |a|_{IB}^{∗}, for every a ∈ A(I). However, in case |a|_{IB}^{∗} = ρ^{c} for some real number c > 0, we may argue as in the proof of (i) to deduce the sought inequality. We may therefore assume that |a|_{IB}^{∗} ≤ ρ^{−c} for some real number c > 0, and we have to show that |a|_{I}^{∗} ≤ ρ^{−c} as well. Define n(k) as in the proof of (i), for every k ∈ ℕ. Notice that, if n(k) > 0 for some k ∈ ℕ, we have |a^{k}|_{IB} ≤ ρ, whence |a|_{IB}^{∗} ≤ ρ^{1/k}, in which case the assertion is already known. Hence, we may further assume that n(k) ≤ 0 for every k ∈ ℕ, in which case

lim_{k→+∞} −n(k)/k ≤ c  and a^{k} · (IB)^{−n(k)} ⊂ B  for every k ∈ ℕ.

Pick any integers p, q > 0 such that c < p/q; it follows easily that

a^{nq} · (IB)^{np−1} ⊂ B for every sufficiently large n ∈ ℕ.

Fix such a sufficiently large n ∈ ℕ; we deduce

x(k_{•}) := a^{nq} · t_{1}^{k_{1}} ⋯ t_{r}^{k_{r}} ∈ IB for every k_{•} := (k_{1}, …, k_{r}) ∈ ℕ^{r} with k_{1} + ⋯ + k_{r} = np

in which case |x(k_{•})|_{IB}^{∗} < 1, and therefore |x(k_{•})|_{I}^{∗} < 1 as well, by the foregoing, for every such k_{•}. Then we may find m ∈ ℕ such that

a^{nmq} · t_{1}^{mk_{1}} ⋯ t_{r}^{mk_{r}} ∈ A_{0} for every k_{•} := (k_{1}, …, k_{r}) ∈ ℕ^{r} with k_{1} + ⋯ + k_{r} = np.

However, let J ⊂ A_{0} be the ideal generated by t_{1}^{m}, …, t_{r}^{m}, and notice that

I^{(m−1)(r+nps)+1} ⊂ J^{nps} for every s ∈ ℕ.

We conclude that

a^{nmqs} · I^{(m−1)(r+nps)+1} ⊂ A_{0} for every s ∈ ℕ

from which we see that

lim_{s→+∞} −ν_{I}(a^{nmqs})/(nmqs) ≤ lim_{s→+∞}((m − 1)(r + nps) + 1)/(nmqs) < p/q

and since the rational number p/q can be taken arbitrary close to c, the sought inequality follows. □

Definition 9.1.12. Let F be a field.

(i) We say that a subring V ⊂ F is a valuation ring of F if the following holds. For every x ∈ F^{×} we have either x ∈ V or x^{−1} ∈ V.

(ii) Let (A, 𝔪_{A}), (B, 𝔪_{B}) be two local subrings of F. We say that B dominates A if A ⊂ B and A ∩ 𝔪_{B} = 𝔪_{A}.

Remark 9.1.13. (i) Let K be any field; notice that any semi-norm, and especially, any valuation on K is necessarily a norm.

(ii) Let | · |_{K} be any such valuation, and let K^{+} := (K, | · |_{K})^{+}, as in remark 9.1.4(v). Then it is easily seen that K^{+} is a valuation ring of K. Also, the invertible elements of of K^{+} are precisely those x ∈ K such that |x|_{K} = 1, and the set {x ∈ K | |x|_{K} < 1} is the unique maximal ideal of K^{+}; especially, K^{+} is a local domain.

(iii) Furthermore, K^{+} is integrally closed. Indeed, let x ∈ K be integral over K^{+}, so that x^{n} + y_{1}x^{n−1} + ⋯ + y_{n} = 0 for some y_{1}, …, y_{n} ∈ K^{+}, and suppose, by way of contradiction, that x ∉ K^{+}, i.e. that |x|_{K} > 1; it follows easily that |x^{n}|_{K} > |y_{i}x^{n−i}|_{K} for every i = 1, …, n, so that remark 9.1.4(iii) yields |x^{n} + y_{1}x^{n−1} + ⋯ + y_{n}|_{K} = |x^{n}|_{K} ≠ 0, which is absurd.

(iv) Conversely, if V is any valuation ring of K, set Γ := K^{×}/V^{×}. Thus, Γ is an abelian group, and we define an ordering on Γ as follows. For given classes \overline{x}, \overline{y} ∈ Γ, we declare that \overline{x} ≤ \overline{y} if and only if y^{−1}x ∈ V. Then it is easily seen that (Γ, ≤) is an ordered abelian group, and the projection K^{×} → Γ is the restriction to K^{×} of a well defined valuation | · |_{K} of K, such that V = (K, | · |_{K})^{+}, so the maximal ideal 𝔪_{V} of V and the subset V^{×} are also described as in (ii) (details left to the reader).

(v) If the valuation | · |_{K} is a surjective map, we say that (K, | · |_{K}) is a valued field. In view of (iv), we see that a valuation ring of K is the same as the datum of the equivalence class (in the sense of definition 9.1.3(vi)) of a valued field (K, | · |_{K}).

(vi) Let A be any ring, | · | and | · |′ two valuations on A; in light of (v), we see that the following conditions are equivalent :

(a) the valuations | · | and | · |′ are equivalent

(b) | · | and | · |′ have the same support and (κ(| · |), | · |_{κ})^{+} = (κ(| · |′), | · |′_{κ})^{+}

(c) for every a, b ∈ A we have |a| ≤ |b| if and only if |a|′ ≤ |b|′. (vii) Let (K, | · |) be any valued field, and Γ the value group of | · |. There is an inclusion-

reversing bijection

Spec Γ →^{∼} Spec K^{+}

that assigns to every convex subgroup Δ ⊂ Γ the prime ideal

𝔭(Δ) := {x ∈ K^{+} | γ > |x| for every γ ∈ Δ}

and whose inverse assigns, to every prime ideal 𝔭 ⊂ K^{+}, the convex subgroup

Δ(𝔭) := {γ ∈ Γ | γ, γ^{−1} > |x| for every x ∈ 𝔭}.

Indeed, it is easily seen that 𝔭(Δ) is a prime ideal, and we check that Δ(𝔭) is a convex subgroup. First, say that γ, γ′ ∈ Δ(𝔭); if either γ > 1 or γ′ > 1, it is clear that γ · γ′ > |x| for every x ∈ 𝔭. If γ, γ′ ≤ 1, we may find y, y′ ∈ K^{+} such that γ = |y| and γ′ = |y′|, and by assumption, neither y nor y′ lie in 𝔭, so also y · y′ ∉ 𝔭, and therefore γ · γ′ > |x|_{K} for every x ∈ 𝔭. Hence, Δ(𝔭) is a subgroup of Γ, and the convexity of Δ(𝔭) is obvious. Next, set 𝔮 := 𝔭(Δ); directly from the definitions we see that Δ ⊂ Δ(𝔮). Conversely, let γ ∈ Γ^{+} \ Δ; then γ = |y| for some y ∈ K^{+}, and since γ < δ for every δ ∈ Δ, we get y ∈ 𝔮, so that γ ∉ Δ_{𝔮}. Hence, Δ = Δ(𝔮), and especially, the rule 𝔭 ↦ Δ(𝔭) is surjective from Spec K^{+} to Spec Γ; but it is easily seen that this rule is also injective, whence the contention.

Definition 9.1.14. Let (K, | · |_{K}) be any valued field, and Γ the value group of | · |_{K}.

(i) For every γ ∈ Γ^{+}, set U_{γ} := {x ∈ K^{+} | |x|_{K} < γ}. The valuation topology of K^{+} is the unique linear topology 𝒯_{K^{+}} that admits the family {U_{γ} | γ ∈ Γ^{+}} as a fundamental system of open neighborhoods of 0.

(ii) In view of remark 8.3.2(i) there exists a unique ring topology 𝒯_{K} on K such that : (a) K^{+} is an open subring of K

(b) 𝒯_{K} induces on K^{+} the valuation topology 𝒯_{K^{+}}.

We call 𝒯_{K} the valuation topology of K. (iii) We say that (K, | · |_{K}) is a Tate valued field, if the valuation topology of K is f-adic and

not discrete. (iv) Let A be any ring, and v any valuation on A. We say that v is a Tate valuation if the

residual valued field (κ(v), \overline{v}) is a Tate valued field.

Remark 9.1.15. The class of Tate valuations was originally introduced by R.Huber in [98], where they are called microbial valuations.

Proposition 9.1.16. With the notation of definition 9.1.14, let (K^{+∧}, 𝒯_{K^{+}}^{∧}) (resp. (K^{∧}, 𝒯_{K}^{∧})) be the completion of the topological ring (K^{+}, 𝒯_{K^{+}}) (resp. (K, 𝒯_{K})). We have :

(i) Let 𝒯 ′ be any topology on K^{+} that is linear and separated. Then either 𝒯 ′ = 𝒯_{K^{+}}, or else 𝒯 ′ is the discrete topology.

(ii) (K, | · |_{K}) is a Tate valued field if and only if (K, 𝒯_{K}) is a Tate topological ring.

(iii) K^{+∧} is a valuation ring and 𝒯_{K^{+}}^{∧} is the valuation topology on K^{+∧}.

(iv) K^{∧} is the field of fractions of K^{+}, and 𝒯_{K}^{∧} is the valuation topology on K^{∧}.

(v) The valuation | · |_{K} : K → Γ_{∘} extends uniquely to a valuation | · |_{K}^{∧} : K^{∧} → Γ_{∘} whose valuation ring is K^{+∧}.

Proof. (i): Let x ∈ K^{+} \ {0} be any element; since 𝒯 ′ is separated, there exists an ideal I ⊂ K^{+} that is open for the topology 𝒯 ′, and such that x ∉ I. Then I ⊂ U_{γ}, with γ := |x|_{K} (notation of definition 9.1.14(i)), so 𝒯 ′ is finer than 𝒯_{K^{+}}. If 𝒯 ′ is not the discrete topology, there exists y ∈ I \ {0}, and therefore U_{δ} ⊂ I, with δ := |y|_{K}, so in this case 𝒯 ′ = 𝒯_{K^{+}}.

(ii): Suppose that 𝒯_{K} is f-adic; then, since K^{+} is a bounded subset of K, the valuation topology 𝒯_{K^{+}} must be adic with a finitely generated ideal of adic definition I (proposition 8.3.18(ii)). Such I is necessarily principal, and I ≠ 0, if 𝒯_{K} is not discrete; in this case, any generator of I is a topologically nilpotent unit in K. Conversely, if a ∈ K is a topologically nilpotent unit for the valuation topology of K, then a ∈ K^{+}, and it is easily seen that the valuation topology of K^{+} agrees with the K^{+}a-adic topology, so 𝒯_{K^{+}} is adic and 𝒯_{K} is f-adic.

(iii): Let a_{•} := (a_{i} | i ∈ I) be any Cauchy net in K (indexed by a filtered ordered set I); for every γ ∈ Γ, pick i(γ) ∈ I such that |a_{j} − a_{k}|_{K} < γ for every j, k ≥ i(γ). Suppose first that |a_{i(γ)}|_{K} ≤ γ for every γ ∈ Γ; then |a_{j}|_{K} ≤ γ for every j ≥ i(γ) and every γ ∈ Γ, i.e. a_{•} converges to 0, and we set |a_{•}|_{K}^{∧} := 0. Otherwise, there exists δ ∈ Γ such that |a_{i(δ)}|_{K} > δ and then |a_{j}|_{K} = δ for every j ≥ i(δ), in which case we set |a_{•}|_{K}^{∧} := |a_{i(δ)}|_{K}. It is easily seen that |a_{•}|_{K}^{∧} depends only on the class of a_{•} in K^{∧}, and if b_{•} is any other Cauchy sequence, then

|(a_{i}b_{i} | i ∈ I)|_{K}^{∧} = |a_{•}|_{K}^{∧} · |b_{•}|_{K}^{∧} and |(a_{i} + b_{i} | i ∈ I)|_{K}^{∧} ≤ max(|a_{•}|_{K}^{∧}, |b_{•}|_{K}^{∧})

(details left to the reader). It follows already that K^{∧} is a domain. Moreover, we have |a_{•}|_{K}^{∧} ≤ 1 if and only if the class of a_{•} is equivalent to the class of a Cauchy net in K^{+}. Next, suppose that 1 ≥ |a_{•}|_{K}^{∧} ≥ |b_{•}|_{K}^{∧} > 0, in which case there exists i_{0} ∈ I such that

|a_{j}|_{K} ≥ |b_{j}|_{K}  and |a_{j}|_{K} = β := |a_{•}|_{K}^{∧}  for every j ≥ i_{0}

and set v_{j} := 1 for every j ∈ I such that j < i_{0} and v_{j} := a_{j}^{−1}b_{j} for every j ≥ i_{0}. We claim that v_{•} := (v_{j} | j ∈ I) is a Cauchy net in K^{+} : indeed, for any γ ∈ Γ, pick i(γ) ∈ I such that |a_{j} −a_{k}|_{K}, |b_{j} −b_{k}|_{R} < γ for every j, k ≥ i(γ); a simple calculation shows that |v_{j} −v_{k}| < γ ·β^{2} whenever j, k ≥ max(i_{0}, i(γ)), as required. Now, since (v_{j}a_{j} | j ∈ I) and b_{•} represent the same element in K^{+∧}, it follows that the class of a_{•} divides the class of b_{•} in K^{+∧} (the quotient is the class of v_{•}). Lastly, (i) easily implies that 𝒯_{K^{+}}^{∧} is the valuation topology on K^{+∧}, whence the assertion.

(v): Furthermore, notice that if the class x of a_{•} in K^{∧} is not 0, we have x = a_{i(δ)} · y, with y an invertible element of K^{+∧}, so we must have |x|′ = |a_{i(δ)}|_{K} for every valuation | · |′ on K^{∧} that extends | · |_{K}.

(iv): By the same token, we have x^{−1} = a_{i(δ)}^{−1} · y^{−1}, which shows that K is a field, and concludes the proof. □

Proposition 9.1.17. Let (K, |·|_{K}) be a valued field such that the valuation ring K^{+} is henselian, K^{s} a separable closure of K, and (K^{∧}, | · |_{K}^{∧}, 𝒯_{K}^{∧}) the completion of (K, | · |_{K}, 𝒯_{K}). Then :

(i) The valuation ring K^{∧+} of K^{∧} is henselian.

(ii) The ring K^{∧s} := K^{∧} ⊗_{K} K^{s} is a field, and is a separable closure of K^{∧}.

(iii) Especially, we have a natural group isomorphism :

Gal(K^{s}/K) →^{∼} Gal(K^{∧s}/K^{∧}) σ ↦ K^{∧} ⊗_{K} σ.

Proof. (i): The assertion follows immediately from the more general :

Claim 9.1.18. Let (A_{•}, J_{•}) := (A_{λ}, J_{λ}) | λ ∈ Ob(Λ)) be a system of henselian pairs indexed by any small category, with transition morphisms f_{φ} : A_{λ} → A_{μ} such that f_{φ}(J_{λ}) ⊂ J_{μ} for every morphism φ : λ → μ of Λ. Let A (resp. J) be the limit of the system A_{•} (resp. of the system J_{•}); then the pair (A, J) is henselian.

Proof of the claim. Let 𝒜 be the category whose objects are all the pairs (B, I), where B is a ring and I ⊂ B an ideal, and whose morphisms (B, I) → (B′, I′) are all the ring homomorphisms f : B → B′ with f(I) ⊂ I′. Denote by 𝒜^{h} the full subcategory of 𝒜 whose objects are the henselian pairs. Then the inclusion functor 𝒜^{h} → 𝒜 admits a left adjoint, which assigns to every pair (B, I) its henselization (B^{h}, I^{h}). It is easily seen that 𝒜 is complete, so the claim follows directly from corollary 1.3.26. ♢

(ii): Let K^{a} be an algebraic closure of K containing K^{s} and contained in an algebraic closure K^{∧a} of K^{∧}, and notice that the valuation | · |_{K} extends uniquely to a valuation | · |_{E} on every subextension E ⊂ K^{a}, since K^{+} is henselian ([75, Rem.6.1.12(iv)]). Especially, for every such E and every K-algebra homomorphism σ : E → K^{a} we must have

|a|_{E} = |σ(a)|_{K^{a}} for every a ∈ E.

Claim 9.1.19. (Krasner’s lemma) With the foregoing notation, let a ∈ K^{s}, b ∈ K^{a}, and suppose that for every homomorphism σ : K[a] → K^{s} of K-algebras with σ(a) ≠ a we have

|b − a|_{K^{s}} < |σ(a) − a|_{K^{s}}.

Then K[a] ⊂ K[b].

Proof of the claim. It suffices to show that for every K[b]-algebra homomorphism τ : K[a, b] → K^{a} we have τ(a) = a. However, suppose that τ(a) ≠ a for such a map τ; we deduce that

|b − a|_{K^{a}} < |τ(a) − a|_{K^{s}} and |b − τ(a)|_{K^{a}} = |τ(b − a)|_{K^{a}} = |b − a|_{K^{a}}

whence :

|τ(a) − a|_{K^{s}} = |τ(a) − b + b − a|_{K^{s}} ≤ max(|τ(a) − b|_{K^{a}}, |b − a|_{K^{a}}) < |τ(a) − a|_{K^{s}}

a contradiction. ♢

Let now K^{∧} ⊂ E^{∧} be a non-trivial finite Galois extension with E^{∧} ⊂ K^{∧a}, and pick a ∈ E^{∧} such that E^{∧} = K^{∧}[a]. Let P (X) ∈ K^{∧}[X] be the minimal polynomial of a over K^{∧}; in light of (i) and [75, Rem.6.1.12(iv)], the valuation |·|_{K}^{∧} of K^{∧} extends uniquely to a valuation of |·|_{E}^{∧} of E^{∧}; with this notation, we set

G := Gal(E^{∧}/K^{∧}) γ := min(|σ(a) − a|_{E}^{∧} | σ ∈ G \ 1_{E^{∧}}) ∈ |E^{∧}|_{E}^{∧}.

Say that P = X^{n} + c_{1}X^{n−1} + ⋯ + c_{n} and pick a polynomial Q(X) ∈ K[X] of degree n, say Q = X^{n} + d_{1}X^{n−1} + ⋯ + d_{n} such that

|c_{i} − d_{i}|_{K}^{∧} < γ^{n}/|a|_{E}^{n−i} for i = 1, …, n.

It follows easily that |Q(a)|_{E}^{∧} < γ^{n}. Then, say that Q(X) = ∏_{i=1}^{n}(X −b_{i}) for some b_{1}, …, b_{n} ∈ K^{a}; we deduce that |a − b| < γ for some b ∈ {b_{1}, …, b_{n}}, in which case claim 9.1.19 implies that E^{∧} ⊂ K^{∧}[b], and since P and Q have the same degree, we see that Q is irreducible over K^{∧} and E^{∧} = K^{∧}[b]. Especially, b is separable over K^{∧}, so the roots of Q are all distinct, and therefore b is also separable over K. Summing up, we have already shown that K^{∧}K^{s} ⊂ K^{∧a} is a separable closure of K^{∧}. It remains to check that the natural map K^{∧} ⊗_{K} K^{s} → K^{∧}K^{s} is injective. To this aim, let K ⊂ E be any finite Galois extension with E ⊂ K^{s}, and denote by tr_{E/K} : E → K the K-linear trace map, so that tr_{E/K}(a) = ∑_{σ∈Gal(E/K)} σ(a). Clearly tr_{E/K} is continuous for the valuation topologies 𝒯_{E} and 𝒯_{K} of E and K; on the other hand, since E is separable over K, the trace map induces a perfect pairing E⊗_{K}E → K : x⊗y ↦ tr_{E/K}(xy) for every x, y ∈ E. Hence, let x_{1}, …, x_{n} be a basis of the K-vector space E, and y_{1}, …, y_{n} ∈ E the unique elements such that tr_{E/K}(x_{i}y_{j}) = δ_{ij} for every i, j = 1, …, n. There follow Klinear isomorphisms

K^{⊕n} ⟶^{φ} E ⟶^{ψ} K^{⊕n} such that ψ ∘ φ = 1_{K^{⊕n}}

where φ(a_{1}, …, a_{n}) := ∑_{i=1}^{n} a_{i}x_{i} and ψ(b) := (tr_{E/K}(by_{1}), …, tr_{E/K}(by_{n})) for every b ∈ E and every (a_{1}, …, a_{n}) ∈ K^{⊕n}. Clearly both φ and ψ are continuous for the topology 𝒯_{E} and the product topology on K^{⊕n}, hence they are both homeomorphisms. Let (E^{∧}, 𝒯_{E}^{∧}) be the completion of (E, 𝒯_{E}); we deduce a commutative diagram of K^{∧}-linear maps

\xymatrix{K^{∧} ⊗_{K} E \ar[r]^{K^{∧}⊗_{K}ψ} \ar[d]_{μ} & K^{∧} ⊗_{K} K^{⊕n} \ar[d] \\ E^{∧} \ar[r]^{ψ^{∧}} & (K^{∧})^{⊕n}}

where ψ^{∧} is the completion of ψ and μ is the multiplication map; especially both horizontal arrows are bijections, and the same holds for the right vertical arrow. Then μ is bijective as well, whence the contention.

(iii) is an immediate consequence of (ii). □

Remark 9.1.20. The left adjoint property of the topological henselization A ↦ A^{h} implies that for every f-adic ring A, the identity map 1_{A^{h}} is the unique continuous A-algebra endomorphism of A^{h}. However, A^{h} in general will admit also non-continuous A-algebra endomorphisms. For instance, let K be an algebraically closed field of characteristic zero, and set

F_{0} := K[ℚ_{+}] = K[T^{1/n} | n ∈ ℕ \ {0}] F := Frac F_{0}.

Let also v : F_{0} → ℚ_{∘} be the valuation of F_{0} associated with the inclusion map ℚ_{+} → ℚ as in example 9.1.5 (where ℚ is endowed with its standard ordering). Then F^{+} := κ(v)^{+} is a valuation ring of F, and (F, v) is a Tate valued field. Clearly the value group of v is ℚ and the residue field is K. Let 𝔪_{F} ⊂ F^{+} be the maximal ideal, and (F^{+h}, 𝔪_{F}^{h} ) the henselization of the pair (F^{+}, 𝔪); then F^{+h} is a valuation ring with the same value group ℚ, and the same residue field K ([75, Lemma 6.2.5]). Moreover, the topological henselization F^{h} of the Tate valued field F is the field of fractions of F^{+h}. It then follows easily from [75, Prop.6.2.12] that F^{h} is algebraically closed; on the other hand, it is easily seen that F is not algebraically closed, so there exists a non-trivial automorphism of F^{h} that fixes F.

Theorem 9.1.21. Let K be a field, A ⊂ K a subring, and 𝔭 ⊂ A a prime ideal of A. Then there exists a valuation ring (V, 𝔪_{V}) of K such that

A ⊂ V  and  𝔭 = 𝔪_{V} ∩ A.

Proof. We may replace A by A_{𝔭}, and assume from start that A is a local ring. Denote by ℱ the set of all subrings B of K containing A and such that 1 ∉ 𝔭B; we endow ℱ with the partial ordering given by inclusion of subrings. Then A ∈ ℱ, and if 𝒢 is a non-empty totally ordered subset of ℱ, then ⋃_{B∈𝒢} B still lies in ℱ. By Zorn’s lemma, it follows that ℱ admits a maximal element V. First, we claim that V is local. Indeed, by assumption 1 ∉ 𝔭V, so 𝔭V is contained in a maximal ideal 𝔪 of V, and 1 ∉ 𝔭V_{𝔪}, so V_{𝔪} ∈ ℱ, and therefore V = V_{𝔪}, since V is maximal. Let 𝔪_{V} be the maximal ideal of V; by construction, 𝔭 ⊂ 𝔪_{V} ∩ A, and since 𝔭 is maximal in A, we get 𝔭 = 𝔪_{V} ∩ A. It remains only to check that V is a valuation ring of K. Thus, say that x ∈ K and x ∉ V; by maximality of V, we have V [x] ∉ ℱ, so 1 ∈ 𝔭V [x], i.e. there exists an identity of the form 1 = a_{0} + a_{1}x + ⋯ + a_{n}x^{n} for some a_{0}, …, a_{n} ∈ 𝔭V ⊂ 𝔪_{V}. Since 1 − a_{0} ∈ V^{×}, we deduce a relation of the form

(9.1.22)  1 = b_{1}x + ⋯ + b_{n}x^{n}  where b_{1}, …, b_{n} ∈ 𝔪_{V}.

Likewise, if x^{−1} ∉ V, we find an identity of the form

(9.1.23)  1 = c_{1}x^{−1} + ⋯ + c_{m}x^{−m}  where c_{1}, …, c_{m} ∈ 𝔪_{V}

and we may assume that m, n ∈ ℕ are the minimal integers for which there exist identities (9.1.22) and (9.1.23). Up to swapping the roles of x and x^{−1}, we may also assume that n ≥ m. In this case we deduce

1 = b_{1}x + ⋯ + b_{n−1}x^{n−1} + b_{n} · (c_{1}x^{n−1} + ⋯ + c_{m}x^{n−m})

which is another identity of the type (9.1.22), but with a strictly smaller value of n, a contradiction. We conclude that either x ∈ V or x^{−1} ∈ V, as required. □

9.1.24. Let K be any field, and denote by ℒ_{K} the set of all local subrings of K. We endow ℒ_{K} with a partial ordering, by declaring that, for every A, B ∈ ℒ_{K} we have A ≤ B if and only if B dominates A (see definition 9.1.12(ii)).

Corollary 9.1.25. With the notation of (9.1.24), the following holds :

(i) For every A ∈ ℒ_{K} there exists a maximal element of ℒ_{K} that dominates A. (ii) An element of ℒ_{K} is maximal if and only if it is a valuation ring of K.

Proof. (i): Let (B_{i} | i ∈ I) be a non-empty totally ordered subset of ℒ_{K} such that A ≤ B_{i} for every i ∈ I, and set B := ⋃_{i∈I} B_{i}. It is easily seen that B ∈ ℒ_{K}, and A ≤ B. Then the assertion follows from Zorn’s lemma.

(ii): Let A be any maximal element of ℒ_{K}; by theorem 9.1.21there exists a valuation ring V of K that dominates A, so A = V. Conversely, if (V, 𝔪_{V}) is a valuation ring of K and (A, 𝔪_{A}) any element of ℒ_{K} that dominates V, then we must have A = V : indeed, if the latter fails, there exists a ∈ A \ V, so that a^{−1} ∈ 𝔪_{V} ⊂ 𝔪_{A}, which is absurd. □

9.1.26. Henceforth, and until the end of this section, we let (K, | · |) be a given valued field, whose valuation ring (resp. maximal ideal, resp. residue field, resp. value group) shall be denoted K^{+} (resp. 𝔪_{K}, resp. κ, resp. Γ). The following result is due to Nagata ([134, Th.3]).

Proposition 9.1.27. Let A be an essentially finitely presented K^{+}-algebra (definition 7.3.52), and M a finitely generated K^{+}-flat A-module. Then M is a finitely presented A-module.

Proof. Let us write A = S^{−1}B for some finitely presented K^{+}-algebra B, and some multiplicative subset S ⊂ B; then we may find a finitely generated B-module M_{B} with an isomorphism φ : S^{−1}M_{B} →^{∼} M of A-modules. Let M′_{B} be the image of M_{B} in M_{B} ⊗_{K^{+}} K; since M is K^{+}-flat, φ induces an isomorphism S^{−1}M′_{B} → M. It then suffices to show that M′_{B} is a finitely presented B-module; hence we may replace A by B and M by M′_{B}, and assume that A is a finitely presented K^{+}-algebra.

We are further easily reduced to the case where A = K^{+}[T_{1}, …, T_{n}] is a free polynomial K^{+}-algebra. Pick a finite system of generators Σ for M; define increasing filtrations Fil_{•}A and Fil_{•}M on A and M, by letting Fil_{k}A be the K^{+}-submodule of all polynomials P (T_{1}, …, T_{n}) ∈ A of total degree ≤ k, and setting Fil_{k}M := Fil_{k}A · Σ ⊂ M for every k ∈ ℕ. We consider the Rees algebra R(\underline{A})_{•} and the Rees module R(\underline{M})_{•} associated with these filtrations as in definition 7.9.1(iii,iv). Say that Σ is a subset of cardinality N; we obtain an A-linear surjection φ : A^{⊕N} → M, and we set M′_{k} := (Fil_{k}A)^{⊕N} ∩ Ker φ for every k ∈ ℕ. Notice that the resulting graded K^{+}-module M′_{•} := ⨁_{k∈ℕ} M′_{k} is actually a R(\underline{A})_{•}-module and we get a short exact sequence of graded R(\underline{A})_{•}-modules

C_{•} : 0 → M′_{•} → R(\underline{A})_{•}^{⊕N} → R(\underline{M})_{•} → 0.

Notice also that the K^{+}-modules R(\underline{A})_{k}^{⊕N} and R(\underline{M})_{k} are torsion-free and finitely generated, hence they are free of finite rank (see [75, Rem.6.1.12(ii)]); then the same holds for M′_{k}, for every k ∈ ℕ. It follows that the complex C_{•} ⊗_{K^{+}} κ is still short exact. On the other hand, R(\underline{A})_{•} is a K^{+}-algebra of finite type (see example 7.9.5), hence R := R(\underline{A})_{•} ⊗_{K^{+}} κ is a noetherian ring; therefore, M′_{•} ⊗_{K^{+}} κ is a graded R-module of finite type, say generated by the system of homogenenous elements \{\overline{x}_{1}, …, \overline{x}_{t}\}. Lift these elements to a system Σ′ := \{x_{1}, …, x_{t}\} of homogeneous elements of M′_{•}, and denote by M′′_{•} ⊂ M′_{•} the R(\underline{A})_{•}-submodule generated by Σ′. By construction, M′′_{k} ⊗_{K^{+}} κ = M′_{k} ⊗_{K^{+}} κ, hence M′′_{k} = M′_{k} for every k ∈ ℕ, by Nakayama’s lemma. In other words, Σ′ is a system of generators for M′_{•}; it follows easily that the image of Σ′ in Ker φ is also a system of generator for the latter A-module, and the proposition follows. □

Corollary 9.1.28. Every essentially finitely presented K^{+}-algebra is a coherent ring.

Proof. One reduces easily to the case of a free polynomial K^{+}-algebra K^{+}[T_{1}, …, T_{n}], in which case the assertion follows immediately from proposition 9.1.27 : the details shall be left to the reader. □

Lemma 9.1.29. If A if a finitely presented K^{+}-algebra, the subset Min A ⊂ Spec A of minimal prime ideals is finite.

Proof. We begin with the following :

Claim 9.1.30. The assertion holds when A is a flat K^{+}-algebra.

Proof of the claim. Indeed, in this case, by the going-down theorem ([126, Th.9.5]) the minimal primes lie in Spec A ⊗_{K^{+}} K, which is a noetherian ring. ♢

In general, A is defined over a ℤ-subalgebra R ⊂ K^{+} of finite type. Let L ⊂ K be the field of fractions of R, and set L^{+} := L ∩ K^{+}; then L^{+} is a valuation ring of finite rank, and there exists a finitely presented L^{+}-algebra A′ such that A ≃ A′ ⊗_{L^{+}} K^{+}. For every prime ideal 𝔭 ⊂ L^{+}, let κ(𝔭) be the residue field of L_{𝔭}^{+}, and set :

A′(𝔭) := A′ ⊗_{L^{+}} κ(𝔭) K^{+}(𝔭) := K^{+} ⊗_{L^{+}} κ(𝔭).

Since Spec L^{+} is a finite set, it suffices to show that the subset Min A⊗_{L^{+}} κ(𝔭) is finite for every 𝔭 ∈ Spec L^{+}. However, A ⊗_{L^{+}} κ(𝔭) ≃ A′(𝔭) ⊗_{κ(𝔭)} K^{+}(𝔭), so this ring is a flat K^{+}(𝔭)-algebra of finite presentation. Let I be the nilradical of K^{+}(𝔭); since K^{+}(𝔭)/I is a valuation ring, claim 9.1.30applies with A replaced by A ⊗_{K^{+}} K^{+}(𝔭)/I, and concludes the proof. □

Lemma 9.1.31. Let f : A → B be a ring homomorphism of essentially finite type, M a Bmodule, and C a B-algebra. The following holds :

(i) If M is finitely presented as an A-module, then it is finitely presented as a B-module. (ii) If C is finitely presented as a B-module, then it is finitely presented as a B-algebra. (iii) If C is finitely presented (resp. essentially finitely presented) as an A-algebra, then it

is finitely presented (resp. essentially finitely presented) as a B-algebra.

Proof. (i): For any ring homomorphism g : R → S and any S-module N, let us denote by N_{[g]} the R-module obtained from N by restriction of scalars along g. Set C := B ⊗_{A} B, and let ∇ : C → B be the ring homomorphism such that ∇(b ⊗ b′) := bb′ for every b, b′ ∈ B; notice that if B is a localization of A[b_{1}, …, b_{k}], then I := Ker(∇) is the ideal generated by (b_{i} ⊗ 1 − 1 ⊗ b_{i} | i = 1, …, k). Let also B ⟶^{j_{1}} C ⟵^{j_{2}} B be the ring homomorphisms such that j_{1}(b) := b ⊗ 1 and j_{2}(b) := 1 ⊗ b for every b ∈ B. Since M_{[f]} is finitely presented, the same holds for the B-module B ⊗_{A} M_{[f]} = (C_{[j_{2}]} ⊗_{B} M)_{[j_{1}]}. The short exact sequence of C-modules

Σ := (0 → I → C ⟶^{∇} B → B_{[∇]} → 0)

induces a right exact sequence of B-modules

(Σ_{[j_{2}]} ⊗_{B} M)_{[j_{1}]} := (I_{[j_{2}]} ⊗_{B} M → B ⊗_{A} M_{[f]} → (B_{[∇∘j_{2}]} ⊗_{B} M)_{[j_{1}]} → 0)

and clearly ∇∘j_{2} = 1_{B}, so that (B_{[∇∘j_{2}]} ⊗_{B} M)_{[j_{1}]} = M. Moreover, if M_{[f]} = Ax_{1} +⋯+Ax_{n}, then the image of I_{[j_{2}]}⊗_{B}M in B⊗_{A}M_{[f]} is the B-submodule generated by (b_{i}⊗x_{j}−1⊗b_{i}x_{j} | i = 1, …, k; j = 1, …, n), whence the assertion.

(ii): Let c_{1}, …, c_{n} be a finite system of generators of the B-module C, with c_{1} = 1; we let

φ : B[X_{1}, …, X_{n}] → C

be the unique surjective homomorphism of B-algebras such that φ(X_{i}) = c_{i} for i = 1, …, n. For every i, j = 1, …, n with i ≤ j there exist b_{ij1}, …, b_{ijn} ∈ B such that c_{i}c_{j} = ∑_{k=1}^{n} b_{ijk}c_{k}, and we let J ⊂ B[X_{1}, …, X_{n}] be the ideal generated by

{X_{i}X_{j} − ∑_{k=1}^{n} b_{ijk}X_{k} | 1 ≤ i ≤ j ≤ n} ∪ {1 − X_{1}}.

Clearly φ factors through a surjective map \overline{φ} : B[X_{1}, …, X_{n}]/J → C. Moreover, it is easily seen that B[X_{1}, …, X_{n}] = J + N, where N ⊂ B[X_{1}, …, X_{n}] is the free B-submodule of rank n generated by X_{1}, …, X_{n}. The restriction N → C of \overline{φ} is then a surjective B-linear map, whose kernel is therefore a B-module of finite type N′ ⊂ N; it follows that Ker(φ) = J + N′ is an ideal of finite type, whence the assertion.

(iii): Suppose first that C is essentially finitely presented as an A-algebra, i.e. for some k ∈ ℕ there exists a surjective homomorphism of A-algebras φ : A[X_{1}, …, X_{k}] → C′ such that Ker(φ) is a finitely generated ideal, and C = S^{−1}C′ for some subset S ⊂ A[X_{1}, …, X_{k}]. Let A′ := S^{−1}A[X_{1}, …, X_{k}], B′ := S^{−1}B[X_{1}, …, X_{k}], and denote by ψ : B′ → C be the unique homomorphism of B-algebras such that ψ(X_{i}) = φ(X_{i}) for i = 1, …, k, and g : A′ → B′ the unique homomorphism of A-algebras such that g(X_{i}) = X_{i} for i = 1, …, k. Then g is essentially of finite presentation, and C is an A′-module of finite presentation, by restriction of scalars along ψ ∘ g, so it is a B′-module of finite presentation via ψ, by (i); hence, C is a finitely presented B′-algebra, by (ii), whence the assertion, in this case. Lastly, if C is finitely presented as an A-algebra, we may assume take S = {1} in the foregoing, so that A′ = A[X_{1}, …, X_{k}] and B′ = B[X_{1}, …, X_{k}], and the same argument then shows that C is a finitely presented B-algebra. □

Lemma 9.1.32. Let (A, 𝔪_{A}) be a local ring, K^{+} → A a local and essentially finitely presented ring homomorphism. Then there exists a local morphism φ : V → A of essentially finitely presented K^{+}-algebras, such that the following holds :

(i) V is a valuation ring, its maximal ideal 𝔪_{V} is generated by the image of 𝔪_{K}, and the map K^{+} → V induces an isomorphism Γ →^{∼} Γ_{V} of value groups.

(ii) φ induces a finite field extension V/𝔪_{V} → A/𝔪_{A}.

Proof. Set X := Spec A, S := Spec K^{+} and let x ∈ X be the closed point. Pick a_{1}, …, a_{d} ∈ A whose classes in κ(x) form a transcendence basis over κ(s). The system (a_{i} | i ≤ d) defines a factorization of the morphism Spec φ : X → S as a composition X ⟶^{g} Y := 𝔸_{S}^{d} ⟶^{h} S, such that ξ := g(x) is the generic point of h^{−1}(s) ⊂ Y. The morphism g is essentially finitely presented (lemma 9.1.31(iii)) and moreover :

Claim 9.1.33. The stalk 𝒪_{Y,ξ} is a valuation ring with value group Γ.

Proof of the claim. Indeed, set B := K^{+}[T_{1}, …, T_{d}]; one sees easily that V := 𝒪_{X,ξ} is the valuation ring of the Gauss valuation | · |_{B} : Frac(B) → Γ ∪ {0} such that

| ∑_{α∈ℕ^{d}} a_{α}T^{α}|_{B} = max{|a_{α}| | α ∈ ℕ^{d}}

(where a_{α} ∈ K^{+} and T^{α} := T_{1}^{α_{1}} ⋯ T_{r}^{α_{r}} for all α := (α_{1}, …, α_{d}) ∈ ℕ^{d}, and a_{α} = 0 except for finitely many α ∈ ℕ^{d}). ♢

In view of claim 9.1.33, to conclude the proof it suffices to notice that κ(x) is a finite extension of κ(ξ). □

Proposition 9.1.34. Let φ : K^{+} → A and ψ : A → B be two essentially finitely presented ring homomorphisms (definition 7.3.52). Then :

(i) If ψ is integral, B is a finitely presented A-module.

(ii) If A is local, φ is local and flat, and A/𝔪_{K}A is a field, then A is a valuation ring and φ induces an isomorphism Γ →^{∼} Γ_{A} of value groups.

Proof. (i): The assumption means that there exist a finitely presented A-algebra C and a multiplicative system T ⊂ C such that B = T^{−1}C. Let I := ⋃_{t∈T} Ann_{C}(t). Then I is the kernel of the localization map C → B; the latter is integral by hypothesis, hence for every t ∈ T there is a monic polynomial P (X) ∈ C[X], say of degree n, such that P (t^{−1}) = 0 in B, hence t′t^{n} · P (t^{−1}) = 0 in C, for some t′ ∈ T, i.e. t′(1 − ct) = 0 holds in C for some c ∈ C, in other words, the image of t is already a unit in C/I, so that B = C/I. Then lemma 8.1.64 says that V (I) ⊂ Spec C is closed under generizations, hence V (I) is open, by corollary 8.1.52 and lemma 9.1.29, and finally I is finitely generated, by lemma 8.1.61(ii). So B is a finitely presented A-algebra. Then (i) follows from the well known :

Claim 9.1.35. Let R be any ring, S a finitely presented and integral R-algebra. Then S is a finitely presented R-module.

Proof of the claim. Let us pick a presentation : S = R[x_{1}, …, x_{n}]/(f_{1}, …, f_{m}). By assumption, for every i ≤ n we can find a monic polynomial P_{i}(T) ∈ R[T] such that P_{i}(x_{i}) = 0 in S. Let us set S′ := R[x_{1}, …, x_{n}]/(P_{1}(x_{1}), …, P_{n}(x_{n})); then S′ is a free R-module of finite rank, and there is an obvious surjection S′ → S of R-algebras, the kernel of which is generated by the images of the polynomials f_{1}, …, f_{m}. The claim follows. ♢

(ii): By lemma 9.1.32, we may assume from start that the residue field of A is a finite extension of κ. We can write A = C_{𝔭} for a K^{+}-algebra C of finite presentation, and a prime ideal 𝔭 ⊂ C containing 𝔪_{K}; under the stated assumptions, A/𝔪_{K}A is a finite field extension of κ, hence 𝔭 is a maximal ideal of C, and moreover 𝔭 is isolated in the fibre g^{−1}(s) of the structure morphism g : Z := Spec C → Spec K^{+}. It then follows from [65, Ch.IV, Cor.13.1.4] that, up to shrinking Z, we may assume that g is a quasi-finite morphism ([60, Ch.II, §6.2]). Let K^{h+} be the henselization of K^{+}, and set C^{h} := C ⊗_{K^{+}} K^{h+}. There exists precisely one prime ideal 𝔮 ⊂ C^{h} lying over 𝔭, and C_{𝔮}^{h} is a flat, finite and finitely presented K^{h+}-algebra by [66, Ch.IV, Th.18.5.11], so C_{𝔮}^{h} is henselian, by [66, Ch.IV, Prop.18.5.6(i)], and then C_{𝔮}^{h} is the henselization of A, since the natural map A → C_{𝔮}^{h} is ind-étale and induces an isomorphism C_{𝔮}^{h} ⊗_{K^{+}} κ ≃ κ(x). Finally, C_{𝔮}^{h} is also a valuation ring with value group Γ, by virtue of [75, Lemma 6.1.13 and Rem.6.1.12(vi)]. To conclude the proof of (ii), it now suffices to remark :

Claim 9.1.36. Let V be a valuation ring, φ : R → V a faithfully flat morphism. Then R is a valuation ring and φ induces an injection φ_{Γ} : Γ_{R} → Γ_{V} of the respective value groups.

Proof of the claim. Clearly R is a domain, since it is a subring of V. To show that R is a valuation ring, it then suffices to prove that, for any two ideals I, J ⊂ R, we have either I ⊂ J or J ⊂ I. However, since V is a valuation ring, we know already that either I · V ⊂ J · V or J · V ⊂ I · V; since φ is faithfully flat, the assertion follows. By the same token, we deduce that an ideal I ⊂ R is equal to R if and only if I · V = V, which implies that φ_{Γ} is injective (see [75, §6.1.11]). □

9.2. Huber’s theory of the valuation spectrum. In this section we present the first elements of Huber’s theory of the valuation spectrum of an arbitrary ring, for which the original reference is his Habilitationschrift [98]. Let A be any ring. We denote by

Spv A

the set of equivalence classes of valuations on A. For every a, b ∈ A we set

R_{A}(\frac{a}{b}) := \{v ∈ Spv A | v(a) ≤ v(b) ≠ 0\}

and more generally, with every n ∈ ℕ and every a_{1}, …, a_{n}, b_{1}, …, b_{n} ∈ A we associate the rational subset

R_{A}(\frac{a_{1}}{b_{1}} , …, \frac{a_{n}}{b_{n}} ) := ⋂_{i=1}^{n} R_{A}(\frac{a_{i}}{b_{i}} )

and we denote ℬ_{A} the boolean algebra generated by the system of rational subsets of Spv A. Explicitly, every element of ℬ_{A} is a finite union of subsets of the form

{v ∈ Spv A | v(a_{1}) < v(b_{1}), …, v(a_{n}) < v(b_{n}), v(c_{1}) ≤ v(d_{1}), …, v(c_{m}) ≤ v(d_{m})}

where n, m ∈ ℕ are arbitrary integers, and a_{1}, b_{1}, …, a_{n}, b_{n}, c_{1}, d_{1}, …, c_{m}, d_{m} ∈ A are arbitrary elements.

Definition 9.2.1. For every ring A, we endow Spv A with the topology 𝒯_{A} with basis given by the rational subsets. The pair (Spv A, 𝒯_{A}) will be called the valuation spectrum of A.

Theorem 9.2.2. For every ring A, the topological space (Spv A, 𝒯_{A}) is spectral, and ℬ_{A} is the set of all constructible subsets of Spv A.

Proof. We endow Spv A with the topology 𝒯_{cons} with basis ℬ_{A}. Endow as well the set {0, 1} with its discrete topology, and the set 𝒫 := {0, 1}^{A×A} with the product topology. We attach to every valuation v of A a mapping φ_{v} : A × A → {0, 1}, by declaring that φ_{v}(a, b) = 1 if and only if v(a) ≥ v(b). If v, w are any two valuations on A, then it is clear that φ_{v} = φ_{w} if and only if v is equivalent to w, so that the rule v ↦ φ_{v} yields a well defined injective map

φ : Spv A → 𝒫

and it is easily seen that the topology on Spv A induced by 𝒫 via φ agrees with 𝒯_{cons}.

Claim 9.2.3. The image of φ is the subset of 𝒫 of all mappings f : A × A → {0, 1} fulfilling the following conditions for every a, b, c ∈ A :

(a) we have either f(a, b) = 1 or f(b, a) = 1

(b) if f(a, b) = f(b, c) = 1, then f(a, c) = 1
(c) we have either f(a, a + b) = 1 or f(b, a + b) = 1
(d) if f(a, b) = 1, then f(ac, bc) = 1
(e) if f(ac, bc) = 1 and f(0, c) = 0, then f(a, b) = 1
(f) f(0, 1) = 0.

Proof of the claim. It is clear that φ_{v} fulfills these conditions for every v ∈ Spv A. Conversely, let f be a mapping fulfilling conditions (a)–(f); from (f), (a) and (d) we deduce :

(g) f(x, 0) = 1 for every x ∈ A. Set 𝔭 := {a ∈ A | f(0, a) = 1}. We show that 𝔭 is an ideal of A. Indeed, from (a) it follows that 0 ∈ 𝔭. Next, say that a, b ∈ 𝔭; from (c) we may assume that f(a, a + b) = 1, and then (b) implies that a + b ∈ 𝔭. Lastly, if a ∈ 𝔭 and b ∈ A, then (d) implies that ab ∈ 𝔭, as required. Moreover, (e) and (f) imply that 𝔭 is a prime ideal, and we set κ := Frac A/𝔭. We notice :

(h) f(a + x, a) = f(a, a + x) = 1 for every a ∈ A and x ∈ 𝔭. Indeed, suppose that f(a + x, a) = 0 for such a and x; then (c) implies that f(−x, a) = 1, and combining with (b) we get f(0, a) = 1. On the other hand, (g) says that f(a + x, 0) = 1, and combining again with (b) we derive f(a + x, a) = 1, a contradiction. To derive the second identity, set b := a + x, so that f(b − x, b) = 1, by the foregoing, as required.

We can now show that f factors through a mapping

\overline{f} : A/𝔭 × A/𝔭 → \{0, 1\}.

Indeed, let a, b ∈ A and c ∈ 𝔭 be arbitrary elements, and assume first that f(a, b) = 1; we need to show that f(a+c, b) = 1, and this follows from (b) and (h). Similarly, we get f(a, b+c) = 1.

Lastly, if f(a, b) = 0 we must have f(a + c, b) = 0, for otherwise the foregoing would give f(a, b) = f(a+c−c, b) = f(a+c, b) = 1, a contradiction; similarly, we see that f(a, b+c) = 0 in this case. This shows that \overline{f} is well defined, and by construction we have

(i) \overline{f}(0, x) = 0 for every x ∈ A/𝔭 \ \{0\}. Now, for every x ∈ κ, write x = a^{−1}b for some a, b ∈ A/𝔭, with a ≠ 0; we say that x is f-positive if \overline{f}(a, b) = 1. From (i) and (e) it follows easily that this condition does not depend on the choice of a and b, and we let V ⊂ κ be the subset of all f-positive elements. Suppose that x, y ∈ V, and write x = a^{−1}b, y = a^{−1}b′ for some a, b, b′ ∈ A/𝔭 with a ≠ 0; this means that \overline{f}(a, b) = \overline{f}(a, b′) = 1, and by virtue of (c) we may assume that \overline{f}(b, b + b′) = 1. Then (b) implies that \overline{f}(a, b + b′) = 1, so x + y ∈ V. Likewise, (d) implies \overline{f}(aa′, ba′) = 1 and \overline{f}(ba′, bb′) = 1, and then (b) yields f(aa′, bb′) = 1, so xy ∈ V. Also, we have either \overline{f}(−1, 1) = 1 or \overline{f}(1, −1) = 1 by (a), so −1 ∈ V; summing up, we conclude that V is a subring of κ, and (a) implies that V is a valuation ring of κ. Now, let w be the valuation of κ attached to V as in remark 9.1.13(iv), and set v := w ∘ π, where π : A → κ is the natural map; it is easily seen that φ_{v} = f, whence the claim. ♢

Claim 9.2.3implies that the image of φ is a closed subset of the quasi-compact and separated topological space 𝒫, so (Spv A, 𝒯_{cons}) is quasi-compact and separated as well. Now, it is clear that every rational subset of Spv A is open and closed in the topology 𝒯_{cons}. Moreover, it follows easily from remark 9.1.13(vi) that (Spv A, 𝒯_{A}) is a T_{0} topological space. Then the theorem follows from proposition 8.1.43. □

Remark 9.2.4. (i) Let A be any ring. Quite generally, we have

R_{A}(\frac{f_{1}}{f_{0}} , …, \frac{f_{n}}{f_{0}} ) ∩ R_{A}(\frac{g_{1}}{g_{0}} , …, \frac{g_{m}}{g_{0}} ) = R_{A}(\frac{f_{i}g_{j}}{f_{0}g_{0}} | i = 0, …, n, j = 0, …, m)

for every sequences f_{•} := (f_{0}, …, f_{n}) and g_{•} := (g_{0}, …, g_{m}) of elements of A.

(ii) Every ring homomorphism f : A → B induces a mapping

Spv f : Spv B → Spv A v ↦ v ∘ f

and obviously we have

(Spv f)^{−1}R_{A}(\frac{a}{b}) = R_{B}(\frac{f(a)}{f(b)}) for every a, b ∈ A

so that Spv f is a continuous spectral map (Spv B, 𝒯_{B}) → (Spv A, 𝒯_{A}) (remark 8.1.20(iii)). (iii) Moreover, we have a natural support map

σ_{A} : Spv A → Spec A v ↦ Ker v

and notice that

σ_{A}^{−1}(Spec A_{s}) = R_{A}(\frac{s}{s}) for every s ∈ A

so also σ_{A} is spectral. Furthermore, it is easily seen that σ_{A} restricts to a homeomorphism

(Spv A)_{0} →^{∼} Spec A

where (Spv A)_{0} is the subset of all rank zero valuations on A, endowed with the topology induced by the inclusion map into Spv A. Notice as well that, by remark 9.1.4(iv,v), we have a natural homeomorphism

(9.2.5)  σ_{A}^{−1}(𝔭) →^{∼} Spv κ(𝔭)  for every 𝔭 ∈ Spec A

where the fibre σ_{A}^{−1}(𝔭) is endowed with the topology induced by Spv A via the inclusion map.

(iv) Let S ⊂ A be any multiplicative system, and i : A → S^{−1}A the localization map; taking into account remark 9.1.4(iv), it is easily seen that Spv i is injective, and

Im Spv i = ⋂_{s∈S} R_{A}(\frac{s}{s}).

Moreover, we have

R_{S^{−1}A}(\frac{s^{−1}a}{t^{−1}b}) = (Spv i)^{−1}R_{A}(\frac{at}{bs}) for every a, b ∈ R and s, t ∈ S

so the topology of Spv S^{−1}A is induced by the topology of Spv A, via the mapping Spv i, and therefore the latter identifies Spv S^{−1}A with a pro-constructible subset of Spv A.

(v) For any ring A, we set

Spv^{+}A := {v ∈ Spv A | v(a) ≤ 1 for every a ∈ A}.

Hence, Spv^{+}A is a pro-constructible subset of Spv A, and we endow it with the topology induced by the inclusion map into Spv A, so that Spv^{+}A is a spectral space (theorem 9.2.2 and corollary 8.1.42). For every v ∈ Spv^{+}A, let also

σ_{A}^{+}(v) := {a ∈ A | v(a) < 1}.

It is easily seen that σ_{A}^{+}(v) is a prime ideal of A, so we get a well defined map

σ_{A}^{+} : Spv^{+}A → Spec A.

Notice that σ_{A}^{+} is surjective : indeed, for every prime ideal 𝔭, the trivial valuation v_{𝔭} with support equal to 𝔭 lies in Spv^{+}A, and σ_{A}^{+}(v_{𝔭}) = 𝔭. Moreover, we have

(σ_{A}^{+})^{−1}(Spec A_{s}) = R_{A}(\frac{1}{s}) ∩ Spv^{+}A for every s ∈ A

so σ_{A}^{+} is continuous and spectral (corollary 8.1.42). In many references, the prime ideal σ_{A}^{+}(v) is called the center of the valuation v. Notice also that, for every ring homomorphism f : A → B, the map Spv f restricts to a continuous spectral map

Spv^{+}f : Spv^{+}B → Spv^{+}A.

9.2.6. Let X be any scheme. A valuation of X is a pair (x, v) consisting of a point x ∈ X and the equivalence class of a valuation v of the residue field κ(x); then we say that x is the support of (x, v). We denote by

Spv X

the set of valuations of X, and we endow it with the coarsest topology 𝒯_{X} containing the subsets

U(\frac{a}{b}) := \{(x, v) ∈ Spv X | x ∈ U, v(a(x)) ≤ v(b(x)) ≠ 0\}

where U ranges over all the open subsets of X, and a, b are arbitrary elements of 𝒪_{X}(U) (and a(x), b(x) are the images of a and b in κ(x)). The topological space (Spv X, 𝒯_{X}) is called the valuation spectrum of X. Any morphism of schemes f : Y → X induces a mapping

Spv f : Spv Y → Spv X (y, w) ↦ (f(y), w ∘ f_{y}^{♭})

where f_{y}^{♭} : κ(f(y)) → κ(y) is the ring homomorphism induced by the morphism of structure sheaves f^{♭} : 𝒪_{X} → f_{∗}𝒪_{Y} associated with f. Notice that

(Spvf)^{−1}U(\frac{a}{b}) = f^{−1}U(\frac{f_{U}^{♭}(a)}{f_{U}^{♭}(b)})

for every U, a and b as in the foregoing, so Spv f is a continuous mapping. In the same vein, consider the support map

σ_{X} : Spv X → X (x, v) → x.

Clearly σ_{X}^{−1}U = U(\frac{1}{1}) for every open subset U ⊂ X, so σ_{X} is continuous as well.

Remark 9.2.7. The construction of (9.2.6) generalizes that of (9.2) : indeed, let A be any ring; we have a natural mapping

(9.2.8)  Spv A → Spv(Spec A)  v ↦ (Ker v, \overline{v})

where, for any valuation v on A, we let \overline{v} be the residual valuation of v (see remark 9.1.4(v)). In light of remarks 9.1.13(vi) and 9.2.4(iv) it is easily seen that (9.2.8) is a homeomorphism (details left to the reader). Taking into account theorem 9.2.2, we see that Spv X is locally spectral for every scheme X, and if X is quasi-compact and quasi-separated, then Spv X is spectral (lemma 8.1.15(iv)). Moreover, remark 9.2.4(ii) and proposition 8.1.16(iii) imply that Spv f is a spectral map, for every morphism of schemes f : Y → X.

Proposition 9.2.9. Let f : A → B and g : A → A′ be two ring homomorphisms, and set B′ := A′ ⊗_{A} B. Then the induced continuous map of topological spaces

φ : Spv B′ → Spv A′ ×_{SpvA} Spv B

is surjective.

Proof. Let v′ ∈ Spv A′ and w ∈ Spv B be two elements such that v := Spv(g)(v′) = Spv(f)(w), and denote by

\overline{v} ∈ Spv κ(v)  \overline{v}′ ∈ Spv κ(v′)  \overline{w} ∈ Spv κ(w)

the residual valuations of v, v′, and w; the maps f and g induce field extensions κ(v) → κ(v′) and κ(v) → κ(w), whence a ring homomorphism h : B′ → C := κ(v′) ⊗_{κ(v)} κ(w). Suppose that u ∈ Spv C is an element whose image in Spv κ(v′) ×_{Spv κ(v)} Spv κ(w) equals the pair (\overline{v}′, \overline{w}); then φ ∘ Spv(h)(u) = (v′, w). Thus, we are reduced to the case where A, A′ and B are three fields. To ease notation, set A^{+} := κ(v)^{+}, and define likewise A^{′+} and B^{+}; let as well B^{′+} := A^{′+} ⊗_{A^{+}} B^{+}. Since the maps A^{+} → A^{′+} and A^{+} → B^{+} are flat, it is easily seen that the induced map B^{′+} → B′ is injective. Pick any prime ideal 𝔭 ∈ Spec B^{′+} whose image in Spec A^{′+} (resp. in Spec B^{+}) is the (unique) closed point; the induced map B_{𝔭}^{′+} → B′_{𝔭} is still injective, and especially, B′_{𝔭} ≠ 0, so we may find a maximal ideal 𝔪 of B′_{𝔭}, and we denote by κ(𝔪) the residue field of 𝔪. The image of B_{𝔭}^{′+} in κ(𝔪) is a local ring D, and by corollary 9.1.25 we may find a valuation ring (V, 𝔪_{V}) of κ(𝔪) that dominates D. Lastly, V yields a well defined point u ∈ Spv B′, and a simple inspection shows that φ(u) = (v′, w), as required. □

9.2.10. We wish next to investigate the specializations in the valuation spectrum. Thus, let A be any ring, v : A → Γ_{∘} a valuation. The convex subgroup of Γ_{v} generated by Im (v) \ Γ_{∘}^{+} is called the characteristic subgroup of v, and shall be denoted

cΓ_{v}.

Now, let Δ ⊂ Γ be any convex subgroup, and π : Γ → Γ/Δ the projection; we associate with Δ two mappings

v_{Δ} : A → (Γ/Δ)_{∘}  a ↦ π_{∘} ∘ v(a)
v^{Δ} : A → Δ_{∘}  a ↦ \{ \begin{matrix}v(a) & if v(a) ∈ Δ \\ 0 & otherwise. \end{matrix}

Lemma 9.2.11. With the notation of (9.2.10), the following holds :

(i) v_{Δ} is a valuation on A.

(ii) v^{Δ} is a valuation if and only if cΓ_{v} ⊂ Δ.

Proof. (i) is clear. For (ii), suppose first that cΓ_{v} ⊂ Δ, pick any a, b ∈ A, and let us show that v(a + b)^{Δ} ≤ max(v(a)^{Δ}, v(b)^{Δ}). This is clear if both v(a), v(b) ∈ Δ_{∘}, so we may assume that v(b) ∉ Δ_{∘}, in which case notice that v(b) < 1, since cΓ_{v} ⊂ Δ. Now, if v(a) ∈ Δ, then we must have v(b) < v(a), as Δ is convex; therefore, v(a + b) = v(a) (remark 9.1.4(iii)), and the contention follows easily in this case. Lastly, if neither of v(a), v(b) lies in Δ, the foregoing yields v(a), v(b) < 1, hence v(a + b) < 1, and moreover we may assume that v(a + b) ≤ v(a); then v(a + b) cannot lie in Δ, as the latter is convex, so v(a + b)^{Δ} = 0, and the assertion follows also in this case. Next, let us check that v(ab)^{Δ} = v(a)^{Δ} · v(b)^{Δ}. Again, this is clear if both v(a), v(b) ∈ Δ_{∘}, hence suppose that v(b) ∉ Δ_{∘}; if v(a) ∈ Δ_{∘}, then it follows easily that v(ab) ∉ Δ, in which case the assertion holds. If neither of v(a), v(b) is in Δ_{∘}, we have already remarked that v(a), v(b) < 1, hence v(ab) < v(a) < 1, so v(ab) cannot lie in Δ, as the latter is convex. Thus, the assertion holds also in this case.

Conversely, suppose that cΓ_{v} ⊄ Δ; then there exists a ∈ A such that v(a) > 1 but v(a) ∉ Δ. Set b := 1 − a; it follows that v(b) = v(a), and v(a + b) = 1 ∈ Δ, so v(a + b)^{Δ} = 1 but v(a)^{Δ} = v(b)^{Δ} = 0, and therefore v^{Δ} is not a semi-norm. □

Example 9.2.12. In the situation of example 9.1.5, suppose that A is a domain and P is integral with P^{gp} torsion-free, so that v_{φ} is a valuation on A[P], and let Δ ⊂ Γ be a convex subgroup.

(i) Lemma 9.2.11(ii) implies that v_{φ}^{Δ} is a valuation on A[P] if and only if φ(P) ⊂ Γ_{∘}^{+} ∪ Δ. Moreover, if the latter condition holds, the mapping

φ^{Δ} : P → Γ_{∘} x ↦ \{ \begin{matrix}φ(x) & if φ(x) ∈ Δ \\ 0 & otherwise. \end{matrix}

is a morphism of monoids, and v_{φ}^{Δ} = v_{φ^{Δ}}.

(ii) Furthermore, (v_{φ})_{Δ} = v_{π∘φ}, where π : Γ → Γ/Δ is the projection.

Remark 9.2.13. With the notation of (9.2.10), we notice :

(i) Let \overline{v} : κ(v) → Γ_{∘} be the residual valuation of v, and κ(v)^{+} the valuation ring of \overline{v}; the correspondence of remark 9.1.13(vii) assigns to Δ a prime ideal 𝔭(Δ) ⊂ κ(v)^{+}, and we have

Ker v_{Δ} = Ker v κ(v_{Δ})^{+} = κ(v)_{𝔭(Δ)}^{+} = \overline{v}^{−1}(Γ^{+} · Δ_{∘}).

(ii) Let π : A → κ(v) be the natural map; then cΓ_{v} ⊂ Δ if and only if π(A) ⊂ κ(v)_{𝔭(Δ)}^{+}. (iii) Suppose now that cΓ_{v} ⊂ Δ; then :

κ(v^{Δ}) ⊂ Frac κ(v)^{+}/𝔭(Δ) and κ(v^{Δ})^{+} = κ(v^{Δ}) ∩ κ(v)^{+}/𝔭(Δ).

(iv) The valuation v_{Δ} is a generization of v in Spv A. Indeed, if a, b ∈ A are any two elements such that v(a) ≤ v(b), then clearly v_{Δ}(a) ≤ v_{Δ}(b), whence the claim.

(v) Also, if cΓ_{v} ⊂ Δ, then v^{Δ} is a specialization of v in Spv A: indeed, if v^{Δ}(a) ≤ v^{Δ}(b), it is easily seen that v(a) ≤ v(b).

Definition 9.2.14. Let A be a ring, and v, w ∈ Spv A any two elements.

(i) We say that w is a primary specialization of v in Spv A, if there exists a convex subgroup Δ ⊂ Γ_{v} containing cΓ_{v}, and such that w is (equivalent to) v^{Δ}. In this case, we also say that v is a primary generization of w.

(ii) We say that w is a secondary generization of v if there exists a convex subgroup Δ ⊂ Γ_{v} such that w is (equivalent to) v_{Δ}. In this case, we also say that v is a secondary specialization of w.

(iii) We say that w is a generalized primary specialization of v, if w is either a primary specialization of v, or else cΓ_{v} = {1} and w is a trivial (i.e. rank zero) valuation of A with Ker v^{{1}} ⊂ Ker w. In this case, w is a specialization of v^{{1}} : see remark 9.2.4(iii).

Recall that specializations of points define a partial ordering on Spv A: see remark 8.1.45(ii). Let v be any semi-norm on A, and I ⊂ A any ideal; for the following lemma, we shall say that I is v-convex, if the following holds. For every a ∈ A such that there exists b ∈ I with v(a) ≤ v(b), we have a ∈ I as well.

Lemma 9.2.15. Let A be any ring, v, w, u ∈ Spv A any three elements. We have :

(i) If w is a primary specialization of v and u is a primary specialization of w, then u is a primary specialization of v.

(ii) If w is a secondary generization of v and u is a secondary generization of w, then u is

a secondary generization of v. (iii) Suppose that w is a generization of v. Then w is a secondary generization of v if and

only if Ker w = Ker v. (iv) The primary specializations (resp. the secondary generizations) of v form a totally

ordered subset of the partially ordered set Spv A.

(v) The supports of the primary specializations of v are the v-convex prime ideals of A.

Proof. (i) and (ii) are clear from the explicit description in (9.2.10).

(iii): Suppose that w is a generization of v whose support equals that of v, and let \overline{v} and \overline{w} be the respective residual valuations on κ := κ(v) = κ(w); from (9.2.5) we deduce that \overline{w} is a generization of \overline{v} in Spv κ, which means that κ(\overline{v})^{+} ⊂ κ(\overline{w})^{+}. Then, let Δ ⊂ Γ_{v} be the image of (κ(w)^{+})^{×}; it follows easily that Δ is a convex subgroup, and taking into account remark 9.1.13(iv), we see that Γ_{w} is naturally identified with Γ_{v}/Δ, and w = v_{Δ}.

(iv): Indeed, the explicit description of (9.2.10) implies more precisely that the partially ordered set of secondary generizations of v is naturally identified with Spec Γ_{v}, where the latter is totally ordered by inclusion (see definition 9.1.1(ii)). Likewise, we have a surjective orderpreserving map from Spec Γ_{v}/cΓ_{v} onto the set of primary specializations of v.

(v): Let Δ ⊂ Γ_{v} be a convex subgroup containing cΓ_{v}; let also a ∈ A and b ∈ Ker(v^{Δ}) with v(a) ≤ v(b). Then v(b) ∉ Δ, and in particular v(b) < 1, since cΓ_{v} ⊂ Δ; since Δ is convex, it follows that v(a) ∉ Δ, hence a ∈ Ker(v^{Δ}). This shows that Ker(v^{Δ}) is v-convex. Conversely, suppose that the prime ideal 𝔭 ⊂ A is v-convex. Hence, v(x) ≠ 0 for every x ∈ A \ 𝔭, and we let Δ ⊂ Γ_{v} be the subgroup generated by v(A \ 𝔭). We claim that γ > v(x) for every γ ∈ Δ and every x ∈ 𝔭. Indeed, suppose that this condition fails for some γ ∈ Δ and x ∈ 𝔭; since A \ 𝔭 is a submonoid of (A, ·), we may write γ = v(a) · v(b)^{−1} for some a, b ∈ A \ 𝔭, and we get v(a) ≤ v(bx), which is absurd, since 𝔭 is v-convex. Now, let Δ′ ⊂ Γ_{v} be the smallest convex subgroup containing Δ; it is easily seen that we still have γ > v(x) for every γ ∈ Δ′ and every x ∈ 𝔭. On the other hand, by construction we have v(A \ 𝔭) ⊂ Δ′, so 𝔭 = {x ∈ A | γ > v(x) for every γ ∈ Δ′}. Moreover, say that a ∈ A and v(a) ≥ 1; then a ∈ A \ 𝔭, since 𝔭 is v-convex, and consequently v(a) ∈ Δ′, i.e. the characteristic subgroup of v lies in Δ′, so that v^{Δ′} ∈ Spv A, and to conclude, it suffices to remark that Ker v^{Δ′} = 𝔭. □

Proposition 9.2.16. Let A be any ring, v a valuation of A, and 𝔭 ⊂ A a prime ideal. We have :

(i) If 𝔭 ⊂ Ker v, there exists a primary generization w of v with Ker w = 𝔭. Especially, the map σ_{A} of remark 9.2.4(iii) is generizing.

(ii) If v ∈ Spv^{+}A, then every primary specialization and every primary generization of v in Spv A lies also in Spv^{+}A, and if w is any primary specialization (or generization) of v, then σ_{A}^{+}(w) = σ_{A}^{+}(v) (notation of remark 9.2.4(v)).

(iii) If v ∈ Spv^{+}A and 𝔭 contains the center of v, there exists a secondary specialization w ∈ Spv^{+}A of v with center equal to 𝔭. Especially, the map σ_{A}^{+} is specializing.

Proof. (i): The valuation v factors through the projection A → A′ := A/𝔭 and a valuation v′ of A′; after replacing A and v by A′ and v′, we may then assume that A is a domain and 𝔭 = 0. In this case, let 𝔮 := Ker v ⊂ A; according to theorem 9.1.21 there exists a valuation ring (V, 𝔪_{V}) of K := Frac A such that A ⊂ V and 𝔮 = 𝔪_{V} ∩ A; by the same token, there exists a valuation ring (\overline{W} , 𝔪\overline{W}) of V/𝔪_{V} such that \overline{W} ∩ κ(𝔮) = κ(v)^{+}. Let π_{V} : V → V/𝔪_{V} be the projection; then W := π_{V}^{−1}(\overline{W}) is a valuation ring of K; let w_{K} be the unique valuation of K with κ(w_{K})^{+} = W, and w the restriction of w_{K} to the subring A. Notice that 𝔪_{V} = Ker(W → \overline{W}) is a prime ideal of W; then the bijection of remark 9.1.13(vii) assigns to 𝔪_{V} a convex subgroup Δ(𝔪_{V}) such that 𝔪_{V} = {x ∈ W | γ < w_{K}(x) for every γ ∈ Δ(𝔪_{V})}. Especially, we see that 𝔪_{W} is a w_{K}-convex ideal, hence 𝔮 is w-convex. Let Δ ⊂ Γ_{w} be the convex subgroup generated by w(A \ 𝔮); the proof of lemma 9.2.15(v) shows that cΓ_{w} ⊂ Δ and Ker(w^{Δ}) = 𝔮. We need to check that w^{Δ} is equivalent to v. Notice first that for every a, b ∈ A we have :

(9.2.17)  w(a) ≤ w(b) ⇔ a ∈ bW  and v(a) ≤ v(b) ⇔ π_{V} (a) ∈ π_{V} (b)\overline{W} .

Now, suppose that w^{Δ}(a) ≤ w^{Δ}(b), and let us show that v(a) ≤ v(b). Indeed, if w^{Δ}(b) ≠ 0, we have w^{Δ} ∈ R_{A}(\frac{a}{b}), whence w ∈ R_{A}(\frac{a}{b}), so w(a) ≤ w(b), and thus v(a) ≤ v(b), in light of (9.2.17). If w^{Δ}(b) = 0, we have as well w^{Δ}(a) = 0, and then v(a) = v(b) = 0, since Ker(v) = Ker(w^{Δ}). Conversely, suppose that v(a) ≤ v(b); if v(b) = 0, we see as in the foregoing that w^{Δ}(a) = w^{Δ}(b) = 0. Thus, let v(b) ≠ 0, and suppose by contradiction that w^{Δ}(a) > w^{Δ}(b); we deduce that w^{Δ}(a), w^{Δ}(b) ≠ 0, hence a, b ≠ 0 and w(a) < w(b). Let 𝔪_{W} = π_{V}^{−1}(𝔪\overline{W}) be the maximal ideal of W; we get b ∈ a · 𝔪_{W}, so that π_{V} (b) ∈ π_{V} (a) · 𝔪\overline{W}, i.e. v(a) > v(b), against our assumption. We may now apply remark 9.1.13(vi) to conclude.

(ii) is clear from the definitions.

(iii): Let V ⊂ κ(v) be the valuation ring of the residual valuation of v, and 𝔪_{V} the maximal ideal of V; then v induces a ring homomorphism π : A → V such that 𝔮 := π^{−1}𝔪_{V} is the center of v. We deduce an injective ring homomorphism \overline{A} := A/𝔮 → κ := V/𝔪_{V}, and we denote by \overline{𝔪} ⊂ \overline{A} the image of 𝔪. By corollary 9.1.25, there exists a valuation ring \overline{W} of κ that dominates the image of \overline{A}\overline{𝔪}, and we denote by W ⊂ V the preimage of \overline{W}. Then W is a valuation ring of κ(v), and it corresponds to a secondary specialization of v in Spv^{+}A. □

Proposition 9.2.18. Let A be any ring, and v ∈ Spv A any element. Then, every specialization of v is a secondary specialization of a generalized primary specialization of v.

Proof. Let w be a specialization of v. If cΓ_{v} = {1} and v(a) ≥ 1 for every a ∈ A \ Ker w, then Ker v^{{1}} ⊂ Ker w, so the trivial semi-norm u with support Ker w is a generalized primary specialization of v, and w is a secondary specialization of u. We may therefore assume that either cΓ_{v} ≠ {1}, or v(a) < 1 for some a ∈ A \ Ker w.

Claim 9.2.19. Let a, b ∈ A be any two elements with v(a) ≤ v(b), w(a) ≠ 0 and w(b) = 0. Then v(a) = v(b) ≠ 0.

Proof of the claim. The assumptions imply that w ∈ R_{A}(\frac{b}{a}), and since v is a generization of w, it follows that v ∈ R_{A}(\frac{b}{a}) as well, whence the claim. ♢

Claim 9.2.20. Ker w is v-convex.

Proof of the claim. Indeed, let x, y ∈ A be two elements with

(9.2.21)  v(x) ≤ v(y)  and w(y) = 0

and suppose, by way of contradiction, that w(x) ≠ 0; by claim 9.2.19it follows that

(9.2.22)  v(x) = v(y) ≠ 0.

Consider first the case where cΓ_{v} ≠ {1}, and pick a ∈ A with v(a) > 1; combining with (9.2.21) we get v(x) ≤ v(ay) and w(ay) = 0, so claim 9.2.19 yields v(ay) = v(x), which contradicts (9.2.22). Next, if v(a) < 1 for some a ∈ A such that w(a) ≠ 0, we get v(ax) ≤ v(y) and w(ax) ≠ 0, again by combining with (9.2.21), and claim 9.2.19yields v(ax) = v(y), again a contradiction. ♢

By claim 9.2.20 and lemma 9.2.15(iv,v), there exists a unique primary specialization u of v with Ker u = Ker w, and in view of lemma 9.2.15(iii), it suffices to show that w is a specialization of u. To this aim, let a, b ∈ A be any two elements such that w ∈ R_{A}(\frac{a}{b}); since v is a generization of w, we have v ∈ R_{A}(\frac{a}{b}) as well, and since u is a primary specialization of v, we deduce u(a) ≤ u(b). Lastly, since Ker u = Ker w and w(b) ≠ 0, we get u(b) ≠ 0, so u ∈ R_{A}(\frac{a}{b}), and the assertion follows. □

Lemma 9.2.23. Let A be a ring, v, w ∈ Spv A be two elements, such that w is a primary specialization of v. We have :

(i) For every secondary specialization v′ of v there exists a unique secondary specialization w′ of w such that w′ is a primary specialization of v′.

(ii) For every secondary specialization w′ of w there exists a secondary specialization v′ of

v such that w′ is a primary specialization of v′. (iii) For every secondary generization v′ of v there exists a unique secondary generization

w′ of w such that w′ is a generalized primary specialization of v′. (iv) For every secondary generization w′ of w there exists a secondary generization v′ of v

such that w′ is a primary specialization of v′.

Proof. (i): The uniqueness of w′ follows from lemma 9.2.15(iii,iv). For the existence, let Δ′ be a convex subgroup of Γ_{v′} such that v = v′_{Δ′}, and Σ a convex subgroup of Γ_{v} = Γ_{v′}/Δ′ containing cΓ_{v} and such that w = v^{Σ}. Denote by Σ′ the unique convex subgroup of Γ_{v′} such that Δ′ ⊂ Σ′ and Σ′/Δ′ = Σ; then cΓ_{v′} ⊂ Σ′ and we may take w′ := (v′)^{Σ′}.

(ii): By remark 9.2.13(ii), there exists a prime ideal 𝔭 ⊂ κ(v)^{+} such that κ(w) ⊂ K := Frac κ(v)^{+}/𝔭 and κ(w)^{+} = κ(w) ∩ K^{+}, where K^{+} := κ(v)^{+}/𝔭. On the other hand, κ(w′) = κ(w), and κ(w′)^{+} ⊂ κ(w)^{+}. Let 𝔪_{K}, 𝔪_{w} and 𝔪_{w′} be the maximal ideals of respectively K^{+}, κ(w)^{+} and κ(w′)^{+}; it follows easily that 𝔪_{K} ∩ κ(w) = 𝔪_{w} ⊂ 𝔪_{w′}. Then 𝔪_{w} is an ideal of κ(w′)^{+}, and we have injective ring homomorphisms

B := κ(w′)^{+}/𝔪_{w} → κ(w)^{+}/𝔪_{w} → K^{+}/𝔪_{K}.

According to corollary 9.1.25, we may then find a valuation ring \overline{V} of K^{+}/𝔪_{K} that dominates B. Since B is a valuation ring, we easily deduce that B = \overline{V} ∩ Frac B. Let V ⊂ K^{+} be the preimage of \overline{V}; then V is a valuation ring of K such that κ(w′)^{+} = V ∩ κ(w′) (details left to the reader). Let V′ ⊂ κ(v)^{+} be the preimage of V; then V′ is a valuation ring of κ(v), and we let v′ be the unique valuation of A with Ker v′ = Ker v and κ(v′)^{+} = V′; it is easily seen that v′ is the sought secondary specialization of v.

(iii): The uniqueness of w′ follows from lemma 9.2.15(iii,iv), and the existence follows from proposition 9.2.18.

(iv): Let Δ be a convex subgroup of Γ_{v} containing cΓ_{v} and such that w = v^{Δ}, and Σ a convex subgroup of Δ such that w′ = w_{Σ}. Set v′ := v_{Σ}; then Δ′ := Δ/Σ is a convex subgroup of Γ_{v′} = Γ_{v}/Σ that contains cΓ_{v′}, and w′ = (v′)^{Δ′}. □

Proposition 9.2.24. Let A be a ring, and v ∈ Spv A any element. Then every specialization of v is a primary specialization of a secondary specialization of v.

Proof. Let w be a specialization of v. If w is a secondary specialization of a primary specialization of v, the assertion follows from lemma 9.2.23(ii). In light of proposition 9.2.18, we may then assume that the characteristic subgroup of v is trivial and Ker v^{{1}} ⊂ Ker w. In this case, proposition 9.2.16(i) yields a primary generization w′ of w with Ker w′ = Ker v^{{1}}. Clearly w′ is a secondary specialization of v^{{1}}, so lemma 9.2.23(ii) ensures that we may find a secondary specialization v′ of v which is also a primary generization of w′, and then w is a primary specialization of v′ (lemma 9.2.15(i)). □

Lemma 9.2.25. Let f : A → B be any ring homomorphism, and v, w ∈ Spv B any two elements. We have :

(i) If w is a primary (resp. secondary) specialization of v, then Spv(f)(w) is a primary (resp. secondary) specialization of Spv(f)(v).

(ii) If Spec f is an open immersion, and Spv(f)(w) is a primary (resp. secondary) specialization of Spv(f)(v), then w is a primary (resp. secondary) specialization of v.

(iii) Spv f restricts to a surjection from the set of secondary specializations (resp. generiza-

tions) of v onto the set of secondary specializations (resp. generizations) of Spv(f)(v). (iv) Suppose that w is a primary specialization of v, and denote by P (resp. Q) the set

of primary specializations of v in Spv B (resp. of Spv(f)(v) in Spv A) that are also primary generizations of w in Spv B (resp. of Spv(f)(w) in Spv A). Then Spv f restricts to a surjection P → Q.

Proof. (i) is immediate from the definitions.

(ii): Set v′ := Spv(f)(v); if Spec f is an open immersion, f induces an isomorphism κ(v′) →^{∼} κ(v), and similarly for w. The assertion is an immediate consequence.

(iii): Clearly, f induces an injective map i : Γ_{v′} → Γ_{v}, and Spec i is surjective, by remark 9.1.2(v), so Spv f maps surjectively the secondary generizations of v onto the ones of v′. Next, let w′ be a secondary specialization of v′ in Spv A; then κ(w′) = κ(v′) and κ(w′)^{+} is a valuation ring of κ(v′) contained in κ(v′)^{+}. Let 𝔪 ⊂ κ(v)^{+} be the maximal ideal, so that 𝔪′ := 𝔪∩κ(v′) is the maximal ideal of κ(v′)^{+} and a prime ideal of κ(w′)^{+}; then K^{+} := κ(w′)^{+}/𝔪′ is a valuation ring of K := κ(v′)^{+}/𝔪′, and by corollary 9.1.25there exists a valuation ring V of κ(v)^{+}/𝔪 that dominates K^{+}. The preimage of V in κ(v)^{+} is a valuation ring of κ(v) and the corresponding valuation w of B is a secondary specialization of v with Spv(f)(w) = w′.

(iv): Again, since the induced morphism of ordered groups i is injective, the assertion follows easily from remark 9.1.2(v). □

Theorem 9.2.26. Let f : A → B be a flat ring homomorphism, and v ∈ Spv B any element. Then the following holds :

(i) Spv f restricts to a surjection from the set of primary generizations of v in Spv B to the set of primary generizations of Spv(f)(v) in Spv A.

(ii) Spv f is generizing.

Proof. (i): Let t be any primary generization of s := Spv(f)(v). According to remark 9.2.13(i), there exists a prime ideal 𝔭 ⊂ κ(t)^{+} such that

• the image of the natural map A → κ(t) lies in A′ := κ(t)_{𝔭}^{+}

• κ(s) is naturally identified with a subfield of the residue field \overline{κ} := A′/𝔭A′ of A′

• under this identification, we have κ(s)^{+} = κ(s) ∩ (κ(t)^{+}/𝔭). With this notation, let h : A → A′ the resulting map, and

\overline{t} : κ(t) = Frac A′ → Γ_{t∘} and \overline{s} : \overline{κ} → Γ_{s∘}

be respectively the residual valuation of t and the valuation of \overline{κ} such that κ(\overline{s})^{+} = κ(t)^{+}/𝔭. Then, \overline{t} and \overline{s} yield elements t′ and respectively s′ of Spv A′ such that

s = Spv(h)(s′) and t = Spv(h)(t′).

Let f′ : A′ → B′ := A′⊗_{A}B be the induced map; by proposition 9.2.9we may find v′ ∈ Spv B′ whose image in Spv A′ (resp. in Spv B) equals s′ (resp. v), and in light of lemma 9.2.25(i), it suffices to exhibit a primary generization w′ of v′ in Spv B′ with Spv(f′)(w′) = t′. However, since f′ is flat, there exists a prime ideal 𝔯 ⊂ 𝔮 := Ker v′ in B′, such that f^{′−1}𝔯 = 0 ([126, Th.9.5]); we set B′′ := B′_{𝔮}/𝔯B′_{𝔮} and we let f′′ : A′ → B′′ be the resulting map. There exists a unique valuation v′′ of B′′ whose image in Spv B′ equals v′, and it suffices to exhibit a primary generization w′′ of v′′ in Spv B′′ such that Spv(f′′)(w′′) = t′. To this aim, set K′′ := Frac B′′; by corollary 9.1.25 we may find a valuation ring V of K′′ that dominates B′′. Let κ′′ be the residue field of V, and denote by R the image of κ(v′′)^{+} in κ′′; we may find a valuation ring V′ of κ′′ that dominates R (corollary 9.1.25). Denote by V′′ ⊂ V the preimage of V′. Then V′′ is a valuation ring of K′′, and we may take for w′′ the unique valuation of B′′ with κ(w′′)^{+} = V′′.

(ii) is an immediate consequence of (i), lemma 9.2.25(iii) and proposition 9.2.24. □

Theorem 9.2.27. Let f : A → B be an integral ring homomorphism, v ∈ Spv B any element. The following holds :

(i) If v′ ∈ Spv B is a specialization of v with Spv(f)(v′) = Spv(f)(v), then v = v′. (ii) Spv f restricts to a bijection from the set of primary specializations of v in Spv B to

the set of primary specializations of Spv(f)(v) in Spv A. (iii) Spv f is specializing. (iv) If f is injective, Spv f is surjective.

(v) If A is integral and normal, B is integral and f is injective, then Spv f restricts to a surjection from the set of primary generizations of v in Spv B onto the set of primary generizations of Spv(f)(v) in Spv A.

Proof. Set w := Spv(f)(v), and let Γ_{v} and Γ_{w} be the value groups of v and w.

(i): Let 𝔭 and 𝔮 be the supports of v and respectively v′; under the current assumptions, we have f^{−1}𝔭 = f^{−1}𝔮, and 𝔮 is a specialization of 𝔭 in Spec B; then 𝔮 = 𝔮′, by [126, Th.9.3(ii)]. Taking into account proposition 9.2.24, we conclude that v′ is a secondary specialization of v. However, since f is integral, it induces an isomorphism Γ_{v} ⊗_{ℤ} ℚ →^{∼} Γ_{w} ⊗_{ℤ} ℚ. On the other hand, we have a natural bijection between Spec Γ_{v} and the set S_{v} of secondary specializations of v in Spv B, and likewise for the set S_{w} of secondary specializations of w; taking into account remark 9.1.2(v), we deduce that f induces a bijection S_{v} →^{∼} S_{w}, whence the assertion.

(ii): We remark :

Claim 9.2.28. Under the assumptions of the theorem, let b ∈ B be any element. Then there exists a ∈ A such that v(b) ≤ w(a).

Proof of the claim. Since f is integral, we may find a_{1}, …, a_{n} ∈ A such that b^{n} + a_{1}b^{n−1} + ⋯ + a_{n} = 0. It follows that v(b)^{n} ≤ v(b)^{i} · w(a_{n−i}) for some i < n. If v(b) = 0, there is nothing to prove; otherwise, we get v(b)^{n−i} ≤ w(a_{n−i}). If v(b) ≤ 1, there is nothing to prove; otherwise, we deduce that v(b) ≤ w(a_{n−i}), as required. ♢

From claim 9.2.28it follows easily that cΓ_{w} = cΓ_{v} ∩ Γ_{w}, whence the assertion.

(iii) follows immediately from (ii), proposition 9.2.24and lemma 9.2.25(iii).

(iv): Notice first that Spec f is surjective, under these assumptions. Indeed, let 𝔭 ⊂ A be any prime ideal, and pick a minimal prime ideal 𝔮 ⊂ A contained in 𝔭; the induced ring homomorphism f_{𝔮} : A_{𝔮} → B_{𝔮} is still injective, so B ≠ 0; now, if 𝔮′ ⊂ B_{𝔮} is any prime ideal, then f_{𝔮}^{−1}(𝔮′) = 𝔮, and by [126, Th.9.4(i)] there exists a specialization 𝔭′ of 𝔮′ in Spec B such that Spec (f)(𝔭′) = 𝔭. Now, let w ∈ Spv A be any element, and 𝔭′ ∈ Spec B any prime ideal such that f^{−1}𝔭′ = 𝔭 := Ker w; it follows easily from theorem 9.1.21that there exists a valuation w′ of the residue field κ(𝔭′) whose restriction to κ(𝔭) is equivalent to v.

(v): Let w′ be a primary generization of w in Spv A. We need to exhibit a primary generization v′ of v in Spv B such that Spv(f)(v′) = w′. To this aim, pick a normal algebraic field extension E of Frac A containing Frac B, and denote by C the integral closure of A in E. By (iv), we may find w′′, v′′ ∈ Spv C such that w′′_{|A} = w′ and v′′_{|B} = v. By (ii), we may find a primary specialization t of w′′ in Spv C such that t_{|A} = w. By construction, t_{|A} = v′′_{|A}, so there exists an automorphism h of the A-algebra C such that Spv(h)(t) = v′′ ([34, Ch.V, §2, no.3, Prop.6 and Ch.VI, §8, no.6, Cor.1]). It is easily seen that v′ := Spv(h)(w′′)_{|B} will do. □

Theorem 9.2.29. Let f : A → B be a finitely presented ring homomorphism, and T ⊂ Spv B any constructible subset. Then Spv(f)(T) is a constructible subset of Spv A.

Proof. We may assume that there exist elements a_{1}, b_{1}, …, a_{n}, b_{n}, c_{1}, d_{1}, …, c_{m}, d_{m} ∈ B such that T is the set of all v ∈ Spv B with v(a_{i}) < v(b_{i}) and v(c_{j}) ≤ v(d_{j}) for every i = 1, …, n and j = 1, …, m. We remark :

Claim 9.2.30. The theorem holds if Spec f is a closed immersion.

Proof of the claim. Indeed, in this case f is a surjective map, whose kernel is a finitely generated ideal I ⊂ A. Pick a finite system x_{1}, …, x_{r} ∈ A of generators of I, and for every i = 1, …, n, j = 1, …, m choose elements a′_{i}, b′_{i}, c′_{j}, d′_{j} ∈ A whose images equal respectively a_{i}, b_{i}, c_{j}, d_{j} in B. Then, Spv(f)(T) is the constructible subset of Spv A of all valuations v of A such that v(x_{k}) = 0 for every k = 1, …, r, and v(a′_{i}) < v(b′_{i}) and v(c′_{j}) ≤ v(d′_{j}) for every i = 1, …, n and j = 1, …, m. ♢

Claim 9.2.31. (i) We may assume that A and B are ℤ-algebras of finite type.

(ii) For every ℤ-algebra R of finite type, Spv^{+}R is a rational subset of Spv R (notation of remark 9.2.4(v)).

Proof of the claim. (i): Indeed, we may write A as the union of the filtered system (A_{λ} | λ ∈ Λ) of its ℤ-subalgebras of finite type, and then there exists λ ∈ Λ such that B = B_{λ} ⊗_{A_{λ}} A for a ring homomorphism f_{λ} : A_{λ} → B_{λ} of finite type. Moreover, after replacing λ by a larger index, we may assume that there exist elements a′_{1}, b′_{1}, …, a′_{n}, b′_{n}, c′_{1}, d′_{1}, …, c′_{m}, d′_{m} ∈ B_{λ} such that a′_{i} ⊗ 1 = a_{i}, b_{i} = b′_{i} ⊗ 1, c_{j} = c′_{j} ⊗ 1 and d_{j} = d′_{j} ⊗ 1 in B, for every i = 1, …, n and j = 1, …, m. We then let T′ ⊂ Spv B_{λ} be the constructible subset of all v ∈ Spv B_{λ} such that v(a′_{i}) < v(b′_{i}) and v(c′_{j}) ≤ v(d′_{j}) for every i = 1, …, n and j = 1, …, m. There follows a commutative diagram of topological spaces

\xymatrix{Spv B \ar[r]^{π_{B}} \ar[d]_{Spv f} & Spv B_{λ} \ar[d]^{Spv f_{λ}} \\ Spv A \ar[r]^{π_{A}} & Spv A_{λ}}

clearly T = π_{B}^{−1}T′, and it follows easily from proposition 9.2.9that

π_{A}^{−1}(Spv(f_{λ})(T′)) = Spv(f)(T).

Since π_{A} is spectral (remark 9.2.4(ii)), we are then reduced to showing that the theorem holds for the map f_{λ}.

(ii): Pick a finite system x_{1}, …, x_{r} of elements of R such that R = ℤ[x_{1}, …, x_{r}]; then it is easily seen that Spv^{+}R = R_{A}(\frac{x_{1}}{1} , …, \frac{x_{r}}{1} ). ♢

Henceforth, we assume that A and B are ℤ-algebras of finite type, and we set

C := B[X_{i}, Y_{j} | i = 1, …, n, j = 1, …, m]/I

where I is the ideal generated by the system (a_{i} −X_{i}b_{i}, c_{j} −Y_{j}d_{j} | i = 1, …, n, j = 1, …, m). We let T′ ⊂ Spv C be the constructible subset of all w ∈ Spv C such that

0 < w(b_{i})  w(X_{i}) < 1  w(Y_{j}) ≤ 1  for 1 ≤ i ≤ n and 1 ≤ j ≤ m

and denote by g : B → C the natural ring homomorphism.

Claim 9.2.32. Spv(g)(T′) = T.

Proof of the claim. Let w ∈ T′ be any element, and set v := Spv(g)(w); we may regard w as a valuation of B[X_{1}, …, X_{n}, Y_{1}, …, Y_{m}] such that w(b_{i}) > 0 and w(a_{i}) = w(X_{i}) · w(b_{i}) for every i = 1, …, n, and w(c_{j}) = w(Y_{j}) · w(d_{j}) for every j = 1, …, m. It follows easily that v ∈ T. Conversely, let v ∈ T be any element, and set D := C ⊗_{B} κ(v). Moreover, set Σ := {1 ≤ j ≤ m | v(d_{j}) = 0}. A simple inspection yields an isomorphism

h : D →^{∼} κ(v)[Y_{j} | j ∈ Σ]

of κ(v)-algebras, such that :

• h(X_{i}) = h(a_{i}) · h(b_{i})^{−1} for i = 1, …, n

• h(Y_{j}) = Y_{j} for j ∈ Σ
• h(Y_{j}) = h(c_{j}) · h(d_{j})^{−1} for every j ∈ {1, …, m} \ Σ.

Let (V, 𝔪_{V}) be any valuation ring of Frac D containing κ(v)^{+}[Y_{j} | j ∈ Σ] and such that 𝔪_{V} ∩ κ(v) is the maximal ideal of κ(v)^{+} (theorem 9.1.21); then V corresponds to a valuation \overline{w} of D whose restriction to κ(v) is equivalent to the residual valuation \overline{v} of v. So, \overline{w} induces a valuation w of C with Spv(g)(w) = v, and by construction we have w(X_{i}) = v(a_{i}) · v(b_{i})^{−1} < 1 for every i = 1, …, n; likewise, we get w(Y_{j}) = v(c_{j}) · v(d_{j})^{−1} ≤ 1 for j ∈ Σ, and w(Y_{j}) ≤ 1 for j ∈ {1, …, m} \ Σ. The claim follows. ♢

In view of claim 9.2.32, we may replace B by C, and T by T′, after which we may assume: (C) There exist finite sequences a_{•} := (a_{1}, …, a_{n}), b_{•} := (b_{0}, b_{1}, …, b_{m}) of elements of B

such that

T = R_{B}(\frac{b_{0}}{b_{0}} , \frac{b_{1}}{1} , …, \frac{b_{m}}{1} ) \ (R_{B}(\frac{1}{a_{1}} ) ∪ ⋯ ∪ R_{B}(\frac{1}{a_{n}} )).

We shall argue by induction on the dimension of B. Since B is noetherian, Spec B has finitely many irreducible components Z_{1}, …, Z_{k}, and

Spv B = ⋃_{i=1}^{k} Spv Z_{i}

(notation of (9.2.6)); moreover, since the inclusion map Z_{i} → Spec B is spectral, the subset Spv Z_{i} ∩ T is constructible in Spv Z_{i}, for every i = 1, …, k. Thus, it suffices to show that the restriction Spv Z_{i} → Spv A of Spv f maps constructible subsets to constructible subsets, for every i = 1, …, k. We may therefore assume from start that Spec B is also irreducible and reduced, i.e. that B is a domain. Then, the topological closure W of Spec(f)(Spec B) in Spec A is irreducible (lemma 8.1.3(v)); by virtue of claim 9.2.30, it suffices to show that the induced map Spv B → Spv W sends constructible subsets to constructible subsets. We may then assume as well that A is a domain, and f is injective. In this situation, if dim B = 0, we see that A and B are finite fields, and the assertion is obvious. Hence, we suppose henceforth that d := dim B > 0, and that the theorem has already been shown for every ℤ-algebra B of finite type of dimension < d.

Pick finite subsets I ⊂ A and J ⊂ B such that A = ℤ[I] and B = A[J] and I ∩ J = ∅. For every pair of subsets I′ ⊂ I and J′ ⊂ J consider the constructible open subsets

\begin{matrix}S_{A}(I′) := & R_{A}(\frac{x}{1} | x ∈ I′) ∩ R_{A}(\frac{1}{x} | x ∈ I \ I′) ⊂ Spv A \\ S_{B}(I′, J′) := & R_{B}(\frac{x}{1} | x ∈ I′ ∪ J′) ∩ R_{B}(\frac{1}{x} | x ∈ (I ∪ J) \ (I′ ∪ J′)) ⊂ Spv B. \end{matrix}

Clearly Spv A = ⋃_{I′⊂I} S_{A}(I′), and likewise for Spv B, so it suffices to show that the subset Spv(f)(T ∩S_{B}(I′, J′)) is constructible in S_{A}(I′), for any such I′ and J′ (lemma 8.1.19(iii,x.a)). Now, for given I′ and J′, let :

• A_{1} := ℤ[I′ ∪ {x^{−1} | x ∈ I \ I′}] ⊂ Frac A
• B_{1} := A_{1}[J′ ∪ {x^{−1} | x ∈ J \ J′} ∪ a_{•} ∪ b_{•}] ⊂ Frac B
• A_{2} := A · A_{1} ⊂ Frac A and B_{2} := B · B_{1} ⊂ Frac B.

The natural inclusions maps A → A_{2} ← A_{1} and B → B_{2} ← B_{1} are localizations, so dim B_{1} = dim B_{2} = d ([65, Ch.IV, Prop.10.6.1(ii)]), and f extends to an injective ring homomorphism f_{2} : A_{2} → B_{2}, which in turn restricts to a map f_{1} : A_{1} → B_{1}. So we get a commutative diagram of continuous maps

\xymatrix{Spv B_{1} \ar[d]_{Spv f_{1}} & Spv B_{2} \ar[l] \ar[r] \ar[d]_{Spv f_{2}} & SpvB \ar[d]^{Spv f} \\ Spv A_{1} & Spv A_{2} \ar[l] \ar[r] & SpvA}

whose horizontal arrows are quasi-compact open immersions. Moreover, S_{A}(I′) and S_{B}(I′, J′) lie respectively in the images of Spv A_{2} and Spv B_{2}, and their images in Spv A_{1} and Spv B_{1} are the subsets

Z_{A}(I′) := {v ∈ Spv^{+}A_{1} | v(x^{−1}) ≠ 0 for every x ∈ I \ I′}
Z_{B}(I′, J′) := {v ∈ Spv^{+}B_{1} | v(x^{−1}) ≠ 0 for every x ∈ (I ∪ J) \ (I′ ∪ J′)}.

Hence, we may replace A by A_{1}, B by B_{1}, and suppose that

T = (R_{B}(\frac{b_{0}}{b_{0}} ) \ R_{B}(\frac{1}{b_{1}} ) ∪ ⋯ ∪ R_{B}(\frac{1}{b_{n}} )) ∩ Spv^{+}B for certains b_{0}, …, b_{n} ∈ B.

Next, by [86, Partie I, Th.5.2.2] we may find an integer r > 0 and elements t_{1}, …, t_{r} ∈ A \ {0} such that, for every i = 1, …, r the inclusion map of subrings of B′_{i} := B[t_{i}^{−1}]

A_{i} := A[\frac{t_{1}}{t_{i}} , …, \frac{t_{r}}{t_{i}} ] → B_{i} := B[\frac{t_{1}}{t_{i}} , …, \frac{t_{r}}{t_{i}} ]

is flat. We consider the constructible subsets

S_{0} := \{v ∈ Spv B | v(t_{i}) = 0 for i = 1, …, r\}  S_{i} := R_{B}(\frac{t_{1}}{t_{i}} , …, \frac{t_{r}}{t_{i}} )  (i = 1, …, r).

Since Spv B = ⋃_{i=0}^{r} S_{i}, it suffices to show that Spv(f)(T ∩ S_{i}) is constructible in Spv A for every i = 0, …, r. However, T ∩ S_{0} is a constructible subset of Spv B/I, where I ⊂ B is the ideal generated by t_{1}, …, t_{r}; since dim B/I < dim B, the inductive assumption yields the assertion for i = 0. For i = 1, …, r, set A′_{i} := A[t_{i}^{−1}]; we get a commutative diagram of continuous maps

\xymatrix{Spv B′_{i} \ar[r]^{φ′_{i}} \ar[d] & Spv B_{i} \ar[r]^{φ_{i}} \ar[d] & Spv B \ar[d] \\ Spv A′_{i} \ar[r]^{ψ′_{i}} & Spv A_{i} \ar[r]^{ψ_{i}} & Spv A}

and we notice that the maps φ′_{i}, ψ′_{i}, ψ_{i} ∘ ψ′_{i} and φ_{i} ∘ φ′_{i} are quasi-compact open immersions, and T ∩ S_{i} = φ_{i}(T_{i}), for the constructible subsets of Spv B′_{i}

T_{i} := (R_{B_{i}}(\frac{t_{i}b_{0}}{t_{i}b_{0}} ) \ (R_{B_{i}}(\frac{1}{b_{1}} ) ∪ ⋯ ∪ R_{B_{i}}(\frac{1}{b_{n}} ))) ∩ Spv^{+}B_{i} ⊂ φ′_{i}(Spv B′_{i}).

In light of lemma 8.1.19(iii,x.a), it then suffices to show that the image of T_{i} is constructible in Spv A_{i}. Thus, we may replace A, B, b_{0} and T respectively by A_{i}, B_{i}, t_{i}b_{0} and T_{i}, and assume furthermore that f is a flat ring homomorphism.

Let φ : Spec B/b_{0}B → Spec A be the restriction of Spec f; there exists a non-empty affine open subset U ⊂ Spec A such that the restriction φ^{−1}U → U of φ is a flat morphism of schemes ([64, Ch.IV, Th.6.9.1]). For every 𝔭 ∈ U, we have a short exact sequence of A-modules

ℰ : 0 → B_{𝔭} → B_{𝔭} → B_{𝔭}/b_{0}B_{𝔭} → 0

and the induced sequence ℰ ⊗_{A} κ(𝔭) is still exact, since B_{𝔭}/b_{0}B_{𝔭} is a flat A_{𝔭}-module (notation of definition 4.9.17(i)). In other words, we have :

(D) The image of b_{0} in B ⊗_{A} κ(𝔭) is a regular element, for every 𝔭 ∈ U. Set

Z := Spec A \ U Z_{B} := Spec(f)^{−1}Z U_{B} := Spec(f)^{−1}U.

Then Spv Z is a constructible closed subset of Spv A and Z_{B} is an affine scheme of dimension < d; also, T′ := T ∩ Spv Z_{B} is a constructible subset of Spv Z_{B}, so our inductive assumption says that Spv(f)(T′) is constructible in Spv A. It remains therefore only to show that the subset Spv(f)(T ∩ Spv U_{B}) is constructible in Spv A.

Let J ⊂ B be the ideal generated by b_{1}, …, b_{n}; by [63, Ch.IV, Th.1.8.4], the subset W := Spec(f)(Spec B/J) is constructible in Spec A, hence

Spv^{+}W := (σ_{A}^{+})^{−1}(W)

is constructible in Spv^{+}A (remark 9.2.4(v)). In light of claim 9.2.31(ii), to conclude the proof, it then suffices to show :

Claim 9.2.33. Under the current assumptions, we have

Spv(f)(T ∩ Spv U_{B}) = Spv^{+}W ∩ Spv U.

Proof of the claim. A simple inspection shows that Spv(f)(T) ⊂ Spv^{+}W, so it remains only to check that Spv^{+}W ∩ Spv U ⊂ Spv(f)(T ∩ Spv U_{B}). Now, let v ∈ Spv^{+}W ∩ Spv U be any element, and let π : A → κ(v) be the induced map. Since v ∈ Spv^{+}A, we have Im π ⊂ V := κ(v)^{+}, and we set B′ := B ⊗_{A} V. Let also 𝔪_{V} and η_{V} be respectively the closed point and the generic point of Spec V; since v ∈ Spv U, the image of η_{V} in Spec A lies in U. By construction, there exists 𝔭 ∈ W such that Spec f(𝔭) is the center of v, i.e. the image of 𝔪_{V} in Spec A (see remark 9.2.4(v)). Then we may find 𝔭′ ∈ Spec B′ whose images in Spec B and Spec V equal respectively 𝔭 and 𝔪_{V}. We pick a minimal prime ideal 𝔮′ of B′ contained in 𝔭′. Since f is flat, the same holds for the induced map V → B ⊗_{A} V, and therefore the image of 𝔮′ in Spec V equals η_{V} ([126, Th.9.5]); consequently the image of 𝔮′ in Spec B lies in U_{B}. Let \overline{b}_{0} ∈ B′ be the image of b_{0}; from condition (D) and [126, Th.6.1(ii), Th.6.5(iii)], we deduce that

(9.2.34)  \overline{b}_{0} ∉ 𝔮′.

Choose a valuation ring V′ of the residue field κ(𝔮′) that dominates the image of the local ring B′_{𝔭′} (corollary 9.1.25). Then, V′ corresponds to a point v′ ∈ Spv B, and we notice that Spv f(v′) = v : indeed, the induced map V → κ(𝔮′) is injective, and V′ dominates the image of this map, whence the assertion. To conclude the proof of the claim, it suffices to check that v′ ∈ T ∩ Spv U_{B}. However, (9.2.34) already implies that v′(b_{0}) ≠ 0. Moreover, since V′ contains the image of B, we have v(b_{i}) ≤ 1 for every i = 1, …, m, and since the image of 𝔭′ lies in the maximal ideal of V, we have v(b_{i}) < 1 for every i = 1, …, n. Thus, v ∈ T. Lastly, since the image of 𝔮′ in Spec B lies in U_{B}, we get v ∈ Spv U_{B} as well, as needed. □

Remark 9.2.35. Our proof of theorem 9.2.29employs crucially Gruson and Raynaud’s flattening technique ([86]). On the other hand, Huber in [98] recovers this theorem rather easily via “elimination of quantifiers” for the first order theory of algebraically closed valued fields with non-trivial valuation. However, the latter is of course a deep model theoretic result, so Huber’s proof is hardly more elementary than ours.

We wish next to explain how the valuation spectrum can be applied to the study of integral closures of rings and ideals. We begin with the following :

Definition 9.2.36. Let f : A → B be any ring homomorphism, I ⊂ A any ideal. The integral closure of I in B is the set denoted

i.c.(I, B)

and consisting of all x ∈ B that satisfy, for some n ∈ ℕ, an equation of the type :

(9.2.37)  x^{n} + f(a_{1}) · x^{n−1} + ⋯ + f(a_{n}) = 0  where a_{j} ∈ I^{j} for j = 1, …, n.

Notice that for I = A, the set i.c.(A, B) is the usual integral closure of A in B. We have :

Lemma 9.2.38. In the situation of definition 9.2.36, let b ∈ B be any element.

(i) The following conditions are equivalent : (a) b ∈ i.c.(I, B).

(b) bU^{−1} ∈ i.c.(R(A, I)_{•}, R(B, B)_{•}) (notation of example 7.9.3). (ii) Especially, i.c.(I, B) is an ideal of i.c.(A, B). (iii) For every prime ideal 𝔭 ∈ Spec B, let \overline{b}_{𝔭} ∈ κ(𝔭) denote the image of b (notation of

definition 4.9.17(i)). The following conditions are equivalent :

(a) b ∈ i.c.(I, B).

(b) \overline{b}_{𝔭} ∈ i.c.(I, κ(𝔭)) for every minimal prime ideal 𝔭 of B.

Proof. Suppose that b satisfies an identity (9.2.37), and set T := U^{−1} to ease notation; we get

(bT)^{n} + T f(a_{1}) · (bT)^{n−1} + ⋯ + T^{n}f(a_{n}) = 0

where T^{j}f(a_{j}) ∈ R(A, I)_{•} for j = 1, …, n. Conversely, suppose the bT is integral over R(A, I)_{•}, so that we have a monic polynomial P (X) = X^{n} + u_{1}X^{n−1} + ⋯ + u_{n}, with coefficients in R(A, I)_{•}, such that P (bT) = 0; for every j = 1, …, n, denote by v_{j} the component of u_{j} in degree n (for the grading of R(A, I)_{•}); since bT is homogeneous of degre 1, it follows that (bT)^{n} + v_{1} · (bT)^{n−1} + ⋯ + v_{n} = 0, and letting T = 1, we deduce that b is integral over I. This proves (i), and assertion (ii) is an immediate consequence.

(iii): Clearly (iii.a)⇒(iii.b). Hence, suppose that (iii.b) holds, and for every minimal prime ideal 𝔭 of B, pick a monic polynomial P_{𝔭}(X) ∈ A[X], say of degree n_{𝔭}, such that P_{𝔭}(\overline{b}_{𝔭}) = 0, and whose coefficient in degree n_{𝔭} − j lies in I^{j}, for j = 1, … n_{𝔭}. For every such 𝔭, let V_{𝔭} ⊂ Spec B denote the subset of all prime ideals containing P_{𝔭}(b). Hence, Z := ⋃_{𝔭∈Min(B)} V_{𝔭} contains the set Min(B) of minimal prime ideals of B; but each V_{𝔭} is a constructible closed subset of Spec B, hence Z = Spec B, and there exists a finite subset S ⊂ Min(B) such that ⋃_{𝔭∈S} V_{𝔭} = Spec B (theorem 8.1.34(i)). Set P := ∏_{𝔭∈S} P_{𝔭}; it follows that P (b) lies in the nilpotent ideal of B, so P^{r}(b) = 0 for a sufficiently large r ∈ ℕ. However, let N be the degree of P^{r}; it is easily seen that the coefficient in degree N −j of P^{r} lies in I^{j}, for every j = 1, …, N (details left to the reader), whence (iii.a). □

Concerning integral closures of ideals, much more can be found in the treatise [102], a comprehensive reference for this subject. For our purposes, proposition 9.2.42hereafter will suffice. For its proof, we shall employ the technical lemma 9.2.40, which shall be applied also later to the study of the adic spectrum of a topological ring, and for this reason, is stated here in wider generality than needed for our immediate purposes in this section.

9.2.39. Let B be any ring, A ⊂ B a subring, I an ideal of A, and Σ, Σ′ two sets; denote by 𝒫_{I} the subset of the free polynomial A-algebra

R[Z] where R := A[X_{i}, Y_{j} | i ∈ Σ, j ∈ Σ′]

consisting of all polynomials P fulfilling the following conditions :

• We have P = Z^{n} + ∑_{k=1}^{n} Z^{n−k} · P_{k}(X_{•}, Y_{•}) for some P_{1}, …, P_{n} ∈ R such that P_{k} is homogeneous of degree k for every k = 1, …, n (for the standard ℕ-grading on R such that X_{i}, Y_{j} ∈ deg_{1} R for every i ∈ Σ and j ∈ Σ′).

• For every k = 1, …, n, denote by P_{k}(0, Y_{•}) the image of P_{k} by the evaluation map R → A[Y_{j} | j ∈ Σ′] such that X_{i} ↦ 0 for every i ∈ Σ; then P_{k}(0, Y_{•}) ∈ I[Y_{i} | j ∈ Σ′].

Set as well

T := {v ∈ Spv B | v(a) ≤ 1 and v(b) < 1 for every a ∈ A and b ∈ I}
T_{0} := {v ∈ T | Ker v is a minimal prime ideal of B}.

Notice that T is a pro-constructible subset of Spv B, and it contains all the primary generizations of its points.

Lemma 9.2.40. In the situation of (9.2.39), suppose moreover that I is contained in the Jacobson radical of A, and let also (g_{i} | i ∈ Σ) and (h_{j} | j ∈ Σ′) be two systems of elements of B. Then for every f ∈ B the following conditions are equivalent :

(a) For every v ∈ T with v(f) ≠ 0 there exists either i ∈ Σ such that v(f) ≤ v(g_{i}) or else j ∈ Σ′ such that v(f) < v(h_{j}).

(b) For every v ∈ T_{0} with v(f) ≠ 0 there exists either i ∈ Σ such that v(f) ≤ v(g_{i}) or else j ∈ Σ′ such that v(f) < v(h_{j}).

(c) There exists a polynomial P ∈ 𝒫_{I} such that P (f, g_{•}, h_{•}) = 0 in B.

Proof. (c)⇒(a): Let P ∈ 𝒫_{I} such that (c) holds, and write P = Z^{n} + ∑_{k=1}^{n} Z^{n−k} · P_{k}(X_{•}, Y_{•}) for some P_{1}, …, P_{n} ∈ R. Fix v ∈ T; then there exists k ≤ n such that

v(f^{n}) ≤ v(f^{n−k} · P_{k}(g_{•}, h_{•})).

If v(f) = 0, there is nothing to show; otherwise, we deduce that v(f^{k}) ≤ v(P_{k}(g_{•}, h_{•})). By assumption, P_{k} is homogeneous of degree k, so there exist two systems of non-negative integers ν_{•} := (ν_{i} | i ∈ Σ), μ_{•} := (μ_{j} | j ∈ Σ′), and an element a ∈ A such that the monomial aX_{•}^{ν_{•}}Y_{•}^{μ_{•}} appears in P_{k} and

v(f^{k}) ≤ v(a) · v(g_{•}^{ν_{•}}) · v(h_{•}^{μ_{•}}).

Especially, set |ν_{•}| := ∑_{i∈Σ} ν_{i}, and define likewise |μ_{•}|; then |ν_{•}| + |μ_{•}| = k. Suppose first that |ν_{•}| > 0; if there exists i ∈ I with v(f) ≤ v(g_{i}), we are done. Otherwise, we have v(f^{|ν_{•}|}) > v(g_{•}^{ν_{•}}), and therefore v(f^{|μ_{•}|}) < v(h_{•}^{μ_{•}}), since v(a) ≤ 1. In this case, we easily get v(f) < v(h_{j}) for some j ∈ Σ′. Lastly, if |ν_{•}| = 0, by assumption we must have a ∈ I, so that v(a) < 1, and therefore v(f^{k}) < v(h_{•}^{μ_{•}}), so we conclude as in the previous case.

(a)⇒(b) is obvious.

(b)⇒(c): We suppose that (c) fails, and we exhibit v ∈ T_{0} such that v(f) ≠ 0 and v(g_{i}) < v(f), v(h_{j}) ≤ v(f) for every i ∈ Σ and j ∈ Σ′. To this aim, consider the subring

A′ := A[g_{i}/f, h_{j}/f | i ∈ Σ, j ∈ Σ′] ⊂ B′ := B[f^{−1}]

and the ideal J_{0} of A′ generated by the system (g_{i}/f | i ∈ Σ); set J := J_{0} + IA′.

Claim 9.2.41. If (c) fails, we have J ≠ A′.

Proof of the claim. If J = A′, we may find an expression of the form

1 = ∑_{i∈Σ} \frac{g_{i}}{f} · F_{i}(\frac{g_{•}}{f} , \frac{h_{•}}{f} ) + ∑_{k=1}^{r} a_{k} · G_{k}(\frac{g_{•}}{f} , \frac{h_{•}}{f} ) in A′

for systems (F_{i}(X_{•}, Y_{•}) | i ∈ Σ) and (G_{k}(X_{•}, Y_{•}) | k = 1, …, r) of elements of R, with F_{i} = 0 except for finitely many values of i ∈ Σ, and a system (a_{1}, …, a_{r}) of elements of I. We may then multiply both sides of this identity by f^{n+1} for a sufficiently large n ∈ ℕ, and obtain an identity in A of the form P (f, g_{•}, h_{•}) = 0, with

P (Z, X_{•}, Y_{•}) = Z^{n+1} − ∑_{i∈Σ} Z^{n} · X_{i} · F_{i}(X_{•}/Z, Y_{•}/Z) − ∑_{k=1}^{r} Z^{n+1} · a_{k} · G_{k}(X_{•}/Z, Y_{•}/Z).

A simple inspection shows that the coefficient of the monomial Z^{n+1} appearing in P is of the form 1 − a for some a ∈ I : namely, a is the sum of the terms a_{k} such that G_{k} ∈ A. However, 1 − a ∈ A^{×}, since I is contained in the Jacobson radical of A; to conclude the proof, it suffices to observe that (1 − a)^{−1} · P ∈ 𝒫_{I} : details left to the reader. ♢

By virtue of claim 9.2.41, we may find a maximal ideal 𝔪 of A′ containing J, and we let 𝔭 ⊂ 𝔪 be any minimal prime ideal of A′; then A′_{𝔭} ⊂ B′_{𝔭} and we pick any minimal prime ideal 𝔮 of B′_{𝔭}. Clearly 𝔮 ∩ A′_{𝔭} = 𝔭A′_{𝔭}, so the induced map C := A′/𝔭 → D := B′_{𝔭}/𝔮 is still injective. Let \overline{𝔪} ⊂ C be the image of 𝔪; by corollary 9.1.25, we may find a valuation ring V of Frac D that dominates C\overline{𝔪}. The valuation ring V corresponds to an element v ∈ T_{0} with the required properties. □

Proposition 9.2.42. Let i : A → B be an injective ring homomorphism, J any ideal of A. Set

𝒱_{B/A} := Spv(i)^{−1}(Spv^{+}A) 𝒱_{B/A}^{0} := {v ∈ 𝒱_{B/A} | ker v is a minimal prime ideal of B}.

For every v ∈ 𝒱_{B/A}, let also π_{v} : B → κ(v) be the natural map. Then we have :

i.c.(J, B) = ⋂_{v∈𝒱_{B/A}}π_{v}^{−1}(J · κ(v)^{+}) = ⋂_{v∈𝒱_{B/A}^{0}}π_{v}^{−1}(J · κ(v)^{+}).

Proof. By applying lemma 9.2.40with I = 0, (g_{i} | i ∈ Σ) any (non-empty) set of generators for J, and Σ′ = ∅, we see that i.c.(J, B) is the set of elements f ∈ B such that the following holds. For every v ∈ 𝒱_{B/A} (respectively, for every v ∈ 𝒱_{B/A}^{0}) there exists i ∈ Σ with v(f) ≤ v(g_{i}). The proposition is an immediate consequence. □

Remark 9.2.43. For every ring homomorphism f : A → B, every ideal I ⊂ A, and every q ∈ ℚ_{+}, write q = m/n for integers m, n ∈ ℕ, and let

i.c.(I, B, q) := {b ∈ B | b^{n} ∈ i.c.(I^{m}, B)}.

(i) Define 𝒱_{B/A} := Spv(f)^{−1}(Spv^{+}A). We claim that i.c.(I, B, q) is the set of all b ∈ B fulfilling the following condition. For every v ∈ 𝒱_{B/A} there exists x ∈ I such that v(b) ≤ v(f(x))^{q} (see remark 9.1.2(ii)). Indeed, by proposition 9.2.42, we have :

i.c.(I, B, q) = {b ∈ B | π_{v}(b^{n}) ∈ I^{m} · κ(v)^{+} for every v ∈ 𝒱_{B/A}}.

The latter condition can be restated as follows. For every v ∈ 𝒱_{B/A} there exists x ∈ I^{m} · κ(v)^{+} such that v(b^{n}) ≤ v(x). We may write such x as a sum ∑_{i=1}^{k} f(x_{i}) · y_{i} for certain x_{i} ∈ I^{m}, y_{i} ∈ V, and then v(x) ≤ v(f(x_{i})) for at least an index i ≤ r. By the same token, we may assume that x_{i} is a product a_{1}a_{2} ⋯ a_{m} of elements of I, and then v(x_{i}) ≤ v(f(a_{j})^{m}) for at least one index j ≤ m.

(ii) Especially, (i) shows that i.c.(I, B, q) depends only on q (and not on the integers m, n). Notice as well that

(9.2.44)  i.c.(I^{m}, B) = i.c.(I, B, m)  for every m ∈ ℕ.

Moreover, a simple inspection of the definition reveals that i.c.(I, B, q) ⊂ i.c.(A, B), and then the criterion of (i) also implies that i.c.(I, B, q) is an ideal of i.c.(A, B), for every q ∈ ℚ_{+}. Furthermore, we have

(9.2.45)  i.c.(I, B, q) = i.c.(i.c.(I, A), B, q)  for every q ∈ ℚ_{+}.

Indeed, since I ⊂ i.c.(I, A), we have obviously i.c.(I, B, q) ⊂ i.c.(i.c.(I, A), B, q). Conversely, if b ∈ i.c.(i.c.(I, A), B, q) and v ∈ 𝒱_{B/A} is any element, (i) says that there exists x ∈ i.c.(I, A) such that v(b) ≤ v(f(x))^{q}; by the same token, there exists y ∈ I such that v(x) ≤ v(y), so that v(b) ≤ v(f(y))^{q}, which shows that b ∈ i.c.(I, B, q). Next, let us set

\overline{i.c.}(I, B, r) := ⋂_{\begin{matrix}q∈ℚ_{+} \\ q<r \end{matrix}} i.c.(I, B, q) for every r ∈ ℝ_{>0}

and define as well \overline{i.c.}(I, B, 0) := i.c.(A, B). Denote also by ℛ_{B/A} the set of all real valued valuations | · | on B that lie in 𝒱_{B/A}. We have :

Lemma 9.2.46. With the notation of remark 9.2.43(ii), let r ∈ ℝ_{+} be any real number, b ∈ B any element, and denote by A[b] the A-subalgebra of B generated by b. The following conditions are equivalent :

(a) b ∈ \overline{i.c.}(I, B, r).
(b) |b| ≤ sup_{x∈I} |f(x)|^{r} for every element | · | of ℛ_{A[b]/A}.

Proof. (a)⇒(b): The assumption means that b ∈ i.c.(I, A[b], q) for every non-negative rational number q < r. By remark 9.2.43(i), this implies that, for every | · | in ℛ_{A[b]/A} and every such q, there exists x ∈ I such that |b| ≤ |f(x)|^{q}. Assertion (b) is an immediate consequence.

(b)⇒(a): Suppose first that r = 0. In this case, we have |b| ≤ 1 for every ℛ_{A[b]/A}, and we need to show that b ∈ i.c.(A, B). Suppose, that the latter fails; then, by remark 9.2.43(i), there exists a valuation |·| in 𝒱_{A[b]/A} such that |b| > 1. It is easily seen that the characteristic subgroup of | · | equals Δ := O(|b|) (notation of (9.2.10) and remark 9.1.2(vii)), hence | · |^{Δ} is still an element of 𝒱_{A[b]/A} and |b|^{Δ} > 1 (lemma 9.2.11(ii)). We may therefore assume that the value group of | · | equals Δ, and | · |′ := | · |_{o(|b|)} is then a valuation of rank one (lemma 9.2.11(i) and remark 9.1.2(vii)). Lastly, | · |′ is equivalent to an element of ℛ_{A[b]/A}, by remark 9.1.2(vi), and by construction we have |b|′ > 1, a contradiction.

Next, suppose that q > 0, in which case – by remark 9.2.43(i) – it suffices to show that, for every | · | in 𝒱_{A[b]/A} and every non-negative rational number q < r, there exists x ∈ I such that |b| ≤ |f(x)|^{q}. However, by the foregoing case we know already that |b| ≤ 1 for every such | · |.

• If |b| = 0, there is nothing to prove, so suppose now that 0 < |b| < 1, and set Δ := O(|b|), Δ′ := o(|b|); notice also that the characteristic subgroup of | · | is {1}, hence | · |^{Δ} lies still in 𝒱_{A[b]/A} (lemma 9.2.11(ii)) and |b|^{Δ} = |b|. The same then holds for | · |′ := | · |_{Δ′}^{Δ} (lemma 9.2.11(i)), and as in the foregoing, we see that the latter valuation is equivalent to an element of ℛ_{A[b]/A}. Moreover, since |b| ∉ Δ′, we still have |b|′ < 1. Assumption (b) then implies that |b|′ ≤ sup_{x∈I} |f(x)|^{′r}. Explicitly, for every real number ε > 0 there exists x_{ε} ∈ I such that |f(x_{ε})|^{′r} > |b|′ − ε. Then |f(x_{ε})|^{′q} > t_{ε} := (|b|′ − ε)^{r/q}, and since q/r < 1 and |b|′ < 1, we may find ε small enough, so that t_{ε} > |b|′, and consequently |f(x_{ε})|^{′q} > |b|′. It follows that |f(x_{ε})|^{q} > |b|, as required.

• Lastly, suppose that |b| = 1. In this case, we must exhibit some x ∈ I with |f(x)| = 1. Suppose that no such element exist, and let π : A[b] → κ := κ(| · |) be the projection (notation of remark 9.1.4(v)); in the current situation, the image of π lies in the valuation ring κ^{+} of the induced valuation | · |_{κ}, and the assumption means that π(I) lies in the maximal ideal 𝔪 of κ^{+}. Then, denote by | · |′_{κ} the trivial valuation on κ^{+}/𝔪; the composition of | · |′_{κ} with the induced projection A → κ^{+}/𝔪 gives an element | · |′_{A} of ℛ_{A[b]/A} such that |b|′_{A} = 1 and |f(x)|′_{A} = 0 for every x ∈ I, contradicting assumption (b). □

For future reference, we point out the following “approximation lemma” for integral closures:

Lemma 9.2.47. Let A be any ring, a_{•} := (a_{1}, …, a_{n}) and b_{•} := (b_{1}, …, b_{n}) two sequences of elements of A, and q > 1 any rational number such that

b_{i} − a_{i} ∈ i.c.(I, A, q) for every i = 1, …, n.

Denote by I ⊂ A (resp. J ⊂ A) the ideal generated by a_{•} (resp. by b_{•}), and suppose that : (a) either, the radical of I equals the radical of J

(b) or else, I is contained in the Jacobson radical of A. Then i.c.(I, A) = i.c.(J, A).

Proof. Since i.c.(I, A, q) ⊂ i.c.(I, A), it is clear that i.c.(J, A) ⊂ i.c.(I, A). To show the converse inclusion, set

r_{v} := max(v(a_{1}), …, v(a_{n}))  s_{v} := max(v(b_{1}), …, v(b_{n}))  for every v ∈ Spv A^{+}.

By virtue of proposition 9.2.42, it suffices to check that s_{v} ≥ r_{v} for every such v. Now, suppose first that (a) holds, and say that r_{v} = v(a_{i}) for some i ≤ n; under our assumptions, remark 9.2.43(i) implies that v(b_{j} − a_{j}) ≤ r_{v}^{q} for every j = 1, …, n. If r_{v} = 1, the ideal I is not contained in the center of v (see remark 9.2.4(v)), and then the same holds for J, so that s_{v} = 1 as well. If r_{v} = 0, we get v(b_{j} − a_{j}) = 0 for every j ≤ n, whence s_{v} = 0 as well. Lastly, if 0 < s_{v} < 1, we get s_{v}^{q} < s_{v}, so that v(a_{i}) = v(b_{i}), and thus s_{v} ≥ r_{v} again.

Next, suppose that (b) holds, and let v ∈ Spv^{+}A be any valuation; again, we need to show that s_{v} ≥ r_{v}, and by virtue of proposition 9.2.16(iii) we may assume that the center 𝔭 of v is a maximal ideal of A; especially I ⊂ 𝔭, in which case r_{v} < 1, and we argue as in the previous case, to conclude. □

We shall continue in section 9.3 the study of the class of ideals considered in remark 9.2.43. We shall be especially interested in the case where A = B is a perfect topological 𝔽_{p}-algebra: see lemma 9.3.74, which in turn shall be useful for our investigation of the finer topological properties of the ring W (A) of Witt vectors over A.

9.3. Witt vectors. This reviews the ring of p-typical Witt vectors associated with an arbitrary ring; the basic reference is [34, Ch.IX, §1], though our treatment is self-contained and pays greater attention to the topological aspects of the theory.

Henceforth we fix a prime number p and we set 𝔽_{p} := ℤ/pℤ.

Definition 9.3.1. Let M be the free monoid with basis {X_{i}, Y_{i} | i ∈ ℕ}, and consider the morphism of monoids

deg : M → ℕ^{⊕2}  such that deg(X_{i}) := (p^{i}, 0) deg(Y_{i}) := (0, p^{i})  for every i ∈ ℕ.

Let P ∈ ℤ[M] = ℤ[X_{i}, Y_{i} | i ∈ ℕ] be any polynomial; we can write uniquely P = ∑_{i=1}^{k} n_{i}P_{i} for integers n_{1}, …, n_{k} ∈ ℤ \ {0} and distinct monomials P_{1}, …, P_{k} ∈ M, and we say that :

• P is bihomogeneous of degree (a, b) ∈ ℕ^{⊕2}, if deg(P_{i}) = (a, b) for every i = 1, …, k • P is homogeneous of total degree d ∈ ℕ, if deg(P_{i}) = (a_{i}, b_{i}) with integers a_{i}, b_{i} ∈ ℕ

such that a_{i} + b_{i} = d, for every i = 1, …, k.

9.3.2. Following [34, Ch.IX, §1, n.1] one defines, for every n ∈ ℕ, the n-th Witt polynomial

ω_{n}(X_{0}, …, X_{n}) := ∑_{i=0}^{n} p^{i}X_{i}^{p^{n−i}} ∈ ℤ[X_{0}, …, X_{n}].

Notice the inductive relations :

(9.3.3)  \begin{matrix}ω_{n+1}(X_{0}, …, X_{n+1}) = & ω_{n}(X_{0}^{p}, …, X_{n}^{p}) + p^{n+1}X_{n+1} \\ = & X_{0}^{p^{n+1}} + p · ω_{n}(X_{1}, …, X_{n+1}). \end{matrix}

Let A be any ring; for every n ∈ ℕ, and every \underline{a} := (a_{n} | n ∈ ℕ) ∈ A^{ℕ}, the element

ω_{n}(\underline{a}) := ω_{n}(a_{0}, …, a_{n}) ∈ A

is called the n-th ghost component of \underline{a}, and we consider the ghost map

ω_{A} : A^{ℕ} → A^{ℕ} \underline{a} ↦ (ω_{n}(\underline{a}) | n ∈ ℕ).

Lemma 9.3.4. Let A be a ring, (J_{n} | n ∈ ℕ) a sequence of ideals of A, such that

pJ_{n} + J_{n}^{p} ⊂ J_{n+1} ⊂ J_{n} for every n ∈ ℕ.

Let also x, y ∈ A be any two elements, and \underline{a}, \underline{b} ∈ A^{ℕ} any two sequences. The following holds : (i) If x ≡ y (mod J_{0}), we have x^{p^{n}} ≡ y^{p^{n}} (mod J_{n}) for every n ∈ ℕ. (ii) If a_{i} ≡ b_{i} (mod J_{m}) for every i = 0, …, n, then ω_{i}(\underline{a}) ≡ ω_{i}(\underline{b}) (mod J_{m+i}) for every

i = 0, …, n.

Proof. (i) is left to the reader, and (ii) follows easily from (i). □

Proposition 9.3.5. With the notation of (9.3.2), we have :

(i) If p is a regular (resp. invertible) element in A, then ω_{A} is injective (resp. bijective). (ii) Suppose that A admits a ring endomorphism φ : A → A such that

φ(a) ≡ a^{p} (mod pA) for every a ∈ A.

Then Im ω_{A} = \{\underline{b} ∈ A^{ℕ} | φ(b_{n}) ≡ b_{n+1} (mod p^{n+1}A) for every n ∈ ℕ\}. Especially, Im ω_{A} is a subring of A^{ℕ}.

Proof. (i): Indeed, (9.3.3) shows that the condition ω_{n}(\underline{a}) = \underline{b} is equivalent to the identities

b_{0} = a_{0}  and b_{n} = ω_{n−1}(a_{0}^{p}, …, a_{n−1}^{p}) + p^{n}a_{n}  for every n ≥ 1

whence the assertion.

(ii): An easy induction argument reduces the assertion to the following :

Claim 9.3.6. Let n > 0 be any integer, and a_{0}, …, a_{n−1}, b_{n} ∈ A any sequence of elements. Set b_{n−1} := ω_{n−1}(a_{0}, …, a_{n−1}). Then the following conditions are equivalent :

(a) There exists a_{n} ∈ A such that b_{n} = ω_{n}(a_{0}, …, a_{n}).

(b) φ(b_{n−1}) ≡ b_{n} (mod p^{n}A).

Proof of the claim. We apply lemma 9.3.4(ii) to the filtration (p^{n}A | n ∈ ℕ), to deduce that

ω_{n}(a_{0}, …, a_{n−1}, x) ≡ ω_{n−1}(a_{0}^{p}, …, a_{n−1}^{p}) (mod p^{n}A) (by (9.3.3))
≡ ω_{n−1}(φ(a_{0}), …, φ(a_{n−1})) (mod p^{n}A)
≡ φ(b_{n−1})  (mod p^{n}A)

for every x ∈ A. The claim is an immediate consequence. □

9.3.7. Take now A := ℤ[X_{i}, Y_{i} | i ∈ ℕ], and let φ : A → A be the endomorphism such that φ(X_{i}) = X_{i}^{p} and φ(Y_{i}) = Y_{i}^{p} for every i ∈ ℕ. Clearly φ fulfills the condition of proposition 9.3.5(ii); from proposition 9.3.5and (9.3.3) we deduce that there exist polynomials

S_{n}, P_{n} ∈ ℤ[X_{0}, …, X_{n}, Y_{0}, …, Y_{n}] I_{n} ∈ ℤ[X_{0}, …, X_{n}] F_{n} ∈ ℤ[X_{0}, …, X_{n+1}]

uniquely characterized, for every n ∈ ℕ, by the identities :

\begin{matrix}ω_{n}(S_{0}, …, S_{n}) = & ω_{n}(X_{0}, …, X_{n}) + ω_{n}(Y_{0}, …, Y_{n}) \\ ω_{n}(P_{0}, …, P_{n}) = & ω_{n}(X_{0}, …, X_{n}) · ω_{n}(Y_{0}, …, Y_{n}) \\ ω_{n}(I_{0}, …, I_{n}) = & − ω_{n}(X_{0}, …, X_{n}) \\ ω_{n}(F_{0}, …, F_{n}) = & ω_{n+1}(X_{0}, …, X_{n+1}). \end{matrix}

Remark 9.3.8. (i) Notice that for every n ∈ ℕ, the polynomial ω_{n} is bihomogeneous of degree (p^{n}, 0). By a simple induction argument, it follows easily that S_{n} is homogeneous of total degree p^{n}, and P_{n} (resp. I_{n}, resp. F_{n}) is bihomogeneous (resp. homogeneous) of degree (p^{n}, p^{n}) (resp. p^{n}, resp. p^{n+1}), for every n ∈ ℕ.

(ii) For example, a simple calculation yields the formulae :

\begin{matrix}S_{0} = & X_{0} + Y_{0} & S_{1} = & X_{1} + Y_{1} − ∑_{i=1}^{p−1} \frac{1}{p}\binom{p}{i}X_{0}^{i}Y_{0}^{p−i} \\ P_{0} = & X_{0}Y_{0} & P_{1} = & X_{0}^{p}Y_{1} + X_{1}Y_{0}^{p} + pX_{1}Y_{1} \\ I_{0} = & − X_{0} & I_{1} = & − X_{1} − ε · X_{0}^{p} \\ F_{0} = & X_{0}^{p} + pX_{1} & F_{1} = & (1 − p^{p−1})X_{1}^{p} + pX_{2} − ∑_{i=1}^{p−1} \frac{1}{p}\binom{p}{i}X_{0}^{pi}(pX_{1})^{p−i}. \end{matrix}

where ε = 1 if p = 2, and ε = 0 if p > 2. Also, since ω_{n}(0, …, 0, X_{n}) = p^{n}X, we see that

(9.3.9)  \begin{matrix}S_{n}(0, …, 0, X_{n}, 0, …, 0, Y_{n}) = & X_{n} + Y_{n} \\ P_{n}(0, …, 0, X_{n}, 0, …, 0, Y_{n}) = & p^{n}X_{n}Y_{n} \end{matrix}  for every n ∈ ℕ.

(iii) A simple induction shows that :

(9.3.10)  F_{n} ≡ X_{n}^{p} (mod pℤ[X_{0}, …, X_{n+1}])  for every n ∈ ℕ.

Indeed, for n = 0 the assertion is clear from (ii). Suppose that n > 0, and that the assertion is already known for every integer < n, and to ease notation, set A := ℤ[X_{0}, …, X_{n+1}] and ω_{n}(F) := ω_{n}(F_{0}, …, F_{n}); from (9.3.3) we get

ω_{n}(F) ≡ ω_{n}(X_{0}^{p}, …, X_{n}^{p}) ≡ ω_{n−1}(X_{0}^{p^{2}}, …, X_{n−1}^{p^{2}}) + p^{n}X_{n}^{p} (mod p^{n+1}A)

On the other hand, the inductive assumption, together with lemma 9.3.4(i,ii) implies that

ω_{n}(F) ≡ ω_{n−1}(F_{0}^{p}, …, F_{n−1}^{p}) + p^{n}F_{n} ≡ ω_{n−1}(X_{0}^{p^{2}}, …, X_{n−1}^{p^{2}}) + p^{n}F_{n} (mod p^{n+1}A)

therefore p^{n}F_{n} ≡ p^{n}X_{n}^{p} (mod p^{n+1}A), and the assertion follows.

(iv) Proposition 9.3.5(ii) seems to be due to B.Dwork and J.Dieudonné (see [121, §VII.4]). Together with (9.3.7), it yields a simplification of the original argument of E.Witt for the construction of the rings that bear his name, as we proceed now to explain.

9.3.11. Let now (A, +, ·, 𝒯 ) be an arbitrary topological ring (see definition 8.3.1(i)). We endow A^{ℕ} with the topology 𝒯_{W (A)} defined as the product of the topologies 𝒯 on each copy of A, and for every \underline{a} := (a_{n} | n ∈ ℕ), \underline{b} := (b_{n} | n ∈ ℕ) in A^{ℕ}, we set :

\begin{matrix}S_{A}(\underline{a}, \underline{b}) := & (S_{n}(a_{0}, …, a_{n}, b_{0}, …, b_{n}) | n ∈ ℕ) \\ P_{A}(\underline{a}, \underline{b}) := & (P_{n}(a_{0}, …, a_{n}, b_{0}, …, b_{n}) | n ∈ ℕ) \\ I_{A}(\underline{a}) := & (I_{n}(a_{0}, …, a_{n}) | n ∈ ℕ). \end{matrix}

There follow identities :

\begin{matrix}ω_{A}(S_{A}(\underline{a}, \underline{b})) = & ω_{A}(\underline{a}) + ω_{A}(\underline{b}) \\ ω_{A}(P_{A}(\underline{a}, \underline{b})) = & ω_{A}(\underline{a}) · ω_{A}(\underline{b}) \\ ω_{A}(I_{A}(\underline{a})) = & − ω_{A}(\underline{a}) \end{matrix}

for every \underline{a}, \underline{b} ∈ A^{ℕ}. Moreover, for any other topological ring B, and any continuous ring homomorphism ψ : B → A, let ψ^{ℕ} : B^{ℕ} → A^{ℕ} be the induced continuous map; clearly we have the identities

(9.3.12) S_{A} ∘ (ψ^{ℕ} × ψ^{ℕ}) = ψ^{ℕ} ∘ S_{B} P_{A} ∘ (ψ^{ℕ} × ψ^{ℕ}) = ψ^{ℕ} ∘ P_{B} I_{A} ∘ ψ^{ℕ} = ψ^{ℕ} ∘ I_{B}.

Theorem 9.3.13. With the notation of (9.3.11), the set A^{ℕ}, endowed with the addition law S_{A} : A^{ℕ} × A^{ℕ} → A^{ℕ} and product law P_{A} : A^{ℕ} × A^{ℕ} → A^{ℕ} is a commutative topological ring, whose zero element and unit element are respectively the sequences

\underline{0}_{A} := (0, 0, …)  and  \underline{1}_{A} := (1, 0, 0, …).

Moreover, the opposite −\underline{a} (that is, with respect to the addition law S_{A}) of any \underline{a} ∈ A^{ℕ}, is the element I_{A}(\underline{a}).

Proof. It is clear that S_{A}, P_{A} and I_{A} are continuous mapping for the topology 𝒯_{W (A)}, hence it remains only to prove that (A^{ℕ}, S_{A}, P_{A}) is a ring. Pick any surjective ring homomorphism ψ : B → A, with B = ℤ[M] for some free monoid M, and endow B with the discrete topology; then ψ^{ℕ} is also surjective, and in light of (9.3.12), it suffices to prove the theorem for B. Also, the p-Frobenius endomorphism of M induces an endomorphism φ of B that lifts the Frobenius endomorphism of B/pB. We may then replace A by B, and assume from start that p is a regular element of A, and φ : A → A is a given lift of the Frobenius endomorphism of A/pA. In this case, proposition 9.3.5(i,ii) implies that ω_{A} maps bijectively A^{ℕ} onto a subring of A^{ℕ}, and the identities of (9.3.7) tell us that the ring structure induced on A^{ℕ} via this identification, is precisely the one given by the addition law S_{A} and multiplication law P_{A}, and also the zero element and the unit are the required ones, since ω_{A}(\underline{0}_{A}) = 0 ∈ A^{ℕ} and ω_{A}(\underline{1}_{A}) = 1 ∈ A^{ℕ}. By the same token, it is clear that −\underline{a} is computed by I_{A}(\underline{a}) in the resulting ring. □

Definition 9.3.14. Let (A, 𝒯 ) be any topological ring. The topological ring

(A^{ℕ}, S_{A}, P_{A}, \underline{0}_{A}, \underline{1}_{A}, 𝒯_{W (A)})

is called the ring of Witt vectors associated with A, and is denoted by W (A, 𝒯 ). If no ambiguities are likely to arise, we often omit reference to 𝒯 , and write simply W (A).

9.3.15. By construction, the ghost map is a continuous ring homomorphism

ω_{A} : W (A) → A^{ℕ}

(where A^{ℕ} is endowed with termwise addition and multiplication). Henceforth, the addition and multiplication of elements \underline{a}, \underline{b} ∈ W (A) shall be denoted simply in the usual way : \underline{a} + \underline{b} and \underline{a} · \underline{b}, and the neutral element of addition and multiplication shall be denoted respectively 0 and 1. The rule A ↦ W (A) is a functor from the category of topological rings (and continuous ring homomorphisms) to itself; indeed, if ψ : A → B is any continuous ring homomorphism, the map ψ^{ℕ} yields a continuous ring homomorphism

W (ψ) : W (A) → W (B) \underline{a} ↦ (ψ(a_{n}) | n ∈ ℕ).

and moreover one has the identity :

ψ^{ℕ} ∘ ω_{A} = ω_{B} ∘ W (ψ).

Remark 9.3.16. (i) More generally, if A is not necessarily unital ring, the proof of theorem 9.3.13shows that the datum (A^{ℕ}, S_{A}, P_{A}, \underline{0}_{A}) is a non-unital ring W (A), that we shall also call the ring of Witt vectors associated with A. Then clearly the rule A ↦ W (A) extends to an endofunctor of the category of (associative, commutative) non-unital rings.

(ii) If A is a unital ring, and f : A → B, g : A → C two unital A-algebras, the natural ring homomorphisms B → B ⊗_{A} C ← C induce a map of W (A)-algebras

(9.3.17)  W B ⊗_{W A} W C → W (B ⊗_{A} C).

Namely, it is the unique W (A)-linear map such that \underline{b} ⊗ \underline{c} ↦ (P_{n}(\underline{b} ⊗ 1, 1 ⊗ \underline{c}) | n ∈ ℕ) for every \underline{b} ∈ W (B) and \underline{c} ∈ W (C), with \underline{b} ⊗ 1 := (b_{n} ⊗ 1 | n ∈ ℕ), and likewise for 1 ⊗ \underline{c}.

(iii) More generally, let A be a unital ring; a non-unital A-algebra is the datum of an Amodule B and a (commutative, associative) non-unital ring structure on B such that a · (bb′) = (ab) · b′ for every a ∈ A and every b, b′ ∈ B. To every non-unital A-algebra B, we attach a unital A-algebra whose underlying A-module is A ⊕ B, whose structure map A → A ⊕ B is the natural inclusion, and with

(a, b) · (a′, b′) := (aa′, ab′ + a′b + bb′).

(iv) With this terminology, we may then extend (ii) to the case where B and C are non-unital A-algebras : in this case, the homomorphism (9.3.17) of non-unital rings is obtained as follows. Recall that the polynomial P_{n} is bihomogeneous of degree (p^{n}, p^{n}) for every n ∈ ℕ (remark 9.3.8(i)); hence there exists N ∈ ℕ such that P_{n} = ∑_{i=1}^{N} r_{i}P_{n,i}, with r_{i} ∈ ℤ, and where each P_{n,i} is a monomial of bidegree (p^{n}, p^{n}), i.e. there exists M_{i} ∈ ℕ with P_{n,i} = ∏_{j=0}^{M_{i}} X_{j}^{d_{j}}Y_{j}^{e_{j}} and ∑_{j=0}^{M_{i}} d_{j}p^{j} = ∑_{j=0}^{M_{i}} e_{j}p^{j} = p^{n}. Then we set Q_{n}(\underline{b}, \underline{c}) := ∑_{i=1}^{N} r_{i}P_{n,i}(\underline{b}, \underline{c}), with P_{n,i}(\underline{b}, \underline{c}) := ∏_{j=0}^{M_{i}} b_{j}^{d_{j}} ⊗∏_{j=0}^{M_{i}} c_{j}^{e_{j}} for every i = 1, …, N and \underline{b}, \underline{c} as in (ii); lastly, (9.3.17) is given by the rule : \underline{b}⊗\underline{c} ↦ (Q_{n}(\underline{b}, \underline{c}) | n ∈ ℕ). In order to check that (9.3.17) is a map of non-unital rings, endow B′ := A⊕B and C′ := A⊕C of the unital A-algebra structures described in (iii), and notice that (9.3.17) is the restriction of the map of unital W (A)-algebras W B′ ⊗_{W A} W C′ → W (B′ ⊗_{A} C′) of (ii).

9.3.18. Next, one defines two continuous mappings W (A) → W (A), by the rule :

\begin{matrix}F_{A}(\underline{a}) := & (F_{n}(a_{0}, …, a_{n+1}) | n ∈ ℕ) \\ V_{A}(\underline{a}) := & (0, a_{0}, a_{1}, …). \end{matrix}

F_{A} and V_{A} are often called, respectively, the Frobenius and Verschiebung maps ([34, Ch.IX, §1, n.5]). Consider also the mappings f_{A}, v_{A} : A^{ℕ} → A^{ℕ} such that

f_{A}(\underline{a}) := (a_{1}, a_{2}, …)  v_{A}(\underline{a}) := (0, pa_{0}, pa_{1}, …)  for every \underline{a} := (a_{0}, a_{1}, …) ∈ A^{ℕ}.

Then f_{A} is a continuous ring endomorphism of A^{ℕ}, and v_{A} is a continuous endomorphism of the additive group of A^{ℕ}, and the basic identity characterizing (F_{n} | n ∈ ℕ) can be written as :

(9.3.19)  ω_{A} ∘ F_{A} = f_{A} ∘ ω_{A}.

On the other hand, (9.3.3) yields the identity :

(9.3.20)  ω_{A} ∘ V_{A} = v_{A} ∘ ω_{A}.

Let ψ : A → B be a continuous ring homomorphism; directly from the definitions we obtain :

(9.3.21)  W (ψ) ∘ F_{A} = F_{B} ∘ W (ψ)  W (ψ) ∘ V_{A} = V_{B} ∘ W (ψ).

Proposition 9.3.22. With the notation of (9.3.18), we have :

(i) The map F_{A} is a continuous endomorphism of the ring W (A), and V_{A} is a continuous endomorphism of the additive group underlying W (A).

(ii) Moreover one has the identities :

(9.3.23)

F_{A} ∘ V_{A} = p · 1_{W (A)}
V_{A}(\underline{a} · F_{A}(\underline{b})) = V_{A}(\underline{a}) · \underline{b} for every \underline{a}, \underline{b} ∈ W (A).

(iii) F_{A} is a lifting of the Frobenius endomorphism of W (A)/pW (A), i.e. :

F_{A}(\underline{a}) ≡ \underline{a}^{p} (mod pW (A)) for every \underline{a} ∈ W (A).

Proof. (i): Arguing as in the proof of theorem 9.3.13, we reduce to the case where p is a regular element in A and the Frobenius endomorphism of A/pA lifts to a ring endomorphism φ : A → A, in which case ω_{A} is an injective ring homomorphism (proposition 9.3.5(i)). Then the assertion follows immediately from (9.3.19) and (9.3.20). By the same token, we also get the identities of (ii). Likewise, (iii) is reduced to the identity

f_{A}(\underline{a}) − \underline{a}^{p} ∈ p · Im ω_{A} for every \underline{a} := (a_{n} | n ∈ ℕ) ∈ Im ω_{A}

(here \underline{a}^{p} = (a_{n}^{p} | n ∈ ℕ) denotes the p-power map for the ring structure of A^{ℕ}). Now, we have

(9.3.24) φ(a_{n}) ≡ a_{n}^{p} (mod pA)  φ(a_{n}) ≡ a_{n+1} (mod p^{n+1}A)  for every n ∈ ℕ

by proposition 9.3.5(ii), hence a_{n}^{p} − a_{n+1} ∈ pA for every n ∈ ℕ. Set b_{n} := p^{−1} · (a_{n}^{p} − a_{n+1}); again by proposition 9.3.5(ii), it suffices to check that φ(b_{n}) ≡ b_{n+1} (mod p^{n+1}A), that is

φ(a_{n}^{p} − a_{n+1}) ≡ a_{n+1}^{p} − a_{n+2} (mod p^{n+2}A) for every n ∈ ℕ.

However, from (9.3.24) and lemma 9.3.4(i) we deduce that φ(a_{n})^{p} ≡ a_{n+1}^{p} (mod p^{n+2}A), whence the contention : details left to the reader. □

9.3.25. Let n ∈ ℕ be any integer; it follows easily from (9.3.23) that

V_{n}(A) := Im V_{A}^{n}

is an ideal of W (A), and we define the ring of n-truncated Witt vectors of A as the quotient

W_{n}(A) := W (A)/V_{n}(A)

which we endow with the topology 𝒯_{W_{n}(A)} induced by the projection π_{n} : W (A) → W_{n}(A).

Remark 9.3.26. (i) More generally, if A is a not necessarily unital ring, then V_{n}(A) is a welldefined ideal of the non-unital ring W (A) (see remark 9.3.16(i)). Hence, the quotient W_{n}(A) is a well-defined non-unital ring in this case, for every n ∈ ℕ.

(ii) Likewise, in the situation of remark 9.3.16(iii), we have a well-defined homomorphism of non-unital A-algebras W_{n}B ⊗_{W_{n}A} W C → W_{n}(B ⊗_{A} C), for every n ∈ ℕ.

Lemma 9.3.27. With the notation of (9.3.25), we have :

(i) \underline{a} = (a_{0}, …, a_{m−1}, 0, …) + V_{A}^{m} ∘ f_{A}^{m}(\underline{a}) for every \underline{a} ∈ W (A) (where the addition is taken in the additive group of W (A)).

(ii) The projection W (A) → A^{n} : (a_{i} | i ∈ ℕ) ↦ (a_{0}, …, a_{n−1}) factors through π_{n} and a bijection W_{n}(A) →^{∼} A^{n}, and 𝒯_{W_{n}(A)} corresponds to the product topology of A^{n}, under this identification.

Proof. (i): Arguing as in the proof of theorem 9.3.13, we may reduce to the case where p is a regular element in A, in which case ω_{A} is an injective ring homomorphism, and therefore it suffices to show that

ω_{A}(\underline{a}) = ω_{A}(a_{0}, …, a_{m−1}, 0, …) + ω_{A} ∘ V_{A}^{m} ∘ f_{A}^{m}(\underline{a}).

The latter follows by a simple inspection : details left to the reader. Assertion (ii) follows easily from (i). □

Remark 9.3.28. (i) In view of lemma 9.3.27(ii), we get therefore an inverse system of topological rings (W_{n}(A) | n ∈ ℕ), whose limit is W (A). Clearly the ghost components descend to well-defined continuous ring homomorphisms :

\overline{ω}_{m} : W_{n}(A) → A for every m < n.

Especially, the map \overline{ω}_{0} : W_{1}(A) → A is an isomorphism of topological rings.

(ii) A direct inspection of the construction shows that the endofunctor W of the category of topological rings commutes with all limits.

(iii) In the same vein, if the topological ring A is the limit of a system (A_{i} | i ∈ I) of topological rings, then F_{A} is the limit of the corresponding system of map (F_{A_{i}} | i ∈ I), and likewise for V_{A}.

(iv) Let A be any topological ring, I any ideal of the monoid (A, ·), and r ∈ ℕ any integer. In light of remark 9.3.8(i), it is easily seen that the sets

W (I, r) := {(a_{n} | n ∈ ℕ) ∈ W (A) | a_{n} ∈ I^{p^{n}}A for every n ≤ r} and W (I) := ⋂_{n∈ℕ} W (I, n)

are ideals of W (A) : details left to the reader. Moreover, we have the inclusions :

(9.3.29)  W (I^{c}, r) ⊂ W (I, r)^{c} and W (I^{c}) ⊂ W (I)^{c}

(where, for any topological space X and every subset S ⊂ X we denote by S^{c} the topological closure of S in X). Indeed, by definition we have W (I, r) := ∏_{n∈ℕ} J_{n}, with J_{n} := I^{p^{n}}A for every n ≤ r and J_{n} := A for n > r, so that W (I, r)^{c} = ∏_{n∈ℕ} J_{n}^{c}, and it suffices to recall that (I^{c})^{p^{n}} ⊂ (I^{p^{n}})^{c} for every n ∈ ℕ. The same argument applies to W (I).

Especially, if I^{p^{n}}A is a closed ideal in the topology of A for every n ∈ ℕ, then it follows that W (I, r) is a closed ideal of W (A), and the same holds also for W (I).

Furthermore, if I, J are any two ideals of the monoid (A, ·), and r ∈ ℕ is any integer, it follows easily from remark 9.3.8(i) that

W (I, r) · W (J, r) ⊂ W (IJ, r) and W (I) · W (J) ⊂ W (IJ).

We warn the reader that the notation W (I) might also refer to the non-unital Witt ring associated to the non-unital ring I, as defined in remark 9.3.26; these two meanings of W (I) agree if and only if I^{2} = I. However, in practice the ambiguity should be harmless, since non-unital Witt rings only intervene once in our text, in the statement and proof of proposition 14.7.18.

Proposition 9.3.30. Let A be a topological ring, and n ∈ ℕ. We have :

(i) With the notation of remark 9.3.28(i), the ring homomorphism

π_{n} : W_{n+1}(A) → A^{n+1} \underline{a} ↦ (\overline{ω}_{0}(\underline{a}), \overline{ω}_{1}(\underline{a}), …, \overline{ω}_{n}(\underline{a}))

is integral with nilpotent kernel. (ii) p^{n}A^{n+1} ⊂ Im π_{n} ⊂ {(x_{0}, …, x_{n}) ∈ A^{n+1} | x_{i} ≡ x_{0}^{p^{i}} (mod pA) for i = 1, …, n}.

Proof. (i): Denote by B ⊂ A^{n+1} the integral closure of the image of π_{n}. Clearly B contains the n + 1 canonical idempotents e_{0}, …, e_{n} of A^{n+1}. Next, a simple inspection shows that

π_{n}(ae_{0}) = (a, a^{p}, ⋯, a^{p^{n}}) for every a ∈ A.

It follows that a^{p^{i}}e_{i} = e_{i} · π_{n}(ae_{0}) ∈ B, and then also ae_{i} ∈ B, for every a ∈ A and every i ≤ n. This implies that B = A^{n+1}, i.e. π_{n} is integral, as asserted. Next, the sequence of ideals (V_{i}(A) | i ∈ ℕ) induces a descending filtration (\overline{V}_{i} | i = 0, …, n) on W_{n+1}(A), and we notice:

Claim 9.3.31. For every i = 0, …, n − 1, we have an isomorphism of W_{n+1}(A)-modules

A →^{∼} \overline{V}_{i}/\overline{V}_{i+1} a ↦ ae_{i} (mod \overline{V}_{i+1})

for the W_{n+1}(A)-module structure on A induced by \overline{ω}_{i}.

Proof of the claim. It is clear that this map is bijective. To show that it is also W_{n+1}(A)-linear, notice that P_{j}(X_{0}, …, X_{j}, 0, …, 0) = 0 for every j < i (remark 9.3.8(i)), so that

ω_{i}(0, …, 0, P_{i}(X_{0}, …, X_{i}, 0, …, 0, Y_{i})) = ω_{i}(X_{0}, …, X_{i}) · ω_{i}(0, …, 0, Y_{i})

by (9.3.7). This translates as the identity

p^{i} · P_{i}(X_{0}, …, X_{i}, 0, …, 0, Y_{i}) = ω_{i}(X_{0}, …, X_{i}) · p^{i}Y_{i}

whence P_{i}(X_{0}, …, X_{i}, 0, …, 0, Y_{i}) = ω_{i}(X_{0}, …, X_{i}) · Y_{i}, which implies the claim. ♢

Set I := Ker π_{n}; from claim 9.3.31 we deduce that I · \overline{V}_{i} ⊂ \overline{V}_{i+1} for every i = 0, …, n, whence I^{n+1} = 0 in W_{n+1}(A), which concludes the proof of (i).

(ii): The second inclusion follows by direct inspection of the Witt polynomials. To show the first inclusion, consider any sequence (a_{0}, …, a_{n}) ∈ A^{n+1}; we need to find a solution (x_{0}, …, x_{n}) ∈ W_{n+1} for the system of polynomial equations :

x_{0}^{p^{i}} + px_{1}^{p^{i−1}} + ⋯ + p^{i}x_{i} = p^{n}a_{i} for i = 0, …, n

and we claim that there exists a solution that fulfills as well the further condition : x_{i} ∈ p^{n−i}A for i = 0, …, n. The proof proceeds by a simple induction on i ≤ n, which we leave to the reader. □

Corollary 9.3.32. Let A be any topological ring such that p^{k}A = 0 for some k ∈ ℕ. Then Ker (\overline{ω}_{0} : W_{n+1}A → A) is nilpotent for every n ∈ ℕ.

Proof. Let \underline{a} ∈ W_{n+1}A be an element such that \overline{ω}_{0}(\underline{a}) = 0, and define π_{n} as in proposition 9.3.30(i); in view of (9.3.3), we see that π_{n}(\underline{a}) = (0, pb_{1}, …, pb_{n}) for some b_{1}, …, b_{n} ∈ A; hence \underline{a}^{k} ∈ Ker π_{n}, and it suffices to invoke proposition 9.3.30(i) to conclude. □

Lemma 9.3.33. Let (A, 𝒯 ) be a topological ring, with separated completion (A^{∧}, 𝒯 ^{∧}).

(i) If the topology 𝒯 is discrete, the topology 𝒯_{W (A)} agrees with the linear topology defined by the filtration (V_{m}(A) | m ∈ ℕ).

(ii) The topology 𝒯 is separated (resp. complete) if and only if the same holds for 𝒯_{W (A)}. (iii) If 𝒯 is a linear topology, then the same holds for 𝒯_{W (A)}. (iv) Denote also by W (A, 𝒯 )^{∧} the separated completion of W (A, 𝒯 ). The natural map

W (A, 𝒯 ) → W (A^{∧}, 𝒯 ^{∧}) factors uniquely via an isomorphism of topological rings

W (A, 𝒯 )^{∧} →^{∼} W (A^{∧}, 𝒯 ^{∧}).

Proof. (i) follows easily from lemma 9.3.27(ii) : details left to the reader.

(iii): Let (I_{λ} | λ ∈ Λ) be a fundamental system of open ideals of A, and for every λ ∈ Λ and every r ∈ ℕ, denote by π_{λ,r} : W (A) → W_{r}(A/I_{λ}) the projection; then it is easily seen that the family of ideals (Ker π_{λ,r} | λ ∈ Λ, r ∈ ℕ) is a fundamental system of open neighborhoods of 0 ∈ W (A). The assertion is an immediate consequence.

(iv) In light of corollary 8.2.16(ii), we are reduced to checking that W_{n}(A^{∧}) is the separated completion of W_{n}(A). The latter is established by induction on n ∈ ℕ, by applying proposition 8.2.13(i,v) to the exact sequences of topological groups :

0 → V_{n}(A)/V_{n+1}(A) → W_{n+1}(A) → W_{n}(A) → 0

where V_{n}(A)/V_{n+1}(A) is endowed with the topology induced by the inclusion into W_{n+1}(A), and is thus isomorphic to (A, 𝒯 ), by virtue of claim 9.3.31.

(ii): 𝒯 is separated (resp. complete) if and only if the completion map A → A^{∧} is injective (resp. surjective), if and only if the induced map W (A, 𝒯 ) → W (A^{∧}, 𝒯 ^{∧}) is injective (resp. surjective), and by virtue of (iv), the latter holds if and only if the completion map W (A, 𝒯 ) → W (A, 𝒯 )^{∧} is injective (resp. surjective), whence the assertion. □

9.3.34. Teichmüller mapping. The projection ω_{0} admits a continuous set-theoretic section

τ_{A} : A → W (A) a ↦ (a, 0, 0, …).

For every a ∈ A, we call τ_{A}(a) the Teichmüller representative of a. Clearly, for any continuous ring homomorphism ψ : A → B we have :

(9.3.35)  τ_{B} ∘ ψ = W (ψ) ∘ τ_{A}.

Proposition 9.3.36. With the notation of (9.3.34), we have :

(i) τ_{A}(a) · \underline{b} = (a^{p^{n}}b_{n} | n ∈ ℕ) for every a ∈ A and \underline{b} := (b_{n} | n ∈ ℕ) ∈ W (A).

(ii) Moreover, τ_{A} is a multiplicative map, i.e. the following identity holds :

τ_{A}(a) · τ_{A}(b) = τ_{A}(a · b) for every a, b ∈ A

(iii) \underline{a} = ∑_{n=0}^{∞} V_{A}^{n}(τ_{A}(a_{n})) for every \underline{a} := (a_{n} | n ∈ ℕ) ∈ W (A), where the convergence of the series is relative to the topology 𝒯_{W (A)}.

Proof. (i): Since P_{n} is bihomogeneous of degree (p^{n}, p^{n}) (remark 9.3.8(i)), we have

P_{n}(X_{0}, 0, …, 0, Y_{0}, …, Y_{n}) = X_{0}^{p^{n}} · P_{n}(1, 0, …, 0, Y_{0}, …, Y_{n}) = X_{0}^{p^{n}} · Y_{n}

where the second identity follows after recalling that \underline{1}_{A} := (1, 0, …) is the unit of W (A). The assertion follows immediately. (ii) is a special case of (i), and (iii) follows easily from lemma 9.3.27(i). □

Example 9.3.37. (i) For any n ∈ ℕ, let (a_{0}, …, a_{n}) ∈ W_{n+1}(ℤ) be the sequence that represents the element p^{n}. Since \overline{ω}_{i} is a ring homomorphism, we have

p^{n} = a_{0}^{p^{i}} + pa_{1}^{p^{i−1}} + ⋯ + p^{i}a_{i} for every i = 0, …, n.

A simple induction then shows that

a_{i} ≡ p^{n−i} (mod p^{n−i+1}) for every i = 0, …, n.

(ii) Let A be any topological ring, n, i ∈ ℕ any integers; set

k := min(i, n)  and \overline{V}_{n,j}(A) := V_{j}(A)/V_{n+1}(A)  for every j ≤ n.

Let also \overline{V} : W_{n}(A) → W_{n+1}(A) be the ℤ-linear map induced by V. We claim that

∑_{j=0}^{k} p^{i−j}\overline{V}_{n,j}(A) = J_{i,n} := \{(a_{0}, …, a_{n}) ∈ W_{n+1}(A) | a_{j} ∈ p^{i−j}A for every j ≤ k\}.

For the proof, we argue by induction on n and i, and notice first that J_{i,n} is an ideal of W_{n+1}(A) for every n and i, due to remark 9.3.8(i). If n = 0 or i = 0, there is nothing to prove. Suppose then that n, i > 0 and that the assertion is known for all strictly smaller values of n and i; from (i) we see that p^{i}W_{n+1}(A) ⊂ J_{i,n}, and the inductive assumption shows that

∑_{j=1}^{k} p^{i−j}\overline{V}_{j}(A) = \overline{V} (∑_{j=0}^{k−1} p^{i−j−1}\overline{V}_{n−1,j}(A)) = \overline{V} (J_{i−1,n−1}) ⊂ J_{i,n}

whence ∑_{j=0}^{k} p^{i−j}\overline{V}_{n,j}(A) ⊂ J_{i,n}. Conversely, say that \underline{a} := (a_{0}, …, a_{n}) ∈ J_{i,n}, so that a_{0} = p^{i}b_{0} for some b_{0} ∈ A; due to lemma 9.3.27(i) and proposition 9.3.36(i) we may write

\underline{a} = τ(a_{0}) + \overline{V} (a_{1}, …, a_{n}) = τ(b_{0}) · (p^{i}, 0, …, 0) + \overline{V} (a_{1}, …, a_{n})

and notice that \overline{V} (a_{1}, …, a_{n}) ∈ \overline{V} (J_{i−1,n−1}) ⊂ ∑_{j=1}^{k} p^{i−j}\overline{V}_{j,n}(A). Also, from (i) we see that (p^{i}, 0, …, 0) = p^{i} + \overline{V} (b_{1}, …, b_{n}), where again (b_{1}, …, b_{n}) ∈ J_{i−1,n−1}. Summing up, we get J_{i,n} ⊂ ∑_{j=0}^{k} p^{i−j}\overline{V}_{j,n}(A), as required.

(iii) Especially, (ii) implies that for every ring A we have

p^{i}W_{n+1}(A) ⊂ J_{i,n} ⊂ p^{i−n}W_{n+1}(A) for every i, n ∈ ℕ with i ≥ n.

If 𝒯_{p} denotes the p-adic topology on A, it follows that the topology of W_{n+1}(A, 𝒯_{p}) agrees with the p-adic topology of the underlying ring W_{n+1}(A), for every n ∈ ℕ.

Example 9.3.38. (i) Let S ⊂ ℤ be any subset, and A any ring (which we may regard as a discrete topological ring). Then the localization map j : A → S^{−1}A induces a ring isomorphism

S^{−1}W_{n+1}(A) →^{∼} W_{n+1}(S^{−1}A) for every n ∈ ℕ.

Indeed, let us show first that multiplication by every s ∈ S is a bijection on W_{n+1}(S^{−1}A). To this aim, we consider the descending filtration (\overline{V}_{i}A | i = 0, …, n) as in the proof of proposition 9.3.30(i); we are then easily reduced to checking that multiplication by s is bijective on \overline{V}_{i}(A)/\overline{V}_{i+1}(A) for i = 0, …, n − 1. But according to claim 9.3.31, the latter quotient is isomorphic to S^{−1}A, for the W_{n+1}(S^{−1}A)-module structure induced on S^{−1}A by \overline{ω}_{i}, whence the contention. Thus, W_{n+1}(j) extends to a well defined ring homomorphism λ : S^{−1}W_{n+1}A → W_{n+1}(S^{−1}A). To see that the latter is an isomorphism, we notice that λ is a homomorphism of filtered rings for the filtrations S^{−1}\overline{V}_{•}A on S^{−1}W_{n+1}A and \overline{V}_{•}(S^{−1}A) on W_{n+1}(S^{−1}A); then it suffices to check that the associated graded ring homomorphism gr_{•}λ is bijective, and the latter assertion follows again from claim 9.3.31.

(ii) If p ∈ A^{×}, a simple inspection shows that ghost map ω_{A} : W (A) → A^{ℕ} is a ring homomorphism, and likewise, W_{n+1}A →^{∼} A^{n+1} in this case, for every n ∈ ℕ. Combining with (i), we deduce that for every ring A, the localization A → A[p^{−1}] induces a ring isomorphism

W_{n+1}(A)[p^{−1}] →^{∼} (A[p^{−1}])^{n+1} for every n ∈ ℕ.

(iii) In the same vein, let S ⊂ A be any multiplicative subset, and denote by τ_{A}(S) ⊂ W_{n+1}A the image of the subset {τ_{A}(s) | s ∈ S} ⊂ W (A) (notation of (9.3.34)). Then the localization map j : A → S^{−1}A induces a ring isomorphism

λ : τ_{A}(S)^{−1}W_{n+1}(A) →^{∼} W_{n+1}(S^{−1}A) for every n ∈ ℕ.

Indeed, from proposition 9.3.36(ii) we see that the image of τ_{A}(S) is invertible in W_{n+1}(S^{−1}A), so λ is well defined. Next, proposition 9.3.36(i) easily implies that λ is surjective. Lastly, the kernel of λ is τ_{A}(S)^{−1}Ker W_{n+1}(j); but clearly Ker W_{n+1}(j) is the ideal of all sequences \underline{a} := (a_{0}, …, a_{n}) such that for every i = 0, …, n there exists s_{i} ∈ S with s_{i}a_{i} = 0. Then τ_{A}(s_{0} ⋯ s_{n}) · \underline{a} = 0 in W_{n+1}A; this shows that Ker λ = 0, and concludes the proof.

9.3.39. Let now A be a topological 𝔽_{p}-algebra. In this case (9.3.10) yields the identity :

(9.3.40)  F_{A}(\underline{a}) = (a_{n}^{p} | n ∈ ℕ)  for every \underline{a} := (a_{n} | n ∈ ℕ) ∈ W (A).

As an immediate consequence we get :

(9.3.41) p · \underline{a} = V_{A} ∘ F_{A}(\underline{a}) = F_{A} ∘ V_{A}(\underline{a}) = (0, a_{0}^{p}, a_{1}^{p}, …)  for every \underline{a} ∈ W (A).

Especially, we have

(9.3.42)  p = (0, 1, 0, …)  in W (A).

Moreover, let Φ_{A} : A → A be the Frobenius endomorphism; (9.3.40) also implies the identity :

(9.3.43)  F_{A} ∘ τ_{A} = τ_{A} ∘ Φ_{A}.

Proposition 9.3.44. Let A be any 𝔽_{p}-algebra. We have :

(i) The p-adic topology on W (A) agrees with the V_{1}(A)-adic topology, and both are finer than the topology 𝒯 ′ defined by the filtration (V_{n}(A) | n ∈ ℕ). (ii) Moreover, W (A) is also complete and separated for the p-adic topology. (iii) W (A)^{×} = ω_{0}^{−1}(A^{×}). (iv) V^{i}(τ_{A}(b)) · V^{j}(τ_{A}(b)) = V^{i+j}(τ_{A}(a^{p^{j}}b^{p^{i}})) for every a, b ∈ A and every i, j ∈ ℕ.

Proof. (i): It is clear from (9.3.41) that the p-adic topology on W (A) is finer than both 𝒯 ′ and the V_{1}-adic topology. Conversely, for every \underline{a}, \underline{b} ∈ W (A) we have

V_{A}(\underline{a}) · V_{A}(\underline{b}) = V_{A}(\underline{a} · F_{A}V_{A}(\underline{b})) = p · V_{A}(\underline{a} · \underline{b})

whence V_{1}^{2} ⊂ pA, which says that the V_{1}-adic topology is finer than the p-adic topology.

(ii): Since W (A) is complete and separated for the linear topology 𝒯 ′, the assertion follows from lemma 8.3.12.

(iii): Clearly W (A)^{×} ⊂ ω_{0}^{−1}(A^{×}). For the converse, notice that the ideal V_{1}(A) is contained in the Jacobson radical ideal of W (A), due to (i),(ii) and remark 8.3.10(v). Therefore, the induced map Spec ω_{0} : Spec A → Spec W (A) restricts to a bijection on the sets of maximal ideals of A and W (A), and the assertion follows easily.

(iv): Without loss of generality, we may assume that i ≤ j; then we compute

V^{i}(τ_{A}(b)) · V^{j}(τ_{A}(b)) = V^{i}(τ_{A}(a) · F^{i}V^{j}(τ_{A}(b)))  (by (9.3.23))
= V^{i}(τ_{A}(a) · p^{i} · V^{j−i}(τ_{A}(b))) (again by (9.3.23))
= p^{i} · V^{i}(τ_{A}(a) · V^{j−i}(τ_{A}(b))) (by proposition 9.3.22(i))
= p^{i} · V^{i}(V^{j−i}(τ_{A}(a^{p^{j−i}}b)))  (by proposition 9.3.36(i))
= p^{i} · V^{j}(τ_{A}(a^{p^{j−i}}b))  (by (9.3.41))
= V^{i+j}(τ_{A}(a^{p^{j}}b^{p^{i}}))

as stated. □

9.3.45. Suppose additionally that A is perfect, i.e. that Φ_{A} is an automorphism of the topological ring A (especially, it is a homeomorphism). It follows from (9.3.40) that F_{A} is an automorphism of W (A). Hence, in view of (9.3.23) and (9.3.41) :

p^{n} · W (A) = V_{n}(A) for every n ∈ ℕ.

So, the p-adic filtration coincides with the filtration (V_{n}(A) | n ∈ ℕ) and with the V_{1}(A)-adic filtration. Especially, the 0-th ghost component descends to an isomorphism

\overline{ω}_{0} : W (A)/pW (A) →^{∼} A.

Also, the identity of proposition 9.3.36(iii) can be written in the form :

(9.3.46)  \underline{a} = ∑_{n=0}^{∞} p^{n} · τ_{A}(a_{n}^{p^{−n}})  for every \underline{a} := (a_{n} | n ∈ ℕ) ∈ W (A).

Proposition 9.3.47. Let A be a reduced 𝔽_{p}-algebra, \underline{a} := (a_{n} | n ∈ ℕ) ∈ W (A) any element.

(i) Consider the following conditions : (a) The ideal J := ∑_{n∈ℕ} a_{n}A is not contained in any minimal prime ideal of A.

(b) \underline{a} is a regular element of W (A).

Then (a)⇒(b), and if A has only finitely many minimal prime ideals, (b)⇒(a). (ii) W (A) is reduced, and if A is a domain, the same holds for W (A). (iii) If A is a perfect field, then W (A) is a complete discrete valuation ring of mixed char-

acteristic (0, p), maximal ideal pW (A), and residue field A. (iv) If A ⊂ B is an inclusion of perfect 𝔽_{p}-algebras, then W (A) ⊂ W (B), and

p^{n}W (A) = W (A) ∩ p^{n}W (B) for every n ∈ ℕ.

(v) If A is perfect, and ∑_{i=0}^{k} a_{i}A = A for some k ∈ ℕ, we have

\underline{a}W (A) ∩ p^{n}W (A) = p^{n−k} · (\underline{a}W (A) ∩ p^{k}W (A)) for every n ∈ ℕ

and moreover, the ideal \underline{a}W (A) is a closed subset for the p-adic topology on W (A).

Proof. Notice that the assertions are independent of the topology of A and B, hence we may assume that both these topologies are discrete.

(iii): We know already that W (A) is p-adically complete, by proposition 9.3.44(ii). Also, it follows easily from (9.3.41) that p is regular in W (A), and taking into account proposition 9.3.44(iii) we see that every element of W (A) can be written uniquely in the form p^{n} · \underline{u}, for some n ∈ ℕ and an invertible element \underline{u} ∈ W (A). The assertion follows immediately.

(ii): Suppose first that A is a domain, and we prove that the same holds for W (A). To this aim, let B be any perfect field containing A; since W (A) is a subring of W (B), it suffices to check that W (B) is a domain, which is clear from (iii).

Next, let Min A be the set of all minimal prime ideals of A; if A is reduced, the natural map j : A → B := ∏_{𝔭∈Min A} A_{𝔭} is injective, and therefore W (A) is a subring of W (B) = ∏_{𝔭∈Min A} W (A_{𝔭}). To prove that W (A) is reduced, it then suffices to show that the same holds for every factor W (A_{𝔭}), but this is already known, since each such A_{𝔭} is a field.

(i): Suppose that (a) holds; arguing as in the proof of (ii), we reduce to checking that the image of \underline{a} is regular in W (A_{𝔭}), for every 𝔭 ∈ Min A. But under condition (a), it is clear that the image of \underline{a} does not vanish in W (A_{𝔭}) for any such 𝔭. Thus, in this case (ii) implies that (b) holds. Conversely, suppose that Min A = {𝔭_{1}, …, 𝔭_{r}} is a finite set, and say that J ⊂ 𝔭_{1}. Pick any f ∈ ⋂_{i=2}^{r} 𝔭_{i} \ 𝔭_{1}; since j is injective, it is easily seen that Ann_{A}(f) = 𝔭_{1} (details left to the reader). By proposition 9.3.36(i), we then see that \underline{a} · τ_{A}(f) = 0, which contradicts (b).

(iv): It is clear that W (A) ⊂ W (B), and also that V_{n}(A) = W (A) ∩ V_{n}(B), so the assertion follows from (9.3.45).

(v): Say that x\underline{a} = p^{n}y for some x, y ∈ W (A); by (i), p is regular in W (A), so for the proof of the stated equality it suffices to show that x ∈ p^{n−k}W (A). However, we have already noticed that the map j is injective, and therefore p^{k}W (A) = W (A) ∩ p^{k}W (B), by (iv). Hence, it suffices to show that the image of x lies in p^{n−k}W (A_{𝔭}) for every such 𝔭. Moreover, A_{𝔭} is a perfect field for every 𝔭 ∈ Min A, and ∑_{i=0}^{k} a_{i} · A_{𝔭} = A_{𝔭}, so we may replace from start A by any A_{𝔭}, in which case W (A) is a discrete valuation ring with p as uniformizer, by (iii), and notice that by assumption the elements a_{0}, …, a_{k} cannot all vanish, therefore

\underline{a} ∉ p^{k+1}W (A) and x · \underline{a} ∈ p^{n}W (A)

whence the assertion : details left to the reader. Lastly, the foregoing implies that the p-adic topology on \underline{a}W (A) agrees with the topology induced from the p-adic topology on W (A). On the other hand, (i) says that \underline{a} is regular on W (A), hence scalar multiplication by a induces a bijection W (A) →^{∼} \underline{a}W (A) that identifies the p-adic topology of W (A) with the p-adic topology of \underline{a}W (A). Since W (A) is p-adically complete and separated (proposition 9.3.44(ii)), it follows that the same holds for \underline{a}W (A). Summing up, \underline{a}W (A) is complete and separated for the topology induced from the p-adic topology of W (A), and consequently it is a closed subset in W (A), for this topology. □

Example 9.3.48. Let (A, 𝒯 ) be a topological 𝔽_{p}-algebra.

(i) If A is perfect and 𝒯 is the discrete topology, then it follows from (9.3.45)and proposition 9.3.44(i) that 𝒯_{W (A)} agrees with the p-adic topology.

(ii) By inspecting the proof of theorem 8.2.8(i), it is easily seen that if (A, 𝒯 ) is perfect, the same holds for its separated completion.

(iii) For instance, we have a natural isomorphism of topological rings :

ℤ_{p} →^{∼} W (𝔽_{p})

(where 𝔽_{p} is endowed with the discrete topology, and ℤ_{p} with the p-adic topology). Indeed,

(i) says that the topology of W (𝔽_{p}) agrees with the p-adic topology, hence the unique ring homomorphism ℤ → W (𝔽_{p}) extends to a unique continuous map ψ : ℤ_{p} → W (𝔽_{p}). Since p is a regular element of W (𝔽_{p}) (proposition 9.3.47(i)), the map is injective. Moreover, if we endow both rings with their p-adic filtrations, the induced map gr_{•}ψ : gr_{•}ℤ_{p} → gr_{•}W (𝔽_{p}) on associated graded rings is easily seen to be an isomorphism. Then the assertion follows from [34, Ch.III, §2, n.8, Cor.2].

(iv) It follows from (iii) that the structure map β : 𝔽_{p} → A induces a natural structure W (β) of topological ℤ_{p}-algebra on W (A).

(v) Let A := 𝔽_{p}[T^{1/p^{∞}}], the universal perfect 𝔽_{p}-algebra in one generator (notation of example 4.8.55(iii)), and endow A with its discrete topology. Then W (A) is the p-adic completion

ℤ_{p}{T^{1/p^{∞}}}

of ℤ_{p}[T^{1/p^{∞}}]. The latter can be described as the ring of all power series

(9.3.49)  ∑_{n∈ℕ} a_{n}T^{λ_{n}}

where (λ_{n} | n ∈ ℕ) is any sequence of elements of ℕ[1/p], and (a_{n} | n ∈ ℕ) is any sequence of elements of ℤ_{p} that converges p-adically to 0. For the proof, we argue as in (iii): we have a unique map of ℤ_{p}-algebras ψ : ℤ_{p}[T^{1/p^{∞}}] → W (A) such that ψ(T^{λ}) := τ_{A}(T^{λ}) for every λ ∈ ℕ[1/p] (for the ℤ_{p}-algebra structure on W (A) provided by (iv)). Then ψ extends by continuity to a unique map of topological ℤ_{p}-algebras ψ^{∧} : ℤ_{p}{T^{1/p^{∞}}} → W (A), and if we endow both rings with their p-adic filtrations, it is easily seen that the resulting map of associated graded rings gr^{•}ψ^{∧} is bijective, so the same holds for ψ^{∧}, again by [34, Ch.III, §2, n.8, Cor.3].

9.3.50. The lifting property. In the situation of lemma 9.3.4, endow A (resp. A/J_{n}, for every n ∈ ℕ) with the linear topology 𝒯 defined by the system of ideals (J_{n} | n ∈ ℕ) (resp. with the discrete topology) and let π_{n} : A → A/J_{n} and π′_{n} : A/J_{n} → A/J_{0} be the natural projections, for every n ∈ ℕ. Let also R be a topological 𝔽_{p}-algebra, and \overline{φ} : R → A/J_{0} a continuous ring homomorphism. We say that a continuous mapping φ_{n} : R → A/J_{n} is a lifting of \overline{φ} if we have:

φ_{n}(x^{p}) = φ_{n}(x)^{p} for every x ∈ R, and π′_{n} ∘ φ_{n} = \overline{φ}.

Lemma 9.3.51. With the notation of (9.3.50), the following holds :

(i) If φ_{n} and φ′_{n} are two liftings of \overline{φ}, then φ_{n} and φ′_{n} agree on R^{p^{n}}.

(ii) If R is a perfect topological 𝔽_{p}-algebra, there exists a unique lifting φ_{n} : R → A/J_{n}

of \overline{φ}. We have φ_{n}(1) = 1 and φ_{n}(xy) = φ_{n}(x) · φ_{n}(y) for every x, y ∈ R. (iii) Let R be as in (ii), and suppose that (A, 𝒯 ) is complete and separated. Then :

(a) There exists a unique continuous mapping φ : R → A such that

π_{0} ∘ φ = \overline{φ}  and  φ(x^{p}) = φ(x)^{p} for every x ∈ R.

(b) Moreover, φ(1) = 1 and φ(xy) = φ(x) · φ(y) for every x, y ∈ R.

(c) If furthermore, A is an 𝔽_{p}-algebra, then φ is a homomorphism of topological rings.

Proof. (i) is an easy consequence of lemma 9.3.4(i).

(ii): The uniqueness follows from (i). For the existence, pick any mapping σ : A/J_{0} → A/J_{n} such that π′_{n} ∘ σ = 1_{A/J_{0}}, and notice that the mapping ψ := σ ∘ \overline{φ} : R → A/J_{n} is continuous, and π′_{n} ∘ ψ = \overline{φ}. We let :

φ_{n}(x) := ψ(x^{p^{−n}})^{p^{n}} for every x ∈ R.

Using lemma 9.3.4(i) one verifies easily that φ_{n} does not depend on the choice of ψ. Especially, define ψ′ : R → A/J_{n} by the rule : x ↦ ψ(x^{p^{−1}})^{p} for every x ∈ R. Clearly π′_{n} ∘ ψ′ = \overline{φ} as well, hence :

φ_{n}(x^{p}) = ψ′(x^{p^{1−n}})^{p^{n}} = ψ(x^{p^{−n}})^{p^{n+1}} = φ_{n}(x)^{p} for every x ∈ R

as claimed. If we choose σ so that σ(1) = 1, we obtain φ_{n}(1) = 1. Finally, since \overline{φ} is a ring homomorphism we have ψ(x) · ψ(y) ≡ ψ(xy) (mod J_{0}) for every x, y ∈ R. Hence

ψ(x)^{p^{n}} · ψ(y)^{p^{n}} ≡ ψ(xy)^{p^{n}} (mod J_{n})

(again by lemma 9.3.4(i)) and finally φ_{n}(x^{p^{n}}) · φ_{n}(y^{p^{n}}) = φ_{n}((xy)^{p^{n}}), which implies the last stated identity, since R is perfect.

(iii): The existence and uniqueness of φ follow from (ii). It remains only to show that φ(x) + φ(y) = φ(x + y) in case A is an 𝔽_{p}-algebra. It suffices to check the latter identity on the projections onto A/J_{n}, for every n ∈ ℕ, in which case one argues analogously to the foregoing proof of the multiplicative property for φ_{n} : the details shall be left to the reader. □

Proposition 9.3.52. In the situation of (9.3.50), suppose that R is a perfect topological 𝔽_{p}- algebra and (A, 𝒯 ) is complete and separated. Then we have :

(i) For every n ∈ ℕ there exists a unique ring homomorphism v_{n} : W_{n+1}(A/J_{0}) → A/J_{n} such that the following diagram commutes :

\xymatrix{W_{n+1}(A) \ar[r]^{^{ω^{n}}} \ar[d]_{W_{n+1}(π_{0})} & A \ar[d]^{π_{n}} \\ W_{n+1}(A/J_{0}) \ar[r]^{v_{n}} & A/J_{n}.}

(ii) Let u_{n} := v_{n} ∘ W_{n+1}(\overline{φ}) ∘ \overline{F}_{R}^{−n}, where \overline{F}_{R} : W_{n+1}(R) → W_{n+1}(R) is induced by the Frobenius automorphism of W (R). Then we have a commutative diagram :

\xymatrix{W_{n+2}(R) \ar[r]^{u_{n+1}} \ar[d] & A/J_{n+1} \ar[d]^{ϑ} \\ W_{n+1}(R) \ar[r]^{u_{n}} & A/J_{n}}  for every n ∈ ℕ

where the vertical maps are the natural projections. (iii) There exists a unique ring homomorphism u such that the following diagram commutes:

\xymatrix{W (R) \ar[r]^{u} \ar[d]_{_{ω_{0}}} & A \ar[d]^{π_{0}} \\ R \ar[r]^{\overline{φ}} & A/J_{0}.}

Furthermore, u is continuous for the topology 𝒯_{W (R)}, and we have :

(9.3.53)  u(\underline{a}) = ∑_{n∈ℕ} p^{n} · φ(a_{n}^{p^{−n}})  for every \underline{a} := (a_{n} | n ∈ ℕ) ∈ W (R)

with φ : R → A the unique continuous mapping characterized as in lemma 9.3.51(iii).

Proof. (i): We have to check that ω_{n}(a_{0}, …, a_{n}) ∈ J_{n} whenever a_{0}, …, a_{n} ∈ J_{0}, which is clear from lemma 9.3.4(ii).

(ii): Since \overline{F}_{R} is an automorphism, it boils down to verifying :

Claim 9.3.54. ϑ ∘ v_{n+1}(\overline{φ}(a_{0}), …, \overline{φ}(a_{n+1})) = v_{n}(\overline{φ}(F_{0}(a_{0}, a_{1})), …, \overline{φ}(F_{n}(a_{0}, …, a_{n+1}))) for every (a_{0}, …, a_{n+1}) ∈ W_{n+1}(A).

Proof of the claim. Directly from (9.3.21) we derive :

(\overline{φ}(F_{0}(a_{0}, a_{1})), …, \overline{φ}(F_{n}(a_{0}, …, a_{n+1}))) = (F_{0}(\overline{φ}(a_{0}), \overline{φ}(a_{1})), …, F_{n}(\overline{φ}(a_{0}), …, \overline{φ}(a_{n}))).

Hence, it suffices to show that

ϑ ∘ v_{n+1}(b_{0}, …, b_{n+1}) = v_{n}(F_{0}(b_{0}, b_{1}), …, F_{n}(b_{0}, …, b_{n+1}))

for every (b_{0}, …, b_{n+1}) ∈ W_{n+1}(A/J_{0}). The latter identity follows from (9.3.19). ♢

(iii): We take u := lim_{n∈ℕ} u_{n}, where the maps u_{n} are as in (ii), for every n ∈ ℕ. With this definition, it is clear that u is continuous for the topology 𝒯_{W (R)}. Next we remark that π_{0} ∘ u ∘ τ_{R} = \overline{φ} ∘ ω_{0} ∘ τ_{R} = \overline{φ}, hence :

(9.3.55)  u ∘ τ_{R} = φ

by lemma 9.3.51(iii.a). Now (9.3.53) holds by (9.3.46), (9.3.55) and the continuity of u. □

Remark 9.3.56. (i) Let 𝒩_{p} be the full subcategory of ℤ-TopAlg whose objects are the discrete topological rings in which p is nilpotent. For every perfect topological 𝔽_{p}-algebra E and every topological ring W, we consider the functors

\overline{H}_{E}, H_{W} : 𝒩_{p} → Set

such that

\overline{H}_{E}(A) := Hom_{ℤ-TopAlg}(E, A/pA) H_{W} (A) := Hom_{ℤ-TopAlg}(W, A)

for every A ∈ Ob(𝒩_{p}) (where A/pA is endowed with the discrete topology); to any continuous ring homomorphism A → A′ of objects of 𝒩_{p}, the functor \overline{H}_{E} assigns the mapping

\overline{H}_{E}(φ) : H_{E}(A) → H_{E}(A′) (\overline{ψ} : E → A/pA) ↦ (φ ⊗_{ℤ} 𝔽_{p}) ∘ \overline{ψ}

and likewise, H_{W} (φ) : H_{W} (A) → H_{W} (A′) assigns to any ψ : A → A′ the map φ ∘ ψ. Then proposition 9.3.52(iii) says that we have a natural isomorphism of functors

(9.3.57)  H_{W (E)} →^{∼} \overline{H}_{E}

that assigns to every A ∈ Ob(𝒩_{p}) the bijection H_{W (E)}(A) →^{∼} \overline{H}_{E}(A) : φ ↦ φ ⊗_{ℤ} 𝔽_{p}.

(ii) Suppose that E is a perfect topological 𝔽_{p}-algebra whose topology is linear, complete and separated. Then the existence of an isomorphism of functors (9.3.57) characterizes W (E) up to natural isomorphism, in the category of complete and separated topological rings whose topology is linear and coarser than the p-adic topology. Indeed, suppose that W′ is any other such topological ring, endowed with an isomorphism of functors H_{W′} →^{∼} \overline{H}_{E}. There follows an isomorphism of functors ω : H_{W (E)} →^{∼} H_{W′}. Now, let I ⊂ W (E) be any open ideal; the quotient W (E)/I is an object of 𝒩_{p}, and the projection π_{I} : W (E) → W (E)/I is an element of H_{W (E)}(W (E)/I), so we get a continuous ring homomorphism ω(π_{I}) : W′ → W (E)/I. Moreover, if J ⊂ I is another open ideal, the projection π_{I,J} : W (E)/J → W (E)/I is a morphism of 𝒩_{p}, and we have

π_{I,J} ∘ ω(π_{J}) = H_{W′}(π_{I,J})(ω(π_{J})) = ω(H_{W (E)}(π_{I,J})(π_{J})) = ω(π_{I})

by the naturality of ω. Hence, we have a well defined inverse system (ω(π_{I}) | I ⊂ W (E)) indexed by the cofiltered system of open ideals of W (E); the limit of this system is a well defined continuous ring homomorphism α : W′ → W (E). Swapping the roles of W′ and W (E), the inverse natural transformation ω^{−1} : H_{W′} →^{∼} H_{W (E)} yields likewise a continuous ring homomorphism β : W (E) → W′, as W′ is complete and separated by assumption.

It remains to check that α and β are mutually inverse isomorphisms. To this aim, let I′ ⊂ W′ be any open ideal, π′_{I′} : W′ → W′/I′ the projection, and set I := Ker (ω^{−1}(π′_{I′}) : W (E) → W′/I′), so that ω^{−1}(π′_{I′}) factors as the composition of π_{I} and a continuous ring homomorphism τ : W (E)/I → W′/I′. Notice that

\begin{matrix}π′_{I′} ∘ β ∘ α = & ω^{−1}(π′_{I′}) ∘ α = τ ∘ π_{I} ∘ α = τ ∘ ω(π_{I}) = H_{W′}(τ)(ω(π_{I})) \\ = & ω(H_{W (E)}(τ)(π_{I})) = ω(τ ∘ π_{I}) = ω(ω^{−1}(π′_{I′})) = π′_{I′} \end{matrix}

which implies that β∘α = 1_{W′}. Likewise one checks that β∘α = 1_{W (E)}, whence the contention.

(iii) Let E, E′ and E′′ be three perfect topological 𝔽_{p}-algebras whose topologies are linear, complete and separated, and E → E′, E → E′′ two continuous ring homomorphisms. It is clear that (E′ ⊗_{E} E′′, 𝒯_{E′,E′′}^{⊗}) is a perfect topological 𝔽_{p}-algebra (notation of (8.3.7)), hence the same holds for E′ ˆ⊗_{E}E′′ (example 9.3.48(ii)), and in light of (8.3.7) and (i), we get natural isomorphisms of functors

H_{W (E′ ˆ⊗_{E}E′′)} →^{∼} H_{W (E′)} ×_{H_{W (E)}} H_{W (E′′)} ⟵^{∼} H_{W (E′) ˆ⊗_{W (E)}W (E′′)}

whence, by (ii), a natural isomorphism of topological rings

(9.3.58)  W (E′ ˆ⊗_{E}E′′) →^{∼} W (E′) ˆ⊗_{W (E)}W (E′′).

Proposition 9.3.59. Let A be any discrete and reduced 𝔽_{p}-algebra. We have :

(i) W (A), endowed with its p-adic topology, is a flat and topologically free ℤ_{p}-module. (ii) If A is perfect, and (g_{λ} | λ ∈ Λ) is any basis of the 𝔽_{p}-vector space underlying A, then (τ_{A}(g_{λ}) | λ ∈ Λ) is a topological basis of the topological ℤ_{p}-module underlying W (A).

Proof. The element p is regular in W (A), by proposition 9.3.47(i), so W (A) is torsion-free, hence flat, as a ℤ_{p}-module, for its natural ℤ_{p}-module structure as in example 9.3.48(iv). In order to check both (i) and (ii), it then suffices to show the following general

Claim 9.3.60. Let R be any artinian ring, κ (resp. 𝔪) the residue field (resp. the maximal ideal) of R, and M any flat R-module. Let also (f_{λ} | λ ∈ Λ) be any system of elements of M, whose image in M_{0} := M ⊗_{R} κ is a basis for the latter κ-vector space. Then the induced R-linear map

ψ : R^{⊕Λ} → M  e_{λ} ↦ f_{λ}  for every λ ∈ Λ

is an isomorphism (here (e_{λ} | λ ∈ Λ) is the canonical basis of R^{⊕Λ}).

Proof of the claim. Endow R^{⊕Λ} and M with the 𝔪-adic filtrations, and let gr_{•}R^{⊕Λ} and gr_{•}M be the associated graded κ-vector spaces. By virtue of [126, Th.22.3], the natural κ-linear map

𝔪^{n}/𝔪^{n+1} ⊗_{κ} M_{0} → gr_{n}M

is an isomorphism for every n ∈ ℕ. This easily implies that the κ-linear map gr_{•} : gr_{•}R^{⊕Λ} → gr_{•}M induced by ψ, is an isomorphism for every n ∈ ℕ. Then the assertion follows from [34, Ch.III, §2, n.8, Cor.3]. □

9.3.61. Teichmüller representative of a sum. We shall need a combinatorial identity expressing the Teichmüller representative of a finite sum of elements of a perfect topological 𝔽_{p}-algebra A, as a series of terms, each of which is itself a Teichmüller representative. To state the result, let

Σ_{n}^{(k)} := {(σ_{0}, …, σ_{k}) ∈ p^{−n}ℕ^{k+1} \ p^{1−n}ℕ^{k+1} | σ_{0} + ⋯ + σ_{k} = 1} for every k, n ∈ ℕ.

and set Σ^{(k)} := ⋃_{n∈ℕ} Σ_{n}^{(k)} for every k ∈ ℕ. Also for every a := (a_{0}, …, a_{k}) ∈ A^{k+1} and every σ := (σ_{0}, …, σ_{k}) ∈ Σ^{(k)} set

a^{σ} := a_{0}^{σ_{0}} ⋯ a_{k}^{σ_{k}}

(notice that the fractional powers a_{i}^{σ_{i}} are well defined, since A is perfect).

Proposition 9.3.62. For every integer k ∈ ℕ the following holds :

(i) There exists a mapping Σ^{(k)} → ℤ_{p} : σ ↦ c_{σ} such that

τ_{A}(a_{0} + ⋯ + a_{k}) = ∑_{n∈ℕ} p^{n} · ∑_{σ∈Σ_{n}^{(k)}} c_{σ} · τ_{A}(a^{σ})

for every perfect 𝔽_{p}-algebra A and every a := (a_{0}, …, a_{k}) ∈ A^{k+1}.
(ii) c_{σ} = 1 for every σ ∈ Σ_{0}^{(k)}.

Proof. To begin with, let us show the following :

Claim 9.3.63. With the notation of the proposition, for every n ∈ ℕ we have :

(τ_{A}(a_{0})^{1/p^{n}} + ⋯ + τ_{A}(a_{k})^{1/p^{n}})^{p^{n}} ≡ τ_{A}(a_{0} + ⋯ + a_{k}) (mod p^{n}W (A)).

Proof of the claim. Since τ_{A} is a splitting of ω_{0}, for every n ∈ ℕ we have

τ_{A}(a_{0})^{1/p^{n}} + ⋯ + τ_{A}(a_{k})^{1/p^{n}} ≡ τ_{A}(a_{0} + ⋯ + a_{k})^{1/p^{n}} (mod pW (A))

and then the claim follows immediately from lemma 9.3.4(i). ♢

Now, consider first the case where A = 𝔽_{p}[T_{0}^{1/p^{∞}}, …, T_{k}^{1/p^{∞}}] is the universal perfect 𝔽_{p}- algebra in the variables T_{0}, …, T_{k} (see example 4.8.55(iii)). We endow A with the discrete topology. Set T_{•} := (T_{0}, …, T_{k}), and notice that the system of monomials (T_{•}^{μ} | μ ∈ ℕ[1/p]^{k+1}) is a basis for the 𝔽_{p}-vector space underlying A, so the system (τ_{A}(T_{•})^{μ} | μ ∈ ℕ[1/p]^{k+1}) is a topological basis of the topologically free ℤ_{p}-module underlying W (A) (proposition 9.3.59(ii)). Hence, the image of the same system in W (A)/p^{n}W (A) is a basis for the latter free ℤ/p^{n}ℤmodule, for every n ∈ ℕ. Consider the image S_{n} of S := τ_{A}(T_{0} + ⋯ + T_{k}) in W (A)/p^{n}W (A); from claim 9.3.63 we see that S_{n} is uniquely a sum of products c_{μ}^{(n)}T_{•}^{μ} with c_{k}^{(n)} ∈ ℤ/p^{n}ℤ and T_{•}^{μ} a monomial of the above type, with the further condition that

c_{μ}^{(n)} = 0 unless μ ∈ ⋃_{i=0}^{n} Σ_{i}^{(k)}.

Moreover, the uniqueness of the resulting expression means that the image of c_{μ}^{(m)} in ℤ/p^{n}ℤ agrees with c_{μ}^{(n)}, for every such μ and every m ≥ n. Summing up, we get the sought identity for S, and (ii) follows as well. Next, if A is any perfect 𝔽_{p}-algebra, there exists a unique (continuous) map of 𝔽_{p}-algebras

ψ : 𝔽_{p}[T_{0}^{1/p^{∞}}, …, T_{k}^{1/p^{∞}}] → A such that ψ(T_{i}) = a_{i} for i = 0, …, k.

Since the sought identity is already known for T_{•}, we deduce it for a, by virtue of (9.3.35). □

9.3.64. Let A be any perfect 𝔽_{p}-algebra, \underline{b} := (b − n | n ∈ ℕ) ∈ W (A) any element, and set

Ann_{A}(\underline{b}) := \{a ∈ A | τ_{A}(a) · \underline{b} = 0\}.

Corollary 9.3.65. With the notation of (9.3.64), the subset Ann_{A}(\underline{b}) is a radical ideal of A.

Proof. Let a ∈ Ann_{A}(\underline{b}) for some n ∈ ℕ. We show first that a^{μ} ∈ Ann_{A}(\underline{b}) for every μ ∈ ℕ[1/p]. Indeed, in view of proposition 9.3.36(i), the assumption means that a^{p^{n}}b_{n} = 0 for every n ∈ ℕ, and therefore a^{p^{n−k}}b_{n}^{p^{−k}} = 0 for every k, n ∈ ℕ, since A is perfect. Hence a^{p^{n−k}}b_{n} = 0 as well, for every n, k ∈ ℕ, which means that τ_{A}(a^{p^{−k}}) · \underline{b} = 0 for every k ∈ ℕ (again by proposition 9.3.36(i)); the assertion is a straightforward consequence.

It then remains only to check that Ann_{A}(\underline{b}) is an ideal of A. However, if a, a′ ∈ Ann_{A}(\underline{b}) and x ∈ A is any element, it is clear that ax ∈ Ann_{A}(\underline{b}), and proposition 9.3.62 easily implies that a + a′ ∈ Ann_{A}(\underline{b}) as well, as required. □

In the same vein, though the Teichmüller mapping is not a ring homomorphism, one can attach to it a continuous map on Zariski spectra, as explained in the following proposition.

Proposition 9.3.66. Let E be any perfect 𝔽_{p}-algebra; the following holds :

(i) For every 𝔭 ∈ Spec W (E), the subset τ_{E}^{−1}𝔭 is a prime ideal of E.

(ii) The resulting map

Spec τ_{E} : Spec W (E) → Spec E 𝔭 ↦ τ_{E}^{−1}𝔭

is continuous and spectral. (iii) Let 𝔭 ∈ Spec W (E) be any prime ideal, set 𝔮 := Spec τ_{E}(𝔭) ∈ Spec E, and let

π_{𝔭} : W (E) → κ(𝔭), π_{𝔮} : E → κ(𝔮) be the projections. If 𝔭 is a closed subset in the p-adic topology of W (E), then τ_{E} induces a morphism of multiplicative monoids

τ_{𝔭} : κ(𝔮) → κ(𝔭)  such that  τ_{𝔭} ∘ π_{𝔮} = π_{𝔭} ∘ τ_{E}.

(iv) Let E′ be any other perfect 𝔽_{p}-algebra, and φ : E → E′ any continuous ring homomorphism. Then the resulting diagram commutes :

\xymatrix{Spec W (E′) \ar[r]^{Spec τ_{E′}} \ar[d]_{Spec W (φ)} & Spec E′ \ar[d]^{Spec φ} \\ Spec W (E) \ar[r]^{Spec τ_{E}} & Spec E.}

Proof. (i): Say that x, y ∈ E and τ_{E}(xy) ∈ 𝔭; since τ_{E} is a multiplicative map, it follows easily that either x or y lies in τ_{E}^{−1}𝔭. By the same token, if x ∈ τ_{E}^{−1}𝔭 and y ∈ E, then xy ∈ τ_{E}^{−1}𝔭. Next, suppose that x, y ∈ τ_{E}^{−1}𝔭; by proposition 9.3.62, we may write τ_{E}(x + y) as a p-adically convergent series ∑_{n∈ℕ} p^{n} · z_{n}, where each z_{n} is a finite ℤ_{p}-linear combination of terms of the form τ_{E}(x^{λ}y^{1−λ}), with λ, 1−λ ∈ ℕ[1/p]. It follows easily that z_{n} = τ_{E}(x^{1/p})·z′_{n}+τ_{E}(y^{1/p})·z′′_{n}, for suitable z′_{n}, z′′_{n} ∈ W (E). Summing up, we get

τ_{E}(x + y) = a · τ_{E}(x^{1/p}) + b · τ_{E}(y^{1/p})  where a := ∑_{n∈ℕ} c_{n}z′_{n}  b := ∑_{n∈ℕ} c_{n}z′′_{n}.

However, by the foregoing we already know that x^{1/p}, y^{1/p} ∈ τ_{E}^{−1}𝔭, so τ_{E}(x + y) ∈ 𝔭, whence the contention.

(ii): Let f ∈ E any element; directly from the definition we get

(9.3.67)  (Spec τ_{E})^{−1}(Spec E_{f}) = Spec A_{τ_{E}(f)}

which says that Spec τ_{E} is continuous and spectral.

(iii): We check first that τ_{E} descends to a morphism \overline{τ} : E/𝔮 → W (E)/𝔭 of multiplicative monoids. To this aim, let x ∈ E and y ∈ 𝔮 be any two elements; arguing as in the proof of (i), we see that d := τ_{E}(x + y) − τ_{E}(x) can be written as p-adically convergent series ∑_{n∈ℕ} d_{n}, where d_{n} ∈ p^{n}τ_{E}(y^{1/p^{n}})W (E) for every n ∈ ℕ. We have already remarked that τ_{E}(y^{1/p^{n}}) ∈ 𝔭 for every n ∈ ℕ, and since 𝔭 is p-adically closed in W (E), it follows that d ∈ 𝔭 as well, whence the claim. Now, by construction \overline{τ}^{−1}(0) = \{0\}, so \overline{τ} extends uniquely to a morphism of monoids τ_{𝔭} as sought.

(iv) follows immediately from (9.3.35). □

9.3.68. Angular powers. For the further investigation of the ring of Witt vectors W (A) of a given perfect topological 𝔽_{p}-algebra A, it is useful to consider special types of ideals of A and W (A) defined by certain combinatorial conditions that we proceed to explain.

• First, consider any p-perfect monoid P, i.e. such that the p-Frobenius endomorphism Φ_{P} of P is bijective. A basic example of p-perfect monoid is ℕ[1/p] := S_{p}^{−1}ℕ, the localization of ℕ at its multiplicative subset S_{p} := {p^{k} | k ∈ ℕ}. Notice that this condition on P means that for every x ∈ P and every λ ∈ ℕ[1/p], the fractional power x^{λ} is well defined : indeed, we may write λ = p^{−n}a for some a, n ∈ ℕ and set x^{λ} := Φ_{P}^{−n}(x^{a}). It is easily seen that this definition is independent of the choice of a and n.

• For any ideal I ⊂ P and every λ = p^{−n}a ∈ ℤ[1/p] we define the angular power :

I^{⟨λ⟩} := ⋃_{r∈ℕ} Φ_{P}^{−n−r}(I^{p^{r}a})  if λ > 0, and I^{⟨λ⟩} := P  if λ ≤ 0.

Notice that I^{⟨λ⟩} is an increasing union of ideals of P, and a simple inspection shows that the definition does not depend on the choice of a and n. Explicitly, if λ ≥ 0, the ideal I^{⟨λ⟩} is the subset of all products of the form a_{1}^{μ_{1}} ⋯ a_{k}^{μ_{k}}, where k ∈ ℕ is any integer, a_{1}, …, a_{k} are arbitrary elements of I, and μ_{1}, …, μ_{k} are arbitrary rational numbers in ℕ[1/p] such that ∑_{i=1}^{k} μ_{i} = λ. Moreover, if S ⊂ I is any system of generators for I, then I^{⟨λ⟩} is generated by the set of all products as above, where a_{1}, …, a_{k} ∈ S.

Lemma 9.3.69. In the situation of (9.3.68), the following holds :

(i) If (I_{j} | j ∈ J) is any filtered family of ideals of P, we have

(⋃_{j∈J} I_{j})^{⟨λ⟩} = ⋃_{j∈J} I_{j}^{⟨λ⟩} for every λ ∈ ℕ[1/p].

(ii) For every ideal I ⊂ P, every n ∈ ℕ and every λ, μ ∈ ℕ[1/p] we have (a) (I^{n})^{⟨λ⟩} = (I^{⟨λ⟩})^{n} = I^{⟨nλ⟩}.

(b) I^{⟨λ⟩}I^{⟨μ⟩} = I^{⟨λ+μ⟩}.

(c) (I^{⟨λ⟩})^{⟨μ⟩} = I^{⟨λμ⟩}. (iii) I^{⟨λ⟩}J^{⟨λ⟩} = (IJ)^{⟨λ⟩} for every ideals I, J ⊂ P and every λ ∈ ℕ[1/p]. (iv) Suppose that I ⊂ P is a non-empty finitely generated ideal generated by r elements of

P. Then I^{⟨λ⟩} ⊂ I^{n} for every n ∈ ℕ such that λ ≥ n + r − 1.

Proof. (i): This is clear from the explicit description of the angular power furnished in (9.3.68). (ii.a): Indeed, say that λ = p^{k}a for some a, n ∈ ℕ; we have

(I^{n})^{⟨λ⟩} = ⋃_{r∈ℕ} Φ_{P}^{−r−k}(I^{nap^{r}}) = I^{⟨nλ⟩} = ⋃_{r∈ℕ}(Φ_{P}^{−r−k}(I^{ap^{r}})^{n} = (I^{⟨λ⟩})^{n}.

(ii.b): Clearly I^{⟨λ⟩}I^{⟨μ⟩} ⊂ I^{⟨λ+μ⟩}. For the converse inclusion, say that a_{1}^{t_{1}} ⋯ a_{k}^{t_{k}} ∈ I^{⟨λ+μ⟩}, for given a_{1}, …, a_{k} ∈ I and rational numbers t_{1}, …, t_{k} in ℕ[1/p] such that ∑_{i=1}^{k} t_{i} = λ + μ. If μ = 0, there is nothing to prove; otherwise, let j < k be the largest integer such that s := ∑_{i=0}^{j} t_{i} ≤ λ. Then t_{j+1} > t′_{j+1} := λ − s, and if we set t′′_{j+1} := t_{j+1} − t′_{j+1} we get

a_{1}^{t_{1}} ⋯ a_{j+1}^{t′_{j+1}} ∈ I^{⟨λ⟩} a_{j+1}^{t′′_{j+1}}a_{j+2}^{t_{j+2}} ⋯ a_{k}^{t_{k}} ∈ I^{⟨μ⟩}

whence the contention.

(ii.c): Say that λ = p^{−n}a and μ = p^{−m}b; taking into account (ii.a) we may compute

(I^{⟨λ⟩})^{⟨μ⟩} = ⋃_{r∈ℕ} Φ_{P}^{−m−r}((I^{⟨λ⟩})^{bp^{r}}) = ⋃_{r∈ℕ} Φ_{P}^{−m−r}((I^{bp^{r}})^{⟨λ⟩}) = ⋃_{r∈ℕ} Φ_{P}^{−m−r}(⋃_{s∈ℕ} Φ_{P}^{−n−s}(I^{abp^{r+s}}))

whence the sought identity.

(iii): Indeed, for λ = p^{−n}a we may compute

I^{⟨λ⟩}J^{⟨λ⟩} = ⋃_{r∈ℕ} Φ_{P}^{−n−r}(I^{ap^{r}}) · ⋃_{r∈ℕ} Φ_{P}^{−n−r}(J^{ap^{r}}) = ⋃_{r∈ℕ} Φ_{P}^{−n−r}(I^{ap^{r}}J^{ap^{r}})

whence the contention.

(iv): Let (x_{1}, …, x_{r}) be such a system of generators for I, so that I^{⟨λ⟩} is generated by all the products of the form x := x_{1}^{t_{1}} ⋯ x_{r}^{t_{r}}, where t_{1}, …, t_{r} ∈ ℕ[1/p] and t_{1} + ⋯ + t_{r} = λ. Then, for every such product and every i = 1, …, r, let a_{i} ∈ ℕ be the unique integer such that t_{i} ∈ [a_{i}, a_{i} + 1[, and set s := a_{1} + ⋯ + a_{r}; clearly x ∈ I^{s}. Suppose now that λ ≥ n + r − 1; if t_{i} ∈ ℕ for every i = 1, …, r, it follows that s = λ ≥ n, since r > 0. Lastly, if there exists i ≤ r such that t_{i} ∉ ℕ, then s + r > λ, whence s ≥ n again, and the assertion follows. □

Remark 9.3.70. (i) Let A be a ring, P a p-perfect monoid, as defined in (9.3.68), and u : P → (A, ·) a morphism of monoids. Then, for every ideal I ⊂ P and every λ ∈ ℤ[1/p] we may consider the ideal I^{⟨λ⟩}A of the ring A, generated by u(I^{⟨λ⟩}). We also define :

I^{⌊r⌋}A := ⋂_{\begin{matrix}μ∈ℤ[1/p] \\ μ<r \end{matrix}} I^{⟨μ⟩}A  I^{⌈s⌉}A := ⋃_{\begin{matrix}μ∈ℤ[1/p] \\ μ>s \end{matrix}} I^{⟨μ⟩}A  for every r ∈ ℝ.

From lemma 9.3.69(ii.a,ii.c) we deduce immediately :

(I^{⟨λ⟩}A)^{n} = (I^{⟨λ⟩})^{n}A = I^{⟨nλ⟩}A  I^{⟨λ⟩}I^{⟨μ⟩}A = I^{⟨λ+μ⟩}A  I^{⟨λ⟩}J^{⟨λ⟩}A = (IJ)^{⟨λ⟩}A

for every ideals I, J ⊂ P and every λ, μ ∈ ℕ[1/p] and every n ∈ ℕ.

(ii) If A is a perfect 𝔽_{p}-algebra, we may take for u the identity map of the multiplicative monoid P := (A, ·). In this case, notice that every ideal J of the ring A is also an ideal of the underlying monoid P, so the angular powers of J are well defined ideals of P; moreover :

(IA)^{⟨λ⟩}A = I^{⟨λ⟩}A for every λ ∈ ℕ[1/p] and every ideal I ⊂ P

(details left to the reader). Combining with lemma 9.3.69(ii.c) we get

(I^{⟨λ⟩}A)^{⟨μ⟩}A = I^{⟨λμ⟩}A for every λ ∈ ℕ[1/p] and every ideal I ⊂ P.

Furthermore, suppose that I is an ideal of the ring A generated (in the usual ring-theoretic sense) by a subset S ⊂ A consisting of r elements, and let J_{S} be the ideal of P generated by S; then I^{n} = J_{S}^{n}A and I^{⟨λ⟩}A = J_{S}^{⟨λ⟩}A for every n ∈ ℕ and λ ∈ ℕ[1/p], so lemma 9.3.69(iv) yields

I^{⟨λ⟩}A ⊂ I^{n} for every n ∈ ℕ and every λ ∈ ℕ[1/p] such that λ ≥ n + r − 1.

(iii) If A is as in (ii), we may also take

P := (A, ·) and u := τ_{A} : P → W (A).

If a_{•} := (a_{j} | j ∈ J) is any system of elements of A generating an ideal I of the monoid P, we may regard we shall also use the notation

[a_{•}]^{⟨λ⟩} := I^{⟨λ⟩}W (A).

(iv) For instance, if A is as in (ii), from lemma 9.3.69(ii.a) and (9.3.46) we see that

(9.3.71)  W (I^{⟨λ⟩}, r) = {∑_{n=0}^{∞} p^{n} · τ_{A}(b_{n}) | b_{n} ∈ I^{⟨λ⟩}A for every n ≤ r}

for every ideal I ⊂ A, every λ ∈ ℕ[1/p] and every r ∈ ℕ (notation of remark 9.3.28(iv)), and likewise we can describe W (I^{⟨λ⟩}). Moreover, for every such I, notice that

(9.3.72) p^{k}W (A) ∩ W (I^{⟨λ⟩}) = p^{k}W (I^{⟨λ⟩})  for every λ ∈ ℕ[1/p] and every k ∈ ℕ.

Indeed, say that \underline{b} := (b_{n} | n ∈ ℕ) ∈ p^{k}W (A) ∩ W (I^{⟨λ⟩}); in view of lemma 9.3.69(ii.a), this means that b_{n} = 0 for n = 0, …, k − 1, and b_{n} ∈ I^{⟨p^{n}λ⟩} for every integer n ≥ k. Clearly \underline{c} := (b_{n+k}^{1/p^{k}} | n ∈ ℕ) ∈ W (I^{⟨λ⟩}), and p^{k} · \underline{c} = \underline{b}, whence the claim. As an immediate consequence, we deduce that the 0-th ghost map ω_{0} induces an A-linear isomorphism

(9.3.73)  W (I^{⟨λ⟩})/pW (I^{⟨λ⟩}) →^{∼} I^{⟨λ⟩}A  for every λ ∈ ℕ[1/p].

Lemma 9.3.74. Let A be any perfect 𝔽_{p}-algebra, and I, J ⊂ A two ideals. We have : (i) I^{⌊r⌋}A = \overline{i.c.}(I, A, r) for every r ∈ ℝ_{+} (notation of remark 9.2.43). (ii) Suppose that I and J are finitely generated, and let (a_{1}, …, a_{n}) (resp. (b_{1}, …, b_{n})) be a finite system of generators for I (resp. J). Let also q > 1 be a rational number with

b_{i} − a_{i} ∈ i.c.(I, A, q) for every i = 1, …, n

and suppose furthermore that

(a) either, the radical of I equals the radical of J

(b) or else, I is contained in the Jacobson radical of A.

Then I^{⟨1⟩}A = J^{⟨1⟩}A.

Proof. (i): We are easily reduced to showing that

I^{⟨λ⟩}A ⊂ i.c.(I, A, λ) ⊂ I^{⟨λ′⟩}A for every λ′ ∈ ℕ[1/p] such that λ′ < λ.

To this aim, a direct inspection of the definition shows that

Φ_{A}(i.c.(I, A, λ)) = i.c.(I, A, pλ)  and Φ_{A}(I^{⟨λ⟩}A) = I^{⟨pλ⟩}A  for every λ ∈ ℕ[1/p]

so that we may replace λ by p^{n}λ for a suitable n ∈ ℕ, after which we may assume that λ ∈ ℕ. Moreover, since the subset S := {kλ/p^{n} | k, n ∈ ℕ} is dense in ℕ[1/p], we may assume that λ′ ∈ S. In this case, taking into account (9.2.44) and lemma 9.3.69(ii.a), we may further replace I by I^{⟨λ⟩}A, λ by 1, and λ′ by λ′/λ, so we reduce to checking that

I^{⟨1⟩}A ⊂ i.c.(I, A) ⊂ I^{⟨μ⟩}A for every μ ∈ ℕ[1/p] such that μ < 1.

To this aim, say that x ∈ I^{⟨1⟩}; by definition, this means that x^{p^{n}} ∈ I^{p^{n}} for some n ∈ ℕ, whence x ∈ i.c.(I, A). Next, let x ∈ i.c.(I, A), so that x^{n} ∈ ∑_{i=0}^{n−1} x^{i}I^{n−i} for some n ∈ ℕ.

Claim 9.3.75. x^{m} ∈ ∑_{i=0}^{n−1} x^{i}I^{m−i} for every m ≥ n.

Proof of the claim. We argue by induction on m − n. If m = n, there is nothing to prove, so assume that the claim is already known for some m ≥ n. In this case, we get

x^{m+1} ∈ ∑_{i=1}^{n} x^{i}I^{m+1−i} = x^{n}I^{m+1−n} + ∑_{i=1}^{n−1} x^{i}I^{m+1−i}.

Moreover, x^{n}I^{m+1−n} ∈ ∑_{i=0}^{n−1} x^{i}I^{m+1−i}, so the claim holds for m + 1. ♢

It follows from claim 9.3.75 that x^{p^{k}} ∈ I^{p^{k}+1−n}, whence x ∈ I^{⟨1−(n−1)/p^{k}⟩} for every k ∈ ℕ such that p^{k} ≥ n, whence the contention.

(ii): Under the stated conditions, lemma 9.2.47 says that i.c.(I, A) = i.c.(J, A). Together with (9.2.45), we deduce that

i.c.(I, A, q) = i.c.(J, A, q).

Hence J ⊂ I + i.c.(I, A, q) ⊂ I^{⟨1⟩}A and likewise I ⊂ J + i.c.(J, A, q) ⊂ J^{⟨1⟩}A, whence the assertion. □

9.3.76. Let A be a perfect 𝔽_{p}-algebra, a_{•} := (a_{1}, …, a_{k}) a finite system of elements of A, and I ⊂ A (resp. 𝒥 ⊂ W (A)) the ideal generated by a_{•} (resp. by τ_{A}(a_{1}), …, τ_{A}(a_{k})). Set

ℐ := pW (A) + 𝒥  W (λ) := W (I^{⟨λ⟩})  for every λ ∈ ℕ[1/p]

(notation of remark 9.3.28(iv)). Lastly, denote by 𝒯_{A} (resp. 𝒯_{W (A)}) the topology of A (resp. of W (A, 𝒯_{A})). We complement the generalities of lemma 9.3.69with the following:

Proposition 9.3.77. In the situation of (9.3.76), the following holds :

(i) [a_{•}]^{⟨λ⟩} ⊂ W (λ) ⊂ [a_{•}]^{⟨λ′⟩} for every λ′ < λ in ℕ[1/p] (notation of remark 9.3.70(iii)). (ii) If 𝒯_{A} agrees with the I-adic topology, 𝒯_{W (A)} agrees with the ℐ -adic topology.

Proof. (ii): Notice that τ_{A}(b^{μ}) = τ_{A}(b)^{μ} for every b ∈ A and every μ ∈ ℕ[1/p]. Taking into account (9.3.71) and proposition 9.3.62, we deduce that

W (λ, r) := W (I^{⟨λ⟩}, r) = p^{r+1}W (A) + [a_{•}]^{⟨λ⟩} every r ∈ ℕ

which, in light of lemma 9.3.69(iv), shows that the linear topology induced by the cofiltered system of ideals (W (λ, r) | r ∈ ℕ, λ ∈ ℕ[1/p]) agrees with the ℐ -adic topology. But by the same token we see that the latter family is a fundamental system of open neighborhoods of zero in W (A), whence the contention.

(i): Since W (λ, r) is an open ideal in the p-adic topology of W (A) for every r ∈ ℕ, we see that W (λ) is a closed ideal in W (A) for the p-adic topology of W (A). On the other hand, from (9.3.72) it follows that the p-adic topology of W (λ) agrees with the topology induced from the p-adic topology of W (A). Taking into account proposition 9.3.44(ii), we deduce that W (λ) is complete and separated for its p-adic topology, and then (9.3.73) and claim 9.3.60 imply that W (λ) is a topologically free ℤ_{p}-module for this topology, and any lifting of an 𝔽_{p}-basis of I^{⟨λ⟩}A yields a topological basis. However, clearly we may find :

• a subset S ⊂ ℕ[1/p]^{⊕k} consisting of sequences (μ_{1}, …, μ_{k}) such that μ_{1}+⋯+μ_{k} = λ • a system of elements (b_{μ} | μ ∈ S) of A such that the family (b_{μ} · a^{μ} | μ ∈ S) is an

𝔽_{p}-basis of I^{⟨λ⟩}A, where a^{μ} := a_{1}^{μ_{1}} ⋯ a_{k}^{μ_{k}} for every μ ∈ S

so that the system (τ_{A}(b_{μ} · a^{μ}) | μ ∈ S) is a topological ℤ_{p}-basis of W (λ). Now, clearly [a_{•}]^{⟨λ⟩} ⊂ W (λ); to check the second inclusion, let \underline{w} ∈ W (λ) be any element. By construction, there exists a unique system (d_{μ} | μ ∈ S) of elements of ℤ_{p} such that

\underline{w} = ∑_{μ∈S} d_{μ} · τ_{A}(b_{μ} · a^{μ})

where the series converges in the p-adic topology of W (A), and the set {μ ∈ S | d_{μ} ∉ p^{n}ℤ_{p}} is finite for every n ∈ ℕ. Pick any N ∈ ℕ such that λ − λ′ ≥ kp^{−N}, and for every r ∈ ℝ define

\overline{r} ∈ p^{−N}ℤ  r^{∗} ∈ [0, p^{−N}[  fulfilling the condition : r = \overline{r} + r^{∗}.

Then, for every μ ∈ S let also \overline{μ} := (\overline{μ}_{1}, …, \overline{μ}_{k}), μ^{∗} := (μ_{1}^{∗}, …, μ_{k}^{∗}), and set

S′ := {σ ∈ p^{−N}ℕ^{⊕k} | λ ≥ σ_{1} + ⋯ + σ_{k} > λ′}.

With this notation, it is easily seen that \overline{μ} ∈ S′ for every μ ∈ S. Hence, let also

c_{σ} := ∑_{\begin{matrix}μ∈S \\ \overline{μ}=σ \end{matrix}} d_{μ} · τ_{A}(b_{μ} · a^{μ^{∗}}) for every σ ∈ S′

(notice that this series converges p-adically in W (A), so c_{σ} is well defined); we get

\underline{w} = ∑_{σ∈S′} c_{σ} · τ_{A}(a^{σ}).

To conclude, it suffices to remark that S′ is a finite set, and a^{σ} ∈ I^{⟨λ′⟩}A for every σ ∈ S′. □

Example 9.3.78. Let A := 𝔽_{p}[T^{1/p^{∞}}], endow A with its T -adic topology 𝒯_{T}, and let (A^{∧}, 𝒯_{T}^{∧}) be the completion of (A, 𝒯_{T}). According to lemma 9.3.33(iv), the topological ring W (A^{∧}, 𝒯_{T}^{∧}) is the completion of W (A, 𝒯_{T}). In light of example 9.3.48(v) and proposition 9.3.77(ii), the latter is naturally isomorphic to the (p, T)-adic completion of ℤ_{p}{T^{1/p^{∞}}}, or equivalently, the (p, T)-adic completion of ℤ_{p}[T^{1/p^{∞}}]. Explicitly, this is the ring of all power series (9.3.49) where (λ_{n} | n ∈ ℕ) is any sequence of elements of ℕ[1/p], and (a_{n} | n ∈ ℕ) is any sequence of elements of ℤ_{p}, with the property that

lim_{n→∞} v_{p}(a_{n}) + λ_{n} = ∞

(where v_{p} : ℤ_{p} → ℕ ∪ {∞} is the p-adic valuation).

Corollary 9.3.79. Let (Λ, ≤) be a filtered set with an initial element λ_{0}, and A_{•} := (A_{λ} | λ ∈ Λ) a system of perfect 𝔽_{p}-algebras; let I_{0} ⊂ A_{λ_{0}} be an ideal of finite type. Endow A_{λ_{0}} and the colimit A of A_{•} with their I_{0}-adic topologies 𝒯_{0}, respectively 𝒯_{A}; also, endow the colimit 𝒲 of the induced system of rings (W (A_{λ}) | λ ∈ Λ) with the unique 𝒲 -linear topology 𝒯_{𝒲} such that the natural map W (A_{λ_{0}}, 𝒯_{0}) → (𝒲 , 𝒯_{𝒲} ) is adic. Then the induced map of separated completions :

(𝒲 , 𝒯_{𝒲} )^{∧} → W ((A, 𝒯_{A})^{∧})

is an isomorphism of topological rings.

Proof. Let a_{1}, …, a_{k} be a finite set of generators of I_{0}, and denote by ℐ_{0} ⊂ W (A_{λ_{0}}) the ideal generated by p, τ_{A_{λ_{0}}}(a_{1}), …, τ_{A_{λ_{0}}}(a_{k}). Then the topologies of W (A_{λ_{0}}, 𝒯_{0}) and of W (A, 𝒯_{A}) are ℐ_{0}-adic, by proposition 9.3.77(ii). Moreover, the natural map W (A, 𝒯_{A})^{∧} → W ((A, 𝒯_{A})^{∧}) is an isomorphism of topological rings (lemma 9.3.33(iv)). Clearly the natural continuous map map φ : (𝒲 , 𝒯_{𝒲} ) → W (A, 𝒯_{A}) has dense image; thus, we are reduced to checking that the ℐ_{0}-adic topology of W (A) induces the ℐ_{0}-adic topology on 𝒲 via φ (theorem 8.2.8(iii)).

However, the proof of proposition 9.3.77(ii) shows that the ℐ_{0}-adic topology on W (A) admits the fundamental system of open ideals (W (I_{0}^{⟨μ⟩}A, r) | μ ∈ ℕ[1/p], r ∈ ℕ), (see remark 9.3.70(iv)), and likewise for the ℐ_{0}-adic topology on W (A_{λ_{0}}). Now, 𝒲 is the set of equivalence classes [λ, \underline{w}] of pairs (λ, \underline{w}) with λ ∈ Λ and \underline{w} := (w_{n} | n ∈ ℕ) ∈ W (A_{λ}); we have φ([λ, \underline{w}]) ∈ W (I_{0}^{⟨μ⟩}A, r) if and only if there exists λ′ ≥ λ such that the image of w_{i} lies in I_{0}^{⟨μ⟩}A_{λ′} for i = 0, …, r, and the latter holds if and only if [λ, μ] ∈ W (I_{0}^{⟨μ⟩}, r) · 𝒲 , whence the assertion. □

9.3.80. Semi-norms on the Witt ring. Let A be any topological ring, | · |_{A} a real-valued seminorm on A (see definition 9.1.3(v)). For any non-empty subset S ⊂ A we let

|S|_{A} := sup_{a∈S} |a|_{A} ∈ ℝ_{+} ∪ {+∞}.

Also, for every real number ρ ∈ [0, 1] we consider the mapping

| · |_{ρ} : W (A) → ℝ_{+} ∪ {+∞} (a_{n} | n ∈ ℕ) ↦ sup_{n∈ℕ} |a_{n}|_{A}^{p^{−n}} · ρ^{n}.

For instance, notice that |0|_{ρ} = 0 and |1|_{ρ} = 1; more generally, we have |τ_{A}(a)|_{ρ} = |a|_{A} for every a ∈ A. We let

W (A, ρ) := \{\underline{a} ∈ W (A) | |\underline{a}|_{ρ} ∈ ℝ_{+}\}.

Lemma 9.3.81. With the notation of (9.3.80), the following holds :

(i) W (A, ρ) is an additive subgroup of W (A) for every ρ ∈ [0, 1].

(ii) Both W (A, 0) and W (A, 1) are subrings of W (A).

(iii) |\underline{a} − \underline{b}|_{ρ} ≤ max(|\underline{a}|_{ρ}, |\underline{b}|_{ρ}) for every \underline{a}, \underline{b} ∈ W (A).

(iv) |\underline{a} · \underline{b}|_{ρ_{1}ρ_{2}} ≤ |\underline{a}|_{ρ_{1}} · |\underline{b}|_{ρ_{2}} for every \underline{a}, \underline{b} ∈ W (A) and every ρ_{1}, ρ_{2} ∈ [0, 1].

(v) ρ^{p} · |F_{A}(\underline{a})|_{ρ^{p}} ≤ |\underline{a}|_{ρ}^{p} for every \underline{a} ∈ W (A) and every ρ ∈ [0, 1].

(vi) |V_{A}(\underline{a})|_{ρ^{1/p}} = ρ^{1/p} · |\underline{a}|_{ρ}^{1/p} for every \underline{a} ∈ W (A) and every ρ ∈ [0, 1].

(vii) |τ_{A}(a) · \underline{b}|_{ρ} ≤ |a|_{A} · |\underline{b}|_{ρ} for every a ∈ A and every \underline{b} ∈ W (A), and if | · |_{A} is a valuation, this inequality is actually an equality.

Proof. Let \underline{a} := (a_{n} | n ∈ ℕ), \underline{b} := (b_{n} | n ∈ ℕ) be any two elements of W (A), and suppose that M := max(|\underline{a}|_{ρ}, |\underline{b}|_{ρ}) ∈ ℝ_{+}. This condition means that

|a_{n}|_{A}, |b_{n}|_{A} ≤ (M/ρ^{n})^{p^{n}} for every n ∈ ℕ.

Set \underline{c} := \underline{a} + \underline{b}, \underline{d} := \underline{a} · \underline{b} and recall that \underline{c} = (c_{n} | n ∈ ℕ) (resp. \underline{d} = (d_{n} | n ∈ ℕ)), where c_{n} = S_{n}(a_{0}, …, a_{n}, b_{0}, …, b_{n}) (resp. d_{n} = P_{n}(a_{0}, …, a_{n}, b_{0}, …, b_{n}) : notation of (9.3.7)) for every n ∈ ℕ. By construction, S_{n} is a ℤ-linear combination of monomials Q(X_{0}, …, X_{n}, Y_{0}, …, Y_{n}) of total degree p^{n} (remark 9.3.8(i)); we claim that

r := |Q(a_{0}, …, a_{n}, b_{0}, …, b_{n})|_{A} ≤ (M/ρ^{n})^{p^{n}} for every such Q.

Indeed, say that Q = ∏_{i=0}^{n} X_{i}^{n_{i}}Y_{i}^{m_{i}}; then ∑_{i=0}^{n}(n_{i} + m_{i})p^{i} = p^{n}, and therefore

r ≤ ∏_{i=0}^{n}(M/ρ^{i})^{(n_{i}+m_{i})p^{i}} = M^{p^{n}} · ρ^{−t} where t := ∑_{i=0}^{n} i · (n_{i} + m_{i})p^{i} ≤ np^{n}.

Since ρ ≤ 1, the claim follows. Lastly, arguing with the polynomials I_{n} of (9.3.7) that are bihomogeneous of degree (p^{n}, 0), we get as well | − \underline{a}|_{ρ} ≤ |\underline{a}|_{ρ}, and therefore | − \underline{a}|_{ρ} = |\underline{a}|_{ρ} for every \underline{a} ∈ W (A). Assertions (i) and (iii) are immediate consequences.

(iv): Likewise, P_{n} is a ℤ-linear combination of monomials R of bidegree (p^{n}, p^{n}), and a similar calculation shows that

|R(a_{0}, …, a_{n}, b_{0}, …, b_{n})|_{A} ≤ |\underline{a}|_{ρ_{1}} · |\underline{b}|_{ρ_{2}} · (ρ_{1}ρ_{2})^{−np^{n}} for every such R

(details left to the reader), whence the assertion. (ii) now follows from (iii) and (iv).

(v) is proven in the same way : for every n ∈ ℕ one estimates |F_{n}(a_{0}, …, a_{n+1})| by writing F_{n} as a sum of bihomogeneous monomials of degree (p^{n+1}, 0) : the details are left to the reader.

(vi): Set M := |\underline{a}|_{ρ}, fix a real number ε > 0, and pick n ∈ ℕ such that |a_{n}|_{A}^{p^{−n}} · ρ^{n} > M − ε. Then we have

|a_{k−1}|_{A} ≤ (M/ρ^{k−1})^{p^{k−1}} = (M^{1/p})^{p^{k}} · ρ^{−(k−1)p^{k−1}} for every k > 0

which is equivalent to

|a_{k−1}|_{A}^{p^{−k}} · ρ^{k/p} ≤ (ρM)^{1/p} for every k > 0.

On the other hand, we have |a_{n}|_{A}^{p^{−n−1}} · ρ^{(n+1)/p} > (ρ(M − ε))^{1/p} whence the contention. (vii) follows directly from proposition 9.3.36(i). □

Proposition 9.3.82. In the situation of (9.3.80), suppose that A is an 𝔽_{p}-algebra. We have: (i) |F_{A}(\underline{a})|_{ρ^{p}} = |\underline{a}|_{ρ}^{p} for every \underline{a} ∈ W (A) and every ρ ∈ [0, 1]. (ii) W (A, ρ) is a subring of W (A), and | · |_{ρ} restricts to a semi-norm on W (A, ρ). (iii) If moreover | · |_{A} is a valuation, the same holds for the restriction of | · |_{ρ} to W (A, ρ). (iv) If A is perfect and | · |_{A} is a valuation, we have

|∑_{n∈ℕ} p^{n} · τ_{A}(b_{n})|_{ρ} = sup_{n∈ℕ} |b_{n}|_{A} · ρ^{n} for every sequence (b_{n} | n ∈ ℕ) of elements of A.

Proof. Assertion (i) (resp. (iv)) follows easily from (9.3.40) (resp. from (9.3.46)).

(iii): For ρ = 0, we have |\underline{a}|_{ρ} = |a_{0}|_{A} for every \underline{a} := (a_{n} | n ∈ ℕ) ∈ W (A), so the assertion is clear in this case. Hence, suppose that ρ ∈]0, 1], and let \underline{a}, \underline{a}′ ∈ W (A, ρ) be any two elements. It is clear that \underline{a} ∈ W (A, ρ − ε) for every ε ∈ [0, ρ], and moreover

(9.3.83)  lim_{ε→0} |\underline{a}|_{ρ−ε} = |\underline{a}|_{ρ}.

Furthermore, it is easily seen that, if ε > 0, there exist minimal N, N′ ∈ ℕ such that

|a_{N}|_{A}^{p^{−N}} · (ρ − ε)^{N} = |\underline{a}|_{ρ−ε} and |a′_{N′}|_{A}^{p^{−N′}} · (ρ − ε)^{N′} < |\underline{a}′|_{ρ−ε}.

We have to check that |\underline{a} · \underline{a}′|_{ρ} = |\underline{a}|_{ρ} · |\underline{a}′|_{ρ}, and in light of (9.3.83) it suffices to show that |\underline{a} · \underline{a}′|_{ρ−ε} = |\underline{a}|_{ρ−ε} · |\underline{a}′|_{ρ−ε} for every ε ∈ [0, ρ]. We may then replace ρ by ρ − ε, and assume from start that N, N′ are the minimal integers such that

|a_{N}|_{A}^{p^{−N}} · ρ^{N} = |\underline{a}|_{ρ} and |a′_{N′}|_{A}^{p^{−N′}} · ρ^{N′} < |\underline{a}′|_{ρ}.

With this notation, write \underline{a} = \underline{b} + V_{A}^{N}(\underline{c}), with \underline{b} := (a_{0}, …, a_{N−1}, 0, …), \underline{c} := (a_{N}, a_{N+1}, …). By remark 9.1.4(iii) we have

|\underline{b}|_{ρ} < |\underline{a}|_{ρ} and therefore |\underline{a}|_{ρ} = |V_{A}^{N}(\underline{c})|_{ρ}

from which we deduce that |\underline{b}·\underline{a}′|_{ρ} < |\underline{a}|_{ρ} ·|\underline{a}′|_{ρ}, so – again by remark 9.1.4(iii) – we are reduced to checking that |V_{A}^{N}(\underline{c})|_{ρ} · |\underline{a}′|_{ρ} = |V_{A}^{N}(\underline{c}) · \underline{a}′|_{ρ}. But proposition 9.3.22(ii) and lemma 9.3.81(v) yield

|V_{A}^{N}(\underline{c}) · \underline{a}′|_{ρ} = |V_{A}^{N}(\underline{c} · F_{A}^{N}(\underline{a}′))|_{ρ} = |\underline{c} · F_{A}^{N}(\underline{a}′)|_{ρ^{p^{N}}}^{1/p^{N}} · ρ^{N}

and on the other hand

|V_{A}^{N}(\underline{c})|_{ρ} = |\underline{c}|_{ρ^{p^{N}}}^{1/p^{N}} · ρ^{N} and |F_{A}^{N}(\underline{a}′)|_{ρ^{p^{N}}}^{1/p^{N}} = |\underline{a}′|_{ρ}

by lemma 9.3.81(v) and (i). Say that \underline{a}′′ := F_{A}^{N}(\underline{a}′); then notice that – in view of (9.3.40) – the integer N′ is still the smallest one such that

|a′′_{N′}|_{A}^{p^{−N′}} · ρ^{N′} = |\underline{a}′′|_{ρ}.

Summing up, we may replace \underline{a} and \underline{a}′ by \underline{c} and respectively \underline{a}′′, and ρ by ρ^{N}, and assume from start that N = 0, i.e. that |a_{0}|_{A} = |\underline{a}|_{ρ}. Likewise, by repeating the same considerations on \underline{a}′, we may assume that |a′_{0}|_{A} = |\underline{a}′|_{ρ}. Lastly, notice that

|\underline{a} · \underline{a}′|_{ρ} ≥ |a_{0}a′_{0}|_{A} = |a_{0}|_{A} · |a′_{0}|_{A} = |\underline{a}|_{ρ} · |\underline{a}′|_{ρ}

and the converse inequality is already known, by (ii), whence the contention.

(ii): Let \underline{a} := (a_{n} | n ∈ ℕ) and \underline{b} := (b_{n} | n ∈ ℕ) be any two elements of W (A, ρ); we have to show that |\underline{a} · \underline{b}|_{ρ} ≤ |\underline{a}|_{ρ} · |\underline{b}|_{ρ}. However, from propositions 9.3.36(iii) and 9.3.44(iv) we get

\underline{a} · \underline{b} ≡ ∑_{i+j=n} V_{A}^{i+j}(τ_{A}(a^{p^{j}}b^{p^{i}})) (mod V_{A}^{n+1}) for every n ∈ ℕ

and in view of lemma 9.3.81(iii) we are then easily reduced to the case where \underline{a} = V_{A}^{i}(τ_{A}(x)) and \underline{b} = V_{A}^{j}(τ_{A}(y)) for some i, j ∈ ℕ and x, y ∈ A. We compute :

|V_{A}^{i+j}(τ_{A}(x^{p^{j}}y^{p^{i}}))|_{ρ} = ρ^{i+j} · |τ_{A}(x^{p^{j}}y^{p^{i}})|_{ρ^{p^{i+j}}}^{1/p^{i+j}}  (by lemma 9.3.81(vi))
= ρ^{i+j} · |x^{p^{j}}y^{p^{i}}|_{A}^{1/p^{i+j}}
≤ ρ^{i+j} · |x|_{A}^{1/p^{i}} · |y|_{A}^{1/p^{j}}
= |V_{A}^{i}(τ_{A}(x))|_{ρ} · |V_{A}^{j}(τ_{A}(y))|_{ρ} (again by lemma 9.3.81(vi))

as required. □

Remark 9.3.84. (i) In spite of the inequalities of lemma 9.3.81(vii) and proposition 9.3.82(iv), the map τ_{A} : A → W (A, ρ) is not necessarily continuous for the topology on W (A, ρ) induced by the semi-norm | · |_{ρ}, unless ρ = 0.

(ii) Normed Witt vectors similar to those introduced in (9.3.80) have been previously considered in the literature : see e.g. [53], [113], [127].

Lemma 9.3.85. Let E be a perfect topological 𝔽_{p}-algebra, I ⊂ E any ideal, and denote by ℛ_{E} the set of all real-valued valuations on E with |E| = 1 (notation of (9.3.80)). For every r, r′ ∈ ℝ_{+} we have :

(i) |I^{⌊r⌋}|_{E} = |I|_{E}^{r} for every | · |_{E} in ℛ_{E}.

(ii) (I^{⌊r⌋})^{⌊r′⌋} = I^{⌊rr′⌋}.

(iii) I^{⌊r⌋} = (I^{⌊r⌋})^{⟨1⟩}.

(iv) W (I^{⌊r⌋}E) = \{\underline{a} ∈ W (E) | |\underline{a}|_{1} ≤ |I|^{r} for every | · | in ℛ_{E} \}.

Proof. (i): Clearly |I^{⌊r⌋}|_{E} ≤ |I|_{E}^{r}. For the converse inequality, fix any real number ε > 0, and pick y ∈ I such that |y|_{E}^{r} > |I|_{E}^{r} − ε. Since ℕ[1/p] is dense in ℝ_{+}, we may find λ ∈ ℕ[1/p] such that λ ≥ r and |y|_{E}^{λ} > |I|_{E}^{r} − ε. It follows that |y^{λ}| = |y|^{λ} ≤ |I|^{r} for every | · | in ℛ_{E}, and |y^{λ}|_{E} > |I|_{E}^{r} − ε, so that |I^{⌊r⌋}| > |I|_{E}^{r} − ε, as needed.

(ii): Indeed, by lemmata 9.2.46and 9.3.74(i), the ideal (I^{⌊r⌋})^{⌊r′⌋} (resp. I^{⌊rr′⌋}) is the set of all x ∈ E such that |x| ≤ |I^{⌊r⌋}|^{r′} (resp. |x| ≤ |I|^{rr′}) for every | · | in ℛ_{E}. But from (i) we get |I^{⌊r⌋}|^{r′} = |I|^{rr′} for every such | · |, whence the assertion.

(iii): This is obvious for r = 0, and for r > 0, we have I^{⌊r⌋} ⊂ (I^{⌊r⌋})^{⟨1⟩} ⊂ (I^{⌊r⌋})^{⌊1⌋} by lemma 9.3.74(i), and then the assertion follows from (ii).

(iv) follows by combining (iii), (9.3.71) and proposition 9.3.82(iv). □

9.4. Fontaine rings. In this section we revisit and expand Fontaine and Wintenberger’s theory of the fields of norms of a local field of characteristic zero, which we frame as a special case of a simple and very general construction that endows every topological 𝔽_{p}-algebra A with a universal perfect cover E(A) → A; indeed, E shall be a right adjoint to the inclusion of the subcategory of perfect topological 𝔽_{p}-algebras into the category of all topological 𝔽_{p}-algebras. In section 16.1, this functor will play an important role in the study of perfectoid rings.

Definition 9.4.1. (i) A topological monoid is the datum (P, μ, 𝒯 ) of a monoid (P, μ) and a topology 𝒯 on the set P, such that the composition law μ : P × P → P of P is a continuous map (where P × P is endowed with the product topology). A morphism of topological monoids (P, μ, 𝒯 ) → (P′, μ′, 𝒯 ′) is a morphism of monoids P → P′ that is continuous for the topologies 𝒯 and 𝒯 ′. Clearly the topological monoids and their morphisms form a category

TopMon.

(ii) Let p be any prime integer. We say that (P, μ, 𝒯 ) is a p-perfect topological monoid if the p-Frobenius endomorphism p_{P} : P → P is an isomorphism of topological monoids (see definition 4.8.40(ii)). This means that P is uniquely p-divisible, and p_{P} is a homeomorphism for the topology 𝒯 . We denote by

p-TopMon

the full subcategory of TopMon whose objects are the p-perfect topological monoids.

Remark 9.4.2. Using the criterion of proposition 1.2.22(i) is easily seen that TopMon is a complete category : indeed, any family ((P_{i}, 𝒯_{i}) | i ∈ I) of topological monoids admits a product P, which is just the product P of the underlying topological spaces, endowed with the unique composition law such that all the projections P → P_{i} are morphisms of monoids. Also, if f, g : (P, 𝒯 ) → (P′, 𝒯 ′) is any pair of morphisms in TopMon, the equalizer of f and g is the equalizer of the underlying morphisms of monoids, endowed with the topology induced from 𝒯 : details left to the reader. Likewise, one checks that TopMon is also cocomplete.

Proposition 9.4.3. The inclusion functor p-TopMon → TopMon admits both a right and a left adjoint, denoted respectively

E : TopMon → p-TopMon  and  E^{∗} : TopMon → p-TopMon.

Proof. Let (P, 𝒯 ) be any topological monoid; we let ((P_{n}, 𝒯_{n}); φ_{n} : P_{n+1} → P_{n} | n ∈ ℕ) be the system of topological monoids such that (P_{n}, 𝒯_{n}) := (P, 𝒯 ) and φ_{n} is the p-Frobenius endomorphism of P for every n ∈ ℕ. In light of remark 9.4.2we may set

(E(P), 𝒯_{E(P)}) := lim_{n∈ℕ} (P_{n}, 𝒯_{n}) (E^{∗}(P), 𝒯_{E(P)}^{∗}) := colim_{n∈ℕ} (P_{n}, 𝒯_{n}).

So E(P) is the set of all sequences (a_{n} | n ∈ ℕ) of elements a_{n} ∈ P such that a_{n} = a_{n+1}^{p} for every n ∈ ℕ. Any morphism of topological monoids f : P → Q induces a morphism of topological monoids

E(f) : E(P) → E(Q) (a_{n} | n ∈ ℕ) ↦ (f(a_{n}) | n ∈ ℕ).

Moreover, clearly the Frobenius endomorphism p_{E(P)} is an automorphism of E(P) with inverse f given explicitly by the rule :

f(a_{n} | n ∈ ℕ) = (a_{n+1} | n ∈ ℕ) for every (a_{n} | n ∈ ℕ) ∈ E(P).

In other words, f is the limit of the system of morphisms (1_{P} : P_{n} → P_{n+1} | n ∈ ℕ), so it is a continuous map, and therefore p_{E(P)} is a homeomorphism. Furthermore, the projection to P_{0} defines a natural morphism of topological monoids

(9.4.4)  \overline{u}_{P} : E(P) → P  (a_{n} | n ∈ ℕ) ↦ a_{0}

and if P is already p-perfect, then clearly \overline{u}_{P} is an isomorphism of topological monoids. Lastly, let f : Q → P be any morphism of topological monoids, such that Q is p-perfect. There follows a well defined morphism f^{†} := E(f) ∘ \overline{u}_{Q}^{−1} : Q → E(P) of topological monoids. Conversely, to any g : Q → E(P) we may attach the morphism g^{†} := \overline{u}_{P} ∘ g : Q → P. It is easily seen that the rules f ↦ f^{†} and g ↦ g^{†} are mutually inverse bijections, which concludes the construction of the right adjoint.

Next, let us check that the inverse g of p_{E^{∗}(P)} is continuous. Indeed, any element α ∈ E^{∗}(P) is the class of a pair (a, n), for some n ∈ ℕ and some a ∈ P_{n}, and g(α) is then the class of (a, n + 1); in other words, g is the colimit of the system of morphisms (1_{P} : P_{n} → P_{n+1} | n ∈ ℕ), whence the contention. This shows that E^{∗}(P) is an object of p-TopMon, and obviously the rule P ↦ E^{∗}(P) yields a well defined functor as sought. Furthermore, the map j_{0} : P → E^{∗}(P) defines a natural transformation which is an isomorphism in case P is already p-perfect. Now, the definition of an adjunction between E^{∗} and the inclusion functor is, mutatis mutandis, as in the foregoing case : the details shall be left to the reader. □

Remark 9.4.5. (i) By construction, the natural transformation P ↦ \overline{u}_{P} defined in (9.4.4) is the counit of the adjunction exhibited by proposition 9.4.3. Likewise, the unit of this adjunction is the transformation Q ↦ \overline{u}_{Q}^{−1} for every p-perfect topological monoid Q.

(ii) Taking into account the triangular identities of (1.1.13), we deduce from (i) the identity

E(\overline{u}_{P}) = \overline{u}_{E(P)} : E(E(P)) → E(P)

and this map is an isomorphism of p-perfect topological monoids.

(iii) For every topological monoid P, we have

E(P)^{×} = \overline{u}_{P}^{−1}(P^{×}).

Indeed, the inclusion E(P)^{×} ⊂ \overline{u}_{P}^{−1}(P^{×}) is obvious. For the converse, suppose that x := (x_{n} | n ∈ ℕ) ∈ E(P) and \overline{u}_{P} (x) ∈ P^{×}. This means that x_{0} ∈ P^{×}, and since x_{n+1}^{p} = x_{n} for every n ∈ ℕ, it follows that x_{n} ∈ P^{×} for every n ∈ ℕ, so x ∈ E(P)^{×}.

(iv) Likewise, it is easily seen that an element x ∈ E(P) is regular if and only if the same holds for \overline{u}_{P} (x) (see example 4.8.36(i)).

Example 9.4.6. Endow the abelian group ℚ_{p}/ℤ_{p} (resp. ℚ_{p}) with its discrete topology (resp. with its usual p-adic topology); then there exists a natural isomorphism E(ℚ_{p}/ℤ_{p}) →^{∼} ℚ_{p}, that identifies the map \overline{u}_{ℚ_{p}/ℤ_{p}} with the projection ℚ_{p} → ℚ_{p}/ℤ_{p} (verification left to the reader).

9.4.7. Consider now the category of topological 𝔽_{p}-algebras

𝔽_{p}-TopAlg

(where 𝔽_{p} is endowed with its discrete topology : see definition 8.3.1(iii)). If R is an object of 𝔽_{p}-TopAlg, the p-Frobenius of the multiplicative monoid (R, ·) is the Frobenius endomorphism Φ_{R} of the ring R, and a simple inspection of the proof of proposition 9.4.3 reveals that the composition law of E(R, ·, 𝒯 ) is the multiplication map for a natural structure of topological ring. Moreover, for any morphism of topological 𝔽_{p}-algebras f : R → S, the map E(f) : E(R) → E(S) is a morphism of topological 𝔽_{p}-algebras. In other words, E lifts to a functor

(9.4.8)  E : 𝔽_{p}-TopAlg → 𝔽_{p}-TopAlg_{perf}

to the full subcategory of 𝔽_{p}-TopAlg whose objects are the perfect topological 𝔽_{p}-algebras (see (9.3.45)), and (9.4.8) commutes with the forgetful functors (R, +, ·, 𝒯 ) ↦ (R, ·, 𝒯 ) to topological monoids. Furthermore, the map \overline{u}_{R} of (9.4.4) is a morphism of topological 𝔽_{p}-algebras, so the adjunction exhibited in the proof of proposition 9.4.3 lifts to an adjunction between the functor E of (9.4.8) and the forgetful functor 𝔽_{p}-TopAlg_{perf} → 𝔽_{p}-TopAlg.

The same argument shows that E^{∗} lifts to a left adjoint to the forgetful functor

E^{∗} : 𝔽_{p}-TopAlg → 𝔽_{p}-TopAlg_{perf}.

Remark 9.4.9. (i) Suppose that (R, 𝒯 ) is a topological 𝔽_{p}-algebra whose topology 𝒯 is linear. Then it is easily seen that a fundamental system of open neighborhoods of 0 ∈ E(R, 𝒯 ) is given by the intersections of finitely many elements of the family of ideals

Φ_{E(R)}^{n}(\overline{u}_{R}^{−1}(I)) for every n ∈ ℕ and every open ideal I ⊂ R.

Especially, the topology 𝒯_{E} of E(R) is also linear. Moreover, if 𝒯 is separated (resp. and complete) then the same holds for the topology of E(R), since the latter is a closed subset of R^{ℕ}, for the product topology : details left to the reader.

(ii) In the situation of (i), suppose moreover that I ⊂ R is an ideal such that 𝒯 agrees with the I-adic topology on R, and set ℐ := \overline{u}_{R}^{−1}(I). Since Φ_{E(R)}^{n}(ℐ) ⊂ ℐ^{p^{n}} ⊂ \overline{u}_{R}^{−1}(I^{p^{n}}), we see that 𝒯_{E} is the linear topology defined by the system of ideals (Φ_{E(R)}^{n}(ℐ) | n ∈ ℕ). Especially, if ℐ is finitely generated, 𝒯_{E} agrees with the ℐ -adic topology.

(iii) Suppose that the Frobenius endomorphism Φ_{R} of the topological 𝔽_{p}-algebra R is surjective. Then clearly the same holds for \overline{u}_{R}. Moreover, if Φ_{R} is open and surjective, the same holds for \overline{u}_{R}. Indeed, any open subset of E(R) is a union of subsets of the form U := E(R) ∩ ∏_{n∈ℕ} U_{n}, where each U_{n} is open in R and there exists k ∈ ℕ such that U_{n} = A for every n > k; with this notation, since Φ_{R} is surjective, we have

\overline{u}_{R}(U) = Φ_{R}^{k}(U_{k} ∩ Φ_{R}^{−1}U_{k−1} ∩ ⋯ ∩ Φ_{R}^{−k}U_{0})

and this set is open in R, when Φ_{R} is an open map.

(iv) By virtue of proposition 1.3.25(iii), both the functor E for topological monoid and the one for topological 𝔽_{p}-algebras commute with all limits. By the same token, the existence of the left adjoint E^{∗} implies that the forgetful functor p-TopMon → TopMon also commutes with limits, and likewise for perfect topological 𝔽_{p}-algebras.

(v) Let A be any 𝔽_{p}-algebra, I ⊂ A a finitely generated ideal, and denote by 𝒯_{I} (resp. 𝒯_{d}) the I-adic topology (resp. the discrete topology) on A. Then (A, 𝒯_{I}) is a perfect topological 𝔽_{p}-algebra if and only if the same holds for (A, 𝒯_{d}). Indeed, suppose that the Frobenius endomorphism Φ_{A} is bijective on A; in order to show that Φ_{A} is an automorphism of (A, 𝒯 ), it suffices to prove that for every k ∈ ℕ there exists n(k) ∈ ℕ such that I^{n(k)} ⊂ Φ(I^{n}); but the latter is clear, since I is finitely generated (details left to the reader).

Theorem 9.4.10. Let (A, 𝒯_{A}) be any complete and separated topological ring whose topology 𝒯_{A} is linear and coarser than the p-adic topology. Let also I ⊂ A be a given ideal that is topologically nilpotent for the topology 𝒯_{A}, and π : A → A/I the projection. Endow A/I with the quotient topology 𝒯_{A/I} induced by 𝒯_{A} via π, and denote by ((A/I)^{∧}, 𝒯_{A/I}^{∧}) the completion of (A/I, 𝒯_{A/I}), and by j : A → (A/I)^{∧} the completion map. Suppose moreover that :

(a) either, I is a closed subset for the topology 𝒯_{A}

(b) or else, I is finitely generated, and a closed subset for the p-adic topology of A. Then the maps

E(π) : E(A, 𝒯_{A}) → E(A/I, 𝒯_{A/I}) E(j) : E(A/I, 𝒯_{A/I}) → E((A/I)^{∧}, 𝒯_{A/I}^{∧})

are isomorphisms of topological monoids.

Proof. Let (J_{λ} | λ ∈ Λ) be a family of ideals of A which is a fundamental system of open neighborhoods of 0 for the topology 𝒯_{A}. Suppose first that I is closed for the topology 𝒯_{A}, and set π^{∧} := j ∘ π : A → (A/I)^{∧}; in this case, j is injective, so the same holds for E(j), and to prove the theorem it suffices to check that E(π^{∧}) is an isomorphism. However, we have natural isomorphisms

A →^{∼} lim_{λ∈Λ} A/J_{λ} (A/I)^{∧} →^{∼} lim_{λ∈Λ} A/(I + J_{λ})

whence, by remark 9.4.9(iv), induced isomorphisms of topological monoids

E(A) →^{∼} lim_{λ∈Λ} E(A/J_{λ}) E((A/I)^{∧}) →^{∼} lim_{λ∈Λ} E(A/(I + J_{λ})).

Thus, it suffices to show that the induced map E(A/J_{λ}) → E(A/(I + J_{λ})) is an isomorphism of topological monoids for every λ ∈ Λ. In this case, we may then replace A by A/J_{λ}, and assume from start that the topology 𝒯_{A} is discrete, hence I is a nilpotent ideal.

Claim 9.4.11. The theorem holds when the topology 𝒯_{A} is discrete and I = pA.

Proof of the claim. In this case, the p-adic filtration is finite, hence the homomorphism of topological 𝔽_{p}-algebras \overline{u}_{A/pA} : E(A/pA) → A/pA lifts to a morphism of monoids v : E(A/pA) → A (lemma 9.3.51(iii)). Explicitly, say that p^{k}A = 0 for some k ∈ ℕ, let \overline{a}_{•} := (\overline{a}_{n} | n ∈ ℕ) ∈ E(A/pA) be any element, and pick a_{k} ∈ A whose image in A/pA equals \overline{a}_{k}; then lemma 9.3.4(i) shows that v(\overline{a}_{•}) = a_{k}^{p^{k}}. This description easily implies that v is continuous (for the discrete topology on A). Therefore v corresponds, by adjunction, to a unique morphism v^{†} := E(v)∘\overline{u}_{E(A/pA)}^{−1} : E(A/pA) → E(A) of p-perfect topological monoids. We have

E(π) ∘ v^{†} = E(π ∘ v) ∘ \overline{u}_{E(A/pA)}^{−1} = E(\overline{u}_{A/pA}) ∘ \overline{u}_{E(A/pA)}^{−1} = 1_{E(A/pA)}

by remark 9.4.5(ii). Lastly, the foregoing explicit description of v implies that v^{†}∘E(π) = 1_{E(A)} (details left to the reader), and the claim follows. ♢

Now, we have a commutative diagram of topological monoids

\xymatrix{E(A) \ar[r]^{E(π′)} \ar[d]_{E(π)} & E(A/pA) \ar[d]^{E(π′′)} \\ E(A/I) \ar[r]^{E(π′′′)} & E(A/(pA + I))}

where π′, π′′, π′′′ are the projections. By claim 9.4.11 we know already that E(π′) and E(π′′′) are isomorphisms, hence it suffices to show that the same holds for E(π′′). Thus, we may further replace A by A/pA, and assume from start that A is also an 𝔽_{p}-algebra. In this case, the I-adic filtration on A fulfills the conditions of lemma 9.3.4, hence lemma 9.3.51(iii) applies and says that the map \overline{u}_{A/I} : E(A/I) → A/I lifts to a morphism of monoids w : E(A/I) → A, and arguing as in the proof of claim 9.4.11 we see that w is continuous, so it yields, again by adjunction, a morphism of topological monoids w^{†} : E(A/I) → E(A), which is seen to be an inverse for E(π), by the same argument as in the proof of claim 9.4.11. This concludes the proof, in case I is closed for the topology 𝒯_{A}.

For the general case, denote by \overline{I} ⊂ A the topological closure of A (for the topology 𝒯_{A}), endow A/\overline{I} with the quotient topology 𝒯_{A/\overline{I}} induced by the projection A → A/\overline{I}, and let ((A/\overline{I})^{∧}, 𝒯_{A/\overline{I}}^{∧}) be the completion of (A/\overline{I}, 𝒯_{A/\overline{I}}); denote also by π′ : (A/I)^{∧} → (A/\overline{I})^{∧} and \overline{π} : A → (A/\overline{I})^{∧} the induced maps; we deduce morphisms of topological monoids E(A) → E((A/I)^{∧}) → E((A/\overline{I})^{∧}) whose composition equals E(\overline{π}). Notice that \overline{I} is still topological nilpotent, hence E(\overline{π}) is an isomorphism, by the foregoing case. Also π′ is an isomorphism, so the same holds for E(π′) as well. We set v := E(\overline{π})^{−1} ∘ E(π′) : E((A/I)^{∧}) → E(A). Clearly v ∘ E(π) = 1_{E(A)}, so it remains only to check that v is bijective, or equivalently, that the same holds for E(π). To this aim, let 𝒯′_{A} be the (I + pA)-adic topology on A, and notice that 𝒯′_{A} is still complete and separated (lemma 8.3.12); moreover, clearly both I and pA are topologically nilpotent ideals in (A, 𝒯′_{A}). Furthermore, notice that

⋂_{n∈ℕ}(I + (I + pA)^{n}) = ⋂_{n∈ℕ}(I + p^{n}A) = I

since I is closed in the p-adic topology of A; thus, I is closed in the topology 𝒯′_{A}, so the foregoing case shows that the map E(π) : E(A, 𝒯′_{A}) → E(A/I) is indeed bijective. □

Remark 9.4.12. Keep the situation of theorem 9.4.10.

(i) For every (\overline{a}_{n} | n ∈ ℕ) ∈ E(A/I) and every n ∈ ℕ choose a representative a_{n} ∈ A for the class \overline{a}_{n}. By inspecting the proof, we see that the inverse of E(π) is given by the rule :

(9.4.13)  (\overline{a}_{n} | n ∈ ℕ) ↦ ( lim_{k→∞} a_{n+k}^{p^{k}} | n ∈ ℕ)

where the convergence is relative to the topology 𝒯_{A}.

(ii) Suppose moreover, that p ∈ I; then E(A/I) is a topological 𝔽_{p}-algebra, and the isomorphism E(π) can be used to transfer to E(A) the ring structure of E(A/I). In light of (i), we get the following expressions for the addition and multiplication laws on E(A) such that E(π) becomes an isomomorphism of topological rings :

\begin{matrix}(a_{n} | n ∈ ℕ) + (b_{n} | n ∈ ℕ) := & ( lim_{k→+∞}(a_{n+k} + b_{n+k})^{p^{k}} | n ∈ ℕ) \\ (a_{n} | n ∈ ℕ) · (b_{n} | n ∈ ℕ) := & (a_{n}b_{n} | n ∈ ℕ) \end{matrix}

where the limit is computed in the topology 𝒯_{A}.

(iii) In the situation of (ii), suppose furthermore that A is a domain (resp. a field). Then theorem 9.4.10 implies immediately that the same holds for E(A/I) (and of course also for E(A), for its ring structure as in (ii)).

Corollary 9.4.14. Let (A, 𝒯 ) be a perfect topological 𝔽_{p}-algebra, I ⊂ A an open and topologically nilpotent ideal, and denote by (A^{∧}, 𝒯 ^{∧}) the separated completion of (A, 𝒯 ). Then there is a natural isomorphism of topological 𝔽_{p}-algebras

ω : (A^{∧}, 𝒯 ^{∧}) →^{∼} E(A/I)  such that  π = ω ∘ \overline{u}_{A/I}

where π : A^{∧} → A/I is the natural map.

Proof. Let I^{∧} be the topological closure of the image of I in A^{∧}, and endow B := A^{∧}/I^{∧} with the quotient topology induced by the projection π : A^{∧} → B; notice that the natural map A/I → B induces an isomorphism φ : A/I →^{∼} B^{∧} on the respective separated completions. Then example 9.3.48(iii) and theorem 9.4.10imply that the composition

A^{∧} ⟶^{\overline{u}_{A^{∧}}^{−1}} E(A^{∧}) ⟶^{E(π)} E(B) ⟶^{E(φ)} E(A/I)

is an isomorphism, whence the corollary. □

9.4.15. Let now (R, | · |_{R}) be any valuation ring with value group Γ_{R} and residue characteristic p. Notice that the p-Frobenius map of Γ_{R} is injective, so we may regard E(Γ_{R}) as a subgroup of Γ_{R}. To ease notation, set E := E(R/pR); also, let Δ := Γ_{R} in case R is an 𝔽_{p}-algebra, and otherwise let Δ := O(|p|_{R}) (notation of remark 9.1.2(vii)). We define a mapping

| · |_{E} : E → Γ_{E∘} where Γ_{E} := E(Γ_{R}) ∩ Δ

as follows. Let \underline{a} := (a_{n} | n ∈ ℕ) ∈ E be any element, and for every n ∈ ℕ choose a representative ã_{n} ∈ R for the class a_{n} ∈ R/pR. If |ã_{n}|_{R} ≤ |p|_{R} for every n ∈ ℕ, then we set |\underline{a}|_{E} := 0; if there exists n ∈ ℕ such that |ã_{n}| > |p|_{R}, then we set |\underline{a}|_{E} := |ã_{n}|_{R}^{p^{n}}. One verifies easily that the definition is independent of all the choices : the details shall be left to the reader.

Lemma 9.4.16. In the situation of (9.4.15), the pair (E, | · |_{E}) is a valuation ring.

Proof. The bijection of remark 9.1.13(vii) attaches to Δ the prime ideal 𝔭(Δ) = ⋂_{n∈ℕ} p^{n}R, and we set R′ := R/𝔭(Δ). Since the set of ideals of R′ is naturally identified with the set of those ideals of R that contain 𝔭(Δ), it follows easily that R′ is a valuation ring, and then it is clear that its valuation is induced by the primary specialization | · |_{R}^{Δ} of | · |_{R} (notation of (9.2.10)). Notice moreover that R′/pR′ = R/pR, so that E(R′/pR′) = E. Summing up, we may replace R by R′ after which we may assume from start that Γ_{R} = Δ, and that the p-adic topology on R is separated. Moreover, notice that in this case the p-adic topology on R agrees with either the valuation topology (see definition 9.1.14) or with the discrete topology; hence the p-adic completion R^{∧} of R is still a valuation ring (proposition 9.1.16(iii)). Then, since R^{∧}/pR^{∧} = R/pR, we may replace R by R^{∧}, and assume from start that R is p-adically complete, in which case remark 9.4.12(iii) already shows that E is a domain. Moreover, in this situation the isomorphism E(R) →^{∼} E of theorem 9.4.10 identifies the map | · |_{E} with the mapping

| · |_{E(R)} : E(R) → Γ_{R∘} (a_{n} | n ∈ ℕ) ↦ |a_{0}|_{R}.

With this notation, clearly |x_{•} · y_{•}|_{E(R)} = |x_{•}|_{E(R)} · |y_{•}|_{E(R)} for every x_{•}, y_{•} ∈ E(R). By the same token, it is clear that x_{•} divides y_{•} in E(R) if and only if |x_{•}|_{E(R)} ≥ |y_{•}|_{E(R)} so (E, | · |_{E}) is a valuation ring. □

The last topics of this section are the graded versions of the functors W and E.

9.4.17. Let (Γ, +, 0) be a monoid, p ∈ ℕ a prime integer, (A, \underline{B}) a topological ring with Γ-graded structure, and suppose that the p-Frobenius endomorphism of Γ is injective. The assumption implies that the natural map Γ → E^{∗}(Γ) is injective (notation of proposition 9.4.3), hence we may regard \underline{B} naturally as an E^{∗}(Γ)-graded ℤ-algebra. We define

W (\underline{B}) := ⨁_{γ∈E^{∗}(Γ)} gr_{γ}W (\underline{B})  where gr_{γ}W (\underline{B}) := ∏_{n∈ℕ} gr_{p^{n}γ}B  for every γ ∈ E^{∗}(Γ).

From remark 9.3.8(i) it follows easily that, for every γ, μ ∈ Γ, the addition and multiplication laws of W (B) restrict respectively to maps

gr_{γ}W (\underline{B}) × gr_{γ}W (\underline{B}) → gr_{γ}W (\underline{B}) gr_{γ}W (\underline{B}) × gr_{μ}W (\underline{B}) → gr_{γ+μ}W (\underline{B}).

We may therefore define an addition law on W (\underline{B}) by taking the direct sum of the addition laws on each Γ-graded summand. Likewise, the map x ↦ −x on W (B) restricts to a map gr_{γ}W (\underline{B}) → gr_{γ}W (\underline{B}) for every γ ∈ Γ, so that W (\underline{B}) is an abelian group with this addition law. Next, clearly 1 ∈ gr_{0}W (\underline{B}), so we obtain a ring structure on W (\underline{B}), by extending linearly the multiplication map already defined on each graded term. Thus, the multiplication map of W (\underline{B}) agrees with that of W (B) on each pair of homogeneous elements, but it will be usually different, on non-homogeneous elements.

Lemma 9.4.18. With the notation of (9.4.17), the induced map

φ : W (\underline{B}) → W (B) (b_{γ} | γ ∈ E^{∗}(Γ)) ↦ ∑_{γ∈E^{∗}(Γ)} b_{γ}

is injective (again, notice that φ agrees with the inclusion map on homogeneous terms, but it usually differs from it on non-homogeneous elements).

Proof. For every r ∈ ℕ and γ ∈ E^{∗}(Γ) define gr_{γ}W_{r}(\underline{B}) as the image of gr_{γ}W (\underline{B}) into W_{r}(B), and set

W_{r}(\underline{B}) := ⨁_{γ∈E^{∗}(Γ)} gr_{γ}W_{r}(\underline{B}).

We shall show, more precisely, that the resulting map φ_{r} : W_{r}(\underline{B}) → W_{r}(B) is a bijection. We argue by induction on r. For r = 1, the assertion follows by a simple inspection. Next, suppose that r ≥ 0 and that φ_{r} is bijective; we deduce a commutative ladder with exact rows :

\xymatrix{0 \ar[r] & ⨁_{γ∈E^{∗}(Γ)} gr_{p^{r}γ}B \ar[r] \ar[d]_{ψ_{r}} & W_{r+1}(\underline{B}) \ar[r] \ar[d]_{φ_{r+1}} & W_{r}(\underline{B}) \ar[r] \ar[d]^{φ_{r}} & 0 \\ 0 \ar[r] & W_{r+1}(B) ∩ \overline{V}_{r} \ar[r] & W_{r+1}(B) \ar[r] & W_{r}(B) \ar[r] & 0}

where \overline{V}_{r} denotes the image of V_{r}(B) in W_{r+1}(B). It then suffices to check that ψ_{r} is a bijection; but (9.3.9) implies that ψ_{r} is none else than the restriction of the additive map V^{r}, and since E^{∗}(Γ) is p-perfect, the source of ψ_{r} is just B, whence the contention. □

Due to lemma 9.4.18, we may regard W (\underline{B}) as a subring of W (B). We also see that W_{r}(B) inherits from \underline{B} a natural E^{∗}(Γ)-graded ℤ-algebra structure, for every r ∈ ℕ. Moreover, gr_{0}W (\underline{B}) = W (gr_{0}B), so W (\underline{B}) is an E^{∗}(Γ)-graded W (gr_{0}B)-algebra. Set

W (A, \underline{B}) := (W (A), W (\underline{B}))

and let also E^{∗}(Γ)_{(p)} be the object of the category E^{∗}(Γ)/Mnd given by the p-Frobenius automorphism p_{E^{∗}(Γ)} : E^{∗}(Γ) → E^{∗}(Γ)_{(p)}.

Proposition 9.4.19. With the notation of (9.4.17), the following holds :

(i) The pair W (A, \underline{B}) is an E^{∗}(Γ)-graded structure on W (A) (see definition 8.5.1(i)). (ii) The map ω_{0} is a morphism of topological rings with E^{∗}(Γ)-graded structures

ω_{0} : W (A, \underline{B}) → (A, \underline{B}).

(iii) The Frobenius and Verschiebung maps of W (A) (see (9.3.18)) induce respectively a morphism of topological rings with E^{∗}(Γ)-graded structures

F_{A} : W (A, \underline{B}) → (W (A), E^{∗}(Γ) ×_{E^{∗}(Γ)_{(p)}} W (\underline{B}))

and a continuous map of E^{∗}(Γ)-graded topological abelian groups

V_{A} : E^{∗}(Γ) ×_{E^{∗}(Γ)_{(p)}} W (\underline{B}) → W (\underline{B})

(notation of definition 7.6.1(iv)).

Proof. (i): Indeed, a direct inspection shows that gr_{γ}W (\underline{B}) is a closed subset in W (A, 𝒯_{A}), for every γ ∈ E^{∗}(Γ). Next, notice that the topology of W (B) agrees with the topology induced by W (A) via the inclusion map W (B) → W (A). Hence, in order to check that the topology induced by W (A) on W (\underline{B}) is linear and defined by a system of open graded ideals, it suffices to show that the same holds for the topology induced by W (B). However, let (J_{λ} | λ ∈ Λ) be any fundamental system of open graded ideals of B, and set 𝒥_{λ,r} := Ker (W (B) → W_{r}(B/J_{λ})) for every λ ∈ Λ and r ∈ ℕ; the proof of lemma 9.3.33(iii) shows that (𝒥_{λ,r} | λ ∈ Λ; r ∈ ℕ) is a fundamental system of open ideals of W (B), and we are reduced to checking that 𝒥_{λ,r} ∩ W (\underline{B}) is a graded ideal of W (\underline{B}), for every λ ∈ Λ and r ∈ ℕ. To this aim, we consider the commutative diagram

\xymatrix{W (\underline{B}) \ar[r]^{φ} \ar[d] & W (B) \ar[d] \\ W_{r}(\underline{B}/J_{λ}) \ar[r]^{φ_{r}} & W_{r}(B/J_{λ})}

where φ_{r} is the isomorphism defined as in the proof of lemma 9.4.18, and the vertical arrows are the natural projections. We deduce that 𝒥_{λ,r} ∩ W (\underline{B}) is the kernel of the projection W (\underline{B}) → W_{r}(\underline{B}/J_{λ}), and the assertion follows easily. It remains to check that W (\underline{B}) is a dense subset of W (A), but this is clear, since φ_{r} is bijective for every r ∈ ℕ.

(ii) follows by a direct inspection of the construction.

(iii) follows easily from remark 9.3.8(i). □

We have the following graded version of proposition 9.3.52:

Proposition 9.4.20. Let p ∈ ℕ be a prime integer, Γ a monoid whose p-Frobenius endomorphism is injective, (R, \underline{S}) and (A, \underline{B}) two topological rings with Γ-structures, and suppose that:

(a) R is a perfect topological 𝔽_{p}-algebra, and A is complete and separated.

(b) \underline{B} admits a fundamental system (J_{n} | n ∈ ℕ) of graded open ideals such that pJ_{n} +

J_{n}^{p} ⊂ J_{n+1} ⊂ J_{n} for every n ∈ ℕ. Then, for every morphism \overline{φ} : (R, \underline{S}) → (A, \underline{B}) of topological rings with Γ-graded structures there exists a unique morphism u : W (R, \underline{S}) → (A, \underline{B}) of topological rings with Γ-graded structures that makes commute the diagram

\xymatrix{W (R, \underline{S}) \ar[r]^{u} \ar[d]_{_{ω_{0}}} & (A, \underline{B}) \ar[d]^{π} \\ (R, \underline{S}) \ar[r]^{\overline{φ}} & (A, \underline{B})/J_{0}}

where π is the natural projection.

Proof. We have to show that the unique continuous ring homomorphism u : W (R) → A provided by proposition 9.3.52(iii) restricts to a map of Γ-graded rings W (\underline{S}) → \underline{B}. However, recall that gr_{γ}B is a closed subset of A, so it is complete and separated for the topology induced by \underline{B}, for every γ ∈ Γ, and hence also for the p-adic topology (lemma 8.3.12). Then, let φ : R → A be the unique continuous mapping characterized as in lemma 9.3.51(iii); by inspecting (9.3.53), we are reduced to checking that φ(gr_{γ}S) ⊂ gr_{γ}B for every γ ∈ Γ. To this aim, pick any map ψ : R → A such that :

(a) π ∘ ψ = \overline{φ}

(b) ψ(gr_{γ}S) ⊂ gr_{γ}B for every γ ∈ Γ

and recall that φ(r) = lim_{n→+∞} ψ(r^{p^{−n}})^{p^{n}} for every r ∈ R. The assertion follows easily. □

9.4.21. Let Γ be a monoid, P a Γ-graded monoid. Clearly E(P) is naturally an E(Γ)-graded monoid. Moreover, let \underline{B} := (B, gr_{•}B) be a Γ-graded topological ℤ-algebra; we define the Γ-graded monoid

B^{∗} := ∐_{γ∈Γ} gr_{γ}B

whose composition law is induced by the multiplication map of B. The natural morphism of Γ-graded monoids B^{∗} → B (which is not injective, unless Γ = {0}) induces a morphism of E(Γ)-graded monoids

E(B^{∗}) → E(B).

Furthermore, if B is an 𝔽_{p}-algebra we may define the E(Γ)-graded subring of E(B)

E(\underline{B}) := ⨁_{γ∈E(Γ)} gr_{γ}E(B^{∗})

which we endow with the topology induced from E(B), via the inclusion map E(\underline{B}) → E(B), and the rule \underline{B} ↦ E(\underline{B}) defines a functor from the category of Γ-graded topological 𝔽_{p}-algebras to the category of perfect E(Γ)-graded topological 𝔽_{p}-algebras. It is easily seen that this functor is right adjoint to the forgetful functor \underline{C} ↦ \underline{C}_{/Γ} induced by \overline{u}_{Γ} : E(Γ) → Γ.

Proposition 9.4.22. Let (Γ, +, 0) be a monoid, (A, \underline{B}) a topological 𝔽_{p}-algebra with Γ-graded structure, such that A is complete and separated, and the p-Frobenius endomorphism p_{Γ} of Γ is injective. We have :

(i) E(A, \underline{B}) := (E(A), E(\underline{B})) is an E(Γ)-graded structure on E(A).

(ii) The pair (\overline{u}_{A}, \overline{u}_{Γ}) is a morphism of topological 𝔽_{p}-algebras with graded structures :

\overline{u}_{A} : E(A, \underline{B}) → (A, \underline{B}).

(iii) Let Γ′ be another monoid whose p-Frobenius endomorphism is injective, and Γ → Γ′ any morphism of monoids. Then the identity map of E(A) induces an isomorphism of topological rings with Γ′-graded structures

E(A, \underline{B}, Γ)_{/Γ′} →^{∼} E((A, \underline{B}, Γ)_{/Γ′}).

Proof. Indeed, the definition immediately shows that gr_{γ}E(B) is closed in the topology of E(A) for every γ ∈ Γ. Next, notice that – under the current assumptions – we may regard E(Γ) as a submonoid of Γ, and for every graded ideal J ⊂ B and every r ∈ ℕ, set

J^{∗} := ∐_{γ∈Γ} gr_{γ}J and E(\underline{J}) := ⨁_{γ∈Γ} gr_{γ}E(J^{∗}).

So, E(\underline{J}) is a graded ideal of E(\underline{B}), and from remark 9.4.9(i) it follows that the topology of E(A) induces on E(\underline{B}) the linear topology defined by the cofiltered system of graded ideals (Φ_{E(\underline{B})}^{r}E(J) | J ⊂ B; r ∈ ℕ), where J ranges over all the open graded ideals of B. It remains to check that E(\underline{B}) is dense in E(A). To this aim, let \underline{a} := (a_{n} | n ∈ ℕ) ∈ E(A) be any element, and for every finite subset S ⊂ E(Γ) and every n ∈ ℕ, set S(n) := p_{Γ}^{−n}(S); for every such S, we define the element

\underline{a}_{S} := (a_{n,S(n)} | n ∈ ℕ) ∈ E(\underline{B})

where the component a_{n,S(n)} ∈ ⨁_{γ∈S(n)} gr_{γ}B is defined as in remark 8.5.2(iii).

Claim 9.4.23. (i) a_{n,γ} = 0 for every γ ∈ Γ \ E(Γ) and every n ∈ ℕ.

(ii) (a_{n,S(n)} | S ⊂ E(Γ)) is a Cauchy net in B whose limit is a_{n}, for every n ∈ ℕ.

(iii) a_{n,S(n)} = a_{n+1,S(n+1)}^{p}, for every n ∈ ℕ and every finite subset S ⊂ E(Γ).

Proof of the claim. For every n ∈ ℕ, let (a_{n,γ} | γ ∈ Γ) ∈ ∏_{γ∈Γ} B_{γ} be the sequence of canonical projections of a_{n} (see remark 8.5.2(iii)); in view of lemma 8.5.6(i) we see that a_{n,pγ} = a_{n+1,γ}^{p} for every γ ∈ Γ, and a_{n,γ} = 0 if γ ∈ Γ\p_{Γ}(Γ). Assertions (i) and (iii) are immediate consequences.

(ii) follows immediately from (i) and proposition 8.5.3(ii). ♢ From claim 9.4.23(ii,iii) we conclude that the system (\underline{a}_{S} | S ⊂ E(Γ)) is a Cauchy net in

E(\underline{B}) whose limit is \underline{a}, as required.

(ii) is clear by inspecting the constructions.

(iii): Indeed, by example 8.5.7(iii), any such morphism will be necessarily an isomorphism, and conversely, consider the natural morphism (A, \underline{B}, Γ) → (A, \underline{B}, Γ)_{/Γ′} which induces a morphism E(A, \underline{B}, Γ) → E((A, \underline{B}, Γ)_{/Γ′}) which in turns factors uniquely through a morphism as sought, again by example 8.5.7(iii). □

9.4.24. Let (A, \underline{B}) be a complete and separated topological ring with Γ-graded structure, whose topology is linear and coarser than the p-adic topology. Endow A/pA with the quotient topology induced by the projection π_{A} : A → A/pA, and let (A/pA)^{∧} be the separated completion of A/pA; by theorem 9.4.10, the map π_{A} induces an isomorphism of topological monoids

E(π_{A}) : E(A) →^{∼} E(A/pA)

and the completion map j : A/pA → (A/pA)^{∧} induces an isomorphism of topological rings

E(j) : E(A/pA) → E((A/pA)^{∧}).

Especially, E(A/pA) is a complete and separated topological ring whose topology is linear (remark 9.4.9(i)), and we may transfer the ring structure of E(A/pA) onto E(A) (see remark 9.4.12(ii)), after which the latter can also be regarded as a complete and separated topological ring with a linear topology. Furthermore, suppose that the p-Frobenius map of Γ is injective, so that the topological rings with E(Γ)-structures

((A/pA)^{∧}, \underline{B}_{0}) := ((A, \underline{B})/pB)^{∧} and E((A/pA)^{∧}, \underline{B}_{0})

are well defined, by proposition 9.4.22(i) (and remark 8.5.2(iv)), and we may transfer as well this graded structure onto E(A) and E(A/pA). We denote by

E(A, \underline{B}) and E(A/pA, \underline{B}/p\underline{B})

the resulting topological rings with E(Γ)-graded structures, whose underlying topological rings are respectively E(A) and E(A/pA). Define B′ as in remark 8.5.2(ii), and notice that pB = pB′ ∩ B and pA ⊂ A ∩ pB′, so that

(9.4.25)  pB = pA ∩ B

and therefore the natural map E(B/pB) → E(A/pA) is injective.

Proposition 9.4.26. In the situation of (9.4.24), let also \underline{C} (resp. \underline{C}_{0}) be the Γ-graded dense subring of E(A) (resp. of E(A/pA)). With the notation of (9.4.21) we have :

\underline{C}_{0} = E(\underline{B}/p\underline{B})  and  \underline{C} = E(\underline{B}).

Proof. Notice that – as explained in remark 8.5.4(ii) – since A is complete and separated, gr_{γ}B_{0} is the maximal separated quotient of gr_{γ}(B/pB), for every γ ∈ Γ; it is also a closed subset of the complete ring (A/pA)^{∧}, so it is the separated completion of gr_{γ}(B/pB).

Now, from (9.4.25) we deduce that the canonical γ-projection of (A, \underline{B}) induces a continuous B_{0}/pB_{0}-linear map π_{γ} : A/pA → gr_{γ}(B/pB) for every γ ∈ Γ, and in view of the injectivity of p_{Γ}, proposition 8.5.3(ii) and lemma 8.5.6(i) easily imply that π_{pγ}(a^{p}) = π_{γ}(a)^{p} in the Γ-graded ℤ-algebra B/pB. To ease notation, set D_{γ} := gr_{γ}E(\underline{B}) and D_{0,γ} := gr_{γ}E(\underline{B}/p\underline{B}) for every γ ∈ E(Γ); there follows, for every such γ, a continuous E(gr_{0}B)-linear map

π_{E,γ} : E(A/pA) → D_{0,γ} (a_{n} | n ∈ ℕ) ↦ (π_{γ/p^{n}}(a_{n}) | n ∈ ℕ).

Let also π_{γ}^{∧} : (A/pA)^{∧} → gr_{γ}B_{0} be the completion of π_{γ}, for every γ ∈ Γ; we obtain likewise a continuous E(gr_{0}B_{0})-linear map for every γ ∈ E(Γ)

π_{E,γ}^{∧} : E((A/pA)^{∧}) → gr_{γ}E(\underline{B}_{0}) (a_{n} | n ∈ ℕ) ↦ (π_{γ/p^{n}}^{∧}(a_{n}) | n ∈ ℕ).

However, π_{E,γ}^{∧} is none else than the canonical γ-projection of E((A/pA)^{∧}, \underline{B}_{0}); indeed, it is easily seen that it coincides with the latter map on the dense subring E(\underline{B}_{0}), and gr_{γ}E(\underline{B}_{0}) is separated, whence the claim. Moreover, by construction we have a commutative diagram

\xymatrix{E(A/pA) \ar[r]^{∏_{γ∈E(Γ)} π_{E,γ}} \ar[d]_{E(j)} & ∏_{γ∈E(Γ)} D_{0,γ} \ar[d]^{∏_{γ∈E(Γ)} i_{γ}} \\ E((A/pA)^{∧}) \ar[r]^{∏_{γ∈E(Γ)} π_{E,γ}^{∧}} & ∏_{γ∈E(Γ)} gr_{γ}E(\underline{B}_{0})}

where i_{γ} : D_{0,γ} → gr_{γ}E(\underline{B}_{0}) is the restriction of E(j), for every γ ∈ Γ. Then, a simple inspection of this diagram shows that gr_{γ}C_{0} ⊂ D_{0,γ} for every γ ∈ Γ, and the converse inclusion is obvious. Lastly, it is clear that E(π_{A})(D_{γ}) ⊂ D_{0,γ} for every γ ∈ E(Γ). To show the converse inclusion, let \overline{b}_{•} := (\overline{b}_{n} | n ∈ ℕ) be any element of D_{0,γ}, and for every n ∈ ℕ pick b_{n} ∈ gr_{γ/p^{n}}B whose image in gr_{γ/p^{n}}(B/pB) agrees with \overline{b}_{n}; according to remark 9.4.12(i), the element E(π_{A})^{−1}(\overline{b}_{•}) is the sequence (b′_{n} | n ∈ ℕ) with b′_{n} := lim_{k→∞} b_{n+k}^{p^{k}} for every n ∈ ℕ, where the convergence is relative to the topology of A. But since gr_{γ}B is closed in A for every γ ∈ Γ, we see that b′_{n} ∈ gr_{γ/p^{n}}B for every n ∈ ℕ, whence the claim. □

9.5. Divided power modules and algebras. This section is a review of the theory of divided power modules and algebras.

Definition 9.5.1. Let A be a ring, k ∈ ℕ an integer, and \underline{M} := (M_{1}, …, M_{k}) an object of the abelian category (A-Mod)^{k}, i.e. any sequence of A-modules. Let also Q be another A-module, and \underline{d} := (d_{1}, …, d_{k}) ∈ ℕ^{⊕k} any sequence of k integers.

(i) We denote by F\underline{M}, G_{Q} : A-Alg → Set the functors given respectively by the rules :

B ↦ (M_{1} × ⋯ × M_{k}) ⊗_{A} B and B ↦ Q ⊗_{A} B for every A-algebra B.

Also, let M_{i}^{∨} := Hom_{A}(M_{i}, A) for every i = 1, …, k, and set :

Sym_{A}^{•}(\underline{M}^{∨}) := ⨂_{i=1}^{k} Sym_{A}^{•}(M_{i}^{∨})

which we view as a ℕ^{⊕k}-graded A-algebra, with the grading given by the rule :

Sym\underline{n} ; A(\underline{M}^{∨}) := ⨂_{i=1}^{k} Sym_{A}^{n_{i}}(M_{i}^{∨}) for every \underline{n} := (n_{1}, …, n_{k}) ∈ ℕ^{⊕k}.

(ii) A homogeneous multipolynomial law of degree \underline{d} on the sequence of A-modules \underline{M}, with values in Q, is a natural transformation λ : F\underline{M} → G_{Q}, which we write also

λ : \underline{M} ⇝ Q

such that for every A-algebra B we have :

(9.5.2) λ_{B}(b_{•}x_{•}) = b_{1}^{d_{1}} ⋯ b_{k}^{d_{k}} · λ_{B}(x_{•})  for every b_{•} ∈ B^{⊕k} and every x_{•} ∈ F\underline{M}(B)

where b_{•}x_{•} := (b_{1}x_{1}, …, b_{k}x_{k}) (and where we let b^{0} := 1 for every b ∈ B). (iii) The set of all homogeneous multipolynomial laws \underline{M} ⇝ Q of degree \underline{d} is denoted

Pol\underline{d} ; A(\underline{M}, Q).

Remark 9.5.3. (i) Let \underline{f} := (f_{i} : M′_{i} → M_{i} | i = 1, …, k) be a sequence of A-linear maps, and set \underline{M}′ := (M′_{1}, …, M′_{k}); let also g : Q → Q′ be another A-linear map. Suppose that λ : \underline{M} ⇝ Q is a homogeneous multipolynomial law of degree d; then we obtain a law

g ∘ λ ∘ \underline{f} : \underline{M}′ ⇝ Q′

of the same type, by the rule :

B ↦ (g ⊗_{A} B) ∘ λ_{B} ∘ ((f_{1} × ⋯ × f_{k}) ⊗_{A} B) : (M′_{1} × ⋯ × M′_{k}) ⊗_{A} B → Q′ ⊗_{A} B

for every A-algebra B. This shows that the rule (\underline{M}, Q) ↦ Pol\underline{d} ; A(\underline{M}, Q) is functorial in both arguments. However, it is not a priori obvious that this functor takes values in (essentially) small sets. The latter assertion will nevertheless be proven shortly.

(ii) Suppose that \underline{M}′ := (M′_{1}, …, M′_{k′}) is another sequence of A-modules, Q′ another Amodule, and \underline{d}′ := (d′_{1}, …, d′_{k′}) ∈ ℕ^{⊕k′} another sequence of integers. If we have two homogeneous polynomial laws λ : \underline{M} ⇝ Q and λ′ : \underline{M}′ ⇝ Q′ of degrees respectively \underline{d} and \underline{d}′, we can form the tensor product

λ ⊗ λ′ : (\underline{M}, \underline{M}′) ⇝ Q ⊗_{A} Q′

which is the homogeneous multipolynomial law of degree (\underline{d}, \underline{d}′) given by the rule :

(λ ⊗ λ′)_{B}(x_{1}, …, x_{k}, x′_{1}, …, x′_{k′}) := λ_{B}(x_{1}, …, x_{k}) ⊗ λ′_{B}(x′_{1}, …, x′_{k′})

for every A-algebra B and every (x_{1}, …, x′_{k′}) ∈ (M_{1} × ⋯ × M′_{k′}) ⊗_{A} B.

(iii) Let B be any A-algebra, and Q a B-module, which we also regard as an A-module, by restriction of scalars. Set also N := M_{1} × ⋯ × M_{k}, to ease notation. We have a bijection :

(9.5.4)  Pol\underline{d} ; B(B ⊗_{A} \underline{M}, Q) →^{∼} Pol\underline{d} ; A(\underline{M}, Q)

natural with respect to B-linear maps Q → Q′ and morphisms \underline{M}′ → \underline{M} in (A-Mod)^{k}. Namely, to every homogeneous polynomial law λ : B ⊗_{A} \underline{M} ⇝ Q of degree \underline{d}, we assign the homogeneous polynomial law λ^{†} : \underline{M} ⇝ Q that makes commute the diagrams :

\xymatrix{C ⊗_{A} N \ar[rr]^{λ_{C}^{†}} \ar[d]_{φ⊗_{A}N} &  & C ⊗_{A} Q \ar[d] \\ C ⊗_{A} B ⊗_{A} N \ar[r]^{∼} & (C ⊗_{A} B) ⊗_{B} (B ⊗_{A} N) \ar[r]^{λ_{C⊗_{A}B}} & (C ⊗_{A} B) ⊗_{B} Q}

whose right vertical arrow is the standard isomorphism, and where φ : C → C ⊗_{A} B is given by the rule : c ↦ c ⊗ 1 for every c ∈ C, for all A-algebras C. Indeed, the inverse of (9.5.4) is the mapping that assigns to every homogeneous polynomial law μ : M ⇝ Q of degree \underline{d}, the homogeneous polynomial law μ^{†} : B ⊗_{A} \underline{M} ⇝ Q that makes commute the diagrams :

\xymatrix{C ⊗_{A} N \ar[r]^{μ_{C}} \ar[d] & C ⊗_{A} Q \ar[d] \\ C ⊗_{B} (B ⊗_{A} N) \ar[r]^{μ_{C}^{†}} & C ⊗_{B} Q}  for every B-algebra C

whose left vertical arrow is the standard isomorphism, and whose right vertical arrow is the B-linear map such that c ⊗ q ↦ c ⊗ q, for every c ∈ C and q ∈ Q. We leave to the reader the verification of the identities : (λ^{†})^{†} = λ and (μ^{†})^{†} = μ for every such λ and μ.

9.5.5. In the situation of definition 9.5.1, let also ℬ be any full subcategory of A-Alg such that Ob(ℬ) contains all the free polynomial A-algebras of finite type, and denote by j_{ℬ} : ℬ → A-Alg the inclusion functor. We consider also the set :

Pol\underline{d} ; A(\underline{M}, Q)_{ℬ}

of all natural transformations λ : F\underline{M} ∘ j_{ℬ} ⇒ G_{Q} ∘ j_{ℬ} verifying condition (9.5.2) for every B ∈ Ob(ℬ).

Proposition 9.5.6. With the notation of (9.5.5), the restriction map :

Pol\underline{d} ; A(\underline{M}, Q) → Pol\underline{d} ; A(\underline{M}, Q)_{ℬ} λ ↦ λ ∗ j_{ℬ}

is bijective for every k ∈ ℕ, every \underline{d} ∈ ℕ^{⊕k}, every ring A, and all A-modules M_{1}, …, M_{k}, Q.

Proof. Let ℬ_{0} (resp. ℬ_{1}) be the full subcategory of A-Alg whose objects are the free polynomial A-algebras of finite type (resp. the free polynomial A-algebras on arbitrary sets of indeterminates). To ease notation, set N := M_{1} × ⋯ × M_{k}; we show first :

Claim 9.5.7. The proposition holds for ℬ = ℬ_{1}.

Proof of the claim. Let us consider, for every A-algebra B, the standard simplicial resolution P_{B}[•] → B arising from the cotriple associated to the forgetful functor A-Alg → Set and its left adjoint (see example 7.10.25(i)); hence the augmentation ε_{B} : P_{B}[0] → B is a surjective map of A-algebras, and the set underlying B is the coequalizer of the maps of A-algebras ∂_{0}, ∂_{1} : P_{B}[1] → P_{B}[0]. If λ : \underline{M} ⇝ Q is any homogeneous polynomial law of degree \underline{d}, the commutativity of the diagram

(9.5.8)  \xymatrix{P_{B}[1] ⊗_{A} N \ar[r]^{λ_{P_{B} [1]}} \ar[d]_{∂_{0}⊗_{A}N} \ar[d]^{∂_{1}⊗_{A}N} & P_{B}[1] ⊗_{A} Q \ar[d]_{∂_{0}⊗_{A}Q} \ar[d]^{∂_{1}⊗_{A}Q} \\ P_{B}[0] ⊗_{A} N \ar[r]^{λ_{P_{B} [0]}} \ar[d]_{ε_{B}⊗_{A}N} & P_{B}[0] ⊗_{A} Q \ar[d]^{ε_{B}⊗_{A}Q} \\ B ⊗_{A} N \ar[r]^{λ_{B}} & B ⊗_{A} Q}

and the surjectivity of ε_{B} ⊗_{A} N, already show that λ_{P_{B}[0]} determines λ_{B}, whence the injectivity of our restriction map (as P_{B}[0] is a free polynomial A-algebra). Next, let λ : F\underline{M} ∘ j_{ℬ_{1}} ⇒ G_{Q} ∘ j_{ℬ_{1}} be an element of Pol\underline{d} ; A(\underline{M}, Q)_{ℬ_{1}} ; since the tensor product is right exact, it is easily seen that the map ε_{B} ⊗_{A} N identifies the set B ⊗_{A} N with the coequalizer (in the category of sets) of the pair of maps ∂_{0} ⊗_{A} N, ∂_{1} ⊗_{A} N : P_{B}[1] ⊗_{A} N → P_{B}[0] ⊗_{A} N (details left to the reader). Since P_{B}[0] and P_{B}[1] are free polynomial A-algebras, in this situation the top square subdiagram of (9.5.8) is well defined and commutes, so there exists exactly one map of sets λ_{B} that makes commute also the bottom square subdiagram. It is easily seen that such λ_{B} fulfills (9.5.2), since the same holds for λ_{P_{B}[0]}, and since both ε_{B} and ε_{B} ⊗_{A} N are surjective. Hence, we obtain a well defined map λ_{B} as required, for every A-algebra B, and it remains to check that the rule B ↦ λ_{B} defines a natural transformation F\underline{M} ⇒ G_{Q}, i.e. that for every map f : B → B′ of A-algebra, we have (f ⊗_{A} Q) ∘ λ_{B} = λ_{B′} ∘ (f ⊗_{A} N). To this aim, we are easily reduced to showing the same identity for the map g := f ∘ ε_{B} : P_{B}[0] → B′; but since P_{B}[0] is a free polynomial A-algebra, we may find a map of A-algebras h : P_{B}[0] → P_{B′}[0] such that g = ε_{B′} ∘ h. Then, by a simple diagram chase, we are further reduced to checking the sought identity for the map h; but the latter holds by assumption, since both P_{B}[0] and P_{B′}[0] are free polynomial A-algebras. ♢

Next, we show that the proposition holds for ℬ = ℬ_{0}. In view of claim 9.5.7, we are reduced to checking that the corresponding restriction map Pol\underline{d} ; A(\underline{M}, Q)_{ℬ_{1}} → Pol\underline{d} ; A(\underline{M}, Q)_{ℬ_{0}} is bijective. To this aim, let us fix, for every free polynomial A-algebra B, a fitered system (B_{i} | i ∈ I_{B}) of free polynomial A-subalgebras of B whose colimit is B. Now, if λ is any element of Pol\underline{d} ; A(\underline{M}, Q)_{ℬ_{1}} , it is easily seen that λ_{B} must be the colimit of the induced system of maps of sets (λ_{B_{i}} | i ∈ I_{B}). This shows already that our restriction map is injective. Lastly, let λ be any element of Pol\underline{d} ; A(\underline{M}, Q)_{ℬ_{0}} ; then conversely we may define, for every free polynomial A-algebra B, a map λ_{B} : F\underline{M}B → G_{Q}B, by taking the colimit of the system of maps (λ_{B_{i}} | i ∈ I_{B}). It is easily seen that this map λ_{B} fulfills condition (9.5.2), for every such B : the details shall be left to the reader. It remains to check that the rule B ↦ λ_{B} defines a natural transformation. Hence, let f : B → B′ be a map of free polynomial A-algebras; we need to show that (f ⊗_{A} Q) ∘ λ_{B} = λ_{B′} ∘ (f ⊗_{A} N). However, let (g_{i} : B_{i} → B | i ∈ I_{B}) and (g′_{j} : B′_{j} → B′ | j ∈ I_{B′}) be the universal cocones; we are easily reduced to checking that the same identity holds for the map f ∘ g_{i} : B_{i} → B′, for every i ∈ I_{B}. But for every such i we may find j ∈ I_{B′} such that f ∘ g_{i} factors through g′_{j} and a map h : B_{i} → B′_{j} of A-algebras, and we are further reduced to checking the sought identity for the map h. The latter holds by assumption, since B_{i}, B′_{j} ∈ Ob(ℬ_{0}).

Now, let ℬ be any full subcategory of A-Alg containing ℬ_{0}; by the foregoing, it is clear that the restriction map Pol\underline{d} ; A(\underline{M}, Q) → Pol\underline{d} ; A(\underline{M}, Q)_{ℬ} is injective. Lastly, let λ be any element of Pol\underline{d} ; A(\underline{M}, Q)_{ℬ} , and denote by λ_{0} its image in Pol\underline{d} ; A(\underline{M}, Q)_{ℬ_{0}} ; by the foregoing, λ_{0} is the restriciton of some λ′ ∈ Pol\underline{d} ; A(\underline{M}, Q), and it remains to check that λ′_{B} = λ_{B} for every B ∈ Ob(ℬ). However, for every such B, and every x ∈ F\underline{M}B, we may find B_{0} ∈ Ob(ℬ_{0}), a map f : B_{0} → B of A-algebras, and y ∈ F\underline{M}B_{0} with f ⊗_{A} M(y) = x; by construction, λ_{B}(x) = λ_{B_{0}}(y) = λ′_{B_{0}}(y) = λ′_{B}(x), as required. □

Lemma 9.5.9. In the situation of definition 9.5.1, suppose that the A-module M_{i} is free of finite rank, for every i = 1, …, k. Then, there exists a natural bijection

Pol\underline{d} ; A(\underline{M}, Q) →^{∼} Q ⊗_{A} Sym\underline{d} ; A(\underline{M}^{∨}).

Proof. Indeed, fix bases (e_{i1}, …, e_{ir_{i}}) for each A-module M_{i}, and denote by (e_{i1}^{∗}, …, e_{ir_{i}}^{∗}) the dual basis of M_{i}^{∨}. For every i = 1, …, k there is a natural (injective) map of A-algebras

Sym_{A}^{•}(M_{i}^{∨}) → Sym_{A}^{•}(\underline{M}^{∨}) : s ↦ 1 ⊗ ⋯ ⊗ s ⊗ ⋯ ⊗ 1

allowing to view naturally the e_{ij}^{∗} as elements of Sym_{A}^{•}(\underline{M}^{∨}). Recall also that Sym_{A}^{•}(M_{i}^{∨}) is the free A-algebra A[e_{i1}^{∗}, …, e_{ir_{i}}^{∗}] on the set e_{i•}^{∗}, for every i = 1, …, k. Let now λ : \underline{M} ⇝ Q be a given homogeneous multipolynomial law of degree \underline{d}, and set

P_{λ}(e_{••}^{∗}) := λ_{Sym_{A}^{•}(\underline{M}^{∨})}(∑_{j=1}^{r_{1}} e_{1j} ⊗ e_{1j}^{∗}, …, ∑_{j=1}^{r_{k}} e_{kj} ⊗ e_{kj}^{∗}) ∈ Q ⊗_{A} Sym_{A}^{•}(\underline{M}^{∨}).

Notice that the terms in parenthesis are elements of M_{i} ⊗_{A} M_{i}^{∨} ⊂ M_{i} ⊗_{A} Sym_{A}^{•}(M_{i}^{∨}) that do not depend on the chosen bases, since they correspond to the identity automorphisms of M_{i}, under the natural identification M_{i} ⊗_{A} M_{i}^{∨} →^{∼} End_{A}(M_{i}). Now, let B be any A-algebra, and b_{••} := (b_{ij} | i = 1, …, k; j = 1, …, r_{i}) any sequence of elements of B. By considering the unique map of A-algebras φ_{b_{••}} : Sym_{A}^{•}(\underline{M}^{∨}) → B given by the rule : e_{ij}^{∗} ↦ b_{ij} for every i = 1, …, k and every j = 1, …, r_{i}, it is easily seen that

λ_{B}(∑_{j=1}^{r_{1}} b_{1j}e_{1j}, …, ∑_{j=1}^{r_{k}} b_{kj}e_{kj}) = P_{λ}(b_{••}) := (Q ⊗_{A} φ_{b_{••}})(P_{λ}(e_{••})) ∈ Q ⊗_{A} B.

In other words, λ is completely determined by the polynomial P_{λ}(e_{••}^{∗}) with coefficients in Q. Next, take B := Sym_{A}^{•}(\underline{M}^{∨})[Y_{1}, …, Y_{k}]; the homogeneity condition on λ implies that

P_{λ}(Y_{i}e_{i•}^{∗} | i = 1, …, k) = λ_{B}(Y_{1} · ∑_{j=1}^{r_{1}} e_{1j} ⊗ e_{1j}^{∗}, …, Y_{k} · ∑_{j=1}^{r_{k}} e_{kj} ⊗ e_{kj}^{∗}) = Y_{1}^{d_{1}} ⋯ Y_{k}^{d_{k}} · P (e_{••}^{∗})

which means that P_{λ} ∈ Q ⊗_{A} Sym\underline{d} ; A(\underline{M}^{∨}). Conversely, it is clear that any element of Q ⊗_{A} Sym\underline{d} ; A(\underline{M}^{∨}) yields a homogeneous law of degree \underline{d}, whence the lemma. □

Lemma 9.5.10. Let f_{•}, g_{•} : \underline{M}′′ → \underline{M}′ be two morphisms of (A-Mod)^{k}, and p_{•} : \underline{M}′ → \underline{M} the projection of \underline{M}′ onto \underline{M} := Coequal(f_{•}, g_{•}). Suppose that for i = 1, …, k, the map

h_{i} : M′′_{i} → M′_{i} ×_{(p_{i},p_{i})} M′_{i}  x ↦ (f_{i}(x), g_{i}(x))  for every x ∈ M′′_{i}

is surjective. Then the induced map of sets

Pol\underline{d} ; A(\underline{M}, Q) → \xymatrix{Equal (Pol\underline{d} ; A(\underline{M}′, Q) \ar[r] \ar[r] & Pol\underline{d} ; A(\underline{M}′′, Q))}

is bijective for every A-module Q and every \underline{d} ∈ ℕ^{⊕k} (notation of example 1.2.16(ii)).

Proof. For i = 1, …, k, let q_{i} : M′_{i} ⊕ M′_{i} → M_{i} be the map such that (x, y) ↦ p_{i}(x − y) for every x, y ∈ M′_{i}. Then q_{i} is a surjection with kernel M′_{i}×_{M_{i}}M′_{i}. If B is any A-algebra, it follows that the image of B ⊗_{A} (\underline{M}′ ×\underline{M} \underline{M}′) in B ⊗_{A} (\underline{M}′ ⊕ \underline{M}′) is (B ⊗_{A} \underline{M}′) ×_{B⊗_{A}\underline{M}} (B ⊗_{A} \underline{M}′), and on the other hand B ⊗_{A} h_{i} is still surjective for every i = 1, …, k, so the same holds for each of the induced maps h_{B,i} : B ⊗_{A} M′′_{i} → (B ⊗_{A} M′_{i}) ×_{B⊗_{A}M_{i}} (B ⊗_{A} M′_{i}), and clearly B ⊗_{A} p_{•} is the projection of B ⊗_{A} \underline{M}′ onto the coequalizer of B ⊗_{A} f_{•} and B ⊗_{A} g_{•}.

Now, let λ : \underline{M}′ → Q be any homogeneous polynomial law of degree \underline{d}, and B any Aalgebra. Since each h_{B,i} is surjective, it follows that λ_{B} factors through a unique map of sets μ_{B} : F\underline{M}B → B ⊗_{A} Q, and an easy inspection shows that the resulting rule : B ↦ μ_{B} is a homogeneous polynomial law of degree \underline{d}. □

Remark 9.5.11. (i) Lemma 9.5.9 shows especially that, if M_{1}, …, M_{k} are free A-modules of finite rank, the rule Q ↦ Pol\underline{d} ; A(\underline{M}, Q) yields a functor with values in essentially small sets, and it is clear that the sum of two homogeneous multipolynomial laws of degree \underline{d} is still a law of the same type; likewise, if we multiply such a law by an element of A, we get a new law of the same type. Up to isomorphism, we may therefore assume that this is a functor A-Mod → A-Mod.

(ii) Let M be a free A-module of finite rank r, and fix any basis e_{1}, …, e_{r} of M. The symmetric group S_{d} acts on the set T := {1, …, r}^{d} by permutations :

σ(i_{1}, …, i_{d}) := (i_{σ^{−1}(1)}, …, i_{σ^{−1}(d)}) for every σ ∈ S_{d} and every i_{•} ∈ T.

For every such i_{•}, let [i_{•}] ∈ S_{d}\T be the class of i_{•}, and denote by e_{[i_{•}]} ∈ Sym_{A}^{d}M be the class of e_{i_{•}} := e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{d}}. Clearly, if i_{•}, j_{•} ∈ T and [i_{•}] = [j_{•}], then e_{[i_{•}]} = e_{[j_{•}]}, so this notation is unambiguous, and it is easily seen that the system (e_{[i_{•}]} | [i_{•}] ∈ S_{d}\T) is a basis of Sym_{A}^{d}M (details left to the reader). Moreover, let N be any other free A-module of finite rank s, and fix as well a basis f_{1}, … f_{s} of N; recall that we have a natural A-linear isomorphism

(9.5.12)  ω_{M,N} : M^{∨} ⊗_{A} N^{∨} →^{∼} (M ⊗_{A} N)^{∨}.

Namely, denote by e_{1}^{∗}, …, e_{r}^{∗} and f_{1}^{∗}, …, f_{s}^{∗} the corresponding dual bases; then (e_{i} ⊗ f_{j} | i = 1, …, r; j = 1, …, s) is a basis of M ⊗_{A} N, and we let likewise ((e_{i} ⊗ f_{j}]^{∗} | i = 1, …, r; j = 1, …, s) be its dual basis. With this notation, we have

ω_{M,N}(e_{i}^{∗} ⊗ f_{j}^{∗}) = (e_{i} ⊗ f_{j})^{∗} for every i = 1, …, r and j = 1, …, s.

It is easily seen that this isomorphism is independent of the choice of bases e_{•} and f_{•}. (iii) The lemma can be rephrased by saying that, under the stated assumptions, the A-module

Γ\underline{d} ; A(\underline{M}) := (Sym\underline{d} ; A\underline{M}^{∨})^{∨}

represents the functor A-Mod → A-Mod such that Q ↦ Pol\underline{d} ; A(\underline{M}, Q) for every A-module Q.

(iv) The rule (A, M) ↦ Γ\underline{d} ; A(\underline{M}) is functorial for sequences \underline{M} of free A-modules of finite rank. Indeed, let \underline{φ} : \underline{M} → \underline{N} be a morphism of such sequences. For every degree \underline{d} of length k, the induced map Pol\underline{d} ; A(\underline{N}, Q) → Pol\underline{d} ; A(\underline{M}, Q) corresponds to a map

(9.5.13)  Sym\underline{d} ; A(\underline{N}^{∨}) ⊗_{A} Q → Sym\underline{d} ; A(\underline{M}^{∨}) ⊗_{A} Q

that can be worked out as follows. Pick bases (e_{ij} | j = 1, …, r_{i}) for N_{i} and (f_{ij} | j = 1, …, s_{i}) for M_{i}, for every i = 1, …, k, and denote as usual by (e_{ij}^{∗} | j = 1, …, r_{i}), (f_{ij}^{∗} | j = 1, …, s_{i}) the respective dual bases. Say that λ : N ⇝ Q is a given homogeneous law of degree \underline{d}, and let P_{λ}(e_{••}^{∗}) ∈ Sym\underline{d} ; A(\underline{N}^{∨}) ⊗_{A} Q be the corresponding homogeneous polynomial. By construction, the polynomial P_{λ∘\underline{φ}} ∈ Sym\underline{d} ; A(\underline{M}^{∨}) ⊗_{A} Q corresponding to λ ∘ φ is calculated as

P_{λ∘\underline{φ}}(f_{••}^{∗}) := (λ ∘ \underline{φ})_{Sym_{A}^{•}(\underline{M}^{∨})}(∑_{j=1}^{s_{i}} f_{ij} ⊗ f_{ij}^{∗} | i = 1, …, k)

= λ_{Sym_{A}^{•}(\underline{M}^{∨})}(∑_{j=1}^{s_{i}} φ_{i}(f_{ij}) ⊗ f_{ij}^{∗} | i = 1, …, k)

= λ_{Sym_{A}^{•}(\underline{M}^{∨})}(∑_{j=1}^{r_{i}} e_{ij} ⊗ φ_{i}^{∨}(e_{ij}^{∗}) | i = 1, …, k)
= P_{λ}(φ_{i}^{∨}(e_{i•}^{∗} | i = 1, …, k)).

In other words, (9.5.13) equals Sym\underline{d} ; A(\underline{φ}^{∨}) ⊗_{A} 1_{Q}, so if we let

Γ\underline{d} ; A(\underline{φ}) := Sym\underline{d} ; A(\underline{φ}^{∨})^{∨}

we obtain :

• a functor Γ\underline{d} ; A(−) from the full subcategory (A-Mod^{o})_{ff}^{k} of (A-Mod^{o})^{k} consisting of all sequences of free A-modules of finite rank, into the category of A-modules

• for every A-module Q, a well defined isomorphism of functors on this subcategory

(Q, \underline{M}) ↦ (Pol\underline{d} ; A(\underline{M}, Q) →^{∼} Hom_{A}(Γ\underline{d} ; A\underline{M}, Q) : A-Mod × (A-Mod^{o})_{ff}^{k} → A-Mod

(v) Notice that for every sequence of A-modules \underline{M} := (M_{1}, …, M_{k}), every A-multinear map φ : M_{1} × ⋯ × M_{k} → Q induces, for every A-algebra B, a B-multilinear map φ_{B} : (B ⊗_{A} M_{1}) × ⋯ × (B ⊗_{A} M_{k}) → B ⊗_{A} Q, and the rule : B ↦ φ_{B} for every such B is obviously a homogeneous polynomial law of degree 1_{k} := (1, …, 1). We obtain therefore a natural transformation of functors :

(9.5.14)  Hom_{A}(M_{1} ⊗_{A} ⋯ ⊗_{A} M_{k}, Q) → Pol_{A}^{1_{k}}(\underline{M}, Q)  for every A-module Q.

In case each M_{i} is free of finite rank, this transformation must come from an A-linear map

(9.5.15)  Γ_{A}^{1_{k}}(\underline{M}) → M_{1} ⊗_{A} ⋯ ⊗_{A} M_{k}.

On the other hand, by construction, and in view of (9.5.12), we have a natural identification

Γ_{A}^{1_{k}}(\underline{M}) = (M_{1}^{∨} ⊗_{A} ⋯ ⊗_{A} M_{k}^{∨})^{∨} →^{∼} M_{1} ⊗_{A} ⋯ ⊗_{A} M_{k}

and by inspecting the proof of lemma 9.5.9, it is easily seen that this latter map coincides with (9.5.15) : the detailed verification shall be left to the reader.

The following result extends the above observations to arbitrary sequences \underline{M} of A-modules.

Theorem 9.5.16. Let \underline{M} be any sequence as in definition 9.5.1(i). We have :

(i) Pol\underline{d} ; A(\underline{M}, Q) is an essentially small set, for every A-module Q.

(ii) There exists a functor

(9.5.17)  Γ\underline{d} ; A : (A-Mod)^{k} → A-Mod

with a natural isomorphism of A-module valued functors

Pol\underline{d} ; A(−, Q) →^{∼} Hom_{A}(Γ\underline{d} ; A(−), Q) for every A-module Q.

Proof. Suppose first that the A-modules M_{1}, …, M_{k} are finitely presented, and for i = 1, …, k, pick an A-linear surjection f_{i} : L_{i} → M_{i}, with L_{i} free of finite rank. Let g_{i} : L_{i} ⊕ L_{i} → M_{i} be the map given by the rule : (l, l′) ↦ f_{i}(l − l′) for every (l, l′) ∈ L_{i} ⊕ L_{i}; since M_{i} is finitely presented, Ker g_{i} is finitely generated, for every i = 1, …, k, so we may find a free A-module L′_{i} of finite rank, and a surjective A-linear map L′_{i} → Ker g_{i}. By composing with the inclusion Ker g_{i} → L_{i} ⊕ L_{i} and the two projections L_{i} ⊕ L_{i} → L_{i}, we deduce a presentation of M_{i} :

(9.5.18)  \xymatrix{Coker (h_{i}^{1}, h_{i}^{2} : L′_{i} \ar[r] \ar[r] & L_{i}) →^{∼} M_{i}}

such that the induced map L′_{i} → L_{i} ×_{M_{i}} L_{i} is surjective. Set \underline{L} := (L_{1}, …, L_{k}), \underline{L}′ := (L′_{1}, …, L′_{k}) and \underline{h}^{j} := (h_{1}^{j}, …, h_{k}^{j}) for j = 1, 2. By lemma 9.5.10, the presentations (9.5.18) induce a presentation of sets :

Pol\underline{d} ; A(\underline{M}, Q) →^{∼} \xymatrix{Equal (Pol\underline{d} ; A(\underline{L}, Q) \ar[r] \ar[r] & Pol\underline{d} ; A(\underline{L}′, Q))} for every A-module Q.

Combining with lemma 9.5.9, it follows easily that the theorem holds for \underline{M}, with

\xymatrix{Γ\underline{d} ; A(\underline{M}) := Coequal (Γ\underline{d} ; A(\underline{h}^{1}), Γ\underline{d} ; A(\underline{h}^{2}) : Γ\underline{d} ; A(\underline{L}′) \ar[r] \ar[r] & Γ\underline{d} ; A(\underline{L}))}

(details left to the reader). Lastly, let \underline{M} be an arbitrary sequence. We may then find a filtered system of sequences (\underline{M}_{σ} | σ ∈ Σ) consisting of finitely presented A-modules and with Alinear transition maps, whose colimit is isomorphic to \underline{M}. Since such colimits are preserved by arbitrary base changes, and commute with the forgetful functor to sets, it follows easily that the system induces an isomorphism of sets :

Pol\underline{d} ; A(\underline{M}, Q) →^{∼} lim_{σ∈Σ} Pol\underline{d} ; A(\underline{M}_{σ}, Q) for every A-module Q

(details left to the reader). In view of the foregoing, the theorem then holds for \underline{M}, with

Γ\underline{d} ; A(\underline{M}) := colim_{σ∈Σ} Γ\underline{d} ; A(\underline{M}_{σ}).

Again, we invite the reader to spell out the details. □

9.5.19. The identity map of Γ\underline{d} ; A\underline{M} corresponds to a homogeneous multipolynomial law

λ\underline{d} ; \underline{M} : \underline{M} ⇝ Γ\underline{d} ; A\underline{M}

such that every other law \underline{M} ⇝ Q homogeneous of the same degree, factors uniquely through λ\underline{d} ; \underline{M} and an A-linear map Γ\underline{d} ; A\underline{M} → Q (in the sense explained in remark 9.5.3(i)). Especially, if \underline{M}′ ∈ Ob((A-Mod)^{k′}) is another sequence, and \underline{d}′ ∈ ℕ^{⊕k′} any sequence of integers, the tensor product λ\underline{d} ; \underline{M} ⊗ λ\begin{matrix}\underline{d} & ′ \\ \underline{M} & ′ \end{matrix} (remark 9.5.3(ii)) factors uniquely through λ_{\underline{M},\underline{M}′}^{\underline{d},\underline{d}′} and an A-linear map

(9.5.20)  Γ_{A}^{\underline{d},\underline{d}′}(\underline{M}, \underline{M}′) → Γ\underline{d} ; A(\underline{M}) ⊗_{A} Γ_{A}^{\underline{d}′}(\underline{M}′).

Furthermore, in the situation of remark 9.5.3(iii), the bijection (9.5.4) translates as a bijection :

Hom_{B}(Γ\underline{d} ; B(B ⊗_{A} \underline{M}), Q) →^{∼} Hom_{A}(Γ\underline{d} ; A\underline{M}, Q) →^{∼} Hom_{B}(B ⊗_{A} Γ\underline{d} ; A\underline{M}, Q)

natural with respect to B-linear maps Q → Q′ and morphisms \underline{M}′ → \underline{M} in (A-Mod)^{k}. Such a natural bijection then must come from an isomorphism of B-modules

(9.5.21)  B ⊗_{A} Γ\underline{d} ; A(\underline{M}) →^{∼} Γ\underline{d} ; B(B ⊗_{A} \underline{M})  1 ⊗ λ\underline{d} ; \underline{M}(m) ↦ λ\underline{d} ; B⊗_{A}\underline{M}(1 ⊗ m)

which is natural with respect to morphisms in (A-Mod)^{k} and maps B → B′ of A-algebras.

Corollary 9.5.22. With the notation of (9.5.19), we have :

(i) The functor (9.5.17) commutes with all filtered colimits.

(ii) The map (9.5.20) is an isomorphism of A-modules, for every \underline{M}, \underline{M}′, \underline{d} and \underline{d}′.

(iii) If the A-modules M_{1}, …, M_{k} are all free (resp. finitely presented, resp. finitely generated, resp. flat), then the same holds for Γ\underline{d} ; A(\underline{M}).

(iv) In the situation of lemma 9.5.10, the induced map

\xymatrix{Coequal (Γ\underline{d} ; A(f_{•}), Γ\underline{d} ; A(g_{•}) : Γ\underline{d} ; A(\underline{M}′′) \ar[r] \ar[r] & Γ\underline{d} ; A(\underline{M}′))} →^{∼} Γ\underline{d} ; A(\underline{M})

is an isomorphism of A-modules.

(v) For every \underline{M} := (M_{1}, …, M_{k}) ∈ Ob((A-Mod)^{k}), the natural transformation (9.5.14) is an isomorphism of functors, and therefore it induces a natural A-linear isomorphism:

ω1_{k} ; \underline{M} : Γ_{A}^{1_{k}}(\underline{M}) →^{∼} M_{1} ⊗_{A} ⋯ ⊗_{A} M_{k}.

(vi) Set 0_{k} := (0, …, 0) ∈ ℕ^{⊕k}; we have a natural isomorphism of functors :

Γ_{A}^{0_{k}}(\underline{M}) →^{∼} A for every \underline{M} ∈ Ob((A-Mod)^{k})

that identifies λ0_{k} ; \underline{M} with the polynomial law \underline{M} ⇝ A : x_{•} ↦ 1 for all B ∈ Ob(A-Alg) and all x_{•} ∈ F\underline{M}(B), and identifies Γ_{A}^{0_{k}}f with 1_{A}, for every morphism f of (A-Mod)^{k}.

Proof. (i): It suffices to check that the natural map

Pol\underline{d} ; A(colim_{σ∈Σ} \underline{M}_{σ}, Q) → lim_{σ∈Σ} Pol\underline{d} ; A(\underline{M}_{σ}, Q)

is an isomorphism, for every filtered system of A-modules (\underline{M}_{σ} | σ ∈ Σ) and every A-module Q; the latter assertion is obvious (details left to the reader).

(iii): The assertion concerning free modules of finite rank, and finitely generated or finitely presented modules follows directly from the explicit construction in lemma 9.5.9 and theorem 9.5.16. The assertion for flat modules follows from the assertion for free modules of finite rank and from (i), by means of Lazard’s theorem [120, Ch.I, Th.1.2]. Moreover, from remark 9.5.11(iv), we see that if φ : \underline{M} → \underline{N} is a morphism of sequences of free A-modules of finite rank, such that φ_{i} is a split injection for every i = 1, …, n, then Γ\underline{d} ; A(φ) is also a split injective map of free A-modules; from this, and from (i), it follows easily that Γ\underline{d} ; A transforms sequences of free A-modules (of possibly infinite rank) into free A-modules : details left to the reader.

(iv) follows formally from lemma 9.5.10: details left to the reader.

(ii): Let t : Γ_{A}^{\underline{d},\underline{d}′}(\underline{M}, \underline{M}′) → Q be an A-linear map. To t we attach an A-linear map

t^{∗} : Γ\underline{d} ; A(\underline{M}) ⊗_{A} Γ_{A}^{\underline{d}′}(\underline{M}′) → Q

as follows. Set τ := t∘λ_{\underline{M},\underline{M}′}^{\underline{d},\underline{d}′} : (\underline{M}, \underline{M}′) ⇝ Q (notation of remark 9.5.3(i)). To every A-algebra B, the law τ attaches the mapping

B ⊗_{A} \underline{M} → Pol_{B}^{\underline{d}′}(B ⊗_{A} \underline{M}′, B ⊗_{A} Q) x ↦ τ_{B,x}

where (τ_{B,x})_{C}(y) := τ_{C}(1 ⊗ x, y) for every x ∈ B ⊗_{A} \underline{M}, every B-algebra C and every y ∈ C ⊗_{A} \underline{M}′. In light of (iii), the latter is the same as a mapping

B ⊗_{A} \underline{M} → Hom_{B}(B ⊗_{A} Γ_{A}^{\underline{d}′}(\underline{M}′), B ⊗_{A} Q) x ↦ t_{B,x}

and notice that, for every x ∈ B ⊗_{A} \underline{M}, the B-linear map t_{B,x} is characterized by the identity

(9.5.23)  t_{B,x}(λ_{\underline{M}′,B}^{\underline{d}′}(y)) = (τ_{B,x})_{B}(y)  for every y ∈ B ⊗_{A} \underline{M}′.

We deduce a compatible system of mappings

B ⊗_{A} Γ_{A}^{\underline{d}′}(\underline{M}′) → Pol\underline{d} ; B(B ⊗_{A} \underline{M}, B ⊗_{A} Q) : y ↦ τ_{B,y}^{∗} for every A-algebra B

where (τ_{B,y}^{∗})_{C}(x) := t_{C,x}(1 ⊗ y) for every B-algebra C, every x ∈ C ⊗_{A} \underline{M}, and every y ∈ B ⊗_{A} Γ_{A}^{\underline{d}′}(\underline{M}′). The C-linearity of t_{C,x} implies that these latter mappings are B-linear (for every A-algebra B), and then they are the same as a compatible system of B-linear maps

(9.5.24)  B ⊗_{A} Γ_{A}^{\underline{d}′}(\underline{M}′) → Hom_{B}(B ⊗_{A} Γ\underline{d} ; A(\underline{M}), B ⊗_{A} Q) : y ↦ t_{B,y}^{∗}

for every A-algebra B. Notice that, for every y ∈ B ⊗_{A} Γ_{A}^{\underline{d}′}(\underline{M}′), the B-linear map t_{B,y}^{∗} is characterized by the identity :

(9.5.25)  t_{B,y}^{∗}(λ\underline{d} ; \underline{M},B(x)) = (τ_{B,y}^{∗})_{B}(x)  for every x ∈ B ⊗_{A} \underline{M}.

Obviously, the datum of a compatible system (9.5.24) is the same as that of a map t^{∗} as sought.

Claim 9.5.26. t^{∗} ∘ (9.5.20) = t.

Proof of the claim. It suffices to show that t^{∗} ∘ (9.5.20) ∘ λ_{\underline{M},\underline{M}′}^{\underline{d},\underline{d}′} = τ. However, recall that

(9.5.20)∘λ_{\underline{M},\underline{M}′}^{\underline{d},\underline{d}′} = λ\underline{d} ; \underline{M} ⊗λ\begin{matrix}\underline{d} & ′ \\ \underline{M} & ′ \end{matrix}, so we are reduced to checking that t^{∗} ∘(λ\underline{d} ; \underline{M} ⊗λ\begin{matrix}\underline{d} & ′ \\ \underline{M} & ′ \end{matrix}) = τ. Hence, let B be any A-algebra, and m ∈ B ⊗_{A} \underline{M}, m′ ∈ B ⊗_{A} \underline{M}′ any two elements. To ease notation, set x := λ\underline{d} ; \underline{M}(m) and y := λ\begin{matrix}\underline{d} & ′ \\ \underline{M} & ′ \end{matrix}(m′). We compute :

(t^{∗} ∘ (λ\underline{d} ; \underline{M} ⊗ λ\begin{matrix}\underline{d} & ′ \\ \underline{M} & ′ \end{matrix}))_{B}(m, m′) = (B ⊗_{A} t^{∗})(x ⊗ y)
= t_{B,y}^{∗}(x)
= (τ_{B,y}^{∗})_{B}(m)  (by (9.5.25))
= t_{B,m}(y)
= (τ_{B,m})_{B}(m′)  (by (9.5.23))
= τ_{B}(m, m′)

as claimed. ♢

Especially, if we let t be the identity map of Γ_{A}^{\underline{d},\underline{d}′}(\underline{M}, \underline{M}′), claim 9.5.26shows that the resulting t^{∗} is a left inverse for (9.5.20). Now, if all the modules M_{1}, …, M′_{k′} are free of finite rank, then lemma 9.5.9shows that both of the A-modules appearing in (9.5.20) are free and have the same rank, so the assertion follows, in this case.

Next, suppose that each A-module M_{i} and M′_{i} is finitely presented; pick for i = 1, …, k a finite presentation for M_{i} as in (9.5.18), and let \underline{L} := (L_{1}, …, L_{k}) and \underline{L}′ := (L′_{1}, …, L′_{k}). By (iv) and naturality of (9.5.20), we get an induced commutative diagram with exact rows :

\xymatrix{Γ\underline{d} ; A(\underline{L}′, \underline{M}′) \ar[r] \ar[r] \ar[d] & Γ\underline{d} ; A(\underline{L}, \underline{M}′) \ar[r] \ar[d] & Γ\underline{d} ; A(\underline{M}, \underline{M}′) \ar[r] \ar[d] & 0 \\ Γ\underline{d} ; A(\underline{L}′) ⊗_{A} Γ\underline{d} ; A(\underline{M}′) \ar[r] \ar[r] & Γ\underline{d} ; A(\underline{L}) ⊗_{A} Γ\underline{d} ; A(\underline{M}′) \ar[r] & Γ\underline{d} ; A(\underline{M}) ⊗_{A} Γ\underline{d} ; A(\underline{M}′) \ar[r] & 0}

so it suffices to prove the assertion for (\underline{L}, \underline{M}′) and (\underline{L}′, \underline{M}′). Arguing likewise with a presentation for \underline{M}′, we reduce to the case where \underline{M} and \underline{M}′ consist of free A-modules of finite rank, which has already been treated. Lastly, if \underline{M} and \underline{N} are arbitrary sequences, we may present them as filtered colimits of sequences of finitely presented A-modules; since tensor products commute with such colimits, we are done, by the foregoing case (details left to the reader).

(v): First, by a direct inspection of the constructions we get a commutative diagram :

\xymatrix{Γ_{A}^{1_{k}}(\underline{M}) \ar[rr] \ar@{-->}[dr]_{ω1_{k} ; \underline{M}} &  & Γ_{A}^{1}M_{1} ⊗_{A} ⋯ ⊗_{A} Γ_{A}^{1}M_{k} \ar[dl]^{ω_{M_{1}}^{1}⊗_{A}···⊗_{A}ω_{M_{k}}^{1}} \\  & M_{1} ⊗_{A} ⋯ ⊗_{A} M_{k}}

whose top horizontal arrow is obtained from the isomorphisms (9.5.20) provided by (ii). Hence, we are reduced to the case where k = 1, so \underline{M} is a single A-module M. Next, by virtue of (i), we may assume that M is an A-module of finite presentation, in which case we may pick a presentation (9.5.18) of M by free A-modules of finite rank, and by virtue of (iv) we are then further reduced to the case where M is free of finite rank, in which case the assertion is already known by remark 9.5.11(v).

(vi): Let λ : \underline{M} ⇝ Q be any homogeneous polynomial law of degree 0_{k}; then :

λ_{B}(x_{•}) = 0^{0} ⋯ 0^{0} · λ_{B}(x_{•}) = λ_{B}(0x_{1}, …, 0x_{k}) = λ_{B}(0, …, 0)

for every A-algebra B and every x_{•} ∈ F\underline{M}(B). In other words, λ_{B} is the constant map with value λ_{A}(0, …, 0) for every such B. We have then a natural isomorphism of functors :

Pol_{A}^{0_{k}}(\underline{M}, Q) →^{∼} Q →^{∼} Hom_{A}(A, Q) for every A-module Q

which must come from an isomorphism Γ_{A}^{0_{k}} \underline{M} →^{∼} A, and the stated identifications of λ0_{k} ; \underline{M} and Γ_{A}^{0_{k}}f follow by direct inspection. □

Remark 9.5.27. (i) Let \underline{M} and \underline{d} be as in (9.5.19), suppose that all the A-modules M_{i} are free of finite rank, and set S(\underline{M}) := Sym_{A}^{•}(\underline{M}^{∨}). Let Q be another A-module, and λ : \underline{M} ⇝ Q a homogeneous multipolynomial law of degree \underline{d}. From lemma 9.5.9 we may extract the following calculation for the A-linear map λ^{†} : Γ\underline{d} ; A(\underline{M}) → Q corresponding to λ. Set

P_{λ} := λ_{S(\underline{M})}(1_{M_{1}}, …, 1_{M_{k}}) ∈ Γ\underline{d} ; A(\underline{M})^{∨} ⊗_{A} Q.

Then λ^{†} is the image of P_{λ} under the natural identification Γ\underline{d} ; A(\underline{M})^{∨}⊗_{A}Q →^{∼} Hom_{A}(Γ\underline{d} ; A(\underline{M}), Q).

(ii) Let \underline{M}, \underline{M}′, \underline{d} and \underline{d}′ be as in (9.5.19), suppose that all the A-modules M_{i} and M′_{i} are free of finite rank, and to ease notation let

λ := λ\underline{d} ; \underline{M}  λ′ := λ\begin{matrix}\underline{d} & ′ \\ \underline{M} & ′ \end{matrix}  λ := λ ⊗ λ′.

By (i), we compute

P_{λ} = λ_{S(\underline{M},\underline{M}′)}(1_{M_{1}}, …, 1_{M_{k}}) ⊗ λ′_{S(\underline{M},\underline{M}′)}(1_{M′_{1}}, …, 1_{M′_{k′}}).

However, λ_{S(\underline{M},\underline{M}′)}(1_{M_{1}}, …, 1_{M_{k}}) is the image of P_{λ} under the A-linear map induced by the A-algebra homomorphism S(\underline{M}) → S(\underline{M}, \underline{M}′) = S(\underline{M})⊗_{A}S(\underline{M}′) given by the rule s ↦ s⊗1, for every s ∈ S(\underline{M}). Likewise we determine λ′_{S(\underline{M},\underline{M}′)}(1_{M′_{1}}, …, 1_{M′_{k′}}), and we conclude that P_{λ} = P_{λ} ⊗ P_{λ}′, whence λ^{†} = λ^{†} ⊗ λ^{′†}, under the natural identification

End_{A}(Γ_{A}^{\underline{d},\underline{d}′}(\underline{M}, \underline{M}′)) →^{∼} End_{A}(Γ\underline{d} ; A(\underline{M})) ⊗_{A} End_{A}(Γ_{A}^{\underline{d}′}(\underline{M}′)).

But by definition, (λ\underline{d} ; \underline{M})^{†} is the identity automorphism of Γ\underline{d} ; A(\underline{M}), for every sequence \underline{M}, and every degree \underline{d}. We see therefore that :

(λ\underline{d} ; \underline{M} ⊗ λ\begin{matrix}\underline{d} & ′ \\ \underline{M} & ′ \end{matrix})^{†} = (λ_{\underline{M},\underline{M}′}^{\underline{d},\underline{d}′})^{†}

which translates as saying that the A-linear map (9.5.20) is none else than the transpose of the natural isomorphism

Sym\underline{d} ; A(\underline{M}^{∨}) ⊗_{A} Sym_{A}^{\underline{d}′}(\underline{M}′^{∨}) →^{∼} Sym_{A}^{\underline{d},\underline{d}′}(\underline{M}^{∨}, \underline{M}′^{∨}).

This gives an alternative proof of corollary 9.5.22(ii), in the case of free A-modules.

9.5.28. Let A be a ring, and d ∈ ℕ. For every A-module M, the symmetric group S_{d} acts on ⊗_{A}^{d}M by permuting the tensor factors :

σ(m_{1} ⊗⋯⊗m_{d}) := m_{σ^{−1}(1)} ⊗⋯⊗m_{σ^{−1}(d)} for every σ ∈ S_{d} and every m_{1}, …, m_{d} ∈ M

(here ⊗_{A}^{0}M = A for every such M, and S_{0} is the trivial group with one element). We set :

TS_{A}^{d}M := (⊗_{A}^{d}M)^{S_{d}}.

Clearly every A-linear map f : M → M′ induces an A-linear map ⊗_{A}^{d}(f) : ⊗_{A}^{d}M → ⊗_{A}^{d}M′ such that m_{1} ⊗ ⋯ m_{d} ↦ f(m_{1}) ⊗ ⋯ ⊗ f(m_{d}) for every m_{1}, …, m_{d} ∈ M, and ⊗_{A}^{d}(f) restricts to a map TS_{A}^{d}(f) : TS_{A}^{d}M → TS_{A}^{d}M′, so we get a well defined functor

TS_{A}^{d} : A-Mod → A-Mod for every d ∈ ℕ and every ring A.

Moreover, notice that the A-linear map

⊗_{A}^{d}M → TS_{A}^{d}M x ↦ ∑_{σ∈S_{d}} σ(x)

factors through an A-linear map

α_{M}^{d} : Sym_{A}^{d}M → TS_{A}^{d}M.

Let furthermore β_{M}^{d} : TS_{A}^{d}M → Sym_{A}^{d}M be the restriction of the projection ⊗_{A}^{d}M → Sym_{A}^{d}M.

Lemma 9.5.29. (i) For every flat A-algebra B, the natural map

B ⊗_{A} TS_{A}^{d}M → TS_{B}^{d}(B ⊗_{A} M)

is an isomorphism.

(ii) If M is free of finite rank r, then TS_{A}^{d}M is a free direct factor of ⊗_{A}^{d}M of rank \binom{r+d−1}{d}. (iii) α_{M}^{d} ∘ β_{M}^{d} = d! · 1_{TS_{A}^{d}M} and β_{M}^{d} ∘ α_{M}^{d} = d! · 1_{Sym_{A}^{d}M}. (iv) Let e ∈ ℕ be another integer; if M is a flat A-module, the same holds for TS_{A}^{d}M, and

there exists a subgroup W ⊂ S_{de} such that the natural map

TS_{A}^{e}(TS_{A}^{d}M) → ⊗_{A}^{e}(⊗_{A}^{d}M) →^{∼} ⊗_{A}^{de}M

identifies TS_{A}^{e}(TS_{A}^{d}M) with the submodule (⊗_{A}^{de}M)^{W} of ⊗_{A}^{de}M.

Proof. (i) and (iii) shall be left to the reader.

(ii): Fix a basis e_{1}, …, e_{r} of M; then ⊗_{A}^{d}M is free with basis (e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{d}} | i_{1}, …, i_{d} ∈ {1, …, j}). For every i_{•} ∈ T := {1, …, r}^{d}, let also S_{d}(i_{•}) ⊂ T be the orbit of i_{•} under the S_{d}-action defined as in remark 9.5.11(ii); moreover, set e_{i_{•}} := e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{d}} for every such i_{•}. Choose a set of representatives Δ for the quotient S_{d}\T; it is then easily seen that TS_{A}^{d}M is free with basis :

[e_{i_{•}}] := ∑_{j_{•}∈S_{d}(i_{•})}e_{j_{•}} for all i_{•} ∈ Δ

and the assertion follows by a direct computation that we leave to the reader.

(iv): In order to check that TS_{A}^{d}M is a flat A-module, since the functor TS_{A}^{d} commutes with filtered colimits, by Lazard’s theorem ([120, Ch.I, Th.1.2]) we are reduced to the case where M is a free A-module of finite rank, in which case the assertion follows from (ii). Let S_{d,e} be the group of permutations of Σ := {1, …, d} × {1, …, e} (of course, this is isomorphic to S_{de}). We have injective group homomorphisms :

S_{d}^{e} ⟶^{f} S_{d,e} ⟵^{g} S_{e}

where f assigns to every (σ_{1}, …, σ_{e}) ∈ S_{d}^{e} the permutation such that (a, b) ↦ (σ_{b}(a), b) for every (a, b) ∈ Σ, and g assigns to every ψ ∈ S_{e} the permutation such that (a, b) ↦ (a, ψ(b)) for every such (a, b). Clearly Im(f)∩Im(g) = {1_{Σ}}, and for every ψ ∈ S_{e}, the inner automorphism of S_{d,e} given by conjugation by g(ψ) leaves Im(f) invariant. Thus, we can combine f and g into an injective group homomorphism

W := S_{d}^{e} ⋊ S_{e} → S_{d,e} (σ, ψ) ↦ f(σ) ∘ g(ψ).

Claim 9.5.30. For i = 1, 2, let M_{i} be an A-module with a finite group Γ_{i} of A-linear automorphisms. If M_{i} and M_{3−i}^{Γ_{3−i}} are flat A-modules for either i = 1 or i = 2, the natural map

M_{1}^{Γ_{1}} ⊗_{A} M_{2}^{Γ_{2}} → (M_{1} ⊗_{A} M_{2})^{Γ_{1}×Γ_{2}}

is an isomorphism.

Proof of the claim. Say that M_{2} and M_{1}^{Γ_{1}} are flat A-modules. We have :

(M_{1} ⊗_{A} M_{2})^{Γ_{1}×Γ_{2}} = ((M_{1} ⊗_{A} M_{2})^{Γ_{1}×1_{M_{2}}})^{1_{M_{1}}×Γ_{2}}

and the natural map (M_{1} ⊗_{A} M_{2})^{Γ_{1}×1_{M_{2}}} → M_{1}^{Γ_{1}} ⊗_{A} M_{2} (resp. (M_{1}^{Γ_{1}} ⊗_{A} M_{2})^{1_{M_{1}}×Γ_{2}} → M_{1}^{Γ_{1}} ⊗_{A} M_{2}^{Γ_{2}}) is an isomorphism, since M_{2} (resp. M_{1}^{Γ_{1}}) is a flat A-module. The assertion follows in this case; one argues likewise in case M_{1} and M_{2}^{Γ_{2}} are flat A-modules. ♢

Since TS_{A}^{d}M is a flat A-module, by claim 9.5.30, and a simple induction on e, we compute :

TS_{A}^{e}(TS_{A}^{d}M) = (⊗_{A}^{e}(⊗_{A}^{d}M)^{S_{d}})^{S_{e}} = ((⊗_{A}^{e}(⊗_{A}^{d}M))^{S_{d}^{e}})^{S_{e}} = ((⊗_{A}^{e}(⊗_{A}^{d}M))^{W}

as stated. □

9.5.31. For every d > 0 and every A-module M set M^{(d)} := (M, …, M) ∈ Ob((A-Mod)^{d}), and for every \underline{e} ∈ ℕ^{⊕d} let |\underline{e}| := e_{1} + ⋯ + e_{k}; we have a natural transformation of functors :

Pol\underline{e} ; A(M^{(d)}, Q) → Pol_{A}^{|\underline{e}|}(M, Q) for every A-module Q

that assigns to every homogeneous polynomial law λ : M^{(d)} ⇝ Q of degree \underline{e} the polynomial law λ^{†} : M ⇝ Q such that

λ_{B}^{†}(x) = λ_{B}(x, …, x) for every A-algebra B and every x ∈ B ⊗_{A} M.

In view of corollary 9.5.22(ii), this transformation then comes from an A-linear map

(9.5.32)  Γ_{A}^{|\underline{e}|}M → Γ\underline{e} ; AM^{(d)} →^{∼} Γ_{A}^{e_{1}}M ⊗_{A} ⋯ ⊗_{A} Γ_{A}^{e_{d}}M

which is characterized as the unique A-linear map such that

(λ\underline{e} ; M^{(d)})_{B}(x) ↦ (λ_{M}^{e_{1}})_{B}(x) ⊗ ⋯ ⊗ (λ_{M}^{e_{d}})_{B}(x)

for every A-algebra B and every x ∈ B ⊗_{A} M.

9.5.33. Consider next the special case where \underline{e} = (e, …, e) for some e ∈ ℕ; then, every σ ∈ S_{d} determines a natural isomorphism of functors :

Pol\underline{e} ; A(M^{(d)}, Q) →^{∼} Pol\underline{e} ; A(M^{(d)}, Q)

that assigns to every λ as in the foregoing, the polynomial law σ(λ) : M^{(d)} ⇝ Q such that

σ(λ)_{B}(x_{•}) = λ_{B}(x_{σ(1)}, …, x_{σ(d)}) for every A-algebra B and every x_{•} ∈ (B ⊗_{A} M)^{⊕d}.

We obtain in this fashion a natural S_{d}-action on the functor Pol\underline{e} ; A(M^{(d)}, −), which clearly translates as the permutation action on Γ\underline{e} ; A(M^{(d)}) →^{∼} ⊗_{A}^{d}(Γ_{A}^{e}M), as defined in (9.5.28). Now, obviously for every homogeneous polynomial law λ : M^{(d)} ⇝ Q and every σ ∈ S_{d} we have σ(λ)^{†} = λ^{†}. This implies that (9.5.32) factors through an A-linear map :

t_{M}^{d,e} : Γ_{A}^{de}M → TS_{A}^{d}(Γ_{A}^{e}M) for every e ∈ ℕ and every d > 0.

By corollary 9.5.22(v), the map t_{M}^{d,1} can be regarded as an A-linear map

t_{M}^{d} : Γ_{A}^{d}M → TS_{A}^{d}M

and in view of lemma 9.5.29(i) and proposition 9.5.6, the corresponding homogeneous polynomial law τ_{M}^{d} : M ⇝ TS_{A}^{d}M of degree d is characterized by the identities :

(τ_{M}^{d})_{B}(x) = x⊗⋯⊗x ∈ TS_{B}^{d}(B⊗_{A}M) for every flat A-algebra B and every x ∈ B ⊗_{A} M.

Proposition 9.5.34. Let e, d ∈ ℕ with d > 0, and M a flat A-module.

(i) The map t_{M}^{d} is an isomorphism.

(ii) We have a commutative diagram :

\xymatrix{Γ_{A}^{de}M \ar[r]^{t_{M}^{d,e}} \ar[d]_{t_{M}^{de}} & TS_{A}^{d}(Γ_{A}^{e}M) \ar[d]^{TS_{A}^{d}(t_{M}^{e})} \\ TS_{A}^{de}M \ar[r]^{j} & TS_{A}^{d}(TS_{A}^{e}M)}

where j is the inclusion map (we identify TS_{A}^{d}(TS_{A}^{e}M) with (⊗_{A}^{de}M)^{W} as in lemma 9.5.29(iv)). (iii) There exists a natural A-linear map α_{M}^{d,e} : TS_{A}^{d}(Γ_{A}^{e}M) → Γ_{A}^{de}M such that :

α_{M}^{d,e} ∘ t_{M}^{d,e} = \frac{(de)!}{(e!)^{d} · d!} · 1_{Γ_{A}^{de}M}.

Proof. (i): Since both Γ_{A}^{d} and TS_{A}^{d} commute with filtered colimits of A-modules, Lazard’s theorem ([120, Ch.I, Th.1.2]) reduces to the case where M is free of finite rank r; fix a basis e_{1}, …, e_{r} of M. By (9.5.12) and a simple induction, we obtain a natural A-linear isomorphism:

(9.5.35)  ⊗_{A}^{d}(M^{∨}) →^{∼} (⊗_{A}^{d}M)^{∨}  e_{i_{1}}^{∗} ⊗ ⋯ ⊗ e_{i_{d}}^{∗} ↦ (e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{d}})^{∗}.

For such M, the split inclusion map TS_{A}^{d}M → ⊗_{A}^{d}M induces a surjection

(9.5.36)  (⊗_{A}^{d}M)^{∨} → (TS_{A}^{d}M)^{∨}

and taking into account remark 9.5.11(iii), it is easily seen that the composition of (9.5.35) and (9.5.36) factors through an A-linear surjection

(9.5.37)  (Γ_{A}^{d}M)^{∨} →^{∼} Sym_{A}^{d}(M^{∨}) → (TS_{A}^{d}M)^{∨}.

By lemma 9.5.29(ii), the A-modules Sym_{A}^{d}(M^{∨}) and (TS_{A}^{d}M)^{∨} are free of the same rank, so (9.5.37) is an isomorphism, and it suffices to check that the transpose of t_{M}^{d} is an inverse of (9.5.37). However, set B := Sym_{A}^{•}(M^{∨}), and recall that t_{M}^{d} is the image of P (e_{•}^{∗}) := (τ_{M}^{d})_{B}(1_{M}) ∈ Sym_{A}^{d}(M^{∨}) ⊗_{A} TS_{A}^{d}M under the natural identification :

Sym_{A}^{d}(M^{∨}) ⊗_{A} TS_{A}^{d}M →^{∼} Hom_{A}(Sym_{A}^{d}(M^{∨})^{∨}, TS_{A}^{d}M).

Set T := {1, …, r}^{d}, and fix a set of representatives Δ for S_{d}\T; we compute :

P (e_{•}^{∗}) = (τ_{M}^{d})_{B}(∑_{i=1}^{r} e_{i}^{∗} ⊗ e_{i}) = ∑_{i_{•}∈T} e_{i_{•}}^{∗} ⊗ e_{i_{•}} = ∑_{i_{•}∈Δ} e_{[i_{•}]}^{∗} ⊗ [e_{i_{•}}]

where ([e_{i_{•}}] | i_{•} ∈ Δ) is the basis of TS_{A}^{d}M exhibited in the proof of lemma 9.5.29(ii), and e_{[i_{•}]}^{∗} ∈ Sym_{A}^{d}(M^{∨}) is defined as in remark 9.5.11(ii), for every i_{•} ∈ Δ. Hence, P (e_{•}^{∗}) corresponds to the A-linear map t_{M}^{d} : Sym_{A}^{d}(M^{∨})^{∨} → TS_{A}^{d}M such that t_{M}^{d}(l) = ∑_{i_{•}∈Δ} l(e_{[i_{•}]}^{∗}) · [e_{i_{•}}] for every A-linear form l : Sym_{A}^{d}(M^{∨}) → A. Consequently, (t_{M}^{d})^{∨} : (TS_{A}^{d}M)^{∨} → Sym_{A}^{d}(M^{∨}) is the map such that (t_{M}^{d})^{∨}(l′) = ∑_{i_{•}∈Δ} l′([e_{i_{•}}]) · e_{[i_{•}]}^{∗} for every A-linear form l′ : TS_{A}^{d}M → A. Lastly, let ([e_{i_{•}}]^{∗} | i_{•}Δ) be the dual of the basis ([e_{i_{•}}] | i_{•} ∈ Δ); a simple inspection shows that (9.5.37) is given by the rule : e_{[i_{•}]}^{∗} ↦ [e_{i_{•}}]^{∗} for every i_{•} ∈ Δ, and on the other hand (t_{M}^{d})^{∨}([e_{i_{•}}]^{∗}) = e_{[i_{•}]}^{∗} for every such i_{•}, as required.

(ii): It suffices to compare the two corresponding homogeneous polynomial laws M ⇝ TS_{A}^{d}(TS_{A}^{e}M); which amounts to showing that j ∘ τ_{M}^{de} = TS^{e}(t_{M}^{e}) ∘ τ_{M}^{d,e}, where τ_{M}^{d,e} : M ⇝ TS_{A}^{d}(Γ_{A}^{e}M) is the homogeneous polynomial law corresponding to t_{M}^{d,e}. Then the assertion follows by inspecting the characterizations of t_{M}^{d,e}, t_{M}^{de} and t_{M}^{e} given in (9.5.31) and (9.5.33).

(iii): Let a_{M}^{d,e} : (⊗_{A}^{de}M)^{W} → TS_{A}^{de}M be the A-linear map such that : x ↦ ∑_{σ∈W \S_{d}} σ(x) for every x ∈ (⊗_{A}^{de}M)^{W}. It is easily seen that a_{M}^{d,e} ∘ j = \frac{(de)!}{(e!)^{d}·d!} · 1_{TS_{A}^{de}M}, where j : TS_{A}^{de}M → (⊗_{A}^{de}M)^{W} is the inclusion map. By virtue of lemma 9.5.29(iv), the maps TS_{A}^{d}(t_{M}^{e}) and t_{M}^{de} identify a_{M}^{d,e} with an A-linear map α_{A}^{d,e} : TS_{A}^{d}(Γ_{A}^{e}M) → Γ_{A}^{de}M, and the sought identity follows directly from (ii). □

9.5.38. Let \underline{M} and \underline{M}′ be two sequences of A-modules, of the same length k. We set

\underline{M} ⊗_{A} \underline{M}′ := (M_{1} ⊗_{A} M′_{1}, …, M_{k} ⊗_{A} M′_{k})

For every \underline{d} ∈ ℕ^{⊕k} we have a natural transformation of functors

Pol\underline{d} ; A(\underline{M} ⊗_{A} \underline{M}′, Q) → Pol_{A}^{\underline{d},\underline{d}}((\underline{M}, \underline{M}′), Q) for every A-module Q

that assigns to every homogeneous polynomial law λ : \underline{M} ⊗_{A} \underline{M}′ ⇝ Q of degree \underline{d} the law λ^{†} : (\underline{M}, \underline{M}′) ⇝ Q given by the rule :

λ_{B}^{†}(m_{•}, m′_{•}) := λ_{B}(m_{1} ⊗ m′_{1}, …, m_{k} ⊗ m′_{k})

for every A-algebra B and every (m_{•}, m′_{•}) ∈ F_{\underline{M},\underline{M}′}(B). This natural transformation must come from a map of A-modules Γ_{A}^{\underline{d},\underline{d}}(\underline{M}, \underline{M}′) → Γ\underline{d} ; A(\underline{M} ⊗_{A} \underline{M}′) which, by virtue of corollary 9.5.22(ii), is the same as a map natural in both \underline{M} and \underline{M}′ :

φ\underline{d} ; \underline{M},\underline{M}′ : Γ\underline{d} ; A(\underline{M}) ⊗_{A} Γ\underline{d} ; A(\underline{M}′) → Γ\underline{d} ; A(\underline{M} ⊗_{A} \underline{M}′).

Remark 9.5.39. (i) By direct inspection, we see that the map B ⊗_{A} φ\underline{d} ; \underline{M},\underline{M}′ is identified with φ\underline{d} ; B⊗_{A}\underline{M},B⊗_{A}\underline{M}′ for every A-algebra B, under the natural isomorphisms (9.5.21).

(ii) Under the identifications of (i), the map φ\underline{d} ; \underline{M},\underline{M}′ is characterized as the unique A-linear map such that

(λ\underline{d} ; \underline{M})_{B}(m) ⊗ (λ\underline{d} ; \underline{M}′)_{B}(m′) ↦ (λ\underline{d} ; \underline{M}⊗_{A}\underline{M}′)_{B}(m ⊗ m′)

for every A-algebras B, every m ∈ B⊗_{A}(M_{1}×⋯×M_{k}) and every m′ ∈ B⊗_{A}(M′_{1}×⋯×M′_{k}).

9.5.40. Let \underline{M} and \underline{M}′ be two sequences of A-modules, of the same length k, and \underline{e}, \underline{e}′ two degrees of length k. If λ : (\underline{M}, \underline{M}′) ⇝ Q is any homogeneous law of degree (\underline{e}, \underline{e}′), then we can also regard λ as a homogeneous multipolynomial law of degree \underline{e} + \underline{e}′ on the sequence \underline{M} ⊕ \underline{M}′ of length k. In this way, we obtain a natural mapping

(9.5.41)  Pol_{A}^{\underline{e},\underline{e}′}((\underline{M}, \underline{M}′), Q) → Pol_{A}^{\underline{e}+\underline{e}′}(\underline{M} ⊕ \underline{M}′, Q)  for every A-module Q

which, by transposition, corresponds to a unique A-linear map

(9.5.42)  Γ_{A}^{\underline{e}+\underline{e}′}(\underline{M} ⊕ \underline{M}′) → Γ_{A}^{\underline{e},\underline{e}′}(\underline{M}, \underline{M}′).

Fix \underline{d} ∈ ℕ^{⊕k}; in view of corollary 9.5.22(ii), we see that the sum of the maps (9.5.42) for all pairs (\underline{e}, \underline{e}′) with \underline{e} + \underline{e}′ = \underline{d}, is a well defined A-linear map

Δ\underline{d} ; \underline{M},\underline{M}′ := ⨁_{\underline{e}+\underline{e}′=\underline{d}} Δ_{\underline{M},\underline{M}′}^{\underline{e},\underline{e}′} : Γ\underline{d} ; A(\underline{M} ⊕ \underline{M}′) → ⨁_{\underline{e}+\underline{e}′=\underline{d}} Γ\underline{e} ; A(\underline{M}) ⊗_{A} Γ_{A}^{\underline{e}′}(\underline{M}′).

Proposition 9.5.43. The map Δ\underline{d} ; \underline{M},\underline{M}′ is an isomorphism, for every \underline{M}, \underline{M}′, and every degree \underline{d}.

Proof. Arguing as in the proof of corollary 9.5.22(ii), we are easily reduced to the case where \underline{M} and \underline{M}′ are sequences of free A-modules of finite rank. In this case, (9.5.42) can be computed as in remark 9.5.27(i); namely, it corresponds to the element

P ∈ Sym_{A}^{\underline{e}+\underline{e}′}((\underline{M} ⊕ \underline{M}′)^{∨}) ⊗_{A} Sym_{A}^{\underline{e},\underline{e}′}(\underline{M}^{∨}, \underline{M}′^{∨})^{∨}

determined as follows. Take Q := Γ_{A}^{\underline{e},\underline{e}′}(\underline{M}, \underline{M}′), and let λ : \underline{M} ⊕ \underline{M}′ ⇝ Γ_{A}^{\underline{e},\underline{e}′}(\underline{M}, \underline{M}′) be the image of λ_{\underline{M},\underline{M}′}^{\underline{e},\underline{e}′} under (9.5.41). To ease notation, for every sequence \underline{N} of A-modules, let S(\underline{N}) := Sym_{A}^{•}(\underline{N}^{∨}); then

P = λ_{S(\underline{M}⊕\underline{M}′)}(1_{M_{1}⊕M′_{1}}, …, 1_{M_{k}⊕M′_{k}}).

Thus, we come down to evaluating the image of 1_{M_{i}⊕M′_{i}} under the natural identification

(M_{i} ⊕ M′_{i}) ⊗_{A} S(M_{i} ⊕ M′_{i}) →^{∼} (M_{i} ⊗_{A} S(M_{i} ⊕ M′_{i})) × (M′_{i} ⊗_{A} S(M_{i} ⊕ M′_{i})).

for every i = 1, …, k. We easily find that 1_{M_{i}⊕M′_{i}} ↦ (p_{i}, p′_{i}) under this identification, where p_{i} : M_{i} ⊕ M′_{i} → M_{i} and p′_{i} : M_{i} ⊕ M′_{i} → M′_{i} are the natural projections; so finally

(9.5.44)  P = (λ_{\underline{M},\underline{M}′}^{\underline{e},\underline{e}′})_{S(\underline{M}⊕\underline{M}′)}(p_{1}, …, p_{k}, p′_{1}, …, p′_{k}).

Pick a basis (e_{i1}, …, e_{ir_{i}}) (resp. (e′_{i1}, …, e′_{ir′_{i}})) for each M_{i} (resp. M′_{i}), let (e_{i1}^{∗}, …, e_{ir_{i}}^{∗}) (resp. (e_{i1}^{′∗}, …, e_{ir′_{i}}^{′∗}))) be the dual basis, and for every = 1, …, k and every j = 1, …, r_{i} (resp. j = 1, …, r′_{i}) denote by ε_{ij} (resp. ε′_{ij}) the element of (M_{i} ⊕ M′_{i})^{∨} that agrees with e_{ij}^{∗} on M_{i} and that vanishes on M′_{i} (resp. that agrees with e_{ij}^{′∗} on M′_{i} and vanishes on M_{i}). Notice that

p_{i} = ∑_{j=1}^{r_{i}} ε_{ij}^{∗} ⊗ e_{ij}  p′_{i} = ∑_{j=1}^{r′_{i}} ε_{ij}^{′∗} ⊗ e′_{ij}  for every i = 1, …, k.

With this notation, it follows that the right-hand side of (9.5.44) is the image of

(λ_{\underline{M},\underline{M}′}^{\underline{e},\underline{e}′})_{S(\underline{M},\underline{M}′)}(1_{M_{1}}, …, 1_{M_{k}}, 1_{M′_{1}}, …, 1_{M′_{k}}) = 1_{Sym_{A}^{\underline{e},\underline{e}′}(\underline{M}^{∨},\underline{M}^{′∨})}

under the natural map

(9.5.45)  S(\underline{M}, \underline{M}′) → S(\underline{M} ⊕ \underline{M}′) : e_{ij}^{∗} ↦ ε_{ij}^{∗} e_{ij}^{′∗} ↦ ε_{ij}^{′∗}.

In other words, the sought P is none else than the restriction of (9.5.45) to the direct factor Sym_{A}^{\underline{e},\underline{e}′}(\underline{M}^{∨}, \underline{M}′^{∨}), and this restriction is an isomorphism onto a direct factor of the A-module Sym_{A}^{\underline{e}+\underline{e}′}(\underline{M} ⊕ \underline{M}′). By transposition, we see that (9.5.42) is identified with the projection onto a natural direct factor of Γ_{A}^{\underline{e}+\underline{e}′}(\underline{M} ⊕\underline{M}′), and a simple inspection shows that the sum of all these projections is the identity map of the latter A-module, whence the proposition. □

Remark 9.5.46. (i) Let \underline{M}, \underline{M}′, Q and λ be as in (9.5.40). The image of λ under (9.5.41) is characterized as the unique law φ : \underline{M} ⊕ \underline{M}′ ⇝ Q of degree \underline{e} + \underline{e}′ such that φ(m, m′) = λ_{B}(m, m′) for every A-algebra B and every (m, m′) ∈ B⊗_{A}(\underline{M} ⊕\underline{M}′). In turns, φ corresponds to the unique A-linear map f : Γ_{A}^{\underline{e}+\underline{e}′}(\underline{M} ⊕ \underline{M}′) → Q such that (B ⊗_{A} f)(λ_{\underline{M}⊕\underline{M}′}^{\underline{e}+\underline{e}′})_{B}(m, m′) = λ_{B}(m, m′) for every B and (m, m′) as above. Especially, if we take λ := λ_{\underline{M},\underline{M}′}^{\underline{e},\underline{e}′}, we see that (9.5.42) is characterized as the unique A-linear map such that

(λ_{\underline{M}⊕\underline{M}′}^{\underline{e}+\underline{e}′})_{B}(m, m′) ↦ (λ_{\underline{M},\underline{M}′}^{\underline{e},\underline{e}′})_{B}(m, m′)

for every B and (m, m′) as above. Lastly, it follows that Δ_{\underline{M},\underline{M}′}^{\underline{e},\underline{e}′} is characterized as the unique A-linear map such that

(λ_{\underline{M}⊕\underline{M}′}^{\underline{e}+\underline{e}′})_{B}(m, m′) ↦ (λ\underline{e} ; \underline{M})_{B}(m) ⊗ (λ\begin{matrix}\underline{e} & ′ \\ \underline{M} & ′ \end{matrix})_{B}(m′)

for every A-algebra B and every (m, m′) ∈ B ⊗_{A} (\underline{M} ⊕ \underline{M}′).

(ii) In the same vein, let f : \underline{M} → \underline{N} be any morphism in (A-Mod)^{k}, and \underline{d} ∈ ℕ^{⊕k} any degree. Then we see that the induced map

Γ\underline{d} ; A(f) : Γ\underline{d} ; A(\underline{M}) → Γ\underline{d} ; A(\underline{N})

is characterized as the unique A-linear map such that

(λ\underline{d} ; \underline{M})_{B}(m) ↦ (λ\underline{d} ; \underline{N})_{B}(f(m))

for every A-algebra B, and every m ∈ B ⊗_{A} \underline{M} (details left to the reader).

9.5.47. Let \underline{M}, \underline{M}′, \underline{M}′′ be three sequences of A-modules of length k, and \underline{d}, \underline{d}′, \underline{d}′′ three degrees, also of lengths k. We obtain a diagram of A-linear maps :

\xymatrix{Γ_{A}^{\underline{e}+\underline{e}′+\underline{e}′′}(\underline{M} ⊕ \underline{M}′ ⊕ \underline{M}′′) \ar[r]^{Δ_{\underline{M},\underline{M}′⊕\underline{M}′′}^{\underline{e},\underline{e}′+\underline{e}′′}} \ar[d]_{Δ_{\underline{M}⊕\underline{M}′,\underline{M}′′}^{\underline{e}+\underline{e}′,\underline{e}′′}} & Γ\underline{e} ; A(\underline{M}) ⊗_{A} Γ_{A}^{\underline{e}′+\underline{e}′′}(\underline{M}′ ⊕ \underline{M}′′) \ar[d]^{Γ\underline{e} ; A(\underline{M})⊗_{A}Δ_{\underline{M}′,\underline{M}′′}^{\underline{e}′,\underline{e}′′}} \\ Γ_{A}^{\underline{e}+\underline{e}′}(\underline{M} ⊕ \underline{M}′) ⊗_{A} Γ_{A}^{\underline{e}′′}(\underline{M}′′) \ar[r]^{Δ_{\underline{M},\underline{M}′}^{\underline{e},\underline{e}′}⊗_{A}Γ_{A}^{\underline{e}′′}(\underline{M}′′)} & Γ\underline{e} ; A(\underline{M}) ⊗_{A} Γ_{A}^{\underline{e}′}(\underline{M}′) ⊗_{A} Γ_{A}^{\underline{e}′′}(\underline{M}′′).}

Proposition 9.5.48. The diagram of (9.5.47) commutes.

Proof. By remark 9.5.46(i), the map Δ_{\underline{M},\underline{M}′⊕\underline{M}′′}^{\underline{e},\underline{e}′+\underline{e}′′} is characterized as the unique one such that

(λ_{\underline{M}⊕\underline{M}′}^{\underline{e}+\underline{e}′+\underline{e}′′})_{B}(m, m′, m′′) ↦ (λ\underline{e} ; \underline{M})_{B}(m) ⊗ (λ_{\underline{M}⊕\underline{M}′}^{\underline{e}′+\underline{e}′′})_{B}(m′, m′′)

for every A-algebra B and every (m, m′, m′′) ∈ B ⊗_{A} (\underline{M} ⊕ \underline{M}′ ⊕ \underline{M}′′), and Δ_{\underline{M}′,\underline{M}′′}^{\underline{e}′,\underline{e}′′} is the unique A-linear map such that

(λ_{\underline{M}′⊕\underline{M}′′}^{\underline{e}′+\underline{e}′′})_{B}(m′, m′′) ↦ (λ\begin{matrix}\underline{e} & ′ \\ \underline{M} & ′ \end{matrix})_{B}(m′) ⊗ (λ\begin{matrix}\underline{e} & ′′ \\ \underline{M} & ′′ \end{matrix})_{B}(m′′)

for every A-algebra B and every (m′, m′′) ∈ B ⊗_{A} (\underline{M}′ ⊕ \underline{M}′′). Therefore, the composition of the top horizontal and right vertical arrows in the diagram is the unique A-linear map such that

(λ_{\underline{M}⊕\underline{M}′}^{\underline{e}+\underline{e}′+\underline{e}′′})_{B}(m, m′, m′′) ↦ (λ\underline{e} ; \underline{M})_{B}(m) ⊗ (λ\begin{matrix}\underline{e} & ′ \\ \underline{M} & ′ \end{matrix})_{B}(m′) ⊗ (λ\begin{matrix}\underline{e} & ′′ \\ \underline{M} & ′′ \end{matrix})_{B}(m′′)

for every B and (m, m′, m′′) as above. The same calculation can be carried out for the composition of the other two arrows, and clearly one obtains the same result. □

9.5.49. Let \underline{M} be any sequence of A-modules of length k, and denote by s : \underline{M} ⊕ \underline{M} → \underline{M} the addition map : (m, m′) ↦ m + m′ for every m, m′ ∈ \underline{M}. By proposition 9.5.43, we have a unique system of A-linear maps

μ\underline{e},\underline{e}′ ; \underline{M} : Γ\underline{e} ; A(\underline{M}) ⊗_{A} Γ_{A}^{\underline{e}′}(\underline{M}) → Γ_{A}^{\underline{e}+\underline{e}′}(\underline{M}) for every \underline{e}, \underline{e}′ ∈ ℕ^{⊕k}

such that, for every \underline{d} ∈ ℕ^{⊕k}, the resulting diagram :

(9.5.50)  [figure: Γ\underline{d} ; A(\underline{M}^{⊕2}); Δ\underline{d} ; \underline{M},\underline{M}; Γ\underline{d} ; A(s); ⨁_{\underline{e}+\underline{e}′=\underline{d}} Γ\underline{e} ; A(\underline{M}) ⊗_{A} Γ_{A}^{\underline{e}′}(\underline{M}); ⨁_{\underline{e}+\underline{e}′=\underline{d}} μ\underline{e},\underline{e}′ ; \underline{M}; Γ\underline{d} ; A(\underline{M}); Γ\underline{d} ; A(\underline{M})]

commutes. Consider now, for every \underline{e}, \underline{e}′, \underline{e}′′ ∈ ℕ^{⊕k} the diagram of A-linear maps :

[figure: Γ\underline{d} ; A(\underline{M}^{⊕3}); Δ\underline{d} ; \underline{M}^{⊕2},\underline{M}; Γ\underline{d} ; A(s⊕1\underline{M}); ⨁_{\underline{d}′+\underline{e}′′=\underline{d}}Γ_{A}^{\underline{d}′}(\underline{M}^{⊕2}) ⊗_{A} Γ_{A}^{\underline{e}′′}\underline{M}; ⨁_{\underline{d}′+\underline{e}′′=\underline{d}} Δ_{\underline{M},\underline{M}}^{\underline{d}′} ⊗_{A}Γ_{A}^{\underline{e}′′}\underline{M}; ⨁_{\underline{d}′+\underline{e}′′=\underline{d}}Γ_{A}^{\underline{d}′}(s)⊗_{A}Γ_{A}^{\underline{e}′′}\underline{M}; ⨁_{\underline{e}+\underline{e}′+\underline{e}′′=\underline{d}} Γ\underline{e} ; A\underline{M} ⊗_{A}Γ_{A}^{\underline{e}′} \underline{M} ⊗_{A}Γ_{A}^{\underline{e}′′} \underline{M}; ⨁_{\underline{e}+\underline{e}′+\underline{e}′′=\underline{d}}μ\underline{e},\underline{e}′ ; \underline{M}⊗_{A}Γ_{A}^{\underline{e}′′} \underline{M}; Γ\underline{d} ; A(\underline{M}^{⊕2}); Δ\underline{d} ; \underline{M},\underline{M}; Γ\underline{d} ; A(s); ⨁_{\underline{d}′+\underline{e}′′=\underline{d}}Γ_{A}^{\underline{d}′} \underline{M} ⊗_{A} Γ_{A}^{\underline{e}′′}\underline{M}; ⨁_{\underline{d}′+\underline{e}′′=\underline{d}}μ\underline{d}′,\underline{e}′′ ; \underline{M}; ⨁_{\underline{d}′+\underline{e}′′=\underline{d}}Γ_{A}^{\underline{d}′} \underline{M} ⊗_{A} Γ_{A}^{\underline{e}′′}\underline{M}; ⨁_{\underline{d}′+\underline{e}′′=\underline{d}}μ\underline{d}′,\underline{e}′′ ; \underline{M}; Γ\underline{d} ; A\underline{M}; Γ\underline{d} ; A\underline{M}; Γ\underline{d} ; A\underline{M}]

whose top left square subdiagram commutes by naturality of Δ\underline{d}, and whose other three square subdiagrams commute by definition. We also have a similar diagram, whose two top horizontal arrows are Δ\underline{d} ; \underline{M},\underline{M}^{⊕2} and ⨁_{\underline{e}+\underline{d}′=\underline{d}} Γ\underline{e} ; A\underline{M} ⊗_{A} Δ_{\underline{M},\underline{M}}^{\underline{d}′}, and whose two left (resp. right) vertical

arrows are Γ\underline{d} ; A(1\underline{M} ⊕ s) and Γ\underline{d} ; A(s) (resp. ⨁_{\underline{e}+\underline{e}′+\underline{e}′′=\underline{d}} Γ\underline{e} ; A\underline{M} ⊗_{A} μ\underline{e}′,\underline{e}′′ ; \underline{M} and ⨁_{\underline{e}+\underline{d}′=\underline{d}} μ\underline{e},\underline{d}′ ; \underline{M}). On

the other hand, clearly s ∘ (s ⊕ 1\underline{M}) = s ∘ (1\underline{M} ⊕ s); combining with proposition 9.5.48, we finally conclude that the diagram

\xymatrix{Γ\underline{e} ; A(\underline{M}) ⊗_{A} Γ_{A}^{\underline{e}′}(\underline{M}) ⊗_{A} Γ_{A}^{\underline{e}′′}(\underline{M}) \ar[r]^{μ\underline{e},\underline{e}′ ; \underline{M}⊗_{A}Γ_{A}^{\underline{e}′′}(\underline{M})} \ar[d]_{Γ\underline{e} ; A(\underline{M})⊗_{A}μ\underline{e}′,\underline{e}′′ ; \underline{M}} & Γ_{A}^{\underline{e}+\underline{e}′}(\underline{M}) ⊗_{A} Γ_{A}^{\underline{e}′′}(\underline{M}) \ar[d]^{μ\underline{e}+\underline{e}′,\underline{e}′′ ; \underline{M}} \\ Γ\underline{e} ; A(\underline{M}) ⊗_{A} Γ_{A}^{\underline{e}′+\underline{e}′′}(\underline{M}) \ar[r]^{μ\underline{e},\underline{e}′+\underline{e}′′ ; \underline{M}} & Γ_{A}^{\underline{e}+\underline{e}′+\underline{e}′′}(\underline{M})}

commutes, for every \underline{e}, \underline{e}′, \underline{e}′′ ∈ ℕ^{⊕k}. The foregoing shows that the maps μ•• ; \underline{M} define on

Γ_{A}^{•}(\underline{M}) := ⨁_{\underline{d}∈ℕ^{⊕k}} Γ\underline{d} ; A(\underline{M}).

a natural structure of associative ℕ^{⊕k}-graded A-algebra. We call Γ_{A}^{•}(\underline{M}) the divided power envelope of \underline{M}. Moreover, let σ : \underline{M} ⊕\underline{M} → \underline{M} ⊕\underline{M} be the map such that σ(m, m′) := (m′, m) for all m, m′ ∈ \underline{M}; remark 9.5.46(i) yields a commutative diagram :

\xymatrix{Γ_{A}^{\underline{e}+\underline{e}′}(\underline{M} ⊕ \underline{M}) \ar[r]^{Δ_{\underline{M},\underline{M}}^{\underline{e},\underline{e}′}} \ar[d]_{Γ_{A}^{\underline{e}+\underline{e}′}(σ)} & Γ\underline{e} ; A(\underline{M}) ⊗_{A} Γ_{A}^{\underline{e}′}(\underline{M}) \ar[d]^{ω\underline{e},\underline{e}′ ; \underline{M}} \\ Γ_{A}^{\underline{e}+\underline{e}′}(\underline{M} ⊕ \underline{M}) \ar[r]^{Δ_{\underline{M},\underline{M}}^{\underline{e}′,\underline{e}}} & Γ_{A}^{\underline{e}′}(\underline{M}) ⊗_{A} Γ\underline{e} ; A(\underline{M})}

for every \underline{e}, \underline{e}′ ∈ ℕ^{⊕k}, where ω\underline{e},\underline{e}′ ; \underline{M} is the isomorphism that swaps the two tensor factors. Since obviously s ∘ σ = s, it follows easily that :

μ\underline{e}′,\underline{e} ; \underline{M} ∘ ω\underline{e},\underline{e}′ ; \underline{M} = μ\underline{e},\underline{e}′ ; \underline{M} for every \underline{e}, \underline{e}′ ∈ ℕ^{⊕k}

which means that the algebra Γ_{A}^{•}(\underline{M}) is commutative. It is customary to use the notation

m^{[\underline{d}]} := (λ\underline{d} ; \underline{M})_{B}(m) ∈ B ⊗_{A} \underline{M}

for every A-algebra B, every m ∈ B ⊗_{A} \underline{M}, and every \underline{d} ∈ ℕ^{⊕k}. With this notation, the commutativity of (9.5.50) amounts to the identity

(9.5.51)  (a + b)^{[\underline{d}]} = ∑_{\underline{e}+\underline{e}′=\underline{d}} a^{[\underline{e}]} · b^{[\underline{e}′]}

for every A-algebra B, every a, b ∈ B ⊗_{A} \underline{M}, and every \underline{d} ∈ ℕ^{⊕k}. Clearly, the above construction yields a well defined functor

(9.5.52)  (A-Mod)^{k} → A-Alg : \underline{M} ↦ Γ_{A}^{•}(\underline{M})

and notice that, for every morphism f : \underline{M} → \underline{N} of sequences, we have

(Γ_{A}^{•}f)(a^{[\underline{d}]}) = (fa)^{[\underline{d}]} = (Γ_{A}^{•}f(a))^{[\underline{d}]}

for every degree \underline{d}, every A-algebra B, and every a ∈ B ⊗_{A} \underline{M}.

Theorem 9.5.53. With the notation of (9.5.49), we have :

(i) For every A-algebra B, there exists a natural isomorphism of ℕ^{⊕k}-graded algebras :

B ⊗_{A} Γ_{A}^{•}(\underline{M}) →^{∼} Γ_{B}^{•}(B ⊗_{A} \underline{M}).

(ii) The functor (9.5.52) commutes with filtered colimits.

(iii) For any two sequences of A-modules \underline{M} and \underline{M}′ of the same length k, there exists a natural isomorphism of ℕ^{⊕k}-graded A-algebras :

Γ_{A}^{•}(\underline{M} ⊕ \underline{M}′) →^{∼} Γ_{A}^{•}(\underline{M}) ⊗_{A} Γ_{A}^{•}(\underline{M}′)

(with the ℕ^{⊕k}-grading of the tensor product defined in the obvious way).

Proof. (i) and (ii) follow immediately from the natural isomorphisms (9.5.21) and corollary 9.5.22(i), and (iii) follows directly from proposition 9.5.43. □

9.5.54. Consider the special case of a sequence \underline{M} consisting of a single A-module M. From (9.5.51), by evaluating the expression (2m)^{[d]} = 2^{d} · m^{[d]}, a simple induction on d shows that

(9.5.55)  d! · m^{[d]} = m^{d}  in Γ_{A}^{•}(M), for every m ∈ M.

Corollary 9.5.56. In the situation of (9.5.54), we have in Γ_{A}^{•}(M) the identity :

m^{[i]} · m^{[j]} = \binom{i + j}{i} · m^{[i+j]} for every m ∈ M and every i, j ∈ ℕ.

Proof. By considering the A-linear map f : A → M such that f(1) = m, we reduce to the case where M = A and m = 1. By considering the unique ring homomorphism ℤ → A we reduce further – in light of theorem 9.5.53(i) – to the case where A = M = ℤ. However, Γ_{ℤ}^{•}(ℤ) is a torsion-free ℤ-module, by lemma 9.5.9, so the identity follows easily from (9.5.55). □

Corollary 9.5.57. Let M be an A-module, and N ⊂ M any submodule. The induced map

Γ_{A}^{•}M → Γ_{A}^{•}(M/N)

is surjection of A-algebras, with kernel generated by the image of Γ_{A}^{+}N := ⨁_{d>0} Γ_{A}^{d}N.

Proof. From corollary 9.5.22(iv) we get an isomorphism of A-algebras

\xymatrix{Coequal (Γ_{A}^{•}(j_{1}), Γ_{A}^{•}(j_{2}) : Γ_{A}^{•}(M ×_{M/N} M) \ar[r] \ar[r] & Γ_{A}^{•}M))} →^{∼} Γ_{A}^{•}(M/N)

where j_{1}, j_{2} : M ×_{M/N} M → M are the natural maps. However, the isomorphism of A-modules M ⊕N →^{∼} M ×_{M/N} M : (x, y) ↦ x+y identifies j_{1} (resp. j_{2}) with the map M ⊕N → M such that (x, y) ↦ x + y (resp. (x, y) ↦ x). Let also i_{1} : M → M ⊕ N and i_{2} : N → M ⊕ N be the inclusion maps to the first and respectively second direct summand, so that j_{2}∘i_{1} = j_{1}∘i_{1} = 1_{M}, j_{2} ∘ i_{2} = 0, and l := j_{1} ∘ i_{2} : N → M is the inclusion map. Now, by direct inspection we see that Γ_{A}^{d}(i_{1}) maps Γ_{A}^{d}M to the direct summand Γ_{A}^{d}M ⊗_{A} Γ_{A}^{0}N →^{∼} Γ_{A}^{(d,0)}(M, N), for every d ∈ ℕ, and the resulting map Γ_{A}^{d}M → Γ_{A}^{(d,0)}(M, N) is characterized as the unique one such that x^{[d]} ↦ (x, 0)^{[d,0]} for every A-algebra B and every x ∈ B ⊗_{A} M. On the other hand, by construction we have (x, y)^{[d,0]} ↦ x^{[d]} ⊗ y^{[0]} = x^{[d]} ⊗ 1 for every A-algebra B and every (x, y) ∈ B ⊗_{A} (M ⊕ N), under the natural identification B ⊗_{A} Γ_{A}^{0}N →^{∼} B (corollary 9.5.22(vi) and remark 9.5.46(i)). Likewise we may characterize Γ_{A}^{d}(i_{2}) for every d ∈ ℕ; hence, Γ_{A}^{•}(i_{1}) and Γ_{A}^{•}(i_{2}) are the maps of A-algebras :

Γ_{A}^{•}M → Γ_{A}^{•}M ⊗_{A} Γ_{A}^{•}N x ↦ x ⊗ 1. Γ_{A}^{•}N → Γ_{A}^{•}M ⊗_{A} Γ_{A}^{•}N y ↦ 1 ⊗ y.

Thus, let d ∈ ℕ and z ∈ Γ_{A}^{d}(M ⊕ N), and write z = ∑_{i=0}^{d} x_{i} ⊗ y_{d−i} with x_{i} ∈ Γ_{A}^{i}M and y_{i} ∈ Γ_{A}^{i}N for i = 0, …, d. It follows that

Γ^{•}(j_{1})(z) = ∑_{i=0}^{d} x_{i} · Γ_{A}^{d−i}(l)(y_{d−i}) and Γ^{•}(j_{2})(z) = x_{d} · Γ_{A}^{0}(l)(y_{0})

so finally, the kernel of the projection onto Γ_{A}^{•}(M/N) is the A-module generated by all products x · Γ_{A}^{i}(l)(y), for every i ∈ ℕ, every j ∈ ℕ \ {0}, every x ∈ Γ_{A}^{i}M and every y ∈ Γ_{A}^{j}N. □

9.5.58. For every d, e ∈ ℕ with d > 0, we have a natural triangle of A-linear maps :

\xymatrix{Γ_{A}^{de}M \ar[rr]^{t_{M}^{d,e}} &  & TS_{A}^{d}(Γ_{A}^{e}M) \ar[dl]^{β_{M}^{d,e}} \\  & Sym_{A}^{d}(Γ_{A}^{e}M) \ar@{-->}[ul]^{γ_{M}^{d,e}}}  for every A-module M

where β_{M}^{d,e} is the restriction of the projection ⊗_{A}^{d}(Γ_{A}^{e}M) → Sym_{A}^{d}(Γ_{A}^{e}M), the map γ_{M}^{d,e} is induced by the multiplication map of the A-algebra Γ_{A}^{•}M, and t_{M}^{d,e} is defined as in (9.5.33).

Remark 9.5.59. If e = 1, the diagram of (9.5.58) becomes the triangle of A-linear maps :

\xymatrix{Γ_{A}^{d}M \ar[rr]^{t_{M}^{d}} &  & TS_{A}^{d}M \ar[dl]^{β_{M}^{d}} \\  & Sym_{A}^{d}M \ar@{-->}[ul]^{γ_{M}^{d}}}  for every A-module M

where t_{M}^{d} and β_{M}^{d} are defined as in (9.5.33) and (9.5.28), and with γ_{M}^{d} := γ_{M}^{d,1}.

Proposition 9.5.60. (i) With the notation of (9.5.58), we have :

γ_{M}^{d,e} ∘ β_{M}^{d,e} ∘ t_{M}^{d,e} = \frac{(de)!}{(e!)^{d}} · 1_{Γ_{A}^{de}M}.

(ii) If M is a flat A-module, let α_{M}^{d,e} be the map of proposition 9.5.34(iii); we have :

γ_{M}^{d,e} ∘ β_{M}^{d,e} = d! · α_{M}^{d,e}.

(iii) Define the map α_{M}^{d} as in (9.5.28); with the notation of remark 9.5.59, we have :

γ_{M}^{d} ∘ β_{M}^{d} ∘ t_{M}^{d} = d! · 1_{Γ_{A}^{d}M}  t_{M}^{d} ∘ γ_{M}^{d} = α_{M}^{d}  β_{M}^{d} ∘ t_{M}^{d} ∘ γ_{M}^{d} = d! · 1_{Sym_{A}^{d}M}.

Proof. (i): It suffices to check that the homogeneous polynomial law M ⇝ Γ_{A}^{de}M corresponding to γ_{M}^{d,e} ∘ β_{M}^{d,e} ∘ t_{M}^{d,e} equals \frac{(de)!}{(e!)^{d}} · λ_{M}^{de}, i.e. that for every A-algebra B we have :

γ_{M}^{d,e} ∘ β_{M}^{d,e} ∘ t_{M}^{d,e}((λ_{M}^{de})_{B}(m)) = \frac{(de)!}{(e!)^{d}} · (λ_{M}^{de})_{B}(m) for every m ∈ B ⊗_{A} M.

By proposition 9.5.6, it even suffices to check the above identity for all flat A-algebras B; then, by considering the A-linear map A → M such that 1 ↦ m, and by invoking the naturality of the maps of (9.5.58), and their compatibility with the flat base change A → B ((9.5.21) and lemma 9.5.29(i)) we are then reduced to the case where M = A = B and m = 1. However, recall that if M is a flat A-module, the isomorphisms t_{M}^{de} and t_{M}^{e} identify t_{M}^{d,e} with the inclusion map TS_{A}^{de}M → (⊗_{A}^{de}M)^{W} (where W ⊂ S_{d} is the subgroup (S_{d})^{e} ⋉ S_{e}), and β_{M}^{d,e} with the restriction TS_{A}^{d}(TS_{A}^{e}M) → Sym_{A}^{d}(TS_{A}^{e}M) of the projection map ⊗_{A}^{d}(TS_{A}^{e}M) → Sym_{A}^{d}(TS_{A}^{e}M) (proposition 9.5.34); hence, we are reduced to checking that :

(9.5.61)  t_{A}^{de} ∘ γ_{A}^{d,e} ∘ Sym_{A}^{d}(t_{A}^{e})^{−1}(1 ⊗ ⋯ ⊗ 1) = \frac{(de)!}{(e!)^{d}} · (1 ⊗ ⋯ ⊗ 1).

Now, let M be more generally a free A-module of rank r, fix a basis e_{1}, …, e_{r} of M, and let e_{1}^{∗}, …, e_{r}^{∗} be the dual basis. Let also p_{1}, p_{2} : M ⊕ M → M be the two projections, and set ε_{j,i}^{∗} := e_{i}^{∗} ∘ p_{j} : M ⊕ M → A for j = 1, 2 and i = 1, …, r. Moreover, set :

l(i_{•}) := k for every k ∈ ℕ and every i_{•} ∈ T_{k} := {1, …, r}^{k}.

Let e_{[i_{•}]}^{∗} ∈ Sym_{A}^{k}(M^{∨}) be the class of e_{i_{1}}^{∗} ⊗ ⋯ ⊗ e_{i_{k}}^{∗}, for every i_{•} ∈ T_{k}, and define likewise ε_{[j,i_{•}]}^{∗} ∈ Sym_{A}^{k}((M ⊕ M)^{∨}) for j = 1, 2. Notice that Sym_{A}^{d}((M ⊕ M)^{∨}) admits the basis :

(ε_{[1,i_{•}]}^{∗} ⊗ ε_{[2,j_{•}]}^{∗} | l(i_{•}) + l(j_{•}) = d) for every d ∈ ℕ.

The proof of proposition 9.5.43 identifies Δ_{M,M}^{k,k′} : Γ_{A}^{k+k′}(M ⊕ M) → Γ_{A}^{k,k′}(M, M) with the transpose of the A-linear map

Sym_{A}^{(k,k′)}(M^{∨}, M^{∨}) → Sym_{A}^{k+k′}((M ⊕ M)^{∨}) e_{[i_{•}]}^{∗} ⊗ e_{[j_{•}]}^{∗} ↦ ε_{[1,i_{•}]}^{∗} ⊗ ε_{[2,j_{•}]}^{∗}

for every k, k′ ∈ ℕ. Furthermore, for every d ∈ ℕ choose a set of representatives Δ_{d} for the quotient S_{d}\T_{d}, define the basis ([e_{i_{•}}] | i_{•} ∈ Δ_{d}) of TS_{A}^{d}M as in the proof of lemma 9.5.29(ii), and let ([e_{i_{•}}]^{∗} | i_{•} ∈ Δ_{d}) be its dual basis. The proof of proposition 9.5.34 shows that the transpose of t_{M}^{d} is the A-linear map

(TS_{A}^{d}M)^{∨} →^{∼} Sym_{A}^{d}(M^{∨})  [e_{i_{•}}]^{∗} ↦ e_{[i_{•}]}^{∗}  for every i_{•} ∈ Δ_{d}.

Notice as well that (ε_{1,i}^{∗}, ε_{2,i}^{∗} | i = 1, …, r) is a basis of (M ⊕ M)^{∨}, and let (ε_{1,i}, ε_{2,i} | i = 1, …, r) be the dual basis of M ⊕ M; for every d ∈ ℕ and every a_{•} ∈ T′_{d} := ({1, 2} × {1, …, r})^{d} we set ε_{a_{•}} := ε_{a_{1}} ⊗ ⋯ ⊗ ε_{a_{d}} ∈ ⊗_{A}^{d}(M ⊕ M). Then, for every k, k′ ∈ ℕ with d = k + k′, and every pair of sequences i_{•} ∈ T_{k}, j_{•} ∈ T_{k′}, let ⟨i_{•}, j_{•}⟩ ∈ T′_{d} be the sequence ((1, i_{1}), …, (1, i_{k}), (2, j_{1}), … (2, j_{k′})); we may choose a set of representatives Δ′_{d} for the quotient S_{d}\T′_{d}, consisting of sequences of this form, and we obtain as in the proof of lemma 9.5.29(ii), a basis ([ε_{⟨i_{•},j_{•}⟩}] | ⟨i_{•}, j_{•}⟩ ∈ Δ′_{d}) of TS_{A}^{d}(M ⊕ M). Summing up, we conclude that the isomorphisms t_{M}^{d}, t_{M}^{k} and t_{M}^{k′} identify Δ_{M,M}^{k,k′} with the A-linear map

TS_{A}^{d}(M ⊕ M) → TS_{A}^{k}(M) ⊗_{A} TS_{A}^{k′}(M) [ε_{⟨i_{•},j_{•}⟩}] ↦ \{ \begin{matrix}[e_{i_{•}}] ⊗ [e_{j_{•}}] & if l(i_{•}) = k \\ 0 & otherwise. \end{matrix}

Next, let s : M ⊕ M → M be the addition map; by naturality of the maps t_{•}^{d}, we get a commutative diagram :

\xymatrix{Γ_{A}^{d}(M ⊕ M) \ar[r]^{t_{M⊕M}^{d}} \ar[d]_{Γ_{A}^{d}(s)} & TS_{A}^{d}(M ⊕ M) \ar[d]^{TS_{A}^{d}(s)} \ar[r] & ⊗_{A}^{d}(M ⊕ M) \ar[d]^{⊗_{A}^{d}(s)} \\ Γ_{A}^{d}M \ar[r]^{t_{M}^{d}} & TS_{A}^{d}M \ar[r] & ⊗_{A}^{d}M}

whose unlabeled horizontal arrows are the inclusion maps. It is easily seen that ⊗_{A}^{d}(s) is given by the rule : ε_{⟨i_{•},j_{•}⟩} ↦ e_{(i_{•},j_{•})} for every ⟨i_{•}, j_{•}⟩ ∈ T′_{d}, with (i_{•}, j_{•}) := (i_{1}, …, i_{k}, j_{1}, …, j_{k′}) (where k := l(i_{•}) and k′ := l(j_{•})). Then, for every k ∈ ℕ and every sequence i_{•} ∈ T_{k}, set

X(i_{•}, u) := {t ∈ {1, …, k} | i_{t} = u} for every u = 1, …, r.

Let N_{u}(i_{•}) be the cardinality of X(i_{•}, u), and for every i_{•} ∈ T_{k} and j_{•} ∈ T_{k′} define

C(i_{•}, j_{•}) := ∏_{u=1}^{r} \binom{N_{u}(i_{•}) + N_{u}(j_{•})}{N_{u}(i_{•})}.

A direct calculation that we leave to the reader then shows that TS_{A}^{d}(s) is given by the rule :

[ε_{⟨i_{•},j_{•}⟩}] ↦ C(i_{•}, j_{•}) · [e_{(i_{•},j_{•})}] for every ⟨i_{•}, j_{•}⟩ ∈ Δ′_{d}.

In view of the foregoing, we deduce that the isomorphisms t_{M}^{d}, t_{M}^{k} and t_{M}^{k′} identify the map μ_{M}^{k,k′} of (9.5.49) with the map

TS_{A}^{k}M ⊗_{A} TS_{A}^{k′}M → TS_{A}^{d}M such that [e_{i_{•}}] ⊗ [e_{j_{•}}] ↦ C(i_{•}, j_{•}) · [e_{(i_{•},j_{•})}]

for every i_{•} ∈ Δ_{k} and j_{•} ∈ Δ_{k′}. Notice also that

N_{u}(i_{•}) + N_{u}(j_{•}) = N_{u}(i_{•}, j_{•}) for every u = 1, …, r.

Hence, for every d, e ∈ ℕ with d ≥ 2, let i_{1,•}, …, i_{d,•} ∈ T_{e}, and set

\binom{a}{b_{1}, …, b_{d}} := \frac{a!}{b_{1}! ⋯ b_{d}!} for every a, b_{1}, …, b_{d} ∈ ℕ.

Define also inductively the sequence (i_{1,•}, …, i_{k,•}) ∈ T_{ke} for every k = 2, …, d as follows. For k = 2 it is already defined; for k > 2, we set (i_{1,•}, …, i_{k,•}) := ((i_{1,•}, …, i_{k−1,•}), i_{k,•}). With this notation, we conclude that the map t_{M}^{de} ∘ γ_{M}^{d,e} ∘ Sym_{A}^{d}(t_{A}^{e})^{−1} is given by the rule :

(9.5.62)  [e_{i_{1},•}] ⊗ ⋯ ⊗ [e_{i_{d},•}] ↦ ∏_{u=1}^{r} \binom{N_{u}(i_{1,•}, …, i_{d,•})}{N_{u}(i_{1}), …, N_{u}(i_{d})} · [e_{(i_{1,•},...,i_{d,•})}].

The sought identity (9.5.61) follows at once.

(ii): As usual, we are easily reduced to the case where M is free of some finite rank r, in which case the isomorphism TS_{A}^{d}(t_{M}^{e}) and lemma 9.5.29(iv) identify again β_{M}^{d,e} with a map :

(9.5.63)  (⊗_{A}^{de}M)^{W} →^{∼} TS_{A}^{d}(TS_{A}^{e}M) → Sym_{A}^{d}(TS_{A}^{e}M).

Now, fix again a basis e_{1}, …, e_{r} of M; for every i_{••} := (i_{1,•}, …, i_{d,•}) ∈ T_{e}^{d} = T_{de}, let W (i_{••}) be the orbit of i_{••} under the natural W -action on T_{de}, and set

[e_{i_{••}}]_{W} := ∑_{j_{••}∈W (i_{••})} e_{j_{••}}

(where, as usual, e_{j_{••}} := e_{j_{1,1}} ⊗ ⋯ ⊗ e_{j_{d,e}} ∈ ⊗_{A}^{de}M for every j_{••} ∈ T_{e}^{d}). Fix also a set Δ_{d,e} of representatives for the quotient W \T_{e}^{d}; it is easily seen that (⊗_{A}^{de}M)^{W} admits the basis ([e_{i_{••}}]_{W} | i_{••} ∈ Δ_{d,e}). Moreover, let [i_{1,•}], …, [i_{d,•}] ∈ S_{e}\T_{e} be the classes of the sequences i_{1,•}, …, i_{d,•}, and for every set Σ, let c(Σ) be the cardinality of Σ. With this notation, a simple inspection shows that (9.5.63) is given by the rule :

[e_{(i_{1,•},...,i_{d,•})}]_{W} ↦ c(S_{d}([i_{1,•}], …, [i_{d,•}])) · e_{[i_{1,•}]} ⊗ ⋯ ⊗ e_{[i_{d,•}]} for every i_{••} ∈ Δ_{d,e}

where S_{d}([i_{1,•}], …, [i_{d,•}]) denotes the orbit of ([i_{1,•}], …, [i_{d,•}]) ∈ (S_{e}\T_{e})^{d} under the natural S_{d}-action. Combining with (9.5.62), we see that γ_{M}^{d,e} ∘ β_{M}^{d,e} is naturally identified with the Alinear map

(9.5.64)  (⊗_{A}^{de}M)^{W} → TS_{A}^{de}M

given by the rule :

[e_{i_{••}}]_{W} ↦ c(S_{d}([i_{1,•}], …, [i_{d,•}])) · ∏_{u=1}^{r} \binom{N_{u}(i_{••})}{N_{u}(i_{1}), …, N_{u}(i_{d})} · [e_{i_{••}}] for every i_{••} ∈ Δ_{d,e}.

Let S_{de}(i_{••}) be the orbit of i_{••} under the natural S_{de}-action on T_{de}, and notice that :

[e_{i_{••}}] = ∑_{j_{••}∈W \S_{de}(i_{••})}[e_{j_{••}}]_{W}  for every i_{••} ∈ T_{d}^{e}

where W \S_{de}(i_{••}) is the quotient of S_{de}(i_{••}) under the natural W -action. Notice as well that the cardinalities of the stabilizers of i_{••} under the S_{de}-action and the S_{e}^{d}-action are respectively:

\frac{(de)!}{c(S_{de}(i_{••}))} = ∏_{u=1}^{r} N_{u}(i_{••})!  and  \frac{(e!)^{d}}{c(S_{e}^{d}(i_{••}))} = ∏_{u=1}^{r} ∏_{j=1}^{d} N_{u}(i_{j,•})!

whence :

\frac{(de)!}{(e!)^{d}} · \frac{c(S_{e}^{d}(i_{••}))}{c(S_{de}(i_{••}))} = ∏_{u=1}^{r} \binom{N_{u}(i_{••})}{N_{u}(i_{1}), …, N_{u}(i_{d})}.

On the other hand, we have : S_{d}([i_{1,•}], …, [i_{d,•}]) = S_{e}^{d}\W (i_{••}), whence :

c(S_{d}([i_{1,•}], …, [i_{d,•}])) = \frac{c(W (i_{••}))}{c(S_{e}^{d}(i_{••}))}.

Summing up, we see that (9.5.64) is given by the rule :

[e_{i_{••}}]_{W} ↦ \frac{(de)!}{(e!)^{d}} · \frac{c(W (i_{••}))}{c(S_{de}(i_{••}))} · ∑_{j_{••}∈W \S_{de}(i_{••})}[e_{j_{••}}]_{W}  for every i_{••} ∈ Δ_{d,e}.

However, for every i_{••} ∈ T_{de} we may compute :

\begin{matrix}\frac{c(W (i_{••}))}{c(S_{de}(i_{••}))} · ∑_{j_{••}∈W \S_{de}(i_{••})}[e_{j_{••}}]_{W} = \frac{1}{c(S_{de}(i_{••}))} · ∑_{j_{••}∈S_{de}(i_{••})}[e_{j_{••}}]_{W} = & \frac{1}{c(S_{de})} · ∑_{σ∈S_{de}}[e_{σ(i_{••})}]_{W} \\ = & \frac{c(W)}{c(S_{de})} · ∑_{σ∈W \S_{de}}[e_{σ(i_{••})}]_{W} \end{matrix}

and notice that c(W)/c(S_{de}) = \frac{(e!)^{d}·d!}{(de)!}. The sought identity now follows by a direct inspection of the definition of α_{M}^{d,e}.

(iii): The first stated identity is a special case of (i). Next, recall that β_{M}^{d} is the restriction of the projection π_{M}^{d} : ⊗_{A}^{d}M → Sym_{A}^{d}M; the second identity will follow immediately from :

Claim 9.5.65. t_{M}^{d} ∘ γ_{M}^{d} ∘ π_{M}^{d}(x) = ∑_{σ∈S_{d}} σ(x) for every x ∈ ⊗_{A}^{d}M.

Proof of the claim. We are easily reduced to the case where M is free of some finite rank r, and as usual we fix a basis e_{1}, …, e_{r}. In this case, π_{M}^{d} is the A-linear map such that e_{i_{•}} ↦ e_{[i_{•}]} for every i_{•} ∈ T_{d}. Moreover, by the foregoing, the map t_{M}^{d} ∘ γ_{M}^{d} is given by the rule :

e_{[i_{•}]} ↦ ∏_{u=1}^{r} N_{u}(i_{•})! · [e_{i_{•}}] = ∑_{σ∈S_{d}} e_{σ(i_{•})} for every i_{•} ∈ T_{d}.

The claim follows at once. ♢ The third stated identity follows from the second one and lemma 9.5.29(iii). □

9.5.66. Let now A be a ring, d ∈ ℕ, and for every A-algebra B and every B-module M, let μ_{B,M} : B ⊗_{A} M → B be the scalar multiplication map, i.e. the A-linear map such that b ⊗ m ↦ bm for every b ∈ B and every m ∈ M. We deduce an A-linear map

μ_{B,M}^{d} : Γ_{A}^{d}(B) ⊗_{A} Γ_{A}^{d}(M) ⟶^{φ_{B,M}^{d}} Γ_{A}^{d}(B ⊗_{A} M) ⟶^{Γ_{A}^{d}(μ_{B,M})} Γ_{A}^{d}(M)

for every such B and M, where φ_{B,M}^{d} is defined as in (9.5.38). By virtue of remarks 9.5.39(ii) and 9.5.46(ii), the system of such maps is uniquely characterized by the identities :

(9.5.67)  μ_{B,M}^{d}(b^{[d]} ⊗ m^{[d]}) = (bm)^{[d]}

for every A-algebra B, every B-module M, and every b ∈ B and m ∈ M. Especially, taking M := B, it follows easily that μ_{B,B}^{d} furnishes the multiplication map for a well defined Aalgebra structure on Γ_{A}^{d}B, for every such B and d. Moreover, μ_{B,M}^{d} is the scalar multiplication map for a well defined Γ_{A}^{d}B-module structure on Γ_{A}^{d}M, for every B-module M. Furthermore, if f : B → C is any map of A-algebras, then clearly the same holds for Γ_{A}^{d}(f), and if g : M → N is any morphism of B-modules, then Γ_{A}^{d}(g) is a Γ_{A}^{d}B-linear map. Summing up, we obtain for every d ∈ ℕ, every ring A, and every A-algebra B, natural functors :

Γ_{A}^{d} : A-Alg → A-Alg Γ_{A}^{d} : B-Mod → Γ_{A}^{d}B-Mod.

Notice also that TS_{A}^{d}B is an A-subalgebra of the A-algebra ⊗_{A}^{d}B.

Proposition 9.5.68. (i) The natural bijection (9.5.21) is an isomorphism of C-algebras

C ⊗_{A} Γ_{A}^{d}B →^{∼} Γ_{C}^{d}(C ⊗_{A} B) for every A-algebra C.

(ii) The map t_{B}^{d} : Γ_{A}^{d}B → TS_{A}^{d}B of (9.5.33) is an integral radicial morphism of A-algebras, and Ker(t_{B}^{d}) lies in the nilradical of Γ_{A}^{d}B, for every d ∈ ℕ and every A-algebra B.

(iii) If B is an A-algebra of finite type, the same holds for Γ_{A}^{d}B.

(iv) Let f : B_{1} → B_{2} be an integral map of A-algebras; we have :

(a) The maps TS_{A}^{d}f and Γ_{A}^{d}f are integral.

(b) If moreover f is radicial, the same holds for Γ_{A}^{d}f.

(c) if Ker f lies in the nilradical of B_{1}, then Ker (Γ_{A}^{d}f) lies in the nilradical of Γ_{A}^{d}B_{1}.

Proof. (i) follows easily from the characterization (9.5.67) of the map μ_{B,B}^{d}.

(ii): Again, the characterization (9.5.67) of μ_{B,B}^{d} easily implies that t_{B}^{d} is a map of A-algebras. Next, let us check that for every algebraically closed field K and every ring homomorphism A → K, the map t_{B}^{d} induces a bijection :

(9.5.69)  Hom_{A-Alg}(TS_{A}^{d}B, K) →^{∼} Hom_{A-Alg}(Γ_{A}^{d}B, K).

To this aim, set B_{K} := K ⊗_{A} B, and consider the commutative diagram

\xymatrix{K ⊗_{A} Γ_{A}^{d}B \ar[r]^{K⊗_{A}t_{B}^{d}} \ar[d] & K ⊗_{A} TS_{A}^{d}B \ar[d]^{φ} \\ Γ_{K}^{d}(B_{K}) \ar[r]^{t_{B_{K}}^{d}} & TS_{K}^{d}B_{K}}

whose left vertical arrow is the isomorphism of (i), and where φ is the restriction of the natural isomorphism K ⊗_{A} (⊗_{A}^{d}B) →^{∼} ⊗_{K}^{d}B_{K}. On the one hand, (9.5.69) is naturally identified with the analogous map Hom_{K-Alg}(TS_{K}^{d}B_{K}, K) →^{∼} Hom_{K-Alg}(Γ_{K}^{d}B_{K}, K) induced by K ⊗_{A} t_{B}, and on the other hand, t_{B_{K}}^{d} is an isomorphism, by proposition 9.5.34(i); hence, we are reduced to checking that φ induces a bijection

Hom_{K-Alg}(TS_{K}^{d}B_{K}, K) →^{∼} Hom_{K-Alg}(K ⊗_{A} TS_{A}^{d}B, K) →^{∼} Hom_{A-Alg}(TS_{A}^{d}B, K).

To this aim, we consider the commutative diagram :

\xymatrix{TS_{A}^{d}B \ar[r]^{j_{B}} \ar[d] & ⊗_{A}^{d}B \ar[d] \\ TS_{K}^{d}B_{K} \ar[r]^{j_{B_{K}}} & ⊗_{K}^{d}B_{K}}

where j_{B} and j_{B_{K}} are the inclusion maps, which are integral, and according to [34, Ch.5, §2, n.2, Cor. of Th.2] induce bijections

Hom_{A-Alg}(⊗_{A}^{d}B, K)/G →^{∼} Hom_{A-Alg}(TS_{A}^{d}B, K)

Hom_{K-Alg}(⊗_{K}^{d}B_{K}, K)/G →^{∼} Hom_{K-Alg}(TS_{K}^{d}B_{K}, K)

for the S_{d}-actions on ⊗_{A}^{d}B and on ⊗_{K}^{d}B_{K} as in (9.5.28). It then suffices to remark that the natural map ⊗_{A}^{d}B → ⊗_{K}^{d}B_{K} induces a bijection Hom_{K-Alg}(⊗_{K}^{d}B_{K}, K) →^{∼} Hom_{A-Alg}(⊗_{A}^{d}B, K). This shows already that t_{B}^{d} is radicial and that Ker t_{B}^{d} lies in the nilradical of Γ_{A}^{d}B. To conclude, it suffices to check that j_{B} ∘ t_{B}^{d} is integral. To this aim, choose a surjective map of A-algebras P → B from a free polynomial A-algebra; since the induced map ⊗_{A}^{d}P → ⊗_{A}^{d}B is surjective, it suffices to check the assertion for the composition of t_{P}^{d} and the inclusion map TS_{A}^{d}P → ⊗_{A}^{d}P. But the latter is integral, and t_{P}^{d} is an isomorphism, by proposition 9.5.34(i).

(iii): Let f : A[T_{•}] → B be a surjection of A-algebras, from a free polynomial A-algebra A[T_{•}] of finite type. As Γ_{A}^{d}f is surjective (corollary 9.5.22(iv)), it suffices to check the assertion for Γ_{A}^{d}A[T_{•}]. Next, the isomorphism A ⊗_{ℤ} Γ_{ℤ}^{d}ℤ[T_{•}] →^{∼} Γ_{A}^{d}A[T_{•}] of lemma 9.5.29(i) reduces to the case where A = ℤ, and especially, A is noetherian. Then, notice that ⊗_{ℤ}^{d}ℤ[T_{•}] is a ℤ-algebra of finite type, and by (ii) and proposition 9.5.34(i), Γ_{ℤ}^{d}ℤ[T_{•}] is isomorphic to TS_{A}^{d}ℤ[T_{•}]. Since the ⊗_{ℤ}^{d}ℤ[T_{•}] is integral over TS_{A}^{d}ℤ[T_{•}] is, the assertion follows from [12, Prop.7.8].

(iv): In light of (ii) we have a commutative diagram of A-algebras :

\xymatrix{Γ_{A}^{d}B_{1} \ar[r]^{t_{B_{1}}^{d}} \ar[d]_{Γ_{A}^{d}f} & TS_{A}^{d}B_{1} \ar[d]^{TS_{A}^{d}f} \ar[r]^{j_{1}} & ⊗_{A}^{d}B_{1} \ar[d]^{⊗_{A}^{d}f} \\ Γ_{A}^{d}B_{2} \ar[r]^{t_{B_{2}}^{d}} & TS_{A}^{d}B_{2} \ar[r]^{j_{2}} & ⊗_{A}^{d}B_{2}}

where j_{1} and j_{2} are the inclusion maps. In order to check that TS_{A}^{d}f is integral, it suffices then to show that the same holds for (⊗_{A}^{d}f)∘j_{1}; however, j_{1} is integral, and the same holds for ⊗_{A}^{d}f, since f is integral, whence the assertion. Combining with (ii), we deduce that (TS_{A}^{d}f) ∘ t_{B_{1}}^{d} is integral; hence, let x ∈ Γ_{A}^{d}B_{2} be any element, and pick a monic polynomial P ∈ (Γ_{A}^{d}B_{1})[T] such that P (t_{B_{2}}x) = 0. Set Q := P − T, and notice that Q(x) ∈ Ker(t_{B_{2}}^{d}); by (ii), it follows that Q(x)^{n} = 0 for some n ∈ ℕ, so x is integral over Γ_{A}^{d}B_{1}. This completes the proof of (iv.a).

Next, recall that f is radicial (resp. that Ker f lies in the nilradical 𝒩 of B_{1}) if and only if, for every algebraically closed field K, the induced map Hom_{ℤ-Alg}(B_{2}, K) → Hom_{ℤ-Alg}(B_{1}, K) is injective (resp. surjective). It follows easily that if f is radicial (resp. if Ker f ⊂ 𝒩 ), then the same holds for ⊗_{A}^{d}f, and arguing as in the proof of (ii), we deduce that the same holds also for TS_{A}^{d}f. By the same token, (ii) implies that t_{B_{i}}^{d} induces a bijection Hom_{ℤ-Alg}(TS_{A}^{d}B_{i}, K) →^{∼} Hom_{ℤ-Alg}(Γ_{A}^{d}B_{i}, K) for i = 1, 2 and every algebraically closed field K. Assertions (iv.b) and (iv.c) follow immediately. □

Definition 9.5.70. With the notation of (9.5.66), we call a norm map of degree d for the Aalgebra B, any map of A-algebras

Γ_{A}^{d}B → A.

Remark 9.5.71. Let B be any A-algebra, d ∈ ℕ, and ν : Γ_{A}^{d}B → A a norm map of degree d.

(i) In light of (9.5.67), we see that the datum of ν is equivalent to that of a system of morphisms of multiplicative monoids

N_{C} : C ⊗_{A} B → C for every A-algebra C

natural with respect to maps of A-algebras C → C′, and such that

N_{C}(c ⊗ b) = c^{d} · N_{C}(1 ⊗ b) for every c ∈ C and b ∈ B.

Indeed, from ν we obtain such a compatible system N_{•} by the rule :

N_{C} := (C ⊗_{A} ν) ∘ (λ_{B}^{d})_{C} for every A-algebra C

and conversely, every such compatible system N_{•} yields an A-linear map ν : Γ_{A}^{d}B → A, by the universal property of Γ_{A}^{d}B, and the condition that N_{C} is a map of multiplicative monoids implies easily that ν is a map of A-algebras (details left to the reader).

(ii) The isomorphism of proposition 9.5.68(i) identifies naturally C ⊗_{A} ν with a norm map ν_{C} : Γ_{C}^{c}(C ⊗_{A} B) → C of degree d. Indeed, if N_{•} (resp. N_{•}^{C}) is the compatible system of morphisms of multiplicative monoids associated with ν (resp. with ν_{C}), then N_{•}^{C} is the just the restriction of N_{•} to the category of C-algebras.

Example 9.5.72. (i) Let A be a ring, B an A-algebra whose underlying A-module is projective of constant finite rank d. Then, for every other A-algebra C, the C-module underlying the Calgebra C ⊗_{A} B is projective of constant finite rank d. Every x ∈ C ⊗_{A} B induces a C-linear endomorphism μ_{x} : C ⊗_{A} B → C ⊗_{A} B such that x′ ↦ xx′ for every x′ ∈ C ⊗_{A} B. The determinant N_{C}(x) := det(μ_{x}) is an element of C, and the construction of N_{C} : C ⊗_{A} B → C is natural for maps C → C′ of A-algebras, so we get a compatible system N_{•} of maps of sets for every A-algebra C. In view of remark 9.5.71(i), N_{•} corresponds to a unique norm map

ν_{B/A} : Γ_{A}^{d}B → A such that (C ⊗_{A} ν_{B/A})(x^{[d]}) = det(μ_{x})

for every A-algebra C and every x ∈ C ⊗_{A} B.

(ii) Let A be a normal domain, K its field of fractions, and B the integral closure of A in a finite field extension E of K, of degree d. Then, every free polynomial A-algebra A[T_{•}] is again a normal domain, and B[T_{•}] = B ⊗_{A} A[T_{•}] is the integral closure of A[T_{•}] in the finite field extension E(T_{•}) of K(T_{•}). With this notation, the norm map N_{K[T_{•}]} : E[T_{•}] = K[T_{•}]⊗_{K} E → K[T_{•}] is well defined as in (i), and we claim that N_{K[T_{•}]}(B[T]) ⊂ A[T]. Indeed, recall that det(μ_{x}) = (−1)^{d} · χ_{x}(0), where χ_{x} ∈ K(T_{•})[X] is the characteristic polynomial of the endomorphism μ_{x} of E(T_{•}), for every x ∈ E(T_{•}); let also m_{x} ∈ K(T_{•})[X] be the minimal polynomial of x over K(T_{•}). Since χ_{x} and m_{x} have the same set of roots Σ, it suffices to check that every root of m_{x} is integral over A[T_{•}], when x ∈ B[T_{•}]. However, let L be any normal extension of K(T_{•}) containing E(T_{•}); then Σ is the orbit of x under the action of the Galois group Gal(L/K(T_{•})), whence the assertion. Thus, for every free polynomial A-algebra A[T_{•}] the restriction of N_{K[T_{•}]} is a well defined morphism of multiplicative monoids

N′_{A[T_{•}]} : B[T_{•}] → A[T_{•}]

and it is clear that the rule A[T_{•}] ↦ N′_{A[T_{•}]} is natural for maps of free polynomial A-algebras, since the same holds for the rule K[T_{•}] ↦ N_{K[T_{•}]}. By virtue of proposition 9.5.6, it follows that N′_{•} extends uniquely to a well defined homogeneous polynomial law N′ : B ⇝ A of degree d, and it is easily seen that N′_{C} is still a morphism of multiplicative monoids, for every A-algebra C (details left to the reader); whence finally, a norm map of degree d for the A-algebra B :

ν′_{B/A} : Γ_{A}^{d}B → A.

(iii) Let A be a ring, G a finite group of automorphisms of A, and H ⊂ G a subgroup with [G : H] = d. For every A^{G}-algebra B, we get an induced action of G on B ⊗_{A^{G}} A, and if B is a flat A^{G}-algebra, the natural maps :

B → (B ⊗_{A^{G}} A)^{G} B ⊗_{A^{G}} A^{H} → (B ⊗_{A^{G}} A)^{H}

are isomorphisms (details left to the reader). Fix a system of representatives Σ ⊂ G for the classes of G/H; for every flat A^{G}-algebra B, we then get a map of multiplicative monoids :

N′′_{B} : B ⊗_{A^{G}} A^{H} → B x ↦ ∏_{g∈Σ} g(x)

which is clearly natural for every map of flat A^{G}-algebras. By proposition 9.5.6, the rule B ↦ N′′_{B} then extends uniquely to a well defined homogeneous polynomial law N′′ : A^{H} ⇝ A^{G} of degree d, whence again a norm map of degree d for the A^{G}-algebra A^{H} :

ν′′_{A^{H}/A^{G}} : Γ_{A^{G}}^{d}A^{H} → A^{G}.

(iv) The following example generalizes all the previous ones. Consider a ring A, an injective ring homomorphism A → A′, a projective A′-module P of constant rank d, and a map of associative A-algebras ρ : B → End_{A′}(P′). For every flat A-algebra C, let moreover

ρ_{C} := C ⊗_{A} ρ : C ⊗_{A} B → C ⊗_{A} End_{A′}(P′) →^{∼} End_{C⊗_{A}A′}(C ⊗_{A} P′)

and suppose that N′′′_{C}(b) := det(ρ_{C}(b)) ∈ C for every such C and every b ∈ C ⊗_{A} B. By proposition 9.5.6, the rule : C ↦ N′′′_{C} for every such C extends uniquely to a homogeneous polynomial law N′′′ : B ⇝ A of degree d, and clearly N_{C} is a morphism of multiplicative monoids for every flat A-algebra C, hence also for any A-algebra C (details left to the reader); whence finally a norm map of degree d

ν′′′_{P,ρ} : Γ_{A}^{d}B → A.

9.5.73. Let A be a ring, B an A-algebra, d ∈ ℕ, and ν : Γ_{A}^{d}B → A a norm map of degree d. Let also N : B ⇝ A be the homogeneous polynomial law of degree d associated with ν, as in remark 9.5.71(i). We set

χ_{b}(T) := N_{C[T]}(T − b) ∈ C[T] for every A-algebra C and every b ∈ C ⊗_{A} B.

Let moreover ν_{C[T]} := C[T] ⊗_{A} Γ_{A}^{d}B → C[T] for every such C, and notice that

χ_{b}(T) = ν_{C[T]}((T − b)^{[d]})

and (T − b)^{[d]} = ∑_{i=0}^{d} T^{[i]} · (−b)^{[d−i]} = ∑_{i=0}^{d} T^{i} · 1^{[i]} · (−b)^{[d−i]} in C[T] ⊗_{A} Γ_{A}^{•}B, by virtue of (9.5.51). Since (−b)^{[0]} = 1, and ν_{C[T]}(1^{[d]}) = N_{C[T]}(1) = 1, we may compute :

χ_{b}(T) = ∑_{i=0}^{d} T^{i} · ν_{C[T]}(1^{[i]} · (−b)^{[d−i]}) = T^{d} + ∑_{i=0}^{d−1} T^{i} · ν_{C[T]}(1^{[i]} · (−b)^{[d−i]}).

We get therefore a well defined homogeneous polynomial law of degree d :

B ⇝ B  b ↦ χ_{b}(b) ∈ C ⊗_{A} B  for every A-algebra C and every b ∈ C ⊗_{A} B

and we define the characteristic map of ν as the associated A-linear map χ^{†} : Γ_{A}^{d}B → B.

Definition 9.5.74. With the notation of (9.5.73), we say that ν is a Cayley-Hamilton norm map if χ^{†} = 0, i.e. if χ_{b}(b) = 0 for every A-algebra C and every b ∈ C ⊗_{A} B.

Example 9.5.75. (i) In the situation of example 9.5.72(iv), notice that, by the theorem of Cayley-Hamilton, we have ρ_{C}(χ_{b}(b)) = 0 for every flat A-algebra C and every b ∈ C ⊗_{A} B. Thus, if ρ_{C} is injective for every such C, the norm map ν′′′_{P,ρ} is Cayley-Hamilton.

(ii) Especially, the norm maps ν_{B/A}, ν′_{B/A} and ν′′_{A^{H}/A^{G}} of example 9.5.72(i,ii,iii) are Cayley-Hamilton.

Remark 9.5.76. (i) Let φ : B′ → B be a map of A-algebras, ν : Γ_{A}^{d}B → A a norm map of degree d, and set ν′ := ν ∘ Γ_{A}^{d}(φ). Let also χ^{†} and χ^{′†} be the characteristic maps of ν, respectively ν′. By a direct inspection of the constructions we get a commutative diagram :

\xymatrix{Γ_{A}^{d}B′ \ar[r]^{χ^{′†}} \ar[d]_{Γ_{A}^{d}φ} & B′ \ar[d]^{φ} \\ Γ_{A}^{d}B \ar[r]^{χ^{†}} & B}

(details left to the reader). In particular, if φ is surjective and ν′ is Cayley-Hamilton, the same holds for ν, since Γ_{A}^{d}(φ) is surjective (corollary 9.5.22(iv)).

(ii) In particular, let ν : Γ_{A}^{d}B → A be a norm map of degree d, and I ⊂ B the ideal generated by the image of the characteristic map χ^{†} of ν; let also π : B → B/I be the projection, and suppose that ν factors through Γ_{A}^{d}(π) and a norm map \overline{ν} : Γ_{A}^{d}(B/I) → A. Then, if \overline{χ}^{†} denotes the characteristic map of \overline{ν}, we have \overline{χ}^{†}∘Γ_{A}^{d}(π) = π∘χ^{†} = 0, by (i), whence \overline{χ}^{†} = 0, since Γ_{A}^{d}(π) is a surjection (corollary 9.5.22(iv)). In other words, in this situation, \overline{ν} is Cayley-Hamilton.

(iii) In the situation of (i), let also C be any A-algebra, and ν_{C} : Γ_{C}^{d}(C ⊗_{A} B) → C the norm map induced by ν as in remark 9.5.71(ii). Then we get a commutative diagram of C-modules :

\xymatrix{C ⊗_{A} Γ_{A}^{d}B \ar[rr]^{∼} \ar[dr]_{C⊗_{A}χ^{†}} &  & Γ_{C}^{d}(C ⊗_{A} B) \ar[dl]^{χ^{C†}} \\  & C ⊗_{A} B}

where χ^{C†} is the characteristic map of ν_{C}, and the top horizontal arrow is the isomorphism of proposition 9.5.68(i). Especially, if ν is a Cayley-Hamilton norm map, the same holds for ν_{C}.

Theorem 9.5.77. Let ν : Γ_{A}^{d}B → A be a norm map of degree d, and I ⊂ B the ideal generated by the image of the characteristic map χ^{†} of ν. Then ν is the composition of the projection Γ_{A}^{d}B → Γ_{A}^{d}(B/I) and a Cayley-Hamilton norm map \overline{ν} : Γ_{A}^{d}(B/I) → A.

Proof. By remark 9.5.76(ii), it suffices to show that ν factors through the projection onto Γ_{A}^{d}(B/I) and a norm map \overline{ν} : Γ_{A}^{d}(B/I) → A. We first deal with the following special case :

Claim 9.5.78. The theorem holds if A is an algebraically closed field.

Proof of the claim. In this case, B is trivially a flat A-algebra, so ν is naturally identified with a map of A-algebras TS_{A}^{d}B → A (propositions 9.5.68(ii) and 9.5.34(i)). The latter extends to a map of A-algebras ⊗_{A}^{d}B → A, which is the same as a system of d maps of A-algebras ν_{1}, …, ν_{d} : B → A. Let ρ : B → End_{A}(A^{⊕d}) be the morphism of associative A-algebras that maps each b ∈ B to the diagonal d × d matrix ρ(b) with ρ(b)_{ii} = ν′_{i}(b) for i = 1, …, d. Working out the definitions, we see that ν corresponds to the homogeneous polynomial law B ⇝ A arising from ρ, as in example 9.5.72(iv). Let J := Ker ρ, so that ρ factors through the projection π : B → B/J and a map of associative A-algebras ρ′ : B/J → End_{A}(A^{⊕d}); then ρ′ induces, again by example 9.5.72(iv), a norm map ν′ : Γ_{A}^{d}(B/J) → A such that ν = ν′ ∘Γ_{A}^{d}(π), and ν′ is Cayley-Hamilton, by virtue of example 9.5.75(i). By remark 9.5.76(i), it follows that I ⊂ J, whence the required factorization of ν. ♢

Now, choose a free polynomial A-algebra P, a surjective map φ : P → B of A-algebras, and set ν^{P} := ν ∘ Γ_{A}^{d}(φ) : Γ_{A}^{d}P → A; remark 9.5.76(i) yields the identity :

φ ∘ χ^{P †} = χ^{†} ∘ Γ_{A}^{d}(φ)

where χ^{†} : Γ_{A}^{d}B → B and χ^{P †} : Γ_{A}^{d}P → P are the characteristic maps of ν and ν^{P}. Let J ⊂ P be the ideal generated by the image of χ^{P †}; since Γ_{A}^{d}(φ) is surjective (corollary 9.5.22(iv)), it follows that φ(J) = I. Let \overline{φ} : P/J → B/I be the map induced by φ, and π : P → P/J the projection; it follows easily from corollary 9.5.57 that Ker Γ_{A}^{d}(\overline{φ}) is the image of Ker Γ_{A}^{d}(φ), under the map Γ_{A}^{d}(π) : Γ_{A}^{d}P → Γ_{A}^{d}(P/J). Now, suppose that ν^{P} factors through Γ_{A}^{d}(π) and a norm map \overline{ν}^{P} : Γ_{A}^{d}(P/J) → A; then we deduce that ν factors through the projection Γ_{A}^{d}B → Γ_{A}^{d}(B/I) and a norm map \overline{ν} : Γ_{A}^{d}(B/I) → A.

Thus, it suffices to show the theorem for P, and we may assume from start that B = A[T_{•}] is a free polynomial A-algebra. In this case, set A_{0} := Γ_{ℤ}^{d}(ℤ[T_{•}]); taking into account the isomorphism of proposition 9.5.68(i), we get a norm map of degree d :

ν_{0} : Γ_{A_{0}}^{d}(A_{0}[T_{•}]) →^{∼} A_{0} ⊗_{ℤ} Γ_{ℤ}^{d}(ℤ[T_{•}]) = A_{0} ⊗_{ℤ} A_{0} ⟶^{μ_{A_{0}}} A_{0}

where μ_{A_{0}} is the multiplication map of the ring A_{0}. By the same token, ν is naturally identified with a map of A-algebras A ⊗_{ℤ} A_{0} → A and we get therefore an induced A_{0}-algebra structure on A (indeed, the set of norm maps of degree d of the A-algebra A[T_{•}] is in natural bijection with the set of all such A_{0}-algebra structures). Then, a simple inspection yields a commutative diagram of A-algebras :

\xymatrix{A ⊗_{A_{0}} Γ_{A_{0}}^{d}(A_{0}[T_{•}]) \ar[rr]^{ω} \ar@{-->}[dr]_{A⊗_{A_{0}}ν_{0}} &  & Γ_{A}^{d}(A[T_{•}]) \ar[dl]^{ν} \\  & A}

where ω is again the natural isomorphism of proposition 9.5.68(i). Let χ_{0}^{†} : Γ_{A_{0}}^{d}(A_{0}[T_{•}]) → A_{0} be the characteristic map of ν_{0}, and I_{0} ⊂ A_{0}[T_{•}] the ideal generated by the image of χ_{0}^{†}; from remark 9.5.76(iii) we see that the image of A ⊗_{A_{0}} I_{0} in A[T_{•}] is the ideal I. Let π_{0} : A_{0}[T_{•}] → A_{0}[T_{•}]/I_{0} be the projection; by corollary 9.5.57, the kernel of Γ_{A_{0}}^{d}(π_{0}) is the image of Γ_{A_{0}}^{d}(I_{0}), so the kernel of ν is the image of A ⊗_{A_{0}} Γ_{A_{0}}^{d}(I_{0}) ≃ Γ_{A}^{d}(A ⊗_{A_{0}} I_{0}), and the latter is also the kernel of the projection Γ_{A}^{d}(A[T_{•}]) → Γ_{A}^{d}(A[T_{•}]/I), again by corollary 9.5.57. Summing up, we see that if ν_{0} factors through Γ_{A_{0}}^{d}(π_{0}) and a norm map \overline{ν}_{0} : Γ_{A_{0}}^{d}(A_{0}[T_{•}]/I_{0}) → A_{0}, then ν is the composition of the projection onto Γ_{A}^{d}(A[T_{•}]/I) and a norm map \overline{ν} : Γ_{A}^{d}(A[T_{•}]/I) → A.

Thus, we are further reduced to the case where A = A_{0}, and in particular A is a domain, since the same clearly holds for TS_{A_{0}}^{d}(A_{0}[T_{•}]) (propositions 9.5.68(ii) and 9.5.34(i)). Next, let K be an algebraic closure of the field of fractions of A, and set B_{K} := K ⊗_{A} B and ν_{K} := K ⊗_{A} ν. Since Γ_{A}^{d}B is a flat A-module (corollary 9.5.22(iii)), the induced map Γ_{A}^{d}B → K ⊗_{A} Γ_{A}^{d}B →^{∼} Γ_{K}^{d}B_{K} is injective, and ν_{K} is naturally identified with a norm map Γ_{K}^{d}B_{K} → K. Let χ_{K}^{†} be the characteristic map of ν_{K}; arguing as in the foregoing, we see that the ideal of B_{K} generated by the image of χ_{K}^{†} is I_{K} := K ⊗_{A} I. Hence, if ν_{K} is the composition of the projection Γ_{K}^{d}B_{K} → Γ_{K}^{d}(B_{K}/I_{K}) and a norm map \overline{ν}_{K} : Γ_{K}^{d}(B_{K}/I_{K}) → K, then ν factors through the projection onto Γ_{A}^{d}(B/I) and a norm map Γ_{A}^{d}(B/I) → A. Thus, it suffices to show the theorem for the norm map ν_{K}, and this is already known by claim 9.5.78. □

Remark 9.5.79. Theorem 9.5.77 admits a non-commutative generalization in the theory of determinants of algebras : see [47, Lemma 1.21].

9.6. Regular rings. In a later section, we shall need a criterion for the regularity of a certain type of extensions of regular local rings. Such a criterion is stated incorrectly in [63, Ch.0, Th.22.5.4]; our first task is to supply a corrected version of loc.cit.

Proposition 9.6.1. Let K be a field, and L a field extension of K. We have :

(i) H_{i}𝕃_{L/K} = 0 for every i > 1.

(ii) If L is a separable extension of K, then H_{i}𝕃_{L/K} = 0 for every i > 0.

(iii) (Cartier’s equality). If L is an extension of finite type of K, then

dim_{L} H_{1}𝕃_{L/K} < +∞  and  dim_{L} Ω_{L/K}^{1} − dim_{L} H_{1}𝕃_{L/K} = tr. deg[L : K].

Proof. (ii): By [103, Ch.II, (1.2.3.4)], we may assume that L is an extension of K of finite type, in which case L is a separable algebraic extension of a subfield L_{0} ⊂ L, which is a purely transcendantal extension of K of finite type. Since L is an étale L_{0}-algebra, and since L_{0} is a localization of a smooth K-algebra, we then have

(9.6.2)  𝕃_{L/L_{0}} →^{∼} 0  and 𝕃_{L_{0}/K} →^{∼} Ω_{L/K_{0}}^{1}[0]  in D(L-Mod)

([103, Ch.III, Prop.3.1.1 and Prop.3.1.2] and [103, Ch.II, Cor.2.3.1.1]). However, the tower of field extensions K ⊂ L_{0} ⊂ L yields a distinguished triangle :

(9.6.3)  𝕃_{L/L_{0}}[1] → 𝕃_{L_{0}/K} ⊗_{L_{0}} L → 𝕃_{L/K} → 𝕃_{L/L_{0}}  in D(L-Mod)

([103, Ch.II, Prop.2.1.2]) whence the assertion.

(i): Let K_{0} be the smallest subfield of K; so K_{0} is either ℚ or 𝔽_{p}, for some prime integer p, and in either case, it is a perfect field. By (i), we then have H_{i}𝕃_{K/K_{0}} = H_{i}𝕃_{L/K_{0}} = 0 for every i > 0. However, the tower of field extensions K_{0} ⊂ K ⊂ L yields a distinguished triangle :

𝕃_{L/K}[1] → 𝕃_{K/K_{0}} ⊗_{K} L → 𝕃_{L/K_{0}} → 𝕃_{L/K}

([103, Ch.II, Prop.2.1.2]) whence the assertion.

(iii): Suppose first that L is a separable extension of K, and pick L_{0} ⊂ L as in the proof of (ii), so L_{0} = Frac K[T_{1}, …, T_{r}], with r := tr. deg[L : K]. From (9.6.2) and (9.6.2), we get :

dim_{L} Ω_{L/K}^{1} = dim_{L} H_{0}𝕃_{L/K} = dim_{L_{0}} Ω_{L_{0}/K}^{1} = dim_{L_{0}} L_{0} ⊗_{K[T_{1},...,T_{r}]} Ω_{K[T_{1},...,T_{r}]/K}^{1} = r

which establishes the sought identity, as in this case we have H_{1}𝕃_{L/K} = 0, by (ii).

Next, suppose that K is of finite type over its smallest subfield K_{0}. We have H_{1}𝕃_{L/K_{0}} = 0, by (ii); then the tower of field extensions K_{0} → K → L yields an exact sequence

0 → H_{1}𝕃_{L/K} → Ω_{K/K_{0}}^{1} ⊗_{K} L → Ω_{L/K_{0}}^{1} → Ω_{L/K}^{1} → 0

([103, Ch.II, Prop.2.1.2]), and on the other hand :

dim_{K} Ω_{K/K_{0}}^{1} = tr. deg[K : K_{0}]  and  dim_{L} Ω_{L/K_{0}}^{1} = tr. deg[L : K_{0}]

by the foregoing case. The assertion follows straightforwardly, in this case.

Lastly, let K be an arbitrary field, and write K as the union of the filtered family (K_{λ} | λ ∈ Λ) of its subfields that are finitely generated over K_{0}. Choose x_{1}, …, x_{t} ∈ L such that K(x_{1}, …, x_{t}) = L. Let φ : K[X_{1}, …, X_{t}] → L be the map of K-algebras given by the rule : X_{i} ↦ x_{i} for i = 1, …, t. Then I := Ker φ is a finitely generated ideal, so we may find λ ∈ Λ and an ideal I_{λ} ⊂ K_{λ}[X_{1}, …, X_{t}] such that I = I_{λ} ⊗_{K_{λ}} K (details left to the reader). Denote by L_{λ} the field of fractions of K_{λ}[T_{1}, …, T_{t}]/I_{λ}; it is easily seen that L_{λ} ⊗_{K_{λ}} K is a domain, and its field of fractions is a K-algebra naturally isomorphic to L. Especially, we have tr. deg[L_{λ} : K_{λ}] = tr. deg[L : K], and on the other hand, there is a natural isomorphism :

𝕃_{L_{λ}/K_{λ}} ⊗_{L_{λ}} L →^{∼} 𝕃_{L/K} in D^{−}(L-Mod)

([103, Ch.II, Prop.2.2.1, Cor.2.3.1.1]). Then the sought identity for the extension K ⊂ L follows from the same identity for the extension K_{λ} ⊂ L_{λ}. The latter is already known, by the previous case. Likewise, the argument shows that dim_{L} H_{1}𝕃_{L/K} < +∞. □

9.6.4. Consider a local ring A, with maximal ideal 𝔪_{A}, and residue field k_{A} := A/𝔪_{A} of characteristic p > 0. From [63, Ch.0, Th.20.5.12(i)] we get a complex of k_{A}-vector spaces

(9.6.5)  0 → 𝔪_{A}/(𝔪_{A}^{2} + pA) ⟶^{d_{A}} Ω_{A/ℤ}^{1} ⊗_{A} k_{A} → Ω_{k_{A}/ℤ}^{1} → 0.

Lemma 9.6.6. The complex (9.6.5) is exact.

Proof. Set (A_{0}, 𝔪_{A_{0}}) := (A/pA, 𝔪_{A}/pA) and 𝔽_{p} := ℤ/pℤ; the induced sequence of ring homomorphisms 𝔽_{p} → A_{0} → k_{A} yields a distinguished triangle ([103, Ch.II, §2.1.2.1]) :

𝕃_{A_{0}/𝔽_{p}} ⊗^{L}_{A_{0}} k_{A} → 𝕃_{k_{A}/𝔽_{p}} → 𝕃_{k_{A}/A_{0}} → 𝕃_{A_{0}/𝔽_{p}} ⊗^{L}_{A_{0}} k_{A}[1].

However, we have H_{i}𝕃_{k_{A}/𝔽_{p}} = 0 for every i > 0, by proposition 9.6.1(ii). We deduce a short exact sequence

0 → H_{1}𝕃_{k_{A}/A_{0}} → Ω_{A_{0}/𝔽_{p}}^{1} ⊗_{A_{0}} k_{A} → Ω_{k_{A}/𝔽_{p}}^{1} → 0

which, under the natural identifications

Ω_{k_{A}/ℤ}^{1} →^{∼} Ω_{k_{A}/𝔽_{p}}^{1}  Ω_{A/ℤ}^{1} ⊗_{A} k_{A} →^{∼} Ω_{A_{0}/𝔽_{p}}^{1} ⊗_{A_{0}} k_{A}  H_{1}𝕃_{k_{A}/A_{0}} →^{∼} 𝔪_{A_{0}}/𝔪_{A_{0}}^{2}

([103, Ch.III, Cor.1.2.8.1]) becomes (9.6.5), up to replacing d_{A} by −d_{A} ([103, Ch.III, Prop.1.2.9]; details left to the reader). □

9.6.7. Keep the situation of (9.6.4), and define

A_{2} := A ×_{k_{A}} W_{2}(k_{A})

where W_{2}(k_{A}) is the ring of 2-truncated Witt vectors of k_{A}, as in (9.3.25), which is augmented over k_{A}, via the ghost component map \overline{ω}_{0} : W_{2}(k_{A}) → k_{A}. Notice that

V_{1}(k_{A}) := Ker \overline{ω}_{0}

is an ideal of W_{2}(k_{A}) such that V_{1}(k_{A})^{2} = 0; especially, it is naturally a k_{A}-vector space, with addition and scalar multiplication given respectively by the rules :

(0, a) + (0, b) := (0, a + b) x · (0, a) := (x, 0) · (0, a) = (0, x^{p} · a)

for every a, b, x ∈ k_{A}. In other words, the map

(9.6.8)  V_{1}(k_{A}) → k_{A}^{1/p}  (0, a) ↦ a^{1/p}

is an isomorphism of k_{A}-vector spaces, and we also see that V_{1}(k_{A}) is a one-dimensional k_{A}^{1/p}- vector space, with scalar multiplication given by the rule : a·(0, b) := (0, a^{p}b) for every a ∈ k_{A}^{1/p} and b ∈ k_{A}. By construction, we have a natural exact sequence of A_{2}-modules :

(9.6.9)  0 → V_{1}(k_{A}) → A_{2} ⟶^{π} A → 0.

Especially, A_{2} is a local ring, and π is a local ring homomorphism inducing an isomorphism on residue fields.

9.6.10. It is easily seen that the rule (A, 𝔪_{A}) ↦ A_{2} defines a functor on the category p-Local of local rings with residue field of characteristic p > 0, and local ring homomorphisms, to the category of (commutative, unital) rings. Hence, let us set

(A, 𝔪_{A}) ↦ \overline{Ω}_{A} := Ω_{A_{2}/ℤ}^{1} ⊗_{A_{2}} A.

It follows that the rule (A, 𝔪_{A}) ↦ (A, \overline{Ω}_{A}) yields a functor

p-Local → Alg.Mod

to the category whose objects are the pairs (B, M) where B is a ring, and M is a B-module (the morphisms (B, M) → (B′, M′) are the pairs (φ, ψ) where φ : B → B′ is a ring homomorphism, and ψ : B′ ⊗_{B} M → M is a B′-linear map : cp. [75, Def.2.5.22(ii)]).

In view of (9.6.9) and [63, Ch.0, Th.20.5.12(i)] we get an exact sequence of A-modules

V_{1}(k_{A}) → \overline{Ω}_{A} → Ω_{A/ℤ}^{1} → 0

whence, after tensoring with k_{A}, a sequence of k_{A}-vector spaces

(9.6.11)  0 → V_{1}(k_{A}) ⟶^{j_{A}} \overline{Ω}_{A} ⊗_{A} k_{A} ⟶^{ρ} Ω_{A/ℤ}^{1} ⊗_{A} k_{A} → 0

which is right exact by construction.

Proposition 9.6.12. With the notation of (9.6.10), we have :

(i) If p ∉ 𝔪_{A}^{2}, then the sequence (9.6.11) is exact.

(ii) If p ∈ 𝔪_{A}^{2}, then Ker j_{A} = {(0, a^{p}) | a ∈ k_{A}}. Especially, the isomorphism (9.6.8) identifies Ker j_{A} with the subfield k_{A} of k_{A}^{1/p}.

Proof. In view of lemma 9.6.6, the kernel of ρ is naturally identified with the kernel of the natural map

𝔪_{A_{2}}/(𝔪_{A_{2}}^{2} + pA_{2}) → 𝔪_{A}/(𝔪_{A}^{2} + pA).

On the other hand, it is easily seen that 𝔪_{A_{2}} = 𝔪_{A} ⊕ V_{1}(k_{A}), and under this identification, the multiplication law of A_{2} restricts on 𝔪_{A_{2}} to the mapping given by the rule : (a, b) · (a′, b′) := (aa′, 0) for every a, a′ ∈ 𝔪_{A} and b, b′ ∈ V_{1}(k_{A}). There follows a natural isomorphism

(9.6.13)  𝔪_{A_{2}}/𝔪_{A_{2}}^{2} →^{∼} (𝔪_{A}/𝔪_{A}^{2}) ⊕ V_{1}(k_{A})

which identifies the map 𝔪_{A_{2}}/𝔪_{A_{2}}^{2} → 𝔪_{A}/𝔪_{A}^{2} deduced from π (notation of (9.6.9)), with the projection on the first factor. Moreover, by inspecting the definitions, we easily get a commutative diagram

\xymatrix{V_{1}(k_{A}) \ar[r]^{j_{A}} \ar[d] & \overline{Ω}_{A} ⊗_{A} k_{A} \\ 𝔪_{A_{2}}/𝔪_{A_{2}}^{2} \ar[r] & 𝔪_{A_{2}}/(𝔪_{A_{2}}^{2} + pA_{2}) \ar[u]_{d_{A_{2}}}}

whose bottom horizotal arrow is the quotient map, and whose left vertical arrow is the inclusion map of the direct summand V_{1}(k_{A}) resulting from (9.6.13).

The map A_{2} → 𝔪_{A_{2}}/𝔪_{A_{2}}^{2} given by the rule : a ↦ pa (mod 𝔪_{A_{2}}^{2}) factors (uniquely) through a k_{A}-linear map

t_{A_{2}} : k_{A} → 𝔪_{A_{2}}/𝔪_{A_{2}}^{2} a (mod 𝔪_{A}) ↦ pa (mod 𝔪_{A_{2}}^{2})

and likewise we may define a map t_{A} : k_{A} → 𝔪_{A}/𝔪_{A}^{2}. The snake lemma then gives an induced map ∂ : Ker t_{A} → V_{1}(k_{A}), and in view of the foregoing, the proposition follows from :

Claim 9.6.14. (i) If p ∉ 𝔪_{A}^{2}, then t_{A} is injective.

(ii) If p ∈ 𝔪_{A}^{2}, then Im ∂ = {(0, a^{p}) | a ∈ k_{A}}.

Proof of the claim. (i) is obvious.

(ii): By virtue of (9.3.23), we have p = (p, (0, 1)) in A_{2}, and if (a, y) ∈ A_{2} is any element, then p · (a, y) = (pa, (0, a^{p})), so in case p ∈ 𝔪_{A}^{2}, the map t_{A_{2}} is given by the rule :

a ↦ (pa, (0, \overline{a}^{p})) = (0, (0, \overline{a}^{p})) ∈ (𝔪_{A}/𝔪_{A}^{2}) ⊕ V_{1}(k_{A}) for every a ∈ A.

By the same token, in this case t_{A} is the zero map. The claim follows straightforwardly. □

9.6.15. Keep the notation of (9.6.10), and assume that p ∉ 𝔪_{A}^{2}, so (9.6.11) is a k_{A}-extension of Ω_{A/ℤ}^{1} ⊗_{A} k_{A} by V_{1}(k_{A}), by virtue of proposition 9.6.12; hence, (9.6.11) ⊗_{k_{A}} k_{A}^{1/p} is a k_{A}^{1/p}- extension of the corresponding k_{A}^{1/p}-vector spaces. Recall now that V_{1}(k_{A}) is naturally a k_{A}^{1/p}- vector space of dimension one, and let ψ_{A} : V_{1}(k_{A}) ⊗_{k_{A}} k_{A}^{1/p} → V_{1}(k_{A}) be the scalar multiplication. By push out along ψ_{A}, we obtain therefore an extension ψ_{A} ∗ (9.6.11) ⊗_{k_{A}} k_{A}^{1/p} fitting into a commutative ladder with exact rows :

[figure: 0; V_{1}(k_{A}) ⊗_{k_{A}} k_{A}^{1/p}; ψ_{A}; \overline{Ω}_{A} ⊗_{A} k_{A}^{1/p}; _{ψ_{A}}; Ω_{A/ℤ}^{1} ⊗_{A} k_{A}^{1/p}; 0; 0; V_{1}(k_{A}); j_{A}; Ω_{A}; Ω_{A/ℤ}^{1} ⊗_{A} k_{A}^{1/p}; 0]

(see [75, §2.5.5]). We consider now the mapping :

d_{A} : A → Ω_{A}  a ↦ ψ_{A}(d(a, τ_{k_{A}}(\overline{a})) ⊗ 1)  for every a ∈ A

where :

• \overline{a} ∈ k_{A} is the image of a in k_{A}
• τ_{k_{A}} is the Teichmüller mapping (see (9.3.34)), so τ_{k_{A}}(\overline{a}) = (\overline{a}, 0) ∈ W_{2}(k_{A})
• d : A_{2} → Ω_{A_{2}/ℤ} is the universal derivation of A_{2}.

Since τ_{k_{A}} is multiplicative, the map d_{A} satisfies Leibniz’s rule, i.e. we have

d_{A}(ab) = \overline{a} · d_{A}(b) + \overline{b} · d_{A}(a) for every a, b ∈ A.

However, d_{A} is not quite a derivation, since additivity fails for τ_{k_{A}}, hence also for d_{A}. Instead, from remark 9.3.8(ii), and recalling that (p − 1)! = −1 in 𝔽_{p}, we get the identity :

\begin{matrix}τ_{k_{A}}(\overline{a} + \overline{b}) = & τ_{k_{A}}(\overline{a}) + τ_{k_{A}}(\overline{b}) − ∑_{i=1}^{p−1}(0, \frac{\overline{a}^{i}}{i!} · \frac{\overline{b}^{p−i}}{(p − i)!}) \\ = & τ_{k_{A}}(\overline{a}) + τ_{k_{A}}(\overline{b}) − ∑_{i=1}^{p−1} \frac{\overline{a}^{i/p}}{i!} · \frac{\overline{b}^{1−i/p}}{(p − i)!} · p \end{matrix}

for every \overline{a}, \overline{b} ∈ k_{A}. On the other hand, notice that

\begin{matrix}ψ_{A}(d(0, x · p) ⊗ 1) = & j_{A} ∘ ψ_{A}(x · p ⊗ 1) \\ = & x · j_{A} ∘ ψ_{A}(p ⊗ 1) \\ = & x · ψ_{A}(d(0, p) ⊗ 1) \\ = & x · ψ_{A}(d(p − (p, 0)) ⊗ 1) \\ = & − x · d_{A}(p) \end{matrix}

for every x ∈ k_{A}^{1/p}, whence :

d_{A}(a + b) = d_{A}(a) + d_{A}(b) + ∑_{i=1}^{p−1} \frac{\overline{a}^{i/p}}{i!} · \frac{\overline{b}^{1−i/p}}{(p − i)!} · d_{A}(p) for every a, b ∈ A.

9.6.16. Especially, notice we do have d_{A}(a + b) = d_{A}(a) + d_{A}(b) in case either a or b lies in 𝔪_{A}. Hence, d_{A} restricts to an additive map 𝔪_{A} → Ω_{A}, and Leibniz’s rule implies that the latter descends to a k_{A}-linear map

\overline{d}_{A} : 𝔪_{A}/𝔪_{A}^{2} → Ω_{A}.

Proposition 9.6.17. Let (A, 𝔪_{A}) be a local ring with residue field k_{A} := A/𝔪_{A} of characteristic p > 0, and such that p ∉ 𝔪_{A}^{2}. Then there exists a natural exact sequence of k_{A}^{1/p}-vector spaces :

0 → 𝔪_{A}/𝔪_{A}^{2} ⊗_{k_{A}} k_{A}^{1/p} ⟶^{\overline{d}_{A}⊗k_{A}^{1/p}} Ω_{A} → Ω_{k_{A}/ℤ}^{1} ⊗_{k_{A}} k_{A}^{1/p} → 0.

Proof. By inspecting the constructions in (9.6.15), we obtain a commutative ladder of k_{A}-vector spaces, with exact rows :

\xymatrix{0 \ar[r] & \frac{pA}{𝔪_{A}^{2} ∩ pA} ⊗_{k_{A}} k_{A}^{1/p} \ar[r] \ar[d]_{i} & \frac{𝔪_{A}}{𝔪_{A}^{2}} ⊗_{k_{A}} k_{A}^{1/p} \ar[r] \ar[d]^{\overline{d}_{A}⊗_{k_{A}}k_{A}^{1/p}} & \frac{𝔪_{A}}{𝔪_{A}^{2} + pA} ⊗_{k_{A}} k_{A}^{1/p} \ar[r] \ar[d]^{d_{A}⊗_{k_{A}}k_{A}^{1/p}} & 0 \\ 0 \ar[r] & V_{1}(k_{A}) \ar[r]^{j_{A}} & Ω_{A} \ar[r] & Ω_{A/ℤ}^{1} ⊗_{A} k_{A}^{1/p} \ar[r] & 0}

such that d_{A} ⊗_{k_{A}} k_{A}^{1/p} is injective with cokernel naturally isomorphic to Ω_{k_{A}/ℤ}^{1} ⊗_{k_{A}} k_{A}^{1/p} (lemma 9.6.6). By the snake lemma, it then suffices to show that i is an isomorphism; but since both the source and target of the latter map are one-dimensional k_{A}^{1/p}-vector spaces, we come down to checking that i is not the zero map. But a simple inspection yields i(p ⊗ 1) = (0, 1). □

9.6.18. We wish next to show how the rule A ↦ Ω_{A} extends to a functor

p-Local → Alg.Mod (A, 𝔪_{A}) ↦ (k_{A}^{1/p}, Ω_{A}).

Namely, if p ∉ 𝔪_{A}^{2}, we let Ω_{A} be the k_{A}^{1/p}-module introduced in (9.6.15), and if p ∈ 𝔪_{A}^{2}, we set

Ω_{A} := Ω_{A/ℤ}^{1} ⊗_{A} k_{A}^{1/p}.

To any local ring homomorphism φ : A → B we attach the k_{A}^{1/p}-linear map

Ω_{φ} : Ω_{A} → Ω_{B}

defined as follows :

• In case p ∉ 𝔪_{B}^{2}, then obviously p ∉ 𝔪_{A}^{2}. Denote by \overline{φ} : k_{A}^{1/p} → k_{B}^{1/p} the map induced by φ. By (9.6.10), we have an induced A-linear map \overline{Ω}_{φ} ⊗_{A} \overline{φ} : \overline{Ω}_{A} ⊗_{A} k_{A}^{1/p} → \overline{Ω}_{B} ⊗_{B} k_{B}^{1/p}. Also, the naturality of the ghost component map \overline{ω}_{0} yields a k_{A}-linear map V_{1}(φ) : V_{1}(k_{A}) → V_{1}(k_{B}), and indeed, V_{1}(φ) is even k_{A}^{1/p}-linear. Moreover, for every a ∈ k_{A} and b ∈ k_{A}^{1/p} we may compute

\begin{matrix}ψ_{B}(\overline{Ω}_{φ}(j_{A}(0, a)) ⊗ \overline{φ}(b)) = & ψ_{B}(j_{B}(V_{1}(φ)(0, a)) ⊗ \overline{φ}(b)) \\ = & j_{B} ∘ ψ_{B}(V_{1}(φ)(0, a) ⊗ \overline{φ}(b)) \\ = & j_{B} ∘ V_{1}(φ) ∘ ψ_{A}((0, a) ⊗ b) \end{matrix}

so the pair (ψ_{B} ∘ (\overline{Ω}_{φ} ⊗ \overline{φ}), j_{B} ∘ V_{1}(φ)) determines a unique k_{A}^{1/p}-linear map Ω_{φ} : Ω_{A} → Ω_{B} as required.

• In case p ∈ 𝔪_{A}^{2}, then p ∈ 𝔪_{B}^{2}, and we set Ω_{φ} := Ω_{φ}^{1} ⊗_{A} \overline{φ}, where Ω_{φ}^{1} : Ω_{A/ℤ}^{1} → Ω_{B/ℤ}^{1} is the natural A-linear map.

• Lastly, if p ∉ 𝔪_{A}^{2} but p ∈ 𝔪_{B}^{2}, we let Ω_{φ} be the composition of the natural projection Ω_{A} → Ω_{A}^{1} ⊗_{A} k_{A}^{1/p} and Ω_{φ}^{1} ⊗_{A} \overline{φ}.

If φ′ : B → C is another morphism in p-Local, it is easy to verify that Ω_{φ′∘φ} = Ω_{φ′} ∘ Ω_{φ}, so we do get a functor as sought (details left to the reader).

9.6.19. In the same vein, we may extend the map d_{A} to a natural transformation of functors. Namely, if p ∉ 𝔪_{A}^{2}, we let d_{A} be the map introduced in (9.6.15), and if p ∈ 𝔪_{A}^{2}, we let d_{A} : A → Ω_{A} be the map induced by the universal derivation A → Ω_{A/ℤ}^{1}. Then it is easily seen that every local ring homomorphism φ : A → B induces a commutative diagram

\xymatrix{A \ar[r]^{^{d^{A}}} \ar[d]_{φ} & Ω_{A} \ar[d]^{Ω_{φ}} \\ B \ar[r]^{_{d_{B}}} & Ω_{B}.}

Proposition 9.6.20. Let (A, 𝔪_{A}), (B, 𝔪_{B}) be two local rings with residue characteristic p > 0, and φ : A → B a local ring homomorphism. We have :

(i) Suppose that φ is formally smooth for the 𝔪_{A}-adic topology on A and the 𝔪_{B}-adic topology on B. Then φ and Ω_{φ} induce injective maps

(𝔪_{A}/𝔪_{A}^{2}) ⊗_{k_{A}} k_{B} → 𝔪_{B}/𝔪_{B}^{2} Ω_{A} ⊗_{k_{A}^{1/p}} k_{B}^{1/p} → Ω_{B}.

(ii) Suppose that (a) 𝔪_{A} · B = 𝔪_{B}.

(b) The induced residue field extension k_{A} → k_{B} is algebraic and separable.

(c) φ is flat.

Then Ω_{φ} induces an isomorphism of k_{B}^{1/p}-vector spaces :

Ω_{A} ⊗_{A} B →^{∼} Ω_{B}.

(iii) If B = A/𝔪_{A}^{2} and φ is the natural projection, then Ω_{φ} is an isomorphism.

(iv) The functor Ω_{•} and the natural transformation d_{•} commute with all filtered colimits.

Proof. (ii): Assumption (a) and the flatness of φ imply that the induced map

(𝔪_{A}/𝔪_{A}^{2}) ⊗_{k_{A}} k_{B} → 𝔪_{B}/𝔪_{B}^{2}

is an isomorphism ([126, Th.22.3]), so p ∈ 𝔪_{A}^{2} if and only if p ∈ 𝔪_{B}^{2}. On the other hand, under assumption (b), we have

(9.6.21)  k_{B}^{1/p} = k_{A}^{1/p} · k_{B}.

Now, consider first the case where p ∈ 𝔪_{A}^{2}; then Ω_{φ} = Ω_{φ}^{1} ⊗_{A} \overline{φ} (notation of (9.6.18)), so the assertion follows from (9.6.21) together with the following

Claim 9.6.22. If p ∈ 𝔪_{A}^{2}, then Ω_{φ}^{1} induces a k_{B}-linear isomorphism

Ω_{A/ℤ}^{1} ⊗_{A} k_{B} → Ω_{B/ℤ}^{1} ⊗_{B} k_{B}.

Proof of the claim. Under the standing assumptions, (9.6.4) gives a commutative ladder with short exact rows

(9.6.23)  \xymatrix{0 \ar[r] & (𝔪_{A}/𝔪_{A}^{2}) ⊗_{k_{A}} k_{B} \ar[r] \ar[d] & Ω_{A/ℤ}^{1} ⊗_{A} k_{B} \ar[r] \ar[d] & Ω_{k_{A}/ℤ}^{1} ⊗_{k_{A}} k_{B} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & 𝔪_{B}/𝔪_{B}^{2} \ar[r] & Ω_{B/ℤ}^{1} ⊗_{B} k_{B} \ar[r] & Ω_{k_{B}/ℤ}^{1} \ar[r] & 0}

and we have already remarked that the left vertical arrow is an isomorphism; the same holds for the right vertical arrow, due to assumption (b). The claim follows. ♢

The case where p ∉ 𝔪_{B}^{2} is treated likewise : one argue as in the proof of claim 9.6.22, except that, instead of appealing to (9.6.5), one invokes proposition 9.6.17 and the naturality of \overline{d}_{•} established in (9.6.19) : the details shall be left to the reader.

(iii): For the case where p ∈ 𝔪_{A}^{2}, we consider again the induced ladder (9.6.23) : under the standing assumptions, clearly the first and third vertical arrows are isomorphisms, so the same holds for the middle one.

For the case where p ∉ 𝔪_{A}^{2}, one argues again likewise : the ladder (9.6.23) is replaced by the corresponding ladder for Ω_{•}, provided by proposition 9.6.17 and the naturality of \overline{d}_{•} : the details shall be left to the reader.

(iv): Let ((A_{λ}, 𝔪_{λ}) | λ ∈ Λ) be a filtered system of local rings and local ring homomorphisms. Notice that if p ∈ 𝔪_{λ}^{2} for some λ ∈ Λ, then p ∈ 𝔪_{μ}^{2} for every μ ≥ λ in Λ, and since Λ is filtered, we may then replace Λ with Λ/λ, and assume that p ∈ 𝔪_{μ} for every μ ∈ Λ. In this case, the contention follows straightforwardly from the corresponding assertion for the functor Ω_{•}^{1} and the universal derivation d_{•}.

It remains to consider the case where p ∉ 𝔪_{λ} for every λ ∈ Λ. For this, one applies proposition 9.6.17, which reduces again the contention to the previous case (details left to the reader).

(i): We consider first the following special case :

Claim 9.6.24. Let n ∈ ℕ be any integer, and 𝔮 ⊂ A[T_{1}, …, T_{n}] any prime ideal such that 𝔮 ∩ A = 𝔪_{A}. Set R := A[T_{1}, …, T_{n}]_{𝔮} and denote by 𝔪_{R} and k_{R} respectively the maximal ideal and the residue field of R. Then :

(i) dim_{k_{R}} 𝔪_{R}/𝔪_{R}^{2} = n + dim_{k_{A}} 𝔪_{A}/𝔪_{A}^{2}.

(ii) The natural map γ : (𝔪_{A}/𝔪_{A}^{2}) ⊗_{k_{A}} k_{R} → 𝔪_{R}/𝔪_{R}^{2} is injective.

(iii) The natural map Ω_{A/ℤ}^{1} ⊗_{A} R → Ω_{R/ℤ}^{1} is injective, and its image is a direct summand of Ω_{R/ℤ}^{1}.

(iv) The natural map Ω_{A} ⊗_{k_{A}^{1/p}} k_{R}^{1/p} → Ω_{R} is injective.

Proof of the claim. (i): By proposition 9.6.1(i), we have H_{i}𝕃_{k_{R}/k_{A}} = 0 for every i > 1; then the transitivity triangle for the cotangent complex relative to the maps A → k_{A} → k_{R} yields a short exact sequence of k_{R}-vector spaces :

0 → H_{1}𝕃_{k_{A}/A} ⊗_{k_{A}} k_{R} ⟶^{α} H_{1}𝕃_{k_{R}/A} → H_{1}𝕃_{k_{R}/k_{A}} → 0

([103, Ch.II, §2.1.2.1]). Likewise, since H_{i}𝕃_{R/A} = 0 for every i > 0 ([103, Ch.II, Cor.1.2.6.3]), the sequence of maps A → R → k_{R} yields an exact sequence of k_{R}-vector spaces :

0 → H_{1}𝕃_{k_{R}/A} ⟶^{β} H_{1}𝕃_{k_{R}/R} → Ω_{R/A}^{1} ⊗_{R} k_{R} → Ω_{k_{R}/A} → 0.

However, we have natural isomorphisms

H_{1}𝕃_{k_{A}/A} →^{∼} 𝔪_{A}/𝔪_{A}^{2} H_{1}𝕃_{k_{R}/R} →^{∼} 𝔪_{R}/𝔪_{R}^{2}

([103, Ch.III, Cor.1.2.8.1]), and clearly dim_{k_{R}} Ω_{R/A} ⊗_{R} k_{R} = n. Thus :

dim_{k_{A}} 𝔪_{A}/𝔪_{A}^{2} = dim_{k_{R}} H_{1}𝕃_{k_{R}/A} − dim_{k_{R}} H_{1}𝕃_{k_{R}/k_{A}}
dim_{k_{R}} 𝔪_{R}/𝔪_{R}^{2} = dim_{k_{R}} H_{1}𝕃_{k_{R}/A} + n − dim_{k_{R}} Ω_{k_{R}/k_{A}}.

Taking into account the identity

dim_{k_{R}} H_{1}𝕃_{k_{R}/k_{A}} = dim_{k_{R}} Ω_{k_{R}/k_{A}}

provided by proposition 9.6.1(iii), the assertion follows.

(ii): Notice that the composition of α and β yields an injective map (𝔪_{A}/𝔪_{A}^{2}) ⊗_{k_{A}} k_{R} → 𝔪_{R}/𝔪_{R}^{2} so it suffices to check that this composition equals γ. However, let

Σ : H_{1}𝕃_{k_{R}/R} ⟶^{d} H_{0}k_{R} ⊗_{R} 𝕃_{R/A}

be the boundary map of the transitivity triangle

(9.6.25)  k_{R} ⊗_{R} 𝕃_{R/A} → 𝕃_{k_{R}/A} → 𝕃_{k_{R}/R} → k_{R} ⊗_{R} σ𝕃_{R/A}

arising from the sequence A → R → k_{R}; we regard Σ as a complex placed in degrees [−1, 0]. Then, clearly, the triangle τ^{≥−1}(9.6.25) is naturally isomorphic in D(k_{R}-Mod) to the triangle

(9.6.26)  k_{R} ⊗_{R} H_{0}𝕃_{R/A}[0] → Σ → H_{1}𝕃_{k_{R}/R}[1] → k_{R} ⊗_{R} H_{0}𝕃_{R/A}[1].

According to [103, III.1.2.9.1], the complex Σ is naturally isomorphic to the complex

Θ : 𝔪_{R}/𝔪_{R}^{2} ⟶^{−d_{R/A}} k_{R} ⊗_{R} Ω_{R/A}^{1}

(placed in degrees [−1, 0]) where d_{R/A} is induced by the universal derivation R → Ω_{R/A}^{1}. Likewise, we have natural isomorphisms

(9.6.27)  τ^{≥−1}𝕃_{k_{R}/R} →^{∼} 𝔪_{R}/𝔪_{R}^{2}[1]  τ^{≥−1}𝕃_{R/A} →^{∼} Ω_{R/A}^{1}[0]

and under these identifications, 9.6.26 is the obvious triangle deduced from Θ. Especially, the map β is naturally identified with the inclusion map Ker d_{R/A} → 𝔪_{R}/𝔪_{R}^{2}.

Likewise, there exists a natural identification

(9.6.28)  τ^{≥−1}𝕃_{k_{A}/A} →^{∼} 𝔪_{A}/𝔪_{A}^{2}[1]  in D(k_{A}-Mod)

as well as a natural map of complexes

(9.6.29)  k_{R} ⊗_{k_{A}} (𝔪_{A}/𝔪_{A}^{2})[1] → Θ

deduced from γ. To conclude, it suffices to check that the map

(9.6.30)  τ^{≥−1}k_{R} ⊗_{A} 𝕃_{k_{A}/A} → τ^{≥−1}𝕃_{k_{R}/A}

coming from the transitivity triangle for the sequence A → k_{A} → k_{R}, corresponds to the morphism (9.6.29), under the identification (9.6.28) and the previous identification of τ^{≥−1}𝕃_{k_{R}/A} with Θ. To this aim, it suffices to compare the maps obtained by applying to these two morphisms the functor Ext_{k_{R}}^{1}(−, M[0]), for arbitrary k_{R}-modules M. Now, recall that there exists a natural isomorphism

Ext_{k_{R}}^{1}(τ^{≥−1}𝕃_{k_{R}/R}, M[0]) →^{∼} Exal_{R}(k_{R}, M) for every k_{R}-module M.

As explained in [103, III.1.2.8], under the identification (9.6.27), this becomes the following k_{R}-linear isomorphism

(9.6.31)  Hom_{k_{R}}(𝔪_{R}/𝔪_{R}^{2}, M) →^{∼} Exal_{R}(k_{R}, M)  φ ↦ φ ∗ U

(notation of [75, §2.5.5]), where

U : 0 → 𝔪_{R}/𝔪_{R}^{2} → R/𝔪_{R}^{2} → k_{R} → 0

is the natural extension. There is a natural surjection

Hom_{k_{R}}(𝔪_{R}/𝔪_{R}^{2}, M) → Ext_{k_{R}}^{1}(Θ, M[0])

and the foregoing implies that the natural isomorphism

Ext_{k_{R}}^{1}(τ^{≥−1}𝕃_{k_{R}/A}, M[0]) →^{∼} Exal_{A}(k_{R}, M)

is also realized as in (9.6.31): given a class c in Ext_{k_{R}}^{1}(Θ, M|0]), take an arbitrary representative φ : 𝔪_{R}/𝔪_{R}^{2} → M, and the correspondence associates with c the class of the push out φ ∗ U.

By the same token, we have a natural isomorphism

Ext_{k_{R}}^{1}(τ^{≥−1}k_{R} ⊗_{k_{A}} 𝕃_{k_{A}/A}, M(0]) →^{∼} Exal_{A}(k_{A}, M) for every k_{R}-module M

and on the one hand, the map Ext_{k_{R}}^{1}((9.6.30), M[0]) is identified naturally with the map

Exal_{A}(k_{R}, M) → Exal_{A}(k_{A}, M)

given by pull back along the inclusion map ι : k_{A} → k_{R}. On the other hand, by [103, III.1.2.8], the identification (9.6.27) induces the isomorphism

Hom_{k_{A}}(𝔪_{A}/𝔪_{A}^{2}, M) →^{∼} Exal_{A}(k_{A}, M) φ′ ↦ φ′ ∗ U′

where

U′ : 0 → 𝔪_{A}/𝔪_{A}^{2} → A/𝔪_{A}^{2} → k_{A} → 0

is the natural extension. So finally, the assertion boils down to the identity

(φ ∘ γ) ∗ U′ = φ ∗ U ∗ ι in Exal_{A}(k_{A}, M)

for every k_{R}-module M and every φ : 𝔪_{R}/𝔪_{R}^{2} → M. We leave the verification as an exercise for the reader.

(iii) is a standard calculation (more precisely, the complement of Ω_{A/ℤ}^{1} ⊗_{A} R in Ω_{R/ℤ}^{1} is the free R-module generated by dT_{1}, …, dT_{n}).

(iv): If p ∈ 𝔪_{A}^{2}, the assertion follows from (iii). Suppose then that p ∉ 𝔪_{A}^{2}. By inspecting the constructions, we get a natural commutative ladder of k_{R}^{1/p}-vector spaces

\xymatrix{0 \ar[r] & V_{1}(k_{A}) ⊗_{k_{A}^{1/p}} k_{R}^{1/p} \ar[r] \ar[d] & Ω_{A} ⊗_{k_{A}^{1/p}} k_{R}^{1/p} \ar[r] \ar[d] & Ω_{A/ℤ}^{1} ⊗_{A} k_{R}^{1/p} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & V_{1}(k_{R}) \ar[r] & Ω_{R} \ar[r] & Ω_{R/ℤ}^{1} ⊗_{R} k_{R}^{1/p} \ar[r] & 0}

whose central vertical arrow is the map of the claim. However, it is easily seen that the left vertical arrow is an isomorphism, so the assertion follows from (iii). ♢

Now, pick a free polynomial A-algebra C and a surjective map C → B of A-algebras; let 𝔮 ⊂ C be the preimage of 𝔪_{B}, set R := C_{𝔮}, and let 𝔪_{R} be the maximal ideal of R. Since B is formally smooth over A for the topologies of the maximal ideals, the induced surjection R/𝔪_{R}^{2} → B/𝔪_{B}^{2} admits a section B/𝔪_{B}^{2} → R/𝔪_{R}^{2} which is also a local map of A-algebras. In view of (iii), there follow commutative diagrams of k_{B}-linear and respectively k_{B}^{1/p}-linear maps

\xymatrix{(𝔪_{A}/𝔪_{A}^{2}) ⊗_{k_{A}} k_{B} \ar[dr] \ar[rr] &  & 𝔪_{B}/𝔪_{B}^{2} \ar[dl] & Ω_{A} ⊗_{k_{A}} k_{B}^{1/p} \ar[rr] \ar[dr] &  & Ω_{B} \ar[dl] \\  & 𝔪_{R}/𝔪_{R}^{2} &  &  & Ω_{R}}

which reduce the assertion to the case where B = R. In this case, write C as the filtered colimit of the system (C_{λ} | λ ∈ Λ) of its free polynomial A-subalgebras of finite type; for each λ ∈ Λ, let 𝔮_{λ} := 𝔮 ∩ C_{λ} and R_{λ} := (C_{λ})_{𝔮_{λ}}. Clearly R is the filtered colimit of the system (R_{λ} | λ ∈ Λ), so the assertion follows from (iv) and claim 9.6.24(ii). □

9.6.32. Let (A, 𝔪_{A}) be any object of p-Local, and k_{A} := A/𝔪_{A}. Consider a finite sequence f_{1}, …, f_{n} of elements of A, and e_{1}, …, e_{n} ∈ ℕ with e_{i} > 1 for every i = 1, …, n. Set

C := A[T_{1}, …, T_{n}]/(T_{1}^{e_{1}} − f_{1}, …, T_{n}^{e_{n}} − f_{n}).

Fix a prime ideal 𝔫 ⊂ C such that 𝔫 ∩ A = 𝔪_{A}, and let B := C_{𝔫}. So the induced map A → B is a local ring homomorphism; we denote by 𝔪_{B} the maximal ideal of B, and set k_{B} := B/𝔪_{B}. Also, let ν := dim_{k_{A}} E, where E ⊂ Ω_{A} is the k_{A}^{1/p}-vector space spanned by d_{A}f_{1}, …, d_{A}f_{n}.

Theorem 9.6.33. In the situation of (9.6.32), suppose moreover that :

(a) f_{i} ∈ 𝔪_{A}, for every i ≤ n such that p does not divide e_{i}.

(b) 𝔪_{A}/𝔪_{A}^{2} is a finite dimensional k_{A}-vector space.

Then 𝔪_{B}/𝔪_{B}^{2} is a finite dimensional k_{B}-vector space, and we have :

dim_{k_{B}} 𝔪_{B}/𝔪_{B}^{2} = n + dim_{k_{A}} 𝔪_{A}/𝔪_{A}^{2} − ν.

Proof. Let 𝔮 ⊂ A[T_{1}, …, T_{n}] be the preimage of 𝔫, set R := A[T_{1}, …, T_{n}]_{𝔮}, and denote by 𝔭 the maximal ideal of R, and by F ⊂ 𝔭/𝔭^{2} the k_{B}-vector space spanned by T_{1}^{e_{1}}−f_{1}, …, T_{n}^{e_{n}}−f_{n}. Clearly, we have a short exact sequence

0 → F → 𝔭/𝔭^{2} → 𝔪_{B}/𝔪_{B}^{2} → 0.

Suppose first that p ∉ 𝔪_{A}^{2}, in which case p ∉ 𝔭^{2}, by claim 9.6.24(ii), hence Ω_{R} is defined by (9.6.15). On the other hand, notice that d(T_{i}^{e_{i}}) = d(T_{i}^{e_{i}} − f_{i}) + d(f_{i}) in Ω_{R}, since T_{i}^{e_{i}} − f_{i} ∈ 𝔭. Now, if e_{i} is a multiple of p, Leibniz’s rule yields d(T_{i}^{e_{i}}) = e_{i} · T_{i}^{e_{i}−1}d(T_{i}) = 0. If e_{i} is not a multiple of p, we have f_{i} ∈ 𝔪_{A} by assumption; hence T_{i} ∈ 𝔭 and therefore d(T_{i}^{e_{i}}) = 0 again, since e_{i} > 1. In either case, we find

d(f_{i}) = −\overline{d}(T_{i}^{e_{i}} − f_{i}) in Ω_{R}, for every i = 1, …, n.

In view of proposition 9.6.17, it follows that

dim_{k_{B}} 𝔪_{B}/𝔪_{B}^{2} = dim_{k_{B}} 𝔭/𝔭^{2} − dim_{k_{B}^{1/p}} E′

where E′ ⊂ Ω_{R} is the k_{B}^{1/p}-vector space spanned by d(f_{1}), …, d(f_{n}). Then the assertion follows from claim 9.6.24(i,iv).

Lastly, suppose p ∈ 𝔪_{A}^{2}, so that p ∈ 𝔭^{2} as well. Arguing as in the foregoing case, we see that dim_{k_{B}} F equals the dimension of the k_{B}-vector subspace of Ω_{R/ℤ}^{1} ⊗_{R} k_{B} spanned by df_{1}, …, df_{n}, and in view of claim 9.6.24(iii), the latter equals ν, whence the contention. □

Corollary 9.6.34. In the situation of theorem 9.6.33, the following conditions are equivalent : (a) A is a regular local ring, and ν = n.

(b) B is a regular local ring.

Proof. Suppose first that (b) holds. Since the map A → B is faithfully flat, it is easily seen that A is noetherian, and then [63, Ch.0, Prop.17.3.3(i)] shows already that A is a regular local ring. Moreover, C is clearly a finite A-algebra, therefore dim_{k_{B}} 𝔪_{B}/𝔪_{B}^{2} = dim B ≤ dim A = dim_{k_{A}} 𝔪_{A}/𝔪_{A}^{2}. Theorem 9.6.33then implies that ν = n, so (a) holds.

Next, suppose that (a) holds. We apply theorem 9.6.33 as in the foregoing, to deduce that dim B = dim A = dim_{k_{B}} 𝔪_{B}/𝔪_{B}^{2}, whence (b) : details left to the reader. □

Remark 9.6.35. (i) Keep the notation of corollary 9.6.34. In [63, Ch.0, Th.22.5.4] it is asserted that condition (b) is equivalent to the following :

(a’) A is regular and the space E ⊂ Ω_{A/ℤ}^{1} ⊗_{A} k_{A} spanned by df_{1}, …, df_{n} has dimension n. When p ∈ 𝔪_{A}^{2}, this condition (a’) agrees with our condition (a), so in this case of course we

do have (a’)⇔(b). However, the latter equivalence fails in general, in case p ∉ 𝔪_{A}^{2} : the mistake is found in [63, Ch.0, Rem.22.4.8], which is false. The implication (a’)⇒(b) does remain true in all cases : this is easily deduced from theorem 9.6.33, since the image of d_{A}(f) in Ω_{A/ℤ}^{1}⊗_{k_{A}} k_{A}^{1/p} agrees with df ⊗ 1, for every f ∈ A. The proof of loc.cit. is correct for p ∈ 𝔪_{A}^{2}, which is the only case that is used in the proof of corollary 9.6.34.

(ii) Moreover, in the situation of corollary 9.6.34, suppose that B is a regular local ring. Then the sequence

(d_{B}f_{1}^{1/e_{1}}, …, d_{B}f_{n}^{1/e_{n}})

spans a k_{B}^{1/p}-vector subspace of Ω_{B} of dimension n. Indeed, consider the ring

C := B[T_{1}, …, T_{n}]/(T_{1}^{p} − f_{1}^{1/e_{1}}, …, T_{n}^{p} − f_{n}^{1/e_{n}}).

Corollary 9.6.34 applies to the extension A ⊂ C, the sequence (f_{1}, …, f_{n}), and the sequence of integers (pe_{1}, …, pe_{n}), so C is a regular local ring. But the same corollary applies as well to the extension B ⊂ C, the sequence (f_{1}^{1/e_{1}}, …, f_{n}^{1/e_{n}}), and the sequence of integers (p, …, p), and yields the assertion.

(iii) Let us say that the sequence (f_{1}, …, f_{n}) is maximal in A if

(d_{A}f_{1}, …, d_{A}f_{n})

is a basis of the k_{A}^{1/p}-vector space Ω_{A}. Then, in the situation of (ii), we claim that the sequence (f_{1}, …, f_{n}) is maximal in A if and only if the sequence (f_{1}^{1/e_{1}}, …, f_{n}^{1/e_{n}}) is maximal in B. Indeed, under the current assumptions we have dim_{k_{A}} Ω_{k_{A}/ℤ}^{1} = dim_{k_{B}} Ω_{k_{B}/ℤ}^{1}, since these integers are equal to the transcendence degree of k_{A} (and k_{B}) over 𝔽_{p}, and on the other hand dim_{k_{A}} 𝔪_{A}/𝔪_{A}^{2} = dim_{k_{B}} 𝔪_{B}/𝔪_{B}^{2}, since A and B are regular local rings of the same dimension; then the assertion follows from (ii), lemma 9.6.6and proposition 9.6.17(details left to the reader).

(iv) Likewise, consider any morphism φ : A → B in Local fulfilling conditions (a),(b) and (c) of proposition 9.6.20(ii). Then clearly the sequence (f_{1}, …, f_{n}) is maximal in A if and only if the sequence (φ(f_{1}), …, φ(f_{n})) is maximal in B.

9.6.36. Let p > 0 be a prime integer, and A an 𝔽_{p}-algebra. Denote by Φ_{A} : A → A the Frobenius endomorphism of A, given by the rule : a ↦ a^{p} for every a ∈ A. For every Amodule M, we let M_{(Φ)} be the A-module obtained from M via restriction of scalars along the map Φ_{A} (that is, a · m := a^{p}m for every a ∈ A and m ∈ M). Notice that Φ_{A} is an A-linear map A → A_{(Φ)}. Theorem 9.6.37, and part (i) of the following theorem 9.7.26are due to E.Kunz.

Theorem 9.6.37. Let A be a noetherian local 𝔽_{p}-algebra. Then the following conditions are equivalent :

(i) A is regular.

(ii) Φ_{A} is a flat ring homomorphism.

(iii) There exists n > 0 such that Φ_{A}^{n} is a flat ring homomorphism.

Proof. (i)⇒(ii): Let A^{∧} be the completion of A, and f : A → A^{∧} the natural map. Clearly

f ∘ Φ_{A} = Φ_{A^{∧}} ∘ f.

Since f is faithfully flat, it follows that Φ_{A} is flat if and only if the same holds for Φ_{A^{∧}}, so we may replace A by A^{∧}, and assume from start that A is complete, hence A = k[[T_{1}, …, T_{d}]], for a field k of characteristic p, and d = dim A ([63, Ch.0, Th.19.6.4]). Then, it is easily seen that

Φ_{A}(A) = A^{p} = k^{p}[[T_{1}^{p}, …, T_{d}^{p}]].

Set B := k[[T_{1}^{p}, …, T_{d}^{p}]]; the ring A is a free B-module (of rank p^{d}), hence it suffices to check that the inclusion map A^{p} → B is flat. However, denote by 𝔪 the maximal ideal of A^{p}; clearly B is an 𝔪-adically ideal-separated A-module (see [126, p.174, Def.]), hence it suffices to check that B/𝔪^{k}B is a flat A^{p}/𝔪^{k}-module for every k > 0 ([126, Th.22.3]). The latter is clear, since k[T_{1}^{p}, …, T_{d}^{p}] is a flat k^{p}[T_{1}^{p}, …, T_{d}^{p}]-module.

(ii)⇒(iii) is obvious.

(iii)⇒(i): Notice first that Spec Φ_{A}^{n} is the identity map on the topological space underlying Spec A; especially, Φ_{A}^{n} is flat if and only if it is faithfully flat, and the latter condition implies that Φ_{A}^{n} is injective. We easily deduce that if (iii) holds, then A is reduced. Now, consider quite generally, any finite system x_{•} := (x_{1}, …, x_{t}) of elements of A, and let I ⊂ A be the ideal generated by x_{•}; we shall say that x_{•} is a system of independent elements, if I/I^{2} is a free A/I-module of rank n. We show first the following :

Claim 9.6.38. Let y, z, x_{2}, …, x_{t} be a family of elements of A, such that x_{•} := (yz, x_{2}, …, x_{t}) is a system of independent elements, and denote by J ⊂ A the ideal generated by x_{•}. We have :

(i) (y, x_{2}, …, x_{t}) is a system of independent elements of A.

(ii) If length_{A}A/J is finite, then

length_{A}A/J = length_{A}A/(y, x_{2}, …, x_{t}) + length_{A}A/(z, x_{2}, …, x_{t}).

Proof of the claim. (i): Suppose a_{1}y +a_{2}x_{2} +⋯+a_{t}x_{t} = 0 is a linear relation with a_{1}, …, a_{t} ∈ A, and let I ⊂ A be the ideal generated by y, x_{2}, …, x_{t}. We have to show that a_{1}, …, a_{t} ∈ I. However, as a_{1}yz + a_{2}zx_{2} + ⋯ + a_{t}zx_{t} = 0, it follows by assumption, that a_{1} lies in J ⊂ I. Write a_{1} = b_{1}yz + b_{2}x_{2} + ⋯ + b_{t}x_{t}; then

b_{1}y^{2}z + (a_{2} + b_{2}y)x_{2} + ⋯ + (a_{t} + b_{t}y)x_{t} = 0.

Therefore a_{i} + b_{i}y ∈ J for i = 2, …, t, and therefore a_{2}, …, a_{t} ∈ I, as required. (ii): It suffices to show that the natural map of A-modules :

A/(z, x_{2}, ⋯, x_{t}) → I/J a ↦ ay + J

is an isomorphism. However, the surjectivity is immediate. To show the injectivity, suppose that ay ∈ J, i.e. ay = b_{1}yz + b_{2}x_{2} + ⋯ + b_{t}x_{t} for some b_{1}, …, b_{t} ∈ A; we deduce that (b_{1}z − a)yz + b_{2}zx_{2} + ⋯ + b_{t}zx_{t} = 0, hence a − b_{1}z ∈ J by assumption, so a lies in the ideal generated by z, x_{2}, …, x_{t}, as required. ♢

Now, set q := p^{n}, and A_{ν} := A^{q^{ν}} ⊂ A for every integer ν > 0; pick a minimal system x_{•} := (x_{1}, …, x_{t}) of generators of the maximal ideal 𝔪_{A} of A, and notice that, since A is reduced, Φ_{A}^{nν} induces an isomorphism A → A_{ν}, hence x_{•}^{(ν)} := (x_{1}^{q^{ν}}, …, x_{t}^{q^{ν}}) is a minimal system of generators for the maximal ideal 𝔪_{ν} of A_{ν}. Set as well I_{ν} := 𝔪_{ν}A; since the inclusion map A_{ν} → A is flat by assumption for every ν > 0, we have a natural isomorphism of Amodules

(𝔪_{ν}/𝔪_{ν}^{2}) ⊗_{A_{ν}} A →^{∼} I_{ν}/I_{ν}^{2}.

On the other hand, set k_{ν} := A_{ν}/𝔪_{ν}; by Nakayama’s lemma, dim_{k_{ν}} 𝔪_{ν}/𝔪_{ν}^{2} = t, so I_{ν}/I_{ν}^{2} is a free A-module of rank t, i.e. x_{•}^{(ν)} is an independent system of elements of A. From claim 9.6.38(ii) and a simple induction, we deduce that

(9.6.39)  length_{A}A/I_{ν} = length_{A^{∧}}A^{∧}/I_{ν}A^{∧} = q^{νt}  for every ν > 0

(where the first equality holds, since I_{ν} is an open ideal in the 𝔪_{A}-adic topology of A). According to [63, Ch.0, Th.19.9.8] (and its proof), A^{∧} contains a field isomorphic to k_{0} := A/𝔪_{A}, and the inclusion map k_{0} → A^{∧} extends to a surjective ring homomorphism k_{0}[[X_{1}, …, X_{t}]] → A^{∧}, such that X_{i} ↦ x_{i} for i = 1, …, t. Denote by J the kernel of this surjection; in view of (9.6.39), we have

length_{A^{∧}}k_{0}[[X_{1}, …, X_{t}]]/(J, X_{1}^{q^{ν}}, …, X_{t}^{q^{ν}}) = q^{νt}

which means that J ⊂ (X_{1}^{q^{ν}}, …, X_{t}^{q^{ν}}) for every ν > 0. We conclude that J = 0, and A^{∧} = k_{0}[[X_{1}, …, X_{t}]] is regular, so the same holds for A. □

As an application, let us point out the following combinatorial version of Kunz’s theorem :

Theorem 9.6.40. Let P be a monoid and k > 1 an integer; suppose that P^{♯} is fine and the Frobenius endomorphism k_{P} : P → P is flat (definition 4.8.40(ii)). Then P^{♯} is a free monoid.

Proof. First, we remark that k_{P}^{♯} : P^{♯} → P^{♯} is still flat (corollary 6.1.49(i)). Moreover, k_{P}^{♯} is injective. Indeed, suppose that x^{k} = y^{k} · u for some x, y ∈ P and u ∈ P^{×}; from theorem 6.1.42 we deduce that there exist b_{1}, b_{2}, t ∈ P such that

b_{1}x = b_{2}y  1 = b_{1}^{k}t  u = b_{2}^{k}t.

Especially, b_{1}, b_{2} ∈ P^{×}, so the images of x and y agree in P^{♯}. Hence, we may replace P by P^{♯}, and assume that k_{P} is flat and injective, in which case ℤ[k_{P}] : ℤ[P] → ℤ[P] is flat (theorem 6.2.3), integral and injective, hence it is faithfully flat. Now, let R be the colimit of the system of rings (R_{n} | n ∈ ℕ), where R_{n} := ℤ[P], and the transition map R_{n} → R_{n+1} is ℤ[k_{P}] for every n ∈ ℕ. The induced map j : R_{0} → R is still faithfully flat; moreover, let p be any prime divisor of k, and notice that j ∘p_{P} = j (where p_{P} is the p-Frobenius map). It follows that p_{P} is flat and injective as well, so 𝔽_{p}[p_{P}] : 𝔽_{p}[P] → 𝔽_{p}[P] is a flat ring homomorphism (again, by theorem 6.2.3), and then the same holds for the induced map 𝔽_{p}[[p_{P}]] : 𝔽_{p}[[P]] → 𝔽_{p}[[P]]. By theorem 9.6.37, we deduce that 𝔽_{p}[[P]] is a regular local ring, with maximal ideal 𝔪 := 𝔽_{p}[[𝔪_{P}]]; notice that the images of the elements of 𝔪_{P} \𝔪_{P}^{2} yield a basis for the 𝔽_{p}-vector space 𝔪/𝔪^{2}. Say that 𝔪_{P} \𝔪_{P}^{2} = {x_{1}, …, x_{s}}; it follows that the continuous ring homomorphism

𝔽_{p}[[T_{1}, …, T_{s}]] → 𝔽_{p}[[P]]  T_{i} ↦ x_{i}  for i = 1, …, s

is an isomorphism. From the discussion of (6.4.25), we immediately deduce that P ≃ ℕ^{⊕s}, as required. □

Next, we wish to present a characterization of regular local rings via the cotangent complex, borrowed from [3], which shall be used in the following section on excellent rings.

Lemma 9.6.41. Let A be a ring, f := (f_{1}, …, f_{n}) a completely secant sequence of elements of A, that generates an ideal I ⊂ A, and set A_{0} := A/I. Then there is a natural isomorphism

𝕃_{A_{0}/A} →^{∼} I/I^{2}[1] in D(A_{0}-Mod)

and I/I^{2} is a free A_{0}-module of rank n.

Proof. The sequence f is quasi-regular, by proposition 7.8.15, so the second assertion follows from remark 7.8.16(ii). Next, set R := ℤ[T_{1}, …, T_{n}], and let u : R → A be the unique ring homomorphism such that u(T_{i}) = a_{i} for i = 1, …, n. We get a cocartesian diagram of rings :

\xymatrix{R \ar[r]^{u} \ar[d]_{p} & A \ar[d] \\ ℤ \ar[r] & A_{0}}

with p the standard augmentation such that p(T_{1}) = ⋯ = p(T_{n}) = 0; also, Tor_{i}^{R}(ℤ, A_{0}) = 0 for every i > 0, by lemma 7.8.14(i). By [103, Ch.II, Prop.2.2.1] there follows a natural isomorphism:

A_{0} ⊗_{ℤ} 𝕃_{ℤ/R} →^{∼} 𝕃_{A_{0}/A} in D(A_{0}-Mod).

Thus, we are reduced to prove the sought isomorphism for the ring R and the regular sequence (T_{1}, …, T_{n}), in which case I = Ker p, and R/I = ℤ. However, by [103, Ch.II, Prop.2.1.2], the sequence of ring homomorphisms ℤ → R ⟶^{p} ℤ induces a distinguished triangle

𝕃_{ℤ/R}[−1] → 𝕃_{R/ℤ} ⊗_{R} ℤ → 𝕃_{ℤ/ℤ} → 𝕃_{ℤ/R} in D(ℤ-Mod)

and to conclude, it suffices to recall that 𝕃_{ℤ/ℤ} →^{∼} 0, and 𝕃_{R/ℤ} →^{∼} Ω_{R/ℤ}^{1}[0], by virtue of [103, Ch.II, Prop.1.2.4.4]. □

Proposition 9.6.42. Let (A, 𝔪) be a local noetherian ring, a ∈ 𝔪, and A_{0} := A/aA. The following conditions are equivalent :

(a) a is a regular element of A.

(b) H_{2}𝕃_{A_{0}/A} = 0, and aA/a^{2}A is a free A_{0}-module of rank one.

Proof. For any ring R, any non-invertible element x ∈ R, and any n ∈ ℕ, set R_{n} := R/x^{n+1}R, and consider the R_{0}-linear map

β_{x,n} : R_{0} → x^{n}R/x^{n+1}R (a modxR) ↦ (x^{n}a modx^{n+1}R).

Notice that the sequence (x) is R-quasi-regular if and only if β_{x,n} is an isomorphism for every n ∈ ℕ (see definition 7.8.13). We remark :

Claim 9.6.43. Suppose that R is a noetherian local ring, denote by κ_{R} the residue field of R, and let n > 0 be any given integer. The following conditions are equivalent :

(c) β_{x,n} is an isomorphism.

(d) x^{n} ≠ 0 and Tor_{1}^{R_{0}}(x^{n}R/x^{n+1}R, κ_{R}) = 0.

(e) x^{n} ≠ 0 and the surjection R_{0} → κ_{R} induces a surjective map

H_{2}(𝕃_{R_{n−1}/R} ⊗_{R_{n−1}} R_{0}) → H_{2}(𝕃_{R_{n−1}/R} ⊗_{R_{n−1}} κ_{R}).

Proof of the claim. It is easily seen that (c)⇒(d).

Conversely, if (d) holds, notice that x^{n}R/x^{n+1}R ≠ 0, as ⋂_{n∈ℕ} x^{n}R = 0 ([126, Th.8.10(i)]). Hence κ_{R} ⊗_{R} β_{x,n} is an isomorphism of one-dimensional κ_{R}-vector spaces. On the other hand, under assumption (d), the natural map κ_{R} ⊗_{R} Ker β_{x,n} → Ker(κ_{R} ⊗_{R} β_{x,n}) is an isomorphism. By Nakayama’s lemma, we conclude that Ker β_{x,n} = 0, i.e. (c) holds.

Next, recall that H_{0}(𝕃_{R_{n−1}/R}) = 0 for every n ∈ ℕ ([103, Ch.II, Prop.1.2.4.2]); then the Künneth spectral sequence ([163, Th.5.6.4]) yields natural isomorphisms of R_{0}-modules

H_{1}(𝕃_{R_{n−1}/R} ⊗_{R_{n−1}} M) →^{∼} x^{n}R/x^{2n}R ⊗_{R_{n−1}} M →^{∼} x^{n}R/x^{n+1}R ⊗_{R_{0}} M

for every R_{0}-module M ([103, Ch.III, Cor.1.2.8.1]). Denote by 𝔪_{0} ⊂ R_{0} the maximal ideal; there follows a left exact sequence

0 → Tor_{1}^{R_{0}}(x^{n}R/x^{n+1}R, κ_{R}) → H_{1}(𝕃_{R_{n−1}/R} ⊗_{R_{n−1}} 𝔪_{0}) → H_{1}(𝕃_{R_{n−1}/R} ⊗_{R_{n−1}} R_{0})

which shows that (d)⇔(e) (details left to the reader). ♢

Claim 9.6.44. Let f : R → R′ be a ring homomorphism, and x′ := f(x). Suppose that β_{x,n} and β_{x′,n} are bijections, for some n ∈ ℕ, and set R′_{n} := R′/x′^{n+1}R′. Then the induced morphism

𝕃_{R_{0}/R_{n}} ⊗_{R_{0}} R′_{0} → 𝕃_{R′_{0}/R′_{n}}

is an isomorphism in D(R′_{0}-Mod).

Proof of the claim. If n = 0, there is nothing to prove, hence assume that n > 0. We remark that, for every i = 0, …, n, we have an exact complex

R_{n} ⟶^{x^{i+1}} R_{n} ⟶^{x^{n−i}} R_{n}.

Indeed, for i = 0, this results immediately from the assumption that Ker β_{n,x} = 0. Suppose that i > 0, and that the assertion is known for i − 1; then, if yx^{n−i} = x^{n+1}z for some y, z ∈ R, the inductive hypothesis implies that y = x^{i}u for some u ∈ R. It follows that x^{n}(u − zx) = 0, and then u − zx ∈ xR, again since Ker β_{x,n} = 0; thus, u ∈ xR, and y ∈ x^{i+1}R, as asserted.

We deduce that the R_{n}-module R_{0} admits a free resolution

Σ : ⋯ → R_{n} ⟶^{x} R_{n} ⟶^{x^{n}} R_{n} ⟶^{x} R_{n} → R_{0}.

The same argument applies to R′ and its element x′, and yields a corresponding free resolution

Σ′ of the R′_{n}-module R′_{0}. Clearly Σ′ ⊗_{R_{n}} R′_{n} = Σ, i.e. the natural morphism R_{0} ⊗^{L}_{R_{n}} R′_{n} → R′_{0} is an isomorphism in D(R′_{0}-Mod). The claim then follows from [103, Ch.II, Prop.2.2.1]. ♢

With these preliminaries, we may now return to the situation of the proposition : first, lemma 9.6.41says that (a)⇒(b). For the converse, in light of proposition 7.8.15, it suffices to show that β_{a,n} : A_{0} → a^{n}A/a^{n+1}A is bijective for every n ∈ ℕ. We shall argue by induction on n.

For n = 0, there is nothing to prove. Assume that n > 0, and that the assertion is known for n − 1. Let 𝔫 ⊂ A[T] be the (unique) maximal ideal containing T, and set B := A[T]_{𝔫}. We let f : B → A be the map of A-algebras given by the rule : T ↦ a. Define as usual B_{n} := B/T^{n+1}B and A_{n} := A/a^{n+1}A for every n ∈ ℕ. Clearly β_{T,n−1} : B_{0} → T^{n−1}B/T^{n}B is bijective, and the same holds for β_{a,n−1}, by inductive assumption. Then, claim 9.6.44 says that the induced morphism 𝕃_{B_{0}/B_{n−1}} ⊗_{B_{0}} A_{0} → 𝕃_{A_{0}/A_{n−1}} is an isomorphism in D(A_{0}-Mod). Denote by κ the residue field of A and B; by considering the Künneth spectral sequence

E_{pq}^{2} := Tor_{p}^{A_{0}}(H_{q}𝕃_{A_{0}/A}, κ) ⇒ H_{p+q}(𝕃_{A_{0}/A} ⊗_{A_{0}} κ)

([163, Th.5.6.4]) we get H_{2}(𝕃_{A_{0}/A} ⊗_{A_{0}} κ) = 0, by virtue of (b). Consequently, the commutative diagram of ring homomorphisms

\xymatrix{B \ar[r] \ar[d] & B_{n−1} \ar[r] \ar[d] & B_{0} \ar[d] \\ A \ar[r] & A_{n−1} \ar[r] & A_{0}}

induces a commutative ladder with exact rows ([103, Ch.II, Prop.2.1.2]) :

\xymatrix{H_{3}(𝕃_{B_{0}/B_{n−1}} ⊗_{B_{0}} κ) \ar[r] \ar[d] & H_{2}(𝕃_{B_{n−1}/B} ⊗_{B_{n−1}} κ) \ar[d] \ar[r] & H_{2}(𝕃_{B_{0}/B} ⊗_{B_{0}} κ) \ar[d] \\ H_{3}(𝕃_{A_{0}/A_{n−1}} ⊗_{A_{0}} κ) \ar[r] & H_{2}(𝕃_{A_{n−1}/A} ⊗_{A_{n−1}} κ) \ar[r] & 0}

whose left vertical arrow is an isomorphism. It follows that the central vertical arrow is surjective. Consider now the commutative diagram induced by the maps B_{0} → A_{0} → κ :

(9.6.45)  \xymatrix{H_{2}(𝕃_{B_{n−1}/B} ⊗_{B_{n−1}} B_{0}) \ar[r] \ar[d] & H_{2}(𝕃_{B_{n−1}/B} ⊗_{B_{n−1}} κ) \ar[d] \\ H_{2}(𝕃_{A_{n−1}/A} ⊗_{A_{n−1}} A_{0}) \ar[r] & H_{2}(𝕃_{A_{n−1}/A} ⊗_{A_{n−1}} κ).}

We have just seen that the right vertical arrow of (9.6.45) is surjective, and the same holds for its top horizontal arrow, in light of claim 9.6.43. Thus, finally, the bottom horizontal arrow is surjective as well, so β_{a,n} is an isomorphism (claim 9.6.43), and the proposition is proved. □

Theorem 9.6.46. Let (A, 𝔪) be a local ring, and I ⊂ 𝔪 an ideal; denote by κ the residue field of A, and set A_{0} := A/I. Consider the following conditions :

(a) Every minimal system of generators of I is a regular sequence of elements of A.

(b) I is generated by a regular sequence of A.

(c) The natural morphism 𝕃_{A_{0}/A} → I/I^{2}[1] is an isomorphism in D(A_{0}-Mod), and I/I^{2} is a flat A_{0}-module.

(d) H_{n}(𝕃_{A_{0}/A} ⊗_{A_{0}} κ) = 0 for every n ≥ 2.

(e) H_{2}(𝕃_{A_{0}/A} ⊗_{A_{0}} κ) = 0. Then (a)⇒(b)⇒(c)⇒(d) ⇒(e), and if A is noetherian, (e)⇒(a).

Proof. Clearly (a)⇒(b) and (d)⇒(e); moreover, (b)⇒(c), by lemma 9.6.41 and proposition 7.8.15. To see that (c)⇒(d), we invoke the Künneth spectral sequence ([163, Th.5.6.4]) :

(9.6.47)  E_{pq}^{2} := Tor_{p}^{A_{0}}(H_{q}𝕃_{A_{0}/A}, κ) ⇒ H_{p+q}(𝕃_{A_{0}/A} ⊗_{A_{0}} κ).

Lastly, let us check that (e)⇒(a) when A is noetherian. Thus, let (a_{1}, …, a_{n}) be a minimal system of generators for I; recall that n = dim_{κ} I ⊗_{A} κ. We argue by induction on n. If n = 0, there is nothing to show. If n = 1, recall that H_{0}𝕃_{A_{0}/A} = 0 and H_{1}𝕃_{A_{0}/A} = I/I^{2} ([103, Ch.II, Prop.1.2.4.2 and Ch.III, Cor.1.2.8.1]); then from the spectral sequence (9.6.47) we deduce that

Tor_{1}^{A_{0}}(I/I^{2}, κ) = E_{1,1}^{2} = E_{1,1}^{∞} = 0

so I/I^{2} is a flat A_{0}-module, by the local flatness criterion ([126, Th.22.3]). Then, again from (9.6.47) it follows easily that

H_{2}(𝕃_{A_{0}/A}) ⊗_{A_{0}} κ = E_{0,2}^{2} = E_{0,2}^{∞} = 0

and by [103, Ch.II, Cor.2.3.7], the A_{0}-module H_{2}𝕃_{A_{0}/A} is of finite type, so that H_{2}𝕃_{A_{0}/A} = 0, by Nakayama’s lemma; we may then conclude with proposition 9.6.42.

Next, suppose that n > 1, and that the assertion is already known when dim_{κ} I ⊗_{A} κ < n. Set B := A/a_{1}A and J := IB. The ring homomorphisms A → B → A_{0} induce exact sequences

0 → H_{2}(𝕃_{A_{0}/B} ⊗_{A_{0}} κ) → H_{1}(𝕃_{B/A} ⊗_{B} κ) → I ⊗_{A} κ → J ⊗_{B} κ → 0
H_{3}(𝕃_{A_{0}/B} ⊗_{A_{0}} κ) → H_{2}(𝕃_{B/A} ⊗_{B} κ) → 0

([103, Ch.II, Prop.2.1.2 and Ch.III, Cor.1.2.8.1]). However, clearly J admits a generating system of length n − 1, hence n′ := dim_{κ} J ⊗_{B} κ < n. But dim_{κ} H_{1}(𝕃_{B/A} ⊗_{B} κ) = 1, so we have necessarily n′ = n − 1 and H_{2}(𝕃_{A_{0}/B} ⊗_{A_{0}} κ) = 0. By inductive assumption, the sequence (\overline{a}_{2}, …, \overline{a}_{n}) of the images in B of (a_{2}, …, a_{n}), is then regular. By virtue of lemma 9.6.41 and proposition 7.8.15, it follows that H_{3}(𝕃_{A_{0}/B} ⊗_{A_{0}} κ) = 0, whence, H_{2}(𝕃_{B/A} ⊗_{B} κ) = 0, so a_{1} is a regular element in A, and finally, (a_{1}, …, a_{n}) is a regular sequence, as stated. □

Corollary 9.6.48. Let (A, 𝔪) be a local noetherian ring, with residue field κ. The following conditions are equivalent :

(a) A is regular.

(b) The natural morphism 𝕃_{κ/A} → 𝔪/𝔪^{2}[1] is an isomorphism in D(κ-Mod).

(c) H_{2}𝕃_{κ/A} = 0.

Proof. It follows immediately, by invoking theorem 9.6.46with I := 𝔪. □

We conclude this section with a desingularization result which was first proved in [19, Chap.II Prop.3.2] and can also be found in [118, Lemma 3.10]; it is moreover closely related to [123, Prop.1.2.8].

9.6.49. Let (A, 𝔪) be a local noetherian ring, 𝔭 ⊂ A a prime ideal with dim A/𝔭 = 1. We define inductively as follows, an inverse system of A-schemes :

⋯ → X_{2} ⟶^{π_{1}} X_{1} ⟶^{π_{0}} X_{0}

and for every n ∈ ℕ a reduced subscheme C_{n} ⊂ X_{n}, with π_{n}(C_{n+1}) = C_{n}. Set X_{0} := Spec A and C_{0} := Spec A/𝔭. For every n ∈ ℕ, let Z_{n} ⊂ C_{n} be the pre-image of 𝔪, endowed with its reduced subscheme structure, and ℐ_{n} ⊂ 𝒪_{X_{n}} the ideal defining Z_{n}; we let π_{n} : X_{n+1} → X_{n} be the blow-up of ℐ_{n}, and C_{n+1} ⊂ X_{n+1} the strict transform of C_{n}. For every n ∈ ℕ, let also i_{n} : C_{n} → X_{n} be the closed immersion, and set 𝒥_{n} := Ker(𝒪_{X_{n}} → i_{n∗}𝒪_{C_{n}}).

Proposition 9.6.50. In the situation of (9.6.49), suppose that the normalization (A/𝔭)^{ν} of A/𝔭 is a finite A/𝔭-algebra. Then there exists n_{0} ∈ ℕ such that for every n ≥ n_{0} we have:

(i) C_{n} is a regular scheme.

(ii) 𝒥_{n,z}^{m}/𝒥_{n,z}^{m+1} is a torsion-free 𝒪_{C_{n},z}-module, for every z ∈ Z_{n} and every m ∈ ℕ.

(iii) If moreover, A_{𝔭} is a regular ring, the same holds for 𝒪_{X_{n},z}, for every z ∈ Z_{n}.

Proof. (i): Recall that the restriction \overline{π}_{n} : C_{n+1} → C_{n} of π_{n} is the blowup of the ideal \overline{ℐ} _{n} ⊂ 𝒪_{C_{n}} defining Z_{n}, for every n ∈ ℕ; we notice :

Claim 9.6.51. \overline{π}_{n} is a finite morphism, for every n ∈ ℕ.

Proof of the claim. Clearly, C_{n+1} is an integral scheme, and \overline{π}_{n} is projective and induces an isomorphism \overline{π}_{n}^{−1}(C_{n} \ Z_{n}) →^{∼} C_{n} \ Z_{n}. Let y ∈ C_{n+1} be any point with z := \overline{π}_{n}(y) ∈ Z_{n}; the induced map 𝒪_{C_{n},z} → 𝒪_{C_{n+1},y} is injective, and induces an isomorphism of the respective fields of fractions. Moreover, dim 𝒪_{C_{n+1},y} ≥ 1, hence the transcendence degree of the residue field extension κ(z) → κ(y) is 0, by [126, Th.15.5]. By [126, Th.5.6], this implies that \overline{π}_{n}^{−1}(z) is a κ(z)-scheme of dimension 0 and of finite type, for every z ∈ Z_{n}, so \overline{π}_{n}^{−1}(x) is a finite set, for every x ∈ C_{n}. Then the claim follows from [65, Ch.IV, Th.8.11.1]. ♢

From claim 9.6.51, we have C_{n} = Spec R_{n}, for some finite A/𝔭-algebra R_{n} lying in the normalization (A/𝔭)^{ν} of A/𝔭. Since A/𝔭 is noetherian and (A/𝔭)^{ν} is a finite A/𝔭-algebra, it follows that there exists n_{0} ∈ ℕ such that \overline{π}_{n} is an isomorphism for every n ≥ n_{0}. The latter means that \overline{ℐ} _{n} is an invertible ideal for every such n, whence the assertion.

(ii,iii): By (i) we find k ∈ ℕ such that C_{k} = Spec(A/𝔭)^{ν}. We then get an inverse system

⋯ → X_{k}(z) ×_{X_{k}} X_{k+2} → X_{k}(z) ×_{X_{k}} X_{k+1} → X_{k}(z) for every z ∈ Z_{k}

(notation of definition 4.9.17(iii)) whose transition morphism X_{k}(z)×_{X_{k}} π_{k+n} is the blow-up of the restriction of ℐ_{k+n}, for every n ∈ ℕ. Since Z_{k} has finitely many points, it suffices therefore to prove the assertions for A replaced by A_{z} := 𝒪_{X_{k},z} and 𝔭 replaced by the prime ideal 𝔭_{z} ⊂ A_{z} such that Spec A_{z}/𝔭_{z} = C_{k}(z), for every z ∈ Z_{k}. Thus, we may assume that A/𝔭 is regular, and \overline{π}_{n} is an isomorphism C_{n+1} →^{∼} C_{n}, so that Z_{n} has a unique point z_{n}, for every n ∈ ℕ. Let

A_{n} := 𝒪_{X_{n},z_{n}}  𝔪_{n} := ℐ_{n,z_{n}}  𝔭_{n} := 𝒥_{n,z_{n}}  for every n ∈ ℕ

so that X_{n}(z_{n}) = Spec A_{n}. For given n ∈ ℕ, let x_{1}, …, x_{k} be a finite system of generators of 𝔭_{n}, and x_{0} ∈ 𝔪_{n} any element whose image in A_{n}/𝔭_{n} generates the ideal 𝔪_{n}/𝔭_{n}, so that the system x_{0}, …, x_{k} generates the ideal 𝔪_{n}. We have already remarked that X_{n}(z_{n}) ×_{X_{n}} π_{n} is the blow-up of the ideal (ℐ_{n})_{|X_{n}(z_{n})}; hence, X_{n}(z_{n}) ×_{X_{n}} X_{n+1} admits the affine open covering U_{0} ∪ ⋯ ∪ U_{k}, where U_{i} := Spec B_{i}, with B_{i} := A_{n}[\frac{x_{0}}{x_{i}} , …, \frac{x_{k}}{x_{i}}] for every i = 0, …, k.

Claim 9.6.52. (i) C_{n+1} ⊂ U_{0}.

(ii) x_{1}/x_{0}, …, x_{k}/x_{0} generate a prime ideal 𝔮_{n} ⊂ B_{0}.

(iii) 𝔫_{n} := B_{0}x_{0} + 𝔮_{n} is a maximal ideal of B_{0}.

(iv) The induced maps A_{n}/𝔭_{n} → B_{0}/𝔮_{n} → A_{n+1}/𝔭_{n+1} are isomorphisms.

Proof of the claim. (i): The universal property of the blow-up X_{n}(z_{n}) ×_{X_{n}} π_{n} says that for i = 0, …, k, every morphism of schemes f : T → X_{n}(z_{n}) such that f^{−1}(ℐ_{n})_{|X_{n}(z_{n})} · 𝒪_{T} is invertible and generated by the image of x_{i}, factors uniquely through U_{i}. Especially, the projection X_{n}(z_{n}) ×_{X_{n}} C_{n+1} → X_{n}(z_{n}) factors through U_{0}, whence (i).

(ii,iv): From (i) we get a commutative diagram of schemes

\xymatrix{X_{n}(z_{n}) ×_{X_{n}} C_{n+1} \ar[r] \ar[d] & U_{0} \ar[d] \\ X_{n}(z_{n}) ×_{X_{n}} C_{n} \ar[r] & X_{n}(z_{n})}

whose horizontal arrows are closed immersions, and whose left vertical arrow is an isomorphism. Hence, the induced morphism

(9.6.53)  X_{n}(z_{n}) ×_{X_{n}} C_{n+1} → U_{0} ×_{X_{n}} C_{n} = Spec B_{0}/𝔭_{n}B_{0}

is a closed immersion. However, X_{n}(z_{n}) ×_{X_{n}} C_{n+1} = Spec A_{n+1}/𝔭_{n+1}, and notice that x_{0} annihilates the ideal 𝔮_{n}/𝔭_{n}B_{0} of B_{0}/𝔭_{n}B_{0}; since x_{0} is a regular element of A_{n+1}/𝔭_{n+1}, it follows that the map B_{0}/𝔭_{n}B_{0} → A_{n+1}/𝔭_{n+1} corresponding to (9.6.53) factors through a surjection f : B_{0}/𝔮_{n} → A_{n+1}/𝔭_{n+1}. On the other hand, the unique map of A_{n}-algebras A_{n} → B_{0}/𝔮_{n} factors through a surjection g : A_{n}/𝔭_{n} → B_{0}/𝔮_{n}, and by assumption f ∘ g is an isomorphism, whence both (ii) and (iv).

(iii): From (iv) we get surjections A_{n}/𝔪_{n} → B_{0}/𝔫_{n+1} → A_{n+1}/𝔪_{n+1}, whence the claim. ♢ In light of claim 9.6.52(ii,iv), and since A_{n+1} is a localization of B_{0}, we see that 𝔭_{n+1} is

generated by the images of x_{1}/x_{0}, …, x_{k}/x_{0}; whence, for every m ∈ ℕ, an A_{n}/𝔭_{n}-linear map

φ_{n,m} : 𝔭_{n}^{m}/𝔭_{n}^{m+1} → 𝔭_{n+1}^{m}/𝔭_{n+1}^{m+1} (y mod 𝔭_{n}^{m+1}) ↦ (x_{0}^{−m}y mod 𝔭_{n+1}^{m+1})

whose image contains a set of generators for the A_{n+1}/𝔭_{n+1}-module 𝔭_{n+1}^{m}/𝔭_{n+1}^{m+1}. Since the natural map A_{n}/𝔭 → A_{n+1}/𝔭_{n+1} is an isomorphism, it follows that φ_{n,m} is surjective, for every m ∈ ℕ. Moreover, set R_{n} := ⨁_{m∈ℕ} 𝔭_{n}^{m}/𝔭_{n}^{m+1} for every n ∈ ℕ; it is easily seen that the direct sum of the maps φ_{n,m} yields a surjective map of graded A_{n}/𝔭_{n}-algebras :

φ_{n} : R_{n} → R_{n+1}.

Claim 9.6.54. (i) A_{n}[x_{0}^{−1}] ⊗_{A_{n}} φ_{n} is an isomorphism.

(ii) Ann_{R_{n}}(𝔪_{n}) ⊂ Ker φ_{n}.

Proof of the claim. (i): We have already remarked that A_{n+1} is a localization of B_{0}, and 𝔭_{n+1} = 𝔮_{n}A_{n+1}; hence 𝔭_{n+1}^{m}/𝔭_{n+1}^{m+1} = A_{n+1} ⊗_{B_{0}} 𝔮_{n}^{m}/𝔮_{n}^{m+1}, so that φ_{n,m} is a composition :

𝔭_{n}^{m}/𝔭_{n}^{m+1} ⟶^{α_{m}} 𝔮_{n}^{m}/𝔮_{n}^{m+1} ⟶^{β_{m}} 𝔭_{n+1}^{m}/𝔭_{n+1}^{m+1} for every m ∈ ℕ

where α_{m} is given by the rule : (y mod 𝔭_{n}^{m+1}) ↦ (x_{0}^{−m}y mod 𝔮_{n}^{m+1}), for every y ∈ 𝔭_{n}^{m}, and β_{m} is the localization. Since the localization B_{0} → A_{n+1} induces an isomorphism B_{0}/𝔮_{n} →^{∼} A_{n+1}/𝔭_{n+1} (claim 9.6.52(iii)), the map β_{m} is bijective for every m ∈ ℕ. Thus, we need to check that A_{n}[x_{0}^{−1}] ⊗_{A_{n}} α_{m} is bijective for every m ∈ ℕ. However, clearly A_{n}[1/x_{0}] = B_{0}[1/x_{0}], and 𝔭_{n}[1/x_{0}] = 𝔮_{n}[1/x_{0}] in B_{0}[1/x_{0}], whence the assertion.

(ii): We come down to checking that the annihilator of x_{0} in 𝔭_{m}^{n}/𝔭_{m}^{n+1} lies in Ker φ_{m,n} for every m ∈ ℕ. Hence, let y ∈ 𝔭_{m}^{n} whose class \overline{y} ∈ 𝔭_{m}^{n}/𝔭_{m}^{n+1} is annihilated by x_{0}; the latter means that there exist polynomials P, Q ∈ A_{n}[X_{1}, …, X_{k}] homogeneous of degree n and respectively n + 1, such that y = P (x_{1}, …, x_{k}), and x_{0}y = Q(x_{1}, …, x_{k}). It follows that

x_{0}^{m+1}P (x_{1}/x_{0}, …, x_{k}/x_{0}) = x_{0}^{m+1}Q(x_{1}/x_{0}, …, x_{k}/x_{0}) in B_{0}

whence t := P (x_{1}/x_{0}, …, x_{k}/x_{0}) = Q(x_{1}/x_{0}, …, x_{k}/x_{0}) in B_{0}, since x_{0} is regular in this ring. But α_{m}(\overline{y}) is precisely the class of t in 𝔮_{m}^{n}/𝔮_{m}^{n+1}, so α_{m}(\overline{y}) = 0, whence the assertion. ♢

Now, set J_{r} := Ann_{R_{0}}(𝔪_{0}^{r}) for every r ∈ ℕ; since R_{0} is noetherian, there exists r ∈ ℕ such that J_{r} = J_{r+1}. Also, clearly Ann_{R_{n}}(𝔪_{n}^{i}) = Ann_{R_{n}}(𝔪_{0}^{i}) for every i, n ∈ ℕ. In light of claim 9.6.54, it then follows easily that Ker (φ_{n} ∘ ⋯ ∘ φ_{0}) = J_{r} for every n ≥ r, and R_{0}/J_{r} →^{∼} R_{n} is then a torsion-free A_{0}/𝔭_{0}-algebra for every n ≥ r, whence (ii).

It follows in particular that 𝔭_{n}^{m}/𝔭_{n}^{m+1} is a free A_{n}/𝔭_{n}-module of finite rank for every n ≥ r and every m ∈ ℕ, since it is torsion-free, and since A_{n}/𝔭_{n} is a discrete valuation ring. According to [66, Ch.IV, Cor.19.1.2], proposition 7.8.15 and remark 7.8.16(i,ii), in order to show (iii) it then suffices to check that the natural map of graded A_{n}/𝔭_{n}-algebras

ψ : Sym_{A_{n}}^{•}(𝔭_{n}/𝔭_{n}^{2}) → R_{n}

is an isomorphism for every n ≥ r. Now, by the same token, Sym_{A_{n}}^{m}(𝔭_{n}/𝔭_{n}^{2}) is a free A_{n}/𝔭_{n}- module of finite rank, for every n ≥ r and every m ∈ ℕ; Since ψ is surjective, it then suffices to check that these modules have the same rank, for every such n and m. However, notice that the structure map A → A_{n} induces an isomorphism A_{𝔭} →^{∼} (A_{n})_{𝔭_{n}}, for every n ∈ ℕ, hence (A_{n})_{𝔭_{n}} is a regular ring for every such n, under the assumptions of (iii). Invoking again [66, Ch.IV, Cor.19.1.2], proposition 7.8.15 and remark 7.8.16, we deduce that the localization ψ_{𝔭_{n}} of ψ is an isomorphism for every n ≥ r, whence the contention. □

Definition 9.6.55. Let A be a noetherian ring. We say that A is a Nagata ring, if for every 𝔭 ∈ Spec A and every finite field extension κ(𝔭) ⊂ L, the integral closure of A/𝔭 in L is a finite A-module. (Such rings are called universally japanese in [63, Ch.0, Déf.23.1.1].)

Corollary 9.6.56. Let (A, 𝔪) be a Nagata local domain such that the regular locus of Spec A is an open subset. Then there exist a regular local domain (A′, 𝔪′) and an injective local ring homomorphism A → A′ of essentially finite type, such that :

(i) The induced map Frac(A) →^{∼} Frac(A′) is an isomorphism. (ii) The induced map A/𝔪 → A′/𝔪′ is a finite field extension.

Proof. Recall that the regular locus U of X := Spec A is the set of all 𝔭 ∈ X such that A_{𝔭} is a regular ring. If U = Spec A, we may take A′ = A; hence, suppose that U is a proper subset of X. Since A is noetherian, we may find 𝔭 ∈ U such that U ∩Spec A/𝔭 = {𝔭}. Especially, {𝔭} is an open subset of Spec A/𝔭, hence there exists f ∈ A \ 𝔭 such that Spec (A/𝔭)_{f} = {𝔭}, so that (A/𝔭)_{f} is a field. By [63, Ch.0, Cor.16.3.3], it follows that dim A/𝔭 = 1. Then the assertion follows immediately from proposition 9.6.50(iii) and claim 9.6.51. □

9.7. Regular morphisms and excellent rings. Recall that a morphism of schemes f : X → Y is called regular, if it is flat, and for every y ∈ Y, the fibre f^{−1}(y) is locally noetherian and geometrically regular ([64, Ch.IV, Déf.6.8.1]).

Lemma 9.7.1. Let f : X → Y and g : Y → Z be two morphisms of locally noetherian schemes. We have :

(i) If f and g are regular, then the same holds for g ∘ f. (ii) If g ∘ f is regular, and f is faithfully flat, then g is regular.

Proof. (i): Clearly h := g ∘ f is flat. Let z ∈ Z be any point, and K any finite extension of κ(z). Set

X′ := h^{−1}(z) ×_{κ(z)} K and Y′ := g^{−1}(z) ×_{κ(z)} K.

It is easily seen that the induced morphism f′ : X′ → Y′ is regular. Moreover, for every y ∈ Y′, the local ring 𝒪_{Y′,y′} is regular, since g is regular. Then the assertion follows from :

Claim 9.7.2. Let A → B be a flat and local ring homomorphism of local noetherian rings. Denote by 𝔪_{A} ⊂ A the maximal ideal, and suppose that both A and B_{0} := B/𝔪_{A}B are regular. Then B is regular.

Proof of the claim. On the one hand, dim B = dim A + dim B_{0} ([63, Ch.IV, Cor.6.1.2]). On the other hand, let a_{1}, …, a_{n} be a minimal generating system for 𝔪_{A}, and b_{1}, …, b_{m} a system of elements of the maximal ideal 𝔪_{B} of B, whose images in B_{0} is a minimal generating system for 𝔪_{B}/𝔪_{A}B. By Nakayama’s lemma, it is easily seen that the system a_{1}, …, a_{n}, b_{1}, …, b_{m} generates the ideal 𝔪_{B}. Since A and B_{0} are regular, n = dim A and m = dim B, so n + m = dim B, and the claim follows. ♢

(ii): Clearly g is flat. Then the assertion follows easily from [63, Ch.0, Prop.17.3.3(i)] : details left to the reader. □

Definition 9.7.3. Let A be a noetherian ring.

(i) We say that A is a G-ring, if the formal fibres of Spec A are geometrically regular, i.e. for every 𝔭 ∈ Spec A, the natural morphism Spec A_{𝔭}^{∧} → Spec A_{𝔭} from the spectrum of the 𝔭-adic completion of A, is regular : see [64, Ch.IV, §7.3.13].

(ii) We say that A is quasi-excellent, if A is a G-ring, and moreover the following holds. For every prime ideal 𝔭 ⊂ A, and every finite radicial extension K′ of the field of fractions K of B := A/𝔭, there exists a finite B-subalgebra B′ of K′ such that the field of fractions of B′ is K′, and the regular locus of Spec B′ is an open subset (the latter is the set of all prime ideals 𝔮 ⊂ B′ such that B′_{𝔮} is a regular ring).

(iii) We say that A is catenary if any two saturated chains (𝔭_{0} ⊂ ⋯ ⊂ 𝔭_{n}), (𝔮_{0} ⊂ ⋯ ⊂ 𝔮_{m}) of prime ideals of A, with 𝔭_{0} = 𝔮_{0} and 𝔭_{n} = 𝔮_{m}, have the same length (so n = m) ([63, Ch.0, (16.1.4)]). We say that A is universally catenary if every A-algebra of finite type is catenary.

(iv) We say that A is excellent, if it is quasi-excellent and universally catenary ([64, Ch.IV, Déf.7.8.2]).

Lemma 9.7.4. Let A be a noetherian ring.

(i) If A is quasi-excellent, then A is a Nagata ring (see definition 9.6.55). (ii) If A is a G-ring, then every A-algebra of essentially finite type is a G-ring. (iii) Suppose that the natural morphism Spec A_{𝔪}^{∧} → Spec A_{𝔪} is regular for every maximal

ideal 𝔪 ⊂ A. Then A is a G-ring. (iv) If A is a local G-ring, then A is quasi-excellent.

(v) If A is a complete local ring, then A is excellent.

Proof. (i): This is [64, Ch.IV, Cor.7.7.3].

(ii): The assertion for localizations is obvious, and the assertion on A-algebras of finite type follows from [64, Ch.IV, Th.7.4.4(ii)].

(iv) follows from [64, Ch.IV, Th.6.12.7, Prop.7.3.18, Th.7.4.4(ii)].

(v): In light of (iv), it suffices to remark that every complete noetherian local ring is universally catenary ([64, Ch.IV, Prop.5.6.4] and [63, Ch.0, Th.19.8.8(i)]) and is a G-ring ([63, Ch.0, Th.22.3.3, Th.22.5.8, and Prop.19.3.5(iii)]).

(iii): Let us remark, more generally :

Claim 9.7.5. Let φ : A → B be a faithfully flat ring homomorphism of noetherian rings, such that f := Spec φ is regular. If B is a G-ring, the same holds for A.

Proof of the claim. In light of (ii), we easily reduce to the case where both A and B are local, φ is a local ring homomorphism, and it suffices to show that the natural morphism π_{A} : Spec A^{∧} → Spec A is regular (where A^{∧} is the completion of A). Consider the commutative diagram :

(9.7.6)  \xymatrix{Spec B^{∧} \ar[r]^{f^{∧}} \ar[d]_{π_{B}} & Spec A^{∧} \ar[d]^{π_{A}} \\ Spec B \ar[r]^{f} & Spec A.}

By assumption, π_{B} is a regular morphism; Then the same holds for f ∘ π_{B} = π_{A} ∘ f^{∧} (lemma 9.7.1(i)). However, it is easily seen that the induced map φ^{∧} : A^{∧} → B^{∧} is still a local ring homomorphism, hence f^{∧} is faithfully flat, so the claim follows from lemma 9.7.1(ii). ♢

Now, in order to prove (iii), it suffices to check that A_{𝔪} is a G-ring for every maximal ideal 𝔪 ⊂ A. In view of our assumption, the latter assertion follows from claim 9.7.5and (v). □

Proposition 9.7.7. Let A be a ring, B a noetherian A-algebra of finite (Krull) dimension, n ∈ ℕ an integer, and suppose that H_{k}(𝕃_{B/A} ⊗_{B} κ(𝔭)) = 0 for every prime ideal 𝔭 ⊂ B and every k = n, …, n + dim B. Then H_{n}(𝕃_{B/A} ⊗_{B} M) = 0 for every B-module M.

Proof. Let us start out with the following more general :

Claim 9.7.8. Let A be a ring, B a noetherian A-algebra, 𝔭 ∈ Spec B a prime ideal, n ∈ ℕ an integer, and suppose that the following two conditions hold :

(a) H_{n}(𝕃_{B/A} ⊗_{B} κ(𝔭)) = 0.

(b) H_{n+1}(𝕃_{B/A} ⊗_{B} B/𝔮) = 0 for every proper specialization 𝔮 of 𝔭 in Spec B.

Then, H_{n}(𝕃_{B/A} ⊗_{B} B/𝔭) = 0.

Proof of the claim. Let b ∈ B \ 𝔭 be any element, and set M := B/(𝔭 + bB). Since B is noetherian, M admits a finite filtration M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{n} := M such that, for every i = 0, …, n−1, the subquotient M_{i+1}/M_{i} is isomorphic to B/𝔮, for some proper specialization 𝔮 of 𝔭 ([126, Th.6.4]). From (b), and a simple induction, we deduce that H_{n+1}(𝕃_{B/A}⊗_{B}M) = 0. Whence, by considering the short exact sequence of B-modules

0 → B/𝔭 ⟶^{b} B/𝔭 → M → 0

we see that scalar multiplication by b is an injective map on the B-module H_{n}(𝕃_{B/A} ⊗_{B} B/𝔭). Since this holds for every b ∈ B \ 𝔭, we conclude that the natural map

H_{n}(𝕃_{B/A} ⊗_{B} B/𝔭) → H_{n}(𝕃_{B/A} ⊗_{B} B/𝔭) ⊗_{B} B_{𝔭} = H_{n}(𝕃_{B/A} ⊗_{B} κ(𝔭))

is injective. Then the assertion follows from (a). ♢

Now, let 𝔭 ⊂ B be any prime ideal; the assumption, together with claim 9.7.8 and a simple induction on d := dim B/𝔭, shows that

H_{k}(𝕃_{B/A} ⊗_{B} B/𝔭) = 0 for every k = n, …, n + dim B − d.

For any B-module M, set H(M) := H_{n}(𝕃_{B/A}⊗_{B} M). Especially, we get H(B/𝔭) = 0, for any prime ideal 𝔭 ⊂ B. Since B is noetherian, it follows easily that H(M) = 0 for any B-module M of finite type (details left to the reader). Next, if M is arbitrary, we may write it as the union of the filtered family (M_{i} | i ∈ I) of its submodules of finite type; since H(M) is the colimit of the induced system (H(M_{i}) | i ∈ I), we see that H(M) = 0, as sought. □

Corollary 9.7.9. Let A → B be a homomorphism of noetherian rings. Then the following conditions are equivalent :

(a) Ω_{B/A}^{1} is a flat B-module, and H_{i}𝕃_{B/A} = 0 for every i > 0.

(b) Ω_{B/A}^{1} is a flat B-module, and H_{1}𝕃_{B/A} = 0.

(c) The induced morphism of schemes f : Spec B → Spec A is regular.

(d) H_{1}(𝕃_{B/A} ⊗_{B} κ(x)) = 0 for every x ∈ Spec B.

Proof. Set X := Spec B and Y := Spec A; let x ∈ X, and y := f(x). We remark :

Claim 9.7.10. The following conditions are equivalent :

(e) the map on stalks 𝒪_{Y,y} → 𝒪_{X,x} is formally smooth for the preadic topologies defined by the maximal ideals.

(f) f is flat at the point x, and the κ(y)-algebra 𝒪_{f^{−1}(y),x} is geometrically regular.

(g) H_{1}(𝕃_{B/A} ⊗_{B} κ(x)) = 0.

(h) H_{i}(𝕃_{B/A} ⊗_{B} κ(x)) = 0 for every i > 0.

Proof of the claim. The equivalence (e)⇔(f) follows from [63, Ch.0, Th.19.7.1]. Next, to ease notation, set A′ := 𝒪_{X,x} and B′ := 𝒪_{Y,y}; we have a natural identification

𝕃_{B/A} ⊗_{B} κ(x) →^{∼} 𝕃_{B′/A′} ⊗_{B′} κ(x)

([103, Ch.II, Cor.2.3.1.1]) from which we get (e)⇔(g), by virtue of proposition 8.6.30. Obviously, (h)⇒(g). Conversely, suppose that (g) holds, and set A′′ := κ(x) and B′′ := A′′ ⊗_{A′} B′; by the foregoing, we know already that B′ is flat A′-algebra, so we get an isomorphism

𝕃_{B′/A′} ⊗_{B′} κ(x) →^{∼} 𝕃_{B′′/A′′} ⊗_{B′′} κ(x)

([103, Ch.II, Prop.2.2.1]). Moreover, B′′ is a regular local ring; in view of corollary 9.6.48, the sequence of ring homomorphisms A′′ → B′′ → κ(x) then yields an isomorphism

H_{i}(𝕃_{B′′/A′′} ⊗_{B′′} κ(x)) →^{∼} H_{i}𝕃_{κ(x)/A′′} for every i > 1

([103, Ch.II, Prop.2.1.2]). Combining with proposition 9.6.1(i), we get (h). ♢

From claim 9.7.10, we already see that (a)⇒(b)⇒(c)⇔(d). Lastly, suppose that (d) holds; in order to show that Ω_{B/A}^{1} is a flat B-module, it suffices to check that Tor_{1}^{B}(H_{0}𝕃_{B/A}, M) = 0 for every B-module M. To this aim, recall the Künneth spectral sequence

E_{pq}^{2} := Tor_{p}^{B}(H_{q}𝕃_{B/A}, M) ⇒ H_{p+q}(𝕃_{B/A} ⊗_{B} M).

It is easily seen that E_{1,0}^{2} = E_{1,0}^{∞}, so we are reduced to proving that H_{1}(𝕃_{B/A} ⊗_{B} M) = 0 for every such M. Then, proposition 9.7.7further reduces to checking that H_{i}(𝕃_{B/A} ⊗_{B} κ(x)) = 0 for every i > 0. The latter follows from (d) and claim 9.7.10. Likewise, taking M := B, and arguing with proposition 9.7.7 and claim 9.7.10, we also deduce that H_{i}𝕃_{B/A} = 0 for every i > 0, whence (a). □

Corollary 9.7.11. Let A be a local noetherian ring, A^{∧} the completion of A. Then the following conditions are equivalent :

(a) A is quasi-excellent.

(b) Ω_{A^{∧}/A}^{1} is a flat A^{∧}-module, and H_{1}𝕃_{A^{∧}/A} = 0.

(c) Ω_{A^{∧}/A}^{1} is a flat A^{∧}-module, and H_{i}𝕃_{A^{∧}/A} = 0 for every i > 0.

Proof. Taking into account lemma 9.7.4(iii), this is a special case of corollary 9.7.9. □

Proposition 9.7.12. Let p > 0 be a prime integer, A a regular local and excellent 𝔽_{p}-algebra, B a local noetherian A-algebra, 𝔪_{B} the maximal ideal of B, and M a B-module of finite type. Then the B-module Ω_{A/𝔽_{p}}^{1} ⊗_{A} M is separated for the 𝔪_{B}-preadic topology.

Proof. Let φ : A → B be the structure morphism, and set 𝔭 := φ^{−1}𝔪_{B}; the localization A_{𝔭} is still excellent (lemma 9.7.4(ii,iv)) and regular, and clearly Ω_{A_{𝔭}/𝔽_{p}}^{1} ⊗_{A_{𝔭}} M = Ω_{A/𝔽_{p}}^{1} ⊗_{A} M, hence we may replace A by A_{𝔭}, and assume that φ is local. Let A^{∧} and B^{∧} be the completions of A and B, set M^{∧} := B^{∧} ⊗_{B} M, and notice that both of the natural maps 𝔽_{p} → A and A → A^{∧} are regular. In view of corollary 9.7.9, it follows that both of the natural B-linear maps

Ω_{A/𝔽_{p}}^{1} ⊗_{A} M → Ω_{A/𝔽_{p}}^{1} ⊗_{A} M^{∧} Ω_{A/𝔽_{p}}^{1} ⊗_{A} M → Ω_{A^{∧}/𝔽_{p}}^{1} ⊗_{A^{∧}} M^{∧}

are injective (to see the injectivity of the second map, one applies the transitivity triangle arising from the sequence of ring homomorphisms 𝔽_{p} → A → A^{∧} : details left to the reader). Thus, we may assume that A is complete. Now, for every ring R, and every integer m ∈ ℕ, set

(9.7.13)  R_{(m)} := R[[T_{1}, …, T_{m}]]  R_{⟨m⟩} := R[[T_{1}^{p}, …, T_{m}^{p}]] ⊂ R_{(m)}.

With this notation, we have an isomorphism A →^{∼} κ_{(d)} of 𝔽_{p}-algebras, where κ is the residue field of A, and d := dim A ([63, Ch.0, Th.19.6.4]).

Claim 9.7.14. Let K be any field of characteristic p, and m ∈ ℕ any integer. We have :

(i) There exists a cofiltered system (K^{λ} | λ ∈ Λ) of subfields of K such that [K : K^{λ}] is finite for every λ ∈ Λ, and ⋂_{λ∈Λ} K^{λ} = K^{p}.

(ii) For every system (K^{λ} | λ ∈ Λ) fulfilling the condition of (i), the following holds : (a) Ω_{K_{(m)}/K_{⟨m⟩}^{λ}}^{1} is a free K_{(m)}-module of finite rank, for every λ ∈ Λ, and the rule

M ↦ Ω_{m}(M) := lim_{λ∈Λ} (Ω_{K_{(m)}/K_{⟨m⟩}^{λ}}^{1} ⊗_{K_{(m)}} M)

defines an exact functor K_{(m)}-Mod → K_{(m)}-Mod.

(b) The natural map

η_{m}(M) : Ω_{K_{(m)}/𝔽_{p}}^{1} ⊗_{K_{(m)}} M → Ω_{m}(M)

is injective for every K_{(m)}-module M.

(c) Let F (resp. F^{λ}) denote the field of fractions of K_{(m)} (resp. of K_{⟨m⟩}^{λ}, for every λ ∈ Λ); then ⋂_{λ∈Λ} F^{λ} = F^{p}.

Proof of the claim. (i) and (ii.c) follow from [63, Ch.0, Prop.21.8.8] (and its proof).

(ii.a): For given λ ∈ Λ, say that x_{1}, …, x_{r} is a p-basis of K over K^{λ}; then it is easily seen that x_{1}, …, x_{r}, T_{1}, …, T_{m} is a p-basis of K_{(m)} over K_{⟨m⟩}^{λ} (see [63, Ch.0, Déf.21.1.9]). According to [63, Ch.0, Cor.21.2.5], it follows that Ω_{K_{(m)}/K_{⟨m⟩}^{λ}}^{1} is the free K_{(m)}-module of finite type with

basis dx_{1}, …, dx_{r}, dT_{1}, …, dT_{m}. Moreover, say that K^{μ} ⊂ K^{λ}, and let x_{r+1}, …, x_{s} be a pbasis of K^{λ} over K^{μ}; then x_{1}, …, x_{s} is a p-basis of K over K^{μ} ([63, Ch.0, Lemme 21.1.10]), so the induced map

Ω_{K_{(m)}/K_{⟨m⟩}^{μ}}^{1} → Ω_{K_{(m)}/K_{⟨m⟩}^{λ}}^{1}

is a projection onto a direct factor, and the assertion follows easily.

(ii.b): We are easily reduced to the case where M is a K_{(m)}-module of finite type, and in light of (ii.a), we may further assume that M is a cyclic K_{(m)}-module. Next, we remark that, due to (ii.c), the natural map

Ω_{F/𝔽_{p}}^{1} → lim_{λ∈Λ} Ω_{F/F^{λ}}^{1}

is injective ([63, Ch.0, Th.21.8.3]); in other words, η_{m}(F) is injective. In order to show the injectivity of η_{m}(M), it then suffices to check that the functor M ↦ Ω_{K_{(m)}/𝔽_{p}}^{1} ⊗_{K_{(m)}} M is exact. The latter holds by virtue of corollary 9.7.9, since the (unique) morphism of schemes Spec K_{(m)} → Spec 𝔽_{p} is obviously regular. This completes the proof for m = 0. Suppose now that m > 0, and that the injectivity of η_{n}(M) is already known for every n < m and every K_{(n)}-module M. By the foregoing, it remains to check that η_{m}(K_{(m)}/I) is injective, for every non-zero ideal I ⊂ K_{(m)}. Pick any non-zero f ∈ I, and set R := K_{(m−1)}; according to [34, Ch.VII, n.7, Lemme 3] and [34, Ch.VII, n.8, Prop.6], there exist an automorphism σ of the ring K_{(m)}, and elements g ∈ R[T_{m}], u ∈ K_{(m)}^{×} such that σ(f) = u·g, and g = T_{m}^{d}+a_{1}T_{m}^{d−1}+⋯+a_{d} for some d ≥ 0 and certain elements a_{1}, …, a_{d} of the maximal ideal of R. However, set M′ := K_{(m)}/σ(I); in view of the commutative diagram of 𝔽_{p}-modules :

\xymatrix{Ω_{K_{(m)}/𝔽_{p}}^{1} ⊗_{K_{(m)}} M \ar[r]^{η_{m}(M)} \ar[d] & Ω_{m}(M) \ar[d] \\ Ω_{K_{(m)}/𝔽_{p}}^{1} ⊗_{K_{(m)}} M′ \ar[r]^{η_{m}(M′)} & Ω_{m}(M′)}

(whose vertical arrows are induced by σ) we see that η_{m}(M) is injective, if and only if the same holds for η_{m}(M′). Hence, we may replace I by σ(I), and assume that g ∈ I. In this case, set also R^{λ} := K_{⟨m−1⟩}^{λ} for every λ ∈ Λ, and notice that the natural maps

R[T_{m}]/g^{p}R[T_{m}] → K_{(m)}/g^{p}K_{(m)} R^{λ}[T_{m}^{p}]/g^{p}R^{λ}[T_{m}^{p}] → K_{⟨m⟩}^{λ}/g^{p}K_{⟨m⟩}^{λ}

are bijective; there follows a commutative diagram of K_{(m)}-modules :

\xymatrix{Ω_{R[T_{m}]/𝔽_{p}}^{1} ⊗_{R[T_{m}]} M \ar[r] \ar[d]_{α^{λ}⊗_{R[Tm]}M} & Ω_{K_{(m)}/𝔽_{p}}^{1} ⊗_{K_{(m)}} M \ar[d]^{η_{m}^{λ}⊗_{K_{(m)}}M} \\ Ω_{R[T_{m}]/R^{λ}}^{1} ⊗_{R[T_{m}]} M \ar[r] & Ω_{K_{(m)}/K_{⟨m⟩}^{λ}}^{1} ⊗_{K_{(m)}} M}

whose horizontal arrows are isomorphisms, and η_{m}(M) = lim_{λ∈Λ} η_{m}^{λ} ⊗_{K_{(m)}} M. On the other hand, for every λ ∈ Λ we have a commutative ladder of R[T_{m}]-modules with exact rows :

Σ_{λ} : [figure: 0; Ω_{R/𝔽_{p}}^{1} ⊗_{R} R[T_{m}]; η_{m−1}^{λ}⊗_{R}R[T_{m}]; Ω_{R[T_{m}]/𝔽_{p}}^{1}; α^{λ}; Ω_{R[T_{m}]/R}^{1}; 0; 0; Ω_{R/R^{λ}}^{1} ⊗_{R} R[T_{m}]; Ω_{R[T_{m}]/R^{λ}}^{1}; Ω_{R[T_{m}]/R}^{1}; 0]

and notice that the rows of Σ_{λ} ⊗_{R[T_{m}]} M are still short exact, for every λ ∈ Λ. By inductive assumption, lim_{λ∈Λ} η_{m−1}^{λ} ⊗_{R} M is an injective map; we deduce that the same holds for lim_{λ∈Λ} α^{λ} ⊗_{R[T]} M, and the claim follows. ♢

Take K := κ, and pick any cofiltered system (K^{λ} | λ ∈ Λ) as provided by claim 9.7.14(i); in light of claim 9.7.14(ii.b), it now suffices to show that the resulting Ω_{d}(M) is a separated Bmodule, for every noetherian κ_{(d)}-algebra B and every B-module M of finite type. However, Ω_{d}(M) is a submodule of ∏_{λ∈Λ}(Ω_{K_{(m)}/K_{⟨m⟩}^{λ}}^{1} ⊗_{K_{(m)}} M), hence we are reduced to checking that each direct factor of the latter B-module is separated. But in view of claim 9.7.14(ii.a), we see that each such factor is a finite direct sum of copies of M, so finally we come down to the assertion that M is separated for the 𝔪_{B}-adic topology, which is well known. □

Theorem 9.7.15. Let φ : A → B be a local ring homomorphism of local noetherian rings. Suppose that A is quasi-excellent, and φ is formally smooth for the preadic topologies defined by the maximal ideals. Then Spec φ is regular.

Proof. This is the main result of [4]. We begin with the following general remark :

Claim 9.7.16. Let R be a local noetherian ring, 𝔪_{R} ⊂ R the maximal ideal, H : R-Mod → R-Mod an additive functor, and suppose that

(a) H is R-linear, i.e. H(t · 1_{M}) = t · H(1_{M}) for every R-module M, and every t ∈ R.

(b) H is semi-exact, i.e. for any short exact sequence 0 → M′ → M → M′′ → 0 of R-modules, the induced sequence H(M′) → H(M) → H(M′′) is exact.

(c) H commutes with filtered colimits.

(d) H(M) is separated for the 𝔪_{R}-adic topology, for every R-module M of finite type.

(e) H(R/𝔪_{R}) = 0. Then H(M) = 0 for every R-module M.

Proof of the claim. Since H commutes with filtered colimits, it suffices to show that H(M) = 0 for every finitely generated R-module M, and since H is semi-exact, a simple induction reduces further to the case where M is a cyclic R-module. Let now ℱ be the family of all ideals I of R such that H(R/I) ≠ 0, and suppose, by way of contradiction, that ℱ ≠ ∅; pick a maximal element J of ℱ, and set M := R/J. By assumption, J ≠ 𝔪_{R}; thus, let t ∈ R be a noninvertible element with t ∉ J; we get an exact sequence

H(M) ⟶^{t} H(M) → H(B_{(m+n)}/(J + tB_{(m+n)}))

whose third term vanishes, by the maximality of J. On the other hand, ⋂_{n∈ℕ} t^{n}H(M) = 0, since H(M) is separated. Thus H(M) = 0, contradicting the choice of J, and the claim follows. ♢

Denote by κ the residue field of A, and for every m, n ∈ ℕ, let φ_{m,n} : A_{(m)} → B_{(m+n)} be the composition of φ_{(m)} : A_{(m)} → B_{(m)} (the T -adic completion of φ ⊗_{A} A[T_{1}, …, T_{m}]) with the natural inclusion map B_{(m)} → B_{(m+n)} (notation of (9.7.13)).

Claim 9.7.17. In the situation of the theorem, suppose furthermore that A is either a field or a complete discrete valuation ring of mixed characteristic. Then, the morphism Spec φ_{m,n} is regular for every m, n ∈ ℕ.

Proof of the claim. Set

H(M) := H_{1}(𝕃_{B_{(m+n)}/A_{(m)}} ⊗_{B_{(m+n)}} M).

We shall consider separately three different cases :

• Suppose first that A is a field of characteristic p > 0. According to corollary 9.7.9, it suffices to show that H(M) vanishes for every B_{(m+n)}-module M. Notice that the natural map 𝔽_{p} → B_{(m+n)} is regular; from the distinguished triangle ([103, Ch.II, Prop.2.1.2])

𝕃_{A_{(m)}/𝔽_{p}} ⊗_{A_{(m)}} B_{(m+n)} → 𝕃_{B_{(m+n)}/𝔽_{p}} → 𝕃_{B_{(m+n)}/A_{(m)}} → 𝕃_{A_{(m)}/𝔽_{p}} ⊗_{A_{(m)}} B_{(m+n)}[1]

and corollary 9.7.9, we deduce an injective B_{(m+n)}-linear map

H(M) → Ω_{A_{(m)}/𝔽_{p}}^{1} ⊗_{A_{(m)}} M

from which it follows that if M is a B_{(m+n)}-module of finite type, H(M) is a separated B_{(m+n)}- module, for the preadic topology defined by the maximal ideal of B_{(m+n)} (proposition 9.7.12). Since φ is formally smooth, φ_{m,n} is also formally smooth for the preadic topologies defined by the maximal ideals of A_{(m)} and B_{(m+n)}; from proposition 8.6.30, we see that H(M) = 0, if M is the residue field of B_{(m+n)}. Then the assertion follows from claim 9.7.16.

• Next, suppose that A is either a field of characteristic zero, or a complete discrete valuation ring of mixed characteristic (so either κ = A, or else κ is a field of positive characteristic). According to corollary 9.7.9, it suffices to show that H(M) vanishes for M = κ(𝔮), where 𝔮 ⊂ B_{(m+n)} is any prime ideal. Fix such 𝔮, and set 𝔭 := 𝔮 ∩ A_{(m)}; if 𝔭 = 0, then M is a K-algebra, where K is the field of fractions of A_{(m)}; now, K is a field of characteristic zero, and B′ := B_{(m+n)} ⊗_{A_{(m)}} K is a regular local K-algebra, so the induced morphism Spec B′ → Spec K is regular; since H(M) = H_{1}(𝕃_{B′/K} ⊗_{B′} M), the assertion follows from corollary 9.7.9. Notice that this argument applies especially to the case where m = 0; for the general case, we argue by induction on m. Hence, suppose that n ∈ ℕ, m > 0, and that the assertion is already known for φ_{n,m−1}.

Consider first the case where A is a discrete valuation ring, and 𝔭 contains the maximal ideal of A, and set \overline{B} := B ⊗_{A} κ. Since φ_{m,n} is flat, and since M is a \overline{B}_{(m+n)}-module, we have a natural isomorphism

H(M) →^{∼} H_{1}(𝕃_{\overline{B}_{(m+n)}/κ_{(m)}} ⊗_{\overline{B}_{(m+n)}} M).

Then the sought vanishing follows from the foregoing, since \overline{B} is a formally smooth κ-algebra (for the preadic topology of its maximal ideal).

Lastly, suppose that either A is a field, or 𝔭 does not contain the maximal ideal of A (and 𝔭 ≠ 0). In either of these two cases, we may find f ∈ 𝔭 whose image in κ_{(m)} is not zero. According to [34, Ch.VII, n.7, Lemme 3] and [34, Ch.VII, n.8, Prop.6], we may find an automorphism σ of the A-algebra A_{(m)} and elements g ∈ A_{(m−1)}[T_{m}], u ∈ A_{(m)}^{×} such that σ(f) = u · g, and g = T_{m}^{d} + a_{1}T_{m}^{d−1} + ⋯ + a_{d} for some d ≥ 0 and certain elements a_{1}, …, a_{d} of the maximal ideal of A_{(m−1)}. Denote by σ′ : B_{(m)} →^{∼} B_{(m)} the T -adic completion of σ ⊗_{A} B, and let σ_{B} : B_{(m+n)} →^{∼} B_{(m+n)} be the T -adically continuous automorphism that restricts to σ′ on B_{(m)}, and such that σ_{B}(T_{i}) = T_{i} for i = m+1, …, m+n. Set M′ := B_{(m+n)}/σ_{B}(𝔮); by construction, we have a commutative diagram of A-algebras

\xymatrix{A_{(m)} \ar[r]^{φ_{m,n}} \ar[d]_{σ} & B_{(m+n)} \ar[d] \ar[d]^{σ_{B}} \\ A_{(m)} \ar[r]^{φ_{m,n}} & B_{(m+n)}}

inducing an isomorphism

𝕃_{B_{(m+n)}/A_{(n)}} ⊗_{B_{(m+n)}} M →^{∼} 𝕃_{B_{(m+n)}/A_{(n)}} ⊗_{B_{(m+n)}} M′ in D(A-Mod).

Thus, we may replace M by M′, and assume from start that g ∈ 𝔭. In this case, set B′ := B[[T_{m+1}, …, T_{m+n}]], and notice that both of the natural maps

A_{(m−1)}[T_{m}]/gA_{(m−1)}[T_{m}] → A_{(m)}/gA_{(m)} B′_{(m−1)}[T_{m}]/gB′_{(m−1)}[T_{m}] → B_{(m+n)}/gB_{(m+n)}

are isomorphisms. Since both φ_{m,n} and the map A_{(m−1)}[T_{m}] → B′_{(m−1)}[T_{m}] induced by φ are flat ring homomorphisms, there follows a natural isomorphism of B_{(m+n)}-modules :

\begin{matrix}𝕃_{B_{(m+n)}/A_{(n)}} ⊗_{B_{(m+n)}} M →^{∼} & 𝕃_{B′_{(m−1)}[T_{m}]/A_{(m−1)}[T_{m}]} ⊗_{B′_{(m−1)}[T_{m}]} M \\ →^{∼} & 𝕃_{B′_{(m−1)}/A_{(m−1)}} ⊗_{B′_{(m−1)}} M \end{matrix}

([103, Ch.II, Prop.2.2.1]). However, the resulting map A_{(m−1)} → B′_{(m−1)} is none else than φ_{m−1,n}, up to a relabeling of the variables; the vanishing of H(M) then follows from the inductive assumption (and from corollary 9.7.9). ♢

Claim 9.7.18. In the situation of the theorem, suppose furthermore that A and B are complete, and let M be a B-module of finite type. Then H_{1}(𝕃_{B/A} ⊗_{B} M) is a B-module of finite type.

Proof of the claim. Let \overline{f} : A_{0} → κ be a surjective ring homomorphism, with A_{0} a Cohen ring ([63, Ch.0, Th.19.8.6(ii)]), and set \overline{B} := B ⊗_{A} κ; in light of [63, Ch.0, Lemme 19.7.1.3], there exists a flat local, complete and noetherian A_{0}-algebra B_{0} fitting into a cocartesian diagram :

\xymatrix{A_{0} \ar[r]^{ψ} \ar[d]_{\overline{f}} & B_{0} \ar[d]^{_{\overline{f}_{B}}} \\ κ \ar[r] & \overline{B}.}

Denote by κ_{B} the residue field of B; there follow natural isomorphisms of B-modules :

H_{1}(𝕃_{B/A} ⊗_{B} κ_{B}) →^{∼} H_{1}(𝕃_{\overline{B}/κ} ⊗\overline{B} κ_{B}) →^{∼} H_{1}(𝕃_{B_{0}/A_{0}} ⊗_{B_{0}} κ_{B})

([103, Ch.II, Prop.2.2.1]) which, according to proposition 8.6.30, imply that ψ is formally smooth (for the preadic topologies defined by the maximal ideals). By [63, Ch.0, Th.19.8.6(i)], \overline{f} lifts to a ring homomorphism f : A_{0} → A, whence a commutative diagram

\xymatrix{A_{0} \ar[r]^{ψ} \ar[d]_{φ∘f} & B_{0} \ar[d]^{_{\overline{f}_{B}}} \\ B \ar[r] & \overline{B}.}

Then, by [63, Ch.0, Cor.19.3.11], the map \overline{f}_{B} lifts to a ring homomorphism f_{B} : B_{0} → B. Notice now that both f and f_{B} are local maps and induce isomorphisms on the residue fields; it follows easily that, for suitable m, n ∈ ℕ, they extend to surjective maps g and g_{B} fitting into a commutative diagram

\xymatrix{A_{0,(m)} \ar[r]^{ψ_{(m,n)}} \ar[d]_{g} & B_{0,(m+n)} \ar[d]^{g_{B}} \\ A \ar[r]^{φ} & B}

whence exact sequences ([103, Ch.II, Prop.2.1.2])

\begin{matrix}H_{1}(𝕃_{B_{0,(m+n)}/A_{0,(m)}} ⊗_{B_{0,(m+n)}} M) → H_{1}(𝕃_{B/A_{0,(m)}} ⊗_{B} M) → H_{1}(𝕃_{B/B_{0,(m+n)}} ⊗_{B} M) \\ H_{1}(𝕃_{B/A_{0,(m)}} ⊗_{B} M) → H_{1}(𝕃_{B/A} ⊗_{B} M) → Ω_{A/A_{0,(m)}}^{1} ⊗_{A} M = 0. \end{matrix}

However, claim 9.7.17 applies to ψ, and together with corollary 9.7.9, it implies that the first module of the first of these sequences vanishes; on the other hand, it is easily seen that the third B-module of the same sequence is finitely generated, so the same holds for the middle term. By inspecting the second exact sequence, the claim follows. ♢

We may now conclude the proof of the theorem : let A^{∧} and B^{∧} be the completions of A and B; since φ is formally smooth, the same holds for its completion φ^{∧} : A^{∧} → B^{∧}. First, we show that Spec φ^{∧} is regular; to this aim, it suffices to check that H^{∧}(M) := H_{1}(𝕃_{B^{∧}/A^{∧}} ⊗_{B^{∧}} M) vanishes for every B^{∧}-module M (corollary 9.7.9). However, we know already that H^{∧}(M) is a B^{∧}-module of finite type, if the same holds for M (claim 9.7.18); especially, for such M, H^{∧}(M) is separated for the adic topology of B^{∧} defined by the maximal ideal. Moreover, H^{∧}(M) = 0, if M is the residue field of B^{∧} (proposition 8.6.30). Then the assertion follows from claim 9.7.16. Lastly, the natural morphism Spec A^{∧} → Spec A is regular by assumption, hence the same holds for the induced morphism Spec B^{∧} → Spec A (lemma 9.7.1(i)). Finally, since B^{∧} is a faithfully flat B-algebra, we conclude that Spec φ is regular, by virtue of lemma 9.7.1(ii). □

Proposition 9.7.19. Let A be a noetherian local ring, and φ : A → B an ind-étale local ring homomorphism. Then :

(i) A is quasi-excellent if and only if the same holds for B.

(ii) If A is excellent, the same holds for B.

Proof. Set f := Spec φ, and f^{∧} := Spec φ^{∧}, where φ^{∧} : A^{∧} → B^{∧} is the map of complete local rings obtained from φ. Notice first that, for every y ∈ Spec A, and every x ∈ f^{−1}(y), the stalk 𝒪_{f^{−1}y,x} is an ind-étale κ(y)-algebra, i.e. is a separable algebraic extension of κ(y), hence f^{−1}(y) is geometrically regular, and therefore f is regular and faithfully flat. Moreover, f is the limit of a cofiltered system of formally étale morphisms, hence it is formally étale; a fortiori, B is also formally smooth over A for the preadic topologies, so B^{∧} is formally smooth over A^{∧} for the preadic topologies, and consequently f^{∧} is a regular morphism (theorem 9.7.15).

(i): If B is quasi-excellent, claim 9.7.5 and lemma 9.7.4(iv) imply that A is quasi-excellent. Next, assume that A is quasi-excellent; we have a commutative diagram (9.7.6), in which π_{A} is a regular morphism, and then the same holds for π_{A} ∘ f^{∧} = f ∘ π_{B} (lemma 9.7.1(i)). Now, let y ∈ Spec B be any point, and set z := f(y); we know that (f ∘ π_{B})^{−1}(z) is a geometrically regular affine scheme, so write it as the spectrum of a noetherian ring C. The stalk E := 𝒪_{f^{−1}(z),y} is a separable algebraic field extension of κ(z), hence π_{B}^{−1}(y) ≃ Spec C ⊗_{B} E, the spectrum of a localization of C, so it is again geometrically regular, i.e. π_{B} is a regular morphism, and we conclude by lemma 9.7.4(iii,iv).

(ii): By (i), it remains only to show that B is universally catenary, provided the same holds for A. To this aim, it suffices to apply [66, Ch.IV, Lemma 18.7.5.1]. □

Proposition 9.7.20. Let p > 0 be a prime integer, and A a noetherian 𝔽_{p}-algebra whose Frobenius endomorphism Φ_{A} is a finite map (see (9.6.36)), and for every prime ideal 𝔭 ⊂ A, set κ(𝔭) := A_{𝔭}/𝔭A_{𝔭} (the residue field of the point 𝔭 ∈ Spec A). Then

dim A_{𝔭}/𝔮A_{𝔭} = dim_{κ(𝔭)} Ω_{κ(𝔭)/𝔽_{p}}^{1} − dim_{κ(𝔮)} Ω_{κ(𝔮)/𝔽_{p}}^{1}

for every pair of prime ideals 𝔮 ⊂ 𝔭 ⊂ A.

Proof. For every 𝔽_{p}-algebra R, let Φ_{R} : R → R be the Frobenius endomorphism. We notice :

Claim 9.7.21. Let A be an 𝔽_{p}-algebra such that Φ_{A} is a finite map. We have :

(i) For every A-algebra B of essentially finite type, Φ_{B} is finite as well.

(ii) Suppose moreover that A is noetherian, I ⊂ A is any ideal, and let A_{I}^{∧} be the I-adic completion of A. Then :

(a) Φ_{A_{I}^{∧}} = 1_{A_{I}^{∧}} ⊗_{A} Φ_{A} is finite, and Ω_{A_{I}^{∧}/𝔽_{p}}^{1} = A_{I}^{∧} ⊗_{A} Ω_{A/𝔽_{p}}^{1}.

(b) If A is reduced, the same holds for A_{I}^{∧}. (iii) Suppose that (A, 𝔪_{A}) is a local noetherian domain, and denote by K (resp. κ) the field

of fractions (resp. the residue field) of A. Then

(9.7.22)  dim_{K} Ω_{K/𝔽_{p}}^{1} = dim_{κ} Ω_{κ/𝔽_{p}}^{1} + dim A.

Proof of the claim. (i): Suppose first that B = A[X]; then it is easily seen that Φ_{B} = Φ_{A} ⊗_{𝔽_{p}} Φ_{𝔽_{p}[X]}, whence the contention, in this case. By an easy induction, we deduce that the claim holds as well for any free polynomial A-algebra of finite type. Next, let I ⊂ A be any ideal; since Φ_{A}(I) ⊂ I, it is easily seen that Φ_{A/I} = Φ_{A} ⊗_{A} A/I, so Φ_{A/I} is finite, and therefore the assertion holds for any A-algebra of finite type. Lastly, let S ⊂ A be any multiplicative subset; since Φ_{A}(S) ⊂ S, it is easily seen that

(9.7.23)  Φ_{S^{−1}A} = S^{−1}Φ_{A}

and the assertion follows.

(ii.a): Set B := A_{(Φ)}; by assumption, Φ_{A} : A → B is a finite A-linear map, hence its I-adic completion (Φ_{A})_{I}^{∧} : A_{I}^{∧} → B_{I}^{∧} equals 1_{A_{I}^{∧}} ⊗_{A} Φ_{A} ([126, Th.8.7]). Say that I is generated by r elements of A; then it is easily seen that I^{(p−1)r+1} ⊂ Φ_{A}(I) ⊂ I, so we have a natural A_{I}^{∧}-linear identification on I-adic completions :

(9.7.24)  B_{I}^{∧} →^{∼} (A_{I}^{∧})_{(Φ)}

and under this identification, (Φ_{A})_{I}^{∧} = Φ_{A_{I}^{∧}}, whence the first assertion of (ii.a). Next, we notice that (9.7.24) yields a natural identification Ω_{A_{I}^{∧}/𝔽_{p}}^{1} = Ω_{B_{I}^{∧}/𝔽_{p}}^{1}, and on the other hand, the sequence of ring homomorphisms

𝔽_{p} → A_{I}^{∧} ⟶^{Φ_{A^{∧}_{I}}} B_{I}^{∧}

yields natural identifications :

Ω_{B_{I}^{∧}/𝔽_{p}}^{1} = Ω_{B_{I}^{∧}/A_{I}^{∧}}^{1} = A_{I}^{∧} ⊗_{A} Ω_{B/A}^{1} = A_{I}^{∧} ⊗_{A} Ω_{B/𝔽_{p}}^{1} = B_{I}^{∧} ⊗_{B} Ω_{B/𝔽_{p}}^{1} = A_{I}^{∧} ⊗_{A} Ω_{A/𝔽_{p}}^{1}

where the last equality is induced by (9.7.24) and the identity map A →^{∼} B. This completes the proof of the second assertion.

(ii.b): Since A is reduced, Φ_{A} is injective, and then the same holds for its completion (Φ_{A})_{I}^{∧}. But we have seen that the latter is naturally identified with Φ_{A_{I}^{∧}}, whence the claim.

(iii): Notice that Ω_{K/𝔽_{p}}^{1} is a K-vector space of finite dimension, since its dimension equals [K : K^{p}] ([63, Ch.0, Th.21.4.5]), and the latter is finite, by (i); the same argument applies to the κ-vector space Ω_{κ/𝔽_{p}}^{1}. Denote by δ(A) the difference between the left and right hand-side of (9.7.22); we have to show that δ(A) = 0. Let A^{∧} be the 𝔪_{A}-adic completion of A, and 𝔭 ⊂ A^{∧} any minimal prime ideal such that dim A^{∧}/𝔭 = d := dim A. In view of (ii.b), L := (A^{∧})_{𝔭} is a field; taking into account (ii.a), we get

Ω_{L/𝔽_{p}}^{1} = L ⊗_{A^{∧}} Ω_{A^{∧}/𝔽_{p}}^{1} = L ⊗_{A} Ω_{A/𝔽_{p}}^{1} = L ⊗_{K} Ω_{K/𝔽_{p}}^{1}.

Hence dim_{L} Ω_{L/𝔽_{p}}^{1} = dim_{K} Ω_{K/𝔽_{p}}^{1}, so we see that

(9.7.25)  δ(A) = δ(A^{∧}/𝔭)  for any prime ideal 𝔭 ⊂ A^{∧} such that dim A^{∧}/𝔭 = d.

We may then replace A by A^{∧}/𝔭, after which we may assume that A is a complete local domain. In this case, A contains a field mapping isomorphically onto κ, and there is a finite map of κalgebras A′ := κ[[T_{1}, …, T_{d}]] → A ([126, Th.29.4(iii)]). Denote by K′ the field of fractions of A′, and notice that [K : K′] = [K : K^{p}] · [K^{p} : K′] = [K : K^{p}] · [K^{p} : K′^{p}]; on the other hand, Φ_{K} induces an isomorphism K →^{∼} K^{p}, hence have as well [K : K′] = [K^{p} : K′^{p}], so [K : K^{p}] = [K′ : K′^{p}] and finally :

dim_{K′} Ω_{K′/𝔽_{p}}^{1} = dim_{K} Ω_{K/𝔽_{p}}^{1}

([63, Ch.0, Cor.21.2.5]). Since dim A′ = dim A, we conclude that δ(A) = δ(A′). Hence, it suffices to check that δ(A′) = 0. Furthermore, set B := κ[T_{1}, …, T_{d}], and let 𝔪 ⊂ B be the maximal ideal generated by T_{1}, …, T_{d}; we have A′ = B_{𝔪}^{∧}, so (9.7.25) further reduces to showing that δ(B_{𝔪}) = 0, which shall be left as an exercise for the reader. ♢

Now, in view of claim 9.7.21(i), we may replace A by A_{𝔭}/𝔮A_{𝔭}, and assume from start that A is a local domain, that 𝔮 = 0 and that 𝔭 is the maximal ideal; in this case, the sought identity is given by claim 9.7.21(iii). □

Theorem 9.7.26. Let p > 0 be a prime integer, and A a noetherian 𝔽_{p}-algebra. We have : (i) If the Frobenius endomorphism Φ_{A} of A is a finite map, then A is excellent. (ii) Conversely, suppose that A is a local Nagata ring, with residue field k, and that [k : k^{p}]

is finite. Then Φ_{A} is a finite ring homomorphism.

Proof. (i): We reproduce the proof from [125, Appendix, Th.108].

Claim 9.7.27. If Φ_{A} is finite, then A is quasi-excellent.

Proof of the claim. We check first the openness condition for the regular loci. To this aim, in light of claim 9.7.21(i), we may assume that A is an integral domain, and it suffices to prove that the regular locus of Spec A is an open subset. Set B := A_{(Φ)} (notation of (9.6.36)), and let 𝔭 ⊂ A be any prime ideal; by theorem 9.6.37 and (9.7.23), the ring A_{𝔭} is regular if and only if (Φ_{A})_{𝔭} is a flat ring homomorphism, if and only if B_{𝔭} is a flat A_{𝔭}-module. Since, by assumption, B is a finite A-module, the latter holds if and only if B_{𝔭} is a free A_{𝔭}-module. Then, the assertion follows from [126, Th.4.10].

Next, we show that A is a G-ring. By claim 9.7.21(i), we may assume that A is local, and we let A^{∧} be the completion of A. From claim 9.7.21(ii.a) we see that the natural map

A_{(Φ)} ⊗^{L}_{A} A^{∧} → (A^{∧})_{(Φ)}

is an isomorphism in D(A-Mod). Then the assertion follows from [75, Lemma 6.5.13(i)] and corollary 9.7.11. ♢

Now, it follows easily from proposition 9.7.20and claim 9.7.21(i), that A is universally catenary; combining with claim 9.7.27, we obtain (i).

(ii): The assertion to prove is that A_{(Φ)} is a finite A-module. To this aim, denote by I ⊂ A the nilradical ideal; we remark :

Claim 9.7.28. It suffices to show that (A/I)_{(Φ)} is a finite A-module.

Proof of the claim. Indeed, suppose that (A/I)_{(Φ)} is a finite A-module; we shall deduce, by induction on s, that (A/I^{s})_{(Φ)} is a finite A-module, for every s > 0. Since A is noetherian, we have I^{t} = 0 for t > 0 large enough, so the claim will follow. The assertion for s = 1 is our assumption. Suppose therefore that s > 1, and that we already know the assertion for s − 1. Notice that I^{s}/I^{s−1} is a finite A/I-module, hence a finite (A/I)_{(Φ)}-module, by our assumption. Since (A/I^{s−1})_{(Φ)} is a finite A-module, we deduce easily that the assertion holds for s, as required. ♢

In view of claim 9.7.28, we may replace A by A/I (which is obviously still a Nagata ring), and assume from start that A is reduced. In this case, let {𝔭_{1}, …, 𝔭_{t}} be the set of minimal prime ideals of A; we have a commutative diagram of ring homomorphisms :

\xymatrix{A \ar[r]^{Φ_{A}} \ar[d] & A \ar[d] \\ ∏_{i=1}^{t} A/𝔭_{i} \ar[r]^{∏_{i=1}^{t} Φ_{A/𝔭_{i}}} & ∏_{i=1}^{t} A/𝔭_{i}}

whose vertical arrows are injective and finite maps. Suppose now that (A/𝔭_{i})_{(Φ)} is a finite A/𝔭_{i}- module for every i = 1, …, t. Then ∏_{i=1}^{t}(A/𝔭_{i})_{(Φ)} is a finite A-module, and therefore the same holds for its A-submodule A_{(Φ)}. Thus, we are reduced to checking the sought assertion for each of the quotients A/𝔭_{i} (which are still Nagata rings), and hence we may assume from start that A is a domain. In this case, denote by K the field of fractions of A; by the definition of Nagata ring, we are further reduced to checking that Φ_{K} : K → K is a finite field extension, i.e. that [K : K^{p}] is finite. Denote A^{∧} the completion of A; we remark :

Claim 9.7.29. (i) The endomorphism Φ_{A^{∧}} is a finite map.

(ii) Spec K ⊗_{A} A^{∧} is a geometrically reduced K-scheme.

Proof of the claim. (i): Indeed, A^{∧} is a quotient of a power series ring B := k[[T_{1}, …, T_{r}]] ([63, Ch.0, Th.19.8.8(i)]), hence it suffices to show that Φ_{B} is finite. However, B^{p} = k^{p}[[T_{1}^{p}, …, T_{r}^{p}]], and [k : k^{p}] is finite by assumption, whence the claim.

(ii): This is [64, Ch.IV, Th.7.6.4]. ♢ Let 𝔭 ⊂ A^{∧} be any minimal prime ideal; it follows from claim 9.7.29(ii) that L := (A^{∧})_{𝔭} is

a separable field extension of K ([64, Ch.IV, Prop.4.6.1]), so the natural map

Ω_{K/𝔽_{p}}^{1} ⊗_{K} L → Ω_{L/ℤ}^{1}

is injective ([63, Ch.0, Cor.20.6.19(i)]); on the other hand, claim 9.7.29(i) implies that [L : L^{p}] is finite, i.e. Ω_{L/𝔽_{p}}^{1} is a finite-dimensional L-vector space ([63, Ch.0, Th.21.4.5]). We conclude that Ω_{K/𝔽_{p}}^{1} is a finite-dimensional K-vector space, whence the contention, again by loc.cit. □

Part (i) of proposition 9.7.31 was first observed in [14, Th.1.1]. Also, the class of ring homomorphisms introduced by definition 9.7.30(ii) has been extensively studied in [13], in which one can find most of the results that follow, and much more.

Definition 9.7.30. Let f : (A, 𝔪_{A}) → (B, 𝔪_{B}) be a local ring homomorphism of local noetherian rings, and denote by B^{∧} the 𝔪_{B}-adic completion of B.

(i) A Cohen factorization of f is the datum of a complete local noetherian ring (C, 𝔪_{C}) and ring homomorphisms g : A → C, h : C → B^{∧} such that C/𝔪_{A}C is a regular ring, g is faithfully flat, h is surjective, and f = h ∘ g.

(ii) We say that f is a homomorphism of complete intersection if it admits a Cohen factorization as in (i), such that Ker h is generated by a regular sequence of elements of 𝔪_{C}.

Proposition 9.7.31. Let f : (A, 𝔪_{A}) → (B, 𝔪_{B}) be a local homomorphism of complete local noetherian rings, and denote by κ_{A} and κ_{B} the residue fields of A and B. The following holds :

(i) f admits a Cohen factorization.

(ii) If H_{1}𝕃_{κ_{B}/κ_{A}} is a finite dimensional κ_{B}-vector space, we may find a Cohen factorization A ⟶^{g} (C, 𝔪_{C}) ⟶^{h} B of f, where g is formally smooth for the 𝔪_{A}-adic and 𝔪_{C}-adic topologies.

Proof. (i): Let p be the characteristic of κ_{A} and κ_{B}. If p = 0, we have fields k_{A} ⊂ A, k_{B} ⊂ B such that the induced maps k_{A} → κ_{A} and k_{B} → κ_{B} are isomorphisms ([126, Th.28.3]); if p > 0, we have complete discrete valuation rings V_{A}, V_{B} and local ring homomorphisms V_{A} → A, V_{B} → B inducing isomorphisms V_{A}/pV_{A} →^{∼} κ_{A}, V_{B}/pV_{B} →^{∼} κ_{B} ([126, Th.29.2]). In the first case, set R_{0} := ℚ, R_{A} := k_{A} and R_{B} := k_{B}; in the second case, set R_{0} := ℤ_{pℤ}, R_{A} := V_{A}, and R_{B} := V_{B}. In either case, let k_{A} and k_{B} be the residue fields of R_{A} and R_{B}, and notice that R_{A} and R_{B} are formally smooth R_{0}-algebras for the p-adic topologies ([126, Th.26.9 and Th.28.10]). Fix finite systems of generators x_{1}, …, x_{n} for 𝔪_{A}, and y_{1}, …, y_{m} for 𝔪_{B}; we may assume that m ≥ n, and f(x_{i}) = y_{i} for i = 1, …, n. Set

A_{0} := R_{A}[[X_{1}, …, X_{n}]] B_{0} := R_{B}[[Y_{1}, …, Y_{m}]]

and let 𝔪_{A_{0}} ⊂ A_{0}, 𝔪_{B_{0}} ⊂ B_{0} be the maximal ideals; we deduce surjective maps

φ_{A} : A_{0} → A φ_{B} : B_{0} → B

such that φ_{A}(X_{i}) = x_{i} and φ_{B}(Y_{j}) = y_{j} for every i = 1, …, n and j = 1, …, m. Obviously J := Ker φ_{B} ⊂ 𝔪_{B_{0}}, and B_{0} is a complete local ring, so that B_{0} is J-adically complete and separated (lemma 8.3.12). We then get a commutative diagram of rings :

\xymatrix{R_{0} \ar[r]^{i_{B}} \ar[d]_{i_{A}} & R_{B} \ar[r]^{j_{B}} & B_{0} \ar[d]^{φ_{B}} \\ R_{A} \ar[r]^{j_{A}} & A \ar[r]^{f} & B}

and by [63, Ch.0, Cor.19.3.11] there exists a local ring homomorphism

ψ : R_{A} → B_{0}  such that ψ ∘ i_{A} = j_{B} ∘ i_{B}  and φ_{B} ∘ ψ = f ∘ j_{A}.

Then ψ extends uniquely to a local ring homomorphism

g_{0} : A_{0} → B_{0} such that g_{0}(X_{1}) = Y_{1}, …, g_{0}(X_{n}) = Y_{n}.

As f is continuous for the 𝔪_{A}-adic and 𝔪_{B}-adic topologies, the resulting diagram commutes :

\xymatrix{A_{0} \ar[r]^{φ_{A}} \ar[d]_{g_{0}} & A \ar[d]^{f} \\ B_{0} \ar[r]^{φ_{B}} & B.}

Set C := B_{0} ⊗_{A_{0}} A; then f factors through g := g_{0} ⊗_{A_{0}} A : A → C and a map h : C → B.

Claim 9.7.32. g_{0} is a flat ring homomorphism.

Proof of the claim. Set X_{•} := (X_{1}, …, X_{n}) and Y_{•} := (Y_{1}, …, Y_{n}); as X_{•} is a regular sequence in A_{0}, the Koszul complex K(X_{•}, A_{0}) is a resolution of the quotient R_{A} of A_{0} (proposition 7.8.7); by the same token, B_{0}⊗_{A_{0}} K_{•}(X_{•}, A_{0}) = K_{•}(Y_{•}, B_{0}) is a resolution of B_{0}⊗_{A_{0}} R_{A}, since Y_{•} is a regular sequence in B_{0}. This shows that Tor_{1}^{A_{0}}(B_{0}, R_{A}) = 0, and clearly B_{0} ⊗_{A_{0}} R_{A} is a flat R_{A}-algebra; then the assertion follows from the local flatness criterion ([126, Th.22.3]). ♢

Clearly g_{0} is a local map, hence it is faithfully flat, by claim 9.7.32, so the same holds for g. Moreover, C/𝔪_{A}C = B_{0} ⊗_{A_{0}} k_{A} = k_{B}[[Y_{n+1}, …, Y_{m}]] is a regular ring, and clearly h is surjective, as the same holds for φ_{B}. Then A ⟶^{g} C ⟶^{h} B is the sought Cohen factorization of f.

(ii): According to [63, Ch.0, Th.19.7.1, Th.19.7.2, Th.22.2.6] there exists a complete local ring (C_{0}, 𝔪_{C_{0}}), and a local ring homomorphism g_{0} : A → C_{0} formally smooth for the 𝔪_{A}-adic and 𝔪_{C_{0}}-adic topologies, with an isomorphism of κ_{A}-algebras C_{0}/𝔪_{C_{0}} →^{∼} κ_{B}. Let π_{B} : B → κ_{B} and π_{C_{0}} : C → κ_{B} be the natural projections; we obtain a commutative diagram :

\xymatrix{A \ar[r]^{f} \ar[d]_{g_{0}} & B \ar[d]^{π_{B}} \\ C_{0} \ar[r]^{π_{C_{0}}} & κ_{B}}

and by [63, Ch.0, Cor.19.3.11] there exists a ring homomorphism h_{0} : C_{0} → B such that π_{B} ∘ h_{0} = π_{C_{0}} and h_{0} ∘ g_{0} = f. Pick y_{1}, …, y_{m} as in the proof of (i), set C := C_{0}[[Y_{1}, …, Y_{m}]], let 𝔪_{C} ⊂ C be the maximal ideal, and h : C → B the unique continuous ring homomorphism such that h_{|C_{0}} = h_{0} and h(Y_{i}) = y_{i} for i = 1, …, m. Then h is surjective, and C is a formally smooth A-algebra ([63, Prop.19.3.5(i)]), for the 𝔪_{A}-adic and 𝔪_{C}-adic topologies. □

Proposition 9.7.33. Let f : (A, 𝔪_{A}) → (B, 𝔪_{B}) be a local ring homomorphism of local noetherian rings, and k_{B} the residue field of B. The following conditions are equivalent :

(a) For every Cohen factorization A ⟶^{g} C ⟶^{h} B^{∧} of f, the kernel of h is generated by a regular sequence.

(b) f is a homomorphism of complete intersection.

(c) H_{i}(𝕃_{B/A} ⊗_{B} k_{B}) = 0 for every i > 1.

(d) H_{2}(𝕃_{B/A} ⊗_{B} k_{B}) = 0.

Proof. Obviously (a)⇒(b) and (c)⇒(d).

(b)⇒(c): First, since B^{∧} is a flat B-algebra, by [103, Ch.II, Prop.2.2.1] we have :

𝕃_{B^{∧}/B} ⊗_{B^{∧}} k_{B} →^{∼} 𝕃_{k_{B}/k_{B}} →^{∼} 0 in D(k_{B}-Mod).

Next, from the sequence of maps A ⟶^{f} B → B^{∧} we get a distinguished triangle :

𝕃_{B/A} ⊗_{B} k_{B} ⟶^{β} 𝕃_{B^{∧}/A} ⊗_{B^{∧}} k_{B} → 𝕃_{B^{∧}/B} ⊗_{B^{∧}} k_{B} = 0

([103, Ch.II, Prop.2.1.2]), so β is an isomorphism in D(k_{B}-Mod). We may therefore replace B by B^{∧}, and assume from start that B is complete and separated. In this case, from a given Cohen factorization A ⟶^{g} C ⟶^{h} B, we get a distinguished triangle

𝕃_{C/A} ⊗_{C} k_{B} → 𝕃_{B/A} ⊗_{B} k_{B} → 𝕃_{B/C} ⊗_{B} k_{B}

and if Ker h is generated by a regular sequence, we have as well H_{i}(𝕃_{B/C} ⊗_{B} k_{B}) = 0 for every i > 1, by theorem 9.6.46. We are thus reduced to checking that H_{i}(𝕃_{C/A} ⊗_{C} k_{B}) = 0 for every i > 1. Let also k_{A} be the residue field of A, and set C′ := k_{A} ⊗_{A} C; since C is a flat A-algebra, invoking again [103, Ch.II, Prop.2.2.1] we get a natural identification :

(9.7.34)  𝕃_{C/A} ⊗_{C} k_{B} →^{∼} 𝕃_{C′/k_{A}} ⊗_{C′} k_{B}.

Hence it suffices to show that H_{i}(𝕃_{C′/k_{A}} ⊗_{C′} k_{B}) = 0 for every i > 1. Recall that the smallest subfield k_{0} of k_{A} is perfect, so C′ is a geometrically regular k_{0}-algebra, whence :

(9.7.35)  H_{i}(𝕃_{C′/k_{0}} ⊗_{C′} k_{B}) = 0  for every i > 0

(claim 9.7.10). By the same token, we have H_{i}(𝕃_{k_{A}/k_{0}}) = 0 for every i > 0. Lastly, the sequence of ring homomorphisms k_{0} → k_{A} → C′ yields a distinguished triangle :

𝕃_{k_{A}/k_{0}} ⊗_{k_{A}} k_{B} → 𝕃_{C′/k_{0}} ⊗_{C′} k_{B} → 𝕃_{C′/k_{A}} ⊗_{C′} k_{B}

from which, combining with the foregoing, we get the sought vanishing.

(d)⇒(a): Arguing as in the foregoing, we may assume that B = B^{∧}; then, a given Cohen factorization A → C → B yields an exact sequence

H_{2}(𝕃_{B/A} ⊗_{B} k_{B}) → H_{2}(𝕃_{B/C} ⊗_{B} k_{B}) ⟶^{∂} H_{1}(𝕃_{C/A} ⊗_{C} k_{B})

([103, Ch.II, Prop.2.1.2]) and in view of theorem 9.6.46, we are reduced to checking that the boundary map ∂ is the zero map. To this aim, we consider the commutative diagram of rings :

[figure: ℤ; C; B; A; C; B]

from which, by the functoriality property of the transitivity triangle ([103, Ch.II, (2.1.1)]), we deduce a commutative diagram of k_{B}-modules :

[figure: H_{2}(𝕃_{B/C} ⊗_{B} k_{B}); ∂; H_{1}(𝕃_{C/ℤ} ⊗_{C} k_{B}); φ; H_{2}(𝕃_{B/C} ⊗_{B} k_{B}); ∂; H_{1}(𝕃_{C/A} ⊗_{C} k_{B})]

which further reduces to checking that φ is the zero map. Now, define k_{A}, k_{0} and C′ as in the foregoing; we get a commutative diagram of k_{B}-modules :

\xymatrix{H_{1}(𝕃_{C/ℤ} ⊗_{C} k_{B}) \ar[r] \ar[d]_{φ} & H_{1}(𝕃_{C′/k_{0}} ⊗_{C′} k_{B}) \ar[d] \\ H_{1}(𝕃_{C/A} ⊗_{C} k_{B}) \ar[r] & H_{1}(𝕃_{C′/k_{A}} ⊗_{C′} k_{B})}

whose bottom horizontal arrow is bijective, by virtue of the natural identification (9.7.34), and whose top horizontal arrow is the zero map, by (9.7.35), whence the assertion. □

Corollary 9.7.36. Let (A, 𝔪_{A}) ⟶^{f} (B, 𝔪_{B}) ⟶^{g} (C, 𝔪_{C}) be two local homomorphisms of noetherian local rings. The following holds :

(i) If f and g are of complete intersection, the same holds for g ∘ f. (ii) If g and g ∘ f are of complete intersection, the same holds for f. (iii) If f is formally smooth for the 𝔪_{A}-adic and 𝔪_{B}-adic topologies, and g∘f is of complete

intersection, then g is of complete intersection. (iv) If f is formally smooth for the 𝔪_{A}-adic and 𝔪_{B}-adic topologies, or else surjective with

kernel generated by a regular sequence of 𝔪_{A}, then f is of complete intersection.

Proof. (i): Let k_{C} be the residue field of C; we consider the induced distinguished triangle :

(9.7.37)  𝕃_{B/A} ⊗_{B} k_{C} → 𝕃_{C/A} ⊗_{C} k_{C} → 𝕃_{C/B} ⊗_{C} k_{C}

([103, Ch.II, Prop.2.1.2]). By proposition 9.7.33, the assumptions of (i) amount to the vanishing of H_{2}(𝕃_{B/A} ⊗_{B} k_{C}) and H_{2}(𝕃_{C/B} ⊗_{C} k_{C}), whence the vanishing of H_{2}(𝕃_{C/A} ⊗_{C} k_{C}), which yields the assertion, again by proposition 9.7.33.

(ii): We argue likewise with (9.7.37) : by proposition 9.7.33, the assumptions imply that H_{2}(𝕃_{C/A} ⊗_{C} k_{C}) = H_{3}(𝕃_{C/B} ⊗_{C} k_{C}) = 0, whence H_{2}(𝕃_{B/A} ⊗_{B} k_{C}) = 0, which yields the assertion, again by proposition 9.7.33.

(iii): By proposition 9.7.33, the assumption on g ∘ f yields H_{2}(𝕃_{C/A} ⊗_{C} k_{C}) = 0, and by claim 9.7.10, the assumption on f yields H_{1}(𝕃_{B/A} ⊗_{B} k_{C}) = 0. Then, from (9.7.37) we get H_{2}(𝕃_{C/B} ⊗_{C} k_{C}) = 0, which yields the assertion, as usual by proposition 9.7.33.

(iv): It follows immediately from theorem 9.6.46, claim 9.7.10, and proposition 9.7.33. □

Lemma 9.7.38. Let f : (A, 𝔪_{A}) → (B, 𝔪_{B}) be a local ring homomorphism of complete intersection of local noetherian rings, M an A-module of finite type, and L_{•} a complex of free A-modules of finite rank. The following holds :

(i) If H_{n}(B ⊗_{A} L_{•}) = 0 for some n ∈ ℤ, then H_{n}(L_{•}) = 0.

(ii) If Tor_{1}^{A}(M, B) = 0, then Tor_{i}^{A}(M, B) = 0 for every i > 0.

Proof. (i): Pick a Cohen factorization A ⟶^{g} C ⟶^{h} B^{∧} of f, so that Ker h is generated by a regular sequence of C; since g is faithfully flat, it suffices to check that H_{n}(C ⊗_{A} L_{•}) = 0, and since B^{∧} is a flat B-algebra, we have H_{n}(B^{∧} ⊗_{A} L_{•}) = 0. Thus, we may replace A by C, B by B^{∧}, and L_{•} by C ⊗_{A} L_{•}, and assume from start that f is surjective, and its kernel is generated by a regular sequence x_{•} := (x_{1}, …, x_{k}) of 𝔪_{A}. Next, arguing by induction on the length k of x_{•}, we are easily reduced to showing the following. If x ∈ 𝔪_{A} is a regular element, and H_{n}(L_{•}/xL_{•}) = 0, then H_{n}(L_{•}) = 0. However, since L_{i} is a free A-module for every i ∈ ℤ, we have a short exact sequence of complexes :

0 → L_{•} ⟶^{x·1_{L•}} L_{•} → L_{•}/xL_{•} → 0

whence a short exact sequence of A-modules :

H_{n}(L_{•}) ⟶^{x·1_{Hn(L•)}} H_{n}(L_{•}) → H_{n}(L_{•}/xL_{•}) = 0.

But H_{n}(L_{•}) is an A-module of finite type, so H_{n}(L_{•}) = 0, by Nakayama’s lemma. (ii): Since B^{∧} is a faithfully flat B-algebra, we have a natural identification

B^{∧} ⊗_{B} Tor_{i}^{A}(M, B) →^{∼} Tor_{i}^{A}(M, B^{∧}) for every i ∈ ℕ

and it suffices to check that B^{∧}⊗_{B}Tor_{i}^{A}(M, B) = 0 for every i > 0. Moreover, the induced map A → B^{∧} is still of complete intersection, by corollary 9.7.36(i,iv). Thus, we may replace B by B^{∧}, and assume that B is 𝔪_{B}-adically complete, in which case f admits a Cohen factorization (A, 𝔪_{A}) ⟶^{g} (C, 𝔪_{C}) ⟶^{h} (B, 𝔪_{B}) such that Ker h is generated by a regular sequence of 𝔪_{C}, and since g is faithfully flat, we have a natural identification

C ⊗_{A} Tor_{i}^{A}(M, B) →^{∼} Tor_{i}^{C}(C ⊗_{A} M, B) for every i ∈ ℕ.

Again, it suffices to check that C ⊗_{A} Tor_{i}^{A}(M, B) = 0 for every i > 0, and h is of complete intersection, by corollary 9.7.36(iv). Thus, we may replace f by h and M by A^{∧} ⊗_{A} M, and assume that f is surjective, with kernel generated by a regular sequence x_{•} := (x_{1}, …, x_{k}) of 𝔪_{A}. Now, in view of lemma 7.8.14(ii), proposition 7.8.15and remark 7.8.16(i), the assumption of (ii) is equivalent to the vanishing of H_{1}(x_{•}, M). By the same token, the latter implies that x_{•} is completely secant on M. Then the assertion follows, by invoking again lemma 7.8.14(ii). □

9.7.39. Let ℒ be the category whose objects are the local noetherian rings, and whose morphisms are the local ring homomorphisms. For every (B, 𝔪_{B}) ∈ Ob(ℒ), denote by

Hom_{ℒ} (−, B) : ℒ → Set

the presheaf on ℒ represented by (B, 𝔪_{B}), and for every k ∈ ℕ, set B_{k} := B/𝔪_{B}^{k+1}; the projection π_{k} : B → B_{k} induces a natural transformation

π_{k∗} : Hom_{ℒ} (−, B) → Hom_{ℒ} (−, B_{k}) (f : A → B) ↦ (π_{k} ∘ f : A → B_{k}).

Proposition 9.7.40. With the notation of (9.7.39), for every (B, 𝔪_{B}) ∈ Ob(ℒ) there exists n ∈ ℕ, and a family of subsets

(Σ_{B,A} ⊂ Hom_{ℒ} (A, B_{n}) | (A, 𝔪_{A}) ∈ Ob(ℒ))

such that for every (A, 𝔪_{A}) ∈ Ob(ℒ) we have :

π_{n∗,A}^{−1}(Σ_{B,A}) = {f ∈ Hom_{ℒ} (A, B) | f is a morphism of complete intersection}.

Proof. By claim 7.9.18, for every i > 0 we have c_{i} ∈ ℕ such that the projection B_{n+c_{i}} → B_{n} induces the zero map

Tor_{i}^{B}(B_{n+c_{i}}, B_{1}) → Tor_{i}^{B}(B_{n}, B_{1}) for every n ∈ ℕ.

Notice that B_{0} = k_{B}, the residue field of B. Then, by [3, Ch.X, Cor.13], for every i > 0 there exists d_{i} ∈ ℕ such that the projection B_{n+d_{i}} → B_{n} induces the zero map

H_{i}(𝕃_{B_{n+d_{i}}/B} ⊗_{B} k_{B}) → H_{i}(𝕃_{B_{n}/B} ⊗_{B} k_{B}) for every n ∈ ℕ.

Now, for every n ∈ ℕ and every morphism f : (A, 𝔪_{A}) → (B, 𝔪_{B}) of ℒ, the sequence of ring homomorphisms A ⟶^{f} B ⟶^{π_{n}} B_{n} induces a distinguished triangle ([103, Ch.II, Prop.2.1.2])

𝕃_{B/A} ⊗_{B} k_{B} → 𝕃_{B_{n}/A} ⊗_{B} k_{B} → 𝕃_{B_{n}/B} ⊗_{B} k_{B}

whence, for every i ∈ ℕ, an exact sequence of k_{B}-modules

H_{i+1}(𝕃_{B_{n}/B} ⊗_{B} k_{B}) → H_{i}(𝕃_{B/A} ⊗_{B} k_{B}) ⟶^{φ_{i,n}} H_{i}(𝕃_{B_{n}/A} ⊗_{B} k_{B}) → H_{i}(𝕃_{B_{n}/B} ⊗_{B} k_{B}).

Taking into account lemma 7.8.19, it follows easily that the the induced inverse systems

Ker φ_{i,•} := (Ker φ_{i,n} | n ∈ ℕ) and Coker φ_{i,•} := (Coker φ_{i,n} | n ∈ ℕ)

are uniformly essentially zero, for every i > 0. Say that e and e′ are the steps of Ker φ_{2,•} and respectively Coker φ_{2,•} (see definition 7.8.18(ii)), and notice that these steps depend only on B (and not on A). Then, a simple diagram chase shows that φ_{2,n} is injective for every n ≥ e, and :

Im(φ_{2,e}) = Im(ψ_{e,e′} : H_{2}(𝕃_{B_{e+e′}/A} ⊗_{B} k_{B}) → H_{2}(𝕃_{B_{e}/A} ⊗_{B} k_{B}))

so that φ_{2,e} induces an isomorphism H_{2}(𝕃_{B/A} ⊗_{B} k_{B}) →^{∼} Im(ψ_{e,e′}). Now, if g : A → B is another local map, denote by B′ the A-algebra whose underlying ring is B, and whose structure map is g; as in the foregoing, we set B′_{n} := B′/𝔪_{B}^{n+1} for every n ∈ ℕ. Then, from the sequence of maps A ⟶^{π_{e+e′}∘g} B′_{e+e′} → B′_{e} we get a corresponding map

ψ′_{e,e′} : H_{2}(𝕃_{B′_{e+e′}/A} ⊗_{B′} k_{B}) → H_{2}(𝕃_{B′_{e}/A} ⊗_{B′} k_{B})

and since the steps e and e′ depend only on the underlying ring B = B′, we deduce as well an isomorphism

H_{2}(𝕃_{B′/A} ⊗_{B′} k_{B}) →^{∼} Im(ψ′_{e,e′}).

Lastly, if π_{e+e′} ∘ f = π_{e+e′} ∘ g, then ψ_{e,e′} = ψ′_{e,e′}, whence an isomorphism

H_{2}(𝕃_{B′/A} ⊗_{B′} k_{B}) →^{∼} H_{2}(𝕃_{B′/A} ⊗_{B′} k_{B})

and especially, g is of complete intersection, if and only if the same holds for f, again by proposition 9.7.33. Hence, the proposition holds with n := e + e′. □

Theorem 9.7.41. Let (A, 𝔪_{A}) be a quasi-excellent local ring, (B, 𝔪_{B}) an excellent henselian local ring, and f : A → B a local ring homomorphism. Let k_{A} and k_{B} be the residue fields of A and B, and suppose that H_{1}𝕃_{k_{B}/k_{A}} is a finite dimensional k_{B}-vector space. Then B is the colimit of a filtered system (B_{λ} | λ ∈ Λ) of local A-algebras of essentially finite type, such that:

(i) the transition map B_{λ} → B_{μ} is local of complete intersection, for every μ ≥ λ in Λ (ii) the universal cocone (B_{λ} → B | λ ∈ Λ) consists of local maps of complete intersection.

Moreover, if the 𝔪_{B}-adic completion B^{∧} of B is a domain, then B_{λ} is a domain for every λ ∈ Λ.

Proof. Assertion (i) follows from (ii) and corollary 9.7.36(ii). Hence, let us show (ii).

• Suppose first that B is 𝔪_{B}-adically complete and separated. Let A^{∧} be the 𝔪_{A}-adic completion of A, and j_{A} : A → A^{∧} the completion map; by proposition 9.7.31(ii), the completion f^{∧} : A^{∧} → B of f admits a Cohen factorization A^{∧} ⟶^{g^{∧}} (C, 𝔪_{C}) ⟶^{h} B, where g^{∧} is formally smooth for the 𝔪_{A}-adic and 𝔪_{C}-adic topologies. Then the same holds for g := g^{∧}∘j_{A} : A → C, and since A is quasi-excellent, the induced morphism Spec C → Spec A is regular (theorem 9.7.15). Then, by [138, Th.1.8], C is the colimit of a filtered system (C_{λ} | λ ∈ Λ) of smooth Aalgebras. For every λ ∈ Λ, let 𝔭_{λ} ⊂ C_{λ} be the preimage of 𝔪_{C}; clearly C is also the filtered colimit of the induced system of local A-algebras of essentially finite type (D_{λ} := C_{λ,𝔭_{λ}} | λ ∈ Λ), with local transition maps. By corollary 9.7.36(iv), the map g is of complete intersection, and by corollary 9.7.36(iii), the same then holds for each map D_{λ} → C of the universal cocone. Now, pick an exact complex of C-modules of finite type

Σ_{•}  :  C^{⊕m} ⟶^{u} C^{⊕n} ⟶^{v} C ⟶^{h} B.

We may then find λ ∈ Λ, and D_{λ}-linear maps

D_{λ}^{⊕m} ⟶^{u_{λ}} D_{λ}^{⊕n} ⟶^{v_{λ}} D_{λ} such that C ⊗_{D_{λ}} u_{λ} = u and C ⊗_{D_{λ}} v_{λ} = v

and we set u_{μ} := D_{μ} ⊗_{D_{λ}} u_{λ}, v_{μ} := D_{μ} ⊗_{D_{λ}} v_{λ}, and B_{λ} := Coker v_{λ} for every μ ∈ Λ with μ ≥ λ. Since v ∘ u = 0, we may find then also find μ ≥ λ such that v_{μ} ∘ u_{μ} = 0. After replacing Λ by a cofinal subset, we thus obtain a compatible system of complexes :

Σ_{λ,•}  :  D_{λ}^{⊕m} ⟶^{u_{λ}} D_{λ}^{⊕n} ⟶^{v_{λ}} D_{λ} ⟶^{h_{λ}} B_{λ}  for every λ ∈ Λ

whose colimit is Σ_{•}, and clearly the induced filtered system (B_{λ} | λ ∈ Λ), whose colimit is B, consists of local A-algebras, and its transition maps are local homomorphisms of A-algebras. By construction, Σ_{•} = C ⊗_{D_{λ}} Σ_{λ,•}, so Σ_{λ,•} is exact for every λ ∈ Λ, by lemma 9.7.38(i). It follows easily that :

Tor_{1}^{D_{λ}}(B_{λ}, C) = 0 for every λ ∈ Λ

and then we have Tor_{i}^{D_{λ}}(B_{λ}, C) = 0 for every λ ∈ Λ and every i > 0 (lemma 9.7.38(ii)). By [103, Ch.II, Prop.2.2.1] we deduce, for every λ ∈ Λ, a natural identification

𝕃_{B/B_{λ}} ⊗_{B} k_{B} →^{∼} 𝕃_{C/D_{λ}} ⊗_{C} k_{B} in D(k_{B}-Mod)

and taking into account proposition 9.7.33, we finally obtain (ii).

• Suppose next that (B, 𝔪_{B}) is henselian, and let j_{B} : B → B^{∧} be the completion map. By the foregoing, B^{∧} is the colimit of a filtered system (B_{λ} | λ ∈ Λ) of local A-algebras of essentially finite type, and such that the universal cocone

(g_{λ} : B_{λ} → B^{∧} | λ ∈ Λ)

consists of maps of complete intersection. For every k ∈ ℕ, let π_{k} : B^{∧} → B^{∧}/𝔪_{B}^{k+1}B^{∧} be the projection, and for every λ ∈ Λ, pick n_{λ} ∈ ℕ such that every local ring homomorphism h : B_{λ} → B^{∧} with π_{n_{λ}} ∘ h = π_{n_{λ}} ∘ g_{λ} is of complete intersection (proposition 9.7.40).

Let then A-Alg_{eft} be the full subcategory of A-Alg whose objects are the A-algebras of essentially finite type, and φ : A-Alg_{eft} → A-Alg the inclusion functor; moreover, let 𝒞 be the full subcategory of φA-Alg_{eft}/B whose set of objects consists of all the maps of A-algebras h : B_{λ} → B, where λ ∈ Λ is any element, and π_{n_{λ}} ∘ j_{B} ∘ h = π_{n_{λ}} ∘ g_{λ}. For any such h, the map j_{B} ∘ h is of complete intersection, due to the choice of n_{λ}, and the same holds for j_{B} (corollary 9.7.36(iv)), and then also for h, by corollary 9.7.36(ii). It is also easily seen that the category φA-Alg_{eft}/B is filtered, and that B represents the colimit of the source functor

φA-Alg_{eft}/B → A-Alg (g : C → B) ↦ C

By virtue of lemma 1.5.7(iii.b) and proposition 1.5.21(ii), we are thus reduced to checking :

Claim 9.7.42. For every A-algebra of essentially finite type C, and every morphism g : C → B of A-algebras, there exist a morphism h : B_{λ} → B of 𝒞, and a morphism of A-algebras l : C → B_{λ} such that h ∘ l = g.

Proof of the claim. Since A is noetherian, C is an A-algebra of essentially of finite presentation; hence, we may find λ ∈ Λ and a morphism l : C → B_{λ} of A-algebras such that g_{λ} ∘ l = j_{B} ∘ g. We regard B and B_{λ} as C-algebras, via the maps g and respectively l, and B^{∧} as a B-algebra, via j_{B}; the maps j_{B} and g_{λ} then define a map of B-algebras t : B ⊗_{C} B_{λ} → B^{∧}, and notice that B ⊗_{C} B_{λ} is a B-algebra of essentially finite type. According to [138, Th.1.3], we may then find a map of B-algebras t′ : B ⊗_{C} B_{λ} → B such that π_{n_{λ}} ∘ j_{B} ∘ t′ = π_{n_{λ}} ∘ t. It is easily seen that the resulting map h : B_{λ} → B with h(b) := t′(1 ⊗ b) for every b ∈ B_{λ}, will do. ♢

• Lastly, suppose that B^{∧} is a domain, and for any given λ ∈ Λ, choose a Cohen factorization B_{λ} ⟶^{g_{λ}} B′_{λ} ⟶^{h_{λ}} B^{∧}, with I := Ker h_{λ} generated by a regular sequence; thus, the graded ring gr_{•}B′_{λ} associated with the I-adic filtration on B′_{λ} is isomorphic to a polynomial algebra B^{∧}[T_{1}, …, T_{n}] (proposition 7.8.15). Hence gr_{•}B′_{λ} is a domain, and since the I-adic filtration is separated on B′_{λ} ([126, Th.8.10(i)]), B′_{λ} is a domain; on the other hand, g_{λ} is faithfully flat, hence injective, so B_{λ} is a domain. □

9.8. Normalization, weak normalization and p-root closure. Recall that for an inclusion of rings A ⊂ B, we denote by i.c.(A, B) the integral closure of A into B (see definition 7.6.3(i)); recall moreover that

(9.8.1)  i.c.(S^{−1}A, S^{−1}B) = S^{−1}i.c.(A, B)  for every subset S ⊂ A.

More generally, let X be a scheme, 𝒜 ⊂ ℬ an inclusion of quasi-coherent 𝒪_{X}-algebras; then the rule : U ↦ 𝒞 (U) := i.c.(𝒜 (U), ℬ(U)) for every affine open subset U ⊂ X defines a subpresheaf 𝒞 ⊂ ℬ on the site of affine open subsets of X, and (9.8.1) easily implies that 𝒞 is a sheaf on this site, so it extend uniquely on the Zariski site of X to a quasi-coherent 𝒪_{X}-subalgebra of ℬ, which we call the integral closure of 𝒜 in ℬ, and denote

i.c.(𝒜 , ℬ).

Also, for every ring A, we let 𝒩_{A} be the nilradical of A, and set A_{red} := A/𝒩_{A}, the maximal reduced quotient of A. Every ring homomorphism f : A → A′ induces a ring homomorphism f_{red} : A_{red} → A′_{red}. Recall that

𝒩_{S^{−1}A} = S^{−1}A  and S^{−1}(A_{red}) = (S^{−1}A)_{red}  For every subset S ⊂ A.

Arguing as in the foregoing, we deduce that for every quasi-coherent 𝒪_{X}-algebra 𝒜, the rule : U ↦ 𝒩_{𝒜 (U)} for every affine open subset U ⊂ X extends uniquely to a quasi-coherent ideal

𝒩_{𝒜} ⊂ 𝒜

called the nilradical of 𝒜. The 𝒪_{X}-algebra 𝒜_{red} := 𝒜/𝒩_{𝒜} is quasi-coherent, and every morphism φ : 𝒜 → 𝒜′ of quasi-coherent 𝒪_{X}-algebras induces a morphism φ_{red} : 𝒜_{red} → 𝒜′_{red}.

Lemma 9.8.2. Let A be a Krull domain, F the field of fractions of A, B a flat A-algebra, and suppose that B ⊗_{A} κ(𝔭) is reduced, for every prime ideal 𝔭 ∈ Spec A of height one. Then B is integrally closed in B ⊗_{A} F.

Proof. See [126, §12] for the basic generalities on Krull domains; especially, [126, Th.12.6] asserts that if A is a Krull domain, the natural sequence of A-modules :

ℰ : 0 → A → F → ⨁_{ht 𝔭=1} F/A_{𝔭} → 0

is short exact, where 𝔭 ∈ Spec A ranges over the prime ideals of height one. By flatness, ℰ ⊗_{A}B is still exact, hence B = ⋂_{ht 𝔭=1} B ⊗_{A} A_{𝔭} (where the intersection takes place in B ⊗_{A} F). We are therefore reduced to the case where A is a discrete valuation ring. Let t denote a chosen generator of the maximal ideal of A, and suppose that x ∈ B ⊗_{A} F is integral over B, so that x^{n} + b_{1}x^{n−1} + ⋯ + b_{n} = 0 in B ⊗_{A} F, for some b_{1}, …, b_{n} ∈ B; let also r ∈ ℕ be the minimal integer such that we have x = t^{−r}b for some b ∈ B. We have to show that r = 0; to this aim, notice that b^{n} + t^{r}b_{1}b^{n−1} + ⋯ + t^{rn}b_{n} = 0 in B; if r > 0, it follows that the image \overline{b} of b in B/tB satisfies the identity : \overline{b}^{n} = 0, therefore \overline{b} = 0, since by assumption, B/tB is reduced. Thus b = tb′ for some b′ ∈ B, and x = t^{1−r}b′, contradicting the minimality of r. □

The following result generalizes [64, Ch.IV, Prop.6.14.4] and [66, ChIV, Prop.18.12.15].

Proposition 9.8.3. Let f : X → S be a flat morphism of schemes with geometrically reduced fibres, ℬ_{1} ⊂ ℬ_{2} two quasi-coherent 𝒪_{S}-algebras, and ℬ_{3} := i.c.(ℬ_{1}, ℬ_{2}). Suppose that :

(a) either f is locally finitely presented,

(b) or else, S is locally noetherian. Then f^{∗}ℬ_{3} = i.c.(f^{∗}ℬ_{1}, f^{∗}ℬ_{2}) and f^{∗}(ℬ_{1})_{red} = f^{∗}(ℬ_{1,red}).

Proof. The assertion is local on both S and X, hence we may assume that S = Spec A and X = Spec A′, for a ring A and an A-algebra A′. Then also ℬ_{1} and ℬ_{2} are the quasi-coherent algebras associated with A-algebras B_{1} ⊂ B_{2}, and ℬ_{3} is the quasi-coherent algebra associated with B_{3} := i.c.(B_{1}, B_{2}). Suppose first that condition (a) holds, and write A as the colimit of a filtered system (A_{λ} | λ ∈ Λ) of noetherian subrings; we remark :

Claim 9.8.4. Let A_{•} := (A_{λ} | λ ∈ Λ) be a filtered system of rings, A the colimit of A_{•}, and X → S := Spec A a flat morphism of schemes of finite presentation, with geometrically reduced fibres. Then for some λ ∈ Λ there exists a flat morphism of schemes with geometrically reduced fibres X_{λ} → S_{λ}, and an isomorphism of S-schemes S ×_{S_{λ}} X_{λ} →^{∼} X.

Proof of the claim. By [65, Ch.IV, Th.8.8.2(ii)], we may find λ ∈ Λ, a finitely presented morphism f_{λ} : X_{λ} → S_{λ}, and an isomorphism of S-schemes : X →^{∼} X_{λ} ×_{S_{λ}} S. We set S_{μ} := Spec A_{μ}, X_{μ} := X_{λ} ×_{S_{λ}} S_{μ}, and f_{μ} := S_{μ} ×_{S_{λ}} f_{λ} : X_{μ} → S_{μ} for every μ ∈ Λ with μ ≥ λ; after replacing Λ by a cofinal subset, we may then assume that f_{μ} is defined for every μ ∈ Λ, and moreover that each such f_{μ} is flat ([65, Ch.IV, Cor.11.2.6.1]). For every μ ∈ Λ, let Z_{μ} ⊂ S_{μ} be the subset of all s ∈ S_{μ} such that the fibre f_{μ}^{−1}(s) is not geometrically reduced; by [65, Ch.IV, Th.9.7.7], Z_{μ} is a constructible subset of S_{μ}. By assumption, we have

⋂_{μ∈Λ} p_{μ}^{−1}Z_{μ} = ∅

and it is clear that p_{μ}^{−1}Z_{μ} ⊂ p_{λ}^{−1}Z_{λ} whenever μ ≥ λ. Then, Z_{μ} = ∅ for some μ ∈ Λ ([63, Ch.IV, Prop.1.8.2, Cor.1.9.8]), hence we may replace Λ by a still smaller cofinal subset, and achieve that all the f_{λ} have geometrically reduced fibres. ♢

By claim 9.8.4, we find λ ∈ Λ and a flat ring homomorphism φ_{0} : A_{λ} → A′_{0} such that : • all the fibres of Spec(φ_{0}) are geometrically reduced

• there exists an isomorphism of A-algebras A′ →^{∼} A ⊗_{A_{λ}} A′_{0}. We need to show that A′ ⊗_{A} B_{3} = i.c.(A′ ⊗_{A} B_{1}, A′ ⊗_{A} B_{2}) and A′ ⊗_{A} B_{1,red} = (A′ ⊗_{A} B_{1})_{red}. However, in view of the natural identifications : A′ ⊗_{A} B_{i} →^{∼} A′_{0} ⊗_{A_{λ}} B_{i} for i = 1, 2, 3, we may replace A and A′ by A_{λ} and A′_{0}, and assume that A is noetherian, i.e. that condition (b) holds. For the first identity, arguing as in the proof of [64, Ch.IV, Prop.6.14.4 and Cor.6.14.5], we reduce to the following two separate cases :

(a’) B_{1} is a domain of finite type over A and B_{2} is a finite extension of Frac(B_{1})

(b’) B_{1} and B_{2} are fields, and B_{1} is algebraically closed in B_{2} (so B_{3} = B_{1} in this case). In case (a’) holds, B_{3} is a Krull domain ([133, Th.33.10]), and the assertion follows from lemma 9.8.2. In case (b’), we may replace A by B_{1} and A′ by A′ ⊗_{A} B_{1}, and assume that A is a field, A′ is a geometrically reduced A-algebra, B := B_{2} is a field extension of A such that A is algebraically closed in B, and we need to check that A′ is integrally closed in A′ ⊗_{A} B. Now, pick a basis (b_{i} | i ∈ I) for the A-vector space B, with b_{i_{0}} = 1 for some i_{0} ∈ I, and say that ∑_{i∈J} a_{i} ⊗b_{i} ∈ A′ ⊗_{A} B is integral over A′ (for some finite subset J ⊂ I); we need to check that a_{i} = 0 for every i ∈ J \ {i_{0}}. However, since A′ is reduced, it suffices to check that the image of a_{i} vanishes in κ(𝔭), for every minimal prime ideal 𝔭 of A′. Thus, we may further assume that A′ is a field, in which case it is a separable (not necessarily algebraic) extension of A, since it is a geometrically reduced A-algebra. Then, A′ ⊗_{A} B is a domain, according to [39, Ch.V, §17, n.2, Cor.], and A′ is algebraically closed in Frac(A′ ⊗_{A} B), by [39, Ch.V, §17, Exerc.2(b)].

For the second identity, notice that A′ ⊗_{A} 𝒩_{B_{1}} ⊂ 𝒩_{A′⊗_{A}B_{1}}; hence it suffices to check that if B_{1} is reduced, the same holds for A′ ⊗_{A} B_{1}. Moreover, we may assume that B_{1} is an A-algebra of finite type, in which case B_{1} is noetherian, and it has finitely many minimal prime ideals 𝔮_{1}, …, 𝔮_{n}, and since B_{1} is reduced, the natural map B_{1} → ∏_{i=1}^{n} B_{𝔮_{i}} is injective. Then, the same holds for the induced map A′ ⊗_{A} B_{1} → ∏_{i=1}^{n} A′ ⊗_{A} B_{𝔮_{i}}. However, B_{𝔮_{i}} is a field, and A′ ⊗_{A} B_{𝔮_{i}} is reduced for every i = 1, …, n, since A′ has geometrically reduced fibres over A; the assertion follows. □

Definition 9.8.5. Let f : X′ → X be a morphism of schemes, ξ′ a geometric point of X′ localized at x′ ∈ X′, and set ξ := f(ξ′). We shall say that f is pro-smooth at the point x′ if the induced map of strictly henselian local rings

𝒪_{X(ξ),ξ} → 𝒪_{X′(ξ′),ξ′}

is ind-smooth (i.e. a filtered colimit of smooth ring homomorphisms).

Corollary 9.8.6. Let f : X′ → X be a morphism of schemes, ξ′ a geometric point of X′, ℛ ⊂ 𝒮 a monomorphism of quasi-coherent 𝒪_{X}-algebras, and

φ : f^{∗}i.c.(ℛ, 𝒮) → i.c.(f^{∗}ℛ, f^{∗}𝒮)

the induced morphism. Suppose that f is pro-smooth at the support x′ of ξ′; then we have : (i) f is flat at the point x′. (ii) φ_{ξ′} is an isomorphism.

Proof. (i): Set x := f(x′), ξ := f(ξ′). We have a commutative diagram of local S-schemes

\xymatrix{X′(ξ′) \ar[r]^{f_{(ξ′)}} \ar[d]_{ψ′} & X(ξ) \ar[d]^{ψ} \\ X′(x′) \ar[r]^{f_{(x′)}} & X(x)}

whose vertical arrows are pro-étale morphisms, and whose top horizontal arrow is pro-smooth by assumption. Especially, since ψ′ is faithfully flat, the same holds for f_{(x′)}, i.e. f is flat at x′.

(ii): We may assume that X = Spec A for some ring A. Let g : X′(ξ′) → X′ be the natural morphism. It suffices to show that g^{∗}φ is an isomorphism. However, g^{∗}i.c.(f^{∗}ℛ, f^{∗}𝒮) = i.c.(g^{∗}f^{∗}ℛ, g^{∗}f^{∗}𝒮) by proposition 9.8.3, and g^{∗}φ is the induced morphism

(f ∘ g)^{∗}i.c.(ℛ, 𝒮) → i.c.((f ∘ g)^{∗}ℛ, (f ∘ g)^{∗}𝒮).

Notice that f ∘ g is induced by an ind-smooth ring homomorphism A → 𝒪_{X′,ξ′}, so we may invoke again proposition 9.8.3to conclude. □

9.8.7. Recall that a morphism of schemes φ : X → Y is called universally injective or radicial if for every field K, the induced map on K-valued points X(K) → Y (K) is injective ([59, Ch.I, Def.3.5.4]). It is easily seen that the latter holds if and only if φ is injective and the residue field extension κ(x) → κ(φ(x)) is purely inseparable for every x ∈ X. In the rest of this section we study certain classes of ring homomorphisms f : A → B such that Spec f is radicial, in which case we just say that f is radicial. We shall see that f is radicial if and only if the kernel I_{f} of the multiplication law B ⊗_{A} B → B lies in the nilradical 𝒩_{B⊗_{A}B} (corollary 9.8.13(i)).

Lemma 9.8.8. (i) Let S be a scheme, and φ : Y → X and φ′ : Y′ → X′ two surjective morphisms of S-schemes. Then φ ×_{S} φ′ : Y ×_{S} Y′ → X ×_{S} X′ is surjective.

(ii) Let f:A→B be a ring homomorphism such that Spec f is surjective; then Ker f ⊂ 𝒩_{A}. (iii) Let A be a ring, f : B → C and f′ : B′ → C′ two integral homomorphisms of A-

algebras with Ker f ⊂ 𝒩_{B} and Ker f′ ⊂ 𝒩_{B′}. Then f ⊗_{A} f′ : B ⊗_{A} B′ → C ⊗_{A} C′ is integral and its kernel lies in 𝒩_{B⊗_{A}B′}.

Proof. (i): (See [59, Ch.I, Prop.3.5.2(i)]). We have φ ×_{S} φ′ = (φ ×_{S} X′) ∘ (Y ×_{S} φ′), so it suffices to show that both φ ×_{S} X′ and Y ×_{S} φ′ are surjective. But notice that the natural isomorphism Y ×_{S} X′ →^{∼} Y ×_{X} (X ×_{S} X′) identifies φ ×_{S} X′ with φ ×_{X} (X ×_{S} X′), and similarly for Y ×_{S} φ′; hence, we are reduced to checking if φ : Y → X is a surjective morphism of schemes, and ψ : Z → X is any X-scheme, then the base change φ ×_{X} Z : Y ×_{X} Z → Z is surjective. Thus, let y ∈ Y and z ∈ Z with x := φ(y) = ψ(z), and let κ(z) ← κ(x) → κ(y) be the induced residue field extensions; it suffices to check that Spec κ(y)×_{Spec κ(x)}Spec κ(z) ≠ ∅, which is clear, since κ(y) ⊗_{κ(x)} κ(z) ≠ 0.

(ii): Let a ∈ Ker f, and 𝔭 ∈ Spec A; by assumption there exists 𝔮 ∈ Spec B with 𝔭 = f^{−1}𝔮, so that a ∈ 𝔭, whence the assertion.

(iii): Obviously f ⊗_{A} f′ is integral. Since (B ⊗_{A} B′)_{red} = (B_{red} ⊗_{A_{red}} B′_{red})_{red}, and similarly for (C ⊗_{A} C′)_{red}, we may replace f and f′ by f_{red} and f′_{red}, and assume from start that f and f′ are injective. Then, by the going up theorem, both Spec f and Spec f′ are surjective, hence the same holds for Spec f ⊗_{A} f′, by (i), and then the assertion follows from (ii). □

The following proposition 9.8.9is borrowed from [20, Exp.XII, Lemma 2.6], and its corollary 9.8.13(ii) can also be easily deduced from [161, Th.1].

Proposition 9.8.9. Let f : A → B be an injective and finite ring homomorphism of noetherian rings. Then f is the composition of a finite sequence of injective ring homomorphisms

B_{n} := A ⟶^{f_{n}} B_{n−1} ⟶^{f_{n−1}} B_{n−2} → ⋯ → B_{0} := B

such that the induced morphism of schemes Spec f_{i} : Spec B_{i} → B_{i+1} is an effective epimorphism in the category of schemes, for every i = 1, …, n.

Proof. The assertion means that f_{i} identifies B_{i+1} with the equalizer of the two natural ring homomorphisms (g_{i,k} : B_{i} → B_{i} ⊗_{B_{i+1}} B_{i} | k = 1, 2), with g_{i,1}(b) := b ⊗ 1 and g_{i,2}(b) := 1 ⊗ b for i = 1, …, n and every b ∈ B_{i+1}. Then let us define inductively B_{0} := B, and B_{i+1} := Ker(h_{i,1} − h_{i,2}) for every i ∈ ℕ, where h_{i,1}, h_{i,2} : B_{i} → B_{i} ⊗_{A} B_{i} are the analogous maps.

Claim 9.8.10. For every i ∈ ℕ, the induced map B_{i} ⊗_{A} B_{i} → B_{i} ⊗_{B_{i+1}} B_{i} is an isomorphism.

Proof of the claim. The map in question is clearly surjective. We compute in B_{i} ⊗_{A} B_{i} :

cb ⊗ b′ = (c ⊗ 1) · (b ⊗ b′) = (1 ⊗ c) · (b ⊗ b′) = b ⊗ cb′ for every b, b′ ∈ B_{i} and c ∈ B_{i+1}

which easily implies that the map in question is also injective. ♢

Due to claim 9.8.10, it remains only to check that B_{n} = A for sufficiently large n ∈ ℕ. To this aim, set M_{i} := B_{i}/A, and let S_{i} ⊂ Spec A be the support of M_{i}, for every i ∈ ℕ. Then S_{i+1} ⊂ S_{i} for every i ∈ ℕ, and since A is noetherian, there exists n ∈ ℕ such that S := S_{n} = S_{i} for every i ≥ n. Thus, it suffices to check that S = ∅. If the latter fails, let 𝔭 be a maximal point of the closed subset S; a simple induction on i ∈ ℕ shows that (B_{i+1})_{𝔭} is the equalizer of the localizations (h_{i,1})_{𝔭}, (h_{i,2})_{𝔭} : B_{i,𝔭} ⊗_{A_{𝔭}} B_{i,𝔭}, for every i ∈ ℕ. Hence, we may replace f by its localization f_{𝔭} : A_{𝔭} → B_{𝔭}, and assume from start that (A, 𝔭) is local, and the support of M_{n} is the maximal ideal of A, hence the A-module M_{n} has finite length, so there exists i ≥ n such that M_{i} = M_{i+1}, i.e. B_{i} = B_{i+1}. Set κ(𝔭) := A/𝔭 and B_{i}(𝔭) := B_{i}/𝔭B_{i}; it follows that each of the maps B_{i} → B_{i} ⊗_{A} B_{i} is an isomorphism, hence the same holds for the induced maps B_{i}(𝔭) → B_{i}(𝔭) ⊗_{κ(𝔭)} B_{i}(𝔭), so dim_{κ(𝔭)} B_{i}(𝔭) = 1, and therefore f : A → B_{i} induces an isomorphism κ(𝔭) →^{∼} B_{i}(𝔭). By Nakayama’s lemma, we conclude that A = B_{i}, whence M_{i} = 0 and S_{i} = ∅, a contradiction. □

9.8.11. Let p ∈ ℤ be a prime integer. For every 𝔽_{p}-algebra R, we let R^{perf} be the inductive limit of the system of 𝔽_{p}-algebras (R_{n} | n ∈ ℕ) with R_{n} := R for every n ∈ ℕ, and with transition map given by the Frobenius endomorphism Φ_{R} : R_{n} → R_{n+1} of R, for every such n. The rule R ↦ R^{perf} yields a left adjoint for the forgetful functor from the category of perfect 𝔽_{p}-algebras to the category of all 𝔽_{p}-algebras. Moreover, by virtue of [75, Th.3.5.13(ii)], any étale homomorphism R → S of 𝔽_{p}-algebras induces an isomorphism

(9.8.12)  R^{perf} ⊗_{R} S →^{∼} S^{perf}.

Especially, for every subset Σ ⊂ R, we have

(Σ^{−1}R)^{perf} = Σ^{−1}R^{perf}.

Corollary 9.8.13. Let f : A → B be a ring homomorphism.

(i) The following conditions are equivalent :

(a) f is a radicial homomorphism.

(b) I_{f} ⊂ 𝒩_{B⊗_{A}B} (notation of (9.8.7)). (ii) Let moreover p ∈ ℕ be a prime integer, and suppose that A and B are 𝔽_{p}-algebras, and

f is integral and injective. Then (a) and (b) are also equivalent to :

(c) f induces an isomorphism f^{perf} : A^{perf} →^{∼} B^{perf}.

Proof. (a)⇔(b): Set X := Spec A, Y := Spec B, and let π : Y ×_{X} Y → X be the projection. With this notation, Spec B ⊗_{A} B/I_{f} is the diagonal closed subset Δ_{Y/X} ⊂ Y ×_{X} Y. Suppose then that (b) holds; it follows already that Δ_{Y/X} = Y ×_{X} Y, so φ := Spec f : Y → X is injective. Moreover, let y ∈ Y be any point, set x := φ(y), and let κ(x) and κ(y) be the residue fields of x and respectively y; then Z(y) := Spec κ(y)⊗_{κ(x)} κ(y) = π^{−1}(y) consists of a unique point of Y ×_{X} Y, which means that the field extension κ(x) ⊂ κ(y) is algebraic and purely inseparable, therefore (a) holds. Conversely, if (a) holds, then φ is injective, and Z(y) consists of a unique point, so that Δ_{Y/X} = Y ×_{X} Y, whence (b).

(b)⇔(c): Since the natural map R^{perf} → (R_{red})^{perf} is an isomorphism for every 𝔽_{p}-algebra R, and since f is radicial if and only if the same holds for the induced map f_{red} : A_{red} → B_{red}, we may assume that A and B are reduced, in which case (c) means that for every b ∈ B there exists n ∈ ℕ with b^{p^{n}} ∈ A. Now, let us write B as the filtered colimit of the system (f_{λ} : A → B_{λ} | λ ∈ Λ) of its finite A-subalgebras. It follows easily that (c) holds for f if and only if it holds for every f_{λ}. On the other hand, Spec f is the limit of the system of maps (Spec f_{λ} | λ ∈ Λ), and each Spec f_{λ} is surjective; it follows easily that Spec f is injective if and only if the same holds for each Spec f_{λ}. Moreover, it is clear that for any field K, and any filtered system E_{•} := (E_{λ} | λ ∈ Λ) of field extension of K, the colimit of E_{•} is algebraic and purely inseparable over K if and only if the same holds for every E_{λ}; summing up, we see that f is radicial if and only if the same holds for every f_{λ}, hence, in view of (i), it suffices to show the equivalence (b)⇔(c) for each f_{λ}, and we may assume that f is a finite ring homomorphism.

Next, let (A_{λ} | λ ∈ Λ) be the filtered system of all ℤ-subalgebras of A of finite type; also, choose any finite subset Σ ⊂ B such that B = A[Σ]. Notice that (c) holds if and only if for every b ∈ Σ there exists n ∈ ℕ such that b^{p^{n}} ∈ A; then the latter condition holds if and only if there exists λ ∈ Λ and n ∈ ℕ such that b^{p^{n}} ∈ A_{λ} for every b ∈ Σ. Moreover, there exists a cofinal subset Λ′ ⊂ Λ such that the restriction f_{λ} : A_{λ} → B_{λ} := A_{λ}[Σ] is a finite ring homomorphism for every λ ∈ Λ′. Furthermore, recall that I_{f} is the ideal generated by (b ⊗ 1 − 1 ⊗ b | b ∈ Σ), and likewise for the ideal I_{f_{λ}}, for every λ ∈ Λ. It follows that I_{f} ⊂ 𝒩_{B⊗_{A}B} if and only if I_{f_{λ}} ⊂ 𝒩_{B_{λ}⊗_{A_{λ}}B_{λ}} for some λ ∈ Λ′. Summing up, it then suffices to check the equivalence (b)⇔(c) for each map f_{λ}, and therefore we may assume that A and B are noetherian.

Next, pick a decomposition f = f_{1} ∘ ⋯ ∘ f_{n} as in proposition 9.8.9; since Spec f_{i} is a surjection for every i = 1, …, n, clearly f is radicial if and only if the same holds for each f_{i}. Similarly, f^{perf} = f_{1}^{perf} ∘ ⋯ ∘ f_{n}^{perf} is an isomorphism if and only if f_{i}^{perf} is an isomorphism for every i = 1, …, n. Thus, it suffices to show the equivalence (b)⇔(c) for each f_{i}, and therefore we may assume that Spec f is an effective epimorphism of schemes. Now, (b) says that for every b ∈ B there exists n ∈ ℕ with

(b ⊗ 1 − 1 ⊗ b)^{p^{n}} = b^{p^{n}} ⊗ 1 − 1 ⊗ b^{p^{n}} = 0.

By assumption, the latter holds if and only if b^{p^{n}} ∈ A, whence the contention. □

Remark 9.8.14. (i) Let f : A → B an injective ring homomorphism, and denote by ℰ the set of all A-subalgebras B′ ⊂ B such that the induced map f′ : A → B′ is integral and radicial; recall that if B′ = A[Σ] for a subset Σ ⊂ B′, then I_{f′} is generated by the system (b⊗1−1⊗b | b ∈ Σ).

(ii) In view of corollary 9.8.13(i), it follows easily that if B′, B′′ ∈ ℰ, then also B′ · B′′ ∈ ℰ. (iii) Moreover, for every system (B_{λ} | λ ∈ Λ) of elements of ℰ filtered by inclusion, we have

⋃_{λ∈Λ} B_{λ} ∈ ℰ.

(iv) Furthermore, if B′ ∈ ℰ, and B′′ ⊂ B′ is any A-subalgebra, then B′′ ∈ ℰ as well.

Definition 9.8.15. From remark (9.8.14)(ii,iii), it follows that ℰ is a filtered set, and the subring C := ⋃_{B′∈ℰ} B′ is the largest element of ℰ; we call C the weak normalization of A in B.

Remark 9.8.16. (i) Consider a commutative diagram of rings :

\xymatrix{A \ar[r]^{i} \ar[d]_{g_{A}} & B \ar[d]^{g_{B}} \\ A′ \ar[r]^{i′} & B′}

where i and i′ are injective ring homomorphisms. Denote by C (resp. C′) the weak normalization of A in B (resp. of A′ in B′). Then g_{B}(C) ⊂ C′. Indeed, let x ∈ C; by remark 9.8.14(iv), the induced map A → A[x] is radicial, hence x ⊗ 1 − 1 ⊗ x is nilpotent in A[x] ⊗_{A} A[x], so that its image g_{B}(x) ⊗ 1 − 1 ⊗ g_{B}(x) is nilpotent in A′[x] ⊗_{A′} A′[x], and therefore the induced map A′ → A′[x] is radicial, whence the claim.

(ii) Let f : A → B be an injective and integral ring homomorphism; let j_{1} (resp. j_{2}) be the ring homomorphism B → B ⊗_{A} B given by the rule : b ↦ b ⊗ 1 (resp. b ↦ 1 ⊗ b) for every b ∈ B. For i = 1, 2 let also j′_{i} : B → (B ⊗_{A} B)_{red} be the composition of j_{i} with the projection B ⊗_{A} B → (B ⊗_{A} B)_{red}. Then the weak normalization of A in B is the equalizer of j′_{1} and j′_{2}. Indeed, if b ∈ B and A[b] is a radicial A-algebra, then b ⊗ 1 − 1 ⊗ b is nilpotent in A[b] ⊗_{A} A[b], hence also in B ⊗_{A} B, so j′_{1}(b) = j′_{2}(b). Conversely, if the latter identity holds, b ⊗ 1 − 1 ⊗ b is nilpotent in B ⊗_{A} B; but notice that the map A[b] → B is still injective and integral, so the kernel of the induced map A[b] ⊗_{A} A[b] → B ⊗_{A} B lies in the nilradical (lemma 9.8.8(iii)). It follows that b⊗1−1⊗b is nilpotent in A[b]⊗_{A} A[b], so A[b] is a radicial A-algebra, as required.

Lemma 9.8.17. Let f_{•} := (f_{λ} : A_{λ} → B_{λ} | λ ∈ Λ) be a filtered system of injective ring homomorphisms; denote by f : A → B the colimit of f_{•}, and for every λ ∈ Λ let C_{λ} be the weak normalization of A_{λ} in B_{λ}. Then, the system of rings (B_{λ} | λ ∈ Λ) restricts to a filtered system of rings C_{•} := (C_{λ} | λ ∈ Λ), whose colimit C is the weak normalization of A in B.

Proof. Let f′ : A → C be the map induced by f, and for every λ ∈ Λ, let f′_{λ} : A_{λ} → C_{λ} be the map induced by f_{λ}. By corollary 9.8.13(i) we know that I_{f′_{λ}} ⊂ 𝒩_{C_{λ}⊗_{A_{λ}}C_{λ}}; on the other hand, it is easily seen that I_{f′} is the colimit of the system of ideals (I_{f′_{λ}} | λ ∈ Λ), hence I_{f′} ⊂ 𝒩_{C⊗_{A}C}. Moreover, since C_{λ} is an integral A_{λ}-algebra for every λ ∈ Λ, it is easily seen that C is an integral A-algebra. Invoking again corollary 9.8.13(i), we deduce that C lies in the weak normalization C′ of A in B. Conversely, if x ∈ C′, then in particular x lies in the integral closure C′′ of A in B, hence we find λ ∈ Λ and x_{λ} ∈ B_{λ} whose class in B agrees with x, and such that x_{λ} is integral over A_{λ}; by corollary 9.8.13(i), we have x ⊗ 1 − 1 ⊗ x ∈ 𝒩_{C′⊗_{A}C′}, and therefore x ⊗ 1 − 1 ⊗ x ∈ 𝒩_{C′′⊗_{A}C′′} as well. Hence, after replacing λ by some λ′ ≥ λ, we may assume that x_{λ} ⊗ 1 − 1 ⊗ x_{λ} ∈ 𝒩_{C′′_{λ}⊗_{A_{λ}}C′′_{λ}}, where C′′_{λ} denotes the integral closure of A_{λ} in B_{λ}. Now, the inclusion A_{λ}[x] → C′′_{λ} is integral and injective, hence the kernel of the induced ring homomorphism A_{λ}[x] ⊗_{A_{λ}} A_{λ}[x] → C′′_{λ} ⊗_{A_{λ}} C′′_{λ} lies in the nilradical (lemma 9.8.8(iii)). Summing up, we conclude that x_{λ} ⊗ 1 − 1 ⊗ x_{λ} is already nilpotent in A_{λ}[x] ⊗_{A_{λ}} A_{λ}[x], so x_{λ} ∈ C_{λ}, and finally x ∈ C. This shows that C′ = C, as required. □

Proposition 9.8.18. Let f : A → B be an injective ring homomorphism, C the weak normalization of A in B, and A′ a flat A-algebra such that the induced morphism Spec A′ → Spec A has geometrically reduced fibres. Suppose that :

• either A′ is a finitely presented A-algebra • or else, A is noetherian.

Then A′ ⊗_{A} f : A′ → B′ := A′ ⊗_{A} B is an injective ring homomorphism, and the weak normalization of A′ in B′ is A′ ⊗_{A} C.

Proof. Since A′ is a flat A-algebra, A′ ⊗_{A} f is still injective, as stated. Now, let A^{ν} (resp. A^{′ν}) be the integral closure of A in B (resp. of A′ in B′); then A^{′ν} = A′ ⊗_{A} A^{ν} (proposition 9.8.3). We may then replace B by A^{ν}, and assume that B is an integral A-algebra. In this case, notice that for every A-algebra R we have (A′ ⊗_{A} R)_{red} = A′ ⊗_{A} R_{red} (proposition 9.8.3); then the assertion follows easily from remark 9.8.16(ii). □

9.8.19. Let f : A → B be an integral and injective ring homomorphism, I ⊂ A an ideal; set X := Spec A, Y := Spec B, Z := Spec A/I, U := X \ Z, and notice that φ := Spec f : Y → X is surjective by the going up theorem, hence the same holds for its restriction φ^{−1}Z → Z, and therefore the kernel of A/I ⊗_{A} f : A/I → B/IB lies in the nilradical (lemma 9.8.8(ii)), i.e. the induced map (A/I)_{red} → (B/IB)_{red} is injective and integral as well.

Lemma 9.8.20. In the situation of (9.8.19), suppose that φ restricts to a radicial morphism of schemes φ_{|U} : φ^{−1}U → U, and let C (resp. \overline{C}) be the weak normalization of A in B (resp. of (A/I)_{red} in (B/IB)_{red}). Then we have :

C = B ×_{(B/IB)_{red}} \overline{C}.

Proof. By remark 9.8.16(i), the projection B → (B/IB)_{red} maps C into \overline{C}, whence the inclusion C ⊂ C′ := B ×_{(B/IB)_{red}} \overline{C}. For the converse inclusion, we need to show that the induced map g : A → C′ is radicial. To this aim, it suffices to check that the same holds for A/I ⊗_{A} g : A/I → C′/IC and for A[a^{−1}] ⊗_{A} g : A[a^{−1}] → C′[a^{−1}], for every a ∈ I. However, for any such a, the localization C′[a^{−1}] is the fibre product B[a^{−1}] ×_{(B/IB)_{red}[a^{−1}]} \overline{C}[a^{−1}], and since (B/IB)_{red}[a^{−1}] = \overline{C}[a^{−1}] = 0, we see that C′[a^{−1}] = B[a^{−1}]; but f induces a radicial map A[a^{−1}] → B[a^{−1}], since φ_{|U} is radicial. It remains thus only to check that C/IC is a radicial A/I-algebra; to this aim, notice that the kernel of the surjection B → (B/IB)_{red} is the radical J of the ideal IB ⊂ B. Then J is also the kernel of the surjection C′ → \overline{C}; on the other hand, since IB ⊂ J ⊂ C, we have I^{2}B ⊂ IC ⊂ IB ⊂ J, and clearly J is also the radical of the ideal I^{2}B of C. Hence, C/IC is a radicial A/I-algebra if and only if the same holds for the A/I-algebra C/J; but C/J = \overline{C}, whence the contention. □

Proposition 9.8.21. Let B be a ring, A ⊂ B a subring, and G a finite group of automorphisms of B such that g(A) = A for every g ∈ G. Denote by C (resp. D) the integral closure (resp. the weak normalization) of A in B. Then we have :

(i) C^{G} is the integral closure of A^{G} in B^{G}.

(ii) D^{G} is the weak normalization of A^{G} in B^{G}.

Proof. (i): Clearly i.c.(A^{G}, B^{G}) ⊂ C^{G}. To show the reverse inclusion, let x ∈ C^{G}, so that x ∈ B^{G}, and x is integral over A; but since G is a finite group, A is integral over A^{G} (details left to the reader), hence x is integral over A^{G}, as required.

(ii): Let D′ be the weak normalization of A^{G} in B^{G}; we have D′ ⊂ D by remark 9.8.16(i), and then obviously D′ ⊂ D^{G}. For the reverse inclusion, we come down to checking that the inclusion map A^{G} → D^{G} is radicial. To this aim, notice first :

Claim 9.8.22. (i) g(D) = D for every g ∈ G.

(ii) Let K be any algebraically closed field, and E ⊂ B any subring such that g(E) = E for every g ∈ G. Then the set Hom_{ℤ-Alg}(E, K) inherits from E a natural G-action, and the map Hom_{ℤ-Alg}(E, K) → Hom_{ℤ-Alg}(E^{G}, K) : φ ↦ φ_{|A^{G}} factors through a bijection

Hom_{ℤ-Alg}(E, K)/G →^{∼} Hom_{ℤ-Alg}(E^{G}, K).

Proof of the claim. (i): We apply remark 9.8.16(i) with A′ := A, B′ := B, g_{B} := g, g_{A} := g_{|A}, and with i and i′ the inclusion maps; then we get g(D) ⊂ D, and similarly g^{−1}(D) ⊂ D, whence the claim.

(ii): The injectivity follows easily from [34, Ch.5, §2, n.2, Cor. of Th.2], and the surjectivity is clear, since the inclusion map E^{G} → E is integral. ♢

Now, the radicial map A → D induces a bijection Hom_{ℤ-Alg}(D, K) →^{∼} Hom_{ℤ-Alg}(A, D) for every algebraically closed field K; in view of claim 9.8.22(i), this map is obviously Gequivariant, hence it induces a bijection on the quotients by the respective G-actions. Then, claim 9.8.22(ii) implies that the map A^{G} → D^{G} also induces a bijection Hom_{ℤ-Alg}(D^{G}, K) →^{∼} Hom_{ℤ-Alg}(A^{G}, D) for every such K, whence the contention. □

Proposition 9.8.23. In the situation of remark 9.8.16(i), let D (resp. D′) be the integral closure of A in B (resp. of A′ in B′), and suppose that we have norm maps ν_{A} : Γ_{A}^{d}A′ → A and ν_{B} : Γ_{B}^{d}B′ → B of some degree d ∈ ℕ, that make commute the induced diagram :

\xymatrix{Γ_{A}^{d}A′ \ar[r] \ar[d]_{ν_{A}} & Γ_{B}^{d}B′ \ar[d]^{ν_{B}} \\ A \ar[r]^{i} & B}

(see definition 9.5.70). Then there exist unique norm maps ν_{C} : Γ_{C}^{d}C′ → C and ν_{D} : Γ_{D}^{d}D′ → D of degree d, that make commute the resulting diagram :

\xymatrix{Γ_{A}^{d}A′ \ar[r] \ar[d]_{ν_{A}} & Γ_{C}^{d}C′ \ar[r] \ar[d]_{ν_{C}} & Γ_{D}^{d}D′ \ar[r] \ar[d]^{ν_{D}} & Γ_{B}^{d}B′ \ar[d]^{ν_{B}} \\ A \ar[r] & C \ar[r] & D \ar[r] & B.}

Moreover, if ν_{B} is Cayley-Hamilton, the same holds for ν_{C} and ν_{D} (see definition 9.5.74).

Proof. We need to show that that the image of the composition of ν_{B} with the natural map Γ_{D}^{d}D′ → Γ_{B}^{d}B′ lies in D; to this aim, it suffices to check that the natural map Γ_{A}^{d}A′ → Γ_{D}^{d}D′ is integral. However, the latter is the composition of the integral map Γ_{A}^{d}A′ → D ⊗_{A} Γ_{A}^{d}D′ (proposition 9.5.68(iv.a)), the isomorphism D⊗_{A}Γ_{A}^{d}D′ →^{∼} Γ_{D}^{d}(D⊗_{A}D′) (proposition 9.5.68(i)) and the surjection Γ_{D}^{d}(D ⊗_{A} D′) → Γ_{D}^{d}D′ (corollary 9.5.22(iv)), whence the contention.

Likewise, we need to check that the image of the composition of ν_{B} with the natural map Γ_{C}^{d}C′ → Γ_{B}^{d}B′ lies in C. To this aim, notice that the natural map Γ_{A}^{d}A′ → Γ_{A}^{d}C′ is integral, radicial, and its kernel lies in the nilradical of Γ_{A}^{d}A′ (proposition 9.5.68(iv.b,iv.c)); the same holds for the natural map Γ_{A}^{d}C′ → C ⊗_{A} Γ_{A}^{d}C′ →^{∼} Γ_{C}^{d}(C ⊗_{A} C′). By the same token, since the natural map C ⊗_{A} C′ → C′ is surjective, and its kernel lies in the nilradical of C ⊗_{A} C′, then the same holds for the induced map Γ_{C}^{d}(C ⊗_{A} C′) → Γ_{C}^{d}C′. Summing up, we conclude that the natural map Γ_{A}^{d}A′ → Γ_{C}^{d}C′ is integral, radicial, and its kernel J lies in the nilradical of Γ_{A}^{d}A′; we have therefore a commutative diagram of rings :

\xymatrix{Γ_{A}^{d}(A′)/J \ar[r] \ar[d] & Γ_{C}^{d}C′ \ar[d] \\ A \ar[r]^{i} & B}

whose top horizontal arrow is integral, radicial and injective. Then the contention follows from remark 9.8.16(i). Lastly, let χ_{B}^{†}, χ_{C}^{†}, χ_{D}^{†} be the characteristic maps of ν_{B}, ν_{C} and ν_{D} (see (9.5.73)); by inspecting the definitions, we get a commutative diagram :

\xymatrix{Γ_{C}^{d}C′ \ar[r] \ar[d]_{χ_{C}^{†}} & Γ_{D}^{d}D′ \ar[r] \ar[d]_{χ_{D}^{†}} & Γ_{B}^{d}B′ \ar[d]^{χ_{B}^{†}} \\ C′ \ar[r] & D′ \ar[r] & B′}

whence the last assertion of the proposition. □

Definition 9.8.24. (i) Let R be a ring, S ⊂ R a subring, and p > 0 a prime integer. We say that S is p-root closed in R, if for every x ∈ R such that x^{p} ∈ S, we have x ∈ S.

(ii) Let R be a ring and p > 0 a prime integer; if (S_{λ} | λ ∈ Λ) is any family of p-root closed subrings of R, then clearly ⋂_{λ∈Λ} S_{λ} is also p-root closed in R. Hence, for every subring S ⊂ R there exists a smallest p-root closed subring S′ of R containing S. We say that S′ is the p-root closure of S in R.

Remark 9.8.25. (i) Let R be a ring, S ⊂ R a subring; consider the increasing countable system (S_{n} | n ∈ ℕ) of subrings of R defined inductively as follows : S_{0} := S, and S_{n+1} := S_{n}[Σ_{n}] for every n ∈ ℕ, where Σ_{n} := {x ∈ R | x^{p} ∈ S_{n}}. Then it is easily seen that ⋃_{n∈ℕ} S_{n} is the p-root closure of S in R.

(ii) In the situation of remark 9.8.16(i), let D (resp. D′) be the p-root closure of A in B (resp. of A′ in B′). Define the system of subrings (A_{n} | n ∈ ℕ) of B (resp. (A′_{n} | n ∈ ℕ) of B′) as in (i); a simple induction shows that g_{B}(A_{n}) ⊂ A′_{n} for every n ∈ ℕ, so g_{B} restricts to a ring homomorphism D → D′.

Lemma 9.8.26. (i) Let R be a ring, a ∈ R a regular element such that pR ⊂ a^{p}R. Then R is p-root closed in R[a^{−1}] if and only if the Frobenius endomorphism of R/a^{p}R induces an injective ring homomorphism \overline{Φ} : R/aR → R/a^{p}R.

(ii) Let R be a topological ring, S ⊂ R an open subring, T the p-root closure of S in R; endow S and T with the topologies induced by R, and denote by R^{∧}, S^{∧} and T^{∧} the respective completions. Then T^{∧} is the p-root closure in R^{∧} of S^{∧}.

(iii) Let R_{•} := (R_{λ} | λ ∈ Λ) be a system of rings, with transition morphisms f_{λμ} : R_{λ} → R_{μ} for every λ, μ ∈ Λ with μ ≥ λ. For every λ ∈ Λ, let also S_{λ} ⊂ R_{λ} be a subring, such that f_{λμ}(S_{λ}) ⊂ S_{μ} for every such λ, μ. Denote by D_{λ} ⊂ R_{λ} the p-root closure of S_{λ} in R_{λ}, for every λ ∈ Λ. The following holds :

(a) Let R and S be the limits of R_{•} and respectively of S_{•} := (S_{λ} | λ ∈ Λ). Then the limit of the induced system D_{•} := (D_{λ} | λ ∈ Λ) is the p-root closure of S in R.

(b) Suppose moreover that Λ is filtered, and let R′ and S′ be the colimits of R_{•} and respec-

tively of S_{•}. Then the colimit of D_{•} is the p-root closure of S′ in R′. (iv) Let B be a ring, A ⊂ B a subring, C the p-root closure of A in B, and S ⊂ A a

multiplicative subset. Then S^{−1}C is the p-root closure of S^{−1}A in S^{−1}B.

Proof. (i): Suppose first that \overline{Φ} is injective; let x ∈ R[a^{−1}] be any element such that x^{p} ∈ R, and suppose, by way of contradiction, that x ∉ R. There exists a smallest m ∈ ℕ such that y := a^{m}x ∉ R and a^{m+1}x ∈ R. Hence, y^{p} = a^{pm}x^{p} ∈ R, and therefore (a^{m+1}x)^{p} = a^{p}y^{p} ∈ a^{p}R. Thus, the class of a^{m+1}x lies in Ker \overline{Φ}, i.e. a^{m+1}x = az for some z ∈ R; since a is regular, it follows that z = a^{m}x ∈ R, contradicting the choice of m.

Conversely, suppose that R is integrally closed in R[a^{−1}], and let x ∈ R be an element whose class \overline{x} ∈ R/aR lies in Ker \overline{Φ}; this means that there exists y ∈ R such that x^{p} = a^{p}y. Hence, (x/a)^{p} ∈ R, so that x/a ∈ R, and therefore \overline{x} = 0.

(ii): Notice that S^{∧} · T = T^{∧}, hence T^{∧} lies in the p-root closure of S^{∧} in R^{∧}. Hence, we are reduced to checking that S is p-root closed in R if and only if S^{∧} is p-root closed in R^{∧}. Suppose first that S is p-root closed in R, and let x ∈ R^{∧} be any element such that x^{p} ∈ S^{∧}. Denote by j : R → R^{∧} the completion map. Pick a Cauchy net (x_{λ} | λ ∈ Λ) in R whose limit is x; then (x_{λ}^{p} | λ ∈ Λ) is a Cauchy net converging to x^{p}, and since S^{∧} is open in R^{∧} (corollary 8.2.17(i)), we may replace Λ by a cofinal subset, and assume that j(x_{λ})^{p} ∈ S^{∧} for every λ ∈ Λ. Then x_{λ}^{p} ∈ S (corollary 8.2.17(ii)), whence x_{λ} ∈ S for every λ ∈ Λ, and finally x ∈ S^{∧}, as required. Conversely, if S^{∧} is p-root closed in R^{∧}, and x ∈ R is any element with x^{p} ∈ S, we have j(x)^{p} ∈ S^{∧}, so j(x) ∈ S, whence x ∈ S, again by corollary 8.2.17(ii).

The proof of (iii) shall be left to the reader.

(iv): Let b ∈ B, s ∈ S such that (b/s)^{p} ∈ S^{−1}C; it follows easily that t^{p}b^{p} ∈ C for some t ∈ S, hence tb ∈ C, and consequently b/s ∈ S^{−1}C. This shows that S^{−1}C is p-root closed in S^{−1}B, hence it contains the p-root closure C′ of S^{−1}A in S^{−1}B. On the other hand, by remark 9.8.25, the localization B → S^{−1}B restricts to a ring homomorphism C → C′; since C′ is an S^{−1}A-algebra, we must then have S^{−1}C ⊂ C′. □

9.8.27. Let X be a scheme, 𝒜 → ℬ a monomorphism of quasi-coherent 𝒪_{X}-algebras; for every affine open subset U ⊂ X, let 𝒞 (U) (resp. 𝒟(U)) be the weak normalization (resp. the proot closure) of 𝒜 (U) in ℬ(U); by remark 9.8.16(i), the rule : U ↦ 𝒞 (U) (resp. U ↦ 𝒟(U)) defines a subpresheaf of 𝒪_{X}-algebras 𝒞 ⊂ ℬ (resp. 𝒟 ⊂ ℬ) on the site of affine open subsets of X, and it follows easily from proposition 9.8.18(resp. from lemma 9.8.26(iv)) that 𝒞 (resp. 𝒟) is a sheaf on this site, so it extends uniquely to a subsheaf of ℬ on the Zariski site of X; moreover, 𝒞 and 𝒟 are quasi-coherent 𝒪_{X}-algebras. We call 𝒞 and 𝒟 respectively the weak normalization and the p-root closure of 𝒜 in ℬ.

Proposition 9.8.28. Let f : A → B be an injective ring homomorphism, p ∈ ℕ a prime integer, and suppose that f induces an isomorphism A[1/p] →^{∼} B[1/p]. Let A^{ν} (resp. C_{0}, resp. C_{1}) be the integral closure (resp. the weak normalization, resp. the p-root closure) of A in B; denote by (A/pA)_{red} ⟶^{j} (A^{ν}/pA^{ν})_{red} ⟵^{π} A^{ν} the natural maps, and set

C_{2} :={b ∈ B | there exists n ∈ ℕ such that b^{p^{n}} ∈ A}
D :={x ∈ (A^{ν}/pA^{ν})_{red} | there exists n ∈ ℕ such that x^{p^{n}} ∈ Im(j)}.

Then j is injective, and C_{0} = C_{1} = C_{2} = π^{−1}D.

Proof. One argues as in (9.8.19) to see that j is injective.

Now, the inclusion C_{2} ⊂ C_{1} is obvious, and the identity C_{0} = π^{−1}D follows easily from lemma 9.8.20. By remark 9.8.25, we may write C_{1} as the increasing union of the family A_{0} ⊂ A_{1} ⊂ A_{2} ⊂ ⋯ of subrings of B, defined inductively by A_{0} := A, and A_{i+1} := A_{i}[Σ_{i}] for every i ∈ ℕ, where Σ_{i} := {b ∈ B | b^{p} ∈ A_{i}}. By remark 9.8.14(iii), in order to check that C_{1} ⊂ C_{0}, it will therefore suffice to show that the inclusion map A_{i} → A_{i+1} is radicial for every i ∈ ℕ. However, notice that A_{i}[1/p] = A_{i+1}[1/p] for every such i, hence we are reduced to checking that the induced map A_{i}/pA_{i} → A_{i+1}/pA_{i+1} is radicial, which is clear by construction.

Lastly, let us show that C_{0} ⊂ C_{2}. To this aim, let us write f as the colimit of a filtered system of injective ring homomorphisms (f_{λ} : A_{λ} → B_{λ} | λ ∈ Λ), where A_{λ} and B_{λ} are ℤ-subalgebras of finite type of A and respectively B, for every λ ∈ Λ; since f induces an isomorphism A[1/p] →^{∼} B[1/p], we may also assume that f_{λ} induces an isomorphism A_{λ}[1/p] →^{∼} B_{λ}[1/p] for every λ ∈ Λ (details left to the reader). For every such λ, let C_{2,λ} be the subset of all b ∈ B_{λ} such that b^{p^{n}} ∈ A_{λ} for some n ∈ ℕ, and let C_{0,λ} be the weak normalization of A_{λ} in B_{λ}. Clearly C_{2} is the filtered union of the resulting system (C_{2,λ} | λ ∈ Λ); on the other hand, C_{0} is the colimit of the system of subrings (C_{0,λ} | λ ∈ Λ), by lemma 9.8.17. Hence, we are reduced to checking that C_{0,λ} ⊂ C_{2,λ} for every λ ∈ Λ, and we may assume that A and B are noetherian.

Let now b ∈ C_{0}, and set B′ := A[b]; we have to check that b^{p^{n}} ∈ A for some n ∈ ℕ, and obviously A[1/p] = B′[1/p], so we may replace B by B′ and suppose that f is finite and radicial. In this case, pick a decomposition f = f_{1} ∘ ⋯ ∘ f_{n} as in proposition 9.8.9; again, it suffices to check the assertion for each f_{i}, hence we may assume that Spec f is an effective epimorphism of schemes (and still radicial). Thus, for every b ∈ B the element c := b⊗1−1⊗b is nilpotent in B ⊗_{A} B (corollary 9.8.13(i)), and moreover notice that the two natural maps B[1/p] → B ⊗_{A} B[1/p] are isomorphisms, since A[1/p] = B[1/p], hence the same holds for the multiplication law B ⊗_{A} B[1/p] → B[1/p], and the therefore image of c vanishes in B ⊗_{A} B[1/p], i.e. p^{k}c = 0 in B ⊗_{A} B for some k ∈ ℕ, and after replacing k by a larger integer, we may assume that c^{k} = 0 as well. We write

b^{p^{n}} ⊗ 1 = 1 ⊗ b^{p^{n}} + ∑_{j=1}^{p^{n}} \binom{p^{n}}{j}(1 ⊗ b^{p^{n}−j}) · c^{j} = 1 ⊗ b^{p^{n}} + ∑_{j=1}^{k−1} \binom{p^{n}}{j}(1 ⊗ b^{p^{n}−j}) · c^{j}

and recall that the p-adic valuation of \binom{p^{n}}{j} equals n − v_{p}(j) for j = 1, …, p^{n}, where v_{p}(j) denotes the p-adic valuation of j. In particular, with n := 2k, we see that \binom{p^{n}}{j} is divisible by p^{k} for every j = 1 …, k − 1, so finally b^{p^{n}} ⊗ 1 = 1 ⊗ b^{p^{n}} in B ⊗_{A} B, and consequently b^{p^{n}} ∈ A, since Spec f is an effective epimorphism. □

Proposition 9.8.29. Let us resume the situation of remark 9.8.16(i), and let a ∈ A be any element and n ∈ ℕ any integer; also, denote by A^{ν} (resp. A^{′ν}) the integral closure of A in B (resp. of A′ in B′). The following holds :

(i) If g_{A} and g_{B} are injective, aB′ ⊂ Im g_{B} and a^{n}A′ ⊂ Im g_{A}, then :

aA^{′ν} ⊂ g_{B}(A^{ν})  and  aC′ ⊂ g_{B}(C).

(ii) If g_{A} and g_{B} are surjective, and aKer g_{B} = 0, then :

a · g_{B}^{−1}A^{′ν} ⊂ A^{ν}  and  a · g_{B}^{−1}C′ ⊂ C.

(iii) If g_{A} and g_{B} are surjective, and Ker g_{B} ⊂ 𝒩_{B} (notation of (9.8.7)), then :

g_{B}^{−1}A^{′ν} = A^{ν}  and  g_{B}^{−1}C′ = C.

Proof. (i): Let b ∈ A^{′ν}, and pick any monic polynomial P := X^{m}+a′_{1}X^{m−1}+⋯+a′_{m} ∈ A′[X] such that P (b^{n}) = 0. Hence a^{mn} · P (b^{n}) = 0 = (ab)^{mn} + a^{n}a′_{1}(ab)^{n(m−1)} + ⋯ + a^{mn}a′_{m} = 0, and by assumption a^{n}a′_{1}, a^{2n}a′_{2}, …, a^{mn}a′_{m} ∈ g_{A}(A) and ab ∈ g_{B}(B), so ab ∈ g_{B}(A^{ν}), whence the first stated inclusion. Next, set C_{0} := g_{A}(A) + aC′; clearly C′_{0} is an A-subalgebra of g_{B}(B), and we need to check that the induced map A → C_{0} is radicial.

Claim 9.8.30. The kernel and cokernel of the induced map C_{0}⊗_{A}C_{0} → C′⊗_{A′}C′ are annihilated by a^{n+2}.

Proof of the claim. By consruction, the cokernel of the inclusion j : C_{0} → C′ is annihilated by a; since the kernel of C′ ⊗_{A} j is a quotient of Tor_{1}^{A}(C′, C′/C_{0}), it follows that both the kernel and cokernel of C′ ⊗_{A} j are annihilated by a. By the same token, the kernel and cokernel of (C′ ⊗_{A} C_{0}) ⊗_{C_{0}} j : C′ ⊗_{A} C_{0} → C′ ⊗_{A} C′ are annihilated by a as well. Lastly, the natural map π : C′⊗_{A}C′ → C′′⊗_{A′} C′ is clearly surjective, and its kernel is generated by the elements of the form a′x⊗y−x⊗a′y, for all a′ ∈ A′ and all x, y ∈ C′. By assumption, a^{n}(a′x⊗y−x⊗a′y) = 0 in C′ ⊗_{A} C′, so a^{n}Ker π = 0. The claim follows easily. ♢

Let c ∈ aC′ be any element; then c ⊗ 1 − 1 ⊗ c is nilpotent in C′ ⊗_{A′} C′ (corollary 9.8.13(i)); by claim 9.8.30, it follows that there exists N ∈ ℕ with a^{n+2}(c ⊗ 1 − 1 ⊗ c)^{N} = 0 in C_{0} ⊗_{A} C_{0}. Then, say that c = ad for some d ∈ C′; it follows that (c ⊗ 1 − 1 ⊗ c)^{3} = a^{3}d^{3} ⊗ 1 − 3a^{2}d^{2} ⊗ ad + 3ad ⊗ a^{2}d^{2} + 1 ⊗ a^{3}d^{3} is divisible by a in C_{0} ⊗_{A} C_{0}, and finally (c ⊗ 1 − 1 ⊗ c)^{N+3n+6} = 0 in C_{0} ⊗_{A} C_{0}, whence the contention.

(ii): Let b and P be as in the foregoing; suppose that b = g_{B}(c) for some c ∈ B, and pick a_{1}, …, a_{m} ∈ A with g_{A}(a_{i}) = a′_{i} for i = 1, …, m. Set Q := X^{m}+a_{1}X^{m−1}+⋯+a_{m} ∈ A[X]; then aQ(c) = 0, and it follows easily that ac ∈ A^{ν}, whence the first inclusion of (ii). Next, set C_{0} := A+a·g_{B}^{−1}C′; we need to show that the resulting ring homomorphism A → C_{0} is radicial. However, notice that the restriction g_{C} : C_{0} → C of g_{B} has both kernel and cokernel annihilated by a; then by lemma 14.1.55 we find an A-linear map h : C → C_{0} with h ∘ g_{C} = a^{2}1_{C_{0}} and g_{C} ∘ h = a^{2} · 1_{C}. Hence (h ⊗_{A} h) ∘ (g_{C} ⊗_{A} g_{C}) = a^{4}1_{C_{0}} and (g_{C} ⊗_{A} g_{C}) ∘ (h ⊗_{A} h) = a^{4}1_{C}, so both the kernel and cokernel of g_{C} ⊗_{A} g_{C} : C_{0} ⊗_{A} C_{0} → C ⊗_{A} C = C ⊗_{A′} C are annihilated by a^{4}. We may now argue as in the proof of (i), to conclude that c ⊗ 1 − 1 ⊗ c is nilpotent in C_{0} ⊗_{A} C_{0}, for every c ∈ a · g_{B}^{−1}C′, whence the contention.

(iii): Let b, c, P and Q be as in the proof of (ii); since Ker g_{B} ⊂ 𝒩_{B}, we have Q(c)^{n} = 0 for some n ∈ ℕ, so c ∈ A^{ν}, whence the first identity of (iii). Next, set C_{0} := g_{B}^{−1}C′; we need to check that the map A → C_{0} is radicial, or equivalently, that the same holds the induced map on reduced quotients A_{red} → (C_{0})_{red}. However, g_{B} induces an isomorphism (C_{0})_{red} →^{∼} (C′)_{red}, and by definition the map A′_{red} → (C′)_{red} is radicial; since g_{A} induces a surjection A_{red} → A′_{red}, the assertion follows easily. □

10. Cohomology and local cohomology of sheaves

10.1. Cohomology of topoi and topological spaces.

Definition 10.1.1. Let X be a topological space, ℱ a sheaf on X.

(i) We say that ℱ is flabby if the restriction map ℱ (X) → ℱ (U) is surjective for every open subset U ⊂ X.

(ii) We say that ℱ is qc-flabby if the restriction map ℱ (V) → ℱ (U) is a surjection

whenever U ⊂ V are quasi-compact open subsets of X. (iii) The support of ℱ is the subset :

Supp ℱ := {x ∈ X | ℱ_{x} ≠ ∅} ⊂ X.

(iv) On the other hand, if ℱ is an abelian sheaf, we define the support of ℱ as the subset :

Supp ℱ := {x ∈ X | ℱ_{x} ≠ 0} ⊂ X.

Lemma 10.1.2. Let X be a topological space, (X_{λ} | λ ∈ Λ) a family of open subsets of X with X = ⋃_{λ∈Λ} X_{λ}, and ℱ a sheaf on X. For every λ ∈ Λ, let ℱ_{λ} be the restriction of ℱ to X_{λ}. Then ℱ is flabby if and only if ℱ_{λ} is flabby for every λ ∈ Λ.

Proof. It is easily seen that if ℱ is flabby, the same holds for every ℱ_{λ}. Conversely, suppose that ℱ_{λ} is flabby for every λ ∈ Λ; let U ⊂ X be any open subset, and σ ∈ ℱ (U). For every subset Λ′ ⊂ Λ set X_{Λ′} := ⋃_{λ∈Λ′} X_{λ}, and denote by 𝒫 the set of all pairs (Λ′, σ′) where Λ′ is a subset of Λ, and σ′ ∈ ℱ (U ∪ X_{Λ′}) is any section whose restriction to U agrees with σ. We endow 𝒫 with the partial order such that (Λ′, σ′) ≤ (Λ′′, σ′′) if and only if Λ′ ⊂ Λ′′ and the restriction of σ′′ to U ∪ X_{Λ′} agrees with σ′. It is clear that if ((Λ_{i}, σ_{i}) | i ∈ I) is any totally ordered subset in 𝒫, then there exists (Λ′, σ′) in 𝒫 with Λ′ = ⋃_{i∈I} Λ_{i} and such that (Λ_{i}, σ_{i}) ≤ (Λ′, σ′) for every i ∈ I. Also, (∅, σ) ∈ 𝒫. By Zorn’s lemma, 𝒫 admits then a maximal element (Λ′, σ′), and it suffices to check that Λ′ = Λ. Thus, suppose that λ ∈ Λ\Λ′; set Λ′′ := Λ′ ∪ {λ}, and let σ′′ be the restriction of σ′ to U_{λ} := X_{λ} ∩ (U ∪ X_{Λ′}). Since ℱ_{λ} is flabby, there exists σ_{λ} ∈ X_{λ} whose restriction to U_{λ} agrees with σ′′. Then there exists τ ∈ ℱ (U ∪X_{Λ′′}) whose restriction to U ∪ X_{Λ′} agrees with σ′ and whose restriction to X_{λ} agrees with σ_{λ}. By construction, we have (Λ′′, τ) > (Λ′, σ′), a contradiction. □

10.1.3. Let X be a topological space, and consider a short exact sequence

Σ  :  0 → ℱ′ → ℱ → ℱ′′ → 0

of abelian sheaves on X. For every open subset U ⊂ X, the sequence Σ induces a complex :

Σ(U)  :  0 → ℱ′(U) → ℱ (U) → ℱ′′(U) → 0.

Lemma 10.1.4. In the situation of (10.1.3), the following holds :

(i) If ℱ′ is flabby, the sequence Σ(U) is short exact. (ii) If both ℱ and ℱ′ are flabby, the same holds for ℱ′′. (iii) Suppose that X is locally coherent and quasi-separated. Then :

(a) If ℱ′ is qc-flabby and U is quasi-compact, the sequence Σ(U) is short exact.

(b) If both ℱ and ℱ′ are qc-flabby, the same holds for ℱ′′.

Proof. (i): Let s′′ ∈ ℱ′′(U) be any section; we need to show that s′′ is the image of an element of ℱ (U). To this aim, let 𝒫 be the set of all pairs (V, t), where V ⊂ U is an open subset, and t ∈ ℱ (V) is a section whose image in ℱ′′(V) agrees with the restriction of s′′. Then 𝒫 is partially ordered by declaring that (V, t) ≤ (V′, t′) if and only if V ⊂ V′ and t is the restriction of t′ to V. Clearly, if ((V_{λ}, t_{λ}) | λ ∈ Λ) is a totally ordered subset of 𝒫, then there exists a unique section t on V := ⋃_{λ∈Λ} V_{λ} such that (V, t) ∈ 𝒫 and (V_{λ}, t_{λ}) ≤ (V, t) for every λ ∈ Λ. Moreover, 𝒫 ≠ ∅, since (∅, 0) ∈ 𝒫. By Zorn’s lemma, 𝒫 admits therefore a maximal element (V, s), and it suffices to check that V = U. Thus, suppose that x ∈ U \ V; by assumption, we may find an open neighborhood V′ of x in U, and t ∈ ℱ (V′) with (V′, t) ∈ 𝒫. Set W := V ∩ V′, and let t_{W} (resp. s_{W}) be the restriction of t (resp. of s) to W; then s_{W} − t_{W} ∈ ℱ′(W). Since ℱ′ is flabby, the same holds for its restriction ℱ′_{|V} (lemma 10.1.2), hence we find r ∈ ℱ′(V′) whose restriction to W agrees with s_{W} − t_{W}; set t′ := r + t ∈ ℱ (V′). Set V′′ := V ∪ V′; clearly the restriction of t′ to W agrees with s_{W}, hence there exists u ∈ ℱ (V′′) whose restriction to V and V′ agrees respectively with s and t′. By construction, we have (V′′, t′) ∈ 𝒫 and (V′′, t′) > (V, s), a contradiction.

(ii): In view of (i), we get a commutative ladder with short exact rows :

\xymatrix{0 \ar[r] & ℱ′(X) \ar[r] \ar[d]_{ρ′} & ℱ (X) \ar[r] \ar[d]^{ρ} & ℱ′′(X) \ar[r] \ar[d]^{ρ′′} & 0 \\ 0 \ar[r] & ℱ′(U) \ar[r] & ℱ (U) \ar[r] & ℱ′′(U) \ar[r] & 0}

whence, by the snake lemma, a surjection : Coker ρ → Coker ρ′′. But Coker ρ = 0, since ℱ is flabby; so Coker ρ′′ = 0, whence the assertion.

The proof of (iii.a) is a variant of that of (i). Indeed, we may replace X by U, and thereby assume that X is quasi-compact. Then we have to check that every section s′′ ∈ ℱ′′(X) is the image of an element of ℱ (X). However, we can find a finite covering (U_{i} | i = 1, …, n) of X, consisting of quasi-compact open subsets, such that s′′_{|U_{i}} is the image of a section s_{i} ∈ ℱ (U_{i}). For every k ≤ n, let V_{k} := U_{1} ∪ ⋯ ∪ U_{k}; we show by induction on k that s′′_{|V_{k}} is in the image of ℱ (V_{k}); the lemma will follow for k = n. For k = 1 there is nothing to prove. Suppose that k > 1 and that the assertion is known for all j < k; hence we can find t ∈ ℱ (V_{k−1}) whose image is s′′_{|V_{k−1}}. The difference u := (t − s_{k})_{|U_{k}∩V_{k−1}} lies in the image of ℱ′(U_{k} ∩ V_{k−1}); since U_{k} ∩ V_{k−1} is quasi-compact and ℱ′ is qc-flabby, u extends to a section of ℱ′(U_{k}). We can then replace s_{k} by s_{k} + u, and assume that s_{k} and t agree on U_{k} ∩ V_{k−1}, whence a section on V_{k} = U_{k} ∪ V_{k−1} with the sought property. Assertion (iii.b) is proved as (ii). □

Lemma 10.1.5. Let f : Y → X be a continuous map of topological spaces, ℱ a sheaf on X, and 𝒢 a sheaf on Y. We have :

(i) If f is an open immersion and ℱ is qc-flabby on X, then f^{∗}ℱ is qc-flabby on Y. (ii) If 𝒢 is flabby on Y, then f_{∗}𝒢 is flabby on X, and R^{p}f_{∗}𝒢 = 0 for every p > 0. (iii) If f is quasi-compact and 𝒢 is qc-flabby on Y, then f_{∗}𝒢 is qc-flabby on X. (iv) If X and Y are locally coherent, f is quasi-compact and quasi-separated, and 𝒢 is

abelian and qc-flabby, then R^{p}f_{∗}𝒢 = 0 for every p > 0.

(v) Let 𝒜 be any sheaf of rings on X. Every injective 𝒜 -module is flabby.

(vi) If X is locally noetherian, every qc-flabby sheaf on X is flabby.

Proof. (i) and (iii) are trivial.

(ii): Clearly f_{∗}𝒢 is flabby, and it remains to check that R^{p}f_{∗}𝒢 = 0 for every p > 0. To this aim, in view of remark 7.3.31(vi) and lemma 10.1.4(ii), it suffices to show that for every short exact sequence 0 → 𝒢′ → 𝒢 → 𝒢′′ → 0 of flabby abelian sheaves on Y, the resulting sequence 0 → f_{∗}𝒢′ → f_{∗}𝒢 → f_{∗}𝒢′′ → 0 is still short exact; the latter follows from 10.1.4(i).

(iv): The assertion is local on X, so we may assume that X is quasi-compact and quasiseparated, and then Y is also quasi-compact. Remark 7.3.31(vi) and lemma 10.1.4(iii.b) reduce to showing that, for every short exact sequence 0 → 𝒢′ → 𝒢 → 𝒢′′ → 0 of qc-flabby abelian sheaves on Y, the resulting sequence 0 → (f_{∗}𝒢′)_{x} → (f_{∗}𝒢)_{x} → (f_{∗}𝒢′′)_{x} → 0 is short exact, for every x ∈ X. However, every such x has a fundamental system of open neighborhoods consisting of coherent open subsets of X, so we come down to checking that the induced sequence 0 → 𝒢′(f^{−1}U) → 𝒢 (f^{−1}U) → 𝒢′′(f^{−1}U) → 0 is short exact, for every coherent open neighborhood U of x. But under the current assumptions, f^{−1}U is quasi-compact and quasi-separated, so we conclude by lemma 10.1.4(iii.a).

(v): Let j : U → X be an open immersion, and ℐ any injective 𝒜 -module; the counit of adjunction j_{!}j^{∗}𝒜 → 𝒜 is a monomorphism of 𝒜 -modules, so it induces a surjection

φ : Hom_{𝒜} (𝒜 , ℐ) → Hom_{𝒜} (j_{!}j^{∗}𝒜 , ℐ).

We have a natural identification : Hom_{𝒜} (𝒜 , ℐ) →^{∼} ℐ (X); by adjunction, we have also a natural identification : Hom_{𝒜} (j_{!}j^{∗}𝒜 , ℐ) →^{∼} Hom_{j^{∗}𝒜} (j^{∗}𝒜 , j^{∗}ℐ) →^{∼} ℐ (U). Under these identifications, φ corresponds to the restriction map ℐ (X) → ℐ (U), whence the contention.

(vi): Let U ⊂ X be any open subset; we have to check that the restriction map ℱ (X) → ℱ (U) is surjective. To this aim, fix s ∈ ℱ (U), and let S be the set of all pairs (V, s_{V}) where V ⊂ X is an open subset containing U, and s_{V} ∈ ℱ (V) is a section whose restriction to U agrees with s. We endow S with a partial ordering, by declaring that (V, s_{V}) ≤ (V′, s_{V′}) if V ⊂ V′ and the restriction of s_{V′} to V agrees with s_{V}. A standard application of Zorn’s lemma shows that S admits a maximal element (V, s_{V}), and it suffices to show that V = X. However, if the latter fails, there exists a noetherian open subset U′ ⊂ X which does not lie in V, and U′ ∩ V is quasi-compact (remark 8.1.6(iv)); since ℱ is qc-flabby, we may then find s_{U′} ∈ ℱ (U′) whose restriction to U′ ∩ V agrees with the restriction of s_{V}. Then there exists a unique s_{U′∪V} ∈ ℱ (U′ ∩ V) whose restriction to U′ and V agrees with s_{U′} and respectively s_{V}; the pair (U′ ∪ V, s_{U′∪V}) lies in S, contradicting the maximality of (V, s_{V}). □

10.1.6. Let Λ be a small cofiltered category. We call a system of topological spaces indexed by Λ, any functor (notation of (8.1.7))

X_{•} : Λ → Top  λ ↦ X_{u}  (λ ⟶^{u} μ) ↦ (φ_{u} : X_{λ} → X_{μ})

We also call a compatible system of sheaves on X_{•} the datum of a family (ℱ_{λ} | λ ∈ Ob(Λ)), where ℱ_{λ} is a sheaf on X_{λ} for every λ ∈ Ob(Λ), together with transition maps :

β_{u} : φ_{u}^{−1}ℱ_{λ} → ℱ_{μ} for every morphism u : μ → λ in Λ

such that β_{1_{λ}} = 1_{ℱ_{λ}} for every λ ∈ Ob(Λ), and for any two morphisms ν ⟶^{v} μ ⟶^{u} λ of Λ, the following diagram commutes :

\xymatrix{φ_{ν}^{−1} ∘ φ_{u}^{−1}ℱ_{λ} \ar[r]^{∼} \ar[d]_{φ_{ν}^{−1}β_{u}} & φ_{u∘v}^{−1}ℱ_{λ} \ar[d]^{β_{u∘v}} \\ φ_{ν}^{−1}ℱ_{μ} \ar[r]^{β_{ν}} & ℱ_{ν}.}

Lemma 10.1.7. With the notation of (10.1.6), let X be the limit of X_{•}, and (φ_{λ} : X → X_{λ} | λ ∈ Λ) a universal cone. Suppose that either one of the following conditions holds :

(a) X_{•} is a constant functor with value X (so X_{λ} = X for every λ ∈ Ob(Λ), and φ_{u} = 1_{X} for every morphism u of Λ), and X is quasi-compact.

(b) X_{λ} is a sober and quasi-compact topological space, for every λ ∈ Λ. Then the natural map

colim_{λ∈Ob(Λ^{o})} Γ(X_{λ}, ℱ_{λ}) → Γ(X, ℱ) with ℱ := colim_{λ∈Ob(Λ^{o})} φ_{λ}^{∗}ℱ_{λ}

is injective.

Proof. By virtue of proposition 1.5.21(ii), we may assume that Λ is a cofiltered partially ordered set, and we let (Λ^{o}, ≤) be the opposite ordering on Λ (see example 1.1.6(iii)). We then come down to proving the following assertion. Let λ ∈ Λ^{o} be any index, s, s′ ∈ Γ(X_{λ}, ℱ_{λ}) two sections whose images agree in Γ(X, ℱ), and for every μ ∈ Λ^{o} such that μ ≥ λ, denote by s_{μ} and s′_{μ} the images of s and s′ in Γ(X_{μ}, ℱ_{μ}); then there exists an index μ ≥ λ, such that s_{μ} = s′_{μ}. However, set

Z_{μ} := {x ∈ X_{μ} | s_{μ,x} ≠ s′_{μ,x}} for every μ ∈ Λ^{o} such that μ ≥ λ

and endow Z_{μ} with the topology induced from X_{μ}. Now Z_{μ} is a closed subset of X_{μ} for every μ ≥ λ, and it is easily seen that ψ_{u}(Z_{μ}) ⊂ Z_{ν} for every morphism u : μ ≥ ν. Thus, (Z_{μ} | μ ≥ λ) is a cofiltered system of topological spaces, and the assumption on s and s′ implies that the limit of this system is empty. If condition (a) holds, each Z_{μ} is a closed subset of the quasi-compact space X, so we deduce that Z_{μ} = ∅ for some μ ≥ λ, whence s_{μ} = s′_{μ} for such index μ. If (b) holds, each Z_{μ} is a sober space (lemma 8.1.3(iii)), and we conclude again that Z_{μ} = ∅ for some μ ≥ λ, due to proposition 8.1.23(ii.a). □

10.1.8. Let Λ be a small cofiltered category; we consider the datum consisting of:

• Two systems X_{•} and Y_{•} of topological spaces indexed by Λ, such that X_{λ} and Y_{λ} are locally spectral for every λ ∈ Λ, with quasi-compact and quasi-separated transition maps :

(10.1.9)  φ_{u} : X_{μ} → X_{λ}  ψ_{u} : Y_{μ} → Y_{λ}  for every morphism u : μ → λ in Λ

• A compatible system of sheaves ℱ_{•} on Y_{•}, as defined in (10.1.6).

• A natural transformation g_{•} : X_{•} ⇒ Y_{•}, i.e. a system g_{•} := (g_{λ} : Y_{λ} → X_{λ} | λ ∈ Ob(Λ)) of quasi-compact and quasi-separated continuous maps, such that

g_{λ} ∘ ψ_{u} = φ_{u} ∘ g_{μ} for every morphism u : μ → λ.

The limits of X_{•}, Y_{•} and g_{•} are representable by topological spaces and continuous maps

X := lim_{Λ} X_{•}  Y := lim_{Λ} Y_{•}  g := lim_{Λ} g_{•} : Y → X

and the universal cones consist of continuous spectral maps

φ_{λ} : X → X_{λ}  ψ_{λ} : Y → Y_{λ}  for every λ ∈ Λ.

Proposition 10.1.10. In the situation of (10.1.8), the following holds :

(i) If Y_{λ} is spectral for every λ ∈ Ob(Λ), the natural map

colim_{λ∈Ob(Λ^{o})} Γ(Y_{λ}, ℱ_{λ}) → Γ(Y, ℱ) with ℱ := colim_{λ∈Ob(Λ^{o})} ψ_{λ}^{∗}ℱ_{λ}

is a bijection. (ii) If ℱ_{•} is a system of abelian sheaves, then the natural morphisms :

colim_{λ∈Ob(Λ^{o})} φ_{λ}^{∗}R^{i}g_{λ∗}ℱ_{λ} → R^{i}g_{∗}ℱ

are isomorphisms for every i ∈ ℕ.

Proof. By virtue of proposition 1.5.21(ii), we may assume that Λ is a cofiltered partially ordered set, and we let (Λ^{o}, ≤) be the opposite ordering on Λ (see example 1.1.6(iii)). The injectivity of the map in (i) is already known, by lemma 10.1.7. Let us then verify the surjectivity. In case Y = ∅, we know that Y_{λ} = ∅ for some λ ∈ Λ^{o}, so the assertion holds. Hence, we may assume that Y is not empty. Now, let s ∈ Γ(Y, ℱ) be any section; since

ℱ_{y} = colim_{λ∈Λ^{o}} ℱ_{λ,ψ_{λ}(y)} for every y ∈ Y

we may find, for every y ∈ Y, an index λ(y) ∈ Λ^{o}, a quasi-compact open neighborhood U_{y} of ψ_{λ(y)}(y) in Y_{λ,y} and a section s(y) ∈ Γ(U_{y}, ℱ_{λ}) such that the image of s(y) in ℱ_{y} equals s_{y}. Then, for each y ∈ Y, we may find a quasi-compact open neighborhood U′_{y} of y in ψ_{λ(y)}^{−1}U_{y} such that the image of s(y) in U′_{y} agrees with the restriction of s. The family (U′_{y} | y ∈ Y) is an open covering of Y, and since Y is quasi-compact (theorem 8.1.34(i)), we may find a finite (non-empty) subset S ⊂ Y such that (U′_{y} | y ∈ S) already covers Y. Then, for each y ∈ S we may find an index μ(y) ≥ λ(y) and a quasi-compact open subset U′′_{y} of ψ_{μ(y)λ(y)}^{−1}U_{λ(y)} such that ψ_{μ(y)}^{−1}U′′_{y} = U′_{y} (corollary 8.1.40(ii.a)). Next, since Λ is cofiltered, we may find μ ∈ Λ^{o} such that μ ≥ μ(y) for every y ∈ S, and after replacing U′′_{y} by ψ_{μ,μ(y)}^{−1}U′′_{y}, and s(y) by its image in Γ(ψ_{μ,μ(y)}^{−1}U′′_{y}, ℱ_{μ}) for every y ∈ S, we may assume that μ(y) is a single index μ, independent of y ∈ S. Now, since ⋃_{y∈S} ψ_{μ}^{−1}U′′_{y} = Y, there exists ν ≥ μ such that ⋃_{y∈S} ψ_{νμ}^{−1}U′′_{y} = Y_{μ} (corollary 8.1.40(ii.b)), so we may replace μ by ν, each U′′_{y} by its preimage in Y_{ν}, and s(y) by its image in Γ(ψ_{νμ}^{−1}U′′_{y}, ℱ_{ν}), and further assume that (U′′_{y} | y ∈ S) is a covering of Y_{μ}. Lastly, notice that – by construction – for every y, y′ ∈ S, the images of s(y) and s(y′) agree in Γ(ψ_{μ}^{−1}(U_{y} ∩ U_{y′}), ℱ). Since the injectivity of the map in (i) has already been established, we deduce that, for every y, y′ ∈ S there exists λ(y, y′) ≥ μ such that the images of s(y) and s(y′) agree in Γ(ψ_{λ(y,y′),μ}^{−1}(U′′_{y} ∩ U′′_{y′}), ℱ_{λ(y,y′)}). Since Λ is cofiltered, we may then find λ ∈ Λ^{o} such that λ ≥ λ(y, y′) for every y, y′ ∈ S. Now, let V_{y} := ψ_{λμ}^{−1}U′′_{y} and denote by s′(y) the image of s(y) in Γ(V_{y}, ℱ_{λ}), for every y ∈ S; by construction we have s′(y)_{|V_{y}∩V_{y′}} = s′(y)_{|V_{y}∩V_{y′}} for every y, y′ ∈ S, so the system (s′(y) | y ∈ S) corresponds to a well defined section s′ ∈ Γ(Y_{λ}, ℱ_{λ}), whose image in Γ(Y, ℱ) is finally the original section s.

(ii): We notice :

Claim 10.1.11. Suppose that ℱ_{λ} is qc-flabby for every λ ∈ Λ^{o}. Then the same holds for ℱ.

Proof of the claim. Indeed, let U ⊂ V be any two quasi-compact open subsets of Y; we need to show that the restriction map

Γ(V, ℱ) → Γ(U, ℱ)

is onto. By corollary 8.1.40(ii) we can find λ ∈ Λ^{o} and quasi-compact open subsets U_{λ} ⊂ V_{λ} ⊂ Y_{λ} such that U = ψ_{λ}^{−1}U_{λ}, and likewise for V. Let us set U_{μ} := ψ_{μλ}^{−1}U_{λ} for every μ ∈ Λ^{o} such that μ ≥ λ, and likewise define V_{μ}. Then, up to replacing Λ^{o} by the cofinal subset {μ ∈ Λ^{o} | μ ≥ λ}, we may assume that U_{μ} ⊂ V_{μ} are defined for every μ ∈ Λ^{o}. By (i) the natural map

colim_{λ∈Λ^{o}} Γ(U_{λ}, ℱ_{λ}) → Γ(U, ℱ)

is bijective for every n ∈ ℕ, and likewise for V, whence the claim. ♢

Now, we pick for each λ ∈ Λ^{o} an injective resolution ℱ_{λ} → ℐ_{λ}^{•}, having care to construct ℐ_{λ}^{•} functorially, so that ℱ_{•} extends to a compatible system of complexes (ℱ_{λ} → ℐ_{λ}^{•} | λ ∈ Λ^{o}). From claim 10.1.11, we know that ℐ^{•} := colim_{λ∈Λ^{o}} ψ_{λ}^{∗}ℐ_{λ}^{•} is a complex of qc-flabby sheaves; then lemma 10.1.5(iv) and remark 7.3.31(vi) yield a natural isomorphism :

g_{∗}ℐ^{•} →^{∼} Rg_{∗}ℱ in D(ℤ_{X}-Mod).

To conclude, let V → X be any quasi-compact open immersion, which as usual we see as the limit of a system (V_{λ} → X_{λ} | λ ∈ Λ) of quasi-compact open immersions; we come down to checking that the natural map

Γ(V, colim_{λ∈Λ^{o}} φ_{λ}^{∗}g_{λ∗}ℐ_{λ}^{•}) → Γ(V, g_{∞∗}ℐ^{•}) = Γ(g^{−1}V, ℐ^{•})

is an isomorphism of complexes. However, g^{−1}V is the limit of the system (g_{λ}^{−1}V_{λ} | λ ∈ Λ) of spectral topological spaces and quasi-compact quasi-separated maps, so (i) naturally identifies both complexes with

colim_{λ∈Λ^{o}} Γ(g_{λ}^{−1}V_{λ}, ℐ_{λ}^{•})

whence the contention. □

Remark 10.1.12. In the situation of (10.1.8), suppose that X_{•} and Y_{•} are two systems of schemes, that the systems φ_{•} and ψ_{•} of (10.1.9) consist of affine morphisms of schemes, and g_{•} is a system of quasi-compact and quasi-separated morphisms of schemes. By [65, Ch.IV, Prop.8.2.3] the limits (in the category of schemes) of the systems X_{•}, Y_{•} (resp. g_{•}) are representable by schemes (resp. by a morphism of schemes) whose underlying topological spaces (resp. continuous map) agrees with X and Y (resp. g), and the resulting φ_{λ}, ψ_{λ} are affine morphisms of schemes, for every λ ∈ Ob(Λ). Hence, proposition 10.1.10 applies especially to such inverse systems.

10.1.13. Let (X, 𝒜) be a ringed space, and denote by ℤ_{X} the sheaf of rings associated with the constant presheaf with values ℤ; then ℤ_{X}-Mod is the category of abelian sheaves on X, and the natural morphism ℤ_{X} → 𝒜 induces a forgetful functor

φ_{𝒜} : D(𝒜 -Mod) → D(ℤ_{X}-Mod).

Furthermore, let f : (Y, ℬ) → (X, 𝒜) be a morphism of ringed spaces. For every ℬ-module ℱ, the direct image f_{∗}ℱ is naturally an 𝒜 -module, and we notice that the derived functor Rf_{∗} : D^{+}(ℬ-Mod) → D^{+}(𝒜 -Mod) commutes with the forgetful functors φ_{𝒜} , φ_{ℬ} ; indeed, every injective ℬ-module is a flabby ℤ_{Y} -module (lemma 10.1.5(v)), and the latter are acyclic for the functor f_{∗} : ℤ_{Y} -Mod → ℤ_{X}-Mod (lemma 10.1.5(ii)), so the assertion follows easily.

One defines as in example 7.1.16(v) a total Hom cochain complex, which is a functor :

ℋ om_{𝒜}^{•} : C(𝒜 -Mod) × C(𝒜 -Mod)^{o} → C(𝒜 -Mod)

on the category of complexes of 𝒜 -modules. Recall that, for any two complexes M_{•}, N^{•}, and any object U of X, the group of n-cocycles in ℋ om_{𝒜}^{•} (M_{•}, N^{•})(U) is naturally isomorphic to Hom_{C(𝒜_{|U} -Mod)} (M_{•|U}, N_{|U}^{•}[n]), and H^{n}ℋ om_{𝒜}^{•} (M_{•}, N^{•})(U) is naturally isomorphic to the group of homotopy classes of maps M_{•|U} → N_{|U}^{•}[n] (see example 7.1.15(i)). We also set :

(10.1.14)  Hom_{𝒜}^{•} := Γ ∘ ℋ om_{𝒜}^{•} : C(𝒜 -Mod) × C(𝒜 -Mod)^{o} → C(Γ(𝒜)-Mod).

The bifunctor ℋ om_{𝒜}^{•} admits a right derived functor :

Rℋ om_{𝒜}^{•} : D^{+}(𝒜 -Mod) × D(𝒜 -Mod)^{o} → D(𝒜 -Mod)

for whose construction we refer to [163, §10.7]. Likewise, one has a derived functor RHom_{𝒜}^{•} for (10.1.14), and there are natural isomorphisms of 𝒜 -modules :

(10.1.15)  H^{i}RHom_{𝒜}^{•} (M^{•}, N^{•}) →^{∼} Hom_{D^{+}(𝒜 -Mod)} (M^{•}, N^{•}[i])

for every i ∈ ℤ, and every bounded below complexes M^{•} and N^{•} of 𝒜 -modules.

Lemma 10.1.16. Let (X, 𝒜) be a ringed space, ℬ an 𝒜 -algebra. Then :

(i) If ℐ is an injective 𝒜 -module, ℋ om_{𝒜} (ℱ , ℐ) is flabby for every 𝒜 -module ℱ, and ℋ om_{𝒜} (ℬ, ℐ) is an injective ℬ-module. (ii) There is a natural isomorphism of bifunctors :

RΓ ∘ Rℋ om_{𝒜}^{•} →^{∼} RHom_{𝒜}^{•} .

(iii) The forgetful functor D^{+}(ℬ-Mod) → D^{+}(𝒜 -Mod) admits the right adjoint :

D^{+}(𝒜 -Mod) → D^{+}(ℬ-Mod) : K^{•} ↦ Rℋ om_{𝒜}^{•} (ℬ, K^{•}).

Proof. (i) : Let j : U ⊂ X be any open immersion; a section s ∈ ℋ om_{𝒜} (ℱ , ℐ)(U) is a map of 𝒜_{|U}-modules s : ℱ_{|U} → ℐ_{|U}. We deduce a map of 𝒜 -modules j_{!}s : j_{!}ℱ_{|U} → j_{!}ℐ_{|U} → ℐ; since ℐ is injective, j_{!}s extends to a map ℱ → ℐ, as required. Next, let ℬ be an 𝒜 -algebra; we recall the following :

Claim 10.1.17. The functor

𝒜 -Mod → ℬ-Mod : ℱ ↦ ℋ om_{𝒜} (ℬ, ℱ)

is right adjoint to the forgetful functor.

Proof of the claim. We have to exhibit a natural bijection

Hom_{𝒜} (N, M) →^{∼} Hom_{ℬ} (N, Hom_{𝒜} (ℬ, M))

for every 𝒜 -module M and ℬ-module N. This is given by the following rule. To an 𝒜 - linear map t : N → M one assigns the ℬ-linear map t′ : N → Hom_{𝒜} (ℬ, M) such that t′(x)(b) = t(x · b) for every x ∈ N(U) and b ∈ ℬ(V), where V ⊂ U are any two open subsets of X. To describe the inverse of this transformation, it suffices to remark that t(x) = t′(x)(1) for every local section x of N. ♢

Since the forgetful functor is exact, the second assertion of (i) follows immediately from claim 10.1.17. Assertion (ii) follows from (i) and [163, Th.10.8.2].

(iii): Let K^{•} (resp. L^{•}) be a bounded below complex of ℬ-modules (resp. of injective 𝒜 -modules); then :

Hom_{D^{+}(𝒜 -Mod)} (K^{•}, L^{•}) = H^{0}Hom_{C(𝒜 -Mod)}^{•} (K^{•}, L^{•})
= H^{0}Hom_{C(ℬ-Mod)}^{•} (K^{•}, ℋ om_{𝒜} (ℬ, L^{•}))
= Hom_{D^{+}(ℬ-Mod)} (K^{•}, ℋ om_{𝒜} (ℬ, L^{•}))

where the last identity follows from (i) and (10.1.15). □

Theorem 10.1.18 (Trivial duality). Let f : (Y, ℬ) → (X, 𝒜) be a morphism of ringed spaces. For every M_{•} ∈ Ob(D^{−}(𝒜 -Mod)) and N^{•} ∈ Ob(D^{+}(ℬ-Mod)), there are natural isomorphisms :

(i) RHom_{𝒜}^{•} (M_{•}, Rf_{∗}N^{•}) →^{∼} RHom_{ℬ}^{•} (Lf^{∗}M_{•}, N^{•}) in D^{+}(𝒜 (X)-Mod).

(ii) Rℋ om_{𝒜}^{•} (M_{•}, Rf_{∗}N^{•}) →^{∼} Rf_{∗}Rℋ om_{ℬ}^{•} (Lf^{∗}M_{•}, N^{•}) in D^{+}(𝒜 -Mod).

Proof. One applies lemma 10.1.16(ii) to deduce (i) from (ii).

(ii) : By standard adjunctions, for any two complexes M_{•} and N^{•} as in the proposition, we have a natural isomorphism of total Hom cochain complexes :

(10.1.19)  ℋ om_{𝒜}^{•} (M_{•}, f_{∗}N^{•}) →^{∼} f_{∗}ℋ om_{ℬ}^{•} (f^{∗}M_{•}, N^{•}).

Now, let us choose a flat resolution P_{•} →^{∼} M_{•} of 𝒜 -modules, and an injective resolution N^{•} →^{∼} I^{•} of ℬ-modules. In view of (10.1.19) and lemma 10.1.16(i) we have natural isomorphisms :

Rf_{∗}Rℋ om_{ℬ}^{•} (Lf^{∗}M_{•}, N^{•}) →^{∼} f_{∗}ℋ om_{ℬ}^{•} (f^{∗}P_{•}, I^{•}) ←^{∼} ℋ om_{𝒜}^{•} (P_{•}, f_{∗}I^{•})

in D^{+}(𝒜 -Mod). It remains to show that the natural map

(10.1.20)  ℋ om_{𝒜}^{•} (P_{•}, f_{∗}I^{•}) → Rℋ om_{𝒜}^{•} (P_{•}, f_{∗}I^{•})

is an isomorphism. However, we have two spectral sequences :

\begin{matrix}E_{pq}^{1} := H_{p}ℋ om_{𝒜}^{•} (P_{q}, f_{∗}I^{•}) ⇒ & H_{p+q}ℋ om_{𝒜}^{•} (P_{•}, f_{∗}I^{•}) \\ F_{pq}^{1} := H_{p}Rℋ om_{𝒜}^{•} (P_{q}, f_{∗}I^{•}) ⇒ & H_{p+q}Rℋ om_{𝒜}^{•} (P_{•}, f_{∗}I^{•}) \end{matrix}

and (10.1.20) induces a natural map of spectral sequences :

(10.1.21)  E_{pq}^{1} → F_{pq}^{1}

Consequently, it suffices to show that (10.1.21) is an isomorphism for every p, q ∈ ℕ, and therefore we may assume that P_{•} consists of a single flat 𝒜 -module placed in degree zero. A similar argument reduces to the case where I_{•} consists of a single injective ℬ-module sitting in degree zero. To conclude, it suffices to show :

Claim 10.1.22. Let P be a flat 𝒜 -module, I an injective ℬ-module. Then the natural map :

ℋ om_{𝒜} (P, f_{∗}I)[0] → Rℋ om_{𝒜}^{•} (P, f_{∗}I)

is an isomorphism.

Proof of the claim. R^{i}ℋ om_{𝒜}^{•} (P, f_{∗}I) is the sheaf associated with the presheaf on X :

U ↦ R^{i}Hom_{𝒜_{|U}}^{•} (P_{|U}, f_{∗}I_{|U}).

Since I_{|U} is an injective ℬ_{|U}-module, it suffices therefore to show that R^{i}Hom_{𝒜}^{•} (P, f_{∗}I) = 0 for i > 0. However, recall that there is a natural isomorphism :

R^{i}Hom_{𝒜}^{•} (P, f_{∗}I) ≃ Hom_{D(𝒜 -Mod)} (P, f_{∗}I[−i]).

Since the homotopy category Hot(𝒜 -Mod) admits a left calculus of fractions, we deduce a natural isomorphism :

(10.1.23)  R^{i}Hom_{𝒜}^{•} (P, f_{∗}I) ≃ colim_{E_{•}→P} Hot_{𝒜 -Mod} (E_{•}, f_{∗}I[−i])

where the colimit ranges over the family of quasi-isomorphisms E_{•} → P. We may furthermore restrict the colimit in (10.1.23) to the subfamily of all such E_{•} → P where E_{•} is a bounded above complex of flat 𝒜 -modules, since this subfamily is cofinal. Every such map E_{•} → P induces a commutative diagram :

\xymatrix{Hom_{𝒜} (P, f_{∗}I[−i]) \ar[r] \ar[d] & Hot_{𝒜 -Mod} (E_{•}, f_{∗}I[−i]) \ar[d] \\ Hom_{ℬ} (f^{∗}P, I[−i]) \ar[r] & Hot_{ℬ-Mod} (f^{∗}E_{•}, I[−i])}

whose vertical arrows are isomorphisms. Since E_{•} and P are complexes of flat 𝒜 -modules, the induced map f^{∗}E_{•} → f^{∗}P is again a quasi-isomorphism; therefore, since I is injective, the bottom horizontal arrow is an isomorphism, hence the same holds for the top horizontal one, and the claim follows easily. □

10.1.24. Suppose that ℱ is an abelian group object in a given topos X, and let (Fil^{n}ℱ | n ∈ ℕ) be a descending filtration by abelian subobjects of ℱ, such that Fil^{0}ℱ = ℱ. Set ℱ^{n} := ℱ /Fil^{n+1}ℱ for every n ∈ ℕ, and denote by gr^{•}ℱ the graded abelian object associated with the filtered object Fil^{•}ℱ; then we have :

Lemma 10.1.25. In the situation of (10.1.24), fix q ∈ ℕ and suppose that the natural map :

ℱ → R lim_{k∈ℕ} ℱ^{k}

is an isomorphism in D(ℤ_{X}-Mod). Then, the following conditions are equivalent : (a) H^{q}(X, Fil^{n}ℱ) = 0 for every n ∈ ℕ.

(b) The natural map H^{q}(X, Fil^{n}ℱ) → H^{q}(X, gr^{n}ℱ) vanishes for every n ∈ ℕ.

Proof. Obviously we have only to show that (b) implies (a). Moreover, let us set :

(Fil^{n}ℱ)^{k} := Fil^{n}ℱ /Fil^{k+1}ℱ for every n, k ∈ ℕ with k ≥ n.

There follows, for every n ∈ ℕ, a short exact sequence of inverse systems of sheaves :

0 → ((Fil^{n}ℱ)^{k} | k ≥ n) → (ℱ^{k} | k ≥ n) → ℱ^{n−1} → 0

(where the right-most term is the constant inverse system which equals ℱ^{n−1} in all degrees, with transition maps given by the identity morphisms); whence a distinguished triangle :

R lim_{k∈ℕ} (Fil^{n}ℱ)^{k} → R lim_{k∈ℕ} ℱ^{k} → ℱ^{n−1}[0] → R lim_{k∈ℕ} (Fil^{n}ℱ)^{k}[1]

which, together with our assumption on ℱ, easily implies that the natural map Fil^{n}ℱ → R lim_{k∈ℕ}(Fil^{n}ℱ)^{k} is an isomorphism in D(ℤ_{X}-Mod). Summing up, we may replace the datum (ℱ , Fil^{•}ℱ) by (Fil^{n}ℱ , Fil^{•+n}ℱ), and reduce to the case where n = 0.

The inverse system (ℱ^{n} | n ∈ ℕ) defines an abelian group object of the topos X^{ℕ} (see [75, §7.3.4]); whence a spectral sequence:

(10.1.26)  E_{2}^{pq} := lim_{n∈ℕ}^{p} H^{q}(X, ℱ^{n}) ⇒ H^{p+q}(X^{ℕ}, ℱ^{•}) ≃ H^{p+q}(X, R lim_{n∈ℕ} ℱ^{n}).

By [163, Cor.3.5.4] we have E_{2}^{pq} = 0 whenever p > 1, and, in view of our assumptions, (10.1.26) decomposes into short exact sequences :

0 → lim_{n∈ℕ}^{1} H^{q−1}(X, ℱ^{n}) ⟶^{α_{q}} H^{q}(X, ℱ) ⟶^{β_{q}} lim_{n∈ℕ} H^{q}(X, ℱ^{n}) → 0 for every q ∈ ℕ

where β_{q} is induced by the natural system of maps (β_{q,n} : H^{q}(X, ℱ) → H^{q}(X, ℱ^{n}) | n ∈ ℕ).

Claim 10.1.27. If (b) holds, then β_{q} = 0.

Proof of the claim. Suppose that (b) holds; we consider the long exact cohomology sequence associated with the short exact sequence :

0 → Fil^{n+1}ℱ → Fil^{n}ℱ → gr^{n}ℱ → 0

to deduce that the natural map H^{q}(X, Fil^{n+1}ℱ) → H^{q}(X, Fil^{n}ℱ) is onto for every n ∈ ℕ; hence the same holds for the natural map H^{q}(X, Fil^{n}ℱ) → H^{q}(X, ℱ). By considering the long exact cohomology sequence attached to the short exact sequence :

(10.1.28)  0 → Fil^{n}ℱ → ℱ → ℱ^{n} → 0

we then deduce that β_{q,n} vanishes for every n ∈ ℕ, which implies the claim. ♢ Claim 10.1.29. If (b) holds, then the inverse system (H^{q−1}(X, ℱ^{n}) | n ∈ ℕ) has surjective transition maps.

Proof of the claim. By considering the long exact cohomology sequence associated with the short exact sequence : 0 → gr^{n}ℱ → ℱ^{n+1} → ℱ^{n} → 0, we are reduced to showing that the boundary map ∂_{q−1} : H^{q−1}(X, ℱ^{n}) → H^{q}(X, gr^{n}ℱ) vanishes for every n ∈ ℕ. However, comparing with (10.1.28), we find that ∂_{q−1} factors through the natural map H^{q}(X, Fil^{n}ℱ) → H^{q}(X, gr^{n}ℱ), which vanishes if (b) holds. ♢

The lemma follows from claims 10.1.27and 10.1.29, and [163, Lemma 3.5.3]. □

Our next result generalizes to arbitrary coherent spaces a classical theorem from [84], where it was stated only for noetherian spaces. We need first to introduce a general truncation construction for abelian sheaves, that shall be useful also in section 11.2.

10.1.30. Let (X, 𝒜) be a ringed topological space, φ : V → X a locally closed immersion, denote by \overline{V} the topological closure of V in X, and set ∂V := \overline{V} \V, which is also a closed subset of X. Let also i : \overline{V} → X and j : V → \overline{V} be the induced closed and respectively open immersions. We set

φ_{!} := i_{∗} ∘ j_{!} : φ^{∗}𝒜 -Mod → 𝒜 -Mod t_{V} := φ_{!} ∘ φ^{∗} : 𝒜 -Mod → 𝒜 -Mod

and we call φ_{!} (resp. t_{V}) the functor of extension by zero from V to X (resp. of truncation outside V). Notice that we have natural identifications

(10.1.31)  t_{V} ℱ_{x} →^{∼} \{ \begin{matrix}ℱ_{x} & if x ∈ V \\ 0 & otherwise. \end{matrix}

Lemma 10.1.32. In the situation of (10.1.30), set W := X \V, denote by ψ : W → X the resulting locally closed immersion, and let ℱ be an 𝒜 -module such that ψ^{∗}ℱ = 0. Then there exists a unique natural isomorphism

f_{ℱ} : t_{V} ℱ →^{∼} ℱ

such that f_{ℱ ,x} is the natural identification (10.1.31), for every x ∈ V.

Proof. The uniqueness of f_{ℱ} is clear. For the existence, set 𝒢 := i^{∗}ℱ; it suffices to exhibit natural isomorphisms

ℱ →^{∼} i_{∗}𝒢 and j_{!}φ^{∗}ℱ →^{∼} 𝒢

which induce the corresponding natural identifications on the stalks. So we may assume that φ is either an open or closed immersion. In case φ is open, we obtain a natural map f : φ_{!}ℱ → ℱ as follows. Recall that φ_{!}ℱ is the sheaf associated with the presheaf ℋ such that ℋ (U) = ℱ (U) for U ⊂ V, and ℋ (U) = 0 otherwise; then clearly there is a natural monomorphism ℋ → ℱ in the category of presheaves on X, and the latter factors uniquely through a map as sought. By construction, f_{x} is the identity endomorphism of ℱ_{x}, for every x ∈ V, and f_{x} is also trivially an isomorphism for x ∉ V, since in this case the stalks of both source and target of f_{x} vanish. So f is indeed an isomorphism with the required properties.

Lastly, suppose φ is a closed immersion; for any open subset T of V, let C_{T} be the cofiltered set of all open subsets U of X such that U ∩ V = T, and set

φ^{−1}ℋ (T) := colim_{U∈C_{T}^{o}} ℋ (U) for every presheaf of abelian groups ℋ on X .

The rule T ↦ φ^{−1}ℋ (T) for every open subset T of V yields a well defined abelian presheaf on V, functorial in ℋ (the resulting functor φ^{−1} from abelian presheaves on X to abelian presheaves on V is left adjoint to the direct image functor φ_{∗} : cp. remark 1.3.6(i)). Now, suppose that ℋ is a 𝒜 -module such that ℋ_{|X\V} = 0; in this case, it is easily seen that φ^{−1}ℋ is a φ^{∗}𝒜 -module, and moreover the transition map ℋ (U) → ℋ (U′) is an isomorphism, for every U, U′ ∈ C_{T} such that U′ ⊂ U. Especially, this applies with ℋ := ℱ, so we get a natural isomorphism

f_{U} : ℱ (U) →^{∼} i^{−1}ℱ (U ∩ V) = i_{∗}i^{−1}ℱ (U) for every open subset U ⊂ X

and the system of such maps f_{U} is the sought isomorphism ℱ →^{∼} i_{∗}i^{∗}ℱ. □

Proposition 10.1.33. Let V and V′ be two locally closed subsets of X. The following holds : (i) We have a unique isomorphism of functors :

τ^{V,V′} : t_{V} ∘ t_{V′} →^{∼} t_{V ∩V′}

which induces, for every 𝒜 -module ℱ and every x ∈ V ∩ V′ a commutative diagram

(10.1.34)  \xymatrix{t_{V} ∘ t_{V′}ℱ_{x} \ar[r]^{τ_{x}^{V,V′}} \ar[d]_{α} & t_{V ∩V′′}ℱ_{x} \ar[d]^{β} \\ t_{V′}ℱ_{x} \ar[r]^{γ} & ℱ_{x}}

where α, β and γ are the natural identifications of (10.1.31). (ii) Let V′′ ⊂ X be a third locally closed subset. Then the diagram

\xymatrix{t_{V} ∘ t_{V′} ∘ t_{V′′} \ar[r]^{t_{V} ∗τ^{V′,V′′}} \ar[d]_{τ^{V,V′}∗t_{V′′}} & t_{V} ∘ t_{V′∩V′′} \ar[d]^{τ^{V,V′∩V′′}} \\ t_{V ∩V′} ∘ t_{V′′} \ar[r]^{τ^{V ∩V′,V′′}} & t_{V ∩V′∩V′′}}

commutes. (iii) Suppose that :

(a) either V ⊂ V′ and the resulting immersion V → V′ is closed

(b) or else V′ ⊂ V and the resulting immersion V′ → V is open. Then there exists a unique morphism of functors

c^{V,V′} : t_{V′} → t_{V}

which induces, for every 𝒜 -module ℱ and every x ∈ V ∩ V′, a commutative diagram

[figure: t_{V′}ℱ_{x}; c_{x}^{V,V′}; t_{V} ℱ_{x}; ℱ_{x}]

whose downward arrows are the natural identifications (10.1.31). (iv) In the situation of (iii), let V′′ be a third locally closed subset of X. Then the resulting

diagram

\xymatrix{t_{V′′} ∘ t_{V′} \ar[r]^{t_{V′′}∗c^{V,V′}} \ar[d]_{τ^{V′′,V′}} & t_{V′′} ∘ t_{V} \ar[d]^{τ^{V′′,V}} \\ t_{V′′∩V′} \ar[r]^{c^{V′′∩V,V′′∩V′}} & t_{V′′∩V}}

commutes.

(v) Let V ⊂ V′ ⊂ V′′ (resp. V′′ ⊂ V′ ⊂ V) be a chain of closed (resp. open) immersions of locally closed subsets of X. Then

c^{V,V′} ∘ c^{V′,V′′} = c^{V,V′′}.

Proof. (i): The uniqueness is clear. For the existence, set W := V ∩ V′′, and let φ : W → X and ψ : X \W → X be the locally closed immersions; by inspecting the definition, we find natural isomorphisms

φ^{∗} ∘ t_{V} ∘ t_{V′}ℱ →^{∼} φ^{∗}ℱ ⟵^{∼} φ^{∗}t_{W} ℱ for every 𝒜 -module ℱ

which induce on stalks the isomorphisms φ^{∗}(γ ∘ α) and respectively φ^{∗}β. There follows an isomorphism φ_{!}f : t_{W} ∘ t_{V} ∘ t_{V′}ℱ →^{∼} t_{V′′} ∘ t_{V′′}ℱ. On the other hand, it is easily seen that ψ^{∗} ∘ t_{V} ∘ t_{V′}ℱ and ψ^{∗} ∘ t_{W} ℱ both vanish, so lemma 10.1.32 naturally identifies φ_{!}f with an isomorphism t_{V} ∘ t_{V′}ℱ →^{∼} t_{W} ℱ with the sought property.

(ii): The commutativity of the diagram can be checked on stalks, and then it follows easily from the commutativity of (10.1.34) : details left to the reader.

(iii): Again, the uniqueness of c^{V,V′} is clear. For the existence, consider first case (a), and write V = V′ ∩ Z for some closed immersion φ : Z → X. The unit of the adjunction (φ^{∗}, φ_{∗}) yields a natural morphism η : ℱ → t_{Z}ℱ, whence a morphism t_{V′}η : t_{V′}ℱ → t_{V′} ∘ t_{Z}ℱ, which, by (i) is naturally identified with a morphism t_{V′}ℱ → t_{V} ℱ fulfilling the stated condition. A similar argument works in case (b) : we write V′ = V ∩ U for some open immersion ψ : U ⊂ X and we consider the counit ε : t_{U}ℱ → ℱ of the adjunction (ψ_{!}, ψ^{∗}); then (i) naturally identifies t_{V} ε with a morphism with the sought properties.

(iv) and (v) can be checked on the stalks : the details shall be left to the reader. □

Theorem 10.1.35. Let X be any coherent space, ℱ any abelian sheaf on X. Then : (i) H^{i}(X, ℱ) = 0 for every i > dim X. (ii) If X is spectral, we have more precisely

H^{i}(X, ℱ) = 0 for every i > d_{ℱ} := sup(dim \overline{\{x\}} | x ∈ Supp ℱ) .

Proof. (Here, for any subset T ⊂ X, we denote by \overline{T} the topological closure of T in X).

Claim 10.1.36. Let d ∈ ℕ be any integer, and suppose that (i) holds for every spectral space X of dimension ≤ d. Then (ii) holds for every spectral space X and every ℤ_{X}-module ℱ such that d_{ℱ} ≤ d.

Proof of the claim. Indeed, let 𝒮 be the set consisting of all pairs (U, s), where U ⊂ X is a quasi-compact open subset, and s ∈ ℱ (U) is any element. For any (U, s) ∈ 𝒮, let Supp(s) := {x ∈ U | s_{x} ≠ 0}, notice that Supp(s) is a closed subset of U, and denote by i_{s} : Supp(s) → U and j_{U} : U → X the resulting closed and respectively open immersions; the section s is the same as a morphism of ℤ_{U}-modules ℤ_{U} → j_{U}^{∗}ℱ which factors through the epimorphism ℤ_{U} → i_{s∗}ℤ_{Supp(s)}. By adjunction, there follows a unique morphism of ℤ_{X}- modules φ_{U,s} : j_{U!}i_{s∗}ℤ_{Supp(s)} → ℱ. For any finite subset S ⊂ 𝒮, we let

φ_{S} : ⨁_{(U,s)∈S} j_{U!}i_{s∗}ℤ_{Supp(s)} → ℱ

be the unique map of ℤ_{X}-modules whose restriction to j_{U!}i_{s∗}ℤ_{Supp(s)} agrees with φ_{U,s}, for every (U, s) ∈ S. It is easily seen that ℱ is the colimit of the filtered system of its subsheaves ℱ_{S} := Im φ_{S}, for S ranging over all finite subsets of 𝒮. By proposition 10.1.10(i) we deduce that H^{i}(X, ℱ) is the filtered colimit of the system of abelian groups (H^{i}(X, ℱ_{S}) | S ⊂ 𝒮); it then suffices to show the assertion for each subsheaf ℱ_{S}. Thus, we may assume from start that there exists a finite subset S ⊂ 𝒮 with ℱ = ℱ_{S}. In this case, we set

Z := \overline{Supp ℱ} = ⋃_{(U,s)∈S} \overline{Supp(s)}

and we let i_{Z} : Z → X be the closed immersion. Clearly we have a natural isomorphism

H^{i}(X, ℱ) →^{∼} H^{i}(Z, i_{Z}^{∗}ℱ) for every i ∈ ℕ

and Z is still a spectral space, by corollary 8.1.42. Hence, we may replace X by Z and ℱ by i_{Z}^{∗}ℱ, and assume from start that Z = X (and clearly, we still have ℱ = ℱ_{S}). In view of our assumption, it then suffices to show that

dim Z = sup(dim \overline{\{x\}} | x ∈ ⋃_{(U,s)∈S} Supp(s)).

To this aim, in light of remark 8.1.2(v), we are reduced to checking that dim \overline{Supp(s)} = sup(dim \overline{\{x\}} | x ∈ Supp(s)) for every (U, s) ∈ S. However, notice that Supp(s) is proconstructible in X, hence \overline{Supp(s)} is the set of all specializations in X of the points of Supp(s) (proposition 8.1.47(i)). Now the contention follows from remark 8.1.45(iii). ♢

(i): Let S_{X} be the sober space associated with X, and f_{X} : X → S_{X} the unit of adjunction (proposition 8.1.8); by remark 8.1.11, the space S_{X} is spectral; moreover the functor

f_{X}^{∗} : ℤ_{S_{X}}-Mod → ℤ_{X}-Mod

is an equivalence of categories, and the induced map Γ(S_{X}, ℱ) → Γ(X, f_{X}^{∗}ℱ) is a bijection for every ℤ_{S_{X}}-module ℱ. It follows easily that the induced map

H^{i}(S_{X}, ℱ) → H^{i}(X, f_{X}^{∗}ℱ)

is an isomorphism as well; thus, we may replace X by S_{X}, and assume from start that X is spectral.

We argue by induction on d := dim X. If d = 0, X is a boolean space (see example 8.1.48), so every sheaf on X is qc-flabby, and the assertion follows from lemma 10.1.5(iv). We may then assume that d > 0 and that (i) is already known for every spectral space of dimension < d. Suppose, by way of contradiction, that H^{d+1}(X, ℱ) ≠ 0 for some ℤ_{X}-module ℱ, and pick any element c ≠ 0 in this cohomology group. For any closed subset Z ⊂ X, let i_{Z} : Z → X be the corresponding closed immersion, denote by 𝒵 the system of all closed subsets Z ⊂ X such that the image of c does not vanish in H^{d+1}(Z, i_{Z}^{∗}ℱ), and endow 𝒵 with the partial ordering given by inclusion. Notice that each Z ∈ 𝒵 is also a spectral space (corollary 8.1.42), so we must have dim Z = d, by inductive assumption.

Claim 10.1.37. (i) The partially ordered set 𝒵 admits minimal elements.

(ii) Every minimal element of 𝒵 is an irreducible closed subset of X.

Proof of the claim. (i): Let (Z_{i} | i ∈ I) be any totally ordered non-empty subset of 𝒵; by Zorn’s lemma, it suffices to check that ⋂_{i∈I} Z_{i} ∈ 𝒵. However, notice that the inclusion maps Z_{i} → Z_{j} are spectral for every i, j ∈ I with Z_{i} ⊂ Z_{j} (lemma 8.1.19(x.e) and remark 8.1.20(iv)), so the latter assertion follows directly from proposition 10.1.10(i).

(ii): Let Z be any minimal element of 𝒵, and suppose, by way of contradiction, that Z = Z_{1} ∪ Z_{2} for two non-empty closed subsets Z_{1}, Z_{2} strictly contained in Z; notice that Z_{1} and Z_{2} are also spectral spaces (corollary 8.1.42). Pick any non-empty quasi-compact open subset U ⊂ Z \ Z_{1}, and set Z′_{1} := Z \ U, Z′_{2} := \overline{U}. Since U ⊂ Z_{2}, we have Z′_{2} ⊂ Z_{2} and Z_{1} ⊂ Z′_{1}. Moreover, both Z′_{1} (resp. Z′_{2}) is the set of all specializations of the points of Z \ U (resp. of U); it follows that Z′_{3} := Z′_{1} ∩ Z′_{2} has dimension < d : indeed, any totally ordered sequence x_{1} < ⋯ < x_{k} of elements of (Z′_{3}, ≤) (with ordering given by specialization) can be extended to a totally ordered sequence of (Z′_{2}, ≤), by adding a generization x_{k+1} of x_{k} with x_{k+1} ∈ U.

On the other hand, consider the sequence

(10.1.38)  0 → t_{Z′_{3}}ℱ → t_{Z′_{1}}ℱ ⊕ t_{Z′_{2}}ℱ → t_{Z}ℱ → 0.

whose second and third arrows are given by the natural transformations provided by proposition 10.1.33(iii); by considering the stalks over the points of Z, it is easily seen that (10.1.38) is short exact, whence an exact sequence of abelian groups

(10.1.39) H^{d}(Z′_{3}, i_{Z′_{3}}^{∗}ℱ) → H^{d+1}(Z, i_{Z}^{∗}ℱ) ⟶^{β} H^{d+1}(Z′_{1}, i_{Z′_{1}}^{∗}ℱ) ⊕ H^{d+1}(Z′_{2}, i_{Z′_{2}}^{∗}ℱ).

However, the first term of (10.1.39) vanishes by inductive assumption, hence β is injective; especially the image of c cannot vanish in both H^{d+1}(Z′_{1}, i_{Z′_{1}}^{∗}ℱ) and H^{d+1}(Z′_{2}, i_{Z′_{2}}^{∗}ℱ), contradicting the minimality of Z. ♢

By claim 10.1.37, we may replace X by a minimal element of 𝒵, and assume from start that X is irreducible. In this case, denote by η ∈ X the maximal point, and let j_{η} : {η} → X be the inclusion map; there follows a natural map of ℤ_{X}-modules f : ℱ → j_{η∗}j_{η}^{∗}ℱ such that f_{η} is an isomorphism. Since j_{η}^{∗}ℱ is obviously qc-flabby on {η}, lemma 10.1.5(iii,iv) implies that H^{i}(X, j_{η∗}j_{η}^{∗}ℱ) = 0 for every i > 0. We deduce an exact sequence of abelian groups

(10.1.40)  H^{d+1}(X, Ker f) → H^{d+1}(X, ℱ) → H^{d+1}(X, Im f)

as well as an isomorphism of abelian groups

H^{d}(X, Coker f) →^{∼} H^{d+1}(X, Im f)

and by construction we have d_{Ker,f}, d_{Coker f} < d. From the inductive assumption and claim 10.1.36, it follows that the first and third terms of (10.1.40) vanish, so the same holds for the middle one, and the proof is concluded. □

Remark 10.1.41. A different proof of (a slight improvement of) theorem 10.1.35 is found in [147]. See also proposition 10.4.7(ii). Alternative proofs for both theorem 10.1.35 and proposition 10.4.7(ii.b,ii.c) are also found in [107, Tag 0A3G].

10.2. Čech cohomology. This section is a review of the standard constructions of resolutions via Čech complexes, for general topoi. We also include two classical applications to the cohomology of quasi-coherent modules on a scheme.

10.2.1. Let T be a topos, and 𝔘 := (U_{i} | i ∈ I) a family of objects of T; pick any final object 1_{T} of T, and set as well (notation of example 4.7.8(iii))

Y := ∐_{i∈I} U_{i}  U := ⋃_{i∈I} Im(U_{i} → 1_{T})  Z := CU.

According to example 4.7.8(i,iii) there exist natural morphisms of topoi

j_{U} : T/U → T  π : T/Y → T  i : Z → T.

Moreover, the functor π^{∗} : T → T/Y admits a left adjoint π_{!} : T/Y → T. Furthermore, let ℤ_{T} (resp. ℤ_{X}, for every object X of T) be the constant ring object of T (resp. of T/X) associated with the ring ℤ. The functor π^{∗} : ℤ_{T} -Mod → ℤ_{Y} -Mod admits a left adjoint

π_{!} : ℤ_{Y} -Mod → ℤ_{T} -Mod.

Then, according to (7.10.3), the adjoint pair (π_{!}, π^{∗}) determines a cotriple (⊥^{𝔘}, η, μ), which in turns yields a functor

ℤ_{T} -Mod → ŝ.ℤ_{T} -Mod ℱ ↦⊥_{•}^{𝔘} ℱ .

(notation of definition 7.4.1(iv)). This construction can be described explicitly as follows. We attach to I the simplicial set I^{•} which, in every degree n ∈ ℕ consists of the set of all mappings [n] → I, where [n] := {0, …, n}; i.e., this is the cartesian power I^{n+1} of I, whose elements are all the ordered sequences \underline{t} := (t_{0}, …, t_{n}) of elements of I. Every morphism φ : [n] → [m] in the simplicial category Δ induces a map φ^{∗} : I^{m+1} → I^{n+1} by the rule :

\underline{t} ↦ \underline{t} ∘ φ := (t_{φ0}, …, t_{φn}) for every \underline{t} ∈ I^{m+1}

and it is clear that the rules : [n] ↦ I^{n+1} and φ ↦ φ^{∗} yield a well defined functor Δ^{o} → Set. For every \underline{t} ∈ I^{n+1}, we fix as well an object U\underline{t} of T that represents the product U_{t_{0}} × ⋯ × U_{t_{n}}, and denote by j\underline{t} : T/U\underline{t} → T the induced morphism. Notice that every φ : [n] → [m] in Δ and every \underline{t} ∈ I^{m+1} induces a well defined morphism in T :

ν_{φ}^{(\underline{t})} : U\underline{t} → U_{φ^{∗}(\underline{t})}

namely, the unique morphism whose composition with the projection on the i-th factor of U_{φ^{∗}(\underline{t})} agrees with the projection onto the φ(i)-th factor of U\underline{t}, for every i = 0, …, n. Then ν_{φ}^{(\underline{t})} in turns induces a morphism

u_{φ}^{(\underline{t})} : j_{\underline{t}!}j∗ ; \underline{t}ℱ → j_{φ^{∗}(\underline{t})!}j_{φ^{∗}(\underline{t})}^{∗}ℱ .

With this notation, we have

⊥_{n}^{𝔘} ℱ := ⨁_{\underline{t}∈I^{n+1}} j_{\underline{t}!}j∗ ; \underline{t}ℱ for every n ∈ ℕ

and every morphism φ : [n] → [m] in Δ corresponds to the unique morphism of ℤ_{T} -modules u_{φ} :⊥_{m}^{𝔘} ℱ →⊥_{n}^{𝔘} ℱ that makes commute the diagram

\xymatrix{j_{\underline{t}!}j∗ ; \underline{t}ℱ \ar[r]^{u_{φ}^{(\underline{t})}} \ar[d] & j_{φ^{∗}(\underline{t})!}j_{φ^{∗}(\underline{t})}^{∗}ℱ \ar[d] \\ ⊥_{m}^{𝔘} ℱ \ar[r]^{u_{φ}} & ⊥_{n}^{𝔘} ℱ} for every \underline{t} ∈ I^{m+1}.

where the vertical arrows are the inclusion maps. Moreover, ⊥_{−1}^{𝔘} ℱ = ℱ, and the augmentation ε :⊥_{0}^{𝔘} ℱ →⊥_{−1}^{𝔘} (𝔘) is the sum of the natural morphisms j_{i!}j_{i}^{∗}ℱ → ℱ given by the counit for the adjunction (j_{i!}, j_{i}^{∗}), for every i ∈ I.

Definition 10.2.2. (i) With the notation of (10.2.1), the augmented Čech resolution associated with 𝔘 and the ℤ_{T} -module ℱ is the unnormalized chain complex of ℤ_{T} -modules

(R_{•}(𝔘, ℱ), d_{•})

obtained from the augmented simplicial complex ⊥_{•}^{𝔘} (𝔘, ℤ_{T}) (see definition 7.4.24(i)). (ii) To ease notation, we shall set also

R_{•}(𝔘) := R_{•}(𝔘, ℤ_{T}) and \overline{R}_{•}(𝔘) := τ_{≤0}R_{•}(𝔘).

Then the augmented Čech complex of ℱ relative to the covering 𝔘 of U is the cochain complex

C^{•}(𝔘_{•}, ℱ) := Hom_{ℤ_{T}}^{•} (R_{•}(𝔘), ℱ [0]).

Hence, R_{•}(𝔘) is an object of C^{≤1}(ℤ_{T} -Mod), and C^{•}(𝔘_{•}, ℱ) is an object of C^{≥−1}(ℤ-Mod).

Lemma 10.2.3. With the notation of definition 10.2.2, we have :

(i) For every ℤ_{T} -module ℱ, the complex of ℤ_{U}-modules j_{U}^{∗}R_{•}(𝔘, ℱ) is acyclic.

(ii) The natural projection ℤ_{T} → i_{∗}ℤ_{Z} induces a quasi-isomorphism :

R_{•}(𝔘) →^{∼} i_{∗}ℤ_{Z}[−1] in C^{−}(ℤ_{T} -Mod).

(iii) Suppose that 𝔘 is a covering of T (i.e. that U = 1_{T} and Z = ∅_{T}). Then the differential d_{0} of R_{•}(𝔘) induces a quasi-isomorphism

\overline{R}_{•}(𝔘) →^{∼} ℤ_{T} [0] in C^{−}(ℤ_{T} -Mod).

Proof. (i): The natural morphism Y → U is an epimorphism, hence it is a covering in the canonical topology of T, and therefore it suffices to check that π^{∗}R_{•}(𝔘, ℱ) is acyclic. But proposition 7.10.5(i) says that π^{∗}(⊥_{•}^{𝔘} ℱ) is homotopically trivial. Then the assertion follows from corollary 7.4.67(ii).

Assertions (ii) and (iii) are immediate consequences of (i). □

10.2.4. Keep the notation of (10.2.1), and choose any total ordering on I. We let I_{alt}^{n+1} ⊂ I^{n+1} be the subset of all injective order-preserving mappings [n] → I, for every n ∈ ℕ, and we set

R_{−1}^{alt}(𝔘, ℱ) := ℱ  R_{n}^{alt}(𝔘, ℱ) := ⨁_{\underline{t}∈I_{alt}^{n+1}} j_{\underline{t}!}j∗ ; \underline{t}ℱ  for every n ∈ ℕ.

Obviously φ^{∗}(I_{alt}^{m+1}) ⊂ I_{alt}^{n+1} for every injective morphism φ : [n] → [m] of Δ, so the differential d_{n} of R_{•}(𝔘, ℱ) restricts to a well defined morphism of ℤ_{T} -modules R_{n}^{alt}(𝔘, ℱ) → R_{n−1}^{alt}(𝔘, ℱ) for every n ∈ ℕ, and we obtain therefore a subcomplex

(R_{•}^{alt}(𝔘, ℱ), d_{•}).

We set as well R_{•}^{alt}(𝔘) := R_{•}^{alt}(𝔘, ℤ_{T}), and to every ℤ_{T} -module ℱ we assign its alternating augmented Čech complex relative to the covering 𝔘 of U, which is the cochain complex

C_{alt}^{•}(𝔘_{•}, ℱ) := Hom_{ℤ}^{•}(R_{•}^{alt}(𝔘), ℱ [0]).

We shall also consider the alternating variant of the Čech resolution

\overline{R}_{•}^{alt}(𝔘) := τ_{≤0}R_{•}^{alt}(𝔘).

10.2.5. For every n ∈ ℕ, denote by S_{n+1} the permutation group of [n]; then, for any \underline{t} ∈ I_{alt}^{n+1}, and any σ ∈ S_{n+1}, the sequence \underline{t} ∘ σ is an element of I^{n+1} such that U_{\underline{t}∘σ} = U\underline{t}, and there exists a unique isomorphism μ\underline{t} : U_{\underline{t}∘σ} →^{∼} U\underline{t} whose composition with the projection on the i-th factor of U\underline{t} agrees with the projection on the σ(i)-th factor of U_{\underline{t}∘σ}, for every i ∈ [n]. We set

Alt^{(\underline{t}∘σ)} := sign(σ) · v\underline{t} : j_{(\underline{t}∘σ)!}ℤ_{U_{\underline{t}∘σ}} → j_{\underline{t}!}ℤ_{U\underline{t}}

where v\underline{t} is the isomorphism induced by μ\underline{t}, and where sign(σ) denotes the signature of σ. Then, for every n ∈ ℕ there exists a unique morphism of ℤ_{T} -modules

Alt_{n}^{𝔘} : R_{n}(𝔘) → R_{n}^{alt}(𝔘)

whose kernel contains the direct summands j_{\underline{t}!}ℤ_{U\underline{t}} such that the mapping \underline{t} : [n] → I is not injective, and that makes commute the diagram

\xymatrix{j_{(\underline{t}∘σ)!}ℤ_{U_{\underline{t}∘σ}} \ar[r]^{^{Alt^{(\underline{t}∘σ)}}} \ar[d] & j_{\underline{t}!}ℤ_{U\underline{t}} \ar[d] \\ R_{n}(𝔘) \ar[r]^{_{Alt_{𝔘}_{n}}} & R_{n}^{alt}(𝔘)}  for every \underline{t} ∈ I_{n}^{alt}

(where the vertical arrows are the inclusion maps).

Proposition 10.2.6. In the situation of (10.2.5), denote by ι_{•}^{𝔘} : R_{•}^{alt}(𝔘) → R_{•}(𝔘) the inclusion map, and suppose that U_{i} is a subobject of 1_{T}, for every i ∈ I. Then the following holds :

(i) The system (Alt_{n−1}^{𝔘} | n ∈ ℕ) defines an epimorphism of chain complexes

Alt_{•}^{𝔘} : R_{•}(𝔘) → R_{•}^{alt}(𝔘)

whose kernel is independent of the choice of ordering on I. (ii) Alt_{•}^{𝔘} ∘ ι_{•}^{𝔘} = 1_{R_{•}^{alt}(𝔘)}. and ι_{•}^{𝔘} ∘ Alt_{•}^{𝔘} is homotopically equivalent to 1_{R_{•}(𝔘)}. (iii) Especially, ι_{•}^{𝔘} and Alt_{•}^{𝔘} are mutually inverse isomorphisms in D(ℤ_{X}-Mod).

(iv) The natural projection ℤ_{T} → i_{∗}ℤ_{Z} induces a quasi-isomorphism :

R_{•}^{alt}(𝔘) →^{∼} i_{∗}ℤ_{Z}[−1] in C^{−}(ℤ_{T} -Mod).

(v) Suppose that 𝔘 is a covering of T (i.e. that U = 1_{T} and Z = ∅_{T}). Then the differential d_{0} of R_{•}^{alt}(𝔘) induces a quasi-isomorphism

\overline{R}_{•}^{alt}(𝔘) →^{∼} ℤ_{T} [0] in C^{−}(ℤ_{T} -Mod).

Proof. (i): Fix an integer n > 0, a sequence \underline{s} ∈ I^{n+1}, and let c ≤ n + 1 be the cardinality of \{s_{0}, …, s_{n}\}. It suffices to check that the compositions d_{n}∘Alt_{n}^{𝔘} and Alt_{n−1}^{𝔘}∘d_{n} agree on j_{\underline{s}!}ℤ_{U\underline{s}}. In case c < n, it is easily seen that both these compositions vanish on such direct summand. In case c = 1 and n = 0, the assertion follows by a direct inspection. In case c = n + 1 > 1, we may find a unique permutation σ ∈ S_{n+1} and a unique \underline{t} ∈ I_{alt}^{n+1} such that \underline{s} = \underline{t} ∘ σ, and we have to show that we have a commutative diagram

\xymatrix{j_{(\underline{t}∘σ)!}ℤ_{U_{\underline{t}∘σ}} \ar[r]^{^{Alt^{(\underline{t}∘σ)}}} \ar[d]_{d_{\underline{t}∘σ}} & j_{\underline{t}!}ℤ_{U\underline{t}} \ar[d]^{d\underline{t}} \\ R_{n−1}(𝔘) \ar[r]^{_{Alt{𝔘 \atop n−1}}} & R_{n−1}(𝔘′)}

where d\underline{t} and d_{\underline{t}∘σ} denote the restrictions of the differentials of R_{•}^{alt}(𝔘) and respectively R_{•}(𝔘). To this aim, for every i = 0, …, n denote by β_{1}^{(i)} the unique permutation of [n] such that

β_{1}^{(i)}(σ(i)) = σ(i) and β_{1}^{(i)}(σ(j)) < β_{1}^{(i)}(σ(k)) for every j, k ∈ [n] \ {i} with j < k.

Also, we define a third permutation as follows :

• If σ(i) < i, let β_{2}^{(i)} be the cycle (i, i − 1, …, σ(i)) of length i − β(i) + 1
• If σ(i) > i, let β_{2}^{(i)} be the cycle (i, i + 1, …, σ(i)) of length σ(i) − i + 1
• If σ(i) = i, let β_{2}^{(i)} be the identity map of [n].

It is easily seen that in either cases

σ^{−1} = β_{2}^{(i)} ∘ β_{1}^{(i)}  and sign(β_{2}^{(i)}) = (−1)^{i−σ(i)}  for every i = 0, …, n.

Now, for every i = 0, …, n let ε_{i} : [n−1] → [n] be the i-th face map (see (7.4.6)); we compute:

Alt_{n−1}^{𝔘} ∘ d_{\underline{t}∘σ} = Alt_{n−1}^{𝔘} ∘ (∑_{i=0}^{n}(−1)^{i} · u_{ε_{i}}^{(\underline{t}∘σ)})

= ∑_{i=0}^{n}(−1)^{i} · sign(β_{1}^{(i)}) · u_{ε_{i}}^{(\underline{t}∘σ)}

= ∑_{i=0}^{n}(−1)^{σ(i)} · sign(σ) · u_{ε_{i}}^{(\underline{t}∘σ)}

= ∑_{i=0}^{n}(−1)^{i} · sign(σ) · u_{ε_{i}}^{(\underline{t})}
= sign(σ) · d\underline{t}

as required. Lastly, if c = n, the morphism d_{n} ∘ Alt^{(\underline{t}∘σ)} vanishes, and it remains to check that Alt_{n−1}^{𝔘} ∘ d\underline{s} = 0 as well. However, in this situation there exist exactly two distinct indices i, j ∈ [n] such that s_{i} = s_{j}, and it follows already that Alt_{n−1}^{𝔘} ∘ u_{ε_{k}}^{(\underline{s})} = 0 for every k ≠ i, j.

Moreover, clearly there exists a unique \underline{t} ∈ I_{alt}^{n} such that \{t_{0}, …, t_{n−1}\} = \{s_{0}, …, s_{n}\}, and notice that the diagram :

\xymatrix{U\underline{s} \ar[r]^{ν_{ε_{i}}^{(\underline{s})}} \ar[d]_{ν_{ε_{j}}^{(\underline{s})}} & U_{ε_{i}^{∗}(\underline{s})} \ar[d]^{μ_{ε^{∗}_{i}(\underline{s})}} \\ U_{ε_{j}^{∗}(\underline{s})} \ar[r]^{μ_{ε^{∗}_{j}(\underline{s})}} & U\underline{t}}

commutes (this is the only place where we use our assumption on the objects U_{i}). There follows a commutative diagram

\xymatrix{j_{\underline{s}!}ℤ_{U\underline{s}} \ar[r]^{u_{ε_{i}}^{(\underline{s})}} \ar[d]_{u_{ε_{j}}^{(\underline{s})}} & j_{ε_{i}^{∗}(\underline{s})!}ℤ_{U_{ε^{∗}_{i}(\underline{s})}} \ar[d]^{v_{ε^{∗}_{i}(\underline{s})}} \\ j_{ε_{j}^{∗}(\underline{s})!}ℤ_{U_{ε^{∗}_{j}(\underline{s})}} \ar[r]^{v_{ε^{∗}_{j}(\underline{s})}} & U\underline{t}}

Now, let β_{i} (resp. β_{j}) be the unique permutation of [n] that fixes i (resp. j) and such that \underline{s} ∘ β_{i} (resp. \underline{s} ∘ β_{j}) restricts to an order-preserving map on [n] \ \{i\} (resp. on [n] \ \{j\}). Arguing as in the foregoing, we see that β_{i} = γ ∘ β_{j} for a cyclic permutation γ of length |j − i|, so that

ρ := (−1)^{i} · sign(β_{i}) = (−1)^{j+1} · sign(β_{j}).

Denote by 1\underline{t} the identity automorphism of j_{\underline{t}!}ℤ_{U\underline{t}}. Summing up, we get

Alt_{n−1}^{𝔘} ∘ d\underline{s} = (−1)^{i} · Alt_{n−1}^{𝔘} ∘ u_{ε_{i}}^{(\underline{s})} + (−1)^{j} · Alt_{n−1}^{𝔘} ∘ u_{ε_{j}}^{(\underline{s})}
= (−1)^{i} · sign(β_{i}) · v_{ε_{i}^{∗}(\underline{s})} ∘ u_{ε_{i}}^{(\underline{s})} + (−1)^{j} · sign(β_{j}) · v_{ε_{j}^{∗}(\underline{s})} ∘ u_{ε_{j}}^{(\underline{s})}
= ρ · (v_{ε_{i}^{∗}(\underline{s})} ∘ u_{ε_{i}}^{(\underline{s})} − v_{ε_{j}^{∗}(\underline{s})} ∘ u_{ε_{j}}^{(\underline{s})})
= 0

and the proof of (i) is concluded.

(ii): Let V be a universe such that T is V-small, and set T′ := T_{V}^{∧}; also, let J be the canonical topology on T. Then T′ is a V-topos, and the Yoneda imbedding h : T → T′ factors through an equivalence T →^{∼} (T, J)_{U}^{∼} and the inclusion functor (T, J)_{U}^{∼} → T′. Set U′_{i} := h_{U_{i}} ∈ Ob(T′) for every i ∈ I, U′ := ⋃_{i∈I} Im(U′_{i} → 1_{T′}), and 𝔘′ := (U′_{i} | i ∈ I). Hence U′ is a presheaf on T whose associated sheaf is isomorphic to h_{U}. Let j_{U′} : T′/U′ → T′ be the induced morphism of topoi; according to lemma 10.2.3, the chain complex j_{U′}^{∗}R_{•}(𝔘′) is acyclic, hence the same holds for j_{U′!}j_{U′}^{∗}R_{•}(𝔘′), and there follows a commutative diagram of complexes of ℤ_{T′}-modules :

[figure: \overline{R}_{•}(𝔘′); τ_{≤0}(ι_{•}^{𝔘′}∘Alt_{•}^{𝔘′}); j_{U′!}ℤ_{U′}[0]; \overline{R}_{•}(𝔘′); j_{U′!}ℤ_{U′}[0]]

whose horizontal arrows are induced by the differential d_{0} of R_{•}(𝔘′), and therefore are isomorphisms in the category D(ℤ_{T′}-Mod), so that

τ_{≤0}(ι_{•}^{𝔘′} ∘ Alt_{•}^{𝔘′}) = 1_{\overline{R}_{•}(𝔘′)} in D(ℤ_{T′}-Mod).

However, from the explicit description of (10.2.1) it is easily seen that \overline{R}_{•}(𝔘′) is a complex of projective ℤ_{T′}-modules, hence we have as well τ_{≤0}(ι_{•}^{𝔘′} ∘ Alt_{•}^{𝔘′}) = 1_{\overline{R}_{•}(𝔘′)} in Hot(ℤ_{T′}-Mod), by theorem 7.3.25(iii), and then it follows as well that ι_{•}^{𝔘′} ∘ Alt_{•}^{𝔘′} = 1_{R_{•}(𝔘′)} in Hot(ℤ_{T′}-Mod). Next, notice that R_{•}(𝔘′) is a complex of abelian presheaves on T, whose associated complex of sheaves agrees with R_{•}(𝔘), and likewise the morphism of abelian sheaves associated with ι_{•}^{𝔘′} ∘ Alt_{•}^{𝔘′} agrees with ι_{•}^{𝔘} ∘ Alt_{•}^{𝔘}. The assertion follows.

(iii) follows directly from (ii). Likewise, (iv) and (v) are immediate from (iii) and lemma 10.2.3(ii,iii). □

Remark 10.2.7. (i) In light of proposition 10.2.6(iv) we shall call R_{•}^{alt}(𝔘) the alternating augmented Čech resolution associated with 𝔘. This name can be justified as follows. Notice that for every ℤ_{T} -module ℱ we have natural identifications

(10.2.8)  C^{n}(𝔘, ℱ) →^{∼} ∏_{\underline{t}∈I^{n+1}} ℱ (U\underline{t})  C_{alt}^{n}(𝔘, ℱ) →^{∼} ∏_{\underline{t}∈I_{alt}^{n+1}} ℱ (U\underline{t})

for every n ∈ ℕ (notation of definition 10.2.2(ii)). Under these identifications, C^{•}(𝔘, ℱ) becomes the cochain complex

(10.2.9)  0 → Γ(ℱ) → ∏_{t∈I} ℱ (U_{t}) → ⋯ → ∏_{\underline{t}∈I^{n+1}} ℱ (U\underline{t}) → ⋯

whose differential d^{n} is given by the following rule. First, d^{−1} assigns to each global section s ∈ Γ(ℱ) the system of its restrictions (s_{|U_{t}} | t ∈ I). In case n ≥ 0, we have

d^{n}(s_{•})\underline{t} := ∑_{k=0}^{n+1}(−1)^{k+n+1} · (s_{\underline{t}∘ε_{k}})_{|U\underline{t}} for every \underline{t} ∈ I^{n+2} and every s_{•} ∈ C^{n}(𝔘, ℱ) .

(ii) Likewise, under the natural identifications of (i), the alternating augmented Čech complex C_{alt}^{•}(𝔘, ℱ) becomes the cochain complex

0 → Γ(ℱ) → ∏_{t∈I} ℱ (U_{t}) → ⋯ → ∏_{\underline{t}∈I_{alt}^{n+1}} ℱ (U\underline{t}) → ⋯

whose differentials are given by the same expressions as in (i).

(iii) Under the identification of (i), the morphisms R_{n}(𝔘) → ℱ that factor through Alt_{n}^{𝔘} correspond to the system of sections f_{•} := (f\underline{t} | \underline{t} ∈ I^{n+1}) with the following properties :

• f\underline{t} = 0 whenever the map \underline{t} : [n] → I is not injective.
• f_{\underline{t}∘σ} = sign(σ) · f\underline{t} for every injective map \underline{t} : [n] → I and every σ ∈ S_{n+1}.

Thus, under the identification (10.2.8), the submodule Hom_{ℤ_{X}}(R_{n}^{alt}(𝔘), ℱ) corresponds to the subgroup of all alternating systems f_{•} of sections of ℱ.

10.2.10. In the situation of (10.2.1), consider another family 𝔘′ := (U′_{i′} | i′ ∈ I′) of objects of T, indexed by a totally ordered set I′, and set U′ := ⋃_{i′∈I′} Im(U′_{i′} → 1_{T}) and Z′ := CU′. Also, for every n ∈ ℕ, and every \underline{s} ∈ I^{′n+1} set as usual U ′ ; \underline{s} := U′_{s_{0}} × ⋯ × U′_{s_{n}}, and denote by j′ ; \underline{s} : T/U ′ ; \underline{s} → T the induced functor. Suppose that 𝔘 is a refinement of 𝔘′, i.e. there exists a mapping (that does not necessarily respect the orderings)

τ : I → I′  such that Hom_{T} (U_{i}, U′_{τ(i)}) ≠ ∅  for every i ∈ I

and pick a morphism ϑ_{i} : U_{i} → U′_{τ(i)} for every i ∈ I. Then, for every n ∈ ℕ and every \underline{t} := (t_{0}, …, t_{n}) ∈ I^{n+1}, we set τ(\underline{t}) := (τ(t_{0}), …, τ(t_{n})) ∈ I^{′n+1}, and notice that the morphism ϑ\underline{t} := ϑ_{t_{0}} × ⋯ × ϑ_{t_{n}} : U\underline{t} → U′_{τ(\underline{t})} induces a morphism of ℤ_{T} -modules

φ\underline{t} : j_{\underline{t}!}ℤ_{U\underline{t}} → j′_{τ(\underline{t})!}ℤ_{U′_{τ(\underline{t})}}.

For every n ∈ ℕ, denote by φ_{n} : R_{n}(𝔘) → R_{n}(𝔘′) the unique morphism of ℤ_{T} -modules that makes commute the diagram

\xymatrix{j_{\underline{t}!}ℤ_{U\underline{t}} \ar[r]^{φ_{(\underline{t})}} \ar[d] & j′_{τ(\underline{t})!}ℤ_{U′_{τ(\underline{t})}} \ar[d] \\ R_{n}(𝔘) \ar[r]^{φ_{n}} & R_{n}(𝔘′)} for every \underline{t} ∈ I^{n+1}.

whose vertical arrows are the inclusion maps. Let as well φ_{−1} := 1_{ℤ_{T}}. With this notation, it is clear that the system (φ_{n−1} | n ∈ ℕ) is a morphism of augmented simplicial complexes R_{•}(𝔘) → R_{•}(𝔘′), and we denote

φ_{•} : R_{•}(𝔘) → R_{•}(𝔘′)

the associated morphism of chain complexes.

10.2.11. Suppose next that U′_{i′} is a subobject of 1_{T}, for every i′ ∈ I′; then, on account of proposition 10.2.6(i) we may define

φ_{•}^{alt} := Alt_{•}^{𝔘′} ∘ φ_{•} ∘ ι_{•}^{𝔘} : R_{•}^{alt}(𝔘) → R_{•}^{alt}(𝔘′).

The terms φ_{n}^{alt} can be described explicitly : first, it is clear that φ_{−1}^{alt} := 1_{ℤ_{T}}. Next, if n ∈ ℕ and \underline{t} ∈ I_{alt}^{n+1} is any sequence, let

φalt ; \underline{t} : j_{\underline{t}!}ℤ_{U\underline{t}} → R_{n}^{alt}(𝔘′)

be the map given by the following rule. If τ(\underline{t}) is also an injective sequence [n] → I′, let σ ∈ S_{n+1} be the unique permutation such that τ(\underline{t}) ∘ σ ∈ I_{alt}^{′n+1}, and set φalt ; \underline{t} := sign(σ) · φ_{(\underline{t})} : j_{\underline{t}!}ℤ_{U\underline{t}} → ℤ_{U′_{τ(\underline{t})∘σ}} ⊂ R_{n}^{alt}(𝔘′). In case τ(\underline{t}) is not injective, we let φalt ; \underline{t} be the zero morphism. Then φ_{n}^{alt} is the sum of the φalt ; \underline{t}, for \underline{t} ranging over all elements of I_{alt}^{n+1}. In case also U_{i} is a subobject of 1_{T} for every i ∈ I, it is easily seen that we get a commutative diagram

\xymatrix{R_{•}(𝔘) \ar[r]^{^{Alt^{𝔘}^{•}}} \ar[d]_{φ_{•}} & R_{•}^{alt}(𝔘) \ar[d]^{φ_{•}^{alt}} \\ R_{•}(𝔘′) \ar[r]^{_{Alt{𝔘′ \atop •}}} & R_{•}^{alt}(𝔘′).}

Lemma 10.2.12. In the situation of (10.2.10), suppose that τ′ : I → I′ is another mapping with Hom_{T} (U_{i}, U′_{τ′(i)}) ≠ ∅ for every i ∈ I. Pick a system of morphisms (ϑ′_{i} : U_{i} → U′_{τ′(i)} | i ∈ I), and denote by φ′_{•} : R_{•}(𝔘) → R_{•}(𝔘′) the associated morphisms of chain complexes. We have :

(i) There exists a homotopy from φ_{•} to φ′_{•}.

(ii) If U′_{i′} is a subobject of 1_{T} for every i′ ∈ I′, there is also a homotopy from φ_{•}^{alt} to φ_{•}^{′alt}.

Proof. Clearly it suffices to show (i). To this aim, for every i ∈ I let (ϑ_{i}, ϑ′_{i}) : U_{i} → U′_{τ(i)}×U′_{τ′(i)} be the unique morphism whose composition with the projection on the first (resp. second) factor equals ϑ_{i} (resp. ϑ′_{i}), and for every n ∈ ℕ, every k = 0, …, n, and every \underline{t} ∈ I^{n+1}, set

\begin{matrix}τ(k, \underline{t}) := & (τ(t_{0}), …, τ(t_{k}), τ′(t_{k}), …, τ′(t_{n})) ∈ I^{n+2} \\ ϑ(k) ; \underline{t} := & ϑ_{t_{0}} × ⋯ × ϑ_{t_{k−1}} × (ϑ_{t_{k}}, ϑ′_{t_{k}}) × ϑ′_{t_{k+1}} × ⋯ × ϑ′_{t_{n}} : U\underline{t} → U′_{τ(k,\underline{t})}. \end{matrix}

Then ϑ(k) ; \underline{t} induces a natural morphism h(k) ; \underline{t} : j_{\underline{t}!}ℤ_{U\underline{t}} → j_{τ(k,\underline{t})!}ℤ_{U′_{τ(k,\underline{t})}} and we set

h\underline{t} := ∑_{k=0}^{n}(−1)^{k} · h(k) ; \underline{t} : j_{\underline{t}!}ℤ_{U\underline{t}} → R_{n+1}(𝔘′).

Clearly there exists a unique morphism of ℤ_{T} -modules h_{n} : R_{n}(𝔘) → R_{n+1}(𝔘′) whose restriction to j_{\underline{t}!}ℤ_{U\underline{t}} agrees with h\underline{t} for every \underline{t} ∈ I^{n+1}. Let also h_{−1} : R_{−1}(𝔘) → R_{0}(𝔘′) be the zero morphism; a direct calculation shows that the system (h_{n−1} | n ∈ ℕ) yields the sought homotopy : the details shall be left to the reader. □

Remark 10.2.13. (i) In the situation of (10.2.11), consider the case where U_{i_{0}} = 1_{T} for some i_{0} ∈ I. Then 𝔘 can be refined by itself in two different ways : we may take for τ : I → I the identity map (with ϑ_{i} := 1_{U_{i}} for every i ∈ I), and we have also the map τ′ : I → I such that τ′(i) = i_{0} for every i ∈ I (and then there is a unique choice of ϑ_{i}). Obviously, the morphism φ_{•} : R_{•}^{alt}(𝔘) → R_{•}^{alt}(𝔘) associated with τ is the identity map. For the morphism φ′_{•} : R_{•}^{alt}(𝔘) → R_{•}^{alt}(𝔘) associated with τ′ it is easily seen that

φ′_{−1} = 1_{ℤ_{T}}  and φ′_{n} = 0  for every n > 0

and φ′_{0} is the sum of the natural morphisms j_{t!}ℤ_{U_{t}} → j_{i_{0}!}ℤ_{U_{i_{0}}} = ℤ_{T}. The homotopy h_{•} from φ_{•} to φ′_{•} furnished by lemma 10.2.12can be described as follows. First, we may suppose that i_{0} is the maximal element of I. A simple inspection shows that h_{−1} : R_{−1}^{alt}(𝔘) → R_{0}^{alt}(𝔘) is the zero map. Next, for every n ∈ ℕ and every \underline{t} ∈ I_{alt}^{n+1}, notice that U\underline{t} = U_{(\underline{t},i_{0})}, where (\underline{t}, i_{0}) denotes the sequence (t_{0}, …, t_{n}, i_{0}); it follows that we may write

h_{n} = ∑_{\underline{t}∈I_{alt}^{n+1}} ρ(\underline{t}) · h\underline{t}

where h\underline{t} : j\underline{t}ℤ_{U\underline{t}} → j_{(\underline{t},i_{0})!}ℤ_{U_{(\underline{t},i_{0})}} is the identity map, and ρ(\underline{t}) equals 0 if t_{n} = i_{0}, and equals 1 otherwise.

(ii) In the situation of (i), we deduce a homotopy from 1_{R_{•}^{alt}(𝔘)} to the zero map. Indeed, let h′_{−1} : ℤ_{T} → R_{0}(𝔘) be the composition of the identity map ℤ_{T} → j_{i_{0}!}ℤ_{U_{i_{0}}} and the inclusion map j_{i_{0}!}ℤ_{U_{i_{0}}} → R_{0}(𝔘). Let also h′_{n} := h_{n} for every n ∈ ℕ. A simple inspection shows that

φ′_{0} = h′_{−1} ∘ d_{0}

(where d_{0} : R_{0}(𝔘) → ℤ_{T} is the differential of R_{•}(𝔘)). It follows easily that (h′_{n−1} | n ∈ ℕ) is the sought homotopy.

(iii) Let ℱ be any ℤ_{T} -module. The homotopy h′_{•} of (i) induces a homotopy from the identity automorphism of the augmented Čech complex C_{alt}^{•}(𝔘, ℱ) to its zero endomorphism. Under the natural identifications of remark 10.2.7(ii), this homotopy h^{•} can be described as follows.

• h^{0} : C^{0}(𝔘, ℱ) → Γ(ℱ) is the projection onto the factor ℱ (U_{i_{0}}) = Γ(ℱ).

• For every n ≥ 0, the map h^{n+1} : C^{n+1}(𝔘, ℱ) → C^{n}(𝔘, ℱ) is the projection that sends to zero every factor ℱ (U_{t_{0},...,t_{n}}) with t_{n} < i_{0}, and whose restriction to every factor

ℱ (U_{t_{0},...,t_{n−1},i_{0}}) is the identity map ℱ (U_{t_{0},...,t_{n−1},i_{0}}) →^{∼} ℱ (U_{t_{0},...,t_{n−1}}).

Remark 10.2.14. (i) In the situation of (10.2.4), it is clear that the rule ℱ ↦ R_{•}^{alt}(𝔘, ℱ) defines a functor from ℤ_{T} -modules to chain complexes of ℤ_{T} -modules. The latter then extends naturally to a functor from the category of chain complexes of ℤ_{T} -modules to the category of double complexes of ℤ_{T} -modules :

C(ℤ_{T} -Mod) → C_{2}(ℤ_{T} -Mod) ℱ_{•} ↦ R_{•}^{alt}(𝔘, ℱ_{•}).

Likewise, ⊥_{•}^{𝔘} extends to a functor from augmented simplicial ℤ_{T} -modules to augmented bisimplicial ℤ_{T} -modules :

ŝ.ℤ_{T} -Mod → ŝ.(ŝ.ℤ_{T} -Mod) ℱ_{•} ↦⊥_{•}^{𝔘} ℱ_{•}.

(ii) Let now 𝔘′ := (U′_{i′} | i′ ∈ I′) be another family of objects of T. Then we may consider the augmented bisimplicial ℤ_{T} -module

⊥_{••}^{𝔘,𝔘′}:=⊥_{•}^{𝔘′} (⊥_{•}^{𝔘} ℤ_{T})

and we notice that

(⊥_{••}^{𝔘,𝔘′})^{Δ} =⊥_{•}^{𝔘×𝔘′} where 𝔘 × 𝔘′ := (U_{i} × U′_{i′} | (i, i′) ∈ I × I′).

In case both U_{i} and U′_{i′} are subobjects of 1_{T} for every i ∈ I and i′ ∈ I′, we may also define the chain double complex

R_{••}^{alt}(𝔘, 𝔘′) := R_{•}^{alt}(𝔘′, (R_{•}^{alt}(𝔘))

and we notice that (notation of example 7.1.16(i))

R_{••}^{alt}(𝔘, 𝔘′) = R_{•}^{alt}(𝔘) ⊠_{A} R_{•}^{alt}(𝔘′).

10.2.15. In the situation of (10.2.1), let 𝒜 be any T -ring; we consider the functor

𝒜 -Mod → C(Γ(𝒜)-Mod) ℱ ↦ \overline{C}^{•}(𝔘, ℱ) := Hom_{ℤ}(\overline{R}_{•}(𝔘), ℱ [0])

that assigns to every 𝒜 -module ℱ its Čech complex, and we define as well the Čech cohomology functor of ℱ relative to the covering 𝔘, by setting :

H^{i}(𝔘, ℱ) := H^{i}\overline{C}^{•}(𝔘, ℱ) for every i ∈ ℕ.

Suppose now that 𝔘 is a covering of T; since ℱ represents a sheaf for the canonical topology on T, we easily see that the system of restriction maps (Γ(ℱ) → ℱ (U_{i}) | i ∈ I) induces a natural isomorphism

(10.2.16)  Γ(ℱ) →^{∼} H^{0}(𝔘, ℱ).

On the other hand, let K^{•} be any bounded below complex of 𝒜 -modules; lemma 10.2.3(iii) yields natural isomorphisms in D^{+}(𝒜 -Mod) and in D^{+}(Γ(𝒜)-Mod)

Rℋ om_{ℤ}^{•}(\overline{R}_{•}(𝔘), K^{•}) →^{∼} R\underline{Γ}K^{•} RHom_{ℤ}^{•}(\overline{R}_{•}(𝔘), K^{•}) →^{∼} RΓK^{•}.

Hence, after fixing an injective resolution K^{•} →^{∼} ℐ^{•} we get a natural isomorphism

RΓK^{•} →^{∼} Tot \overline{C}^{•}(𝔘, ℐ^{•}) in D^{+}(Γ(𝒜)-Mod) .

Thus, for every p ∈ ℤ let also define a presheaf ℋ ^{p}(K^{•}) on T by the rule

ℋ ^{p}(K^{•})(U) := R^{p}Γ(U, K^{•}) for every object U of T.

We may regard ℋ ^{p}(K^{•}) as an object of the V-topos T_{V}^{∧}, for some universe V such that T is Vsmall, and the family 𝔘 can be regarded as a system of objects of T_{V}^{∧}, via the Yoneda imbedding T → T_{V}^{∧}. Then, the Čech cohomology functors of ℋ ^{p}(K^{•}) relative to the covering 𝔘 are also well defined, and we deduce a 2-spectral sequence

(10.2.17)  E(𝔘)_{2}^{p,q} := H^{p}(𝔘, ℋ ^{q}(K^{•})) ⇒ R^{p+q}ΓK^{•}.

Notice also that

ℋ ^{0}(ℱ [0]) = ℱ for every ℤ_{T} -module ℱ

whence, natural maps

Ψ^{p}(𝔘, ℱ) : H^{p}(𝔘, ℱ) → R^{p}Γℱ for every p ∈ ℕ

such that Ψ^{0}(𝔘, ℱ) is always an isomorphism (in fact, it is the inverse of the isomorphism (10.2.16)), and Ψ^{1}(𝔘, ℱ) is always an injective map.

10.2.18. In case U_{i} is a subobject of 1_{T} for every i ∈ I, we may repeat the considerations of (10.2.15) with the alternating Čech complex

\overline{C}_{alt}^{•}(𝔘, ℱ) := Hom_{ℤ}(\overline{R}_{•}^{alt}(𝔘), ℱ [0]).

Namely, let us set

H_{alt}^{i}(𝔘, ℱ) := H^{i}\overline{C}_{alt}^{•}(𝔘, ℱ) for every i ∈ ℕ.

Then, proposition 10.2.6(ii) (together with example 7.1.25(ii)) yields a natural isomorphism

H_{alt}^{i}(𝔘, ℱ) →^{∼} H^{i}(𝔘, ℱ) for every i ∈ ℕ

and if 𝔘 is a covering of T, obviously we get therefore a natural isomorphism

(10.2.19)  RΓK^{•} →^{∼} Tot \overline{C}_{alt}^{•}(𝔘, ℐ^{•})  in D^{+}(Γ(𝒜)-Mod) .

for any injective resolution K^{•} →^{∼} ℐ^{•}, and a 2-spectral sequence

(10.2.20)  F (𝔘)_{2}^{p,q} := H_{alt}^{p}(𝔘, ℋ ^{q}(K^{•})) ⇒ R^{p+q}ΓK^{•}.

Taking K^{•} := ℱ [0], there follows a natural map

Ψ_{alt}^{p}(𝔘, ℱ) : H_{alt}^{p}(𝔘, ℱ) → R^{p}Γℱ for every p ∈ ℕ

and again, Ψ_{alt}^{0}(𝔘, ℱ) is an isomorphism, whereas Ψ_{alt}^{1}(𝔘, ℱ) is always an injective map. The following immediate corollary is often useful :

Corollary 10.2.21. In the situation of (10.2.15), suppose furthermore that

ℋ ^{q}(ℱ [0])(U\underline{t}) = 0 for every integer q > 0 and every \underline{t} ∈ I^{q}.

Then we have :

(i) The map Ψ^{p}(𝔘, ℱ) is an isomorphism for every p ∈ ℕ.

(ii) If U_{i} is a subobject of 1_{T} for every i ∈ ℕ, then the map Ψ_{alt}^{p}(𝔘, ℱ) is an isomorphism for every p ∈ ℕ.

Proof. The assumption of (i) (resp. (ii)) implies that E(𝔘)_{2}^{p,q} = 0 (resp. F (𝔘)_{2}^{p,q} = 0) for every q > 0, whence both contentions. □

10.2.22. Let now C := (𝒞 , J) be a small site whose finite non-empty products and fibre products are representable. Then, the finite products of 𝒞 /X are representable, for every X ∈ Ob(𝒞); for every such X, we denote by J_{X} the topology on 𝒞 /X induced by J (see (4.7.2)).

Now, let S be either C or (𝒞 /X, J_{X}) for some X ∈ Ob(𝒞), and denote by 𝒥 (S) the set of all families of objects of S that generate a sieve covering the final object of T := S^{∼}, for the canonical topology on T. Then 𝒥 (S) is endowed with a partial ordering, by declaring that 𝔘 ≤ 𝔘′ if and only if the sieve generated by 𝔘 contains the sieve generated by 𝔘′, for every 𝔘, 𝔘′ ∈ 𝒥 (S). Notice that this is equivalent to saying that the family 𝔘′ is a refinement of 𝔘, when we regard 𝔘 and 𝔘′ as families of objects of the topos T. Especially, when 𝔘 ≤ 𝔘′, the discussion of (10.2.10) yields a map of complexes φ_{𝔘,𝔘′} : R_{•}(𝔘′) → R_{•}(𝔘), and lemma 10.2.12 says that the homotopy class of φ_{𝔘,𝔘′} depends only on 𝔘 and 𝔘′, so we get a well defined functor

R_{•} : 𝒥 (S)^{o} → Hot(ℤ_{T} -Mod) 𝔘 ↦ R_{•}(𝔘).

Now, let ℱ be any abelian sheaf on S. We then get, for every i ∈ ℕ, an induced functor

H^{i}(−, ℱ) : 𝒥 (S) → ℤ-Mod 𝔘 ↦ H^{i}(𝔘, ℱ).

Notice now that 𝒥 (S) is a small and filtered category; we may then define

Ȟ^{i}(S, ℱ) := colim_{𝔘∈𝒥 (S)} H^{i}(𝔘, ℱ)

and clearly the rule ℱ ↦ Ȟ^{i}(S, ℱ) yields a well defined functor

Ȟ^{i}(S, −) : ℤ_{T} -Mod → ℤ-Mod for every i ∈ ℕ.

In case S = (𝒞 /X, J_{X}), we shall denote this functor by Ȟ(X, −). By the same token, it is also clear that we get as well a natural morphism of 2-spectral sequences

E(𝔘)_{2}^{••} → E(𝔘′)_{2}^{••} whenever 𝔘 ≤ 𝔘′

for every bounded below complex K^{•} of ℤ_{T} -modules whence, after taking colimits, a 2-spectral sequence

E_{2}^{p,q} := Ȟ^{p}(S, ℋ ^{q}(K^{•})) ⇒ R^{p+q}ΓK^{•}

that, in turn, gives us again natural maps, for every ℤ_{T} -module ℱ

Ψ̌^{p}(S, ℱ) : Ȟ^{p}(S, ℱ) → R^{p}Γℱ for every p ∈ ℕ

which, in case S = (𝒞 /X, J_{X}), shall be denoted simply Ψ̌^{p}(X, ℱ).

Lemma 10.2.23. In the situation of (10.2.22), let ℱ be any ℤ_{T} -module, and take K^{•} := ℱ [0]. Then we have :

(i) E_{2}^{0,q} = 0 for every q > 0.

(ii) Both Ψ̌^{0}(S, ℱ) and Ψ̌^{1}(S, ℱ) are isomorphisms, and Ψ̌^{2}(S, ℱ) is injective.

Proof. (i): Fix any injective resolution ℱ → ℐ^{•}, and let \overline{s} ∈ E_{2}^{0,q} be any element; by definition, there exists a family (U_{i} | i ∈ I) of objects of S that generates a sieve covering the final object of S^{∼}, such that \overline{s} is represented by an element

\overline{s}_{•} ∈ Ker(∏_{i∈I} H^{q}(ℐ^{•}(U_{i})) ⟶^{d_{v}^{0,q}} ∏_{(i,j)∈I^{2}} H^{q}(ℐ^{•}(U_{(i,j)}))

where d_{v}^{0,q} is the differential of the Čech complex in degree 0, as in remark 10.2.7(i). In turn, the class \overline{s}_{i} ∈ H^{q}(ℐ^{•}(U_{i})) is represented by a section s_{i} ∈ ℐ^{q}(U_{i}) for every i ∈ I, such that d_{h}^{0,q}(s_{i}) = 0, where d_{h}^{0,q} : ℐ^{q}(U_{i}) → ℐ^{q+1}(U_{i}) denotes the differential in degree q of the complex ℐ^{•}. Since the latter is exact in every degree q > 0, it follows that for every i ∈ I there exist a family (U_{i,λ} → U_{i} | λ ∈ Λ_{i}) of objects of 𝒞 /U_{i} that generates a sieve covering the final object of (𝒞 /U_{i}, J_{U_{i}}), and for every λ ∈ Λ_{i} a section s_{i,λ} ∈ ℐ^{q−1}(U_{i,λ}) such that d_{h}^{0,q−1}(s_{i,λ}) = s_{i|U_{i,λ}}. Set 𝔘 := ⋃_{i∈I}{U_{i,λ} | λ ∈ Λ_{i}}; then 𝔘 lies in 𝒥 (S), and it is easily seen that the image of \overline{s} vanishes in H^{0}(𝔘, ℋ ^{q}(ℱ)), whence the contention.

(ii) is an immediate consequence of (i). □

Theorem 10.2.24 (Cartan). In the situation of (10.2.22), let ℱ be any ℤ_{T} -module such that

Ȟ^{i}(X, ℱ) = 0 for every X ∈ Ob(𝒞) and every i > 0.

Then we have :

(i) The map Ψ̌^{q}(C, ℱ) is an isomorphism for every q ∈ ℕ.

(ii) More precisely, the map Ψ^{q}(𝔘, ℱ) is an isomorphism for every q ∈ ℕ and every family 𝔘 of objects of 𝒞 that covers the final object of T.

Proof. (i): We argue by induction on q ∈ ℕ, and notice that the assertion for q ≤ 1 is already known without any assumption on ℱ, by lemma 10.2.23(ii). Thus, let i ≥ 2 and suppose that the assertion of the theorem is already known for every q < i, every site C and every abelian sheaf ℱ on C. Notice that if X ∈ Ob(𝒞) and φ : Y → X is any object of 𝒞 /X, we have a natural isomorphism of categories

(𝒞 /X)/φ →^{∼} 𝒞 /Y

and the topology J_{φ} induced on (𝒞 /X)/φ by J_{X} agrees, under this identification, with the topology J_{Y}. Therefore, the site 𝒞 /X fulfills the assumptions of the theorem, and by inductive assumption we deduce that Ψ̌^{q}(X, ℱ) is an isomorphism for every q < i, and therefore R^{q}Γ(X, ℱ) = 0 for every X ∈ Ob(𝒞) and every q = 1, …, i − 1. Summing up, we get ℋ ^{q}(ℱ) = 0 whenever 1 ≤ q < i, whence E_{2}^{pq} = 0 for every q = 1, …, i − 1 (notation of (10.2.22)). We also know that E_{2}^{0q} = 0 for every q > 0, by lemma 10.2.23(i); in this situation, it is easily seen that E_{2}^{i,0} = E_{∞}^{i0} and E_{∞}^{i−p,p} = 0 for every p > 0, and then Ψ̌^{i}(X, ℱ) is indeed an isomorphism.

(ii) is an immediate consequence of (i) and corollary 10.2.21(i). □

10.2.25. We wish now to apply the foregoing results to the cohomology of quasi-coherent modules on a scheme. To begin with, let A be a ring, M an A-module; set X := Spec A and denote by ℳ the quasi-coherent 𝒪_{X}-module arising from M. Let also f := (f_{1}, …, f_{r}) be a sequence of elements of A, and for every integer n ≥ −1 and every injective order-preserving map

\underline{t} : [n] → Σ := \{1, …, r\} k ↦ t_{k}

let A\underline{t} := A[f_{t_{0}}^{−1} ⋯ f_{t_{n}}^{−1}] and U\underline{t} := Spec A\underline{t}; we can describe as follows the A\underline{t}-module ℳ (U\underline{t}). For every such \underline{t}, consider the system of A-modules ((M (k) ; \underline{t}, φ_{k}) | k ∈ ℕ) such that M (k) ; \underline{t} := M for every k ∈ ℕ, and the map φ_{k} : M (k) ; \underline{t} → M (k+1) ; \underline{t} is the scalar multiplication by f_{t_{0}} ⋯ f_{t_{n}} if n ≥ 0, and it is 1_{M} when n = −1. Then we have a natural identification

(10.2.26)  ℳ (U\underline{t}) →^{∼} colim_{k∈ℕ} M (k) ; \underline{t}  for every n ≥ −1 and every \underline{t} : [n] → Σ.

Moreover, for every face map ε_{i} : [n − 1] → [n] in Δ, we can describe as follows the restriction map ρ_{i,\underline{t}} : ℳ (U_{\underline{t}∘ε_{i}}) → ℳ (U\underline{t}). We consider the morphism of directed systems

ρ_{i}^{(•)} : M_{ε_{i}^{∗}(\underline{t})}^{(•)} → M (•) ; \underline{t} such that ρ_{i}^{(k)} := f_{t_{i}}^{k} · 1_{M} for every k ∈ ℕ.

Then, under the identifications (10.2.26), the map ρ_{i,\underline{t}} corresponds to the map

colim_{k∈ℕ} ρ_{i}^{(k)} : colim_{k∈ℕ} M_{\underline{t}∘ε_{i})}^{(k)} → colim_{k∈ℕ} M (k) ; \underline{t}.

For every n ≥ −1 let Σ_{alt}^{n+1} be the set of all injective order-preserving maps [n] → Σ, and set

D_{(k)}^{n} := Hom_{Set}(Σ_{alt}^{n+1}, M) for every k ∈ ℕ

which we endow with the A-module structure inherited from M; consider also the map

d_{(k)}^{n} : D_{(k)}^{n} → D_{(k)}^{n+1} (μ : Σ_{alt}^{n+1} → M) ↦ (\underline{t} ↦ ∑_{i=0}^{n+1}(−1)^{i} · f_{t_{i}}^{k} · μ(\underline{t} ∘ ε_{i}))

for every k ∈ ℕ. Then the system (D_{(k)}^{•}, d_{(k)}^{•}) is a well defined complex of A-modules, for every k ∈ ℕ. Lastly, for every k ∈ ℕ we consider the morphism of complexes

D_{(k)}^{•} → D_{(k+1)}^{•} (μ : Σ_{alt}^{n+1} → M) ↦ (\underline{t} ↦ f_{t_{0}} ⋯ f_{t_{n}} · μ(\underline{t})).

Summing up, and comparing with remark 10.2.7(i) we obtain a natural identification

C_{alt}^{•}(𝔘, ℳ) →^{∼} colim_{k∈ℕ} D_{(k)}^{•} where 𝔘 := (U_{1}, …, U_{r}).

On the other hand, notice that, for every n ≥ −1, the free A-module Λ_{A}^{n+1}A^{⊕r} admits the standard basis (e_{t_{0}} ∧ ⋯ ∧ e_{t_{n}} | \underline{t} ∈ Σ_{alt}^{n+1}), so we have as well a natural identification

Hom_{A}(Λ_{A}^{n+1}A^{⊕r}, M) →^{∼} D_{(k)}^{n} for every n ≥ −1

and a direct inspection reveals that, under this latter identification, the differential d_{(k)}^{n} of D_{(k)}^{•} corresponds to the differential Hom_{A}(d_{f^{k},n}, M), where f^{k} := (f_{1}^{k}, …, f_{r}^{k}), and d_{f^{k},n} is the differential in degree n of the Koszul complex of the sequence f^{k} (see remark 7.8.1(ii)). Thus, we get finally a natural isomorphism of complexes of A-modules

(10.2.27)  C_{alt}^{•}(𝔘, ℳ) →^{∼} colim_{k∈ℕ} K^{•}(f^{k}, M)[1]

where the transition maps K^{•}(f^{k}, M) → K^{•}(f^{k+1}, M) are the morphisms φ_{f} of (7.8.20). As a corollary, we get the following classical result of Grothendieck :

Theorem 10.2.28. In the situation of (10.2.25), the following holds :

(i) The natural map M[0] → RΓ ℳ is an isomorphism in D(𝒪_{X}-Mod).

(ii) Let Y be any separated scheme, ℱ^{•} a bounded below complex of quasi-coherent 𝒪_{Y} - modules, and 𝔘 any affine open covering of Y. Then the natural map

Tot \overline{C}_{alt}^{•}(𝔘, ℱ^{•}) → RΓℱ^{•}

is an isomorphism in D^{+}(𝒪_{Y} (Y)-Mod). (iii) Especially, if Y is a separated scheme that can be covered by r affine open subsets,

then H^{i}(Y, ℱ) = 0 for every i ≥ r, and every quasi-coherent 𝒪_{Y} -module ℱ.

Proof. (i): Since the affine open subsets of X of the form Spec A[f^{−1}] for f ∈ A are a basis of the topology of X that is closed under finite intersections, theorem 10.2.24 reduces to showing that

H^{0}(𝔘, ℳ) = M and H^{i}(𝔘, ℳ) = 0 for every i > 0

where 𝔘 := (U_{i}, …, U_{r}) is the finite affine open covering of X associated with a sequence f := (f_{1}, …, f_{r}) as in (10.2.25). However, the condition X = ⋃_{i=1}^{r} U_{i} is equivalent to saying that the ideal generated by the system f is A, in which case the same holds for the ideal generated by f^{k}, for every k ∈ ℕ, and then lemma 7.8.2(iii) says that the complex H^{•}(f^{k}, M) is homotopically trivial for every k ∈ ℕ. Taking into account (10.2.27), the contention follows.

(ii): Pick any resolution ℱ^{•} →^{∼} ℐ^{•} by a bounded below complex of injective 𝒪_{Y} -modules; in view of (10.2.19), it suffices to show that the natural map of double complexes

\overline{C}_{alt}^{•}(𝔘, ℱ^{•}) → \overline{C}_{alt}^{•}(𝔘, ℐ^{•})

induces a quasi-isomorphism on total complexes. However, notice that the open subset U\underline{t} ⊂ Y is affine, for every n ∈ ℕ and every \underline{t} ∈ I_{alt}^{n+1} (notation of (10.2.4)); hence, it suffices to check that the induced map ℱ^{•}(V) → ℐ^{•}(V) is a quasi-isomorphism, for every affine open subset V ⊂ Y. To this aim, consider the spectral sequence

E_{1}^{p,q} := H^{p}(V, ℱ^{q}) ⇒ H^{p+q}ℐ^{•}(V).

Since V is affine, we have E_{1}^{p,q} = 0 for every p > 0, due to (i); on the other hand, the differential d_{1}^{0,q} : E_{1}^{0,q} → E_{1}^{0,q+1} is nothing else than the differential d^{q}(V) : ℱ^{q}(V) → ℱ^{q+1}(V), for every q ∈ ℤ, whence the claim.

(iii): Indeed, if 𝔘 is such a covering, then the complex \overline{C}_{alt}^{•}(𝔘, ℱ) is an object of the category C^{[0,r−1]}(𝒪_{Y} (Y)-Mod), so the assertion follows from (ii). □

We conclude this section with an application of Čech cohomology to the computation of sheaf cohomology after change of base scheme; much more can be found in [62, Ch.III, §6].

10.2.29. Namely, consider a diagram of schemes

\xymatrix{X_{2} \ar[r] \ar[d]_{π_{2}} & X_{1} \ar[r] \ar[d]_{π_{1}} & X_{0} \ar[d]^{π_{0}} \\ S_{2} \ar[r] & S_{1} \ar[r]^{φ} & S_{0}}

whose two square subdiagram are cartesian, and bounded above complexes of quasi-coherent 𝒪_{X_{0}}-modules ℱ_{0}^{•} and quasi-coherent 𝒪_{S_{i}}-modules 𝒢_{i}^{•} for i = 1, 2. Then, for every q ∈ ℕ there exists a quasi-coherent 𝒪_{X_{1}}-module

𝒯or_{q}^{S_{0}}(𝒢_{1}^{•}, ℱ_{0}^{•})

characterized, up to unique isomorphism, by the following properties ([62, Ch.III, §6.5]) :

(a) For every affine open subsets V_{0} ⊂ S_{0}, V ⊂ S_{1} ×_{S_{0}} V_{0} and U ⊂ X_{0} ×_{S_{0}} V_{0}, there exists

an isomorphism of 𝒪_{X_{1}}(V ×_{S_{0}} U)-modules

ω_{V,U} : Γ(V ×_{S_{0}} U, 𝒯or_{q}^{S_{0}}(𝒢_{1}^{•}, ℱ_{0}^{•})) →^{∼} H_{q}(𝒢_{1}^{•}(V) ⊗^{L}_{𝒪_{S_{0}}(V_{0})} ℱ_{0}^{•}(U)).

(b) For every V_{0}, V, U as in (a) and every inclusion of affine open subsets V′_{0} ⊂ V_{0}, V′ ⊂ V ∩ (S_{1} ×_{S_{0}} V′_{0}) and U′ ⊂ U ∩ (X_{0} ×_{S_{0}} V′_{0}), the resulting diagram commutes :

[figure: Γ(V ×_{S_{0}} U, 𝒯or_{q}^{S_{0}}(𝒢_{1}^{•}, ℱ_{0}^{•})); ω_{V,U}; H_{q}(𝒢_{1}^{•}(V) ⊗^{L}_{𝒪_{S_{0}}(V_{0})} ℱ_{0}^{•}(U)); Γ(V′ ×_{S_{0}} U′, 𝒯or_{q}^{S_{0}}(𝒢_{1}^{•}, ℱ_{0}^{•})); ω_{V′,U′}; H_{q}(𝒢_{1}^{•}(V′) ⊗^{L}_{𝒪_{S_{0}}(V′_{0})} ℱ_{0}^{•}(U′))]

whose left vertical arrow is the restriction map of the sheaf 𝒯or_{q}^{S_{0}}(𝒢_{1}^{•}, ℱ_{0}^{•}), and whose

right vertical arrow is the map H_{q}(ρ′ ⊗^{L}_{ρ} ρ′′) associated with the restriction maps ρ : 𝒪_{S_{0}}(V_{0}) → 𝒪_{S_{0}}(V′_{0}), ρ′ : 𝒢^{•}(V) → 𝒢^{•}(V′) and ρ′′ : ℱ_{0}^{•}(U) → ℱ_{0}^{•}(U′).

Proposition 10.2.30. In the situation of (10.2.29), the following holds :

(i) There exists a natural (homological) 2-spectral sequence

𝒯or_{p}^{S_{1}}(𝒢_{2}^{•}, 𝒯or_{q}^{S_{0}}(𝒪_{S_{1}}[0], ℱ_{0}^{•})) ⇒ 𝒯or_{p+q}^{S_{0}}(𝒢_{2}^{•}, ℱ_{0}^{•}).

(ii) Suppose that π_{0} is quasi-compact and separated, ℱ_{0} is a bounded complex, and both S_{1} and S_{0} are affine. Set A_{i} := 𝒪_{S_{i}}(S_{i}) for i = 0, 1. Then there exist two natural spectral sequences :

E_{2}^{pq} := H^{−p}(X_{1}, 𝒯or_{q}^{S_{0}}(𝒢^{•}, ℱ_{0}^{•})) ⇒ H_{p+q}(RΓ𝒢^{•} ⊗^{L}_{A_{0}} RΓℱ_{0}^{•})
F_{2}^{pq} := H_{p}(RΓ𝒢^{•} ⊗^{L}_{A_{0}} H^{−q}(X_{0}, ℱ_{0}^{•})[0]) ⇒ H_{p+q}(RΓ𝒢^{•} ⊗^{L}_{A_{0}} RΓℱ_{0}^{•}).

Proof. (i): This is obtained from the standard change of base ring spectral sequence, which is natural in all arguments, and therefore globalizes immediately to the sheaf-theoretic situation considered here : the details shall be left to the reader.

(ii): Under our assumptions, X_{0} is quasi-compact and separated, hence we may find a finite affine covering 𝔘 of X_{0}, and by theorem 10.2.28(ii), the alternating Čech complex for 𝔘 computes the cohomology of ℱ_{•}^{•}; especially the complex RΓℱ_{0}^{•} is bounded. Moreover, we may find a bounded above Cartan-Eilenberg resolution ℒ^{•} →^{∼} φ_{∗}𝒢^{•} consisting of quasi-coherent

flat 𝒪_{S_{0}}-modules, and H_{i}(RΓ𝒢^{•} ⊗^{L}_{A_{0}} RΓℱ_{0}^{•}) is naturally isomorphic to the quasi-coherent 𝒪_{S_{1}}- module associated with the A_{1}-module

H_{i}((Tot \overline{C}_{alt}^{•}(𝔘, ℱ_{0}^{•})) ⊗_{A_{0}} ℒ^{•}(S_{0}))

for every i ∈ ℤ. Then the first spectral sequence is the standard spectral sequence attached to the double complex (Tot \overline{C}_{alt}^{•}(𝔘, ℱ_{0}^{•})) ⊠_{A_{0}} ℒ^{•}(S_{0}); similarly one obtains the second spectral sequence : details left to the reader. □

10.3. Quasi-coherent modules. For any scheme X, we denote by

𝒪_{X}-Mod  𝒪_{X}-Mod_{qcoh}  𝒪_{X}-Mod_{coh}  𝒪_{X}-Mod_{lfft}

the category of all (resp. of quasi-coherent, resp. of coherent, resp. of locally free of finite type) 𝒪_{X}-modules. Recall that there is a natural functor

(10.3.1)  𝒪_{X}(X)-Mod → 𝒪_{X}-Mod_{qcoh}  M ↦ M^{∼}

that assigns to every 𝒪_{X}(X)-module the quasi-coherent module M^{∼} such that

M^{∼}(U) := M ⊗_{𝒪_{X}(X)} 𝒪_{X}(U) for every affine open subset U ⊂ X.

This functor is an equivalence, if X is affine. We notice :

Corollary 10.3.2. Let A be a ring, M^{•} (resp. N_{•}) a bounded below (resp. above) complex of A-modules. Set X := Spec A and denote by M^{•∼} (resp. N_{•}^{∼}) the associated complex of quasi-coherent 𝒪_{X}-modules. We have :

(i) The natural map

RHom_{A}^{•}(N_{•}, M^{•}) → RHom_{𝒪_{X}}^{•} (N_{•}^{∼}, M^{•∼})

is an isomorphism in D^{+}(A-Mod). (ii) If A is coherent and N_{•} is a complex of finitely presented A-modules, then the natural

morphism

RHom_{A}^{•}(N_{•}, M^{•})^{∼} → Rℋ om_{𝒪_{X}}^{•} (N_{•}^{∼}, M^{•∼})

is an isomorphism in D(𝒪_{X}-Mod).

Proof. (i): We apply the trivial duality theorem 10.1.18to the unique morphism

f : (X, 𝒪_{X}) → ({pt}, A)

of ringed spaces, where {pt} denotes the one-point space. Since f is flat and all quasi-coherent 𝒪_{X}-modules are f_{∗}-acyclic, the assertion follows easily.

(ii): Due to (i), it suffices to show the following assertion. For every f ∈ A, the natural map

RHom_{A}^{•}(N_{•}, M^{•}) ⊗_{A} A_{f} → RHom_{A_{f}}^{•} (N_{•,f}, M_{f}^{•})

is an isomorphism in D(A_{f}-Mod). To this aim, notice that, since A is coherent, we may find a resolution P_{•} of N_{•} consisting of free A-modules of finite type (details left to the reader); then we come down to checking that the natural map Hom_{A}(P_{i}, M^{j}) ⊗_{A} A_{f} → Hom_{A_{f}} (P_{i,f}, M_{f}^{j}) is an isomorphism, which is obvious. □

We denote by

D(𝒪_{X}-Mod)_{qcoh} D(𝒪_{X}-Mod)_{coh}

the full triangulated subcategory of D(𝒪_{X}-Mod) consisting of the complexes K^{•} such that H^{i}K^{•} is a quasi-coherent (resp. coherent) 𝒪_{X}-module for every i ∈ ℤ. As usual, we shall use also the variants D^{+}(𝒪_{X}-Mod)_{qcoh} (resp. D^{−}(𝒪_{X}-Mod)_{qcoh}, resp. D^{b}(𝒪_{X}-Mod)_{qcoh}, resp. D^{[a,b]}(𝒪_{X}-Mod)_{qcoh}) consisting of all objects of D(𝒪_{X}-Mod)_{qcoh} whose cohomology vanishes in sufficiently large negative degree (resp. sufficiently large positive degree, resp. outside a bounded interval, resp. outside the interval [a, b]), and likewise for the corresponding subcategories of D(𝒪_{X}-Mod)_{coh}. Obviously, (10.3.1) induces a natural functor

D(𝒪_{X}(X)-Mod) → D(𝒪_{X}-Mod)_{qcoh} M^{•} ↦ M^{•∼}.

Proposition 10.3.3. Let f : Y → X be a flat morphism of schemes, with X locally coherent (see definition 8.1.58(i)), and K_{•} ∈ Ob(D^{−}(𝒪_{X}-Mod)_{coh}) any complex. We have :

(i) Rℋ om_{𝒪_{X}}^{•} (K_{•}, L^{•}) ∈ Ob(D^{+}(𝒪_{X}-Mod)_{coh}) for every L^{•} ∈ Ob(D^{+}(𝒪_{X}-Mod)_{coh}).

(ii) For every L^{•} ∈ Ob(D^{+}(𝒪_{X}-Mod)) the natural morphism

(10.3.4)  f^{∗}Rℋ om_{𝒪_{X}}^{•} (K_{•}, L^{•}) → Rℋ om_{𝒪_{Y}}^{•} (f^{∗}K_{•}, f^{∗}L^{•})

is an isomorphism in D^{+}(𝒪_{X}-Mod)).

Proof. (i): The assertion is local on X, so we may assume that X = Spec A, for some coherent ring A. In this case, we may find a bounded below (resp. above) complex M^{•} (resp. N_{•}) of finitely presented A-modules, with isomorphisms K^{•} →^{∼} M^{•∼}, L_{•} →^{∼} N_{•}^{∼} in D(𝒪_{X}-Mod). By virtue of corollary 10.3.2(ii), we are then reduced to checking that RHom_{A}^{•}(N_{•}, M^{•}) lies in D(A-Mod_{coh}). But this is easily seen, since we may find a resolution of N_{•} by a bounded above complex of free A-modules of finite type (details left to the reader).

(ii): Again, the question is local on X, so we may assume that K_{•} admits a resolution by a bounded above complex P_{•} of free 𝒪_{X}-modules of finite type. Since f is flat, we have natural convergent spectral sequences

\begin{matrix}E_{1}^{pq} := f^{∗}R^{p}ℋ om_{𝒪_{X}}^{•} (P_{q}, L^{•}) ⇒ & f^{∗}R^{p+q}ℋ om_{𝒪_{X}}^{•} (K_{•}, L^{•}) \\ F_{1}^{pq} := R^{p}ℋ om_{𝒪_{Y}}^{•} (f^{∗}P_{q}, f^{∗}L^{•}) ⇒ & R^{p+q}ℋ om_{𝒪_{Y}}^{•} (f^{∗}K_{•}, f^{∗}L^{•}) \end{matrix}

as well as a morphism of spectral sequences E_{•}^{••} → F_{•}^{••}, such that H^{•}(10.3.4) is a morphism of filtered 𝒪_{Y} -modules, for the two finite filtrations induced by these spectral sequences on their abutments. It then suffices to check that the induced morphism E_{∞}^{pq} → F_{∞}^{pq} is an isomorphism for every p, q ∈ ℤ, and this in turn will follow, if we show that the morphism E_{1}^{pq} → F_{1}^{pq} is an isomorphism; but the latter assertion is obvious. □

Corollary 10.3.5. Let f : Y → X be a quasi-compact and quasi-separated morphism of schemes, ℱ a flat quasi-coherent 𝒪_{X}-module, and 𝒢 any 𝒪_{Y} -module. Then the natural map

(10.3.6)  ℱ ⊗_{𝒪_{X}} Rf_{∗}𝒢 → Rf_{∗}(f^{∗}ℱ ⊗_{𝒪_{Y}} 𝒢)

is an isomorphism in D(𝒪_{X}-Mod).

Proof. The assertion is local on X, hence we may assume that X = Spec A for some ring A, and ℱ = M^{∼} for some flat A-module M. By [120, Ch.I, Th.1.2], M is the colimit of a filtered family of free A-modules of finite rank; in view of proposition 10.1.10(ii), we may then assume that M = A^{⊕n} for some n ≥ 0, in which case the assertion is obvious. □

Remark 10.3.7. Notice that, for an affine morphism f : Y → X, the map (10.3.6) is an isomorphism for any quasi-coherent 𝒪_{X}-module ℱ and any quasi-coherent 𝒪_{Y} -module 𝒢. The details shall be left to the reader.

Corollary 10.3.8. Consider a cartesian diagram of schemes

\xymatrix{Y′ \ar[r]^{g′} \ar[d]_{f′} & Y \ar[d]^{f} \\ X′ \ar[r]^{g} & X}

such that f is quasi-compact and quasi-separated, and g is flat. Then the natural map

(10.3.9)  g^{∗}Rf_{∗}𝒢 → Rf′_{∗}g′^{∗}𝒢

is an isomorphism in D(𝒪_{X′}-Mod)_{qcoh}, for every quasi-coherent 𝒪_{Y} -module 𝒢.

Proof. We easily reduce to the case where both X and X′ are affine, hence g is an affine morphism. In this case, it suffices to show that g_{∗}(10.3.9) is an isomorphism, since Rf_{∗}𝒢 lies in D(𝒪_{X}-Mod)_{qcoh} and Rf′_{∗}g′^{∗}𝒢 lies in D(𝒪_{X′}-Mod)_{qcoh} ([2, Th.5.6]). However, we have a commutative diagram

\xymatrix{g_{∗}(g^{∗}Rf_{∗}𝒢) \ar[rr]^{α} \ar[d] &  & Rf_{∗}𝒢 ⊗_{𝒪_{X}} g_{∗}𝒪_{X′} \ar[d] \\ g_{∗}(Rf′_{∗}g′^{∗}𝒢) \ar[r]^{∼} & Rf_{∗}g′_{∗}(g′^{∗}𝒢) \ar[r]^{β} & Rf_{∗}(𝒢 ⊗_{𝒪_{Y}} g′_{∗}𝒪_{Y′})}

whose left vertical arrow is g_{∗}(10.3.9) and whose right vertical arrow is the natural isomorphism provided by corollary 10.3.5(applied to the flat 𝒪_{Y} -module ℱ := g′_{∗}𝒪_{Y′}). Also, α and β are the natural maps obtained as in [59, Ch.0, (5.4.10)]; it is easily seen that these are isomorphisms, for all affine morphisms g and g′. The assertion follows. □

Remark 10.3.10. Corollary 10.3.8 is also proved in [2, Th.6.7]. Notice that the corollary fails if 𝒢 is not quasi-coherent. For a counterexample, let R := k[x, y]_{𝔪}, where k is a field and 𝔪 ⊂ k[x, y] is the maximal ideal generated by x and y; i.e. R is the local ring at the origin of the affine plane over k. Let R^{∧} be the completion of R, X := Spec R, X′ := Spec R^{∧}, and set Y := X \{𝔪R} and Y′ := Y ×_{X} X′. Let moreover 𝔭 ∈ Y be the prime ideal generated by x, and 𝔭′ its inverse image in Y′; take for 𝒢 the 𝒪_{Y} -module obtained by extension by zero of the restriction to {𝔭} of the structure sheaf of 𝒪_{Y}. With this notation, suppose that the map (10.3.9) induces an isomorphism on the stalks at the closed points; in cohomological degree zero, this comes down to saying that the natural map f : (R^{∧})_{𝔭} → (R^{∧})_{𝔭′} is an isomorphism. However, notice that the restriction R_{𝔭} → (R^{∧})_{𝔭′} of f induces a homeomorphism Spec (R^{∧})_{𝔭′} →^{∼} Spec R_{𝔭}. On the other hand, we can exhibit a height one prime ideal 𝔮 ∈ Spec (R^{∧})_{𝔭} whose image in Spec R_{𝔭} is the generic point. Indeed, let a(X) ∈ x · k[[x]] be a power series that is transcendental over k(x); the kernel of the surjective continuous map of k[[x]]-algebras g : R^{∧} = k[[x, y]] → k[[x]] such that g(y) := a(x) is the ideal I generated by y − a(x), hence the latter is a prime ideal of R^{∧}, and we take 𝔮 := I_{𝔭}. To see that the preimage of 𝔮 in Spec R_{𝔭} is the generic point, it suffices to notice that the restriction R_{𝔭} → k[[x]] of g is injective, as a(x) is transcendental over k(x).

10.3.11. There are obvious forgetful functors:

ι_{X} : 𝒪_{X}-Mod_{qcoh} → 𝒪_{X}-Mod Rι_{X} : D(𝒪_{X}-Mod_{qcoh}) → D(𝒪_{X}-Mod)_{qcoh}

and we wish to exhibit right adjoints to these functors. To this aim, suppose first that X is affine; then we may consider the functor:

qcoh_{X} : 𝒪_{X}-Mod → 𝒪_{X}-Mod_{qcoh} ℱ ↦ ℱ (X)^{∼}

If 𝒢 is a quasi-coherent 𝒪_{X}-module, then clearly qcoh_{X}𝒢 ≃ 𝒢; moreover, for any other 𝒪_{X}- module ℱ, there is a natural bijection:

Hom_{𝒪_{X}} (𝒢 , ℱ) →^{∼} Hom_{𝒪_{X}(X)} (𝒢 (X), ℱ (X)).

It follows easily that qcoh_{X} is the sought right adjoint.

10.3.12. Slightly more generally, let U be quasi-affine, i.e. a quasi-compact open subset of an affine scheme, and choose a quasi-compact open immersion j : U → X into an affine scheme X. In this case, we may define

qcoh_{U} : 𝒪_{U}-Mod → 𝒪_{X}-Mod ℱ ↦ (qcoh_{X}j_{∗}ℱ)_{|U}.

Since j_{∗} : 𝒪_{U}-Mod → 𝒪_{X}-Mod is right adjoint to j^{∗} : 𝒪_{X}-Mod → 𝒪_{U}-Mod, we have a natural isomorphism: Hom_{𝒪_{U}} (𝒢 , ℱ) →^{∼} Hom_{𝒪_{X}} (j_{∗}𝒢 , j_{∗}ℱ) for every 𝒪_{U}-modules 𝒢 and ℱ. Moreover, if 𝒢 is quasi-coherent, the same holds for j_{∗}𝒢 ([59, Ch.I, Cor.9.2.2]), whence a natural isomorphism 𝒢 ≃ qcoh_{U}𝒢, by the foregoing discussion for the affine case. Summing up, this shows that qcoh_{U} is a right adjoint to ι_{U}, and especially it is independent, up to unique isomorphism, of the choice of j.

10.3.13. Next, suppose that X is quasi-compact and quasi-separated. We choose a finite covering 𝔘 := (U_{i} | i ∈ I) of X, consisting of affine open subsets, and set U := ∐_{i∈I} U_{i}, the X-scheme which is the disjoint union (i.e. categorical coproduct) of the schemes U_{i}. We denote by 𝔘_{•} the simplicial covering such that 𝔘_{n} := U ×_{X} ⋯ ×_{X} U, the (n + 1)-th power of U, with the face maps given by the natural projections, and degeneracies induced by the diagonal map U → U ×_{X} U; let also π_{n} : 𝔘_{n} → X be the natural morphism, for every n ∈ ℕ. Clearly we have π_{n−1} ∘ ∂_{i} = π_{n} for every face morphism ∂_{i} : 𝔘_{n} → 𝔘_{n−1}. The simplicial scheme 𝔘_{•} (with the Zariski topology on each scheme 𝔘_{n}) can also be regarded as a fibred topos over the category Δ^{o} (notation of [75, §2.2]); then the datum 𝒪_{𝔘_{•}} := (𝒪_{𝔘_{n}} | n ∈ ℕ) consisting of the structure sheaves on each 𝔘_{n} and the natural maps ∂_{i}^{∗}𝒪_{𝔘_{n−1}} → 𝒪_{𝔘_{n}} for every n > 0 and every i = 0, …, n (and similarly for the degeneracy maps), defines a ring in the associated topos Top(𝔘_{•}) (see [75, §3.3.15]). We denote by 𝒪_{𝔘_{•}}-Mod the category of 𝒪_{𝔘_{•}}-modules in the topos Top(𝔘_{•}). The family (π_{n} | n ∈ ℕ) induces a morphism of topoi

π_{•} : Top(𝔘_{•}) → s.X

where s.X is the topos Top(X_{•}) associated with the constant simplicial scheme X_{•} (with its Zariski topology) such that X_{n} := X for every n ∈ ℕ and such that all the face and degeneracy maps are 1_{X}. Clearly the objects of s.X are nothing else than the cosimplicial Zariski sheaves on X. Especially, if we view a 𝒪_{X}-module ℱ as a constant cosimplicial 𝒪_{X}-module, we may define the augmented cosimplicial Čech 𝒪_{X}-module

(10.3.14)  ℱ → 𝒞^{•}(𝔘, ℱ) := π_{•∗} ∘ π_{•}^{∗}ℱ

associated with ℱ and the covering 𝔘; in every degree n ∈ ℕ this is defined by the rule :

𝒞^{n}(𝔘, ℱ) := π_{n∗}π_{n}^{∗}ℱ

and the coface operators ∂^{i} are induced by the face morphisms ∂_{i} in the obvious way.

Lemma 10.3.15. (i) The augmented complex (10.3.14) is aspherical for every 𝒪_{X}-module ℱ. (ii) If ℱ is an injective 𝒪_{X}-module, then (10.3.14) is homotopically trivial.

Proof. (See also e.g. [79, Th.5.2.1].) The proof relies on the following alternative description of the cosimplicial Čech 𝒪_{X}-module. Consider the adjoint pair :

\xymatrix{(π^{∗}, π_{∗}) : 𝒪_{𝔘}-Mod \ar[r] & 𝒪_{X}-Mod \ar[l]}

arising from our covering π : 𝔘 → X. Let (⊤ := π_{∗} ∘ π^{∗}, η, μ) be the associated triple (see (7.10.3)), ℱ any 𝒪_{X}-module; we leave to the reader the verification that the resulting augmented cosimplicial complex ℱ → ⊤^{•}ℱ – as defined in (7.10.2) – is none else than the augmented Čech complex (10.3.14). Thus, for every 𝒪_{X}-module ℱ, the augmented complex π^{∗}ℱ → π^{∗}𝒞^{•}(𝔘, ℱ) is homotopically trivial (proposition 7.10.5); since π is a covering morphism, (i) follows. Furthermore, by the same token, the augmented complex π_{∗}𝒢 → 𝒞^{•}(𝔘, π_{∗}𝒢) is homotopically trivial for every 𝒪_{𝔘}-module 𝒢; especially we may take 𝒢 := π^{∗}ℐ, where ℐ is an injective 𝒪_{X}-module. On the other hand, when ℐ is injective, the unit of adjunction ℐ → ⊤ℐ is split injective; hence the augmented complex ℐ → 𝒞^{•}(𝔘, ℐ) is a direct summand of the homotopically trivial complex ⊤ℐ → 𝒞^{•}(𝔘, ⊤ℐ), and (ii) follows. □

Notice now that the schemes 𝔘_{n} are quasi-affine for every n ∈ ℕ, hence the functors qcoh_{𝔘_{n}} are well defined as in (10.3.12), and indeed, the rule : (ℱ_{n} | n ∈ ℕ) ↦ (qcoh_{𝔘_{n}}ℱ_{n} | n ∈ ℕ) yields a functor :

qcoh_{𝔘_{•}} : 𝒪_{𝔘_{•}}-Mod → 𝒪_{𝔘_{•}}-Mod.

This suggests to introduce a quasi-coherent cosimplicial Čech 𝒪_{X}-module :

q𝒞^{•}(𝔘, ℱ) := π_{•∗} ∘ qcoh_{𝔘_{•}} ∘ π_{•}^{∗}ℱ

for every 𝒪_{X}-module ℱ (regarded as a constant cosimplicial module in the usual way). More plainly, this is the cosimplicial 𝒪_{X}-module such that :

q𝒞^{n}(𝔘, ℱ) := π_{n∗} ∘ qcoh_{𝔘_{n}} ∘ π_{n}^{∗}ℱ for every n ∈ ℕ.

According to [59, Ch.I, Cor.9.2.2], the 𝒪_{X}-modules q𝒞^{n}(𝔘, ℱ) are quasi-coherent for all n ∈ ℕ. Finally, we define the functor :

qcoh_{X} : 𝒪_{X}-Mod → 𝒪_{X}-Mod_{qcoh} ℱ ↦ \xymatrix{Equal(q𝒞^{0}(𝔘, ℱ) \ar[r]^{∂^{0}} \ar[r]_{∂^{1}} & q𝒞^{1}(𝔘, ℱ))}

Proposition 10.3.16. (i) In the situation of (10.3.13), the functor qcoh_{X} is right adjoint to ι_{X}.

(ii) Let Y be any other quasi-compact and quasi-separated scheme, f : X → Y any morphism. Then the induced diagram of functors:

\xymatrix{𝒪_{X}-Mod \ar[r]^{f_{∗}} \ar[d]_{_{qcoh_{X}}} & 𝒪_{Y} -Mod \ar[d]^{_{qcoh_{Y}}} \\ 𝒪_{X}-Mod_{qcoh} \ar[r]^{f_{∗}} & 𝒪_{Y} -Mod_{qcoh}}

commutes up to a natural isomorphism of functors.

Proof. For every n ∈ ℕ and every 𝒪_{𝔘_{n}}-module ℋ , the counit of the adjunction yields a natural map of 𝒪_{𝔘_{n}}-modules: qcoh_{𝔘_{n}}ℋ → ℋ . Taking ℋ to be π_{n}^{∗}ℱ on 𝔘_{n} (for a given 𝒪_{X}-module ℱ), these maps assemble to a morphism of cosimplicial 𝒪_{X}-modules :

(10.3.17)  q𝒞^{•}(𝔘, ℱ) → 𝒞^{•}(𝔘, ℱ)

and it is clear that (10.3.17)is an isomorphism whenever ℱ is quasi-coherent. Let now φ : 𝒢 → ℱ be a map of 𝒪_{X}-modules, with 𝒢 quasi-coherent; after applying the natural transformation (10.3.17) and forming equalizers, we obtain a commutative diagram :

\xymatrix{qcoh_{X}𝒢 \ar[r]^{∼} \ar[d]_{qcoh_{X}φ} & 𝒢 \ar[d]^{φ} \\ qcoh_{X}ℱ \ar[r] & ℱ}

from which we see that the rule: φ ↦ qcoh_{X}φ establishes a natural injection:

(10.3.18)  Hom_{𝒪_{X}} (𝒢 , ℱ) → Hom_{𝒪_{X}} (𝒢 , qcoh_{X}ℱ)

and since Hom_{𝒪_{X}} (𝒢 , (10.3.17)) is an isomorphism of cosimplicial 𝒪_{X}-modules, (10.3.18) is actually a bijection, whence (i).

(ii) is obvious, since both qcoh_{Y} ∘f_{∗} and f_{∗} ∘qcoh_{X} are right adjoint to f^{∗} ∘ι_{Y} = ι_{X} ∘f^{∗}. □

Remark 10.3.19. The right adjoint functor qcoh_{X} was first constructed in [20, Exp.II, Lemma 3.2], for X quasi-compact and quasi-separated; a construction valid for all schemes X is found in [107, Tag 0781].

10.3.20. In the situation of (10.3.13), the functor qcoh_{X} is left exact, since it is a right adjoint, hence it gives rise to a left derived functor :

Rqcoh_{X} : D^{+}(𝒪_{X}-Mod) → D^{+}(𝒪_{X}-Mod_{qcoh}).

Proposition 10.3.21. Let X be a quasi-compact and quasi-separated scheme. Then : (i) Rqcoh_{X} is right adjoint to Rι_{X}. (ii) Suppose moreover, that X is semi-separated, i.e. such that the diagonal morphism X → X ×_{ℤ} X is affine. Then the unit of the adjunction (Rι_{X}, Rqcoh_{X}) is an isomorphism of functors.

Proof. (i): The exactness of the functor ι_{X} implies that qcoh_{X} preserves injectives; the assertion is a formal consequence : the details shall be left to the reader.

(ii): Let ℱ^{•} be any complex of quasi-coherent 𝒪_{X}-modules; we have to show that the natural map ℱ^{•} → Rqcoh_{X}ℱ^{•} is an isomorphism. Using a Cartan-Eilenberg resolution ℱ^{•} →^{∼} ℐ^{••} we deduce a spectral sequence ([163, §5.7])

E_{1}^{pq} := R^{p}qcoh_{X}H^{q}ℱ^{•} ⇒ R^{p+q}qcoh_{X}ℱ^{•}

which easily reduces to checking the assertion for the cohomology of ℱ^{•}, so we may assume from start that ℱ^{•} is a single 𝒪_{X}-module placed in degree zero. Let us then choose an injective resolution ℱ →^{∼} ℐ^{•} (that is, in the category of all 𝒪_{X}-modules); we have to show that H^{p}qcoh_{X}ℐ^{•} = 0 for p > 0. We deal first with the following special case :

Claim 10.3.22. Assertion (i) holds if X is affine.

Proof of the claim. Indeed, in this case, the chosen injective resolution of ℱ yields a long exact sequence 0 → ℱ (X) → ℐ^{•}(X), and therefore a resolution qcoh_{X}ℱ := ℱ (X)^{∼} → qcoh_{X}ℐ^{•} := ℐ^{•}(X)^{∼}. ♢

For the general case, we choose any affine covering 𝔘 of X and we consider the cochain complex of cosimplicial complexes q𝒞^{•}(𝔘, ℐ^{•}).

Claim 10.3.23. For any injective 𝒪_{X}-module ℐ, the augmented complex:

qcoh_{X}ℐ → q𝒞^{•}(𝔘, ℐ)

is homotopically trivial.

Proof of the claim. It follows easily from proposition 10.3.16(ii) that

q𝒞^{•}(𝔘, ℱ) ≃ qcoh_{X}(𝒞^{•}(𝔘, ℱ))

for any 𝒪_{X}-module ℱ. Then the claim follows from lemma 10.3.15(ii). ♢ We have a spectral sequence :

E_{1}^{pq} := H^{p}q𝒞^{•}(𝔘, ℐ^{q}) ⇒ Tot^{p+q}(q𝒞^{•}(𝔘, ℐ^{•}))

and it follows from claim 10.3.23that E_{1}^{pq} = 0 whenever p > 0, and E^{0q} = qcoh_{X}ℐ^{q} for every q ∈ ℕ, so the spectral sequence E^{••} degenerates, and we deduce a quasi-isomorphism

(10.3.24)  qcoh_{X}ℐ^{•} →^{∼} Tot^{•}(q𝒞^{•}(𝔘, ℐ^{•})).

On the other hand, for fixed q ∈ ℕ, we have q𝒞^{q}(𝔘, ℐ^{•}) = π_{n∗}qcoh_{𝔘_{n}}π_{n}^{∗}ℐ^{•}; since X is semiseparated, 𝔘_{n} is affine, so the complex qcoh_{𝔘_{n}}π_{n}^{∗}ℐ^{•} is a resolution of qcoh_{𝔘_{n}}π_{n}^{∗}ℱ = π_{n}^{∗}ℱ, by claim 10.3.22. Furthermore, π_{n} : 𝔘_{n} → X is an affine morphism, so q𝒞^{q}(𝔘, ℐ^{•}) is a resolution of π_{n∗}π_{n}^{∗}ℱ. Summing up, we see that Tot^{•}(q𝒞^{•}(𝔘, ℐ^{•})) is quasi-isomorphic to 𝒞^{•}(𝔘, ℱ), which is a resolution of ℱ, by lemma 10.3.15(i). Combining with (10.3.24), we deduce (ii). □

Theorem 10.3.25. Let X be a quasi-compact semi-separated scheme. The forgetful functor

Rι_{X} : D^{+}(𝒪_{X}-Mod_{qcoh}) → D^{+}(𝒪_{X}-Mod)_{qcoh}

is an equivalence of categories, whose quasi-inverse is the restriction of Rqcoh_{X}.

Proof. By proposition 10.3.21(ii) we know already that the composition Rqcoh_{X} ∘ Rι_{X} is a self-equivalence of D^{+}(𝒪_{X}-Mod_{qcoh}). For every complex ℱ^{•} in D^{+}(𝒪_{X}-Mod), the counit of adjunction ε_{ℱ^{•}} : Rι_{X} ∘ Rqcoh_{X}ℱ^{•} → ℱ^{•} can be described as follows. Pick an injective resolution α : ℱ^{•} → ℐ^{•}; then ε_{ℱ^{•}} is defined by the diagram :

\xymatrix{qcoh_{X}ℐ^{•} \ar[r]^{ε^{•}} & ℐ^{•} & ℱ^{•} \ar[l]_{α}}

where, for each n ∈ ℕ, the map ε^{n} : qcoh_{X}ℐ^{n} → ℐ^{n} is the counit of the adjoint pair (ι_{X}, qcoh_{X}). It suffices then to show that ε^{•} is a quasi-isomorphism, when ℐ^{•} is an object of D^{+}(𝒪_{X}-Mod)_{qcoh}. To this aim, we may further choose ℐ^{•} of the form Tot^{•}(ℐ^{••}), where ℐ^{••} is a Cartan-Eilenberg resolution of ℱ^{•} (see [163, §5.7]). We then deduce a spectral sequence :

E_{2}^{pq} := R^{p}qcoh_{X}H^{q}ℱ^{•} ⇒ R^{p+q}qcoh_{X}ℱ^{•}.

The double complex ℐ^{••} also gives rise to a similar spectral sequence F_{2}^{pq}, and clearly F_{2}^{pq} = 0 whenever p > 0, and F^{0q} = H^{q}ℱ^{•}. Furthermore, the counit of adjunction ε^{••} : qcoh_{X}ℐ^{••} → ℐ^{••} induces a morphism of spectral sequences ω^{pq} : E_{2}^{pq} → F_{2}^{pq}. Consequently, in order to prove that the ε_{ℱ^{•}} is a quasi-isomorphism, it suffices to show that ω^{pq} is an isomorphism for every p, q ∈ ℕ. This comes down to the assertion that R^{p}qcoh_{X}𝒢 = 0 for every quasicoherent 𝒪_{X}-module 𝒢, and every p > 0. However, we have 𝒢 = Rι_{X}𝒢, so that Rqcoh_{X}𝒢 = Rqcoh_{X} ∘ Rι_{X}𝒢, and then the contention follows from proposition 10.3.21(ii). □

Remark 10.3.26. Theorem 10.3.25appeared in [158, Prop.B.16] and [20, Exp.II, Prop.3.5] (the latter is stated for separated schemes, but the proof works in the semi-separated case as well). The theorem holds also for any locally noetherian scheme X, by virtue of [87, Ch.II, Cor.7.19] (whose proof uses the structure of injective 𝒪_{X}-modules, rather than the functor qcoh_{X}).

Lemma 10.3.27. Let X be a quasi-compact and quasi-separated scheme, U a quasi-compact open subset of X, ℋ a quasi-coherent 𝒪_{X}-module, 𝒢 a finitely presented quasi-coherent 𝒪_{U}- module, and φ : 𝒢 → ℋ_{|U} an 𝒪_{U}-linear map. Then:

(i) There exist a finitely presented quasi-coherent 𝒪_{X}-module ℱ on X, and a 𝒪_{X}-linear map ψ : ℱ → ℋ , such that ℱ_{|U} = 𝒢 and ψ_{|U} = φ.

(ii) Especially, every finitely presented quasi-coherent 𝒪_{U}-module extends to a finitely presented quasi-coherent 𝒪_{X}-module.

Proof. (i): Let (V_{i} | i = 1, …, n) be a finite affine open covering of X. For every i = 0, …, n, let us set U_{i} := U ∪ V_{1} ∪ ⋯ ∪ V_{i}; we construct, by induction on i, a family of finitely presented quasi-coherent 𝒪_{U_{i}}-modules ℱ_{i}, and morphisms ψ_{i} : ℱ_{i} → ℋ_{|U_{i}} such that ℱ_{i+1|U_{i}} = ℱ_{i} and ψ_{i+1|U_{i+1}} = ψ_{i} for every i < n. For i = 0 we have U_{0} = U, and we set ℱ_{0} := 𝒢, ψ_{0} := φ. Suppose that i < n and that ℱ_{i} and ψ_{i} have already been given. Since X is quasi-separated, the same holds for U_{i+1}, and the immersion j : U_{i} → U_{i+1} is quasi-compact; it follows that j_{∗}ℱ_{i} and j_{∗}ℋ_{|U_{i}} are quasi-coherent 𝒪_{U_{i}}-modules ([59, Ch.I, Prop.9.4.2(i)]). We let

ℳ := j_{∗}ℱ_{i} ×_{j_{∗}ℋ_{|U_{i}}} ℋ_{|U_{i+1}}.

Then ℳ is a quasi-coherent 𝒪_{U_{i+1}}-module admitting a map ℳ → ℋ_{|U_{i+1}}, and such that ℳ_{|U_{i}} = ℱ_{i}. We can then find a filtered family of quasi-coherent 𝒪_{V_{i+1}}-modules of finite presentation (ℳ_{λ} | λ ∈ Λ), whose colimit is ℳ_{|V_{i+1}}. Since ℱ_{i} is finitely presented and U_{i} ∩ V_{i+1} is quasicompact, there exists λ ∈ Λ such that the induced morphism β : ℳ_{λ|U_{i}∩V_{i+1}} → ℱ_{i|U_{i}∩V_{i+1}} is an epimorphism. Notice moreover that the restriction of 𝒦 := Ker(ℳ_{λ} → ℳ_{|V_{i+1}}) to U_{i} ∩ V_{i+1} coincides with Ker(β); since Ker(β) is of finite type, and since U_{i} ∩ V_{i+1} is quasi-compact, there exists an 𝒪_{V_{i+1}}-submodule of finite type 𝒩 of 𝒦 such that 𝒩_{|U_{i}∩V_{i+1}} = Ker(β). Set \overline{ℳ} _{λ} := ℳ_{λ}/𝒩 ; then β descends to an isomorphism \overline{β} : \overline{ℳ} _{λ} →^{∼} ℱ_{i|U_{i}∩V_{i+1}}, and the map ℳ_{λ} → ℳ_{|V_{i+1}} factors through \overline{ℳ} _{λ}. We can thus define ℱ_{i+1} by gluing ℱ_{i} and \overline{ℳ} _{λ} along \overline{β}; likewise, ψ_{i} and the induced map \overline{ℳ} _{λ} → ℳ_{|V_{i+1}} → ℋ_{|V_{i+1}} glue to a map ψ_{i+1} as required. Clearly the pair (ℱ := ℱ_{n}, ψ := ψ_{n}) is the sought extension of (𝒢 , φ).

(ii): Let 0_{X} be the final object in the category of 𝒪_{X}-modules; to extend a finitely presented quasi-coherent 𝒪_{U}-module 𝒢, it suffices to apply (i) to the unique map 𝒢 → 0_{X|U}. □

For ease of reference, we point out the following simple consequence of lemma 10.3.27.

Corollary 10.3.28. Let U be a quasi-affine scheme, ℰ a locally free 𝒪_{U}-module of finite type. Then we may find integers n, m ∈ ℕ and a left exact sequence of 𝒪_{U}-modules :

0 → ℰ → 𝒪_{U}^{⊕m} → 𝒪_{U}^{⊕n}.

Proof. Set ℰ^{∨} := ℋ om_{𝒪_{U}} (ℰ , 𝒪_{U}), and notice that ℰ^{∨} is a locally free 𝒪_{U}-module of finite type, especially it is quasi-coherent and of finite presentation. By assumption, U is a quasicompact open subset of an affine scheme X, hence ℰ^{∨} extends to a quasi-coherent 𝒪_{X}-module ℱ of finite presentation (lemma 10.3.27(ii)); we have ℱ = F^{∼}, for some finitely presented A-module F; we choose a presentation of F as the cokernel of an A-linear map A^{⊕n} → A^{⊕m}, for some m, n ∈ ℕ, whence a presentation of ℰ^{∨} as the cokernel of a morphism 𝒪_{U}^{⊕n} → 𝒪_{U}^{⊕m}, and after dualizing again, we get the sought left exact sequence. □

10.3.29. Let X be a scheme; for any quasi-coherent 𝒪_{X}-module ℱ, we consider the full subcategory C/ℱ of the category 𝒪_{X}-Mod_{qcoh}/ℱ whose objects are all the maps φ : 𝒢 → ℱ such that 𝒢 is a finitely presented 𝒪_{X}-module (notation of (1.1.24)). We denote by

ι_{ℱ} : C/ℱ → 𝒪_{X}-Mod_{qcoh} (𝒢 → ℱ) ↦ 𝒢

the restriction of the source functor. The first observation is:

Lemma 10.3.30. C/ℱ is a filtered category.

Proof. Since C/ℱ always admits the initial object 0 → ℱ, its set of objects is non-empty. Thus, according to remark 1.2.21(i), it suffices to check that C/ℱ is directed and satisfies the coequalizing condition. However, say that φ_{i} : 𝒢_{i} → ℱ (i = 1, 2) are any two objects of C/ℱ; then obviously there is a unique morphism 𝒢_{1} ⊕ 𝒢_{2} → ℱ whose restriction to 𝒢_{i} agrees with φ_{i} for i = 1, 2.

Next, suppose that β_{1}, β_{2} : 𝒢 → 𝒢′ are two morphisms of finitely presented 𝒪_{X}-modules and ψ : 𝒢′ → ℱ an object of C/ℱ such that ψ ∘ β_{1} = ψ ∘ β_{2}. Set 𝒢′′ := Coker (β_{1} − β_{2}); it is easily seen that 𝒢′′ is a finitely presented 𝒪_{X}-module and ψ factors through a unique morphism 𝒢′′ → ℱ, whence the lemma. □

Proposition 10.3.31. With the notation of (10.3.29), suppose that X is quasi-compact and quasi-separated. Then, for any quasi-coherent 𝒪_{X}-module ℱ, the induced map

colim_{C/ℱ}ι_{ℱ} → ℱ

is an isomorphism.

Proof. Let ℱ′ be such colimit; clearly there is a natural map ℱ′ → ℱ, and we have to show that it is an isomorphism. To this aim, we can check on the stalk over the points x ∈ X, hence we come down to showing :

Claim 10.3.32. Let ℱ be any quasi-coherent 𝒪_{X}-module. Then :

(i) For every s ∈ ℱ_{x} there exist a finitely presented quasi-coherent 𝒪_{X}-module 𝒢, a morphism φ : 𝒢 → ℱ and t ∈ 𝒢_{x} such that φ_{x}(t) = s.

(ii) For every map φ : 𝒢 → ℱ with 𝒢 finitely presented and quasi-coherent, and every s ∈ Ker φ_{x}, there exists a commutative diagram of quasi-coherent 𝒪_{X}-modules :

[figure: 𝒢; φ; ψ; ℋ; ℱ]

with ℋ finitely presented and ψ_{x}(s) = 0.

Proof of the claim. (i): Let U ⊂ X be an open subset such that s extends to a section s_{U} ∈ ℱ (U); we deduce a map φ_{U} : 𝒪_{U} → ℱ_{|U} by the rule a ↦ a · s_{U} for every a ∈ 𝒪_{U}(U). In view of lemma 10.3.27(i), the pair (φ_{U}, 𝒪_{U}) extends to a pair (φ, 𝒢) with the sought properties.

(ii): We apply (i) to the quasi-coherent 𝒪_{X}-module Ker φ, to find a finitely presented quasicoherent 𝒪_{X}-module 𝒢′, a morphism β : 𝒢′ → Ker φ and t ∈ ℋ_{x} such that β_{x}(t) = s. Then we let ψ : 𝒢 → ℋ := Coker(𝒢′ ⟶^{β} Ker φ → 𝒢) be the natural map. By construction, ℋ is finitely presented, ψ_{x}(s) = 0 and clearly φ factors through ψ. □

10.3.33. Let X be a coherent scheme, U ⊂ X a quasi-compact open subset, a, b ∈ ℤ any two integers with a ≤ b, and ℋ ^{•} any object of D^{[a,b]}(𝒪_{X}-Mod)_{qcoh} such that ℋ_{|U}^{•} lies in D^{[a,b]}(𝒪_{U}-Mod)_{coh}. Let also ℱ_{i} ⊂ H^{i}ℋ ^{•} be a quasi-coherent 𝒪_{X}-submodule of finite type, for every i = a, …, b.

Proposition 10.3.34. In the situation of (10.3.33), the following holds :

(i) There exists an object 𝒢^{•} of D^{[a,b]}(𝒪_{X}-Mod)_{coh}, with a morphism φ^{•} : 𝒢^{•} → ℋ ^{•} in D(𝒪_{X}-Mod)_{qcoh} such that :

(a) φ_{|U}^{•} is an isomorphism.

(b) ℱ_{i} ⊂ Im H^{i}φ^{•} for every i = a, …, b. (ii) If X is noetherian, we can find φ^{•} as in (i) such that H^{i}φ^{•} is a monomorphism, for

every i = a, …, b.

Proof. To start out, for any quasi-coherent 𝒪_{X}-module ℱ such that ℱ_{|U} is a coherent 𝒪_{U}- module, define the category C/ℱ as in (10.3.29), and consider the full subcategory C′/ℱ (resp. C′′/ℱ) of C/ℱ whose objects are all the maps φ : 𝒢 → ℱ such that φ_{|U} is an isomorphism (resp. an epimorphism).

Claim 10.3.35. C′/ℱ and C′′/ℱ are filtered categories.

Proof of the claim. The proof of lemma 10.3.30 applies verbatim to show that both C′/ℱ and C′′/ℱ satisfy the coequalizing condition of definition 1.2.19(v), and it also proves that C′′/ℱ is directed. Moreover, lemma 10.3.27(i) says that the sets of objects of these two categories are non-empty, which already implies that C′′/ℱ is filtered, by remark 1.2.21(i). By the same token, it remains only to check that C′/ℱ satisfies the coequalizing condition. Hence, let φ_{i} : 𝒢_{i} → ℱ (i = 1, 2) be any two objects of C′/ℱ, and set 𝒢′ := 𝒢_{1} ×_{ℱ} 𝒢_{2}; then 𝒢′ is a quasi-coherent 𝒪_{X}-module, and 𝒢′_{|U} is isomorphic to ℱ_{|U}, especially it is coherent. Then pick any object ψ : 𝒢′′ → 𝒢′ of the category C′′/𝒢′, and denote by ψ_{i} : 𝒢′′ → 𝒢_{i} (i = 1, 2) the two morphisms induced by ψ; by construction we have φ_{1}∘ψ_{1} = φ_{2}∘ψ_{2}. Hence, set 𝒢 := 𝒢_{1}⨿_{𝒢′′}𝒢_{2}; we get a unique morphism φ : 𝒢 → ℱ such that both φ_{1} and φ_{2} factor through φ and φ_{|U} is an isomorphism. Since 𝒢 is a finitely presented 𝒪_{X}-module, we are done. ♢

Claim 10.3.36. With the foregoing notation, the following holds :

(i) C′/ℱ is a cofinal subcategory of C/ℱ.

(ii) For every object K^{•} of D^{b}(𝒪_{X}-Mod)_{coh} and every c ∈ ℤ, the natural map

(10.3.37)  colim_{C/ℱ}Hom_{D(𝒪_{X}-Mod)} (K^{•}, ι_{ℱ} [c]) → Hom_{D(𝒪_{X}-Mod)} (K^{•}, ℱ [c])

is an isomorphism (notation of (10.3.29)).

Proof of the claim. (i): Since U is quasi-compact, it is easily seen that C′′/ℱ is cofinal in C/ℱ, so we are reduced to checking that C′/ℱ is cofinal in C′′/ℱ. Hence, let ψ : 𝒢′ → ℱ be any object of C′′/ℱ; since 𝒪_{X} is coherent, 𝒦 := Ker ψ_{|U} is a coherent 𝒪_{U}-module, therefore we may extend 𝒦 to a coherent 𝒪_{X}-module 𝒦 ′ and the identity map of 𝒦 to a morphism ψ′ : 𝒦 ′ → Ker ψ of 𝒪_{X}-modules (lemma 10.3.27(i)). Set 𝒢 := 𝒢′/ψ′(𝒦 ′); the induced map 𝒢 → ℱ is an object of C′/ℱ, whence the contention.

(ii): We have two spectral sequences

E_{2}^{pq} := colim_{C/ℱ}R^{p}Γ ∘ R^{q}ℋ om_{𝒪_{X}}^{•} (K^{•}, ι_{ℱ} [c]) ⇒ colim_{C/ℱ}Hom_{D(𝒪_{X}-Mod)} (K^{•}, ι_{ℱ} [c + p + q])
F_{2}^{pq} := R^{p}Γ ∘ R^{q}ℋ om_{𝒪_{X}}^{•} (K^{•}, ℱ [c]) ⇒ Hom_{D(𝒪_{X}-Mod)} (K^{•}, ℱ [c + p + q])

and a morphism of spectral sequences E_{•}^{••} → F_{•}^{••}, such that (10.3.37) is a morphism of filtered abelian groups, for the two finite filtrations induced by these spectral sequences on their abutments. Since the functors R^{p}Γ commute with filtered colimits, we deduce that it suffices to show that the natural morphism

colim_{C/ℱ}R^{q}ℋ om_{𝒪_{X}}^{•} (K^{•}, ι_{ℱ} [c])) → R^{q}ℋ om_{𝒪_{X}}^{•} (K^{•}, ℱ [c])

is an isomorphism for every q ∈ ℤ. Then, a standard dévissage argument further reduces to the case where K^{•} is concentrated in a single degree, so we come down to checking that the functor ℱ ↦ R^{q}ℋ om_{𝒪_{X}}^{•} (𝒢 , ℱ) commutes with filtered colimits of quasi-coherent 𝒪_{X}-modules, for every coherent 𝒪_{X}-module 𝒢 and every q ∈ ℤ. To this aim we may assume that X is affine, in which case – since 𝒪_{X} is coherent – 𝒢 admits a resolution ℒ^{•} consisting of free 𝒪_{X}-modules of finite rank; the latter are acyclic for the functor ℋ om_{𝒪_{X}} , so we are left with the assertion that the functor ℱ ↦ H^{q}ℋ om_{𝒪_{X}} (ℒ^{•}, ℱ) commutes with filtered colimits, which is clear. ♢

Claim 10.3.38. The proposition holds if a = b.

Proof of the claim. From proposition 10.3.31 and claim 10.3.36(i) we deduce that, for every x ∈ X we may find an object φ : 𝒢 → H^{a}ℋ such that ℱ_{a,x} ⊂ Im φ_{x}, and therefore ℱ_{a|U_{x}} ⊂ Im φ_{|U_{x}} for some open neighborhood U_{x} of x in X. Since X is quasi-compact and C/ℱ is filtered, assertion (i) follows easily. Next, suppose that a morphism φ : 𝒢 → H^{a}ℋ has already been exhibited that fulfills conditions (i.a) and (i.b), and set 𝒢′ := Im φ; to show assertion (ii), it suffices to remark that if X is noetherian, 𝒢′ is still a coherent 𝒪_{X}-module. ♢

We proceed now by induction on b − a, the case where a = b having been covered by claim 10.3.38. Thus, suppose that a < b, and that the proposition is already known for every object of D^{[a+1,b]}(𝒪_{X}-Mod)_{qcoh} fulfilling the stated condition. We have a distinguished triangle

ℱ [−a] → ℋ ^{•} → τ^{≥a+1}ℋ ^{•} ⟶^{α^{•}} ℱ [1 − a] with ℱ := H^{a}ℋ ^{•}

in D(𝒪_{X}-Mod)_{qcoh} (where τ^{≥a+1} denotes the truncation functor : see remark 7.3.14(iv)). By inductive assumption, we may find an object 𝒢′^{•} of D^{[a+1,b]}(𝒪_{X}-Mod)_{coh} and a morphism φ′^{•} : 𝒢′^{•} → τ^{≥a+1}ℋ ^{•} in D(𝒪_{X}-Mod)_{qcoh} which restricts to an isomorphism on U and such that ℱ_{i} ⊂ Im H^{i}φ′^{•} for every i = a + 1, …, b. There follows a morphism of distinguished triangles :

[figure: ℱ [−a]; ℋ ′^{•}; β^{•}; 𝒢′^{•}; φ′^{•}; ℱ [1 − a]; ℱ [−a]; ℋ ^{•}; τ^{≥a+1}ℋ ^{•}; α^{•}; ℱ [1 − a]]

for some object ℋ ′^{•} of D^{[a,b]}(𝒪_{X}-Mod)_{qcoh}, and clearly β^{•} restricts to an isomorphism on U and H^{a}β^{•} is an isomorphism. We may then replace ℋ ^{•} by ℋ ′^{•}, and assume that τ^{≥a+1}ℋ ^{•} lies in D^{[a+1,b]}(𝒪_{X}-Mod)_{coh}.

By claim 10.3.36, we may find an object ψ : 𝒢 → ℱ of C′/ℱ such that α^{•} factors through ψ[1 − a] and a morphism ω^{•} : τ^{≥a+1}ℋ ^{•} → 𝒢 [1 − a] in D(𝒪_{X}-Mod)_{coh}. Moreover, by claim 10.3.38we may assume that ℱ_{a} ⊂ Im ψ, and also that ψ is a monomorphism, if X is noetherian. Then Cone ω^{•}[−1] lies in D^{[a,b]}(𝒪_{X}-Mod)_{coh}, and the induced morphism Cone ω^{•}[−1] → ℋ ^{•} will do. □

Corollary 10.3.39. Let X be a coherent scheme, U ⊂ X a quasi-compact open subset. Then the induced functor

D^{[a,b]}(𝒪_{X}-Mod)_{coh} → D^{[a,b]}(𝒪_{U}-Mod)_{coh}

is essentially surjective, for every a, b ∈ ℤ with a ≤ b.

Proof. Let j : U → X be the open immersion, ℱ^{•} an object of D^{[a,b]}(𝒪_{U}-Mod)_{coh}. By [61, Ch.III, Prop.1.4.10, Cor.1.4.12] and a standard spectral sequence argument, it is easily seen that

ℋ ^{•} := τ^{≥a} ∘ τ^{≤b}Rj_{∗}ℱ^{•}

lies in D^{[a,b]}(𝒪_{X}-Mod)_{qcoh} and ℋ_{|U}^{•} = ℱ^{•}. Then the assertion follows immediately from proposition 10.3.34(i). □

10.4. Depth and cohomology with supports. This section introduces and studies local cohomology and the closely related notion of depth, in the context of arbitrary schemes (whereas the usual references restrict to the case of locally noetherian schemes).

10.4.1. To begin with, let (X, 𝒜) be any ringed topological space, i : Z → X a closed immersion, set U := X \Z and denote by j : U → X the resulting open immersion. One defines the functor

\underline{Γ}_{Z} : 𝒜 -Mod → 𝒜 -Mod ℱ ↦ Ker (ℱ → j_{∗}j^{∗}ℱ)

as well as its composition with the global section functor

Γ_{Z} : 𝒜 -Mod → 𝒜 (X)-Mod ℱ ↦ Γ(X, \underline{Γ}_{Z}ℱ).

It is clear that \underline{Γ}_{Z} and Γ_{Z} are left exact functors, hence they give rise to right derived functors

R\underline{Γ}_{Z} : D^{+}(𝒜 -Mod) → D^{+}(𝒜 -Mod) RΓ_{Z} : D^{+}(𝒜 -Mod) → D^{+}(𝒜 (X)-Mod)

such that the natural morphism \underline{Γ}_{Z}ℱ → R^{0}\underline{Γ}_{Z}ℱ is an isomorphism, for every 𝒜 -module ℱ (and likewise for R^{0}Γ_{Z}). Moreover, suppose that ℱ → ℐ^{•} is a resolution of ℱ by a complex of injective 𝒜 -modules; since injective sheaves are flabby (lemma 10.1.5(v)), we obtain a short exact sequence of complexes

0 → \underline{Γ}_{Z}ℐ^{•} → ℐ^{•} → j_{∗}j^{∗}ℐ^{•} → 0

whence a natural exact sequence of 𝒜 -modules :

(10.4.2)  0 → \underline{Γ}_{Z}ℱ → ℱ → j_{∗}j^{∗}ℱ → R^{1}\underline{Γ}_{Z}ℱ → 0  for every 𝒜 -module ℱ

and natural isomorphisms :

(10.4.3)  R^{q−1}j_{∗}j^{∗}ℱ →^{∼} R^{q}\underline{Γ}_{Z}ℱ  for all q > 1.

Lemma 10.4.4. In the situation of (10.4.1), let ℬ → 𝒜 be any morphism of sheaves of rings on X. The following holds :

(i) Every flabby 𝒜 -module is \underline{Γ}_{Z}-acyclic.

(ii) The functor R\underline{Γ}_{Z} commutes with the forgetful functor D^{+}(𝒜 -Mod) → D^{+}(ℬ-Mod). (iii) Suppose that X is locally coherent and quasi-separated, and that Z is a constructible

closed subset of X. Then :

(a) Every qc-flabby 𝒜 -module is \underline{Γ}_{Z}-acyclic.

(b) For every i ∈ ℕ the functor R^{i}\underline{Γ}_{Z} commutes with filtered colimits of 𝒜 -modules.

Proof. (i): First of all, if ℱ is flabby on X, the natural morphism ℱ → j_{∗}j^{∗}ℱ is obviously an epimorphism; together with (10.4.2), this implies that R^{1}\underline{Γ}_{Z}ℱ = 0 when ℱ is flabby. Next, (10.4.3), together with the fact that flabby 𝒜 -modules are acyclic for direct image functors such as j_{∗} (lemma 10.1.5(ii)), implies that R^{i}\underline{Γ}_{Z}ℱ = 0 also for i > 1, whence the contention.

Assertion (iii.a) is checked in the same way : first one notices that, under the stated assumptions, the morphism ℱ → j_{∗}j^{∗}ℱ is an epimorphism also for all qc-flabby sheaves on X, which again gives the vanishing of R^{1}\underline{Γ}_{Z}ℱ; for the higher derived functors one then appeals to (10.4.3) and lemma 10.1.5(iv).

(iii.b): Suppose first that i > 1; in this case, the assertion follows from (10.4.3) and the following more general :

Claim 10.4.5. In the situation of (iii), the functor R^{i}j_{∗} commutes with filtered colimits, for every i ∈ ℕ.

Proof of the claim. The claim can be checked at the stalks at the points x ∈ X, hence we may assume that X is quasi-compact, in which case the same holds for U. Now, on the one hand, R^{q}j_{∗}j^{∗}ℱ is the sheaf associated with the presheaf given by the rule : U′ ↦ H^{q}(U ∩ U′, ℱ) for every open subset U′ ⊂ X, so (R^{q}j_{∗}j^{∗}ℱ)_{x} is also the stalk at x of this presheaf. On the other hand, x admits a fundamental system of open neighborhoods consisting of quasi-compact open subsets, and if U′ is such a neighborhood of x, the subset U′ ∩ U is also quasi-compact, and the functor ℱ ↦ H^{q}(U ∩ U′, ℱ) commutes with filtered colimits, by proposition 10.1.10(ii). The claim follows by combining these two observations. ♢

In case i ≤ 1, we argue similarly with (10.4.2), which reduces to checking that both functors ℱ ↦ ℱ_{x} and ℱ ↦ (j_{∗}j^{∗}ℱ)_{x} commute with filtered colimits; this is trivial for the former functor, and for the latter it is a special case of claim 10.4.5.

Lastly, (ii) is an immediate consequence of (i) and remark 7.3.31(vi), since any injective 𝒜 -module is a flabby ℬ-module (lemma 10.1.5(v)). □

10.4.6. In the situation of (10.4.1), let X′ be the amalgamated sum in the following cocartesian diagram of topological spaces :

\xymatrix{U \ar[r]^{j} \ar[d]_{j} & X \ar[d]^{j′_{1}} \\ X \ar[r]^{j′_{2}} & X′}

and let π : X′ → X be the unique continuous map with π ∘ j′_{i} = 1_{X} for i = 1, 2. We notice :

Proposition 10.4.7. In the situation of (10.4.6), let also ℱ be any sheaf on X. Then we have : (i) If ℱ is flabby, the same holds for π^{∗}ℱ. (ii) If moreover ℱ is an 𝒜 -module, the following holds :

(a) We have a natural isomorphism in D^{+}(𝒜 (X)-Mod) :

RΓℱ ⊕ RΓ_{Z}ℱ →^{∼} RΓ(π^{∗}ℱ).

(b) If X is coherent and Z is constructible in X, then R^{i}Γ_{Z}ℱ = 0 for every i > dim X.

(c) If X is spectral and Z is constructible in X, define d_{ℱ} as in theorem 10.1.35(ii). Then

R^{i}Γ_{Z}ℱ = 0 for every i > d_{ℱ} .

Proof. (i): Set V_{i} := j′_{i}(X) for i = 1, 2; then X′ admits the open covering X′ = V_{1} ∪ V_{2}, and the restriction of π^{∗}ℱ is flabby on V_{1} and V_{2}. Then the assertion follows from lemma 10.1.2.

(ii.a): For i = 1, 2 and every 𝒜 -module 𝒢, we have a natural isomorphism of 𝒜 (X)-modules

ψ_{𝒢 ,i} : Γ(X, ℱ) →^{∼} Γ(V_{i}, 𝒢).

Moreover, for every (σ, τ) ∈ Γ𝒢 ⊕Γ_{Z}𝒢 there exists a unique section φ_{𝒢} (σ, τ) ∈ Γ(π^{∗}𝒢) whose restriction to V_{1} agrees with ψ_{𝒢 ,1}(σ) and whose restriction to V_{2} agrees with ψ_{𝒢_{2}}(σ + τ) (details left to the reader). The rule (σ, τ) ↦ φ_{𝒢} (σ, τ) yields a natural isomorphism

φ_{𝒢} : Γ𝒢 ⊕ Γ_{Z}𝒢 →^{∼} Γ(π^{∗}𝒢).

Now, let ℱ → 𝒢^{•} be a resolution of ℱ by a complex of injective 𝒜 -modules; by lemma 10.1.5(v), the complex π^{∗}𝒢^{•} is a resolution of π^{∗}ℱ by flabby π^{∗}𝒜 -modules. Combining with lemma 10.4.4(i) and remark 7.3.31(vi), we deduce that φ_{𝒢^{•}} : Γ𝒢^{•} ⊕ Γ_{Z}𝒢^{•} →^{∼} Γ(π^{∗}𝒢^{•}) is the sought natural isomorphism.

Lastly, if X is coherent (resp. spectral) and Z is constructible, the open subsets V_{1} and V_{2} are coherent (resp. spectral) and retro-compact in X′; by lemma 8.1.15(iv), in this case X′ is coherent (resp. X′ is spectral, and it is easily seen that d_{π^{∗}ℱ} = d_{ℱ} ). Then (ii.b) and (ii.c) follow directly from (ii.a) and theorem 10.1.35. □

10.4.8. In the situation of (10.1.8), suppose that the maps g and g_{λ}, for every λ ∈ Ob(Λ) are open immersions, and set

Z := X \Y  Z_{λ} := X_{λ}\Y_{λ}  for every λ ∈ Ob(Λ).

Consider also a compatible system (𝒢_{λ} | λ ∈ Ob(Λ)) of sheaves on X_{•}, where 𝒢_{λ} is a ℤ_{X_{λ}}- module for every λ ∈ Λ, with transition maps

φ_{u}^{−1}𝒢_{λ} → 𝒢_{μ} for every morphism u : μ → λ in Λ

and set

𝒢 := colim_{λ∈Ob(Λ)} φ_{λ}^{∗}𝒢_{λ}.

Notice that for every λ ∈ Ob(Λ) and every morphism u : μ → λ in Λ we have

φ_{u}^{−1}Z_{λ} ⊂ Z_{μ} and φ_{λ}^{−1}Z_{λ} ⊂ Z.

There follows a system of natural morphisms

τ_{u} : φ_{u}^{∗}\underline{Γ}_{Z_{λ}}𝒢_{λ} → \underline{Γ}_{Z_{μ}}𝒢_{μ} and τ_{λ} : φ_{λ}^{∗}\underline{Γ}_{Z_{λ}}𝒢_{λ} → \underline{Γ}_{Z}𝒢

amounting to a co-cone τ_{•} with vertex \underline{Γ}_{Z}𝒢.

Proposition 10.4.9. In the situation of (10.4.8), the co-cone τ_{•} induces a natural isomorphism

colim_{λ∈Ob(Λ)} φ_{λ}^{∗} R^{i}\underline{Γ}_{Z_{λ}}𝒢_{λ} →^{∼} R^{i}\underline{Γ}_{Z}𝒢 for every i ∈ ℕ.

Proof. From proposition 10.1.10(ii) we deduce a natural isomorphism

colim_{λ∈Ob(Λ)} φ_{λ}^{∗} \underline{Γ}_{Z_{λ}}𝒢_{λ} = colim_{λ∈Ob(Λ)} φ_{λ}^{∗} Ker (𝒢_{λ} → g_{λ∗}g_{λ}^{∗}𝒢_{λ})
→^{∼} Ker (𝒢 → g_{∗}g^{∗}𝒢) = \underline{Γ}_{Z}𝒢 .

Now, let us choose a compatible system of injective resolutions 𝒢_{λ} → ℐ_{λ}^{•}, so that the transition maps for the system (𝒢_{λ} | λ ∈ Ob(Λ)) extend to a system of transition morphisms of complexes (ℐ_{λ}^{•} | λ ∈ Ob(Λ)). On the one hand, claim 10.1.11says that

ℐ^{•} := colim_{λ∈Ob(Λ)} φ_{λ}^{∗}ℐ_{λ}^{•}

is a qc-flabby resolution of 𝒢. On the other hand, the foregoing yields a natural isomorphism

colim_{λ∈Ob(Λ)} φ_{λ}^{∗}R^{i}\underline{Γ}_{Z_{λ}}𝒢_{λ} →^{∼} H^{i}\underline{Γ}_{Z}ℐ^{•}.

Then the assertion follows from lemma 10.4.4(iii.a). □

Corollary 10.4.10. Let (X, 𝒜) be a ringed topological space with X locally spectral, Z ⊂ X a constructible closed subset, x ∈ Z any point, and denote by j_{x} : X(x) → X the inclusion map (notation of definition 8.1.44(iii)). We have a natural isomorphism of functors

j_{x}^{∗}R\underline{Γ}_{Z} →^{∼} R\underline{Γ}_{Z(x)} in D^{+}(j_{x}^{∗}𝒜 -Mod) .

Proof. We may assume that X is spectral, in which case X(x) is the limit of the cofiltered system of all quasi-compact open neighborhoods of x in X, and if U is any such neighborhood, the inclusion map U → X is quasi-compact and quasi-separated. The sought morphism φ^{•} is then deduced from a co-cone constructed as in (10.4.8), and it suffices to check that H^{i}φ^{•} is an isomorphism on the underlying ℤ_{X(x)}-modules for every i ∈ ℤ, so the assertion is reduced to proposition 10.4.9. □

10.4.11. In the situation of (10.4.1), notice the natural isomorphism of 𝒜 -modules :

ℋ om_{ℤ}(i_{∗}ℤ_{Z}, ℱ) →^{∼} \underline{Γ}_{Z}ℱ for every 𝒜 -module ℱ

whence a natural isomorphism of 𝒜 (X)-modules :

Hom_{ℤ}(i_{∗}ℤ_{Z}, ℱ) →^{∼} Γ_{Z}ℱ for every 𝒜 -module ℱ .

Now, let 𝔘 := (U_{i} | i ∈ I) be any open covering of U, indexed by a set I, and fix any total ordering on I. By proposition 10.2.6(iv), we get natural isomorphisms in D^{+}(𝒜 -Mod) and D^{+}(𝒜 (X)-Mod)

Rℋ om_{ℤ}^{•}(R_{•}^{alt}(𝔘)[1], K^{•}) →^{∼} R\underline{Γ}_{Z}K^{•} RHom_{ℤ}^{•}(R_{•}^{alt}(𝔘)[1], K^{•}) →^{∼} RΓ_{Z}K^{•}

for every bounded below complex K^{•} of 𝒜 -modules. These isomorphisms allow us to compute RΓ_{Z}K^{•} as a Čech cohomology functor, as follows. Clearly, the rule : ℱ ↦ C_{alt}^{•}(𝔘, ℱ) defines a functor 𝒜 -Mod → C(𝒜 (X)-Mod) (notation of definition 10.2.2(ii)). Now, pick any resolution K^{•} →^{∼} ℐ^{•} by a bounded below complex of injective 𝒜 -modules; the foregoing yields a natural isomorphism :

RΓ_{Z}K^{•} →^{∼} Tot C_{alt}^{•}(𝔘, ℐ^{•})[−1] in D^{+}(𝒜 (X)-Mod) .

10.4.12. We define a bifunctor

ℋ om_{Z}^{•} := \underline{Γ}_{Z} ∘ ℋ om_{𝒜}^{•} : C(𝒜 -Mod) × C(𝒜 -Mod)^{o} → C(𝒜 -Mod).

The bifunctor ℋ om_{Z}^{•} admits a right derived functor :

Rℋ om_{Z}^{•} : D^{+}(𝒜 -Mod) × D(𝒜 -Mod)^{o} → D(𝒜 -Mod).

The construction can be outlined as follows. First, for a fixed complex K_{•} of 𝒜 -modules, one can consider the right derived functor of the functor 𝒢 ↦ ℋ om_{Z}^{•}(K_{•}, 𝒢), which is denoted Rℋ om_{Z}^{•}(K_{•}, −) : D^{+}(𝒜 -Mod) → D(𝒜 -Mod). Next, one verifies that every quasi-isomorphism K_{•} → K′_{•} induces an isomorphism of functors : Rℋ om_{Z}^{•}(K′_{•}, −) →^{∼} Rℋ om_{Z}^{•}(K_{•}, −), hence the natural transformation K_{•} ↦ Rℋ om_{Z}^{•}(K_{•}, −) factors through D(𝒜 -Mod), and this is the sought bifunctor Rℋ om_{Z}^{•}. In case Z = X, one recovers the functor Rℋ om_{𝒜}^{•} of (10.1.13).

Lemma 10.4.13. In the situation of (10.4.1), the following holds :

(i) The functor \underline{Γ}_{Z} is right adjoint to i_{∗}i^{∗}. More precisely, for any two 𝒜 -modules ℱ and 𝒢 we have natural isomorphisms :

(a) ℋ om_{𝒜} (ℱ , \underline{Γ}_{Z}𝒢) →^{∼} ℋ om_{𝒜} (i_{∗}i^{∗}ℱ , 𝒢).

(b) Rℋ om_{𝒜}^{•} (ℱ , R\underline{Γ}_{Z}𝒢) →^{∼} Rℋ om_{𝒜}^{•} (i_{∗}i^{∗}ℱ , 𝒢).

(ii) If ℱ is an injective (resp. flabby) 𝒜 -module on X, then the same holds for \underline{Γ}_{Z}ℱ.

(iii) There are natural isomorphisms of bifunctors :

Rℋ om_{Z}^{•} →^{∼} R\underline{Γ}_{Z} ∘ Rℋ om_{𝒜}^{•} →^{∼} Rℋ om_{𝒜}^{•} (−, R\underline{Γ}_{Z}−)

(iv) If W ⊂ X is any other closed subset, there is a natural isomorphism of functors :

R\underline{Γ}_{W ∩Z} →^{∼} R\underline{Γ}_{W} ∘ R\underline{Γ}_{Z}.

(v) If f : (Y, ℬ) → (X, 𝒜) is any morphism of ringed spaces, there is a natural isomorphism of functors :

Rf_{∗} ∘ R\underline{Γ}_{f^{−1}Z} →^{∼} R\underline{Γ}_{Z} ∘ Rf_{∗} : D^{+}(ℬ-Mod) → D^{+}(𝒜 -Mod).

(vi) There is a natural isomorphism of bifunctors on D^{+}(𝒜 -Mod)^{o} × D^{+}(𝒜 -Mod) :

Rℋ om_{Z}^{•}(R\underline{Γ}_{Z}−, −) →^{∼} Rℋ om_{𝒜}^{•} (R\underline{Γ}_{Z}−, −).

Proof. (i): To establish the isomorphism (a), one uses the short exact sequence

0 → j_{!}j^{∗}ℱ → ℱ → i_{∗}i^{∗}ℱ → 0

to show that any map ℱ → \underline{Γ}_{Z}𝒢 factors through ℋ := i_{∗}i^{∗}ℱ. Conversely, since j_{∗}j^{∗}ℋ = 0, it is clear that every map ℋ → 𝒢 must factor through \underline{Γ}_{Z}𝒢. The isomorphism (b) is derived easily from (a) and (ii).

(iii): According to [163, Th.10.8.2], the first isomorphism in (iii) is deduced from lemmata 10.4.4(i) and 10.1.16(ii). The second isomorphism in (iii) follows from [163, Th.10.8.2], and assertion (ii) concerning injective sheaves.

(ii): Suppose first that ℱ is injective; let 𝒢_{1} → 𝒢_{2} a monomorphism of 𝒜 -modules and φ : 𝒢_{1} → \underline{Γ}_{Z}ℱ any 𝒜 -linear map. By (i) we deduce a map i_{∗}i^{∗}𝒢_{1} → ℱ, which extends to a map i_{∗}i^{∗}𝒢_{2} → ℱ, by injectivity of ℱ. Finally, (i) again yields a map 𝒢_{2} → \underline{Γ}_{Z}ℱ extending φ.

Next, if ℱ is flabby, let V ⊂ X be any open subset, and s ∈ \underline{Γ}_{Z}ℱ (V); since s_{|V \Z} = 0, we can extend s to a section s′ ∈ \underline{Γ}_{Z}ℱ (V ∪ (X \Z)). Since ℱ is flabby, s′ extends to a section s′′ ∈ ℱ (X); however, by construction s′′ vanishes on the complement of Z.

(iv): Clearly \underline{Γ}_{W ∩Z} = \underline{Γ}_{W} ∘ \underline{Γ}_{Z}, so the claim follows easily from (ii) and [163, Th.10.8.2]. (v): Since flabby ℬ-modules are acyclic for f_{∗} and \underline{Γ}_{f^{−1}Z} (lemmata 10.1.5(ii) and 10.4.4(i)),

one can apply (ii) and [163, Th.10.8.2] to identify both functors with R(\underline{Γ}_{Z} ∘ f_{∗}).

(vi): Let I^{•} and J^{•} be two bounded below complexes of injective 𝒜 -modules. Clearly any morphism \underline{Γ}_{Z}I^{•} → J^{•} vanishes on U, whence the assertion. □

Definition 10.4.14. Let X be a topological space, and K^{•} ∈ D^{+}(ℤ_{X}-Mod).

(i) The depth of K^{•} along Z is

depth_{Z}K^{•} := sup\{n ∈ ℤ | R^{i}\underline{Γ}_{Z}K^{•} = 0 for all i < n\} ∈ ℤ ∪ \{+∞\}.

From lemma 10.4.13(ii,iv) and the Grothendieck spectral sequence ([126, Th.5.8.3]), we see that

(10.4.15)  depth_{W} K^{•} ≥ depth_{Z}K^{•}  whenever W ⊂ Z.

(ii) Let Φ := (Z_{λ} | λ ∈ Λ) be a family of closed subsets of X, and suppose that Φ is cofiltered by inclusion. Due to (10.4.15) it is reasonable to define the depth of K^{•} along Φ as

depth_{Φ}K^{•} := sup {depth_{Z_{λ}}K^{•} | λ ∈ Λ} ∈ ℤ ∪ {+∞}.

10.4.16. Let now X be a scheme and i : Z → X a closed immersion of schemes, such that Z is constructible in X. Then the open immersion j : X\Z → X is quasi-compact and separated, hence the functors R^{q}j_{∗} preserve the subcategories of quasi-coherent modules, for every q ∈ ℕ ([61, Ch.III, Prop.1.4.10]). In light of (10.4.3) we deduce that R^{q}\underline{Γ}_{Z} restricts to a functor :

R^{q}\underline{Γ}_{Z} : 𝒪_{X}-Mod_{qcoh} → 𝒪_{X}-Mod_{qcoh} for every q ∈ ℕ.

Lemma 10.4.17. In the situation of (10.4.16), the following holds :

(i) The functor R\underline{Γ}_{Z} restricts to a triangulated functor :

R\underline{Γ}_{Z} : D^{+}(𝒪_{X}-Mod)_{qcoh} → D^{+}(𝒪_{X}-Mod)_{qcoh}

(notation of (10.3.11)). (ii) Let f : Y → X be an affine morphism of schemes, K^{•} any object of D^{+}(𝒪_{Y} -Mod)_{qcoh}.

Then we have the identity :

depth_{f^{−1}Z}K^{•} = depth_{Z}Rf_{∗}K^{•}.

(iii) Let f : Y → X be a flat morphism of schemes, K^{•} any object of D^{+}(𝒪_{X}-Mod)_{qcoh}. Then the natural map

f^{∗}R\underline{Γ}_{Z}K^{•} → R\underline{Γ}_{f^{−1}Z}f^{∗}K^{•}

is an isomorphism in D^{+}(𝒪_{Y} -Mod)_{qcoh}. Especially, we have

depth_{f^{−1}Z}f^{∗}K^{•} ≥ depth_{Z}K^{•}

and if f is faithfully flat, the inequality is actually an equality. (iv) Let K^{•} be any object of D^{+}(𝒪_{X}-Mod), and ℱ a flat quasi-coherent 𝒪_{X}-module.

Then the natural map

ℱ ⊗_{𝒪_{X}} R\underline{Γ}_{Z}K^{•} → R\underline{Γ}_{Z}(ℱ ⊗_{𝒪_{X}} K^{•})

is an isomorphism in D(𝒪_{X}-Mod).

Proof. (i): Indeed, if K^{•} is an object of D^{+}(𝒪_{X}-Mod)_{qcoh}, we have a spectral sequence :

E_{2}^{pq} := R^{p}\underline{Γ}_{Z}H^{q}K^{•} ⇒ R^{p+q}\underline{Γ}_{Z}K^{•}.

(This spectral sequence is obtained from a Cartan-Eilenberg resolution of K^{•} : see e.g. [163, §5.7].) Hence R^{•}\underline{Γ}_{Z}K^{•} admits a finite filtration whose subquotients are quasi-coherent; then the lemma follows from [61, Ch.III, Prop.1.4.17].

(ii): To start with, a spectral sequence argument as in the foregoing shows that Rf_{∗} restricts to a functor D^{+}(𝒪_{Y} -Mod)_{qcoh} → D^{+}(𝒪_{X}-Mod)_{qcoh}, and moreover we have natural identifications : R^{i}f_{∗}K^{•} →^{∼} f_{∗}H^{i}K^{•} for every i ∈ ℤ and every object K^{•} of D^{+}(𝒪_{Y} -Mod)_{qcoh}. In view of lemma 10.4.13(v), we derive natural isomorphisms : R^{i}\underline{Γ}_{Z}Rf_{∗}K^{•} →^{∼} Rf_{∗}R^{i}\underline{Γ}_{f^{−1}Z}K^{•} for every object K^{•} of D^{+}(𝒪_{Y} -Mod)_{qcoh}. The assertion follows easily.

(iii): A spectral sequence argument as in the proof of (i) allows us to assume that K^{•} = ℱ [0] for some quasi-coherent 𝒪_{X}-module ℱ. In this case, let j_{X} : X\Z → X and j_{Y} : Y \f^{−1}Z → Y be the open immersions; considering the exact sequences f^{∗}(10.4.2) and f^{∗}(10.4.3), we reduce to showing that the natural map f^{∗}Rj_{X∗}j_{X}^{∗}ℱ → Rj_{Y ∗}j_{Y}^{∗} f^{∗}ℱ is an isomorphism in D^{+}(𝒪_{Y} -Mod)_{qcoh}. The latter assertion follows from corollary 10.3.8.

(iv): The map in question is obtained as follows. We may assume that K^{•} is a complex of injective 𝒪_{X}-modules, and let i : \underline{Γ}_{Z}K^{•} → K^{•} be the inclusion map; clearly ℱ ⊗_{𝒪_{X}} i factors through the inclusion map \underline{Γ}_{Z}(ℱ ⊗_{𝒪_{X}} K^{•}) → ℱ ⊗_{𝒪_{X}} K^{•}, whence the sought map ω_{ℱ} : ℱ ⊗_{𝒪_{X}} \underline{Γ}_{Z}K^{•} → \underline{Γ}_{Z}(ℱ ⊗_{𝒪_{X}} K^{•}). Now, the assertion is local on X, hence we may assume that X = Spec A for some ring A, and ℱ = F^{∼} for some flat A-module F. Then F is the colimit of a filtered system (L_{λ} | λ ∈ Λ) of free A-modules of finite rank ([120, Ch.I, Th.1.2]), and by virtue of lemma 10.4.4(iii.b), the map ω_{ℱ} is the colimit of the system of induced maps (ω_{L_{λ}^{∼}} | λ ∈ Λ). Thus, we are reduced to the case where F is a free A-module of finite rank, in which case the assertion is clear. □

Proposition 10.4.18. Let X be an affine scheme, Z ⊂ X any closed subset, ℱ^{•} a bounded below complex of quasi-coherent 𝒪_{X}-modules, and 𝔘 := (U_{t} | t ∈ I) a family of affine open subsets of X such that ⋃_{t∈I} U_{t} = X \Z. There exists a natural isomorphism

RΓ_{Z}ℱ^{•} →^{∼} Tot C_{alt}^{•}(𝔘, ℱ^{•})[−1] in D^{+}(𝒪_{X}(X)-Mod)

(where C_{alt}^{•} denotes the augmented alternating Čech complex).

Proof. Pick any resolution ℱ^{•} →^{∼} ℐ^{•} by a bounded below complex of injective 𝒪_{X}-modules; by virtue of (10.4.11) it suffices to show that the natural map of double complexes

C_{alt}^{•}(𝔘, ℱ^{•}) → C_{alt}^{•}(𝔘, ℐ^{•})

induces a quasi-isomorphism on total complexes. To this aim, we can argue as in the proof of theorem 10.2.28(ii) : the details shall be left to the reader (the condition that X is affine, is needed here, since we are dealing with the augmented Čech complex, whereas in the proof of theorem 10.2.28(ii) only the ordinary version intervenes, so for the proof of the latter it is only required that the intersection of any finite system of members of 𝔘 is affine). □

10.4.19. Let X be a scheme; for every x ∈ X, we consider the cofiltered family Φ(x) of all non-empty constructible closed subsets of X(x) := Spec 𝒪_{X,x}; clearly ⋂_{Z∈Φ(x)} Z = {x}. Let K^{•} be any object of D^{+}(ℤ_{X}-Mod); we are interested in the quantities :

δ(x, K^{•}) := depth_{{x}}K^{•}(x) and δ′(x, K^{•}) := depth_{Φ(x)}K^{•}(x)

(notation of definition 4.9.17(iii)). Especially, we wish to compute the depth of a complex K^{•} along a closed subset Z ⊂ X, in terms of the local invariants δ(x, K^{•}) or δ′(x, K^{•}), evaluated at the points x ∈ Z. This shall be achieved by theorem 10.4.21. To begin with, we remark :

Lemma 10.4.20. With the notation of (10.4.19), we have :

(i) δ(x, K^{•}) ≥ δ′(x, K^{•}) for every x ∈ X and every K^{•} ∈ Ob(D^{+}(ℤ_{X}-Mod)).

(ii) If the topological space |X| underlying X is locally noetherian, the inequality of (i) is

actually an equality. (iii) If ℱ is a flat quasi-coherent 𝒪_{X}-module, then δ′(x, ℱ) ≥ δ′(x, 𝒪_{X}).

Proof. (i) follows easily from (10.4.15). Likewise, (ii) follows likewise from (10.4.15) and the fact that if |X| is locally noetherian, {x} is the smallest element of the family Φ(x).

(iii): According to [120, Ch.I, Th.1.2], the 𝒪_{X(x)}-module ℱ (x) is the colimit of a filtered system (ℒ_{λ} | λ ∈ Λ) of free 𝒪_{X(x)}-modules of finite rank. In view of lemma 10.4.4(iii.b), it is easily seen that

depth_{Z}ℱ ≥ inf_{λ∈Λ} depth_{Z}ℒ_{λ} = depth_{Z}𝒪_{X}

for every closed constructible subset Z ⊂ X. The assertion follows. □

Theorem 10.4.21. With the notation of (10.4.19), let Z ⊂ X be any closed constructible subset, K^{•} any object of D^{+}(ℤ_{X}-Mod). Then

depth_{Z}K^{•} = inf {δ(x, K^{•}) | x ∈ Z} = inf {δ′(x, K^{•}) | x ∈ Z}.

Proof. Let d := depth_{Z}K^{•}; in light of corollary 10.4.10, it is clear that δ(x, K^{•}) ≥ d for all x ∈ Z, hence, in order to prove the first identity it suffices to show :

Claim 10.4.22. Suppose d < +∞. Then there exists x ∈ Z such that R^{d}Γ_{{x}}K^{•}(x) ≠ 0.

Proof of the claim. By definition of d, we can find an open subset V ⊂ X and a non-zero section s ∈ Γ(V, R^{d}\underline{Γ}_{Z}K^{•}). The support of s is a closed subset S ⊂ Z. Let x be a maximal point of S. From lemma 10.4.13(iv) we deduce a spectral sequence

E_{2}^{pq} := R^{p}\underline{Γ}_{\{x\}}R^{q}\underline{Γ}_{Z∩X(x)}K^{•}(x) ⇒ R^{p+q}\underline{Γ}_{\{x\}}K^{•}(x)

and corollary 10.4.10implies that E_{2}^{pq} = 0 whenever q < d, therefore :

R^{d}\underline{Γ}_{\{x\}}K^{•}(x) ≃ R^{0}\underline{Γ}_{\{x\}}R^{d}\underline{Γ}_{Z∩X(x)}K^{•}(x).

However, the image of s in R^{d}\underline{Γ}_{Z∩X(x)}K^{•}(x) is supported precisely at x, as required. ♢

Finally, since Z is constructible, Z ∩ X(x) ∈ Φ(x) for every x ∈ X, hence δ′(x, K^{•}) ≥ d. Then the second identity follows from the first and lemma 10.4.20(i). □

Corollary 10.4.23. Let Z ⊂ X be a closed and constructible subset, ℱ a ℤ_{X}-module, and j : X \ Z → X the induced open immersion. Then the natural map ℱ → j_{∗}j^{∗}ℱ is an isomorphism if and only if δ(x, ℱ) ≥ 2, if and only if δ′(x, ℱ) ≥ 2 for every x ∈ Z. □

10.4.24. Let now X be an affine scheme, say X := Spec A for some ring A, and Z ⊂ X a constructible closed subset. In this case, we wish to show that the depth of a complex of quasi-coherent 𝒪_{X}-modules along Z can be computed in terms of Ext functors on the category of A-modules. This is the content of the following :

Proposition 10.4.25. In the situation of (10.4.24), let N be any finitely presented A-module such that Supp N = Z, and M^{•} any bounded below complex of A-modules. Then :

depth_{Z}M^{•∼} = sup {n ∈ ℤ | Ext_{A}^{j}(N, M^{•}) = 0 for all j < n}.

Proof. (M^{•∼} is the complex of quasi-coherent 𝒪_{X}-modules determined by M^{•}, as in (10.3)).

Claim 10.4.26. There is a natural isomorphism in D^{+}(A-Mod) :

RHom_{A}^{•}(N, M^{•}) →^{∼} RHom_{𝒪_{X}}^{•} (N^{∼}, R\underline{Γ}_{Z}M^{•∼}).

Proof of the claim. Let i : Z → X be the closed immersion. The assumption on N implies that N^{∼} = i_{∗}i^{∗}N^{∼}; hence lemma 10.4.13(i.b) yields a natural isomorphism

(10.4.27)  RHom_{𝒪_{X}}^{•} (N^{∼}, M^{•∼}) →^{∼} RHom_{𝒪_{X}}^{•} (N^{∼}, R\underline{Γ}_{Z}M^{•∼}).

The claim follows by combining corollary 10.3.2(i) and (10.4.27). ♢

From claim 10.4.26 we see already that Ext_{A}^{j}(N, M^{•}) = 0 whenever j < depth_{Z}M^{•∼}. Suppose that depth_{Z}M^{•∼} = p < +∞; in this case, claim 10.4.26gives an isomorphism :

Ext_{A}^{p}(N, M^{•}) ≃ Hom_{𝒪_{X}} (N^{∼}, R^{p}\underline{Γ}_{Z}M^{•∼}) ≃ Hom_{A}(N, R^{p}Γ_{Z}M^{•∼})

where the last isomorphism holds by lemma 10.4.17. To conclude the proof, we have to exhibit a non-zero map from N to Q := R^{p}Γ_{Z}M^{•∼}. Let F_{0}(N) ⊂ A denote the Fitting ideal of N; this is a finitely generated ideal, whose zero locus coincides with the support of N. More precisely, Ann_{A}(N)^{r} ⊂ F_{0}(N) ⊂ Ann_{A}(N) for all sufficiently large integers r > 0. Let now x ∈ Q be any non-zero element, and f_{1}, …, f_{k} a finite system of generators for F_{0}(N); since x vanishes on X \Z, the image of x in Q_{f_{i}} is zero for every i ≤ k, i.e. there exists n_{i} ≥ 0 such that f_{i}^{n_{i}}x = 0 in Q. It follows easily that F_{0}(N)^{n} ⊂ Ann_{A}(x) for a sufficiently large integer n, and therefore x defines a map φ : A/F_{0}(N)^{n} → Q by the rule : a ↦ ax. According to [75, Lemma 3.2.21], we can find a finite filtration 0 = J_{m} ⊂ ⋯ ⊂ J_{1} ⊂ J_{0} := A/F_{0}(N)^{n} such that each J_{i}/J_{i+1} is quotient of a direct sum of copies of N. Let s ≤ m be the smallest integer such that J_{s} ⊂ Ker φ. By restriction, φ induces a non-zero map J_{s−1}/J_{s} → Q, whence a non-zero map N^{(S)} → Q, for some set S, and finally a non-zero map N → Q, as required. □

Remark 10.4.28. Notice that the existence of a finitely presented A-module N with Supp N = Z is equivalent to the constructibility of Z.

10.4.29. In the situation of (10.4.24), let I ⊂ A be a finitely generated ideal such that V (I) = Z, and M^{•} a bounded below complex of A-modules. Then it is customary to set :

depth_{I}M^{•} := depth_{Z}M^{•∼}

and the depth along Z of M^{•∼} is also called the I-depth of M^{•}. Moreover, if A is a local ring with maximal ideal 𝔪_{A}, we shall often use the standard notation :

depth_{A}M^{•} := depth_{{𝔪_{A}}}M^{•∼}

and this invariant is often called briefly the depth of M^{•}. With this notation, theorem 10.4.21 can be rephrased as the identity :

(10.4.30)  depth_{I}M^{•} = inf_{𝔭∈V (I)} depth_{A_{𝔭}}M_{𝔭}^{•}.

10.4.31. Depth sensitivity of the Koszul complex. Let now f := (f_{1}, …, f_{r}) be a finite system of generators of the ideal I ⊂ A, and Z := Spec_{,}A/I. In this generality, the Koszul complex K^{•}(f) of remark 7.8.1(ii) is not a resolution of the A-module A/I, and thus H^{•}(f, M) (for an A-module M) does not necessarily agree with Ext_{A}^{•}(A/I, M). Nevertheless, we have :

Proposition 10.4.32. (i) In the situation of (10.4.31), let M^{•} ∈ Ob(D^{+}(A-Mod)); then :

depth_{I}M^{•} = d := sup {i ∈ ℤ | H^{j}(f, M^{•}) = 0 for all j < i}.

(ii) If d < +∞, there are natural A-linear isomorphisms :

H^{d}(f, M^{•}) →^{∼} Hom_{A}(A/I, R^{d}Γ_{Z}M^{•∼}) →^{∼} Ext_{A}^{d}(A/I, M^{•}).

(iii) Moreover we have a natural isomorphism in D(A-Mod) :

colim_{m∈ℕ} K^{•}(f^{m}, M^{•}) →^{∼} RΓ_{Z}M^{•∼}

where the transition maps in the colimit are the maps φ_{f^{n}}^{•} of (7.8.20).

Proof. (i): Let B := ℤ[t_{1}, …, t_{r}] → A be the ring homomorphism defined by the rule : t_{i} ↦ f_{i} for i = 1, …, r; we denote ψ : X → Y := Spec B the induced morphism, J ⊂ B the ideal generated by the system t := (t_{i} | i = 1, …, r), and set W := V (J) ⊂ Y. From lemma 10.4.13(v) we deduce a natural isomorphism in D(𝒪_{Y} -Mod) :

(10.4.33)  ψ_{∗}R\underline{Γ}_{Z}M^{•∼} →^{∼} R\underline{Γ}_{W} ψ_{∗}M^{•∼}.

Hence we are reduced to showing :

Claim 10.4.34. d = depth_{J}ψ_{∗}M^{•}.

Proof of the claim. Notice that t is a regular sequence in B, hence Ext_{B}^{•}(B/J, ψ_{∗}M^{•}) ≃ H^{•}(t, ψ_{∗}M^{•}); by proposition 10.4.25, there follows the identity :

depth_{J}ψ_{∗}M^{•} = sup {n ∈ ℕ | H^{i}(t, ψ_{∗}M^{•}) = 0 for all i < n}.

Then the assertion follows from lemma 7.8.2(iv). ♢ (ii): From lemma 10.4.13(i.b) and corollary 10.3.2(i) we derive a natural isomorphism :

RHom_{B}^{•}(B/J, ψ_{∗}M^{•}) →^{∼} RHom_{𝒪_{Y}}^{•} ((B/J)^{∼}, R\underline{Γ}_{W} ψ_{∗}M^{•∼}).

However, due to (10.4.33) and claim 10.4.34we may compute :

R^{d}Hom_{𝒪_{Y}}^{•} ((B/J)^{∼}, R\underline{Γ}_{W} ψ_{∗}M^{•∼}) →^{∼} Hom_{B}(B/J, R^{d}Γ_{W} ψ_{∗}M^{•∼}) →^{∼} Hom_{A}(A/I, R^{d}Γ_{Z}M^{•∼}).

The first claimed isomorphism easily follows. Similarly, one applies lemma 10.4.13(i.b) and corollary 10.3.2(i) to compute Ext_{A}^{•}(A/I, M^{•}), and deduces the second isomorphism of (ii) using (i). Assertion (iii) follows immediately from (10.2.27) and proposition 10.4.18. □

Corollary 10.4.35. Let A be a noetherian ring, I ⊂ A an ideal, M^{•} an object of D^{+}(A-Mod), and set Z := Spec A/I. We have a natural isomorphism

colim_{n∈ℕ} R^{i}Hom_{A}^{•}(A/I^{n}, M^{•}) →^{∼} R^{i}Γ_{Z}M^{•∼} for every i ∈ ℤ.

Proof. This is obtained by combining proposition 10.4.32(iii) and remark 7.8.45. □

Corollary 10.4.36. Let A be a noetherian ring, M an A-module, and f := (f_{1}, …, f_{r}) a finite sequence of elements of A that generates an ideal I ⊂ A. Suppose that :

(a) depth_{A_{𝔭}}(M_{𝔭}) ≥ r for every 𝔭 ∈ Z := Spec A/I.

(b) Z has pure codimension r in Spec A. Then H^{i}(f, M) = 0 for every i < r, and for every 𝔭 ∈ Z we have :

depth_{A_{𝔭}}(M_{𝔭}/IM_{𝔭}) = depth_{A_{𝔭}}(M_{𝔭}) − r  and  dim A_{𝔭}/I_{𝔭} = dim(A_{𝔭}) − r.

Proof. The stated vanishing for H^{i}(f, M) follows immediately from (10.4.30) and proposition 10.4.32(i). Next, we have dim A_{𝔭}/I_{𝔭} ≥ dim(A_{𝔭}) − r for every 𝔭 ∈ Z, by [126, Th.13.6(ii)], and the opposite inequality follows easily from (b).

For a given 𝔭 ∈ Z, set δ_{𝔭} := depth_{A_{𝔭}}M_{𝔭}; pick a sequence f′ := (f_{r+1}, …, f_{r+n}) in A whose image in A_{𝔭} generates 𝔭A_{𝔭}, and set f′′ := (f_{1}, …, f_{r+n}) and \overline{M} := M/IM. By proposition 10.4.32(i) and lemma 7.8.2(vi), if δ_{𝔭} = +∞, we have H_{i}(f′′, M_{𝔭}) = 0 for every i ∈ ℕ, and otherwise r ≤ δ_{𝔭} ≤ n, and we have

H_{n+r−δ_{𝔭}}(f′′, M_{𝔭}) ≠ 0  and H_{i}(f′′, M_{𝔭}) = 0  for every i > n + r − δ_{𝔭}.

By the same token, H_{i}(f, M_{𝔭}) = 0 for every i > 0, i.e. the natural map K_{•}(f, M_{𝔭}) → \overline{M}_{𝔭}[0] is an isomorphism in D(A_{𝔭}-Mod), so we get isomorphisms :

K_{•}(f′′, M_{𝔭}) →^{∼} K_{•}(f′)⊗_{A_{𝔭}} K_{•}(f, M_{𝔭}) →^{∼} K_{•}(f′)⊗_{A_{𝔭}} \overline{M}_{𝔭}[0] = K_{•}(f′, \overline{M}_{𝔭}) in D(A_{𝔭}-Mod).

Summing up, we see that, if δ_{𝔭} = +∞, then H_{i}(f′, \overline{M}_{𝔭}) = 0 for every i ∈ ℕ, and otherwise :

H^{j}(f′, \overline{M}_{𝔭}) = H_{n−j}(f′, \overline{M}_{𝔭}) = H_{n−j}(f′′, M_{𝔭}) = 0 whenever n − j > n + r − δ_{𝔭}

and H^{δ_{𝔭}−r}(f′, \overline{M}_{𝔭}) = H_{n+r−δ_{𝔭}}(f′, \overline{M}_{𝔭}) ≠ 0, which implies that depth_{A_{𝔭}}(\overline{M}_{𝔭}) = δ_{𝔭} − r, again by proposition 10.4.32(i). □

Corollary 10.4.37. In the situation of (10.4.29), let B be a faithfully flat A-algebra. Then :

depth_{IB}B ⊗_{A} M^{•} = depth_{I}M^{•}.

Proof. It is a special case of lemma 10.4.17(iii). Alternatively, one remarks that

K^{•}(g, M^{•} ⊗_{A} B) ≃ K^{•}(g, M^{•}) ⊗_{A} B for every finite sequence of elements g in A

which allows us to apply proposition 10.4.32(iii). □

Theorem 10.4.38. Let A → B be a local homomorphism of local rings, and suppose that the maximal ideals 𝔪_{A} and 𝔪_{B} of A and B are finitely generated. Let also M be an A-module and N a B-module which is flat over A. Then we have :

depth_{B}(M ⊗_{A} N) = depth_{A}M + depth_{B/𝔪_{A}B}(N/𝔪_{A}N).

Proof. Let f := (f_{1}, …, f_{r}) and \overline{g} := (\overline{g}_{1}, …, \overline{g}_{s}) be finite systems of generators for 𝔪_{A} and respectively \overline{𝔪}_{B}, the maximal ideal of B/𝔪_{A}B. We choose an arbitrary lifting of \overline{g} to a finite system g of elements of 𝔪_{B}; then (f, g) is a system of generators for 𝔪_{B}. We have a natural identification of complexes of B-modules :

K_{•}(f, g) →^{∼} Tot_{•}(K_{•}(f) ⊗_{A} K_{•}(g))

whence natural isomorphisms :

K^{•}((f, g), M ⊗_{A} N) →^{∼} Tot^{•}(K^{•}(f, M) ⊗_{A} K^{•}(g, N)).

A standard spectral sequence, associated with the filtration by rows, converges to the cohomology of this total complex, and since N is a flat A-module, K^{•}(g, N) is a complex of flat A-modules, so that the E_{1} term of this spectral sequence is found to be :

E_{1}^{pq} ≃ H^{q}(f, M) ⊗_{A} K^{p}(g, N)

and the differentials d_{1}^{pq} : E_{1}^{pq} → E_{1}^{p+1,q} are induced by the differentials of the complex K^{•}(g, N). Set κ_{A} := A/𝔪_{A}; notice that H^{q}(f, M) is a κ_{A}-vector space, hence

H^{q}(f, M) ⊗_{A} H^{•}(g, N) ≃ H^{•}(H^{q}(f, M) ⊗_{κ_{A}} K^{•}(\overline{g}, N/𝔪_{A}N))

and consequently :

E_{2}^{pq} ≃ H^{q}(f, M) ⊗_{κ_{A}} H^{p}(\overline{g}, N/𝔪_{A}N).

Especially :

(10.4.39)  E_{2}^{pq} = 0 if either p < d := depth_{A}M or q < d′ := depth_{B/𝔪_{A}B}(N/𝔪_{A}N).

Hence H^{i}((f, g), M ⊗_{A} N) = 0 for all i < d + d′ and moreover, if d and d′ are finite, H^{d+d′}((f, g), M ⊗_{A} N) ≃ E_{2}^{dd′} ≠ 0. Now the sought identity follows readily from proposition 10.4.32(i). □

Remark 10.4.40. Let d and d′ be as in (10.4.39), and suppose that d and d′ are finite. By inspection of the proof, we see that – under the assumptions of theorem 10.4.38 – a natural isomorphism has been exhibited :

Ext_{A}^{d+d′}(B/𝔪_{B}, M ⊗_{A} N) ≃ Ext_{A}^{d}(A/𝔪_{A}, M) ⊗_{A} Ext_{B/𝔪_{A}B}^{d′}(B/𝔪_{B}, N/𝔪_{A}N).

Definition 10.4.41. Let (A, 𝔪) be a local ring, and M_{•} ∈ Ob(D^{b}(A-Mod)). The residual Tor

dimension of M_{•} is rTor.dim_{A}(M_{•}) := sup{i ∈ ℤ | H_{i}(M_{•} ⊗^{L}_{A} A/𝔪) ≠ 0} ∈ ℤ ∪ {±∞}.

Remark 10.4.42. (i) If A is a noetherian local ring, and M is any A-module of finite type, then rTor.dim_{A}(M) equals the projective dimension of M : see [126, §19, Lemma 1].

(ii) Our next result generalizes [15, Th.4.1].

Theorem 10.4.43. Let A be a local noetherian ring, and M_{•} ∈ Ob(D^{b}(A-Mod)). Then : (i) depth_{A}(M_{•}) = +∞ if and only if rTor.dim_{A}(M_{•}) = −∞. (ii) If rTor.dim_{A}(M_{•}) ∈ ℤ, we have the Auslander-Buchsbaum identity :

depth_{A}(M_{•}) + rTor.dim_{A}(M_{•}) = depth_{A}A.

Proof. (ii): Let d := dim A, and pick a system of parameters f of A, i.e. a sequence (f_{1}, …, f_{d}) of d elements of A that generates a primary ideal. Using a Cartan-Eilenberg resolution of the Koszul complex K_{•}(f) (see remark 7.8.1(ii)), we get a spectral sequence (see [163, §5.7])

E_{pq}^{2} := H_{p}(M_{•} ⊗^{L}_{A} H_{q}(f, A)) ⇒ H_{p+q}(f, M_{•}).

Now, by proposition 10.4.32(i) and lemma 7.8.2(v), we have :

depth_{A}A = d − e_{A} where e_{A} := inf{i ∈ ℤ | H_{j}(f, A) = 0 for all i > j}
depth_{A}M_{•} = d − e_{M_{•}} where e_{M_{•}} := inf{i ∈ ℤ | H_{j}(f, M_{•}) = 0 for all i > j}.

Thus, we have to show that if t_{M_{•}} := rTor.dim_{A}(M_{•}) ∈ ℤ, then

(10.4.44)  e_{M_{•}} = e_{A} + t_{M_{•}}.

Claim 10.4.45. For every A-module N ≠ 0 of finite length, we have :

H_{t_{M•}}(M_{•} ⊗^{L} N) ≠ 0  and H_{i}(M_{•} ⊗^{L} N) = 0  for every i > t_{M_{•}}.

Proof of the claim. There exists a finite filtration 0 := N_{0} ⊂ N_{1} ⊂ ⋯ ⊂ N_{n−1} ⊂ N_{n} := N such that N_{i+1}/N_{i} is isomorphic to A/𝔪 for i = 0, …, n − 1. Hence, by a simple induction on the length n of the filtration, we are reduced to checking the following assertion : let Σ_{•} := (0 → N′ → N → A/𝔪 → 0) be a short exact sequence of A-modules of finite length. The

induced map H_{i}(M_{•} ⊗^{L}_{A} N′) → H_{i}(M_{•} ⊗^{L}_{A} N) is an isomorphism for every i ≥ t_{M_{•}}. The latter follows easily, y considering the long exact homology sequence induced by the short exact

sequence of complexes M_{•} ⊗^{L}_{A} Σ_{•}. ♢

Now, since A is noetherian and f generates a primary ideal, H_{q}(f, A) is a module of finite length for every q ∈ ℕ (lemma 7.8.2(i)). Claim 10.4.45 implies that 0 ≠ E_{t_{M•},e_{A}}^{2}, and E_{pq}^{2} = 0 whenever either p > t_{M_{•}} or q > e_{A}, so that E_{t_{M•},e_{A}}^{2} = E_{t_{M•},e_{A}}^{∞}, whence (10.4.44).

(i): If t_{M_{•}} ∈ ℤ, then (10.4.44) implies that e_{M_{•}} ∈ ℤ, since e_{A} ∈ ℕ; then in this case depth_{A}M_{•} ∈ ℤ. If t_{M_{•}} = −∞, claim 10.4.45 says that E_{pq}^{2} = 0 for every p, q ∈ ℤ, whence H_{i}(f, M) = 0 for every i ∈ ℤ, i.e. depth_{A}(M_{•}) = +∞. Lastly, by proposition 10.4.32(iii), this condition holds if and only if colim_{m∈ℕ} K^{•}(f^{m}, M^{•}) = 0 in D^{b}(A-Mod). Set

K_{•} := colim_{m∈ℕ} K_{•}(f^{m}) T_{m,•} := t^{≤−1}K_{•}(f^{m}) for every m ∈ ℕ T_{•} := colim_{m∈ℕ} T_{m,•}.

Then, in turn the latter means that M^{•} ⊗_{A} K_{•} = 0 in D^{b}(A-Mod) (lemma 7.8.2(v)). Set k := A/𝔪; the short exact of complexes 0 → A[0] → K_{•} → T_{•} → 0 induces a distinguished triangle in D^{−}(A-Mod) :

M_{•} ⊗^{L}_{A} k[0] → M_{•} ⊗^{L}_{A} K_{•} ⊗^{L}_{A} k[0] → M_{•} ⊗^{L}_{A} T_{•} ⊗^{L}_{A} k[0] → M_{•} ⊗^{L}_{A} k[1].

Notice that the transition maps T_{m,•} → T_{m+1,•} induce the zero maps T_{m,•} ⊗_{A} k → T_{m+1,•} ⊗_{A} k

for every m ∈ ℕ; hence T_{•} ⊗^{L}_{A} k[0] = 0 in D(A-Mod), and therefore M_{•} ⊗^{L}_{A} T_{•} ⊗^{L}_{A} k[0] = 0 as

well. Summing up, we conclude that if depth_{A}M_{•} = +∞, then M_{•} ⊗^{L}_{A}k[0] = 0 in D(A-Mod), i.e. t_{M^{•}} = −∞. □

10.4.46. Let f : X → S be any morphism of schemes. Recall that an 𝒪_{X}-module ℱ is said to be f-flat at a point x ∈ X if ℱ_{x} is a flat 𝒪_{S,f(x)}-module. We say that ℱ is f-flat over a point s ∈ S if ℱ is f-flat at all points of f^{−1}(s). Finally, we say that ℱ is f-flat if ℱ is f-flat at all the points of X ([59, Ch.0, §6.7.1]).

Corollary 10.4.47. Let f : X → S be a finitely presented morphism of schemes, ℱ a f-flat quasi-coherent 𝒪_{X}-module of finite presentation, 𝒢 any quasi-coherent 𝒪_{S}-module, x ∈ X any point, i : f^{−1}f(x) → X the natural morphism. Then :

δ′(x, ℱ ⊗_{𝒪_{X}} f^{∗}𝒢) = δ′(x, i^{∗}ℱ) + δ′(f(x), 𝒢).

Proof. Set s := f(x), and denote by

j_{s} : S(s) := Spec 𝒪_{S,s} → S and f_{s} : X(s) := X ×_{S} S(s) → S(s)

the induced morphisms. By inspecting the definitions, one checks easily that

δ′(x, ℱ ⊗_{𝒪_{X}} f^{∗}𝒢) = δ′(x, ℱ_{|X(s)} ⊗_{𝒪_{X(s)}} f_{s}^{∗}𝒢 (s)) and δ′(s, 𝒢) = δ′(s, 𝒢 (s))

(notation of definition 4.9.17(iii)). Thus, we may replace S by S(s) and X by X(s), and assume from start that S is a local scheme and s its closed point. Clearly we may also assume that X is affine and finitely presented over S. Then we can write S as the limit of a cofiltered family (S_{λ} | λ ∈ Λ) of local noetherian schemes, with local transition maps, and we may assume that f : X → S is a limit of a cofiltered family (f_{λ} : X_{λ} → S_{λ} | λ ∈ Λ) of morphisms of finite type, such that X_{μ} = X_{λ} ×_{S_{λ}} S_{μ} whenever μ ≥ λ. Likewise, we may descend ℱ to a family (ℱ_{λ} | λ ∈ Λ) of finitely presented quasi-coherent 𝒪_{X_{λ}}-modules, such that ℱ_{μ} = φ_{μλ}^{∗}ℱ_{λ} for every μ ≥ λ (where φ_{μλ} : X_{μ} → X_{λ} is the natural morphism). Furthermore, up to replacing Λ by some coinitial subset, we may assume that ℱ_{λ} is f_{λ}-flat for every λ ∈ Λ ([65, Ch.IV, Cor.11.2.6.1(ii)]). For every λ ∈ Λ, denote by ψ_{λ} : S → S_{λ} the natural morphism, so that s_{λ} := ψ_{λ}(s) is the closed point of S_{λ}; for every constructible closed subset Z_{λ} ⊂ S_{λ}, the set Z := ψ_{λ}^{−1}Z_{λ} is constructible and closed in S, and according to lemma 10.4.17(ii) we have depth_{Z}𝒢 = depth_{Z_{λ}}ψ_{λ∗}𝒢. On the other hand, every closed constructible subset Z ⊂ S is of the form ψ_{λ}^{−1}Z_{λ} for some λ ∈ Λ and some Z_{λ} closed constructible in S_{λ} ([65, Ch.IV, Cor.8.3.11]), so that :

(10.4.48)  δ′(s, 𝒢) = sup{δ′(s_{λ}, ψ_{λ∗}𝒢) | λ ∈ Λ}.

Similarly, for every λ ∈ Λ let x_{λ} be the image of x in X_{λ} and denote by i_{λ} : f_{λ}^{−1}(s_{λ}) → X_{λ} the natural morphism. Notice that f_{λ}^{−1}(s_{λ}) is a scheme of finite type over Spec κ(s_{λ}), especially it is noetherian, hence δ′(x_{λ}, i_{λ}^{∗}ℱ_{λ}) = δ(x_{λ}, i_{λ}^{∗}ℱ_{λ}), by lemma 10.4.20(ii); by the same token we have as well the identity : δ′(x, i^{∗}ℱ) = δ(x, i^{∗}ℱ). Denote by g_{λ} : f^{−1}(s) → f_{λ}^{−1}(s_{λ}) and g_{λ,x} : Spec 𝒪_{f^{−1}(s),x} → Spec 𝒪_{f_{λ}^{−1}(s_{λ}),x_{λ}} the induced morphisms.

Claim 10.4.49. There exists λ ∈ Λ such that g_{μ,x}^{−1}(x_{μ}) = {x} for every μ ≥ λ.

Proof of the claim. Let Z be the topological closure of {x} in f^{−1}(s); we may find λ ∈ Λ and a closed subset Z_{λ} ⊂ f_{λ}^{−1}(s_{λ}) such that Z = g_{λ}^{−1}Z_{λ} ([65, Ch.IV, Cor.8.3.11]). Since Z is irreducible and g_{λ} is surjective, it is easily seen that Z_{λ} is irreducible. Moreover, the topological closure Z′_{λ} of {x_{λ}} in f_{λ}^{−1}(s_{λ}) lies in Z_{λ}, and g_{λ}^{−1}Z′_{λ} contains x; thus, Z_{λ} = Z′_{λ}. It follows that g_{λ,x}^{−1}(x_{λ}) = {x}, whence the assertion. ♢

Since g_{λ,x} is faithfully flat, from claim 10.4.49 and corollary 10.4.37 we see that, after replacing Λ by a coinitial subset, we may assume :

(10.4.50)  δ′(x_{λ}, i_{λ}^{∗}ℱ_{λ}) = δ′(x, i^{∗}ℱ)  for every λ ∈ Λ.

Claim 10.4.51. Let K be a field, X and Y two K-schemes, and set Z := X ×_{Spec K} Y. Let also p : Z → X and q : Z → Y be the projections, ℳ any 𝒪_{X}-module, and 𝒩 any 𝒪_{Y} -module. Then we have :

Supp (p^{∗}ℳ ⊗_{𝒪_{Z}} q^{∗}𝒩 ) = p^{−1}(Supp ℳ) ∩ q^{−1}(Supp 𝒩 ).

Proof of the claim. To ease notation, set 𝒫 := p^{∗}ℳ ⊗_{𝒪_{Z}} q^{∗}𝒩 . Let z ∈ Z be any point with x := p(z) ∈ Supp ℳ and y := q(z) ∈ Supp 𝒩 ; it suffices to show that 𝒫_{z} ≠ 0. Now, 𝒪_{Z,z} is a localization of 𝒪_{X,x} ⊗_{K} 𝒪_{Y,y}, at a prime ideal 𝔭, and 𝒫_{z} = (ℳ_{x} ⊗_{K} 𝒩_{y})_{𝔭}. We come down therefore to the following situation. Let A and B be two local K-algebras, 𝔪_{A} ⊂ A, 𝔪_{B} ⊂ B the respective maximal ideals, M ≠ 0 an A-module, N ≠ 0 a B-modules, and 𝔭 ⊂ A ⊗_{K} B a prime ideal such that

(10.4.52)  𝔪_{A} ⊗_{K} B + A ⊗_{K} 𝔪_{B} ⊂ 𝔭.

We have to check that (M ⊗_{K} N)_{𝔭} ≠ 0. To this aim, we may reduce easily to the case where M and N are cyclic modules, in which case there exist a surjective A-linear map M → κ(A) := A/𝔪_{A} and a surjective B-linear map N → κ(B) := B/𝔪_{B}; we are then further reduced to showing that (κ(A) ⊗_{K} κ(B))_{𝔭} ≠ 0, which is ensured by (10.4.52). ♢

Next, notice that the local scheme X(x) is the limit of the cofiltered system of local schemes (X_{λ}(x_{λ}) | λ ∈ Λ). Let Z_{λ} ⊂ X_{λ}(x_{λ}) be a closed constructible subset and set Z := φ_{λ}^{−1}Z_{λ}, where φ_{λ} : X(x) → X_{λ}(x_{λ}) is the natural morphism. Let Y_{λ} be the fibre product in the cartesian diagram of schemes :

\xymatrix{Y_{λ} \ar[r]^{q_{λ}} \ar[d]_{p_{λ}} & X_{λ}(x_{λ}) \ar[d] \\ S \ar[r]^{ψ_{λ}} & S_{λ}.}

The morphism φ_{λ} factors through a unique morphism of S-schemes \overline{φ}_{λ} : X(x) → Y_{λ}, and if y_{λ} ∈ Y_{λ} is the image of the closed point x of X(x), then \overline{φ}_{λ} induces a natural identification X(x) →^{∼} Y_{λ}(y_{λ}). To ease notation, let us set :

ℋ := ℱ ⊗_{𝒪_{X}} f^{∗}𝒢 and ℋ_{λ} := ℱ_{λ} ⊗_{𝒪_{X_{λ}}} f_{λ}^{∗}ψ_{λ∗}𝒢 .

In view of corollary 10.4.10, there follows a natural isomorphism :

(10.4.53)  R\underline{Γ}_{Z}ℋ (x) →^{∼} \overline{φ}_{λ}^{∗}R\underline{Γ}_{q_{λ}^{−1}(Z_{λ})}(q_{λ}^{∗}ℱ_{λ}(x_{λ}) ⊗_{𝒪_{Y_{λ}}} p_{λ}^{∗}𝒢).

On the other hand, applying the projection formula (see remark 10.3.7) we get :

(10.4.54)  q_{λ∗}(q_{λ}^{∗}ℱ_{λ}(x_{λ}) ⊗_{𝒪_{Y_{λ}}} p_{λ}^{∗}𝒢) ≃ ℋ_{λ}(x_{λ}).

Combining (10.4.53), (10.4.54) and lemma 10.4.13(v) we deduce :

R\underline{Γ}_{Z}ℋ (x) ≃ φ_{λ}^{∗}R\underline{Γ}_{Z_{λ}}ℋ_{λ}(x_{λ}).

Notice that the foregoing argument applies also in case S is replaced by some S_{μ} for some μ ≥ λ (and consequently X is replaced by X_{μ}); we arrive therefore at the inequality :

(10.4.55)  δ′(x_{λ}, ℋ_{λ}) ≤ δ′(x_{μ}, ℋ_{μ}) ≤ δ′(x, ℋ )  whenever μ ≥ λ.

Claim 10.4.56. δ′(x, ℋ ) = sup {δ′(x_{λ}, ℋ_{λ}) | λ ∈ Λ}.

Proof of the claim. Due to (10.4.55) we may assume – again after replacing Λ by a cofinal subset – that δ′(x_{λ}, ℋ_{λ}) is a constant d ∈ ℕ independent of λ. Especially, R^{d}\underline{Γ}_{Z_{λ}}ℋ_{λ}(x_{λ}) ≠ 0 for every λ ∈ Λ and every closed constructible subset Z_{λ} ⊂ X_{λ}(x_{λ}). We have to show that R^{d}\underline{Γ}_{Z}ℋ (x) ≠ 0 for arbitrarily small constructible closed subsets Z ⊂ X(x). However, given such Z, there exist λ ∈ Λ and Z_{λ} closed constructible in X_{λ}(x_{λ}) such that Z = φ^{−1}(Z_{λ}) ([65, Ch.IV, Cor.8.3.11]). Say that:

S = Spec A  S_{λ} = Spec A_{λ}  X_{λ}(x_{λ}) = Spec B_{λ}.

Hence Y_{λ} ≃ Spec A ⊗_{A_{λ}} B_{λ}. Let also F_{λ} (resp. G) be a B_{λ}-module (resp. A-module) such that F_{λ}^{∼} ≃ ℱ_{λ} (resp. G^{∼} ≃ 𝒢). Then ℋ_{|Y_{λ}} ≃ (F_{λ} ⊗_{A_{λ}} G)^{∼}. Let 𝔪_{λ} ⊂ B_{λ} and 𝔫_{λ} ⊂ A_{λ} be the maximal ideals. Up to replacing Z by a smaller subset, we may assume that Z_{λ} = V (𝔪_{λ}). To ease notation, set :

E_{1} := Ext_{A_{λ}}^{a}(κ(s_{λ}), G) E_{2} := Ext_{B_{λ}/𝔫_{λ}B_{λ}}^{b}(κ(x_{λ}), F_{λ}/𝔫_{λ}F_{λ})

with a = δ′(s_{λ}, ψ_{λ∗}𝒢) and b = d − a. Proposition 10.4.32(i),(ii) and remark 10.4.40say that

0 ≠ E_{3} := Ext_{B_{λ}}^{d}(κ(x_{λ}), F_{λ} ⊗_{A_{λ}} G) ≃ E_{1} ⊗_{κ(s_{λ})} E_{2}.

To conclude, it suffices to show that y_{λ} ∈ Supp E_{3}. However, claim 10.4.51says that

Supp E_{3} = p_{λ}^{−1}(Supp E_{1}) ∩ q_{λ}^{−1}(Supp E_{2}).

Now, p_{λ}(y_{λ}) = s, and clearly s ∈ Supp E_{1}, since G is an A-module and s is the closed point of S; likewise, q_{λ}(y_{λ}) = x_{λ} is the closed point of X_{λ}(x_{λ}), therefore it is in the support of E_{2}. ♢ We can now conclude the proof : indeed, from theorem 10.4.38we derive :

δ′(x_{λ}, ℋ_{λ}) = δ′(x, i_{λ}^{∗}ℱ_{λ}) + δ′(f(x), ψ_{λ∗}𝒢) for every λ ∈ Λ

and then the corollary follows from (10.4.48), (10.4.50) and claim 10.4.56. □

10.4.57. Consider now a finitely presented morphism of schemes f : X → Y, and a sheaf of ideals ℐ ⊂ 𝒪_{X} of finite type. Let x ∈ X be any point, and set y := f(x); notice that f^{−1}(y) is an algebraic κ(y)-scheme, so that the local ring 𝒪_{f^{−1}(y),x} is noetherian. We let ℐ (y) := i_{y}^{−1}ℐ · 𝒪_{f^{−1}(y)}, which is a coherent sheaf of ideals of the fibre f^{−1}(y). The fibrewise ℐ -depth of X over Y at the point x is the integer:

depth_{ℐ ,f} (x) := depth_{ℐ (y)_{x}} 𝒪_{f^{−1}(y),x}.

In case ℐ = 𝒪_{X}, the fibrewise ℐ -depth shall be called the fibrewise depth at the point x, and shall be denoted by depth_{f}(x). In view of (10.4.30) we have the identity:

(10.4.58)  depth_{ℐ ,f} (x) = inf\{depth_{f}(z) | z ∈ V (ℐ (y)) and x ∈ \overline{\{z\}}\}.

10.4.59. The fibrewise ℐ -depth can be computed locally as follows. For a given x ∈ X, set y := f(x) and let U ⊂ X be any affine open neighborhood of x such that ℐ_{|U} is generated by a finite family f := (f_{i})_{1≤i≤r}, where f_{i} ∈ ℐ (U) for every i ≤ r. Then proposition 10.4.32(i) implies that :

(10.4.60) depth_{ℐ ,f} (x) = sup{n ∈ ℕ | H^{i}(K^{•}(f, 𝒪_{X}(U))_{x} ⊗_{𝒪_{Y,y}} κ(y)) = 0 for all i < n}.

Proposition 10.4.61. In the situation of (10.4.57), for every integer d ∈ ℕ we define the subset:

L_{ℐ} (d) := {x ∈ X | depth_{ℐ ,f} (x) ≥ d}.

Then the following holds:

(i) L_{ℐ} (d) is a constructible subset of X (see definition 8.1.13(v)).

(ii) Let Y′ → Y be any morphism of schemes, set X′ := X ×_{Y} Y′, and let g : X′ → X, f′ : X′ → Y′ be the induced morphisms. Then:

L_{g^{∗}ℐ} (d) = g^{−1}L_{ℐ} (d) for every d ∈ ℕ.

Proof. (ii): The identity can be checked on the fibres, hence we may assume that Y = Spec k and Y′ = Spec k′ for some fields k ⊂ k′. Let x′ ∈ X′ be any point and x ∈ X its image; since the map 𝒪_{X,x} → 𝒪_{X′,x′} is faithfully flat, corollary 10.4.37implies:

depth_{g^{∗}ℐ ,f′} (x′) = depth_{ℐ ,f} (x)

whence (ii).

(i): We may assume that Y = Spec A, X = Spec B, ℐ = I^{∼}, where A is a ring, B is a finitely presented A-algebra, and I ⊂ B is a finitely generated ideal, for which we choose a finite system of generators f := (f_{i})_{1≤i≤r}. We may also assume that A is reduced. Suppose first that A is noetherian; then, for every i ∈ ℕ and y ∈ Y let us set:

N_{ℐ} (i) = ⋃_{y∈Y} Supp H^{i}(K^{•}(f, B) ⊗_{A} κ(y)).

Taking into account (10.4.30) and (10.4.60), we deduce:

L_{ℐ} (d) = ⋂_{i=0}^{d−1}(X \N_{ℐ} (i)).

It suffices therefore to show that each subset N_{ℐ} (i) is constructible. For every j ∈ ℕ we set:

Z^{j} := Ker(d^{j} : K^{j}(f, B) → K^{j+1}(f, B)) B^{j} := Im(d^{j−1} : K^{j−1}(f, B) → K^{j}(f, B)).

Using [65, Ch.IV, Cor. 8.9.5], one deduces easily that there exists an affine open subscheme U ⊂ Y, say U = Spec A′ for some flat A-algebra A′, such that A′ ⊗_{A} Z^{•}, A′ ⊗_{A} B^{•} and A′ ⊗_{A} H^{•}(K^{•}(f, B)) are flat A′-modules. By noetherian induction, we can then replace Y by U and X by X ×_{Y} U, and assume from start that Z^{•}, B^{•} and H^{•}(K^{•}(f, B)) are flat Amodules. In such case, taking homology of the complex K^{•}(f, B) commutes with any base change; therefore:

N_{ℐ} (i) = ⋃_{y∈Y} Supp H^{i}(f, B) ⊗_{A} κ(y) = Supp H^{i}(f, B)

whence the claim, since the support of a B-module of finite type is closed in X.

Finally, for a general ring A, we can find a noetherian subalgebra A′ ⊂ A, an A′-algebra B′ of finite type and a finitely generated ideal I′ ⊂ B′ such that B = A ⊗_{A′} B and I = I′B. Let ℐ′ be the sheaf of ideals on X′ := Spec B′ determined by I′; by the foregoing, L_{ℐ′} (d) is a constructible subset of X′. Thus, the assertion follows from (ii), and the fact that a morphism of schemes is continuous for the constructible topology ([63, Ch.IV, Prop.1.8.2]). □

10.5. Depth and associated primes.

Definition 10.5.1. Let X be a scheme, ℱ a quasi-coherent 𝒪_{X}-module.

(i) Let x ∈ X be any point; we say that x is associated with ℱ if there exists f ∈ ℱ_{x} such that the radical of the annihilator of f in 𝒪_{X,x} is the maximal ideal of 𝒪_{X,x}. If x is a point associated with ℱ and x is not a maximal point of Supp ℱ, we say that x is an imbedded point for ℱ. We shall denote :

Ass ℱ := {x ∈ X | x is associated with ℱ }.

(ii) We say that the 𝒪_{X}-module ℱ satisfies condition S_{1} if every associated point of ℱ is

a maximal point of X. (iii) Likewise, if X is affine, say X = Spec A, and M is any A-module, we denote by

Ass_{A} M ⊂ X (or just Ass M, if the notation is not ambiguous) the set of prime ideals associated with the 𝒪_{X}-module M^{∼} arising from M. An associated (resp. imbedded) point of M^{∼} is also called an associated (resp. imbedded) prime ideal of M.

We say that M satisfies condition S_{1} if the same holds for the 𝒪_{X}-module M^{∼}. (iv) Let x ∈ X be a point, 𝒢 a quasi-coherent 𝒪_{X}-submodule of ℱ. We say that 𝒢 is a

x-primary submodule of ℱ if Ass ℱ /𝒢 = {x}. We say that 𝒢 is a primary submodule of ℱ if there exists a point x ∈ X such that 𝒢 is x-primary.

(v) We say that a submodule 𝒢 of the 𝒪_{X}-module ℱ admits a primary decomposition if there exist primary submodules 𝒢_{1}, …, 𝒢_{n} ⊂ ℱ such that 𝒢 = 𝒢_{1} ∩ ⋯ ∩ 𝒢_{n}.

Remark 10.5.2. (i) Our definition of associated point is borrowed from [86, Partie I, Déf.3.2.1], and it corresponds to Bourbaki’s notion of weakly associated prime ([34, Ch.IV, §1, Exerc.17]), also called weak Bourbaki prime in other works. Bourbaki’s definition of associated prime, in [34, Ch.IV, §1, no.1] (which is the same as in [85, Exp.III, Déf.1.1]), agrees with ours for modules over noetherian rings (see lemma 10.5.3(ii)), but in general the two notions diverge.

(ii) For a noetherian ring A and a finitely generated A-module M, our condition S_{1} is the same as in [89]. It also agrees with that of [64, Ch.IV, Déf.5.7.2], in case Supp M = Spec A.

Lemma 10.5.3. Let A be a ring, and M any A-module. The following holds :

(i) Ass M is the set of all 𝔭 ∈ Spec A with the following property. There exists m ∈ M, such that 𝔭 is a maximal point of the closed subset Supp(m) (i.e. 𝔭 is the preimage of a minimal prime ideal of the ring A/Ann_{A}(m)).

(ii) If 𝔭 ∈ Ass M, and 𝔭 is finitely generated, there exists m ∈ M with 𝔭 = Ann_{A}(m). (iii) The natural map M → ∏_{𝔭∈Ass M} M_{𝔭} is injective.

Proof. (i): Indeed, suppose that m ∈ M \ {0} is any element, and 𝔭 is a maximal point of V (Ann_{A}(m)); denote by m_{𝔭} ∈ M_{𝔭} := M ⊗_{A} A_{𝔭} the image of m. Then :

(10.5.4)  Ann_{A_{𝔭}}(m_{𝔭}) = Ann_{A}(m) ⊗_{A} A_{𝔭}.

Hence A_{𝔭}/Ann_{A_{𝔭}}(m_{𝔭}) = (A/Ann_{A}(m)) ⊗_{A} A_{𝔭}, and the latter is by assumption a local ring of Krull dimension zero; it follows easily that the radical of Ann_{A_{𝔭}}(m_{𝔭}) is 𝔭A_{𝔭}, i.e. 𝔭 ∈ Ass_{A}M.

Conversely, suppose that 𝔭 ∈ Ass_{A}M; then there exists m_{𝔭} ∈ M_{𝔭} such that the radical of Ann_{A_{𝔭}}(m_{𝔭}) equals 𝔭A_{𝔭}. We may assume that m_{𝔭} is the image of some m ∈ M; then (10.5.4) implies that V (Ann_{A}(m)) ∩ Spec A_{𝔭} = V (Ann_{A_{𝔭}}(m_{𝔭})) = {𝔭}, so 𝔭 is a maximal point of V (Ann_{A}(m)).

(ii): Suppose 𝔭 ∈ Ass M is finitely generated; by definition, there exists y ∈ M_{𝔭} such that the radical of I := Ann_{A_{𝔭}}(y) equals 𝔭A_{𝔭}. Since 𝔭 is finitely generated, some power of 𝔭A_{𝔭} is contained in I. Let t ∈ ℕ be the smallest integer such that 𝔭^{t}A_{𝔭} ⊂ I, and pick any nonzero element x in 𝔭^{t−1}A_{𝔭}y; then Ann_{A_{𝔭}}(x) = 𝔭A_{𝔭}. After clearing some denominators, we may assume that x ∈ M. Let a_{1}, …, a_{r} be a finite set of generators for 𝔭; we may then find t_{1}, …, t_{r} ∈ A\𝔭 such that t_{i}a_{i}x = 0 in M, for every i ≤ r. Set x′ := t_{1} · . . . · t_{r}x; then 𝔭 ⊂ Ann_{A}(x′); however Ann_{A_{𝔭}}(x′) = 𝔭A_{𝔭}, therefore 𝔭 = Ann_{A}(x′), whence the contention.

(iii): The assertion means that for every m ∈ M \ {0} there exists 𝔭 ∈ Ass M such that the image of m does not vanish in M_{𝔭}. But according to (i), Ass M contains every maximal point 𝔭 of Supp(m). □

Lemma 10.5.5. Let X be a scheme, ℱ a quasi-coherent 𝒪_{X}-module. Then :

(i) Ass ℱ ⊂ Supp ℱ.

(ii) Ass ℱ_{|U} = U ∩ Ass ℱ for every open subset U ⊂ X.

(iii) ℱ = 0 if and only if Ass ℱ = ∅.

Proof. (i) and (ii) are obvious. To show (iii), we may assume that X is affine, according to (ii). Then the assertion follows immediately from lemma 10.5.5(iii). □

Proposition 10.5.6. Let X be a scheme, ℱ a quasi-coherent 𝒪_{X}-module. We have : (i) Ass ℱ = {x ∈ X | δ(x, ℱ) = 0}. (ii) If 0 → ℱ′ → ℱ → ℱ′′ is an exact sequence of quasi-coherent 𝒪_{X}-modules, then :

Ass ℱ′ ⊂ Ass ℱ ⊂ Ass ℱ′ ∪ Ass ℱ′′.

(iii) If x ∈ X is any point, and 𝒢_{1}, …, 𝒢_{n} (for some n > 0) are x-primary submodules of

ℱ, then 𝒢_{1} ∩ ⋯ ∩ 𝒢_{n} is also x-primary. (iv) Let f : X → Y be a finite morphism of schemes. Then

(a) The natural map :

⨁_{x∈f^{−1}(y)} Γ_{{x}}ℱ_{|X(x)} → Γ_{{y}}f_{∗}ℱ_{|Y (y)}

is a bijection for all y ∈ Y.

(b) Ass f_{∗}ℱ = f(Ass ℱ).

(v) Suppose that ℱ is the union of a filtered family (ℱ_{λ} | λ ∈ Λ) of quasi-coherent 𝒪_{X}-

submodules. Then :

⋃_{λ∈Λ} Ass ℱ_{λ} = Ass ℱ .

(vi) Let f : Y → X be a flat morphism of schemes, and suppose that the topological space |X| underlying X is locally noetherian. Then Ass f^{∗}ℱ ⊂ f^{−1}Ass ℱ.

Proof. (i) and (v) follow directly from the definitions.

(ii): Consider, for every point x ∈ X the induced exact sequence of 𝒪_{X(x)}-modules :

0 → \underline{Γ}_{\{x\}}ℱ′_{x} → \underline{Γ}_{\{x\}}ℱ_{x} → \underline{Γ}_{\{x\}}ℱ′′_{x}.

Then :

Supp \underline{Γ}_{\{x\}}ℱ′_{x} ⊂ Supp \underline{Γ}_{\{x\}}ℱ_{x} ⊂ Supp \underline{Γ}_{\{x\}}ℱ′_{x} ∪ Supp \underline{Γ}_{\{x\}}ℱ′′_{x}

which, in light of (i), is equivalent to the contention.

(iii): One applies (ii) to the natural injection : ℱ /(𝒢_{1} ∩ ⋯ ∩ 𝒢_{n}) → ⊕_{i=1}^{n}ℱ /𝒢_{i}.

(iv): We may assume that Y is a local scheme with closed point y ∈ Y. Let x_{1}, …, x_{n} ∈ X be the finitely many points lying over y; for every i, j ≤ n, we let

π_{j} : X(x_{j}) → Y and π_{ij} : X(x_{i}) ×_{X} X(x_{j}) → Y

be the natural morphisms. To ease notation, denote also by ℱ_{i} (resp. ℱ_{ij}) the pull back of of ℱ to X(x_{j}) (resp. to X(x_{i}) ×_{X} X(x_{j})). The induced morphism U := X(x_{1}) ⨿ ⋯ ⨿ X(x_{n}) → X is faithfully flat, so descent theory yields an exact sequence of 𝒪_{Y} -modules :

(10.5.7)  \xymatrix{0 \ar[r] & f_{∗}ℱ \ar[r] & ∏_{j=1}^{n} π_{j∗}ℱ_{j} \ar[r]^{p_{1}^{∗}} \ar[r]_{p_{2}^{∗}} & ∏_{i,j=1}^{n} π_{ij∗}ℱ_{ij}.}

where p_{1}, p_{2} : U ×_{X} U → U are the natural morphisms.

Claim 10.5.8. The induced maps :

Γ_{{y}}p_{1}^{∗}, Γ_{{y}}p_{2}^{∗} : ∏_{j=1}^{n} Γ_{{y}}π_{j∗}ℱ_{j} → ∏_{i,j=1}^{n} Γ_{{y}}π_{ij∗}ℱ_{ij}

coincide.

Proof of the claim. Indeed, it suffices to verify that they coincide after projecting onto each factor 𝒢_{ij} := Γ_{{y}}π_{ij∗}ℱ_{ij}. But this is clear from definitions if i = j. On the other hand, if i ≠ j, the image of π_{ij} in Y does not contain y, so the corresponding factor 𝒢_{ij} vanishes. ♢

From (10.5.7) and claim 10.5.8we deduce that the natural map

Γ_{{y}}f_{∗}ℱ → ∏_{j=1}^{n} Γ_{{y}}π_{j∗}ℱ_{j}

is a bijection. Clearly Γ_{{y}}π_{j∗}ℱ_{j} = Γ_{{x}}ℱ_{j}, so both assertions follow easily.

(vi): Let x ∈ X be any point; since |X| is locally noetherian, the subset {x} is closed and constructible in X(x), and then f^{−1}(x) is closed and constructible in Y ×_{X} X(x). Hence δ(y, f^{∗}ℱ) ≥ δ(x, ℱ) for every y ∈ f^{−1}(x) (lemma 10.4.17(iii) and theorem 10.4.21). Then the assertion follows from (i). □

Corollary 10.5.9. In the situation of corollary 10.4.47, suppose furthermore that |S| is a locally noetherian topological space. Then we have :

Ass ℱ ⊗_{𝒪_{X}} f^{∗}𝒢 = ⋃_{s∈Ass 𝒢} Ass ℱ ⊗_{𝒪_{S,s}} κ(s).

Proof. Under the stated assumptions, it is easily seen that, for every x ∈ X (resp. s ∈ S), the subset {x} (resp. {s}) is constructible in X(x) (resp. in S(s)). Then the assertion follows immediately from proposition 10.5.6(i) and theorem 10.4.21. □

Remark 10.5.10. (i) Actually, it can be shown that the extra assumption on |S| in corollary 10.5.9is superflous : see [86, Part I, Prop.3.4.3].

(ii) In the situation of proposition 10.5.6, one could also consider the set of points x ∈ X such that δ′(x, ℱ) = 0. This set contains Ass ℱ, by proposition 10.5.6(i) and lemma 10.4.20(i).

Such points are called attached primes or strong Krull primes of ℱ in some literature : see e.g. [68], [67] (though the terminology “attached prime” is used also for a different, unrelated notion: see the footnote on page 404 of [67]).

Corollary 10.5.11. Let X be a quasi-compact and quasi-separated scheme, ℱ a quasi-coherent 𝒪_{X}-module and 𝒢 ⊂ ℱ a quasi-coherent submodule. Then :

(i) For every quasi-compact open subset U ⊂ X and every quasi-coherent primary 𝒪_{U}- submodule 𝒩 ⊂ ℱ_{|U}, with 𝒢_{|U} ⊂ 𝒩 , there exists a quasi-coherent primary 𝒪_{X}-submodule ℳ ⊂ ℱ such that ℳ_{|U} = 𝒩 and 𝒢 ⊂ 𝒩 .

(ii) 𝒢 admits a primary decomposition if and only if there exists a finite open covering X = U_{1} ∪ ⋯ ∪ U_{n} consisting of quasi-compact open subsets, such that the submodules 𝒢_{|U_{i}} ⊂ ℱ_{|U_{i}} admit primary decompositions for every i = 1, …, n.

Proof. Say that 𝒩 is x-primary for some point x ∈ U, and set Z := X \U. According to [59, Ch.I, Prop.9.4.2], we can extend 𝒩 to a quasi-coherent 𝒪_{X}-submodule ℳ_{1} ⊂ ℱ; up to replacing ℳ_{1} by ℳ_{1} + 𝒢, we may assume that 𝒢 ⊂ ℳ_{1}. Since (ℱ /ℳ_{1})_{|U} = ℱ_{|U}/𝒩 , it follows from lemma 10.5.5(ii) that Ass ℱ /ℳ_{1} ⊂ \{x\} ∪ Z. Let \overline{ℳ} := \underline{Γ}_{Z}(ℱ /ℳ_{1}), and denote by ℳ the preimage of \overline{ℳ} in ℱ. There follows a short exact sequence :

0 → \overline{ℳ} → ℱ /ℳ_{1} → ℱ /ℳ → 0.

Clearly R^{1}\underline{Γ}_{Z} \overline{ℳ} = 0, whence a short exact sequence

0 → \underline{Γ}_{Z} \overline{ℳ} → \underline{Γ}_{Z}(ℱ /ℳ_{1}) → \underline{Γ}_{Z}(ℱ /ℳ) → 0.

We deduce that depth_{Z}(ℱ /ℳ) > 0; then theorem 10.4.21and proposition 10.5.6(i) show that Z ∩ Ass (ℱ /ℳ) = ∅, so that ℳ is x-primary, as required.

(ii): We may assume that a covering X = U_{1} ∪ ⋯ ∪ U_{n} is given with the stated property. For every i ≤ n, let 𝒢_{|U_{i}} = 𝒩_{i1} ∩ ⋯ ∩ 𝒩_{ik_{i}} be a primary decomposition; by (i) we may extend every 𝒩_{ij} to a primary submodule ℳ_{ij} ⊂ ℱ containing 𝒢. Then 𝒢 = ⋂_{i=1}^{n} ⋂_{j=1}^{k_{i}} ℳ_{ij} is a primary decomposition of 𝒢. □

10.5.12. Let A, B be two local rings, with A a domain, f : A → B a local ring homomorphism, M an f-flat B-module of finite presentation. Denote by K (resp. κ) the field of fractions (resp. the residue field) of A, and set

B_{0} := B ⊗_{A} κ  B_{K} := B ⊗_{A} K  M_{0} := M ⊗_{A} κ  M_{K} := M ⊗_{A} K.

Proposition 10.5.13. In the situation of (10.5.12), suppose that :

(a) either, A and B are both noetherian rings

(b) or else, f is essentially of finite presentation. Let 𝔭 ∈ Ass_{B_{K}}M_{K} be any point, and denote by V (𝔭) the topological closure of {𝔭} in Spec B. Then all the maximal points of V (𝔭) ∩ Spec B_{0} lie in Ass_{B_{0}}M_{0}.

Proof. We consider first case (a). Via the inclusion map Spec B_{K} → Spec B we may regard 𝔭 as a prime ideal of B, and Ass_{B_{K}}M_{K} as a subset of Ass_{B}M; by lemma 10.5.3(ii), there exists a cyclic B-submodule N of M such that Ann_{B}N = 𝔭. Denote by 𝔪_{A} the maximal ideal of A, endow M with its 𝔪_{A}-adic filtration Fil^{•}M, and N with the filtration Fil^{•}N induced by Fil^{•}M. Let also gr^{•}M and gr^{•}N be the graded B_{0}-modules associated with these filtrations on M and N. Since M is f-flat, we have a B_{0}-linear isomorphism

gr^{i}M →^{∼} (𝔪_{A}^{i}/𝔪_{A}^{i+1}) ⊗_{κ} M_{0} for every i ∈ ℕ

which implies that Ass_{B}(gr^{i}M) = Ass_{B}M_{0} for every i ∈ ℕ. From this, proposition 10.5.6(ii) and an easy induction on i ∈ ℕ, we deduce that

Ass_{B}(N/Fil^{i}N) ⊂ Ass_{B}(M/Fil^{i}M) ⊂ Ass_{B}M_{0} for every i ∈ ℕ.

On the other hand, by the Artin-Rees lemma ([126, Th.8.5]), there exist i, j ∈ ℕ such that

𝔪_{A}^{j}N ⊂ Fil^{i}N ⊂ 𝔪_{A}N

so that Supp N/𝔪_{A}N = Supp N/Fil^{i}N. However, lemma 10.5.3(i) implies that all the maximal points of Supp N/Fil^{i}N lie in Ass_{B}(N/Fil^{i}N). To conclude, it suffices to remark that the support of N/𝔪_{A}N equals V (𝔭) ∩ Spec B_{0}.

Next, suppose that (b) holds, and say that B = C_{𝔮} for some finitely presented A-algebra C and some prime ideal 𝔮 ⊂ C whose preimage in A is the maximal ideal. We write A as the colimit of a filtered system (A_{λ} | λ ∈ Λ) of noetherian local subrings, such that the transition maps A_{μ} → A_{λ} are local ring homomorphisms for every λ, μ ∈ Λ with λ ≥ μ; then there exist μ ∈ Λ and an A_{μ}-algebra C_{μ} of finite type with an isomorphism of C_{μ} ⊗_{A_{μ}} A → C of A-algebras. For every λ ≥ μ, let C_{λ} := C_{μ} ⊗_{A_{μ}} A_{λ}, denote by 𝔮_{λ} ⊂ C_{λ} the preimage of 𝔮 under the induced map C_{λ} → C, and set B_{λ} := (C_{λ})_{𝔮_{λ}}. Clearly B is the colimit of the resulting filtered system of rings (B_{λ} | λ ∈ Λ, λ ≥ μ), and the induced maps A_{λ} → B_{λ} are local ring homomorphisms, for every λ ≥ μ; therefore we may assume that there exists a finitely generated B_{μ}-module M_{μ} with a B-linear isomorphism M_{μ} ⊗_{B_{μ}} B →^{∼} M, and we may likewise define M_{λ} := M_{μ} ⊗_{B_{μ}} B_{λ} for every λ ≥ μ. In view of [65, Ch.IV, Cor.11.2.6.1(i)], we may further assume that M_{λ} is a flat A_{λ}-module, for every λ ≥ μ, and after replacing Λ by a cofinal subset, we may assume that μ is the initial element of the partially ordered set Λ. Then, to ease notation, denote by K_{λ} (resp. κ_{λ}) the field of fractions (resp. the residue field) of A_{λ}, and set

\begin{matrix}B_{K,λ} := & B_{λ} ⊗_{A_{λ}} K_{λ} & M_{K,λ} := & M_{λ} ⊗_{A_{λ}} K_{λ} & C_{K,λ} := & C_{λ} ⊗_{A_{λ}} K_{λ} \\ B_{0,λ} := & B_{λ} ⊗_{A_{λ}} κ_{λ} & M_{0,λ} := & M_{λ} ⊗_{A_{λ}} κ_{λ} \end{matrix} for every λ ∈ Λ.

Notice that the resulting map f_{λ} : C_{K,λ} → C_{K} := C ⊗_{A} K induces an isomorphism

C_{K,λ} ⊗_{K_{λ}} K →^{∼} C_{K}.

So, f_{λ} is a flat ring homomorphism, hence the natural morphism φ_{K,λ} : Spec B_{K} → Spec B_{K,λ} is also flat, and proposition 10.5.6(vi) yields

(10.5.14)  Ass_{B_{K}}M_{K} ⊂ φ_{K,λ}^{−1}Ass_{B_{K,λ}}(M_{K,λ})  for every λ ∈ Λ.

Now, say that 𝔭 ∈ Ass_{B_{K}}M_{K}, and let 𝔭_{λ} ⊂ B_{K,λ} be the preimage of 𝔭, for every λ ∈ Λ; we have 𝔭_{λ} ∈ Ass_{B_{K,λ}}(M_{K,λ}) by virtue of (10.5.14), so in this situation the case (a) of the proposition applies and shows that

Z_{λ} := Max(Spec B_{0,λ}/𝔭_{λ}B_{0,λ}) ⊂ Z′_{λ} := Ass_{B_{λ}}(M_{0,λ}) for every λ ∈ Λ

(where, for a scheme Z, we have denoted Max(Z) the set of maximal points of Z). Notice moreover, that the natural morphism φ_{0,λ} : Spec B_{0} → Spec B_{0,λ} is also flat, so we have

Max(Spec B_{0}/𝔭_{λ}B_{0}) = ⋃_{z∈Z_{λ}} Max(φ_{0,λ}^{−1}(z)) for every λ ∈ Λ

by the going down theorem ([126, Th.9.5]). On the other hand, from [64, Ch.IV, Prop.4.2.7] we have

⋃_{z∈Z′_{λ}} Max(φ_{0,λ}^{−1}(z)) = (Spec B_{0}) ∩ Ass_{B_{0,λ}⊗_{k_{λ}}κ}(M_{0,λ} ⊗_{κ_{λ}} κ) = Ass_{B_{0}}(M_{0})

for every λ ∈ Λ. Summing up, we are reduced to checking that there exists λ ∈ Λ such that the surjection B_{0}/𝔭_{λ}B_{0} → B_{0}/𝔭B_{0} is an isomorphism, i.e. such that 𝔭_{λ}B_{0} = 𝔭B_{0}. However, clearly 𝔭B_{0} is the filtered union of the system of subideals (𝔭_{λ}B_{0} | λ ∈ Λ); but B_{0} is a noetherian ring, so the claim is obvious. □

Lemma 10.5.15. Let X be a quasi-separated and quasi-compact scheme, ℱ a quasi-coherent 𝒪_{X}-module of finite type. Then the submodule 0 ⊂ ℱ admits a primary decomposition if and only if the following conditions hold:

(a) Ass ℱ is a finite set.

(b) For every x ∈ Ass ℱ there is a x-primary ideal I ⊂ 𝒪_{X,x} such that the natural map

Γ_{{x}}ℱ_{x} → ℱ_{x}/Iℱ_{x}

is injective.

Proof. In view of corollary 10.5.11(i) we are reduced to the case where X is an affine scheme, say X = Spec A, and ℱ = M^{∼} for an A-module M of finite type. Suppose first that 0 admits a primary decomposition :

(10.5.16)  0 = ⋂_{i=1}^{k} N_{i}.

Then the natural map M → ⊕_{i=1}^{k}M/N_{i} is injective, hence

(10.5.17)  Ass M ⊂ Ass ⨁_{i=1}^{k} M/N_{i} ⊂ ⋃_{i=1}^{k} Ass M/N_{i}

by proposition 10.5.6(ii), and this shows that (a) holds. Next, if N_{i} and N_{j} are 𝔭-primary for the same prime ideal 𝔭 ⊂ A, we may replace both of them by their intersection (proposition 10.5.6(iii)). Proceeding in this way, we achieve that the N_{i} appearing in (10.5.16) are primary submodules for pairwise distinct prime ideals 𝔭_{1}, …, 𝔭_{k} ⊂ A. By (10.5.17) we have Ass M ⊂ {𝔭_{1}, …, 𝔭_{k}}. Suppose that 𝔭_{1} ∉ Ass M, and set Q := Ker (M → ⊕_{i=2}^{k}M/N_{i}). For every j > 1 we have Γ_{{𝔭_{j}}}(M/N_{1})^{∼} = 0, therefore :

Γ_{{𝔭_{j}}}Q_{𝔭_{j}}^{∼} = Ker (Γ_{{𝔭_{j}}}M_{𝔭_{j}}^{∼} → ⊕_{i=1}^{k}(M/N_{i})_{𝔭_{j}}^{∼}) = 0.

Hence Ass Q = ∅, by proposition 10.5.6(i), and then Q = 0 by lemma 10.5.5(iii). In other words, we can omit N_{1} from (10.5.16), and still obtain a primary decomposition of 0; iterating this argument, we may achieve that Ass M = {𝔭_{1}, …, 𝔭_{k}}. After these reductions, we see that

Γ_{{𝔭_{j}}}(M/N_{i})_{𝔭_{j}}^{∼} = 0 whenever i ≠ j

and consequently :

Ker (φ_{j} : Γ_{{𝔭_{j}}}M_{𝔭_{j}}^{∼} → (M/N_{j})_{𝔭_{j}}) = 0 for every j = 1, …, k.

Now, for given j ≤ k, let \overline{f}_{1}, …, \overline{f}_{n} be a system of non-zero generators for M/N_{j}; by assumption I_{i} := Ann_{A}(\overline{f}_{i}) is a 𝔭_{j}-primary ideal for every i ≤ n. Hence 𝔮_{j} := Ann_{A}(M/N_{j}) = ⋂_{i=1}^{n} I_{i} is 𝔭_{j}-primary as well; since φ_{j} factors through (M/𝔮_{j}M)_{𝔭_{j}}, we see that (b) holds.

Conversely, suppose that (a) and (b) hold. Say that Ass M = {𝔭_{1}, …, 𝔭_{k}}; for every j ≤ k we choose a 𝔭_{j}-primary ideal 𝔮_{j} such that the map Γ_{{𝔭_{j}}}M_{𝔭_{j}}^{∼} → M_{𝔭_{j}}/𝔮_{j}M_{𝔭_{j}} is injective. Clearly N_{j} := Ker (M → M_{𝔭_{j}}/𝔮_{j}M_{𝔭_{j}}) is a 𝔭_{j}-primary submodule of M; moreover, the induced map φ : M → ⊕_{j=1}^{k}M/N_{j} is injective, since Ass Ker φ = ∅ (proposition 10.5.6(ii) and lemma 10.5.5(iii)). In other words, 0 = ⋂_{j=1}^{k} N_{j} is a primary decomposition of 0. □

Proposition 10.5.18. Let X be a quasi-compact and quasi-separated scheme, and 0 → ℱ′ → ℱ → ℱ′′ → 0 a short exact sequence of quasi-coherent 𝒪_{X}-modules of finite type such that

(a) Ass ℱ′ ∩ Supp ℱ′′ = ∅.

(b) The submodules 0 ⊂ ℱ′ and 0 ⊂ ℱ′′ admit primary decompositions. Then the submodule 0 ⊂ ℱ admits a primary decomposition.

Proof. The assumptions imply that Ass ℱ′ and Ass ℱ′′ are finite sets, (lemma 10.5.15), hence the same holds for Ass ℱ (proposition 10.5.6(ii)). Given any point x ∈ X and any x-primary ideal I ⊂ 𝒪_{X,x}, we may consider the commutative diagram with exact rows:

[figure: 0; Γ_{{x}}ℱ′_{x}; α; Γ_{{x}}ℱ_{x}; β; Γ_{{x}}ℱ′′_{x}; γ; Tor_{1}^{𝒪_{X,x}} (I, ℱ′′_{x}); ℱ′_{x}/Iℱ′_{x}; ℱ_{x}/Iℱ_{x}; ℱ′′_{x}/Iℱ′′_{x}]

Now, if x ∈ Ass ℱ′, assumption (a) implies that Γ_{{x}}ℱ′′_{x} = 0 = Tor_{1}^{𝒪_{X,x}} (I_{x}, ℱ′′_{x}), and by (b) and lemma 10.5.15 we can choose I such that α is injective; a little diagram chase then shows that β is injective as well. Similarly, if x ∈ Ass ℱ′′, we have Γ_{{x}}ℱ′_{x} = 0 and we may choose I such that γ is injective, which implies again that β is injective. To conclude the proof it suffices to apply again lemma 10.5.15. □

Proposition 10.5.19. Let Y be a quasi-compact and quasi-separated scheme, f : X → Y a finite morphism and ℱ a quasi-coherent 𝒪_{X}-module of finite type. Then the 𝒪_{X}-submodule 0 ⊂ ℱ admits a primary decomposition if and only if the same holds for the 𝒪_{Y} -submodule 0 ⊂ f_{∗}ℱ.

Proof. Under the stated assumptions, we can apply the criterion of lemma 10.5.15. To start out, it is clear from proposition 10.5.6(iv.b) that Ass ℱ is finite if and only if Ass f_{∗}ℱ is finite. Next, suppose that 0 ⊂ ℱ admits a primary decomposition, let y ∈ Ass f_{∗}ℱ be any point and set f^{−1}(y) := {x_{1}, …, x_{n}}; for every j ≤ n we can find an x_{j}-primary ideal I_{j} ⊂ 𝒪_{X} such that the map Γ_{{x_{j}}}ℱ_{x_{j}} → ℱ_{x_{j}}/I_{j}ℱ_{x_{j}} is injective. Let I be the kernel of the natural map 𝒪_{Y} → f_{∗}(𝒪_{X}/(I_{1} ∩⋯∩I_{n})). Then I is a quasi-coherent y-primary ideal of 𝒪_{Y} and we deduce a commutative diagram :

(10.5.20)  [figure: ⨁_{j=1}^{n} Γ_{{x_{j}}}(ℱ_{x_{j}}); α; Γ_{{y}}(f_{∗}ℱ_{y}); β; ⨁_{j=1}^{n} ℱ_{x_{j}}/Iℱ_{x_{j}}; f_{∗}ℱ_{y}/If_{∗}ℱ_{y}]

whose horizontal arrows are isomorphisms, in view of proposition 10.5.6(iv.a), and where α is injective by construction. It follows that β is injective, so condition (b) of lemma 10.5.15 holds for the stalk f_{∗}ℱ_{y}, and since y ∈ Ass f_{∗}ℱ is arbitrary, we see that 0 ⊂ f_{∗}ℱ admits a primary decomposition. Conversely, suppose that 0 ⊂ f_{∗}ℱ admits a primary decomposition; then for every y ∈ Y we can find a quasi-coherent y-primary ideal I ⊂ 𝒪_{Y} such that the map β of (10.5.20) is injective; hence α is injective as well, and again we deduce easily that 0 ⊂ ℱ admits a primary decomposition. □

Proposition 10.5.21. Let X and ℱ be as in lemma 10.5.15, i : Z → X a closed constructible immersion, and U := X \Z. Suppose that :

(a) The 𝒪_{U}-submodule 0 ⊂ ℱ_{|U} and the 𝒪_{Z}-submodule 0 ⊂ i^{∗}ℱ admit primary decompositions.

(b) The natural map \underline{Γ}_{Z}ℱ → i_{∗}i^{∗}ℱ is injective. Then the 𝒪_{X}-submodule 0 ⊂ ℱ admits a primary decomposition.

Proof. We shall verify that conditions (a) and (b) of lemma 10.5.15 hold for ℱ. To check condition (a), it suffices to remark that U ∩ Ass ℱ = Ass ℱ_{|U} (which is obvious) and that Z ∩ Ass ℱ ⊂ Ass i^{∗}ℱ, which follows easily from our current assumption (b).

Next we check that condition (b) of loc.cit. holds. This is no problem for the points x ∈ U ∩ Ass ℱ, so suppose that x ∈ Z ∩ Ass ℱ. Moreover, we may also assume that X is affine, say X = Spec A, so that Z = V (J) for some ideal J ⊂ A. Due to proposition 10.5.19 we know that 0 ⊂ 𝒢 := i_{∗}i^{∗}ℱ admits a primary decomposition, hence we can find an x-primary ideal I ⊂ A such that the natural map Γ_{{x}}𝒢 → 𝒢_{x}/I𝒢_{x} ≃ ℱ_{x}/(I + J)ℱ_{x} is injective. Clearly I + J is again a x-primary ideal; since Z is closed and constructible, corollary 10.4.10and our assumption (b) imply that the natural map Γ_{{x}}ℱ → Γ_{{x}}𝒢 is injective, hence the same holds for the map Γ_{{x}}ℱ → ℱ_{x}/(I + J)ℱ_{x}, as required. □

10.5.22. Let now A be a ring, 𝔭 ⊂ A any prime ideal, and n ≥ 1 any integer; if A/𝔭^{n} does not admit imbedded primes, then the ideal 𝔭^{n} is 𝔭-primary. In the presence of imbedded primes, this fails. For instance, we have the following :

Example 10.5.23. Let k be a field, k[x, y] the free polynomial algebra in indeterminates x and y; consider the ideal I := (xy, y^{2}), and set A := k[x, y]/I. Let \overline{x} and \overline{y} be the images of x and y in A; then 𝔭 := (\overline{y}) ⊂ A is a prime ideal, and 𝔭^{2} = 0. However, 𝔪 := Ann_{A}(\overline{y}) is the maximal ideal generated by \overline{x} and \overline{y}, so the ideal 0 ⊂ A is not 𝔭-primary.

There is however a natural sequence of 𝔭-primary ideals naturally attached to 𝔭. To explain this, let us remark, more generally, the following :

Lemma 10.5.24. Let A be a ring, 𝔭 ⊂ A a prime ideal. Denote by φ_{𝔭} : A → A_{𝔭} the localization map. The rule :

I ↦ φ_{𝔭}^{−1}I

induces a bijection from the set of 𝔭A_{𝔭}-primary ideals of A_{𝔭}, to the set of 𝔭-primary ideals of A.

Proof. Suppose that I ⊂ A_{𝔭} is 𝔭A_{𝔭}-primary; since the natural map A/φ_{𝔭}^{−1}I → A_{𝔭}/I is injective, it is clear that φ_{𝔭}^{−1}I is 𝔭-primary. Conversely, suppose that J ⊂ A is 𝔭-primary; we claim that J = φ_{𝔭}^{−1}(J_{𝔭}). Indeed, by assumption (and by lemma 10.5.5(iii)) we have (A/J)_{𝔮} = 0 whenever 𝔮 ≠ 𝔭; it follows easily that the localization map A/J → A_{𝔭}/J_{𝔭} is an isomorphism, whence the contention. □

Definition 10.5.25. Keep the notation of lemma 10.5.24; for every n ≥ 0 one defines the n-th symbolic power of 𝔭, as the ideal :

𝔭^{(n)} := φ_{𝔭}^{−1}(𝔭^{n}A_{𝔭}).

By lemma 10.5.24, the ideal 𝔭^{(n)} is 𝔭-primary for every n ≥ 1. More generally, for every A-module M, let φ_{M,𝔭} : M → M_{𝔭} be the localization map; then one defines the 𝔭-symbolic filtration on M, by the rule :

Fil_{𝔭}^{(n)}M := φ_{M,𝔭}^{−1}(𝔭^{n}M_{𝔭}) for every n ≥ 0.

The filtration Fil_{𝔭}^{(•)}M induces a (linear) topology on M, called the 𝔭-symbolic topology.

More generally, if Σ ⊂ Spec A is any subset, we define the Σ-symbolic topology on M, as the coarsest linear topology 𝒯_{Σ} on M such that Fil_{𝔭}^{(n)}M is an open subset of 𝒯_{Σ}, for every 𝔭 ∈ Σ and every n ≥ 0. If Σ is finite, it is induced by the Σ-symbolic filtration, defined by the submodules :

Fil_{Σ}^{(n)}M := ⋂_{𝔭∈Σ} Fil_{𝔭}^{(n)}M for every n ≥ 0.

10.5.26. Let A be a ring, M an A-module, I ⊂ A an ideal. We shall show how to characterize the finite subsets Σ ⊂ Spec A such that the Σ-symbolic topology on M agrees with the I-preadic topology (see theorem 10.5.35). Hereafter, we begin with a few preliminary observations. First, suppose that A is noetherian; then it is easily seen that, for every prime ideal 𝔭 ⊂ A, and every 𝔭A_{𝔭}-primary ideal I ⊂ A_{𝔭}, there exists n ∈ ℕ such that 𝔭^{n}A_{𝔭} ⊂ I. From lemma 10.5.24 we deduce that every 𝔭-primary ideal of A contains a symbolic power of 𝔭; i.e., every 𝔭-primary ideal is open in the 𝔭-symbolic topology of A. More generally, let Σ ⊂ Spec A be any subset, M a finitely generated A-module, and N ⊂ M a submodule; from the existence of a primary decomposition for N ([126, Th.6.8]), we see that N is open in the Σ-symbolic topology of M, whenever Ass M/N ⊂ Σ. Especially, if Ass M/I^{n}M ⊂ Σ for every n ∈ ℕ, then the Σ-symbolic topology is finer than the I-preadic topology on M. On the other hand, clearly I^{n}M ⊂ Fil_{𝔭}^{(n)}M for every 𝔭 ∈ Supp M/IM and every n ∈ ℕ, so the I-preadic topology on M is finer than the Supp M/IM-symbolic topology. Summing up, if we have :

Σ_{0}(M) := ⋃_{n∈ℕ} Ass M/I^{n}M ⊂ Σ ⊂ Supp M/IM

then the Σ-symbolic topology on M agrees with the I-preadic topology. Notice that the above expression for Σ_{0}(M) is a union of finite subsets ([126, Th.6.5(i)]), hence Σ_{0}(M) is – a priori – at most countable; in fact, we shall show that Σ_{0}(M) is finite. Indeed, for every n ∈ ℕ, set :

gr_{I}^{n}A := I^{n}/I^{n+1} gr_{I}^{n}M := I^{n}M/I^{n+1}M.

Then gr_{I}^{•}A := ⊕_{n∈ℕ}gr_{I}^{n}A is naturally a graded A/I-algebra, and gr_{I}^{•}M := ⊕_{n∈ℕ}gr_{I}^{n}M is a graded gr_{I}^{•}A-module. Let ψ : Spec gr_{I}^{•}A → Spec A/I be the natural morphism, and set :

Σ(M) := ψ(Ass_{gr_{I}^{•}A}(gr_{I}^{•}M)).

Lemma 10.5.27. With the notation of (10.5.26), we have :

Ass_{A/I}(gr_{I}^{n}M) ⊂ Σ(M) for every n ∈ ℕ.

Proof. To ease notation, set A_{0} := A/I and B := gr_{I}^{•}A. Suppose that 𝔭 ∈ Ass_{A_{0}}(gr_{I}^{n}M); by lemma 10.5.3(i), there exists m ∈ gr_{I}^{n}M such that 𝔭 is the preimage of a minimal prime ideal of A_{0}/Ann_{A_{0}}(m). However, if we regard m as a homogeneous element of the B-module gr_{I}^{•}M, we have the obvious identity :

Ann_{A_{0}}(m) = A_{0} ∩ Ann_{B}(m) ⊂ B

from which we see that the induced map

A_{0}/Ann_{A_{0}}(m) → B/Ann_{B}(m)

is injective, hence ψ restricts to a dominant morphism V (Ann_{B}(m)) → V (Ann_{A_{0}}(m)). Especially, there exists 𝔮 ∈ V (Ann_{B}(m)) with ψ(𝔮) = 𝔭; up to replacing 𝔮 by a generization, we may assume that 𝔮 is a maximal point of V (Ann_{B}(m)), hence 𝔮 is an associated prime for gr_{I}^{•}M, again by lemma 10.5.3(i). □

10.5.28. An easy induction, starting from lemma 10.5.27, shows that Ass_{A}M/I^{n}M ⊂ Σ(M), for every n ∈ ℕ, therefore Σ_{0}(M) ⊂ Σ(M). However, if A is noetherian, the same holds for gr_{I}^{•}A (since the latter is a quotient of an A/I-algebra of finite type), hence Σ(M) is finite, provided M is finitely generated ([126, Th.6.5(i)]), and a fortiori, the same holds for Σ_{0}(M).

Remark 10.5.29. Another proof of the finiteness of Σ_{0}(M) can be found in [42].

10.5.30. Next, we wish to show that actually there exists a smallest subset Σ_{min}(M) ⊂ Spec A such that the Σ_{min}(M)-symbolic topology on M agrees with I-preadic topology; after some simple reductions, this boils down to the following assertion. Let Σ ⊂ Spec A be a finite subset, and 𝔭, 𝔭′ ∈ Σ two elements, such that the Σ-symbolic topology on M agrees with both the Σ \{𝔭}-symbolic topology and the Σ\{𝔭′}-symbolic topology; then these topologies agree as well with the Σ \{𝔭, 𝔭′}-symbolic topology. Indeed, given any subset Σ′ ⊂ Spec A with 𝔭 ∈ Σ′, for the Σ′-symbolic and the Σ′ \{𝔭}-symbolic topologies to agree, it is necessary and sufficient that, for every n ∈ ℕ there exists m ∈ ℕ such that :

Fil_{Σ′\{𝔭}}^{(m)}M ⊂ Fil_{𝔭}^{(n)}M or, what is the same : Fil_{Σ′\{𝔭}}^{(m)}M_{𝔭} ⊂ 𝔭^{n}M_{𝔭}.

In the latter inclusion, we may then replace Σ′ \{𝔭} by the smaller subset Σ′ ∩ Spec A_{𝔭} \{𝔭}, without changing the two terms. Suppose now that 𝔭′ ∉ Spec A_{𝔭}; then if we apply the above, first with Σ′ := Σ, and then with Σ′ := Σ \{𝔭′}, we see that the Σ-symbolic topology agrees with the Σ\{𝔭}-symbolic topology, if and only if the Σ\{𝔭′}-symbolic topology agrees with the Σ\{𝔭, 𝔭′}-symbolic topology, whence the contention. In case 𝔭′ ∈ Spec A_{𝔭}, we may assume that 𝔭 ≠ 𝔭′, otherwise there is nothing to prove; then we shall have 𝔭 ∉ Spec A_{𝔭′}, so the foregoing argument still goes through, after reversing the roles of 𝔭 and 𝔭′.

10.5.31. Finally, theorem 10.5.35 will characterize the subset Σ_{min}(M) as in (10.5.28). To this aim, for every prime ideal 𝔭 ⊂ A, let A_{𝔭}^{∧} denote the 𝔭-adic completion of A; we set :

Ass_{A}(I, M) := \{𝔭 ∈ Spec A | there exists 𝔮 ∈ Ass_{A_{𝔭}^{∧}}M ⊗_{A}A_{𝔭}^{∧} such that \sqrt{𝔮 + IA_{𝔭}^{∧}} = 𝔭A_{𝔭}^{∧}\}

(where, for any ideal J ⊂ A_{𝔭}^{∧} we denote by \sqrt{J} ⊂ A_{𝔭}^{∧} the radical of J, so the above condition selects the points 𝔮 ∈ Spec A_{𝔭}^{∧}, such that \overline{\{𝔮\}} ∩ V (IA_{𝔭}^{∧}) = \{𝔭A_{𝔭}^{∧}\}).

Lemma 10.5.32. Let A be a noetherian ring, M an A-module. Then we have :

(i) depth_{A_{𝔭}}M ⊗_{A}A_{𝔭} = depth_{A_{𝔭}^{∧}}M ⊗_{A}A_{𝔭}^{∧} for every 𝔭 ∈ Spec A.

(ii) Ass_{A}(0, M) = Ass_{A}M.

(iii) Ass_{A}(I, M) ⊂ V (I) ∩ Supp M.

(iv) Suppose that M is the union of a filtered family (M_{λ} | λ ∈ Λ) of A-submodules. Then :

Ass_{A}(I, M) = ⋃_{λ∈Λ} Ass_{A}(I, M_{λ}).

(v) Ass_{A}(I, M) contains the maximal points of V (I) ∩ Supp M.

Proof. (iii) is immediate, and in view of [126, Th.8.8], (i) is a special case of corollary 10.4.37.

(ii): By definition, Ass_{A}(0, M) consists of all the prime ideals 𝔭 ⊂ A such that 𝔭A_{𝔭}^{∧} is an associated prime ideal of M ⊗_{A}A_{𝔭}^{∧}; in light of proposition 10.5.6(i), the latter condition holds if and only if depth_{A_{𝔭}^{∧}}M ⊗_{A} A_{𝔭}^{∧} = 0, hence if and only if depth_{A_{𝔭}}M ⊗_{A} A_{𝔭} = 0, by (i); to conclude, one appeals again to proposition 10.5.6(i).

(iv): In view of proposition 10.5.6(v), we have Ass_{A_{𝔭}^{∧}}M⊗_{A}A_{𝔭}^{∧} = ⋃_{λ∈Λ} Ass_{A_{𝔭}^{∧}}M_{λ}⊗_{A}A_{𝔭}^{∧}; the contention is an immediate consequence.

(v): In view of (iv), we are easily reduced to the case where M is a finitely generated Amodule, in which case Supp M = V (J) for some ideal J ⊂ A. Hence, suppose that 𝔭 ⊂ A is a maximal point of Spec A/(I + J), in other words, the preimage of a minimal prime ideal of A/(I + J); notice that Supp M ⊗_{A}A_{𝔭}^{∧} = V (JA_{𝔭}^{∧}). Hence we have (J + I)A_{𝔭}^{∧} ⊂ 𝔮 + IA_{𝔭}^{∧} for every 𝔮 ∈ Supp M ⊗_{A}A_{𝔭}^{∧}, so the radical of 𝔮 + IA_{𝔭}^{∧} equals 𝔭A_{𝔭}^{∧}, as required. □

Lemma 10.5.33 (Chevalley’s theorem). Let (A, 𝔪) be an 𝔪-adically complete local noetherian ring, M a finitely generated A-module, and (Fil^{n}M | n ∈ ℕ) a descending filtration consisting of A-submodules of M. Then the following conditions are equivalent :

(a) ⋂_{n∈ℕ} Fil^{n}M = 0.

(b) For every i ∈ ℕ there exists n ∈ ℕ such that Fil^{n}M ⊂ 𝔪^{i}M.

Proof. Clearly (b)⇒(a), hence suppose that (a) holds. For every i, n ∈ ℕ, set

J_{i,n} := Im(Fil^{n}M → M/𝔪^{i}M).

For given i ∈ ℕ, the A-module M/𝔪^{i}M is artinian, hence there exists n ∈ ℕ such that J_{i} := J_{i,n} = J_{i,n′} for every n′ ≥ n. Assertion (b) then follows from the following :

Claim 10.5.34. If (a) holds, J_{i} = 0 for every i ∈ ℕ.

Proof of the claim. By inspecting the definition, it is easily seen that the natural A/𝔪^{i+1}- linear map J_{i+1} → J_{i} is surjective for every i ∈ ℕ, hence we are reduced to showing that J := lim_{i∈ℕ} J_{i} vanishes. However, J is naturally a submodule of lim_{i∈ℕ} M/𝔪^{i}M ≃ M, and if x ∈ M lies in J, then we have x ∈ Fil^{n}M + 𝔪^{i}M for every i, n ∈ ℕ. Since Fil^{n}M is a closed subset for the 𝔪-adic topology of M ([126, Th.8.10(i)]), we have ⋂_{i∈ℕ}(Fil^{n}M + 𝔪^{i}M) = Fil^{n}M, for every n ∈ ℕ, hence x ∈ ⋂_{n∈ℕ} Fil^{n}M, which vanishes, if (a) holds. □

The following theorem generalizes [145, Ch.VIII, §5, Cor.5] and [88, Prop.7.1], and is closely related to [148, Th.3.2]; see also [51, Th.3.6].

Theorem 10.5.35. Let A be a noetherian ring, I ⊂ A an ideal, M a finitely generated Amodule, and Σ ⊂ Spec A/I a finite subset. Then the Σ-symbolic topology on M agrees with the I-preadic topology if and only if Ass_{A}(I, M) ⊂ Σ.

Proof. Let 𝔭 ∈ Ass_{A}(I, M) \ Σ, and suppose, by way of contradiction, that the Σ-symbolic topology agrees with the I-adic topology, i.e. for every n ∈ ℕ there exists m ∈ ℕ such that :

Fil_{Σ}^{(m)}M ⊂ I^{n}M.

Let X := Spec A_{𝔭} and U := X \{𝔭}; after localizing at the prime 𝔭, we deduce the inclusion :

(10.5.36)  Fil_{Σ∩U}^{(m)}M_{𝔭} ⊂ I^{n}M_{𝔭}

(cp. the discussion in (10.5.28)). Let also ℳ := M^{∼}, the quasi-coherent 𝒪_{X}-module associated to M; clearly we have I^{m}M ⊂ Fil_{𝔮}^{(m)}M for every 𝔮 ∈ Spec A/I, hence (10.5.36) implies the inclusion :

{x ∈ M_{𝔭} | x_{|U} ∈ I^{m}ℳ (U)} ⊂ I^{n}M_{𝔭}.

Let A_{𝔭}^{∧} (resp. M_{𝔭}^{∧}) be the 𝔭-adic completion of A (resp. of M), f : X^{∧} := Spec A_{𝔭}^{∧} → X the natural morphism, and set U^{∧} := f^{−1}U = X^{∧} \ {𝔭A_{𝔭}^{∧}}. Since f is faithfully flat, and U is quasi-compact, we deduce that :

(10.5.37)  {x ∈ M_{𝔭}^{∧} | x_{|U^{∧}} ∈ I^{m}f^{∗}ℳ (U^{∧})} ⊂ I^{n}M_{𝔭}^{∧}.

On the other hand, by assumption there exists 𝔮 ∈ Ass_{A_{𝔭}^{∧}}M_{𝔭}^{∧} such that \overline{\{𝔮\}}∩V (IA_{𝔭}^{∧}) = \{𝔭A_{𝔭}^{∧}\}, hence we may find x ∈ M_{𝔭}^{∧} whose support is \overline{\{𝔮\}} (lemma 10.5.3(ii)). It follows easily that the image of x vanishes in f^{∗}ℳ /I^{m}f^{∗}ℳ (U) for all m ∈ ℕ, i.e. x_{|U} ∈ I^{m}f^{∗}ℳ (U) for every m ∈ ℕ, hence x ∈ I^{n}M_{𝔭}^{∧} for every n ∈ ℕ, in view of (10.5.37). However, M_{𝔭}^{∧} is separated for the 𝔭-adic topology, a fortiori also for the I-preadic topology, so x = 0, a contradiction.

Next, let 𝔭 ∈ Σ\Ass_{A}(I, M), set Σ′ := Σ\{𝔭}, and suppose that the Σ-symbolic topology agrees with the I-adic topology; we have to prove that the latter agrees as well with the Σ′- symbolic topology. This amounts to showing that, for every n ∈ ℕ there exists m ∈ ℕ such that :

Fil_{Σ′}^{(m)}M ⊂ Fil_{𝔭}^{(n)}M or, what is the same : Fil_{Σ′}^{(m)}M_{𝔭} ⊂ 𝔭^{n}M_{𝔭}.

We may write :

Fil_{Σ′}^{(m)}M_{𝔭} = {x ∈ M_{𝔭} | x_{|U} ∈ (Fil_{Σ′}^{(m)}M_{𝔭})^{∼}(U)} = {x ∈ M_{𝔭} | x_{|U} ∈ (Fil_{Σ}^{(m)}M_{𝔭})^{∼}(U)}

from which it is clear that the contention holds if and only if, for every n ∈ ℕ there exists m ∈ ℕ such that :

{x ∈ M_{𝔭} | x_{|U} ∈ I^{m}ℳ (U)} ⊂ 𝔭^{n}M_{𝔭}.

Arguing as in the foregoing, we see that the latter condition holds if and only if :

Fil^{m}M_{𝔭}^{∧} := {x ∈ M_{𝔭}^{∧} | x_{|U^{∧}} ∈ I^{m}f^{∗}ℳ (U^{∧})} ⊂ 𝔭^{n}M_{𝔭}^{∧}.

In view of lemma 10.5.33, we are then reduced to showing the following :

Claim 10.5.38. ⋂_{m∈ℕ} Fil^{m}M_{𝔭}^{∧} = 0.

Proof of the claim. Let x be an element in this intersection; for any 𝔮 ∈ U^{∧} ∩ V (IA_{𝔭}^{∧}), we have x_{𝔮} ∈ ⋂_{m∈ℕ} 𝔮^{m}(M_{𝔭}^{∧})_{𝔮}, hence x_{𝔮} = 0, by [126, Th.8.10(i)]. In other words, Supp(x) ∩ U ∩ V (I) = ∅. Suppose that x ≠ 0, and let 𝔮 be any maximal point of Supp(x); by lemma 10.5.3(i), 𝔮 is an associated prime, and the foregoing implies that {𝔮} ∩ V (IA_{𝔭}^{∧}) = {𝔭A_{𝔭}^{∧}}, which contradicts the assumption that 𝔭 ∉ Ass_{A}(I, M). □

Corollary 10.5.39. In the situation of theorem 10.5.35, Ass_{A}(I, M) is a finite set.

Proof. We have already found a finite subset Σ ⊂ Spec A/I such that the Σ-symbolic topology on M agrees with the I-preadic topology (see (10.5.28)). The contention then follows straightforwardly from theorem 10.5.35. □

Example 10.5.40. Let k be a field with char k ≠ 2, and let C ⊂ 𝔸_{k}^{2} := Spec k[X, Y] be the nodal curve cut by the equation Y^{2} = X^{2} + X^{3}, so that the only singularity of C is the node at the origin p := (0, 0) ∈ C. Let R := k[X, Y, Z], A := R/(Y^{2} − X^{2} − X^{3}); denote by π : 𝔸_{k}^{3} := Spec R → 𝔸_{k}^{2} the linear projection which is dual to the inclusion k[X, Y] → R, so that D := π^{−1}C = Spec A. We define a morphism φ : 𝔸_{k}^{1} := Spec k[T] → D by the rule : T ↦ (T^{2} − 1, T (T^{2} − 1), T) (i.e. φ is dual to the homomorphism of k-algebras such that X ↦ T^{2} − 1, Y ↦ T (T^{2} − 1) and Z ↦ T). Let C′ ⊂ D be the image of φ, with its reduced subscheme structure. It is easy to check that the restriction of π maps C′ birationally onto C, so there are precisely two points p′_{0}, p′_{1} ∈ C′ lying over p. Let 𝔫 := I(C′) ⊂ A, the prime ideal which is the generic point of the (irreducible) curve C′. We claim that the 𝔫preadic topology on A does not agree with the 𝔫-symbolic topology. To this aim – in view of theorem 10.5.35 – it suffices to show that {p′_{0}, p′_{1}} ⊂ Ass_{A}(𝔫, A). However, for any closed point 𝔭 ∈ π^{−1}(p), the 𝔭-adic completion A_{𝔭}^{∧} admits two distinct minimal primes, corresponding to the two branches of the nodal conic C at the node p, and the corresponding irreducible components of B := Spec A_{𝔭}^{∧} meet along the affine line V (Z). To see this, we may suppose that 𝔭 = (X, Y, Z), hence A_{𝔭}^{∧} ≃ k[[X, Y, Z]]/(Y^{2} − X^{2}(1 + X)), and notice that the latter is isomorphic to k[[S, Y, Z]]/(Y^{2} − S^{2}), via the isomorphism that sends Y ↦ Y, Z ↦ Z and S ↦ X(1+X)^{1/2} (the assumption on the characteristic of k ensures that 1+X admits a square root in k[[X]]). Now, say that 𝔭 = p′_{0}; then C′_{𝔭} := C′ ∩ B is contained in only one of the two irreducible components of B. Let 𝔮 ∈ B be the minimal prime ideal whose closure does not contain C′_{𝔭}; then 𝔮 ∈ AssA_{𝔭}^{∧} and \overline{\{𝔮\}} ∩ C′_{𝔭} = \{𝔭A_{𝔭}^{∧}\}, therefore p′_{0} ∈ Ass_{A}(𝔫, A), as stated.

Example 10.5.41. Let A be an excellent normal ring, I ⊂ A any ideal, and set Z := V (I). Then Ass_{A}(I, A) is the set Max(Z) of all maximal points of Z. Indeed, Max(Z) ⊂ Ass_{A}(I, A) by lemma 10.5.32(v). Conversely, suppose 𝔭 ∈ Ass_{A}(I, A); the completion A_{𝔭}^{∧} is still normal ([126, Th.32.2(i)]), and therefore its only associated prime is 0, so the assumption means that the radical of IA_{𝔭}^{∧} is 𝔭A_{𝔭}^{∧}. Equivalently, dim A_{𝔭}^{∧}/IA_{𝔭}^{∧} = 0, so dim A_{𝔭}/IA_{𝔭} = 0, which is the contention.

Definition 10.5.42. Let X be a noetherian scheme, Y ⊂ X a closed subset, 𝔛 the formal completion of X along Y ([59, Ch.I, §10.8]), and f : 𝔛 → X the natural morphism of locally ringed spaces. We say that the pair (X, Y) satisfies the Lefschetz condition, if for every open subset U ⊂ X with Y ⊂ U, and every locally free 𝒪_{U}-module ℰ of finite type, the natural map

Γ(U, ℰ) → Γ(𝔛, f^{∗}ℰ)

is an isomorphism. In this case, we also say that Lef(X, Y) holds. (Cp. [85, Exp.X, §2].)

Lemma 10.5.43. In the situation of definition 10.5.42, suppose that Lef(X, Y) holds, and let U ⊂ X be any open subset such that Y ⊂ U. Then :

(i) The functor :

𝒪_{U}-Mod_{lfft} → 𝒪_{𝔛}-Mod_{lfft} : ℰ ↦ f^{∗}ℰ

is fully faithful (notation of (10.3)). (ii) Denote by 𝒪_{U}-Alg_{lfft} the category of 𝒪_{U}-algebras, whose underlying 𝒪_{U}-module is lo-

cally free of finite type, and define likewise 𝒪_{𝔛}-Alg_{lfft}. Then the functor :

𝒪_{U}-Alg_{lfft} → 𝒪_{𝔛}-Alg_{lfft} : 𝒜 ↦ f^{∗}𝒜

is fully faithful.

Proof. (i): Let ℰ and ℱ be any two locally free 𝒪_{U}-modules of finite type. We have :

Hom_{𝒪_{U}} (ℰ , ℱ) = Γ(U, ℋ om_{𝒪_{U}} (ℰ , ℱ))

and likewise we may compute Hom_{𝒪_{𝔛}} (f^{∗}ℰ , f^{∗}ℱ). However, the natural map :

f^{∗}ℋ om_{𝒪_{U}} (ℰ , ℱ) → ℋ om_{𝒪_{𝔛}} (f^{∗}ℰ , f^{∗}ℱ)

is an isomorphism of 𝒪_{𝔛}-modules. The assertion follows.

(ii): An object of 𝒪_{U}-Alg_{lfft} is a locally free 𝒪_{U}-module 𝒜 of finite type, together with morphisms 𝒜 ⊗_{𝒪_{U}} 𝒜 → 𝒜 and 1_{𝒜} : 𝒪_{U} → 𝒜 of 𝒪_{U}-modules, fulfilling the usual unitarity, commutativity and associativity conditions. An analogous description holds for the objects of 𝒪_{𝔛}-Alg_{lfft}, and for the morphisms of either category. Since 𝒜 ⊗_{𝒪_{U}} 𝒜 is again locally free of finite type, the assertion follows easily from (i) : the details are left to the reader. □

Lemma 10.5.44. Let A be a noetherian ring, I ⊂ A an ideal, U ⊂ Spec A an open subset, 𝔘 the formal completion of U along U ∩ V (I). Consider the following conditions :

(a) Lef(U, U ∩ V (I)) holds.

(b) The natural map ρ_{U} : Γ(U, 𝒪_{U}) → Γ(𝔘, 𝒪_{𝔘}) is an isomorphism.

(c) The natural map ρ : A → Γ(𝔘, 𝒪_{𝔘}) is an isomorphism. Then (c)⇒(b)⇔(a), and (c) implies that A is I-adically complete.

Proof. Clearly (a)⇒(b), hence we assume that (b) holds, and we show (a). Let V ⊂ U be an open subset with U ∩ V (I) ⊂ V and ℰ a coherent locally free 𝒪_{V} -module. As A is noetherian, V is quasi-compact, so we may find a left exact sequence P_{•} := (0 → ℰ → 𝒪_{V}^{⊕m} → 𝒪_{V}^{⊕n}) of 𝒪_{V} -modules (corollary 10.3.28). Since the natural map of locally ringed spaces f : 𝔘 → U is flat, the sequence f^{∗}P is still left exact. Since the global section functors are left exact, there follows a ladder of left exact sequences :

Γ(V, P_{•}) → Γ(𝔘, f^{∗}P_{•})

which reduces the assertion to the case where ℰ = 𝒪_{V}; the latter is covered by the following :

Claim 10.5.45. The natural map ρ_{V} : Γ(V, 𝒪_{V}) → Γ(𝔘, 𝒪_{𝔘}) is an isomorphism.

Proof of the claim. The isomorphism ρ_{U} factors through ρ_{V}, hence the latter is a surjection. Suppose that s ∈ Ker ρ_{V}, and s ≠ 0; then we may find x ∈ V such that the image s_{x} of s in 𝒪_{V,x} does not vanish. Moreover, we may find a ∈ A whose image a(x) in κ(x) does not vanish, and such that as is the restriction of an element of A; especially, as ∈ Γ(U, 𝒪_{U}), and clearly the image of as in Γ(V, 𝒪_{V}) lies in Ker ρ_{V}. Therefore, as = 0 in Γ(U, 𝒪_{U}); however the image as_{x} of as in 𝒪_{U,x} is non-zero by construction, a contradiction. This shows that ρ_{V} is injective, whence the claim. ♢

Finally, suppose that (c) holds; arguing as in the proof of claim 10.5.45, one sees that ρ_{U} is an isomorphism. Moreover, since Γ(𝔘, 𝒪_{𝔘}) ≃ lim_{n∈ℕ} Γ(𝔘, 𝒪_{𝔘}/I^{n}𝒪_{𝔘}), the morphism ρ factors through the natural A-linear map i : A → A^{∧} to the I-adic completion of A. The composition with ρ^{−1} yields an A-linear left inverse s : A^{∧} → A to i. Set N := Ker s; clearly s is surjective, hence A^{∧} ≃ A ⊕ N. It follows easily that N/I^{n}N = 0 for every n ∈ ℕ, especially N ⊂ ⋂_{n∈ℕ} I^{n}A^{∧}. Therefore N = 0, since A^{∧} is separated for the I-adic topology. □

Proposition 10.5.46. Let φ : A → B be a flat homomorphism of noetherian rings, I ⊂ A an ideal, U ⊂ Spec B an open subset. Set f := Spec φ : Spec B → Spec A, and assume that :

(a) B is complete for the IB-adic topology.

(b) For every x ∈ V (I), we have : {y ∈ f^{−1}(x) | δ(y, 𝒪_{f^{−1}(x)}) = 0} ⊂ U.

(c) For every x ∈ Ass_{A}(I, A), we have : {y ∈ f^{−1}(x) | δ(y, 𝒪_{f^{−1}(x)}) ≤ 1} ⊂ U. Then Lef(U, U ∩ V (IB)) holds.

Proof. Set Σ := Ass_{A}(I, A), and let 𝔘 be the formal completion of U along V (IB); by theorem 10.5.35, the I-preadic topology on A agrees with the Σ-symbolic topology. Let also 𝒥 be the family consisting of all ideals J ⊂ A such that Ass A/J ⊂ Σ; it follows that the natural maps :

B → lim_{J∈𝒥} B/JB Γ(𝔘, 𝒪_{𝔘}) → lim_{J∈𝒥} Γ(U, 𝒪_{U}/J𝒪_{U})

are isomorphisms (see the discussion in (10.5.26)). In view of lemma 10.5.44, we are then reduced to showing :

Claim 10.5.47. The natural map B/JB → Γ(U, 𝒪_{U}/J𝒪_{U}) is an isomorphism for every J ∈ 𝒥.

Proof of the claim. Let f : Y := Spec B/JB → X := Spec A/J be the induced morphism; in view of corollary 10.4.23, it suffices to prove that δ(y, 𝒪_{Y}) ≥ 2 whenever y ∈ Y \U. Thus, set x := f(y); by corollary 10.4.47we have :

δ(y, 𝒪_{Y}) = δ(y, 𝒪_{f^{−1}(x)}) + δ(x, 𝒪_{X}).

Now, if δ(x, 𝒪_{X}) = 1, notice that f(Y) ⊂ V (J) ⊂ V (I), by lemmata 10.5.3(i) and 10.5.32(iii); hence (b) implies the contention in this case. Lastly, if δ(x, 𝒪_{X}) = 0, then x ∈ Ass A/J by proposition 10.5.6, hence we use assumption (c) to conclude. □

10.6. Cohomology of projective schemes. We begin by recalling a few generalities on graded algebras and their homogeneous prime spectra; next we define the blow up of a scheme along a quasi-coherent sheaf of ideals, and we prove some basic results on the higher direct images of quasi-coherent modules under blow up morphisms.

10.6.1. Let A := ⊕_{n∈ℕ}A_{n} be a ℕ-graded ring, and set A_{+} := ⊕_{n>0}A_{n}, which is an ideal of A. Following [60, Ch.II, (2.3.1)], one denotes by Proj A the set consisting of all graded prime ideals of A that do not contain A_{+}, and one endows Proj A with the topology induced from the Zariski topology of Spec A. For every homogeneous element f ∈ A_{+}, set :

D_{+}(f) := D(f) ∩ Proj A

where as usual, D(f) := Spec A_{f} ⊂ Spec A. Clearly :

(10.6.2)  D_{+}(fg) = D_{+}(f) ∩ D_{+}(g)  for every homogeneous f, g ∈ A_{+}.

The system of open subsets D_{+}(f), for f ranging over the homogeneous elements of A_{+}, is a basis of the topology of Proj A ([60, Ch.II, Prop.2.3.4]), and obviously any system of homogeneous generators (f_{λ} | λ ∈ Λ) for the ideal A_{+} yields an open covering

Proj A = ⋃_{λ∈Λ} D_{+}(f_{λ}).

For any homogeneous element f ∈ A_{+}, let also A_{(f)} ⊂ A_{f} be the subring consisting of all elements of degree zero (for the natural ℤ-grading of A_{f}); in other words :

A_{(f)} := ∑_{k∈ℕ} A_{k} · f^{−k} ⊂ A_{f}.

The topological space Proj A carries a sheaf of rings 𝒪, with isomorphisms of ringed spaces :

ω_{f} : (D_{+}(f), 𝒪_{|D_{+}(f)}) →^{∼} Spec A_{(f)} for every homogeneous f ∈ A_{+}.

and the system of isomorphisms ω_{f} is compatible, in an obvious way, with the inclusions :

j_{f,g} : D_{+}(fg) ⊂ D_{+}(f)

as in (10.6.2), and with the natural homomorphisms A_{(f)} → A_{(fg)}. Especially, the locally ringed space (Proj A, 𝒪) is a separated scheme.

Example 10.6.3. For any d ∈ ℕ, and any ring R, take A := R[T_{0}, …, T_{d}], endowed with its standard ℕ-grading such that gr_{n}A is the R-module generated by the monomials of total degree n, for every n ∈ ℕ. The scheme Proj A is the projective d-dimensional space over Spec R, denoted

ℙ_{R}^{d}.

According to (10.6.1), it admits the standard affine covering ℙ_{R}^{d} = ⋃_{i=0}^{d} D_{+}(T_{i}). Set as well

τ_{ij} := T_{j}/T_{i} for every i, j = 0, …, n.

Then clearly A_{i} := A_{(T_{i})} = R[τ_{ij} | j = 0, …, d] is a free polynomial R-algebra in d variables, so that U_{i} := D_{+}(T_{i}) is isomorphic to the d-dimensional affine space 𝔸_{R}^{d} over Spec R, for every i = 0, …, d. For i ≠ j, the intersection U_{ij} := D_{+}(T_{i}T_{j}) = U_{i} ∩ U_{j} corresponds, under these identifications, to the open subsets

Spec A_{i}[τ_{ij}^{−1}] ⊂ U_{i} and Spec A_{j}[τ_{ji}^{−1}] ⊂ U_{j}

so we get a commutative diagram of isomorphisms of R-algebras

\xymatrix{ & 𝒪_{ℙ_{R}^{d}}(U_{ij}) \ar@{-->}[dl] \ar@{-->}[dr] \\ A_{i}[τ_{ij}^{−1}] \ar[rr] &  & A_{j}[τ_{ji}^{−1}]}

whose downward arrows are induced by the maps ω_{T_{i}} and ω_{T_{j}} of (10.6.1), and where the horizontal arrow is given by the rule :

(10.6.4)  τ_{ik} ↦ τ_{jk} · τ_{ji}^{−1}  for every k = 0, …, d.

10.6.5. Let A′ := ⊕_{n∈ℕ}A′_{n} be another ℕ-graded ring, and φ : A → A′ a homomorphism of graded rings (i.e. φ(A_{n}) ⊂ A′_{n} for every n ∈ ℕ). Following [60, Ch.II, (2.8.1)], we let :

G(φ) := Proj A′ \ V (φ(A_{+})).

This open subset of Proj A′ is also the same as the union of all the open subsets of the form D_{+}(φ(f)), where f ranges over the homogeneous elements of A_{+}. The restriction to G(φ) of Spec φ : Spec A′ → Spec A, is a continuous map^{a}φ : G(φ) → Proj A. Moreover, we have the identity :

aφ^{−1}(D_{+}(f)) = D_{+}(φ(f)) for every homogeneous f ∈ A_{+}.

Furthermore, the homomorphism φ induces a homomorphism φ_{(f)} : A_{(f)} → A′_{(φ(f))}, whence a morphism of schemes :

Φ_{f} : D_{+}(φ(f)) → D_{+}(f).

Let g ∈ A_{+} be another homogeneous element; it is easily seen that :

j_{f,g} ∘ Φ_{fg} = (Φ_{f})_{|D_{+}(φ(fg))}.

It follows that the locally defined morphisms Φ_{f} glue to a unique morphism of schemes :

Proj φ : G(φ) → Proj A

such that the diagram of schemes :

(10.6.6)  \xymatrix{Spec A′_{(φ(f))} \ar[d]_{ω_{φ(f)}} \ar[r]^{Spec φ_{(f)}} & Spec A_{(f)} \ar[d]^{ω_{f}} \\ D_{+}(φ(f)) \ar[r]^{(Proj φ)_{|D_{+}(φ(f))}} & D_{+}(f)}

commutes for every homogeneous f ∈ A_{+} ([60, Ch.II, Prop.2.8.2]). Lastly, we remark that, for every homogeneous element f′ ∈ A′_{+}, the open subset D_{+}(f′) lies in G(φ) if and only if f′ lies in the radical of the ideal of A′ generated by φ(A_{+}) ([60, Ch.II, Cor.2.3.15]). Especially, G(φ) = Proj A′ whenever φ(A_{+}) generates the ideal A′_{+}.

10.6.7. To ease notation, set Y := Proj A. Let M := ⊕_{n∈ℤ}M_{n} be a ℤ-graded A-module (i.e. A_{k} · M_{n} ⊂ M_{k+n} for every k ∈ ℕ and n ∈ ℤ); for every homogeneous f ∈ A_{+}, denote by M_{(f)} ⊂ M_{f} the submodule consisting of all elements of degree zero (for the natural ℤgrading of M_{f}). Clearly M_{(f)} is an A_{(f)}-module in a natural way, whence a quasi-coherent 𝒪_{D_{+}(f)}-module M_{(f)}^{∼}; these modules glue to a unique quasi-coherent 𝒪_{Y} -module M^{∼} ([60, Ch.II, Prop.2.5.2]). Especially, for every n ∈ ℤ, let A(n) be the ℤ-graded A-module such that A(n)_{k} := A_{n+k} for every k ∈ ℤ (with the convention that A_{k} = 0 if k < 0). We set :

𝒪_{Y} (n) := A(n)^{∼}.

Any element f ∈ A_{n} induces a natural isomorphism of D_{+}(f)-modules :

𝒪_{Y} (n)_{|D_{+}(f)} →^{∼} 𝒪_{D_{+}(f)}

([60, Ch.II, Prop.2.5.7]). Hence, on the open subset

U_{n}(A) := ⋃_{f∈A_{n}} D_{+}(f)

the sheaf 𝒪_{Y} (n) restricts to an invertible 𝒪_{U_{n}(A)}-module. Especially, if A_{1} generates the ideal A_{+}, the 𝒪_{Y} -modules 𝒪_{Y} (n) are invertible, for every n ∈ ℤ.

Example 10.6.8. Resume the notation of example 10.6.3. A direct inspection of the definitions shows that A(n)_{(T_{i})} is the A_{i}-module generated by all fractions of the form T_{0}^{r_{0}}T_{1}^{r_{1}} ⋯ T_{d}^{r_{d}}T_{i}^{−k}, for every k ∈ ℕ and every (r_{0}, …, r_{d}) ∈ ℕ^{⊕d+1} such that r_{0} + r_{1} + ⋯ + r_{d} = n + k. In other words, A(n)_{(T_{i})} is the A_{i}-submodule of A(n)_{T_{i}} generated by T_{i}^{n}. We see then directly that this is a free A_{i}-module of rank one. Moreover, we get a commutative diagram of isomorphisms

\xymatrix{ & 𝒪_{ℙ_{R}^{d}}(n)(U_{ij}) \ar@{-->}[dl] \ar@{-->}[dr] \\ T_{i}^{n}A_{i}[τ_{ij}^{−1}] \ar[rr] &  & T_{j}^{n}A_{j}[τ_{ji}^{−1}]}

whose downward arrows are induced by the restriction maps of the sheaf 𝒪_{ℙ_{R}^{d}}(n), and where the horizontal map is the A_{i}[τ_{ij}^{−1}]-linear map given by the rule

T_{i}^{n} ↦ T_{j}^{n} · τ_{ji}^{n}

and where the A_{i}[τ_{ij}^{−1}]-module structure of T_{j}^{n}A_{j}[τ_{ji}^{−1}] is defined by restriction of scalars along the isomorphism A_{i}[τ_{ij}^{−1}] →^{∼} A_{j}[τ_{ji}^{−1}] as in (10.6.4).

10.6.9. In the situation of (10.6.5), let M := ⊕_{n∈ℤ}M_{n} be a ℤ-graded A-module. Then M′ := M ⊗_{A} A′ is a ℤ-graded A′-module, with the grading defined by the rule :

M′_{n} := ∑_{j+k=n} Im(M_{j} ⊗_{ℤ} A′_{k} → M′) for every n ∈ ℤ

([60, Ch.II, (2.1.2)]). Then [60, Ch.II, Prop.2.8.8] yields a natural morphism of 𝒪_{G(φ)}-modules:

ν_{M} : (Proj φ)^{∗}M^{∼} → (M′)_{|G(φ)}^{∼}.

Moreover, set :

G_{1}(φ) := ⋃_{f∈A_{1}} D_{+}(φ(f))

and notice that G_{1}(φ) ⊂ U_{1}(A′) ∩ G(φ); by inspecting the proof of loc.cit. we see that the restriction ν_{M|G_{1}(φ)} is an isomorphism. Especially, ν_{M} is an isomorphism whenever A_{1} generates the ideal A_{+}. It is also easily seen that G_{1}(φ) = U_{1}(A′) if φ(A_{+}) generates A′_{+}.

For any f ∈ A_{1}, the restriction (ν_{M})_{|D_{+}(φ(f))} can be described explicitly : namely, we have natural identifications

ω_{f}^{∗}(M_{|D_{+}(f)}^{∼}) →^{∼} M_{(f)}^{∼} ω_{φ(f)}^{∗}(M′)_{|D_{+}(φ(f))}^{∼} →^{∼} (M′)_{(φ(f))}^{∼}

and in view of (10.6.6), the morphism (ν_{M})_{|D_{+}(φ(f))} is induced by the A′_{(φ(f))}-linear map :

M_{(f)} ⊗_{A_{(f)}} A′_{(φ(f))} → M′_{(φ(f))}

given by the rule :

(m_{k} · f^{−k}) ⊗ (a′_{j} · φ(f)^{−j}) ↦ (m_{k} ⊗ a′_{j}) · φ(f)^{−j−k} for all k, j ∈ ℤ, m_{k} ∈ M_{k}, a′_{j} ∈ A′_{j}.

10.6.10. The foregoing results can be improved somewhat, in the following special situation. Let R → R′ be a ring homomorphism, A a ℕ-graded R-algebra (hence the structure morphism R → A is a ring homomorphism R → A_{0}); the ring A′ := R′ ⊗_{R} A is naturally a ℕ-graded R′-algebra, and the induced map φ : A → A′ is a homomorphism of graded rings. In this case, obviously φ(A_{+}) generates the ideal A′_{+}, hence G(φ) = Proj A′, and indeed, Proj φ induces an isomorphism of Spec R′-schemes :

Y′ →^{∼} Spec R′ ×_{Spec R} Y

where again Y := Proj A and Y′ := Proj A′. Moreover, for every ℤ-graded A-module M, the corresponding morphism ν_{M} is an isomorphism, regardless of whether or not A_{1} generates A_{+} ([60, Ch.II, Prop.2.8.10]). Especially, ν_{A(n)} is a natural identification ([60, Ch.II, Cor.2.8.11]) :

(Proj φ)^{∗}𝒪_{Y} (n) →^{∼} 𝒪_{Y′}(n) for every n ∈ ℤ.

10.6.11. Keep the notation of (10.6.1), and for every integer d > 0, set

A_{n}^{(d)} := A_{nd} for every n ∈ ℕ, and A^{(d)} := ⨁_{n∈ℕ} A_{n}^{(d)}.

Clearly A^{(d)} is an ℕ-graded ring, and a subring of A, but the inclusion map j : A^{(n)} → A is not a homomorphism of ℕ-graded rings. However, for every homogeneous element f ∈ A_{+}, the map j induces a natural identification of A-algebras :

(10.6.12)  A_{(f^{d})}^{(d)} →^{∼} A_{(f)}

which in turn yields a natural isomorphism of A_{0}-schemes ([60, Ch.II, Prop.2.4.7(i)])

Ω^{(d)} : Proj A →^{∼} Proj A^{(d)} for every d > 0.

To ease notation, we shall let Y := Proj A and Y^{(d)} := Proj A^{(d)} for every integer d > 0. Likewise, if M is any ℤ-graded A-module, we may consider that ℤ-graded A^{(d)}-module M^{(d)} whose homogeneous direct summand of degree n equals M_{nd}, for every n ∈ ℤ. Then the inclusion map M^{(d)} → M induces natural identifications

(10.6.13)  M_{(f^{d})}^{(d)} → M_{(f)}  for every homogeneous element f of A.

Indeed, it is easily seen that this map is surjective, and it is injective, since it is the restriction of the injective map M_{f^{d}}^{(d)} → M_{f^{d}}. Moreover, if we endow M_{(f)} with the A_{(f^{d})}^{(d)}-module structure

induced by (10.6.12), it is easily seen that (10.6.13) is an isomorphism of A_{(f^{d})}^{(d)}-modules. Thus, we obtain a natural isomorphism of quasi-coherent 𝒪_{Y^{(d)}}-modules

M^{(d)∼} →^{∼} Ω_{∗}^{(d)}M^{∼}.

Next, for every n ∈ ℤ, let M(n) be the ℤ-graded A-module such that M(n)_{k} := M_{n+k} for every k ∈ ℤ (with A-module structure deduced from that of M, in the obvious way). Clearly

M(nd)^{(d)} = M^{(d)}(n) for every n ∈ ℤ and every d > 0

whence an induced isomorphism of quasi-coherent 𝒪_{Y^{(d)}}-modules

M^{(d)}(n)^{∼} →^{∼} Ω_{∗}^{(d)}M(nd)^{∼} for every n ∈ ℤ and every d > 0.

Especially, we have a natural identification

(10.6.14)  𝒪_{Y^{(d)}}(n) →^{∼} Ω_{∗}^{(d)}𝒪_{Y} (nd)  for every n ∈ ℤ and every d > 0.

10.6.15. Let X be a scheme, 𝒜 := ⊕_{n∈ℕ}𝒜_{n} a ℕ-graded quasi-coherent 𝒪_{X}-algebra on the Zariski site of X; we let 𝒜_{+} := ⊕_{n>0}𝒜_{n}. According to [60, Ch.II, Prop.3.1.2], there exists an X-scheme π : Proj 𝒜 → X, with natural isomorphisms of U-schemes :

ψ_{U} : U ×_{X} Proj 𝒜 →^{∼} Proj 𝒜 (U)

for every affine open subset U ⊂ X, and the system of isomorphisms ψ_{U} is compatible, in an obvious way, with inclusions U′ ⊂ U of affine open subsets. Especially, π is a separated morphism. For any integer d > 0, every f ∈ Γ(X, 𝒜_{d}) defines an open subset D_{+}(f) ⊂ Proj 𝒜, such that :

D_{+}(f) ∩ π^{−1}U = D_{+}(f_{|U}) ⊂ Proj 𝒜 (U) for every affine open subset U ⊂ X.

10.6.16. To ease notation, set Y := Proj 𝒜, and let again π : Y → X be the natural morphism. Let ℳ := ⊕_{n∈ℤ}ℳ_{n} be a ℤ-graded 𝒜 -module, quasi-coherent as a 𝒪_{X}-module; for every affine open subset U ⊂ X, the graded 𝒜 (U)-module ℳ (U) yields a quasi-coherent 𝒪_{π^{−1}U}-module ℳ_{U}^{∼}, and every inclusion of affine open subsets U′ ⊂ U induces a natural isomorphism of 𝒪_{π^{−1}U′}-modules : ℳ_{U|U′}^{∼} →^{∼} ℳ_{U′}^{∼}. Therefore the locally defined modules ℳ_{U}^{∼} glue to a well defined quasi-coherent 𝒪_{Y} -module ℳ^{∼}.

Especially, for every n ∈ ℤ, denote by 𝒜 (n) the ℤ-graded 𝒜 -module such that 𝒜 (n)_{k} := 𝒜_{n+k} for every k ∈ ℤ, with the convention that 𝒜_{n} = 0 whenever n < 0. We set:

𝒪_{Y} (n) := 𝒜 (n)^{∼} and ℳ^{∼}(n) := ℳ^{∼} ⊗_{𝒪_{Y}} 𝒪_{Y} (n).

Denote by U_{n}(𝒜) ⊂ Y the union of the open subsets U_{n}(𝒜 (U)), for U ranging over the affine open subsets of X; from the discussion in (10.6.7), it clear that the restriction 𝒪_{Y} (n)_{|U_{n}(𝒜)} is an invertible 𝒪_{U_{n}(𝒜)} -module. This open subset can be described as follows. For every x ∈ X, let :

(10.6.17)  𝒜_{n}(x) := 𝒜_{n,x} ⊗_{𝒪_{X,x}} κ(x)  and set 𝒜 (x) := ⊕_{n∈ℕ}𝒜_{n}(x)

which is a ℕ-graded κ(x)-algebra; then :

(10.6.18)  U_{n}(𝒜) = {y ∈ Y | 𝒜_{n}(π(y)) ⊄ 𝔭(y)}

where 𝔭(y) ⊂ 𝒜 (π(y)) denotes the prime ideal corresponding to the point y.

10.6.19. Moreover, for every ℤ-graded 𝒜 -module ℳ, and every n ∈ ℤ, there exists a natural morphism of 𝒪_{Y} -modules :

(10.6.20)  ℳ^{∼}(n) → ℳ (n)^{∼}

where ℳ (n) is the ℤ-graded 𝒜 -module given by the rule : ℳ (n)_{k} := ℳ_{n+k} for every k ∈ ℕ ([60, Prop.3.2.16]). The restriction of (10.6.20) to the open subset U_{1}(𝒜) is an isomorphism ([60, Ch.II, Cor.3.2.8]). Especially, we have natural morphisms of 𝒪_{Y} -modules :

(10.6.21)  𝒪_{Y} (n) ⊗_{𝒪_{Y}} 𝒪_{Y} (m) → 𝒪_{Y} (n + m)  for every n, m ∈ ℤ

([60, Ch.II, Prop.3.2.6]) whose restrictions to U_{1}(𝒜) are isomorphisms. Furthermore, we have a natural morphism ℳ_{0} → π_{∗}ℳ^{∼} of 𝒪_{X}-modules ([60, Ch.II, (3.3.2.1)]), whence, by adjunction, a morphism of 𝒪_{Y} -modules :

(10.6.22)  π^{∗}ℳ_{0} → ℳ^{∼}.

Applying (10.6.22) to the modules ℳ_{n} = ℳ (n)_{0}, and taking into account the isomorphism (10.6.20), we deduce a natural morphism of 𝒪_{U_{1}(𝒜)} -modules :

(10.6.23)  (π^{∗}ℳ_{n})_{|U_{1}(𝒜)} → ℳ^{∼}(n)_{|U_{1}(𝒜)}

which can be described as follows. Let U ⊂ X be any affine open subset; for every f ∈ 𝒜_{1}(U), the restriction of (10.6.23) to D_{+}(f) ⊂ π^{−1}U is given by the morphisms

ℳ_{n}(U) ⊗_{𝒪_{X}(U)} 𝒜 (U)_{(f)} → ℳ (n)(U)_{(f)} := ∑_{k∈ℤ} ℳ_{k+n}(U) · f^{−k} ⊂ ℳ (U)_{f}.

induced by the scalar multiplication ℳ_{n} ⊗_{𝒪_{X}} 𝒜_{k} → ℳ_{n+k}. Especially, we have natural morphisms of 𝒪_{Y} -modules :

(10.6.24)  π^{∗}𝒜_{n} → 𝒪_{Y} (n)  for every n ∈ ℕ

whose restrictions to U_{1}(𝒜) are epimorphisms. An inspection of the definition also shows that the diagram of 𝒪_{Y} -modules :

(10.6.25)  \xymatrix{π^{∗}𝒜_{n} ⊗_{𝒪_{Y}} π^{∗}𝒜_{m} \ar[r] \ar[d] & π^{∗}𝒜_{n+m} \ar[d] \\ 𝒪_{Y} (n) ⊗_{𝒪_{Y}} 𝒪_{Y} (m) \ar[r] & 𝒪_{Y} (n + m)}

commutes for every n, m ∈ ℕ, where the top horizontal arrow is induced by the graded multiplication 𝒜_{n} ⊗_{𝒪_{X}} 𝒜_{m} → 𝒜_{n+m}, the vertical arrows are the maps (10.6.24), and the bottom horizontal arrow is the map (10.6.21).

10.6.26. Next, let 𝒜′ := ⊕_{n∈ℕ}𝒜′_{n} be another ℕ-graded quasi-coherent 𝒪_{X}-algebra on the Zariski site of X, and φ : 𝒜 → 𝒜′ a morphism of graded 𝒪_{X}-algebras; for every affine open subset U ⊂ X, we deduce a morphism φ_{U} : 𝒜 (U) → 𝒜′(U) of graded 𝒪_{X}(U)-algebras, whence an open subset G(φ_{U}) ⊂ Proj 𝒜′(U) as in (10.6.5). If V ⊂ U is a smaller affine open subset, the natural isomorphism

V ×_{U} Proj 𝒜′(U) →^{∼} Proj 𝒜′(V)

induces an identification V ×_{U} G(φ_{U}) →^{∼} G(φ_{V}), hence there exists a well defined open subset G(φ) ⊂ Proj 𝒜′ such that the morphisms Proj φ_{U} glue to a unique morphism of X-schemes :

Proj φ : G(φ) → Proj 𝒜 .

If 𝒜′_{+} is generated – locally on X – by φ(𝒜_{+}), we have G(φ) = Proj 𝒜′.

Moreover, if ℳ is a ℤ-graded quasi-coherent 𝒜 -module, the morphisms ν_{ℳ (U)} assemble into a well defined morphism of 𝒪_{G(φ)}-modules :

ν_{ℳ} : (Proj φ)^{∗}ℳ^{∼} → (ℳ′)_{|G(φ)}^{∼}

where the grading of ℳ′ := ℳ ⊗_{𝒜} 𝒜′ is defined as in (10.6.9). Likewise, the union of the subsets G_{1}(φ_{U}), for U ranging over the affine open subsets of X, is an open subset :

(10.6.27)  G_{1}(φ) ⊂ U_{1}(𝒜′) ∩ G(φ)

such that the restriction ν_{ℳ |G_{1}(φ)} is an isomorphism. Especially, set Y′ := Proj 𝒜′; we have a natural morphism :

(10.6.28)  ν_{𝒜 (n)} : (Proj φ)^{∗}𝒪_{Y} (n) → 𝒪_{Y′}(n)_{|G(φ)}

which is an isomorphism, if 𝒜_{1} generates 𝒜_{+} locally on X. Again, we have G_{1}(φ) = U_{1}(𝒜′) whenever φ(𝒜_{+}) generates 𝒜′_{+}, locally on X.

10.6.29. The discussion in (10.6.10) implies that any morphism of schemes f : X′ → X induces a natural isomorphism of X′-schemes ([60, Ch.II, Prop.3.5.3]) :

(10.6.30)  Proj f^{∗}𝒜 →^{∼} X′ ×_{X} Proj 𝒜

and the description (10.6.18) implies that (10.6.30) restricts to an isomorphism :

U_{n}(f^{∗}𝒜) →^{∼} X′ ×_{X} U_{n}(𝒜) for every n ∈ ℕ.

Furthermore, set Y′ := Proj f^{∗}𝒜, and let π_{Y} : Y′ → Y be the morphism deduced from (10.6.30); the discussion in (10.6.10) implies as well that, for any ℤ-graded quasi-coherent 𝒜 -module ℳ, there is a natural isomorphism :

(f^{∗}ℳ)^{∼} →^{∼} π_{Y}^{∗} ℳ^{∼}

([60, Ch.II, Prop.3.5.3]). Especially, f induces a natural identification ([60, Ch.II, Cor.3.5.4]) :

(10.6.31)  𝒪_{Y′}(n) →^{∼} π_{Y}^{∗} 𝒪_{Y} (n)  for every n ∈ ℤ.

10.6.32. Keep the notation of (10.6.16), and let 𝒞_{X} be the category whose objects are all the pairs (ψ : Z → X, ℒ), where ψ is a morphism of schemes and ℒ is an invertible 𝒪_{Z}-module on the Zariski site of Z; the morphisms (ψ : Z → X, ℒ) → (ψ′ : Z′ → X, ℒ′) are the pairs (β, h), where β : Z → Z′ is a morphism of X-schemes, and h : β^{∗}ℒ′ →^{∼} ℒ is an isomorphism of 𝒪_{Z}-modules (with composition of morphisms defined in the obvious way). Consider the functor:

F_{𝒜} : 𝒞_{X}^{o} → Set

which assigns to any object (ψ, ℒ) of 𝒞_{X}, the set consisting of all homomorphisms of graded 𝒪_{Z}-algebras :

g : ψ^{∗}𝒜 → Sym_{𝒪_{Z}}^{•} ℒ

which are epimorphisms on the underlying 𝒪_{Z}-modules (here Sym_{𝒪_{Z}}^{•} ℒ denotes the symmetric 𝒪_{Z}-algebra on the 𝒪_{Z}-module ℒ); on a morphism (β, h) as in the foregoing, and an element g′ ∈ F_{𝒜} (ψ′, ℒ′), the functor acts by the rule :

F_{𝒜} (β, h)(g′) := (Sym_{𝒪_{Z}}^{•} h) ∘ β^{∗}g′.

Lemma 10.6.33. The object (π : U_{1}(𝒜) → X, 𝒪_{Y} (1)_{|U_{1}(𝒜)} ) of 𝒞_{X} represents the functor F_{𝒜} .

Proof. Given an object (ψ : Z → X, ℒ) of 𝒞_{X}, and g ∈ F_{𝒜} (ψ, ℒ), set :

ℙ(ℒ) := Proj (Sym_{𝒪_{Z}}^{•} ℒ).

According to [60, Ch.II, Cor.3.1.7, Prop.3.1.8(iii)], the natural morphism π_{Z} : ℙ(ℒ) → Z is an isomorphism, and clearly in this situation the natural maps (10.6.24) are isomorphisms :

(10.6.34)  π_{Z}^{∗}ℒ^{⊗n} →^{∼} 𝒪_{ℙ(ℒ)} (n)  for every n ∈ ℕ.

On the other hand, since g is an epimorphism, we have G(g) = ℙ(ℒ); taking (10.6.30) into account, we deduce a morphism of Z-schemes :

Proj g : ℙ(ℒ) → Y′ := Z ×_{X} Proj 𝒜

which is the same as a morphism of X-schemes :

ℙ(g) : Z → Proj 𝒜 .

We need to show that the image of ℙ(g) lies in the open subset U_{1}(𝒜); to this aim, we may assume that both X and Z are affine, say X = Spec R, Z = Spec S, in which case 𝒜 is the quasi-coherent algebra associated to a ℕ-graded R-algebra A, ℒ is the invertible module associated to a projective rank one S-module L, and g : S ⊗_{R} A → Sym_{S}^{•}L is a surjective homomorphism of R-algebras. Then locally on Z, ℒ is generated by elements of the form g(1⊗t), for some local sections t of 𝒜_{1}, and up to replacing Z by an affine open subset, we may assume that t ∈ A_{1} is an element such that t′ := g(1 ⊗ t) generates the free S-module L. In this situation, we have ℙ(ℒ) = D_{+}(t′), and ℙ(g) is the same as the morphism Φ_{t} : D_{+}(t′) → D_{+}(t) (notation of (10.6.5)); especially the image of ℙ(g) lies in U_{1}(𝒜), as required.

Moreover, the isomorphism π_{Z} : ℙ(ℒ) →^{∼} Z is induced by the natural identification:

(10.6.35)  S = S[t′]_{(t′)}.

From this description, we also can extract an explicit expression for Φ_{t}; namely, it is induced by the map of R-algebras :

A_{(t)} → S such that a_{k} · t^{−k} ↦ g(1 ⊗ a_{k}) · t^{′−k}

for every k ∈ ℕ, and every a_{k} ∈ A_{k}. Next, letting n := 1 in (10.6.28) and (10.6.31), we obtain a natural isomorphism of 𝒪_{ℙ(ℒ)} -modules :

𝒪_{ℙ(ℒ)} (1) →^{∼} (Proj g)^{∗}𝒪_{Y′}(1) →^{∼} (Proj g)^{∗} ∘ π_{Y}^{∗} 𝒪_{Y} (1) →^{∼} π_{Z}^{∗} ∘ ℙ(g)^{∗}𝒪_{Y} (1)

(notice that, since by assumption g is an epimorphism, we have G_{1}(g) = U_{1}(Sym_{𝒪_{Z}}^{•} ℒ) = ℙ(ℒ), hence ν_{ψ^{∗}𝒜 (1)} is an isomorphism). Composition with (10.6.34) yields an isomorphism :

h(g) : ℙ(g)^{∗}𝒪_{Y} (1) →^{∼} ℒ

of 𝒪_{Z}-modules, whence a morphism in 𝒞_{X}

(ℙ(g), h(g)) : (ψ, ℒ) → (π_{|U_{1}(𝒜)} , 𝒪_{Y} (1)_{|U_{1}(𝒜)} ).

In case X and Z are affine, and ℒ is associated to a free module L, generated by an element of the form t′ := g(1 ⊗ t) as in the foregoing, we can describe explicitly h(g); namely, a direct inspection of the construction shows that in this case h(g) is induced by the map of S-modules

S ⊗_{A_{(t)}} A(1)_{(t)} → L : s ⊗ a_{k} · t^{1−k} ↦ s · g(1 ⊗ a_{k}) · (t′)^{1−k} for every s ∈ S, a_{k} ∈ A_{k}.

Conversely, let β : Z → U_{1}(𝒜) be a morphism of X-schemes, and h : β^{∗}𝒪_{Y} (1)_{|U_{1}(𝒜)} →^{∼} ℒ an isomorphism of 𝒪_{Z}-modules. In view of the natural isomorphisms (10.6.21), we deduce, for every n ∈ ℕ, an isomorphism :

h^{⊗n} : β^{∗}𝒪_{Y} (n)_{|U_{1}(𝒜)} →^{∼} ℒ^{⊗n}.

Combining with the epimorphisms (10.6.24) :

ω_{n} : (π^{∗}𝒜_{n})_{|U_{1}(𝒜)} → 𝒪_{Y} (n)_{|U_{1}(𝒜)}

we may define the epimorphism of 𝒪_{Z}-modules :

(10.6.36)  g(β, h) := ⨁_{n∈ℕ} h^{⊗n} ∘ β^{∗}(ω_{n}) : ψ^{∗}𝒜 → Sym_{𝒪_{Z}}^{•} ℒ

which, in view of (10.6.25), is a homomorphism of graded 𝒪_{Z}-algebras, i.e. g(β, h) ∈ F (ψ, ℒ). This homomorphism can be described explicitly, locally on Z : namely, say again that X = Spec R, Z = Spec S, ℒ = L^{∼} for a free S-module of rank one, and 𝒜 = A^{∼} for some ℕ-graded R-algebra A; suppose moreover that the image of β lies in an open subset D_{+}(t) ⊂ U_{1}(𝒜), for some t ∈ A_{1}. Then β comes from a ring homomorphism β^{♮} : A_{(t)} → S, h is an S-linear isomorphism S ⊗_{A_{(t)}} A(1)_{(t)} →^{∼} L, and t′ := h(1 ⊗ t) is a generator of L; moreover, ω_{n} is the epimorphism deduced from the map :

A_{n} ⊗_{R} A_{(t)} → A(n)_{(t)} : a_{n} ⊗ b_{k} · t^{−k} ↦ a_{n}b_{k} · t^{−k} for every a_{n} ∈ A_{n}, b_{k} ∈ A_{k}

By inspecting the construction, we see therefore that g is the direct sum of the morphisms :

g_{n} : S ⊗_{R} A_{n} → L^{⊗n} : s ⊗ a_{n} ↦ s · β^{♮}(a_{n} · t^{−n}) · t^{′⊗n} for every s ∈ S, a_{n} ∈ A_{n}.

Finally, it is easily seen that the natural transformations :

(10.6.37)  g ↦ (ℙ(g), h(g))  and (β, h) ↦ g(β, h)

are inverse to each other : indeed, the verification can be made locally on Z, hence we may assume that X and Z are affine, and ℒ is free, in which case one may use the explicit formulae provided above. □

Definition 10.6.38. Let X be a scheme, ℐ a quasi-coherent sheaf of ideals of 𝒪_{X}.

(i) The positive Rees algebra of ℐ is the quasi-coherent ℕ-graded 𝒪_{X}-algebra

R^{+}(ℐ)_{•} := ⨁_{n∈ℕ} ℐ^{n}

with multiplication law deduced in the obvious way from that of 𝒪_{X} (cp. definition 7.9.1(iii)).

(ii) The blowing up of ℐ is the X-scheme

Proj R^{+}(ℐ)_{•} → X.

Remark 10.6.39. (i) With the notation of definition 10.6.38, let π : Y → X be the blowing up of ℐ. Notice that ℐ is an invertible 𝒪_{X}-module if and only if it is generated, locally on X, by a regular section of 𝒪_{X}, and then the natural map

Sym_{𝒪_{X}}^{n} ℐ → ℐ^{n}

is an isomorphism, for every n ∈ ℕ. Taking into account [60, Ch.II, Cor.3.1.7, Prop.3.1.8(iii)], it follows that if ℐ is an invertible ideal, then the blowing up of ℐ is an isomorphism.

(ii) Let X′ → X be any flat morphism of schemes; in light of (10.6.29), we get a natural isomorphism from X′ ×_{X} Y → X′ to the blowing up of the quasi-coherent ideal ℐ 𝒪_{X′} of 𝒪_{X′}.

(iii) Especially, let U ⊂ X be the complement in X of the closed subscheme Spec 𝒪_{X}/ℐ. From (i) and (ii) we deduce that the restriction π^{−1}U → U of π is an isomorphism.

(iv) Notice that R^{+}(ℐ)_{•} is generated, locally on X, by its degree one direct summand ℐ, hence 𝒪_{Y} (1) is an invertible 𝒪_{Y} -module. On the other hand, a simple inspection shows that

R^{+}(ℐ)_{•}(n) = ℐ^{n} · R^{+}(ℐ)_{•} for every n ∈ ℕ

whence a natural identification

𝒪_{Y} (n) →^{∼} ℐ^{n} · 𝒪_{Y} for every n ∈ ℕ.

Proposition 10.6.40. With the notation of definition 10.6.38, the blowing up Y → X of ℐ is characterized, up to unique isomorphism of X-schemes, by the following two conditions :

(i) The sheaf of ideals ℐ · 𝒪_{Y} is an invertible 𝒪_{Y} -module.

(ii) For every X-scheme Z such that ℐ · 𝒪_{Z} is an invertible 𝒪_{Z}-module, there exists a unique morphism Z → Y of X-schemes.

Proof. The uniqueness up to unique isomorphism of an X-scheme π : Y → X fulfilling conditions (i) and (ii) is clear. It remains to check that these conditions hold for Y := Proj R^{+}(ℐ)_{•}.

However, condition (i) follows immediately from remark 10.6.39(iv). Next, let f : Z → X be as in (ii), and set ℒ := ℐ · 𝒪_{Z}; the induced map f^{∗}𝒪_{X} → 𝒪_{Z} induces an epimorphism of 𝒪_{Z}-modules

φ_{n} : f^{∗}ℐ^{n} → ℐ^{n}𝒪_{Z} →^{∼} Sym_{𝒪_{Z}}^{n} ℒ for every n ∈ ℕ

and it is clear that the system (φ_{n} | n ∈ ℕ) amounts to a morphism of ℕ-graded 𝒪_{Z}-algebras

φ_{•} : f^{∗}R^{+}(ℐ)_{•} → Sym_{𝒪_{Z}}^{•} ℒ

which in turn corresponds to a morphism g : Z → Y of X-schemes, by virtue of lemma 10.6.33. Lastly, let g′ : Z → Y be any other morphism of X-schemes, and denote by U ⊂ X (resp. U′ ⊂ Z) the complement in X (resp. in Z) of the closed subset Spec 𝒪_{X}/ℐ (resp. Spec 𝒪_{Z}/ℐ 𝒪_{Z}); clearly f(U′) ⊂ U, hence g and g′ both restrict to morphisms U′ → π^{−1}U of X-schemes. Then remark 10.6.39(iii) implies that

g_{|U′} = g′_{|U′}.

Next, since ℐ 𝒪_{Z} is generated, locally on Z, by a regular section of 𝒪_{Z}, the open subset U′ is schematically dense in Z ([65, Ch.IV, Déf.11.10.2]), and on the other hand, we know that π is a separated morphism (see (10.6.15)); in view of [65, Ch.IV, Prop.11.10.1] we deduce that g = g′, which concludes the verification of (ii). □

10.6.41. We may generalize as follows remark 10.6.39(ii). Let X → X_{0} be a morphism of schemes, ℐ_{0} ⊂ 𝒪_{X_{0}} a quasi-coherent ideal, and set ℐ := ℐ_{0}𝒪_{X}; let also π_{0} : Y_{0} → X_{0} and π : Y → X be the blowing up morphisms of ℐ_{0} and resepctively ℐ. By proposition 10.6.40 there exists a unique morphism of schemes ψ : Y → Y_{0} that makes commute the diagram

(10.6.42)  \xymatrix{Y \ar[r]^{ψ} \ar[d]_{π} & Y_{0} \ar[d]^{π_{0}} \\ X \ar[r] & X_{0}.}

Then we have :

Lemma 10.6.43. The morphism of X-schemes \overline{ψ} : Y → Y′ := Y_{0} ×_{X_{0}} X induced by ψ is a closed immersion, and identifies Y with Spec 𝒪_{Y′}/𝒥, where

𝒥 := ⋃_{n∈ℕ} Ann_{𝒪_{Y′}} (ℐ^{n}).

Proof. Let Z → X be any morphism of schemes such that ℐ 𝒪_{Z} is an invertible 𝒪_{Z}-module; since ℐ = ℐ_{0}𝒪_{X}, proposition 10.6.40 yields a unique morphism of X_{0}-schemes Z → Y_{0}, so there exists a unique morphism of X-schemes g : Z → Y′ as well. Next, since ℐ 𝒪_{Z} is invertible, it is easily seen that the induced morphism 𝒪_{Y′} → g_{∗}𝒪_{Z} factors through 𝒪_{Y′}/𝒥, whence a unique morphism of X-schemes Z → Y′′ := Spec 𝒪_{Y′}/𝒥. Lastly, by construction ℐ 𝒪_{Y′} is a locally principal ideal of 𝒪_{Y′}, and then it is clear that ℐ 𝒪_{Y′′} is an invertible 𝒪_{Y′′}- module. Summing up, we see that the morphism Y′′ → X enjoys the universal property that characterizes the blowing up of ℐ, whence the lemma. □

10.6.44. Let A be a ring, r ≥ 1 an integer, f := (f_{1}, …, f_{r}) a sequence of elements of A, and I ⊂ A the ideal generated by f. We view A as an algebra over the ring A_{0} := ℤ[T_{1}, …, T_{r}], via the ring homomorphism

g : A_{0} → A  T_{i} ↦ f_{i}  i = 1, …, r

and denote by I_{0} ⊂ A_{0} the ideal generated by (T_{1}, …, T_{r}). Set also X := Spec A, X_{0} := Spec A_{0}, let ℐ ⊂ 𝒪_{X} (resp. ℐ_{0} ⊂ 𝒪_{X_{0}}) be the quasi-coherent sheaf of ideals arising from I (resp. from I_{0}), and π : Y → X (resp. π_{0} : Y_{0} → X_{0}) the blowing up of ℐ (resp. of ℐ_{0}). Clearly ℐ_{0}𝒪_{Y} = ℐ, whence a commutative diagram (10.6.42), with bottom horizontal arrow given by Spec g : X → X_{0}. Furthermore, notice that we have systems of monomorphisms of invertible 𝒪_{Y} -modules :

(10.6.45)  𝒪_{Y_{0}}(n + 1) → 𝒪_{Y_{0}}(n)  𝒪_{Y} (n + 1) → 𝒪_{Y} (n)  for every n ∈ ℕ

corresponding to the inclusion maps ℐ^{n+1}𝒪_{Y_{0}} → ℐ^{n}𝒪_{Y_{0}} and ℐ^{n+1}𝒪_{Y} → ℐ^{n}𝒪_{Y} under the natural identifications of remark 10.6.39(iv).

Proposition 10.6.46. With the notation of (10.6.44), suppose moreover that the sequence f is completely secant. Then, for every n ∈ ℕ we have :

(i) The natural map I^{n} → H^{0}(Y, 𝒪_{Y} (n)) is an isomorphism.

(ii) H^{p}(Y, 𝒪_{Y} (n)) = 0 for every p > 0.

Proof. We consider first the ring A_{0} and the sequence f_{0} := (T_{1}, …, T_{r}); we endow A_{0} with its standard ℕ-grading, such that gr_{n}A_{0} is the ℤ-module generated by the monomials of total degree n, for every n ∈ ℕ. Let also R_{0}^{+} ⊂ A_{0}[V] be the Rees algebra associated with the I_{0}-adic filtration of A_{0}, so that Y_{0} = Proj R_{0}^{+} (see definition 7.9.1(iii); but since here we have a descending filtration on A_{0}, we let V := U^{−1}, so gr_{n}R_{0}^{+} := V^{n}I_{0}^{n} for every n ∈ ℕ). We consider the morphism of ℕ-graded ℤ-algebras

h : A_{0} → R_{0}^{+}  T_{i} ↦ V T_{i}  i = 1, …, r.

Notice that φ(A_{0+}) generates the ideal R_{0+}^{+} of R_{0}^{+}, hence φ := Proj h is well defined on the whole of Y_{0} (see (10.6.5)), and we get a commutative diagram of morphisms of schemes

\xymatrix{Y_{0} \ar[r]^{φ} \ar[d]_{π_{0}} & ℙ_{ℤ}^{r−1} \ar[d] \\ X_{0} \ar[r] & Spec ℤ.}

For every i = 1, …, r, set U_{i} := D_{+}(T_{i}) ⊂ ℙ_{ℤ}^{r−1}, and U′_{i} := φ^{−1}U_{i}. Recall that there are natural isomorphisms ω_{i} : 𝒪_{ℙ_{ℤ}^{r−1}}(U_{i}) →^{∼} A_{i} := ℤ[τ_{ij} | j = 1, …, r] (notation of example 10.6.3); we wish to give a corresponding description of the A_{i}-algebra B_{i} := 𝒪_{Y_{0}}(U′_{i}). To this aim, notice that, by definition, every element of B_{i} ⊂ ℤ[T_{1}, …, T_{r}, V, (V T_{i})^{−1}] is a finite sum of terms of the form

(10.6.47) (V T_{1})^{α_{1}} ⋯ (V T_{r})^{α_{r}} · (V T_{i})^{−k} · P (T_{1}, …, T_{r}) = τ_{i1}^{α_{1}} ⋯ τ_{ir}^{α_{r}} · P (T_{i}τ_{i1}, …, T_{i}τ_{ir})

where k ∈ ℕ is any integer, P ∈ A any polynomial, and α_{1} + ⋯ + α_{r} = k; i.e.

B_{i} = A_{i}[T_{i}] for every i = 1, …, r.

Likewise, for every n ∈ ℕ, the elements of R_{0}^{+}(n)_{(T_{i})} are the finite sums of terms (10.6.47) where k ∈ ℕ is any integer, P ∈ A any polynomial, and α_{1} + ⋯ + α_{r} = n + k; i.e.

R_{0}^{+}(n)_{(T_{i})} = (V T_{i})^{n}B_{i} for every i = 1, …, r.

For every i ≠ j, set also U_{ij} := U_{i} ∩ U_{j} and U′_{ij} := φ^{−1}U_{ij}, and recall that the isomorphisms ω_{i} induce isomorphisms ω_{ij} : 𝒪_{ℙ_{ℤ}^{r−1}}(U_{ij}) →^{∼} A_{i}[τ_{ij}^{−1}], such that the composition

ψ_{ij} := ω_{ji} ∘ ω_{ij}^{−1} : A_{i}[τ_{ij}^{−1}] →^{∼} A_{j}[τ_{ji}^{−1}]

is given by the rule (10.6.4). There follow natural identifications 𝒪_{Y_{0}}(U′_{ij}) →^{∼} B_{i}[τ_{ij}^{−1}], whence a commutative diagram

\xymatrix{A_{i}[τ_{ij}^{−1}] \ar[r]^{ψ_{ij}} \ar[d] & A_{j}[τ_{ji}^{−1}] \ar[d] \\ B_{i}[τ_{ij}^{−1}] \ar[r]^{ψ′_{ij}} & B_{j}[τ_{ji}^{−1}]}

whose vertical arrows are induced by the map 𝒪_{ℙ_{ℤ}^{r−1}} → φ_{∗}𝒪_{Y_{0}} associated with φ, and where ψ′_{ij} is the isomorphism given by the rule :

T_{i} ↦ T_{j} · τ_{ji}.

Likewise, we get a commutative diagram

\xymatrix{ & 𝒪_{Y_{0}}(n)(U′_{ij}) \ar[dl] \ar@{-->}[dr] \\ (V T_{i})^{n}B_{i}[τ_{ij}^{−1}] \ar[rr] &  & (V T_{j})^{n}B_{j}[τ_{ji}^{−1}]}

whose downward arrows are induced by the restriction maps of the sheaf 𝒪_{Y_{0}}(n), and where the horizontal map is the B_{i}[τ_{ij}^{−1}]-linear map given by the rule

(V T_{i})^{n} ↦ (V T_{j})^{n} · τ_{ji}^{n}

and where the B_{i}[τ_{ij}^{−1}]-module structure of (V T_{j})^{n}B_{j}[τ_{ji}^{−1}] is defined by restriction of scalars along the isomorphism ψ′_{ij}. Comparing with example 10.6.8 we deduce a natural isomorphism of 𝒪_{ℙ_{ℤ}^{r−1}}-modules :

φ_{∗}𝒪_{Y_{0}}(n) →^{∼} ⨁_{k∈ℕ} 𝒪_{ℙ_{ℤ}^{r−1}}(n + k) for every n ∈ ℕ.

On the other hand, from [61, Ch.III, Prop.2.1.12] we get isomorphisms of ℕ-graded ℤ-modules:

H^{p}(ℙ_{ℤ}^{r−1}, ⨁_{k∈ℕ} 𝒪_{ℙ_{ℤ}^{r−1}}(k)) = \{ \begin{matrix}0 & for every p > 0 \\ A & for p = 0 \end{matrix}

Now, since φ is affine, we have

H^{p}(Y_{0}, 𝒪_{Y_{0}}(n)) = H^{p}(ℙ_{ℤ}^{r−1}, φ_{∗}𝒪_{Y_{0}}(n)) for every n ∈ ℕ

and assertion (ii) already follows. We also deduce an isomorphism of ℕ-graded ℤ-modules

H^{0}(Y_{0}, 𝒪_{Y_{0}}(n)) →^{∼} H^{0}(ℙ_{ℤ}^{r−1}, ⨁_{k∈ℕ} 𝒪_{ℙ_{ℤ}^{r−1}}(n + k)) = I_{0}^{n}

where gr_{k}I_{0}^{n} := gr_{n+k}A, for every k ∈ ℕ. To conclude the proof in this case, it remains only to check that this identification is the inverse of the natural map of (i). To this aim, we consider the induced commutative diagram

\xymatrix{H^{0}(Y_{0}, 𝒪_{Y_{0}}(n)) \ar[r] \ar[d] & H^{0}(ℙ_{ℤ}^{r−1}, φ_{∗}𝒪_{Y_{0}}(n)) \ar[d] \\ H^{0}(U′_{i}, 𝒪_{Y_{0}}(n)) \ar[r] & H^{0}(U_{i}, φ_{∗}𝒪_{Y_{0}}(n)).}

Now, let T^{α} := T_{1}^{α_{1}} ⋯ T_{r}^{α_{r}} ∈ gr_{k}I^{n} be any monomial (for any k ∈ ℕ); by inspecting the foregoing notation, we see that the image of T^{α} in H^{0}(U′_{i}, 𝒪_{Y_{0}}(n)) equals (V T_{i})^{n}T_{i}^{k}τ_{i,1}^{α_{1}} ⋯ τ_{i,r}^{α_{r}}, which maps to the section T_{i}^{n+k}τ_{i,1}^{α_{1}} ⋯ τ_{i,r}^{α_{r}} of H^{0}(U_{i}, φ_{∗}𝒪_{Y_{0}}(n)). But the latter is also the image of T^{α} under the restriction map H^{0}(ℙ_{ℤ}^{r−1}, 𝒪_{ℙ^{r−1}}(n + k)) → H^{0}(U_{i}, 𝒪_{ℙ^{r−1}}(n + k)), whence the contention.

Next, let A, f and I as in (10.6.44), and we define likewise R^{+} ⊂ A[V] as the positive Rees algebra associated with the I-adic filtration of A; we consider the homomorphism of ℕ-graded ℤ-algebras

G : R_{0}^{+} → R^{+}  V^{n}x ↦ V^{n}g(x)  for every n ∈ ℕ and every x ∈ I_{0}^{n}.

Notice that G is even a morphism of ℕ-graded A_{0}-algebras, for the A_{0}-algebra structure on A and R^{+} induced by g. There follows a morphism of ℕ-graded A-algebras

h′ : A ⊗_{A_{0}} R_{0}^{+} → R^{+}

whose restriction gr_{n}h′ is induced by the natural map A ⊗_{A_{0}} I_{0}^{n} → I^{n}, for every n ∈ ℕ. The latter is an isomorphism for every n ∈ ℕ, by virtue of lemma 7.8.23, and it is clear that the morphism ψ of (10.6.44) equals Proj h′; in view of (10.6.10), we deduce that the diagram (10.6.42) is cartesian under the current assumptions, and moreover h′ induces natural isomorphisms of 𝒪_{Y} -modules

ψ^{∗}𝒪_{Y_{0}}(n) →^{∼} 𝒪_{Y} (n) for every n ∈ ℕ.

Denote by 𝔘′ the affine open covering (U′_{i} | i = 1, …, r) of Y_{0}, and set ψ^{−1}𝔘′ := (ψ^{−1}U′_{i} | i = 1, …, r). there follows a natural isomorphism of alternating Čech complexes

(10.6.48)  \overline{C}_{alt}^{•}(𝔘′, 𝒪_{Y_{0}}(n)) ⊗_{A_{0}} A →^{∼} \overline{C}_{alt}^{•}(ψ^{−1}𝔘′, 𝒪_{Y} (n))  for every n ∈ ℕ.

Claim 10.6.49. The natural map \overline{C}_{alt}^{•}(𝔘′, 𝒪_{Y_{0}}(n)) ⊗^{L}_{A_{0}} A → \overline{C}_{alt}^{•}(𝔘′, 𝒪_{Y_{0}}(n)) ⊗_{A_{0}} A is an isomorphism in D(A-Mod), for every n ∈ ℕ.

Proof of the claim. We consider the standard spectral sequence

E_{1}^{pq} := Tor_{−q}^{A_{0}}(\overline{C}_{alt}^{p}(𝔘′, 𝒪_{Y_{0}}(n)), A) ⇒ H^{p+q}(\overline{C}_{alt}^{•}(𝔘′, 𝒪_{Y_{0}}(n)) ⊗^{L}_{A_{0}} A)

and notice that it suffices to check that E_{1}^{pq} = 0 for every p ≠ 0, since in that case the abutment is naturally isomorphic to the cohomology of the complex (E_{1}^{0,•}, d_{1}^{0,•}), which is the same as \overline{C}_{alt}^{•}(𝔘′, 𝒪_{Y_{0}}(n)) ⊗_{A_{0}} A. However, with the foregoing notation, we see that \overline{C}_{alt}^{q}(𝔘′, 𝒪_{Y_{0}}(n)) is a direct sum of finitely many A_{0}-modules, each of which is a localization of B_{i}, for some i ≤ r, so we are reduced to checking that Tor_{p}^{A_{0}}(B_{i}, A) = 0 for i = 1, …, r and every p > 0. But in turn, B_{i} is the degree 0 summand of a ℤ-graded A_{0}-module, which is a localization of R_{0}^{+}, so we are further reduced to checking that Tor_{p}^{A_{0}}(R_{0}^{+}, A) = 0 for every p > 0, or equivalently, that Tor_{p}^{A_{0}}(I_{0}^{n}, A) = 0 for every p > 0 and every n ∈ ℕ. The latter follows from lemma 7.8.23. ♢

Combining (10.6.48), claim 10.6.49and theorem 10.2.28(ii), we get a natural isomorphism :

\overline{C}_{alt}^{•}(𝔘′, 𝒪_{Y_{0}}(n)) ⊗^{L}_{A_{0}} A →^{∼} RΓ(Y, 𝒪_{Y} (n)) in D(A-Mod), for every n ∈ ℕ.

By the same token, we have a natural isomorphism \overline{C}_{alt}^{•}(𝔘′, 𝒪_{Y_{0}}(n)) →^{∼} RΓ(Y_{0}, 𝒪_{Y_{0}}(n)) in D(A_{0}-Mod), whence a standard spectral sequence for every n ∈ ℕ

E(n)_{2}^{pq} := Tor_{−p}^{A_{0}}(H^{q}(Y_{0}, 𝒪_{Y_{0}}(n)), A) ⇒ H^{p+q}(\overline{C}_{alt}^{•}(𝔘′, 𝒪_{Y_{0}}(n)) ⊗^{L}_{A_{0}} A)

and the previous case tells us that E(n)_{2}^{pq} = 0 for every q > 0 and every n ∈ ℕ, and moreover E(n)_{2}^{p0} = Tor_{−p}^{A_{0}}(I_{0}^{n}, A) for every p ∈ ℤ and every n ∈ ℕ. Lastly, by invoking again lemma 7.8.23 we see that E(n)_{2}^{p0} = 0 whenever p ≠ 0, and E(n)_{2}^{00} = I_{0}^{n}A = I^{n} for every n ∈ ℕ, which concludes the proof of the proposition. □

Theorem 10.6.50. With the notation of (10.6.44), the following conditions are equivalent : (a) The ring A satisfies condition (a)_{f}^{un}.

(b) The inverse system (H^{p}(Y, 𝒪_{Y} (n)) | n ∈ ℕ) deduced from the system of maps (10.6.45) is uniformly essentially zero for every p > 0, and the same holds for the kernel and cokernel of the natural morphism of inverse systems (I^{n} → H^{0}(Y, 𝒪_{Y} (n)) | n ∈ ℕ).

(c) There exists n ∈ ℕ such that I^{n} · H^{p}(Y, 𝒪_{Y}) = 0 for every p > 0, and I^{n} annihilates the kernel and cokernel of the natural map A → H^{0}(Y, 𝒪_{Y}).

Proof. Define I_{0} ⊂ A, the positive Rees algebras R_{0}^{+}, R^{+}, and the morphism π_{0} : Y_{0} → X_{0} := Spec A_{0} as in the proof of proposition 10.6.46. We set Y′ := Y_{0} ×_{X_{0}} X and define 𝒥 ⊂ 𝒪_{Y′}, ψ : Y → Y and \overline{ψ} : Y → Y′ as in lemma 10.6.43, so that 𝒥 is the kernel of the induced map \overline{ψ}^{♯} : 𝒪_{Y′} → \overline{ψ}_{∗}𝒪_{Y}. We shall consider as well the condition :

(d) There exists k ∈ ℕ such that I^{k}𝒥 = I^{k} · 𝒯or_{i}^{X_{0}}(𝒪_{Y_{0}}, 𝒪_{X}) = 0 for every i > 0.

Claim 10.6.51. If condition (a) holds, there exists k ∈ ℕ such that

I^{k} · Tor_{i}^{A_{0}}(I_{0}^{n}, A) = I^{k} · Ker(I_{0}^{n} ⊗_{A_{0}} A → I^{n}) = 0 for every n ∈ ℕ and every i > 0.

Proof of the claim. For every integer p > 0, let k(p) be the step of the uniformly essentially zero system (Tor_{p}^{A_{0}}(A_{0}/I_{0}^{n}, A) | n ∈ ℕ). For p > 1 (resp. for p = 1), the integer k(p) is also the step of the uniformly essentially zero system (T_{p−1}^{n} := Tor_{p−1}^{A_{0}}(I_{0}^{n}, A) | n ∈ ℕ) (resp. the uniformly essentially zero system (K^{n} := Ker(I_{0}^{n} ⊗_{A_{0}} A → I^{n}) | n ∈ ℕ)). Now, for every a ∈ I_{0}^{k(p)}, scalar multiplication by a on I_{0}^{n} factors through the inclusion map I_{0}^{n+k(p)} → I_{0}^{n}, so it induces the zero map on T_{p−1}^{n} (resp. on K^{n}) for every n ∈ ℕ. Lastly, recall that the homological dimension of A_{0} equals r + 1, so T_{p}^{n} = 0 for every p > r + 1. We may therefore choose k := max(k(1), …, k(r + 1)). ♢

(a)⇒(d): Indeed, recall that we have a natural 𝒪_{Y′}(U ×_{X_{0}} X)-linear isomorphism

Γ(U ×_{X_{0}} X, 𝒯or_{i}^{X_{0}}(𝒪_{Y_{0}}, 𝒪_{X})) →^{∼} T_{i}(U) := Tor_{i}^{A_{0}}(H^{0}(𝒪_{Y_{0}}(U), A)

for every affine open subsets U ⊂ Y_{0}. Since Y′ is quasi-compact, and taking into account lemma 10.6.43, it will then suffice to find k ∈ ℕ and a finite open covering U_{•} := (U_{λ} | λ ∈ Λ) of Y_{0} such that

• I^{k} · T_{i}(U_{λ}) = 0 for every i > 0 and every λ ∈ Λ

• I^{k} · Ker(\overline{ψ}_{U_{λ}}^{♯} : 𝒪_{Y_{0}}(U_{λ}) ⊗_{A_{0}} A → 𝒪_{Y} (\overline{ψ}^{−1}U_{λ})) = 0 for every λ ∈ Λ.

However, we may choose U_{•} such 𝒪_{Y_{0}}(U_{λ}) is a direct summand of a graded localization of R_{0}^{+}, for every λ ∈ Λ; then T_{i}(U_{λ}) is a direct summand of a localization of Tor_{i}^{A_{0}}(R_{0}^{+}, A), and \overline{ψ}_{U_{λ}}^{♯} is a direct summand of a localization of the natural map R_{0}^{+} ⊗_{A_{0}} A → R^{+}. Thus, the assertion follows from claim 10.6.51.

(b)⇒(c): The natural identification ℐ^{n}𝒪_{Y} →^{∼} 𝒪_{Y} (n) of remark 10.6.39(iv) yields exact cohomology sequences

H^{p}(Y, 𝒪_{Y} (n)) ⟶^{α_{n}^{p}} H^{p}(Y, 𝒪_{Y}) → H^{p}(Y, 𝒪_{Y} /ℐ^{n}𝒪_{Y}) for every p, n ∈ ℕ.

However, assumption (b) implies that for every p > 0 there exists k ∈ ℕ such that α_{n}^{p} is the zero map, for every n ≥ k; hence, for such k the term H^{p}(Y, 𝒪_{Y}) is a submodule of H^{p}(Y, 𝒪_{Y} /ℐ^{k}𝒪_{Y}), which is obviously annihilated by I^{k}. By the same token we get a commutative ladder with exact rows for every n ∈ ℕ :

\xymatrix{0 \ar[r] & I^{n} \ar[r] \ar[d]_{β_{n}} & A \ar[r] \ar[d]^{γ_{n}} & A/I^{n} \ar[r] \ar[d]^{δ_{n}} & 0 \\ 0 \ar[r] & H^{0}(Y, 𝒪_{Y} (n)) \ar[r] & H^{0}(Y, 𝒪_{Y}) \ar[r] & H^{0}(Y, 𝒪_{Y} /ℐ^{n}𝒪_{Y})}

whence an exact sequence

0 → Ker β_{n} ⟶^{τ_{n}} Ker γ_{n} → Ker δ_{n} → Coker β_{n} ⟶^{σ_{n}} Coker γ_{n} → Coker δ_{n}.

However, assumption (b) implies that there exists n ∈ ℕ such that τ_{n} and σ_{n} both vanish; since Ker δ_{n} and Coker δ_{n} are both annihilated by I^{n}, we conclude that (c) holds.

Next, we show that (d) implies both (a) and (b). To this aim, we remark :

Claim 10.6.52. Suppose that condition (d) holds; then we have :

(i) The inverse system (𝒯_{n}^{i} := 𝒯or_{i}^{X_{0}}(𝒪_{Y_{0}}(n), 𝒪_{X}) | n ∈ ℕ) induced by the system of morphisms (10.6.45) is uniformly essentially zero, for every i > 0.

(ii) Likewise, the inverse system (𝒥 (n) := Ker(𝒪_{Y′}(n) → \overline{ψ}_{∗}𝒪_{Y} (n)) | n ∈ ℕ) is uniformly essentially zero.

Proof of the claim. (i): There exists an affine open covering (U_{1}, …, U_{r}) of Y_{0} such that

(ℐ 𝒪_{Y_{0}})_{|U_{i}} = f_{i}𝒪_{Y_{0}} for every i = 1, …, r

and f_{i} is a regular element of 𝒪_{Y_{0}}(U_{i}), whence an identification

(ℐ^{n}𝒪_{Y_{0}})_{|U_{i}} = f_{i}^{n}𝒪_{U_{i}} →^{∼} 𝒪_{U_{i}} for every i = 1, …, r.

Under this identification, the restriction 𝒪_{Y_{0}}(n + 1)_{|U_{i}} → 𝒪_{Y_{0}}(n)_{|U_{i}} of the morphism (10.6.45) corresponds to the scalar multiplication by f_{i}. Hence, the inverse system ((𝒯_{n}^{i})_{|U_{i}} | n ∈ ℕ) is naturally identified with the inverse system (𝒯_{n}^{′i} | n ∈ ℕ) with 𝒯_{n}^{′i} := 𝒯or_{i}^{X_{0}}(𝒪_{U_{i}}, 𝒪_{X}) for every i, n ∈ ℕ, with transition maps given by scalar multiplication by f_{i}, whence (i).

For the proof of (ii) one argues in the same way, using the fact that I^{k}𝒥 = 0 for some k ∈ ℕ whose existence is ensured by condition (d) : the details shall be left to the reader. ♢

We consider now the inverse systems of spectral sequences provided by proposition 10.2.30

(10.6.53) E(n)_{2}^{pq} := H^{p}(Y_{0} ×_{X_{0}} X, 𝒯or_{−q}^{X_{0}}(𝒪_{Y_{0}}(n), 𝒪_{X})) ⇒ H^{p+q}(A ⊗^{L}_{A_{0}} RΓ𝒪_{Y_{0}}(n))

where, as usual, the transition maps E(n + 1)_{2}^{pq} → E(n)_{2}^{pq} are induced by the morphisms (10.6.45). Clearly E_{2}^{pq} = 0 if either p < 0 or q > 0, and claim 10.6.52(i) implies that the inverse system (E(n)_{2}^{pq} | n ∈ ℕ) is uniformly essentially zero whenever q < 0. It follows already that the inverse system (E(n)_{∞}^{pq} | n ∈ ℕ) is uniformly essentially zero whenever p + q < 0. On the other hand, proposition 10.6.46shows that

H^{−i}(A ⊗^{L}_{A_{0}} RΓ𝒪_{Y_{0}}(n)) = Tor_{i}^{A_{0}}(A, I_{0}^{n}) for every i ∈ ℤ.

Summing up, and taking into account lemma 7.8.19, a simple induction shows that the system (Tor_{i}^{A_{0}}(A, I_{0}^{n}) | n ∈ ℕ) is uniformly essentially zero for every i > 0. Notice also that E(n)_{2}^{p0} = H^{p}(Y′, 𝒪_{Y′}(n) for every n, p ∈ ℕ.

Claim 10.6.54. (i) The inverse system (E(n)_{s}^{p0}) | n ∈ ℕ) is uniformly essentially zero, for every p > 0 and every s ≥ 2.

(ii) The kernel and cokernel of the natural morphism of inverse systems

h_{•} : (A ⊗_{A_{0}} I_{0}^{n} | n ∈ ℕ) → (E(n)_{2}^{00} | n ∈ ℕ)

are both uniformly essentially zero.

Proof of the claim. (i): We argue by descending induction on p : if p ≥ r, we have E(n)_{2}^{p0} = 0 for every n ∈ ℕ, by theorem 10.2.28(iii), hence the claim trivially holds in this case. Thus, suppose that q < r, and that the claim is already known for every p > q and every s ≥ 2. We show, by descending induction on s ≥ 2, that (E(n)_{s}^{q0} | n ∈ ℕ) is uniformly essentially zero. Since E(n)_{2}^{q−s,s−1} = 0 for every s ≥ 2, we have E(n)_{s}^{q−s,s−1} = E(n)_{s}^{q+s,1−s} = 0 as well, whenever q + s ≥ r, and therefore E(n)_{s}^{q0} = E(n)_{∞}^{q0} = 0 for every p > 0 and every s ≥ r. Thus, suppose that 2 ≤ t < r, and that the assertion is known for every s > t; we consider the exact sequence of inverse systems

0 → (E(n)_{t+1}^{q0} | n ∈ ℕ) → (E(n)_{t}^{q0} | n ∈ ℕ) → (E(n)_{t}^{q+t,1−t} | n ∈ ℕ).

Since 1 − t < 0, the third term is uniformly essentially zero, and the same holds for the first term, by inductive assumption; by lemma 7.8.19, we deduce that the same holds for the middle term, as required.

(ii): The map h_{n} is the composition of the projection h′_{n} : A ⊗_{A_{0}} I_{0}^{n} → E(n)_{∞}^{00} with the injective map h′′_{n} : E(n)_{∞}^{00} → E(n)_{2}^{00} deduced from the spectral sequences (10.6.53). But since the inverse system (E(n)_{2}^{p,−p} | n ∈ ℕ) is uniformly essentially zero for every p > 0, and vanishes for p ≥ r, the same holds for the inverse systems (E(n)_{∞}^{p,−p} | n ∈ ℕ), and consequently the kernel of h′_{•} is uniformly essentially zero. Likewise, it is easily seen that the cokernel of h′′_{•} is uniformly essentially zero. ♢

Define 𝒥 (n) as in claim 10.6.52(ii), and set J(n)^{p} := H^{p}(Y′, 𝒥 (n) for every n, p ∈ ℕ; we get an exact sequence of inverse systems

(J(n)^{p} | n ∈ ℕ) → (E(n)_{2}^{p0} | n ∈ ℕ) ⟶^{k_{•}^{p}} (H^{p}(Y, 𝒪_{Y} (n)) | n ∈ ℕ) → (J(n)^{p+1} | n ∈ ℕ)

for every p ∈ ℕ, and claims 10.6.52(ii) and 10.6.54(i) imply that the first and third terms are uniformly essentially zero, whenever p > 0. Therefore, the same holds for the middle term. Lastly, we consider the commutative diagram

\xymatrix{(A ⊗_{A_{0}} I_{0}^{n} | n ∈ ℕ) \ar[r]^{m_{•}} \ar[d]_{h_{•}} & (I^{n} | n ∈ ℕ) \ar[d]^{l_{•}} \\ (E(n)_{2}^{00} | n ∈ ℕ) \ar[r]^{k_{•}^{0}} & (H^{0}(Y, 𝒪_{Y} (n)) | n ∈ ℕ)}

where m_{•} is given by the natural surjections, and h_{•} is defined as in claim 10.6.54. Since the inverse system (J(n)^{p} | n ∈ ℕ) is uniformly essentially zero for p = 0, 1, we see that the kernel and cokernel of k_{•}^{0} are both uniformly essentially zero. In view of claim 10.6.54(ii), it follows immediately that the kernel of m_{•} is uniformly essentially zero, and the same holds for the kernel and cokernel of l_{•}, which concludes the verification of both (a) and (b).

Lastly, we check that (c) implies both (a) and (b). To this aim we consider, for every n ∈ ℕ, the spectral sequences provided by proposition 10.2.30(ii)

\begin{matrix}E(n)_{2}^{pq} := & Tor_{−p}^{A_{0}}(I_{0}^{n}, H^{q}(Y, 𝒪_{Y})) ⇒ H^{p+q}(I_{0}^{n} ⊗^{L}_{A_{0}} RΓ𝒪_{Y}) \\ F (n)_{2}^{pq} := & H^{p}(Y, 𝒯or_{−q}^{X_{0}}(ℐ_{0}^{n}, 𝒪_{Y})) ⇒ H^{p+q}(I_{0}^{n} ⊗^{L}_{A_{0}} RΓ𝒪_{Y}). \end{matrix}

Moreover, for every ∈ ℕ denote by

φ_{n} : 𝒯or_{0}^{X_{0}}(𝒪_{Y_{0}}, ℐ_{0}^{n}) = π_{0}^{∗}ℐ_{0}^{n} → ℐ_{0}^{n}𝒪_{Y_{0}} = 𝒪_{Y_{0}}(n)

the natural morphism, and notice that ψ^{∗}π_{0}^{∗}ℐ_{0}^{n} = 𝒯or_{0}^{X_{0}}(ℐ_{0}^{n}, 𝒪_{Y}).

Claim 10.6.55. (i) The inverse system (Ker φ_{n} | n ∈ ℕ) is uniformly essentially zero, and the same holds for the inverse system (𝒯or_{q}^{X_{0}}(ℐ_{0}^{n}, 𝒪_{Y}) | n ∈ ℕ), for every q > 0.

(ii) The inverse system (H^{i}(I_{0}^{n} ⊗^{L}_{A_{0}} RΓ𝒪_{Y}) | n ∈ ℕ) is uniformly essentially zero for i ≠ 0. (iii) The spectral sequences E(n)_{2}^{••} induce, for every p ∈ ℤ, an epimorphism of inverse

systems (E(n)_{2}^{p0} | n ∈ ℕ) → (E(n)_{∞}^{p0} | n ∈ ℕ) whose kernel is uniformly essentially zero.

(iv) The spectral sequences F (n)_{2}^{••} induce, for every p ∈ ℤ, a monomorphism of inverse systems (F (n)_{∞}^{p0} | n ∈ ℕ) → (F (n)_{2}^{p0} | n ∈ ℕ) whose cokernel is uniformly essentially zero.

(v) The inverse systems (E(n)_{2}^{p,−p} | n ∈ ℕ) and (F (n)_{2}^{−p,p} | n ∈ ℕ) are uniformly essentially zero for p < 0, and vanish identically for p > 0.

Proof of the claim. (i): We look at the spectral sequence

G(n)_{pq}^{2} := 𝒯or_{p}^{Y_{0}}(𝒪_{Y} , 𝒯or_{q}^{X_{0}}(𝒪_{Y_{0}}, ℐ_{0}^{n})) ⇒ 𝒯or_{p+q}^{X_{0}}(ℐ_{0}^{n}, 𝒪_{Y})

of proposition 10.2.30(i). Since Y_{0} is a noetherian scheme, corollary 7.9.22(iii) implies that the inverse system (𝒯or_{q}^{X_{0}}(𝒪_{Y_{0}}, ℐ_{0}^{n}) | n ∈ ℕ) is uniformly essentially zero for every q > 0, so the same holds for the inverse system (G(n)_{2}^{pq} | n ∈ ℕ), for every p ∈ ℕ and every q > 0. By the same token, the inverse system (Ker φ_{n} | n ∈ ℕ) is uniformly essentially zero; since the functor

𝒯or_{p}^{Y_{0}}(𝒪_{Y} , −) : D^{−}(𝒪_{Y_{0}}-Mod_{qcoh}) → D^{−}(𝒪_{Y} -Mod_{qcoh})

is triangulated for every p ∈ ℤ, we get a short exact sequence

0 → 𝒯or_{p}^{Y_{0}}(𝒪_{Y} , Ker φ_{n}) → G(n)_{p0}^{2} → 𝒯or_{p}^{Y_{0}}(𝒪_{Y} , 𝒪_{Y_{0}}(n)) → 0 for every p, n ∈ ℕ

which shows that the inverse system (G(n)_{p0}^{2} | n ∈ ℕ) is uniformly essentially zero for every p > 0. The assertion now follows by a simple induction, using lemma 7.8.19.

(ii): Condition (c) implies that the inverse system (E(n)_{2}^{pq} | n ∈ ℕ) is uniformly essentially zero for q > 0, and clearly it vanishes identically when either q < 0 or p > 0. It also vanishes identically for q ≥ r, due to theorem 10.2.28(iii). Summing up, we see that the filtration

induced by the spectral sequence E(n)_{2}^{••} on its abutment H^{i} := H^{i}(I_{0}^{n} ⊗^{L}_{A_{0}} RΓ𝒪_{Y}) is finite for every i ∈ ℤ, and the inverse systems of its graded subquotients (E(n)_{∞}^{pq} | n ∈ ℕ) are uniformly essentially zero whenever p + q > 0, whence the assertion in case i > 0, after invoking as usual lemma 7.8.19. For i < 0, we argue likewise, using the spectral sequences F (n)_{2}^{••} : indeed, we see again easily that the filtration induced on the abutment H^{i} is finite for every i ∈ ℤ; moreover, if p < 0 the inverse system (F (n)_{2}^{pq} | n ∈ ℕ) is identically zero, and it is uniformly essentially zero if q < 0, by virtue of (i), whence the assertion.

(iii): For p > 0 we have E(n)_{2}^{p0} = 0 for every n ∈ ℕ, so the assertion is trivial in this case. For p ≤ 0, notice that E(n)_{s}^{p+s,1−s} = 0 for every s ≥ 2, hence E(n)_{s}^{p0} is a quotient of E(n)_{2}^{p0} for every s ≥ 2, whence a surjection E(n)_{2}^{p0} → E(n)_{∞}^{p0} as sought. Moreover, notice that the inverse system (E(n)_{2}^{p−s,s−1} | n ∈ ℕ) is uniformly essentially zero for every s ≥ 2, due to condition (c) and lemma 7.8.24, and it vanishes identically for s ≥ r. Then the same holds for the inverse system (E(n)_{s}^{p−s,s−1} | n ∈ ℕ), for every s ≥ 2, and the assertion follows easily.

(iv): For p < 0 we have F (n)_{2}^{p0} = 0 for every n ∈ ℕ, so the assertion is trivial in this case. For p ≥ 0, notice that F (n)_{s}^{p−s,s−1} = 0 for every s ≥ 2, hence F (n)_{s}^{p0} is a submodule of F (n)_{2}^{p0} for every s ≥ 2, whence a injection F (n)_{∞}^{p0} → F (n)_{2}^{p0} as sought. Moreover, notice that the inverse system (F (n)_{2}^{p+s,1−s} | n ∈ ℕ) is uniformly essentially zero for every s ≥ 2, due to (i), and it vanishes identically for s ≥ r. Then the same holds for the inverse system (F (n)_{s}^{p+s,1−s} | n ∈ ℕ), for every s ≥ 2, and the assertion follows easily.

(v): The assertion for p > 0 is obvious, and the case p < 0 for (E(n)_{2}^{p,−p} | n ∈ ℕ) follows from condition (c) and lemma 7.8.24, and for (F (n)_{2}^{−p,p} | n ∈ ℕ) follows from (i). ♢

Assumption (c) implies that the natural map A → H^{0}(Y, 𝒪_{Y}) induces a morphism of inverse systems

(Tor_{−p}^{A_{0}}(I_{0}^{n}, A) | n ∈ ℕ) → (E(n)_{2}^{p0} | n ∈ ℕ) for every p ∈ ℤ

whose kernel and cokernel are uniformly essentially zero. Combining with claim 10.6.55(ii,iii), it follows already that the inverse system (Tor_{p}^{A_{0}}(A/I_{0}^{n}, A) | n ∈ ℕ) is uniformly essentially zero for every p > 1. Next, since 𝒪_{Y_{0}}(n) is a flat 𝒪_{Y_{0}}-module for every n ∈ ℕ, we have an exact sequence of 𝒪_{Y} -modules

0 → ψ^{∗}Ker φ_{n} → 𝒯or_{0}^{X_{0}}(ℐ_{0}^{n}, 𝒪_{Y}) → ψ^{∗}𝒪_{Y_{0}}(n) = 𝒪_{Y} (n) → 0

whence exact sequences

H^{p}(Y, ψ^{∗}Ker φ_{n}) → F (n)_{2}^{p0} → H^{p}(Y, 𝒪_{Y} (n)) → H^{p+1}(Y, ψ^{∗}Ker φ_{n})

for every p ∈ ℤ and every n ∈ ℕ. Combining with claim 10.6.55(ii,iv), we deduce that the inverse system (H^{p}(Y, 𝒪_{Y} (n)) | n ∈ ℕ) is uniformly essentially zero for every p > 0.

Lastly, claim 10.6.55(v) implies that (E(n)_{∞}^{p,−p} | n ∈ ℕ) and (F (n)_{∞}^{−p,p} | n ∈ ℕ) vanish identically for every p > 0, and are uniformly essentially zero inverse systems for p < 0. It follows that the spectral sequences E(n)_{2}^{••} and F (n)_{2}^{••} induce respectively a monomorphism and an epimorphism

(E(n)_{∞}^{00} | n ∈ ℕ) → (H^{0}(I_{0}^{n} ⊗^{L}_{A_{0}} RΓ𝒪_{Y}) | n ∈ ℕ) → (F (n)_{∞}^{00} | n ∈ ℕ)

and the kernel and cokernel of the composition of these morphisms are uniformly essentially zero inverse systems. Taking into account claim 10.6.55(iv) we get therefore a commutative diagram of inverse systems

[figure: (E(n)_{2}^{00} | n∈ℕ); (E(n)_{∞}^{00} | n ∈ ℕ); (F (n)_{∞}^{00} | n ∈ ℕ); (I_{0}^{n} ⊗_{A_{0}} H^{0}(Y, 𝒪_{Y}) | n ∈ ℕ); β_{•}; (H^{0}(Y, 𝒪_{Y} (n) | n ∈ ℕ); (F (n)_{2}^{00} | n ∈ ℕ)]

all whose arrows, except possibly β_{•}, have uniformly essentially zero kernels and cokernels, but then also β_{•} does.

Claim 10.6.56. β_{•} is the map induced by the identification ω_{n} : ℐ^{n}𝒪_{Y} →^{∼} 𝒪_{Y} (n) of remark 10.6.39(iv) and the natural map ψ^{∗}π_{0}^{∗}ℐ_{0}^{n} → ℐ^{n}𝒪_{Y}.

Proof of the claim. Indeed, by construction, the map F (n)_{2}^{00} → H^{0}(Y, 𝒪_{Y} (n)) is obtained from the identification ω_{n} and the natural isomorphism 𝒯or_{0}^{X_{0}}(𝒪_{Y_{0}}, ℐ_{0}^{n}) →^{∼} ψ^{∗}π_{0}^{∗}ℐ_{0}^{n}, so we are reduced to checking that the map

E(n)_{2}^{00} = I_{0}^{n} ⊗_{A_{0}} H^{0}(Y, 𝒪_{Y}) → F (n)_{2}^{00} = H^{0}(Y, ψ^{∗}π_{0}^{∗}ℐ_{0}^{n})

deduced from the foregoing commutative diagram, agrees with the standard one that assigns to any element x ⊗ s its class in H^{0}(Y, ψ^{∗}π_{0}^{∗}ℐ_{0}^{n}). To this aim, set U_{i} := D_{+}(f_{i}) ⊂ Y for i = 1, …, r, and let Y′′ := ⨿_{i=1}^{r}U_{i}, the affine X-scheme given by the disjoint union of the affine open subsets U_{1}, …, U_{r}. We have an obvious morphism ν : Y′′ → Y of schemes, that restricts to the inclusion map on each open subset U_{i} of Y′′, and let also ψ′′ := ψ ∘ν : Y′′ → Y_{0}. We have as well two systems of spectral sequences E′′(n)_{2}^{••} and F′′(n)_{2}^{••}, obtained as in the foregoing, after replacing Y by Y′′ and 𝒪_{Y} by 𝒪_{Y′′}. By functoriality of the spectral sequences, there follows a commutative diagram

\xymatrix{E(n)_{2}^{00} = I_{0}^{n} ⊗_{A_{0}} H^{0}(Y, 𝒪_{Y}) \ar[r] \ar[d] & F (n)_{2}^{00} = H^{0}(Y, ψ^{∗}π_{0}^{∗}ℐ_{0}^{n}) \ar[d] \\ E′′(n)_{2}^{00} = I_{0}^{n} ⊗_{A_{0}} H^{0}(Y′′, 𝒪_{Y′′}) \ar[r] & F′′(n)_{2}^{00} = H^{0}(Y′′, ψ^{′′∗}π_{0}^{∗}ℐ_{0}^{n})}

and notice that the right vertical arrow is injective. Thus, it suffices to check that the bottom horizontal arrow is the expected map. However, we have E′′(n)_{2}^{00} = E′′(n)_{∞}^{00} = E′′(n)_{∞}^{0} = F′′(n)_{∞}^{0} = F′′(n)_{∞}^{00} = F′′(n)_{2}^{00} for every n ∈ ℕ, and a direct inspection shows that the bottom horizontal map is the identity map, whence the claim. ♢

From claim 10.6.56 and condition (c), it follows already that the natural morphism of inverse systems (I_{0}^{n} ⊗_{A_{0}} A | n ∈ ℕ) → (H^{0}(Y, 𝒪_{Y} (n)) | n ∈ ℕ) has uniformly essentially zero kernel and cokernel. But the latter factors as the composition of the epimorphism of inverse systems (I_{0}^{n} ⊗_{A_{0}} A | n ∈ ℕ) → (I^{n} | n ∈ ℕ) and the natural morphism (I^{n} | n ∈ ℕ) → (H^{0}(Y, 𝒪_{Y} (n)) | n ∈ ℕ). It follows that the kernels and cokernels of both of the latter morphisms are uniformly essentially zero, and the proof of the theorem is concluded. □

Remark 10.6.57. By direct inspection of the proof, we obtain the following refinement of theorem 10.6.50. For every r ∈ ℕ there exists an integer ν(r) such that, if the ring A satisfies condition (a)_{f}^{un} with step ≤ r, then the inverse system (H^{p}(Y, 𝒪_{Y} (n)) | n ∈ ℕ) is uniformly essentially zero with step ≤ ν(r), and the same holds for the kernel and cokernel of the inverse system (I^{n} → H^{0}(Y, 𝒪_{Y} (n)) | n ∈ ℕ). The detailed verification shall be left to the reader.

11. Duality theory

11.1. Duality for quasi-coherent modules. Let f : X → Y be a morphism of schemes; theorem 10.1.18falls short of proving that Lf^{∗} is a left adjoint to Rf_{∗}, since the former functor is defined on bounded above complexes, while the latter is defined on complexes that are bounded below. This deficency has been overcome by N.Spaltenstein’s paper [154], where he shows how to extend the usual constructions of derived functors to unbounded complexes. On the other hand, in many cases one can also construct a right adjoint to Rf_{∗}. This is the subject of Grothendieck’s duality theory. We shall collect the statements that we need from that theory, and refer to the original source [87] for the rather elaborate proofs.

11.1.1. For any morphism of schemes f : X → Y, let \overline{f} : (X, 𝒪_{X}) → (Y, f_{∗}𝒪_{X}) be the corresponding morphism of ringed spaces. The functor

\overline{f}_{∗} : 𝒪_{X}-Mod → f_{∗}𝒪_{X}-Mod

admits a left adjoint \overline{f}^{∗} defined as usual ([59, Ch.0, §4.3.1]). In case f is affine, \overline{f} is flat and the unit and counit of the adjunction restrict to isomorphisms of functors :

1 → \overline{f}_{∗} ∘ \overline{f}^{∗} : f_{∗}𝒪_{X}-Mod_{qcoh} → f_{∗}𝒪_{X}-Mod_{qcoh}
\overline{f}^{∗} ∘ \overline{f}_{∗} → 1 : 𝒪_{X}-Mod_{qcoh} → 𝒪_{X}-Mod_{qcoh}.

on the corresponding subcategories of quasi-coherent modules ([60, Ch.II, Prop.1.4.3]). It follows easily that, on these subcategories, \overline{f}^{∗} is also a right adjoint to \overline{f}_{∗}.

Lemma 11.1.2. Keep the notation of (11.1.1), and suppose that f is affine. Then the functor :

\overline{f}^{∗} : D^{+}(f_{∗}𝒪_{X}-Mod)_{qcoh} → D^{+}(𝒪_{X}-Mod)_{qcoh}

is right adjoint to the functor R\overline{f}_{∗} : D^{+}(𝒪_{X}-Mod)_{qcoh} → D^{+}(f_{∗}𝒪_{X}-Mod)_{qcoh}. Moreover, the unit and counit of the resulting adjunction are isomorphisms of functors.

Proof. By trivial duality (theorem 10.1.18), \overline{f}^{∗} is left adjoint to R\overline{f}_{∗} on D^{+}(𝒪_{X}-Mod); it suffices to show that the unit (resp. and counit) of this latter adjunction are isomorphisms for every object K^{•} of D^{+}(𝒪_{X}-Mod)_{qcoh} (resp. L^{•} of D^{+}(f_{∗}𝒪_{X}-Mod)_{qcoh}). Concerning the unit, we can use a Cartan-Eilenberg injective resolution of \overline{f}^{∗}K^{•}, and a standard spectral sequence, to reduce to the case where K^{•} = ℱ [0] for a quasi-coherent f_{∗}𝒪_{X}-module ℱ; however, the natural map \overline{f}_{∗} ∘ \overline{f}^{∗}ℱ → R\overline{f}_{∗} ∘ \overline{f}^{∗}ℱ is an isomorphism, so the assertion follows from (11.1.1). □

Remark 11.1.3. (i) Even when both X and Y are affine, neither the unit nor the counit of adjunction in (11.1.1) is an isomorphism on the categories of all modules. For a counterexample concerning the unit, consider the case of a finite injection of domains A → B, where A is local and B is semi-local (but not local); let f : X := Spec B → Spec A be the corresponding morphism, and denote by ℱ the f_{∗}𝒪_{X}-module supported on the closed point, with stalk equal to B. Then one verifies easily that \overline{f}_{∗} ∘ \overline{f}^{∗}ℱ is a direct product of finitely many copies of ℱ, indexed by the closed points of X. Concerning the counit, keep the same morphism f, let U := X \\{x\}, where x is a closed point and set ℱ := j_{!}𝒪_{U}; then it is clear that \overline{f}^{∗} ∘ \overline{f}_{∗}ℱ is supported on the complement of the closed fibre of f.

(ii) Similarly, one sees easily that \overline{f}^{∗} is not a right adjoint to \overline{f}_{∗} on the category of all f_{∗}𝒪_{X}- modules.

(iii) Incidentally, the analogous functor \overline{f}_{ét}^{∗} defined on the étale (rather than Zariski) site is a right adjoint to \overline{f}_{ét∗} : 𝒪_{X,ét}-Mod → f_{∗}𝒪_{X,ét}-Mod, and on these sites the unit and counit of the adjunctions are isomorphisms for all modules.

11.1.4. For the construction of the right adjoint f^{!} to Rf_{∗}, one considers first the case where f factors as a composition of a finite morphism followed by a smooth one, in which case f^{!} admits a corresponding decomposition. Namely :

• In case f is smooth, we shall consider the functor :

f^{♯} : D(𝒪_{Y} -Mod) → D(𝒪_{X}-Mod) K^{•} ↦ f^{∗}K^{•} ⊗_{𝒪_{X}} Λ_{𝒪_{X}}^{n} Ω_{X/Y}^{1} [n]

where n is the locally constant relative dimension function of f.

• In case f is finite, we shall consider the functor :

f^{♭} : D^{+}(𝒪_{Y} -Mod) → D^{+}(𝒪_{X}-Mod) K^{•} ↦ \overline{f}^{∗}Rℋ om_{𝒪_{Y}}^{•} (f_{∗}𝒪_{X}, K^{•}).

If f is any quasi-projective morphism, such factorization can always be found locally on X, and one is left with the problem of patching a family of locally defined functors ((f_{|U_{i}})^{!} | i ∈ I), corresponding to an open covering X = ⋃_{i∈I} U_{i}. Since such patching must be carried out in the derived category, one has to take care of many cumbersome complications.

11.1.5. Recall that a finite morphism f : X → Y is said to be pseudo-coherent if f_{∗}𝒪_{X} is a pseudo-coherent 𝒪_{Y} -module. This condition is equivalent to the pseudo-coherence of f in the sense of [20, Exp.III, Déf.1.2].

Lemma 11.1.6. Let f : X → Y be a finite morphism of schemes. Then :

(i) If f is finitely presented, the functor

ℱ ↦ \overline{f}^{∗}ℋ om_{𝒪_{Y}} (f_{∗}𝒪_{X}, ℱ)

is right adjoint to f_{∗} : 𝒪_{X}-Mod_{qcoh} → 𝒪_{Y} -Mod_{qcoh}. (ii) If f is pseudo-coherent, the functor

K^{•} ↦ \overline{f}^{∗}Rℋ om_{𝒪_{Y}}^{•} (f_{∗}𝒪_{X}, K^{•}) : D^{+}(𝒪_{Y} -Mod)_{qcoh} → D^{+}(𝒪_{X}-Mod)_{qcoh}

is right adjoint to

Rf_{∗} : D^{+}(𝒪_{X}-Mod)_{qcoh} → D^{+}(𝒪_{Y} -Mod)_{qcoh}

(notation of (10.3)).

Proof. (i): Under the stated assumptions, f_{∗}𝒪_{X} is a finitely presented 𝒪_{Y} -module (by claim 9.1.35), hence the functor

ℱ ↦ ℋ om_{𝒪_{Y}} (f_{∗}𝒪_{X}, ℱ) : 𝒪_{Y} -Mod → f_{∗}𝒪_{X}-Mod

preserves the subcategories of quasi-coherent modules, hence it restricts to a right adjoint for the forgetful functor f_{∗}𝒪_{X}-Mod_{qcoh} → 𝒪_{Y} -Mod_{qcoh} (claim 10.1.17). Then the assertion follows from (11.1.1).

(ii) is analogous : since f_{∗}𝒪_{X} is pseudo-coherent, the functor

K^{•} ↦ Rℋ om_{𝒪_{Y}}^{•} (f_{∗}𝒪_{X}, K^{•}) : D^{+}(𝒪_{Y} -Mod) → D^{+}(f_{∗}𝒪_{X}-Mod)

preserves the subcategories of complexes with quasi-coherent homology, hence it restricts to a right adjoint for the forgetful functor D^{+}(f_{∗}𝒪_{X}-Mod)_{qcoh} → D^{+}(𝒪_{Y} -Mod)_{qcoh}, by lemma 10.1.16(iii). To conclude, it then suffices to apply lemma 11.1.2. □

Proposition 11.1.7. Suppose that X ⟶^{f} Y ⟶^{g} Z ⟶^{h} W are morphisms of schemes. Then :

(i) If f and g are finite and f is pseudo-coherent, there is a natural isomorphism of functors

on D^{+}(𝒪_{Z}-Mod) :

ξ_{f,g} : (g ∘ f)^{♭} →^{∼} f^{♭} ∘ g^{♭}.

(ii) If f and g ∘ f are finite and pseudo-coherent, and g is smooth of bounded fibre dimension, there is a natural isomorphism of functors on D^{+}(𝒪_{Z}-Mod) :

ζ_{f,g} : (g ∘ f)^{♭} →^{∼} f^{♭} ∘ g^{♯}.

(iii) If f, g and h ∘ g are finite and pseudo-coherent, and h is smooth of bounded fibre dimension, then the diagram of functors on D^{+}(𝒪_{W} -Mod) :

\xymatrix{(h ∘ g ∘ f)^{♭} \ar[r]^{ζ_{g∘f,h}} \ar[d]_{ξ_{f,h∘g}} & (g ∘ f)^{♭} ∘ h^{♯} \ar[d]^{ξ_{f,g}∘h^{♯}} \\ f^{♭} ∘ (h ∘ g)^{♭} \ar[r]^{f^{♭}(ζ_{g,h})} & f^{♭} ∘ g^{♭} ∘ h^{♯}}

commutes. (iv) If, moreover

\xymatrix{Y \ar[r]^{g} \ar[d]_{h′} & Z \ar[d]^{h} \\ W′ \ar[r]^{j} & W}

is a cartesian diagram of schemes, such that h is smooth of bounded fibre dimension and j is finite and pseudo-coherent, then there is a natural isomorphism of functors :

ϑ_{j,h} : g^{♭} ∘ h^{♯} →^{∼} h^{′♯} ∘ j^{♭}.

Proof. This is [87, Ch.III, Prop.6.2, 8.2, 8.6]. We check only (i). There is a natural commutative diagram of ringed spaces :

\xymatrix{(X, 𝒪_{X}) \ar[r]^{f} \ar@{-->}[dr]_{\overline{f}} & (Y, 𝒪_{Y}) \ar[r]^{g} \ar[dr]_{\overline{g}} & (Z, 𝒪_{Z}) \\  & (Y, f_{∗}𝒪_{X}) \ar[dr]_{φ} \ar[u]_{α} & (Z, g_{∗}𝒪_{Y}) \ar[u]_{γ} \\  &  & (Z, (g ∘ f)_{∗}𝒪_{X}) \ar[u]_{β}}

where :

(11.1.8)  φ ∘ \overline{f} = \overline{g ∘ f}.

By claim 10.1.17, the (forgetful) functors α_{∗}, β_{∗} and γ_{∗} admit right adjoints :

α^{♭} : 𝒪_{Y} -Mod → f_{∗}𝒪_{X}-Mod  : ℱ ↦ ℋ om_{𝒪_{Y}} (f_{∗}𝒪_{X}, ℱ)
β^{♭} : g_{∗}𝒪_{Y} -Mod → (g ∘ f)_{∗}𝒪_{X}-Mod  : ℱ ↦ ℋ om_{g_{∗}𝒪_{Y}} ((g ∘ f)_{∗}𝒪_{X}, ℱ)
γ^{♭} : 𝒪_{Z}-Mod → g_{∗}𝒪_{Y} -Mod  : ℱ ↦ ℋ om_{𝒪_{Z}} (g_{∗}𝒪_{Y} , ℱ).

Likewise, (γ ∘ β)_{∗} admits a right adjoint (γ ∘ β)^{♭} and the natural identification

γ_{∗} ∘ β_{∗} →^{∼} (γ ∘ β)_{∗}

induces a natural isomorphisms of functors :

(11.1.9)  (γ ∘ β)^{♭} := ℋ om_{𝒪_{Z}} ((g ∘ f)_{∗}𝒪_{X}, −) →^{∼} β^{♭} ∘ γ^{♭}.

(Notice that, in general, there might be several choices of such natural transformations (11.1.9), but for the proof of (iii) – as well as for the construction of f^{!} for more general morphisms f – it is essential to make an explicit and canonical choice.)

Claim 11.1.10. (i) α_{∗} ∘ φ^{∗} = \overline{g}^{∗} ∘ β_{∗}.

(ii) The natural commutative diagram of sheaves :

\xymatrix{\overline{g}^{−1}g_{∗}𝒪_{Y} \ar[r] \ar[d] & 𝒪_{Y} \ar[d] \\ \overline{g}^{−1}(g ∘ f)_{∗}𝒪_{X} \ar[r] & f_{∗}𝒪_{X}}

is cocartesian.

Proof of the claim. (i) is an easy consequence of (ii). Assertion (ii) can be checked on the stalks, hence we may assume that Z = Spec A, Y = Spec B and X = Spec C are affine schemes. Let 𝔭 ∈ Y be any prime ideal, and set 𝔮 := g(𝔭) ∈ Z; then (\overline{g}^{−1}g_{∗}𝒪_{Y})_{𝔭} = B_{𝔮} (the A_{𝔮}-module obtained by localizing the A-module B at the prime 𝔮). Likewise, (\overline{g}^{−1}(g ∘ f)_{∗}𝒪_{X})_{𝔭} = C_{𝔮} and (f_{∗}𝒪_{X})_{𝔭} = C_{𝔭}. Hence the claim boils down to the standard isomorphism : C_{𝔮} ⊗_{B_{𝔮}} B_{𝔭} ≃ C_{𝔭}. ♢

Let I^{•} be a bounded below complex of injective 𝒪_{Z}-modules; by lemma 10.1.16(i), γ^{♭}I^{•} is a complex of injective g_{∗}𝒪_{Y} -modules; taking into account (11.1.9) we deduce a natural isomorphism of functors on D^{+}(𝒪_{Z}-Mod) :

(11.1.11)  R(γ ∘ β)^{♭} →^{∼} R(β^{♭} ∘ γ^{♭}) = Rβ^{♭} ∘ Rγ^{♭}.

Furthermore, (11.1.8) implies that \overline{g ∘ f}^{∗} = \overline{f}^{∗} ∘ φ^{∗}. Combining with (11.1.11), we see that the sought ξ_{f,g} is a natural transformation :

\overline{f}^{∗} ∘ φ^{∗} ∘ Rβ^{♭} ∘ Rγ^{♭} → \overline{f}^{∗} ∘ Rα^{♭} ∘ \overline{g}^{∗} ∘ Rγ^{♭}

of functors on D^{+}(𝒪_{Z}-Mod). Hence (i) will follow from :

Claim 11.1.12. (i) There exists a natural isomorphism of functors :

g_{∗}𝒪_{Y} -Mod → f_{∗}𝒪_{X}-Mod : φ^{∗} ∘ β^{♭} →^{∼} α^{♭} ∘ \overline{g}^{∗}.

(ii) The natural transformation :

R(α^{♭} ∘ \overline{g}^{∗}) → Rα^{♭} ∘ \overline{g}^{∗}

is an isomorphism of functors : D^{+}(g_{∗}𝒪_{Y} -Mod) → D^{+}(f_{∗}𝒪_{X}-Mod).

Proof of the claim. First we show how to construct a natural transformation as in (i); this is the same as exhibiting a map of functors : α_{∗} ∘ φ^{∗} ∘ β^{♭} → \overline{g}^{∗}. In view of claim 11.1.10(i), the latter can be defined as the composition of \overline{g}^{∗} and the counit of adjunction β_{∗} ∘ β^{♭} → 1_{g_{∗}𝒪_{Y} -Mod} .

Next, recall the natural transformation :

(11.1.13) \overline{g}^{∗}ℋ om_{g_{∗}𝒪_{Y}} (ℱ , 𝒢) → ℋ om_{𝒪_{Y}} (\overline{g}^{∗}ℱ , \overline{g}^{∗}𝒢) for every g_{∗}𝒪_{Y} -modules ℱ and 𝒢

(defined so as to induce the pull-back map on global Hom functors). Notice that the natural map \overline{g}^{∗}\overline{g}_{∗}ℱ → ℱ (counit of adjunction) is an isomorphism for every quasi-coherent 𝒪_{Y} -module ℱ. Especially, if ℱ = g_{∗}𝒜 for a quasi-coherent 𝒪_{Y} -module 𝒜, then (11.1.13) takes the form :

(11.1.14)  \overline{g}^{∗}ℋ om_{g_{∗}𝒪_{Y}} (\overline{g}_{∗}𝒜 , 𝒢) → ℋ om_{𝒪_{Y}} (𝒜 , \overline{g}^{∗}𝒢).

Furthermore, by inspecting the definitions (and the proof of claim 10.1.17), one verifies easily that the map φ^{∗} ∘β^{♭}(𝒢) → α^{♭} ∘\overline{g}^{∗}(𝒢) constructed above is the same as the map (11.1.14), taken with 𝒜 = f_{∗}𝒪_{X}. Notice that, since f is pseudo-coherent, f_{∗}𝒪_{X} is even a finitely presented 𝒪_{Y} -module; thus, in order to conclude the proof of (i), it suffices to show that (11.1.14) is an isomorphism for every finitely presented 𝒪_{Y} -module 𝒜 and every g_{∗}𝒪_{Y} -module 𝒢. To this aim, we may assume that Z and Y are affine; then we may find a presentation \underline{ℰ} := (𝒪_{Y}^{⊕p} → 𝒪_{Y}^{⊕q} → 𝒜 → 0). We apply the natural transformation (11.1.14) to \underline{ℰ}, thereby obtaining a commutative ladder with left exact rows; then the five-lemma reduces the assertion to the case where 𝒜 = 𝒪_{Y}, which is obvious. To prove assertion (ii) we may again assume that Y and Z are affine, in which case we can find a resolution L_{•} → f_{∗}𝒪_{X} consisting of free 𝒪_{Y} -modules of finite rank. For a given bounded below complex I^{•} of injective g_{∗}𝒪_{Y} -modules, choose a resolution \overline{g}^{∗}I^{•} → J^{•} consisting of injective 𝒪_{Y} -modules. We deduce a commutative ladder :

\xymatrix{\overline{g}^{∗}ℋ om_{g_{∗}𝒪_{Y}}^{•} (\overline{g}_{∗} ∘ f_{∗}𝒪_{X}, I^{•}) \ar[r]^{μ_{1}} \ar[d]_{λ_{1}} & ℋ om_{𝒪_{Y}}^{•} (f_{∗}𝒪_{X}, \overline{g}^{∗}I^{•}) \ar[r]^{μ_{3}} \ar[d] & ℋ om_{𝒪_{Y}}^{•} (f_{∗}𝒪_{X}, J^{•}) \ar[d]^{λ_{2}} \\ \overline{g}^{∗}ℋ om_{g_{∗}𝒪_{Y}}^{•} (\overline{g}_{∗}L_{•}, I^{•}) \ar[r]^{μ_{2}} & ℋ om_{𝒪_{Y}}^{•} (L_{•}, \overline{g}^{∗}I^{•}) \ar[r]^{μ_{4}} & ℋ om_{𝒪_{Y}}^{•} (L_{•}, J^{•}).}

Since \overline{g} is flat, λ_{1} is a quasi-isomorphism, and the same holds for λ_{2}. The maps μ_{1} and μ_{2} are of the form (11.1.14), hence they are quasi-isomorphisms, by the foregoing proof of (i). Finally, it is clear that μ_{4} is a quasi-isomorphism as well, hence the same holds for μ_{3} ∘ μ_{1}, and (ii) follows. □

11.1.15. Let now f : X → Y be an embeddable morphism, i.e. such that it can be factored as a composition f = g ∘ h where h : X → Z is a finite pseudo-coherent morphism, g : Z → Y is smooth and separated, and the fibres of g have bounded dimension. One defines :

f^{!} := h^{♭} ∘ g^{♯}.

Lemma 11.1.16. Let X ⟶^{f} Y ⟶^{g} Z ⟶^{h} W be three embeddable morphisms of schemes, such that the compositions g ∘ f, h ∘ g and h ∘ g ∘ f are also embeddable. Then we have :

(i) There exists a natural isomorphism of functors

ψ_{g,f} : (g ∘ f)^{!} →^{∼} f^{!} ∘ g^{!}.

Especially, f^{!} is independent – up to a natural isomorphism of functors – of the choice of factorization as a finite morphism followed by a smooth morphism.

(ii) The diagram of functors :

\xymatrix{(h ∘ g ∘ f)^{!} \ar[r]^{ψ_{h,g∘f}} \ar[d]_{ψ_{h∘g,f}} & (g ∘ f)^{!} ∘ h^{!} \ar[d]^{ψ_{g,f} ∘h^{!}} \\ f^{!} ∘ (h ∘ g)^{!} \ar[r]^{f^{!}(ψ_{h,g})} & f^{!} ∘ g^{!} ∘ h^{!}}

commutes.

Proof. The proof is a complicated verification, starting from proposition 11.1.7. See [87, Ch.III, Th.8.7] for details. □

11.1.17. Let X ⟶^{f} Y ⟶^{g} Z be two finite and pseudo-coherent morphisms of schemes. The map f^{♮} : 𝒪_{Y} → f_{∗}𝒪_{X} induces a natural transformation :

(11.1.18)  ℋ om_{𝒪_{Z}}^{•} (g_{∗}(f^{♮}), −) : β_{∗} ∘ (γ ∘ β)^{♭} → γ^{♭}

(notation of the proof of proposition 11.1.7(i)) and we wish to conclude this section by exhibiting another compatibility involving (11.1.18) and the isomorphism ξ_{f,g}. To this aim, we consider the composition of isomorphism of functors on D^{+}(𝒪_{Z}-Mod)_{qcoh} :

(11.1.19)  Rf_{∗} ∘ \overline{g ∘ f}^{∗} ⟶^{[figure: 1]} α_{∗} ∘ R\overline{f}_{∗} ∘ \overline{f}^{∗} ∘ φ^{∗} ⟶^{[figure: 2]} α_{∗} ∘ φ^{∗} ⟶^{[figure: 3]} \overline{g}^{∗} ∘ β_{∗}

where :

[figure: 1]is deduced from (11.1.8) and the decomposition f = α ∘ \overline{f}.

[figure: 2]is induced from the counit of adjunction \overline{ε} : R\overline{f}_{∗} ∘ \overline{f}^{∗} → 1 provided by lemma 11.1.2. [figure: 3]is the identification of claim 11.1.10.

We define a morphism of functors as a composition :

(11.1.20)  Rf_{∗} ∘ (g ∘ f)^{♭} = Rf_{∗} ∘ \overline{g ∘ f}^{∗} ∘ R(γ ∘ β)^{♭} ⟶^{[figure: 4]} \overline{g}^{∗} ∘ β_{∗} ∘ R(γ ∘ β)^{♭} ⟶^{[figure: 5]} g^{♭}

where :

[figure: 4]is the isomorphism (11.1.19) ∘ R(γ ∘ β)^{♭}.

[figure: 5]is the composition of \overline{g}^{∗} and the morphism β_{∗} ∘ R(γ ∘ β)^{♭} → Rγ^{♭} deduced from (11.1.18).

We have a natural diagram of functors on D^{+}(𝒪_{Z}-Mod)_{qcoh} :

(11.1.21) [figure: R(g ∘ f)_{∗} ∘ (g ∘ f)^{♭}; 6; R(g∘f)_{∗}(ξ_{f,g}); Rg_{∗} ∘ Rf_{∗} ∘ (g ∘ f)^{♭}; 9; Rg_{∗}∘Rf_{∗}(ξ_{f,g}); Rg_{∗} ∘ g^{♭}; R(g ∘ f)_{∗} ∘ (f^{♭} ∘ g^{♭}); 7; Rg_{∗} ∘ Rf_{∗} ∘ (f^{♭} ∘ g^{♭}); Rg_{∗} ∘ (Rf_{∗} ∘ f^{♭}) ∘ g^{♭}; 8]

where :

[figure: 6]and [figure: 7]are induced by the natural isomorphism R(g ∘ f)_{∗} →^{∼} Rg_{∗} ∘ Rf_{∗}.
[figure: 8]is induced by the counit of the adjunction ε : Rf_{∗} ∘ f^{♭} → 1.
[figure: 9]is Rg_{∗} ∘ (11.1.20).

Lemma 11.1.22. In the situation of (11.1.17), diagram (11.1.21) commutes.

Proof. The diagram splits into left and right subdiagrams, and clearly the left subdiagram commutes, hence it remains to show that the right one does too; to this aim, it suffices to consider the simpler diagram :

\xymatrix{Rf_{∗} ∘ (g ∘ f)^{♭} \ar[r] \ar[d]_{Rf_{∗}(ξ_{f,g})} & g^{♭} \\ Rf_{∗} ∘ f^{♭} ∘ g^{♭} \ar@{-->}[ur]}

whose horizontal arrow is (11.1.20), and whose upward arrow is deduced from the counit ε. However, the counit ε, used in [figure: 8], can be expressed in terms of the counit \overline{ε} : R\overline{f}_{∗}∘\overline{f}^{∗} → 1, used in [figure: 2], therefore we are reduced to considering the diagram of functors on D^{+}(𝒪_{Z}-Mod)_{qcoh} :

[figure: α_{∗} ∘ R\overline{f}_{∗} ∘ \overline{f}^{∗} ∘ φ^{∗} ∘ R(γ ∘ β)^{♭}; a; Rf_{∗}(ξ_{f,g}); α_{∗} ∘ φ^{∗} ∘ R(γ ∘ β)^{♭}; c; e; \overline{g}^{∗} ∘ β_{∗} ∘ R(γ ∘ β)^{♭}; 5; α_{∗} ∘ φ^{∗} ∘ Rβ^{♭} ∘ Rγ^{♭}; d; α_{∗} ∘ R\overline{f}_{∗} ∘ \overline{f}^{∗} ∘ Rα^{♭} ∘ g^{♭}; b; α_{∗} ∘ Rα^{♭} ∘ \overline{g}^{∗} ∘ Rγ^{♭}; f; g^{♭} = \overline{g}^{∗} ∘ Rγ^{♭}]

where :

[figure: a] and [figure: b]are induced from \overline{ε}.

[figure: c] is induced by the isomorphism (11.1.11).

[figure: d] is induced by the isomorphism of claim 11.1.12(i).

[figure: e] is induced by the identity of claim 11.1.10. [figure: f] is induced by the counit α_{∗} ∘ Rα^{♭} → 1.

Now, by inspecting the construction of ξ_{f,g} one checks that the left subdiagram of the latter diagram commutes; moreover, from claim 11.1.12(ii) (and from lemma 10.1.16(i)), one sees that the right subdiagram is obtained by evaluating on complexes of injective modules the analogous diagram of functors on 𝒪_{Z}-Mod :

\xymatrix{α_{∗} ∘ φ^{∗} ∘ (γ ∘ β)^{♭} \ar[r]^{e} \ar[d]_{c} & \overline{g}^{∗} ∘ β_{∗} ∘ (γ ∘ β)^{♭} \ar[dd]^{5} \\ α_{∗} ∘ φ^{∗} ∘ β^{♭} ∘ γ^{♭} \ar[d]_{d} \\ α_{∗} ∘ α^{♭} ∘ \overline{g}^{∗} ∘ γ^{♭} \ar[r]^{f} & \overline{g}^{∗} ∘ γ^{♭}.}

Furthermore, by inspecting the proof of claim 11.1.12(i), one verifies that f ∘ d is the same as the composition

α_{∗} ∘ φ^{∗} ∘ β^{♭} ∘ γ^{♭} ⟶^{[figure: i]} \overline{g}^{∗} ∘ β_{∗} ∘ β^{♭} ∘ γ^{♭} ⟶^{[figure: j]} \overline{g}^{∗} ∘ γ^{♭}

where [figure: j]is deduced from the counit β_{∗} ∘ β^{♭} → 1, and [figure: i]is deduced from the identity of claim 11.1.10. Hence we come down to showing that the diagram :

[figure: β_{∗} ∘ (γ ∘ β)^{♭}; β_{∗}(11.1.9); (11.1.18); γ^{♭}; β_{∗} ∘ β^{♭} ∘ γ^{♭}; k]

commutes in g_{∗}𝒪_{Y} -Mod, where [figure: k]is deduced from the counit β_{∗} ∘ β^{♭} → 1. This is an easy verification, that shall be left to the reader. □

11.2. Cousin complexes. In this section we study a special type of complexes of abelian sheaves, called Cousin complexes, defined on a large class of topological spaces. These complexes play an important role in the further development of duality theory for locally coherent schemes : in many cases the techniques explained in this section will allow to exhibit canonical representatives for the dualizing complexes that shall be investigated at length in section 11.3.

11.2.1. We resume the notation of (10.1.30), so (X, 𝒜) is a ringed topological space, φ : V → X a locally closed immersion, we denote by \overline{V} the topological closure of V in X, and we set ∂V := \overline{V} \V. In (10.4.1) we have introduced the functor Γ_{\overline{V}} of sections with support in \overline{V}, and its sheaf-valued variant \underline{Γ}\overline{V}. We now extend these definitions to the case of a general locally closed immersion, by setting

\underline{Γ}_{V} := φ_{∗} ∘ φ^{∗} ∘ \underline{Γ}\overline{V} : 𝒜 -Mod → 𝒜 -Mod Γ_{V} := Γ ∘ \underline{Γ}_{V} : 𝒜 -Mod → 𝒜 (X)-Mod.

The functor \underline{Γ}_{V} admits a derived functor

R\underline{Γ}_{V} : D^{+}(𝒜 -Mod) → D^{+}(𝒜 -Mod)

obtained as usual by evaluating on injective resolutions of given bounded below complexes.

Remark 11.2.2. (i) We have the following explicit description of \underline{Γ}_{V}. Let k : V → X\∂V and l : X \∂V → X be respectively the closed and open immersions. Since φ = l ∘ k, there exist natural isomorphisms

\underline{Γ}_{V} →^{∼} φ_{∗} ∘ k^{∗} ∘ l^{∗} ∘ \underline{Γ}\overline{V} →^{∼} l_{∗} ∘ k_{∗} ∘ k^{∗} ∘ \underline{Γ}_{V} ∘ l^{∗} →^{∼} l_{∗} ∘ \underline{Γ}_{V} ∘ l^{∗}

where the last map is induced by the natural identification k_{∗} ∘ k^{∗} ∘ \underline{Γ}_{V} →^{∼} \underline{Γ}_{V} provided by lemma 10.1.32. There follows a natural isomorphism

\underline{Γ}_{V} ℱ (S) →^{∼} \underline{Γ}\overline{V} ℱ (S\∂V) for every open subset S of X.

(ii) Notice that ∂V = ∅ if φ is a closed immersion; in this case, (i) says that the functor defined in (11.2.1) is naturally isomorphic to its namesake introduced in (10.4.1), so the current notation is consistent with that of section 10.4.

(iii) For a general locally closed subset V and any 𝒜 -module ℱ, the natural monomorphism ι_{ℱ} : \underline{Γ}\overline{V} ℱ → ℱ induces a natural transformation

(11.2.3)  \underline{Γ}_{V} ℱ → φ_{∗}φ^{∗}ℱ

which is a monomorphism for every such ℱ, since the functor φ_{∗}φ^{∗} is left exact. Notice that if V is open in X, the map φ^{∗}ι_{ℱ} is an isomorphism, so the same holds for (11.2.3).

Proposition 11.2.4. In the situation of (11.2.1), the functor \underline{Γ}_{V} is right adjoint to t_{V}.

Proof. For φ a closed immersion, remark 11.2.2(ii) reduces the assertion to lemma 10.4.13(i), since in this case φ_{!} = φ_{∗}. The assertion is also clear in case φ is an open immersion, since in this case \underline{Γ}_{V} is naturally isomorphic to the functor φ_{∗} ∘ φ^{∗}, by remark 11.2.2(iii). Notice that V = \overline{V} ∩ (X \∂V). By proposition 10.1.33(i) (and by remark 1.1.17(i)) we are reduced to exhibiting a natural isomorphism of functors

\underline{Γ}_{V} →^{∼} \underline{Γ}_{X\∂V} ∘ \underline{Γ}\overline{V} .

But the existence of such an isomorphism is asserted already in remark 11.2.2(i). □

11.2.5. For our discussion, it will be important to know that in fact there exists a canonical adjunction for the pair (t_{V} , \underline{Γ}_{V}). In view of the proof of proposition 11.2.4 (and of remark 1.1.17(i)), it suffices to exhibit a canonical pair (η^{V} , ε^{V}) of units and counits for this adjunction, in case φ is either an open or a closed immersion.

• Suppose first that φ is a closed immersion. Then we want natural transformations

η_{ℱ}^{V} : ℱ → \underline{Γ}_{V} ∘ φ_{∗} ∘ φ^{∗}ℱ ε_{ℱ}^{V} : φ_{∗} ∘ φ^{∗} ∘ \underline{Γ}_{V} ℱ → ℱ

related by the triangular identities of (1.1.13). However, remark 11.2.2(ii) yields a natural isomorphism η_{1} : \underline{Γ}_{V} φ_{∗}φ^{∗}ℱ →^{∼} φ_{∗}φ^{∗}ℱ, and on the other hand, we have a canonical unit η_{2} : ℱ → φ_{∗}φ^{∗}ℱ for the adjunction (φ^{∗}, φ_{∗}), so we take η_{ℱ}^{V} := η_{1}^{−1} ∘ η_{2}. Likewise, let ε_{1} : \underline{Γ}_{V} ℱ → ℱ be the natural monomorphism; we take ε_{ℱ}^{V} := ε_{1}∘f_{ℱ} , where f_{ℱ} : t_{V} ∘\underline{Γ}_{V} ℱ →^{∼} \underline{Γ}_{V} ℱ is the isomorphism provided by lemma 10.1.32. The triangular identities may then be checked on the stalks : we leave the details to the reader.

• Next, if φ is an open immersion, we need to find natural transformations

η_{ℱ}^{V} : ℱ → φ_{∗} ∘ φ^{∗} ∘ φ_{!} ∘ φ^{∗}ℱ ε_{ℱ}^{V} : φ_{!} ∘ φ^{∗} ∘ φ_{∗} ∘ φ^{∗}ℱ → ℱ

again related by the same triangular identities. We take for η_{ℱ}^{V} (resp. ε_{ℱ}^{V} ) the composition of the natural unit ℱ → φ_{∗}φ^{∗}ℱ of the adjunction (φ^{∗}, φ^{∗}) (resp. the counit φ_{!} ∘ φ^{∗}ℱ → ℱ of the adjunction (φ_{!}, φ^{∗})), with the isomorphism

φ_{∗}φ^{∗}ℱ →^{∼} φ_{∗}φ^{∗}φ_{!}φ^{∗}ℱ ( resp. φ_{!}φ^{∗}φ_{∗}φ^{∗}ℱ →^{∼} φ_{!}φ^{∗}ℱ )

deduced from the natural identification of φ^{∗} ∘ φ_{!} (resp. of φ^{∗} ∘ φ_{∗}) with the identity functor of φ^{∗}𝒜 -Mod. The triangular identities are as usual checked on the stalks.

11.2.6. Due to proposition 11.2.4, and the subsequent observations in (11.2.5), all the constructions of proposition 10.1.33have natural counterparts for the functors \underline{Γ}_{V}. Namely :

• If V and V′ are any two locally closed subsets of X, there is a natural isomorphism

β_{V,V′} : \underline{Γ}_{V} ∘ \underline{Γ}_{V′} →^{∼} \underline{Γ}_{V ∩V′}

and if V′′ ⊂ X is any other locally closed subset, the diagram of functors

(11.2.7)  \xymatrix{\underline{Γ}_{V} ∘ \underline{Γ}_{V′} ∘ \underline{Γ}_{V′′} \ar[r]^{\underline{Γ}_{V} ∗β_{V′,V′′}} \ar[d]_{β_{V,V′}∗\underline{Γ}_{V′′}} & \underline{Γ}_{V} ∘ \underline{Γ}_{V′∩V′′} \ar[d]^{β_{V,V′∩V′′}} \\ \underline{Γ}_{V ∩V′} ∘ \underline{Γ}_{V′′} \ar[r]^{β_{V ∩V′,V′′}} & _{\underline{Γ}_{V ∩V′∩V′′}}}

commutes.

• Moreover, suppose that the pair (V, V′) satisfies either of the conditions (a), (b) of proposition 10.1.33(iii); then there exists a natural morphism

γ_{V,V′} : \underline{Γ}_{V} → \underline{Γ}_{V′}.

which corresponds to c^{V,V′} under the natural adjunction described in (11.2.5), and if V′′ is any third locally closed subset of X, we get a commutative diagram

\xymatrix{\underline{Γ}_{V′′} ∘ \underline{Γ}_{V} \ar[r]^{\underline{Γ}_{V′′}∗γ_{V,V′}} \ar[d]_{β_{V′′,V}} & \underline{Γ}_{V′′} ∘ \underline{Γ}_{V′} \ar[d]^{β_{V′′,V′}} \\ _{\underline{Γ}_{V′′∩V}} \ar[r]^{γ_{V′′∩V,V′′∩V′}} & \underline{Γ}_{V′′∩V′}.}

• Lastly, in the situation of proposition 10.1.33(v) we have the identity :

γ_{V′,V′′} ∘ γ_{V,V′} = γ_{V,V′′} : \underline{Γ}_{V} → \underline{Γ}_{V′′}.

Remark 11.2.8. Let X be a topological space, j : U → X an open immersion, φ : V → X and φ′ : V′ → X two locally closed immersions fulfilling either of conditions (a),(b) of proposition 10.1.33(iii). The transformations γ_{V,V′} can also be described explicitly, thanks to the following considerations.

(i) First, notice that the functors t_{•} on 𝒜 -modules, the unit η^{•} and the counit ε^{•} are all compatible with restriction to U, i.e. we have identities of functors on j^{∗}𝒜 -modules and of morphisms between such functors :

j^{∗} ∗ t_{V} = t_{U∩V} j^{∗} ∗ η^{V} = ε^{V ∩U} j^{∗} ∗ ε^{V} = η^{V ∩U} for every locally closed subset V ⊂ X.

This is obvious for the truncation functors, and for the units and counits it follows by a simple inspection of the definition. Then, the uniqueness properties of the transformations τ^{••} and c^{••} imply that the same compatibility with restriction to U holds for the latter.

(ii) It follows formally from (i) that the isomorphisms β_{••} and the transformations γ_{••} are likewise compatible with restriction to U. On the other hand, notice that the restriction map

Hom_{𝒜 -Mod} (t_{V′}ℱ , 𝒢) → Hom_{j^{∗}𝒜 -Mod} ((t_{V′}ℱ)_{|U}, 𝒢_{|U})

is a bijection for every ℱ , 𝒢 ∈ Ob(𝒜 -Mod), whenever V′ ⊂ U. We conclude that the restriction map

Hom_{𝒜 -Mod} (ℱ , \underline{Γ}_{V′}𝒢) → Hom_{j^{∗}𝒜 -Mod} (ℱ_{|U}, \underline{Γ}_{V′∩U}𝒢_{|U})

is also bijective when V′ ⊂ U. Letting ℱ := \underline{Γ}_{V} 𝒢, we deduce that γ_{V,V′} is determined by its restriction to U, which is just γ_{V ∩U,V′∩U}.

(iii) Now, suppose first that ψ : V → V′ is a closed immersion, and denote by \overline{V} and \overline{V}′ the topological closures in X of V and respectively V′; notice that the unit of the adjunction (ψ^{∗}, ψ_{∗}) induces an isomorphism

(11.2.9)  φ′^{∗} ∘ Γ\overline{V} →^{∼} ψ_{∗} ∘ ψ^{∗} ∘ φ′^{∗} ∘ Γ\overline{V}

since (φ′^{∗} ∘ Γ\overline{V} ℱ)_{|V′\V} = 0 for every 𝒜 -module ℱ (lemma 10.1.32). We claim that γ_{V,V′} is the composition

φ_{∗} ∘ φ^{∗} ∘ \underline{Γ}\overline{V} →^{∼} φ′_{∗} ∘ ψ_{∗} ∘ ψ^{∗} ∘ φ′^{∗} ∘ \underline{Γ}\overline{V} →^{∼} φ′_{∗} ∘ φ′^{∗} ∘ \underline{Γ}\overline{V} → φ′_{∗} ∘ φ′^{∗} ∘ \underline{Γ}_{\overline{V}_{′}}.

where the second map is the inverse of (11.2.9) and the third is deduced from γ_{\overline{V} ,\overline{V}′} : \underline{Γ}\overline{V} → \underline{Γ}_{\overline{V}_{′}}. Indeed, notice that the foregoing construction is compatible with restriction to U; taking into account (iii), we may then replace X by its open subset X \ ∂V′, and assume that both φ and φ′ are closed immersions. In this case, the assertion follows by direct inspection of the definition of the unit and counit of the adjunction (t_{•}, \underline{Γ}_{•}), and of the definition of t_{•}, in the case of closed immersions.

(iv) Likewise, if V′ is an open subset of V, pick a factorization of φ : V → X as the composition of a closed immersion i : V → U and an open immersion j : U → X. There is a unique open subset U′ ⊂ U such that V′ = U′ ∩V; denote by j′ : U′ → X the open immersion, and for any given 𝒜 -module ℱ, let η_{ℱ} : ℱ → j′_{∗}j′^{∗}ℱ be the unit of adjunction; according to remark 11.2.2, we may regard η_{ℱ} as a natural transformation

η_{ℱ} : ℱ → \underline{Γ}_{U′}ℱ

and we claim that γ_{V,V′} is the composition

\underline{Γ}_{V} ⟶^{\underline{Γ}_{V} ∗η} \underline{Γ}_{V} ∘ \underline{Γ}_{U′} ⟶^{β_{V,U′}} \underline{Γ}_{V′}.

For the proof, we may argue as in (iii) to see that γ_{V,V′} is determined by its restriction to U′, and since the proposed construction is compatible with restriction to U′, we are then reduced to the case where U′ = U = X and therefore V = V′. However, clearly c^{V,V} is the identity endomorphism of t_{V}, so it remains only to prove that the proposed map is the identity of \underline{Γ}_{V}, if V = V′. The latter can be checked easily, whence the claim. We could have used the above construction as a definition for γ_{V,V′} in the case of open immersions, but the method we followed has the advantage of dispensing us from verifying that the resulting transformation is independent of the chosen factorization of φ.

11.2.10. Let V ⊂ V′ be a closed immersion of locally closed subsets of a topological space X; from remark 11.2.8(iii), it is clear that γ_{V,V′} is a monomorphism. Moreover, we have :

Lemma 11.2.11. In the situation of (11.2.10), the sequence

Σ_{V,V′} : 0 → \underline{Γ}_{V} ℱ ⟶^{γ_{V,V′}} \underline{Γ}_{V′}ℱ ⟶^{γ_{V′,V′\V}} \underline{Γ}_{V′\V} ℱ

is left exact for every 𝒜 -module ℱ.

Proof. Indeed, say that V′ = U ∩ Z for some closed subset Z and open subset U; then V = \overline{V} ∩ U, with \overline{V} the topological closure of V in X, and in light of (11.2.6) we see that there is a natural isomorphism Σ_{V,V′} →^{∼} \underline{Γ}_{U}(Σ_{\overline{V} ,Z}). However, \underline{Γ}_{U} is a left exact functor (since it admits a left adjoint), so it suffices to check that Σ_{\overline{V} ,Z} is left exact. Thus, we may assume from start that V and V′ are closed in X. By the same token, we see that Σ_{V,V′} ≃ \underline{Γ}_{V′}(Σ_{V,X}), so we may further assume that V′ = X, in which case the sequence reduces to the initial segment of (10.4.2). □

Proposition 11.2.12. With the notation of (11.2.1), the following holds :

(i) If ℱ is a flabby 𝒜 -module, the same holds for \underline{Γ}_{V} ℱ.

(ii) Every flabby 𝒜 -module is \underline{Γ}_{V} -acyclic.

(iii) If ℬ → 𝒜 is any morphism of sheaves of rings on X, the functor R\underline{Γ}_{V} commutes with

the forgetful functor D^{+}(𝒜 -Mod) → D^{+}(ℬ-Mod). (iv) If φ′ : V′ → X is any other locally closed immersion, there exists a natural isomor-

phism of functors :

R\underline{Γ}_{V ∩V′} →^{∼} R\underline{Γ}_{V} ∘ R\underline{Γ}_{V′} D^{+}(𝒜 -Mod) → D^{+}(𝒜 -Mod).

(v) Suppose that X is locally spectral and quasi-separated, and V is a constructible locally closed subset of X, then we have :

(a) Every qc-flabby 𝒜 -module is \underline{Γ}_{V} -acyclic.

(b) For every i ∈ ℕ, the functor R^{i}\underline{Γ}_{V} commutes with filtered colimits of 𝒜 -modules.

(c) For every x ∈ V, denote by j_{x} : X(x) → X the inclusion map (notation of definition 8.1.44(iii)). Then there exists a natural isomorphism

j_{x}^{∗}R\underline{Γ}_{V} →^{∼} R\underline{Γ}_{V (x)} in D^{+}(j_{x}^{∗}𝒜 -Mod) .

Proof. (i): By lemmata 10.4.13(ii) and 10.4.4(i), we know already that both assertions (i) and (ii) hold in case φ is a closed immersion; for the general case, we may then assume that φ is an open immersion. However, in this case both φ_{∗} and φ^{∗} take flabby sheaves to flabby sheaves (lemmata 10.1.2and 10.1.5(ii)), and flabby sheaves are acyclic for both of these functors (lemma 10.1.5(ii)), so the assertion follows from remark 11.2.2(iii).

(iii) is clear from (ii), as every injective 𝒜 -module is a flabby ℬ-module (lemma 10.1.5(v)). (iv) follows formally from (i), (ii) and (11.2.6) : details left to the reader. (v.a): Since the assertion is local on X, we may assume that X is spectral, in which case

V is the intersection of a closed constructible subset and an open constructible subset. By (iv) and lemma 10.1.5(iii), it then suffices to consider separately the cases where V is either open or closed in X. By lemma 10.4.4(iii.b) we know already the assertions in case V is a closed constructible subset of X, and the case where V is open follows easily from lemma 10.1.5(i,iv).

(v.b): We reduce again to the case where V is the intersection of a closed constructible subset Z and an open constructible subset U. Since – by (ii) – the functor \underline{Γ}_{Z} transforms injective 𝒜 -modules into \underline{Γ}_{U}-acyclic 𝒜 -modules, we have a convergent spectral sequence

R^{i}\underline{Γ}_{U} ∘ R^{j}\underline{Γ}_{Z}ℱ ⇒ R^{i+j}\underline{Γ}_{V} ℱ .

It then suffices to consider separately the case where V is either open or closed. These cases follow respectively from lemma 10.4.4(iii.b) and claim 10.4.5.

(v.c): Arguing as in the proof of (v.b), we reduce to consider separately the cases where V is either open or closed. For V closed, the assertion is corollary 10.4.10. For V open, the assertion follows from proposition 10.1.10(ii) and remark 11.2.2(iii), by arguing as in the proof of corollary 10.4.10: details left to the reader. □

Remark 11.2.13. (i) Using proposition 11.2.12, the various compatibilities that have been found so far beween the functors \underline{Γ}_{V} extend straightforwardly to their derived extensions. Especially, diagram (11.2.7) still commutes, after replacing the functors \underline{Γ}_{•} by their derived extensions everywhere, and the natural transformations β_{••} by their derived versions, provided by proposition 11.2.12(iii).

(ii) Likewise, if V ⊂ V′ (resp. V′ ⊂ V) is any closed (resp. open) immersion of locally closed subsets of X, by deriving the morphisms γ_{V,V′} of (11.2.6) we get a natural transformation

R\underline{Γ}_{V} → R\underline{Γ}_{V′} ( resp. R\underline{Γ}_{V′} → R\underline{Γ}_{V} ) in D^{+}(𝒜 -Mod)

and the compatibilities between the functors γ_{••} and β_{••} explicited in (11.2.6) hold, mutatis mutandis, also for the functors R\underline{Γ}_{•}.

Lemma 11.2.14. In the situation of (11.2.10), let ℱ be any flabby 𝒜 -module. Then, for every open subset U ⊂ X, the sequence

0 → \underline{Γ}_{V} ℱ (U) → \underline{Γ}_{V′}ℱ (U) → \underline{Γ}_{V′\V} ℱ (U) → 0

induced by the sequence Σ_{V,V′} of lemma 11.2.11, is short exact.

Proof. We may assume that U = X, and it suffices to show that the map g : Γ_{V′}ℱ → Γ_{V′\V} ℱ deduced from γ_{V′,V′\V} is surjective, whenever ℱ is flabby. To this aim, write V′ \ V = U′ ∩ V′ for some open subset U′ ⊂ X; then g is naturally identified with the map

Γ_{X} ∘ \underline{Γ}_{V′}ℱ → Γ_{U′} ∘ \underline{Γ}_{V′}ℱ

deduced from γ_{X,U′}. But \underline{Γ}_{V′}ℱ is still flabby (proposition 11.2.12(i)) so we are reduced to the case where V′ = X, in which case g is naturally identified with the restriction map ℱ (X) → ℱ (X \V), whence the contention. □

11.2.15. Let (X, 𝒜) be a ringed topological space. A family of supports on X is a set Φ consisting of closed subsets of X, such that :

• ∅ ∈ Φ.

• If Z ∈ Φ and Z′ ⊂ Z is any closed subset, then Z′ ∈ Φ as well.

• If Z_{1}, Z_{2} ∈ Φ, then Z_{1} ∪ Z_{2} ∈ Φ as well.

Now, let Φ and Φ′ be any two families of supports on X; we let Φ|Φ′ be the set whose elements are all the pairs \underline{Z} := (Z, Z′), where Z and Z′ are arbitrary elements of Φ and respectively Φ′, such that Z′ ⊂ Z. We endow Φ|Φ′ with a partial ordering, by declaring that, for given pairs \underline{Z}_{1} := (Z_{1}, Z′_{1}), \underline{Z}_{2} := (Z_{2}, Z′_{2}) of Φ|Φ′, we have \underline{Z}_{1} ≤ \underline{Z}_{2} if Z_{1} ⊂ Z_{2} and Z′_{1} ⊂ Z′_{2}. It is easily seen that Φ|Φ′ is filtered. To every object \underline{Z} ∈ Φ|Φ′ we assign the functor \underline{Γ}_{Z\Z′}, and to any pair (\underline{Z}_{1}, \underline{Z}_{2}) of elements of Φ|Φ′ with \underline{Z}_{1} ≤ \underline{Z}_{2}, we attach the composition

(11.2.16)  \underline{Γ}_{Z_{1}\Z′_{1}} ⟶^{γ_{Z_{1}\Z′_{1},Z_{2}\Z′_{1}}} \underline{Γ}_{Z_{2}\Z′_{1}} ⟶^{γ_{Z_{2}\Z′_{2},Z_{2}\Z′_{1}}} \underline{Γ}_{Z_{2}\Z′_{2}}

(notation of (11.2.6)). We then define the functor

\underline{Γ}_{Φ|Φ′} := colim_{(Z,Z′)∈Φ|Φ′} \underline{Γ}_{Z\Z′} : 𝒜 -Mod → 𝒜 -Mod

where the transition maps in the colimit are the maps (11.2.16), whence a derived functor

R\underline{Γ}_{Φ|Φ′} : D^{+}(𝒜 -Mod) → D^{+}(𝒜 -Mod)

and we have a natural isomorphism of functors on 𝒜 -Mod :

R^{i}\underline{Γ}_{Φ|Φ′} →^{∼} colim_{(Z,Z′)∈Φ|Φ′} R^{i}\underline{Γ}_{Z\Z′} for every i ∈ ℕ.

11.2.17. Keep the notation of (11.2.15). As a special case, we may take Φ′ := {∅}, and we let

\underline{Γ}_{Φ} := \underline{Γ}_{Φ|\{∅\}} : 𝒜 -Mod → 𝒜 -Mod.

If Φ and Φ′ are arbitrary family of supports, we have maps of partially ordered sets

Φ ⟵ Φ|Φ′ → Φ′ : Z ←∣ (Z, Z′) ↦ Z′.

Suppose moreover that Φ′ ⊂ Φ; then both of these maps are cofinal, by lemma 1.5.7(i), and with lemma 11.2.10and proposition 11.2.12(ii), we deduce a short exact sequence of complexes

(11.2.18)  0 → \underline{Γ}_{Φ′}ℐ → \underline{Γ}_{Φ}ℐ → \underline{Γ}_{Φ|Φ′}ℐ → 0

for every flabby 𝒜 -module ℐ, whence a distinguished triangle of functors on D^{+}(𝒜 -Mod) :

R\underline{Γ}_{Φ′} → R\underline{Γ}_{Φ} → R\underline{Γ}_{Φ|Φ′} → R\underline{Γ}_{Φ′}[1].

11.2.19. Keep the notation of (11.2.15). Suppose now that X is sober and locally noetherian, and moreover that we are given a weak codimension function on X, i.e. a map

c : X → ℤ

such that c(x) > c(y) whenever x ≠ y and x is a specialization of y in X. To the weak codimension function c we attach a system (Φ^{p} | p ∈ ℕ) of families of supports, by the rule :

Φ^{p} := {Z ⊂ X | Z is closed in X, and c(z) ≥ p for every z ∈ Z} for every p ∈ ℕ.

For every x ∈ X, let also j_{x} : X(x) → X and i_{x} : {x} → X be the inclusion maps (notation of definition 8.1.44(iii)). Moreover, let us set Φ^{p}(x) := {Z ∩ X(x) | Z ∈ Φ^{p}}, for every p ∈ ℤ.

Lemma 11.2.20. In the situation of (11.2.19), the following holds for every x ∈ X :

(i) The topological space X(x) is spectral, noetherian and finite dimensional, and the function c is bounded on X(x).

(ii) There exists a natural isomorphism of functors on D^{+}(j_{x}^{∗}𝒜 -Mod) :

j_{x}^{∗}R\underline{Γ}_{Φ^{p}} →^{∼} R\underline{Γ}_{Φ^{p}(x)} for every p ∈ ℤ.

(iii) (Φ^{p}(x) | p ∈ ℤ) is the system of families of supports on X(x) attached to the weak codimension function c_{|X(x)}.

Proof. (i): Since the topology of X(x) is induced from that of X, it is clear that X(x) is noetherian, and we have already seen that it is spectral (remark 8.1.45(i)). Especially, X(x) admits finitely many maximal points (remark 8.1.45(iii)); if η_{1}, …, η_{k} are these maximal points, set C := max(c(η_{1}), …, c(η_{k})). Then it is clear that the restriction to X(x) of the function c takes its values in the range C, C + 1, …, c(x), and it follows easily that the length of any descending sequence of points in (X(x), ≤) is bounded by c(x) − C.

(ii) follows easily from proposition 11.2.12(v.c), since every closed subset of a locally noetherian topological space is constructible.

(iii): If Z is any closed subset of X(x), denote by \overline{Z} the topological closure of Z in X; since X(x) is a pro-constructible subset of X, the same holds for Z = X(x) ∩ \overline{Z}. Then \overline{Z} is the set of specializations in X of the points of Z (proposition 8.1.47(i)). Especially, if c(z) ≥ p for every z ∈ Z, then c(z) ≥ p for every z ∈ \overline{Z}, and then clearly Z ∈ Φ^{p}(x). The assertion follows immediately. □

Proposition 11.2.21. In the situation of (11.2.19), we have a natural isomorphism of functors

R^{i}\underline{Γ}_{Φ^{p}|Φ^{p+1}} →^{∼} ⨁_{c(x)=p} j_{x∗} ∘ R^{i}\underline{Γ}_{\{x\}} ∘ j_{x}^{∗} for every p ∈ ℤ and i ∈ ℕ.

Proof. Since the assertion is local on X, we may assume that X is noetherian. We consider first the case where i = 0, and we fix p ∈ ℤ. Let Z ∈ Φ^{p} be any element; recall that Z has a finite number of irreducible components Z_{1}, …, Z_{k} (proposition 8.1.4). Let η_{i} be the generic point of Z_{i}, so that c(η_{i}) ≥ p for every i = 1, …, k. For every i, j = 1, …, k with i ≠ j, set Z_{ij} := Z_{i} ∩ Z_{j}; then, for every such i, j, every irreducible component of T of Z_{ij} and every t ∈ T, we must have c(t) ≥ c(η_{T}) > c(η_{i}), if η_{T} denotes the generic point of T. We conclude that Z_{ij} ∈ Φ^{p+1} for every i, j = 1, … k with i ≠ j. Set also

Z^{>p} := ⋃_{c(η_{i})>p} Z_{i} ∪ ⋃_{i≠j} Z_{ij} and Ψ^{p+1} := {(Z, Z′) ∈ Φ^{p}|Φ^{p+1} | Z^{>p} ⊂ Z′ ⊂ Z}.

Since the subset Ψ^{p} is clearly cofinal in Φ^{p}|Φ^{p+1}, we deduce a natural isomorphism

\underline{Γ}_{Φ^{p}|Φ^{p+1}} →^{∼} colim_{(Z,Z′)∈Ψ^{p}} \underline{Γ}_{Z\Z′}.

However, say that (Z, Z′) ∈ Ψ^{p}; we notice that the closed subset V := Z \Z′ of X \Z′ is the disjoint union of its irreducible components V_{1}, …, V_{t}, hence the topology induced by X on V is the coproduct (i.e. disjoint union) of the topologies induced by X on V_{1}, …, V_{t}, and there follows a natural isomorphism

(11.2.22)  \underline{Γ}_{V} →^{∼} ⨁_{i=1}^{t} \underline{Γ}_{V_{i}}.

Moreover, say that \underline{Z}_{1}, \underline{Z}_{2} ∈ Ψ^{p} and \underline{Z}_{1} ≤ \underline{Z}_{2}, and denote by V_{1}, …, V_{s} (resp. W_{1}, …, W_{t}) the irreducible components of V := Z_{1}\Z′_{1} (resp. of W := Z_{2}\Z′_{2}). A simple inspection of the definitions reveals that s ≤ t and – after reordering of the W_{i} – the subset W_{i} is open in V_{i} for every i = 1, …, s. Moreover, under the isomorphism (11.2.22), the transition map \underline{Γ}_{V} → \underline{Γ}_{W} is identified with the composition

⨁_{i=1}^{s} \underline{Γ}_{V_{i}} \underrightarrow{⨁_{i=1}^{s} γ_{V_{i},W_{i}}} ⨁_{i=1}^{s} \underline{Γ}_{W_{i}} → ⨁_{i=1}^{t} \underline{Γ}_{W_{i}}

(where the second map is the natural monomorphism). Hence, let Δ^{p} be the set of all locally closed subsets V of X fulfilling the following condition. There exists x ∈ X with c(x) = p, and V is a non-empty open subset of the topological closure \overline{\{x\}} of \{x\} in X. We endow Δ^{p} with a partial ordering, by declaring that, for any V, V′ ∈ Δ^{p} we have V ≤ V′ if V′ ⊂ V (notice that Δ^{p} is usually not a filtered partially ordered set); from the foregoing, we get a natural isomorphism

\underline{Γ}_{Φ|Φ} →^{∼} colim_{V ∈Δ^{p}} \underline{Γ}_{V}

where, for every V, V′ ∈ Δ^{p} such that V ≤ V′, the transition map \underline{Γ}_{V} → \underline{Γ}_{V′} is again given by the transformation γ_{V,V′} corresponding to the open immersion V′ → V.

Next, for every x ∈ X such that c(x) = p, set Δ_{x} := {V ∈ Δ^{p} | x ∈ V }, and notice that the partially ordered set Δ^{p} is the coproduct (i.e. the disjoint union) of its partially ordered subsets Δ_{x}. We deduce a natural isomorphism

\underline{Γ}_{Φ|Φ} →^{∼} ⨁_{c(x)=p} colim_{V ∈Δ_{x}} \underline{Γ}_{V} .

Thus, we come down to checking the following :

Claim 11.2.23. For every p ∈ ℤ and every x ∈ X such that c(x) = p, there exists a natural isomorphism of functors

colim_{V ∈Δ_{x}^{p}} \underline{Γ}_{V} →^{∼} j_{x∗} ∘ \underline{Γ}_{\{x\}} ∘ j_{x}^{∗}.

Proof of the claim. Let Σ_{x} be the set of all quasi-compact open neighborhoods of x in X, and endow Σ_{x} with the partial ordering such that, for every U, U′ ∈ Σ_{x} we have U ≤ U′ if U′ ⊂ U; then Σ_{x} is filtered, and there follows a morphism of partially ordered sets

Σ_{x} → Δ_{x} U ↦ U ∩ \overline{\{x\}}

and using lemma 1.5.7(i), it is easily seen that this map is cofinal. Now, for every U ∈ Σ_{x}, let j_{U} : U → X be the open immersion; we have a natural identification

\underline{Γ}_{U∩\overline{\{x\}}} →^{∼} j_{U∗} ∘ j_{U}^{∗} ∘ \underline{Γ}\overline{\{x\}}

whence a natural isomorphism

colim_{V ∈Δ_{x}} \underline{Γ}_{V} = colim_{U∈Σ_{x}} j_{U∗} ∘ j_{U}^{∗} ∘ \underline{Γ}\overline{\{x\}}.

However, X(x) is the limit of the filtered system of spectral spaces Σ_{x}, so the claim follows easily from proposition 10.1.10(ii) (details left to the reader). ♢

Next, let i ∈ ℕ be arbitrary, let ℱ be any 𝒜 -module, and pick any resolution ℱ → ℐ^{•} consisting of injective 𝒜 -modules; by the foregoing case, we get a natural isomorphism of 𝒜 -modules

R^{i}\underline{Γ}_{Φ^{p}|Φ^{p+1}}ℱ →^{∼} ⨁_{c(x)=p} H^{i}(j_{x∗} ∘ \underline{Γ}_{\{x\}} ∘ j_{x}^{∗}ℐ^{•}).

However, on the one hand, j^{∗}ℐ^{•} is a complex of qc-flabby j^{∗}𝒜 -modules (claim 10.1.11); on the other hand, qc-flabby abelian sheaves are acyclic for the functor \underline{Γ}_{\{x\}} (lemmata 10.4.4(iii.b) and 8.1.15(vi)), so we get a natural isomorphism

H^{i}(j_{x∗} ∘ \underline{Γ}_{\{x\}} ∘ j_{x}^{∗}ℐ^{•}) →^{∼} H^{i}(j_{x∗} ∘ R\underline{Γ}_{\{x\}} ∘ j_{x}^{∗}ℱ) for every i ∈ ℕ and every x ∈ X.

Lastly, notice that R\underline{Γ}_{\{x\}}∘j_{x}^{∗}ℱ is a complex of j^{∗}𝒜 -modules supported on \{x\}, and the functor j_{x∗} is exact on the full subcategory of j^{∗}𝒜 -Mod consisting of such sheaves; thus, we have a natural isomorphism

H^{i}(j_{x∗} ∘ R\underline{Γ}_{\{x\}} ∘ j_{x}^{∗}ℱ) →^{∼} j_{x∗} ∘ R^{i}\underline{Γ}_{\{x\}} ∘ j_{x}^{∗}ℱ for every i ∈ ℕ and every x ∈ X.

Summing up, we obtain the proposition. □

Definition 11.2.24. Let (X, 𝒜) be a ringed topological space, with X sober and locally noetherian, and c a weak codimension function on X.

(i) We say that an object K^{•} of D^{+}(𝒜 -Mod) is a c-Cohen-Macaulay complex, if we have

R^{i}\underline{Γ}_{\{x\}}K_{|X(x)}^{•} = 0 for every x ∈ X and every integer i ≠ c(x).

We denote by

𝒜 -CM^{c}

the full subcategory of D^{+}(𝒜 -Mod) whose objects are the c-Cohen-Macaulay complexes.

(ii) We say that an object C^{•} of C^{+}(𝒜 -Mod) is a Cousin complex, if there exists a family (M_{x} | x ∈ X) where M_{x} is an i_{x}^{∗}𝒜 -module for every x ∈ X (notation of (11.2.19)), such that

C^{p} = ⨁_{c(x)=p} i_{x∗}M_{x} for every p ∈ ℤ.

The full subcategory of C^{+}(𝒜 -Mod) whose objects are the Cousin complexes is denoted

𝒜 -Cousin.

Lemma 11.2.25. In the situation of definition 11.2.24, we have :

(i) R\underline{Γ}_{\{x\}} ∘ j_{x}^{∗} ∘ i_{y∗} = 0 for every x, y ∈ X such that x ≠ y.

(ii) If C^{•} is a Cousin complex of 𝒜 -modules, then C^{p} is qc-flabby, for every p ∈ ℤ.

Proof. (i): Notice that any i_{y}^{∗}𝒜 -module M is trivially qc-flabby, so the same holds for i_{y∗}M (lemma 10.1.5(iv)), and therefore also for j_{x}^{∗}j_{y∗}M (claim 10.1.11). Since qc-flabby 𝒜 -modules are acyclic for \underline{Γ}_{\{x\}} (lemma 10.4.4(iii.a)), we conclude that R\underline{Γ}_{\{x\}}j_{x}^{∗}i_{y∗} is the derived functor of \underline{Γ}_{\{x\}}j_{x}^{∗}i_{y∗}; but it is easily seen that the latter vanishes, whence the assertion.

(ii): We have just seen that i_{∗}M is qc-flabby, for every i_{x}^{∗}𝒜 -module M; on the other hand, all direct sums of qc-flabby 𝒜 -modules are qc-flabby (claim 10.1.11), whence the contention. □

11.2.26. As j_{x}^{∗} and R^{i}\underline{Γ}_{\{x\}} commute with all direct sums (proposition 11.2.12(v.b)), lemma 11.2.25(i) implies that the localization C^{+}(𝒜 -Mod) → D^{+}(𝒜 -Mod) restricts to a functor

F : 𝒜 -Cousin → 𝒜 -CM^{c}.

Theorem 11.2.27. The functor F of (11.2.26) is an equivalence of categories.

Proof. We construct a quasi-inverse to F, as follows. First, we endow every object C^{•} of C(𝒜 -Mod), with a descending filtration Fil_{Φ}^{•}C^{•}, by the rule

Fil_{Φ}^{p}C^{•} := \underline{Γ}_{Φ^{p}}C^{•}

where (Φ^{p} | p ∈ ℤ) is the system of families of supports associated with the weak codimension function c, as in (11.2.19). We also denote by gr_{Φ}^{•}C^{•} the resulting associated graded object of C(𝒜 -Mod). Next, for every object K^{•} of D^{+}(𝒜 -Mod) we choose a quasi-isomorphism β_{K}^{•} : K^{•} →^{∼} ℐ_{K}^{•}, with ℐ_{K}^{•} a bounded below complex of injective 𝒜 -modules. Following (7.2.14), we denote E(ℐ_{K})_{•}^{••} the spectral sequence arising from the filtered complex (ℐ_{K}^{•}, Fil_{Φ}^{•}ℐ_{K}^{•}). In light of (11.2.18) we have a natural identification

gr_{Φ}^{p}ℐ_{K}^{•} →^{∼} \underline{Γ}_{Φ^{p}|Φ^{p+1}}ℐ_{K}^{•}

and combining with remark 7.2.16(ii), we get a natural isomorphism

E(ℐ_{K})_{1}^{pq} →^{∼} R^{p+q}\underline{Γ}_{Φ^{p}|Φ^{p+1}}K^{•}.

Taking into account proposition 11.2.21we deduce that

G(K)^{•} := (E(ℐ_{K})_{1}^{•0}, d(ℐ_{K})_{1}^{•0})

is a Cousin complex, for every q ∈ ℤ. Moreover, every morphism φ^{•} : K^{•} → L^{•} in the category D^{+}(𝒜 -Mod) can be represented by a unique morphism ℐ_{φ}^{•} : ℐ_{K}^{•} → ℐ_{L}^{•} in Hot(𝒜 -Mod)

(notation of definition 7.1.4(ii)), and notice that ℐ_{φ}^{•} is a morphism of filtered complexes; in light of (7.2.23) we deduce a well defined morphism G(φ)^{•} : G(K)^{•} → G(L)^{•}. In this way, we obtain a functor

G : D^{+}(𝒜 -Mod) → 𝒜 -Cousin

and we claim that the restriction of G to the subcategory 𝒜 -CM^{c} is the sought quasi-inverse.

We check first that G ∘ F is naturally isomorphic to the identity endofunctor of 𝒜 -Cousin. To this aim, let us remark :

Claim 11.2.28. Let K^{•} be any c-Cohen-Macaulay complex. We have :

(i) E(ℐ_{K})_{1}^{pq} = 0 for every p, q ∈ ℤ with q ≠ 0.

(ii) If K^{•} = F C^{•} for a Cousin complex C^{•}, the map β_{K}^{•} induces a natural isomorphism

gr_{Φ}^{p}C^{•} →^{∼} E(ℐ_{K})_{1}^{p0}[−p] →^{∼} C^{p}[−p] for every p ∈ ℤ.

Proof of the claim. (i): Indeed, from proposition 11.2.21we get a natural isomorphism

(11.2.29)  H^{p+q}(gr_{Φ}^{p}ℐ_{C}^{•}) →^{∼} ⨁_{c(x)=p} j_{x∗} ∘ R^{p+q}\underline{Γ}_{\{x\}}K_{|X(x)}^{•}

which vanishes for q ≠ 0, since K^{•} is c-Cohen-Macaulay.

(ii): Let C^{•} and (M_{x} | x ∈ X) be as in definition 11.2.24(ii); from (11.2.18) and lemma 11.2.25(ii), it follows that the natural map

gr_{Φ}^{•}C^{•} → \underline{Γ}_{Φ^{p}|Φ^{p+1}}C^{•}

is an isomorphism in C^{+}(𝒜 -Mod). Also, combining with claim 10.1.11, we see that C_{|X(x)}^{•} is a complex of qc-flabby j_{x}^{∗}𝒜 -modules. By lemma 10.4.4(iii.a) we deduce that the natural map \underline{Γ}_{\{x\}}C_{|X(x)}^{•} → R\underline{Γ}_{\{x\}}C_{|X(x)}^{•} is an isomorphism in D^{+}(𝒜 -Mod), for every x ∈ X. Moreover, since the functor \underline{Γ}_{\{x\}} commutes with arbitrary direct sums (lemma 10.4.4(iii.b)), lemma 11.2.25(i) says that

\underline{Γ}_{\{x\}}C_{|X(x)}^{•} = j_{x}^{∗}i_{x∗}M_{x}[−c(x)] for every x ∈ X.

Taking into account (11.2.29) (and again, proposition 11.2.21), the assertion follows. ♢

Now, let C^{•} be a cousin complex, and set D^{•} := GF (C)^{•}. From claim 11.2.28 we obtain a natural isomorphism

gr_{Φ}^{p}ℐ_{C}^{•} →^{∼} C^{p}[−p] for every p ∈ ℤ.

We need to show that, under this identification, the differential d_{D}^{p} of D^{p} := H^{p}(gr^{p}ℐ_{C}^{•}) corresponds to the differential d_{C}^{p} of C^{•}. However, on the one hand we have a commutative ladder

\xymatrix{0 \ar[r] & gr_{Φ}^{p+1}C^{•} \ar[r] \ar[d] & Fil_{Φ}^{p}C^{•}/Fil_{Φ}^{p+2}C^{•} \ar[r] \ar[d] & gr_{Φ}^{p}C^{•} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & gr_{Φ}^{p+1}ℐ_{C}^{•} \ar[r] & Fil_{Φ}^{p}ℐ_{C}^{•}/Fil_{Φ}^{p+2}ℐ_{C}^{•} \ar[r] & gr_{Φ}^{p}ℐ_{C}^{•} \ar[r] & 0}

and claim 11.2.28(ii) says that both the first and third vertical arrows are isomorphisms, so the top row of is isomorphic to the bottom row. Moreover, we see that the top row is the natural short exact sequence

0 → C^{p+1}[−p − 1] → t^{≤p+1}t^{≥p}C^{•} → C^{p}[−p] → 0

where t^{≤p+1} and t^{≥p} are the brutal truncation functors of (7.1.1). Then it is clear that the boundary map H^{p}(C^{p}[−p]) → H^{p+1}(C^{p+1}[−p − 1]) in degree p associated with this short exact sequence, is none else than the differential d_{C}^{p}. Now the contention follows from the naturality of the boundary map, together with remark 7.2.16(ii).

Next, let K^{•} be any c-Cohen-Macaulay complex, and set

(𝒥_{K}, Fil^{•}𝒥_{K}) := Déc (ℐ_{K}, Fil_{Φ}^{•}ℐ_{K}^{•})

(notation of (7.2.24)). By proposition 7.2.26(i,ii) and claim 11.2.28(i), we have a natural epimorphism

u^{•} : E(𝒥_{K})_{0}^{•} → E(ℐ_{K})_{1}^{•} in C^{+}(𝒜 -Mod)

which induces an isomorphism in D^{+}(𝒜 -Mod). Moreover, from claim 11.2.28(i) we see that the subcomplex (E(𝒥_{K})_{0}^{p,n−p} | n ∈ ℤ) lies in Ker u^{•}, for every p ≠ 0, so u^{•} restricts to an isomorphism

(11.2.30)  E(𝒥_{K})_{0}^{0,•} = gr^{0}𝒥_{K}^{•} →^{∼} G(K)^{•}  in D^{+}(𝒜 -Mod)

(see remark 7.2.16(i)). We point out :

Claim 11.2.31. Fil^{1}𝒥_{K}^{•} and 𝒥_{K}^{•}/Fil^{0}𝒥_{K}^{•} are acyclic.

Proof of the claim. The assertion can be checked on the stalks over the points of X; taking into account lemma 11.2.20(ii,iii), we may then replace X by X(x), for any point x ∈ X, and assume from start that c is bounded on X (lemma 11.2.20(i)). In this case, it is clear that the filtration Fil_{Φ}^{•}ℐ_{K}^{•} is finite, for every complex K^{•} of 𝒜 -modules, and then the same holds for the filtration Fil^{•}𝒥_{K}^{•} (see (7.2.24)). In this situation, a simple induction argument reduces to checking that the associated subquotients gr^{p}𝒥_{K}^{•} are acyclic, for every p ≠ 0. However, combining remarks 7.2.9(i) and 7.2.16(ii), proposition 7.2.26(ii) and claim 11.2.28(i) we may compute

H^{q}gr^{p}𝒥_{K}^{•} →^{∼} E(𝒥_{K})_{1}^{p,q−p} →^{∼} E(ℐ_{K})_{2}^{p+q,−p} = 0 for every p, q ∈ ℤ with p ≠ 0

whence the claim. ♢ Claim 11.2.31yields a natural isomorphism

gr^{0}𝒥_{K}^{•} →^{∼} 𝒥_{K}^{•} = ℐ_{K}^{•} in D^{+}(𝒜 -Mod)

and combining with (11.2.30), we finally get an isomorphism

K^{•} →^{∼} F G(K)^{•} in D^{+}(𝒜 -Mod) .

Now the theorem follows from proposition 1.1.20(i). □

11.3. Duality over coherent schemes. If X is a locally coherent scheme (definition 8.1.58(i)) and ℱ is an 𝒪_{X}-module, we define the dual 𝒪_{X}-module

ℱ^{∨} := ℋ om_{𝒪_{X}} (ℱ , 𝒪_{X}).

Moreover, for every morphism of 𝒪_{X}-modules φ : ℱ → 𝒢, we denote by φ^{∨} : 𝒢^{∨} → ℱ^{∨} the induced (transpose) morphism. As usual, there is a natural morphism of 𝒪_{X}-modules:

β_{ℱ} : ℱ → ℱ^{∨∨}.

Definition 11.3.1. Let X be a locally coherent scheme, and ℱ an 𝒪_{X}-module.

(i) We say that ℱ is reflexive at a point x ∈ X if there exists an open neighborhood U ⊂ X of x such that ℱ_{|U} is a coherent 𝒪_{U}-module, and β_{ℱ |U} is an isomorphism.

(ii) We say that ℱ is reflexive if it is reflexive at all points of X.

(iii) We denote by 𝒪_{X}-Rflx the full subcategory of the category 𝒪_{X}-Mod, consisting of all

the reflexive 𝒪_{X}-modules. It contains 𝒪_{X}-Mod_{lfft} as a full subcategory (see (10.3)). (iv) Suppose that X = Spec R for a coherent ring R, and ℱ = M^{∼} for an R-module M.

Then we say that M is a reflexive R-module if ℱ is a reflexive 𝒪_{X}-module.

Lemma 11.3.2. Let X be a locally coherent scheme, x ∈ X any point, and ℱ a coherent 𝒪_{X}-module. We have :

(i) ℱ^{∨} is a coherent 𝒪_{X}-module.

(ii) The following conditions are equivalent: (a) ℱ is reflexive at the point x.

(b) ℱ_{x} is a reflexive 𝒪_{X,x}-module.

(c) The map β_{ℱ ,x} : ℱ_{x} → (ℱ^{∨∨})_{x} is an isomorphism.

Proof. (i): The assertion is local on X, hence we may assume that there exists a presentation 𝒪_{X}^{⊕n} → 𝒪_{X}^{⊕m} → ℱ → 0, whence – after taking duals – a left exact sequence 0 → ℱ^{∨} → 𝒪_{X}^{⊕m} → 𝒪_{X}^{⊕n}. Since 𝒪_{X} is coherent, the assertion follows.

(ii) shall be left to the reader. □

Lemma 11.3.3. Let X be a reduced locally coherent scheme, and ℱ a quasi-coherent 𝒪_{X}- module of finite type. Endow the set Max X of maximal points of X with the topology induced by X, and consider the following conditions :

(a) There exists a quasi-compact open immersion j : U → X with dense image, such that j^{∗}ℱ is a locally free 𝒪_{U}-module.

(b) There exists a quasi-compact open immersion j : U → X with dense image, such that j^{∗}ℱ is a coherent 𝒪_{U}-module.

(c) The rank function of ℱ :

X → ℕ x ↦ dim_{κ(x)} κ(x) ⊗_{𝒪_{X,x}} ℱ_{x}

restricts to a locally constant function on Max X.

(d) ℱ^{∨} is a reflexive 𝒪_{X}-module and β_{ℱ}^{∨} ∘ β_{ℱ^{∨}} = 1_{ℱ^{∨}} . Then (a)⇒(b)⇒(c)⇒(d) and if X is coherent we have (c)⇒(a) as well.

Proof. Let us check first that (c)⇒(a) in case X is coherent. Indeed, notice that j is quasicompact if and only if U is retro-compact in X, and the latter holds if and only if U is open and constructible in X (lemma 8.1.19(iv.b,v.c)). It follows that the existence of j with the sought properties can be checked locally on X, hence we may assume that X is affine. By proposition 10.3.34(i) we may find a coherent 𝒪_{X}-module 𝒢 and an 𝒪_{X}-linear epimorphism φ : 𝒢 → ℱ. Now, fix any η ∈ Max X and write 𝒦 := Ker φ as the filtered union of the system (𝒦_{i} | i ∈ I) of its quasi-coherent 𝒪_{X}-submodules of finite type; since 𝒪_{X,η} is a field, there exists i ∈ I such that 𝒦_{i,η} = 𝒦_{η}. Thus, after replacing 𝒢 by 𝒢 /𝒦_{i}, we may assume that φ_{η} is an isomorphism. Moreover, by [126, Th.4.10] we may find an affine open neighborhood U_{η} of η in X, an integer r ∈ ℕ and an isomorphism 𝒢_{|U_{η}} →^{∼} 𝒪_{U_{η}}^{⊕r}, and by assumption (c) we may also assume that the rank function of ℱ is constant on Max U_{η}. In this case, φ_{τ} is an isomorphism for every τ ∈ Max U_{η}, so 𝒦_{τ} = 0 for every such point τ. Since X is reduced and 𝒢_{|U_{η}} is a locally free 𝒪_{U_{η}}-module, it follows that 𝒦_{|U_{η}} = 0. By proposition 8.1.59, we may then find a finite subset Σ ⊂ Max X such that ⋃_{η∈Σ} Max U_{η} = Max X, and it is easily seen that the open subset U := ⋃_{η∈Σ} U_{η} fulfills condition (a).

(a)⇒(b) is obvious.

(b)⇒(c): Notice first that any U as in (b) contains Max X, by proposition 8.1.47(iii). Then, since X is reduced, [126, Th.4.10] implies that, for every η ∈ Max X, we may find an affine open neighborhood U_{η} of η in X such that ℱ_{|U_{η}} is a locally free 𝒪_{U_{η}}-module, whence (c).

(c)⇒(d): Let us show first that ℱ^{∨} is coherent, under condition (c). The assertion is local on X, hence we may assume that X is affine, in which case, as we have already seen, (c) implies that there exists a quasi-compact open immersion j : U → X fulfilling condition (a), and we also know that any such U contains Max X. By proposition 10.3.34(i), we may then find a coherent 𝒪_{X}-module 𝒢 and an 𝒪_{X}-linear epimorphism φ : 𝒢 → ℱ such that j^{∗}φ is an isomorphism. There follows a left exact sequence 0 → ℱ^{∨} → 𝒢^{∨} → (Ker φ)^{∨}, and 𝒢^{∨} is coherent, by lemma 11.3.2(i). Since Max X ⊂ U, we have (Ker φ)_{η} = 0 for every η ∈ Max X, and since 𝒪_{X} is reduced, it follows easily that (Ker φ)^{∨} = 0, and the claim follows. Now, since X is reduced, the only associated points of 𝒪_{X} are the maximal points of X (i.e. 𝒪_{X} has no imbedded points). It follows easily that, for every quasi-coherent 𝒪_{X}-module 𝒢, the dual 𝒢^{∨} satisfies condition S_{1} (see definition 10.5.1(ii)). Next, rather generally, let ℳ be any 𝒪_{X}-module; directly from the definitions one derives the identity:

(11.3.4)  β_{ℳ}^{∨} ∘ β_{ℳ^{∨}} = 1_{ℳ^{∨}} .

It remains therefore only to show that β_{ℱ}^{∨} is a right inverse for β_{ℱ^{∨}} when ℱ^{∨} is coherent. Since ℱ^{∨∨∨} satisfies condition S_{1}, it suffices to check that, for every maximal point ξ, the induced map on stalks

β_{ℱ ,ξ}^{∨} : ℱ_{ξ}^{∨∨∨} → ℱ_{ξ}^{∨}

is a right inverse for β_{ℱ^{∨},ξ} . However, since 𝒪_{X,ξ} is a field, β_{ℱ ,ξ}^{∨} is a linear map of 𝒪_{X,ξ}-vector spaces of the same finite dimension, hence it is an isomorphism, in view of (11.3.4). □

Remark 11.3.5. The following observation is often useful. Suppose that X is a locally coherent scheme, ℱ an 𝒪_{X}-module, reflexive at a given point x ∈ X. We can then choose a presentation 𝒪_{X,x}^{⊕n} → 𝒪_{X,x}^{⊕m} → ℱ_{x}^{∨} → 0, and after dualizing we deduce a left exact sequence

(11.3.6)  0 → ℱ_{x} → 𝒪_{X,x}^{⊕m} ⟶^{u} 𝒪_{X,x}^{⊕n}.

Especially, if 𝒪_{X,x} satisfies condition S_{1} (in the sense of definition 10.5.1(iii)), then the same holds for ℱ_{x}. For the converse, suppose additionally that X is reduced, and let x ∈ X be a point for which there exists a left exact sequence such as (11.3.6); then lemma 11.3.3 says that ℱ is reflexive at x : indeed, ℱ_{x} ≃ (Coker u^{∨})^{∨}.

Lemma 11.3.7. (i) Let f : X → Y be a flat morphism of locally coherent schemes. The induced functor 𝒪_{Y} -Mod → 𝒪_{X}-Mod restricts to a functor

f^{∗} : 𝒪_{Y} -Rflx → 𝒪_{X}-Rflx.

(ii) Let X_{0} be a quasi-compact and quasi-separated scheme, (X_{λ} | λ ∈ Λ) a cofiltered family of coherent X_{0}-schemes with flat transition morphisms ψ_{λμ} : X_{λ} → X_{μ} such that the structure morphisms X_{λ} → X_{0} are affine, and set X := lim_{λ∈Λ} X_{λ}. Then:

(a) X is coherent.

(b) the natural functor: 2-colim_{λ∈Λ^{o}} 𝒪_{X_{λ}}-Rflx → 𝒪_{X}-Rflx is an equivalence. (iii) Suppose that f is surjective, and let ℱ be any coherent 𝒪_{Y} -module. Then ℱ is reflexive

if and only if f^{∗}ℱ is a reflexive 𝒪_{X}-module.

Proof. (i) follows easily from [75, Lemma 2.4.29(i.a)].

(ii): For every λ ∈ Λ, denote by ψ_{λ} : X → X_{λ} the natural morphism. Let U ⊂ X be a quasicompact open subset, u : 𝒪_{U}^{⊕n} → 𝒪_{U} any morphism of 𝒪_{U}-modules; by corollary 8.1.40(ii.a) there exist λ ∈ Λ and a quasi-compact open subset U_{λ} ⊂ X_{λ} such that U = ψ^{−1}(U_{λ}). By [65, Ch.IV, Th.8.5.2(i)] we may then suppose that u descends to a homomorphism u_{λ} : 𝒪_{U_{λ}}^{⊕n} → 𝒪_{U_{λ}}, whose kernel is of finite type, since X_{λ} is coherent. Since the transition morphisms are flat, we have Ker u = ψ_{λ}^{∗}(Ker u_{λ}), whence (ii.a). Next, using [65, Ch.IV, Th.8.5.2] one sees easily that the functor of (ii.b) is fully faithful and moreover, every reflexive 𝒪_{X}-module ℱ descends to a coherent 𝒪_{X_{λ}}-module ℱ_{λ} for some λ ∈ Λ. For every μ ≥ λ let ℱ_{μ} := ψ_{μλ}^{∗}ℱ_{λ}; since β_{ℱ} is an isomorphism, loc.cit. shows that β_{ℱ_{μ}} is already an isomorphism for some μ ≥ λ, whence (ii.b).

(iii): By virtue of (i), we may assume that f^{∗}ℱ is reflexive, and we need to show that the same holds for ℱ. However, the natural map f^{∗}(ℱ^{∨∨}) → (f^{∗}ℱ)^{∨∨} is an isomorphism (proposition 10.3.3(ii)), hence f^{∗}β_{ℱ} = β_{f^{∗}ℱ} is an isomorphism. Since f is faithfully flat, we deduce that β_{ℱ} is an isomorphism, as stated. □

Proposition 11.3.8. Let X be a coherent scheme, U ⊂ X a quasi-compact open subset. Then:

(i) If X is reduced, the restriction functor

𝒪_{X}-Rflx → 𝒪_{U}-Rflx

is essentially surjective. (ii) Let ℱ be a reflexive 𝒪_{X}-module. If δ′(x, 𝒪_{X}) ≥ 2 for all x ∈ X \ U, the natural map

ℱ → j_{∗}j^{∗}ℱ

is an isomorphism. Especially, the restriction functor of (i) is an equivalence.

Proof. (i): Given a reflexive 𝒪_{U}-module ℱ, lemma 10.3.27(ii) says that we can find a finitely presented quasi-coherent 𝒪_{X}-module 𝒢 extending ℱ^{∨}; since X is coherent, 𝒢 is a coherent 𝒪_{X}-module, hence the same holds for 𝒢^{∨}, which extends ℱ and is reflexive in light of lemma 11.3.3.

(ii): Notice that the first assertion easily implies that the restriction functor of (i) is fully faithful, so the second assertion follows from the first together with (i).

Next, since the first assertion is local on X, we can suppose that there exists a left exact sequence 0 → ℱ → 𝒪_{X}^{⊕m} → 𝒪_{X}^{⊕n} (see remark 11.3.5). Since the functor j_{∗} is left exact, it then suffices to prove the contention for the sheaves 𝒪_{X}^{⊕m} and 𝒪_{X}^{⊕n}, and thus we may assume from start that ℱ = 𝒪_{X}. Then, since X\U is constructible, corollary 10.4.23applies and yields the assertion. □

Corollary 11.3.9. Let X be a locally coherent scheme, f : X → S a flat, locally finitely presented morphism, j : U → X a quasi-compact open immersion, ℱ a reflexive 𝒪_{X}-module. Suppose that

(a) depth_{f}(x) ≥ 1 for every point x ∈ X \U, and

(b) depth_{f}(x) ≥ 2 for every maximal point η of S and every x ∈ (X \U) ∩ f^{−1}(η).

(c) δ′(s, 𝒪_{S}) > 0 for every non-maximal point s ∈ S. Then the natural morphism ℱ → j_{∗}j^{∗}ℱ is an isomorphism.

Proof. Since f is flat, corollary 10.4.47and our assumptions imply that δ′(x, 𝒪_{X}) ≥ 2 for every x ∈ X \U, so the assertion follows from proposition 11.3.8(ii). □

11.3.10. Let X be any scheme. Recall that the rank of an 𝒪_{X}-module ℱ of finite type, is the upper semicontinuous function:

rk ℱ : X → ℕ x ↦ dim_{κ(x)} ℱ_{x} ⊗_{𝒪_{X,x}} κ(x).

Clearly, if ℱ is a is locally free 𝒪_{X}-module of finite type, rk ℱ is a continuous function on X. The converse holds, provided X is a reduced scheme. Moreover, if ℱ is of finite presentation, rk ℱ is a constructible function and there exists a dense open subset U ⊂ X such that rk ℱ restricts to a continuous function on U. We denote by Pic X the full subcategory of 𝒪_{X}-Mod_{lfft} consisting of all the objects whose rank is constant equal to one (i.e. the invertible 𝒪_{X}-modules). In case X is locally coherent, we shall also consider the category Div X of generically invertible 𝒪_{X}-modules, defined as the full subcategory of 𝒪_{X}-Rflx consisting of all objects which are locally free of rank one on a dense open subset of X. If X is locally coherent, Pic X is a full subcategory of DivX.

Remark 11.3.11. (i) Let A be any integral domain, and set X = Spec A. Classically, one has a notion of reflexive fractional ideal of A (see [126, p.80] or example 6.4.38). Suppose now that A is also coherent, in which case we have the notion of reflexive A-module of definition 11.3.1(iv). We claim that these two notions overlap on the subclass of reflexive fractional ideals of finite type: more precisely, let Div(A) be the full subcategory of A-Mod whose objects are the finitely generated reflexive fractional ideals of A. Then the essential image of the functor

(11.3.12)  Div(A) → 𝒪_{X}-Mod  M ↦ M^{∼}

is the category Div X, and (11.3.12) yields an equivalence of Div(A) with the latter category. Indeed, let ℱ be any generically invertible 𝒪_{X}-module, and set I := ℱ (X); if K denotes the field of fractions of A, then dim_{K} I ⊗_{A} K = 1. Let us then fix a K-linear isomorphism I ⊗_{A} K →^{∼} K, and notice that the induced A-linear map I → K is injective, since ℱ is S_{1} (remark 11.3.5). We may then view I as a finitely generated A-submodule of K, and then it is clear that I is a fractional ideal of A. Moreover, on the one hand ℱ^{∨} is the coherent 𝒪_{X}-module Hom_{A}(I, A)^{∼}; on the other hand, the natural map Hom_{A}(I, A) → Hom_{K}(I ⊗_{A} K, K) = K is injective, and its image is the fractional ideal I^{−1} (see (6.4.35)). We easily deduce that I is a reflexive fractional ideal, and conversely it is easily seen that every 𝒪_{X}-module in the essential image of (11.3.12) is generically invertible; since this functor is also obviously fully faithful, the assertion follows.

(ii) Let A be a coherent integral domain, and denote by coh.Div(A) the set of all coherent reflexive fractional ideals of A. It is easily seen that coh.Div(A) is a submonoid of Div(A), for the natural monoid structure introduced in example 6.4.38. More generally, if X is a coherent integral scheme, we may define a sheaf of monoids 𝒟iv_{X} on X, as follows. First, to any affine open subset U ⊂ X, we assign the monoid 𝒟iv_{X}(U) := coh.Div(𝒪_{X}(U)). For each inclusion j : U′ ⊂ U of affine open subset, notice that the restriction map 𝒪_{X}(U) → 𝒪_{X}(U′) is a flat ring homomorphism, hence it gives a flat morphism of monoids 𝒪_{X}(U) \ {0} → 𝒪_{X}(U′) \ {0}. We then have an induced map of monoids 𝒟iv_{X}(j) : 𝒟iv_{X}(U) → 𝒟iv_{X}(U′), by virtue of lemma 6.4.45(iv). It is easily seen that the resulting presheaf 𝒟iv_{X} is a sheaf on the site of all affine open subsets of X. By [59, Ch.0, §3.2.2], the latter extends uniquely to a sheaf of monoids on X, which we denote again 𝒟iv_{X}. We set

Div(X) := 𝒟iv_{X}(X).

If X is normal and locally noetherian (or more generally, if X is a Krull scheme, i.e. 𝒪_{X}(U) is a Krull ring, for every affine open subset U ⊂ X) then 𝒟iv_{X} is an abelian sheaf, and Div(X) is an abelian group (proposition 6.4.42(i,iii)).

11.3.13. For future reference, it is useful to recall some preliminaries concerning the determinant functors defined in [119]. Let X be a scheme. We denote by gr.Pic X the category of graded invertible 𝒪_{X}-modules. An object of gr.Pic X is a pair (L, α), where L is an invertible 𝒪_{X}-module and α : X → ℤ is a continuous function. A homomorphism h : (L, α) → (M, β) is a homomorphism of 𝒪_{X}-modules h : L → M such that h_{x} = 0 for every x ∈ X with α(x) ≠ β(x). We denote by gr.Pic^{∗}X the subcategory of gr.Pic X with the same objects, and whose morphisms are the isomorphisms in gr.Pic X. Notice that gr.Pic X is a tensor category : the tensor product of two objects (L, α) and (M, β) is the pair (L, α) ⊗ (M, β) := (L ⊗_{𝒪_{X}} M, α + β). We denote by 𝒪_{X}-Mod_{lfft}^{∗} the category whose objects are the locally free 𝒪_{X}-modules of finite type, and whose morphisms are the 𝒪_{X}-linear isomorphisms. The determinant is the functor:

det : 𝒪_{X}-Mod_{lfft}^{∗} → gr.Pic^{∗}X F ↦ (Λ_{𝒪_{X}}^{rk F}F, rk F).

Let D(𝒪_{X}-Mod)_{perf} be the category of perfect complexes of 𝒪_{X}-modules; recall that, by definition, every perfect complex is locally isomorphic to a bounded complex of locally free 𝒪_{X}- modules of finite type. The category D(𝒪_{X}-Mod)_{perf}^{∗} is the subcategory of D(𝒪_{X}-Mod)_{perf} with the same objects, and whose morphisms are the isomorphisms (i.e. the quasi-isomorphisms of complexes). The main theorem of chapter 1 of [119] can be stated as follows.

Lemma 11.3.14. ([119, Th.1]) With the notation of (11.3.13) there exists, for every scheme X, an extension of the determinant functor to a functor:

det : D(𝒪_{X}-Mod)_{perf}^{∗} → gr.Pic X.

These determinant functors commute with every base change. □

Proposition 11.3.15. Let X be a regular scheme. Then every reflexive generically invertible 𝒪_{X}-module is invertible.

Proof. The question is local on X, hence we may assume that X is affine. Let ℱ be a generically invertible 𝒪_{X}-module, and U ⊂ X a dense open subset such that ℱ_{|U} is invertible. Denote by Z_{1}, …, Z_{t} the irreducible components of Z := X \U whose codimension in X equals one, and for every i = 1, …, t, let η_{i} be the maximal point of Z_{i}, and set A_{i} := 𝒪_{X,η_{i}}. Since ℱ is S_{1} (remark 11.3.5), the stalk ℱ_{η_{i}} is a torsion-free A_{i}-module of finite type for every i ≤ t; however, A_{i} is a discrete valuation ring, hence ℱ_{η_{i}} is a free A_{i}-module, necessarily of rank one, for i = 1, …, t. Since ℱ is coherent, it follows that there exists an open neighborhood U_{i} of η_{i} in X, such that ℱ_{|U_{i}} is a free 𝒪_{|U_{i}}-module. Hence, we may replace U by U ∪ U_{1} ∪ ⋯ ∪ U_{t} and assume that every irreducible component of Z has codimension > 1, therefore δ′(x, 𝒪_{X}) ≥ 2 for every x ∈ Z. Now, ℱ [0] is a perfect complex by Serre’s theorem ([163, Th.4.4.16]), so the invertible 𝒪_{X}-module det ℱ is well defined (lemma 11.3.14). Let j : U → X be the open immersion; in view of proposition 11.3.8, we deduce natural isomorphisms

ℱ →^{∼} j_{∗}j^{∗}ℱ →^{∼} j_{∗}det(j^{∗}ℱ [0]) →^{∼} j_{∗}j^{∗}det ℱ [0] ←^{∼} det ℱ [0]

and the assertion follows. □

We wish now to introduce a notion of duality better suited to derived categories of 𝒪_{X}- modules (over a scheme X). Hereafter we only carry out a preliminary investigation of such derived duality – the full development of which, will be the task of section 11.5.

Definition 11.3.16. Let X be a locally coherent scheme. A complex ω^{•} in D^{b}(𝒪_{X}-Mod)_{coh} (notation of (10.3)) is called dualizing if it fulfills the following two conditions :

(a) The functor :

𝒟 : D(𝒪_{X}-Mod)^{o} → D(𝒪_{X}-Mod) C^{•} ↦ Rℋ om_{𝒪_{X}}^{•} (C^{•}, ω^{•})

restricts to a duality functor : 𝒟 : D^{b}(𝒪_{X}-Mod)_{coh}^{o} → D^{b}(𝒪_{X}-Mod)_{coh}.

(b) The natural transformation : η_{C^{•}} : C^{•} → 𝒟 ∘ 𝒟(C^{•}) restricts to a biduality isomorphism of functors on the category D^{b}(𝒪_{X}-Mod)_{coh}.

Remark 11.3.17. Let X be a locally coherent scheme, ω^{•} an object of D^{b}(𝒪-Mod)_{coh}, and define the functor 𝒟 as in definition 11.3.16.

(i) A standard dévissage argument shows that ω^{•} is dualizing on X if and only if 𝒟(ℱ [0]) lies in D^{b}(𝒪_{X}-Mod)_{coh}, and the biduality map ℱ [0] → 𝒟 ∘ 𝒟(ℱ [0]) is an isomorphism for every coherent 𝒪_{X}-module ℱ.

(ii) Suppose that the complex ω^{•} is dualizing on X. From the natural identification ω^{•} →^{∼} Rℋ om_{𝒪_{X}}^{•} (𝒪_{X}[0], ω^{•}), we deduce a natural isomorphism

Rℋ om_{𝒪_{X}}^{•} (ω^{•}, ω^{•}) →^{∼} 𝒪_{X}[0] in D^{b}(𝒪_{X}-Mod) .

Example 11.3.18. Suppose that X is a noetherian regular scheme (i.e. all the stalks 𝒪_{X,x} are regular rings). In light of Serre’s theorem [163, Th.4.4.16], every object of D^{b}(𝒪_{X}-Mod)_{coh} is a perfect complex. It follows easily that the complex 𝒪_{X}[0] is dualizing. For more general schemes, the structure sheaf does not necessarily work, and the existence of a dualizing complex is a delicate issue. On the other hand, one may ask to what extent a complex is determined by the properties (a) and (b) of definition 11.3.16. Clearly, if ω^{•} is dualizing on X, then so is any other complex of the form ω^{•} ⊗_{𝒪_{X}} ℒ, where ℒ is an invertible 𝒪_{X}-module. Also, any shift of ω^{•} is again dualizing. Conversely, the following proposition 11.3.25says that any two dualizing complexes are related in such manner, up to quasi-isomorphism.

Lemma 11.3.19. Let (X, 𝒪_{X}) be any locally ringed space, and P^{•} and Q^{•} two objects of D^{−}(𝒪_{X}-Mod) with a quasi-isomorphism :

P^{•} ⊗^{L}_{𝒪_{X}} Q^{•} →^{∼} 𝒪_{X}[0].

Then there exist an invertible 𝒪_{X}-module ℒ, a continuous function σ : |X| → ℤ and quasiisomorphisms:

P^{•} →^{∼} ℒ [σ]  and  Q^{•} →^{∼} ℒ^{−1}[−σ].

Proof. It suffices to verify that, locally on X, the complexes P^{•} and Q^{•} are of the required form; indeed in this case X will be a disjoint union of open sets U_{n} on which H^{•}P^{•} is concentrated in degree n and H^{•}Q^{•} is concentrated in degree −n. Note that 𝒪_{X}-Mod is equivalent to the product of the categories 𝒪_{U_{n}}-Mod and that the derived category of a product of abelian categories is the product of the derived categories of the factors.

Claim 11.3.20. Let A be a (commutative) local ring, K^{•} and L^{•} two objects of D^{−}(A-Mod) with a quasi-isomorphism

(11.3.21)  K^{•} ⊗^{L}_{A} L^{•} →^{∼} A[0].

Then there exist s ∈ ℤ and quasi-isomorphisms :

K^{•} →^{∼} A[s] L^{•} →^{∼} A[−s].

Proof of the claim. Set :

i_{0} := max(i ∈ ℤ | (H^{i}K^{•}) ≠ 0) and j_{0} := max(i ∈ ℤ | (H^{i}L^{•}) ≠ 0).

We may assume that K^{i} = 0 for every i > i_{0}, and that L^{•} is a bounded above complex of free A-modules. Then we may find a filtered system (K_{λ}^{•} | λ ∈ Λ) of complexes of A-modules bounded from above, such that

• H^{i_{0}}(K_{λ}^{•}) is a finitely generated A-module, for every λ ∈ Λ;

• the colimit of the system (K_{λ}^{•} | λ ∈ Λ) (in the category of complexes of A-modules) is

isomorphic to K^{•}. From (11.3.21) we get an isomorphism H^{0}(K^{•} ⊗_{A} L^{•}) →^{∼} A, and it follows easily that the natural map H^{0}(K_{λ}^{•} ⊗_{A} L^{•}) → H^{0}(K^{•} ⊗_{A} L^{•}) is surjective for some λ ∈ Λ. For such λ, we may then find a morphism in D^{−}(A-Mod) :

(11.3.22)  A[0] → K_{λ}^{•} ⊗^{L}_{A} L^{•}

whose composition with the natural map K_{λ}^{•} ⊗^{L}_{A} L^{•} → K^{•} ⊗^{L}_{A} L^{•} is the inverse of (11.3.21). Hence

(11.3.22) ⊗^{L}_{A} K^{•} : K^{•} → (K_{λ}^{•} ⊗^{L}_{A} L^{•}) ⊗^{L}_{A} K^{•} →^{∼} K_{λ}^{•} ⊗^{L}_{A} (L^{•} ⊗^{L}_{A} K^{•}) →^{∼} K_{λ}^{•}

is a right inverse of the natural morphism K_{λ}^{•} → K^{•} in D^{−}(A-Mod). Especially, the induced map H^{i_{0}}K_{λ}^{•} → H^{i_{0}}K^{•} is surjective, i.e. H^{i_{0}}K^{•} is a finitely generated A-module. Likewise, we see that H^{j_{0}}L^{•} is a finitely generated A-module. Now, notice that

H^{k}(K^{•} ⊗^{L}_{A} L^{•}) ≃ \{ \begin{matrix}H^{i_{0}}K^{•} ⊗_{A} H^{j_{0}}L^{•} & for k = i_{0} + j_{0} \\ 0 & for k > i_{0} + j_{0}. \end{matrix}

From Nakayama’s lemma it follows easily that H^{i_{0}+j_{0}}(K^{•} ⊗^{L}_{A} L^{•}) ≠ 0, and then our assumptions imply that i_{0} + j_{0} = 0 and H^{i_{0}}K^{•} ⊗_{A} H^{j_{0}}L^{•} ≃ A. One deduces easily that H^{i_{0}}K^{•} ≃ A ≃ H^{j_{0}}L^{•} (see e.g. [75, Lemma 4.1.5]). Furthermore, we can find a complex K_{1}^{•} in D^{<i_{0}}(A-Mod) (resp. L_{1}^{•} in D^{<j_{0}}(A-Mod)) such that :

K^{•} ≃ A[−i_{0}] ⊕ K_{1}^{•} (resp. L^{•} ≃ A[−j_{0}] ⊕ L_{1}^{•})

whence a quasi-isomorphism :

φ : A[0] →^{∼} K^{•} ⊗^{L}_{A} L^{•} →^{∼} A[0] ⊕ K_{1}^{•}[−j_{0}] ⊕ L_{1}^{•}[−i_{0}] ⊕ (K_{1}^{•} ⊗^{L}_{A} L_{1}^{•}).

However, by construction φ^{−1} restricts to an isomorphism on the direct summand A[0], therefore K_{1}^{•} ≃ 0 ≃ L_{1}^{•} in D(A-Mod), and the claim follows. ♢

Now, for any point x ∈ X, let i_{x} : {x} → X be the inclusion map, and set K_{x}^{•} := i_{x}^{∗}K^{•} for every complex K^{•} of 𝒪_{X}-modules. Notice that if K^{•} is a complex of flat 𝒪_{X}-modules, then K_{x}^{•} is a complex of flat 𝒪_{X,x}-modules ([9, Exp.V, Prop.1.6(1)]), therefore the rule K^{•} ↦ K_{x}^{•} yields a well-defined functor D^{−}(𝒪_{X}-Mod) → D^{−}(𝒪_{X,x}-Mod), and moreover we have a natural isomorphism

(11.3.23)  K_{x}^{•} ⊗^{L}_{𝒪_{X,x}} L_{x}^{•} →^{∼} (K^{•} ⊗^{L}_{𝒪_{X}} L^{•})_{x}

for every objects K^{•}, L^{•} of D^{−}(𝒪_{X}-Mod). Especially, under the current assumptions, and in view of claim 11.3.20, we may find s ∈ ℤ, and an isomorphism of 𝒪_{X,x}-modules

(11.3.24)  H^{s}P_{x}^{•} ⊗_{𝒪_{X,x}} H^{−s}Q_{x}^{•} →^{∼} H^{0}(P_{x}^{•} ⊗^{L}_{𝒪_{X,x}} Q_{x}^{•}) →^{∼} 𝒪_{X,x}.

Since 𝒪_{X,x} is local, we may thus find a_{x} ∈ H^{s}P_{x}^{•} and b_{x} ∈ H^{−s}Q_{x}^{•} such that (11.3.24) maps a_{x} ⊗ b_{x} to 1. Then a_{x} and b_{x} extend to local sections

a ∈ Γ(U, Ker(d : P^{s} → P^{s−1})) b ∈ Γ(U, Ker(d : Q^{−s} → P^{−s−1}))

on some neighborhood U ⊂ X of x, and after shrinking U, we may assume that a ⊗ b gets mapped to 1, under the induced morphism of 𝒪_{U}-modules

H^{s}P_{|U}^{•} ⊗_{𝒪_{U}} H^{−s}Q_{|U}^{•} → H^{0}(P^{•} ⊗^{L}_{𝒪_{X}} Q^{•})_{|U} →^{∼} 𝒪_{U}.

Then we obtain a well defined morphism in D^{−}(𝒪_{U}-Mod)

φ : 𝒪_{U}[s] → P_{|U}^{•} (resp. ψ : 𝒪_{U}[−s] → Q_{|U}^{•})

by the rule : t ↦ t · a (resp. t ↦ t · b) for every local section t of 𝒪_{U}. Again by claim 11.3.20 and (11.3.23) we deduce that φ_{y} : 𝒪_{U,y}[s] → P_{y}^{•} is a quasi-isomorphism for every y ∈ U (and likewise for ψ_{y}); . i.e. φ and ψ are the sought isomorphisms in D^{−}(𝒪_{U}-Mod). □

Proposition 11.3.25. Suppose that ω_{1}^{•} and ω_{2}^{•} are two dualizing complexes for the locally coherent scheme X. Then there exist an invertible 𝒪_{X}-module ℒ and a continuous function σ : |X| → ℤ such that

ω_{2}^{•} ≃ ω_{1}^{•} ⊗_{𝒪_{X}} ℒ [σ] in D^{b}(𝒪_{X}-Mod)_{coh} .

Proof. Denote by 𝒟_{1} and 𝒟_{2} the duality functors associated with ω_{1} and respectively ω_{2}. By assumption, we can find complexes P^{•}, Q^{•} in D^{b}(𝒪_{X}-Mod)_{coh} such that ω_{2} ≃ 𝒟_{1}(P^{•}) and ω_{1} ≃ 𝒟_{2}(Q^{•}), and therefore

𝒟_{2}(ℱ^{•}) ≃ Rℋ om_{𝒪_{X}}^{•} (ℱ^{•}, 𝒟_{1}(P^{•})) ≃ 𝒟_{1}(ℱ^{•} ⊗^{L}_{𝒪_{X}} P^{•})

for every object ℱ^{•} of D^{b}(𝒪_{X}-Mod)_{coh} ([163, Th.10.8.7]).

Claim 11.3.26. Let C^{•} be an object of D^{−}(𝒪_{X}-Mod)_{coh}, such that 𝒟_{1}(C^{•}) is in D^{b}(𝒪_{X}-Mod). Then C^{•} is in D^{b}(𝒪_{X}-Mod)_{coh}.

Proof of the claim. For given m, n ∈ ℕ, the natural maps: τ^{≤−n}C^{•} ⟶^{α} C^{•} ⟶^{β} τ^{≥−m}C^{•} induce morphisms

𝒟_{1}(τ^{≥−m}C^{•}) ⟶^{𝒟_{1}(β)} 𝒟_{1}(C^{•}) ⟶^{𝒟_{1}(α)} 𝒟_{1}(τ^{≤−n}C^{•})

Say that ω^{•} ≃ τ_{≥a}ω^{•} for some integer a ∈ ℕ. Then 𝒟_{1}(τ^{≤−n}C^{•}) lies in D^{≥n+a}(𝒪_{X}-Mod). Since 𝒟_{1}(C^{•}) is bounded, it follows that 𝒟_{1}(β) = 0 for n large enough. Consider now the commutative diagram :

\xymatrix{τ^{≥−n}C^{•} \ar[r]^{β∘α} \ar[d] & τ^{≥−m}C^{•} \ar[d]^{η} \\ 𝒟_{1} ∘ 𝒟_{1}(τ^{≥−n}C^{•}) \ar[r]^{𝒟_{1}∘𝒟_{1}(β∘α)} & 𝒟_{1} ∘ 𝒟_{1}(τ^{≥−m}C^{•})}

Since τ^{≥−m}C^{•} is a bounded complex, η is an isomorphism in D(𝒪_{X}-Mod), so β ∘ α = 0 whenever n is large enough. Clearly this means that C^{•} is bounded, as claimed. ♢

Applying claim 11.3.26 to C^{•} := ℱ^{•} ⊗^{L}_{𝒪_{X}} P^{•} (which is in D^{−}(𝒪_{X}-Mod)_{coh}, since X is coherent) we see that the latter is a bounded complex, and by reversing the roles of ω_{1} and ω_{2} it

follows that the same holds for ℱ^{•} ⊗^{L}_{𝒪_{X}} Q^{•}. We then deduce isomorphisms in D^{b}(𝒪_{X}-Mod)_{coh}

𝒟_{1} ∘ 𝒟_{2}(ℱ^{•}) ≃ ℱ^{•} ⊗^{L}_{𝒪_{X}} P^{•} and 𝒟_{2} ∘ 𝒟_{1}(ℱ^{•}) ≃ ℱ^{•} ⊗^{L}_{𝒪_{X}} Q^{•}.

Letting ℱ^{•} := 𝒪_{X}[0] we derive :

𝒪_{X}[0] ≃ 𝒟_{2} ∘ 𝒟_{1} ∘ 𝒟_{1} ∘ 𝒟_{2}(𝒪_{X}[0]) ≃ 𝒟_{2} ∘ 𝒟_{1}(P^{•}) ≃ P^{•} ⊗^{L}_{𝒪_{X}} Q^{•}.

Then lemma 11.3.19 says that P^{•} ≃ ℰ [τ] for an invertible 𝒪_{X}-module ℰ and a continuous function τ : |X| → ℤ. Consequently : ω_{2} ≃ ℰ^{∨}[−τ] ⊗_{𝒪_{X}} ω_{1}, so the proposition holds with ℒ := ℰ^{∨} and σ := −τ. □

Lemma 11.3.27. Let f : X → Y be a morphism of locally coherent schemes, and ω_{Y}^{•} a dualizing complex on Y. We have :

(i) If f is finite and finitely presented, then f^{!}ω_{Y}^{•} is dualizing on X.

(ii) If Y is coherent and f is a quasi-compact open immersion, f^{∗}ω_{Y}^{•} is dualizing on X.

Proof. (i): Denote by \overline{f} : (X, 𝒪_{X}) → (Y, f_{∗}𝒪_{X}) the morphism of ringed spaces deduced from f. For any object C^{•} of D^{−}(𝒪_{X}-Mod)_{coh} we have natural isomorphisms :

𝒟(C^{•}) := Rℋ om_{𝒪_{X}} (C^{•}, f^{!}ω_{Y}^{•}) →^{∼} Rℋ om_{𝒪_{X}} (C^{•}, \overline{f}^{∗}Rℋ om_{𝒪_{Y}} (f_{∗}𝒪_{X}, ω_{Y}^{•}))

→^{∼} \overline{f}^{∗}Rℋ om_{f_{∗}𝒪_{X}} (f_{∗}C^{•}, Rℋ om_{𝒪_{Y}} (f_{∗}𝒪_{X}, ω_{Y}^{•}))

→^{∼} \overline{f}^{∗}Rℋ om_{𝒪_{Y}} (f_{∗}C^{•}, ω_{Y}^{•}).

Hence, if C^{•} is in D^{b}(𝒪_{X}-Mod)_{coh}, the same holds for 𝒟(C^{•}); especially, this applies to f^{!}ω_{Y} ≃ 𝒟(𝒪_{X}[0]), and we can compute :

\begin{aligned}𝒟 ∘ 𝒟(C^{•}) & →^{∼} \overline{f}^{∗}Rℋ om_{𝒪_{Y}} (Rℋ om_{𝒪_{Y}} (f_{∗}C^{•}, ω_{Y}^{•}), ω_{Y}^{•}) \\ & →^{∼} \overline{f}^{∗}f_{∗}C^{•} →^{∼} C^{•} \end{aligned}

and by inspecting the definitions, one verifies that the resulting natural transformation C^{•} → 𝒟 ∘ 𝒟(C^{•}) is the biduality isomorphism. The claim follows.

(ii): Since 𝒪_{Y} is coherent, the natural map

f^{∗}Rℋ om_{𝒪_{Y}} (𝒢 [0], ω_{Y}^{•}) → Rℋ om_{𝒪_{X}} (f^{∗}𝒢 [0], f^{∗}ω_{Y}^{•})

is an isomorphism in D^{b}(𝒪_{X}-Mod)_{coh}, for every 𝒪_{Y} -module 𝒢. Then the assertion follows easily from lemma 10.3.27(ii) and remark 11.3.17(i). □

Proposition 11.3.28. Let A be a noetherian local ring, κ its residue field, M^{•} a bounded below complex of A-modules of finite type, and set X := Spec A. We have :

(i) The following conditions are equivalent (a) There exists c ∈ ℤ such that RHom_{A}^{•}(κ[0], M^{•}) ≃ κ[c].

(b) The complex of 𝒪_{X}-modules M^{•∼} arising from M^{•} is dualizing on X. (ii) If the equivalent conditions of (i) hold, then M^{•} has finite injective dimension.

Proof. (i): We show first that (a)⇒(b). To this aim, in view of remark 11.3.17(i) and corollary 10.3.2(ii), it suffices to check :

Claim 11.3.29. Assume (a). Then, for every finitely generated A-module N, we have : (i) D^{•}(N) := RHom_{A}^{•}(N[0], M^{•}) ∈ D^{≤−c}(A-Mod) (notation of (7.1)). (ii) The natural map

N[0] → DD^{•}(N) := RHom_{A}^{•}(D^{•}(N), M^{•})

is an isomorphism.

Proof of the claim. We first show the claim for N = κ, in which case (i) holds by assumption. To check (ii), let M^{•} →^{∼} I^{•} be a resolution consisting of a bounded below complex of injective A-modules, so that

D^{•}(κ[0], M^{•}) →^{∼} Hom_{A}^{•}(κ[0], I^{•}) →^{∼} I^{•}[𝔪]

where 𝔪 ⊂ A is the maximal ideal, and I^{k}[𝔪] denotes the submodule of 𝔪-torsion elements in I^{k}, for every k ∈ ℤ. Under these isomorphisms, it is easily seen that the biduality map of (ii) is identified with the unique one

κ → H := Hot_{A-Mod}(I^{•}[𝔪], I^{•})

that sends 1 ∈ κ to the inclusion map j^{•} : I^{•}[𝔪] → I^{•} (details left to the reader). Now, it is clear that any morphism I^{•}[𝔪] → I^{•} in Hot(A-Mod) factors through j^{•}, and on the other hand, our assumption implies that H ≃ κ. Hence, pick a morphism f^{•} : I^{•}[𝔪] → I^{•} representing a generator for the A-module H, and write f^{•} = j^{•} ∘ g^{•} for some endomorphism g^{•} of I^{•}[𝔪]; in other words, f^{•} is the image of j^{•} under the A-linear map

Hot_{A-Mod}(g^{•}, I^{•}) : H → H

so the class of j^{•} cannot vanish in H, and (ii) follows in this case.

Next, we shall argue by induction on d := dim Supp N. If d = 0, then N is an A-module of finite length, in which case we argue by induction on the length l of N. If l = 1, we have N ≃ κ, so the assertions are already known. Suppose l > 1, and that both (i) and (ii) are already known for all A-modules of length < d; we may find an A-submodule N′ ⊂ N such that both N′ and N′′ := N/N′ have length < d. From the inductive assumption for N′ and N′′, and the induced distiguished triangle

D^{•}(N′′) → D^{•}(N) → D^{•}(N′) → D^{•}(N′′)[1]

we deduce that (i) holds for N. Likewise, since (ii) is known for both N′ and N′′, using the 5-lemma we deduce easily that the same holds also for N.

Lastly, suppose that d > 0, and both (i) and (ii) are already known for all A-modules of finite type whose support has dimension < d. Let N′ := Γ_{{𝔪}}N; both (i) and (ii) are already known for N′, so the same dévissage argument as in the foregoing reduces to showing the claim for N/N′, i.e. we may assume that 𝔪 ∉ Ass N. Thus, let t ∈ 𝔪 be any element such that the scalar multiplication map t · 1_{N} is injective, so we have a short exact sequence

0 → N ⟶^{t^{k}} N → N_{k} := N/t^{k}N → 0 for every k > 0.

Notice that dim Supp N_{k} < d : indeed, if 𝔭 is a minimal element of Supp N, then 𝔭 ∈ Ass N ([126, Th.6.5(iii)]), hence t ∉ 𝔭, and therefore 𝔭 ∉ Supp N_{k}. By inductive assumption, both (i) and (ii) hold for N_{k} (for every k > 0), and by considering the induced distinguished triangle

D^{•}(N_{k}) → D^{•}(N) ⟶^{t^{k}} D^{•}(N) → D^{•}(N_{k})[1]

we deduce that scalar multiplication by t is an epimorphism on H^{j}D^{•}(N) whenever j ≥ −c; but the latter is an A-module of finite type, so it must vanish, by Nakayama’ lemma, and we conclude that (i) holds for N. Furthermore, by the same token we get an exact sequence

H^{i}DD^{•}(N) ⟶^{t^{k}} H^{i}DD^{•}(N) → 0 for every i ≠ 0

whence H^{i}DD(N) = 0 for i ≠ 0, again by Nakayama’s lemma. Lastly, for every k > 0 consider the ladder with exact rows :

\xymatrix{0 \ar[r] & N \ar[r]^{t^{k}} \ar[d]_{α} & N \ar[r] \ar[d]^{α} & N_{k} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & H^{0}DD^{•}(N) \ar[r]^{t^{k}} & H^{0}DD^{•}(N) \ar[r] & H^{0}DD^{•}(N_{k}) \ar[r] & 0}

whose right-most vertical arrow is an isomorphism, by inductive assumption. A simple diagram chase then yields

H^{0}DD^{•}(N) = t^{k} · H^{0}DD^{•}(N) + α(N)

so α is surjective, by Nakayama’s lemma. To show the injectivity of α, let x ∈ N be any nonzero element, and choose k > 0 such that x ∉ t^{k}N, so the image of x does not vanish in N_{k}, whence necessarily α(x) ≠ 0, and the claim follows. ♢

(b)⇒(a): Let i : Spec κ → X be the closed immersion; from corollary 10.3.2(ii) we know that i^{!}M^{•∼} is the complex arising from the complex of κ-modules RHom_{A}(κ[0], M^{•}). On the other hand, i^{!}M^{•∼} is dualizing on Spec κ, by virtue of lemma 11.3.27(i); obviously, κ[0]^{∼} is also dualizing on the same scheme, so the assertion follows from proposition 11.3.25.

(ii): We notice more precisely :

Claim 11.3.30. Suppose that the conditions of (i) hold for M^{•}, and moreover that H^{0}M^{•} ≠ 0 and H^{i}M^{•} = 0 for every i < 0. Then c ≤ 0, and the injective dimension of M^{•} equals −c.

Proof of the claim. Under the assumptions of the claim, it is clear that c ≤ 0. Pick a quasiisomorphism M^{•} →^{∼} I^{•}, where I^{•} is an object of C^{[0,−c]}(A-Mod) such that I^{j} is an injective A-module for every j < −c. Notice that I^{0} ≠ 0, since H^{0}M^{•} does not vanish. Let N be any A-module; a standard argument shows that

Ext_{A}^{1}(N, I^{c}) ≃ R^{1−c}Hom_{A}^{•}(N[0], M^{•})

and the latter vanishes if N is finitely generated, by virtue of claim 11.3.29(i). It follows that I^{c} is injective as well (see e.g. [126, Th.B3]), so the injective dimension of M^{•} is ≤ −c. But it is also ≥ −c, due to condition (i.a). ♢

After replacing M^{•} by M^{•}[a] for a suitable a ∈ ℤ, the assumptions of claim 11.3.30 can obviously be fulfilled, whence (ii). □

Let us recall the following :

Definition 11.3.31. Let X be a noetherian scheme.

(i) We say that X is Gorenstein, if 𝒪_{X}[0] is dualizing on X.

(ii) If X = Spec A is affine and Gorenstein, we also say that A is a Gorenstein ring.

Remark 11.3.32. (i) In view of example 11.3.18, every regular noetherian ring is Gorenstein.

(ii) Let A be any local noetherian ring, and κ the residue field of A. In light of proposition 11.3.28(i) (and of corollary 10.3.2(ii)), we see that A is Gorenstein if and only if there exists c ∈ ℕ such that RHom_{A}^{•}(κ[0], A[0]) ≃ κ[c].

11.3.33. Let A → B be a flat homomorphism of local noetherian rings, κ_{A} (resp. κ_{B}) the residue field of A (resp. of B), and K^{•} a bounded complex of A-modules of finite type. Set X := Spec A, Y := Spec B, denote by f : Y → X the resulting morphism of local schemes, and let K^{•∼} be the complex of 𝒪_{X}-modules arising from K^{•}. We have :

Proposition 11.3.34. In the situation of (11.3.33), the following conditions are equivalent : (a) f^{∗}K^{•∼} is dualizing on Y.

(b) K^{•∼} is dualizing on X and B ⊗_{A} κ_{A} is a Gorenstein ring.

Proof. Set C := B⊗_{A}κ_{A}, F := Spec C and Z := Spec κ_{B}, denote i_{1} : Z → F and i_{2} : F → Y the closed immersions, and let i := i_{2} ∘ i_{1}. In light of proposition 11.3.28(i) (and of corollary 10.3.2(ii)) we see that (a) holds if and only if there exists c ∈ ℤ such that i^{!}(f^{∗}K^{•∼}) ≃ 𝒪_{Z}[c]. However, recall that i^{!} ≃ i_{1}^{!} ∘ i_{2}^{!} (proposition 11.1.7(i)); invoking again proposition 11.3.28(i) we deduce that (a) holds if and only if i_{2}^{!}(f^{∗}K^{•∼}) is dualizing on F. The latter is the complex of 𝒪_{F} -modules arising from

(11.3.35) RHom_{B}^{•}(C[0], B ⊗_{A} K^{•}) ≃ B ⊗_{A} L^{•} where L^{•} := RHom_{A}^{•}(κ_{A}[0], K^{•})

(proposition 10.3.3(ii) and corollary 10.3.2(ii)). Hence, suppose that (b) holds; from proposition 11.3.28(i) we conclude that i_{2}^{!}(f^{∗}K^{•∼}) ≃ 𝒪_{F} [c] for some c ∈ ℤ; the latter is dualizing on Z by assumption, whence (a). Conversely, suppose that (a) holds; from (11.3.35) we see that B⊗_{A}L^{•} is a bounded complex of C-modules whose cohomology is a free C-module in every degree. We recall the following :

Claim 11.3.36. Let R be any ring, M^{•} a bounded complex of R-modules, such that H^{i}M^{•} is a projective R-module, for every i ∈ ℤ. Then we have an isomorphism

M^{•} →^{∼} ⨁_{i∈ℤ}(H^{i}M^{•})[−i] in D(R-Mod).

Proof of the claim. We argue by induction on the cardinality c_{M} of S_{M} := {i ∈ ℤ | H^{i}M^{•} ≠ 0}. If c_{M} = 0, the assertion is obvious. Suppose that c_{M} > 0, and that the assertion is known for every complex P^{•} as in the claim, with c_{P} < c_{M}. Let a ∈ ℤ be the largest element of S_{M}; after replacing M^{•} by its truncation τ^{≤a}M^{•}, we may assume that M^{i} = 0 for every i > a. Then H^{a}M^{•} ≃ Coker (d^{a−1} : M^{a−1} → M^{a}), and since H^{a}M^{•} is a projective R-module, we may find an R-linear isomorphism M →^{∼} (H^{a}M^{•}) ⊕ Im d^{a−1}. Denote by P^{•} the complex such that P^{i} := M^{i} for every integer i < a, and such that P^{a} := Im d^{a−1}, with differentials induced by those of M^{•}; then M^{•} ≃ (H^{a}M^{•})[−a] ⊕ P^{•} in D(R-Mod), and c_{P} < c_{M}, so the assertion is known for P^{•}, and the claim follows. ♢

In light of claim 11.3.36 and remark 11.3.17(ii), we easily see that B ⊗_{A} L^{•} ≃ C[a] in D^{b}(C-Mod) for some a ∈ ℤ (details left to the reader), and then clearly L^{•} ≃ k_{A}[c] in D^{b}(A-Mod). Taking into account proposition 11.3.28(i), assertion (b) follows. □

Proposition 11.3.37. Let X be a noetherian scheme, 𝒦 ^{•} an object of D^{b}(𝒪_{X}-Mod)_{coh}. The following conditions are equivalent :

(a) 𝒦 ^{•} is dualizing on X.

(b) 𝒦 ^{•}(x) is dualizing on X(x), for every x ∈ X (notation of definition 4.9.17(iii)).

Proof. (b)⇒(a): We notice the following :

Claim 11.3.38. Let X be a noetherian scheme, 𝒦 ^{•} an object of D^{b}(𝒪_{X}-Mod)_{coh}, and x ∈ X any point. Suppose that the complex of 𝒪_{X(x)}-modules 𝒦 ^{•}(x) has finite injective dimension. Then, for every ℱ^{•} ∈ D^{b}(𝒪_{X,x}-Mod)_{coh} there exists an open neighborhood U of x in X such that Rℋ om_{𝒪_{X}}^{•} (ℱ^{•}, 𝒦 ^{•})_{|U} lies in D^{b}(𝒪_{U}-Mod)_{coh}.

Proof of the claim. The question is local on X, so we may assume that X is affine, say X = Spec A for a noetherian ring A; also, by theorem 10.3.25 we may assume that 𝒦 ^{•} arises from a bounded complex K^{•} of A-modules of finite type (details left to the reader). Then, remark 11.3.17(i) and corollary 10.3.2(ii) reduce to checking that for every A-module M there exists an open neighborhood U of x in X, such that the A-modules T^{i}(M) := H^{i}RHom_{A}^{•}(M[0], K^{•}) have support in X \ U, for every sufficiently large i ∈ ℤ.

We shall argue by induction on the dimension d of the support of M; if d = 0, there is nothing to show, hence suppose that d > 0, and that the assertion is already known for every Amodule whose support has dimension strictly less than d. By [126, 6.4] we may further reduce to the case where M = A/𝔭, where 𝔭 ⊂ A is a prime ideal. Then, pick a finite system of generators t_{1}, …, t_{n} ∈ A for 𝔭, and set L_{•} := K_{•}(t_{1}, …, t_{n}) (notation of remark 7.8.1(ii)). Now, H_{0}L_{•} ≃ A/𝔭, and H_{i}L_{•} = 0 for every i > n; moreover, lemma 7.8.2(ii) says that H_{i}L_{•} is an A/𝔭-module of finite type, for every i = 0, …, n, and vanishes for i ∈ ℤ outside this range; therefore, we may find f ∈ A \ 𝔭 such that (H_{i}L_{•})_{f} is a free (A/𝔭)_{f}-module of some finite rank b(i) ∈ ℕ, for every i ∈ ℤ ([126, Th.4.10(ii)]), and there exists an integer m ≤ n such that b(m) ≠ 0 and b(i) = 0 for every i > m. By choosing an injective resolution of K^{•}, we get a spectral sequence

E_{2}^{pq} := R^{p}Hom_{A}^{•}((H_{q}L_{•})[0], K^{•}) ⇒ R^{p+q}Hom_{A}^{•}(L_{•}, K^{•})

with differentials d_{r}^{pq} : E_{r}^{pq} → E_{r}^{p−r+1,q+r} for every p, q ∈ ℤ and every r ≥ 2. Taking into account corollary 10.3.2(ii), we see that

(11.3.39)  (E_{2}^{pq})_{f} ≃ (E_{2}^{p,0})_{f}^{⊕b(q)}  for every p ∈ ℤ.

By the same token, we also get a natural isomorphism

E_{2}^{p0}(x) := E_{2}^{p0} ⊗_{A} 𝒪_{X,x} ≃ R^{p}Hom_{𝒪_{X(x)}}^{•} ((A/𝔭)^{∼}[0](x), 𝒦 ^{•}(x)) for every p ∈ ℤ.

Since 𝒦 ^{•}(x) has finite injective dimension, it follows that there exists an integer r such that E_{2}^{p,0}(x) = 0 whenever p > r. Since E_{2}^{pq} is an A-module of finite type for every p, q ∈ ℤ, we deduce that there exists an open neighborhood U′ of x in X, such that, for every p = r + 1, …, r + n, the support of E_{2}^{p,0} is contained in X \ U′ ([126, Th.4.10(i)]). In this case, the support of (E_{2}^{pq})_{f} is also contained in X \ U′, for every (p, q) ∈ ℤ^{⊕2} with p = r + 1, …, r + n.

Now, let 𝔮 ∈ U′ ∩ Spec A_{f} be any prime ideal, and suppose that (E_{2}^{p,0})_{𝔮} ≠ 0 for some integer p > r + n. From (11.3.39), we deduce that (E_{2}^{p,m})_{𝔮} ≠ 0 as well, and it is easily seen that (E_{∞}^{p,m})_{𝔮} = (E_{2}^{p,m})_{𝔮}. Especially

(11.3.40)  R^{p+m}Hom_{A}^{•}(L_{•}, K^{•}) ≠ 0.

However, L_{•} is also a complex of free A-modules, therefore

RHom_{A}^{•}(L_{•}, K^{•}) ≃ Hom_{A}^{•}(L_{•}, K^{•}) in D(A-Mod).

Thus, if we represent K^{•} via an object of C^{[a,b]}(A-Mod) (for some a, b ∈ ℤ with a ≤ b : notation of (7.1)), then RHom_{A}^{•}(L_{•}, K^{•}) is represented by an object of C^{[a,b+n]}(A-Mod), and consequently, R^{i}Hom_{A}^{•}(L_{•}, K^{•}) = 0 for i > b + n. Taking into account (11.3.40), we conclude that p ≤ p + m ≤ b + n, so that

(11.3.41)  T^{i}(M)_{𝔮} = 0  whenever i > n + max(b, r) and 𝔮 ∈ U′ ∩ Spec A_{f}.

On the other hand, from the short exact of A-modules

0 → M ⟶^{f} M → M/fM → 0

we see that T^{i}(M)/fT^{i}(M) ⊂ T^{i+1}(M/fM) for every i ∈ ℤ; since the support of M/fM has dimension < d, the inductive assumption tells us that there exist N ∈ ℕ and an open neighborhood U′′ of x in X such that the support of T^{i+1}(M/fM) lies in X \ U′′, for every i > N. From this, Nakayama’s lemma implies that

T^{i}(M)_{𝔮} = 0 for every 𝔮 ∈ Spec A/fA and every i > N.

Combining with (11.3.41), we conclude that U := U′ ∩ U′′ will do. ♢

Since X is quasi-compact, assumption (b), claim 11.3.38 and proposition 11.3.28(ii) imply that the rule : ℱ^{•} ↦ 𝒟(ℱ^{•}) := Rℋ om_{𝒪_{X}}^{•} (ℱ^{•}, 𝒦 ^{•}) takes objects of D^{b}(𝒪_{X}-Mod)_{coh} to objects of the same category. Then, corollary 10.3.2(ii) and assumption (b) imply that the natural map ℱ [0](x) → 𝒟 ∘ 𝒟(ℱ [0])(x) is an isomorphism in D^{b}(𝒪_{X(x)}-Mod) for every x ∈ X and every coherent 𝒪_{X}-module ℱ, whence (a).

(a)⇒(b): Let x ∈ X be any point, and ℳ any coherent 𝒪_{X(x)}-module; we may extend ℳ to a coherent 𝒪_{U}-module on some affine open neighborhood U of x in X, and then the latter can be further extended to a coherent 𝒪_{X}-module ℱ (lemma 10.3.27(ii)). According to corollary 10.3.2(ii), the natural map 𝒟(ℱ [0])(x) → Rℋ om_{𝒪_{X(x)}}^{•} (ℳ [0], 𝒦 ^{•}(x)) is an isomorphism in D^{b}(𝒪_{X(x)}-Mod), whence (b). □

Corollary 11.3.42. Let f : Y → X be a morphism of finite type between affine noetherian schemes, and ω^{•} a dualizing complex on X. Then f^{!}ω^{•} is a dualizing complex on Y.

Proof. We may factor f as the composition of a closed immersion g : Y → Z followed by a smooth and affine morphism h : Z → X. In light of lemmata 11.3.27(i) and 11.1.16(i), it then suffices to check that h^{!}ω^{•} is dualizing on Z, so we may assume from start that f is smooth, in which case we are reduced to showing that f^{∗}ω^{•} is dualizing on Y. Let y ∈ Y be any point, set x := f(y), and denote by f_{y} : Y (y) → X(x) the morphism induced by f; taking into account proposition 11.3.37, we are further reduced to checking that f_{y}^{∗}(ω_{|X(x)}^{•}) is dualizing on Y (y). However, the latter assertion follows easily from proposition 11.3.34, remark 11.3.32(i) and [126, Th.28.7]. □

Corollary 11.3.43. Let A be a noetherian ring, x ∈ X := Spec A any point, and K^{•} any object of D^{b}(𝒪_{X}-Mod)_{coh}. Suppose that the following holds :

(a) Every irreducible reduced closed subscheme Z ⊂ X containing x admits a dense open subset which is Gorenstein (see definition 11.3.31(i)).

(b) K^{•}(x) is dualizing on X(x). Then there exists an open neighborhood U of x in X, such that K_{|U}^{•} is dualizing on U.

Proof. We begin with the following :

Claim 11.3.44. For every reduced and irreducible closed subscheme Z of X containing x, there exists a dense open subset U_{Z} of Z such that K^{•}(z) is dualizing on X(z), for every z ∈ U_{Z}.

Proof of the claim. Let i_{Z} : Z → X be the closed immersion, and set i_{Z(x)} := i_{Z} ×_{X} X(x) : Z(x) → X(x); in view of proposition 10.3.3(ii), we easily see that there is a natural isomorphism

i_{Z(x)}^{!}K^{•}(x) →^{∼} (i_{Z}^{!}K^{•})(x) in D(𝒪_{Z(x)}-Mod) .

Hence, due to (b) and lemma 11.3.27, (i_{Z}^{!}K^{•})(x) is dualizing on Z(x). Denote by η_{Z} the generic point of Z; it follows that (i_{Z}^{!}K^{•})(η_{Z}) is dualizing on Z(η_{Z}) (proposition 11.3.37), and therefore it is isomorphic to 𝒪_{Z(η_{Z})}[c] in D(𝒪_{Z(η_{Z})}-Mod), due to (a). On the other hand, claim 11.3.38 implies that there exists an open neighborhood U′_{Z} of x in Z, such that (i_{Z}^{!}K^{•})_{|U′_{Z}} lies in D^{b}(𝒪_{U′_{Z}}-Mod)_{coh}. It then follows easily that there exists a smaller open subset U_{Z} ⊂ U′_{Z} of Z, such that (i_{Z}^{!}K^{•})_{|U_{Z}} ≃ 𝒪_{U_{Z}}[c] in D(𝒪_{U′_{E}}-Mod) (details left to the reader). In view of (a), we may then further shrink U_{Z}, and assume that (i_{Z}^{!}K^{•})_{|U_{Z}} is dualizing on U_{Z}. Now, let y ∈ U_{Z} be any point, denote by i_{1} : Spec κ(y) → Z(y) and i_{2} : Z(y) → X(y) the closed immersions, and set i_{y} := i_{2} ∘ i_{1}. By proposition 11.3.37, the complex i_{2}^{!}K^{•}(y) ≃ (i_{Z}^{!}K^{•})(y) is dualizing on Z(y), so i_{y}^{!}K^{•}(y) ≃ i_{1}^{!}(i_{2}^{!}K^{•}(y)) is dualizing on Spec κ(y) (lemma 11.3.27(i)), and consequently K^{•}(y) is dualizing on X(y) (proposition 11.3.28(i)). The claim follows. ♢

Denote by 𝒵 the set of all reduced and irreducible closed subschemes of X containing x, and for every Z ∈ 𝒵, let U_{Z} be the largest open subset of Z such that K^{•}(y) is dualizing on X(y) for every y ∈ U_{Z}. The subset U := ⋃_{Z∈𝒵} U_{Z} is ind-constructible in X, and it is clearly generizing (proposition 11.3.37). Thus, U is open in X ([63, Ch.IV, Th.1.10.1]), and to conclude the proof, it suffices – again, by virtue of proposition 11.3.37 – to check that x ∈ U. Suppose that the latter fails, and let Z ⊂ X be the closure of {x} in X, endowed with its reduced subscheme structure; by assumption, Z ∩ U = ∅; on the other hand, Z ∈ 𝒵, so that U_{Z} ≠ ∅, by claim 11.3.44. The contradiction proves the assertion. □

Remark 11.3.45. Let A be a noetherian ring, and set X := Spec A.

(i) If X admits a dualizing complex, then every irreducible reduced closed subscheme Z ⊂ X admits a dense open subset U which is Gorenstein. Indeed, let η_{Z} be the generic point of Z; we know that Z admits a dualizing complex ω_{Z}^{•} (lemma 11.3.27(i)), hence ω_{Z}^{•}(η_{Z}) is dualizing on Z(η_{Z}) (proposition 11.3.37). But then there exists an isomorphism 𝒪_{Z(η_{Z})}[c] →^{∼} ω^{•}(η_{Z}) in D^{b}(𝒪_{Z(η_{Z})}-Mod) for some c ∈ ℤ (proposition 11.3.25), and any such isomorphism extends to an isomorphism 𝒪_{U}[c] →^{∼} (ω_{Z}^{•})_{|U} on some open subset U ⊂ Z; since (ω_{Z}^{•})_{|U} is dualizing on U (lemma 11.3.27(ii)), the assertion follows.

(ii) The observation in (i) shows that the converse of corollary 11.3.43 holds as well : if x ∈ X is any point, and U any open neighborhood of x in X such that U admits a dualizing complex, then conditions (a) and (b) of corollary 11.3.43holds for x.

(iii) If A is local and X admits a dualizing complex, then the formal fibres of A are Gorenstein, i.e., if we denote A^{∧} the completion of A and set X^{∧} := Spec A^{∧}, then X^{∧} ×_{X} Spec κ(x) is a Gorenstein scheme, for every x ∈ X. Indeed, notice that A^{∧} ⊗_{A} A/𝔭 is the completion of A/𝔭, for every 𝔭 ∈ Spec A, and Spec A/𝔭 admits a dualizing complex, if X does (lemma 11.3.27(i)); hence we may replace A by its image in κ(x) and assume from start that A is a local noetherian domain, and x the generic point of X. Let ω^{•} be a dualizing complex on X, and f : X^{∧} → X the natural morphism; by proposition 11.3.34, the complex f^{∗}ω^{•} is dualizing on X^{∧}. On the other hand, by (i) there exists a non-empty open subset U ⊂ X which is Gorenstein, and therefore ω_{|U}^{•} ≃ 𝒪_{U}[c] for some c ∈ ℤ. Due to lemma 11.3.27(ii), it follows that f^{−1}U is Gorenstein as well, and then the same holds for f^{−1}(x), by proposition 11.3.37.

(iv) If A is local and complete, then X admits a dualizing complex. Indeed, in this case A is a quotient of a regular local ring ([126, Th.29.4(ii)]), so the assertion follows from lemma 11.3.27(i) and example 11.3.18.

Corollary 11.3.46. Let X be any noetherian scheme, ω^{•} a dualizing complex on X. We have : (i) For every x ∈ X there exists a unique c ∈ ℤ such that

J(x) := R^{c}Γ_{{x}}ω_{|X(x)}^{•} ≠ 0.

(ii) For every x ∈ X, the A-module J(x) is the injective hull of the residue field κ(x).

Proof. In view of proposition 11.3.37, we may assume that X is local and x is its closed point; say that X = Spec A for some local noetherian ring A, in which case ω^{•} is the 𝒪_{X}-module arising from a bounded complex M^{•} of A-modules of finite type. We notice :

Claim 11.3.47. Let 𝔪 ⊂ A be the maximal ideal. There exists a unique c ∈ ℤ such that

(i) T^{i}(N) := R^{i}Hom_{A}^{•}(N, M^{•}) = 0 for every A-module N of finite type supported at {x} and every i ≠ c.

(ii) The map T^{c}(A/𝔪^{n}) → T^{c}(A/𝔪^{n+1}) induced by the projection A/𝔪^{n+1} → A/𝔪^{n} is injective, for every n ∈ ℕ.

Proof of the claim. (i): By proposition 11.3.28(i), we know already that there exists c ∈ ℕ such that T^{i}(A/𝔪) = 0 if and only if i ≠ c. We must therefore check that the claim holds for this value of c. Now, let N be any A-module of finite type supported at {x}; we may find a finite filtration N_{0} := 0 ⊂ N_{1} ⊂ ⋯ ⊂ N_{k} consisting of A-submodules, such that N_{j+1}/N_{j} ≃ A/𝔪 for every i = 0, …, k − 1. We deduce short exact sequences

T^{i}(A/𝔪) → T^{i}(N_{j+1}) → T^{i}(N_{j}) for every i ∈ ℕ and every j = 0, …, k − 1

whence the contention, by a simple induction on j.

(ii) is similar : arguing as in the foregoing, we get an exact sequence

T^{c−1}(𝔪^{n+1}/𝔪^{n}) → T^{c}(A/𝔪^{n}) → T^{c}(A/𝔪^{n+1})

whence the assertion. ♢ On the other hand, corollary 10.4.35yields a natural identification :

(11.3.48)  J(x) →^{∼} colim_{n∈ℕ} T^{c}(A/𝔪^{n})

and then assertion (i) follows proposition 11.3.28(i) and claim 11.3.47.

(ii): Claim 11.3.47also implies that T^{c} is an exact functor on the full subcategory of A-Mod whose objects are the A-modules supported at {x}. By proposition 7.11.31, remark 7.11.32(i,ii) and (11.3.48) we deduce already that J(x) is an injective A-module. Next, let κ(x)^{∼} be the quasi-coherent 𝒪_{X}-module arising from κ(x); we have natural identifications

Hom_{A}(κ(x), J(x))[c] →^{∼} RHom_{𝒪_{X}}^{•} (κ(x)^{∼}[0], R\underline{Γ}_{\{x\}}ω^{•})  (by corollary 10.3.2(i))
→^{∼} RΓ_{{x}} ∘ Rℋ om_{𝒪_{X}}^{•} (κ(x)^{∼}[0], ω^{•})  (by lemma 10.4.13(iii))
→^{∼} RHom_{𝒪_{X}}^{•} (κ(x)^{∼}[0], ω^{•})  (as Supp κ(x) = {x})
→^{∼} RHom_{A}^{•}(κ(x)[0], M^{•})  (by corollary 10.3.2(i))
→^{∼} κ(x)[c]  (by proposition 11.3.28(i)).

Then the assertion follows immediately from theorem 7.11.22. □

11.3.49. Let X be a noetherian scheme that admits a dualizing complex ω^{•}. According to corollary 11.3.46, we get a well defined function

c_{X} : |X| → ℤ

by assigning to each x ∈ X the unique c_{X}(x) ∈ ℤ such that R^{c_{X}(x)}Γ_{{x}}ω_{|X(x)}^{•} does not vanish. With this notation, we may state :

Lemma 11.3.50. In the situation of (11.3.49), let x, y ∈ X be any two points, such that x is an immediate specialization of y. Then

c_{X}(x) = c_{X}(y) + 1.

Proof. By proposition 11.3.37 we may replace X by X(x), and ω by ω_{|X(x)}, and assume from start that X is local, and x is its closed point. Let Z ⊂ X be the topological closure of {y}, endowed with its reduced closed subscheme structure. If we denote i : Z → X the corresponding closed immersion, then i^{!}ω is dualizing on Z (lemma 11.3.27(i)). Fix z ∈ Z, let i_{z} : Z(z) → X(z) be the induced closed immersion, and set c := c_{X}(z); we get natural isomorphisms :

RΓ_{{z}}(i^{!}ω^{•})_{|Z(z)} →^{∼} RΓ_{{z}} ∘ Rℋ om_{𝒪_{X(z)}}^{•} (i_{z,∗}𝒪_{Z(y)}, ω_{|X(z)}^{•})  (by proposition 10.3.3(ii))
→^{∼} RHom_{𝒪_{X(z)}}^{•} (i_{z,∗}𝒪_{Z(y)}, R\underline{Γ}_{\{z\}}ω_{|X(z)}^{•})  (by lemma 10.4.13(iii))
→^{∼} Hom_{𝒪_{X(z)}}^{•} (i_{z,∗}𝒪_{Z(y)}, R^{c}\underline{Γ}_{\{z\}}ω_{|X(z)}^{•})[−c]  (by corollary 11.3.46)

which shows that

c_{Z}(z) = c_{X}(z)

provided c_{Z} is defined via i^{!}ω^{•}. Hence, we may replace X by Z, and assume additionally that X = Spec A, where A is a one-dimensional local noetherian domain, and y is the generic point of X. In this case, ω^{•} is the complex of 𝒪_{X}-module arising from a bounded complex M^{•} of A-modules of finite type, and clearly

R^{c_{X}(y)}Γ_{{y}}ω_{|X(y)}^{•} →^{∼} H^{c_{X}(y)}(ω_{y}^{•}) ≃ H^{c_{X}(y)}(M^{•}) ⊗_{A} K

where K is the field of fractions of A. To compute c_{X}(x), we apply proposition 10.4.32(iii); to this aim, let f be any non-zero element of the maximal ideal of A; we get

R^{i}Γ_{{x}}ω^{•} →^{∼} colim_{n∈ℕ} H^{i}(f^{n}, M^{•}) for every i ∈ ℤ

(notation of remark 7.8.1(iii)). By inspecting the definitions, we find a ladder with exact rows

[figure: H^{i}M^{•}; f^{n}; H^{i}M^{•}; f; H^{i+1}(f^{n}, M^{•}); H^{i}M^{•}; f^{n+1}; H^{i}M^{•}; H^{i+1}(f^{n+1}, M^{•})]  for every i ∈ ℤ and n ∈ ℕ

whence – after taking colimits – an exact sequence

H^{i}M^{•} ⟶^{α_{i}} (H^{i}M^{•}) ⊗_{A} K ⟶^{β_{i}} R^{i+1}Γ_{{x}}M^{•} for every i ∈ ℤ

where α_{i} is induced by the inclusion map A → K. However, if (H^{i}M^{•}) ⊗_{A} K ≠ 0, clearly α_{i} cannot be surjective, so Im β_{i} ≠ 0, and the lemma follows. □

Example 11.3.51. Let K be any field, and A a K-algebra of finite type. Set S := Spec K, X := Spec A, and denote by f : X → S the structure morphism. Since 𝒪_{S}[0] is dualizing on S, the complex ω^{•} := f^{!}𝒪_{S}[0] is dualizing on X (corollary 11.3.42). In this case, the function c_{X} associated with ω_{X} as in (11.3.49) can be determined explicitly as follows. First, we may find a closed immersion i : X → Y := 𝔸_{S}^{n} for some n, and if p : Y → S is the structure morphism, the proof of lemma 11.3.50 shows that c_{X} = c_{Y} ∘ i, where c_{Y} is the function of the same type associated with g^{!}𝒪_{S}[0]. Hence, we may assume that X = 𝔸_{S}^{n}, in which case ω^{•} ≃ 𝒪_{X}[n]. Then X is regular ([126, Th.28.7]), so for every x ∈ X we may compute :

c_{X}(x) = depth_{{x}}𝒪_{X(x)} − n = n − tr. deg(κ(x)/K) − n = −tr. deg(κ(x)/K).

Theorem 11.3.52. Let A be a noetherian ring, and suppose that X := Spec A admits a dualizing complex. Then A is universally catenary.

Proof. In light of corollary 11.3.42, it suffices to show that A is catenary. But this follows easily from lemma 11.3.50(details left to the reader). □

11.3.53. In the situation of (11.3.49), lemma 11.3.50 shows that c_{X} is a weak codimension function, with the terminology of (11.2.19). Then corollary 11.3.46(i) means precisely that ω^{•} is a c_{X}-Cohen-Macaulay complex of 𝒪_{X}-modules on X (see definition 11.2.24(i)). By theorem 11.2.27, we may then find a Cousin complex of 𝒪_{X}-modules ℛ_{ω}^{•}, unique up to isomorphism of complexes, such that ω^{•} is isomorphic to ℛ_{ω}^{•} in D^{+}(𝒪_{X}-Mod). We call ℛ_{ω}^{•} the residual complex arising from ω^{•}. By corollary 11.3.46(ii) we see that

ℛ_{ω}^{p} = ⨁_{c_{X}(x)=p} E(x)^{∼} for every p ∈ ℤ

where E(x) is an injective hull of the A-module κ(x), for every x ∈ X (notation of (10.3)).

11.3.54. For every coherent scheme X, consider the category

Dual_{X}

whose objects are the pairs (U, ω_{U}^{•}), where U ⊂ X is an open subset, and ω_{U}^{•} is a dualizing complex on X. The morphisms (U, ω_{U}^{•}) → (U′, ω_{U′}^{•}) are the pairs (j, β), where j : U → U′ is an inclusion map of open subsets of X, and β : j^{∗}ω_{U′}^{•} →^{∼} ω_{U}^{•} is an isomorphism in D^{b}(𝒪_{U}-Mod). Let X_{Zar} denote the full subcategory of Sch/X whose objects are the (Zariski) open subsets of X; it follows easily from lemma 11.3.27(ii), that the forgetful functor

(11.3.55)  Dual_{X} → X_{Zar}  (U, ω_{U}^{•}) ↦ U

is a fibration (see definition 3.1.2(ii)) With this notation, we have :

Proposition 11.3.56. Every descent datum for the fibration (11.3.55) is effective.

Proof. Let ((U_{i}, ω_{i}^{•}); β_{ij} | i, j = 1, …, n) be a descent datum for the fibration (11.3.55); this means that each (U_{i}, ω_{i}^{•}) is an object of Dual_{X}, and if let U_{ij} := U_{i} ∩ U_{j} for every i, j ≤ n, then

β_{ij} : ω_{i|U_{ij}}^{•} →^{∼} ω_{j|U_{ij}}^{•}

are isomorphisms in D^{b}(𝒪_{U_{ij}}-Mod), fulfilling a suitable cocycle condition (see (3.5.22)).

We shall show, by induction on k = 1, …, n, that there exists a dualizing complex ω_{X_{k}}^{•} on X_{k} := U_{1} ∪ ⋯ ∪ U_{k}, such that the descent datum ((U_{i}, ω_{i}^{•}); β_{ij} | i, j = 1, …, k) is isomorphic to the descent datum relative to the family (U_{i} | i = 1, …, k) determined by (X_{k}, ω_{X_{k}}^{•}).

For k = 1, there is nothing to prove. Next, suppose that k > 1, and ω_{X_{k−1}}^{•} with the sought properties has already been constructed. If k − 1 = n, we are done; otherwise, let V_{k} := X_{k} ∩ U_{k+1}, set 𝔘_{k} := (U_{i,k+1} | i = 1, …, k), and C^{•} := Rℋ om_{𝒪_{V_{k}}}^{•} (ω_{X_{k}|V_{k}}^{•}, ω_{k+1|V_{k}}^{•}). The system of morphisms (β_{i,k+1} | i = 1, …, k) determines a class c_{k} in the Čech cohomology group H^{0}(𝔘_{k}, C^{•}). However, we have a spectral sequence (see lemma 10.3.15) :

E_{2}^{pq} := H^{p}(𝔘_{k}, H^{q}C^{•}) ⇒ H^{p+q}RHom_{𝒪_{V_{k}}}^{•} (ω_{X_{k}|V_{k}}^{•}, ω_{k+1|V_{k}}^{•}).

Claim 11.3.57. E_{2}^{pq} = 0 for every q < 0.

Proof of the claim. More precisely, we check that H^{q}C^{•} = 0 for every q < 0. Indeed, by lemma 11.3.27(ii), that both ω_{X_{k}|V_{k}}^{•} and ω_{k+1|V_{k}}^{•} are dualizing on V_{k}. Moreover, on U_{i,k+1} they restrict to isomorphic objects of D^{b}(𝒪_{U_{i,k+1}}-Mod), for every i = 1, …, k. It follows easily from proposition 11.3.25that there exists an invertible 𝒪_{V_{k}}-module ℒ and an isomorphism

ω_{X_{k}|V_{k}}^{•} →^{∼} ω_{k+1|V_{k}}^{•} ⊗_{𝒪_{V_{k}}} ℒ in D^{b}(𝒪_{V_{k}}-Mod) .

Therefore, the biduality map induces an isomorphism

Rℋ om_{𝒪_{V_{k}}}^{•} (ω_{X_{k}|V_{k}}^{•}, ω_{k+1|V_{k}}^{•}) →^{∼} ℒ [0]

whence the claim. ♢

Claim 11.3.57 implies that c_{k} corresponds to an element of H^{0}RHom_{𝒪_{V_{k}}}^{•} (ω_{X_{k}|V_{k}}^{•}, ω_{k+1|V_{k}}^{•}), and by construction, it is clear that this global section is an isomorphism c_{k} : ω_{X_{k}|V_{k}}^{•} →^{∼} ω_{k+1|V_{k}}^{•} in D^{b}(𝒪_{V_{k}}-Mod). By definition, c_{k} is represented by a diagram of quasi-isomorphisms :

(11.3.58)  ω_{X_{k}|V_{k}}^{•} ← T^{•} → ω_{k+1|V_{k}}^{•}  in C(𝒪_{V_{k}}-Mod)

for some complex T^{•}. Let T_{!} be the 𝒪_{X_{k+1}}-module obtained as extension by zero of T, and define likewise (ω_{X_{k}}^{•})_{!} and (ω_{k+1}^{•})_{!}. We let ω_{X_{k+1}}^{•} be the cone of the map of complexes

T_{!} → (ω_{X_{k}}^{•})_{!} ⊕ (ω_{k+1}^{•})_{!}

deduced from (11.3.58). An easy inspection shows that this complex is dualizing on X_{k+1}, and it fulfills the stated condition. □

Remark 11.3.59. The proof of proposition 11.3.56 is a special case of a general technique for “glueing perverse sheaves” developed in [18].

In the study of duality for derived categories of modules, the role of reflexive modules is taken by the more general class of Cohen-Macaulay modules. There is a version of this theory for noetherian regular schemes, and a relative variant, for smooth morphisms. Let us begin by recalling the following :

Definition 11.3.60. Let (A, 𝔪_{A}) be a local ring, M an A-module.

(i) Let ℳ be the set of all finitely generated A-submodules M. The dimension of M is

dim_{A} M := sup(dim Supp M′ | M′ ∈ ℳ).

(ii) Suppose that the open subset Spec A \ {𝔪_{A}} is quasi-compact, and depth_{A} M < +∞. If dim_{A} M = depth_{A} M, we say that M is a Cohen-Macaulay A-module. The category

A-CM

is the full subcategory of A-Mod consisting of all finitely presented Cohen-Macaulay A-modules. For every n ∈ ℕ, we let A-CM_{n} be the full subcategory of A-CM whose objects are the Cohen-Macaulay A-modules of dimension n.

(iii) If A is a Cohen-Macaulay A-module, we say that A is a Cohen-Macaulay local ring. (iv) Let X be a scheme, ℱ a quasi-coherent 𝒪_{X}-module. We say that ℱ is a Cohen-

Macaulay 𝒪_{X}-module, if ℱ_{x} is a Cohen-Macaulay 𝒪_{X,x}-module, for every x ∈ X. We say that X is a Cohen-Macaulay scheme, if 𝒪_{X} is a Cohen-Macaulay 𝒪_{X}-module.

(v) Let B be another local ring, φ : A → B a local ring homomorphism. The category

φ-CM

of φ-Cohen-Macaulay modules is the full subcategory of B-Mod whose objects are all the finitely presented B-modules N that are φ-flat, and such that N/𝔪_{A}N is a Cohen-Macaulay B-module. For every n ∈ ℕ, we let φ-CM_{n} be the full subcategory of φ-CM whose objects are the φ-Cohen-Macaulay modules N with dim_{B} N/𝔪_{A}N = n.

(vi) Let f : X → Y be a locally finitely presented morphism of schemes, ℱ a quasicoherent 𝒪_{X}-module, x ∈ X any point, and set y := f(x). We say that ℱ is f-Cohen-Macaulay at the point x, if ℱ_{x} is a f_{x}^{♮}-Cohen-Macaulay module. We say that f is Cohen-Macaulay at the point x, if 𝒪_{X} is f-Cohen-Macaulay at the point x (cp. [64, Ch.IV, Déf.6.8.1]).

(vii) Let f and ℱ be as in (vi). We say that ℱ is f-Cohen-Macaulay (resp. that f is Cohen-Macaulay) if ℱ (resp. 𝒪_{X}) is f-Cohen-Macaulay at every point x ∈ X.

(viii) Let f and ℱ be as in (vi). The f-Cohen-Macaulay locus of ℱ is the subset

CM(f, ℱ) ⊂ X

consisting of all x ∈ X such that f is Cohen-Macaulay at x. The subset CM(f, 𝒪_{X}) is also called the Cohen-Macaulay locus of f and is denoted briefly CM(f).

Remark 11.3.61. Let (A, 𝔪_{A}) be any local ring; set X := Spec A, and for every A-module M, let M^{∼} be the quasi-coherent 𝒪_{X}-module arising from M. Then, with the notation of theorem 10.1.35(i), we have

dim_{A} M = d_{M^{∼}} for every A-module M.

For the proof, let ℳ be the set of all finitely generated A-submodules of M. Clearly d_{M^{∼}} = sup(d_{M^{′∼}} | M′ ∈ ℳ), so we are reduced to the case where M is finitely generated. Then Supp M is a closed subset of X, and dim_{A} M = dim(Supp M); moreover, clearly Supp M = Supp M^{∼}, and the assertion folows easily.

Lemma 11.3.62. Let f : X → Y be a locally finitely presented morphism of schemes, and ℱ a finitely presented 𝒪_{X}-module such that Supp ℱ = X. We have :

(i) CM(f, ℱ) is an open subset of X,

(ii) The restriction CM(f) → Y of f is locally equidimensional.

(iii) Let g : X′ → X be another morphism of schemes, x′ ∈ X′ a point such that g is étale

at x′. Then g(x′) ∈ CM(f, ℱ) if and only if x′ ∈ CM(f ∘ g, g^{∗}ℱ). (iv) If x ∈ CM(f), then δ′(x, 𝒪_{X}) = δ′(f(x), 𝒪_{Y})+dim 𝒪_{f^{−1}(fx),x} (notation of (10.4.19)).

Proof. (i) and (ii) follow easily from [65, Ch.IV, Prop.15.4.3] and [64, Ch.IV, Prop.2.3.4].

(iii) follows from corollary 10.4.37. Lastly, (iv) is an immediate consequence of corollary 10.4.47. □

Remark 11.3.63. (i) The terminology “locally equidimensional” follows [66, Err_{IV}35], which modifies the terminology “equidimensional” of [65, Ch.IV, Déf.13.3.2]. Hence, a morphism of schemes f : X → Y locally of finite type is locally equidimensional, if it verifies the equivalent conditions of [65, Ch.IV, Prop.13.3.1] at every point x ∈ X. However, the caveat here is that the said conditions are not quite equivalent as stated : they become equivalent if one omits the words “contenant y” in line 2 of loc.cit. (with unchanged proof). So, the correct definition of “locally equidimensional” is in terms of the cited proposition thus amended.

(ii) A counterexample to [65, Ch.IV, Prop.13.3.1] is given by the identity morphism X → X of a scheme X with a point x such that the union of the irreducible components of X passing through x does not contain an open neighborhood of x in X (e.g. take X whose underlying topological space is zero-dimensional and not discrete); then 1_{X} satisfies condition b) of [65, Ch.IV, Prop.13.3.1], but does not satisfy conditions a),a′) and a′′).

Proposition 11.3.64. Let A be a regular local ring of dimension d, and M a finitely generated Cohen-Macaulay A-module. Then :

(i) Ext_{A}^{i}(M, A) = 0 for every i ≠ c := d − dim M.

(ii) The A-module 𝒟(M) := Ext_{A}^{c}(M, A) is Cohen-Macaulay.

(iii) The natural map M → Ext^{c}(𝒟(M), A) is an isomorphism, and Supp 𝒟(M) =

Supp M. (iv) For every n ∈ ℕ, we have an equivalence of categories :

𝒟 : A-CM_{n} → A-CM_{n}^{o} N ↦ Ext_{A}^{d−n}(N, A).

Proof. (i): According to [63, Ch.0, Prop.17.3.4], the projective dimension of M equals c, so we may find a minimal free resolution 0 → L_{c} → ⋯ → L_{1} → L_{0} → M for M of length c. Hence, we need only prove the sought vanishing for every i < c. Set X := Spec A, Z := Supp M ⊂ X. By virtue of proposition 10.4.25, it suffices to show that depth_{Z}𝒪_{X} ≥ c. In light of (10.4.30), this comes down to showing that 𝒪_{X,z} is a local ring of depth ≥ c, for every z ∈ Z. The latter holds, since 𝒪_{X,z} is a regular local ring of dimension ≥ c ([63, Ch.0, Cor.16.5.12]).

(ii): From (i) we deduce that L_{•}^{∨} := (L_{0}^{∨} → ⋯ → L_{n}^{∨}), together with its natural augmentation L_{n}^{∨} → 𝒟(M), is a free resolution of 𝒟(M); especially, the projective dimension of 𝒟(M) is ≤ c, and therefore

depth_{A}𝒟(M) ≥ dim M

again by [63, Ch.0, Prop.17.3.4]. On the other hand, it follows easily from [163, Prop.3.3.10] that Supp 𝒟(M) ⊂ Supp M, so 𝒟(M) is Cohen-Macaulay.

(iii): From the proof of (ii) we see that RHom_{A}(𝒟(M), A) is computed by the complex L_{•}^{∨∨} = L_{•}, whence the first assertion. Invoking again [163, Prop.3.3.10], we deduce that Supp M ⊂ Supp 𝒟(M); since the converse inclusion is already known, these two supports coincide.

(iv) follows straightforwardly from (ii) and (iii). □

Corollary 11.3.65. Let X be a regular noetherian scheme, and j : Y → X a closed immersion. We have :

(i) Y admits a dualizing complex ω_{Y}^{•}.

(ii) If Y is a Cohen-Macaulay scheme, then we may find a dualizing complex ω_{Y}^{•} that is concentrated in degree 0. Moreover, H^{0}(ω_{Y}^{•}) is a Cohen-Macaulay 𝒪_{Y} -module.

Proof. Indeed, lemma 11.3.27(i) shows that the complex ω_{Y}^{•} := j^{∗}Rℋ om_{𝒪_{X}} (j_{∗}𝒪_{Y} , 𝒪_{X}) will do. According to proposition 11.3.64(i,ii), this complex fulfills the condition of (ii), up to a suitable shift. □

11.3.66. Let f : X → S be a smooth quasi-compact morphism of schemes, ℱ a finitely presented quasi-coherent 𝒪_{X}-module, x ∈ X any point, and s := f(x). Set

A := 𝒪_{S,s}  B := 𝒪_{X,x}  F := ℱ_{x}  B_{0} := B ⊗_{A} κ(s)  F_{0} := F ⊗_{A} κ(s).

Theorem 11.3.67. In the situation of (11.3.66), suppose that F is a f_{x}^{♮}-Cohen-Macaulay module. Then we may find an open neighborhood U ⊂ X of x in X such that the following holds:

(i) There exists a finite resolution of the 𝒪_{U}-module ℱ_{|U}, of length n := dim B_{0}−dim_{B} F_{0}

Σ_{•} : 0 → ℒ_{n} → ℒ_{n−1} → ⋯ → ℒ_{0} → ℱ_{|U} → 0

consisting of free 𝒪_{U}-modules of finite rank. Moreover, Σ_{•} is universally 𝒪_{S}-exact, i.e. for every coherent 𝒪_{S}-module 𝒢, the complex Σ_{•} ⊗_{𝒪_{X}} f^{∗}𝒢 is still exact.

(ii) The complex K^{•} := Rℋ om_{𝒪_{U}}^{•} (ℱ_{|U}, 𝒪_{U}) is concentrated in degree n.

(iii) The natural map ℱ_{|U} → Rℋ om_{𝒪_{U}} (K^{•}, 𝒪_{U}) is an isomorphism in D(𝒪_{U}-Mod).

(iv) The B-module G := H^{n}K_{x}^{•} is f_{x}^{♮}-Cohen-Macaulay, and Supp G = Supp F.

Proof. (i): According to proposition 7.3.53, the B-module F admits a minimal free resolution

Σ_{x,•} : ⋯ → L_{2} ⟶^{d_{2}} L_{1} ⟶^{d_{1}} L_{0} ⟶^{ε} F → 0

that is universally A-exact; especially, d_{i}(L_{i}) is a flat A-module, for every i ∈ ℕ. It also follows that Σ_{x,•} ⊗_{A} κ(s) is a minimal free resolution of the B_{0}-module F_{0}. Since the latter is Cohen-Macaulay, and B_{0} is a regular local ring ([66, Ch.IV, Th.17.5.1]), the projective dimension of the B_{0}-module F_{0} equals n ([63, Ch.0, Prop.17.3.4]), therefore d_{n}(L_{n}) ⊗_{A} κ(s) is a free B_{0}- module, so d_{n}(L_{n}) is a flat B-module (lemma 7.11.35); then we deduce that it is actually a free B-module, as it is finitely presented. Since Σ_{x,•} is minimal, we conclude that L_{i} = 0 for every i > n. We may now extend Σ_{x,•} to a finite resolution Σ_{•} of ℱ_{|U} by free 𝒪_{U}-modules on some open neighborhood U of x, and after replacing U by a smaller neighborhood of U, we may assume that ℱ_{|U} is f_{|U}-flat ([65, Ch.IV, Th.11.3.1]), hence Σ_{•} is universally 𝒪_{S}-exact, as stated.

(ii): Let L_{•} := (L_{n} → ⋯ → L_{0}) be the complex obtained after omitting F from the resolution Σ_{x,•} (where L_{0} is placed in degree 0). Then K^{•} is isomorphic to L_{•}^{∨} := Hom_{B}(L_{•}, B). On the other hand, the universal exactness property of Σ_{x,•} implies that

Hom_{B_{0}}(L_{•} ⊗_{A} κ(s), B_{0}) = L_{•}^{∨} ⊗_{A} κ(s)

computes RHom_{B_{0}}^{•}(F_{0}, B_{0}). In view of proposition 11.3.64(i), we have Ext_{B_{0}}^{i}(F_{0}, B_{0}) = 0 for every i ≠ n. From this, a repeated application of [65, Ch.IV, Prop.11.3.7] shows that L_{•}^{∨} is concentrated in degree n as well, and after shrinking U, we may assume that (ii) holds (details left to the reader).

(iii): Let ℒ_{•} := (ℒ_{n} → ⋯ → ℒ_{0}) be the complex obtained by omitting ℱ_{|U} form the resolution Σ_{•}; the proof of (iii) shows that Rℋ om_{𝒪_{U}} (K^{•}, 𝒪_{U}) is computed by ℒ_{•}^{∨∨} = ℒ_{•}, whence the contention.

(iv): By the same token, [65, Ch.IV, Prop.11.3.7] implies that Coker d_{i}^{∨} is a flat A-module, for every i = 1, …, n. Especially, G is a flat A-module, and the complex L_{•}^{∨}[−n], with its natural augmentation L_{n}^{∨} → G, is a universally A-exact and free resolution of the B-module G. Especially, the projective dimension of G is ≤ n, therefore

(11.3.68)  depth_{B}G ≥ dim_{B} F_{0}

by [63, Ch.0, Prop.17.3.4]. From (iii) we also see that the induced map

F_{𝔭} → Ext_{B_{𝔭}}^{n}(G_{𝔭}, B_{𝔭})

is an isomorphism, for every prime ideal 𝔭 ⊂ B; therefore Supp F ⊂ Supp G. Symmetrically, the same argument yields Supp G ⊂ Supp F. Hence the supports of F and G agree. Lastly, combining with (11.3.68) we see that G is Cohen-Macaulay. □

11.3.69. Keep the notation of (11.3.66), and let φ := f_{x}^{♮} : A → B. Theorem 11.3.67(iii,iv) implies that, for every n ∈ ℕ, the functor

𝒟_{φ} : φ-CM_{n} → φ-CM_{n}^{o} M ↦ Ext_{B}^{dim B_{0}−n}(M, B)

is an equivalence, and the natural map M → 𝒟_{φ} ∘ 𝒟_{φ}(M) is an isomorphism, for every φ-Cohen-Macaulay module M. We shall see later also a relative variant of corollary 11.3.65, in a more special situation (see proposition 11.5.5).

11.4. Schemes over a valuation ring. Throughout this section, (K, |·|) is a valued field, whose valuation ring (resp. maximal ideal, resp. residue field, resp. value group) shall be denoted K^{+} (resp. 𝔪_{K}, resp. κ, resp. Γ). Also, we let

S := Spec K^{+}  and S_{/b} := Spec K^{+}/bK^{+}  for every b ∈ 𝔪_{K}

(so S_{/0} = S) and we denote by s := Spec κ (resp. by η := Spec K) the closed (resp. generic) point of S. More generally, for every S-scheme X we let as well

X_{/b} := X ×_{S} S_{/b} for every b ∈ 𝔪_{K}.

A basic fact, that follows immediately from corollary 9.1.28, is that every finitely presented S-scheme is coherent (this can also be deduced easily from [86, Part I, Th.3.4.6]).

Lemma 11.4.1. Let X be an irreducible scheme, and f : X → S a flat morphism of schemes of finite type, x ∈ X, and y := f(x). The following holds :

(i) The scheme f^{−1}(y) is equidimensional of dimension dim f^{−1}(η).

(ii) The local ring 𝒪_{f^{−1}(y),x} is equidimensional.

Proof. (i): After replacing K by the residue field κ(y), and K^{+} by its image in κ(y) (which is a valuation ring of κ(y) : see [126, Th.10.1(iii)]), the assertion follows from [65, Ch.IV, Cor.13.1.6 and Lemme 14.3.10].

(ii) follows easily from (i) and [64, Ch.IV, Prop.5.2.1]. □

Proposition 11.4.2. Let f : X → S be a flat and finitely presented morphism, ℱ a coherent 𝒪_{X}-module, x ∈ X any point, y := f(x), and suppose that f^{−1}(y) is a regular scheme (this holds e.g. if f is a smooth morphism). Let also n := dim 𝒪_{f^{−1}(y),x}. Then:

(i) proj.dim_{𝒪_{X,x}} ℱ_{x} ≤ n + 1 (here we set proj.dim_{𝒪_{X,x}} 0 := 0). (ii) If ℱ is f-flat at the point x, then proj.dim_{𝒪_{X,x}} ℱ_{x} ≤ n. (iii) If ℱ is reflexive at the point x, then proj.dim_{𝒪_{X,x}} ℱ_{x} ≤ max(0, n − 1). (iv) If f has regular fibres, and ℱ is reflexive and generically invertible (see (11.3.10)),

then ℱ is invertible.

(v) All the fibres of the induced morphism f_{x} : X(x) → S(y) are regular and irreducible.

Proof. To start out, we may assume that X is affine, and then it follows easily from lemma 11.3.7(ii.a) that 𝒪_{X} is coherent.

(ii): Since 𝒪_{X} is coherent, we can find a possibly infinite resolution

⋯ → E_{2} → E_{1} → E_{0} → ℱ_{x} → 0

by free 𝒪_{X,x}-modules E_{i} (i ∈ ℕ) of finite rank; set E_{−1} := ℱ_{x} and L := Im(E_{n} → E_{n−1}). It suffices to show that the 𝒪_{X,x}-module L is free. For every 𝒪_{X,x}-module M we shall let M(y) := M ⊗_{𝒪_{X,x}} 𝒪_{f^{−1}(y),x}. Since L and ℱ_{x} are torsion-free, hence flat K^{+}-modules, the induced sequence of 𝒪_{f^{−1}(y),x}-modules:

0 → L(y) → E_{n−1}(y) → ⋯ → E_{1}(y) → E_{0}(y) → ℱ_{x}(y) → 0

is exact; since f^{−1}(y) is a regular scheme, L(y) is a free 𝒪_{f^{−1}(y),x}-module of finite rank. Since L is also flat as a K^{+}-module, [65, Ch.IV, Prop.11.3.7] and Nakayama’s lemma show that any set of elements of L lifting a basis of L(y), is a free basis of L.

(i): Locally on X we can find an epimorphism 𝒪_{X}^{⊕n} → ℱ, whose kernel 𝒢 is again a coherent 𝒪_{X}-module, since 𝒪_{X} is coherent. Clearly it suffices to show that 𝒢 admits, locally on X, a finite free resolution of length ≤ n, which holds by (ii), since 𝒢 is f-flat.

(iii): Suppose ℱ is reflexive at x; by remark 11.3.5we can find a left exact sequence

0 → ℱ_{x} → 𝒪_{X,x}^{⊕m} ⟶^{α} 𝒪_{X,x}^{⊕n}.

It follows that proj.dim_{𝒪_{X,x}} ℱ_{x} = max(0, proj.dim_{𝒪_{X,x}} (Coker α) − 2) ≤ max(0, n − 1), by (i).

(iv): Suppose that ℱ is generically invertible, let j : U → X be the maximal open immersion such that j^{∗}ℱ is an invertible 𝒪_{U}-module, and set Z := X \U. Under the current assumptions, X is reduced; it follows easily that U is the set where rk ℱ = 1, and that j is quasi-compact (since the rank function is constructible : see (11.3.10)). It follows from (iii) that Z ∩ f^{−1}(y) has codimension ≥ 2 in f^{−1}(y), for every y ∈ S. Hence the conditions of corollary 11.3.9 are fulfilled, and furthermore ℱ [0] is a perfect complex by (ii), so the invertible 𝒪_{X}-module det ℱ is well defined (lemma 11.3.14). Then (iv) follows from the natural isomorphisms :

ℱ →^{∼} j_{∗}j^{∗}ℱ →^{∼} j_{∗}det(j^{∗}ℱ [0]) →^{∼} j_{∗}j^{∗}det ℱ [0] ←^{∼} det ℱ [0].

(v): Let y′ ∈ S(y) be any point; we need to check that f_{x}^{−1}(y′) is an irreducible regular scheme, and after replacing K^{+} by a quotient, we may assume that y′ is the generic point of S; after further replacing K^{+} by a localization, we may also assume that y is the closed point of S. Set A := 𝒪_{X,x}; by assumption, f_{x}^{−1}(y) is a regular scheme, hence 𝔭 := 𝔪_{K}A is a prime ideal of A, so that A_{𝔭} is a valuation ring (proposition 9.1.34(ii)), and especially, it is a domain. Moreover, [65, Ch.IV, Prop.11.3.7] implies that for every h ∈ A \ 𝔭, scalar multiplication by h is injective on A; thus, the localization map A → A_{𝔭} is injective as well, so A is a domain. This already proves that f_{x}^{−1}(y′) = Spec K ⊗_{K^{+}} A is irreducible. Next, let 𝔮 ⊂ A be any prime ideal with 𝔮 ∩ K^{+} = {0}, and M any A_{𝔮}-module of finite type; pick a finitely presented A-module N with an isomorphism N_{𝔮} →^{∼} M. By (i), there exists a finite resolution L_{•} of N, consisting of free A-modules of finite rank; then A_{𝔮} ⊗_{A} L_{•} is a finite resolution of M by free A_{𝔮}-modules of finite rank. By Serre’s theorem [126, Th.19.2], this shows that A_{𝔮} is regular, as required. □

Proposition 11.4.3. Let X be a scheme, ℰ a locally free 𝒪_{X}-module of finite rank > 1 at every point of X, and denote by π : ℙ(ℰ) → X the structure morphism of the projective X-scheme associated with ℰ. Then the map

H^{0}(X, ℤ) ⊕ Pic X → Pic ℙ(ℰ) (r, ℒ) ↦ 𝒪_{ℙ(ℰ)}(r) ⊗ π^{∗}ℒ

is an isomorphism of abelian groups.

Proof. To begin with, let us recall the following well known :

Claim 11.4.4. Let f : Y → T be a proper morphism of noetherian schemes, ℱ a coherent f-flat 𝒪_{Y} -module, t ∈ T a point, p ∈ ℕ an integer, and denote i_{t} : f^{−1}(t) → Y the natural immersion. Suppose that H^{p+1}(f^{−1}(t), i_{t}^{∗}ℱ) = 0. Then the natural map

(R^{p}f_{∗}ℱ)_{t} → H^{p}(f^{−1}(t), i_{t}^{∗}ℱ)

is surjective.

Proof of the claim. In light of [61, Ch.III, Prop.1.4.15], we easily reduce to the case where T is a local scheme, say T = Spec A for some noetherian local ring A, and t is the closed point of T. Set k := κ(t), and for every q ∈ ℕ, consider the functor

F_{−q} : A-Mod → A-Mod M ↦ H^{0}(T, R^{p}f_{∗}(f^{∗}M^{∼} ⊗_{𝒪_{Y}} ℱ))

(notation of (10.3)). Since ℱ is f-flat, the system (F_{−q} | q ∈ ℕ) defines a homological functor. The assertion is that F_{−p−1}(k) = 0, in which case [62, Ch.III, Cor.7.5.3] (together with [61, Ch.III, Th.3.2.1 and Th.4.1.5]) says that F_{−p−1}(M) = 0 for every A-module M. Hence, F_{−p−1} is trivially an exact functor, and it follows that the natural map F_{−p}(A) → F_{−p}(k) is surjective ([62, Ch.III, Prop.7.5.4]), which is the claim. ♢

Now, the injectivity of the stated map is clear (details left to the reader). For the surjectivity, let 𝒢 be an invertible 𝒪_{ℙ(ℰ)}-module, x ∈ X any point, and U ⊂ X an affine open neighborhood of x in X such that ℰ_{|U} is a free 𝒪_{X}-module of constant rank n + 1 (for some n ∈ ℕ \ {0}), so that we have an isomorphism of U-schemes ℙ_{U}^{n} →^{∼} ℙ(ℰ_{|U}), where ℙ_{U}^{n} denotes the projective scheme of relative dimension n over U. Write U as the limit of a filtered system (U_{λ} | λ ∈ Λ) of affine noetherian schemes, and for every λ ∈ Λ, let p_{λ} : ℙ_{U}^{n} → ℙ_{U_{λ}}^{n} be the induced morphism, and π_{λ} : ℙ_{U_{λ}}^{n} → U_{λ} the projection; for some μ ∈ Λ, we may find an invertible 𝒪_{ℙ_{Uμ}^{n}}-module 𝒢_{μ}

with an isomorphism 𝒢_{|ℙ_{U}^{n}} →^{∼} p_{μ}^{∗}𝒢_{μ}. Let y ∈ U_{μ} be the image of x, and i_{y} : ℙ_{κ(y)}^{n} → ℙ_{U_{μ}}^{n} the natural immersion; we have i_{y}^{∗}𝒢_{μ} ≃ 𝒪_{ℙ_{κ(y)}^{n}}(r) for some r ∈ ℤ. Set ℱ := 𝒢_{μ}(−r), and notice that H^{1}(ℙ_{κ(y)}^{n}, i_{y}^{∗}ℱ) = 0; moreover, H := H^{0}(ℙ_{κ(y)}^{n}, i_{y}^{∗}ℱ) is a one-dimensional κ(y)-vector space. From claim 11.4.4, it follows that there exist an open neighborhood V_{μ} of y in U_{μ}, and a section s ∈ H^{0}(π_{μ}^{−1}V_{μ}, ℱ) mapping to a generator of H. The section s defines a map of 𝒪_{ℙ_{Vμ}^{n}}-modules φ : 𝒪_{ℙ_{Vμ}^{n}}(r) → 𝒢_{μ|π_{μ}^{−1}V_{μ}}. Furthermore, since i_{y}^{∗}ℱ ≃ 𝒪_{ℙ_{κ(y)}^{n}}, it is easily seen that φ restricts to an isomorphism on some open subset of the form π_{μ}^{−1}W_{μ}, where W_{μ} ⊂ V_{μ} is an affine open neighborhood of y in U_{μ}. Then W := W_{μ} ×_{U_{μ}} U is an affine open neighborhood of x in X, and φ induces an isomorphism 𝒪_{ℙ_{W}^{n}} (r) →^{∼} 𝒢_{|π^{−1}W} of 𝒪_{ℙ_{𝕎}^{n}}-modules.

Summing up, since x ∈ X is arbitrary, we may find an affine open covering X = ⋃_{i∈I} U_{i}, and for every i ∈ I an integer r_{i} and an isomorphism φ_{i} : 𝒪_{ℙ(ℰ_{i})}(r_{i}) →^{∼} 𝒢_{|ℙ(ℰ_{i})} of 𝒪_{ℙ(ℰ_{i})}-modules, with ℰ_{i} := ℰ_{|U_{i}}. Then, for every i, j ∈ I such that U_{ij} := U_{i} ∩ U_{j} ≠ ∅, the composition of the restriction of φ_{i} followed by the restriction of φ_{j}^{−1}, is an isomorphism of 𝒪_{ℙ(ℰ_{ij})}-modules

φ_{ij} : 𝒪_{ℙ(ℰ_{ij})}(r_{i}) →^{∼} 𝒪_{ℙ(ℰ_{ij})}(r_{j}) where ℰ_{ij} := ℰ_{|U_{ij}} .

This already implies that r_{i} = r_{j} whenever U_{ij} ≠ ∅, hence the rule : x ↦ r_{i} if x ∈ U_{i} yields a well defined continuous map r : X → ℤ. We are therefore easily reduced to the case where r is constant on X, and after tensoring by 𝒪_{ℙ(ℰ)}(−r), we may even assume that r = 0.

In this case, each φ_{i} is an isomorphism φ_{i} : π_{i}^{∗}ℒ_{i} →^{∼} 𝒢_{|ℙ(ℰ_{i})}, where π_{i} : ℙ(ℰ_{i}) → U_{i} denotes the restriction of π, and ℒ_{i} is an invertible 𝒪_{U_{i}}-module. Then, for each i, j ∈ I let also π_{ij} : ℙ(ℰ_{ij}) → U_{ij} be the restriction of π; we get unique isomorphisms of 𝒪_{ℙ(ℰ_{ij})}-modules

φ_{ij} : π_{ij}^{∗}ℒ_{i|U_{ij}} →^{∼} π_{ij}^{∗}ℒ_{j|U_{ij}}  such that φ_{j|U_{ij}} ∘ φ_{ij} = φ_{i|U_{ij}}  for all i, j ∈ I

whence a unique system of isomorphisms of 𝒪_{U_{ij}}-modules :

ψ_{ij} : ℒ_{i|U_{ij}} →^{∼} ℒ_{j|U_{ij}}  such that φ_{ij} = π_{ij}^{∗}ψ_{ij}  for every i, j ∈ I.

Now, for every i, j, k ∈ I let U_{ijk} := U_{ij} ∩ U_{k} and ℰ_{ijk} := ℰ_{|U_{ijk}}; it is easily seen that

φ_{jk|ℙ(ℰ_{ijk})} ∘ φ_{ij|ℙ(ℰ_{ijk})} = φ_{ik|ℙ(ℰ_{ijk})} whence ψ_{jk|U_{ijk}} ∘ ψ_{ij|U_{ijk}} = ψ_{ik|U_{ijk}}.

Hence, there exists an invertible 𝒪_{X}-module ℒ with a system of isomorphisms :

(λ_{i} : ℒ_{|U_{i}} →^{∼} ℒ_{i} | i ∈ I)  such that ψ_{ij} ∘ λ_{i|U_{ij}} = λ_{j|U_{ij}}  for every i, j ∈ I.

It follows that the system of induced isomorphisms :

φ_{i} ∘ π_{i}^{∗}(λ_{i}) : (π^{∗}ℒ)_{|ℙ(ℰ_{i})} →^{∼} 𝒢_{|ℙ(ℰ_{i})}

glues to yield an isomorphism of 𝒪_{ℙ(ℰ)}-modules π^{∗}ℒ →^{∼} 𝒢, and that concludes the proof. □

Corollary 11.4.5. Let f : X → S be a smooth morphism of finite presentation, set 𝔾_{m,X} := X ×_{S} Spec K^{+}[T, T^{−1}], and denote by π : 𝔾_{m,X} → X the natural morphism. Then the map

Pic X → Pic 𝔾_{m,X} ℒ ↦ π^{∗}ℒ

is an isomorphism.

Proof. To start out, the map is injective, since π admits a section. Next, let j : 𝔾_{m,X} → ℙ_{X}^{1} be the natural open immersion, and denote again by π : ℙ_{X}^{1} → X the natural projection. According to theorem 10.3.25 and proposition 11.4.2(i), every invertible 𝒪_{𝔾_{m,X}}-module extends to an invertible 𝒪_{ℙ_{X}^{1}}-module (namely the determinant of a coherent extension). Since j^{∗}𝒪_{ℙ_{X}^{1}}(n) ≃ 𝒪_{𝔾_{m,X}}, the claim follows from proposition 11.4.3. □

Remark 11.4.6. The proof of proposition 11.4.3 follows [131, p.20] and [132, Lecture 13, Prop.3]. Of course, statements related to corollary 11.4.5 abound in the literature, and more general results are available: see e.g. [16] and [156].

Proposition 11.4.7. Let f : X → S be a smooth morphism of finite type, j : U → X an open immersion, and Z := X \U; suppose that X and U are quasi-compact and quasi-separated. Then:

(i) Every invertible 𝒪_{U}-module extends to an invertible 𝒪_{X}-module.

(ii) Suppose furthermore, that: (a) For every point y ∈ S, the codimension of Z ∩ f^{−1}(y) in f^{−1}(y) is ≥ 1, and (b) The codimension of Z ∩ f^{−1}(η) in f^{−1}(η) is ≥ 2. Then the restriction functors:

(11.4.8)  𝒪_{X}-Rflx → 𝒪_{U}-Rflx  Pic X → Pic U

are equivalences.

Proof. (i): First let us notice :

Claim 11.4.9. X is a finite disjoint union of normal integral schemes.

Proof of the claim. Since X is quasi-compact, the assertion is easily reduced to the case where X = Spec A, for a smooth K^{+}-algebra A. Then A is integrally closed in A_{K} := K ⊗_{K^{+}} A, by proposition 9.8.3, and A_{K} is smooth over K, so it is a product of finitely many normal domains. The idempotents corresponding to this decomposition are integral over A, hence they lie in A. Thus, A is itself a product of finitely many normal domains, which is the claim. ♢

In view of claim 11.4.9, we may assume from start that X is integral, and then U is either empty (in which case there is nothing to prove), or dense in X, in which case the assertion follows from propositions 11.3.8(i) and 11.4.2(iv).

(ii): We begin with the following:

Claim 11.4.10. Under the assumptions of (ii), the functors (11.4.8) are faithful.

Proof of the claim. Since f is smooth, all the stalks of 𝒪_{X} are reduced ([66, Ch.IV, Prop.17.5.7]). Since every point of Z is specialization of a point of U, the claim follows easily from remark 11.3.5. ♢

Under assumptions (a) and (b), the conditions of corollary 11.3.9 are fulfilled, and one deduces that (11.4.8) are full functors, hence fully faithful, in view of claim 11.4.10. The essential surjectivity of the restriction functor for reflexive 𝒪_{X}-modules is already known, by proposition 11.3.8(i). □

Lemma 11.4.11. Let f : X → S be a flat, finitely presented morphism, and denote by U ⊂ X the maximal open subset such that the restriction f_{|U} : U → S is smooth. Suppose that the following two conditions hold:

(a) For every point y ∈ S, the fibre f^{−1}(y) is geometrically reduced.

(b) The generic fibre f^{−1}(η) is geometrically normal. Then the restriction functor induces an equivalence of categories (notation of (11.3.10)) :

(11.4.12)  Div X →^{∼} Pic U.

Proof. Let ℱ be any generically invertible reflexive 𝒪_{X}-module; it follows from proposition 11.4.2(iv) that ℱ_{|U} is an invertible 𝒪_{U}-module, so the functor (11.4.12) is well defined. Let y ∈ S be any point; condition (a) says in particular that the fibre f^{−1}(y) is generically smooth over κ(y) ([126, Th.28.7]). Then it follows from [66, Ch.IV, Th.17.5.1] that U ∩ f^{−1}(y) is a dense open subset of f^{−1}(y), and if x ∈ f^{−1}(y)\U, then the depth of 𝒪_{f^{−1}(y),x} is ≥ 1, since the latter is a reduced local ring of dimension ≥ 1. Similarly, condition (b) and Serre’s criterion [126, Ch.8, Th.23.8] say that for every x ∈ f^{−1}(η)\U, the local ring 𝒪_{f^{−1}(η),x} has depth ≥ 2. Furthermore, [65, Ch.IV, Prop.9.9.4] implies that the open immersion j : U → X is quasicompact; summing up, we see that all the conditions of corollary 11.3.9 are fulfilled, so that ℱ = j_{∗}j^{∗}ℱ for every reflexive 𝒪_{X}-module ℱ. This already means that (11.4.12) is fully faithful. To conclude, it suffices to invoke proposition 11.3.8(i). □

11.4.13. Let K(T) be the fraction field of the free polynomial K-algebra K[T]. For every γ ∈ Γ one can define an extension of |·| to a Gauss valuation |·|_{0,γ} : K(T) → Γ ([75, Ex.6.1.4(iii)]). If f(T) := ∑_{i=0}^{d} a_{i}T^{i} is any polynomial, then |f(T)|_{0,γ} = max{|a_{i}| · γ^{i} | i = 0, …, d}. We let V (γ) be the valuation ring of | · |_{0,γ}, and set

R(γ) := K[T, T^{−1}] ∩ V (1) ∩ V (γ).

Since K[T, T^{−1}] is a Dedekind domain, R(γ) is an intersection of valuation rings of the field K(T), hence it is a normal domain. By inspecting the definition we see that R(γ) consists of all the elements of the form f(T) := ∑_{i=−n}^{n} a_{i}T^{i}, such that |a_{i}| ≤ 1 and |a_{i}| · γ^{i} ≤ 1 for every i = −n, …, n. Suppose now that γ ≤ 1, and choose c ∈ K^{+} with |c| = γ; then every such f(T) can be written uniquely in the form ∑_{i=0}^{n} a_{i}T^{i} + ∑_{j=1}^{n} b_{j}(cT^{−1})^{j}, where a_{i}, b_{j} ∈ K^{+} for every i, j ≤ n. Conversely, every such expression yields an element of R(γ). In other words, we obtain a surjection of K^{+}-algebras K^{+}[X, Y] → R(γ) by the rule: X ↦ T, Y ↦ cT^{−1}. Obviously the kernel of this map contains the ideal (XY − c), and we leave to the reader the verification that the induced map

(11.4.14)  K^{+}[X, Y]/(XY − c) → R(γ)

is indeed an isomorphism.

11.4.15. Throughout the following discussion, we shall assume that γ ≤ 1. Every δ ∈ Γ with γ ≤ δ ≤ 1, determines a prime ideal 𝔭(δ) := {f ∈ R(γ) | |f|_{0,δ} < 1} ⊂ R(γ), such that 𝔪_{K}R(γ) ⊂ 𝔭(δ). Then it is easy to see that R(γ)_{𝔭(δ)} ⊂ V (δ), and moreover :

R(γ)_{𝔭(γ)} = V (γ) R(γ)_{𝔭(1)} = V (1)

since V (1) (resp. V (γ)) is already a localization of K^{+}[T] (resp. of K^{+}[cT^{−1}]). In case γ < 1, (11.4.14) implies that R(γ) ⊗_{K^{+}} κ ≃ κ[X, Y]/(XY), and it follows easily that 𝔭(1) and 𝔭(γ) correspond to the two minimal prime ideals of κ[X, Y]/(XY). In case γ = 1, we have R(1) ⊗_{K^{+}} κ ≃ κ[X, X^{−1}], and again 𝔭(1) corresponds to the generic point of Spec κ[X, X^{−1}]. Notice that the natural morphism

f_{γ} : 𝕋_{K}(γ) := Spec R(γ) → S

restricts to a smooth morphism f_{γ}^{−1}(η) → Spec K; moreover the closed fibre f_{γ}^{−1}(s) is geometrically reduced. Notice also that 𝕋_{K}(γ) ×_{S} Spec E^{+} ≃ 𝕋_{E}(γ) for every extension of valued fields K ⊂ E. In the following, we will write just 𝕋(γ) in place of 𝕋_{K}(γ), unless we have to deal with more than one base ring.

Proposition 11.4.16. Keep the notation of (11.4.15), and let g : X → 𝕋(γ) be an étale morphism, ℱ a coherent 𝒪_{X}-module. Set h := f_{γ} ∘ g : X → S and denote by i_{s} : h^{−1}(s) → X the natural morphism. Then ℱ is reflexive at the point x ∈ h^{−1}(s) if and only if the following three conditions hold:

(a) ℱ is h-flat at the point x.

(b) ℱ_{x} ⊗_{K^{+}} K is a reflexive 𝒪_{X,x} ⊗_{K^{+}} K-module.

(c) The 𝒪_{h^{−1}(s),x}-module i_{s}^{∗}ℱ_{x} satisfies condition S_{1} (see definition 10.5.1(iii)).

Proof. Suppose that ℱ is reflexive at the point x; then it is easy to check that (a) and (b) hold. We prove (c): by remark 11.3.5we can find a left exact sequence

0 → ℱ_{x} →^{α} 𝒪_{X,x}^{⊕m} →^{β} 𝒪_{X,x}^{⊕n}.

Then Im β is a flat K^{+}-module, since it is a submodule of the flat K^{+}-module 𝒪_{X,x}^{⊕n}; hence α ⊗_{K^{+}} 1_{κ} : i_{s}^{∗}ℱ_{x} → 𝒪_{h^{−1}(s),x}^{⊕m} is still injective, so we are reduced to showing that h^{−1}(s) is a reduced scheme, which follows from [66, Ch.IV, Prop.17.5.7] and the fact that f_{γ}^{−1}(s) is reduced.

Conversely, suppose that conditions (a)–(c) hold.

Claim 11.4.17. Let ξ be the generic point of an irreducible component of h^{−1}(s). Then: (i) ℱ_{x} is a torsion-free 𝒪_{X,x}-module. (ii) 𝒪_{X,ξ} is a valuation ring. (iii) Suppose that the closure of ξ contains x. Then ℱ_{ξ} is a free 𝒪_{X,ξ}-module of finite rank.

Proof of the claim. (i): By (a), the natural map ℱ_{x} → ℱ_{x} ⊗_{K^{+}} K is injective; since (b) implies that ℱ_{x} ⊗_{K^{+}} K is a torsion-free 𝒪_{X,x}-module, the same must then hold for ℱ_{x}.

(ii) follows from proposition 9.1.34(ii).

(iii): Suppose that x ∈ \overline{\{ξ\}}. We derive easily from (i) that ℱ_{ξ} is a torsion-free 𝒪_{X,ξ}-module, so the assertion follows from (ii) and [34, Ch.VI, §3, n.6, Lemma 1]. ♢

By (b), the morphism β_{ℱ,x} ⊗_{K^{+}} 1_{K} : ℱ_{x} ⊗_{K^{+}} K → ℱ_{x}^{∨∨} ⊗_{K^{+}} K is an isomorphism (notation of (11.3)); since ℱ is h-flat at x, we deduce easily that β_{ℱ,x} is injective and C := Coker β_{ℱ,x} is a torsion K^{+}-module. To conclude, it remains only to show:

Claim 11.4.18. C is a flat K^{+}-module.

Proof of the claim. In view of lemma 7.11.35, it suffices to show that Tor_{1}^{K^{+}}(C, κ(s)) = 0. However, from the foregoing we derive a left exact sequence

\xymatrix{0 \ar[r] & Tor_{1}^{K^{+}}(C, κ(s)) \ar[r] & ℱ_{x} ⊗_{K^{+}} κ(s) \ar[r]^{β_{ℱ,x} ⊗_{K^{+}}κ(s)} & ℱ_{x}^{∨∨} ⊗_{K^{+}} κ(s).}

We are thus reduced to showing that β_{ℱ,x} ⊗_{K^{+}} κ(s) is an injective map. In view of condition (c), it then suffices to prove that β_{ℱ,ξ} is an isomorphism, whenever ξ is the generic point of an irreducible component of h^{−1}(s) containing x. The latter assertion holds by virtue of claim 11.4.17(iii). □

11.4.19. For a given ρ ∈ Γ, let us pick a ∈ K \ {0} such that |a| = ρ; we define the fractional ideal I(ρ) ⊂ K[T, T^{−1}] as the R(γ)-submodule generated by T and a. The module I(ρ) determines a quasi-coherent 𝒪_{𝕋(γ)}-module ℐ (ρ).

Lemma 11.4.20. With the notation of (11.4.19):

(i) ℐ (ρ) is a reflexive 𝒪_{𝕋(γ)}-module for every ρ ∈ Γ. (ii) There exists a short exact sequence of R(γ)-modules:

0 → I(ρ^{−1}γ) → R(γ)^{⊕2} → I(ρ) → 0.

Proof. To start with, let a ∈ K \ {0} with |a| = ρ.

Claim 11.4.21. If either ρ ≥ 1 or ρ ≤ γ, then I(ρ) and I(ρ^{−1}γ) are rank one, free R(γ)- modules.

Proof of the claim. If ρ ≥ 1 (resp. ρ ≤ γ) then ρ^{−1}γ ≤ γ (resp. ρ^{−1}γ ≥ 1), hence it suffices to show that I(ρ) is free of rank one, in both cases. Suppose first that ρ ≥ 1; in this case, multiplication by a^{−1} yields an isomorphism of R(γ)-modules I(ρ) →^{∼} R(γ). Next, suppose that ρ ≤ γ. Then I(ρ) is the ideal (T, a), where a = c · b for some b ∈ K^{+} and |c| = γ. Therefore I(ρ) = (T, T · (cT^{−1}) · b) = T · (1, cT^{−1}b) = TR(γ), and again I(ρ) is a free R(γ)-module of rank one. ♢

In view of claim 11.4.21, we may assume that γ < ρ < 1. Let (e_{1}, e_{2}) be the canonical basis of the free R(γ)-module R(γ)^{⊕2}; we consider the R(γ)-linear surjection π : R(γ)^{⊕2} → I(ρ) determined by the rule: e_{1} ↦ T, e_{2} ↦ a. Clearly Ker π contains the submodule M(ρ) ⊂ R(γ)^{⊕2} generated by:

f_{1} := ae_{1} − Te_{2} and f_{2} := cT^{−1}e_{1} − ca^{−1}e_{2}.

Let 𝒩 be the quasi-coherent 𝒪_{𝕋(γ)}-module associated with N := R(γ)^{⊕2}/M(ρ).

Claim 11.4.22. With the foregoing notation:

(i) N is a flat K^{+}-module.

(ii) K^{+}[c^{−1}] ⊗_{K^{+}} N is a free K^{+}[c^{−1}] ⊗_{K^{+}} R(γ)-module of rank one.

(iii) K ⊗_{K^{+}} M(ρ) = K ⊗_{K^{+}} Ker π.

Proof of the claim. (i): We let gr_{•}R(γ) be the T-adic grading on R(γ) (i.e. gr_{i}R(γ) = T^{i} · K ∩ R(γ) for every i ∈ ℤ), and we define a compatible grading on R(γ)^{⊕2} by setting: gr_{i}R(γ)^{⊕2} := (gr_{i−1}R(γ)·e_{1})⊕(gr_{i}R(γ)·e_{2}) for every i ∈ ℤ. Since f_{1} and f_{2} are homogeneous elements, we deduce by restriction a grading gr_{•}M(ρ) on M(ρ), and a quotient grading gr_{•}N on N, whence a short exact sequence of graded K^{+}-modules:

0 → gr_{•}M(ρ) ⟶^{gr_{•}j} gr_{•}R(γ)^{⊕2} → gr_{•}N → 0.

However, by inspecting the definitions, it is easy to see that gr_{•}j is a split injective map of free K^{+}-modules, hence gr_{•}N is a free K^{+}-module, and then the same holds for N.

(ii) is easy and shall be left to the reader.

(iii): Similarly, one checks easily that K ⊗_{K^{+}} I(ρ) is a free K ⊗_{K^{+}} R(γ)-module of rank one; then, by (ii) the quotient map K ⊗_{K^{+}} N → K ⊗_{K^{+}} I(ρ) is necessarily an isomorphism, whence the assertion. ♢

Claim 11.4.23. 𝒩 is a reflexive 𝒪_{𝕋(γ)}-module.

Proof of the claim. Since 𝒩 is coherent, it suffices to show that 𝒩_{x} is a reflexive 𝒪_{𝕋(γ),x}- module, for every x ∈ 𝕋(γ) (lemma 11.3.2(ii)). Let y := f_{γ}(x); we may then replace 𝒩 by its restriction to 𝕋(γ) ×_{S} S(y), which allows to assume that y = s is the closed point of S. In this case, we can apply the criterion of proposition 11.4.16 to the morphism f_{γ} : 𝕋(γ) → S. We already know from claim 11.4.22(i) that 𝒩 is f_{γ}-flat. Moreover, by claim 11.4.22(ii) we see that the restriction of 𝒩 to f_{γ}^{−1}(η) is reflexive. As γ < ρ < 1, by inspecting the definition and using the presentation (11.4.14), we deduce an isomorphism :

κ ⊗_{K^{+}} N ≃ \overline{R}^{⊕2}/(Xe_{2}, Ye_{1}) ≃ (\overline{R}/X\overline{R}) ⊕ (\overline{R}/Y\overline{R})

where \overline{R} := κ[X, Y]/(XY) ≃ κ ⊗_{K^{+}} R(γ). Thus, the \overline{R}-module κ ⊗_{K^{+}} N satisfies condition S_{1}, whence the claim. ♢

It follows from claim 11.4.22(i,iii) that the quotient map N → I(ρ) is an isomorphism, so ℐ (ρ) is reflexive, by claim 11.4.23. Next, let us define an R(γ)-linear surjection π′ : R(γ)^{⊕2} → M(ρ) by the rule: e_{i} ↦ f_{i} for i = 1, 2. One checks easily that Ker π′ contains the submodule generated by the elements ca^{−1}e_{1} − T e_{2} and ce_{1} − aT e_{2}, and the latter is none else than the module M(ρ^{−1}γ), according to our notation (notice that γ < ρ^{−1}γ < 1). We deduce a surjection of torsion-free R(γ)-modules I(ρ^{−1}γ) →^{∼} R(γ)^{⊕2}/M(ρ^{−1}γ) → M(ρ), which induces an isomorphism after tensoring by K, therefore I(ρ^{−1}γ) →^{∼} M(ρ), which establishes (ii). □

11.4.24. Let 𝕋(γ)_{sm} ⊂ 𝕋(γ) be the largest open subset which is smooth over S. Set S_{γ} := Spec K^{+}/cK^{+}; it is easy to see that f_{γ}^{−1}(S \S_{γ}) ⊂ 𝕋(γ)_{sm}, and for every y ∈ S_{γ}, the difference f_{γ}^{−1}(y) \ 𝕋(γ)_{sm} consists of a single point.

Proposition 11.4.25. Let Δ(γ) ⊂ Γ be the smallest convex subgroup containing γ. Then there is a natural isomorphism of groups:

Pic 𝕋(γ)_{sm} →^{∼} Δ(γ)/γ^{ℤ}.

Proof. We consider the affine covering of 𝕋(γ)_{sm} consisting of the two open subsets

U := Spec K^{+}[T, T^{−1}] and V := Spec K^{+}[cT^{−1}, c^{−1}T]

with intersection U∩V = Spec K^{+}[c^{−1}, T, T^{−1}]. We notice that both U and V are S-isomorphic to 𝔾_{m,S}, and therefore

(11.4.26)  Pic U = Pic V = 0

by corollary 11.4.5. From (11.4.26), a standard computation yields a natural isomorphism:

Pic 𝕋(γ)_{sm} →^{∼} 𝒪_{𝕋(γ)}(U)^{×}\𝒪_{𝕋(γ)}(U ∩ V)^{×}/𝒪_{𝕋(γ)}(V)^{×}.

(Here, for a ring A, the notation A^{×} means the invertible elements of A.) However, 𝒪_{𝕋(γ)}(U)^{×} = (K^{+})^{×} · (cT^{−1})^{ℤ}, 𝒪_{𝕋(γ)}(U ∩ V)^{×} = (K^{+}[c^{−1}])^{×} · T^{ℤ} and 𝒪_{𝕋(γ)}(V) = (K^{+})^{×} · T^{ℤ}, whence the contention. □

11.4.27. Proposition 11.4.25 establishes a natural bijection between the set of isomorphism classes of invertible 𝒪_{𝕋(γ)_{sm}}-modules and the set :

]γ, 1] := {ρ ∈ Γ | γ < ρ < 1} ∪ {1}.

On the other hand, lemma 11.4.11 yields a natural bijection between Pic 𝕋(γ)_{sm} and the set of isomorphism classes of generically invertible reflexive 𝒪_{𝕋(γ)}-modules. Furthermore, lemma 11.4.20provides already a collection of such reflexive modules, and by inspection of the proof, we see that the family of sheaves ℐ (ρ) is really parametrized by the subset ]γ, 1] (since the other values of ρ correspond to free 𝒪_{𝕋(γ)}-modules of rank one). The two parametrizations are essentially equivalent. Indeed, let a ∈ K \ {0} be any element such that ρ := |a| ∈]γ, 1]. With the notation of the proof of proposition 11.4.25, we can define isomorphisms

φ : 𝒪_{U} →^{∼} ℐ (ρ)_{|U} ψ : 𝒪_{V} →^{∼} ℐ (ρ)_{|V}

by letting: φ(1) := T and ψ(1) := a. To verify that φ is an isomorphism, it suffices to remark that T is a unit on U, so ℐ (ρ)_{|U} = T𝒪_{U} = 𝒪_{U}. Likewise, on V we can write T = a · (a^{−1}c) · (c^{−1}T), so ℐ (ρ)_{|V} = a𝒪_{V}, and ψ is an isomorphism. Hence ℐ (ρ) is isomorphic to the (unique) 𝒪_{𝕋(γ)}-module whose global sections consist of all the pairs (s_{U}, s_{V}) ∈ 𝒪_{U}(U) × 𝒪_{V} (V), such that T^{−1}s_{U|U∩V} = a^{−1}s_{V |U∩V}. Clearly, under the bijection of proposition 11.4.25, the invertible sheaf ℐ (ρ)_{|𝕋(γ)_{sm}} corresponds to the class of ρ in Δ(γ)/γ^{ℤ}. In particular, this shows that the reflexive 𝒪_{𝕋(γ)}-modules ℐ (ρ) are pairwise non-isomorphic for ρ ∈]γ, 1], and that every reflexive generically invertible 𝒪_{𝕋(γ)}-module is isomorphic to one such ℐ (ρ).

11.4.28. When γ < 1 and t ∈ 𝕋(γ) is the singular point of the closed fibre, the discussion of (11.4.27) also applies to describe the set \overline{coh.Div}(𝒪_{𝕋(γ),t}) of isomorphism classes of coherent reflexive fractional ideals of 𝒪_{𝕋(γ),t} (remark 11.3.11(ii)). Indeed, any such module M extends to a reflexive 𝒪_{𝕋(γ)}-module (first one uses lemma 11.3.7(ii.b) to extend M to some quasi-compact open subset U ⊂ 𝕋(γ), and then one may extend to the whole of 𝕋(γ), via proposition 11.3.8)(i). Hence M ≃ ℐ (ρ)_{t} for some ρ ∈]γ, 1]. It follows already that coh.Div(𝒪_{𝕋(γ),t}) is naturally an abelian group, with multiplication law given by the rule :

(M, N) ↦ M ⊙ N := j_{∗}j^{∗}(M ⊗_{𝒪_{𝕋(γ),t}} N) for any two classes M, N ∈ Div(𝒪_{𝕋(γ),t})

where j : 𝕋(γ)_{sm} ∩ Spec 𝒪_{𝕋(γ),t} → Spec 𝒪_{𝕋(γ),t} is the natural open immersion. Indeed, M ⊙ N is reflexive (by proposition 11.3.8(i) and corollary 11.3.9), and the composition law ⊙ is clearly associative and commutative, with 𝒪_{𝕋(γ),t} as neutral element; moreover, for any ρ ∈ Δ(γ), the class of ℐ (ρ)_{t} admits the inverse j_{∗}((j^{∗}ℐ (ρ)_{t})^{∨}). Furthermore, the modules ℐ (ρ)_{t} are pairwise non-isomorphic for ρ ∈]γ, 1]. Indeed, using the group law ⊙, the assertion follows once we know that ℐ (ρ)_{t} is not trivial, whenever ρ ∈]γ, 1[. However, from the presentation of lemma 11.4.20(ii) one sees that ℐ (ρ)_{t} ⊗_{𝒪_{𝕋(γ),t}} κ(t) is a two-dimensional κ(t)-vector space, so everything is clear. Moreover, a simple inspection shows that the composition law ⊙ thus defined, agrees with the composition law of the monoid coh.Div(𝒪_{T (γ),t}) given in remark 11.3.11(ii). Summing, we get a natural group isomorphism :

\overline{coh.Div}(𝒪_{𝕋(γ),t}) →^{∼} Δ(γ)/γ^{ℤ}.

The following theorem generalizes this classification to reflexive modules of arbitrary generic rank.

Theorem 11.4.29. Let g : X → 𝕋(γ) be a pro-étale morphism, x ∈ X any point, M a reflexive 𝒪_{X,x}-module. Then there exist ρ_{1}, …, ρ_{n} ∈]γ, 1] and an isomorphism of 𝒪_{X,x}-modules:

M →^{∼} ⨁_{i=1}^{n} g^{∗}ℐ (ρ_{i})_{x}.

Proof. Using lemma 11.3.7(ii.b), we are easily reduced to the case where g is étale. Set t := g(x); first of all, if t ∈ 𝕋(γ)_{sm}, then X is smooth over S at the point t, and consequently M (resp. ℐ (ρ)_{t}) is a free 𝒪_{X,x}-module (resp. 𝒪_{𝕋(γ),t}-module) of finite rank (by proposition 11.4.2(iii)), so the assertion is obvious in this case. Hence we may assume that γ < 1 and t is the unique point in the closed fibre of 𝕋(γ) \ 𝕋(γ)_{sm}. Next, let K^{sh+} be the strict henselization of K^{+}; denote by h : 𝕋_{K^{sh}}(γ) → 𝕋_{K}(γ) the natural map, and set X′ := X ×_{𝕋_{K}(γ)} 𝕋_{K^{sh}}(γ). Choose also a point x′ ∈ X′ lying over x, and let t′ ∈ 𝕋_{K^{sh}}(γ) be the image of x′; then t′ is the unique point of 𝕋_{K^{sh}}(γ) with h(t′) = t. We have a commutative diagram of ring homomorphisms:

\xymatrix{𝒪_{𝕋(γ),t} \ar[r]^{g_{x}^{♮}} \ar[d]_{h_{t}^{♮}} & 𝒪_{X,x} \ar[d] \\ 𝒪_{𝕋_{K^{sh}}(γ),t′} \ar[r] & 𝒪_{X′,x′}}

whence an essentially commutative diagram of functors:

\xymatrix{𝒪_{𝕋(γ),t}-Rflx \ar[r]^{g_{x}^{∗}} \ar[d]_{h_{t}^{∗}} & 𝒪_{X,x}-Rflx \ar[d]^{β} \\ 𝒪_{𝕋_{K^{sh}}(γ),t′}-Rflx \ar[r]^{α} & 𝒪_{X′,x′}-Rflx.}

Since K and K^{sh} have the same value group, the discussion in (11.4.28) shows that h_{t}^{∗} induces bijections on the isomorphism classes of generically invertible modules. On the other hand, proposition 7.11.12implies that the functor β induces injections on isomorphism classes. Consequently, in order to prove the theorem, we may replace the pair (X, x) by (X′, x′), and assume from start that K = K^{sh}.

In terms of the presentation 11.4.14 we can write 𝕋(γ) ×_{S} Spec κ(s) = Z_{1} ∪ Z_{2}, where Z_{1} (resp. Z_{2}) is the reduced irreducible component on which X (resp. Y) vanishes. By inspecting the definitions, it is easy to check that Z_{1} ≃ 𝔸_{κ}^{1} ≃ Z_{2} as κ-schemes. Let ξ_{i} be the generic point of Z_{i}, for i = 1, 2; clearly {t} = Z_{1} ∩ Z_{2}, hence W_{i} := X ×_{𝕋(γ)} Z_{i} is non-empty and étale over Z_{i}, so 𝒪_{W_{i},x} is an integral domain, hence g^{−1}(ξ_{i}) contains exactly one point ζ_{i} that specializes to x, for both i = 1, 2. To ease notation, let us set A := 𝒪_{X,x}. Then Spec A ⊗_{K^{+}} κ consists of exactly three points, namely x, ζ_{1} and ζ_{2}. Set M(ζ_{i}) := M ⊗_{A} κ(ζ_{i}), and notice that n := dim_{κ(ζ_{1})} M(ζ_{1}) = dim_{κ(ζ_{2})} M(ζ_{2}), since M restricts to a locally free module over (Spec A)_{sm}, the largest essentially smooth open S-subscheme of Spec A, which is connected. We choose a basis \overline{e}_{1}, …, \overline{e}_{n} (resp \overline{e}′_{1}, …, \overline{e}′_{n}) for M(ζ_{1})^{∨} (resp. M(ζ_{2})^{∨}), which we can then lift to a system of sections e_{1}, …, e_{n} ∈ M_{ζ_{1}}^{∨} := M^{∨} ⊗_{A} 𝒪_{X,ζ_{1}} (and likewise we construct a system e′_{1}, …, e′_{n} ∈ M_{ζ_{2}}^{∨}). Let 𝔭_{i} ⊂ A be the prime ideal corresponding to ζ_{i} (i = 1, 2); after multiplication by an element of A \ 𝔭_{i}, we may assume that e_{1}, …, e_{n} ∈ M^{∨} (and likewise for e′_{1}, …, e′_{n}). Finally, we set e′′_{i} := Ye_{i} + Xe′_{i} for every i = 1, …, n; it is clear that the system (e′′_{1}, …, e′′_{n}) induces bases of M(ζ_{i})^{∨} for both i = 1, 2. We wish to consider the map:

j : M → A^{⊕n} m ↦ (e′′_{1}(m), …, e′′_{n}(m)).

Set C := Coker j, I := Ann_{A}C, and B := A/I.

Claim 11.4.30. (i) The maps M_{ζ_{i}} → 𝒪_{X,ζ_{i}}^{⊕n} induced by j are isomorphisms.

(ii) Ker j = Ker j ⊗_{A} 1_{κ} = 0.

(iii) B is a finitely presented K^{+}-module, and C is a free K^{+}-module of finite rank.

Proof of the claim. (i): Using Nakayama’s lemma, one deduces easily that these maps are surjective; since M_{ζ_{i}} is a free 𝒪_{X,ζ_{i}}-module of rank n, they are also necessarily injective.

(ii): Since 𝒪_{X,x} is normal, {0} is its only associated prime; then the injectivity of j (resp. of j ⊗_{A} 1_{κ}) follows from (i), and the fact that M (resp. M ⊗_{A} κ) satisfies condition S_{1}, by remark 11.3.5(resp. by proposition 11.4.16).

(iii): First of all, since A is coherent, I is a finitely generated ideal of A. Let f : Spec B → Spec K^{+} be the natural morphism. Since C is a finitely presented A-module, its support Z is a closed subset of Spec A. From (i) we see that Z ∩ Spec A ⊗_{K^{+}} κ ⊂ {x}; on the other hand, Z is also the support of the closed subscheme Spec B of Spec A. Therefore

(11.4.31)  f^{−1}(s) ∩ Spec B ⊂ {x}.

Since K^{+} is henselian, it follows easily from (11.4.31) and [66, Ch.IV, Th.18.5.11(c′′)] that B is a finite K^{+}-algebra of finite presentation, hence also a finitely presented K^{+}-module (claim 9.1.35). Since C is a finitely presented B-module, we conclude that C is finitely presented as K^{+}-module, as well. From (ii) we deduce a short exact sequence: 0 → M → A^{⊕n} → C → 0, and then the long exact Tor sequence yields: Tor_{1}^{A}(C, κ) = 0 (cp. the proof of claim 11.4.18). We conclude by [34, Ch.II, §3, n.2, Cor.2 of Prop.5]. ♢

Claim 11.4.32. The composed morphism: Spec B → Spec A → 𝕋(γ) is a closed immersion.

Proof of the claim. Let 𝔭 ⊂ R(γ) be the maximal ideal corresponding to t, so that 𝒪_{𝕋(γ),t} = R(γ)_{𝔭}. From claim 11.4.30(iii) we see that the natural morphism ψ : R(γ)_{𝔭} → B is finite. Moreover, A/𝔭A ≃ R(γ)/𝔭, hence ψ ⊗_{R(γ)} 1_{R(γ)/𝔭} is a surjection. By Nakayama’s lemma we deduce that ψ is already a surjection, i.e. the induced morphism Spec B → Spec R(γ)_{𝔭} is a closed immersion. Let J_{𝔭} := Ker ψ, J := R(γ) ∩ J_{𝔭} and D := R(γ)/J. We are reduced to showing that the induced map D → D_{𝔭} is an isomorphism. However, let e ∈ R(γ) \ 𝔭; since D_{𝔭} ≃ B is finite over K^{+}, we can find a monic polynomial P [T] ∈ K^{+}[T] such that P (e^{−1}) = 0, therefore an identity of the form 1 = e · Q(e) holds in D_{𝔭} for some polynomial Q(T) ∈ K^{+}[T]. But then the same identity holds already in the subring D, i.e. the element e is invertible in D, and the claim follows. ♢

Now, C is a finitely generated B-module, hence also a finitely generated R(γ)-module, due to claim 11.4.32. We construct a presentation of C in the following way. First of all, we have a short exact sequence of R(γ) ⊗_{K^{+}} R(γ)-modules:

\underline{E} : 0 → Δ → R(γ) ⊗_{K^{+}} R(γ) →^{μ} R(γ) → 0

where μ is the multiplication map. The homomorphism R(γ) → R(γ) ⊗_{K^{+}} R(γ) : a ↦ 1 ⊗ a fixes an R(γ)-module structure on every R(γ) ⊗_{K^{+}} R(γ)-module (the right R(γ)-module structure), and clearly \underline{E} is split exact, when regarded as a sequence of R(γ)-modules via this homomorphism. Moreover, in terms of the presentation (11.4.14), the R(γ) ⊗_{K^{+}} R(γ)-module Δ is generated by the elements X ⊗ 1 − 1 ⊗ X and Y ⊗ 1 − 1 ⊗ Y. Let n be the rank of the free K^{+}-module C (claim 11.4.30(iii)); there follows an exact sequence

\underline{E} ⊗_{R(γ)} C : 0 → Δ ⊗_{R(γ)} C → R(γ)^{⊕n} → C → 0

which we may and do view as a short exact sequence of R(γ)-modules, via the left R(γ)- module structure induced by the restriction of scalars R(γ) → R(γ) ⊗_{K^{+}} R(γ) : a ↦ a ⊗ 1. The elements X, Y ∈ R(γ) act as K^{+}-linear endomorphisms on C; one can then find bases (b_{i} | i = 1, …, n) and (b′_{i} | i = 1, …, n) of C, and elements a_{1}, …, a_{n} ∈ K^{+} \ {0} such that Xb_{i} = a_{i}b′_{i} for every i ≤ n. Since XY = c in R(γ), it follows that Yb′_{i} = ca_{i}^{−1}b_{i} for every i ≤ n. With this notation, it is clear that Δ ⊗_{R(γ)} C, with its left R(γ)-module structure, is generated by the elements:

X ⊗ b_{i} − 1 ⊗ a_{i}b′_{i} and Y ⊗ b′_{i} − 1 ⊗ ca_{i}^{−1}b_{i} (i = 1, …, n).

For every i ≤ n, let F_{i} be the R(γ)-module generated freely by elements (ε_{i}, ε′_{i}), and Δ_{i} ⊂ F_{i} the submodule generated by Xε_{i}−a_{i}ε′_{i} and Yε′_{i}−ca_{i}^{−1}ε_{i}. Moreover, let us write b′_{i} = ∑_{j=1}^{n} u_{ij}b_{j} with unique u_{ij} ∈ K^{+}, let F be the free R(γ)-module with basis (e_{i} | i = 1, …, n), and define φ : F → ⊕_{i=1}^{n}F_{i} by the rule: e_{i} ↦ ε′_{i} − ∑_{j=1}^{n} u_{ij}ε_{j} for every i ≤ n. We deduce a right exact sequence of R(γ)-modules:

F ⊕ ⨁_{i=1}^{n} Δ_{i} →^{ψ_{1}} ⨁_{i=1}^{n} F_{i} →^{ψ_{2}} C → 0

where:

ψ_{1}(f, d_{1}, …, d_{n}) = φ(f) + (d_{1}, …, d_{n}) for every f ∈ F and d_{i} ∈ Δ_{i}
ψ_{2}(ε_{i}) = b_{i} and ψ_{2}(ε′_{i}) = b′_{i}  for every i = 1, …, n.

Claim 11.4.33. ψ_{1} is injective.

Proof of the claim. Let L be the field of fractions of R(γ); since the domain of ψ_{1} is a torsionfree K^{+}-module, it suffices to verify that ψ ⊗_{R(γ)} 1_{L} is injective. However, on the one hand C ⊗_{R(γ)} L = 0, and on the other hand, each Δ_{i} ⊗_{R(γ)} L is an L-vector space of dimension one, so the claim follows by comparing dimensions. ♢

By inspecting the definitions and the proof of lemma 11.4.20, one sees easily that

(11.4.34)  Δ_{i} ≃ I(|a_{i}^{−1}c|) for every i ≤ n.

Moreover, by remark (7.3.50)(iii), there exist p ∈ ℕ and an R(γ)-linear isomorphism:

(11.4.35)  Ker ψ_{2} →^{∼} R(γ)^{⊕p} ⊕ Syz_{R(γ)}^{1}C.

On the other hand, remark (7.3.50)(iii) and claim 11.4.30(ii) also shows that there exist q ∈ ℕ and an A-linear isomorphism:

(11.4.36)  M →^{∼} A^{⊕q} ⊕ Syz_{A}^{1}C.

Combining (11.4.35) and (11.4.36) and using lemma 7.3.51, we deduce an R(γ)^{h}-linear isomorphism:

(R(γ)^{h})^{⊕q} ⊕ (R(γ)^{h} ⊗_{R(γ)} Ker ψ_{2}) →^{∼} (R(γ)^{h})^{⊕p+q} ⊕ (R(γ)^{h} ⊗_{R(γ)} Syz_{R(γ)}^{1}C)
→^{∼} (R(γ)^{h})^{⊕p+q} ⊕ Syz_{R(γ)^{h}}^{1}(R(γ)^{h} ⊗_{R(γ)} C)
→^{∼} (R(γ)^{h})^{⊕p+q} ⊕ (R(γ)^{h} ⊗_{A} Syz_{A}^{1}C)
→^{∼} (R(γ)^{h})^{⊕p} ⊕ (R(γ)^{h} ⊗_{A} M).

By claim 11.4.33 and (11.4.34) it follows that (R(γ)^{h})^{⊕p} ⊕ (R(γ)^{h} ⊗_{A} M) is a direct sum of modules of the form R(γ)^{h}⊗_{R(γ)}I(ρ_{i}), for various ρ_{i} ∈ Γ (recall that I(1) = R(γ)). Notice that every I(ρ_{i}) is generically of rank one, hence indecomposable. Then it follows from corollary 7.11.11 that R(γ)^{h} ⊗_{A} M is a direct sum of various indecomposable R(γ)^{h}-modules of the form R(γ)^{h} ⊗_{A} g^{∗}ℐ (ρ_{i})_{x}. Finally, we apply proposition 7.11.12 to conclude the proof of the theorem. □

The rest of this section shall be concerned with some results that hold in the special case where the valuation of K has rank one.

Theorem 11.4.37. Suppose that K is a valued field of rank one. Let f : X → S be a finitely presented morphism, ℱ a coherent 𝒪_{X}-module. Then :

(i) Every coherent proper submodule 𝒢 ⊂ ℱ admits a primary decomposition.

(ii) Ass ℱ is a finite set.

Proof. By lemma 10.5.15, assertion (ii) follows from (i). Using corollary 10.5.11(ii) we reduce easily to the case where X is affine, say X = Spec A for a finitely presented K^{+}-algebra A, and ℱ = M^{∼}, 𝒢 = N^{∼} for some finitely presented A-modules N ⊂ M ≠ 0. By considering the quotient M/N, we further reduce the proof to the case where N = 0. Let us choose a closed imbedding i : X → Spec B, where B := K^{+}[T_{1}, …, T_{r}] is a free polynomial K^{+}-algebra; by proposition 10.5.19, the submodule 0 ⊂ ℱ admits a primary decomposition if and only if the submodule 0 ⊂ i_{∗}ℱ does. Thus, we may replace A by B, and assume that A = K^{+}[T_{1}, …, T_{r}], in which case we shall argue by induction on r. Let ξ denote the maximal point of f^{−1}(s); by claim 9.1.33, V := 𝒪_{X,ξ} is a valuation ring with value group Γ. Then [75, Lemma 6.1.14] says that

M_{ξ} ≃ V^{⊕m} ⊕ (V/b_{1}V) ⊕ ⋯ ⊕ (V/b_{k}V)

for some m ∈ ℕ and elements b_{1}, …, b_{k} ∈ 𝔪_{K} \{0}. After clearing some denominators, we may then find a map φ : M′ := A^{⊕m} ⊕ (A/b_{1}A) ⊕ ⋯ ⊕ (A/b_{k}A) → M whose localization φ_{ξ} is an isomorphism.

Claim 11.4.38. Ass A/bA = {ξ} whenever b ∈ 𝔪_{K} \{0}.

Proof of the claim. Clearly Ass A/bA ⊂ f^{−1}(s). Let x ∈ f^{−1}(s) be a non-maximal point. Thus, the prime ideal 𝔭 ⊂ A corresponding to x is not contained in 𝔪_{K}A, i.e. there exists a ∈ 𝔭 such that |a|_{A} = 1. Suppose by way of contradiction, that x ∈ Ass A/bA; then we may find c ∈ A such that c ∉ bA but a^{n}c ∈ bA for some n ∈ ℕ. The conditions translate respectively as the inequalities :

|c|_{A} > |b|_{A} and |a^{n}|_{A} · |c|_{A} = |a^{n}c|_{A} ≤ |b|_{A}

which are incompatible, since |a^{n}|_{A} = 1. ♢

Since (Ker φ)_{ξ} = 0, claim 11.4.38 and proposition 10.5.6(ii) imply that Ass Ker φ = ∅; therefore Ker φ = 0, by lemma 10.5.5(iii). Moreover, the submodule N_{1} := A^{⊕m} (resp. N_{2} := (A/b_{1}A) ⊕ ⋯ ⊕ (A/b_{k}A)) of M′ is either ξ-primary (resp. {0}-primary), or else equal to M′; in the first case, 0 ⊂ M′ admits the primary decomposition 0 = N_{1} ∩ N_{2}, and in the second case, either 0 is primary, or else M′ = 0 has the empty primary decomposition. Now, if r = 0 then M = M′ and we are done. Suppose therefore that r > 0 and that the theorem is known for all integers < r. Set M′′ := Coker φ; clearly ξ ∉ Supp M′′, so by proposition 10.5.18 we are reduced to showing that the submodule 0 ⊂ M′′ admits a primary decomposition. We may then replace M by M′′ and assume from start that ξ ∉ Supp ℱ. Let ℱ^{t} ⊂ ℱ be the K^{+}-torsion submodule; clearly ℱ /ℱ^{t} is a flat K^{+}-module, hence it is a finitely presented 𝒪_{X}-module (proposition 9.1.27), so ℱ^{t} is a finitely generated 𝒪_{X}-module, by [37, §1, n.4, Prop.6]. Hence we may find a non-zero a ∈ 𝔪_{K} such that aℱ^{t} = 0, i.e. the natural map ℱ^{t} → ℱ /aℱ is injective. Denote by i : Z := V (Ann ℱ /aℱ) → X the closed immersion.

Claim 11.4.39. There exists a finite morphism g : Z → 𝔸_{S}^{r−1}.

Proof of the claim. Since ξ ∉ Supp ℱ, we have I ⊄ 𝔪_{K}A, i.e. we can find b ∈ I such that |b|_{A} = 1. Set W := V (a, b) ⊂ X; it suffices to exhibit a finite morphism W → 𝔸_{S}^{r−1}. By [33, §5.2.4, Prop.2], we can find an automorphism σ : A/aA → A/aA such that σ(b) = u · b′, where u is a unit, and b′ ∈ (K^{+}/aK^{+})[T_{1}, …, T_{r}] is of the form b′ = T_{r}^{k} + ∑_{j=0}^{k−1} a_{j}T_{r}^{j}, for some a_{0}, …, a_{k−1} ∈ (K^{+}/aK^{+})[T_{1}, …, T_{r−1}]. Hence the ring A/(aA + bA) = A/(aA + b′A) is finite over K^{+}[T_{1}, …, T_{r−1}], and the claim follows. ♢ Claim 11.4.40. The 𝒪_{X}-submodule 0 ⊂ ℱ /aℱ admits a primary decomposition.

Proof of the claim. Let 𝒢 := (ℱ /aℱ)_{|Z}. By our inductive assumption the submodule 0 ⊂ g_{∗}𝒢 on 𝔸_{S}^{r−1} admits a primary decomposition; a first application of proposition 10.5.19 then shows that the submodule 0 ⊂ 𝒢 on Z admits a primary decomposition, and a second application proves the same for the submodule 0 ⊂ i_{∗}𝒢 = ℱ /aℱ. ♢

Finally, let j : Y := V (a) ⊂ X be the closed immersion, and set U := X \Y. By construction, the natural map \underline{Γ}_{Y} ℱ → j_{∗}j^{∗}ℱ is injective. Furthermore, U is an affine noetherian scheme, so [126, Th.6.8] ensures that the 𝒪_{U}-submodule 0 ⊂ ℱ_{|U} admits a primary decomposition. The same holds for j^{∗}ℱ, in view of claim 11.4.40. To conclude the proof, it remains only to invoke proposition 10.5.21. □

Corollary 11.4.41. Let K be a valued field of rank one, f : X → S a finitely presented morphism, ℱ a coherent 𝒪_{X}-module, and \underline{ℐ} := (ℐ_{λ} | λ ∈ Λ) a cofiltered family such that :

(a) ℐ_{λ} ⊂ 𝒪_{X} is a coherent ideal for every λ ∈ Λ.

(b) ℐ_{λ} · ℐ_{μ} ∈ \underline{ℐ} whenever ℐ_{λ}, ℐ_{μ} ∈ \underline{ℐ} . Then the following holds :

(i) ℱ_{λ} := Ann_{ℱ} (ℐ_{λ}) ⊂ ℱ is a submodule of finite type for every λ ∈ Λ.

(ii) There exists λ ∈ Λ such that ℱ_{μ} ⊂ ℱ_{λ} for every μ ∈ Λ.

Proof. We easily reduce to the case where X is affine, say X = Spec A with A finitely presented over K^{+}, and for each λ ∈ Λ the ideal I_{λ} := Γ(X, ℐ_{λ}) ⊂ A is finitely generated.

(i): For given λ ∈ Λ, let f_{1}, …, f_{k} be a finite system of generators of I_{λ}; then ℱ_{λ} is the kernel of the map φ : ℱ → ℱ^{⊕k} defined by the rule : m ↦ (f_{1}m, …, f_{k}m), which is finitely generated because A is coherent.

(ii): By theorem 11.4.37 we can find primary submodules 𝒢_{1}, …, 𝒢_{n} ⊂ ℱ such that 𝒢_{1} ∩ ⋯ ∩ 𝒢_{n} = 0; for every i ≤ n and λ ∈ Λ set ℋ_{i} := ℱ /𝒢_{i} and ℋ_{i,λ} := Ann_{ℋ_{i}} (ℐ_{λ}). Since the natural map ℱ → ⊕_{i=1}^{n}ℋ_{i} is injective, we have :

ℱ_{λ} = ℱ ∩ (ℋ_{1,λ} ⊕ ⋯ ⊕ ℋ_{n,λ}) for every λ ∈ Λ.

It suffices therefore to prove that, for every i ≤ n, there exists λ ∈ Λ such that ℋ_{i,μ} = ℋ_{i,λ} for every μ ∈ Λ. Say that ℋ_{i} is 𝔭-primary, for some prime ideal 𝔭 ⊂ A; suppose now that there exists λ such that I_{λ} ⊂ 𝔭; since I_{λ} is finitely generated, we deduce that I_{λ}^{n}ℋ_{i} = 0 for n ∈ ℕ large enough; from (b) we see that ℐ_{λ}^{n} ∈ \underline{ℐ} , so (ii) holds in this case. In case I_{λ} ⊄ 𝔭 for every λ ∈ Λ, we have Ann_{ℋ_{i}} (ℐ_{λ}) = 0 for every λ ∈ Λ, so (ii) holds in this case as well. □

Corollary 11.4.42. Suppose A is a valuation ring with value group Γ_{A}, and φ : K^{+} → A is an essentially finitely presented local homomorphism from a valuation ring K^{+} of rank one. Then:

(i) If the valuation of K is not discrete, φ induces an isomorphism Γ →^{∼} Γ_{A}. (ii) If Γ ≃ ℤ and φ is flat, φ induces an inclusion Γ ⊂ Γ_{A}, and (Γ_{A} : Γ) is finite.

Proof. Suppose first that Γ ≃ ℤ; then A is noetherian, hence Γ_{A} is discrete of rank one as well, and the assertion follows easily. In case Γ is not discrete, we claim that A has rank ≤ 1. Indeed, suppose by way of contradiction, that the rank of A is higher than one, and let 𝔪_{A} ⊂ A be the maximal ideal; then we can find a, b ∈ 𝔪_{A} \{0} such that a^{−i}b ∈ A for every i ∈ ℕ. Let us consider the A-module M := A/bA; we notice that Ann_{M}(a^{i}) = a^{−i}bA form a strictly increasing sequence of ideals, contradicting corollary 11.4.41(ii). Next we claim that φ is flat. Indeed, suppose this is not the case; then A is a κ-algebra. Let 𝔭 ⊂ A be the maximal ideal; by lemma 9.1.32, we may assume that A/𝔭 is a finite extension of κ. Now, choose any finitely presented quotient \overline{A} of A supported at 𝔭; it follows easily that \overline{A} is integral over K^{+}, hence it is a finitely presented K^{+}-module by proposition 9.1.34(i), and its annihilator contains 𝔪_{K}, which is absurd in view of [75, Lemma 6.1.14]. Next, since Γ is not discrete, one sees easily that 𝔪_{K}A is a prime ideal, and in light of the foregoing, it must then be the maximal ideal. Now the assertion follows from proposition 9.1.34(ii). □

11.4.43. Let K be a valued field of rank one, B a finitely presented K^{+}-algebra, and consider a pair \underline{A} := (A, Fil_{•}A) consisting of a B-algebra and a B-algebra filtration on A; let also M be a finitely generated A-module, and Fil_{•}M a good \underline{A}-filtration on M (see definition 7.9.4). By definition, this means that the Rees module R(\underline{M})_{•} of the filtered \underline{A}-module \underline{M} := (M, Fil_{•}M) is finitely generated over the graded Rees B-algebra R(\underline{A})_{•}. We have :

Proposition 11.4.44. In the situation of (11.4.43), suppose that A is a finitely presented Balgebra, M is a finitely presented A-module, and Fil_{•}A is a good filtration. Then :

(i) R(\underline{A})_{•} is a finitely presented B-algebra, and R(\underline{M})_{•} is a finitely presented R(\underline{A})_{•}- module.

(ii) If furthermore, Fil_{•}A is a positive filtration (see definition 7.9.4(ii)), then Fil_{i}M is a finitely presented B-module, for every i ∈ ℤ.

Proof. By lemma 7.9.8, there exists a finite system of generators m := (m_{1}, …, m_{n}) of M, and a sequence of integers k := (k_{1}, …, k_{n}) such that Fil_{•}M is of the form (7.9.7).

(i): Consider first the case where A is a free B-algebra of finite type, say A = B[t_{1}, …, t_{p}], such that Fil_{•}A is the good filtration associated with the system of generators t := (t_{1}, …, t_{p}) and the sequence of integers r := (r_{1}, …, r_{p}), and moreover M is a free A-module with basis m. Then R(\underline{A})_{•} is also a free B-algebra of finite type (see example 7.9.5). Moreover, for every j ≤ n, let M_{j} ⊂ M be the A-submodule generated by m_{j}, and denote by Fil_{•}M_{j} the good \underline{A}-filtration associated with the pair (\{m_{j}\}, \{k_{j}\}); clearly Fil_{•}M = Fil_{•}M_{1} ⊕ ⋯ ⊕ Fil_{•}M_{n}, therefore R(\underline{M})_{•} = R(\underline{M}_{1})_{•} ⊕ ⋯ ⊕ R(\underline{M}_{n})_{•}, where \underline{M}_{j} := (M_{j}, Fil_{•}M_{j}) for every j ≤ n. Obviously each R(\underline{A})_{•}-module R(\underline{M}_{j})_{•} is free of rank one, so the proposition follows in this case.

Next, suppose that A is a free B-algebra of finite type, and M is arbitrary. Let F be a free A-module of rank n, e := (e_{1}, …, e_{n}) a basis of F, and define an A-linear surjection φ : F → M by the rule e_{j} ↦ m_{j} for every j ≤ n. Then φ is even a map of filtered \underline{A}- modules, provided we endow F with the good \underline{A}-filtration Fil_{•}F associated with the pair (e, k). More precisely, let N := Ker φ; then the filtration Fil_{•}M is induced by Fil_{•}F, meaning that Fil_{i}M := (N + Fil_{i}F)/N for every i ∈ ℤ. Obviously the natural map R(\underline{F})_{•} → R(\underline{M})_{•} is surjective, and its kernel is the Rees module R(\underline{N})_{•} corresponding to the filtered \underline{A}-module \underline{N} := (N, Fil_{•}N) with Fil_{i}N := N ∩ Fil_{i}F for every i ∈ ℤ. Let x := (x_{1}, …, x_{s}) be a finite system of generators of N, and choose a sequence of integers j := (j_{1}, …, j_{s}) such that x_{i} ∈ Fil_{j_{i}}N for every i ≤ s; denote by \underline{L} the filtered \underline{A}-module associated as in (7.9.6), with the A-module N and the pair (x, j). Thus, R(\underline{L})_{•} is a finitely generated graded R(\underline{A})_{•}-submodule of R(\underline{N})_{•}. To ease notation, set \overline{N} := R(\underline{N})_{•}/R(\underline{L})_{•} and \overline{F} := R(\underline{F})_{•}/R(\underline{L})_{•}. Recall (definition 7.9.1(iii)) that R(\underline{A})_{•} is a graded B-subalgebra of A[U, U^{−1}]; then we have :

Claim 11.4.45. \overline{N} = ⋃_{n∈ℕ} Ann\overline{F} (U^{n}).

Proof of the claim. It suffices to consider a homogeneous element y := U^{i}z ∈ R(\underline{F})_{i}, for some some z ∈ Fil_{i}F. Suppose that U^{n}y ∈ R(\underline{L})_{i+n}; especially, U^{i+n}z ∈ R(\underline{N})_{i+n}, so z ∈ N, hence y ∈ R(\underline{N})_{i}. Conversely, suppose that y ∈ R(\underline{N})_{i}; write z = x_{1}a_{1} + ⋯ + x_{s}a_{s} for some a_{1}, …, a_{s} ∈ A. Say that a_{r} ∈ Fil_{b_{r}}A for every r ≤ s, and set l := max(i, b_{1} + j_{1}, …, b_{s} + j_{s}). Then U^{l−i}y = U^{l}z ∈ R(\underline{L})_{l}. ♢

By the foregoing, R(\underline{F})_{•} is finitely presented over R(\underline{A})_{•}. It follows that \overline{N} is finitely generated (corollary 11.4.41), hence R(\underline{N})_{•} is a finitely generated R(\underline{A})_{•}-module, so finally R(\underline{M})_{•} is finitely presented over R(\underline{A})_{•}.

Lastly, consider the case of an arbitrary finitely presented B-algebra A, endowed with the good B-algebra filtration associated with a system of generators x := (x_{1}, …, x_{p}) and the sequence of integers r. We map the free B-algebra P := B[t_{1}, …, t_{p}] onto A, by the rule: t_{j} ↦ x_{j} for every j ≤ p. This is even a map of filtered algebras, provided we endow P with the good filtration Fil_{•}P associated with the pair (t, r); more precisely, Fil_{•}P induces the filtration Fil_{•}A on A, hence Fil_{•}A is a good \underline{P} -filtration. By the foregoing case, we then deduce that R(\underline{A})_{•} is a finitely presented R(\underline{P})_{•}-module (where \underline{P} := (P, Fil_{•}P)), hence also a finitely presented K^{+}-algebra. Moreover, M is a finitely presented P -module, and clearly Fil_{•}M is good when regarded as a \underline{P} -filtration, hence R(\underline{M})_{•} is a finitely presented R(\underline{P})_{•}-module, so also a finitely presented R(\underline{A})_{•}-module.

(ii): The positivity condition implies that R(\underline{A})_{0} = B; then, taking into account (i), the assertion is just a special case of proposition 7.6.11(iii). □

Theorem 11.4.46 (Artin-Rees lemma). Let K be a valued field of rank one, A an essentially finitely presented K^{+}-algebra, I ⊂ A an ideal of finite type, M a finitely presented A-module, N ⊂ M a finitely generated submodule. Then there exists an integer c ∈ ℕ such that :

I^{n}M ∩ N = I^{n−c}(I^{c}M ∩ N) for every n ≥ c.

Proof. We reduce easily to the case where A is a finitely presented K^{+}-algebra; then, with all the work done so far, we only have to repeat the argument familiar from the classical noetherian case. Indeed, let us define a filtered K^{+}-algebra \underline{A} := (A, Fil_{•}A) by the rule : Fil_{n}A := I^{−n} if n ≤ 0, and Fil_{n}A := A otherwise; also let \underline{M} := (M, Fil_{•}M) be the filtered \underline{A}-module such that Fil_{i}M := M · Fil_{i}A for every i ∈ ℤ. We endow Q := M/N with the filtration Fil_{•}Q induced from Fil_{•}M, so that the natural projection M → M/N yields a map of filtered \underline{A}-modules \underline{M} → \underline{Q} := (Q, Fil_{•}Q). By proposition 11.4.44, there follows a surjective map of finitely presented graded R(\underline{A})_{•}-modules : π_{•} : R(\underline{M})_{•} → R(\underline{Q})_{•}, whose kernel in degree k ≤ 0 is the A-module I^{−k}M ∩ N. Hence we can find a finite (non-empty) system f_{1}, …, f_{r} of generators for the R(\underline{A})_{•}-module Ker π_{•}; clearly we may assume that each f_{i} is homogeneous, say of degree d_{i}; we may also suppose that d_{i} ≤ 0 for every i ≤ r, since Ker π_{0} generates Ker π_{n}, for every n ≥ 0. Let d := min (d_{i} | i = 1, …, r); by inspecting the definitions, one verifies easily that

R(\underline{A})_{i} · Ker π_{j} ⊂ R(\underline{A})_{i+j−d} · Ker π_{d} whenever i + j ≤ d and 0 ≥ j ≥ d.

Therefore Ker π_{d+k} = R(\underline{A})_{k} · Ker π_{d} for every k ≤ 0, so the assertion holds with c := −d. □

11.4.47. In the situation of theorem 11.4.46, let M be any finitely presented A-module; we denote by M^{∧} the I-adic completion of M, which is an A^{∧}-module.

Corollary 11.4.48. With the notation of (11.4.47), the following holds :

(i) The functor M ↦ M^{∧} is exact on the category of finitely presented A-modules. (ii) For every finitely presented A-module M, the natural map M ⊗_{A} A^{∧} → M^{∧} is an

isomorphism. (iii) A^{∧} is flat over A. If additionally I ⊂ rad A (the Jacobson radical of A), then A^{∧} is

faithfully flat over A.

Proof. (i): Let N ⊂ M be any injection of finitely presented A-modules; by theorem 11.4.46, the I-adic topology on M induces the I-adic topology on N. Then the assertion follows from proposition 8.2.13(i,v).

(ii): Choose a presentation A^{⊕p} → A^{⊕q} → M → 0. By (i) we deduce a commutative diagram with exact rows :

\xymatrix{A^{⊕p} ⊗_{A} A^{∧} \ar[r] \ar[d] & A^{⊕q} ⊗_{A} A^{∧} \ar[r] \ar[d] & M ⊗_{A} A^{∧} \ar[r] \ar[d] & 0 \\ (A^{⊕p})^{∧} \ar[r] & (A^{⊕q})^{∧} \ar[r] & M^{∧} \ar[r] & 0}

and clearly the two left-most vertical arrows are isomorphisms. The claim follows.

(iii): The first assertion means that the functor M ↦ M ⊗_{A} A^{∧} is exact; this follows from (i) and (ii), via a standard reduction to the case where M is finitely presented. Suppose next that I ⊂ rad A; to conclude, it suffices to show that the image of the natural map Spec A^{∧} → Spec A contains the maximal spectrum Max A ([126, Th.7.3]). Since the natural map Max A/I → Max A is a bijection, the latter assertion follows from the :

Claim 11.4.49. For every finitely presented A-module M, the natural map i_{M} : M → M^{∧} induces an isomorphism M/IM →^{∼} M^{∧}/IM^{∧}.

Proof of the claim. From (i) we get a natural identification : M^{∧}/(IM)^{∧} →^{∼} (M/IM)^{∧} →^{∼} M/IM, whose inverse is the map induced by i_{M}. However, by (ii) the image of (IM)^{∧} in M^{∧} is the same as the image of IM ⊗_{A} A^{∧}, which is the same as the image of IM^{∧}. □

Theorem 11.4.50. Let K be a valued field of rank one, B → A a map of finitely presented K^{+}-algebras. Then :

(i) If M is an ω-coherent A-module, M is ω-coherent as a B-module.

(ii) If J is a coh-injective B-module, and I ⊂ B is a finitely generated ideal, we have : (a) Hom_{B}(A, J) is a coh-injective A-module.

(b) ⋃_{n∈ℕ} Ann_{J}(I^{n}) is a coh-injective B-module. (iii) If (J_{n}, φ_{n} | n ∈ ℕ) is a direct system consisting of coh-injective B/I^{n}-modules J_{n} and

B-linear maps φ_{n} : J_{n} → J_{n+1} (for every n ∈ ℕ), then colim_{n∈ℕ} J_{n} is a coh-injective B-module.

Proof. (See (7.11.25) for the generalities on coh-injective and ω-coherent modules.)

(i): We reduce easily to the case where M is finitely presented over A. Let x_{1}, …, x_{p} be a system of generators for the B-algebra A, and m_{1}, …, m_{n} a system of generators for the Amodule M. We let \underline{A} := (A, Fil_{•}A), where Fil_{•}A is the good B-algebra filtration associated with the pair x := (x_{1}, …, x_{p}) and r := (1, …, 1); likewise, let \underline{M} be the filtered \underline{A}-module defined by the good \underline{A}-filtration on M associated with m := (m_{1}, …, m_{n}) and k := (0, …, 0) (see definition 7.9.4). Then claim follows easily after applying proposition 11.4.44(ii) to the filtered B-algebra \underline{A} and the filtered \underline{A}-module \underline{M}.

(ii.a): Let N ⊂ M be two coherent A-modules, and φ : N → Hom_{B}(A, J) an A-linear map. According to claim 10.1.17, φ corresponds by adjunction to a unique B-linear map \overline{φ} : N → J; on the other hand by (i), M, and M/N are ω-coherent B-modules, hence \overline{φ} extends to a Blinear map \overline{ψ} : M → J (lemma 7.11.27). Under the adjunction, \overline{ψ} corresponds to an A-linear extension ψ : M → Hom_{B}(A, J) of φ.

(iii): Let M ⊂ N be two finitely presented B-modules, f : M → J := colim_{n∈ℕ} J_{n} a B-linear map. Since M is finitely generated, f factors through a map f_{n} : M → J_{n} and the natural map J_{n} → J, provided n is large enough ([75, Prop.2.3.16(ii)]). By theorem 11.4.46 there exists c ∈ ℕ such that I^{n+c}N ∩ M ⊂ I^{n}M. Hence

f_{n+c} := φ_{n+c−1} ∘ ⋯ ∘ φ_{n} ∘ f_{n} : M → J_{n+c}

factors through a unique B/I^{n+c}-linear map \overline{f}_{n+c} : \overline{M} := M/(I^{n+c}N ∩M) → J_{n+c}; since I^{n+c} is finitely generated, \overline{M} is a coherent submodule of the coherent B-module \overline{N} := N/I^{n+c}N, therefore \overline{f}_{n+c} extends to a map \overline{g} : \overline{N} → J_{n+c}. The composition of \overline{g} : \overline{N} → J_{n+c} with the projection N → \overline{N} and the natural map J_{n+c} → J, is the sought extension of f.

(ii.b): Letting A := B/I^{n} in (ii.a), we deduce that J_{n} := Ann_{J}(I^{n}) is a coh-injective B/I^{n}- module, for every n ∈ ℕ. Then the assertion follows from (iii). □

11.5. Local duality. Throughout this section we let (K, | · |) be a valued field of rank one. We shall continue to use the general notation of (11.4).

11.5.1. Let A be finitely presented K^{+}-algebra, I ⊂ A an ideal generated by a finite system f := (f_{i} | i = 1, …, r), and denote by i : Z := V (I) → X := Spec A the natural closed immersion. Let also ℐ ⊂ 𝒪_{X} be the ideal arising from I. For every n ≥ 0 there is a natural epimorphism

i_{∗}i^{−1}𝒪_{X} → 𝒪_{X}/ℐ^{n}

whence natural morphisms in D(A-Mod) :

(11.5.2)  RHom_{𝒪_{X}}^{•} (𝒪_{X}/ℐ^{n}, ℱ^{•}) → RHom_{𝒪_{X}}^{•} (i_{∗}i^{−1}𝒪_{X}, ℱ^{•}) →^{∼} RΓ_{Z}ℱ^{•}

for any bounded below complex ℱ^{•} of 𝒪_{X}-modules.

Theorem 11.5.3. In the situation of (11.5.1), let M^{•} be any object of D^{+}(A-Mod). Then (11.5.2) induces natural isomorphisms :

colim_{n∈ℕ} Ext_{A}^{i}(A/I^{n}, M^{•}) →^{∼} R^{i}Γ_{Z}M^{•∼} for every i ∈ ℕ.

Proof. For ℱ^{•} := M^{•∼}, trivial duality (theorem 10.1.18) identifies the source of (11.5.2) with RHom_{A}^{•}(A/I^{n}, M^{•}); then one takes cohomology in degree i and forms the colimit over n to define the sought map. Next, by usual spectral sequence arguments, we may reduce to the case where M^{•} is a single A-module M sitting in degree zero (see e.g. the proof of proposition 10.3.21(i)). By inspecting the definitions, one verifies easily that the morphism thus defined is the composition of the isomorphism of proposition 10.4.32(iii) and the map (7.8.41) (see the proof of proposition 10.4.32(iii)); then it suffices to show that the inverse system (H_{i}K_{•}(f^{n}) | n ∈ ℕ) is essentially zero when i > 0 (lemma 7.8.42). By lemma 7.8.44, this will in turn follow, provided the following holds. For every finitely presented quotient B of A, and every b ∈ B, there exists p ∈ ℕ such that Ann_{B}(b^{q}) = Ann_{B}(b^{p}) for every q ≥ p. This latter assertion is a special case of corollary 11.4.41. □

Corollary 11.5.4. In the situation of theorem 11.5.3, we have natural isomorphisms :

colim_{n∈ℕ} Ext_{A}^{i}(I^{n}, M^{•}) →^{∼} H^{i}(X \Z, M^{•∼}) for every i ∈ ℕ.

Proof. Set U := X \ Z. We may assume that M^{•} is a complex of injective A-modules, in which case the sought map is obtained by taking colimits over the direct system of composed morphisms :

Hom_{A}(I^{n}, M^{•}) ⟶^{β_{n}} Γ(U, M^{•}) → RΓ(U, M^{•∼})

where β_{n} is induced by the identification (I^{n})_{|U}^{∼} = 𝒪_{U} and the natural isomorphism :

Hom_{𝒪_{U}} (𝒪_{U}, M^{•∼}) ≃ Γ(U, M^{•∼}).

The complex RΓ(U, M^{•∼}) is computed by a Cartan-Eilenberg injective resolution M^{•∼} →^{∼} ℳ^{•} of 𝒪_{X}-modules, and then the usual arguments allow to reduce to the case where M^{•} consists of a single injective A-module M placed in degree zero. Finally, from the short exact sequence 0 → I^{n} → A → A/I^{n} → 0 we deduce a commutative ladder with exact rows :

[figure: 0; Hom_{A}(A/I^{n}, M); α_{n}; M; Hom_{A}(I^{n}, M); β_{n}; 0; 0; Γ_{Z}M^{∼}; M; Γ(U, M^{∼}); R^{1}Γ_{Z}M^{∼}; 0]

where α_{n} is induced from (11.5.2). From theorem 11.5.3 it follows that colim_{n∈ℕ} α_{n} is an isomorphism, and R^{i}Γ_{Z}M^{∼} vanishes for all i > 0; hence colim_{n∈ℕ} β_{n} is an isomorphism, and the contention follows. □

Proposition 11.5.5. Let X ⟶^{f} Y ⟶^{g} S be two finitely presented morphisms of schemes. (i) If g is smooth, 𝒪_{Y} [0] is a dualizing complex on Y. (ii) If g is smooth and f is a closed immersion, then X admits a dualizing complex ω_{X}^{•}. (iii) If X and Y are affine and ω_{Y}^{•} is a dualizing complex on Y, then f^{!}ω_{Y}^{•} is dualizing on

X (notation of (11.1.15)). (iv) Every finitely presented quasi-separated S-scheme admits a dualizing complex.

Proof. (i) is an immediate consequence of proposition 11.4.2(i).

(ii): As in the proof of corollary 11.3.65, this follows directly from (i) and lemma 11.3.27(i). (iii): Let us choose a factorization f = p_{Y} ∘i where i : X → 𝔸_{Y}^{n} is a finitely presented closed

immersion, and p_{Y} : 𝔸_{Y}^{n} → Y the smooth projection onto Y. In view of lemma 11.3.27(i) and (11.1.15), it suffices to prove the assertion for the morphism p_{Y}. To this aim, we pick a closed finitely presented immersion h : Y → 𝔸_{S}^{m} and consider the fibre diagram :

\xymatrix{𝔸_{Y}^{n} \ar[r]^{h′} \ar[d]_{p_{Y}} & 𝔸_{S}^{n+m} \ar[d]^{p_{S}} \\ Y \ar[r]^{h} & 𝔸_{S}^{m}.}

By (i) we know that the scheme 𝔸_{S}^{m} admits a dualizing complex ω^{•}, and then lemma 11.3.27(i) says that h^{!}ω^{•} is dualizing as well. By proposition 11.3.25we deduce that ω_{Y}^{•} ≃ ℒ [σ]⊗_{𝒪_{Y}} h^{!}ω^{•} for some invertible 𝒪_{Y} -module ℒ and some continuous function σ : |Y | → ℤ. Since p_{Y} is smooth, we can compute: p_{Y}^{!} ω_{Y}^{•} ≃ (p_{Y}^{!} ∘ h^{!}ω^{•}) ⊗_{𝒪_{𝔸^{n}_{Y}}} p_{Y}^{∗} ℒ [σ], hence p_{Y}^{!} ω_{Y}^{•} is dualizing if and only if the same holds for p_{Y}^{!} ∘h^{!}ω^{•}. By proposition 11.1.7(iv), the latter complex is isomorphic to h^{′♭} ∘ p_{S}^{!}ω^{•} and again using lemma 11.3.27(i) we reduce to checking that p_{S}^{!}ω^{•} is dualizing, which is clear from (i).

(iv): Let f : X → S be a finitely presented morphism, with X quasi-separated. If X is affine, (iii) says that f^{!}𝒪_{S}[0] is dualizing on X. In the general case, let (U_{i} | i = 1, …, n) be a finite covering of X consisting of affine open subsets; for each i, j, k = 1, …, n, denote by f_{i} : U_{i} → S the restriction of f, set U_{ij} := U_{i} ∩ U_{j}, U_{ijk} := U_{ij} ∩ U_{k}, and let g_{ij} : U_{ij} → U_{i} be the inclusion map. We know that f_{i}^{!}𝒪_{S}[0] is dualizing on U_{i}, for every i = 1, …, n; moreover, for every i, j = 1, …, n there exists a natural isomorphism

ψ_{ij} : g_{ij}^{∗}f_{i}^{!}𝒪_{S}[0] →^{∼} g_{ji}^{∗}f_{j}^{!}𝒪_{S}[0]

fulfilling the cocycle condition

ψ_{jk|U_{ijk}} ∘ ψ_{ij|U_{ijk}} = ψ_{ik|U_{ijk}} for every i, j, k = 1, …, n

(lemma 11.1.16). In other words, ((U_{i}, f_{i}^{!}𝒪_{S}[0]); ψ_{ij} | i, j = 1, …, n) is a descent datum for the fibration (11.3.55). Then the assertion follows from proposition 11.3.56. □

Example 11.5.6. (i) For given b ∈ 𝔪_{K}, let i_{b} : S_{/b} → S be the closed immersion. If b ≠ 0, a simple computation yields a natural isomorphism in D(𝒪_{S_{/b}}-Mod) :

i_{b}^{!}𝒪_{S} →^{∼} 𝒪_{S_{/b}}[−1].

By proposition 11.5.5(i,iii), we deduce that 𝒪_{S_{/b}}[0] is a dualizing complex on S_{/b}, for every b ∈ 𝔪_{K}.

(ii) Next, let f : X → S_{/b} be an affine finitely presented Cohen-Macaulay morphism, of constant fibre dimension n. Then ω_{X}^{•} := f^{!}𝒪_{S_{/b}}[0] is a dualizing complex on X, by (i) and proposition 11.5.5(iii). Moreover, ω_{X}^{•} is concentrated in degree −n, and H^{−n}ω_{X}^{•} is a finitely presented f-Cohen-Macaulay 𝒪_{X}-module. Indeed, let i : X → Y := 𝔸_{S_{/b}}^{m} be a closed immersion of S_{/b}-schemes, and denote by g : Y → S_{/b} the projection. Fix any x ∈ X, let y := i(x) and set

d_{x} := dim 𝒪_{f^{−1}(fx),x} d_{y} := dim 𝒪_{g^{−1}(gy),y}.

We have a natural isomorphism in D(𝒪_{X}-Mod)

ω_{X}^{•} →^{∼} i^{!}𝒪_{Y} [m] →^{∼} i^{∗}Rℋ om_{𝒪_{Y}}^{•} (i_{∗}𝒪_{X}, 𝒪_{Y} [m])

(lemma 11.1.16(i)), and by assumption i_{∗}𝒪_{X} is a g-Cohen-Macaulay 𝒪_{Y} -module; by theorem 11.3.67(ii), it follows that ω_{X,x}^{•} is concentrated in degree d_{y} −d_{x}−m, and its cohomology in that degree is f-Cohen-Macaulay, as asserted. Lastly, since the fibres of f and g are equidimensional ([64, Ch.IV, Prop.5.2.1] and lemma 11.3.62(ii)), it is easily seen that d_{y} − d_{x} = m − n (details left to the reader), whence the claim.

11.5.7. For any finitely presented morphism f : X → S we consider the map :

d : |X| → ℤ x ↦ tr. deg(κ(x)/κ(f(x))) + dim \overline{\{f(x)\}}.

Lemma 11.5.8. With the notation of (11.5.7), let x, y ∈ X, and suppose that x is a specialization of y. We have :

(i) x is an immediate specialization of y if and only if d(y) = d(x) + 1.

(ii) If X is irreducible, d(x) − d(y) = dim X(y) − dim X(x).

(iii) X is catenary and of finite Krull dimension.

(iv) If f is flat, then dim 𝒪_{X,x} = dim 𝒪_{f^{−1}(fx),x} + dim 𝒪_{S,f(x)}.

(v) If f is Cohen-Macaulay at the point x, then 𝒪_{X,x} is equidimensional.

Proof. (i): In case f(x) = f(y), the assertion follows from [64, Ch.IV, Prop.5.2.1]. Hence, suppose that f(x) = s, f(y) = η; let Z be the topological closure of {y} in X, and endow Z with its reduced subscheme structure; notice that Z is an S-scheme of finite type. By assumption, X is quasi-compact and quasi-separated, hence {y} is a pro-constructible subset of X, and therefore Z is the set of all specializations of y in X ([63, Ch.IV, Th.1.10.1]). Especially, y is the unique maximal point of Z_{η} := Z ∩ f^{−1}(η); also, if x is an immediate specialization of y, then x is a maximal point of Z_{s} := Z ∩ f^{−1}(s), and by lemma 11.4.1(i) and [64, Ch.IV, Prop.5.2.1], the latter implies that d(y) = d(x) + 1.

Conversely, suppose that d(y) = d(x) + 1; from lemma 11.4.1(i) and [64, Ch.IV, Prop.5.2.1] we deduce that x is a maximal point of Z_{s}; then x is an immediate specialization of y in X, since otherwise x would be an immediate specialization in X of a proper specialization y′ of y in Z_{η}, and in this case, by the foregoing we would get d(y′) = d(x) + 1, i.e. d(y) = d(y′), contradicting [64, Ch.IV, Prop.5.2.1].

(iii) It is easily seen that X has finite Krull dimension. Now, consider any sequence y_{0}, …, y_{n} of points of X, with y_{0} := y, y_{n} := x, and such that y_{i+1} is an immediate specialization of y_{i}, for i = 0, …, n − 1; from (i) we deduce that n = d(y) − d(x), especially n is independent of the chosen chain of specializations, so X is catenary.

(ii): We reduce easily to the case where y is the maximal point of X, in which case we may argue as in the proof of (iii) (details left to the reader).

(iv): If f(x) = η, the identity is [64, Ch.IV, Cor.6.1.2], hence we may assume that f(x) = s, and we need to check that

dim X(x) = dim 𝒪_{f^{−1}(s),x} + 1.

Now, notice that – by the flatness assumption – every irreducible component of X intersect f^{−1}(η), and it is therefore itself a flat S-scheme (with its reduced subscheme structure). Since f^{−1}(η) is a noetherian scheme, it also follows that the set of irreducible components of X is finite; thus, let X_{1}, …, X_{n} be the reduced irreducible components of X containing x, and f_{i} : X_{i} → S (i = 1, …, n) the corresponding restrictions of f. Then

dim 𝒪_{X,x} = max_{1≤i≤n} dim 𝒪_{X_{i},x}  and  dim 𝒪_{f^{−1}(s),x} = max_{1≤i≤n} dim 𝒪_{f_{i}^{−1}(s),x}

so that it suffices to show the stated identity for each of the morphisms f_{1}, …, f_{n}. Hence, we may assume that X is irreducible, and we let z be the maximal point of X, and y any maximal generization of x in f^{−1}(s). By lemma 11.4.1(i), f^{−1}(s) is equidimensional, and we have

dim 𝒪_{f^{−1}(s),x} = dim f^{−1}(s) − d(x)  and  dim f^{−1}(s) = d(y)

([64, Ch.IV, Cor.5.2.3]). Arguing as in the proof of (i), we see that z is an immediate generization of y, hence d(z) = d(y) + 1. So, we come down to showing that dim X(x) = d(z) − d(x); the latter follows from (ii), since dim X(z) = 0.

(v): By lemma 11.3.62(ii), the morphism f is equidimensional at the point x. Especially, f(u) = η for every maximal point u of Spec 𝒪_{X,x}, and there exist an open neighborhood U of x in X and an integer e ∈ ℕ, such that f^{−1}f(y) is equidimensional of dimension e, for every y ∈ U. Since f is finitely presented, it follows that

tr. deg(κ(y)/κ(f(y))) = e

for every y ∈ U such that y is maximal in f^{−1}f(y) ([126, Th.5.6]). Hence, d(u) = e + 1 for every maximal point u of Spec 𝒪_{X,x}, and especially, for every such u, the difference d(x)−d(u) is independent of u; in light of (i), the assertion follows. □

Lemma 11.5.9. (i) The K^{+}-module K/K^{+} is coh-injective.

(ii) Let A be a finitely presented K^{+}-algebra, I ⊂ A a finitely generated ideal, J a cohinjective K^{+}-module. Then the A-module

J_{A} := colim_{n∈ℕ} Hom_{K^{+}}(A/I^{n}, J)

is coh-injective.

Proof. (i): It suffices to show that Ext_{K^{+}}^{1}(M, K/K^{+}) = 0 whenever M is a finitely presented K^{+}-module. This is clear when M = K^{+}, and then [75, Ch.6, Lemma 6.1.14] reduces to the case where M = K^{+}/aK^{+} for some a ∈ 𝔪_{K} \{0}; in that case we can compute using the free resolution K^{+} ⟶^{a} K^{+} → M, and the claim follows easily.

(ii): According to theorem 11.4.50(ii.a), the A-module J′_{A} := Hom_{K^{+}}(A, J) is coh-injective. However, J_{A} = ⋃_{n∈ℕ} Ann_{J′_{A}}(I^{n}), so the assertion follows from theorem 11.4.50(ii.b). □

Theorem 11.5.10. Let f : X → S be a finitely presented affine morphism. Then for every point x ∈ X, the 𝒪_{X,x}-module

J(x) := R^{1−d(x)}Γ_{{x}}(f^{!}𝒪_{S}[0])_{|X(x)}

is coh-injective, and we have a natural isomorphism in D(𝒪_{X,x}-Mod) :

RΓ_{{x}}(f^{!}𝒪_{S}[0])_{|X(x)} ≃ J(x)[d(x) − 1].

Proof. Fix x ∈ X and set d := d(x).

Claim 11.5.11. The theorem holds in case f(x) = η.

Proof of the claim. Indeed, let f_{η} : f^{−1}(η) → S(η) := Spec K be the restriction of f; according to lemma 11.1.16(i), f_{η}^{!}𝒪_{S(η)}[0] is the restriction of f^{!}𝒪_{S}[0], so the claim follows immediately from example 11.3.51. ♢

Hence, suppose that f(x) = s, the closed point of S, and say that X = Spec A.

Claim 11.5.12. We may assume that x is closed in X, hence that κ(x) is finite over κ(s).

Proof of the claim. Arguing as in the proof of lemma 9.1.32, we can find a factorization of the morphism f as a composition X ⟶^{g} Y := 𝔸_{S}^{d} ⟶^{h} S, such that ξ := g(x) is the generic point of h^{−1}(s) ⊂ Y, the morphism g is finitely presented, and the stalk 𝒪_{Y,ξ} is a valuation ring. Moreover, let g_{y} := g ×_{Y} 1_{Y (y)} : X(y) := X ×_{Y} Y (y) → Y (y); we have a natural isomorphism

(f^{!}𝒪_{S}[0])_{|X(y)} ≃ g_{y}^{!}𝒪_{Y (y)}[d].

Then we may replace f by g_{y}, and K^{+} by 𝒪_{Y,ξ}, whence the claim. ♢

Hence, suppose now that x is closed in X, let 𝔭 ⊂ A be the maximal ideal corresponding to x, and \overline{𝔭} the image of 𝔭 in \overline{A} := A ⊗_{K^{+}} κ; we choose a finite system of elements b_{1}, …, b_{r} ∈ 𝒪_{X}(X) whose images in \overline{A} generate \overline{𝔭}. Pick any non-zero a ∈ 𝔪_{K}, and let I ⊂ A be the ideal generated by the system (a, b_{1}, …, b_{r}), and ℐ ⊂ 𝒪_{X} the corresponding coherent ideal; clearly V (I) = {x}, hence theorem 11.5.3yields a natural isomorphism :

(11.5.13)  colim_{n∈ℕ} R^{i}Hom_{𝒪_{X}}^{•} (𝒪_{X}/ℐ^{n}, f^{!}𝒪_{S}[0]) →^{∼} R^{i}Γ_{{x}}f^{!}𝒪_{S}[0]  for every i ∈ ℤ

where the transition maps in the colimit are induced by the natural maps 𝒪_{X}/ℐ^{n} → 𝒪_{X}/ℐ^{m}, for every n ≥ m. However, from lemmata 11.1.6(ii), 11.1.16(i) and corollary 10.3.2(i) we deduce as well natural isomorphisms :

RHom_{𝒪_{X}}^{•} (𝒪_{X}/ℐ^{n}, f^{!}𝒪_{S}[0]) →^{∼} RHom_{𝒪_{S}}^{•} (f_{∗}𝒪_{X}/ℐ^{n}, 𝒪_{S}) →^{∼} RHom_{K^{+}}^{•}(A/I^{n}, K^{+}).

We wish to compute these Ext groups by means of lemma 7.11.27(iii); to this aim, let us remark first that A/I^{n} is an integral K^{+}-algebra for every n ∈ ℕ, hence it is a finitely presented torsion K^{+}-module, according to proposition 9.1.34(i). We may then use the coh-injective resolution K^{+}[0] → (0 → K → K/K^{+} → 0) (lemmata 11.5.9(i), 7.11.27(iii)) to compute :

Ext_{K^{+}}^{i}(A/I^{n}, K^{+}) = \{ \begin{matrix}J_{n} := Hom_{K^{+}}(A/I^{n}, K/K^{+}) & if i = 1 \\ 0 & otherwise. \end{matrix}

Now the theorem follows from lemma 11.5.9(i) and theorem 11.4.50(ii.a),(iii).

We could also appeal directly to lemma 11.5.9(ii), provided we already knew that the foregoing natural identifications transform the direct system whose colimit appears in (11.5.13), into the direct system (J_{n} | n ∈ ℕ) whose transition maps are induced by the natural maps A/I^{n} → A/I^{m}, for every n ≥ m. For the sake of completeness, we check this latter assertion.

For every n ≥ m, let j_{n} : X_{n} := Spec A/I^{n} → X and j_{mn} : X_{m} → X_{n} be the natural closed immersions. We have a diagram of functors :

\xymatrix{j_{m∗} ∘ j_{mn}^{!} ∘ (f ∘ j_{n})^{!} \ar[r]^{ζ_{1}} \ar@{-->}[dr]^{ε_{1}} & j_{n∗} ∘ (j_{mn∗} ∘ j_{mn}^{!}) ∘ j_{n}^{!} ∘ f^{!} \ar[dr]^{ε_{2}} & j_{m∗} ∘ j_{m}^{!} ∘ f^{!} \ar[l]_{ξ_{2}} \ar[d]^{α_{mn}} \\ j_{m∗} ∘ (f ∘ j_{m})^{!} \ar[u]^{ξ_{1}} \ar[r]^{β_{mn}} & j_{n∗} ∘ (f ∘ j_{n})^{!} \ar[r]^{ζ_{2}} & j_{n∗} ∘ j_{n}^{!} ∘ f^{!}}

where :

• ζ_{1} and ζ_{2} are induced by the natural isomorphism ψ_{f,j_{n}} : (f ∘ j_{n})^{!} →^{∼} j_{n}^{!} ∘ f^{!} of lemma 11.1.16(i).

• ξ_{1} (resp. ξ_{2}) is induced by the isomorphism ψ_{f∘j_{n},j_{mn}} (resp. ψ_{j_{n},j_{mn}}).

• ε_{1} and ε_{2} are induced by the counit of adjunction j_{mn∗} ∘ j_{mn}^{!} → 1_{D^{+}(𝒪_{Xn}-Mod)} .

• β_{mn} (resp. α_{mn}) is induced by the natural map (f ∘ j_{n})_{∗}𝒪_{X_{n}} → (f ∘ j_{m})_{∗}𝒪_{X_{m}} (resp.

by the map j_{n∗}𝒪_{X_{n}} → j_{m∗}𝒪_{X_{m}}).

It follows from lemma 11.1.22that the two triangular subdiagrams commute, and it is also clear that the same holds for the inner quadrangular subdiagram. Moreover, lemma 11.1.16(ii) yields a commutative diagram

(11.5.14)  \xymatrix{(f ∘ j_{m})^{!} \ar[d]_{ψ_{f∘jn,jmn}} \ar[r]^{ψ_{f,jm}} & j_{m}^{!} ∘ f^{!} \ar[d]^{ψ_{jn,jmn}∘f^{!}} \\ j_{mn}^{!} ∘ (f ∘ j_{n})^{!} \ar[r]^{j_{mn}^{!}(ψ_{f,jn})} & j_{mn}^{!} ∘ j_{n}^{!} ∘ f^{!}}

such that j_{m∗}(11.5.14) is the diagram :

\xymatrix{j_{m∗} ∘ (f ∘ j_{m})^{!} \ar[d]_{ξ_{1}} \ar[r]^{j_{m∗}(ψ_{f,jm})} & j_{m∗} ∘ j_{m}^{!} ∘ f^{!} \ar[d]^{ξ_{2}} \\ j_{m∗} ∘ j_{mn}^{!} ∘ (f ∘ j_{n})^{!} \ar[r]^{ζ_{1}} & j_{m∗} ∘ j_{mn}^{!} ∘ j_{n}^{!} ∘ f^{!}.}

We then arrive at the following commutative diagram :

[figure: RHom_{𝒪_{S}}^{•} (𝒪_{X}/ℐ^{m}, 𝒪_{S}); RΓ(ψ_{f,jm}); RΓ(β_{mn}); RHom_{𝒪_{X}}^{•} (𝒪_{X}/ℐ^{m}, f^{!}𝒪_{S}[0]); RΓ(α_{mn}); RHom_{𝒪_{S}}^{•} (𝒪_{X}/ℐ^{n}, 𝒪_{S}); RΓ(ψ_{f,jn}); RHom_{𝒪_{X}}^{•} (𝒪_{X}/ℐ^{n}, f^{!}𝒪_{S}[0]).]

Now, the maps RΓ(α_{mn}) are the transition morphisms of the inductive system whose colimit appears in (11.5.13), hence we may replace the latter by the inductive system formed by the maps RΓ(β_{mn}). Combining with corollary 10.3.2(i), we finally deduce natural A-linear isomorphisms

colim_{n∈ℕ} Ext_{K^{+}}^{i}(A/I^{n}, K^{+}) →^{∼} R^{i}Γ_{{x}}f^{!}𝒪_{S}[0]

where the transition maps in the colimit are induced by the natural maps A/I^{n} → A/I^{m}, for every n ≥ m. Our assertion is an immediate consequence. □

11.5.15. Let A be a local ring, set X := Spec A let x ∈ X be the closed point, and suppose that

(a) either, A is noetherian and X admits a dualizing complex

(b) or else, A is essentially of finite presentation over K^{+}. Notice that X admits a dualizing complex in case (b) as well. Indeed, in that case one can find a finitely presented affine S-scheme Y and a point y ∈ Y such that X ≃ Spec 𝒪_{Y,y}; by proposition 11.5.5(iv), Y admits a dualizing complex ω_{Y}^{•}, and since every coherent 𝒪_{X}-module extends to a coherent 𝒪_{Y} -module, one verifies easily that the restriction of ω_{Y}^{•} is dualizing for X (cp. the proof of proposition 11.3.37). Hence, let ω^{•} be a dualizing complex for X. In case (a) (resp. in case (b)), it follows easily from corollary 11.3.46 (resp. from theorem 11.5.10 and propositions 11.5.5(iii) and 11.3.25), that there exists a unique c ∈ ℤ such that

J(x) := R^{c}Γ_{{x}}ω^{•} ≠ 0.

Moreover, J(x) is a coh-injective A-module, hence by lemma 7.11.27(iii), we obtain a well defined functor

(11.5.16)  D : D^{b}(A-Mod) → D^{b}(A-Mod)^{o} : C^{•} ↦ Hom_{A}^{•}(C^{•}, J(x)).

Furthermore, let D_{{x}}^{b}(A-Mod_{coh}) be the full subcategory of D^{b}(A-Mod_{coh}) consisting of all complexes C^{•} such that Supp H^{•}C^{•} ⊂ {x}. We have the following :

Corollary 11.5.17 (Local duality). In the situation of (11.5.15), the following holds : (i) The functor (11.5.16) restricts to an equivalence of categories :

D_{x} : D_{{x}}^{b}(A-Mod_{coh}) →^{∼} D_{{x}}^{b}(A-Mod_{coh})^{o}

and the natural transformation C_{•} → D_{x} ∘ D_{x}(C_{•}) is an isomorphism of functors. (ii) For every i ∈ ℤ there exists a natural isomorphism of functors :

R^{i}Γ_{{x}} ∘ 𝒟 →^{∼} D ∘ R^{c−i}Γ : D^{b}(𝒪_{X}-Mod)_{coh} → A-Mod^{o}

where 𝒟 is the duality functor corresponding to ω^{•} (see definition 11.3.16). (iii) Let c and 𝒟 be as in (ii), I ⊂ A a finitely generated ideal such that V (I) = {x}, and

denote by A^{∧} the I-adic completion of A. Then for every i ∈ ℤ there exists a natural isomorphism of functors :

D ∘ R^{i}Γ_{{x}} ∘ 𝒟 →^{∼} A^{∧} ⊗_{A} R^{c−i}Γ : D^{b}(𝒪_{X}-Mod)_{coh} → A^{∧}-Mod_{coh}^{o}.

Proof. Let C_{•} be any object of D_{{x}}^{b}(A-Mod_{coh}); we denote by i : {x} → X the natural closed immersion. Obviously C_{•}^{∼} = i_{∗}i^{−1}C_{•}^{∼} (notation of (10.3)), therefore :

Hom_{A}^{•}(C_{•}, J(x)) →^{∼} RHom_{A}^{•}(C_{•}, J(x))  by lemma 7.11.27(iii)
→^{∼} RHom_{𝒪_{X}}^{•} (C_{•}^{∼}, R\underline{Γ}_{\{x\}}ω^{•}[c]) by theorem 10.1.18
→^{∼} RHom_{𝒪_{X}}^{•} (i_{∗}i^{−1}C_{•}^{∼}, ω^{•}[c])  by lemma 10.4.13(i.b)

→^{∼} RHom_{𝒪_{X}}^{•} (C_{•}^{∼}, ω^{•}[c])

which easily implies (i). Next, we compute, for any object ℱ^{•} of D^{b}(𝒪_{X}-Mod)_{coh} :

R^{i}Γ_{{x}}𝒟(ℱ^{•}) →^{∼} R^{i}Hom_{𝒪_{X}}^{•} (ℱ^{•}, RΓ_{{x}}ω^{•})  by lemma 10.4.13(iii)

→^{∼} R^{i}Hom_{𝒪_{X}}^{•} (ℱ^{•}, J(x)[−c])

→^{∼} R^{i}Hom_{A}^{•}(RΓℱ^{•}, J(x)[−c])  by theorem 10.1.18
→^{∼} Hom_{A}(R^{c−i}Γℱ^{•}, J(x))  by lemma 7.11.27(iii)

whence (ii). Finally, let ℱ^{•} be any object of D^{b}(𝒪_{X}-Mod)_{coh}; if A is as in case (b) of (11.5.15), we compute :

A^{∧} ⊗_{A} R^{c−i}Γℱ^{•} →^{∼} lim_{n∈ℕ} (A/I^{n}) ⊗_{A} R^{c−i}Γℱ^{•}  by corollary 11.4.48(ii)
→^{∼} lim_{n∈ℕ} D ∘ D((A/I^{n}) ⊗_{A} R^{c−i}Γℱ^{•})  by (i)
→^{∼} lim_{n∈ℕ} D(Hom_{A}(A/I^{n}, R^{i}Γ_{{x}}𝒟(ℱ^{•})))  by (ii)

→^{∼} D(colim_{n∈ℕ} Hom_{A}(A/I^{n}, R^{i}Γ_{{x}}𝒟(ℱ^{•})))

→^{∼} D(R^{i}Γ_{{x}}𝒟(ℱ^{•}))

so (iii) holds as well. If A is as in case (a) of (11.5.15), the same proof works : instead of corollary 11.4.48(ii), one has just to appeal to [126, Th.8.7 and Th.8.8]. □

Corollary 11.5.18. Let (A, 𝔪) be a local noetherian ring, complete and separated for its 𝔪adic topology, ω^{•} a dualizing complex for X := Spec A, and c ∈ ℤ the unique integer such that J := R^{c}Γ_{{𝔪}}ω^{•} ≠ 0. Let also M^{•} be a bounded complex of A-modules, and M^{•∼} the induced complex of quasi-coherent 𝒪_{X}-modules. Then we have a natural isomorphism

Ext_{A}^{c−i}(M^{•}, ω^{•}) →^{∼} Hom_{A}(R^{i}Γ_{{𝔪}}M^{•∼}, J) for every i ∈ ℤ.

Proof. We may find a filtered system (M_{λ}^{•} | λ ∈ Λ) of bounded complexes of A-modules, such that M_{λ}^{n} is a submodule of finite type of M^{n}, for every n ∈ ℤ, and such that M^{•} = colim_{λ∈Λ} M_{λ}^{•} in C^{b}(A-Mod) (details left to the reader). There follows a natural isomorphism of A-modules :

Hom_{A}(R^{i}Γ_{{𝔪}}M^{•∼}, J) →^{∼} Hom_{A}(colim_{λ∈Λ} R^{i}Γ_{{𝔪}}M_{λ}^{•∼}, J) →^{∼} lim_{λ∈Λ} Hom_{A}(R^{i}Γ_{{𝔪}}M_{λ}^{•∼}, J).

Taking into account corollary 8.3.53, we are then reduced to proving the assertion for each M_{λ}^{•}; but in this case, the sought isomorphism follows immediately from corollary 11.5.17(iii). □

Definition 11.5.19. Let A and J(x) be as in (11.5.15), and E an A-module. We say that E is finitely copresented if there exist a finitely presented A-module M and an A-linear isomorphism

E →^{∼} Hom_{A}(M, J(x)).

Lemma 11.5.20. Let A be as in (11.5.15), and I ⊂ A a finitely generated ideal, whose radical is the maximal ideal of A. For every A-module E, the following conditions are equivalent :

(a) E is finitely copresented and finitely generated.

(b) E is finitely copresented and I^{k}E = 0 for some k ∈ ℕ.

(c) E is finitely presented and I^{k}E = 0 for some k ∈ ℕ.

Proof. To ease notation, for every ideal L ⊂ A and every A-module M set

D(M) := Hom_{A}(M, J(x)) and M[L] := Ann_{M}(L).

Claim 11.5.21. Let E be any finitely copresented A-module. Then E[I^{k}] is a finitely presented A-module for every k ∈ ℕ, and E = ⋃_{k∈ℕ} E[I^{k}].

Proof of the claim. We may assume that E = D(M) for some finitely presented A-module M; then clearly every element of E is annihilated by some power of I, since the same holds for J(x). Moreover, we have a natural identification

E[I^{n}] →^{∼} D(M/I^{n}M) for every n ∈ ℕ.

But M/I^{k}M is finitely presented and supported at {x}, so D(M/I^{k}M) is finitely presented as well, by corollary 11.5.17(i). ♢

It follows immediately from claim 11.5.21that (a)⇒(b)⇒(c).

(c)⇒(a): If E is finitely presented and I^{k}E = 0, then the same holds for D(E), by corollary 11.5.17(i), and moreover D ∘ D(E) = E, by the same token, whence (a). □

Proposition 11.5.22. Let A be as in (11.5.15), and f : E → N a homomorphism of A-modules, with E finitely copresented and N finitely presented. Then we have :

(i) Ker f is a finitely copresented A-module.

(ii) Im f and Coker f are finitely presented A-modules.

Proof. Clearly, it suffices to show the assertions concerning Ker f and Im f. To this aim, pick any finitely generated ideal I ⊂ A such that Supp A/I = {x} (where x denotes the closed point of Spec A); by corollary 11.4.41(ii) there exists n ∈ ℕ such that N[I^{n}] = N[I^{m}] for every integer m ≥ n. In view of claim 11.5.21, we see that there exists n ∈ ℕ such that Im f = (Im f)[I^{n}]. Taking into account corollary 11.4.41(i), we may then replace N by N[I^{n}], and assume from start that I^{n}N = 0, so N is also finitely copresented, by lemma 11.5.20.

Now, more generally, let E and F be two finitely copresented A-modules, and say that E = D(M), F = D(P), where D is defined as in the proof of lemma 11.5.20; in view of claim 11.5.21and corollary 11.5.17(i) we get natural identifications :

Hom_{A}(E, F) →^{∼} lim_{n∈ℕ} Hom_{A}(E[I^{n}], F [I^{n}])
→^{∼} lim_{n∈ℕ} Hom_{A}(P/I^{n}P, M/I^{n}M)
→^{∼} lim_{n∈ℕ} Hom_{A}(P, M/I^{n}M)
→^{∼} lim_{n∈ℕ} Hom_{A}(P, M^{∧})

where M^{∧} denotes the I-adic completion of M. Recall that the natural map M → M^{∧} is injective (corollary 11.4.48(ii,iii)), so the same holds for the induced map

(11.5.23)  Hom_{A}(P, M) → Hom_{A}(E, F)  f ↦ D(f).

Claim 11.5.24. With the foregoing notation, we have :

(i) Let g : E → F be any morphism. If g lies in the image of (11.5.23), then Ker g, Coker g and Im g are all finitely copresented A-modules.

(ii) If either I^{k}M = 0 or I^{k}P = 0 for some k ∈ ℕ, then (11.5.23) is an isomorphism.

Proof of the claim. (i): Indeed, say that g is the image of h : P → M under (11.5.23); since J(x) is coh-injective, we get natural identifications

Ker g →^{∼} D(Coker h)  Coker g →^{∼} D(Ker h)  Im g →^{∼} D(Im h)

whence the contention.

(ii): If I^{k}M = 0 for some k ∈ ℕ, then the natural map M → M^{∧} is an isomorphism, and the assertion follows. If I^{k}P = 0, then any morphism P → M^{∧} factors through M^{∧}[I^{k}] (notation of the proof of lemma 11.5.20); however, corollaries 11.4.41(i) and 11.4.48(i) yield natural identifications

M[I^{n}] →^{∼} (M[I^{n}])^{∧} →^{∼} M^{∧}[I^{n}]

whence the contention (details left to the reader). ♢

Now, letting F := N in claim 11.5.24, we deduce that both Ker f and Im f are finitely copresented. Lastly, since I^{n} · Im f = 0, lemma 11.5.20 says that Im f is finitely presented, as sought. □

Theorem 11.5.25. In the situation of (11.5.15), let I ⊂ A be any finitely generated ideal, C^{•} any object of D^{b}(𝒪_{X}-Mod_{coh}), and set U := X\{x}. For every i ∈ ℤ, the following conditions are equivalent :

(a) I · H^{i}(U, C^{•}) is a finitely presented A-module.

(b) I · H^{i}(U, C^{•}) is a finitely generated A-module.
(c) I · R^{i+1}Γ_{{x}}C^{•} is a finitely presented A-module.
(d) I · R^{i+1}Γ_{{x}}C^{•} is a finitely generated A-module.
(e) The radical of Ann_{A}(I · R^{i+1}Γ_{{x}}C^{•}) contains the maximal ideal of A.
(f) I · R^{i}Γ_{{y}}C_{|X(y)}^{•} = 0 for every closed point y ∈ U.

Proof. Let ω^{•} be a dualizing complex for X and y ∈ X any point; then ω_{|X(y)}^{•} is a dualizing complex for X(y), and according to (11.5.15), there is a unique integer c(y) such that J(y) := R^{c(y)}Γ_{{y}}ω_{|X(y)}^{•} ≠ 0. If A is as in case (b) (resp. as in case (a)) of (11.5.15), we may invoke theorem 11.5.10and lemma 11.5.8(i) (resp. lemma 11.3.50) to see that

(11.5.26)  c(y) + 1 = c := c(x) ⇔ y is a closed point of U.

By corollary 11.5.17(i,ii), the rule

M ↦ D_{y}(M) := Hom_{A}(M, J(y)) for every 𝒪_{X,y}-module M

restricts to an equivalence 𝒪_{X,y}-Mod_{coh} →^{∼} 𝒪_{X,y}-Mod_{coh}^{o}, and there is a natural isomorphism

(11.5.27)  R^{i}Γ_{{y}}C_{|X(y)}^{•} →^{∼} D_{y}(H^{c(y)−i}(X(y), (𝒟C^{•})_{|X(y)}))  in 𝒪_{X,y}-Mod

where 𝒟 is the duality functor corresponding to ω^{•}. Notice that a · 1_{D_{y}M} = D_{y}(a · 1_{M}) for every a ∈ 𝒪_{X,y}, so (11.5.27) and (11.5.26) imply that (f) holds if and only if

(11.5.28)  I · H^{c−i−1}(X(y), (𝒟C^{•})_{|X(y)}) = 0  for every closed point y of U.

However, if A is as in case (b) (resp. as in case (a)) of (11.5.15), lemma 11.5.8(iii) (resp. theorem 11.3.52) says that every point of U specializes to a closed point, so (11.5.28) holds if and only if I · H^{c−i−1}(𝒟C^{•})_{|U} = 0. Since 𝒟C^{•} is a complex of coherent 𝒪_{X}-module, the latter condition is equivalent to saying that there exists an ideal I′ whose radical is the maximal ideal of A, and such that I′I · H^{c−i−1}(𝒟C^{•}) = 0. By invoking (11.5.27) with y := x, we see that this holds if and only if I′I · R_{{x}}^{i+1}C^{•} = 0. Summing up, we have shown that (e)⇔(f).

Claim 11.5.29. The A-module I · R^{i}Γ_{{x}}C^{•} is finitely copresented, for every i ∈ ℤ.

Proof of the claim. Notice first that, due to (11.5.27), the A-module E := R^{i}Γ_{{x}}C^{•} is finitely copresented, so the same holds for E^{⊕n}, for every n ∈ ℕ. Since I is finitely generated, it is the image of a morphism f : A^{⊕n} → A, for some n ∈ ℕ, and then IE is naturally identified with the image of f ⊗_{A} 1_{E}. Say that E = D_{x}(M) for a finitely presented A-module M; then clearly f ⊗ 1_{E} = D_{x}(f^{∨} ⊗_{A} 1_{M}), where f^{∨} : A → A^{⊕n} is the transpose of f. Now the assertion follows from claim 11.5.24(i). ♢

Taking into account claim 11.5.29and lemma 11.5.20, we deduce that (c)⇔(d)⇔(e).

Obviously, (a)⇒(b). Next, consider the exact sequence

H^{i}(X, C^{•}) ⟶^{α_{i}} H^{i}(U, C^{•}) ⟶^{β_{i}} R^{i+1}Γ_{{x}}C^{•} ⟶^{γ_{i+1}} H^{i+1}(X, C^{•})

(see (10.4.1)). Assume (b), which implies that I · Im β_{i} is a finitely generated A-module, so there exists an ideal I′ ⊂ A whose radical is the maximal ideal, and such that I′I · Im β_{i} = 0. On the other hand, by claim 11.5.29 and proposition 11.5.22 we know that Im γ_{i+1} is a finitely presented A-module, hence there exists an ideal I′′ ⊂ A whose radical is the maximal ideal, and such that I′′ · Im γ_{i+1} = 0. We conclude that I′′I′I · R^{i+1}Γ_{{x}}C^{•} = 0, whence (b)⇒(e).

Lastly, we suppose that (e) holds, and we show that (a) follows. To this aim, set M := I · H^{i}(U, C^{•}) ∩ Im α_{i}, and consider the induced ladder with exact rows

(11.5.30)  \xymatrix{0 \ar[r] & M \ar[r] \ar[d] & I · H^{i}(U, C^{•}) \ar[r] \ar[d] & I · Ker γ_{i+1} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & Coker γ_{i} \ar[r]^{_{\overline{α}_{i}}} & H^{i}(U, C^{•}) \ar[r] & Ker γ_{i+1} \ar[r] & 0.}

From (11.5.27) we know that Ker γ_{i+1} is finitely copresented and arguing as in the proof of claim 11.5.29, we conclude that the same holds for I · Ker γ_{i+1}. But (e) implies that the latter A-module is annihilated by a finitely generated ideal whose radical is the maximal ideal of A, so it is finitely presented, by lemma 11.5.20. We are thus reduced to checking that M is a finitely presented A-module. Now, set N := Ker (\overline{α}_{i} ⊗_{A} 1_{A/I}) and notice the short exact sequence

0 → I · Coker γ_{i} → M → N → 0.

By proposition 11.5.22(ii) and (11.5.27), the A-module Coker γ_{i} is finitely presented, so the same holds for I · Coker γ_{i}, because A is coherent and I is finitely generated. So, we are further reduced to showing that N is finitely presented. However, N is naturally identified with the image of the map

Tor_{1}^{A}(Ker γ_{i+1}, A/I) → (Coker γ_{i}) ⊗_{A} A/I

induced by the bottom row of (11.5.30). On the other hand, we have already noticed that Coker γ_{i} is finitely presented, so the same holds for (Coker γ_{i}) ⊗_{A} A/I; also, proposition 11.5.22(i) and (11.5.27) say that Ker γ_{i+1} is finitely copresented. Taking into account proposition 11.5.22(ii) it finally suffices to show :

Claim 11.5.31. If Q is any finitely copresented A-module and P any finitely presented Amodule, the A-module Tor_{i}^{A}(Q, P) is finitely copresented as well, for every i ∈ ℕ.

Proof of the claim. Since I is finitely generated and A is coherent, we may find a resolution F_{•} →^{∼} P [0] consisting of free A-modules of finite type (details left to the reader), and then Tor_{i}^{A}(Q, P) ≃ H_{i}(F_{•} ⊗_{A} Q), so the assertion follows easily from claim 11.5.24(i) (details left to the reader). □

Proposition 11.5.32. In the situation of (11.5.15), let d ∈ ℕ be any integer, and suppose additionally that :

(a’) If A is as in case (a), then A is regular of dimension d + 1.

(b’) If A is as in case (b), the structure morphism f : X → S is flat, f(x) = s and f^{−1}(s)

is a regular scheme of dimension d. Let j : U := X \{x} → X be the open immersion, ℱ a flat quasi-coherent 𝒪_{U}-module. Then the following two conditions are equivalent :

(c) Γ(U, ℱ) is a flat A-module.

(d) δ(x, j_{∗}ℱ) ≥ d + 1.

Proof. If d = 0, then A is a valuation ring (proposition 9.1.34(ii)), and U is the spectrum of the field of fractions of A, in which case both (c) and (d) hold trivially. Hence we may assume that d ≥ 1. Suppose now that (c) holds, and set F := Γ(U, ℱ). In view of (10.4.3), we need to show that H^{i}(U, ℱ) = 0 whenever 1 ≤ i ≤ d − 1. If F^{∼} denotes the 𝒪_{X}-module determined by F, then F_{|U}^{∼} = ℱ. Moreover, by [120, Ch.I, Th.1.2], F is the colimit of a filtered family (L_{i} | i ∈ I) of free A-modules of finite rank, hence H^{i}(U, ℱ) = colim_{i∈I} H^{i}(U, L_{i}^{∼}) by proposition 10.1.10(ii), so we are reduced to the case where ℱ = 𝒪_{U}, and therefore j_{∗}ℱ = 𝒪_{X} (corollary 11.3.9). In case (a’), assertion (d) follows already, by virtue of [126, Th.17.8]. So, suppose that (b’) holds; since f^{−1}(s) is regular, we have δ(x, 𝒪_{f^{−1}(s)}) = d; then, since the topological space underlying X is noetherian, lemma 10.4.20(ii) and corollary 10.4.47 imply that δ(x, 𝒪_{X}) ≥ d + 1, which is (d).

Conversely, suppose that (d) holds; we consider first the case (b’), and we shall derive (c) by induction on d, the case d = 0 having already been dealt with. Let 𝔪_{A} (resp. \overline{𝔪}_{A}) be the maximal ideal of A (resp. of \overline{A} := A/𝔪_{K}A). Suppose then that d ≥ 1 and that the assertion is known whenever dim f^{−1}(s) < d. Pick \overline{t} ∈ \overline{𝔪}_{A}\\overline{𝔪}_{A}^{2}, and let t ∈ 𝔪_{A} be any lifting of \overline{t}. Since f^{−1}(s) is regular, \overline{t} is a regular element of \overline{A}, so that t is regular in A and the induced morphism g : X′ := Spec A/tA → S is flat ([65, Ch.IV, Th.11.3.8]). Let j′ : U′ := U ∩ X′ → X′ be the restriction of j; our choice of \overline{t} ensures that g^{−1}(s) is a regular scheme, so the pair (X′, ℱ /tℱ_{|U′}) fulfills the conditions of the proposition, and dim g^{−1}(s) = d − 1. Furthermore, since ℱ is a flat 𝒪_{U}-module, the sequence 0 → ℱ ⟶^{t} ℱ → ℱ /tℱ → 0 is short exact. Assumption (d) means that

(11.5.33)  H^{i}(U, ℱ) = 0  whenever 1 ≤ i ≤ d − 1.

Therefore, from the long exact cohomology sequence we deduce that H^{i}(U, ℱ /tℱ) = 0 for 1 ≤ i ≤ d − 2, i.e.

(11.5.34)  δ(x, j′_{∗}ℱ /tℱ) ≥ d.

The same sequence also yields a left exact sequence :

(11.5.35)  0 → H^{0}(U, ℱ) ⊗_{A} A/tA ⟶^{α} H^{0}(U, ℱ /tℱ) → H^{1}(U, ℱ).

Claim 11.5.36. H^{0}(U, ℱ) ⊗_{A} A/tA is a flat A/tA-module.

Proof of the claim. By (11.5.34) and our inductive assumption, H^{0}(U, ℱ /tℱ) is a flat A/tAmodule. In case d = 1, proposition 9.1.34(ii) shows that A/tA is a valuation ring, and then the claim follows from (11.5.35), and [34, Ch.VI, §3, n.6, Lemma 1]. If d > 1, (11.5.33) implies that α is an isomorphism, so the claim holds also in such case. ♢

Set V := Spec A[t^{−1}] = X \ V (t) ⊂ U; since ℱ_{|V} is a flat 𝒪_{V} -module, the A[t^{−1}]- module H^{0}(U, ℱ) ⊗_{A} A[t^{−1}] ≃ H^{0}(V, ℱ_{|V}) is flat. Moreover, since t is regular on both A and H^{0}(U, ℱ), an easy calculation shows that Tor_{i}^{A}(A/tA, H^{0}(U, ℱ)) = 0 for i > 0. Then the contention follows from claim 11.5.36and [75, Lemma 5.2.1].

Lastly, in case (a’) holds, one picks any element t ∈ 𝔪_{A} \𝔪_{A}^{2} and argues in the same way (with some simplifications : details left to the reader). □

Remark 11.5.37. The special case of proposition 11.5.32 where A is regular local ring and ℱ is a locally free 𝒪_{U}-module of finite rank has been studied in detail in [97].

Lemma 11.5.38. Let A be a noetherian ring, M an A-module endowed with a finite filtration M_{0} := 0 ⊂ M_{1} ⊂ ⋯ ⊂ M_{k} := M, and denote by gr^{•}M the associated graded A-module. Suppose that, for every i = 1, …, k, there exists an ideal I_{i} ⊂ A such that I_{i} · gr^{i}M is an A-module of finite type. Then ∏_{i=1}^{k} I_{i}M is an A-module of finite type.

Proof. A simple induction on the length of the filtration reduces the assertion to the following

Claim 11.5.39. Let A be a noetherian ring, I, J ⊂ A two ideals, and

0 → M_{1} → M_{2} → M_{3} → 0

a short exact sequence of A-modules, such that IM_{1} and JM_{3} are finitely generated. Then IJM_{2} is finitely generated.

Proof of the claim. Indeed, the induced map N := JM_{2}/(M_{1} ∩ JM_{2}) → JM_{3} is injective, so N is finitely generated, and then the same holds for I ⊗_{A} N; but the latter maps surjectively onto IJM_{2}/(IM_{1} ∩ IJM_{2}) ≃ (IJM_{2} + IM_{1})/IM_{1}, so the latter is also of finite type, and therefore the same holds for IJM_{2} + IM_{1}, whence the lemma. □

11.5.40. Let A be a noetherian ring, ω^{•} a dualizing complex on X := Spec A, and I ⊂ A any ideal. Let also U ⊂ X be any open subset, and K^{•} any object of D^{b}(𝒪_{U}-Mod)_{coh}. We denote :

• \overline{U} the topological closure of U in X

• Z := X \U and ∂U := \overline{U} \ U, endowed with the reduced closed subscheme structures • U_{1} := \{x ∈ U | there exists y ∈ ∂U which is an immediate specialization of x in \overline{U}\} • 𝒟_{U} : D^{b}(𝒪_{U}-Mod)_{coh}^{o} → D^{b}(𝒪_{U}-Mod)_{coh} the duality functor associated with ω_{|U}^{•}

• n(K^{•}) the cardinality of {i ∈ ℤ | H^{i}(𝒟_{U}K^{•}) ≠ 0}

• d(K^{•}) := min(n(K^{•}), 1 + dim ∂U)

• ∂U(x) := ∂U ∩ X(x) and δ(x) := dim ∂U(x) for every x ∈ X

• e(x) := min(1 + δ(y) | y is a specialization of x in \overline{U}) for every x ∈ U_{1}. Here we use the convention that dim ∂U = −1 if ∂U = ∅, and likewise we define the dimension of ∂U(y), for any y ∈ X such that this set is empty. On the other hand, the dimension of ∂U can be infinite, in the current setting. Notice that the integer n(K^{•}) depends only on K^{•}, and does not depend on the choice of ω^{•}.

Lemma 11.5.41. For every x ∈ ∂U, every closed point of U ∩ X(x) is an immediate generization of x in X(x). Especially, every such point lies in U_{1}.

Proof. Fix any x ∈ ∂U, set A := 𝒪_{X,x}, and for any closed point z of U ∩ X(x), denote by 𝔭_{z} ⊂ A the prime ideal corresponding to z, so that 𝔭_{z} is strictly contained in the maximal ideal 𝔪_{A} of A. Let also I ⊂ A be any ideal such that Spec A/I = Z(x), so I ⊄ 𝔭_{z} by construction. Set B := A/𝔭_{z}, and denote by 𝔪_{B}, \overline{I} ⊂ B the images of respectively 𝔪_{A} and I. It suffices to show that dim B = 1; suppose by contradiction that the latter fails, and pick any f ∈ \overline{I} \ \{0\}.

Claim 11.5.42. If dim B > 1, there exists a prime ideal \overline{𝔮} ⊂ B of height 1, such that f ∉ \overline{𝔮}.

Proof of the claim. Clearly, every prime ideal of height one containing f is a maximal point of Supp B/fB, hence it lies in Ass B/fB (lemma 10.5.3(i)); therefore, the set Σ of prime ideals of B of height one containing f is finite ([126, Th.6.5(i)]). If Σ is empty, Krull’s Hauptidealsatz ([126, Th.13.5]) implies that f must be invertible, in which case the claim is obvious. If Σ contains a single prime ideal \overline{𝔮}_{1}, notice that \overline{𝔮}_{1} ≠ 𝔪_{B}, since dim B > 1, and pick any g ∈ 𝔪_{B} \ 𝔮_{1}; again by Krull’s theorem we know that there exists a prime ideal \overline{𝔮} ⊂ B of height one containing g, and by construction \overline{𝔮} ≠ \overline{𝔮}_{1}, whence the claim, in this case. Otherwise, say that Σ = \{\overline{𝔮}_{1}, ⋯, \overline{𝔮}_{n}\} for some n ≥ 2. For every i, j ≤ n with i ≠ j we may find an element g_{ij} ∈ 𝔮_{i} \ 𝔮_{j}, and we set g := ∑_{j=1}^{n} ∏_{i≠j} g_{ij}. It is easily seen that g ∈ 𝔪_{B} \ (\overline{𝔮}_{1} ∪ ⋯ ∪ \overline{𝔮}_{n}), so we may apply again Krull’s theorem to exhibit a prime ideal \overline{𝔮} which cannot lie in Σ, since it contains g. ♢

Let \overline{𝔮} be as in claim 11.5.42, and denote by 𝔮 ⊂ A the preimage of \overline{𝔮}; then 𝔮 corresponds to a point y ∈ U ∩ X(x) which is different from x, and is a proper specialization of z in X(x), which is impossible, since by assumption z is closed in U. □

Theorem 11.5.43. In the situation of (11.5.40), let i ∈ ℤ be any integer, and consider the following conditions :

(a) I · R^{i}Γ_{{x}}K_{|U(x)}^{•} = 0 for every x ∈ U_{1}.

(b) I^{d(K^{•})} · H^{i}(U, K^{•}) is an A-module of finite type.
(c) I · H^{j}(U, K^{•}) is an A-module of finite type, for every j ≤ i.
(d) I^{e(x)} · R^{j}Γ_{{x}}K_{|U(x)}^{•} = 0 for every x ∈ U_{1} and every j ≤ i.

Then (a)⇒(b) and (c)⇒(d).

Proof. (a)⇒(b): By corollary 10.3.39, we may find an object L^{•} of D^{b}(𝒪_{X}-Mod)_{coh} with an isomorphism L_{|U}^{•} →^{∼} K^{•} in D^{b}(𝒪_{U}-Mod)_{coh}, whence an exact sequence

H^{i}(X, L^{•}) → H^{i}(U, K^{•}) → R^{i+1}Γ_{Z}L^{•} → H^{i+1}(X, L^{•})

(see (10.4.1)) whose first and last terms are finitely generated A-modules. By an easy diagram chase, it follows that condition (b) is equivalent to

(e) I^{d(K^{•})} · R^{i+1}\underline{Γ}_{Z}L^{•} is a coherent 𝒪_{X}-module. It will therefore suffice to show that (a)⇒(e). However, from lemma 10.4.13(iii) we get a natural isomorphism

R\underline{Γ}_{Z}L^{•} →^{∼} Rℋ om_{Z}^{•}(𝒟_{X}L^{•}, ω^{•})

where 𝒟_{X} is the dualizing functor on D^{b}(𝒪_{X}-Mod)_{coh} defined by ω^{•}. By choosing an injective resolution for ω^{•}, we obtain a spectral sequence

E_{2}^{pq} := R^{p}ℋ om_{Z}^{•}(H^{−q}(𝒟_{X}L^{•})[0], ω^{•}) ⇒ R^{p+q}ℋ om_{Z}^{•}(𝒟_{X}L^{•}, ω^{•}).

Now, for every q ∈ ℤ, define the 𝒪_{X}-module \overline{H}^{−q} by the short exact sequence

0 → \underline{Γ}_{Z}H^{−q}(𝒟_{X}L^{•}) → H^{−q}(𝒟_{X}L^{•}) → \overline{H}^{−q} → 0.

There results an exact sequence (11.5.44)

F^{pq} := R^{p}ℋ om_{Z}^{•}(\overline{H}^{−q}[0], ω^{•}) → E_{2}^{pq} → G^{pq} := R^{p}ℋ om_{Z}^{•}(\underline{Γ}_{Z}H^{−q}(𝒟_{X}L^{•})[0], ω^{•})

Claim 11.5.45. For every p, q ∈ ℤ we have :

(i) E_{2}^{pq} and F^{pq} are quasi-coherent 𝒪_{X}-modules.

(ii) G^{pq} and \overline{H}^{−q} are coherent 𝒪_{X}-modules.

Proof of the claim. The assertions about E_{2}^{pq} and F^{pq} follow immediately from lemmata 10.4.17(i) and 10.4.13(iii) and corollary 10.3.2(ii), and the assertion for \overline{H}^{−q} follows from lemma 10.4.17(i). Next, notice that the natural map

\underline{Γ}_{Z}H^{−q}(𝒟_{X}L^{•}) → R\underline{Γ}_{Z} ∘ \underline{Γ}_{Z}H^{−q}(𝒟_{X}L^{•})

is an isomorphism in D(𝒪_{X}-Mod), whence an isomorphism of 𝒪_{X}-modules

(11.5.46)  G^{pq} →^{∼} R^{p}ℋ om_{𝒪_{X}}^{•} (\underline{Γ}_{Z}H^{−q}(𝒟_{X}L^{•})[0], ω^{•})

(lemma 10.4.13(vi)). But since X is noetherian, \underline{Γ}_{Z}H^{−q}(𝒟_{X}L^{•}) is a coherent 𝒪_{X}-module, so the same holds for the target of (11.5.46) (proposition 10.3.3(i)), and the claim follows. ♢

Claim 11.5.47. In order to show (e), it suffices to check that :

(i) I · F^{pq} = 0 for every p, q ∈ ℤ such that p + q = i + 1.

(ii) F_{x}^{pq} = 0 for every x ∈ X and every p ∈ ℤ such that p ∉ [c_{X}(x) − dim ∂U(x), c_{X}(x)].

Proof of the claim. Say that 𝒟_{X}L^{•} ∈ D^{[a,b]}(𝒪_{X}-Mod) for some a, b ∈ ℤ with a ≤ b; to begin with, we show that – under assumptions (i) and (ii) of the claim – there exist for every q = a, …, b and every p = i + 1 − b, …, i + 1 − a and x ∈ X, integers ν(q), μ(p, x) ∈ ℕ such that I^{ν(q)}F^{pq} = 0 and I^{μ(p,x)}F_{x}^{pq} = 0, and moreover

(11.5.48)  ∑_{q=a}^{b} ν(q) ≤ n(K^{•})  ∑_{p=i+1−b}^{i+1−a} μ(p, x) ≤ 1 + δ(x) for every x ∈ X.

Indeed, if q ∈ [a, b] is an index such that H^{−q}(𝒟_{U}K^{•}) = 0, it follows that H^{−q}(𝒟_{X}L^{•})_{|U} = 0, so \overline{H}^{−q} = 0, therefore F^{pq} = 0 for every p ∈ ℤ, and we may take ν(q) = 0 in this case. But assumption (i) says that in any case we may take ν(q) ≤ 1 for every q = a, …, b. This already suffices to achieve the sought upper bound for the sum of the values ν(q). The other upper bound is treated similarly : we may in any case take μ(p, x) ≤ 1, due to condition (i), and moreover condition (ii) says that we may take μ(p, x) = 0 for every p ∈ ℤ such that p ∉ [c_{X}(x) − dim ∂U(x), c_{X}(x)], whence the contention.

Next, recall that M := R^{i+1}\underline{Γ}_{Z}L^{•} admits a finite filtration Fil^{•}M whose associated graded 𝒪_{X}-module is ⨁_{p+q=i+1} E_{∞}^{pq}, and by construction, each term E_{∞}^{pq} is a quasi-coherent subquotient of E_{2}^{pq} (claim 11.5.45(i)), so Fil^{q}M is a quasi-coherent 𝒪_{X}-module as well, for every q = a, …, b. Using claims 11.5.39 and 11.5.45(i,ii) we deduce that I^{ν(q)}E_{2}^{pq} is a coherent 𝒪_{X}- module for every q = a, …, b, so the same holds for I^{ν(q)}E_{∞}^{pq}, and then (11.5.48) and lemma 11.5.38imply that I^{n(K^{•})}M is a coherent 𝒪_{X}-module.

Set δ := dim ∂U ∈ ℕ ∪ {∞}; if δ ≥ n(K^{•}) − 1, the proof is complete; otherwise δ ∈ ℕ, and it remains only to check that I^{1+δ}M is also a coherent 𝒪_{X}-module. To this aim, notice that (11.5.44) induces a short exact sequence

0 → F_{∞}^{pq} → E_{∞}^{pq} → G_{∞}^{pq} → 0

where F_{∞}^{pq} (resp. G_{∞}^{pq}) is a certain quasi-coherent subquotient of F^{pq} (resp. of G^{pq}); from claim 11.5.45(ii) it follows that G_{∞}^{pq} is a coherent 𝒪_{X}-module, and the foregoing implies that I^{μ(p,x)}F_{∞,x}^{pq} = 0 for every x ∈ X and every p = i + 1 − b, …, i + 1 − a. We may then find, for every q = a, …, b a coherent 𝒪_{X}-submodule N^{q} ⊂ Fil^{q}M such that the induced morphism Fil^{q}M → G_{∞}^{pq} restricts to an epimorphism N^{q} → G_{∞}^{pq}. Set N := ∑_{q=a}^{b} N^{q} ⊂ M, and P := M/N, endow P with the filtration Fil^{•}P induced by the filtration of M, and denote by gr^{•}P the associated graded 𝒪_{X}-module. By construction, we get an epimorphism ψ_{q} : E_{∞}^{pq} → gr^{q}P for every q = a, …, b, whose kernel contains the image of N^{q} in E_{∞}^{pq}. Consequently, the restriction of ψ_{q} to F_{∞}^{pq} is still an epimorphism, so that I^{μ(p,x)}gr^{i+1−p}P_{x} = 0 for every x ∈ X and every p = i + 1 − b, …, i + 1 − a. In view of (11.5.48) we conclude that I^{1+δ(x)}P_{x} = 0 for every x ∈ X, and since clearly δ(x) ≤ δ for every x ∈ X, it follows that I^{1+δ}P = 0. Lastly, since N is a coherent 𝒪_{X}-module, it suffices to invoke claim 11.5.39to conclude the proof. ♢

Let ℛ^{•} be the residual complex arising from ω^{•}, introduced in (11.3.53). With this notation, F^{pq} is a subquotient of the A-module ℋ om_{𝒪_{X}} (\overline{H}^{−q}, \underline{Γ}_{Z}ℛ^{p}). From lemma 10.4.4(iii.b), we get

\underline{Γ}_{Z}ℛ^{p} = ⨁_{x∈X_{p}∩Z} E(x)^{∼}. with X_{p} := \{x ∈ X | c_{X}(x) = p\} for every p ∈ ℤ

where c_{X} : |X| → ℤ is the weak codimension function associated with ω^{•} as in (11.3.49) and E(x) is an injective hull of the A-module κ(x), for every x ∈ X (notation of (10.3)). Hence, denote by X_{p}^{∗} the set of all specializations in X of the points of X_{p}; we deduce

(11.5.49)  Supp \underline{Γ}_{Z}ℛ^{p} = Z ∩ X_{p}^{∗}

(we caution the reader that X_{p} is usually not a pro-constructible subset of X, so X_{p}^{∗} is usually strictly contained in the topological closure of X_{p} in X). Moreover, since \overline{H}^{−q} is a coherent 𝒪_{X}-module (claim 11.5.45(ii)) and \underline{Γ}_{Z}ℛ^{p} is quasi-coherent (lemma 10.4.17(i)), we also see that

(11.5.50)  ℋ om_{𝒪_{X}} (\overline{H}^{−q}, \underline{Γ}_{Z}ℛ^{p})_{x} ≃ Hom_{𝒪_{X,x}} (\overline{H}_{x}^{−q}, (\underline{Γ}_{Z}ℛ^{p})_{x})  for every x ∈ X

([75, Lemma 2.4.29(i.a)]). Combining (11.5.49) and (11.5.50) we conclude that

Supp F^{pq} ⊂ Z ∩ X_{p}^{∗} ∩ Supp \overline{H}^{−q}.

From this, we can already show condition (ii) of claim 11.5.47 : indeed, say that F_{x}^{pq} ≠ 0; then x ∈ X_{p}^{∗}, so c_{X}(x) ≥ p, and on the other hand, by construction we have \underline{Γ}_{Z} \overline{H}^{−q} = 0, so Supp H^{−q} is the topological closure in X of U ∩ Supp \overline{H}^{−q}, therefore x ∈ ∂U and furthermore there exists a generization z of x in X with c_{X}(z) = p, so z ∈ X_{p} ∩ ∂U(x), which forces the bound c_{X}(x) ≤ p + dim ∂U(x), as required.

Lastly, to get condition (i) of claim 11.5.47 it suffices to check assertions (i) and (ii) of the following :

Claim 11.5.51. (i) Supp (I · F^{pq}) ⊂ Supp ℋ om_{𝒪_{X}} (I · \overline{H}^{−q}, \underline{Γ}_{Z}ℛ^{p}) for every p, q ∈ ℤ. (ii) ℋ om_{𝒪_{X}} (I · \overline{H}^{−q}, \underline{Γ}_{Z}ℛ^{p}) = 0 whenever p + q = i + 1. (iii) Z ∩ X_{p} ∩ Supp (I · \overline{H}^{−q}) = ∅ whenever p + q = i + 1.

Proof of the claim. (i): Consider arbitrary x ∈ X, a ∈ I and f ∈ F_{x}^{pq}; by virtue of (11.5.50), the element f is represented by an 𝒪_{X,x}-linear map φ : \overline{H}_{x}^{−q} → \underline{Γ}_{Z}ℛ_{x}^{p}, and therefore af is represented by aφ; the latter obviously factors through the submodule I ·\overline{H}_{x}^{−q}, so the contention follows from the natural identification

(11.5.52) ℋ om_{𝒪_{X}} (I · \overline{H}^{−q}, \underline{Γ}_{Z}ℛ^{p})_{x} ≃ Hom_{𝒪_{X,x}} (I · \overline{H}_{x}^{−q}, (\underline{Γ}_{Z}ℛ^{p})_{x})  for every x ∈ X

which is proven as (11.5.50).

(ii): Let x ∈ X be any point, and φ : I · \overline{H}_{x}^{−q} → (\underline{Γ}_{Z}ℛ^{p})_{x} any 𝒪_{X,x}-linear map; taking into account (11.5.52), we are reduced to showing that φ = 0. Now, notice that

(\underline{Γ}_{Z}ℛ^{p})_{x} = ⨁_{y∈Z(x)_{p}} E(y) where Z(x)_{p} := X_{p} ∩ Z ∩ X(x).

If Z(x)_{p} = ∅, we are done; otherwise, for every y ∈ Z(x)_{p} denote by p_{y} : (\underline{Γ}_{Z}ℛ^{p})_{x} → E(y) the natural projection; it suffices to check that p_{y} ∘ φ = 0 for every such y. However, p_{y} ∘ φ factors through an 𝒪_{X,y}-linear map I · \overline{H}_{y}^{−q} → (\underline{Γ}_{Z}ℛ^{p})_{y}, so we may replace x by y, and assume from start that x ∈ X_{p} ∩ Z, in which case the contention will follow from assertion (iii) of the claim.

(iii): Fix any point y ∈ X_{p} ∩ Z. We have already noticed that Supp \overline{H}^{−q} is the topological closure in X of U ∩ Supp \overline{H}^{−q}; we are then reduced to showing that I · \overline{H}_{z}^{−q} = 0 for every closed point z of X(y) ∩ U. But for any such z, we have an isomorphism

(11.5.53)  H^{−q}(𝒟_{U}K^{•})_{z} →^{∼} \overline{H}_{z}^{−q}

of 𝒪_{X,z}-modules. On the other hand, corollary 11.5.17(iii) yields an isomorphism of 𝒪_{X,z}- modules

(11.5.54)  𝒪_{X,z}^{∧} ⊗_{𝒪_{X,z}} H^{−q}(𝒟_{U}K^{•})_{z} →^{∼} D_{z} ∘ R^{q+c_{X}(z)}Γ_{{z}}K_{|U(z)}^{•}

where 𝒪_{X,z}^{∧} denotes the completion of the ring 𝒪_{X,z} and D_{z} denotes the local duality functor corresponding to the point z. We remark now that

(11.5.55)  c_{X}(z) = c_{X}(y) − 1 = p − 1.

Indeed, in view of lemma 11.3.50, identity (11.5.55)asserts that y is an immediate specialization of z in X(y), and this is already known from lemma 11.5.41.

Finally, let a ∈ I be any element; taking into account (11.5.53), we are reduced to checking that scalar multiplication by a is the zero endomorphism of H^{−q}(𝒟_{U}K^{•})_{z}, which – by virtue of (11.5.54) and (11.5.55) – is equivalent to showing that scalar multiplication by a is the zero endomorphism on R^{i}Γ_{{z}}K_{|U(z)}^{•}, and the latter is ensured by our condition (a). ♢

(c)⇒(d): Let x ∈ U_{1} be any point, pick any immediate specialization y ∈ ∂U of x in X such that 1 + dim ∂U(y) = e(x), let U(y) := U ∩ X(y), and notice that the natural map

H^{j}(U, K^{•}) ⊗_{A} 𝒪_{X,y} → H^{j}(U(y), K_{|U(y)}^{•})

is an isomorphism for every j ∈ ℤ, so we may replace X by X(y), U by U(y) and K^{•} by K_{|U(y)}^{•}, and assume from start that A is a local ring, and y is the closed point of X. In this situation, we need to show that I^{e}R^{j}Γ_{{x}}K_{|U(x)}^{•} = 0, with e := 1 + dim ∂U and every j ≤ i. Then, let V := X \ {y}, denote j : U → V the induced open immersion, and set

L^{•} := τ^{≤i}Rj_{∗}K^{•}

(where τ^{≤i} denotes the usual truncation functor), so that L ∈ Ob(D^{b}(𝒪_{V} -Mod)_{qcoh}) and L_{|U}^{•} = τ^{≤i}K^{•}. Notice also that the natural morphism of 𝒪_{X}-modules

H^{j}(U, K^{•})^{∼} → H^{j}L^{•}

is an isomorphism for every j ≤ i. Taking into account proposition 10.3.34(ii), it follows that we may find a morphism φ^{•} : C^{•} → L^{•} in D^{b}(𝒪_{X}-Mod) such that

• C^{•} ∈ Ob(D^{b}(𝒪_{X}-Mod)_{coh}).

• The induced morphism C_{|U}^{•} → τ^{≤i}K_{|U}^{•} is an isomorphism in D^{b}(𝒪_{U}-Mod)_{coh}.

• H^{j}φ^{•} : H^{j}C^{•} → H^{j}L^{•} is a monomorphism for every j ∈ ℤ, and its image contains

the 𝒪_{X}-submodule I · H^{j}(U, K^{•})^{∼}, for every j ≤ i.

Set D^{•} := Cone φ; by construction, for every j ∈ ℤ we have

(11.5.56) Supp H^{j}L^{•} ⊂ \overline{U} ∩ V Supp H^{j}D^{•} ⊂ Z ∩ Supp H^{i}L^{•} ⊂ ∂U ∩ V I ·H^{j}D^{•} = 0.

Now, consider the spectral sequence

E_{2}^{pq} := H^{p}(V, H^{q}D^{•}) ⇒ H^{p+q}(V, D^{•}).

It follows from (11.5.56) that E_{2}^{pq} = 0 for every p > dim ∂U ∩ V = e − 2, and I · E_{2}^{pq} = 0 for every p, q ∈ ℤ, whence

I^{e−1}H^{j}(V, D^{•}) = 0 for every j ∈ ℤ.

On the other hand, we have an exact sequence of A-modules

H^{j−1}(V, D^{•}) → H^{j}(V, C^{•}) → H^{j}(V, L^{•}) = H^{j}(U, K^{•}) for every j ≤ i.

In view of lemma 11.5.38, we conclude that I^{e}H^{j}(V, C^{•}) is an A-module of finite type, whence I^{e} · R^{j}Γ_{{x}}C_{|U(x)}^{•} = 0 for every j ≤ i (theorem 11.5.25). To conclude the proof, it now suffices to observe that the natural map R^{j}Γ_{{x}}C_{|U(x)}^{•} → R^{j}Γ_{{x}}K_{|U(x)}^{•} is an isomorphism, for every j ≤ i. □

The following corollary recovers the results of [85, Exp.VIII, §2].

Corollary 11.5.57. In the situation of (11.5.40), for every i ∈ ℤ we have :

(i) If R^{i}Γ_{{x}}K^{•} = 0 for every x ∈ U_{1}, then H^{i}(U, K^{•}) is an A-module of finite type.

(ii) The following conditions are equivalent : (a) R^{j}Γ_{{x}}K^{•} = 0 for every x ∈ U_{1} and every j ≤ i.

(b) H^{j}(U, K^{•}) is an A-module of finite type for every j ≤ i.

Proof. It is the special case of theorem 11.5.43with I := A. □

11.6. Hochster’s theorem and Stanley’s theorem. The two main results of this section are theorems 11.6.35 and 11.6.43, which were proved originally respectively by Hochster in [91], and by Stanley in [155]. Our proofs follow in the main the posterior methods presented by Bruns and Herzog in [43] (which in turns, partially rely on some ideas of Danilov). These results shall be used in section 12.5, in order to show that regular log schemes are Cohen-Macaulay. We begin with some preliminaries from algebraic topology, which we develop only to the extent that is required for our special purposes: a much more general theory exists, and is well known to experts (see e.g. [124, Ch.IX]).

For any topological space X, and any subset T ⊂ X, we denote by \overline{T} the topological closure of T in X, endowed with the topology induced from X. We let H_{•}(X) (resp. H_{•}(X, T)) be the singular homology groups of X (resp. of the pair (X, T)). Also, for every n ∈ ℕ fix a Banach norm ∥ · ∥ on ℝ^{n}, and for every real number ρ > 0, let 𝔹^{n}(ρ) := {v ∈ ℝ^{n} | ∥v∥ < ρ}, and set 𝕊^{n−1} := \overline{𝔹}^{n}(1) \ 𝔹^{n}(1) (so 𝕊^{−1} = ∅).

Definition 11.6.1. (i) A finite regular cell complex (or briefly : a cell complex) is the datum of a topological space X, together with a finite filtration

X^{−1} = ∅ ⊂ X^{0} ⊂ X^{1} ⊂ X^{2} ⊂ ⋯ ⊂ X^{k} = X

consisting of closed subspaces, such that for every i = 0, …, k we have a decomposition

X^{i} \ X^{i−1} = ⋃_{λ∈Λ_{i}} e_{λ}^{i}

where :

(a) Λ_{i} is a finite set, and for every λ ∈ Λ_{i} there exists a homeomorphism

f_{λ} : \overline{𝔹}^{i}(1) →^{∼} \overline{e}_{λ}^{i} such that f_{λ}^{−1}(e_{λ}^{i}) = 𝔹^{i}(1).

(b) e_{λ}^{i} ∩ e_{μ}^{i} = ∅ for any two distinct indices λ, μ ∈ Λ_{i}.

(c) \overline{e}_{λ}^{i} \ e_{λ}^{i} ⊂ X^{i−1} for every λ ∈ Λ_{i}. (ii) The smallest k ∈ ℕ such that X^{k} = X is called the dimension of X^{•}, and is denoted dim X^{•}. For every i = 0, …, k, the subsets e_{λ}^{i} are called the i-dimensional cells of X^{•}.

(iii) A subcomplex of the cell complex X^{•} is a closed subset Y ⊂ X which is a union of cells of X. Then the filtration such that Y^{i} := Y ∩ X^{i} for every i ≥ −1 defines a cell complex structure Y^{•} on Y.

Remark 11.6.2. With the notation of definition 11.6.1(i), notice that X^{0} is a (finite) discrete topological space. Moreover, X is a Hausdorff space : indeed, more generally, the quotient of a compact Hausdorff space by a closed equivalence relation is Hausdorff, so in particular a space is Hausdorff, if it can be covered by finitely many closed compact Hausdorff subspaces.

Lemma 11.6.3. With the notation of definition 11.6.1, we have :

(i) For every i = 0, …, dim X and q ∈ ℤ, we have natural isomorphisms of abelian groups :

H_{q}(X^{i}, X^{i−1}) →^{∼} \{ \begin{matrix}ℤ^{Λ_{i}} & if q = i \\ 0 & otherwise. \end{matrix}

(ii) More precisely, let (b_{λ} | λ ∈ Λ_{i}) be the canonical basis of the free ℤ-module ℤ^{Λ_{i}}; for every λ ∈ Λ_{i}, the isomorphism of (i) maps the image of the induced map

H_{i}f_{λ} : H_{i}(\overline{𝔹}^{i}(1), 𝕊^{i−1}) → H_{i}(X^{i}, X^{i−1})

isomorphically onto the direct summand generated by b_{λ}.

Proof. (i): This is obvious for i = 0. For i = 1, …, k := dim X^{•} and every λ ∈ Λ_{i}, set

Y_{λ}^{i} := f_{λ}(\overline{𝔹}^{i}(1/2))  Y^{i} := ⋃_{λ∈Λ_{i}} Y_{λ}^{i}  A^{i} := ⋃_{λ∈Λ_{i}}\{f_{λ}(0)\}

and for any q ∈ ℤ, consider the natural group homomorphisms

(11.6.4) \xymatrix{H_{q}(\overline{𝔹}^{i}(1/2), \overline{𝔹}^{i}(1/2) \\{0\}) \ar[r]^{α′} \ar[d]_{γ} & H_{q}(\overline{𝔹}^{i}(1), \overline{𝔹}^{i}(1) \\{0\}) \ar[d] & H_{q}(\overline{𝔹}^{i}(1), 𝕊^{i−1}) \ar[l]_{β′} \ar[d] \\ H_{q}(Y^{i}, Y^{i} \ A^{i}) \ar[r]^{α} & H_{q}(X^{i}, X^{i} \ A^{i}) & H_{q}(X^{i}, X^{i−1}). \ar[l]_{β}}

The map α is an isomorphism, by excision; the same holds for β, since X^{i−1} is a deformation retract of X^{i} \ A^{i}. However, for every λ ∈ Λ_{i} we have natural isomorphisms

H_{q}(Y_{λ}^{i}, Y_{λ}^{i} \ A) →^{∼} \{ \begin{matrix}ℤ & if q = i \\ 0 & otherwise. \end{matrix}

whence the contention.

(ii): Take q := i in (11.6.4); then clearly γ is a monomorphism whose image is the direct summand H_{i}(Y_{λ}^{i}, Y_{λ}^{i} \ A); on the other hand, arguing as in the foregoing, we see that both α′ and β′ are isomorphisms. The assertion follows easily. □

Remark 11.6.5. (i) In the situation of lemma 11.6.3, notice that there is a natural bijection from Λ_{i} to the set of connected components π_{0}(X^{i} \X^{i−1}) of X^{i} \X^{i−1}, for every i = 0, …, dim X.

(ii) The direct sum decomposition of H_{i}(X^{i}, X^{i−1}) provided by lemma 11.6.3(i) depends only on the filtration X^{•} (and not on the choice of homeomorphisms f_{λ}). Indeed, this is clear, since the map H_{i}f_{λ} factors as a composition :

H_{i}(\overline{𝔹}^{i}(1), 𝕊^{i−1}) →^{∼} H_{i}(\overline{e}_{λ}^{i}, \overline{e}_{λ}^{i} \ e_{λ}^{i}) ⟶^{l_{λ}} H_{i}(X^{i}, X^{i−1})

where l_{λ} is the homomorphism induced by the obvious map of pairs (\overline{e}_{λ}^{i}, \overline{e}_{λ}^{i} \e_{λ}^{i}) → (X^{i}, X^{i−1}). (iii) In view of (ii), the maps l_{λ} admit natural left inverse homomorphisms

p_{λ} : H_{i}(X^{i}, X^{i−1}) → H_{i}(\overline{e}_{λ}^{i}, \overline{e}_{λ}^{i} \ e_{λ}^{i})

such that

(11.6.6)  p_{μ} ∘ l_{λ} = 0  for every λ, μ ∈ Λ_{i} with λ ≠ μ.

These maps can be described as follows. For any λ ∈ Λ_{i}, the natural map of pairs (X^{i}, X^{i−1}) → (X^{i}, X^{i} \ e_{λ}^{i}) induces a homomorphism

m_{λ} : H_{i}(X^{i}, X^{i−1}) → H_{i}(X^{i}, X^{i} \ e_{λ}^{i}).

Arguing by excision as in the proof of lemma 11.6.3(ii), it is easily seen that q_{λ} := m_{λ} ∘ l_{λ} is an isomorphism (details left to the reader), and we set p_{λ} := q_{λ}^{−1} ∘ m_{λ}. Obviously p_{λ} is left inverse to l_{λ}, and in oder to check (11.6.6) it suffices to show that m_{μ}∘l_{λ} = 0 for every μ ≠ λ. However, m_{μ} ∘ l_{λ} is the homomorphism arising from the map of pairs (\overline{e}_{λ}^{i}, \overline{e}_{λ}^{i} \ e_{λ}^{i}) → (X^{i}, X^{i} \ e_{μ}^{i}), so the assertion follows, after remarking that \overline{e}_{λ}^{i} ⊂ X^{i} \ e_{μ}^{i}.

11.6.7. We attach to any cell complex X^{•} a complex of abelian groups 𝒞_{•}(X^{•}), as follows. • For i < 0 and for i > k := dim X^{•}, we set 𝒞_{i}(X^{•}) := 0, and for i = 1, …, k we let

𝒞_{i}(X^{•}) := H_{i}(X^{i}, X^{i−1}).

Sometimes we write just 𝒞_{i}(X), unless it is useful to stress which filtration X^{•} on X we are considering. For every i > 0, the differential d_{i} : 𝒞_{i}(X^{•}) → 𝒞_{i−1}(X^{•}) is the composition

H_{i}(X^{i}, X^{i−1}) ⟶^{∂_{i}} H_{i−1}(X^{i−1}) ⟶^{j_{i−1}} H_{i−1}(X^{i−1}, X^{i−2})

where ∂_{i} is the boundary operator of the long exact homology sequence associated with the pair (X^{i}, X^{i−1}), and j_{i−1} is the homomorphism induced by the obvious map of pairs (X^{i−1}, ∅) → (X^{i−1}, X^{i−2}). In order to check that d_{i} ∘ d_{i+1} = 0 for every i ∈ ℤ, recall that every element of H_{i+1}(X^{i+1}, X^{i}) is the class \overline{c} of a singular (i + 1)-chain c of X^{i+1}, whose boundary ∂_{i+1}c is a singular i-chain of X^{i}; then d_{i+1}(\overline{c}) is the class of ∂_{i+1}c in H_{i}(X^{i}, X^{i−1}), and d_{i} ∘ d_{i+1}(\overline{c}) is the class of ∂_{i} ∘ ∂_{i+1}c in H_{i−1}(X^{i−1}, X^{i−2}), so it vanishes.

• It is also useful to consider an augmented version of the above complex; namely, let us set

\overline{𝒞} _{−1}(X^{•}) := ℤ  and \overline{𝒞} _{i}(X^{•}) := 𝒞_{i}(X^{•})  for every i ≠ −1

with differential d_{0} given by the rule : d_{0}(b_{λ}^{0}) = 1 for every λ ∈ Λ_{0}.

Proposition 11.6.8. With the notation of (11.6.7), there exist natural isomorphisms of abelian groups

H_{q}𝒞_{•}(X^{•}) →^{∼} H_{q}(X) for every q ∈ ℕ.

Proof. For every topological space T, let C_{•}(T) denote the complex of singular chains of T. The filtration X^{•} induces a finite filtration of complexes

C_{•}(X^{0}) ⊂ C_{•}(X^{1}) ⊂ C_{•}(X^{2}) ⊂ ⋯ ⊂ C_{•}(X)

whence a convergent spectral sequence

E_{pq}^{1} := H_{p+q}(X^{p}, X^{p−1}) ⇒ H_{p+q}(X)

(see [163, Th.5.5.1]). By direct inspection, it is easily seen that the differential d_{p,0}^{1} : E_{p,0}^{1} → E_{p−1,0}^{1} agrees with the differential d_{p} of 𝒞_{•}(X^{•}), for every p ∈ ℕ. On the other hand, lemma 11.6.3shows that this spectral sequence degenerates, whence the contention. □

11.6.9. Keep the notation of definition 11.6.1, and set Λ_{0} := X^{0}. For every i = 0, …, dim X and every λ ∈ Λ_{i}, the abelian group H_{i,λ} := H_{i}(\overline{e}_{λ}^{i}, \overline{e}_{λ}^{i} \ e_{λ}^{i}) is free of rank one; if i > 0, we fix one of the two generators of this group, and we denote by b_{λ}^{i} its image in 𝒞_{i}(X^{•}) (see remark 11.6.5(ii)). For i = 0, each e_{λ}^{0} is a point, hence H_{0,λ} admits a canonical identification with ℤ, and we let b_{λ}^{0} be the image of 1, under the resulting map ℤ →^{∼} H_{0,λ} → 𝒞_{0}(X^{•}). The system of classes (b_{λ}^{i} | 0 ≤ i ≤ dim X, λ ∈ Λ_{i}) is called an orientation for X^{•}. We may write

d_{i}(b_{λ}^{i}) = ∑_{μ∈Λ_{i−1}}[b_{λ}^{i} : b_{μ}^{i−1}]b_{μ}^{i−1} for every i ≤ dim X^{•} and every λ ∈ Λ_{i}

for a system of uniquely determined integers [b_{λ}^{i} : b_{μ}^{i−1}], called the incidence numbers of the cells e_{λ}^{i} and e_{μ}^{i−1} (relative to the chosen orientation of X^{•}). With this notation, we may state :

Lemma 11.6.10. The incidence numbers of a cell complex X^{•} fulfill the following conditions : (i) ∑_{μ∈Λ_{i−1}}[b_{λ}^{i} : b_{μ}^{i−1}] · [b_{μ}^{i−1} : b_{ν}^{i−2}] = 0 for every i ≥ 2, every λ ∈ Λ_{i} and every ν ∈ Λ_{i−2}. (ii) Let λ ∈ Λ_{1} be any index, and say that \overline{e}_{λ}^{1} \ e_{λ}^{1} = e_{μ}^{0} ∪ e_{ρ}^{0}. Then [b_{λ}^{1} : b_{μ}^{0}] + [b_{λ}^{1} : b_{ρ}^{0}] = 0.

Proof. Condition (i) translates the identity d_{i−1} ∘ d_{i} = 0 for the differential d_{•} of the complex 𝒞_{•}(X^{•}). The identity of (ii) follows by a simple inspection of the definition of d_{1} : details left to the reader. □

11.6.11. The generalities of the previous paragraphs shall be applied to the following situation. Let (V, σ) be a strictly convex polyhedral cone such that ⟨σ⟩ = V, and set d := dim_{ℝ} V. We attach to σ a cell complex C_{σ}^{•} as follows. Fix u_{0} ∈ σ^{∨} such that σ ∩ Ker u_{0} = {0} (corollary 6.3.14), let σ^{∘} be the topological interior of σ (in V) and set

C_{σ} := σ ∩ u_{0}^{−1}(1) C_{σ}^{∘} := σ^{∘} ∩ u_{0}^{−1}(1).

Let v_{1}, …, v_{k} be a minimal system of generators for σ; we may assume that u_{0}(v_{i}) = 1 for i = 1, …, k, in which case

(11.6.12)  C_{σ} = {∑_{i=1}^{k} λ_{i}v_{i} | λ_{1}, …, λ_{k} ≥ 0 and ∑_{i=1}^{k} λ_{i} = 1}

which shows that C_{σ} is a compact topological space (with the topology inherited from V).

Lemma 11.6.13. There exists a homeomorphism \overline{𝔹}^{d−1}(1) →^{∼} C_{σ} that maps 𝔹^{d−1}(1) onto C_{σ}^{∘}.

Proof. Set v_{0} := k^{−1} · (v_{1} + ⋯ + v_{k}) and W := Ker u_{0}. It suffices to show that there exists a homeomorphism \overline{𝔹}^{d−1}(1) →^{∼} D_{σ} := C_{σ} − v_{0} ⊂ W that maps 𝔹^{d−1}(1) onto D_{σ}^{∘} := C_{σ}^{∘} − v_{0}. However, pick a minimal system u_{1}, …, u_{t} of generators of σ^{∨} (corollary 6.3.12(i)); since σ spans V, for every i = 1, …, t there exists j ≤ k such that u_{i}(v_{j}) > 0. Hence – after replacing the u_{i} by suitable scalar multiples – we may assume that u_{i}(v_{0}) = 1 for every i = 1, …, t, and then lemma 6.3.2implies that

D_{σ} = {v ∈ W | u_{i}(v) ≥ −1 for every i = 1, …, t}.

Also, proposition 6.3.11(i) implies that

D_{σ}^{∘} = {v ∈ W | u_{i}(v) > −1 for every i = 1, …, t}.

For every v ∈ W, set μ(v) := min(u_{i}(v) |i = 1, …, t). It is easily seen that μ(v) < 0 for every v ∈ W \ {0}; indeed, if μ(w) ≥ 0, then the subset {λw | λ ∈ ℝ_{+}} lies in D_{σ}, and since the latter is compact, it follows that w = 0. Moreover we have

(11.6.14)  (ℝ_{+}w) ∩ D_{σ} = {λw | λ ∈ [0, −1/μ(w)]}  for every w ∈ W \ {0}.

Fix a Banach norm ∥ · ∥_{V} on V, and consider the mapping

φ : D_{σ} → W such that φ(0) := 0 and φ(v) := −\frac{μ(v)}{∥v∥} · v for v ≠ 0.

It is easily seen that φ is injective; since μ is a continuous mapping, the same follows for φ, and since D_{σ} is compact, we conclude that φ induces a homeomorphism D_{σ} →^{∼} φ(D_{σ}). However, from (11.6.14) it follows that

φ(D_{σ}) = {v ∈ W | ∥v∥ ≤ 1} and φ(D_{σ}^{∘}) = {v ∈ W | ∥v∥ < 1}

whence the claim. □

11.6.15. Keep the notation of (11.6.11); we consider the finite filtration C_{σ}^{•} of C_{σ}, defined as follows. For every i = −1, …, d − 1, we let C_{σ}^{i} ⊂ C_{σ} be the union of the subsets τ ∩ u_{0}^{−1}(1), where τ ranges over the (finite) set of all faces of σ of dimension i + 1. Clearly C_{σ}^{−1} = ∅, C_{σ}^{d−1} = C_{σ}, and C_{σ}^{i} is a closed subset of C_{σ}, for every i = 0, …, d − 1. Moreover, it follows easily from lemma 11.6.13 and proposition 6.3.11(i) that the datum of C_{σ} and its filtration C_{σ}^{•} is a cell complex. With the notation of definition 11.6.1, the indexing set Λ_{i} can be taken to be the set of all (i + 1)-dimensional faces of σ, for every i = 0, …, d − 1 : indeed, if τ is such a face, denote by τ^{∘} the relative interior of τ (see example 6.3.16(iii)); then it is clear that e_{τ}^{i} := τ^{∘} ∩ C_{σ} ≠ ∅ and \overline{e}_{τ}^{i} = τ ∩ C_{σ}.

Remark 11.6.16. (i) Notice that the cell complex C_{σ}^{•} is independent – up to homeomorphism – of the choice of u_{0}. Indeed, say that u′_{0} ∈ σ^{∨} is any other linear form such that σ∩Ker u′_{0} = {0}, and let C′_{σ} := σ ∩ u_{0}^{′−1}(1). We define a homeomorphism ψ : C′_{σ} →^{∼} C_{σ}, by the rule :

v ↦ u_{0}(v)^{−1} · v for every v ∈ C′_{σ}.

It is easily seen that ψ restricts to homeomorphisms C_{σ}^{′i} →^{∼} C_{σ}^{i} for every i = 0, …, d − 1.

(ii) For any integer i = 0, …, d − 3, and any cells e_{τ}^{i}, e_{λ}^{i+2} of C_{σ}^{•} such that e_{τ}^{i} ⊂ \overline{e}_{λ}^{i+2}, there exist exactly two (i + 1)-dimensional cells e_{μ}^{i+1}, e_{ρ}^{i+1} such that e_{τ}^{i} ⊂ \overline{e}_{μ}^{i+2} ∩ \overline{e}_{ρ}^{i+1} and e_{μ}^{i+1} ∪ e_{ρ}^{i+1} ⊂ \overline{e}_{λ}^{i+2} : indeed, this assertion is a direct translation of claim 6.3.9(ii).

Proposition 11.6.17. With the notation of (11.6.9) and (11.6.15), the following holds for every i = 0, …, d − 2 :

(i) If τ ∈ Λ_{i+1} and μ ∈ Λ_{i} is not a facet of τ, we have [b_{τ}^{i+1} : b_{μ}^{i}] = 0. (ii) If τ ∈ Λ_{i+1} and μ ∈ Λ_{i} is a facet of τ, then [b_{τ}^{i+1} : b_{μ}^{i}] ∈ {1, −1}. (iii) In the situation of remark 11.6.16(ii), we have :

[b_{λ}^{i+2} : b_{μ}^{i+1}] · [b_{μ}^{i+1} : b_{τ}^{i}] + [b_{λ}^{i+2} : b_{ρ}^{i+1}] · [b_{ρ}^{i+1} : b_{τ}^{i}] = 0.

Proof. (i): Consider the commutative diagram

(11.6.18)  \xymatrix{H_{i+1}(\overline{e}_{τ}^{i+1}, \overline{e}_{τ}^{i+1} \ e_{τ}^{i+1}) \ar[r]^{∂} \ar[d]_{l_{τ}} & H_{i}(\overline{e}_{τ}^{i+1} \ e_{τ}^{i+1}) \ar[r]^{j_{τ}} \ar[d]^{g} & H_{i}(C_{σ}^{i}, C_{σ}^{i} \ e_{μ}^{i}) \\ H_{i+1}(C_{σ}^{i+1}, C_{σ}^{i}) \ar[r]^{∂′} & H_{i}(C_{σ}^{i}) \ar[r]^{j_{i}} & H_{i}(C_{σ}^{i}, C_{σ}^{i−1}) \ar[u]_{m_{μ}}}

where l_{τ} and m_{μ} are defined as in remark 11.6.5(ii,iii) and g is induced by the inclusion map \overline{e}_{τ}^{i+1} \ e_{τ}^{i+1} → C_{σ}^{i}, the maps ∂ and ∂′ are the boundary operators of the long exact sequences attached to the pairs (\overline{e}_{τ}^{i+1}, \overline{e}_{τ}^{i+1} \ e_{τ}^{i+1}) and (C_{σ}^{i+1}, C_{σ}^{i}), and j_{i} (resp. j_{τ}) is deduced from the obvious map of pairs (\overline{e}_{τ}^{i+1} \ e_{τ}^{i+1}, ∅) → (C_{σ}^{i}, C_{σ}^{i} \ e_{μ}^{i}) (resp. (C_{σ}^{i}, ∅) → (C_{σ}^{i}, C_{σ}^{i−1})). In light of remark 11.6.5(iii), we need to check that m_{μ} ∘ j_{i−1} ∘ ∂′ ∘ l_{τ} = 0, and to this aim, it suffices to show that j_{τ} = 0. However, the assumption implies that μ ∩ τ is a (proper) face of both μ and τ; this translates as the identity (\overline{e}_{τ}^{i+1} \ e_{τ}^{i+1}) ∩ e_{μ}^{i} = ∅, whence the claim.

(ii): Clearly C_{τ}^{•} is a cell subcomplex of C_{σ}^{•}, and e_{τ}^{i+1} and e_{μ}^{i} are cells of this subcomplex. Moreover, any orientation for C_{σ}^{•} restricts to an orientation for C_{τ}^{•}, and the resulting incidence number of e_{τ}^{i+1} and e_{μ}^{i} is the same for either cell complex. Thus, we may replace σ by τ, in which case the map g of (11.6.18) is the identity.

Claim 11.6.19. If σ = τ and i > 0, both ∂ and k_{μ} := m_{μ} ∘ j_{i} in (11.6.18) are isomorphisms.

Proof of the claim. For ∂, we use the long exact homology sequence of the pair (\overline{e}_{τ}^{i+1}, \overline{e}_{τ}^{i+1}\e_{τ}^{i+1}): since i ≥ 0, lemma 11.6.13implies that H_{i+1}(\overline{e}_{τ}^{i+1}) = 0, so ∂ is injective; since i > 0, the same argument shows that ∂ is surjective. Next, k_{μ} is the homomorphism deduced from the long exact homology sequence of the pair (C_{τ}^{i}, C_{τ}^{i} \ e_{μ}^{i}), and we remark that C_{τ}^{i} \ e_{μ}^{i} is contractible : indeed, e_{μ}^{i} is homeomorphic to 𝔹^{i}(1) (lemma 11.6.13), hence C_{τ}^{i} \e_{μ}^{i} is a retraction of C_{τ}^{i} \{x}, for any x ∈ e_{μ}^{i}, and the latter is homeomorphic to ℝ^{i} (again by lemma 11.6.13). If i > 1, we deduce already that k_{μ} is an isomorphism. For i = 1, the same argument shows that k_{μ} is injective, so its cokernel is a cyclic torsion group that injects into H_{0}(C_{τ}^{1} \ e_{μ}^{1}) ≃ ℤ, hence it must vanish as well. ♢

Since (11.6.18) commutes, claim 11.6.19 yields the assertion, in case i > 0. If i = 0, C_{τ}^{1} is isomorphic to \overline{𝔹}^{1}(1), and e_{μ}^{0} is one of the two points of \overline{𝔹}^{1}(1) \ 𝔹^{1}(1), so the assertion can be checked easily, by inspecting the definitions.

(iii) follows from (i), lemma 11.6.10(i) and remark 11.6.16(ii). □

The following result says that the properties of proposition 11.6.17 completely characterize the incidence numbers of C_{σ}^{•}.

Proposition 11.6.20. Keep the notation of (11.6.15), and consider a system of integers

(β_{λμ}^{i} | i = 1, …, d − 1, λ ∈ Λ_{i}, μ ∈ Λ_{i−1})

such that :

(a) If λ ∈ Λ_{i} and μ ∈ Λ_{i−1} is not a facet of λ, then β_{λμ}^{i} = 0.

(b) If λ ∈ Λ_{i} and μ ∈ Λ_{i−1} is a facet of λ, then β_{λμ}^{i} ∈ {1, −1}.

(c) If μ and τ are the two 1-dimensional facets of the 2-dimensional face λ of σ, then

β_{λμ}^{1} + β_{λτ}^{1} = 0.

(d) Let λ ∈ Λ_{i+1} and μ ∈ Λ_{i−1} be two faces, such that μ is a face of λ, and denote by τ and ρ the two facets of λ that contain μ. Then

β_{λτ}^{i+1}β_{τμ}^{i} + β_{λρ}^{i+1}β_{ρμ}^{i} = 0.

Then there exists a unique orientation of C_{σ}^{•}

(b_{λ}^{i} | i = 0, …, d − 1, λ ∈ Λ_{i})

such that for every i = 1, …, d − 1 we have :

(11.6.21)  [b_{λ}^{i} : b_{μ}^{i−1}] = β_{λμ}^{i}  for every λ ∈ Λ_{i}, μ ∈ Λ_{i−1}.

Proof. We construct the orientation classes b_{λ}^{i} fulfilling condition (11.6.21), by induction on i. For i = 0, the condition is empty, and the classes b_{λ}^{0} are prescribed by (11.6.9). For i = 1, and a given λ ∈ Λ_{1}, denote by τ and ρ the two faces of λ; clearly the condition [b_{λ}^{1} : b_{τ}^{0}] = β_{λτ}^{1} (and assumption (b)) determines b_{λ}^{1} univocally, and in view of (c) and lemma 11.6.10(ii), for this choice of orientation of e_{λ}^{1} we have as well [b_{λ}^{1} : b_{ρ}^{0}] = β_{λρ}^{1}. Moreover, if μ ∈ Λ_{0} \ {τ, ρ}, then (11.6.21) is verified as well, by virtue of (a) and proposition 11.6.17(i).

Now, suppose that i > 1 and that we have already constructed orientation classes b_{μ}^{j} as sought, for every j < i and every μ ∈ Λ_{j}. Let λ ∈ Λ_{i} be any face, and fix a facet τ of λ; again, the identity

(11.6.22)  [b_{λ}^{i} : b_{τ}^{i}] = β_{λτ}^{i}

determines b_{λ}^{i}, and it remains to check that – with this choice of b_{λ}^{i} – condition (11.6.21) holds for every μ ∈ Λ_{i−1} \ {τ}, and by virtue of (a) and proposition 11.6.17(i), it suffices to consider the facets μ of λ. However, notice that the system of orientation classes (b_{μ}^{j} | j = 0, …, i − 1, μ ⊂ λ) amounts to an orientation of C_{λ}^{i−1} (regarded as a cell subcomplex of C_{σ}^{•}), and the incidence numbers for the complex (C_{λ}^{i−1})^{•} relative to these classes agree with the incidence numbers of C_{σ}^{•}, relative to the same classes. Especially, the sums

c := ∑_{μ∈Λ_{i−1}} β_{λμ}^{i}b_{μ}^{i−1} c′ := ∑_{μ∈Λ_{i−1}}[b_{λ}^{i} : b_{μ}^{i−1}]b_{μ}^{i−1}

are well defined elements of 𝒞_{i−1}(C_{λ}^{i−1}), and in fact :

d_{i−1}(c) = ∑_{μ∈Λ_{i−1}} β_{λμ}^{i} · d_{i−1}(b_{μ}^{i−1})

= ∑_{μ∈Λ_{i−1}} β_{λμ}^{i} · ∑_{ρ∈Λ_{i−2}}[b_{μ}^{i−1} : b_{ρ}^{i−2}]b_{ρ}^{i−2}

= ∑_{μ∈Λ_{i−1}} ∑_{ρ∈Λ_{i−2}} β_{λμ}^{i}β_{μρ}^{i−1}b_{ρ}^{i−2}  (by inductive assumption)
= 0  (by (d))

and a similar calculation yields d_{i−1}(c′) = 0 as well. However, C_{λ}^{i−1} is homeomorphic to 𝕊^{i−1} (lemma 11.6.13), hence

Ker(d_{i−1} : 𝒞_{i−1}(C_{λ}^{i−1}) → 𝒞_{i−2}(C_{λ}^{i−1})) ≃ ℤ

which, in view of (b), implies that c = ±c′. Taking into account (11.6.22), we see that actually c = c′, whence the claim. The uniqueness of the orientation fulfilling condition (11.6.21) can be checked easily by induction on i : the details shall be left to the reader. □

11.6.23. Let now L be a free abelian group of finite rank d, and (L_{ℝ}, σ) a strictly convex L-rational polyhedral cone (see (6.3.20)), such that ⟨σ⟩ = L_{ℝ}; set

P := L ∩ σ  F_{λ} := L ∩ λ  P_{λ} := F_{λ}^{−1}P  for every face λ of σ

so P and its localization P_{λ} are fine and saturated monoids (proposition 6.3.22(i)), and F_{λ} is a face of P, for every such λ. Let R be any ring, and if λ, μ ⊂ σ are any two faces with λ ⊂ μ, denote by

j_{λμ} : R[P_{λ}] → R[P_{μ}]

the natural localization map; notice that if 0 ⊂ σ is the unique 0-dimensional face, then P_{0} = P, and j_{0μ} is the localization map R[P] → R[P_{μ}]. We attach to P the complex \overline{𝒞} _{P}^{•} of R[P]- modules such that :

\overline{𝒞} _{P}^{0} := R[P]  and \overline{𝒞} _{P}^{i} := ⨁_{λ∈Λ_{i−1}} R[P_{λ}]  for every i = 1, …, dim P

with differentials given by the rule :

d^{i}(x_{λ}) := ∑_{\begin{matrix}μ∈Λ_{i} \\ λ⊂μ \end{matrix}}[b_{μ}^{i} : b_{λ}^{i−1}] · j_{λμ}(x_{λ}) for every i ≥ 1, λ ∈ Λ_{i−1} and x_{λ} ∈ R[P_{λ}]

and

d^{0}(x) := ∑_{μ∈Λ_{0}} j_{0μ}(x) for every x ∈ R[P]

where (b_{τ}^{j} | j = 0, … dim σ − 1, τ ∈ Λ_{j}) is a chosen orientation for C_{σ}^{•}. Taking into account lemma 11.6.10(ii) and proposition 11.6.17(iii), it is easily seen that d^{i+1} ∘d^{i} = 0 for every i ≥ 0 (essentially, the differential d^{i} is the transpose of the differential d_{i−1} of \overline{𝒞} _{•}(C_{σ}^{•})).

Set as well X := Spec R[P], and notice that Z := Spec R⟨P/𝔪_{P} ⟩ ≃ Spec R is a closed subset of X. For any R[P]-module M, we denote as usual by M^{∼} the quasi-coherent 𝒪_{X}- module arising from M.

Theorem 11.6.24. With the notation of (11.6.23), for every R[P]-module M we have a natural isomorphism

RΓ_{Z}M^{∼} →^{∼} M ⊗_{R[P]} \overline{𝒞} _{P}^{•} in D^{+}(R[P]-Mod).

Proof. For every face λ of σ, set U_{λ} := Spec R[P_{λ}], and denote by g_{λ} : U_{λ} → X the open immersion. We consider the chain complex ℛ_{•} of ℤ_{X}-modules such that :

ℛ_{0} := ℤ_{X}  ℛ_{i} := ⨁_{λ∈Λ_{i−1}} g_{λ!}ℤ_{U_{λ}}  for every i = 1, …, dim P.

The differential d_{1} : ℛ_{1} → ℛ_{0} is just the sum of the natural morphisms g_{λ!}ℤ_{U_{λ}} → ℤ_{X}, for λ ranging over the one-dimensional faces of σ. For i > 1, the differential d_{i} is the sum of the maps

d_{λ} := ∑_{\begin{matrix}μ∈Λ_{i} \\ μ⊂λ \end{matrix}}[b_{λ}^{i} : b_{μ}^{i−1}] · d_{λμ!} : g_{λ!}ℤ_{U_{λ}} → ℛ_{n−1} for every λ ∈ Λ_{i−1}

where d_{λμ} : g_{λ!}ℤ_{U_{λ}} → g_{μ!}ℤ_{U_{μ}} ⊂ ℛ_{n−1} is induced by the inclusion U_{λ} ⊂ U_{μ}. With this notation, a simple inspection of the definitions yields a natural identification

(11.6.25)  M ⊗_{R[P]} \overline{𝒞} _{P}^{•} →^{∼} Hom_{ℤ}^{•}(ℛ_{•}, M^{∼}[0])  in C(R[P]-Mod).

On the other hand, let λ_{1}, …, λ_{n} be the one-dimensional faces of σ; then λ_{i} is L-rational (see (6.3.20)), and F_{λ_{i}} is a fine, sharp and saturated monoid of dimension one (propositions 6.3.22(i) and 6.4.9(ii)), so F_{λ_{i}} ≃ ℕ for i = 1, …, n (theorem 6.4.18(ii)). For each i = 1, …, n, let y_{i} be the unique generator of F_{λ_{i}}, and denote by I ⊂ P the ideal generated by y_{1}, …, y_{n}; we remark

Claim 11.6.26. The radical of I is 𝔪_{P}.

Proof of the claim. For any x ∈ 𝔪_{P}, pick a subset S ⊂ {1, …, n} such that x = ∑_{i∈S} a_{i}y_{i}, with a_{i} > 0 for every i ∈ S. Let N ∈ ℕ be large enough such that Na_{i} ≥ 1 for every i ∈ S, and pick b_{i} ∈ ℕ such that Na_{i} ≥ b_{i} ≥ 1 for every i ∈ S. It follows that Nx − ∑_{i∈S} b_{i}y_{i} ∈ P, and the claim follows. ♢

Now, let i : Z → X be the closed immersion; we have :

Claim 11.6.27. The natural map ℤ_{X} → i_{∗}ℤ_{Z} induces an isomorphism

(11.6.28)  ℛ_{•} →^{∼} i_{∗}ℤ_{Z}[0]  in D^{+}(ℤ_{X}-Mod).

Proof of the claim. The assertion can be checked on the stalks at the points of X, hence let x ∈ X be any such point; if x ∈ Z, claim 11.6.26 easily implies that (ℛ_{•})_{x} is concentrated in degree zero, and then (11.6.28)_{x} is clearly an isomorphism of complexes of ℤ-modules. If x ∉ Z, let 𝔭 ⊂ R[P] be the prime ideal corresponding to x, and λ ⊂ σ the unique face such that P \ F_{λ} = 𝔭 ∩ P; a simple inspection of the construction shows that

(ℛ_{•})_{x} →^{∼} \overline{𝒞} _{•}(C_{λ}^{•})

(notation of (11.6.7)). But proposition 11.6.8and lemma 11.6.13imply that \overline{𝒞} _{•}(C_{λ}^{•}) is acyclic, whence the claim. ♢

In view of (11.6.25) and claim 11.6.27, we are reduced to showing that the natural map

Hom_{ℤ}^{•}(ℛ_{•}, M^{∼}[0]) → RHom_{ℤ}^{•}(ℛ_{•}, M^{∼}[0])

is an isomorphism in D^{+}(R[P]-Mod). This can be done by a spectral sequence argument, along the lines of proposition 10.4.18(which indeed includes a special case of the situation we are considering here, namely the case where P is a free monoid : the details shall be left to the reader). □

11.6.29. Notice that \overline{𝒞} _{P}^{•} is a complex of L-graded R-modules, hence its cohomology is Lgraded as well, and we wish next to compute the graded terms gr_{•}H^{•}(\overline{𝒞} _{P}^{•}). To this aim, we make the following :

Definition 11.6.30. Let V be a finite dimensional ℝ-vector space, X ⊂ V any subset, and z ∈ V any point. Then :

(a) We say that a point x ∈ X is visibile from z, if {tz + (1 − t)x | 0 ≤ t ≤ 1} ∩ X = {x}.

(b) We say that a subset S ⊂ X is visibile from z, if every point of S is visible from z.

Lemma 11.6.31. In the situation of (11.6.15), let z ∈ V \ C_{σ} be any point, and denote by S the set of points of C_{σ} that are visible from z. Then :

(i) S is a subcomplex of C_{σ}^{•}.

(ii) S (with the topology induced from V) is homeomorphic to \overline{𝔹}^{e}(1), for e ∈ \{d−2, d−1\}.

Proof. Pick a system of generators ρ_{1}, …, ρ_{n} for σ^{∨}, and choose ρ_{0} ∈ σ^{∨} so that C_{σ} = σ ∩ ρ_{0}^{−1}(1); we may assume that, for some integer k ≤ n we have

ρ_{i}(z) < 0 if and only if 1 ≤ i ≤ k.

Say that y ∈ C_{σ}, so that ρ_{i}(y) ≥ 0 for every i = 1, …, n, and set y_{t} := tz + (1 − t)y for every t ∈ [0, 1]; then clearly ρ_{i}(y_{t}) ≥ 0 for every i = k + 1, …, n. Now, if ρ_{0}(z) ≠ 1, we get ρ_{0}(y_{t}) ≠ 1 for every t ≠ 0, therefore the whole of C_{σ} is visible from z, in which case (i) is trivial, and (ii) follows from 11.6.13. Hence we may assume that ρ_{0}(z) = 1, so ρ_{0}(y_{t}) = 1 for every t ∈ [0, 1]. Suppose now that ρ_{i}(y) > 0 for some i ≤ k; then there exists a unique t_{i} ∈]0, 1[ such that ρ_{i}(y_{t_{i}}) = 0. Hence, if s := min(ρ_{i}(y) | i = 1, …, k) > 0, let t := min(t_{i} | i = 1, …, k); it follows that ρ_{i}(y_{t}) ≥ 0 for every i = 1, …, n, so y_{t} ∈ C_{σ}, which says that y is not visible from z. Conversely, if s = 0, then it follows easily that y is visible from z. We conclude that

S = C_{σ} ∩ ⋃_{i=1}^{k} Ker ρ_{i}

which shows (i). Next, set W := Ker ρ_{0}, and denote by τ_{z} : ρ_{0}^{−1}(1) →^{∼} W the translation map given by the rule : x ↦ x − z for every x ∈ ρ_{0}^{−1}(1). To conclude, it suffices to check that S′ := τ_{z}(S) is homeomorphic to \overline{𝔹}^{d−2}(1). To this aim, denote by λ the convex cone in W generated by τ_{z}(C_{σ}). Explicitly, if v_{1}, …, v_{k} ∈ σ have been chosen so that (11.6.12) holds, then λ is the cone generated by v_{1} − z, …, v_{k} − z; especially, λ is a polyhedral cone, and it is easily seen that ⟨λ⟩ = W. Moreover, λ is strictly convex; indeed, otherwise there exist real numbers a_{1}, …, a_{k} ≥ 0, with a_{i} > 0 for at least an index i ≤ k, such that ∑_{i=1}^{k} a_{i} · (v_{i} − z) = 0, i.e. ∑_{i=1}^{k} a_{i}v_{i} = (∑_{i=1}^{k} a_{i}) · z, which is absurd, since z ∉ σ. Pick u ∈ λ^{∨} such that λ ∩ Ker u = 0, and set C_{λ} := λ ∩ u^{−1}(1); by lemma 11.6.13, the subset C_{λ} is homeomorphic to \overline{𝔹}^{d−2}(1). Lastly, let π : W \ Ker u → u^{−1}(1) be the radial projection (so π(w) is the intersection point of ℝw with u^{−1}(1), for every w ∈ W \ Ker u). It is easily seen that π maps S′ bijectively onto C_{λ}, so the restriction of π is a homeomorphism S′ →^{∼} C_{λ}, as required. □

Lemma 11.6.32. In the situation of (11.6.23), let λ ⊂ σ be any face. For every l ∈ L_{ℝ} we have:

(i) The set of points of σ that are visible from l is a union of faces of σ. (ii) Suppose l ∈ L, and endow R[P_{λ}] with its natural L-grading. Then

gr_{l}R[P_{λ}] = \{ \begin{matrix}Rl & if λ ⊂ σ is not visible from l \\ 0 & otherwise. \end{matrix}

Proof. Let ρ_{1}, …, ρ_{n} be a system of generators of σ^{∨}; we may assume that, for some integer k ≤ n we have

ρ_{i}(l) < 0 if and only if 1 ≤ i ≤ k.

Arguing as in the proof of lemma 11.6.31, we check easily that the set of points of σ visible from l equals S := σ ∩ ⋃_{i=1}^{k} Ker ρ_{i}, which already shows (i).

(ii): Suppose first that λ ⊂ σ is not visible from l; then there exists x ∈ λ \ S, and since F_{λ} generates λ (see (6.3.20)), we may assume that x ∈ F_{λ} (details left to the reader). A simple inspection then shows that there exists a sufficiently large N ∈ ℕ such that l +Nx ∈ σ, whence l ∈ R[P_{λ}], and so gr_{l}R[P_{λ}] = Rl. Conversely, if the latter identity holds, then there exists x ∈ F_{λ} such that l + x ∈ P, whence ρ_{i}(x) > 0 for i = 1, …, k, so x is not visible from l. □

11.6.33. In the situation of (11.6.23), let us fix a linear form u_{0} ∈ σ^{∨} such that σ∩Ker u_{0} = 0, and define σ^{∘} and C_{σ} as in (11.6.11). For any l ∈ L_{ℝ}, denote by S_{l} the set of points of σ that are visible from l, so S_{l} is a union of faces of σ, by lemma 11.6.32(i), and therefore C_{l} := S_{l} ∩ C_{σ} is a subcomplex of C_{σ}^{•}.

Proposition 11.6.34. With the notation of (11.6.33), suppose l ∈ L. Then the following holds : (i) If −l ∈ σ^{∘}, the complex of R-modules gr_{l}\overline{𝒞} _{P}^{•} is isomorphic to R[−d]. (ii) If −l ∉ σ^{∘}, there is a natural isomorphism of complexes of R-modules :

gr_{l}\overline{𝒞} _{P}^{•} →^{∼} Hom_{ℤ}^{•}(\overline{𝒞} _{•}(C_{σ}^{•})/\overline{𝒞} _{•}(C_{l}^{•}), R[−1]).

Moreover, in this case, both \overline{𝒞} _{•}(C_{σ}^{•}) and \overline{𝒞} _{•}(C_{l}^{•}) are acyclic complexes.

Proof. (i): If −l ∈ σ^{∘}, then l ∈ P_{λ} if and only if λ = σ, so the assertion is clear.

(ii): The sought identification of complexes follows from lemma 11.6.32(ii), by a direct inspection of the constructions (and indeed, this holds even if −l ∈ σ^{∘} : details left to the reader). Moreover, it is already known from proposition 11.6.8and lemma 11.6.13that \overline{𝒞} _{•}(C_{σ}^{•}) is acyclic.

• Next, if l ∈ P, then clearly S_{l} = ∅, so the assertion for \overline{𝒞} _{•}(C_{l}^{•}) is trivial in this case.

• Thus, suppose that l ∉ P; if furthermore −l ∉ P, then the convex cone τ generated by P and l is still strictly convex, so we may find u_{1} ∈ τ^{∨} such that u_{1}(l) = 1 and σ∩Ker u_{1} = 0. Set C′_{σ} := σ ∩u_{1}^{−1}(1) and C′_{l} := S_{l} ∩C′_{σ}. In remark 11.6.16(i) we have exhibited a homeomorphism C_{σ}^{′•} →^{∼} C_{σ}^{•} that preserves the respective cell complex structures, and a simple inspection shows that this homeomorphism maps C′_{l} onto C_{l}. On the other hand, it is easily seen that C′_{l} is also the set of points of C′_{σ} that are visible from l (indeed, the segment that joins any point of C′_{σ} to l lies in u_{1}^{−1}(1), so its intersection with σ equals its intersection with C′_{σ} : details left to the reader). By virtue of lemma 11.6.31(ii) (and proposition 11.6.8), it follows that \overline{𝒞} _{•}(C_{l}^{′•}) is acyclic, so the same holds for \overline{𝒞} _{•}(C_{l}^{•}).

• Lastly, suppose −l ∈ P \ σ^{∘}; we let ρ_{1}, …, ρ_{n} be a system of generators of σ^{∨}, and k ≤ n an integer such that ρ_{i}(l) < 0 if and only if 1 ≤ i ≤ k. Denote by τ^{∨} (resp. μ^{∨}) the convex cone in L_{ℝ}^{∨} generated by (ρ_{k+1}, …, ρ_{n}) (resp. by (−ρ_{1}, …, −ρ_{k})), and let τ (resp. μ) be the dual of τ^{∨} (resp. of μ^{∨}) in L_{ℝ}; with this notation, we have l ∈ τ ∩ μ^{∘}. Especially, τ ∩ μ^{∘} ≠ ∅, and since τ^{∘} is dense in τ (by proposition 6.3.11(i)), we deduce that τ^{∘} ∩ μ^{∘} ≠ ∅ as well. Pick z ∈ τ^{∘} ∩ μ^{∘}; by inspecting the proof of lemma 11.6.32(i), it is easily seen that S_{l} = S_{z}. But by construction, −z ∉ σ, therefore – arguing as in the previous case – we conclude that \overline{𝒞} _{•}(C_{z}^{•}) = \overline{𝒞} _{•}(C_{l}^{•}) is acyclic. □

Theorem 11.6.35 (Hochster). Let R be a Cohen-Macaulay noetherian ring, P a fine, sharp and saturated monoid. Then R[P] is a Cohen-Macaulay ring.

Proof. In view of [126, p.181, Cor.] we may assume that R is a field. We argue by induction on d := dim P. If d = 0, we have P = 0, and there is nothing to show. Suppose that d > 0 and that the assertion is already known for every Cohen-Macaulay ring R and every monoid as above, of dimension < d. Set L := P^{gp}, and let σ ⊂ L_{ℝ} be the unique convex polyhedral cone such that P = L ∩ σ. The ideal 𝔫 := R[𝔪_{P}] is maximal in R[P], and proposition 11.6.34 and theorem 11.6.24show that

(11.6.36)  depth R[P]_{𝔫} = dim P.

On the other hand, we have the more general :

Claim 11.6.37. Let F be a field, P a fine monoid, A an integral domain which is an F -algebra of finite type, and F′ the field of fractions of A. Then we have :

(i) For every maximal ideal 𝔪 ⊂ A, the Krull dimension of A_{𝔪} equals the transcendence degree of F′ over F.

(ii) For every maximal ideal 𝔪 ⊂ F [P], the Krull dimension of F [P]_{𝔪} equals rk_{ℤ} P^{gp}.

Proof of the claim. (i): This is a straightforward consequence of [126, Th.5.6].

(ii): Choose an isomorphism : P^{gp} →^{∼} L ⊕ T, where T is the torsion subgroup of P^{gp}, and L is a free abelian group of finite rank; there follows an induced isomorphism (4.8.52) :

F [P^{gp}] →^{∼} F [L] ⊗_{F} F [T].

Let B be the maximal reduced quotient of F [P^{gp}] (so the kernel of the projection F [P^{gp}] → B is the nilradical); we deduce that B is a direct product of the type ∏_{i=1}^{n} F′_{i}[L], where each F′_{i} is a finite field extension of F. By (i), the Krull dimension of F′_{i}[L]_{𝔪} equals r := rk_{ℤ}P^{gp} for every maximal ideal 𝔪 ⊂ F′_{i}[L], hence every irreducible component of Spec B has dimension r. Let also C be the maximal reduced quotient of F [P]; the natural map C → B is an injective localization, obtained by inverting a finite system of generators of P, hence the induced morphism Spec B → Spec C is an open immersion with dense image. Let Z be any (reduced) irreducible component of Spec C; again by (i) it follows that every non-empty open subset of Z has dimension equal to dim Z, so necessarily the latter equals r. ♢

From (11.6.36), corollary 6.4.12(i) and claim 11.6.37(ii) we see already that R[P]_{𝔫} is a Cohen-Macaulay ring. Next, let λ_{1}, …, λ_{k} be the one-dimensional faces of σ, and define P_{λ_{i}} as in (11.6.23), for every i = 1, …, k. In light of claim 11.6.26, we have

Spec R[P] \ {𝔫} = ⋃_{i=1}^{k} Spec R[P_{λ_{i}}]

so it remains to check that R[P_{λ_{i}}] is Cohen-Macaulay for every i = 1, …, k. However, we may find a decomposition P_{λ_{i}} →^{∼} Q_{i} × G_{i}, where G_{i} is a free abelian group of finite type, and Q_{i} is a fine, sharp and saturated monoid of dimension d − 1 (lemma 6.2.10). Set S_{i} := R[G_{i}]; then R[P_{λ_{i}}] = S_{i}[Q_{i}], and S_{i} is a Cohen-Macaulay ring ([126, Th.17.7]), so the claim follows by inductive assumption. □

11.6.38. In the situation of (11.6.23), we endow R[P] with its natural L-grading, and denote by \underline{A} := (R[P], gr_{•}R[P]) the resulting L-graded R-algebra. Let (M, gr_{•}M) be an L-graded \underline{A}-module (see definition 7.6.1(ii)); we set

M^{†} := ⨁_{l∈L} Hom_{R}(gr_{l}M, R)

which is naturally an R-submodule of M^{∗} := Hom_{R}(M, R) : indeed, any R-linear map gr_{l}M → R yields a linear form M → R, after composition with the projection M → gr_{l}M. Notice that M^{∗} is naturally an R[P]-module; namely for any linear form f : M → R and any x ∈ P, one defines x · f : M → R by the rule : x · f(m) := f(xm) for every m ∈ M. Then, it is easily seen that M^{†} is an R[P]-submodule of M^{∗}; more precisely, we have

(11.6.39)  x · Hom_{R}(gr_{l}M, R) ⊂ Hom_{R}(gr_{l−x}M, R)  for every x ∈ P.

In light of (11.6.39), it is convenient to define an L-grading on M^{†} by the rule :

gr_{l}M^{†} := Hom_{R}(gr_{−l}M, R) for every l ∈ L

and then (M^{†}, gr_{•}M^{†}) is naturally an L-graded \underline{A}-module.

Remark 11.6.40. (i) Clearly, the rule M ↦ M^{†} yields a functor from the category \underline{A}-Mod of L-graded \underline{A}-modules to \underline{A}-Mod^{o}.

(ii) Let \underline{A}-Mod_{rflx} denote the full subcategory of \underline{A}-Mod whose objects are all the L-graded \underline{A}-modules (M, gr_{•}M) such that gr_{l}M is a projective R-module of finite type, for every l ∈ L. A simple inspection of the definition shows that the functor (−)^{†} of (i) restricts to an equivalence of categories :

\underline{A}-Mod_{rflx} → (\underline{A}-Mod_{rflx})^{o}.

Example 11.6.41. If S ⊂ L is any P -submodule, notice that L \ (−S) is also a P -submodule of L, and set

S^{†} := L/(L \ (−S))

where the quotient is a pointed P -module, as in remark 4.8.17(iii). Explicitly, S^{†} is the set (−S) ∪ {0_{S^{†}}} (where the zero element 0_{S^{†}} should not be confused with the neutral element 0 of the abelian group L); the P -module structure on S^{†} is determined by the rule :

x · s := \{ \begin{matrix}x + s & if x + s ∈ −S \\ 0_{S^{†}} & otherwise \end{matrix} for every s ∈ −S.

Then it is easily seen that there exists a natural isomorphism of L-graded \underline{A}-modules :

R[S]^{†} →^{∼} R⟨S^{†}⟩

(notation of (6.1.31)). On the other hand, the natural map R[S] → (R[S]^{∗})^{∗} induces an isomorphism R[S] →^{∼} (R[S]^{†})^{†} whence an isomorphism of L-graded \underline{A}-modules

(11.6.42)  R[S] →^{∼} R⟨S^{†}⟩^{†}.

Theorem 11.6.43 (Stanley, Danilov). In the situation of (11.6.23), set P^{∘} := L ∩ σ^{∘}. We have : (i) There exists a natural isomorphism :

RHom_{R[P]}^{•}(R, R[P^{∘}]) →^{∼} R[−d] in D(R[P]-Mod)

(here R is regarded as an R[P]-module, via the augmentation map R[P] → R). (ii) If R is a Gorenstein noetherian ring, the complex of coherent 𝒪_{X}-modules R[P^{∘}]^{∼}[0]

is dualizing on X.

Proof. (i): From proposition 11.6.34we deduce a map of complexes of L-graded \underline{A}-modules

φ^{•} : \overline{𝒞} _{P}^{•} → R⟨(P^{∘})^{†}⟩[−d]

(notation of example 11.6.41) with gr_{l}φ^{•} a quasi-isomorphism of complexes of R-modules, for every l ∈ L. Since gr_{l}\overline{𝒞} _{P}^{•} is a finite dimensional R-vector space for every l ∈ L, there follows – in view of (11.6.42) – a quasi-isomorphism of complexes of L-graded \underline{A}-modules

(11.6.44)  R[P^{∘}][0] →^{∼} (\overline{𝒞} _{P}^{•})^{†}[−d].

whence a convergent spectral sequence

E_{1}^{pq} := Ext_{R[P]}^{p}(R, (\overline{𝒞} _{P}^{d−q})^{†}) ⇒ Ext_{R[P]}^{p+q}(R, R[P^{∘}]).

Hence, the assertion follows immediately from the :

Claim 11.6.45. With the foregoing notation, we have :

(i) E_{1}^{pq} = 0 whenever q < d.

(ii) E_{1}^{0d} ≃ R, and E_{1}^{pd} = 0 for every p > 0.

Proof of the claim. (i): Let λ ⊂ σ be any face with dim λ > 0; it suffices to check that E^{i} := Ext_{R(P]}^{i}(R, R[P_{λ}]^{†}) = 0 for every i ∈ ℕ. To this aim, pick any x ∈ F_{λ} \ {0}; since R[P_{λ}]^{†} = R⟨P_{λ}^{†}⟩ (example 11.6.41), we see that scalar multiplication by x is an automorphism on R[P_{λ}]^{†}, hence also on E^{i}. On the other hand, scalar multiplication by x is the zero endomorphism on R, hence also on E^{i}, and the claim follows.

(ii): Pick a resolution F_{•} of the P -graded ℤ[P]-module ℤ := ℤ[P]/ℤ[𝔪_{P}] as in remark 7.6.18(iii), so that F_{n} is a free ℤ[P]-module of finite type for every n ∈ ℕ, and the differential d_{n} : F_{n} → F_{n−1} is a morphism of P -graded ℤ[P]-modules, for every n > 0. Since the augmented complex F_{•} → ℤ is flat as a complex of ℤ-modules, the complex of R[P]-modules R ⊗_{ℤ} F_{•} is still a free resolution of the R[P]-module R, and we get natural isomorphisms

E_{1}^{pd} →^{∼} H^{p}Hom_{R[P]}^{•}(R ⊗_{ℤ} F_{•}, R[P]^{†}[0]) →^{∼} H^{p}((R ⊗_{ℤ} F_{•})^{†}) for every p ∈ ℕ.

(where the differentials of the complex (R⊗_{ℤ}F_{•})^{†} are the maps d_{•}^{†}). However, remark 11.6.40(ii) implies that the complex (R ⊗_{ℤ} F_{•} → R)^{†} is still acyclic, whence the contention. ♢

(ii): To ease notation, set S := Spec R and ω_{P} := R[P^{∘}]^{∼}; in view of proposition 11.3.37, it suffices to check that the complex of coherent 𝒪_{X(x)}-modules ω_{P} (x)[0] is dualizing on X(x), for every x ∈ X (notation of definition 4.9.17(iii)). Hence, fix any such point x, and let 𝔭 ⊂ R[P] be the prime ideal corresponding to x; after replacing R by its localization at the prime ideal 𝔭 ∩ R, we may assume that R is a local ring, and the structure morphism X → S maps x to the closed point of S, corresponding to the maximal ideal 𝔪_{R} of R. Denote by λ ⊂ σ the unique face such that P \ F_{λ} = 𝔭 ∩ P, so that x ∈ U_{λ} := Spec R[P_{λ}]. We may find a decomposition P →^{∼} F_{λ}^{gp} × Q, where Q is also a fine, sharp and saturated monoid (lemma 6.2.10), whence an isomorphism of S-schemes

U_{λ} →^{∼} Spec R[F_{λ}^{gp}] ×_{S} Y where Y := Spec R[Q]

and by construction, F^{−1}(𝔭 ∩ P) is the maximal ideal of P_{λ}, so the induced projection p : U_{λ} → Y maps x to the maximal ideal R[𝔪_{Q}] + 𝔪_{R}[Q] ⊂ R[Q]. Let τ ⊂ Q_{ℝ}^{gp} be the unique polyhedral cone such that Q = τ ∩ Q^{gp}, set Q^{∘} := Q ∩ τ^{∘}, and define the coherent 𝒪_{Y} -module ω_{Q} := R[Q^{∘}]^{∼}. It is easily seen that there is a natural identification

ω_{P |U_{λ}} →^{∼} p^{∗}ω_{Q}.

In view of proposition 11.3.34 and remark 11.3.32(i), it then suffices to check that ω_{Q}(p(x)) is dualizing on Y (p(x)). Thus, we may replace P by Q, and assume from start that 𝔭 = R[𝔪_{P}] + 𝔪_{R}[P] ⊂ R[P]. In this case, set S_{0} := Spec R/𝔪_{R}, and denote by i_{1} : S_{0} → S and i_{2} : S → X(S) the closed immersions (where i_{2} corresponds to the ring homomorphism R(P]_{𝔭} → R deduced from augmentation map); according to (i) and corollary 10.3.2(ii), we have a natural identification

i_{2}^{!}ω_{P} (x)[0] ≃ 𝒪_{S}[−d].

On the other hand, since R is Gorenstein, 𝒪_{S}[−d] is a dualizing complex on S; from lemma 11.3.27(i) and proposition 11.1.7(i) it follows that i_{1}^{!}𝒪_{S}[−d] →^{∼} (i_{2} ∘ i_{1})^{!}ω_{P} (x)[0] is dualizing on S_{0}, and therefore it is isomorphic to 𝒪_{S_{0}}[c] for some c ∈ ℤ. Lastly, proposition 11.3.28(i) and corollary 10.3.2(ii) now say that ω_{P} (x)[0] is dualizing on X(x). □

Remark 11.6.46. Suppose that R is a field; then example 7.11.33 shows that (\overline{𝒞} _{P}^{0} )^{†} is the injective hull of the R[P]-module R. In this case, claim 11.6.45(ii) is therefore a special case of claim 7.11.24. For a general R, the R[P]-module (\overline{𝒞} _{P}^{0} )^{†} shall not be injective, but it can be viewed as a graded variant of the injective hull construction.

12. Logarithmic geometry

12.1. Log topoi. Henceforth, all topoi under consideration will be locally ringed and with enough points, and all morphisms of topoi will be morphisms of locally ringed topoi (see (4.9.13)). The purpose of this restriction is to insure that we obtain the right notions, when we specialize to the case of schemes.

Definition 12.1.1. Let T := (T, 𝒪_{T}) be a locally ringed topos.

(i) A pre-log structure on T is the datum of a pair

(\underline{M}, α)

where \underline{M} is a T -monoid, and α : \underline{M} → 𝒪_{T} is a morphism of T -monoids, called the structure map of \underline{M}, and where the monoid structure on 𝒪_{T} is induced by the multiplication law (hence, by the multiplication in the ring 𝒪_{T} (U), for every object U of T).

(ii) A morphism (\underline{M}, α) → (\underline{N}, β) of pre-log structures on T, is a map γ : \underline{M} → \underline{N} of T -monoids, such that β ∘ γ = α. We denote by

pre-log_{T}

the category of pre-log structures on T. (iii) A pre-log structure (\underline{M}, α) on T is a log structure if α restricts to an isomorphism:

α^{−1}𝒪_{T}^{×} →^{∼} \underline{M}^{×} →^{∼} 𝒪_{T}^{×}.

The datum of a locally ringed topos (T, 𝒪_{T}) and a log structure on T is also called, for short, a log topos. We denote by

log_{T}

the full subcategory of pre-log_{T} consisting of all log structures on T. When there is no danger of ambiguity, we shall often omit mentioning explicitly the map α, and therefore only write \underline{M} to denote a pre-log or a log structure.

12.1.2. The category log_{T} admits an initial object, namely the log structure (𝒪_{T}^{×}, j), where j is the natural inclusion; this is called the trivial log structure. log_{T} admits a final object as well: this is (𝒪_{T} , 1_{𝒪_{T}} ). A morphism of locally ringed topoi f : T → S induces a pair of adjoint functors :

(12.1.3)  f^{∗} : pre-log_{S} → pre-log_{T}  f_{∗} : pre-log_{T} → pre-log_{S}.

Namely, let (\underline{M}, α) (resp. (\underline{N}, β)) be a pre-log structure on S (resp. on T) and

f^{♭} : 𝒪_{S} → f_{∗}𝒪_{T} f^{♯} : f^{−1}𝒪_{S} → 𝒪_{T}

the natural morphisms (corresponding to each other under the adjunction (f^{−1}, f_{∗}) that defines the morphism f); then

f^{−1}\underline{M} ⟶^{f^{−1}α} f^{−1}𝒪_{S} ⟶^{f^{♯}} 𝒪_{T}

defines f^{∗}(\underline{M}, α), and f_{∗}(\underline{N}, β) is the pair (\underline{P} , γ), where \underline{P} is the fibre product in the cartesian diagram

\xymatrix{\underline{P} \ar[r]^{γ} \ar[d] & 𝒪_{S} \ar[d]^{f^{♭}} \\ f_{∗}\underline{N} \ar[r]^{f_{∗}β} & f_{∗}𝒪_{T} .}

Lemma 12.1.4. Let T be a topos with enough points, φ : ℳ → 𝒩 a morphism of integral T -monoids inducing isomorphisms ℳ^{×} →^{∼} 𝒩 ^{×} and ℳ^{♯} →^{∼} 𝒩 ^{♯}. Then φ is an isomorphism.

Proof. This can be checked on the stalks, hence we are reduced to the corresponding assertions for a morphism M → N of monoids. Moreover, M^{♯} is just the set-theoretic quotient of M by the translation action of M^{×} (lemma 4.8.31(iii)), so the assertion is straightforward, and shall be left as an exercise for the reader. □

Definition 12.1.5. Let γ : (\underline{M}, α) → (\underline{N}, β) be a morphism of pre-log structures on the locally ringed topos T, and ξ a T -point.

(i) We say that (\underline{M}, α) is integral (resp. saturated) if \underline{M} is an integral (resp. integral and saturated) T -monoid.

(ii) We say that γ is flat (resp. saturated) at the point ξ, if γ_{ξ} : \underline{M}_{ξ} → \underline{N}_{ξ} is a flat morphism of monoids (resp. a saturated morphism of integral monoids) (see remark 4.8.23(vi)).

(iii) We say that γ is flat (resp. saturated), if γ is a flat morphism of T -monoids (resp. a saturated morphism of integral T -monoids) (see definition 6.2.28). In view of proposition 4.8.26 (resp. corollary 6.2.29), this is the same as saying that γ is flat (resp. saturated) at every T -point.

12.1.6. The forgetful functor :

log_{T} → pre-log_{T} : \underline{M} ↦ \underline{M}^{pre-log}

admits a left adjoint :

pre-log_{T} → log_{T} : (\underline{M}, α) ↦ (\underline{M}, α)^{log}

such that the resulting diagram :

(12.1.7)  \xymatrix{α^{−1}(𝒪_{T}^{×}) \ar[r] \ar[d] & \underline{M} \ar[d] \\ 𝒪_{T}^{×} \ar[r] & _{\underline{M}_{log}}}

is cocartesian in the category of pre-log structures. One calls \underline{M}^{log} the log structure associated to \underline{M}. From this description, it is easily seen that the counit of adjunction

(12.1.8)  (\underline{M}^{pre-log})^{log} → \underline{M}

is an isomorphism for every log structure \underline{M} on T.

Composing with the adjunction (12.1.3), we deduce a pair of adjoint functors :

f^{∗} : log_{S} → log_{T} f_{∗} : log_{T} → log_{S}

for any map of locally ringed topoi f : T → S. Explicitly, if \underline{N} is any log structure on S, then f^{∗}\underline{N} is the push-out in the cocartesian diagram of T -monoids :

\xymatrix{f^{∗}𝒪_{S}^{×} \ar[r] \ar[d] & f^{∗}(\underline{N}^{pre-log}) \ar[d] \\ 𝒪_{T}^{×} \ar[r] & f^{∗}\underline{N}}

and on the other hand, if \underline{M} is any log structure on T, then it is easily seen that the pre-log structure f_{∗}(\underline{M}^{pre-log}) is actually a log structure (this can be checked on the stalks over the points of T). It follows easily that the induced map of T -monoids :

(12.1.9)  f^{∗}(\underline{N}^{♯}) → (f^{∗}\underline{N})^{♯}

is an isomorphism. As a consequence, for every point ξ of T, the natural map \underline{N}_{f(ξ)} → (f^{∗}\underline{N})_{ξ} is a local morphism of monoids.

Remark 12.1.10. The category log_{T} admits arbitrary colimits : indeed, since the counit (12.1.8) is an isomorphism, it suffices to construct such colimits in the category of pre-log structures, and then apply the functor (−) ↦ (−)^{log} which preserves colimits, since it is a left adjoint.

12.1.11. Let (\underline{M}, α) be a pre-log structure on T. The morphism α extends to a unique morphism of pointed T -monoids α_{∘} : \underline{M}_{∘} → 𝒪_{T}, whence a new pre-log structure

(\underline{M}, α)_{∘} := (\underline{M}_{∘}, α_{∘}).

Clearly, (\underline{M}, α) is a log structure if and only if the same holds for (\underline{M}, α)_{∘}. More precisely, for any pre-log structure \underline{M} there is a natural isomorphism of log structures :

(\underline{M}_{∘})^{log} →^{∼} (\underline{M}^{log})_{∘}.

Furthermore, for any morphism f : T → S of topoi, we have natural isomorphisms of pre-log (resp. log) structures

f^{∗}(\underline{N}_{∘}) →^{∼} (f^{∗}\underline{N})_{∘} f_{∗}(\underline{M}_{∘}) →^{∼} (f_{∗}\underline{M})_{∘}

for every pre-log (resp. log) structure \underline{N} on S and \underline{M} on T (details left to the reader).

Example 12.1.12. (i) Let T → S be a morphism of topoi. Since the initial object of a category is the empty coproduct, it follows formally that the inverse image f^{∗}(𝒪_{S}^{×}, j) of the trivial log structure on S, is the trivial log structure on T.

(ii) Let T be a topos, and j_{U} : T/U → T an open subtopos (see example 4.7.8(i)). Consider the subsheaf of monoids \underline{M} ⊂ 𝒪_{T} such that :

\underline{M}(V) := \{s ∈ 𝒪_{T} (V) | s_{|U×V} ∈ 𝒪_{T}^{×}(U × V)\} for every object V of T.

Then it is easily seen that the natural map \underline{M} → 𝒪_{T} is a log structure on T. This log structure is (naturally isomorphic to) the extension j_{U∗}𝒪_{U}^{×} of the trivial log structure on T/U.

(iii) Let U be any object of the topos T, and \underline{M} a log structure on T. Since 𝒪_{T/U} = (𝒪_{T})_{|U}, it is easily seen that the natural morphism of pre-log structures

(\underline{M}^{pre-log})_{|U} → (\underline{M}_{|U})^{pre-log}

is an isomorphism.

(iv) Let β : \underline{M} → 𝒪_{T} be a pre-log structure on a topos T. Then β^{−1}(0) ⊂ \underline{M} is an ideal, and β factors uniquely through the natural map \underline{M} → \underline{M}/β^{−1}(0), and a pre-log structure

(\underline{M}, β)_{red} := (\underline{M}/β^{−1}(0), \overline{β})

called the reduced pre-log structure associated to \underline{M}. As usual, we shall often write just \underline{M}_{red} instead of (\underline{M}, β)_{red}. We say that β is reduced if the induced morphism of pre-log structures \underline{M} → \underline{M}_{red} is an isomorphism.

Suppose now that \underline{M} is a log structure; then it is easily seen that the same holds for \underline{M}_{red}. More precisely, since the tensor product is right exact (see (4.8.19)), for any pre-log structure \underline{M} the natural morphism of log structures

(\underline{M}_{red})^{log} → (\underline{M}^{log})_{red}

is an isomorphism.

Lemma 12.1.13. Let γ : (\underline{M}, α) → (\underline{N}, β) be a morphism of pre-log structures on T. We have: (i) If \underline{M} is integral (resp. saturated), then the same holds for \underline{M}^{log}. (ii) The unit of adjunction \underline{M} → \underline{M}^{log} is a flat morphism. (iii) If γ is flat (resp. saturated) at a T -point ξ, the same holds for the induced morphism

γ^{log} : \underline{M}^{log} → \underline{N}^{log} of log structures.

(iv) Especially, if γ is flat (resp. saturated), the same holds for γ^{log}.

Proof. In view of lemma 4.8.46(ii) and proposition 4.8.26, both (i) and (ii) can be checked on stalks. Taking into account lemma 4.8.46(i), we are reduced to showing the following. Let P be a monoid, A a ring, β : P → (A, ·) a morphism of monoids; then the natural map P → P′ := P ⊗_{β^{−1}A^{×}} A^{×} is flat, and if P is integral (resp. saturated), the same holds for P′. The first assertion follows easily from example 6.1.23(vi), and the second follows from remark 6.2.5(i) (resp. from corollary 6.2.25(ix) and proposition 6.2.26).

(iii): The map γ^{log} can be factored as the composition of

γ′ := γ ⊗_{β^{−1}𝒪_{T}^{×}} 𝒪_{T}^{×} : \underline{N}^{log} → \underline{P} := \underline{M} ⊗_{β^{−1}𝒪_{T}^{×}} 𝒪_{T}^{×}

and the natural unit of adjunction γ′′ : \underline{P} → \underline{P}^{log} = \underline{M}^{log}. If γ_{ξ} is flat, the same clearly holds for γ′_{ξ}, and (ii) says that γ′′ is flat, hence γ_{ξ}^{log} is flat in this case. Lastly, suppose that γ_{ξ} is saturated, and we wish to show that γ_{ξ}^{log} is saturated. Set P := α_{ξ}^{−1}𝒪_{T,ξ}^{×} and Q := β_{ξ}^{−1}𝒪_{T,ξ}^{×}. Then the induced map (P^{−1}\underline{M}_{ξ})^{♯} → (Q^{−1}\underline{N}_{ξ})^{♯} is saturated (lemma 6.2.12(ii,iii)). But the latter is the same as the morphism (γ_{ξ}^{log})^{♯}, and then also γ_{ξ} is saturated, again by lemma 6.2.12(iii). □

Lemma 12.1.14. Let f : T′ → T be a morphism of topoi, ξ a T′-point, and γ : (\underline{M}, α) → (\underline{N}, β) a morphism of integral log structures on T. The following conditions are equivalent :

(a) γ is flat (resp. saturated) at the T -point f(ξ).

(b) f^{∗}γ is flat (resp. saturated) at the T′-point ξ.

(c) γ_{ξ}^{♯} is a flat (resp. saturated) morphism of monoids.

Proof. The equivalence of (a) and (c) follows from corollary 6.1.49 (resp. lemma 6.2.12(ii)). By the same token, (b) holds if and only if (f^{∗}γ)_{ξ}^{♯} is flat (resp. saturated); in light of the isomorphism (12.1.9), the latter condition is equivalent to (c). □

12.1.15. For any locally ringed topos T, let us write the objects of Mnd/Γ(T, 𝒪_{T}) in the form (M, φ), where M is any monoid, and φ : M → Γ(T, 𝒪_{T}) a morphism of monoids. There is an obvious global sections functor :

Γ(T, −) : pre-log_{T} → Mnd/Γ(T, 𝒪_{T}) : (\underline{N}, α) ↦ (Γ(T, \underline{N}), Γ(T, α))

which admits a left adjoint :

Mnd/Γ(T, 𝒪_{T}) → pre-log_{T} : (M, φ) ↦ (M, φ)_{T} := (M_{T} , φ_{T}).

Indeed, M_{T} is the constant sheaf on (T, C_{T}) with value M, and φ_{T} is the composition of the map of constant sheaves M_{T} → Γ(T, 𝒪_{T})_{T} induced by φ, with the natural map Γ(T, 𝒪_{T})_{T} → 𝒪_{T}. Again, we shall often just write M_{T} to denote this pre-log structure.

After taking associated log structures, we deduce a left adjoint :

(12.1.16)  Mnd/Γ(T, 𝒪_{T}) → log_{T} : (M, φ) ↦ M_{T}^{log} := (M, φ)_{T}^{log}

to the global sections functor. M_{T}^{log} is called the constant log structure associated to (M, φ).

Definition 12.1.17. Let T be a locally ringed topos, (\underline{M}, α) a log structure on T.

(i) A chart for \underline{M} is an object (P, β) of Mnd/Γ(T, 𝒪_{T}), together with a map of prelog structures ω_{P} : (P, β)_{T} → \underline{M}, inducing an isomorphism on the associated log structures. (Notation of (12.1.15).)

(ii) We say that a chart (P, β) is finite (resp. integral, resp. fine, resp. saturated) if P is a finitely generated (resp. integral, resp. fine, resp. integral and saturated) monoid.

(iii) Let φ : \underline{M} → \underline{N} be a morphism of log structures on T. A chart for φ is the datum of

charts :

ω_{P} : (P, β)_{T} → \underline{M} and ω_{Q} : (Q, γ)_{T} → \underline{N}

for \underline{M}, respectively \underline{N}, and a morphism of monoids ϑ : Q → P, fitting into a commutative diagram :

\xymatrix{Q_{T} \ar[r]^{ϑ_{T}^{log}} \ar[d]_{ω_{Q}} & P_{T}^{log} \ar[d]^{ω_{P}} \\ \underline{N} \ar[r]^{φ} & \underline{M}.}

We say that such a chart is finite (resp. integral, resp. fine, resp. saturated) if the monoids P and Q are finitely generated (resp. integral, resp. fine, resp. integral and saturated). We say that the chart is flat (resp. saturated), if ϑ is a flat morphism of monoids (resp. a saturated morphism of integral monoids).

(iv) We say that \underline{M} is quasi-coherent (resp. coherent) if there exist a covering family (U_{λ} → 1_{T} | λ ∈ Λ) of the final object 1_{T} in (T, C_{T}), and for every λ ∈ Λ, a chart (resp. a finite chart) (P_{λ}, β_{λ}) for \underline{M}_{|U_{λ}}.

(v) We say that (\underline{M}, α) is quasi-fine (resp. fine) if it is integral and quasi-coherent (resp. and coherent).

(vi) Let ξ be any T -point. We say that a chart (P, β) is local (resp. sharp) at the point ξ, if the morphism β_{ξ} : P → 𝒪_{T,ξ} is local (resp. if P is sharp and β_{ξ} is local).

Lemma 12.1.18. Let f : T → S be a morphism of locally ringed topoi, \underline{Q} a log structure on S, and ξ any point of S. The following holds :

(i) If \underline{Q} is quasi-coherent (resp. coherent, resp. integral, resp. saturated, resp. quasi-fine, resp. fine), then the same holds for f^{∗}\underline{Q}.

(ii) Suppose that \underline{Q} is an integral log structure. Then \underline{Q}^{♯} is an integral S-monoid, and \underline{Q} is

saturated if and only if \underline{Q}^{♯} is a saturated S-monoid. (iii) Suppose that \underline{Q} is quasi-coherent. Then \underline{Q} is integral (resp. integral and saturated,

resp. fine, resp. fine and saturated) if and only if there exist a covering family (U_{λ} → 1_{S} | λ ∈ Λ) of the final object of S, and for every λ ∈ Λ, an integral (resp. integral and saturated, resp. fine, resp. fine and saturated) chart (P_{λ})_{U_{λ}} → \underline{Q}_{|U_{λ}}.

(iv) If \underline{P} is any coherent log structure on S, and ω : \underline{P}_{ξ} → \underline{Q}_{ξ} is a map of monoids, then :

(a) There exist a neighborhood U of ξ and a morphism ϑ : \underline{P}_{|U} →^{∼} \underline{Q}_{|U} of log structures, such that ϑ_{ξ} = ω.

(b) Moreover, for any two morphisms ϑ, ϑ′ : \underline{P}_{|U} →^{∼} \underline{Q}_{|U} with the property of (a), we may find a smaller neighborhood V → U of ξ such that ϑ_{|V} = ϑ′_{|V}.

(c) Especially, if \underline{Q} is also coherent, and ω is an isomorphism, we may find ϑ and a

small enough U as in (a), such that ϑ is an isomorphism.

(v) If M is a finitely generated monoid, and ω : M → 𝒪_{S,ξ} a morphism of monoids, then we may find a neighborhood U of ξ and a morphism ϑ : M_{U} → 𝒪_{U} of S/U-monoids, such that ϑ_{ξ} = ω.

Proof. (i): If \underline{Q} is the constant log structure associated to a map of monoids α : Q → Γ(S, 𝒪_{S}), then f^{∗}\underline{Q} is the constant log structure associated to Γ(S, f^{♮}) ∘ α (where f^{♮} : 𝒪_{S} → f_{∗}𝒪_{T} is the natural map). The assertions concerning quasi-coherent or coherent log structures are a straightforward consequence. Next, suppose that \underline{Q} is integral (resp. saturated); we wish to show that f^{∗}\underline{Q} is integral (resp. saturated). To this aim, let \underline{M} := f^{∗}(\underline{Q}^{pre-log}); by lemma 4.8.45(i), \underline{M} is an integral (resp. saturated) T -monoid; then the assertion follows from lemma 12.1.13(i).

(ii): By lemma 4.8.46(ii) the assertion can be checked on stalks. Hence, suppose that \underline{Q} is integral; then \underline{Q}_{ξ} is integral by loc.cit., consequently, the same holds for \underline{Q}_{ξ}/\underline{Q}_{ξ}^{×} (lemma 4.8.38). The second assertion follows from lemma 6.2.9(ii).

(iii): Suppose first that \underline{Q} is quasi-coherent and integral. Hence, there is a covering family (U_{λ} → 1_{S} | λ ∈ Λ), and for every λ ∈ Λ a monoid M_{λ}, a pre-log structure α_{λ} : (M_{λ})_{U_{λ}} → 𝒪_{U_{λ}}, and an isomorphism ((M_{λ})_{U_{λ}}, α_{λ})^{log} →^{∼} \underline{Q}_{|U_{λ}}; whence a cocartesian diagram of S-monoids, as in (12.1.7) :

(12.1.19)  \xymatrix{\underline{N} := α_{λ}^{−1}(𝒪_{U_{λ}}^{×}) \ar[r] \ar[d] & (M_{λ})_{U_{λ}} \ar[d] \\ 𝒪_{U_{λ}}^{×} \ar[r] & \underline{Q}_{|U_{λ}}.}

The induced diagram (12.1.19)^{int} of integral S-monoids is still cocartesian. Moreover, since \underline{Q}_{|U_{λ}} is integral, α_{λ} factors through a unique map β_{λ} : ((M_{λ})_{U_{λ}})^{int} → \underline{Q}_{|U_{λ}} → 𝒪_{U_{λ}}, and the morphism in S underlying the induced morphism of S-monoids \underline{N}^{int} → \underline{N}′ := β_{λ}^{−1}(𝒪_{U_{λ}}^{×}) is an epimorphism (this can be checked easily on the stalks). Furthermore, ((M_{λ})_{U_{λ}})^{int} ≃ (M_{λ}^{int})_{U_{λ}} (see (4.8.49)). It follows that the natural map

\underline{Q}_{|U_{λ}} → 𝒪_{U_{λ}}^{×} ⨿\underline{N}′ (M_{λ}^{int})_{U_{λ}} ≃ (M_{λ}^{int})_{U_{λ}}^{log}

is an isomorphism, so the claim holds with P_{λ} := M_{λ}^{int}. If Q is fine, we can find M_{λ} as above which is also finitely generated, in which case the resulting P_{λ} shall be fine.

Suppose additionally, that \underline{Q} is saturated. By the previous case, we may then find a covering family (U_{λ} → e_{S} | λ ∈ Λ), and for every λ ∈ Λ an integral monoid M_{λ}, a pre-log structure α_{λ} : (M_{λ})_{U_{λ}} → 𝒪_{U_{λ}}, and an isomorphism ((M_{λ})_{U_{λ}}, α_{λ})^{log} →^{∼} \underline{Q}_{|U_{λ}}; whence a cocartesian diagram (12.1.19) consisting of integral S-monoids. The induced diagram (12.1.19)^{sat} is still cocartesian; one may then argue as in the foregoing, to obtain a natural isomorphism : \underline{Q}_{|U_{λ}} →^{∼} (M_{λ}^{sat})_{U_{λ}}^{log}. Furthermore, if M_{λ} is finitely generated, the same holds for M_{λ}^{sat} (corollary 6.4.1(ii)), hence the chart thus obtained shall be fine and saturated, in this case.

Conversely, if a family (P_{λ} | λ ∈ Λ) of integral (resp. saturated) monoids can be found fulfilling the condition of (iii), then (P_{λ})_{U_{λ}} is an integral (resp. saturated) pre-log structure on T/U_{λ} (example 4.8.47(ii)), hence the same holds for its associated log structure \underline{Q}_{|U_{λ}} (lemma 12.1.13(i)), and thus also for \underline{Q} (lemma 4.8.46(ii), example 4.7.12, and example 12.1.12(iii)). Moreover, if each P_{λ} is fine, then \underline{Q} is fine as well.

(iv.a): We may assume that \underline{P} admits a finite chart α : M_{S} → \underline{P}, for some finitely generated monoid M, denote by β : \underline{Q} → 𝒪_{S} the structure map of \underline{Q}, and set ω′ := ω ∘ α_{ξ} : M → \underline{Q}_{ξ}. According to lemma 6.1.7(ii), the morphism ω′ factors through a map ω′′ : M → Γ(U′, \underline{Q}), for some neighborhood U′ of ξ. By adjunction, ω′′ determines a morphism of S/U′-monoids ψ : M_{U′} → \underline{Q}_{|U′} whence a morphism of pre-log structures

(12.1.20)  (M_{U′}, β_{|U′} ∘ ψ) → (\underline{Q}_{|U′}, β_{|U′}).

Let us make the following general observation :

Claim 12.1.21. Let N be a finitely generated monoid, F any S-monoid, f, g : N_{S} → F two morphisms of S-monoids, such that f_{ξ} = g_{ξ}. Then there exists a neighborhood U of ξ in S such that f_{|U} = g_{|U}.

Proof of the claim. By adjunction, the morphisms f and g correspond to unique maps of monoids Γ(f), Γ(g) : N → Γ(S, F); since N is finitely generated and f_{ξ} = g_{ξ}, we may find a neighborhood U of ξ such that the maps N → F (U) induced by Γ(f) and Γ(g) coincide. Again by adjunction, we deduce a unique morphism of S/U-monoids N_{U} → F_{|U}, which by construction is just the restriction of both f and g. ♢

Let γ : \underline{P} → 𝒪_{S} be the structure map of \underline{P}; we apply claim 12.1.21with S replaced by S/U′, to deduce that there exists a small enough neighborhood U → U′ of ξ such that the restriction (γ ∘α)_{|U} : M_{|U} → 𝒪_{S|U} agrees with β_{|U} ∘ψ_{|U}. Then it is clear that the morphism of log structure associated to (12.1.20)_{|U} yields the sought extension ϑ of ω.

(iv.b): By assumption we have the identity : ϑ_{ξ} = ϑ′_{ξ}; however, any morphism of log structures \underline{P}_{|U} → \underline{Q}_{|U} is already determined by its restriction to the image of any finite local chart M_{U} → \underline{P}_{|U}. Hence the assertion follows from claim 12.1.21.

(iv.c): We apply (iv.a) to ω^{−1} to deduce the existence of a morphism σ : \underline{Q}_{|U} → \underline{P}_{|U} such that σ_{ξ} = ω^{−1} on some neighborhood U of ξ. Hence, (ϑ ∘ σ)_{ξ} = 1_{\underline{Q}_{ξ}} and (σ ∘ ϑ)_{ξ} = 1_{\underline{P}_{ξ}}. By (iv.b), these identities persist on some smaller neighborhood.

(v): The proof is similar to that of (iv.a), though simpler : we leave it as an exercise for the reader. □

Definition 12.1.22. (i) A morphism (T, \underline{M}) → (S, \underline{N}) of topoi with pre-log (resp. log) structures, is a pair f := (f, log f) consisting of a morphism of locally ringed topoi f : T → S, and a morphism

log f : f^{∗}\underline{N} → \underline{M}

of pre-log structures (resp. log structures) on T. We say that f is log flat (resp. saturated) if log f is a flat (resp. saturated) morphism of pre-log structures.

(ii) Let (f, log f) as in (i) be a morphism of log topoi, ξ a T -point; we say that f is strict at the point ξ, if log f_{ξ} is an isomorphism. We say that f is strict, if it is strict at every T -point.

(iii) A chart for φ is the datum of charts

ω_{P} : (P, β)_{T} → \underline{M} and ω_{Q} : (Q, γ)_{S} → \underline{N}

for \underline{M}, respectively \underline{N}, and a morphism of monoids ϑ : Q → P, such that (f^{∗}ω_{Q}, ω_{P} , ϑ) is a chart for the morphism log f. We say that such a chart (ω_{Q}, ω_{P} , ϑ) is finite (resp. fine, resp. flat, resp. saturated) if the same holds for the corresponding chart (f^{∗}ω_{Q}, ω_{P} , ϑ) of log f.

Remark 12.1.23. (i) Let f : (T, \underline{M}) → (S, \underline{N}) be a morphism of log topoi, g : T′ → T a morphism of topoi, and f′ : (T′, g^{∗}\underline{M}) → (S, \underline{N}) the composition of f and the natural morphism of log topoi (T′, g^{∗}\underline{M}) → (T, \underline{M}); let also ξ be a T′-point. Then f′ is strict at the point ξ if and only if f is strict at the point g(ξ). Indeed, f′ is strict at ξ if and only if (log f′)_{ξ}^{♯} is an isomorphism (lemma 12.1.4), if and only if (log f)_{g(ξ)}^{♯} is an isomorphism (by (12.1.9)), if and only if log f_{g(ξ)} is an isomorphism (again by lemma 12.1.4).

(ii) For any log topos (T, \underline{M}), let us set (T, \underline{M})_{∘} := (T, \underline{M}_{∘}). Then (T, \underline{M})_{∘} is a log topos (see (12.1.11)), and clearly, every morphism f : (T, \underline{M}) → (S, \underline{N}) of log topoi extends naturally to a morphism f_{∘} : (T, \underline{M})_{∘} → (S, \underline{N})_{∘} of log topoi.

12.1.24. The 2-category of log topoi admits arbitrary 2-limits indexed by usual categories. Indeed, if 𝒯 := ((T_{λ}, \underline{M}_{λ}) | λ ∈ Λ) is any pseudo-functor (from a small category Λ, to the category of log topoi), the 2-limit of 𝒯 is the pair (T, \underline{M}), where T is the 2-limit of the system of ringed topoi (T_{λ} | λ ∈ Λ), and \underline{M} is the colimit of the induced system of log structures (p_{λ}^{∗}\underline{M}_{λ} | λ ∈ Λ) on T, where p_{λ} : T → T_{λ} denotes the natural projection, for every λ ∈ Λ (see remark 12.1.10).

12.1.25. Consider a 2-cartesian diagram of log topoi :

\xymatrix{(T′, \underline{M}′) \ar[r]^{g} \ar[d]_{f′} & (T, \underline{M}) \ar[d]^{f} \\ (S′, \underline{N}′) \ar[r]^{h} & (S, \underline{N}).}

The following result yields a relative variant of the isomorphism (12.1.9) :

Lemma 12.1.26. In the situation of (12.1.25), the morphism

g^{∗}Coker(log f) → Coker(log f′)

induced by log g is an isomorphism of T′-monoids.

Proof. Indeed, denote by β : \underline{M} → 𝒪_{T} and γ : \underline{N}′ → 𝒪_{S′} the log structures of T and S′. Fix any T′-point ξ, let ξ := g(ξ′), and set

P := \underline{M}_{ξ} ⊗_{\underline{N}_{f(ξ)}} \underline{N}′_{f′(ξ′)} and ρ := β_{ξ} ⊗ γ_{f′(ξ′)} : P → 𝒪_{T′}.

Then \underline{M}′_{ξ′} = P ⊗_{ρ^{−1}𝒪_{T′,ξ′}} 𝒪_{T′,ξ′}^{×}, and it is easily seen that ρ^{−1}𝒪_{T′,ξ′} = 𝒪_{T,ξ}^{×} ⊗_{𝒪_{S,f(ξ)}^{×}} 𝒪_{S′,f′(ξ′)}^{×}.

Therefore (\underline{M}′_{ξ′})^{♯} = P/ρ^{−1}𝒪_{T′,ξ′} = \underline{M}_{ξ}^{♯} ⊗\underline{N}{♯ \atop f(ξ)} \underline{N}′_{f′(ξ′)}^{♯}, and

Coker(log f′_{ξ′}) = Coker(\underline{N}′_{f′(ξ′)}^{♯} → \underline{M}_{ξ}^{♯} ⊗\underline{N}{♯ \atop f(ξ)} \underline{N}′_{f′(ξ′)}^{♯}) = Coker(\underline{N}_{f(ξ)}^{♯} → \underline{M}_{ξ}^{♯})

whence the contention. □

12.1.27. We consider now a special situation that we will encounter in proposition 12.1.30. Namely, let Q be a monoid, and H ⊂ Q^{×} a subgroup. Let also G be an abelian group, ρ : G → Q^{gp} a group homomorphism, and set :

H_{ρ} := G ×_{Q^{gp}} H Q_{ρ} := G ×_{Q^{gp}} Q.

The natural inclusion H → Q and the projection Q_{ρ} → Q determine a unique morphism :

(12.1.28)  Q_{ρ} ⊗_{H_{ρ}} H → Q.

Lemma 12.1.29. In the situation of (12.1.27), suppose furthermore that the composition :

G ⟶^{ρ} Q^{gp} → (Q/H)^{gp}

is surjective. Then (12.1.28) is an isomorphism.

Proof. Set G′ := G ⊕ H, and let ρ′ : G′ → Q^{gp} be the unique group homomorphism that extends ρ and the natural map H → Q → Q^{gp}. Under the standing assumptions, ρ′ is clearly surjective. Define Q_{ρ′} and H_{ρ′} as in (12.1.27); there is a natural isomorphism of monoids : Q_{ρ′} →^{∼} Q_{ρ} ⊕ H, inducing an isomorphism H_{ρ′} →^{∼} H_{ρ} ⊕ H, and defined as follows. To every g ∈ G, h ∈ H, q ∈ Q such that [(g, h), q] ∈ Q_{ρ′}, we assign the element [(g, h^{−1}q), a] ∈ Q_{ρ} ⊕ H (details left to the reader). Under this isomorphism, the projection H_{ρ′} → H is identified with the map H_{ρ} ⊕ H → H given by the rule : (h_{1}, h_{2}) ↦ π_{H}(h_{1}) · h_{2}, where π_{H} : H_{ρ} → H is the projection. It then follows that the natural map :

Q_{ρ} ⊗_{H_{ρ}} H → Q_{ρ′} ⊗_{H_{ρ′}} H

is an isomorphism. Thus, we may replace G and ρ by G′ and ρ′, which allows to assume from start that ρ is surjective. However, we have natural isomorphisms :

Ker(Q_{ρ} ⟶^{π_{Q}} Q) ≃ Ker ρ ≃ Ker(H_{ρ} ⟶^{π_{H}} H).

Moreover, the set underlying Q (resp. H) is the set-theoretic quotient of the set Q_{ρ} (resp. H_{ρ}) by the translation action of Ker ρ; hence the natural maps Q_{ρ}/Ker π_{Q} → Q and H_{ρ}/Ker π_{H} → H are isomorphisms (lemma 4.8.31(ii)). We can then compute :

Q_{ρ} ⊗_{H_{ρ}} H ≃ (Q_{ρ}/Ker π_{Q}) ⊗_{H_{ρ}/Ker π_{H}} H ≃ Q ⊗_{H} H ≃ Q

as stated. □

Proposition 12.1.30. Let T be a locally ringed topos, ξ any T -point, and \underline{M} a coherent (resp. fine) log structure on T. Suppose that G is a finitely generated abelian group with a group homomorphism G → \underline{M}_{ξ}^{gp} such that the induced map G → (\underline{M}^{♯})_{ξ}^{gp} is surjective. Set

P := G ×\underline{M}_{ξ}^{gp} \underline{M}_{ξ}.

Then the induced morphism P → \underline{M}_{ξ} extends to a finite (resp. fine) chart P_{U} → \underline{M}_{|U} on some neighborhood U of ξ.

Proof. We begin with the following :

Claim 12.1.31. Let Y be any locally ringed topos, ξ a Y -point, and α : Q_{Y} → 𝒪_{Y} the constant pre-log structure associated to a map of monoids ϑ : Q → Γ(Y, 𝒪_{Y}), where Q is finitely generated. Set S := α_{ξ}^{−1}𝒪_{Y,ξ}^{×} ⊂ Q_{Y,ξ} = Q. Then :

(i) S and S^{−1}Q are finitely generated monoids.

(ii) There exists a neighborhood U of ξ such that α_{|U} factors as the composition of the natural map of sheaves of monoids j_{U} : Q_{U} → (S^{−1}Q)_{U}, and a (necessarily unique) pre-log structure α_{S} : (S^{−1}Q)_{U} → 𝒪_{U}.

(iii) The induced map of log structures j_{U}^{log} : Q_{U}^{log} → (S^{−1}Q)_{U}^{log} is an isomorphism.

(iv) α_{S,ξ}^{−1}(𝒪_{U,ξ}^{×}) = (S^{−1}Q)^{×} is a finitely generated group.

Proof of the claim. (i) follows from lemma 6.1.20(iv).

(ii): Since 𝒪_{Y,ξ}^{×} is the filtered colimit of the groups Γ(U, 𝒪_{U}^{×}), where U ranges over the neighborhoods of ξ, lemma 6.1.7(ii) and (i) imply that the induced map S → 𝒪_{Y,ξ}^{×} factors through Γ(U, 𝒪_{Y}^{×}) for some neighborhood U of ξ. Then the composition of ϑ and the natural map Γ(Y, 𝒪_{Y}) → Γ(U, 𝒪_{U}) extends to a unique map S^{−1}Q → Γ(U, 𝒪_{U}), whence the sought pre-log structure α_{S} on U.

(iii): Let \underline{N} be any log structure on U; it is clear that every morphism of pre-log structures Q_{U} → \underline{N} factors uniquely through (S^{−1}Q)_{U}, whence the contention.

(iv): Indeed, by construction we have : α_{S,ξ}^{−1}(𝒪_{U,ξ}^{×}) = S^{gp}. ♢ Let Y be a neighborhood of ξ such that \underline{M}_{|Y} admits a finite local chart α : Q_{Y} → 𝒪_{Y}; we

lift ξ to some Y -point, ξ_{Y}, and choose a neighborhood U ∈ Ob(T/Y) of ξ_{Y}, as provided by claim 12.1.31. We may then replace T by U, ξ by ξ_{Y} and α by the chart α_{S} of claim 12.1.31(ii), which allows to assume from start that S := α_{ξ}^{−1}(𝒪_{T,ξ}^{×}) is a finitely generated group. Moreover, let H := Ker(S → 𝒪_{T,ξ}^{×}); clearly α_{ξ} : Q → 𝒪_{T,ξ} factors through the quotient Q′ := Q/H, hence we may find a neighborhood U of ξ such that α_{|U} factors through a (necessarily unique) map of pre-log structures α_{H} : Q′_{U} → 𝒪_{U}. Furthermore, if \underline{N} is any log structure on U, every map of pre-log structures Q_{U} → \underline{N} factors through Q′_{U}, so that α_{H} is a chart for \underline{M}_{|U}. We may therefore replace T by U and α by α_{H}, which allows to assume additionally, that α_{ξ} is injective on the subgroup S. Now, for any finitely generated subgroup H ⊂ 𝒪_{T,ξ}^{×} with S ⊂ H, we set \underline{M}_{ξ,H} := H ⨿_{S} Q; clearly, the monoids \underline{M}_{ξ,H} are finitely generated, and moreover :

\underline{M}_{ξ} = colim_{S⊂H⊂𝒪_{T,ξ}^{×}} \underline{M}_{ξ,H}.

Furthermore, we deduce a natural sequence of maps of monoids :

(12.1.32)  \{1\} → \underline{M}_{ξ,H} ⟶^{φ_{H}} \underline{M}_{ξ} ⟶^{ψ_{H}} 𝒪_{T,ξ}^{×}/H → \{1\}.

Claim 12.1.33. For every subgroup H as above, the sequence (12.1.32) is exact, i.e. \underline{M}_{ξ,H} is the kernel of ψ_{H}, and 𝒪_{T,ξ}^{×}/H is the cokernel of φ_{H}.

Proof of the claim. By lemma 4.8.29(iii), the assertion concerning Ker ψ_{H} can be verified on the underlying map of sets; however, lemma 4.8.31(ii) says that the set \underline{M}_{ξ} is the set-theoretic quotient (Q × 𝒪_{T,ξ}^{×})/S, for the natural translation action of S, and a similar description holds for \underline{M}_{ξ,H}, therefore Ker φ_{H} is the set-theoretic quotient (H × Q)/S, as required.

The assertion concerning Coker φ_{H} holds by general categorical nonsense. ♢ Let ε : \underline{M}_{ξ} → \underline{M}_{ξ}^{gp} be the natural map; claim 12.1.33 implies that the sequence of abelian

groups (12.1.32)^{gp} is right exact, and then a little diagram chase shows that :

(12.1.34)  ε^{−1}(Im φ_{H}^{gp}) = \underline{M}_{ξ,H}  whenever S ⊂ H ⊂ 𝒪_{T,ξ}^{×} .

Since G is finitely generated, we may find H as above, large enough, so that \underline{M}_{ξ,H}^{gp} contains the image of G. In view of (12.1.34), we deduce that the natural map

G ×\underline{M}_{ξ,H}^{gp} \underline{M}_{ξ,H} → P

is an isomorphism, so P is finitely generated, by corollary 6.4.2; moreover P is integral whenever \underline{M}_{ξ} is. Then, lemma 12.1.18(iv.a) implies that the natural map P → \underline{M}_{ξ} extends to a morphism of log structures ϑ : P_{U}^{log} → \underline{M}_{|U} on some neighborhood U of ξ. It remains to show that ϑ restricts to a chart for \underline{M}_{|V}, on some smaller neighborhood V of ξ. To this aim, it suffices to show that the map of stalks ϑ_{ξ} is an isomorphism (lemma 12.1.18(iv.c)). The latter assertion follows from lemma 12.1.29. □

Proposition 12.1.30 is the basis of several frequently used tricks that allow to construct “good” charts for a given coherent log structure (and for a morphism of such structures), or to “improve” given charts. We conclude this section with a selection of these tricks.

Corollary 12.1.35. Let T be a topos, ξ a T -point, \underline{M} a fine log structure on T. Then there exist a neighborhood U of ξ in T, and a chart P_{U} → \underline{M}_{|U} such that :

(a) P^{gp} is a free abelian group of finite rank.

(b) The induced morphism of monoids P → 𝒪_{T,ξ} is local.

Proof. Choose a group homomorphism G := ℤ^{⊕n} → \underline{M}_{ξ}^{gp} (for some integer n ≥ 0), such that the induced map G → (\underline{M}^{♯})_{ξ}^{gp} is surjective, and set P := G ×\underline{M}_{ξ}^{gp} \underline{M}_{ξ}. By proposition 12.1.30, the induced map P → \underline{M}_{ξ} extends to a chart P_{U} → \underline{M}_{|V}, for some neighborhood U of ξ. According to example 4.8.36(v), P^{gp} is a subgroup of G, whence (a). Next, claim 12.1.31 implies that, after replacing P by some localization (which does not alter P^{gp}), and U by a smaller neighborhood of ξ, we may achieve (b) as well. □

Corollary 12.1.36. Let T be a topos, ξ a T -point, \underline{M} a coherent log structure on T, and suppose that \underline{M}_{ξ} is integral and saturated. Then we have :

(i) There exist a neighborhood U of ξ in T, and a fine and saturated chart P_{U} → \underline{M}_{|U} which is sharp at the point ξ.

(ii) Especially, there exists a neighborhood U of ξ in T, such that \underline{M}_{|U} is a fine and saturated log structure.

Proof. (i): By lemma 6.2.10, we may find a decomposition \underline{M}_{ξ} = P × \underline{M}_{ξ}^{×}, for a sharp submonoid P ⊂ \underline{M}_{ξ}. Set G := P^{gp}; we deduce an isomorphism G →^{∼} \underline{M}_{ξ}^{gp}/\underline{M}_{ξ}^{×}, and clearly P = G ×\underline{M}_{ξ}^{gp} \underline{M}_{ξ}. By proposition 12.1.30, it follows that the induced map P → \underline{M}_{ξ} extends to a chart β : P_{U} → \underline{M}_{|U} on a neighborhood U of ξ. By construction, β is sharp at the T -point ξ; moreover, since \underline{M}_{ξ} is saturated, it is easily seen that the same holds for P. Finally, P is finitely generated, by corollary 6.4.2.

(ii): This follows immediately from (i) and lemma 12.1.18(iii). □

Theorem 12.1.37. Let T be a locally ringed topos, ξ a T -point, f : \underline{M} → \underline{N} a morphism of coherent (resp. fine) log structures on T. Then :

(i) There exists a neighborhood U of ξ, such that f_{|U} admits a finite (resp. fine) chart. (ii) More precisely, given a finite (resp. fine) chart ω_{P} : P_{T} → \underline{M}, we may find a neighborhood U of ξ, a finite (resp. fine) monoid Q, and a chart of f_{|U} of the form

(ω_{P |U}, ω_{Q} : Q_{U} → \underline{N}_{|U}, ϑ : P → Q).

(iii) Moreover, if f is a flat (resp. saturated) morphism of fine log structures and (ω_{P} , ω_{Q}, ϑ) is a fine chart for f, then we may find a neighborhood U of ξ, a localization map j : Q → Q′, and a flat (resp. saturated) and fine chart for f_{|U} of the form (ω_{P |U}, ω_{Q′}, j∘ϑ), such that

(a) ω_{Q|U} = ω_{Q′} ∘ j_{U}.

(b) The chart ω_{Q′} is local at the point ξ.

Proof. Up to replacing T by T/U′_{i} for a covering (U′_{i} → 1_{T} | i ∈ I) of the final object, we may assume that we have finite (resp. fine) charts ω_{P} : P_{T} → \underline{M} and Q′_{T} → \underline{N} (lemma 12.1.18(iii)), whence a morphism of pre-log structures :

ω : P_{T} ⟶^{ω_{P}} \underline{M} ⟶^{f} \underline{N}.

Let ξ be any T -point; there follow maps of monoids φ : P → \underline{M}_{ξ} → \underline{N}_{ξ} and ψ : Q′ → \underline{N}_{ξ}. Set G := (P ⊕ Q′)^{gp}, and apply proposition 12.1.30 to the group homomorphism G → \underline{N}_{ξ}^{gp} induced by φ and ψ; for Q := G ×\underline{N}_{ξ}^{gp} \underline{N}_{ξ}, we obtain a finite (resp. fine) local chart Q_{U} → \underline{N}_{|U} on some neighborhood U of ξ. Then, φ and the natural map P → G determine a unique map P → Q, whence a morphism ω′ : P_{U} → Q_{U} → \underline{N}_{|U} of pre-log structures; by construction, ω′_{ξ} : P = P_{U,ξ} → \underline{N}_{ξ} is none else than φ. By lemma 12.1.18(iv.b), we may then find a smaller neighborhood V → U of ξ such that ω′_{|V} = ω_{|V}. This proves (i) and (ii).

Next, we suppose that f is flat (resp. saturated) and both \underline{M}, \underline{N} are fine, and we wish to show (iii). In view of claim 12.1.31, we may find – after replacing T by a neighborhood of ξ – a fine chart for f of the form (ω_{P′}, ω_{Q′}, ϑ′), such that :

• P′ and Q′ are localizations of P and Q, and ϑ′ is induced by ϑ;

• ω_{P} = ω_{P′} ∘ (j_{P})_{T} and ω_{Q} = ω_{Q′} ∘ (j_{Q})_{T}, where j_{P} : P → P′ and j_{Q} : Q → Q′ are the localization maps;

• the induced maps Q^{′♯} → \underline{M}_{ξ}^{♯} and P^{′♯} → \underline{N}_{ξ}^{♯} are isomorphisms. Now, by proposition 4.8.26(resp. corollary 6.2.29), the map f_{ξ} is flat (resp. saturated), hence the same holds for the induced map \underline{M}_{ξ}^{♯} → \underline{N}_{ξ}^{♯}, by corollary 6.1.49(i) (resp. by lemma 6.2.12(iii)). Then the map P^{′♯} → Q^{′♯} induced by ϑ′ is flat (resp. saturated) as well, so the same holds for ϑ′, by corollary 6.1.49(ii) (resp. again by lemma 6.2.12(iii)). Finally, j_{P} ∘ ϑ′ : P → Q′ is flat (resp. saturated), by example 6.1.23(iii) (resp. by lemma 6.2.12(i)), and (ω_{P} , ω_{Q′}, j_{P} ∘ ϑ′) is a chart for f with the sought properties. □

Corollary 12.1.38. Let T be a topos, ξ a T -point, and φ : \underline{M} → \underline{N} a saturated morphism of fine and saturated log structures on T. We have :

(i) There exist a neighborhood U of ξ, and a fine and saturate chart (ω_{P} , ω_{Q}, ϑ : P → Q) of φ_{|U}, such that ω_{P} and ω_{Q} are sharp at the point ξ.

(ii) More precisely, suppose that (ω_{P} , ω_{Q}, ϑ : P → Q) is a fine and saturated chart for φ, such that \underline{M} is sharp at the point ξ, and ω_{Q} is local at ξ. Then there exists a section σ : Q^{♯} → Q of the projection Q → Q^{♯}, such that (ω_{P} , ω_{Q} ∘ σ_{T} , ϑ^{♯}) is a chart for φ.

Proof. After replacing T by some neighborhood of ξ, we may assume that \underline{M} admits a chart ω_{P} : P_{T} → \underline{M} which is fine, saturated, and sharp at the point ξ (corollary 12.1.36(i)). Then, by theorem 12.1.37(iii), we may find a neighborhood U of ξ, and a fine and saturated chart (ω_{P |U}, ω_{Q}, ϑ : P → Q) for φ_{|U}, such that ω_{Q} is local at ξ, so that ϑ is also a local morphism. Hence, it suffices to show assertion (ii).

(ii): We notice the following :

Claim 12.1.39. Let ϑ : P → Q a local and saturated morphism of fine and saturated monoids, and suppose that P is sharp. Then there exists a section σ : Q^{♯} → Q of the projection Q → Q^{♯}, such that ϑ(P) lies in the image of σ.

Proof of the claim. Pick an isomorphism β : Q →^{∼} Q^{♯} × Q^{×} as in lemma 6.2.10, and denote by ψ : P → Q^{×} the composition of ϑ with the induced projection Q → Q^{×}. The morphism ϑ^{♯} is still local and saturated (lemma 6.2.12(iii)), hence corollary 6.2.32(ii) implies that ϑ^{♯gp} extends to an isomorphism P^{gp} ⊕ L →^{∼} Q^{♯gp}, where L is a free abelian group of finite rank. Thus, we may extend ψ^{gp} to a group homomorphism ψ′ : Q^{♯gp} → Q^{×}. Define an automorphism α of Q^{♯} × Q^{×}, by the rule : (x, g) ↦ (x, g · ψ′(x)^{−1}). The restriction σ : Q^{♯} → Q of (α ∘ β)^{−1} will do. ♢

With the notation of claim 12.1.39 it is easily seen that ω_{Q} ∘ σ_{T} is still a chart for \underline{N}, hence (ω_{P} , ω_{Q} ∘ σ_{T} , ϑ^{♯}) is a chart for φ as required. □

Corollary 12.1.40. Let f : (T, \underline{M}) → (S, \underline{N}) a morphism of log topoi with coherent (resp. fine) log structures, and suppose that \underline{N} admits a finite (resp. fine) chart ω_{Q} : Q_{S} → \underline{N}. Let also ξ be any T -point; we have :

(i) There exist a neighborhood U of ξ, and a finite (resp. fine) chart (ω_{Q|U}, ω_{P} , ϑ : Q → P) for the morphism f_{|U}.

(ii) Moreover, if \underline{M} and \underline{N} are fine and f is log flat (resp. saturated) then, on some neighborhood U of ξ, we may also find a chart (ω_{Q|U}, ω_{P} , ϑ) which is flat (resp. saturated) and fine.

Proof. This is an immediate consequence of theorem 12.1.37. □

12.2. Log schemes. We specialize now to the case of a scheme X. Whereas in [75, §6.4] we considered only pre-log structures on the Zariski site of a scheme, hereafter we shall treat uniformly the categories of log structures on the topoi X_{ét}^{∼} and X_{Zar}^{∼} (notation of (4.9.13)).

12.2.1. Henceforth, we choose τ ∈ {ét, Zar} (see (4.9.19)), and whenever we mention a topology on a scheme X, it will be implicitly meant that this is the topology X_{τ} (unless explicitly stated otherwise). Let X be a scheme; a pre-log structure (resp. a log structure) on X is a pre-log structure (resp. a log structure) on the topos X_{τ}^{∼}. The datum of a scheme X and a log structure on X is called briefly a log scheme. It is known that a morphism of schemes X → Y is the same as a morphism of locally ringed topoi X_{τ}^{∼} → Y_{τ}^{∼}, hence we may define a morphism of log schemes (X, \underline{M}) → (Y, \underline{N}) as a morphism of log topoi (X_{τ}^{∼}, \underline{M}) → (Y_{τ}^{∼}, \underline{N}) (and likewise for morphisms of schemes with pre-log structures). We denote by pre-log_{τ} (resp. log_{τ}) the category of schemes with pre-log structures (resp. of log schemes) on the chosen topology τ. We denote by :

int.log_{τ} sat.log_{τ} qcoh.log_{τ} coh.log_{τ}

the full subcategory of the category log_{τ}, consisting of all log schemes with integral (resp. integral and saturated, resp. quasi-coherent, resp. coherent) log structures.

A scheme with a quasi-fine (resp. fine, resp. quasi-fine and saturated, resp. fine and saturated) log structure is called, briefly, a quasi-fine log scheme (resp. a fine log scheme, resp. a qfs log scheme, resp. a fs log scheme), and we denote by

qf.log_{τ}  f.log_{τ}  qfs.log_{τ}  fs.log_{τ}

the full subcategory of log_{τ} consisting of all quasi-fine (resp. fine, resp. qfs, resp. fs) log schemes on the topology τ. In case it is clear (or indifferent) which topology we are dealing with, we will usually omit the subscript τ. There is an obvious (forgetful) functor :

F : log → Sch

to the category of schemes, and it is easily seen that this functor is a fibration. For every scheme X, we denote by log_{X} the fibre category F^{−1}(X) i.e. the category of all log structures on X (or log_{X_{τ}}, if we need to specify the topology τ). The same notation shall be used also for the various subcategories : so for instance we shall write int.log_{X} for the full subcategory of all integral log structures on X. Moreover, we shall say that the log scheme (X, \underline{M}) is locally noetherian if the underlying scheme X is locally noetherian.

12.2.2. Most of the forthcoming assertions hold in both the étale and Zariski topoi, with the same proof. However, it may occasionally happen that the proof of some assertion concerning X_{τ}^{∼} (for τ ∈ {ét, Zar}), is easier for one choice or the other of these two topologies; in such cases, it is convenient to be able to change the underlying topology, to suit the problem at hand. This is sometimes possible, thanks to the following general considerations.

The morphism of locally ringed topoi ũ of (4.9.15) induces a pair of adjoint functors :

ũ^{∗} : log_{Zar} → log_{ét}  ũ_{∗} : log_{ét} → log_{Zar}

as well as analogous adjoint pairs for the corresponding categories of sheaves of monoids (resp. of pre-log structures) on the two sites. It follows formally that ũ^{∗} sends constant log structures to constant log structures, i.e. for every scheme X, and every object M := (M, φ) of Mnd/Γ(X, 𝒪_{X}) we have a natural isomorphism :

ũ^{∗}(X_{Zar}, M_{X_{Zar}}^{log}) ≃ (X_{ét}, M_{X_{ét}}^{log}).

More generally, lemma 12.1.18(i) shows that ũ^{∗} preserves the subcategories of quasi-coherent (resp. coherent, resp. integral, resp. fine, resp. fine and saturated) log structures.

Proposition 12.2.3. (i) The functor ũ^{∗} on log structures is faithful and conservative. (ii) The functor ũ^{∗} restricts to a fully faithful functor :

ũ^{∗} : int.log_{Zar} → int.log_{ét}.

(iii) Let (X_{ét}, \underline{M}) be any log scheme. Then the counit of adjunction ũ^{∗}ũ_{∗}(X_{ét}, \underline{M}) → (X_{ét}, \underline{M}) is an isomorphism if and only if the same holds for the counit of adjunction ũ^{∗}ũ_{∗}(\underline{M}^{♯}) → \underline{M}^{♯}.

Proof. (i): Let (X_{Zar}, \underline{M}) be any log structure; set (X_{ét}, \underline{M}_{ét}) := ũ^{∗}(X, \underline{M}), and denote by ũ_{X}^{♮} : ũ^{∗}𝒪_{X_{Zar}} → 𝒪_{X_{ét}} the natural map of structure rings. Since ũ is a morphism of locally ringed topoi, we have :

(ũ_{X}^{♮})^{−1}𝒪_{X_{ét}}^{×} = ũ^{∗}(𝒪_{X_{Zar}}^{×}).

It follows easily that \underline{M}_{ét} is the push-out in the cocartesian diagram :

\xymatrix{ũ^{∗}\underline{M}^{×} \ar[r]^{α} \ar[d] & 𝒪_{X_{ét}}^{×} \ar[d] \\ _{ũ_{∗}\underline{M}} \ar[r]^{β} & \underline{M}_{ét}.}

However, for every geometric point ξ of X, the natural map 𝒪_{X,|ξ|} → 𝒪_{X,ξ} is faithfully flat, hence α_{ξ} is injective, and therefore also β_{ξ}, in light of lemma 4.8.31(ii). The faithfulness of ũ^{∗} is an easy consequence. Next, let f : \underline{M} → \underline{N} be a morphism of log structures on X_{Zar}, set (X_{ét}, \underline{N}_{ét}) := ũ^{∗}(X, \underline{N}), and suppose that ũ^{∗}f : \underline{M}_{ét} → \underline{N}_{ét} is an isomorphism; we wish to show that f is an isomorphism. However, β induces an isomorphism of monoids :

(12.2.4)  ũ^{∗}(\underline{M}^{♯}) →^{∼} \underline{M}_{ét}^{♯}

and likewise for \underline{N}; it follows already that f induces an isomorphism \underline{M}^{♯} →^{∼} \underline{N}^{♯}. To conclude, it suffices to invoke lemma 12.1.4.

(ii): Let us suppose that \underline{M} is integral. According to [28, Prop.3.4.1], it suffices to show that the unit of adjunction (X, \underline{M}) → ũ_{∗}(X_{ét}, \underline{M}_{ét}) is an isomorphism. Now, from the isomorphism (12.2.4) we deduce a commutative diagram :

\xymatrix{\underline{M}^{♯} \ar[r] \ar[d] & (ũ_{∗}\underline{M}_{ét})^{♯} \ar[d] \\ ũ_{∗}ũ^{∗}(\underline{M}^{♯}) \ar[r] & ũ_{∗}(\underline{M}_{ét}^{♯})}

whose bottom arrow is an isomorphism, and whose left vertical arrow is an isomorphism as well, by lemma 4.9.27(iii) (and again [28, Prop.3.4.1]). We claim that also the right vertical arrow is an isomorphism. Indeed, since ũ_{∗} is left exact, the latter arrow is a monomorphism, hence it suffices to show it is an epimorphism; however, since \underline{M}_{ét} is an integral log structure (lemma 12.1.18(i)), it is easily seen that the projection \underline{M}_{ét} → \underline{M}_{ét}^{♯} is a 𝒪_{X_{ét}}^{×}-torsor (in the topos X_{ét}^{∼}/\underline{M}_{ét}^{♯}). Then the contention follows from the exact sequence of pointed sheaves (4.9.12), and the vanishing result of lemma 4.9.27(iv).

Summing up, we conclude that the top horizontal arrow in the above diagram is an isomorphism, so the assertion follows from lemma 12.1.4.

(iii): Set (X_{ét}, (ũ_{∗}\underline{M})_{ét}) := ũ^{∗}ũ_{∗}(X_{ét}, \underline{M}). To begin with, lemma 4.9.27(iv) easily implies that the natural morphism (ũ_{∗}\underline{M})^{♯} → ũ_{∗}(\underline{M}^{♯}) is an isomorphism; together with the general isomorphism (12.1.9), this yields a short exact sequence of X_{ét}-monoids :

0 → 𝒪_{X_{ét}}^{×} → (ũ_{∗}\underline{M})_{ét} → ũ^{∗}ũ_{∗}(\underline{M}^{♯}) → 0

which easily implies the assertion : the details shall be left to the reader. □

We shall prove later on some more results in the same vein (see corollary 12.2.24).

12.2.5. Arguing as in (12.1.24), we see easily that all finite limits are representable in the category of log schemes. The rule X ↦ (X, 𝒪_{X}^{×}) defines a fully faithful inclusion of the category of schemes into the category of log schemes. Hence, we shall regard a scheme as a log scheme with trivial log structure. Especially, if (X, \underline{M}) is any log scheme, and Y → X is any morphism of schemes, we shall often use the notation :

(12.2.6)  Y ×_{X} (X, \underline{M}) := (Y, 𝒪_{Y}^{×}) ×_{(X,𝒪_{X}^{×})} (X, \underline{M}).

Especially, if ξ is any τ-point of X, the localization of (X, \underline{M}) at ξ is the log scheme

(X(ξ), \underline{M}(ξ)) := X(ξ) ×_{X} (X, \underline{M})

(see definition 4.9.17(ii,iii)). If τ = ét, this operation is also called the strict henselization of X at ξ.

Definition 12.2.7. (i) For every integer n ∈ ℕ, we have the subset :

(X, \underline{M})_{n} := \{x ∈ X | dim \underline{M}_{ξ} ≤ n for every τ-point ξ → X localized at x\}.

By remark 6.1.11(i), (X, \underline{M})_{0} consists of all x ∈ X such that \underline{M}_{ξ} = 𝒪_{X,ξ}^{×} for every τ-point ξ of X localized at x; this subset is called the trivial locus of (X, \underline{M}), and is also denoted (X, \underline{M})_{tr}.

(ii) If f : (X, \underline{M}) → (Y, \underline{N}) is a morphism of log schemes, we denote by Str(f) ⊂ X the strict locus of f, which is the subset consisting of all points x ∈ X such that f is strict at every τ-point localized at x (see definition 12.1.22(ii)).

Remark 12.2.8. Let f : (X, \underline{M}) → (Y, \underline{N}) be any morphism of log schemes.

(i) Since log f induces local morphisms on stalks, it is easily seen that f restricts to a map

f_{tr} : (X, \underline{M})_{tr} → (Y, \underline{N})_{tr}.

(ii) Especially, we have (X, \underline{M})_{tr} ⊂ Str(f).

12.2.9. Let \underline{X} := (X_{i} | i ∈ I) be a cofiltered family of quasi-separated schemes, with affine transition morphisms f_{φ} : X_{j} → X_{i}, for every morphism φ : j → i in I. Denote by X the limit of \underline{X}, and for each i ∈ I let π_{i} : X → X_{i} be the natural projection.

Lemma 12.2.10. In the situation of (12.2.9), let 𝒳 := ((X_{i}, \underline{M}_{i}) | i ∈ I) be a cofiltered system of log schemes, with transition morphisms (f_{φ}, log f_{φ}) : (X_{i}, \underline{M}_{i}) → (X_{j}, \underline{M}_{j}) for every morphism φ : i → j in I. We have :

(i) The limit of the system 𝒳 exists in the category log.

(ii) Let (X, \underline{M}) denote the limit of the system 𝒳 . If X_{i} is quasi-compact and quasiseparated for every i ∈ I, then the natural map

colim_{i∈I} Γ(X_{i}, \underline{N}_{i}) → Γ(X, \underline{N})

is an isomorphism.

Proof. (i): Let X be the limit of the system of schemes \underline{X}, and endow X with the sheaf of monoids \underline{M} := colim_{i∈I} π_{i}^{∗}\underline{M}_{i}, where π^{∗} is the pull-back functor for sheaves of monoids (see (4.8.43)), and the transition maps π_{j}^{∗}\underline{M}_{j} → π_{i}^{∗}\underline{M}_{i} are induced by the morphisms log f_{φ} : f_{φ}^{∗}\underline{M}_{j} → \underline{M}_{i}, for every φ : i → j in I. Then the structure maps of the log structures \underline{M}_{i} induce a well defined morphism of X-monoids \underline{M} → 𝒪_{X}, and we claim that the resulting scheme with pre-log structure (X, \underline{M}) is actually a log scheme. Indeed, the assertion can be checked on the stalks, and notice that, for every τ-point ξ of X we have a natural identification

𝒪_{X,ξ} →^{∼} colim_{i∈I} 𝒪_{X_{i},π_{i}(ξ_{i})}.

(This is clear for τ = Zar, and for τ = ét one uses [66, Ch.IV, Prop.18.8.18(ii)]); it then suffices to invoke lemma 4.8.46(i). Lastly, it is easily seen that (X, \underline{M}) is a limit of the system 𝒳 : the details shall be left to the reader.

(ii): In view of the explicit construction in (i), the assertion for τ = Zar follows immediately from proposition 10.1.10, and for τ = ét it follows from [9, Exp.VII, Rem.5.14]. □

Example 12.2.11. Let X be a scheme. For the following example we choose to work with the étale topology X_{ét} on X. A divisor on X is a closed subscheme D ⊂ X which is regularly embedded in X and of codimension 1 ([66, Ch.IV, Déf.19.1.3, §21.2.12]). Suppose moreover that X is noetherian, let D ⊂ X be a divisor, and denote by (D_{i} | i ∈ I) the irreducible reduced components of D. We say that D is a strict normal crossings divisor, if :

• 𝒪_{X,x} is a regular ring, for every x ∈ D.

• D is a reduced subscheme.

• For every subset J ⊂ I, the (scheme theoretic) intersection ⋂_{j∈J} D_{j} is regular of pure

codimension ♯J in X. A closed subscheme D of a noetherian scheme X is a normal crossings divisor if, for every x ∈ X there exists an étale neighborhood f : U → X of x such that f^{−1}D is a strict normal crossings divisor in U.

Suppose that D is a normal crossings divisor of a noetherian scheme X, and let j : U := X\D → X be the natural open immersion. We claim that the log structure j_{∗}𝒪_{U}^{×} is fine (this is the direct image of the trivial log structure on U_{ét} : see example 12.1.12(ii)). To see this, let ξ be any geometric point of X localized at a point of D; up to replacing X by an étale neighborhood of ξ, we may assume that D is a strict normal crossings divisor; we can assume as well that X is affine and small enough, so that the irreducible components (D_{λ} | λ ∈ Λ) are of the form V (I_{λ}), where I_{λ} ⊂ A := Γ(X, 𝒪_{X}) is a principal divisor, say generated by an element x_{λ} ∈ A, for every λ ∈ Λ. We claim that j_{∗}𝒪_{U}^{×} is the constant log structure associated to the pre-log structure :

α : ℕ_{X}^{(Λ)} → 𝒪_{X} : e_{λ} ↦ x_{λ}

where (e_{λ} | λ ∈ Λ) is the standard basis of ℕ^{(Λ)}. Indeed, let S ⊂ Λ be the largest subset such that the image of ξ lies in D_{S} := ⋂_{λ∈S} D_{λ}, we have x_{λ} ∈ 𝒪_{X,x}^{×} for all λ ∉ S, so that the pushout of the induced diagram of stalks 𝒪_{X,ξ}^{×} ← α^{−1}𝒪_{X,ξ}^{×} → ℕ^{(Λ)} is the same as the push-out P_{S} of the analogous diagram 𝒪_{X,ξ}^{×} ← α_{S}^{−1}𝒪_{X,ξ}^{×} → ℕ^{(S)}, where α_{S} : ℕ_{X}^{(S)} → 𝒪_{X} is the restriction of α. Suppose that a ∈ 𝒪_{X,ξ} and a is invertible on X(ξ)\D_{S}; the minimal associated primes of A/(a) are all of height one, and they must therefore be found among the prime ideals Ax_{λ}, with λ ∈ S. It follows easily that a is of the form u · ∏_{λ∈S} x_{λ}^{k_{λ}} for certain k_{λ} ∈ ℕ and u ∈ 𝒪_{X,ξ}^{×}. Therefore, the natural map β_{ξ} : P_{S} → (j_{∗}𝒪_{U}^{×})_{ξ} is surjective. Moreover, the family (x_{λ} | λ ∈ S) is a regular system of parameters of 𝒪_{X,ξ} ([63, Ch.0, Prop.17.1.7]), therefore the natural map Sym_{κ(ξ)}^{n}(𝔪_{ξ}/𝔪_{ξ}^{2}) → 𝔪_{ξ}^{n}/𝔪_{ξ}^{n+1} is an isomorphism for every n ∈ ℕ (here 𝔪_{ξ} ⊂ 𝒪_{X,ξ} is the maximal ideal); it follows easily that β_{ξ} is also injective.

Example 12.2.12. Suppose that X is a regular noetherian scheme, and D ⊂ X a divisor on X; let U := X \D. If D is not a normal crossings divisor, the log structure \underline{M} := j_{∗}𝒪_{U}^{×} on X_{ét} is not necessarily fine. For a counterexample, let K be an algebraically closed field, C ⊂ 𝔸_{K}^{2} a nodal cubic; take D ⊂ X := 𝔸_{K}^{3} to be the (reduced, affine) cone over the cubic C, with vertex x_{0} ∈ 𝔸^{3}, and pick a geometric point ξ localized at x_{0}. It is easily seen that, away from the vertex, D is a normal crossings divisor, hence \underline{M}_{|X\\{x_{0}\}} is a fine log structure on X \\{x_{0}\}, by example 12.2.11. More precisely, let y_{0} ∈ C be the unique singular point, L ⊂ D the line spanned by x_{0} and y_{0}, and η a geometric point localized at the generic point of L. By inspecting the argument in example 12.2.11, we find that :

\underline{M}_{η} ≃ ℕ^{⊕2} ⊕ 𝒪_{X,η}^{×}

(indeed, an isomorphism is obtained by choosing a, b ∈ 𝒪_{C,y_{0}} such that V (a) and V (b) are the two branches of the cubic C in an étale neighborhood of y_{0}). On the other hand, let 𝔭 ⊂ A := K[T_{1}, T_{2}, T_{3}] be the prime ideal corresponding to x_{0}, and I ⊂ A_{𝔭} the ideal defining the closed subscheme X(x_{0})∩D in X(x_{0}); we claim that I ·𝒪_{X,ξ} is still a prime ideal, necessarily of height one. Indeed, let B := A_{𝔭}/I, and denote by A^{∧} (resp. B^{∧}) the 𝔭-adic completion of A_{𝔭} (resp. of B); then B^{∧} is also the completion of the reduced ring 𝒪_{X,ξ}/I, hence it suffices to show that Spec B^{∧} is irreducible. However, we may assume that C ⊂ Spec K[T_{1}, T_{2}] is the affine cubic defined by the ideal J ⊂ K[T_{1}, T_{2}] generated by T_{1}^{3} − T_{2}^{2} + T_{1}T_{2}. Then I is generated by the element f := T_{1}^{3} − T_{2}^{2}T_{3} + T_{1}T_{2}T_{3}; also, 𝔭 = (T_{1}, T_{2}, T_{3}), so that A^{∧} ≃ K[[T_{1}, T_{2}, T_{3}]] and B^{∧} ≃ A^{∧}/(f). Suppose Spec B^{∧} is not irreducible. This means that V (f) ⊂ Spec A^{∧} is a union V (f) = Z_{1} ∪ ⋯ ∪ Z_{n} of n ≥ 2 irreducible components Z_{i}. Since A^{∧} is a local regular ring, each such irreducible component Z_{i} is a divisor, defined by some principal prime ideal 𝔮_{i} in A^{∧}. Let a_{i} be a generator for 𝔮_{i}; then f admits factorizations of the form f = a_{i}b_{i} for some non-invertible b_{i} ∈ A^{∧}. Fix some i, and set a := a_{i}, b := b_{i}; since f is homogeneous of degree 3, we must have a ∈ 𝔭^{k}\𝔭^{k+1} for either k = 1 or k = 2 (k ≠ 0 since a is not a unit, and k ≠ 3, since b is not a unit); then b ∈ 𝔭^{3−k}\𝔭^{4−k}. Write a = a′ + a′′ and b = b′ + b′′, where a′′ ∈ 𝔭^{k+1}, b′′ ∈ 𝔭^{4−k} and a′ (resp. b′) is homogeneous of degree k (resp. 3 − k). Then f = ab = a′b′ + c, where c ∈ 𝔭^{4} and a′b′ is homogeneous of degree 3. This means that f = a′b′ is a factorization of f in A. However, f is irreducible in A, a contradiction. (Instead of this elementary argument, one can appeal to [133, Th.43.20], which runs as follows. If R is a local domain, then there is a natural bijection between the set of minimal prime ideals of the henselization R^{h} of R and the set of maximal ideals of the normalization of R in its ring of fractions. In our case, the normalization D^{ν} of D is the cone over the normalization of C, hence the only point of D^{ν} lying over x_{0} is the vertex of D^{ν}.)

It follows that any choice of a generator of I yields an isomorphism :

\underline{M}_{ξ} ≃ ℕ ⊕ 𝒪_{X,ξ}^{×}.

Suppose now that – in an étale neighborhood V of ξ – the log structure \underline{M} is associated to a pre-log structure α : P_{V} → 𝒪_{V}, for some monoid P; hence \underline{M}_{|V} is the push-out of the diagram 𝒪_{V}^{×} ← α^{−1}(𝒪_{V}^{×}) → P_{V}, whence isomorphisms :

P/α_{ξ}^{−1}(𝒪_{X,ξ}^{×}) ≃ \underline{M}_{ξ}/𝒪_{X,ξ}^{×} ≃ ℕ P/α_{η}^{−1}(𝒪_{X,η}^{×}) ≃ \underline{M}_{η}/𝒪_{X,η}^{×} ≃ ℕ^{⊕2}.

But clearly α_{ξ}^{−1}(𝒪_{X,ξ}^{×}) ⊂ α_{η}^{−1}(𝒪_{X,η}^{×}), so we would have a surjection of monoids ℕ → ℕ^{⊕2}, which is absurd.

On the other hand, we remark that the log structure j_{∗}𝒪_{U}^{×} on the Zariski site X_{Zar} is fine : indeed, one has a global chart ℕ_{X_{Zar}} → j_{∗}𝒪_{U}^{×}, provided by the equation defining the divisor D.

12.2.13. Let R be a ring, M a monoid, and set S := Spec R[M]. The unit of adjunction ε_{M} : M → R[M] can be regarded as an object (M, ε_{M}) of Mnd/Γ(S, 𝒪_{S}), whence a constant log structure M_{S}^{log} on S (see (12.1.15)). The rule

M ↦ Spec(R, M) := (S, M_{S}^{log})

is clearly functorial in M. Namely, to any morphism λ : M → N of monoids, we attach the morphism of log schemes

Spec(R, λ) := (Spec R[λ], λ_{Spec R[N]}^{log}) : Spec(R, N) → Spec(R, M).

Likewise, if P is a pointed monoid, Spec R⟨P ⟩ is a closed subscheme of Spec R[P], and we may define

Spec⟨R, P ⟩ := Spec(R, P) ×_{Spec R[P]} Spec R⟨P ⟩.

Lastly, if M is a non-pointed monoid, notice the natural isomorphism of log schemes

Spec⟨R, M_{∘}⟩ →^{∼} Spec(R, M)_{∘}.

(Notation of remark 12.1.23(ii) : the details shall be left to the reader.)

Lemma 12.2.14. With the notation of (12.2.13), let a ∈ M be any element, and set M_{a} := S_{a}^{−1}M, where S_{a} := {a^{n} | n ∈ ℕ}. Then U_{a} := Spec R[M_{a}] is an open subscheme of S, and the induced morphism of log schemes :

Spec(R, M_{a}) → Spec(R, M) ×_{S} U_{a}

is an isomorphism.

Proof. Let β_{S} : M_{S} → 𝒪_{S} and β_{U_{a}} : (M_{a})_{U_{a}} → 𝒪_{U_{a}} be the natural charts, and denote by φ : M → M_{a} the localization map. For every τ-point ξ of U_{a}, we have the identity :

β_{S,ξ} = β_{U_{a},ξ} ∘ φ : M → 𝒪_{S,ξ}.

Let Q := β_{U_{a},ξ}^{−1}𝒪_{U_{a},ξ}^{×}. The assertion is a straightforward consequence of the following :

Claim 12.2.15. The induced commutative diagram of monoids :

\xymatrix{φ^{−1}Q \ar[r] \ar[d] & M \ar[d] \\ Q \ar[r] & M_{a}}

is cocartesian.

Proof of the claim. Let b ∈ M, and suppose that β_{U_{a},ξ}(a^{−1}b) ∈ 𝒪_{U_{a},ξ}^{×}. Since β_{U_{a},ξ}(a) ∈ 𝒪_{U_{a},ξ}^{×}, we deduce that the same holds for β_{U_{a},ξ}(b), i.e. b ∈ φ^{−1}Q. Let Q′ := φ(φ^{−1}Q); we conclude that Q = S_{a}^{−1}Q′, the submonoid of M_{a} generated by Q′ and a^{−1}. The claim follows easily. □

12.2.16. In the same vein, let X be a R-scheme, and (M, φ) any object of Mnd/Γ(X, 𝒪_{X}). The map φ induces, via the adjunction of (4.8.50), a homomorphism of R-algebras R[M] → Γ(X, 𝒪_{X}), whence a map of schemes f : X → S := Spec R[M], inducing a morphism

(12.2.17)  X ×_{S} Spec(R, M) → (X, (M, φ)_{X}^{log})

of log schemes.

Lemma 12.2.18. In the situation of (12.2.16), we have :

(i) The map (12.2.17) is an isomorphism.

(ii) The log scheme Spec(R, M) represents the functor

log → Set : (Y, \underline{N}) ↦ Hom_{ℤ-Alg}(R, Γ(Y, N)) × Hom_{Mnd}(P, Γ(Y, \underline{N})).

Proof. (i): The log structure f^{∗}(M_{S}^{log}) of Spec(R, M) ×_{S} X represents the functor :

F : log_{X} → Set : \underline{N} ↦ Hom_{Mnd/Γ(S,𝒪_{S})} ((M, ε_{M}), Γ(S, f_{∗}\underline{N})).

However, if \underline{N} is any log structure on X, the pre-log structure (f_{∗}\underline{N})^{pre-log} is the same as f_{∗}(\underline{N}^{pre-log}) (see [75, (6.4.8)]). From the explicit construction of direct images for pre-log structures, and since the global sections functor is left exact (because it is a right adjoint), we deduce a cartesian diagram of monoids :

\xymatrix{Γ(S, f_{∗}\underline{N}) \ar[r] \ar[d] & Γ(S, 𝒪_{S}) \ar[d] \\ Γ(X, \underline{N}) \ar[r] & Γ(X, 𝒪_{X}).}

It follows easily that F is naturally isomorphic to the functor given by the rule :

\underline{N} ↦ Hom_{Mnd/Γ(X,𝒪_{X})} ((M, φ), Γ(X, \underline{N})).

The latter is of course the functor represented by (M, φ)_{X}^{log}.

(ii) can now be deduced formally from (i), or proved directly by inspecting the definitions. □

12.2.19. From (12.2.13) it is also clear that the rule (R, M) ↦ Spec(R, M) defines a functor

ℤ-Alg^{o} × Mnd^{o} → log

which commutes with fibre products; namely, say that

(R′, M′) ← (R, M) → (R′′, M′′)

are two morphisms of ℤ-Alg × Mnd; then there is a natural isomorphism of log schemes :

Spec(R′ ⊗_{R} R′′, M′ ⊗_{M} M′′) →^{∼} Spec(R′, M′) ×_{Spec(R,M)} Spec(R′′, M′′).

For the proof, one compares the universal properties characterizing these log schemes, using lemma 12.2.18(ii) : details left to the reader.

12.2.20. Let X be a scheme, \underline{M} a sheaf of monoids on X_{τ}. We say that \underline{M} is locally constant if there exist a covering family (U_{λ} → X | λ ∈ Λ) and for every λ ∈ Λ, a monoid P_{λ} and an isomorphism of sheaves of monoids \underline{M}_{|U_{λ}} ≃ (P_{λ})_{U_{λ}}. We say that \underline{M} is weakly constructible if, for every affine open subset U ⊂ X we can find finitely many locally closed constructible subsets Z_{1}, …, Z_{n} ⊂ U such that :

• U = Z_{1} ∪ ⋯ ∪ Z_{n}

• \underline{M}_{|Z_{i}} is a locally constant sheaf of monoids, for every i = 1, …, n.

If moreover, \underline{M}_{ξ} is a finitely generated monoid for every τ-point ξ of X, we say that \underline{M} is constructible. If φ : \underline{M} → \underline{N} is a morphism of constructible sheaves of monoids on X_{τ}, then it is easily seen that Ker φ, Coker φ and Im φ are constructible (corollary 6.4.3(i)). Moreover, constructibility is preserved under coequalizers; if \underline{M} is weakly constructible with integral stalks \underline{M}_{ξ} at every τ-point ξ of X, then a fibre product \underline{M}′×\underline{M} \underline{M}′′ is constructible if the same holds for both \underline{M}′ and \underline{M}′′; furthermore, under the same assumptions for \underline{M}, an equalizer Equal(φ, φ′ : \underline{N} → \underline{M}) is constructible if the same holds for \underline{N} (corollaries 6.4.2and 6.4.3(ii)).

If X is locally noetherian and both \underline{M} and \underline{N} are weakly constructible, then the same holds for Ker φ, Coker φ and Im φ.

Suppose that \underline{M} is a sheaf of monoids on X_{τ}, and for every x ∈ X choose a τ-point \overline{x} localized at x; the rank of \underline{M} is the function

rk\underline{M} : X → ℕ ∪ \{∞\} x ↦ dim_{ℚ} \underline{M} gp ; \overline{x} ⊗_{ℤ} ℚ.

It is clear from the definitions that the rank function of a constructible sheaf of monoid is constructible on X.

Lemma 12.2.21. Let X be a scheme, φ : \underline{Q} → \underline{Q}′ a morphism of coherent log structures on X_{τ}. Then :

(i) The sheaves \underline{Q}^{♯} and Coker φ are constructible.

(ii) (X, \underline{Q})_{n} is an open subset of X, for every integer n ≥ 0. (See definition 12.2.7(i).) (iii) The rank functions of \underline{Q}^{♯} and Coker φ are (constructible and) upper semicontinuous.

Proof. (i): Suppose that \underline{Q} admits a finite chart α : M_{X} → \underline{Q}. Pick a finite system of generators Σ ⊂ M, and for every S ⊂ Σ, set :

Z_{S} := ⋂_{s∈S} D(s) ∩ ⋂_{t∈Σ\S} V (t)

where, as usual D(s) (resp. V (s)) is the open (resp. closed) subset of the points x ∈ X such that the image s(x) ∈ κ(x) of s is invertible (resp. vanishes). Clearly each Z_{S} is a constructible subset of X, and their union equals X. Moreover, for every S ⊂ Σ, and every τ-point ξ supported on Z_{S}, the submonoid N_{ξ} := α_{ξ}^{−1}(𝒪_{X,ξ}^{×}) ⊂ M is a face of M (lemma 6.1.20(i)), hence it is the submonoid ⟨S⟩ generated by Σ ∩ N_{ξ} = S (lemma 6.1.20(ii)). It follows easily that \underline{Q}_{|Z_{S}}^{♯} ≃ (M/⟨S⟩)_{|Z_{S}}.

More generally, suppose that \underline{Q} is coherent. We may assume that X is affine. Then, by the foregoing, we may find a finite set Λ and a covering family (f_{λ} : U_{λ} → X | λ ∈ Λ) of X_{τ}, such that U_{λ} is affine and \underline{Q}_{|U_{λ}}^{♯} is a constructible sheaf of monoids on (U_{λ})_{τ}, for every λ ∈ Λ. Since f_{λ} is finitely presented, it maps constructible subsets to constructible subsets ([66, Ch.IV, Th.1.8.4]); if τ = Zar, it follows easily that the restriction of \underline{Q}^{♯} to f_{λ}(U_{λ})_{τ} is constructible, therefore \underline{Q}^{♯} is constructible. For τ = ét we remark :

Claim 12.2.22. Let f : Y → X a surjective morphism of schemes, locally of finite presentation, and ℱ a sheaf of monoids on X_{ét}. Then ℱ is constructible if and only if the same holds for f_{ét}^{∗}ℱ.

Proof of the claim. This can be proved as [10, Exp.IX, Prop.2.8]. ♢

Hence, also for τ = ét we see that \underline{Q}^{♯} is constructible. Next, notice that Coker φ = Coker φ^{♯}; by the foregoing, \underline{Q}^{′♯} is constructible as well, so the same holds for Coker φ.

(ii,iii): Let x ∈ X be any point, and ξ a τ-point of X localized at x; by theorem 12.1.37(i) we may find a neighborhood f : U → X of ξ in X_{τ} and a finite chart (ω_{P} , ω_{P′}, ϑ) for φ_{|U}. By claim 12.1.31, we may assume, after replacing U by a smaller neighborhood of ξ, that ω_{P} and ω_{P′} are local at the point ξ, in which case \underline{Q}_{ξ}^{♯} = P^{♯} and Coker φ_{ξ} = Coker ϑ. Let r : X → ℕ (resp. r′ : U → ℕ) denote the rank function of \underline{Q}^{♯} (resp. of \underline{Q}_{|U}^{♯}); then it is clear that r′ = r ∘ f. On the other hand, for every y ∈ U and every τ-point η of U localized at y, the stalk \underline{Q}_{η}^{♯} is a quotient of P^{♯}, hence r′(y) ≤ r(x) and dim \underline{Q}_{η} ≤ dim \underline{Q}_{ξ}. Since f is an open mapping, this shows (ii), and also that the rank function of \underline{Q}^{♯} is upper semicontinuous. Likewise, let s (resp. s′) denote the rank function of Coker φ (resp. Coker φ_{|U}); then s′ = s ∘ f, and Coker φ_{η} is a quotient of Coker ϑ for every τ-point η of U; the latter implies that s′(y) ≤ s(x) for every y ∈ U, which shows that s is upper semicontinuous. □

Remark 12.2.23. By lemma 12.2.21(ii), if \underline{Q} is any coherent log structure on a scheme X, then (X, \underline{Q})_{n} is naturally an open subscheme of X for every n ∈ ℕ, and even naturally a log scheme, with the log structure inherited via the open immersion in X. Henceforth, for every such n and \underline{Q}, the notation (X, \underline{Q})_{n} will refer to either the underlying open subscheme of X, or to the corresponding open log subscheme of (X, \underline{Q}), depending on the context.

Corollary 12.2.24. Let X be a scheme, \underline{M} a log structure on X_{Zar}, and set

(X_{ét}, \underline{M}_{ét}) := ũ^{∗}(X_{Zar}, \underline{M}).

Then \underline{M} is integral (resp. coherent, resp. fine, resp. fine and saturated) if and only if the same holds for \underline{M}_{ét}.

Proof. It has already been remarked that (X_{ét}, \underline{M}_{ét}) is coherent (resp. fine, resp. fine and saturated) whenever the same holds for (X_{Zar}, \underline{M}); furthermore, the proof of proposition 12.2.3(i) shows that the natural map on stalks \underline{M}_{|ξ|} → \underline{M}_{ét,ξ} is injective for every geometric point ξ of X, therefore \underline{M} is integral whenever the same holds for \underline{M}_{ét}.

Next, we suppose that \underline{M}_{ét} is coherent, and we wish to show that \underline{M} is coherent.

Let x ∈ X be any point, ξ a geometric point localized at x. By assumption there exists a finitely generated monoid P′, with a morphism α : P′ → \underline{M}_{ét,ξ} inducing an isomorphism :

P′ ⊗_{β^{−1}\underline{M}_{ét,ξ}^{×}} \underline{M}_{ét,ξ}^{×} → \underline{M}_{ét,ξ} ≃ \underline{M}_{x} ⊗\underline{M}_{x}^{×} 𝒪_{X,ξ}^{×}.

It follows easily that we may find a finitely generated submonoid Q ⊂ \underline{M}_{x}, such that the image of α lies in (Q · \underline{M}_{x}^{×}) ⊗\underline{M}_{x}^{×} 𝒪_{X,ξ}^{×}, and therefore the natural map :

(Q · \underline{M}_{x}^{×}) ⊗\underline{M}_{x}^{×} 𝒪_{X,ξ}^{×} → \underline{M}_{ét,ξ}

is surjective. Then lemma 4.8.31(ii) implies that Q · \underline{M}_{x}^{×} = \underline{M}_{x}, in other words, the induced map Q → \underline{M}_{x}^{♯} = \underline{M}_{ét,ξ}^{♯} is surjective. Set

P := Q^{gp} ×\underline{M}_{ét,ξ}^{gp} \underline{M}_{ét,ξ}.

In this situation, proposition 12.1.30 tells us that the induced map β_{ξ} : P → \underline{M}_{ét,ξ} extends to an isomorphism of log structures β^{log} : P_{U_{ét}}^{log} → \underline{M}_{ét|U_{ét}} on some étale neighborhood U → X of ξ. On the other hand, it is easily seen that the diagram of monoids :

\xymatrix{\underline{M}_{x} \ar[r] \ar[d] & \underline{M}_{x}^{gp} \ar[d] \\ _{\underline{M}_{ét,ξ}} \ar[r] & _{\underline{M}{gp \atop ét,ξ}}}

is cartesian, therefore β_{ξ} factors uniquely through a morphism β′_{x} : P → \underline{M}_{x}. The latter extends to a morphism of log structures β^{′ log} : P_{U′_{Zar}}^{log} → \underline{M}_{|U′_{Zar}} on some (Zariski) open neighborhood U′ of x in X (lemma 12.1.18(iv.a),(v)). By inspecting the construction, we find a commutative diagram of monoids :

[figure: (ũ^{∗}P_{U′_{Zar}}^{log})_{ξ}; (ũ^{∗}β^{′ log})_{ξ}; P_{U_{ét},ξ}^{log}; β_{ξ}^{log}; (ũ^{∗}\underline{M})_{ξ}; _{\underline{M}_{ét,ξ}}]

(where ũ^{∗} is the functor on log structures of (12.2.2)) whose horizontal arrows and right vertical arrow are isomorphisms; it follows that (ũ^{∗}β^{′ log})_{ξ} is an isomorphism as well, therefore ũ^{∗}β^{′ log} restricts to an isomorphism on some smaller étale neighborhood f : U′′ → U′ of ξ (lemma 12.1.18(iv.c)). Since f is an open morphism, we deduce that the restriction of (ũ^{∗}β^{′ log})_{ξ} is already an isomorphism on f(U′′)_{ét}, and f(U′′) is a (Zariski) open neighborhood of x in X. Finally, in light of proposition 12.2.3(i), we conclude that the restriction of β^{′ log} is an isomorphism on f(U′′)_{Zar}, so \underline{M} is coherent, as stated.

Lastly, we suppose that \underline{M}_{ét} is fine and saturated, and we wish to show that \underline{M} is saturated. However, the assertion can be checked on the stalks, hence let ξ be any geometric point of X; the proof of proposition 12.2.3 shows that the natural map \underline{M}_{|ξ|}^{♯} → \underline{M}_{ét,ξ}^{♯} is bijective, so the assertion follows from lemma 6.2.9(ii). □

Corollary 12.2.25. The category coh.log admits all finite limits. More precisely, the limit (in the category of log structures) of a finite system of coherent log structures, is coherent.

Proof. Let Λ be a finite category, 𝒳 := ((X_{λ}, \underline{M}_{λ}) | λ ∈ Λ) an inverse system of schemes with coherent log structures indexed by Λ. Denote by Y the limit of the system (X_{λ} | λ ∈ Λ) of underlying schemes, and by π_{λ} : Y → X_{λ} the natural morphism, for every λ ∈ Λ. It is easily seen that the limit of 𝒳 is naturally isomorphic to the limit of the induced system 𝒴 := ((Y, π_{λ}^{∗}\underline{M}_{λ}) | λ ∈ Λ), and in view of lemma 12.1.18(i), we may therefore replace 𝒳 by 𝒴 , and assume that 𝒳 is a finite inverse system in the category log_{X}, for some scheme X (especially, the underlying maps of schemes are 1_{X}, for every morphism λ → μ in Λ).

It suffices then to show that the push-out of two morphisms g : \underline{N} → \underline{M}, h : \underline{N} → \underline{M}′ of coherent log structures on X, is coherent. However, notice that the assertion is local on the site X_{τ}, hence we may assume that \underline{N} admits a finite chart Q_{X} → \underline{N}. Then, thanks to theorem 12.1.37(ii), we may further assume that both f and g admit finite charts of the form Q_{X} → P_{X} and respectively Q_{X} → P′_{X}, for some finitely generated monoids P and P′. We deduce a natural map (P ⨿_{Q} P′)_{X} →^{∼} P_{X} ⨿_{Q_{X}} P′_{X} → \underline{M} ⨿\underline{N} \underline{M}′, which is the sought finite chart. □

Lemma 12.2.26. In the situation of (12.2.9), suppose that X_{i} is quasi-compact and quasiseparated for every i ∈ I, and that there exist i ∈ I, and a coherent log structure \underline{N}_{i} on X_{i}, such that the log structure \underline{N} := π_{i}^{∗}\underline{N}_{i} on X_{τ} has a finite chart β : Q_{X} → \underline{N}. Then there exist a morphism φ : j → i in I and a chart β_{j} : Q_{X_{j}} → \underline{N}_{j} := f_{φ}^{∗}\underline{N}_{i} for \underline{N}_{j}, such that π_{j}^{∗}β_{j} = β.

Proof. After replacing I by I/i, we may assume that \underline{N}_{j} is well defined for every j ∈ I, and i is the final object of I. In this case, notice that (X, \underline{N}) is the limit of the cofiltered system of log schemes ((X_{j}, \underline{N}_{j}) | j ∈ I). We begin with the following :

Claim 12.2.27. In order to prove the lemma, it suffices to show that, for every τ-point ξ of X there exist j(ξ) ∈ I, a neighborhood U_{ξ} → X_{j(ξ)} of π_{j(ξ)}(ξ) in X_{j(ξ),τ}, and a chart β_{j(ξ)} : Q_{U_{ξ}} → \underline{N}_{j(ξ)|U_{ξ}} such that (1_{U_{ξ}} ×_{X_{j(ξ)}} π_{j(ξ)})^{∗}β_{j(ξ)} = β.

Proof of the claim. Clearly we may assume that U_{ξ} is an affine scheme, for every τ-point ξ of X. Under the stated assumptions, X is quasi-compact, hence we may find a finite set {ξ_{1}, …, ξ_{n}} of τ-points of X, such that (U_{ξ_{k}} ×_{X_{j(ξ_{k})}} X → X | k = 1, …, n) is covering in X_{τ}. Since I is cofiltered, we may then find j ∈ I and morphisms φ_{k} : j → j(ξ_{k}) for every k = 1, …, n. After replacing each β_{j(ξ_{k})} by f_{φ_{k}}^{∗}β_{j(ξ_{k})}, we may assume that j(ξ_{k}) = j for every k ≤ n.

In this case, set β_{k} := β_{j(ξ_{k})}, and U_{k} := U_{ξ_{k}} for every k ≤ n; let also U_{kl} := U_{k} ×_{X_{j}} U_{l}, U_{kl}^{∼} := U_{kl} ×_{X_{j}} X for every k, l ≤ n, and denote by π_{kl} : U_{kl}^{∼} → U_{kl} the natural projection. Hence, for every k, l ≤ n we have a morphism of U_{kl}-monoids :

β_{kl} := β_{k|U_{kl}} : Q_{U_{kl}} → \underline{N}_{j|U_{kl}}

and by construction we have π_{kl}^{∗}β_{kl} = π_{lk}^{∗}β_{lk} under the natural identification : U_{kl}^{∼} →^{∼} U_{lk}^{∼}. Notice now that U_{kl} is quasi-compact and quasi-separated for every k, l ≤ n, hence the natural map

colim_{i∈I} Γ(U_{kl} ×_{X_{j}} X_{i}, \underline{N}_{i}) → Γ(U_{kl}^{∼}, \underline{N})

is an isomorphism (lemma 12.2.10(ii)). On the other hand, β_{kl} is given by a morphism of monoids b_{kl} : Q → Γ(U_{kl}, \underline{N}_{j}), and likewise π_{kl}^{∗}β_{kl} is given by the morphism Q → Γ(U_{kl}^{∼}, \underline{N}) obtained by composiiton of b_{kl} and the natural map Γ(U_{kl}, \underline{N}_{j}) → Γ(U_{kl}^{∼}, \underline{N}). By lemma 6.1.7(ii), we may then find a morphism j′ → j in I such that the following holds. Set

V_{k} := U_{k} ×_{X_{j}} X_{j′}  V_{kl} := V_{k} ×_{X_{j′}} V_{l}  for every k, l ≤ n

and let p_{k} : V_{k} → U_{k} be the projection for every k ≤ n; let also β′_{k} := p_{k}^{∗}β_{k}, which is a chart Q_{V_{k}} → \underline{N}_{j′|V_{k}} for the restriction of \underline{N}_{j′}. Then

(12.2.28)  β′_{k|V_{kl}} = β′_{l|V_{kl}}  for every k, l ≤ n

under the natural identification V_{kl} →^{∼} V_{lk}. By construction, the system of morphisms (V_{k} ×_{X_{j′}} X → X | k = 1, …, n) is covering in X_{τ}; after replacing j′ by a larger index, we may then assume that the system of morphisms (V_{k} → X_{j′} | k = 1, …, n) is covering in X_{j′,τ} ([65, Ch.IV, Th.8.10.5]). In this case, (12.2.28) implies that the local charts β′_{k} glue to a well defined chart β_{j′} : Q_{X_{j′}} → \underline{N}_{j′}, and a direct inspection shows that we have indeed π_{j′}^{∗}β_{j′} = β. ♢

Now, denote by α : Q_{X} → 𝒪_{X} the composition of β and the structure map of \underline{N}, and let ξ be a τ-point of X; we have a natural isomorphism

\underline{N}_{ξ} = \underline{N}_{i,π_{i}(ξ)} ⊗\underline{N}_{i,π_{i}(ξ)}^{×} 𝒪_{X,ξ}^{×} →^{∼} colim_{j∈I} \underline{N}_{i,π_{i}(ξ)} ⊗\underline{N}_{i,π_{i}(ξ)}^{×} 𝒪_{X_{j},π_{j}(ξ_{j})}^{×} →^{∼} colim_{j∈I} \underline{N}_{j,π_{j}(ξ)}

([66, Ch.IV, Prop.18.8.18(ii)]). It follows that β_{ξ} and α_{ξ} factor through morphisms of monoids β_{ξ,j} : Q → \underline{N}_{j,π_{j}(ξ)} and α_{ξ,j} : Q → 𝒪_{X_{j},π_{j}(ξ)} for some j ∈ I (lemma 6.1.7(ii)), and again we may replace I by I/j, after which we may assume that β_{ξ,j} and α_{ξ,j} are defined for every j ∈ I. Then α_{ξ,j} extends to a pre-log structure α_{j} : Q_{U_{j}} → 𝒪_{U_{j}} on some neighborhood U_{j} → X_{j} of π_{j}(ξ) in X_{j,τ} (lemma 12.1.18(v)), and we may also assume that β_{ξ,j} extends to a morphism of pre-log structures β_{j} : (Q_{U_{j}}, α_{j}) → \underline{N}_{j|U_{j}} (lemma 12.1.18(iv.a)). Notice as well that

\underline{N}_{j,ξ}^{♯} ≃ \underline{N}_{ξ}^{♯} and β_{ξ,j}^{−1}\underline{N}_{j,π_{j}(ξ)}^{×} = β_{ξ}^{−1}\underline{N}_{ξ}^{×} for every j ∈ I

and since β_{ξ} induces an isomorphism Q/β_{ξ}^{−1}\underline{N}_{ξ}^{×} →^{∼} \underline{N}_{ξ}^{♯}, we deduce that β_{j,ξ} (which is the same as β_{ξ,j}) induces an isomorphism Q/α_{j,ξ}^{−1}𝒪_{X_{j},π_{j}(ξ)}^{×} →^{∼} \underline{N}_{j,ξ}^{♯} for every j ∈ I. In turn, it then follows from lemma 12.1.4that β_{j,ξ} induces an isomorphism (Q_{U_{j}}, α_{j})_{π_{j}(ξ)}^{log} →^{∼} \underline{N}_{j,π_{j}(ξ)}.

Next, by lemma 12.1.18(iv.b,c) we may find, for every j ∈ I, a neighborhood U′_{j} → U_{j} of ξ in X_{τ}, such that the restriction (Q_{U′_{j}}, α_{j|U′_{j}}) → \underline{N}_{j|U′_{j}} of β_{j} is a chart for \underline{N}_{j|U′_{j}}.

By construction, the morphism ((1_{U_{j}} ×_{X_{j}} π_{j})^{∗}β_{j})_{ξ} : Q → \underline{N}_{ξ} is the same as β_{ξ}, so by lemma 12.1.18(iv.b) we may find a neighborhood V_{j} → U′_{j} ×_{X_{j}} X of ξ in X_{τ}, such that

((1_{U_{j}} ×_{X_{j}} π_{j})^{∗}β_{j})_{|V_{j}} = β_{|V_{j}} and ((1_{U_{j}} ×_{X_{j}} π_{j})^{∗}α_{j})_{|V_{j}} = α_{|V_{j}}.

Claim 12.2.29. In the situation of (12.2.9), let Y → X be an object of the site X_{τ}, with Y quasi-compact and quasi-separated. We have :

(i) There exist i ∈ I, an object Y_{i} → X_{i} of X_{i,τ}, and an isomorphism of X-schemes

Y →^{∼} Y_{i} ×_{X_{i}} X.

(ii) Moreover, if Y → X is covering in X_{τ}, then we may find i ∈ I and Y_{i} → X_{i} as in (i), which is covering in X_{i,τ}.

Proof of the claim. (i) is obtained by combining [65, Ch.IV, Th.8.8.2(ii)] and [66, Ch.IV, Prop.17.7.8(ii)] (and [65, Ch.IV, Cor.8.6.4] if τ = Zar). Assertion (ii) follows from (i) and [65, Ch.IV, Th.8.10.5]. ♢

By claim 12.2.29(i), we may assume that V_{j} = U′′_{j} ×_{X_{j}} X for some neighborhood U′′_{j} → U′_{j} of π_{j}(ξ) in X_{j,τ}. To conclude, it suffices to invoke claim 12.2.27. □

Proposition 12.2.30. In the situation of (12.2.9), suppose that X_{i} is quasi-compact and quasiseparated for every i ∈ I. Then the natural functor :

(12.2.31)  2-colim_{I} coh.log_{X_{i}} → coh.log_{X}

is an equivalence.

Proof. To begin with, we show :

Claim 12.2.32. The functor (12.2.31) is faithful. Namely, for a given i ∈ I, let \underline{M}_{i} and \underline{N}_{i} be two coherent log structures on X_{i}, and f_{i}, g_{i} : \underline{M}_{i} → \underline{N}_{i} two morphisms, such that π_{i}^{∗}f_{i} agrees with π_{i}^{∗}g_{i}. Then there exists a morphism ψ : j → i in I such that f_{ψ}^{∗}f_{i} = f_{ψ}^{∗}g_{i}.

Proof of the claim. Set \underline{M} := π_{i}^{∗}\underline{M}_{i}, and define likewise the coherent log structure \underline{N} on X. For any τ-point ξ of X, pick a neighborhood U_{ξ} → X_{i} of π_{i}(ξ) in X_{i}, and a finite chart β : P_{U_{ξ}} → \underline{M}_{i|U_{ξ}} for the restriction of \underline{M}_{i}. The morphisms f_{i|U_{ξ}} and g_{i|U_{ξ}} are determined by the induced maps φ := Γ(U_{ξ}, f_{i}) ∘ β and γ := Γ(U_{ξ}, g_{i}) ∘ β, and the assumption means that the composition of φ and the natural map Γ(U_{ξ}, \underline{N}_{i}) → Γ(U_{ξ} ×_{X_{i}} X, \underline{N}) equals the composition of γ with the same map.

It then follows from lemmata 12.2.10(ii) and 6.1.7(ii) that there exists a morphism ψ : i(ξ) → i in I such that the composition of φ with the natural map Γ(U_{ξ}, \underline{N}_{i}) → Γ(U_{ξ} ×_{X_{i}} X_{i(ξ)}, f_{ψ}^{∗}\underline{N}) equals the composition of γ with the same map; in other words, if we set U′_{ξ} := U_{ξ} ×_{X_{i}} X_{i(ξ)}, we have f_{ψ}^{∗}f_{i|U′_{ξ}} = f_{ψ}^{∗}g_{i|U′_{ξ}}. Next, since X is quasi-compact, we may find finitely many τ-points ξ_{1}, …, ξ_{n} such that the family (U′_{ξ_{k}} ×_{X_{i(ξ_{k})}} X → X) is covering in X_{τ}. Then, by [65, Ch.IV, Th.8.10.5] we may find j ∈ I and morphisms ψ_{k} : j → i(ξ_{k}) in I, for k = 1, …, n, such that the induced family (U′′_{k} := U′_{ξ_{k}} ×_{X_{i(ξ_{k})}} X_{j} → X_{j}) is covering in X_{j,τ}. By construction we have

f_{ψ_{k}∘ψ}^{∗}f_{i|U′′_{k}} = f_{ψ_{k}∘ψ}^{∗}g_{i|U′′_{k}} for k = 1, …, n

therefore f_{ψ_{k}∘ψ}^{∗}f_{i} = f_{ψ_{k}∘ψ}^{∗}g_{i}, as required. ♢

Claim 12.2.33. The functor (12.2.31) is full. Namely, let i ∈ I and \underline{M}_{i}, \underline{N}_{i} as in claim 12.2.32, and suppose that f : π_{i}^{∗}\underline{M}_{i} → π_{i}^{∗}\underline{N}_{i} is a given morphism of log structures; then there exist a morphism ψ : j → i in I, and a morphism of log structures f_{j} : f_{ψ}^{∗}\underline{M}_{i} → f_{ψ}^{∗}\underline{N}_{i} such that π_{j}^{∗}f_{j} = f.

Proof of the claim. Indeed, by theorem 12.1.37(i) we may find a covering family (U_{λ} → X | λ ∈ Λ) in X_{τ}, and for each λ ∈ Λ a finite chart

β_{λ} : P_{λ,U_{λ}} → \underline{M}_{|U_{λ}} γ_{λ} : Q_{λ,U_{λ}} → \underline{N}_{|U_{λ}} ϑ_{λ} : P_{λ} → Q_{λ}

for the restriction f_{|U_{λ}}. Clearly we may assume that each U_{λ} is affine, and since X is quasicompact, we may assume as well that Λ is a finite set; in this case, claim 12.2.29 implies that there exist a morphism j → i in I, a covering family (U_{j,λ} → X_{j} | λ ∈ Λ), and isomorphism of X-schemes U_{λ} →^{∼} U_{j,λ} ×_{X_{j}} X for every λ ∈ Λ. Next, lemma 12.2.26 says that, after replacing j by some larger index, we may assume that for every λ ∈ Λ there exist charts

β_{j,λ} : P_{λ,U_{j,λ}} → \underline{M}_{j} := f_{ψ}^{∗}\underline{M}_{i} and γ_{j,λ} : Q_{λ,U_{j,λ}} → \underline{N}_{j} := f_{ψ}^{∗}\underline{N}_{i}

with π_{j}^{∗}β_{j,λ} = β_{λ} and π_{j}^{∗}γ_{j,λ} = γ_{λ}. Set f_{j,λ} := ϑ_{λ,U_{j,λ}}^{log} : \underline{M}_{j} → \underline{N}_{j}; by construction we have :

(1_{U_{j,λ}} ×_{X_{j}} π_{j})^{∗}f_{j,λ} = f_{|U_{λ}} for every λ ∈ Λ.

Next, for every λ, μ ∈ Λ, let U_{j,λμ} := U_{j,λ} ×_{X_{j}} U_{j,μ}; we deduce, for every λ, μ ∈ Λ, two morphisms of log structures f_{j,λ|U_{j,λμ}}, f_{j,μ|U_{λμ}} : \underline{M}_{j|U_{λμ}} → \underline{N}_{j|U_{λμ}}, which agree after pull back to U_{j,λμ}×_{X_{j}} X. By applying claim 12.2.32to the cofiltered system of schemes (U_{j,λμ}×_{X_{j}} X_{j′} | j′ ∈ I/j), we may then achieve – after replacing j by some larger index – that f_{j,λ|U_{j,λμ}} = f_{j,μ|U_{λμ}}, in which case the system (f_{j,λ} | λ ∈ Λ) glues to a well defined morphism f_{j} as sought. ♢

Finally, let us show that (12.2.31) is essentially surjective. Indeed, let \underline{M} be a coherent log structure on X; let us pick a covering family 𝒰 := (U_{λ} → X | λ ∈ Λ) and finite charts β_{λ} : P_{λ,U_{λ}} → \underline{M}_{|U_{λ}} for every λ ∈ Λ. As in the foregoing, we may assume that each U_{λ} is affine, and Λ is a finite set, in which case, according to claim 12.2.29(ii) we may find i ∈ I and a covering family (U_{i,λ} → X_{i} | λ ∈ Λ) with isomorphisms of X-schemes U_{i,λ} ×_{X_{i}} X →^{∼} U_{λ} for every λ ∈ Λ; for every j ∈ I/i and every λ ∈ Λ, let us set U_{j,λ} := U_{i,λ}×_{X_{i}} X_{j}. The composition α_{λ} : P_{λ,U_{λ}} → 𝒪_{U_{λ}} of β_{λ} and the structure map of \underline{M}_{|U_{λ}} is determined by a morphism of monoids

P_{λ} → Γ(U_{λ}, 𝒪_{U_{λ}}) = colim_{i∈I/i} Γ(U_{j,λ}, 𝒪_{U_{j,λ}}).

Then, as usual, lemma 6.1.7(ii) implies that there exists j ∈ I/i such that α_{λ} descends to a pre-log structure P_{λ,U_{j,λ}} → 𝒪_{U_{j,λ}} on U_{j,λ}, whose associated log structure we denote by \underline{M}_{j,λ}. For every λ, μ ∈ Λ, let U_{j,λμ} := U_{j,λ} ×_{X_{j}} U_{j,μ}; by construction we have isomorphisms

(12.2.34) (1_{U_{j,λμ}} ×_{X_{j}} π_{j})^{∗}\underline{M}_{j,λ|U_{j,λμ}} →^{∼} (1_{U_{j,μλ}} ×_{X_{j}} π_{j})^{∗}\underline{M}_{j,μ|U_{j,μλ}}  for every λ, μ ∈ Λ.

By applying claim 12.2.33 to the cofiltered system (U_{j,λμ} ×_{X_{j}} X_{j′} | j′ ∈ I/j), we can then obtain – after replacing j by a larger index – isomorphisms ω_{λμ} : \underline{M}_{j,λ|U_{j,λμ}} →^{∼} M_{j,μ|U_{j,μλ}} for every λ, μ ∈ Λ, whose pull-back to U_{j,λμ} ×_{X_{j}} X are the isomorphisms (12.2.34). Lastly, in view of claim 12.2.32, we may achieve – after further replacement of j by a larger index – that the system 𝒟 := (\underline{M}_{j,λ}, ω_{λμ} | λ, μ ∈ Λ) is a descent datum for the fibration F of (12.2.1), whose pull-back to X is isomorphic to the natural descent datum for \underline{M}, associated to the covering family 𝒰 . Then 𝒟 glues to a log structure \underline{M}_{j}, such that π_{j}^{∗}\underline{M}_{j} ≃ \underline{M}. □

Corollary 12.2.35. In the situation of (10.1.8), suppose that X_{λ} is quasi-compact for every λ ∈ Ob(Λ), and that we have a morphism of log schemes with coherent log structures

(g, log g) : (Y, \underline{N}) → (X, \underline{M}).

Then there exist λ ∈ Λ, and a morphism of log schemes with coherent log structures

(g_{λ}, log g_{λ}) : (Y_{λ}, \underline{N}_{λ}) → (X_{λ}, \underline{M}_{λ})  such that  log g = ψ_{λ}^{∗} log g_{λ}.

Proof. The assertion is an immediate consequence of proposition 12.2.30. □

Corollary 12.2.36. In the situation of (10.1.8), let 0 ∈ Ob(Λ) be an index such that \underline{N}_{0} and \underline{M}_{0} are coherent log structures on Y_{0}, and respectively X_{0}, and (g_{0}, log g_{0}) : (Y_{0}, \underline{N}_{0}) → (X_{0}, \underline{M}_{0}) a morphism of log schemes. Suppose also that X_{λ} is quasi-compact (as well as quasi-separated) for every λ ∈ Ob(Λ); then we have :

(i) If the morphism of log schemes

(g, ψ_{0}^{∗} log g_{0}) : Y ×_{Y_{0}} (Y_{0}, \underline{N}_{0}) → X ×_{X_{0}} (X_{0}, \underline{M}_{0})

admits a chart (ω, ω′, ϑ : P → Q), there exists a morphism u : λ → 0 in Λ such that the morphism of log schemes

(g_{λ}, ψ_{u}^{∗} log g_{0}) : Y_{λ} ×_{Y_{0}} (Y_{0}, \underline{N}_{0}) → X_{λ} ×_{X_{0}} (X_{0}, \underline{M}_{0})

admits a chart (ω_{λ}, ω′_{λ}, ϑ). (ii) If (g, ψ_{0}^{∗} log g_{0}) is a log flat (resp. saturated) morphism of fine log schemes, there exists a morphism u : λ → 0 in Λ such that (g_{λ}, ψ_{u}^{∗} log g_{0}) is a log flat (resp. saturated) morphism of fine log schemes.

Proof. (i): Under the current assumptions, X and Y are quasi-compact. In view of lemma 12.2.26, we may then find a morphism v : μ → 0, such that ω and ω′ descend to charts ω_{v} : P_{X_{μ}} → φ_{v}^{∗}\underline{M}_{0} and ω′_{v} : Q_{Y_{μ}} → ψ_{v}^{∗}\underline{N}_{0}. We deduce two morphisms of pre-log structures P_{Y_{μ}} → ψ_{v}^{∗}\underline{N}_{0}, namely

β_{1} := ψ_{v}^{∗}(log g_{0}) ∘ g_{μ}^{∗}ω_{v} and β_{2} := ω′_{v} ∘ ϑ_{Y_{μ}}

and by construction we have ψ_{μ}^{∗}β_{1}^{log} = ψ_{μ}^{∗}β_{2}^{log}. By proposition 12.2.30 it follows that there exists w : λ → μ such that ψ_{w}^{∗}β_{1}^{log} = ψ_{w}^{∗}β_{2}^{log}. The latter means that (φ_{w}^{∗}ω_{v}, ψ_{w}^{∗}ω′_{v}, ϑ) is a chart for the morphism of log schemes (g_{λ}, ψ_{v∘w}^{∗} log g_{0}) : (Y_{λ}, ψ_{v∘w}^{∗}\underline{N}_{0}) → (X_{λ}, φ_{v∘w}^{∗}\underline{M}_{0}), i.e. the claim holds with u := v ∘ w.

(ii): Suppose therefore that (g, ψ_{0}^{∗} log g_{0}) is a log flat (resp.saturated) morphism, and let 𝒰 := (U_{i} → X | i ∈ I) be a covering family for X_{τ}, such that U_{i} ×_{X_{0}} (X_{0}, \underline{M}_{0}) admits a finite (resp. fine) chart, and U_{i} is affine for every i ∈ I. Since X is quasi-compact, we may assume that I is a finite set, and then there exists λ ∈ Λ such that 𝒰 descends to a covering family 𝒰_{λ} := (U_{λ,i} → X_{λ} | i ∈ I) for X_{λ,τ} (claim 12.2.29(ii)). After replacing Λ by Λ/λ, we may assume that λ = 0, in which case we set U_{λ,i} := U_{0,i} ×_{X_{0}} X_{λ} for every object λ of Λ, and every i ∈ I. Clearly, it suffices to show that there exists u : λ → 0 such that U_{λ,i} ×_{X_{λ}} (g_{λ}, ψ_{u}^{∗} log g_{0}) is flat (resp. saturated) for every i ∈ I. Set Y′_{λ,i} := Y_{λ} ×_{X_{λ}} U_{λ,i} for every λ ∈ Λ; we may then replace the cofiltered system \underline{X} and \underline{Y}, by respectively (U_{λ,i} | λ ∈ Λ) and (Y′_{λ,i} | λ ∈ Λ), which allows to assume from start, that X ×_{X_{0}} (X_{0}, \underline{M}_{0}) admits a finite (resp. fine) chart. In this case, lemma 12.2.26allows to further reduce to the case where \underline{M}_{0} admits a finite (resp. fine) chart.

Then, by theorem 12.1.37(iii), we may find a covering family 𝒱 := (V_{j} → Y | j ∈ J) for Y_{τ}, consisting of finitely many affine schemes V_{j}, and for every j ∈ J, a flat (resp. saturated) and fine chart for the induced morphism V_{j} ×_{Y_{0}} (Y_{0}, \underline{N}_{0}) → X ×_{X_{0}} (X_{0}, \underline{M}_{0}). As in the foregoing, after replacing Λ by some category Λ/λ, we may assume that 𝒱 descends to a covering family 𝒱_{0} := (V_{0,j} → Y_{0} | j ∈ J) for Y_{0,τ}, in which case we set V_{λ,j} := V_{0,j} ×_{Y_{0}} Y_{λ} for every λ ∈ Λ. Clearly, it suffices to show that there exists λ ∈ Λ such that the induced morphism V_{λ,j} ×_{Y_{0}} (Y_{0}, \underline{N}_{0}) → X_{λ} ×_{X_{0}} (X_{0}, \underline{M}_{0}) is log flat (resp. saturated). Thus, we may replace \underline{Y} by the cofiltered system (V_{λ,j} | λ ∈ Λ), which allows to assume that (g, ψ_{0}^{∗} log g_{0}) admits a flat (resp. saturated) and fine chart. In this case, the assertion follows from (i). □

Proposition 12.2.37. The inclusion functors :

qf.log → qcoh.log qfs.log → qf.log admit right adjoints :

qcoh.log → qf.log : (X, \underline{M}) ↦ (X, \underline{M})^{qf} qf.log → qfs.log : (X, \underline{M}) ↦ (X, \underline{M})^{qfs}.

Proof. Let (X, \underline{M}) be a scheme with quasi-coherent (resp. quasi-fine) log structure. We need to construct a morphism of log schemes

φ : (X, \underline{M})^{qf} → (X, \underline{M}) (resp. φ : (X, \underline{M})^{qfs} → (X, \underline{M}))

such that (X, \underline{M})^{qf} (resp. (X, \underline{M})^{qfs}) is a quasi-fine (resp. qfs) log scheme, and the following holds. Every morphism of log schemes ψ : (Y, \underline{N}) → (X, \underline{M}) with (Y, \underline{N}) quasi-fine (resp. qfs), factors uniquely through φ. To this aim, suppose first that \underline{M} admits a chart (resp. a quasi-fine chart) α : P_{X} → \underline{M}. By lemma 12.2.18(i), α determines an isomorphism

(X, \underline{M}) →^{∼} Spec(ℤ, P) ×_{Spec ℤ[P]} X.

Since \underline{N} is integral (resp. and saturated), the morphism (Y, \underline{N}) → Spec(ℤ, P) induced by ψ factors uniquely through Spec(ℤ, P^{int}) (resp. Spec(ℤ, P^{sat})) (lemma 12.2.18(ii)). Taking into account lemma 12.1.18(iii), it follows easily that we may take

(X, \underline{M})^{qf} := Spec(ℤ, P^{int}) ×_{ℤ[P]} X (X, \underline{M})^{qfs} := Spec(ℤ, P^{sat}) ×_{ℤ[P]} X

Next, notice that the universal property of (X′, \underline{M}′) := (X, \underline{M})^{qf} (resp. of (X′, \underline{M}′) := (X, \underline{M})^{qfs}) is local on X_{τ} : namely, suppose that (X′, \underline{M}′) has already been found, and let U → X be an object of X_{τ}, with a morphism (Y, \underline{N}) → (U, \underline{M}_{|U}) from a quasi-fine (resp. qfs) log scheme; there follows a unique morphism

(Y, \underline{N}) → (U, \underline{M}_{|U}) ×_{(X,\underline{M})} (X′, \underline{M}′) →^{∼} U ×_{X} (X′, \underline{M}′)

(notation of (12.2.6)). Thus (U, \underline{M}_{|U})^{qf} ≃ U ×_{X} (X, \underline{M})^{qf} (resp. (U, \underline{M}_{|U})^{qfs} ≃ U ×_{X} (X, \underline{M})^{qfs},) since the latter is a quasi-fine (resp. qfs) log scheme. Therefore, for a general quasi-coherent (resp. quasi-fine) log structure \underline{M}, choose a covering family (U_{λ} → X | λ ∈ Λ) such that (U_{λ}, \underline{M}_{|U_{λ}}) admits a chart for every λ ∈ Λ; it follows that the family 𝒰 := ((U_{λ}, \underline{M}_{|U_{λ}})^{qf} | λ ∈ Λ), together with the natural isomorphisms :

U_{μ} ×_{X} (U_{λ}, \underline{M}_{|U_{λ}})^{qf} ≃ U_{λ} ×_{X} (U_{μ}, \underline{M}_{|U_{μ}})^{qf} for every λ, μ ∈ Λ

is a descent datum for the fibred category over X_{τ}, whose fibre over any object U → X is the category of affine U-schemes endowed with a log structure (resp. likewise for the family 𝒰 := ((U_{λ}, \underline{M}_{|U_{λ}})^{qfs} | λ ∈ Λ)). Using faithfully flat descent ([82, Exp.VIII, Th.2.1]), one sees that 𝒰 comes from a quasi-fine (resp. qfs) log scheme which enjoys the sought universal property. □

Remark 12.2.38. (i) By inspecting the proof of proposition 12.2.37, we see that the quasifine log scheme associated to a coherent log scheme, is actually fine, and the qfs log scheme associated to a fine log scheme, is a fs log scheme (one applies corollary 6.4.1(i)). Hence we obtain functors

coh.log → f.log : (X, \underline{M}) ↦ (X, \underline{M})^{f} f.log → fs.log : (X, \underline{M}) ↦ (X, \underline{M})^{fs}

which are right adjoint to the inclusion functors f.log → coh.log and fs.log → f.log.

(ii) Notice as well that, for every log scheme (X, \underline{M}) with quasi-coherent (resp. fine) log structure, the morphism of schemes underlying the counit of adjunction (X, \underline{M})^{qf} → (X, \underline{M}) (resp. (X, \underline{M})^{fs} → (X, \underline{M})) is a closed immersion (resp. is finite).

(iii) Furthermore, let f : Y → X be any morphism of schemes; the proof of proposition 12.2.37also yields a natural isomorphism of (Y, f^{∗}\underline{M})-schemes :

(Y, f^{∗}\underline{M})^{qf} →^{∼} Y ×_{X} (X, \underline{M})^{qf} (resp. (Y, f^{∗}\underline{M})^{qfs} →^{∼} Y ×_{X} (X, \underline{M})^{qfs}).

(iv) Let (X, \underline{M}) be a quasi-fine log scheme, and suppose that X is a normal, irreducible scheme, and (X, \underline{M})_{tr} is a dense subset of X. Denote by X^{qfs} the scheme underlying (X, \underline{M})^{qfs}; then we claim that the projection X^{qfs} → X (underlying the counit of adjunction) admits a natural section :

σ_{X} : X → X^{qfs}.

The naturality means that if f : (X, \underline{M}) → (Y, \underline{N}) is any morphism of quasi-fine log schemes, where Y is also normal and irreducible, and (Y, \underline{N})_{tr} is dense in Y, then the induced diagram of schemes :

(12.2.39)  \xymatrix{X \ar[r]^{σ_{X}} \ar[d]_{f} & X^{qfs} \ar[d]^{f^{qfs}} \\ Y \ar[r]^{σ_{Y}} & Y^{qfs}}

commutes, and therefore it is cartesian, by virtue of (iii). Indeed, suppose first that X is affine, say X = Spec A for some normal domain A, and \underline{M} admits an integral chart, given by a morphism β : P → A, for some integral monoid P; we have to exhibit a ring homomorphism P^{sat} ⊗_{P} A → A, whose composition with the natural map A → P^{sat} ⊗_{P} A is the identity of A. The latter is the same as the datum of a morphism of monoids P^{sat} → A whose restriction to P agrees with β. However, since the trivial locus of \underline{M} is dense in X, the image of P in A does not contain 0, hence β extends to a group homomorphism β^{gp} : P^{gp} → Frac(A)^{×}; since A is integrally closed in Frac(A), we have β^{gp}(P^{sat}) ⊂ A, as required. Next, suppose that U → X is an object of X_{τ}, with U also affine and irreducible; then U is normal and the trivial locus of (U, \underline{M}_{|U}) is dense in U. Thus, the foregoing applies to (U, \underline{M}_{|U}) as well, and by inspecting the constructions we deduce a natural identification :

σ_{U} = 1_{U} ×_{X} σ_{X}.

Lastly, for a general (X, \underline{M}), we can find a covering family (U_{λ} → X | λ → Λ) in X_{τ}, such that U_{λ} is affine, and (U_{λ}, \underline{M}_{|U_{λ}}) admits an integral chart for every λ ∈ Λ; proceeding as above, we obtain a system of morphisms (σ_{λ} : U_{λ} → U^{qfs} | λ ∈ Λ), as well as natural identifications :

1_{U_{μ}} ×_{X} σ_{λ} = 1_{U_{λ}} ×_{X} σ_{μ} for every λ, μ ∈ Λ.

In other words, we have defined a descent datum for the category fibred over X_{τ}, whose fibre over any object U → X is the category of all morphisms of schemes U → U^{qfs}. By invoking faithfully flat descent ([82, Exp.VIII, Th.2.1]), we see that this descent datum yields a morphism σ_{X} : X → X^{qfs} such that 1_{U_{λ}} ×_{X} σ_{X} = σ_{λ} for every λ ∈ Λ. The verification that σ_{X} is a section of the projection X^{qfs} → X, and that (12.2.39) commutes, can be carried out locally on X_{τ}, in which case we can assume that \underline{M} admits a chart as above, and one can check explicitly these assertions, by inspecting the constructions.

12.3. Logarithmic differentials and smooth morphisms. In this section we introduce the logarithmic version of the usual sheaves of relative differentials, and we study some special classes of morphisms of log schemes.

Definition 12.3.1. Let (X, \underline{M} ⟶^{α} 𝒪_{X}) and (Y, \underline{N} ⟶^{β} 𝒪_{Y}) be two schemes with pre-log structures, and f : (X, \underline{M}) → (Y, \underline{N}) a morphism of schemes with pre-log structures. Let also ℱ be an 𝒪_{X}-module. An f-linear derivation of \underline{M} with values in ℱ is a pair (∂, log ∂) consisting of maps of sheaves :

∂ : 𝒪_{X} → ℱ log ∂ : log \underline{M} → ℱ

such that :

• ∂ is a derivation (in the usual sense).

• log ∂ is a morphism of sheaves of (additive) monoids on X_{τ}.

• ∂ ∘ f^{♮} = 0 and log ∂ ∘ log f = 0.
• ∂ ∘ α(m) = α(m) · log ∂(m) for every object U of X_{τ}, and every m ∈ \underline{M}(U).

The set of all f-linear derivations with values in f shall be denoted by :

Der_{(Y,\underline{N})}((X, \underline{M}), ℱ).

The f-linear derivations shall also be called (Y, \underline{N})-linear derivations, when there is no danger of ambiguity.

12.3.2. The set Der_{(Y,\underline{N})}((X, \underline{M}), ℱ) is clearly functorial in ℱ, and moreover, for any object U of X_{τ}, any s ∈ 𝒪_{X}(U), and any f-linear derivation (∂, log ∂), the restriction (s·∂_{|U}, s·log ∂_{|U}) is an element of Der_{(Y,\underline{N})}((U, \underline{M}_{|U}), ℱ), hence the rule U ↦ Der_{(Y,\underline{N})}((U, \underline{M}_{|U}), ℱ) defines an 𝒪_{X}-module :

𝒟er_{(Y,\underline{N})}((X, \underline{M}), ℱ).

In case \underline{M} = 𝒪_{X}^{×} and \underline{N} = 𝒪_{Y}^{×} are the trivial log structures, the f-linear derivations of \underline{M} are the same as the usual f-linear derivations, i.e. the natural map

(12.3.3)  𝒟er_{(Y,𝒪_{Y}^{×})} ((X, 𝒪_{X}^{×}), ℱ) → 𝒟er_{Y} (X, ℱ)

is an isomorphism. In the category of usual schemes, the functor of derivations is represented by the module of relative differentials. This construction extends to the category of schemes with pre-log structures. Namely, let us make the following :

Definition 12.3.4. Let (X, \underline{M} ⟶^{α} 𝒪_{X}) and (Y, \underline{N} ⟶^{β} 𝒪_{Y}) be two schemes with pre-log structures, and f : (X, \underline{M}) → (Y, \underline{N}) a morphism of schemes with pre-log structures. The sheaf of logarithmic differentials of f is the 𝒪_{X}-module :

Ω_{X/Y}^{1} (log \underline{M}/\underline{N}) := (Ω_{X/Y}^{1} ⊕ (𝒪_{X} ⊗_{ℤ} log \underline{M}^{gp}))/R

where R is the 𝒪_{X}-submodule generated locally on X_{τ} by local sections of the form : • (dα(a), −α(a) ⊗ log a) with a ∈ \underline{M}(U)

• (0, 1 ⊗ log a) with a ∈ Im((f^{−1}\underline{N})(U) → \underline{M}(U))

where U ranges over all the objects of X_{τ} (here we use the notation of (6.1)). For arbitrary a ∈ \underline{M}(U), the class of (0, 1 ⊗ log a) in Ω_{X/Y}^{1} (log \underline{M}/\underline{N}) shall be denoted by d log a.

12.3.5. It is easy to verify that Ω_{X/Y}^{1} (log \underline{M}/\underline{N}) represents the functor

ℱ ↦ Der_{(Y,\underline{N})}((X, \underline{M}), ℱ)

on 𝒪_{X}-modules. Consequently, (12.3.3) translates as a natural isomorphism of 𝒪_{X}-modules :

(12.3.6)  Ω_{X/Y}^{1} →^{∼} Ω_{X/Y}^{1} (log 𝒪_{X}^{×}/𝒪_{Y}^{×}).

Furthermore, let us fix a scheme with pre-log structure (S, \underline{N}), and define the category :

pre-log/(S, \underline{N})

as in (1.1.1). Also, let Mod.pre-log/(S, \underline{N}) be the category whose objects are all the pairs ((X, \underline{M}), ℱ), where (X, \underline{M}) is a (S, \underline{N})-scheme, and ℱ is an 𝒪_{X}-module. The morphisms

((X, \underline{M}), ℱ) → ((Y, \underline{N}), 𝒢)

in Mod.pre-log/(S, \underline{N}) are the pairs (f, φ) consisting of a morphism f : (X, \underline{M}) → (Y, \underline{N}) of (S, \underline{N})-schemes, and a morphism φ : f^{∗}𝒢 → ℱ of 𝒪_{X}-modules. With this notation, we claim that the rule :

(12.3.7)  (X, \underline{M}) ↦ ((X, \underline{M}), Ω_{X/S}^{1}(log \underline{M}/\underline{N}))

defines a functor pre-log/(S, \underline{N}) → Mod.pre-log/(S, \underline{N}). Indeed, slightly more generally, consider a commutative diagram of schemes with pre-log structures :

(12.3.8)  \xymatrix{(X, \underline{M}) \ar[r]^{g} \ar[d]_{f} & (X′, \underline{M}′) \ar[d]^{f′} \\ (S, \underline{N}) \ar[r]^{h} & (S′, \underline{N}′).}

An 𝒪_{X}-linear map :

(12.3.9)  g^{∗}Ω_{X′/S′}^{1}(log \underline{M}′/\underline{N}′) ⟶^{dg} Ω_{X/S}(log \underline{M}/\underline{N})

is the same as a natural transformation of functors :

(12.3.10)  𝒟er_{(S,\underline{N})}((X, \underline{M}), ℱ) → 𝒟er_{(S′,\underline{N}′)}((X′, \underline{M}′), g_{∗}ℱ)

on all 𝒪_{X}-modules ℱ. The latter can be defined as follows. Let (∂, log ∂) be an (S, \underline{N})-linear derivation of \underline{M} with values in ℱ; then we deduce morphisms :

∂′ : 𝒪_{X′} ⟶^{g^{♮}} g_{∗}𝒪_{X} ⟶^{g_{∗}∂} g_{∗}ℱ log ∂′ : \underline{M}′ ⟶^{log g} g_{∗}\underline{M} ⟶^{g_{∗} log ∂} g_{∗}ℱ

and it is easily seen that (∂′, log ∂′) is a (S′, \underline{N}′)-linear derivation of \underline{M}′ with values in g_{∗}ℱ.

12.3.11. Consider two morphisms (X, \underline{M}) ⟶^{f} (Y, \underline{N}) ⟶^{g} (Z, \underline{P}) of schemes with pre-log structures. A direct inspection of the definitions shows that :

𝒟er_{(Y,\underline{N})}((X, \underline{M}), ℱ) = Ker(𝒟er_{(Z,\underline{P})}((X, \underline{M}), ℱ) → 𝒟er_{(Z,\underline{P})}((Y, \underline{N}), f_{∗}ℱ))

for every 𝒪_{X}-module ℱ, whence a right exact sequence of 𝒪_{X}-modules :

(12.3.12)  f^{∗}Ω_{Y/Z}^{1}(log \underline{N}/\underline{P}) ⟶^{df} Ω_{X/Z}^{1}(log \underline{M}/\underline{P}) → Ω_{X/Y}^{1} (log \underline{M}/\underline{N}) → 0

extending the standard right exact sequence for the usual sheaves of relative differentials.

Proposition 12.3.13. Suppose that the diagram (12.3.8) is cartesian. Then the map (12.3.9) is an isomorphism.

Proof. If (12.3.8) is cartesian, X is the scheme X′ ×_{S′} S, and \underline{M} is the push-out of the diagram:

f^{−1}\underline{N} ← (f′ ∘ g)^{−1}\underline{N}′ ⟶^{φ} g^{∗}\underline{M}′.

Suppose now that log ∂ : \underline{M}′ → g_{∗}ℱ and ∂ : 𝒪_{X′} → g_{∗}\underline{ℱ} define a f′-linear derivation of \underline{M}′. By adjunction, we deduce morphisms α : g^{−1}𝒪_{X′} → ℱ and β : g^{−1}\underline{M}′ → ℱ. By construction, we have β ∘ φ = 0, hence β extends uniquely to a morphism log ∂′ : \underline{M} → ℱ such that log ∂′ ∘ log f = 0. Likewise, α extends by linearity to a unique f-linear derivation ∂ : 𝒪_{X} → ℱ. One checks easily that (∂′, log ∂′) is a f-linear derivation of \underline{M}, and that every f-linear derivation of \underline{M} with values in ℱ is obtained in this fashion. □

12.3.14. The functor (12.3.7) admits a left adjoint. Indeed, let ((X, \underline{M} ⟶^{α} 𝒪_{X}), ℱ) be any object of Mod.pre-log/(S, \underline{N}); we define an (S, \underline{N})-scheme (X ⊕ ℱ , \underline{M} ⊕ ℱ) as follows. X ⊕ ℱ is the spectrum of the 𝒪_{X}-algebra 𝒪_{X} ⊕ ℱ, whose multiplication law is given by the rule :

(s, t) · (s′, t′) := (ss′, st′ + s′t) for every local section s of 𝒪_{X} and t of ℱ .

Likewise, we define a composition law on the sheaf \underline{M} ⊕ ℱ, by the rule :

(m, t) · (m′, t′) := (mm′, α(m) · t′ + α(m′) · t) for every local section m of \underline{M} and t of ℱ .

Then \underline{M} ⊕ ℱ is a sheaf of monoids, and α extends to a pre-log structure α ⊕ 1_{ℱ} : \underline{M} ⊕ ℱ → 𝒪_{X} ⊕ ℱ. The natural map 𝒪_{X} → 𝒪_{X} ⊕ ℱ is a morphism of algebras, whence a natural map of schemes π : X ⊕ ℱ → X, which extends to a morphism of schemes with pre-log structures (X ⊕ ℱ , \underline{M} ⊕ ℱ) → (X, \underline{M}), by letting log π : π^{∗}\underline{M} → \underline{M} ⊕ ℱ be the map induced by the natural monomorphism (notice that π^{∗} induces an equivalence of sites X_{τ} →^{∼} (X ⊕ ℱ)_{τ}).

Now, let (Y, \underline{P}) be any (S, \underline{N})-scheme, and :

φ : ℱ := ((X, \underline{M}), ℱ) → Ω := ((Y, \underline{P}), Ω_{Y/S}^{1}(log \underline{P} /\underline{N}))

a morphism in Mod.pre-log/(S, \underline{N}). By definition, φ consists of a morphism f : (X, \underline{M}) → (Y, \underline{P}) and an 𝒪_{X}-linear map f^{∗}Ω_{Y/S}^{1}(log \underline{P} /\underline{N}) → ℱ, which is the same as a (S, \underline{N})-linear derivation :

∂ : 𝒪_{Y} → f_{∗}ℱ log ∂ : \underline{P} → f_{∗}ℱ .

In turns, the latter yields a morphism of (S, \underline{N})-schemes :

(12.3.15)  (Y ⊕ f_{∗}ℱ , \underline{P} ⊕ f_{∗}ℱ) → (Y, \underline{P})

determined by the map of algebras :

𝒪_{Y} → 𝒪_{Y} ⊕ f_{∗}ℱ s ↦ (s, ∂s) for every local section s of 𝒪_{Y}

and the map of monoids :

\underline{P} ↦ \underline{P} ⊕ f_{∗}ℱ p ↦ (p, log ∂p) for every local section p of \underline{P}.

Finally, we compose (12.3.15) with the natural morphism

(X ⊕ ℱ , \underline{M} ⊕ ℱ) → (Y ⊕ f_{∗}ℱ , \underline{P} ⊕ f_{∗}ℱ)

that extends f, to obtain a morphism g_{φ} : (X ⊕ ℱ , \underline{M} ⊕ ℱ) → (Y, \underline{P}). We leave to the reader the verification that the rule φ ↦ g_{φ} establishes a natural bijection :

Hom_{Mod.pre-log/(S,\underline{N})}(ℱ , Ω) →^{∼} Hom_{pre-log/(S,\underline{N})}((X ⊕ ℱ , \underline{M} ⊕ ℱ), (Y, \underline{P})).

12.3.16. In the situation of definition 12.3.4, let (∂, log ∂) be an f-linear derivation of \underline{M} with values in an 𝒪_{X}-module ℱ. Consider the map :

∂′ : 𝒪_{X}^{×} → ℱ : u ↦ u^{−1} · ∂u for all local sections u of 𝒪_{X}^{×} .

By definition, ∂′∘α : α^{−1}\underline{M} → ℱ agrees with the restriction of log ∂; in view of the cocartesian diagram (12.1.7), we deduce that log ∂ extends uniquely to an f-linear derivation log ∂^{log} of \underline{M}^{log}. Let f^{log} : (X, \underline{M}^{log}) → (Y, \underline{N}^{log}) be the map deduced from f; a similar direct verification shows that log ∂^{log} is a f^{log}-linear derivation. There follow natural identifications :

(12.3.17)  Ω_{X/Y}^{1} (log(\underline{M}/\underline{N})) = Ω_{X/Y}^{1} (log(\underline{M}^{log}/\underline{N})) = Ω_{X/Y}^{1} (log(\underline{M}^{log}/\underline{N}^{log})).

Moreover, if \underline{M} and \underline{N} are log structures, the natural map :

(12.3.18)  𝒪_{X} ⊗_{ℤ} log \underline{M}^{gp} → Ω_{X/Y}^{1} (log(\underline{M}/\underline{N}))  a ⊗ b ↦ a · d log(b)

is an epimorphism. Indeed, we have da = d(a + 1) for every local section a ∈ 𝒪_{X}(U) (for any étale X-scheme U), and locally on X_{τ}, either a or 1 + a is invertible in 𝒪_{X} (this holds certainly on the stalks, hence on appropriate small neighborhoods U′ → U); hence da lies in the image of (12.3.18).

Example 12.3.19. Let R be a ring, φ : N → M be any map of monoids, and set :

S := Spec R  S[M] := Spec R[M]  S[N] := Spec R[N].

Also, let f : Spec(R, M) → Spec(R, N) be the morphism of log schemes induced by φ (see (12.2.13)). With this notation, we claim that (12.3.18) induces an isomorphism :

𝒪_{S[M]} ⊗_{ℤ} Coker φ^{gp} →^{∼} Ω_{S[M]/S[N]}^{1}(log M_{S[M]}^{log}/N_{S[N]}^{log})

To see this, we may use (12.3.12) to reduce to the case where N = {1}. Next, notice that the functor

Mnd^{o} → pre-log/(S, 𝒪_{S}^{×}) : M ↦ Spec(R, M)

commutes with limits, and the same holds for the functor (12.3.7), since the latter is a right adjoint. Hence, we may assume that M is finitely generated; then lemma 6.1.7(i) further reduces to the case where M = ℕ^{⊕n} for some integer n ≥ 0, and even to the case where n = 1. Set X := S[ℕ]; it is easy to see that a S-linear derivation of ℕ_{X}^{log} with values in an 𝒪_{X}-module ℱ, is completely determined by a map of additive monoids ℕ → Γ(X, ℱ), and the latter is the same as an 𝒪_{X}-linear map 𝒪_{X} → ℱ, whence the contention.

Lemma 12.3.20. Let f : (X, \underline{M}) → (Y, \underline{N}) be a morphism of schemes with quasi-coherent log structures. Then :

(i) The 𝒪_{X}-module Ω_{X/Y}^{1} (log \underline{M}/\underline{N}) is quasi-coherent.

(ii) If \underline{M} is coherent, X is noetherian, and f : X → Y is locally of finite type, then

Ω_{X/Y}^{1} (log \underline{M}/\underline{N}) is a coherent 𝒪_{X}-module. (iii) If both \underline{M} and \underline{N} are coherent, and f : X → Y is locally of finite presentation, then

Ω_{X/Y}^{1} (log \underline{M}/\underline{N}) is a quasi-coherent 𝒪_{X}-module of finite presentation.

Proof. Applying the right exact sequence (12.3.12) to the sequence (X, \underline{M}) ⟶^{f} (Y, \underline{N}) → (Y, 𝒪_{Y}^{×}), we may easily reduce to the case where \underline{N} = 𝒪_{Y}^{×} is the trivial log structure on Y. In this case, \underline{N} admits the chart given by the unique map of monoids : \{1\} → Γ(Y, 𝒪_{Y}), and f admits the chart \{1\} → P, whenever \underline{M} admits a chart P → Γ(X, 𝒪_{X}).

Hence, everything follows from the following assertion, whose proof shall be left to the reader. Suppose that φ : A → B is a ring homomorphism, \underline{M} (resp. \underline{N}) is the constant log structure on X := Spec B (resp. on Y := Spec A) associated to a map of monoids α : P → B (resp. β : Q → A), and f : (X, \underline{M}) → (Y, \underline{N}) is defined by φ admits a chart ϑ : Q → P. Then Ω_{X/Y}^{1} (log \underline{M}/\underline{N}) is the quasi-coherent 𝒪_{X}-module L^{∼}, associated to the B-module L := (Ω_{B/A}^{1} ⊕ (B ⊗_{ℤ} P^{gp}))/R, where R is the submodule generated by the elements of the form (0, 1 ⊗ log ϑ(q)) for all q ∈ Q, and those of the form (dα(m), −α(m) ⊗ log m), for all m ∈ M. □

12.3.21. Let us fix a log scheme (Y, \underline{N}). To any pair of (Y, \underline{N})-schemes X := (X, \underline{M}), X′ := (X′, \underline{M}′), we attach a contravariant functor

ℋ_{Y} (X′, X) : (X′_{τ})^{o} → Set

by assigning, to every object U of X′_{τ}, the set of all morphisms (U, \underline{M}′_{|U}) → (X, \underline{M}) of (Y, \underline{N})- schemes. It is easily seen that ℋ_{Y} (X′, X) is a sheaf on X′_{τ}. Any morphism φ : (X′′, \underline{M}′′) → (X′, \underline{M}′) (resp. ψ : (X, \underline{M}) → (X′′, \underline{M})) of (Y, \underline{N})-schemes induces a map of sheaves :

φ^{∗} : φ^{∗}ℋ_{Y} (X′, X) → ℋ_{Y} (X′′, X) (resp. ψ_{∗} : ℋ_{Y} (X′, X) → ℋ_{Y} (X′, X′′))

in the obvious way.

Definition 12.3.22. With the notation of (12.3.21) :

(i) We say that a morphism i : (T′, \underline{L′}) → (T, \underline{L}) of log schemes is a closed immersion (resp. an exact closed immersion) if the underlying morphism of schemes T′ → T is a closed immersion, and log i : i^{∗}\underline{L} → \underline{L′} is an epimorphism (resp. an isomorphism) of T′_{τ}-monoids.

(ii) We say that a morphism i : (T′, \underline{L′}) → (T, \underline{L}) of log schemes is an exact nilpotent immersion if i is an exact closed immersion, and the ideal ℐ := Ker(𝒪_{T} → i_{∗}𝒪_{T′}) is nilpotent.

(iii) We say that a morphism f : (X, \underline{M}) → (Y, \underline{N}) of log schemes is formally smooth (resp. formally unramified, resp. formally étale) if, for every exact nilpotent immersion i : T′ → T of fine (Y, \underline{N})-schemes, the induced map of sheaves i^{∗} : i^{∗}ℋ_{Y} (T, X) → ℋ_{Y} (T′, X) is an epimorphism (resp. a monomorphism, resp. an isomorphism).

(iv) We say that a morphism f : (X, \underline{M}) → (Y, \underline{N}) of log schemes is smooth (resp. unramified, resp. étale) if the underlying morphism X → Y is locally of finite presentation, and f is formally smooth (resp. formally unramified, resp. formally étale).

Example 12.3.23. Let (S, \underline{P}) be a log scheme, (f, log f) : (X, \underline{M}) → (Y, \underline{N}) be a morphism of S-schemes, such that Y is a separated S-scheme. The pair (1_{(X,\underline{M})}, (f, log f)) induces a morphism

Γ_{f} : (X, \underline{M}) → (X′, \underline{M}′) := (X, \underline{M}) ×_{S} (Y, \underline{N})

the graph of f. Then it is easily seen that Γ_{f} is a closed immersion of log schemes. Indeed, the morphism of schemes underlying Γ_{f} is a closed immersion ([59, Ch.I, 5.4.3]) and it is easily seen that the morphism log Γ_{f} : Γ_{f}^{∗} \underline{M}′ → \underline{M} is an epimorphism on the underlying sheaves of sets, so it is a fortiori an epimorphism of X_{τ}-monoids.

Proposition 12.3.24. Let f : (X, \underline{M}) → (Y, \underline{N}), g : (Y, \underline{N}) → (Z, \underline{P}), and h : (Y′, \underline{N}′) → (Y, \underline{N}) be morphisms of log schemes. Denote by P either one of the properties : ”formally smooth”, ”formally unramified”, ”formally étale”, ”smooth”, ”unramified”, ”étale”. The following holds :

(i) If f and g enjoy the property P, then the same holds for g ∘ f. (ii) If (f, log f) enjoys the property P, then the same holds for

(f, log f) ×_{(Y,\underline{N})} (Y′, \underline{N}′) : (X, \underline{M}) ×_{(Y,\underline{N})} (Y′, \underline{N}′) → (Y′, \underline{N}′).

(iii) Let (j_{λ} : U_{λ} → X | λ ∈ Λ) be a covering family in X_{τ}; endow U_{λ} with the log structure (\underline{M})_{|U_{λ}} and suppose that f_{λ} := (f ∘ j_{λ}, (log f)_{|U_{λ}}) : (U_{λ}, (\underline{M})_{|U_{λ}}) → (Y, \underline{N}) enjoys the property P, for every λ ∈ Λ. Then f enjoys the property P as well.

(iv) An open immersion of log schemes is étale.

(v) A closed immersion of log schemes is formally unramified.

Proof. (i), (ii) and (iv) are left to the reader.

(iii): To begin with, if each f_{λ} is locally of finite presentation, the same holds for f, by [66, Ch.IV, lemme 17.7.5], hence we may assume that P is either ”formally smooth” or ”formally unramified”. (Clearly, the case where P is ”formally étale” will follow.)

Now, let i : T′ → T be an exact nilpotent immersion of (Y, \underline{N})-schemes, and ξ a τ-point of T′. By inspecting the definitions, it is easily seen that the stalk ℋ_{Y} (T, X)_{ξ} is the union of the images of the stalks ℋ_{Y} (T, U_{λ})_{ξ}, for every λ ∈ Λ, and likewise for ℋ_{Y} (T′, X)_{ξ}. It readily follows that f is formally smooth whenever all the f_{λ} are formally smooth.

Lastly, suppose that all the f_{λ} are formally unramified, and let s_{ξ}, t_{ξ} ∈ ℋ_{Y} (T, X)_{i(ξ)} be two sections whose images in ℋ_{Y} (T′, X)_{ξ} agree; after replacing T by a neighborhood of i(ξ) in T_{τ}, we may assume that s_{ξ}, t_{ξ} are represented by two (Y, \underline{N})-morphisms s, t : T → (X, \underline{M}) such that s ∘ i = t ∘ i. Choose λ ∈ Λ such that U_{λ} is a neighborhood of s ∘ i(ξ) = t ∘ i(ξ); this means that there exist a neighborhood p′ : U′ → T′ of ξ, and a morphism s_{U′} : U′ → U_{λ} lifiting s ∘ i (and thus, also t ∘ i). Then we may find a neighborhood p : U → T of i(ξ) which identifies p′ with p×_{T} 1_{T′} ([66, Ch.IV, Th.18.1.2]), and since j_{λ} isétale, we may furthermore find morphisms s_{U}, t_{U} : U → U_{λ} such that j_{λ} ∘ s_{U} = s ∘ i = t ∘ i = j_{λ} ∘ t_{U}. Set i_{U} := i ×_{T} 1_{U} : U′ → U; by construction, s_{U} and t_{U} yield two sections of i_{U}^{∗}ℋ_{Y} (U, U_{λ})_{ξ}, whose images in ℋ_{Y} (U′, U_{λ})_{ξ} coincide. Since f_{λ} is formally unramified, it follows that – up to replacing U by a neighborhood of i(ξ) in U_{τ} – we must have s_{U} = t_{U}, so s_{ξ} = t_{ξ}, and we conclude that f is formally unramified.

(v): Consider a commutative diagram of log schemes :

\xymatrix{(T′, \underline{L}′) \ar[r]^{h′} \ar[d]_{i} & (X, \underline{M}) \ar[d]^{f} \\ (T, \underline{L}) \ar[r]^{h} & (Y, \underline{N})}

where f is a closed immersion, and i is an exact closed immersion. We are easily reduced to showing that there exists at most a morphism (g, log) : (T, \underline{L}) → (X, \underline{M}) such that f ∘ g = h and h′ = g ∘ i. Since f : X → Y is a closed immersion of schemes, there exists at most one morphism of schemes g : T → X lifting h and extending h′ ([66, Ch.IV, Prop.17.1.3(i)]). Hence we may assume that such a g is given, and we need to check that there exists at most one morphism log g : g^{∗}\underline{M} → \underline{L} whose composition with g^{∗}(log f) : h^{∗}\underline{N} → g^{∗}\underline{M} equals log h. However, by assumption log f is an epimorphism, hence the same holds for g^{∗}(log f) (proposition 1.3.25(iv)), whence the contention. □

Corollary 12.3.25. Let f and g be as proposition 12.3.24. We have :

(i) If g ∘ f is formally unramified, the same holds for f.

(ii) If g ∘ f is formally smooth (resp. formally étale) and g is formally unramified, then f

is formally smooth (resp. formally étale). (iii) Suppose that g is formally étale. Then f is formally smooth (resp. formally unramified,

resp. formally étale) if and only if the same holds for g ∘ f.

Proof. (i): Let Y = ⋃_{λ∈Λ} be an affine open covering of Y, and for each λ ∈ Λ, let f_{λ} : U_{λ} ×_{Y} (X, \underline{M}) → (U_{λ}, \underline{N}_{|U_{λ}}) be the restriction of f; in light of proposition 12.3.24(iii) it suffices to show that each f_{λ} is formally unramified, and on the other hand, the restriction g ∘ f_{λ} : U_{λ} ×_{Y} (X, \underline{M}) → (Z, \underline{P}) of g ∘ f is formally unramified, by proposition 12.3.24(i),(iv). It follows that we may replace f and g respectively by f_{λ} and g_{λ}, which allows to assume that Y is affine, especially separated, so that g is a separated morphism of schemes. In such situation, one may – in view of example 12.3.23 – argue as in the proof of [66, Ch.IV, Prop.17.1.3] : the details shall be left to the reader.

(ii) is a formal consequence of the definitions (cp. the proof of [66, Ch.IV, Prop.17.1.4]), and (iii) follows from (ii) and proposition 12.3.24(i). □

Proposition 12.3.26. Let f : (X, \underline{M}) → (Y, \underline{N}) be a morphism of log schemes, and i an exact closed immersion of (Y, \underline{N})-schemes, defined by an ideal ℐ := Ker(𝒪_{T} → i_{∗}𝒪_{T′}) with ℐ^{2} = 0. For any global section s : T′ → ℋ_{Y} (T′, X), denote by 𝒯_{s} the morphism :

i^{∗}ℋ_{Y} (T, X) ×_{ℋ_{Y} (T′,X)} T′ → T′

deduced from i^{∗} : i^{∗}ℋ_{Y} (T, X) → ℋ_{Y} (T′, X). Let also U ⊂ T′ be the image of 𝒯_{s} (i.e. the subset of all t′ ∈ T′ such that 𝒯_{s,ξ} ≠ ∅ for every τ-point ξ localized at t′), and suppose that U ≠ ∅. We have :

(i) U is an open subset of T′, and 𝒯_{s|U} is a torsor for the abelian sheaf

𝒢 := ℋ om_{𝒪_{T′}} (s^{∗}Ω_{X/Y}^{1} (log(\underline{M}/\underline{N})), ℐ)_{|U}.

(ii) If f is a smooth morphism of log schemes with coherent log structures, we have : (a) The 𝒪_{X}-module Ω_{X/Y}^{1} (log(\underline{M}/\underline{N})) is locally free of finite type.

(b) If T′ is affine, 𝒯_{s} is a trivial 𝒢 -torsor.

Proof. (i): To any τ-point ξ of U, and any two given local sections h and g of 𝒯_{s,ξ}, we assign the f-linear derivation of \underline{M}_{s(ξ)} with values in s_{∗}ℐ_{ξ} given by the rule :

a ↦ h^{∗}(a) − g^{∗}(a)  for every a ∈ 𝒪_{X,s(ξ)}
d log m ↦ log u(m) := u(m) − 1 for every m ∈ \underline{M}_{s(ξ)}

where u(m) is the unique local section of Ker(𝒪_{T,ξ} → i_{∗}𝒪_{T′,ξ}) such that

log(g(m) · u(m)) = log h(m).

We leave to the reader the laborious, but straightforward verification that the above map is well-defined, and yields the sought bijection between 𝒯_{s,ξ} and 𝒢_{ξ}.

(ii.b) is standard : first, since f is smooth, we have U = T′; by lemma 12.3.20(iii), the 𝒪_{X}- module Ω_{X/Y}^{1} (log(\underline{M}/\underline{N})) is quasi-coherent and finitely presented, hence 𝒢 is quasi-coherent; however, the obstruction to gluing local sections of a 𝒢 -torsor lies in H^{1}(T_{τ}, 𝒢) (see (4.9.11)); the latter vanishes whenever T is affine.

(ii.a) We may assume that X is affine, say X = Spec A; then Ω_{X/Y}^{1} (log(\underline{M}/\underline{N})) is the quasicoherent 𝒪_{X}-module arising from a finitely presented A-module Ω (lemma 12.3.20(iii)). Let I be any A-module, and set :

T_{I} := X ⊕ I^{∼} \underline{L}_{I} := \underline{M} ⊕ I^{∼}

(notation of (12.3.14)). Let i : X → T_{I} (resp. π : T_{I} → X) be the natural closed immersion (resp. the natural projection); then (T_{I}, \underline{L}_{I}) is a fine log scheme, and π (resp. i) extends to a morphism of log schemes (π, log π) : (T_{I}, \underline{L}_{I}) → (X, \underline{M}) (resp. (i, log i) : (X, \underline{M}) → (T_{I}, \underline{L}_{I})) with log π : π^{∗}\underline{M} → \underline{L}_{I} (resp. log i : i^{∗}\underline{L}_{I} → \underline{M}) induced by the obvious inclusion (resp. projection) map. Notice that (i, log i) is an exact immersion and (I^{∼})^{2} = 0, hence (i) says that the set 𝒯 (T_{I}) of all morphisms g : (T_{I}, \underline{L}_{I}) → (X, \underline{M}) such that

f ∘ g = f ∘ π g ∘ i = 1_{(X,\underline{M})}

is in bijection with Hom_{A}(Ω, I). Moreover, any map φ : I → J of A-modules induces a morphism i_{φ} : (T_{J}, \underline{L}_{J}) → (T_{I}, \underline{L}_{I}) of log schemes, and if φ is surjective, i_{φ} is an exact nilpotent immersion. Furthermore, we have a commutative diagram of sets :

\xymatrix{𝒯 (T_{I}) \ar[r]^{i_{φ}^{∗}} \ar[d] & 𝒯 (T_{J}) \ar[d] \\ Hom_{A}(Ω, I) \ar[r]^{φ_{∗}} & Hom_{A}(Ω, J)}

whose vertical arrows are bijections, and where i_{φ}^{∗} (resp. φ_{∗}) is given by the rule g ↦ g ∘ i_{φ}, (resp. ψ ↦ φ ∘ ψ). Assertion (ii.b) implies that ι_{φ}^{∗} is surjective when f is smooth and φ is surjective, hence the same holds for φ_{∗}, i.e. Ω is a projective A-module, as stated. □

Corollary 12.3.27. Let (f, log f) : (X, \underline{M}) → (Y, \underline{N}) be a morphism of log schemes. We have:

(i) If \underline{M} and \underline{N} are fine log structures, and f is strict (see definition 12.1.22(ii)), then (f, log f) is smooth (resp. étale) if and only if the underlying morphism of schemes f : X → Y is smooth (resp. étale).

(ii) Suppose that (f, log f) is a smooth (resp. étale) morphism of log schemes on the Zariski sites of X and Y, and either :

(a) \underline{M} and \underline{N} are both integral log structures,

(b) or else \underline{N} is coherent (on Y_{Zar}) and \underline{M} is fine (on X_{Zar}). Then the induced morphism of log schemes on étale sites :

ũ^{∗}(f, log f) := (f, ũ_{X}^{∗} log f) : ũ_{X}^{∗}(X, \underline{M}) → ũ_{Y}^{∗} (Y, \underline{N})

is smooth (resp. étale). (Notation of (12.2.2).) (iii) Suppose that (f, log f) is a morphism of log schemes on the Zariski sites of X and Y,

and \underline{M} is an integral log structure on X_{Zar}. Suppose also that ũ^{∗}(f, log f) is smooth (resp. étale). Then the same holds for (f, log f).

Proof. (i): Suppose first that (f, log f) is smooth (resp. étale). Let i : T′ → T be a nilpotent immersion of affine schemes, defined by an ideal ℐ ⊂ 𝒪_{T} such that ℐ^{2} = 0; let s : T′ → X and t : T → Y be morphisms of schemes such that f ∘ s = t ∘ i. By lemma 12.1.18(i), the log structures \underline{L} := t^{∗}\underline{N} and \underline{L}′ := s^{∗}\underline{M} are fine, and by choosing the obvious maps log s and log t, we deduce a commutative diagram :

(12.3.28)  \xymatrix{(T′, \underline{L′}) \ar[r]^{s} \ar[d]_{i} & (X, \underline{M}) \ar[d]^{f} \\ (T, \underline{L}) \ar[r]^{t} & (Y, \underline{N}).}

Then proposition 12.3.26(ii.b) says that there exists a morphism (resp. a unique morphism) of schemes g : T → X such that g ∘ i = s and f ∘ g = t, i.e. f is smooth (resp. étale).

The converse is easy, and shall be left as an exercise for the reader.

(ii): Suppose first that (f, log f) is smooth, \underline{N} is coherent and \underline{M} is fine. By proposition 12.3.24(iii), the assertion to prove is local on X_{ét}, hence we may assume that f admits a chart, given by a morphism of finitely generated monoids P → P′ and commutative diagrams :

P′_{X_{Zar}} → \underline{M} ω : P_{Y_{Zar}} → \underline{N}

(theorem 12.1.37(i)). Now, consider a commutative diagram of log schemes on étale sites :

(12.3.29)  \xymatrix{(T′, \underline{L′}) \ar[r]^{s} \ar[d]_{i} & ũ_{X}^{∗}(X, \underline{M}) \ar[d]^{ũ^{∗}f} \\ (T, \underline{L}) \ar[r]^{t} & ũ_{Y}^{∗} (Y, \underline{N})}

where i is an exact nilpotent immersion of fine log schemes, defined by an ideal ℐ ⊂ 𝒪_{T} such that ℐ^{2} = 0. Since the assertion to prove is local on T_{ét}, we may assume that T is affine and – in view of theorem 12.1.37(ii) – that t admits a chart, given by a morphism of finitely generated monoids φ : P → Q, and commutative diagrams :

(12.3.30)  \xymatrix{P_{T_{ét}} = t^{∗}P_{Y_{ét}} \ar[r]^{t^{∗}u_{Y}^{∗} ω} \ar[d]_{φ_{T}} & t^{∗}ũ_{Y}^{∗} \underline{N} \ar[d]^{log t} & P \ar[r] \ar[d]_{φ} & Γ(Y, 𝒪_{Y}) \ar[d]^{t^{♮}} \\ Q_{T_{ét}} \ar[r]^{β} & \underline{L} & Q \ar[r]^{ψ} & Γ(T, 𝒪_{T})}

Since i is an exact closed immersion, it follows that the morphism :

Q_{T′_{ét}} ⟶^{i^{∗}β} i^{∗}\underline{L} ⟶^{log i} \underline{L}′

is a chart for \underline{L}′. Especially, (T′, \underline{L}′) is isomorphic to (T′_{ét}, Q_{T′}^{log}), the constant log structure deduced from the morphism i^{♮} ∘ ψ : Q → Γ(T′, 𝒪_{T′}). The latter is also the log scheme ũ_{T′}^{∗}(T_{Zar}, Q_{T′}^{log}). From proposition 12.2.3(ii) we deduce that there exists a unique morphism s_{Zar} : (T_{Zar}, Q_{T′}^{log}) → (X, \underline{M}), such that ũ^{∗}s_{Zar} = s. On the other hand, by inspecting (12.3.30) we find that there exists a unique morphism t_{Zar} : (T_{Zar}, Q_{T}^{log}) → (Y, \underline{N}) such that ũ^{∗}t_{Zar} = t.

These morphisms can be assembled into a diagram :

(12.3.31)  \xymatrix{(T′_{Zar}, Q_{T′}^{log}) \ar[d]_{i_{Zar}} \ar[r]^{s_{Zar}} & (X, \underline{M}) \ar[d]^{f} \\ (T_{Zar}, Q_{T}^{log}) \ar[r]^{t_{Zar}} & (Y, \underline{N})}

where i_{Zar} is an exact closed immersion. By construction, the diagram ũ^{∗}(12.3.31) is naturally isomorphic to (12.3.29); especially, (12.3.31) commutes (proposition 12.2.3(i)). Now, since f is smooth, proposition 12.3.26(ii.b) implies that there exists a morphism v : (T_{Zar}, Q_{T}^{log}) → (X, \underline{M}) such that f ∘ v = t_{Zar} and v ∘ i_{Zar} = s_{Zar}; then ũ^{∗}v provides an appropriate lifting of t, which allows to conclude that ũ^{∗}(f, log f) is smooth.

Next, suppose that (f, log f) is étale (and we are still in case (b) of the corollary); then there exists a unique morphism v with the properties stated above. However, from proposition 12.2.3(i),(ii) we deduce easily that the natural map :

ℋ_{Y} ((T_{Zar}, Q_{T}^{log}), (X, \underline{M}))(T) → ℋ_{ũ^{∗}(Y,\underline{N})}((T, \underline{L}), ũ^{∗}(X, \underline{M}))(T)

is a bijection, and the same holds for the analogous map for T′. In view of the foregoing, this shows that the map ℋ_{ũ^{∗}(Y,\underline{N})}((T, \underline{L}), ũ^{∗}(X, \underline{M}))(T) → ℋ_{ũ^{∗}(Y,\underline{N})}((T′, \underline{L}′), ũ^{∗}(X, \underline{M}))(T′) is bijective, whenever t admits a chart and T is affine. We easily conclude that ũ^{∗}(f, log f) is étale.

The case where both \underline{N} and \underline{M} are both integral, is similar, though easier : we may assume that \underline{L} admits a chart, in which case (T, \underline{L}) is of the form ũ^{∗}(T_{Zar}, Q_{T}^{log}) for some finite integral monoid Q; then (T′, \underline{L}′) admits a similar description, and again, by appealing to proposition 12.2.3(ii) we deduce that (12.3.29) is of the form ũ^{∗}(12.3.31), in which case we conclude as in the foregoing.

Conversely, if ũ^{∗}(f, log f) is smooth, consider again a commutative diagram (12.3.28) of log schemes on Zariski sites, with i an exact closed immersion, defined by a sheaf of ideals ℐ ⊂ 𝒪_{T} such that ℐ^{2} = 0. By applying everywhere the pull-back functors from Zariski to étale sites, we deduce a commutative diagram ũ^{∗}(12.3.28) of log schemes on étale sites, and it is easy to see that ũ^{∗}(i, log i) is again an exact nilpotent immersion. According to proposition 12.3.26(ii.b), after replacing T by any affine open subset, we may find a morphism of log schemes g : ũ_{T}^{∗} (T, \underline{L}) → ũ_{X}^{∗}(X, \underline{M}), such that g ∘ ũ^{∗}i = ũ^{∗}s and ũ^{∗}f ∘ g = t. By proposition 12.2.3(i),(ii) there exists a unique morphism g′ : (T, \underline{L}) → (X, \underline{M}) such that ũ^{∗}g′ = g, and necessarily g′ ∘ i = s and f ∘ g′ = t. We conclude that (f, log f) is smooth.

Finally, if ũ^{∗}(f, log f) is étale, the morphism g exhibited above is unique, and therefore the same holds for g′, so (f, log f) is étale as well. □

Proposition 12.3.32. In the situation of (12.3.11), suppose that the log structures \underline{M}, \underline{N}, \underline{P} are coherent, and consider the following conditions :

(a) f is smooth (resp. étale).

(b) df is a locally split monomorphism (resp. an isomorphism). Then (a)⇒(b), and if g ∘ f is smooth, then (b)⇒(a).

Proof. We may assume that the schemes under consideration are affine, say X = Spec A, Y = Spec B, Z = Spec C; then (12.3.12) amounts to an exact sequence of A-modules :

A ⊗_{B} Ω(g) ⟶^{df} Ω(g ∘ f) ⟶^{ω} Ω(f) → 0.

On the other hand, let i : (T′\underline{L}′) → (T, \underline{L}) be an exact closed immersion of (Y, \underline{N})-schemes, defined by an ideal ℐ ⊂ 𝒪_{T} with ℐ^{2} = 0. Suppose that s : T′ → ℋ_{Y} (T′, X) is a global section, i.e. a given morphism of (Y, \underline{N})-schemes (T′\underline{L}′) → (X, \underline{M}). In this situation, we have a natural sequence of morphisms :

ℋ_{1} := ℋ_{Y} (T, X) ⟶^{α} ℋ_{2} := ℋ_{Z}(T, X) ⟶^{f_{∗}} ℋ_{3} := ℋ_{Z}(T, Y)

where α is the obvious monomorphism. Let us consider the pull-back of these sheaves along the global section s :

𝒯 := T′ ×_{ℋ_{Y} (T′,X)} i^{∗}ℋ_{1}  𝒯 ′ := T′ ×_{ℋ_{Z}(T′,X)} i^{∗}ℋ_{2}  𝒯 ′′ := T′ ×_{ℋ_{Z}(T′,Y)} i^{∗}ℋ_{3}.

By proposition 12.3.26(i), any choice of a global section of 𝒯 (T) determines a commutative diagram of sets :

(12.3.33) \xymatrix{Hom_{A}(Ω(f), ℐ (T)) \ar[r]^{ω^{∗}} \ar[d] & Hom_{A}(Ω(g ∘ f), ℐ (T)) \ar[r]^{df^{∗}} \ar[d] & Hom_{B}(Ω(g), ℐ (T)) \ar[d] \\ 𝒯 (T) \ar[r] & 𝒯 ′(T) \ar[r]^{f_{∗}} & 𝒯 ′′(T)}

whose vertical arrows are bijections.

In order to show that df is a locally split monomorphism (resp. an isomorphism), it suffices to show that the map df^{∗} := Hom_{A}(df, I) is surjective for every A-module I. To this aim, we take (T, L) := (T_{I}, \underline{L}_{I}), (T′, \underline{L}′) := (X, \underline{M}), and let i be the nilpotent immersion defined by the ideal I ⊂ A ⊕ I, as in the proof of proposition 12.3.26(ii.a). If ℐ ⊂ 𝒪_{T} is the corresponding sheaf of ideals, then ℐ (T) = I; moreover, the inclusions A → A ⊕ I and \underline{M} ⊂ \underline{M} ⊕ ℐ determine a morphism h : (T_{I}, \underline{L}_{I}) → (X, \underline{M}), which is a global section of 𝒯 (T). Having made these choices, consider the resulting diagram (12.3.33) : if f is étale, f_{∗} is an isomorphism, and when f is smooth, f_{∗} is surjective (proposition 12.3.26(ii.b)), whence the contention.

For the converse, we may suppose that df is a split monomorphism, hence df^{∗} in (12.3.33)is a split surjection. We have to show that 𝒯 (T) is not empty, and by assumption (and proposition 12.3.26(ii.b)) we know that 𝒯 ′(T) ≠ ∅. Choose any h̃ ∈ 𝒯 ′(T); then t and f_{∗}h̃ are two elements of 𝒯 ′′(T), so we may find φ ∈ Hom_{B}(Ω(g), ℐ (T)) such that φ + f_{∗}h̃ = t (where the sum denotes the action of Hom_{B}(Ω(g), ℐ (T)) on its torsor 𝒯 ′′). Then we may write φ = ψ∘df for some ψ : Ω(g ∘ f) → ℐ (Y), and it follows easily that ψ + h̃ lies in 𝒯 (T). Finally, if df is an isomorphism, we have Ω(f) = 0, hence 𝒯 (T) contains exactly one element. □

Proposition 12.3.34. Let R be a ring, φ : P → Q a morphism of finitely generated monoids, such that Ker φ^{gp} and the torsion subgroup of Coker φ^{gp} (resp. Ker φ^{gp} and Coker φ^{gp}) are finite groups whose orders are invertible in R. Then, the induced morphism

Spec(R, φ) : Spec(R, Q) → Spec(R, P)

is smooth (resp. étale).

Proof. To ease notation, set

f := Spec(R, φ)  (X, \underline{M}) := Spec(R, Q)  (Y, \underline{N}) := Spec(R, P).

Clearly f is finitely presented. We have to show that, for every commutative diagram like (12.3.28) with i an exact nilpotent immersion of fine log schemes, there is, locally on T_{τ} at least one morphism (resp. a unique morphism) h : (T, \underline{L}) → (X, \underline{M}) such that f ∘ h = h ∘ i.

Let ℐ := Ker(𝒪_{T} → i_{∗}𝒪_{T′}); by considering the ℐ -adic filtration on 𝒪_{T}, we reduce easily to the case where ℐ^{2} = 0, and then we may embed ℐ in \underline{L} via the morphism :

ℐ → 𝒪_{T}^{×} ⊂ \underline{L} x ↦ 1 + x.

(Here ℐ is regarded as a sheaf of abelian groups, via its addition law.) Since i is exact, the natural morphism \underline{L}/ℐ → i_{∗}\underline{L}′ is an isomorphism, whence a commutative diagram :

(12.3.35)  \xymatrix{\underline{L} \ar[r]^{log i} \ar[d] & i_{∗}\underline{L}′ \ar[d] \\ _{\underline{L}_{gp}} \ar[r]^{(log i)^{gp}} & i_{∗}(\underline{L}′)^{gp} ≃ \underline{L}^{gp}/ℐ}

and since \underline{L} is integral, one sees easily that (12.3.35) is cartesian (it suffices to consider the stalks over the τ-points).

• On the other hand, suppose first that both Ker φ^{gp} and Coker φ^{gp} are finite groups whose order is invertible in R, hence in ℐ; then a standard diagram chase shows that we may find a unique map g : P_{T}^{gp} → L^{gp} of abelian sheaves that fits into a commutative diagram :

(12.3.36)  \xymatrix{Q_{T}^{gp} \ar[r] \ar[d]_{φ_{T}^{gp}} & t^{∗}\underline{N}^{gp} \ar[r]^{(log t)^{gp}} & \underline{L}^{gp} \ar[d]^{(log i)^{gp}} \\ P_{T}^{gp} \ar[r] \ar[urr]^{g} & i_{∗}s^{∗}\underline{M}^{gp} \ar[r]^{i_{∗}(log s)^{gp}} & i_{∗}(\underline{L}′)^{gp}.}

• More generally, we may write : Coker φ^{gp} ≃ G ⊕ H, where H is a finite group with order invertible in R, and G is a free abelian group of finite rank. The direct summand G lifts to a direct summand G′ ⊂ P^{gp}. Extend φ^{gp} to a map ψ : Q^{gp} ⊕ G′ → P^{gp}, by the rule : (x, g) ↦ φ^{gp}(x)·g. Given any τ-point ξ of T, we may extend the morphism Q^{gp} = Q_{T,ξ}^{gp} → \underline{L}_{ξ}^{gp} in (12.3.36)_{ξ}, to a map ω : Q^{gp} ⊕ G′ → \underline{L}_{ξ}^{gp} whose composition with (log i)_{ξ}^{gp} agrees with the composition of ψ and the bottom map P^{gp} = P_{T,ξ}^{gp} → i_{∗}(\underline{L}′)_{ξ}^{gp} of (12.3.36)_{ξ}. By the usual arguments, ω extends to a map of abelian sheaves ϑ : (Q^{gp} ⊕ G′)_{U} → (\underline{L}^{gp})_{|U} on some neighborhood U → T of ξ, and if U is small enough, the composition (log i)_{|U}^{gp} ∘ ϑ agrees with the composition of ψ_{U} and the bottom map β : P_{U}^{gp} → i_{∗}(\underline{L}^{′gp})_{|U} of (12.3.36)_{|U}. We may then replace T by U, and since Ker ψ = Ker φ^{gp}, and Coker ψ = H, the same diagram chase as in the foregoing shows that we may again find a morphism g : P_{T}^{gp} → \underline{L}^{gp} fitting into a commutative diagram :

\xymatrix{(Q^{gp} ⊕ G′)_{T} \ar[r]^{ϑ} \ar[d]_{ψ_{T}} & \underline{L}^{gp} \ar[d]^{(log i)^{gp}} \\ P_{T}^{gp} \ar[r]^{β} \ar[ur]^{g} & i_{∗}\underline{L}^{′gp}}

In either case, in view of (12.3.35), the morphisms

P_{T} → P_{T}^{gp} ⟶^{g} \underline{L}^{gp} and P_{T} → i_{∗}s^{∗}\underline{M} ⟶^{i_{∗} log s} i_{∗}\underline{L}′

determine a unique morphism P_{T} → \underline{L}, which induces a morphism of log schemes (T, \underline{L}) → (X, \underline{M}) with the sought property. □

Theorem 12.3.37. Let f : (X, \underline{M}) → (Y, \underline{N}) be a morphism of fine log schemes. Assume we are given a fine chart β : Q_{Y} → \underline{N} of \underline{N}. Then the following conditions are equivalent :

(a) f is smooth (resp. étale).

(b) There exist a covering family (g_{λ} : U_{λ} → X | λ ∈ Λ) in X_{ét}, and for every λ ∈ Λ, a fine chart (β, (P_{λ})_{U_{λ}} → g_{λ}^{∗}\underline{M}, φ_{λ} : Q → P_{λ}) of the induced morphism of log schemes

f_{|U_{λ}} := (f ∘ g_{λ}, g_{λ}^{∗} log f) : (U_{λ}, g_{λ}^{∗}\underline{M}) → (Y, \underline{N})

such that :

(i) Ker φ_{λ}^{gp} and the torsion subgroup of Coker φ_{λ}^{gp} (resp. Ker φ_{λ}^{gp} and Coker φ_{λ}^{gp}) are finite groups of orders invertible in 𝒪_{U_{λ}}.

(ii) The natural morphism of Y -schemes p_{λ} : U_{λ} → Y ×_{Spec ℤ[Q]} Spec ℤ[P_{λ}] is étale.

Proof. Suppose first that τ = ét, so \underline{M} and \underline{N} are log structures on étale sites. Then the log structure g_{λ}^{∗}\underline{M} on (U_{λ})_{ét} is just the restriction of \underline{M}, and g_{λ}^{∗} log f is the restriction of log f to (U_{λ})_{ét}. In case τ = Zar, the log structure g_{λ}^{∗}\underline{M} can be described as follows. Form the log scheme (X, \underline{M}′) := ũ_{X}^{∗}(X, \underline{M}) (notation of (12.2.2)), take the restriction \underline{M}′_{|U_{λ}} of \underline{M}′ to (U_{λ})_{ét}, and push forward to the Zariski site to obtain ũ_{X∗}(\underline{M}′_{|U_{λ}}) = g_{λ}^{∗}\underline{M} (by proposition 12.2.3(ii)). Since f is smooth (resp. étale) if and only if ũ^{∗}f is (corollary 12.3.27(ii),(iii)), we conclude that the assertion concerning f holds if and only if the corresponding assertion for ũ^{∗}f does. Hence, it suffices to consider the case of log structures on étale sites. Set S := Spec ℤ[Q].

(b)⇒(a): Taking into account lemma 12.2.18(i), we deduce a commutative diagram of log schemes :

\xymatrix{(U_{λ}, \underline{M}_{|U_{λ}}) \ar[r]^{∼} \ar[d]_{g_{λ}} & Spec(ℤ, P_{λ}) ×_{Spec ℤ[P_{λ}]} U_{λ} \ar[r]^{p_{λ}} & Spec(ℤ, P_{λ}) ×_{Spec ℤ[Q]} Y \ar[d]^{π_{λ}} \\ (X, \underline{M}) \ar[r]^{f} & (Y, \underline{N}) \ar[r]^{∼} & Spec(ℤ, Q) ×_{Spec ℤ[Q]} Y.}

It follows by corollary 12.3.27(i) (resp. by propositions 12.3.24(ii) and 12.3.34) that p_{λ} (resp. π_{λ}) is smooth. Hence f ∘ g_{λ} is smooth, by proposition 12.3.24(i). Finally, f is smooth, by proposition 12.3.24(iii).

(a)⇒(b): Suppose that f is smooth, and fix a geometric point ξ of X. Since (12.3.18)_{ξ} is a surjection, we may find elements t_{1}, …, t_{r} ∈ \underline{M}_{ξ} such that (d log t_{i} | i = 1, …, r) is a basis of the free 𝒪_{X,ξ}-module Ω_{X/Y}^{1} (log \underline{M}/\underline{N})_{ξ} (proposition 12.3.26(ii.a)). Moreover, the kernel of (12.3.18)_{ξ} is generated by sections of the form :

• 1 ⊗ log a where a ∈ N′ := Im(log f_{ξ} : \underline{N}_{f(ξ)} → \underline{M}_{ξ})

• ∑_{j=1}^{s} α(m_{i}) ⊗ log m_{i} where m_{1}, …, m_{s} ∈ \underline{M}_{ξ} and ∑_{j=1}^{s} dα(m_{j}) = 0 in Ω_{X/Y}^{1}

whence a well-defined 𝒪_{X,ξ}-linear map :

(12.3.38)  Ω_{X/Y}^{1} (log \underline{M}/\underline{N})_{ξ} → κ(ξ) ⊗_{ℤ} (\underline{M}_{ξ}^{gp}/(\underline{M}_{ξ}^{×} · N′)) : d log a ↦ 1 ⊗ a.

Consider the map of monoids :

φ : P_{1} := ℕ^{⊕r} ⊕ Q → \underline{M}_{ξ}

which is given by the rule : e_{i} ↦ t_{i} on the canonical basis e_{1}, …, e_{r} of ℕ^{⊕r}, and on the summand Q it is given by the map Q ⟶^{β_{ξ}} \underline{N}_{f(ξ)} ⟶^{log f_{ξ}} \underline{M}_{ξ}. Since (12.3.38) is a surjection, we see that the same holds for the induced map

κ(ξ) ⊗_{ℤ} P_{1}^{gp} ⟶^{1_{κ(ξ)}⊗_{ℤ}φ^{gp}} κ(ξ) ⊗_{ℤ} \underline{M}_{ξ}^{gp} → κ(ξ) ⊗_{ℤ} (\underline{M}_{ξ}^{gp}/\underline{M}_{ξ}^{×}).

It follows that the cokernel of the map \overline{φ} : P_{1}^{gp} → \underline{M}_{ξ}^{gp}/\underline{M}_{ξ}^{×} induced by φ^{gp}, is a finite group (lemma 12.2.21(i)) annihilated by an integer n which is invertible in 𝒪_{X,ξ}. Let m_{1}, …, m_{s} ∈ \underline{M}_{ξ}^{gp} be finitely many elements, whose images in \underline{M}_{ξ}^{gp}/\underline{M}_{ξ}^{×} generate Coker \overline{φ}; therefore we may find u_{1}, …, u_{s} ∈ \underline{M}_{ξ}^{×}, and x_{1}, …, x_{s} ∈ P_{1}^{gp}, such that m_{i}^{n} · u_{i} = \overline{φ}(x_{i}) for every i ≤ s. However, since 𝒪_{X,ξ} is strictly henselian, \underline{M}_{ξ}^{×} ≃ 𝒪_{X,ξ}^{×} is n-divisible, hence we may find v_{1}, …, v_{s} in \underline{M}_{ξ}^{×} such that u_{i} = v_{i}^{n} for i = 1, …, n. Define group homomorphisms :

ℤ^{⊕s} ⟶^{γ} ℤ^{⊕s} ⊕ P_{1}^{gp} ⟶^{δ} \underline{M}_{ξ}^{gp}

by the rules : γ(e_{i}) = (−ne_{i}, x_{i}) and δ(e_{i}, y) = m_{i}v_{i} · φ(y) for every i = 1, …, s and every y ∈ P_{1}^{gp}. It is easily seen that δ factors through a group homomorphism h : G := Coker γ → \underline{M}_{ξ}; moreover the natural map P_{1}^{gp} → G is injective, and its cokernel is annihilated by n; furthermore, the induced map G → \underline{M}_{ξ}^{♯} is surjective. Let P := h^{−1}\underline{M}_{ξ}. Then, the natural map Q^{gp} → P^{gp} is injective, and the torsion subgroup of P^{gp}/Q^{gp} is annihilated by n. We deduce that the rule : x ↦ d log h(x) for every x ∈ P^{gp}, induces an isomorphism :

(12.3.39)  𝒪_{X,ξ} ⊗_{ℤ} (P^{gp}/Q^{gp}) →^{∼} Ω_{X/Y}^{1} (log \underline{M}/\underline{N})_{ξ}.

It follows that we may find an étale neighborhood U → X of ξ, such that (12.3.39) extends to an isomorphism of 𝒪_{U}-modules :

(12.3.40)  𝒪_{U} ⊗_{ℤ} (P^{gp}/Q^{gp}) →^{∼} Ω_{U/Y}^{1} (log \underline{M}_{|U}/\underline{N}).

Next, proposition 12.1.30says that, after replacing U by a smaller étale neighborhood of ξ, the restriction h_{|P} : P → \underline{M}_{ξ} extends to a chart P_{U} → \underline{M}_{|U}, whence a strict morphism

p : (U, \underline{M}_{|U}) → (Y′, P_{Y′}^{log}) := (Y, \underline{N}) ×_{Spec(ℤ,Q)} Spec(ℤ, P).

as sought. Taking into account corollary 12.3.27(i), it remains only to show :

Claim 12.3.41. p is étale.

Proof of the claim. By proposition 12.3.13and example 12.3.19, we have natural isomorphisms

Ω_{Y′/Y}^{1} (log P_{Y′}^{log}/\underline{N}) →^{∼} 𝒪_{Y′} ⊗_{ℤ} (P^{gp}/Q^{gp}).

In view of (12.3.40), it follows that the map

dp : p^{∗}Ω_{Y′/Y}^{1} (log P_{Y′}^{log}/\underline{N}) → Ω_{U/Y}^{1} (log \underline{M}_{|U}/\underline{N})

is an isomorphism, and then the claim follows from proposition 12.3.32. □

Corollary 12.3.42. Keep the notation of theorem 12.3.37. Suppose that f is a smooth morphism of fs log schemes, and that Q is fine, sharp and saturated. Then there exist a covering family (g_{λ} : U_{λ} → X) in X_{ét}, and fine and saturated charts (β, (P_{λ})_{U_{λ}} → g_{λ}^{∗}\underline{M}, φ_{λ} : Q → P_{λ}) of f_{|U_{λ}} fulfilling conditions (b.i) and (b.ii) of the theorem, and such that moreover φ_{λ} is injective, and P_{λ}^{×} is a torsion-free abelian group, for every λ ∈ Λ.

Proof. Notice that, under the stated assumptions, Q^{gp} is a torsion-free abelian group; hence theorem 12.3.37 already implies the existence of a covering family (g_{λ} : U_{λ} → X) in X_{ét}, and of fine charts (β, ω′_{λ} : (P′_{λ})_{U_{λ}} → g_{λ}^{∗}\underline{M}, φ′_{λ} : Q → P′_{λ}) such that φ′_{λ} is injective and Coker(φ′_{λ})^{gp} is a finite group of order invertible in 𝒪_{U_{λ}}, for every λ ∈ Λ. Since g_{λ}^{∗}\underline{M} is saturated (lemma 12.1.18(i)), it is clear that ω′_{λ} factors through a morphism (P′_{λ})_{U_{λ}}^{sat} → g_{λ}^{∗}\underline{M} of pre-log structures, which is again a chart, so we may assume that P′_{λ} is fine and saturated, for every λ ∈ Λ (corollary 6.4.1(ii)), in which case we may find an isomorphism of monoids P′_{λ} = P_{λ} × G, where P_{λ}^{×} is torsion-free, and G is a finite group (lemma 6.2.10). Let d be the order of Coker(φ′_{λ})^{gp}, and denote by φ_{λ} the composition of φ′_{λ} and the projection P′_{λ} → P_{λ}; since Q^{gp} is a torsion-free abelian group, we deduce a short exact sequence :

(12.3.43)  0 → G → Coker(φ′_{λ})^{gp} → Coker φ_{λ}^{gp} → 0

and notice that φ_{λ} is also injective. We may assume that U_{λ} and Y are affine, say U_{λ} = Spec B_{λ} and Y = Spec A, and since d is invertible in 𝒪_{U_{λ}}, we reduce easily – via base change by a finite morphism Y′ → Y – to the case where A contains the subgroup μ_{d} ⊂ \overline{ℚ}^{×} of d-th power roots of 1. The chart ω′_{λ} determines a morphism of monoids P′_{λ} → B_{λ}, and the map f^{♯} : A → B_{λ} factors through the natural ring homomorphism

A → A ⊗_{R[Q]} R[P′_{λ}] →^{∼} A ⊗_{R[Q]} (R[P_{λ}] ⊗_{R} R[G]) where R := ℤ[d^{−1}, μ_{d}].

On the other hand, let Γ := Hom_{ℤ}(G, μ_{d}); then we have a natural decomposition

R[G] ≃ ∏_{χ∈Γ} e_{χ}R[G]

where e_{χ} is the idempotent of R[G] defined as in (14.6.11) (cp. the proof of theorem 14.6.23(i)); each factor is a ring isomorphic to R, whence a corresponding decomposition of U_{λ} as a disjoint union of Y -schemes U_{λ} = ∐_{χ∈Γ} U_{λ,χ}. We are then further reduced to the case where U_{λ} = U_{λ,χ} for some character χ of G. In view of (12.3.43), it is easily seen that the composition

Q^{gp} ⟶^{(φ′_{λ})^{gp}} (P′_{λ})^{gp} → G ⟶^{χ} μ_{d}

extends to a well defined group homomorphism \overline{χ} : P_{λ}^{gp} → μ_{d}, whence a map \overline{χ}_{U_{λ}} : P_{λ,U_{λ}} → μ_{d,U_{λ}} ⊂ g_{λ}^{∗}\underline{M} of sheaves on U_{λ,τ}. Define ω_{λ} : P_{λ,U_{λ}} → g_{λ}^{∗}\underline{M} by the rule : s ↦ ω′_{λ}(s) · \overline{χ}_{U_{λ}}(s) for every local section s of P_{λ,U_{λ}}. It is easily seen that ω_{λ} is again a chart for g_{λ}^{∗}\underline{M} (e.g. one may apply lemma 12.1.4). Lastly, a direct inspection shows that (β, ω_{λ}, φ_{λ}) is a chart of f_{|U_{λ}} with the sought properties. □

12.3.44. Let (Y_{i} | i ∈ I) be a cofiltered system of quasi-compact and quasi-separated schemes, with affine transition morphisms, and suppose that 0 is an initial object of the indexing category I. Suppose also that g_{0} : X_{0} → Y_{0} is a finitely presented morphism of schemes. Let X_{i} := X_{0} ×_{Y_{0}} Y_{i} for every i ∈ I, and denote g_{i} : X_{i} → Y_{i} the induced morphism. Let also g : X → Y be the limit of the family of morphisms (g_{i} | i ∈ I).

Corollary 12.3.45. In the situation of (12.3.44), suppose that (g, log g) : (X, \underline{M}) → (Y, \underline{N}) is a smooth morphism of fine log schemes. Then there exist i ∈ I, and a smooth morphism (g_{i}, log g_{i}) : (X_{i}, \underline{M}_{i}) → (Y_{i}, \underline{N}_{i}) of fine log schemes, such that log g = π_{i}^{∗} log g_{i}.

Proof. First, using corollary 12.2.35, we may find i ∈ I such that (g, log g) descends to a morphism (g_{i}, log g_{i}) of log schemes with coherent log structures. After replacing I by I/i, we may then suppose that i = 0, in which case we set (X_{i}, \underline{M}_{i}) := X_{i} × (X_{0}, \underline{M}_{0}), and define likewise (Y_{i}, \underline{N}_{i}) for every i ∈ I.

Next, arguing as in the proof of corollary 12.2.36(ii), we may assume that \underline{N}_{0} admits a fine chart. In this case, also \underline{N} admits a fine chart, and then we my find a covering family 𝒰 := (U_{λ} → X | λ ∈ Λ) for X_{τ}, such that the induced morphism (U_{λ}, \underline{M}_{|U_{λ}}) → (Y, \underline{N}) admits a chart fulfilling conditions (i) and (ii) of theorem 12.3.37. Moreover, under the current assumptions, X is quasi-compact, hence we may assume that Λ is a finite set, in which case there exists i ∈ I such that 𝒰 descends to a covering family (U_{i,λ} → X_{i} | λ ∈ Λ) for X_{i,τ} (claim 12.2.29(ii)), and as usual, we may then reduce to the case where i = 0. It then suffices to show that there exists i ∈ I such that the induced morphism U_{0,λ} ×_{X_{0}} (X_{i}, \underline{M}_{i}) → (Y_{i}, \underline{N}_{i}) is smooth (proposition 12.3.24(iii)). Thus, we may replace the system (X_{i} | i ∈ I) by the system of schemes (U_{0,λ} ×_{X_{0}} X_{i} | i ∈ I), and assume from start that (g, log g) admits a fine chart fulfilling conditions (i) and (ii) of theorem 12.3.37. In this case, corollary 12.2.36(i) and [65, Ch.IV, Prop.17.7.8(ii)] imply more precisely that there exists i ∈ I such that the morphism (X_{i}, \underline{M}_{i}) → (Y_{i}, \underline{N}_{i}) fulfills conditions (i) and (ii) of theorem 12.3.37, whence the contention. □

12.4. Logarithmic blow up of a coherent ideal. This section introduces the logarithmic version of the scheme-theoretic blow up of a coherent ideal, which will be exhibited as the logarithmic homogeneous spectrum of a certain graded algebra, naturally attached to any sheaf of ideals in a log structure.

12.4.1. We shall consider first the logarithmic counterparts of the notions introduced in section 10.6. To begin with, let X be any scheme, N a monoid, and \underline{P} a (commutative) N-graded monoid of the topos X_{τ} (see definition 4.8.8). Notice that \underline{P}^{×} = ∐_{n∈N}(\underline{P}^{×} ∩ \underline{P}_{n}), hence the sheaf of invertible sections of a N-graded monoid on X is a N-graded abelian sheaf.

Definition 12.4.2. Let X be a scheme, and β : \underline{M} → 𝒪_{X} a log structure on X_{τ}.

(i) A ℕ-graded 𝒪_{(X,\underline{M})}-algebra is a datum (𝒜 , \underline{P} , α, β_{𝒜} ) consisting of a ℕ-graded 𝒪_{X}- algebra 𝒜 := ⊕_{n∈ℕ}𝒜_{n} (on the site X_{τ}), a graded monoid \underline{P} on X_{τ}, and a commutative diagram :

\xymatrix{\underline{M} \ar[r]^{α} \ar[d]_{β} & \underline{P} \ar[d]^{β_{𝒜}} \\ 𝒪_{X} \ar[r] & 𝒜}

where β_{𝒜} restricts to a morphism of graded monoids \underline{P} → ∐_{n∈ℕ} 𝒜_{n} (and the composition law on the target is induced by the multiplication law of 𝒜), α is a morphism of monoids \underline{M} → (\underline{P})_{0}, and the bottom map is the natural morphism 𝒪_{X} → 𝒜_{0}. We say that the ℕ-graded 𝒪_{(X,\underline{M})}-algebra (𝒜 , \underline{P} , α, β_{𝒜} ) is quasi-coherent, if 𝒜 is a quasi-coherent 𝒪_{X}-algebra.

(ii) A morphism (g, log g) : (𝒜 , \underline{P} , α, β_{𝒜} ) → (𝒜′, \underline{P}′, α′, β′_{𝒜} ) of ℕ-graded 𝒪_{(X,\underline{M})}-alge-bras is a commutative diagram :

\xymatrix{\underline{P} \ar[d]_{β_{𝒜}} \ar[r]^{log g} & \underline{P}′ \ar[d]^{β′_{𝒜}} \\ 𝒜 \ar[r]^{g} & 𝒜′}

where log g is a morphism of ℕ-graded monoid such that log g ∘ α = α′, and g is a morphism of 𝒪_{Z}-algebras.

12.4.3. Let M be a monoid, S an M-module; we say that an element s ∈ S is invertible, if the translation map M → S : m ↦ m · s (for all m ∈ M) is an isomorphism. It is easily seen that the subset S^{×} consisting of all invertible elements of S, is naturally an M^{×}-module.

Let \underline{M} be a sheaf of monoids on X_{τ}, and 𝒩 an \underline{M}-module; by restriction of scalars, 𝒩 is naturally an \underline{M}^{×}-module. We define the \underline{M}^{×}-submodule 𝒩 ^{×} ⊂ 𝒩 , by the rule :

𝒩 ^{×}(U) := 𝒩 (U)^{×} for every object U of X_{τ}.

Conversely, for any \underline{M}^{×}-module 𝒫, the extension of scalars 𝒩 ⊗\underline{M}^{×} \underline{M} defines a functor \underline{M}^{×}-Mod → \underline{M}-Mod. It is easily seen that the latter restricts to an equivalence from the full subcategory \underline{M}^{×}-Inv of \underline{M}^{×}-torsors to the subcategory (\underline{M}-Inv)^{×} of \underline{M}-Mod whose objects are all invertible \underline{M}-modules, and whose morphisms are the isomorphisms of \underline{M}-modules (see definition 4.8.6(iv)); the functor 𝒩 ↦ 𝒩 ^{×} provides a quasi-inverse (\underline{M}-Inv)^{×} → \underline{M}^{×}-Inv.

Especially, if (X, \underline{M}) is a log scheme, we see that the category of invertible \underline{M}-modules is equivalent to that of invertible 𝒪_{X}^{×}-modules, hence also to that of invertible 𝒪_{X}-modules.

If φ : (Z, \underline{N}) → (X, \underline{M}) is a morphism of log schemes, and 𝒩 a \underline{M}-module, we let :

(12.4.4)  φ^{∗}𝒩 := φ^{−1}𝒩 ⊗_{φ^{−1}\underline{M}} \underline{N}.

Clearly, φ^{∗}𝒩 is an invertible \underline{N}-module, whenever 𝒩 is an invertible \underline{M}-module.

12.4.5. Keep the notation of definition 12.4.2(i); by faithfully flat descent, the restriction of 𝒜 to the Zariski site of X is again a quasi-coherent 𝒪_{X}-algebra, which we denote again by 𝒜. We may then set Y := Proj 𝒜, and let π : Y → X be the natural morphism. The composition of π^{−1}β_{𝒜} and the morphism (10.6.24), yields a map on the site Y_{τ} :

(12.4.6)  π^{−1}\underline{P}_{n} → 𝒪_{Y} (n)  for every n ∈ ℕ.

Set 𝒪_{Y} (•) := ∐_{n∈ℤ} 𝒪_{Y} (n); it is easily seen that the morphisms (10.6.21) induce a natural ℤ-graded monoid structure on 𝒪_{Y} (•), and the coproduct of the maps (12.4.6) amounts to a morphism of graded monoids :

ω_{•} : π^{−1}\underline{P} → 𝒪_{Y} (•).

We let \underline{Q} be the push-out in the cocartesian diagram :

(12.4.7)  \xymatrix{ω_{•}^{−1}(𝒪_{Y} (•)^{×}) \ar[r] \ar[d] & π^{−1}\underline{P} \ar[d] \\ 𝒪_{Y} (•)^{×} \ar[r] & \underline{Q}.}

Clearly \underline{Q} is naturally a ℤ-graded monoid, in such a way that all the arrows in (12.4.7) are morphisms of ℤ-graded monoids. For every n ∈ ℤ, let \underline{Q}_{n} be the degree n subsheaf of \underline{Q}; the map ω_{•} and the natural inclusion 𝒪_{Y} (•)^{×} → 𝒪_{Y} (•) determine a unique morphism \underline{Q} → 𝒪_{Y} (•), whose restriction in degree zero is a pre-log structure :

β_{𝒜}^{∼} : \underline{Q}_{0} → 𝒪_{Y} .

Clearly α induces a unique morphism α^{∼} : π^{−1}\underline{M} → \underline{Q}_{0}, such that the diagram of monoids :

\xymatrix{π^{−1}\underline{M} \ar[r]^{α^{∼}} \ar[d]_{π^{−1}β} & \underline{Q}_{0} \ar[d]^{β_{𝒜}^{∼}} \\ π^{−1}\underline{𝒪}_{X} \ar[r]^{π^{♮}} & 𝒪_{Y}}

commutes. Denote by \underline{P}^{∼} the log structure associated to β_{𝒜}^{∼}; the homogeneous spectrum of the quasi-coherent ℕ-graded algebra (𝒜 , \underline{P} , α, β_{𝒜} ) is defined as the (X, \underline{M})-scheme :

Proj(𝒜 , \underline{P}) := (Y, \underline{P}^{∼}).

We also let :

U_{1}(𝒜 , \underline{P}) := U_{1}(𝒜) ×_{Y} Proj(𝒜 , \underline{P}).

Furthermore, for every n ∈ ℤ we have a natural morphism of \underline{Q}_{0}-monoids :

(12.4.8)  \underline{Q}_{0} ⊗_{𝒪_{Y}^{×}} 𝒪_{Y} (n)^{×} → \underline{Q}_{n}

and it is easily seen that (12.4.8)_{|U_{1}(𝒜)} is an isomorphism. We set :

\underline{P}^{∼}(n) := (\underline{P}^{∼} ⊗_{\underline{Q}_{0}} \underline{Q}_{n})_{|U_{1}(𝒜)} for every n ∈ ℤ.

Hence (12.4.8) induces a natural isomorphism :

(12.4.9)  (\underline{P}^{∼} ⊗_{𝒪_{Y}^{×}} 𝒪_{Y} (n)^{×})_{|U_{1}(𝒜)} →^{∼} \underline{P}^{∼}(n).

Especially, \underline{P}^{∼}(n) is an invertible \underline{P} _{|U_{1}(𝒜)}^{∼} -module, for every n ∈ ℤ. From (12.4.9), we also deduce natural isomorphisms of \underline{P} _{|U_{1}(𝒜)}^{∼} -modules :

(12.4.10)  \underline{P}^{∼}(n) ⊗\underline{P}^{∼} \underline{P}^{∼}(m) →^{∼} \underline{P}^{∼}(n + m)  for every n, m ∈ ℤ

and of 𝒪_{U_{1}(𝒜)} -modules :

(12.4.11)  \underline{P}^{∼}(n) ⊗\underline{P}^{∼} 𝒪_{U_{1}(𝒜)} →^{∼} 𝒪_{Y} (n)_{|U_{1}(𝒜)}  for every n ∈ ℤ.

Additionally, the morphism π^{−1}\underline{P}_{n} → \underline{Q}_{n} deduced from (12.4.7), yields a natural map of \underline{P} _{|U_{1}(𝒜)}^{∼} -modules :

(12.4.12)  λ_{n} : (π^{∗}\underline{P}_{n})_{|U_{1}(𝒜)} → \underline{P}^{∼}(n)  for every n ∈ ℕ.

Example 12.4.13. Let (Z, γ : \underline{N} → 𝒪_{Z}) be a log scheme, ℒ an invertible \underline{N}-module, and set :

β_{𝒜 (ℒ)} := Sym• ; \underline{N}ℒ ⊗\underline{N} γ : Sym• ; \underline{N}ℒ → 𝒜 (ℒ) := Sym_{𝒪_{Z}}^{•} (ℒ ⊗\underline{N} 𝒪_{Z})

which is a morphisms of ℕ-graded monoids (notation of example 4.8.10). Clearly 𝒜 (ℒ) is also a ℕ-graded quasi-coherent 𝒪_{Z}-algebra. Denote also :

α_{ℒ} : \underline{N} → Sym• ; \underline{N}ℒ

the natural morphism that identifies \underline{N} to Sym0 ; \underline{N}ℒ; then the datum

(𝒜 (ℒ), Sym• ; \underline{N}ℒ , α_{ℒ} , β_{𝒜 (ℒ)} )

is a quasi-coherent 𝒪_{(Z,\underline{N})}-algebra, and a direct inspection of the definitions shows that the induced morphism of log schemes :

(12.4.14)  π_{(Z,\underline{N})} : ℙ(ℒ) := Proj(𝒜 (ℒ), Sym• ; \underline{N}ℒ) → (Z, \underline{N})

is an isomorphism. Furthermore, we have natural isomorphisms as in (10.6.34) :

π_{(Z,\underline{N})}^{∗}(ℒ^{⊗n} ⊗\underline{N} 𝒪_{Z}) →^{∼} 𝒪_{ℙ(ℒ)} (n) for every n ∈ ℤ.

Let \underline{P} _{ℒ}^{∼} → 𝒪_{ℙ(ℒ)} be the log structure of ℙ(ℒ); there follows a natural identification :

(12.4.15)  π_{(Z,\underline{N})}^{∗}ℒ^{⊗n} →^{∼} \underline{P} _{ℒ}^{∼} ⊗_{𝒪_{ℙ(ℒ)}^{×}} 𝒪_{ℙ(ℒ)} (n)^{×} →^{∼} \underline{P} _{ℒ}^{∼} (n)  for every n ∈ ℤ

where the last isomorphism is (12.4.9), in view of the fact that U_{1}(𝒜 (ℒ)) = Proj 𝒜 (ℒ).

Example 12.4.16. (i) Let (Z, \underline{N}) be a log scheme, and n ∈ ℕ any integer. We define an ℕ-grading on ℕ^{⊕n}, by setting

gr^{k}ℕ^{⊕n} := {a_{•} := (a_{1}, …, a_{n}) | a_{1} + ⋯ + a_{n} = k} for every k ∈ ℕ.

We then define the ℕ-graded monoid

Sym• ; \underline{N} \underline{N}^{⊕n} := ℕ_{Z}^{⊕n} × \underline{N}

whose ℕ-grading is deduced in the obvious way from the foregoing grading of ℕ^{⊕n}. The log structure γ : \underline{N} → 𝒪_{Z} extends naturally to a map of ℕ-graded Z-monoids :

Sym• ; \underline{N}γ^{⊕n} : Sym• ; \underline{N} \underline{N}^{⊕n} → Sym_{𝒪_{Z}}^{•} 𝒪_{Z}^{⊕n}.

Namely, if e_{1}, …, e_{n} is the canonical basis of the free 𝒪_{Z}-module 𝒪_{Z}^{⊕n}, then Sym_{𝒪_{Z}}^{k} 𝒪_{Z}^{⊕n} is a free 𝒪_{Z}-module with basis

{e_{a_{•}} := e_{1}^{a_{1}} ⋯ e_{n}^{a_{n}} | a_{•} ∈ gr^{k}ℕ^{⊕n}}

and Sym• ; \underline{N}γ^{⊕n} is given by the rule : (a_{•}, x) ↦ γ(x)·e_{a_{•}} for every a_{•} ∈ ℕ^{⊕n}, every τ-open subset of Z, and every section x ∈ \underline{N}(U). Clearly Sym• ; \underline{N}γ^{⊕n} defines an ℕ-graded 𝒪_{(Z,\underline{N})}-algebra

Sym_{(𝒪_{Z},\underline{N})}^{•} (𝒪_{Z}, \underline{N})^{⊕n} := (Sym_{𝒪_{Z}}^{•} 𝒪_{Z}^{⊕n}, Sym• ; \underline{N} \underline{N}^{⊕n}) for every n ∈ ℕ.

We set :

ℙ_{(Z,\underline{N})}^{n} := Proj Sym_{(𝒪_{Z},\underline{N})}^{•} (𝒪_{Z}, \underline{N})^{⊕n+1}

and we call it the projective n-dimensional space over (Z, \underline{N}).

(ii) Denote by \underline{N}^{∼} the log structure of ℙ_{(Z,\underline{N})}^{n}, and by \underline{N}^{∼}(k) the \underline{N}^{∼}-modules defined as in (12.4.9), for every k ∈ ℤ. By simple inspection we get a commutative diagram of monoids :

\xymatrix{Γ(Z, Symk ; \underline{N} \underline{N}^{⊕n+1}) \ar[r]^{Γ(Z,Symk ; \underline{N} γ^{⊕n+1})} \ar[d] & Γ(Z, Sym_{𝒪_{Z}}^{k} 𝒪_{Z}^{⊕n+1}) \ar[d] \\ Γ(ℙ_{(Z,\underline{N})}^{n}, \underline{N}^{∼}(k)) \ar[r] & Γ(ℙ_{(Z,\underline{N})}^{n}, 𝒪_{ℙ_{(Z,\underline{N})}^{n}}(k))}  for every k ∈ ℕ

whose right vertical arrow is an isomorphism. Especially, the natural basis of the free \underline{N}(Z)- module Γ(Z, Sym1 ; \underline{N} \underline{N}^{⊕n+1}) = \underline{N}(Z)^{⊕n+1} yields a distinguished system of n + 1 elements

ε_{0}, …, ε_{n} ∈ Γ(ℙ_{(Z,\underline{N})}^{n}, \underline{N}^{∼}(1)).

On the other hand, we have as well the distinguished system of global sections

T_{0}, …, T_{n} ∈ Γ(ℙ_{(Z,\underline{N})}^{n}, 𝒪_{ℙ_{(Z,\underline{N})}^{n}}(1))

corresponding to the natural basis of Γ(Z, Sym_{𝒪_{Z}}^{1} 𝒪_{Z}^{⊕n+1}) = 𝒪_{Z}(Z)^{⊕n+1}. For each i = 0, …, n, the largest open subset U_{i} ⊂ ℙ_{(Z,\underline{N})}^{n} such that T_{i} ∈ Γ(U_{i}, 𝒪_{ℙ_{(Z,\underline{N})}^{n}}(1)^{×}) is the complement of the hyperplane where T_{i} vanishes. Moreover, notice that

T_{i}^{−1}T_{j} ∈ \underline{N}^{∼}(U_{i}) for every i, j = 0, …, n.

With this notation, the isomorphism (12.4.9) yields the identification :

ε_{j} = T_{i}^{−1}T_{j} ⊗ T_{i}  on U_{i}  for every i, j = 0, …, n

from which we also see that, for every i = 0, …, n, the open subset U_{i} is the largest such that ε_{i} ∈ Γ(U_{i}, \underline{N}^{∼}(1)^{×}). By the same token, we obtain :

(12.4.17)  (ℙ_{(Z,\underline{N})}^{n}, \underline{N}^{∼})_{tr} = U_{0} ∩ ⋯ ∩ U_{n} ≃ 𝔾_{m,Z}^{n}.

12.4.18. Let (Z, \underline{N}) be a log scheme, (φ, log φ) : (𝒜 , \underline{P}) → (𝒜′, \underline{P}′) a morphism of quasicoherent ℕ-graded 𝒪_{(Z,\underline{N})}-algebras. We let (notation of (10.6.26)) :

G(φ, log φ) := G(φ) ×_{Proj 𝒜′} Proj(𝒜′, \underline{P}′).

Denote also by π : Y := Proj 𝒜 → Z and π′ : Y′ := Proj 𝒜′ → Z′ the natural projections; there follows, on the one hand, a morphism of ℕ-graded monoids :

(12.4.19)  π^{′−1}(log φ) : (Proj φ)^{−1}(π^{−1}\underline{P}) → (π^{′−1}\underline{P}′)_{|G(φ)}

and on the other hand, a morphism of ℤ-graded monoids :

(12.4.20)  (Proj φ)^{−1}𝒪_{Y} (•)^{×} → 𝒪_{Y′}(•)_{|G(φ)}^{×}

deduced from (10.6.28). Define the ℤ-graded monoid \underline{Q} on Y_{τ} as in (12.4.7), and the analogous ℤ-graded monoid \underline{Q}′ on Y′_{τ}. Then (12.4.19) and (12.4.20) determine a unique morphism of ℤ-graded monoids :

ϑ : (Proj φ)^{−1}\underline{Q} → \underline{Q}′_{|G(φ)}

and by construction, the restriction of ϑ in degree zero is a morphism of pre-log structures :

ϑ_{0} : (Proj φ)^{∗}((\underline{Q})_{0}, β_{𝒜}^{∼}) → ((\underline{Q}′)_{0}, β_{𝒜′}^{∼})

whence a morphism of (Z, \underline{N})-schemes :

Proj(φ, log φ) : G(φ, log φ) → Proj(𝒜 , \underline{P}).

Moreover, on the one hand, (10.6.28) induces an isomorphism of 𝒪_{Y′}^{×}-modules :

ν_{𝒜 (n)}^{×} : (𝒪_{Y′}^{×} ⊗_{(Proj φ)^{−1}𝒪_{Y}^{×}} (Proj φ)^{−1}𝒪_{Y} (n)^{×})_{|G_{1}(φ)} →^{∼} 𝒪_{Y′}(n)_{|G_{1}(φ)}^{×} for every n ∈ ℤ

(notation of (10.6.27)). On the other hand, for every n ∈ ℤ, the morphism of (Proj φ)^{−1}(\underline{Q})_{0}- modules ϑ_{n} determines a morphism of \underline{P}^{′∼}-modules :

(12.4.21)  ϑ_{n}^{∼} : Proj(φ, log φ)^{∗}\underline{P}^{∼}(n)_{|G_{1}(φ)} → \underline{P}^{′∼}(n)_{|G_{1}(φ)}

and by inspecting the construction, it is easily seen that the isomorphism (12.4.9) (and the corresponding one for \underline{P}^{′∼}(n)) identifies ϑ_{n}^{∼} with ν_{𝒜 (n)}^{×} ⊗_{𝒪_{Y′}^{×}} \underline{P}^{′∼}; especially, ϑ_{n}^{∼} is an isomorphism.

12.4.22. Let ψ : (Z′, \underline{N}′) → (Z, \underline{N}) be a morphism of log schemes, and (𝒜 , \underline{P} , α, β_{𝒜} ) a ℕ-graded quasi-coherent 𝒪_{(Z,\underline{N})}-algebra. We may view \underline{P} as a \underline{N}-module, via the morphism α, hence we may form the \underline{N}′-module ψ^{∗}\underline{P}, as in (12.4.4). Moreover, by remark (6.1.25)(i), ψ^{∗}\underline{P} is a ℕ-graded sheaf of monoids on Z′_{τ}, such that

(12.4.23)  ψ^{∗}(𝒜 , \underline{P}) := (ψ^{∗}𝒜 , ψ^{∗}\underline{P} , ψ^{∗}α, ψ^{∗}β_{𝒜} )

is a ℕ-graded quasi-coherent 𝒪_{(Z′,\underline{N}′)}-algebra, and in view of the isomorphism (6.1.26), we obtain a natural isomorphism of (Z′, \underline{N}′)-schemes:

(12.4.24)  Proj ψ^{∗}(𝒜 , \underline{P}) →^{∼} (Z′, \underline{N}′) ×_{(Z,\underline{N})} Proj(𝒜 , \underline{P}).

Furthermore, denote by π_{(𝒜 ,\underline{P})} : U_{1}(ψ^{∗}(𝒜 , \underline{P})) → U_{1}(𝒜 , \underline{P}) the morphism deduced from (12.4.24), and by π_{Y} : Y′ := Proj ψ^{∗}𝒜 → Y := Proj 𝒜 the underlying morphism of schemes. From (10.6.31) we obtain natural isomorphisms :

(𝒪_{Y′}^{×} ⊗_{π_{Y}^{−1}𝒪_{Y}^{×}} π_{Y}^{−1}𝒪_{Y} (n)^{×})_{|U_{1}(ψ^{∗}𝒜)} →^{∼} 𝒪_{Y′}(n)_{|U_{1}(ψ^{∗}𝒜)}^{×} for every n ∈ ℤ

and the latter induce natural identifications :

(12.4.25)  π_{(𝒜 ,\underline{P})}^{∗} \underline{P}^{∼}(n) →^{∼} (ψ^{∗}\underline{P})^{∼}(n)  for every n ∈ ℤ.

12.4.26. Keep the notation of (12.4.5), and let log 𝒞_{(X,\underline{M})} be the category whose objects are the pairs ((Z, \underline{N}), ℒ), where (Z, \underline{N}) is a (X, \underline{M})-scheme, and ℒ is an invertible \underline{N}-module. The morphisms ((Z, \underline{N}), ℒ) → ((Z′, \underline{N}′), ℒ′) are the pairs (φ, h), where φ : (Z, \underline{N}) → (Z′, \underline{N}′) is a morphism of (X, \underline{M})-schemes, and h : φ^{∗}ℒ′ →^{∼} ℒ is an isomorphism of \underline{N}-modules (with composition of morphisms defined in the obvious way). There is an obvious forgetful functor :

p : log 𝒞_{(X,\underline{M})} → 𝒞_{X} : ((Z, \underline{N}), ℒ) ↦ (Z, ℒ ⊗\underline{N} 𝒪_{Z})

and the functor F_{𝒜} can be lifted to a functor :

F_{(𝒜 ,\underline{P})} : log 𝒞_{(X,\underline{M})} → Set

which assigns to any object ((Z, γ : \underline{N} → 𝒪_{Z}), ℒ) the set consisting of all morphisms of ℕ-graded quasi-coherent 𝒪_{(Z,\underline{N})}-algebras :

\xymatrix{ψ^{∗}\underline{P} \ar[r]^{log g} \ar[d]_{ψ^{∗}β_{𝒜}} & Sym• ; \underline{N}ℒ \ar[d]^{β_{𝒜 (ℒ)}} \\ ψ^{∗}𝒜 \ar[r]^{g} & 𝒜 (ℒ)}

where ψ : (Z, \underline{N}) → (X, \underline{M}) is the structural morphism, and g is an epimorphism on the underlying 𝒪_{Z}-modules.

Proposition 12.4.27. (U_{1}(𝒜 , \underline{P}), \underline{P}^{∼}(1)) ∈ Ob(log 𝒞_{(X,\underline{M})}) represents the functor F_{(𝒜 ,\underline{P})} .

Proof. Given ((Z, \underline{N}) ⟶^{ψ} (X, \underline{M}), ℒ) ∈ Ob(log 𝒞_{(X,\underline{M})}) and (g, log g) ∈ F_{(𝒜 ,\underline{P})} ((Z, \underline{N}), ℒ), define ℙ(ℒ) as in (12.4.14); there follows a morphism of (Z, \underline{N})-schemes :

Proj(g, log g) : ℙ(ℒ) → Proj ψ^{∗}(𝒜 , \underline{P}).

In view of (12.4.14) and (12.4.24), this is the same as a morphism of (X, \underline{M})-schemes :

ℙ(g, log g) : (Z, \underline{N}) → Proj(𝒜 , \underline{P})

and arguing as in the proof of lemma 10.6.33, we see that the image of ℙ(g, log g) lands in U_{1}(𝒜 , \underline{P}). Next, combining (12.4.15), (12.4.21) and (12.4.25), we deduce a natural isomorphism :

π_{(Z,\underline{N})}^{∗} ∘ ℙ(g, log g)^{∗}\underline{P}^{∼}(1) →^{∼} Proj(g, log g)^{∗} ∘ π_{(𝒜 ,\underline{P})}^{∗} \underline{P}^{∼}(1)

→^{∼} Proj(g, log g)^{∗}(ψ^{∗}\underline{P})^{∼}(1)

→^{∼} \underline{P} _{ℒ}^{∼} (1)

→^{∼} π_{(Z,\underline{N})}^{∗}ℒ

whence an isomorphism h(g, log g) : ℙ(g, log g)^{∗}\underline{P}^{∼}(1) →^{∼} ℒ, and the datum :

(ℙ(g, log g), h(g, log g)) : ((Z, \underline{N}), ℒ) → (U_{1}(𝒜 , \underline{P}), \underline{P}^{∼}(1))

is a well defined morphism of log 𝒞_{(X,\underline{M})}.

Conversely, let φ := (β, log β) : (Z, \underline{N}) → U_{1}(𝒜 , \underline{P}) be a morphism of (X, \underline{M})-schemes, and h : φ^{∗}\underline{P}^{∼}(1) →^{∼} ℒ an isomorphism of \underline{N}-modules. In view of (12.4.10), we deduce an isomorphism :

h^{⊗n} : φ^{∗}\underline{P}^{∼}(n) →^{∼} ℒ^{⊗n} for every n ∈ ℤ.

Combining with (12.4.12), we may define the map of \underline{N}-modules :

log ĝ(φ, h) := ⨁_{n∈ℕ} h^{⊗n} ∘ β^{∗}(λ_{n}) : ψ^{∗}\underline{P} → Sym• ; \underline{N}ℒ .

On the other hand, in view of (12.4.11), we have an isomorphism of 𝒪_{Z}-modules :

h ⊗\underline{N} 𝒪_{Z} : β^{∗}𝒪_{Y} (1)_{|U_{1}(𝒜)} →^{∼} φ^{∗}\underline{P}^{∼}(1) ⊗\underline{N} 𝒪_{Z} →^{∼} ℒ ⊗\underline{N} 𝒪_{Z}

(where, as usual, Y := Proj 𝒜). We let (notation of (10.6.36)) :

ĝ(φ, h) := g(β, h ⊗\underline{N} 𝒪_{Z})

and notice that the pair (ĝ(φ, h), log ĝ(φ, h)) is an element of F_{(𝒜 ,\underline{P})} ((Z, \underline{N}), ℒ). Summing up, we have exhibited two natural transformations :

ϑ : F_{(𝒜 ,\underline{P})} ⇒ Hom_{log 𝒞_{(X,\underline{M})}}(−, (U_{1}(𝒜 , \underline{P}), \underline{P}^{∼}(1))) (g, log g) ↦ ℙ(g, log g)
σ : Hom_{log 𝒞_{(X,\underline{M})}}(−, (U_{1}(𝒜 , \underline{P}), \underline{P}^{∼}(1))) ⇒ F_{(𝒜 ,\underline{P})}  (φ, h) ↦ (ĝ(φ, h), log ĝ(φ, h))

and it remains to show that these transformations are isomorphisms of functors. However, the latter fit into an essentially commutative diagram of natural transformations :

\xymatrix{F_{(𝒜 ,\underline{P})} \ar@{=>}[r]^{ϑ} \ar@{=>}[d] & Hom_{log 𝒞_{(X,\underline{M})}}(−, (U_{1}(𝒜 , \underline{P}), \underline{P}^{∼}(1))) \ar@{=>}[r]^{σ} \ar@{=>}[d] & F_{(𝒜 ,\underline{P})} \ar@{=>}[d] \\ F_{𝒜} ∘ p \ar@{=>}[r] & Hom_{𝒞_{X}} (p(−), (U_{1}(𝒜), 𝒪_{Y} (1)_{|U_{1}(𝒜)} )) \ar@{=>}[r] & F_{𝒜} ∘ p}

whose bottom line is given by the natural transformations (10.6.37). Moreover, given an isomorphism h of invertible \underline{N}-modules, the discussion in (12.4.3) leads to the identity :

(12.4.28)  h = (h ⊗\underline{N} 𝒪_{Z})^{×} ⊗_{𝒪_{Z^{×}}} \underline{N}.

Likewise, we have natural identifications :

Symn ; \underline{N}ℒ = (Sym_{𝒪_{Z}}^{n} ℒ ⊗\underline{N} 𝒪_{Z})^{×} ⊗_{𝒪_{Z}^{×}} \underline{N} for every n ∈ ℤ

which show that β_{𝒜} and g determine uniquely log g. This – and an inspection of the proof of lemma 10.6.33 – already implies that σ ∘ ϑ is the identity automorphism of the functor F_{(𝒜 ,\underline{P})} . Finally, let φ = (β, log β) and h as in the foregoing, so that (φ, h) is a morphism in log 𝒞_{(X,\underline{M})}; to conclude we have to show that (φ′, h′) := ϑ ∘ σ(φ, h) equals (φ, h). Say that φ′ := (β′, log β′); by the above (and by the proof of lemma 10.6.33) we already know that β = β′, and (12.4.28) implies that h = h′. Hence, it remains only to show that log β = log β′, which can be checked directly on the stalks over the τ-points of Z : we leave the details to the reader. □

Example 12.4.29. (i) Let ψ : (Z′, \underline{N}′) → (Z, \underline{N}) be a morphism of log schemes, n ∈ ℕ any integer, and ℙ_{(Z,\underline{N})}^{n} the n-dimensional projective space over (Z, \underline{N}), defined as in example 12.4.16. A simple inspection of the definitions yields a natural isomorphism of 𝒪_{(Z′,\underline{N}′)}-algebras

Sym_{(𝒪_{Z′},\underline{N}′)}^{•} (𝒪_{Z′}, \underline{N}′)^{⊕n} →^{∼} ψ^{∗}Sym_{(𝒪_{Z},\underline{N})}^{•} (𝒪_{Z}, \underline{N})^{⊕n} for every n ∈ ℕ

whence a natural isomorphism of (Z′, \underline{N}′)-schemes :

ℙ_{(Z′,\underline{N}′)}^{n} →^{∼} (Z′, \underline{N}′) ×_{(Z,\underline{N})} ℙ_{(Z,\underline{N})}^{n} for every n ∈ ℕ.

(ii) Let ℒ be any invertible \underline{N}-module; notice that a morphism of ℕ-graded monoids

Sym• ; \underline{N} \underline{N}^{⊕n} → Sym• ; \underline{N}ℒ

which is the identity map in degree zero, is the same as the datum of a sequence

(β_{i} : \underline{N} → ℒ | i = 1, , …, n)

of morphisms of \underline{N}-modules, and the latter is the same as a sequence (b_{1}, …, b_{n}) of global sections of ℒ. Since Sym_{𝒪_{Z}}^{1} 𝒪_{Z}^{⊕n+1} generates Sym_{𝒪_{Z}}^{•} 𝒪_{Z}^{n+1}, proposition 12.4.27 and (i) imply that ℙ_{(Z,\underline{N})}^{n} represents the functor log 𝒞_{(Z,\underline{N})} → Set that assigns to any pair ((X, \underline{M}), ℒ) the set of all sequences (b_{0}, …, b_{n}) of global sections of ℒ. The bijection

(12.4.30)  Hom_{log 𝒞_{(Z,\underline{N})}}(((X, \underline{M}), ℒ), (ℙ_{(Z,\underline{N})}^{n}, \underline{N}^{∼}(1))) →^{∼} Γ(X, ℒ)^{n+1}

can be explicited as follows. Let (φ, h) : ((X, \underline{M}), ℒ) → (ℙ_{(Z,\underline{N})}^{n}, \underline{N}^{∼}(1)) be a given morphism in log 𝒞_{(Z,\underline{N})}; then h : φ^{∗}\underline{N}^{∼}(1) → ℒ is an isomorphism of \underline{M}-modules, which induces a map on global sections

Γ(h) : Γ(ℙ_{(Z,\underline{N})}^{n}, \underline{N}^{∼}(1))) → Γ(X, ℒ).

Now, \underline{N}^{∼}(1) admits a distiguished system of global sections ε_{0}, …, ε_{n} (example 12.4.16(ii)), and the bijection (12.4.30) assigns to (φ, h) the sequence (Γ(h)(ε_{0}), …, Γ(h)(ε_{n})).

(iii) Given a sequence b_{•} := (b_{0}, …, b_{n}) as in (ii), denote by

f_{b_{•}} : (X, \underline{M}) → ℙ_{(Z,\underline{N})}^{n}

the corresponding morphism. A direct inspection of the definitions shows that the formation of f_{b_{•}} is compatible with arbitrary base changes h : (Z′, \underline{N}′) → (Z, \underline{N}). Namely, set (X′, \underline{M}′) := (Z′, \underline{N}′) ×_{(Z,\underline{N})} (X, \underline{M}), let g : (X′, \underline{M}′) → (X, \underline{M}) be the induced morphism, ℒ′ := g^{∗}ℒ, and suppose that ℒ′ is also invertible (which always holds, if \underline{M}′ is an integral log structure); the sequence b_{•} pulls back to a corresponding sequence b′_{•} := (b′_{0}, …, b_{n}) of global sections of ℒ′, and there follows a cartesian diagram of log schemes :

\xymatrix{(X′, \underline{M}′) \ar[r]^{f_{b′_{•}}} \ar[d]_{g} & ℙ_{(Z′,\underline{N}′)}^{n} \ar[d]^{ℙ_{h}^{n}} \\ (X, \underline{M}) \ar[r]^{f_{b•}} & ℙ_{(Z,\underline{N})}^{n}.}

The details shall be left to the reader.

Definition 12.4.31. Let (X, \underline{M}) be a log scheme, and ℐ ⊂ \underline{M} an ideal of \underline{M} (see (3.7.22)).

(i) Let f : (Y, \underline{N}) → (X, \underline{M}) be a morphism of log schemes; then f^{−1}ℐ is an ideal in the sheaf of monoids f^{−1}\underline{M}; we let :

ℐ \underline{N} := log f(f^{−1}ℐ) · \underline{N}

which is the smallest ideal of \underline{N} containing the image of f^{−1}ℐ. (ii) A logarithmic blow up of the ideal ℐ is a morphism of log schemes

φ : (X′, \underline{M}′) → (X, \underline{M})

which enjoys the following universal property. The ideal ℐ \underline{M}′ is invertible, and every morphism of log schemes (Y, \underline{N}) → (X, \underline{M}) such that ℐ \underline{N} ⊂ \underline{N} is invertible, factors uniquely through φ.

Remark 12.4.32. (i) Keep the notation of definition 12.4.31. By the usual general nonsense, it is clear that the blow up (X′, \underline{M}′) is determined up to unique isomorphism of (X, \underline{M})-schemes.

(ii) Moreover, let f : Y → X be a morphism of schemes. Then we claim that the natural projection :

(Y′, \underline{M}′_{Y}) := Y ×_{X} (X′, \underline{M}′) → (Y, \underline{M}_{Y}) := Y ×_{X} (X, \underline{M})

is a logarithmic blow up of ℐ \underline{M}_{Y}, provided ℐ \underline{M}′_{Y} is an invertible ideal; especially, this holds whenever \underline{M}′ is an integral log structure (lemma 12.1.18(i)). The proof is left as an exercise for the reader.

12.4.33. Let (X, \underline{M}) be a log scheme, ℐ ⊂ \underline{M} an ideal; we shall show the existence of the logarithmic blow up of ℐ, under fairly general conditions. To this aim, we introduce the graded blow up 𝒪_{X}-algebra :

B(X, \underline{M}, 𝒥) := ⨁_{n∈ℕ} ℐ^{n} ⊗\underline{M} 𝒪_{X}

where ℐ^{n} ⊗\underline{M} 𝒪_{X} is the sheaf associated to the presheaf U ↦ ℐ^{n}(U) ⊗_{\underline{M}(U)} 𝒪_{X}(U) on X_{τ}. Here ℐ^{0} := \underline{M}, and for every n > 0 we let ℐ^{n} be the sheaf associated to the presheaf U ↦ ℐ (U)^{n} on X_{τ}. The graded multiplication law of the blow up 𝒪_{X}-algebra is induced by the multiplication ℐ^{n} × ℐ^{m} → ℐ^{n+m}, for every n, m ∈ ℕ.

The natural map ℐ^{n} → ℐ^{n} ⊗\underline{M} 𝒪_{X} induces a morphism of sheaves of monoids :

∐_{n∈ℕ} ℐ^{n} → B(X, \underline{P} , 𝒥).

The latter defines a ℕ-graded 𝒪_{(X,\underline{M})}-algebra, which we denote ℬ(X, \underline{M}, ℐ).

12.4.34. Suppose first that ℐ is invertible; then it is easily seen that, locally on X_{τ}, ℐ is generated by a regular local section (see example 4.8.36(i)), hence the same holds for the power ℐ^{n}, for every n ∈ ℕ. Therefore ℐ^{n} is a locally free \underline{M}-module of rank one, and we have a natural isomorphism :

ℬ(X, \underline{M}, ℐ) →^{∼} (𝒜 (ℐ), Sym• ; \underline{M}ℐ)

(notation of example (12.4.13)). It follows that in this case, the natural projection :

π_{(X,\underline{M},ℐ)} : Proj ℬ(X, \underline{M}, ℐ) → (X, \underline{M})

is an isomorphism of log schemes.

12.4.35. The formation of ℬ(X, \underline{M}, ℐ) is obviously functorial with respect to morphisms of log schemes; more precisely, such a morphism f : (Y, \underline{N}) → (X, \underline{M}) induces a morphism of ℕ-graded 𝒪_{(Y,\underline{N})}-algebras:

ℬ(f, ℐ) : f^{∗}ℬ(X, \underline{M}, ℐ) → ℬ(Y, \underline{N}, ℐ \underline{N})

(notation of (12.4.23)) which is an epimorphism on the underlying 𝒪_{Y} -modules. Moreover, if g : (Z, \underline{Q}) → (Y, \underline{N}) is another morphism, we have the identity :

(12.4.36)  ℬ(f ∘ g, ℐ) = ℬ(g, ℐ \underline{N}) ∘ g^{∗}ℬ(f, ℐ).

Furthermore, the construction of the blow up algebra is local for the topology of X_{τ} : if U → X is any object of X_{τ}, we have a natural identification

ℬ(U, \underline{M}_{|U}, ℐ_{|U}) →^{∼} ℬ(X, \underline{M}, ℐ)_{|U}.

In the presence of charts for the log structure \underline{M}, we can give a handier description for the blow up algebra; namely, we have the following :

Lemma 12.4.37. Let X be a scheme, α : \underline{P} → 𝒪_{X} a pre-log structure, β : \underline{P} → \underline{P}^{log} the natural morphism of pre-log structures. Let also ℐ ⊂ \underline{P} be an ideal, and set ℐ \underline{P}^{log} := β(ℐ) · \underline{P}^{log} (which is the ideal of \underline{P}^{log} generated by the image of ℐ). Then :

(i) There is a natural isomorphism of graded 𝒪_{X}-algebras :

⨁_{n∈ℕ} ℐ^{n} ⊗\underline{P} 𝒪_{X} →^{∼} B(X, \underline{P}^{log}, ℐ \underline{P}^{log}).

(ii) Especially, suppose (X, \underline{M}) is a log scheme that admits a chart β : P_{X} → \underline{M}, let I ⊂ P be an ideal, and set I\underline{M} := β(I_{X}) · \underline{M}. Then ℬ(X, \underline{M}, I\underline{M}) is a ℕ-graded quasi-coherent 𝒪_{(X,\underline{M})}-algebra.

(iii) In the situation of (ii), set (S, P_{S}^{log}) := Spec(ℤ, P) (see (12.2.13)), and denote by f : (X, \underline{M}) → (S, P_{S}^{log}) the natural morphism. Then the map :

ℬ(f, IP_{S}^{log}) : f^{∗}ℬ(S, P_{S}^{log}, IP_{S}^{log}) → ℬ(X, \underline{M}, I\underline{M})

is an isomorphism of ℕ-graded 𝒪_{(X,\underline{M})}-algebras.

Proof. (i): Since the functor (4.8.51) commutes with all colimits, we have a natural isomorphism of sheaves of rings on X_{τ} :

ℤ[\underline{P}^{log}] →^{∼} ℤ[\underline{P}] ⊗_{ℤ[α^{−1}𝒪_{X}^{×}]} ℤ[𝒪_{X}^{×}].

We are therefore reduced to showing that the natural morphism

ℤ[ℐ^{n}] ⊗_{ℤ[α^{−1}𝒪_{X}^{×}]} ℤ[𝒪_{X}^{×}] → ℐ^{n}ℤ[\underline{P}^{log}]

is an isomorphism for every n ∈ ℕ. The latter assertion can be checked on the stalks over the τ-points of X; to this aim, we invoke the more general :

Claim 12.4.38. Let G be an abelian group, φ : H → P′ and ψ : H → G two morphisms of monoids, I ⊂ P′ an ideal. Then the natural map

(12.4.39)  ℤ[I] ⊗_{ℤ[H]} ℤ[G] → Iℤ[P′ ⊗_{H} G]

is an isomorphism.

Proof of the claim. Recall that ℤ[I] = Iℤ[P′], and the map (12.4.39) is induced by the natural identification: ℤ[P′] ⊗_{ℤ[H]} ℤ[G] →^{∼} ℤ[P′ ⊗_{H} G] (see (4.8.52)). Especially, (12.4.39) is clearly surjective, and it remains to show that it is also injective. To this aim, notice first that ψ factors through the unit of adjunction η : H → H^{gp}; the morphism ℤ[η] : ℤ[H] → ℤ[H^{gp}] = H^{−1}ℤ[H] is a localization map (see (4.8.53)), especially it is flat, hence (12.4.39) is injective when G = H^{gp} and ψ = η. It follows easily that we may replace H by H^{gp}, P′ by P′ ⊗_{H} H^{gp}, I by I · (P′ ⊗_{H} H^{gp}), and therefore assume that H is a group. Let L := ψ(H); arguing as in the foregoing, we may consider separately the case where ψ is the surjection H → L and the case where ψ is the injection i : L → G. However, one sees easily that ℤ[i] : ℤ[L] → ℤ[G] is a flat morphism, hence it suffices to consider the case where ψ is a surjective group homomorphism. Set K := Ker ψ; we have a natural identification : P′ ⊗_{H} G ≃ P′ ⊗_{K} {1}, hence we may further reduce to the case where G = {1}. Then the contention is that the augmentation map ℤ[K] → ℤ induces an isomorphism ω : ℤ[I] ⊗_{ℤ[K]} ℤ →^{∼} Iℤ[P′/K]. From lemma 4.8.31(ii), we derive easily that Iℤ[P′/K] = ℤ[I/K], where I/K is the set-theoretic quotient of I for the Kaction deduced from φ. However, any set-theoretic section I/K → I of the natural projection I → I/K yields a well-defined surjection ℤ[I/K] → ℤ[I] ⊗_{ℤ[K]} ℤ whose composition with ω is the identity map. The claim follows. ♢

(ii): Let U be an affine object of X_{τ}, say U = Spec A; from (i), we see that ℬ(X, \underline{M}, I\underline{M})_{|U} is the quasi-coherent 𝒪_{U}-algebra associated to the A-algebra ⊕_{n∈ℕ}ℤ[I^{n}] ⊗_{ℤ[P]} A.

(iii): In view of (i) we know already that ℬ(f, IP_{S}^{log}) induces an isomorphism on the underlying 𝒪_{X}-algebras; hence, by lemma 12.2.18(i), it remains to show that ℬ(f, IP_{S}^{log}) induces an isomorphism :

f^{∗}(I^{n}P_{S}^{log}) →^{∼} I^{n}f^{∗}(P_{S}^{log}) for every n ∈ ℕ.

Let γ : f^{−1}P_{S}^{log} → f^{−1}𝒪_{S} → 𝒪_{X} be the natural map; after replacing I by I^{n}, we come down to showing that the natural map :

f^{−1}(IP_{S}^{log}) ⊗_{γ^{−1}𝒪_{X}^{×}} 𝒪_{X}^{×} → I · (f^{−1}P_{S}^{log} ⊗_{γ^{−1}𝒪_{X}^{×}} 𝒪_{X}^{×})

is an isomorphism. This assertion can be checked on the stalks over the τ-points of X; if x is such a point, let G := 𝒪_{X,x}^{×}, H := γ_{x}^{−1}G and P′ := P_{S,f(x)}^{log}. The map under consideration is the natural morphism of P′-modules :

ω : (IP′) ⊗_{H} G → I(P′ ⊗_{H} G)

and it suffices to show that ℤ[ω] is an isomorphism; however, the latter is none else than (12.4.39), so we may appeal to claim 12.4.38to conclude. □

12.4.40. We wish to generalize lemma 12.4.37(ii) to log structures that do not necessarily admit global charts. Namely, suppose now that \underline{M} is a quasi-coherent log structure on X, and ℐ ⊂ \underline{M} is a coherent ideal (see definition 4.8.6(v)). For every τ-point ξ of X, we may then find a neighborhood U of ξ in X_{τ}, a chart β : P_{U} → \underline{M}_{|U}, and local sections s_{1}, …, s_{n} ∈ ℐ (U) which form a system of generators for ℐ_{|U}. We may then write s_{i,ξ} = u_{i} · β(x_{i}) for certain x_{1}, …, x_{n} ∈ P and u_{1}, …, u_{n} ∈ 𝒪_{X,ξ}^{×}. Up to shrinking U, we may assume that u_{1}, …, u_{n} ∈ 𝒪_{X}^{×}(U), and it follows that ℐ_{|U} = I ·\underline{M}_{|U}, where I ⊂ P is the ideal generated by x_{1}, …, x_{n}. In other words, locally on X_{τ}, the datum (X, \underline{M}, ℐ) is of the type considered in lemma 12.4.37(ii); therefore the blow up 𝒪_{X}-algebra ℬ(X, \underline{M}, ℐ) is quasi-coherent. We may then consider the natural projection :

(12.4.41)  π_{(X,\underline{M},ℐ)} : Bl_{ℐ} (X, \underline{M}) := Proj ℬ(X, \underline{M}, ℐ) → (X, \underline{M}).

Next, let f : (Y, \underline{N}) → (X, \underline{M}) be a morphism of log schemes; it is easily seen that ℐ \underline{N} is a coherent ideal of \underline{N}, hence ℬ(Y, \underline{N}, ℐ \underline{N}) is quasi-coherent as well, and since the map ℬ(f, ℐ) of (12.4.35) is an epimorphism on the underlying 𝒪_{Y} -modules, we have :

G(ℬ(f, ℐ)) = Bl_{ℐ \underline{N}} (Y, \underline{N})

(notation of (12.4.18)) whence a closed immersion of (Y, \underline{N})-schemes :

Proj ℬ(f, ℐ) : Bl_{ℐ \underline{N}} (Y, \underline{N}) → (Y, \underline{N}) ×_{(X,\underline{M})} Bl_{ℐ} (X, \underline{M})

([60, Ch.II, Prop.3.6.2(i)] and (12.4.24)), which is the same as a morphism of (X, \underline{M})-schemes :

φ(f, ℐ) : Bl_{ℐ \underline{N}} (Y, \underline{N}) → Bl_{ℐ} (X, \underline{M}).

Moreover, if g : (Z, \underline{Q}) → (Y, \underline{N}) is another morphism, (12.4.36) induces the identity :

(12.4.42)  φ(f, ℐ) ∘ φ(g, ℐ \underline{N}) = φ(f ∘ g, ℐ).

Example 12.4.43. In the situation of lemma 12.4.37(iii), notice that 𝒪_{S} is a flat ℤ[P_{S}]-algebra, and therefore :

B(S, P_{S}^{log}, IP_{S}^{log}) = ⨁_{n∈ℕ} I^{n}𝒪_{S}.

Moreover, (12.4.24) specializes to a natural isomorphism of (X, \underline{M})-schemes :

(12.4.44)  Bl_{I\underline{M}}(X, \underline{M}) →^{∼} X ×_{S} Bl_{IP_{S}^{log}}(S, P_{S}^{log}).

We wish to give a more explicit description of the log structure of Bl_{IP_{S}^{log}}(S, P_{S}^{log}). To begin

with, recall that S′ := Proj B(S, P_{S}^{log}, IP_{S}^{log}) admits a distinguished covering by (Zariski) affine open subsets : namely, for every a ∈ I, consider the localization

P_{a} := T_{a}^{−1}P where T_{a} := {a^{n} | n ∈ ℕ}

and let Q_{a} ⊂ P_{a} be the submonoid generated by the image of P and the subset {a^{−1}b | b ∈ I}; then

S′ = ⋃_{a∈I} Spec ℤ[Q_{a}].

Hence, let us set

U_{a} := Spec(ℤ, Q_{a}) for every a ∈ I.

We claim that these locally defined log structures glue to a well defined log structure \underline{Q} on the whole of S′_{τ}. Indeed, let a, b ∈ I; we have

U_{a,b} := Spec ℤ[Q_{a}] ∩ Spec ℤ[Q_{b}] = Spec ℤ[Q_{a} ⊗_{P} Q_{b}]

and it is easily seen that Q_{a} ⊗_{P} Q_{b} = Q_{a}[b^{−1}a], i.e. the localization of Q_{a} obtained by inverting its element a^{−1}b, and this is of course the same as Q_{b}[a^{−1}b]. Then lemma 12.2.14 implies that the log structures of U_{a} and U_{b} agree on U_{a,b}, whence the contention. It is easy to check that the resulting log scheme is precisely Bl_{IP_{S}^{log}}(S, P_{S}^{log}) : the details shall be left to the reader.

Proposition 12.4.45. Let (X, \underline{M}) be a log scheme with quasi-coherent log structure, and ℐ ⊂ \underline{M} a coherent ideal. Then, the morphism (12.4.41) is a logarithmic blow up of ℐ.

Proof. Let f : (Y, \underline{N}) → (X, \underline{M}) be a morphism of log schemes, and suppose that ℐ \underline{N} is an invertible ideal of \underline{N}; in this case, we have already remarked (see (12.4.34)) that the projection π_{(Y,\underline{N},ℐ \underline{N})} : Bl_{ℐ \underline{N}} (Y, \underline{N}) → (Y, \underline{N}) is an isomorphism. We deduce a morphism :

(12.4.46)  φ(f, ℐ) ∘ π_{(Y,\underline{N},ℐ \underline{N})}^{−1} : (Y, \underline{N}) → Bl_{ℐ} (X, \underline{M}).

To conclude, it remains to show that (12.4.46) is the only morphism of log schemes whose composition with π_{(X,\underline{M},ℐ)} equals f. The latter assertion can be checked locally on X_{τ}, hence we may assume that \underline{M} admits a chart P_{X} → \underline{M}, such that ℐ = I\underline{M} for some finitely generated ideal I ⊂ P. In this case, in view of (12.4.44), the set of morphisms of (X, \underline{M})-schemes (Y, \underline{N}) → Bl_{ℐ} (X, \underline{M}) is in natural bijection with the set of (S, P_{S}^{log})-morphisms (Y, \underline{N}) → B := Bl_{IP_{S}^{log}}(S, P_{S}^{log}) (notation of example 12.4.43). In other words, we may assume that

(X, \underline{M}) = (S, P_{S}^{log}), and ℐ = IP_{S}^{log}. Then f is determined by log f, i.e. by a map β : P → \underline{N}(Y). Let a_{1}, …, a_{k} be a system of generators for I; for each i = 1, …, k, we let U_{i} ⊂ Y be the subset of all y ∈ Y for which there exists a τ-point ξ of Y with |ξ| = y and

(12.4.47)  a_{i}\underline{N}_{ξ} = I\underline{N}_{ξ}.

Notice that if y ∈ U_{i}, then (12.4.47) holds for every τ-point ξ of Y localized at y (details left to the reader).

Claim 12.4.48. The subset U_{i} is open in Y_{i} for every i = 1, …, k, and Y = ⋃_{i=1}^{k} U_{i}.

Proof of the claim. Say that y ∈ U_{i}, and let ξ be a τ-point of Y localized at y, such that (12.4.47) holds. This means that, for every j = 1, …, k, there exists u_{j} ∈ \underline{N}_{ξ} such that a_{j} = u_{j}a_{i}. Then, we may find a τ-neighborhood h : U′ → of ξ such that this identity persists in \underline{N}(U′); thus, h(U′) ⊂ U_{i}. Since h is an open map, this shows that U_{i} is an open subset.

Next, let ξ be any τ-point of Y; by assumption, we have I\underline{N}_{ξ} = b\underline{N}_{ξ} for some b ∈ \underline{N}_{ξ}; this means that for every i = 1, …, k there exists u_{i} ∈ \underline{N}_{ξ} such that a_{i} = u_{i}b. Since a_{1}, …, a_{k} generate I, we must have u_{i} ∈ \underline{N}_{ξ}^{×} for at least an index i ≤ k, in which case |ξ| ∈ U_{i}, and this shows that the U_{i} cover the whole of Y, as claimed. ♢

It is easily seen that, for every i = 1, …, k, any morphism (U_{i}, \underline{N}_{|U_{i}}) → B of (S, P_{S}^{log})- schemes factors through the open immersion Spec (Z, Q_{a_{i}}) → B (where Q_{a}, for an element a ∈ P, is defined as in example 12.4.43). Conversely, by construction β extends to a unique morphism of monoids Q_{a_{i}} → \underline{N}(U_{i}). Summing up, there exists at most one morphism of (S, P_{S}^{log})-schemes (U_{i}, \underline{N}_{|U_{i}}) → B. In light of claim 12.4.48, the proposition follows. □

12.4.49. Keep the notation of proposition 12.4.45; by inspecting the construction, it is easily seen that the log structure \underline{M}′ of Bl_{ℐ} (X, \underline{M}) is quasi-coherent, and if \underline{M} is coherent (resp. quasi-fine, resp. fine), then the same holds for \underline{M}′. However, simple examples show that \underline{M}′ may fail to be saturated, even in cases where (X, \underline{M}) is a fs log scheme. Due to the prominent role played by fs log schemes, it is convenient to introduce the special notation :

sat.Bl_{ℐ} (X, \underline{M}) := (Bl_{ℐ} (X, \underline{M}))^{qfs}

for the saturated logarithmic blow up of a coherent ideal ℐ in a quasi-fine log structure \underline{M} (notation of proposition 12.2.37). Clearly the projection sat.Bl_{ℐ} (X, \underline{M}) → (X, \underline{M}) is a final object of the category of saturated (X, \underline{M})-schemes in which the preimage of ℐ is invertible. If (X, \underline{M}) is a fine log scheme, sat.Bl_{ℐ} (X, \underline{M}) is a fs log scheme. Moreover, for any morphism of schemes f : Y → X, let \underline{M}_{Y} := f^{∗}\underline{M}; from remarks (12.4.32)(ii) and 12.2.38(iii), we deduce a natural isomorphism of (Y, \underline{M}_{Y})-schemes :

(12.4.50)  sat.Bl_{ℐ \underline{M}_{Y}} (Y, \underline{M}_{Y}) →^{∼} Y ×_{X} sat.Bl_{ℐ} (X, \underline{M}).

Theorem 12.4.51. Let (X, \underline{M}) be a quasi-fine log scheme with saturated log structure, ℐ ⊂ \underline{M} an ideal, ξ a τ-point of X. Suppose that, in a neighborhood of ξ, the ideal ℐ is generated by at most two sections, and denote :

φ : Bl_{ℐ} (X, \underline{M}) → (X, \underline{M}) (resp. φ_{sat} : sat.Bl_{ℐ} (X, \underline{M}) → (X, \underline{M}))

the logarithmic (resp. saturated logarithmic) blow up of ℐ. Then :

(i) If ℐ_{ξ} is an invertible ideal of \underline{M}_{ξ}, the natural morphisms

φ^{−1}(ξ) → Spec κ(ξ) φ_{sat}^{−1}(ξ) → Spec κ(ξ)

are isomorphisms. (ii) Otherwise, φ^{−1}(ξ) is a κ(ξ)-scheme isomorphic to ℙ_{κ(ξ)}^{1}; furthermore, the same holds

for the reduced fibre φ_{sat}^{−1}(ξ)_{red}, provided \underline{M} is fine.

Proof. After replacing X by a τ-neighborhood of ξ, we reduce to the case where \underline{M} admits an integral and saturated chart α : P_{X} → \underline{M} (lemma 12.1.18(iii)), and if \underline{M} is a fs log structure, we may also assume that the chart α is fine and sharp at ξ (corollary 12.1.36(i)). Furthermore, we may assume that ℐ is generated by at most two elements of \underline{M}(X), and if ℐ_{ξ} is principal, we may assume that the same holds for ℐ. In the latter case, since \underline{M} is integral, ℐ is invertible, hence φ and φ_{sat} are isomorphisms, so (i) follows already.

Now, suppose that ℐ_{ξ} is not invertible, and let a′, b′ ∈ \underline{M}(X) be a system of generators for ℐ; we can write a′ = α(a) · u, b′ = α(b) · v for some a, b ∈ P and u, v ∈ κ^{×}. Set t := a^{−1}b, and let J ⊂ P be the ideal generated by a and b; clearly J\underline{M} = ℐ, and P a, P b ≠ J.

Consider first the case where X = Spec κ, where κ is a field (resp. a separably closed field, in case τ = ét). In this situation, a pre-log structure on X_{τ} is the same as a morphism of monoids β : P → κ, the associated log structure is the induced map of monoids

β^{log} : P ⊗_{P_{0}} κ^{×} → κ where P_{0} := β^{−1}κ^{×}

and α is the natural map P → P ⊗_{P_{0}} κ^{×}. After replacing P by its localization P_{0}^{−1}P, we may also assume that P_{0} = P^{×}. Let

(S, P_{S}^{log}) := Spec(κ, P)  J^{∼} := JP_{S}^{log}  (Y, \underline{N}) := Bl_{J^{∼}}(S, P_{S}^{log}).

Denote by ε : κ[P] → κ the homomorphism of κ-algebras induced by β via the adjunction (4.8.50), and set I := Ker ε. In view of lemma 12.4.37(i) and (12.4.50), we have natural cartesian diagrams of κ-schemes :

(12.4.52)  \xymatrix{φ^{−1}(ξ) \ar[r] \ar[d] & (Y, \underline{N}) \ar[d] & φ_{sat}^{−1}(ξ) \ar[r] \ar[d] & (Y, \underline{N})^{sat} \ar[d] \\ |ξ| \ar[r]^{Spec ε} & S & |ξ| \ar[r]^{Spec ε} & S.}

On the other hand, (a, b) can be regarded as a pair of global sections of J^{∼}\underline{N}, so the universal property of example 12.4.29(ii) yields a morphism of (S, P_{S}^{log})-schemes :

f_{(a,b)} : (Y, \underline{N}) → ℙ_{(S,P_{S}^{log})}^{1}.

In light of example 12.4.29(i), the assertion concerning φ^{−1}(ξ) will then follow from the :

Claim 12.4.53. The morphism |ξ| ×_{S} f_{(a,b)} is an isomorphism of κ-schemes.

Proof of the claim. Let Q_{a} ⊂ P^{gp} (resp. Q_{b} ⊂ P^{gp}) be the submonoid generated by P and t (resp. by P and t^{−1}); by inspecting example 12.4.43, we see that Y is covered by two affine open subsets :

U_{a} := Spec κ[Q_{a}] U_{b} := Spec κ[Q_{b}]

and U_{a} ∩ U_{b} = Spec κ[Q_{a} ⊗_{P} Q_{b}]. On the other hand, ℙ_{(S,P_{S}^{log})}^{1} is covered as well by two affine open subsets U′_{0} and U′_{∞}, both isomorphic to Spec κ[P ⊕ ℕ], and such that U′_{0} ∩ U′_{∞} = Spec κ[P ⊕ ℤ], as usual. Moreover, a direct inspection shows that f_{(a,b)} restricts to morphisms

U_{a} → U′_{0} U_{b} → U′_{∞}

induced respectively by the maps of κ-algebras

ω_{0} : κ[P ⊕ ℕ] → κ[Q_{a}] ω_{∞} : κ[P ⊕ ℕ] → κ[Q_{b}]

such that ω_{0}(x, n) = x · t^{n} and ω_{∞}(x, n) = x · t^{−n} for every (x, n) ∈ P ⊕ ℕ. We show that \overline{ω}_{0} := ω_{0} ⊗_{κ[P]} κ[P]/I is an isomorphism; the same argument will apply also to ω_{∞}, so the claim shall follow.

Indeed, clearly the κ[P]-algebra κ[Q_{a}] is generated by t, hence ω_{0} is surjective, and then the same holds for \overline{ω}_{0}. Next, set H_{a} := I · κ[Q_{a}]; it is easily seen that H_{a} consists of all sums of the form ∑_{j=0}^{n} c_{j}t^{j}, for arbitrary n ∈ ℕ, with c_{j} ∈ I for every j = 0, …, n. Clearly, an element p(T) ∈ κ[ℕ] = κ[T] lies in Ker \overline{ω}_{0} if and only if p(t) ∈ H_{a}, so we come down to the following assertion. Let c_{0}, …, c_{n} ∈ κ[P] such that

(12.4.54)  ∑_{j=0}^{n} c_{j}t^{j} = 0

in κ[Q_{a}]; then c_{j} lies in the ideal κ[β^{−1}(0)] of κ[P], for every j = 0, …, n. Since P is integral, (12.4.54) is equivalent to the identity : ∑_{j=0}^{n} c_{j}a^{n−j}b^{j} = 0 in κ[P]. For every x ∈ P, denote by π_{x} : κ[P] → κ the κ-linear projection such that π_{x}(x) = 1, and π_{x}(y) = 0 for every y ∈ P \{x}.

Suppose, by way of contradiction, that c_{i} ∉ κ[β^{−1}(0)] for some i ≤ n, hence π_{x}(c_{i}) ≠ 0 for some x ∈ P_{0}; since P_{0} = P^{×}, we may replace c_{j} by x^{−1}c_{j}, for every j ≤ n, and assume that π_{1}(c_{i}) ≠ 0, hence π_{a^{n−i}b^{i}}(c_{i}a^{n−i}b^{i}) ≠ 0 (again, using the assumption that P is integral). Thus, there exists j ≠ i with j ≤ n, such that π_{a^{n−i}b^{i}}(c_{j}a^{n−j}b^{j}) ≠ 0, and we may then find an element c ∈ P such that π_{a^{n−i}b^{i}}(ca^{n−j}b^{j}) = 1, i.e. ca^{n−j}b^{j} = a^{n−i}b^{i}; up to swapping the roles of a and b, we may assume that i > j, in which case we may write t^{i−j} = c; since P is saturated, it follows that t ∈ P, hence J is generated by a, which is excluded. ♢

Next, we assume that \underline{M} is a fs log structure (and X is still Spec κ), and we consider the morphism φ_{sat}. As already remarked, we may assume that α is sharp at ξ, and P fine and saturated; the sharpness condition amounts to saying that β(x) = 0 for every x ∈ P \ {1}, therefore I is the augmentation ideal of the graded κ-algebra κ[P]. By inspecting the proof of proposition 12.2.37, we see that (Y, \underline{N})^{sat} is covered by two affine open subsets

U_{a}^{fs} := Spec κ[Q_{a}^{sat}] U_{b}^{fs} := Spec κ[Q_{b}^{sat}]

and U_{a}^{fs} ∩ U_{b}^{fs} = Spec κ[Q_{a}^{sat} ⊗_{P} Q_{b}^{sat}]. Since J is not principal, we have t ∉ P, and since P is saturated, we deduce that t is not a torsion element of P^{gp}; as the latter is a finitely generated abelian group, it follows that we may find a unimodular element u ∈ P^{gp} such that t lies in the submonoid ℕu ⊂ P^{gp} generated by u; this condition means that t = u^{k} for some k ∈ ℕ, and ℕu is not properly contained in another rank one free submonoid of P^{gp}. Write u = a′^{−1}b′ for some a′, b′ ∈ P, let J′ ⊂ P be the ideal generated by a′ and b′, and R_{a′} (resp. R_{b′}) the submonoid of P^{gp} generated by P and u (resp. by P and u^{−1}); clearly R_{a′}^{sat} = Q_{a}^{sat}, and R_{b′}^{sat} = Q_{b}^{sat}. Denote by \underline{N}′ the log structure of (Y, \underline{N})^{sat}, and 𝒥′ := J′\underline{N}′; it is easily seen that

𝒥′_{|U_{a}^{fs}} = a′\underline{N}′_{|U_{a}^{fs}} 𝒥′_{|U_{b}^{fs}} = b′\underline{N}′_{|U_{b}^{fs}}

hence 𝒥′ is invertible, and example 12.4.29(ii) yields a morphism of (S, P_{S}^{log})-schemes

f_{(a′,b′)} : (Y, \underline{N})^{sat} → ℙ_{(S,P_{S}^{log})}^{1}.

In light of example 12.4.29(i), the assertion concerning φ_{sat}^{−1}(ξ) will then follow from the :

Claim 12.4.55. (|ξ| ×_{S} f_{(a′,b′)})_{red} : φ_{sat}^{−1}(ξ)_{red} → ℙ_{κ}^{1} is an isomorphism of κ-schemes.

Proof of the claim. As in the proof of claim 12.4.53, the morphism f_{(a′,b′)} restricts to morphisms U_{a}^{fs} → U′_{0} and U_{b}^{fs} → U′_{∞} induced by maps of κ-algebras :

ω_{0} : κ[P ⊕ ℕ] → κ[Q_{a}^{sat}] ω_{∞} : κ[P ⊕ ℕ] → κ[Q_{b}^{sat}]

such that ω_{0}(x, n) = x · u^{n} and ω_{∞}(x, n) = x · u^{−n} for every (x, n) ∈ P ⊕ ℕ, and again, it suffices to show that

(ω_{0} ⊗_{κ[P]} κ[P]/I)_{red} : κ[T] → (κ[R_{a}^{sat}]/Iκ[R_{a}^{sat}])_{red}

is an isomorphism (where for any ring A, we denote A_{red} be the maximal reduced quotient of A, i.e. A_{red} := A/nil(A), where nil(A) is the nilradical of A). We break the latter verification in two steps : first, let us check that the map

\overline{ω}′_{0} : κ[T] → κ[R_{a}]/Iκ[R_{a}] p(T) ↦ p(u) (mod Iκ[R_{a}])

is an isomorphism. Indeed, \overline{ω}′_{0} is induced by the map of monoids φ : ℕ → R_{a} such that n ↦ u^{n} for every n ∈ ℕ; if t = u^{k}, the map φ fits into the cocartesian diagram :

\xymatrix{ℕ \ar[r]^{ψ} \ar[d]_{_{k_{ℕ}}} & Q_{a} \ar[d]^{j} \\ ℕ \ar[r]^{φ} & R_{a}}

where k_{ℕ} is the k-Frobenius map, ψ is given by the rule : n ↦ t^{n} for every n ∈ ℕ, and j is the natural inclusion. Hence :

\overline{ω}′_{0} = (κ[ψ] ⊗_{κ[P]} κ[P]/I) ⊗_{κ[T^{k}]} κ[T].

However, the proof of claim 12.4.53 shows that κ[ψ] ⊗_{κ[P]} κ[P]/I is an isomorphism, whence the contention. Lastly, let show that the natural map

\overline{ω}′′_{0} : κ[R_{a}]/Iκ[R_{a}] → (κ[R_{a}^{sat}]/Iκ[R_{a}^{sat}])_{red}

is an isomorphism. Indeed, it is clear that the natural map ω′′_{0} : κ[R_{a}] → κ[R_{a}^{sat}] is integral and injective, hence Spec ω′′_{0} is surjective; therefore Spec \overline{ω}′′_{0} is still surjective and integral. However, the foregoing shows that κ[R_{a}]/Iκ[R_{a}] is reduced, so we deduce that \overline{ω}′′_{0} is injective. To show that \overline{ω}′′_{0} is surjective, it suffices to show that the classes of the generating system R_{a}^{sat} ⊂ κ[R_{a}^{sat}] lie in the image of \overline{ω}′′_{0}. Hence, let x ∈ R_{a}^{gp}, with x^{m} ∈ R_{a} for some m > 0, so that x^{m} = y · u^{n} for some n ∈ ℕ and y ∈ P. If y ≠ 1, we have y ∈ I, hence the image of x^{m} vanishes in κ[R_{a}^{sat}]/Iκ[R_{a}^{sat}], and the image of x vanishes in the reduced quotient; finally, if y = 1, the identity x^{m} = u^{n} implies that m divides n, since u is unimodular; hence x = u^{n/m} and the image of x agrees with \overline{ω}′′_{0}(u^{n/m}). ♢

Finally, let us return to a general quasi-fine log scheme (X, \underline{M}); the theorem will follow from the more precise :

Claim 12.4.56. In the situation of the theorem, suppose moreover that (a) \underline{M} admits a saturated chart α : P_{X} → \underline{M}

(b) ℐ = J\underline{M}, where J ⊂ P is an ideal generated by two elements a, b ∈ P

(c) ℐ_{ξ} is not invertible. Then we have :

(i) There exists a morphism of (X, \underline{M})-schemes : Bl_{ℐ} (X, \underline{M}) → ℙ_{(X,\underline{M})}^{1} inducing an isomorphism of κ(ξ)-schemes φ^{−1}(ξ) →^{∼} ℙ_{κ(ξ)}^{1}.

(ii) If furthermore, P is fine (and saturated) and α is sharp at ξ, then there exists a morphism of (X, \underline{M})-schemes : sat.Bl_{ℐ} (X, \underline{M}) → ℙ_{(X,\underline{M})}^{1} inducing an isomorphism of κ(ξ)- schemes φ_{sat}^{−1}(ξ)_{red} →^{∼} ℙ_{κ(ξ)}^{1}.

Proof of the claim. (i): Denote by \underline{N} the log structure of Bl_{ℐ} (X, \underline{M}); the elements a, b define global sections of the invertible \underline{N}-module ℐ \underline{N}, and we claim that the corresponding morphism of (X, \underline{M})-schemes f_{(a,b)} : Bl_{ℐ} (X, \underline{M}) → ℙ_{(X,\underline{M})}^{1} will do. Indeed, set (|ξ|, \underline{M}_{ξ}) := |ξ| ×_{X} (X, \underline{M}), and recall that there exists natural isomorphisms

|ξ| ×_{X} ℙ_{(X,\underline{M})}^{1} →^{∼} ℙ_{(|ξ|,\underline{M}_{ξ})}^{1} |ξ| ×_{X} Bl_{ℐ} (X, \underline{M}) →^{∼} Bl_{ℐ \underline{M}_{ξ}} (|ξ|, \underline{M}_{ξ})

(example 12.4.29(i) and remark 12.4.32(ii)). Denote by \underline{N}_{ξ} the log structure of Bl_{ℐ \underline{M}_{ξ}} (|ξ|, \underline{M}_{ξ}); by example 12.4.29(iii), the base change

|ξ| ×_{X} f : Bl_{ℐ \underline{M}_{ξ}} (|ξ|, \underline{M}_{ξ}) → ℙ_{(|ξ|,\underline{M}_{ξ})}^{1}

is the unique morphism f_{(\overline{a},\overline{b})} of (|ξ|, \underline{M}_{ξ})-schemes corresponding to the pair (\overline{a}, \overline{b}) of global sections of ℐ \underline{N}_{ξ} obtained by pulling back the pair (a, b). Therefore, in order to check that |ξ|×_{X} f is an isomorphism, we may replace from start (X, \underline{M}) by (|ξ|, \underline{M}_{ξ}) (whose log structure is still quasi-fine, by lemma 12.1.18(i), and assume that X = Spec κ, where κ is a field (resp. a separably closed field, in case τ = ét), in which case the assertion is just claim 12.4.53.

(ii): Denote by \underline{N}′ the log structure of sat.Bl_{ℐ} (X, \underline{M}), define a′, b′ and J′ as in the foregoing, and set again 𝒥′ := J′\underline{N}′. Again, it is easily seen that 𝒥′ is an invertible \underline{N}′-module, and the pair (a′, b′) yields a morphism f_{(a′,b′)} : sat.Bl_{ℐ} (X, \underline{M}) → ℙ_{(X,\underline{M})}^{1} which fulfills the sought condition. Indeed, denote by \underline{N}′_{ξ} the log structure of sat.Bl_{ℐ \underline{M}_{ξ}} (|ξ|, \underline{M}_{ξ}); in light of (12.4.50) and example 12.4.29(iii), the base change

|ξ| ×_{X} f_{(a′,b′)} : sat.Bl_{ℐ \underline{M}_{ξ}} (|ξ|, \underline{M}_{ξ}) → ℙ_{(|ξ|,\underline{M}_{ξ})}^{1}

is the unique morphism f_{(\overline{a}′,\overline{b}′)} of (|ξ|, \underline{M}_{ξ})-schemes corresponding to the pair (\overline{a}′, \overline{b}′) of global sections of 𝒥′\underline{N}′_{ξ} obtained by pulling back the pair (a′, b′). Thus, the assertion is just claim 12.4.55. □

12.5. Regular log schemes. In this section we introduce the logarithmic version of the classical regularity condition for locally noetherian schemes. This theory is essentially due to K.Kato ([112]), and we mainly follow his exposition, except in a few places where his original arguments are slightly flawed, in which cases we supply the necessary corrections.

Lemma 12.5.1. Let A be a noetherian local ring, N an A-module of finite type, α : P → A and β : Q → A two morphisms of pointed monoids, with P^{♯} and Q^{♯} both fine. Suppose that α and β induce the same constant log structure on Spec A (see (12.1.15)). Then N is α-flat if and only if it is β-flat.

Proof. Let ξ be a τ-point localized at the closed point of X := Spec A, set B := 𝒪_{X,ξ} and let φ : A → B be the natural map. Let also M be the push-out of the diagram of monoids P ← (φ ∘ α)^{−1}B^{×} → B^{×} deduced from α; then M ≃ P_{X,ξ}^{log}, the stalk at the point ξ of the constant log structure on X_{τ} associated to α. Since φ is faithfully flat, it is easily seen that N is α-flat if and only if N ⊗_{A} B is φ ∘ α-flat.

Hence we may replace A by B, α by φ ∘ α, N by N ⊗_{A} B, and Q by M, after which we may assume that Q = P ⨿_{α^{−1}(A^{×})} A^{×}; especially, there exists a morphism of monoids γ : P → Q such that α = β ∘ γ, and moreover β is a local morphism.

Next, set S := α^{−1}(A^{×}); clearly γ extends to a morphism of monoids γ′ : S^{−1}P → Q, and α and β ∘ γ′ induce the same constant log structure on X_{τ}. Arguing as in the proof of proposition 6.4.31, we see that N is P -flat if and only if it is S^{−1}P -flat. Hence, we may replace P by S^{−1}P, which allows to assume that γ induces an isomorphism P ⨿_{P^{×}} A^{×} →^{∼} Q, therefore also an isomorphism P^{♯} →^{∼} Q^{♯}. The latter implies that 𝔪_{Q} = 𝔪_{P} Q; moreover, notice that the morphism of monoids P^{×} → A^{×} is faithfully flat, so the natural map :

Tor_{1}^{ℤ⟨P ⟩}(N, ℤ⟨P/𝔪_{P} ⟩) → Tor_{1}^{ℤ⟨Q⟩}(N, ℤ⟨(P/𝔪_{P}) ⊗_{P} Q⟩) → Tor_{1}^{ℤ⟨Q⟩}(N, ℤ⟨Q/𝔪_{Q}⟩)

is an isomorphism. The assertion follows. □

Lemma 12.5.2. Let M be an integral monoid, A a ring, φ : M → A a morphism of monoids, and set S := Spec A. Suppose that A is φ-flat. Then the log structure (M, φ)_{S}^{log} on S_{τ} is the subsheaf of monoids of 𝒪_{S} generated by 𝒪_{S}^{×} and the image of M.

Proof. To ease notation, set \underline{M} := (M, φ)_{S}^{log}; let ξ be any τ-point of S. Then the stalk \underline{M}_{ξ} is the push-out of the diagram : 𝒪_{S,ξ}^{×} ← φ_{ξ}^{−1}(𝒪_{S,ξ}^{×}) → M where φ_{ξ} : M → 𝒪_{S,ξ} is deduced from φ. Hence \underline{M}_{ξ} is generated by 𝒪_{S,ξ}^{×} and the image of M, and it remains only to show that the structure map \underline{M}_{ξ} → 𝒪_{S,ξ} is injective. Therefore, let a, b ∈ M and u, v ∈ 𝒪_{S,ξ}^{×} such that :

(12.5.3)  φ_{ξ}(a) · u = φ_{ξ}(b) · v.

We come down to showing :

Claim 12.5.4. There exist c, d ∈ M such that :

φ_{ξ}(c), φ_{ξ}(d) ∈ 𝒪_{S,ξ}^{×}  ac = bd  φ_{ξ}(c) · v = φ_{ξ}(d) · u.

Proof of the claim. Let 𝔪_{ξ} ⊂ 𝒪_{S,ξ} be the maximal ideal, and set 𝔭 := φ_{ξ}^{−1}(𝔪_{ξ}), so that φ_{ξ} extends to a local morphism φ_{𝔭} : M_{𝔭} → 𝒪_{S,ξ}. Since 𝒪_{S,ξ} is a flat A-algebra, 𝒪_{S,ξ} is φ-flat, and consequently it is faithfully φ_{𝔭}-flat (lemma 6.1.36). Then, assumption (12.5.3) leads to the identity:

aM_{𝔭} ⊗_{M_{𝔭}} 𝒪_{S,ξ} = φ_{ξ}(a) · 𝒪_{S,ξ} = φ_{ξ}(b) · 𝒪_{S,ξ} = bM_{𝔭} ⊗_{M_{𝔭}} 𝒪_{S,ξ}

whence aM_{𝔭} = bM_{𝔭}, by faithful φ_{𝔭}-flatness. It follows that there exist x, y ∈ M_{𝔭} such that ax = b and by = a, hence axy = a, which implies that xy = 1, since M_{𝔭} is an integral module. The latter means that there exist c, d ∈ M \𝔭 such that ac = bd in M. We deduce easily that

φ_{ξ}(a) · φ_{ξ}(c) · v = φ_{ξ}(a) · φ_{ξ}(d) · u.

Thus, to complete the proof, it suffices to show that φ_{ξ}(a) is regular in 𝒪_{S,ξ}. However, the morphism of M-modules μ_{a} : M → M : m ↦ am (for all m ∈ M) is injective, hence the same holds for the map μ_{a} ⊗_{M} 𝒪_{S,ξ} : 𝒪_{S,ξ} → 𝒪_{S,ξ}, which is just multiplication by φ(a). □

12.5.5. Let (X, \underline{M}) be a locally noetherian log scheme, with coherent log structure (on the site X_{τ}, see (12.2.1)), and let ξ be any τ-point of X. We denote by I(ξ, \underline{M}) ⊂ 𝒪_{X,ξ} the ideal generated by the image of the maximal ideal of \underline{M}_{ξ}, and we set :

d(ξ, \underline{M}) := dim 𝒪_{X,ξ}/I(ξ, \underline{M}) + dim \underline{M}_{ξ}.

Lemma 12.5.6. In the situation of (12.5.5), suppose furthermore that (X, \underline{M}) is a fs log scheme. Then we have the inequality :

dim 𝒪_{X,ξ} ≤ d(ξ, \underline{M}).

Proof. According to corollary 12.1.36(i), there exist a neighborhood U → X of ξ in X_{τ}, and a fine and saturated chart α : P_{U} → \underline{M}_{|U}, which is sharp at the point ξ. Especially, P ≃ \underline{M}_{ξ}^{♯}, therefore dim P = dim \underline{M}_{ξ} (corollary 6.4.12(ii)). Notice that 𝒪_{X,ξ} is a noetherian local ring (this is obvious for τ = Zar, and follows from [66, Ch.IV, Prop.18.8.8(iv)] for τ = ét), hence to conclude it suffices to apply corollary 6.4.27(i) to the induced map of monoids P → 𝒪_{X,ξ}. □

Definition 12.5.7. Let (X, \underline{M}) be a locally noetherian fs log scheme, ξ a τ-point of X.

(i) We say that (X, \underline{M}) is regular at the point ξ, if the following holds : (a) the inequality of lemma (12.5.6) is actually an equality, and

(b) the local ring 𝒪_{X,ξ}/I(ξ, \underline{M}) is regular. (ii) We denote by (X, \underline{M})_{reg} ⊂ X the set of points x such that (X, \underline{M}) is regular at any

(hence all) τ-points of X localized at x. (iii) We say that (X, \underline{M}) is regular, if (X, \underline{M})_{reg} = X. (iv) Let K be a field, and X a K-scheme. We say that the K-log scheme (X, \underline{M}) is geo-

metrically regular, if E×_{K}(X, \underline{M}) is regular, for every finite field extension E of K.

Remark 12.5.8. (i) Certain constructions produce log structures \underline{M} → 𝒪_{X} that are morphisms of pointed monoids. It is then useful to extend the notion of regularity to such log structures. We shall say that (X, \underline{M}) is a pointed regular log scheme, if there exists a log structure \underline{N} on X, such that \underline{M} = \underline{N}_{∘} (notation of (12.1.11)), and (X, \underline{N}) is a regular log scheme.

(ii) Likewise, if K is a field, X a K-scheme, and \underline{M} a log structure on X, we shall say that the K-log scheme (X, \underline{M}) is geometrically pointed regular, if \underline{M} = \underline{N}_{∘} for some log structure \underline{N} on X, such that (X, \underline{N}) is geometrically regular.

12.5.9. Let (X, \underline{M}) be a locally noetherian fs log scheme, ξ a τ-point of X, and 𝒪_{X,ξ}^{∧} the completion of 𝒪_{X,ξ}. The next result is the logarithmic version of the classical characterization of complete regular local rings ([126, Th.29.7 and Th.29.8]).

Theorem 12.5.10. With the notation of (12.5.9), the log scheme (X, \underline{M}) is regular at the point ξ if and only if there exist :

(a) a complete regular local ring (R, 𝔪_{R}), and a local ring homomorphism R → 𝒪_{X,ξ}^{∧};

(b) a fine and saturated chart P_{X(ξ)} → \underline{M}(ξ) which is sharp at the closed point ξ of X(ξ), such that the induced continuous ring homomorphism

R[[P]] → 𝒪_{X,ξ}^{∧}

is an isomorphism if 𝒪_{X,ξ} contains a field, and otherwise it is a surjection, with kernel generated by an element ϑ ∈ R[[P]] whose constant term lies in 𝔪_{R}\𝔪_{R}^{2}.

Proof. Suppose first that (a) and (b) hold. If R contains a field, then it follows that

dim 𝒪_{X,ξ} = dim 𝒪_{X,ξ}^{∧} = dim R + dim P

by corollary 6.4.27(ii); moreover, in this case I(ξ, \underline{M}) = 𝔪_{P} 𝒪_{X,ξ}, hence 𝒪_{X,ξ}/I(ξ, \underline{M}) = 𝒪_{X,ξ}/𝔪_{P} 𝒪_{X,ξ}, whose completion is 𝒪_{X,ξ}^{∧}/𝔪_{P} 𝒪_{X,ξ}^{∧} ≃ R so that 𝒪_{X,ξ}/I(ξ, \underline{M}) is regular ([63, Ch.0, Prop.17.3.3(i)]). Furthermore,

dim R = dim 𝒪_{X,ξ}^{∧}/𝔪_{P} 𝒪_{X,ξ}^{∧} = dim 𝒪_{X,ξ}/I(ξ, \underline{M})

([126, Th.15.1]). Hence (X, \underline{M}) is regular at the point ξ. If R does not contain a field, we obtain dim 𝒪_{X,ξ} = dim R + dim P − 1. On the other hand, let ϑ_{0} be the image of ϑ in 𝔪_{R}; then we have 𝒪_{X,ξ}^{∧}/𝔪_{P} 𝒪_{X,ξ}^{∧} ≃ R/ϑ_{0}R, which is regular of dimension dim R − 1, and again we invoke [63, Ch.0, Prop.17.3.3(i)] to see that (X, \underline{M}) is regular at ξ.

Conversely, suppose that (X, \underline{M}) is regular at ξ. Suppose first that 𝒪_{X,ξ} contains a field; then we may find a field k ⊂ 𝒪_{X,ξ}^{∧} mapping isomorphically to the residue field of 𝒪_{X,ξ}^{∧} ([126, Th.28.3]). Pick a sequence (t_{1}, …, t_{r}) of elements of 𝒪_{X,ξ} whose image in the regular local ring 𝒪_{X,ξ}/I(ξ, \underline{M}) forms a regular system of parameters. Let also P a fine saturated monoid for which there exist a neighborhood U → X of ξ and a chart P_{U} → \underline{M}_{|U}, sharp at the point ξ. There follows a map of monoids α : P → 𝒪_{X,ξ}, and necessarily the image of 𝔪_{P} lies in the maximal ideal of 𝒪_{X,ξ}, and generates I(ξ, \underline{M}). We deduce a continuous ring homomorphism

k[[P × ℕ^{⊕r}]] → 𝒪_{X,ξ}^{∧}

which extends α, and which maps the generators T_{1}, …, T_{r} of ℕ^{⊕r} onto respectively t_{1}, …, t_{r}. This map is clearly surjective, and by comparing dimensions (using corollary 6.4.27(ii)) one sees that it is an isomorphism. Then the theorem holds in this case, with R := k[[ℕ^{⊕r}]].

Next, if 𝒪_{X,ξ} does not contain a field, then its residue characteristic is a positive integer p, and we may find a complete discrete valuation ring V ⊂ 𝒪_{X,ξ}^{∧} whose maximal ideal is pV, and such that V/pV maps isomorphically onto the residue field of 𝒪_{X,ξ}^{∧} ([126, Th.29.3]). Again, we choose a morphism of monoids α : P → 𝒪_{X,ξ} as in the foregoing, and a sequence (t_{1}, …, t_{r}) of elements of 𝒪_{X,ξ} lifting a regular system of parameters for 𝒪_{X,ξ}/I(ξ, \underline{M}), by means of which we define a continuous ring homomorphism

φ : V [[P × ℕ^{⊕r}]] → 𝒪_{X,ξ}^{∧}

as in the previous case. Again, it is clear that φ is surjective. The image of the ideal J generated by the maximal ideal of P × ℕ^{⊕r} is the maximal ideal of 𝒪_{X,ξ}^{∧}; in particular, there exists x ∈ J such that ϑ := p − x lies in Ker φ. If we let R := V [[ℕ^{⊕r}]], it is clear that ϑ ∈ 𝔪_{R} \ 𝔪_{R}^{2}.

Claim 12.5.11. Let A be a ring, π a regular element of A such that A/πA is an integral domain, P a fine and sharp monoid. Let also ϑ be an element of A[[P]] whose constant term is π (see (6.4.25)). Then A[[P]]/(ϑ) is an integral domain.

Proof of the claim. To ease notation, set A_{0} := A/πA, B := A[[P]] and C := B/ϑB. Choose a decomposition (6.4.26), and set Fil_{n}^{γ}B := ∏_{i≥n} A[[gr_{i}^{γ}P]] for every n ∈ ℕ. Fil_{•}^{γ}B is a separated filtration by ideals of B, and we may consider the induced filtration Fil_{•}^{γ}C on C. First, we remark that Fil_{•}^{γ}C is also separated. This comes down to checking that

⋂_{n≥0} ϑB + Fil_{n}^{γ}B = ϑB.

To this aim, suppose that, for a given x ∈ B we have identities of the type x = ϑy_{n} + z_{n}, with y_{n} ∈ B and z_{n} ∈ Fil_{n}^{γ}B for every n ∈ ℕ. Then, since π is regular, an easy induction shows that gr_{i}^{γ}(y_{n}) = gr_{i}^{γ}(y_{m}) whenever n, m > i, and moreover x = ϑ · ∑_{i∈ℕ} gr_{i}^{γ}(y_{i+1}), which shows the contention. It follows that, in order to show that C is a domain, it suffices to show that the same holds for the graded ring gr_{•}^{γ}C associated to Fil_{•}^{γ}C. However, notice that :

Fil_{n}^{γ}B ∩ ϑB = ϑ · Fil_{n}^{γ}B for every n ∈ ℕ.

(Indeed, this follows easily from the fact that π is a regular element : the verification shall be left to the reader). Hence, we may compute :

gr_{n}^{γ}C = \frac{Fil_{n}^{γ}B + ϑB}{Fil_{n+1}^{γ} + ϑB} ≃ \frac{Fil_{n}^{γ}B}{Fil_{n+1}^{γ} + ϑFil_{n}^{γ}} ≃ A[gr_{n}^{γ}P]/ϑA[gr_{n}^{γ}P] ≃ A_{0}[gr_{n}^{γ}P].

Thus, gr_{•}^{γ}A ≃ A_{0}[P], which is a domain, since by assumption A_{0} is a domain. ♢

From claim 12.5.11(ii) we deduce that R[[P]]/(ϑ) is an integral domain. Then, again by comparing dimensions, we see that φ factors through an isomorphism R[[P]]/(ϑ) →^{∼} 𝒪_{X,ξ}^{∧}. □

Remark 12.5.12. Resume the notation of (12.5.9), and suppose that 𝒪_{X,ξ}/I(ξ, \underline{M}) is a regular local ring. Let P_{X(ξ)} → \underline{M}(ξ) be a chart as in theorem 12.5.10(b), and α : P → 𝒪_{X,ξ} the corresponding morphism of monoids. Moreover, if 𝒪_{X,ξ} contains a field, let V denote a coefficient field of 𝒪_{X,ξ}^{∧}, and otherwise, let V be a complete discrete valuation ring whose maximal ideal is generated by p, the residue characteristic of 𝒪_{X,ξ}. In either case, pick a ring homomorphism V → 𝒪_{X,ξ}^{∧} inducing an isomorphism of V/pV onto the residue field of 𝒪_{X,ξ}. Let as well t_{1}, …, t_{r} ∈ 𝒪_{X,ξ} be any sequence of elements whose image in 𝒪_{X,ξ}/I(ξ, \underline{M}) forms a regular system of parameters, and extend the map α to a morphism of monoids P × ℕ^{⊕r} → 𝒪_{X,ξ}, by the rule : e_{i} ↦ t_{i}, where e_{1}, …, e_{r} is the natural basis of ℕ^{⊕r}. Then by inspecting the proof of theorem 12.5.10, we see that the induced continuous ring homomorphism :

V [[P × ℕ^{⊕r}]] → 𝒪_{X,ξ}^{∧}

is always surjective, and if V is not a field, its kernel contains an element ϑ whose constant term in V is ϑ_{0} = p. If V is a field (resp. if V is a discrete valuation ring) then (X, \underline{M}) is regular at the point ξ, if and only if this map is an isomorphism (resp. if and only if the kernel of this map is generated by ϑ).

Corollary 12.5.13. Let (X, \underline{M}) be a regular log scheme. Then the scheme X is normal and Cohen-Macaulay.

Proof. Let x ∈ X be any point, and ξ a τ-point localized at x; we have to show that 𝒪_{X,x} is Cohen-Macaulay; in light of [66, Ch.IV, Cor.18.8.13(a)] (when τ = ét), it suffices to show that the same holds for 𝒪_{X,ξ}. Then [126, Th.17.5] further reduces to showing that the completion 𝒪_{X,ξ}^{∧} is Cohen-Macaulay; the latter follows easily from theorems 12.5.10 and 11.6.35(i). Next, in order to prove that X is normal, it suffices to show that 𝒪_{X,x} is regular, whenever x has codimension one in X ([126, Th.23.8]). Again, by [66, Ch.IV, Cor.18.8.13(c)] (when τ = ét) and [63, Ch.0, Prop.17.1.5], we reduce to showing that 𝒪_{X,ξ}^{∧} is regular for such a point x. However, for a point of codimension one we have r := dim \underline{M}_{ξ} ≤ 1. If r = 0, then I(ξ, \underline{M}) = \{0\}, hence 𝒪_{X,ξ} is regular. Lastly, if r = 1, we see that \underline{M}_{ξ}^{♯} ≃ ℕ (theorem 6.4.18(iii)); consequently, there exist a regular local ring R and an isomorphism 𝒪_{X,ξ}^{∧} ≃ R[[ℕ]]/(ϑ), where ϑ = 0 if 𝒪_{X,ξ} contains a field, and otherwise the constant term of ϑ lies in 𝔪_{R} \ 𝔪_{R}^{2}. Since R[[ℕ]] is again regular, the assertion follows in either case. □

Corollary 12.5.14. Let i : (X′, \underline{M}′) → (X, \underline{M}) be an exact closed immersion of regular log schemes (see definition 12.3.22(i)). Then the underlying morphism of schemes X′ → X is a regular closed immersion.

Proof. Let ξ be a τ-point of X, and denote by J the kernel of i_{ξ}^{♮} : 𝒪_{X,ξ} → 𝒪_{X′,ξ}. To ease notation, let as well A := 𝒪_{X,ξ}/I(ξ, \underline{M}) and A′ := 𝒪_{X′,ξ}/I(ξ, \underline{M}′) Since log i : i^{∗}\underline{M} → \underline{M}′ is an isomorphism, i_{ξ}^{♮} induces an isomorphism :

𝒪_{X,ξ}/(J + I(ξ, \underline{M})) →^{∼} A′.

Since A and A′ are regular, there exists a sequence of elements t_{1}, …, t_{k} ∈ J, whose image in A forms the beginning of a regular system of parameters and generate the kernel of the induced map A → A′ ([63, Ch.0, Cor.17.1.9]). Extend this sequence by suitable elements of 𝒪_{X,ξ}, to obtain a sequence (t_{1}, …, t_{r}) whose image in A is a regular system of parameters. We deduce a surjection φ : V [[P × ℕ^{⊕r}]] → 𝒪_{X,ξ}^{∧} as in remark 12.5.12, where V is either a field or a complete discrete valuation ring. Denote by (e_{1}, …, e_{r}) the natural basis of ℕ^{⊕r}. Now, suppose first that V is a complete discrete valuation ring; then Ker φ is generated by an element ϑ ∈ V [[P × ℕ^{⊕r}]], whose constant term is a uniformizer in V; we deduce easily from claim 12.5.11that (e_{1}, …, e_{r}, ϑ) is a regular sequence in V [[P × ℕ^{⊕r}]]; hence the same holds for the sequence (ϑ, e_{1}, …, e_{r}), in view of [126, p.127, Cor.]. This implies that (t_{1}, …, t_{r}) is a regular sequence in 𝒪_{X,ξ}^{∧}, hence (t_{1}, …, t_{k}) is a regular sequence in 𝒪_{X,ξ}, which is the contention. The case where V is a field is analogous, though simpler : the details shall be left to the reader. □

Remark 12.5.15. In the situation of remark 12.5.12, suppose that P is a free monoid of finite rank d, let {f_{1}, …, f_{d}} ⊂ 𝒪_{X,ξ} be the image of the (unique) basis of P, and for every i ≤ d let Z_{i} ⊂ X be the zero locus of f_{i}; then ⋃_{i=1}^{d} Z_{i} is a strict normal crossings divisor in the sense of example 12.2.11. The proof is analogous to that of corollary 12.5.14 : the sequence (f_{1}, …, f_{d}, ϑ) is regular in V [[P ⊕ ℕ^{⊕r}]], hence also its permutation (ϑ, f_{1}, …, f_{d}) is regular, whence the claim (the details shall be left to the reader).

Proposition 12.5.16. Resume the situation of (12.5.9). We have :

(i) The log scheme (X, \underline{M}) is regular at the τ-point ξ if and only if the following two conditions hold :

(a) The ring 𝒪_{X,ξ}/I(ξ, \underline{M}) is regular.

(b) There exists a morphism of monoids P → 𝒪_{X,ξ} from a fine monoid P, whose associated constant log structure on X(ξ) is the same as \underline{M}(ξ), and such that 𝒪_{X,ξ} is P -flat.

(ii) Moreover, if the equivalent conditions of (i) hold, and Q_{X} → \underline{M}(ξ) is any fine chart, then 𝒪_{X,ξ} is Q-flat, for the induced map of monoids Q → 𝒪_{X,ξ}.

Proof. (i): Suppose first that (X, \underline{M}) is regular at the point ξ; then by definition (a) holds. Next, we may find a fine monoid P and a local morphism α : P → A := 𝒪_{X,ξ} whose associated constant log structure on X(ξ) is the same as \underline{M}(ξ), and a regular local ring R, with a ring homomorphism R → A such that the induced continuous map R[[P]] → A^{∧} fulfills condition (b) of theorem 12.5.10 (where A^{∧} is the completion of A). Since the completion map φ : A → A^{∧} is faithfully flat, A is α-flat if and only if A^{∧} is φ ∘ α-flat; in light of proposition 6.1.39(i), it then suffices to show that, for every ideal I ⊂ P, the natural map

A^{∧} ⊗^{L}_{P} P/I → A^{∧} ⊗_{P} P/I

is an isomorphism in D^{−}(A^{∧}-Mod) (notation of (6.1.33)). To this aim we remark :

Claim 12.5.17. For any ideal I ⊂ P, the natural morphism

R[P] ⊗^{L}_{P} P/I → R⟨P/I⟩

is an isomorphism in D^{−}(R[P]-Mod).

Proof of the claim. We consider the change of rings spectral sequence for Tor :

E_{ij}^{2} := Tor_{i}^{ℤ[P]}(Tor_{j}^{ℤ}(R, ℤ[P]), ℤ⟨P/I⟩) ⇒ Tor_{i+j}^{ℤ}(R, ℤ⟨P/I⟩)

([163, Th.5.6.6]). Clearly E_{ij}^{2} = 0 whenever j > 0, whence isomorphisms :

Tor_{i}^{ℤ⟨P ⟩}(R⟨P ⟩, ℤ⟨P/I⟩) →^{∼} Tor_{i}^{ℤ}(R, ℤ⟨P/I⟩)

for every i ∈ ℕ. The claim follows easily. ♢ In light of claim 12.5.17we deduce natural isomorphisms :

A^{∧} ⊗^{L}_{P} P/I →^{∼} A^{∧} ⊗^{L}_{R[P]} (R[P] ⊗^{L}_{P} P/I) →^{∼} A^{∧} ⊗^{L}_{R[P]} R⟨P/I⟩

in D^{−}(A^{∧}-Mod). Now, if A contains a field, we have A^{∧} ≃ R[[P]], which is a flat R[P]-algebra ([126, Th.8.8]), and the contention follows. If A does not contain a field, the complex

0 → R[[P]] ⟶^{ϑ} R[[P]] → A^{∧} → 0

is a R[P]-flat resolution of A^{∧}. Since R[[P]]/IR[[P]] ≃ lim_{n∈ℕ} R⟨P/(I ∪ 𝔪_{P}^{n})⟩, we come down to the following :

Claim 12.5.18. Let ϑ ∈ R[P] be any element whose constant term ϑ_{0} is a regular element in R. Then, for every ideal I ⊂ P, the image of ϑ in R⟨P/I⟩ is a regular element.

Proof of the claim. In view of (6.4.22), P can be regarded as a graded monoid P = ∐_{n∈ℕ} P_{n}, so R[P] is a graded algebra, and I = ∐_{n∈ℕ}(I ∩ P_{n}) is a graded ideal. Thus, ⟨P/I⟩ is a graded R-algebra as well, and the claim follows easily (details left to the reader). ♢

Conversely, suppose that conditions (a) and (b) hold. By virtue of lemma 12.5.1, we may assume that P is fine, sharp and saturated, and that the map P → 𝒪_{X,ξ} is local. According to remark 12.5.12, we may find a regular local ring R of dimension ≤ 1 + dim A/𝔪_{P} A, and a surjective ring homomorphism :

φ : R[[P]] → A^{∧}

Suppose first that dim R = 1 + dim A/𝔪_{P} A; then Ker φ contains an element ϑ whose constant term ϑ_{0} lies in 𝔪_{R} \ 𝔪_{R}^{2}, and φ induces an isomorphism R_{0} := R/ϑ_{0}R →^{∼} A^{∧}/𝔪_{P} A^{∧}. We have to show that φ induces an isomorphism \overline{φ} : R[[P]]/(ϑ) →^{∼} A^{∧}. To this aim, we consider the 𝔪_{P} -adic filtrations on these rings; for the associated graded rings we have :

gr_{n}R[[P]]/(ϑ) ≃ R_{0}⟨𝔪_{P}^{n} /𝔪_{P}^{n+1}⟩ gr_{n}A^{∧} = 𝔪_{P}^{n} A^{∧}/𝔪_{P}^{n+1}A^{∧}.

Hence gr_{n}\overline{φ} is the natural map A^{∧}/𝔪_{P} A^{∧} ⊗_{P} (𝔪_{P}^{n} /𝔪_{P}^{n+1}) → 𝔪_{P}^{n} A^{∧}/𝔪_{P}^{n+1}A^{∧}, and the latter is an isomorphism, since A^{∧} is P -flat. The assertion follows in this case. The remaining case where dim R = dim A/𝔪_{P} A is similar, but easier, so shall be left to the reader, as an exercise.

(ii) follows immediately from (i) and lemma 12.5.1. □

Corollary 12.5.19. Let (X, \underline{M}) be a log scheme, ξ a τ-point of X, and suppose that (X, \underline{M}) is regular at ξ. Then the following conditions are equivalent :

(a) 𝒪_{X,ξ} is a regular local ring.

(b) \underline{M}_{ξ}^{♯} is a free monoid of finite rank.

Proof. (b)⇒(a) : Indeed, it suffices to show that 𝒪_{X,ξ}^{∧} is regular ([63, Ch.0, Prop.17.3.3(i)]); by theorem 12.5.10, the latter is isomorphic to either R[[P]] or R[[P]]/ϑR[[P]], where R is a regular local ring, P := \underline{M}_{ξ}, and the constant term of ϑ lies in 𝔪_{R} \ 𝔪_{R}^{2}. Since, by assumption, P is a free monoid of finite rank, it is easily seen that rings of the latter kind are regular ([63, Ch.0, Cor.17.1.8]).

(a)⇒(b) : By proposition 12.5.16(ii), R := 𝒪_{X,ξ} is P -flat, for a morphism P := \underline{M}_{ξ}^{♯} → R that induces the log structure \underline{M}(ξ) on X(ξ). It follows that

𝔪_{P} R/𝔪_{P}^{2} R = (𝔪_{P} /𝔪_{P}^{2} ) ⊗_{P} R = R^{⊕r}

where r := rk_{P^{×}}^{∘}𝔪_{P} /𝔪_{P}^{2} is the cardinality of 𝔪_{P} \ 𝔪_{P}^{2} . In view of [66, Ch.IV, Prop.16.9.3, Cor.16.9.4, Cor.19.1.2], we deduce that the image of 𝔪_{P} \ 𝔪_{P}^{2} is a regular sequence of R of length r, hence dim P = dim R − dim R/𝔪_{P} R = r, since (X, \underline{M}) is regular at ξ. Then the assertion follows from proposition 6.4.23. □

Corollary 12.5.20. Let (X, \underline{M}) be a regular log scheme, ξ any τ-point of X, and 𝔭 ⊂ \underline{M}_{ξ} an ideal. We have :

(i) If 𝔭 is a prime ideal, 𝔭𝒪_{X,ξ} is a prime ideal of 𝒪_{X,ξ}, and ht 𝔭 = ht 𝔭𝒪_{X,ξ}.

(ii) If 𝔭 is a radical ideal, 𝔭𝒪_{X,ξ} is a radical ideal of 𝒪_{X,ξ}.

Proof. To ease notation, set A := 𝒪_{X,ξ}. Pick a chart β : P_{X(ξ)} → \underline{M}(ξ) with P fine, sharp and saturated (corollary 12.1.36(i)); by proposition 12.5.16(ii), the ring A is P -flat for the resulting map P → A. Let 𝔮 := β^{−1}𝔭 ⊂ P; then 𝔭A = 𝔮A.

(i): Since 𝔭 is a prime ideal, 𝔮 is a prime ideal of P, and in order to prove that 𝔭A is a prime ideal, it suffices therefore to show that the completion (A/𝔮A)^{∧} of A/𝔮A is an integral domain. However, remark 12.5.12implies that (A/𝔮A)^{∧} ≃ B/ϑB, where B := R[[P \𝔮]], with (R, 𝔪_{R}) a regular local ring, and ϑ is either zero, or else it is an element whose constant term ϑ_{0} ∈ R lies in 𝔪_{R}\𝔪_{R}^{2}. The assertion is obvious when ϑ = 0, and otherwise it follows from claim 12.5.11. Next, by going down (corollary 6.4.34), we see that :

ht 𝔭A ≥ ht 𝔭.

On the other hand, notice that we have a natural identification Spec \underline{M}_{ξ} →^{∼} Spec P, especially ht 𝔭 = ht 𝔮, hence dim A/𝔭A = dim(A/𝔮A)^{∧} = dim R + dim(P \𝔮) − ε, where ε is either 0 or 1 depending on whether R does or does not contain a field (corollary 6.4.27(ii)). Thus :

ht 𝔭A ≤ dim A − dim A/𝔭A = dim P − dim(P \𝔮) = ht 𝔭

(corollary 6.4.12(iii)), which completes the proof.

(ii): In this case, 𝔮 is a radical ideal of P, so it can be written as a finite intersection of prime ideals of P (lemmata 6.1.16 and 6.1.20(iii)); then the assertion follows from (i) and lemma 6.1.37(details left to the reader). □

Lemma 12.5.21. Let X be a scheme, \underline{M} a fs log structure on X_{Zar}, and ξ a geometric point of X. Then (X, \underline{M}) is regular at the point |ξ| if and only if ũ_{X}^{∗}(X, \underline{M}) is regular at ξ.

Proof. Set (Y, \underline{N}) := ũ_{X}^{∗}(X, \underline{M}) (notation of (12.2.2) : of course Y = X, but the sheaf 𝒪_{Y} is defined on the site X_{ét}, hence B := 𝒪_{Y,ξ} is the strict henselization of A := 𝒪_{X,|ξ|}), and let α : \underline{M}_{|ξ|} → B be the induced morphism of monoids; since α is local, it is easily seen that \underline{N}_{ξ} is isomorphic to the push-out of the diagram \underline{M}_{|ξ|} ← \underline{M}_{|ξ|}^{×} → B^{×}, especially dim \underline{M}_{|ξ|} = dim \underline{N}_{ξ}. Moreover, I(ξ, \underline{N}) = I(|ξ|, \underline{M})B, so that B_{0} := B/I(ξ, \underline{N}) is the strict henselization of A_{0} := A/I(|ξ|, \underline{M}) ([66, Ch.IV, Prop.18.6.8]); hence A_{0} is regular if and only if the same holds for B_{0} ([66, Ch.IV, Cor.18.8.13]). Finally, dim A = dim B and dim A_{0} = dim B_{0} ([126, Th.15.1]). The lemma follows. □

12.5.22. Let (Y, \underline{N}) be a log scheme, \overline{y} a τ-point of Y, and α : Q_{Y}^{log} → \underline{N} a fine and saturated chart which is sharp at \overline{y}. Let also φ : Q → P be an injective morphism of monoids, with P fine and saturated, and such that P^{×} is a torsion-free abelian group. Define (X, \underline{M}) as the fibre product in the cartesian diagram of log schemes :

(12.5.23)  \xymatrix{(X, \underline{M}) \ar[r]^{f} \ar[d] & (Y, \underline{N}) \ar[d]^{h} \\ Spec(ℤ, P) \ar[r]^{Spec(ℤ,φ)} & Spec(ℤ, Q)}

where h is induced by α; especially, h is strict.

Lemma 12.5.24. In the situation of (12.5.22), let \overline{x} be any τ-point of X such that f(\overline{x}) = \overline{y}, and suppose that (Y, \underline{N}) is regular at \overline{y}. Then (X, \underline{M}) is regular at \overline{x}.

Proof. To begin with, we show :

Claim 12.5.25. The natural map :

𝒪_{X,\overline{x}} ⊗^{L}_{P} P/I → 𝒪_{X,\overline{x}} ⊗_{P} P/I

is an isomorphism in D^{−}(𝒪_{X,\overline{x}} -Mod), for every ideal I ⊂ P (notation of (6.1.33)).

Proof of the claim. It suffices to show that this map is an isomorphism in D^{−}(𝒪_{Y,\overline{y}} -Mod), and in the latter category we have a commutative diagram :

(12.5.26)  \xymatrix{(𝒪_{Y,\overline{y}} ⊗^{L}_{Q} P) ⊗^{L}_{P} P/I \ar[d] \ar[rr] &  & 𝒪_{Y,\overline{y}} ⊗^{L}_{Q} P/I \ar[d] \\ (𝒪_{Y,\overline{y}} ⊗_{Q} P) ⊗^{L}_{P} P/I \ar[r] & (𝒪_{Y,\overline{y}} ⊗_{Q} P) ⊗_{P} P/I \ar[r]^{∼} & 𝒪_{Y,\overline{y}} ⊗_{Q} P/I.}

Since 𝒪_{X,\overline{x}} is a localization of 𝒪_{Y,\overline{y}} ⊗_{Q} P, we are reduced to showing that the bottom arrow of (12.5.26) is an isomorphism. However, the top arrow of (12.5.26) is always an isomorphism. Moreover, on the one hand, since (Y, \underline{N}) is regular at \overline{y}, the ring 𝒪_{Y,\overline{y}} is Q-flat (proposition 12.5.16(ii)), and on the other hand, since φ is injective, P/I is an integral Q-module, so the two vertical arrows are isomorphisms as well, and the claim follows. ♢

Claim 12.5.27. 𝒪_{X,\overline{x}}/I(\overline{x}, \underline{M}) is a regular ring.

Proof of the claim. Let β : P → A := 𝒪_{X,\overline{x}} be the morphism deduced from h, and set :

S := β^{−1}(𝒪_{X,\overline{x}}^{×}) I(\overline{x}, P) := P \S.

Then the ℤ[P]-algebra A is a localization of the ℤ[P]-algebra

B := S^{−1}(𝒪_{Y,\overline{y}} ⊗_{Q} P)

and it suffices to show that B/I(\overline{x}, \underline{M}) is regular.

It is easily seen that I(\overline{x}, \underline{M}) = I(\overline{x}, P)A and φ(𝔪_{Q}) ⊂ I(\overline{x}, P); on the other hand, since α is sharp at the point \overline{y}, we have I(\overline{y}, \underline{N}) = 𝔪_{Q}𝒪_{Y,\overline{y}}, and Q \ 𝔪_{Q} = \{1\}. Let p be the residue characteristic of 𝒪_{Y,\overline{y}}; there follow isomorphisms of ℤ_{(p)}-algebras :

B/I(\overline{x}, \underline{M})B ≃ S^{−1}𝒪_{Y,\overline{y}} ⊗_{Q} P/I(\overline{x}, P) ≃ 𝒪_{Y,\overline{y}}/I(\overline{y}, \underline{N}) ⊗_{ℤ_{(p)}} ℤ_{(p)}[S^{gp}].

By assumption, 𝒪_{Y,\overline{y}}/I(\overline{y}, \underline{N}) is regular, hence we are reduced to showing that ℤ_{(p)}[S^{gp}] is a smooth ℤ_{(p)}-algebra ([66, Ch.IV, Prop.17.5.8(iii)]). However, under the current assumptions P^{gp} is a free abelian group of finite rank, hence the same holds for S^{gp}, and the contention follows easily. ♢

In light of proposition 6.1.40(i), claims 12.5.25and 12.5.27assert that conditions (a) and (b) of proposition 12.5.16(i) are satisfied for the τ-point \overline{x} of (X, \underline{M}), so the latter is regular at \overline{x}, as stated. □

Theorem 12.5.28. Let f : (X, \underline{M}) → (Y, \underline{N}) be a smooth morphism of locally noetherian fs log schemes, ξ a τ-point of X, and suppose that (Y, \underline{N}) is regular at the point f(ξ). Then (X, \underline{M}) is regular at the point ξ.

Proof. In case τ = Zar, lemma 12.5.21 and corollary 12.3.27(ii) reduce the assertion to the corresponding one for ũ^{∗}f. Hence, we may assume that τ = ét. Next, since the assertion is local on X_{τ}, we may assume that \underline{N} admits a fine and saturated chart α : Q_{Y}^{log} → \underline{N} which is sharp at f(ξ) (corollary 12.1.36(i)); then, by corollary 12.3.42, we may further assume that there exist an injective morphism of fine and saturated monoids φ : Q → P, such that P^{×} is torsion-free, and a cartesian diagram as in (12.5.23). Then the assertion follows from lemma 12.5.24. □

Corollary 12.5.29. Let f : (X, \underline{M}) → (Y, \underline{N}) be a smooth morphism of locally noetherian fine log schemes. Then, for every y ∈ (Y, \underline{N})_{tr}, the log scheme Spec κ(y) ×_{Y} (X, \underline{M}) is geometrically regular.

Proof. The trivial log structure on Spec κ(y) is obviously saturated, so the same holds for the log structure of Spec κ(y) ×_{Y} (X, \underline{M}). Hence, the assertion is an immediate consequence of theorem 12.5.28. □

In the same vein we have :

Proposition 12.5.30. Let f : (X, \underline{M}) → (Y, \underline{N}) be a strict morphism of locally noetherian fs log schemes, ξ a τ-point of X, and suppose that the morphism of schemes f_{ξ} : X(ξ) → Y (f(ξ)) is flat. The following holds :

(i) If (X, \underline{M}) is regular at the point ξ, then (Y, \underline{N}) is regular at the point f(ξ).

(ii) Conversely, if (Y, \underline{N}) is regular at the point f(ξ), and the fibre f_{ξ}^{−1}(f(ξ)) is a regular scheme (notation of (4.9.20)), then (X, \underline{M}) is regular at the point ξ.

Proof. Set A := 𝒪_{Y,f(ξ)} and B := 𝒪_{X,ξ}, let g : A → B be the induced ring homomorphism, pick a fine chart β : P → A for the log structure \underline{N}(f(ξ)), and denote by I ⊂ A the ideal generated by the image of the maximal ideal of \underline{N}_{f(ξ)}.

(i): By proposition 12.5.16(ii), the local ring B/IB is regular, g ∘ β : P → B is a fine chart for the log structure \underline{M}(ξ), and B is (g ∘ β)-flat. Since the induced map A/I → B/IB is flat and local, also A/I is regular ([63, Ch.0, Prop.17.3.3(i)]), and since g is faithfully flat, A is β-flat (remark 6.1.35(iii)). Then the assertion follows from proposition 12.5.16(i).

(ii): By proposition 12.5.16(ii), the local ring A/I is regular, and A is β-flat. Since g is flat, the same holds for the induced map A/I → B/IB; it follows that also B/IB is regular ([63, Ch.0, Prop.17.3.3(ii)]), the induced map P → B is a chart for \underline{M}(ξ) and B is (g ∘ β)-flat, whence the contention, again by proposition 12.5.16(i). □

Theorem 12.5.31. Let (X, \underline{M}) be a locally noetherian fs log scheme. Then the subset (X, \underline{M})_{reg} is closed under generization.

Proof. Let ξ be a τ-point of X, with support x ∈ (X, \underline{M})_{reg}, and let η be a generization of ξ, whose support is a strict generization y of x. We have to show that (X, \underline{M}) is regular at the point η. Since the assertion is local on X, we may assume that (X, \underline{M}) admits a fine and saturated chart β : P_{X} → \underline{M}, sharp at the point ξ (corollary 12.1.36(i)). Let α : P → 𝒪_{X,η} be the morphism deduced from β_{η}, and set 𝔭 := α^{−1}𝔪_{η}, where 𝔪_{η} ⊂ 𝒪_{X,η} is the maximal ideal. We consider the cartesian diagram of log schemes :

\xymatrix{(X′, \underline{M}′)_{∘} \ar[r] \ar[d] & (X, \underline{M}) \ar[d]^{g} \ar[d] \\ Spec⟨ℤ, P/𝔭⟩ \ar[r] & Spec(ℤ, P)}

where g is induced by β (notation of (12.2.13)). Clearly ξ and η induce τ-points on X′, which we denote by the same names. We have natural identifications:

𝒪_{X,ξ}/I(ξ, \underline{M}) →^{∼} 𝒪_{X′,ξ}/I(ξ, \underline{M}′)  \underline{M}′_{η} = 𝒪_{X′,η}^{×}  𝒪_{X,η}/I(η, \underline{M}) →^{∼} 𝒪_{X′,η}

([66, Ch.IV, Prop.18.6.8]). Moreover, from proposition 12.5.16(ii) we know that 𝒪_{X,ξ} is P -flat; then corollary 6.4.34yields the inequality :

dim 𝒪_{X′,ξ} = dim 𝒪_{X,ξ} ⊗_{P} P/𝔭 ≥ dim 𝒪_{X,ξ} − ht(𝔭) = dim 𝒪_{X,ξ}/I(ξ, \underline{M}) + dim P/𝔭

in other words : dim 𝒪_{X′,ξ} = d(ξ, \underline{M}′) (notation of definition 12.5.7), so (X′, \underline{M}′) is regular at ξ. By the same token, 𝒪_{X,η} is P_{𝔭}-flat; from proposition 12.5.16(i), it follows that (X, \underline{M}) is regular at η if and only if the same holds for (X′, \underline{M}′).

Hence we may replace (X, \underline{M}) by (X′, \underline{M}′), and P by P \𝔭, after which we may assume that y ∈ (X, \underline{M})_{tr}. In this case :

(12.5.32)  α(P) ⊂ 𝒪_{X,η}^{×}

and we have to show that 𝒪_{X,η} is a regular local ring, or equivalently, that 𝒪_{X,y} is regular ([66, Ch.IV, Cor.18.8.13(c)]). Now, if P = 0, then 𝒪_{X,x} is regular, and the assertion follows from [126, Th.19.3]; thus, we may assume that P ≠ 0.

Denote by Y the topological closure of y in X, endowed with its reduced subscheme structure. According to [60, Ch.II, Prop.7.1.7] (and [66, Ch.IV, Prop.18.8.8(iv)] in case τ = ét), we may find a local injective ring homomorphism j : 𝒪_{Y,ξ} → V, where V is a discrete valuation ring. Let also \overline{β} : P → 𝒪_{Y,ξ} be the morphism deduced from β. Then (12.5.32) implies that j ∘ \overline{β}(P) ⊂ V \\{0\}, hence j ∘ \overline{β} extends to a homomorphism of groups P^{gp} → K^{×}, where K^{×} := (V \{0})^{gp} is the multiplicative group of the field of fractions of V; after composition with the valuation K^{×} → ℤ of V, there follows a group homomorphism

φ : P^{gp} → ℤ.

Notice also that j ∘ \overline{β} is a local morphism, since the same holds for j and \overline{β}; consequently, φ(P^{gp}) ≠ {0}. Set Q := φ^{−1}ℕ; then φ(Q) is a non-trivial submonoid of ℕ, and dim Q = dim Q/Ker φ = dim φ(Q) = 1. Set

T := Spec V (S, \underline{Q}) := Spec(ℤ, Q).

Notice that Q is saturated and fine (corollary 6.4.2), so there exists an isomorphism :

ℤ^{⊕r} × ℕ →^{∼} Q for some r ∈ ℕ

(theorem 6.4.18(iii)); the latter determines a chart :

(12.5.33)  ℕ_{S} → \underline{Q}

which is sharp at every τ-point of S localized outside the trivial locus Spec(ℤ, Q)_{tr}. We consider the cartesian diagram of log schemes :

\xymatrix{(X′, g^{′∗}\underline{Q}) \ar[r]^{g′} \ar[d]_{f′} & Spec(ℤ, \underline{Q}) \ar[d]^{f} \\ (X, \underline{M}) \ar[r]^{g} & Spec(ℤ, P)}

where f is the morphism of log schemes induced by the inclusion map ψ : P → Q. Notice that f is a smooth morphism (proposition 12.3.34); moreover, the restriction

f_{tr} : Spec(ℤ, Q)_{tr} → Spec(ℤ, P)_{tr}

of f, is just the morphism Spec ℤ[ψ^{gp}] : Spec ℤ[Q^{gp}] → Spec ℤ[P^{gp}], hence it is an isomorphism of schemes. It follows that f′ is a smooth morphism (proposition 12.3.24(ii)), and its restriction f′_{tr} to the trivial loci, is an isomorphism.

The homomorphism j induces a morphism h : T → X, such that the closed point of T maps to x and the generic point maps to y. By construction g ∘ h lifts to a morphism of schemes h′ : T → S, and the pair (h, h′) determines a morphism T → X′. Let x′, y′ ∈ X′ be the images of respectively the closed point and the generic point of T, and choose τ-points ξ′ and η′ localized at x′ and respectively y′; then the image of x′ in X is the point x, therefore (X′, g^{′∗}\underline{Q}) is regular at the point ξ′, by theorem 12.5.28. Furthermore, g′(ξ′) lies outside Spec(ℤ, Q)_{tr}, hence (12.5.33) induces a chart ℕ_{X′} → g^{′∗}\underline{Q}, which is sharp at ξ′. In light of corollary 12.5.19, we deduce that 𝒪_{X′,x′} is a regular ring, and then the same holds also for 𝒪_{X′,y′} ([63, Ch.0, Cor.17.3.2]). However, y′ lies in the trivial locus of (X′, g^{′∗}\underline{Q}), and its image in X is y, so by the foregoing the natural map 𝒪_{X,y} → 𝒪_{X′,y′} is an isomorphism, and the contention follows. □

Remark 12.5.34. Notice that the proof of [60, Chap.II, Prop.7.1.7] (that is invoked in the proof of theorem 12.5.31) is slightly incorrect : indeed, with the notation of the proof of loc.cit., it is implicitly assumed that B/𝔪B ≠ 0, which may fail, e.g. take A := k[[x_{2}]] (for any field k), whose maximal ideal is generated by x_{1} := x_{2}^{2} and x_{2}, so that B = k((x_{2})) in this case. The proof can be amended by considering the scheme X := Spec A and the blow-up π : E → X of the coherent ideal 𝔪𝒪_{X}; since π is proper and surjective, we see that there exists i ∈ {1, …, n} such the image of the restriction Spec A[x_{1}/x_{i}, …, x_{n}/x_{i}] → X of π contains 𝔪, and the argument then carries through with B := A[x_{1}/x_{i}, …, x_{n}/x_{i}]. See also [102, Th.6.4.3].

12.5.35. Let (X, \underline{M}) be any log scheme, ξ a τ-point of X. There follows a continuous map

ψ_{ξ} : X(ξ) → T_{ξ} := Spec \underline{M}_{ξ}

that sends the closed point of X(ξ) to the closed point of T_{ξ}. For every point 𝔭 ∈ T_{ξ}, let \overline{\{𝔭\}} be the topological closure of {𝔭} in T_{ξ}; then

X(𝔭) := ψ_{ξ}^{−1}\overline{\{𝔭\}}

is a closed subset of X(ξ), which we endow with its reduced subscheme structure, and we set

(X(𝔭), \underline{M}(𝔭)) := ((X(ξ), \underline{M}(ξ)) ×_{X} X(𝔭))_{red}

(notation of example 12.1.12(iv)). Notice that U_{𝔭} := ψ_{ξ}^{−1}(𝔭) is an open subset of X(𝔭), for every 𝔭 ∈ T_{ξ}. We call the family

(U_{𝔭} | 𝔭 ∈ T_{ξ})

of locally closed subschemes of X(ξ), the logarithmic stratification of (X(ξ), \underline{M}(ξ)). For instance, U_{∅} = (X(\overline{x}), \underline{M}(\overline{x}))_{tr}. More generally, it is clear from the definition that

(12.5.36)  U_{𝔭} = (X(𝔭), \underline{M}(𝔭))_{tr}  for every 𝔭 ∈ T_{ξ}.

Corollary 12.5.37. In the situation of (12.5.35), suppose that (X, \underline{M}) is regular at ξ. Then : (i) The log scheme (X(𝔭), \underline{M}(𝔭)) is pointed regular, for every 𝔭 ∈ T_{ξ}. (ii) The scheme X(𝔭) is irreducible, and its codimension in X equals the height of 𝔭 in T_{ξ},

for every 𝔭 ∈ T_{ξ}. (iii) The scheme U_{𝔭} is regular and irreducible, for every 𝔭 ∈ T_{ξ}.

Proof. (i): By theorem 12.5.31, it suffices to show that (X(𝔭), \underline{M}(𝔭)) is pointed regular at ξ, for every 𝔭 ∈ T_{ξ}. However, say that X(ξ) = Spec A, and let P → \underline{M}(ξ) be a fine and saturated chart, sharp at the τ-point ξ. Then X(𝔭) = Spec A_{0}, where A_{0} := A/𝔭A, and \underline{M}(𝔭) = \underline{N}_{∘}, where \underline{N} is the fs log structure deduced from the induced map β : P \𝔭 → A_{0}. By proposition 12.5.16(ii), the ring A is P -flat; then A_{0} is β-flat, and the assertion follows from proposition 12.5.16(i).

Next, (ii) is a rephrasing of corollary 12.5.20(i), and (iii) follows from (i),(ii) and (12.5.36), by virtue of corollary 12.5.19. □

Proposition 12.5.38. Let (X, β : \underline{M} → 𝒪_{X}) be a regular log scheme, set U := (X, \underline{M})_{tr}, and denote by j : U → X the open immersion. Then the morphism β induces identifications:

\underline{M} →^{∼} j_{∗}𝒪_{U}^{×} ∩ 𝒪_{X} \underline{M}^{gp} →^{∼} j_{∗}𝒪_{U}^{×}.

Proof. Notice that the scheme X is normal (corollary 12.5.13), hence both j_{∗}𝒪_{U}^{×} and 𝒪_{X} are subsheaves of the sheaf i_{∗}𝒪_{X_{0}}, where X_{0} is the subscheme of maximal points of X, and i : X_{0} → X is the natural morphism; so we may intersect these two sheaves inside the latter.

In view of lemma 12.5.2and proposition 12.5.16(ii), we know already that β is injective, and clearly the image of β lands in j_{∗}𝒪_{U}^{×}, so it remains only to show that β (resp. β^{gp}) induces an epimorphism onto j_{∗}𝒪_{U}^{×} ∩ 𝒪_{X} (resp. onto j_{∗}𝒪_{U}^{×}). The assertions can be checked on the stalks, hence let ξ be any τ-point of X; to begin with, we show :

Claim 12.5.39. The induced map β_{ξ}^{gp} : \underline{M}_{ξ}^{gp} → (j_{∗}𝒪_{U}^{×})_{ξ} is a surjection.

Proof of the claim. Set A := 𝒪_{X,ξ}; notice that (X(ξ), \underline{M}(ξ))_{tr} is the complement of the union of the finitely many closed subsets of the form Z_{𝔭} := Spec A/𝔭A, where 𝔭 ⊂ \underline{M}_{ξ} runs over the prime ideals of height one. Due to corollary 12.5.20(i), each Z_{𝔭} is an irreducible divisor in Spec A. Now, let s ∈ (j_{∗}𝒪_{U}^{×})_{ξ}; then the divisor of s is of the form ∑_{ht𝔭=1} n_{𝔭}[𝔭], for some n_{𝔭} ∈ ℤ. By lemmata 12.1.18(ii) and 12.2.21(i), \underline{M}_{ξ}^{♯} is a fine and saturated monoid; then lemma 6.4.39 and proposition 6.4.50 say that the fractional ideal I := ⋂_{ht 𝔭=1} 𝔪_{𝔭}^{n_{𝔭}} of \underline{M}_{ξ} is reflexive, and therefore the fractional ideal IA of A is reflexive as well (lemma 6.4.45(ii)). Since A is normal, by considering the localizations of I𝒪_{X,ξ} at the prime ideals of height one of A, we deduce easily that IA = sA ([34, Ch.VII, §4, n.2, Cor. du Th.2]). Thus, we may write s = ∑_{i=1}^{n} β_{ξ}^{gp}(x_{i})a_{i} for certain a_{i} ∈ A and x_{i} ∈ I (for i = 1, …, n). Then we must have β_{ξ}^{gp}(x_{i})a_{i} ∉ s · 𝔪_{ξ} for at least one index i ≤ n (where 𝔪_{ξ} ⊂ A is the maximal ideal); for such i, it follows that s^{−1} · β_{ξ}^{gp}(x_{i}) ∈ A^{×}, whence s ∈ \underline{M}_{ξ}^{gp}, as required. ♢

Now, let s ∈ (j_{∗}𝒪_{U}^{×})_{ξ} ∩ A; by claim 12.5.39, we may find x ∈ \underline{M}_{ξ}^{gp} such that β_{ξ}^{gp}(x) = s. To conclude, it suffices to show that x ∈ \underline{M}_{ξ}. By theorem 6.4.18(i), we are reduced to showing that x ∈ (\underline{M}_{ξ})_{𝔭} for every prime ideal 𝔭 ⊂ \underline{M}_{ξ} of height one. Set 𝔮 := 𝔭A; then 𝔮 is a prime ideal of height one (corollary 12.5.20(i)), and β_{ξ} extends to a well defined map of monoids β_{𝔭} : (\underline{M}_{ξ})_{𝔭} → A_{𝔮}. Furthermore, A_{𝔮} is a discrete valuation ring (corollary 12.5.13), whose valuation we denote v : A_{𝔮} \ \{0\} → ℕ. In view of theorem 6.4.18(ii), we see that x ∈ (\underline{M}_{ξ})_{𝔭} if and only if (v ∘ β_{𝔭})^{gp}(x) ≥ 0. However, clearly v(s) ≥ 0, whence the contention. □

Remark 12.5.40. Let reg.log_{τ} denote the full subcategory of log_{τ} whose objects are the regular log schemes. As an immediate consequence of proposition 12.5.38(and of remark 12.2.8(i)) we see that the forgetful functor F of (12.2.1) restricts to a fully faithful functor

reg.log_{τ} → Open (X, \underline{M}) ↦ ((X, \underline{M})_{tr} → X)

where Open is full subcategory of Morph(Sch) whose objects are the open immersions.

12.5.41. Let (X, \underline{M}) be a quasi-compact and regular log scheme. Set Z := X \ (X, \underline{M})_{tr}; then Z is a closed subset of X (lemma 12.2.21(ii)), and we endow it with its reduced subscheme structure. Denote by ω_{X} ⊂ 𝒪_{X} the sheaf of ideals corresponding to Z. We have :

Theorem 12.5.42. In the situation of (12.5.41), the complex ω_{X}[0] is dualizing on X. (See definition 11.3.16.)

Proof. In light of proposition 11.3.37, we may assume that X is local, say X = Spec A, for a local noetherian ring A. Next, pick any τ-point ξ localized at the closed point of X; by virtue of proposition 11.3.34 and [66, Ch.IV, Prop.18.8.8(iv) and Prop.18.8.12(ii)], we may further reduce to the case where (X, \underline{M}) = (X(ξ), \underline{M}(ξ)). Let X^{∧} := Spec A^{∧}, where A^{∧} denotes the completion of the local ring A, and denote by f : X^{∧} → X the natural morphism; by applying again proposition 11.3.34, we reduce to checking that f^{∗}ω_{X}[0] is dualizing on X^{∧}. However, theorem 12.5.10 says that there exist a local ring homomorphism R → A^{∧}, with R a complete regular local ring, and a fine and saturated chart P_{X} → \underline{M}, sharp at the closed point of X, such that the induced continuous ring homomorphism φ : R[[P]] → A^{∧} is an isomorphism if A contains a field, and otherwise it is a surjection, whose kernel is generated by a regular element ϑ ∈ R[[P]]. With this notation, a simple inspection shows that Z is the union of the closed subsets X(𝔭) ⊂ X, where 𝔭 ⊂ P ranges over all prime ideals 𝔭 ≠ ∅ (notation of (12.5.35)). Then, corollary 12.5.20(i) says that ω_{X} is the intersection of the ideals of 𝒪_{X} of the form 𝔭𝒪_{X}, where 𝔭 ⊂ P is an arbitrary non-empty prime ideal. Denote by P^{∘} ⊂ P the intersection of all the (finitely many) non-empty prime ideals of P; lemma 6.1.37 and proposition 12.5.16(ii) imply that ω_{X} = P^{∘} · 𝒪_{X} (details left to the reader), and since f is a flat morphism, we deduce that f^{∗}ω_{X} = P^{∘} · 𝒪_{X^{∧}}. The ideal P^{∘} can also be described as follows. Set V := P_{ℝ}^{gp}, denote by σ the unique strictly convex polyhedral cone such that P = σ ∩ P^{gp}, and let σ^{∘} be the topological interior of σ in V; then it is easily seen that P^{∘} = P ∩ σ^{∘}.

Set Y := Spec R[P], ω_{Y} := P^{∘} · 𝒪_{Y}, and let y ∈ Y be the image of the closed point of Y^{∧} := Spec R[[P]], under the natural morphism g : Y^{∧} → Y. Theorem 11.6.43(ii) says that ω_{Y} [0] is dualizing on Y; then, since g is flat, propositions 11.3.34 and 11.3.37 imply that g^{∗}ω_{Y} [0] is dualizing on Y^{∧}.

Now, suppose that A contains a field; then φ induces an isomorphism X^{∧} →^{∼} Y^{∧} which identifies f^{∗}ω_{X} with g^{∗}ω_{Y}, whence the contention.

Lastly, if A does not contain a field, φ induces a regular closed immersion i : X^{∧} → Y^{∧}, so i^{!}(g^{∗}ω_{Y} [0]) is dualizing on X^{∧} (lemma 11.3.27(i)). The latter is the complex of 𝒪_{X^{∧}}- modules arising from the complex of A^{∧}-modules L^{•} := RHom_{R[[P]]}^{•}(A^{∧}[0], R[[P^{∘}]]) (corollary 10.3.2(ii); here R[[P^{∘}]] := P^{∘} · R[[P]]). However, A^{∧}[0] is naturally isomorphic (in D(R[[P]]-Mod)) to the Koszul complex K_{•}(ϑ) (notation of remark 7.8.1(ii) : this is a bounded complex of free R[[P]]-modules), so we see that

L^{•} ≃ (R[[P^{∘}]] ⊗_{R[[P]]} A^{∧})[−1].

However, we have natural A^{∧}-linear isomorphisms :

R[[P^{∘}]] ⊗_{R[[P]]} A^{∧} →^{∼} R[P^{∘}] ⊗_{R[P]} A^{∧}  (since g is flat)
→^{∼} ℤ[P^{∘}] ⊗_{ℤ[P]} A ⊗_{A} A^{∧}
→^{∼} (P^{∘} · A) ⊗_{A} A^{∧}  (by proposition 12.5.16(ii))

which shows that i^{!}(g^{∗}ω_{Y} [0]) ≃ f^{∗}ω_{X}[−1], and concludes the proof of the theorem. □

12.6. Resolution of singularities of regular log schemes. Most of this section concerns results that are special to the class of log schemes over the Zariski topology; these are then applied to étale fs log structures, after we have shown that every such log structure admits a logarithmic blow up which descends to the Zariski topology (proposition 12.6.52). The same statement – with an unnecessary restriction to regular log schemes – can be found in the article [135] by W.Niziol : see theorem 5.6 of loc.cit. We mainly follow her treatment, except for fleshing out some details, and correcting some inaccuracies.

Therefore, we let here τ = Zar, and all log structures considered in this section until (12.6.48) are defined on the Zariski sites of their underlying schemes.

12.6.1. Let f : (Y, \underline{N}) → (X, \underline{M}) be any morphism of log schemes; we remark that f is a morphism of monoidal spaces, i.e. for every y ∈ Y, the map \underline{M}_{f(y)} → \underline{N}_{y} induced by log f, is local. Indeed, it has been remarked in (12.1.6) that the natural map \underline{M}_{f(y)} → (f^{∗}\underline{M})_{y} is local, and on the other hand, a section s ∈ (f^{∗}\underline{M})_{y} is invertible if and only if its image in 𝒪_{Y,y} is invertible, if and only if log f_{y}(s) is invertible in \underline{N}_{y}, whence the contention. We shall denote

f^{♯} : (Y, \underline{N})^{♯} → (X, \underline{M})^{♯}

the morphism of sharp monoidal spaces induced by f in the obvious way (i.e. the underlying continuous map is the same as the continuous map underlying f, and log f^{♯} : f^{∗}\underline{M}^{♯} → \underline{N}^{♯} is (log f)^{♯} : see definition 6.5.1).

12.6.2. We let 𝒦 be the category whose objects are all data of the form \underline{X} := ((X, \underline{M}), F, ψ), where (X, \underline{M}) is a log scheme, F is a fan, and ψ : (X, \underline{M})^{♯} → F is a morphism of sharp monoidal spaces, such that log ψ : ψ^{∗}𝒪_{F} → \underline{M}^{♯} is an isomorphism. The morphisms :

((Y, \underline{N}), F′, ψ′) → \underline{X}

in 𝒦 are all the pairs (f, φ), where f : (Y, \underline{N}) → (X, \underline{M}) is a morphism of log schemes, and φ : F′ → F is a morphism of fans, such that the diagram

\xymatrix{(Y, \underline{N})^{♯} \ar[r]^{f^{♯}} \ar[d]_{ψ′} & (X, \underline{M})^{♯} \ar[d]^{ψ} \\ F′ \ar[r]^{φ} & F}

commutes. Especially, notice that lemma 12.1.4implies the identity :

(12.6.3)  Str(f) = ψ^{′−1}Str(φ).

(notation of definitions 6.5.1(ii) and 12.2.7(ii)). We shall say that an object \underline{X} is locally noetherian, if the same holds for the scheme X. Likewise, a morphism (f, φ) in 𝒦 is quasi-compact, (resp. quasi-separated, resp. separated, resp. locally of finite type, resp. of finite type) if the same holds for the morphism of schemes underlying f. We say that f is étale, if the same holds for the morphism of log schemes underlying f. Also, the trivial locus of \underline{X} is defined as the subset (X, \underline{M})_{tr} of X.

12.6.4. There is an obvious (forgetful) functor :

F : 𝒦 → Sch ((X, \underline{M}), F, ψ) ↦ X

which is a fibration. More precisely, every morphism of schemes S′ → S induces a base change functor (notation of (1.1.27)) :

F 𝒦 /S → F 𝒦 /S′ \underline{X} ↦ S′ ×_{S} \underline{X}

unique up to natural isomorphism of functors. Namely, for \underline{X} := ((X, \underline{M}), F, ψ) one lets

S′ ×_{S} \underline{X} := (S′ ×_{S} (X, \underline{M}), F, ψ′)

where ψ′ := ψ ∘ π^{♯}, and π : S′ ×_{S} (X, \underline{M}) → (X, \underline{M}) is the natural projection.

Example 12.6.5. (i) Let P be a monoid, R a ring, and set (S, P_{S}^{log}) := Spec(R, P) (see (12.2.13)). The unit of adjunction ε_{P} : P → R[P] determines a unique morphism of sharp monoidal spaces

ψ_{P} : Spec(R, P)^{♯} → T_{P} := (Spec P)^{♯}

(proposition 6.5.6), and we claim that \underline{S} := (Spec(R, P), T_{P} , ψ_{P}) is an object of 𝒦 .

The assertion can be checked on the stalks, hence let ξ be a point of S; then ξ corresponds to a prime ideal 𝔭_{ξ} ⊂ R[P], and by inspecting the definitions, we see that ψ_{P} (ξ) = ε_{P}^{−1}(𝔭_{ξ}) = 𝔮_{ξ} := P ∩ 𝔭_{ξ} ∈ Spec P. Again, a direct inspection shows that the morphism

(log ψ_{P})_{ξ} : 𝒪_{T_{P} ,𝔮_{ξ}} → (P_{S}^{log})_{ξ}^{♯}

is none else than the natural isomorphism P_{𝔮_{ξ}}^{♯} →^{∼} P_{S,ξ}^{log}/𝒪_{S,ξ}^{×} deduced from ε_{P} and the natural identifications

P_{S,ξ}^{log} = 𝒪_{S,ξ}^{×} ⊗_{P\𝔮_{ξ}} P = 𝒪_{S,ξ}^{×} ⊗_{P_{𝔮_{ξ}}^{×}} P_{𝔮_{ξ}}.

(ii) The construction of \underline{S} is clearly functorial in P. Namely, say that λ : P → Q is a morphism of monoids, and set S′ := Spec R[Q], T_{Q} := (Spec Q)^{♯}. There follows a morphism

(12.6.6)  \underline{S}′ := (Spec(R, Q), T_{Q}, ψ_{Q}) → \underline{S}

in 𝒦 , whose underlying morphism of log schemes is Spec(R, λ), and whose underlying morphism of fans is just (Spec λ)^{♯}.

12.6.7. Example 12.6.5 can be globalized to more general log schemes, at least under some additional assumptions. Namely, let (X, \underline{M}) be a regular log scheme. For every point x of X, let 𝔪_{x} ⊂ 𝒪_{X,x} be the maximal ideal; we set :

F (X) := \{x ∈ X | I(x, \underline{M}) = 𝔪_{x}\}

(notation of (12.5.5)), and we endow F (X) with the topology induced from X. The fan of (X, \underline{M}) is the sharp monoidal space

F (X, \underline{M}) := (F (X), \underline{M}_{|F (X)})^{♯}.

We wish to show that F (X, \underline{M}) is indeed a fan. Let U ⊂ X be any open subset; to begin with, it is clear that F (U, \underline{M}_{|U}) is naturally an open subset of F (X, \underline{M}); hence the contention is local on X, so we may assume that X is affine, say X = Spec A for a noetherian ring A, and that \underline{M} admits a finite chart P_{X} → \underline{M}; denote by β : P → A the induced morphism of monoids. Let now ξ be a point of X, and 𝔭_{ξ} ⊂ A the corresponding prime ideal; set 𝔮_{ξ} := β^{−1}𝔭_{ξ} ∈ Spec P. By inspecting the definitions, it is easily seen that I(ξ, \underline{M}) = 𝔮_{ξ}A_{𝔭_{ξ}}. Therefore, for any prime ideal 𝔮 ⊂ P, let V (𝔮)_{max} be the finite set consisting of all the maximal points of the closed subset V (𝔮) := Spec A/𝔮A (i.e. the minimal prime ideals of A/𝔮A); in light of corollary 12.5.20(i), it follows easily that

(12.6.8)  F (X) = ⋃_{𝔮∈Spec P} V (𝔮)_{max}

especially, F (X) is a finite set (lemma 6.1.20(iii)). Let t ∈ F (X) be any element, and denote by U(t) ⊂ F (X) the subset of all generizations of t in F (X); as a corollary, we see that U(t) is an open subset of F (X). Moreover, (12.6.8) also implies that :

U(t) = ⋃_{𝔮∈Spec P}(Spec 𝒪_{X,t}/𝔮𝒪_{X,t})_{max}.

However, corollary 12.5.20(i) says that Spec 𝒪_{X,t}/𝔮𝒪_{X,t} is irreducible for every 𝔮 ∈ Spec P, therefore the set U(t) is naturally identified with a subset of Spec P. Furthermore, arguing as in example 12.6.5(i) we find a natural isomorphism P_{𝔮_{t}}^{♯} →^{∼} 𝒪_{F (X,\underline{M}),t}. Moreover, if t′ ∈ U(t) is any other point, the specialization map 𝒪_{F (X,\underline{M}),t} → 𝒪_{F (X,\underline{M}),t′} corresponds – under the above isomorphism – to the natural morphism P_{𝔮_{t}}^{♯} → P_{𝔮′_{t}}^{♯} induced by the localization map P_{𝔮_{t}} → P_{𝔮′_{t}}. This shows that the open monoidal subspace (U(t), \underline{M}_{|U(t)}) is naturally isomorphic to (Spec P_{𝔮_{t}})^{♯}, hence F (X, \underline{M}) is a fan, as stated.

Remark 12.6.9. (i) The discussion in (12.6.7) shows more precisely that if :

(a) (U, \underline{M}) is a log scheme with U = Spec A affine,

(b) there exists a morphism β : P → A from a finitely generated monoid P, inducing the log structure \underline{M} on U, and

(c) β(𝔮)A is a prime ideal for every 𝔮 ∈ Spec P then F (U, \underline{M}) is naturally identified with (Spec P)^{♯}. In this situation, denote by

f_{β} : (U, \underline{M})^{♯} → T_{P} := (Spec P)^{♯}

the morphism of sharp monoidal spaces deduced from β (proposition 6.5.6); by inspecting the definitions, we see that the associated map log f_{β} : f_{β}^{∗}𝒪_{T_{P}} → \underline{M}^{♯} is an isomorphism. Via the foregoing natural identification, there results a morphism π_{U} : (U, \underline{M})^{♯} → F (U, \underline{M}) which can be described without reference to P. Indeed, let x ∈ U be any point, and 𝔭 ∈ Spec A the corresponding prime ideal; by inspecting the definitions we find that

π_{U}(x) = I(x, \underline{M}) which is the largest prime ideal in Spec A_{𝔭} ∩ F (U, \underline{M}).

Especially, π_{U}(x) is a generization of x, and the inverse of (log π_{U})_{x} : 𝒪_{F (U,\underline{M}),π_{U} (x)} →^{∼} \underline{M}_{x}^{♯} is induced by the specialization map \underline{M}_{x} → \underline{M}_{π_{U} (x)} (which induces an isomorphism on the associated sharp quotient monoids).

(ii) Finally, it follows easily from corollary 12.5.20(i) and [65, Ch.IV, Cor.8.4.3], that every regular log scheme (X, \underline{M}) admits an affine open covering X = ⋃_{i∈I} U_{i}, such that each (U_{i}, \underline{M}_{|U_{i}}) fulfills conditions (a)–(c) above, and the intrinsic description in (i) shows that the morphisms π_{U_{i}} glue to a well defined morphism π_{X} : (X, \underline{M})^{♯} → F (X, \underline{M}) of sharp monoidal spaces, such that the datum

𝒦 (X, \underline{M}) := ((X, \underline{M}), F (X, \underline{M}), π_{X})

is an object of 𝒦 .

(iii) Notice that π_{X}^{−1}(t) is an irreducible locally closed subset of X of codimension equal to the height of t, for every t ∈ F (X, \underline{M}); also π_{X}^{−1}(t) is a regular scheme, for its reduced subscheme structure. Indeed, we have already observed that t is the unique maximal point of π_{X}^{−1}(t), and then the assertion follows immediately from corollary 12.5.37(ii,iii). Furthermore, the inclusion map j : F (X, \underline{M}) → X is a continuous section of π_{X}, and notice that j ∘π_{X}(U) = j^{−1}U for every open subset U ⊂ X, since j maps every t ∈ F (X, \underline{M}) to the unique maximal point of π_{X}^{−1}(t). It follows easily that π_{X} is an open map.

12.6.10. Denote by 𝒦_{int} the full subcategory of 𝒦 whose objects are the data ((X, \underline{M}), F, ψ) such that \underline{M} is an integral log structure, and F is an integral fan. There is an obvious functor

𝒦_{int} → int.Fan : ((X, \underline{M}), F, ψ) ↦ F

to the category of integral fans, which shall be used to construct useful morphisms of log schemes, starting from given morphisms of fans. This technique rests on the following three results :

Lemma 12.6.11. Let ((X, \underline{M}), F, ψ) be an object of 𝒦_{int}, with F locally fine. Then, for each point x ∈ X there exist an open neighborhood U ⊂ X of x, a fine chart Q_{U} → \underline{M}_{|U} (for some fine monoid Q depending on x), and an isomorphism of monoids Q^{♯} →^{∼} 𝒪_{F,ψ(x)}.

Proof. The assertion is local on X, hence we may assume that F = (Spec P)^{♯} for a fine monoid P. In this case, ψ is determined by the corresponding map

\overline{β} : P_{X} → \underline{M}^{♯}.

Indeed, for any x ∈ X, the point ψ(x) is the prime ideal \overline{β}_{x}^{−1}(𝔪_{x}) ⊂ P, where 𝔪_{x} ⊂ \underline{M}_{x}^{♯} is the maximal ideal; moreover :

𝒪_{F,ψ(x)} = P/\overline{S}_{x} where \overline{S}_{x} := \overline{β}_{x}^{−1}(1)

and – under this identification – the isomorphism log ψ_{x} : 𝒪_{F,ψ(x)} →^{∼} \underline{M}_{x}^{♯} is deduced from \overline{β}_{x} in the obvious way.

Now, let x ∈ X be any point; after replacing X by the open subset ψ^{−1}U(ψ(x)), we may assume that P = 𝒪_{F,ψ(x)} (notation of (6.5.16)), and by assumption log ψ_{x} : P → \underline{M}_{x}^{♯} is an isomorphism. Pick a surjection α : ℤ^{⊕r} → P^{gp}, and let Q be the pull-back in the cartesian diagram :

\xymatrix{Q \ar[r] \ar[d] & ℤ^{⊕r} \ar[d]^{(log ψ_{x})^{gp}∘α} \\ _{\underline{M} ♯x} \ar[r] & \underline{M}_{x}^{gp}/\underline{M}_{x}^{×}.}

By construction, α restricts to a morphism of monoids ϑ : Q → P, inducing an isomorphism Q^{♯} →^{∼} P, whence an isomorphisms of fans (Spec P)^{♯} →^{∼} (Spec Q)^{♯}, and Q is fine, by corollary 6.4.2. Moreover, the choice of a lifting ℤ^{⊕r} → \underline{M}_{x}^{gp} of (log ψ_{x})^{gp} ∘ α determines a morphism

β_{x} : Q → \underline{M}_{x}^{♯} ×_{\underline{M}_{x}^{gp}/\underline{M}_{x}^{×}} \underline{M}_{x}^{gp} = \underline{M}_{x}

which lifts log ψ_{x} ∘ ϑ (here it is needed that \underline{M} is integral). Next, after replacing X by an open neighborhood of x, we may assume that β_{x} extends to a map of pre-log structures β : Q_{X} → \underline{M} lifting \overline{β} (lemma 12.1.18(iv.b),(v)). It remains only to show that β is a chart for \underline{M}, which can be checked on the stalks. Thus, let y ∈ X be any point, and set S_{y} := β_{y}^{−1}\underline{M}_{y}^{×}; with the foregoing notation, the stalk Q_{X,y}^{log} of the induced log structure is naturally isomorphic to (S_{y}^{−1}Q × 𝒪_{X,y}^{×})/S_{y}^{gp}, therefore

(Q_{X,y}^{log})^{♯} ≃ Q/S_{y} ≃ P/\overline{S}_{y}

and the induced map P/\overline{S}_{y} → \underline{M}_{y}^{♯} is again deduced from \overline{β}_{y}, so it is an isomorphism; then the same holds for β_{y}^{log} : Q_{X,y}^{log} → \underline{M}_{y} (lemma 12.1.4). □

Lemma 12.6.12. Let μ : P → P′ be a morphism of integral monoids such that μ^{gp} is surjective, φ : (Spec P′)^{♯} → (Spec P)^{♯} the induced morphism of affine fans, and denote by

λ : P → Q := P^{gp} ×_{P^{′gp}} P′

the map of monoids determined by μ and the unit of adjunction P → P^{gp}. Then :

(i) The natural projection Q → P′ induces an isomorphism ω : (Spec P′)^{♯} → (Spec Q)^{♯} of affine fans, such that (Spec λ)^{♯} ∘ ω = φ.

(ii) The induced morphism (12.6.6) in 𝒦_{int} (with R := ℤ) is int.Fan-cartesian.

(iii) If P and P′ are fine, the morphism (12.6.6) is étale.

Proof. (See (3.1) for generalities concerning inverse images and cartesian morphisms relative to a functor.) Notice first that, since μ^{gp} is surjective, the projection Q → P′ induces an isomorphism Q^{♯} →^{∼} P^{′♯}, whence (i).

(ii): Notice that the log structure of Spec(ℤ, Q) is integral, by lemma 12.1.18(iii). Next, set

F′ := (Spec P′)^{♯}.

Define \underline{S}, \underline{S}′ as in example 12.6.5(ii) (with R := ℤ), let g : ((Y, \underline{N}), F′′, ψ_{Y}) → \underline{S} be any morphism of 𝒦_{int}, and φ′ : F′′ → F′ a morphism of integral fans, such that the image of g in int.Fan equals φ ∘ φ′; we must show that g factors through a unique morphism h : ((Y, \underline{N}), F′′, ψ_{Y}) → \underline{S}′, whose image in int.Fan equals φ′. As usual, we may reduce to the case where Y = Spec A is affine, F′′ = Spec P′′ is an affine fan for a sharp integral monoid P′′, and ψ_{Y} is given by a map of sheaves P′′_{Y} → \underline{N}^{♯}. In such situation, the morphism (Y, \underline{N}) → Spec(ℤ, P) underlying g is determined by a morphism of monoids P → \underline{N}(Y), or which is the same, a map of sheaves γ_{Y} : P_{Y} → \underline{N}; likewise, φ′ is given by a morphism of monoids P′ → P′′, and composing with ψ_{Y}, we get a map of sheaves α_{Y} : P′_{Y} → \underline{N}^{♯}. Finally, the condition that g lies over φ ∘ φ′ translates as the commutativity of the following diagram of sheaves :

\xymatrix{P_{Y}^{gp} \ar[r] \ar[d]_{γ_{Y}^{gp}} & P_{Y}^{′gp} \ar[d] & P′_{Y} \ar[l] \ar[d]^{α_{Y}} \\ _{\underline{N}_{gp}} \ar[r] & \underline{N}^{gp}/\underline{N}^{×} & \underline{N}^{♯}. \ar[l]}

There follows a unique morphism of sheaves :

(12.6.13)  Q_{Y} → \underline{N}^{gp} ×_{\underline{N}^{gp}/\underline{N}^{×}} \underline{N}^{♯} = \underline{N}

(here it is needed that \underline{N} is integral) such that the diagram :

[figure: P_{Y}; γ_{Y}; Q_{Y}; P′_{Y}; α_{Y}; \underline{N}; \underline{N}; _{\underline{N}_{♯}}]

commutes. Then (12.6.13) determines a morphism of log schemes h_{Y} : (Y, \underline{N}) → Spec(ℤ, Q), such that the pair (h_{Y} , φ′) is the unique morphism h in 𝒦_{int} with the sought properties.

(iii): If both P and P′ are fine, so is Q (corollary 6.4.2), and by construction, λ^{gp} is an isomorphism. Then the assertion follows from theorem 12.3.37. □

Proposition 12.6.14. Let \underline{X} := ((X, \underline{M}), F, ψ) be an object of 𝒦_{int}. Let also φ : F′ → F be an integral partial subdivision, with F locally fine and F′ integral. We have :

(i) (X, \underline{M}) is a fine log scheme.

(ii) If F is saturated, (X, \underline{M}) is a fs log scheme.

(iii) \underline{X} admits an inverse image over φ (relative to the functor of (12.6.10)).

(iv) If φ is finite, then the cartesian morphism (f, φ) : φ^{∗}\underline{X} → \underline{X}, is quasi-compact.

(v) If F′ is locally fine, the morphism (f, φ) is quasi-separated and étale.

Proof. (i): This is just a restatement of lemma 12.6.11.

(ii): In light of (i), we only have to show that \underline{M}_{x} is a saturated monoid, for every x ∈ X. Since by assumption \underline{M}_{x}^{♯} is saturated, the assertion follows from lemma 6.2.9(ii).

Claim 12.6.15. In order to show (iii)–(v), we may assume that :

(a) F = (Spec P)^{♯} for a fine monoid P, and X is an affine scheme.

(b) The map of global sections P → Γ(X, \underline{M}^{♯}) determined by ψ, comes from a morphism of pre-log structures β : P_{X} → \underline{M} which is a fine chart for \underline{M}.

Proof of the claim. To begin with, suppose that ((X′, \underline{M}′), F′, ψ′) is the sought preimage of \underline{X}; let U ⊂ X be any open subset, and V ⊂ F an open subset such that ψ(U) ⊂ V. Then it is easily seen that the object

φ^{∗}\underline{X} ×\underline{X} (U, V) := ((f^{−1}U, \underline{M}′_{|f^{−1}U}), φ^{−1}V, ψ′_{|f^{−1}U})

is a preimage of ((U, \underline{M}_{|U}), V, ψ_{|U}) over the restriction φ^{−1}V → V of φ. Now, suppose that we have found an affine open covering X = ⋃_{i∈I} U_{i}, and for every i ∈ I an affine open subset V_{i} ⊂ F with ψ(U_{i}) ⊂ V_{i}, such that the object \underline{U}_{i} := ((U_{i}, \underline{M}_{|U_{i}}), V_{i}, ψ_{|U_{i}}) admits a preimage over the restriction φ_{i} : φ^{−1}V_{i} → V_{i} of φ; then the foregoing implies that there are natural isomorphisms:

φ_{i}^{∗}\underline{U}_{i} ×_{\underline{U}_{i}} (U_{ij}, V_{ij}) →^{∼} φ_{j}^{∗}\underline{U}_{j} ×_{\underline{U}_{j}} (U_{ij}, V_{ij})

for every i, j ∈ I, where U_{ij} := U_{i} ∩ U_{j} and V_{ij} := V_{i} ∩ V_{j}. Thus, we may glue all these inverse images along these isomorphisms, to obtain the sought inverse image of \underline{X}.

Moreover, if φ is finite, the same will hold for the restrictions φ_{i}, and if each cartesian morphism φ_{i}^{∗}\underline{U}_{i} → \underline{U}_{i} is quasi-compact, the same will hold also for (f, φ) ([58, Ch.I, §6.1]). Likewise, if each morphism φ_{i}^{∗}\underline{U}_{i} → \underline{U}_{i} is étale and quasi-separated, then the same will hold for (f, φ) ([58, Ch.I, Prop.6.1.11] and proposition 12.3.24(iii)).

Therefore, we may replace X by any U_{i}, and F by the corresponding V_{i}, which reduces the proof of (iii)–(v) to the case where condition (a) is fulfilled. Lastly, in light of lemma 12.6.11we may suppose that the open subsets U_{i} are small enough, so that also condition (b) is fulfilled. ♢

In view of claim 12.6.15, we shall assume henceforth that conditions (a) and (b) are fulfilled. Let \underline{S} := (Spec(ℤ, P), T_{P} , ψ_{P}) be the object of 𝒦_{int} considered in example 12.6.5(i) (with R := ℤ); in this situation, β determines a morphism of schemes f_{β} : X → S, and in view of lemma 12.2.14 we have \underline{X} ≃ X ×_{S} \underline{S} (notation of (12.6.4)). Therefore, if we find an inverse image \underline{S}′ for \underline{S} over φ, the object X ×_{S} \underline{S}′ will provide the sought inverse image of \underline{X}. Thus, in order to show (iii)–(v), we may further reduce to the case where \underline{X} = \underline{S} ([58, Ch.I, Prop.6.1.5(iii), Prop.6.1.9(iii)] and proposition 12.3.24(ii)).

Claim 12.6.16. In order to prove (iii)–(v), we may assume that F′ is affine.

Proof of the claim. Indeed, say that F′ = ⋃_{i∈I} V_{i} is an open covering, and let φ_{i} : V_{i} → T_{P} be the restriction of φ; suppose that we have found, for each i ∈ I, an inverse image φ_{i}^{∗}\underline{S} of \underline{S} over φ_{i}; again, it follows easily that an inverse image for \underline{S} over φ can be constructed by gluing the objects φ_{i}^{∗}\underline{S}. This already implies that, in order to show (iii), we may assume that F′ is affine.

Now, suppose that φ is finite, so we may find an open covering as above, such that furthermore each V_{i} is affine, and I is a finite set. Suppose that each of the corresponding morphisms φ_{i}^{∗}\underline{S} → \underline{S} is quasi-compact; by the foregoing, the schemes underlying the objects φ_{i}^{∗}\underline{S} give a finite open covering of the scheme underlying φ^{∗}\underline{S}, and then it is clear that (iv) holds.

Next, suppose furthermore that each morphism φ_{i}^{∗}\underline{S} → \underline{S} is étale. It follows that the morphism φ^{∗}\underline{S} → \underline{S} is also étale (proposition 12.3.24(ii)); then, since S is noetherian, the scheme underlying φ^{∗}\underline{S} is locally noetherian ([58, Ch.I, Prop.6.2.2]), and therefore the morphism φ^{∗}\underline{S} → \underline{S} is quasi-separated ([58, Ch.I, Cor.6.1.13]). ♢

In view of claim 12.6.16, we shall assume that F′ = (Spec P′)^{♯} is affine as well, for a sharp and integral P′, and that φ is given by a morphism λ : P → P′ inducing a surjection on the associated groups (details left to the reader). Then assertions (iii) and (iv) are now straightforward consequences of lemma 12.6.12(i,ii), and (v) follows from lemma 12.6.12(iii), after one remarks that, when F′ is fine, one may choose for P′ a fine monoid. □

Remark 12.6.17. Keep the assumptions of proposition 12.6.14, and let t ∈ F_{0} a point of F of height zero (see (6.5.16)). Notice that the inclusion map j_{t} : {t} → F is an open immersion, hence the fibre X_{t} := ψ^{−1}(t) ⊂ X is open; indeed, it is clear from the definitions, that ψ^{−1}(F_{0}) is precisely the trivial locus of \underline{X}. Moreover, let t′ ∈ φ^{−1}(t) since the group homomorphism 𝒪_{F,t}^{gp} → 𝒪_{F′,t′}^{gp} is surjective, we see that t′ is of height zero in F′, and φ restricts to an isomorphism of fans (\{t′\}, 𝒪_{F,t′}) →^{∼} (\{t\}, 𝒪_{F,t}). Set \underline{X}_{t} := j_{t}^{∗}\underline{X} (whose underlying scheme is X_{t}), and define likewise φ^{∗}\underline{X}_{t′} := (φ ∘ j_{t′})^{∗}\underline{X} (whose underlying scheme is an open subset of the trivial locus of φ^{∗}\underline{X}). We deduce that :

• The trivial locus of φ^{∗}\underline{X} is the preimage of (X, \underline{M})_{tr}.

• For every t′ ∈ F′_{0}, the restriction of (f, φ) : φ^{∗}\underline{X}_{t′} → \underline{X}_{t} is an isomorphism.

Corollary 12.6.18. In the situation of proposition 12.6.14, let ((X′, \underline{M}′), F′, ψ′) := φ^{∗}\underline{X}. The following holds :

(i) If F′ = F^{sat}, and φ : F^{sat} → F is the counit of adjunction, then (X′, \underline{M}′) = (X, \underline{M})^{fs}, and f is the counit of adjunction.

(ii) If φ is the blow up of a coherent ideal ℐ of 𝒪_{F}, then f is the blow up of ℐ\underline{M}, the

unique ideal of \underline{M} whose image in \underline{M}^{♯} equals ψ^{−1}ℐ. (iii) Suppose moreover, that (X, \underline{M}) is regular, F′ is saturated , and \underline{X} = 𝒦 (X, \underline{M})

(notation of remark 12.6.9(iii)). Then (X′, \underline{M}′) is regular, and φ^{∗}\underline{X} is isomorphic to 𝒦 (X′, \underline{M}′).

Proof. To start with, we remark that the assertions are local on X. Indeed, this is clear for (i), and for (iii) it follows easily from remark 12.6.9(i,,ii); concerning (ii), suppose that X = ⋃_{i∈I} U_{i} is an open covering, such that φ^{∗}(U_{i}, \underline{M}_{|U_{i}}) is the blow up of the ideal ℐ\underline{M}_{|U_{i}}, for every i ∈ I. For every i, j ∈ I set U_{ij} := U_{i} ∩ U_{j}; by the universal property of the blow up, there are unique isomorphisms of (U_{ij}, \underline{M}_{|U_{ij}})-schemes :

U_{ij} ×_{U_{i}} φ^{∗}(U_{i}, \underline{M}_{|U_{i}}) →^{∼} U_{ij} ×_{U_{j}} φ^{∗}(U_{j}, \underline{M}_{|U_{j}}).

Then both φ^{∗}(X, \underline{M}) and the blow up of ℐ\underline{M} are necessarily obtained by gluing along these isomorphisms, so they are isomorphic.

Thus, we may assume that F = (Spec P)^{♯} for some fine monoid P, and (X, \underline{M}) = X ×_{S} (S, P_{S}^{log}) for some morphism of schemes X → S, where as usual (S, P_{S}^{log}, F, ψ_{P}) is defined as in example 12.6.5(i). In this case, (i) follows from remarks 12.2.38(iii), 6.5.8(ii) and lemma 12.6.12(ii).

Likewise, remark 12.4.32(ii) allows to reduce (ii) to the case where \underline{X} = (S, P_{S}^{log}, F, ψ_{P}), and ℐ = I^{∼} for some ideal I ⊂ P; in which case we conclude by inspecting the explicit description in example 12.4.43.

(iii): From proposition 12.6.14(ii,v) and theorem 12.5.28 we already see that (X′, \underline{M}′) is regular. Next, we are easily reduced to the case where X is affine, say X = Spec A, and F′ = (Spec Q)^{♯}, for some saturated monoid Q, and by lemma 12.6.12, we may assume that φ is induced by a morphism of monoids λ : P → Q such that λ^{gp} is an isomorphism, and (X′, \underline{M}′) = φ^{∗}\underline{X} = X ×_{S} \underline{S}′, where \underline{S}′ := (Spec(ℤ, Q), F′, ψ_{Q}) is defined as in example 12.6.5(ii). Let 𝔮 ⊂ Q be any prime ideal, and set 𝔭 := 𝔮 ∩ P; in light of remark 12.6.9(i), it then suffices to show that Q/𝔮 ⊗_{P} A = Q/𝔮 ⊗_{P/𝔭} A/𝔭A is an integral domain. To this aim, let us remark :

Claim 12.6.19. Let λ : P → Q be a morphism of fine monoids, such that λ^{gp} is an isomorphism, F ⊂ Q any face, and denote λ_{F} : F ∩ P → P the inclusion map. Then :

(i) The natural map Coker λ_{F}^{gp} → P^{gp}/(F ∩ P)^{gp} is injective.

(ii) If moreover, P is saturated, then Coker λ_{F}^{gp} is a free abelian group of finite rank.

Proof of the claim. The map of (i) is obtained via the snake lemma, applied to the ladder of abelian groups :

\xymatrix{0 \ar[r] & (F ∩ P)^{gp} \ar[r] \ar[d]_{λ_{F}^{gp}} & P^{gp} \ar[r] \ar[d]^{λ^{gp}} & P^{gp}/(F ∩ P)^{gp} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & F^{gp} \ar[r] & Q^{gp} \ar[r] & Q^{gp}/F^{gp} \ar[r] & 0.}

taking into account that both Ker λ^{gp} and Coker λ^{gp} vanish. Then (i) is obvious, and (ii) comes down to checking that P^{gp}/(F ∩ P)^{gp} is torsion-free, in case P is fine and saturated. But since F ∩ P is a face of P, the latter assertion is an easy consequence of proposition 6.4.9. ♢

Now, set F := Q \ 𝔮; we come down to checking that B := F ⊗_{F ∩P} A/𝔭A is an integral domain, and remark 12.6.9(i) tells us that A/𝔭A is a domain. However, A/𝔭A is (F ∩ P)-flat (proposition 12.5.16(ii)), hence the natural map

B → C := F^{gp} ⊗_{(F ∩P)^{gp}} Frac(A/𝔭A)

is injective; on the other hand, claim 12.6.19(ii) implies that C = A[Coker λ_{F}^{gp}] is a free (polynomial) A-algebra, whence the contention. □

Example 12.6.20. In the situation of example 6.5.27, take T := Spec P for a fine monoid P, and define \underline{S} as in example 12.6.5(i). The k-Frobenius map k_{P} (example 6.5.10(i)) induces an endomorphism k\underline{S} := (Spec(R, k_{P}), k_{T}) of \underline{S} in 𝒦 . By proposition 12.6.14(iii) and example 6.5.27, there exists a unique morphism \underline{g} fitting into a commutative diagram of 𝒦_{int} :

(12.6.21)  \xymatrix{φ^{∗}\underline{S} \ar[r] \ar[d]_{\underline{g}} & \underline{S} \ar[d]^{_{k\underline{S}}} \\ φ^{∗}\underline{S} \ar[r] & \underline{S}}

(where the horizontal arrows are the cartesian morphisms). Say that φ^{∗}\underline{S} = ((Y, \underline{N}), F, ψ); then we have \underline{g} = (g, k_{F}) for a unique endomorphism g := (g, log g) of the log scheme (Y, \underline{N}). Let U := Spec P′ ⊂ F be any open affine subset ; since k_{F} is the identity on the underlying topological spaces, g restricts to an endomorphism g_{|ψ^{−1}U} of ψ^{−1}U ×_{Y} (Y, \underline{N}). In view of lemma 12.6.12, the latter log scheme is of the form \underline{S}′ as in example 12.6.5(ii), with Q := P^{gp} ×_{P^{′gp}} P′. Then g_{|ψ^{−1}U} is induced by an endomorphism ν of Q, fitting into a commutative diagram :

\xymatrix{P \ar[r] \ar[d]_{_{k_{P}}} & Q \ar[d]^{ν} \\ P \ar[r] & Q}

whose horizontal arrows are the natural injections. Since Q ⊂ P^{gp}, it is clear that ν = k_{Q}. Especially, g : Y → Y is a finite morphism of schemes. Furthermore, for every point y ∈ Y, we have a commutative diagram of monoids :

(12.6.22)  \xymatrix{_{\underline{N}_{g(y)}} \ar[d]_{log g_{y}} \ar[r] & \underline{N}_{g(y)}^{♯} \ar[d]_{log g_{y}^{♯}} & 𝒪_{F,ψ(y)} \ar[l]_{∼} \ar[d]^{k_{ψ(y)}:=(log k_{F})_{ψ(y)}} \\ _{\underline{N}_{y}} \ar[r] & _{\underline{N} ♯y} & 𝒪_{F,ψ(y)}. \ar[l]_{∼}}

12.6.23. Let (K, |·|) be a valued field, Γ and K^{+} respectively the value group and the valuation ring of | · |; denote by 1 ∈ Γ the neutral element, and by Γ_{+} ⊂ Γ the submonoid consisting of all elements ≤ 1. Set S := Spec K^{+}; we consider the log scheme (S, 𝒪_{S}^{∗}), where 𝒪_{S}^{∗} ⊂ 𝒪_{S} is the subsheaf such that 𝒪_{S}^{∗}(U) := 𝒪_{S}(U)\{0} for every open subset U ⊂ S. To this log scheme we associate the object of 𝒦_{int} :

f(K, | · |) := ((S, 𝒪_{S}^{∗}), Spec Γ_{+}, ψ_{Γ})

where ψ_{Γ} is the morphism of monoidal spaces arising from the isomorphism

Γ_{+} →^{∼} (K^{+}\{0})/(K^{+})^{×}

deduced from the valuation | · |. It is well known that ψ_{Γ} is a homeomorphism (see e.g. [75, §6.1.26]). We shall use objects of this kind to state some separation and properness criteria for log schemes whose existence is established via proposition 12.6.14. To this aim, we need to digress a little, to prove the following auxiliary results, which are refinements of the standard valuative criteria for morphisms of schemes.

12.6.24. Let f : X → Y be a morphism of schemes, R an integral ring, and K the field of fractions of R. Let us denote by X(R)_{max} ⊂ X(R) the set of morphisms Spec R → X which map Spec K to a maximal point of X. There follows a commutative diagram of sets :

(12.6.25)  \xymatrix{X(R)_{max} \ar[r] \ar[d] & X(K)_{max} \ar[d] \\ Y (R) \ar[r] & Y (K).}

Proposition 12.6.26. Let f : X → Y be a morphism of schemes. The following conditions are equivalent :

(a) f is separated.

(b) f is quasi-separated, and for every valued field (K, | · |), the map of sets :

(12.6.27)  X(K^{+})_{max} → X(K)_{max} ×_{Y (K)} X(K^{+})_{max}

deduced from diagram (12.6.25) (with R := K^{+}, the valuation ring of |·|), is injective.

Proof. (a) ⇒ (b) by the valuative criterion of separation ([58, Ch.I, Prop.5.5.4]). Conversely, we shall show that if (b) holds, then the assumptions of the criterion of loc.cit. are fulfilled. Indeed, let (L, | · |_{L}) be any valued field, with valuation ring L^{+}, and suppose we have two morphisms σ_{1}, σ_{2} : Spec L^{+} → X whose restrictions to Spec L agree, and such that f ∘ σ_{1} = f ∘ σ_{2}.

Let s, η ∈ Spec L^{+} be respectively the closed point and the generic point, set x := σ_{1}(η) = σ_{2}(η) ∈ X, and denote by φ : 𝒪_{X,x} → L the ring homomorphism corresponding to the restriction of σ_{1} (and σ_{2}); then x admits two specializations x_{i} := σ_{i}(s) ∈ X (for i = 1, 2) such that φ sends the image A_{i} ⊂ 𝒪_{X,x} of the specialization map 𝒪_{X,x_{i}} → 𝒪_{X,x}, into L^{+}, and the maximal ideal of A_{i} into the maximal ideal 𝔪_{L} of L^{+}, for both i = 1, 2. Moreover, y := f(x_{1}) = f(x_{2}).

Denote by B ⊂ 𝒪_{X,x} the smallest subring containing A_{1} and A_{2}, and set 𝔭 := B ∩ φ^{−1}𝔪_{L}. Then φ(B_{𝔭}) ⊂ L^{+} as well, and B_{𝔭} dominates both A_{1} and A_{2}. Now, let t ∈ X be a maximal point which specializes to x; by [58, Ch.I, Prop.5.5.2] we may find a valued field (E, | · |_{E}) with a local ring homomorphism 𝒪_{X,t} → E, such that the valuation ring E^{+} of | · |_{E} dominates the image of the specialization map 𝒪_{X,x} → 𝒪_{X,t}. Let κ(E) be the residue field of E^{+}, and \overline{B}_{𝔭} ⊂ κ(E) the image of B_{𝔭}. By the same token, we may find a valuation ring V ⊂ κ(E) with fraction field κ(E), and dominating \overline{B}_{𝔭}; then it is easily seen that the preimage K^{+} ⊂ E^{+} of V is a valuation ring with field of fractions K = E, which dominates the image of B_{𝔭} in 𝒪_{X,t} ([126, Th.10.1(iv)]). Hence, K^{+} dominates the images of 𝒪_{X,x_{i}}, for both i = 1, 2, and therefore, also the image of 𝒪_{Y,y}; in other words, in this way we obtain two elements in X(K^{+})_{max} whose images agree in Y (K^{+}), and whose restrictions Spec K → X coincide. By assumption, these two K^{+}-points must then coincide, especially x_{1} = x_{2}, and therefore σ_{1} = σ_{2}, as required. □

Proposition 12.6.28. Let f : X → Y be a quasi-compact morphism of schemes. The following conditions are equivalent :

(a) f is universally closed.

(b) For every valued field (K, | · |), the corresponding map (12.6.27) is surjective.

Proof. (a) ⇒ (b) by the valuative criterion of [58, Ch.I, Prop.5.5.8]. Conversely, we will show that (b) implies that the conditions of loc.cit. are fulfilled. Hence, let (L, | · |_{L}) be any valued field, with valuation ring L^{+}, and suppose that we have a morphism σ : Spec L → X, whose composition with f extends to a morphism Spec L^{+} → Y; we have to show that σ extends to a morphism Spec L^{+} → X. Denote by η, s ∈ Spec L^{+} respectively the generic and the closed point, let x ∈ X be the image of η, and y ∈ Y the image of s. Then σ corresponds to a ring homomorphism σ^{♮} : 𝒪_{X,x} → L, and L^{+} dominates the image of the map 𝒪_{Y,y} → 𝒪_{X,x} determined by f. Moreover, σ^{♮} factors through the residue field κ(x) of 𝒪_{X,x}. Then κ(x)^{+} := κ(x) ∩ L^{+} is a valuation ring with fraction field κ(x), and we are reduced to showing that there exists a specialization x′ ∈ X of x, such that f(x′) = y, and such that κ(x)^{+} dominates the image of the specialization map 𝒪_{X,x′} → 𝒪_{X,x}.

Let now t ∈ X be a maximal point which specializes to x; by [58, Ch.I, Prop.5.5.8] we may find a valued field (E, | · |_{E}) with a local ring homomorphism 𝒪_{X,t} → E, whose valuation ring E^{+} dominates the image of the specialization map 𝒪_{X,x} → 𝒪_{X,t}. Let κ(E) be the residue field of E^{+}; the induced map 𝒪_{X,x} → κ(E) factors through κ(x), and by [126, Th.10.2] we may find a valuation ring V ⊂ κ(E) with field of fractions κ(E), which dominates κ(x)^{+}. The preimage K^{+} ⊂ E^{+} of V is a valuation ring with field of fractions K = E ([126, Th.10.1]). By construction, K^{+} dominates the image of 𝒪_{X,y}, in which case assumption (b) says that there exists a specialization x′ of x such that f(x′) = y, and such that K^{+} dominates the image of the specialization map 𝒪_{X,x′} → 𝒪_{X,t}. A simple inspection then shows that κ(x)^{+} dominates the image of 𝒪_{X,x′} in 𝒪_{X,x}, as required. □

Corollary 12.6.29. Let f : X → Y be a morphism of schemes which is quasi-separated and of finite type. Then the following conditions are equivalent :

(a) f is proper.

(b) For every valued field (K, | · |), the corresponding diagram (12.6.25) (with R := K^{+} the valuation ring of | · |) is cartesian.

Proof. It is immediate from propositions 12.6.26and 12.6.28. □

12.6.30. We are now ready to return to log schemes. Resume the situation of (12.6.23), let \underline{X} := ((X, \underline{M}), F, ψ) be any object of 𝒦 , and denote by α : \underline{M} → 𝒪_{X} the structure map of \underline{M}.

Suppose we are given a morphism σ : S → X of schemes, and we ask whether there exists a morphism of log structures β : σ^{∗}\underline{M} → 𝒪_{S}^{∗}, such that the pair (σ, β) is a morphism of log schemes (S, 𝒪_{S}^{∗}) → (X, \underline{M}). By definition, this holds if and only if the composition

\overline{β} : σ^{∗}\underline{M} ⟶^{σ^{∗}α} σ^{∗}𝒪_{X} ⟶^{σ^{♮}} 𝒪_{S}

factors through 𝒪_{S}^{∗}. Moreover, in this case the factorization is unique, so that σ determines β uniquely. Let η ∈ S be the generic point; we claim that the stated condition is fulfilled, if and only if t := σ(η) lies in (X, \underline{M})_{tr} (see definition 12.2.7(i)). Indeed, if t lies in the trivial locus, \underline{M}_{t} = 𝒪_{X,t}^{×}, so certainly the image of \overline{β}_{t} : \underline{M}_{t} → 𝒪_{S,η} lies in 𝒪_{S,η}^{∗} = K^{×}. Now, if s ∈ S is any other point, the composition

\underline{M}_{σ(s)} ⟶^{\overline{β}_{σ(s)}} 𝒪_{S,s} → 𝒪_{S,η} = K

factors through \underline{M}_{t}, hence its image lies in K^{×} ∩ 𝒪_{S,s} = 𝒪_{S,s}^{∗}, whence the contention. Conversely, if \overline{β} factors through 𝒪_{S}^{∗}, it follows especially that the image of the stalk of the structure map \underline{M}_{t} → 𝒪_{X,t} lies in the preimage of 𝒪_{S,η}^{∗} = 𝒪_{S,η}^{×}, and the latter is just 𝒪_{X,t}^{×}, so t lies in the trivial locus.

Next, let f : (S, 𝒪_{S}^{∗}) → (X, \underline{M}) be any morphism of log schemes. We claim that there exists a unique morphism of fans φ := (φ, log φ) : Spec Γ_{+} → F, such that the pair (f, φ) is a morphism f(K, | · |) → \underline{X} in 𝒦 . Indeed, since ψ_{Γ} is a homeomorphism, there exists a unique continuous map φ on the underlying topological spaces, such that φ ∘ ψ_{Γ} = ψ ∘ f, and for the same reason, the map log f^{♯} : f^{∗}\underline{M}/𝒪_{S}^{×} → 𝒪_{S}^{∗}/𝒪_{S}^{×} is of the form ψ_{Γ}^{∗}(log φ) for a unique morphism of sheaves log φ as sought. Then log φ will be a local morphism, since the same holds for log f (see (12.6.1)).

Summing up, we have shown that the natural map

Hom_{𝒦} (f(K, | · |), \underline{X}) → X(K^{+}) : (σ, φ) ↦ σ

is injective, and its image is the set of all the morphisms S → X which map η into (X, \underline{M})_{tr}.

Proposition 12.6.31. Let \underline{X} := ((X, \underline{M}), F, ψ) be an object of 𝒦_{int}, and φ : F′ → F an integral partial subdivision, with F and F′ locally fine. We have :

(i) If the induced map F′(ℕ) → F (ℕ) is injective, the cartesian morphism φ^{∗}\underline{X} → \underline{X} is separated.

(ii) If φ is a proper subdivision, the cartesian morphism φ^{∗}\underline{X} → \underline{X} is proper, and induces an isomorphism of schemes (φ^{∗}\underline{X})_{tr} →^{∼} (X, \underline{M})_{tr}.

Proof. The assertions are local on X (cp. the proof of claim 12.6.15), so we may assume – by lemma 12.6.11 – that \underline{M} admits a chart P_{X} → \underline{M}, and F = (Spec P)^{♯}. In this case, let S := Spec ℤ[P], and denote by \underline{S} the object of 𝒦_{int} attached to P, as in example 12.6.5(i) (with R := ℤ); then (X, \underline{M}) is isomorphic to X ×_{S} (S, P_{S}^{log}), and if f_{S} : φ^{∗}\underline{S} → \underline{S} is the cartesian morphism over φ, then the cartesian morphism f : φ^{∗}\underline{X} → \underline{X} is given by the pair (1_{X} ×_{S} f_{S}, φ) (see the proof of proposition 12.6.14(iii)). Thus, we may replace \underline{X} by \underline{S} ([58, Ch.I, Prop.5.3.1(iv)]), in which case lemma 12.6.12 shows that the log scheme (X′, \underline{M}′) underlying φ^{∗}\underline{S} admits an open covering consisting of affine log schemes of the form Spec(ℤ, Q), for a fine monoid Q. Notice that, for such Q we have :

Spec(ℤ, Q)_{tr} = ℤ[Q^{gp}]

which is a dense open subset of Spec ℤ[Q].

(i): According to proposition 12.6.14(v), the morphism f is quasi-separated, so we may apply the criterion of proposition 12.6.26. Indeed, let (K, | · |) be any valued field, and suppose that σ_{i} ∈ X′(K^{+})_{max} (for i = 1, 2) are two sections, whose images in X′(K)_{max} ×_{S(K)} S(K^{+}) coincide; we have to show that σ_{1} = σ_{2}. However, we have just seen that the maximal points of X′ lie in (X′, \underline{M}′)_{tr}, so the discussion of (12.6.30) shows that both σ_{i} extend uniquely to morphisms σ′_{i} : f(K, | · |) → φ^{∗}\underline{S} in 𝒦 , and it suffices to show that σ′_{1} = σ′_{2}. By definition, the datum of σ′_{i} is equivalent to the datum of a morphism σ′′_{i} : f(K, | · |) → \underline{S}, and a morphism of fans φ′_{i} : Spec Γ_{+} → F′. Again by (12.6.30), the morphisms σ′′_{1} and σ′′_{2} agree if and only if they induce the same morphisms of schemes; the latter holds by assumption, since σ_{1} and σ_{2} yield the same element of S(K^{+}). On the other hand, in view of (b) and proposition 6.5.24, the elements φ_{1}, φ_{2} ∈ F′(Γ_{+}) coincide if and only if their images in F (Γ_{+}) coincide; but again, this last condition holds since the images of both σ_{i} agree in S(K^{+}).

(ii): In view of (i) and proposition 12.6.14(iv),(v) we know already that f is separated and of finite type, so we may apply the criterion of corollary 12.6.29. Hence, let σ ∈ S(K^{+}) be a section, and x ∈ X′(K)_{max} a K-rational point such that f(x) is the image of σ in S(K); in view of (i), it suffices to show that σ lifts to a section σ̃ ∈ X′(K^{+})_{max}, whose image in X′(K) is x. Since x ∈ (X′, \underline{M}′)_{tr}, remark 12.6.17 implies that f(x) ∈ (S, P_{S}^{log})_{tr}, and then the discussion in (12.6.30) says that σ underlies a unique morphism (σ′, β) : f(K, | · |) → \underline{S}. By proposition 6.5.26, the element β ∈ F (Γ_{+}) lifts to an element β′ ∈ F′(Γ_{+}), and finally the pair ((σ, β), β′) determines a unique morphism f(K, | · |) → φ^{∗}\underline{S}, whose underlying morphism of schemes is the sought σ̃. Lastly, notice that the map F′({1}) → F ({1}) induced by φ, is bijective by propositions 6.5.24 and 6.5.26 (where {1} is the monoid with one element), which means that φ restricts to a bijection on the points of height zero; then remark 12.6.17 implies the second assertion of (ii). □

Theorem 12.6.32. Let (X, \underline{M}) be a regular log scheme. Then there exists a smooth morphism of log schemes f : (X′, \underline{M}′) → (X, \underline{M}), whose underlying morphism of schemes is proper and birational, and such that X′ is a regular scheme. More precisely, f restricts to an isomorphism of schemes f^{−1}X_{reg} → X_{reg} on the preimage of the open locus of regular points of X.

Proof. We use the object \underline{X} := ((X, \underline{M}), F (X, \underline{M}), π_{X}) attached to (X, \underline{M}) as in remark 12.6.9(ii). Indeed, it is clear that F (X, \underline{M}) is locally fine and saturated, hence theorem 6.6.31 yields an integral, proper, simplicial subdivision φ : F′ → F (X, \underline{M}) which restricts to an isomorphism φ^{−1}F (X, \underline{M})_{sim} →^{∼} F (X, \underline{M})_{sim}. Take (f, φ) : φ^{∗}\underline{X} → \underline{X} to be the cartesian morphism over φ, and denote by (X′, \underline{M}′) the log scheme underlying φ^{∗}\underline{X}; it follows already from proposition 12.6.31(ii) that f is proper on the underlying schemes. Next, corollary 12.5.19 shows that X_{reg} is π^{−1}F (X, \underline{M})_{sim}, so f restricts to an isomorphism f^{−1}X_{reg} →^{∼} X_{reg}. Furthermore, f is étale, by proposition 12.6.14(v), hence the log scheme (X′, \underline{M}′) is regular (theorem 12.5.28). Finally, again by corollary 12.5.12we see that X′ is regular. □

12.6.33. Let now (Y, \underline{N}) be a regular log scheme, such that F (Y, \underline{N}) is affine (notation of remark 12.6.9(ii)), say isomorphic to (Spec P)^{♯}, for some fine, sharp and saturated monoid P. Let I ⊂ P be an ideal generated by two elements a, b ∈ P, and denote by f : (Y′, \underline{N}′) → (Y, \underline{N}) the saturated blow up of the ideal I\underline{N} of \underline{N} (see (12.4.49)). Set U′ := (Y′, \underline{N}′)_{tr}, U := (Y, \underline{N})_{tr}, and denote j : U → Y, j′ : U′ → Y′ the open immersions. In this situation we have :

Lemma 12.6.34. (i) H^{1}(Y′, \underline{N}′^{♯gp}) = 0.

(ii) Suppose moreover, that R^{1}j′_{∗}𝒪_{U′}^{×} = 0. Then R^{1}j_{∗}𝒪_{U}^{×} = 0.

Proof. (i): Since \underline{N}′^{♯gp} = π_{X}^{∗}𝒪_{F (Y′,\underline{N}′)}^{gp}, claim 12.6.39(ii) and remark 12.6.9(iii) reduce to showing

(12.6.35)  H^{1}(F (Y′, \underline{N}′), 𝒪_{F (Y′,\underline{N}′)}^{gp}) = 0.

However, by corollary 12.6.18(iii), the fan F (Y′, \underline{N}′) is the saturated blow up of the ideal I𝒪_{F (Y,\underline{N})} of 𝒪_{F (Y,\underline{N})}, hence it admits the affine covering

F (Y′, \underline{N}′) = (Spec P [a^{−1}b]^{sat})^{♯} ∪ (Spec P [b^{−1}a]^{sat})^{♯}.

Notice now that every affine fan is a local topological space, hence the left hand-side of (12.6.35) is computed by theČech cohomology of 𝒪_{F (Y′,\underline{N}′)}^{gp} relative to this covering (theorem 10.2.24(ii)).

However, the intersection of the two open subsets is (Spec P [a^{−1}b, b^{−1}a])^{♯}, and clearly the restriction map

H^{0}(Spec P [a^{−1}b]^{sat}, 𝒪_{F (Y′,\underline{N}′)}^{gp}) → H^{0}(Spec P [a^{−1}b, b^{−1}a], 𝒪_{F (Y′,\underline{N}′)}^{gp})

is surjective. The assertion is an immediate consequence.

(ii): Let y ∈ Y be any point; according to (12.4.50), the morphism

f ×_{Y} Y (y) : (Y′, \underline{N}′) ×_{Y} Y (y) → (Y (y), \underline{N}(y))

is the saturated blow up of the ideal I\underline{N}(y); on the other hand, let U(y) := U ×_{Y} Y (y), U′(y) := U′ ×_{Y} Y (y) and denote by j_{y} : U(y) → Y (y) and j′_{y} : U′(y) → Y′ ×_{Y} Y (y) the open immersions; in light of proposition 10.1.10(ii), it suffices to show that R^{1}j_{y∗}𝒪_{U(y)} = 0, and the assumption implies that R^{1}j′_{y∗}𝒪_{U′(y)} = 0. Summing up, we may replace Y be Y (y), and assume from start that Y is local, and y is its closed point. From the assumption we get :

H^{1}(Y′, j′_{∗}𝒪_{U′}^{×}) = H^{1}(U′, 𝒪_{Y′}^{×}) = Pic U′.

On the other hand, recall that \underline{N}′^{♯gp} = j′_{∗}𝒪_{U′}^{×}/𝒪_{Y′}^{×} (proposition 12.5.38); combining with (i), we deduce that the natural map

Pic Y′ → Pic U′

is surjective. Set Y′_{0} := f^{−1}(y) ⊂ Y′, endow Y′_{0} with its reduced subscheme structure, and let i : Y′_{0} → Y′ be the closed immersion. If I is an invertible ideal of P, then f is an isomorphism, in which case the assertion is obvious. We may then assume that I is not invertible, in which case claim 12.4.56(ii) says that there exists a morphism of (Y, \underline{N})-schemes h : (Y′, \underline{N}′) → ℙ_{(Y,\underline{N})}^{1} inducing an isomorphism of κ(y)-schemes

(12.6.36)  (h ×_{Y} Spec κ(y))_{red} : Y′_{0} →^{∼} ℙ_{κ(y)}^{1}.

Let us remark :

Claim 12.6.37. Let S be a noetherian local scheme, s the closed point of S, and f : X → S a proper morphism of schemes. Suppose that dim X(s) ≤ 1, and H^{1}(X(s), 𝒪_{X(s)}) = 0. Then the natural map

Pic X → Pic X(s)

is injective.

Proof of the claim. Say that S = Spec A for a local ring A, and denote by 𝔪_{A} ⊂ A the maximal ideal. For every k ∈ ℕ, set S_{n} := Spec A/𝔪_{A}^{k+1}, and let i_{n} : X_{n} := X ×_{S} S_{n} → X be the closed immersion. Let ℒ be any invertible 𝒪_{X}-module, and suppose that i_{0}^{∗}ℒ ≃ 𝒪_{X_{0}}; we have to show that ℒ ≃ 𝒪_{X}. We notice that, for every k ∈ ℕ, the natural map

H^{0}(X_{k+1}, i_{k+1}^{∗}ℒ) → H^{0}(X_{k}, i_{k}^{∗}ℒ)

is surjective : indeed, its cokernel is an A-submodule of H^{1}(X_{0}, 𝔪_{A}^{k}ℒ /𝔪_{A}^{k+1}ℒ), and since dim X_{0} ≤ 1, the natural map

(𝔪_{A}^{k}/𝔪_{A}^{k+1}) ⊗_{κ(s)} H^{1}(X_{0}, i_{0}^{∗}ℒ) →^{∼} H^{1}(X_{0}, (𝔪_{A}^{k}/𝔪_{A}^{k+1}) ⊗_{κ(s)} ℒ) → H^{1}(X_{0}, 𝔪_{A}^{k}ℒ /𝔪_{A}^{k+1}ℒ)

is surjective; on the other hand, our assumptions imply that H^{1}(X_{0}, i_{0}^{∗}ℒ) = 0, whence the contention. Let A^{∧} be the 𝔪_{A}-adic completion of A; taking into account [61, Ch.III, Th.4.1.5], we deduce that the natural map

H^{0}(X, ℒ) ⊗_{A} A^{∧} → H := H^{0}(X_{0}, i^{∗}ℒ)

is a continuous surjection, for the 𝔪_{A}-adic topologies. Since H is a discrete space for this topology, and since the image of H^{0}(X, ℒ) is dense in the 𝔪_{A}-adic topology of H^{0}(X, ℒ) ⊗_{A} A^{∧}, we conclude that the restriction map H^{0}(X, ℒ) → H is surjective as well. Let \overline{s} ∈ H be a global section of i_{0}^{∗}ℒ whose image in i_{0}^{∗}ℒ_{x} is a generator of the latter A_{0}-module, for every x ∈ X_{0}, and pick s ∈ H^{0}(X, ℒ) whose image in H equals \overline{s}. It remains only to check that, for every x ∈ X, the A-module ℒ_{x} is generated by the image s_{x} of s. However, since X is proper, every x ∈ X specializes to a point of X_{0}, hence we may assume that x ∈ X_{0}, in which case one concludes easily, by appealing to Nakayama’s lemma (details left to the reader). ♢

Combining claim 12.6.37and [122, Prop.11.1(i)], we see that the induced map

i^{∗} : Pic Y′ → Pic Y′_{0}

is injective. Let now ℒ be any invertible 𝒪_{U}-module; we have to show that ℒ extends to an invertible 𝒪_{Y} -module (which is then isomorphic to 𝒪_{Y}). However, notice that f restricts to an isomorphism g : U′ →^{∼} U, hence g^{∗}ℒ is an invertible 𝒪_{U′}-module, and by the foregoing there exists an invertible 𝒪_{Y′}-module ℒ′ such that ℒ′_{|U′} ≃ g^{∗}ℒ. In light of the isomorphism (12.6.36), there exists an invertible 𝒪_{ℙ_{Y}^{1}} -module ℒ′′ such that i^{∗}ℒ′ ≃ i^{∗}h^{∗}ℒ′′. Therefore

ℒ′ ≃ h^{∗}ℒ′′.

Now, on the one hand, claim 12.6.37 implies that ℒ′′ = 𝒪_{ℙ_{Y}^{1}} (n) for some n ∈ ℕ; on the other hand, (12.4.17) implies that h restricts to a morphism of schemes h_{tr} : U′ → 𝔾_{m,Y}, and ℒ′′_{|𝔾_{m,Y}} = 𝒪_{ℙ_{Y}^{1}} (n)_{|𝔾_{m,Y}} = 𝒪_{𝔾_{m,Y}}, so finally g^{∗}ℒ = 𝒪_{U′}, hence ℒ = 𝒪_{U}, whence the contention. □

The following result complements proposition 12.5.38.

Theorem 12.6.38. Let (X, \underline{M}) be any regular log scheme, set U := (X, \underline{M})_{tr} and denote by j : U → X the open immersion. Then we have :

R^{1}j_{∗}𝒪_{U}^{×} = 0.

Proof. We begin with the following general :

Claim 12.6.39. Let π : T_{1} → T_{2} be a continuous open and surjective map of topological spaces, such that π^{−1}(t) is an irreducible topological space (with the subspace topology) for every t ∈ T_{2}. Then, we have :

(i) For every sheaf S on T_{2}, the natural map S → π_{∗}π^{∗}S is an isomorphism.

(ii) For every abelian sheaf S on T_{2}, the natural map S[0] → Rπ_{∗}π^{∗}S is an isomorphism in D(ℤ_{T_{2}}-Mod).

Proof of the claim. (i): Since π is open, π^{∗}S is the sheaf associated to the presheaf : U ↦ S(πU), for every open subset U of T_{1}. We show, more precisely, that this presheaf is already a sheaf; since π is surjective, the claim shall follow immediately. Now, let U ⊂ T_{1} be an open subset, and (U_{i} | i ∈ I) a family of open subsets of X covering U; for every i, j ∈ I, set U_{ij} := U_{i} ∩ U_{j}. It suffices to show that S(πU) is the equalizer of the two maps :

\xymatrix{∏_{i∈I} S(πU_{i}) \ar[r] \ar[r] & ∏_{i,j∈I} S(πU_{ij}).}

Since S is a sheaf, the latter will hold, provided we know that πU_{i} ∩ πU_{j} = πU_{ij} for every i, j ∈ I. Hence, let t ∈ πU_{i} ∩ πU_{j}; this means that π^{−1}(t) ∩ U_{i} ≠ ∅ and π^{−1}(t) ∩ U_{j} ≠ ∅. Since π^{−1}(t) is irreducible, we deduce that π^{−1}(t) ∩ U_{ij} ≠ ∅, as required.

(ii): The proof of (i) also shows that, for every flabby abelian sheaf J on T_{2}, the abelian sheaf π^{∗}J is flabby on T_{1}. Hence, if S is any abelian sheaf on T_{2}, we obtain a flabby resolution of π^{∗}S of the form π^{∗}J_{•}, by taking a flabby resolution S → J_{•} of S on T_{2}. According to remark 7.3.31(vi), there is a natural isomorphism

π_{∗}π^{∗}J →^{∼} Rπ_{∗}π^{∗}S

in D(ℤ_{T_{2}}-Mod). Then the assertion follows from (i). ♢

After these preliminaries, let us return to the log scheme (X, \underline{M}), and its associated object \underline{X} := ((X, \underline{M}), F (X, \underline{M}), π_{X}). The assertion to prove is local on X, hence we may assume that F (X, \underline{M}) = (Spec P)^{♯}, for some sharp, fine and saturated monoid P. Next, by theorem 12.6.32(and its proof) there exists an integral proper simplicial subdivision φ : F′ → F (X, \underline{M}), such that the log scheme (X′, \underline{M}′) underlying φ^{∗}\underline{X} is regular, X′ is regular, and the morphism X′ → X restricts to an isomorphism U′ := (X′, \underline{M}′)_{tr} → U. In this situation, we may find a further subdivision φ′ : F′′ → F′ such that both φ′ and φ ∘ φ′ are compositions of saturated blow up of ideals generated by at most two elements of P (example 6.6.15(iii)). Say that φ ∘ φ′ = φ_{r} ∘ φ_{r−1} ⋯ ∘ φ_{1}, where each φ_{i} is a saturated blow up of the above type. By proposition 12.6.14(iii), we deduce a sequence of morphisms of log schemes

(X_{1}, \underline{M}_{1}) ⟶^{g_{1}} ⋯ → (X_{r−1}, \underline{M}_{r−1}) ⟶^{g_{r−1}} (X_{r}, \underline{M}_{r}) ⟶^{g_{r}} (X_{r+1}, \underline{M}_{r+1}) := (X, \underline{M})

each of which is the blow up of a corresponding ideal, and by the same token, φ′ induces a morphism g : (X_{1}, \underline{M}_{1}) → (X′, \underline{M}′) of (X, \underline{M})-schemes. For every i = 1, ⋯, r + 1, set U_{i} := (X_{i}, \underline{M}_{i})_{tr}, and let j_{i} : U_{i} → X_{i} be the open immersion; especially, U_{r+1} = U. We shall show, by induction on i, that

(12.6.40)  R^{1}j_{i∗}𝒪_{U_{i}}^{×} = 0  for i = 1, …, r + 1.

Notice first that the stated vanishing translates the following assertion. For every x ∈ X_{i} and every invertible 𝒪_{U_{i}}-module ℒ, there exists an open neighborhood U_{x} of x in X_{i} such that ℒ_{|U_{x}∩U_{i}} extends to an invertible 𝒪_{U_{x}}-module. However, it follows immediately from propositions 11.3.8and 11.3.15, that every invertible 𝒪_{U′}-module extends to an invertible 𝒪_{X′}-module. Since g restricts to an isomorphism U_{1} = g^{−1}U′ →^{∼} U′, we easily deduce that every invertible 𝒪_{U_{1}}-module extends to an invertible 𝒪_{X_{1}}-module (namely : if ℒ is such a module, extend g_{|U_{1}∗}ℒ to an invertible 𝒪_{X′}-module, and pull the extension back to X_{1}, via g^{∗}). Summing up, we see that (12.6.40) holds for i = 1.

Next, suppose that (12.6.40) has already been shown to hold for a given i ≤ r; by lemma 12.6.34(ii), it follows that (12.6.40) holds for i + 1, so we are done. □

12.6.41. Let X be a scheme, \underline{M} a fine log structure on the Zariski site of X, x a point of X, and notice that every fractional ideal of \underline{M}_{x} is finitely generated (lemma 6.4.39(iv)). Say that X(x) = Spec A; in view of lemma 6.4.45(ii) there is a natural map of abelian groups

Div(\underline{M}_{x}) → Div(A).

Composing with the map I ↦ I^{∼} as in (11.3.12), we get, by virtue of loc.cit., a map

(12.6.42)  Div(\underline{M}_{x}) → Div X(x).

Corollary 12.6.43. With the notation of (12.6.41), suppose furthermore that (X, \underline{M}) is regular at the point x. Then (12.6.42) induces an isomorphism

\overline{Div}(\underline{M}_{x}) →^{∼} \overline{Div}(X(x)).

Proof. By virtue of proposition 6.4.55(ii) (and remark 11.3.11(i)), the map under investigation is already known to be injective. To show surjectivity, let K be the field of fractions of A, and L ⊂ K any reflexive fractional ideal of A; we may then regard ℒ := L^{∼} as a coherent 𝒪_{X}- submodule of i_{∗}𝒪_{X_{0}}, where i : X_{0} → X is the inclusion map of the subscheme X_{0} := Spec K. By proposition 11.3.15, the 𝒪_{U}-module ℒ_{|U} is invertible, hence it extends to an invertible 𝒪_{X(x)}- module ℒ′, by virtue of theorem 12.6.38. Since X(x) is local, ℒ′ is a free 𝒪_{X(x)}-module, and therefore ℒ_{|U} ≃ 𝒪_{U}. Thus, pick a ∈ ℒ (U) ⊂ K which generates ℒ_{|U}; after replacing L by a^{−1}L, we may assume that ℒ_{|U} = 𝒪_{U} as subsheaves of i_{∗}𝒪_{X_{0}}. Let Σ be the set of points of height one of X(x) contained in X(x) \ U; for each y ∈ Σ, the maximal ideal 𝔪_{y} of 𝒪_{X(x),y} is generated by a single element a_{y}, and there exists k_{y} ∈ ℤ such that a_{y}^{k_{y}} generates the 𝒪_{X(x),y}- submodule ℒ_{y} of K. To ease notation, let P := \underline{M}_{x}, and denote ψ : X(x) → Spec P the natural continuous map; also, let 𝔪_{ψ(y)} be the maximal ideal of the localization P_{ψ(y)} for every y ∈ X(x), and set

I := ⋂_{y∈Σ} 𝔪_{ψ(y)}^{k_{y}}.

In light of lemma 6.4.39(i,ii) and proposition 6.4.50, it is easily seen that I/P^{×} is a reflexive fractional ideal of the fine and saturated monoid P^{♯}, so I is a reflexive fractional ideal of P. Then IA ⊂ K is a reflexive fractional ideal of A. Set ℒ′′ := (IA)^{∼} ⊂ i_{∗}𝒪_{X_{0}}; then ℒ′′_{|U} = ℒ_{|U} and ℒ′′_{y} = ℒ_{y} for every y ∈ Σ. It follows that, for every y ∈ Σ, there exists an open neighborhood U_{y} of y in X(x) such that ℒ′′_{|U_{y}} = ℒ_{|U_{y}}. Let U′ := U ∪ ⋃_{y∈Σ} U_{y}, and denote by j′ : U′ → X(x) the open immersion. Notice that δ′(y, 𝒪_{X}) ≥ 2 for every y ∈ X \ U′ (corollary 12.5.13). In light of proposition 11.3.8(ii) (and remark 11.3.11(i)), we deduce

ℒ = j′_{∗}ℒ_{|U′} = j′_{∗}ℒ′′_{|U′} = ℒ′′

whence the contention. □

12.6.44. Let \underline{M} be a log structure on the Zariski site of a local scheme X, such that (X, \underline{M}) is a regular log scheme. Let x ∈ X be the closed point, say that X = Spec B for some local ring B, and let β : P → B a chart for \underline{M} which is sharp at x. As usual, if M is any B-module, we denote M^{∼} the quasi-coherent 𝒪_{X}-module arising from M.

Theorem 12.6.45. In the situation of (12.6.44), suppose as well that dim P = 2 and dim X = 2. Then every indecomposable reflexive 𝒪_{X}-module is isomorphic to (IB)^{∼}, for some reflexive fractional ideal I of P. (Notation of (6.4.44).)

Proof. Set Q := Div_{+}(P) and define φ : P → Q as in example 6.4.64. The chart β defines a morphism ψ : (X, \underline{M}) → Spec(ℤ, P), and we let (X′, \underline{M}′) be the fibre product in the cartesian diagram

\xymatrix{(X′, \underline{M}′) \ar[r] \ar[d]_{f} & Spec(ℤ, Q) \ar[d]^{Spec(ℤ,φ)} \\ (X, \underline{M}) \ar[r]^{ψ} & Spec(ℤ, P).}

Arguing as in the proof of claim 13.3.37, it is easily seen that X′ is a local scheme and (X′, \underline{M}′) is regular. Moreover, X′ is a regular scheme, since Q is a free monoid (corollary 12.5.19), and f is a finite morphism of Kummer type (lemma 13.3.6).

Claim 12.6.46. The 𝒪_{X}-module f_{∗}𝒪_{X′} is isomorphic to a finite direct sum (I_{1}B ⊕ ⋯ ⊕ I_{k}B)^{∼}, where I_{1}, …, I_{k} are reflexive fractional ideals of P, and 𝒪_{X} is a direct summand of f_{∗}𝒪_{X′}.

Proof of the claim. In view of example 6.4.64, we see that f_{∗}𝒪_{X′} = (Q⊗_{P} B)^{∼} is the direct sum of the 𝒪_{X}-modules (gr_{γ}Q ⊗_{P} B)^{∼}, where γ ranges over the elements of Q^{gp}/P^{gp}, and gr_{•}Q denotes the φ-grading of Q. But for each such γ, the natural map gr_{γ}Q ⊗_{P} B → gr_{γ}Q · B is an isomorphism, since B is P -flat (proposition 12.5.16(ii)). This shows the first assertion, and the second is clear as well, since gr_{0}Q = P. ♢

Denote by x′ the closed point of X′; from corollary 12.5.20(i) we see that U := (X, \underline{M})_{1} is the complement of \{x\} and U′ := (X′, \underline{M}′)_{1} is the complement of \{x′\} (notation of definition 12.2.7(i)). Let also j : U → X and j′ : U′ → X′ be the open immersions.

Claim 12.6.47. (i) The restriction g : U′ → U of f is a flat morphism of schemes. (ii) The functor j^{∗} : 𝒪_{X}-Rflx → 𝒪_{U}-Rflx is an equivalence (see definition 11.3.1(iii)).

Proof of the claim. (i): It suffices to check that the restriction Spec(ℤ, Q)_{1} → Spec(ℤ, P)_{1} of Spec(ℤ, φ) is flat. However, we have the affine open covering

Spec(ℤ, P)_{1} = Spec ℤ[P_{𝔭_{1}}] ∪ Spec ℤ[P_{𝔭_{2}}]

where 𝔭_{1}, 𝔭_{2} ⊂ P are the two prime ideals of height one (see example 6.4.19(i)). Hence, we are reduced to showing that the morphism of log schemes underlying

Spec(ℤ, φ_{𝔭_{i}}) : Spec(ℤ, Q_{𝔭_{i}}) → Spec(ℤ, P_{𝔭_{i}})

is flat for i = 1, 2. However, it is clear that Q_{𝔭_{i}} is an integral P_{𝔭_{i}}-module, hence it suffices to check that Q_{𝔭_{i}} is a flat P_{𝔭_{i}}-module, for i = 1, 2 (proposition 6.1.52), or equivalently, that Q_{𝔭_{i}}^{♯} is a flat P_{𝔭_{i}}^{♯}-module (corollary 6.1.49(ii)). The latter assertion follows immediately from the discussion of (6.4.61).

(ii): From proposition 11.3.8 we see that j^{∗} is full and essentially surjective. Moreover, it follows from remark 11.3.5 that every reflexive 𝒪_{X}-module is S_{1}, so j^{∗} is also faithful (details left to the reader). ♢

In light of claim 12.6.47(ii), it suffices to show that every indecomposable reflexive 𝒪_{U}- module ℱ is isomorphic to (IB)_{|U}^{∼}, for some reflexive fractional ideal I of P. However, for such ℱ, claim 12.6.47(i) and lemma 11.3.7(i) imply that g^{∗}ℱ is a reflexive 𝒪_{U′}-module. From proposition 11.3.8 and corollary 10.4.23 we deduce that δ′(x′, j′_{∗}g^{∗}ℱ) ≥ 2, so j′_{∗}g^{∗}ℱ is a free 𝒪_{X′}-module of finite rank ([63, Ch.0, Prop.17.3.4]), and finally, g^{∗}ℱ is a free 𝒪_{U′}-module of finite rank. Taking into account claim 12.6.46, it follows that g_{∗}g^{∗}ℱ = ℱ ⊗_{𝒪_{U}} g_{∗}𝒪_{U′} is a direct sum of 𝒪_{U}-modules of the type (IB)_{|U}^{∼}, for various I ∈ Div(P); moreover, ℱ is a direct summand of g_{∗}g^{∗}ℱ. Then, we may find a decomposition g_{∗}g^{∗}ℱ = ℱ_{1} ⊕ ⋯ ⊕ ℱ_{t} such that ℱ_{i} is an indecomposable 𝒪_{U}-module for i = 1, …, t, and ℱ_{1} = ℱ (details left to the reader : notice that – since reflexive 𝒪_{U}-modules are S_{1} – the length t of such a decomposition is bounded by the dimension of the κ(η)-vector space (g_{∗}g^{∗}ℱ)_{η}, where η is the maximal point of X). On the other hand, notice that

End_{𝒪_{U}} ((IB)_{|U}^{∼}) = End_{B}(IB)  (claim 12.6.47(ii))
= (IB : IB)  (by (6.4.53))
= (I : I)B  (lemma 6.4.45(i))
= B  (proposition 6.4.55(i))

for every I ∈ Div(P). Now the contention follows from theorem 7.11.3. □

12.6.48. Let now (X_{ét}, \underline{M}) be a quasi-coherent log scheme on the étale site of X. Pick a covering family (U_{λ} | λ ∈ Λ) of X in X_{ét}, and for every λ ∈ Λ, a chart P_{λ,U_{λ}} → \underline{M}_{|U_{λ}}. The latter induce isomorphisms of log schemes (U_{λ}, \underline{M}_{|U_{λ}}) →^{∼} ũ^{∗}(U_{λ,Zar}, P_{U_{λ,Zar}}^{log}) (see (12.2.2)); in other words, every quasi-coherent log structure on X_{ét} descends, locally on X_{ét}, to a log structure on the Zariski site. However, (X_{ét}, \underline{M}) may fail to descend to a unique log structure on the whole of X_{Zar}. Our present aim is to show that, at least under a few more assumptions on \underline{M}, we may find a blow up (X′_{ét}, \underline{M}′) → (X_{ét}, \underline{M}) such that (X′_{ét}, \underline{M}′) descends to a log structure on X′_{Zar}. To begin with, for every λ ∈ Λ let

T_{λ} := (Spec P_{λ})^{♯} and S_{λ} := Spec ℤ[P_{λ}].

Also, let \underline{S}_{λ} := (Spec(ℤ, P_{λ}), T_{λ}, ψ_{P_{λ}}) be the object of 𝒦 attached to P_{λ}, as in example 12.6.5(i). From the isomorphism

(U_{λ,Zar}, ũ_{∗}\underline{M}_{|U_{λ}}) →^{∼} U_{λ} ×_{S_{λ}} Spec(ℤ, P_{λ})

we deduce an object

\underline{U}_{λ} := U_{λ} ×_{S_{λ}} \underline{S}_{λ} = ((U_{λ,Zar}, ũ_{∗}\underline{M}_{|U_{λ}}), T_{λ}, ψ_{λ}).

Suppose now that \underline{M} is a fs log structure; then we may choose for each P_{λ} a fine and saturated monoid (lemma 12.1.18(iii)). Next, suppose that X is quasi-compact; in this case we may assume that Λ is a finite set, hence 𝒮 := {P_{λ} | λ ∈ Λ} is a finite set of monoids, and consequently we may choose a finite sequence of integers \underline{c}(𝒮) fulfilling the conditions of (6.6.25) relative to the category 𝒮 -Fan. Then, for every λ ∈ Λ we have a well defined integral roof ρ_{λ} : T_{λ}(ℚ_{+}) → ℚ_{+}, and we denote by f_{λ} : T (ρ_{λ}) → T_{λ} the corresponding subdivision. There follows a cartesian morphism

f_{λ}^{∗}\underline{U}_{λ} → \underline{U}_{λ}

whose underlying morphism of log schemes is a saturated blow up of the ideal ℐ_{ρ_{λ}} ũ_{∗}\underline{M}_{|U_{λ}} (proposition 6.6.13and corollary 12.6.18). Next, for λ, μ ∈ Λ, set U_{λμ} := U_{λ} ×_{X} U_{μ} and let

\underline{U}_{λμ} := ((U_{λμ}, ũ_{∗}\underline{M}_{|U_{λ}μ}), T_{λ}, ψ_{λμ})

where ψ_{λμ} is the composition of ψ_{λ} and the projection U_{λμ} → U_{λ}. Denote by (U_{λ}^{∼}, \underline{M}_{λ}^{∼}) (resp. (U_{λμ}^{∼}, \underline{M}_{λμ}^{∼})) the log scheme underlying f_{λ}^{∗}\underline{U}_{λ} (resp. f_{λ}^{∗}\underline{U}_{λμ}). Also, for any λ, μ, γ ∈ Λ, let π_{λμγ} : U_{λμ} ×_{X} U_{γ} → U_{λμ} be the natural projection.

Lemma 12.6.49. In the situation of (12.6.48), we have :

(i) There exists a unique isomorphism of log schemes g_{λμ} : (U_{λμ}^{∼}, \underline{M}_{λμ}^{∼}) →^{∼} (U_{μλ}^{∼}, \underline{M}_{μλ}^{∼}) fitting into a commutative diagram :

[figure: (U_{λμ}^{∼}, \underline{M}_{λμ}^{∼}); g_{λμ}; (U_{μλ}^{∼}, \underline{M}_{μλ}^{∼}); U_{λμ}; U_{μλ}]

whose vertical arrows are the saturated blow up morphisms. (ii) There exist natural isomorphisms of 𝒪_{U_{λμ}}-modules

ω_{λμ} : g_{λμ}^{∗}𝒪_{U_{μλ}}(1) →^{∼} 𝒪_{U_{λμ}}(1)

such that (π_{μγλ}^{∗}ω_{μλ}) ∘ (π_{λμγ}^{∗}ω_{λμ}) = π_{λγμ}^{∗}ω_{λγ} for every λ, μ, γ ∈ Λ.

Proof. (i): By the universal property of the saturated blow up, it suffices to show that :

(12.6.50)  ℐ_{ρ_{λ}} ũ_{∗}\underline{M}_{|U_{λμ}} = ℐ_{ρ_{μ}} ũ_{∗}\underline{M}_{|U_{μλ}}  on U_{λμ} = U_{μλ}.

The assertion is local on U_{λμ}, hence let x ∈ U_{λμ} be any point; we get an isomorphism of ũ_{∗}\underline{M}_{x}-monoids :

𝒪_{T_{λ},ψ_{λμ}(x)} →^{∼} 𝒪_{T_{μ},ψ_{μλ}(x)}

whence an isomorphism of fans U(ψ_{λμ}(x)) →^{∼} U(ψ_{μλ}(x)) (notation of (6.5.16)). Therefore, the subset U(x) := ψ_{λμ}^{−1}U(ψ_{λμ}(x)) ∩ ψ_{μλ}^{−1}U(ψ_{μλ}(x)) is an open neighborhood of x in U_{λμ}, and both ψ_{λμ} and ψ_{μλ} factor through the same morphism of monoidal spaces :

ψ(x) : (U(x), (ũ_{∗}\underline{M}^{♯})_{|U(x)}) → F (x) := (Spec ũ_{∗}\underline{M}_{x})^{♯}

and open immersions F (x) → T_{λ} and F (x) → T_{μ}. It then follows from (6.6.25) that the preimage of ℐ_{ρ_{λ}} on F (x) agrees with the preimage of ℐ_{ρ_{μ}}, whence the contention.

(ii): By inspecting the definitions, it is easily seen that the epimorphism (10.6.24) identifies naturally 𝒪_{U_{λμ}^{∼}}(1) to the ideal ℐ_{ρ_{λμ}}𝒪_{U_{λμ}^{∼}} of 𝒪_{U_{λμ}^{∼}} generated by the image of ℐ_{ρ_{λ}} ũ_{∗}\underline{M}_{|U_{λμ}}. Hence the assertion follows again from (12.6.50). □

12.6.51. Lemma 12.6.49implies that

((U_{λ}^{∼}, 𝒪_{U_{λ}^{∼}}(1)), (g_{λμ}, ω_{λμ}) | λ, μ ∈ Λ)

is a descent datum – relative to the faithfully flat and quasi-compact morphism ∐_{λ∈Λ} U_{λ} → X – for the fibred category of schemes endowed with an ample invertible sheaf. According to [82, Exp.VIII, Prop.7.8], such a datum is effective, hence it yields a projective morphism π : X^{∼} → X together with an ample invertible sheaf 𝒪_{X^{∼}}(1) on X^{∼}, with isomorphisms g_{λ} : X^{∼} ×_{X} U_{λ} →^{∼} U_{λ}^{∼} of U_{λ}-schemes and π_{λ}^{∗}𝒪_{X^{∼}}(1) →^{∼} g_{λ}^{∗}𝒪_{U_{λ}^{∼}}(1) of invertible modules.

Then the datum (ũ^{∗}\underline{M}_{λ}^{∼}, ũ^{∗} log g_{λμ} | λ, μ ∈ Λ) likewise determines a unique sheaf of monoids \underline{M}^{∼} on X_{ét}^{∼}, and the structure maps of the log structures \underline{M}_{λ} glue to a well defined morphism of sheaves of monoids \underline{M}^{∼} → 𝒪_{X_{ét}^{∼}}, so that (X_{ét}^{∼}, \underline{M}^{∼}) is a log scheme, and the projection π extends to a morphism of log schemes (π, log π) : (X_{ét}^{∼}, \underline{M}^{∼}) → (X_{ét}, \underline{M}).

Proposition 12.6.52. In the situation of (12.6.51), the counit of adjunction

ũ^{∗}ũ_{∗}(X_{ét}^{∼}, \underline{M}^{∼}) → (X_{ét}^{∼}, \underline{M}^{∼})

is an isomorphism.

Proof. (This is the counit of the adjoint pair (ũ^{∗}, ũ_{∗}) of (12.1.6), relating the categories of log structures on X_{Zar}^{∼} and X_{ét}^{∼}.) Recall that there exist natural epimorphisms (ℚ_{+}^{⊕d})_{T (ρ_{λ})} → 𝒪_{T (ρ_{λ}),ℚ} (see (6.6.27)), which induce epimorphisms of U_{λ,Zar}^{∼}-monoids

ϑ_{λ} : (ℚ_{+}^{⊕d})_{U_{λ,Zar}^{∼}} → (\underline{M}_{λ}^{∼})_{ℚ}^{♯} for every λ ∈ Λ.

The compatibility with open immersions expressed by (6.6.28) implies that the system of maps (ũ^{∗}ϑ_{λ} | λ ∈ Λ) glues to a well defined epimorphism of X_{ét}^{∼}-monoids :

ϑ : (ℚ_{+}^{⊕d})_{X_{ét}^{∼}} → (\underline{M}^{∼})_{ℚ}^{♯}.

In view of lemma 4.9.27(ii), it follows that the counit of adjunction :

ũ^{∗}ũ_{∗}(\underline{M}^{∼})_{ℚ}^{♯} → (\underline{M}^{∼})_{ℚ}^{♯}

is an isomorphism. By applying again lemma 4.9.27(ii) to the monomorphism (\underline{M}^{∼})^{♯} → (\underline{M}^{∼})_{ℚ}^{♯}, we deduce that also the counit

ũ^{∗}ũ_{∗}(\underline{M}^{∼})^{♯} → (\underline{M}^{∼})^{♯}

is an isomorphism. Then the assertion follows from proposition 12.2.3(iii). □

Corollary 12.6.53. Let (X_{ét}, \underline{M}) be a quasi-compact regular log scheme. Then there exists a smooth morphism of log schemes (X′_{ét}, \underline{M}′) → (X_{ét}, \underline{M}) whose underlying morphism of schemes is proper and birational, and such that X′ is a regular scheme. More precisely, f restricts to an isomorphism (X′_{ét}, \underline{M}′)_{tr} → (X_{ét}, \underline{M})_{tr} on the trivial loci.

Proof. Given such (X_{ét}, \underline{M}), we construct first the morphism π : (X_{ét}^{∼}, \underline{M}^{∼}) → (X_{ét}, \underline{M}) as in (12.6.51). Since π ×_{X} 1_{U_{λ}} is proper for every λ ∈ Λ, if follows that π is proper ([64, Ch.IV, Prop.2.7.1]). Likewise, notice that each morphism U_{λ}^{∼} → U_{λ} induces an isomorphism (U_{λ}^{∼}, \underline{M}_{λ}^{∼})_{tr} →^{∼} (U_{λ}, ũ_{∗}\underline{M}_{|U_{λ}})_{tr} (remark 12.6.17). It follows easily that π restricts an isomorphism on the trivial loci. Hence, we may replace (X_{ét}, \underline{M}) by (X_{ét}^{∼}, \underline{M}_{ét}^{∼}). Then, by corollary 12.3.27(ii) and proposition 12.6.52we are further reduced to showing that there exists a proper morphism π′ : (X′_{Zar}, \underline{M}′) → ũ_{∗}(X_{ét}^{∼}, \underline{M}^{∼}) with X′ regular, restricting to an isomorphism on the trivial loci. However, in light of lemma 12.5.21(and again, proposition 12.6.52), the sought π′ is provided by the more precise theorem 12.6.32. □

12.7. Local properties of the fibres of a smooth morphism. Resume the situation of example 12.6.5(ii), and to ease notation set φ := (Spec λ)^{♯}, and (f, log f) := Spec(R, λ). Suppose now, that λ : P → Q is an integral, local and injective morphism of fine monoids. Then f : S′ → S is flat and finitely presented (theorem 6.2.3). Moreover :

Lemma 12.7.1. In the situation of (12.7), for every point s ∈ S, the fibre f^{−1}(s) is either empty, or else it is pure-dimensional, of dimension

dim f^{−1}(s) = dim Q − dim P = rk_{ℤ} Coker λ^{gp}.

Proof. To begin with, notice that λ^{−1}Q^{×} = P^{×}, whence the second stated identity, in view of corollary 6.4.12(i). To prove the first stated identity, we easily reduce to the case where R is a field. Notice that the image of f is an open subset U ⊂ S ([64, Ch.IV, Th.2.4.6]), especially U (resp. S′) is pure-dimensional of dimension rk_{ℤ}P^{gp} (resp. rk_{ℤ}Q^{gp}) by claim 11.6.37(ii). Hence, fix any closed point s ∈ S, and set X := f^{−1}(s). From [64, Ch.IV, Cor.6.1.2] we deduce that, for every closed point s′ ∈ X, the Krull dimension of 𝒪_{X,s′} equals r := rk_{ℤ}Coker λ^{gp}. More precisely, say that Z ⊂ X is any irreducible component; we may find a closed point s′ ∈ Z which does not lie on any other irreducible component of X, and then the foregoing implies that the dimension of Z equals r, as stated.

Next, let s ∈ U be any point, and denote K the residue field of 𝒪_{U,s}, and π : R[P] → K the natural map. Let y ∈ Spec K[P] be the K-rational closed point such that a(y) = π(a) for every a ∈ P; then the image of y in S equals s, and if we let f_{K} := Spec K[λ], we have an isomorphism of K-schemes f_{K}^{−1}(y) →^{∼} f^{−1}(s). The foregoing shows that f_{K}^{−1}(y) is puredimensional of dimension r, hence the same holds for f^{−1}(s). □

Now, let us fix s ∈ S, such that f^{−1}(s) ≠ ∅, and suppose that ψ_{P} (s) = 𝔪_{P} is the closed point of T_{P}. For every 𝔮 ∈ φ^{−1}(𝔪_{P}), the closure \overline{\{𝔮\}} of \{𝔮\} in T_{Q} is the image of the natural map Spec Q/𝔮 → T_{Q}. We deduce natural isomorphisms of schemes :

S_{0} := ψ_{P}^{−1}\{𝔪_{𝔭}\} →^{∼} Spec R⟨P/𝔪_{P} ⟩ S′_{𝔮} := ψ_{Q}^{−1}\overline{\{𝔮\}} →^{∼} Spec R⟨Q/𝔮⟩

under which, the restriction f_{𝔮} : S′_{𝔮} → S_{0} of f is identified with Spec R⟨λ_{𝔮}⟩, where λ_{𝔮} : P/𝔪_{P} → Q/𝔮 is induced by λ. The latter is an integral and injective morphism as well (corollary 6.1.51). Explicitly, set F := Q\𝔮, and let λ_{F} : P^{×} → F be the restriction of λ; then λ_{𝔮} = (λ_{F})_{∘}, and lemma 12.7.1yields the identity :

(12.7.2)  dim f_{𝔮}^{−1}(s) = dim Q/𝔮 = dim Q − ht 𝔮.

Also, notice that T_{Q} is a finite set under the current assumptions (lemma 6.1.20(iii)), and clearly

f^{−1}(s) = ⋃_{𝔮∈Max(φ^{−1}𝔪_{P})} f_{𝔮}^{−1}(s)

where, for a topological space T, we denote by Max(T) the set of maximal points of T. Therefore, for every irreducible component Z of f^{−1}(s) there must exist 𝔮 ∈ Max(φ^{−1}𝔪_{P}) such that Z ⊂ f_{𝔮}^{−1}(s), and especially,

(12.7.3)  dim f^{−1}(s) = dim f_{𝔮}^{−1}(s).

Conversely, we claim that (12.7.3) holds for every 𝔮 ∈ Max(φ^{−1}𝔪_{P}). Indeed, suppose that dim f_{𝔮}^{−1}(s) < dim f^{−1}(s) for one such 𝔮, and let Z be an irreducible component of f_{𝔮}^{−1}(s); let also Z′ be an irreducible component of f^{−1}(s) containing Z. By the foregoing, there exists 𝔮′ ∈ Max(φ^{−1}𝔪_{P}) with Z′ ⊂ f_{𝔮′}^{−1}(s). Set 𝔮′′ := 𝔮 ∪ 𝔮′; then clearly 𝔮′′ ∈ φ^{−1}(𝔪_{P}), and

\overline{\{𝔮′′\}} = \overline{\{𝔮\}} ∩ \overline{\{𝔮′\}}.

Especially, Z ⊂ f_{𝔮′′}^{−1}(s); however, it follows from (12.7.2) that dim f_{𝔮′′}^{−1}(s) < dim f_{𝔮}^{−1}(s) (since ht 𝔮 < ht 𝔮′′); but this is absurd, since dim Z = dim f_{𝔮}^{−1}(s) (lemma 12.7.1). The same counting argument also shows that every maximal point of f^{−1}(s) gets mapped necessarily to a maximal point of φ^{−1}(𝔪_{P}); in other words, we have shown that ψ_{Q} restricts to a surjective map :

Max(f^{−1}s) → Max(φ^{−1}𝔪_{P}).

More precisely, let s′ ∈ f^{−1}(s) be a point such that ψ_{Q}(s′) = 𝔪_{Q}, the closed point of T_{Q}. Then clearly s′ ∈ f_{𝔮}^{−1}(s) for every 𝔮 ∈ Max(φ^{−1}𝔪_{P}), and it follows that the foregoing surjection restricts further to a surjective map :

(12.7.4)  f_{s′}^{−1}(s) → Max(φ^{−1}𝔪_{P}).

On the other hand, since log ψ_{Q} is an isomorphism, we have

(Q_{S′}^{log})_{\overline{s}′}^{♯} = 𝒪_{T_{Q},𝔮} = Q_{𝔮}^{♯} for every s′ ∈ ψ_{Q}^{−1}(𝔮)

(where \overline{s}′ denotes any τ-point of S′ localized at s′); explicitly, if F = Q\𝔮, then Q_{𝔮}^{♯} = Q/F; likewise, we get (P_{S}^{log})♯ ; \overline{s} = P/φ^{−1}F. Taking into account (12.7.2), we deduce :

dim f^{−1}(f(s′)) − dim f_{𝔮}^{−1}(f(s′)) = rk_{ℤ} Coker (log f)_{\overline{s}′}^{gp} for every s′ ∈ ψ_{Q}^{−1}(𝔮).

Thus, for every n ∈ ℕ, let

U_{n} := \{s′ ∈ S′ | rk_{ℤ} Coker (log f)_{\overline{s}′}^{gp} = n\}.

The foregoing implies that for every s ∈ S, the subset U_{0} ∩ f^{−1}(s) is open and dense in f^{−1}(s), and U_{n} ∩ f^{−1}(s) is either empty, or else it is a subset of pure codimension n in f^{−1}(s).

12.7.5. In the situation of (12.7), suppose moreover that log f is saturated; notice that this condition holds if and only if log φ is saturated, if and only if the same holds for λ (lemma 6.2.12(iii)). Then, corollary 6.2.32(ii) says that Coker(log f^{♯})_{\overline{s}′}^{gp} is torsion-free for every s′ ∈ S′; especially, Coker(log f^{♯})_{\overline{s}′}^{gp} vanishes for every s′ ∈ U_{0}. Then corollary 6.2.32(i) implies that log f_{\overline{s}′}^{♯} is an isomorphism for every s′ ∈ U_{0}, in which case the same holds for log f\overline{s}′ (lemma 12.1.4); in other words, U_{0} is the strict locus of f (see definition 12.2.7(ii)).

Lemma 12.7.6. Let K be an algebraically closed field of characteristic p, and χ : P → (K, ·) a local morphism of monoids. Let also λ : P → Q be as in (12.7.5). We have :

(i) The K-algebra Q ⊗_{P} K is Cohen-Macaulay.

(ii) If moreover, either p = 0, or else the order of the torsion subgroup of Γ := Coker λ^{gp} is not divisible by p, then Q ⊗_{P} K is reduced (i.e. its nilradical is trivial).

Proof. (i): Since χ is a local morphism, we have χ^{−1}(0) = 𝔪_{P}, and χ is determined by its restriction P^{×} → K^{×}, which is a homomorphism of abelian groups. Notice that K^{×} is divisible, hence it is injective in the category of abelian groups; since the unit of adjunction P → P^{sat} is local (lemma 6.2.9(iii)), it follows that χ extends to a local morphism χ′ : P^{sat} → K. Notice that Q ⊗_{P} K = Q^{sat} ⊗_{P^{sat}} K (lemma 6.2.12(iv)), hence we may replace P by P^{sat} and Q by Q^{sat}, which allows to assume from start that Q is saturated. In this case, by theorem 11.6.35(i), both K[P] and K[Q] are Cohen-Macaulay; since K[λ] is flat, theorem 10.4.38 and [64, Ch.IV, Cor.6.1.2] imply that Q ⊗_{P} K is Cohen-Macaulay as well.

(ii): Let Q = ⨁_{γ∈Γ} Q_{γ} be the λ-grading of Q (see remark 6.2.5(iii)). Under the current assumptions, λ is exact (lemma 6.2.30(ii)), consequently Q_{0} = P (remark 6.2.5(v)). Moreover, Q_{γ} is a finitely generated P -module, for every γ ∈ Γ (corollary 6.4.5), hence either Q_{γ} = ∅, or else Q_{γ} is a free P -module of rank one, say generated by an element u_{γ} (remark 6.2.5(iv)). Furthermore, Q_{γ}^{k} = Q_{kγ} for every integer k > 0 and every γ ∈ Γ (proposition 6.2.31). Thus, whenever Q_{γ} ≠ ∅, the element u_{γ} ⊗1 is a basis of the K-vector space Q_{γ} ⊗_{P} K, and (u_{γ} ⊗1)^{k} is a basis of Q_{kγ} ⊗_{P} K, for every k > 0; especially, u_{γ} ⊗ 1 is not nilpotent. In view of proposition 7.6.28(ii), the contention follows. □

12.7.7. Let f : (X, \underline{M}) → (Y, \underline{N}) be a smooth and log flat morphism of fine log schemes. For every n ∈ ℕ, let U(f, n) ⊂ X be the subset of all x ∈ X such that rk_{ℤ}Coker (log f ♯ ; \overline{x})^{gp} = n, for any τ-point \overline{x} localized at x. By lemma 12.2.21(iii), U(f, n) is a locally closed subset (resp. an open subset) of X, for every n > 0 (resp. for n = 0).

Theorem 12.7.8. In the situation of (12.7.7), we have :

(i) f is a flat morphism of schemes.

(ii) For all y ∈ Y, every connected component of f^{−1}(y), is pure dimensional, and f^{−1}(y)∩ U_{n} is either empty, or else it has pure codimension n in f^{−1}(y), for every n ∈ ℕ.

(iii) Moreover, if f is saturated, we have :

(a) The strict locus Str(f) of f is open in X, and Str(f) ∩ f^{−1}(y) is a dense subset of f^{−1}(y), for every y ∈ Y.

(b) f^{−1}(y) is geometrically reduced and Cohen-Macaulay, for every y ∈ Y.

(c) Coker(log f ♯ ; \overline{x})^{gp} is a free abelian group of finite rank, for every τ-point \overline{x} of X.

Proof. Let ξ be any τ-point of X; according to corollary 12.1.35, there exist a neighborhood V of ξ, and a fine chart P_{V} → \underline{N}_{|V}, such that P^{gp} is a free abelian group of finite rank, and the induced morphism of monoids P → 𝒪_{Y,ξ} is local.

By lemma 12.1.14, [64, Ch.IV, Th.2.4.6, Prop.2.5.1] and [66, Ch.IV, Prop.17.5.7], it suffices to prove the theorem for the morphism f ×_{Y} V : (X, \underline{M}) ×_{Y} V → (V, \underline{N}_{|V}). Hence we may assume from start that \underline{N} admits a fine chart β : P_{Y} → \underline{N}, such that P^{gp} is torsion-free and the induced map P → 𝒪_{Y,ξ} is local.

By theorem 12.3.37, remark 12.1.23(i), [64, Ch.IV, Th.2.4.6] and [66, Ch.IV, Prop.17.5.7] (and again lemma 12.1.14) we may further assume that f admits a fine chart (β, ω_{Q} : Q_{X} → \underline{M}, λ), such that λ is injective, the torsion subgroup of Coker λ^{gp} is a finite group whose order is invertible in 𝒪_{X}, and the induced morphism of schemes X → Y ×_{Spec ℤ[Q]} Spec ℤ[P] is étale. Moreover, by theorem 12.1.37(iii), we may assume – after replacing Q by a localization, and X by a neighborhood of ξ in X_{τ} – that the morphism λ : P → Q is integral (resp. saturated, if f is saturated), and the morphism Q → 𝒪_{X,ξ} induced by ω_{P,ξ} is local, so λ is local as well.

In this case, the same sort of reduction as in the foregoing shows that, in order to prove (i) and (ii), it suffices to consider a morphism f as in (12.7), for which these assertions have already been established. Likewise, in order to show (iii), it suffices to consider a morphism f as in (12.7.5). For such a morphism, assertions (iii.a) and (iii.c) are already known, and (iii.b) is an immediate consequence of lemma 12.7.6. □

Theorem 12.7.8(iii) admits the following partial converse :

Proposition 12.7.9. Let f : (X, \underline{M}) → (Y, \underline{N}) be a smooth and log flat morphism of fs log schemes, such that f^{−1}(y) is geometrically reduced, for every y ∈ Y. Then f is saturated.

Proof. Fix a τ-point ξ of X; the assertion can be checked on stalks, hence we may assume that \underline{N} admits a fine and saturated chart β : P_{Y} → \underline{N} (lemma 12.1.18(iii)), such that the induced morphism α_{P} : P → 𝒪_{Y,f(ξ)} is local (claim 12.1.31). Then, by corollary 12.3.42, we may find an étale morphism g : U → X and a τ-point ξ′ of U with g(ξ′) = ξ, such that the induced morphism of log schemes f_{U} : (U, g^{∗}\underline{M}) → (Y, \underline{N}) admits a fine and saturated chart (β, ω_{Q} : Q_{U} → g^{∗}\underline{M}, λ), where λ is injective, and the induced ring homomorphism

(12.7.10)  Q ⊗_{P} 𝒪_{Y,f(ξ)} → 𝒪_{U,ξ′}

is étale. By [66, Ch.IV, Prop.17.5.7], the fibres of f_{U} are still geometrically reduced, hence we are reduced to the case where U = X and ξ = ξ′. Furthermore, after replacing Q by a localization, and X by a neighborhood of ξ, we may assume that the map α_{Q} : Q → 𝒪_{X,ξ} induced by ω_{Q,ξ} is local and λ is integral (theorem 12.1.37(iii) and lemma 6.2.9(i)). Lastly, let K be the residue field of 𝒪_{Y,f(ξ)}; our assumption on f^{−1}(y) means that the ring A := 𝒪_{X,ξ}⊗_{𝒪_{Y,f(ξ)}} K is reduced.

We shall apply the criterion of proposition 6.2.31. Thus, let Q = ⨁_{γ∈Γ} Q_{γ} be the λ-grading of Q and notice as well that λ is a local morphism (since the same holds for α_{P} and α_{Q}), therefore it is exact (lemma 6.2.30(ii)); consequently Q_{0} = P (remark 6.2.5(v)). Moreover, Q_{γ} is a finitely generated P -module, for every γ ∈ Γ (corollary 6.4.5), hence either Q_{γ} = ∅, or else Q_{γ} is a free P -module of rank one.

We have to prove that Q_{γ}^{k} = Q_{kγ} for every integer k > 0 and every γ ∈ Γ. In case Q_{γ} = ∅, this is the assertion that Q_{kγ} = ∅ as well. However, since Q is saturated, the same holds for Γ′ := (λP)^{−1}Q/(λP)^{gp} (lemma 6.2.9(i,ii)), so it suffices to remark that Γ′ ⊂ Γ is precisely the submonoid consisting of all those γ ∈ Γ such that Q_{γ} ≠ ∅.

Therefore, fix a generator u_{γ} for every P -module Q_{γ} ≠ ∅, and by way of contradiction, suppose that there exist γ ∈ Γ′ and k > 0 such that u_{γ}^{k} does not generate the P -module Q_{kγ}; this means that there exists a ∈ 𝔪_{P} such that u_{γ}^{k} = a · u_{kγ}. Now, notice that the induced morphism of monoids β : P → K is local, especially β(a) = 0, therefore (u_{γ} ⊗ 1)^{k} = 0 in the K-algebra Q ⊗_{P} K. Denote by I ⊂ Q ⊗_{P} K the annihilator of u_{γ} ⊗ 1, and notice that, since (12.7.10) is flat, IA is the annihilator of the image u′ of u_{γ} ⊗ 1 in A.

On the other hand, it is easily seen that I is the graded ideal generated by (u_{μ} ⊗ 1 | μ ∈ Γ′′) where Γ′′ ⊂ Γ′ is the subset of all μ such that u_{γ} · u_{μ} is not a generator of the P -module Q_{γ+μ}. Clearly u_{μ} ∉ Q^{×} for every μ ∈ Γ′′, therefore the image of u_{μ} ⊗ 1 in A lies in the maximal ideal. Therefore IA ≠ A, i.e. u′ is a non-zero nilpotent element, a contradiction. □

12.7.11. Let (X, \underline{M}) be any log scheme, and \overline{x} any geometric point, localized at a point x ∈ X. Suppose that y ∈ X is a generization of x, and \overline{y} a geometric point localized at y, and assume first that the log structure \underline{M} is defined on the étale site X_{ét}; then, arguing as in (4.9.23), we may extend uniquely any strict specialization morphism X(\overline{y}) → X(\overline{x}) to a morphism of log schemes (X(\overline{y}), \underline{M}(\overline{y})) → (X(\overline{x}), \underline{M}(\overline{x})) fitting into a commutative diagram

\xymatrix{(X(\overline{y}), \underline{M}(\overline{y})) \ar[r] \ar[d] & (X(y), \underline{M}(y)) \ar[d] \\ (X(\overline{x}), \underline{M}(\overline{x})) \ar[r] & (X(x), \underline{M}(x))}

whose right vertical arrow is induced by the natural isomorphism

(X(y), \underline{M}(y)) →^{∼} (X(x), \underline{M}(x)) ×_{X(x)} X(y).

A simple inspection shows that the induced morphism

Γ(X(\overline{x}), \underline{M}(\overline{x}))^{♯} → Γ(X(\overline{y}), \underline{M}(\overline{y}))^{♯}

is naturally identified with the morphism \underline{M} ♯ ; \overline{x} → \underline{M} ♯ ; \overline{y} obtained from the specialization map \underline{M}\overline{x} → \underline{M}\overline{y}. In case \underline{M} is defined on the Zariski site X_{Zar}, we may apply the foregoing construction to the log scheme ũ^{∗}(X, \underline{M}), to obtain again the strictly local log schemes (X(\overline{x}), \underline{M}(\overline{x})) and (X(\overline{y}), \underline{M}(\overline{y})) a corresponding commutative diagram, as in the foregoing case.

12.7.12. Let g : (X, \underline{M}) → (Y, \underline{N}) be a morphism of log schemes, \overline{x} any geometric point of X, and set \overline{y} := g(\overline{x}). The log structures \underline{M} → 𝒪_{X} and \underline{N} → 𝒪_{Y}, and the map log g\overline{x} induce a commutative diagram of continuous maps

(12.7.13)  \xymatrix{X(\overline{x}) \ar[r]^{g\overline{x}} \ar[d]_{ψ\overline{x}} & Y (\overline{y}) \ar[d]^{ψ\overline{y}} \\ _{Spec \underline{M}\overline{x}} \ar[r]^{φ\overline{x}} & _{Spec \underline{N}\overline{y}}}

(notation of 4.9.20), and notice that ψ\overline{x} (resp. ψ\overline{y}) maps the closed point of X(\overline{x}) (resp. of Y (\overline{y})) to the closed point t\overline{x} ∈ Spec \underline{M}\overline{x} (resp. t\overline{y} ∈ Spec \underline{N}\overline{y}).

Proposition 12.7.14. In the situation of (12.7.12), suppose that g is a smooth morphism of fine log schemes, and moreover :

(a) either g is a saturated morphism

(b) or (X, \underline{M}) is a fs log scheme. Then the following holds :

(i) The map ψ\overline{x} restricts to a bijection :

Max(g−1 ; \overline{x}(\overline{y})) →^{∼} Max(φ−1 ; \overline{x}(t\overline{y})).

(ii) For every irreducible component Z of g−1 ; \overline{x}(\overline{y}), set

(Z, \underline{M}(Z)) := (X(\overline{x}), \underline{M}(\overline{x})) ×_{X(\overline{x})} Z.

Then the κ(\overline{y})-log scheme (Z, \underline{M}(Z)_{red}) is geometrically pointed regular. (Notation and terminology of example 12.1.12(iv) and remark 12.5.8(ii).)

Proof. By corollary 12.1.35, there exist a neighborhood U → Y of \overline{y}, and a fine chart β : P_{U} → \underline{N}_{|U} such that β\overline{y} is a local morphism, and P^{gp} is torsion-free. Now, let

g\overline{x} := (g\overline{x}, log g\overline{x}) : (X(\overline{x}), \underline{M}(\overline{x})) → (Y (\overline{y}), \underline{N}(\overline{y}))

be the morphism of log schemes induced by g, as in 12.7.11; by theorem 12.3.37, we may find a fine chart for g\overline{x} of the type (i∗ ; \overline{y}β, ω, λ : P → Q), where λ is injective, and the order of the torsion subgroup of Coker λ^{gp} is invertible in 𝒪_{X,x}. Moreover, set R := 𝒪_{Y (\overline{y}),\overline{y}}; then the induced map X(\overline{x}) → Spec Q ⊗_{P} R is pro-étale, and – after replacing Q by some localization – we may assume that ω\overline{x} : Q → 𝒪_{X(\overline{x}),\overline{x}} is local (claim 12.1.31), hence the same holds for λ. Furthermore, under assumption (a) (resp. (b)), we may also suppose that Q is saturated, by lemmata 6.2.9(ii) and 12.1.18(ii) (resp. that λ is saturated, by theorem 12.1.37(iii)).

Let us now define f : S′ → S as in (12.7); notice that ω\overline{x} induces a closed immersion Y (\overline{y}) → S, and we have a natural identification of Y (\overline{y})-schemes :

Spec Q ⊗_{P} R = Y (\overline{y}) ×_{S} S′.

Denote by \overline{s} the image of \overline{y} in S, and \overline{s}′ the image of \overline{x} in Y (\overline{y}) ×_{S} S′ ⊂ S′; there follows an isomorphism of Y (\overline{y})-schemes :

X(\overline{x}) →^{∼} Y (\overline{y}) ×_{S(\overline{s})} S′(\overline{s}′)

([66, Ch.IV, Prop.18.8.10]). Moreover, our chart induces isomorphisms :

Spec \underline{M}\overline{x} →^{∼} T_{Q} Spec \underline{N}\overline{y} →^{∼} T_{P}

which identify φ\overline{x} to the map φ : T_{Q} → T_{P} of (12.7). In view of these identifications, we see that (12.7.13) is the restriction to the closed subset X(\overline{x}), of the analogous diagram :

\xymatrix{S′(\overline{s}′) \ar[r]^{f_{\overline{s}′}} \ar[d]_{ψ_{\overline{s}′}} & S(\overline{s}) \ar[d]^{ψ\overline{s}} \\ T_{Q} \ar[r]^{φ} & T_{P} .}

We may thus assume that X = S′, Y = S and g = f. Moreover, let s (resp. s′) be the support of \overline{s} (resp. of \overline{s}′); the morphism π : S′(\overline{s}′) → S′(s′) is flat, hence it restricts to a surjection

Max(f_{\overline{s}′}^{−1}\overline{s}) → Max(f_{s′}^{−1}s).

In order to prove (i), it suffices then to show that the map Max(f_{\overline{s}′}^{−1}\overline{s}) → Max(φ^{−1}𝔪_{P}), defined as the composition of the foregoing map and the surjection (12.7.4), is injective. This boils down to the assertion that, for every 𝔮 ∈ Max(φ^{−1}𝔪_{P}) the point s′ lies on a unique irreducible component of the fibre π^{−1}(f_{𝔮,s′}^{−1}s). However, let \overline{β} : P → κ(s) be the composition of the chart P → R and the projection R → κ(s); then f_{𝔮}^{−1}(s) = Spec Q/𝔮 ⊗_{P} κ(s). Since π is pro-étale, the assertion will follow from [66, Ch.IV, Prop.17.5.7] and corollary 12.5.13, together with :

Claim 12.7.15. The log scheme W_{𝔮} := Spec⟨ℤ, Q/𝔮⟩ ×_{Spec ℤ[P]} Spec κ(s) is geometrically pointed regular.

Proof of the claim. To ease notation, set F := Q \ 𝔮; by assumption λ^{−1}F = P^{×}, so that

W_{𝔮} = (W′_{𝔮})_{∘} where W′_{𝔮} := Spec(ℤ, F) ×_{Spec(ℤ,P^{×})} Spec κ(s)

(notation of (12.2.13); notice that the log structures of Spec(ℤ, P^{×}) and Spec κ(s) are trivial). Moreover, let λ_{F} : P^{×} → F be the restriction of λ; then λ_{F}^{gp} is injective, and

Coker λ_{F}^{gp} ⊂ Coker λ^{gp}

(corollary 6.2.33(i)), hence the order of the torsion subgroup of Cokerλ_{F}^{gp} is invertible in 𝒪_{S,s}. Moreover, if λ is saturated, then the same holds for λ_{F} (corollary 6.2.33(ii)), and it is easily seen that if Q is saturated, the same holds for F. Consequently, the morphism Spec(ℤ, λ_{F}) is smooth (proposition 12.3.34). The same then holds for the morphism W′_{𝔮} → Spec κ(s) obtained after base change of Spec(ℤ, λ_{F}) along the morphism h : Spec(ℤ, P^{×}) → Spec κ(s) induced by \overline{β} (proposition 12.3.24(ii)).

Now we notice that under either of the assumptions (a) or (b), W′_{𝔮} is a fs log scheme. Indeed, under assumption (b), this follows by remarking that Spec κ(s) is trivially a fs log scheme, and Spec(ℤ, λ_{F}) is saturated. Under assumption (a), Spec(ℤ, F) is a fs log scheme, and it suffices to observe that h is a strict morphism. Lastly, since the morphism W′_{𝔮} → Spec κ(s) is obviously saturated, we apply corollary 12.5.29to conclude. ♢

(ii): In light of the foregoing, we see that, for any irreducible component Z of g−1 ; \overline{x}(\overline{y}), there exists a unique 𝔮(Z) ∈ Max(φ^{−1}𝔪_{P}) such that Z is isomorphic to the strict henselization of f_{𝔮(Z)}^{−1}(s), at the geometric point \overline{s}′. Notice now that the log structure of W_{𝔮(Z)} is reduced, by virtue of claim 12.7.15 and proposition 12.5.38; then, a simple inspection of the definitions shows that (Z, \underline{M}(Z)_{red}) is isomorphic to the strict henselization W_{𝔮(Z)}(\overline{s}′). Invoking again claim 12.7.15, we deduce the contention. □

12.7.16. In the situation of (12.7.12), suppose that Y is a normal scheme, and (Y, \underline{N})_{tr} is a dense subset of Y. Let \overline{η} be a geometric point of Y (\overline{y}), localized at the generic point η, and

(U_{𝔮} | 𝔮 ∈ Spec \underline{M}\overline{x})

the logarithmic stratification of (X(\overline{x}), \underline{M}(\overline{x})) (see (12.5.35)). Notice that ψ\overline{x}(g^{−1}(η)) lies in the preimage Σ ⊂ Spec \underline{M}\overline{x} of the maximal point ∅ of Spec \underline{N}\overline{y}; therefore, g−1 ; \overline{x}(η) is the union of the subsets U_{𝔮} ×_{Y} |η|, for all 𝔮 ∈ Σ.

Proposition 12.7.17. In the situation of (12.7.16), suppose that g is a smooth morphism of fine log schemes. Then the following holds :

(i) X is a normal scheme.

(ii) The scheme g−1 ; \overline{x}(\overline{η}) is normal and irreducible.

(iii) For every 𝔮 ∈ Σ, the κ(η)-scheme U_{𝔮} ×_{Y} |η| is non-empty, geometrically normal and

geometrically irreducible, of pure dimension dim g−1 ; \overline{x}(η) − ht 𝔮. (iv) Especially, set W := (X(\overline{x})\U_{∅}) ×_{Y} |\overline{η}|; then ψ\overline{x} induces a bijection :

Max(W) →^{∼} {𝔮 ∈ Σ | ht𝔮 = 1}.

(v) For every w ∈ Max(W), the stalk 𝒪_{g−1 ; \overline{x}(\overline{η}),w} is a discrete valuation ring.

Proof. Set R := 𝒪_{Y (\overline{y}),\overline{y}}; arguing as in the proof of proposition 12.7.14, we may find :

• a local, flat and saturated morphism λ : P → Q of fine monoids, such that the order of the torsion subgroup of Coker λ^{gp} is invertible in R;

• local morphisms of monoids P → R, Q → 𝒪_{X(\overline{x}),\overline{x}} which are charts for the log structures deduced from \underline{N} and respectively \underline{M}, and such that the induced morphism of Y (\overline{y})-schemes X(\overline{x}) → Spec Q ⊗_{P} R is pro-étale.

By [66, Ch.IV, Prop.17.5.7], we may then assume that (X, \underline{M}) (resp. (Y, \underline{N})) is the scheme Spec Q ⊗_{P} R (resp. Spec R), endowed with the log structure deduced from the natural map Q → Q ⊗_{P} R (resp. the chart P → R), and g is the natural projection. Suppose first that R is excellent, and let R′ be the normalization of R in a finite extension K′ of Frac(R); then R′ is also strictly local and noetherian, and if y′ denotes the closed point of Y′ := Spec R′, then the residue field extension κ(y) ⊂ κ(y′) is purely inseparable. Set

(X′, \underline{M}′) := (X, \underline{M}) ×_{Y} Y′ (Y′, \underline{N}′) := (Y, \underline{N}) ×_{Y} Y′

and let g′ : (X′, \underline{M}′) → (Y′, \underline{N}′) be the induced morphism of log schemes; it follows especially that the restriction g^{′−1}(y′) → g^{−1}(y) is a homeomorphism on the underlying topological spaces. Hence, there is a geometric point \overline{x}′ of X′, unique up to isomorphism, whose image in X agrees with \overline{x}, and we easily deduce an isomorphism of Y -schemes ([66, Ch.IV, Prop.18.8.10])

(12.7.18)  X′(\overline{x}′) →^{∼} X(\overline{x}) ×_{Y} Y′.

Let η′ be the generic point of Y′; by assumption, \underline{N}′ is trivial in a Zariski neighborhood of η′, hence (X′, \underline{N}′) ×_{Y′} |η′| is a fs log schemes (since g is saturated), and then the same log scheme is also regular (theorem 12.5.28), therefore g^{′−1}(η′) is a normal scheme (corollary 12.5.13). On the other hand, R′ is a Krull domain ([133, Th.33.10]), and g′ is flat with reduced fibres (theorem 12.7.8(iii.b)), so X′ is a noetherian normal scheme (lemma 9.8.2), consequently the same holds for X′(\overline{x}′) ([66, Ch.IV, Prop.18.8.12(i)]); especially, the latter is irreducible, so the same holds for X′(\overline{x}′) ×_{Y′} |η′|. In view of (12.7.18), it follows that X(\overline{x}) ×_{Y} |η′| is also normal and irreducible. Since K′ is arbitrary, this completes the proof of (i) and (ii), in this case.

Next, if R is any normal ring, we may write R as the union of a filtered family (R_{i} | i ∈ I) of excellent normal local subrings ([64, Ch.IV, (7.8.3)(ii,vi)]). For each i ∈ I, denote by \overline{y}_{i} the image of \overline{y} in Spec R_{i}; then the strict henselization R_{i}^{sh} of R_{i} at \overline{y}_{i} is also a subring of R, so we may replace R_{i} by R_{i}^{sh}, which allows to assume that each R_{i} is strictly local, normal and noetherian ( [66, Ch.IV, Prop.18.8.8(iv), Prop.18.8.12(i)]). Up to replacing I by a cofinal subset, we may assume that the image of P lies in R_{i}, for every i ∈ I. For each i ∈ I, set X_{i} := Spec Q ⊗_{P} R_{i}, Y_{i} := Spec R_{i}, and endow X_{i} (resp. Y_{i}) with the log structure \underline{M}_{i} (resp. \underline{N}_{i}) deduced from the natural map Q → Q ⊗_{P} R_{i} (resp. P → R_{i}). There follows a system of morphisms of log schemes g_{i} : (X_{i}, \underline{M}_{i}) → (Y_{i}, \underline{N}_{i}) for every i ∈ I, whose limit is the morphism g. Moreover, since (Y, \underline{N})_{tr} is dense in Y, the image of P lies in R\\{0\}, hence it lies in R_{i} \\{0\} for every i ∈ I, and the latter means that (Y_{i}, \underline{N}_{i})_{tr} is dense in Y_{i}, for every i ∈ I. Let \overline{η}_{i} (resp. \overline{x}_{i}) be the image of \overline{η} (resp. of \overline{x}) in Y_{i} (resp. in X_{i}); moreover, for each i ∈ I, let x_{i} ∈ X_{i} be the support of \overline{x}_{i}. By the previous case, we know that g_{i,\overline{x}_{i}}^{−1}(\overline{η}_{i}) is normal and irreducible. However, X (resp. g−1 ; \overline{x}(\overline{η})) is the limit of the system of schemes (X_{i} | i ∈ I) (resp.

(g_{i,\overline{x}_{i}}^{−1}(\overline{η}_{i}) | i ∈ I)), so (i) and (ii) follow. (Notice that the colimit of a filtered system of integral (resp. normal) domains, is an integral (resp. normal) domain : exercise for the reader.)

(iii): For every 𝔮 ∈ Spec Q = Spec \underline{M}\overline{x}, set X_{𝔮} := Spec Q/𝔮 ⊗_{P} R; since the chart Q → 𝒪_{X(\overline{x}),\overline{x}} is local, it is clear that x ∈ X_{𝔮} for every such 𝔮, and :

X_{𝔮}(\overline{x}) = ⋃_{𝔭∈Spec Q/𝔮} U_{𝔭}.

If φ(𝔮) = ∅, the induced map P → Q/𝔮 is still flat (corollary 6.1.51), hence the projection X_{𝔮}(\overline{x}) → Y is a flat morphism of schemes, especially g−1 ; \overline{x}(η) ∩ X_{𝔮}(\overline{x}) ≠ ∅. Furthermore, the subset X_{𝔮}(\overline{x}) is pure-dimensional, of codimension ht 𝔮 in g−1 ; \overline{x}(η), by (12.7.2) and [64, Ch.IV, Cor.6.1.4]. It follows that U_{𝔮} is a dense open subset of X_{𝔮}(\overline{x}). To conclude, it remains only to show that each X_{𝔮}(\overline{x}) is geometrically normal and geometrically irreducible; however, let j_{𝔮} : X_{𝔮} → X be the closed immersion; the induced morphism of log schemes g_{𝔮} : (X_{𝔮}, j_{𝔮}^{∗}\underline{M}) → (Y, \underline{N}) is also smooth, hence the assertion follows from (ii).

(iv) is a straightforward consequence of (iii).

(v): Notice that A := 𝒪_{g−1 ; \overline{x}(\overline{η}),w} is ind-étale over the noetherian ring Q ⊗_{P} κ(\overline{η}), hence its strict henselization is noetherian, and then A itself is noetherian ([66, Ch.IV, Prop.18.8.8(iv)]). Since X is normal, and w is a point of height one in g−1 ; \overline{x}(\overline{η}), we conclude that A is a discrete valuation ring. □

13. Étale coverings of schemes and log schemes

13.1. Acyclic morphisms of schemes. For any scheme X, we shall denote by :

Cov(X)

the category whose objects are the finite étale morphisms E → X; the morphisms (E → X) → (E′ → X) are the X-morphisms of schemes E → E′. By faithfully flat descent, Cov(X) is naturally equivalent to the subcategory of X_{ét}^{∼} consisting of all locally constant constructible sheaves. If f : X → Y is any morphism of schemes, and φ : E → Y is an object Cov(Y), then f^{∗}φ := φ ×_{Y} X : E ×_{Y} X → X is an object of Cov(X); more precisely, we have a fibration :

(13.1.1)  Cov → Sch

over the category of schemes, whose fibre, over any scheme X, is the category Cov(X).

Lemma 13.1.2. Let f be a morphism of schemes, and suppose that either one of the following conditions holds :

(a) f is integral and surjective.

(b) f is faithfully flat.

(c) f is proper and surjective. Then f is of universal 2-descent for the fibred category (13.1.1).

Proof. This is [9, Exp.VIII, Th.9.4]. □

Lemma 13.1.3. In the situation of definition 10.5.42, let U ⊂ X be any open subset with Y ⊂ U. If Lef(X, Y) holds, the closed immersion j : Y → U induces a fully faithful functor

j^{∗} : Cov(U) → Cov(Y).

Proof. Let Cov(𝔛) be the full subcategory of 𝒪_{𝔛}-Alg_{lfft} consisting of all finite étale 𝒪_{𝔛}-algebras (notation of lemma 10.5.43(ii)); the category Cov(U) is a full subcategory of 𝒪_{U}-Alg_{lfft}, so lemma 10.5.43(ii) already implies that the functor Cov(U) → Cov(𝔛) is fully faithful, hence we are reduced to showing that the functor :

Cov(𝔛) → Cov(Y) 𝒜 ↦ Spec 𝒜 ⊗_{𝒪_{𝔛}} 𝒪_{Y}

is fully faithful. To this aim, let ℐ ⊂ 𝒪_{X} be the ideal defining the closed immersion Y ⊂ X; consider the direct system of schemes

(Y_{n} := Spec 𝒪_{X}/ℐ^{n+1} | n ∈ ℕ)

and let Cov(Y_{•}) be the category consisting of all direct systems (E_{n} | n ∈ ℕ) of schemes, such that E_{n} is finite étale over Y_{n}, and such that the transition maps E_{n} → E_{n+1} induce isomorphisms E_{n} →^{∼} E_{n+1} ×_{Y_{n+1}} Y_{n} for every n ∈ ℕ. The morphisms in Cov(Y_{•}) are the morphisms of direct systems of schemes. We have a natural fully faithful functor :

Cov(𝔛) → Cov(Y_{•}) 𝒜 ↦ (Spec 𝒜 /ℐ^{n+1}𝒜 | n ∈ ℕ)

([59, Ch.I, Cor.10.6.10(ii)]). Finally, the functor Cov(Y_{•}) → Cov(Y) given by the rule : (E_{n} | n ∈ ℕ) → E_{0} is an equivalence, by [66, Ch.IV, Th.18.1.2]. The claim follows. □

13.1.4. Consider a cofiltered system X_{•} := (X_{λ} | λ ∈ Λ) of quasi-compact and quasiseparated schemes, with affine transition morphisms; let X be the limit of X_{•}, and for every λ ∈ Λ, let p_{λ} : X → X_{λ} be the natural morphism. The functors p_{λ}^{∗} : Cov(X_{λ}) → Cov(X) define a pseudo-cocone in the 2-category Cat, whence a functor :

(13.1.5)  2-colim_{λ∈Λ} Cov(X_{λ}) → Cov(X).

Lemma 13.1.6. In the situation of (13.1.4), the functor (13.1.5) is an equivalence.

Proof. It is a rephrasing of [65, Ch.IV, Th.8.8.2, Th.8.10.5] and [66, Ch.IV, Prop.17.7.8(ii)]. □

Lemma 13.1.7. Let X be a scheme, j : U → X an open immersion with dense image, and f : X′ → X an integral surjective and radicial morphism. The following holds :

(i) The morphism f induces an equivalence of sites

f^{∗} : X′_{ét} → X_{ét}

and the functor f^{∗} : Cov(X) → Cov(X′) is an equivalence. (ii) The functor j^{∗} : Cov(X) → Cov(U) is faithful. (iii) Suppose that X is reduced and normal, and moreover :

(a) either X has finitely many maximal points

(b) or else, j is a quasi-compact open immersion.

Then j^{∗} is fully faithful, and its essential image consists of all the objects φ of Cov(U) such that φ ×_{X} X(\overline{x}) lies in the essential image of the pull-back functor :

Cov(X(\overline{x})) → Cov(X(\overline{x}) ×_{X} U)

for every geometric point \overline{x} of X. (Notation of definition 4.9.17(ii).) (iv) Furthermore, if X is locally noetherian and regular, and X \U has codimension ≥ 2

in X, then j^{∗} is an equivalence.

Proof. (i) follows from [9, Exp.VIII, Th.1.1].

(ii): Indeed, let φ : E → X and φ′ : E′ → X be any two finite étale morphisms, and f, g : E → E′ two morphisms of X-schemes such that f ×_{X} U = g ×_{X} U; let Δ_{E′} → E′ ×_{X} E′ be the open and closed diagonal immersion, (f, g) : E → E′ ×_{X} E′ the morphism deduced from f and g, and set D := (f, g)^{−1}Δ_{E′}. Then D is the largest open subset of E′ such that f_{|D} = g_{|D}; on the other hand, by assumption φ^{−1}U ⊂ D, and since φ is an open map, φ^{−1}U is dense in E. Lastly, D is also a closed subset of E, so D = E, whence the claim.

(iii): Choose a covering X = ⋃_{i∈I} V_{i} consisting of affine open subsets, let

X′ := ∐_{i∈I} V_{i} X′′ := X′ ×_{X} X′

and denote by g : X′ → X the induced morphism; set also j′_{i} := j ×_{X} V_{i} for every i ∈ I. By lemma 13.1.2, g is of universal 2-descent for the fibred category Cov. On the other hand, the induced open immersion j′′ : U ×_{X} X′′ → X′′ has dense image, hence the corresponding functor j′′^{∗} is faithful, by (ii). By corollary 3.5.30(ii), the full faithfulness of j^{∗} follows from the full faithfulness of the pull-back functor j′^{∗} corresponding to the open immersion j′ := j×_{X} X′. The latter holds if and only if the same holds for all the pull-back functors j_{i}^{′∗}. Hence, we may replace X by V_{i}, and assume from start that X is affine, say X = Spec A. Let E → X and E′ → X be two objects of Cov(X), and h : E ×_{X} U → E′ ×_{X} U a morphism, and write E = Spec B, E′ = Spec B′ for finite étale A-algebras B and B′; we have to check that h extends to a morphism E → E′.

• To this aim, consider first the case where X has finitely many maximal points η_{1}, …, η_{s}. Under the current assumptions, A is the product of s domains, and its total ring of fractions Frac A is the product of fields κ(η_{1}) × ⋯ × κ(η_{s}). The restrictions h_{η_{i}} := h ×_{U} X(η_{i}) : E(η_{i}) → E′(η_{i}) induce a map of Frac A-algebras

h_{η}^{♮} := ∏_{i=1}^{s} h_{η_{i}}^{♮} : B′ ⊗_{A} Frac A → B ⊗_{A} Frac A.

On the other hand, by proposition 9.8.3, B (resp. B′) is the normalization of A in B ⊗_{A} Frac A (resp. in B′ ⊗_{A} Frac A). It follows that h_{η}^{♮} restricts to a map B′ → B, and the corresponding morphism E → E′ is necessarily an extension of h. This shows that j^{∗} is fully faithful in this case.

• Next, suppose that assumption (b) holds, and set ℰ := B^{∼}, ℰ′ := B′^{∼}; then ℰ and ℰ′ are étale 𝒪_{X}-algebras and locally free 𝒪_{X}-modules of finite type, and h corresponds to a morphism h^{♯} : ℰ → ℰ′. Under the current assumptions, j_{∗}𝒪_{U} is a quasi-coherent 𝒪_{X}-algebra, and 𝒪_{X,x} is integrally closed in (j_{∗}𝒪_{U})_{x}, for every x ∈ X ([75, Prop.8.2.31(i)]). Likewise, j_{∗}j^{∗}ℰ = ℰ ⊗_{𝒪_{X}} j_{∗}𝒪_{U}, so ℰ′_{x} is the integral closure of 𝒪_{X,x} in (j_{∗}j^{∗}ℰ)_{x}, for every x ∈ X. Lastly, ℰ_{x} is integral over 𝒪_{X,x}, so the map (j_{∗}j^{∗}h^{♯})_{x} : (j_{∗}j^{∗}ℰ)_{x} → (j_{∗}j^{∗}ℰ′)_{x} restricts to a map ℰ_{x} → ℰ′_{x}, and the assertion follows.

To proceed, we make the following general remark.

Claim 13.1.8. Let Z be a scheme, V_{0} ⊂ Z an open subset, and φ : E → V_{0} an object of Cov(V_{0}). Suppose that, for every open subset V ⊂ Z containing V_{0}, the pull-back functor j_{V}^{∗} : Cov(V) → Cov(V_{0}) is fully faithful. Let ℱ be the family consisting of all the data (V, ψ, α) where V ⊂ X is any open subset with V_{0} ⊂ V, ψ : E_{V} → V is a finite étale morphism, and α : ψ^{−1}V_{0} →^{∼} E is an isomorphism of V_{0}-schemes. ℱ is preordered (see example 1.1.6(iii)) by the relation such that (V, ψ, α) ≥ (V′, ψ′, α′) if and only if V′ ⊂ V and there is an isomorphism β : ψ^{−1}V′ →^{∼} E_{V′} of V′-schemes, such that α′ ∘j_{V′}^{∗}(β) = α. Then the partially ordered quotient ℱ′ of ℱ admits a supremum (see example 1.1.16). (Notice that ℱ is a proper class, but it can be replaced by a subset that meets every isomorphism class.)

Proof of the claim. Using Zorn’s lemma, it is easily seen that every element of ℱ′ can be dominated by a maximal element, and it remains to show that any two maximal elements (V, ψ, α) and (V′, α′, ψ) of ℱ are isomorphic; to see this, set V′′ := V ∩ V′ : by assumption, the isomorphism α′^{−1} ∘ α : ψ^{−1}V_{0} →^{∼} ψ′^{−1}V_{0} extends to an isomorphism of V′′-schemes ψ^{−1}V′′ →^{∼} ψ′^{−1}V′′, using which one can glue E_{V} and E_{V′} to obtain a datum (V ∪ V′, ψ′′, α′′) which is larger than both our maximal elements, hence it is isomorphic to both. ♢

Claim 13.1.9. In the situation of claim 13.1.8, suppose that Z is reduced and normal, V_{0} is dense in Z, and either one of the assumptions (iii.a) or (iii.b) hold for Z and the open immersion V_{0} → Z. Let (V_{max}, ψ, α) be a supremum for ℱ, and \overline{z} a geometric point of Z, such that φ ×_{Z} Z(\overline{z}) extends to a finite étale covering of Z(\overline{z}); then the support of \overline{z} lies in V_{max}.

Proof of the claim. By lemma 13.1.6, there exist an étale neighborhood g : Y → Z of \overline{z}, with Y affine, a finite étale morphism φ_{Y} : E_{Y} → Y, and an isomorphism h : φ ×_{V_{0}} Y ≃ φ_{Y} ×_{Z} V_{0}. We have a natural essentially commutative diagram :

\xymatrix{Cov(gY) \ar[r]^{α} \ar[d] & Desc(Cov, g) \ar[r] \ar[d] & Cov(Y) \ar[d] \\ Cov(V_{0} ∩ gY) \ar[r]^{β} & Desc(Cov, g ×_{Z} V_{0}) \ar[r] & Cov(Y ×_{Z} V_{0})}

where α and β are equivalences, by lemma 13.1.2. Moreover, let Y′ := Y ×_{Z} Y and Y′′ := Y′ ×_{Z} Y; it is easily seen that if assumption (iii.a) (resp. (iii.b)) holds for Z and V_{0}, the same holds for Y′ and Y′ ×_{Z} V_{0}, and also for Y′′ and Y′′ ×_{Z} V_{0}. By the foregoing, it follows that the functors

Cov(Y′) → Cov(Y′ ×_{Z} V_{0}) and Cov(Y′′) → Cov(Y′′ ×_{Z} V_{0})

are fully faithful, hence the right square subdiagram is 2-cartesian (corollary 3.5.30(iii)). Thus, the datum (φ ×_{Z} gY, φ_{Y} , h) determines an object φ′ of Cov(gY), together with an isomorphism φ′ ×_{Z} V_{0} ≃ φ×_{Z} gY, which we may use to glue φ and φ′ to a single object φ′′ of Cov(V_{0} ∪gY). The claim follows. ♢

The foregoing shows that the assumptions of claim 13.1.8are fulfilled, with Z := X, V_{0} := U and any object φ of Cov(U), hence there exists a largest open subset U_{max} ⊂ X over which φ extends. However, claim 13.1.9shows that U_{max} = X, so the proof of (iii) is complete.

(iv): In view of (iii) and [66, Ch.IV, Cor.18.8.13], we are reduced to the case where X is a regular local scheme, and it suffices to show that j^{∗} is essentially surjective. We argue by induction on the dimension n of X \U. If n = 0, then X \U is the closed point, in which case it suffices to invoke the Zariski-Nagata purity theorem ([85, Exp.X, Th.3.4(i)]). Suppose n > 0 and that the assertion is already known for smaller dimensions. Let φ be a given finite étale covering of U, and x a maximal point of X \U; then X(x)\U = {x}, so φ_{|U∩X(x)} extends to a finite étale morphism φ_{x} over X(x). In turns, φ_{x} extends to an affine open neighborhood V ⊂ X of X, and up to shrinking V, this extension φ′ agrees with φ on U ∩ V, by lemma 13.1.6. Hence we can glue φ and φ′, and replace U by U ∪ V. Repeating the procedure for every maximal point of X \U, we reduce the dimension of X \U; then we conclude by the inductive assumption. □

Definition 13.1.10. Let f : X → S be a morphism of schemes, \overline{x} a geometric point of X localized at x ∈ X; set s := f(x), \overline{s} := f(\overline{x}), and denote by f\overline{x} : X(\overline{x}) → S(\overline{s}) the morphism of strictly local schemes induced by f. Let also 𝕃 ⊂ ℕ be any non-empty set of prime numbers.

(i) Let also P be any property of schemes (such as : connected, irreducible, integral, normal, …); we shall say that the Milnor fibres of f at the point x have property P, if the scheme f −1 ; \overline{x}(ξ) has property P, for every strict geometric point ξ of S(\overline{s}) (see definition 4.9.17(i)).

(ii) We say that f is locally (−1)-acyclic at the point x, if the Milnor fibres of f at the point

x are non-empty. (iii) We say that f is locally 0-acyclic at the point x, if the Milnor fibres of f at the point x

are connected (and in particular, non-empty). (iv) We say that a group G is a finite 𝕃-group if G is finite and all the primes dividing

the order of G lie in 𝕃. We say that G is an 𝕃-group if it is a filtered union of finite 𝕃-groups.

(v) We say that f is locally 1-aspherical for 𝕃 at the point x, if we have :

H^{1}(f −1 ; \overline{x}(ξ)_{ét}, G) = \{1\}

for every strict geometric point ξ of S(\overline{s}), and every 𝕃-group G (where 1 denotes the trivial G-torsor).

(vi) We say that f is locally (−1)-acyclic (resp. locally 0-acyclic, resp. locally 1-aspherical for 𝕃), if f is locally (−1)-acyclic (resp. locally 0-acyclic, resp. locally 1-aspherical for 𝕃) at every point of X.

(vii) We say that f is (−1)-acyclic (resp. 0-acyclic) if the unit of adjunction : ℱ → f_{∗}f^{∗}ℱ is a monomorphism (resp. an isomorphism) for every sheaf ℱ on S_{ét}.

See section 4.1 for generalities about G-torsors for a group object G on a topos T. Here we shall be mainly concerned with the case where T is the étale topos X_{ét}^{∼} of a scheme X, and G is representable by a group scheme, finite and étale over X. In this case, using faithfully flat descent one can show that any G-torsor is representable by a principal G-homogeneous space, i.e. a finite, surjective, étale morphism E → X with a G-action G → Aut_{X}(E) such that the induced morphism of X-schemes

G × E → E ×_{X} E

is an isomorphism.

13.1.11. If G_{X} is the constant X_{ét}^{∼}-group arising from a finite group G and X is non-empty and connected, right G_{X}-torsors are also understood as G-valued characters of theétale fundamental group of X. Indeed, let ξ be a geometric point of X; recall ([82, Exp.V, §7]) that π := π_{1}(X_{ét}, ξ) is defined as the automorphism group of the fibre functor

F_{ξ} : Cov(X) → Set (E ⟶^{f} X) ↦ f^{−1}ξ.

We endow π with its natural profinite topology, as in (3.6.7), so that F_{ξ} can be viewed as an equivalence of categories

F_{ξ} : Cov(X) → π-Set.

Lemma 13.1.12. In the situation of (13.1.11), there exists a natural bijection of pointed sets :

H^{1}(X_{ét}, G_{X}) →^{∼} H_{cont}^{1}(π, G)

from the pointed set of right G_{X}-torsors, to the first non-abelian continuous cohomology group of P with coefficients in G (see (3.6.1)).

Proof. Let f : E → X be a right G_{X}-torsor, and fix a geometric point s ∈ F_{ξ}(E); given any σ ∈ π, there exists a unique g_{s,σ} ∈ G such that

s · g_{s,σ} = σ_{E}(s).

Any g ∈ G determines a X-automorphism g_{E} : E → E, and by definition, the automorphism F_{ξ}(g_{E}) on F_{ξ}(E) commutes with the left action of any element τ ∈ π; however F_{ξ}(g_{E}) is just the right action of g on F_{ξ}(E), hence we may compute :

s · g_{s,τ} · g_{s,σ} = (τ_{E}(s)) · g_{s,σ} = τ_{E}(s · g_{s,τ}) = τ_{E} · σ_{E}(s)

so the rule σ ↦ g_{s,σ} defines a group homomorphism ρ_{s,f} : π → G which is clearly continuous. We claim that the conjugacy class of ρ_{s,f} does not depend on the choice of s. Indeed, if s′ ∈ F_{ξ}(E) is another choice, there exists a (unique) element h ∈ G such that h(s) = s′; arguing as in the foregoing we see that σ_{E} commutes with the right action of h on F_{ξ}(E). In other words, σ_{E}(s) = h^{−1} ∘ σ_{E}(s′), so that g_{s′,σ} = h ∘ g_{s,σ} ∘ h^{−1}.

Therefore, denote by ρ_{f} the conjugacy class of ρ_{s,f}; we claim that ρ_{f} depends only on the isomorphism class of the G_{X}-torsor E. Indeed, any isomorphism t : E →^{∼} E′ of right G_{X}- torsors induces a bijection F_{ξ}(t) : F_{ξ}(E) →^{∼} F_{ξ}(E′), equivariant for the action of G, and for any σ ∈ π we have F_{ξ}(t) ∘ σ_{E} = σ_{E′} ∘ F_{ξ}(t), whence the assertion.

Conversely, given a continuous group homomorphism ρ : π → G, let us endow G with the induced left π-action, and right G-action. Then G is an object of π-Set, to which there corresponds a finite étale morphism E_{ρ} → X, with an isomorphism E_{ρ} ×_{X} |ξ| →^{∼} G of sets with left π-action. Since the right action of G is π-equivariant, we have a corresponding G-action by X-automorphisms on E_{ρ}, so E_{ρ} is G-torsor, and its image under the map of the lemma is clearly the conjugacy class of ρ.

Finally, in order to show that the map of the lemma is injective, it suffices to prove that, for any right G_{X}-torsor (f : E → X, φ : E × G → E) and any s ∈ F_{ξ}(E), there exists an isomorphism of right G_{X}-torsors E_{ρ_{s,f}} →^{∼} E. However, s and ρ_{s,f} determine an identification of sets with left π-action :

(13.1.13)  G →^{∼} F_{ξ}(E)

whence an isomorphism t : E_{ρ_{s,f}} →^{∼} E in Cov(X). Moreover, (13.1.13) also identifies the right G-action on F_{ξ}(E) to the natural right G-action on G; the latter is π-equivariant, hence it induces a right G-action φ′ : E × G → E, such that t is G-equivariant. To conclude, it suffices to show that φ = φ′. In view of [66, Ch.IV, Cor.17.4.8], the latter assertion can be checked on the stalks over the geometric point ξ, where it holds by construction. □

Remark 13.1.14. (i) In the situation of (13.1.11), let E → X be any right G_{X}-torsor, and ρ_{E} : π → G the corresponding representation. Then the proof of lemma 13.1.12shows that the left π-action on F_{ξ}(E) is isomorphic to the left π-action on G induced by ρ_{E}; especially, ρ_{E} is surjective if and only if π acts transitively on F_{ξ}(E), if and only if the scheme E is connected (since a decomposition of E into connected components corresponds to a decomposition of F_{ξ}(E) into orbits for the π-action).

(ii) Let φ : G′ → G be a homomorphism of finite groups, and

H_{cont}^{1}(π, φ) : H_{cont}^{1}(π, G′) → H_{cont}^{1}(π, G)

the induced map. Denote by r (resp. l) the right (resp. left) translation action of G on itself. Let also E′ → X be a principal G′-homogeneous space, given by a map ρ : G′ → Aut_{X}(E′), and denote by c′ ∈ H_{cont}^{1}(π, G′) the class of E′. Then the class c := H_{cont}^{1}(π, φ)(c) can be described geometrically as follows. The scheme E′ × G admits an obvious right G-action, induced by r. Moreover, it admits as well a right G′-action : namely, to any element g ∈ G′, we assign the X-automorphism ρ_{g} × l_{φ(g^{−1})} : E′ × G →^{∼} E′ × G. Set E := (E × G)/G′; it is easily seen that E is a principal G-homogeneous space, and its class is precisely c (the detailed verification shall be left as an exercise for the reader).

(iii) Consider a commutative diagram of schemes

\xymatrix{E′ \ar[rr]^{g} \ar[dr]_{f′} &  & E \ar[dl]^{f} \\  & X}

where f (resp. f′) is a G-torsor (resp. a G′-torsor) for a given finite group G (resp. G′). Denote by ρ : G → Aut_{X}(E) and ρ′ : G′ → Aut_{X}(E′) the respective actions. We have :

(a) Suppose that there exists a group homomorphism φ : G′ → G, such that ρ ∘ φ = g ∘ ρ′. Then g induces a G′-equivariant map

F_{ξ}(E′) → Res(φ)F_{ξ}(E)

whence an isomorphism (E′ × G)/G′ →^{∼} G of G-torsors. Hence, let c (resp. c′) denote any representative of the equivalence class of E (resp. E′) in H_{cont}^{1}(π, G) (resp.

H_{cont}^{1}(π, G′)); in view of (ii), it follows that

H^{1}(π, φ)(c′) = c.

In other words, the induced diagram of continuous group homomorphisms

\xymatrix{ & π_{1}(X_{ét}, ξ) \ar[dl]_{c′} \ar[dr]^{c} \\ G′ \ar[rr]^{φ} &  & G}

commutes, up to composition with an inner automorphism of G. (Details left to the reader.)

(b) If E is connected, a group homomorphism φ : G → G′ fulfilling the condition of (a) exists and is unique up to composition with an inner automorphism of G. Indeed, fix any e′ ∈ F_{ξ}(E′) and let e := f(e′); if g′ ∈ G′, define φ(g′) as the unique g ∈ G such that f(e′ · g′) = e · g; also, in view of (i) we may pick σ_{g′} ∈ π_{1}(X, ξ) such that σ_{g′} · e′ = e′ · g′, and notice that σ_{g′} · e = e · φ(g′) for every g′ ∈ G′. Now, if h′ ∈ G′ is any other element, we may compute :

f(e′ · g′h′) = f(σ_{g′} · e′ · h′) = σ_{g′} · f(e′ · h′) = σ_{g′} · e · φ(h′) = e · φ(g′) · φ(h′)

whence φ(g′h′) = φ(g′) · φ(h′), as required.

Lemma 13.1.15. Let f : X → Y be a morphism of schemes, ℱ, 𝒢 two sheaves on Y_{ét}. Then : (i) If ℱ is locally constant and constructible, the natural map :

ϑ_{f} : f^{∗}ℋ om_{Y_{ét}^{∼}}(ℱ , 𝒢) → ℋ om_{X_{ét}^{∼}}(f^{∗}ℱ , f^{∗}𝒢)

is an isomorphism. (ii) If f is 0-acyclic, it induces a fully faithful functor

f^{∗} : Cov(Y) → Cov(X) : (E → Y) ↦ (E ×_{Y} X → X).

Proof. (i): Suppose we have a cartesian diagram of schemes :

\xymatrix{X′ \ar[r]^{g′} \ar[d]_{f′} & X \ar[d]^{f} \\ Y′ \ar[r]^{g} & Y.}

Then, according to (4.8.2), we have a natural isomorphism :

ϑ_{g′} ∘ g^{′∗}ϑ_{f} ⇒ ϑ_{f′} ∘ f^{′∗}ϑ_{g}

(an invertible 2-cell, in the terminology of (2.1)). Now, if g – and therefore g′ – is a covering morphism, ϑ_{g} and ϑ_{g′} are isomorphisms, and g^{′∗}ϑ_{f} is an isomorphism if and only if the same holds for ϑ_{f}. Summing up, in this case ϑ_{f} is an isomorphism if and only if the same holds for ϑ_{f′}. Thus, we may choose g such that g^{∗} is a constant sheaf, and after replacing f by f′, we may assume that ℱ = S_{Y} is the constant sheaf associated with a finite set S. Since the functors

ℋ om_{Y_{ét}^{∼}}(−, 𝒢) : (Y_{ét}^{∼})^{o} → Y_{ét}^{∼} and f^{∗} : Y_{ét}^{∼} → X_{ét}^{∼}

are left exact, we may further reduce to the case where S = {1} is the set with one element, in which case ℱ = 1_{Y} is the final object of Y_{ét}^{∼}, and f^{∗}ℱ = 1_{X} is the final object of X_{ét}^{∼}. Moreover, we have a natural identification :

ℋ om_{Y_{ét}^{∼}}(1_{Y} , 𝒢) →^{∼} 𝒢 : σ ↦ σ(1)

and likewise for ℋ om_{X_{ét}^{∼}}(1_{Y} , f^{∗}𝒢). Using the foregoing characterization, it is easily checked that, under these identifications, ϑ_{f} is the identity map of f^{∗}𝒢, whence the claim.

(ii): It has already been remarked that Cov(Y) is equivalent to the category of locally constant constructible sheaves on Y_{ét}, and likewise for Cov(X). Let E and F be two objects of Cov(Y); we have natural bijections :

Hom_{Cov(X)}(f^{∗}E, f^{∗}F) →^{∼} Γ(X, ℋ om_{X_{ét}^{∼}}(f^{∗}E, f^{∗}F))
→^{∼} Γ(Y, f_{∗}f^{∗}ℋ om_{Y_{ét}^{∼}}(E, F)) by (i)
→^{∼} Γ(Y, ℋ om_{Y_{ét}^{∼}}(E, F))  since f is 0-acyclic
→^{∼} Hom_{Cov(Y)}(E, F)

as stated. □

Lemma 13.1.16. Let f : X → S be a quasi-compact morphism of schemes, and suppose that : (a) f is locally (−1)-acyclic.

(b) For every strict geometric point ξ of S, the induced morphism f_{ξ} : f^{−1}(ξ) → |ξ| is

0-acyclic (i.e. f has non-empty geometrically connected fibres). Then f is 0-acyclic.

Proof. Let ℱ be a sheaf on S_{ét}. For every strict geometric point ξ of S, we have a commutative diagram :

(13.1.17)  \xymatrix{ℱ_{ξ} \ar[r]^{ε_{ξ}} \ar[d]_{α} & (f_{∗}f^{∗}ℱ)_{ξ} \ar[d] \\ Γ(|ξ|, ξ^{∗}ℱ) \ar[r]^{f_{ξ}^{∗}} & Γ(f^{−1}(ξ), f_{ξ}^{∗} ∘ ξ^{∗}ℱ)}

where ε : ℱ → f_{∗}f^{∗}ℱ is the unit of adjunction. The map α is an isomorphism, and the same holds for f_{ξ}^{∗}, since f_{ξ} is 0-acyclic. Hence ε_{ξ} is injective, which shows already that f is (−1)-acyclic. It remains to show that ε_{ξ} is surjective. Hence, let t ∈ (f_{∗}f^{∗}ℱ)_{ξ} be any section. From (13.1.17) we see that there exists a section t′ ∈ ℱ_{ξ} such that the images of t and ε_{ξ}(t′) agree on Γ(f^{−1}(ξ), f_{ξ}^{∗} ∘ ξ^{∗}ℱ). We may find an étale neighborhood g : U → S of ξ, such that t′ (resp. t) extends to a section t′_{U} ∈ ℱ (U) (resp. t_{U} ∈ Γ(X ×_{S} U, f^{∗}ℱ)). Let X_{U} := X ×_{S} U, f_{U} := f ×_{S} U : X_{U} → U, and for every geometric point \overline{x} of X_{U}, denote by f ∗ ; \overline{x} : ℱ_{f_{U} (\overline{x})} → f_{U}^{∗}ℱ\overline{x} the natural isomorphism (4.9.22). We set

V := \{x ∈ X_{U} | t_{U,\overline{x}} = f ∗ ; \overline{x}(t′_{U,f(\overline{x})})\}

where \overline{x} is any geometric point of X localized at x, and t_{U,\overline{x}} ∈ f^{∗}ℱ\overline{x} (resp. t′_{U,f(\overline{x})} ∈ ℱ_{f_{U} (\overline{x})}) denotes the image of t_{U} (resp. of t′_{U}). Clearly V is an open subset of X_{U}, and we have :

Claim 13.1.18. (i) V = f_{U}^{−1}f_{U}(V).

(ii) f_{U}(V) ⊂ U is an open subset.

Proof of the claim. (i): Given a point u ∈ U, choose a strict geometric point \overline{u} localized at u, and set \overline{s} := g(\overline{u})^{st}; by assumption, the morphism f\overline{s} : f^{−1}(\overline{s}) → |\overline{s}| is 0-acyclic, hence the image of t_{U} in Γ(f^{−1}(\overline{s}), f ∗ ; \overline{s} ∘ \overline{s}^{∗}ℱ) is of the form f ∗ ; \overline{s}t′′, for some t′′ ∈ ℱ\overline{s}. It follows that V ∩ f_{U}^{−1}(u) is either the whole of f_{U}^{−1}(u) or the empty set, according to whether t′′ agrees or not with the image of t′_{U} in ℱ\overline{s} = g^{∗}ℱ\overline{u}.

(ii): The subset X \V is closed, especially pro-constructible; since f is quasi-compact, we deduce that f_{U}(X\V) is a pro-constructible subset of U (corollary 8.1.50). It then follows from (i) that f_{U}(V) is ind-constructible, hence we are reduced to showing that f_{U}(V) is closed under generizations (proposition 8.1.47(ii)). To this aim, since V is open, it suffices to show that f_{U} is generizing, i.e. that the induced maps X_{U}(x) → U(u) are surjective, for every u ∈ U and every x ∈ f_{U}^{−1}(u). However, choose a geometric point \overline{x} localized at x, and let \overline{u} := f_{U}(\overline{x}); since the natural maps X_{U}(\overline{x}) → X_{U}(x) and U(\overline{u}) → U(u) are surjective, it suffices to show that the same holds for the map f_{U,\overline{x}} : X_{U}(\overline{x}) → U(\overline{u}). The image of \overline{x} (resp. \overline{U}) in X (resp. in S) is a geometric point which we denote by the same name; since the natural maps X_{U}(\overline{x}) → X(\overline{x}) and U(\overline{u}) → S(\overline{u}) are isomorphisms, we are reduced to showing that f\overline{x} : X(\overline{x}) → S(\overline{u}) is surjective, which holds, since f is locally (−1)-acyclic. ♢

Set W := f_{U}(V); in view of claim 13.1.18, W is an étale neighborhood of ξ, and the natural map ℱ (W) → f^{∗}ℱ (U) sends the restriction t′_{U|W} of t′_{U} to the restriction t_{U|V} of t_{U}, whence the claim. □

13.1.19. Consider now a cartesian diagram of schemes :

(13.1.20)  \xymatrix{X′ \ar[r]^{g′} \ar[d]_{f′} & X \ar[d]^{f} \\ S′ \ar[r]^{g} & S}

where g is a local morphism of strictly local schemes, and denote by s (resp. by s′) the closed point of S (resp. of S′). Let x′ ∈ f^{′−1}(s′) be any point, \overline{x}′ a geometric point of X′ localized at x′, and set x := g′(x′), \overline{x} := g′(\overline{x}′). Then g′ induces a morphism of S′-schemes :

(13.1.21)  X′(\overline{x}′) → X(\overline{x}) ×_{S} S′.

Lemma 13.1.22. In the situation of (13.1.19), suppose that g is an integral morphism. Then :

(i) The induced morphism f^{′−1}(s′) → f^{−1}(s) induces a homeomorphism on the underlying topological spaces. (ii) (13.1.21) is an isomorphism.

Proof. If g is integral, κ(s′) is a purely inseparable algebraic extension of κ(s), hence the morphism T′ := Spec κ(s′) → T := Spec κ(s) is radicial, and the same holds for the induced morphisms :

f^{′−1}(s′) →^{∼} f^{−1}(s) ×_{T} T′ → f^{−1}(s) f_{x}^{−1}(s) ×_{T} T′ → f_{x}^{−1}(s)

([59, Ch.I, Prop.3.5.7(ii)]). Especially (i) holds, and therefore the natural map X′(x′) → X(x) ×_{S} S′ is an isomorphism; we see as well that f_{x}^{−1}(s) ×_{T} T′ is a local scheme. Then the assertion follows from [66, Ch.IV, Rem.18.8.11]. □

Proposition 13.1.23. Let f : X → S and g : S′ → S be morphisms of schemes, with g quasifinite, and set X′ := X ×_{S} S′. Suppose that f is locally (−1)-acyclic (resp. locally 0-acyclic); then the same holds for f′ := f ×_{S} S′ : X′ → S′.

Proof. Let \overline{s}′ be any geometric point of S′, and set \overline{s} := g(\overline{s}′). Denote by s ∈ S (resp. s′ ∈ S′) the support of \overline{s} (resp. \overline{s}′); then f is locally (−1)-acyclic at the points of f^{−1}(s), if and only if f\overline{s} := f ×_{S} S(\overline{s}) : X ×_{S} S(\overline{s}) → S(\overline{s}) enjoys the same property at the points of f −1 ; \overline{s}(\overline{s}). Likewise, f′ is locally (−1)-acyclic (resp. locally 0-acyclic) at the points of f^{′−1}(s′), if and only if f′_{\overline{s}′} : X′ ×_{S′} S′(\overline{s}′) → S′(\overline{s}′) enjoys the same property at the points of f −1 ; \overline{s}(\overline{s}). Hence, we may replace g by g\overline{s}′ : S′(\overline{s}′) → S(\overline{s}), and f by the induced morphism X ×_{S} S(\overline{s}) → S(\overline{s}), which allows to assume that g is finite ([66, Ch.IV, Th.18.5.11]), hence integral. Let ξ′ (resp. \overline{x}′) be any strict geometric point of S′ (resp. of f^{′−1}(s′)), and let ξ := g(ξ′)^{st}, (resp. let \overline{x} be the image of \overline{x}′ in X); we have natural morphisms :

f_{\overline{x}′}^{′−1}(ξ′) ⟶^{α} X(\overline{x}) ×_{S} ξ′ ⟶^{β} f −1 ; \overline{x}(ξ).

However, α is an isomorphism, by lemma 13.1.22(ii), and β is a radicial morphism, since the field extension κ(ξ) ⊂ κ(ξ′) is purely inseparable ([59, Ch.I, Prop.3.5.7(ii)]). The claim follows. □

Lemma 13.1.24. (i) Let S be a strictly local scheme, s ∈ S the closed point, f : X → S a morphism of schemes, \overline{x} (resp. ξ) a strict geometric point of f^{−1}(s) (resp. of S). We may find :

(a) A cartesian diagram (13.1.20), with S′ strictly local, irreducible and normal.

(b) A strict geometric point \overline{x}′ of f^{′−1}(s′) with g′(\overline{x}′)^{st} = \overline{x}.

(c) A strict geometric point ξ′ of S′ localized at the generic point η′ of S′, with g(ξ′)^{st} = ξ, and such that (13.1.21) induces an isomorphism :

(13.1.25)  f_{\overline{x}′}^{′−1}(ξ′) →^{∼} f −1 ; \overline{x}(ξ).

(ii) Moreover, if S is noetherian, we may find S′ as in (i), such that 𝒪_{S′}(S′) is a Krull domain. (iii) Alternatively, we may find S′ as in (i), with separably closed residue field κ(η′).

Proof. (i): Denote by Z ⊂ S the closure of the image of ξ, endow Z with its reduced subscheme structure, set Y := X ×_{S} Z ⊂ X, and let h\overline{x} : Y (\overline{x}) → Z be the natural morphism. Then Z is a strictly local scheme ([66, Ch.IV, Prop.18.5.6(i)]). Moreover, the closed immersion Y → X induces an isomorphism Y (\overline{x}) →^{∼} X(\overline{x}) ×_{S} Z of Z-schemes (lemma 13.1.22(ii)). By construction, ξ factors through a strict geometric point ξ′ of Z, and we deduce an isomorphism h−1 ; \overline{x}(ξ′) →^{∼} f −1 ; \overline{x}(ξ) of Z-schemes. Thus, we may replace (S, X, ξ) by (Z, Y, ξ′), and assume that S is the spectrum of a strictly local domain, and ξ is localized at the generic point of S.

Claim 13.1.26. Let A be a strictly local domain, and A^{ν} the normalization of A in an algebraic extension K of F := Frac A. Then A^{ν} is strictly local.

Proof of the claim. A^{ν} is the union of a filtered family (A_{λ} | λ ∈ Λ) of finite A-subalgebras of F; since A is henselian, each A_{λ} is a product of henselian local rings, hence it is a local henselian ring, so the same holds for A^{ν}. Moreover, the residue field of A^{ν} is separably closed, since it is an algebraic extension of the residue field of A, which is separably closed by assumption. ♢

Say that S = Spec A, and denote by A^{ν} the normalization of A in its field of fractions F. By claim 13.1.26, A^{ν} is strictly henselian, so we may fulfill condition (a) by taking S′ := Spec A^{ν}. Condition (b) holds as well, due to lemma 13.1.22(i). Finally, it is clear that ξ lifts to a unique strict geometric point ξ′ of S′, and it follows from lemma 13.1.22(ii) that (13.1.25) is an isomorphism, as required.

(ii): A direct inspection of the proof of (i) reveals that if S is noetherian, the scheme S′ exhibited is the spectrum of the normalization of a noetherian domain; the assertion then follows from [133, Th.33.10].

(iii): Let A_{K}^{ν} be the normalization of A in a separable closure K of F; the proof of (i) applies as well with S′ := Spec A_{K}^{ν}, whence the assertion. □

13.1.27. Consider now a cofiltered family 𝒮 := (S_{λ} | λ ∈ Λ) of affine schemes. Denote by S the limit of 𝒮, and suppose moreover that Λ admits a final element 0 ∈ Λ. Let f_{0} : X_{0} → S_{0} be a finitely presented morphism of schemes, and set :

X_{λ} := X_{0} ×_{S_{0}} S_{λ}  f_{λ} := f_{0} ×_{S_{0}} S_{λ} : X_{λ} → S_{λ}  for every λ ∈ Λ.

Set as well X := X_{0} ×_{S_{0}} S and f := f_{0} ×_{S_{0}} S : X → S. Let x ∈ X be any point, and s := f(x). Let also ξ be a strict geometric point of S. We deduce a compatible system of points x_{λ} := p′_{λ}(x) ∈ X_{λ}, whence a cofiltered system of local schemes

𝒳 := (X_{λ}(x_{λ}) | λ ∈ Λ).

Moreover, we get a compatible system of strict geometric points (ξ_{λ} := p_{λ}(ξ)^{st} | λ ∈ Λ), with :

ξ →^{∼} lim_{λ∈Λ} ξ_{λ}.

Choose a geometric point \overline{x} of X localized at x, and set likewise \overline{x}_{λ} := p′_{λ}(\overline{x}); then 𝒳 lifts to a system 𝒳 ^{sh} := (X_{λ}(\overline{x}_{λ}) | λ ∈ Λ), whose limit is naturally isomorphic to X(\overline{x}) ([66, Ch.IV, Prop.18.8.18(ii)]). Furthermore, 𝒳 ^{sh} induces a natural isomorphism of κ(ξ)-schemes :

(13.1.28)  f −1 ; \overline{x}(ξ) →^{∼} lim_{λ∈Λ} f_{λ,\overline{x}_{λ}}^{−1}(ξ_{λ})

where, as usual, f\overline{x} : X(\overline{x}) → S (resp. f_{λ,\overline{x}_{λ}} : X_{λ}(\overline{x}_{λ}) → S_{λ}) is deduced from f (resp. from f_{λ}). These remarks, together with the following lemma 13.1.29, and the previous lemma 13.1.24, will allow in many cases, to reduce the study of the fibres of f\overline{x}, to the case where the base S is strictly local, excellent and normal.

Lemma 13.1.29. Let S be a strictly local normal scheme. Then there exists a cofiltered family 𝒮 := (S_{λ} | λ ∈ Λ) consisting of strictly local normal excellent schemes, such that :

(a) S is isomorphic to the limit of 𝒮.

(b) The natural morphism S → S_{λ} is dominant for every λ ∈ Λ.

Proof. Say that S = Spec A, and write A as the union of a filtered family 𝒜 := (A_{λ} | λ ∈ Λ) of excellent noetherian local subrings, which we may assume to be normal, by [64, Ch.IV, (7.8.3)(ii),(vi)]. Proceeding as in (13.1.27), we choose a compatible family of geometric points \overline{s}_{λ} localized at the closed points of Spec A_{λ}, for every λ ∈ Λ; using these geometric points, we lift 𝒜 to a filtered family (A_{λ}^{sh} | λ ∈ Λ) of strict henselizations, whose colimit is naturally isomorphic to A. Moreover, each A_{λ}^{sh} is noetherian, normal and excellent ([66, Ch.IV, Prop.18.8.8(iv), Prop.18.8.12(i)] and proposition 9.7.19(ii)). Let η be the generic point of S, h_{λ} : S → S_{λ} := Spec A_{λ}^{sh} the natural morphism, and η_{λ}^{sh} := h_{λ}(η) for every λ ∈ Λ. The cofiltered system (S_{λ} | λ ∈ Λ) fulfills condition (a). Moreover, by construction, the image of η_{λ}^{sh} in Spec A_{λ} is the generic point η_{λ}; then η_{λ}^{sh} is the generic point of S_{λ}, since the latter is the only point of S_{λ} lying over η_{λ}. Hence (b) holds as well. □

Part (ii) of the following proposition improves upon [10, Exp.XV, Th.4.1].

Proposition 13.1.30. Let f : X → S be a flat morphism of schemes. We have :

(i) f is locally (−1)-acyclic.

(ii) Suppose moreover, that f has geometrically reduced fibres, and : (a) either f is locally finitely presented,

(b) or else, S is locally noetherian.

Then f is locally 0-acyclic. (iii) Furthermore, if f is locally finitely presented and has geometrically normal fibres, then

the Milnor fibres of f at every point of X are geometrically normal and irreducible.

Proof. Let x ∈ X be any point, set s := f(x), choose a geometric point \overline{x} of X localized at x, set \overline{s} := f(\overline{x}), and let ξ be any strict geometric point of S(\overline{s}).

(i): If f is flat, the induced morphism f\overline{x} : X(\overline{x}) → S(\overline{s}) is faithfully flat; especially, f\overline{x} is surjective ([126, Th.7.3(i)]).

(ii),(iii): Set X′ := X ×_{S} S(\overline{s}). The natural morphism X(\overline{x}) → X factors uniquely through a morphism of S(\overline{s})-schemes j : X(\overline{x}) → X′, and if we denote by \overline{x}′ the image in X′ of \overline{x}, then j induces an isomorphism of S(\overline{s})-schemes : X(\overline{x}) →^{∼} X′(\overline{x}′). Hence we may replace S by S(\overline{s}), and assume that S is strictly local, when (a) holds, and even strictly local and noetherian, when (b) holds. By lemma 13.1.24, we are further reduced to the case where S = S(\overline{s}) = Spec A is strictly local and normal, ξ is localized at the generic point of S, and :

(a’) either f is finitely presented, (b’) or else, A is a (not necessarily noetherian) Krull domain.

Claim 13.1.31. (i) In case (b’) holds, f −1 ; \overline{x}(ξ) is connected.

(ii) If (a’) holds and f has geometrically normal fibres, then f −1 ; \overline{x}(ξ) is normal and irreducible.

Proof of the claim. Let F be the field of fractions of A; the field κ(ξ) is algebraic over F, hence, in order to show (i), it suffices to check that X(\overline{x}) ×_{S} Spec K is connected for every finite field extension F ⊂ K ([65, Ch.IV, Prop.8.4.1(ii)]). Likewise, in order to show (ii), it suffices to check that X(\overline{x}) ×_{S} Spec K is normal and irreducible for every such K. Let A_{K} be the normalization of A in K, set S_{K} := Spec A_{K}, X_{K} := X ×_{S} S_{K}, let f_{K} : X_{K} → S_{K} be the induced morphism, and pick any geometric point \overline{x}_{K} of X_{K} whose image in X agrees with \overline{x}. By claim 13.1.26, S_{K} is strictly local, and if (b’) holds, then A_{K} is again a Krull domain ([34, Ch.VII, §1, n.8, Prop.12]). Moreover, lemma 13.1.22(ii) yields a natural identification :

X_{K}(\overline{x}_{K}) ×_{S_{K}} Spec K →^{∼} X(\overline{x}) ×_{S} Spec K.

Clearly, if f has geometrically reduced (resp. geometrically normal) fibres, the same holds for f_{K}, hence we may replace f by f_{K}, and reduce to checking the claim for K = F.

Now, B := 𝒪_{X,\overline{x}} is clearly a flat A-algebra, and notice that the geometric fibres f\overline{x} are cofiltered limits of schemes that are étale over the fibres of f; if the fibres of f are geometrically reduced (resp. geometrically normal), it follows that the same holds for the fibres of f\overline{x}. In case A is a Krull domain, B is integrally closed in B ⊗_{A} F (lemma 9.8.2); especially these two rings have the same idempotents, whence (i).

Lastly, if the assumptions of (ii) hold, then we know already that the fibres of f\overline{x} are geometrically normal, so that X(\overline{x}) is normal, by [65, Ch.IV, Prop.11.3.13(ii)], and therefore B ⊗_{A} F is a normal domain, whence (ii). ♢

Claim 13.1.31yields already (iii), and also shows that (ii) holds when (b’) holds.

Finally, let us check that (ii) holds when (a’) holds. By lemma 13.1.29, the scheme S is the limit of a cofiltered family (S_{λ} | λ ∈ Λ) of strictly local excellent and normal schemes, such that the natural morphisms p_{λ} : S → S_{λ} are dominant. By claim 9.8.4, there exists λ ∈ Λ and a flat morphism of schemes f_{λ} : X_{λ} → S_{λ} with geometrically reduced fibres, with an isomorphism of S-schemes S ×_{S_{λ}} X_{λ} →^{∼} X; then, for every μ ∈ Λ with μ ≥ λ, set X_{μ} := S_{μ} ×_{S_{λ}} X_{λ} and f_{μ} := S_{μ} ×_{S_{λ}} f_{λ} : X_{μ} → S_{μ}. After replacing Λ by a coinitial subset, we may assume that f_{μ} is defined for every λ ∈ Λ. For every such λ, let x_{λ} ∈ X_{λ} be the image of x. Arguing as in (13.1.27), we obtain a compatible system of strict geometric points ξ_{λ} of S_{λ} (resp. \overline{x}_{λ} of X_{λ}), such that p_{λ}(ξ) factors through ξ_{λ}; whence an isomorphism (13.1.28). Thus, f −1 ; \overline{x}(ξ) is reduced if and only if f_{λ,\overline{x}_{λ}}^{−1}(ξ_{λ}) is reduced for every sufficiently large λ ∈ Λ ([65, Ch.IV, Prop.8.7.2]). Furthermore, since p_{λ} is dominant, ξ_{λ} is localized at the generic point of S_{λ}, for every λ ∈ Λ. Thus, we are reduced to the case where S = S(s) is the spectrum of a strictly local noetherian normal domain A, and ξ is localized at the generic point of S; since such A is a Krull domain ([126, Th.12.4(i)]), this is covered by claim 13.1.31. □

Remark 13.1.32. In the statement and proof of proposition 13.1.30(iii) we use a definition of geometrically normal scheme (over a given field) which is more general than that introduced originally in [64, Ch.IV, Def.6.7.6] (which includes some unnecessary noetherian assumptions), and agrees e.g. with the definition in [107, Tag 0380, Tag 038M].

Example 13.1.33. (i) Let A be an excellent local ring, and A^{∧} the completion of A. Then the natural morphism :

f : Spec A^{∧} → Spec A

is locally 0-acyclic. Indeed, this follows from proposition 13.1.30(ii) (and from the excellence assumption, which includes the geometric regularity of the formal fibres of A).

(ii) Suppose additionally, that A is strictly local. Then f is 0-acyclic. To see this, we apply the criterion of lemma 13.1.16 : indeed, since f is flat, it is (−1)-acyclic (proposition 13.1.30(i)); it remains to show that f has geometrically connected fibres, and since A^{∧} is strictly local ([66, Ch.IV, Prop.18.5.14]), this is the same as showing that f is locally 0-acyclic at the closed point of Spec A^{∧}, which has already been remarked in (i).

(iii) More generally, f is 0-acyclic whenever A is excellent and henselian. Indeed, in this case the argument of (ii) again reduces to showing that f has geometrically connected fibres. However, consider the natural commutative diagram :

(13.1.34)  \xymatrix{Spec (A^{∧})^{sh} \ar[r] \ar[d]_{f^{sh}} & Spec A^{∧} \ar[d]^{f} \\ Spec A^{sh} \ar[r] & Spec A.}

Since A is henselian, A^{sh} is the colimit of a filtered family of finite étale and local A-algebras. Since A and A^{∧} have the same residue field, it follows easily that A^{∧} ⊗_{A} A^{sh} is the colimit of a filtered family of finite étale and local A^{∧}-algebras, hence it is strictly henselian, and therefore (13.1.34) is cartesian, especially the geometric fibres of f are connected if and only if the same holds for the geometric fibres of f^{sh}, and the latter are reduced (even regular), since A is excellent. Hence, we come down to showing that f^{sh} is locally 0-acyclic at the closed point of Spec (A^{∧})^{sh}, which holds again by proposition 13.1.30(ii).

Proposition 13.1.35. Let f : X → S be a morphism of schemes locally of finite type, and suppose that either one of the following conditions holds :

(a) For some d ∈ ℕ, the fibres of f are all of pure dimension d.

(b) f is quasi-flat, in the sense of [64, Ch.IV, (2.3.3)] i.e. there exists a quasi-coherent 𝒪_{X}- module ℱ of finite type which is S-flat and with Supp ℱ = X (notation of definition 10.1.1(iv)).

Then the Milnor fibres of f at every point of X have finitely many irreducible components.

Proof. Suppose first that (a) holds. We may assume that X = Spec B is affine, and by lemma 13.1.24(iii), we may also assume that S = Spec A is strictly local and normal, and the field of fractions F of A is separably closed. In this situation, we need to check that X(\overline{x}) ×_{S} Spec F has finitely many irreducible components, for every geometric point \overline{x} of X whose image in S is localized at the maximal ideal 𝔪 ⊂ A. By assumption, B/𝔪B is of pure dimension d; by the Nother normalization lemma, we may then find a finite map κ(𝔪)[T_{1}, …, T_{d}] → B/𝔪B of κ(𝔪)-algebras, and lift it to a map of A-algebras φ : A[T_{1}, …, T_{d}] → B which will be quasifinite at the support x ∈ Spec B of \overline{x}. After replacing X by an affine open neighborhood of x, we may therefore assume that the morphism g : X → 𝔸_{S}^{d} induced by φ is quasi-finite ([65, Ch.IV, Cor.13.1.4]). Then, since X_{F} := X ×_{S} Spec F is of pure dimension d, it follows easily that the restriction g_{F} : X_{F} → 𝔸_{F}^{d} of g is maximizing (see definition 8.1.46(ii)). Set \overline{y} := g(\overline{x}), let h : X(\overline{x}) → 𝔸_{S}^{d}(\overline{y}) be the induced morphism, and consider the commutative diagram of schemes :

\xymatrix{X(\overline{x}) ×_{S} Spec F \ar[r] \ar[d]_{h×_{S}Spec K} & X_{F} \ar[d]^{g_{F}} \\ 𝔸_{S}^{d}(\overline{y}) ×_{S} Spec F \ar[r] & 𝔸_{F}^{d} .}

Notice that both horizontal arrows are flat morphisms, hence they are generizing, and the fibres of the bottom horizontal arrows are 0-dimensional; since g_{F} is maximizing, it then follows that the same holds for h ×_{S} Spec K. On the other hand, h is a finite morphism, by Zariski’s main theorem, so it has finite fibres, and 𝔸_{S}^{d}(\overline{y}) ×_{S} Spec K is an integral scheme, by proposition 13.1.30(iii), whence the assertion.

Next, suppose that (b) holds; again, we fix a geometric point \overline{x} of X, and we are easily reduced to the case where S is strictly local, X is affine, and the image \overline{s} of \overline{x} in S is localized at the closed point. Then we may find a closed immersion i : X → Y := 𝔸_{S}^{n}, for some n ∈ ℕ; according to [86, Part I, Th.3.4.1], the 𝒪_{Y} -module i_{∗}ℱ is finitely presented in an affine open neighborhood U of the support of \overline{y} := i(\overline{x}), hence we may replace Y by U and X by X ×_{Y} U, and assume that i_{∗}ℱ is a finitely presented Y -scheme. Let ℐ ⊂ 𝒪_{Y} be the 0-th Fitting ideal of i_{∗}ℱ, set X′ := Spec 𝒪_{Y} /ℐ, and let g : X′ → S be the induced projection; then X′ is a closed subscheme of Y, and i(X) = Supp_{Y} (i_{∗}ℱ) = X′, hence the maximal reduced subschemes of X and X′ are S-isomorphic. It follows easily that the S(\overline{s})-schemes X(\overline{x}) and X′(\overline{y}) have isomorphic maximal reduced subschemes. Hence, for every strict geometric point ξ of S(\overline{s}), the fibres f −1 ; \overline{x}(ξ) and g−1 ; \overline{x}(ξ) are homeomorphic. Thus, we may replace f : X → S by g : X′ → S, and assume from start that X is a finitely presented S-scheme.

Claim 13.1.36. There exist finitely many closed subschemes Z_{1}, …, Z_{k} ⊂ X such that X = Z_{1} ∪ ⋯ ∪ Z_{k}, and integers d_{1}, …, d_{k} ∈ ℕ such that for every i = 1, …, k, all the fibres of the restriction Z_{i} → S of f are of pure dimension d_{i}.

Proof of the claim. Say that S = Spec A, X = Spec B and ℱ = M^{∼} for a finitely presented A-algebra B and an A-flat B-module M of finite presentation. Then we may find a noetherian subring A_{0} ⊂ A, an A_{0}-algebra B_{0} of finite type, and a B_{0}-module M_{0} with an isomorphism of A-algebras A ⊗_{A_{0}} B_{0} →^{∼} B and of B-modules B ⊗_{B_{0}} M_{0} →^{∼} M. Moreover, we may assume that M_{0} is a flat A_{0}-module ([65, Ch.IV, Cor.11.2.6.1]), and that Supp_{B_{0}}M_{0} = Spec B_{0} ([65, Ch.IV, Cor.8.3.3]). In view of [64, Ch.IV, Cor.4.1.4], we are easily reduced to checking the assertion for the corresponding morphism Spec B_{0} → Spec A_{0}. The latter is provided by [65, Ch.IV, Prop.12.1.1.5]. ♢

To a decomposition X = Z_{1}∪⋯∪Z_{k} as in claim 13.1.36, there corresponds a decomposition X(\overline{x}) = Z_{1}(\overline{x}) ∪ ⋯ ∪ Z_{k}(\overline{x}) for every geometric point \overline{x} of X, hence it suffices to check the assertion for each of the restrictions Z_{i} → S of f, so we may assume that condition (a) holds, in which case the assertion is already known. □

For future use, we point out the following

Proposition 13.1.37. Let g : X → Y be a flat morphism of excellent noetherian schemes, with X strictly local, and Y normal. Let U ⊂ X be an open subset, and Z ⊂ Y a closed subscheme. Suppose that :

(i) g^{−1}(z) ⊂ U for every maximal point z of Z.

(ii) U ∩ g^{−1}(z) is a dense open subset of g^{−1}(z), for every z ∈ Z.

(iii) The fibres g^{−1}(z) are reduced, for every z ∈ Z. Then the induced functor Cov(U) → Cov(U ×_{Y} Z) is fully faithful.

Proof. Indeed, say that X = Spec B, Y = Spec A, Z = V (I) for some ideal I ⊂ A, and denote by B^{∧} the 𝔪_{B}-adic completion of the local ring B (where 𝔪_{B} ⊂ B denotes the maximal ideal). Let also f : Spec B^{∧} → Y be the induced morphism, and U^{∧} ⊂ Spec B^{∧} the preimage of U. In light of example 13.1.33(ii) and lemma 13.1.15(ii), it suffices to show that the induced functor Cov(U^{∧}) → Cov(U^{∧} ×_{Y} Z) is fully faithful. In view of lemma 13.1.3, we are further reduced to checking that conditions (a)–(c) of proposition 10.5.46 hold for the induced ring homomorphism φ : A → B^{∧}, the open subset U^{∧}, and the ideal I. However, by example 10.5.41, we have Ass_{A}(I, A) = Max(Z), hence (c) follows trivially from our assumption (i). Next, since B^{∧} is a faithfully flat B-algebra, assumption (ii) implies that U^{∧} ∩f^{−1}(z) is a dense open subset of f^{−1}(z), for every z ∈ Z. Moreover, since B is excellent, the natural morphism Spec B^{∧} → X is regular, so the same holds for the induced morphism f^{−1}(z) → g^{−1}(z), and then our assumption (iii) implies – together with [126, Th.32.3(i)] – that f^{−1}(z) is reduced, for every z ∈ Z, whence condition (b). Lastly, we check condition (a), i.e. we show that B^{∧} is I-adically complete. Indeed, let C be the I-adic completion of B^{∧}; the natural map B^{∧} → C is injective, and it admits a left inverse, constructed as follows. Let \underline{a} := (a_{n} | n ∈ ℕ) be a given sequence of elements of B^{∧}, which is Cauchy for the I-adic topology; then \underline{a} is also Cauchy for the 𝔪_{B}-adic topology, and it is easily seen that the limit l of \underline{a} in the 𝔪_{B}-adic topology depends only on the class [\underline{a}] of \underline{a} in C, so we get a well defined ring homomorphism λ : C → B^{∧} by the rule : [\underline{a}] ↦ l, and clearly λ is the sought left inverse. It remains to check that λ is injective; thus, suppose that l = 0, and that [\underline{a}] ≠ 0; this means that there exists N ∈ ℕ such that a_{n} ∉ I^{N}, for every n ∈ ℕ. Now, the induced sequence (\overline{a}_{n} | n ∈ ℕ) of elements of B^{∧}/I^{N} is stationary, and on the other hand, it converges 𝔪_{B}-adically to 0; therefore \overline{a}_{n} = 0 for every sufficiently large n ∈ ℕ, a contradiction. □

13.1.38. Let A be a noetherian normal ring, and endow the A-algebra A[[t]] with its t-adic topology. Let

φ : 𝔛 := Spf A[[t]] → X := Spec A[[t]]  π : X → S := Spec A  i : S → X

be respectively the natural morphism of locally ringed spaces, the natural projection, and the closed immersion determined by the ring homomorphism A[[t]] → A given by the rule : f(t) ↦ f(0), for every f(t) ∈ A[[t]]. Let also U_{0} ⊂ Spec A be an open subset, U := π^{−1}U_{0} and 𝔘 := φ^{−1}U. Finally, denote by ℰ a locally free 𝒪_{U}-module of finite rank, and set ℰ^{∧} := φ_{|𝔘}^{∗}ℰ, which is a locally free 𝒪_{𝔘}-module of finite rank.

Lemma 13.1.39. In the situation of (13.1.38), suppose that S \U_{0} has codimension ≥ 2 in S. Then:

(i) The natural map

Γ(U, ℰ) → Γ(𝔘, ℰ^{∧})

is an isomorphism of A[[t]]-modules. (ii) The restriction i_{|U_{0}} : U_{0} → U of i induces an equivalence :

i_{|U_{0}}^{∗} : Cov(U) → Cov(U_{0}) : (E → U) ↦ (E ×_{U} U_{0} → U_{0}).

Proof. (i): To begin with, set Z := S\U_{0}; since the morphism π is flat, hence generizing ([126, Th.9.5]), the closed subset X \U = π^{−1}Z has codimension ≥ 2 in X. Since A and A[[t]] are both normal, we deduce :

(13.1.40)  depth_{X\V} 𝒪_{X} ≥ 2  depth_{Z}𝒪_{S} ≥ 2

(theorem 10.4.21and [126, Th.23.8]); therefore (corollary 10.4.23) :

(13.1.41)  Γ(U, 𝒪_{X}) = Γ(X, 𝒪_{X}) = A[[t]].

Next, the short exact sequences of 𝒪_{X}-modules :

0 → i_{∗}𝒪_{S} → 𝒪_{X}/t^{n+1}𝒪_{X} → 𝒪_{X}/t^{n}𝒪_{X} → 0 for every n ∈ ℕ

induce exact sequences

(13.1.42)  R^{j}Γ_{Z}i_{∗}𝒪_{S} → R^{j}Γ_{Z}𝒪_{X}/t^{n+1}𝒪_{X} → R^{j}Γ_{Z}𝒪_{X}/t^{n}𝒪_{X}  for every n, j ∈ ℕ.

Then (13.1.40) and (13.1.42) yield inductively :

depth_{Z}𝒪_{X}/t^{n}𝒪_{X} ≥ 2 for every n ∈ ℕ

and again corollary 10.4.23implies :

(13.1.43)  Γ(U, 𝒪_{X}/t^{n}𝒪_{X}) = A[t]/t^{n}A[t]  for every n ∈ ℕ.

Since U is quasi-compact, we may find a left exact sequence P := (0 → ℰ → 𝒪_{U}^{⊕m} → 𝒪_{U}^{⊕n}) of 𝒪_{U}-modules (corollary 10.3.28). Since φ is a flat morphism of locally ringed spaces, the sequence φ^{∗}P is still left exact. Since the global section functors are left exact, we are then reduced to the case where ℰ = 𝒪_{U}. Then we may write :

ℰ^{∧} = 𝒪_{𝔘} = lim_{n∈ℕ} 𝒪_{U}/t^{n}𝒪_{U}

where, for each n ∈ ℕ, we regard 𝒪_{U}/t^{n}𝒪_{U} as a sheaf of (pseudo-discrete) rings on 𝔘 = V (t) ⊂ U. The functor Γ(𝔘, −) is a right adjoint, hence commutes with limits, and we deduce an isomorphism :

Γ(𝔘, ℰ^{∧}) →^{∼} lim_{n∈ℕ} Γ(U, 𝒪_{U}/t^{n}𝒪_{U}).

(This is even a homeomorphism, provided we view the target as a limit of rings with the discrete topology.) Taking (13.1.43) into account, we obtain Γ(𝔘, ℰ^{∧}) = A[[t]] which, together with (13.1.41), implies the contention.

(ii): Notice that (i) and lemma 10.5.44imply that Lef(U, i(U_{0})) holds (see definition 10.5.42). Since the pull-back functor π_{|U}^{∗} : Cov(U_{0}) → Cov(U) is a right quasi-inverse to i_{|U_{0}}^{∗}, the latter is essentially surjective. The full faithfulness is a special case of lemma 13.1.3. □

13.2. Local asphericity of smooth morphisms of schemes. Let S be a strictly local scheme, s ∈ S the closed point, f : X → S a smooth morphism, \overline{x} any geometric point of f^{−1}(s), and denote by f\overline{x} : X(\overline{x}) → S the induced morphism of strictly local schemes. For any open subset U ⊂ S we have a base change functor :

(13.2.1)  f ∗ ; \overline{x} : Cov(U) → Cov(f −1 ; \overline{x}U)  (E → U) ↦ (E ×_{U} f −1 ; \overline{x}U).

Theorem 13.2.2. In the situation of (13.2), we have :

(i) The functor (13.2.1) is fully faithful.

(ii) Suppose moreover that S is excellent and normal, and that S\U has codimension ≥ 2 in S. Then (13.2.1) is an equivalence of categories.

Proof. (i): In view of lemma 13.1.15(ii) it suffices to show that f\overline{x} is 0-acyclic (since in that case, the same will obviously hold also for its restriction f −1 ; \overline{x}U → U). To begin with, f\overline{x} is locally (−1)-acyclic, by proposition 13.1.30(i), hence it remains only to show that f is locally 0-acyclic at the point x (lemma 13.1.16). The latter assertion follows from proposition 13.1.30(ii) and [66, Ch.IV, Th.17.5.1].

(ii): In light of (i), it suffices to show that (13.2.1) is essentially surjective, under the assumptions of (ii). We argue by induction on the relative dimension n of f. Let x ∈ X be the support of \overline{x}. We may find an open neighborhood U ⊂ X of x, and an étale morphism of S-schemes φ : U → 𝔸_{S}^{n} ([66, Ch.IV, Cor.17.11.4]). Let \overline{x}′ := φ(\overline{x}); there follows an isomorphism of S-schemes : X(\overline{x}) →^{∼} 𝔸_{S}^{n}(\overline{x}′), hence we may assume from start that X = 𝔸_{S}^{n}, and f is the natural projection. Especially, the theorem holds for n = 0. Suppose then, that n > 0, and that the theorem is already known when the relative dimension is < n. Write f = h ∘ g, where

g : X ≃ 𝔸_{S}^{n−1} ×_{S} 𝔸_{S}^{1} → 𝔸_{S}^{1} and h : 𝔸_{S}^{1} → S

are the natural projections; set \overline{x}_{1} := g(\overline{x}), and U_{1} := h_{\overline{x}_{1}}^{−1}U, (where h_{\overline{x}_{1}} : S_{1} := 𝔸_{S}^{1}(\overline{x}_{1}) → S is the morphism induced by h). We have S_{1}\U_{1} = h_{\overline{x}_{1}}^{−1}(S\U), and since flat maps are generizing ([126, Th.9.5]) we easily see that the codimension of S_{1} \U_{1} in S_{1} equals the codimension of S\U in S. From our inductive assumption, we deduce that the base change functor Cov(U_{1}) → Cov(f −1 ; \overline{x}U) is essentially surjective, and hence it suffices to show that the same holds for the functor Cov(U) → Cov(U_{1}). Thus, we are reduced to the case where X = 𝔸_{S}^{1}. Suppose now, that E → f −1 ; \overline{x}U is a finite étale morphism; we can write f\overline{x} as the limit of a cofiltered family of smooth morphisms (f_{λ} : Y_{λ} → S | λ ∈ Λ), where each Y_{λ} is an affine étale 𝔸_{S}^{1}-scheme. Then f −1 ; \overline{x}U is the limit of the family (Y_{λ} ×_{S} U | λ ∈ Λ). By [65, Ch.IV, Th.8.8.2(ii), Th.8.10.5] and [66, Ch.IV, Prop.17.7.8], we may find a λ ∈ Λ, a finite étale morphism E_{λ} → Y_{λ} ×_{S} U and an isomorphism of f −1 ; \overline{x}U-schemes : E_{λ} ×_{Y_{λ}} 𝔸_{S}^{1}(\overline{x}) →^{∼} E. Denote by y ∈ Y_{λ} the image of the closed point of 𝔸_{S}^{1}(\overline{x}), and by \overline{y} the geometric point of Y_{λ} obtained as the image of \overline{x} (the latter is viewed naturally as a geometric point of 𝔸_{S}^{1}(\overline{x})); by construction, y lies in the closed fibre Y_{0} := Y_{λ} ×_{S} Spec κ(s), which is an étale 𝔸_{κ(s)}^{1}-scheme, and we may therefore find a specialization z ∈ Y_{0} of y, with z a closed point. Pick a geometric point \overline{z} of Y_{0} localized at z, and a strict specialization map Y_{λ}(\overline{z}) → Y_{λ}(\overline{y}) as in (4.9.23); there follows a commutative diagram :

\xymatrix{𝔸_{S}^{1}(\overline{x}) ≃ Y_{λ}(\overline{y}) \ar[r] \ar[d] & Y_{λ}(\overline{z}) \ar[d] \\ Y_{λ}(y) \ar[r] & Y_{λ}(z).}

The finite étale covering E_{λ} ×_{Y_{λ}} Y_{λ}(y) → Y_{λ}(y) ×_{S} U lies in the essential image of the functor

Cov(Y_{λ}(z) ×_{S} U) → Cov(Y_{λ}(y) ×_{S} U) C ↦ C ×_{Y_{λ}(z)} Y_{λ}(y).

It follows that E → f −1 ; \overline{x}U lies in the essential image of the functor

Cov(f_{λ,\overline{z}}^{−1}U) → Cov(f −1 ; \overline{x}U) C ↦ C ×_{Y (\overline{z})} Y (\overline{y}) ≃ C ×_{Y (\overline{z})} 𝔸_{S}^{1}(\overline{x})

and therefore it suffices to show that the pull-back functor Cov(U) → Cov(f −1 ; \overline{z}U) is essentially surjective. In other words, we may replace x by z, and assume throughout that x is a closed point of 𝔸_{S}^{1}.

Claim 13.2.3. Under the current assumptions, we may find a strictly local normal scheme T, with closed point t, a finite surjective morphism g : T → S, and a finite morphism of Spec κ(s)- schemes :

Spec κ(t) → Spec κ(x).

Proof of the claim. Since κ(x) is a finite extension of κ(s), it is generated by finitely many algebraic elements u_{1}, … u_{n}, and an easy induction allows to assume that n = 1. In this case, one constructs first a scheme T′ by taking any lifting of the minimal polynomial of u_{1} : for the details, see e.g. [61, Ch.0, (10.3.1.2.)], which shows that the resulting T is local, finite and flat over S, so T′ maps surjectively onto S. Next, we may replace T′ by its maximal reduced subscheme, which is still strictly local and finite over S. Next, since S is excellent, the normalization (T′)^{ν} of T′ is finite over S ([64, Ch.IV, Scholie 7.8.3(vi)]); let T be any irreducible component of (T′)^{ν}; by [66, Ch.IV, Prop.18.8.10], T fulfills all the sought conditions. ♢

Choose g : T → S as in claim 13.2.3; since the residue field extension κ(s) → κ(t) is algebraic and purely inseparable, there exists a unique point x′ ∈ 𝔸_{S}^{1}(\overline{x}) ×_{S} T lying over t, and we may find a unique strict geometric point \overline{x}′ of 𝔸_{S}^{1}(\overline{x}) ×_{S} T localized at x′, and lying over \overline{x}. In view of [66, Ch.IV, Prop.18.8.10], there follows a natural isomorphism of T -schemes :

𝔸_{S}^{1}(\overline{x}) ×_{S} T →^{∼} 𝔸_{T}^{1} (\overline{x}′).

Denote by f\overline{x}′ := f\overline{x} ×_{S} T : 𝔸_{T}^{1} (\overline{x}′) → T the natural projection, and set U_{T} := g^{−1}U; since the morphism g : T → S is generizing ([126, Th.9.4(ii)]), it is easily seen that T \U_{T} has codimension ≥ 2 in T.

Let F : Cov → Sch be the fibred category (13.1.1). We have a natural essentially commutative diagram of categories :

(13.2.4)  \xymatrix{Cov(U) \ar[r] \ar[d] & Desc(F, g ×_{S} U) \ar[d]^{δ} \\ Cov(f −1 ; \overline{x}U) \ar[r] & Desc(F, g ×_{S} f −1 ; \overline{x}U)}

where, for any morphism of schemes h, we have denoted by Desc(F, h) the category of descent data for the fibred category F, relative to the morphism h.

According to lemma 13.1.2, the morphism g is of universal 2-descent for the fibred category F, so the horizontal arrows in (13.2.4) are equivalences. Hence, the theorem will follow, once we know that δ is essentially surjective. However, we have :

Claim 13.2.5. (i) Set U′_{T} := U_{T} ×_{S} T and U′′_{T} := U′_{T} ×_{S} T. The pull-back functors :

Cov(U′_{T}) → Cov(𝔸_{T}^{1} (\overline{x}′) ×_{T} U′_{T}) Cov(U′′_{T}) → Cov(𝔸_{T}^{1} (\overline{x}′) ×_{T} U′′_{T})

are fully faithful.

(ii) Suppose that the pull-back functor

Cov(U_{T}) → Cov(f_{\overline{x}′}^{−1}U_{T})

is essentially surjective. Then the same holds for the functor δ.

Proof of the claim. (i): Let \overline{z}′′ be any geometric point of X′′ := 𝔸_{T}^{1} ×_{S} T ×_{S} T whose strict image in 𝔸_{T}^{1} is \overline{x}′, and let \overline{z}′ be the image of \overline{z}′′ in X′ := 𝔸_{T}^{1} ×_{S} T; by lemma 13.1.22(ii), the natural morphisms :

X′(\overline{z}′) → 𝔸_{T}^{1} (\overline{x}′) ×_{S} T X′′(\overline{z}′′) → 𝔸^{1}(\overline{x}′) ×_{S} T ×_{S} T

are isomorphisms (notice that T ×_{S} T is also strictly local). Then the claim follows from assertion (i) of the theorem, applied to the projections X′ → T ×_{S} T and X′′ → T ×_{S} T ×_{S} T.

(ii): Recall that an object of Desc(F, g ×_{S} f_{\overline{x}′}^{−1}U) consists of a finite étale morphism E′_{T} → f_{\overline{x}′}^{−1}U_{T} and a X′-isomorphism β′ : E′_{T} ×_{S} T →^{∼} T ×_{S} E′_{T} fulfilling a cocycle condition on E ×_{S} T ×_{S} T. By assumption, E_{T} descends to a finite étale morphism E_{T} → U_{T}; then (i) implies that β′ descends to a U′_{T} -isomorphism β : E_{T} ×_{S} T →^{∼} T ×_{S} E_{T}, and the cocycle identity for β′ descends to a cocycle identity for β. ♢

In view of claim 13.2.5, we may replace (S, U, x) by (T, U_{T} , x′), and therefore assume that x is a κ(s)-rational point of 𝔸_{κ(s)}^{1}. In this case, any choice of coordinate t on 𝔸_{S}^{1} yields a section σ\overline{x} : S → 𝔸_{S}^{1}(\overline{x}) of the natural projection, such that σ\overline{x}(s) = x. To conclude the proof of the theorem, it suffices to show that the pull-back functor :

Cov(f −1 ; \overline{x}U) ⟶^{σ∗ ; \overline{x}} Cov(U)

is fully faithful.

Say that S = Spec A; then the scheme 𝔸_{S}^{1}(\overline{x}) is the spectrum of A\{t\}, the henselization of A[t] along the ideal 𝔪{t} generated by t and the maximal ideal 𝔪 of A. Let A^{∧} (resp. A{t}^{∧}) be the 𝔪-adic (resp. 𝔪{t}-adic) completion of A (resp. of A{t}), and notice the natural isomorphism:

A{t}^{∧}/tA{t}^{∧} →^{∼} A^{∧}

(indeed, it is easy to check that A{t}^{∧} ≃ A^{∧}[[t]]), whence a natural diagram of schemes :

\xymatrix{X^{∧} := Spec A\{t\}^{∧} \ar[r]^{g′} \ar[d]_{π} & Spec A\{t\} \ar[d]_{f\overline{x}} \\ S^{∧} := Spec A^{∧} \ar[r]^{g} \ar[u]_{σ} & Spec A \ar[u]_{σ\overline{x}}}

(where π is the natural projection) whose horizontal arrows commute with both the downward arrows and the upward ones. Set U^{∧} := g^{−1}U; by example 13.1.33(ii) and lemma 13.1.15(ii), the pull-back functors

g^{∗} : Cov(U) → Cov(U^{∧}) g^{′∗} : Cov(f −1 ; \overline{x}U) → Cov(π^{−1}U^{∧})

are fully faithful. Consequently, we are easily reduced to showing that the pull-back functor : Cov(π^{−1}U^{∧}) ⟶^{σ^{∗}} Cov(U^{∧}) is an equivalence. The latter holds by lemma 13.1.39(ii). □

Example 13.2.6. As an application of theorem 13.2.2, suppose that K ⊂ E is an extension of separably closed fields, V_{K} a geometrically normal and strictly local K-scheme, U ⊂ V_{K} an open subset, and ξ a geometric point of V_{E} := V_{K} ×_{K} E, whose image in V_{K} is supported on the closed point. Then the induced functor

Cov(U) → Cov(U ×_{V_{K}} V_{E}(ξ))

is fully faithful, and it is an equivalence in case V_{K} is excellent and V_{K} \U has codimension ≥ 2 in V_{K}. Indeed, let K^{a} (resp. E^{a}) be an algebraic closure of K (resp. E), and choose a homomorphism K^{a} → E^{a} extending the inclusion of K into E. Then both V_{K^{a}} := V_{K} ×_{K} K^{a} and V_{E^{a}}(ξ) := V_{E}(ξ) ×_{E} E^{a} are still normal and strictly local (lemma 13.1.22(ii)), and the induced functors

Cov(U) → Cov(U ×_{K} K^{a}) Cov(U ×_{V_{K}} V_{E}(ξ)) → Cov(U ×_{V_{K}} V_{E^{a}}(ξ))

are equivalences (lemma 13.1.7(i)). It then suffices to show that the induced functor

Cov(U ×_{K} K^{a}) → Cov(U ×_{V_{K}} V_{E^{a}}(ξ))

has the asserted properties. Hence, we may replace K by K^{a} and E by E^{a}, and assume from start that K ⊂ E is an extension of algebraically closed fields. In this case, E can be written as the colimit of a filtered family (R_{λ} | λ ∈ Λ) of smooth K-algebras; correspondingly, V_{E} is the limit of a cofiltered system (V_{λ} | λ ∈ Λ) of smooth V_{K}-schemes, and – by lemma 13.1.6 – Cov(U ×_{V_{K}} V_{E}(ξ)) is the 2-colimit of the system of categories

Cov(U ×_{V_{K}} V_{λ}(ξ_{λ})) (λ ∈ Λ)

(where, for each λ ∈ Λ, we denote by ξ_{λ} the image of ξ in V_{λ}). Now the contention follows directly from theorem 13.2.2.

Theorem 13.2.7. Let f : X → S is a smooth morphism of schemes, 𝕃 ⊂ ℕ be a set of primes, and suppose that all the elements of 𝕃 are invertible in 𝒪_{S}. Then f is 1-aspherical for 𝕃.

Proof. Let \overline{x} be any geometric point of X, \overline{s} := f(\overline{x}), and \overline{η} a strict geometric point of S(\overline{s}). To ease notation, set T := X(\overline{x}), let f\overline{x} : T → S(\overline{s}) be the natural map, and T\overline{η} := f −1 ; \overline{x}(\overline{η}); we have to show that H^{1}(T_{\overline{η},ét}, G) = \{1\} for every 𝕃-group G. Arguing as in the proof of proposition 13.1.30, we reduce to the case where S = S(\overline{s}). Then, by lemma 13.1.24, we can further assume that S is normal and \overline{η} is localized at the generic point η of S. By lemma 13.1.29, S is the limit of a cofiltered system (S_{λ} | λ ∈ Λ) of strictly local, normal and excellent schemes, with dominant transition maps, and as usual, after replacing Λ by a coinitial subset, we may assume that f (resp. \overline{η}) descends to a compatible system of morphisms (f_{λ} : X_{λ} → S_{λ} | λ ∈ Λ), (resp. of strict geometric points \overline{η}_{λ} localized at the generic point of S_{λ}). By [66, Ch.IV, Prop.17.7.8(ii)], there exists λ ∈ Λ such that f_{μ} is smooth for every μ ≥ λ. Then, in view of [9, Exp.VII, Rem.5.14] and the isomorphism (13.1.28), we may replace f by f_{λ}, and \overline{η} by \overline{η}_{λ}, and assume from start that S is strictly local, normal and excellent, and G is a finite 𝕃-group.

Let (φ : E\overline{η} → T\overline{η}, ρ : G → Aut_{T\overline{η}}(E\overline{η})) be a principal G-homogeneous space; we come down to showing that E\overline{η} has a section T\overline{η} → E\overline{η}. By [66, Ch.IV, Prop.17.7.8(ii)] and [65, Ch.IV, Th.8.8.2(ii), Th.8.10.5], we may find a finite separable extension κ(η) ⊂ L, and a principal Ghomogeneous space

φ_{L} : E_{L} → T_{L} := T ×_{S} Spec L ρ_{L} : G → Aut_{T_{L}}(E_{L})

such that

φ = φ_{L} ×_{Spec L} Spec κ(\overline{η}) ρ = ρ_{L} ×_{Spec L} Spec κ(\overline{η}).

Say that S = Spec A, denote by A_{L} the normalization of A in L, and set S_{L} := Spec A_{L}. Then S_{L} is again normal and excellent ([64, Ch.IV, (7.8.3)(ii,vi)]), and the residue field of A_{L} is an algebraic extension of the residue field of A, hence it is separably closed, so S_{L} is strictly local as well. Thus, we may replace S by S_{L}, and assume that E\overline{η} descends to a principal G-homogeneous space E_{η} → T_{η} := f −1 ; \overline{x}(η) on T_{η}. Next, we may write η as the limit of the filtered system of affine open subsets of S, so that – by the same arguments – we find an affine open subset U ⊂ S and a principal G-homogeneous space E_{U} → T_{U} := f −1 ; \overline{x}U, with a G-equivariant isomorphism of T_{η}-schemes : E_{U} ×_{T_{U}} T_{η} →^{∼} E_{η}. Denote by D_{1}, …, D_{n} the irreducible components of S\U which have codimension one in S, and for every i ≤ n, endow D_{i} with its reduced closed subscheme structure, and set D′_{i} := D_{i} ×_{S} T. Let also η_{T} be the generic point of T.

Claim 13.2.8. For given i ≤ n, let y be the generic point of D_{i}. We have:

(i) T and E_{U} are noetherian normal schemes, and T (y) := T ×_{S} S(y) is regular.

(ii) D′_{i} is an integral scheme of pure codimension one in T, and D′(y) := D′_{i} ×_{D_{i}} D_{i}(y) is

regular (and irreducible). (iii) Let also z ∈ D′_{i} be the generic point, and 𝔪_{y} (resp. 𝔪_{z}) the maximal ideal of 𝒪_{S,y} (resp.

of 𝒪_{T,z}); then 𝒪_{S,y} and 𝒪_{T,z} are discrete valuation rings, and 𝔪_{y} · 𝒪_{T,z} = 𝔪_{z}. (iv) Let t ∈ A be any element such that t · 𝒪_{S,y} = 𝔪_{y}. Then there exist an integer m > 0

such that (m, char κ(s)) = 1, a finite étale covering

E_{y} → T (y)[t^{1/m}] := T (y) ×_{S} Spec A[T]/(T^{m} − t)

and an isomorphism of T (y)[t^{1/m}] ×_{T} T_{U}-schemes :

E_{y} ×_{T} T_{U} →^{∼} E_{U} ×_{T} T (y)[t^{1/m}].

Proof of the claim. (i): Since S is normal by assumption, the assertion for T and E_{U} follows from [66, Ch.IV, Prop.17.5.7, Prop.18.8.12(i), Prop.18.8.8(iv)]. Let u ∈ T (y) be any point; denote by w ∈ X the image of u. Since 𝒪_{S,y} is a discrete valuation ring, 𝒪_{X,w} is a regular local ring ([66, Ch.IV, Prop.17.5.8(iii)]); moreover, the natural map 𝒪_{X,w} → 𝒪_{T (y),u} induces isomorphisms on strict henselizations, therefore 𝒪_{T (y),u} is regular ([66, Ch.IV, Prop.18.8.13]).

(ii): By virtue of proposition 13.1.30(ii), the geometric generic fibre of the induced morphism D′_{i} → D_{i} is connected, hence the same holds for its generic fibre. But arguing as in the proof of (i), we see that this generic fibre is noetherian and regular, so it is irreducible and reduced. Since moreover the morphism D′_{i} → D_{i} is flat, it follows easily that D′_{i} is irreducible and reduced as well. Lastly, D′_{i} is of pure codimension one in T, by virtue of [64, Ch.IV, Cor.6.1.4].

(iii): From (i) and (ii) we see that 𝒪_{S,y} and 𝒪_{T,z} are discrete valuation rings; since moreover D′_{i} is reduced, the same holds for 𝒪_{T,z}/𝔪_{y}𝒪_{T,z}, so 𝔪_{y} · 𝒪_{T,z} = 𝔪_{z}.

(iv): By (iii) we have t·𝒪_{T,z} = 𝔪_{z}. Notice that T (z)×_{T} T_{U} = T (η_{T}), and E_{η_{T}} := E_{U} ×_{T} T (z) is a disjoint union of spectra of finite separable extensions L_{1}, …, L_{k} of κ(η_{T}). Moreover, E_{η_{T}} is a principal G-homogeneous space over T (η_{T}), i.e. every L_{j} is a Galois extension of κ(η_{T}), with Galois group G_{j} := Gal(L_{j}/κ(η_{T})) ⊂ G. Since G is an 𝕃-group, the same holds for G_{j}, hence E_{U} ×_{T} T (y) is tamely ramified along the divisor \overline{\{z\}} (the topological closure of \{z\} in T (y)), and the assertion follows from Abhyankar’s lemma [82, Exp.XIII, Prop.5.2]. ♢

Claim 13.2.9. There exist :

(a) a finite dominant morphism S′ → S, such that both S′ and T′ := T ×_{S} S′ are strictly local and normal;

(b) an open subset U′ ⊂ S′, such that S′\U′ has codimension ≥ 2 in S′;

(c) a finite étale morphism E′ → T′_{U′} := T ×_{S} U′, with an isomorphism of T′_{U′}-schemes :

E′ ×_{T} T_{η} ≃ E_{η} ×_{T} T′_{U′}.

Proof of the claim. For every i ≤ n, let y_{i} be the maximal point of D_{i}, and choose t_{i} ∈ A whose image in 𝒪_{S,y_{i}} generates the maximal ideal. Choose also m_{i} ∈ ℕ with (m_{i}, char κ(s)) = 1 and such that there exists a finite étale covering E_{i} → T (y_{i})[t_{i}^{1/m_{i}}] extending the étale covering E_{U} ×_{T} T (y_{i})[t_{i}^{1/m_{i}}] (claim 13.2.8(iv)). Fix an algebraic closure K of Frac A, pick for each i = 1, …, n an m_{i}-th root τ_{i} of t_{i} in K, and let S′ be the normalization of Spec A[τ_{1}, …, τ_{n}]. Then S′ is finite over S, hence it is excellent ([64, Ch.IV, (7.8.3)(ii,vi)]), and strictly local (cp. the proof of lemma 13.1.24). Set E′_{η} := E_{η} ×_{S} S′, T′ := T ×_{S} S′; since the geometric fibres of f\overline{x} are connected (proposition 13.1.30(ii)), the same holds for the geometric fibres of the induced morphism T′ → S′, therefore T′ is connected, and then it is also strictly local, by the usual arguments. Notice also that T′ is the limit of a cofiltered family of smooth S′-schemes, hence it is reduced and normal ([66, Ch.IV, Prop.17.5.7]). Say that E′_{η} = Spec C, T = Spec B, T′ = Spec B′, and let C′ be the integral closure of B′ in C. Notice that C ⊗_{B} κ(η_{T}) is a finite product of finite separable extensions of the field B′ ⊗_{B} κ(η_{T}), and consequently the natural morphism φ′ : E_{T′} := Spec C′ → T′ is finite ([126, §33, Lemma 1]). Define :

U′ := {y ∈ S′ | φ′ ×_{S′} S′(y) : E_{T′}(y) → T′(y) is étale}.

Let now y ∈ U′ any point; then S′(y) is the limit of the cofiltered family (U_{λ} | λ ∈ Λ) of affine open neighborhoods of y in S′, and φ′ ×_{S′} S′(y) the limit of the system of morphisms (φ′_{λ} := φ′ ×_{S′} U_{λ} | λ ∈ Λ); we may then find λ ∈ Λ such that φ′_{λ} is étale ([66, Ch.IV, Prop.17.7.8(ii)]), hence U_{λ} ⊂ U′, which shows that U′ is open. Furthermore, from [66, Ch.IV, Prop.17.5.7] it follows that E_{U} ×_{T} T′ is normal, whence an isomorphism of T′-schemes :

E_{T′} ×_{S} U ≃ E_{U} ×_{T} T′

(cp. the proof of lemma 13.1.7(iii)) especially, U ×_{S} S′ ⊂ U′. Likewise, by construction we have natural morphisms : T′(y_{i}) → T (y_{i})[t_{i}^{1/m_{i}}], and using the fact that all the schemes in view are normal we deduce isomorphisms of T′(y_{i})-schemes :

E_{T′}(y_{i}) →^{∼} E_{i} ×_{T (y_{i})[t_{i}^{1/m_{i}}]} T′(y_{i}).

Thus, U′ contains all the points of S′ of codimension ≤ 1, since the image in S of any such point lies in U ∪ {y_{1}, …, y_{n}}. The morphism E′ := E_{T′} ×_{S′} U′ → T′_{U′} fulfills conditions (a)-(c). ♢

Now, choose S′ → S, U′ ⊂ S′, and E′ → T′_{U′} as in claim 13.2.9; since the corresponding T′ is local, there exists a unique point x′ ∈ X′ := X ×_{S} S′ lying over x; pick a geometric point \overline{x}′ of X′ localized at x′, and lying over \overline{x}; it then follows from [66, Ch.IV, Prop.18.8.10] that the natural morphism X′(\overline{x}′) → T′ is an isomorphism. In such situation, theorem 13.2.2 says that there exists a finite étale covering E → U′ with an isomorphism of T′_{U′}-schemes : E ×_{U′} T′_{U′} →^{∼} E′, whence an isomorphism of T\overline{η}-schemes :

E\overline{η} ≃ E(\overline{η}) ×_{Spec κ(\overline{η})} T\overline{η}.

Since κ(\overline{η}) is separably closed, the étale morphism E(\overline{η}) → Spec κ(\overline{η}) admits a section, hence the same holds for φ, as claimed. □

13.2.10. Let f : X → S be a morphism of schemes, and j : U ⊂ X an open immersion, and set U_{η} := U ∩ f^{−1}(η) for every η ∈ S_{max}, where S_{max} ⊂ S denotes the subset of all maximal points of S. We deduce a natural essentially commutative diagram of functors :

𝒟(S, f, U) : \xymatrix{Cov(X) \ar[r]^{j^{∗}} \ar[d]_{∏_{η} ι_{η}^{∗}} & Cov(U) \ar[d]^{∏_{η} ι_{η|U}^{∗}} \\ ∏_{η∈S_{max}} Cov(f^{−1}η) \ar[r]^{∏_{η} j_{η}^{∗}} & ∏_{η∈S_{max}} Cov(U_{η})}

where j_{η} : U_{η} → f^{−1}(η) is the restriction of j and ι_{η} : f^{−1}(η) → X is the natural immersion. Let us say that U ⊂ X is fibrewise dense, if f^{−1}(s) ∩ U is dense in f^{−1}(s), for every s ∈ S. Then we have :

Theorem 13.2.11. In the situation of (13.2.10), suppose that f is smooth, and U is fibrewise dense. The following holds :

(i) The restriction functor j^{∗} is fully faithful. (ii) The diagram 𝒟(S, f, U) is 2-cartesian. (iii) If furthermore, f^{−1}S_{max} ⊂ U, then j^{∗} is an equivalence.

Proof. Assertion (ii) means that the functors j^{∗} and ι_{η}^{∗} induce an equivalence (j, ι_{•})^{∗} from Cov(X) to the category 𝒞 (X, U) of data

(13.2.12)  \underline{E} := (φ, (ψ_{η}, α_{η} | η ∈ S_{max}))

where φ (resp. ψ_{η}) is an object of Cov(U) (resp. of Cov(f^{−1}η), for every η ∈ S_{max}), and α_{η} : φ ×_{U} Spec κ(η) →^{∼} ψ_{η} ×_{f^{−1}η} U_{η} is an isomorphism of U-schemes, for every η ∈ S_{max} (see example 3.3.12(ii)). On the basis of this description, it is easily seen that (i),(ii)⇒(iii). Furthermore, we remark :

Claim 13.2.13. (i) If j^{∗} is fully faithful, then the same holds for (j, ι_{•})^{∗}.

(ii) For every open subset U′ ⊂ X containing U, suppose that : (a) The pull-back functor Cov(U′) → Cov(U) is fully faithful.

(b) If f^{−1}S_{max} ⊂ U′, the pull-back functor Cov(X) → Cov(U′) is an equivalence. Then assertion (ii) holds.

Proof of the claim. (i): Since f^{−1}η is a normal (even regular) scheme ([66, Ch.IV, Prop.17.5.7]), the pull-back functors ι_{η}^{∗} are fully faithful (lemma 13.1.7(iii)); the assertion is an immediate consequence.

(ii): In light of (i), it remains only to check that (j, ι_{•})^{∗} is essentially surjective. Thus, let φ : E → U be a finite étale morphism, such that i_{η}^{∗}φ extends to a finite étale morphism φ′_{η} : E′_{η} → f^{−1}(η), for every maximal point η ∈ S. By claim 13.1.8, there is a largest open subset U_{max} containing U, over which φ extends to a finite étale morphism φ_{max}. To conclude, we have to show that U_{max} = X. However, for any maximal point η, let i_{η} : f^{−1}(η) → X(η) be the natural closed immersion. By lemma 13.1.7(i), i_{η}^{∗} is an equivalence, hence we may find a finite étale morphism φ′_{(η)} : E′(η) → X(η) such that i_{η}^{∗}φ′_{(η)} ≃ φ′_{η}. By the same token, we also see that E′(η) ×_{X(η)} U(η) is U(η)-isomorphic to E ×_{U} U(η).

Next, S(η) is the limit of the filtered system 𝒱 of all open subsets V ⊂ S with η ∈ V, hence lemma 13.1.6ensures that we may find V ∈ 𝒱 and an object φ′_{V} : E′_{V} → f^{−1}V of Cov(f^{−1}V) such that φ′_{V} ×_{V} S(η) ≃ φ′_{(η)}, and after shrinking V, we may also assume (again by lemma 13.1.6) that E′_{V} ×_{X} U is U-isomorphic to E ×_{S} V. Hence we may glue E′ and E along the common intersection, to deduce a finite étale morphism E′ → U′ := U ∪ f^{−1}V that extends φ. It follows that f^{−1}(η) ⊂ f^{−1}V ⊂ U_{max}. Since η is arbitrary, (b) implies that the pull-back functor Cov(X) → Cov(U_{max}) is an equivalence, especially φ lies in the essential image of j^{∗}, as claimed. ♢

Claim 13.2.14. (i) Suppose that S is noetherian and normal, and X is separated. Then (i) holds. (ii) If furthermore, S is also excellent, then (ii) holds as well.

Proof of the claim. (i): Under the assumptions of the claim, X is normal and noetherian ([66, Ch.IV, Prop.17.5.7]), so (i) follows from lemma 13.1.7(iii), which also says – more generally – that assumption (a) of claim 13.2.13(ii) holds in this case, hence in order to show (ii) it suffices to check that assumption (b) of claim 13.2.13(ii) holds whenever U ∪ f^{−1}S_{max} ⊂ U′ ⊂ X, especially X\U′ has codimension ≥ 2 in X. Suppose first that S is regular; then the same holds for X ([66, Ch.IV, Prop.17.5.8]), and the contention follows from lemma 13.1.7(iv).

In the general case, let S_{reg} ⊂ S be the regular locus, which is open since S is excellent, and contains all the points of codimension ≤ 1, by Serre’s normality criterion ([64, Ch.IV, Th.5.8.6]). Consider the restriction f^{−1}S_{reg} → S_{reg} of f, and the fibrewise dense open immersion j_{reg} : U′ ∩ f^{−1}S_{reg} ⊂ f^{−1}S_{reg}; by the foregoing, the functor j_{reg}^{∗} is an equivalence, hence we are easily reduced to showing that the functor Cov(X) → Cov(U′ ∪ f^{−1}S_{reg}) is an equivalence, i.e. we may assume that V := f^{−1}S_{reg} ⊂ U′. Moreover, since the full faithfulness of j^{∗} is already known, we only need to show that any finite étale morphism φ : E → U′ extends to an object of Cov(X). To this aim, by lemma 13.1.7(iii), it suffices to prove that φ×_{U′} (X(\overline{x})×_{X} U′) extends to an object of Cov(X(\overline{x})), for every geometric point \overline{x} of X. Let \overline{s} := f(\overline{x}), and denote by s ∈ S the support of \overline{s}; by assumption, we may find a geometric point ξ of f^{−1}(s), whose support lies in U′ ∩ f^{−1}(s), and a strict specialization morphism X(ξ) → X(\overline{x}). There follows an essentially commutative diagram :

\xymatrix{Cov(X(\overline{x})) \ar[r]^{ρ} \ar[d]_{δ} & Cov(X(\overline{x}) ×_{X} V) \ar[d]^{γ} & Cov(S(\overline{s}) ×_{S} S_{reg}) \ar[l]_{α} \ar[dl]^{β} \\ Cov(X(ξ)) \ar[r]^{τ} & Cov(X(ξ) ×_{X} V)}

where α and β are both equivalences, by theorem 13.2.2(ii); hence γ is an equivalence as well. Moreover, both Cov(X(\overline{x})) and Cov(X(ξ)) are equivalent to the category of finite sets, and δ is obviously an equivalence. By construction, γ(φ ×_{U′} (X(ξ) ×_{X} V)) lies in the essential image of τ, hence φ ×_{U′} (X(ξ) ×_{X} V) lies in the essential image of ρ, so say it is isomorphic to ρ(φ′) for some object φ′ of Cov(X(\overline{x})). Using (i) (and [66, Ch.IV, Prop.18.8.12]) one checks easily that φ′ ×_{X} U′ ≃ φ ×_{U′} (X(ξ) ×_{X} U′), whence the contention. ♢

Claim 13.2.15. Let m ∈ ℕ be any integer. Assertions (i) and (ii) hold if S and X are affine schemes of finite type over Spec ℤ, the fibres of f have pure dimension m, and furthermore :

(13.2.16)  dim f^{−1}(s)\U < m  for every s ∈ S.

Proof of the claim. Indeed, in this situation, S admits finitely many maximal points, hence the normalization morphism S^{ν} → S is integral and surjective. Set :

S_{2} := S^{ν} ×_{S} S^{ν}  U_{1} := U ×_{S} S^{ν}  U_{2} := U ×_{S} S_{2}.

Let β : X_{1} := X ×_{S} S^{ν} → X, f_{1} : X_{1} → S^{ν} and j_{2} : U_{2} → X_{2} := X ×_{S} S_{2} be the induced morphisms; clearly f_{1}^{−1}(s′) has pure dimension m for every s′ ∈ S^{ν}, and from (13.2.16) we deduce that dim f_{1}^{−1}(s′)\U_{1} < m, especially, U_{1} is dense in every fibre of f_{1}; by the same token, U_{2} is dense in X_{2}. Then j_{2}^{∗} is faithful (lemma 13.1.7(ii)), and lemma 13.1.2and corollary 3.5.30(ii) imply that j^{∗} is fully faithful, provided the same holds for the functor j_{1}^{∗} : Cov(X_{1}) → Cov(U_{1}). In other words, in order to prove assertion (i), we may replace (f, U) by (f_{1}, U_{1}), which allows to assume that S is an affine normal scheme, and then it suffices to invoke claim 13.2.14, to conclude.

Concerning assertion (ii) : by the foregoing, we already know that j^{∗} is fully faithful, so the same holds for (j, ι_{•})^{∗} (claim 13.2.13(i)). To show that (j, ι_{•})^{∗} is essentially surjective, let \underline{E} be an object as in (13.2.12) of the category 𝒞 (X, U); the normalization morphism induces a bijection S_{max}^{ν} →^{∼} S_{max} : η^{ν} ↦ η, and clearly κ(η^{ν}) = κ(η) for every η ∈ S_{max}, whence a datum

\underline{E}^{ν} := (φ_{1} := φ ×_{U} U_{1}, (ψ_{η}, α_{η} | η^{ν} ∈ S_{max}^{ν}))

of the analogous category 𝒞 (X_{1}, U_{1}); by claim 13.2.14(ii), we may find φ′_{1} ∈ Ob(Cov(X_{1})) and an isomorphism α : φ′_{1} ×_{X_{1}} U_{1} →^{∼} φ_{1}. Let U_{3} := U_{2} ×_{U} U_{1}, and denote by j_{3} : U_{3} → X_{3} := X_{2} ×_{X} X_{1} the natural open immersion; by the foregoing, we know already that both j_{2}^{∗} and j_{3}^{∗} are fully faithful; then corollary 3.5.30(iii) says that the natural essentially commutative diagram :

\xymatrix{Desc(Cov, β) \ar[r] \ar[d] & Desc(Cov, β ×_{X} U) \ar[d] \\ Cov(X_{1}) \ar[r] & Cov(U_{1})}

is 2-cartesian. Thus, let ρ : Cov(U) → Desc(Cov, β ×_{X} U) be the functor defined in (3.5.22); it follows that the datum (φ′_{1}, ρ(φ), α) comes from a descent datum (φ′_{1}, ω) in Desc(Cov, β). By lemma 13.1.2, the latter descends to an object φ′ of Cov(X), and by construction we have (j, ι_{•})^{∗}φ′ = \underline{E}, as required. ♢

Next, we consider assertions (i) and (ii) in case where both X and S are affine. We may find an affine open covering X = V_{0} ∪ ⋯ ∪ V_{n} such that the fibres of f_{|V_{i}} : V_{i} → fV_{i} are of pure dimension i, for every i = 0, …, n ([66, Ch.IV, Prop.17.10.2]). For i = 0, …, n, let j_{i} : V_{i} ∩ U → V_{i} be the induced open immersion; we have natural equivalences of categories :

Cov(X) →^{∼} ∏_{i=0}^{n} Cov(V_{i}) Cov(U) →^{∼} ∏_{i=0}^{n} Cov(V_{i} ∩ U)

which induce a natural identification : j^{∗} = j_{0}^{∗} × ⋯ × j_{n}^{∗}. It follows j^{∗} is fully faithful if and only if the same holds for every j_{i}^{∗}, and moreover 𝒟(S, f, U) decomposes as a product of n diagrams 𝒟(S, f_{|V_{i}}, U ∩ V_{i}). Hence we may replace f by f_{|V_{m}}, for any m ≤ n, after which we may also assume that all the fibres f have the same pure dimension m. In that case, notice that the assumption on U is equivalent to (13.2.16). Next, say that X = Spec A, and let I ⊂ A be an ideal such that V (I) = X \U; we may write I as the union of the filtered family (I_{λ} | λ ∈ Λ) of its finitely generated subideals. Set U_{λ} := X\V (I_{λ}) for every λ ∈ Λ; it follows that U = ⋃_{λ∈Λ} U_{λ}. For every λ ∈ Λ, set

Z_{λ} := {s ∈ S | dim f^{−1}(s)\U_{λ} < m}.

Claim 13.2.17. (i) Z_{λ} is a constructible subset of S, for every λ ∈ Λ.

(ii) We have Z_{λ} ⊂ Z_{μ} whenever μ ≥ λ, and moreover S = ⋃_{λ∈Λ} Z_{λ}.

Proof of the claim. (i): Let V_{λ} := {x ∈ X | dim_{x} f^{−1}(f(x)) \ U_{λ} = m}; according to [65, Ch.IV, Prop.9.9.1], every V_{λ} is a constructible subset of X, hence f(V_{λ}) is a constructible subset of S ([63, Ch.IV, Th.1.8.4]), so the same holds for Z_{λ} = S\f(V_{λ}).

(ii): Let μ, λ ∈ Λ, such that μ ≥ λ; then it is clear that f^{−1}(s)\U_{λ} ⊂ f^{−1}(s)\U_{μ} for every s ∈ S; using (13.2.16), the claim follows easily. ♢

Claim 13.2.17 and [63, Ch.IV, Cor.1.9.9] imply that Z_{λ} = S for every sufficiently large λ ∈ Λ. Hence, after replacing Λ by a cofinal subset, we may assume that all the open subsets U_{λ} are fibrewise dense. We have a natural essentially commutative diagram :

\xymatrix{Cov(U) \ar[r] \ar[d] & 2-lim_{λ∈Λ} Cov(U_{λ}) \ar[d] \\ ∏_{η∈S_{max}} Cov(U_{η}) \ar[r] & ∏_{η∈S_{max}} 2-lim_{λ∈Λ} Cov(f^{−1}(η) ∩ U_{λ})}

whose horizontal arrows are equivalences (notation of definition 2.5.1(i)); it follows formally that j^{∗} is fully faithful, provided the same holds for all the pull-back functors Cov(X) → Cov(U_{λ}), and likewise, 𝒟(S, f, U) is 2-cartesian, provided the same holds for all the diagrams 𝒟(S, f, U_{λ}). Hence, we may replace U by U_{λ}, and assume that U is constructible, and (13.2.16) still holds.

Next, we may write S as the limit of a cofiltered family (S_{λ} | λ ∈ Λ) of affine schemes of finite type over Spec ℤ, and f as the limit of a cofiltered family f_{•} := (f_{λ} : X_{λ} → S_{λ} ∈ Λ) of affine finitely presented morphisms, such that :

• The natural morphism g_{λ} : S → S_{λ} is dominant for every λ ∈ Λ.

• f_{λ} is smooth for every λ ∈ Λ ([66, Ch.IV, Prop.17.7.8(ii)]), and f_{μ} = f_{λ} ×_{S_{λ}} S_{μ}

whenever μ ≥ λ. Furthermore, we may find λ ∈ Λ such that U = U_{λ} ×_{S_{λ}} S ([65, Ch.IV, Cor.8.2.11]), so that j is the limit of the cofiltered system of open immersions (j_{μ} : U_{μ} := U_{λ} ×_{S_{λ}} S_{μ} → X_{λ} | μ ≥ λ), and after replacing Λ by a cofinal subset, we may assume that j_{μ} is defined for every μ ∈ Λ. For every λ ∈ Λ and n ∈ ℕ, let X_{λ,n} ⊂ X_{λ} be the open and closed subset consisting of all x ∈ X_{λ} such that dim_{x} f^{−1}f(x) = n; clearly f_{•} restricts to a cofiltered family f_{•,m} := (f_{λ|X_{λ,m}} : X_{λ,m} → S_{λ} | λ ∈ Λ), whose limit is again f. Hence we may replace X_{λ} by X_{λ,m}, and assume that the fibres of f_{λ} have pure dimension m, for every λ ∈ Λ. For every λ ∈ Λ, let :

Z′_{λ} := {s ∈ S_{λ} | dim f_{λ}^{−1}(s)\U_{λ} = m}.

and endow Z′_{λ} with its constructible topology 𝒯_{λ}; since Z′_{λ} is a constructible subset of S_{λ} ([65, Ch.IV, Prop.9.9.1]), (Z′_{λ}, 𝒯_{λ}) is a compact topological space, and due to (13.2.16), we have :

lim_{λ∈Λ} Z′_{λ} = ∅.

Then [41, Ch.I, §9, n.6, Prop.8] implies that Z′_{λ} = ∅ for every sufficiently large λ ∈ Λ. Set :

Cov(X_{•}) := 2-colim_{μ≥λ} Cov(X_{μ}) Cov(U_{•}) := 2-colim_{μ≥λ} Cov(U_{μ}).

(See definition 2.5.1(ii).) There follows an essentially commutative diagram of categories:

(13.2.18)  \xymatrix{Cov(X) \ar[r] \ar[d]_{j^{∗}} & Cov(X_{•}) \ar[d]^{j_{•}^{∗}} \\ Cov(U) \ar[r] & Cov(U_{•})}

where j_{•}^{∗} is the 2-colimit of the system of pull-back functors j_{μ}^{∗} : Cov(X_{μ}) → Cov(U_{μ}). In light of lemma 13.1.6, the horizontal arrows of (13.2.18) are equivalences, so j^{∗} will be fully faithful, provided the same holds for the functors j_{μ}^{∗}, for every large enough μ ∈ Λ.

Hence, in order to prove assertion (i) when X and S are affine, we may assume that S is of finite type over Spec ℤ, the fibres of f have pure dimension m, and (13.2.16) holds, which is the case covered by claim 13.2.15.

Concerning assertion (ii), since the morphism g_{μ} is dominant, for every η′ ∈ (S_{μ})_{max} we may find η ∈ S_{max} such that g_{μ}(η) = η′. Denote by :

h : f^{−1}η → f_{μ}^{−1}η′ and j′_{μ} : (U_{μ})_{η′} := U_{μ} ∩ f_{μ}^{−1}η′ → f_{μ}^{−1}η′

the natural morphisms. With this notation, we have the following :

Claim 13.2.19. The induced essentially commutative diagram :

\xymatrix{Cov(f_{μ}^{−1}η′) \ar[r]^{j_{μ}^{′∗}} \ar[d]_{h^{∗}} & Cov((U_{μ})_{η′}) \ar[d]^{h_{U}^{∗}} \\ Cov(f^{−1}η) \ar[r]^{j_{η}^{∗}} & Cov(U_{η})}

is 2-cartesian.

Proof of the claim. The pair (h^{∗}, j_{μ}^{′∗}) induces a functor (h, j′_{μ})^{∗} from Cov(f_{μ}^{−1}η′) to the category of data of the form (φ, φ′, α), where φ′ (resp φ) is a finite étale covering of (U_{μ})_{η′} (resp. of f^{−1}η) and α : φ ×_{f^{−1}η} U_{η} →^{∼} φ′ ×_{η′} η is an isomorphism in Cov(U_{η}), and the contention is that (h, j′_{μ})^{∗} is an equivalence. The full faithfulness of the functors j_{μ}^{′∗} and j_{η}^{∗} (lemma 13.1.7(iii)) easily implies the full faithfulness of (h, j′_{μ})^{∗}. To prove that (h, j′_{μ})^{∗} is essentially surjective, amounts to showing that if φ′ : E′ → (U_{μ})_{η′} is a finite étale morphism and

φ′′ := φ′ ×_{η′} η : E′′ := E′ ×_{η′} η → U_{η}

extends to a finite étale morphism φ : E → f^{−1}η, then φ′ extends to a finite étale covering of f_{μ}^{−1}η′. Now, let L be the maximal purely inseparable extension of κ(η′) contained in κ(η). Since the induced morphism η′′ := Spec L → Spec κ(η′) is radicial, the base change functors

Cov(f_{μ}^{−1}η′) → Cov((f_{μ}^{−1}η′) ×_{η′} η′′) Cov((U_{μ})_{η′}) → Cov((U_{μ})_{η′} ×_{η′} η′′)

are equivalences (lemma 13.1.7(i)). Thus, we may replace η′ by η′′, and assume that the field extension κ(η′) ⊂ κ(η) is separable, hence the induced morphism Spec κ(η) → Spec κ(η′) is regular ([34, Ch.VIII, §7, no.3, Cor.1]), and then the same holds for the morphism h ([64, Ch.IV, Prop.6.8.3(iii)]). Given φ′ as above, set 𝒜 := (j′_{μ} ∘ φ′)_{∗}𝒪_{E′}; then 𝒜 is a quasi-coherent 𝒪_{f_{μ}^{−1}η′}-algebra, and we may define the quasi-coherent 𝒪_{f_{μ}^{−1}η′}-algebra ℬ as the integral closure of 𝒪_{f_{μ}^{−1}η′} in 𝒜. By [64, Ch.IV, Prop.6.14.1], h^{∗}ℬ is the integral closure of 𝒪_{f^{−1}η} in h^{∗}𝒜 = j_{η∗}∘ h_{U}^{∗}(φ′′_{∗}𝒪_{E′′}) = j_{η∗}(φ_{∗}𝒪_{E}). By proposition 9.8.3, it then follows that h^{∗}ℬ = φ^{∗}𝒪_{E}, therefore ℬ is a finite étale 𝒪_{f_{μ}^{−1}η′}-algebra ([66, Ch.IV, Prop.17.7.3(ii)] and [64, Ch.IV, Prop.2.7.1]). The claim follows. ♢

By the foregoing, we already know that j^{∗} is fully faithful, hence the same holds for (j, ι_{•})^{∗} (claim 13.2.13(i)). To show that (j, ι_{•})^{∗} is essentially surjective, consider any \underline{E} as in (13.2.12); we may find μ ∈ Λ, and a finite étale morphism φ′ : E_{μ} → U_{μ} such that φ = φ′ ×_{U_{μ}} U, whence objects φ′_{η′} := φ′ ×_{U_{μ}} (U_{μ})_{η′} in Cov((U_{μ})_{η′}), for every η′ ∈ (S_{μ})_{max}. By construction, we have φ′_{η′} ×_{η′} η ≃ ψ_{η} ×_{f^{−1}η} U_{η} for every η′ ∈ (S_{μ})_{max} and every η ∈ S_{max} such that g_{μ}(η) = η′. Then claim 13.2.19 shows that, for every η′ ∈ (S_{μ})_{max} there exists an object ψ′_{η′} of Cov(f_{μ}^{−1}η′) with isomorphisms :

ψ′_{η′} ×_{f_{μ}^{−1}η′} f^{−1}η ≃ ψ_{η} α_{η′} : ψ′_{η′} ×_{f_{μ}^{−1}η′} (U_{μ})_{η′} →^{∼} φ′_{η′}.

Therefore, the datum \underline{E}_{μ} := (φ′, (ψ′_{η′}, α′_{η′} | η′ ∈ (S_{μ})_{max})) is an object of the 2-limit of the diagram of categories

Cov(U_{μ}) ⟵^{j_{μ}^{∗}} Cov(X_{μ}) \underrightarrow{∏_{η′} ι_{η′}^{∗}} ∏_{η′∈(S_{μ})_{max}} Cov(f_{μ}^{−1}η′)

(where ι_{η′} : f_{μ}^{−1}η′ → X_{μ} is the natural immersion, for every η′ ∈ (S_{μ})_{max}). By claim 13.2.15, the datum \underline{E}_{μ} comes from an object φ_{μ} of Cov(X_{μ}). Let φ′′ be the image of φ′_{μ} in Cov(X); by construction we have (j, ι_{•})^{∗}φ′′ = \underline{E}, as required.

This conclude the proof of (i) and (ii), in case X and S are affine. To deal with the general case, let X = ⋃_{i∈I} V_{i} be a covering consisting of affine open subschemes, and for every i ∈ I, let fV_{i} = ⋃_{λ∈Λ_{i}} S_{iλ} be an affine open covering of the open subscheme fV_{i} ⊂ S; set also V_{iλ} := V_{i} ∩ f^{−1}S_{iλ} for every i ∈ I and λ ∈ Λ_{i}. The restrictions f_{|V_{iλ}} : V_{iλ} → S_{iλ} are smooth morphisms; moreover, the image of the open immersion

j_{iλ} := j_{|U∩V_{iλ}} : U ∩ V_{iλ} → V_{iλ}

is dense in every fibre of f_{|V_{i}}. The induced morphism

(13.2.20)  β : X′ := ∐_{i∈I} ∐_{λ∈Λ_{i}} V_{iλ} → X

is faithfully flat, hence of universal 2-descent for (13.1.1); moreover, it is easily seen that X′′ := X′ ×_{X} X′ is separated, and j′′ := j ×_{X} X′′ is a dense open immersion, hence j′′^{∗} is faithful (lemma 13.1.7(ii)). Then, by corollary 3.5.30(ii), j^{∗} is fully faithful, provided the pull-back functor Cov(X′) → Cov(X′ ×_{X} U) is fully faithful, i.e. provided the same holds for the functors j_{iλ}^{∗} : Cov(V_{iλ}) → Cov(V_{iλ} ∩ U). However, each V_{iλ} is affine ([59, Ch.I, Prop.5.5.10]), hence assertion (i) is already known for the morphisms f_{|V_{iλ}} and the open subsets U ∩ V_{iλ}; this concludes the proof of (i).

To show (ii), we use the criterion of claim 13.2.13(ii) : indeed, assumption (a) is already known, hence we are reduced to showing that assertion (iii) holds. To this aim, we consider again the morphism β of (13.2.20), and denote by f′′ : X′′ → S the induced morphism. Clearly f′′ is smooth, and j′′ is an open immersion, such that f′′^{−1}(s) ×_{X} U is dense in f^{′′−1}(s), for every s ∈ S; then assertion (i) implies that j′′^{∗} is fully faithful. Moreover it is easily seen that X′′′ := X′′ ×_{X} X′ is separated, and j ×_{X} X′′′ is a dense open immersion, so j′′′^{∗} is faithful (lemma 13.1.7(ii)), and therefore corollary 3.5.30(ii) reduces to showing that the pullback functor Cov(X′) → Cov(X′ ×_{X} U) is an equivalence, or – what is the same – that this holds for the pull-back functors j_{iλ}^{∗}, which is already known. □

13.2.21. We consider now the local counterpart of theorem 13.2.11. Namely, suppose that f : X → S is a smooth morphism, let \overline{x} be any geometric point of X, set \overline{s} := f(\overline{x}), and let s ∈ S be the support of \overline{s}. Then f induces the morphism f\overline{x} : X(\overline{x}) → S(\overline{s}), and the latter has geometrically irreducible fibres by proposition 13.1.30(iii). For every open immersion j : U → X(\overline{x}), we may consider the diagram 𝒟(S(\overline{x}), f\overline{x}, U) as in (13.2.10).

Theorem 13.2.22. In the situation of (13.2.21), suppose that U contains the generic point of

f −1 ; \overline{x}(s). Then :

(i) j^{∗} : Cov(X(\overline{x})) → Cov(U) is fully faithful.

(ii) The diagram 𝒟(S(\overline{x}), f\overline{x}, U) is 2-cartesian.

Proof. (i): To begin with, since f\overline{x} is generizing ([126, Th.9.5]), and S is local, every fibre of f\overline{x} has a point that specializes to the generic point η_{s} of f −1 ; \overline{x}(s); since the fibres are irreducible, it follows that the generic point of every fibre specializes to η_{s}. Therefore U is fibrewise dense in X(\overline{x}), and moreover it is connected. Now, the category Cov(X) is equivalent to the category of finite sets, hence every object in the essential image of j^{∗} is (isomorphic to) a finite disjoint union of copies of U; since U is connected, the morphisms of U-schemes between two such objects E and E′ are in natural bijection with the set-theoretic mappings π_{0}(E) → π_{0}(E′) of their sets of connected components, whence the assertion.

(ii): Let us write U as the union of a filtered family (U_{λ} | λ ∈ Λ) of constructible open subsets of X(\overline{x}); up to replacing Λ by a cofinal subset, we may assume that η_{s} ∈ U_{λ} for every λ ∈ Λ. Arguing as in the proof of theorem 13.2.11, we see that 𝒟(S(\overline{x}), f\overline{x}, U) is the 2-limit of the system of diagrams 𝒟(S(\overline{x}), f\overline{x}, U_{λ}), hence it suffices to show the assertion with U = U_{λ}, for every λ ∈ Λ, which allows to assume that U is quasi-compact. Next, arguing as in the proof of proposition 13.1.30, we are reduced to the case where S = S(\overline{s}). We may write X(\overline{x}) as the limit of a cofiltered system (X_{λ} | λ ∈ Λ) of affine schemes, étale over X, and for λ ∈ Λ large enough, we may find an open subset U_{λ} ⊂ X_{λ} such that U = U_{λ} ×_{X_{λ}} X(\overline{x}) ([65, Ch.IV, Cor.8.2.11]). For every μ ≥ λ, set U_{μ} := U_{λ} ×_{X_{λ}} X_{μ}, and denote by f_{μ} : X_{μ} → S the natural morphism. Suppose first that S is irreducible; then, from lemma 13.1.6 it is easily seen that 𝒟(S, f\overline{x}, U) is the 2-colimit of the system of diagrams 𝒟(S, f_{μ}, U_{μ}), so the assertion follows from theorem 13.2.11(ii) (more generally, this argument works whenever S_{max} is a finite set, since filtered 2-colimits of categories commute with finite products).

In the general case, let φ : E → U be a finite étale morphism, and suppose that φ_{η} := φ ×_{X(\overline{x})} f −1 ; \overline{x}(η) extends to an object ψ_{η} of Cov(f −1 ; \overline{x}η), for every η ∈ S_{max}. The assertion boils down to showing that φ extends to an object φ′ of Cov(X(\overline{x})). To this aim, for every η ∈ S_{max}, let Z_{η} → S be the closed immersion of the topological closure of η in S (which we endow with its reduced scheme structure); set also Y_{η} := X ×_{S} Z_{η}. Then Z_{η} is a strictly local scheme ([66, Ch.IV, Prop.18.5.6(i)]), and \overline{x} factors through the closed immersion Y → X, which induces an isomorphism of Z-schemes :

Y_{η}(\overline{x}) →^{∼} X(\overline{x}) ×_{S} Z_{η}

(lemma 13.1.22(ii)). By the foregoing case, φ ×_{S} Z_{η} extends to an object \overline{ψ}_{η} of Cov(Y_{η}(\overline{x})). However, Z_{η} is the limit of the cofiltered system (Z_{η,i} | i ∈ I(η)) consisting of the constructible closed subschemes of S that contain Z_{η}. By lemma 13.1.6, it follows that we may find i ∈ I(η) and an object \overline{ψ}_{η,i} of Cov(X(\overline{x})×_{S} Z_{η,i}) whose image in Cov(Y_{η}(\overline{x})) is isomorphic to \overline{ψ}_{η}, and if i is large enough, \overline{ψ}_{η,i} ×_{X(\overline{x})} U agrees with φ ×_{S} Z_{η,i} in Cov(U ×_{S} Z_{η,i}). For each η, η′ ∈ S_{max}, choose i ∈ I(η), i′ ∈ I(η′) with these properties, and to ease notation, set :

X′_{η} := X ×_{S} Z_{η,i}  X′′_{ηη′} := X′_{η} ×_{S} Z_{η′,i′}  φ′_{η} := \overline{ψ}_{η,i}

and denote by α_{η} : φ′_{η} ×_{X(\overline{x})} U →^{∼} φ ×_{S} Z_{η,i} the given isomorphism. As in the foregoing, we notice that \overline{x} factors through X′_{η}, and the closed immersion X′_{η} → X induces an isomorphism X′_{η}(\overline{x}) →^{∼} X(\overline{x}) ×_{S} Z_{η,i} of Z_{η,i}-schemes. According to [63, Ch.IV, Cor.1.9.9], we may then find a finite subset T ⊂ S_{max} such that the induced morphism :

β : X_{1} := ∐_{η∈T} X′_{η}(\overline{x}) → X(\overline{x})

is surjective. Set X_{2} := X_{1} ×_{X(\overline{x})} X_{1} and X_{3} := X_{2} ×_{X(\overline{x})} X_{1}; notice that X_{2} is the disjoint union of schemes of the form X(\overline{x}) ×_{S} Z_{η,i} ×_{S} Z_{η′,i′}, for η, η′ ∈ T, and again, the latter is naturally isomorphic to X′′_{ηη′}(\overline{x}), for a unique lifting of the geometric point \overline{x} to a geometric point of X′′_{ηη′}. Similar considerations can be repeated for X_{3}, and in light of (i), we deduce that the pull-back functors :

Cov(X_{i} ×_{X(\overline{x})} U) → Cov(X_{i}) i = 1, 2, 3

are fully faithful, in which case corollary 3.5.30(iii) says that the essentially commutative diagram of categories :

\xymatrix{Desc(Cov, β) \ar[r] \ar[d] & Desc(Cov, β ×_{X(\overline{x})} U) \ar[d] \\ ∏_{η∈T} Cov(X′_{η}(\overline{x})) \ar[r] & ∏_{η∈T} Cov(X′_{η}(\overline{x}) ×_{X(\overline{x})} U)}

is 2-cartesian. Let ρ : Cov(U) → Desc(Cov, β ×_{X(\overline{x})} U) be the functor defined in (3.5.22); it follows that the datum ((φ′_{η}, α_{η} | η ∈ T), ρ(φ)) comes from a descent datum (φ′_{1}, ω) in Desc(Cov, β). By lemma 13.1.2, the latter descends to an object φ′ of Cov(X(\overline{x})), and by construction we have j^{∗}φ′ = φ, as required. □

13.3. Étale coverings of log schemes. We resume the general notation of (12.2.1), especially, we choose implicitly τ to be either the Zariski or étale topology, and we shall omit further mention of this choice, unless the omission might be a source of ambiguities.

13.3.1. Let \underline{Y} := ((Y, \underline{N}), T, ψ) be an object of 𝒦_{int} (see (12.6.10) : especially τ = Zar here), φ : T′ → T an integral proper subdivision of the fan T (definition 6.5.22(ii),(iii)), and suppose that both T and T′ are locally fine and saturated. Set ((Y′, \underline{N}′), T′, ψ′) := φ^{∗}\underline{Y} (proposition 12.6.14(iii)), and let f : Y′ → Y be the morphism of schemes underlying the cartesian morphism φ^{∗}\underline{Y} → \underline{Y}.

Proposition 13.3.2. In the situation of (13.3.1), the following holds :

(i) For every geometric point ξ of Y, the fibre f^{−1}(ξ) is connected (and hence, non-empty). (ii) The functor f^{∗} : Cov(Y) → Cov(Y′) is an equivalence.

Proof. (i): The assertion is local on Y, hence we may assume that T = (Spec P)^{♯} for a fine, sharp and saturated monoid P. In this case, we may find a subdivision φ′ : T′′ → T′ such that both φ′ and φ ∘ φ′ are compositions of saturated blow up of ideals generated by at most two elements of P (example 6.6.15(iii)). We are reduced to showing the assertion for the morphisms φ′ and φ ∘ φ′, after which, we may further assume that φ is the saturated blow up of an ideal generated by two elements of Γ(T, 𝒪_{T}), in which case the assertion follows from the more precise theorem 12.4.51.

(ii): First, we claim that the assertion is local on the Zariski topology of Y. Indeed, let Y = ⋃_{i∈I} U_{i} be a Zariski open covering, and set U_{ij} := U_{i} ∩ U_{j} for every i, j ∈ I; according to corollary 3.5.30(ii) and lemma 13.1.2, it suffices to prove the contention for the objects (U_{i} ×_{Y} (Y, \underline{N}), T, ψ_{|U_{i}}) and (U_{ij} ×_{Y} (Y, \underline{N}), T, ψ_{|U_{ij}}), and the morphism φ. Hence, we may again suppose that T = Spec P, for a monoid P as in (i).

Arguing as in the proof of (i), we may next reduce to the case where φ is the saturated blow up of an ideal generated by two elements of Γ(T, 𝒪_{T}). Now, the morphism of schemes f induces a morphism of topoi Y_{ét}^{′∼} → Y_{ét}^{∼} which we denote again by f; then, assertion (i) and the proper base change theorem [10, Exp.XII, Th.5.1(i)] imply that the unit of adjunction ℱ → f_{∗}f^{∗}ℱ is an isomorphism for every sheaf ℱ on Y, so f^{∗} is fully faithful (proposition 1.1.20(iii)). Next, if φ : E′ → Y′ is a finite étale covering, Y′ decomposes into a disjoint union of open and closed subsets Y′ = Y′_{0} ∪⋯∪Y′_{k}, such that (φ_{∗}𝒪_{E′})_{|Y′_{r}} is locally free of rank r, for every r = 0, …, k. Set Y_{r} := f(Y′_{r}) for r = 0, …, k; since the fibres of f are connected, we see that Y′_{r} = f^{−1}Y_{r} for every such r. Moreover, since f is a closed map and the topology of Y is induced from that of Y′ via f, we see that Y_{r} is a closed subset of Y for r = 0, …, k, so Y = Y_{0} ∪ ⋯ ∪ Y_{k} is a partition of Y by open and closed subsets. After replacing Y by Y_{r}, we may then assume that φ_{∗}𝒪_{E} is locally free of rank r, and notice that the isomorphism classes of étale coverings of this type are classified by the pointed set H^{1}(Y′_{ét}, S_{r,Y′}), where S_{r} is the symmetric group on r elements. Thus, it suffices to show that the induced map :

(13.3.3)  H^{1}(Y_{ét}, S_{r,Y}) → H^{1}(Y′_{ét}, S_{r,Y′})

is a bijection. However, theorem 4.9.9yields the exact sequence of pointed sets :

{1} → H^{1}(Y_{ét}, f_{∗}S_{r,Y′}) ⟶^{u} H^{1}(Y′_{ét}, S_{r,Y′}) → H^{0}(Y_{ét}, R^{1}f_{∗}S_{r,Y′}).

On the other hand, assertion (i) and the proper base change theorem [10, Exp.XII, Th.5.1(i)] imply that the unit of adjunction S_{r,Y} → f_{∗}f^{∗}S_{r,Y} = f_{∗}S_{r,Y′} is an isomorphism, hence u is naturally identified to (13.3.3), and we are reduced to showing that the natural morphism

τ_{f,S_{r,Y′}} : 1_{Y_{ét}^{∼}} → R^{1}f_{∗}S_{r,Y′}

is an isomorphism (notation of (4.9.3)). The latter can be checked on the stalks, and in view of [10, Exp.XII, Cor.5.2(ii)] we are reduced to showing that H^{1}(f^{−1}(ξ)_{ét}, S_{r}) = {1} for every geometric point ξ of Y. However, according to theorem 12.4.51, the reduced geometric fibre f^{−1}(ξ)_{red} is either isomorphic to |ξ| (in which case the contention is trivial), or else it is isomorphic to the projective line ℙ_{κ(ξ)}^{1}, in which case – in view of lemma 13.1.7(i) – it suffices to show that every finite étale morphism E → ℙ_{κ(ξ)}^{1} admits a section, which is well known. □

The class of étale morphisms of log schemes was introduced in section 12.3 : its definition and its main properties parallel those of the corresponding notion for schemes, found in [63, Ch.IV, §17]. In the present section this theme is further advanced : we will consider the logarithmic analogue of the classical notion of étale covering of a scheme. To begin with, we make the following :

Definition 13.3.4. (i) Let φ : T → S be a morphism of fans. We say that φ is of Kummer type, if the map (log φ)_{t} : 𝒪_{S,φ(t)} → 𝒪_{T,t} is of Kummer type, for every t ∈ T (see definition 6.4.58).

(ii) Let f : (Y, \underline{N}) → (X, \underline{M}) be a morphism of log schemes.

(a) We say that f is of Kummer type, if for every τ-point ξ of Y, the morphism of monoids (log f)_{ξ} : f^{∗}\underline{M}_{ξ} → \underline{N}_{ξ} is of Kummer type.

(b) A Kummer chart for f is the datum of charts

ω_{P} : P_{Y} → \underline{N} ω_{Q} : Q_{X} → \underline{M}

and a morphism of monoids ϑ : Q → P such that (ω_{P} , ω_{Q}, ϑ) is a chart for f (see definition 12.1.17(iii)), and ϑ is of Kummer type.

Remark 13.3.5. (i) Let f : P → Q be a morphism of monoids of Kummer type, with P integral and saturated. It follows easily from lemma 6.4.59(iii,v), that the induced morphism of fans (Spec f)^{♯} : (Spec Q)^{♯} → (Spec P)^{♯} is of Kummer type.

(ii) In the situation of definition 13.3.4(ii), it is easily seen that f is of Kummer type, if and only if the morphism of Y -monoids f^{∗}\underline{M}_{ξ}^{♯} → \underline{N}_{ξ}^{♯} deduced from log f, is of Kummer type for every τ-point ξ of Y.

(iii) In the situation of definition 13.3.4(ii.b), suppose that the chart (ω_{P} , ω_{Q}, ϑ) is of Kummer type, and Q is integral and saturated. It then follows easily from (i), (ii) and example 12.6.5(ii), that f is of Kummer type. In the same vein, we have the following :

Lemma 13.3.6. Let f : (Y, \underline{N}) → (X, \underline{M}) be a morphism of log schemes with coherent log structures, ξ a τ-point of Y, and suppose that :

(a) \underline{M}_{f(ξ)} is fine and saturated.

(b) The morphism (log f)_{ξ} : \underline{M}_{f(ξ)} → \underline{N}_{ξ} is of Kummer type.

Then there exists a (Zariski) open neighborhood U of |ξ| in Y, such that the restriction f_{|U} : (U, \underline{N}_{|U}) → (X, \underline{M}) of f is of Kummer type.

Proof. By corollary 12.1.36(i) and theorem 12.1.37(ii), we may find a neighborhood U′ → Y of ξ in Y_{τ}, and a finite chart (ω_{P} , ω_{Q}, ϑ) for the restriction f_{|U′}, with Q fine and saturated. Set

S_{P} := ω_{P,ξ}^{−1}\underline{N}_{ξ}^{×} S_{Q} := ω_{Q,f(ξ)}^{−1}\underline{M}_{f(ξ)}^{×} P′ := S_{P}^{−1}P Q′ := S_{Q}^{−1}Q.

According to claim 12.1.31(iii) we may find neighborhoods U′′ → U′ of ξ in Y_{τ}, and V → X of f(ξ) in X_{τ}, such that the charts ω_{P |U′′} and ω_{Q|V} extend to charts

ω_{P′} : P′_{U′′} → \underline{N}_{|U′′} ω_{Q′} : Q′_{V} → \underline{M}_{|V} .

Clearly ϑ extends as well to a unique morphism ϑ′ : Q′ → P′, and after shrinking U′′ we may assume that the restriction f_{|U′′} : (U′′, \underline{N}_{|U′′}) → (X, \underline{M}) factors through a morphism f′ : (U′′, \underline{N}_{|U′′}) → (V, \underline{M}_{|V}), in which case it is easily seen that the datum (ω_{P′}, ω_{Q′}, ϑ′) is a chart for f′. Notice as well that Q′ is still fine and saturated (lemma 6.2.9(i)), and by claim 12.1.31(iv) the maps ω_{P′} and ω_{Q′} induce isomorphisms :

P^{′♯} →^{∼} \underline{N}_{ξ}^{♯} Q^{′♯} →^{∼} \underline{M}_{f(ξ)}^{♯}.

Our assumption (b) then implies that the map Q^{′♯} → P^{′♯} deduced from ϑ′ is of Kummer type, and then the morphism of fans Spec ϑ′ : Spec P′ → Spec Q′ is of Kummer type as well (remark 13.3.5(i)). However, let

\overline{ω}_{P′} : (U′′, \underline{N}_{|U′′}^{♯}) → (Spec P′)^{♯} \overline{ω}_{Q′} : (V, \underline{M}_{|V}^{♯}) → (Spec Q′)^{♯}

be the morphisms of monoidal spaces deduced from ω_{P′} and ω_{Q′}; we obtain a morphism

(f′, Spec ϑ′) : (U′′, \underline{N}_{|U′′}, (Spec P′)^{♯}, \overline{ω}_{P′}) → (V, \underline{M}_{|V} , (Spec Q′)^{♯}, \overline{ω}_{Q′})

in the category 𝒦 of (12.6.2), which – in view of remark 13.3.5(ii) – shows that f′ is of Kummer type. This already concludes the proof in case τ = Zar, and for τ = ét it suffices to remark that the image of U′′ in Y is a Zariski open neighborhood U of ξ, such that the restriction f_{|U} is of Kummer type. □

We shall use the following criterion :

Proposition 13.3.7. Let k be a field, f : P → Q an injective morphism of fine monoids, with P sharp. Suppose that the scheme Spec k⟨Q/𝔪_{P} Q⟩ admits an irreducible component of Krull dimension 0. We have :

(i) f is of Kummer type, and k⟨Q/𝔪_{P} Q⟩ is a finite k-algebra.

(ii) If moreover, Q^{×} is a torsion-free abelian group, then Q is sharp, and k⟨Q/𝔪_{P} Q⟩ is a local k-algebra with k as residue field.

Proof. Notice that the assumption means especially that Q/𝔪_{P} Q ≠ {1}, so f is a local morphism. Set I := rad(𝔪_{P} Q) (notation of definition 6.1.8(ii)), and let 𝔭_{1}, …, 𝔭_{n} ⊂ Q be the minimal prime ideals containing I; then I = 𝔭_{1} ∩ ⋯ ∩ 𝔭_{n}, by lemma 6.1.16. Clearly the natural closed immersion Spec k⟨Q/I⟩ → Spec k⟨Q/𝔪_{P} Q⟩ is a homeomorphism; on the other hand, say that 𝔮 ⊂ k⟨Q⟩ is a prime ideal containing I; then 𝔮 ∩ Q is a prime ideal of Q containing I, hence 𝔭_{i} ⊂ 𝔮 for some i ≤ n, i.e. Spec k⟨Q/I⟩ is the union of its closed subsets Spec k⟨Q/𝔭_{i}Q⟩, for i = 1, …, n. The Krull dimension of each irreducible component of Spec k⟨Q/𝔭_{i}⟩ = Spec k[Q\𝔭_{i}] equals

rk_{ℤ}(Q\𝔭)^{×} + dim Q\𝔭 = rk_{ℤ}Q^{×} + d − ht(𝔭_{i}) where d := dim Q

(claim 11.6.37(ii) and corollary 6.4.12(i,ii)). Our assumption then implies that Q^{×} is a finite group, and ht(𝔭_{i}) = d, i.e. 𝔭_{i} = 𝔪_{Q}, for at least an index i ≤ n, and therefore I = 𝔪_{Q}. Furthermore, since 𝔪_{Q} is finitely generated, we have 𝔪_{Q}^{n} ⊂ 𝔪_{P} Q for a sufficiently large integer n > 0, and then it follows easily that k⟨Q/𝔪_{P} Q⟩ is a finite k-algebra. If Q^{×} is torsion-free, then we also deduce that Q is sharp, and moreover the maximal ideal of k⟨Q/𝔪_{P} Q⟩ generated by 𝔪_{Q} is nilpotent, hence the latter k-algebra is local.

Let now 𝔮 ⊂ Q be any prime ideal, and pick x ∈ 𝔪_{Q} \𝔮; the foregoing implies that there exists an integer r > 0 such that x^{r} = f(p_{𝔮})q for some p_{𝔮} ∈ 𝔪_{P} and q ∈ Q, hence f(p_{𝔮}) ∉ 𝔮, and f(p_{𝔮}) is not invertible in Q, since f is local. If now 𝔮 has height d − 1, it follows that f(p_{𝔮}) is a generator of (Q\𝔮)_{ℝ}, which is an extremal ray of the polyhedral cone Q_{ℝ}, and every such extremal ray is of this form (proposition 6.4.9); furthermore, the latter cone is strictly convex, since Q^{×} is finite. Hence the set S := (f(p_{𝔮}) | ht(𝔮) = d − 1) is a system of generators of the polyhedral cone Q_{ℝ} (see (6.3.15)). Let Q′ ⊂ Q be the submonoid generated by S; then S_{ℚ} = S_{ℝ} ∩ Q_{ℚ} = Q_{ℚ} (proposition 6.3.22(iii)), i.e. f is of Kummer type. □

13.3.8. Let f : (Y_{Zar}, \underline{N}) → (X_{Zar}, \underline{M}) be a morphism of log schemes with Zariski log structures; it follows easily from the isomorphism (12.1.9) and remark 13.3.5(ii) that f is of Kummer type if and only if ũ^{∗}f : (X_{ét}, ũ^{∗}\underline{M}) → (Y_{ét}, ũ^{∗}\underline{N}) is a morphism of Kummer type between schemes with étale log structures (notation of (12.2.2)). Suppose now that \underline{M} is an integral and saturated log structure on X_{Zar}, and denote by s.Kum(X_{Zar}, \underline{M}) (resp. s.Kum(X_{ét}, ũ^{∗}\underline{M})) the full subcategory of sat.log/(X_{Zar}, \underline{M}) (resp. of sat.log/(X_{ét}, ũ^{∗}\underline{M}) whose objects are all the morphisms of Kummer type. In view of the foregoing (and of lemma 12.1.18(i)), we see that ũ^{∗} restricts to a functor :

(13.3.9)  s.Kum(X_{Zar}, \underline{M}) → s.Kum(X_{ét}, ũ^{∗}\underline{M}).

Lemma 13.3.10. The functor (13.3.9) is an equivalence.

Proof. By virtue of proposition 12.2.3(ii) we know already that (13.3.9) is fully faithful, hence we only need to show its essential surjectivity. Thus, let f : (Y_{ét}, \underline{N}) → (X_{ét}, ũ^{∗}\underline{M}) be a morphism of Kummer type. By remark 13.3.5(ii), we know that log f induces an isomorphism

ũ^{∗}f^{∗}\underline{M}_{ℚ}^{♯} →^{∼} f^{∗}ũ^{∗}\underline{M}_{ℚ}^{♯} →^{∼} \underline{N}_{ℚ}^{♯}

(notation of (6.3.20)). Since \underline{N} is integral and saturated, the natural map \underline{N}^{♯} → \underline{N}_{ℚ}^{♯} is a monomorphism, so that the counit of adjunction ũ^{∗}ũ_{∗}\underline{N}^{♯} → \underline{N}^{♯} is an isomorphism (lemma 4.9.27(ii)), and then the same holds for the counit of adjunction ũ^{∗}ũ_{∗}(Y, \underline{N}) → (Y, \underline{N}) (proposition 12.2.3(iii)). □ Proposition 13.3.11. Let f : (Y, \underline{N}) → (X, \underline{M}) be a morphism of fs log schemes. The following conditions are equivalent:

(a) Every geometric point of X admits an étale neighborhood U → X such that Y_{U} := U ×_{X} Y decomposes as a disjoint union Y_{U} = ⋃_{i=1}^{n} Y_{i} of open and closed subschemes (for some n ∈ ℕ), and we have :

(i) Each restriction Y_{i} ×_{Y} (Y, \underline{N}) → U ×_{X} (X, \underline{M}) of f ×_{X} 1_{U} admits a fine, saturated Kummer chart (ω_{P_{i}}, ω_{Q_{i}}, ϑ_{i}) such that P_{i} and Q_{i} are sharp, and the order of Coker ϑ_{i}^{gp} is invertible in 𝒪_{U}.

(ii) The induced morphism of U-schemes Y_{i} → U ×_{Spec ℤ[P_{i}]} Spec ℤ[Q_{i}] is an isomor-

phism, for every i = 1, …, n.

(b) f is étale, and the morphism of schemes underlying f is finite. Proof. (a) ⇒ (b): Indeed, it is easily seen that the morphism Spec ℤ[ϑ_{i}] is finite, hence the same holds for the restriction Y_{i} → U of f, in view of (a.ii), and then the same holds for f ×_{X} 1_{U}, so finally f is finite on the underlying schemes ([64, Ch.IV, Prop.2.7.1]), and it is étale by the criterion of theorem 12.3.37.

(b) ⇒ (a): Arguing as in the proof of theorem 12.3.37, we may reduce to the case where τ = ét. Suppose first that both X and Y are strictly local. In this case, \underline{M} admits a fine and saturated chart ω_{P} : P_{X} → \underline{M}, sharp at the closed point (corollary 12.1.36(i)). Moreover, f admits a chart (ω_{P} , ω_{Q} : Q_{Y} → \underline{N}, ϑ : P → Q), for some fine monoid Q such that Q^{×} is torsion-free; also ϑ is injective, the induced morphism of X-schemes

g : Y → X′ := X ×_{Spec ℤ[P]} Spec ℤ[Q]

is étale, and the order of Coker ϑ^{gp} is invertible in 𝒪_{X} (corollary 12.3.42). Since f is finite, g is also closed ([60, Ch.II, Prop.6.1.10]), hence its image Z is an open and closed local subscheme, finite over X. Let k be the residue field of the closed point of X; it follows that X′ ×_{X} Spec k ≃ Spec k⟨Q/𝔪_{P} Q⟩ admits an irreducible component of Krull dimension zero (namely, the intersection of Z with the fibre of X′ over the closed point of X), in which case the criterion of proposition 13.3.7(ii) ensures that ϑ is of Kummer type and Q is sharp, hence X′ is finite over X, and moreover X′ ×_{X} Spec k is a local scheme with k as residue field. Since X is strictly henselian, it follows that X′ itself is strictly local, and therefore g is an isomorphism, so the proposition is proved in this case.

Let now X be a general scheme, and ξ a geometric point of X; denote by X(ξ) the strict henselization of X at the point ξ, and set Y (ξ) := X(ξ) ×_{X} Y. Since f is finite, Y (ξ) decomposes as the disjoint union of finitely many open and closed strictly local subschemes Y_{1}(ξ), …, Y_{n}(ξ). Then we may find an étale neighborhood U → X of ξ, and open and closed subschemes Y_{1}, …, Y_{n} of Y ×_{X} U, with isomorphisms of X(ξ)-schemes Y_{i}(ξ) →^{∼} Y_{i} ×_{U} X(ξ), for every i = 1, …, n ([65, Ch.IV, Cor.8.3.12]). We may then replace X by U, and we reduce to proving the proposition for each of the restrictions Y_{i} → U of f ×_{X} 1_{U}; hence we may assume that Y (ξ) is strictly local. By the foregoing case, we may find a chart

(13.3.12)  P_{X(ξ)} → \underline{M}(ξ)  Q_{Y (ξ)} → \underline{N}(ξ)  ϑ : P → Q

of f ×_{X} 1_{X(ξ)}, with ϑ of Kummer type, such that P and Q are sharp, the order d of Coker ϑ^{gp} is invertible in 𝒪_{X(ξ)}, and the induced morphism of X(ξ)-schemes Y (ξ) → X(ξ) ×_{Spec ℤ[P]} Spec ℤ[Q] is an isomorphism. By corollary 12.2.36we may find an étale neighborhood U → X of ξ such that (13.3.12) extends to a chart for f ×_{X} 1_{U}. After shrinking U, we may assume that d is invertible in 𝒪_{U}. Lastly, after further shrinking of U, we may ensure that the induced morphism of U-schemes U ×_{X} Y → U ×_{Spec ℤ[P]} Spec ℤ[Q] is an isomorphism ([65, Ch.IV, Cor.8.8.2.4]). □

Definition 13.3.13. (i) Let f : (Y, \underline{N}) → (X, \underline{M}) be a morphism of fs log schemes. We say that f is an étale covering of (X, \underline{M}), if f fulfills the equivalent conditions (a) and (b) of proposition 13.3.11. We denote by

Cov(X, \underline{M})

the full subcategory of log/(X, \underline{M}), whose objects are the étale coverings of (X, \underline{M}).

(ii) Let K be a field, E a finite separable field extension of K, and w a discrete valuation of E; set v := w_{|K}, and denote by Γ_{v} and κ(v) (Γ_{w} and κ(w)) the value group of v (resp. of w) and the residue field of the valuation ring K^{+} of v (resp. of the valuation ring E^{+} of w). We say that the ring extension K^{+} ⊂ E^{+} is tamely ramified if the ramification index

e(w/v) := (Γ_{w} : Γ_{v})

(which is known to be finite) is invertible in κ(v), and the residue field extension κ(v) → κ(w) is separable.

Remark 13.3.14. (i) Notice that all morphisms in Cov(X, \underline{M}) are étale coverings, in light of corollary 12.3.25(ii).

(ii) Moreover, Cov(X, \underline{M}) is a Galois category (see [82, Exp.V, Déf.5.1]), and if ξ is any geometric point of (X, \underline{M})_{tr}, we obtain a fibre functor for this category, by the rule : f ↦ f^{−1}(ξ), for every étale covering f of (X, \underline{M}). (Details left to the reader.) We shall denote by

π_{1}((X, \underline{M})_{ét}, ξ)

the corresponding fundamental group.

Example 13.3.15. Let (f, log f) : (Y, \underline{N}) → (X, \underline{M}) be an étale covering of a regular log scheme (X, \underline{M}), and suppose that X is strictly local of dimension 1 and Y is connected (hence strictly local as well). Let x ∈ X (resp. y ∈ Y) be the closed point; it follows that 𝒪_{X,x} is a strictly henselian discrete valuation ring (corollary 12.5.13). The same holds for 𝒪_{Y,y} in view of theorem 12.5.28 and [66, Ch.IV, Prop.18.5.10]. In case dim \underline{M}_{x} = 0, the log structures \underline{M} and \underline{N} are trivial, so f : Y → X is an étale morphism of schemes. Otherwise we have dim \underline{M}_{x} = 1 = dim \underline{N}_{y} (lemma 6.4.59(i)), and then \underline{M}_{x}^{♯} ≃ ℕ ≃ \underline{N}_{y}^{♯} (theorem 6.4.18(ii)); also, the choice of a chart for f as in proposition 13.3.11, induces an isomorphism

𝒪_{X,x} ⊗\underline{M} ♯_{x} \underline{N}_{y}^{♯} →^{∼} 𝒪_{Y,y}

where the map \underline{M}_{x}^{♯} → \underline{N}_{y}^{♯} is the N-Frobenius map of ℕ, where N > 0 is an integer invertible in 𝒪_{X,x}. Moreover, notice that the image of the maximal ideal of \underline{M}_{x} generates the maximal ideal of 𝒪_{X,x} (and likewise for the image of \underline{N}_{y} in 𝒪_{Y,y}), so the structure map of \underline{M} induces an isomorphism \underline{M}_{x}^{♯} →^{∼} Γ_{+}, onto the submonoid of the value group (Γ, ≤) of 𝒪_{X,x} consisting of all elements ≤ 1 (and likewise for \underline{N}_{y}^{♯}). In other words, 𝒪_{Y,y} is obtained from 𝒪_{X,x} by adding the N-th root of a uniformizer. It then follows that the ring homomorphism 𝒪_{X,x} → 𝒪_{Y,y} is a finite tamely ramified extension of discrete valuation rings.

Definition 13.3.16. Let X be a normal scheme, and Z → X a closed immersion such that : (a) Z is constructible in X.

(b) Z is a union of irreducible closed subsets of codimension one in X.

(c) For every maximal point z ∈ Z, the stalk 𝒪_{X,z} is a discrete valuation ring. Set U := X\Z, let f : U′ → U be an étale covering, and g : X′ → X the normalization of X in U′. Notice that g^{−1}Z is a union of irreducible closed subsets of codimension one in X′ (by the going down theorem [126, Th.9.4(ii)]). Notice also that for every z ∈ Max Z, the induced map A := 𝒪_{X,z} → B := g_{∗}𝒪_{X′,z} is a finite ring homomorphism, since A is a noetherian domain, and since Frac(A) ⊗_{A} B is an étale Frac(A)-algebra ([126, §33, Lemma 1]). It follows easily that, for every geometric point ξ of X′ localized at a maximal point of g^{−1}Z, the induced extension of strictly henselian discrete valuation rings :

(13.3.17)  𝒪_{X,g(ξ)} → 𝒪_{X′,ξ}

is finite (notation of definition 4.9.17(iii).

(i) We say that f is tamely ramified along Z if, for every geometric point ξ of X′ localized at a maximal point of g^{−1}Z, the corresponding extension (13.3.17) is tamely ramified. Hence, let Γ_{g(ξ)} → Γ_{ξ} be the induced map of value groups; then the ramification index of f at ξ

e_{ξ}(f) := (Γ_{ξ} : Γ_{g(ξ)})

is invertible in the residue field κ(ξ), and g induces an isomorphism κ(g(ξ)) →^{∼} κ(ξ).

(ii) Suppose that f is tamely ramified along Z, and that the set of maximal points of Z is finite. Then the ramification index of f along Z is the least common multiple e_{Z}(f) of the ramification indices e_{ξ}(f), where ξ ranges over the set of all geometric points of X′ supported at a maximal point of g^{−1}Z.

(iii) We denote by Tame(X, U) the full subcategory of Cov(U) whose objects are the étale coverings of U that are tamely ramified along Z.

Lemma 13.3.18. Let K, K′, L be subfields of a given field, and V_{L′} a discrete valuation ring of L′ := K′L; for every subfield E ⊂ L′ set V_{E} := E ∩ V_{L′}, and for every pair E ⊂ E′ of such subfields, let e(E′/E) be the ramification index of the extension V_{E} ⊂ V_{E′}. We have :

(i) Suppose moreover that :

(a) V_{K} ≠ K.

(b) K′ is a finite separable extension of K, and the extension V_{K} ⊂ V_{K′} is tamely ramified. Then the extension V_{L} ⊂ V_{L′} is tamely ramified, and e(L′/L) divides e(K′/K).

(ii) Suppose that :

(a) The extension V_{L} ⊂ V_{L′} is tamely ramified, and both V_{K} and V_{L} are strictly henselian.

(b) [K′ : K] = [L′ : L] < +∞ and e(L′/K′) = 1. Then the extension V_{K} ⊂ V_{K′} is tamely ramified, e(K′/K) = e(L′/L), and e(L/K) = 1.

Proof. Let κ_{K} be the residue field of V_{K}, and define likewise the residue fields κ_{K′}, κ_{L} and κ_{L′}. (i): To begin with, let us remark :

Claim 13.3.19. In order to show (i), we may assume that V_{K} and V_{L} are strictly henselian.

Proof of the claim. Let us fix a separable closure Ω of κ_{L′}, and let V_{K}^{sh}, V_{L}^{+sh}, V_{K′}^{+sh}, V_{L′}^{+sh} be the strict henselizations of V_{K}, V_{L}, V_{K′} and V_{L′} relative to the corresponding maps to Ω. According to [75, Lemma 6.2.5], all these are discrete valuation rings, and we denote by K^{sh}, L^{sh}, K^{′sh} and L^{′sh} the respective fields of fractions; also, the ramification indices e(K^{sh}/K), e(L^{sh}/L), e(K^{′sh}/K′) and e(L^{′sh}/L′) of the extensions V_{K} ⊂ V_{K}^{sh}, V_{L} ⊂ V_{L}^{sh}, V_{K′} ⊂ V_{K′}^{sh} and V_{L′} ⊂ V_{L′}^{sh} all equal one. Moreover, using the explicit construction of strict henselizations of loc.cit., it is easily seen that K^{′sh} = K′ · K^{sh} and L^{′sh} = L′ · L^{sh} (details left to the reader), and furthermore the residue field of V_{K}^{sh} is a separable closure of κ_{K}, and likewise for the residue fields of V_{L}^{sh}, V_{K′}^{sh} and V_{L′}^{sh}. Then it follows easily that if the extension V_{L}^{sh} ⊂ V_{L′}^{sh} is tamely ramified, the same holds for the extension V_{L} ⊂ V_{L′}, and we have both e(L^{′sh}/L^{sh}) = e(L′/L) and e(K^{′sh}/K^{sh}) = e(K′/K). Thus, it suffices to prove assertion (i) for the discrete valuation ring V_{L′}^{sh} and the fields K^{sh}, K^{′sh}, L^{sh}, L^{′sh}. ♢

Henceforth, we thus assume that V_{K} and V_{L} are strictly henselian; since K′ is separable over K, it is clear that L′ is separable over L, and V_{K′} (resp. V_{L′}) is strictly henselian as well, and is the integral closure of V_{K} in K′ (resp. of V_{L} in L′); then V_{K′} is a finite V_{K}-module ([126, §33, Lemma 1]). Next, let π_{K} (resp. π_{K′}, resp. π_{L}) be a uniformizer for V_{K} (resp. for V_{K′}, resp. for V_{L}); the assumption means that π_{K} = π_{K′}^{e} · a for some a ∈ V_{K′}^{×}, and e := e(K′/K) is invertible in the residue field of V_{K}. Since V_{K′} is strictly henselian, we have a = b^{e} for some b ∈ V_{K′}, and after replacing π_{K′} by b · π_{K′}, we may assume that π_{K} = π_{K′}^{e}; since κ_{K} = κ_{K′}, we have :

V_{K′} = V_{K} + π_{K′}V_{K′} = V_{K} + π_{K′}V_{K} + π_{K′}^{2}V_{K′} = ⋯ = V_{K} + π_{K′}V_{K} + ⋯ + π_{K′}^{e−1}V_{K} + π_{K}V_{K′}

whence V_{K′} = V_{K}[π_{K′}], by Nakayama’s lemma; therefore K′ = K(π_{K′}), and K′ is the splitting field of X^{e}−π_{K} over K. It follows that L′ = L(π_{K′}) is the splitting field of the same polynomial over L, and especially, it is a finite Galois extension of L. Moreover, the injective map

Gal(L′/L) → L^{×} σ ↦ σ(π_{K′})/π_{K′}

is a character with values in the e-torsion subgroup of L^{×}; hence [L′ : L] divides e. Recalling that [L′ : L] = e(L′/L) · [κ_{L′} : κ_{L}] ([34, Ch.VI, §8, no.5, Cor.1]), the assertion follows.

(ii): The assumptions imply that κ_{L} = κ_{L′}; then, [34, Ch.VI, §8, no.5, Cor.1] yields :

e(K′/K) · [κ_{K′} : κ_{K}] · e(L/K) = [K′ : K] · e(L/K) = [L′ : L] · e(L/K)
= e(L′ : L) · e(L/K)
= e(L′ : K)

= e(L′/K′) · e(K′/K) = e(K′/K)

whence [κ_{K′} : κ_{K}] = e(L/K) = 1 and e(K′/K) = e(L′/L). □

Proposition 13.3.20. Let φ : X′ → X be a maximizing morphism of normal schemes (see definition 8.1.46(ii)), Z ⊂ X a closed subset, and set Z′ := φ^{−1}Z. Suppose that :

(a) The closed immersions Z → X and Z′ → X′ satisfy conditions (a),(b),(c) of definition 13.3.16.

(b) φ restricts to a maximizing map Z′ → Z. Set moreover U := X \ Z and U′ := X′ \ Z′. We have :

(i) φ^{∗} : Cov(U) → Cov(U′) restricts to a functor

Tame(X, U) → Tame(X′, U′).

(ii) Suppose that Max Z and Max Z′ are finite sets. Then, for any object f : V → U of

Tame(X, U), the index e_{Z′}(φ^{∗}f) divides e_{Z}(f). (iii) Suppose that X is regular and noetherian, and let j : U → X be the open immer-

sion. Then the essential image of the pull-back functor j^{∗} : Cov(X) → Tame(X, U) consists of the objects f : V → U such that e_{Z}(f) = 1.

Proof. (i,ii): Let V → U be an object of Tame(X, U), and denote by g : W → X (resp. g′ : W′ → X′) the normalization of X in V (resp. of X′ in V ×_{U} U′). Let w′ be a geometric point of W′ localized at a maximal point of g^{′−1}Z′, and denote by w (resp. x′, resp. x) the image of w′ in W (resp. in X′, resp. in X); in order to check (i) and (ii), we have to show that the ramification index of g′ at the geometric point w′ is invertible in κ(w′), and the residue field extension κ(x′) ⊂ κ(w′) is trivial. To this aim, in view of proposition 9.8.3, we may replace X′ by X′(x′), X by X(x), W by W (w), and assume from start that X, X′ and W are strictly local, say X = Spec A, X′ = Spec B and W = Spec C for some strictly henselian discrete valuation rings A, B, C. Then, by construction, we have :

Frac 𝒪_{W′,w′} = Frac(B) · Frac(C).

hence the assertion follows from lemma 13.3.18.

(iii): Let f : V → U be an object of Tame(X, U). By claim 13.1.8, there exists a largest open subset U_{max} ⊂ X containing U, and such that f is the restriction of an étale covering f_{max} : V′ → U_{max}; now, if e_{Z}(f) = 1, every point of codimension one of X\U lies in U_{max}, by claim 13.1.9. Hence X \U has codimension ≥ 2 in X, in which case lemma 13.1.7(iv) implies that X = U_{max}, as required. □

Remark 13.3.21. Let X be a normal scheme, and Z → X a closed immersion verifying conditions (a),(b),(c) of definition 13.3.16. Then the assumptions (a) and (b) of proposition 13.3.20 are fulfilled, notably, when φ is finite and maximizing, or when φ is flat and X′ is locally noetherian ([126, Th.9.4(ii), Th.9.5]).

13.3.22. Let now \underline{X} := (X_{i} | i ∈ I) be a cofiltered system of quasi-compact and quasiseparated normal schemes, such that, for every morphism φ : i → j in the indexing category I, the corresponding transition morphism f_{φ} : X_{i} → X_{j} is maximizing and affine. Suppose also, that for every i ∈ I, there exists a closed immersion Z_{i} → X_{i}, fulfilling conditions (a),(b),(c) of definition 13.3.16, such that Max Z_{i} is a finite set, and for every morphism φ : i → j of I, we have :

• Z_{i} = f_{φ}^{−1}Z_{j}.

• The corresponding morphism f_{φ} restricts to a maximizing map Z_{i} → Z_{j}. Let also X be the limit of \underline{X}, and Z the limit of the system (Z_{i} | i ∈ I), and suppose furthermore that the closed immersion Z → X fulfills as well conditions (a),(b),(c) of definition 13.3.16.

Proposition 13.3.23. In the situation of (13.3.22), we have :

(i) The induced morphism π_{i} : Z → Z_{i} is maximizing for every i ∈ I.

(ii) The universal cone (X → X_{i} | i ∈ I) induces an equivalence of categories :

2-colim_{I} Tame(X_{i}, X_{i}\Z_{i}) → Tame(X, X \Z).

Proof. (i): More precisely, we shall prove that there is a natural homeomorphism :

Max Z →^{∼} lim_{i∈I} Max Z_{i}.

(Notice the each Max Z_{i} is a discrete finite set, hence this will show that Max Z is a profinite topological space.) Indeed, suppose that z := (z_{i} | i ∈ I) is a maximal point of Z. For every i ∈ I, let T_{i} ⊂ Max Z_{i} be the subset of maximal generizations of z_{i} in Z_{i}. It is easily seen that f_{φ}T_{i} ⊂ T_{j} for every morphism φ : i → j in I. Clearly T_{i} is a finite non-empty set for every i ∈ I, hence the limit T of the cofiltered system (T_{i} | i ∈ I) is non-empty. However, any point of T is a generization of z in Z, hence it must coincide with z. The assertion follows easily.

(ii): To begin with, proposition 13.3.20(i) shows that, for every morphism φ : i → j in I, the transition morphism f_{φ} induces a pull-back functor Tame(X_{j}, X_{j}\Z_{j}) → Tame(X_{i}, X_{i}\Z_{i}), so the 2-colimit in (ii) is well-defined, and combining (i) with proposition 13.3.20(i) we obtain indeed a well-defined functor from this 2-colimit to Tame(X, X \Z).

The full faithfulness of the functor of (ii) follows from lemma 13.1.6. Next, let g : V → X\Z be an object of Tame(X, X \Z); invoking again lemma 13.1.6, we may descend g to an étale covering g_{j} : V_{j} → X_{j} \Z_{j}, for some j ∈ I; after replacing I by I/j, we may assume that j is the final object of I and we may define g_{i} := f_{φ}^{∗}(g_{j}) for every φ : i → j in I. To conclude the proof, it suffices to show that there exists i ∈ I such that g_{i} is tamely ramified along Z_{i}.

Now, let \overline{g}_{i} : \overline{V}_{i} → X_{i} be the normalization of X_{i} in V_{i} ×_{X_{j}} X_{i}, for every i ∈ I, and \overline{g} : \overline{V} → X the normalization of X in V. Given a geometric point \overline{v} localized at a maximal point of \overline{g}^{−1}Z, let \overline{v}_{i} (resp. \overline{z}_{i}) be the image of \overline{v} in \overline{V}_{i} (resp. in Z_{i}), and z_{i} ∈ Z_{i} the support of \overline{z}_{i}, for every i ∈ I. Let also \overline{z} be the image of \overline{v} in Z, and z ∈ Z the support of \overline{z}. Then

𝒪_{X,\overline{z}} = colim_{i∈I} 𝒪_{X_{i},\overline{z}_{i}} 𝒪_{\overline{V} ,\overline{v}} = colim_{i∈I} 𝒪_{\overline{V}_{i},\overline{v}_{i}}

and Frac 𝒪_{\overline{V} ,\overline{v}} = Frac(𝒪_{X,\overline{z}}) · Frac(𝒪_{\overline{V}_{i},\overline{v}_{i}}) for every i ∈ I ([66, Ch.IV, Prop.18.8.18(ii)]), and it is easily seen that there exists i ∈ I such that the ramification index of the extension 𝒪_{\overline{V}_{i},\overline{v}_{i}} ⊂ 𝒪_{\overline{V} ,\overline{v}} equals 1, and such that [Frac 𝒪_{\overline{V}_{i},\overline{v}_{i}} : Frac 𝒪_{X_{i},\overline{z}_{i}}] = [Frac 𝒪_{\overline{V} ,\overline{v}} : Frac 𝒪_{X,\overline{z}}]. Moreover, for every sufficiently large i, the induced map \overline{g}^{−1}(z) → \overline{g}_{i}^{−1}(z_{i}) is a bijection. Then, by lemma 13.3.18(ii), for every such index i, the finite extension 𝒪_{X_{i},\overline{z}_{i}} → 𝒪_{\overline{V}_{i},\overline{v}_{i}} is already tamely ramified, and since I is cofiltered, it follows that there exists i ∈ I such that the induced morphism \overline{V}_{i} ×_{X_{i}} X_{i}(\overline{z}_{i}) → X_{i}(\overline{z}_{i}) is already tamely ramified. However, notice that π_{i}^{−1}(z_{i}) is open in Max Z, for every i ∈ I, and every z_{i} ∈ Max Z_{i}. Therefore, we may find a finite subset I_{0} ⊂ I, and for every i ∈ I_{0} a subset T_{i} ⊂ Max Z_{i}, such that :

• Max Z = ⋃_{i∈I_{0}} π_{i}^{−1}(T_{i}).

• For every geometric point \overline{z}_{i} localized in T_{i}, the morphism \overline{V}_{i} ×_{X_{i}} X_{i}(\overline{z}_{i}) → X_{i}(\overline{z}_{i}) is

tamely ramified. Since I is cofiltered, we may find k ∈ I with morphisms φ_{i} : k → i for every i ∈ I_{0}; after replacing T_{i} by f_{φ_{i}}^{−1}(T_{i}) for every i ∈ I_{0}, we may then assume that I_{0} = {k}, so that Max Z = π_{k}^{−1}(T_{k}). It follows that Max Z_{i} = f_{φ}^{−1}T_{k} for some i ∈ I and some φ : i → k. Then it is clear that g_{i} is tamely ramified along Z_{i}, as required. □

13.3.24. Let X be a scheme, U ⊂ X a connected open subset, ξ any geometric point of U, and N > 0 an integer which is invertible in 𝒪_{U}; then the N-Frobenius map N of 𝒪_{U}^{×} gives a Kummer exact sequence of abelian sheaves on X_{ét} :

0 → μ_{N,U} → 𝒪_{U}^{×} ⟶^{N} 𝒪_{U}^{×} → 0

(where μ_{N,U} is the N-torsion subsheaf of 𝒪_{U_{ét}}^{×} : see [10, Exp.IX, §3.2]). Suppose now that μ_{N,U} is a constant sheaf on U_{ét}; this means especially that μ_{N} := (μ_{N,U})_{ξ} is a cyclic group of order N. Indeed, μ_{N} certainly contains such a subgroup (since N is invertible in 𝒪_{U}), so denote by ζ one of its generators; then every u ∈ μ_{N} satisfies the identity 0 = u^{N} − 1 = ∏_{i=1}^{N}(u − ζ^{i}), and each factor u_{ζ}^{i} of this decomposition vanishes on a closed subset U_{i} of U(ξ); clearly U_{i}∩U_{j} = ∅ for i ≠ j, so we get a decomposition of U(ξ) as a disjoint union of open and closed subsets U(ξ) = U_{1} ∪ ⋯ ∪ U_{N} such that u = ζ^{i} on U_{i}, for every i ≤ N; since U(ξ) is connected, it follows that U(ξ) = U_{i} for some i, i.e. ζ generates μ_{N}. There follows a natural map :

∂_{N} : Γ(U_{ét}, 𝒪_{U}^{×}) → H^{1}(U_{ét}, μ_{N,U}) →^{∼} Hom_{cont}(π_{1}(U_{ét}, ξ), μ_{N})

(lemma 13.1.12). Recall the geometric interpretation of ∂_{N} : a given u ∈ 𝒪_{U}^{×}(U) is viewed as a morphism of schemes u : U → 𝔾_{m}, where 𝔾_{m} := Spec ℤ[ℤ] is the standard multiplicative group scheme; we let U′ be the fibre product in the cartesian diagram of schemes :

(13.3.25)  \xymatrix{U′ \ar[r] \ar[d]_{φ_{u}} & 𝔾_{m} \ar[d]^{Spec ℤ[N_{ℤ}]} \\ U \ar[r]^{u} & 𝔾_{m}.}

Then φ_{u} is a torsor under the U_{ét}-group μ_{N,U}, and to such torsor, lemma 13.1.12attaches a well defined continuous group homomorphism as required.

If M > 0 is any integer dividing N, a simple inspection yields a commutative diagram :

(13.3.26)  \xymatrix{Γ(U_{ét}, 𝒪_{X}^{×}) \ar[r]^{∂_{N}} \ar@{-->}[dr]_{∂_{M}} & Hom_{cont}(π_{1}(U_{ét}, ξ), μ_{N}) \ar[d]^{π_{N,M}} \\  & Hom_{cont}(π_{1}(U_{ét}, ξ), μ_{M})}

where π_{N,M} is induced by the map μ_{N} → μ_{M} given by the rule : x ↦ x^{N/M} for every x ∈ μ_{N}.

13.3.27. Let now X be a strictly local scheme, x the closed point of X, and denote by p the characteristic exponent of the residue field κ(x) (so p is either 1 or a positive prime integer). Let also β : \underline{M} → 𝒪_{X} be a log structure on X_{ét}, take U := (X, \underline{M})_{tr}, suppose that U ≠ ∅, and fix a geometric point ξ of U; for every integer N > 0 with (N, p) = 1, we get a morphism of monoids :

(13.3.28)  \underline{M}_{x} ⟶^{β_{x}} Γ(U_{ét}, 𝒪_{X}^{×}) ⟶^{∂_{N}} Hom_{cont}(π_{1}(U_{ét}, ξ), μ_{N})

whose kernel contains \underline{M}_{x}^{N}, the image of the N-Frobenius endomorphism of \underline{M}_{x}. Notice that the N-Frobenius map of 𝒪_{X,x}^{×} is surjective : indeed, since the residue field κ(x) is separably closed, and (N, p) = 1, all polynomials in κ(x)[T] of the form T^{N} − u (for u ≠ 0) split as a product of distinct monic polynomials of degree 1; since 𝒪_{X,x} is henselian, the same holds for all polynomials in 𝒪_{X,x}[T] of the form T^{N} − u, with u ∈ 𝒪_{X,x}^{×}. It follows that (13.3.28) factors through a natural map:

\underline{M}_{x}^{♯gp} → Hom_{cont}(π_{1}(U_{ét}, ξ), μ_{N})

which is the same as a group homomorphism :

(13.3.29)  \underline{M}_{x}^{♯gp} × π_{1}(U_{ét}, ξ) → μ_{N}.

In view of the commutative diagram (13.3.26), it is easily seen that the pairings (13.3.29), for N ranging over all the positive integers with (N, p) = 1, assemble into a single pairing :

\underline{M}_{x}^{gp} × π_{1}((X, \underline{M})_{tr,ét}, ξ) → ∏_{ℓ≠p} ℤ_{ℓ}(1)

(where ℓ ranges over the prime numbers different from p, and ℤ_{ℓ}(1) := lim_{n∈ℕ} μ_{ℓ^{n}}). The latter is the same as a group homomorphism :

(13.3.30)  π_{1}((X, \underline{M})_{tr,ét}, ξ) → \underline{M}_{x}^{gp∨} ⊗_{ℤ} ∏_{ℓ≠p} ℤ_{ℓ}(1).

13.3.31. Let f : (X, \underline{M}) → (Y, \underline{N}) be a morphism of log schemes, such that both X and Y are strictly local, and f maps the closed point x of X to the closed point y of Y. Then f restricts to a morphism of schemes f_{tr} : (X, \underline{M})_{tr} → (Y, \underline{N})_{tr} (remark 12.2.8(i)). Fix again a geometric point ξ of (X, \underline{M})_{tr}; we get a diagram of group homomorphisms :

\xymatrix{π_{1}((X, \underline{M})_{tr,ét}, ξ) \ar[r]^{π_{1}(f_{tr},ξ)} \ar[d] & π_{1}((Y, \underline{N})_{tr,ét}, f(ξ)) \ar[d] \\ \underline{N}_{y}^{gp∨} ⊗_{ℤ} ∏_{ℓ≠p} ℤ_{ℓ}(1) \ar[r] & \underline{M}_{x}^{gp∨} ⊗_{ℤ} ∏_{ℓ≠p} ℤ_{ℓ}(1)}

whose vertical arrows (resp. bottom horizontal arrow) are the maps (13.3.30) (resp. is induced by (log f_{x})^{gp∨}), and by inspecting the constructions, it is easily seen that this diagram commutes.

13.3.32. Suppose now that (X, \underline{M}) as in (13.3.27) is a fs log scheme, so that P := \underline{M}_{x}^{♯} is fine and saturated; in this case, we wish to give a second construction of the pairing (13.3.29).

• Namely, for every integer N > 0 set S_{P} := Spec ℤ[P] and consider the finite morphism of schemes g_{N} : S_{P} → S_{P} induced by the N-Frobenius endomorphism of P. Notice that S_{P} contains the dense open subset U_{P} := Spec ℤ[1/N, P^{gp}], and it is easily seen that the restriction g_{N|U_{P}} : U_{P} → U_{P} of g_{N} is an étale morphism. Let τ ∈ U_{P} be any geometric point; then τ corresponds to a ring homomorphism ℤ[1/N, P^{gp}] → κ := κ(τ), which is the same as a character χ_{τ} : P^{gp} → κ^{×}. Likewise, any τ′ ∈ g_{N}^{−1}(τ) is determined by a character χ_{τ′} : P^{gp} → κ^{×} extending χ_{τ}, i.e. such that χ_{τ′}(x^{N}) = χ_{τ}(x) for every x ∈ P^{gp}. Notice that every character χ_{τ} as above admits at least one such extension χ_{τ′}, since κ is separably closed, and N is invertible in κ. Let C_{N} be the cokernel of the N-Frobenius endomorphism of P^{gp}; there follows a short exact sequence of finite abelian groups :

0 → Hom_{ℤ}(C_{N}, κ^{×}) → Hom_{ℤ}(P^{gp}, κ^{×}) ⟶^{N} Hom_{ℤ}(P^{gp}, κ^{×}) → 0.

Especially, we see that the fibre g_{N}^{−1}(τ) is naturally a torsor under the group Hom_{ℤ}(C_{N}, κ^{×}) = Hom_{ℤ}(P^{gp}, μ_{N}(κ)), where μ_{N}(κ) ⊂ κ^{×} is the N-torsion subgroup. Hence g_{N|U_{P}} is an étale Galois covering of U_{P}, whose Galois group is naturally isomorphic to Hom_{ℤ}(P^{gp}, μ_{N}(κ)); therefore, g_{N|U_{P}} is classified by a well defined continuous representation :

(13.3.33)  π_{1}(U_{P,ét}, τ) → Hom_{ℤ}(P^{gp}, μ_{N}(κ))

(lemma 13.1.12). The corresponding action of Hom_{ℤ}(P^{gp}, μ_{N}(κ)) on U_{P} can be extracted from the construction : namely, let χ : P^{gp} → μ_{N}(κ) be a given character; by definition, the action of χ sends the geometric point τ to the geometric point τ′ whose character χ_{τ′} : P^{gp} → κ^{×} is given by the rule : x ↦ χ(x) · χ_{τ} for every x ∈ P^{gp}. Consider the automorphism ρ_{χ} of the ℤ-algebra ℤ[P^{gp}] given by the rule x ↦ χ(x) · x for every x ∈ P^{gp}, and notice that ρ(τ) = τ′; since the fibre functors are faithful on étale coverings, we conclude that χ acts as Spec ρ_{χ} on U_{P}.

• Moreover, if λ : Q → P is any morphism of fine and saturated monoids, clearly we have a commutative diagram

\xymatrix{U_{P} \ar[r]^{Spec ℤ[λ]_{|U_{P}}} \ar[d]_{g_{N|U_{P}}} & U_{Q} \ar[d]^{g_{N|U_{Q}}} \\ U_{P} \ar[r]^{Spec ℤ[λ]_{|U_{P}}} & U_{Q}}

whence, by remark 13.1.14(iii.a), a well defined group homomorphism

(13.3.34)  Hom_{ℤ}(P^{gp}, μ_{N}) → Hom_{ℤ}(Q^{gp}, μ_{N})

that makes commute the induced diagram :

\xymatrix{π_{1}(U_{P,ét}, ξ) \ar[r] \ar[d] & π_{1}(U_{Q}, ξ′) \ar[d] \\ Hom_{ℤ}(P^{gp}, μ_{N}) \ar[r] & Hom_{ℤ}(Q^{gp}, μ_{N})}

whose vertical arrows are the maps (13.3.33). By inspecting the constructions, it is easily seen that (13.3.34) is just the map Hom_{ℤ}(λ^{gp}, μ_{N}).

• Next, by corollary 12.1.36(i) the projection \underline{M}_{x} → P admits a splitting

α : P → \underline{M}_{x}

(which defines a sharp chart for \underline{M}); if N is invertible in 𝒪_{X}, this splitting induces a morphism of schemes X → S_{P}, which restricts to a morphism U → U_{P}. If we let τ be the image of ξ, we deduce a continuous group homomorphism π_{1}(U_{ét}, ξ) → π_{1}(U_{P,ét}, τ) ([82, Exp.V, §7]), whose composition with (13.3.33) yields a continuous map :

(13.3.35)  π_{1}(U_{ét}, ξ) → Hom_{ℤ}(P^{gp}, μ_{N}(κ)).

We claim that the pairing P^{gp} × π_{1}(U_{ét}, ξ) → μ_{N}(κ) arising from (13.3.35) agrees with (13.3.29), under the natural identification μ_{N}(κ) →^{∼} μ_{N}. Indeed, for given s ∈ P, let j_{s} : ℤ → P^{gp} be the map such that j_{s}(n) := s^{n} for every n ∈ ℤ; by composing (13.3.35) with Hom_{ℤ}(j_{s}, μ_{N}(κ)), we obtain a map π_{1}(U_{ét}, ξ) → Hom_{ℤ}(ℤ, μ_{N}(κ)) →^{∼} μ_{N}(κ); in view of the foregoing, it is easily seen that the corresponding μ_{N}(κ)-torsor is precisely φ_{α(s)}, in the notation of (13.3.25), whence the contention.

Proposition 13.3.36. In the situation of (13.3.27), suppose that (X, \underline{M}) is a regular log scheme. Then (13.3.30) is a surjection.

Proof. From the discussion of (13.3.32), we are reduced to showing that the morphism (13.3.35) is surjective, for every N > 0 such that (N, p) = 1. The latter comes down to showing that the corresponding torsor g_{N|U_{P}} ×_{U_{P}} U : U′ → U is connected (remark 13.1.14(i)). However, the map α : P → \underline{M}_{x} induces a morphism of log schemes ψ : (X, \underline{M}) → Spec(ℤ, P) (see (12.2.13)); we remark the following :

Claim 13.3.37. Slightly more generally, for any Kummer morphism ν : P → Q of monoids, with Q fine, sharp and saturated, define (X_{ν}, \underline{M}_{ν}) as the fibre product in the cartesian diagram:

(13.3.38)  \xymatrix{(X_{ν}, \underline{M}_{ν}) \ar[r] \ar[d]_{f_{ν}} & Spec(ℤ, Q) \ar[d]^{Spec(ℤ,ν)} \\ (X, \underline{M}) \ar[r]^{ψ} & Spec(ℤ, P)}

Then (X_{ν}, \underline{M}_{ν}) is regular, and X_{ν} is strictly local.

Proof of the claim. By lemma 12.5.24, (X_{ν}, \underline{M}_{ν}) is regular at every point of the closed fibre X_{ν} ×_{X} Spec κ(x), and since f_{ν} is a finite morphism, every point of X_{ν} specializes to a point of the closed fibre, therefore (X_{ν}, \underline{M}_{ν}) is regular (theorem 12.5.31). Moreover, X_{ν} is connected, provided the same holds for the closed fibre; in turn, this follows immediately from proposition 13.3.7(ii). Then X_{ν} is strictly local, by [66, Ch.IV, Prop.18.5.10]. ♢

From claim 13.3.37we see that X_{ν} is normal (corollary 12.5.13) for every such ν, especially this holds for the N-Frobenius endomorphism of P, in which case U′ is an open subset of X_{ν}; since the latter is connected, the same then holds for U′. □

Example 13.3.39. (i) In the situation of example 12.6.20, let X → S be any morphism of schemes, set \underline{X} := X ×_{S} \underline{S}, denote by (X, \underline{M}) the log scheme underlying \underline{X}, and define (X_{(k)}, \underline{M}_{(k)}) as the fibre product in the cartesian diagram of log schemes :

\xymatrix{(X_{(k)}, \underline{M}_{(k)}) \ar[r]^{π_{(k)}} \ar[d]_{_{k_{X}}} & Spec(R, P) \ar[d]^{Spec(R,k_{P})} \\ (X, \underline{M}) \ar[r]^{π} & Spec(R, P)}

where π is the natural projection. Set as well

\underline{X}_{(k)} := ((X_{(k)}, \underline{M}_{(k)}), T_{P} , ψ_{P} ∘ π_{(k)}^{♯})

which is an object of 𝒦_{int}; by proposition 12.6.14(iii), diagram (6.5.28) (with T := T_{P}) underlies a commutative diagram in 𝒦_{int} :

\xymatrix{φ^{∗}\underline{X}_{(k)} \ar[r] \ar[d]_{_{\underline{g}_{X}}} & \underline{X}_{(k)} \ar[d]^{(k_{X},k_{T_{P}})} \\ φ^{∗}\underline{X} \ar[r] & \underline{X}}

whose horizontal arrows are cartesian. Clearly this diagram is isomorphic to X ×_{S} (12.6.21); we deduce that the morphism g_{X} of log schemes underlying \underline{g}_{X} is finite, and of Kummer type; by proposition 13.3.11, g_{X} is even an étale covering, if k is invertible in 𝒪_{X}.

(ii) Suppose furthermore, that (X, \underline{M}) is regular; by claim 13.3.37it follows that (X_{(k)}, \underline{M}_{(k)}) is regular as well, and then the same holds for the log schemes (X_{φ}, \underline{M}_{φ}) and (Y, \underline{N}) underlying respectively φ^{∗}\underline{X} and φ^{∗}\underline{X}_{(k)} (proposition 12.6.14(v) and theorem 12.5.28). Let now y ∈ Y be a point of height one in Y, lying in the closed subset Y \(Y, \underline{N})_{tr}, and set x := g_{X}(y); arguing as in example 13.3.15, we see that \underline{N}_{y}^{♯} and \underline{M}_{φ,x}^{♯} are both isomorphic to ℕ, and the induced map 𝒪_{X_{φ},x} → 𝒪_{Y,y} is an extension of discrete valuation rings, whose corresponding extension of valued fields is finite. Denote by Γ_{x} → Γ_{y} the associated extension of value groups; in view of (12.6.22) we see that the ramification index (Γ_{y} : Γ_{x}) equals k.

13.3.40. Resume the situation of (13.3.32). Every (finite) discrete quotient map

ρ : \underline{M}_{x}^{gp∨} ⊗_{ℤ} ∏_{ℓ≠p} ℤ_{ℓ}(1) → G

corresponds, via composition with (13.3.30), to a G_{U}-torsor, which can be explicitly constructed as follows. Set P := \underline{M}_{x}^{♯}, pick a splitting α as in (13.3.32), choose an integer N > 0 large enough, so that (N, p) = 1, and ρ factors through a group homomorphism

\overline{ρ} : Hom_{ℤ}(P^{gp}, μ_{N}(κ)) → G

and set H := Ker \overline{ρ}. Now, via (13.3.30), the quotient G′ := Hom_{ℤ}(P^{gp}, μ_{N}(κ)) corresponds to the G′_{U}-torsor on U obtained from g_{N} : U_{P} → U_{P} via pull-back along the morphism U → U_{P} given by the chart α (notation of (13.3.32)). According to remark 13.1.14(ii), the sought G_{U} is therefore isomorphic to the one obtained from the quotient \overline{g}_{N} : U_{P} /H → U_{P}, via pull-back along the same morphism. To exhibit such quotient, consider the exact sequence:

0 → Hom_{ℤ}(G, κ^{×}) → P^{gp} ⊗_{ℤ} ℤ/Nℤ → Hom_{ℤ}(H, κ^{×}) → 0.

Define Q^{gp} ⊂ P^{gp} as the kernel of the induced map P^{gp} → Hom_{ℤ}(H, κ^{×}), and set Q := Q^{gp} ∩ P. By construction, the N-Frobenius endomorphism of P factors through an injective map ν : P → Q and the inclusion map j : Q → P, so g_{N} factors as a composition :

U_{P} → U_{Q} ⟶^{h} U_{P} .

The maps on geometric points U_{P} (κ) → U_{Q}(κ) → U_{P} (κ) induced by g_{N} and h correspond to j^{gp∗} and respectively ν^{gp∗} in the resulting commutative diagram

[figure: 0; 0; H; Hom_{ℤ}(P^{gp}, μ_{N}(κ)); G; 0; 0; H; Hom_{ℤ}(P^{gp}, κ^{×}); j^{gp∗}; Hom_{ℤ}(Q^{gp}, κ^{×}); ν^{gp∗}; 0; Hom_{ℤ}(P^{gp}, κ^{×}); 0]

whose rows and column are exact. Hence, let τ and χ_{τ} : P^{gp} → κ^{×} be as in (13.3.32); the fibre h^{−1}(τ) corresponds to the set of all characters χ_{τ′} : Q^{gp} → κ^{×} whose restriction χ_{τ′} ∘ ν^{gp} to P^{gp} agrees with χ_{τ}. Then, with the notation of (13.3.38), we conclude that the restriction (f_{ν})_{tr} : (X_{ν}, \underline{M}_{ν})_{tr} → U is the sought G_{U}-torsor. Notice as well that f_{ν} is an étale covering of (X, \underline{M}).

• Here is another handier description of the same submonoid Q of P. Notice that ρ is the same as a group homomorphism

ρ^{†} : P^{gp∨} → Hom_{ℤ}(_{∏ℓ≠p}ℤ_{ℓ}(1), G)

and let L := Ker ρ^{†}. Fix a generator ζ_{N} of μ_{N}(κ); by definition

Q^{gp} = \{x ∈ P^{gp} | t(x) ⊗ ξ = 0 for all t ∈ P^{gp∨} and ξ ∈ μ_{N} such that \overline{ρ}(t ⊗ ξ) = 0\}
= \{x ∈ P^{gp} | t(x) ⊗ ζ_{N} = 0 for all t ∈ P^{gp∨} such that \overline{ρ}(t ⊗ ζ_{N}) = 0\}
= {x ∈ P^{gp} | t(x) ∈ Nℤ for all t ∈ L}.

On the other hand, notice that the N-Frobenius of P^{gp∨} factors through a map β : P^{gp∨} → L and the inclusion map i : L → P^{gp∨}, and β∘i is the N-Frobenius map of L. Let ω : P^{gp} →^{∼} (P^{gp∨})^{∨} be the natural isomorphism. We may then write

Q^{gp} = {x ∈ P^{gp} | ω(x)(t) ∈ Nℤ for all t ∈ L}
= {x ∈ P^{gp} | ω(x) ∘ i ∈ N · L^{∨} = i^{∨} ∘ β^{∨}(L^{∨})}
= ω^{−1}(Im β^{∨}).

In other words, ω induces a natural isomorphism Q^{gp} →^{∼} L^{∨}, that identifies β^{∨} with the map j^{gp} : Q^{gp} → P^{gp}, and then necessarily also the map i^{∨} with ν. Now, the image of Q inside L^{∨} can be recovered just as L^{∨} ∩ P_{ℚ}^{∨∨} (the intersection here takes place in L_{ℚ}^{∨}, which contains (P^{gp∨})^{∨}, via the injective map i^{∨} : details left to the reader).

13.3.41. Our chief supply of tamely ramified coverings comes from the following source. Let f : (Y, \underline{N}) → (X, \underline{M}) be an étale morphism of log schemes, whose underlying morphism of schemes is finite, and suppose that (X, \underline{M}) is regular. Then (Y, \underline{N}) is regular, and both X and Y are normal schemes (theorem 12.5.28 and corollary 12.5.13). Moreover, lemma 6.4.59(ii) and proposition 13.3.11imply that (Y, \underline{N})_{tr} = f^{−1}(X, \underline{M})_{tr}, hence the restriction of f

f_{tr} : (Y, \underline{N})_{tr} → (X, \underline{M})_{tr}

is a finite étale morphism of schemes (corollary 12.3.27(i)). Furthermore, it follows easily from corollary 12.5.20(i) that the closed subset X \(X, \underline{M})_{tr} is a union of irreducible closed subsets of codimension 1 in X (and the union is locally finite on the Zariski topology of X); the same holds also for Y \(Y, \underline{N})_{tr}, especially Y is the normalization of (Y, \underline{N})_{tr} over X. Finally, example 13.3.15 implies that f_{tr} is tamely ramified along X \(X, \underline{M})_{tr}. It is then clear that the rule f ↦ f_{tr} defines a functor :

F_{(X,\underline{M})} : Cov(X, \underline{M}) → Tame(X, (X, \underline{M})_{tr}).

It follows easily from remark 12.5.40 that F_{(X,\underline{M})} is fully faithful; therefore, any choice of a geometric point ξ of (X, \underline{M})_{tr} determines a surjective group homomorphism :

(13.3.42)  π_{1}((X, \underline{M})_{tr,ét}, ξ) → π_{1}((X, \underline{M})_{ét}, ξ)

([82, Exp.V, Prop.6.9]).

Proposition 13.3.43. In the situation of (13.3.27), suppose that (X, \underline{M}) is a regular log scheme. Then the map (13.3.30) factors through (13.3.42), and induces an isomorphism :

(13.3.44)  π_{1}((X, \underline{M})_{ét}, ξ) →^{∼} \underline{M} gp∨ ; \overline{x} ⊗_{ℤ} ∏_{ℓ≠p} ℤ_{ℓ}(1).

Proof. The discussion of (13.3.40) shows that (13.3.30) factors through (13.3.42), and proposition 13.3.36 implies that (13.3.44) is surjective. Next, let f : (Y, \underline{N}) → (X, \underline{M}) be an étale covering; according to proposition 13.3.11, f admits a Kummer chart (ω_{P} , ω_{Q}, ϑ) with Q fine, sharp and saturated, such that the order k of Coker ϑ^{gp} is invertible in 𝒪_{Y}. Moreover, the induced morphism of X-schemes Y → X ×_{Spec ℤ[P]} Spec ℤ[Q] is an isomorphism. With this notation, denote by G the cokernel of the induced group homomorphism Hom_{ℤ}(ϑ^{gp}, k_{P^{gp}}); there follows a map ρ : P^{gp} ⊗_{ℤ} ∏_{ℓ≠p} ℤ_{ℓ}(1) → G, and by inspecting the construction, one may check that the corresponding G_{U}-torsor, constructed as in (13.3.40), is isomorphic to f ×_{X} 1_{U} (details left to the reader). This shows the injectivity of (13.3.44), and completes the proof of the proposition. □

The following result is the logarithmic version of the classical Abhyankar’s lemma.

Theorem 13.3.45. The functor F_{(X,\underline{M})} is an equivalence.

Proof. First we show how to reduce to the case where τ = ét.

Claim 13.3.46. Let (X_{Zar}, \underline{M}) be a regular log structure on the Zariski site of X, and suppose that the theorem holds for ũ^{∗}(X_{Zar}, \underline{M}). Then the theorem holds for (X_{Zar}, \underline{M}) as well.

Proof of the claim. Let g : V → (X_{Zar}, \underline{M})_{tr} be an étale covering whose normalization over X is tamely ramified along X \ (X_{Zar}, \underline{M})_{tr}. By assumption, there exists an étale covering f : (Y_{ét}, \underline{N}) → ũ^{∗}(X_{Zar}, \underline{M}) such that f_{tr} = g; by lemma 13.3.10, we may then find a morphism of log schemes of Kummer type f_{Zar} : ũ_{∗}(Y_{ét}, \underline{N}) → (X_{Zar}, \underline{M}) such that ũ^{∗}f_{Zar} = f, therefore f_{Zar} is an étale covering of (X_{Zar}, \underline{M}) (corollary 12.3.27(iii)), and clearly (f_{Zar})_{tr} = g, so the functor F_{(X_{Zar},\underline{M})} is essentially surjective. Full faithfulness for the same functor is derived formally from the full faithfulness of the functor F_{ũ^{∗}(X_{Zar},\underline{M})}, and that of the functor (13.3.9) : details left to the reader. ♢

Henceforth we assume that τ = ét.

Claim 13.3.47. Let g : V → (X, \underline{M})_{tr} be an object of Tame(X, (X, \underline{M})_{tr}). Then g lies in the essential image of F_{(X,\underline{M})} if (and only if) there exists anétale covering family (U_{λ} → X | λ ∈ Λ) of X, such that g ×_{X} U_{λ} lies in the essential image of F_{(U_{λ},\underline{M}_{|U_{λ}})}, for every λ ∈ Λ.

Proof of the claim. We have to exhibit a finite étale covering f : (Y, \underline{N}) → (X, \underline{M}) such that f_{tr} = g. However, given such f, theorem 12.5.28 and corollary 12.5.13 imply that Y is the normalization of V over X, and proposition 12.5.38 says that \underline{N} = j_{∗}𝒪_{V}^{×} ∩ 𝒪_{Y}, where j : V → Y is the open immersion; then log f : f^{∗}\underline{M} → \underline{N} is completely determined by f, hence by g. Thus, we come down to showing that :

(i) \underline{N} := j_{∗}𝒪_{V}^{×} ∩ 𝒪_{Y} is a regular log structure on the normalization Y of V over X.

(ii) The unique morphism f : (Y, \underline{N}) → (X, \underline{M}) is an étale covering.

However, [126, §33, Lemma 1] implies that Y is finite over X. To show (ii), it then suffices to prove that each restriction f_{λ} := f ×_{X} 1_{U_{λ}} is étale (proposition 12.3.24(iii)). Likewise, (i) holds, provided the restriction \underline{N}_{|Y_{λ}} is a regular log structure on Y_{λ} := Y ×_{X} U_{λ}, for every λ ∈ Λ. Let j_{λ} : V_{λ} := V ×_{X} U_{λ} → Y_{λ} be the induced open immersion. It is clear that \underline{N}_{|Y_{λ}} = j_{λ∗}𝒪_{V_{λ}}^{×} ∩ 𝒪_{Y_{λ}}, and Y_{λ} is the normalization of V_{λ} over U_{λ} (proposition 9.8.3); moreover, (U_{λ}, \underline{M}_{|U_{λ}}) is again regular, so our assumption implies that (Y_{λ}, \underline{N}_{|Y_{λ}}) is the unique regular log structure on Y_{λ} whose trivial locus is V_{λ}, and that f_{λ} is indeed étale. ♢

Claim 13.3.48. If F_{(X(ξ),\underline{M}(ξ))} is essentially surjective for every geometric point ξ of X, the same holds for F_{(X,\underline{M})}.

Proof of the claim. Indeed, let g : V → (X, \underline{M})_{tr} be an object of Tame(X, (X, \underline{M})_{tr}), and ξ any geometric point of X. By claim 13.3.47 it suffices to find an étale neighborhood U → X of ξ, such that g ×_{X} 1_{U} lies in the essential image of F_{(U,\underline{M}_{|U})}. Denote by Y the normalization of X in V, which is a finite X-scheme ([126, §33, Lemma 1]). The scheme Y (ξ) := Y ×_{X} X(ξ) decomposes as a finite disjoint union of strictly local open and closed subschemes Y_{1}(ξ), …, Y_{n}(ξ), and we may find an étale neighborhood U → X of ξ, and a decomposition of Y ×_{X} U by open and closed subschemes Y_{1}, …, Y_{n}, with isomorphisms of X(ξ)-schemes Y_{i}(ξ) →^{∼} Y_{i} ×_{X} X(ξ) for every i = 1, …, n ([65, Ch.IV, Cor.8.3.12]). We are then reduced to showing that all the restrictions Y_{i} → U lie in the essential image of F_{(U,\underline{M}_{|U})}. Thus, we may replace X by U, and Y by any Y_{i}, after which we may assume that Y (ξ) is strictly local. Proposition 12.5.16 and theorem 12.5.31 imply that (X(ξ), \underline{M}(ξ)) is a regular log scheme. Clearly g_{ξ} := g ×_{X} 1_{X(ξ)} is an object of Tame(X(ξ), (X(ξ), \underline{M}(ξ))_{tr}), so by assumption there exists a finite étale covering h : (Z, \underline{N}) → (X(ξ), \underline{M}(ξ)) such that h_{tr} = g_{ξ}. By theorem 12.5.28 and corollary 12.5.13 we know that Z is a normal scheme, and we deduce that Z = Y (ξ) ([66, Ch.IV, Prop.17.5.7]).

According to proposition 13.3.11, the morphism h admits a fine and saturated Kummer chart (ω_{P} , ω_{Q}, ϑ : P → Q), such that the order d of Coker ϑ^{gp} is invertible in 𝒪_{X,ξ}, and such that the induced map Y (ξ) → X(ξ) ×_{Spec P} Spec Q is an isomorphism. By proposition 12.2.30, there exist an étale neighborhood U → X of ξ, and a coherent log structure \underline{N}′ on Y′ := Y ×_{X} U with an isomorphism Y (ξ) ×_{Y′} (Y′, \underline{N}′) →^{∼} (Y (ξ), \underline{N}). Moreover, let h′ : Y′ → U be the projection; after shrinking U, the map log h descends to a morphism of log structures h^{′∗}\underline{M}_{|U} → \underline{N}′, whence a morphism (h′, log h′) : (Y′, \underline{N}′) → (U, \underline{M}_{|U}) of log schemes, such that h′ ×_{U} 1_{X(ξ)} = h. After further shrinking U, we may also assume that h′ admits a Kummer chart (ω′_{P} , ω′_{Q}, ϑ) (corollary 12.2.36), that d is invertible in 𝒪_{U}, and that the induced morphism Y′ → U ×_{Spec P} Spec Q is an isomorphism ([65, Ch.IV, Cor.8.8.24]). Then h′ is an étale covering, by proposition 13.3.11, and by construction, F_{(U,\underline{M}_{|U})}(h′) = g ×_{X} 1_{U}, as desired. ♢

Claim 13.3.49. Assume that X is strictly local, denote by x the closed point of X, set U := (X, \underline{M})_{tr}, and P := \underline{M}_{x}^{♯}. Let h : V → U be any connected non-empty étale covering, tamely ramified along X \U; then h lies in the essential image of F_{(X,\underline{M})}, provided there exist a fine, sharp and saturated monoid Q, and a morphism ν : P → Q of Kummer type, such that

h_{ν} := h ×_{X} X_{ν} : V ×_{X} X_{ν} → U ×_{X} X_{ν}

admits a section (notation of (13.3.38)).

Proof of the claim. Suppose first that the order of Coker ν^{gp} is invertible in 𝒪_{X}; in this case, f_{ν} is an étale covering of log schemes (proposition 13.3.11), hence (f_{ν})_{tr} is an étale covering of the scheme U. Then, by composing a section of h_{ν} with the projection V ×_{X} X_{ν} → V we deduce a morphism U ×_{X} X_{ν} → V of étale coverings of U. Such morphism shall be open and closed, hence surjective, since V is connected; hence the π_{1}(U, ξ)-set h^{−1}(ξ) will be a quotient of f_{ν}^{−1}(ξ), on which π_{1}(U, ξ) acts through (13.3.30), as required. Next, for a general morphism ν of Kummer type, let L ⊂ Q^{gp} be the largest subgroup such that ν^{gp}(P^{gp}) ⊂ L, and (L : P^{gp}) is invertible in 𝒪_{X}. Set Q′ := L ∩ Q, and notice that Q^{′gp} = L. Indeed, every element of L can be written in the form x = b^{−1}a, for some a, b ∈ Q; then choose n > 0 such that b^{n} ∈ νP, write x = b^{−n} · (b^{n−1}a) and remark that b^{−n}, b^{n−1}a ∈ Q′. The morphism ν factors as the composition of ν′ : P → Q′ and ψ : Q′ → Q, and therefore f_{ν} factors through a morphism f_{ψ} : X_{ν} → X_{ν′}. In view of the previous case, we are reduced to showing that the morphism h_{ν′} already admits a section, hence we may replace (X, \underline{M}) by (X_{ν′}, \underline{M}_{ν′}) (which is still regular and strictly local, by claim 13.3.37), h by h_{ν′}, and ν by ψ, after which we may assume that the order of Coker ν^{gp} is p^{m} for some integer m > 0, where p > 0 is the characteristic of the residue field κ(x), and then we need to show that h already admits a section.

Next, using again claim 13.3.37 and an easy induction, we may likewise reduce to the case where m = 1. Say that X = Spec A, and suppose first that A is a 𝔽_{p}-algebra (where 𝔽_{p} is the finite field with p elements), so that X_{ν} = Spec A ⊗_{𝔽_{p}[P]} 𝔽_{p}[Q]; it is easily seen that the ring homomorphism 𝔽_{p}[ν] : 𝔽_{p}[P] → 𝔽_{p}[Q] is invertible up to Φ, in the sense of [75, Def.3.5.8(i)]. Especially, Spec 𝔽_{p}[ν] is integral, surjective and radicial, hence the same holds for f_{ν}, and therefore the morphism of sites :

f_{ν}^{∗} : X_{ét} → X_{ν,ét}

is an equivalence of categories (lemma 13.1.7(i)). It follows that in this case, h admits a section if and only if the same holds for h_{ν}.

Next, suppose that the field of fractions K of A has characteristic zero; we may write X_{ν} ×_{X} Spec K = Spec K_{ν} and V ×_{X} Spec K = Spec E for two field extensions K_{ν} and E of K, such that [K_{ν} : K] = p, and the section of h_{ν} yields a map E → K_{ν} of K-algebras. Therefore we have either E = K (in which case V = U, and then we are done), or else E = K_{ν}, in which case V = (X_{ν}, \underline{M}_{ν})_{tr}, since both these schemes are normal and finite over U.

Hence, let us assume that V = (X_{ν}, \underline{M}_{ν}), and pick any point η ∈ X of codimension one, whose residue field κ(η) has characteristic p. Then X_{ν} ×_{X} X(η) = Spec B, where B := 𝒪_{X,η} ⊗_{ℤ[P]} ℤ[Q], and \overline{B} := B ⊗_{A} κ(η) = κ(η) ⊗_{𝔽_{p}[P]} 𝔽_{p}[Q]. Since the map 𝔽_{p}[P] → 𝔽_{p}[Q] is invertible up to Φ, we easily deduce that \overline{B} is local, and its residue field is a purely inseparable extension of κ(η), say of degree d. Therefore f_{ν}^{−1}η consists of a single point η′, and 𝒪_{X_{ν},η′} ≃ B is a normal and finite 𝒪_{X,η}-algebra, hence it is a discrete valuation ring. Let e denote the ramification index of the extension 𝒪_{X,η} → 𝒪_{X_{ν},η′}; then ed = p ([34, Ch.VI, §8, n.5, Cor.1]). Suppose that e = p; in view of [75, Lemma 6.2.5], the ramification index of the induced extension of strict henselizations 𝒪_{X,η}^{sh} → 𝒪_{X_{ν},η′}^{sh} is still equal to p, which is impossible, since h is tamely ramified along X \U. In case e = 1, we must have d = p, and since the residue field κ(η)^{s} of 𝒪_{X,η}^{sh} is a separable closure of κ(η), it follows easily that the residue field of 𝒪_{X_{ν},η′}^{sh} must be a purely inseparable extension of κ(η)^{s} of degree p, which again contradicts the tameness of h. ♢

Claim 13.3.50. If (X, \underline{M}) = (X, \underline{M})_{2}, then F_{(X,\underline{M})} is essentially surjective (notation of definition 12.2.7(i)).

Proof of the claim. By claim 13.3.48, we may assume that X is strictly henselian. In this case, let U, P and h : V → U be as in claim 13.3.49. By assumption d := dim P ≤ 2, and we have to show that h ×_{X} X_{ν} admits a section, for a suitable Kummer morphism ν : P → Q (notation of (13.3.38)). If d = 0, then P = {1}, in which case U = X is strictly local, so its fundamental group is trivial, and there is nothing to prove. In case d = 1, then P ≃ ℕ (theorem 6.4.18(ii)), in which case X is a regular scheme (corollary 12.5.19), and X \U is a regular divisor (remark 12.5.15) and then the assertion follows from the classical Abhyankar’s lemma ([82, Exp.XIII, Prop.5.2]). For d = 2, we may find e_{1}, e_{2} ∈ P, and an integer N > 0 such that

ℕe_{1} ⊕ ℕe_{2} ⊂ P ⊂ P′ := ℕ\frac{e_{1}}{N} ⊕ ℕ\frac{e_{2}}{N}

(see example 6.4.19(i)). Especially, the inclusion ν : P → P′ is a morphism of Kummer type. Since a composition of morphisms of Kummer type is obviously of Kummer type, claims 13.3.49 and 13.3.37 imply that we may replace (X, \underline{M}) by (X_{ν}, \underline{M}_{ν}) and h by h_{ν} (notation of (13.3.38)), after which we may assume that P is isomorphic to ℕ^{⊕2}. In this case, X is again regular and X \U is a strict normal crossings divisor, so the assertion follows again from [82, Exp.XIII, Prop.5.2]. ♢

Claim 13.3.51. In the situation of (13.3.1), suppose that (Y, \underline{N}) is regular. Then the functor f_{tr}^{∗} : Cov(Y, \underline{N})_{tr} → Cov(Y′, \underline{N}′)_{tr} restricts to an equivalence :

(13.3.52)  Tame(Y, (Y, \underline{N})_{tr}) →^{∼} Tame(Y′, (Y′, \underline{N}′)_{tr}).

Proof of the claim. Arguing as in the proof of proposition 13.3.2, we may reduce to the case where φ : T′ → T is the saturated blow up of an ideal generated by two elements of Γ(T, 𝒪_{T}).

Notice that (f, log f) is an étale morphism (proposition 12.6.14(v)), and it restricts to an isomorphism f_{tr} : (Y′, \underline{N}′)_{tr} →^{∼} (Y, \underline{N})_{tr} (remark 12.6.17). Especially, (Y′, \underline{N}′) is regular, and f_{tr}^{∗} is trivially an equivalence from the étale coverings of (Y, \underline{N})_{tr} to those of (Y′, \underline{N}′)_{tr}.

Let g : V → (Y, \underline{N})_{tr} be an object of Tame(Y, (Y, \underline{N})_{tr}). By claim 13.3.50 and lemma 12.5.21, the functor F_{ũ^{∗}(Y,\underline{N})_{2}} is essentially surjective, and then the same holds for F_{(Y,\underline{N})_{2}}, in view of claim 13.3.46; hence we may find an étale covering (W, \underline{Q}) → (Y, \underline{N})_{2}, and an isomorphism V →^{∼} (Y, \underline{N})_{tr} ×_{Y} W of (Y, \underline{N})_{tr}-schemes. On the other hand, example 6.5.56(ii) shows that φ restricts to a morphism T′_{1} → T_{2} (notation of (6.5.16)), consequently f restricts to a morphism (Y′, \underline{N}′)_{1} → (Y, \underline{N})_{2}. There follows a well defined étale covering :

(Y′, \underline{N}′)_{1} ×_{(Y,\underline{N})_{2}} (W, \underline{Q}) → (Y′, \underline{N}′)_{1}.

whose image under F_{(Y′,\underline{N}′)_{1}} is f_{tr}^{∗}(g). However, (Y′, \underline{N}′)_{1} contains all the points of height one of Y′\(Y′, \underline{N}′)_{tr} (corollary 12.5.20(i)), hence f_{tr}^{∗}(g) is tamely ramified along Y′\(Y′, \underline{N}′)_{tr} (see (13.3.41)). This shows that (13.3.52) is well defined, and clearly this functor is fully faithful, since the same holds for f_{tr}^{∗}. Lastly, let y ∈ Y \ (Y, \underline{N})_{tr} be any point of height one; by proposition 12.6.31(ii) and corollary 12.6.29, there exists a point y′ ∈ Y′ \ (Y′, \underline{N}′)_{tr} with f(y′) = y, and since f restricts to an isomorphism f_{tr}, the corresponding map β : 𝒪_{Y,y} → 𝒪_{Y′,y′} is local, injective, and induces an isomorphism on the respective fields of fractions. Since 𝒪_{Y,y} is a discrete valuation ring, it follows easily that β is an isomorphism, and therefore y′ is a point of height one in Y′. Consequently, the equivalence (f_{tr}^{−1})^{∗} : Cov(Y′, \underline{N}′)_{tr} →^{∼} Cov(Y, \underline{N})_{tr} induced by the isomorphism of schemes f_{tr}^{−1} restricts to a well defined functor Tame(Y′, (Y′, \underline{N}′)_{tr}) → Tame(Y, (Y, \underline{N})_{tr}) which yields a quasi-inverse for (13.3.52). ♢

Suppose again that (X, \underline{M}) is strictly henselian, define P, U and h : V → U as in claim 13.3.49, and set T_{P} := (Spec P)^{♯}; it is easily seen that the counit of adjunction

ũ^{∗}ũ_{∗}(X, \underline{M}) → (X, \underline{M})

is an isomorphism (cp. (12.6.48)), hence ũ_{∗}(X, \underline{M}) is regular (lemma 12.5.21). Let

\underline{X} := (ũ_{∗}(X, \underline{M}), T_{P} , π_{X})

be the object of 𝒦 arising from ũ_{∗}(X, \underline{M}), as in (12.6.7); by theorem 12.6.32 (and its proof), we may find an integral proper simplicial subdivision φ : F → T_{P}, such that the morphism of schemes underlying (f, φ) : φ^{∗}\underline{X} → \underline{X} is a resolution of singularities for X. For every integer k ≥ 1 we define (X_{(k)}, \underline{M}_{(k)}), (X_{φ}, \underline{M}_{φ}), (Y, \underline{N}), g_{X} and k_{X} as in example 13.3.39 : this makes sense, since, in the current situation, \underline{X} is isomorphic to X ×_{S} \underline{S} (where \underline{S} is defined as in example 12.6.5(i)). Moreover, by the same token, the morphism of schemes Y → X_{(k)} is a resolution of singularities.

Set as well U_{φ} := (X_{φ}, \underline{M}_{φ})_{tr} and U_{(k)} := (X_{(k)}, \underline{M}_{(k)})_{tr}; by claim 13.3.51 and proposition 13.3.20(i), we have an essentially commutative diagram of functors :

\xymatrix{Tame(X, U) \ar[r]^{f_{tr}^{∗}} \ar[d]_{_{k_{∗}_{X}}} & Tame(X_{φ}, U_{φ}) \ar[d]^{g_{X}^{∗}} \\ Tame(X_{(k)}, U_{(k)}) \ar[r] & Tame(Y, (Y, \underline{N})_{tr})}

whose horizontal arrows are equivalences. Let k be the ramification index of h_{φ} along X_{φ}\U_{φ}; in light of claim 13.3.49, we are reduced to showing that k_{X}^{∗}(h) admits a section. Set V_{φ} := V ×_{U} U_{φ} and h_{φ} := f_{tr}^{∗}(h) : V_{φ} → U_{φ}; then it suffices to show that g_{X}^{∗}(h_{φ}) has a section.

Let \overline{V}_{φ} → X_{φ} be the normalization of h_{φ} over X_{φ}, and \overline{W} → Y the normalization of \overline{V}_{φ} ×_{X_{φ}} Y in V_{φ} ×_{X_{φ}} Y. Choose a geometric point w of \overline{W} whose support has height one; denote by y (resp. v, resp. x) the image of w in Y (resp. in \overline{V}_{φ}, resp. in X_{φ}), and suppose that the support of x lies in X_{φ} \ U_{φ}. Denote by K(x) the field of fractions of the strictly henselian local ring 𝒪_{X,x}, and define likewise K(w), K(y) and K(v). There follows a commutative diagram of inclusions of valued fields :

\xymatrix{K(x) \ar[r] \ar[d] & K(y) \ar[d] \\ K(v) \ar[r] & K(w)}

such that K(w) is the compositum of K(y) and K(v) : cp. the proof of proposition 13.3.20(i).

Example 13.3.39(ii) shows that the ramification index of 𝒪_{Y,y} over 𝒪_{X_{φ},x} is k, which is a multiple of the ramification index of 𝒪_{\overline{V}_{φ},v} over 𝒪_{X_{φ},x}. It then follows (e.g. from [75, Claim 6.2.15]) that K(v) ⊂ K(y), and therefore K(w) = K(y); this shows that the ramification index of g_{X}^{∗}(h_{φ}) along Y \(Y, \underline{N})_{tr} equals 1, therefore g_{X}^{∗}(h_{φ}) is the restriction of an étale covering \overline{h} of Y (proposition 13.3.20(iii)). By proposition 13.3.2(ii), \overline{h} is the pull-back of an étale covering of X_{(k)}. Since X_{(k)} is strictly local (claim 13.3.37), it follows that \overline{h} admits a section, hence the same holds for g_{X}^{∗}(h_{φ}), as required. □

Remark 13.3.53. A proof of theorem 13.3.45 similar to the one given here can be found in [130, §2.3].

13.3.54. As an application of theorem 13.3.45, we shall determine the fundamental group of the “punctured” scheme obtained by removing the closed point from a strictly local regular log scheme (X, \underline{M}) (with non-trivial log structure) of dimension ≥ 2. Indeed, let x ∈ X be the closed point, and set r := dim \underline{M}_{x}. Also, let \overline{y} be any geometric point of X, localized at a point y ∈ U := X \ {x}, and ξ a geometric point of X localized at the maximal point. Let p (resp. p′) denote the characteristic exponent of κ(x) (resp. of κ(y)). Pick any lifting of ξ to a geometric point ξ_{y} of X(\overline{y}); since (X(\overline{y}), \underline{M}(\overline{y})) is regular (theorem 12.5.31), according to proposition 13.3.43, the vertical arrows of the commutative diagram in (13.3.31) factor through the surjections (13.3.42), and we get a commutative diagram of group homomorphisms :

\xymatrix{π_{1}((X(\overline{y}), \underline{M}(\overline{y}))_{ét}, ξ_{y}) \ar[r]^{φ_{y}} \ar[d] & π_{1}((X, \underline{M})_{ét}, ξ) \ar[d] \\ \underline{M} gp∨ ; \overline{y} ⊗_{ℤ} ∏_{ℓ≠p′} ℤ_{ℓ}(1) \ar[r] & \underline{M}_{x}^{gp∨} ⊗_{ℤ} ∏_{ℓ≠p} ℤ_{ℓ}(1)}

whose vertical arrows are the isomorphisms (13.3.44), and whose top (resp. bottom) horizontal arrow is induced by the natural morphism X(\overline{y}) → X (resp. by the specialization map \underline{M}_{x} → \underline{M}\overline{y}). Now, by corollary 12.5.13, the scheme U is connected and normal, hence the restriction functor Cov(U) → Tame(X, (X, \underline{M})_{tr}) is fully faithful (lemma 13.1.7(iii)); by [82, Exp.V, Prop.6.9] and theorem 13.3.45, it follows that the induced group homomorphism

(13.3.55)  π_{1}((X, \underline{M})_{ét}, ξ) → π_{1}(U_{ét}, ξ)

is surjective (see (13.3.41)). Clearly, the image of φ_{y} lies in the kernel of (13.3.55). Especially:

(U, \underline{M}_{|U})_{r} ≠ ∅ ⇒ π_{1}(U_{ét}, ξ) = \{1\}

since, for any geometric point \overline{y} of (X, \underline{M})_{r}, the specialization map \underline{M}_{x} → \underline{M}\overline{y} is an isomorphism (notation of definition 12.2.7(i)). Thus, suppose that (X, \underline{M})_{r} = \{x\}, set P := \underline{M}_{x}^{♯}, and denote by ψ : X → Spec P the natural continuous map; we have \underline{M} ♯ ; \overline{y} = P_{ψ(y)}, and if we let F_{y} := P \ ψ(y), we get a short exact sequence of free abelian groups of finite rank :

(13.3.56)  0 → F_{y}^{gp} → \underline{M}_{x}^{♯gp} → \underline{M} ♯gp ; \overline{y} → 0.

It is easily seen that ψ is surjective; hence, for any 𝔭 ∈ Spec P of height r −1, pick y ∈ ψ^{−1}(𝔭). With this choice, we have F_{y}^{gp} = ℤ; considering (13.3.56)^{∨}, we deduce that π_{1}(U_{ét}, ξ) is a quotient of ℤ̂′(1) := ∏_{ℓ≠p} ℤ_{ℓ}(1). More precisely, let (Spec P)_{r−1} be the set of prime ideals of P of height r − 1; then we have :

Theorem 13.3.57. If (X, \underline{M})_{r} = \{x\}, we have a natural identification :

(13.3.58)  \frac{P^{∨}}{∑_{𝔭∈(Spec P)_{r−1}} P_{𝔭}^{∨}} ⊗_{ℤ} ℤ̂′(1) →^{∼} π_{1}(U_{ét}, ξ)

and these two abelian groups are cyclic of finite order prime to p.

Proof. The foregoing discussion already yields a natural surjective map as stated; it remains only to check the injectivity, and to verify that the source is a cyclic finite group.

However, since d ≥ 2, and (X, \underline{M})_{r} = \{x\}, corollary 12.5.20(i) implies that r ≥ 2, in which case (Spec P)_{r−1} must contain at least two distinct elements, say 𝔭 and 𝔮. Let F := P \ 𝔭; the image H of (P_{𝔮})^{♯gp∨} in F^{gp∨} is clearly a non-trivial subgroup, hence F^{gp∨}/H is a cyclic finite group, and then the same holds for the source of (13.3.58).

Next, it follows easily from lemma 13.1.7(iii) that (13.3.55) identifies π_{1}(U_{ét}, ξ) with the quotient of π_{1}((X, \underline{M})_{ét}, ξ) by the sum of the images of the maps φ_{y}, for y ranging over all the points of U. However, it is clear that the same sum is already spanned by the sum of the images of the φ_{y} such that y ∈ (X, \underline{M})_{r−1}; whence the theorem. □

Example 13.3.59. Let P ⊂ ℕ^{⊕2} be the submonoid of all pairs (a, b) such that a + b ∈ 2ℕ. Clearly P is fine and saturated of dimension 2. Let K be any field; the K-scheme X := Spec K[P] is the singular quadric in 𝔸_{K}^{3} cut by the equation XY − Z^{2} = 0. It is easily seen that Spec(K, P)_{2} consists of a single point x (the vertex of the cone); let \overline{x} be a geometric point of X localized at x, and U ⊂ X(\overline{x}) the complement of the closed point. Then U is a normal K-scheme of dimension 1, and we have a natural isomorphism

π_{1}(U, ξ) →^{∼} ℤ/2ℤ.

Indeed, a simple inspection shows that P admits exactly two prime ideals of height one, namely 𝔭 := P \ (2ℕ ⊕ {0}) and 𝔮 := P \ ({0} ⊕ 2ℕ). Then, P_{𝔭} = {(a, b) ∈ ℤ ⊕ ℕ | a + b ∈ 2ℕ}, and similarly P_{𝔮} is a submonoid of ℕ ⊕ ℤ. The quotients P_{𝔭}^{♯} and P_{𝔮}^{♯} are both isomorphic to ℕ, and are both generated by the class of (1, 1). Let φ : P_{𝔭}^{♯} → ℤ be a map of monoids; then the image of φ in P^{gp∨} is the unique map of monoids P → ℤ given by the rule : (2, 0) ↦ 0, (1, 1) ↦ φ(1, 1), (0, 2) ↦ 2φ(1, 1). Likewise, a map ψ : P_{𝔮}^{♯} → ℤ gets sent to the morphism P → ℤ such that (2, 0) ↦ 2ψ(1, 1), (1, 1) ↦ ψ(1, 1), and (0, 2) ↦ 0. We see therefore that (P_{𝔭})^{♯gp∨}+(P_{𝔮})^{♯gp∨} is a subgroup of index two in P^{gp∨}, and the contention follows from theorem 13.3.57.

13.4. Local acyclicity of smooth morphisms of log schemes. In this section we consider a smooth and saturated morphism f : (X, \underline{M}) → (Y, \underline{N}) of fine log schemes. We fix a geometric point \overline{x} of X, localized at a point x, and let \overline{y} := f(\overline{x}). We shall suppose as well that Y is strictly local and normal, that (Y, \underline{N})_{tr} is a dense open subset of Y, and that \overline{y} is localized at the closed point y of Y. Let \overline{η} be a strict geometric point of Y, localized at the generic point η; to ease notation, set :

U := f −1 ; \overline{x}(η)  U_{tr} := (X(\overline{x}), \underline{M}(\overline{x}))_{tr} ∩ U  \overline{U} := U ×_{|η|} |\overline{η}|  \overline{U}_{tr} := U_{tr} ×_{|η|} |\overline{η}|

and notice that U_{tr} is a dense open subset of U, by virtue of proposition 12.7.17(ii,iv). Choose a geometric point ξ of \overline{U}_{tr}, and let ξ′ be the image of ξ in U_{tr}. There follows a short exact sequence of topological groups :

1 → π_{1}(\overline{U}_{tr,ét}, ξ) → π_{1}(U_{tr,ét}, ξ′) ⟶^{α} π_{1}(|η|_{ét}, \overline{η}) → 1.

On the other hand, let p be the characteristic exponent of κ := κ(\overline{y}); for every integer e such that (e, p) = 1, the discussion of (13.3.27) yields a commutative diagram :

(13.4.1)  \xymatrix{_{\underline{N}\overline{y}} \ar[r] \ar[d]_{log f\overline{x}} & κ(η)^{×} \ar[r] \ar[d] & Hom_{cont}(π_{1}(|η|_{ét}, \overline{η}), μ_{e}(κ)) \ar[d]^{Hom_{cont}(α,μ_{e}(κ))} \\ _{\underline{M}\overline{x}} \ar[r] & Γ(U_{tr}, 𝒪_{U}^{×}) \ar[r] & Hom_{cont}(π_{1}(U_{tr,ét}, ξ′), μ_{e}(κ))}

such that the composition of the two top (rep. bottom) horizontal arrows factors through \underline{N} ♯ ; \overline{y} (resp. \underline{M} ♯ ; \overline{x}). There follows a system of natural group homomorphisms :

(13.4.2)  Coker (log f)gp ; \overline{x} → Hom_{cont}(π_{1}(\overline{U}_{tr,ét}, ξ), μ_{e}(κ))  where (e, p) = 1

which assemble into a group homomorphism :

(13.4.3)  π_{1}(\overline{U}_{tr,ét}, ξ) → Coker (log f gp ; \overline{x})^{∨} ⊗_{ℤ} ∏_{ℓ≠p} ℤ_{ℓ}(1).

13.4.4. We wish to give a second description of the map (13.4.3), analogous to the discussion in (13.3.32). To this aim, let R be a ring; for any monoid P, and any integer e > 0, denote by e_{P} : P → P the e-Frobenius map of P. Let λ : P → Q be a local morphism of finitely generated monoids. For any integer e > 0, we get a commutative diagram of monoids :

(13.4.5)  \xymatrix{P \ar[r]^{^{e^{P}}} \ar[d]_{λ} & P \ar[d]^{λ_{e}} \ar[dr]^{λ} \\ Q \ar[r]^{μ′} & Q′ \ar[r]^{_{e_{Q|P}}} & Q}

whose square subdiagram is cocartesian, and such that e_{Q|P} ∘ μ′ = e_{Q}. The latter induces a commutative diagram of log schemes :

\xymatrix{Spec(R, Q) \ar[r]^{g_{Q|P}} \ar[dr]_{g} & Spec(R, Q′) \ar[r]^{g′} \ar[d]_{g_{e}} & Spec(R, Q) \ar[d]^{g} \\  & Spec(R, P) \ar[r]^{g_{P}} & Spec(R, P)}

whose square subdiagram is cartesian, and such that g_{Q} := g′ ∘ g_{Q|P} is the morphism induced by e_{Q}. Also, by example 12.6.5, we have a commutative diagram of monoidal spaces :

\xymatrix{Spec(R, Q) \ar[r]^{g_{Q|P}} \ar[d]_{ψ_{Q}} & Spec(R, Q′) \ar[d]^{ψ_{Q′}} \ar[r]^{g_{e}} & Spec(R, P) \ar[d]^{ψ_{P}} \\ T_{Q} \ar[r]^{φ} & T_{Q′} \ar[r] & T_{P}}

(where φ := (Spec e_{Q|P})^{♯}) which determines morphisms in the category 𝒦 :

(Spec(R, Q), T_{Q}, ψ_{Q}) → (Spec(R, Q′), T_{Q′}, ψ_{Q′}) → (Spec(R, P), T_{P} , ψ_{P}).

13.4.6. Now, suppose that the closed point 𝔪_{Q} of T_{Q} lies in the strict locus of (Spec λ)^{♯} (which just means that λ^{♯} is an isomorphism). Notice that the functor M ↦ M^{♯} commutes with colimits (since it is a left adjoint); taking into account lemma 6.1.13, we deduce that the square subdiagram of (13.4.5)^{♯} is still cocartesian, and therefore e_{Q|P}^{♯} is an isomorphism, so the same holds for φ.

More generally, let 𝔮 ∈ Spec Q be any prime ideal in the strict locus of (Spec λ)^{♯}; set 𝔭 := λ^{−1}𝔮, 𝔯 := φ(𝔮), and recall that T_{Q_{𝔮}} := Spec Q_{𝔮} is naturally an open subset of T_{Q} (see (6.5.16)). A simple inspection shows that the restriction T_{Q_{𝔮}} → T_{Q′_{𝔯}} of φ is naturally identified with the morphism of affine fans (Spec e_{Q_{𝔮}|P_{𝔭}})^{♯}. Since 𝔮 is the closed point of T_{Q_{𝔮}}, the foregoing shows that T_{Q_{𝔮}} lies in the strict locus of φ; in other words, Str(φ) is an open subset of T_{Q}, and we have

(13.4.7)  Str((Spec λ)^{♯}) ⊂ Str(φ).

Moreover, recall that Spec e_{Q} : T_{Q} → T_{Q} is the identity on the underlying topological space (see example 6.5.10(i)), and by construction it factors through φ, so the latter is injective on the underlying topological spaces. Especially, Str(φ) = φ^{−1}φ(Str(φ)), and therefore

(13.4.8)  Str(g_{Q|P}) = ψ_{Q}^{−1}(Str(φ)) = g_{Q|P}^{−1} (ψ_{Q′}^{−1}φ(Str(φ))).

Hence, set g := Spec(R, λ); from (13.4.7) and (12.6.3) we obtain :

Str(g) ⊂ Str(g_{Q|P})

and together with (13.4.8) we deduce that :

• Str(g_{Q|P}) is an open subset of Spec R[Q].

• ψ_{Q′}^{−1}φ(Str(φ)) is a locally closed subscheme of Spec R[Q′].

• The restriction Str(g) → ψ_{Q′}^{−1}φ(Str(φ)) of g_{Q|P} is a finite morphism.

Lastly, suppose that e is invertible in R; in this case, the morphisms g_{P} and g_{Q} are étale (proposition 12.3.34), so the same holds for g_{Q|P} (corollary 12.3.25(iii)). From corollary 12.3.27(i) we deduce that the restriction Str(g) → ψ_{Q′}^{−1}φ(Str(φ)) of g_{Q|P} is a (finite) étale covering.

13.4.9. In the situation of (13.4.4), suppose additionally, that R is a ℤ[1/e, μ_{e}]-algebra (where μ_{e} is the e-torsion subgroup of ℂ^{×}), P and Q are fine monoids, and λ is integral, so that Q′ is also fine. Set

G_{P} := Hom_{ℤ}(P^{gp}, μ_{e})  G_{Q} := Hom_{ℤ}(Q^{gp}, μ_{e})  G_{Q|P} := Hom_{ℤ}(Coker λ^{gp}, μ_{e}).

Notice that the trivial locus Spec(R, P)_{tr} is the open subset Spec R[P^{gp}], and likewise for Spec(R, Q)_{tr} and Spec(R, Q′)_{tr}; therefore, (13.4.5)^{gp} induces a cartesian diagram of schemes

\xymatrix{Spec(R, Q′)_{tr} \ar[r]^{g′_{tr}} \ar[d]_{g_{e,tr}} & Spec(R, Q)_{tr} \ar[d]^{g_{tr}} \\ Spec(R, P)_{tr} \ar[r]^{g_{P,tr}} & Spec(R, P)_{tr}.}

Fix a geometric point τ′_{Q} of Spec(R, Q′)_{tr}, and let τ_{Q} := g′(τ′_{Q}), τ_{P} := g(τ_{Q}), τ′_{P} := g_{e}(τ′_{Q}). It was shown in (13.3.32) that g_{P}^{−1}(τ_{P}) is a G_{P} -torsor, so g_{P,tr} is a Galois étale covering, corresponding to a continuous representation (13.3.33) into G_{P}. Hence g^{′−1}(τ_{Q}) is a G_{P} -torsor, and g′_{tr} is a Galois étale covering, whose corresponding representation of π_{1}(Spec(R, Q)_{tr,ét}, τ_{Q}) is obtained by composing (13.3.33) with the natural continuous group homomorphism

π_{1}(g_{tr}, τ_{Q}) : π_{1}(Spec(R, Q)_{tr,ét}, τ_{Q}) → π_{1}(Spec(R, P)_{tr,ét}, τ_{P}).

By the same token, g_{Q}^{−1}(τ_{Q}) is a G_{Q}-torsor, so also g_{Q,tr} is a Galois étale covering, and a simple inspection shows that the surjection

g_{Q}^{−1}(τ_{Q}) → g^{′−1}(τ_{Q})

induced by g_{Q|P} is G_{Q}-equivariant, for the G_{Q}-action on the target obtained from the map

Hom_{ℤ}(λ^{gp}, μ_{e}) : G_{Q} → G_{P} .

The situation is summarized by the commutative diagram of continuous group homomorphisms

\xymatrix{π_{1}(Spec(R, Q′)_{tr,ét}, τ′_{Q}) \ar[r] \ar[d]_{π_{1}(g_{e,tr},τ′_{Q})} & π_{1}(Spec(R, Q)_{tr,ét}, τ_{Q}) \ar[r] \ar[d]_{π_{1}(g_{tr},τ_{Q})} & G_{Q} \ar[d]^{Hom_{ℤ}(λ^{gp},μ_{e})} \\ π_{1}(Spec(R, P)_{tr,ét}, τ′_{P}) \ar[r] & π_{1}(Spec(R, P)_{tr,ét}, τ_{P}) \ar[r] & G_{P}}

whose horizontal right-most arrows are the maps (13.3.33). Consequently, g_{Q|P}^{−1} (τ′_{Q}) is a G_{Q|P} - torsor, and g_{Q|P,tr} is a Galois étale covering, classified by a continuous group homomorphism

(13.4.10)  Ker π_{1}(g_{e,tr}, τ′_{Q}) → Ker π_{1}(g_{tr}, τ_{Q}) → G_{Q|P} .

13.4.11. Let us return to the situation of (13.4), and assume additionally, that both (X, \underline{M}) and (Y, \underline{N}) are fs log schemes. Take R := 𝒪_{Y,\overline{y}} in (13.4.4); by corollary 12.3.42 and theorem 12.1.37(iii), we may assume that there exist

• a local and saturated morphism λ : P → Q of fine and saturated monoids, such that P is sharp, Q^{×} is a free abelian group of finite type, say of rank r, and Ass_{ℤ} Coker λ^{gp} does not contain the characteristic exponent of κ(\overline{y});

• a morphism of schemes π : Y → Spec R[P], which is a section of the projection Spec R[P] → Y, such that

(Y, \underline{N}) = Y ×_{Spec R[P]} Spec(R, P) (X, \underline{M}) = Y ×_{Spec R[P]} Spec(R, Q)

and f is obtained by base change from the morphism g := Spec(R, λ). Moreover, the induced chart Q_{X} → \underline{M} shall be local at the geometric point \overline{x}.

By claim 12.1.39, we may further assume that the projection Q → Q^{♯} admits a section σ : Q^{♯} → Q, such that λ(P) lies in the image of σ. In this case, g factors through the morphism Spec(R, P) → Spec(R, Q^{♯}) induced by λ^{♯}, and σ induces an isomorphism of log schemes :

Spec(R, Q) = 𝔾_{m,Y}^{⊕r} ×_{Y} Spec(R, Q^{♯})

(where 𝔾_{m,Y}^{⊕r} denotes the standard torus of rank r over Y). In this situation, X is smooth over Y ×_{Spec R[P]} Spec R[Q^{♯}], and more precisely

(13.4.12)  (X, \underline{M}) = 𝔾_{m,Y}^{⊕r} ×_{Spec R[P]} Spec(R, Q^{♯}).

Summing up, after replacing Q by Q^{♯}, we may assume that Q is also sharp, and (13.4.12) holds with Spec(R, Q) instead of Spec(R, Q^{♯}). Moreover, the image of x in T_{Q} is the closed point 𝔪_{Q}.

Let e > 0 be an integer which is invertible in R; by inspecting the definition, it is easily seen that there exists a finite separable extension K_{e} of κ(η) = Frac(R), such that the normalization Y_{e} of Y in Spec K_{e} fits into a commutative diagram of schemes :

\xymatrix{Y_{e} \ar[r] \ar[d]_{π_{e}} & Y \ar[d]^{π} \\ Spec R[P] \ar[r]^{g_{P}} & Spec R[P]}

(whose top horizontal arrow is the obvious morphism); namely, π is defined by some morphism of monoids β : P → R, and one takes for K_{e} any subfield of κ(\overline{η}) containing κ(η) and the e-th roots of the elements of β(P). Set (Y_{e}, \underline{N}_{e}) := Y_{e} ×_{Spec R[P]} Spec(R, P), and define log schemes (X_{e}, \underline{M}_{e}), (X′_{e}, \underline{M}′_{e}) so that the two square subdiagrams of the diagram of log schemes

\xymatrix{(X′_{e}, \underline{M}′_{e}) \ar[r]^{h_{e}} \ar[d] & (X_{e}, \underline{M}_{e}) \ar[r] \ar[d] & 𝔾_{m,Y}^{⊕r} ×_{Y} (Y_{e}, \underline{N}_{e}) \ar[d]^{π′_{e}} \\ Spec(R, Q) \ar[r]^{g_{Q|P}} & Spec(R, Q′) \ar[r]^{g_{e}} & Spec(R, P)}

are cartesian (here π′_{e} is the composition of π_{e} and the projection 𝔾_{m,Y}^{⊕r} ×_{Y} Y_{e} → Y_{e}), whence a commutative diagram :

(13.4.13)  \xymatrix{(X′_{e}, \underline{M}′_{e}) \ar[r]^{h_{e}} \ar@{-->}[dr]_{f′_{e}} & (X_{e}, \underline{M}_{e}) \ar[d]^{f_{e}} \\  & (Y_{e}, \underline{N}_{e}).}

Notice that both f_{e} and f′_{e} are smooth and saturated morphisms of fine log schemes. Also, by construction Y_{e} is strictly local, and (Y_{e}, \underline{N}_{e})_{tr} is a dense subset of Y_{e}. In other words, f_{e} and f′_{e} are still of the type considered in (13.4). Moreover, h_{e} is étale, since the same holds for g_{Q|P}, and the discussion in (13.4.6) shows that the restriction

Str(f′_{e}) → X_{e}

is an étale morphism of schemes. Furthermore, the discussion in (13.4.9) shows that

h_{e,tr} : (X′_{e}, \underline{M}′_{e})_{tr} → (X_{e}, \underline{M}_{e})_{tr}

is a Galois étale covering.

13.4.14. More precisely, notice that λ^{♯} = log f ♯ ; \overline{x}; combining with (13.4.10), we deduce a continuous group homomorphism :

(13.4.15)  Ker π_{1}(f_{e,tr}, ξ′_{e}) → Hom_{ℤ}(Coker (log f)gp ; \overline{x}, μ_{e}(κ))

where ξ′_{e} is the image in X_{e} of the geometric point ξ. The geometric point \overline{y} lifts uniquely to a geometric point \overline{y}_{e} of Y_{e}, localized at the closed point y_{e}, and the pair (\overline{x}, \overline{y}_{e}) determines a unique geometric point \overline{x}_{e} such that f_{e}(\overline{x}_{e}) = \overline{y}_{e}. Also, since the field extension κ(y) → κ(y_{e}) is purely inseparable, it is easily seen that the induced map f_{e}^{−1}(y_{e}) → f^{−1}(y) is a homeomorphism; there follows an isomorphism of X(\overline{x})-schemes ([66, Ch.IV, Prop.18.8.10]) :

X_{e}(\overline{x}_{e}) →^{∼} X(\overline{x}) ×_{Y} Y_{e}.

Let η_{e} be the generic point of Y_{e}; by construction, \overline{η} lifts to a geometric point \overline{η}_{e} of Y_{e}, localized at η_{e}, and we have continuous group homomorphisms :

(13.4.16)  π_{1}(\overline{U}_{tr}, ξ) →^{∼} Ker (π_{1}(U_{e,tr}, ξ′_{e}) → π_{1}(η_{e}, |\overline{η}_{e}|)) → Ker π_{1}(f_{e,tr}, ξ′_{e}).

The composition of (13.4.15) and (13.4.16) is a continuous group homomorphism

(13.4.17)  π_{1}(\overline{U}_{tr}, ξ) → Hom_{ℤ}(Coker (log f)gp ; \overline{x}, μ_{e}(κ))

whence, finally, a pairing

Coker (log f)gp ; \overline{x} × π_{1}(\overline{U}_{tr,ét}, ξ) → μ_{e}(κ).

We claim that this pairing agrees with the one deduced from (13.4.2). Indeed, by tracing back through the constructions, we see that (13.4.17) is the homomorphism arising from the Galois covering of \overline{U}_{tr}, which is obtained from g_{Q|P}, after base change along the composition

\overline{U}_{tr} → X_{e} → Spec R[Q′].

On the other hand, the discussion of (13.3.32) shows that the homomorphism π_{1}(U_{tr,ét}, ξ′) → G_{Q} arising from the bottom row of (13.4.1), classifies the Galois covering C → \overline{U}_{tr} obtained from g_{Q}, by base change along the same map. By the same token, the top row of (13.4.1) corresponds to the G_{P} -Galois covering C′ → |η| obtained by base change of g_{P} along the composition |η| → Y → S_{P}. The map log f\overline{x} induces a morphism of schemes C → C′ ×_{|η|} U_{tr}, and (13.4.2) corresponds to the G_{Q|P} -torsor obtained from a fibre of this morphism. Evidently, this torsor is isomorphic to g_{Q|P}^{−1} (τ′_{Q}), whence the contention.

13.4.18. In the situation of (13.4), recall that there is a natural bijection between the set of maximal points of f −1 ; \overline{x}(\overline{y}), and the set Σ of maximal points of the closed fibre of the induced map

(13.4.19)  Spec \underline{M}\overline{x} → Spec \underline{N}\overline{y}

(proposition 12.7.14). For every 𝔮 ∈ Σ, denote by η_{𝔮} the corresponding maximal point of f −1 ; \overline{x}(\overline{y}), choose a geometric point \overline{η}_{𝔮} localized at η_{𝔮}, and let X(\overline{η}_{𝔮}) be the strict henselization of X(\overline{x}) at \overline{η}_{𝔮}. Also, set

U_{𝔮} := U ×_{X(\overline{x})} X(\overline{η}_{𝔮}) \overline{U}_{𝔮} := U_{𝔮} ×_{|η|} |\overline{η}|

and notice that \overline{U}_{𝔮} is an irreducible normal scheme. Notice as well that f induces a strict morphism (X(\overline{η}_{𝔮}), \underline{M}(\overline{η}_{𝔮})) → (Y, \underline{N}) (theorem 12.7.8(iii.a)), and therefore the log structure of U_{𝔮} ×_{X(\overline{η}_{𝔮})} (X(\overline{η}_{𝔮}), \underline{M}(\overline{η}_{𝔮})) is trivial.

Recall that Z := \overline{U} \ \overline{U}_{tr} is a finite union of irreducible closed subsets of codimension one in \overline{U}, and 𝒪_{\overline{U},z} is a discrete valuation ring, for each maximal point z ∈ Z (proposition 12.7.17(iv,v)); especially, the category Tame(\overline{U}, \overline{U}_{tr}) is well defined (definition 13.3.16(iii)). We denote

Tame(f, \overline{x})

the full subcategory of Tame(\overline{U}, \overline{U}_{tr}) consisting of all the coverings C → \overline{U}_{tr} such that, for every 𝔮 ∈ Σ, the induced covering

C ×_{\overline{U}_{tr}} \overline{U}_{𝔮} → \overline{U}_{𝔮}

is trivial (i.e. C ×_{\overline{U}_{tr}} \overline{U}_{𝔮} is a disjoint union of copies of \overline{U}_{𝔮}). It is easily seen that Tame(f, \overline{x}) is a Galois category (see [82, Exp.V, Déf.5.1]), and we obtain a fibre functor for this category, by restriction of the usual fibre functor φ ↦ φ^{−1}(ξ) defined on all étale coverings φ of \overline{U}_{tr}; we denote by π_{1}(\overline{U}_{tr}/Y_{ét}, ξ) the corresponding fundamental group. According to [82, Exp.V, Prop.6.9], the fully faithful inclusion Tame(f, \overline{x}) → Cov(\overline{U}_{tr}) induces a continuous surjective group homomorphism

(13.4.20)  π_{1}(\overline{U}_{tr,ét}, ξ) → π_{1}(\overline{U}_{tr}/Y_{ét}, ξ).

Proposition 13.4.21. The map (13.4.3) factors through (13.4.20), and the induced group homomorphism :

(13.4.22)  π_{1}(\overline{U}_{tr}/Y_{ét}, ξ) → Coker (log f gp ; \overline{x})^{∨} ⊗_{ℤ} ∏_{ℓ≠p} ℤ_{ℓ}(1).

is surjective.

Proof. Let σ_{Y} : Y → Y^{qfs} be the natural morphism of schemes exhibited in remark 12.2.38(iv), and set

(Y, \underline{N}′) := Y ×_{Y^{qfs}} (Y, \underline{N})^{qfs} (X, \underline{M}′) := (Y, \underline{N}′) ×_{(Y,\underline{N})} (X, \underline{M}).

Since f is saturated, both (X, \underline{M}′) and (Y, \underline{N}′) are fs log schemes (see remark 12.2.38(i)); also, the morphism of schemes underlying the induced morphism of log schemes f′ : (X, \underline{M}′) → (Y, \underline{N}′), agrees with that underlying f. Moreover, by construction we have \underline{N} ′♯ ; \overline{z} = (\underline{N} ♯ ; \overline{z})^{sat} for every geometric point \overline{z} of Y, especially (Y, \underline{N}′)_{tr} = (Y, \underline{N})_{tr}. Likewise, \underline{M} ′♯ ; \overline{z} = (\underline{M} ♯ ; \overline{z})^{sat} (lemma 6.2.12(iii,iv)), therefore Str(f′) = Str(f), and especially, (X, \underline{M}′)_{tr} = (X, \underline{M})_{tr}. Furthermore, notice that the the natural map

(13.4.23)  Coker (log f)gp ; \overline{x} → Coker (log f′)gp ; \overline{x}

is surjective, and its kernel is a quotient of (\underline{M} sat ; \overline{x})^{×}/\underline{M} × ; \overline{x}, especially it is a torsion subgroup. However, the cokernel of (log f)gp ; \overline{x} equals the cokernel of (log f ♯ ; \overline{x})^{gp}, hence it is torsion-free (corollary 6.2.32(ii)), so (13.4.23) is an isomorphism. Thus, we may replace \underline{N} (resp. \underline{M}) by \underline{N}′ (resp. \underline{M}′), and assume from start that f is a smooth, saturated morphism of fs log schemes.

In this case, in light of the discussion of (13.4.14), it suffices to prove that (13.4.17) is a surjection, and that it factors through π_{1}(\overline{U}_{tr}/Y_{ét}, ξ). To prove the surjectivity comes down to showing that \overline{U}_{tr} ×_{X_{e}} X′_{e} is a connected scheme. However, let x_{e} be the support of \overline{x}_{e}, and notice that ψ_{P} ∘ π_{e}(y_{e}) = 𝔪_{P}, the closed point of T_{P}. Since x maps to the closed point of T_{Q}, we deduce easily that the image of x_{e} in T_{Q′} is the closed point 𝔪_{Q′}, i.e. x_{e} lies in the closed subscheme X_{e} ×_{S′_{Q}} Spec κ⟨Q′/𝔪_{Q′}⟩. Notice as well that e_{Q|P} is of Kummer type (see definition 6.4.58); by proposition 13.3.7(ii), it follows that there exists a unique geometric point \overline{x}′_{e} of X′_{e} lying over \overline{x}_{e}, whence an isomorphism of X_{e}(\overline{x}_{e})-schemes ([66, Ch.IV, Prop.18.8.10])

X′_{e}(\overline{x}′_{e}) →^{∼} X′_{e} ×_{X_{e}} X_{e}(\overline{x}_{e}).

Hence \overline{U}_{tr} ×_{X_{e}} X′_{e} is an open subset of |\overline{η}_{e}| ×_{Y_{e}} X′_{e}(\overline{x}_{e}), and the latter is an irreducible scheme, by proposition 12.7.17(ii). We also deduce that the induced morphism

h_{e,\overline{x}} : (X′_{e}(\overline{x}′_{e}), \underline{M}′_{e}(\overline{x}′_{e})) → (X_{e}(\overline{x}_{e}), \underline{M}_{e}(\overline{x}_{e}))

is a finite étale covering of log schemes. From the discussion in (13.4.11), we see that the restriction of h_{e,\overline{x}}

Str(h_{e,\overline{x}}) → X_{e}(\overline{x}_{e})

is an étale morphism, and Str(h_{e,\overline{x}}) contains the strict locus of the induced morphism

f′_{e,\overline{x}} : (X′_{e}(\overline{x}′_{e}), \underline{M}′_{e}(\overline{x}′_{e})) → (Y_{e}, \underline{N}_{e}).

Since the field extension κ(\overline{y}) → κ(\overline{y}_{e}) is purely inseparable, \overline{η}_{𝔮} lifts uniquely to a geometric point η_{e,𝔮} ∈ X_{e}(\overline{x}_{e}), and as usual we deduce that the strict henselization X_{e}(\overline{η}_{𝔮}) of X_{e}(\overline{x}_{e}) at \overline{η}_{e,𝔮} is isomorphic, as an X_{e}(\overline{x}_{e})-scheme, to X_{e}(\overline{x}_{e}) ×_{X(\overline{x})} X(\overline{η}_{𝔮}). Moreover, if η_{e,𝔮} is the support of \overline{η}_{e,𝔮}, a simple inspection shows that the fibre h_{e,\overline{x}}^{−1}(η_{e,𝔮}) consists of maximal points of f_{e,\overline{x}}^{′−1}(y_{e}). By theorem 12.7.8(iii.a), every point of h_{e,\overline{x}}^{−1}(η_{e,𝔮}) lies in Str(f′_{e,\overline{x}}), therefore the induced morphism X′_{e} ×_{X_{e}} X_{e}(\overline{η}_{𝔮}) → X_{e}(\overline{η}_{𝔮}) is finite and étale. Taking into account (13.3.41), we conclude that the étale covering X′_{e} ×_{X_{e}} \overline{U}_{tr} → \overline{U}_{tr} is an object of Tame(f, \overline{x}), whence the proposition. □

13.4.24. Say that Y = Spec R; for every algebraic field extension K of κ(η) = Frac(R), let R_{K} be the normalization of R in K, set |η_{K}| := Spec K and

Y_{K} := Spec R_{K} (Y_{K}, \underline{N}_{K}) := Y_{K} ×_{Y} (Y, \underline{N}) (X_{K}, \underline{M}_{K}) := Y_{K} ×_{Y} (X, \underline{M}).

Moreover, let f_{K} : (X_{K}, \underline{M}_{K}) → (Y_{K}, \underline{N}_{K}) be the induced morphism, and \overline{y}_{K} any geometric point localized at the closed point y_{K} of the strictly local scheme Y_{K}; since the extension κ(\overline{y}) → κ(\overline{y}_{K}) is purely inseparable, there exists a unique geometric point \overline{x}_{K} of X_{K} lifting \overline{x}, and we have an isomorphism of (X(\overline{x}), \underline{M}(\overline{x}))-schemes :

(X_{K}(\overline{x}_{K}), \underline{M}_{K}(\overline{x}_{K})) →^{∼} X_{K} ×_{X} (X(\overline{x}), \underline{M}(\overline{x})) = Y_{K} ×_{Y} (X(\overline{x}), \underline{M}(\overline{x})).

Clearly the morphism f_{K} is again of the type considered in (13.4.18); especially, the maximal points of f_{K,\overline{x}_{K}}^{−1}(\overline{y}_{K}) are in natural bijection with the elements of Σ, and it is natural to denote

U_{K} := U ×_{Y} |η_{K}|  U_{K,tr} := U_{tr} ×_{Y} |η_{K}|  U_{K,𝔮} := U_{K,tr} ×_{X(\overline{x})} X(\overline{η}_{𝔮})

for every 𝔮 ∈ Σ. Then, let \overline{η}_{K,𝔮} be the unique geometric point of f_{K,\overline{x}_{K}}^{−1}(\overline{y}_{K}) lying over \overline{η}_{𝔮}, and η_{K,𝔮} the support of \overline{η}_{K,𝔮}; as usual, we have

(13.4.25)  X_{K}(\overline{η}_{K,𝔮}) = X(\overline{η}_{𝔮}) ×_{Y} Y_{K}

hence the above notation is consistent with the one introduced for the original morphism f. Furthermore, if z is any maximal point of \overline{U}\\overline{U}_{tr}, the image z_{K} of z in U_{K} is a maximal point of U_{K}\U_{K,tr}, and since the induced map 𝒪_{U_{K},z_{K}} → 𝒪_{\overline{U},z} is faithfully flat, proposition 12.7.17(iv,v) easily implies that 𝒪_{U_{K},z_{K}} is a discrete valuation ring. We may then denote

Tame(f, \overline{x}, K)

the full subcategory of Tame(U_{K}, U_{K,tr}), consisting of those objects C → U_{K,tr}, such that the induced covering C ×_{U_{K,tr}} U_{K,𝔮} → U_{K,𝔮} is trivial, for every 𝔮 ∈ Σ. We have a natural functor

(13.4.26)  2-colim_{K} Tame(f, \overline{x}, K) → Tame(f, \overline{x})

where the 2-colimit ranges over the filtered family of all finite separable extensions K of κ(η).

Lemma 13.4.27. The functor (13.4.26) is an equivalence.

Proof. Let \overline{h} : \overline{C} → \overline{U}_{tr} be an object of the category Tame(f, \overline{x}). According to [66, Ch.IV, Prop.17.7.8(ii)] and [65, Ch.IV, Prop.8.10.5], we may find a finite extension K of κ(η), such that \overline{h} descends to a finite étale morphism

h_{K} : C_{K} → U_{K,tr}.

Let C′_{K} (resp. \overline{C}′) denote the normalization of C_{K} (resp. of \overline{C}) over U_{K} (resp. over \overline{U}). Since the morphism |\overline{η}| → |η_{K}| is pro-étale, we have \overline{C}′ = C′_{K} ×_{|η_{K}|} |\overline{η}| (proposition 9.8.3), and it follows easily that C_{K} is tamely ramified along the divisor U_{K} \U_{K,tr}. From (13.4.25) we get a natural isomorphism :

\overline{U}_{𝔮} →^{∼} U_{K,𝔮} ×_{|η_{K}|} |\overline{η}|.

Thus, after replacing K by a larger finite separable extension of κ(η), we may assume that the induced morphism C_{K} ×_{U_{K,tr}} U_{K,𝔮} → U_{K,𝔮} is a trivial étale covering, for every 𝔮 ∈ Σ.

This shows that (13.4.26) is essentially surjective; likewise one shows the full faithfulness : the details shall be left to the reader. □

13.4.28. In the situation of (13.4.24), let K be an algebraic extension of κ(η), and

h : C → U_{K,tr}

any object of Tame(U_{K}, U_{K,tr}); denote by C′ the normalization of X_{K}(\overline{x}_{K}) in C, and let h′ : C′ → X_{K}(\overline{x}_{K}) be the induced morphism of schemes. We claim that there exists a largest non-empty open subset

E(h) ⊂ X_{K}(\overline{x}_{K})

such that the restriction h^{′−1}E(h) → E(h) of h′ is étale. Indeed, in any case, h′ restricts to an étale morphism on a dense open subset containing U_{K,tr}, and there exists a largest open subset E′ ⊂ C′ such that h′_{|E′} is étale (claim 13.1.8 and lemma 13.1.7(iii)). Then it is easily seen that E(h) := X_{K}(\overline{x}_{K})\h′(C′\E′) will do.

Lemma 13.4.29. With the notation of (13.4.28), the category Tame(f, \overline{x}, K) is the full subcategory of Tame(U_{K}, U_{K,tr}) consisting of those objects h : C → U_{K,tr} such that E(h) contains the maximal points of X_{K}(\overline{x}_{K}) ×_{Y_{K}} |y_{K}|.

Proof. In view of claim 13.1.9, this characterization is a rephrasing of the definition of the category Tame(f, \overline{x}, K). □

13.4.30. Keep the notation of (13.4.28), and suppose that h is an object of Tame(f, \overline{x}, K). Fix 𝔮 ∈ Σ; then lemma 13.4.29says that η_{K,𝔮} ∈ E(h). Thus, we obtain a functor

Tame(f, \overline{x}, K) → Cov(|η_{K,𝔮}|)  :  C ↦ C′ ×_{X_{K}(\overline{x}_{K})} |η_{K,𝔮}|.

However, the natural morphism |η_{K,𝔮}| → |η_{𝔮}| is radicial, hence it induces an equivalence

Cov(|η_{𝔮}|) →^{∼} Cov(|η_{K,𝔮}|)

(lemma 13.1.7(i)). Combining these two functors in the special special case where K := κ(\overline{η}), we get a functor

(13.4.31)  Tame(f, \overline{x}) → Cov(|η_{𝔮}|) : (C → \overline{U}_{tr}) ↦ (C_{|η_{𝔮}} → |η_{𝔮}|).

Now, the rule φ ↦ φ^{−1}(\overline{η}_{𝔮}) yields a fibre functor for the Galois category Cov(|η_{𝔮}|); by composition with (13.4.31), we deduce a fibre functor for Tame(f, \overline{x}), whose group of automorphisms we denote π_{1}(\overline{U}_{tr}/Y_{ét}, \overline{η}_{𝔮}). Also, set F (𝔮) := \underline{M}\overline{x}\𝔮, and notice that the structure map \underline{M}\overline{x} → 𝒪_{X(\overline{x}),\overline{x}} induces a group homomorphism

(13.4.32)  F (𝔮)^{gp} → κ(η_{𝔮})^{×}.

Lemma 13.4.33. With the notation of (13.4.30), we have :

(i) The natural map Coker (log f\overline{x}) → F (𝔮) induces a commutative diagram of groups

\xymatrix{π_{1}(|η_{𝔮}|_{ét}, \overline{η}_{𝔮}) \ar[r] \ar[d]_{α} & π_{1}(\overline{U}_{tr}/Y_{ét}, \overline{η}_{𝔮}) \ar[r] & π_{1}(\overline{U}_{tr}/Y_{ét}, ξ) \ar[d]^{β} \\ F (𝔮)^{gp∨} ⊗_{ℤ} ∏_{ℓ≠p} ℤ_{ℓ}(1) \ar[rr]^{γ} &  & Coker (log f gp ; \overline{x})^{∨} ⊗_{ℤ} ∏_{ℓ≠p} ℤ_{ℓ}(1)}

where β is (13.4.22), and α is deduced from (13.4.32), as in the discussion of (13.3.27). (ii) α is surjective, and γ is an isomorphism.

Proof. (i): The proof amounts to unwinding the definitions, and shall be left as an exercice for the reader. Notice that the second arrow on the top row is only well-defined up to inner automorphisms, but since the groups on the bottom row are abelian, the ambiguity does not affect the statement.

(ii): Notice that log f\overline{x} restricts to a map of monoids 𝒪_{Y,\overline{y}}^{×} → F (𝔮), which induces an isomorphism Coker(log f)^{♯} →^{∼} F (𝔮)^{♯}; we deduce that γ is an isomorphism. Next, let Z be the topological closure of η_{𝔮} in X(\overline{x}), and endow Z with its reduced subscheme structure; set also (Z, \underline{M}(Z)) := Z ×_{X(\overline{x})} (X, \underline{M}). The map α factors as a composition

π_{1}(|η_{𝔮}|_{ét}, \overline{η}_{𝔮}) → π_{1}((Z, \underline{M}(Z))_{tr,ét}, \overline{η}_{𝔮}) → F (𝔮)^{gp∨} ⊗_{ℤ} ∏_{ℓ≠p} ℤ_{ℓ}(1)

where the first map is surjective, by lemma 13.1.7(ii). Lastly, notice that \underline{M}(Z)_{red,\overline{x}} = F (𝔮)_{∘}; by propositions 13.3.43 and 12.7.14(ii), it follows that the second map is surjective as well, so the proof of (ii) is complete. □

13.4.34. In the situation of (12.3.44), suppose that Y_{i} is a strictly local normal scheme for every i ∈ I, and the transition morphisms Y_{j} → Y_{i} are local and dominant, for every morphism i → j in I. Let \overline{x} be a geometric point of X, and denote by \overline{x}_{i} the image of \overline{x} in X_{i}, for every i ∈ I. Suppose that the image \overline{y} of \overline{x} in Y is localized at the closed point. Also, let \overline{η} be a strict geometric point of Y, localized at the generic point η_{i}, and denote by \overline{η}_{i} (resp. \overline{y}_{i}) the strict image of \overline{η} (resp. \overline{y}) in Y_{i} (see definition 4.9.17(v)).

Lemma 13.4.35. In the situation of (13.4.34), suppose that (g, log g) : (X, \underline{M}) → (Y, \underline{N}) is a smooth and saturated morphism of fine log schemes. Then there exist i ∈ I, and a smooth and saturated morphism (g_{i}, log g_{i}) : (X_{i}, \underline{M}_{i}) → (Y_{i}, \underline{N}_{i}) of fine log schemes, such that log g = π_{i}^{∗} log g_{i}.

Proof. By corollary 12.3.45, we can descend (g, log g) to a smooth morphism (g_{i}, log g_{i}) of fine log schemes, and after replacing I by I/i, we may assume that i = 0. Then the contention follows from corollary 12.2.36(ii). □

13.4.36. Keep the situation of (13.4.34), and suppose that (g_{0}, log g_{0}) : (X_{0}, \underline{M}_{0}) → (Y_{0}, \underline{N}_{0}) is a smooth and saturated morphism of fine log schemes; set

(X_{i}, \underline{M}_{i}) := X_{i} ×_{X_{0}} (X_{0}, \underline{M}_{0}) (Y_{i}, \underline{N}_{i}) := Y_{i} ×_{Y_{0}} (Y_{0}, \underline{Y}_{0})

and denote (g_{i}, log g_{i}) : (X_{i}, \underline{M}_{i}) → (Y_{i}, \underline{N}_{i}) the induced morphism of log schemes, for every i ∈ I. Also, let (g, log g) : (X, \underline{M}) → (Y, \underline{N}) be the limit of the system of morphisms ((g_{i}, log g_{i}) | i ∈ I). These are morphisms of the type considered in (13.4), so we may define U_{i} := g_{i,\overline{x}_{i}}^{−1}(η_{i}), and introduce likewise the schemes U_{i,tr}, \overline{U}_{i} and \overline{U}_{i,tr} as in (13.4). Moreover, set Z_{i} := \overline{U}_{i}\\overline{U}_{i,tr} for every i ∈ I; clearly Z_{i} = Z_{j} ×_{X_{j}(\overline{x}_{j})} X_{i}(\overline{x}_{i}) for every morphism i → j in I. Also, each Z_{i} is a finite union of irreducible subsets of codimension one, and for every i → j in I, the transition morphisms X_{i} → X_{j} restrict to maps

Max Z_{i} → Max Z_{j} Max X_{i}(\overline{x}_{i}) ×_{Y_{i}} |\overline{y}_{i}| → Max X_{j}(\overline{x}_{j}) ×_{Y_{j}} |\overline{y}_{j}|

Combining proposition 13.3.23(ii) and lemma 13.4.29, we deduce a fully faithful functor

(13.4.37)  2-colim_{i∈I} Tame(g_{i}, \overline{x}_{i}) → Tame(g, \overline{x}).

Lemma 13.4.38. The functor (13.4.37) is an equivalence.

Proof. It remains only to show the essential surjectivity. Hence, let h be a given object of Tame(g, \overline{x}); by proposition 13.3.23(ii), we know that there exists j ∈ I such that h descends to an étale covering h_{j} : V_{j} → \overline{U}_{j,tr}, tamely ramified along Z_{j}, and after replacing I by I/i, we may assume that j is the final object of I, and define h_{i} := \overline{U}_{i,tr} ×_{\overline{U}_{j,tr}} h_{j} for every i → j in I. Now, let E′ ⊂ E(h) be a constructible open subset containing the maximal points of X(\overline{x}) ×_{Y} |\overline{y}|. For every i ∈ I, let \overline{Y}_{i} be the normalization of Y_{i} in Spec κ(\overline{η}_{i}), and set \overline{X}_{i} := X_{i}(\overline{x}_{i}) ×_{Y_{i}} \overline{Y}_{i}; according to [65, Ch.IV, Th.8.3.11], there exists i ∈ I such that E′ descends to a constructible open subset E′_{i} ⊂ \overline{X}_{i}, and then necessarily E′_{i} contains all the maximal points of X_{i}(\overline{x}_{i}) ×_{Y_{i}} |\overline{y}_{i}|. As usual, we may assume that i is the final object, so E′_{i} is defined for every i ∈ I. Lastly, since h extends to an étale covering on E′, we see that h_{i} extends to an étale covering of E′_{k}, for some k ∈ I ([66, Ch.IV, Prop.17.7.8(i)]). In view of lemma 13.4.29, the contention follows. □

Theorem 13.4.39. The map (13.4.22) is an isomorphism.

Proof. Arguing as in the proof of proposition 13.4.21, we may assume that both (X, \underline{M}) and (Y, \underline{N}) are fs log schemes, and in view of proposition 13.4.21, we need only show that (13.4.22) is injective. This comes down to the following assertion. For every object \overline{h} : \overline{C} → \overline{U}_{tr} of the category Tame(f, \overline{x}), the induced action of π_{1}(\overline{U}_{tr}, ξ) on \overline{h}^{−1}(ξ) factors through the quotient Coker (log f gp ; \overline{x})^{∨} ⊗_{ℤ} ∏_{ℓ≠p} ℤ_{ℓ}(1).

• By lemma 13.4.27, there exist a finite separable extension K of κ(η), and an object h : C_{K} → U_{K,tr} of Tame(f, \overline{x}, K), with an isomorphism C_{K} ×_{U_{K,tr}} \overline{U}_{tr} →^{∼} \overline{C} of \overline{U}_{tr}-schemes. Since log f\overline{x} = log f_{K,\overline{x}_{K}}, the theorem will hold for the morphism f and the point \overline{x}, if and only if it holds for f_{K} and the point \overline{x}_{K}.

Claim 13.4.40. The theorem holds if Y is noetherian of dimension one.

Proof of the claim. In this case, Y is the spectrum of a strictly henselian discrete valuation ring R, and the same then holds for Y_{K}. Hence, we may replace throughout f by f_{K}, and assume from start that there exists an object h : C → U_{tr} of Tame(f, \overline{x}, κ(η)), with an isomorphism \overline{C} →^{∼} C ×_{U_{tr}} \overline{U}_{tr} of \overline{U}_{tr}-schemes. Endow Y with the fine log structure \underline{N}′ such that Γ(Y, \underline{N}′) = R \\{0\}; since (Y, \underline{N})_{tr} is dense in Y, we have a well defined morphism of log schemes π : (Y, \underline{N}′) → (Y, \underline{N}), which is the identity on the underlying schemes. Set (X, \underline{M}′) := (Y, \underline{N}′) ×_{(Y,\underline{N})} (X, \underline{M}). Then (Y, \underline{N}′) is a regular log scheme, and consequently the same holds for (X, \underline{M}′), by theorem 12.5.28.

Furthermore, π trivially restricts to a strict morphism on the open subset |η|, hence the induced morphism (X, \underline{M}′) ×_{Y} |η| → (X, \underline{M}) ×_{Y} |η| is an isomorphism, especially U_{tr} is the trivial locus of (X(\overline{x}), \underline{M}′(\overline{x})) ×_{Y} |η|. However, it is easily seen that (X(\overline{x}), \underline{M}′(\overline{x}))_{tr} does not intersect the closed fibre f −1 ; \overline{x}(\overline{y}), so finally U_{tr} = (X(\overline{x}), \underline{M}′(\overline{x}))_{tr}.

From theorem 13.3.45, we deduce that h extends to an étale covering of (X(\overline{x}), \underline{M}′(\overline{x})). Then, arguing as in (13.4) we get a commutative diagram of groups :

\xymatrix{π_{1}(\overline{U}_{tr,ét}, ξ) \ar[r] \ar[d] & Coker (log f gp ; \overline{x})^{∨} ⊗_{ℤ} ∏_{ℓ≠p} ℤ_{ℓ}(1) \ar[d] \\ π_{1}(U_{tr,ét}, ξ′) \ar[r]^{α} & \underline{M} gp∨ ; \overline{x} ⊗_{ℤ} ∏_{ℓ≠p} ℤ_{ℓ}(1)}

whose top horizontal arrow is (13.4.3), and whose right vertical arrow is deduced from the projection \underline{M} gp ; \overline{x} → Coker (log f gp ; \overline{x}). Lastly, proposition 13.3.43shows that the natural action of π_{1}(U_{tr,ét}, ξ′) on h^{−1}(ξ′) factors through α, so the claim follows. ♢

• Next, suppose that Y is an arbitrary normal, strictly local scheme. The discussion in (13.4.14) implies that, in order to prove the theorem, it suffices to find an integer e > 0, a (X, \underline{M})-scheme (X′_{e}, \underline{M}′_{e}) as in (13.4.11), and a geometric point \overline{x}′_{e} of X′_{e} lying over \overline{x}, such that h ×_{X(\overline{x})} X′_{e}(\overline{x}′_{e}) is a trivial covering. To this aim, we write Y as the limit of a cofiltered system (Y_{i} | i ∈ I) of strictly local excellent and normal schemes (lemma 13.1.29), and we denote by η_{i} the generic point of Y_{i}, for every i ∈ I. By lemma 13.4.35, we may then descend (f, log f) to a smooth and saturated morphism (f_{i}, log f_{i}) : (X_{i}, \underline{M}_{i}) → (Y_{i}, \underline{N}_{i}), for some i ∈ I, and as usual, we may assume that i is the final object of I. Let \overline{x}_{i} be the image of \overline{x} in X_{i}; by lemmata 13.4.38and 13.4.27, the object h of Tame(f, \overline{x}, κ(η)) descends to an object h_{i} of Tame(f_{i}, \overline{x}_{i}, K), for some i ∈ I, and some finite separable extension K of κ(η_{i}). It suffices therefore to find e > 0, a (X_{i}, \underline{M}_{i})-scheme (X′_{i,e}, \underline{M}′_{i,e}), and a geometric point \overline{x}′_{i,e} of X′_{i,e} lying over \overline{x}_{i}, such that h_{i} ×_{X_{i}(\overline{x}_{i})} X′_{i,e}(\overline{x}′_{i,e}) is a trivial covering. In other words, we may replace throughout Y by Y_{i,K}, and assume from start that Y is excellent, and \overline{h} descends to an object h : C → U_{tr} of Tame(f, \overline{x}, κ(η)).

• By [60, Ch.II, Prop.7.1.7] (see also remark 12.5.34) we may find a discrete valuation ring V and a local injective morphism R → V inducing an isomorphism on the respective fields of fractions. Let V^{sh} be the strict henselization of V (at a geometric point whose support is the closed point), and set

Y := Spec V^{sh} (Y, \underline{N}) := Y ×_{Y} (Y, \underline{N}) (X, \underline{M}) := Y ×_{Y} (X, \underline{M}).

Also, let f : (X, \underline{M}) → (Y, \underline{N}) be the induced morphism. Denote by \overline{y} a geometric point localized at the closed point y of Y; also, pick any geometric point \overline{x} of X, whose image in X is \overline{x}; the induced morphism

(13.4.41)  (X(\overline{x}), \underline{M}(\overline{x})) → (X(\overline{x}), \underline{M}(\overline{x}))

restricts to a flat morphism f−1 ; \overline{x}(\overline{y}) → f −1 ; \overline{x}(\overline{y}) and from proposition 12.7.14, we see that the latter induces a bijection between the sets of maximal points of the two fibres. On the other hand, let \overline{η}_{V} denote a geometric point localized at the generic point η_{V} of Y; then (13.4.41) restricts to an ind-étale morphism f−1 ; \overline{x}(η_{V}) → f −1 ; \overline{x}(η). Hence, set

U_{tr} := (X(\overline{x}), \underline{M}(\overline{x}))_{tr} ×_{Y} |η_{V}|.

From lemma 13.4.29, it follows easily that the covering C ×_{U_{tr}} U_{tr} → U_{tr} is an object of Tame(f, \overline{x}, κ(η_{V})).

For any integer e > 0 invertible in R, pick a (Y, \underline{N})-scheme (Y_{e}, \underline{N}_{e}) as in (13.4.11), so that we may define the étale morphism (X′_{e}, \underline{M}′_{e}) → (X_{e}, \underline{M}_{e}) of (Y_{e}, \underline{N}_{e})-schemes as in (13.4.13). Notice that the morphism (X′_{e}, \underline{M}′_{e}) → (Y_{e}, \underline{N}_{e}) is again of the type considered in (13.4), and there exists, up to isomorphism, a unique geometric point \overline{x}′_{e} of X′_{e} lifting \overline{x}; moreover, for any geometric point \overline{y}_{e} supported at y_{e}, the induced map

Spec \underline{M}′_{e,\overline{x}′_{e}} → Spec \underline{N}′_{e,\overline{y}_{e}}

is naturally identified with (13.4.19). Likewise, pick a (Y, \underline{N})-scheme (Y_{e}, \underline{N}_{e}) in the same fashion, and denote by η_{e} (resp. η_{V,e}) the generic point of Y_{e} (resp. of Y_{e}), and by y_{e} ∈ Y_{e} (resp. y_{e} ∈ Y_{e}) the closed point. We may choose Y_{e} so that κ(η_{V,e}) contains κ(η_{e}), in which case we have a strict morphism

(Y_{e}, \underline{N}_{e}) → (Y_{e}, \underline{N}_{e})

of log schemes, and we may set (X′_{e}, \underline{M}′_{e}) := Y_{e} ×_{Y_{e}} (X′_{e}, \underline{M}′_{e}). Again, there exists, up to isomorphism, a unique geometric point \overline{x}′_{e} of X′_{e} lifting \overline{x}, and by claim 13.4.40, we may assume that both e and κ(η_{V,e}) have been chosen large enough, so that the base change

h ×_{X(\overline{x})} X′_{e}(\overline{x}′_{e})

shall be a trivial étale covering. Hence, we may replace Y by Y_{e}, Y by Y_{e}, X by X′_{e}, and h by h ×_{X(\overline{x})} X′_{e}(\overline{x}′_{e}), and assume from start that C ×_{U_{tr}} U_{tr} is a trivial covering of U_{tr}. The theorem will follow, once we show that – in this case – h is a trivial étale covering.

Let C′ (resp. C′) be the normalization of X(\overline{x}) in C (resp. of X(\overline{x}) in C ×_{U_{tr}} U_{tr}), and define E(h) as in (13.4.28). Then C′ is a trivial étale covering of X(\overline{x}), and since X(\overline{x}) is excellent, the induced morphism h′ : C′ → X(\overline{x}) is finite.

Claim 13.4.42. For every maximal point η_{𝔮} of f −1 ; \overline{x}(\overline{y}), the induced covering C_{|η_{𝔮}} → |η_{𝔮}| is trivial (notation of (13.4.31)).

Proof of the claim. Let Z_{𝔮} denote the topological closure of \{η_{𝔮}\} in X(\overline{x}), and endow Z_{𝔮} with its reduced subscheme structure; then E_{𝔮} := E(h) ∩ Z_{𝔮} is non-empty (lemma 13.4.29), and geometrically normal (proposition 12.7.14(ii) and corollary 12.5.13). Also, (13.4.41) induces an isomorphism of κ(y)-schemes

E_{𝔮} ×_{X(\overline{x})} X(\overline{x}) →^{∼} E_{𝔮} ×_{κ(y)} κ(y).

Moreover, Z_{𝔮} is strictly local, and Z_{𝔮} ×_{X(\overline{x})} X(\overline{x}) is the strict henselization of Z_{𝔮} ×_{κ(y)} κ(y) at the point x. Furthermore, the morphism h′′ := h′ ×_{X(\overline{x})} E_{𝔮} is an étale covering of E_{𝔮}, and h′′ ×_{κ(y)} κ(y) is naturally identified with the restriction of C′ to the subscheme E_{𝔮} ×_{X(\overline{x})} X(\overline{x}) ([66, Ch.IV, Prop.17.5.7]), hence it is a trivial covering. By example 13.2.6, it follows that h′′ is trivial as well. Since C_{|η_{𝔮}} is the restriction of h′′ to |η_{𝔮}|, the claim follows. ♢

Clearly (13.4.41) maps each stratum U_{𝔮} of the logarithmic stratification of X(\overline{x}), to the corresponding stratum U_{𝔮} of the logarithmic stratification of X(\overline{x}) (see (12.5.35)). More precisely, since (13.4.41) ×_{Y} |η| is ind-étale, proposition 12.7.17(iii) implies that the generic point of U_{𝔮} ×_{Y} |η_{V}| gets mapped to the generic point of U_{𝔮} ×_{Y} |η|. We conclude that E(h) contains the generic point of every stratum U_{𝔮} ×_{Y} |η|.

Claim 13.4.43. X(\overline{x}) ×_{Y} |η| ⊂ E(h).

Proof of the claim. Notice first that (X, \underline{M}) ×_{Y} |η| is a regular log scheme (corollary 12.5.29).

For any geometric point ξ of X(\overline{x}), denote by (X(ξ), \underline{M}(ξ)) the strict henselization of (X(\overline{x}), \underline{M}(\overline{x})) at ξ, and set C′(ξ) := C′ ×_{X(\overline{x})} X(ξ). Now, suppose that the support of ξ lies in the stratum U_{𝔮} ×_{Y} |η|, and let ξ_{𝔮} be a geometric point localized at the generic point of U_{𝔮}. By assumption, C′(ξ) is a finite X(ξ)-scheme, tamely ramified along the non-trivial locus of (X(ξ), \underline{M}(ξ)). Likewise, C′(ξ_{𝔮}) is tamely ramified along the non-trivial locus of (X(ξ_{𝔮}), \underline{M}(ξ_{𝔮})). Pick any strict specialization map (X(ξ_{𝔮}), \underline{M}(ξ_{𝔮})) → (X(ξ), \underline{M}(ξ)) (see (12.7.11)); it induces a functor

(13.4.44)  Cov(X(ξ), \underline{M}(ξ)) → Cov(X(ξ_{𝔮}), \underline{M}(ξ_{𝔮}))

and theorem 13.3.45implies that C′(ξ) is an object of the source of (13.4.44), which is mapped, under this functor, to the object C′(ξ_{𝔮}). By proposition 13.3.43, for any geometric point ξ of X(\overline{x}) ×_{Y} |η|, the category Cov(X(ξ), \underline{M}(ξ)) is equivalent to the category of finite sets with a continuous action of (\underline{M}_{ξ})^{gp∨} ⊗_{ℤ} ∏_{ℓ≠p} ℤ_{ℓ}(1). On the other hand, clearly \underline{M}(\overline{x})^{♯} restricts to a constant sheaf of monoids on (U_{𝔮})_{τ}. In view of (13.3.54), we deduce that (13.4.44) is an equivalence; lastly, we have seen that the induced morphism C′(ξ_{𝔮}) → X(ξ_{𝔮}) is étale, i.e. is a trivial covering, therefore the same holds for the morphism C′(ξ) → X(ξ), and consequently the support of ξ lies in E(h) (claim 13.1.9). Since ξ is arbitrary, the assertion follows. ♢

Claim 13.4.45. There exists a non-empty open subset U_{Y} ⊂ Y such that X(\overline{x}) ×_{Y} U_{Y} ⊂ E(h).

Proof of the claim. Since X(\overline{x}) is a noetherian scheme, E(h) is a constructible open subset, hence Z := X(\overline{x})\E(h) is a constructible closed subset of X(\overline{x}). The subset f\overline{x}(Z) is proconstructible ([63, Ch.IV, Prop.1.9.5(vii)]) and does not contain η (by claim 13.4.43), hence neither does its topological closure W ([63, Ch.IV, Th.1.10.1]). It is easily seen that U_{Y} := Y \ W will do. ♢

Claim 13.4.46. Let U_{Y} be as in claim 13.4.45. We have :

(i) There exists an irreducible closed subset Z of Y of dimension one, such that Z ∩ (Y, \underline{N})_{tr} ∩ U_{Y} ≠ ∅.

(ii) For any Z as in (i), the induced functor Cov(E(h)) → Cov(Z×_{Y} E(h)) is fully faithful.

Proof of the claim. (i): More generally, let (A, 𝔪) be any local noetherian domain of Krull dimension d ≥ 1, and W ⊂ Spec A a proper closed subset; we show that there exists an irreducible closed subset Z ⊂ Spec A of dimension one, not contained in W. To this aim, we may assume that W = Spec A/fA for some f ∈ 𝔪\{0}; let 𝔫 be any maximal ideal of A[f^{−1}], and 𝔭 := A ∩ 𝔫. Since A/𝔭[f^{−1}] is a field, [63, Ch.0, Cor.16.3.3] implies that Z := Spec A/𝔭 will do.

(ii): It suffices to check that conditions (i)–(iii) of proposition 13.1.37 hold for Z and the open subset E(h). However, condition (i) is immediate, since the generic point η_{Z} of Z lies in U_{Y}. Likewise, condition (ii) holds trivially for the fibre over the point η_{Z}, so it suffices to consider the fibre over the closed point y of Z, in which case the assertion is just lemma 13.4.29. Lastly, condition (iii) follows directly from theorem 12.7.8(iii.b) and [66, Ch.IV, Prop.18.8.10, 18.8.12(i)]. ♢

Let Z be as in claim 13.4.46(i), and endow Z with its reduced subscheme structure. Let also Z′ be the normalization of Z; then both Z and Z′ are strictly local, and the morphism Z′ → Z is radicial and surjective, hence the induced functor

Cov(Z ×_{Y} E(h)) → Cov(Z′ ×_{Y} E(h))

is an equivalence (lemma 13.1.7(i)). Taking into account claim 13.4.46, we are thus reduced to showing that the morphism

(Z′ ×_{Y} E(h)) ×_{X(\overline{x})} C′ → Z′ ×_{Y} E(h)

is a trivial étale covering. However, let η_{Z′} be the generic point of Z′, and set

(Z′, \underline{N}′) := Z′ ×_{Y} (Y, \underline{N}) (X′, \underline{M}′) := Z′ ×_{Y} (X, \underline{M}).

The open subset (Z′, \underline{N}′)_{tr} is dense in Z′, by virtue of claim 13.4.46(i), so the induced morphism f′ : (X′, \underline{M}′) → (Z′, \underline{N}′) is still of the type considered in (13.4), the geometric point \overline{x} lifts uniquely (up to isomorphism) to a geometric point \overline{x}′ of X′, and h ×_{Y} Z′ is an object of Tame(f′, \overline{x}′, κ(η_{Z′})) (proposition 13.3.20(i)). Hence, we may replace from start (X, \underline{M}) by (X′, \underline{M}′), (Y, \underline{N}) by (Z′, \underline{N}′), h by h ×_{Y} Z′, after which, we may assume that Y is noetherian and of dimension one. Moreover, taking into account claim 13.4.42, we may assume that the induced covering C_{|η_{𝔮}} → |η_{𝔮}| is trivial, for every maximal point η_{𝔮} of f −1 ; \overline{x}(\overline{y}), and it remains to show that h is trivial under these assumptions.

To this aim, we look at the corresponding commutative diagram of groups, provided by lemma 13.4.33(i) : with the notation of loc.cit., we see that in the current situation, β is an isomorphism as well, by claim 13.4.40, therefore lemma 13.4.33(ii) says that the group homomorphism π_{1}(|η_{𝔮}|_{ét}, \overline{η}_{𝔮}) → π_{1}(\overline{U}_{tr}/Y_{ét}, ξ) is surjective, for any maximal point η_{𝔮} of f −1 ; \overline{x}(\overline{y}). From this, we deduce that \overline{h} is a trivial covering, and therefore there exists an étale covering C_{Y} → |η| with an isomorphism C →^{∼} C_{Y} ×_{|η|} U_{tr} ([82, Exp.IX, Th.6.1]). Denote by C_{Y}^{ν} the normalization of Y in C_{Y}. Also, set E′ := E(h) ⊂ Str(f\overline{x}). In light of theorem 12.7.8(iii.a) and lemma 13.4.29, it is easily seen that the restriction E′ → Y of f\overline{x} is surjective; the latter is also a smooth morphism of schemes (corollary 12.3.27(i)). It follows that C_{Y}^{ν} ×_{Y} E′ is the normalization of E′ in C ([66, Ch.IV, Prop.17.5.7]), especially, it is an étale covering of E′. We then deduce that C_{Y}^{ν} is already a (trivial) étale covering of Y ([66, Ch.IV, Prop.17.7.1(ii)]), and then clearly h must be a trivial covering as well. □

Remark 13.4.47. Theorem 13.4.39 is the local acyclicity result that gives the name to this section. However, the title is admittedly not self-explanatory, and its full justification would require the introduction of a more advanced theory of the log-étale site, that lies beyond the bounds of this treatise. In rough terms, we can try to describe the situation as follows. In lieu of the standard strict henselization, one should consider a suitable notion of strict log henselization for points of the log-étale topoi associated with log schemes. Then, for f : X → Y as in (13.4) with saturated log structures on both X and Y, and log-étale points x̃ of X with image ỹ in Y, one should look, not at our f\overline{x}, but rather at the induced morphism fx̃ of strict log henselizations (of X at x̃ and of Y at ỹ). The (suitably defined) log geometric fibres of fx̃ will be the log Milnor fibres of f at the log-étale point x̃, and one can state for such fibres an acyclicity result : namely, the prime-to-p quotients of their (again, suitably defined) log fundamental groups vanish. The proof proceeds by reduction to our theorem 13.4.39, which, with hindsight, is seen to supply the essential geometric information encoded in the more sophisticated log-étale language.

14. The almost purity toolbox

The sections of this rather heterogeneous chapter are each devoted to a different subject, and are linked to each other only very loosely, if they are at all. They have been lumped here together, because they each contribute a distinct self-contained little theory, that will find application in one step or other of the proof of the almost purity theorem or of its applications in chapter 17. The exception is section 14.8 : it studies a class of rings more general than the measurable algebras introduced in section 14.5; the results of section 14.8 will not be used elsewhere in this treatise, but they might be interesting for other purposes.

Section 14.4 develops the yoga of almost pure pairs (see definition 14.4.1(i)); the relevance to the almost purity theorem is clear, since the latter establishes the almost purity of certain pairs (X, Z) consisting of a scheme X and a closed subscheme Z ⊂ X. This section provides the means to perform various kinds of reductions in the proof of the almost purity theorem, allowing to replace the given pair (X, Z) by more tractable ones.

Section 14.5 introduces measurable (and more generally, ind-measurable) K^{+}-algebras, for K^{+} a fixed valuation ring of rank one : see definitions 14.5.3(ii) and 14.5.64. For modules over a measurable algebra, one can define a well-behaved real-valued normalized length. This length function is non-negative, and additive for short exact sequences of modules. Moreover, the length of an almost zero module vanishes (for the standard almost structure associated with K^{+}). Conversely, under some suitable assumptions, a module of normalized length zero will be almost zero.

Lastly, section 14.6 studies some questions concerning the formation of quotients of affine almost schemes under a finite group action.

14.1. Non-flat almost structures. This section contains some material that complements the generalities of [75, §2.4, §2.5] : indeed, whereas many of the preliminaries in loc.cit. make no assumptions on the basic setup (V, 𝔪) that underlies the whole discussion, for the more advanced results it is usually required that 𝔪̃ := 𝔪 ⊗_{V} 𝔪 is a flat V -module. We shall show how, with more work, one can remove this condition (or at least, weaken it significantly) and still recover most of the useful almost ring theory of [75]. This extension shall be applied in a later section, in order to state and prove the most general case of almost purity.

14.1.1. We begin with a simple observation, that completes the discussion of [75, §2.1.6].

Proposition 14.1.2. Let V be a reduced ring, and consider ideals 𝔪, J_{1}, …, J_{k} ⊂ V such that: (a) 𝔪 is a filtered union of principal ideals.

(b) J_{1} ⊗_{V} ⋯ ⊗_{V} J_{k} is a flat V -module. Then the following holds :

(i) The natural map 𝔪 ⊗_{V} 𝔪 → 𝔪^{2} is an isomorphism.

(ii) If the set Min V of minimal prime ideals of V is a quasi-compact subset of Spec V, then the natural map J_{1} ⊗_{V} ⋯ ⊗_{V} J_{k} → J_{1} ⋯ J_{k} is an isomorphism.

Proof. (i): We are easily reduced to checking the following. For every x ∈ V, the natural map μ : V x ⊗_{V} V x → V x^{2} is an isomorphism. Now, I_{n} := Ann_{V} (x^{n}) for n = 1, 2, and notice that I_{1} ⊂ I_{2}; the map μ is naturally identified with the map

V/I_{1} ⊗_{V} V/I_{1} → V/I_{2} \overline{a} ⊗ \overline{b} ↦ \overline{ab}.

The latter is the composition of the natural isomorphism V/I_{1} ⊗_{V} V/I_{1} →^{∼} V/I_{1} and the projection V/I_{1} → V/I_{2}, so we are reduced to checking that I_{1} = I_{2}. But if x^{2}y = 0 for some y ∈ V, then (xy)^{2} = 0, hence xy = 0, since V is reduced, and so y ∈ I_{1}, whence the contention.

(ii): Set L := J_{1} ⊗_{V} ⋯ ⊗_{V} J_{k} and J := J_{1} ⋯ J_{k}; the natural map φ : L → J is surjective, so it remains to check that it is injective, under the assumptions of (ii). However, notice that, for every 𝔭 ∈ Y := Min V, the localization V_{𝔭} is a field, since V is reduced; it follows easily that φ_{𝔭} : L_{𝔭} → J_{𝔭} is an isomorphism for every such 𝔭. We are then reduced to checking that the natural map ψ_{L} : L → ∏_{𝔭∈Min V} L_{𝔭} is injective. However, endow Y with the topology induced by X := Spec V via the inclusion map i : Y → X, and denote by L^{∼} the quasi-coherent 𝒪_{X}-module arising from L. Then ψ_{L} is a composition of maps

L ⟶^{ψ′_{L}} Γ(Y, i^{∗}L^{∼}) ⟶^{ψ′′_{L}} ∏_{𝔭∈Y} L_{𝔭}^{∼} = ∏_{𝔭∈Y} L_{𝔭}

and ψ′′_{L} is obviously injective. Thus, it remains to show that ψ′_{L} is injective. But, by Lazard’s theorem [120, Ch.I,Th.1.2], L is the colimit of a filtered system (F_{λ} | λ ∈ Λ) of free V -modules of finite rank, so L^{∼} is the colimit of the induced filtered system (F_{λ}^{∼} | λ ∈ Λ) of quasicoherent 𝒪_{X}-modules. By lemma 10.1.7(i), we are then reduced to checking that the natural map F_{λ} → Γ(Y, i^{∗}F_{λ}^{∼}) is injective for every λ ∈ Λ. For the latter, it suffices to check that the natural map ψ_{V} : V → Γ(Y, i^{∗}𝒪_{X}) is injective; but the composition of ψ_{V} with the natural map Γ(Y, i^{∗}𝒪_{X}) → ∏_{𝔭∈Y} 𝒪_{X,𝔭} = ∏_{𝔭∈Y} V_{𝔭} is injective, since V is reduced, whence the claim. □

Remark 14.1.3. As a corollary, we see that if (V, 𝔪) is a basic setup (in the sense of [75, §2.1.1]), such that V is a reduced ring, then the natural map 𝔪̃ → 𝔪 is an isomorphism, provided that either (a) 𝔪 is a filtered union of principal ideal, or else (b) 𝔪̃ is a flat V -module, and Min V is a quasi-compact subset of Spec V.

14.1.4. Let now (V, 𝔪) be any basic setup in the sense of [75, §2.1.1], and R any V -algebra. For every interval I ⊂ ℕ, we have a localization functor

(14.1.5)  C^{I}(R-Mod) → C^{I}(R^{a}-Mod)  K^{•} ↦ K^{•a}

from complexes of R-modules, to complexes of R^{a}-modules, which is obviously exact, hence it induces a derived localization functor :

(14.1.6)  D^{I}(R-Mod) → D^{I}(R^{a}-Mod).

The functor (14.1.5) admits a left (resp. right) adjoint

(14.1.7)  C^{I}(R^{a}-Mod) → C^{I}(R-Mod)  K^{•} ↦ K_{!}^{•}  ( resp. K^{•} ↦ K_{∗}^{•} )

defined by applying termwise to K^{•} the functor M ↦ M_{!} (resp. M ↦ M_{∗}) for R^{a}-modules given by [75, §2.2.10, §2.2.21]. However, if 𝔪̃ is not flat, the functor M ↦ M_{!} is obviously not exact, and the localization functor R-Mod → R^{a}-Mod does not send injectives to injectives (cp. [75, Cor.2.2.24]). This makes it trickier to deal with constructions in the derived category; for instance, if 𝔪̃ is flat, we get a left adjoint to (14.1.6), simply by deriving trivially the exact functor M ↦ M_{!}. This fails in the general case, but we shall see later that a suitable derived version of the construction of M_{!} is still available.

14.1.8. For any interval I ⊂ ℕ, let Σ_{I} be the multiplicative set of morphisms φ in D^{I}(R-Mod) such that φ^{a} is an isomorphism in D^{I}(R^{a}-Mod); arguing as in the proof of [163, Prop.10.4.4], one sees that Σ_{I} is right locally small, hence the localized category Σ_{I}^{−1}D^{I}(R-Mod) has small Hom-sets (proposition 1.6.16(ii)) which are independent (up to natural isomorphism) of the choice of universe. Obviously the derived localization functor factors through a natural functor

(14.1.9)  Σ_{I}^{−1}D^{I}(R-Mod) → D^{I}(R^{a}-Mod).

Lemma 14.1.10. (i) For every interval I, the functor (14.1.9) is an equivalence.

(ii) For every K^{•}, L^{•} ∈ Ob(D^{I}(R-Mod)) the induced R^{a}-linear morphism

Hom_{D^{I}(R-Mod)}(K^{•}, L^{•})^{a} → Hom_{D^{I}(R^{a}-Mod)}(K^{•a}, L^{•a})^{a}

is an isomorphism.

Proof. (i): A proof is sketched in [75, §2.4.9], in case 𝔪̃ is flat, but in fact this assumption is superfluous. Indeed, since the unit of adjunction M → M_{!}^{a} is an isomorphism ([75, Prop.2.2.23(ii)]), it is clear that the functor K^{•} ↦ K_{!}^{•} of (14.1.7) descends to a functor

(14.1.11)  D^{I}(R^{a}-Mod) → Σ_{I}^{−1}D^{I}(R-Mod)

such that the composition (14.1.9) ∘ (14.1.11) is naturally isomorphic to the identity automorphism of D^{I}(R^{a}-Mod). Likewise, a simple inspection shows that (14.1.11) ∘ (14.1.9) is naturally isomorphic to the identity of Σ_{I}^{−1}D^{I}(R-Mod), whence the contention.

(ii): The R-module Hom_{Σ_{I}^{−1}D^{I}(R-Mod)}(K^{•}, L^{•}) is calculated as the colimit of the system of R-modules (Hom_{D^{I}(R-Mod)}(K^{′•}, L^{•}) | φ^{•} : K^{′•} → K^{•}), where φ^{•} ranges over the elements of Σ_{I} with target K^{•}. For such a morphism φ^{•}, we have 𝔪 · H^{i}(Cone φ^{•}) = 0 for every i ∈ ℤ.

Claim 14.1.12. Let C^{•} be any object of D^{I}(R-Mod) such that (H^{i}C^{•})^{a} = 0 for every i ∈ ℤ. Then Hom_{D^{I}(R-Mod)}(C^{•}, L^{•})^{a} = 0 for every L^{•} ∈ Ob(D^{I}(R-Mod)).

Proof of the claim. Set H_{n,k} := Hom_{D^{I}(R-Mod)}(C^{•}, (τ^{≥−n}L^{•})[k]) for every n, k ∈ ℕ; the natural morphism L^{•} → lim_{n∈ℕ} τ^{≥−n}L^{•} is an isomorphism in C(R-Mod), so we have a short exact sequence

0 → lim_{n∈ℕ}^{1}H_{n,−1} → Hom_{D(R-Mod)}(C^{•}, L^{•}) → lim_{n∈ℕ} H_{n,0} → 0

(proposition 7.3.45(ii)). Hence, it suffices to check that H_{n,0}^{a} = H_{n,−1}^{a} = 0 for every n ∈ ℕ, and we are reduced to the case where L^{•} is bounded below.

Likewise, set H′_{n} := Hom_{D(R-Mod)}(τ^{≤n}C^{•}, L^{•}) for every n, k ∈ ℤ; since the natural morphism colim_{n∈ℕ} τ^{≤n}C^{•} → C^{•} is an isomorphism in C(R-Mod), we have an isomorphism

Hom_{D(R-Mod)}(C^{•}, L^{•}) →^{∼} lim_{n∈ℕ} H′_{n}

(proposition 7.3.45(i)), so we may assume that C^{•} is a bounded above complex. In this situation, say that L^{•} ∈ Ob(D^{≥a}(R-Mod)) and C^{•} ∈ Ob(D^{≤b}(R-Mod)) for some a, b ∈ ℤ; we may assume that a ≤ b, and we show, by induction on n, that (H′_{n})^{a} = 0 for every n ≥ a − 1. The claim will follow for n = b. Indeed, notice that

H′_{r} = Hom_{D(R-Mod)}(τ^{≤r}C^{•}, τ^{≤r}L^{•}) for every r ∈ ℤ

by virtue of remark 7.3.14(iv), so the assertion is clear for n = a − 1. Next, suppose that the assertion is already known for every integer < n; from the short exact of complexes 0 → τ^{≤n−1}C^{•} → τ^{≤n}C^{•} → (H^{n}C^{•})[−n] → 0 we deduce the exact sequence

Hom_{D(R-Mod)}((H^{n}C^{•})[−n], L^{•}) → H′_{n} → H′_{n−1}

(remark 7.3.35). But since 𝔪 · H^{n}C^{•} = 0, the first term in this exact sequence is annihilated by 𝔪, and the same holds for the third term, by inductive assumption; then it holds for the middle term as well, as required. ♢

By claim 14.1.12, we have Hom_{D(R-Mod)}(Cone φ^{•}, L^{•})^{a} = 0; by considering the long exact Hom_{D(R-Mod)}(−, L^{•})-sequence associated with the distinguished triangle K^{′•} → K^{•} → Cone φ, we deduce an R^{a}-linear isomorphism

Hom_{D^{I}(R-Mod)}(φ^{•}, L^{•})^{a} : Hom_{D^{I}(R-Mod)}(K^{′•}, L^{•})^{a} →^{∼} Hom_{D^{I}(R-Mod)}(K^{•}, L^{•})^{a}

(see remark 7.3.35). In light of (i), the assertion follows straightforwardly. □

14.1.13. We sketch a few generalities on derived tensor products, that will be applied to construct useful objects in various derived categories. Recall first that the tensor product −⊗_{R} − on R-modules descends to a bifunctor −⊗_{R^{a}} − ([75, §2.2.6, §2.2.12]), and if M is a flat R-module, then M^{a} is a flat R^{a}-module ([75, Lemma 2.4.7]). It follows that the category R^{a}-Mod has enough flat objects, so every bounded above complex of R^{a}-modules admits a bounded above flat resolution. Now, given bounded above complexes K^{•}, L^{•} of R^{a}-modules, set

K^{•} ⊗^{L}_{R^{a}} L^{•} := (K_{!}^{•} ⊗^{L}_{R} L_{!}^{•})^{a}

which is a well defined object of D(R^{a}-Mod).

• We claim that this rule yields a well defined functor

− ⊗^{L}_{R^{a}} − : D^{−}(R^{a}-Mod) × D^{−}(R^{a}-Mod) → D^{−}(R^{a}-Mod).

Indeed, suppose φ^{•} : K_{1}^{•} → K_{2}^{•} is a morphism in C(R^{a}-Mod), inducing an isomorphism in D(R^{a}-Mod), and set C^{•} := Cone (φ_{!}^{•}); clearly, C^{•a} = 0 in D(R^{a}-Mod). Now, pick any flat

bounded above resolution P^{•} → L_{!}^{•}, so that K_{i!}^{•} ⊗_{R} P^{•} computes K_{i!}^{•} ⊗^{L}_{R} L_{!}^{•}, for i = 1, 2. We get natural isomorphism :

Cone(φ^{•} ⊗^{L}_{R^{a}} L^{•}) →^{∼} (C^{•} ⊗_{R} P^{•})^{a} →^{∼} C^{•a} ⊗_{R^{a}} P^{•a} = 0.

so the derived tensor product depends only on the image of K^{•} in D^{−}(R^{a}-Mod); likewise for the argument L^{•}, whence the contention.

• Next, suppose that P^{•} → K^{•} is a bounded above resolution, with P^{•} a complex of flat R^{a}-modules; we claim that there is a natural isomorphism in D(R^{a}-Mod)

K^{•} ⊗^{L}_{R^{a}} L^{•} →^{∼} P^{•} ⊗_{R^{a}} L^{•}.

Indeed, pick any bounded above flat resolution Q^{•} → L_{!}^{•}; we have natural isomorphisms

K^{•} ⊗^{L}_{R^{a}} L^{•} →^{∼} (K_{!}^{•} ⊗_{R} Q^{•})^{a} →^{∼} K^{•} ⊗_{R^{a}} Q^{•a} →^{∼} P^{•} ⊗_{R^{a}} Q^{•a}

in D(R^{a}-Mod), where the last holds, since Q^{•a} is a complex of flat R^{a}-modules. Finally, the induced map

P^{•} ⊗_{R^{a}} Q^{•a} → P^{•} ⊗_{R^{a}} L^{•}

is also an isomorphism in D(R^{a}-Mod), since P^{•} is a complex of flat R^{a}-modules, so the claim follows.

• Notice that, for any K^{•} ∈ Ob(D^{−}(R-Mod)) and any flat resolution P^{•} → K^{•}, the induced morphism P^{•a} → K^{•a} is a flat resolution; it follows that, for any L^{•} ∈ Ob(D(R-Mod)) we get a natural isomorphism

(14.1.14)  (K^{•} ⊗^{L}_{R} L^{•})^{a} →^{∼} K^{•a} ⊗^{L}_{R^{a}} L^{•a}  in D(R^{a}-Mod).

Remark 14.1.15. Clearly, for the derived tensor products of R^{a}-modules, one has the same commutativity and associativity isomorphisms as the ones detailed in remark 7.3.39(i) for usual modules, as well as the vanishing properties of lemma 7.3.42(ii).

We are now ready to return to the question of the existence of adjoints to derived localization. The key point is the following :

Lemma 14.1.16. In the situation of (14.1.4), let K^{•} be any complex of R-modules, i ∈ ℤ any integer and suppose that :

(a) K^{•} ∈ Ob(D^{≤i}(R-Mod))

(b) K^{•a} ∈ Ob(D^{≤i−1}(R^{a}-Mod)).

Then 𝔪 ⊗^{L}_{V} K^{•} ∈ Ob(D^{≤i−1}(R-Mod)).

Proof. We apply the standard spectral sequence

E_{pq}^{2} := Tor_{p}^{V} (𝔪, H^{q}K^{•}) ⇒ H^{q−p}(𝔪 ⊗^{L}_{V} K^{•}).

Indeed, (a) says that H^{q}K^{•} = 0 for every q > i, and (b) says that (H^{i}K^{•})^{a} = 0, and therefore 𝔪_{V} ⊗_{V} H^{i}K^{•} = 0 ([75, Rem.2.1.4(i)]). In either case, we conclude that E_{pq}^{2} = 0 whenever q − p ≥ i, whence the lemma. □

14.1.17. Now, let us define inductively :

𝔐_{0}^{•} := V [0] and 𝔐_{i+1}^{•} := 𝔪 ⊗^{L}_{V} 𝔐_{i}^{•} for every i ∈ ℕ.

A simple induction shows that 𝔐_{i}^{•} ∈ D^{≤0}(V -Mod) for every i ∈ ℕ, so all these derived tensor product are well defined in D^{≤0}(V -Mod). Moreover, from the short exact sequence of V -modules

Σ : 0 → 𝔪 → V → V/𝔪 → 0

we obtain a distiguished triangle

𝔐_{i}^{•} ⊗^{L}_{V} Σ : 𝔐_{i+1}^{•} → 𝔐_{i}^{•} → 𝔐_{i}^{•} ⊗^{L}_{V} (V/𝔪) → 𝔐_{i+1}^{•}[1] for every i ∈ ℕ.

Especially, we get an inverse system of morphisms in D^{≤0}(V -Mod) :

⋯ → 𝔐_{i+1}^{•} ⟶^{π_{i}^{•}} 𝔐_{i}^{•} ⟶^{π_{i−1}^{•}} 𝔐_{i−1}^{•} → ⋯ → 𝔐_{0}^{•} := V [0].

Also, from (14.1.14) we deduce natural isomorphisms in D^{≤0}(V^{a}-Mod) :

(14.1.18)  𝔐_{i}^{•a} →^{∼} V^{a}[0]  for every i ∈ ℕ

and under these identifications, the morphism π_{i}^{•a} corresponds to the identity automorphism of V^{a}[0] (details left to the reader). Furthermore, we deduce the following derived version of [75, Rem.2.1.4(i)] :

Proposition 14.1.19. Let a, b ∈ ℤ be any integers with a ≤ b. We have :

(i) For every K^{•} ∈ Ob(D^{[a,b]}(R-Mod)), the following conditions are equivalent : (a) K^{•a} ≃ 0 in D^{[a,b]}(R^{a}-Mod).

(b) τ^{≥a}(𝔐_{b−a+1}^{•} ⊗^{L}_{V} K^{•}) ≃ 0 in D^{[a,b]}(R-Mod).

(ii) For every morphism φ^{•} : K^{•} → L^{•} in D^{≤b}(R-Mod), the following conditions are equivalent :

(a) φ^{•a} is an isomorphism in D^{[a,b]}(R^{a}-Mod).

(b) τ^{≥a}(𝔐_{b−a+2}^{•} ⊗^{L}_{V} φ^{•}) is an isomorphism in D^{[a,b]}(R-Mod).

Proof. (i): From (14.1.18), it is easily seen that (b)⇒(a). The other direction follows straightforwardly from lemma 14.1.16, via an easy descending induction on b.

(ii): Again, the direction (b)⇒(a) is immediate from (14.1.18). For the other direction, denote by C^{•} the cone of φ^{•}; then C^{•} ∈ Ob(D^{[a−1,b]}(R-Mod)), and C^{•a} ≃ 0 in D^{[a−1,b]}(R^{a}-Mod).

From (i) we deduce that τ^{≥a−1}(𝔐_{b−a+2}^{•} ⊗^{L}_{V} C^{•}) ≃ 0 in D^{[a−1,b]}(R-Mod). Since the derived tensor product is a triangulated functor, the assertion follows easily : details left to the reader.

□

Corollary 14.1.20. With the notation of (14.1.17), the morphism

τ^{≥2−i}π_{i}^{•} : τ^{≥2−i}𝔐_{i+1}^{•} → τ^{≥2−i}𝔐_{i}^{•}

is an isomorphism in D^{[2−i,0]}(V -Mod) for every integer i ≥ 2.

Proof. By construction, we have a natural isomorphism :

Cone(τ^{≥2−i}π_{i}^{•}) →^{∼} τ^{≥1−i}(𝔐_{i}^{•} ⊗^{L}_{V} (V/𝔪)) in D^{[1−i,0]}(V -Mod)

in light of which, the assertion is an immediate consequence of proposition 14.1.19(i). □

Proposition 14.1.21. In the situation of (14.1.4), we have :

(i) The localization functor D^{+}(R-Mod) → D^{+}(R^{a}-Mod) admits the right adjoint :

(14.1.22) D^{+}(R^{a}-Mod) → D^{+}(R-Mod) : K^{•} ↦ K_{[∗]}^{•} := RHom_{R^{a}}^{•}(R^{a}[0], K^{•}).

(ii) Let a, b, i ∈ ℤ be any three integers with b ≥ a and i ≥ b − a + 2. The localization functor D^{[a,b]}(R-Mod) → D^{[a,b]}(R^{a}-Mod) admits the left adjoint :

D^{[a,b]}(R^{a}-Mod) → D^{[a,b]}(R-Mod) : K^{•} ↦ K_{[!]}^{•} := τ^{≥a}(𝔐_{i}^{•} ⊗^{L}_{V} K_{[∗]}^{•}).

and the right adjoint

D^{[a,b]}(R^{a}-Mod) → D^{[a,b]}(R-Mod) : K^{•} ↦ τ^{≤b}K_{[∗]}^{•}.

(iii) For every K^{•} ∈ Ob(D^{+}(R^{a}-Mod)) (resp. L^{•} ∈ Ob(D^{[a,b]}(R^{a}-Mod))) the counit of adjunction is an isomorphism

(K_{[∗]}^{•})^{a} →^{∼} K^{•} ( resp. (τ^{≤b}L_{[∗]}^{•})^{a} →^{∼} L^{•} ).

(iv) For every L^{•} ∈ Ob(D^{[a,b]}(R^{a}-Mod)), the unit of adjunction is an isomorphism

L^{•} → (L_{[!]}^{•})^{a}.

Proof. (i): This is analogous to lemma 10.1.16(iii). Recall the construction : we know that the category R^{a}-Mod admits enough injectives ([75, 2.2.18]), hence (14.1.22) can be represented by I_{∗}^{•}, where K^{•} →^{∼} I^{•} is any injective resolution of K^{•}, and I_{∗}^{•} is obtained by applying termwise to I^{•} the functor M ↦ M_{∗} of [75, §2.2.10]. Indeed, taking into account [75, Cor.2.2.19] we get natural isomorphisms :

Hom_{D^{+}(R-Mod)}(L^{•}, I_{∗}^{•}) →^{∼} H^{0}Hom_{R}^{•}(L^{•}, I_{∗}^{•})
→^{∼} H^{0}Hom_{R^{a}}^{•}(L^{•a}, I^{•})
→^{∼} Hom_{D(R^{a}-Mod)}(L^{•a}, I^{•})
→^{∼} Hom_{D(R^{a}-Mod)}(L^{•a}, K^{•})

for every bounded below complex L^{•} of R-modules.

(iii) follows by direct inspection of the definitions, taking into account that, for every R^{a}- module M, the counit of adjunction (M_{∗})^{a} → M is an isomorphism ([75, Prop.2.2.14(iii)]) : details left to the reader.

(ii): We consider first the assertion concerning the right adjoint : let K^{•} ∈ D^{[a,b]}(R^{a}-Mod); we may find an injective resolution K^{•} →^{∼} I^{•} such that I^{j} = 0 for every j < a, in which case I_{∗}^{•} ∈ D^{≥a}(R-Mod), and τ^{≤b}I_{∗}^{•} represents τ^{≤b}K_{[∗]}^{•} in D^{[a,b]}(R-Mod). Then, in view of (i), the assertion is reduced to remark 7.3.14(iv). For the left adjoint, let us set

ωL^{•} := τ^{≥a}(𝔐_{i}^{•} ⊗^{L}_{V} L^{•}) for every L^{•} ∈ Ob(D^{[a,b]}(R-Mod)).

Then ωL^{•} is naturally an object of D^{[a,b]}(R-Mod), as explained in remark 7.3.39(ii), and likewise for K_{[!]}^{•}, if K^{•} is any object of D^{[a,b]}(R^{a}-Mod). We begin with the following :

Claim 14.1.23. Let K^{•}, L^{•} ∈ Ob(D^{[a,b]}(R-Mod)) be any two objects. Then the natural map

Hom_{D^{[a,b]}(R-Mod)}(ωK^{•}, L^{•}) → Hom_{D^{[a,b]}(R^{a}-Mod)}(K^{•a}, L^{•a})

is an isomorphism.

Proof of the claim. Arguing as in [75, §2.2.2], and taking into account lemma 14.1.10(i), we reduce to showing that for any K^{•} ∈ D^{[a,b]}(R-Mod), the natural morphism

φ_{K^{•}} : ωK^{•} → K^{•}

is initial in the full subcategory of D^{[a,b]}(R-Mod)/K^{•} whose objects are the morphisms ψ : L^{•} → K^{•} that lie in Σ_{[a,b]}. However, for any such ψ, we have a commutative diagram in D^{[a,b]}(R-Mod) :

\xymatrix{ωL^{•} \ar[r]^{φ_{L^{•}}} \ar[d] & L^{•} \ar[d]^{ψ} \\ ωK^{•} \ar[r]^{φ_{K^{•}}} & K^{•}}

whose left vertical arrow is an isomorphism, by proposition 14.1.19(ii). There follows a morphism φ_{K^{•}} → ψ, and we have to check that this is the unique morphism from φ_{K^{•}} to ψ. However, say that α, β : φ_{K^{•}} → ψ are two such morphisms; then their difference is a morphism γ := α − β : ωK^{•} → L^{•} such that ψ ∘ γ = 0, so γ factors through a morphism \overline{γ} : ωK^{•} → Cone ψ[−1]. Set C^{•} := τ^{≤b}Cone ψ[−1]; then C^{•} ∈ D^{[a,b]}(R-Mod), and according to remark 7.3.14(iv), \overline{γ} lifts uniquely to a morphism ωK^{•} → C^{•} that we denote again \overline{γ}. We deduce a commutative diagram

\xymatrix{ω ∘ ωK^{•} \ar[r] \ar[d]_{φ_{ω_{K^{•}}}} & ωC^{•} \ar[d]^{φ_{C^{•}}} \\ ωK^{•} \ar[r]^{\overline{γ}} & C^{•}.}

Now, by construction C^{•a} = 0, therefore ωC^{•} = 0 (proposition 14.1.19(i)); on the other hand, φ_{ω_{K^{•}}} is an isomorphism, by proposition 14.1.19(ii). We conclude that \overline{γ} = 0, whence α = β, as sought. ♢

Assertion (ii.a) is an immediate consequence of (iii) and claim 14.1.23; from this, also (iv) is immediate : details left to the reader. □

Remark 14.1.24. Let a, b, i ∈ ℤ be any three integers such that a ≤ b and i ≥ b − a + 2. From propositions 14.1.21(ii.a,iii) and 14.1.19(ii) we deduce a natural isomorphism

τ^{≥a}(𝔐_{i}^{•} ⊗^{L}_{V} K^{•}) →^{∼} (K^{•a})_{[!]} for every K^{•} ∈ Ob(D^{[a,b]}(R-Mod))

(details left to the reader), which allows to compute (K^{•a})_{[!]} purely in terms of K^{•} and operations within D(R-Mod). It turns out that an analogous isomorphism is available also for K_{[∗]}^{•} : this is contained in the following

Lemma 14.1.25. Let a, b, i ∈ ℕ be any integers such that a ≤ b and i ≥ b − a + 2. For every K^{•} ∈ Ob(D^{[a,b]}(R-Mod)), we have a natural isomorphism :

τ^{≤b}RHom_{V}^{•} (𝔐_{i}^{•}, K^{•}) →^{∼} τ^{≤b}K_{[∗]}^{•a}.

Proof. Theorem 10.1.18(ii) yields a natural isomorphism

RHom_{V}^{•} (𝔐_{i}^{•}, K^{•}) →^{∼} RHom_{R}^{•}(𝔐_{i}^{•} ⊗^{L}_{V} R[0], K^{•}).

To compute the right-hand side, we may fix an injective resolution K^{•} →^{∼} I^{•}; the complex I^{•a} is not necessarily injective, but we can find an injective resolution φ : I^{•a} →^{∼} J^{•} (in the category of bounded below complexes of R^{a}-modules). In view of (14.1.14) and (14.1.18), the morphism φ induces a natural transformation

ψ : RHom_{R}^{•}(𝔐_{i}^{•} ⊗^{L}_{V} R[0], K^{•}) → RHom_{R^{a}}^{•}(R^{a}[0], K^{•a}) = (K^{•a})_{[∗]}

and it suffices to show that, for every j ≤ b, the map

H^{j}ψ : Hom_{D(R-Mod)}(𝔐_{i}^{•} ⊗^{L}_{V} R[0], K^{•}[j]) → Hom_{D(R^{a}-Mod)}(R^{a}[0], K^{•a}[j])

is an isomorphism. However, by remark 7.3.14(iv), the latter is the same as a map

(14.1.26) Hom_{D(R-Mod)}(𝔐_{i}^{•} ⊗^{L}_{V} R[0], τ^{≤0}K^{•}[j]) → Hom_{D(R^{a}-Mod)}(R^{a}[0], τ^{≤0}K^{•a}[j])

and a direct inspection shows that (14.1.26) agrees with the map arising in claim 14.1.23, for every j ≤ b. Especially, for every such j, the map (14.1.26) is an isomorphism, as sought. □

Proposition 14.1.27. Let a, b, c ∈ ℤ be any three integers such that a ≤ b, and K^{•}, L^{•} any two objects of D^{[a,b]}(R-Mod)). Suppose that

(a) Hom_{D(R-Mod)}(K^{•}, X[−j]) = 0 for all j ∈ [c, b] and all R-modules X with X^{a} = 0.

(b) Hom_{D(R-Mod)}(Y [−j], L^{•}) = 0 for all j ∈ [a, c] and all R-modules Y with Y^{a} = 0.

Then the natural map

(14.1.28)  Hom_{D(R-Mod)}(K^{•}, L^{•}) → Hom_{D(R^{a}-Mod)}(K^{•a}, L^{•a})

is an isomorphism.

Proof. We start out with the following observation :

Claim 14.1.29. Consider the following conditions :

(a’) Hom_{D(R-Mod)}(K^{•}, X^{•}) = 0 for every X^{•} ∈ Ob(D^{[c,b]}(R-Mod)) such that X^{•a} = 0. (b’) Hom_{D(R-Mod)}(Y^{•}, L^{•}) = 0 for every Y^{•} ∈ Ob(D^{[a,c]}(R-Mod)) such that Y^{•a} = 0.

Then (a)⇔(a’) and (b)⇔(b’).

Proof of the claim. Obviously (a’)⇒(a). For the converse, one argues by decreasing induction on c ≤ b. Indeed, the case c = b is immediate. Then, suppose that the sought equivalence has already been established for some d ≤ b; if X^{•} ∈ D^{[d−1,b]} and X^{•a} = 0, and if we know that (a) holds with c := d − 1, we set H^{•} := H^{c}X^{•}[−c], and consider the distinguished triangle

H^{•} → X^{•} → τ^{≥d}X^{•} → H^{•}[1].

By inductive assumption, we have Hom_{D(R-Mod)}(K^{•}, τ^{≥d}X^{•}) = 0, and (a) says that

Hom_{D(R-Mod)}(K^{•}, H^{•}) = 0.

It then follows that Hom_{D(R-Mod)}(K^{•}, X^{•}) = 0, which shows that the equivalence holds for c.

The proof of the equivalence (b)⇔(b’) is wholly analogous. ♢ Fix an integer i ≥ b − a + 2, and set

C^{•} := Cone(π_{0}^{•} ∘ ⋯ ∘ π_{i}^{•} : 𝔐_{i}^{•} → V [0])

(notation of (14.1.17)); notice that C^{•} ∈ D^{≤1}(R-Mod), and C^{•a} = 0. By virtue of condition (b), claim 14.1.29and remark 7.3.14(iv), it follows that :

R^{j}Hom_{R}^{•}(C^{•}, L^{•}) = Hom_{D(R-Mod)}(τ^{≥a}(C^{•}[−j]), L^{•}) = 0 for every j < c

In other words, D^{•} := RHom_{R}^{•}(C^{•}, L^{•}) ∈ D^{≥c}(R-Mod), and clearly D^{•a} = 0; also notice the induced distinguished triangle

Σ : D^{•} → L^{•} → RHom_{R}^{•}(𝔐_{i}^{•}, L^{•}) → D^{•}[−1].

Now, condition (a), claim 14.1.29and remark 7.3.14(iv) imply that

Hom_{D(R-Mod)}(K^{•}, D^{•}[j]) = Hom_{D(R-Mod)}(K^{•}, τ^{≤b}D^{•}[j]) = 0 for every j ≤ 0

whence, by considering the distinguished triangle RHom_{R}^{•}(K^{•}, Σ), natural isomorphisms

Hom_{D(R-Mod)}(K^{•}, L^{•}) →^{∼} Hom_{D(R-Mod)}(K^{•}, RHom_{R}^{•}(𝔐_{i}^{•}, L))
→^{∼} Hom_{D(R-Mod)}(𝔐_{i}^{•} ⊗^{L}_{V} K^{•}, L^{•})  (by [163, Th.10.8.7])
→^{∼} Hom_{D(R-Mod)}(τ^{≥a}𝔐_{i}^{•} ⊗^{L}_{V} K^{•}, L^{•}) (by remark 7.3.14(iv))
→^{∼} Hom_{D(R^{a}-Mod)}(K^{•a}, L^{•a})  (by claim 14.1.23)

whose composition, after a simple inspection, is seen to agree with the map (14.1.28). □

Remark 14.1.30. (i) For every interval I ⊂ ℕ, denote by

Φ_{I} : D^{I}(R/𝔪R-Mod) → D^{I}(R-Mod)

the forgetful functor. It follows easily from lemma 10.1.16(iii) and remark 7.3.14(iv), that, for every interval [a, b], the functor Φ_{[a,b]} admits the right adjoint

D^{[a,b]}(R-Mod) → D^{[a,b]}(R/𝔪R-Mod) K^{•} ↦ Ψ_{[a,b]}^{r}K^{•} := τ^{≤b}RHom_{R}^{•}(R/𝔪R[0], K^{•}).

(ii) Likewise, theorem 10.1.18and remark 7.3.14(iv) imply that Φ_{[a,b]} admits the left adjoint

D^{[a,b]}(R-Mod) → D^{[a,b]}(R/𝔪R-Mod) K^{•} ↦ Ψ_{[a,b]}^{l}K^{•} := τ^{≥a}(K^{•} ⊗_{R} R/𝔪R[0]).

(iii) Consider as well the following two conditions :

(a”) Hom_{D(R-Mod)}(K^{•}, Φ_{[c,b]}X^{•}) = 0 for every X^{•} ∈ Ob(D^{[c,b]}(R/𝔪R-Mod)).

(b”) Hom_{D(R-Mod)}(Φ_{[a,c]}Y^{•}, L^{•}) = 0 for every Y^{•} ∈ Ob(D^{[a,c]}(R/𝔪R-Mod)). Then, arguing as in the proof of claim 14.1.29 it is easily seen that condition (a) of proposition 14.1.27is equivalent to (a”), and condition (b) is equivalent to (b”).

Proposition 14.1.31. Let a, b ∈ ℤ be any two integers such that a ≤ b. For every object K^{•} of D^{[a,b]}(R-Mod), the following conditions are equivalent :

(a) K^{•} lies in the essential image of the left adjoint functor X^{•} ↦ X_{[!]}^{•}.

(b) K^{•} ⊗^{L}_{R} R/𝔪R[0] ∈ Ob(D^{<a−1}(R/𝔪R-Mod)).

Proof. We start out with the following :

Claim 14.1.32. We may assume that V = R.

Proof of the claim. Clearly condition (b) does not depend on the underlying ring V. It suffices then to remark that condition (a) depends only on the basic setup (R, 𝔪R) (as opposed to the original basic setup (V, 𝔪)). Indeed, notice that there is a natural equivalence

Ω : R^{a}-Mod →^{∼} (R, 𝔪R)^{a}-Mod

(where R^{a} denotes, as in the foregoing, the image of R in the category of (V, 𝔪)-algebras), and the induced equivalence of the respective derived categories fits into an essentially commutative diagram

\xymatrix{ & D(R-Mod) \ar[dl] \ar[dr] \\ D(R^{a}-Mod) \ar[rr]^{D(Ω)} &  & D((R, 𝔪R)^{a}-Mod)}

whose downward arrows are the forgetful functors. Especially, the left (resp. right) adjoints of these two forgetful functors share the same essential images. ♢

Henceforth, we assume that V = R (and therefore, 𝔪 = 𝔪R). Fix an integer i ≥ b − a + 2,

set L^{•} := τ^{≥a−1}(𝔐_{i}^{•} ⊗^{L}_{V} K^{•}), and notice first that, taking into account proposition 14.1.21(iv) and remark 14.1.24, condition (a) is equivalent to :

(c) The morphism π_{0}^{•} ∘ ⋯ ∘ π_{i−1}^{•} : 𝔐_{i} → V [0] induces an isomorphism τ^{≥a}L^{•} → K^{•}. (c)⇒(b): Indeed, set H := H^{a−1}L^{•}; if (c) holds, we have a distinguished triangle

H[a − 1] → L^{•} → K^{•} → H[a]

whence a distinguished triangle in D(V/𝔪-Mod)

(14.1.33) H[a − 1] ⊗^{L}_{V} V/𝔪[0] → L^{•} ⊗^{L}_{V} V/𝔪[0] → K^{•} ⊗^{L}_{V} V/𝔪[0] → H[a] ⊗^{L}_{V} V/𝔪[0].

However, we have natural isomorphisms

τ^{≥a−1}(L^{•} ⊗^{L}_{V} V/𝔪[0]) →^{∼} τ^{≥a−1}((𝔐_{i}^{•} ⊗^{L}_{V} K^{•}) ⊗^{L}_{V} V/𝔪[0])  (by lemma 7.3.42(ii))
→^{∼} τ^{≥a−1}(𝔐_{i}^{•} ⊗^{L}_{V} (K^{•} ⊗^{L}_{V} V/𝔪[0]))  (by remark 7.3.39(i))
→^{∼} τ^{≥a−1}(𝔐_{i}^{•} ⊗^{L}_{V} τ^{≥a−1}(K^{•} ⊗^{L}_{V} V/𝔪[0])) (by lemma 7.3.42(ii))
→^{∼} 0  (by proposition 14.1.19(i))

whence (b), after considering the distinguished triangle τ^{≥a−1}(14.1.33).

(b)⇒(a): We remark

Claim 14.1.34. Condition (b) is equivalent to condition (a”) of remark 14.1.30(iii), with c := a − 1.

Proof of the claim. Indeed, (b) holds if and only if Ψ_{[a−1,b]}^{l}K^{•} = 0 in D^{[a−1,b]}(R/𝔪R-Mod) (notation of remark 14.1.30(ii)). If the latter condition holds, then clearly (a”) holds with c := a − 1. Conversely, if (a”) holds for this value of c, then Hom_{D(R/𝔪R-Mod)}(Ψ_{[a−1,b]}^{l}K^{•}, X^{•}) = 0 for every X^{•} ∈ D^{[a−1,b]}(R/𝔪R-Mod); especially, the identity automorphism of Ψ_{[a−1,b]}^{l}K^{•} factors through 0, so Ψ_{[a−1,b]}^{l}K^{•} vanishes. ♢

From claim 14.1.34 and remark 14.1.30(iii) we deduce that if (b) holds, condition (a) of proposition 14.1.27 holds for c := a − 1, and condition (b) of the same proposition holds trivially for this value of c, for every L^{•} ∈ D^{[a;b]}(R-Mod). We conclude that the natural map

Hom_{D(R-Mod)}(K^{•}, L^{•}) → Hom_{D(R^{a}-Mod)}(K^{•a}, L^{•a}) →^{∼} Hom_{D(R-Mod)}(K_{[!]}^{•a}, L^{•})

is an isomorphism, for every L^{•} ∈ D^{[a;b]}(R-Mod), whence (a). □

Proposition 14.1.35. Let a, b ∈ ℤ be any two integers such that a ≤ b. For every L^{•} ∈ Ob(D^{[a,b]}(R-Mod)), the following conditions are equivalent :

(a) L^{•} lies in the essential image of the right adjoint functor X^{•} ↦ τ^{≤b}X_{[∗]}^{•}.

(b) RHom_{R}^{•}(R/𝔪R[0], L^{•}) ∈ Ob(D^{>b+1}(R/𝔪R-Mod)).

Proof. By the same argument as in the proof of claim 14.1.32, we reduce to the case where V = R. Next, fix i ∈ ℕ such that i ≥ b − a + 2, define

K^{•} := RHom_{V}^{•} (𝔐_{i}^{•}, L^{•}) 𝔓_{i} := 𝔐_{i} ⊗^{L}_{V} V/𝔪[0]

and notice that

(14.1.36)  τ^{≥a−b−1}𝔓_{i} = 0  in D^{[a−b−1,0]}(V/𝔪-Mod)

due to proposition 14.1.19(i). Moreover, in view of proposition 14.1.21(iii) and lemma 14.1.25, condition (a) is equivalent to :

(c) The morphism π_{0}^{•} ∘ ⋯ ∘ π_{i−1}^{•} : 𝔐_{i} → V [0] induces an isomorphism L^{•} →^{∼} τ^{≤b}K^{•}. (c)⇒(b): We argue as in the proof of proposition 14.1.31; namely, set H := H^{b+1}K^{•}; if (c)

holds, we obtain a distinguished triangle

(14.1.37)  H[−b − 2] → L^{•} → τ^{≤b+1}K^{•} → H[−b − 1]

and by considering the induced distinguished triangle τ^{≤b+1}RHom_{V}^{•} (V/𝔪[0], (14.1.37)), we reduce to observing that

τ^{≤b+1}RHom_{V}^{•} (V/𝔪[0], τ^{≤b+1}K^{•}) →^{∼} τ^{≤b+1}RHom_{V}^{•} (V/𝔪[0], K^{•})  (by lemma 7.3.42)
→^{∼} τ^{≤b+1}RHom_{V}^{•} (𝔓_{i}^{•}, L^{•})  (by [163, Th.10.8.7])
→^{∼} τ^{≤b+1}RHom_{V}^{•} (τ^{≥a−b−1}𝔓_{i}^{•}, L^{•}) (by lemma 7.3.42)
→^{∼} 0  (by (14.1.36)).

(b)⇒(a): Again, we proceed as in the proof of the corresponding assertion in proposition 14.1.31; namely, arguing as in the proof of claim 14.1.34, we see that condition (b) is equivalent to condition (b”) of remark 14.1.30(iii), with c := b + 1. Hence, if (b) holds, condition (b) of proposition 14.1.27 holds for c := b + 1, and notice that condition (a) of loc.cit. holds trivially for this value of c, for every K^{•} ∈ D^{[a;b]}(R-Mod). We conclude that the natural map

Hom_{D(R-Mod)}(K^{•}, L^{•}) → Hom_{D(R^{a}-Mod)}(K^{•a}, L^{•a}) →^{∼} Hom_{D(R-Mod)}(K^{•}, τ^{≤b}L_{[∗]}^{•a})

is an isomorphism, for every K^{•} ∈ D^{[a;b]}(R-Mod), whence (a). □

14.1.38. Let A be any V^{a}-algebra; recall that the localization functor V -Alg → V^{a}-Alg admits a left adjoint R ↦ R_{!!}, whose restriction to the subcategory of A_{!!}-algebras yields a left adjoint for the localization functor A_{!!}-Alg → A-Alg ([75, Prop.2.2.29]). In [75], we have studied the deformation theory of A-algebras by means of this left adjoint, under the assumption that 𝔪̃ is V -flat; here we wish to show that the same can be repeated in the current setting, if one appeals instead to the results of the foregoing paragraphs. To begin with, we remark :

Proposition 14.1.39. Let A → B be a morphism of V^{a}-algebras, N a B_{!!}-module. We have: (i) If the unit of adjunction N → N_{∗}^{a} is injective, the natural map

(14.1.40)  Exal_{A_{!!}}(B_{!!}, N) → Exal_{A}(B, N^{a})

is a bijection (notation of [75, §2.5.7]). (ii) If N^{a} = 0, then Exal_{A_{!!}}(B_{!!}, N) = 0.

Proof. (i): Consider any square-zero extension of A-algebras

Σ : 0 → N^{a} → E ⟶^{φ} B → 0.

There follows a square-zero extension of A_{!!}-algebras

Σ_{!!} : 0 → N_{!}^{a}/Ker φ_{!!} → E_{!!} → B_{!!} → 0

Under the stated assumption, the counit of adjunction N_{!}^{a} → N factors uniquely through a Blinear map g_{φ} : N_{!}^{a}/Ker φ_{!!} → N; then g_{φ} ∗ Σ_{!!} (defined as in [75, §2.5.5]) yields an element of Exal_{A_{!!}}(B_{!!}, N) whose image under (14.1.40) equals the class of Σ. Conversely, if

Ω : 0 → N → F ⟶^{ψ} B_{!!} → 0

is a square-zero extension of A_{!!}-algebras, then by adjunction we get a natural map

Ω_{!!}^{a} → Ω

which in turns, by simple inspection, induces an isomorphism g_{ψ^{a}} ∗ Ω_{!!}^{a} →^{∼} Ω in the category of square-zero A_{!!}-algebra extensions of B_{!!} (details left to the reader). The assertion follows.

(ii): Suppose N^{a} = 0, and let Ω be as in the foregoing; it follows that ψ^{a} : F^{a} → B is an isomorphism of A-algebras. By adjunction, the morphism (ψ^{a})^{−1} corresponds to a map of A_{!!}-algebras φ : B_{!!} → E, and it is easily seen that ψ ∘ φ is the identity automorphism of B_{!!}, whence the assertion. □

Definition 14.1.41. Let f : A → B be a morphism of V^{a}-algebras. We set

𝕃_{B/A}^{a} := (𝕃_{B_{!!}/A_{!!}})^{a}

which is a simplicial complex of B-modules that we call the almost cotangent complex of f.

Remark 14.1.42. (i) In case 𝔪̃ is a flat V -module, we have introduced in [75, Def.2.5.20] a simplicial B_{!!}-module 𝕃_{B/A}. Notice that the notation of loc.cit. agrees with the current one: indeed, [75, Prop.8.1.7(ii)] shows that complex (𝕃_{B/A})^{a} obtained by applying the derived localization functor to 𝕃_{B/A} is naturally isomorphic (in D(s.B-Mod)) to the complex of definition 14.1.41.

(ii) Depending on the context, we will want to regard 𝕃_{B/A} either as a simplicial object, or as a cochain complex, via the Dold-Kan isomorphism ([163, Th.8.4.1]). The resulting slight notational ambiguity should not be a source of confusion.

Theorem 14.1.43. In the situation of definition 14.1.41, let N be any B-module. Then there are natural isomorphisms

Der_{A}(B, N) →^{∼} Ext_{B}^{0}(𝕃_{B/A}^{a}, N)
Exal_{A}(B, N) →^{∼} Ext_{B}^{1}(𝕃_{B/A}^{a}, N).

(Notation of [75, Def.2.5.22(i)]; so, here we view 𝕃_{B/A}^{a} as an object of D^{≤0}(B-Mod).)

Proof. The first isomorphism follows easily from [103, II.1.2.4.2] and the natural isomorphism

(14.1.44)  Ω_{B_{!!}/A_{!!}} →^{∼} (Ω_{B/A})_{!}

proved in [75, Lemma 2.5.29] (the proof in loc.cit. does not use the assumption that 𝔪̃ is V -flat). Clearly we have

(14.1.45) Hom_{D(B_{!!}-Mod)}(Y [0], N_{∗}[0]) = 0  for every B_{!!}-module Y such that Y^{a} = 0.

On the other hand, we have :

Claim 14.1.46. Hom_{D(B_{!!}-Mod)}(𝕃_{B_{!!}/A_{!!}}, X^{•}) = 0 for every X^{•} ∈ Ob(D^{[0,1]}(B_{!!}-Mod)) such that X^{•a} = 0.

Proof of the claim. For every X^{•} ∈ Ob(D^{[0,1]}(B_{!!}-Mod)) we have a distinguished triangle

H^{0}X^{•}[0] → X^{•} → H^{1}X^{•}[−1] → (H^{0}X^{•})[1]

which reduces to considering the cases where X^{•} = M[j] for some almost zero B_{!!}-module M, and j = 0, −1. The case where j = −1 follows from [103, III.1.2.3] and proposition 14.1.39(ii). The case where j = 0 follows easily from [103, II.1.2.4.2] and (14.1.44) : details left to the reader. ♢

Now, (14.1.45) says that L^{•} := N_{∗}[0] fulfills condition (b) of proposition 14.1.27, and claim 14.1.46says that K^{•} := 𝕃_{B_{!!}/A_{!!}} fulfills condition (a), so the natural map

Ext_{B_{!!}}^{1}(𝕃_{B_{!!}/A_{!!}}, N_{∗}) → Ext_{B}^{1}(𝕃_{B/A}^{a}, N)

is an isomorphism. Taking into account proposition 14.1.39(i) and [103, III.1.2.3], the theorem follows. □

14.1.47. For the further study the almost cotangent complex, we shall need some preliminaries concerning the derived functors of certain non-additive functors. This material generalizes the results of [75, §8.1], that were obtained under the assumption that 𝔪̃ is V -flat.

Lemma 14.1.48. Let (V, 𝔪) be any basic setup, R a simplicial V -algebra, n ∈ ℕ an integer, M and N two R-modules such that H_{i}M = H_{i}N = 0 for every i ≥ n. The following holds :

(i) If M^{a} = 0 in D(R^{a}-Mod), then a · 1_{M} = 0 in D(R-Mod), for every a ∈ 𝔪.

(ii) If φ : M → N is a morphism of R-modules such that φ^{a} in an isomorphism in D(R^{a}-Mod), then for every a ∈ 𝔪 we may find a morphism ψ : N → M in D(R-Mod), such that ψ ∘ φ = a · 1_{M} and φ ∘ ψ = a · 1_{N} in D(R-Mod).

Proof. (i): For every R-module X, set (notation of remark 7.10.31(iii))

τ^{≤−1}X := σ ∘ ωX

According to [103, I.3.2.1.9(ii)], there exists a natural sequence of morphisms

(14.1.49)  τ^{≤−1}X → X → s.H_{0}(X) → σ(τ^{≤−1}X)  in D(R-Mod)

whose induced sequence of normalized complexes is a distinguished triangle in D(V -Mod) (i.e. a distinguished triangle of D(R-Mod), in the terminology of [103, I.3.2.2.4], and in view of [103, I.3.2.2.5]). Let now n and M be as in the lemma; we argue by induction on n. The case where n = 0 is trivial, so suppose that n > 0, and that the assertion has already been proven for all almost zero R-modules N such that H_{i}N = 0 for every i ≥ n − 1. Especially, for N := τ^{≤1}X and P := s.H_{0}(M) we have a · 1_{ωN} = 0 and a · 1_{P} = 0 in D(R-Mod), for every a ∈ 𝔪. Since the adjunction σ ∘ ωN → N is an isomorphism in D(R-Mod) ([103, I.3.2.1.10]), we deduce that

Hom_{D(R-Mod)}(M, N)^{a} = 0 = Hom_{D(R-Mod)}(M, P)^{a}

whence End_{D(R-Mod)}(M)^{a} = 0, by virtue of (14.1.49) (with X := M) and [103, I.3.2.2.10]. The assertion follows.

(ii): Set C := Cone φ; according to [103, I.3.2.2], we have a distinguished triangle

(14.1.50)  M ⟶^{φ} N → C → σM  in D(R-Mod)

whence – by [103, I.3.2.2.10] – an exact sequence of V -modules

Hom_{D(R-Mod)}(N, M) ⟶^{α} End_{D(R-Mod)}(N) ⟶^{β} Hom_{D(R-Mod)}(N, C).

Now, let us write a = ∑_{i=1}^{n} a_{i}b_{i} for some a_{1}, b_{1}, …, a_{n}, b_{n} ∈ 𝔪; the assumption on φ implies that C^{a} = 0 in D(R^{a}-Mod), therefore (i) yields β(a_{i} · 1_{N}) = a_{i} · β(1_{N}) = 0, so there exists a morphism ψ_{i} : N → M in D(R-Mod) such that α(ψ_{i}) = a_{i} · 1_{N}, i.e. φ ∘ ψ_{i} = a_{i} · 1_{N} for every i = 1, …, n. Likewise, by considering the long exact sequence 𝔼xt_{R}^{•}((14.1.50), M) provided by [103, I.3.2.2.10] we find, for every i = 1, …, n, a morphism ψ′_{i} : N → M such that ψ′_{i} ∘ φ = b_{i} · 1_{M}. Thus,

a_{i} · ψ′_{i} = ψ′_{i} ∘ φ ∘ ψ_{i} = b_{i} · ψ_{i} for every i = 1, …, n

and a simple computation shows that ψ := ∑_{i=1}^{n} b_{i} · ψ_{i} will do. □

Remark 14.1.51. Before considering non-additive functors, let us see how to define derived tensor products in D(R^{a}-Mod), for any simplicial V -algebra R. We proceed as in (14.1.13) : for given R^{a}-modules M, N, set

M ⊗^{ℓ}_{R^{a}} N := (M_{!} ⊗^{ℓ}_{R} N_{!})^{a}.

(i) We claim that this rule yields a well defined functor

− ⊗^{ℓ}_{R^{a}} − : D(R^{a}-Mod) × D(R^{a}-Mod) → D(R^{a}-Mod).

Indeed, say that φ : M → M′ is a quasi-isomorphism of R^{a}-modules, and set C := Cone(φ_{!}).

We need to check that (φ_{!} ⊗^{ℓ}_{R} N)^{a} is a quasi-isomorphism, for any R-module N; in light of

remark 7.10.31(iv), it then suffices to show that (C ⊗^{ℓ}_{R} N)^{a} = 0; but the latter R^{a}-module may be computed as

(C⊗_{R} ⊥_{•}^{R} N)^{a} = C^{a} ⊗_{R^{a}} (⊥_{•}^{R} N)^{a}

whence the claim, since C^{a} = 0 in R^{a}-Mod.

(ii) Next, suppose that M is a flat R^{a}-module; then we claim that the natural morphism of R^{a}-modules

M ⊗^{ℓ}_{R^{a}} N → M ⊗_{R^{a}} N

is a quasi-isomorphism, for every R^{a}-module N. Indeed, by Eilenberg-Zilber’s theorem 7.4.50, the assertion comes down to checking that the augmented simplicial R^{a}-module

(M_{!}⊗_{R} ⊥_{•}^{R} N_{!})^{a} → M ⊗_{R^{a}} N

is aspherical. But for every k ∈ ℕ, the k-th column of the latter is isomorphic to the augmented R^{a}-module

M[k] ⊗_{R^{a}[k]} (⊥_{•}^{R[k]} N_{!}[k])^{a} → M[k] ⊗_{R^{a}[k]} N[k]

which is aspherical, since M[k] is a flat R^{a}[k]-module, whence the assertion.

(iii) Just as in (14.1.13), the foregoing immediately implies yields a natural isomorphism

(M ⊗^{ℓ}_{R} N)^{a} →^{∼} M^{a} ⊗^{ℓ}_{R^{a}} N^{a} in D(R^{a}-Mod)

for any two R-modules M and N (details left to the reader).

(iv) Furthermore, we get suspension and loop functors σ and ω for R^{a}-modules, by the rule :

σM := (σM_{!})^{a} and ωM := (ωM_{∗})^{a} for every R^{a}-module M

from which it follows that σ is left adjoint to ω. Then, it is clear that the assertions of remark 7.10.31(iii) hold as well for these functors.

(v) Likewise, we define the cone of a morphism φ : M → N of R^{a}-modules, by the rule

Cone φ := (Cone φ_{!})^{a}

and then the assertion of remark 7.10.31(iv) holds also for morphisms of R^{a}-modules.

(vi) In view of (iv) and (v), it is then easy to check that also lemma 7.10.32 holds verbatim for A := V, and any two R^{a}-modules X, Y.

Remark 14.1.52. (i) In the same vein, we may define derived tensor products of R^{a}-algebras, for any simplicial V -algebra R. Namely, if S and S′′ are any two R^{a}-algebras, we set

S ⊗^{ℓ}_{R^{a}} S′ := (S_{!!} ⊗^{ℓ}_{R} S′_{!!})^{a}

(see (14.1.38)), where ⊗^{ℓ}_{R} denotes the derived tensor product for R-algebras, defined in example 7.10.25. In view of remarks 7.10.31(ii) and 14.1.51(i), it is easily seen that this rule defines a functor

− ⊗^{ℓ}_{R^{a}} − : D(R^{a}-Alg) × D(R^{a}-Alg) → D(R^{a}-Alg)

and moreover, the formation of these tensor products commutes with the forgetful functor D(R^{a}-Alg) → D(R^{a}-Mod).

(ii) Moreover, they are computed by arbitrary flat resolutions : if S (or S′) is a flat R^{a}- algebras, then the natural morphism

S ⊗^{ℓ}_{R^{a}} S′ → S ⊗_{R^{a}} S′

is an isomorphism in D(R^{a}-Alg).

(iii) Furthermore, if R^{a} → S is a given morphism of simplicial V^{a}-algebras, we obtain a well defined functor

D(R^{a}-Alg) → D(S-Alg) S′ ↦ S ⊗^{ℓ}_{R^{a}} S′.

Namely, given an R^{a}-algebra S′, we pick a resolution P → S′ with P a flat R^{a}-algebra, and endow S ⊗_{R^{a}} P with its natural S-algebra structure, which is independent, up to natural isomorphism, of the choice of P. All the verifications are exercises for the reader.

Definition 14.1.53. Let V be a ring, A a V -algebra, d∈ℕ and T:A-Mod→A-Mod a functor. We say that T is V -homogeneous of degree d if we have

T (a · 1_{M}) = a^{d} · 1_{T M} for every A-module M and every a ∈ V .

Remark 14.1.54. Let (V, 𝔪) be a basic setup, A a V -algebra, d ∈ ℕ, and T : A-Mod → A-Mod a V -homogeneous functor of degree d. Suppose that either d ≤ 1 or else 𝔪 fulfills condition (B) of [75, §2.1.6].

(i) Let also φ : M → N be a homomorphism of A-modules such that φ^{a} : M^{a} → N^{a} is an isomorphism (for the almost structure given by (V, 𝔪)); arguing as in the proof of lemma 14.1.48(ii), it is easily seen that, for every a ∈ 𝔪 there exists a B-linear map ψ : N → M such that φ ∘ ψ = a · 1_{N} and ψ ∘ φ = a · 1_{M}. Then, the V -homogeneity property of T implies that (T φ)^{a} is an isomorphism as well (details left to the reader).

(ii) Hence, T induces a well defined functor

T^{a} : A^{a}-Mod → A^{a}-Mod  M^{a} ↦ (T M)^{a}  (φ^{a} : M^{a} → N^{a}) ↦ (T φ)^{a}.

For every cardinality c, let ℳ_{c}(A^{a}) be the set of isomorphism classes of A^{a}-modules which admit a set of generators of cardinality ≤ c. We endow ℳ_{c}(A^{a}) with the uniform structure as in [75, Def.2.3.1(i)]. Let ω be an infinite cardinality, such that 𝔪 is generated by at most ω elements; then there exists a cardinality ω′ ≥ ω such that the isomorphism class of T^{a}M lies in ℳ_{ω′}(A^{a}), for every A^{a}-module M whose isomorphism class lies in ℳ_{ω}(A^{a}); thus, T^{a} induces a map

ℳ_{ω,ω′}(T^{a}) : ℳ_{ω}(A^{a}) → ℳ_{ω′}(A^{a}) N ↦ T^{a}N.

Lemma 14.1.55. Let (V, 𝔪) be a basic setup, A a V -algebra, I, J ⊂ A two ideals, and φ : M′ → M a morphism of A^{a}-modules with I · Ker φ = J · Coker φ = 0. Then there exists an A^{a}-linear morphism λ : I ⊗_{A} J ⊗_{A} M → M′ with

φ ∘ λ = μ_{I,J} ⊗_{A} 1_{M}  and  λ ∘ (I ⊗_{A} J ⊗_{A} φ) = μ_{I,J} ⊗_{A} 1_{M′}

where μ_{I,J} : I ⊗_{A} J → A is the multiplication law : a ⊗ b ↦ ab for every a ∈ I and b ∈ J.

Proof. Let M′ ⟶^{\overline{φ}} M_{0} := Im φ ⟶^{i} M be the natural factorization of φ; then for every a ∈ I and b ∈ J we have A^{a}-linear morphisms

ψ_{a} : M_{0} → M′ and μ_{b} : M → M_{0} such that ψ_{a}∘\overline{φ} = a·1_{M′} and i∘μ_{b} = b·1_{M}.

Let us check that \overline{φ} ∘ ψ_{a} = a · 1_{M_{0}}; since \overline{φ} is an epimorphism, it suffices to show that \overline{φ} ∘ ψ_{a} ∘ \overline{φ} = (a · 1_{M_{0}}) ∘ \overline{φ}, which is clear. Likewise, let us check that μ_{b} ∘ i = b · 1_{M_{0}}; since i is a monomorphism, it suffices to show that i ∘ μ_{b} ∘ i = i ∘ b · 1_{M_{0}}, which is clear. For every a ∈ I and b ∈ J set λ_{a,b} := ψ_{a} ∘ μ_{b} : M → M′; we deduce that

φ ∘ λ_{a,b} = i ∘ \overline{φ} ∘ ψ_{a} ∘ μ_{b} = i ∘ (a · 1_{M_{0}}) ∘ μ_{b} = i ∘ μ_{b} ∘ (a · 1_{M}) = ab · 1_{M}

and a similar computation yields : λ_{a,b} ∘ φ = ab · 1_{M′}. Thus, we obtain a map

I × J × M → M′ (a, b, x) ↦ λ_{a,b}(x).

A simple inspection shows that this map is A-trilinear, hence it factors uniquely through an Alinear map λ : I ⊗_{A} J ⊗_{A} M → M′, and the foregoing easily implies that λ fulfills the required identities. □

Proposition 14.1.56. In the situation of remark 14.1.54the following holds :

(i) If T commutes with filtered colimits, and for every free A-module L of finite rank, (T L)^{a} is a flat A^{a}-module (resp. is the zero A^{a}-module), then for every flat A^{a}-module M, the A^{a}-module T^{a}M is flat (resp. is the zero A^{a}-module).

(ii) If for every free A-module L of finite rank (resp. for every free A-module L), (T L)^{a} is an almost projective A^{a}-module, then for every almost projective and almost finitely generated A^{a}-module M (resp. for every almost projective A^{a}-module M), the A^{a}- module T^{a}M is almost projective.

(iii) The map ℳ_{ω,ω′}(T^{a}) of remark 14.1.54(ii) is uniformly continuous.

Proof. (i): Say that M = N^{a} for an A-module N, and let ℱ be the category whose objects are the pairs (C, φ), where C is a finitely presented A-module and φ : C → N is an A-linear map; the morphisms ψ : (C, φ) → (C′, φ′) in ℱ are the A-linear maps ψ : C → C′ such that φ′ ∘ ψ = φ, with the obvious composition law. We have a functor

F : ℱ → A-Mod  (C, φ) ↦ C  ((C, φ) ⟶^{ψ} (C′, φ′)) ↦ (C ⟶^{ψ} C′)

as well as a natural co-cone β : F ⇒ c_{N} given by the rule : (C, φ) ↦ φ. It is easily seen that ℱ is filtered, and β is a universal co-cone : the details are left to the reader. Since T commutes with filtered colimits, T N is isomorphic to the colimit of the functor T ∘F : ℱ → A-Mod, and T ∗ β : T ∘ F ⇒ c_{T N} is again a universal co-cone. We need to show that 𝔪Tor_{1}^{A}(T N, X) = 0 for every A-module X (resp. that 𝔪T N = 0), and by the foregoing it then suffices to check that for every (C, φ) ∈ Ob(ℱ), the induced map

λ := Tor_{1}^{A}(T φ, X) : Tor_{1}^{A}(T C, X) → Tor_{1}^{A}(T N, X)

is almost zero (resp. that T φ is almost zero). However, since N^{a} is a flat A^{a}-module, [75, Lemma 2.4.17] implies that for every a ∈ 𝔪 there exists a free A-module L of finite rank and A-linear maps φ′ : C → L, φ′′ : L → M with a · φ = φ′′ ∘ φ′. Hence a^{d} · T φ = T (φ′′) ∘ T (φ′′), and therefore a^{d} · λ^{a} = 0, since by assumption (T L)^{a} is a flat A^{a}-module (resp. and therefore a^{d} · (T φ)^{a} = 0, since (T L)^{a} = 0); the contention follows, since we have either d ≤ 1, or else 𝔪 fulfills condition (B).

(ii): Suppose first that for every free A-module L of finite rank, (T L)^{a} is almost projective, and let M be an almost projective and almost finitely generated A-module. According to [75, 2.4.15], for every a ∈ 𝔪 there exists a free A-module L of finite rank and A^{a}-linear morphisms φ′ : M → L^{a}, φ′′ : L^{a} → M such that a · 1_{M} = φ′′ ∘ φ′. We deduce that

a^{d} · Ext_{A}^{1}(1_{T M}, X)^{a} = Ext_{A}^{1}(T φ′, X)^{a} ∘ Ext_{A}^{1}(T φ′′, X)^{a} = 0

for every A-module X, since Ext_{A}^{1}(T L, X)^{a} = 0 by assumption (and by [75, Rem.2.4.12(i)]). The contention follows, again because either d ≤ 1 or else 𝔪 fulfills condition (B). One argues similarly in case M is almost projective and (T L)^{a} is almost projective for every free A-module L : the details shall be left to the reader.

(iii): Let 𝔪_{0} ⊂ 𝔪 be a subideal of finite type, and φ : M′ → M a morphism of A^{a}-module with 𝔪_{0}Ker φ = 𝔪_{0}Coker φ = 0. Denote by 𝔪_{0}^{(2d)} ⊂ 𝔪_{0} the subideal generated by the system (a^{d} | a ∈ 𝔪_{0}^{2}). By lemma 14.1.55, for every a ∈ 𝔪_{0}^{2} there exists a morphism λ : M → M′ of A^{a}-modules with φ ∘ λ = a · 1_{M} and λ ∘ φ = a · 1_{M′}; therefore

T^{a}(φ) ∘ T^{a}(λ) = a^{d} · 1_{T^{a}M} and T^{a}(λ_{a}) ∘ T^{a}(φ) = a^{d} · 1_{T^{a}M′}

which implies that 𝔪_{0}^{(2d)} · Ker T^{a}(φ) = 𝔪_{0}^{(2d)} · Coker T^{a}(φ) = 0, whence the assertion, since we either have d ≤ 1 or 𝔪 fulfills condition (B). □

Example 14.1.57. Let (V, 𝔪) be a basic setup such that 𝔪 satisfies condition (B); let A be a V -algebra, and P any A-module. According to [37, §9, n.3], for every integer n > 0 we have a complex of A-modules

Σ_{A}^{•}(P)  :  0 → Λ_{A}^{n}P → ⋯ → (Λ_{A}^{j}P) ⊗_{A} (Sym_{A}^{n−j}P) → ⋯ → Sym_{A}^{n}P → 0

which is functorial for morphisms of A-modules, and is acyclic if P is a flat A-module ([37, §9, n.3, Prop.3]). Clearly, for every j ∈ ℕ the functor

T^{j} : A-Mod → A-Mod P ↦ H^{j}(Σ_{A}^{•}(P))

fulfills the conditions of (14.1.58) (with d := n) and T^{j}P = 0 if P is a flat A-module. By proposition 14.1.56(i), we deduce that the induced complex of A^{a}-modules Σ_{A}^{•}(P)^{a} is acyclic, if P^{a} is a flat A^{a}-module. With this observation, all the results of [75, §4.3, §4.4] extend now verbatim to the case where 𝔪 fulfills condition (B) (whereas the proofs in loc.cit. used the stronger condition that 𝔪 is a flat V -module : the details shall be left to the reader).

14.1.58. Let π : V -Alg.Mod → V -Alg be the functor such that π(A, M) := A and π(f, φ) := f for every object (A, M) and every morphism (f, φ) of V -Alg.Mod. We consider a functor

T : V -Alg.Mod → V -Alg.Mod such that π ∘ T = π.

Let also be R a simplicial V -algebra, d ∈ ℕ and suppose that :

• T is V -homogeneous of degree d, i.e. for every V -algebra A, the restriction of T

T_{A} : A-Mod → A-Mod

is V -homogeneous of degree d.

• T_{A} commutes with filtered colimits (cp. (7.10.26)) for every V -algebra A.

• Either d ≤ 1 or else the ideal 𝔪 · H_{0}(R) of H_{0}(R) satisfies condition (B) of [75, §2.1.6].

Remark 14.1.59. (i) As detailed in (7.10.26), the functor T induces a functor

T_{R} : R-Mod → R-Mod (M[n] | n ∈ ℕ) ↦ (T_{R[n]}M[n] | n ∈ ℕ).

(ii) Let T and T′ be two functors as in (14.1.58), V -homogeneous of degrees respectively d and d′; then we get the functor V -homogeneous of degree d + d′

T ⊗ T′ : V -Alg.Mod → V -Alg.Mod (A, M) ↦ (A, T_{A}M ⊗_{A} T′_{A}M)

and it is easily seen that (T ⊗ T′)_{A} commutes with filtered colimits, for every V -algebra A.

(iii) Likewise, if f : T → T′ is a natural transformation of functors fulfilling the conditions of (14.1.58), for the same degree d; then the same holds for the functors Ker f and Coker f.

(iv) In the situation of (14.1.58), let f : H_{0}(R) → B be any morphism of V -algebras. If condition (B) holds for 𝔪 · H_{0}(R), it holds also also for 𝔪B; then remark 14.1.54(ii) implies that T_{B} induces a functor T_{B^{a}} := T_{B}^{a} : B^{a}-Mod → B^{a}-Mod. Especially, the above holds with B := R[p], for any p ∈ ℕ, and f the natural map given by the degeneracies of the simplicial V -algebra R. It is then clear that T_{R} induces a well defined functor

T_{R^{a}} : R^{a}-Mod → R^{a}-Mod.

The following result shows that, in this situation, the construction of left derived functors descends likewise to R^{a}-modules.

Theorem 14.1.60. In the situation of (14.1.58), the following holds :

(i) Let φ : M → N be a morphism of R-modules, and n ∈ ℕ an integer such that H_{i}(φ)^{a} : (H_{i}M)^{a} → (H_{i}N)^{a} is an isomorphism of V^{a}-modules, for every i ≤ n. Then

H_{i}(LT φ)^{a} : H_{i}(LT M)^{a} → H_{i}(LT N)^{a}

is an isomorphism of V^{a}-modules, for every i ≤ n. (ii) Especially, the functor T_{R} induces a well defined left derived functor

LT_{R^{a}} : D(R^{a}-Mod) → D(R^{a}-Mod).

Proof. Clearly (ii) follows from (i).

(i): In light of corollary 7.10.34(i), we may replace M and N by respectively cosk_{n}M and cosk_{n}N, after which, we may assume that H_{i}M = H_{i}N = 0 for every i > n, so φ^{a} is an isomorphism in D(R^{a}-Mod). Next, by proposition 7.10.28, we may replace M and N by their respective standard free resolutions, in which case we are reduced to checking that the induced map (T_{R}φ)^{a} is an isomorphism in D(R^{a}-Mod). Since T is homogeneous and (B) holds for 𝔪 · H_{0}(R), the latter assertion follows straightforwardly from lemma 14.1.48(ii). □

14.1.61. Next, we wish to decide how much of the foregoing theory can be salvaged, when we drop condition (B). We will concentrate on the functors that are relevant to the later study of the cotangent complex. Thus, henceforth, for every integer d ∈ ℕ, we shall denote by T^{d} one of the three standard functors

Sym^{d}, Λ^{d}, Γ^{d} : V -Alg.Mod → V -Alg.Mod

(namely, the symmetric and exterior d-th power functors, and the d-th divided power functor). Notice that the functor T^{d} fulfills the conditions of (14.1.58), for every d ∈ ℕ. Moreover, in all three cases we have natural identifications :

(14.1.62)  T^{1} →^{∼} 1_{V -Alg.Mod}.

In general, we can no longer expect that T^{d} descends to almost modules; indeed, consider the following :

Example 14.1.63. Let (V, 𝔪) be as in (14.1.61), and p ∈ ℕ any prime integer. The Frobenius map Φ_{R} : R/pR → R/pR for V -algebras R, can be seen as a homogeneous polynomial law of degree p on the V -module V/pV. Denote by 𝔪^{(p)} ⊂ V the ideal generated by (x^{p} | x ∈ 𝔪). Clearly Φ descends to a polynomial law

\overline{Φ} : V/(pV + 𝔪) → W_{p} := V/(pV + 𝔪^{(p)})

which is still homogeneous of degree p, so it factors through a unique V -linear map

Γ_{A}^{p}(V/(pV + 𝔪)) → W_{p}.

The latter is surjective, since its image contains the class of the unit element of V. Now, the proof of [75, Prop.2.1.7(ii)] shows that – if (B) does not hold for 𝔪 – there exists some prime p such that pV + 𝔪^{(p)} does not contain 𝔪, therefore W_{p}^{a} ≠ 0. This shows that the functor

V -Mod → V^{a}-Mod M ↦ (Γ_{A}^{p}M)^{a}

does not factor through V^{a}-Mod.

However, we will see that example 14.1.63is, in a sense, the worst that can happen.

Lemma 14.1.64. In the situation of (14.1.61), we have :

(i) There are natural transformations

T^{i+j} → T^{i} ⊗ T^{j} → T^{i+j} for every i, j ∈ ℕ

whose composition equals \binom{i+j}{i} · 1_{T^{i+j}}. (Notation of remark 14.1.59(ii).) (ii) For every simplicial V -algebra R, the maps of (i) induce natural transformations

LT_{R}^{i+j} → LT_{R}^{i} ⊗^{ℓ}_{R} LT_{R}^{j} → LT_{R}^{i+j} for every i, j ∈ ℕ

whose composition equals \binom{i+j}{i} · 1_{LT_{R}^{i+j}}.

Proof. (i): For T = Λ, the sought morphism Λ^{i+j} → Λ^{i} ⊗ Λ^{j} is the one denoted Δ_{i,j} in [75, §4.3.20], and the morphism Λ^{i} ⊗ Λ^{j} → Λ^{i+j} is given by the rule : x ⊗ y ↦ x ∧ y, for every (A, M) ∈ Ob(V -Alg.Mod), every x ∈ Λ_{A}^{i}M and every y ∈ Λ_{A}^{j}M. The sought identity follows easily from the explicit formula given in [75, (4.3.21)] : details left to the reader.

For T = Sym, the natural transformation Sym^{i+j} → Sym^{i} ⊗ Sym^{j} is given by a similar formula; namely, for every object (A, M) of V -Alg.Mod, every sequence of elements x_{1}, …, x_{i+j} ∈ M, and every subset I ⊂ {1, …, i + j}, set x_{I} := ∏_{k∈I} x_{k} ∈ Sym_{A}^{k}M (where the multiplication is formed in the graded ring Sym_{A}^{•}M); one checks easily that the rule

x_{1} ⋯ x_{i+j} ↦ ∑_{I,J} x_{I} ⊗ x_{J}

defines a well defined A-linear map on Sym_{A}^{i+j}M (where the sum ranges over all the partitions (I, J) of {1, …, i + j} such that the cardinality of I equals i). Then one defines a map Sym_{A}^{i}M ⊗_{A} Sym_{A}^{j}M → Sym_{A}^{i+j}M by the rule : u ⊗ v ↦ u · v for every u ∈ Sym_{A}^{i}M and v ∈ Sym_{A}^{j}M. Again, the sought identity is verified by direct computation.

If T = Γ, then for every object (A, M) of V -Alg.Mod we define a homogeneous polynomial law M ⇝ Γ_{A}^{i}M ⊗_{A} Γ_{A}^{j}M of degree i + j, by the rule : x ↦ x^{[i]} ⊗ x^{[j]} for every A-algebra B, and every x ∈ B ⊗_{A} M. This law yields a transformation Γ^{i+j} → Γ^{i} ⊗ Γ^{j} as sought. Next, the multiplication of the graded ring functor Γ^{•} gives a transformation Γ^{i} ⊗ Γ^{j} → Γ^{i+j}. The composition of these two transformations is characterized as the unique transformation ψ of Γ^{i+j} such that ψ_{(A,M)}(x^{[i+j]}) = x^{[i]} · x^{[j]} for every object (A, M) of V -Alg.Mod, and every x ∈ M. Then, by corollary 9.5.56we must have ψ_{(A,M)} = \binom{i+j}{i}1_{M}, as required.

(ii): Recall that the three functors Λ, Γ and Sym transform free A-modules into free Amodules, for every V -algebra A. Therefore, the sought natural transformation is none else than the map

T_{R}^{i+j}(⊥_{•}^{R} M) → T_{R}^{i}(⊥_{•}^{R} M) ⊗_{R} T_{R}^{j}(⊥_{•}^{R} M) → T_{R}^{i+j}(⊥_{•}^{R} M)

given by (i), for any R-module M. □

Proposition 14.1.65. In the situation of (14.1.61), let p ∈ ℕ be a prime integer, R a simplicial V ⊗_{ℤ} ℤ_{(p)}-algebra, M an R-module such that M^{a} = 0 in D(R^{a}-Mod). Then

(LT_{R}^{d}M)^{a} = 0 in D(R^{a}-Mod), for every d ∈ ℕ such that (p, d) = 1.

Proof. In light of corollary 7.10.34(i), we may replace M by cosk_{i}M, in which case lemma 14.1.48(i) implies that a · 1_{M} = 0 for every a ∈ 𝔪. Now, we apply lemma 14.1.64(ii) with i = 1 and j = d − 1 (notice that j ≥ 0, since d > 0); in view of (14.1.62), there result natural transformations

LT_{R}^{d}M → M ⊗_{R} LT_{R}^{d−1}M → LT_{R}^{d}M

whose composition is d · 1_{LT_{R}^{d}M}. Since the image of d is invertible in R, it follows easily that a · 1_{LT_{R}^{d}M} = 0 for every a ∈ 𝔪, whence the assertion. □

Theorem 14.1.66. Let d, n, p ∈ ℕ be any three integers, with d > 0 and p a prime. Let also R be a simplicial V ⊗_{ℤ} ℤ_{(p)}-algebra, M an R-module, and suppose that

(a) H_{i}M = 0 for every i < n.

(b) M^{a} = 0 in D(R^{a}-Mod).

Then we have :

(i) H_{i}(LΓ_{R}^{d}M)^{a} = 0 for every i < n.

(ii) H_{i}(LΛ_{R}^{d}M)^{a} = 0 for every i < n + p − 1.

(iii) H_{i}(LSym_{R}^{d}M)^{a} = 0 for every i < n + 2(p − 1).

Proof. (i) is just a special case of corollary 7.10.34(ii), and for d < p, assertions (ii) and (iii) follow from the more general proposition 14.1.65, hence we may assume that d ≥ p. For every j = 0, …, d, set

F^{j} := Γ^{j} ⊗ Λ^{d−j} G^{j} := Λ^{j} ⊗ Sym^{d−j}.

By remark 14.1.59(ii), these functors are homogeneous of degree d, and fulfill the conditions of (14.1.58). Moreover, since Γ, Λ and Sym transform free modules into free modules, we have natural isomorphisms of functors :

(14.1.67) LF_{R}^{j} →^{∼} LΓ_{R}^{j} ⊗^{ℓ}_{R} LΛ_{R}^{d−j} LG_{R}^{j} →^{∼} LΛ_{R}^{j} ⊗^{ℓ}_{R} LSym_{R}^{d−j}  for every j = 0, …, d

(details left to the reader).

(ii): According to [103, I.4.3.1.7], there is a natural complex of functors :

(14.1.68)  0 → F_{R}^{d} ⟶^{∂_{d−1}} F_{R}^{d−1} → ⋯ → F_{R}^{1} ⟶^{∂_{0}} F_{R}^{0} → 0

which is exact on flat R-modules. Set Z_{p} := Ker ∂_{p−1}; due to remark 14.1.59(iii) we may consider the derived functor LZ_{p}, and from corollary 7.10.34(ii) and assumption (a), we get

(14.1.69)  H_{i}(LZ_{p}M) = 0  for every i < n.

The evaluation of (14.1.68) on (⊥_{•}^{R} M)^{Δ} gives an exact sequence of R-modules :

0 → LZ_{p}M → LF_{R}^{p−1}M → ⋯ → LF_{R}^{1}M → LF_{R}^{0}M → 0.

Notice as well, that

(14.1.70)  (LF_{R}^{j}M)^{a} = 0  in D(R^{a}-Mod) for j = 1, …, p − 1

in view of (14.1.67), remark 14.1.51(iii), and proposition 14.1.65. The assertion now follows from (14.1.69) and claim 7.10.50.

(iii): According to [103, I.4.3.1.7], there is a natural complex of functors :

Σ : 0 → G_{R}^{d} ⟶^{∂_{d−1}} G_{R}^{d−1} → ⋯ → G_{R}^{1} ⟶^{∂_{0}} G_{R}^{0} → 0.

Set Z′_{p} := Ker ∂_{p−1}; due to remark 14.1.59(iii) we may consider the derived functor LZ′_{p}, and since Σ is exact on flat R-modules, we obtain two exact sequences of R-modules :

\begin{matrix}Σ′ : & 0 → LG_{R}^{d}M → LG_{R}^{d−1}M → ⋯ → LG_{R}^{p}M → LZ′_{p}M → 0 \\ Σ′′ : & 0 → LZ′_{p}M → LG_{R}^{p−1}M → ⋯ → LG_{R}^{1}M → LG_{R}^{0}M → 0. \end{matrix}

On the one hand, in light of (ii), remark 14.1.51(iii,vi) and (14.1.67), we see that

cosk_{n+p−1}(LG_{R}^{j}M)^{a} = 0 in D(R^{a}-Mod) for every j = 1, …, d.

By applying claim 7.10.50to the exact sequence cosk_{n+p−1}Σ′^{a}, we deduce that H_{i}(LZ′_{p}M)^{a} = 0 for every i < n + p − 1. On the other hand, (14.1.67) and proposition 14.1.65imply that

(LG_{R}^{j}M)^{a} = 0 in D(R^{a}-Mod) for j = 1, …, p − 1

so the assertion follows, after applying claim 7.10.50to the exact sequence Σ′′. □

14.1.71. Let now φ : R → S be any morphism of simplicial V^{a}-algebras; pick any resolution ρ : P → S with P a flat R-algebra, and consider the composition

(14.1.72)  S ⊗_{R} P ⟶^{S⊗_{R}ρ} S ⊗_{R} S ⟶^{μ_{S}} S

where μ_{S} is the multiplication law of S. Then (14.1.72) represents a morphism

Δ(φ) : S ⊗^{ℓ}_{R} S → S in D(R-Alg)

which is independent (up to unique isomorphism) of the choice of P. Notice that if φ is an

isomorphism in D(s.V^{a}-Alg), then the same holds for φ ⊗^{ℓ}_{R} φ : R → S ⊗^{ℓ}_{R} S; also, we have

φ = Δ(φ) ∘ (φ ⊗^{ℓ}_{R} φ) in D(s.V^{a}-Alg).

More generally, set C := Cone φ; then, by considering the section S ⊗^{ℓ}_{R} φ of Δ(φ), we get a natural decomposition

S ⊗^{ℓ}_{R} S →^{∼} S ⊕ (S ⊗^{ℓ}_{R} C) in D(R-Mod)

which identifies Δ(φ) with the natural projection, whence a natural isomorphism

(14.1.73)  Cone Δ(φ) →^{∼} σS ⊗^{ℓ}_{R} C.

Thus, say that C = σ^{k}C′ in D(R-Mod), for some R-module C′ with H_{0}C′ ≠ 0; there follows a distinguished triangle in D(R-Mod)

σ^{k}C′ → σ^{k}S ⊗^{ℓ}_{R} C′ → σ^{2k}C′ ⊗^{ℓ}_{R} C′ → σ^{k+1}C′.

Especially, if k ≥ 2, then H_{k}C = H_{k}(S ⊗^{ℓ}_{R} C), and we see that, in this case, Cone Δ(φ) = σ^{k+1}C′′ in D(R-Mod), for some C′′ such that H_{0}C′′ ≠ 0. However, it is clear from (14.1.73) that Cone Δ^{2}(φ) = σ^{2}C′′ for some object C′′ of D(R-Mod). We conclude that if Δ^{n}(φ) is an isomorphism in D(R-Mod) for some n ≥ 2, then the same holds already for Δ^{2}(φ).

Definition 14.1.74. In the situation of (14.1.71), we say that φ is a weakly étale morphism, if Δ^{2}(φ) is an isomorphism in D(R-Alg).

14.1.75. Let φ : R′ → R be any morphism of simplicial V^{a}-algebras, and S, T two Ralgebras. Since the standard resolution F_{R}(S) := F_{R_{!!}}(S_{!!})^{a} of example 7.10.25 is clearly functorial in both R and S, we have a natural map

F_{R′}(S)^{Δ} → F_{R}(S)^{Δ}

of simplicial R′-algebras, whence a natural morphism

(14.1.76)  S ⊗^{ℓ}_{R′} T → F_{R}(S)^{Δ} ⊗_{R′} T → S ⊗^{ℓ}_{R} T.

Lemma 14.1.77. In the situation of (14.1.75), suppose that φ is a quasi-isomorphism. Then the same holds for (14.1.76).

Proof. Let ψ : F_{R′}(S)^{Δ} ⊗_{R′} R → F_{R}(S)^{Δ} be the natural morphism; by construction, (14.1.76) factors as the composition of ψ ⊗_{R} 1_{T} and the natural isomorphism

F_{R′}(S)^{Δ} ⊗_{R′} T →^{∼} (F_{R′}(S)^{Δ} ⊗_{R′} R) ⊗_{R} T.

Since both F_{R′}(S)^{Δ} ⊗_{R′} R and F_{R}(S)^{Δ} are flat R-algebras, remark 14.1.52(ii) then reduces to checking that ψ is a quasi-isomorphism. Since the natural maps β : F_{R}(S)^{Δ} → S and α : F_{R′}(S)^{Δ} → F_{R′}(Δ) ⊗_{R′} R are quasi-isomorphisms, it suffices to show that the composition β ∘ ψ ∘ α is a quasi-isomorphism. But the latter is none else than the standard resolution F_{R′}(S) → S, whence the lemma. □

Proposition 14.1.78. Let φ : R → S and φ′ : R′ → S′ be two morphisms of simplicial V -algebras, and suppose we have a commutative diagram in D(R-Alg)

(14.1.79)  \xymatrix{R \ar[r]^{φ} \ar[d]_{ψ} & S \ar[d]^{ψ′} \\ R′ \ar[r]^{φ′} & S′}

where ψ and ψ′ are isomorphisms (in D(R-Alg)). Then φ^{a} is weakly étale if and only if the same holds for φ′^{a}.

Proof. Denote by Hot(V -Alg) the homotopy category of simplicial V -algebras, and recall that the multiplicative system of quasi-isomorphisms in Hot(V -Alg) admits a right calculus of fractions ([103, I.3.1.8(ii)]). We begin with the following special case :

Claim 14.1.80. Suppose that ψ and ψ′ are also morphisms of simplicial V -algebras, and that (14.1.79) commutes in the category Hot(V -Alg). Then the proposition holds.

Proof of the claim. Indeed, in this situation it follows easily from lemma 14.1.77 that φ^{a} (resp. φ′^{a}) is weakly étale if and only if the same holds for ψ′^{a} ∘ φ^{a} (resp. for φ′^{a} ∘ ψ^{a}), so we are reduced to the case where R = R′, S = S′, and φ, φ′ : R → S are two homotopic morphisms of simplicial V -algebras. In this case, there exist morphisms of simplicial V -algebras φ′′ : R → S′′ and d_{0}, d_{1} : S′′ → S, such that d_{0} and d_{1} are quasi-isomorphisms, and φ = d_{0} ∘ φ′′, φ′ = d_{1} ∘ φ′′ (see [103, I.2.3.2]). Then the claim follows by applying repeatedly lemma 14.1.77. ♢

Next, there exist a simplicial V -algebra S′′ and morphisms β : S′′ → S and γ : S′′ → S′ of simplicial V -algebras, such that both β and γ are quasi-isomorphisms, and ψ′ = γ ∘ β^{−1} in D(s.V -Alg). Moreover, there exist morphisms φ′′ : R′′ → S′′ and ψ′′ : R′′ → R such that ψ′′ is a quasi-isomorphism, and the resulting diagram

\xymatrix{R′′ \ar[r]^{φ′′} \ar[d]_{ψ′′} & S′′ \ar[d]^{β} \\ R \ar[r]^{φ} & S}

commutes in Hot(V -Alg). By claim 14.1.80, we may then replace φ by φ′′ and ψ′ by β, after which we may assume that ψ′ is a morphism of simplicial V -algebras.

Likewise, we may find morphisms β′ : R′′ → R and γ′ : R′′ → R′ of simplicial V -algebras that are quasi-isomorphisms, and such that ψ = γ′ ∘ β′^{−1} in D(s.V -Alg); again by lemma 14.1.77, we see that φ^{a} is weakly étale if and only if the same holds for φ^{a} ∘ β^{a}, so we may replace φ by φ ∘ β, ψ by ψ ∘ γ, and assume from start that also ψ is a map of simplicial V -algebras.

In this situation, by applying once again lemma 14.1.77 we see that φ^{a} (resp. φ′^{a}) is weakly étale if and only if the same holds for ψ′^{a} ∘ φ^{a} (resp. for φ′^{a} ∘ ψ^{a}). Therefore, we may assume from start that R = R′, S = S′, and φ, φ′ : R → S are two maps of simplicial V -algebras that represent the same morphism in D(s.V -Alg). In this case, we may find a morphism ϑ : R′′′ → R of simplicial V -algebras, such that ϑ is a quasi-isomorphism, and φ ∘ ϑ, φ′ ∘ ϑ : R′′′ → S are homotopic maps; then the assertion follows from claim 14.1.80 (and again, from lemma 14.1.77: details left to the reader). □

Corollary 14.1.81. Let φ : R → S and ψ : S → T be any two morphisms of simplicial V^{a}-algebras. We have :

(i) If φ and ψ are weakly étale, the same holds for ψ ∘ φ. (ii) If φ and ψ ∘ φ are weakly étale, the same holds for ψ. (iii) If φ is weakly étale, and R → R′ is any morphism of simplicial V^{a}-algebras, then

φ′ := R′ ⊗^{ℓ}_{R} φ is weakly étale. (iv) φ is weakly étale if and only if the same holds for Δ(φ).

Proof. (iv) is immediate from the discussion of (14.1.71).

(iii): Endow S′ := R′ ⊗^{ℓ}_{R} S with the R′-algebra structure deduced from its left tensor factor; then the assertion is an immediate consequence of the following more general

Claim 14.1.82. There exists a commutative diagram in D(R′-Alg)

\xymatrix{S′ ⊗^{ℓ}_{R′} S′ \ar[rr] \ar[dr]_{Δ(φ′)} &  & R′ ⊗^{ℓ}_{R} (S ⊗^{ℓ}_{R} S) \ar[dl]^{R′⊗^{ℓ}_{R}Δ(φ)} \\  & S′}

whose horizontal arrow is an isomorphism.

Proof of the claim. Pick any resolution ρ : P → S with P a flat R-algebra; then R′ ⊗_{R} P

represents R′ ⊗^{ℓ}_{R} S, and we have a natural isomorphism of R′-algebras

(R′ ⊗_{R} P) ⊗_{R′} (R′ ⊗_{R} P) →^{∼} R′ ⊗_{R} (P ⊗_{R} P).

In view of remark 7.10.31(i), the source of this map represents S′ ⊗^{ℓ}_{R′} S′, and the target repre-

sents R′ ⊗^{ℓ}_{R} (S ⊗^{ℓ}_{R} S). Under these identifications, the morphism R′ ⊗^{ℓ}_{R} Δ(φ) becomes the map 1_{R′}⊗_{R}(μ_{S}∘(ρ⊗_{R}ρ)) (notation of (14.1.71)), whereas Δ(φ′) is represented by the multiplication map μ_{R′⊗_{R}P} = 1_{R′} ⊗_{R} μ_{P} of R′ ⊗_{R} P. The claim follows straightforwardly. ♢

(i): Pick a resolution P → S_{!!} with P a flat R_{!!}-algebra, and a resolution Q → T_{!!} with Q a flat P -algebra; in light of proposition 14.1.78 it suffices to show that the resulting morphism R → Q^{a} is weakly étale, so we may replace S by P^{a} and T by Q^{a}, and assume from start that both φ and ψ are flat morphisms. Due to (iv), it then suffices to check that Δ(ψ∘φ) : T ⊗_{R} T → T is weakly étale; however, the latter can be factored as the composition

(14.1.83)  T ⊗_{R} T →^{∼} T ⊗_{S} (S ⊗_{R} S) ⊗_{S} T ⟶^{T ⊗_{S}Δ(φ)⊗_{S}T} T ⊗_{S} T ⟶^{Δ(ψ)} T

and by (iv) the maps Δ(ψ) and Δ(φ) are weakly étale, so the same holds for T ⊗_{S} Δ(φ) ⊗_{S} T, in view of (iii). We may therefore replace ψ by Δ(ψ) and φ by T ⊗_{S} Δ(φ) ⊗_{S} T, after which, we may assume that Δ(ψ) is an isomorphism in D(s.A-Alg). In this case, the factorization (14.1.83) makes it clear that Δ(ψ ∘ φ) is weakly étale, as stated.

(ii): Set S′ := S ⊗^{ℓ}_{R} S and T′ := S ⊗^{ℓ}_{R} T; we may factor ψ as the composition

S ⟶^{S⊗^{ℓ}_{R}(ψ∘φ)} T′ ⟶^{T′⊗^{ℓ}_{S′}Δ(φ)} T

so the assertion follows from (i), (iii) and (iv). □

Theorem 14.1.84. Let φ : R → S be a morphism of simplicial V -algebras, c, p ∈ ℕ two integers, with p a prime, and suppose that φ^{a} is weakly étale. We have :

(i) If the ideal 𝔪 · H_{0}(S) of H_{0}(S) satisfies condition (B) of [75, §2.1.6], then

𝕃_{S/R}^{a} = 0 in D(S^{a}-Mod).

(ii) If S is a ℤ_{(p)}-algebra and H_{i}𝕃_{S/R} = 0 for every i < c, then

H_{i}𝕃_{S/R}^{a} = 0 for every i < c + 2p − 1.

Proof. We begin with the following :

Claim 14.1.85. We may assume that both Δ(φ^{a}) and H_{0}(φ) are isomorphisms.

Proof of the claim. To ease notation, set S′ := S ⊗^{ℓ}_{R} S, and consider the sequence of morphisms

(14.1.86)  S ⟶^{1_{S} ⊗^{ℓ}_{R}φ} S′ ⟶^{μ_{B}} S

where μ_{S} is the composition of the multiplication map S ⊗_{R} S → S and the natural map S′ → S ⊗_{R} S; the transitivity triangle ([103, III.2.1.2]) relative to (14.1.86) yields a natural isomorphism

𝕃_{S′/S} →^{∼} σS ⊗_{S′} 𝕃_{S′/S} →^{∼} σS ⊗_{S′} (S′ ⊗_{S} 𝕃_{S/R}) in D(S-Mod)

where the last isomorphism follows from the base change theorem of [103, III.2.2.1]. Thus, 𝕃_{S′/S} ≃ σ𝕃_{S/R} in D(S-Mod), so assertion (i) (resp. (ii)) holds for φ, provided it holds for Δ(φ), and then the claim follows from corollary 14.1.81(iv). ♢

Henceforth, we assume that both Δ(φ^{a}) and H_{0}(φ) are isomorphisms. Let P → S be a resolution, with P an R-algebra, such that P [n] is a free R[n]-algebra for every n ∈ ℕ; set P′ := S ⊗_{R} P, and recall that there are natural isomorphisms

𝕃_{S/R} →^{∼} S ⊗_{P} Ω_{P/R}^{1} →^{∼} S ⊗_{P′} Ω_{P′/S}^{1}

where Ω_{P/R}^{1} denotes the flat simplicial P -module such that Ω_{P/R}^{1}[n] := Ω_{P [n]/R[n]} (and likewise for Ω_{P′/S}^{1}), with faces and degeneracies deduced from those of P and R (resp. those of P′ and S), in the obvious way. In other words, set

J := Ker(μ_{S} : P′ → S)

(with μ_{S} as in the proof of claim 14.1.85); then 𝕃_{S/R} ≃ J/J^{2} in D(S-Mod), and notice that

(14.1.87)  J^{a} = 0  in D(S^{a}-Mod)

since Δ(φ^{a}) is an isomorphism.

Claim 14.1.88. J is a quasi-regular ideal, and H_{0}J = 0.

Proof of the claim. Since H_{0}(φ) is an isomorphism, it is clear that H_{0}J = 0. Next, for every n ∈ ℕ, the S[n]-algebra P′[n] is free, hence we reduce to showing the following. Let B be any ring, C := B[X_{i} | i ∈ I] any free B-algebra (for any set I), and f : C → B any morphism of B-algebras; then Ker f is a quasi-regular ideal of C. However, for every i ∈ I, set b_{i} := f(X_{i}), and let g : C →^{∼} C be the isomorphism of B-algebras such that g(X_{i}) = X_{i} − a_{i} for every i ∈ I; clearly Ker f is a quasi-regular ideal if and only if the same holds for g^{−1}Ker f. But the latter is the ideal (X_{i} | i ∈ I), so the assertion follows from remark 7.10.44(iii). ♢

(i): We need to show that H_{n}(J/J^{2})^{a} = 0 for every n ∈ ℕ, and we shall argue by induction on n. The assertion for n = 0 is clear from (14.1.87), in view of the exact sequence

H_{n}J → H_{n}(J/J^{2}) → H_{n−1}J^{2} for every n ∈ ℕ.

Hence, suppose that n > 0, and the assertion is already known for every degree < n. The same exact sequence reduces to checking that H_{n−1}(J^{2})^{a} = 0. Now, from claim 14.1.88and theorem 7.10.48, we know that H_{n−1}J^{n} = 0. Therefore, by an easy induction, we are further reduced to showing that H_{n−1}(J^{i}/J^{i+1})^{a} = 0 for every i = 2, …, n − 1. However, on the one hand, proposition 7.10.45(ii.a) says that the natural map

LSym_{S}^{i}(J/J^{2}) → J^{i}/J^{i+1}

is an isomorphism in D(S-Mod), for every i ∈ ℕ; on the other hand, the inductive assumption and theorem 14.1.60(i) imply that H_{j}(LSym^{i}J/J^{2})^{a} = 0 for every j ≤ n, whence the contention.

(ii): We show, by induction on n, that H_{n}(J/J^{2})^{a} = 0 for every n < c + 2p − 1. Arguing as in the foregoing, we may assume that c + 2p − 1 > n > 0, and the sought vanishing is already known in degrees < n, in which case we reduce to checking that H_{n−1}(LSym_{S}^{i}J/J^{2})^{a} = 0 for every i = 2, …, n − 1. Set M := cosk_{n}(s.trunc_{n}J/J^{2}); then H_{j}M = 0 for every j < c, and the inductive assumption implies that M^{a} = 0 in D(S^{a}-Mod), so H_{n−1}(LSym_{S}^{i}M)^{a} = 0, by theorem 14.1.66(iii). Lastly, by corollary 7.10.34(i), we see that

H_{n−1}(LSym_{S}^{i}M) = H_{n−1}(LSym_{S}^{i}J/J^{2}) for every i ∈ ℕ

whence the contention. □

Remark 14.1.89. Let A → B be any morphism of simplicial rings such that the multiplication map μ_{B} : B ⊗_{A} B → B induces an isomorphism H_{0}(μ_{B}) : H_{0}(B) ⊗_{H_{0}(A)} H_{0}(B) →^{∼} H_{0}(B). Pick a resolution φ : P → B such that P [n] is a free A[n]-algebra for every n ∈ ℕ, and set P′ := B ⊗_{A} P, J := Ker (μ_{B} ∘ (B ⊗_{A} φ) : P′ → B). The assumption on B implies that H_{0}J = 0. On the other hand, arguing as in the proof of claim 14.1.88 we see that J is a quasiregular ideal. Taking into account proposition 7.10.45(ii) the spectral sequence associated with the J-adic filtration of P′ can be written as

E_{pq}^{2} := H_{p+q}(LSym_{B}^{q}(𝕃_{B/A})) ⇒ Tor_{p+q}^{A}(B, B).

Though the J-adic filtration is not finite, using proposition 7.2.18(i) and theorem 7.10.48 one sees that this spectral sequence is convergent, and the filtration on its abutment is finite. We will not use this result.

Corollary 14.1.90. Let φ : A → B be a morphism of V^{a}-algebras, p ∈ ℕ a prime integer, and suppose that

(a) V is a ℤ_{(p)}-algebra

(b) the induced morphism s.φ : s.A → s.B of simplicial V^{a}-algebras is weakly étale. Then H_{i}𝕃_{B/A}^{a} = 0 for every i ≤ 2p.

Proof. Notice that condition (a) implies that B_{!!} is a ℤ_{(p)}-algebra; letting c := 0 in theorem 14.1.84(ii), we deduce that H_{i}𝕃_{B/A}^{a} = 0 for i = 0, 1. But in view of claim 14.1.46, the latter implies that H_{i}𝕃_{B_{!!}/A_{!!}} = 0 for i = 0, 1. So, actually the morphism φ fulfills the conditions of theorem 14.1.84(ii), with c = 2, whence the assertion. □

Remark 14.1.91. As an application we can now generalize the deformation theory for almost rings in [75, §3.2] to the case of an arbitrary basic setup (V, 𝔪). Indeed, [75, Lemmata 3.2.1 and 3.26, Prop.3.2.9 and Cor.3.2.11] hold verbatim for any such basic setup, with the same proof : the only difference is that instead of invoking [75, 3.2.1] we must appeal to our theorem 14.1.43 in the explanation of [75, §3.2.7]. Next, let A → B be any weakly étale morphism of V^{a}-algebras; for every prime integer p, let V_{(p)} := ℤ_{(p)} ⊗_{ℤ} V, and denote

j_{p} : (V, 𝔪)^{a}-Alg → (V_{(p)}, 𝔪V_{(p)})^{a}-Alg C ↦ ℤ_{(p)} ⊗_{ℤ} C

the natural functor; with this notation, it is easily seen that

ℤ_{(p)} ⊗_{ℤ} B_{!!} = (j_{p}B)_{!!}

(details left to the reader), and in view of [103, II,2.2.1] we deduce a natural isomorphism

ℤ_{(p)} ⊗_{ℤ} 𝕃_{B_{!!}/A_{!!}} →^{∼} 𝕃_{(j_{p}B)_{!!}/(j_{p}A)_{!!}}.

Corollary 14.1.90 then easily implies that H_{i}𝕃_{B/A}^{a} = 0 for every i ≤ 4. Combining with the results just quoted from [75], we see that the whole of [75, Th.3.2.18] still holds under the current assumptions. In the same vein, we may complement as follows the descent results of [75, §3.4] :

Corollary 14.1.92. Consider a cartesian diagram of V^{a}-algebras

𝒟  :  \xymatrix{A_{0} \ar[r] \ar[d] & A_{1} \ar[d] \\ A_{2} \ar[r] & A_{3}}

such that the morphism A_{1} → A_{3} is an epimorphism on the underlying V^{a}-modules. Then the resulting morphism A_{0} → A_{1} × A_{2} is of universal effective descent for the fibred categories w.Ét of weakly étale morphisms and Ét of étale morphisms.

Proof. Arguing as in the proof of [75, Prop.3.4.33], and using [75, Cor.3.4.22], we see that the morphism A_{0} → A_{1} × A_{2} is of effective descent for the fibred categories w.Ét and Ét. Next, consider any morphism A_{0} → B_{0} of V^{a}-algebras, and set B_{i} := B_{0} ⊗_{A_{0}} A_{i} for i = 1, 2, 3. The resulting diagram B_{0} ⊗_{A_{0}} 𝒟 is not necessarily cartesian; however, we notice :

Claim 14.1.93. The induced morphism f : B_{0} → B′_{0} := B_{1} ×_{B_{3}} B_{2} is an epimorphism on the underlying V^{a}-modules, with nilpotent kernel.

Proof of the claim. Indeed, we have a short exact sequence of A_{0}-modules

0 → A_{0} → A_{1} × A_{2} → A_{3} → 0

inducing a right exact sequence B_{0} → B_{1} × B_{2} → B_{3} → 0, which shows that f is an epimorphism. Newt, for any B-module M we show more generally that

Ann_{B}(M ⊗_{A_{0}} A_{1}) · Ann_{B}(M ⊗_{A_{0}} A_{2}) ⊂ Ann_{B}(M).

Letting M := B, it will then follow easily that (Ker f)^{2} = 0. Now, let x, y ∈ B_{∗} be two almost elements such that x · (M ⊗_{A_{0}} A_{1}) = 0 and y · (M ⊗_{A_{0}} A_{2}) = 0, and let also I := Ker (A_{1} → A_{3}) = Ker (A_{0} → A_{2}); since M ⊗_{A_{0}} A_{2} = M/IM, we deduce that yM ⊂ IM, and on the other hand the natural morphism of A_{0}-modules M ⊗_{A_{0}} A_{1} ⊗_{A_{1}} I →^{∼} M ⊗_{A_{0}} I → IM is an epimorphism, so that x · IM = 0, and the assertion follows. ♢

Now, by the foregoing we known already that the induced morphism B′_{0} → B_{1} × B_{2} is of effective descent for w.Ét and Ét. But claim 14.1.93together with remark 14.1.91implies that the base change functors

B_{0}-w.Ét → B′_{0}-w.Ét and B_{0}-Ét → B′_{0}-Ét

are equivalences of categories; thus, also the induced morphism B_{0} → B_{1} × B_{2} is of effective descent, as required. □

14.1.94. We conclude this section with a few words on the cohomology of sheaves of almost modules. Namely, let (V, 𝔪) again be an arbitrary basic setup, which we view as a basic setup relative to the one-point topos {pt}, in the sense of [75, §3.3]. Let (X, 𝒪_{X}) be a ringed topos, 𝔪_{X} ⊂ 𝒪_{X} an ideal, such that (𝒪_{X}, 𝔪_{X}) is a basic setup relative to the topos X, and suppose also that we are given a morphism of ringed topoi :

π : (X, 𝒪_{X}) → ({pt}, V) such that (π^{∗}𝔪) · 𝒪_{X} ⊂ 𝔪_{X}.

We deduce a natural morphism :

Rπ_{∗} : D^{+}((𝒪_{X}, 𝔪_{X})^{a}-Mod) → D^{+}((V, 𝔪)^{a}-Mod)

which can be constructed as usual, by taking injective resolutions. In case 𝔪̃ is a flat V -module, this is the same as setting

Rπ_{∗}(K^{•}) := (Rπ_{∗}K_{!}^{•})^{a}.

Using [75, Cor.2.2.24] one may verify that the two definitions coincide : the details shall be left to the reader. In many cases, both statements and proofs of results concerning the cohomology of 𝒪_{X}-modules carry over verbatim to 𝒪_{X}^{a}-modules. One sets, as customary :

H^{•}(X, K^{•}) := H^{•}Rπ_{∗}K^{•} for every object K^{•} of D^{+}((𝒪_{X}, 𝔪_{X})^{a}-Mod) .

14.1.95. As an illustration, we consider the following situation. Suppose that :

• f : A → A′ is a map of V -algebras, and set :

φ := Spec f : X′ := Spec A′ → X := Spec A.

• t ∈ A is an element which is regular both in A and in A′, and such that the induced map A/tA → A′/tA′ is an isomorphism.

• U ⊂ X is any open subset containing D(t) := {𝔭 ∈ X | t ∉ 𝔭}, and set U′ := φ^{−1}U.

• ℱ is a quasi-coherent 𝒪_{U}^{a}-module, and set ℱ′ := φ_{|U′}^{∗}ℱ, which is a quasi-coherent

𝒪_{U′}^{a}-module. Then we have natural morphisms of A^{a}-modules :

(14.1.96)  H^{q}(U, ℱ) → H^{q}(U′, ℱ′)  for every q ∈ ℕ.

Lemma 14.1.97. In the situation of (14.1.95), suppose moreover that t is ℱ -regular. Then (14.1.96) is an isomorphism for every q > 0, and induces an isomorphism of A^{′a}-modules :

H^{0}(U, ℱ) ⊗_{A} A′ →^{∼} H^{0}(U′, ℱ′).

Proof. To ease notation, we shall write φ instead of φ_{|U′}. To start out, we remark :

Claim 14.1.98. (i) 𝒯or_{1}^{𝒪_{U}^{a}} (ℱ , 𝒪_{U}^{a}/t𝒪_{U}^{a}) = 0 and Tor_{1}^{A^{a}}(H^{0}(U, ℱ), A/tA) = 0.

(ii) The 𝒪_{U}^{a}-module φ_{∗}ℱ′ = ℱ ⊗_{𝒪_{U}^{a}} φ_{∗}𝒪_{U′}^{a} (resp. the A^{′a}-module H^{0}(U, ℱ) ⊗_{A^{a}} A^{′a}) is t-torsion-free.

Proof of the claim. (i): Since t is regular on 𝒪_{U}, we have a short exact sequence : ℰ := (0 → 𝒪_{U}^{a} → 𝒪_{U}^{a} → 𝒪_{U}^{a}/t𝒪_{U}^{a} → 0), and since t is regular on ℱ, the sequence ℰ ⊗_{𝒪_{U}^{a}} ℱ is still exact; the vanishing of 𝒯or_{1}^{𝒪_{U}^{a}} (ℱ , 𝒪_{U}^{a}/t𝒪_{U}^{a}) is an easy consequence. For the second stated vanishing one argues similarly, using the exact sequence 0 → A → A → A/tA → 0.

(ii): Under the current assumptions, the natural map 𝒪_{U}/t𝒪_{U} → φ_{∗}(𝒪_{U′}/t𝒪_{U′}) is an isomorphism, whence a short exact sequence : ℰ′ := (0 → φ_{∗}𝒪_{U′}^{a} → φ_{∗}𝒪_{U′}^{a} → 𝒪_{U}^{a}/t𝒪_{U}^{a} → 0). In view of (i), the sequence ℰ′ ⊗_{𝒪_{U}^{a}} ℱ is still exact, so ℱ ⊗_{𝒪_{U}^{a}} φ_{∗}𝒪_{U′}^{a} is t-torsion-free. An analogous argument works as well for H^{0}(U, ℱ) ⊗_{A^{a}} A^{′a}. ♢

Claim 14.1.99. For every n > 0, the map A/t^{n}A → A′/t^{n}A′ induced by f is an isomorphism.

Proof of the claim. Indeed, since t is regular on both A and A′, and A/tA →^{∼} A′/tA′, the map of graded rings ⊕_{n∈ℕ}t^{n}A/t^{n+1}A → ⊕_{n∈ℕ}t^{n}A′/t^{n+1}A′ is bijective. Then the claim follows from [34, Ch.III, §2, n.8, Cor.3]. ♢

Let j : D(t) → U be the natural open immersion, and set :

ℱ [t^{−1}] := j_{∗}j^{∗}ℱ ≃ ℱ ⊗_{𝒪_{U}^{a}} j_{∗}𝒪_{D(t)}^{a}.

Notice the natural isomorphisms :

(14.1.100)  j_{∗}j^{∗}φ_{∗}ℱ′ ≃ (ℱ ⊗_{𝒪_{U}^{a}} φ_{∗}𝒪_{U′}^{a}) ⊗_{𝒪_{U}^{a}} j_{∗}𝒪_{D(t)}^{a} ≃ ℱ [t^{−1}] ⊗_{𝒪_{U}^{a}} φ_{∗}𝒪_{U′}^{a}.

Since t is ℱ -regular, the natural map ℱ → ℱ [t^{−1}] is a monomorphism, and the same holds for the corresponding map φ_{∗}ℱ′ → j_{∗}j^{∗}φ_{∗}ℱ′, in view of claim 14.1.98(ii). Moreover, 𝒢 := ℱ [t^{−1}]/ℱ can be written as the increasing union of its subsheaves Ann_{𝒢} (t^{n}) (for all n ∈ ℕ); hence claim 14.1.99implies that the natural map 𝒢 → 𝒢 ⊗_{𝒪_{U}^{a}} φ_{∗}𝒪_{U′}^{a} is an isomorphism. There follows a ladder of short exact sequences :

(14.1.101)  [figure: 0; ℱ; ℱ [t^{−1}]; 𝒢; 0; 0; φ_{∗}ℱ′; ℱ [t^{−1}] ⊗_{𝒪_{U}^{a}} φ_{∗}𝒪_{U′}^{a}; 𝒢; 0.]

On the other hand, since j is an affine morphism, we may compute :

H^{q}(U, ℱ [t^{−1}]) ≃ H^{q}(U, Rj_{∗}j^{∗}ℱ) ≃ H^{q}(D(t), j^{∗}ℱ) ≃ 0 for every q > 0.

Likewise, from (14.1.100) we get :

H^{q}(U, ℱ [t^{−1}] ⊗_{𝒪_{U}^{a}} φ_{∗}𝒪_{U′}^{a}) ≃ 0 for every q > 0.

Thus, in the commutative diagram :

\xymatrix{H^{q−1}(U, 𝒢) \ar[r]^{∂} \ar[d]_{∂′} & H^{q}(U, ℱ) \ar[d] \\ H^{q}(U, φ_{∗}ℱ) \ar[r]^{∼} & H^{q}(U′, ℱ′)}

the boundary maps ∂ and ∂′ are isomorphisms whenever q > 1, and the right vertical arrow is (14.1.96), so the assertion follows already for every q > 1. To deal with the remaining cases with q = 0 or 1, we look at the ladder of exact cohomology sequences:

[figure: 0; H^{0}(U, ℱ); α; H^{0}(D(t), ℱ); β; H^{0}(U, 𝒢); H^{1}(U, ℱ); 0; 0; H^{0}(U′, ℱ′); H^{0}(D(t), ℱ) ⊗_{A^{a}} A^{′a}; β′; H^{0}(U, 𝒢); H^{1}(U′, ℱ′); 0]

deduced from (14.1.101). On the one hand, we remark that the natural inclusion Im β ⊂ Im β′ factors as a composition

Im β ⟶^{ι} M := A^{′a} ⊗_{A^{a}} Im β ⟶^{π} Im β′

where ι is given by the rule : x ↦ 1 ⊗ x, for all x ∈ A_{∗}^{a}, and π is an epimorphism.

On the other hand, the image of β is isomorphic to N := Coker α, and the latter is the increasing union of its submodules Ann_{N}(t^{n}) (for all n ∈ ℕ). Again from claim 14.1.99 we deduce that M = Im β, hence Im β′ = Im β, which shows that (14.1.96) is an isomorphism also for q = 1. By the same token, Ker(α ⊗_{A^{a}} 1_{A^{′a}}) is the increasing union of its t^{n}-torsion submodules (for all n ∈ ℕ), since it is a quotient of Tor_{1}^{A^{a}}(Im β, A^{′a}); hence Ker(α⊗_{A^{a}} 1_{A^{′a}}) = 0, in view of claim 14.1.98(ii). This easily implies the last assertion for q = 0. □

14.2. Inverse systems of almost modules. The considerations of this section expand upon [75, §2.3.14, §2.4.1, Lemmata 2.3.15, 2.4.2 and 2.4.13].

14.2.1. Let V be a ring, (𝔪_{λ} | λ ∈ Λ) a filtered system of ideals of V such that 𝔪_{λ}^{2} = 𝔪_{λ} for every λ, and set 𝔪 := ⋃_{λ∈Λ} 𝔪_{λ}. Then (V, 𝔪) and (V, 𝔪_{λ}) for every λ ∈ Λ are basic setups, and we have an obvious compatible system of localization functors :

(14.2.2)  \xymatrix{(V, 𝔪)^{a}-Mod \ar[r]^{π_{λ}} \ar[dr]_{π_{μ}} & (V, 𝔪_{λ})^{a}-Mod \ar[d]^{π_{λμ}} \\  & (V, 𝔪_{μ})^{a}-Mod}  for every λ ≥ μ in Λ.

Proposition 14.2.3. (i) The compatible system (14.2.2) induces an equivalence of categories

ω : (V, 𝔪)^{a}-Mod →^{∼} 2-lim_{λ∈Λ} (V, 𝔪_{λ})^{a}-Mod.

(ii) If 𝔪_{λ} fulfills condition (B) of [75, §2.1.6] for every λ ∈ Λ, then the same holds for 𝔪.

Proof. (ii) is obvious. To show (i), we construct a quasi-inverse τ for ω as follows. Recall that an object of the above 2-limit is a datum (M_{•}, f_{••}) := ((M_{λ} | λ ∈ Λ), (f_{λμ} : π_{λμ}(M_{λ}) →^{∼} M_{μ} | λ ≥ μ)) consisting of (V, 𝔪_{λ})^{a}-modules M_{λ} for every λ ∈ Λ, and isomorphisms f_{λμ} of (V, 𝔪_{μ})^{a}-modules, for every λ ≥ μ in Λ. Such a datum is required moreover to satisfy the compatibility condition :

f_{μν} ∘ a_{μν}(f_{λμ}) = f_{λν} whenever λ ≥ μ ≥ ν.

However, the discussion of [75, §2.2.2] allows to describe such a datum more concretely as follows. Choose, for every λ ∈ Λ a V -module M′_{λ} whose image in (V, 𝔪_{λ})^{a}-Mod represents M_{λ}; the isomorphism f_{λμ} corresponds then to a unique V -linear isomorphism

f′_{λμ} : 𝔪̃_{μ} ⊗_{V} M′_{λ} →^{∼} 𝔪̃_{μ} ⊗_{V} M′_{μ} whenever λ ≥ μ

(with 𝔪̃_{λ} := 𝔪_{λ} ⊗_{V} 𝔪_{λ} for every λ ∈ Λ) such that

(14.2.4)  f′_{μν} ∘ (𝔪̃_{ν} ⊗_{V} f′_{λμ}) = 𝔪̃_{ν} ⊗_{V} f′_{λν}  whenever λ ≥ μ ≥ ν.

Set M_{λ!} := 𝔪̃_{λ}⊗_{V} M′_{λ}; by composing the inverse of f′_{λμ} with the natural map 𝔪̃_{μ}⊗_{V} M′_{λ} → M_{λ!}, we obtain a well defined V -linear map

f′′_{λμ} : M_{μ!} → M_{λ!} whenever λ ≥ μ

and in light of (14.2.4) we deduce easily that f′′_{λν} = f′′_{μν} ∘ f′′_{λμ} whenever λ ≥ μ ≥ ν (details left to the reader). We now let

τ(M_{•}, f_{••}) := (colim_{λ∈Λ} M_{λ!})^{a} ∈ Ob((V, 𝔪)^{a}-Mod).

If (N_{•}, g_{••}) is another object of the above 2-limit of categories, a morphism (M_{•}, f_{••}) → (N_{•}, g_{••}) is the datum of a system of morphisms h_{λ} : M_{λ} → N_{λ} fulfilling obvious compatibility conditions with respect ot the isomorphisms f_{••} and g_{••}; after choosing representative V -modules N′_{λ} for each N_{λ}, the morphism h_{λ} corresponds to a unique V -linear map h′_{λ} : M_{!} → N_{!} := 𝔪̃_{λ} ⊗_{V} N′_{λ}, for every λ ∈ Λ. It is then clear that the resulting system h′_{•} yields a natural transformation M_{•!} → N_{•!} of direct systems indexed by Λ, whence a well defined morphism of (V, 𝔪)^{a}-modules

τ(h_{•}) := (colim_{λ∈Λ} h′_{λ})^{a} : τ(M_{•}, f_{••}) → τ(N_{•}, g_{••}).

This completes the construction of our functor τ. It is easily seen that the definition of τ is independent of all choices, up to a canonical isomorphism of functors. Especially, if (M_{•}, f_{••}) = ω(M^{a}) for some V -module M, we may assume that M′_{λ} has been chosen equal to M, and f′_{λμ} = 𝔪̃_{λ} ⊗_{V} 1_{M} for every λ, μ ∈ Λ with λ ≥ μ. With these choices, we get a natural isomorphism τ ∘ ω(M^{a}) →^{∼} M^{a} for every (V, 𝔪)^{a}-module M^{a}. Next, for any given datum (M_{•}, f_{••}) as in the foregoing, set (P_{•}, t_{••}) := ω ∘ τ(M_{•}, f_{••}); for every μ ∈ Λ, the subset Λ(μ) := {λ ∈ Λ | λ ≥ μ} is cofinal in Λ, hence P_{μ} is naturally isomorphic to

(colim_{λ∈Λ(μ)} M_{λ!})^{a} →^{∼} colim_{λ∈Λ(μ)}(M_{λ!})^{a} →^{∼} colim_{λ∈Λ(μ)} π_{λμ}(M_{λ})

where the transition morphisms in the latter colimit are given by the isomorphisms f_{λμ}; i.e. is naturally isomorphic to M_{μ}. Lastly, under this identification, it is easily seen that the morphism t_{λμ} corresponds to f_{λμ} whenever λ ≥ μ in Λ (details left to the reader); summing up, we have obtained a natural isomorphism ω ∘ τ(M_{•}, f_{••}) →^{∼} (M_{•}, f_{••}), and the proof is concluded. □

Corollary 14.2.5. With the notation of (14.2.1), for every (V, 𝔪)^{a}-module M and every λ ∈ Λ, denote by M_{λ} the image of M in (V, 𝔪_{λ})^{a}-Mod. Let A be a (V, 𝔪)^{a}-algebra, M an A-module, B an A-algebra, and r ∈ ℕ. Then the following holds :

(i) We have natural isomorphisms of V -modules :

colim_{λ∈Λ} M_{λ!} →^{∼} M_{!} M_{∗} →^{∼} lim_{λ∈Λ} M_{λ∗}.

(ii) The system of localization functors A-Alg → A_{λ}-Alg induces a natural equivalence

A-Alg →^{∼} 2-lim_{λ∈Λ} A_{λ}-Alg.

(iii) The A-module M is flat (resp. faithfully flat, resp. almost projective, resp. almost finitely generated, resp. almost finitely presented) if and only if the same holds for the A_{λ}-module M_{λ}, for every λ ∈ Λ.

(iv) Suppose that 𝔪_{λ} fulfills condition (B) for every λ ∈ Λ. Then the A-module M is almost projective of almost finite rank (resp. of finite rank ≤ r) if and only if the same holds for the A_{λ}-module M_{λ}, for every λ ∈ Λ.

(v) The A-algebra B is flat (resp. faithfully flat, resp. weakly unramified, resp. weakly étale, resp. unramified, resp. étale) if and only if the same holds for the A_{λ}-algebra B_{λ}, for every λ ∈ Λ.

Proof. (i): The assertion for M_{!} is clear. For the assertion concerning M_{∗}, it suffices to notice that for any two (V, 𝔪)^{a}-modules M and N, proposition 14.2.3(i) implies that the natural map

(14.2.6)  Hom_{(V,𝔪)^{a}-Mod}(M, N) → lim_{λ∈Λ} Hom_{(V,𝔪_{λ})^{a}-Mod}(M_{λ}, N_{λ})

is an isomorphism of V -modules.

(ii),(iii): Clearly the functors π_{λ} and π_{λμ} of (14.2.1) are all compatible with tensor products and with the alHom functors; moreover, for every λ ∈ Λ, every short exact sequence of (V, 𝔪_{λ})^{a}-modules is isomorphic to Σ^{a}, for some short exact sequence Σ of V -modules (and likewise for short exact sequences of (V, 𝔪)^{a}-modules : details left to the reader). In view of proposition 14.2.3(i), assertion (ii) follows straightforwardly, and taking into account the isomorphism (14.2.6), we also deduce that the A-module M is flat (resp. almost projective) if and only if the same holds for the A_{λ}-module M_{λ}, for every λ ∈ Λ.

Next, if every M_{λ} is a faithfully flat A_{λ}-module, and X is an A-module with M ⊗_{A} X = 0, it follows that M_{λ} ⊗_{A_{λ}} X_{λ} = 0 for every λ ∈ Λ, whence X_{λ} = 0 for every such λ, and hence X = 0, by proposition 14.2.3(i). Conversely, if M is faithfully flat and λ ∈ Λ, consider any A-module X such that M_{λ} ⊗_{A_{λ}} X_{λ} = 0; then (M ⊗_{A} X)_{λ} = 0, i.e. 𝔪̃_{λ} ⊗ (M_{∗} ⊗_{A_{∗}} X_{∗}) = 0, so that M ⊗_{A} (𝔪̃_{λ} ⊗_{V} X) = 0. By assumption, this implies that 𝔪̃_{λ} ⊗_{V} X = 0, and therefore X_{λ} = 0; this shows that M_{λ} is faithfully flat.

Next, say that A = R^{a} and M = N^{a} for some V -algebra R and some R-module N; suppose that each M_{λ} is almost finitely generated (resp. almost finitely presented), and let 𝔪_{0} ⊂ 𝔪 be any finitely generated subideal. We then find λ ∈ Λ such that 𝔪_{0} ⊂ 𝔪_{λ}, and by applying [75, Cor.2.3.13] to the A_{λ}-module M_{λ} we find a finitely generated (resp. finitely presented) R-module N′ and an R-linear map N′ → N whose kernel and cokernel are annihilated by 𝔪_{0}. Then, by applying [75, Cor.2.3.13] to the A-module M, we deduce the assertion.

(iv): If every M_{λ} is almost projective of almost finite rank, we already know that M is almost projective, and it remains only to check that M is of almost finite rank. To this aim, let ε ∈ 𝔪 be any element, choose λ ∈ Λ such that ε ∈ 𝔪_{λ}^{2}, and write ε = ∑_{i=1}^{n} ε_{i}ε′_{i} for some n ∈ ℕ and ε_{1}, ε′_{1}, …, ε_{n}, ε′_{n} ∈ 𝔪_{λ}; if M_{λ} is of almost finite rank, we may then find j ∈ ℕ such that ε_{i} · Λ_{A_{λ}}^{j}M_{λ} = 0 for i = 1, …, n, in which case it follows easily that ε_{i}ε′_{i} · Λ_{A}^{j}M = 0 for every i = 1, …, n (details left to the reader), whence the assertion. One argues likewise in case each M_{λ} has rank ≤ r : the details shall be left to the reader.

(v) follows immediately from (i), (ii) and (iii). □

Remark 14.2.7. (i) Let V be a ring, 𝔪, 𝔪′ ⊂ V two ideals such that (V, 𝔪) and (V, 𝔪′) are both basic setups, and let 𝔪′′ := 𝔪 · 𝔪′; set as usual 𝔪̃ := 𝔪 ⊗_{V} 𝔪, and define likewise the V -modules 𝔪̃′ and 𝔪̃′′. It is easily seen that (V, 𝔪′′) is a basic setup as well. Moreover, we claim that there exists a natural isomorphism of V -modules :

(14.2.8)  𝔪̃ ⊗_{V} 𝔪̃′ →^{∼} 𝔪̃′′  x ⊗ y ⊗ x′ ⊗ y′ ↦ xx′ ⊗ yy′.

Indeed, let μ : 𝔪 ⊗_{V} 𝔪′ → 𝔪′′ be the multiplication map; we know already that Ker μ is annihilated by both 𝔪 and 𝔪′, hence 𝔪′′ ⊗_{V} μ : 𝔪′′ ⊗_{V} 𝔪 ⊗_{V} 𝔪′ → 𝔪̃′′ is an isomorphism, and likewise for μ ⊗_{V} 𝔪 ⊗_{V} 𝔪′ : 𝔪̃ ⊗_{V} 𝔪̃′ → 𝔪′′ ⊗_{V} 𝔪 ⊗_{V} 𝔪′, so the same holds for their composition, which is (14.2.8).

(ii) Let M be any V -module; denote by (M, 𝔪)^{a} ∈ Ob((V, 𝔪)^{a}-Mod) the image of M, and define likewise (M, 𝔪′)^{a} ∈ Ob((V, 𝔪′)^{a}-Mod) and (M, 𝔪′′)^{a} ∈ Ob((V, 𝔪′′)^{a}-Mod). We deduce from (i) natural V -linear identifications

\begin{matrix}(M, 𝔪′′)_{∗}^{a} →^{∼} ((M, 𝔪)_{∗}^{a}, 𝔪′)_{∗}^{a} →^{∼} ((M, 𝔪′)_{∗}^{a}, 𝔪)_{∗}^{a} \\ (M, 𝔪′′)_{!}^{a} →^{∼} ((M, 𝔪)_{!}^{a}, 𝔪′)_{!}^{a} →^{∼} ((M, 𝔪′)_{!}^{a}, 𝔪)_{!}^{a}. \end{matrix}

14.2.9. Let V be any ring; a V -linear abelian category is the datum of an abelian category 𝒜 together with a V -module structure on Hom_{𝒜} (A, B) for every A, B ∈ Ob(𝒜), such that the composition law of 𝒜 is a V -bilinear map

Hom_{𝒜} (A, B) × Hom_{𝒜} (B, C) → Hom_{𝒜} (A, C) for every A, B, C ∈ Ob(𝒜) .

We say that a functor F : 𝒜 → 𝒜′ between V -linear abelian categories is V -linear if the map

Hom_{𝒜} (A, B) → Hom_{𝒜′} (F A, F B) f ↦ F f

is V -linear, for every A, B ∈ Ob(𝒜).

Remark 14.2.10. Let V be a ring, and 𝒜 , 𝒜′ two V -linear abelian categories.

(i) For every x ∈ V and A ∈ Ob(𝒜) we define the suboobject xA := Im x · 1_{A} ⊂ A. Moreover, if J ⊂ V is an ideal generated by a finite system x_{1}, …, x_{n} ∈ V, we let JA := ∑_{i=1}^{n} x_{i}A ⊂ A. Let us check that the subobject JA does not depend on the choice of a finite system of generators for J : it suffices to show that for every a ∈ J we have aA ⊂ JA; to this aim, say that a = ∑_{i=1}^{n} a_{i}x_{i} for certain a_{1}, …, a_{n} ∈ V, and for every i = 1, …, n denote by A ⟶^{e_{i}} A^{⊕n} ⟶^{p_{i}} A the inclusion into the i-the direct summand and the projection onto the i-th direct factor of A^{⊕n}. Set φ := ∑_{i=1}^{n} x_{i} · p_{i} and ψ := ∑_{i=1}^{n} a_{i} · e_{i}; it is easily seen that Im φ = JA and the V -bilinearity of the composition law implies that φ ∘ ψ = a · 1_{A}. Since Im φ ∘ ψ ⊂ Im φ, the assertion follows.

(ii) Recall that the opposite 𝒜^{o} of the abelian category 𝒜 is abelian; moreover, 𝒜^{o} inherits from 𝒜 an obvious V -linear structure : the reader may spell out the details. Likewise, if F : 𝒜 → 𝒜′ any V -linear functor, then F^{o} : 𝒜^{o} → 𝒜^{′o} is again V -linear.

(iii) Obviously, for every V -algebra A, the category A-Mod is naturally a V -linear abelian category. If (V, 𝔪) is any basic setup, and A any (V, 𝔪)^{a}-algebra, then also A-Mod is a V - linear abelian category.

(iv) Let J ⊂ V be any ideal, and A ∈ Ob(𝒜) any object. We say that JA = 0 if xA = 0 for every x ∈ J; notice that if J is finitely generated, this notation agrees with that of (i). We say that 𝒜 is J-torsion-free if there are no non-zero objects A of 𝒜 with JA = 0. For instance, if (V, 𝔪) is any basic setup, and A any (V, 𝔪)^{a}-algebra, then A-Mod is 𝔪-torsion-free. Clearly, if 𝒜 is J-torsion-free, the same holds for 𝒜^{o}, for its natural V -linear structure.

(v) Let f : A → B be any morphism of 𝒜, and J, J′ ⊂ V two ideals such that J · Ker f = J′ · Coker f = 0 (notation of (iv)). Then for every a ∈ J and b ∈ J′ there exists a morphism

g : B → A such that f ∘ g = ab · 1_{B} and g ∘ f = ab · 1_{A}.

Indeed, write f = e ∘ p, where A ⟶^{p} Im f ⟶^{e} B are the natural morphisms. Since J · Ker p = J · Ker f = 0, the endomorphism a · 1_{A} factors through p and a morphism g_{1} : Im f → A. Likewise, since J′ · Coker e = J′ · Coker f = 0, the endomorphism b · 1_{B} factors through e and a morphism g_{2} : B → Im f. Then g := g_{1} ∘ g_{2} fulfills the stated conditions.

(vi) In the situation of (v), let F : 𝒜 → 𝒜′ be a V -linear functor; we get F f ∘F g = ab·1_{F B} and F g ∘ F f = ab · 1_{F A}. It follows easily that JJ′ · Ker F f = JJ′ · Coker F f = 0.

Definition 14.2.11. Let (V, 𝔪) be a basic setup, 𝒜 a V -linear abelian category, I a small category, and consider a functor

M_{•} : I → 𝒜  i ↦ M_{i}  (j ⟶^{φ} i) ↦ (M_{j} ⟶^{M_{φ}} M_{i}).

(i) We say that M_{•} is essentially zero, if for every i ∈ Ob(I) there exists a morphism φ : j → i in I with M_{φ} = 0.

(ii) We say that M_{•} is almost essentially zero, if for every i ∈ Ob(I) and every subideal 𝔪_{0} ⊂ 𝔪 of finite type there exists a morphism φ : j → i in I with 𝔪_{0} · M_{φ} = 0. We denote

Fun(I, 𝒜)_{a.ess.0}

the full subcategory of Fun(I, 𝒜) whose objects are the almost essentially zero functors.

(iii) We say that M_{•} is almost essentially constant, if there exist L ∈ Ob(𝒜) and a cone π_{•} : c_{L} ⇒ M_{•} inducing almost essentially zero functors

Ker π_{•} : I → A-Mod and Coker π_{•} : I → A-Mod.

(iv) We say that M_{•} has the almost Mittag-Leffler property if for every i ∈ Ob(I) and every subideal 𝔪_{0} ⊂ 𝔪 of finite type there exists a morphism j ⟶^{φ} i in I with :

𝔪_{0} · Im M_{φ} ⊂ Im M_{φ∘ψ} for every morphism ψ : k → j in I.

(v) We say that M_{•} is a Cauchy functor, if for every subideal 𝔪_{0} ⊂ 𝔪 of finite type there exists i ∈ Ob(I) with 𝔪_{0} · Ker M_{φ} = 𝔪_{0} · Coker M_{φ} = 0 for every morphism φ : j → i of I.

We say that M_{•} is null, if it is both a Cauchy functor and almost essentially zero.

(vi) Let P(M_{•}) be either one of the conditions : “M_{•} is essentially zero”, “M_{•} is almost essentially zero”, “M_{•} is almost essentially constant”, “M_{•} has the almost Mittag-Leffler property”, “M_{•} is a Cauchy functor” or “M_{•} is null”. We say that M_{•} has the dual P property if P(M_{•}^{o}) holds, for the opposite functor M_{•}^{o} : I^{o} → 𝒜^{o} (see remark 14.2.10(ii)).

Remark 14.2.12. (i) In the situation of definition 14.2.11, for every M ∈ Ob(𝒜) and every subideal 𝔪_{0} ⊂ 𝔪 of finite type, set M[𝔪_{0}] := ⋂_{x∈𝔪_{0}} Ker x · 1_{M}. Notice that M_{•} is almost essentially zero if and only if the following holds. For every such 𝔪_{0}, the induced functor

M_{•}/M_{•}[𝔪_{0}] : I → 𝒜 i ↦ M_{i}/M_{i}[𝔪_{0}]

is essentially zero.

(ii) Suppose that I is cofiltered and 𝒜 is complete, and let us set :

M_{i}^{△} := ⋂_{φ:j→i} Im M_{φ} for every i ∈ Ob(I).

We claim that M_{ψ} restricts to a morphism in 𝒜

M_{ψ}^{△} : M_{i′}^{△} → M_{i}^{△} for every morphism ψ : i′ → i in I.

Indeed, it suffices to check that M_{ψ}(M_{i′}^{△}) ⊂ Im M_{φ} for every morphism φ : j → i in I. However, since I is cofiltered, for every such φ there exist k ∈ Ob(I) and morphisms φ′ : k → i′ and ψ′ : k → j such that φ∘ψ′ = ψ∘φ′. It follows that M_{ψ}(M_{i′}^{△}) ⊂ M_{ψ}(Im φ′) = M_{φ}(Im ψ′) whence the contention. Next, let τ_{•} : c_{L} ⇒ M_{•} be any cone; notice that

(14.2.13)  Im τ_{i} ⊂ M_{i}^{△}  for every i ∈ Ob(I).

On the other hand, a direct inspection of the definitions shows that

(M_{•}/M_{•}^{△})_{i}^{△} = 0 for every i ∈ Ob(I).

Hence, (14.2.13) implies that every cone c_{L′} ⇒ M_{•}/M_{•}^{△} is the zero morphism, so we must have lim_{I} M_{•}/M_{•}^{△} = 0, and since lim_{I} is a left exact functor (because it is a right adjoint : see (1.3)) we conclude that the inclusion M_{•}^{△} → M_{•} induces an isomorphism :

lim_{I} M_{•}^{△} →^{∼} lim_{I} M_{•}.

(iii) In the situation of (ii), let A be any (V, 𝔪)^{a}-algebra, and take 𝒜 := A-Mod (see remark 14.2.10(iii)); notice then that M_{•} has the almost Mittag-Leffler property if and only if the following holds. For every i ∈ Ob(I), the induced system M_{•/i} := (Im M_{φ} | φ : j → i) of submodules of M_{i} is a Cauchy net of the uniform space ℐ_{A}(M_{i}) defined as in [75, Def.2.3.1]. In this case, by [75, Lemma 2.3.5], the Cauchy sequence M_{•/i} admits a unique limit in ℐ_{A}(M_{i}), and by direct inspection we see that this limit is M_{i}^{△}, since for every subideal 𝔪_{0} ⊂ 𝔪 of finite type there exists a morphism φ : j → i in I such that 𝔪_{0} · Im M_{φ} ⊂ M_{i}^{△}.

(iv) Notice that M_{•} : I → 𝒜 is a Cauchy functor if and only if the following holds. For every subideal 𝔪_{0} ⊂ 𝔪 of finite type there exists i ∈ Ob(I) such that for every pair of morphisms k ⟶^{ψ} j ⟶^{φ} i of I, we have 𝔪_{0} · Ker M_{ψ} = 𝔪_{0} · Coker M_{ψ} = 0. Indeed, the latter condition clearly implies that M_{•} is a Cauchy functor; conversely, suppose that M_{•} is a Cauchy functor, and let 𝔪_{0} ⊂ 𝔪_{1} ⊂ 𝔪 be subideals of finite type with 𝔪_{0} ⊂ 𝔪_{1}^{2}; by assumption, we may find i ∈ Ob(I) such that, for every φ and ψ as in the foregoing, 𝔪_{1} annihilates all but the first and the fourth terms in the induced exact sequence :

0 → Ker M_{ψ} → Ker M_{φ∘ψ} → Ker M_{φ} → Coker M_{ψ} → Coker M_{φ∘ψ} → Coker M_{φ} → 0.

It follows easily that 𝔪_{1} · Ker M_{ψ} = 𝔪_{0} · Coker M_{ψ}, as required. Notice also that every Cauchy functor has the almost Mittag-Leffler property : the details shall be left to the reader.

(v) Suppose again that I is cofiltered. In view of (iv), we deduce that a functor M_{•} : I → 𝒜 is null if and only if the following holds. For every subideal 𝔪_{0} ⊂ 𝔪 of finite type there exists i ∈ Ob(I) such that 𝔪_{0}M_{j} = 0 for every morphism φ : j → i : the details shall be left to the reader. It follows easily that the full subcategory of Fun(I, 𝒜) whose objects are the null functors, is a Serre subcategory : again, we leave the details to the reader.

Lemma 14.2.14. In the situation of definition 14.2.11, the following holds :

(i) The category Fun(I, 𝒜)_{a.ess.0} is a Serre subcategory of Fun(I, 𝒜). Hence

ℒ (I, 𝒜) := Fun(I, 𝒜)/Fun(I, 𝒜)_{a.ess.0}.

is an abelian category whose objects are the functors I → 𝒜.

(ii) If 𝒜 is 𝔪-torsion-free (see remark 14.2.10(iv)), then for every almost essentially zero functor M_{•} : I → A-Mod, we have lim_{I} M_{•} = 0.

(iii) Suppose that I is cofiltered, and 𝒜 is complete and 𝔪-torsion-free. Then the functor

Lim_{I} : Fun(I, 𝒜) → 𝒜

(see (1.3)) factors through the localization Fun(I, 𝒜) → ℒ (I, 𝒜) and a functor

ℒ (I, 𝒜) → 𝒜 M_{•} ↦ lim_{I} M_{•}.

Proof. (i): Consider a short exact sequence of functors from I to A-Mod :

(14.2.15)  0 → M′_{•} ⟶^{f_{•}} M_{•} ⟶^{g_{•}} M′′_{•} → 0.

It is easily seen that if M_{•} is almost essentially zero, the same holds for M′_{•} and M′′_{•}. Conversely, suppose that M′_{•} and M′′_{•} are almost essentially zero, and let i ∈ Ob(I), and 𝔪_{0} ⊂ 𝔪 a finitely generated subideal. By assumption there exists a morphism φ : j → i in I such that 𝔪_{0} · Im M′_{φ} = 0, and also a morphism ψ : k → j such that 𝔪_{0} · Im M′′_{ψ} = 0. It follows that there exists a morphism h : 𝔪_{0}M_{k} → M′_{j} such that f_{j} ∘ h : 𝔪_{0}M_{k} → M_{j} is the restriction of M_{ψ}. Then M_{φ∘ψ}(𝔪_{0}^{2}M_{k}) = 𝔪_{0} · Im(M_{φ} ∘ f_{j} ∘ h) = 𝔪_{0} · Im(f_{i} ∘ M′_{φ} ∘ h) = 0, which shows that M_{•} is almost essentially zero.

(ii): Suppose first that M_{•} is essentially zero. Let X ∈ Ob(𝒜) be any object, and τ_{•} : c_{X} ⇒ M_{•} any cone; by assumption, for evey i ∈ Ob(I) there exists a morphism φ : j → i in I with M_{φ} = 0. Then τ_{i} = M_{φ} ∘ τ_{j} = 0 for every such i, so that τ_{•} = 0, whence the contention.

For a general almost essentially zero functor M_{•}, and every subideal 𝔪_{0} ⊂ 𝔪 of finite type, we have a short exact sequence of functors

0 → M_{•}[𝔪_{0}] → M_{•} → M_{•}/M_{•}[𝔪_{0}] → 0

whose right-most term is essentially zero, by remark 14.2.12(i)), so that lim_{I} M_{•}/M_{•}[𝔪_{0}] = 0 by the previous case. On the other hand, clearly 𝔪_{0} annihilates lim_{I} M_{•}[𝔪_{0}]. The functor Lim_{I} is left exact (since it admits a left adjoint : see (1.3)); it then follows that 𝔪_{0} annihilates as well lim_{I} M_{•}, and since 𝔪_{0} is arbitrary and 𝒜 is 𝔪-torsion-free, the claim follows.

(iii): As Lim_{I} is left exact, in the situation of (14.2.15) we see easily from (ii) that if M′′_{•} is almost essentially zero, then f_{•} induces an isomorphism lim_{I} M′_{•} →^{∼} lim_{I} M_{•}. Thus, it remains only to check that if M′_{•} is almost essentially zero, then g_{•} induces an isomorphism g : lim_{I} M_{•} →^{∼} lim_{I} M′′_{•}. To this aim, we suppose first that M_{•} is the constant functor of value M, and that for every subideal 𝔪_{0} ⊂ 𝔪 of finite type there exists i ∈ Ob(I) such that 𝔪_{0} · M′_{i} = 0. Then, for every morphism φ : j → i in I we have as well 𝔪_{0} · M′_{j} = 0. Hence, for such 𝔪_{0} and i, and for every b ∈ 𝔪_{0} and every φ : j → i, the endomorphism b · 1_{M} : M → M is the composition of g_{j} : M → M′′_{j} and a unique morphism h_{j} : M′′_{j} → M of 𝒜, and it is easily seen that g_{j} ∘ h_{j} = b · 1_{M′′_{j}}. Moreover, the system of morphisms (g_{j} | φ : j → i) amounts to a natural cocone h_{•} : M′′_{•} ∘ s_{i} ⇒ M_{•} ∘ s_{i}, where s_{i} : I/i → I denotes the source functor. Since s_{i} is coinitial (example 1.5.9(i)), we deduce a morphism of 𝒜

h : L′′ := lim_{I} M′′_{•} → M = lim_{I} M_{•}.

By construction, we have g ∘ h = b · 1_{L′′} and h ∘ g = b · 1_{M}. Especially, b annihilates Ker g and Coker g, and since 𝔪_{0} and b are arbitrary, the assertion follows in this case.

Lastly, let M′_{•} be an arbitrary almost essentially zero functor, and notice that

N_{φ} := f_{i}(Im M′_{φ}) = M_{φ}(Ker g_{j}) for every morphism φ : j → i in I.

Hence, for every such φ, the morphism M′′_{φ} is the composition of unique morphisms of 𝒜

h_{φ} : M′′_{j} ⟶^{h_{φ}} Q_{φ} := M_{i}/N_{φ} ⟶^{\overline{g}_{φ}} M′′_{i} such that h_{φ} ∘ g_{j} = π_{φ} ∘ M_{φ} and \overline{g}_{φ} ∘ π_{φ} = g_{i}

where π_{φ} : M_{i} → Q_{i} is the projection. For any fixed i ∈ Ob(I), let c_{M_{i}} : I/i → 𝒜 be the constant functor of value M_{i}; we get a short exact sequence of functors

0 → N_{•} → c_{M_{i}} ⟶^{π_{•}} Q_{•} → 0

where N_{•}, Q_{•} : I/i → 𝒜 are given by the rules : φ ↦ N_{φ} and respectively φ ↦ Q_{φ} for every morphism φ : j → i in I. By construction, for every subideal 𝔪_{0} ⊂ 𝔪 of finite type there exists φ ∈ Ob(I/i) such that 𝔪_{0} · N_{φ} = 0; by the foregoing case, it follows that π_{•} induces an isomorphism

π_{i} : M_{i} →^{∼} Q := lim_{I/i} Q_{•}.

On the other hand, the system (h_{φ} | φ ∈ Ob(I/i)) amounts to a natural transformation h_{•} : M′′_{•} ∘ s_{i} ⇒ Q_{•} (the details are left to the reader), whence an induced morphism of 𝒜

h_{i} : L′′ := lim_{I} M′′_{•} →^{∼} lim_{I/i} M′′_{•} ∘ s_{i} → Q.

In turn, the system (π_{i}^{−1} ∘ h_{i} | i ∈ Ob(I)) amounts to a cone c_{L′′} ⇒ M_{•}, whence an induced morphism of 𝒜

h : L′′ → L := lim_{I} M_{•}.

Claim 14.2.16. g ∘ h = 1_{L′′}.

Proof of the claim. Indeed, let also τ_{•} : c_{L} ⇒ M_{•} and τ′′_{•} : c_{L′′} ⇒ M′′_{•} be the universal cones; it suffices to show that τ′′_{i} ∘ g ∘ h = τ′′_{i} for every i ∈ Ob(I). But we have

τ′′_{i} ∘ g ∘ h = τ′′_{i} = g_{i} ∘ τ_{i} ∘ h = g_{i} ∘ π_{i}^{−1} ∘ h_{i} = \overline{g}_{φ} ∘ π_{φ} ∘ π_{i}^{−1} ∘ h_{i}

for every morphism φ : j → i in I, and on the other hand τ′′_{i} = M′′_{φ} ∘ τ′′_{j} = \overline{g}_{φ} ∘ h_{φ} ∘ τ′′_{j} so we are reduced to checking that

π_{φ} ∘ π_{i}^{−1} ∘ h_{i} = h_{φ} ∘ τ′′_{j} for every morphism φ : j → i in I.

Now, let τ_{•}^{Q} : c_{Q} ⇒ Q_{•} be the universal cone; then π_{φ} ∘ π_{i}^{−1} = τ_{φ}^{Q} for every such φ, and finally, τ_{φ}^{Q} ∘ h_{i} = h_{φ} ∘ τ′′_{j}, as required. ♢

From claim 14.2.16 we see that g is an epimorphism; but it is also a monomorphism, since lim_{I} M′_{•} = 0, and since the functor lim_{I} is left exact. Hence g is an isomorphism, as stated. □

Remark 14.2.17. Keep the situation of definition 14.2.11, and suppose that I is cofiltered.

(i) Consider a short exact sequence (14.2.15) of functors I → 𝒜, such that M′_{•} and M′′_{•} are Cauchy functors. Then M_{•} is also a Cauchy functor. Indeed, let 𝔪_{0} ⊂ 𝔪_{1} ⊂ 𝔪 be two subideals of finite type with 𝔪_{0} ⊂ 𝔪_{1}^{2}. Since I is cofiltered, taking into account remark 14.2.12(iv), we find i ∈ Ob(I) such that 𝔪_{1} annihilates the kernel and cokernel of both M′_{φ} and M′′_{φ}, for every morphism φ : j → i; by the snake lemma, it follows easily that 𝔪_{0}·Ker M_{φ} = 𝔪_{0}·Coker M_{φ} = 0 for every such φ, whence the contention.

(ii) Moreover, if X_{•}, Y_{•} : I → 𝒜 are Cauchy functors, and f_{•} : X_{•} → Y_{•} is any morphism of functors, then Ker f_{•}, Coker f_{•} and Im f_{•} are Cauchy functors. Indeed, let us check first the assertion for Z_{•} := Im f_{•} : since the latter is a subfunctor of Y_{•}, for every subideal 𝔪_{0} ⊂ 𝔪 of finite type we find i ∈ Ob(I) such that for every pair of morphisms k ⟶^{ψ} j ⟶^{φ} i we have 𝔪_{0} · Ker Z_{ψ} = 0 (remark 14.2.12(iv)). Likewise, since Z_{•} is a quotient of X_{•}, for every such

𝔪_{0} there exists i′ ∈ Ob(I) such that for every pair of morphisms k′ ⟶^{ψ′} j′ ⟶^{φ′} i′ we have 𝔪_{0} · Coker Z_{ψ′} = 0. Since I is cofiltered, we find i′′ ∈ Ob(I) with morphisms i ← i′′ → i′, and then 𝔪_{0} annihilates the kernel of cokernel of Z_{λ}, for every morphism λ : j′′ → i′′ of I.

Lastly, by considering the short exact sequences of functors :

0 → Ker f_{•} → X_{•} → Z_{•} → 0 0 → Z_{•} → Y_{•} → Coker f_{•} → 0

and combining with (i), we deduce the assertions for Ker f_{•} and Coker f_{•}.

Proposition 14.2.18. With the notation of definition 14.2.11 and of remark 14.2.12(ii), assume that I is cofiltered, and 𝒜 is complete and 𝔪-torsion-free. Then we have :

(i) If M_{•} : I → 𝒜 has the almost Mittag-Leffler property, the following holds :

(a) M_{φ}^{△} : M_{j}^{△} →M_{i}^{△} is an epimorphism in 𝒜, for every morphism φ : j →i of I.

(b) The inclusions (M_{i}^{△} →M_{i} | i ∈ Ob(I)) define an isomorphism M_{•}^{△} →^{∼}M_{•} in ℒ (I, 𝒜). (ii) For every functor M_{•} : I → 𝒜 the following conditions are equivalent :

(a) M_{•} is almost essentially constant.

(b) M_{•} is isomorphic to a constant functor, in the category ℒ (I, 𝒜) of lemma 14.2.14(i).

(c) Every universal cone c_{L} ⇒ M_{•} is an isomorphism in the category ℒ (I, 𝒜).

(d) M_{•} has the almost Mittag-Leffler property, and M_{•}^{△} is a Cauchy functor. (iii) Every Cauchy functor M_{•} : I → 𝒜 is almost essentially constant.

Proof. (i.a): Pick a morphism k ⟶^{ψ} j such that 𝔪_{0} · Im M_{ψ} ⊂ M_{j}^{△} (remark 14.2.12(iii)). Then

𝔪_{0}M_{i}^{△} ⊂ 𝔪_{0} · Im M_{φ∘ψ} = M_{φ}(𝔪_{0} · Im M_{ψ}) ⊂ M_{φ}^{△}(M_{j}^{△}).

Since 𝔪_{0} is arbitrary, the assertion follows.

(i.b): For every i ∈ Ob(I), let β_{i} : M_{i}^{△} → M_{i} be the inclusion; we need to check that the resulting functor Coker β_{•} : I → 𝒜 is almost essentially zero. Indeed, let i ∈ Ob(I) and 𝔪_{0} ⊂ 𝔪 any subideal of finite type; since M_{•} has the almost Mittag-Leffler property, there exists a morphism φ : j → i in I such that 𝔪_{0} ·Im M_{φ} ⊂ M_{i}^{△}. Hence, 𝔪_{0} annihilates the image of the induced morphism Coker β_{j} → Coker β_{i}, whence the assertion.

(ii.a)⇒(ii.d): Let L ∈ Ob(𝒜) with a cone τ_{•} : c_{L} ⇒ M_{•} such that the functors Ker τ_{•} and Coker τ_{•} are almost essentially zero. This means that for every i ∈ Ob(I) and every subideal 𝔪_{0} ⊂ 𝔪 of finite type, there exists a morphism φ : j → i such that 𝔪_{0} · Im M_{φ} ⊂ Im τ_{i}. But clearly Im τ_{i} ⊂ M_{i}^{△}, so we see that M_{•} has the almost Mittag-Leffler property. By the same token, since 𝔪_{0}M_{i}^{△} ⊂ 𝔪_{0} · Im M_{φ}, we get 𝔪_{0}M_{i}^{△} ⊂ Im τ_{i}, and since 𝔪_{0} is arbitrary, we conclude that M_{i}^{△} = Im τ_{i} for every i ∈ Ob(I). Next, by assumption, for every i ∈ Ob(I) and every subideal 𝔪_{0} ⊂ 𝔪 of finite type there exists a morphism φ : j → i of I such that 𝔪_{0} annihilates the image of the induced morphism Ker τ_{j} → Ker τ_{i}; but obviously the latter is a monomorphism, so the condition means that for every such 𝔪_{0} there exists j ∈ Ob(I) such that 𝔪_{0} · Ker τ_{j} = 0. Now, let ψ : k → j be any morphism of I, and set K := τ_{k}^{−1}(Ker M_{ψ}^{△}); it is easily seen that K ⊂ Ker τ_{j}, hence 𝔪_{0}K = 0, and finally 𝔪_{0} · Ker M_{ψ}^{△} = 0, since we have already established that τ_{k} : L → M_{k}^{△} is an epimorphism.

(ii.d)⇒(ii.b): In light of (i.b), we are reduced to checking that the functor M_{•}^{△} is almost essentially constant. Thus, let M^{△} be the limit of the functor M_{•}^{△}, and τ_{•} : c_{M^{△}} ⇒ M_{•}^{△} the universal cone; let also 𝔪_{0} ⊂ 𝔪 be any subideal of finite type, and pick i ∈ Ob(I) such that 𝔪_{0}·Ker M_{φ}^{△} = 0 for every morphism φ : j → i of I. Recall that the source functor s_{i} : I/i → I is cofinal, so that M^{△} represents also the limit of M_{•}^{△} ∘ s_{i} : I/i → 𝒜. It follows easily that Ker(τ_{i} : M^{△} → M_{i}^{△}) represents the limit of the induced functor

(0 ×_{M_{i}^{△}} M_{•}^{△}) ∘ s_{i} : I → 𝒜 (j ⟶^{φ} i) ↦ Ker M_{φ}^{△} = 0 ×_{M_{i}^{△}} M_{j}^{△}.

But then it is clear that 𝔪_{0} · Ker τ_{i} = 0. Especially, the functor Ker τ_{•} : I → 𝒜 is trivially almost essentially zero. To conclude, we show that τ_{i} is an epimorphism for every i ∈ Ob(I). To this aim, let 𝔪_{0} ⊂ 𝔪 be any subideal of finite type; in view of (i.a), we may assume that 𝔪_{0} · Ker M_{φ}^{△} = 0 for every morphism φ : j → i. Hence, for every b ∈ 𝔪_{0} we have a unique morphism in 𝒜

f_{φ,b} : M_{i}^{△} → M_{j}^{△} such that M_{φ}^{△} ∘ f_{φ,b} = b · 1_{M_{i}^{△}}.

By inspecting the construction, it is easily seen that M_{ψ}^{△} ∘ f_{φ∘ψ,b} = f_{φ,b} for every morphism ψ : k → j in I, hence the rule φ ↦ f_{φ,b} defines a cone c_{M_{i}^{△}} ⇒ M_{•}^{△} ∘ s_{i}, whence a unique morphism of A-modules

h : M_{i}^{△} → M^{△} such that f_{φ,b} = τ_{j} ∘ h for every φ : j → i.

Especially, τ_{i} ∘ h = M_{φ}^{△} ∘ τ_{j} ∘ h = M_{φ}^{△} ∘ f_{φ,b} = b · 1_{M_{i}^{△}}, and since b ∈ 𝔪_{0} is arbitrary, the assertion follows.

(ii.b)⇒(ii.a): It suffices to prove :

Claim 14.2.19. Let M_{•}, M′_{•} : I → 𝒜 be two functors, f_{•} : M_{•} ⇒ M′_{•} a natural transformation, and suppose that f_{•} is an isomorphism in ℒ (I, 𝒜). Then M_{•} is almost essentially constant if and only if the same holds for M′_{•}.

Proof of the claim. Let L and L′ be the limits of M_{•} and respectively M′_{•}; denote by τ_{•} : c_{L} ⇒ M_{•} and τ′_{•} : c_{L′} ⇒ M′_{•} the respective universal cones. There exists a unique morphism f : L → L′ in 𝒜 that makes commute the diagram :

\xymatrix{c_{L} \ar[r]^{c_{f}} \ar[d]_{τ_{•}} & c_{L′} \ar[d]^{τ′_{•}} \\ M_{•} \ar[r]^{f_{•}} & M′_{•}.}

But lemma 14.2.14(iii) implies that f is an isomorphism; as f_{•} is an isomorphism in ℒ (I, 𝒜), it follows that τ_{•} is an isomorphism in ℒ (I, 𝒜) if and only if the same holds for τ′_{•}. ♢

Lastly, clearly (c)⇒(a). Suppose then that (a) holds, and let τ_{•} : c_{L} ⇒ M_{•} be a universal cone; let also μ_{•} : c_{X} ⇒ M_{•} be another cone that is an isomorphism in ℒ (I, 𝒜). Then there exists a unique morphism f : X → L in 𝒜 such that τ_{•} ⊙ c_{f} = μ_{•}. Set μ := lim_{I} μ_{•} and τ := lim_{I} τ_{•}, so that τ ∘f = μ. Now, τ is an isomorphism, and the same holds for μ_{•}, by lemma 14.2.14(iii); hence f is an isomorphism, so the cone μ_{•} is universal, whence (c).

(iii): We know already that M_{•} has the almost Mittag-Leffler property (remark 14.2.12(iv)); moreover, Ker M_{φ}^{△} ⊂ Ker M_{φ} for every morphism φ of I, hence M_{•} fulfills condition (ii.c), whence the contention. □

Proposition 14.2.20. Let 𝒜 and 𝒜′ be V -linear abelian categories, I a small category, F : 𝒜 → 𝒜′ a V -linear functor, and M_{•}, M′_{•} : I → 𝒜 two functors. The following holds :

(i) Let f_{•} : M_{•} ⇒ M′_{•} be a natural transformation. If Ker f_{•} and Coker f_{•} are almost essentially zero (resp. null, resp. dual almost essentially zero, resp. dual null), then the same holds for the kernel and cokernel of F f_{•} : F M_{•} ⇒ F M′_{•}.

(ii) If M_{•} is a Cauchy functor, the same holds for F M_{•}.

(iii) If M_{•} is almost essentially constant (resp. dual almost essentially constant), so is F M_{•}. (iv) Suppose that F is right exact (resp. left exact). If M_{•} has the almost Mittag-Leffler

property (resp. the dual almost Mittag-Leffler property), then the same holds for F M_{•}.

Proof. (ii): The assertion follows immediately from remark 14.2.10(vi).

(i): We can write f_{•} = e_{•} ∘ p_{•}, where e_{•} : I → 𝒜 is a monomorphism of functors, and p_{•} : I → 𝒜 is an epimorphism of functors, and then clearly it suffices to check the stated assertions for e_{•} and p_{•}. Moreover, if the stated assertions are known for all monomorphisms of functors and all V -linear categories 𝒜, they follow for every epimorphism as well, by considering the opposite transformation f_{•}^{o} : M_{•}^{′o} ⇒ M_{•}^{o} and invoking remark 14.2.10(ii).

Thus, let f_{•} be a monomorphism, φ : j → i any morphism in I, and a ∈ V that annihilates the morphism Coker f_{j} → Coker f_{i} induced by M′_{φ}. The latter means that Im(aM′_{φ}) ⊂ Im f_{i}; it follows that there exists a morphism g : M′_{j} → M_{i} in 𝒜 such that f_{i} ∘g = aM′_{φ}, and therefore

f_{i} ∘ g ∘ f_{j} = aM′_{φ} ∘ f_{j} = af_{i} ∘ M_{φ} = f_{i} ∘ (aM_{φ})

whence g ∘ f_{j} = aM_{φ}, since f_{i} is a monomorphism. The induced identities F f_{i} ∘ F g = aF M′_{φ} and F g ∘ F f_{j} = aF M_{φ} imply that Im(aF M′_{φ}) ⊂ Im F f_{i} and aM_{φ}(Ker F f_{j}) = 0. Hence, if Ker f_{•} and Coker f_{•} are almost essentially zero (resp. dual almost essentially zero) then the same holds for Ker F f_{•} and Coker F f_{•}. Combining with (ii), we deduce that if Ker f_{•} and Coker f_{•} are null (resp. dual null), the same follows for Ker F f_{•} and Coker F f_{•}.

(iii) is an immediate consequence of (i).

(iv): As usual, the assertion in case F is left exact will follow from the assertion for the case F is right exact, by considering M_{•}^{o} and F^{o}. Hence, suppose that F is right exact and M_{•} has the almost Mittag-Leffler property. For a given subideal 𝔪_{0} ⊂ 𝔪 of finite type and i ∈ Ob(I), choose a morphism φ : j → i in I such that for every morphism ψ : k → j of I we have 𝔪_{0} · Im M_{φ} ⊂ Im M_{φ∘ψ}. Let C be the cokernel of the morphism \overline{M}_{φ∘ψ} : M_{k} → Im M_{φ} of 𝒜 induced by M_{φ∘ψ}; the assumption means that 𝔪_{0}C = 0. Since F is right exact, F C is the cokernel of F \overline{M}_{φ∘ψ} : F M_{k} → F (Im M_{φ}), and obviously 𝔪_{0}F C = 0. By the same token, M_{φ} : M_{j} → M_{i} factors through epimorphisms F M_{j} → F (Im M_{φ}) → Im F M_{φ}; we conclude that the cokernel of the morphism F M_{k} → Im F M_{φ} induced by F M_{φ∘ψ} is a quotient of F C, and especially is annihilated by 𝔪_{0}. The assertion follows. □

Remark 14.2.21. Keep the notation of definition 14.2.11, and let M_{•} : I → 𝒜 be any functor.

(i) By unwinding the definitions, we find that M_{•} has the dual almost Mittag-Leffler property if and only if the following holds. For every subideal 𝔪_{0} ⊂ 𝔪 of finite type, and every i ∈ Ob(I) there exists a morphism φ : i → j in I such that

𝔪_{0} · Ker M_{ψ∘φ} ⊂ Ker M_{φ} for every morphism ψ : j → k in I.

(ii) The natural isomorphism Fun(I, 𝒜) →^{∼} Fun(I^{o}, 𝒜^{o})^{o} of remark 1.1.19(i) identifies (Fun(I^{o}, 𝒜^{o})_{a.ess.0})^{o} with the full subcategory of Fun(I, 𝒜) whose objects are the dual almost essentially zero functors. Since the opposite of a Serre subcategory is a Serre subcategory, lemma 14.2.14(i) implies that the quotient

𝒞 (I, 𝒜) := Fun(I, 𝒜)/(Fun(I^{o}, 𝒜^{o})_{a.ess.0})^{o}

is again an abelian category, whose objects are the functors I → 𝒜.

(iii) The rest of lemma 14.2.14 dualizes as well : first, part (ii) of the lemma implies that if 𝒜 is 𝔪-torsion-free and M_{•} is dual almost essentially zero, then colim_{I} M_{•} = 0. Next, if I is filtered, and 𝒜 is cocomplete and 𝔪-torsion-free, the functor Colim_{I} of remark 1.3.3(ii) factors through the localization Fun(I, 𝒜) → 𝒞 (I, 𝒜) and a functor

𝒞 (I, 𝒜) → 𝒜 M_{•} ↦ colim_{I} M_{•}.

(iv) Also remark 14.2.12(ii,iii) admits a valid dual : namely, suppose that I is filtered and 𝒜 is cocomplete. We set

M_{i}^{▽} := ⋃_{φ:i→j} Ker M_{φ} ⊂ M_{i} for every i ∈ Ob(I).

Then, arguing as in 14.2.12(ii) one easily checks that M_{φ} restricts to a morphism

M_{ψ}^{▽} : M_{i}^{▽} → M_{i′}^{▽} for every morphism ψ : i → i′ in I

whence an induced functor M_{•}^{▽} : I → 𝒜, and a direct inspection of the definitions shows that

(14.2.22)  (M_{•}^{▽})_{•}^{▽} = M_{•}^{▽}.

Notice moreover that every cocone τ : M_{•} ⇒ c_{L} factors (uniquely) through the projection π_{•} : M_{•} → M_{•}/M_{•}^{▽}. Combining with (14.2.22) we deduce that

colim_{I} M_{•}^{▽} = 0

and furthermore, π_{•} induces an isomorphism in 𝒜 :

colim_{I} M_{•} →^{∼} colim_{I} M_{•}/M_{•}^{▽}.

(v) Under the assumptions of (iv), we deduce that M_{•} has the dual almost Mittag-Leffler property if and only if for every subideal 𝔪_{0} ⊂ 𝔪 of finite type and every i ∈ Ob(I) there exists a morphism φ : i → j such that 𝔪_{0} · M_{i}^{▽} ⊂ Ker M_{φ}. In the case where 𝒜 = A-Mod for a given (V, 𝔪)^{a}-algebra A, it follows that for every i ∈ Ob(I) the system (Ker M_{φ} | φ : i → j) is a Cauchy net in the uniform space ℐ_{A}(M_{i}), and its unique limit is the A-submodule M_{i}^{▽}.

(vi) Suppose that I is filtered, and is 𝒜 is cocomplete and 𝔪-torsion-free. Then :

• By dualizing proposition 14.2.18(ii), we see that the following conditions are equivalent : (a) M_{•} is dual almost essentially constant.

(b) M_{•} is isomorphic to a constant functor, in the category 𝒞 (I, 𝒜) of remark 14.2.21(ii).

(c) M_{•} has the dual almost Mittag-Leffler property, and for every subideal 𝔪_{0} ⊂ 𝔪 of finite type, there exists i ∈ Ob(I) such that 𝔪_{0} annihilates the cokernel of the morphism

(14.2.23)  M_{i}/M_{i}^{▽} → M_{j}/M_{j}^{▽}

induced by M_{φ}, for every morphism φ : i → j in I (notation of remark 14.2.21(iv)).

• Moreover, the dual of proposition 14.2.18(i) states that if M_{•} has the dual almost Mittag-Leffler property, then (14.2.23) is a monomorphism for every morphism φ : i → j in I, and the projections (M_{i} → M_{i}/M_{i}^{▽} | i ∈ Ob(I)) define an isomorphism M_{•} →^{∼} M_{•}/M_{•}^{▽} in 𝒞 (I, 𝒜).

Lemma 14.2.24. In the situation of definition 14.2.11, let A be any (V, 𝔪)^{a}-algebra, M_{•} : ℕ^{o} → A-Mod a functor whose transition morphisms M_{m,n} : M_{m} → M_{n} are epimorphisms for every m, n ∈ ℕ, and set L := lim_{n∈ℕ} M_{n}. Then we have :

(i) The universal cone τ_{•} : c_{L} ⇒ M_{•} is an epimorphism of functors.

(ii) lim_{n∈ℕ}^{1} M_{n} = 0.

(iii) For every almost finitely generated (resp. almost finite projective) A-module N, the cone N ⊗_{A} τ_{•} induces an epimorphism (resp. an isomorphism)

φ_{N} : N ⊗_{A} L → lim_{n∈ℕ} N ⊗_{A} M_{n}.

(iv) More generally, if N is an A-module and a ∈ A_{∗} such that a · 1_{N} factors as a composition of A-linear morphisms N ⟶^{α} N_{0} ⟶^{β} N with N_{0} almost finitely generated (resp. almost finite projective), then a · Coker φ_{N} = 0 (resp. a · Ker φ_{N} = 0).

Proof. (i): Let M_{•!} : ℕ^{o} → A_{∗}-Mod be the composition of M_{•} with the left adjoint (−)_{!} : A-Mod → A_{∗}-Mod of the localization (−)^{a}. Set T := lim_{n∈ℕ} M_{n!}, and let μ_{•} : c_{T} ⇒ M_{•!} be the universal cone; then L = T^{a} and the natural isomorphism M_{•} →^{∼} (M_{•!})^{a} identifies τ_{•} with μ_{•}^{a}. Thus, it suffices to check that μ_{•} is an epimorphism of functors. However, the transition morphisms (M_{m,n})_{!} : M_{m!} → M_{n!} are surjective, so the assertion is clear.

(ii): Recall that for every functor M_{•} : ℕ^{o} → A-Mod (resp. N_{•} : ℕ^{o} → A_{∗}-Mod) the right derived functor R lim_{ℕ^{o}} M_{•} (resp. R lim_{ℕ^{o}} N_{•}) is computed by a natural two-term complex Π(M_{•}) (resp. Π(N_{•})) : see [163, §3.5]. With this notation, we have Π(M_{•}) = Π(M_{•!})^{a}, since the localization (−)^{a} commutes with all products. However, lim_{n∈ℕ}^{1} M_{n!} = 0, since the transition morphisms (M_{m,n})_{!} are surjective ([163, Lemma 3.5.3]). The assertion follows.

(iii): Suppose first that N is finitely generated; hence there exists r ∈ ℕ and a short exact sequence 0 → K → A^{⊕r} → N → 0 of A-modules. For every n ∈ ℕ, let C_{n} ⊂ A^{⊕r} ⊗_{A} M_{n} be the image of K ⊗_{A} M_{n}; we get an inverse system (C_{n} | n ∈ ℕ) whose transition morphisms are also epimorphisms (details left to the reader). By virtue of (ii), there follows an epimorphism of A-modules

lim_{n∈ℕ} A^{⊕r} ⊗_{A} M_{n} = A^{⊕r} ⊗_{A} L ⟶^{ψ} lim_{n∈ℕ} N ⊗_{A} M_{n}.

Lastly, it is easily seen that ψ factors through φ_{N} and the induced morphism A^{⊕r} ⊗_{A} L → N ⊗_{A} L, whence the contention in this case. Next, consider the functor

T : A_{∗}-Mod → A_{∗}-Mod X ↦ (Coker φ_{X^{a}})_{∗}

that assigns to every A-linear morphism f : N → N′ the induced map (Coker φ_{N^{a}})_{∗} → (Coker φ_{N^{′a}})_{∗} of A_{∗}-modules. It is easily seen that T is V -homogeneous of degree 1 (see definition 14.1.53), and the associated functor T^{a} : A-Mod → A-Mod is given by the rule: N ↦ Coker φ_{N} for every A-module N (notation of remark 14.1.54(ii)). By the foregoing, T^{a}N = 0 whenever N is finitely generated; however, on the one hand, for every suitable pair of infinite cardinalities ω, ω′ the associated map ℳ_{ω,ω′}(T^{a}) : ℳ_{ω}(A) → ℳ_{ω′}(A) is uniformly continuous (proposition 14.1.56(iii)), and on the other hand, it is easily seen that the subset {0} is closed in the topology of the uniform space ℳ (A). Hence, T^{a}N = 0 for every almost finitely generated A-module, as stated.

Next, suppose that N is almost finite projective; then for every a ∈ 𝔪 there exists r ∈ ℕ and A-linear morphisms N ⟶^{α} A^{⊕r} ⟶^{β} N whose composition is a · 1_{N} ([75, Lemma 2.4.15]). There follows a commutative diagram :

\xymatrix{N ⊗_{A} L \ar[r]^{α⊗_{A}L} \ar[d]_{φ_{N}} & A^{⊕r} ⊗_{A} L \ar[r]^{β⊗_{A}L} \ar[d] & N ⊗_{A} L \ar[d]^{φ_{N}} \\ lim_{n∈ℕ} N ⊗_{A} M_{n} \ar[r] & lim_{n∈ℕ} A^{⊕r} ⊗_{A} M_{n} \ar[r] & lim_{n∈ℕ} N ⊗_{A} M_{n}}

whose central arrow is an isomorphism, and we already know that φ_{N} is an epimorphism. By a simple diagram chase we deduce that a · Ker φ_{N} = 0, and since a is arbitrary, we conclude that φ_{N} is an isomorphism in this case, as stated.

(iv): In light of (iii) we have : a · 1_{T N} = T β ∘ T α = 0, i.e. a · Coker φ_{N} = 0. If N_{0} is almost finite projective, consider the functor T′ : A^{a}-Mod → A^{a}-Mod such that T′X := Ker φ_{X} for every A^{a}-module X : by (iii) we have T′N_{0} = 0, whence a · 1_{T′N} = 0. □

Proposition 14.2.25. In the situation of definition 14.2.11, let A be any (V, 𝔪)^{a}-algebra, M_{•} : I → A-Mod any functor, and τ_{•} : c_{L} ⇒ M_{•} and τ′_{•} : M_{•} ⇒ c_{L′} be respectively a universal cone and a universal cocone. We have :

(i) Suppose that I is cofiltered and consider the following conditions :

(a) The functor Coker τ_{•} is almost essentially zero.

(b) M_{•} has the almost Mittag-Leffler property. Then (a)⇒(b), and if there exists a coinitial functor ℕ^{o} → I, then (b)⇒(a).

(ii) Suppose that is I filtered. Then M_{•} has the dual almost Mittag-Leffler property if and only if the functor Ker τ′_{•} is dual almost essentially zero.

(iii) Suppose that I = ℕ^{o}, where ℕ is endowed with its standard total ordering. If M_{•} has the almost Mittag-Leffler property, then lim_{n∈ℕ}^{1} M_{n} = 0.

Proof. (iii): Suppose first that M_{•} is almost essentially zero, and consider, for every finitely generated subideal 𝔪_{0} ⊂ 𝔪, the short exact sequence of functors

Σ_{•}  :  0 → M_{•}[𝔪_{0}] → M_{•} → M_{•}/M_{•}[𝔪_{0}] → 0

whose right-most term is an essentially zero inverse system, by remark 14.2.12(i)), and therefore

lim_{n∈ℕ} M_{n}/M_{n}[𝔪_{0}] = lim_{n∈ℕ}^{1}M_{n}/M_{n}[𝔪_{0}] = 0.

On the other hand, clearly 𝔪_{0} annihilates lim_{n∈ℕ}^{1} M_{n}[𝔪_{0}]. From the long exact cohomology sequence associated with Σ_{•} it follows that 𝔪_{0} annihilates as well lim_{n∈ℕ}^{1} M_{n}. Since 𝔪_{0} is arbitrary, the claim follows in this case.

Next, let M_{•} be an arbitrary functor with the almost Mittag-Leffler property; for every m, n ∈ ℕ with m > n, we let M_{m,n} : M_{m} → M_{n} be the transition morphism. By proposition 14.2.18(i.a), the restriction M_{p,n}^{△} : M_{m}^{△} → M_{n}^{△} of M_{m,n} is an epimorphism for every such n and p. Let us then consider the short exact sequence of inverse systems

Σ′_{•}  :  0 → M_{•}^{△} → M_{•} → M_{•}/M_{•}^{△} → 0.

Proposition 14.2.18(i.b) implies that M_{•}/M_{•}^{△} is almost essentially zero, so lim_{n∈ℕ}^{1} M_{n}/M_{n}^{△} = 0, by the previous case; also, lim_{n∈ℕ}^{1} M_{n}^{△} = 0, by lemma 14.2.24(ii). The assertion follows then, by considering the long exact R lim sequence associated with Σ′_{•}.

(i.a)⇒(i.b): By assumption, for every subideal 𝔪_{0} ⊂ 𝔪 of finite type and every i ∈ Ob(I) there exists a morphism φ : j → i in I such that 𝔪_{0} annihilates the image of the induced morphism Coker τ_{j} → Coker τ_{i}. The latter means that 𝔪_{0} · Im M_{φ} ⊂ Im τ_{i}; but clearly Im τ_{i} ⊂ Im M_{φ∘ψ} for every morphism ψ : k → j in I, whence the contention.

Next, suppose that there exists a coinitial functor λ : ℕ^{o} → I, and that M_{•} has the almost Mittag-Leffler property; notice that τ_{i} factors through a morphism μ_{i} : L → M_{i}^{△} and the inclusion M_{i}^{△} → M_{i}, for every i ∈ Ob(I). There follows a short exact sequence of functors :

Coker μ_{•} → Coker τ_{•} → M_{•}/M_{•}^{△} → 0.

By proposition 14.2.18(i.b), M_{•}/M_{•}^{△} is almost essentially zero, hence L also represents the limit of M_{•}^{△}, and μ_{•} is a universal cone; by lemma 14.2.14(i) it then suffices to show that Coker μ_{•} = 0. Now, μ_{•} ∗ λ : c_{L} ⇒ M_{•}^{△} ∘ λ is still a universal cone (remark 1.5.5(ii,iii)). Since we also know that M_{φ}^{△} is an epimorphism for every morphism φ of I (proposition 14.2.18(i.a)), we are then reduced to checking that μ_{λ(n)} : L → M_{λ(n)}^{△} is an epimorphism for every n ∈ ℕ. The latter follows from lemma 14.2.24(i).

(ii): Suppose that Ker τ′_{•} is dual almost essentially zero; this means that for every subideal 𝔪_{0} ⊂ 𝔪 of finite type, and every i ∈ Ob(I) there exists a morphism φ : i → j in I such that M_{φ}(𝔪_{0} · Ker τ′_{i}) = 0. But we have Ker M_{ψ∘φ} ⊂ Ker τ′_{i} for every morphism ψ : j → k in I; whence 𝔪_{0} · Ker M_{ψ∘φ} ⊂ Ker M_{φ} for every such ψ. Hence M_{•} has the dual Mittag-Leffler property, by remark 14.2.21(i). Conversely, suppose that M_{•} has the dual Mittag-Leffler property; we know that the functor M_{•}^{▽} is almost essentially zero (remark 14.2.21(vi)) hence it suffices to check that M_{i}^{▽} = Ker τ′_{i} for every i ∈ Ob(I). After replacing I by i/I, we may assume that i is an initial object of I, whence an induced cone μ_{•} : c_{M_{i}} ⇒ M_{•}, and notice that M_{i}^{▽} = colim_{I} Ker μ_{•}. On the other hand, colim_{I} μ_{•} = τ′_{i}; since filtered colimits in A-Mod are exact, the assertion follows. □

14.2.26. Pontryagin duality. Recall that for every ring R and every R-module N, we have a natural R-module structure on

N^{∨} := Hom_{ℤ}(N, ℚ/ℤ).

Namely, for every a ∈ R and every ℤ-linear map φ : N → ℚ/ℤ we set a · φ := φ ∘ (a1_{N}). Especially, R^{∨} is naturally an R-module, and there follows a natural R-linear isomorphism :

Hom_{R}(N, R^{∨}) →^{∼} N^{∨}

assigning to every R-linear map φ : N → R^{∨} the ℤ-linear map φ′ : N → ℚ/ℤ such that φ′(n) := φ(n)(1) for every n ∈ N. Its inverse assigns to every ℤ-linear map ψ : N → ℚ/ℤ the homomorphism of R-modules ψ′ : N → R^{∨} that sends every n ∈ N to the ℤ-linear map ψ′(n) : R → ℚ/ℤ such that ψ′(n)(a) := ψ(an) for every a ∈ R. Since ℚ/ℤ is an injective ℤ-module, there results an exact functor

R-Mod → R-Mod^{o} N ↦ N^{∨}.

Moreover, it is easily seen that the natural biduality R-linear map is an injection :

N → N^{∨∨} for every R-module N.

Let now (V, 𝔪) be a basic setup, and A a (V, 𝔪)^{a}-algebra; we consider the associated A-module

J_{A} := ((A_{∗})^{∨})^{a}

and from the foregoing it is easily seen that the Pontryagin duality functor

(−)^{∨} : A-Mod → A-Mod^{o} M ↦ M^{∨} := alHom_{A}(M, J_{A})

is again exact and V -linear, where alHom_{A}(−, −) denotes the bifunctor of almost homomomorphisms defined as in [75, §2.2.11]; indeed, we have a natural isomorphism of A-modules

M^{∨} →^{∼} ((M_{∗})^{∨})^{a} for every A-module M

(details left to the reader). Moreover, it follows that the biduality morphism M → M^{∨∨} is a monomorphism of A-modules; especially, (−)^{∨} is a conservative functor.

Remark 14.2.27. (i) For every small category I, we get a natural extension of Pontryagin duality to functors I → A-Mod. Namely, we have a conservative exact functor :

Fun(I, A-Mod)^{o} → Fun(I^{o}, A-Mod) M_{•} ↦ M_{•}^{∨} := (−)^{∨} ∘ M_{•}.

(ii) For every A-module M, and every A-submodule N ⊂ M, let i_{N} : N → M the inclusion; it is easily seen that the induced map

ℐ_{A}(M) → ℐ_{A}(M^{∨}) N ↦ Ker(i_{N}^{∨} : M^{∨} → N^{∨})

is uniformly continuous, for the uniform structures defined as in [75, Def.2.3.1(i)].

(iii) Then, if I is cofiltered, and M_{•} : I → A-Mod is any functor with the almost Mittag-Leffler property, combining with remarks 14.2.12(iii) and 14.2.21(v) we deduce that the Alinear epimorphism i_{M_{i}^{△}}^{∨} : M_{i}^{∨} → (M_{i}^{△})^{∨} factors through an isomorphism of A-modules

M_{i}^{∨}/(M_{•}^{∨})_{i}^{▽} →^{∼} (M_{i}^{△})^{∨} for every i ∈ Ob(I).

Lemma 14.2.28. With the notation of remark 14.2.27, for every functor M_{•} : I → A-Mod the following holds :

(i) M_{•} is essentially zero (resp. almost essentially zero) if and only if M_{•}^{∨} is dual essentially zero (resp. dual almost essentially zero).

(ii) Suppose that I is cofiltered. Then M_{•} is almost essentially constant (resp. has the almost Mittag-Leffler property) if and only if M_{•}^{∨} is dual almost essentially constant (resp. has the dual almost Mittag-Leffler property).

Proof. (i): By direct inspection we see that if M_{•} is essentially zero (resp. almost essentially zero), then M^{∨} is dual essentially zero (resp. dual almost essentially zero). Likewise, if M_{•} is dual essentially zero (resp. dual almost essentially zero), then M^{∨} is essentially zero (resp. almost essentially zero). From this, the converse assertions follow easily, since the biduality morphism M_{•} → M_{•}^{∨∨} is a monomorphism.

(ii): For a given pair of morphisms k ⟶^{ψ} j ⟶^{φ} i of I, set N := ImM_{φ}/Im M_{φ∘ψ}. Since Pontryagin duality is exact and conservative, for every subideal 𝔪_{0} ⊂ 𝔪 we have :

𝔪_{0}N = 0 ⇔ 𝔪_{0}N^{∨} = 0 ⇔ 𝔪_{0} · Ker(Im M_{φ∘ψ}^{∨} → M_{φ}^{∨}) = 0 ⇔ 𝔪_{0}Ker M_{φ∘ψ}^{∨} ⊂ KerM_{φ}^{∨}.

Taking into account remark 14.2.21(i), we deduce that M_{•} has the almost Mittag-Leffler property if and only if M_{•}^{∨} has the dual almost Mittag-Leffler property.

Next, proposition 14.2.20(ii) implies that if M_{•} is almost essentially constant, then M_{•}^{∨} is dual almost essentially constant. Conversely, if M_{•}^{∨} is dual almost essentially constant, then M_{•}^{∨} has the dual almost Mittag-Leffler property, and for every subideal 𝔪_{0} ⊂ 𝔪 of finite type there exists i ∈ Ob(I) with

(14.2.29) 𝔪_{0} · Coker(M_{i}^{∨}/(M_{•}^{∨})_{i}^{▽} → M_{j}^{∨}/(M_{•}^{∨})_{j}^{▽}) = 0  for every morphism i ⟶^{φ} j

(remark 14.2.21(vi)). By the foregoing, we deduce already that M_{•} has the almost Mittag-Leffler property. Moreover, by remark 14.2.27(iii), condition (14.2.29) is equivalent to :

𝔪_{0} · Coker((M_{i}^{△})^{∨} → (M_{j}^{△})^{∨}) = 0 for every morphism i ⟶^{φ} j.

But Coker((M_{i}^{△})^{∨} → (M_{j}^{△})^{∨}) = (Ker M_{φ}^{△})^{∨}, and in light of proposition 14.2.18(ii), we deduce that M_{•} is almost essentially constant. □

Proposition 14.2.30. Let I be a small cofiltered category, and

0 → M′_{•} → M_{•} → M′′_{•} → 0

a short exact sequence of functors I → A-Mod. We have:

(i) If M′_{•} and M′′_{•} are almost essentially constant (resp. have the almost Mittag-Leffler property), then the same holds for M_{•}.

(ii) If M_{•} has the almost Mittag-Leffler property, the same holds for M′′_{•}.

(iii) If M_{•} has the almost Mittag-Leffler property and M′′_{•} is almost essentially constant, then M′_{•} has the almost Mittag-Leffler property.

(iv) If f_{•} : X_{•} → Y_{•} is a natural transformation between almost essentially constant functors, then Ker f_{•} and Coker f_{•} are almost essentially constant.

Proof. (ii) is clear from the definitions.

(iv): By proposition 14.2.18(ii), the universal cones τ_{•}^{X} : c_{X} ⇒ X_{•} and τ_{•}^{Y} : c_{X} ⇒ Y_{•} are isomorphisms in ℒ (I, A-Mod). Thus, there exists a unique morphism f : X → Y of A-modules that makes commute the diagram

\xymatrix{c_{X} \ar[r]^{c_{f}} \ar[d]_{τ_{•}^{X}} & c_{Y} \ar[d]^{τ_{•}^{Y}} \\ X_{•} \ar[r]^{f_{•}} & Y_{•}}

whose vertical arrows are isomorphism in ℒ (I, A-Mod). Set K := Ker f and C := Coker f; there follow induced isomorphisms c_{K} →^{∼} Ker f_{•} and c_{C} →^{∼} Coker f_{•} in ℒ (I, A-Mod), whence the contention, in light of proposition 14.2.18(ii).

(i): Let L, L′ and L′′ be the colimits of the functors M_{•}^{∨}, M_{•}^{′∨} and respectively M_{•}^{′′∨}; since filtered colimits are exact in the category A-Mod, we get a commutative diagram of functors with exact rows :

\xymatrix{0 \ar[r] & M_{•}^{′′∨} \ar[r] \ar[d] & M_{•}^{∨} \ar[r] \ar[d] & M_{•}^{′∨} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & c_{L′′} \ar[r] & c_{L} \ar[r] & c_{L′} \ar[r] & 0}

whose vertical arrows are the universal cocones. Consider first the case where M′_{•} and M′′_{•} are almost essentially constant; then M_{•}^{′∨} and M_{•}^{′′∨} are dual almost essentially constant (lemma 14.2.28(ii)), and arguing as in the proof of (iv) we deduce that the first and third vertical arrows are isomorphisms in the quotient category 𝒞 (I, A-Mod), hence the same holds for the middle one, by the 5-lemma. Then M_{•}^{•} is dual almost essentially constant, by remark 14.2.21(vi), hence M_{•} is almost essentially constant, again by lemma 14.2.28(ii).

Next, if M′_{•} and M′′_{•} have the almost Mittag-Leffler property, then M_{•}^{′∨} and M_{•}^{′′∨} have the dual almost Mittag-Leffler property (lemma 14.2.28(ii)), hence the kernels of the first and third vertical arrows are dual almost essentially zero (proposition 14.2.25(ii)). Then the same holds for the kernel of the middle vertical arrow, so M_{•}^{∨} has the dual almost Mittag-Leffler property, again by proposition 14.2.25(ii); finally, M_{•} has the almost Mittag-Leffler property, by lemma 14.2.28(ii).

(iii): We consider again the induced commutative ladder with exact rows. Arguing as in the proof of (ii), we see that the kernel of the middle vertical arrow and the cokernel of the left-most vertical arrow are both dual almost essentially zero, hence the same holds for the kernel of the right-most vertical arrow, by the snake lemma. Then M_{•}^{′∨} has the dual almost Mittag-Leffler property, by proposition 14.2.25(ii), and finally, M′_{•} has the almost Mittag-Leffler property, by lemma 14.2.28(ii). □

14.2.31. Let us now consider a basic setup (V, 𝔪), a (small) cofiltered category I, and a functor

I → (V, 𝔪)^{a}-Alg.Mod i ↦ (A_{i}, M_{i})

where (V, 𝔪)^{a}-Alg.Mod is defined as in [75, Def.2.5.22(ii)]. Hence, for every i ∈ Ob(I), the pair (A_{i}, M_{i}) consists of a (V, 𝔪)^{a}-algebra A_{i}, and an A_{i}-module M_{i}. To each morphism φ : j → i of I, the functor assigns a pair (A_{φ}, g_{φ}) : (A_{j}, M_{j}) → (A_{i}, M_{i}), where A_{φ} : A_{j} → A_{i} is a morphism of (V, 𝔪)^{a}-algebras, and g_{φ} : A_{i} ⊗_{A_{j}} M_{j} → M_{i} is an A_{i}-linear morphism. We get two obvious induced functors

A_{•} : I → (V, 𝔪)^{a}-Alg M_{•} : I → (V, 𝔪)^{a}-Mod.

Namely, for every morphism φ as in the foregoing we let M_{φ} : M_{j} → M_{i} be the composition of g_{φ} with the natural A_{j}-linear morphism M_{j} → A_{i} ⊗_{A_{j}} M_{j}. Set also

A := lim_{I} A_{•} and M := lim_{I} M_{•}.

Notice that A_{•} (resp. M_{•}) can also be viewed as a functor with values in A-Alg (resp. A-Mod).

Proposition 14.2.32. With the notation of (14.2.31), the following holds :

(i) If A_{•} : I → A-Mod has the almost Mittag-Leffler property, and g_{φ} is an epimorphism for every morphism φ of I, then M_{•} : I → A-Mod has the almost Mittag-Leffler property.

(ii) Suppose that g_{φ} is an isomorphism for every morphism φ of I. Then, if A_{•} : I → A-Mod is almost essentially constant (resp. is a Cauchy functor), the same holds for M_{•}.

(iii) Moreover, under the assumptions of (i) (resp. (ii)), the natural morphism

α_{i} : A_{i} ⊗_{A} M_{i}^{△} → M_{i} (resp. β_{i} : A_{i} ⊗_{A} M → M_{i} )

is an epimorphism (resp. an isomorphism) for every i ∈ Ob(I).

(iv) Lastly, under the assumptions of (i), if there exists a coinitial functor ℕ^{o} → I, then β_{i} is an epimorphism for every i ∈ Ob(I).

Proof. We fix universal cones τ_{•}^{A} : c_{A} ⇒ A_{•} and τ_{•}^{M} : c_{M} ⇒ M_{•}.

Claim 14.2.33. Assertions (ii) and (iii) hold if A_{•} is a Cauchy functor and g_{φ} is an isomorphism for every morphism φ of I.

Proof of the claim. Indeed, in this case for every subideal 𝔪_{0} ⊂ 𝔪 of finite type we find i ∈ Ob(I) such that for every morphism φ : j → i we have 𝔪_{0} · Ker A_{φ} = 𝔪_{0} · Coker A_{φ} = 0. By virtue of remark 14.2.10(vi), it follows that 𝔪_{0}^{2} annihilates the kernel and cokernel of A_{φ}⊗_{A_{j}} M_{j}, for every such φ. Since g_{φ} is an isomorphism, it follows that 𝔪_{0}^{2}·Ker M_{φ} = 𝔪_{0}^{2}·Coker M_{φ} = 0, for every such φ. This shows that M_{•} is a Cauchy functor.

Next, notice that Ker τ_{•}^{A} and Coker τ_{•}^{A} are also Cauchy functors (remark 14.2.17(ii)), and moreover they are almost essentially zero, since A_{•} is almost essentially constant (proposition 14.2.18(ii,iii)). Hence, Ker τ_{•}^{A} and Coker τ_{•}^{A} are null. The same argument shows that Ker τ_{•}^{M} and Coker τ_{•}^{M} are null, since M_{•} is a Cauchy functor. Moreover, the kernel and cokernel of τ_{•}^{A} ⊗_{A} M are null as well, since the functor − ⊗_{A} M : A-Mod → A-Mod is V -linear (proposition 14.2.20(i)). Since we have :

β_{i} ∘ (τ_{i}^{A} ⊗_{A} M) = τ_{i}^{M} for every i ∈ Ob(I)

and as the null functors form a full Serre subcategory of Fun(I, A-Mod) (remark 14.2.12(v)), it follows that also Ker β_{•} and Coker β_{•} are null functors. Now, let 𝔪_{0} ⊂ 𝔪 be a subideal of finite type, and i ∈ Ob(I); since I is cofiltered, we deduce that there exists a morphism φ : j → i such that 𝔪_{0} · Ker β_{j} = 𝔪_{0} · Coker β_{j} = 0. Since the functor A_{i} ⊗_{A_{j}} − : A_{j}-Mod → A_{i}-Mod is V -linear, it follows that 𝔪_{0}^{2} annihilates the kernel and cokernel of A_{i} ⊗_{A_{j}} β_{j} : A_{i} ⊗_{A} M → A_{i} ⊗_{A_{j}} M_{j} (remark 14.2.10(vi)). But notice that

g_{φ} ∘ (A_{i} ⊗_{A_{j}} β_{j}) = β_{i}.

So finally 𝔪_{0}^{2} · Ker β_{i} = 𝔪_{0}^{2} · Coker β_{i} = 0. Since 𝔪_{0} is arbitrary, we conclude that β_{i} is an isomorphism, as stated. ♢

We can now complete the proof of (ii) and (iii), for the case where A_{•} is almost essentially constant, and g_{φ} is still an isomorphism for every morphism φ of I. To this aim, set

\overline{A}_{φ} := Im A_{φ} and \overline{M}_{φ} := \overline{A}_{φ} ⊗_{A_{j}} M_{j} for every morphism φ : j → i of I.

Let j ⟶^{φ} i and j′ ⟶^{φ′} i′ be two object of Morph(I), and (ψ, ψ′) : φ′ → φ a morphism of Morph(I); by definition, ψ : j′ → j and ψ′ : i′ → i are morphisms in I such that ψ′∘φ′ = φ∘ψ, and then it is clear that A_{ψ′} restricts to a morphism of (V, 𝔪)^{a}-algebras

\overline{A}_{(ψ,ψ′)} : \overline{A}_{φ′} → \overline{A}_{φ}.

Then, M_{ψ} : M_{j′} → M_{j} induces a \overline{A}_{φ}-linear morphism

\overline{g}_{(ψ,ψ′)} : \overline{A}_{φ} ⊗_{\overline{A}_{φ′}} \overline{M}_{φ′} →^{∼} \overline{A}_{φ} ⊗_{A_{j′}} M_{j′} → \overline{M}_{φ}

and it is easily seen that the rules :

φ ↦ (\overline{A}_{φ}, \overline{M}_{φ}) and (φ′ ⟶^{(ψ,ψ′)} φ) ↦ (\overline{A}_{(ψ,ψ′)}, \overline{g}_{(ψ,ψ′)})

for every φ ∈ Ob(Morph(I)) and every morphism (ψ, ψ′) of Morph(I), define a functor

Morph(I) → (V, 𝔪)^{a}-Alg.Mod

whence, as in (14.2.31), two functors

\overline{A}_{•} : Morph(I) → A-Alg \overline{M}_{•} : Morph(I) → A-Mod

and notice also that :

(14.2.34)  \overline{A}_{φ}^{△} = A_{i}^{△}  for every morphism φ : j → i in I.

Next, let ℐ be the category whose objects are the pairs (j ⟶^{φ} i, 𝔪_{0}) where φ is a morphism of I and 𝔪_{0} ⊂ 𝔪 is a subideal of finite type such that :

𝔪_{0}\overline{A}_{φ} ⊂ \overline{A}_{φ}^{△}

(notation of remark 14.2.12(ii)). For every (φ, 𝔪_{0}), (φ′, 𝔪′_{0}) ∈ Ob(ℐ), the set of morphisms (φ′, 𝔪′_{0}) → (φ, 𝔪_{0}) in ℐ is empty if 𝔪_{0} ⊄ 𝔪′_{0}, and otherwise consists of the morphisms φ′ → φ in Morph(I). The composition law is then the same as that of Morph(I), so that the projection (φ, 𝔪_{0}) ↦ φ and the rule : i ↦ 1_{i} for every i ∈ Ob(I) define functors

ℐ ⟶^{π} Morph(I) ⟵^{δ} I.

Claim 14.2.35. The functors π and δ are coinitial.

Proof of the claim. It is easily seen that both ℐ and Morph(I) are cofiltered, so we can apply the (dual of the) criterion of lemma 1.5.7(i). Condition (a) of the lemma is trivially verified by π. To check condition (b) for π, consider any object φ : j → i of Morph(I), any object (φ′ : j′ → i′, 𝔪_{0}) of ℐ, and a pair of morphisms (ψ, ψ′), (μ, μ′) : φ′ → φ. Since I is cofiltered, we find morphisms ν : j′′ → j′ and ν′ : i′′ → i′ in I such that ψ ∘ ν = μ ∘ ν and ψ′ ∘ ν′ = μ′ ∘ ν′. Then we also find k ∈ Ob(I) with morphisms λ : k → j′′ and λ′ : k → i′′ such that ν′ ∘ λ′ = φ′ ∘ ν ∘ λ. Set φ′′ := ν′ ∘ λ′; we deduce a morphism (ν ∘ λ, 1_{i′}) : φ′′ → φ′ in Morph(I) with (ψ, ψ′) ∘ (ν ∘ λ, 1_{i′}) = (μ, μ′) ∘ (ν ∘ λ, 1_{i′}). From (14.2.34) we get \overline{A}_{φ′′}^{△} = \overline{A}_{φ′}^{△}. By assumption we have 𝔪_{0} · \overline{A}_{φ′} ⊂ \overline{A}_{φ′}^{△}, and clearly \overline{A}_{φ′′} ⊂ \overline{A}_{φ′}. Hence 𝔪_{0} · \overline{A}_{φ′′} ⊂ \overline{A}_{φ′′}^{△}, so (φ′′, 𝔪_{0}) ∈ Ob(ℐ), and (ν ∘ λ, 1_{i′}) is a morphism (φ′′, 𝔪_{0}) → (φ′, 𝔪_{0}) in ℐ such that (ψ, ψ′) ∘ π(ν ∘ λ, 1_{i′}) = (μ, μ′) ∘ π(ν ∘ λ, 1_{i′}), as required. The verification of conditions (a) and (b) for the functor δ is easy, and shall be left to the reader. ♢

We consider then the induced functor :

\overline{A}_{•} ∘ π : ℐ → A-Mod (j ⟶^{φ} i, 𝔪_{0}) ↦ \overline{A}_{φ}.

Notice that Im τ_{i}^{A} ⊂ \overline{A}_{φ} for every morphism φ : j → i of I; we deduce a cone

\overline{τ}_{•}^{A} : c_{A} ⇒ \overline{A}_{•} (j ⟶^{φ} i) ↦ (τ_{i}^{A} : A → \overline{A}_{φ})

such that \overline{τ}_{•}^{A} ∗ δ = τ_{•}^{A}. In light of claim 14.2.35, it follows that \overline{τ}_{•}^{A} is a universal cone, and by the same token, the same holds for the cone \overline{τ}_{•}^{A} ∗ π : c_{A} ⇒ \overline{A}_{•} ∘ π.

Claim 14.2.36. \overline{A}_{•} ∘ π is a Cauchy functor.

Proof of the claim. By proposition 14.2.18, the functor A_{•}^{△} : I → A-Mod is Cauchy and A_{•} has the almost Mittag-Leffler property, so A_{φ}^{△} : A_{j}^{△} → A_{i}^{△} is an epimorphism for every morphism φ : j → i in I. Hence, for every subideal 𝔪_{0} ⊂ 𝔪 of finite type there exists i ∈ Ob(I) with 𝔪_{0} · Ker A_{φ}^{△} = 0 for every such φ, and moreover there exists φ : j → i with 𝔪_{0}\overline{A}_{φ} ⊂ A_{i}^{△}. For such φ, we have (φ, 𝔪_{0}) ∈ Ob(ℐ); now, let (ψ, ψ′) : (φ′ : j′ → i′, 𝔪′_{0}) → (φ, 𝔪_{0}) be any morphism in ℐ; this means that 𝔪_{0} ⊂ 𝔪′_{0} and 𝔪′_{0}\overline{A}_{φ′} ⊂ \overline{A}_{φ′}^{△} = A_{i′}^{△}. Also, \overline{A}_{(ψ,ψ′)} : \overline{A}_{φ′} → \overline{A}_{φ} is the restriction of A_{ψ′} : A_{i′} → A_{i}. Therefore

𝔪_{0} · Ker \overline{A}_{(ψ,ψ′)} ⊂ A_{i′}^{△} ∩ Ker A_{ψ′} ⊂ Ker A_{ψ′}^{△}

whence 𝔪_{0}^{2} · Ker \overline{A}_{(ψ,ψ′)} = 0. Lastly, Coker \overline{A}_{(ψ,ψ′)} is a quotient of \overline{A}_{φ}/A_{ψ′}(A_{i′}^{△}) = \overline{A}_{φ}/A_{i}^{△}, which is annihilated by 𝔪_{0}, so 𝔪_{0} · Coker \overline{A}_{(ψ,ψ′)} = 0. ♢

Since the morphisms g_{φ} are isomorphisms for every morphism φ of I, a simple inspection shows that \overline{g}_{(ψ,ψ′)} is an isomorphism for every morphism (ψ, ψ′) of Morph(I). Together with claims 14.2.36, and 14.2.33, we deduce that assertion (iii) holds for the functors \overline{A}_{•} ∘ π and \overline{M}_{•} ∘ π, and moreover \overline{M}_{•} ∘ π is a Cauchy functor.

However, let λ : I → ℐ be the functor given by the rules : i ↦ (1_{i}, 0) and φ ↦ (φ, φ) for every i ∈ Ob(I) and every morphism φ of I. It is easily seen that \overline{A}_{•} ∘ π ∘ λ = A_{•} and \overline{M}_{•} ∘ π ∘ λ = M_{•}. Then assertion (iii) for A_{•} and M_{•} follows straightforwardly. Especially, the functor M_{•} is isomorphic to the functor A_{•} ⊗_{A} M. Since the functor − ⊗_{A} M : A-Mod → A-Mod is V -linear, proposition 14.2.20(iii) implies then that M_{•} is almost essentially constant, and the proof of (iii) is completed, under the assumptions of (ii).

Lastly, suppose that the assumptions of (i) hold; hence g_{φ} is an epimorphism for every morphism φ : j → i in I, and it follows easily that the induced morphism A_{i}⊗_{A_{j}} Im M_{φ} → M_{i} is an epimorphism as well, for every such φ. On the other hand, by assumption for every i ∈ Ob(I) and every subideal 𝔪_{0} ⊂ 𝔪 of finite type there exists φ : j → i such that 𝔪_{0}Im M_{φ} ⊂ M_{i}^{△}. Hence, 𝔪_{0}M_{i} ⊂ Im α_{i}. Since 𝔪_{0} is arbitrary, we conclude that α_{i} is an epimorphism.

(i): By assumption, for every i ∈ Ob(I) and every subideal 𝔪_{0} ⊂ 𝔪 there exists a morphism φ : j → i in I such that for every morphism ψ : k → j in I we have 𝔪_{0} · Im A_{φ∘ψ} ⊂ Im A_{φ}. It then follows that for every such 𝔪_{0}, φ and ψ we have

𝔪_{0} · Im(A_{φ} ⊗_{A_{k}} M_{k}) ⊂ Im(A_{φ∘ψ} ⊗_{A_{k}} M_{k}).

Since g_{ψ} is an epimorphism and g_{φ∘ψ} ∘ (A_{φ∘ψ} ⊗_{A_{k}} M_{k}) = M_{φ∘ψ}, we deduce :

𝔪_{0} · Im M_{φ} = 𝔪_{0} · Im(M_{φ} ∘ g_{ψ}) = 𝔪_{0} · Im(g_{φ∘ψ} ∘ (A_{φ} ⊗_{A_{k}} M_{k})) ⊂ Im M_{φ∘ψ}

which shows that M_{•} has the almost Mittag-Leffler property, as stated.

(iv): Notice that τ_{i}^{M} : M → M_{i} factors through a morphism μ_{i} : M → M_{i}^{△} and the inclusion M_{i}^{△} → M_{i}, for every i ∈ Ob(I). Since M_{•}/M_{•}^{△} is almost essentially zero (proposition 14.2.18(i.b)), it follows that μ_{•} : c_{M} ⇒ M_{•}^{△} is a universal cone. If λ : ℕ^{o} → I is a coinitial functor, the cone μ_{•} ∗ λ is still universal (remark 1.5.5(ii,iii)), and then μ_{λ(n)} : M → M_{λ(n)}^{△} is an epimorphism for every n ∈ ℕ (lemma 14.2.24(i)). Since M_{φ}^{△} is an epimorphism for every i ∈ Ob(I) (proposition 14.2.18(i.a)), we conclude that μ_{i} : M → M_{i}^{△} is an epimorphism for every i ∈ Ob(I). Since we already know that α_{i} is an epimorphism, it follows that the same holds for β_{i}, for every such i. □

14.2.37. Keep the notation of (14.2.31), and consider a second functor

I → (V, 𝔪)^{a}-Alg.Mod i ↦ (A′_{i}, N_{i}).

as well as the corresponding induced functors

A′_{•} : I → (V, 𝔪)^{a}-Alg N_{•} : I → (V, 𝔪)^{a}-Mod

and suppose that A_{•} = A′_{•}. Let also N be the limit of N_{•}. Then we get an induced functor

I → (V, 𝔪)^{a}-Alg.Mod i ↦ (A_{i}, M_{i} ⊗_{A_{i}} N_{i})

and the corresponding induced functor

M_{•} ⊗_{A_{•}} N_{•} : I → (V, 𝔪)^{a}-Mod i ↦ M_{i} ⊗_{A_{i}} N_{i}.

Proposition 14.2.38. In the situation of (14.2.37), suppose that A_{•}, M_{•} and N_{•} are almost essentially constant functors with values in A-Mod. Then the same holds for M_{•} ⊗_{A_{•}} N_{•}, and M ⊗_{A} N represents the limit of M_{•} ⊗_{A_{•}} N_{•}.

Proof. For every i ∈ Ob(I), the V^{a}-linear map

φ_{i} : M_{i} ⊗_{V^{a}} A_{i} ⊗_{V^{a}} N_{i} → M_{i} ⊗_{V^{a}} N_{i} x ⊗ a ⊗ y ↦ ax ⊗ y − x ⊗ ay

yields a natural identification : Coker φ_{i} →^{∼} M_{i} ⊗_{A_{i}} N_{i}. In view of proposition 14.2.30(iv), we are then easily reduced to showing that the two functors :

I → (V, 𝔪)^{a}-Mod  i ↦ M_{i} ⊗_{V^{a}} A_{i} ⊗_{V^{a}} N_{i}  i ↦ M_{i} ⊗_{V^{a}} N_{i}

are both almost essentially constant, and their limits are represented by M ⊗_{V^{a}} A ⊗_{V^{a}} N and respectively M ⊗_{V^{a}}N. Moreover, if both P_{•} := M_{•}⊗_{V^{a}}A_{•} and P_{•}⊗_{V^{a}}N_{•} are almost essentially constant with limits P := M ⊗_{V^{a}} A and respectively P ⊗_{V^{a}} N, then M_{•} ⊗_{V^{a}} A_{•} ⊗_{V^{a}} N_{•} will be almost essentially constant with the sought limit. Hence, we are further reduced to showing the proposition in case A_{•} is the constant functor with A_{i} = V^{a} for every i ∈ Ob(I). Now, let τ_{•} : c_{M} ⇒ M_{•} be the universal cone. We have right exact sequences :

K_{i} := (Ker τ_{i}) ⊗_{V^{a}} N_{i} → Ker(τ_{i} ⊗_{V^{a}} N_{i}) → T_{i} → 0 for every i ∈ Ob(I)

where T_{i} is the cokernel of the natural morphism Tor_{1}^{V^{a}}(M_{i}, N_{i}) → Tor_{1}^{V^{a}}(Coker τ_{i}, N_{i}). Since, by assumption, both Ker τ_{•} and Coker τ_{•} are almost essentially zero, it is easily seen that the same holds for the induced functors K_{•} and T_{•}, and then the same follows for the functor Ker(τ_{•}⊗_{V^{a}}N_{•}), by lemma 14.2.14(i). Likewise, Coker(τ_{•}⊗_{V^{a}}N_{•}) is the functor (Coker τ_{•})⊗_{V^{a}} N_{•}, which is again almost essentially zero. Thus, M_{•} ⊗_{V^{a}} N_{•} is isomorphic to M ⊗_{V^{a}} N_{•} in the category ℒ (I, (V, 𝔪)^{a}-Mod). Lastly, let F : (V, 𝔪)^{a}-Mod → (V, 𝔪)^{a}-Mod be the functor such that F X := M ⊗_{V^{a}} X for every V^{a}-module X; since N_{•} is almost essentially constant, the same holds for F N_{•} = M ⊗_{V^{a}} N_{•}, and its limit is M ⊗_{V^{a}} N, by proposition 14.2.20(i,iii), whence the assertion. □

Theorem 14.2.39. Let (V, 𝔪) be a basic setup, I a small cofiltered category, and consider an almost essentially constant functor A_{•} : I → (V, 𝔪)^{a}-Alg. Denote by A the limit of A_{•}, and let τ_{•}^{A} : c_{A} ⇒ A_{•} be a universal cone; let also M be an A-module, B an A-algebra, r ∈ ℕ and

A_{•}-Mod : I → Cat i ↦ A_{i}-Mod  and  A_{•}-Alg : I → Cat i ↦ A_{i}-Alg

the pseudo-functors naturally induced by A_{•}. The following holds :

(i) The cone τ_{•}^{A} induces equivalences of categories

A-Alg →^{∼} 2-lim_{I} A_{•}-Alg A-Mod →^{∼} 2-lim_{I} A_{•}-Mod.

(ii) The A-module M is almost finitely generated (resp. almost finitely presented, resp. flat, resp. faithfully flat resp. almost projective) if and only if the same holds for the A_{i}-module A_{i} ⊗_{A} M, for every i ∈ Ob(I).

(iii) Suppose that (V, 𝔪) fulfills condition (B) of [75, §2.1.6]. Then M is almost projective of almost finite rank (resp. of finite rank ≤ r) if and only if the same holds for the A_{i}-module A_{i} ⊗_{A} M, for every i ∈ Ob(I).

(iv) The A-algebra B is weakly unramified (resp. unramified, resp. weakly étale, resp. étale) if and only if the same holds for the A_{i}-algebra A_{i} ⊗_{A} B, for every i ∈ Ob(I).

Proof. (i): Let π : (V, 𝔪)^{a}-Alg.Mod → (V, 𝔪)^{a}-Alg be the natural projection functor. Then 2-lim_{I} A_{•}-Mod is the subcategory of Fun(I, (V, 𝔪)^{a}-Alg.Mod) whose objects are the functors

(A_{•}, M_{•}) : I → (V, 𝔪)^{a}-Alg.Mod i ↦ (A_{i}, M_{i})

with π ∘ (A_{•}, M_{•}) = A_{•}, and such that, with the notation of (14.2.31), the morphism g_{φ} : A_{i} ⊗_{A_{j}} M_{j} → M_{i} is an isomorphim of A_{i}-modules, for every morphism φ : j → i in I. The morphisms β_{•} : (A_{•}, M_{•}) → (A_{•}, M′_{•}) are the natural transformations such that π ∗ β_{•} = 1_{A_{•}}.

Under this identification, the functor (A_{•}, A_{•} ⊗_{A} −) : A-Mod → 2-lim_{I} A_{•}-Mod of (i) assigns to every A-module M the functor

(A_{•}, A_{•} ⊗_{A} M) : I → (V, 𝔪)^{a}-Alg.Mod i ↦ (A_{i}, A_{i} ⊗_{A} M).

Conversely, following (14.2.31), with every functor (A_{•}, M_{•}) as in the foregoing, we associate a functor M_{•} : I → A-Mod, and let M be the limit of M_{•}. From proposition 14.2.32(iii) we obtain a natural isomorphism

(A_{•}, A_{•} ⊗_{A} M) →^{∼} (A_{•}, M_{•}).

This shows that (A_{•}, A_{•} ⊗_{A} −) is essentially surjective. Next, let M, N be two A-modules; according to proposition 14.2.20(i), the kernel and cokernel of τ_{•}^{A} ⊗_{A} N : c_{N} ⇒ A_{•} ⊗_{A} N are almost essentially zero, and therefore τ_{•}^{A} ⊗_{A} N is a universal cone. There follow natural isomorphisms

Hom_{A}(M, N) →^{∼} lim_{I} Hom_{A}(M, A_{•} ⊗_{A} N) →^{∼} lim_{I} Hom_{A_{i}}(A_{•} ⊗_{A} M, A_{•} ⊗_{A} N)

and notice that the latter limit is naturally identified with the set of morphisms (A_{•}, A_{•}⊗_{A}M) → (A_{•}, A_{•} ⊗_{A} N) in 2-lim_{I} A_{•}-Mod. Thus (A_{•}, A_{•} ⊗_{A} −) is an equivalence.

(ii): Suppose first that M_{i} := A_{i} ⊗_{A} M is almost finitely generated for every i ∈ Ob(I). Let N_{•} : Λ → A-Mod be any functor from a small filtered category Λ; denote by N the colimit of N_{•}, and let τ_{•}^{N} : N_{•} ⇒ c_{N} be a universal cocone. We have two V -linear functors

F : A-Mod → A-Mod  P ↦ colim_{Λ} alHom_{A}(M, P ⊗_{A} N_{•})
G : A-Mod → A-Mod  P ↦ alHom_{A}(M, P ⊗_{A} N)

and τ_{•}^{N} induces a natural transformation

β_{•} : F ⇒ G P ↦ colim_{Λ} alHom_{A}(M, P ⊗_{A} τ_{•}^{N}) : F P → GP

and according to [75, Prop.2.3.16(i)], it suffices to check that β_{A} is a monomorphism. However, we have a commutative diagram of functors :

\xymatrix{c_{F A} \ar[r]^{c_{β_{A}}} \ar[d]_{F ∗τ_{•}^{A}} & c_{GA} \ar[d]^{G∗τ_{•}^{A}} \\ F ∘ A_{•} \ar[r]^{β_{•}∗A_{•}} & G ∘ A_{•}}

and by propositions 14.2.18(ii) and 14.2.20(i), the vertical arrows of the diagram are both isomorphisms in the category ℒ (I, A-Mod). We claim that β_{A_{i}} : F A_{i} → GA_{i} is a monomorphism for every i ∈ Ob(I). Indeed :

F A_{i} = colim_{Λ} alHom_{A_{i}}(M_{i}, A_{i} ⊗_{A} N_{•}) and GA_{i} = alHom_{A_{i}}(M_{i}, A_{i} ⊗_{A} N)

and since by assumption the A_{i}-module M_{i} is almost finitely generated, the assertion follows from [75, Prop.2.3.16(i)]. We deduce that c_{β_{A}} is also a monomorphism in ℒ (I, A-Mod), i.e. β_{A} is a monomorphism, as required.

One argues likewise, in case M_{i} is almost finitely presented for every i ∈ Ob(I), invoking [75, Prop.2.3.16(ii)] to see that in this case β_{A_{i}} is an isomorphism for every such i, whence β_{A} is an isomorphism, so that M is almost finitely presented, again by [75, Prop.2.3.16(ii)].

Next, suppose that M_{i} is an almost projective A_{i}-module for every i ∈ Ob(I), and let f : N′ → N be an epimorphism of A-modules. We attach to the pair (M, N) the V -linear functor

H_{M,N} : A-Mod → A-Mod P ↦ alHom_{A}(P ⊗_{A} M, P ⊗_{A} N)

and define likewise H_{M,N′} : A-Mod → A-Mod. The morphism f induces a natural transformation H_{M,f} : H_{M,N} ⇛ H_{M,N′}, whence a commutative diagram of functors :

\xymatrix{c_{H_{M,N} (A)} \ar[r]^{c_{H_{M,f} (A)}} \ar[d]_{H_{M,N} ∗τ_{•}^{A}} & c_{H_{M,N′}(A)} \ar[d]^{H_{M,N′}∗τ_{•}^{A}} \\ H_{M,N} ∘ A_{•} \ar[r]^{H_{M,f} ∗A_{•}} & H_{M,N′} ∘ A_{•}}

whose vertical arrows are isomorphisms in ℒ (I, A-Mod), by virtue of propositions 14.2.18(ii) and 14.2.20(i); moreover, H_{M,N′} ∗ τ_{i}^{A} is an epimorphism for every i ∈ Ob(I), since M_{i} is an almost projective A_{i}-module. As in the previous case, we deduce that H_{M,f}(A) is an epimorphism of A-modules, so M is almost projective.

Suppose next that M_{i} is flat for every i ∈ Ob(I), and let f : N → N′ be a monomorphism of A-modules. Set N_{i} := A_{i} ⊗_{A} N and N′_{i} := A_{i} ⊗_{A} N′ for every i ∈ Ob(I), and let \overline{N}_{i} := Im(A_{i} ⊗_{A} f) ⊂ N′_{i}. Clearly the rule : i ↦ \overline{N}_{i} defines a subfunctor \overline{N}_{•} of A_{•} ⊗_{A} N′. By proposition 14.2.20(i), the kernel of A_{•} ⊗_{A} f : A_{•} ⊗_{A} N ⇒ A_{•} ⊗_{A} N′ is almost essentially zero, hence the induced natural transformation π_{•} : A_{•} ⊗_{A} N → \overline{N}_{•} is an isomorphism in ℒ (I, A-Mod). By the same token, the same holds for the natural transformation τ_{•}^{A} ⊗_{A} N : c_{N} ⇒ A_{•} ⊗_{A} N, and therefore also for the composition μ_{•} := π_{•} ⊙ (τ_{•}^{A} ⊗_{A} N) : c_{N} ⇒ \overline{N}_{•}. We obtain therefore a commutative diagram of functors :

\xymatrix{c_{M⊗_{A}N} \ar[r]^{c_{M⊗_{A}f}} \ar[d]_{M⊗_{A}μ_{•}} & c_{M⊗_{A}N′} \ar[d]^{M⊗_{A}τ_{•}^{A}⊗_{A}N} \\ M ⊗_{A} \overline{N}_{•} \ar[r] & M ⊗_{A} A_{i} ⊗_{A} N′}

whose vertical arrows are, as usual, isomorphisms in ℒ (I, A-Mod). Let j_{i} : \overline{N}_{i} → N′_{i} be the inclusion; we have natural isomorphisms M ⊗_{A} \overline{N}_{i} →^{∼} M_{i} ⊗_{A_{i}} \overline{N}_{i} and M ⊗_{A} A_{i} ⊗_{A} N′ →^{∼} M_{i} ⊗_{A_{i}} N′_{i} for every i ∈ Ob(I), that identify the bottom horizontal arrow with the morphism M ⊗_{A_{i}} j_{i}. The latter is a monomorphism for every i ∈ Ob(I), since M_{i} is a flat A_{i}-module; as usual it follows that M ⊗_{A} f is a monomorphism as well, so M is flat.

Lastly, suppose that M_{i} is a faithfully flat A_{i}-module for every i ∈ Ob(I); by the foregoing, we know already that M is a flat A-module. Then, let X be any A-module such that M ⊗_{A} X = 0; it follows that M_{i} ⊗_{A_{i}} (A_{i} ⊗_{A} X) = 0 for every i ∈ Ob(I), whence A_{i} ⊗_{A} X = 0 for every such i. But by the foregoing, X is the limit of the functor A_{•} ⊗_{A} X, whence X = 0; this shows that M is faithfully flat.

(iii): Suppose that M_{i} is an almost projective A_{i}-module of almost finite rank, for every i ∈ Ob(I). By the foregoing, we know already that M is an almost projective A-module, and it remains to check that it is of almost finite rank. To this aim, define the category ℐ and the functors π : ℐ → Morph(I) and \overline{A}_{•} : Morph(I) → A-Alg as in the proof of proposition 14.2.32; recall that π is coinitial, and \overline{A}_{•} ∘ π is a Cauchy functor (claims 14.2.35 and 14.2.36). Moreover, the cone \overline{τ}_{•}^{A} : c_{A} ⇒ \overline{A}_{•} deduced from τ_{•}^{A}, is universal, so the same holds for \overline{τ}_{•}^{A} ∗ π. It follows that Ker (\overline{τ}_{•}^{A} ∗ π) and Coker (\overline{τ}_{•}^{A} ∗ π) are null functors (remark 14.2.17(ii)). Now, let 𝔪_{0} ⊂ 𝔪 be a subideal of finite type; by the foregoing there exists (φ : j → i, 𝔪_{1}) ∈ Ob(ℐ) such that the kernel and cokernel of \overline{τ}_{φ}^{A} : A → \overline{A}_{φ} := Im A_{φ} are both annihilated by 𝔪_{0}. On the other hand, since M_{j} has almost finite rank, there exists n ∈ ℕ such that 𝔪_{0} · Λ_{A_{j}}^{n}M_{j} = 0, and then we have as well 𝔪_{0} · (\overline{A}_{φ} ⊗_{A_{j}} Λ_{A_{j}}^{n}M_{j}) = 0. But by remark 14.2.10(vi), the kernel of

\overline{τ}_{φ}^{A} ⊗_{A} Λ_{A}^{n}M : Λ_{A}^{n}M → \overline{A}_{φ} ⊗_{A} Λ_{A}^{n}M →^{∼} \overline{A}_{φ} ⊗_{A_{j}} Λ_{A_{j}}^{n}M_{j}

is annihilated by 𝔪_{0}^{2}. We conclude that 𝔪_{0}^{3} · Λ_{A}^{n}M = 0, whence the contention. Lastly, if M_{i} has rank ≤ r for every i ∈ Ob(I), the same argument shows that 𝔪 · 𝔪_{0}^{2} · Λ_{A}^{r}M = 0 for every such 𝔪_{0}, i.e. Λ_{A}^{r}M = 0, so M has rank ≤ r.

(iv): Set C := B ⊗_{A} B, and let μ : C → B be the multiplication law of the A-algebra B; the functor C_{•} := C ⊗_{A} A_{•} : I → (V, 𝔪)^{a}-Alg is still almost essentially constant, and C ⊗_{A} τ_{•}^{A} : c_{C} ⇒ C_{•} is an isomorphism in ℒ (I, A-Mod) (proposition 14.2.20(i,iii)). Suppose now that B_{i} := B ⊗_{A} A_{i} is a weakly unramified A_{i}-algebra, i.e. that the morphism μ ⊗_{A} A_{i} : C_{i} → B_{i} is flat for every i ∈ Ob(I). By (ii), it follows that μ is flat, i.e. B is a weakly unramified A-algebra. One argues likewise in case B_{i} is an unramified (resp. weakly étale, resp. étale) A_{i}-algebra for every i ∈ Ob(I). □

The following notion of flatness for almost modules, relative to a given base ring will find application in section 17.5.

Definition 14.2.40. Let (V, 𝔪) be a basic setup, f : A → V a ring homomorphism, and M any V -module; we say that M^{a} is flat over A (resp. faithfully flat over A) if the functor

A-Mod → (V, 𝔪)^{a}-Mod N ↦ M^{a} ⊗_{A} N := (M ⊗_{A} N)^{a}

is exact (resp. is exact and faithful).

Lemma 14.2.41. In the situation of definition 14.2.40, let g : V → B be a ring homomorphism, and N a B-module. The following holds :

(i) If the V^{a}-module B^{a} is faithfully flat over A, the map g ∘ f : A → B is injective. (ii) M^{a} is faithfully flat over A if and only if it is flat over A and (M/𝔫M)^{a} ≠ 0 for every

maximal ideal 𝔫 ⊂ A. (iii) If the V^{a}-module B^{a} is flat (resp. faithfully flat) over A, and N^{a} is a flat (resp. faithfully

flat) B^{a}-module, then N^{a} is flat (resp. faithfully flat) over A.

Proof. (i): Set I := Ker (g ∘ f); by assumption, the inclusion map I → A induces a monomorphism (I ⊗_{A} B)^{a} → B^{a} of V^{a}-modules, i.e. the natural morphism (I ⊗_{A} B)^{a} → (IB)^{a} = 0 is an isomorphism of V^{a}-modules. By assumption, the latter implies that I = 0.

Assertions (ii) and (iii) shall be left to the reader. □

14.2.42. In the situation of definition 14.2.40, suppose that A is noetherian, let I ⊂ A be a given ideal, and set A_{n} := A/I^{n}, V_{n} := V/I^{n}V, for every n ∈ ℕ. Consider an inverse system M_{•} := (M_{n}, φ_{n} | n ∈ ℕ), where M_{n} is a V_{n}-module such that M_{n}^{a} is flat over A_{n}, and φ_{n} : M_{n+1} → M_{n} is a V -linear map, such that the induced inverse system of V^{a}-modules M_{•}^{a} := (M_{n}^{a}, φ_{n}^{a} | n ∈ ℕ) has the almost Mittag-Leffler property (see definition 14.2.11(iv)). Let also M be the limit of M_{•} (in the category V -Mod). The following result refines [107, Tag 0912].

Proposition 14.2.43. With the notation of (14.2.42), the following holds :

(i) M^{a} is flat over A.

(ii) Every A-module N of finite type induces an isomorphism of V^{a}-modules

(N ⊗_{A} M)^{a} →^{∼} lim_{n∈ℕ} (N ⊗_{A} M_{n})^{a}.

Proof. (ii): Let (F_{•}, d_{•}) → N be a resolution of N by free A-modules of finite rank (hence the differential d_{i} : F_{i} → F_{−1} is an A-linear map, for every i > 0, and d_{0} : F_{0} → N is an A-linear surjection). There follows, for every n ∈ ℕ, exact sequences of A-modules :

0 → Tor_{1}^{A}(N, M_{n}) → Coker(d_{2}) ⊗_{A} M_{n} ⟶^{d_{1,n}} F_{0} ⊗_{A} M_{n} → N ⊗_{A} M_{n} → 0

0 → Im(d_{2} ⊗_{A} M_{n}) → F_{1} ⊗_{A} M_{n} → Coker(d_{2}) ⊗_{A} M_{n} → 0

where d_{1,n} is induced by d_{1} ⊗_{A} M_{n}, for every such n. Since M_{n}^{a} is flat over A_{n}, we have a natural isomorphism of V^{a}-modules :

Tor_{1}^{A}(N, M_{n})^{a} →^{∼} (Tor_{1}^{A}(N, A_{n}) ⊗_{A} M_{n})^{a} for every n ∈ ℕ.

Claim 14.2.44. (i) The inverse system (Tor_{1}^{A}(N, A_{n}) | n ∈ ℕ) is essentially zero. (ii) Consider an exact sequence of inverse systems of V^{a}-modules :

K_{••} : (K_{3,n} | n ∈ ℕ) ⟶^{φ_{3,•}} (K_{2,n} | n ∈ ℕ) ⟶^{φ_{2,•}} (K_{1,n} | n ∈ ℕ) ⟶^{φ_{1,•}} (K_{0,n} | n ∈ ℕ) → 0

such that K_{3,•} is almost essentially zero, and K_{2,•} has the almost Mittag-Leffler property. Then K_{••} induces a short exact sequence of V^{a}-modules :

0 → lim_{n∈ℕ} K_{2,n} → lim_{n∈ℕ} K_{1,n} → lim_{n∈ℕ} K_{0,n} → 0.

Proof of the claim. (i): Let K := Ker(d_{0} : F_{0} → N); we have a natural identification :

Tor_{1}^{A}(N, A_{n}) →^{∼} (K ∩ I^{n}F)/(I^{n}K) for every n ∈ ℕ

and by the Artin-Rees lemma there exists c ∈ ℕ such that K ∩ I^{n+c}F_{0} ⊂ I^{n}K for every n ∈ ℕ, whence the assertion.

(ii): Set J_{n} := Im φ_{2,n} for every n ∈ ℕ; we get induced exact sequences of inverse systems :

K_{3,•} → K_{2,•} → J_{•} → 0 0 → J_{•} → K_{2,•} → K_{1,•} → 0

whence, by lemma 14.2.14(iii), an isomorphism of V^{a}-modules :

lim_{n∈ℕ} K_{2,n} →^{∼} lim_{n∈ℕ} J_{n}

and notice that J_{•} has the almost Mittag-Leffler property, as the same holds for K_{2,•} (proposition 14.2.30(ii)); then, proposition 14.2.25(iii) yields a short exact sequence of V^{a}-modules :

0 → lim_{n∈ℕ} J_{n} → lim_{n∈ℕ} K_{1,n} → lim_{n∈ℕ} K_{0,n} → 0

whence the assertion. ♢

From claim 14.2.44(i) we deduce easily that the inverse system (Tor_{1}^{A}(N, M_{n})^{a} | n ∈ ℕ) is also essentially zero. On the other hand, since M_{•}^{a} has the almost Mittag-Leffler property, the same holds for the inverse system of V^{a}-modules (Coker d_{2} ⊗_{A} M_{n})^{a}; Likewise, with proposition 14.2.30(ii)) one checks that the inverse system (Im(d_{2} ⊗_{A} M_{n})^{a} | n ∈ ℕ) has the almost Mittag-Leffler property; then, claim 14.2.44(ii) yields short exact sequences of V^{a}-modules :

0 → lim_{n∈ℕ} (Coker d_{2} ⊗_{A} M_{n})^{a} → lim_{n∈ℕ}(F_{0} ⊗_{A} M_{n})^{a} → lim_{n∈ℕ} (N ⊗_{A} M_{n})^{a} → 0

0 → lim_{n∈ℕ} Im(d_{2} ⊗_{A} M_{n})^{a} → lim_{n∈ℕ} (F_{1} ⊗_{A} M_{n})^{a} → lim_{n∈ℕ} (Coker d_{2} ⊗_{A} M_{n})^{a} → 0.

Lastly, since F_{i} is a free A-module of finite type, clearly we have a natural identification

lim_{n∈ℕ} (F_{i} ⊗_{A} M_{n})^{a} →^{∼} (F_{i} ⊗_{A} M)^{a} for every i ∈ ℕ.

Summing up, we get a right exact sequence of V^{a}-modules

(F_{1} ⊗_{A} M)^{a} ⟶^{(d_{1}⊗_{A}M)^{a}} (F_{0} ⊗_{A} M)^{a} → lim_{n∈ℕ} (N ⊗_{A} M_{n})^{a} → 0

whence the contention.

(i): We need to check that every A-linear injection f : N → N′ of A-modules of finite type induces a monomorphism (f ⊗_{A} M)^{a}. However, for every n ∈ ℕ we have an exact sequence :

Tor_{1}^{A}(Coker f, M_{n}) → N ⊗_{A} M_{n} ⟶^{f⊗_{A}M_{n}} N′ ⊗_{A} M_{n} → Coker f_{n} ⊗_{A} M_{n} → 0

and arguing as in the proof of (i) we see that the inverse system (Tor_{1}^{A}(Coker f, M_{n})^{a} | n ∈ ℕ) is essentially zero, and the inverse system (N ⊗_{A} M_{n} | n ∈ ℕ) has the almost Mittag-Leffler property, therefore claim 14.2.44(ii) implies that the inverse system ((f ⊗_{A} M_{n})^{a} | n ∈ ℕ) induces a monomorphism lim_{n∈ℕ}(N ⊗_{A} M_{n})^{a} → lim_{n∈ℕ}(N′ ⊗_{A} M_{n})^{a}. By (i), the latter is naturally identified with (f ⊗_{A} M)^{a}, whence the asssertion. □

For future reference, let us also point out the following almost analogue of proposition 8.2.13(i,v) :

Lemma 14.2.45. Let (V, 𝔪) be a basic setup, f : M′ → M a morphism of V^{a}-modules, and 𝒯 a linear topology on M. Endow M′ (resp. M′′ := Coker f) with the linear topology 𝒯 ′ (resp. 𝒯 ′′) induced by 𝒯 via f (resp. via the projection π : M → M′′). We have :

(i) f induces a short exact sequence of the respective separated completions :

0 → (M′, 𝒯 ′)^{∧} ⟶^{f^{∧}} (M, 𝒯 )^{∧} ⟶^{π^{∧}} (M′′, 𝒯 ′′)^{∧}.

(ii) If 𝒯 has a countable fundamental system of open submodules, π^{∧} is an epimorphism.

Proof. (i): See [75, §5.3] for generalities concerning linear topologies on V^{a}-modules. By definition, 𝒯 is determined by a cofiltered family (M_{λ} | λ ∈ Λ) of V^{a}-submodules of M, and 𝒯 ′ and 𝒯 ′′ are likewise given by the families (M′_{λ} := f^{−1}M_{λ} | λ ∈ Λ) and (M′′_{λ} := π(M_{λ})) | λ ∈ Λ). We need to check that f induces an exact sequence of V^{a}-modules :

0 → lim_{λ∈Λ} M′/M′_{λ} → lim_{λ∈Λ} M/M_{λ} → lim_{λ∈Λ} M′′/M′′_{λ}.

To this aim, it suffices to show that the induced sequence of V -modules :

0 → lim_{λ∈Λ}(M′/M′_{λ})_{∗} → lim_{λ∈Λ}(M/M_{λ})_{∗} → lim_{λ∈Λ}(M′′/M′′_{λ})_{∗}

is exact. However, f induces a short exact sequence of V^{a}-modules

0 → M′/M′_{λ} → M/M_{λ} → M′′/M′′_{λ} → 0 for every λ ∈ Λ

whence a corresponding short exact sequence of V -modules

0 → (M′/M′_{λ})_{∗} → (M/M_{λ})_{∗} → (M′′/M′′_{λ})_{∗} for every λ ∈ Λ.

The assertion follows immediately.

(ii): Under the stated assumptions, we may suppose that Λ = ℕ^{o}; then the assertion follows from proposition 14.2.25(iii). □

We conclude with two homological applications to usual rings : the techiques developed in this section are employed in the proof of the following propositions, but only trivial almost structures intervene.

Proposition 14.2.46. Let g : A → B be a ring homomorphism, I ⊂ A an ideal generated by a sequence f := (f_{1}, …, f_{r}), and g(f) := (g(f_{1}), …, g(f_{r})). Endow A and B with their I-adic topologies, and suppose that g is adically flat, and that B is complete and separated. We have :

(i) If A satisfies condition (c)_{f}, the ring B satisfies (c)_{g(f)} (notation of (7.8.21)).

(ii) Suppose additionally that also A is complete and separated, that g is adically faithfully flat, and that B satisfies (c)_{g(f)}; then A satisfies (c)_{f}.

Proof. (i): By assumption, the inverse system (H_{i}(f, I^{n}) | n ∈ ℕ) is essentially zero for every i > 0, and we need to check that the same holds for the inverse system (H_{i}(g(f), I^{n}B) | n ∈ ℕ). To this aim, we notice :

Claim 14.2.47. Let R be a ring, J ⊂ R an ideal generated by a finite sequence g, and consider the following conditions :

(a) R satisfies condition (c)_{g} of (7.8.21).

(b) For every i, n ∈ ℕ there exists an integer n′ ≥ n such that for every integer m ≥ n the

natural map

H_{i}(g, J^{n′}/J^{m′}) → H_{i}(g, J^{n}/J^{m})

vanishes for some integer m′ ≥ max(m, n′).

(c) For every i, n ∈ ℕ there exists an integer n′ ≥ n such that the natural map

H_{i}(g, J^{n′}) → H_{i}(g, J^{n}/J^{m})

vanishes for every integer m ≥ n. Then (a)⇒(b)⇒(c), and if R is J-adically complete and separated, we have also (c)⇒(a).

Proof of the claim. (a)⇒(b): By assumption, for every i ∈ ℕ there exists a map φ_{i} : ℕ → ℕ with φ_{i}(n) ≥ n, and such that the induced map H_{i}(g, J^{φ_{i}(n)}) → H_{i}(g, J^{n}) vanishes, for every n ∈ ℕ. Clearly, we may assume that φ_{i}(n) ≥ φ_{i}(m) whenever n ≥ m Set also φ_{−1}(n) := n for every n ∈ ℕ, and define ψ_{i} := φ_{i−1} ∘ ψ_{i} for every i ∈ ℕ. For every i, n, m ∈ ℕ with m ≥ n we get a commutative diagram with exact horizontal rows :

\xymatrix{H_{i}(g, J^{ψ_{i}(m)}) \ar[r] \ar[d]_{α_{0}} & H_{i}(g, J^{ψ_{i}(n)}) \ar[r] \ar[d]^{α_{1}} & H_{i}(g, J^{ψ_{i}(n)}/J^{φ_{i}(m)}) \ar[r] \ar[d]^{α_{2}} & H_{i−1}(g, J^{ψ_{i}(m)}) \ar[d]^{α_{3}} \\ H_{i}(g, J^{φ_{i}(m)}) \ar[r] \ar[d]_{β_{0}} & H_{i}(g, J^{φ_{i}(n)}) \ar[r] \ar[d]^{β_{1}} & H_{i}(g, J^{φ_{i}(n)}/J^{φ_{i}(m)}) \ar[r] \ar[d]^{β_{2}} & H_{i−1}(g, J^{φ_{i}(m)}) \ar[d]^{β_{3}} \\ H_{i}(g, J^{m}) \ar[r] & H_{i}(g, J^{n}) \ar[r] & H_{i}(g, J^{n}/J^{m}) \ar[r] & H_{i−1}(g, J^{m})}

and by construction, the composition β_{k} ∘ α_{k} vanishes for k = 0, 1, 3. By the snake lemma, it follows that β_{2} ∘ α_{2} = 0 as well; then n′ := ψ_{i}(n) and m′ := ψ_{i}(m) will do.

(b)⇒(c) is trivial.

(c)⇒(a): Recall that for every n ∈ ℕ, the natural map J^{n} → lim_{k∈ℕ} J^{n}/J^{n+k} is an isomorphism (remark 8.3.3(iii,iv)); then, according to [163, Th.3.5.8], we have a short exact sequence

L_{i,n} := lim_{k∈ℕ}^{1}H_{i+1}(g, J^{n}/J^{n+k}) → H_{i}(g, J^{n}B) ⟶^{τ_{i,n}} L′_{i,n} := lim_{k∈ℕ} H_{i}(f, J^{n}/J^{n+k}) → 0

for every i, n ∈ ℕ. On the other hand, for every i, n, k ∈ ℕ we have an exact sequence :

H_{i+1}(g, J^{n}) ⟶^{α_{i,n,k}} H_{i+1}(g, J^{n}/J^{n+k}) ⟶^{β_{i,n,k}} H_{i}(g, J^{n+k}) ⟶^{γ_{i,n,k}} H_{i}(g, J^{n}).

But notice that (c) implies that the inverse system H_{i}(f, I^{n+k} | k ∈ ℕ) has the Mittag-Leffler property, for every i, n ∈ ℕ; then, the inverse system (Im γ_{i,n,k} | k ∈ ℕ) is essentially constant, and consequently (Ker γ_{i,n,k} | k ∈ ℕ) has the Mittag-Leffler property (proposition 14.2.30(iii)). The same holds for (Im α_{i,n,k} | k ∈ ℕ) (proposition 14.2.30(ii)), and finally we conclude that (H_{i+1}(g, J^{n}/J^{n+k}) | k ∈ ℕ) has the Mittag-Leffler property, for every i, n ∈ ℕ (proposition 14.2.30(i)). Hence, L_{i,n} = 0 for every i, n ∈ ℕ (proposition 14.2.25(iii)), and τ_{i,n} is an isomorphism. Lastly, (c) implies that for every i, n ∈ ℕ there exists an integer n′ ≥ n such that the induced map H_{i}(g, J^{n′}) → L′_{i,n} vanishes, so the same holds for the natural map H_{i}(g, J^{n′}) → H_{i}(g, J^{n}), whence (a). ♢

We are then reduced to checking that condition (b) of claim 14.2.47 holds for R := B, g := g(f), and J := IB. But since g is adically flat for the I-adic topologies, we have a natural identification

(14.2.48)  B ⊗_{A} H_{i}(f, I^{n}/I^{n+k}) →^{∼} H_{i}(g(f), I^{n}B/I^{n+k}B)  for every i, n, k ∈ ℕ.

On the other hand, by assumption condition (a) of claim 14.2.47holds for R := A, g := f, and J := I, so condition (b) of the claim holds as well for A, f and I, whence the assertion.

(ii): Again, we have the isomorphisms (14.2.48), and by assumption, condition (b) of claim 14.2.47 holds for R := B, g := g(f), and J := IB. Under the current assumptions, it then follows that condition (b) of claim 14.2.47 also holds for R := A, g := f and J := I. Since A is I-adically complete, the assertion follows. □

Proposition 14.2.49. Let g : A → B be a ring homomorphism, f := (f_{1}, …, f_{r}) a finite sequence of elements of A that generates an ideal I ⊂ A, n ∈ ℕ \ {0}, and f′ the image of f in B. Suppose that B/IB is a flat A/I-algebra, and consider the following conditions :

(a) The natural map φ_{i} : B ⊗_{A} H_{i}(f, A) → H_{i}(f′, B) is bijective for i = 0, …, n − 1 and surjective for i = n.

(b) Tor_{i}^{A}(B, M) = 0 for i = 1, …, n, every k ∈ ℕ \ {0}, and every A/I^{k}-module M.

(c) Tor_{i}^{A}(B, A/I) = 0 for i = 1, …, n.

(d) Tor_{1}^{A}(B, A/I) = 0.

(e) g is adically flat for the I-adic topologies of A and B. Then we have :

(i) (a)⇔(b)⇔(c) ⇒(d)⇒(e).

(ii) If (a) holds for all n ≥ 0, and A verifies (c)_{f} of (7.8.21), then B verifies (c)_{f′}.

(iii) If (e) holds and A satisfies (c)_{f}, and B satisfies (c)_{f′}, then (a) holds for all n ≥ 0.

Proof. (d)⇒(e) holds by by [126, Th.22.3], and trivially (b)⇒(c)⇒(d).

(c)⇒(b): The short exact sequence 0 → IM → M → M/IM → 0 induces an exact sequence

Tor_{i}^{A}(B, IM) → Tor_{i}^{A}(B, M) → Tor_{i}^{A}(B, M/IM) for every i ∈ ℕ.

Clearly IM is annihilated by I^{k−1}, and M/IM is annhilated by I; by a simple induction, we are then reduced to the case where k = 1. In this case, we find a set S and an A-linear surjection h : (A/I)^{(S)} → M, whence an exact sequence :

T_{i} := Tor_{i}^{A}(B, (A/I)^{(S)}) → Tor_{i}^{A}(B, M) → Tor_{i−1}^{A}(B, Ker h) for every i > 0

and since the functor Tor_{i}^{A}(B, −) commutes with direct limits and finite sums, we have T_{i} = 0 for i = 1, …, n, due to (a). Hence, we are further reduced to checking that Tor_{1}^{A}(B, M) = 0 for every A/I-module M, and we reduced easily first to the case where M is of finite type, say generated by k elements (since Tor_{1}^{A}(B, −) commutes with direct limits), and then to the case where k = 1, arguing by induction on k : the details are left to the reader. So, we may assume that M = A/J for some ideal J ⊂ A with I ⊂ J. In this case, notice that the short exact sequence 0 → J/I → A/I → A/J → 0 is still exact after tensoring by B, since B/IB is a flat A/I-algebra; so the induced map Tor_{1}^{A}(B, A/I) → Tor_{1}^{A}(B, A/J) is surjective, whence the assertion.

(b)⇒(a): We consider the Künneth spectral sequence (see [163, Th.5.6.4])

E_{pq}^{2} := Tor_{p}^{A}(B, H_{q}(f, A)) ⇒ H_{p+q}(f′, B)

and recall that H_{q}(f, A) is an A/I-module for every q ∈ ℕ (lemma 7.8.2(i)); thus, E_{pq}^{2} = 0 for every p = 1, …, n, due to (b). It follows easily that E_{p,i−p}^{∞} = 0 for p = 1, …, i, and E_{0,i}^{∞} = E_{0,i}^{2}, if i < n. Hence, φ_{i} is bijective for i < n. Lastly, for i = n we get an exact sequence

E_{n+1,0}^{2} ⟶^{∂_{n+1,0}^{n+1}} E_{0,n}^{2} = B ⊗_{A} H_{n}(f, A) → H_{n}(f′, B) → 0

which shows that φ_{n} is surjective.

(a)⇒(c): We argue by induction on n ∈ ℕ \ {0}. If n = 1, notice that E_{1,0}^{∞} = E_{1,0}^{2} = Tor_{1}^{A}(B, A/I), and we have a short exact sequence :

E_{0,1}^{2} = B ⊗_{A} H_{1}(f, A) → H_{1}(f′, B) → E_{1,0}^{∞} → 0.

But (a) implies that E_{1,0}^{∞} = 0, whence (c). Next, let n > 1, and suppose that it is already known that Tor_{i}^{A}(B, A/I) = 0 for i = 1, …, n − 1. By the foregoing, it follows that Tor_{i}^{A}(B, M) = 0 for i = 1, …, n − 1 and every A/I-module M. Hence, E_{pq}^{2} = 0 whenever p = 1, …, n − 1, and therefore E_{pq}^{∞} = 0 for every such p. In particular, E_{0,n−1}^{∞} = H_{n−1}(f′, B), and on the other hand, (a) implies that H_{n−1}(f′, B) = E_{0,n−1}^{2}. It follows that E_{0,n−1}^{r} = E_{0,n−1}^{2} for every r ≥ 2, and especially, the differential ∂_{r,n−r}^{r} : E_{r,n−r}^{r} → E_{0,n−1}^{2} is the zero map for every such r. But notice that E_{n,0}^{∞} = Ker(∂_{n,0}^{n} : E_{n,0}^{2} → E_{0,n−1}^{2}), so finally E_{n,0}^{∞} = E_{n,0}^{2}. The subquotients of the filtration on the abutment H_{n}(f′, B) vanish, except possibly for E_{0,n}^{∞} and E_{n,0}^{∞}, so we arrive at the short exact sequence :

0 → E_{0,n}^{∞} → H_{n}(f′, B) → E_{n,0}^{2} → 0.

where E_{0,n}^{∞} is a quotient of E_{0,n}^{2} = B ⊗_{A} H_{n}(f, A). Since φ_{n} is surjective by assumption, we deduce that E_{n,0}^{2} = 0, i.e. Tor_{n}^{A}(B, A/I) = 0, as required.

(ii): To ease notation, set H_{i}(A) := H_{i}(f, A) and H_{i,n}(A) := H_{i}(f, A/I^{n}) for every i, n ∈ ℕ, and define likewise H_{i}(B) and H_{i,n}(B). The short exact sequences 0 → I^{n} → A → A/I^{n} → 0 and 0 → I^{n}B → B → B/I^{n}B → 0 yield a commutative ladder :

\xymatrix{B⊗_{A}H_{i+1}(A) \ar[r]^{ψ_{i+1,n}} \ar[d]_{φ_{i+1}} & B⊗_{A}H_{i+1,n}(A) \ar[r] \ar[d]_{β_{i+1,n}} & B⊗_{A}H_{i}(f, I^{n}) \ar[r] \ar[d]^{γ_{i,n}} & B⊗_{A}H_{i}(A) \ar[r]^{ψ_{i,n}} \ar[d]^{φ_{i}} & B⊗_{A}H_{i,n}(A) \ar[d]^{β_{i,n}} \\ H_{i+1}(B) \ar[r]^{ψ′_{i+1,n}} & H_{i+1,n}(B) \ar[r] & H_{i}(f′, I^{n}B) \ar[r] & H_{i}(B) \ar[r]^{ψ′_{i,n}} & H_{i,n}(B)}

whose bottom horizontal row is exact, and the same holds for its top horizontal row, since we know already that (a) implies (e), and since H_{i}(f, M) is an A/I-module for every A-module M (lemma 7.8.2(ii)). By the same token, β_{i,n} is bijective for every i, n ∈ ℕ, so the same holds for γ_{i,n}, by the five lemma, whence the assertion.

(iii): Under the assumptions of (iii), the horizontal rows of the commutative ladder in the proof of (ii) are still exact, and both (B ⊗_{A} H_{i}(f, I^{n}) | n ∈ ℕ) and (H_{i}(f′, I^{n}B) | n ∈ ℕ) are essentially zero inverse systems, for every i > 0. The same also holds trivially for i = 0. In light of lemma 14.2.14(iii), it follows that the systems (ψ_{i,n} | n ∈ ℕ) and (ψ′_{i,n} | n ∈ ℕ) induce isomorphisms :

B ⊗_{A} H_{i}(A) →^{∼} lim_{n∈ℕ} B ⊗_{A} H_{i,n}(A) H_{i}(B) →^{∼} lim_{n∈ℕ} H_{i,n}(B).

Moreover, β_{i,n} is still an isomorphism for every n ∈ ℕ, whence (a). □

Corollary 14.2.50. Let (A, 𝔪) be a local noetherian ring, f a system of parameters for A, and g : B → B′ a homomorphism of A-algebras. Endow B and B′ with their 𝔪-adic topologies, and let g (resp. g′) be the image of f in B (resp. in B′). Suppose that g is adically flat; then :

(i) If B satisfies condition (c)_{g} and B′ satisfies condition (c)_{g′} of (7.8.21), we have :

depth_{A}B′ ≥ depth_{A}B.

(ii) If B′ is complete and separated, and if depth_{A}B ≥ dim A, then depth_{A}B′ ≥ dim A. (iii) If B and B′ are complete and separated, g is adically faithfully flat, and depth_{A}B′ ≥

dim A, then depth_{A}B ≥ dim A.

Proof. (i) follows directly from propositions 14.2.49(iii) and 10.4.32(i), and lemma 7.8.2(v).

(ii): The assumptions trivially imply that B satisfies condition (d)_{f} of (7.8.21), so it also satisfies condition (c)_{f} (proposition 7.8.25(i)). Then B′ satisfies condition (c)_{f}, by proposition 14.2.46(i), and we apply (i) to conclude.

(iii): The assumptions trivially imply that B′ satisfies condition (d)_{f} of (7.8.21), so it also satisfies condition (c)_{f} (proposition 7.8.25(i)). Then B satisfies condition (c)_{f}, by proposition 14.2.46(i). Then, by proposition 14.2.49(iii), we have B′ ⊗_{B} H_{i}(g, B) = 0 for every i > 0; but since g is adically faithfully flat, and since H_{i}(f, B) is an A/I-module, we get H_{i}(g, B) = 0 for every i > 0. Then the assertion follows from proposition 10.4.32(i), and lemma 7.8.2(v). □

14.3. Quasi-coherent sheaves of almost modules and almost rings. Throughout this section, we fix a basic setup (V, 𝔪) (see [75, §2.1.1]) and we set S := Spec V; for every S-scheme X we may consider the sheaf 𝒪_{X}^{a} of almost algebras on X, and we refer to [75, §5.5] for the definition of quasi-coherent 𝒪_{X}^{a}-modules and algebras. However, whereas in [75] it was assumed that 𝔪̃ := 𝔪 ⊗_{V} 𝔪 is a flat V -module, in this section the basic setup can be arbitrary, except where it is explicitly said otherwise. This of course requires some care when quoting from [75]; however, it turns out that – thanks to the work done in section 14.1 – most of the results in [75] do extend verbatim to the case of a general setup. The main exception is the theory of the finite and almost finite rank of almost projective modules, that relies on the existence of a well behaved exterior power functor, which is available only if the basic setup satisfies some minimal conditions (see (14.1.61)). In any case, whenever we need to import some theorem from [75], we shall comment on its range of validity.

Definition 14.3.1. Let X be an S-scheme, 𝒜 a quasi-coherent 𝒪_{X}^{a}-algebra, and ℱ an 𝒜 - module which is quasi-coherent as an 𝒪_{X}^{a}-module.

(i) ℱ is said to be an 𝒜 -module of almost finite type (resp. of almost finite presentation, resp flat, resp. faithfully flat) if, for every affine open subset U ⊂ X, the 𝒜 (U)-module ℱ (U) is almost finitely generated (resp. almost finitely presented, resp. flat, resp. faithfully flat).

(ii) ℱ is said to be an almost coherent 𝒜 -module if it is an 𝒜 -module of almost finite type, and for every open subset U ⊂ X, every quasi-coherent 𝒜_{|U}-submodule of ℱ_{|U} of almost finite type, is almost finitely presented.

(iii) We say that ℱ is a torsion-free 𝒪_{X}^{a}-module if we have Ker b·1_{ℱ (U)} = 0, for every affine open subset U ⊂ X and every regular element b ∈ 𝒪_{X}(U).

(iv) Suppose that 𝔪 satisfies condition (B) of [75, §2.1.6]. Then we say that ℱ is an 𝒜 - module of almost finite rank (resp. of finite rank) if, for every affine open subset U ⊂ X, the 𝒜 (U)-module ℱ (U) is almost finitely generated projective of almost finite (resp. finite) rank.

Remark 14.3.2. In the situation of definition 14.3.1:

(i) We denote by

ℰnd_{𝒜} (ℱ)

the 𝒪_{X}-module of 𝒜 -linear endomorphisms of ℱ, defined by the rule : U ↦ End_{𝒜 (U)} (ℱ (U)) for every open subset U ⊂ X. It is easily seen that ℰnd_{𝒜} (ℱ)^{a} is an 𝒜 -module.

(ii) Suppose that ℱ is a flat and almost finitely presented 𝒜 -module. Then there exists a trace morphism

tr_{ℱ /𝒜} : ℰnd_{𝒜} (ℱ) → 𝒜

that, on every affine open subset U ⊂ X, induces the trace morphism tr_{ℱ (U)𝒜 (U)} of the almost finitely generated projective 𝒜 (U)-module ℱ (U) (details left to the reader : see [75, §4.1], which does not depend on any assumption on the basic setup).

14.3.3. Let f : Y → X be any morphism of V -schemes. The usual functor f^{∗} for quasicoherent 𝒪_{X}-module admits a variant for quasi-coherent 𝒪_{X}^{a}-modules; namely, we define

f^{∗} : 𝒪_{X}^{a}-Mod_{qcoh} → 𝒪_{Y}^{a} -Mod_{qcoh} ℱ ↦ (f^{∗}ℱ_{!})^{a}.

If f is also quasi-compact and quasi-separated, we have as well a direct image functor

f_{∗} : 𝒪_{Y}^{a} -Mod_{qcoh} → 𝒪_{X}^{a}-Mod_{qcoh} 𝒢 ↦ (f_{∗}𝒢_{!})^{a}

([59, Ch.I, Prop.9.2.1]). Moreover, if ℱ (resp. 𝒢) is any quasi-coherent 𝒪_{X}-module (resp. 𝒪_{Y} -module), it is easily seen that the natural morphism

f^{∗}(ℱ^{a}) → (f^{∗}ℱ)^{a} resp. f_{∗}(𝒢^{a}) → (f_{∗}𝒢)^{a}

is an isomorphism. It follows that, if Z ⊂ X is any constructible closed subset, there is a well defined functor of sections with support in Z :

\underline{Γ}_{Z} : 𝒪_{X}-Mod_{qcoh} → 𝒪_{X}-Mod_{qcoh} ℱ ↦ (\underline{Γ}_{Z}ℱ_{!})^{a}

(see (10.4.16)) and again, for every quasi-coherent 𝒪_{X}-module ℱ, the natural map

\underline{Γ}_{Z}(ℱ^{a}) → (\underline{Γ}_{Z}ℱ)^{a}

is an isomorphism (details left to the reader).

Lemma 14.3.4. Suppose that 𝔪 fulfills condition (B), and let X be a reduced S-scheme, ℱ a flat quasi-coherent 𝒪_{X}^{a}-module of almost finite type. We have :

(i) If X is reduced and ℱ is an 𝒪_{X}^{a}-module of almost finite presentation, then ℱ is an 𝒪_{X}^{a}-module of almost finite rank.

(ii) If X is integral, then ℱ is an 𝒪_{X}^{a}-module of finite rank.

Proof. (i): We may assume that X is affine, say X = Spec R for a reduced V -algebra R, and ℱ = P^{∼} for a flat and almost finitely presented R^{a}-module P. Then P is almost projective ([75, Prop.2.4.18(ii)]), and we need to check that it has almost finite rank. Thus, let ε ∈ 𝔪 be any element and set R′ := R[ε^{−1}]; then P′ := R′ ⊗_{R^{a}} P is a projective R′-module of finite rank, so there exists n ∈ ℕ such that R′ ⊗_{R} Λ_{R}^{n}P = Λ_{R′}^{n}P′ = 0. Since Λ_{R}^{n}P is also almost finitely presented (see [75, §4.3.1]), we deduce easily that there exists N ∈ ℕ such that ε^{N} · Λ_{R}^{n}P = 0, and since moreover Λ_{R}^{n}P is a flat R^{a}-module and R is reduced, we have

Ann_{Λ_{R}^{n}P} (ε^{N}) = Ann_{R}(ε^{N}) · Λ_{R}^{n}P = Ann_{R}(ε) · Λ_{R}^{n}P

i.e. ε · Λ_{R}^{n}P = 0, whence the contention.

(ii): Let η be the generic point of X; if κ(η)^{a} = 0, we have 𝒪_{X}^{a} = 0 as well, and there is nothing to show. If κ(η)^{a} ≠ 0, the almost structure on κ(η) is the trivial one (the “classical limit” case of [75, Ex.2.1.2(ii)]); in this case, clearly ℱ_{η} is a free κ(η)^{a}-module of finite rank, and let r be this rank. From [75, Prop.2.4.19], it follows easily that ℱ is almost finitely presented. Moreover, the exterior powers of ℱ are still flat 𝒪_{X}^{a}-modules; then the r-th exterior power vanishes, since it vanishes at the generic point of X. □

Lemma 14.3.5. Let f : Y → X a faithfully flat and quasi-compact morphism of S-schemes, 𝒜 a quasi-coherent 𝒪_{X}^{a}-algebra, ℱ a quasi-coherent 𝒜 -module, and r ∈ ℕ. Then :

(i) The 𝒜 -module ℱ is of almost finite type (resp. of almost finite presentation, resp. flat, resp. faithfully flat) if and only if the same holds for the f^{∗}𝒜 -module f^{∗}ℱ.

(ii) Suppose that 𝔪 fulfills condition (B). Then the 𝒜 -module ℱ is of almost finite rank (resp. of finite rank ≤ r) if and only if the same holds for the f^{∗}𝒜 -module f^{∗}ℱ.

(iii) If the f^{∗}𝒜 -module f^{∗}ℱ is almost coherent, then the same holds for the 𝒜 -module ℱ.

Proof. The assertions are local on X, so we may assume that X is affine; then Y admits a finite covering (U_{i} | i ∈ I) consisting of affine open subsets, and we may further reduce to the case where Y is the disjoint union of the schemes U_{i}, so Y is affine as well.

(i): For the conditions “almost finite type”, “almost finite presentation”, and “flat”, the assertion follows from [75, Rem.3.2.26(ii)], which holds for any basic setup. It remains to check that if A → B is a faithfully flat map of V -algebras, and M is any A-module, such that (B ⊗_{A} M)^{a} is a faithfully flat B^{a}-module, then M^{a} is a faithfully flat A^{a}-module. To this aim, let X be any A-module such that (M ⊗_{A} X)^{a} = 0; then (B ⊗_{A} X)^{a} ⊗_{B^{a}} (B ⊗_{A} X)^{a} = 0, and therefore (B ⊗_{A} X)^{a} = 0, i.e. 𝔪̃ ⊗_{V} B ⊗_{A} X = 0, whence 𝔪̃ ⊗_{V} X = 0, i.e. X^{a} = 0. Since we know already that M^{a} is a flat A^{a}-module, the contention follows.

(ii): It suffices to apply [75, Rem.3.2.26(iii)], and recall that exterior powers commute with any base changes : the details shall be left to the reader.

(iii) follows easily from (i). □

Lemma 14.3.6. Let X be any S-scheme.

(i) The full subcategory 𝒪_{X}^{a}-Mod_{acoh} of 𝒪_{X}^{a}-Mod_{qcoh} consisting of all almost coherent modules is abelian and closed under extensions. (More precisely, the embedding 𝒪_{X}^{a}-Mod_{acoh} → 𝒪_{X}^{a}-Mod_{qcoh} is an exact functor.)

(ii) If X is a coherent scheme, then every quasi-coherent 𝒪_{X}^{a}-module of almost finite presentation is almost coherent.

Proof. Both assertions are local on X, hence we may assume that X is affine, say X = Spec R.

(i): Let f : M → N be a morphism of almost coherent R^{a}-modules. It is clear that Coker f is again almost coherent. Moreover, f(M) ⊂ N is almost finitely generated, hence almost finitely presented; then by [75, Lemma 2.3.18], Ker f is almost finitely generated, and so it is almost coherent as well. Similarly, using [75, Lemma 2.3.18] one sees that 𝒪_{X}^{a}-Mod_{acoh} is closed under extensions.

(ii): Let M be an almost finitely presented R^{a}-module; according to [75, Cor.2.3.13], for every finitely generated ideal 𝔪_{0} ⊂ 𝔪 there exist a finitely presented R-module N and an R-linear map φ : N → M_{∗} whose kernel and cokernel are annihilated by 𝔪_{0}. Likewise, if M′ ⊂ M is an almost finitely generated R^{a}-submodule, we may find finitely many almost elements x_{1}, …, x_{n} ∈ M′_{∗} that generate an R-module containing 𝔪_{0}M′_{∗}. Let a_{1}, …, a_{k} be a finite system of generators for 𝔪_{0}; for each i ≤ k and j ≤ n we may find y_{ij} ∈ N such that φ(y_{ij}) = a_{i}x_{j}. Denote by N′ the R-submodule of N generated by (y_{ij} | i ≤ k, j ≤ n). Then φ restricts to an R-linear map N′ → M′_{∗} whose kernel is annihilated by 𝔪_{0} and whose cokernel is annihilated by 𝔪_{0}^{2}. Since R is coherent, N′ is finitely presented, and since 𝔪_{0} is arbitrary, we conclude that M′ is almost finitely presented, as stated. □

Lemma 14.3.7. Let X be an S-scheme, φ : 𝒜 → ℬ a morphism of quasi-coherent 𝒪_{X}^{a}- algebras, and ℬ_{•} := (ℬ_{λ} | λ ∈ Λ) a filtered system of quasi-coherent and faithfully flat 𝒜 -algebras. We have :

(i) φ is faithfully flat if and only if φ is injective and Coker φ is a flat 𝒜 -module.

(ii) The colimit of ℬ_{•} is a faithfully flat 𝒜 -algebra.

Proof. It is easily seen that (i)⇒(ii). In order to show (i), we may assume that X is affine, in which case φ is the morphism of quasi-coherent 𝒪_{X}^{a}-algebras arising from a morphism f : A → B of 𝒪_{X}(X)^{a}-algebras. Now, let M be any A-module; if f is injective, the short exact sequence of A-modules 0 → A → B → Coker f → 0 induces a long exact sequence

0 → Tor_{1}^{A}(B, M) → Tor_{1}^{A}(Coker f, M) → M ⟶^{f⊗_{A}M} B ⊗_{A} M → Coker f ⊗_{A} M → 0.

Then, if Coker f is a flat A-module, we deduce that f ⊗_{A} M is injective and Tor_{1}^{A}(B, M) = 0 for every A-module M, so B is a faithfully flat A-algebra. Conversely, if the latter holds, then f and f ⊗_{M} A are both injective, hence the foregoing short exact sequence shows that Tor_{1}^{A}(Coker f, M) = 0 for every A-module M, so that Coker f is a flat A-module. □

Definition 14.3.8. Let X be an S-scheme, and 𝒜 a quasi-coherent 𝒪_{X}^{a}-algebra.

(i) We say that a quasi-coherent 𝒜 -algebra ℬ is almost finite (resp. weakly unramified, resp. weakly étale, resp. unramified, resp. étale) if, for every affine open subset U ⊂ X, the 𝒜 (U)- algebra ℬ(U) is almost finite (resp. weakly unramified, resp. weakly étale, resp. unramified, resp. étale) .

(ii) We define the 𝒪_{X}-algebra 𝒜_{!!} by the short exact sequence :

𝔪̃ ⊗_{V} 𝒪_{X} → 𝒪_{X} ⊕ 𝒜_{!} → 𝒜_{!!} → 0

analogous to [75, §2.2.25]. This agrees with the definition of [75, §3.3.2], corresponding to the basic setup (𝒪_{X}, 𝔪𝒪_{X}) relative to the Zariski topos of X; in general, this is not the same as forming the algebra 𝒜_{!!} relative to the basic setup (V, 𝔪). The reason why we prefer the foregoing version of 𝒜_{!!}, is explained by the following lemma 14.3.9.

Lemma 14.3.9. Let X be an S-scheme, 𝒜 a quasi-coherent 𝒪_{X}^{a}-algebra. Then 𝒜_{!!} is a quasicoherent 𝒪_{X}-algebra, and if 𝒜 is almost finite, then 𝒜_{!!} is an integral 𝒪_{X}-algebra.

Proof. First, 𝒜_{!} is a quasi-coherent 𝒪_{X}-module ([75, §5.5.4]), hence the same holds for 𝒜_{!!}. It remains to show that if 𝒜 is almost finite, then 𝒜_{!!}(U) = 𝒜 (U)_{!!} is an integral 𝒪_{X}(U)-algebra for every affine open subset U ⊂ X. Since 𝒜 (U)_{!!} = 𝒪_{X}(U) + 𝔪𝒜 (U)_{!!}, we need only show that every element of 𝔪𝒜 (U)_{!!} is integral over 𝒪_{X}(U). However, by adjunction we get a map :

𝒜 (U)_{!!} → A_{U} := 𝒪_{X}^{a}(U)_{∗} + 𝔪𝒜 (U)_{∗}

whose kernel is annihilated by 𝔪, and according to [75, Lemma 5.1.13(i)], A_{U} is integral over 𝒪_{X}^{a}(U)_{∗}. Let a ∈ 𝔪𝒜 (U)_{!!} be any element, and \overline{a} its image in 𝔪𝒜 (U)_{∗}; we can then find b_{0}, …, b_{n} ∈ 𝒪_{X}(U)_{∗} such that \overline{a}^{n+1} + ∑_{i=0}^{n} b_{i}\overline{a}^{i} = 0 in A_{U}. It follows that (εa)^{n+1} + ∑_{i=0}^{n} ε^{n+1−i}b_{i}(εa)^{i} = 0 in 𝒜 (U)_{!!}, for every ε ∈ 𝔪. Since ε^{n+1−i}b_{i} lies in the image of 𝒪_{X}(U) for every i ≤ n, the claim follows easily. □

Remark 14.3.10. With the notation of definition 14.3.8we have :

(i) ℬ is an unramified 𝒜 -algebra if and only if, for every affine open subset U ⊂ X, there exists an idempotent element e_{ℬ(U)/𝒜 (U)} ∈ (ℬ ⊗_{𝒜} ℬ)(U)_{∗}, uniquely characterized by the following conditions :

(a) μ_{ℬ/𝒜} (e_{ℬ(U)/𝒜 (U)} ) = 1, where μ_{ℬ/𝒜} : ℬ ⊗_{𝒜} ℬ → ℬ is the multiplication morphism of the 𝒜 -algebra ℬ.

(b) e_{ℬ(U)/𝒜 (U)} · Ker μ_{ℬ/𝒜} (U) = 0 ([75, Lemma 3.1.4]). It is easily seen that, on U′ ⊂ U, the element e_{ℬ(U)/𝒜 (U)} restricts to e_{ℬ(U′)/𝒜 (U′)} ; hence, the system (e_{ℬ(U)/𝒜 (U)} | U ⊂ X), for U ranging over all the affine open subsets of X, determines a global section

e_{ℬ/𝒜} ∈ Γ(X, ℬ ⊗_{𝒜} ℬ)_{∗}

which we call the diagonal idempotent of the 𝒜 -algebra ℬ (see [75, §5.5.4]).

(ii) In the situation of (i), let 𝒜 → 𝒜′ be any morphism of quasi-coherent 𝒪_{X}^{a}-algebras, and set ℬ′ := 𝒜′ ⊗_{𝒜} ℬ. Then the induced morphism 𝒜′ → ℬ′ is unramified, and in view of the characterization provided by (i), it is easily seen that e_{ℬ′/𝒜′} is the image in ℬ′ ⊗_{𝒜′} ℬ′ of e_{ℬ/𝒜} .

(iii) If ℬ is a flat and almost finitely presented 𝒜 -algebra, there exists a trace form

t_{ℬ/𝒜} : ℬ ⊗_{𝒜} ℬ → 𝒜

that, on each open affine subset U ⊂ X, induces the trace form t_{ℬ(U)/𝒜 (U)} of [75, §4.1.12] (details left to the reader).

(iv) Suppose that ℬ is a flat, unramified and almost finitely presented 𝒜 -algebra. Then the diagonal idempotent and the trace form of ℬ are related by the identity

t_{ℬ/𝒜} (e_{ℬ/𝒜} · (1 ⊗ b)) = b = t_{ℬ/𝒜} (e_{ℬ/𝒜} · (b ⊗ 1))

for every affine open subset U ⊂ X and every b ∈ ℬ(U)_{∗} ([75, Rem.4.1.17]).

Definition 14.3.11. (i) Let X be an S-scheme. For a monomorphism ℛ ⊂ 𝒮 of quasi-coherent 𝒪_{X}-algebras, the integral closure i.c.(ℛ, 𝒮) of ℛ in 𝒮 is a quasi-coherent 𝒪_{X}-algebra. For a monomorphism 𝒜 → ℬ of 𝒪_{X}^{a}-algebras, the integral closure of 𝒜 in ℬ is

i.c.(𝒜 , ℬ) := i.c.(𝒜_{!!}, ℬ_{!!})^{a}

(see also [75, Lemma 8.2.28]). In view of lemma 14.3.9, this is a well defined quasi-coherent 𝒪_{X}^{a}-algebra. It is characterized as the unique 𝒜 -subalgebra of ℬ such that :

i.c.(𝒜 , ℬ)(U) = i.c.(𝒜 (U), ℬ(U))

for every affine open subset U ⊂ X (notation of [75, Def.8.2.27]). We say that 𝒜 is integrally closed in ℬ, if 𝒜 = i.c.(𝒜 , ℬ). We say that ℬ is an integral 𝒜 -algebra if i.c.(𝒜 , ℬ) = ℬ.

(ii) Likewise, the weak normalization 𝒞 and the p-integral closure 𝒟 of the image of 𝒜_{!!} in ℬ_{!!} are quasi-coherent 𝒪_{X}-algebras (see (9.8.27)); then we define the weak normalization (resp. the p-integral closure) of 𝒜 in ℬ as the quasi-coherent 𝒪_{X}^{a}-algebra 𝒞^{a} (resp. 𝒟^{a}).

Remark 14.3.12. (i) In the notation of definition 14.3.11, if ℛ′ is the integral closure of ℛ in 𝒮, then ℛ^{′a} is the integral closure of ℛ^{a} in 𝒮^{a}. Indeed, we may assume that X is affine, in which case the assertion follows from [75, Lemma 8.2.28].

(ii) It follows that ℬ is an integral 𝒜 -algebra if and only if ℬ_{!!} is an integral 𝒜_{!!}-algebra. Indeed, suppose that ℬ is an integral 𝒜 -algebra, so that ℬ = i.c.(𝒜_{!!}, ℬ_{!!})^{a}, whence by adjunction, a morphism of 𝒜_{!!}-algebras ℬ_{!!} → i.c.(𝒜_{!!}, ℬ_{!!}), which must be the identity map.

(iii) Also, if ℛ′′ is the weak normalization of ℛ in 𝒮, then it follows easily from proposition 9.8.29(i,ii) that ℛ^{′′a} is the weak normalization of ℛ^{a} in 𝒮^{a} : the details are left to the reader.

Proposition 14.3.13. Suppose that the ideal 𝔪 fulfills condition (B) of [75, §2.1.6]. Let X be a quasi-compact S-scheme, 𝒜 a quasi-coherent 𝒪_{X}^{a}-algebra, and ℬ a flat 𝒜 -algebra that is almost finitely presented as an 𝒜 -module. Then ℬ(X) is an integral 𝒜 (X)-algebra.

Proof. For every b ∈ ℬ(X)_{∗} and every i ∈ ℕ, set :

c_{i}(b) := tr_{Λ_{𝒜}^{i} ℬ/𝒜} (Λ_{𝒜}^{i} (b1_{ℬ} )) ∈ 𝒜 (X)

(see remark 14.3.2(ii)). In view of [75, Lemma 8.2.28], it suffices to show

Claim 14.3.14. For every b ∈ ℬ(X)_{∗} and a ∈ 𝔪, there exists n ∈ ℕ such that :

(i) c_{i}(ab) = 0 for every i > n.

(ii) ∑_{i=0}^{n}(−ab)^{i+1} · c_{n−i}(ab) = 0.

Proof of the claim. Since X admits a finite affine covering, and since the trace morphism commutes with arbitrary base changes, we are easily reduced to the case where X is affine. In this case, set A := 𝒜 (X) and B := ℬ(X); then B is an almost finitely generated projective A-algebra, and there exists n ∈ ℕ such that a1_{B} factors through A-linear maps u : B → A^{⊕n} and v : A^{⊕n} → B ([75, Lemma 2.4.15]). Thus, ab1_{B} = v ∘ (u ∘ b1_{B}), and we get :

c_{i}(ab) = tr_{Λ_{A}^{i}A^{⊕n}/A}(Λ_{A}^{i}(u ∘ b1_{B} ∘ v)) for every i ∈ ℕ

([75, Lemma 4.1.2(i)]), from which (i) already follows. By the same token, we also obtain :

χ := ∑_{i=0}^{n}(−u ∘ b1_{B} ∘ v)^{i} · c_{n−i}(ab) = 0

([75, Prop.4.4.30]). However, the left-hand side of the identity of (ii) is none else than (−1) · v ∘ χ ∘ (u ∘ b1_{B}); whence the claim. □

The following lemma generalizes [75, Cor. 4.4.31].

Lemma 14.3.15. Let A be a V^{a}-algebra, B ⊂ A a V^{a}-subalgebra, P an almost finitely generated projective A-module, and φ an A-linear endomorphism of P. Suppose that :

(a) B is integrally closed in A.

(b) φ is integral over B_{∗}. Then we have :

(i) tr_{P/A}(φ) ∈ B_{∗}.

(ii) If 𝔪 fulfills condition (B) of [75, §2.1.6], then det(1_{P} + T φ) ∈ B_{∗}[[T]].

Proof. For the meaning of assumption (b), see the proof of [75, Cor. 4.4.31].

(i): For every ε ∈ 𝔪 we may find a free A-module L of finite rank, and A-linear morphisms u : P → L, v : L → P such that v ∘ u = ε1_{P} ([75, Lemma 2.4.15]). It follows that

tr_{L/A}(u ∘ φ ∘ v) = tr_{P/A}(φ ∘ v ∘ u) = ε · tr_{P/A}(φ)

([75, Lemma 4.1.2]). On the other hand, say that the polynomial T^{n} + ∑_{j=0}^{n−1} b_{j}T^{j} ∈ B_{∗}[T] annihilates φ; it is easily seen that T^{n} + ∑_{j=0}^{n−1} b_{j}ε^{n−j}T^{j} annihilates u ∘ φ ∘ v, so the latter is integral over B_{∗} as well, and therefore its trace lies in B_{∗} ([75, Cor.4.4.31 and Rem.8.2.30(i)]). Summing up, we have shown that ε · tr_{P/A}(φ) ∈ B_{∗} for every ε ∈ 𝔪, whence the contention.

(ii): Recall that det(1_{P} + T φ) is the power series in the variable T, whose coefficient in degree i is the trace of Λ_{A}^{i}φ on Λ_{A}^{i}P : see [75, §4.3.1, §4.3.3].

Claim 14.3.16. Let Q be another almost finitely generated projective A-module, ψ be an endomorphism of P that is also integral over B_{∗}. Then the endomorphism φ ⊗_{A} ψ of P ⊗_{A} Q is integral over B_{∗}.

Proof of the claim. The tensor product defines a map of unital associative B_{∗}-algebras

End_{A}(P) ⊗_{B_{∗}} End_{A}(Q) → End_{A}(P ⊗_{A} Q).

By assumption, the subalgebra B_{∗}[φ] ⊂ End_{A}(P) is finite over B_{∗}, and similarly for B_{∗}[ψ] ⊂ End_{A}(Q). Hence, the same holds for the image of B_{∗}[φ] ⊗_{B_{∗}} B_{∗}[ψ] in End_{A}(P ⊗_{A} Q), whence the claim. ♢

It follows easily from claim 14.3.16, that the endomorphism Λ_{A}^{i}φ of Λ_{A}^{i}P is integral over B_{∗}. Hence, we may replace P by Λ_{A}^{i}P, and reduce to showing that the trace of φ lies in B_{∗}, which is known, by (i). □

Lemma 14.3.17. Let A be a V^{a}-algebra, I ⊂ rad(A) a tight ideal (see [75, Def.5.1.5]), and P a flat and almost finitely generated A-module. Let also r ∈ ℕ. We have :

(i) If P/IP is a faithfully flat A-module, then P is a faithfully flat A-module.

(ii) If 𝔪 fulfills condition (B), and P is an almost projective A-module such that P/IP is an A/I-module of almost finite rank (resp. of finite rank ≤ r), then P is an A-module of almost finite rank (resp. of finite rank ≤ r).

Proof. (i): It suffices to show that for every almost finitely generated A-module M ≠ 0, we have M ⊗_{A} P ≠ 0. To this aim, it suffices to check that M ⊗_{A} P/IP ≠ 0; since M ⊗_{A} P/IP = (M ⊗_{A} A/I) ⊗_{A/I} P/IP, and since by assumption P/IP is a faithfully flat A/I-module, we are further reduced to showing that M ⊗_{A} A/I ≠ 0; the latter follows from [75, Lemma 5.1.7].

(ii): Suppose first that P/IP is of finite rank ≤ r, so that Λ_{A/I}^{r}(P/IP) = 0. Hence (Λ_{A}^{r}P)⊗_{A} P/IP = 0, and then [75, Lemma 5.1.7] yields Λ_{A}^{r}P = 0, as sought. In the more general case where P/IP has almost finite rank, pick n ∈ ℕ and a finitely generated subideal 𝔪_{0} ⊂ 𝔪, such that I^{n} ⊂ 𝔪_{0}. Let also 𝔪_{1} ⊂ 𝔪 be any finitely generated subideal such that 𝔪_{0} ⊂ 𝔪_{1}^{n+1}; by assumption, there exists i ∈ ℕ such that 𝔪_{1}Λ_{A/I}^{i}(P/IP) = 0, i.e. 𝔪_{1}Λ_{A}^{i}P ⊂ IΛ_{A}^{i}P. Therefore

𝔪_{0}Λ_{A}^{i}P ⊂ 𝔪_{1}^{n+1}Λ_{A}^{i}P ⊂ I^{n+1}Λ_{A}^{i}P ⊂ 𝔪_{0}IΛ_{A}^{i}P

whence 𝔪_{0}Λ_{A}^{i}P = 0, by [75, Lemma 5.1.7]. Since 𝔪_{0} is arbitrary, we deduce that P has almost finite rank, as claimed. □

Lemma 14.3.18. Let A → B be a morphism of V^{a}-algebras which is a monomorphism on the underlying V^{a}-modules, P an almost finitely generated and almost projective A-module, and r ∈ ℕ. Suppose as well that 𝔪 fulfills condition (B); the following holds :

(i) P is an A-module of almost finite rank (resp. of finite rank ≤ r) if and only if the same holds for the B-module B ⊗_{A} P.

(ii) Suppose that P has almost finite rank. Then P is a faithfully flat A-module if and only if the same holds for the B-module B ⊗_{A} P.

Proof. (i): According to proposition 14.1.56(i), the A-module Λ_{A}^{i}P is flat for every i ∈ ℕ; hence the natural A-linear morphism

Λ_{A}^{i}P → B ⊗_{A} Λ_{A}^{i}P →^{∼} Λ_{B}^{i}(B ⊗_{A} P)

is a monomorphism for every i ∈ ℕ; the assertion follows straightforwardly.

(ii): According to [75, Th.4.3.28] (which holds whenever 𝔪 fulfills condition (B)) we have V^{a}-algebras A_{0}, A′ and an isomorphism of V^{a}-algebras A →^{∼} A_{0} × A′ such that A_{0} ⊗_{A} P = 0 and A′ ⊗_{A} P is a faithfully flat A′-module. Suppose now that B ⊗_{A} P is a faithfully flat Bmodule; since (B ⊗_{A} A_{0}) ⊗_{B} (B ⊗_{A} P) ≃ B ⊗_{A} A_{0} ⊗_{A} P = 0, it follows that B ⊗_{A} A_{0} = 0.

On the other hand, the induced A-linear morphism A_{0} → B ⊗_{A} A_{0} is a monomorphism, since A_{0} is a flat A-module; thus, A_{0} = 0, and consequently P = A′ ⊗_{A} P is faithfully flat. □

Lemma 14.3.19. Let j : U → X be an open quasi-compact immersion of S-schemes, ℬ a quasi-coherent 𝒪_{U}^{a}-algebra, 𝒜 a quasi-coherent 𝒪_{X}^{a}-algebra, and j^{∗}𝒜 → ℬ an étale morphism. Then the induced morphism

𝒜 → j_{∗}j^{∗}𝒜 → j_{∗}ℬ

is flat if and only if it is étale.

Proof. We may assume that j_{∗}ℬ is a flat 𝒜 -algebra, and it suffices to show that j_{∗}ℬ admits a diagonal idempotent as in remark 14.3.10(i). To this aim, let us remark more generally :

Claim 14.3.20. Let f : Y → X be a quasi-compact and quasi-separated morphism, ℱ a flat quasi-coherent 𝒜 -module, and 𝒢 an f^{∗}𝒜 -module, quasi-coherent as an 𝒪_{Y}^{a} -module. Then the natural map

ℱ ⊗_{𝒜} f_{∗}𝒢 → f_{∗}(f^{∗}ℱ ⊗_{f^{∗}𝒜} 𝒢)

is an isomorphism.

Proof of the claim. This is proved just as for 𝒪_{X}-modules. Namely, one reduces easily to the case where X is affine, and one needs to check that the natural map

ℱ (X) ⊗_{𝒜 (X)} 𝒢 (Y) → (f^{∗}ℱ ⊗_{f^{∗}𝒜} 𝒢)(Y)

is an isomorphism. However, by assumption Y can be covered by finitely many affine open subsets U_{1}, …, U_{n}, and the intersection U_{ij} := U_{i} ∩ U_{j} can be covered by finitely many affine open subsets U_{ij1}, …, U_{ijn}, for every i, j = 1, …, n. Since f^{∗}ℱ, 𝒢 and f^{∗}𝒜 are quasicoherent 𝒪_{Y}^{a} -modules, the natural maps

\begin{aligned}ℱ (X) ⊗_{𝒜 (X)} 𝒢 (U_{i}) & → (f^{∗}ℱ ⊗_{f^{∗}𝒜} 𝒢)(U_{i}) \\ ℱ (X) ⊗_{𝒜 (X)} 𝒢 (U_{ijk}) & → (f^{∗}ℱ ⊗_{f^{∗}𝒜} 𝒢)(U_{ijk}) \end{aligned}

are isomorphisms, for every i, j, k = 1, …, n. Then we get :

\begin{matrix}(f^{∗}ℱ ⊗_{𝒜} 𝒢)(Y) = & \xymatrix{Ker(∏_{i=1}^{n} ℱ (X) ⊗_{𝒜 (X)} 𝒢 (U_{i}) \ar[r] \ar[r] & ∏_{i,j,k=1}^{n} ℱ (X) ⊗_{𝒜 (X)} 𝒢 (U_{ijk}))} \\ = & ℱ (X) ⊗_{𝒜 (X)} \xymatrix{Ker(∏_{i=1}^{n} 𝒢 (U_{i}) \ar[r] \ar[r] & ∏_{i,j,k=1}^{n} 𝒢 (U_{ijk}))} \end{matrix}

(since ℱ (X) is a flat 𝒜 (X)-module) whence the claim. ♢

Now, set ℛ := ℬ ⊗_{j^{∗}𝒜} ℬ; since j is quasi-compact and j_{∗}ℬ is a flat 𝒜 -algebra, claim 14.3.20implies that the natural morphism :

j_{∗}ℬ ⊗_{𝒜} j_{∗}ℬ → j_{∗}ℛ

is an isomorphism. Especially, the diagonal idempotent of ℬ extends to a global section of j_{∗}ℬ ⊗_{𝒜} j_{∗}ℬ, and the assertion follows. □

The criterion of lemma 14.3.19 has limited usefulness, since it is not usually known a priori that j_{∗}ℬ is a flat 𝒜 -algebra. In several situations, one can however apply the following variant.

Proposition 14.3.21. Let X be an S-scheme, j : U → X a quasi-compact open immersion, 𝒜 → ℬ a morphism of quasi-coherent 𝒪_{X}^{a}-algebras, and suppose that :

(a) The units of adjunction 𝒜 → j_{∗}j^{∗}𝒜 and ℬ → j_{∗}j^{∗}ℬ are monomorphisms.

(b) 𝒜 is integrally closed in j_{∗}j^{∗}𝒜, and ℬ is an integral 𝒜 -algebra.

(c) j^{∗}ℬ is an étale j^{∗}𝒜 -algebra and an almost finitely presented j^{∗}𝒜 -module.

(d) The diagonal idempotent e_{j^{∗}ℬ/j^{∗}𝒜} is a global section of the subalgebra

Im(ℬ ⊗_{𝒜} ℬ → j_{∗}j^{∗}(ℬ ⊗_{𝒜} ℬ)).

Then ℬ is an étale 𝒜 -algebra and an almost finitely presented 𝒜 -module.

Proof. We may assume that X is affine. Under our assumptions, the restriction map 𝒜 (X)_{∗} → 𝒜 (U)_{∗} is injective, and its image is integrally closed in 𝒜 (U)_{∗} ([75, Rem.8.2.30]); moreover, ℬ(X)_{!!} is an integral 𝒜 (X)_{!!}-algebra (remark 14.3.12(ii)). In view of lemma 14.3.15(i), we deduce a commutative diagram

(14.3.22)  [figure: ℬ ⊗_{𝒜} ℬ; t; 𝒜; j_{∗}j^{∗}(ℬ ⊗_{𝒜} ℬ); j_{∗}(t_{j^{∗}ℬ/j^{∗}𝒜} ); j_{∗}j^{∗}𝒜]

whose vertical arrows are the units of adjunctions, and where t is a uniquely determined 𝒜 - bilinear form. Then we are reduced to showing :

Claim 14.3.23. Let j : U → X be as in the proposition, and 𝒜 → ℬ a morphism of quasicoherent 𝒪_{X}^{a}-algebras fulfilling conditions (a), (c), (d) of the proposition, and such that there exists a bilinear form t making commute diagram 14.3.22. Then ℬ is an étale 𝒜 -algebra and an almost finitely presented 𝒜 -module.

Proof of the claim. The proof proceeds by a familiar argument (see e.g. the proof of [75, Claim 3.5.33]). Namely, for a given ε ∈ 𝔪, we can write

ε · e_{j^{∗}ℬ/j^{∗}𝒜} = ∑_{j=1}^{m} a_{j} ⊗ b_{j} where a_{j}, b_{j} ∈ ℬ(X)_{∗} for every j = 1, …, m.

In light of remark 14.3.10(iv), we deduce

ε · a = ∑_{j=1}^{m} t(a ⊗ a_{j}) · b_{j} for every a ∈ ℬ(X)_{∗}.

Indeed, the identity holds after restriction to U, and assumption (a) implies that the restriction map ℬ(X)_{∗} → ℬ(U)_{∗} is injective. We may then define 𝒜 -linear maps : ℬ ⟶^{φ} 𝒜^{⊕m} ⟶^{ψ} ℬ by the rules :

φ(a) := (t(a ⊗ a_{1}), …, t(a ⊗ a_{m})) ψ(s_{1}, …, s_{m}) := ∑_{j=1}^{m} s_{j}b_{j}

for every open subset U ⊂ X, every a ∈ ℬ(U)_{∗}, and every s_{1}, …, s_{m} ∈ 𝒜 (U)_{∗}. Thus we have ψ ∘φ = ε1_{ℬ} , and since ε is arbitrary, this already proves that ℬ is an 𝒜 -module of almost finite type, so we may apply [75, Lemma 2.4.15 and Prop.2.4.18(i)] to deduce that ℬ is a flat and almost finitely presented 𝒜 -module. Now, assumption (a) and claim 14.3.20imply that the horizontal arrows of the induced diagram

[figure: ℬ ⊗_{𝒜} ℬ; μ_{ℬ/𝒜}; j_{∗}j^{∗}(ℬ ⊗_{𝒜} ℬ); j_{∗}j^{∗}μ_{ℬ/𝒜}; ℬ; j_{∗}j^{∗}ℬ]

are monomorphisms. From the characterization of remark 14.3.10(i), it follows easily that the section e_{j^{∗}ℬ/j^{∗}𝒜} is the diagonal idempotent for the morphism 𝒜 → ℬ, so ℬ is an unramified 𝒜 -algebra, and the proof is concluded. □

Corollary 14.3.24. Let X be an S-scheme, j : U → X a quasi-compact open immersion, 𝒜 → ℬ a morphism of quasi-coherent 𝒪_{U}^{a}-algebras, and suppose that :

(a) ℬ is an étale 𝒜 -algebra and an almost finitely presented 𝒜 -module.

(b) The diagonal idempotent e_{ℬ/𝒜} is a global section of the subalgebra

Im((j_{∗}ℬ) ⊗_{j_{∗}𝒜} (j_{∗}ℬ) → j_{∗}(ℬ ⊗_{𝒜} ℬ)).

Then j_{∗}ℬ is an étale j_{∗}𝒜 -algebra and an almost finitely presented j_{∗}𝒜 -module.

Proof. Indeed, the induced morphism j_{∗}𝒜 → j_{∗}ℬ fulfills conditions (a), (c) and (d) of proposition 14.3.21, and diagram 14.3.22 trivially commutes with t := j_{∗}(t_{ℬ/𝒜} ), so the corollary follow from claim 14.3.23. □

Using lemma 14.3.15 we can also relax one assumption in [75, Prop.8.2.31(i)]; namely, we have the following :

Proposition 14.3.25. Let A ⊂ B be a pair of V^{a}-algebras, such that A = i.c.(A, B). Then, for every étale almost finite projective A-algebra A_{1} we have A_{1} = i.c.(A_{1}, A_{1} ⊗_{A} B).

Proof. Set B_{1} := A_{1} ⊗_{A} B, and suppose that x ∈ B_{1∗} is integral over A_{1∗}. Let e ∈ (A_{1} ⊗_{A} A_{1})_{∗} be the diagonal idempotent of the unramified A-algebra A_{1}; for given ε_{1}, ε_{2}, ε_{3} ∈ 𝔪 we write ε_{1} · e = ∑_{j=1}^{k} c_{i} ⊗ d_{i} for some c_{i}, d_{i} ∈ A_{1∗}. According to remark 14.3.10(iv) and [75, Prop.4.1.8(ii)], we have ∑_{j=1}^{k} c_{i} · Tr_{B_{1}/B}(xd_{i}) = ε · x. On the other hand, ε_{2} · x and ε_{3} · d_{i} are integral over A_{∗} (by [75, Lemma 5.1.13(i)], which holds for arbitrary basic setups); then lemma 14.3.15(i) implies that Tr_{B_{1}/B}(ε_{2}ε_{3} · xd_{i}) ∈ A_{∗} for every i = 1, …, k. Hence ε_{1}ε_{2}ε_{3} · x ∈ A_{1∗}, and since ε_{1}, ε_{2}, ε_{3} are arbitrary, the assertion follows. □

Lemma 14.3.26. Let X be an S-scheme, 𝒜 a quasi-coherent 𝒪_{X}^{a}-algebra. We have :

(i) Let f : Y → X a faithfully flat and quasi-compact morphism. Then 𝒜 is an étale (resp. weakly étale, resp. weakly unramified) 𝒪_{X}^{a}-algebra if and only if f^{∗}𝒜 is an étale (resp. weakly étale, resp. weakly unramified) 𝒪_{Y}^{a} -algebra.

(ii) Suppose that X is integral and 𝒜 is a weakly étale 𝒪_{X}^{a}-algebra. Let η ∈ X be the generic point; then 𝒜 is an étale 𝒪_{X}^{a}-algebra if and only if 𝒜_{η} is an étale 𝒪_{X,η}^{a}-algebra.

(iii) Suppose that X is normal and irreducible, and 𝒜 is integral, torsion-free and unrami-

fied over 𝒪_{X}^{a}. Then 𝒜 is étale over 𝒪_{X}^{a}, and almost finitely presented as an 𝒪_{X}^{a}-module. (iv) Let j : X → Y be an open quasi-compact immersion of S-schemes, with Y normal and

irreducible, and such that j_{∗}𝒪_{X} = 𝒪_{Y}, and suppose that 𝒜 is integral and torsion-free over 𝒪_{X}^{a}. Then j_{∗}𝒜 is an integral 𝒪_{Y}^{a}-algebra.

Proof. (i): Arguing as in the proof of lemma 14.3.5, one reduces easily to the case where X and Y are affine, in which case the assertion follows from [75, §3.4.1].

(ii): Let U ⊂ X be any affine open subset; by assumption 𝒜 (U) is flat over the almost ring B := 𝒜 (U) ⊗_{𝒪_{X}^{a}(U)} 𝒜 (U), and 𝒜 (U) ⊗_{𝒪_{X}^{a}(U)} κ(η)^{a} is almost projective over B ⊗_{𝒪_{X}^{a}(U)} κ(η)^{a}. Then the assertion follows from [75, Prop.2.4.19].

(iii): Let η ∈ X be the generic point. We begin with the following :

Claim 14.3.27. Suppose that V = 𝔪 (the “classical limit” case of [75, Ex.2.1.2(ii)]), and let f : A → B be a local homomorphism of local rings. Then :

(i) f^{a} is weakly étale if and only if f extends to an isomorphism of strict henselizations A^{sh} → B^{sh}.

(ii) Especially, if A is a field and f^{a} is weakly étale, then B is a separable algebraic extension of A.

Proof of the claim. (i): In the classical limit case, a weakly étale morphism is the same as an “absolutely flat” map as defined in [137]; then the assertion is none else than [137, Th.5.2].

(ii): If A is a field, A^{sh} is a separable closure of A; hence (i) implies that B is a subring of A^{sh}, hence it is a subfield of A^{sh}. ♢ Claim 14.3.28. Suppose that V = 𝔪. Let A be a field and f : A → B a ring homomorphism such that f^{a} is weakly étale. Then :

(i) Every finitely generated A-subalgebra of B is finite étale over A (that is, in the usual sense of [66, Ch.IV, §17]).

(ii) f^{a} is étale if and only if f is étale (in the usual sense of [66]).

Proof of the claim. (i): From claim 14.3.27(ii) we see that B is reduced of Krull dimension ≤ 0, and all its residue fields are separable algebraic extensions of A. We consider first the case of a monogenic extension C := A[b] ⊂ B. For every prime ideal 𝔭 ⊂ B, let b_{𝔭} be the image of b in B_{𝔭}; then for every such 𝔭 we may find an irreducible separable monic polynomial P_{𝔭}(T) ∈ A[T] with P_{𝔭}(b_{𝔭}) = 0. This identity persists in an open neighborhood U_{𝔭} ⊂ Spec B of 𝔭, and finitely many such U_{𝔭} suffice to cover Spec B. Thus, we find finitely many P_{𝔭_{1}}(T), …, P_{𝔭_{k}}(T) such that ∏_{i=1}^{k} P_{𝔭_{i}}(b) = 0 holds in B, and after omitting repetitions, we may assume that all these polynomials are distinct. Since C is a quotient of the separable A-algebra A[T]/(∏_{i=1}^{k} P_{𝔭_{i}}(T)), the claim follows in this case. In the general case, we may write C = A[b_{1}, …, b_{n}] for certain b_{1}, …, b_{n} ∈ B. Then Spec C is a reduced closed subscheme of Spec A[b_{1}] ×_{Spec A} ⋯ ×_{Spec A} Spec A[b_{n}]; by the foregoing, the latter is étale over Spec A, hence the same holds for Spec C.

(ii): We may assume that f^{a} is étale, and we have to show that f is étale, i.e. that B is a finitely generated A-module. Hence, let e_{B/A} ∈ B ⊗_{A} B be the diagonal idempotent (see remark 14.3.10(i)); we choose a finitely generated A-subalgebra C ⊂ B such that e_{B/A} is the image of an element e′ ∈ C ⊗_{A} C. Notice that 1 − e′ lies in the kernel of the multiplication map μ_{C/A} : C ⊗_{A} C → C. By (i), the morphism A → C is étale, hence it admits as well a diagonal idempotent e_{C/A} ∈ C ⊗_{A} C; on the other hand, [75, Lemma 3.1.2(v)] says that B^{a} is étale over C^{a}, especially B is a flat C-algebra, hence the natural map C ⊗_{A} C → B ⊗_{A} B is injective. Since 1 − e_{C/A} lies in the kernel of the multiplication map μ_{B/A} : B ⊗_{A} B → B, we have :

e_{B/A}(1 − e_{C/A}) = 0 = e_{C/A}(1 − e′) = e_{C/A}(1 − e_{B/A})

from which it follows easily that e_{B/A} = e_{C/A}. Moreover, the induced morphism Spec B → Spec C has dense image, so it must be surjective, since Spec C is a discrete finite set; therefore B is even a faithfully flat C-algebra. Let J := Ker(B ⊗_{A} B → B ⊗_{C} B); then J is the ideal generated by all elements of the form 1 ⊗ c − c ⊗ 1, where c ∈ C; clearly this is the same as the extension of the ideal I_{C/A} := Ker μ_{C/A}. However, I_{C/A} is generated by the idempotent 1 − e_{C/A} ([75, Cor.3.1.9]), consequently J is generated by 1 − e_{B/A}, i.e. J = Ker μ_{B/A}. So finally, the multiplication map μ_{B/C} : B ⊗_{C} B → B is an isomorphism, whose inverse is the map B → B⊗_{C} B : b ↦ b⊗1. The latter is of the form j⊗_{C} 1_{B}, where j : C → B is the natural inclusion map. By faithfully flat descent we conclude that C = B, whence the claim. ♢

Next, we remark that 𝒜_{η} is a finitely presented 𝒪_{X,η}^{a}-module. Indeed, since K := 𝒪_{X,η} is a field, we have either 𝔪K = 0, in which case the category of K^{a}-modules is trivial and there is nothing to show, or else 𝔪K = K. So we may assume that we are in the “classical limit” case, and then the assertion follows from claim 14.3.28(ii).

Now, set ℬ := 𝒜 ⊗_{𝒪_{X}^{a}} 𝒜, and let j : X(η) → X be the natural morphism; since X is normal and 𝒜 is torsion-free, the units of adjunction 𝒪_{X}^{a} → j_{∗}j^{∗}𝒪_{X(η)} and 𝒜 → j_{∗}j^{∗}𝒜 are monomorphisms. Since 𝒜 is unramified over 𝒪_{X}^{a}, the diagonal idempotent of j^{∗}𝒜 lies in the image of the restriction map ℬ(X)_{∗} → ℬ_{η∗}. In view of these observations, an easy inspection shows that the proof of proposition 14.3.21 carries over verbatim to the current situation, and yields assertion (iii).

(iv): Let 𝒯 ⊂ 𝒜_{!!} be the maximal torsion 𝒪_{X}-subsheaf, and set ℬ := 𝒜_{!!}/𝒯 . Then ℬ is an integral, torsion-free 𝒪_{X}-algebra, by remark 14.3.12(ii), and ℬ^{a} ≃ 𝒜, hence (j_{∗}ℬ)^{a} ≃ j_{∗}𝒜. Let η ∈ Y be the generic point; in light of remark 14.3.12(ii), it then suffices to show :

Claim 14.3.29. Under the assumptions of (v), let ℛ be an integral quasi-coherent and torsionfree 𝒪_{X}-algebra. Then j_{∗}ℛ is an integral 𝒪_{Y} -algebra.

Proof of the claim. Let us write ℛ_{η} as the filtered union of the family (R_{λ} | λ ∈ Λ) of its finite 𝒪_{Y,η}-subalgebras, and for every λ ∈ Λ, let ℛ_{λ} ⊂ ℛ be the quasi-coherent 𝒪_{X}-subalgebra such that ℛ_{λ}(V) = R_{λ} ∩ ℛ(V) for every non-empty open subset V ⊂ X. Then ℛ is the filtered colimit of the family (ℛ_{λ} | λ ∈ Λ), and clearly it suffices to show the claim for every ℛ_{λ}. We may thus assume from start that ℛ_{η} is a finite 𝒪_{Y,η}-algebra. Let ℛ^{ν} be the integral closure of ℛ, i.e. the quasi-coherent 𝒪_{X}-algebra such that ℛ^{ν}(V) is the integral closure of ℛ(V) in ℛ_{η}, for every non empty affine open subset V ⊂ X. Clearly it suffices to show that j_{∗}ℛ^{ν} is integral, hence we may replace ℛ by ℛ^{ν} and assume from start that ℛ is integrally closed. In this case, for every non-empty open subset V ⊂ X the restriction map ℛ(V) → ℛ_{η} induces a bijection between the idempotents of ℛ(V) and those of ℛ_{η}. Especially, ℛ(X) admits finitely many idempotents, and moreover we have a natural decomposition ℛ = ℛ_{1} × ⋯ × ℛ_{k}, as a product of 𝒪_{X}-algebras, such that ℛ_{i,η} is a field for every i ≤ k. It then suffices to show the claim for every j_{∗}ℛ_{i}, and then we may assume throughout that ℛ_{η} is a field. Up to replacing ℛ by its normalization in a finite extension of ℛ_{η}, we may even assume that ℛ_{η} is a finite normal extension of 𝒪_{Y,η}. Hence, let V ⊂ Y be any non-empty affine open subset, and a ∈ ℛ(X ∩ V) any element. Let P (T) be the minimal polynomial of a over the field 𝒪_{Y,η}; we have to show that the coefficients of P (T) lie in 𝒪_{Y} (V) = 𝒪_{X}(X ∩ V), and since Y is normal, it suffices to prove that these coefficients are integral over 𝒪_{X}(W), for every non-empty affine open subset W ⊂ X ∩ V. However, since ℛ_{η} is normal over 𝒪_{Y,η}, such coefficients can be written as some elementary symmetric polynomials of the conjugates of a in ℛ_{η}. Hence, we come down to showing that the conjugates of a are still integral over 𝒪_{X}(W). The latter assertion is clear: indeed, if Q(T) ∈ 𝒪_{X}(W)[T] is a monic polynomial with Q(a) = 0, then we have also Q(a′) = 0 for every conjugate a′ of a. □

Proposition 14.3.30. Let X be a quasi-compact and quasi-separated S-scheme, j : U → X a quasi-compact open immersion, and 𝒜 a quasi-coherent 𝒪_{U}^{a}-algebra, almost finitely presented as an 𝒪_{U}^{a}-module. Then for every finitely generated subideal 𝔪_{0} ⊂ 𝔪, there exist a quasicoherent 𝒪_{X}-algebra ℬ, finitely presented as an 𝒪_{X}-module, and a morphism ℬ_{|U}^{a} → 𝒜 of 𝒪_{U}^{a}-algebras, whose kernel and cokernel are annihilated by 𝔪_{0}.

Proof. Set 𝒞 := i.c.(𝒪_{X}^{a}, j_{∗}𝒜); then 𝒞 is a quasi-coherent 𝒪_{X}^{a}-algebra, and 𝒞_{|U} = 𝒜, by lemma 14.3.9. According to proposition 10.3.31, we may write 𝒞_{!!} as the colimit of a filtered family (ℱ_{i} | i ∈ I) of finitely presented quasi-coherent 𝒪_{X}-modules. Pick a finitely generated subideal 𝔪_{1} ⊂ 𝔪 such that 𝔪_{0} ⊂ 𝔪_{1}^{2}; for every affine open subset U′ ⊂ U, we may find i ∈ I such that 𝔪_{1} · 𝒜_{!!}(U′) ⊂ Im(ℱ_{i}(U′) → 𝒜_{!!}(U′)), and since U is quasi-compact, finitely many such open subsets cover U; hence, we may find i ∈ I such that 𝔪_{1} · 𝒜_{!!} ⊂ Im(ℱ_{i|U} → 𝒜_{!!}); we set ℱ := ℱ_{i}, and let φ : 𝒢 := Sym_{𝒪_{X}}^{•} ℱ → 𝒞_{!!} be the induced morphism of quasi-coherent 𝒪_{X}- algebras. Notice that 𝒢 is finitely presented as an 𝒪_{X}-algebra. Using again proposition 10.3.31 (or [59, Ch.I, Cor.9.4.9]) we may write Ker φ as the colimit of a filtered family (𝒦_{λ} | λ ∈ Λ) of quasi-coherent 𝒪_{X}-submodules of finite type. Fix a finite covering (U′_{i} | i ∈ I) of X consisting of affine open subsets, and for every i ∈ I, let f_{i,1}, …, f_{i,n(i)} be a finite set of generators of ℱ (U′_{i}); we may find monic polynomials P_{i,1}(T), …, P_{i,n(i)}(T) with coefficients in 𝒪_{X}(U′_{i}) such that P_{i,j}(φ(f_{i,j})) = 0 in 𝒞 (U′_{i})_{!!}, for every i ∈ I and every j ≤ n(i). Let λ ∈ Λ be chosen large enough, so that P_{i,j}(f_{i,j}) ∈ 𝒦_{λ}(U′_{i}) for every i ∈ I and every j ≤ n(i). Let also 𝒦′_{λ} ⊂ 𝒢 be the ideal generated by 𝒦_{λ}; then 𝒢′ := 𝒢 /𝒦′_{λ} is an integral 𝒪_{X}-algebra of finite presentation, hence it is finitely presented as an 𝒪_{X}-module, in view of claim 9.1.35. Clearly φ_{|U} descends to a map φ′ : 𝒢′_{|U} → 𝒜_{!!}, whose cokernel is annihilated by 𝔪_{1}, and by [75, Claim 2.3.12 and the proof of Cor.2.3.13] we may find, for every affine open subset U′ ⊂ U, a finitely generated submodule K_{U′} ⊂ Ker (φ′_{U′} : 𝒢′(U′) → 𝒜 (U′)_{!!}) such that 𝔪_{1}^{2} · Ker φ′_{U′} ⊂ K_{U′}. Another invocation of proposition 10.3.31 ensures the existence of a quasi-coherent 𝒪_{X}-submodule of finite type 𝒥 ⊂ Ker φ such that K_{U′} ⊂ 𝒥 (U′) for all the U′ of a finite covering of U; the 𝒪_{X}-algebra ℬ := 𝒢′/𝒥 · 𝒢′ fulfills the required conditions. □

14.3.31. Let (X_{i} | i ∈ I) be a cofiltered system of quasi-compact and quasi-separated Sschemes, with affine transition morphisms h_{φ} : X_{j} → X_{i}, for every morphism φ : j → i in I. Let also U_{i} ⊂ X_{i} be a quasi-compact open subset, for every i ∈ I, such that U_{j} = h_{φ}^{−1}U_{i} for every φ : j → i in I. Set

X := lim_{i∈I} X_{i} U := lim_{i∈I} U_{i}.

and denote by h_{i} : U → X_{i} the natural morphism, for every i ∈ I.

Corollary 14.3.32. In the situation of (14.3.31), let 𝒜 be any flat almost finitely presented 𝒪_{U}^{a}- algebra. Then, for every finitely generated subideal 𝔪_{0} ⊂ 𝔪 there exist i ∈ I, a quasi-coherent 𝒪_{X_{i}}-algebra ℛ, finitely presented as an 𝒪_{X_{i}}-module, and a map of 𝒪_{U}^{a}-algebras f : h_{i}^{∗}ℛ^{a} → 𝒜 such that :

(a) Ker f and Coker f are annihilated by 𝔪_{0}.

(b) For every x ∈ U_{i} and every b ∈ 𝔪_{0}, the map b · 1_{ℛ,x} : ℛ_{x} → ℛ_{x} factors through a free 𝒪_{U_{i},x}-module.

Proof. From [65, Ch.IV, Th.8.3.11] it is easily seen that X is a quasi-compact and quasiseparated S-scheme, and U is a quasi-compact open subset of X. Pick a finitely generated subideal 𝔪_{1} such that 𝔪_{0} ∈ 𝔪_{1}^{2}; according to proposition 14.3.30, we may find a quasi-coherent 𝒪_{X}-algebra ℬ, finitely presented as an 𝒪_{X}-module, and a morphism g : ℬ_{|U}^{a} → 𝒜 of 𝒪_{U}^{a}- algebras whose kernel and cokernel are annihilated by 𝔪_{1}. By standard arguments we deduce

(14.3.33)  𝔪_{1}^{2} · Tor_{𝒪_{X}(U′)}^{1} (M, ℬ(U′)) = 0

for every affine open subset U′ ⊂ U and every 𝒪_{X}(U′)-module M. Now, let us remark, quite generally :

Claim 14.3.34. Let A be any ring, M and N any two A-modules, φ : M → N any A-linear map, and M^{∨} := Hom_{A}(M, A). Then φ factors through a free A-module of finite rank if and only if it lies in the image of the natural map

(14.3.35)  M^{∨} ⊗_{A} N → Hom_{A}(M, N).

Proof of the claim. Suppose first that φ factors as the composition of A-linear maps ψ : M → A^{⊕n} and ψ′ : A^{⊕n} → N, for some n ∈ ℕ. For every i = 1, …, n, let p_{i} : A^{⊕n} → A (resp. e_{i} : A → A^{⊕n}) be the natural epimorphism (resp. monomorphism), and set ψ_{i} := p_{i} ∘ ψ, ψ′_{i} := ψ′ ∘ e_{i}. Then φ = ∑_{i=1}^{n} ψ′_{i} ∘ ψ_{i} and each summand ψ′_{i} ∘ ψ_{i} lies in the image of (14.3.35). Conversely, if φ lies in the image of (14.3.35), we have φ = ∑_{i=1}^{n} ψ′_{i} ∘ ψ_{i}, for some ψ_{1}, …, ψ_{n} ∈ M^{∨} and ψ′_{1}, …, ψ′_{n} ∈ Hom_{A}(A, N) = N, and we may define ψ := ∑_{i=1}^{n} e_{i} ∘ ψ_{i}, ψ′_{i} := ∑_{i=1}^{n} ψ′_{i} ∘ p_{i}. Then φ = ψ′ ∘ ψ. ♢

Now, pick a finite covering (U′_{i,λ} | λ ∈ Λ) of U_{i}, consisting of affine open subsets, and a finite system b_{1}, …, b_{k} of generators of 𝔪_{1}^{2}, and set U′_{λ} := h_{i}^{−1}U′_{i,λ} for every λ ∈ Λ; from claim 14.3.34, (14.3.33) and [75, Lemma 2.4.17] we deduce that for every λ ∈ Λ and every j = 1, …, k there exists n(λ, j) ∈ ℕ such that b_{j} · (1_{ℬ} )_{|U′_{λ}} factors through 𝒪_{U′_{λ}}^{⊕n(λ,j)}. Next, by virtue of [65, Ch.IV, Th.8.5.2] we may assume that ℬ descends to a 𝒪_{X_{i}}-algebra ℛ, finitely presented as an 𝒪_{X_{i}}-module, and such that b_{j} · (1_{ℛ} )_{|U′_{i,λ}} factors through 𝒪_{U′_{i,λ}}^{⊕n(λ,j)} for every j = 1, …, k and every λ ∈ Λ. It follows easily that

𝔪_{1}^{2} · Tor_{𝒪_{X_{i}}(U′_{i,λ})}^{1} (M, ℛ(U′_{i,λ})) = 0

for every λ ∈ Λ and every 𝒪_{X_{i}}(U′_{i,λ})-module M. Then, again [75, Lemma 2.4.17] and claim 14.3.34imply that condition (b) holds for ℛ, and by construction we also get condition (a). □

14.4. Almost pure pairs. Throughout this section, we keep the notation of (14.3), and we assume that 𝔪 fulfills condition (B).

Definition 14.4.1. Let X be an S-scheme, Z ⊂ X a closed subscheme such that U := X \Z is a dense subset of X, and denote by j : U → X the open immersion.

(i) We say that the pair (X, Z) is almost pure relative to (V, 𝔪) (or just almost pure, when the underlying basic setup is clear from the context) if the restriction functor

(14.4.2)  𝒪_{X}^{a}-Ét_{afr} → 𝒪_{U}^{a}-Ét_{afr} : 𝒜 ↦ 𝒜_{|U}

from the category of étale 𝒪_{X}^{a}-algebras of almost finite rank, to the category of étale 𝒪_{U}^{a}-algebras of almost finite rank, is an equivalence.

(ii) We say that the pair (X, Z) is normal if Z is a constructible subset of X, and the natural map 𝒪_{X}^{a} → j_{∗}𝒪_{U}^{a} is a monomorphism, whose image is integrally closed in j_{∗}𝒪_{U}^{a}.

Remark 14.4.3. Let (X, Z) be a pair as in definition 14.4.1, where Z is a constructible subset of X, and set U := X \Z.

(i) Consider the following conditions:

(a) 𝒪_{X,z} is a normal domain, for every z ∈ Z.

(b) the natural map 𝒪_{X} → j_{∗}𝒪_{U} is a monomorphism, and 𝒪_{X} = i.c.(𝒪_{X}, j_{∗}𝒪_{U}).

(c) The pair (X, Z) is normal. Then (a)⇒(b), since (j_{∗}𝒪_{U})_{z} = 𝒪_{X(z)}(U ∩ X(z)) for every z ∈ Z. Also, (b)⇒(c), by virtue of [75, Lemma 8.2.28].

(ii) Let 𝒜 be a quasi-coherent 𝒪_{U}^{a}-algebra. Then j_{∗}𝒜_{!!} is a quasi-coherent 𝒪_{X}-algebra ([59, Ch.I, Prop.9.4.2(i)] and lemma 14.3.9), hence the integral closure 𝒜^{ν} of the image of 𝒪_{X}^{a} in j_{∗}𝒜 is a well defined quasi-coherent 𝒪_{X}^{a}-algebra (see definition 14.3.11). We call 𝒜^{ν} the normalization of 𝒜 over X.

(iii) Let (X, Z) be any normal pair, and U ⊂ X any open subset. Directly from the definition, we see that (U, Z ∩ U) is still a normal pair.

Lemma 14.4.4. Let (X, Z) be a normal pair as in definition 14.4.1(ii), and set U := X\Z. Let 𝒜 be an étale 𝒪_{U}^{a}-algebra whose underlying 𝒪_{U}^{a}-module is almost finitely presented (resp. is of finite rank). The following conditions are equivalent :

(a) The normalization 𝒜^{ν} of 𝒜 over X (see remark 14.4.3(ii)) is a weakly unramified 𝒪_{X}^{a}-algebra.

(b) 𝒜^{ν} is an étale 𝒪_{X}^{a}-algebra, and an almost finitely presented 𝒪_{X}^{a}-module (resp. and an 𝒪_{X}^{a}-module of finite rank).

Proof. Obviously (b)⇒(a), hence suppose that (a) holds, and let ℬ := 𝒜^{ν} ⊗_{𝒪_{X}^{a}} 𝒜^{ν}. We may assume that X is affine, and we set

A := 𝒜^{ν}(X) B := ℬ(X).

By assumption, the multiplication morphism μ : ℬ → 𝒜^{ν} is flat, especially A is a flat Bmodule. Let also B′ be the image of B in ℬ(U); clearly μ(X) : B → A factors through an epimorphism μ′ : B′ → A, therefore A = A ⊗_{B} B′, so A is also a flat B′-module. Pick a finite covering U = U_{1} ∪ ⋯ ∪ U_{k} consisting of affine open subsets of U. The induced morphism

B′ → ℬ(U_{1}) × ⋯ × ℬ(U_{k})

is a monomorphism. On the other hand, notice that A ⊗_{B′} ℬ(U_{i}) = 𝒜 (U_{i}) is an almost finitely presented ℬ(U_{i})-module. It follows that A is an almost finitely generated projective B′-module ([75, Prop. 2.4.18 and 2.4.19]), and therefore the kernel of μ′ is generated by an idempotent e ∈ B′_{∗} ([75, Rem. 3.1.8]). A simple inspection shows that e is necessarily the diagonal idempotent of the unramified 𝒪_{U}^{a}-algebra 𝒜. Then all the assumptions of proposition 14.3.21 are fulfilled, so 𝒜^{ν} is an étale 𝒪_{X}^{a}-algebra, and an almost finitely presented 𝒪_{X}^{a}-module.

Lastly, suppose that 𝒜 is an 𝒪_{U}^{a}-module of finite rank, and pick r ∈ ℕ such that, for every i = 1, …, k, the r-th exterior power of the 𝒪_{U}^{a}(U_{i})-module 𝒜 (U) = A⊗_{𝒪_{X}^{a}(X)} 𝒪_{U}^{a}(U_{i}) vanishes. Since the induced morphism 𝒪_{X}^{a} → 𝒪_{U}^{a}(U_{1}) × ⋯ × 𝒪_{U}^{a}(U_{1}) is a monomorphism, and A is a flat 𝒪_{X}^{a}-module, we deduce that the r-th exterior power of A vanishes as well. Especially, 𝒜^{ν} is an 𝒪_{X}^{a}-module of finite rank. □

Lemma 14.4.5. Let (X, Z) be a normal pair, j : X \Z → X the open immersion, ℬ any étale almost finitely presented 𝒪_{X}^{a}-algebra, and Z′ ⊂ Z any constructible closed subset. We have :

(i) The natural map ℬ → j_{∗}j^{∗}ℬ factors through an isomorphism ℬ →^{∼} (j^{∗}ℬ)^{ν}.

(ii) The restriction functor (14.4.2) is fully faithful.

(iii) The pair (X, Z′) is normal.

(iv) If the pair (X, Z) is almost pure, the same holds for (X, Z′).

Proof. (i): Set U := X \ Z and 𝒜 := ℬ_{|U}; the natural map ℬ → j_{∗}𝒜 factors through a morphism φ : ℬ → 𝒜^{ν} of 𝒪_{X}^{a}-algebras (lemma 14.3.9 and remark 14.3.12(ii)). Fix z ∈ Z, and set R := 𝒪_{X,z}^{a}, R′ := (j_{∗}𝒪_{X}^{a})_{z}. It follows that the stalk ℬ_{z} is an étale R-algebra. Notice that (j_{∗}𝒜)_{z} = R′ ⊗_{R} ℬ_{z}, therefore φ is a monomorphism. Moreover, since ℬ is also an almost finitely presented 𝒪_{X}^{a}-module, we have

ℬ_{z} = i.c.(ℬ_{z}, R′ ⊗_{R} ℬ_{z}) 𝒜_{z}^{ν} = i.c.(R, R′ ⊗_{R} ℬ_{z})

(proposition 14.3.25). On the other hand, ℬ_{z} is an integral R-algebra, therefore ℬ_{z} = 𝒜_{z}^{ν}, and since z is arbitrary, we conclude that φ is an isomorphism, as asserted.

(ii): Let ℬ_{1} and ℬ_{2} be two étale almost finitely presented 𝒪_{X}^{a}-algebra, set 𝒜_{i} := ℬ_{_{i}|U} for i = 1, 2, and let ψ : 𝒜_{1} → 𝒜_{2} be any morphism of 𝒪_{U}^{a}-algebras; by (i), ψ extends uniquely to a morphism ψ^{ν} : ℬ_{1} = 𝒜_{1}^{ν} → 𝒜_{2}^{ν} = ℬ_{2} of 𝒪_{X}^{a}-algebras, whence the contention.

(iii): Set U′ := X \Z′ and let j′ : U′ → X be the open immersion; the natural morphism 𝒪_{X}^{a} → j_{∗}𝒪_{U} factors through the morphism 𝒪_{X}^{a} → j′_{∗}𝒪_{U′}, so the latter is a monomorphism. In order to show that (X, Z′) is normal, it then suffices to check that the image of 𝒪_{X}^{a} is integrally closed in j′_{∗}𝒪_{U′}^{a}, and since 𝒪_{X}^{a} is integrally closed in j_{∗}𝒪_{U}^{a}, we are reduced to proving that the natural morphism j′_{∗}𝒪_{U′}^{a} → j_{∗}𝒪_{U}^{a} is a monomorphism. However, let V ⊂ X be any affine open subset, and write U′ ∩ V = V_{1} ∩ ⋯ ∩ V_{n} for certain affine open subsets V_{1}, …, V_{n} of X; by assumption, the restriction map 𝒪_{X}^{a}(V_{i}) → 𝒪_{X}^{a}(U ∩ V_{i}) is a monomorphism for every i = 1, …, n, so the same holds for the restriction map 𝒪_{X}^{a}(U′ ∩ V) → 𝒪_{X}^{a}(U ∩ V), whence the assertion.

(iv): Suppose that (X, Z) is almost pure; in view of (ii), in order to check that (X, Z′) is almost pure, it suffices to show that every étale 𝒪_{U′}^{a}-algebra 𝒜 of finite rank extends to an étale 𝒪_{X}^{a}-algebra of finite rank. However, the assumption implies that 𝒜_{|U} extends to an 𝒪_{X}^{a}- algebra ℬ as sought, and since (X \Z′, Z \Z′) is normal (remark 14.4.3(iii)), (ii) says that the isomorphism ℬ_{|U} →^{∼} 𝒜 extends to an isomorphism ℬ_{|U′} →^{∼} 𝒜, i.e. ℬ is an extension of 𝒜, as required. □

Lemma 14.4.6. Let (A_{i} | i ∈ ℕ) be a system of V^{a}-algebras, and set A := ∏_{i∈ℕ} A_{i}. Let also P be an A-module, B an A-algebra, and suppose that

(a) lim_{i→∞} Ann_{V^{a}}(A_{i}) = V^{a} for the uniform structure of [75, Def.2.3.1].

(b) For every i ∈ ℕ, the A_{i}-modules P_{i} := P ⊗_{A} A_{i}, B_{i} := B ⊗_{A} A_{i} are almost projective

of finite constant rank equal to i, and B_{i} is an étale A_{i}-algebra. Then P is an almost projective A-module of almost finite rank, and B is an étale A-algebra.

Proof. For every j ∈ ℕ, the finite product P_{≤j} := ∏_{i=1}^{j} P_{i} is an almost projective A-module of finite rank, and clearly the induced morphism π_{j} : P → P_{≤j} is an epimorphism. On the other hand, the (j + 1)-th exterior power of P equals the (j + 1)-th exterior power of Ker π_{j}, and from condition (i) we see that lim_{j→∞} Ann_{V^{a}}(Ker π_{j}) = 0, whence the assertion for P.

It follows already that B is an almost projective A-module. It remains to show that B is an unramified A-algebra, to which aim, we may apply the criterion of [75, Prop.3.1.4].

Claim 14.4.7. Under the assumptions of the lemma, the natural morphism

φ : B ⊗_{A} B → C := ∏_{i∈ℕ} B_{i} ⊗_{A_{i}} B_{i}

is an isomorphism of A-algebras.

Proof of the claim. For every j ∈ ℕ, let π_{j} : C → ∏_{i≤j} B_{i} ⊗_{A} B_{i} be the natural morphism. Then

lim_{j→∞} Ann_{V^{a}}Ker(π_{j} ∘ φ) = V^{a} = lim_{j→∞} Ann_{V^{a}}Ker π_{j}.

The first identity implies that φ is a monomorphism. Next, since π_{j} ∘ φ is an epimorphism, the natural morphism Ker π_{j} → Coker φ is an epimorphism; then the second identity implies that φ is also an epimorphism. ♢

Now, by [75, Prop.3.1.4], for every i ∈ ℕ there exists an idempotent e_{i} ∈ (B_{i} ⊗_{A_{i}} B_{i})_{∗} uniquely characterized by the conditions (i)–(iii) of loc.cit. In view of claim 14.4.7, the sequence (e_{i} | i ∈ ℕ) defines an idempotent in (B ⊗_{A} B)_{∗}, which clearly fulfills the same conditions, whence the contention, again by [75, Prop.3.1.4]. □

Proposition 14.4.8. Let (X, Z) be a normal pair, and set U := X \ Z. Then the following conditions are equivalent :

(a) The pair (X, Z) is almost pure.

(b) The restriction functor

𝒪_{X}^{a}-Ét_{fr} → 𝒪_{U}^{a}-Ét_{fr} : 𝒜 ↦ 𝒜_{|U}

from the category of étale 𝒪_{X}^{a}-algebras of finite rank, to the category of étale 𝒪_{U}^{a}- algebras of finite rank, is an equivalence.

(c) For every étale 𝒪_{U}^{a}-algebra 𝒜 of finite rank, the normalization 𝒜^{ν} of 𝒜 over X is an étale 𝒪_{X}^{a}-algebra of finite rank (see remark 14.4.3(ii)).

(d) For every étale 𝒪_{U}^{a}-algebra 𝒜 of finite rank, 𝒜^{ν} is a weakly unramified 𝒪_{X}^{a}-algebra.

(e) Every étale 𝒪_{U}^{a}-algebra 𝒜 of finite rank extends to an étale almost finite 𝒪_{X}^{a}-algebra.

Proof. The equivalence (b)⇔(c) is already clear from lemma 14.4.5(i,ii).

Next, clearly (c)⇒(d). Conversely, suppose that (d) holds, and let 𝒜 be any étale 𝒪_{U}^{a}-algebra of finite rank, so 𝒜^{ν} is a weakly unramified 𝒪_{X}^{a}-algebra. Then 𝒜^{ν} is actually an étale 𝒪_{X}^{a}- algebra, and an almost finitely presented 𝒪_{X}^{a}-module (lemma 14.4.4). Fix any affine open subset V ⊂ X, and pick a finite covering of U ∩ V consisting of affine open subsets V_{1}, …, V_{k} of U. Let r ∈ ℕ be an integer such that the r-th exterior power of the 𝒪_{X}^{a}(V_{i})-modules 𝒜 (V_{i}) vanish. The induced map

B := 𝒪_{X}^{a}(V) → B′ := 𝒪_{X}^{a}(V_{1}) × ⋯ × 𝒪_{X}^{a}(V_{k})

is a monomorphism, since (X, Z) is a normal pair. Moreover, the r-th exterior power of the B′-module 𝒜^{ν}(V) ⊗_{B} B′ vanishes; since 𝒜^{ν}(V) is a flat B-module, we conclude that the r-th exterior power of 𝒜^{ν}(V) vanishes as well. Especially, 𝒜^{ν} is of finite rank, so (c) holds.

Since (c)⇒(e), we suppose that (e) holds, and deduce that (d) holds as well. Indeed, let ℬ be an almost finite étale 𝒪_{X}^{a}-algebra extending 𝒜. In the foregoing, we have already remarked that ℬ ⊂ 𝒜^{ν}; especially, the diagonal idempotent of 𝒜 lies in the image of the restriction map 𝒜^{ν} ⊗_{𝒪_{X}^{a}} 𝒜^{ν}(X)_{∗} → 𝒜 ⊗_{𝒪_{X}^{a}} 𝒜 (U)_{∗}. Then proposition 14.3.21 implies that 𝒜^{ν} is an étale almost finite 𝒪_{X}^{a}-algebra, as sought.

Obviously, (a)⇒(e); to conclude, it suffices then to check that (c)⇒(a). Thus, let 𝒜 be an étale 𝒪_{U}^{a}-algebra of almost finite rank, V ⊂ U an affine open subset, and set A := 𝒪_{X}^{a}(V), B := 𝒜 (V). According to [75, Th.4.3.28], there exists a decomposition of A as an infinite product of a system of V^{a}-algebras (A_{i} | i ∈ ℕ) fulfilling condition (a) of lemma 14.4.6. Such a decomposition determines a system of idempotent elements e_{V,i} ∈ A_{∗}, for every i ∈ ℕ, such that e_{V,i} · e_{V,j} = 0 whenever i ≠ j, and characterized by the identities e_{V,i}A = A_{i} for every i ∈ ℕ. Since 𝒪_{X∗} is a sheaf ([75, §5.5.4]), condition (ii) ensures that, for any fixed i ∈ ℕ, and V ranging over the affine open subsets of U, the idempotents e_{V,i} glue to a global section e_{i} ∈ 𝒪_{X}^{a}(U)_{∗}, which will still be an idempotent; moreover, the direct factor 𝒜_{i} := e_{i}𝒜 of 𝒜 is an étale 𝒪_{U}^{a}-algebra of finite rank, and the projection 𝒜 → 𝒜_{i} is a morphism of 𝒪_{U}^{a}-algebras. Especially, A and B fulfill as well condition (b) of lemma 14.4.6. By (c), we deduce that the normalization 𝒜_{i}^{ν} of 𝒜_{i} over X is an étale 𝒪_{X}^{a}-algebra of finite rank, for every i ∈ ℕ. On the other hand, notice that the natural morphism

𝒪_{X∗}^{a} → j_{∗}(𝒪_{U∗}^{a}) = (j_{∗}𝒪_{U}^{a})_{∗}

is a monomorphism, and its image is integrally closed in j_{∗}𝒪_{U∗}^{a} ([75, Rem.8.2.30(i)]). It follows easily that e_{i} ∈ 𝒪_{X∗}^{a}(X); hence

e_{i}𝒜_{i}^{ν} = 𝒜_{i}^{ν} for every i ∈ ℕ

and the latter is the integral closure of e_{i}𝒪_{X}^{a} in j_{∗}(𝒜_{i}). Consequently, 𝒜^{ν} = ∏_{i∈ℕ} 𝒜^{ν} ([75, Rem.8.2.30(ii)]). By lemma 14.4.6, it follows that 𝒜^{ν} is an étale 𝒪_{X}^{a}-algebra of almost finite rank. This shows that the functor (14.4.2) is essentially surjective; combining with lemma 14.4.5(ii), we get (a). □

Corollary 14.4.9. Let (X, Z) be a normal pair, and suppose that :

(a) The scheme Z(z) is finite dimensional, for every z ∈ Z.

(b) The pair (X(z), {z}) is almost pure, for every z ∈ Z. Then the pair (X, Z) is almost pure.

Proof. Let 𝒜 be any étale 𝒪_{U}^{a}-algebra of finite rank, and 𝒜^{ν} the normalization of 𝒜 over X. In light of proposition 14.4.8, it suffices to show that 𝒜_{z}^{ν} is a weakly unramified 𝒪_{X,z}^{a}-algebra, for every z ∈ Z. Suppose, by way of contradiction, that the latter assertion fails; in view of condition (a), we may then find a point z ∈ Z, such that 𝒜_{z}^{ν} is not weakly unramified over 𝒪_{X,z}^{a}, but for every proper generization w ∈ Z of z, the 𝒪_{X,w}^{a}-algebra 𝒜_{w}^{ν} is weakly unramified.

Now, set U(z) := U ∩ X(z), V (z) := X(z)\{z}, and let j : V (z) → U be the induced morphism. Notice that (V (z), Z(z)) is a normal pair, and our assumption implies that j^{∗}𝒜^{ν} is a weakly unramified 𝒪_{V (z)}^{a}-algebra. By lemma 14.4.4, it follows that j^{∗}𝒜^{ν} is actually an étale 𝒪_{V (z)}^{a}-algebra of finite rank. Since by assumption, (X(z), {z}) is almost pure, proposition 14.4.8says that 𝒜_{z}^{ν} is an étale 𝒪_{X,z}^{a}-module of finite rank, a contradiction. □

Proposition 14.4.10. Let (X, Z) be a normal pair, and f : X′ → X a morphism of S-schemes; set Z′ := f^{−1}Z, and suppose that :

(a) f(Z′) = Z and f is pro-smooth at every point of Z′ (see definition 9.8.5).

(b) The pair (X′, Z′) is almost pure. Then the pair (X, Z) is almost pure.

Proof. Set U := X \Z, U′ := X′\Z′, and let f_{|U} : U′ → U be the restriction of f. Notice that the open immersion j′ : U′ → X′ is quasi-compact ([59, Ch.I, Prop.6.6.4]), and since f is flat at every point of Z′ (corollary 9.8.6(i)), U′ is dense in X′. Moreover :

Claim 14.4.11. (i) The pair (X′, Z′) is normal.

(ii) More generally, let 𝒜 be any quasi-coherent 𝒪_{U}^{a}-algebra. Then the natural morphism

f^{∗}(𝒜^{ν}) → (f_{|U}^{∗}𝒜)^{ν}

is an isomorphism (notation of remark 14.4.3(ii)).

Proof of the claim. Since f is flat at every point of Z′, and the unit of adjunction 𝒪_{X}^{a} → j_{∗}𝒪_{U}^{a} is a monomorphism, the induced morphism 𝒪_{X′}^{a} → f^{∗}j_{∗}𝒪_{U}^{a} = j′_{∗}𝒪_{U′}^{a} is a monomorphism as well (corollary 10.3.8). Also, (𝒪_{U}^{a})^{ν} = 𝒪_{X}^{a}, since (X, Z) is a normal pair; hence, (ii) implies (i).

(ii): Denote 𝒜_{!!}^{ν} the integral closure in j_{∗}(𝒜_{!!}) of the image of of 𝒪_{X}; by corollary 10.3.8, the natural map

f^{∗}j_{∗}(𝒜_{!!}) → j′_{∗}f_{|U}^{∗}(𝒜_{!!})

is an isomorphism of 𝒪_{X′}-algebras. On the other hand, corollary 9.8.6(ii) implies that the integral closure of the image of 𝒪_{X′} in f^{∗}j_{∗}(𝒜_{!!}) equals f^{∗}(𝒜_{!!}^{ν}). The assertion follows. ♢

Suppose now that 𝒜 is an étale 𝒪_{U}^{a}-algebra of finite rank. Since, by assumption, (X′, Z′) is almost pure, proposition 14.4.8 says that (f_{|U}^{∗}𝒜)^{ν} is an étale 𝒪_{X′}^{a}-algebra of finite rank. By claim 14.4.11and lemma 14.3.26(i) and corollary 9.8.6(ii), we deduce that 𝒜^{ν} is a weakly étale 𝒪_{X}^{a}-algebra. To conclude the proof, it suffices now to invoke proposition 14.4.8. □

Lemma 14.4.12. Let (A, I) be a tight henselian pair (see [75, Def.5.1.9]). Then we have : (i) The base change functor

Cov(Spec A) → Cov(Spec A/I)

is an equivalence (notation of [75, §8.2.22]). (ii) More generally, the base change functor A-Ét → A/I-Ét restricts to an equivalence

A-Ét_{afr} →^{∼} A/I-Ét_{afr}

on the respective full subcategories of étale algebras of almost finite rank.

Proof. In view of [75, Th.5.5.7(iii)], the assertions follow from lemma 14.3.17(ii). □

14.4.13. Let R be a V -algebra, I ⊂ R a principal ideal generated by a regular element, R^{∧} the I-adic completion of R. Set

X := Spec R  X^{∧} := Spec R^{∧}  Z := Spec R/I  Z^{∧} := Spec R^{∧}/IR^{∧}.

Proposition 14.4.14. In the situation of (14.4.13), suppose furthermore that :

(i) There exist n ∈ ℕ and a finitely generated subideal 𝔪_{0} ⊂ 𝔪 such that I^{n} ⊂ 𝔪_{0}R. (ii) The pair (R, I) is henselian.

Then the pair (X, Z) is almost pure if and only if the same holds for the pair (X^{∧}, Z^{∧}).

Proof. Under the current assumptions, the pair (R^{a}, I^{a}) is tight henselian ([75, §5.1.12]). Hence the assertion is a straightforward consequence of [75, Prop.5.4.53] and lemma 14.4.12(i) (details left to the reader). □

14.4.15. Let p ∈ ℤ be a prime integer with pV ⊂ 𝔪, and such that p is a regular element of V, and suppose that V contains an element, denoted p^{1/p}, whose p-th power generates the ideal pV. Let A and B be two flat V -algebras, and for every n ∈ ℕ set V_{n} := V/p^{n+1}V, A_{n} := A/p^{n+1}A, B_{n} := B/p^{n+1}B. Let also

(14.4.16)  A_{0} →^{∼} B_{0}

be a given isomorphism of V_{0}-algebras, and suppose that the Frobenius endomorphisms of V_{0} and A_{0} induce isomorphisms

(14.4.17)  V/p^{1/p}V →^{∼} V_{0}  A/p^{1/p}A →^{∼} A_{0}.

Set X := Spec A, X_{0} := Spec A_{0}, Y := Spec B and Y_{0} := Spec B_{0}.

Corollary 14.4.18. In the situation of (14.4.15), suppose moreover that the pairs (A, pA) and (B, pB) are henselian. Then the pair (X, X_{0}) is almost pure if and only if the same holds for the pair (Y, Y_{0}).

Proof. In view of proposition 14.4.14, it suffices to show that (14.4.16) lifts to an isomorphism A^{∧} →^{∼} B^{∧} between the p-adic completions of A and B. In turns, this reduces to exhibiting a system of isomorphisms (φ_{n} : A_{n} →^{∼} B_{n} | n ∈ ℕ) such that φ_{n} ⊗_{V_{n}} V_{n_{1}} = φ_{n−1} for every n > 0. To this aim, it suffices to show that

𝕃_{A_{n}/V_{n}} = 0 in D(s.A-Mod) for every n ∈ ℕ

([75, Prop.3.2.16]). However, in view of [75, Th.2.5.36], for every n ∈ ℕ, the short exact sequence 0 → p^{n+1}V/p^{n+2}V → V_{n+1} → V_{n} → 0 induces a distinguished triangle in D(s.A-Mod)

𝕃_{A_{0}/V_{0}} → 𝕃_{A_{n+1}/V_{n+1}} → 𝕃_{A_{n}/V_{n}} → σ𝕃_{A_{0}/V_{0}}.

Hence, an easy induction further reduces to checking that 𝕃_{A_{0}/V_{0}} = 0 in D(s.A-Mod). According to [75, Lemma 6.5.13], this will follow, once we have shown that the natural map

V_{0,(Φ)} ⊗^{L}_{V_{0}} A_{0} → A_{0,(Φ)}

is an isomorphism in D(V_{0}-Mod) (notation of loc.cit.). Taking into account the isomorphisms (14.4.17), this holds if and only if the natural map

V/p^{1/p}V ⊗^{L}_{V_{0}} A_{0} → A/p^{1/p}A

is an isomorphism in D(V_{0}-Mod). The latter assertion is clear, since A is a flat V -algebra. □

14.5. Normalized lengths. Let (V, | · |) be any valuation ring, with value group Γ_{V}; according to our general conventions, the composition law of Γ_{V} is denoted multiplicatively; however, sometimes it is convenient to switch to an additive notation. Hence, we adopt the notation :

(log Γ_{V} , ≤)

to denote the ordered group Γ_{V} with additive composition law and whose ordering is the reverse of the original ordering of Γ_{V}. The unit of log Γ_{V} shall be naturally denoted by 0, and we shall extend the ordering of log Γ_{V} by adding a largest element +∞, as customary. Also, we set : log Γ_{V}^{+} := {γ ∈ log Γ_{V} | γ ≥ 0} and log |0| := +∞.

14.5.1. In this section, (K, | · |) denotes a valued field of rank one, with value group Γ. We let κ be the residue field of K^{+}. As usual, we set S := Spec K^{+}, and denote by s (resp. η) the closed (resp. generic) point of S. Let 𝔪_{K} ⊂ K^{+} be the maximal ideal, and set 𝔪 := 𝔪_{K} in case Γ is not discrete, or else 𝔪 := K^{+}, in case Γ ≃ ℤ; in the following, whenever we refer to almost rings or almost modules, we shall assume – unless otherwise stated – that the underlying almost ring theory is the one defined by the standard setup (K^{+}, 𝔪) (see [75, §6.1.15]).

Let A be any K^{+a}-algebra, and c a cardinal number; following [75, §2.3], we denote by ℳ_{c}(A) the set of isomorphism classes of K^{+a}-modules which admit a set of generators of cardinality ≤ c. The set ℳ_{c}(A) carries a natural uniform structure (see [75, Def.2.3.1]), which admits the fundamental system of entourages

(E_{γ} | γ ∈ log Γ^{+} \ {0})

defined as follows. For any b ∈ 𝔪\{0}, we let E_{|b|} be the set of all pairs (M, M′) of elements of ℳ_{c}(A) such that there exist a third A-module N and A-linear morphisms N → M, N → M′ whose kernel and cokernel are annihilated by b.

14.5.2. The aim of this section is to define and study a well-behaved notion of normalized length for torsion modules M over K^{+}-algebras of a fairly general type. This shall be achieved in several steps. Let us first introduce the categories of algebras with which we will be working.

Definition 14.5.3. Let V be any valuation ring, with maximal ideal 𝔪_{V}.

(i) We let V -m.Alg_{0} be the subcategory of V -Alg whose objects are the local and essentially finitely presented V -algebras A whose maximal ideal 𝔪_{A} contains 𝔪_{V} A. The morphisms φ : A → B in K^{+}-m.Alg_{0} are the local maps. Notice that every morphism in V -m.Alg_{0} is an essentially finitely presented ring homomorphism ([63, Ch.IV, Prop.1.4.3(v)]).

Recall as well, that every object of V -m.Alg_{0} is a coherent ring (see (11.4)). More generally, if \underline{A} := (A_{i} | i ∈ I) is any filtered system of essentially finitely presented V -algebras with flat transition morphisms, then the colimit of \underline{A} is still a coherent ring (lemma 11.3.7(ii.a)).

(ii) We say that a local V -algebra A is measurable if it admits an ind-étale local map of V -algebras A_{0} → A, from some object A_{0} of V -m.Alg_{0}. The measurable V -algebras form a category V -m.Alg, whose morphisms are the local maps of V -algebras. As noted above, every measurable V -algebra is a coherent ring.

Lemma 14.5.4. In the situation of definition 14.5.3, let (A, 𝔪_{A}) be any measurable V -algebra. Then the following holds :

(i) A/𝔪_{V} A is a noetherian ring.

(ii) If the valuation of V has finite rank, every 𝔪_{A}-primary ideal of A contains a finitely

generated 𝔪_{A}-primary subideal. (iii) For every A-module M of finite type supported at the closed point of Spec A, the A-

module M/𝔪_{V} M has finite length. (iv) For every finitely generated ideal I ⊂ A, the V -algebra A/I is measurable.

Proof. (i): Let A^{sh} be the strict henselization of A at a geometric point localized at the closed point; then A^{sh}/𝔪_{V} A^{sh} is a strict henselization of A/𝔪_{V} A ([66, Ch.IV, Prop.18.8.10]), and it is therefore also a strict henselization of a V/𝔪_{V} -algebra of finite type. Hence A^{sh}/𝔪_{V} A^{sh} is noetherian, and then the same holds for A/𝔪_{V} A ([66, Ch.IV, Prop.18.8.8(iv)]).

(iii) follows easily from (i) : the details shall be left to the reader.

(ii): If the valuation of V has finite rank, there exists an element t_{0} ∈ 𝔪_{V} that generates a 𝔪_{V} -primary ideal. Let I ⊂ A be a 𝔪_{A}-primary ideal; then t_{0}^{N} ∈ I, for some integer N > 0; moreover, the image \overline{I} of I in A/𝔪_{V} A is finitely generated, by (i). Pick elements t_{1}, …, t_{n} ∈ I whose images in A/𝔪_{V} A form a system of generators of \overline{I}; it follows that t_{0}^{N}, t_{1}, …, t_{n} form a 𝔪_{A}-primary ideal contained in I.

(iv): We may find an ind-étale local morphism A_{0} → A from an object A_{0} of V -m.Alg_{0}, and a finitely generated ideal I_{0} ⊂ A_{0} such that I = I_{0}A. Then A/I_{0} is an object of V -m.Alg_{0} as well, and the induced map A_{0}/I_{0} → A/I is ind-étale. □

14.5.5. Now, suppose that V is both a valuation ring and a flat, measurable K^{+}-algebra, and denote 𝔪_{V} (resp. κ(V), resp. |·|_{V}) the maximal ideal (resp. the residue field, resp. the valuation) of V. We claim that V has rank one, and the ramification index (Γ_{V} : Γ) is finite. Indeed, let us write V as the colimit of a filtered system (V_{i} | i ∈ I) of objects of K^{+}-m.Alg_{0} with essentially étale transition maps; it follows that each V_{i} is a valuation ring of rank one (claim 9.1.36). Also, the transition maps induce isomorphisms on the value groups : indeed, this is clear if K^{+} is a discrete valuation ring, since in that case the same holds for the V_{i}, and the transition maps are unramified by assumption; in the case where Γ is not discrete, the assertion follows from corollary 11.4.42. Therefore, we are reduced to the case where V is an object of K^{+}-m.Alg_{0}, to which corollary 11.4.42applies.

Suppose first that M is a finitely generated torsion V -module; in this case the Fitting ideal F_{0}(M) ⊂ V is well defined, and it is shown in [75, Lemma 6.3.1 and Rem.6.3.5] that the map M ↦ F_{0}(M) is additive on the set of isomorphism classes of finitely generated V -modules, i.e. for every short exact sequence 0 → M_{1} → M_{2} → M_{3} → 0 of such modules, one has :

(14.5.6)  F_{0}(M_{2}) = F_{0}(M_{1}) · F_{0}(M_{3}).

Next, the almost module F_{0}(M)^{a} is an element of the group of fractional ideals Div(V^{a}) defined in [75, §6.1.16], and there is a natural isomorphism

(14.5.7)  Div(V^{a}) ≃ log Γ_{V}^{∧}  I ↦ |I|

where Γ_{V}^{∧} is the completion of Γ_{V} for the uniform structure deduced from the ordering : see [75, Lemma 6.1.19]. Hence we may define :

λ_{V} (M) := (Γ_{V} : Γ) · |F_{0}(M)^{a}| ∈ log Γ_{V}^{∧}.

In view of (14.5.6), we see that :

(14.5.8)  λ_{V} (M′) ≤ λ_{V} (M)  whenever M′ ⊂ M are finitely generated.

More generally, if M is any torsion V -module, we let :

(14.5.9) λ_{V} (M) := sup {λ_{V} (M′) | M′ ⊂ M, M′ finitely generated} ∈ log Γ_{V}^{∧} ∪ {+∞}

which, in view of (14.5.8), agrees with the previous definition, in case M is finitely generated.

14.5.10. Suppose now that the V^{a}-module M^{a} is uniformly almost finitely generated. Then according to [75, Prop.2.3.23] one has a well defined Fitting ideal F_{0}(M^{a}) ⊂ V^{a}, which agrees with F_{0}(M)^{a} in case M is finitely generated.

Lemma 14.5.11. In the situation of (14.5.10), we have :

(i) λ_{V} (M) = (Γ_{V} : Γ) · |F_{0}(M^{a})|.

(ii) If 0 → N_{1} → N_{2} → N_{3} → 0 is any short exact sequence of uniformly almost finitely generated V^{a}-modules, then

|F_{0}(N_{2})| = |F_{0}(N_{1})| + |F_{0}(N_{3})|.

Proof. (ii): It is a translation of [75, Lemma 6.3.1 and Rem.6.3.5(ii)].

(i): Let k be a uniform bound for M^{a}, and denote by ℐ_{k}(M) the set of all submodules of M generated by at most k elements. Set e := (Γ_{V} : Γ); if N ∈ ℐ_{k}(M), then λ_{V} (N) ≤ e·|F_{0}(M^{a})|, due to (ii). By inspecting the definitions, it then follows that

e · |F_{0}(M^{a})| = sup {λ_{V} (N) | N ∈ ℐ_{k}(M)} ≤ λ_{V} (M).

Now, suppose M′ ⊂ M is any submodule generated by, say r elements; for every ε ∈ 𝔪 we can find N ∈ ℐ_{k}(M) such that εM ⊂ N, hence εM′ ⊂ N, therefore λ_{V} (εM′) ≤ λ_{V} (N) ≤ e · |F_{0}(M^{a})|. However, λ_{V} (M′) − λ_{V} (εM′) = λ_{V} (M′/εM′) ≤ λ_{V} (K^{+⊕r}/εK^{+⊕r}) = r · λ_{V} (K^{+}/εK^{+}). We deduce easily that λ_{V} (M′) ≤ e · |F_{0}(M^{a})|, whence the claim. □

Proposition 14.5.12. (i) Let 0 → M_{1} → M_{2} → M_{3} → 0 be a short exact sequence of torsion V -modules. Then :

λ_{V} (M_{2}) = λ_{V} (M_{1}) + λ_{V} (M_{3}).

(ii) λ_{V} (M) = 0 if and only if M^{a} = 0.

(iii) Let M be any torsion V -module. Then : (a) If M is finitely presented, λ_{V} (N) > 0 for every non-zero submodule N ⊂ M.

(b) If (M_{i} | i ∈ I) is a filtered system of submodules of M, then :

λ_{V} (colim_{i∈I} M_{i}) = lim_{i∈I} λ_{V} (M_{i}).

Proof. (ii): By lemma 14.5.11(i) it is clear that λ_{V} (M) = 0 whenever M^{a} = 0. Conversely, suppose that λ_{V} (M) = 0, and let m ∈ M be any element; then necessarily λ_{V} (V m) = 0, and then it follows easily that 𝔪 ⊂ Ann_{V} (m), so M^{a} = 0.

(iii.a): It suffices to show that M does not contain non-zero elements that are annihilated by 𝔪, which follows straightforwardly from [75, Lemma 6.1.14].

(iii.b): Set M := colim_{i∈I} M_{i}. By inspecting the definitions we see easily that λ_{V} (M) ≥ sup{λ_{V} (M_{i}) | i ∈ I}. To show the converse inequality, let N ⊂ M be any finitely generated submodule; we may find i ∈ I such that N ⊂ M_{i}, hence λ_{V} (N) ≤ λ_{V} (M_{i}), and the assertion follows.

(i): Set e := (Γ_{V} : Γ). We shall use the following :

Claim 14.5.13. Every submodule of a uniformly almost finitely generated V^{a}-module is uniformly almost finitely generated.

Proof of the claim. Let N′ ⊂ N, with N uniformly almost finitely generated, and let k be a uniform bound for N; for every ε ∈ 𝔪 we can find N′′ ⊂ N such that N′′ is generated by at most k almost elements and εN ⊂ N′′. Clearly, it suffices to show that N′ ∩ N′′ is uniformly almost finitely generated and admits k as a uniform bound, so we may replace N by N′′ and N′ by N′ ∩ N′′, and assume from start that N is finitely generated. Let us pick an epimorphism φ : (V^{a})^{⊕k} → N; it suffices to show that φ^{−1}(N′) is uniformly almost finitely generated with k as a uniform bound, so we are further reduced to the case where N is free of rank k. Then we can write N′ = L^{a} for some submodule L ⊂ V^{⊕k}; notice that (V^{⊕k}/L)^{a} is almost finitely presented, since it is finitely generated ([75, Prop.6.3.6(i)]), hence N′ is almost finitely generated ([75, Lemma 2.3.18(iii)]). Furthermore, L is the colimit of the family (L_{i} | i ∈ I) of its finitely generated submodules, and each L_{i} is a free V -module ([34, Ch.VI, §3, n.6, Lemma 1]). Necessarily the rank of L_{i} is ≤ k for every i ∈ I, hence Λ_{V}^{k+1}L ≃ colim_{i∈I} Λ_{V}^{k+1}L_{i} = 0, and then the claim follows from [75, Prop.6.3.6(ii)]. ♢

Now, let N ⊂ M_{2} be any finitely generated submodule, \overline{N} ⊂ M_{3} the image of N; by claim 14.5.13, M_{1} ∩ N is uniformly almost finitely generated, hence lemma 14.5.11(i,ii) shows that :

λ_{V} (N) = e · |F_{0}(N^{a})| = e · |F_{0}(M_{1}^{a} ∩ N^{a})| + e · |F_{0}(\overline{N}^{a})| = λ_{V} (M_{1} ∩ N) + λ_{V} (\overline{N}).

Taking the supremum over the family (N_{i} | i ∈ I) of all finitely generated submodules of M_{2} yields the identity :

λ_{V} (M_{2}) = λ_{V} (M_{3}) + sup{λ_{V} (M_{1} ∩ N_{i}) | i ∈ I}.

By definition, λ_{V} (M_{1} ∩ N_{i}) ≤ λ_{V} (M_{1}) for every i ∈ I; conversely, every finitely generated submodule of M_{1} is of the form N_{i} for some i ∈ I, whence the contention. □

Remark 14.5.14. Suppose that Γ ≃ ℤ, and let γ_{0} ∈ log Γ^{+} be the positive generator. Then by a direct inspection of the definition one finds the identity :

λ(M) = (Γ_{V} : Γ) · length_{V} (M) · γ_{0}

for every torsion V -module M. The verification shall be left to the reader.

14.5.15. Let M be a torsion V -module, such that M^{a} is almost finitely generated; we wish now to explain that λ_{V} (M) can also be computed in terms of a suitable sequence of elementary divisors for M^{a}. Indeed, suppose first that M is a finitely presented torsion V -module; then we have a decomposition

(14.5.16)  M = (V/a_{0}V) ⊕ ⋯ ⊕ (V/a_{n}V)  where n := dim_{κ(V)} M/𝔪_{V} M − 1

for certain a_{0}, …, a_{n} ∈ 𝔪_{V} ([75, Lemma 6.1.14]). Clearly γ_{i} := log |a_{i}|_{V} > 0 for every i = 0, …, n, and after reordering we may assume that γ_{0} ≥ ⋯ ≥ γ_{n}; then we may set γ_{i} := 0 for every i > n, and the resulting sequence (γ_{i} | i ∈ ℕ) of elementary divisors of M is independent of the chosen decomposition (14.5.16), since we have more precisely :

(14.5.17)  a_{i}V = Ann_{V} (Λ_{V}^{i+1}M)  for every i ∈ ℕ.

Indeed, from a decomposition (14.5.16) with a_{j}V ⊂ a_{j+1}V for every j < n, we get a V -linear surjection M → (V/a_{i}V)^{⊕i+1}, whence a surjection

N_{i} := Λ_{V}^{i+1}M → Λ_{V}^{i+1}(V/a_{i}V)^{⊕i+1} →^{∼} V/a_{i}V

which shows that J_{i} := Ann_{V} N_{i} ⊂ a_{i}V. For the converse inclusion, notice that N_{i} is generated by the system of all elements of the form ω := x_{0} ∧ ⋯ ∧ x_{i}, where x_{0} ∈ M_{j(0)}, …, x_{i} ∈ M_{j(i)} for some strictly increasing map j : {0, …, i} → {0, …, n}; especially, j(i) ≥ i, whence a_{i}ω = 0, and the assertion follows. Moreover, a simple inspection yields the identity

(14.5.18)  λ_{V} (M) = (Γ_{V} : Γ) · (γ_{0} + ⋯ + γ_{n}).

We regard the sequence (γ_{i} | i ∈ ℕ) as an element of the log Γ_{V}^{∧}-normed space ℓ^{∞}(Γ_{V}^{+}) of bounded sequences of elements of log Γ_{V}^{∧+}, i.e. the set of all sequences \underline{δ} := (δ_{i} | i ∈ ℕ) with

∥\underline{δ}∥ := sup(δ_{i} | i ∈ ℕ) < +∞.

Lemma 14.5.19. Let φ : N → N′ be a map of finitely presented torsion V -modules, and denote by (γ_{i} | i ∈ ℕ) (resp. (γ′_{i} | i ∈ ℕ)) the sequence of elementary divisors of N (resp. N′). Then :

(i) If φ is injective, we have γ_{i} ≤ γ′_{i} for every i ∈ ℕ.

(ii) If φ is surjective, we have γ_{i} ≥ γ′_{i} for every i ∈ ℕ.

Proof. Denote by K_{V} the field of fractions of V, and set P^{†} := Hom_{V} (P, K_{V} /V) for every V -module P. We have a natural bilinear pairing P ⊗_{V} P^{†} → K_{V} /V, whence a V -linear map

ω_{P} : P → (P^{†})^{†}.

Claim 14.5.20. (i) If φ is injective, φ^{†} := Hom_{V} (φ, K_{V} /V) is surjective.

(ii) For every finitely presented V -module P, we have : (a) The map ω_{P} is an isomorphism.

(b) P and P^{†} are isomorphic V -modules.

Proof of the claim. (ii) is left to the reader. (i) follows immediately from lemma 11.5.9(i). ♢

By virtue of claim 14.5.20(ii.b), the V -modules N and N^{†} have the same sequences of elementary divisors. Taking into account claim 14.5.20(i), we deduce that (ii)⇒(i), so it remains only to show that (ii) holds. However, if φ is surjective, the same holds for Λ_{V}^{i+1}φ, and then the assertion follows immediately from (14.5.17). □

Proposition 14.5.21. Let 0 → M′ → M → M′′ → 0 be a short exact sequence of finitely presented torsion V -modules. Then for every i, j, t ∈ ℕ we have :

∑_{k=i+j}^{i+j+t} γ_{k}(M) ≤ ∑_{k=i}^{i+t} γ_{k}(M′) + ∑_{k=j}^{j+t} γ_{k}(M′′).

Proof. Let us choose a decomposition for M as in (14.5.16), with a_{0}V ⊂ a_{1}V ⊂ ⋯ ⊂ a_{n}V, and set \overline{M} := (V/b_{0}V)⊕⋯⊕(V/b_{i+j+t}V), where b_{i} = a_{i+j} for i = 0, …, i+j and b_{i} = a_{i} for i = i+j +1, …, i+j +t. We have an obvious V -linear surjection M → \overline{M}, and we let as well \overline{M}′ be the image of M′ in \overline{M}, and \overline{M}′′ := \overline{M}/\overline{M}′. By construction, we have ∑_{k=i+j}^{i+j+t} γ_{k}(M) = ∑_{k=i+j}^{i+j+t} γ_{k}(\overline{M}), and taking into account lemma 14.5.19(ii), we are easily reduced to checking the stated inequality for the resulting short exact sequence 0 → \overline{M}′ → \overline{M} → \overline{M}′′ → 0. Next, for k = 0, …, i + j + t we consider the V -linear map φ_{k} : V → V/a_{i+j}V that maps 1 to the class of a_{i+j}/b_{k}; clearly φ_{k} factors through a V -linear injection \overline{φ}_{k} : V/b_{k}V → V/a_{i+j}V, and the direct sum of the latter maps is a V -linear injection \overline{φ} : \overline{M} → N := (V/a_{i+j}V)^{⊕i+j+t+1}. Let N′ ⊂ N be the image of \overline{M}′, and set N′′ := N/N′; we notice that

∑_{k=i+j}^{i+j+t}(γ_{k}(N) − γ_{k}(\overline{M})) = |F_{0}(N)| − |F_{0}(\overline{M})| = C := ∑_{k=1}^{t}(log |a_{i+j}|_{V} − log |a_{i+j+k}|_{V}).

Taking into account proposition 14.5.12(i), we deduce that

∑_{k=j}^{j+t}(γ_{k}(N′′) − γ_{k}(\overline{M}′′)) ≤ |F_{0}(N′′)| − |F_{0}(\overline{M}′′)| = C.

Summing up, we are further reduced to checking the sought inequality for the short exact sequence 0 → N′ → N ⟶^{π} N′′ → 0. Let us remark, quite generally :

Claim 14.5.22. Let A be a local ring, M a finitely generated A-module, F, F′ two free Amodules of the same rank r ∈ ℕ, and π : F → M, π′ : F′ → M two A-linear surjections. Then there exists an A-linear isomorphism ω : F →^{∼} F′ such that π′ ∘ ω = π.

Proof of the claim. Let κ be the residue field of A, and set m := dim_{κ}(M ⊗_{A}κ); by Nakayama’s lemma, M admits a minimal system of generators x_{1}, …, x_{m}, and we choose f_{1}, …, f_{m} ∈ F with π(f_{i}) = x_{i} for i = 1, …, m. Next, notice that G := Ker (π ⊗_{A} κ) is a κ-vector space of dimension r −m, and pick a system of elements f_{m+1}, …, f_{r} ∈ F whose image in G is a basis; applying again Nakayama’s lemma, it is easily seen that the system f_{1}, …, f_{r} generates the A-module F, and it is therefore a basis of the latter. Likewise, we may find a basis f′_{1}, …, f′_{r} of F′ such that π′(f_{i}) = x_{i} for i = 1, …, m and π′(f_{i}) = 0 for i = m + 1, …, r; then the unique A-linear map ω : F → F′ such that ω(f_{i}) = f′_{i} for i = 1, …, r will do. ♢

Now, say that N′′ = (V/c_{0}V) ⊕ ⋯ ⊕ (V/c_{p}V) is a minimal decomposition with c_{0}V ⊂ ⋯ ⊂ c_{p}V; clearly we must have p ≤ i + j + t and a_{i+j} ∈ c_{0}V. Set c_{p+1} = ⋯ = c_{i+j+t} = 1, and let π′_{k} : V/a_{i+j+t} → V/c_{k}V be the natural projection, for k = 0, …, i + j + t; the direct sum of the maps π′_{k} is a V -linear surjection π′ : N → N′′, and by virtue of claim 14.5.22 there exists a V -linear automorphism ω of N such that π′ ∘ ω = π. Thus, it suffices to show the stated inequality for the short exact sequence 0 → Q := ω(N′) → N ⟶^{π′} N′′ → 0. But notice that Q = ⨁_{k=0}^{i+j+t} ker π′_{k}, and therefore

γ_{k}(Q) = log |a_{i+j}|_{V} − log |c_{i+j+t−k}|_{V} for k = 0, …, i + j + t.

On the other hand γ_{k}(N) = log |a_{i+j}|_{V} and γ_{k}(N′′) = log |c_{k}|_{V} for k = 0, …, i + j + t; summing up, we conclude the the sought inequality holds, and indeed it is an equality for this last short exact sequence. □

14.5.23. We fix now a large cardinal number ω, and write just ℳ (V^{a}) instead of ℳ_{ω}(V^{a}). Also, for any γ ∈ log Γ_{V}^{+}, set [−γ, γ] := {δ ∈ log Γ_{V} | − γ ≤ δ ≤ γ}. Suppose that N and N′ are two finitely presented V -modules such that (N^{a}, N^{′a}) ∈ E_{δ} for some δ ∈ log Γ_{V}^{+} \ {0}. Say that δ = log |b|_{V} for some b ∈ 𝔪_{V}; by standard arguments, we obtain maps φ : N → N′ and φ′ : N′ → N such that φ′ ∘ φ = b^{4} · 1_{N} and φ′ ∘ φ = b^{4} · 1_{N′}. Let now (γ_{i} | i ∈ ℕ) (resp. (γ′_{i} | i ∈ ℕ)) be the sequence of elementary divisors for N (resp. for N′). Then the sequence of elementary divisors for b^{4}N is (max(0, γ_{i} − 4δ) | i ∈ ℕ), and likewise for b^{4}N′. In view of lemma 14.5.19, we deduce easily that

γ_{i} − γ′_{i} ∈ [−4δ, 4δ] for every i ∈ ℕ.

Consider a torsion almost finitely generated V^{a}-module M, and recall that M is almost finitely presented ([75, Prop.6.3.6(i)]); we then may attach to M a net of elements of ℓ^{∞}(log Γ_{V}^{+}), as follows. For every δ ∈ log Γ_{V}^{+} \ {0}, pick a finitely presented V -module N_{δ} such that (M, N_{δ}^{a}) ∈ E_{δ}, and denote by \underline{γ}_{δ} the sequence of elementary divisors of N_{δ}. The foregoing easily implies that the system (\underline{γ}_{δ} | δ ∈ log Γ_{V}^{+} \ \{0\}) is a net for the uniform structure of ℓ^{∞}(log Γ_{V}^{+}) induced by the norm ∥·∥. However, (ℓ^{∞}(log Γ_{V}^{+}), ∥·∥) is a complete normed space, hence this net converges to a well defined sequence \underline{γ}_{M} := (γ_{i} | i ∈ ℕ) ∈ ℓ^{∞}(log Γ_{V}^{+}). It is easily seen that \underline{γ}_{M} is independent of the chosen net, and defines an invariant which we call the sequence of elementary divisors of M. A simple inspection of the construction shows that the sequence \underline{γ}_{M} is monotonically decreasing, and

lim_{i→+∞} γ_{i} = 0.

14.5.24. Denote again by K_{V} the field of fractions of V; as in the proof of lemma 14.5.19, we consider the functor

V^{a}-Mod → V^{a}-Mod : M ↦ M^{†} := alHom_{V^{a}}(M, K_{V}^{a} /V^{a}).

For every V^{a}-module M we have a natural V^{a}-bilinear pairing

M ⊗_{V^{a}} M^{†} → K_{V}^{a} /V^{a}

which in turns yields a natural transformation

(14.5.25)  M → (M^{†})^{†}.

Together with the constructions of (14.5.23), we may now extend lemma 14.5.19 verbatim to arbitrary almost finitely generated V^{a}-modules.

Proposition 14.5.26. Let φ : M → M′ be a morphism of almost finitely generated torsion V^{a}-modules, and denote by (γ_{i} | i ∈ ℕ) (resp. (γ′_{i} | i ∈ ℕ)) the sequence of elementary divisors of M (resp. M′). Then :

(i) M^{†} is an almost finitely generated torsion V^{a}-module, the morphism (14.5.25) is an isomorphism, and the sequences of elementary divisors of M and M^{†} coincide.

(ii) If φ is a monomorphism, we have γ_{i} ≤ γ′_{i} for every i ∈ ℕ.

(iii) If φ is an epimorphism, we have γ_{i} ≥ γ′_{i} for every i ∈ ℕ.

Proof. (i): Since M is almost finitely presented ([75, Prop.6.3.6(i)]), the assertion follows immediately from [75, Lemma 2.3.7(iii)] and claim 14.5.20(ii).

(iii): To begin with, we remark more generally :

Claim 14.5.27. Let P, Q be two torsion almost finitely generated V^{a}-modules, and δ ∈ log Γ_{V}^{+} any element such that (P, Q) ∈ E_{δ}. Denote by (γ_{i}^{P} | i ∈ ℕ) (resp. (γ_{i}^{Q} | i ∈ ℕ)) the sequence of elementary divisors of P (resp. of Q). Then

γ_{i}^{P} − γ_{i}^{Q} ∈ [−4δ, 4δ] for every i ∈ ℕ.

Proof of the claim. For any ε ∈ log Γ_{V}^{+} pick a torsion finitely presented V -module N such that (P, N^{a}) ∈ E_{ε}, and let (γ_{i}^{N} | i ∈ ℕ) be the sequence of elementary divisors of N. It follows that (Q, N^{a}) ∈ E_{δ+ε}, and from the construction of (14.5.23) we see that

γ_{i}^{N} − γ_{i}^{P} ∈ [−4ε, 4ε]  γ_{i}^{N} − γ_{i}^{Q} ∈ [−4(ε + δ), 4(ε + δ)]  for every i ∈ ℕ

so γ_{i}^{P} − γ_{i}^{Q} ∈ [−8ε − 4δ, 8ε + 4δ] for every i ∈ ℕ. Since ε is arbitrary, the claim follows. ♢ Fix δ ∈ log Γ_{V}^{+} \ {0}, pick finitely presented V -modules N, N′ such that

(N^{a}, M), (N′^{a}, M′) ∈ E_{δ}

and denote by (β_{i} | i ∈ ℕ) (resp. (β′_{i} | i ∈ ℕ)) the sequence of elementary divisors of N (resp. of N′). According to claim 14.5.27we have :

(14.5.28)  γ_{i} − β_{i}, γ′_{i} − β′_{i} ∈ [−4δ, 4δ]  for every i ∈ ℕ.

On the other hand, by the usual arguments we may find V -linear maps f : N → M_{∗} and g : M′_{∗} → N′ such that b^{2} · Coker f = b^{2} · Coker g = 0 for every b ∈ V with log |b|_{V} = δ. Set h := g ∘ ψ_{∗} ∘ f; it follows easily that b^{5} · Coker h = 0. The sequence of elementary divisors of b^{5}N′ is (max(0, β′_{i} − 5δ) | i ∈ ℕ), and since N′′ := Im h is a finitely presented V -module, the sequence (β′′_{i} | i ∈ ℕ) of its elementary divisors satisfies the inequalities

β_{i} ≥ β′′_{i} ≥ β′_{i} − 5δ for every i ∈ ℕ

by lemma 14.5.19(i,ii). Combining with (14.5.28), we deduce that

γ_{i} ≥ γ′_{i} − 13δ for every i ∈ ℕ

and since δ is arbitrary, the assertion follows.

(ii): in light of [75, Lemma 2.3.7(iii)] and claim 14.5.20(i), it is easily seen that φ^{†} is an epimorphism, so the assertion follows formally from (i) and (iii), as in the proof of lemma 14.5.19(i). □

One may ask, to which extent the sequence of elementary divisors of a V^{a}-module M determines the isomorphism class of M. To address this question, we make the following :

Definition 14.5.29. Let (V, 𝔪) be an arbitrary basic setup, A any V^{a}-algebra, M an A-module and ω a (very large) cardinal number, such that ℳ_{ω}(A) contains the isomorphism class of M. The topological space underlying the uniform space ℳ_{ω}(A) is not necessarily separated, but it admits a maximal separated quotient space ℳ_{ω}^{sep}(A). The ω-type of M is the image in ℳ_{ω}^{sep}(A) of the isomorphism class of M. If ω′ is any cardinal larger than ω, clearly the ω-type of M maps to the ω′-type of M, under the natural map ℳ_{ω}^{sep}(A) → ℳ_{ω′}^{sep}(A). For this reason, we shall usually omit explicit mention of ω, and shall call the type of M any one of these invariants.

Lemma 14.5.30. In the situation of definition 14.5.29, suppose that M and M′ are two Amodules of the same type. Then we have :

(i) M is almost finitely presented (resp. almost finitely generated) if and only if the same holds for M′.

(ii) M is a flat (resp. almost projective) A-module if and only if the same holds for M′.

Proof. (i) follows directly from the definitions.

(ii) follows easily from [75, Lemma 2.3.7(iii,iv)] : details left to the reader. □

We may now answer as follows to the foregoing question :

Proposition 14.5.31. Let (V, 𝔪_{V}) be as in (14.5.5), and M, M′ any two almost finitely generated torsion V^{a}-modules. The following conditions are equivalent :

(a) M and M′ have the same type.

(b) The sequences of elementary divisors of M and M′ coincide.

Proof. (a)⇒(b): Condition (a) means that (M, M′) ∈ E_{δ} for every δ ∈ log Γ_{V}^{+} \ {0}, so the assertion is immediate from the construction of the sequences of elementary divisors.

(b)⇒(a): Let (γ_{i} | i ∈ ℕ) be the common sequence of elementary divisors for M and M′. Fix δ as in the foregoing, pick finitely presented V -modules N, N′ such that

(M, N), (M′, N′) ∈ E_{δ}

and let (β_{i} | i ∈ ℕ), respectively (β′_{i} | i ∈ ℕ) be the sequences of elementary divisors of N and N′. From claim 14.5.27we deduce

β_{i} − γ_{,}β′_{i} − γ_{i} ∈ [−4δ, 4δ] for every i ∈ ℕ.

Thus, β_{i} − β′_{i} ∈ [−8δ, 8δ] for every i ∈ ℕ. Say that 8δ = log |a|_{V} for some a ∈ V.

Claim 14.5.32. There exists a V -linear map f : N → N′ whose kernel and cokernel is annihilated by a.

Proof of the claim. By [75, Lemma 6.1.14] we reduce easily to the case where N and N′ are cyclic V -modules, so N = V/bV and N′ = V/b′V, for b, b′ ∈ V \ {0} such that log |b/b′|_{V} ∈ [−2δ, 2δ]. Now, if b ∈ b′V, we let f be the natural projection V/bV → V/b′V. If b′ ∈ bV, we notice that b′b^{−1}1_{N} factors through a map N → N′ with the sought properties (details left to the reader). ♢

Clearly claim 14.5.32 implies that (N, N′) ∈ E_{8δ}, and therefore (M, M′) ∈ E_{10δ}. Since δ is arbitrary, the assertion follows. □

Corollary 14.5.33. Let b ∈ 𝔪_{V} be any non-zero element, M an almost finitely generated V^{a}/bV^{a}-module, and (γ_{i} | i ∈ ℕ) the sequence of elementary divisors of M. The following conditions are equivalent :

(a) M is a flat V^{a}/bV^{a}-module.

(b) There exists n ∈ ℕ such that γ_{i} = log |b|_{V} for every i ≤ n, and γ_{i} = 0 for every i > n.

Proof. For every i ∈ ℕ, pick a_{i} ∈ V with log |a_{i}|_{V} = γ_{i}. According to proposition 14.5.31, the type of M coincides with that of M′ := ⨁_{i∈ℕ} V^{a}/a_{i}V^{a}, and by lemma 14.5.30(ii), we may then assume that M = M′, in which case clearly (b)⇒(a). For the converse, notice that a standard computation gives

Tor_{1}^{V/bV} (V/aV, V/aV) ≃ V/(aV + a^{−1}bV) for every a ∈ V such that log |a|_{V} ≤ log |b|_{V}

from which the assertion follows easily (details left to the reader). □

Proposition 14.5.34. Let (V, 𝔪_{V}) be as in (14.5.5), and M a torsion V -module such that M^{a} is almost finitely generated. Let (γ_{i} | i ∈ ℕ) be the sequence of elementary divisors of M^{a}. Then

λ_{V} (M) = (Γ_{V} : Γ) · ∑_{i∈ℕ} γ_{i}.

Especially, λ_{V} (M) depends only on the type of M^{a}.

Proof. Fix a sequence (a_{k} | k ∈ ℕ) of elements of 𝔪_{V} \ {0} with lim_{k→+∞} log |a_{k}|_{V} = 0. For every k ∈ ℕ, let M_{k} ⊂ M be a finitely generated submodule such that a_{k}M ⊂ M_{k} ([75, Prop. 2.3.10(i)]), and denote also by (γ_{i}^{k} | i ∈ ℕ) the sequence of elementary divisors of M_{k}^{a}. After replacing each M_{k} by ∑_{j=0}^{k} M_{j}, we may also assume that M_{k} ⊂ M_{k+1} for every k ∈ ℕ.

Claim 14.5.35. For every i, k ∈ ℕ we have

4δ_{k} ≥ γ_{i} − γ_{i}^{k} ≥ 0  γ_{i}^{k+1} ≥ γ_{i}^{k}  where δ_{k} := log |a_{k}|.

Proof of the claim. The inequalities γ_{i} ≥ γ_{i}^{k+1} ≥ γ_{i}^{k} follow from proposition 14.5.26(ii). Next, clearly we have (M_{k}^{a}, M^{a}) ∈ E_{δ_{k}} for every k ∈ ℕ, so the upper bound for γ_{i} − γ_{i}^{k} follows from claim 14.5.27. ♢

Set L^{k} := ∑_{i∈ℕ} γ_{i}^{k} and L := ∑_{i∈ℕ} γ_{i}. It follows easily from claim 14.5.35that

lim_{k→+∞} L^{k} = ∑_{i∈ℕ} lim_{k→+∞} γ_{i}^{k} = L.

In light of proposition 14.5.12(iii.b), it then suffices to check the proposition with M replaced by M_{k}, for every k ∈ ℕ. We may therefore assume from start that M is finitely generated, and let g be the cardinality of a finite system of generators for M. We may then find a surjective V - linear map f : F := V^{⊕g} → M, and N := Ker f^{a} is almost finitely generated (claim 14.5.13). Thus, for every k ∈ ℕ we may find a finitely generated V -submodule N_{k} ⊂ N with a_{k}N ⊂ N_{k}, so that (F^{a}/N_{k}^{a}, M) ∈ E_{δ_{k}}. After replacing each N_{k} by ∑_{j=0}^{k} N_{j}, we may also assume that N_{k} ⊂ N_{k+1} for every k ∈ ℕ. Denote by (β_{i}^{k} | i ∈ ℕ) the sequence of elementary divisors of F/N_{k}, and notice that β_{i}^{k} = 0 for every i ≥ g. Taking into account (14.5.18), claim 14.5.27and proposition 14.5.26(iii) we get

λ_{V} (F/N_{k}) = ∑_{i∈ℕ} β_{i}^{k}  4δ_{k} ≥ β_{i}^{k} − γ_{i} ≥ 0  for every i, k ∈ ℕ.

Therefore lim_{k→+∞} λ_{V} (F/N_{k}) = L, and in view of proposition 14.5.12(i), it remains only to check that lim_{k→+∞} λ_{V} (N/N_{k}) = 0, or equivalently, that lim_{k→+∞} λ_{V} (N_{k}) = λ_{V} (N) (proposition 14.5.12(i)). However, set N′ := ⋃_{k∈ℕ} N_{k}; by proposition 14.5.12(iii.b), the latter identity holds if and only λ_{V} (N′) = λ_{V} (N), and since (N/N′)^{a} = 0, this follows from proposition 14.5.12(i,ii). □

14.5.36. In order to deal with general measurable K^{+}-algebras, we introduce hereafter some further notation which shall be standing throughout this section.

• To begin with, any ring homomorphism φ : A → B induces functors

(14.5.37)  φ_{∗} : B-Mod → A-Mod  and φ^{∗} : A-Mod → B-Mod.

Namely, φ_{∗} assigns to any B-module M the A-module φ_{∗}M obtained by restriction of scalars, and φ^{∗}(M) := B ⊗_{A} M.

• For any local ring (A, 𝔪_{A}), we let κ(A) := A/𝔪_{A}, and we denote by s(A) the closed point of Spec A. If A is also a coherent ring, we denote by A-Mod_{coh,{s}} the full subcategory of A-Mod consisting of all the finitely presented A-modules M such that Supp M ⊂ {s(A)}. Notice that the coherence of A implies that A-Mod_{coh,{s}} is an abelian category.

• Lastly, let 𝒜 be any small abelian category; recall that K_{0}(𝒜) is the abelian group defined by generators and relations as follows. The generators are the isomorphism classes [T] of objects T of 𝒜, and the relations are generated by the elements of the form [T_{1}] − [T_{2}] + [T_{3}], for every short exact sequence 0 → T_{1} → T_{2} → T_{3} → 0 of objects of 𝒜. One denotes by K_{0}^{+}(𝒜) ⊂ K_{0}(𝒜) the submonoid generated by the classes [T] of all objects of 𝒜. We shall use the following well known dévissage lemma :

Lemma 14.5.38. Let ι : ℬ ⊂ 𝒜 be an additive exact and fully faithful inclusion of abelian categories, and suppose that :

(a) If T ∈ Ob(ℬ) and T′ is a subquotient of ι(T), then T′ is in the essential image of ι.

(b) Every object T of 𝒜 admits a finite filtration Fil^{•}T such that the associated graded

object gr^{•}T is in the essential image of ι. Then ι induces an isomorphism :

K_{0}(ℬ) →^{∼} K_{0}(𝒜).

Proof. Left to the reader. □

Proposition 14.5.39. Let φ : A → B be a morphism of measurable K^{+}-algebras. We have :

(i) If φ induces a finite field extension κ(A) → κ(B), then the functor φ_{∗} of (14.5.37) restricts to a functor

φ_{∗} : B-Mod_{coh,{s}} → A-Mod_{coh,{s}}

which induces a group homomorphism of the respective K_{0}-groups :

φ_{∗} : K_{0}(B-Mod_{coh,{s}}) → K_{0}(A-Mod_{coh,{s}}).

(ii) If φ induces an integral morphism κ(A) → B/𝔪_{A}B, then length_{B}(B/𝔪_{A}B) is finite, and the functor φ^{∗} of (14.5.37) restricts to a functor

φ^{∗} : A-Mod_{coh,{s}} → B-Mod_{coh,{s}}

and if φ is also a flat morphism, φ^{∗} induces a group homomorphism :

φ^{∗} : K_{0}(A-Mod_{coh,{s}}) → K_{0}(B-Mod_{coh,{s}}).

Proof. Write A (resp. B) as the colimit of a filtered system \underline{A} := (A_{i} | i ∈ I) (resp. \underline{B} := (B_{j} | j ∈ J)) of objects of K^{+}-m.Alg_{0}, with local and essentially étale transition maps. After replacing J (resp. I) by a cofinal subsets, we may assume that the indexing set admits an initial element 0 ∈ J (resp. 0 ∈ I). Furthermore, we may assume that the induced map A_{0} → B factors through a morphism A_{0} → B_{0} in K^{+}-m.Alg_{0}.

(i): Let M be any object of B-Mod_{coh,{s}}. We need to show that φ_{∗}M is finitely presented. We may find j ∈ J and a finitely presented B_{j}-module M_{j}, with an isomorphism M →^{∼} M_{j} ⊗_{B_{j}} B of B-modules. After replacing J by J/j, we may assume that j = 0 is the initial index. Since the natural map B_{0} → B is local and ind-étale, it is easily seen that M_{0} is an object of B_{0}-Mod_{coh,{s}}. Especially, there exists a finitely generated 𝔪_{B_{0}}-primary ideal I ⊂ Ann_{B_{0}}M_{0}. We may then replace the system \underline{B} by (B_{j}/IB_{j} | j ∈ J) and assume that each B_{j} has Krull dimension zero.

Claim 14.5.40. Let φ : A → B be a morphism of measurable K^{+}-algebras inducing a finite residue field extension κ(A) → κ(B), and such that B has dimension zero. Let also M be any finitely presented B-module. Then we may find :

(a) a cocartesian diagram of local maps of K^{+}-algebras

\xymatrix{A_{l} \ar[r]^{φ_{l}} \ar[d] & C_{l} \ar[d] \\ A \ar[r]^{φ} & B}

whose vertical arrows are ind-étale, and where φ_{l} is a morphism in K^{+}-m.Alg_{0} (b) and an object M_{l} of C_{l}-Mod_{coh,{s}}, with an isomorphism M_{l} ⊗_{C_{l}} B →^{∼} M.

Proof of the claim. Define \underline{A} and \underline{B} as in the foregoing. Notice that – under the current assumptions – B_{j} is a henselian ring, for every j ∈ J. Moreover, we may find j ∈ J such that κ(B) is generated by the image of κ(A) ⊗_{κ(B_{0})} κ(B_{j}); after replacing again J by J/j, we may then also assume that κ(B) = κ(A) · κ(B_{0}).

For every i ∈ I, set B′_{i} := A_{i} ⊗_{A_{0}} B_{0}; the natural map B_{0} → B factors through a map B′_{i} → B, and we let 𝔭_{i} ⊂ B′_{i} be the preimage of 𝔪_{B}. Set also C_{i} := B′_{i,𝔭_{i}}, so we deduce a filtered system of local maps (C_{i} → B | i ∈ I), whose limit is a local map ψ : C → B of local ind-étale B_{0}-algebras, which – by construction – induces an isomorphism κ(C) →^{∼} κ(B) on residue fields. It follows easily that ψ is itself ind-étale, so say that ψ is the colimit of a filtered system (ψ_{λ} : C → D_{λ} | λ ∈ Λ) of étale C-algebras. Notice that C is a henselian local ring; in light of [66, Ch.IV, Th.18.5.11] we may then assume that D_{λ} is a local ring and ψ_{λ} is a finite étale map, for every λ ∈ Λ. Clearly the induced residue field extension κ(C) → κ(D_{λ}) is an isomorphism; in view of [66, Ch.IV, Prop.18.5.15] it follows that ψ_{λ} is an isomorphism, for every λ ∈ Λ, so the same holds for ψ.

Notice that the sequence of residue degrees d_{i} := [κ(C_{i}) : κ(A_{i})] is non-increasing, hence there exists l ∈ I such that d_{i} = d := d_{l} for every index i ≥ l. Notice as well that, for i ≥ l, the local algebra C_{i} is also a localization of C′_{i} := A_{i} ⊗_{A_{l}} C_{l}, and the latter is an essentially finitely presented K^{+}-algebras of Krull dimension zero, hence its spectrum is finite and discrete (lemma 9.1.29). Moreover, since the image of the map Spec C_{l} → Spec A_{l} is the closed point, it is clear that the same holds for the image of the induced map Spec C′_{i} → Spec A_{i}. Since the extension κ(A_{l}) → κ(A_{i}) is finite and separable, we conclude that

(14.5.41)  κ(A_{i}) ⊗_{κ(A_{l})} κ(C_{l}) = ∏_{𝔭∈Spec C′_{i}}κ(C′_{i,𝔭}).

However, clearly the left-hand side of (14.5.41) is a κ(A_{i})-algebra of degree d, whereas one of factors of the right-hand side – namely κ(C_{i}) – is already of degree d over κ(A_{i}). Hence Spec C′_{i} contains a single element, i.e. C′_{i} = C_{i} is a local ring, and C = A ⊗_{A_{l}} C_{l}. Summing up, we have obtained the sought cocartesian diagram, and the claim holds with M_{l} := M_{0} ⊗_{B_{0}} C_{l}. ♢

Let M_{l} and φ_{l} be as in claim 14.5.40; then φ_{∗}M is isomorphic to A ⊗_{A_{l}} φ_{l∗}M_{l}, so may replace from start φ by φ_{l}, and assume that φ is a morphism in K^{+}-m.Alg_{0}, with B of Krull dimension zero. In such situation, one sees easily that φ is integral, hence B/I is a finitely presented A-module, by proposition 9.1.34(i), therefore φ_{∗}M is a finitely presented A-module, as required. Lastly, since the functor φ_{∗} is exact, it is clear that it induces a map on K_{0}-groups as stated.

(ii): Under the current assumptions, the induced map κ(A_{0}) → B_{0}/𝔪_{A_{0}}B_{0} is integral and essentially finitely presented, hence it is finite, so B_{0}/𝔪_{A_{0}}B_{0} is a B_{0}-module of finite length; but this is also the length of the B-module B/𝔪_{A}B, whence the first assertion. Next, let M be an object of A-Mod_{coh,{s}}; then φ^{∗}M is a finitely presented B-module; moreover, we may find a 𝔪_{A}-primary ideal I ⊂ A such that M is a A/I-module, hence φ^{∗}M is a B/IB-module. Notice that the induced map Spec B/𝔪_{A}B → Spec B/IB is bijective, and its target is a local scheme of dimension zero (since B/𝔪_{A}B is integral over a field). It follows easily that IB is a 𝔪_{B}-primary ideal, so φ^{∗}M is an object of B-Mod_{coh,{s}}. The last assertion is then a trivial consequence of the exactness of the functor φ^{∗}, when φ is flat. □

Lemma 14.5.42. Let A be any measurable K^{+}-algebra. Then there exists a morphism

V ⟶^{φ} A/I

of measurable K^{+}-algebras, where :

(a) I ⊂ A is a finitely generated 𝔪_{A}-primary ideal.

(b) V is a valuation ring, φ is a finitely presented surjection and the natural map K^{+} → V induces an isomorphism of value groups Γ →^{∼} Γ_{V}.

Proof. Suppose first that A is an object of K^{+}-m.Alg_{0}. In this case, choose an affine finitely presented S-scheme X and a point x ∈ X such that A = 𝒪_{X,x}; next, take a finitely presented closed immersion h : X → Y := 𝔸_{K^{+}}^{n} of S-schemes; set \overline{Y} := 𝔸_{κ}^{n} ⊂ Y, pick elements f_{1}, …, f_{d} ∈ B := 𝒪_{Y,h(x)} whose images in the regular local ring 𝒪_{\overline{Y} ,h(x)} form a regular system of parameters (i.e. a regular sequence that generates the maximal ideal), and let J ⊂ B be the ideal generated by the f_{i}, i = 1, …, d. Let I ⊂ A be any finitely generated 𝔪_{A}-primary ideal containing the image of J. We deduce a surjection φ : V := B/J → A/I, and by construction V/𝔪_{K}V is a field; moreover, the induced map K^{+} → V is flat by virtue of [65, Ch.IV, Th.11.3.8]. It follows that V is a valuation ring with the sought properties, by proposition 9.1.34(ii). Also, φ is finitely presented, by proposition 9.1.34(i).

Next, let A be a general measurable K^{+}-algebra, and write A as the colimit of a filtered system (A_{j} | j ∈ J) of objects of K^{+}-m.Alg_{0}. We may assume that 0 ∈ J is an initial index, and the foregoing case yields an 𝔪_{A_{0}}-primary ideal I_{0} ⊂ A_{0}, and a surjective finitely presented morphism φ_{0} : V_{0} → A_{0}/I_{0} in K^{+}-m.Alg_{0} from a valuation ring V_{0}, such that Γ_{V} = Γ. Notice that A/I_{0} is a henselian ring, hence φ_{0} extends to a ring homomorphism φ_{0}^{h} : V_{0}^{h} → A_{0}/I_{0} from the henselization V_{0}^{h} of V_{0}; more precisely, φ_{0} induces an isomorphism of V_{0}^{h}-algebras :

V_{0}^{h} ⊗_{V_{0}} (A_{0}/I_{0}) →^{∼} A_{0}/I_{0}

so φ_{0}^{h} is still finitely presented. On the one hand, φ_{0}^{h} induces an identification

κ(V_{0}^{h}) = κ(A_{0}).

On the other hand, we have the filtered system of separable field extensions (κ(A_{j}) | j ∈ J), whose colimit is κ(A). There follows a corresponding filtered system (V_{j}^{h} | j ∈ J) of finite étale V_{0}^{h}-algebras, whose colimit we denote V ([66, Ch.IV, Prop.18.5.15]). Then V is a valuation ring, and the map V_{0}^{h} → V induces an isomorphism on value groups. Moreover, the induced isomorphisms κ(V_{j}^{h}) →^{∼} κ(A_{j}) lift uniquely to morphisms of A_{0}-algebras φ_{j}^{h} : V_{j}^{h} → A_{j}/I_{0}A_{j}, for every j ∈ J ([66, Ch.IV, Cor.18.5.12]). Due to the uniqueness of φ_{j}^{h}, we see that the resulting system (φ_{j}^{h} | j ∈ J) is filtered, and its colimit is a morphism φ : V → A/I_{0}A. Moreover, φ_{j}^{h} induces an isomorphism V_{j}^{h} ⊗_{V_{0}^{h}} A_{0}/IA_{0} →^{∼} A_{j}/I_{0}A_{j}, especially φ_{j}^{h} is surjective for every j ∈ J, so the same holds for φ. More precisely, φ_{0}^{h} induces an isomorphism V ⊗_{V_{0}^{h}} (A_{0}/I_{0}) →^{∼} A/I_{0}A, hence φ is still finitely presented. □

Theorem 14.5.43. With the notation of (14.5.36), the following holds :

(i) For every measurable K^{+}-algebra A there is a natural group isomorphism :

λ_{A} : K_{0}(A-Mod_{coh,{s}}) →^{∼} log Γ

which induces an isomorphism K_{0}^{+}(A-Mod_{coh,{s}}) →^{∼} log Γ^{+}. (ii) The family of isomorphisms λ_{A} (for A ranging over the measurable K^{+}-algebras) is

characterized uniquely by the following two properties.

(a) If V is a valuation ring and a flat measurable K^{+}-algebra, then

λ_{V} ([M]) = λ_{V} (M) for every object M of V -Mod_{coh,{s}}

where λ_{V} (M) is defined as in (14.5.2).

(b) Let ψ : A → B be a morphism of measurable K^{+}-algebras inducing a finite residue field extension κ(A) → κ(B). Then

λ_{A}(ψ_{∗}[M]) = [κ(B) : κ(A)] · λ_{B}([M]) for every [M] ∈ K_{0}(B-Mod_{coh,{s}}).

(iii) For every a ∈ K^{+}\{0} and any object M of A-Mod_{coh,{s}} we have : (a) [M] = 0 in K_{0}(A-Mod_{coh,{s}}) if and only if M = 0.

(b) If M is flat over K^{+}/aK^{+}, then :

(14.5.44)  λ_{A}([M]) = |a| · length_{A}(M ⊗_{K^{+}} κ).

(iv) Let ψ : A → B be a flat morphism of measurable K^{+}-algebras inducing an integral map κ(A) → B/𝔪_{A}B. Then :

λ_{B}(ψ^{∗}[M]) = length_{B}(B/𝔪_{A}B) · λ_{A}([M]) for every [M] ∈ K_{0}(A-Mod_{coh,{s}}).

Proof. We start out with the following :

Claim 14.5.45. Let (K, | · |) → (E, | · |) be an extension of valued fields of rank one inducing an isomorphism of value groups, a ∈ K^{+} \ {0} any element, M an E^{+}/aE^{+} module. Then :

(i) M is a flat E^{+}/aE^{+}-module if and only if it is a flat K^{+}/aK^{+}-module.

(ii) M ⊗_{K^{+}} κ = M ⊗_{E^{+}} κ(E).

Proof of the claim. According to [126, Th.7.8], in order to show (i) it suffices to prove that

Tor_{1}^{E^{+}/aE^{+}}(E^{+}/bE^{+}, M) = Tor_{1}^{K^{+}/aK^{+}}(K^{+}/bK^{+}, M)

for every b ∈ K^{+} such that |b| ≥ |a|. The latter assertion is an easy consequence of the faithful flatness of the extension K^{+} → E^{+}. (ii) follows from the identity : 𝔪_{E} = 𝔪_{K}E^{+}, which holds since Γ_{E} = Γ. ♢ Claim 14.5.46. Let f : K^{+} → V be a morphism of measurable K^{+}-algebras, where V is a valuation ring and f induces an isomorphism on value groups. Then λ_{V} (defined by (ii.a)) is an isomorphism and assertion (iii) holds for A = V.

Proof of the claim. According to (ii.a), λ_{V} ([M]) = λ_{V} (M) for every finitely presented torsion V -module. However, every such module M admits a decomposition of the form M ≃ (V/a_{1}V) ⊕ ⋯ ⊕ (V/a_{k}V), with a_{1}, …, a_{k} ∈ 𝔪_{K} ([75, Lemma 6.1.14]). Then by claim 14.5.45(i), M is flat over K^{+}/aK^{+} if and only if M is flat over V/aV, if and only if |a| = |a_{i}| for every i ≤ k. In this case, an explicit calculation shows that λ_{V} (M) = |a| · length(M ⊗_{V} κ(V)), which is equivalent to (14.5.44), in view of claim 14.5.45(ii). Next, we consider the map :

μ : log Γ^{+} → K_{0}(V -Mod_{coh,{s}}) : |a| ↦ [V/aV] for every a ∈ K^{+} \ {0}.

We leave to the reader the verification that μ extends to a group homomorphism well-defined on the whole of Γ, that provides an inverse to λ_{V}. Finally, it is clear that λ_{V} (M) = 0 if and only if M = 0, so also (iii.a) holds. ♢

Claim 14.5.47. Let A be any measurable K^{+}-algebra. For every object N of A-Mod_{coh,{s}} there exist a finite filtration 0 = N_{0} ⊂ ⋯ ⊂ N_{k} = N by finitely presented A-submodules, and elements a_{1}, …, a_{k} ∈ 𝔪 such that N_{i}/N_{i−1} is a flat K^{+}/a_{i}K^{+}-module for every 1 ≤ i ≤ k.

Proof of the claim. Let us find I ⊂ A and φ : V → A/I as in lemma 14.5.42. It suffices to show the claim for the finitely presented A-modules I^{n}N/I^{n+1}N (for every n ∈ ℕ), hence we may assume that N is an A/I-module. Then φ_{∗}N is a finitely presented V -module, hence of the form (V/a_{1}V) ⊕ ⋯ ⊕ (V/a_{k}V) for some a_{i} ∈ 𝔪; we may order the summands so that |a_{i}| ≥ |a_{i+1}| for all i < k. We argue by induction on d(N) := dim_{κ}(N ⊗_{V} κ(V)). If d(N) = 0, then N = 0 by Nakayama’s lemma. Suppose d > 0; we remark that N/a_{1}N is a flat V/a_{1}V -module, hence a flat K^{+}/a_{1}K^{+}-module (claim 14.5.45), and d(a_{1}N) < d(N); the claim follows. ♢

Claim 14.5.48. Let A be any measurable K^{+}-algebra, I ⊂ A a finitely generated 𝔪_{A}-primary ideal, and π_{I} : A → A/I the natural projection. Then the map :

π_{I∗} : K_{0}(A/I-Mod_{coh}) → K_{0}(A-Mod_{coh,{s}})

is an isomorphism.

Proof of the claim. On the one hand we have :

A-Mod_{coh,{s}} = ⋃_{n∈ℕ} A/I^{n}-Mod_{coh}

and on the other hand, in view of lemma 14.5.38, we see that the projections A/I^{n+1} → A/I^{n} induce isomorphisms K_{0}(A/I^{n}-Mod_{coh}) → K_{0}(A/I^{n+1}-Mod_{coh}) for every n > 0, whence the claim. ♢

Let A, I and φ : V → A/I be as in lemma 14.5.42, and π_{I} : A → A/I the natural surjection; taking into account claim 14.5.48, we may let :

(14.5.49)  λ_{A} := λ_{V} ∘ φ_{∗} ∘ π_{I∗}^{−1}

where λ_{V} is given by the rule of (ii.a). In view of claim 14.5.46 we see that λ_{A}([M]) = 0 if and only if M = 0, so (iii.a) follows already.

Claim 14.5.50. The isomorphism λ_{A} is independent of the choice of I, V and φ.

Proof of the claim. Indeed, suppose that J ⊂ A is another ideal and ψ : W → A/J is another surjection from a valuation ring W, fulfilling the foregoing conditions. We consider the commutative diagram

\xymatrix{ & A/I \ar[dr]^{_{\overline{π}_{J}}} & V \ar[l]_{φ} \ar[d]^{φ′} \\ A \ar[ur]^{π_{I}} \ar[dr]_{π_{J}} \ar[rr]^{π_{I+J}} &  & A/(I + J) \\  & A/J \ar[ur]_{_{\overline{π}_{I}}} & W. \ar[l]^{ψ} \ar[u]_{ψ′}}

We compute : φ_{∗} ∘ π_{I∗}^{−1} = φ_{∗} ∘ π_{J∗} ∘ \overline{π}_{J∗}^{−1} ∘ \overline{π}_{I∗}^{−1} = φ′_{∗} ∘ π_{I+J,∗}^{−1}, and a similar calculation shows that ψ_{∗} ∘ π_{J∗}^{−1} = ψ′_{∗} ∘ π_{I+J,∗}^{−1}. We are thus reduced to showing that λ_{V} ∘ φ′_{∗}([N]) = λ_{W} ∘ ψ′_{∗}([N]) for every A/(I + J)-module N. In view of claim 14.5.47 we may assume that N is flat over K^{+}/aK^{+}, for some a ∈ 𝔪, in which case the assertion follows from claim 14.5.46. ♢

Claims 14.5.46 and 14.5.50 show already that (i) holds. Next, let ψ : A → B be as in (ii.b). Choose an ideal I ⊂ A and a surjection φ : V → A/I as in lemma 14.5.42. Let also J ⊂ B be a finitely generated 𝔪_{B}-primary ideal containing ψ(I), and \overline{ψ} : A/I → B/J the induced map.

By inspecting the definitions we see that (ii.b) amounts to the identity : λ_{V} ((\overline{ψ} ∘ φ)_{∗}[M]) = [κ(B) : κ(A)] · λ_{B/J}([M]) for every finitely presented B/J-module M. Furthermore, up to enlarging J, we may assume that there is a finitely presented surjection ξ : W → B/J of K^{+}-algebras, where W is a valuation ring with value group Γ, and then we come down to showing:

λ_{V} ((\overline{ψ} ∘ φ)_{∗}[M]) = [κ(B) : κ(A)] · λ_{W} (ξ_{∗}[M]).

In view of claim 14.5.47we may also assume that M is a flat K^{+}/aK^{+}-module for some a ∈ 𝔪, in which case the identity becomes :

|a| · length_{V} (M ⊗_{K^{+}} κ) = [κ(B) : κ(A)] · |a| · length_{W} (M ⊗_{K^{+}} κ)

thanks to claim 14.5.46. However, the latter is an easy consequence of the identities : κ(A) = κ(V) and κ(B) = κ(W). Next, we show that (iii.b) holds for a general measurable K^{+}-algebra A. Indeed, let M be any object of A-Mod_{coh,{s}}; as usual, we may find a local ind-étale map A_{0} → A from some object A_{0} of K^{+}-m.Alg_{0}, and an object M_{0} of A_{0}-Mod_{coh,{s}} with an isomorphism of A-modules A ⊗_{A_{0}} M_{0} →^{∼} M (cp. the proof of proposition 14.5.39(i)). Since (ii.b) is already proved, we have

λ_{A}([M]) = λ_{A_{0}}([M_{0}]) and length_{A}(M ⊗_{K^{+}} κ) = length_{A_{0}}(M ⊗_{K^{+}} κ).

Therefore, we may replace A by A_{0}, and assume that A is an object of K^{+}-m.Alg_{0}. In this case, by lemma 9.1.32 we may find morphisms f : K^{+} → V and g : V → A in K^{+}-m.Alg_{0} such that V is a valuation ring with value group Γ and g induces a finite extension of residue fields κ(V) → κ(A). Let N be an A-module supported at s(A) and flat over K^{+}/aK^{+}; we may find a finitely generated 𝔪_{A}-primary ideal I ⊂ A such that I ⊂ Ann_{A}(N), and since (ii.b) is already known in general, we reduce to showing that (iii.b) holds for the A/I-module N. However, the induced map \overline{g} : V → A/I is finite and finitely presented (proposition 9.1.34(i)), so another application of (ii.b) reduces to showing that (14.5.44) holds for A := V and M := \overline{g}_{∗}N, in which case the assertion is already known by claim 14.5.46.

(ii.a): Suppose that A is a valuation ring, and let M be any object of A-Mod_{coh,{s}}. The sought assertion is obvious when Γ_{A} = Γ, since in that case we can choose A = V and φ = π_{I} in (14.5.49). However, we know already that the rank of A equals one, and e := (Γ_{A} : Γ) is finite (see (14.5.5)); we can then assume that e > 1, in which case corollary 11.4.42(ii) implies that Γ ≃ ℤ. Then it suffices to check (ii.a) for M = κ(A), which is a (flat) κ-module, so that – by assertion (iii) – one has λ_{A}([M]) = e · γ_{0}, where γ_{0} ∈ log Γ_{A}^{+} is the positive generator. In view of remark 14.5.14, we see that this value agrees with λ_{V} (M), as stated.

(iv): In view of claim 14.5.47, we may assume that M is a flat K^{+}/aK^{+}-module, for some a ∈ 𝔪, and then the same holds for ψ^{∗}M, since ψ is flat. In view of (iii.b), it then suffices to show that :

length_{B}(B ⊗_{A} M) = length_{A}(M) · length_{B}(B/𝔪_{A}B)

for any A-module M of finite length. In turn, this is easily reduced to the case where M = κ(A), for which the identity is obvious. □

14.5.51. Let A be a measurable K^{+}-algebra. The next step consists in extending the definition of λ_{A} to the category A-Mod_{{s}} of arbitrary A-modules M supported at s(A). First of all, suppose that M is finitely generated. Let 𝒞_{M} be the set of isomorphism classes of objects M′ of A-Mod_{coh,{s}} that admit a surjection M′ → M. Then we set :

λ_{A}^{∗}(M) := inf {λ_{A}([M′]) | M′ ∈ 𝒞_{M}} ∈ log Γ^{∧}.

Notice that – by the positivity property of theorem 14.5.43(i) – we have λ_{A}^{∗}(M) = λ_{A}([M]) whenever M is finitely presented. Next, for a general object of A-Mod_{{s}} we let : (14.5.52)

λ_{A}(M) := sup {λ_{A}^{∗}(M′) | M′ ⊂ M and M′ is finitely generated} ∈ log Γ^{∧} ∪ {+∞}.

Lemma 14.5.53. If M is finitely generated, then λ_{A}^{∗}(M) = λ_{A}(M).

Proof. Let M′ ⊂ M be a finitely generated submodule; we have to show that λ^{∗}(M′) ≤ λ^{∗}(M). To this aim, let f : N → M and g : N′ → M′ be two surjections of A-modules with N ∈ 𝒞_{M} and N′ ∈ 𝒞_{M′}; by filtering the kernel of f by the system of its finitely generated submodules, we obtain a filtered system (N_{i} | i ∈ I) of finitely presented quotients of N, with surjective transition maps, such that colim_{i∈I} N_{i} = M. By [75, Prop.2.3.16(ii)] the induced map N′ → M lifts to a map h : N′ → N_{i} for some i ∈ I. Since A is coherent, h(N′) is a finitely presented Amodule with a surjection h(N′) → M′, hence λ_{A}^{∗}(M′) ≤ λ_{A}([h(N′)]) ≤ λ_{A}([N_{i}]) ≤ λ_{A}(N). Since N is arbitrary, the claim follows. □

Proposition 14.5.54. (i) If A is a valuation ring, (14.5.52) agrees with (14.5.9).

(ii) If (M_{i} | i ∈ I) is a filtered system of objects of A-Mod_{{s}} with injective (resp. surjective) transition maps, then :

λ_{A}(colim_{i∈I} M_{i}) = lim_{i∈I} λ_{A}(M_{i})

(resp. provided there exists i ∈ I such that λ_{A}(M_{i}) < +∞). (iii) If ψ : A → B is a morphism of measurable K^{+}-algebras inducing a finite extension

κ(A) → κ(B) of residue fields, then :

λ_{B}(M) = \frac{λ_{A}(ψ_{∗}M)}{[κ(B) : κ(A)]} for every object M of B-Mod_{\{s\}}.

(iv) Let 0 → M_{1} → M_{2} → M_{3} → 0 be a short exact sequence in A-Mod_{{s}}. Then :

λ_{A}(M_{2}) = λ_{A}(M_{1}) + λ_{A}(M_{3}).

(v) Let ψ be a flat morphism of measurable K^{+}-algebras inducing an integral map κ(A) → B/𝔪_{A}B. Then :

λ_{B}(ψ^{∗}M) = length_{B}(B/𝔪_{A}B) · λ_{A}(M) for every object M of A-Mod_{{s}}.

Proof. (i): Set e := (Γ_{V} : Γ). It suffices to check the assertion for a finitely generated A-module M, in which case one has to show the identity :

e · |F_{0}(M)^{a}| = λ_{A}^{∗}(M).

By proposition 14.5.12(i) we have |F_{0}(M′)^{a}| ≥ |F_{0}(M)^{a}| for every M′ ∈ 𝒞_{M}, hence λ_{A}^{∗}(M) ≥ e · |F_{0}(M)^{a}|. On the other hand, let us fix a surjection V^{⊕n} → M, and let us write its kernel in the form K = ⋃_{i∈I} K_{i}, for a filtered system (K_{i} | i ∈ I) of a finitely generated V -submodules; it follows that :

F_{0}(M) = ⋃_{i∈I} F_{0}(V^{⊕n}/K_{i}).

In view of proposition 14.5.12(iii.b) we deduce : e · |F_{0}(M)^{a}| = lim_{i∈I} e · |F_{0}(V^{⊕n}/K_{i})^{a}| ≥ λ_{A}^{∗}(M).

The proof of (ii) in the case where the transition maps are injective, is the same as that of proposition 14.5.12(iii.b).

(iii): Let us write M = ⋃_{i∈I} M_{i} for a filtered family (M_{i} | i ∈ I) of finitely generated B-submodules. By the case already known of (ii) we have : λ_{B}(M) = lim_{i∈I} λ_{B}(M_{i}), and likewise for λ_{A}(ψ_{∗}M), hence we may assume from start that M is a finitely generated Bmodule, in which case the annihilator of M contains a finitely generated 𝔪_{B}-primary ideal J′ ⊂ B. Choose a finitely generated 𝔪_{A}-primary ideal J ⊂ A contained in the kernel of the induced map A → B/J′.

Claim 14.5.55. λ_{B}(M) = λ_{B/J′}(M) and λ_{A}(ψ_{∗}M) = λ_{A/J}(\overline{ψ}_{∗}M).

Proof of the claim. Directly on the definition (and by applying theorem 14.5.43(ii.b) to the surjection B → B/J′) we see that λ_{B}(M) ≤ λ_{B/J′}(M). On the other hand, any surjection of Bmodules M′ → M with M′ finitely presented, factors through the natural map M′ → M′/J′M′, and by theorem 14.5.43(i,ii.b) we have λ_{B}(M′) − λ_{B/J′}(M′/J′M′) = λ_{B}(J′M′) ≥ 0, whence the first stated identity. The proof of the second identity is analogous. ♢

In view of claim 14.5.55we are reduced to proving the assertion for the morphism \overline{ψ} and the B/J′-module M, hence we may replace ψ by \overline{ψ}, and assume from start that A and B have Krull dimension zero.

Claim 14.5.56. Let A be a measurable K^{+}-algebra of Krull dimension zero. Then there exists a morphism φ : V → A of measurable K^{+}-algebras, with V a valuation ring flat over K^{+}, such that the residue field extension κ(V) → κ(A) is finite, and the induced map of value groups Γ → Γ_{V} is an isomorphism.

Proof of the claim. Let A_{0} → A be a local ind-étale map, from an object A_{0} of K^{+}-m.Alg_{0}. By lemma 9.1.32 we may find a K^{+}-flat valuation ring V_{0} in K^{+}-m.Alg_{0} and a local map φ_{0} : V → A_{0}, inducing a finite residue field extension κ(V_{0}) → κ(A_{0}) and an isomorphism on value groups Γ →^{∼} Γ_{V}. Then A_{0} has also Krull dimension zero; especially, it is henselian, hence φ_{0} factors through a morphism φ_{0}^{h} : V_{0}^{h} → A_{0}, from the henselization V_{0}^{h} of V_{0}. Let E ⊂ κ(A) be the largest separable subextension of κ(V_{0}) = κ(V_{0}^{h}) contained in κ(A); there exists a unique (up to unique isomorphism) finiteétale morphism V_{0}^{h} → V′ with an isomorphism κ(V′) →^{∼} E of κ(V_{0})-algebras, and φ_{0}^{h} factors through a morphism V′ → A_{0}. Notice that V′ is still a henselian valuation ring and a measurable K^{+}-algebra, hence we may replace V_{0} by V′, and assume that V_{0} is henselian, and the residue field extension \overline{φ}_{0} : κ(V_{0}) → κ(A_{0}) is purely inseparable. Since A_{0} is henselian, we may write A as the colimit of a filtered system (A_{i} | i ∈ I) of finite étale A_{0}-algebras. Now, on the one hand, \overline{φ}_{0} induces an equivalence from the category of finite étale κ(V_{0})-algebras, to the category of finite étale κ(A_{0})-algebras (lemma 13.1.7(i)). On the other hand, the category of finite étale V_{0}-algebras is equivalent to the category of finite étale κ(V_{0})- algebras, and likewise for A_{0}. Therefore, for every i ∈ I we may find a finite étale morphism V_{0} → V_{i}, unique up to unique isomorphism, inducing an isomorphism of κ(A_{0})-algebras :

(14.5.57)  κ(V_{i}) ⊗_{κ(V_{0})} κ(A_{0}) →^{∼} κ(A_{i})

and the transition maps of residue fields κ(A_{i}) → κ(A_{j}) induce unique maps V_{i} → V_{j} of V_{0}- algebras, compatible with the isomorphisms (14.5.57). Hence, the resulting system (V_{i} | i ∈ I) is filtered, and its colimit is a valuation ring V, which is still a measurable K^{+}-algebra. Moreover, the field extensions κ(V_{i}) → κ(A_{i}) deduced from (14.5.57) lift uniquely to maps of V_{0}-algebras V_{i} → A_{i} ([66, Ch.IV, Cor.18.5.12]); taking colimits, we get finally a map V → A as sought. ♢

Let φ be as in claim 14.5.56; clearly it suffices to prove the sought identity for the two morphisms ψ ∘ φ and φ, so we may replace A by V, and assume from start that A is a valuation ring. Let us set d := [κ(B) : κ(A)]; we deduce :

λ_{B}(M) = inf {d^{−1} · λ_{A}(ψ_{∗}M′) | M′ ∈ 𝒞_{M}} ≥ d^{−1} · λ_{A}(ψ_{∗}M)

by theorem 14.5.43(ii.b). Furthermore, let us choose a surjection B^{⊕k} → M, whose kernel we write in the form K := ⋃_{i∈I} K_{i} where (K_{i} | i ∈ I) is a filtered family of finitely generated Bsubmodules of K. Next, by applying (i), proposition 14.5.12(i,iii.b), and theorem 14.5.43(ii.b) we derive :

λ_{B}(M) ≤ inf {λ_{B}(B^{⊕k}/K_{i}) | i ∈ I} = inf {d^{−1} · λ_{A}(ψ_{∗}(B^{⊕k}/K_{i})) | i ∈ I}
= d^{−1} · (λ_{A}(ψ_{∗}B^{⊕k}) − sup {λ_{A}(ψ_{∗}K_{i}) | i ∈ I})
= d^{−1} · (λ_{A}(ψ_{∗}B^{⊕k}) − λ_{A}(ψ_{∗}K))
= d^{−1} · λ_{A}(ψ_{∗}M)

whence the claim.

(iv): Let (N_{i} | i ∈ I) be the filtered system of finitely generated submodules of M_{2}. For every i ∈ I we have short exact sequences : 0 → M_{1} ∩ N_{i} → N_{i} → \overline{N}_{i} → 0, where \overline{N}_{i} is the image of N_{i} in M_{3}. In view of the case of (ii) already known, we may then replace M_{2} by N_{i}, and thus assume from start that M_{2} is finitely generated, so that we may find a finitely generated 𝔪_{A}-primary ideal J ⊂ A that annihilates M_{2}. By (iii) we have λ_{A}(M_{2}) = λ_{A/J}(M_{2}), and likewise for M_{1} and M_{3}, hence we may replace A by A/J. By claim 14.5.56, we may then find a morphism V → A of measurable K^{+}-algebras with V a valuation ring, inducing a finite residue field extension κ(V) → κ(A); then again (iii) reduces to the case where A = V, to which one may apply (i) and proposition 14.5.12(i) to conclude the proof.

Next we consider assertion (ii) for the case where the transition maps are surjective. We may assume that I admits a smallest element i_{0}; for every i ∈ I let K_{i} denote the kernel of the transition map M_{i_{0}} → M_{i}. We deduce a short exact sequence :

0 → ⋃_{i∈I} K_{i} → M_{i_{0}} → colim_{i∈I} M_{i} → 0

and we may then compute using (iv) and the previous case of (ii) :

λ_{A}(colim_{i∈I} M_{i}) = λ_{A}(M_{i_{0}}) − λ_{A}(⋃_{i∈I} K_{i}) = lim_{i∈I}(λ_{A}(M_{i_{0}}) − λ_{A}(K_{i})) = lim_{i∈I} λ_{A}(M_{i_{0}}/K_{i})

whence the claim.

(v): Since ψ_{∗} is an exact functor which commutes with colimits, we may use (ii) to reduce to the case where M is finitely presented, for which the assertion is already known, in view of theorem 14.5.43(iv). □

Remark 14.5.58. Suppose that the valuation of K is discrete; then one sees easily that theorem 14.5.43(i,iii) still holds (with simpler proof) when A is replaced by any local noetherian K^{+}- algebra, and by inspecting the definition, the resulting map λ_{A} is none else than the standard length function for modules supported on {s(A)}.

Lemma 14.5.59. Let A be a measurable K^{+}-algebra, and M an A-module supported at s(A) with λ_{A}(M) < ∞. Then

{a ∈ K^{+} | log |a| > λ_{A}(M)} ⊂ Ann_{K^{+}}M.

Proof. Using proposition 14.5.54(ii), we easily reduce, first, to the case where M is finitely generated, and second, to the case where M is finitely presented. Pick an ideal I ⊂ A and a valuation ring V mapping onto A/I, as in lemma 14.5.42; by considering the I-adic filtration of M, the additivity properties of λ_{A} allow to further reduce to the case where M is an A/Imodule. Next, by theorem 14.5.43(ii.b) we may replace A by V, and therefore assume that A is a valuation ring whose value group equals Γ. In this case, λ_{A} is computed by Fitting ideals, so the assertion follows easily from [75, Prop.6.3.6(iii)]. □

14.5.60. Let us consider a K^{+}-algebra R_{∞} that is the colimit of an inductive system

(14.5.61)  R_{0} → R_{1} → R_{2} → ⋯

of morphisms of measurable K^{+}-algebras inducing integral ring homomorphisms κ(R_{i}) → R_{i+1}/𝔪_{R_{i}}R_{i} for every i ∈ ℕ. The final step consists in generalizing the definition of normalized length to the category R_{∞}-Mod_{{s}} of R_{∞}-modules supported at the closed point s(R_{∞}) of Spec R_{∞}. To this purpose, we shall axiomatize the general situation in which we can solve this problem. Later we shall see that our axioms are satisfied in many interesting cases.

14.5.62. Hence, let R_{∞} be as in (14.5.60). After fixing an order-preserving isomorphism

(14.5.63)  (ℚ ⊗_{ℤ} log Γ)^{∧} →^{∼} ℝ

we may regard the mappings λ_{A} (for any measurable K^{+}-algebra A) as real-valued functions on A-modules. To ease notation, for every R_{n}-module N supported on {s(R_{n})} we shall write λ_{n}(N) instead of λ_{R_{n}}(N). Notice that, for every such N, and every m ≥ n, the R_{m}-module R_{m} ⊗_{R_{n}} N is supported at {s(R_{m})}, since by assumption the map R_{n} → R_{m}/𝔪_{R_{n}}R_{m} is integral.

Definition 14.5.64. In the situation of (14.5.62), we say that R_{∞} is an ind-measurable K^{+}- algebra, if there exists a sequence of real normalizing factors (d_{n} > 0 | n ∈ ℕ) such that:

(a) For every n ∈ ℕ and every object N of R_{n}-Mod_{coh,{s}}, the sequence :

m ↦ d_{m}^{−1} · λ_{m}(R_{m} ⊗_{R_{n}} N)

converges to an element λ_{∞}(R_{∞} ⊗_{R_{n}} N) ∈ ℝ.

(b) For every m ∈ ℕ, every finitely generated 𝔪_{R_{0}}-primary ideal I ⊂ R_{0}, and every ε > 0 there exists δ(m, ε, I) > 0 such that the following holds. For every n ∈ ℕ and every surjection N → N′ of finitely presented R_{n}/IR_{n}-modules generated by m elements, such that

|λ_{∞}(R_{∞} ⊗_{R_{n}} N) − λ_{∞}(R_{∞} ⊗_{R_{n}} N′)| ≤ δ(m, ε, I)

we have :

d_{n}^{−1} · |λ_{n}(N) − λ_{n}(N′)| ≤ ε.

14.5.65. Assume now that R_{∞} is ind-measurable, and let N be a finitely presented R_{n}-module supported on {s(R_{n})}. The first observation is that λ_{∞}(R_{∞} ⊗_{R_{n}} N) only depends on the R_{∞}- module R_{∞} ⊗_{R_{n}} N. Indeed, suppose that R_{∞} ⊗_{R_{m}} M ≃ R_{∞} ⊗_{R_{n}} N for some m ∈ ℕ and some finitely presented R_{m}-module M; then there exists p ≥ m, n such that R_{p} ⊗_{R_{m}} M ≃ R_{p} ⊗_{R_{n}} N, and then the assertion is clear.

The second observation – contained in the following lemma 14.5.66 – will show that the conditions of definition 14.5.64 impose some non-trivial restrictions on the inductive system (R_{n} | n ∈ ℕ).

Lemma 14.5.66. Let (R_{n}; d_{n} | n ∈ ℕ) be an inductive system of measurable K^{+}-algebras and a sequence of positive reals, fulfilling conditions (a) and (b) of definition 14.5.64. Then:

(i) The natural map

M → R_{m} ⊗_{R_{n}} M

is injective for every n, m ∈ ℕ with m ≥ n and every R_{n}-module M. (ii) Especially, the transition maps R_{n} → R_{n+1} are injective for every n ∈ ℕ. (iii) Suppose that (d′_{n} | n ∈ ℕ) is another sequence of positive reals such that conditions

(a) and (b) hold for the datum (R_{n}; d′_{n} | n ∈ ℕ). Then the sequence (d_{n}/d′_{n} | n ∈ ℕ) converges to a non-zero real number.

Proof. (i): We reduce easily to the case where M is finitely presented over R_{n}. Let N ⊂ Ker (M → R_{m} ⊗_{R_{n}} M) be a finitely generated R_{n}-module; since R_{n} is coherent, N is a finitely presented R_{n}-module. We suppose first that M is in R_{n}-Mod_{coh,{s}}.

Claim 14.5.67. The natural map R_{m} ⊗_{R_{n}} M → R_{m} ⊗_{R_{n}} (M/N) is an isomorphism.

Proof of the claim. On the one hand, the R_{m}-module R_{m} ⊗_{R_{n}} M represents the functor

R_{m}-Mod → Set : Q ↦ Hom_{R_{n}}(M, Q).

On the other hand, the assumption on N implies that Hom_{R_{n}}(M, Q) = Hom_{R_{n}}(M/N, Q) for every R_{m}-module Q, so the claim follows easily. ♢

From claim 14.5.67 we deduce that the natural map R_{∞} ⊗_{R_{n}} M → R_{∞} ⊗_{R_{n}} (M/N) is an isomorphism, and then condition (b) says that λ_{n}(M) = λ_{n}(M/N), hence λ_{n}(N) = 0 and finally N = 0, as stated. Next, suppose M is any finitely presented R_{n}-module, and pick a finitely generated 𝔪_{R_{n}}-primary ideal I ⊂ R_{n}.

Claim 14.5.68. There exists c ∈ ℕ such that N ∩ I^{k+c}M = I^{k}(N ∩ I^{c}M) for every k ≥ 0.

Proof of the claim. We may find a local ind-étale map A → R_{n} of K^{+}-algebras, where A is an object of K^{+}-m.Alg_{0}, and such that I, M and N descend respectively to a finitely generated ideal I_{0} ⊂ A, a finitely presented A-module M_{0}, and a finitely generated submodule N_{0} ⊂ M_{0}. Then theorem 11.4.46ensures the existence of c ∈ ℕ such that N_{0} ∩ I_{0}^{k+c}M_{0} = I_{0}^{k}(N_{0} ∩ I_{0}^{c}M_{0}) for every k ≥ 0. Since R_{n} is a faithfully flat A-algebra, the claim follows. ♢

Pick c ∈ ℕ as in claim 14.5.68; then N/(N ∩ I^{k+c}M) is in the kernel of the natural map M/I^{k+c}M → R_{m}⊗_{R_{n}}(M/I^{k+c}M), hence N = N ∩I^{k+c}M by the foregoing, so that N ⊂ I^{k}N for every k ≥ 0, and finally N = 0 by Nakayama’s lemma.

(ii) is a special case of (i). To show (iii), let us denote by λ′_{∞}(M) the normalized length of any object M of R_{∞}-Mod_{coh,{s}}, defined using the sequence (d′_{n} | n ∈ ℕ). From (b) it is clear that λ_{∞}(M), λ′_{∞}(M) ≠ 0 whenever M ≠ 0. Then, for any such non-zero M, the quotient λ′_{∞}(M)/λ_{∞}(M) is the limit of the sequence (d_{n}/d′_{n} | n ∈ ℕ). □

Remark 14.5.69. (i) There is another situation of interest which leads to a well-behaved notion of normalized length. Namely, suppose that R_{•} := (R_{n} | n ∈ ℕ) is an inductive system of local homomorphisms of local noetherian rings, such that the fibres of the induced morphisms Spec R_{n+1} → Spec R_{n} have dimension zero, and denote by R_{∞} the inductive limit of the system R_{•}. For every n ∈ ℕ, let also λ_{n} be the usual length function on the set of isomorphism classes of finitely generated R_{n}-modules supported on s(R_{n}). Then for every m, n ∈ ℕ with m ≥ n, and every R_{n}-module M of finite length, the R_{m}-module R_{m} ⊗_{R_{n}} M has again finite length, so the analogues of conditions (a) and (b) of (14.5.62) can be formulated (cp. remark 14.5.58), and if these conditions hold for R_{•}, we shall say that R_{∞} is an ind-measurable ring. In such situation, lemma 14.5.66 – as well as the forthcoming lemma 14.5.70 and theorem 14.5.75 – still hold, with simpler proofs : we leave the details to the reader.

(ii) In spite of the uniqueness properties expressed by lemma 14.5.66, we do not know to which extent the normalized length of an ind-measurable K^{+}-algebra depends on the chosen tower of measurable algebras. Namely, suppose that (R_{n} | n ∈ ℕ) and (R′_{n} | n ∈ ℕ) are two such towers, with isomorphic colimit R_{∞}, and suppose that we have found normalizing factors (d_{n} | n ∈ ℕ) (resp. (d′_{n} | n ∈ ℕ)) for the first (resp. second) tower, whence a normalized length λ_{∞} (resp. λ′_{∞}) for R_{∞}-modules. Then we do not know whether the ratio of λ_{∞} and λ′_{∞} is a constant.

Next, for a given finitely generated R_{∞}-module M supported on {s(R_{∞})}, we shall proceed as in (14.5.51) : we denote by 𝒞_{M} the set of isomorphism classes of finitely presented R_{∞}- modules supported on {s(R_{∞})} that admit a surjection M′ → M, and we set

λ_{∞}^{∗}(M) := inf {λ_{∞}(M′) | M′ ∈ 𝒞_{M}} ∈ ℝ.

Directly on the definitions, one checks that λ_{∞}^{∗}(M) = λ_{∞}(M) if M is finitely presented.

Lemma 14.5.70. Let M be a finitely generated R_{∞}-module, Σ ⊂ M any finite set of generators, (N_{i} | i ∈ I) any filtered system of objects of R_{∞}-Mod_{{s}}, with surjective transition maps, such that colim_{i∈I} N_{i} ≃ M. Then :

(i) λ_{∞}^{∗}(M) = lim_{n→∞}d_{n}^{−1} · λ_{n}(ΣR_{n}).

(ii) If every N_{i} is finitely generated, we have :

inf {λ_{∞}^{∗}(N_{i}) | i ∈ I} = λ_{∞}^{∗}(M).

Proof. To start out, we show assertion (ii) in the special case where all the modules N_{i} are finitely presented. Indeed, let us pick any surjection φ : M′ → M with M′ ∈ 𝒞_{M}. We may find i ∈ I such that φ lifts to a map φ_{i} : M′ → N_{i} ([75, Prop.2.3.16(ii)]), and up to replacing I by a cofinal subset, we may assume that φ_{i} is defined for every i ∈ I. Moreover

colim_{i∈I} Coker φ_{i} = Coker φ = 0

hence there exists i ∈ I such that φ_{i} is surjective. It follows easily that λ_{∞}(N_{i}) ≤ λ_{∞}(M′), whence (ii) in this case.

(i): Let k be the cardinality of Σ, ε > 0 any real number, Q ⊂ Ann_{R_{0}}Σ a finitely generated 𝔪_{R_{0}}-primary ideal, S := R_{∞}/QR_{∞} and β : S^{⊕k} → M a surjection that sends the standard basis onto Σ. By filtering Ker β by the system (K_{j} | j ∈ J) of its finitely generated submodules, we obtain a filtered system (N_{j} := S^{⊕k}/K_{j} | j ∈ J) of finitely presented R_{∞}-modules supported on {s(R_{∞})}, with surjective transition maps and colimit isomorphic to M. Let ψ_{j} : N_{j} → M be the natural map; by the foregoing, we may find j_{0} ∈ J such that

(14.5.71)  0 ≤ λ_{∞}(N_{j_{0}}) − λ_{∞}^{∗}(M) ≤ min(δ(k, ε, Q), ε)

(notation of definition 14.5.64(b)). We can then find n ∈ ℕ and a finitely presented R_{n}-module M′_{n} supported on {s(R_{n})} and generated by at most k elements, such that N_{j_{0}} = R_{∞} ⊗_{R_{n}} M′_{n}; we set M′_{m} := R_{m} ⊗_{R_{n}} M_{n} for every m ≥ n and let M_{m} be the image of M′_{m} in M. Up to replacing n by a larger integer, we may assume that M_{m} = ΣR_{m} for every m ≥ n. Choose m ∈ ℕ so that :

(14.5.72)  |λ_{∞}(N_{j_{0}}) − d_{m}^{−1} · λ_{m}(M′_{m})| ≤ ε.

Let (M_{m,i} | i ∈ I) be a filtered system of finitely presented R_{m}-modules supported at {s(R_{m})}, with surjective transition maps, such that colim_{i∈I} M_{m,i} = M_{m}. Arguing as in the foregoing, we show that there exists i ∈ I such that the natural map φ : M′_{m} → M_{m} factors through a surjection φ_{i} : M′_{m} → M_{m,i}, and up to replacing I by a cofinal subset, we may assume that such a surjection φ_{i} exists for every i ∈ I. We obtain therefore a compatible system of surjections of R_{∞}-modules :

N_{j_{0}} ≃ R_{∞} ⊗_{R_{m}} M′_{m} → R_{∞} ⊗_{R_{m}} M_{m,i} → M

and combining with (14.5.71) we find :

|λ_{∞}(N_{j_{0}}) − λ_{∞}(R_{∞} ⊗_{R_{m}} M_{m,i})| ≤ δ(k, ε, Q) for every i ∈ I.

In such situation, condition (b) of definition 14.5.64ensures that :

d_{m}^{−1} · |λ_{m}(M′_{m}) − λ_{m}(M_{m,i})| ≤ ε for every i ∈ I.

Therefore : d_{m}^{−1} · |λ_{m}(M′_{m}) − λ_{m}(M_{m})| ≤ ε, by proposition 14.5.54(ii). Combining with (14.5.72) and again (14.5.71) we obtain :

|λ_{∞}^{∗}(M) − d_{m}^{−1} · λ_{m}(M_{m})| ≤ 3ε

which implies (i).

(ii): With no loss of generality, we may assume that I admits a smallest element i_{0}. Let us fix a surjection F := R_{∞}^{⊕k} → N_{i_{0}}, and for every i ∈ I, let C_{i} denote the kernel of the induced surjection F → N_{i}. We consider the filtered system (D_{j} | j ∈ J) consisting of all finitely generated submodules D_{j} ⊂ F such that D_{j} ⊂ C_{i} for some i ∈ I. It is clear that colim_{j∈J} F/D_{j} ≃ M, hence λ_{∞}^{∗}(M) = inf {λ_{∞}(F/D_{j}) | j ∈ J}, by (i). On the other, by construction, for every j ∈ J we may find i ∈ I such that λ_{∞}^{∗}(N_{i}) ≤ λ_{∞}(F/D_{j}); since clearly λ_{∞}^{∗}(N_{i}) ≥ λ_{∞}^{∗}(M) for every i ∈ I, the assertion follows. □

14.5.73. Let now M be an arbitrary object of the category R_{∞}-Mod_{{s}}. We let :

λ_{∞}(M) := sup {λ_{∞}^{∗}(M′) | M′ ⊂ M and M′ is finitely generated} ∈ ℝ ∪ {+∞}.

Lemma 14.5.74. If M is finitely generated, then λ_{∞}(M) = λ_{∞}^{∗}(M).

Proof. Let N ⊂ M be any finitely generated submodule. We choose finite sets of generators Σ ⊂ N and Σ′ ⊂ M with Σ ⊂ Σ′. In view of proposition 14.5.54(iv) we have λ_{n}(ΣR_{n}) ≤ λ_{n}(Σ′R_{n}) for every n ∈ ℕ, hence λ_{∞}^{∗}(N) ≤ λ_{∞}^{∗}(M), by lemma 14.5.70(i). The contention follows easily. □

Theorem 14.5.75. (i) Let (M_{i} | i ∈ I) be a filtered system of objects of R_{∞}-Mod_{{s}}, and suppose that either :

(a) all the transition maps of the system are injections, or

(b) all the transition maps are surjections and λ_{∞}(M_{i}) < +∞ for every i ∈ I. Then :

λ_{∞}(colim_{i∈I} M_{i}) = lim_{i∈I} λ_{∞}(M_{i}).

(ii) Let 0 → M′ → M → M′′ → 0 be a short exact sequence in R_{∞}-Mod_{{s}}. Then :

λ_{∞}(M) = λ_{∞}(M′) + λ_{∞}(M′′).

(iii) Let M be a finitely presented R_{∞}-module, N ⊂ M a submodule supported at s(R_{∞}). Then λ_{∞}(N) = 0 if and only if N = 0.

Proof. The proof of (i) in case (a) is the same as that of proposition 14.5.54(iii.b). (ii): We proceed in several steps :

• Suppose first that M and M′′ are finitely presented, hence M′ is finitely generated ([75, Lemma 2.3.18(ii)]). We may then find an integer n ∈ ℕ and finitely presented R_{n}-modules M_{n} and M′′_{n} such that M ≃ R_{∞} ⊗_{R_{n}} M_{n} and M′′ ≃ R_{∞} ⊗_{R_{n}} M′′_{n}. For every m ≥ n we set M_{m} := R_{m} ⊗_{R_{n}} M_{n} and likewise we define M′′_{m}. Up to replacing n by a larger integer, we may assume that the given map φ : M → M′′ descends to a surjection φ_{n} : M_{n} → M′′_{n}, and then φ is the colimit of the induced maps φ_{m} := 1_{R_{m}} ⊗_{R_{n}} φ_{n}, for every m ≥ n. Moreover, M′ ≃ colim_{m≥n} Ker φ_{m}, and by the right exactness of the tensor product, the image of Ker φ_{m} generates M′ for every m ≥ n. Furthermore, since the natural maps M_{m} → M_{p} are injective for every p ≥ m ≥ n (lemma 14.5.66), the same holds for the induced maps Ker φ_{m} → Ker φ_{p}. The latter factors as a composition :

Ker φ_{m} ⟶^{α} R_{p} ⊗_{R_{m}} Ker φ_{m} ⟶^{β} Ker φ_{p}

where α is injective (lemma 14.5.66) and β is surjective. In other words, R_{p} · Ker φ_{m} = Ker φ_{p} for every p ≥ m ≥ n. In such situation, lemma 14.5.70(i) ensures that :

λ_{∞}(M′) = lim_{m≥n} d_{m}^{−1} · λ_{m}(Ker φ_{m})

and likewise :

λ_{∞}(M) = lim_{m≥n} d_{m}^{−1} · λ_{m}(M_{m}) λ_{∞}(M′′) = lim_{m≥n} d_{m}^{−1} · λ_{m}(M′′_{m}).

To conclude the proof of (ii) in this case, it suffices then to apply proposition 14.5.54(iv).

• Suppose next that M, M′ (and hence M′′) are finitely generated. We choose a filtered system (M_{i} | i ∈ I) of finitely presented R_{∞}-modules, with surjective transition maps, such that M ≃ colim_{i∈I} M_{i}. After replacing I by a cofinal subset, we may assume that M′ is generated by a finitely generated submodule M′_{i} of M_{i}, for every i ∈ I, and that (M′_{i} | i ∈ I) forms a filtered system with surjective transition maps, whose colimit is necessarily M′; set also M′′_{i} := M_{i}/M′_{i} for every i ∈ I, so that the colimit of the filtered system (M′′_{i} | i ∈ I) is M′′. In view of lemmata 14.5.70(ii) and 14.5.74, we are then reduced to showing the identity :

inf {λ_{∞}^{∗}(M_{i}) | i ∈ I} = inf {λ_{∞}^{∗}(M′_{i}) | i ∈ I} + inf {λ_{∞}^{∗}(M′′_{i}) | i ∈ I}

which follows easily from the previous case.

• Suppose now that M is finitely generated. We let (M′_{i} | i ∈ I) be the filtered family of finitely generated submodules of M′. Then :

M′ ≃ colim_{i∈I} M′_{i} and M′′ ≃ colim_{i∈I} M/M′_{i}.

Hence :

λ_{∞}(M′) = lim_{i∈I} λ_{∞}(M′_{i}) (resp. λ_{∞}(M′′) = lim_{i∈I} λ_{∞}(M/M′_{i}))

by (i.a) (resp. by lemmata 14.5.70(ii) and 14.5.74). However, the foregoing case shows that λ_{∞}(M) = λ_{∞}(M′_{i}) + λ_{∞}(M/M′_{i}) for every i ∈ I, so assertion (ii) holds also in this case.

• Finally we deal with the general case. Let (M_{i} | i ∈ I) be the filtered system of finitely generated submodules of M; we denote by M′′_{i} the image of M_{i} in M′′, and set M′_{i} := M′ ∩ M_{i} for every i ∈ I. By (i.a) we have :

λ_{∞}(M) = lim_{i∈I} λ_{∞}(M_{i})

and likewise for M′ and M′′. Since we already know that λ_{∞}(M_{i}) = λ_{∞}(M′_{i}) + λ_{∞}(M′′_{i}) for every i ∈ I, we are done.

(iii): Let f ∈ N be any element; in view of (ii) we see that λ_{∞}(fR_{∞}) = 0, and it suffices to show that f = 0. However, we may find n ∈ ℕ and a finitely presented R_{n}-module M_{n} such that M ≃ R_{∞} ⊗_{R_{n}} M_{n}; notice that the natural map M_{n} → M is injective, by lemma 14.5.66(i). We may also assume that f is in the image of M_{n}. Let I ⊂ Ann_{R_{n}}(f) be a finitely generated 𝔪_{R_{n}}-primary ideal; after replacing M by M/IM, we may assume that M_{n} is supported at s(R_{n}). In light of (ii), we see that λ_{∞}(M) = λ_{∞}(M/fR_{∞}), hence λ_{n}(M_{n}) = λ_{n}(M_{n}/fR_{n}), due to condition (b) of definition 14.5.64. Hence λ_{n}(fR_{n}) = 0 by proposition 14.5.54(iv), and finally f = 0 by theorem 14.5.43(i,iii.a).

To conclude, we consider assertion (i) in case (b) : set M := colim_{i∈I} M_{i}; it is clear that λ_{∞}(M) ≤ λ_{∞}(M_{i}) ≤ λ_{∞}(M_{j}) whenever i ≥ j, hence

lim_{i∈I} λ_{∞}(M_{i}) = inf {λ_{∞}(M_{i}) | i ∈ I} ≥ λ_{∞}(M).

For the converse inequality, fix ε > 0; without loss of generality, we may assume that I admits a smallest element i_{0}, and we can find a finitely generated submodule N_{i_{0}} ⊂ M_{i_{0}} such that λ_{∞}(M_{i_{0}}) − λ_{∞}(N_{i_{0}}) < ε. For every i ∈ I, let N_{i} ⊂ M_{i} be the image of N_{i_{0}}, and let N ⊂ M be the colimit of the filtered system (N_{i} | i ∈ I); then M_{i}/N_{i} is a quotient of M_{i_{0}}/N_{i_{0}}, and the additivity assertion (ii) implies that λ_{∞}(M_{i}) − λ_{∞}(N_{i}) < ε for every i ∈ I. According to lemma 14.5.70(ii) (and lemma 14.5.74) we have :

λ_{∞}(M) ≥ λ_{∞}(N) = inf {λ_{∞}(N_{i}) | i ∈ I} ≥ inf {λ_{∞}(M_{i}) | i ∈ I} − ε

whence the claim. □

14.5.76. We wish now to show that the definition of normalized length descends to almost modules (see (14.5)). Namely, we have the following :

Proposition 14.5.77. Let M, N be two objects of R_{∞}-Mod_{{s}} such that M^{a} ≃ N^{a}. Then λ_{∞}(M) = λ_{∞}(N).

Proof. Using additivity (theorem 14.5.75(ii)), we easily reduce to the case where M^{a} = 0, in which case we need to show that λ_{∞}(M) = 0. Using theorem 14.5.75(i) we may further assume that M is finitely generated. Then, in view of lemma 14.5.70(i), we are reduced to showing the following :

Claim 14.5.78. Let A be any measurable K^{+}-algebra, and M any object of A-Mod_{{s}} such that M^{a} = 0. Then λ_{A}(M) = 0.

Proof of the claim. Arguing as in the foregoing we reduce to the case where M is finitely generated. Then, let us pick I ⊂ A and φ : V → A/I as in lemma 14.5.42. It suffices to show the assertion for the finitely generated module ⨁_{n∈ℕ} I^{n}M/I^{n+1}M, hence we may assume that I ⊂ Ann_{A}M, in which case, by proposition 14.5.54(iii) we may replace A by V and assume throughout that A is a valuation ring. Then the claim follows from propositions 14.5.54(i) and 14.5.12(ii). □

14.5.79. Proposition 14.5.77 suggests the following definition. We let R_{∞}^{a}-Mod_{{s}} be the full subcategory of R_{∞}^{a}-Mod consisting of all the R_{∞}^{a}-modules M such that M_{!} is supported at s(R_{∞}), in which case we say that M is supported at s(R_{∞}). Then, for every such M we set :

λ_{∞}(M) := λ_{∞}(M_{!}).

With this definition, it is clear that theorem 14.5.75(i,ii) extends mutatis mutandis to almost modules. For future reference we point out :

Lemma 14.5.80. Let 0 → M′ → M → M′′ → 0 be a short exact sequence of R_{∞}^{a}-modules. We have :

(i) If M lies in R_{∞}^{a}-Mod_{{s}}, then λ_{∞}(abM) ≤ λ_{∞}(aM′)+λ_{∞}(bM′′) for every a, b ∈ 𝔪. (ii) If M is almost finitely presented, and M′ is supported at s(R_{∞}), then λ_{∞}(M′) = 0 if

and only if M′ = 0.

Proof. (i): To start out, we deduce a short exact sequence : 0 → bM ∩M′ → bM → bM′′ → 0. Next, let N := Ker(a : bM → bM), and denote by N′ the image of N in bM′′; there follows a short exact sequence : 0 → a(bM ∩ M′) → abM → bM′′/N′ → 0. The claim follows.

(ii): We reduce easily to the case where M′ is a cyclic R_{∞}^{a}-module, say M′ = R_{∞}^{a}x for some x ∈ M_{!}. In this case, let I ⊂ Ann_{R_{0}}(x) be a finitely generated 𝔪_{R_{0}}-primary ideal; we may then replace M by M/IM, and assume that M lies in R_{∞}^{a}-Mod_{{s}} as well. We remark :

Claim 14.5.81. M is almost finitely presented if and only if, for every b ∈ 𝔪 there exist a finitely presented R_{∞}-module N and a morphism M → N^{a} whose kernel and cokernel are annihilated by b. Moreover if M is supported at s(R_{∞}), one can choose N to be supported at s(R_{∞}).

Proof of the claim. The “if” direction is clear. For the “only if” part, we use [75, Cor.2.3.13], which provides us with a morphism φ : N → M_{!} with N finitely presented over R_{∞}, such that b · Ker φ = b · Coker φ = 0. Then b · 1_{N}^{a} factors through a morphism φ′ : Im φ^{a} → N^{a}, and b · 1_{M} factors through a morphism φ′′ : M → Im φ^{a}; the kernel and cokernel of φ′ ∘ φ′′ are annihilated by b^{2}. Finally, suppose that M is supported on s(R_{∞}), and let I ⊂ R_{∞} be any finitely generated ideal such that V (I) = {s(R_{∞})}. By assumption, for every f ∈ I and every m ∈ M_{!} there exists n ∈ ℕ such that f^{n}m = 0; it follows that I^{n} annihilates Im φ for every sufficiently large n ∈ ℕ, and we may then replace N by N/I^{n}N. ♢

Let M and M′ be as in (ii), and choose a morphism φ : M → N^{a} as in claim 14.5.81; by adjunction we get a map ψ : M′_{!} → M_{!} → N with b · 𝔪 · Ker ψ = 0. It follows that λ_{∞}(Im ψ) = 0, hence Im ψ = 0, by theorem 14.5.75(iii). Hence bM′ = 0; since b is arbitrary, the assertion follows. □

Simple examples show that an almost finitely generated (or even almost finitely presented) R_{∞}^{a}-module may fail to have finite normalized length. The useful finiteness condition for almost modules is contained in the following :

Definition 14.5.82. Let M be a R_{∞}^{a}-module supported at s(R_{∞}). We say that M has almost finite length if λ_{∞}(bM) < +∞ for every b ∈ 𝔪.

Lemma 14.5.83. (i) The set of isomorphism classes of R_{∞}^{a}-modules of almost finite length forms a closed subset of the uniform space ℳ (A) (notation of [75, §2.3]).

(ii) Especially, every almost finitely generated R_{∞}^{a}-module supported at s(R_{∞}) has almost finite length.

Proof. Assertion (i) boils down to the following :

Claim 14.5.84. Let a, b ∈ 𝔪, and f : N → M, g : N → M′ morphisms of R_{∞}^{a}-modules, such that the kernel and cokernel of f and g are annihilated by a ∈ 𝔪, and such that λ_{∞}(bM) < +∞. Then λ_{∞}(a^{2}bM′) < +∞.

Proof of the claim. By assumption, Ker f ⊂ Ker a · 1_{N}, whence an epimorphism f(N) → aN; likewise, we have an epimorphism g(N) → aM′. It follows that

λ_{∞}(a^{2}bM′) ≤ λ_{∞}(ab · g(M)) ≤ λ_{∞}(abN) ≤ λ_{∞}(b · f(N)) < +∞

as stated. ♢

(ii) follows from (i) and the obvious fact that every finitely generated R_{∞}-module supported at s(R_{∞}) has finite normalized length. □

14.5.85. For the case of a measurable K^{+}-algebra A of dimension zero, we can show a further Lipschitz type uniform estimate for the normalized length. Namely, for any integer k > 0 define the set ℳ_{k}(A^{a}) with its uniform structure as in (14.5.1); i.e. we have the fundamental system of entourages

(E_{r} | r ∈ ℝ_{>0})

where each E_{r} consists of the pairs (N, N′) such that there exist a third A^{a}-module N′′ and A^{a}- linear maps N′′ → N, N′′ → N′ whose kernel and cokernels are annihilated by any b ∈ K^{+} such that log |b| ≥ r.

Lemma 14.5.86. In the situation of (14.5.85), we have :

|λ_{A^{a}}(N) − λ_{A^{a}}(N′)| ≤ 4k · length_{A}(A/𝔪A) · r

for every r ∈ ℝ_{>0} and every (N, N′) ∈ E_{r}.

Proof. We begin with the following

Claim 14.5.87. For any finitely generated A-module N, and every b ∈ K^{+} \ {0} we have

λ_{A}(N/bN) ≤ length_{A}(N/𝔪N) · log |b|.

Proof of the claim. Choose a map ψ : V → A of measurable K^{+}-algebras, inducing a finite residue field extension κ(V) → κ(A), where V is a K^{+}-flat valuation ring, and the induced map of value groups Γ → Γ_{V} is an isomorphism (claim 14.5.56). Let d := dim_{κ(V)} N/𝔪_{V} N; applying Nakayama’s lemma, we get a surjection (V/bV)^{⊕d} → N/bN of V -modules, for every b ∈ K^{+}. Hence

λ_{A}(N/bN) = \frac{λ_{V} (ψ_{∗}(N/bN))}{[κ(V) : κ(A)]} ≤ \frac{d log |b|}{[κ(V) : κ(A)]} = length_{A}(N/𝔪N) · log |b|

as stated. ♢

Now, suppose that (N, N′) ∈ E_{log |b|} for some b ∈ K^{+}, and pick maps φ : N′′ → N, ψ : N′′ → N′ whose kernel and cokernel are annihilated by b. Especially, if n_{1}, …, n_{k} ∈ N_{∗} (resp. n′_{1}, …, n′_{k} ∈ N′_{∗}) is a system of generators for N (resp. for N′), we may find n′′_{1}, …, n′′_{2k} ∈ N′′_{∗} such that φ(n′′_{i}) = b^{2}n_{i} for i = 1, …, k and ψ(n′′_{i}) = b^{2}n′_{i−k} for i = k + 1, …, 2k. After replacing N′′ by its submodule generated by n′′_{1}, …, n′′_{2k}, we may assume that N′′ ∈ ℳ_{2k}(A), Ker φ is still annihilated by b, but Coker φ is only annihilated by b^{2}. On the one hand, we deduce that

λ_{A^{a}}(N′′) ≥ λ_{A^{a}}(φ(N′′)) ≥ λ_{A^{a}}(b^{2}N) = λ_{A^{a}}(N) − λ_{A^{a}}(N/b^{2}N)

therefore

λ_{A^{a}}(N) − λ_{A^{a}}(N′′) ≤ λ_{A^{a}}(N/b^{2}N).

On the other hand, the map N′′ → N′′ given by the rule : n′′ ↦ bn′′ for every n′′ ∈ N′′_{∗}, factors through φ(N′′), therefore λ_{A^{a}}(N) ≥ λ_{A^{a}}(bN′′) so the same calculation yields the inequality

λ_{A^{a}}(N′′) − λ_{A^{a}}(N) ≤ λ_{A^{a}}(N′′/bN′′).

Taking into account claim 14.5.87(and proposition 14.5.77) we see that

|λ_{A^{a}}(N′′) − λ_{A^{a}}(N)| ≤ 2k · length_{A}(A/𝔪A) · log |b|.

Of course, the same holds with N replaced by N′, and the lemma follows. □

We conclude this section with some basic examples of the situation contemplated in definition 14.5.64.

Example 14.5.88. Suppose that all the transition maps R_{n} → R_{n+1} of the inductive system in (14.5.62) are flat. Then, proposition 14.5.54(v) implies that conditions (a) and (b) hold with:

d_{n} := length_{R_{n}}(R_{n}/𝔪_{R_{0}}R_{n}) for every n ∈ ℕ.

Example 14.5.89. Suppose that p := char κ > 0. Let d ∈ ℕ be any integer, f : X → 𝔸_{K^{+}}^{d} := Spec K^{+}[T_{1}, …, T_{d}] an étale morphism. For every r ∈ ℕ we consider the cartesian diagram of schemes

\xymatrix{X_{r} \ar[r]^{f_{r}} \ar[d]_{ψ_{r}} & 𝔸_{K^{+}}^{d} \ar[d]^{φ_{r}} \\ X \ar[r]^{f} & 𝔸_{K^{+}}^{d}}

where φ_{r} is the morphism corresponding to the K^{+}-algebra homomorphism

φ_{r}^{♮} : K^{+}[T_{1}, …, T_{d}] → K^{+}[T_{1}, …, T_{d}]

defined by the rule: T_{j} ↦ T_{j}^{p^{r}} for j = 1, …, d. For every r, s ∈ ℕ with r ≥ s, ψ_{r} factors through an obvious S-morphism ψ_{rs} : X_{r} → X_{s}, and the collection of the schemes X_{r} and transition morphisms ψ_{sr} gives rise to an inverse system \underline{X} := (X_{r} | r ∈ ℕ), whose inverse limit is representable by an S-scheme X_{∞} ([65, Ch.IV, Prop.8.2.3]). Let g_{∞} : X_{∞} → S (resp. g_{r} : X_{r} → S for every r ∈ ℕ) be the structure morphism, x ∈ g_{∞}^{−1}(s) any point, x_{r} ∈ X_{r} the image of x and R_{r} := 𝒪_{X_{r},x_{r}} for every r ∈ ℕ. Clearly the colimit R_{∞} of the inductive system (R_{r} | r ∈ ℕ) is naturally isomorphic to 𝒪_{X_{∞},x}. Moreover, notice that the restriction g_{r+1}^{−1}(s) → g_{r}^{−1}(s) is a radicial morphism for every r ∈ ℕ. It follows easily that the transition maps R_{r} → R_{r+1} are finite; furthermore, by inspection one sees that φ_{r}^{♮} is flat and finitely presented, so R_{r+1} is a free R_{r}-module of rank p^{d}, for every r ∈ ℕ. Hence, the present situation is a special case of example 14.5.88, and therefore conditions (a) and (b) hold if we choose the sequence of integers (d_{i} | i ∈ ℕ) with

d_{i} := p^{id}/[κ(x_{i}) : κ(x_{0})] for every i ∈ ℕ.

The foregoing discussion then yields a well-behaved notion of normalized length for arbitrary R_{∞}-modules supported at {s(R_{∞})}.

Example 14.5.90. In the situation of example 14.5.89, it is easy to construct R_{∞}^{a}-modules M ≠ 0 such that λ_{∞}(M) = 0. For instance, for d := 1, let x ∈ X_{∞} be the point of the special fibre where T_{1} = 0; then we may take M := R_{∞}/I, where I is the ideal generated by a non-zero element of 𝔪_{K} and by the radical of T_{1}R_{∞}. The verification shall be left to the reader.

14.6. Finite group actions on almost algebras. In this section we fix a basic setup (V, 𝔪) such that 𝔪̃ := 𝔪 ⊗_{V} 𝔪 is a flat V -module (see [75, §2.1.1]), and we consider some descent problems for V^{a}-algebras endowed with a finite group of automorphisms. Hence, the results below overlap with those of [75, §4.5].

14.6.1. Let G be a finite group, A a V^{a}-algebra, and let S := Spec V^{a}, X := Spec A. A right action of G on X is a group homomorphism :

ρ : G → Aut_{V^{a}-Alg}(A)

from G to the group of automorphisms of A. Let G_{S} be the affine group S-scheme defined by G; hence every g ∈ G determines a section g_{S} : S → G_{S} of the structure morphism G_{S} → S, and the resulting morphism:

∐_{g∈G} g_{S} : S ⨿ ⋯ ⨿ S → G_{S}

is an isomorphism of S-schemes. Then ρ can be also regarded as a right action of G_{S} on X, as defined in [75, §3.3.6]. Especially, ρ induces morphisms of S-schemes :

∂_{i} : X × G := X ×_{S} G_{S} → X i = 0, 1

as in loc. cit., and we may define a G-action on an A-module M (covering the given action of G on X) as a morphism of quasi-coherent 𝒪_{X×G}-modules :

β : ∂_{0}^{∗}M → ∂_{1}^{∗}M

fulfilling the conditions of [75, §3.3.7]. One also says that (M, β) is a G-equivariant A-module. We denote by A[G]-Mod the category of all G-equivariant A-modules and G-equivariant Alinear morphisms. Notice that A[G]-Mod is an abelian tensor category : indeed, for any two objects (M, β), (M, β′) we may set (M, β) ⊗_{A} (M′, β′) := (M ⊗_{A} M′, β ⊗_{𝒪_{X×G}} β′).

14.6.2. Likewise, if B is any A-algebra, a G-action on B is a morphism β : ∂_{0}^{∗}B → ∂_{1}^{∗}B of quasi-coherent 𝒪_{X×G}-algebras, such that the pair (B, β) is a G-equivariant A-module. We say that (B, β) is a G-equivariant A-algebra, and we denote by A[G]-Alg the category of such pairs, with G-equivariant morphisms of A-algebras. One verifies easily that the datum (B, β) is the same as a morphism ψ : A → B of V^{a}-algebras, together with a G-action ρ_{B} : G → Aut_{V^{a}-Alg}(B) on the affine scheme Spec B, such that the diagram :

\xymatrix{A \ar[r]^{ψ} \ar[d]_{ρ(g)} & B \ar[d]^{ρ_{B}(g)} \\ A \ar[r]^{ψ} & B}

commutes for every g ∈ G. We shall also consider the full subcategory A[G]-Alg_{fl} (resp. A[G]-w.Ét, resp. A[G]-Ét, resp. A[G]-Ét_{afp}) of all such pairs, where B is a flat (resp. weakly étale, resp. étale, resp. étale and almost finitely presented) A-algebra.

14.6.3. The trivial G-action on X is the map ρ with ρ(g) = 1_{A} for every g ∈ G; this is the same as saying that ∂_{0} = ∂_{1}. If G acts trivially on X, a G-equivariant A-module (M, β) is the same as a group homomorphism \overline{β} : G → Aut_{A}(M) from G to the group of A-linear automorphisms of M. Namely, for every g ∈ G, one lets :

(14.6.4)  \overline{β}(g) := (1_{X} ×_{S} g_{S})^{∗}β

and conversely, to a given map \overline{β} there corresponds a unique pair (M, β) such that (14.6.4) holds.

Under this correspondence, the trivial G-action \overline{β}_{0} (such that \overline{β}(g) = 1_{M} for every g ∈ G) corresponds to the identity morphism β_{0} : ∂_{0}^{∗}M →^{∼} ∂_{1}^{∗}M.

More generally, let (M, β) be a G-action on an A-module M, covering the trivial G-action on X; for every A_{∗}-valued character χ : G → A_{∗}^{×} of G, we let

M_{χ} := ⋂_{g∈G} Ker(\overline{β}(g) − χ(g) · 1_{M}).

The restriction of \overline{β} defines a G-action on M_{χ}, such that the monomorphism M_{χ} ⊂ M is Gequivariant. In the special case where χ is the trivial character, we have M_{χ} = (M, β)^{G}, the largest G-equivariant A-submodule of (M, β) on which β restricts to the trivial G-action (i.e. the submodule fixed by G). When the notation is not ambiguous, we shall often just write M^{G} instead of (M, β)^{G}.

14.6.5. If H ⊂ G is a subgroup, any G-action ρ on X induces by restriction an H-action ρ_{|H}; then the morphisms ∂_{i} : X × H → X are just the restrictions of the corresponding morphisms for G (under the natural closed immersion X × H → X × G). Similarly, a G-action β on an A-module M induces by restriction an H-action β_{|H} on the same module.

Let Y := Spec B be any affine S-scheme. The G-action ρ induces a G-action Y ×_{S} ρ on Y ×_{S} X; namely, Y ×_{S} ρ(g) := 1_{Y} ×_{S} ρ(g) for every g ∈ G. In terms of the group scheme G_{S}, this is the action given by the morphisms :

∂_{Y,i} := 1_{Y} ×_{S} ∂_{i} : (Y ×_{S} X) × G → (Y ×_{S} X) i = 0, 1.

Let π_{X} : Y ×_{S} X → X be the natural morphism; then every G-equivariant A-module (M, β) induces a G-equivariant B ⊗_{V^{a}} A-module π_{X}^{∗}(M, β) := (π_{X}^{∗}M, π_{X}^{∗}β), whose action covers the G-action Y ×_{S} ρ on Y ×_{S} X.

14.6.6. Let us set :

\xymatrix{X/G := Coequal(X × G \ar[r]^{∂_{1}} \ar[r]_{∂_{1}} & X) and X^{⟨g⟩} := Equal(X \ar[r]^{ρ(g)} \ar[r]_{1_{X}} & X)} for every g ∈ G.

In other words, X/G = Spec A^{G}, the subalgebra fixed by G, and X^{⟨g⟩} is the closed subscheme fixed by the subgroup ⟨g⟩ ⊂ G generated by g. Thus, X^{⟨g⟩} is the spectrum of a quotient A/I_{g} of A, where I_{g} ⊂ A is the ideal generated by the almost elements of the form a − ρ(g)(a), for every a ∈ A_{∗}.

Let π : X → X/G be the natural morphism, N any quasi-coherent 𝒪_{X/G}-module; the pullback π^{∗}N is the quasi-coherent 𝒪_{X}-module A ⊗_{A^{G}} N. By construction, there is a natural isomorphism β_{N} : ∂_{0}^{∗}(π^{∗}N) →^{∼} ∂_{1}^{∗}(π^{∗}N) (deduced from the natural isomorphisms of functors ∂_{i}^{∗} ∘ π^{∗} ≃ (π ∘ ∂_{i})^{∗}, for i = 0, 1), and one verifies easily that β_{N} is a G-action on π^{∗}N.

Moreover, let i_{g} : X^{⟨g⟩} → X be the natural closed immersion; if M is any A-module, then i_{g}^{∗}M is the A/I_{g}-module M/I_{g}M. Especially, take M := π^{∗}N; we notice that the restriction ρ_{|⟨g⟩} of the given action ρ, induces the trivial ⟨g⟩-action on X^{⟨g⟩}, and directly from the construction, we see that the natural action β_{N|⟨g⟩} restricts to the trivial ⟨g⟩-action on i_{g}^{∗}M.

We are thus led to the :

Definition 14.6.7. Let G be a finite group, A a V^{a}-algebra, ρ a G-action on X := Spec A.

(i) The category A[G]-Mod_{hor} of A-modules with horizontal G-action is the full subcategory of A[G]-Mod consisting of all pairs (M, β) subject to the following condition. For every g ∈ G, the restriction β_{|⟨g⟩} induces the trivial action on i_{g}^{∗}M.

(ii) We denote by A[G]-Mod_{hor.fl} the full subcategory of A[G]-Mod_{hor} consisting of the

pairs (M, β) as above, such that M is a flat A-module. (iii) Likewise, we denote by A[G]-Alg_{hor} (resp. A[G]-Alg_{hor.fl}, resp. A[G]-w.Ét_{hor}, resp.

A[G]-Ét_{hor}, resp. A[G]-Ét_{hor.afp}) the full subcategory of A[G]-Alg (resp. A[G]-Alg_{fl}, resp. A[G]-w.Ét, resp. A[G]-Ét, resp. A[G]-Ét_{afp}) consisting of all pairs (B, β) which are horizontal, when regarded as G-equivariant A-modules.

Notice that the tensor product of horizontal modules (resp. algebras) is again horizontal. By the foregoing, the rule N ↦ (π^{∗}N, β_{N}) defines a functor :

(14.6.8)  A^{G}-Mod → A[G]-Mod_{hor}.

On the other hand, if (M, β) is any G-equivariant A-module, the pair π_{∗}(M, β) := (π_{∗}M, π_{∗}β) may be regarded as a G-action on π_{∗}M, covering the trivial G-action on Spec A^{G}, i.e. a group homomorphism G → Aut_{A^{G}}(M) (see (14.6.3)). One verifies easily that the functor N ↦ (π^{∗}N, β_{N}) is left adjoint to the functor A[G]-Mod → A^{G}-Mod : (M, β) ↦ π_{∗}(M, β)^{G} (details left to the reader). Hence, also (14.6.8) admits a right adjoint, given by the same rule. Similar assertions hold for the analogous functors :

(14.6.9)  A^{G}-Alg → A[G]-Alg_{hor}

and the variants considered in definition 14.6.7(iii).

Lemma 14.6.10. In the situation of definition 14.6.7, suppose furthermore that the order o(G) of G is invertible in A_{∗}, and let (M, β) be any G-equivariant A-module. Then :

(i) For every A_{∗}^{G}-valued character χ : G → (A_{∗}^{G})^{×}, the natural A^{G}-linear monomorphism π_{∗}M_{χ} → π_{∗}M admits a G-equivariant A^{G}-linear right inverse π_{∗}M → π_{∗}M_{χ}.

(ii) For every A^{G}-module N, the unit of adjunction :

ε_{N} : N → (A ⊗_{A^{G}} N)^{G}

is an isomorphism. (iii) Let Y := Spec B be any affine S-scheme; denote by π_{X} : Y ×_{S} X → X and π_{X/G} :

Y ×_{S} (X/G) → X/G the natural projections. Then the natural morphism :

π_{X/G}^{∗}(M, β)^{G} → (π_{X}^{∗}M, π_{X}^{∗}β)^{G}

is an isomorphism of B ⊗_{V^{a}} A^{G}-modules.

Proof. This is standard : for every χ as in (i), the group algebra A_{∗}[G] admits the central idempotent :

(14.6.11)  e_{χ} := \frac{1}{o(G)} · ∑_{g∈G} χ(g) · g

and M_{χ} = e_{χ}M for any G-equivariant A-module M. Especially, we may take M := N ⊗_{A^{G}} A, and e_{0} the central idempotent associated with the trivial character, in which case e_{0}A = A^{G}, and M = N ⊕ (N ⊗_{A^{G}} (1 − e_{0})A), so all the claims follow easily. □

Definition 14.6.12. Let Γ be any finite abelian group with neutral element 0 ∈ Γ.

(i) A Γ-graded V^{a}-algebra is a pair \underline{A} := (A, gr_{•}A) consisting of a V^{a}-algebra A and a decomposition A = ⨁_{χ∈Γ} gr_{χ}A as a direct sum of V^{a}-modules, such that :

1 ∈ gr_{0}A_{∗} and gr_{χ}A · gr_{χ′}A ⊂ gr_{χ+χ′}A for every χ, χ′ ∈ Γ

(where as usual gr_{χ}A · gr_{χ′}A denotes the image of the restriction gr_{χ}A ⊗_{V^{a}} gr_{χ′}A → A of the multiplication morphism μ_{A}). Especially, gr_{0}A is a V^{a}-subalgebra of A, and every submodule gr_{χ}A is a gr_{0}A-module.

(ii) A Γ-graded \underline{A}-module is a pair \underline{N} := (N, gr_{•}N) consisting of an A-module N and a decomposition N = ⨁_{χ∈Γ} gr_{χ}N as a direct sum of V^{a}-modules, such that :

gr_{χ}A · gr_{χ′}N ⊂ gr_{χ+χ′}N for every χ, χ′ ∈ Γ.

Of course, a morphism of Γ-graded \underline{A}-modules \underline{N} → \underline{N}′ := (N′, gr_{•}N′) is an A-linear morphism N → N′ that respects the gradings.

(iii) For every subgroup Δ ⊂ Γ, let J_{Δ} ⊂ A be the graded ideal generated by ⨁_{χ∉Δ} gr_{χ}A. We say that \underline{N} is horizontal if gr_{χ}(N/J_{Δ}N) := gr_{χ}N/(gr_{χ}N ∩ J_{Δ}N) = 0 for every subgroup Δ ⊂ Γ and every χ ∉ Δ.

(iv) If Δ ⊂ Γ is any subgroup, p : Γ → Γ/Δ the natural projection, and ρ ∈ Γ/Δ any

element, we let :

gr_{ρ}^{Γ/Δ}N := ⨁_{χ∈p^{−1}(ρ)} gr_{χ}N

and set \underline{N}_{Γ/Δ} := (N, gr_{•}^{Γ/Δ}N). Then \underline{A}_{Γ/Δ} is a Γ/Δ-graded V^{a}-algebra, and \underline{N}_{Γ/Δ} is a Γ/Δ-graded \underline{A}_{Δ}-module. Moreover, let also \underline{N}_{|Δ} be the pair consisting of N_{|Δ} := gr_{0}^{Γ/Δ}N together with its decomposition N_{|Δ} = ⨁_{χ∈Δ} gr_{χ}N; then \underline{A}_{|Δ} is a Δ-graded V^{a}-algebra, and \underline{N}_{|Δ} is a Δ-graded \underline{A}_{|Δ}-module.

Proposition 14.6.13. In the situation of definition 14.6.12, let \underline{N} be any Γ-graded \underline{A}-module. Then the following conditions are equivalent :

(a) \underline{N} is horizontal.

(b) The natural morphism A ⊗_{gr_{0}A} gr_{0}N → N is an epimorphism.

Proof. (b)⇒(a): The assertion is obvious for the Γ-graded \underline{A}-module consisting of A ⊗_{gr_{0}A} gr_{0}N and its natural grading deduced from gr_{•}A; however any (graded) quotient of a horizontal module is horizontal, hence the assertion follows also for \underline{N}.

(a)⇒(b): We argue by induction on o(Γ). The first case is covered by the following :

Claim 14.6.14. The proposition holds if o(Γ) is a prime number.

Proof of the claim. Indeed, suppose that \underline{N} is horizontal. We may replace N by N/(A · gr_{0}N), which is a horizontal Γ-graded \underline{A}-module, when endowed with the grading induced from gr_{•}N. Then gr_{0}N = 0, and we have to show that N = 0. By assumption, N ⊂ J_{{0}}N, i.e. :

gr_{χ}N ⊂ ∑_{σ≠0,χ} gr_{σ}A · gr_{χ−σ}N for every χ ≠ 0.

For every n > 0, the symmetric group S_{n} acts on the set (Γ \ {0})^{n} by permutations; we let Q_{n} := (Γ \ \{0\})^{n}/S_{n}, the set of equivalence classes under this action. For every \underline{σ} := (σ_{1}, …, σ_{n}) ∈ Q_{n}, let |\underline{σ}| := ∑_{i=1}^{n} σ_{i}. By an easy induction, it follows that :

gr_{χ}N ⊂ ∑_{\underline{σ}∈Q_{n}} gr_{σ_{1}}A ⋯ gr_{σ_{n}}A · gr_{χ−|\underline{σ}|}N.

Since every element of Γ \ {0} generates Γ, it is also clear that there exists n ∈ ℕ large enough such that every sequence \underline{σ} in Q_{n} admits a subsequence, say \underline{τ} := (τ_{1}, …, τ_{m}) for some m ≤ n, with |\underline{τ}| = χ (details left to the reader). Up to a permutation, we may assume that \underline{τ} is the final segment of \underline{σ}; then we have :

gr_{σ_{1}}A ⋯ gr_{σ_{n}}A · gr_{χ−|\underline{σ}|}N ⊂ gr_{σ_{1}}A ⋯ gr_{σ_{n−m}}A · gr_{0}N = 0

whence the claim. ♢

Next, suppose that the assertion is already known for every subgroup Γ′ ⊂ Γ, every graded Γ′-algebra \underline{B}, and every Γ′-graded \underline{B}-module \underline{P}. We choose a subgroup Γ′ ⊂ Γ such that (Γ : Γ′) is a prime number. We shall use the following :

Claim 14.6.15. Let G be a group, 0 ∈ G the neutral element, H ⊂ G a subgroup. For every subgroup L ⊂ G with H ∩ L = {0}, choose an element g_{L} ∈ G \ L; denote by S the subgroup generated by all these elements g_{L}. Then S ∩ H ≠ {0}.

Proof of the claim. Indeed, if S ∩ H = {0}, we would have g_{S} ∈ S, a contradiction. ♢

Claim 14.6.16. Suppose that \underline{N} is horizontal. Then \underline{N}_{|Γ′} is a horizontal Γ′-graded \underline{A}_{|Γ′}-module.

Proof of the claim. For any given subgroup Δ ⊂ Γ′, let J′_{Δ} ⊂ A_{|Γ′} be the ideal generated by ⨁_{χ∈Γ′\Δ} gr_{χ}A; have to show that gr_{χ}(N/J′_{Δ}N) = 0 for every χ ∈ Γ′ \ Δ. To this aim, we may replace Γ, Γ′, \underline{A}, \underline{A}_{|Γ′}, \underline{N} and \underline{N}_{|Γ′}, by respectively Γ/Δ, Γ′/Δ, \underline{A}_{Γ/Δ}, (\underline{A}_{|Γ′})_{Γ′/Δ}, \underline{N}_{Γ/Δ} and (\underline{N}_{|Γ′})_{Γ′/Δ}, which allows to assume that Δ = \{0\}. In this case, we have to show that :

(14.6.17)  N_{|Γ′} = gr_{0}N + J′_{{0}} · N_{|Γ′}.

Notice that A_{|Γ′} · gr_{0}N = gr_{0}N + J′_{{0}} · gr_{0}N; the quotient P := N/(A · gr_{0}N) carries a unique Γ-grading gr_{•}P such that the projection \underline{N} → \underline{P} := (P, gr_{•}P) is a morphism of Γ-graded \underline{A}-modules (namely, gr_{χ}P := N_{χ}/gr_{χ}A · gr_{0}N for every χ ∈ Γ). Moreover, P is horizontal, gr_{0}P = 0, and (14.6.17) is equivalent to : P_{|Γ′} = J′_{\{0\}} · P_{|Γ′}. Therefore we may replace \underline{N} by \underline{P}, and assume from start that gr_{0}N = 0. Similarly, notice that J′_{\{0\}} · N is a Γ-graded \underline{A}-submodule of \underline{N}, and the pair \underline{Q} consisting of Q := N/(J′_{\{0\}} · N) and its quotient grading, is horizontal.

Moreover, gr_{χ}Q = gr_{χ}(N_{|Γ′}/J′_{{0}}N_{|Γ′}), for every χ ∈ Γ′. Hence, we may replace N by Q, which allows to assume as well that

(14.6.18)  J′_{{0}}N_{|Γ′} = 0

in which case, we are reduced to showing that N = 0. Now, by assumption, for every subgroup H ⊂ Γ we have :

N = N_{|H} + J_{H}N.

Let 𝒞 be the set of all subgroups H ⊂ Γ with H ∩ Γ′ = {0}; we deduce that :

(14.6.19)  gr_{χ}N ⊂ ∑_{σ∉H} gr_{σ}A · gr_{χ−σ}N  for every χ ∈ Γ′ \ {0} and every H ∈ 𝒞 .

Moreover, in view of (14.6.18) we may omit from the sum in (14.6.19) all the elements σ that lie in Γ′; for the remaining elements we have χ − σ ∉ Γ′, hence we may apply claim 14.6.14to the horizontal Γ/Γ′-graded \underline{A}_{Γ/Γ′}-module \underline{N}_{Γ/Γ′}, to deduce that :

gr_{χ−σ}N ⊂ ∑_{δ∈Γ′\{0}} gr_{χ−σ−δ}A · gr_{δ}N.

Therefore :

gr_{χ}N ⊂ ∑_{σ∉H∪Γ′} ∑_{δ∈Γ′\{0}} gr_{σ}A · gr_{χ−σ−δ}A · gr_{δ}N for every χ ∈ Γ′ \ {0} and every H ∈ 𝒞 .

However – again due to (14.6.18) – we may omit from this sum all the terms corresponding to the pairs (σ, δ) with χ ≠ δ, hence we conclude that :

gr_{χ}N ⊂ ∑_{σ∉H∪Γ′} gr_{σ}A · gr_{−σ}A · gr_{χ}N for every χ ∈ Γ′ \ {0} and every H ∈ 𝒞 .

Denote by Σ the set of all mappings \underline{σ} : 𝒞 → Γ such that \underline{σ}(H) ∉ H ∪ Γ′ for every H ∈ 𝒞. Moreover, for every \underline{σ} ∈ Σ, set :

B\underline{σ} := ∏_{H∈𝒞} gr_{\underline{σ}(H)}A · gr_{−\underline{σ}(H)}A

(this is an ideal of gr_{0}A) and let S\underline{σ} ⊂ Γ be the subgroup generated by the image of \underline{σ}. By an easy induction we deduce that :

gr_{χ}N ⊂ ∑_{\underline{σ}∈Σ} Bn ; \underline{σ} · gr_{χ}N for every n > 0.

By claim 14.6.15, for every \underline{σ} ∈ Σ we may find γ(\underline{σ}) ∈ S\underline{σ} ∩ Γ′ \ \{0\}. On the other hand, since Γ is finite and abelian, it is easy to verify that there exists n ∈ ℕ large enough such that

Bn ; \underline{σ} ⊂ gr_{γ(\underline{σ})}A · gr_{−γ(\underline{σ})}A for every \underline{σ} ∈ Σ

(details left to the reader). But (14.6.18) implies that gr_{γ(\underline{σ})}A·gr_{χ}N = 0 whenever χ ∈ Γ′ \\{0\}, so the claim follows. ♢

To conclude, we apply first claim 14.6.14to the horizontal Γ/Γ′-graded \underline{A}_{Γ/Γ′}-module \underline{N}_{Γ/Γ′}, to see that N_{|Γ/Γ′} generates the A-module N, and then claim 14.6.16– together with our inductive assumption – to deduce that gr_{0}N generates that A_{|Γ/Γ′}-module N_{|Γ/Γ′}. The proposition follows. □

14.6.20. Recall that a morphism M → N of A-modules is said to be pure if the natural morphism Q ⊗_{A} M → Q ⊗_{A} N is a monomorphism for every A-module Q. A morphism A → B of V^{a}-algebras is called pure if it is pure when regarded as a morphism of A-modules.

Lemma 14.6.21. Let f : A → B be a pure morphism of V^{a}-algebras, M an A-module. Then : (i) If B ⊗_{A} M is a flat B-module, then M is a flat A-module (i.e. f descends flatness). (ii) If B ⊗_{A} M is almost finitely generated (resp. almost finitely presented) as a B-module, then M is almost finitely generated (resp. almost finitely presented) as an A-module.

Proof. (i): To begin with, we remark :

Claim 14.6.22. Let R be any V -algebra such that A = R^{a}.

(i) A morphism φ : M_{1} → M_{2} of A-modules is pure if and only if the same holds for the induced morphism φ_{!} : M_{1!} → M_{2!} of R-modules.

(ii) A morphism g : A → B′ of V^{a}-algebras is pure if and only if the same holds for the induced morphism g_{!!} : A_{!!} → B′_{!!}.

Proof of the claim. (i): Suppose that φ is pure, and let Q be any R-module. Then Q ⊗_{R} M_{i!} ≃ (Q^{a} ⊗_{A} M_{i})_{!} for i = 1, 2. Since the functor M ↦ M_{!} is exact ([75, Cor.2.2.24(i)]), we deduce that φ_{!} is pure. The converse is easy, and shall be left to the reader.

(ii): From (i) we already see that g is pure whenever g_{!!} is. Next, suppose that g is pure; we may assume that R = A_{!!}, and then (i) says that g_{!} is a pure morphism of A_{!!}-modules. However, the natural diagram of A_{!!}-modules :

\xymatrix{A_{!} \ar[r] \ar[d] & B′_{!} \ar[d] \\ A_{!!} \ar[r] & B′_{!!}}

is cofibred; since tensor products are right exact functors, the claim follows. ♢ The assertion now follows from claim 14.6.22(ii) and [86, Partie II, Lemme 1.2.1]. (ii): Suppose first that B ⊗_{A} M is an almost finitely generated B-module. The assumption

on f implies that Ann_{A}(B ⊗_{A} M) ⊂ Ann_{A}M; then [75, Rem.3.2.26(i)] shows that M is an almost finitely generated A-module.

Finally, we suppose that B ⊗_{A} M is an almost finitely presented B-module, and we wish to show that M is an almost finitely presented A-module. To this aim, let φ : N → N′ be a morphism of A-modules. The assumption on f implies that the natural morphism Ker φ → Ker(1_{B} ⊗_{A} φ) is a monomorphism; especially :

Ann_{A}(Ker(1_{B} ⊗_{A} φ)) ⊂ Ann_{A}(Ker φ).

Then one may easily adapt the proof of [75, Lemma 3.2.25(iii)], to derive the assertion. □

Theorem 14.6.23. In the situation of definition 14.6.7, suppose furthermore that G is abelian and the order o(G) of G is invertible in A_{∗}. Then the following holds :

(i) Let M be any G-equivariant A-module. The G-action on M is horizontal if and only if the counit of adjunction :

η_{M} : A ⊗_{A^{G}} M^{G} → M

is an epimorphism (of A-modules). (ii) The functor (14.6.8) restricts to an equivalence on the full subcategory of flat A^{G}-

modules :

(14.6.24)  A^{G}-Mod_{fl} →^{∼} A[G]-Mod_{hor.fl}.

Proof. (i): For m := o(G), let μ_{m} ⊂ \overline{ℚ}^{×} be the group of m-th roots of 1, and set B := V^{a}[μ_{m}] := (V [T]/(T^{m} − 1))^{a}. Since B is a faithfully flat V^{a}-algebra, lemma 14.6.10(iii) allows to replace A by B ⊗_{V^{a}} A and M by B ⊗_{V^{a}} M, and therefore we may assume from start that μ_{m} ⊂ (A_{∗}^{G})^{×}. Set Γ := Hom_{ℤ}(G, μ_{m}). For every χ ∈ Γ, let e_{χ} ∈ A_{∗}[G] be the central idempotent defined as in (14.6.11). A standard calculation shows that :

∑_{χ∈Γ} e_{χ} = 1.

Hence, every G-equivariant A^{G}-module (N, β) admits the G-equivariant decomposition :

(14.6.25)  N ≃ ⨁_{χ∈Γ} N_{χ}.

Especially, A = ⨁_{χ∈Γ} A_{χ}, and clearly the datum \underline{A} consisting of A and its decomposition, is a Γ-graded V^{a}-algebra. Furthermore, the datum \underline{N} consisting of N and its decomposition (14.6.25) is a Γ-graded \underline{A}-module.

Claim 14.6.26. The functor (N, β) ↦ \underline{N} is an equivalence from A[G]-Mod to the category of Γ-graded \underline{A}-modules.

Proof of the claim. Indeed, if Q is a Γ-graded \underline{A}-module, we may define a G-action on Q by requiring that Q_{χ} = gr_{χ}Q, the χ-graded direct summand of Q. This gives a quasi-inverse for the functor (N, β) ↦ \underline{N}. (Details left to the reader.) ♢

Claim 14.6.27. Let (N, β) be any G-equivariant A-module, and \underline{N} its associated Γ-graded \underline{A}- module. The following conditions are equivalent :

(a) (N, β) is horizontal.

(b) \underline{N} is horizontal.

Proof of the claim. For every g ∈ G, let Δ(g) ⊂ Γ be the subgroup consisting of all χ ∈ Γ such that χ(g) = 1; a direct inspection of the definitions shows that J_{Δ(g)} is the ideal I_{g}, as defined in (14.6.6), and (N, β) is horizontal if and only if N_{χ}/(N_{χ} ∩ I_{g}N) = 0 for every g ∈ G and every χ ∉ Δ(g). This already shows that (b)⇒(a); it also shows that condition (b) holds for the subgroups Δ(g), when (N, β) is horizontal. However, every subgroup of Γ can be written in the form Δ = Δ(g_{1}) ∩ ⋯ ∩ Δ(g_{n}), for appropriate g_{1}, …, g_{n} ∈ G, and then J_{Δ(g_{1})} + ⋯ + J_{Δ(g_{n})} ⊂ J_{Δ}, hence (b) follows for all subgroups. ♢

Assertion (i) now follows from claim 14.6.27and proposition 14.6.13.

(ii): Let (M, β) be any object of A[G]-Mod_{hor.fl}, and denote by L the kernel of the counit η_{M}. Since M is a flat A-module and η_{M} is an epimorphism by (i), it follows that i_{g}^{∗}L is the kernel of i_{g}^{∗}η_{M}, for every g ∈ G (notation of (14.6.6)). Since the category A[G]-Mod_{hor} is abelian, we deduce that the natural G-action on L is horizontal, and then (i) says that L is generated by L^{G}. But lemma 14.6.10(ii) easily implies that L^{G} = 0, so η_{M} is an isomorphism. Next, letting M := A in lemma 14.6.10(i), we deduce easily that the natural morphism A^{G} → A is pure, hence M^{G} is a flat A^{G}-module, by lemma 14.6.21(i). Now the assertion follows from lemma 14.6.10(ii) and [28, Prop.3.4.3]. □

Corollary 14.6.28. In the situation of theorem 14.6.23, the following holds :

(i) The functor (14.6.24) restricts to an equivalence from the subcategory of flat, almost finitely generated (resp. almost finitely presented) A^{G}-modules, onto the subcategory of G-equivariant, flat, horizontal and almost finitely generated (resp. almost finitely presented) A[G]-modules.

(ii) The functor (14.6.9) restricts to an equivalence :

A^{G}-Alg_{fl} → A[G]-Alg_{hor.fl}

and likewise for the subcategories of weakly étale (resp. étale, resp. étale and almost finitely presented) algebras.

Proof. (i) follows from theorem 14.6.23(ii), lemma 14.6.21(ii), and the fact that the morphism A^{G} → A is pure.

(ii): The assertion concerning A^{G}-Alg_{fl} is an immediate consequence of theorem 14.6.23. Next, let (B, β) be an object of A[G]-w.Ét; by the foregoing, B descends to a flat A^{G}-algebra B^{G} with a G-equivariant isomorphism : B ≃ A ⊗_{A^{G}} B^{G}. However, on the one hand, B is – by assumption – a flat B ⊗_{A} B-algebra, and on the other hand, B ⊗_{A} B underlies a flat, horizontal A[G]-algebra with (B ⊗_{A} B)^{G} ≃ B^{G} ⊗_{A^{G}} B^{G}; theorem 14.6.23 then says that B^{G} is a flat B^{G} ⊗_{A^{G}} B^{G}-algebra, whence the assertion for A^{G}-w.Ét. Next, since an almost finitely generated module is almost projective if and only if it is flat and almost finitely presented ([75, Prop.2.4.18]), the assertion for A^{G}-Ét follows from the same assertion for A^{G}-w.Ét and lemma 14.6.21(ii). Likewise, the assertion for étale almost finitely presented A^{G}-algebras follows from the assertion for A^{G}-Ét and lemma 14.6.21(ii). □

Remark 14.6.29. In case the G-action on X is free, i.e. when (∂_{0}, ∂_{1}) : X × G → X ×_{S} X is a monomorphism, corollary 14.6.28also follows from [75, Prop.4.5.25].

14.7. Almost Witt vectors. In this section we show that the construction of the ring of Witt vectors descends to almost rings, and we study some properties of the resulting functor of almost Witt vectors. We begin with some general observation concerning liftings of basic setups (in the sense of [75, §2.1.1]) along ring homomorphisms. These preliminaries shall then be applied to find natural basic setups on truncated rings of Witt vectors of arbitrary rings R, by lifting given setups on R along the 0-th ghost map.

Lemma 14.7.1. Let A be a ring, I ⊂ A a nilpotent ideal, π : A → A_{0} := A/I the projection, and 𝔪_{0} ⊂ A_{0} an ideal such that 𝔪_{0} = 𝔪_{0}^{2}. We have :

(i) There exists a unique ideal 𝔪 ⊂ A such that 𝔪^{2} = 𝔪 and π(𝔪) = 𝔪_{0}.

(ii) 𝔪 is the smallest of the ideals J ⊂ A such that π(J) = 𝔪_{0}.

(iii) 𝔪 fulfills condition (B) of [75, §2.1.6] if and only if the same holds for 𝔪_{0}.

Proof. (i): Say that I^{N} = 0 for some N ∈ ℕ, and set J := π^{−1}𝔪_{0}. We remark :

Claim 14.7.2. J^{2N+1} = J^{2N+2}.

Proof of the claim. Indeed, let a_{1}, …, a_{2N+1} ∈ J; we may find an integer r ≥ 0 and elements b_{1i}, c_{1i}, …, b_{ri}, c_{ri} ∈ 𝔪_{0} for every i = 1, …, 2N + 1, such that π(a_{i}) = ∑_{j=1}^{r} b_{ji}c_{ji}. Lift each b_{ji} and each c_{ji} to elements b′_{ji}, c′_{ji} ∈ J; it follows that x_{i} := a_{i} − ∑_{j=1} b′_{ji}c′_{ji} ∈ I for every i = 1, …, 2N + 1. Thus :

∏_{i=1}^{2N+1} a_{i} = ∏_{i=1}^{2N+1}(x_{i} + ∑_{j=1}^{r} b′_{ji}c′_{ji}) ∈ J^{2(N+1)} + I^{N} = J^{2N+2}

whence the claim. ♢

Set 𝔪 := J^{2N+1}; claim 14.7.2 implies that 𝔪 = 𝔪^{2}, and clearly π(𝔪) = 𝔪_{0}^{2N+1} = 𝔪. Next, let 𝔪′ ⊂ A be any ideal such that 𝔪^{′2} = 𝔪′ and π(𝔪′) = 𝔪_{0}; it follows that

𝔪′ = 𝔪^{′2N+1} ⊂ J^{2N+1} = (𝔪′ + I)^{2N+1} ⊂ 𝔪^{′N+1} = 𝔪′

whence 𝔪′ = 𝔪.

(ii): Let J ⊂ A be any ideal such that π(J) = 𝔪_{0}; it follows that

𝔪 = 𝔪^{N} ⊂ (J + I)^{N} ⊂ J + I^{N} = J.

(iii): It is clear that if 𝔪 fulfills condition (B), then the same holds for 𝔪_{0}. Conversely, if condition (B) holds for 𝔪_{0}, then for every integer k > 1 the system of elements (x^{k} | x ∈ 𝔪) generates an ideal J ⊂ A such that π(J) = 𝔪_{0} and J ⊂ 𝔪. By (ii), we must then have J = 𝔪, which shows that condition (B) holds for 𝔪. □

Let us also point out the following result, which shall not be needed in the sequel :

Lemma 14.7.3. Let (A, 𝔪) be a basic setup and f : B → A a ring homomorphism such that : (a) p^{k}A = 0 for some integer k ∈ ℕ.

(b) 𝔪 fulfills condition (B) of [75, §2.1.6].

(c) f ⊗_{ℤ} 𝔽_{p} is invertible up to Φ^{n} for some n ∈ ℕ, in the sense of [75, Def.3.5.8]. Then there exists a unique basic setup (B, 𝔫) with 𝔫A = 𝔪 and with 𝔫 fulfilling condition (B).

Proof. Using lemma 14.7.1, we are easily reduced to the case where A and B are 𝔽_{p}-algebras, and f is invertible up to Φ^{n}. The latter means that there exists a morphism g : A → B such that g ∘ f = Φ_{B}^{n} and f ∘ g = Φ_{A}^{n}.

Suppose first that A = B and f = Φ_{A}^{n}; in this case, we claim that 𝔫 := 𝔪 will do. Indeed, obviously 𝔫^{2} = 𝔫, and assumption (b) says that Φ_{A}^{n}(𝔪) generates the ideal 𝔪; moreover, if 𝔫′ ⊂ A is another ideal fulfilling these conditions, we see that x^{p} ∈ 𝔪 for every x ∈ 𝔫′, whence 𝔫′ ⊂ 𝔪, since 𝔫′ fulfills condition (B), and conversely, 𝔪 = Φ_{A}^{n}(𝔫′) · A ⊂ 𝔫′.

In the general case, we claim that 𝔫 := g(𝔪) · B will do. Indeed, clearly 𝔫^{2} = 𝔫 and f(𝔫) · A = Φ_{A}^{n}(𝔪) · A = 𝔪, due to assumption (b); it is also easily seen that 𝔫 fulfills condition (B). Moreover, if 𝔫′ ⊂ B is another ideal fulfilling these conditions, it follows that

Φ_{B}^{n}(𝔫′) · B = g(f(𝔫′) · A) · B = g(𝔪) · B = 𝔫

whence 𝔫 = 𝔫′, by the foregoing case. □

14.7.4. Consider now a cartesian diagram of rings

\xymatrix{A \ar[r]^{p_{1}} \ar[d]_{p_{2}} & A_{1} \ar[d]^{π_{1}} \\ A_{2} \ar[r]^{π_{2}} & A_{3}}

such that π_{2} (and hence p_{1}) is surjective.

Proposition 14.7.5. (i) In the situation of (14.7.4), let 𝔪_{1} ⊂ A_{1}, 𝔪_{2} ⊂ A_{2} be two ideals with

𝔪_{1}^{2} = 𝔪_{1}  𝔪_{2}^{2} = 𝔪_{2}  𝔪_{1}A_{3} = 𝔪_{2}A_{3}.

Then there exists a unique ideal 𝔪 ⊂ A such that

(14.7.6)  𝔪^{2} = 𝔪  𝔪A_{1} = 𝔪_{1}  𝔪A_{2} = 𝔪_{2}.

(ii) Moreover, 𝔪 is the smallest of the ideals J ⊂ A such that JA_{1} = 𝔪_{1} and JA_{2} = 𝔪_{2}. (iii) Furthermore, 𝔪 fulfills condition (B) if and only if the same holds for both 𝔪_{1} and 𝔪_{2}.

Proof. (i): We shall regard A as a subset of A_{1} × A_{2}, in the natural fashion. Now, set I := 𝔪_{1} ×_{A_{3}} 𝔪_{2}; then I is an ideal of A, and we remark :

Claim 14.7.7. IA_{1} = 𝔪_{1} and IA_{2} = 𝔪_{2}.

Proof of the claim. Since π_{2} is surjective, it is easily seen that p_{1} restricts to a surjection I → 𝔪_{1}. Next, let a ∈ 𝔪_{2} be any element; we may find an integer r ∈ ℕ and elements x_{1}, …, x_{r} ∈ 𝔪_{1} and y_{1}, …, y_{r} ∈ A_{3} such that π_{2}(a) = ∑_{i=1}^{r} y_{i} · π_{1}(x_{i}). For every i = 1, …, r, pick b_{i} ∈ 𝔪_{2} and c_{i} ∈ A_{2} such that π_{2}(b_{i}) = π_{1}(x_{i}) and π_{2}(c_{i}) = y_{i}; then (x_{i}, b_{i}) ∈ I for i = 1, …, r, and a − ∑_{i=1}^{r} p_{2}(x_{i}, b_{i}) · c_{i} ∈ Ker π_{2}. However, 0 × (𝔪_{2} ∩ Ker π_{2}) ⊂ I, whence IA_{2} = 𝔪_{2}. ♢

Claim 14.7.8. Let J ⊂ A be any ideal such that JA_{1} = 𝔪_{1} and JA_{2} = 𝔪_{2}. Then

(A_{1} ×_{A_{3}} 𝔪_{2}) · (𝔪_{1} ×_{A_{3}} A_{2}) ⊂ J.

Proof of the claim. Indeed, let (a_{1}, a_{2}), (b_{1}, b_{2}) ∈ A be two elements such that a_{2} ∈ 𝔪_{2} and b_{1} ∈ 𝔪_{1}. Since p_{1} is surjective, we may find x ∈ A_{2} such that (b_{1}, x) ∈ J, so π_{2}(x) = π_{1}(b_{1}), and we have

(a_{1}, a_{2}) · (b_{1}, b_{2}) − (a_{1}, a_{2}) · (b_{1}, x) = (a_{1}, a_{2}) · (0, b_{2} − x) = (0, a_{2} · y)

where y := b_{2} − x ∈ Ker π_{2}. We are thus reduced to checking that (0, ab) ∈ J for every a ∈ 𝔪_{2} and b ∈ Ker π_{2}. However, say that a = ∑_{i=1}^{r} p_{2}(c_{i})d_{i} for elements c_{1}, …, c_{r} ∈ J and d_{1}, …, d_{r} ∈ A_{2}; it follows that (0, d_{i}b) ∈ A for every i = 1, …, r, and ∑_{i=1}^{r} c_{i} · (0, d_{i}b) = (0, ab), whence the contention. ♢

From claim 14.7.7 we deduce that I^{3}A_{1} = 𝔪_{1}^{3} = 𝔪_{1}, and likewise, I^{3}A_{2} = 𝔪_{2}; from claim 14.7.8, it then follows that (𝔪_{1}×_{A_{3}} A_{2})·(A_{1}×_{A_{3}} 𝔪_{2}) ⊂ I^{3}. Especially, I^{2} ⊂ I^{3}, so that I^{3} = I^{2}, and therefore the ideal 𝔪 := I^{2} fulfills conditions (14.7.6).

(ii): For every ideal J as in claim 14.7.8 we also get : 𝔪 ⊂ J ⊂ 𝔪_{1} ×_{A_{3}} 𝔪_{2}; if we have as well J^{2} = J, it follows that J ⊂ 𝔪, i.e. J = 𝔪, as required.

(iii): Due to [75, Claim 2.19], it suffices to show, for every prime p, that the following conditions are equivalent :

(a) The A-module 𝔪/p𝔪 is generated by the p-th powers of its elements

(b) For i = 1, 2, the A_{i}-module 𝔪_{i}/p𝔪_{i} is generated by the p-th powers of its elements. However, it is easily seen that (a)⇒(b). For the converse, consider the ideal J ⊂ A generated by the system {x^{p} | x ∈ 𝔪} ∪ {px | x ∈ 𝔪}; since p_{1} is surjective, (b) implies that JA_{1} = 𝔪_{1}. Likewise, JA_{2} = 𝔪_{2} : indeed, since 𝔪_{2}/p𝔪_{2} is generated by the p-th powers of its elements, it suffices to check that for every a ∈ 𝔪_{2} we have a^{p} ∈ JA_{2}. But say that a = ∑_{j=1}^{n} x_{i}a_{i} for some x_{1}, …, x_{n} ∈ 𝔪 and a_{1}, …, a_{n} ∈ A_{2}; then a^{p} = ∑_{i=1} x_{i}^{p}a_{i}^{p} + py for some y ∈ 𝔪_{2}, whence the contention. In view of (ii), we conclude that J = 𝔪, whence (a). □

14.7.9. Henceforth, we fix a prime number p, and the notation W (A) and W_{n+1}(A) will refer to the ring of p-typical Witt vector of section (9.3), for every ring A and every integer n ∈ ℕ. Notice that the functors W and W_{n+1} introduced in section 9.3 are defined on the category of topological rings; however, in this section we shall be interested only in the underlying rings, and the topologies will play no role; if one wishes, one may assume that all the rings in this section carry the discrete topology. We consider first the case where p is nilpotent on A :

Corollary 14.7.10. Let A be any ring such that p^{k}A = 0 for some k ∈ ℕ, and 𝔪 ⊂ A an ideal with 𝔪^{2} = 𝔪. For every n ∈ ℕ we have :

(i) There exists a unique ideal 𝔫_{n+1} ⊂ W_{n+1}A such that

𝔫_{n+1}^{2} = 𝔫_{n+1}  and  \overline{ω}_{0}(𝔫_{n+1}) = 𝔪.

(ii) The image of 𝔫_{n+2} under the projection W_{n+2}A → W_{n+1}A agrees with 𝔫_{n+1}.

Proof. (See remark 9.3.28(i) for the definition of the ring homomorphism \overline{ω}_{0}.)

(i) follows immediately from corollary 9.3.32and lemma 14.7.1.

(ii) follows immediately from (i). □

For a general ring A, we may state :

Proposition 14.7.11. Let A be a ring, n ∈ ℕ an integer, 𝔪 ⊂ A an ideal with 𝔪 = 𝔪^{2} and fulfilling condition (B) of [75, §2.1.6]. Set 𝔫_{n+1} = \{\underline{a} ∈ W_{n+1}A | a_{0}, …, a_{n} ∈ 𝔪\}. We have :

(i) 𝔫_{n+1} is an ideal of W_{n+1}A with 𝔫_{n+1}^{2} = 𝔫_{n+1} and fulfilling condition (B).

(ii) Moreover, 𝔫_{n+1} is the unique ideal with 𝔫_{n+1}^{2} = 𝔫_{n+1} and such that
(14.7.12)  \overline{ω}_{i}(𝔫_{n+1}) · A = 𝔪  for i = 0, …, n.

Proof. We consider the ring homomorphism

π_{n} : W_{n+1}A → A^{n+1}

of proposition 9.3.30(i). By proposition 9.3.30(i), the kernel of the surjection W_{n+1}A → R := Im π_{n} is nilpotent, hence every basic setup (R, 𝔪_{R}) lifts uniquely to a basic setup (W_{n+1}A, 𝔪̃), and 𝔪̃ satisfies condition (B) if and only if the same holds for 𝔪_{R} (lemma 14.7.1(i,iii)). Moreover, by proposition 9.3.30(ii), the subring R of A^{n+1} contains the ideal p^{n+1}A^{n+1}, and lies in the larger subring S ⊂ A^{n+1} consisting of all sequences (a_{0}, …, a_{n}) such that a_{i} ≡ a_{0}^{p^{i}} (mod pA) for i = 0, …, n; there follows a cartesian diagram of rings

(14.7.13)  \xymatrix{R \ar[r] \ar[d] & R_{0} := R/p^{n+1}A^{n+1} \ar[d] \\ S \ar[r] & S_{0} := S/p^{n+1}A^{n+1}}

whose horizontal arrows are surjections. By proposition 14.7.5(i,iii), the datum of (R, 𝔪_{R}) is then equivalent to that of a pair of basic setups (R_{0}, 𝔪_{R_{0}}) and (S, 𝔪_{S}) with 𝔪_{R_{0}}S_{0} = 𝔪_{S}S_{0}. Also, 𝔪_{R} fulfills condition (B) if and only if the same holds for both 𝔪_{R_{0}} and 𝔪_{S}. Next, set A_{0} := A/pA, and notice that we have a cartesian diagram of rings

(14.7.14)  \xymatrix{S \ar[r] \ar[d]_{ψ} & A^{n+1} \ar[d] \\ A_{0} \ar[r]^{φ} & A_{0}^{n+1}}

where φ is given by the rule : x ↦ (x, x^{p}, …, x^{p^{n}}) for every x ∈ A_{0}; the top horizontal arrow is the inclusion map, and the right vertical arrow is the projection. Moreover, ψ is a surjection. Hence – again by virtue of proposition 14.7.5(i) – the datum of (S, 𝔪_{S}) is equivalent to that of a system of basic setups (A, 𝔪_{i}) such that

𝔪_{i}A_{0} = Φ_{A_{0}}^{i}(𝔪_{0}A_{0}) · A_{0} for i = 1, …, n

where Φ_{A_{0}} : A_{0} → A_{0} is the Frobenius endomorphism. Especially, if (A, 𝔪) is a basic setup on A such that 𝔪 fulfills condition (B), then we can take 𝔪_{i} := 𝔪 for every i = 0, …, n, and in this way we obtain a natural basic setup (S, 𝔪_{S}) on S with 𝔪_{S}A^{n+1} = 𝔪^{n+1}, and such that condition (B) still holds for 𝔪_{S}. Furthermore, the restriction of ψ to R factors through a surjection R_{0} → A_{0} whose kernel is also nilpotent, so that the basic setup (A_{0}, 𝔪A_{0}) lifts uniquely to a basic setup (R_{0}, 𝔪_{R_{0}}) with 𝔪_{R_{0}} fulfilling condition (B) (lemma 14.7.1(i)). Lastly, both ideals 𝔪_{R_{0}}S_{0} and 𝔪_{S}S_{0} lift 𝔪A_{0} along the map S_{0} → A_{0} induced by ψ, whose kernel is also nipotent; by invoking again 14.7.1(i), we deduce that 𝔪_{R_{0}}S_{0} = 𝔪_{S}S_{0}. Summing up, we have finally associated with the basic setup (A, 𝔪) a natural basic setup (W_{n+1}A, 𝔪̃) which is the unique one with property (14.7.12), and with 𝔪̃ fulfilling condition (B). To conclude the proof of both (i) and (ii), it then suffices to check that 𝔪̃ = 𝔫_{n+1}. However, it is clear that 𝔫_{n+1} is an ideal of W_{n+1}A enjoying property (14.7.12), since \overline{ω}_{i}(𝔫_{n+1}) is contained in 𝔪 and contains the system (x^{p^{i}} | x ∈ 𝔪), which generates 𝔪, under condition (B). It remains to check that 𝔫_{n+1} = 𝔫_{n+1}^{2}. Now, let us define a descending filtration by subideals of 𝔫_{n+1}, by setting Fil_{i}𝔫_{n+1} := 𝔫_{n+1} ∩ \overline{V}_{i}A, where \overline{V}_{i}A := V_{i}A/V_{n+1}A for every i = 0, …, n. Let also gr_{•}𝔫_{n+1} be the associated graded W_{n+1}-module. We are then further reduced to showing that 𝔫_{n+1} · gr_{i}𝔫_{n+1} = gr_{i}𝔫_{n+1} for i = 0, …, n. However, we have a natural identification of W_{n+1}A-modules : gr_{i}𝔫_{n+1} →^{∼} 𝔪, for the W_{n+1}A-module structure on 𝔪 induced by restriction of scalars along the ghost map \overline{ω}_{i} : W_{n+1}A → A (claim 9.3.31). Since 𝔪^{2} = 𝔪, we then come down to the assertion that \overline{ω}_{i}(𝔫_{n+1}) · A = 𝔪, which was already remarked. □

14.7.15. Let (A, 𝔪) be a basic setup such that either p^{k}A = 0 for some integer k ∈ ℕ, or else such that 𝔪 fulfills condition (B). For every n ∈ ℕ let 𝔫_{n+1} ⊂ W_{n+1}A be the ideal provided by corollary 14.7.10(i), or respectively by proposition 14.7.11; especially, (W_{n+1}A, 𝔫_{n+1}) is also a basic setup. Let also f : B → C be a morphism of A-algebras such that f^{a} : B^{a} → C^{a} is an isomorphism of (A, 𝔪)^{a}-algebras. Then we claim that f induces an isomorphism of (W_{n+1}A, 𝔫_{n+1})^{a}-algebras

(W_{n+1}f)^{a} : (W_{n+1}B)^{a} →^{∼} (W_{n+1}C)^{a} for every n ∈ ℕ.

Indeed, arguing by induction on n ∈ ℕ, we are easily reduced to checking that f induces an isomorphism of (W_{n+1}A)^{a}-modules (V_{n}B/V_{n+1}B)^{a} →^{∼} (V_{n}C/V_{n+1}C)^{a} for every n ∈ ℕ. However, in light of claim 9.3.31we have a commutative diagram of W_{n+1}A-modules

\xymatrix{B \ar[r] \ar[d]_{f} & V_{n}B/V_{n+1}B \ar[d] \\ C \ar[r] & V_{n}C/V_{n+1}C}

whose horizontal arrows are isomorphisms of W_{n+1}A-modules, and where the W_{n+1}A-module structures on B and C are induced by the n-th ghost maps. By assumption, 𝔪 annihilates Ker f and Coker f; since \overline{ω}_{n}(𝔫_{n+1}) ⊂ 𝔪, it follows that 𝔫_{n+1} annihilates these W_{n+1}A-modules as well, whence the contention. Thus, we obtain a well defined functor

W_{n+1} : (A, 𝔪)^{a}-Alg → (W_{n+1}A, 𝔫_{n+1})^{a}-Alg  B^{a} ↦ (W_{n+1}B)^{a}  for every n ∈ ℕ.

Moreover, for every k ≤ n, the ghost component \overline{ω}_{k} : W_{n}B → B induces pull-back functors

\overline{ω}_{k}^{∗} : B^{a}-Alg → W_{n+1}B^{a}-Alg \overline{ω}_{k}^{∗} : B^{a}-Mod → W_{n+1}B^{a}-Mod.

Namely, for every B-algebra C we let \overline{ω}_{k}^{∗}C be the W_{n+1}B-algebra whose underlying ring is C, and whose structure morphism is the composition of \overline{ω}_{k} and the structure morphism B → C of C. Since \overline{ω}_{k}(𝔫_{n+1}) ⊂ 𝔪, it is clear that this functor on B-algebras descends to a well defined functor \overline{ω}_{k}^{∗} on B^{a}-algebras, as stated. Likewise one argues to define the functor \overline{ω}_{k}^{∗} on B^{a}-modules. Furthermore, we have a well defined ideal

\overline{V}_{k}B^{a} := (V_{k}B/V_{n+1}B)^{a} ⊂ W_{n+1}B^{a} for every k = 0, …, n

and claim 9.3.31yields a natural isomorphism of W_{n+1}B^{a}-modules :

(14.7.16)  \overline{V}_{k}B^{a}/\overline{V}_{k+1}B^{a} →^{∼} \overline{ω}_{k}^{∗}B^{a}  for every k = 0, …, n.

14.7.17. In the situation of (14.7.15), suppose that 𝔪 fulfills condition (B), and let B be any A-algebra; denote by

B_{!!}^{a} ⟶^{ε_{B}} B ⟶^{η_{B}} B_{∗}^{a} and (W_{n+1}B)_{!!}^{a} ⟶^{ε_{W_{n+1}B}} W_{n+1}B ⟶^{η_{W_{n+1}B}} (W_{n+1}B)_{∗}^{a}

the units and counits of adjunction. We have :

Proposition 14.7.18. In the situation of (14.7.17), the following holds :

(i) There exist isomorphisms of W_{n+1}A-algebras

ω : W_{n+1}(B_{∗}^{a}) →^{∼} (W_{n+1}B)_{∗}^{a}  and  τ : W_{n+1}(B_{!!}^{a}) →^{∼} (W_{n+1}B)_{!!}^{a}

such that ω ∘ W_{n+1}(η_{B}) = η_{W_{n+1}(B)} and ε_{W_{n+1}(B)} ∘ τ = W_{n+1}(ε_{B}). (ii) Moreover, we have a natural isomorphism of non-unital rings (see remark 9.3.16)

𝔫̃_{n+1} := 𝔫_{n+1} ⊗_{W_{n+1}A} 𝔫_{n+1} →^{∼} W_{n+1}(𝔪̃).

Proof. (i): We consider the commutative diagrams

\xymatrix{W_{n+1}B \ar[r]^{W_{n+1}(η_{B})} \ar[d]_{η_{W_{n+1}B}} & W_{n+1}(B_{∗}^{a}) \ar[d]^{η_{W_{n+1}(B^{a}_{∗})}} & W_{n+1}(B_{!!}^{a})_{!!}^{a} \ar[r]^{(W_{n+1}ε_{B})_{!!}^{a}} \ar[d]_{ε_{W_{n+1}(B{a \atop !!})}} & W_{n+1}(B)_{!!}^{a} \ar[d]^{ε_{W_{n+1}(B)}} \\ (W_{n+1}B)_{∗}^{a} \ar[r]^{(W_{n+1}η_{B})_{∗}^{a}} & W_{n+1}(B_{∗}^{a})_{∗}^{a} & W_{n+1}(B_{!!}^{a}) \ar[r]^{W_{n+1}(ε_{B})} & W_{n+1}(B)}

and notice that (W_{n+1}η_{B})_{∗}^{a} and (W_{n+1}ε_{B})_{!!}^{a} are isomorphisms, by the discussion of (14.7.15). Hence, it suffices to show that η_{W_{n+1}(B_{∗}^{a})} and ε_{W_{n+1}(B_{!!}^{a})} are isomorphisms, which follows from :

Claim 14.7.19. If η_{B} (resp. ε_{B}) is an isomorphism, the same holds for η_{W_{n+1}B} (resp. ε_{W_{n+1}B}).

Proof of the claim. Let gr_{•}(W_{n+1}B) be the graded ring associated with the filtration (\overline{V}_{k}B | k = 0, …, n) of W_{n+1}B; consider also the filtration ((\overline{V}_{k}B^{a})_{∗} | k = 0, …, n) of (W_{n+1}B)_{∗}^{a}, and let gr_{•}(W_{n+1}B)_{∗}^{a} be the associated graded ring. We get a commutative diagram

\xymatrix{ & gr_{i}(W_{n+1}B) \ar[dl]_{gr_{i}(η_{W_{n+1}B})} \ar[dr]^{η_{gr_{i}(W_{n+1}B)}} \\ gr_{i}(W_{n+1}B)_{∗}^{a} \ar[rr] &  & (gr_{i}W_{n+1}B)_{∗}^{a}}

whose bottom horizontal arrow is injective, since the functor (−)_{∗}^{a} is left exact. Here η_{gr_{i}(W_{n+1}B)} is again the unit of adjunction, which, by virtue of (14.7.16), is naturally identified with the unit of adjunction

η_{\overline{ω}_{i}^{∗}B} : \overline{ω}_{i}^{∗}B → (\overline{ω}_{i}^{∗}B)_{∗}^{a} for every i = 0, …, n.

Let now ℬ be the category of basic setups, and ℬ-Alg → ℬ (resp. ℬ^{a}-Alg → ℬ) the fibred and cofibred category of ℬ-algebras (resp. of almost ℬ-algebras), as in [75, §3.5]. The localization ℬ-Alg → ℬ^{a}-Alg admits a right adjoint

(−)_{∗} : ℬ^{a}-Alg → ℬ-Alg ((V, 𝔪), R) ↦ ((V, 𝔪), R_{∗})

which is a ℬ-cartesian functor ([75, §3.5.4]). By (14.7.12), we have the morphism \overline{ω}_{i} : (W_{n+1}A, 𝔫_{n+1}) → (A, 𝔪) in ℬ, whence a commutative diagram in ℬ-Alg :

\xymatrix{((W_{n+1}A, 𝔫_{n+1}), \overline{ω}_{i}^{∗}B) \ar[r] \ar[d]_{η_{\overline{ω}^{∗}_{i}B}} & ((A, 𝔪), B) \ar[d]^{η_{B}} \\ ((W_{n+1}A, 𝔫_{n+1}), (\overline{ω}_{i}^{∗}B)_{∗}^{a}) \ar[r] & ((A, 𝔪), B_{∗}^{a})}

whose top horizontal arrow is given by the identity map of \overline{ω}_{i}^{∗}B, and is therefore a cartesian morphism of ℬ-Alg. Then also the bottom vertical arrow is a cartesian morphism, since (−)_{∗} is a cartesian functor. We conclude that η_{gr_{i}(W_{n+1}B)} is an isomorphism, under our assumptions. Then the same holds for gr_{i}(η_{W_{n+1}B}), for every i = 0, …, n, whence the assertion for η_{W_{n+1}B}.

To show the assertion concerning ε_{W_{n+1}B}, we argue by induction on n ∈ ℕ; the case n = 0 is trivial. Let then n > 0, and suppose that the assertion is already known for ε_{W_{n}B}; recall that for every i ∈ ℕ, the ring (W_{i+1}B)_{!!}^{a} is the cokernel of a map of W_{i+1}A-modules

δ_{i} : 𝔫̃_{i+1} → W_{i+1}A ⊕ (𝔫̃_{i+1} ⊗_{W_{i+1}A} W_{i+1}B) x_{•} ⊗ y_{•} ↦ (x_{•} · y_{•}, x_{•} ⊗ y_{•} ⊗ 1).

We consider the commutative ladder with right exact rows :

\xymatrix{𝔫̃_{n+1} \ar[r]^{δ_{n}} \ar[d]_{α} & W_{n+1}A ⊕ (𝔫̃_{n+1} ⊗_{W_{n+1}A} W_{n+1}B) \ar[r] \ar[d]^{β} & (W_{n+1}B)_{!!}^{a} \ar[r] \ar[d]^{γ} & 0 \\ 𝔫̃_{n} \ar[r]^{δ_{n−1}} & W_{n}A ⊕ (𝔫̃_{n} ⊗_{W_{n}A} W_{n}B) \ar[r] & (W_{n}B)_{!!}^{a} \ar[r] & 0}

whose vertical arrows are induced by the projections π_{n}^{A} : W_{n+1}A → W_{n}A and π_{n}^{B} : W_{n+1}B → W_{n}B (notice that the almost structure used to compute (W_{n+1}B)_{!!}^{a} is the one relative to the basic setup (W_{n+1}A, 𝔫_{n+1}), whereas for (W_{n}B)_{!!}^{a} we use the almost structure relative to the basic setup (W_{n}A, 𝔫_{n})). Since π_{n}^{A}(𝔫_{n+1}) = 𝔫_{n}, by [75, Rem.2.1.4(ii)] we have a natural isomorphism

𝔫̃_{n+1} ⊗_{W_{n+1}A} W_{n}A →^{∼} 𝔫̃_{n} x_{•} ⊗ y_{•} ⊗ a_{•} ↦ a_{•} · π_{n}^{A}(x_{•}) ⊗ π_{n}^{A}(y_{•})

which identifies α with 𝔫̃_{n+1} ⊗_{W_{n+1}A} π_{n}^{A}, and β with π_{n}^{A} ⊕ (𝔫̃_{n+1} ⊗_{W_{n+1}A} π_{n}^{B}). Taking claim 9.3.31into account, we deduce natural W_{n+1}A-linear surjections :

(14.7.20) 𝔫̃_{n+1} ⊗_{W_{n+1}A} \overline{ω}_{n}^{∗}(A) → Ker α  \overline{ω}_{n}^{∗}(A) ⊕ (𝔫̃_{n+1} ⊗_{W_{n+1}A} \overline{ω}_{n}^{∗}(B)) → Ker β.

Then, in light of (14.7.12) and [75, Rem.2.1.4(ii)], we have as well the natural isomorphism

𝔫̃_{n+1} ⊗_{W_{n+1}A} \overline{ω}_{n}^{∗}(A) →^{∼} 𝔪̃ x_{•} ⊗ y_{•} ⊗ a ↦ a · \overline{ω}_{n}(x_{•}) ⊗ \overline{ω}_{n}(y_{•})

which identifies (14.7.20) with W_{n+1}A-linear surjections :

(14.7.21)  \overline{ω}_{n}^{∗}(𝔪̃) → Ker α  \overline{ω}_{n}^{∗}(A ⊕ (𝔪̃ ⊗_{A} B)) → Ker β.

By a direct inspection, we then get a commutative diagram :

\xymatrix{\overline{ω}_{n}^{∗}(𝔪̃) \ar[r]^{\overline{ω}_{n}^{∗}(δ_{0})} \ar[d] & \overline{ω}_{n}^{∗}(A ⊕ (𝔪̃ ⊗_{A} B)) \ar[d] \\ Ker α \ar[r]^{δ′_{n}} & Ker β}

whose vertical arrows are the surjections (14.7.21), and δ′_{n} is the restriction of δ_{n}. Since α and β are surjective, the induced map Coker δ′_{n} → Ker γ is bijective, by the snake lemma, and γ is surjective, so finally we get a right exact sequence of W_{n+1}A-modules :

\overline{ω}_{n}^{∗}(B_{!!}^{a}) ⟶^{ρ} (W_{n+1}B)_{!!}^{a} ⟶^{γ} (W_{n}B)_{!!}^{a} → 0.

We consider then the diagram of W_{n+1}A-modules :

\xymatrix{ & \overline{ω}_{n}^{∗}(B_{!!}^{a}) \ar[r]^{ρ} \ar[d]_{\overline{ω}_{n}^{∗}ε_{B}} & (W_{n+1}B)_{!!}^{a} \ar[r]^{γ} \ar[d]^{ε_{W_{n+1}B}} & (W_{n}B)_{!!}^{a} \ar[r] \ar[d]^{ε_{WnB}} & 0 \\ 0 \ar[r] & \overline{ω}_{n}^{∗}B \ar[r]^{ρ′} & W_{n+1}B \ar[r]^{γ′} & W_{n}B \ar[r] & 0}

where γ′ is the projection, and ρ′ is the natural identification of \overline{ω}_{n}^{∗}B with Ker γ′ given by claim 9.3.31. Thus, the bottom horizontal sequence is exact, and by inductive assumption the first and third vertical arrows are bijective; therefore, in order to conclude the proof, it will suffice to check the diagram commutes. The commutativity of the right square subdiagram is clear. To see the commutativity of the left square subdiagram, let w ∈ \overline{ω}_{n}^{∗}(B_{!!}^{a}) be any element; hence w is the class [a, z] of a pair (a, z) ∈ A ⊕ (𝔪̃ ⊗_{A} B), and we may assume that z = x ⊗ y ⊗ b for some x, y ∈ 𝔪 and b ∈ B. Then, by inspecting the constructions, we see that w′ := ρ(w) is the class [(0, …, 0, a), (0, …, 0, xyb)] of the corresponding pair in W_{n+1}A⊕(𝔫̃_{n+1} ⊗_{W_{n+1}A} W_{n+1}B). On the other hand, \overline{ω}_{n}^{∗}ε_{B}(w) = a+xyb; then, again by claim 9.3.31, we deduce that ε_{W_{n+1}B}(w′) = (0, …, 0, a + xyb) = ρ′(a + xyb), as required. ♢

(ii): For every n ∈ ℕ, we have a natural map of non-unital W_{n+1}A-algebras :

μ_{n} : W_{n+1}(𝔪)⊗_{W_{n+1}A} W_{n+1}(𝔪) → W_{n+1}(𝔪̃) a_{•} ⊗b_{•} ↦ (P_{i}(a_{•} ⊗1, 1⊗b_{•}) | i = 0, …, n)

for the non-unital ring structure of W_{n+1}(𝔪) = 𝔫_{n+1} and W_{n+1}(𝔪̃) induced by those of 𝔪 and respectively 𝔪̃ (remark 9.3.16). We show by induction on n ∈ ℕ, that μ_{n} is an isomorphism; the assertion is trivial for n = 0, since μ_{0} = 1𝔪̃. Thus, let n > 0, and suppose that the assertion is already known for n − 1; we consider the commutative diagram of W_{n+1}A-modules :

\xymatrix{𝔫_{n+1} ⊗_{W_{n+1}A} 𝔫_{n+1} \ar[r]^{μ_{n}} \ar[d]_{α} & W_{n+1}(𝔪̃) \ar[d]^{β} \\ 𝔫_{n} ⊗_{W_{n}A} 𝔫_{n} \ar[r]^{μ_{n−1}} & W_{n}(𝔪̃)}

whose left vertical arrows is induced by the projection W_{n+1}(𝔪) → W_{n}(𝔪), and whose right vertical arrow is likewise the natural projection. Claim 9.3.31 yields a natural isomorphism of W_{n+1}A-modules :

(14.7.22)  \overline{ω}_{n}^{∗}(𝔪̃) →^{∼} Ker β.

Claim 14.7.23. We have a commutative diagram of W_{n+1}A-modules :

\xymatrix{\overline{ω}_{n}^{∗}(𝔪̃) \ar[r]^{\overline{ω}_{n}^{∗}(μ_{0})} \ar[d] & \overline{ω}_{n}^{∗}(𝔪̃) \ar[d] \\ Ker α \ar[r] & Ker β}

whose bottom horizontal arrow is the restriction of μ_{n}, and whose right (resp. left) vertical arrow is (14.7.22) (resp. is (14.7.21)).

Proof of the claim. By construction, the left vertical arrow is characterized as the map such that a · \overline{ω}_{n}(x_{•}) ⊗ \overline{ω}_{n}(y_{•}) ↦ (0, …, 0, a) · (x_{•} ⊗ y_{•}) for every a ∈ A and every x_{•}, y_{•} ∈ 𝔫_{n+1}. On the other hand, the right vertical arrow is the map such that x ⊗ y ↦ (0, …, 0, x ⊗ y) for every x, y ∈ 𝔪. Hence, we need to check the identity :

μ_{n}((0, …, 0, a) · (x_{•} ⊗ y_{•})) = (0, …, 0, a · \overline{ω}_{n}(x_{•}) ⊗ \overline{ω}_{n}(y_{•}))

for every a ∈ A and every x_{•}, y_{•} ∈ 𝔫_{n+1}. To this aim, endow R := A ⊕ 𝔪 with its natural A-algebra structure as in remark 9.3.16(iii), and recall that μ_{n} is the restriction of the map of unital W_{n+1}A-algebras

W_{n+1}R ⊗_{W_{n+1}A} W_{n+1}R → W_{n+1}(R ⊗_{A} R) x_{•} ⊗ y_{•} ↦ (x_{•} ⊗ 1) · (1 ⊗ y_{•}).

Thus, we are reduced to checking that

((V_{R}^{n}(τ_{R}(a)) · x_{•}) ⊗ 1) · (1 ⊗ y_{•}) = V_{R⊗_{A}R}^{n}(τ_{R⊗_{A}R}(a · \overline{ω}_{n}(x_{•}) ⊗ \overline{ω}_{n}(y_{•})))

for every a ∈ A and every x_{•}, y_{•} ∈ W_{n+1}R, where V_{R} and τ_{R} denote respectively the Verschiebung and the Teichmüller maps, and likewise for V_{R⊗_{A}R} and τ_{R⊗_{A}R} (see (9.3.18)). Let also F_{R} be the Frobenius map; using proposition 9.3.22(ii) we compute :

((V_{R}^{n}(τ_{R}(a)) · x_{•}) ⊗ 1) · (1 ⊗ y_{•}) = (V_{R}^{n}(τ_{R}(a) · F_{R}^{n}(x_{•})) ⊗ 1) · (1 ⊗ y_{•})
= V_{R}^{n}((τ_{R}(a) · F_{R}^{n}(x_{•})) ⊗ 1) · (1 ⊗ y_{•})
= V_{R⊗_{A}R}^{n}(((τ_{R}(a) · F_{R}^{n}(x_{•})) ⊗ 1) · (1 ⊗ F_{R}^{n}(y_{•})))

= V_{R⊗_{A}R}^{n}(τ_{R⊗_{A}R}(a) · F_{R⊗_{A}R}^{n}(x_{•} ⊗ 1) · F_{R⊗_{A}R}^{n}(1 ⊗ y_{•})).

Hence, we are reduced to checking that :

\overline{ω}_{n}(x_{•}) ⊗ \overline{ω}_{n}(y_{•}) = \overline{ω}_{0}(F_{R⊗_{A}R}^{n}(x_{•} ⊗ 1) · F_{R⊗_{A}R}^{n}(1 ⊗ y_{•})) in W_{n+1}(R ⊗_{A} R).

The latter follows easily from (9.3.19) : details left to the reader. ♢

Claim 14.7.23 implies that also (14.7.21) is an isomorphism, and the same for μ_{n−1}, by inductive assumption; so the same follows for μ_{n} and the proof is concluded. □

14.7.24. Let now A_{0} be any 𝔽_{p}-algebra, (A_{0}, 𝔪_{0}) any basic setup, and R any A_{0}-algebra. Recall that the Frobenius endomorphism Φ_{R} : R → R induces an endofunctor

Φ_{R}^{k∗} : R-Alg → R-Alg for every k ∈ ℕ

that assigns to every R-algebra S the R-algebra Φ_{R}^{k∗}S whose underlying ring is the same as S, and whose structure morphism is the composition of Φ_{R}^{k} with the structure morphism R → S of S. Moreover, the Frobenius endomorphism Φ_{S}^{k} induces a natural transformation

Φ_{S/R}^{k} : S ⊗_{R} Φ_{R}^{k∗}R → Φ_{R}^{k∗}S  x ⊗ y ↦ x^{p^{k}}y  for every R-algebra S

Let R^{a} be the (A_{0}, 𝔪)^{a}-algebra represented by R; then Φ_{R}^{k∗} descends to an endofunctor

Φ_{R^{a}}^{k∗} : R^{a}-Alg → R^{a}-Alg

that assigns to every R^{a}-algebra S^{a} the R^{a}-algebra Φ_{R^{a}}^{k∗}(S^{a}) := (Φ_{R}^{k∗}S)^{a}, where S is any Ralgebra representing S^{a} : see [75, §3.5.7]. Likewise, the Frobenius endomorphism Φ_{S}^{k} induces a natural transformation

Φ_{S^{a}/R^{a}}^{k} := (Φ_{S/R}^{k})^{a} : (S^{a} ⊗_{R^{a}} Φ_{R^{a}}^{k∗}R^{a}) → Φ_{R^{a}}^{k∗}S^{a} for every R^{a}-algebra S^{a}.

Moreover, notice that Φ_{S^{a}/R^{a}}^{k+1} also equals the composition of Φ_{S^{a}/R^{a}} ⊗_{Φ_{R^{a}}^{∗}R^{a}} Φ_{R^{a}}^{k+1∗}R^{a} :

S^{a} ⊗_{R^{a}} Φ_{R^{a}}^{k+1∗}R^{a} →^{∼} S^{a} ⊗_{R^{a}} Φ_{R^{a}}^{∗}R^{a} ⊗_{Φ_{R^{a}}^{∗}R^{a}} Φ_{R^{a}}^{k+1∗}R^{a} → Φ_{R^{a}}^{∗}S^{a} ⊗_{Φ_{R^{a}}^{∗}R^{a}} Φ_{R^{a}}^{k+1∗}R^{a}

together with Φ_{R^{a}}^{∗}(Φ_{S^{a}/R^{a}}^{k}) : Φ_{R^{a}}^{∗}S^{a} ⊗_{Φ_{R^{a}}^{∗}R^{a}} Φ_{R^{a}}^{k+1∗}R^{a} = Φ_{R^{a}}^{∗}(S^{a} ⊗_{R^{a}} Φ_{R^{a}}^{k∗}R^{a}) → Φ_{R^{a}}^{k+1∗}S^{a}. Especially, if Φ_{S^{a}/R^{a}} is an isomorphism, the same holds for Φ_{S^{a}/R^{a}}^{k}, for every k ∈ ℕ.

Remark 14.7.25. In the situation of (14.7.24), suppose that 𝔪 fulfills condition (B) of [75, §2.1.6], and let f : R → S be any weakly étale morphism of A_{0}^{a}-algebras. Then Φ_{S/R} is an isomorphism. Indeed, the assertion is [75, Th.3.5.13(ii)] in case 𝔪⊗_{A_{0}}𝔪 is a flat A_{0}-module, but by direct inspection one sees easily that the proof of loc.cit. is valid more generally, whenever condition (B) holds.

Theorem 14.7.26. In the situation of (14.7.15), let f : B → C be a flat morphism of A^{a}- algebras; set B_{0} := B/pB, C_{0} := C/pC, and suppose that Φ_{C_{0}/B_{0}} is an isomorphism. Then:

(i) f induces a flat morphism of W_{n+1}A^{a}-algebras

W_{n+1}f : W_{n+1}B → W_{n+1}C for every n ∈ ℕ.

(ii) For every n ∈ ℕ and every i = 0, …, n, the following diagram is cocartesian :

\xymatrix{W_{n+1}B \ar[r]^{W_{n+1}f} \ar[d]_{_{\overline{ω}_{i}}} & W_{n+1}C \ar[d]^{_{\overline{ω}_{i}}} \\ \overline{ω}_{i}^{∗}B \ar[r]^{\overline{ω}_{i}^{∗}f} & \overline{ω}_{i}^{∗}C.}

Proof. We prove first both assertions in the case where p^{k}A = 0 for some integer k ∈ ℕ. For every j ∈ ℕ, and every A^{a}-algebra R set

W_{n,j}R := W_{n+1}(R)/p^{j}\overline{V}_{n}(R) with \overline{V}_{n}(R) := V_{n}R/V_{n+1}R ⊂ W_{n+1}R.

Hence, W_{n,0}R = W_{n}R, and in light of corollary 9.3.32, we see that W_{n,j}R = W_{n+1}R for every sufficiently large j ∈ ℕ. We shall show, by induction on j and n, the following assertions :

(a)_{n,j} The induced map W_{n,j}f : W_{n,j}B → W_{n,j}C is flat for every j, n ∈ ℕ.

(b)_{n,j} The morphism f induces a cocartesian diagram of W_{n+1}B-algebras :

\xymatrix{W_{n,j}B \ar[r] \ar[d]_{W_{n,j}f} & \overline{ω}_{0}^{∗}B \ar[d]^{\overline{ω}_{0}^{∗}f} \\ W_{n,j}C \ar[r] & \overline{ω}_{0}^{∗}C.}

Both assertions are trivial for n = 0 and every j ∈ ℕ. Let then n, j ∈ ℕ be any integers, and suppose that (a)_{n,j} and (b)_{n,j} hold. According to (14.7.16), the kernel K_{n,j} of the projection W_{n,j+1}B → W_{n,j}B is isomorphic, as a W_{n+1}B-module, to \overline{ω}_{n}^{∗}(p^{j}B/p^{j+1}B). Likewise for the kernel K′_{n,j} of the projection W_{n,j+1}C → W_{n,j}C, and the induced morphism K_{n,j} → K′_{n,j} corresponds to the restriction of \overline{ω}_{n}^{∗}(f ⊗_{ℤ} ℤ/p^{j+1}ℤ), under these identifications. By the (almost version of the) local flatness criterion (see [126, Th.22.3]), assertion (a)_{n,j+1} will follow from :

Claim 14.7.27. The induced morphism \overline{ω}_{n}^{∗}(p^{j}B/p^{j+1}B) ⊗_{W_{n,j}B} W_{n,j}C → \overline{ω}_{n}^{∗}(p^{j}C/p^{j+1}C) is an isomorphism.

Proof of the claim. Clearly \overline{ω}_{n}^{∗}(p^{j}B/p^{j+1}B) is a \overline{ω}_{n}^{∗}B_{0}-module and \overline{ω}_{n}^{∗}(p^{j}C/p^{j+1}C) is a C_{0}- module, and notice that \overline{ω}_{n}^{∗}(B_{0}) = \overline{ω}_{0}^{∗} ∘ Φ_{B_{0}}^{n∗}(B_{0}), and likewise for \overline{ω}_{n}^{∗}(C_{0}). Then the stated morphism agrees with the composition

\overline{ω}_{n}^{∗}(p^{j}B/p^{j+1}B) ⊗_{W_{n,j}B} W_{n,j}C →^{∼} \overline{ω}_{0}^{∗}Φ_{B_{0}}^{n∗}(p^{j}B/p^{j+1}B) ⊗_{\overline{ω}_{0}^{∗}B_{0}} \overline{ω}_{0}^{∗}B_{0} ⊗_{W_{n,j}B} W_{n,j}C
→^{∼} \overline{ω}_{0}^{∗}Φ_{B_{0}}^{n∗}(p^{j}B/p^{j+1}B) ⊗_{\overline{ω}_{0}^{∗}B_{0}} \overline{ω}_{0}^{∗}C_{0}
= \overline{ω}_{0}^{∗}(Φ_{B_{0}}^{n∗}(p^{j}B/p^{j+1}B) ⊗_{B_{0}} C_{0})
→^{∼} \overline{ω}_{0}^{∗}(Φ_{B_{0}}^{n∗}(p^{j}B/p^{j+1}B) ⊗_{Φ_{B_{0}}^{n∗}B_{0}} Φ_{B_{0}}^{n∗}B_{0} ⊗_{B_{0}} C_{0})
→^{∼} \overline{ω}_{0}^{∗}(Φ_{B_{0}}^{n∗}(p^{j}B/p^{j+1}B) ⊗_{Φ_{B_{0}}^{n∗}B_{0}} Φ_{B_{0}}^{n∗}C_{0})
= \overline{ω}_{0}^{∗}Φ_{B_{0}}^{n∗}((p^{j}B/p^{j+1}B) ⊗_{B_{0}} C_{0})
→^{∼} \overline{ω}_{0}^{∗}Φ_{B_{0}}^{n∗}(p^{j}C/p^{j+1}C)
= \overline{ω}_{n}^{∗}(p^{j}C/p^{j+1}C)

where the second isomorphism follows from (b)_{n,j}, the fourth follows from our assumption about Φ_{C_{0}/B_{0}}, and the fifth follows from the flatness of f. ♢

Claim 14.7.27also implies that the diagram

\xymatrix{W_{n,j+1}B \ar[r] \ar[d]_{W_{n,j+1}f} & W_{n,j}B \ar[d]^{W_{n,j}f} \\ W_{n,j+1}C \ar[r] & W_{n,j}C}

is cocartesian; combining with (b)_{n,j}, we see that (b)_{n,j+1} holds. Thus, we conclude that (a)_{n,j} and (b)_{n,j} hold for every j ∈ ℕ; for large values of j, we deduce that (a)_{n+1,0} and (b)_{n+1,0} both hold. By induction on n ∈ ℕ, the assertion follows. This concludes the proof of (i) in case p^{k}A = 0. To prove (ii) under the same assumption, we show first that the natural morphism

(14.7.28)  \overline{ω}_{i}^{∗}B_{0} ⊗_{W_{n+1}B} W_{n+1}C → \overline{ω}_{i}^{∗}C_{0}

is an isomorphism. Indeed, notice that \overline{ω}_{i}^{∗}B_{0} = \overline{ω}_{0}^{∗}(Φ_{B_{0}}^{i∗}B_{0}), and likewise for \overline{ω}_{i}^{∗}C_{0}. Then the foregoing morphism is the composition :

\begin{matrix}\overline{ω}_{i}^{∗}B_{0} ⊗_{W_{n+1}B} W_{n+1}C = & \overline{ω}_{0}^{∗}(Φ_{B_{0}}^{i∗}B_{0}) ⊗_{W_{n+1}B} W_{n+1}C \\ →^{∼} & \overline{ω}_{0}^{∗}(Φ_{B_{0}}^{i∗}B_{0}) ⊗_{\overline{ω}_{0}^{∗}B_{0}} \overline{ω}_{0}^{∗}B_{0} ⊗_{W_{n+1}B} W_{n+1}C \\ →^{∼} & \overline{ω}_{0}^{∗}(Φ_{B_{0}}^{i∗}B_{0}) ⊗_{\overline{ω}_{0}^{∗}B_{0}} \overline{ω}_{0}^{∗}C_{0} \\ = & \overline{ω}_{0}^{∗}(Φ_{B_{0}}^{i∗}B_{0} ⊗_{B_{0}} C_{0}) \\ →^{∼} & \overline{ω}_{0}^{∗}(Φ_{B_{0}}^{i∗}C_{0}) \\ = & \overline{ω}_{i}^{∗}C_{0} \end{matrix}

where the second isomorphism follows from the foregoing condition (b)_{n+1,0}, and the third isomorphism follows from our assumption on Φ_{C_{0}/B_{0}}. Now, since p^{k}B = 0, the assertion follows from the isomorphism (14.7.28) together with the following :

Claim 14.7.29. Let g : R → S be a flat morphism of A^{a}-algebras, I ⊂ R a nilpotent ideal, and suppose that g ⊗_{R} R/I is an isomorphism. Then g is an isomorphism.

Proof of the claim. By a simple induction, we are reduced to the case where I^{2} = 0. Then we have the commutative ladder with exact rows :

\xymatrix{0 \ar[r] & I \ar[r] \ar[d] & R \ar[r] \ar[d] & R/I \ar[r] \ar[d] & 0 \\ 0 \ar[r] & I ⊗_{R} S \ar[r] & S \ar[r] & S/IS \ar[r] & 0}

and a natural identification I ⊗_{R} S →^{∼} I ⊗_{R/I} S/IS →^{∼} I. Thus, the first and third vertical arrows are both isomorphisms, and the same then follows for the middle one. ♢

We consider next the case where A is a general ring, and 𝔪 fulfills condition (B). We shall use the following criterion :

Claim 14.7.30. Let (V, 𝔪_{V}) be any basic setup, R a (V, 𝔪_{V})^{a}-algebra and M an R-module. Set

M_{tor} := ⋃_{n∈ℕ} Ann_{M}(p^{n}) R_{tor} := ⋃_{n∈ℕ} Ann_{R}(p^{n}).

Then M is a flat R-module if and only if the following three conditions hold :

(a) M/p^{i}M is a flat R/p^{i}R-module for every i ∈ ℕ.

(b) M ⊗_{R} R[p^{−1}] is a flat R[p^{−1}]-module.
(c) The natural morphism R_{tor} ⊗_{R} M → M_{tor} is an isomorphism.

Proof of the claim. Clearly if M is flat, conditions (a)–(c) hold. Thus, suppose that (a)–(c) hold; according to [75, Lemma 5.2.1] it suffices to prove that

(14.7.31)  Tor_{i}^{R}(M, R/pR) = 0  for i = 1, 2.

However, a standard calculation shows that

Tor_{1}^{R}(M, R/pR) = \frac{Ann_{M}(p)}{Ann_{R}(p)M} and Tor_{2}^{R}(M, R/pR) = Ker (Ann_{R}(p)⊗_{R}M → M).

Let us then consider the commutative ladder :

\xymatrix{0 \ar[r] & Ann_{R}(p) ⊗_{R} M \ar[r] \ar[d] & R_{tor} ⊗_{R} M \ar[r]^{p} \ar[d] & R_{tor} ⊗_{R} M \ar[d] \\ 0 \ar[r] & Ann_{M}(p) \ar[r] & M_{tor} \ar[r]^{p} & M_{tor}}

whose bottom horizontal row is exact, and whose central and right vertical arrows are isomorphisms; in order to verify (14.7.31), it suffices therefore to check that the top horizontal row is also exact; the latter is the inductive limit of the system of morphisms :

Ann_{R}(p) ⊗_{R} M → Ann_{R}(p^{n}) ⊗_{R} M → Ann_{R}(p^{n}) ⊗_{R} M for every n ∈ ℕ.

But clearly Ann_{R}(p^{i}) ⊗_{R} M = Ann_{R}(p^{i}) ⊗_{R/p^{n}R} M/p^{n}M for every i, n ∈ ℕ with i ≤ n; in view of (a), the assertion follows. ♢

Now, example 9.3.38(ii), implies easily that W_{n+1}(C)[p^{−1}] is a flat W_{n+1}(B)[p^{−1}]-algebra. Next, for every k ∈ ℕ let π_{k} : C → C/p^{k}C be the projection; from example 9.3.37(iii) we get

Ker W_{n+1}(π_{n+k}) ⊂ p^{k}W_{n+1}(C) for every k ∈ ℕ

so that π_{n+k} induces an isomorphism of W_{n+1}A^{a}-algebras

(14.7.32)  W_{n+1}(C) ⊗_{ℤ} ℤ/p^{k}ℤ →^{∼} W_{n+1}(C/p^{n+k}C) ⊗_{ℤ} ℤ/p^{k}ℤ  for every k ∈ ℕ.

But, by the foregoing case, we know already that W_{n+1}(C/p^{n+k}C) is a flat W_{n+1}(B/p^{n+k}B)- algebra for every k ∈ ℕ, so we conclude that W_{n+1}(C) ⊗_{ℤ} ℤ/p^{k}ℤ is a flat W_{n+1}(B) ⊗_{ℤ} ℤ/p^{k}ℤalgebra for every k ∈ ℕ. In light of claim 14.7.30, we are therefore reduced to checking that the natural morphism

W_{n+1}(B)_{tor} ⊗_{W_{n+1}B} W_{n+1}C → W_{n+1}(C)_{tor}

is an isomorphism. More precisely, we shall show, by descending induction on i, that :

• the induced morphism (\overline{V}_{i}B)_{tor} ⊗_{W_{n+1}B} W_{n+1}C → (\overline{V}_{i}C)_{tor} is an isomorphism of

W_{n+1}B-modules for every i = 0, …, n + 1

• the induced morphism (\overline{V}_{i+1}B)_{tor} ⊗_{W_{n+1}B} W_{n+1}C → (\overline{V}_{i}B)_{tor} ⊗_{W_{n+1}B} W_{n+1}C is a

monomorphism of W_{n+1}B-modules for i = 0, …, n.

Indeed, both assertions are trivial for i = n + 1. Suppose that both assertions are already known for some strictly positive integer i ≤ n + 1; from the commutative diagram

\xymatrix{(\overline{V}_{i}B)_{tor} ⊗_{W_{n+1}B} W_{n+1}C \ar[r] \ar[d] & (\overline{V}_{i}C)_{tor} \ar[d] \\ (\overline{V}_{i−1}B)_{tor} ⊗_{W_{n+1}B} W_{n+1}C \ar[r] & (\overline{V}_{i−1}C)_{tor}}

we then deduce that the left vertical arrow is a monomorphism, i.e. the second assertion holds for i − 1. Moreover, in order to prove the first assertion for i − 1, it suffices to check that the induced morphism

(\overline{V}_{i−1}B)_{tor}/(\overline{V}_{i}B)_{tor} ⊗_{W_{n+1}B} W_{n+1}C → (\overline{V}_{i−1}C)_{tor}/(\overline{V}_{i−1}C)_{tor}

is an isomorphism of W_{n+1}B-modules. On the other hand, from example 9.3.38(i) we see that W_{n+1}(B)_{tor} is the kernel of the natural morphism W_{n+1}B → W_{n+1}(B[p^{−1}]), and likewise for W_{n+1}(C)_{tor}; taking into account (14.7.16), we deduce a natural isomorphism :

(\overline{V}_{i−1}B)_{tor}/(\overline{V}_{i}B)_{tor} →^{∼} (\overline{ω}_{i−1}^{∗}B)_{tor}

of W_{n+1}B-modules, and likewise for (\overline{V}_{i−1}C)_{tor}/(\overline{V}_{i−1}C)_{tor}. We are then further reduced to checking that the induced morphism

\overline{ω}_{i−1}^{∗}(Ann_{B}(p^{k})) ⊗_{W_{n+1}(B)} W_{n+1}C → \overline{ω}_{i−1}^{∗}Ann_{C}(p^{k})

is an isomorphism for every k ∈ ℕ. However, the latter is the composition of the isomorphisms

\overline{ω}_{i−1}^{∗}(Ann_{B}(p^{k})) ⊗_{W_{n+1}(B)} W_{n+1}C →^{∼} \overline{ω}_{i−1}^{∗}(Ann_{B}(p^{k})) ⊗_{W_{n+1}(B/p^{n+k}B)} W_{n+1}(C/p^{n+k}C)
→^{∼} \overline{ω}_{i−1}^{∗}(Ann_{B}(p^{k})) ⊗\overline{ω}_{i−1}^{∗}_{(B/p^{n+k}B)} \overline{ω}_{i−1}^{∗}(C/p^{n+k}C)
= \overline{ω}_{i−1}^{∗}(Ann_{B}(p^{k}) ⊗_{B} C)
→^{∼} \overline{ω}_{i−1}^{∗}Ann_{C}(p^{k})

where the first isomorphism is due to (14.7.32), the second one follows from the part of assertion (ii) that we have already proved, and the third one follows from the flatness of f.

To conclude the proof of (ii) for a general ring A, we shall need the following :

Claim 14.7.33. Let (V, 𝔪_{V}) be a basic setup, R a (V, 𝔪)^{a}-algebra and φ : M → N a morphism of flat R-modules. Suppose that :

(a) φ ⊗_{ℤ} 𝔽_{p} is an isomorphism of R/pR-modules.

(b) φ ⊗_{ℤ} ℤ[p^{−1}] is an isomorphism of R[p^{−1}]-modules.

Then φ is an isomorphism.

Proof of the claim. Denote by C_{•} the complex of R-modules [M ⟶^{φ} N], say with M placed in degree 0. Consider as well the acyclic complex

D_{•} : 0 → Ann_{R}(p) → R ⟶^{p1_{R}} R → R/pR → 0.

Since C_{•} is flat in every degree, the complex D_{•} ⊗_{R} C_{•} is still exact; the latter is also the total complex of the double complex

0 → E_{•} := Ann_{R}(p) ⊗_{R/pR} (C_{•} ⊗_{ℤ} 𝔽_{p}) → C_{•} ⟶^{p1_{C•}} C_{•} → C_{•} ⊗_{ℤ} 𝔽_{p} → 0.

On the other hand, assumption (a) says that C_{•} ⊗_{ℤ} 𝔽_{p} is an acyclic complex of flat R/pRmodules, hence also E_{•} is acyclic. Summing up, we find that the morphism of complexes p1_{C_{•}} : C_{•} → C_{•} is a quasi-isomorphism, and hence induces isomorphisms p · 1_{H_{i}(C_{•})} : H_{i}(C_{•}) →^{∼} H_{i}(C_{•}) for i = 0, 1. In other words, H_{i}(C_{•}) = H_{i}(C_{•}) ⊗_{ℤ} ℤ[p^{−1}] for i = 0, 1; combining with assumption (b), we conclude that Ker φ = Coker φ = 0, whence the claim. ♢

Now, we need to show that the morphism

φ_{f} : \overline{ω}_{i}^{∗}B ⊗_{W_{n+1}B} W_{n+1}C → \overline{ω}_{i}^{∗}C

resulting from the diagram of (ii) is an isomorphism of W_{n+1}B-modules; however, set as well g := f ⊗_{ℤ} ℤ/p^{n+1}ℤ : B/p^{n+1}B → C/p^{n+1}C; according to the part of (ii) that has already been proved, the corresponding morphism

φ_{g} : \overline{ω}_{i}^{∗}(B/p^{n+1}B) ⊗_{W_{n+1}(B/p^{n+1}B)} W_{n+1}(C/p^{n+1}C) → \overline{ω}_{i}^{∗}(C/p^{n+1}C)

is an isomorphism; on the other hand, in view of (14.7.32) the morphism φ_{f} ⊗_{ℤ} 𝔽_{p} is naturally identified with φ_{g} ⊗_{ℤ} 𝔽_{p}, so φ_{f} ⊗_{ℤ} 𝔽_{p} is an isomorphism as well. Next, recall that the ghost map induces an isomorphism of W_{n+1}A^{a}-algebras

(W_{n+1}B)[p^{−1}] →^{∼} B[p^{−1}]^{n+1} := ∏_{i=0}^{n} \overline{ω}_{i}^{∗}B[p^{−1}]

and likewise for (W_{n+1}C)[p^{−1}] (example 9.3.38(ii)); under these isomorphisms, the morphism \overline{ω}_{i} ⊗_{ℤ} ℤ[p^{−1}] on (W_{n+1}B)[p^{−1}] is identified with the projection π_{i} : B[p^{−1}]^{n+1} → B[p^{−1}] on the (i + 1)-th factor, and likewise for the corresponding morphism on (W_{n+1}C)[p^{−1}]. Thus, φ_{f} ⊗_{ℤ} ℤ[p^{−1}] is identified with the natural isomorphism of B[p^{−1}]^{n+1}-modules

π_{i}^{∗}B[p^{−1}] ⊗_{B[p^{−1}]^{n+1}} C[p^{−1}]^{n+1} →^{∼} π_{i}^{∗}C[p^{−1}].

From claim 14.7.33and (i) we deduce that φ_{f} is an isomorphism, as required. □

Corollary 14.7.34. In the situation of theorem 14.7.26, for every n ∈ ℕ we have : (i) Every B-algebra D induces a cocartesian diagram of W_{n+1}A^{a}-algebras :

\xymatrix{W_{n+1}B \ar[r] \ar[d] & W_{n+1}C \ar[d] \\ W_{n+1}D \ar[r] & W_{n+1}(C ⊗_{B} D).}

(ii) The W_{n+1}B-module W_{n+1}C is almost finitely generated (resp. almost finitely presented, resp. almost projective) if and only if the same holds for the B-module C.

(iii) Suppose that condition (B) of [75, §2.1.6] holds for 𝔪. Then W_{n+1}(f) is étale (resp.

weakly étale) if and only if the same holds for the morphism f. (iv) The projection W_{n+2}C → W_{n+1}C induces an isomorphism of W_{n+1}B-algebras :

W_{n+2}C ⊗_{W_{n+2}B} W_{n+1}B →^{∼} W_{n+1}C.

Proof. (i): Notice that the natural morphism h : W_{n+1}C ⊗_{W_{n+1}B} W_{n+1}D → W_{n+1}(C ⊗_{B} D) restricts to morphisms of W_{n+1}B-modules

W_{n+1}C ⊗_{W_{n+1}B} \overline{V}_{k}D → \overline{V}_{k}(C ⊗_{B} D) for every k = 0, …, n.

Taking into account (14.7.16), we are then reduced to checking that h induces isomorphisms

gr_{k}h : W_{n+1}C ⊗_{W_{n+1}B} \overline{ω}_{k}^{∗}D → \overline{ω}_{k}^{∗}(C ⊗_{B} D) for every k = 0, …, n.

However, a simple inspection shows that gr_{k}h agrees with the composition

W_{n+1}C ⊗_{W_{n+1}B} \overline{ω}_{k}^{∗}D →^{∼} W_{n+1}C ⊗_{W_{n+1}B} \overline{ω}_{k}^{∗}B ⊗_{\overline{ω}_{k}^{∗}B} \overline{ω}_{k}^{∗}D →^{∼} \overline{ω}_{k}^{∗}C ⊗_{\overline{ω}_{k}^{∗}B} \overline{ω}_{k}^{∗}D = \overline{ω}_{k}^{∗}(C ⊗_{B} D)

where the second isomorphism follows from theorem 14.7.26(ii).

(iv): The proof of (i) also shows that the natural morphism

W_{n+2}C ⊗_{W_{n+2}B} \overline{V}_{n+1}B → \overline{V}_{n+1}C

is an isomorphism. The assertion is an immediate consequence.

(ii): Let ψ : W_{n+1}B → B^{n+1} := ∏_{i=0}^{n} \overline{ω}_{i}^{∗}B be the morphism of W_{n+1}A^{a}-algebras whose composition with the projection B^{n+1} → \overline{ω}_{i}^{∗}B is the morphism \overline{ω}_{i}, for i = 0, …, n; denote by R ⊂ B^{n+1} the image of ψ. Let also P be one of the properties : “almost finitely generated”, “almost finitely presented”, or “almost projective”, and suppose that f enjoys P; in view of theorem 14.7.26(i), proposition 9.3.30(i) and [75, Lemma 3.2.25], it suffices to prove that the R-module W_{n+1}C ⊗_{W_{n+1}B} R enjoys property P. Next, by proposition 9.3.30(ii) we know that

p^{n+1}B^{n+1} ⊂ R  and we let R_{0} := R/p^{n+1}B^{n+1}  B_{0}^{n+1} := B^{n+1}/pB^{n+1}.

With this notation, it is easily seen that there exists a well defined morphism of W_{n+1}B-algebras

φ : B_{0} → B_{0}^{n+1}  such that π_{i} ∘ φ = Φ_{B_{0}}^{i} : B_{0} → \overline{ω}_{i}^{∗}B_{0} = Φ_{B_{0}}^{i∗}B_{0}  for i = 0, …, n

where π_{i} : B_{0}^{n+1} → \overline{ω}_{i}^{∗}B_{0} is the projection on the (i + 1)-th factor. Arguing as in the proof of proposition 14.7.11, we get cartesian diagrams

\xymatrix{S \ar[r] \ar[d] & B^{n+1} \ar[d] & R \ar[r] \ar[d] & R_{0} \ar[d] \\ B_{0} \ar[r]^{φ} & B_{0}^{n+1} & S \ar[r] & S_{0} := S/p^{n+1}B^{n+1}.}

In case P is “almost finitely generated” or “almost finitely presented” we may then apply [75, Rem.3.2.26(i) and Lemma 3.4.18(i)] to the second of these cartesian diagrams, and thereby reduce to checking property P for the (S × R_{0})-module W_{n+1}C ⊗_{W_{n+1}B} (S × R_{0}). In case P is the property “almost projective”, we may argue as in the proof of [75, Prop.3.4.21] to achieve the same reduction. But notice as well that the restriction R_{0} → \overline{ω}_{0}^{∗}B_{0} of π_{0} is an epimorphism (on the underlying modules), and has nilpotent kernel (see the proof of proposition 14.7.11), so the R_{0}-module W_{n+1}C ⊗_{W_{n+1}B} R_{0} enjoys P if and only the same holds for the \overline{ω}_{0}^{∗}B_{0}-module W_{n+1}C ⊗_{W_{n+1}B} \overline{ω}_{0}^{∗}B_{0}. Next, by invoking again [75, Rem.3.2.26(i) and Lemma 3.4.18(i)] or the proof of [75, Prop.3.4.21] to the first of the two above diagrams we deduce that the S-module W_{n+1}C ⊗_{W_{n+1}B} S enjoys P if and only if the same holds for the (\overline{ω}_{0}^{∗}B_{0} × B^{n+1})-module W_{n+1}C ⊗_{W_{n+1}B} (\overline{ω}_{0}^{∗}B_{0} × B^{n+1}). Since \overline{ω}_{0}^{∗}B_{0} is a quotient of B^{n+1}, we are finally reduced to checking that the B^{n+1}-module W_{n+1}C ⊗_{W_{n+1}B} B^{n+1} enjoys P; but according to theorem 14.7.26(ii), the latter is isomorphic to the B^{n+1}-module ∏_{i=0}^{n} \overline{ω}_{i}^{∗}C, so the assertion is clear.

(iii): Suppose first that f is weakly étale, so that the multiplication morphism μ_{C/B} : C ⊗_{B} C → C is flat; since it is also an epimorphism on the underlying B-modules, it follows easily that μ_{C/B} is weakly étale. Especially, Φ_{C/C⊗_{B}C} is an isomorphism, by remark 14.7.25 and our assumption on 𝔪. Then theorem 14.7.26 says that W_{n+1}(μ_{C/C⊗_{B}C}) is flat. Lastly, according to (i), the multiplication morphism μ_{W_{n+1}C/W_{n+1}B} of W_{n+1}C factors through W_{n+1}(μ_{C/C⊗_{B}C}) and an isomorphism

(14.7.35)  W_{n+1}C ⊗_{W_{n+1}B} W_{n+1}C →^{∼} W_{n+1}(C ⊗_{B} C).

Hence μ_{W_{n+1}C/W_{n+1}B} is flat, and combining with theorem 14.7.26, we conclude that W_{n+1}f is weakly étale. In case f is étale, we deduce from (ii), the isomorphism (14.7.35), and [75, Prop.2.4.18] that W_{n+1}f is étale as well. Conversely, (iv) implies that if W_{n+1}f is weakly étale (resp. étale), then the same holds for f. □ Remark 14.7.36. (i) It is shown in [104, 1.5.8] that if f : B → C is a homomorphism of 𝔽_{p}- algebras and n ∈ ℕ any integer, then W_{n+1}f is étale (in the usual sense of [66], which includes the condition that f is finitely presented), if and only if the same holds for f. This result can be deduced from corollary 14.7.34(iii) and extended to arbitrary rings, as follows. Resume the notation of the proof of the corollary, and suppose that 𝔪 = A, so we are dealing with the “classical limit” for which almost rings are just usual rings; then, W_{n+1}f is a weakly étale ring homomorphism, and by [75, §3.4.44] we know already that W_{n+1}f is finitely presented if and only if the same holds for W_{n+1}f ⊗_{W_{n+1}B} R. We claim next that W_{n+1}f ⊗_{W_{n+1}B} R is finitely presented if and only if the same holds for W_{n+1}f ⊗_{W_{n+1}B} (R_{0} × S). For the proof, arguing as in [75, §3.4.44] we reduce to checking that the natural morphism R → R_{0} × S is of universal effective descent for the fibred category of weakly étale morphisms of rings. The latter assertion is already known by corollary 14.1.92. By the same token, W_{n+1}f ⊗_{W_{n+1}B} S is finitely presented if and only if the same holds for W_{n+1}f ⊗_{W_{n+1}B}(B^{n+1}×B_{0}). Moreover, the surjective ring homomorphism R_{0} → B_{0} has nilpotent kernel, so again by [75, §3.4.44] we know that W_{n+1}f ⊗_{W_{n+1}B} R_{0} is finitely presented if and only if the same holds for W_{n+1}f ⊗_{W_{n+1}B} B_{0}. Since B_{0} is a quotient of B^{n+1}, we are finally reduced to checking that W_{n+1}f ⊗_{W_{n+1}B} B^{n+1} is finitely presented; but it was observed in the proof of corollary 14.7.34(ii) that the latter is isomorphic to the B^{n+1}-module ∏_{i=0}^{n} \overline{ω}_{i}^{∗}C, so the assertion is clear.

(ii) A version of corollary 14.7.34(iii) for truncated big Witt vectors of usual rings is found in [109, Th.2.4]. Another proof and generalization is given by [31, Th.9.2]. 14.8. Complements : locally measurable algebras. This section studies the global counterpart of the class of measurable algebras introduced in section 14.5. To begin with – and until (14.8.42) – we consider an arbitrary valued field (K, | · |), and we resume the notation of (11.4) and (14.5). Our first result is the following generalization of proposition 9.1.27: Proposition 14.8.1. Let A be a measurable K^{+}-algebra, M a K^{+}-flat and finitely generated A-module. Then M is a finitely presented A-module. Proof. Let Σ be a finite system of generators for M; also let us write A as the colimit of a filtered system (A_{i} | i ∈ I) of finitely presented K^{+}-algebras, with étale transition maps. For every i ∈ I, let M_{i} be the A_{i}-submodule of M generated by Σ; notice that M_{i} is still K^{+}-flat, hence it is a finitely presented A_{i}-module (proposition 9.1.27). We may then write M as the colimit of the filtered system (M_{i}⊗_{A_{i}} A | i ∈ I) of finitely presented A-modules, with surjective transition maps. Moreover, since \overline{A} := A/𝔪_{K}A is noetherian (lemma 14.5.4(i)), there exists i ∈ I such that M_{j} ⊗_{A_{j}} \overline{A} = M/𝔪_{K}M for every j ≥ i. Since the ring homomorphism \overline{A}_{j} := A_{j}/𝔪_{K}A_{j} → \overline{A} is faithfully flat, we deduce that

(14.8.2)  M_{i} ⊗_{A_{i}} \overline{A}_{j} = M_{j} ⊗_{A_{j}} \overline{A}_{j}  for every j ≥ i.

Consider the short exact sequence

C : 0 → N → M_{i} ⊗_{A_{i}} A_{j} → M_{j} → 0.

Since M_{i} is K^{+}-flat, the same holds for N, and the latter is a finitely generated A_{j}-module, since both M_{j} and M_{i} ⊗_{A_{i}} A_{j} are finitely presented; therefore, the sequence C ⊗_{K^{+}} κ is still exact. Taking into account (14.8.2), we see that N/𝔪_{K}N = 0. By Nakayama’s lemma, it follows that N = 0, and finally, M = M_{i} ⊗_{A_{i}} A is finitely presented, as stated. □

Definition 14.8.3. Let A be a K^{+}-algebra. Set X := Spec A, S := Spec K^{+}, and denote by f : X → S the structure morphism. We say that A is locally measurable, if the following holds. For every x ∈ X and every point ξ of X localized at x, the strict henselization of A at ξ is a measurable 𝒪_{S,f(x)}-algebra.

Remark 14.8.4. Let A be a local K^{+}-algebra, A^{sh} the strict henselization of A at a geometric point localized at the closed point, and M an A-module.

(i) Clearly, A is locally measurable if and only if A^{sh} is measurable.

(ii) However, if A^{sh} is measurable, it does not necessarily follow that A is measurable.

(iii) On the other hand, if A is a normal local domain, one can show that A is measurable if and only if the same holds for A^{sh}.

(iv) Suppose that A is local and locally measurable. Since the natural map A → A^{sh} is faithfully flat, and since every measurable K^{+}-algebra is a coherent ring, it is easily seen that A is coherent. If furthermore, the structure map K^{+} → A is local, lemma 14.5.4(i) implies that A/𝔪_{K}A is a noetherian ring.

Remark 14.8.5. Let A be a locally measurable K^{+}-algebra. Then, for every finitely generated ideal I ⊂ A, the K^{+}-algebra A/I is also locally measurable. Indeed, for every geometric point ξ of Spec A, let A_{ξ}^{sh} (resp. (A/I)_{ξ}^{sh}) be the strict henselization of A (resp. of A/I) at ξ. Then the natural map A_{ξ}^{sh}/IA_{ξ}^{sh} → (A/I)_{ξ}^{sh} is an isomorphism for every such ξ ([66, Ch.IV, Prop.18.8.10]), so the assertion follows from lemma 14.5.4(iv).

Definition 14.8.6. Let A be a K^{+}-algebra, M an A-module, and γ ∈ log Γ^{+} any element.

(i) A K^{+}-flattening sequence for the A-module M is a finite sequence \underline{b} := (b_{0}, …, b_{n}) of elements of K^{+} such that

(a) log |b_{i+1}| > log |b_{i}| for every i = 0, …, n − 1, b_{0} = 1 and b_{n} = 0.

(b) b_{i}M/b_{i+1}M is a K^{+}/b_{i}^{−1}b_{i+1}K^{+}-flat module, for every i = 0, …, n − 1.

We say that a K^{+}-flattening sequence \underline{b} for M is minimal, if no proper subsequence of \underline{b} is still flattening for M.

(ii) Say that γ = log |b| for some b ∈ K^{+}, and let b_{M} : M → bM be the map given by the rule : m ↦ bm, for every m ∈ M. We say that γ breaks M if the κ-linear map

b_{M} ⊗_{K^{+}} κ : M ⊗_{K^{+}} κ → bM ⊗_{K^{+}} κ

is not an isomorphism.

Remark 14.8.7. Let A be a K^{+}-algebra, M an A-module.

(i) Suppose that γ ∈ log Γ^{+} breaks M, and say that γ = log |c| for some c ∈ K^{+}. Clearly, for every b ∈ cK^{+}, the map b_{M} factors through c_{M}. We deduce that every γ′ ∈ log Γ^{+} with γ′ ≥ γ also breaks M.

(ii) For given b ∈ K^{+}, suppose that M is a flat K^{+}/bK^{+}-module. Then we claim that no γ < log |b| breaks M. Indeed, for such γ pick c ∈ K^{+} with log |c| = γ, and set W := K^{+}/bK^{+}; notice that the map c_{W} : W → cW induces an isomorphism c_{W} ⊗_{W} 1_{κ} : κ →^{∼} cW ⊗_{W} κ (notation of definition 14.8.6(ii)), whence an isomorphism

1_{M} ⊗_{W} c_{W} ⊗_{W} 1_{κ} : M ⊗_{W} κ →^{∼} M ⊗_{W} (cW ⊗_{W} κ).

Since M is a flat W -module, the natural map M ⊗_{W} cW → cM is an isomorphism, and the resulting isomorphism M ⊗_{W} κ →^{∼} cM ⊗_{W} κ is naturally identified with c_{M}, whence the contention.

(iii) Let \underline{b} := (b_{0}, …, b_{n}) be a sequence of elements of K^{+} fulfilling condition (a) of definition 14.8.6(i), and suppose that \underline{b} admits a subsequence that is K^{+}-flattening for M. Then \underline{b} is K^{+}-flattening for M as well. Indeed, an easy induction reduces the contention to the following. Let b, c ∈ K^{+} be any two elements such that log |c| < log |b|, and suppose that M is K^{+}/bK^{+}- flat; then M/cM is K^{+}/cK^{+}-flat, and cM is K^{+}/c^{−1}bK^{+}-flat. Of this two assertion, the first is trivial; to show the second, recall that M can be written as the colimit of a filtered system of free K^{+}/bK^{+}-modules ([120, Ch.I, Th.1.2]). Then we may assume that M is free, in which case the assertion is easily verified.

Lemma 14.8.8. Let A be a K^{+}-algebra, M a coherent A-module that admits a K^{+}-flattening sequence, and suppose that the Jacobson radical of A contains 𝔪_{K}A. Then we have :

(i) M admits a minimal K^{+}-flattening sequence, unique up to units of K^{+}.

(ii) Let (b_{0}, …, b_{n}) be a minimal K^{+}-flattening sequence for M. For every γ ∈ log Γ^{+} and every i = 0, …, n − 1, the following conditions are equivalent :

(a) γ breaks b_{i}M.

(b) γ ≥ log |b_{i}^{−1}b_{i+1}|. (As usual, we set log |0| := +∞ : see (14.5).)

Proof. It is clear that M admits a minimal K^{+}-flattening sequence (1, b_{1}, …, b_{n−1}, 0), and (i) asserts that every other such minimal sequence is of the type (1, u_{1}b_{1}, …, u_{n−1}b_{n−1}, 0) for some elements u_{1}, …, u_{n−1} ∈ (K^{+})^{×}. However, notice that the condition of (ii) characterize uniquely such a sequence, so we only have to show that (ii) holds. We may also assume that n > 1; indeed, when n = 1, the module M is K^{+}-flat, and the assertion is immediate. Moreover, since M is coherent, the same holds for bM, for every b ∈ K^{+}. Thus, an easy induction reduces to showing the equivalence of conditions (ii.a) and (ii.b) for i = 0.

However, notice that if γ breaks M, then it obviously also breaks M/b_{1}M; in light of remark 14.8.7(ii), it follows already that (a)⇒(b). Therefore, taking into account remark 14.8.7(i), it remains only to show that log |b_{1}| breaks M.

If n = 2, then b_{1}M is a flat K^{+}-module; let N := Ker b_{1,M}. We have already observed that b_{1}M is a finitely presented A-module, hence N is a finitely generated A-module, and N ⊗_{K^{+}} κ is the kernel of b_{1,M} ⊗_{K^{+}} κ. Suppose that log |b_{1}| does not break M; then N ⊗_{K^{+}} κ vanishes, and then N = 0, by Nakayama’s lemma. It follows that M is K^{+}-flat, which contradicts the minimality of the sequence (1, b_{1}, 0).

Next, we consider the case where n > 2 and set M′ := M/b_{2}M, b := b_{1}^{−1}b_{2}; since b_{2}M ⊂ b_{1}𝔪M, it suffices to show that log |b_{1}| breaks M′, hence we may assume that b_{2}M = 0, b_{1}M is a flat K^{+}/bK^{+}-module, and the flattening sequence is (1, b_{1}, b_{2}, 0). We remark :

Claim 14.8.9. If log |b_{1}| does not break M, the map b_{1,M/bM} : M/bM → b_{1}M is an isomorphism of K^{+}/bK^{+}-modules.

Proof of the claim. Indeed, let N := Ker b_{1,M/bM}; since b_{1}M is K^{+}/bK^{+}-flat, N ⊗_{K^{+}} κ is the kernel of b_{1,M/bM} ⊗_{K^{+}} κ, and the latter vanishes if log |b_{1}| does not break M; on the other hand, N is a finitely generated A-module, hence N = 0, by Nakayama’s lemma. ♢

We shall use the following variant of the local flatness criterion :

Claim 14.8.10. Let R be a ring, I_{1}, I_{2} ⊂ A two ideals, and M an R-module such that : (a) I_{1}I_{2} = 0

(b) M/I_{i}M is R/I_{i}-flat for i = 1, 2
(c) the natural map I_{1} ⊗_{R} M/I_{2}M → I_{1}M is an isomorphism.

Then M is a flat R-module.

Proof of the claim. Set I_{3} := I_{1} ∩ I_{2}; to begin with, [75, Lemma 3.4.18] and (b) imply that M/I_{3}M is a flat R/I_{3}-module. We have an obviously commutative diagram

\xymatrix{I_{3} ⊗_{R} M/I_{3}M \ar[rr]^{α} \ar[d]_{β} &  & I_{3}M \ar[d]^{γ} \\ I_{3} ⊗_{R/I_{2}} M/I_{2}M \ar[r]^{δ} & I_{1} ⊗_{R/I_{2}} M/I_{2}M \ar[r]^{τ} & I_{1}M}

of R-linear maps. From (a) we see that I_{3}^{2} = 0, and it follows easily that β is an isomorphism; γ is clearly injective, and the same holds for δ, since M/I_{2}M is a flat R/I_{2}-module; by the same token, τ is an isomorphism. We conclude that α is an isomorphism, and then the local flatness criterion ([126, Th.22.3]) yields the contention. ♢

We shall apply claim 14.8.10 with R := K^{+}/b_{2}K^{+}, I_{1} := b_{1}R, I_{2} := bR. Indeed, condition (a) obviously holds with these choices; by assumption, M/I_{1}M is a R/I_{1}-flat module, and if log |b_{1}| does not break M, claim 14.8.9 implies that M/I_{2}M is a flat R/I_{2}-module. Lastly, condition (c) is equivalent to claim 14.8.9. Summing up, we have shown that if log |b_{1}| does not break M, then M is a flat K^{+}/b_{2}-module, contradicting again the minimality of our flattening sequence. □

Proposition 14.8.11. Let A be a K^{+}-algebra, M a finitely presented A-module, and suppose that either one of the following two conditions holds :

(a) A is an essentially finitely presented K^{+}-algebra.

(b) A is a local and locally measurable K^{+}-algebra. Then M admits a K^{+}-flattening sequence.

Proof. Suppose first that (b) holds, and let A^{sh} be the strict henselization of A at a geometric point localized at the closed point. Since A^{sh} is a faithfully flat A-algebra, it suffices to show that M ⊗_{A} A^{sh} admits a K^{+}-flattening sequence. Hence, we may assume that A is measurable, in which case we may find a finitely presented K^{+}-algebra A_{0} with an ind-étale map A_{0} → A and a finitely presented A_{0}-module M_{0} with an isomorphism M_{0} ⊗_{A_{0}} A →^{∼} M of A-modules. We may then replace A by A_{0}, M by M_{0}, and therefore assume that A is finitely presented over K^{+}, especially, we are reduced to showing the assertion in the case where (a) holds. To this aim, let us remark :

Claim 14.8.12. Let B := ⨁_{n∈ℕ} B_{n} be a ℕ-graded finitely presented K^{+}-algebra with B_{0} = K^{+}, and N := ⨁_{n∈ℕ} N_{n} a ℕ-graded finitely presented B-module. We have :

(i) The K^{+}-module N_{n} is finitely presented, for every n ∈ ℕ.

(ii) For every n ∈ ℕ, let (γ_{n,i} | i ∈ ℕ) be the sequence of elementary divisors of N_{n} (see

(14.5.15)). Then Γ(N) := {γ_{n,i} | n, i ∈ ℕ} is a finite set. (iii) N admits a K^{+}-flattening sequence.

Proof of the claim. (i) is just a special case of proposition 7.6.11(iii).

(ii): Let 𝒬 be the set of all finitely presented graded quotients Q of the B-modules N, for which Γ(Q) is infinite. We have to show that 𝒬 = ∅. However, for every Q ∈ 𝒬, the B ⊗_{K^{+}} κmodule \overline{Q} := Q ⊗_{K^{+}} κ is a quotient of \overline{N} := N ⊗_{K^{+}} κ; since B ⊗_{K^{+}} κ is a noetherian ring, the set \overline{𝒬} := \{\overline{Q} | Q ∈ 𝒬\} admits minimal elements, if it is not empty. In the latter case, we may then replace N by any Q_{0} ∈ 𝒬 such that \overline{Q}_{0} is a minimal element of \overline{𝒬} , and assume that, for every graded quotient Q of N, either \overline{Q} = \overline{N}, or else Γ(Q) is a finite set. Now, for every n ∈ ℕ, let γ_{n} be the minimal non-zero elementary divisor of N_{n}, and pick a_{n} ∈ 𝔪_{K} with log |a_{n}| = γ_{n} (if N_{n} = 0, set a_{n} = 0). Let I ⊂ K^{+} be the ideal generated by {a_{n} | n ∈ ℕ}; it is easily seen that N_{n}/IN_{n} is a free K^{+}/I-module for every n ∈ ℕ, hence N/IN is a K^{+}/I-flat finitely presented B/IB-module. We may then find a ∈ I such that N/aN is already a K^{+}/aK^{+}-flat B/aB-module ([65, Ch.IV, Cor.11.2.6.1]), so N_{n}/aN_{n} is a free K^{+}/aK^{+}-module for every n ∈ ℕ, and we easily deduce that log |a| ≤ γ_{n} for every n ∈ ℕ, i.e. I = aK^{+}. Notice that aN is a finitely presented B-module (corollary 9.1.28); it follows that

Γ(N) = {γ + log |a| | γ ∈ Γ(aN)} ∪ {0}.

Especially, Γ(aN) is an infinite set. On the other hand, there exists some n ∈ ℕ such that dim_{κ}(aN_{n})⊗_{K^{+}} κ < dim_{κ} N_{n}⊗_{K^{+}} κ, so (aN)⊗_{K^{+}} κ is a proper quotient of \overline{N}, a contradiction.

(iii): Let |b_{0}|, …, |b_{n−1}| be the finitely many elements of Γ(N) (for suitable b_{1}, …, b_{n} ∈ K^{+}), and set b_{n} := 0; after permutation, we may assume that b_{i+1} ∈ b_{i}𝔪_{K} for every i = 0, …, n − 1, and b_{0} = 1. Then we claim that (b_{0}, …, b_{n}) is a K^{+}-flattening sequence for N, i.e. b_{i}N_{k}/b_{i+1}N_{k} is K^{+}/b_{i}^{−1}b_{i+1}K^{+}-flat for every i = 0, …, n and every k ∈ ℕ. However, say that (log |c_{j}| | j ∈ ℕ) is the sequence of elementary divisors of N_{k}; we are reduced to checking that (c_{j}K^{+} + b_{i}K^{+})/(c_{j}K^{+} + b_{i+1}K^{+}) is a flat K^{+}/b_{i}^{−1}b_{i+1}K^{+}-module for every j ∈ ℕ. However, by construction we have either c_{j}K^{+} ⊂ b_{i+1}K^{+}, or b_{i}K^{+} ⊂ c_{j}K^{+}; in either case the assertion is clear. ♢

Let now A be an arbitrary essentially finitely presented K^{+}-algebra, and M an arbitrary finitely presented A-module. We easily reduce to the case where A = K^{+}[T_{1}, , …, T_{r}] is a free polynomial K^{+}-algebra. In this case, we define a filtration Fil_{•}A on A, by declaring that Fil_{k}A is the K^{+}-submodule of all polynomials of total degree ≤ k, for every k ∈ ℕ; then R := R(A, Fil_{•}A)_{•} is a free polynomial K^{+}-algebra as well (see example 7.9.5). Let

L_{1} ⟶^{φ} L_{0} → M

be a presentation of M as quotient of free A-modules of finite rank. Let

e := (e_{1}, …, e_{n}) (resp. f := (f_{1}, …, f_{m}))

be a basis of L_{0} (resp. of L_{1}); we endow L_{0} with the good (A, Fil_{•}A)-filtration Fil_{•}L_{0} associated with the pair (e, (1, …, 1)) as in (7.9.6) (this means that e_{i} ∈ Fil_{1}L_{0} for every i = 1, …, n). Also, for every i = 1, …, m, pick j_{i} ∈ ℕ such that φ(f_{i}) ∈ Fil_{j_{i}}L_{0}, and endow L_{1} with the good (A, Fil_{•}A)-filtration Fil_{•}L_{1} associated with the pair (f, (j_{1}, …, j_{m})). Set L′_{i} := R(L_{i}, Fil_{•}L_{i}), and notice that L′_{i} is a free R-module of finite rank, for i = 0, 1. With these choices, φ is a map of filtered A-modules, and there follows an R-linear map of ℕ-graded R-modules R(φ)_{•} : L′_{1} → L′_{0}, whose cokernel is a ℕ-graded finitely presented R-module N_{•}. By inspecting the construction, it is easily seen that the inclusion Fil_{k}L_{0} ⊂ Fil_{k+1}L_{0} induces a K^{+}-linear map N_{k} → N_{k+1}, for every k ∈ ℕ, as well as an isomorphism of K^{+}-modules

colim_{k∈ℕ} N_{k} →^{∼} M.

On the other hand, claim 14.8.12 ensures that N_{•} admits a K^{+}-flattening sequence. We easily deduce that the same sequence is also flattening for M. □

Corollary 14.8.13. Let A be a local and locally measurable K^{+}-algebra, I ⊂ K^{+} any ideal, M a finitely generated A/IA-module, and suppose that the structure map K^{+} → A is local. Then the following conditions are equivalent :

(a) M is a flat K^{+}/I-module.

(b) For every c ∈ K^{+} such that I ⊂ c · 𝔪_{K}, the value log |c| does not break M.

(c) M is a K^{+}/I-flat finitely presented A/IA-module.

Proof. (a)⇒(b): in light of remark 14.8.7(ii), it suffices to remark that M ⊗_{K^{+}} /bK^{+} is a flat K^{+}/bK^{+}-module, for every b ∈ K^{+} such that I ⊂ K^{+}b.

(b)⇒(c): Let us write M as the colimit of a filtered system (M_{λ} | λ ∈ Λ) of finitely presented A-modules, with surjective transition maps. Then each M_{λ} is a coherent A-module, and A/𝔪_{K}A is noetherian (remark 14.8.4(iv)), so we may also assume that the induced maps φ_{λ} : M_{λ}/𝔪_{K}M_{λ} → M/𝔪_{K}M are isomorphisms for every λ ∈ Λ. In view of the commutative diagram

\xymatrix{M_{λ} ⊗_{K^{+}} κ \ar[r]^{c_{M_{λ}}} \ar[d]_{φ_{λ}} & cM_{λ} ⊗_{K^{+}} κ \ar[d] \\ M ⊗_{K^{+}} κ \ar[r]^{c_{M}} & cM ⊗_{K^{+}} κ}

it easily follows that log |c| does not break M_{λ}, for any λ ∈ Λ and any c ∈ K^{+} such that I ⊂ c·𝔪_{K}. Thus, if (b_{0}, …, b_{n}) is the minimal flattening sequence for M_{λ}, we see that b_{1} ∈ I (lemma 14.8.8(ii)), and therefore M_{λ}/IM_{λ} is a flat K^{+}/I-module for every λ ∈ Λ. Now, for λ, μ ∈ Λ with μ ≥ λ, let ψ_{λμ} : M_{λ} → M_{μ} be the transition map, and set N_{λμ} := Ker(ψ_{λμ} ⊗_{K^{+}} K^{+}/I); it follows that N_{λμ} ⊗_{K^{+}} κ is the kernel of ψ_{λμ} ⊗_{K^{+}} κ. But the latter map is an isomorphism, since the same holds for φ_{λ} and φ_{μ}. By Nakayama’s lemma, we deduce that N_{λμ} = 0, therefore M = M_{λ}/IM_{λ} for any λ ∈ Λ, whence (c).

Lastly, (c)⇒(a) is obvious. □

Proposition 14.8.14. Let A be a locally measurable K^{+}-algebra. Suppose that (a) A/𝔪_{K}A is a noetherian ring.

(b) The Jacobson radical of A contains 𝔪_{K}A. Then we have :

(i) A ⊗_{K^{+}} K is a noetherian ring.

(ii) Every finitely generated K^{+}-flat A-module is finitely presented.

(iii) If the valuation of K is discrete, A is noetherian.

Proof. For any K^{+}-algebra B, and any ideal I ⊂ B_{K} := B ⊗_{K^{+}} K, let us set I^{sat} := Ker(B → B_{K}/I), and notice that I^{sat} ⊗_{K^{=}} K = I. To begin with, we remark :

Claim 14.8.15. Let B be a measurable K^{+}-algebra, I ⊂ B_{K} an ideal. Then I^{sat} is a finitely generated ideal of B.

Proof of the claim. Clearly B/I^{sat} is a K^{+}-flat finitely generated B-module, hence it is finitely presented, by proposition 14.8.1; now the claim follows from [75, Lemma 2.3.18(ii)]. ♢

(i): Suppose (I_{k} | k ∈ ℕ) is an increasing sequence of ideals of A ⊗_{K^{+}} K; assumption (a) implies that there exists n ∈ ℕ such that the images of I_{n}^{sat} and I_{m}^{sat} agree in A/𝔪_{K}A, for every m ≥ n. This means that

(14.8.16)  (I_{m}^{sat}/I_{n}^{sat}) ⊗_{K^{+}} κ = 0  for every m ≥ n.

Next, for any geometric point ξ of Spec A, let A_{ξ}^{sh} be the strict henselization of A at ξ; since the natural map A → A_{ξ}^{sh} is flat, I_{k} ⊗_{A} A_{ξ}^{sh} is an ideal of A_{ξ}^{sh} ⊗_{K^{+}} K, and clearly I_{k}^{sat} ⊗_{A} A_{ξ}^{sh} = (I_{k} ⊗_{A} A_{ξ}^{sh})^{sat}, for every k ∈ ℕ. Especially I_{k}^{sat} ⊗_{A} A_{ξ}^{sh} is a finitely generated ideal of A_{ξ}^{sh}, by claim 14.8.15. In view of (14.8.16), and assumption (b), Nakayama’s lemma then says that (I_{m}^{sat}/I_{n}^{sat}) ⊗_{A} A_{ξ}^{sh} = 0 for every m ≥ n. Since ξ is arbitrary, we conclude that I_{m}^{sat}/I_{n}^{sat} = 0 for every m ≥ n, therefore the sequence (I_{k} | k ∈ ℕ) is stationary.

(ii): Let M be a K^{+}-flat and finitely generated A-module, pick an A-linear surjection φ : A^{⊕n} → M, and set N := Ker φ. Since M is K^{+}-flat, N/𝔪_{K}N is the kernel of φ ⊗_{K^{+}} κ, and assumption (a) implies that N/𝔪_{K}N is a finitely generated A/𝔪_{K}A-module. Hence we may find a finitely generated A-submodule N′ ⊂ N such that N = N′ + 𝔪_{K}N. For any geometric point ξ of Spec A, define A_{ξ}^{sh} as in the foregoing; by proposition 14.8.1 (and by [75, Lemma 2.3.18(ii)]), N ⊗_{A} A_{ξ}^{sh} is a finitely generated A_{ξ}^{sh}-module; then (b) and Nakayama’s lemma imply that N′ ⊗_{A} A_{ξ}^{sh} = N ⊗_{A} A_{ξ}^{sh}. Since ξ is arbitrary, it follows that N = N′; especially, M is finitely presented, as stated.

(iii): It suffices to show that every prime ideal of A is finitely generated ([126, Th.3.4]). However, let 𝔭 ⊂ A be such a prime ideal, and fix a generator t of 𝔪_{K}. Suppose first that t ∈ 𝔭; in that case (a) implies that 𝔭/tA is a finitely generated ideal of A/tA, so then clearly 𝔭 is finitely generated as well. Next, in case t ∉ 𝔭, the quotient A/𝔭 is a K^{+}-flat finitely generated A-module, hence it is finitely presented (proposition 14.8.1), so again 𝔭 is finitely generated ([75, Lemma 2.3.18(ii)]). □

Theorem 14.8.17. Let A be a locally measurable K^{+}-algebra, M a finitely presented Amodule, and suppose that :

(a) The valuation of K has finite rank.

(b) For every t ∈ Spec K^{+}, the ring A ⊗_{K^{+}} κ(t) is noetherian. Then we have :

(i) M admits a K^{+}-flattening sequence.

(ii) Ass M is a finite set.

Proof. (i): Set X := Spec A, S := Spec K^{+}, let f : X → S be the structure morphism, and denote by ℳ the quasi-coherent 𝒪_{X}-module associated with M. Also, for every quasi-coherent 𝒪_{X}-module ℱ and every c ∈ K^{+}, let c_{ℱ} : ℱ → cℱ be the unique 𝒪_{X}-linear morphism such that c_{ℱ} (U) = c_{ℱ (U)} for every affine open subset U ⊂ X (notation of definition 14.8.6(ii)). For every x ∈ X, set K_{f(x)}^{+} := 𝒪_{S,f(x)}; then K_{f(x)}^{+} is a valuation ring whose valuation we denote | · |_{f(x)}, and 𝒪_{X,x} is a locally K_{x}^{+}-measurable algebra. By proposition 14.8.11, the 𝒪_{X,x}-module ℳ_{x} admits a K_{f(x)}^{+}-flattening sequence (b_{x,0}, …, b_{x,n}); we wish to show that there exists an open neighborhood U(x) of x in X, such that (b_{x,i}ℳ /b_{x,i+1}ℳ)_{y} is a K^{+}/b_{x,i}^{−1}b_{x,i+1}K^{+}-flat module, for every y ∈ U(x) and every i = 0, …, n − 1. To this aim, we remark :

Claim 14.8.18. Let 𝒩 be a quasi-coherent 𝒪_{X}-module of finite type, x ∈ X a point, and b ∈ K^{+} such that 𝒩_{x} is a flat K^{+}/bK^{+}-module. Then there exists an open neighborhood U ⊂ X of x such that the map

c_{𝒩 ,y} ⊗_{K^{+}} κ(f(y)) : 𝒩_{y} ⊗_{K^{+}} κ(f(y)) → c𝒩_{y} ⊗_{K^{+}} κ(f(y))

is an isomorphism for every y ∈ U and every c ∈ K^{+} with log |c|_{f(y)} < log |b|_{f(y)}.

Proof of the claim. Let z ∈ X(x) be any point, and set t := f(z); under our assumptions, 𝒩_{z} is a flat K^{+}/bK^{+}-module. Therefore, the sequence of 𝒪_{X,z}-modules

0 → c^{−1}b𝒩_{z} ⟶^{j} 𝒩_{z} ⟶^{c_{𝒩 ,z}} c𝒩_{z} → 0

is exact, for every c ∈ K^{+} with log |c| < log |b|, and a fortiori, whenever log |c|_{t} < log |b|_{t}. If the latter inequality holds, the map j ⊗_{K^{+}} κ(t) vanishes, hence c_{𝒩 ,z} ⊗_{K^{+}} κ(t) is injective.

Let now y ∈ X be any point; set u := f(y), pick c ∈ K^{+} with log |c|_{u} < log |b|_{u}, set 𝒦 (u) := Ker(c_{𝒩} ⊗_{K^{+}} κ(u)), and suppose that 𝒦 (u)_{y} ≠ 0. The foregoing shows that in this case y ∉ X(x). On the other hand, by assumption f^{−1}(u) is a noetherian scheme, and 𝒩 (u) := 𝒩 ⊗_{K^{+}} κ(u) is a coherent 𝒪_{f^{−1}(u)}-module, therefore Ass 𝒩 (u) is a finite set ([126, Th.6.5(i)]). In view of proposition 10.5.6(ii), we conclude that Ass 𝒦 (u) is contained in the finite set Ass 𝒩 (u)\X(x). Let Z be the topological closure in X of the (finite) set ⋃_{u∈S} Ass 𝒩 (u)\X(x); taking into account lemma 10.5.5(ii,iii), we see that U := X \Z will do. ♢

Fix x ∈ X and i ≤ n; taking 𝒩 := b_{x,i}ℳ /b_{x,i+1}ℳ and b := b_{x,i}^{−1}b_{x,i+1} in claim 14.8.18, and invoking the criterion of corollary 14.8.13, we obtain an open neighborhood U_{i} of x in X such that 𝒩_{y} is K^{+}/bK^{+}-flat for every y ∈ U_{i}. Clearly the subset U(x) := U_{1} ∩ ⋯ ∩ U_{n} fulfills the sought condition. Next, pick finitely many points x_{1}, …, x_{k} ∈ X and corresponding K^{+}- flattening sequences \underline{b}_{i} for ℳ_{x_{i}}, for each i = 1, …, k, such that U(x_{1})∪⋯∪U(x_{k}) = X; after reordering, the sequence (\underline{b}_{1}, …, \underline{b}_{k}) becomes K^{+}-flattening for M (see remark 14.8.7(iii)).

(ii): Let (b_{0}, …, b_{n}) be a K^{+}-flattening sequence for M; in view of proposition 10.5.6(ii), it suffices to prove that Ass b_{i}M/b_{i+1}M is a finite set, for every i = 0, …, n − 1. Taking into account remark 14.8.4(iv), we are then reduced to showing

Claim 14.8.19. Let 𝒩 be a quasi-coherent 𝒪_{X}-module, and b ∈ K^{+} any element such that 𝒩_{x} is a K^{+}/bK^{+}-flat and finitely presented 𝒪_{X,x}-module for every x ∈ X. Then Ass 𝒩 is finite.

Proof of the claim. For given x ∈ X and any geometric point ξ of X localized at x, let A^{sh} be the strict henselization of A at ξ, and B a local and essentially finitely presented K^{+}-algebra with a local and ind-étale map B → A^{sh}. We may assume that 𝒩_{x} ⊗_{A} A^{sh} descends to a finitely presented K^{+}/bK^{+}-flat B-module N_{B} ([65, Ch.IV, Cor.11.2.6.1(ii)]). Denote by \overline{x} (resp. by x_{B}) the closed point of X(ξ) = Spec A^{sh} (resp. of Spec B); in light of corollary 10.4.37 we have

(14.8.20) x ∈ Ass 𝒩 ⇔ x ∈ Ass_{𝒪_{X,x}} 𝒩_{x} ⇔ \overline{x} ∈ Ass_{A^{sh}}𝒩_{x} ⊗_{A} A^{sh} ⇔ x_{B} ∈ Ass_{B}N_{B}.

On the other hand, it is easily seen that Ass_{K^{+}}K^{+}/bK^{+} consists of only one point, namely the maximal point t of Spec K^{+}/bK^{+}. From corollary 10.5.9, we deduce :

x_{B} ∈ Ass_{B}N_{B} ⇔ x_{B} ∈ Ass_{B}N_{B} ⊗_{K^{+}} κ(t)

and by applying again repeatedly corollary 10.4.37as in (14.8.20), we see that

x_{B} ∈ Ass_{B}N_{B} ⊗_{K^{+}} κ(t) ⇔ x ∈ Ass 𝒩 ⊗_{K^{+}} κ(t).

Summing up, we conclude that Ass𝒩 = Ass𝒩 ⊗_{K^{+}} κ(t), and the latter is a finite set, by [126, Th.6.5(i)]. □

Corollary 14.8.21. Let (K, | · |) be a valued field, and A a locally measurable K^{+}-algebra fulfilling conditions (a) and (b) of theorem 14.8.17. Let also M be a finitely generated Amodule, and 𝔭 ⊂ A any prime ideal such that M_{𝔭} is a finitely presented A_{𝔭}-module. We have:

(i) There exists f ∈ A\𝔭 such that M_{f} is a finitely presented A_{f}-module.

(ii) A is coherent.

Proof. (i): We may find a finitely presented A-module M′ with an A-linear surjection φ : M′ → M such that φ_{𝔭} is an isomorphism. Set M′′ := Ker φ; it follows that Ass M′′ ⊂ Ass M′ \ Spec A_{𝔭}. However, Ass M′ is a finite set (theorem 14.8.17(ii)); therefore the support of Ker φ is contained in a closed subset of X := Spec A that does not contain 𝔭, and the assertion follows.

(ii): Let M be a finitely presented A-module, M′ ⊂ M a finitely generated submodule, and let ℳ′ be the quasi-coherent 𝒪_{X}-module associated with M′. By remark 14.8.4(iv), we know that ℳ′_{x} is a finitely presented 𝒪_{X,x}-module for every x ∈ X; from (i), it follows that ℳ′ is a finitely presented 𝒪_{X}-module, and the assertion follows easily. □

In view of theorem 14.8.17 and corollary 14.8.21, it is interesting to have criteria ensuring that the fibres over Spec K of the spectrum of a locally measurable K^{+}-algebra are noetherian. We present two results in this direction. To state them, let us make first the following :

Definition 14.8.22. Let f : X → Y be a morphism of schemes.

(i) We say that f is absolutely flat if the following holds. For every geometric point ξ of X, the induced morphism f_{ξ} : X(ξ) → Y (f(ξ)) is an isomorphism. (ii) We say that f has finite fibres if the set f^{−1}(y) is finite, for every y ∈ Y. (iii) We say that Y admits a geometrically unibranch stratification if every irreducible

closed subset Z of Y contains a subset U ≠ ∅, open in Z, and geometrically unibranch.

(iv) We say that a ring homomorphism φ : A → B is absolutely flat if Spec φ is absolutely flat.

Remark 14.8.23. In practice, it is often the case that a noetherian scheme admits a geometrically unibranch stratification; for instance, this holds (essentially by definition) for the spectrum of a quasi-excellent noetherian ring (see definition 9.7.3(ii)).

Lemma 14.8.24. Let f : X → Y be an absolutely flat morphism of schemes. Then 𝕃_{X/Y} ≃ 0 in D(𝒪_{X}-Mod).

Proof. We easily reduce to the case where both X and Y are local schemes. Let ξ be a geometric point of X localized at the closed point; then 𝕃_{X(ξ)/X} ≃ 0 in D(𝒪_{X(ξ)}-Mod) ([75, Th.2.5.37]). Hence, by transitivity ([75, Th.2.5.33]), we are reduced to showing that 𝕃_{X(ξ)/Y} ≃ 0 in D(𝒪_{X(ξ)}-Mod). The latter assertion assertion holds, again by virtue of [75, Th.2.5.37], since f_{ξ} is an isomorphism. □

Proposition 14.8.25. Let φ : A → B be a local and absolutely flat morphism of local rings. Assume that either one of the following two conditions holds :

(a) A is a normal local domain.

(b) B is a henselian local ring. Then φ is an ind-étale ring homomorphism.

Proof. Pick a geometric point ξ of Spec B localized at the closed point, and let ξ′ denote the image of ξ in Spec A. Let also A^{sh} (resp. B^{sh}) be the strict henselization of A at ξ′ (resp. of B at ξ). Notice first that A is a normal local domain if and only if the same holds for B ([66, Ch.IV, Prop.18.8.12(i)]). In case (a) holds, let F_{A} (resp. F_{B}) be the field of fractions of A (resp. B). We remark :

Claim 14.8.26. Let R be any normal local domain, and R^{sh} the strict henselization of R at a geometric point localized at the closed point of Spec R. We have :

(i) R = R^{sh} ∩ Frac R (where the intersection takes place in Frac R^{sh}).

(ii) For every field extension F of Frac R contained in Frac R^{sh}, the R-algebra F ∩ R^{sh} is ind-étale.

Proof of the claim. (i): More generally, let C → D be any faithfully flat ring homomorphism, and denote by Frac C (resp. Frac D) the total ring of fractions of C (resp. of D); then it is easily seen that C = D ∩ Frac C, where the intersection takes place in Frac D : the proof shall be left as an exercise for the reader.

(ii): We easily reduce to the case where F is a finite extension of F_{R} := Frac R. Then, let E be a finite Galois extension of F_{R}, contained in a fixed separable closure F^{sep} of F_{R}^{sh} := Frac R^{sh}, and containing F; also, denote by R^{ν} (resp. R_{E}^{ν}) the integral closure of R in F^{sep} (resp. in E). Recall that there exists a unique maximal ideal 𝔭^{sep} ⊂ R^{ν} lying over the maximal ideal of R, such that R^{sh} = (R^{ν})_{𝔭^{sep}}^{I}, where I ⊂ Gal(F^{sep}/F_{R}) is the inertia subgroup associated with 𝔭^{sep} ([141, Ch.X, §2, Th.2]); therefore, F_{R}^{sh} = (F^{sep})^{I}. On the other hand, F = E^{H} for a subgroup H ⊂ Gal(E/F_{R}), and recall that the natural surjection Gal(F^{sep}/F_{R}) → Gal(E/F_{R}) maps I onto the inertia subgroup I_{E} associated with the maximal ideal 𝔭_{E} := 𝔭^{sep} ∩ E ⊂ R_{E}^{ν}. It follows that H contains I_{E}. Set 𝔭_{F} := 𝔭_{E} ∩ F; then R_{F} := (R_{E}^{ν} ∩ F)_{𝔭_{F}} is a faithfully flat and essentially étale R-algebra ([141, Ch.X, §1, Th.1(1)]). Especially, R^{sh} is also a strict henselization of R_{F}, and notice that Frac R_{F} = F, so the assertion follows from (i). ♢

By assumption, φ extends to an isomorphism of local domains A^{sh} →^{∼} B^{sh}; in view of claim 14.8.26(i), we see that B = F_{B} ∩ A^{sh}, and then the proposition follows from claim 14.8.26(ii).

Next, suppose assumption (b) holds. Then, φ extends uniquely to a local ring homomorphism φ^{h} : A^{h} → B from the henselization of A ([66, Ch.IV, Th.18.6.6(ii)]); clearly φ^{h} is still absolutely flat, hence we may replace A by A^{h} and assume that both A and B are henselian. Since φ is absolutely flat, the residue field extension κ(A) → κ(B) is separable and algebraic; then for every finite extension E of κ(A) contained in κ(B) there exists a finite étale local Aalgebra A_{E}, unique up to isomorphism, whose residue field is E ([66, Ch.IV, Prop.18.5.15]). The natural map κ(B) → E ⊗_{κ(A)} κ(B) admits a well defined section, given by the multiplication in κ(B), and the latter extends to a section s_{E} of the induced finite étale ring homomorphism B → A_{E} ⊗_{A} B ([66, Ch.IV, Th.18.5.11]). The composition of s_{E} with the natural map A_{E} → A_{E} ⊗_{A} B is a ring homomorphism A_{E} → B that extends φ. The construction is clearly compatible with inclusion of subextensions E ⊂ E′, hence let A′ be the colimit of the system (A_{E} | E ⊂ κ(B)); summing up, φ extends to a local and absolutely flat map A′ → B. We may therefore replace A by A′ and assume from start that κ(A) = κ(B). In this case, set B′ := A^{sh}⊗_{A}B, and notice that κ(A^{sh})⊗_{κ(A)}κ(B) = κ(A^{sh}), so that B′ is a local ring, ind-étale, faithfully flat over B, and with separably closed residue field. Since B is already henselian, it follows that B′ is a strict henselization of B, and therefore the induced map A^{sh} → B′ is an isomorphism; by faithfully flat descent, we deduce that φ is already an isomorphism, and the proof of the proposition is concluded. □

Lemma 14.8.27. Let A be a locally measurable K^{+}-algebra, and set X := Spec A. Let also f : X → Y be a morphism of K^{+}-schemes with Ω_{X/Y} = 0, and Y a finitely presented K^{+}- scheme. Then f_{ξ} is a finitely presented closed immersion, for every geometric point ξ of X.

Proof. Fix such a geometric point ξ; by definition, we may find a local K^{+}-scheme Z essentially finitely presented, and a pro-étale morphism g : X(ξ) → Z of K^{+}-schemes. We may also assume that the morphism X(ξ) → Y deduced from f factors through g, in which case the resulting morphism h : Z → Y is essentially finitely presented. Under the current assumptions, Ω_{X(ξ)/Y} = 0, and then lemma 14.8.24 easily implies that Ω_{Z/Y} vanishes as well, so h is essentially unramified, and therefore it factors as the composition of a finitely presented closed immersion Z → Z′ followed by an essentially étale morphism Z′ → Y ([66, Ch.IV, Cor.18.4.7]). The lemma is an immediate consequence. □

Proposition 14.8.28. Let A be a local, henselian, and locally measurable K^{+}-algebra, whose structure map K^{+} → A is local. Then A is a measurable K^{+}-algebra.

Proof. Pick a local and essentally finitely presented K^{+}-algebra B and a local and ind-étale map φ : B → A of K^{+}-algebras. Let ξ be a geometric point of Spec A localized at the closed point, and denote by ξ′ the image of ξ in Spec B; also, let A^{sh} (resp. B^{sh}) be the strict henselization of A at ξ (resp. of B at ξ′). Then Ω_{A/B} = 0, hence the induced map φ^{sh} : B^{sh} → A^{sh} is a finitely presented surjection (lemma 14.8.27), especially I := Ker φ^{sh} is a finitely generated ideal, and it follows as well that the residue field extension κ(B) → κ(A) is separable and algebraic. Let B^{h} be the henselization of B, and for every field extension E of κ(B) contained in κ(A)^{sep} = κ(B)^{sep}, let B_{E}^{h} denote the local ind-étale B^{h}-algebra – determined up to isomorphism – whose residue field is E ([66, Ch.IV, Prop.18.5.15]). We may find a finite Galois extension E of κ(B) such that I descends to a finitely generated ideal I_{E} ⊂ B_{E}^{h}. For any automorphism σ ∈ G := Gal(κ(B)^{sep}/κ(B)), let \overline{σ} ∈ G_{E} := Gal(E/κ(B)) be the image of σ. Recall that G (resp. G_{E}) is the group of automorphisms of the B^{h}-algebra B^{sh} (resp. B_{E}^{h}). With this notation, we have the identity

\overline{σ}(I_{E})B^{sh} = σ(I) = I for every σ ∈ G

and since B^{sh} is a faithfully flat B^{h}-algebra, it follows that I_{E} is invariant under the action of G_{E}; by Galois descent, we conclude that I descends to a finitely generated ideal I_{0} ⊂ B^{h}. Set C := B^{h}/I_{0}, and let C^{sh} denote the strict henselization of C at (the unique lifting of) the geometric point ξ′. Since A is henselian, φ extends uniquely to a local homomorphism φ^{h} : B^{h} → A ([66, Ch.IV, Th.18.6.6(ii)]), and it is easily seen that φ^{h} factors through C. By construction, the resulting map C → A is absolutely flat, so the assertion follows from proposition 14.8.25. □

Lemma 14.8.29. Let X be a noetherian scheme that admits a geometrically unibranch stratification, and ℱ a coherent 𝒪_{X_{ét}}-module. Then there exists a partition

X = X_{1} ∪ ⋯ ∪ X_{k}

of X into finitely many disjoint irreducible locally closed subsets, such that the following holds. For every i = 1, …, k, every geometric point \overline{x} of X_{i}, and every generization \overline{u} of \overline{x} in X_{i}, every strict specialization map

s_{\overline{x},\overline{u}} : ℱ\overline{x} → ℱ\overline{u}

is injective (see (4.9.23)).

Proof. Arguing by noetherian induction, it suffices to show that every reduced and irreducible closed subscheme W of X contains a subset U ≠ ∅ that is open in W, and such that every strict specialization map s_{\overline{x},\overline{u}} with \overline{x}, \overline{u} localized in U, is injective. However, W contains an open subset U ≠ ∅ such that

(a) U is geometrically unibranch

(b) U is affine and irreducible

(c) ℱ is normally flat along W at every point of U (see [64, Ch.IV,§6.10.1]). Indeed, (a) holds by assumption; (b) can be easily arranged by shrinking U, since X is noetherian. Lastly, (c) follows from [64, Ch.IV, Prop.6.10.2]. We claim that such U will do. Indeed, let i : W → W be the closed immersion; set

ℐ := Ker(𝒪_{X} → i_{∗}𝒪_{W}) ℛ^{•} := ⨁_{n∈ℕ} ℐ^{n}ℱ /ℐ^{n+1}ℱ

and recall that condition (c) means that ℛ_{u}^{•} is a flat 𝒪_{W,u}-module, for every u ∈ U. Now, for a given s_{\overline{x},\overline{u}} as in the foregoing, denote by x (resp. u) the support of \overline{x} (resp. of \overline{u}) and define descending filtrations by the rule :

Fil^{k}ℱ\overline{x} := ℐ_{x}^{k} · ℱ\overline{x}  Fil^{k}ℱ\overline{u} := ℐ_{u}^{k} · ℱ\overline{u}  for every k ∈ ℕ.

By [126, Th.8.9], both these filtrations are separated, and obviously s_{\overline{x},\overline{u}} is a map of filtered modules; thus, it suffices to show that the induced maps gr^{k}ℱ\overline{x} → gr^{k}ℱ\overline{u} of associated graded modules are injective, for every k ∈ ℕ. However, notice the natural identifications :

gr^{•}ℱ\overline{x} = ℛ_{x}^{•} ⊗_{𝒪_{W,x}} 𝒪_{W_{ét},\overline{x}} gr^{•}ℱ\overline{u} = ℛ_{x}^{•} ⊗_{𝒪_{W,x}} 𝒪_{W_{ét},\overline{u}}.

Then, the normal flatness condition reduces to checking that the induced strict specialization map 𝒪_{W_{ét},\overline{x}} → 𝒪_{W_{ét},\overline{u}} is injective; the latter assertion holds by the following :

Claim 14.8.30. Let W be a reduced, irreducible scheme, \overline{w} a geometric point of W, and suppose that W is unibranch at the support w of \overline{w}. Then, for every generization \overline{u} of \overline{w} in W, every strict specialization map 𝒪_{W_{ét},\overline{w}} → 𝒪_{W_{ét},\overline{u}} is injective.

Proof of the claim. Let W^{ν} be the normalization of W, and \overline{w}^{ν} a geometric point of W^{ν} whose image in X is isomorphic to \overline{w}, and denote by w^{ν} the support of \overline{w}^{ν}. The assumption on w means that the induced morphism W^{ν}(w^{ν}) → W (w) is integral, and the residue field extension κ(w) → κ(w^{ν}) is radicial, hence the natural morphism of W^{ν}(\overline{w}^{ν})-schemes

W (\overline{w}) ×_{W (w)} W^{ν}(w^{ν}) → W^{ν}(\overline{w}^{ν})

is an isomorphism ([66, Ch.IV, Prop.18.8.10]). Since W^{ν}(\overline{w}^{ν}) is a normal local scheme ([66, Prop.18.8.12(i)]), it follows easily that W (\overline{w}) is reduced and irreducible. However, any specialization map is the composition of a localization map, followed by a local ind-étale map of local rings, whence the claim. □

14.8.31. Now, consider a ring homomorphism A → B, with A noetherian. Set Y := Spec A, X := Spec B, and denote by f : X → Y the associated morphism of affine schemes.

Lemma 14.8.32. In the situation of (14.8.31), suppose moreover that f is absolutely flat. Then the following conditions are equivalent :

(a) B is noetherian.

(b) f has finite fibres.

Proof. Let ξ be any geometric point of X, and x (resp. y) the support of ξ (resp. of f(ξ)).

(a)⇒(b): B(y) := B ⊗_{A} κ(y) is a noetherian ind-étale κ(y)-algebra (proposition 14.8.25), so of dimension ≤ 0, and then Spec B(y) is a finite set (proposition 8.1.4). Since x is arbitrary,

(b) follows.

(b)⇒(a): Let I ⊂ B be any ideal, and denote by ℐ the associated quasi-coherent 𝒪_{X}-module. We need to show that ℐ is an 𝒪_{X}-module of finite type. Since f_{ξ} is an isomorphism, we may find a commutative diagram of affine schemes

\xymatrix{X′ \ar[r]^{g} \ar[d]_{h} & Y′ \ar[d] \\ X \ar[r]^{f} & Y}

whose left (resp. right) vertical arrow is an étale neighborhood of ξ (resp. of f(ξ)), and a quasicoherent ideal 𝒥 ⊂ 𝒪_{Y′} such that g^{∗}𝒥 ⊂ h^{∗}ℐ and such that (g^{∗}𝒥)_{x′} = (h^{∗}ℐ)_{x′} for some x′ ∈ h^{−1}(x) (notice that g is a flat morphism). Set ℳ := 𝒪_{Y′}/𝒥, let z′ ∈ X′ an arbitrary point, and set y′ := g(z′); in light of corollary 10.4.37, we have

z′ ∈ Ass g^{∗}ℳ ⇔ z′ ∈ Ass g^{∗}ℳ_{|X′(x′)} ⇔ y′ ∈ Ass ℳ_{|Y′(y′)} ⇔ y′ ∈ Ass ℳ .

In other words, Ass g^{∗}ℳ = g^{−1}Ass ℳ. However, Ass ℳ is finite ([126, Th.6.5(i)]), hence the same holds for Ass g^{∗}ℳ, in view of (b). Now, notice that Ass h^{∗}ℐ /g^{∗}𝒥 ⊂ Ass g^{∗}ℳ \ X′(x′) (proposition 10.5.6(ii)). It follows that there exists an open neighborhood U of x′ in X′ such that (h^{∗}ℐ /g^{∗}𝒥)_{|U} = 0 (lemma 10.5.5(iii)), i.e. (g^{∗}𝒥)_{|U} = h^{∗}ℐ_{|U}; especially, h^{∗}ℐ_{|U} is an 𝒪_{U}- module of finite type. Since h is an open map, we deduce that h(U) is an open neighborhood of x in X, and ℐ_{|h(U)} is an 𝒪_{h(U)}-module of finite type. Since x is arbitrary, the lemma follows. □

14.8.33. For our second criterion, keep the situation of (14.8.31), and suppose additionally that, for every geometric point ξ of X, the induced morphism f_{ξ} : X(ξ) → Y (ξ) is a closed immersion. Under this weaker assumption, it is not necessarily true that conditions (a) and (b) of lemma 14.8.32are equivalent. For instance, we have :

Example 14.8.34. Take A := k[X], the free polynomial algebra over a given infinite field k; also, let (a_{i} | i ∈ ℕ) be a sequence of distinct elements of k. We construct an A-algebra B, as the colimit of the inductive system (B_{i} | i ∈ ℕ) of A-algebras, such that

• B_{i} := k[X] × k^{i+1} (the product of k[x] and i + 1 copies of k, in the category of rings)

• the structure map A → B_{i} is the unique map of k-algebras given by the rule : X ↦ (X, a_{0}, …, a_{i})

• the transition maps B_{i} → B_{i+1} are given by the rule : X ↦ (X, 0, …, 0, a_{i+1}) and (0, e_{i}) ↦ (0, e_{i}, 0) for i = 0, …, i. (Here e_{0}, …, e_{i} is the standard basis of the k-vector space k^{i+1}.)

Then one can check that X := Spec B = Spec k[X] ∪ ℕ, and ℕ is an open subset of X with the discrete topology. Moreover, the induced map Spec B → Y := Spec A restricts to the continuous map ℕ → Y given by the rule i ↦ 𝔭_{i}, for every i ∈ ℕ, where 𝔭_{i} is the prime ideal generated by X − a_{i}. It follows easily that the condition of (14.8.33) is fulfilled; nevertheless, clearly X has infinitely many maximal points, hence its underlying topological space is not noetherian, and a fortiori, B cannot be noetherian.

However, we have the following positive result : Proposition 14.8.35. In the situation of (14.8.33), suppose additionally that Y admits a geometrically unibranch stratification. Then the following conditions are equivalent :

(a) B is noetherian.

(b) The topological space underlying X is noetherian.

(c) For every geometric point ξ of X there exist a neighborhood X′ of ξ in X_{ét}, an unramified Y -scheme Y′, and an absolutely flat morphism X′ → Y′ of Y -schemes, with finite fibres.

Proof. Obviously (a)⇒(b).

(c)⇒(a): Indeed, under assumption (c), we may find finitely many geometric points ξ_{1}, …, ξ_{k} of X, and for every i = 1, …, k, a neighborhood X′_{i} of ξ_{i} in X_{ét}, and a Y -morphism X′_{i} → Y′_{i} with the stated properties, such that moreover, the family (X′_{i} | i = 1, …, k) is an étale covering of X. Furthermore, we may assume that X′_{i} and Y′_{i} are affine for every i ≤ k. In this case, lemma 14.8.32shows that every X′_{i} is noetherian, and then the same holds for X.

(b)⇒(c): Fix a geometric point ξ of X, and let B_{ξ}^{sh} (resp. A_{ξ}^{sh}) denote the strict henselization of B at ξ (resp. of A at f(ξ)); by assumption, we may find a (finitely generated) ideal I ⊂ A_{ξ}^{sh} such that f_{ξ} induces an isomorphism A_{ξ}^{sh}/I →^{∼} B_{ξ}^{sh}. Then we may find an affine étale neighborhood Y′ of f(ξ), say Y′ := Spec A′ for some étale A-algebra A′, and an ideal I′ ⊂ A′ such that I′A_{ξ}^{sh} = I. Next, we may find an affine étale neighborhood X′ := Spec B′ of ξ such that f_{ξ} extends to a morphism g : X′ → Y′, and we may further suppose that the corresponding ring homomorphism A′ → B′ factors through A′/I′, so g factors through a morphism

h : X′ → Z := Spec A′/I′

and the closed immersion Z → Y′. By construction, ξ lifts to a geometric point ξ′ of X′, and h_{ξ′} : X′(ξ′) → Z(h(ξ′)) is an isomorphism ([66, Ch.IV, Prop.18.8.10]). To conclude the proof, it then suffices to exhibit an open subset U ⊂ X′ containing the support of ξ′, and such that the restriction U → Z of h is absolutely flat with finite fibres. Claim 14.8.36. Let φ : W → W′ be a quasi-finite, separated, dominant and finitely presented morphism of reduced, irreducible schemes, and suppose that W′ contains a non-empty geometrically unibranch open subset. Then the same holds for W. Proof of the claim. After replacing W′ by some open subset U′ ⊂ W′, and W by φ^{−1}U′, may assume that W′ is affine and unibranch; then we may also suppose that φ is finite ([65, Ch.IV, Th.8.12.6]), in which case W is affine as well, and φ is surjective.

Let η ∈ W and η′ ∈ W′ be the respective generic points, and denote by E ⊂ κ(η) the maximal subfield that is separable over κ(η′). We may then find a reduced and irreducible scheme W′′, with generic point η′′, such that φ factors as the composition of finite surjective morphisms φ′ : W → W′′, φ′′ : W′′ → W′, and such that κ(η′′) = E. By virtue of [65, Ch.IV, Th.8.10.5] and [66, Ch.IV, Prop.17.7.8(ii)], we may then replace W′ by a non-empty open subset, and assume that φ′ is radicial, and φ′′ is étale. Then W′′ is geometrically unibranch ([64, Ch.IV, Prop.6.15.10]); hence, we may replace W′ by W′′, and reduce to the case where φ is radicial. Let p be the characteristic of κ(η′); if p = 0, φ is birational, in which case the assertion follows from [64, Prop.6.15.5(ii)]. In case p > 0, write W = Spec C, W′ = Spec C′; the induced ring homomorphism C′ → C is finite and injective, and we have C^{p^{n}} ⊂ C′ for n ∈ ℕ large enough. Denote by C^{ν} (resp. C′^{ν}) the normalization of the domain C (resp. of C′); it follows easily that (C^{ν})^{p^{n}} ⊂ C′^{ν}, so the morphism Spec C^{ν} → Spec C′^{ν} is radicial. On the other hand, since W′ is geometrically unibranch, the normalization morphism Spec C′^{ν} → W′ is radicial ([63, Ch.0, Lemme 23.2.2]) and therefore the normalization map Spec C^{ν} → W is radicial as well ([64, Ch.IV, Lemme 6.15.3.1(i)]). Then W is geometrically unibranch, again by [63, Ch.0, Lemme 23.2.2]. ♢

From claim 14.8.36and our assumption on Y, it follows easily that Z admits a geometrically unibranch stratification. The morphism h induces as usual a morphism of étale topoi

(14.8.37)  \xymatrix{(X′)_{ét}^{∼} \ar[r]^{h_{∗}} & Z_{ét}^{∼} \ar[l]^{h^{∗}}}

as well as a morphism h^{♮} : h^{∗}𝒪_{Z_{ét}} → 𝒪_{X′_{ét}} of (X′)_{ét}^{∼}-rings. By construction, for every geometric point τ of X′, the induced map on stalks h_{τ}^{♮} : 𝒪_{Z_{ét},h(τ)} → 𝒪_{X′_{ét},τ} is surjective, and h_{ξ′}^{♮} is a bijection. It follows easily that h_{τ}^{♮} is also bijective for every generization τ of ξ′. Now, choose a partition Z = Z_{1} ∪ ⋯ ∪ Z_{k} as in lemma 14.8.29 (with ℱ := 𝒪_{Z_{ét}}), and for given i ≤ k, suppose that τ and η are two geometric points of h^{−1}Z_{i}, with η a generization of τ. The choice of a strict specialization morphism X′(η) → X′(τ) yields a commutative diagram

(14.8.38)  \xymatrix{𝒪_{Z_{ét},h(τ)} \ar[r]^{h_{τ}^{♮}} \ar[d] & 𝒪_{X′_{ét},τ} \ar[d] \\ 𝒪_{Z_{ét},h(η)} \ar[r]^{h_{η}^{♮}} & 𝒪_{X′_{ét},η}}

whose vertical arrows are strict specialization maps (see remark 4.9.26(i)); in light of lemma 14.8.29, we deduce that h_{τ}^{♮} is injective, whenever the same holds for h_{η}^{♮}. Now, let x′ ∈ X′ be the support of ξ′, and Σ_{i} the set of maximal points of h^{−1}Z_{i}. Condition (b) and proposition 8.1.65 imply that h^{−1}Z_{i} is a noetherian topological space, hence Σ_{i} is a finite set; it follows that the topological closure W of ⋃_{i=1}^{k} Σ_{i} \ X′(x′) in X′ is a closed subset that does not contain x′; by construction, h^{♮} restricts to a monomorphism on U := X′ \ W, i.e. the restriction h_{U} : U → Z of h is absolutely flat, as required. It also follows that the fibres of h_{U} are noetherian topological spaces of dimension zero (cp. the proof of lemma 14.8.32), hence they are finite, and the proof is complete. □

Lemma 14.8.39. Suppose that the valuation of K has finite rank, and let X be a finitely presented K^{+}-scheme, ℱ a coherent 𝒪_{X_{ét}}-module. Then there exists a partition

X = X_{1} ∪ ⋯ ∪ X_{k}

of X into finitely many disjoint irreducible locally closed subsets, such that the following holds. For every i = 1, …, k, every geometric point \overline{x} of X_{i}, and every generization \overline{u} of \overline{x} in X_{i}, every strict specialization map

s_{\overline{x},\overline{u}} : ℱ\overline{x} → ℱ\overline{u}

is injective (see (4.9.23)).

Proof. We easily reduce to the case where X is affine, say X = Spec A, and then ℱ is the coherent 𝒪_{X_{ét}}-module arising from a finitely presented A-module M. By theorem 14.8.17(i), M admits a K^{+}-flattening sequence (b_{0}, …, b_{n}). Then we are further reduced to showing the assertion for the subquotients b_{i}M/b_{i+1}M (that are finitely presented, by corollary 14.8.21(ii)). So, we may assume from start that f : X → S_{0} := Spec K^{+}/bK^{+} is a finitely presented morphism for some b ∈ K^{+}, and ℱ is f-flat. For every t ∈ S_{0}, let

i_{t} : X_{t} := f^{−1}(t) → X

be the locally closed immersion; since X_{t} is an excellent noetherian scheme, we may apply lemma 14.8.29and remark 14.8.23to produce a partition X_{t} = X_{t,1}∪⋯∪X_{t,k} by finitely many disjoint irreducible locally closed subsets such that, for every i = 1, …, k, every geometric point \overline{x} of X_{t,i} and every generization \overline{u} of \overline{x} in X_{t,i}, every specialization map

(i_{t}^{∗}ℱ)\overline{x} → (i_{t}^{∗}ℱ)\overline{u}

is injective. Since |S_{0}| is a finite set, the lemma will then follow from :

Claim 14.8.40. Let g : Y → T be any finitely presented morphism of schemes, 𝒢 a finitely presented, quasi-coherent g-flat 𝒪_{Y} -module, and t ∈ T. Let also \overline{y}, \overline{u} be two geometric points of g^{−1}(t), such that \overline{u} is a generization of \overline{y}. Let s_{\overline{y},\overline{u}} : 𝒢\overline{y} → 𝒢\overline{u} be a strict specialization map, and suppose that s_{\overline{y},\overline{u}} ⊗_{𝒪_{T,t}} κ(t) is injective. Then the same holds for s_{\overline{y},\overline{u}}.

Proof of the claim. Set Y′ := Y (\overline{y}), denote by j : Y′ → Y the natural morphism, and set 𝒢′ := j^{∗}𝒢. The map s_{\overline{y},\overline{u}} is deduced from a morphism Y (\overline{u}) → Y′ of Y -schemes; the latter factors through a faithfully flat morphism Y (\overline{u}) → Y′(u), where u ∈ Y′ is the image of the closed point of Y (\overline{u}). Hence, s_{\overline{y},\overline{u}} is the composition of the specialization map s′ : 𝒢 ′ ; \overline{y} → 𝒢′_{u}, and the injective map 𝒢′_{u} → 𝒢 ′ ; \overline{u}. Our assumption implies that s′ ⊗_{𝒪_{T,t}} κ(t) is injective, and it suffices to show that the same holds for s′. However, Y′ is the limit of a cofiltered system (j_{λ} : Y_{λ} → Y | λ ∈ Λ) of local, essentially étale Y -schemes. Write 𝒢_{λ} := j_{λ}^{∗}𝒢 for every λ ∈ Λ, and notice that the transition morphisms Y_{λ} → Y_{μ} are faithfully flat, for every λ ≥ μ; it follows that 𝒢 ′ ; \overline{y} is the filtered union of the system of modules (G_{λ} := Γ(Y_{λ}, 𝒢_{λ}) | λ ∈ Λ) (proposition 10.1.10(i)); likewise, 𝒢 ′ ; \overline{y} ⊗_{𝒪_{T,t}} κ(t) is the filtered union of the submodules (G_{λ} ⊗_{𝒪_{T,t}} κ(t) | λ ∈ Λ). We are then reduced to checking that all the restrictions s_{λ} : G_{λ} → 𝒢′_{u} of s′ are injective, and we know already that s_{λ} ⊗_{𝒪_{T,t}} κ(t) is injective for every λ ∈ Λ. For every such λ, let u_{λ} ∈ Y_{λ} be the image of u; then s_{λ} factors through the injective map 𝒢_{λ,u_{λ}} → 𝒢′_{u} and the specialization map s′_{λ} : G_{λ} → 𝒢_{λ,u_{λ}}. Consequently, it suffices to show that s′_{λ} is injective, and we know already that the same holds for s′_{λ} ⊗_{𝒪_{T,t}} κ(t). However, 𝒢_{λ,u_{λ}} is a localization Q_{λ}^{−1}G_{λ}, for a multiplicative set Q_{λ} ⊂ Γ(Y_{λ}, 𝒪_{Y_{λ}}), and s′_{λ} is the localization map. The claim therefore boils down to the assertion that, for every λ ∈ Λ and every q ∈ Q_{λ}, the endomorphism q · 1_{G_{λ}} is injective on G_{λ}, and our assumption already ensures that (q · 1_{G_{λ}}) ⊗_{𝒪_{T,t}} κ(t) is injective on G_{λ} ⊗_{𝒪_{T,t}} κ(t). However, let g_{λ} : Y_{λ} → T be the morphism induced by g; by construction, 𝒢_{λ} is a g_{λ}-flat 𝒪_{Y_{λ}}-module, hence the contention follows from [65, Prop.11.3.7]. □

Proposition 14.8.41. Let (K, | · |) be a valued field, and A a locally measurable K^{+}-algebra fulfilling conditions (a) and (b) of theorem 14.8.17. The following conditions are equivalent:

(c) For every geometric point ξ of X := Spec A there exist a neighborhood U of ξ in X_{ét}, and a finitely presented K^{+}-scheme Z with an absolutely flat morphism of K^{+}-schemes X′ → Z.

(d) Ω_{A/K^{+}} is an A-module of finite type.

Proof. (c)⇒(d) follows easily from lemma 14.8.24.

(d)⇒(c): Let a_{1}, …, a_{k} ∈ A be a finite system of elements such that da_{1}, …, da_{k} generate the A-module Ω_{A/K^{+}}. We define a map of K^{+}-algebras A_{0} := K^{+}[T_{1}, …, T_{k}] → A by the rule: T_{i} ↦ a_{i} for i = 1, …, k. Clearly Ω_{A/A_{0}} = 0. Let Z_{0} := Spec A_{0}, and denote by f : X → Z_{0} the induced morphism of schemes. Let A_{ξ}^{sh} (resp. A_{0,ξ}^{sh}) be the strict henselization of A at ξ (resp. of A_{0} at f(ξ)). According to lemma 14.8.27, the induced map A_{0,ξ}^{sh} → A_{ξ}^{sh} is surjective, and its kernel is a finitely generated ideal I ⊂ A_{0,ξ}^{sh}. In this situation, we may argue as in the proof of proposition 14.8.35, to produce an affine étale neighborhood X′ of ξ, a finitely presented affine unramified Z_{0}-scheme Z, and a morphism of Z_{0}-schemes h : X′ → Z such that h_{τ} : X′(τ) → Z(h(τ)) is a closed immersion for every geometric point τ of X′, and h_{ξ′} is an isomorphism for some lifting ξ′ of ξ.

Then we consider the associated morphism of étale topoi as in (14.8.37) and the morphism h^{♮} : h^{∗}𝒪_{Z_{ét}} → 𝒪_{X′_{ét}} of (X′)_{ét}^{∼}-rings. Again, the induced map on stalks h_{τ}^{♮} is surjective for every geometric point τ of X′, and is bijective if τ is a generization of ξ′. We pick a finite partition Z = Z_{1} ∪ ⋯ ∪ Z_{k} as in lemma 14.8.39(for ℱ := 𝒪_{Z}). For any i ≤ k, let τ, η be two geometric points of h^{−1}Z_{i}, such that η is a generization of τ; by considering the commutative diagram (14.8.38), we see again that h_{τ}^{♮} is injective whenever the same holds for h_{η}^{♮}.

Now, condition (b) of theorem 14.8.17 easily implies that h^{−1}Z_{i} is a noetherian topological space, hence its set Σ_{i} of maximal points is finite. Again we let x′ ∈ X′ be the support of ξ′, and W the topological closure of ⋃_{i=1}^{k} Σ_{i} \ X′(x′) in X′, and it is easily seen that the restriction U → Z of h is absolutely flat, so (c) holds. □

14.8.42. Henceforth we restrict to the case where the value group Γ of K is not discrete and of rank one. As usual, we consider the almost structure attached to the standard setup attached to (K, | · |).

Definition 14.8.43. In the situation of (14.8.42), let A be a K^{+a}-algebra and M an A-module.

(i) We say M is an almost noetherian A-module, if every A-submodule of M is almost finitely generated.

(ii) We say that A is an almost noetherian K^{+a}-algebra, if A is an almost noetherian Amodule.

Remark 14.8.44. (i) In the situation of (14.8.42), suppose that A is an almost noetherian K^{+a}- algebra. Then the same argument as in the “classical limit” case shows that every almost finitely generated A-module M is almost noetherian. The details shall be left to the reader.

(ii) The notion of almost noetherian module goes back to the appendix of Kiehl’s article [117]; in our language, he proved in Satz 5.1 that for a rank 1 complete valuation ring V with dense value group, every finitely generated module over a ring of restricted formal series V ⟨X_{1}, ..., X_{n}⟩ (or more generally, over V -algebras of topologically finite type) is almost noetherian.

Theorem 14.8.45. Let A be a locally measurable K^{+}-algebra. Suppose that both A⊗_{K^{+}} κ and A ⊗_{K^{+}} K are noetherian rings. Then A^{a} is an almost noetherian K^{+a}-algebra.

Proof. If the valuation of K is discrete, the assertion is proposition 14.8.14(iii). Hence, we may assume that the valuation of K is not discrete. Moreover, let 𝒬 be the set of all locally measurable K^{+}-algebras B that are quotients of A, and such that B^{a} is not almost noetherian. We have to show that 𝒬 = ∅. However, for every B ∈ 𝒬 the κ-algebra \overline{B} := B ⊗_{K^{+}} κ is a quotient of the noetherian κ-algebra \overline{A} := A ⊗_{K^{+}} κ; it follows that the set \overline{𝒬} := \{\overline{B} | B ∈ 𝒬\} admits minimal elements, if it is not empty. In the latter case, we may then replace A by any quotient B ∈ 𝒬 such that \overline{B} is minimal in \overline{𝒬} , and therefore assume that for every locally measurable quotient B of A, either \overline{B} = \overline{A}, or else B^{a} is almost noetherian.

Let I ⊂ A be any ideal; we have to show that I^{a} is almost finitely generated. By assumption, the image \overline{I} of I in \overline{A} is finitely generated, and the same holds for the ideal I_{K} := I ⊗_{K^{+}} K of A_{K} := A ⊗_{K^{+}} K. Thus, we may find a finitely generated subideal I_{0} ⊂ I whose image in \overline{A} agrees with \overline{I}, and such that I_{0} ⊗_{K^{+}} K = I_{K}. After replacing A by A/I_{0} and I by I/I_{0}, we are then reduced to the case where both \overline{I} and I_{K} vanish. Let J ⊂ A denote the kernel of the localization map A → A_{K}, set S := 1 + 𝔪_{K}A, and let B := S^{−1}A × A_{K}. Clearly B is a faithfully flat A-algebra, and A/J is a K^{+}-flat A-module; therefore (A/J) ⊗_{A} B is a K^{+}-flat B-module of finite type, and since A_{K} is noetherian, proposition 14.8.14(ii) implies that (A/J)⊗_{A} B is finitely presented. Then A/J is finitely presented as well, and therefore J is a finitely generated ideal. We conclude that there exists c ∈ 𝔪_{K} such that cJ = 0. Then notice that J ∩ cA = 0 : indeed, if a ∈ J ∩ cA, we have a = cx for some x ∈ A, and ca = 0, therefore c^{2}x = 0, so x ∈ J, and consequently a = cx = 0. Now, fix b ∈ 𝔪_{K}; since the valuation of K has rank one, and since I ⊂ J, it follows that there exists n ∈ ℕ large enough, so that

I ∩ b^{n}A = 0.

Let i_{0} := max{i ∈ ℕ | I ⊂ b^{i}A}, and set

N := b^{i_{0}}A/(I + b^{i_{0}+1}A) N′ := b^{i_{0}}A/b^{i_{0}+1}A.

Let φ : N′ → N be the natural surjection, and set c_{N}^{∗} := c_{N} ∘ φ : N′ → cN for every c ∈ K^{+} (notation of definition 14.8.6(ii)).

Claim 14.8.46. There exists c ∈ K^{+} with log |c| < log |b| such that c_{N}^{∗} ⊗_{K^{+}} κ is not an isomorphism.

Proof of the claim. Suppose that the claim fails; then it is easily seen that no γ ∈ log Γ^{+} with γ < log |b| breaks N. For every geometric point ξ of Spec A, let A_{ξ}^{sh} denote the strict henselization of A at ξ; since A_{ξ}^{sh} is a flat A-algebra, we have a natural identification

cN ⊗_{A} A_{ξ}^{sh} = c(N ⊗_{A} A^{sh})

of A^{sh}-modules; therefore, no γ < log |b| breaks N ⊗_{A} A_{ξ}^{sh}. By corollary 14.8.13, we deduce that N ⊗_{A} A_{ξ}^{sh} is a K^{+}/bK^{+}-flat and finitely presented A_{ξ}^{sh}-module, for every geometric point ξ. Hence, C_{ξ} := Ker φ ⊗_{A} A_{ξ}^{sh} is a finitely generated A_{ξ}^{sh}-module, and C_{ξ} ⊗_{K^{+}} κ = 0, for every geometric point ξ. By Nakayama’s lemma, it follows that C_{ξ} = 0 for every such ξ, so finally Ker φ = 0, which means that I ⊂ b^{i_{0}+1}A, contradicting the choice of i_{0}. ♢

Let c be as in claim 14.8.46, and set d := cb^{i_{0}}; notice the natural isomorphism of \overline{A}-modules

cN ⊗_{K^{+}} κ →^{∼} \frac{dA}{I ∩ dA} ⊗_{K^{+}} κ.

Let φ : A → dA/(I ∩ dA) be the composition of d_{A} : A → dA and the projection dA → dA/(I ∩dA) (notation of definition 14.8.6(ii)); by construction, φ⊗_{K^{+}} κ is not an isomorphism, hence there exists x ∈ I ∩ dA such that the composition φ_{x} : A → dA/xA of d_{A} and the projection dA → dA/xA induces a map φ_{x} ⊗_{K^{+}} κ with non-trivial kernel. In other words, dA/xA is a cyclic module over a locally measurable quotient B of A such that the projection \overline{A} → \overline{B} is not an isomorphism, so B^{a} is almost noetherian. Set I_{0} := (I ∩ dA)/xA; then I_{0} is a submodule of dA/xA, and consequently I_{0}^{a} is an almost finitely generated B^{a}-module (remark 14.8.44(i)). Then clearly (I ∩ dA)^{a} is an almost finitely generated ideal of A^{a}. But by construction, cI ⊂ I ∩ dA, and b annihilates (I ∩ dA)/cI. Since b is arbitrary, this easily implies that I^{a} is almost finitely generated, as required. □

Corollary 14.8.47. In the situation of theorem 14.8.45, the following holds :

(i) Every almost finitely generated A^{a}-module is almost finitely presented.

(ii) Every flat almost finitely generated A^{a}-module is almost projective of finite rank.

Proof. Assertion (i) is an easy consequence of theorem 14.8.45 and remark 14.8.44(i) : the details shall be left to the reader.

(ii): Let M be a flat and almost finitely generated A^{a}-module; by (i) and [75, Prop.2.4.18(ii)], M is almost projective. It remains to show that there exists n ∈ ℕ such that Λ_{A^{a}}^{n}M = 0, or equivalently, that (Λ_{A}^{n}M_{!})^{a} = 0. However, M_{!} is a flat A-module, so Ass M_{!} ⊂ Ass A; in view of theorem 14.8.17(ii), we may argue by induction on the cardinality c of Ass A, and it suffices to check that, for every 𝔭 ∈ Ass A there exists n ∈ ℕ such that Λ_{A_{𝔭}}^{n}(M_{!})_{𝔭} = 0. If c = 0, we have A = 0, and there is nothing to prove. Suppose that c > 0 and the assertion is known for every locally measurable K^{+}-algebra B such that B ⊗_{K^{+}} K and B ⊗_{K^{+}} κ are noetherian, and such that Ass B has cardinality < c. Especially, for a fixed 𝔭 ∈ Ass A, we can cover Spec A_{𝔭} \ {𝔭} by finitely many affine open subsets Spec B_{1}, …, Spec B_{k}, and then the inductive assumption yields n ∈ ℕ such that Λ_{B_{i}}^{n}(M_{!} ⊗_{A} B_{i}) = Λ_{B_{i}}^{n}(M ⊗_{A^{a}} B_{i}^{a})_{!} = 0 for every i = 1, …, k. In other words, N := Λ_{A_{𝔭}}^{n}(M_{𝔭}) is a flat A_{𝔭}^{a}-module with Supp N_{!} ⊂ {𝔭}.

Let A_{𝔭}^{sh} denote the strict henselization of A at some geometric point localized at 𝔭; on the one hand, the A_{𝔭}-algebra A_{𝔭}^{sh} is faithfully flat, and M_{!} ⊗_{A} A_{𝔭}^{sh} = (M ⊗_{A^{a}} (A_{𝔭}^{sh})^{a})_{!}. On the other hand, exterior powers commute with arbitrary base change; thus, we are reduced to showing :

Claim 14.8.48. Let B be a measurable K^{+}-algebra, and N a flat almost finitely generated B^{a}- module whose support is contained in {s(B)} (notation of (14.5.36)). Then Λ_{B}^{n}N_{!} = 0 for every sufficiently large n ∈ ℕ.

Proof of the claim. By assumption, we may find an essentially finitely presented K^{+}-algebra B_{0} and an ind-étale and faithfully flat map B_{0} → B of K^{+}-algebras. Set X_{0} := Spec B_{0} and X := Spec B; since {s(B_{0})} is a constructible subset of X_{0}, the natural map

B ⊗_{B_{0}} Γ_{{s(B_{0})}}𝒪_{X_{0}} → Γ_{{s(B)}}𝒪_{X}

is an isomorphism (lemma 10.4.17(iii)). On the other hand, there exists a finitely generated s(B_{0})-primary ideal J ⊂ B_{0} such that the natural map Γ_{{s(B_{0})}}𝒪_{X_{0}} → B_{0}/J is injective (lemma 10.5.15and theorem 11.4.37(i)). It follows that the natural map Γ_{{s(B)}}𝒪_{X} → B/JB is injective as well. Now, N_{!} can be written as the colimit of a filtered system (L_{λ} | λ ∈ Λ) of free Bmodules of finite rank; for each λ ∈ Λ, let L_{λ}^{∼} be the quasi-coherent 𝒪_{X}-module arising from L_{λ}, and define likewise N_{!}^{∼}; taking into account lemma 10.4.4(iii.b) we deduce that the natural map

N_{!} = Γ_{{s(B)}}N_{!}^{∼} = colim_{λ∈Λ} Γ_{{s(B)}}L_{λ}^{∼} → colim_{λ∈Λ} L_{λ}/JL_{λ} = N_{!}/JN_{!}

is injective. In other words, N is a B^{a}/JB^{a}-module. We may then replace B by B/JB, and assume from start that B has Krull dimension zero. In this situation, we may find a nilpotent ideal I ⊂ B, a valuation ring V that is a measurable K^{+}-algebra, and a finitely presented surjection V → B/I (lemma 14.5.42). It suffices to find n ∈ ℕ such that (Λ_{B}^{n}N_{!}) ⊗_{B} B/I = 0; hence, we may replace N by N/IN and B by B/I, and assume as well that B = V/bV for some b ∈ V. In this case, the assertion follows easily from corollary 14.5.33. □

15. Continuous valuations and adic spaces

In this chapter we present Huber’s theory of adic spaces.

15.1. Formal schemes. In this section we define a category of topologically ringed spaces that generalize the usual formal schemes from [59]. This generalization was crucial for the previous releases of this work, that followed closely Faltings’s original method for proving the almost purity theorem. In the current release, where we have switched to the new approach invented by Scholze, and based on his theory of perfectoid spaces, formal schemes are still useful, but they play a rather different role : we will employ them to establish some foundational properties of adic spaces (and later, of perfectoid spaces); for such purposes, the standard adic formal schemes of [59] would already suffice, but the generalization that we worked out in the previous releases might be interesting in its own right, and for other applications. Besides, our former treatment included some complements on the cohomology of formal schemes that do not appear explicitly in [61], which will come in handy in section 15.5.

15.1.1. In this section we shall deal with topological rings whose topology is linear, so that 0 ∈ A admits a fundamental system of open neighborhoods I_{•} := (I_{λ} | λ ∈ Λ) consisting of ideals of A (see definition 8.3.1(v)). Then the separated completion A^{∧} of A is also a topological ring of this type : namely, its topology is linear, defined by the system I_{•}^{∧} := (I_{λ}^{∧} | λ ∈ Λ), where I_{λ}^{∧} denotes the topological closure of I_{λ} in A^{∧}, for every λ ∈ Λ.

15.1.2. Keep the notation of (15.1.1). We shall consider topological A-modules (M, 𝒯_{M}) whose topology is A-linear (see definition 8.3.1(iv)); additionally, we shall assume that 𝒯_{M} is coarser than the I_{•}-adic topology, i.e. for every open submodule N ⊂ M, there exists λ ∈ Λ such that I_{λ}M ⊂ N (see remark 8.3.2(ii)). (Notice that [59, Ch.0, §7.7.1] is slightly ambiguous: it is not clear whether all the topological modules considered there are supposed to satisfy the foregoing additional condition.) Then, the topology of the separated completion (M^{∧}, 𝒯_{M}^{∧}) is A^{∧}-linear and coarser than the I_{•}^{∧}-adic topology. Let N be any other topological A-module of this type; notice that the topology 𝒯_{M,N}^{⊗} on M ⊗_{A}N (see (8.3.7)) is also coarser than the I_{•}-adic topology, so the topology of M ˆ⊗_{A}N is coarser than the I_{•}^{∧}-adic topology. We also denote

top.Hom_{A}(M, N)

the A-module of all continuous A-linear maps M → N. Notice the natural identification :

(15.1.3)  top.Hom_{A}(M, N^{∧}) ≃ lim_{N′⊂N} colim_{M′⊂M} Hom_{A}(M/M′, N/N′)

where N′ (resp. M′) ranges over the family of open submodules of N (resp. of M).

Definition 15.1.4. Let X := (𝒳 , J) be any site, and 𝒞 any other category.

(i) A sheaf of topological spaces on X, any sheaf on X with values in the category Top of topological spaces (see definition 5.5.1(ii)). Likewise, a sheaf of topological groups (resp. of topological rings) on X is a sheaf 𝒜 on X with values in the category of topological groups (resp. topological rings).

(ii) Let 𝒜 be a sheaf of topological rings on X. A sheaf of topological 𝒜 -modules – or briefly, a topological 𝒜 -module – is the datum of an 𝒜 -module ℱ, and for every U ∈ Ob(𝒳 ), a topology 𝒯_{U} on ℱ (U), such that \underline{ℱ} (U) := (ℱ (U), 𝒯_{U}) is a topological 𝒜 (U)-module, and the rule U ↦ \underline{ℱ} (U) defines a sheaf of topological groups on X.

(iii) An 𝒜 -linear map φ : ℱ → 𝒢 between topological 𝒜 -modules is said to be continuous if it is a morphism of sheaves of topological groups, i.e. if φ_{U} : ℱ (U) → 𝒢 (U) is a continuous map for every U ∈ Ob(𝒳 ). We denote by :

top.Hom_{𝒜} (ℱ , 𝒢)

the A-module of all continuous 𝒜 -linear morphisms ℱ → 𝒢.

Remark 15.1.5. Keep the notation of definition 15.1.4.

(i) Since the forgetful functor Top → Set commutes with all limits, remark 5.5.2(i,ii) implies that the presheaf (of sets) underlying any sheaf of topological spaces on X is a sheaf (of sets). Likewise, the presheaf underlying any sheaf of topological groups or topological rings is also a sheaf of sets.

(ii) For any sheaf 𝒜 of topological rings on X, and any topological 𝒜 -modules ℱ and 𝒢, consider the presheaf on X defined by the rule :

(15.1.6)  U ↦ top.Hom_{𝒜_{|U}} (ℱ_{|U}, 𝒢_{|U}).

From the discussion of remark 5.5.2(v) we deduce that (15.1.6) is a sheaf on X, which we denote by :

top.ℋ om_{𝒜} (ℱ , 𝒢).

15.1.7. In the situation of (15.1.1), set X_{λ} := Spec A/I_{λ} for every λ ∈ Λ; we define the formal spectrum of A as the colimit of topological spaces :

Spf A := colim_{λ∈Λ} X_{λ}.

Hence, the set underlying Spf A is the filtered union of the X_{λ}, and a subset U ⊂ Spf A is open (resp. closed) in Spf A if and only if U ∩ X_{λ} is open (resp. closed) in X_{λ} for every λ ∈ Λ.

Let I ⊂ A be any open ideal of A; then I contains an ideal I_{λ}, and therefore Spec A/I is a closed subset of Spec A/I_{λ}, hence also a closed subset of Spf A.

15.1.8. We endow Spf A with a sheaf of topological rings as follows. For every λ ∈ Λ, the structure sheaf 𝒪_{X_{λ}} carries a natural pseudo-discrete topology that makes it a sheaf of topological rings (see [59, Ch.0, §3.8]). Let j_{λ} : X_{λ} → Spf A be the closed immersion; we set

𝒪_{Spf A} := lim_{λ∈Λ} j_{λ∗}𝒪_{X_{λ}}

where the limit is taken in the category of sheaves of topological rings ([59, Ch.0, §3.2.6]). By remark 5.5.2(iii), it follows that :

(15.1.9)  𝒪_{Spf A}(U) = lim_{λ∈Λ} 𝒪_{X_{λ}}(U ∩ X_{λ})

for every open subset U ⊂ Spf A. In this equality, the right-hand side is endowed with the topology of the (projective) limit, and the identification with the left-hand side is a homeomorphism. Set X := Spf A; directly from the definitions we get natural maps of locally ringed spaces :

(15.1.10)  (X_{λ}, 𝒪_{X_{λ}}) ⟶^{j_{λ}} (X, 𝒪_{X}) ⟶^{i_{X}} (Spec A^{∧}, 𝒪_{Spec A^{∧}})  for every λ ∈ Λ

where A^{∧} := 𝒪_{X}(X) is the (separated) completion of A, and i_{X} is given by the universal property [58, Ch.I, Prop.1.6.3] of the spectrum of a ring. Clearly the composition i_{X} ∘ j_{λ} is the map of affine schemes induced by the surjection A^{∧} → A/I_{λ}. Therefore i_{X} identifies the set underlying X with the subset ⋃_{λ∈Λ} X_{λ} ⊂ Spec A^{∧}. However, the topology of X is usually strictly finer than the subspace topology on the image of i_{X}. For any f ∈ A^{∧} we let:

𝔇(f) := i_{X}^{−1}D(f)

where as usual, D(f) ⊂ Spec A^{∧} is the open subset consisting of all prime ideals that do not contain f.

15.1.11. Let f : A → B be a continuous map of topological rings whose topologies are linear. For every open ideal J ⊂ B, we have an induced map A/f^{−1}J → B/J, and after taking colimits, a natural continuous map of topologically ringed spaces :

Spf f : (Spf B, 𝒪_{Spf B}) → (Spf A, 𝒪_{Spf A}).

Lemma 15.1.12. In the situation of (15.1.11), we have :

(i) Spf A is a locally and topologically ringed space.

(ii) φ := Spf f : Spf B → Spf A is a morphism of locally and topologically ringed spaces; in particular, the induced map on stalks 𝒪_{Spf A,φ(x)} → 𝒪_{Spf B,x} is a local ring homomorphism, for every x ∈ Spf B.

(iii) Let I ⊂ A be any open ideal, x ∈ Spec A/I ⊂ Spf A any point. The induced map :

(15.1.13)  𝒪_{Spf A,x} → 𝒪_{Spec A/I,x}

is a surjection.

Proof. (i): We need to show that the stalk 𝒪_{X,x} is a local ring, for every x ∈ X := Spf A. Let (I_{λ} | λ ∈ Λ) be a cofiltered fundamental system of open ideals. Then x ∈ X_{μ} := Spec A/I_{μ} for some μ ∈ Λ; let κ(x) be the residue field of the stalk 𝒪_{X_{μ},x}; there follows a ring homomorphism

(15.1.14)  𝒪_{X,x} → (j_{μ∗}𝒪_{X_{μ}})_{x} = 𝒪_{X_{μ},x} → κ(x).

Moreover, for every λ ≥ μ, the closed immersion X_{μ} → X_{λ} induces an identification of κ(x) with the residue field of 𝒪_{X_{λ},x}, so (15.1.14) is independent of μ.

Suppose now that g ∈ 𝒪_{X,x} is mapped to a non-zero element in κ(x); according to (15.1.9)we may find an open subset U ⊂ X, such that g is represented as a compatible system (g_{λ} | λ ≥ μ) of sections g_{λ} ∈ 𝒪_{X_{λ}}(U ∩ X_{λ}). For every λ ≥ μ, let V_{λ} ⊂ X_{λ} denote the open subset of all y ∈ X_{λ} such that g_{λ}(y) ≠ 0 in κ(y). Then V_{η} ∩ X_{λ} = V_{λ} whenever η ≥ λ ≥ μ. It follows that V := ⋃_{λ≥μ} V_{λ} is an open subset of X, and clearly g is invertible at every point of V, hence g is invertible in 𝒪_{X,x}, which implies the contention.

(ii): The assertion is easily reduced to the corresponding statement for the induced morphisms of schemes : Spec B/J → Spec A/f^{−1}J, for any open ideal J ⊂ B. The details shall be left to the reader.

(iii): Indeed, using (15.1.10) with X := Spf A and I_{λ} := I, we see that the natural surjection 𝒪_{Spec A^{∧},x} → 𝒪_{Spec A/I,x} factors through (15.1.13). □

15.1.15. Let A and M be as in (15.1.2), and fix a (cofiltered) fundamental system (M_{λ} | λ ∈ Λ) of open submodules M_{λ} ⊂ M. For every λ ∈ Λ we may find an open ideal I_{λ} ⊂ A such that M/M_{λ} is an A/I_{λ}-module. Let j_{λ} : X_{λ} := Spec A/I_{λ} → X := Spf A be the natural closed immersion (of ringed spaces). We define the topological 𝒪_{X}-module :

M^{∼} := lim_{λ∈Λ} j_{λ∗}(M/M_{λ})^{∼}

where, as usual, (M/M_{λ})^{∼} denotes the quasi-coherent pseudo-discrete 𝒪_{X_{λ}}-module associated with M/M_{λ}, and the limit is formed in the category of sheaves of topological 𝒪_{X}-modules ([59, Ch.0, §3.2.6]). Thus, for every open subset U ⊂ X one has the identity of topological modules

(15.1.16)  M^{∼}(U) = lim_{λ∈Λ} (M/M_{λ})^{∼}(U ∩ X_{λ})

that generalizes (15.1.9), and follows likewise from remark 5.5.2(iii).

To proceed beyond these simple generalities, we need to add further assumptions. The following definition covers all the situations that we shall find in the sequel.

Definition 15.1.17. Let A be a topological ring whose topology is linear.

(i) We say that A is ω-admissible if A is separated and complete, and 0 ∈ A admits a countable fundamental system of open neighborhoods.

(ii) We say that an open subset U ⊂ Spf A is affine if there exist an ω-admissible topological ring B and an isomorphism (Spf B, 𝒪_{Spf B}) →^{∼} (U, 𝒪_{Spf A|U}) of topologically ringed spaces.

(iii) We say that an open subset U ⊂ Spf A is truly affine if U ∩ Spec A/I is an affine open

subset of Spec A/I for every open ideal I ⊂ A. (iv) An affine ω-formal scheme is a topologically and locally ringed space (X, 𝒪_{X}) that is

isomorphic to the formal spectrum of an ω-admissible topological ring.

(v) An ω-formal scheme is a topologically and locally ringed space (X, 𝒪_{X}) that admits an open covering X = ⋃_{i∈I} U_{i} such that, for every i ∈ I, the restriction (U_{i}, 𝒪_{X|U_{i}}) is an affine ω-formal scheme.

(vi) A morphism of ω-formal schemes f : (X, 𝒪_{X}) → (Y, 𝒪_{Y}) is a map of topologically and locally ringed spaces, i.e. a morphism of locally ringed spaces such that the corresponding map 𝒪_{Y} → f_{∗}𝒪_{X} induces continuous ring homomorphisms 𝒪_{Y} (U) → 𝒪_{X}(f^{−1}U), for every open subset U ⊂ Y.

Remark 15.1.18. (i) Let X be any ω-formal scheme, and U ⊂ X any open subset. Using the condition of definition 5.5.1(ii), it is easily seen that :

(a) 𝒪_{X}(U) is a complete and separated topological ring.

(b) If additionally, U = ⋃_{i∈I} U_{i} for a countable family (U_{i} | i ∈ I) of open subsets, such that (U_{i}, 𝒪_{X|U_{i}}) is an affine ω-formal scheme for every i ∈ I (briefly, each U_{i} is an affine open subset of X), then 𝒪_{X}(U) is ω-admissible.

(ii) In [58] one finds a notion of admissible topological ring, and to any such ring A it is assigned an affine formal scheme Spf A. These notions relate to ours as follows. Let us say that a topological ring is c-admissible if it is admissible in the sense of [58, Ch.0, Déf.7.1.2] and 0 ∈ A admits a countable fundamental system of open neighborhoods. Then, an admissible topological ring A is ω-admissible if and only if it is c-admissible. Likewise, if A is admissible, then the formal scheme Spf A defined in [58, Ch.I, Déf.10.1.2] is an ω-formal scheme if and only if A is c-admissible. Indeed, if A is c-admissible, obviously Spf A coincides with the affine ω-formal scheme attached to A as in (15.1.7) and (15.1.8). For the converse, notice that Spf A (in the sense of [58]) is quasi-compact, so if it is an ω-formal scheme we may cover it by finitely many of its affine open subsets U_{1}, …, U_{n}; say that U_{i} = Spf B_{i} for i = 1, …, n, where B_{1}, …, B_{n} are ω-admissible. There follows a natural continuous map ρ : A → B := ∏_{i=1}^{n} B_{i}, and the topology of A agrees with the topology induced by B via ρ (remark 5.5.2(i)); however, clearly 0 ∈ B admits a countable fundamental system of open neighborhoods, so the same holds for A.

(iii) In the same vein, if A is ω-admissible, then Spf A as defined in (15.1.7) and (15.1.8) is a quasi-compact formal scheme in the sense of [58] if and only if A is c-admissible. Indeed, the condition is obviously sufficient; conversely, if Spf A is a quasi-compact formal scheme in the sense of [58], we may cover it by finitely many open subsets U_{1}, …, U_{n} such that, for every i = 1, …, n we have U_{i} = Spf B_{i} for some topological ring that is admissible in the sense of [58], and the topology of A is induced by the natural inclusion map into B := ∏_{i=1}^{n} B_{i}. For each such B_{i}, pick an open and nilpotent ideal I_{i} ⊂ B_{i}; then I := ∏_{i=1}^{n} I_{i} is an open and nilpotent ideal of B, and I ∩ A is open and nilpotent in A, so the latter is c-admissible.

Proposition 15.1.19. Suppose that A is an ω-admissible topological ring. Then :

(i) The truly affine open subsets form a basis for the topology of X := Spf A. (ii) Let I ⊂ A be an open ideal, U ⊂ X a truly affine open subset. Then :

(a) The natural map ρ_{U} : A → A_{U} := 𝒪_{X}(U) induces an isomorphism :

(Spf A_{U}, 𝒪_{Spf A_{U}}) →^{∼} (U, 𝒪_{X|U}).

(b) The topological closure I_{U} of IA_{U} in A_{U} is an open ideal, and the natural map : Spec A_{U}/I_{U} → Spec A/I is an open immersion.

(c) U is affine.

(d) Every open covering of U admits a countable subcovering.

Proof. (i): By assumption, we may find a countable fundamental system (I_{n} | n ∈ ℕ) of open ideals of A, and clearly we may assume that this system is ordered under inclusion (so that I_{n} ⊂ I_{m} whenever n ≥ m). Let x ∈ Spf A be any point, and U ⊂ Spf A an open neighborhood of x. Then x ∈ X_{n} := Spec A/I_{n} for sufficiently large n ∈ ℕ. We are going to exhibit, by induction on m ∈ ℕ, a sequence (f_{m} | m ≥ n) of elements of A, such that :

(15.1.20)  x ∈ 𝔇(f_{p}) ∩ X_{m} = 𝔇(f_{m}) ∩ X_{m} ⊂ U  whenever p ≥ m ≥ n

To start out, we may find f_{n} ∈ A/I_{n} such that x ∈ 𝔇(f_{n}) ∩ X_{n} ⊂ U ∩ X_{n}. Next, let m ≥ n and suppose that f_{m} has already been found; we may write

U ∩ X_{m+1} = X_{m+1} \ V (J) for some ideal J ⊂ A/I_{m+1}.

Let \overline{J} ⊂ A/I_{m} and \overline{f}_{m} ∈ A/I_{m} be the images of J and f_{m}; then V (\overline{J}) ⊂ V (\overline{f}_{m}), hence there exists k ∈ ℕ such that \overline{f}_{m}^{k} ∈ \overline{J}. Pick \overline{f}_{m+1} ∈ J such that the image of \overline{f}_{m+1} in A/I_{m} agrees with \overline{f}_{m}^{k}, and let f_{m+1} ∈ A be any lifting of \overline{f}_{m+1}; with this choice, one verifies easily that (15.1.20) holds for p := m + 1. Finally, the subset :

U′ := ⋃_{m≥n} 𝔇(f_{m}) ∩ X_{m}

is an admissible affine open neighborhood of x contained in U.

(ii.a): For every n, m ∈ ℕ with n ≥ m, we have a closed immersion of affine schemes: U ∩ X_{m} ⊂ U ∩ X_{n}; whence induced surjections :

A_{U,n} := 𝒪_{X_{n}}(U ∩ X_{n}) → A_{U,m} := 𝒪_{X_{m}}(U ∩ X_{m}).

By [59, Ch.0, §3.8.1], A_{U,n} is a discrete topological ring, for every n ∈ ℕ, hence A_{U} ≃ lim_{n∈ℕ} A_{U,n} carries the linear topology that admits the fundamental system of open ideals (Ker (A_{U} → A_{U,n}) | n ∈ ℕ), especially A_{U} is ω-admissible. It follows that the topological space underlying Spf A_{U} is colim_{n∈ℕ}(U ∩ X_{n}), which is naturally identified with U, under Spf ρ_{U}. Likewise, let i : U → X be the open immersion; one verifies easily (e.g. using (15.1.9)) that the natural map :

i^{∗} lim_{n∈ℕ} j_{n∗}𝒪_{X_{n}} → lim_{n∈ℕ} i^{∗}j_{n∗}𝒪_{X_{n}}

is an isomorphism of topological sheaves, which implies the assertion.

(ii.b): We may assume that I_{0} = I. Since U ∩ X_{n} is affine for every n ∈ ℕ, we deduce short exact sequences :

ℰ_{n} := (0 → I · 𝒪_{X_{n}}(X_{n} ∩ U) → 𝒪_{X_{n}}(U ∩ X_{n}) → 𝒪_{X_{0}}(U ∩ X_{0}) → 0)

and lim_{n∈ℕ} ℰ_{n} is the exact sequence :

0 → I_{U} → A_{U} → 𝒪_{X_{0}}(U ∩ X_{0}) → 0.

Since U ∩ X_{0} is an open subset of X_{0}, both assertions follow easily.

(ii.c): It has already been remarked that A_{U} is ω-admissible. More precisely, for every n ∈ ℕ let (I_{n}A_{U})^{c} be the topological closure of I_{n}A_{U} in A_{U}; then the proof of (ii.b) shows that the family of ideals ((I_{n}A_{U})^{c} | n ∈ ℕ) is a fundamental system of open neighborhoods of 0 ∈ A_{U}. Hence the assertion follows from (ii.a).

(ii.d): By definition, U is a countable union of quasi-compact subsets, so the assertion is immediate. □

Corollary 15.1.21. Let X be an ω-formal scheme, U ⊂ X any open subset. Then (U, 𝒪_{X|U}) is an ω-formal scheme. □

Proposition 15.1.22. Let X be an ω-formal scheme, A an ω-admissible topological ring. Then the rule :

(15.1.23)  (f : X → Spf A) ↦ (f^{♮} : A → Γ(X, 𝒪_{X}))

establishes a natural bijection between the set of morphisms of ω-formal schemes X → Spf A and the set of continuous ring homomorphisms A → Γ(X, 𝒪_{X}).

Proof. We reduce easily to the case where X = Spf B for some ω-admissible topological ring B. Let f : X → Y := Spf A be a morphism of ω-formal schemes; we have to show that f = Spf f^{♮}, where f^{♮} : A → B is the map on global sections induced by the morphism of sheaves 𝒪_{Y} → f^{∗}𝒪_{X} that defines f. Let U ⊂ X and V ⊂ Y be two truly affine open subsets, such that f(U) ⊂ V, and let likewise :

f_{U,V}^{♮} : A_{V} := 𝒪_{Y} (V) → B_{U} := 𝒪_{X}(U)

be the map induced by f_{|U}. Using the universal property of [58, Ch.I, Prop.1.6.3], we obtain a commutative diagram of morphisms of locally ringed spaces:

(15.1.24)  \xymatrix{Spec B_{U} \ar[rrr]^{Spec f_{U,V}^{♮}} \ar[ddd] &  &  & Spec A_{V} \ar[ddd] \\  & U \ar[d] \ar[r]^{f_{|U}} \ar[ul]^{i_{U}} & V \ar[d] \ar[ur]_{i_{V}} \\  & X \ar[r]^{f} \ar[dl]_{i_{X}} & Y \ar[dr]^{i_{Y}} \\ Spec B \ar[rrr]^{Spec f^{♮}} &  &  & Spec A}

where i_{X}, i_{Y}, i_{U} and i_{V} are the morphisms of (15.1.10). Since i_{X} and i_{Y} are injective on the underlying sets, it follows already that f and Spf f^{♮} induce the same continuous map of topological spaces. Let now J ⊂ B be any open ideal, I ⊂ f^{♮−1}(J) an open ideal of A, and let J_{U} (resp. I_{V}) be the topological closure of JB_{U} in B_{U} (resp. of IA_{V} in A_{V}). Since the map f_{U,V}^{♮} is continuous, we derive a commutative diagram of schemes :

\xymatrix{Spec B_{U}/J_{U} \ar[r]^{φ} \ar[d]_{α} & Spec A_{V} /I_{V} \ar[d]^{β} \\ Spec B/J \ar[r]^{ψ} & Spec A/I}

where α and β are open immersions, by proposition 15.1.19(ii.b), and φ (resp. ψ) is induced by f_{U,V}^{♮} (resp. by f^{♮}). Let ℐ be a fundamental system of open neighborhoods of 0 ∈ A consisting of ideals, and for every I ∈ ℐ denote by (IA_{V})^{c} the topological closure of IA_{V} in A_{V}; from proposition 15.1.19(ii.b) (and its proof) it follows as well that A_{V} ≃ lim_{I∈ℐ} A_{V} /(IA_{V})^{c}. Summing up, this shows that f_{U,V}^{♮} is determined by f^{♮}, whence the contention. □

Corollary 15.1.25. Let A be an ω-admissible topological ring, U ⊂ Spf A an affine open subset, V ⊂ U a truly affine open subset of X. Then V is a truly affine open subset of U.

Proof. Say that U = Spf B, for some ω-admissible topological ring B; by proposition 15.1.22, the immersion j : U → X is of the form j = Spf φ for a unique map φ : A → B. Let now J ⊂ B be any open ideal, and set I := φ^{−1}J; there follows a commutative diagram of locally ringed spaces :

\xymatrix{U_{0} := Spec B/J \ar[r] \ar[d] & X_{0} := Spec A/I \ar[d] \\ U \ar[r]^{j} & X.}

By assumption, V ∩ X_{0} = Spec C for some A/I-algebra C; it follows that :

V ∩ U_{0} = Spec B/J ⊗_{A/I} C

and since J is arbitrary, the claim follows. □

Remark 15.1.26. (i) Let A be an ω-admissible topological ring, and B, C two topological A-algebras whose topologies are linear (and the structure maps A → B, A → C are continuous). Recall that B ˆ⊗_{A}C represents the product of B and C in the category of ω-admissible A-algebras (see (8.3.7)). By standard arguments, we deduce from proposition 15.1.22 that Spf B ˆ⊗_{A}C represents the product of Spf B and Spf C in the category of ω-formal Spf Aschemes; especially, the category of ω-formal schemes admits arbitrary fibre products.

(ii) We shall say that a topological A-module M is ω-admissible if M is complete and separated, and admits a countable fundamental system of open neighborhoods of 0 ∈ M. Notice that the completion functor N ↦ N^{∧} on topological A-modules is not always “right exact”, in the following sense. Suppose N → N′ is a quotient map; then the induced map N^{∧} → N^{′∧} is not necessarily onto. However, this is the case if N^{∧} is ω-admissible (see proposition 8.2.13(v)). For such A-modules, we have moreover the following :

Lemma 15.1.27. Let A be a topological ring, M, M′, N three ω-admissible topological Amodules, and f : M → M′ a quotient map. Then :

(i) The homomorphism f ˆ⊗_{A}1_{N} : M ˆ⊗_{A}N → M′ ˆ⊗_{A}N is a quotient map.

(ii) Let us endow Ker f with the subspace topology induced from M, and suppose additionally that, for every open ideal I ⊂ A :

(a) The topological closure (IN)^{c} of IN in N is an open submodule of N.

(b) N/(IN)^{c} is a flat A/I-module.

Then the complex :

0 → (Ker f) ˆ⊗_{A}N → M ˆ⊗_{A}N → M′ ˆ⊗_{A}N → 0

is an admissible short exact sequence of topological A^{∧}-modules.

Proof. By assumption, we may find an inverse system of discrete A-modules, with surjective transition maps (M_{n} | n ∈ ℕ) (resp. (N_{n} | n ∈ ℕ)), and an isomorphism of topological Amodules : M ≃ lim_{n∈ℕ} M_{n} (resp. N ≃ lim_{n∈ℕ} N_{n}); let us define M′_{n} := M′ ⨿_{M} M_{n} for every n ∈ ℕ (the cofibred sum of M′ and M_{n} over M). Since f is a quotient map, M′_{n} is a discrete A-module for every n ∈ ℕ, and the natural map : M′ → lim_{n∈ℕ} M′_{n} is a topological isomorphism. We deduce an inverse system of short exact sequences of discrete A-modules :

0 → K_{n} → M_{n} ⊗_{A} N_{n} → M′_{n} ⊗_{A} N_{n} → 0

where K_{n} is naturally a quotient of Ker(M_{n} → M′_{n}) ⊗_{A} N_{n}, for every n ∈ ℕ; especially, the transition maps K_{j} → K_{i} are surjective whenever j ≥ i. Then assertion (i) follows from lemma 8.6.2.

(ii): We may find open ideals I_{n} ⊂ A such that I_{n}N_{n} = I_{n}M_{n} = 0 for every n ∈ ℕ, hence (I_{n}N)^{c} ⊂ Ker(N → N_{n}), and from (a) we deduce that ((I_{n}N)^{c} | n ∈ ℕ) is a fundamental system of open neighborhoods of 0 ∈ N. Due to (b), we obtain short exact sequences :

0 → Ker(M_{n} → M′_{n}) ⊗_{A} N/(I_{n}N)^{c} → M_{n} ⊗_{A} N/(I_{n}N)^{c} → M′_{n} ⊗_{A} N/(I_{n}N)^{c} → 0

for every n ∈ ℕ. Then it suffices to invoke again lemma 8.6.2. □

Lemma 15.1.28. Let A be an ω-admissible topological ring, M a topological A-module, and U ⊂ X := Spf A any affine open subset. Then the following holds :

(i) There is a natural isomorphism of topological 𝒪_{X}(U)-modules :

M^{∼}(U) ≃ M ˆ⊗_{A}𝒪_{X}(U).

(ii) If L is any other topological A-module, the natural map :

top.Hom_{A}(L, M^{∧}) → top.Hom_{𝒪_{X}} (L^{∼}, M^{∼}) φ ↦ φ^{∼}

is an isomorphism. (iii) The functor M ↦ M^{∼} on topological A-modules, is left adjoint to the global sections

functor ℱ ↦ ℱ (X), defined on the category of complete and separated topological 𝒪_{X}-modules and continuous maps.

Proof. (i): First we remark that the assertion holds whenever U is a truly affine open subset of X; the easy verification shall be left to the reader. For a general U, set M_{U} := M ˆ⊗_{A}𝒪_{X}(U), and denote by M_{U}^{∼} the associated 𝒪_{U}-module. Let V ⊂ U be any open subset which is truly affine in X; by corollary 15.1.25, V is truly affine in U as well. We deduce natural isomorphisms:

M^{∼}(V) ≃ M_{U} ˆ⊗_{𝒪_{X}(U)} 𝒪_{X}(V) ≃ M_{U}^{∼}(V)

which – in view of proposition 15.1.19(i) – amount to a natural isomorphism of topological 𝒪_{U}-modules : (M^{∼})_{|U} →^{∼} M_{U}^{∼}. Assertion (i) follows easily.

(iii): Given a continuous map M^{∼} → ℱ, we get a map of global sections M = M^{∼}(X) → ℱ (X). Conversely, suppose f : M → ℱ (X) is a given continuous map; let U ⊂ X be any affine open subset, and f_{U} : M → ℱ (U) the composition of f and the restriction map ℱ (X) → ℱ (U). Then f_{U} extends first – by linearity – to a map M ⊗_{A} 𝒪_{X}(U) → ℱ (U), and second – by continuity – to a map M ˆ⊗_{A}𝒪_{X}(U) → ℱ (U); the latter, in view of (i), is the same as a map f_{U}^{∼} : M^{∼}(U) → ℱ (U). Clearly the rule U ↦ f_{U}^{∼} thus defined is functorial for inclusion of open subsets U ⊂ U′, whence (iii).

(ii) is a straightforward consequence of (iii). □

Proposition 15.1.29. Let A be an ω-admissible topological ring, M an ω-admissible topological A-module, N ⊂ M a submodule, and U ⊂ X := Spf A an affine open subset. Then:

(i) If we endow M/N with the quotient topology, the sequence of 𝒪_{X}-modules :

0 → N^{∼} → M^{∼} → (M/N)^{∼} → 0

is short exact. (ii) The induced sequence

0 → N^{∼}(U) → M^{∼}(U) → (M/N)^{∼}(U) → 0

is short exact and admissible in the sense of (8.6).

Proof. (i): We recall the following :

Claim 15.1.30. Let j : Z′ := Spec R′ → Z := Spec R be an open immersion of affine schemes. Then R′ is a flat R-algebra.

Proof of the claim. The assertion can be checked on the localizations at the prime ideals of R′; however, the induced maps 𝒪_{Z,j(z)} → 𝒪_{Z′,z} are isomorphisms for every z ∈ Z′, so the claim is clear. ♢

Let V ⊂ X be any truly affine open subset. It follows easily from proposition 15.1.19(ii.b) and claim 15.1.30 that 𝒪_{X}(V) fulfills conditions (a) and (b) of lemma 15.1.27(ii). In light of lemma 15.1.28(i), we deduce that the sequence :

0 → N^{∼}(V) → M^{∼}(V) → (M/N)^{∼}(V) → 0

is admissible short exact, whence the contention.

(ii): Clearly the sequence is left exact; using lemmata 15.1.27(i) and 15.1.28(i), we see that it is also right exact, and moreover M^{∼}(U) → (M/N)^{∼}(U) is a quotient map. It remains only to show that the topology on N^{∼}(U) is induced from M^{∼}(U), and to this aim we may assume – thanks to the condition of definition 5.5.1(ii) – that U is a truly affine open subset of X, in which case the assertion has already been observed in the proof of (i). □

Definition 15.1.31. Let X be an ω-formal scheme, and ℱ a topological 𝒪_{X}-module ℱ.

(i) We say that ℱ is quasi-coherent if there exist a covering 𝔘 := (U_{i} | i ∈ I) of X consisting of affine open subsets, and for every i ∈ I, an ω-admissible topological 𝒪_{X}(U_{i})- module M_{i} with an isomorphism ℱ_{|U_{i}} →^{∼} M_{i}^{∼} of topological 𝒪_{U_{i}}-modules (in the sense of definition 15.1.4(ii,iii)).

(ii) We denote by 𝒪_{X}-Mod_{qcoh} the category of quasi-coherent 𝒪_{X}-modules and continuous 𝒪_{X}-linear morphisms.

(iii) We say that a continuous morphism f : ℱ → 𝒢 of quasi-coherent 𝒪_{X}-modules is a quotient map if the induced map f(U) : ℱ (U) → 𝒢 (U) is a quotient map of topological 𝒪_{X}(U)-modules, for every affine open subset U ⊂ X. Remark 15.1.32. (i) The category 𝒪_{X}-Mod_{qcoh} is usually not abelian (see (8.6)); more than that, the kernel (in the category of abelian sheaves) of a continuous map f : ℱ → 𝒢 of quasicoherent 𝒪_{X}-modules may fail to be quasi-coherent. However, using proposition 15.1.29 one may show that Ker f is quasi-coherent whenever f is a quotient map, and in this case Ker f is also the kernel of f in the category 𝒪_{X}-Mod_{qcoh}.

(ii) Furthermore, any (continuous) morphism f in 𝒪_{X}-Mod_{qcoh} admits a cokernel. This can be exhibited as follows. To start out, let us define presheaves ℐ and ℒ by declaring that ℐ (U) ⊂ 𝒢 (U) is the topological closure of the 𝒪_{X}(U)-submodule Im(f(U) : ℱ (U) → 𝒢 (U)), and ℒ (U) := 𝒢 (U)/ℐ (U), which we endow with its natural quotient topology, for every open subset U ⊂ X. Now, suppose that V ⊂ U is an inclusion of sufficiently small affine open subsets of X (so that ℱ_{|U} and 𝒢_{|U} are of the form M^{∼} for some topological 𝒪_{X}(U)- module M); since ℱ (U)ˆ⊗_{𝒪_{X}(U)} 𝒪_{X}(V) = ℱ (V), we see that the image of ℐ (U) in ℐ (V) generates a dense 𝒪_{X}(V)-submodule. On the other hand, by construction the exact sequence ℰ := (0 → ℐ (U) → 𝒢 (U) → ℒ (U) → 0) is admissible, hence the same holds for the sequence ℰ ˆ⊗_{𝒪_{X}(U)} 𝒪_{X}(V) (lemma 15.1.28(i) and proposition 15.1.29(ii)). It follows that ℐ (V) = ℐ (U)ˆ⊗_{𝒪_{X}(U)} 𝒪_{X}(V) and ℒ (V) = ℒ (U)ˆ⊗_{𝒪_{X}(U)} 𝒪_{X}(V). Thus ℐ and ℒ are sheaves of topological 𝒪_{X}-modules on the site C of all sufficiently small affine open subsets of X, and their sheafifications ℐ′ and ℒ′ are the topological 𝒪_{X}-modules obtained as in [59, Ch.0, §3.2.1], by extension of ℐ and ℒ from the site C to the whole topology of X. It follow that ℐ′ and ℒ′ are quasi-coherent 𝒪_{X}-modules; then it is easy to check that ℒ′ is the cokernel of f in the category 𝒪_{X}-Mod_{qcoh} (briefly : the topological cokernel of f), and shall be denoted

top.Coker f.

The sheaf ℐ′ shall be called the topological closure of the image of f, and denoted

\overline{Im}(f).

A morphism f with top.Coker f = 0 shall be called a topological epimorphism.

(iii) Notice also that the natural map Coker f → top.Coker f is an epimorphism of abelian sheaves, hence a continuous morphism of quasi-coherent 𝒪_{X}-modules which is an epimorphism of 𝒪_{X}-modules, is also a topological epimorphism. Proposition 15.1.33. Let X be an affine ω-formal scheme, ℱ a quasi-coherent 𝒪_{X}-module. Then ℱ (X) is an ω-admissible 𝒪_{X}(X)-module, and the natural map

ℱ (X)^{∼} → ℱ

is an isomorphism of topological 𝒪_{X}-modules. Proof. By assumption we may find an affine open covering 𝔘 := (U_{i} | i ∈ I) of X such that, for every i ∈ I, ℱ_{|U_{i}} ≃ M_{i}^{∼} for some 𝒪_{X}(U_{i})-module M_{i}. In view of lemma 15.1.28(i) we may assume – up to replacing 𝔘 by a refinement – that U_{i} is a truly affine subset of X for every i ∈ I. Furthermore, we may write X = ⋃_{n∈ℕ} X_{n} for an increasing countable family of quasi-compact subsets; for each n ∈ ℕ we may then find a finite subset I(n) ⊂ I such that X_{n} ⊂ ⋃_{i∈I(n)} U_{i}, and therefore we may replace I by ⋃_{n∈ℕ} I_{n}, which allows to assume that I is countable.

Next, for every i ∈ I we may find a countable fundamental system of open submodules (M_{i,n} | n ∈ ℕ) of M_{i}, and for every n ∈ ℕ an open ideal J_{i,n} ⊂ A_{i} := 𝒪_{X}(U_{i}) such that N_{i,n} := M_{i}/M_{i,n} is an A_{i}/J_{i,n}-module. Let j_{i,n} : U_{i,n} := Spec A_{i}/J_{i,n} → U_{i} be the natural closed immersion; we may write :

(15.1.34)  ℱ_{|U_{i}} ≃ lim_{n∈ℕ} j_{i,n∗}N_{i,n}^{∼}.

Let also ι_{i,n} : U_{i,n} → X be the locally closed immersion obtained as the composition of j_{i,n} and the open immersion j_{i} : U_{i} → X; we deduce natural maps of 𝒪_{X}-modules :

φ_{i,n} : ℱ → j_{i∗}ℱ_{|U_{i}} → 𝒢_{i,n} := ι_{i,n∗}N_{i,n}^{∼}.

Claim 15.1.35. (i) There exists m ∈ ℕ such that 𝒢_{i,n} is the extension by zero of a quasi-coherent 𝒪_{X_{m}}-module.

(ii) φ_{i,n} is continuous for the pseudo-discrete topology on 𝒢_{i,n}.

Proof of the claim. (i): It is easy to see that U_{i,n} ⊂ X_{m} for m ∈ ℕ large enough; then the assertion follows from ([59, Ch.I, Cor.9.2.2]).

(ii): We need to check that the map φ_{i,n,V} : ℱ (V) → 𝒢_{i,n}(V) is continuous for every open subset V ⊂ X. However, the condition of definition 5.5.1(ii) implies that the assertion is local on X, hence we may assume that V is a truly affine open subset of X, in which case 𝒢_{i,n}(V) is a discrete space. We may factor φ_{i,n,V} as a composition :

ℱ (V) →^{α} ℱ (V ∩ U_{i}) →^{β} N_{i,n}^{∼}(U_{i,n} ∩ V)

where the restriction map α is continuous, and β is continuous for the pseudo-discrete topology on N_{i,n}^{∼}(U_{i,n} ∩ V). However, U_{i} ∩ V is a truly affine open subset of U_{i}; therefore U_{i,n} ∩ V is quasi-compact, and the pseudo-discrete topology on N_{i,n}^{∼} induces the discrete topology on N_{i,n}^{∼}(U_{i,n} ∩ V). The claim follows. ♢

For every finite subset S ⊂ I × ℕ, let φ_{S} : ℱ → 𝒢_{S} := ∏_{(i,n)∈S} 𝒢_{i,n} be the product of the maps φ_{i,n}; according to claim 15.1.35(ii), φ_{S} is continuous for the pseudo-discrete topology on 𝒢_{S}. Hence, for every i ∈ I, the restriction φ_{S|U_{i}} : ℱ_{|U_{i}} → 𝒢_{S|U_{i}} is of the form f_{i,S}^{∼}, for some continuous map f_{i,S} : M_{i} → 𝒢_{S}(U_{i}) (lemma 15.1.28(ii)). It follows easily that (Im φ_{S})_{|U_{i}} is the quasi-coherent 𝒪_{U_{i}}-module (Im f_{i,S})^{∼}, especially Im φ_{S} is quasi-coherent. Notice that 𝒢_{S} is already a quasi-coherent 𝒪_{X_{m}}-module for m ∈ ℕ large enough, hence the same holds for Im φ_{S}; we may therefore find an 𝒪_{X_{m}}(X_{m})-module G_{S} such that G_{S}^{∼} ≃ Im φ_{S}. Furthermore, for every other finite subset S′ ⊂ I × ℕ containing S, we deduce a natural 𝒪_{X}(X)-linear surjection G_{S′} → G_{S}, compatible with compositions of inclusions S ⊂ S′ ⊂ S′′. Let (S_{n} | n ∈ ℕ) be a countable increasing family of finite subsets, whose union is I ×ℕ; we endow G := lim_{n∈ℕ} G_{S_{n}} with the pro-discrete topology. It is then easy to check that G is an ω-admissible 𝒪_{X}(X)- module; furthermore, by construction we get a unique continuous map φ : ℱ → G^{∼} whose composition with the projection onto G_{S_{n}}^{∼} agrees with φ_{S_{n}} for every n ∈ ℕ. In light of (15.1.34) we see easily that φ_{|U_{i}} is an isomorphism of topological 𝒪_{U_{i}}-modules for every i ∈ I, hence the same holds for φ. Then φ necessarily induces an isomorphism ℱ (X) →^{∼} G. □

Corollary 15.1.36. Let X be an ω-formal scheme, (ℱ_{n} | n ∈ ℕ) an inverse system of quasicoherent 𝒪_{X}-modules, whose transition maps are topological epimorphisms. Then lim_{n∈ℕ} ℱ_{n} (with its inverse limit topology) is a quasi-coherent 𝒪_{X}-module.

Proof. We may assume that X is affine; then, for every n ∈ ℕ we have ℱ_{n} = M_{n}^{∼} for some complete and separated 𝒪_{X}(X)-module M_{n} (by proposition 15.1.33), and the transition maps ℱ_{n+1} → ℱ_{n} come from corresponding continuous linear maps f_{n} : M_{n+1} → M_{n}. We choose inductively, for every n ∈ ℕ, a descending fundamental system of open submodules (N_{n,k} | k ∈ ℕ) of M_{n}, such that f_{n}(N_{n+1,k}) ⊂ N_{n,k} for every n, k ∈ ℕ. Set M_{n,k} := M_{n}/N_{n,k} for every n, k ∈ ℕ. By assumption, top.Coker f_{n}^{∼} = 0; hence the induced maps M_{n+1,k}^{∼} → M_{n,k}^{∼} are topological epimorphisms; since M_{n,k}^{∼} is pseudo-discrete (on the closure of its support), it follows that the latter maps are even epimorphisms (of abelian sheaves) so the corresponding maps M_{n+1,k} → M_{n,k} are onto for every n, k ∈ ℕ. Thus :

lim_{n∈ℕ} ℱ_{n} ≃ lim_{n∈ℕ} lim_{k∈ℕ} M_{n,k}^{∼} ≃ lim_{n∈ℕ} M_{n,n}^{∼} ≃ (lim_{n∈ℕ} M_{n,n})^{∼}

and it is clear that the 𝒪_{X}(X)-module lim_{n∈ℕ} M_{n,n} is ω-admissible. □

Theorem 15.1.37. Let X be an ω-formal scheme, ℱ a quasi-coherent 𝒪_{X}-module, 𝔘 := (U_{i} | i ∈ I) an open covering of X, such that the intersection U_{t_{0}} ∩ ⋯ ∩ U_{t_{n}} is affine for every n ∈ ℕ and every (t_{0}, …, t_{n}) ∈ I^{n+1}. Then :

(i) There is a natural isomorphism (notation of (10.2.18))

H_{alt}^{•}(𝔘, ℱ) →^{∼} H^{•}(X, ℱ).

(ii) If moreover X is affine, we have H^{i}(X, ℱ) = 0 for every i > 0.

Proof. (The statements refer to the cohomology of the abelian sheaf underlying ℱ, in other words, we forget the topology of the modules ℱ (U).)

(ii): Say that X = Spf A for an ω-admissible topological ring A. Let U ⊂ X be any truly affine open subset, and 𝔘 := (U_{i} | i ∈ I) a covering of U consisting of truly affine open subsets. By proposition 15.1.33we have ℱ_{|U} ≃ M^{∼}, where M := ℱ (U).

Claim 15.1.38. The augmented alternating Čech complex C_{alt}^{•}(𝔘, M^{∼}) is acyclic.

Proof of the claim. Let (M_{n} | n ∈ ℕ) be a fundamental system of neighborhoods of 0 ∈ M, consisting of open submodules, and for every n ∈ ℕ, choose an open ideal I_{n} ⊂ A such that I_{n}M ⊂ M_{n}. Set X_{n} := Spec A/I_{n} and 𝔘_{n} := (U_{i} ∩ X_{n} | i ∈ I) for every n ∈ ℕ. For every n ∈ ℕ we may consider the augmented alternating Čech complex C_{alt}^{•}(𝔘_{n}, (M/M_{n})^{∼}), and in view of (15.1.16) we obtain a natural isomorphism of complexes :

C_{alt}^{•}(𝔘, M^{∼}) →^{∼} lim_{n∈ℕ} C_{alt}^{•}(𝔘_{n}, (M/M_{n})^{∼}).

We may view the double complex C_{alt}^{•}(𝔘_{•}, (M/M_{•})^{∼}) also as a complex of inverse systems of modules, whose term in degree i ∈ ℕ is C_{alt}^{i}(𝔘_{•}, (M/M_{•})^{∼}). Notice that, for every i ∈ ℕ, all the transition maps of this latter inverse system are surjective. Hence ([163, Lemma 3.5.3]) :

lim_{n∈ℕ}^{q} C_{alt}^{i}(𝔘_{n}, (M/M_{n})^{∼}) = 0 for every q > 0.

In other words, these inverse systems are acyclic for the inverse limit functor. It follows (e.g. by means of [163, Th.10.5.9]) that :

R lim_{n∈ℕ} C_{alt}^{•}(𝔘_{•}, (M/M_{•})^{∼}) ≃ C_{alt}^{•}(𝔘, M^{∼}).

On the other hand, the complexes C_{alt}^{•}(𝔘_{n}, (M/M_{n})^{∼}) are acyclic for every n ∈ ℕ ([61, Ch.III, Th.1.3.1 and Prop.1.4.1]), hence C_{alt}^{•}(𝔘_{•}, (M/M_{•})^{∼}) is acyclic, when viewed as a cochain complex of inverse systems of modules. The claim follows. ♢

Assertion (ii) follows from claim 15.1.38, proposition 15.1.19(i) and theorem 10.2.24(i). Assertion (i) follows from (ii) and corollary 10.2.21(ii). □

Corollary 15.1.39. In the situation of (15.1.15), suppose that A and M are ω-admissible. Then:

lim_{λ∈Λ}^{q} j_{λ∗}(M/M_{λ})^{∼} = 0 for every q > 0.

Proof. For every truly affine open subset U ⊂ X, we have a topos U^{Λ} defined as in [75, §7.3.4], and the cofiltered system ℳ := (j_{λ∗}(M/M_{λ})^{∼} | λ ∈ Λ) defines an abelian sheaf on U^{Λ}. According to loc.cit. there are two spectral sequences :

E_{2}^{pq} := R^{p}Γ(U, lim_{λ∈Λ}^{q} j_{λ∗}(M/M_{λ})^{∼}) ⇒ H^{p+q}(U^{Λ}, ℳ)
F_{2}^{pq} := lim_{λ∈Λ}^{q} R^{p}Γ(U, j_{λ∗}(M/M_{λ})^{∼}) ⇒ H^{p+q}(U^{Λ}, ℳ)

and we notice that F_{2}^{pq} = 0 whenever p > 0 (since U ∩ X_{λ} is affine for every λ ∈ Λ) and whenever q > 0, since the cofiltered system (Γ(U, j_{λ∗}(M/M_{λ})^{∼}) | λ ∈ Λ) has surjective transition maps. One can then argue as in the proof of [75, Lemma 7.3.5] : the sheaf L^{q} := lim_{λ∈Λ}^{q} j_{λ∗}(M/M_{λ})^{∼} is the sheafification of the presheaf : U ↦ H^{q}(U^{Λ}, ℳ) and the latter vanishes by the foregoing. We supply an alternative argument. Since the truly affine open subsets form a basis of X, it suffices to show that E_{2}^{0q} = 0 whenever q > 0. We proceed by induction on q. For q = 1, we look at the differential d_{2}^{01} : E_{2}^{0,1} → E_{2}^{2,0}; by theorem 15.1.37 we have E_{2}^{2,0} = 0, hence E_{2}^{0,1} = E_{∞}^{0,1}, and the latter vanishes by the foregoing. Next, suppose that q > 1, and that we have shown the vanishing of L^{j} for 1 ≤ j < q. It follows that E_{2}^{pj} = 0 whenever 1 ≤ j < q, hence E_{r}^{pj} = 0 for every r ≥ 2 and the same values of j. Consequently :

0 = E_{∞}^{0q} ≃ Ker(d_{q+1}^{0q} : E_{2}^{0q} = E_{q+1}^{0q} → E_{q+1}^{q+1,0}).

However, theorem 15.1.37 implies that E_{r}^{p0} = 0 whenever p > 0 and r ≥ 2, therefore E_{2}^{0q} = E_{∞}^{0q}, i.e. E_{2}^{0q} = 0, which completes the inductive step. □

15.1.40. Let X be an ω-formal scheme, ℱ a quasi-coherent 𝒪_{X}-module. For every affine open subset U ⊂ X, we let Cl_{ℱ} (U) be the set consisting of all closed 𝒪_{X}(U)-submodules of ℱ (U). It follows from proposition 15.1.29(ii) that the rule

U ↦ Cl_{ℱ} (U)

defines a presheaf on the site of all affine open subsets of X. Namely, for an inclusion U′ ⊂ U of affine open subsets, the restriction map Cl_{ℱ} (U) → Cl_{ℱ} (U′) assigns to N ⊂ ℱ (U) the submodule N^{∼}(U′) ⊂ ℱ (U′) (here N^{∼} is a quasi-coherent 𝒪_{U}-module).

Proposition 15.1.41. With the notation of (15.1.40), the presheaf Cl_{ℱ} is a sheaf on the site of affine open subsets of X.

Proof. Let U ⊂ X be an affine open subset, and U = ⋃_{i∈I} U_{i} a covering of U by affine open subsets U_{i} ⊂ X. For every i, j ∈ I we let U_{ij} := U_{i} ∩ U_{j}. Suppose there is given, for every i ∈ I, a closed 𝒪_{X}(U_{i})-submodule N_{i} ⊂ ℱ (U_{i}), with the property that :

N_{ij} := N_{i}^{∼}(U_{ij}) = N_{j}^{∼}(U_{ij}) for every i, j ∈ I

(an equality of topological 𝒪_{X}(U_{ij})-submodules of ℱ (U_{ij})). Then (N_{i}^{∼})_{|U_{ij}} = (N_{j}^{∼})_{|U_{ij}}, by proposition 15.1.33, hence there exist a quasi-coherent 𝒪_{U}-module 𝒩 , and isomorphisms 𝒩_{|U_{i}} →^{∼} N_{i}^{∼}, for every i ∈ I, such that the induced 𝒪_{U_{i}}-linear maps N_{i}^{∼} → ℱ_{|U_{i}} assemble into a continuous 𝒪_{U}-linear morphism φ : 𝒩 → ℱ_{|U}. By construction, we have a commutative diagram of continuous maps with exact rows :

\xymatrix{0 \ar[r] & 𝒩 (U) \ar[r]^{ρ_{𝒩}} \ar[d] & ∏_{i∈I} N_{i} \ar[d] \ar[r] & ∏_{i,j∈I} N_{ij} \ar[d] \\ 0 \ar[r] & ℱ (U) \ar[r]^{ρ_{ℱ}} & ∏_{i∈I} ℱ (U_{i}) \ar[r] & ∏_{i,j∈I} ℱ (U_{ij})}

where the central vertical arrow is a closed immersion. However, the condition of definition 5.5.1(ii) implies that both ρ_{𝒩} and ρ_{ℱ} are admissible monomorphisms (in the sense of (8.6)), and moreover the image of ρ_{𝒩} is a closed submodule, since each N_{ij} is a separated module. Hence also the left-most vertical arrow is a closed immersion, and the assertion follows. □

Definition 15.1.42. Let f : X → Y be a morphism of ω-formal schemes. We say that f is affine (resp. a closed immersion) if there exists a covering Y = ⋃_{i∈I} U_{i} by affine open subsets, such that for every i ∈ I, the open subset f^{−1}U_{i} ⊂ X is an affine ω-formal scheme (resp. is isomorphic, as a U_{i}-scheme, to an ω-formal scheme of the form Spf A_{i}/J_{i}, where A_{i} := 𝒪_{Y} (U_{i}) and J_{i} ⊂ A_{i} is a closed ideal).

Corollary 15.1.43. Let f : X → Y be a morphism of ω-formal schemes. The following conditions are equivalent :

(i) f is an affine morphism (resp. a closed immersion).

(ii) For every affine open subset U ⊂ Y, the preimage f^{−1}U is an affine ω-formal scheme (resp. is isomorphic, as a U-scheme, to the ω-formal scheme Spf A/J, where A := 𝒪_{Y} (U), and J := Ker(A → 𝒪_{X}(f^{−1}U)) ⊂ A is a closed ideal).

Proof. Of course, it suffices to show that (i)⇒(ii). Hence, suppose that f is an affine morphism, and choose an affine open covering Y = ⋃_{i∈I} U_{i} such that f^{−1}U_{i} is an affine ω-formal scheme for every i ∈ I. We may find an affine open covering U = ⋃_{j∈J} V_{j} such that, for every j ∈ J there exists i ∈ I with V_{j} ⊂ I_{i}; moreover, we may assume that J is countable, by proposition 15.1.19(ii.d). The assumption implies that f^{−1}V_{j} is affine for every j ∈ J, and then remark 15.1.18(i.b) says that A := 𝒪_{X}(f^{−1}U) is an ω-admissible topological ring, so there exists a unique morphism g : f^{−1}U → Spf A such that g^{♮} : A → 𝒪_{X}(f^{−1}U) is the identity (proposition 15.1.22), and f_{|f^{−1}U} factors as the composition of g and the morphism h : Spf A → U induced by the natural map B := 𝒪_{Y} (U) → A. It remains to check that g is an isomorphism. To this aim, it suffices to verify that the restriction g_{|V_{j}} : f^{−1}V_{j} → h^{−1}V_{j} is an isomorphism for every j ∈ J. However, it is clear that f_{∗}𝒪_{X} is a quasi-coherent 𝒪_{Y} -module, whence a natural isomorphism of A-algebras :

𝒪_{X}(f^{−1}V_{j}) = f_{∗}𝒪_{X}(V_{j}) →^{∼} A_{j} := Aˆ⊗_{B}𝒪_{Y} (V_{j})

(proposition 15.1.33) as well as an isomorphism f^{−1}V_{j} →^{∼} Spf A_{j}. On the other hand, h^{−1}V_{j} = Spf A_{j}, and by construction g_{|V_{j}} is the unique morphism such that (g_{|V_{j}})^{♮} is the identity map of A_{j}. The assertion follows.

Next, suppose that f is a closed immersion, and choose an affine open covering Y = ⋃_{i∈I} U_{i}, and closed ideals J_{i} ⊂ A_{i} := 𝒪_{Y} (U_{i}) such that we have isomorphisms f^{−1}U_{i} →^{∼} Spf A_{i}/J_{i} for every i ∈ I. Set U_{ij} := U_{i} ∩ U_{j} for every i, j ∈ I. Clearly, J_{i}^{∼}(U_{ij}) = J_{j}^{∼}(U_{ij}) for every i, j ∈ I, hence there exists a unique closed ideal J ⊂ A := 𝒪_{Y} (U) such that :

(15.1.44)  J^{∼}(U_{i}) = J_{i}  for every i ∈ I

(proposition 15.1.41). Especially, we have J𝒪_{Y} (U_{i}) ⊂ J_{i}, whence a unique morphism of Uschemes : g_{i} : f^{−1}U_{i} → Z := Spf A/J, for every i ∈ I. The uniqueness of g_{i} implies in particular that g_{i|U_{ij}} = g_{j|U_{ij}} for every i, j ∈ I, whence a unique morphism g : f^{−1}U → Z of U-schemes. It remains to verify that g is an isomorphism, and to this aim it suffices to check that the restriction g^{−1}(U_{i} ∩Z) → U_{i} ∩Z is an isomorphism for every i ∈ I. The latter assertion is clear, in view of (15.1.44). □

Corollary 15.1.45. Let A be an ω-admissible topological ring, U ⊂ X := Spf A an affine open subset. We have :

(i) The following conditions are equivalent :

(a) U is truly affine.

(b) U ∩ Spec A/I is quasi-compact, for every open ideal I ⊂ A. (ii) Especially, every quasi-compact affine open subset of X is truly affine.

Proof. (i): Obviously (i.a)⇒(i.b). Conversely, let j : Y := Spf A/I → X be the closed immersion, and set A_{U} := 𝒪_{X}(U). On the one hand, we have j_{∗}𝒪_{Y} = (A/I)^{∼}; on the other hand, proposition 15.1.29(ii) gives an admissible short exact sequence of topological A_{U}-modules 0 → I^{∼}(U) → A_{U} → B := (A/I)^{∼}(U) → 0, and notice that the topology of B is discrete, since U is quasi-compact. Thus, (Y ∩ U, (𝒪_{Y})_{|U}) = Spf B = Spec B, by corollary 15.1.43(ii).

(ii) is an immediate consequence of (i). □

Remark 15.1.46. (i) Let A be an ω-admissible topological ring. Then Spf A may well contain affine subsets that are not truly affine. As an example, consider the one point compactification X := ℕ ∪ {∞} of the discrete topological space ℕ (this is the space which induces the discrete topology on its subset ℕ, and such that the open neighborhoods of ∞ are the complements of the finite subsets of ℕ). We choose any field F, which we endow with the discrete topology, and let A be the ring of all continuous functions X → F. We endow A with the discrete topology, in which case Spf A = Spec A, and one can exhibit a natural homeomorphism Spec A →^{∼} X (exercise for the reader). On the other hand, we have an isomorphism of topological rings :

𝒪_{Spf A}(ℕ) = lim_{b∈ℕ} 𝒪_{Spec A}({0, …, b}) ≃ k^{ℕ}

where k^{ℕ} is endowed with the product topology. A verification that we leave to the reader, shows that the natural injective ring homomorphism A → k^{ℕ} induces an isomorphism of ringed spaces

Spf k^{ℕ} →^{∼} (ℕ, 𝒪_{Spf A|ℕ})

hence ℕ ⊂ Spf A is an affine subset. However, ℕ is not an affine subset of Spec A, hence ℕ is not a truly affine open subset of Spf A.

(ii) On the other hand, suppose that A is c-admissible, in the sense of remark 15.1.18(ii), and let U ⊂ Spf A be any open subset. Combining corollary 15.1.45(ii) and remark 15.1.18, we see that U is an affine formal scheme in the sense of [58] if and only if it is an affine and quasi-compact ω-formal scheme, if and only if it is truly affine.

We conclude this section by reviewing briefly a standard method for getting formal schemes out of usual schemes.

15.1.47. Let X be a scheme, ℐ ⊂ 𝒪_{X} a quasi-coherent sheaf of ideals of finite type, and ℱ a quasi-coherent 𝒪_{X}-module. Let also i : X_{0} := Spec 𝒪_{X}/ℐ → X be the closed immersion. For every n ∈ ℕ, set ℱ_{n} := ℱ /ℐ^{n+1}ℱ; we endow the i^{−1}𝒪_{X}-module i^{−1}(ℱ_{n}) with its pseudodiscrete topology. Following [58, Ch.I, Déf.10.8.2], we define the completion of ℱ along X_{0}

ℱ^{∧} := lim_{n∈ℕ} i^{−1}ℱ_{n}

where the limit is taken in the category of sheaves of topological abelian groups. Especially, we may consider the completion 𝒪_{X^{∧}} := 𝒪_{X}^{∧} which is naturally a sheaf of topological rings on X_{0}, and clearly ℱ^{∧} is naturally an 𝒪_{X^{∧}}-module. For every open subset U ⊂ X, set U_{0} := U ∩ X_{0}; notice that, if U ⊂ U′ are two such open subsets with U_{0} = U′_{0}, then the restriction map ℱ_{n}(U′) → ℱ_{n}(U) is an isomorphism for every n ∈ ℕ, since the support of ℱ_{n} lies in X_{0}. There follows a natural identification

i^{−1}ℱ_{n}(U_{0}) →^{∼} ℱ_{n}(U) for every n ∈ ℕ and every open subset U ⊂ X

whence, by virtue of remark 5.5.2(iii) a natural isomorphism of topological groups

ℱ^{∧}(U_{0}) →^{∼} lim_{n∈ℕ} ℱ_{n}(U) for every open subset U ⊂ X.

If moreover U is quasi-compact, then ℱ_{n}(U) is a discrete topological group for every n ∈ ℕ. If U is affine, and A_{U} := 𝒪_{X}(U), I_{U} := ℐ (U), we have ℱ_{n}(U) = A_{U}/I_{U}^{n} ⊗_{A_{U}} ℱ (U) for every n ∈ ℕ, so that

ℱ^{∧}(U_{0}) = ℱ (U)^{∧}

where ℱ (U)^{∧} denotes the I_{U}-adic completion of ℱ (U). Especially, the ringed space

(U_{0}, (𝒪_{X^{∧}})_{|U_{0}})

is an affine ω-formal scheme (see definition 15.1.17(iv)), isomorphic to Spf 𝒪_{X}(U)^{∧}. Thus

X^{∧} := (X_{0}, 𝒪_{X^{∧}})

is an ω-formal scheme called the completion of X along X_{0}, and ℱ^{∧} is a quasi-coherent 𝒪_{X^{∧}}- module (see definition 15.1.31(i)). Furthermore, the system of projections (i^{−1}ℱ → ℱ_{n}) yields a morphism of i^{−1}𝒪_{X}-modules

i^{−1}ℱ → ℱ^{∧}.

Especially, the map ϑ : i^{−1}𝒪_{X} → 𝒪_{X^{∧}} is a morphism of sheaves of rings, and the pair

π_{X} := (i, ϑ) : X^{∧} → X

is a natural morphism of ringed spaces. By inspecting the proof of lemma 15.1.12(i), we see more precisely that π_{X} is a morphism of locally ringed spaces.

Remark 15.1.48. In the situation of (15.1.47), let U ⊂ X^{∧} be any quasi-compact open subset. Then the topology of ℱ (U) is the linear topology defined by the system of submodules

(Γ(U, (ℐ^{n}ℱ)^{∧}) | n ∈ ℕ).

Indeed, the assertion is clear if U is affine, since in this case the ideal I_{U} is finitely generated (see remark 8.3.3(ii,iv)). For the general case, let (U_{λ} | λ ∈ Λ) be a finite affine covering of U, and endow M := ∏_{λ∈Λ} ℱ (U_{λ}) with the product topology; according to remark 5.5.2(i), the topology of ℱ (U) agrees with the topology induced by M via the natural injection ℱ (U) → M. However, the topology of M is I_{U}-adic, and we have ℱ (U) ∩ I_{U}^{n}M = Γ(U, (ℐ^{n}ℱ)^{∧}) for every n ∈ ℕ, whence the contention.

15.1.49. Keep the notation of (15.1.47), and let Y be another scheme, 𝒥 ⊂ 𝒪_{Y} another quasi-coherent sheaf of ideals, and f : X → Y a morphism of schemes such that the associated map of sheaves 𝒪_{Y} → f_{∗}𝒪_{X} restricts to a morphism 𝒥 → f_{∗}ℐ. We may then consider the completion Y^{∧} of Y along Y_{0} := Spec𝒪_{Y} /𝒥. Let also U ⊂ X and V ⊂ Y be two affine open subsets with f(U) ⊂ V, set I_{U} := ℐ (U) ⊂ A_{U} := 𝒪_{X}(U), J_{V} := 𝒥 (V) ⊂ B_{V} := 𝒪_{Y} (V), and endow A_{U} (resp. B_{V}) with its I_{U}-adic (resp. J_{V} -adic) topology; it follows that the resulting map B_{V} → A_{U} is continuous, and therefore we get an induced morphism

f_{U,V}^{∧} : (U_{0}, (𝒪_{X^{∧}})_{|U_{0}}) → (V_{0}, (𝒪_{Y^{∧}})_{|V_{0}}).

If moreover, U′ ⊂ U and V′ ⊂ V is another pair of affine open subsets such that f(U′) ⊂ V′, it is clear that f_{U′,V′}^{∧} agrees with the restriction of f_{U,V}^{∧}. We deduce a well defined morphism of formal schemes

f^{∧} : X^{∧} → Y^{∧}

that makes commute the diagram

\xymatrix{X^{∧} \ar[r]^{f^{∧}} \ar[d]_{π_{X}} & Y^{∧} \ar[d]^{π_{Y}} \\ X \ar[r]^{f} & Y.}

Especially, if f is an open immersion and ℐ = f^{−1}𝒥, then clearly f^{∧} is an open immersion.

15.2. Analytically noetherian rings.

Definition 15.2.1. Let A be an adic topological ring with a finitely generated ideal I of adic definition.

(i) We say that A is analytically noetherian if the following two conditions hold : (a) The analytic locus of Spec A is a noetherian scheme (see definition 8.3.28).

(b) For every finitely generated A-module M, the increasing sequence of submodules

(Ann_{M}(I^{n}) | n ∈ ℕ)

is stationary.

(ii) We say that A is universally analytically noetherian if, for every n ∈ ℕ, the polynomial A-algebra A[X_{1}, …, X_{n}], endowed with its I-adic topology, is analytically noetherian.

(iii) We say that A satisfies the topological Artin-Rees condition, if the following holds. For every A-module M of finite type, and every submodule N ⊂ M, the I-adic topology on N agrees with the topology induced by the I-adic topology of M.

(iv) We say that an A-module M is analytically of finite type, if there exists a submodule N ⊂ M of finite type and an integer n ∈ ℕ such that I^{n}M ⊂ N.

Remark 15.2.2. (i) It is easily seen that condition (b) of definition 15.2.1(i) does not depend on the chosen finitely generated ideal I of adic definition of A. Likewise, the class of A-modules of analytically finite type is determined solely by the topology of A.

(ii) Obviously, every noetherian ring is analytically noetherian when endowed with the adic topology defined by any of its ideals.

(iii) Let A be an analytically noetherian ring, and M an A-module of analytically finite type; then the sequence (Ann_{M}(I^{k}) | k ∈ ℕ) is stationary. Indeed, pick a finitely generated submodule N ⊂ M such that I^{n}M ⊂ N for some n ∈ ℕ; by assumption, there exists t ∈ ℕ such that Ann_{N}(I^{t}) = Ann_{N}(I^{s}) for every s ≥ t. It follows easily that Ann_{M}(I^{k}) = Ann_{M}(I^{t+n}) for every k ≥ t + n.

Lemma 15.2.3. Let A be an adic topological ring that admits a finitely generated ideal I of adic definition. Then we have :

(i) The following conditions are equivalent :

(a) A satisfies the topological Artin-Rees condition.

(b) For every A-module M of finite type, every n ∈ ℕ, and every submodule N ⊂ M

such that I^{n}N = 0, there exists m ∈ ℕ such that N ∩ I^{m}M = 0. (ii) Suppose that the equivalent conditions of (i) hold for A, let M be any A-module of analytically finite type, and N ⊂ M any submodule. Then the I-adic topology on N agrees with the topology induced by the I-adic topology of M.

Proof. (i): Clearly (a)⇒(b). Conversely, let M be an A-module of finite type, N ⊂ M a submodule, and n ∈ ℕ any integer. Set M′ := M/I^{n}N and N′ := N/I^{n}N; clearly I^{n}N′ = 0, so (b) implies that N′∩I^{m}M′ = 0 for some m ∈ ℕ, and the latter means that N ∩I^{m}M ⊂ I^{n}N, whence (a).

(ii): By assumption, there exist n ∈ ℕ and a submodule M′ ⊂ M of finite type such that I^{n}M ⊂ M′; set N′ := M′ ∩ N. Then, for every t ∈ ℕ there exists s ∈ ℕ such that I^{s}M′ ∩ N = I^{s}M′ ∩ N′ ⊂ I^{t}N′ ⊂ I^{t}N. We conclude that I^{s+n}M ∩ N ⊂ I^{t}N, whence the assertion. □

Lemma 15.2.4. Let A be any analytically noetherian (resp. universally analytically noetherian) topological ring, and I ⊂ A any ideal of adic definition. We have :

(i) For every multiplicative subset S ⊂ A, the localization S^{−1}A is analytically noetherian (resp. universally analytically noetherian) for its S^{−1}I-adic topology.

(ii) Let f : A → B be a ring homomorphism, and 𝒯_{B} the IB-adic topology on B. If f is finite (resp. of finite type) Then (B, 𝒯_{B}) is analytically noetherian (resp. universally analytically noetherian).

Proof. (i): Suppose that A is analytically noetherian. Clearly the analytic locus of Spec S^{−1}A is noetherian. Next, let M be any S^{−1}A-module of finite type; pick a finite system of generators x_{•} := (x_{1}, …, x_{k}) for M, and let N be the A-submodule of M generated by x_{•}. Then S^{−1}N = M, and Ann_{M}(S^{−1}I^{n}) = S^{−1}Ann_{N}(I^{n}) for every n ∈ ℕ, whence the assertion. The assertion for the case where A is universally analytically noetherian case follows immediately,

(ii): Again, suppose first that A is analytically noetherian, and f is finite. The analytic locus of Spec B is finite over the analytic locus of Spec A, hence it is noetherian. Next, if M is a Bmodule of finite type, then it is also an A-module of finite type, and Ann_{M}(I^{n}B) = Ann_{M}(I^{n}) for every n ∈ ℕ, whence the assertion. Lastly, suppose that A is universally analytically noetherian and f is of finite type. For every n ∈ ℕ, the A-algebra B[T_{1}, …, T_{n}] is a quotient of a free polynomial A-algebra of finite type, so it is analytically noetherian, by the foregoing case, whence the assertion. □

Proposition 15.2.5. Let A be a ring, I, J ⊂ A two finitely generated ideals, and denote by 𝒯_{I} (resp. 𝒯_{J}, resp. 𝒯_{I+J}) the I-adic (resp. J-adic, resp. (I + J)-adic) topology on A. We have :

(i) If (A, 𝒯_{I}) is analytically noetherian, and 𝒯_{J} is finer than 𝒯_{I}, then (A, 𝒯_{J}) is analytically noetherian.

(ii) If (A, 𝒯_{I}) and (A, 𝒯_{J}) are analytically noetherian, then the same holds for (A, 𝒯_{I+J}). (iii) If (A, 𝒯_{I}) and (A, 𝒯_{J}) satisfy the topological Artin-Rees condition, the same holds for

(A, 𝒯_{I+J}).

Proof. (i): After replacing J by J^{n} for a suitable n ∈ ℕ, we may assume that J ⊂ I, in which case it is clear that the analytic locus U_{I} ⊂ X := Spec A of (A, 𝒯_{I}) contains the analytic locus U_{J} of (A, 𝒯_{J}). Since by assumption U_{I} is noetherian, the same holds for the scheme U_{J}. Next, let M be any A-module of finite type, and for every n ∈ ℕ, denote by ℳ_{n} the quasi-coherent 𝒪_{X}-module associated with M_{n} := Ann_{M}(J^{n}). Since U_{I} is noetherian, there exists t ∈ ℕ such that ℳ_{n|U_{I}} = ℳ_{t|U_{I}} for every n ≥ t. Set M′ := M/M_{t}, and M′_{n} := Ann_{M′}(J^{n}) for every n ∈ ℕ. It is easily seen that

(15.2.6)  M′_{n} = M_{n+t}/M_{t}  for every n ∈ ℕ.

On the other hand, since (ℳ_{n+t}/ℳ_{t})_{|U_{I}} = 0 for every n ∈ ℕ, we get

⋃_{n∈ℕ} M′_{n} = N := ⋃_{n∈ℕ} Ann_{M′}(I^{n}).

By assumption, there exists s ∈ ℕ such that N = Ann_{M′}(I^{s}), and therefore N ⊂ M′_{s} ⊂ N. Combining with (15.2.6) we conclude that M_{n} = M_{s+t} for every n ≥ s + t, whence the assertion.

(ii): Let U_{I} and U_{J} be as in the foregoing, and define U_{I+J} likewise as the analytic locus of (A, 𝒯_{I+J}). Then U_{I+J} = U_{I} ∪ U_{J}; since by assumption U_{I} and U_{J} are noetherian, the same then holds for U_{I+J}. Next, let M be as in the foregoing; by assumption, there exists t ∈ ℕ such that Ann_{M}(I^{n}) = Ann_{M}(I^{t}) and Ann_{M}(J^{n}) = Ann_{M}(J^{t}) for every n ≥ t. It then follows easily that Ann_{M}((I + J)^{n}) = Ann_{M}((I + J)^{2t−1}) for every n ≥ 2t − 1, whence the assertion.

(iii): Let M be any A-module of finite type, N ⊂ M a submodule such that (I + J)^{n}N = 0 for some n ∈ ℕ; in light of lemma 15.2.3(i), it suffices to show that there exists m ∈ ℕ such that N ∩ (I + J)^{m}M = 0. However, again by lemma 15.2.3(i), our assumption on (A, 𝒯_{I}) implies that N ∩ I^{t}M = 0 for some t ∈ ℕ; set M′ := M/I^{t}M and let N′ ⊂ M′ be the image of N. Notice that J^{n}N′ = 0; invoking again lemma 15.2.3(i), our assumption on (A, 𝒯_{J}) implies that N′ ∩J^{s}M′ = 0 for some s ∈ ℕ. We conclude that N ∩(I^{t} +J^{s})M = 0, and then m := t+s−1 will do. □

As a corollary, we get the following weak form of the Artin-Rees lemma :

Corollary 15.2.7. Let (A, 𝒯 ) be an analytically noetherian topological ring, f := (f_{1}, …, f_{n}) a finite sequence of elements of A that generates an ideal of adic definition. We have :

(i) A satisfies the topological Artin-Rees condition.

(ii) A satisfies condition (d)_{f} of (7.8.21).

Proof. For every i = 1, …, n, let 𝒯_{i} be the f_{i}A-adic topology on A; by proposition 15.2.5(i), (A, 𝒯_{i}) is analytically noetherian for every i = 1, …, n. Taking into account lemma 15.2.4(ii), we see that A fulfills the assumptions of lemma 7.8.44, whence (ii).

Next, if the topological Artin-Rees condition holds for each (A, 𝒯_{i}), proposition 15.2.5(iii) and a simple induction show that the same holds for (A, 𝒯 ). Thus, we may assume from start that n = 1 and A admits a principal ideal I = fA of adic definition.

Now, let M be any A-module of finite type, N ⊂ M a submodule, and set M′ := M/N. Pick t ∈ ℕ such that Ann_{M′}(f^{n}) = Ann_{M′}(f^{t}) for every n ≥ t; it follows easily that

N ∩ f^{n+t}M = f^{n}(N ∩ f^{t}M) for every n ∈ ℕ

whence the contention. □

Corollary 15.2.8. Let A be an analytically noetherian ring, I ⊂ A a finitely generated ideal of adic definition, and

0 → M_{1} → M_{2} → M_{3} → 0

a short exact sequence of A-modules, which we endow with their I-adic topologies. We have : (i) M_{2} is analytically of finite type if and only if the same holds for both M_{1} and M_{3}. (ii) Suppose that M_{2} is analytically of finite type. Then the induced sequence of separated

completions

(15.2.9)  0 → M_{1}^{∧} → M_{2}^{∧} → M_{3}^{∧} → 0

is short exact.

Proof. (i): Suppose that M_{2} is analytically of finite type. Then it is easily seen that the same holds for M_{3}. Next, set X := Spec A, so that U := X \ Spec A/I is the analytic locus of X. Let also ℳ_{1} and ℳ_{2} be the quasi-coherent 𝒪_{X}-modules associated with M_{1} and respectively M_{2}. Then ℳ_{2|U} is a quasi-coherent 𝒪_{U}-module of finite type, and since U is noetherian, the same then holds for its submodule ℳ_{1|U}. Hence, we may find a quasi-coherent 𝒪_{X}-submodule 𝒫 ⊂ ℳ_{1} of finite type such that 𝒫_{|U} = ℳ_{1|U}, and we denote by P ⊂ M_{1} the submodule corresponding to 𝒫. Set M′_{1} := M_{1}/P and M′_{2} := M_{2}/P; by construction, the support of M′_{1} lies in Spec A/I, and since A is analytically noetherian and M′_{2} is analytically of finite type, it follows that I^{n}M′_{1} = 0 for some n ∈ ℕ (remark 15.2.2(iii)). Thus, I^{n}M_{1} ⊂ P, which shows that M_{1} is analytically of finite type.

Lastly, suppose that M_{1} and M_{3} are analytically of finite type, and pick n ∈ ℕ and finitely generated submodules N_{i} ⊂ M_{i} such that I^{n}M_{i} ⊂ N_{i} for i = 1, 3. Then we may find a submodule P ⊂ M_{2} of finite type whose image in M_{3} equals N_{3}, and it is easily seen that I^{2n}M_{2} ⊂ P + N_{1}.

(ii): This follows from corollary 15.2.7(i), lemma 15.2.3(ii) and proposition 8.2.13(v). □

Proposition 15.2.10. Let A and I be as in corollary 15.2.8, and M any A-module of analytically finite type. Endow M with its I-adic topology, and denote by A^{∧} and M^{∧} the separated completions of A and M. We have :

(i) The natural map φ_{M} : A^{∧} ⊗_{A} M → M^{∧} is an isomorphism.

(ii) The completion map A → A^{∧} is flat.

Proof. (i): We consider first the case where M is of finite type. Pick any surjective A-linear map ψ : L → M, from a finitely presented A-module L, and endow K := Ker ψ with its I-adic topology. We consider the induced commutative diagram

\xymatrix{0 \ar[r] & A^{∧} ⊗_{A} K \ar[r] \ar[d]_{φ_{K}} & A^{∧} ⊗_{A} L \ar[r] \ar[d]^{φ_{L}} & A^{∧} ⊗_{A} M \ar[r] \ar[d]^{φ_{M}} & 0 \\ 0 \ar[r] & K^{∧} \ar[r] & L^{∧} \ar[r] & M^{∧} \ar[r] & 0}

whose bottom row is short exact, by corollary 15.2.8. If L is free of finite rank, φ_{L} is an isomorphism; then φ_{M} is surjective and φ_{K} is injective. Since we can always find such a surjection φ with L free of finite rank, we conclude already that φ_{M} is surjective for every finitely generated A-module M.

Next, suppose that M is finitely presented and L is still free of finite rank; in this case, K is an A-module of finite type ([75, Lemma 2.3.18(iii)]), so φ_{K} is also surjective, by the foregoing, hence φ_{K} is an isomorphism and therefore the same holds for φ_{M}.

For a general M of finite type, in view of corollary 15.2.8(i) we may find a submodule K′ ⊂ K of finite type such that I^{n}K ⊂ K′. Set L′ := L/K′, and let ψ′ : L′ → M be the surjective map induced by ψ; then L′ is still finitely presented, so we may replace L by L′ and ψ by ψ′, and assume from start that the I-adic topology is discrete on K, in which case the completion map K → K^{∧} is an isomorphism, consequently φ_{K} is again surjective, and we conclude as in the foregoing that φ_{M} is an isomorphism, as required.

Lastly, suppose that M is of analytically finite type, and pick a submodule N ⊂ M of finite type such that I^{n}(M/N) = 0 for some n ∈ ℕ. We consider the analogous commutative diagram

\xymatrix{0 \ar[r] & A^{∧} ⊗_{A} N \ar[r] \ar[d]_{φ_{N}} & A^{∧} ⊗_{A} M \ar[r] \ar[d]^{φ_{M}} & A^{∧} ⊗_{A} (M/N) \ar[r] \ar[d]^{φ_{M/N}} & 0 \\ 0 \ar[r] & N^{∧} \ar[r] & M^{∧} \ar[r] & (M/N)^{∧} \ar[r] & 0}

and it is easily seen that both φ_{M/N} and the completion map M/N → (M/N)^{∧} are isomorphisms. The same holds for φ_{N}, by the foregoing, so finally φ_{M} is an isomorphism.

(ii): Let N → M be any injective homomorphism of A-modules; we need to check that the induced map A^{∧} ⊗_{A} N → A^{∧} ⊗_{A} M is still injective. Since the tensor product commutes with all colimits, we are easily reduced to the case where M and N are A-modules of finite type. In this case, endow M and N with their I-adic topologies; by corollary 15.2.8, the induced map on I-adic completions N^{∧} → M^{∧} is injective. Then it suffices to apply (i) to conclude. □

15.2.11. Let (A, 𝒯 ) be a complete and separated adic topological ring that admits a principal ideal I = Aa of adic definition. Let also M be an A-module whose I-adic topology 𝒯_{M} is complete and separated. For a submodule N ⊂ M let N^{c} be the topological closure of N in M, and for every k ∈ ℕ denote by N_{k} ⊂ M the submodule of all x ∈ M such that a^{k}x ∈ N. Set

N^{s} := ⋃_{k∈ℕ} N_{k}.

Lemma 15.2.12. In the situation of (15.2.11), let N′ ⊂ N ⊂ M be two submodules, such that N′ is dense in N (for the topology 𝒯_{M}) and N = N^{s}. The following holds :

(i) The I-adic topology on N agrees with the topology induced by 𝒯_{M}.

(ii) N^{c} = (N^{c})^{s}.

(iii) If N = N^{c} as well, and N[a^{−1}] is an A[a^{−1}]-module of finite type, then N is analytically of finite type, and N′ = N.

Proof. (i): Since N = N^{s} we have a^{m}N = a^{m}M ∩ N for every m ∈ ℕ, whence the assertion.

(ii): If x ∈ M and ax ∈ N^{c}, then for every n ∈ ℕ there exists z_{n} ∈ M such that ax − a^{n}z_{n} ∈ N, and therefore x − a^{n−1}z_{n} ∈ N for every n ≥ 1, whence x ∈ N^{c}.

(iii): By assumption, there exists a finitely generated submodule L ⊂ N with

N = L^{s} = ⋃_{k∈ℕ}(L_{k})^{c}

(indeed, the first identity holds if we choose L such that N[a^{−1}] = L[a^{−1}], and the second follows, since N = N^{c}). Notice next that we may regard M as a complete metric space, with the metric defined by the rule

d(x, y) := \{ \begin{matrix}0 & if x = y \\ 2^{−b} & if x ≠ y, where b := max(n ∈ ℕ | x − y ∈ a^{n}M). \end{matrix}

By Baire’s category theorem ([41, Ch.IX, §5, n.3, Th.1]), it follows that (L_{k})^{c} is open in N for some k ∈ ℕ (for the topology of N induced by 𝒯_{M}), and therefore it contains a^{m}N for some m ∈ ℕ; thus a^{k+m}N ⊂ a^{k}(L_{k})^{c} ⊂ L^{c}. Combining with (i) we deduce that the I-adic topology of L^{c} agrees with the topology induced by 𝒯_{M}; then by lemma 8.3.22(i) we conclude that L = L^{c}, so a^{k+m}N ⊂ L, which shows that N is analytically of finite type.

Lastly, since N′ is dense in N, from (i) we see that N = N′ + a^{m+k+1}N ⊂ N′ + aL. It follows that L = (N′ ∩ L) + aL, and by Nakayama’s lemma we get L = N′ ∩ L, i.e. L ⊂ N′, so N′ is open in N and thus N = N′. □

Theorem 15.2.13. Let (A, 𝒯 ) be a complete and separated adic topological ring that admits a finitely generated ideal I of adic definition, and suppose that the analytic locus of Spec A is noetherian. Then A is analytically noetherian.

Proof. (This is [71, Th.5.1.2].) Let a_{1}, …, a_{k} be a finite system of elements of A that generates I, and for every i = 1, …, k denote by 𝒯_{i} that Aa_{i}-adic topology on A; taking into account proposition 15.2.5(ii), it suffices to show that (A, 𝒯_{i}) is analytically noetherian for i = 1, …, k. However, since the analytic locus of (A, 𝒯 ) is noetherian, the same holds for the analytic locus of (A, 𝒯_{i}), and the topology 𝒯_{i} is also complete and separated for every i = 1, …, k, by virtue of lemma 8.3.12. We are then reduced to the case where I = Aa is a principal ideal.

Now, let M be an A-module of finite type; we need to show that the sequence of submodules (Ann_{M}(a^{n}) | n ∈ ℕ) is stationary. Let us write M = L/Q for a free A-module L of finite rank, and a submodule Q ⊂ L, and set N := Q^{s} (notation of (15.2.11)). Notice that

N/Q = ⋃_{n∈ℕ} Ann_{M}(a^{n})

so it suffices to show that a^{k}N ⊂ Q for some k ∈ ℕ. Now, obviously N^{s} = N, so N^{c} = (N^{c})^{s} as well, by lemma 15.2.12(ii). Moreover, since A[a^{−1}] is noetherian, N^{c}[a^{−1}] is an A[a^{−1}]- module of finite type, and consequently N^{c} is analytically of finite type, and N = N^{c}, by lemma 15.2.12(iii). It follows that N/Q is also analytically of finite type; so, pick a submodule N′ ⊂ N/Q of finite type and an integer k ∈ ℕ such that a^{k}(N/Q) ⊂ N′; since N′ is finitely generated, there exists as well n ∈ ℕ such that a^{n}N′ = 0, so finally a^{k+n}(N/Q) = 0, as required. □

Next, we globalize definition 15.2.1to schemes as follows.

Definition 15.2.14. Let A be an adic topological ring that has a finitely generated ideal I of adic definition, X an A-scheme, and ℱ a quasi-coherent 𝒪_{X}-module.

(i) We say that X is locally analytically noetherian if every point of X admits an affine open neighborhood U such that the following holds. Endow A_{U} := 𝒪_{X}(U) with the I-adic topology 𝒯_{U}; then (A_{U}, 𝒯_{U}) is analytically noetherian.

(ii) We say that X is analytically noetherian if it is quasi-compact, quasi-separated and locally analytically noetherian.

(iii) We say that ℱ is analytically of finite type if every point of X admits an affine open neighborhood U such that ℱ (U) is an (A_{U}, 𝒯_{U})-module of analytically finite type.

Remark 15.2.15. If A is a universally analytically noetherian ring, then every A-scheme of finite type is analytically noetherian, by virtue of lemma 15.2.4.

Lemma 15.2.16. Let A and I be as in definition 15.2.14. Let also X be a quasi-compact and quasi-separated A-scheme, and ℱ a quasi-coherent 𝒪_{X}-module. We have :

(i) If ℱ is analytically of finite type, there exist an integer n ∈ ℕ and a quasi-coherent 𝒪_{X}-submodule of finite type 𝒢 ⊂ ℱ such that I^{n}ℱ ⊂ 𝒢.

(ii) If ℱ is analytically of finite type and X is analytically noetherian, there exists p ∈ ℕ

such that Ann_{ℱ} (I^{q}) = Ann_{ℱ} (I^{p}) for every integer q ≥ p. (iii) If ℱ is of finite type and X is analytically noetherian, there exist a finitely presented

𝒪_{X}-module 𝒢, an epimorphism φ : 𝒢 → ℱ of 𝒪_{X}-modules, and an integer n ∈ ℕ such that I^{n}Ker φ = 0.

Proof. (i): By assumption, we may find a finite affine open covering (U_{λ} | λ ∈ Λ) of X, and for every λ ∈ Λ a quasi-coherent 𝒪_{U_{λ}}-submodule ℋ_{λ} of finite type of ℱ_{λ} := ℱ_{|U_{λ}} and an integer n_{λ} ∈ ℕ such that I^{n_{λ}}ℱ_{λ} ⊂ ℋ_{λ}. We may also find, for every λ ∈ Λ, a finitely presented quasi-coherent 𝒪_{U_{λ}}-module ℋ′_{λ} with an 𝒪_{U_{λ}}-linear morphism φ_{λ} : ℋ′_{λ} → ℱ_{λ} whose image is ℋ_{λ}. According to lemma 10.3.27(i) there exist a finitely presented quasi-coherent 𝒪_{X}-module 𝒢_{λ} with 𝒢_{λ|U_{λ}} = ℋ′_{λ} and a morphism ψ_{λ} : 𝒢_{λ} → ℱ_{λ} of 𝒪_{X}-modules such that ψ_{λ|U_{λ}} = φ_{λ}. Then we may take 𝒢 := ∑_{λ∈Λ} Im ψ_{λ}.

(ii): Suppose that for every λ ∈ Λ there exists p_{λ} ∈ ℕ such that Ann_{ℱ_{λ}} (I^{q}) = Ann_{ℱ_{λ}} (I^{p_{λ}}) for every q ≥ p_{λ}. Then p := max(p_{λ} | λ ∈ Λ) fulfills the stated condition. Thus, we may suppose that X = Spec A for an analytically noetherian ring A, in which case the assertion follows from remark 15.2.2(iii).

(iii): By assumption, we may find a finite affine covering (U_{λ} | λ ∈ Λ) of X such that A_{λ} := 𝒪_{X}(U_{λ}) is analytically noetherian relative to its IA_{λ}-adic topology, for every λ ∈ Λ. Pick a finitely presented quasi-coherent 𝒪_{U_{λ}}-module ℋ_{λ} and an epimorphism ψ_{λ} : ℋ_{λ} → ℱ_{|U_{λ}} for every λ ∈ Λ. By lemma 10.3.27(i) there exist a finitely presented quasi-coherent 𝒪_{X}- module ℋ′_{λ} with ℋ′_{λ|U_{λ}} = ℋ_{λ} and a morphism of 𝒪_{X}-modules ψ′_{λ} : ℋ′_{λ} → ℱ such that ψ′_{λ|U_{λ}} = ψ_{λ}, for every λ ∈ Λ. Set ℋ := ⨁_{λ∈Λ} ℋ′_{λ}; the sum of the morphisms ψ′_{λ} is an epimorphism ψ : ℋ → ℱ, and ℋ is clearly finitely presented. For every λ ∈ Λ, let also V_{λ} ⊂ U_{λ} be the analytic locus; then V := ⋃_{λ∈Λ} V_{λ} is locally noetherian, quasi-compact and quasi-separated, and therefore 𝒦 := Ker ψ restricts to a finitely presented quasi-coherent 𝒪_{V} - module 𝒦_{|V}. Then, invoking again lemma 10.3.27(i) we find a quasi-coherent finitely presented 𝒪_{X}-module 𝒦 ′ and an 𝒪_{X}-linear morphism 𝒦 ′ → 𝒦 that restricts to an isomorphism on V. Let 𝒦 ′′ ⊂ 𝒦 be the image of 𝒦 ′, and set 𝒢 := ℋ /𝒦 ′′. Clearly ψ factors through an epimorphism φ : 𝒢 → ℱ, and the support of Ker φ lies in X \ V. Lastly, since A_{λ} is analytically noetherian, we may find an integer n_{λ} ∈ ℕ such that I^{n_{λ}} · (Ker φ)_{|V_{λ}} = 0, for every λ ∈ Λ. The assertion then holds with n := max(n_{λ} | λ ∈ Λ). □

Proposition 15.2.17. Let A be an adic topological ring with a finitely generated ideal I of adic definition, B an A-algebra, that we endow with its IB-adic topology 𝒯_{B}. Let also F be any B-module, and denote by ℱ the quasi-coherent 𝒪_{Spec B}-module arising from F. We have :

(i) The following conditions are equivalent : (a) (B, 𝒯_{B}) is analytically noetherian.

(b) Spec B is an analytically noetherian A-scheme. (ii) The following conditions are equivalent :

(a) F is a B-module of analytically finite type.

(b) ℱ is an 𝒪_{Spec B}-module of analytically finite type.

Proof. (i): Clearly (a)⇒(b). Hence, suppose that X := Spec B is analytically noetherian, and let (U_{i} | i = 1, …, k) be a finite affine covering of X such that B_{i} := 𝒪_{X}(U_{i}) is analytically noetherian for its IB_{i}-adic topology for every i = 1, …, k. By virtue of lemma 15.2.4, we may assume that for every i = 1, …, k there exists b_{i} ∈ B such that B_{i} = B[b_{i}^{−1}]. Clearly the analytic locus U of X is the union of the analytic loci of Spec B_{1}, …, Spec B_{k}, so U is noetherian. Next, let M be any B-module of finite type, and for every i = 1, …, k, set M_{i} := B_{i} ⊗_{B} M; choose n ∈ ℕ such that Ann_{M_{i}}(I^{r}B_{i}) = Ann_{M_{i}}(I^{n}B_{i}) for every i = 1, …, n and every r ≥ n. It follows that Ann_{M}(I^{r}) = Ann_{M}(I^{n}) for every r ≥ n, whence the contention.

(ii) follows easily from lemma 15.2.16(i). □

15.2.18. Let A be a universally analytically noetherian ring, I ⊂ A a finitely generated ideal of adic definition, X a quasi-separated A-scheme locally of finite type, and ℱ a quasi-coherent 𝒪_{X}-module. According to (15.1.47) we may consider the completions X^{∧} and ℱ^{∧} of X and ℱ along the closed subscheme X_{0} := Spec 𝒪_{X}/I𝒪_{X} = Spec A/I ×_{Spec A} X, which are respectively a formal scheme endowed with a natural morphism of locally ringed spaces

π : X^{∧} → X

and a quasi-coherent 𝒪_{X^{∧}}-module with a natural map of 𝒪_{X^{∧}}-modules

(15.2.19)  π^{∗}ℱ → ℱ^{∧}.

Proposition 15.2.20. With the notation of (15.2.18), the following holds :

(i) If ℱ is analytically of finite type, the map (15.2.19) is an isomorphism.

(ii) We have H^{i}(U, π^{∗}ℱ) = 0 for every affine quasi-compact open subset U ⊂ X^{∧} and

every i > 0. (iii) If X is affine, the natural map

𝒪_{X^{∧}}(X^{∧}) ⊗_{𝒪_{X}(X)} H^{0}(X, ℱ) → H^{0}(X^{∧}, π^{∗}ℱ)

is an isomorphism. (iv) Let 𝔘 := (U_{i} | i ∈ I) be an open covering of X^{∧}, and suppose that U_{i} ∩ U_{j} is affine for

every i, j ∈ I. Then there is a natural isomorphism (notation of (10.2.18)) :

H_{alt}^{•}(𝔘, π^{∗}ℱ) →^{∼} H^{•}(X^{∧}, π^{∗}ℱ).

Proof. (i): The assertion is local on X, hence we may assume that X is local, say X = Spec B, and ℱ is the quasi-coherent 𝒪_{X}-module attached to the module F := ℱ (X). Let T be any final object in the category of topological spaces (i.e. T is the unique topology over a set that has only one point), and endow T with a structure sheaf 𝒪_{T} by declaring that 𝒪_{T} (T) := B and 𝒪_{T} (∅) := 0. The module F defines an 𝒪_{T} -module F_{T} by the rule : F_{T} (T) := F and F_{T} (∅) := 0. The system of restriction maps B → 𝒪_{X}(U) for U ranging over the open subsets of X, may be viewed as a morphism of ringed spaces

ε : X → T

and with this notation we then have ℱ = ε^{∗}F_{T}. Set as well ε^{∧} := ε ∘ π; there follows a natural isomorphism of 𝒪_{X^{∧}}-modules

π^{∗}ℱ →^{∼} ε^{∧∗}F_{T} .

In other words, π^{∗}ℱ is the sheaf associated to the presheaf given by the rule :

U_{0} ↦ 𝒪_{X^{∧}}(U_{0}) ⊗_{B} F for every open subset U_{0} ⊂ X^{∧} = X_{0}

and the discussion of (15.1.47) shows that (15.2.19) is the morphism associated to the morphism of presheaves given, on every affine open subset U_{0} ⊂ X_{0}, by the natural map

(15.2.21)  𝒪_{X}(U)^{∧} ⊗_{B} F → ℱ (U)^{∧}

where U ⊂ X is any affine open subset such that U_{0} = U ∩X_{0}, and where 𝒪_{X}(U)^{∧} and ℱ (U)^{∧} denote the I-adic completions of 𝒪_{X}(U) and respectively ℱ (U). However, since ℱ is quasicoherent, we have ℱ (U) = 𝒪_{X}(U) ⊗_{B} F; taking into account remark 15.2.15and propositions 15.2.10 and 15.2.17(ii) we deduce that (15.2.21) is an isomorphism for every such U_{0}. Since the affine open subsets are a basis of the topology of X^{∧} that is closed under finite intersections, the assertion follows.

(ii): Since U is quasi-compact, π(U) is contained in a quasi-compact open subset V ⊂ X; we may then replace X by V, and assume from start that X is quasi-compact and quasiseparated. In this case, ℱ is the colimit of the filtered family (ℱ_{λ} | λ ∈ Λ) of its quasi-coherent 𝒪_{X}-submodules of finite type (proposition 10.3.31), hence π^{∗}ℱ is the colimit of the system (π^{∗}ℱ_{λ} | λ ∈ Λ). Since U is a spectral topological space, proposition 10.1.10 then reduces to showing that H^{i}(U, π^{∗}ℱ_{λ}) = 0 for every λ ∈ Λ. Hence, we may assume from start that ℱ is of finite type, in which case π^{∗}ℱ is a quasi-coherent 𝒪_{X^{∧}}-module, by (i), and then the assertion follows from theorem 15.1.37(ii).

(iii): Arguing as in the proof of (ii), we reduce to the case where ℱ is analytically of finite type; then ℱ (X) is an 𝒪_{X}(X)-module of analytically finite type (proposition 15.2.17(ii)), and the discussion of (15.1.47) says that 𝒪_{X^{∧}}(X^{∧}) (resp. ℱ^{∧}(X^{∧})) is the I-adic completion of 𝒪_{X}(X) (resp. of ℱ (X)). Then the assertion follows from (i), lemma 15.1.28(i) and proposition 15.2.10(i).

(iv) follows from (ii), together corollary 10.2.21(ii) and [58, Ch.I, Prop.10.7.2]. □

Theorem 15.2.22. Let A be a universally analytically noetherian ring, X a proper and finitely presented A-scheme, and ℱ a quasi-coherent 𝒪_{X}-module of analytically finite type. Then the A-module H^{i}(X, ℱ) is analytically of finite type for every i ∈ ℕ.

Proof. Let I ⊂ A be any ideal of adic definition. The first observation is the following :

Claim 15.2.23. Let i ∈ ℕ be any integer, and suppose that H^{i}(X, ℱ′) is analytically of finite type for every finitely presented 𝒪_{X}-module ℱ′. Then H^{i}(X, ℱ) is analytically of finite type for every 𝒪_{X}-module ℱ of analytically finite type.

Proof of the claim. Let 𝒢 ⊂ ℱ be an 𝒪_{X}-submodule as in lemma 15.2.16(i), so that I^{n}ℱ ⊂ 𝒢 for some n ∈ ℕ, and denote by j : 𝒢 → ℱ the inclusion morphism. Then, for every a ∈ I^{n} the endomorphism a · 1_{ℱ} factors through an 𝒪_{X}-linear morphism φ_{a} : ℱ → 𝒢. Set H^{i} := H^{i}(X, ℱ); we deduce that a · 1_{H^{i}} = H^{i}(X, j) ∘ H^{i}(X, φ_{a}). Especially, I^{n}H^{i} is contained in the image of H^{i}(X, j). Hence, if H^{i}(X, 𝒢) is analytically of finite type, the same follows for H^{i}. We may then replace ℱ by 𝒢, and assume from start that ℱ is quasi-coherent of finite type. Next, by lemma 15.2.16(iii) and remark 15.2.15 we may find a short exact sequence of quasi-coherent 𝒪_{X}-modules 0 → 𝒦 → ℱ′ → ℱ → 0 such that ℱ′ is finitely presented and I^{n}𝒦 = 0 for some n ∈ ℕ. It follows easily that I^{n} annihilates the cokernel of the induced map H^{i}(X, ℱ′) → H^{i}, whence the claim. ♢

Thus, suppose henceforth that ℱ is finitely presented, and set S := Spec A; according to [65, Ch.IV, Prop.8.9.1(i,ii)] we may assume that there exist a noetherian subring A_{0} ⊂ A, a finitely presented morphism of schemes X_{0} → S_{0} := Spec A_{0} and a coherent 𝒪_{X_{0}}-module ℱ_{0} such that X = S ×_{S_{0}} X_{0} and ℱ = π^{∗}ℱ_{0}, where π : X → X_{0} is the projection. By [65, Ch.IV, Th.8.10.5], we may moreover assume that X_{0} is a proper A_{0}-scheme.

Claim 15.2.24. The 𝒪_{X}-module 𝒯or_{q}^{S_{0}}(𝒪_{S}, ℱ_{0}) is analytically of finite type for every q ∈ ℕ.

Proof of the claim. The assertion is local on X_{0}, so we may assume that X_{0} is and affine and finitely presented S_{0}-scheme; then we may find an integer n ∈ ℕ and a closed immersion i : X_{0} → 𝔸_{S_{0}}^{n}. Set i_{S} := S ×_{S_{0}} i : X → 𝔸_{S}^{n}; according to [62, Ch.III, Prop.6.5.11] there is a natural isomorphism of 𝒪_{X}-modules :

i_{S∗}𝒯or_{q}^{S_{0}}(𝒪_{S}, ℱ_{0}) →^{∼} 𝒯or_{q}^{S_{0}}(𝒪_{S}, i_{∗}ℱ_{0})

so we may replace X_{0} by 𝔸_{S_{0}}^{n} and ℱ_{0} by i_{∗}ℱ_{0}, and assume from start that X_{0} is a flat and affine S_{0}-scheme. In this situation, there exists as well a natural isomorphism of 𝒪_{X}-modules :

𝒯or_{q}^{S_{0}}(𝒪_{S}, ℱ_{0}) →^{∼} 𝒯_{q} := 𝒯or_{q}^{X_{0}}(𝒪_{X}, ℱ_{0}) for every n ∈ ℕ.

Indeed, for every V_{0}, V and U as in the foregoing condition (a) we have an isomorphism

γ_{V,U} : Tor_{q}^{𝒪_{S_{0}}(V_{0})} (𝒪_{S}(V), ℱ_{0}(U)) →^{∼} Tor_{q}^{𝒪_{X_{0}}(U)} (𝒪_{X}(V ×_{S_{0}} U), ℱ_{0}(U))

of 𝒪_{X}(V ×_{S_{0}} U)-modules, such that for every inclusion of open subsets V′_{0} ⊂ V_{0}, V′ ⊂ V ∩ (S ×_{S_{0}} V′_{0}) and U′ ⊂ U ∩ (S ×_{S_{0}} V′_{0}) the resulting diagram commutes :

\xymatrix{Tor_{q}^{𝒪_{S_{0}}(V_{0})} (𝒪_{S}(V), ℱ_{0}(U)) \ar[r]^{γ_{V,U}} \ar[d] & Tor_{q}^{𝒪_{X_{0}}(U)} (𝒪_{X}(V ×_{S_{0}} U), ℱ_{0}(U)) \ar[d] \\ Tor_{q}^{𝒪_{S_{0}}(V′_{0})} (𝒪_{S}(V′), ℱ_{0}(U′)) \ar[r]^{γ_{V′,U′}} & Tor_{q}^{𝒪_{X_{0}}(U′)} (𝒪_{X}(V′ ×_{S_{0}} U′), ℱ_{0}(U′))}

([163, Prop.3.2.9]) whence the assertion. Thus, it suffices to check that 𝒯_{q} is analytically of finite type; however, since X_{0} is affine, ℱ_{0} admits a resolution 𝒫_{•} →^{∼} ℱ [0] consisting of free 𝒪_{X_{0}}-modules of finite type, and 𝒯_{q} is isomorphic to H_{q}(𝒪_{X} ⊗_{𝒪_{X_{0}}} 𝒫_{•}). The claim then follows easily from corollary 15.2.8(i). ♢

Let 𝔘 := (U_{i} | i = 1, …, n) be a finite affine covering of X_{0}; we have natural isomorphisms in D(A_{0}-Mod) and respectively D(A-Mod)

RΓ(X_{0}, ℱ_{0}) →^{∼} \overline{C}_{alt}^{•}(𝔘, ℱ_{0}) RΓ(X, ℱ) →^{∼} A ⊗_{A_{0}} \overline{C}_{alt}^{•}(𝔘, ℱ_{0})

where \overline{C}_{alt}^{•}(𝔘, −) denotes the truncated alternating Čech complex associated with the covering 𝔘 (theorem 10.2.28(ii)). Let us choose a Cartan-Eilenberg projective resolution P^{••} →^{∼} \overline{C}_{alt}^{•}(𝔘, ℱ_{0})[0] such that P^{pq} = 0 whenever q > 0 (see [163, Lemma 5.7.2]); the double complex A ⊗_{A_{0}} P^{••} gives rise to a spectral sequence

E_{2}^{p,−q} := Tor_{q}^{A_{0}}(A, H^{p}\overline{C}_{alt}^{•}(𝔘, ℱ_{0})) ⇒ M_{p−q} := H_{p−q}(A ⊗^{L}_{A_{0}} \overline{C}_{alt}^{•}(𝔘, ℱ_{0}))

as well as a spectral sequence

F_{1}^{p,−q} := Tor_{q}^{A_{0}}(A, \overline{C}_{alt}^{p}(𝔘, ℱ_{0})) ⇒ M_{p−q}

whose differentials d_{1}^{pq} : F_{1}^{pq} → F_{1}^{p+1,q} are induced by those of \overline{C}_{alt}^{•}(𝔘, ℱ_{0}), whence natural isomorphisms for every p, q ∈ ℕ :

(15.2.25)  E_{2}^{p,−q} →^{∼} Tor_{q}^{A_{0}}(A, H^{p}(X_{0}, ℱ_{0}))  F_{2}^{p,−q} →^{∼} H^{p}(X, 𝒯or_{q}^{S_{0}}(𝒪_{S}, ℱ_{0}))

and especially, we have natural isomorphisms

(15.2.26)  F_{2}^{i,0} →^{∼} H^{i}  for every i ∈ ℕ.

Claim 15.2.27. M_{p−q} is analytically of finite type, for every p, q ∈ ℕ.

Proof of the claim. The A_{0}-module H^{p}(X_{0}, ℱ_{0}) is finitely generated for every p ∈ ℕ, because X_{0} is a proper A_{0}-scheme ([61, Ch.III, Th.3.2.1]); since A_{0} is noetherian, we may then find a resolution P_{•} of H^{p}(X_{0}, ℱ_{0}) consisting of free A_{0}-modules of finite type, and there are natural isomorphisms E_{2}^{p,−q} →^{∼} H_{q}(A⊗_{A_{0}} P_{•}); taking into account corollary 15.2.8(i), we conclude that E_{2}^{p,−q} is an A-module of analytically finite type, for every p, q ∈ ℕ. Then, again by corollary 15.2.8(i), a simple induction on r ∈ ℕ shows that E_{r}^{p,−q} is analytically of finite type, for every p, q, r ∈ ℕ. We deduce that the A-module M_{p−q} admits a finite filtration whose subquotients are analytically of finite type, and to conclude it suffices to invoke again corollary 15.2.8(i). ♢

Now the theorem is a special case of the following more general

Claim 15.2.28. F_{r+2}^{p,−q} is an A-module of analytically finite type, for every p, q, r ∈ ℕ.

Proof of the claim. We argue by descending induction on p, and notice that F_{1}^{p,−q} = 0 for every p ≥ n − 1, so the assertion is clear for p ≥ n − 1. Next, let t ≤ n − 1, and suppose that the claim has already been shown for every p, q, r ∈ ℕ with r ≥ 2 and p ≥ t. First we prove, by descending induction on r ≥ 2, that F_{r}^{t−1,0} is analytically of finite type. Indeed, there exists s ∈ ℕ large enough, such that F_{r}^{t−1,0} is a subquotient of M_{t−1} for every r ≥ s, whence the assertion for every r ≥ s, by virtue of claim 15.2.27 and corollary 15.2.8(i). Thus, suppose that 2 < r ≤ s, and we know already that F_{r}^{t−1,0} is analytically of finite type; we have an exact sequence

0 → F_{r}^{t−1,0} → F_{r−1}^{t−1,0} → F_{r−1}^{t−2+r,2−r}

and by inductive assumption we also know that F_{r−1}^{t−2+r,2−r} is analytically of finite type. From corollary 15.2.8(i), it follows that the same holds for F_{r−1}^{t−1,0}, as required. Especially, in light of (15.2.26), the foregoing implies that H^{t−1}(X, ℱ) is analytically of finite type, for every finitely presented 𝒪_{X}-module ℱ, hence the same holds more generally whenever ℱ is analytically of finite type, by virtue of claim 15.2.23. Taking into account claim 15.2.24 and (15.2.25), we deduce that F_{2}^{t−1,−q} is analytically of finite type for every q ∈ ℕ, and finally, the same follows for F_{r}^{t−1,−q}, whenever q, r ∈ ℕ and r ≥ 2, after invoking once more corollary 15.2.8(i). □

15.2.29. In the situation of theorem 15.2.22, let I ⊂ A be any finitely generated ideal of adic definition; we notice that I^{n}ℱ is a quasi-coherent 𝒪_{X}-module of analytically finite type for every ∈ ℕ, and we define a descending filtration Fil_{I}^{•}H^{i} on H^{i} := H^{i}(X, ℱ) by setting

Fil_{I}^{n}H^{i} := Im(ψ_{i,n} : H^{i}(X, I^{n}ℱ) → H^{i}) for every i, n ∈ ℕ.

Corollary 15.2.30. With the notation of (15.2.29), the following holds for every i ∈ ℕ :

(i) The linear topology on H^{i} defined by the descending filtration Fil_{I}^{•}H^{i} agrees with the I-adic topology.

(ii) The system (Ker ψ_{i,n} | n ∈ ℕ) is essentially zero.

Proof. Let us show first that (i)⇒(ii). To ease notation, set K_{n}^{i} := Ker ψ_{i,n} for every i, n ∈ ℕ. For every a ∈ I^{n}, the endomorphism a·1_{ℱ} is the composition of the inclusion map j_{n} : I^{n}ℱ → ℱ and a morphism φ_{a,n} : ℱ → I^{n}ℱ of 𝒪_{X}-modules, and we have as well

φ_{a,n} ∘ j_{n} = a · 1_{I^{n}ℱ}

whence H^{i}(X, φ_{a,n}) ∘ ψ_{i,n} = a · 1_{H^{i}(X,I^{n}ℱ)} . We deduce immediately that

(15.2.31)  I^{n}K_{n}^{i} = 0  for every p, n ∈ ℕ.

Next, notice that

Im(K_{p}^{i} → K_{n}^{i}) = K_{n}^{i} ∩ Im(H^{i}(X, I^{p}ℱ) → H^{i}(X, I^{n}ℱ)) for all integers p ≥ n ≥ 0.

Suppose now that (i) holds for every 𝒪_{X}-module ℱ of analytically finite type; especially, it applies to I^{n}ℱ, and therefore for every k ∈ ℕ we may find an integer p ≥ n such that Im(H^{i}(X, I^{p}ℱ) → H^{i}(X, I^{n}ℱ)) ⊂ I^{k}H^{i}(X, I^{n}ℱ), whence

Im(K_{p}^{i} → K_{n}^{i}) ⊂ K_{n}^{i} ∩ I^{k}H^{i}(X, I^{n}ℱ).

By theorem 15.2.22, we know that H^{i}(X, I^{n}ℱ) is analytically of finite type; by corollary 15.2.7 and lemma 15.2.3(ii), it follows that there exists k ∈ ℕ large enough, such that K_{n}^{i} ∩ I^{k}H^{i}(X, I^{n}ℱ) ⊂ I^{n}K_{n}^{i}, and then the assertion follows from (15.2.31).

(i): Let a_{1}, …, a_{k} be a finite system of generators for I; we argue by induction on k. If k = 1, let p ∈ ℕ such that Ann_{ℱ} (a_{1}^{q}) = Ann_{ℱ} (a_{1}^{p}) for every q ≥ p (lemma 15.2.16(ii)); there follows a commutative diagram of 𝒪_{X}-modules

\xymatrix{a_{1}^{p}ℱ \ar[r] \ar[d] & a_{1}^{q}ℱ \ar[d] \\ ℱ \ar[r]^{a_{1}^{q−p}·1_{ℱ}} & ℱ}

whose top horizontal arrow is an isomorphism, and whose vertical arrows are the inclusion maps. We deduce easily that Fil_{I}^{q}H^{i} = a_{1}^{q−p} · Fil_{I}^{p}H^{i} for every q ≥ p, whence

a_{1}^{q}H^{i} ⊂ Fil_{I}^{q}H^{i} ⊂ a_{1}^{q−p}H^{i}

as required. Next, let k > 1, and set J := Aa_{1} + ⋯ + Aa_{k−1} and L := Aa_{k}. Denote by 𝒯_{I}, 𝒯_{J} and 𝒯_{L} respectively the I-adic, J-adic and L-adic topologies on A.

Fix p ∈ ℕ; it is easily seen that I^{p}H^{i} ⊂ Fil^{p}H^{i}, so it remains only to check that there exists n ∈ ℕ such that Fil^{n}H^{i} ⊂ I^{p}H^{i}. To this aim, notice first that X is also an analytically noetherian (A, 𝒯_{L})-scheme (proposition 15.2.5(i)), and ℱ is also analytically of finite type relative to the topology 𝒯_{L}; since L is principal, it follows that we may find t ∈ ℕ such that

(15.2.32)  Fil_{L}^{t}H^{i} ⊂ L^{p}H^{i}.

Notice also that

J^{r+t} ⊂ I^{r+t} ⊂ J^{r} + L^{t} for every r, t ∈ ℕ

which implies that the I-adic topology on any A/L^{t}-module agrees with the J-adic topology. By the same token, an A/L^{t}-module is an (A, 𝒯_{I})-module of analytically finite type if and only if it is an (A, 𝒯_{J})-module of analytically finite type. Taking into account proposition 15.2.5(i) we deduce that X is also an analytically noetherian (A, 𝒯_{J})-scheme and ℱ /L^{t}ℱ is analytically of finite type relative to the topology 𝒯_{J} as well, for every t ∈ ℕ. Then, theorem 15.2.22 says that H_{t}^{i} := H^{i}(X, ℱ /L^{t}ℱ) is an A-module analytically of finite type relative to the topology 𝒯_{J}, hence also relative to the topology 𝒯_{I}, since it is an A/L^{t}A-module. Moreover, by inductive assumption the linear topology on H_{t}^{i} defined by the filtration Fil_{J}^{•}H_{t}^{i} agrees with the J-adic topology, hence the linear topology on H_{t}^{i} defined by the filtration Fil_{I}^{•}H_{t}^{i} agrees with the I-adic topology, by the foregoing observation.

Now, the image of Fil_{I}^{n}H^{i} in H_{t}^{i} lies in the intersection of Fil_{I}^{n}H_{t}^{i} with the submodule H^{i}/Fil_{L}^{t}H^{i} ⊂ H_{t}^{i}, and the I-adic topology on the latter agrees with the one induced by the I-adic topology of H_{t}^{i} (corollary 15.2.7and lemma 15.2.3(ii)), so

Im(Fil_{I}^{n}H^{i} → H^{i}/Fil_{L}^{t}H^{i}) ⊂ I^{p}(H^{i}/Fil_{L}^{t}H^{i}) for some n ∈ ℕ

i.e. Fil_{I}^{n}H^{i} ⊂ I^{p}H^{i} + Fil_{L}^{t}H^{i}, and combining with (15.2.32), the assertion follows. □

15.2.33. Keep the notation of (15.2.29), and define π : X^{∧} → X and ℱ^{∧} as in (15.2.18). Moreover, for every i, n ∈ ℕ endow H^{i}(X, ℱ) with its I-adic topology and H^{i}(X, ℱ /I^{n}ℱ) with the discrete topology, and denote by A^{∧} and H^{i}(X, ℱ)^{∧} the separated completions of A and respectively H^{i}(X, ℱ).

Corollary 15.2.34. In the situation of (15.2.33), the following holds :

(i) There exist natural A^{∧}-linear isomorphisms

H^{i}(X^{∧}, ℱ^{∧}) →^{∼} lim_{n∈ℕ} H^{i}(X, ℱ /I^{n}ℱ) ⟵^{∼} H^{i}(X, ℱ)^{∧} for every i ∈ ℕ.

(ii) For i = 0, the maps of (i) are even isomorphisms of topological A^{∧}-modules.

(iii) For every quasi-coherent 𝒪_{X}-module 𝒢, there exists a natural isomorphism

H^{i}(X^{∧}, π^{∗}𝒢) →^{∼} A^{∧} ⊗_{A} H^{i}(X, 𝒢) for every i ∈ ℕ.

Proof. (i): Fix a finite affine covering U_{•} := (U_{λ} | λ ∈ Λ) of X, and for every λ ∈ Λ let U_{λ}^{∧} be the completion of U_{λ} along its closed subscheme Spec A/I ×_{Spec A} U, . In light of lemma 15.1.28(i) we have a natural identification of complexes of topological A^{∧}-modules

\overline{C}_{alt}^{•}(U_{•}^{∧}, ℱ^{∧}) →^{∼} lim_{n∈ℕ} \overline{C}_{alt}^{•}(U_{•}, ℱ /I^{n}ℱ)

(where \overline{C}_{alt}^{q}(U_{•}, ℱ /I^{n}ℱ) is endowed with the discrete topology, for every n, q ∈ ℕ). On the other hand, by theorems 15.1.37(i) and 10.2.28(ii), we have natural isomorphisms

H^{i}(X, ℱ /I^{n}ℱ) →^{∼} H^{i}\overline{C}_{alt}^{•}(U_{•}, ℱ /I^{n}ℱ) H^{i}(X^{∧}, ℱ^{∧}) →^{∼} H^{i}\overline{C}_{alt}^{•}(U_{•}^{∧}, ℱ^{∧})

for every i, n ∈ ℕ. Taking into account [163, Th.3.5.8] there follows a short exact sequence

(15.2.35) 0 → lim_{n∈ℕ}^{1} H^{i−1}(X, ℱ /I^{n}ℱ) → H^{i}(X^{∧}, ℱ^{∧}) → lim_{n∈ℕ} H^{i}(X, ℱ /I^{n}ℱ) → 0

of A^{∧}-modules, for every i ∈ ℕ. Set also

F_{k}^{i} := Im(H^{i}(X, ℱ) → H^{i}(X, ℱ /I^{k}ℱ)) for every i, k ∈ ℕ.

We notice :

Claim 15.2.36. lim_{n∈ℕ}^{1} H^{i−1}(X, ℱ /I^{n}ℱ) = 0 for every i ∈ ℕ.

Proof of the claim. By virtue of [163, Prop.3.5.7], it suffices to check that the descending system of submodules

(M_{k,k+n}^{i−1} := Im(H^{i−1}(X, ℱ /I^{k+n}ℱ) → H^{i−1}(X, ℱ /I^{k}ℱ)) | n ∈ ℕ)

is stationary, for every i, k ∈ ℕ. However, the short exact sequence of 𝒪_{X}-modules

0 → I^{n}ℱ → ℱ → ℱ /I^{n}ℱ → 0

induces a commutative diagram of A-modules with exact rows

[figure: H^{i−1}(X, ℱ); H^{i−1}(X, ℱ /I^{n+k}ℱ); H^{i}(X, I^{n+k}ℱ); H^{i}(X, ℱ); H^{i−1}(X, ℱ); H^{i−1}(X, ℱ /I^{k}ℱ); H^{i}(X, I^{k}ℱ); H^{i}(X, ℱ)]

which shows that

F_{k}^{i−1} ⊂ M_{k,k+n}^{i−1} for every i, k, n ∈ ℕ.

On the other hand, corollary 15.2.30(ii) implies that

M_{k,k+n}^{i−1} ⊂ Ker(H^{i−1}(X, ℱ /I^{k}ℱ) → H^{i}(X, I^{k}ℱ)) = F_{k}^{i−1}

for every k ∈ ℕ and every sufficiently large n ∈ ℕ. The claim follows. ♢

From (15.2.35) and claim 15.2.36, we already get the first stated isomorphism. Next, the proof of claim 15.2.36 shows that the system of inclusion maps (F_{n}^{i} → H^{i}(X, ℱ /I^{n}ℱ) | n ∈ ℕ) induces an isomorphism of A^{∧}-modules

lim_{n∈ℕ} F_{n}^{i} →^{∼} lim_{n∈ℕ} H^{i}(X, ℱ /I^{n}ℱ) for every i ∈ ℕ.

But the system (F_{n}^{i} | n ∈ ℕ) is also isomorphic to the system (H^{i}/Fil_{I}^{n}H^{i} | n ∈ ℕ) (notation of (15.2.29)), and lastly, from corollary 15.2.30(i) we obtain a natural isomorphism

lim_{n∈ℕ} H^{i}/Fil_{I}^{n}H^{i} →^{∼} H^{i}(X, ℱ)^{∧}

whence the second stated isomorphism.

(ii): The topology of H^{0}(X, ℱ)^{∧} is I-adic, by remark 8.3.3(ii,iv). On the other hand, the topology of H^{0}(X^{∧}, ℱ^{∧}) is the linear topology given by the system of submodules

(H^{0}(X^{∧}, (I^{n}ℱ)^{∧}) | n ∈ ℕ)

by remark 15.1.48. Denote H^{0}(X, I^{n}ℱ)^{∧} the I-adic completion of H^{0}(X, I^{n}ℱ), for every n ∈ ℕ; by (i) we get a commutative diagram of A^{∧}-modules

(15.2.37)  \xymatrix{H^{0}(X^{∧}, (I^{n}ℱ)^{∧}) \ar[r] \ar[d] & H^{0}(X, I^{n}ℱ)^{∧} \ar[d] \\ H^{0}(X^{∧}, ℱ^{∧}) \ar[r] & H^{0}(X, ℱ)^{∧}}

whose horizontal arrows are isomorphisms, and whose left (resp. right) vertical arrow is induced by the inclusion (I^{n}ℱ)^{∧} ⊂ ℱ^{∧} (resp. is the I-adic completion of the natural map H^{0}(X, I^{n}ℱ) → H^{0}(X, ℱ)). By corollary 15.2.30(i), the topology on H^{0}(X, ℱ)^{∧} given by the filtration (H^{0}(X, I^{n}ℱ)^{∧} | n ∈ ℕ) agrees with the topology given by the filtration ((I^{n}H^{0}(X, ℱ))^{∧} | n ∈ ℕ), and the latter is none else than the I-adic filtration, again by remark 8.3.3(ii,iv). We conclude already that the bottom horizontal arrow of (15.2.37) is an isomorphism of topological A^{∧}-modules. The same holds for the natural map H^{0}(X^{∧}, ℱ^{∧}) → lim_{n∈ℕ} H^{0}(X, ℱ /I^{n}ℱ), by the discussion of (15.1.47).

(iii): Since X is quasi-compact and quasi-separated, 𝒢 is the colimit of the filtered family (𝒢_{λ} | λ ∈ Λ) of its quasi-coherent 𝒪_{X}-submodules of finite type (proposition 10.3.31), hence π^{∗}𝒢 is the colimit of the system (π^{∗}𝒢_{λ} | λ ∈ Λ). Since X^{∧} is a spectral topological space, proposition 10.1.10then reduces to showing the assertion for each 𝒢_{λ}, so we may assume from start that 𝒢 is analytically of finite type, in which case it suffices to apply (i), propositions 15.2.10(i) and 15.2.20(i), and theorem 15.2.22. □

Remark 15.2.38. (i) The assertions of theorem 15.2.22 and of its corollaries 15.2.30 and 15.2.34 hold also in case X is projective but not necessarily finitely presented : indeed, if i : X → ℙ_{A}^{n} is a closed immersion in a projective space over Spec A, then H^{•}(X, ℱ) = H^{•}(ℙ_{A}^{n}, i_{∗}ℱ), and i_{∗}ℱ is clearly an 𝒪_{ℙ_{A}^{n}}-module of analytically finite type, so we are reduced to the case where X = ℙ_{A}^{n}, which is covered by theorem 15.2.22.

(ii) More generally, theorem 15.2.22 and its two corollaries hold for any proper A-scheme X (and any 𝒪_{X}-module ℱ of analytically finite type) : indeed, according to [49, Th.4.1 and Th.4.3] there exists a locally closed immersion i : X → X′ of A-scheme, with X′ proper and finitely presented over Spec A. Since X is proper, i is even a closed immersion ([60, Ch.II, Cor.5.4.3]), so again it suffices to apply the theorem to the 𝒪_{X′}-module i_{∗}ℱ.

(iii) Theorem 15.2.22 and corollary 15.2.34 have also been announced, respectively as theorem C.3.1 and theorem C.3.3 of Chapter 1, in K.Fujiwara and F.Kato’s treatise [72]; however, their proofs have been postponed to a forthcoming article, which has not appeared yet.

Corollary 15.2.39. Let A be a universally analytically noetherian ring, f := (f_{1}, …, f_{r}) a sequence of elements of A that generates an ideal of adic definition. Then A satisfies condition (a)_{f}^{un} of (7.8.21).

Proof. Let X := Spec A, denote by I ⊂ A the ideal generated by f, and by ℐ ⊂ 𝒪_{X} the quasicoherent ideal arising from I. Let also π : Y → X be the blowing up morphism of ℐ. It follows easily from remark 10.6.39(iii) that the support of the A-module H^{p} := H^{p}(Y, 𝒪_{Y}) is contained in Spec A/I, for every p > 0, and the same holds for the kernel and cokernel of the natural map φ : A → H^{0}(Y, 𝒪_{Y}). Combining with theorem 15.2.22 and remark 15.2.2(iii), we deduce that there exists n ∈ ℕ such that I^{n} · H^{p} = 0 for every p > 0, and I^{n} · Ker φ = I^{n} · Coker φ = 0. Then it suffices to invoke theorem 10.6.50. □

15.3. Continuous valuations. As already mentioned in (8.1.54), the study of the valuation spectrum of a topological ring A will lead us to consider certain subsets of Spv A that are not pro-constructible, but nevertheless are spectral spaces, with the topology induced via the inclusion map into Spv A.

15.3.1. As a first step, let A be any ring, I ⊂ A a finitely generated ideal; we attach to I a set of specializations S_{I} of Spv A (see (8.1.54)), by declaring that (v, w) ∈ S_{I} if and only if :

• either, w(I) ≠ {0} and w is a primary specialization of v

• or else, v = w. The S_{I}-admissible specializations in Spv A will be simply called I-admissible, and the S_{I}- closed subsets of Spv A shall be called I-closed. It is clear that the set of I-admissible specializations of Spv A is transitive, and it satisfies condition (S3) of corollary 8.1.57, by lemma 9.2.15(i). We shall show hereafter that it fulfills also conditions (S1) and (S2) of proposition 8.1.55. This shall be achieved in several steps; to start out, for any abelian ordered group Δ, let us say that an element δ ∈ Δ_{∘} is final in Δ if the following holds. For every γ ∈ Δ there exists n ∈ ℕ such that δ^{n} < γ. We remark :

Lemma 15.3.2. In the situation of (15.3.1), let v be a valuation of A with

v(I) ≠ {0}  and  v(I) ∩ cΓ_{v} = ∅.

Then we have :

(i) There exists a smallest convex subgroup Δ of Γ_{v} with v(I) ∩ Δ ≠ ∅.

(ii) cΓ_{v} ⊂ Δ.

(iii) For every i ∈ I, either v(i) < δ for every δ ∈ Δ, or else v(i) is final in Δ.

Proof. (i): Let a_{1}, …, a_{n} be a finite system of generators of I; the stated assumptions on v imply that v(a_{i}) < 1 for every i = 1, …, n, and v(a_{k}) > 0 for at least one index k ≤ n. Let Δ be the convex hull of the subgroup Δ_{0} of Γ_{v} generated by max{v(a_{i}) | i = 1, …, n}. It remains only to check that Δ ⊂ Δ′ for every convex subgroup Δ′ ⊂ Γ_{v} with Δ′ ∩ v(I) ≠ ∅. To this aim, notice first that cΓ_{v} ⊂ Δ′ for every such Δ′ : indeed, otherwise there exists γ ∈ cΓ_{v} such that γ < δ for every δ ∈ Δ′. Then, pick i ∈ I with v(i) ∈ Δ′; it follows that 1 > v(i) > γ, so v(i) ∈ cΓ_{v}, contradicting our assumptions. Now, let i ∈ I be any element with v(i) ∈ Δ′, and write i = a_{1}b_{1} + ⋯ + a_{n}b_{n} for some b_{1}, …, b_{n} ∈ A; since cΓ_{v} ⊂ Δ′ and v(i) ≤ max{v(a_{i}) · v(b_{i}) | i = 1, …, n}, it follows that v(a_{k}) ∈ T for some k ≤ n (details left to the reader). Then clearly Δ_{0} ⊂ Δ′, and the contention follows. Moreover, the argument also shows (ii).

(iii): Let a_{0} ∈ I be any element; the sequence a_{0}, a_{1}, …, a_{n} is also a system of generators for I, and the foregoing shows that Δ is the convex hull of the subgroup generated by max{v(a_{i}) | i = 0, …, n}. The assertion follows immediately. □

Definition 15.3.3. Keep the situation of (15.3.1). To every v ∈ Spv A we attach a convex subgroup

cΓ_{v}(I) ⊂ Γ_{v}

as follows :

• If v(I) = {0}, we set cΓ_{v}(I) := Γ_{v}.

• If v(I) ∩ cΓ_{v} ≠ ∅, we set cΓ_{v}(I) := cΓ_{v}.

• Otherwise, v fulfills the conditions of lemma 15.3.2, and we let cΓ_{v}(I) be the smallest of the convex subgroups Δ such that v(I) ∩ Δ ≠ ∅.

Lemma 15.3.4. Keep the notation of definition 15.3.3, and let a_{1}, …, a_{n} be a system of generators for the ideal I. We have :

(i) The following conditions are equivalent :

(a) Γ_{v} = cΓ_{v}(I).

(b) Γ_{v} = cΓ_{v}, or else v(a) is final in Γ_{v} for every a ∈ I.

(c) Γ_{v} = cΓ_{v}, or else v(a_{i}) is final in Γ_{v}, for every i = 1, …, n. (ii) An element w ∈ Spv A is an I-admissible specialization of v if and only if w = v^{Δ} for

a convex subgroup Δ of Γ_{v} with cΓ_{v}(I) ⊂ Δ. (iii) v has no proper I-admissible specializations if and only if Γ_{v} = cΓ_{v}(I).

Proof. (i): The equivalence of (a) and (b) follows directly from the definitions and lemma 15.3.2(iii). The equivalence of (b) and (c) follows by remarking that the set

{a ∈ A | v(a) = 0 or else v(a) is final in Γ_{v}}

is an ideal in A, provided Γ_{v} ≠ cΓ_{v} (details left to the reader).

(ii) and (iii) are immediate from the definitions. □

Lemma 15.3.5. Let A be a ring, I ⊂ A a finitely generated ideal. We have :

(i) Let also f_{•} := (f_{0}, f_{1}, …, f_{n}) be a finite sequence of elements of A, and J ⊂ A the ideal generated by f_{•}. Suppose that the radical of J contains I. Then the rational subset R_{A}(\frac{f_{1}}{f_{0}} , …, \frac{f_{n}}{f_{0}} ) is I-closed.

(ii) Let v be any valuation of A such that Γ_{v} = cΓ_{v}(I). Then for every open neighborhood U of v in Spv A there exists a finite sequence f_{•} := (f_{0}, f_{1}, …, f_{n}) of elements of A such that :

(a) v ∈ R_{A}(\frac{f_{1}}{f_{0}} , …, \frac{f_{n}}{f_{0}} ) ⊂ U.

(b) I is contained in the radical of the ideal generated by f_{•}.

Proof. (i): Let w ∈ Spv A be an I-admissible specialization of some v ∈ R := R_{A}(\frac{f_{1}}{f_{0}} , …, \frac{f_{n}}{f_{0}} ). If v(I) = {0}, then v = w, and there is nothing to prove. Thus, we may assume v(I) ≠ {0}, in which case w(I) ≠ {0} as well. Since w is a primary specialization of v, we have w(f_{i}) ≤ w(f_{0}) for every i = 1, …, n. Suppose, by way of contradiction, that w ∉ R; then we must have w(f_{0}) = 0, and consequently w(f_{i}) = 0 for every i ≤ n as well. Thus, J ⊂ Ker w, and since the support of w is a prime ideal, it contains also the radical of J; hence, I ⊂ Ker w, a contradiction.

(ii): We may assume that U = R_{A}(\frac{g_{1}}{g_{0}} , …, \frac{g_{k}}{g_{0}} ) for some g_{0}, g_{1}, …, g_{k} ∈ A. Pick also a finite system a_{1}, …, a_{n} of generators of I. We distinguish two cases :

• Suppose first that Γ_{v} = cΓ_{v}. In this case, since v(g_{0}) ≠ 0, there exists d ∈ A such that v(g_{0}) · v(d) = v(g_{0}d) > 1. Then v ∈ R_{A}(\frac{1}{g_{0}d}) ∩ U, and we may take f_{•} := (g_{0}d, …, g_{k}d, 1).

• Lastly, suppose that Γ_{v} ≠ cΓ_{v}. In this case, since v(g_{0}) ≠ 0, lemma 15.3.4(i) implies that there exists r ∈ ℕ with v(g_{0}) ≥ v(a_{i}^{r}) for every i = 1, …, n. Then v ∈ R_{A}(\frac{a_{1}^{r}}{g_{0}} , …, \frac{a_{n}^{r}}{g_{0}} ) ∩ U, and the sequence f_{•} := (g_{0}, …, g_{k}, a_{1}^{r}, …, a_{n}^{r}) will do. □

Definition 15.3.6. Let A be any ring and I ⊂ A any finitely generated ideal.

(i) We set

Spv(A, I) := {v ∈ Spv A | Γ_{v} = cΓ_{v}(I)}

and we endow Spv(A, I) with the topology induced by Spv A via the inclusion map. (ii) The rational subsets of Spv(A, I) are the subsets of the form

Spv(A, I) ∩ R_{A}(\frac{f_{1}}{f_{0}} , …, \frac{f_{n}}{f_{0}} )

where f_{•} := (f_{0}, f_{1}, …, f_{n}) is any sequence of elements of A such that I is contained in the radical of the ideal generated by f_{•}.

Theorem 15.3.7. With the notation of definition 15.3.6, we have :

(i) The topological space Spv(A, I) is spectral, and the retraction

r_{I} : Spv A → Spv(A, I) v ↦ v^{cΓ_{v}(I)}

is spectral. (ii) The rational subsets of Spv(A, I) are constructible in Spv(A, I), and form a basis of

the topology of Spv(A, I) that is closed under finite intersections. (iii) A subset T ⊂ Spv(A, I) is constructible in Spv(A, I) if and only if r_{I}^{−1}T is con-

structible in Spv A.

Proof. (i): From lemma 15.3.5 we see that the set of I-admissible specializations of Spv A fulfills conditions (S1) and (S2) of proposition 8.1.55, whence the assertion.

(ii): Lemma 15.3.5already shows that the rational subsets form a basis ℬ of the topology of Spv(A, I). If R_{A}(\frac{f_{1}}{f_{0}} , …, \frac{f_{n}}{f_{0}} ) and R_{A}(\frac{g_{1}}{g_{0}} , …, \frac{g_{m}}{g_{0}} ) are any two rational open subset of Spv A such that the radicals of the ideals J and J′ generated respectively by f_{•} and g_{•} contain I, then the same holds for the radical of the ideal JJ′; taking into account remark 9.2.4(i), it follows that ℬ is closed under finite intersections.

Lastly, let f_{•} and J be as in the foregoing, with I contained in the radical of J, and set R := R_{A}(\frac{f_{1}}{f_{0}} , …, \frac{f_{n}}{f_{0}} ). By lemma 15.3.5, the set R is I-closed, so r_{I}(R) = R∩Spv(A, I); since R is quasi-compact in Spv A, the subset r_{I}(R) is quasi-compact in Spv(A, I), so R ∩Spv(A, I) is a constructible open subset of Spv(A, I).

(iii) follows from proposition 8.1.55(iii). □

Example 15.3.8. Let A be any ring.

(i) We may take I = 0, in which case cΓ_{v}(I) = Γ_{v} for every v ∈ Spv A, so that Spv(A, 0) = Spv A. Also, the rational subsets of Spv(A, 0) are the same as the rational subsets of Spv A.

(ii) We may also take I = A, in which case cΓ_{v}(I) = cΓ_{v} for every v ∈ Spv A, so that Spv(A, A) = {v ∈ Spv A | Γ_{v} = cΓ_{v}}. The rational subsets of Spv(A, A) can be described as the sets of the form

Spv(A, A) ∩ {v ∈ Spv A | v(f_{1}) ≤ v(f_{0}), …, v(f_{n}) ≤ v(f_{0})}

where f_{0}, f_{1}, …, f_{n} is any sequence of elements with ∑_{i=1}^{n} f_{i}A = A (details left to the reader).

Lemma 15.3.9. Let A be any ring, and I, J ⊂ A two finitely generated ideals. We have :

(i) If I is contained in the radical of J, then Spv(A, J) ⊂ Spv(A, I). The inclusion map Spv(A, J) → Spv(A, I) is not spectral in general. On the other hand, there is a spectral retraction

r_{I,J} : Spv(A, I) → Spv(A, J)  such that  r_{I,J} ∘ r_{I} = r_{J}.

(ii) Especially, the spectral map r_{I,I+J} is well defined, and it coincides with the restriction

of r_{J} to Spv(A, I). (iii) Spv(A, I · J) = Spv(A, I) ∪ Spv(A, J). (iv) Spv(A, I + J) = Spv(A, I) ∩ Spv(A, J).

Proof. (i): Indeed, the assumption easily implies that cΓ_{v}(J) ⊂ cΓ_{v}(I) for every v ∈ Spv A, from which the assertion follows immediately.

(ii): Let v ∈ Spv A be any element, and set w := r_{I}(v), u := r_{J}(w); then u does not have proper J-admissible specializations, nor any proper I-admissible specialization, because any I-admissible specialization of u would be also an I-admissible specialization of w. It follows easily that u is a primary specialization of v that does not have any proper (I +J)-specialization, whence the assertion.

Claim 15.3.10. For every v ∈ Spv A we have :

cΓ_{v}(I + J) = cΓ_{v}(I) ∩ cΓ_{v}(J) and cΓ_{v}(I · J) = cΓ_{v}(I) ∪ cΓ_{v}(J).

Proof of the claim. We consider first the assertion for cΓ_{v}(I · J) : it is easily seen that v(IJ) = {0} if and only if either v(I) = {0} or v(J) = {0}, so the assertion holds in case v(IJ) = {0}. Similarly, v(IJ) ∩ cΓ_{v} ≠ ∅ if and only if both v(I) and v(J) intersect cΓ_{v}, so the assertion holds also in case v(IJ) intersects cΓ_{v}. Lastly, suppose that v(IJ) ≠ {0} and v(IJ) ∩ cΓ_{v} = ∅, and pick a finite system of generators a_{1}, …, a_{n} (resp. b_{1}, …, b_{m}) for I (resp. for J), with

v(a_{1}) = max(v(a_{i}) | i = 1, …, n) and v(b_{1}) = max(v(b_{i}) | i = 1, …, m).

Then the system (a_{i}b_{j} | i ≤ n, j ≤ m) generates IJ, and v(ab) = max(v(a_{i}b_{j}) | i ≤ n, j ≤ m). Hence v(a_{1}) (resp. v(b_{1}), resp. v(a_{1}b_{1})) is final in cΓ_{v}(I) (resp. in cΓ_{v}(J), resp. in cΓ_{v}(IJ)) by lemma 15.3.2(iii). Without loss of generality we may assume that cΓ_{v}(J) ⊂ cΓ_{w}(I), in which case there exists n ∈ ℕ such that v(a_{1}^{n}) ≤ v(b_{1}), hence v(a_{1}b_{1})^{n} ≥ v(a_{1})^{2N}, which shows that cΓ_{v}(IJ) ⊂ cΓ_{v}(I), and the converse inclusion is clear.

We argue similarly for the calculation of cΓ_{v}(I + J) : first, it is clear that v(I + J) = {0} if and only if v(I) = v(J) = {0}. Next, we have v(I + J) ∩ cΓ_{v} ≠ ∅ if and only if (v(I)∪v(J))∩cΓ_{v} ≠ ∅ (details left to the reader). Lastly, if v(IJ) ≠ {0} and v(IJ)∩cΓ_{v} = ∅, pick a_{1}, …, a_{n} and b_{1}, …, b_{m} as in the foregoing; without loss of generality we may assume that v(a_{1}) ≥ v(b_{1}). Since the system a_{1}, …, a_{n}, b_{1}, …, b_{m} generates I + J, we deduce that v(a_{1}) is final in cΓ_{v}(I + J) (lemma 15.3.2(iii)), whence the contention. ♢

Assertion (iii) and (iv) follow immediately from claim 15.3.10: details left to the reader. □

Example 15.3.11. Let A be any ring, I ⊂ A a finitely generated ideal.

(i) Every element of the subset L := {v ∈ Spv A | v(I) = {0}} has no proper I-admissible specializations, so L lies in Spv(A, I). Clearly, L is constructible in Spv A, and r_{I}^{−1}(L) = L, so L is constructible in Spv(A, I), by theorem 15.3.7(iii).

(ii) Let (Spv A)_{0} be the set of trivial valuations of A; from remark 9.2.4(iii) we know already that (Spv A)_{0} is a spectral space, with the topology induced by the inclusion into Spv A. Moreover, if R := R_{A}(\frac{f_{1}}{f_{0}} , …, \frac{f_{n}}{f_{0}} ) is any rational subset of Spv(A, I), then R ∩ (Spv A)_{0} = \{v ∈ (Spv A)_{0} | v(f_{0}) = 1}, which is a constructible (open) subset of (Spv A)_{0}. Hence, the inclusion map (Spv A)_{0} → Spv(A, I) is spectral.

(iii) Pick a finite system a_{1}, …, a_{n} of generators for I, and set

E_{0} := {v ∈ (Spv A)_{0} | v(I) = {0}}
E_{i} := {v ∈ Spv A | v(a_{i}) = 1 and v(a) ≤ 1 for every a ∈ A} for every i = 1, …, n.

Then E_{0}, …, E_{n} are pro-constructible subsets of Spv A that are closed under I-admissible specializations and generizations. We have (Spv A)_{0} = Spv(A, I) ∩ ⋃_{i=0}^{n} E_{i}, and therefore r_{I}^{−1}(Spv A)_{0} = ⋃_{i=0}^{n} E_{i}, by corollary 8.1.57.

We are now ready to introduce the spectrum of continuous valuations of a topological ring.

Definition 15.3.12. Let (A, 𝒯_{A}) be any topological ring, and v a valuation of A.

(i) We endow Γ_{v∘} with a topology 𝒯_{Γ_{v}}, by ruling that a subset U ⊂ Γ_{v∘} is open if either 0 ∉ U, or else there exists δ ∈ Γ_{v} such that {γ ∈ Γ_{v∘} | γ < δ} ⊂ U. Then, we say that v is continuous if it is a continuous map (A, 𝒯_{A}) → (Γ_{v∘}, 𝒯_{Γ_{v}}).

(ii) We set

Cont(A) := {v ∈ Spv A | v is continuous} Cont^{+}(A) := Cont(A) ∩ Spv^{+}(A)

and we endow Cont(A) and Cont^{+}(A) with the topology induced by Spv A.

Remark 15.3.13. Let A be any topological ring, any v any valuation of A.

(i) Endow κ(v) with its valuation topology 𝒯_{v} (see definition 9.1.14(i)). A simple inspection of the definitions shows that the residual valuation \overline{v} : (κ(v), 𝒯_{v}) → Γ_{v∘} of v is continuous. Moreover, v is continuous if and only if the same holds for the natural map π_{v} : A → (κ(v), 𝒯_{v}).

Indeed, clearly if π_{v} is continuous, the same holds for v = \overline{v} ∘ π_{v}. Conversely, suppose that v is continuous; it suffices to check that π_{v} is continuous at the point 0 ∈ A. However, for every δ ∈ Γ_{v} let U_{δ} := {γ ∈ Γ_{v∘} | γ < δ}; then the family (U_{δ} | δ ∈ Γ_{v}) is a fundamental system of open neighborhoods of 0 ∈ Γ_{v∘}, and (\overline{v}^{−1}U_{δ} | δ ∈ Γ_{v}) is a fundamental system of open neighborhoods of 0 ∈ κ(v), whence the contention.

(ii) It follows easily from (i) that the valuation v is continuous if and only if it is continuous at the point 0 ∈ A.

(iii) Let φ : Γ → Γ′ be a morphism of ordered groups, and endow Γ_{∘} and Γ′_{∘} with the topologies 𝒯_{Γ} and 𝒯_{Γ′} described in definition 15.3.12(i). In general, the induced map φ_{∘} : Γ_{∘} → Γ′_{∘} is not necessarily continuous for these topologies; for instance :

• If Γ is a subgroup of Γ′, and φ is the inclusion map, then φ_{∘} is continuous if and only if the convex hull of Γ equals Γ′ (see remark 9.1.2(iii)), and if the latter condition holds, then the topology 𝒯_{Γ} agrees with the topology induced by 𝒯_{Γ′}.

• If Γ′ = Γ/Δ for a proper convex subgroup Δ ⊂ Γ, and φ is the projection, then φ_{∘} is continuous.

• If Γ ≠ {1} and Γ′ = {1}, then φ_{∘} is not continuous. Especially, a secondary generization w of v is continuous if either v = w or else Γ_{w} ≠ {1}, but w may fail to be continuous in case Γ_{v} ≠ {1} and Γ_{w} = {1}. On the other hand, if φ_{∘} ∘ v is a continuous map, then it is easily seen that v is continuous as well.

(iv) Let Δ ⊂ Γ_{v} be any convex subgroup containing cΓ_{v}, denote by Γ_{v∘}^{+} · Δ the smallest submonoid of Γ_{v∘} containing Γ_{v∘}^{+} and Δ, and endow Γ_{v∘}^{+} · Δ with the topology 𝒯_{Γ_{v∘}^{+}·Δ} induced by 𝒯_{Γ_{v}} via the inclusion map. Let also 𝒯_{Δ} be the topology on Δ_{∘} described in definition 15.3.12(i). Then the inclusion map Δ_{∘} → Γ_{v∘}^{+} · Δ admits a unique continuous retraction

ρ : (Γ_{v∘}^{+} · Δ, 𝒯_{Γ_{v∘}^{+}·Δ}) → (Δ_{∘}, 𝒯_{Δ}) such that ρ(γ) = 0 whenever γ ∉ Δ_{∘}.

On the other hand, since cΓ_{v} ⊂ Δ, the map v factors through a (unique) mapping v′ : A → Γ_{v∘}^{+} · Δ, and if v is continuous, the same holds for v′. Clearly ρ ∘ v′ = v^{Δ}, so the latter is continuous as well. Combining with (iii) and proposition 9.2.24, we conclude that every specialization of a continuous valuation is continuous. However, in general Cont(A) is not a closed subset (not even a pro-constructible subset) of Spv A.

(v) Since Spv A is a T_{0} topological space, the same holds for Cont(A). Especially, the specializations in Cont(A) define a partial ordering on Cont(A) : see remark 8.1.45(ii).

(vi) It follows also from (ii) that if Ker v is an open prime ideal, then v is continuous. We shall say that v is non-analytic if Ker v is an open prime ideal. If v is continuous and its support is not an open prime ideal, we shall say that v is analytic. We denote by

Cont(A)_{a} and Cont(A)_{na}

the sets of all analytic and respectively non-analytic valuations of A.

(vii) Let f : A → B be any continuous ring homomorphism of topological rings. Then Spv(f) restricts to continuous maps

Cont(f) : Cont(B) → Cont(A) Cont^{+}(f) : Cont^{+}(B) → Cont^{+}(A).

Indeed, let w be a continuous valuation of B; obviously v := w ∘ f : A → Γ_{w∘} is a continuous map. The value group Γ_{v} of v is a subgroup of Γ_{w}, and v factors uniquely through a valuation v′ : A → Γ_{v} that is obviously equivalent to v; by (iii), the continuity of w implies the continuity of v′, whence the claim. It is then also clear that Cont(f) in turns restricts to a mapping

Cont(f)_{na} : Cont(B)_{na} → Cont(A)_{na}.

Lemma 15.3.14. Let A be any f-adic topological ring, and v ∈ Cont(A). We have : (i) Cont(A) = {w ∈ Spv A | w(a) is final in Γ_{w}, for every a ∈ A^{∘∘}}.

(ii) Suppose that v is analytic, and endow κ(v) with its valuation topology 𝒯_{v}. Then v is a Tate valuation (see definition 9.1.14(iv)), and the natural map

π_{v} : A → (κ(v), 𝒯_{v})

is an f-adic continuous ring homomorphism. (iii) Let w ∈ Cont(A) be any proper primary specialization of v. Then w is non-analytic. (iv) Let w ∈ Cont(A)_{a} be a specialization of v. Then v ∈ Cont(A)_{a} and w is a secondary

specialization of v.

(v) If v is analytic, the totally ordered set of continuous generizations of v admits a maximal element, which is a rank one analytic valuation.

Proof. Let A_{0} ⊂ A be a subring of definition, I ⊂ A_{0} a finitely generated ideal of adic definition, and a_{1}, …, a_{n} a finite system of generators of I; for every w ∈ Spv A we set γ_{w} := max(w(a_{1}), …, w(a_{n}))

(i): The condition is clearly necessary. Conversely, suppose that w ∈ Spv A fulfills this condition; by remark 15.3.13(ii), it suffices to check that, for every δ ∈ Γ_{w} there exists an open neighborhood U_{δ} of 0 in A such that w(a) < δ for every a ∈ U_{δ}. However, we have w(a) < 1 for every a ∈ I, since I ⊂ A^{∘∘}, and we may find k ∈ ℕ such that δ > γ_{w}^{k}. Hence U_{δ} := I^{k+1} will do.

(ii): Denote by \overline{v} the residual valuation of v; if v is analytic, the value γ_{v} lies in Γ_{v}, and if x ∈ κ(v)^{+} is any element with \overline{v}(x) = γ, it follows easily that the valuation topology on κ(v)^{+} agrees with the x-adic topology, so v is a Tate valuation. Next, after replacing A_{0} by the smallest ℤ-subalgebra of A_{0} containing I, we may assume that π_{v}(A_{0}) ⊂ κ(v)^{+}, and notice that π_{v}(I^{n}) generates x^{n} · κ(v)^{+} for every n ∈ ℕ, so that π_{v} is continuous and f-adic (details left to the reader).

(iii): Say that w = v^{Δ} for some convex subgroup Δ strictly contained in Γ_{v}. Then the value γ_{v} cannot lie in Δ, so w(I) = 0, whence the assertion.

(iv): Proposition 9.2.24 says that w is a primary specialization of a secondary specialization u of v, and u is a continuous valuation of A, by remark 15.3.13(iv). In light of (iii), it follows that u = w, whence the contention.

(v): First, from (iv) we see that every continuous generization of v must be a secondary generization and must be analytic; hence, let Δ ⊂ Γ_{v} be the largest convex subgroup that does not contain γ_{v}, and set w := v_{Δ}. It is easily seen that the image \overline{γ}_{v} of γ_{v} in Γ_{w} = Γ_{v}/Δ is still final, and by definition γ_{w} = \overline{γ}_{v}, so w is continuous, by (i). It remains only to check that w does not admit any proper (secondary) generization; however, by construction, every non-trivial convex subgroup Δ′ ⊂ Γ_{w} contains γ_{w}, and on the other hand, γ_{w} cannot lie in any proper convex subgroup of Γ_{w}. Thus, the only convex subgroups of Γ_{w} are {1} and Γ_{w}, whence the assertion. □

Theorem 15.3.15. Let A be any f-adic ring, and J ⊂ A any finitely generated ideal such that Spec A/J is the non-analytic locus of Spec A. Then we have :

(i) Cont(A) = {v ∈ Spv(A, J) | v(a) < 1 for every a ∈ A^{∘∘}}.

(ii) Cont(A) is a closed (in particular, pro-constructible) subset of Spv(A, J). Especially, Cont(A) is a spectral topological space.

Proof. (i): Let A_{0} ⊂ A be a subring of definition, and I ⊂ A_{0} an ideal of adic definition; by lemma 8.3.29(i) we have Spec A/IA = Spec A/J, so that the ideals J and IA have the same radical; from lemma 15.3.9(i) we deduce that Spv(A, J) = Spv(A, IA), so we may assume from start that J = IA. Since I is an open subset of A, for every a ∈ A^{∘∘} we may find n ∈ ℕ such that a^{n} ∈ I; combining with lemma 15.3.14(i) we easily deduce that

Cont(A) = {v ∈ Spv A | v(a) is final in Γ_{v}, for every a ∈ I}.

In light of lemma 15.3.4(i) we then see that

Cont(A) ⊂ {v ∈ Spv(A, J) | v(a) < 1 for every a ∈ I}

and to conclude, it suffices to prove the converse inclusion. Hence, let v ∈ Spv(A, J) be any valuation such that v(a) < 1 for every a ∈ I. If Γ_{v} ≠ cΓ_{v}, lemma 15.3.4(i) implies that either v(a) = 0 or v(a) is final in Γ_{v} for every a ∈ J, and then v ∈ Cont(A), by the foregoing.

Lastly, suppose that Γ_{v} = cΓ_{v}, and let a ∈ I be any element; in this case, it suffices to show that, for every b ∈ A with v(b) ≠ 0, there exists n ∈ ℕ such that v(a)^{n} < v(b)^{−1}. However, we may find n ∈ ℕ such that a^{n}b ∈ I, so v(a^{n}b) < 1 by assumption, whence the contention.

(ii): From (i) we see that Cont(A) = ⋂_{a∈A^{∘∘}}(Spv(A, J)\R_{A}(\frac{1}{a})). Combining with corollary 8.1.42, we obtain the assertion. □

15.3.16. Let A be and J be as in theorem 15.3.15. We say that a subset of Cont(A) is rational, if it can be written as an intersection Cont(A) ∩ R, where R is a rational subset of Spv(A, J) (see definition 15.3.6(ii)). Hence, a rational subset is of the form Cont(A) ∩ R_{A}(\frac{f_{1}}{f_{0}} , …, \frac{f_{n}}{f_{0}} ), where (f_{0}, f_{1}, …, f_{n}) is any sequence of elements of A that generates an open ideal. Especially, the class of rational subsets of Cont(A) does not depend on the choice of the ideal J, and provides a basis of quasi-compact open subsets of Cont(A) that is closed under finite intersections (theorem 15.3.7(ii)).

Corollary 15.3.17. Let f : A → B be an f-adic morphism of f-adic topological rings. We have:

(i) Cont(f) is a spectral map. More precisely, if R is a rational subset of Cont(A), then Cont(f)^{−1}(R) is a rational subset of Cont(B).

(ii) Cont(f) restricts to maps Cont(B)_{a} → Cont(A)_{a} and Cont(B)_{na} → Cont(A)_{na}.

Proof. (i): Indeed, if R = Cont(A) ∩ R_{A}(\frac{a_{1}}{a_{0}} , …, \frac{a_{n}}{a_{0}} ) for a sequence a_{0}, a_{1}, …, a_{n} that generates an open ideal of A, then Cont(f)^{−1}(R) = Cont(B) ∩ R_{A}(\frac{f(a_{1})}{f(a_{0})}, …, \frac{f(a_{n})}{f(a_{0})}), and since f is adic, it is easily seen that the sequence f(a_{0}), f(a_{1}), …, f(a_{n}) generates an open ideal of B.

(ii) is clear. □

Proposition 15.3.18. Let A be an f-adic ring, B ⊂ A an open subring, and denote by i : B → A the inclusion map. Then we have :

(i) Cont(A) = Spv(i)^{−1}Cont(B).

(ii) Cont(i) restricts to a homeomorphism Cont(A)_{a} →^{∼} Cont(B)_{a}.

Proof. (i): Let v be a valuation of A such that w := Spv(i)(v) = v_{|B} is a continuous valuation of B; we have to show that v is a continuous valuation of A, and by remark 15.3.13(vi), we may assume that Ker v is not open. In this case, lemma 8.3.29(iii) implies that κ(v) = κ(w), and therefore v and w have the same value group; the assertion follows immediately.

(ii): Define X_{A} and X_{A}^{∘∘} as in definition 8.3.28; we have

Cont(A)_{a} = Spv(X_{A} \ X_{A}^{∘∘}) ∩ Cont(A)

and likewise for Cont(B)_{a}, so the assertion follows from (i) and lemma 8.3.29(iii). □

Proposition 15.3.19. Let A be a topological ring, A^{∧} the separated completion of A. We have: (i) The completion map i : A → A^{∧} induces a bijective and continuous map

Cont(i) : Cont(A^{∧}) → Cont(A).

(ii) Moreover, with the ordering given by specializations on Cont(A) and Cont(A^{∧}), the map Cont(i) is also an isomorphism of partially ordered sets (see remark 15.3.13(v))

(Cont(A^{∧}), ≤) →^{∼} (Cont(A), ≤).

(iii) Furthermore, Cont(i) restricts to bijections

Cont(A^{∧})_{a} →^{∼} Cont(A)_{a}  and  Cont(A^{∧})_{na} →^{∼} Cont(A)_{na}.

(iv) If A is f-adic, Cont(i) is a homeomorphism.

Proof. (i): For the injectivity of Cont(i) we show the following more precise :

Claim 15.3.20. Let w ∈ Cont(A^{∧}) be any element, and set v := Cont(i)(w). Let Γ_{w} and Γ_{v} be the value groups of w and respectively v. Then we have:

(i) Γ_{w} = Γ_{v} and cΓ_{w} = cΓ_{v}.

(ii) The induced map of residue fields κ(v) → κ(w) has dense image, for the valuation topology of κ(w) (notation of remark 9.1.4(v)).

Proof of the claim. (i): Let a ∈ A^{∧} be any element such that γ := w(a) ∈ Γ_{w}; by assumption, there exists an open neighborhood U_{γ} of 0 in A^{∧} such that w(x) < γ for every x ∈ U_{γ}, hence w(a + U_{γ}) = {γ}, and since a + U_{γ} intersects the image of A in A^{∧}, we deduce that γ ∈ Γ_{v}, whence the claim.

(ii): According to remark 15.3.13(i), the natural map π_{v} : A → κ(v) is continuous for the valuation topology 𝒯_{v} of κ(v), hence it extends uniquely to a continuous ring homomorphism π_{v}^{∧} : A^{∧} → κ(v)^{∧}, where (κ(v)^{∧}, 𝒯_{v}^{∧}) denotes the completion of (κ(v), 𝒯_{v}). However, v extends uniquely to a valuation v^{∧} : κ(v)^{∧} → Γ_{v∘} and 𝒯_{v}^{∧} agrees with the corresponding valuation topology of κ(v)^{∧} (proposition 9.1.16(iv,v)). It follows that v^{∧} ∘ π_{v}^{∧} : A^{∧} → Γ_{v∘} is the unique continuous valuation whose composition with i agrees with v. Consequently, there exists as well a unique inclusion of fields j : κ(w) → κ(v)^{∧} whose composition with the projection π_{w} : A^{∧} → κ(w) agrees with π_{v}^{∧}, and v^{∧} ∘ j is the residual valuation 𝒯_{w} of κ(w). Especially, 𝒯_{w} agrees with the topology induced by 𝒯_{v}^{∧} via j; since the image of κ(v) is dense in κ(v)^{∧} relative to the topology 𝒯_{v}^{∧} (theorem 8.2.8(ii)), the assertion follows. ♢

Now, let w, w′ ∈ Cont(A^{∧}) be two valuations such that w ∘ i is equivalent to w′ ∘ i, and denote by Γ_{w} (resp. Γ_{w′}) the value group of w (resp. of w′). From claim 15.3.20we may assume that Γ_{w} = Γ_{w′} and w ∘ i = w′ ∘ i. Since the image of A is dense in A^{∧} and the topology of Γ_{w} is separated, we conclude that w = w′, as required.

Next, we show that Cont(i) is surjective. Indeed, let v be any continuous valuation of A with value group Γ, so that, for every γ ∈ Γ there exists an open neighborhood U_{γ} of 0 in A such that v(x) < γ for every x ∈ U_{γ}; we need to show that there exists a valuation v^{∧} : A^{∧} → Γ_{∘} such that v = v^{∧} ∘ i. We define v^{∧} as follows. Let 𝒞 := (a_{i} | i ∈ I) be any Cauchy net in A (indexed by some filtered ordered set I); for every γ ∈ Γ, we may then find i(γ) ∈ I such that a_{j} − a_{k} ∈ U_{γ} for every j, k ∈ I with j, k ≥ i(γ). We let 𝒞_{γ} be the Cauchy net (a_{j} | j ∈ I, j ≥ i(γ)) for every such γ. Now, suppose that there exists γ ∈ Γ and a ∈ 𝒞_{γ} with δ := v(a) ≥ γ; then v(b) = δ for every b ∈ 𝒞_{γ}, and in this case we set v^{∧}(𝒞) := δ. Otherwise, we set v^{∧}(𝒞) := 0. It is easily seen that this definition depends only on the equivalence class of 𝒞 in A^{∧}, so it yields a well defined map v^{∧} : A^{∧} → Γ_{∘}, and it is clear that v^{∧} ∘ i = v. We leave to the reader the verification that v^{∧} is indeed a continuous valuation on A^{∧}.

(ii): Let u ∈ Cont(A^{∧}) be any element, and w′ a specialization of u′ := Cont(i)(u) in Cont(A). By proposition 9.2.24 there exists v′ ∈ Spv A that is both a secondary specialization of u′ and a primary generization of w′ in Spv A. By lemma 9.2.25(iii) we may find a secondary specialization v of u in Spv A^{∧} with Spv(i)(v) = v′; then v ∈ Cont(A^{∧}) and v′ ∈ Cont(A) (remark 15.3.13(iv)). By claim 15.3.20, we know that v and v′ have the same value groups and the same characteristic subgroups; hence, we may find a primary specialization w of v in Spv A with w′ = Spv(i)(w), and w ∈ Cont(A) (remark 15.3.13(iv)), as required.

(iii): We have already observed that Cont(i) restricts to a map Cont(A^{∧})_{na} → Cont(A)_{na} (remark 15.3.13(vii)); in light of (i), it then suffices to show that Cont(i) restricts as well to a map Cont(A^{∧})_{a} → Cont(A)_{a}. Thus, let v : A^{∧} → Γ_{∘} be any continuous analytic valuation, and suppose that 𝔭 := Ker v ∘ i is open in A; by continuity, the topological closure i(𝔭)^{c} of i(𝔭) in A^{∧} lies in Ker v. However, i(𝔭)^{c} contains an open subset of A^{∧} (claim 8.2.11(ii)), and therefore it is open, a contradiction.

(iv): From (i), we know already that Cont(i) is continuous and bijective. It then suffices to show that Cont(i) is a closed map, if A is f-adic. However, in this case Cont(A) and Cont(A^{∧}) are spectral topological spaces (theorem 15.3.15(ii)) and Cont(i) is a spectral map (corollary 15.3.17(i) and proposition 8.3.33(ii)), so it suffices to check that Cont(i) is specializing (proposition 8.1.47(i) and corollary 8.1.50(i)). The latter follows immediately from (ii). □

Corollary 15.3.21. Let f : A → B be a continuous ring homomorphism of f-adic rings. Suppose that B is topologically local, and denote by B^{∧} the separated completion of B. Then the following conditions are equivalent :

(a) f is an f-adic map.

(b) The composition A → B^{∧} of f with the completion map B → B^{∧} is f-adic.

(c) Cont(f) restricts to a map Cont(B)_{a} → Cont(A)_{a}.

Proof. (a)⇒(b) since the completion map B → B^{∧} is f-adic (proposition 8.3.33(i,ii)). (b)⇒(c) by corollary 15.3.17(ii) and proposition 15.3.19(i). (c)⇒(a): Let B_{0} be a subring of definition of B, and A_{0} a subring of definition of A such that

A_{0} ⊂ f^{−1}B_{0} (corollary 8.3.19(ii)), and let f_{0} : A_{0} → B_{0} be the restriction of f. From (c) and proposition 15.3.18(ii) we deduce that Cont(f_{0}) restricts to a map Cont(B_{0})_{a} → Cont(A_{0})_{a}, and it suffices to check that f_{0} is f-adic. Moreover, B_{0} is still topologically local (proposition 8.4.2(i)). Thus, we may replace f by f_{0}, and assume from start that B = B^{∘}, so that B^{∘∘} lies in the Jacobson radical of B. Now, define X_{A} and X_{A}^{∘∘} as in definition 8.3.28, and likewise for the ring B; also, set φ := Spec f. By lemma 8.3.29(iv), it suffices to show that φ^{−1}X_{A}^{∘∘} = X_{B}^{∘∘}. However, suppose that the latter fails; then there exists a prime ideal 𝔭 ∈ X_{B} \ X_{B}^{∘∘} with φ(𝔭) ∈ X_{A}^{∘∘}, and we pick any maximal ideal 𝔪 of B containing 𝔭. Set C := (B/𝔭)_{𝔪}, and choose a valuation ring V of Frac C that dominates C (corollary 9.1.25). The valuation ring V corresponds to a point v ∈ Spv B; next, pick any finitely generated ideal of definition I of B, and set w := v^{cΓ_{v}(I)}. Since B^{∘∘} ⊂ 𝔪, theorem 15.3.15(i) implies that w is a continuous valuation of B. Since 𝔭 = Ker v and I is not contained in 𝔭, it follows that I is not contained in the support of w, therefore w ∈ Cont(B)_{a}. However, w′ := Cont(f)(w) is a primary specialization of Spv f(v), and the support of the latter is an open ideal of A, by construction, so w′ ∈ Cont(A)_{na}, which contradicts our assumptions. □

Proposition 15.3.22. Let A be any f-adic topological ring, and denote by A^{sep} the maximal separated quotient of A. The following holds :

(i) Spv A = ∅ if and only if A = 0. (ii) Cont(A) = ∅ if and only if A^{sep} = 0. (iii) Cont(A)_{a} = ∅ if and only if the topology of A^{sep} is discrete.

Proof. (i): Indeed, if A ≠ 0, it has a prime ideal 𝔭, and the trivial valuation on A/𝔭 yields an element of Spv A.

(iii): Let A^{∧} be the separated completion of A; if Cont(A)_{a} = ∅, we have Cont(A^{∧})_{a} = ∅ as well, by proposition 15.3.19(iii), and it suffices to show that the topology of A^{∧} is discrete. Let B ⊂ A^{∧} be a subring of definition, and I ⊂ B a finitely generated ideal of adic definition (proposition 8.3.33(i)); then

(15.3.23)  Cont(B)_{a} = ∅

as well (proposition 15.3.18(ii)), and we remark :

Claim 15.3.24. I is a nilpotent ideal.

Proof of the claim. Let 𝔭 ⊂ B be any prime ideal, and suppose by way of contradiction, that I is not contained in 𝔭. Choose a maximal ideal 𝔪 of C := B/𝔭, and pick a valuation ring V of Frac C that dominates C_{𝔪} (corollary 9.1.25). Let 𝔫 ⊂ B be the maximal ideal such that 𝔫/𝔭 = 𝔪; then V corresponds to a valuation v of B with support equal to 𝔭, and such that v(a) < 1 for every a ∈ 𝔫. Since B is I-adically complete, we have I ⊂ 𝔫 (remark 8.3.10(v)), so it follows easily that v(a) < 1 for every a ∈ B^{∘∘}. Let Γ_{v} be the value group of v, and set w := v^{cΓ_{v}(I)} (notation of definition 15.3.3). Then w ∈ Spv(B, I) and w(a) < 1 for every a ∈ B^{∘∘}, so w is continuous (theorem 15.3.15(i)). Lastly, since v(I) ≠ {0} by construction, we have w(I) ≠ {0} as well, since w is an I-admissible specialization of v. Therefore w ∈ Cont(B)_{a}, which contradicts (15.3.23). ♢

Claim 15.3.24means that the topology of B is discrete, hence the same holds for the topology of A^{∧}, as required.

(ii): From (iii) we know that the topological closure {0}^{c} of the ideal {0} of A is open in A; hence, every prime ideal containing {0}^{c} is open as well, and therefore every valuation with support equal to such a prime ideal would be continuous (remark 15.3.13(vi)); since Cont(A) = ∅, we conclude that no prime ideal of A contains {0}^{c}, i.e. {0}^{c} = A, whence the assertion. □

Lemma 15.3.25. Let A be a topologically local f-adic ring, J ⊂ A an ideal. The following conditions are equivalent :

(a) J is an open ideal.

(b) For every v ∈ Cont(A)_{a} we have J ⊄ Ker v.

(c) For every v ∈ Cont(A)_{a} of rank one, we have J ⊄ Ker v.

Proof. Clearly (a)⇒(b)⇒(c).

Next, suppose that J is not open; to conclude, it suffices to exhibit a rank one continuous analytic valuation on A whose support contains J. Now, let B ⊂ A be a subring of definition, and set J_{B} := B ∩ J; since J is not open in A, the ideal J_{B} cannot be open in B. Moreover, B is also topologically local, by proposition 8.4.2(i).

Claim 15.3.26. It suffices to exhibit a rank one continuous analytic valuation on B whose support contains J_{B}.

Proof of the claim. Indeed, let v : B → Γ_{∘} be such a valuation, and denote by 𝔭 ⊂ B the support of v; with the notation of definition 8.3.28, we have 𝔭 ∈ X_{B} \ X_{B}^{∘∘}, and by lemma 8.3.29(iii) the inclusion map i : B → A induces an isomorphism of schemes φ : X_{A} \ X_{A}^{∘∘} →^{∼} X_{B} \ X_{B}^{∘∘}. Set 𝔮 := φ^{−1}(𝔭); it follows that i extends to a ring isomorphism i_{𝔭} : B_{𝔭} →^{∼} A_{𝔮}. Let also j : A → A_{𝔮} be the localization map; v extends uniquely to a valuation \overline{v} : B_{𝔭} → Γ_{∘}, and composing with i_{𝔭}^{−1} ∘ j we deduce a valuation w : A → Γ_{∘}. Moreover, Cont(i) : Cont(A) → Cont(B) is a homeomorphism (proposition 15.3.18(ii)), and a direct inspection of the definitions yields

Cont(i)^{−1}(v) = w.

Especially, w is continuous, analytic and of rank one. Lastly, the support of \overline{v} contains J_{B,𝔭} = i_{𝔭}^{−1}J_{𝔮}, so the support of w contains J. ♢

Due to claim 15.3.26, we may replace A by B and J by J_{B}, and assume from start that A = A^{∘}; especially, A^{∘∘} lies in the Jacobson radical of A. By lemma 8.3.29(v), there exists x ∈ Spec A/J \ X_{A}^{∘∘}; according to remark 8.1.45(ii) we may find a minimal specialization y of x in Spec A/J \ X_{A}^{∘∘}, and if {y}^{c} denotes the topological closure of {y} in Spec A/J, the closed subset Z := {y}^{c} ∩ X_{A}^{∘∘} of Spec A/J is non-empty, since X_{A}^{∘∘} contains all the maximal ideals of A. We let 𝔭 ⊂ A be any maximal point of Z, and we notice that the image C of A_{𝔭} in κ(y) is a one-dimensional local domain.

Claim 15.3.27. For any one-dimensional local domain (R, 𝔪_{R}) there exists a one-dimensional valuation ring of Frac R that dominates R.

Proof of the claim. By corollary 9.1.25we may find a valuation ring V of Frac R that dominates R. Let ℱ be the set of all prime ideals 𝔮 of V such that 𝔮 ∩ R = 𝔪_{R}; then ℱ is non-empty, and it admits a minimal element 𝔯 := ⋂_{𝔮∈ℱ} 𝔮. It then suffices to check that the localization V_{𝔯} is one-dimensional. However, let 𝔭 ⊂ V_{𝔯} be any non-zero prime ideal, and pick any a ∈ 𝔭 \ {0}; we may write a := b/b′ for some b, b′ ∈ R, and then b ∈ 𝔭 as well, so 𝔭 ∩ R = 𝔪_{R}, in which case 𝔭 must be the maximal ideal of V_{𝔯}, and the assertion follows. ♢

By claim 15.3.27, we may find a one-dimensional valuation ring V of κ(y) that dominates C, and denote by v ∈ Spv A the corresponding valuation. Let also I ⊂ A be any finitely generated ideal of adic definition; by construction, the support of v contains J and is not an open ideal, and v does not admit any proper I-admissible specializations (see (15.3.1)), so v ∈ Spv(A, I). Moreover A^{∘∘} ⊂ 𝔭, so that v(b) < 1 for every b ∈ A^{∘∘}, and consequently v ∈ Cont(A)_{a}. □

15.4. Affinoid rings and affinoid schemes. In this section we present an extension and refinement of Huber’s theory of affinoid rings.

Definition 15.4.1. (i) Let A be any f-adic ring. An open subring B ⊂ A is called a ring of integral elements of A, if B ⊂ A^{∘} and B is integrally closed in A.

(ii) An affinoid ring is a datum (A, A^{+}) consisting of an f-adic ring A and a ring A^{+} of integral elements of A. A morphism of affinoid rings f : (A, A^{+}) → (B, B^{+}) is a continuous ring homomorphism f : A → B such that f(A^{+}) ⊂ B^{+}. Then we say that f is f-adic, if the same holds for the underlying map A → B.

(iii) A quasi-affinoid ring is a datum (A, A^{+}, U) consisting of an affinoid ring (A, A^{+}) and a constructible open subset U ⊂ X_{A} that contains the analytic locus (see definition 8.3.28). A morphism of quasi-affinoid rings f : (A, A^{+}, U) → (B, B^{+}, V) is a morphism f : (A, A^{+}) → (B, B^{+}) of affinoid rings such that Spec f restricts to a morphism of schemes V → U. We say that such f is f-adic, if the same holds for the underlying map A → B.

(iv) A quasi-affinoid scheme is a datum \underline{X} := (X, 𝒯_{X}, A_{X}^{+}) consisting of a (quasi-compact) quasi-affine scheme X (see [60, Ch.II, Déf.5.1.1]), an f-adic topology 𝒯_{X} on A_{X} := 𝒪_{X}(X), and a subring A_{X}^{+} ⊂ A_{X} such that the following holds :

• (A_{X}, A_{X}^{+}) is an affinoid ring.

• For every s ∈ A_{X}^{∘∘}, the open subscheme X_{s} := X ×_{Spec A_{X}} Spec A_{X}[s^{−1}] is affine.

A morphism of quasi-affinoid schemes φ : (X, 𝒯_{X}, A_{X}^{+}) → (Y, 𝒯_{Y} , A_{Y}^{+}) is a morphism of schemes φ : X → Y whose associated morphism of sheaves of rings φ^{♭} : 𝒪_{Y} → φ_{∗}𝒪_{X} induces a morphism of affinoid rings φ_{Y}^{♭} : (𝒪_{Y} (Y), A_{Y}^{+}) → (𝒪_{X}(X), A_{X}^{+}). We say that φ is f-adic if φ_{Y}^{♭} is f-adic. We say that \underline{X} is an affinoid scheme if X is an affine scheme.

(v) Let \underline{A} := (A, A^{+}, U) (resp. \underline{X} := (X, 𝒯_{X}, A_{X}^{+})) be any quasi-affinoid ring (resp. quasiaffinoid scheme). We say that \underline{A} (resp. \underline{X}) is topologically local (resp. topologically henselian, resp. complete, resp. separated) if the underlying f-adic ring A (resp. 𝒪_{X}(X)) enjoys the corresponding property.

Remark 15.4.2. (i) Clearly the affinoid rings, the quasi-affinoid rings, the affinoid schemes and the quasi-affinoid schemes with their morphisms as in definition 15.4.1form categories denoted respectively :

Afd.Ring q.Afd.Ring Afd.Sch q.Afd.Sch

(Afd.Sch is a full subcategory of q.Afd.Sch). Also, we have a natural fully faithful functor

Afd.Ring → q.Afd.Ring (A, A^{+}) ↦ (A, A^{+}, Spec A)

so we may regard Afd.Ring as a full subcategory of q.Afd.Ring.

(ii) By virtue of [60, Ch.II, Prop.5.1.2], we have a natural fully faithful functor

Γ : q.Afd.Sch^{o} → q.Afd.Ring (X, 𝒯_{X}, A_{X}^{+}) ↦ (A_{X}, A_{X}^{+}, X)

(where A_{X} := 𝒪_{X}(X) and X is naturally identified with the image of the open immersion X → Spec A_{X}). Indeed, let s ∈ A_{X}^{∘∘} be any element, and define the open subscheme X_{s} ⊂ X as in definition 15.4.1(iv); by assumption, X_{s} is affine, and on the other hand we have a natural isomorphism of A_{X}-algebras

Γ(X_{s}, 𝒪_{X}) →^{∼} A_{X}[s^{−1}]

by [59, Ch.I, Cor.9.2.2]. Therefore the image of the open immersion Spec A_{X}[s^{−1}] → Spec A_{X} equals X_{s}; especially, Spec A_{X}[s^{−1}] ⊂ X for every s ∈ A_{X}^{∘∘}, so (A_{X}, A_{X}^{+}, X) is indeed a quasiaffinoid ring. Especially, the analytic locus of Spec A_{X} lies in X, and shall also be called the analytic locus of \underline{X}.

(iii) Let (A, A^{+}) be any affinoid ring, B ⊂ A an open subring, and endow B with the topology induced by the inclusion map B → A. Then corollary 8.3.20(i) implies that the pair (B, A^{+} ∩ B) is an affinoid ring.

(iv) Let A be any f-adic ring, and endow R := ℤ ⊕ A^{∘∘} with the multiplication map given by the rule : (n, a) · (m, b) := (nm, ma + nb + ab) for every n, m ∈ ℤ and a, b ∈ A^{∘∘}. Then R is a ring, and we have a unique ring homomorphism f : R → A that restricts to the inclusion map on A^{∘∘}. The integral closure B of the image of f is the smallest ring of integral elements of A. On the other hand, A^{∘} is the largest subring of integral elements of A.

Proposition 15.4.3. (i) The functor Γ of remark 15.4.2(ii) admits a left adjoint :

Spec : q.Afd.Ring → q.Afd.Sch^{o} (A, A^{+}, U) ↦ (U, 𝒯_{U}, A_{U}^{+})

(ii) If f : \underline{A} → \underline{B} is an f-adic morphism of quasi-affinoid rings, then Spec(f) : Spec(\underline{B}) → Spec(\underline{A}) is an f-adic morphism of quasi-affinoid rings.

(iii) The functor Spec is not fully faithful, but it restricts to an equivalence of categories

Spec : Afd.Ring → Afd.Sch^{o}.

Proof. (i): For a quasi-affinoid ring \underline{A} := (A, A^{+}, U), we let 𝒯_{U} be the f-adic topology on A_{U} := 𝒪_{U}(U) provided by proposition 8.3.30(i), and let A_{U}^{+} be the integral closure of A^{+} in A_{U}. Indeed, notice that the restriction map A → A_{U} sends A^{∘} into A_{U}^{∘} (lemma 8.3.24(iii.a)); since A^{+} ⊂ A^{∘}, remark 8.3.10(iii) easily implies that A_{U}^{+} ⊂ A_{U}^{∘}. Next, let \overline{A} := Im(A → A_{U}) and endow \overline{A} with the topology induced by A_{U} via the open inclusion map \overline{A} → A_{U}; since A^{∘∘} is an open subgroup of A, for every s ∈ A_{U}^{∘∘} we may find n ∈ ℕ and t ∈ A^{∘∘} such that s^{n} equals the image of t in \overline{A}. Notice that t annihilates the kernel of the projection π : A → \overline{A}, so Spec π restricts to an isomorphism of schemes Spec \overline{A}[s^{−n}] →^{∼} Spec A[t^{−1}] ⊂ U. It follows that

U ×_{Spec A_{U}} Spec A_{U}[s^{−1}] = U ×_{Spec A} Spec A[t^{−1}] = Spec A[t^{−1}]

so the construction of Spec(\underline{A}) is achieved. Next, let f : \underline{A} → \underline{B} := (B, B^{+}, V) be any morphism of quasi-affinoid rings; set (V, 𝒯_{V} , B_{V}^{+}) := Spec(\underline{B}) and denote by \overline{B} the image of B in B_{V} := 𝒪_{V} (V). We deduce easily from proposition 8.3.30(i) that (a) the subrings \overline{A} and \overline{B} are open in A_{U} and B_{V}, so the induced map φ : A_{U} → B_{V} is continuous if and only the same holds for its restriction \overline{f} : \overline{A} → \overline{B}, and (b) the topologies of \overline{A} and \overline{B} are induced from the surjections A → \overline{A} and B → \overline{B}, so the continuity of \overline{f} follows from that of f. Thus, φ is continuous, and clearly φ(A_{U}^{+}) ⊂ B_{V}^{+}, so φ : Spec(\underline{B}) → Spec(\underline{A}) is a well defined morphism of quasi-affinoid schemes, and we let Spec(f) := φ.

Now, a simple inspection shows that Spec ∘ Γ is the identity automorphism of the category q.Afd.Sch^{o}; on the other hand, for every quasi-affinoid ring \underline{A} := (A, A^{+}, U), the restriction map A → A_{U} determines a natural morphism η\underline{A} : \underline{A} → Γ ∘ Spec(\underline{A}). Lastly, it is easily seen that the pair of natural transformations (ε := 1_{Spec∘Γ}, η) fulfills the triangular identities of (1.1.13), so the assertion follows from proposition 1.1.15(i).

(ii): Indeed, set A_{U} := 𝒪_{U}(U) and B_{V} := 𝒪_{V} (V); by assumption there exist open and bounded subrings A_{0} ⊂ A and B_{0} ⊂ B such that f(A_{0}) ⊂ B_{0}, and f restricts to an adic ring homomorphism A_{0} → B_{0}. Then, let \overline{A}_{0} be the image of A_{0} in A_{U}, and define likewise \overline{B}_{0} ⊂ B_{V}; we know that \overline{A}_{0} and \overline{B}_{0} are open and bounded subrings of A_{U} and respectively B_{V}, for the topologies induced by the latter topological rings, and the image of every ideal of adic definition of A_{0} (resp. of B_{0}) is an ideal of adic definition of \overline{A}_{0} (resp. of \overline{B}_{0}) (proposition 8.3.30(i)). It follows easily that the continuous map A_{U} → B_{V} restricts to an adic ring homomorphism \overline{A}_{0} → \overline{B}_{0}, whence the contention.

(iii) follows easily from the definitions. □

Example 15.4.4. Let \underline{X} := (X, 𝒯_{X}, A_{X}^{+}) be any quasi-affinoid scheme, set A_{X} := 𝒪_{X}(X), and let U ⊂ X be any quasi-compact open subset containing the analytic locus of Spec A_{X}. Let us denote by F_{U} the presheaf on the category q.Afd.Sch such that

F_{U}(\underline{Z}) := \{φ : \underline{Z} → \underline{X} | φ(Z) ⊂ U\} for every quasi-affinoid scheme \underline{Z} := (Z, 𝒯_{Z}, A_{Z}^{+})

so that F_{U} is a sub-presheaf of the presheaf h\underline{X} represented by \underline{X}. Let \underline{A} be the quasi-affinoid ring (A_{X}, A_{X}^{+}, U); we claim that F_{U} is representable by the quasi-affinoid scheme

U ×_{X} \underline{X} := Spec \underline{A}.

Indeed, the identity map of A_{X} yields a morphism of quasi-affinoid rings i : Γ(\underline{X}) → \underline{A}; now, for every quasi-affinoid scheme \underline{Z} := (Z, 𝒯_{Z}, A_{Z}^{+}), the datum of a morphism ψ : \underline{Z} → U ×_{X} \underline{X} of quasi-affinoid schemes is equivalent, by adjunction, to that of a morphism g : \underline{A} → Γ(\underline{Z}) of quasi-affinoid rings, and the composition f := g ∘ i : Γ(\underline{X}) → Γ(\underline{Z}) is a morphism such that Spec(f)(Z) ⊂ U, whence the morphism φ := Spec(f) : \underline{Z} → \underline{X} with φ(Z) ⊂ U. Conversely, any morphism φ : \underline{Z} → \underline{X} with φ(Z) ⊂ U yields a morphism Γ(φ) : Γ(\underline{X}) → Γ(\underline{Z}) that factors uniquely through \underline{A}, whence the contention.

Example 15.4.5. (i) Consider two f-adic morphisms of affinoid rings

\underline{B}′ := (B′, B^{′+}) ⟵^{g} \underline{B} := (B, B^{+}) ⟶^{f} \underline{A} := (A, A^{+})

and endow A′ := B′ ⊗_{B} A with the f-adic topology 𝒯_{A′} characterized by proposition 8.3.34(i), so that the natural maps f′ : B′ → A′ and g_{A} : A → A′ are both f-adic. Let also R ⊂ A′ be the image of B^{′+} ⊗_{B^{+}} A^{+}, and denote by A^{′+} the integral closure of R in A′. Then the datum

\underline{B}′ ⊗\underline{B} \underline{A} := (A′, A^{′+})

is an affinoid ring. Indeed, a simple inspection of the definition of 𝒯_{A′} shows that R is open in A′, and we have g_{A}(A^{+}), f(B^{+}) ⊂ A^{′∘} by lemma 8.3.24(iii), so also R ⊂ A^{′∘}, and then A^{′+} ⊂ A^{′∘} as well, by remark 8.3.10(iv).

(ii) In the situation of (i), let also U_{B} ⊂ X_{B} := Spec B, U_{A} ⊂ X_{A} := Spec A and U_{B′} ⊂ X_{B′} := Spec B′ be three open subsets, such that f and g are morphisms of quasi-affinoid rings

(\underline{B}′, U_{B′}) ⟵^{g} (\underline{B}, U_{B}) ⟶^{f} (\underline{A}, U_{A})

and set U_{A′} := U_{B′} ×_{U_{B}} U_{A}. Then the datum

(\underline{B}′, U_{B′}) ⊗_{(\underline{B},U_{B})} (\underline{A}, U_{A}) := (\underline{B}′ ⊗\underline{B} \underline{A}, U_{A′})

is a quasi-affinoid ring as well. Indeed, let B_{0} ⊂ B be a ring of definition, and I_{0} ⊂ B_{0} an ideal of adic definition; we need to show that Spec A′\Spec A′/I_{0}A′ ⊂ U_{A′}. Set Z_{B′} := X_{B′}\U_{B′} and Z_{A} := X_{A} \ U_{A}; the assertion is equivalent to (Z_{B′} ×_{X_{B}} X_{A}) ∪ (X_{B′} ×_{X_{B}} Z_{A}) ⊂ Spec A′/I_{0}A′. The latter is clear, since by assumption Z_{B′} ⊂ Spec B′/I_{0}B′ and Z_{A} ⊂ Spec A/I_{0}A.

Lemma 15.4.6. With the notation of example 15.4.5, the resulting commutative diagrams

\xymatrix{\underline{B} \ar[r]^{f} \ar[d]_{g} & \underline{A} \ar[d]^{g_{A}} & (\underline{B}, U_{B}) \ar[r]^{f} \ar[d]_{g} & (\underline{A}, U_{A}) \ar[d]^{g_{A}} \\ _{\underline{B}_{′}} \ar[r]^{f′} & \underline{B}′ ⊗\underline{B} \underline{A} & (\underline{B}′, U_{B′}) \ar[r]^{f′} & (\underline{B}′, U_{B′}) ⊗_{(\underline{B},U_{B})} (\underline{A}, U_{A})}

are cocartesian in the categories Afd.Ring and respectively q.Afd.Ring.

Proof. Indeed, consider any two morphisms h : (A, A^{+}) → (C, C^{+}), k : (B, B^{+}) → (C, C^{+}) of affinoid rings such that h∘f = k∘g; by proposition 8.3.34(i) there exists a unique continuous ring homomorphism l : A′ → C such that l ∘ g_{A} = h and l ∘ f′ = k, and it is easily seen that l(A^{′+}) ⊂ C^{+}, i.e. l : (A′, A^{′+}) → (C, C^{+}) is a morphism of affinoid rings, whence the assertion for the left diagram. The assertion for the right diagram is an immediate consequence. □

Example 15.4.7. Consider a quasi-affinoid ring \underline{A} := (A, A^{+}) and any two f-adic morphisms f : B → A, g : B → B′ of f-adic rings.

(i) As a special case of example 15.4.5(i), take for B^{+} and B^{′+} the smallest subrings of integral elements of B and respectively B′ (see remark 15.4.2(iv)) and set \underline{B} := (B, B^{+}) and \underline{B}′ := (B′, B^{′+}); we may regard f and g as morphisms \underline{B} → \underline{A} and \underline{B} → \underline{B}′ respectively. The resulting tensor product \underline{B}′ ⊗\underline{B} \underline{A} shall be simply denoted

B′ ⊗_{B} \underline{A}.

By lemma 15.4.6, the resulting morphisms \underline{A} ⟶^{g_{A}} B′ ⊗_{B} \underline{A} ⟵^{f′} \underline{B}′ enjoy the following universal property. Let (h, k) be any pair consisting of a morphism h : \underline{A} → (C, C^{+}) of affinoid rings, and a continuous ring homomorphism k : B′ → C such that h ∘ f = k ∘ g; then there exists a unique morphism l : B′ ⊗_{B} \underline{A} → (C, C^{+}) of affinoid rings such that l ∘ g_{A} = h and l ∘ f′ = k.

(ii) Likewise, let (\underline{A}, U) be a quasi-affinoid ring. As a special case of example 15.4.5(ii), we may define \underline{B} and \underline{B}′ as in (i), and take U_{B} := Spec B, U_{B′} := Spec B′. There follow morphisms of quasi-affinoid rings f : (\underline{B}, U_{B}) → (\underline{A}, U) and f : (\underline{B}, U_{B}) → (\underline{B}′, U_{B′}). The resulting tensor product (B′ ⊗_{B} \underline{A}, U_{B′}) will be denoted simply

B′ ⊗_{B} (\underline{A}, U)

and notice that U_{B′} = Spec A′×_{Spec A}U. It follows easily from lemma 8.3.29(iv) that this quasiaffinoid ring enjoys the corresponding universal property as in (i). Namely, let (h, k) be any pair consisting of a morphism h : (\underline{A}, U) → (C, C^{+}, V) of quasi-affinoid rings, and k : B′ → C a continuous ring homomorphism such that h ∘ f = k ∘ g; then there exists a unique morphism l : B′ ⊗_{B} (\underline{A}, U) → (C, C^{+}, V) of quasi-affinoid rings such that l ∘ g_{A} = h and l ∘ f′ = k.

Example 15.4.8. Let \underline{Y} → \underline{X} and \underline{Y}′ → \underline{X} be two f-adic morphisms of quasi-affinoid schemes. Then the fibre product \underline{Y} ×\underline{X} \underline{Y}′ is representable in the category q.Afd.Sch. Indeed, the induced morphisms of quasi-affinoid rings Γ(\underline{X}) → Γ(\underline{Y}) and Γ(\underline{X}) → Γ(\underline{Y}′) are f-adic, hence we may form the tensor product \underline{A} := Γ(\underline{Y}) ⊗_{Γ(\underline{X})} Γ(\underline{Y}′) as in example 15.4.5(ii), and in view of lemma 15.4.6, the quasi-affinoid scheme Spec \underline{A} represents the sought fibre product (details left to the reader).

Example 15.4.9. (i) Let (A, A^{+}) be an affinoid ring and B a finite A-algebra. According to remark 8.3.43(iii), the canonical topology 𝒯_{B}^{A} on B is f-adic, and the structure map φ : A → B is f-adic and restricts to a finite map A_{0} → B_{0} of suitable subrings of definition. Let B^{+} be the integral closure of φ(A^{+}) in B. We claim that B^{+} is a ring of integral elements of B. Indeed, B^{+} ⊂ B^{∘} by lemma 8.3.24(iii.a) and remark 8.3.10(iv). To check that B^{+} is open in B, pick a finite system b_{1}, …, b_{n} of generators of the A_{0}-module B_{0}; for every i = 1, …, n there exist m_{i} ∈ ℕ and a polynomial P_{i} = X^{m_{i}} + ∑_{j=0}^{m_{i}−1} a_{ij}X^{j} ∈ A_{0}[X] with P_{i}(b_{i}) = 0. Let I_{0} ⊂ A_{0} be an ideal of adic definition; since A^{+} is open, there exists k ∈ ℕ such that I_{0}^{k}a_{ij} ∈ A^{+} for i = 1, …, n and j = 0, …, m_{i} − 1. Thus, for every c ∈ I_{0}^{k} and i = 1, …, n we have

0 = c^{m_{i}}P_{i}(b_{i}) = Q_{i}(cb_{i}) where Q_{i} := X^{m_{i}} + ∑_{j=0}^{m_{i}−1} c^{m_{i}−j}a_{ij}X^{j} ∈ A^{+}[X].

This shows that I_{0}^{k}B_{0} ⊂ B^{+}, whence the contention. We call (B, B^{+}) the affinoid ring associated with the finite A-algebra B, and we denote it by

B ⊗_{A} (A, A^{+}).

(ii) Thus, we get an f-adic morphism φ : (A, A^{+}) → B ⊗_{A} (A, A^{+}) of affinoid rings. We claim that the morphism of affinoid rings φ enjoys the following universal property. For every morphism f : (A, A^{+}) → (C, C^{+}) of affinoid rings, every ring homomorphism g : B → C such that g ∘ φ = f is a morphism g : (B, B^{+}) → (C, C^{+}) of affinoid rings. Indeed, a simple inspection shows that g(B^{+}) ⊂ C^{+}, and g is continuous by virtue of proposition 8.3.41(ii).

(iii) As usual, the universal property determines the pair (B ⊗_{A} (A, A^{+}), φ) up to unique isomorphism of affinoid rings. Especially, if ψ : B → B′ is any finite ring homomorphism, then ψ ∘ φ : A → B′ is also finite, and it follows easily that

B′ ⊗_{B} (B ⊗_{A} (A, A^{+})) = B′ ⊗_{A} (A, A^{+}).

(iv) Likewise, if \underline{A} := (A, A^{+}, U) is any quasi-affinoid ring, and φ : A → B is as in (i), let us set U_{B} := Spec B ×_{Spec A} U. According to example 15.4.7(ii) we get a quasi-affinoid ring

B ⊗_{A} \underline{A} := (B, B^{+}, U_{B})

(for the topology 𝒯_{B}^{A} on B) and φ is an f-adic morphism of quasi-affinoid rings \underline{A} → B ⊗_{A} \underline{A} enjoying a corresponding universal property that the reader may spell out.

(v) In the same vein, let \underline{X} := (X, 𝒯_{X}, A_{X}^{+}) be a quasi-affinoid scheme and f : X′ → X a finite morphism of schemes; set A_{X} := 𝒪_{X}(X), Y := Spec A_{X} and let i : X → Y be the open immersion and g := i ∘ f : X′ → Y. We have natural morphisms of quasi-coherent 𝒪_{Y} -algebras i^{♭} : 𝒪_{Y} → i_{∗}𝒪_{X} and g^{♭} : i_{∗}𝒪_{X} → g_{∗}𝒪_{X′}, and we let 𝒜 be the integral closure of the image of i_{∗}𝒪_{X} in g_{∗}𝒪_{X′}. Notice that i^{♭} is an isomorphism, and 𝒜 is a quasi-coherent 𝒪_{Y} -algebra; hence 𝒜 is the filtered union of its finite quasi-coherent 𝒪_{Y} -subalgebras. Clearly 𝒜_{|X} = f_{∗}𝒪_{X′} is a finite 𝒪_{X}-algebra; since X is quasi-compact, it follows that there exists a finite quasi-coherent 𝒪_{Y} -subalgebra ℬ ⊂ 𝒜 such that ℬ_{|X} = f_{∗}𝒪_{X′}, and B := ℬ(Y) is a finite A_{X}-algebra. We claim that the quasi-affinoid scheme

X′ ×_{X} \underline{X} := Spec (B ⊗_{A_{X}} Γ(\underline{X})).

is independent, up to unique isomorphism, of the choice of ℬ and enjoys the following universal property. For every morphism of quasi-affinoid schemes h : \underline{Z} := (Z, 𝒯_{Z}, A_{Z}^{+}) → \underline{X}, every morphism k : Z → X′ of schemes such that f ∘ k = h is a morphism of quasi-affinoid schemes \underline{Z} → X′ ×_{X} \underline{X}. Indeed notice that, by construction, the commutative diagram of schemes

\xymatrix{X′ \ar[r]^{j} \ar[d]_{f} & Spec B \ar[d] \\ X \ar[r]^{i} & Y}

is cartesian, hence X′ is the scheme underlying X′ ×_{X} \underline{X}. The morphisms k and j induce homomorphisms of A_{X}-algebras 𝒪_{X′} → 𝒪_{Z}(Z) and B → 𝒪_{X′}; their composition is a homomorphism of A_{X}-algebras u : B → 𝒪_{Z}(Z), and since 𝒪_{Z}(Z) is a topological (A_{X}, 𝒯_{X})-algebra, (iv) implies that u is a morphism of quasi-affinoid rings B ⊗_{A_{X}} Γ(\underline{X}) → Γ(\underline{Z}). By adjunction, u induces a morphism of quasi-affinoid schemes \underline{Z} → X′ ×_{X} \underline{X}, and a simple inspection shows that this morphism is precisely given by k. Then the universal property implies as usual the stated independence of the choice of ℬ. By the same token, we also deduce that if X′′ → X′ is another finite morphism of schemes, we have a natural identification of quasi-affinoid schemes

X′′ ×_{X′} (X′ ×_{X} \underline{X}) →^{∼} X′′ ×_{X} \underline{X}

(details left to the reader). Lastly, notice that the induced projection X′ ×_{X} \underline{X} → \underline{X} is f-adic, by proposition 15.4.3(ii).

(vi) In the situation of (i) (resp. of (iv), resp. of (v)), suppose that (A, A^{+}) (resp. \underline{A}, resp. \underline{X}) is topologically henselian; combining with proposition 8.4.2(iv) we deduce that the same holds for B ⊗_{A} (A, A^{+}) (resp. for B ⊗_{A} \underline{A}, resp. for X′ ×_{X} \underline{X}).

Remark 15.4.10. (i) Denote by Afd.Ring_{comp} (resp. q.Afd.Ring_{comp}) the full subcategory of Afd.Ring (resp. of q.Afd.Ring) whose objects are the complete and separated affinoid (resp. quasi-affinoid) rings. Then the inclusion functor

Afd.Ring_{comp} → Afd.Ring (resp. q.Afd.Ring_{comp} → q.Afd.Ring)

admits a left adjoint, that assigns to every affinoid ring (A, A^{+}) (resp. quasi-affinoid ring (A, A^{+}, U)) its completion, which is the datum

(A, A^{+})^{∧} := (A^{∧}, (A^{+})^{∧}) (resp. (A, A^{+}, U)^{∧} := (A^{∧}, (A^{+})^{∧}, U^{∧}))

consisting of the separated completions of A and A^{+} (resp. and of the open subset U^{∧} := U ×_{Spec A} Spec A^{∧}). Indeed, taking into account proposition 8.3.33(i,iii) and lemma 8.3.6 we see that (A, A^{+})^{∧} = A^{∧} ⊗_{A} (A, A^{+}) and (A, A^{+}, U)^{∧} = A^{∧} ⊗_{A} (A, A^{+}, U), so the sought adjunctions follow from example 15.4.7(i,ii).

(ii) Let Afd.Ring_{hens} (resp. q.Afd.Ring_{hens}) be the full subcategory of Afd.Ring (resp. of q.Afd.Ring) whose objects are the topologically henselian affinoid (resp. quasi-affinoid) rings. Then the inclusion functor

Afd.Ring_{hens} → Afd.Ring (resp. q.Afd.Ring_{hens} → q.Afd.Ring)

admits a left adjoint, that assigns to every affinoid ring (A, A^{+}) (resp. quasi-affinoid ring (A, A^{+}, U)) its topological henselization, which is the datum

(A, A^{+})^{h} := (A^{h}, A^{+h}) (resp. (A^{h}, A^{+h}, U^{h}))

consisting of the topological henselizations of A and A^{+} (resp. and of the open subset U^{h} := U ×_{Spec A} Spec A^{h} : see definition 8.4.13). Indeed, by inspecting the construction of (8.4.9) it is easily seen that A^{+h} identifies naturally with an open subring of A^{h} contained in (A^{h})^{∘}. Moreover, A^{+h} is integrally closed in A^{h}, by corollary 9.8.6(ii). Hence (A, A^{+})^{h} is a topologically henselian affinoid ring, and theorem 8.4.10(ii) easily implies that every morphism (A, A^{+}) → (B, B^{+}) of affinoid rings with (B, B^{+}) topologically henselian, factors uniquely through a morphism (A, A^{+})^{h} → (B, B^{+}). By the same token, we deduce that

(A, A^{+})^{h} = A^{h} ⊗_{A} (A, A^{+})

from which it follows that (A, A^{+}, U)^{h} = A^{h} ⊗_{A} (A, A^{+}, U), whence also the second sought adjunction.

(iii) In the same vein, if (A, A^{+}) (resp. (A, A^{+}, U)) is any affinoid (resp. quasi-affinoid) ring, we may define its topological localization by considering the datum (A, A^{+})_{loc} (resp. (A, A^{+}, U)_{loc}) consisting of the topological localizations A_{loc} and A_{loc}^{+} of A and respectively A^{+}, defined as in (8.4.8) (resp. and of the open subset U ×_{Spec A} Spec A_{loc}). Then it is easily seen that the rule (A, A^{+}) ↦ (A, A^{+})_{loc} (resp. (A, A^{+}, U) ↦ (A, A^{+}, U)_{loc}) yields a left adjoint to the inclusion functor

Afd.Ring_{loc} → Afd.Ring (resp. q.Afd.Ring_{loc} → q.Afd.Ring) from the full subcategory of Afd.Ring (resp. of q.Afd.Ring) whose objects are the topologically local affinoid (resp. quasi-affinoid) rings. Notice also that (A, A^{+})_{loc} = A_{loc} ⊗_{A} (A, A^{+}), and likewise for (A, A^{+}, U)_{loc}.

(iv) Let \underline{X} be any quasi-affinoid scheme; by definition, \underline{X} is topologically local (resp. topologically henselian) if and only if the same holds for the quasi-affinoid ring Γ(\underline{X}). On the other hand, let \underline{A} := (A, A^{+}, U) be any quasi-affinoid ring; then \underline{A} is topologically local (resp. topologically henselian) if and only if the same holds for the quasi-affinoid scheme Spec \underline{A} (notation of proposition 15.4.3). Indeed, set A_{U} := 𝒪_{U}(U), and recall that the image B of A in A_{U} is open in the f-adic topology 𝒯_{U} of A_{U} (proposition 8.3.30(i)). In light of proposition 8.4.2(i,iii), we are then reduced to showing that A is topologically local (resp. topologically henselian) if and only if the same holds for B, where the latter is endowed with the f-adic topology induced by the inclusion map into A_{U}, which is the same as the topology induced by A via the surjection π : A → B. Let U_{A} and U_{B} be the analytic loci of Spec A and respectively Spec B; since U contains U_{A}, it is easily seen that Spec π induces a homeomorphism U_{B} →^{∼} U_{A} (details left to the reader). On the other hand, notice that B^{∘∘} is the radical of the ideal A^{∘∘}B; then the assertion follows from claim 8.4.4and [75, Rem.5.1.10(i)].

(v) For every quasi-affinoid scheme \underline{X}, set

\underline{X}_{loc} := Spec(Γ(\underline{X})_{loc}) \underline{X}^{h} := Spec(Γ(\underline{X})^{h}).

It follows easily from (iii),(iv), remark 15.4.2(ii) and proposition 15.4.3(i) that \underline{X}_{loc} is a topologically local quasi-affinoid scheme endowed with a natural morphism j\underline{X} : \underline{X}_{loc} → \underline{X} with the following property. Every morphism of quasi-affinoid schemes \underline{Y} → \underline{X} with \underline{Y} topologically local, factors uniquely through j\underline{X}. In other words, the rule \underline{X} ↦ \underline{X}_{loc} yields a right adjoint

q.Afd.Sch → q.Afd.Sch_{loc}

to the inclusion functor of the full subcategory q.Afd.Sch_{loc} of topologically local quasi-affinoid schemes. Likewise, \underline{X}^{h} is a topologically henselian quasi-affinoid scheme endowed with a universal morphism h\underline{X} : \underline{X}^{h} → \underline{X}, so the rule \underline{X} ↦ \underline{X}^{h} yields a right adjoint

q.Afd.Sch → q.Afd.Sch_{hens}

to the inclusion functor of the full subcategory q.Afd.Sch_{hens} of topologically henselian quasiaffinoid schemes.

(vi) From (iv), we deduce that the adjunction (Spec, Γ) of proposition 15.4.3(i) restricts to two adjoint pairs of functors

\xymatrix{q.Afd.Ring_{loc} \ar[r]^{Spec} & q.Afd.Sch_{loc}^{o} \ar[l]^{Γ} & q.Afd.Ring_{hens} \ar[r]^{Spec} & q.Afd.Sch_{hens}^{o}. \ar[l]^{Γ}}

Taking into account (v) and remark 1.1.17(i), we then obtain natural isomorphisms

Spec(\underline{A}_{loc}) →^{∼} (Spec \underline{A})_{loc} Spec(\underline{A}_{hens}) →^{∼} (Spec \underline{A})_{hens}

of quasi-affinoid schemes, for every quasi-affinoid ring \underline{A}. On the other hand, we have as well natural identifications

(15.4.11)  Γ(\underline{X}_{loc}) →^{∼} Γ(\underline{X})_{loc}  Γ(\underline{X}^{h}) →^{∼} Γ(\underline{X})^{h}

for every quasi-affinoid scheme \underline{X}. Indeed, if Γ(\underline{X}) = (A, A^{+}, U), we have Γ(\underline{X})_{loc} = (A_{loc}, A_{loc}^{+}, U_{loc}), where U_{loc} = U ×_{Spec A} Spec A_{loc}. Therefore Γ(\underline{X}_{loc}) = (B, B^{+}, U_{loc}), where B := 𝒪_{U_{loc}}(U_{loc}), and with B^{+} equal to the integral closure of A_{loc}^{+} in B. Since the localization map is flat, and since A = 𝒪_{U}(U), the assertion for Γ(\underline{X}_{loc}) then follows from corollary 10.3.8. The same argument applies to Γ(\underline{X}^{h}).

(vii) On the other hand, if \underline{A} := (A, A^{+}, U) is a complete and separated quasi-affinoid ring, the quasi-affinoid scheme Spec \underline{A} is not necessarily complete and separated (essentially, this is because the completion map A → A^{∧} is not necessarily flat, in this generality). Thus, in order to produce a completion functor for quasi-affinoid schemes, we have to proceed more carefully, as explained in the following proposition.

Proposition 15.4.12. Let q.Afd.Sch_{comp} be the full subcategory of q.Afd.Sch whose objects are the complete and separated quasi-affinoid schemes (X, 𝒯_{X}, A_{X}^{+}). Then the inclusion functor

q.Afd.Sch_{comp} → q.Afd.Sch

admits a right adjoint, called the completion functor :

q.Afd.Sch → q.Afd.Sch_{comp} (X, 𝒯_{X}, A_{X}^{+}) ↦ (X, 𝒯_{X}, A_{X}^{+})^{∧}.

Proof. For any given quasi-affinoid scheme (X, 𝒯_{X}, A_{X}^{+}), set A_{X} := 𝒪_{X}(X), and denote by (A_{X}^{∧}, 𝒯_{X}^{∧}) the completion of (A_{X}, 𝒯_{X}). For every ideal I ⊂ A_{X}^{∧}, set as well

V (I) := Spec A_{X}^{∧}/I and X_{I}^{∧} := X ×_{Spec A_{X}} V (I).

We let ℱ be the family of all topologically closed ideals I ⊂ A_{X}^{∧} such that the induced map ρ_{I} : A_{X}^{∧}/I → 𝒪_{V (I)}(X_{I}^{∧}) is injective. Set J := ⋂_{I∈ℱ} I, so that the natural map i : A_{X}^{∧}/J → ∏_{I∈ℱ} A_{X}^{∧}/I is injective, and notice that (∏_{I∈ℱ} ρ_{I}) ∘ i factors naturally through ρ_{J}, therefore the latter is injective, and thus J ∈ ℱ. We endow A′ := A_{X}^{∧}/J with the quotient topology induced by the projection π : A_{X}^{∧} → A′; it is clear that A′ is a complete and separated f-adic ring, and π is f-adic (lemma 8.3.24(iv)), so we may define

(X, 𝒯_{X}, A_{X}^{+})^{∧} := Spec (A′ ⊗_{A_{X}^{∧}} (A_{X}, A_{X}^{+}, X)^{∧}).

Since the restriction map A′ → A′_{X} := 𝒪_{V (J)}(X_{J}^{∧}) is injective and open, the topology of A′_{X} is complete and separated, so the datum (X, 𝒯_{X}, A_{X}^{+})^{∧} is indeed an object of q.Afd.Sch_{comp}.

On the other hand, let (Y, 𝒯_{Y} , A_{Y}^{+}) be any complete and separated quasi-affinoid scheme, and f : (Y, 𝒯_{Y} , A_{Y}^{+}) → (X, 𝒯_{X}, A_{X}^{+}) any morphism of quasi-affinoid schemes; by adjunction, Γ(f) factors uniquely through a morphism of complete quasi-affinoid rings

Γ(f)^{∧} : (A_{X}, A_{X}^{+}, X)^{∧} → Γ(Y, 𝒯_{Y} , A_{Y}^{+})

and we let K be the kernel of the underlying continuous ring homomorphism h : A_{X}^{∧} → A_{Y}. Clearly K is a closed ideal of A_{X}, and Spec h restricts to a morphism of schemes g : Y → V (K); then, the pair (f, g) determines a unique morphism of X-schemes Y → X_{K}^{∧}. Therefore, the resulting map \overline{h} : A_{X}^{∧}/K → A_{Y} is injective by construction, and it factors through ρ_{K}; so ρ_{K} is injective as well, and thus K ∈ ℱ. Hence, h factors uniquely through a continuous ring homomorphism A′ → A_{Y}, so in turns Γ(f)^{∧} factors uniquely through a morphism of quasi-affinoid rings

φ : A′ ⊗_{A_{X}} (A_{X}, A_{X}^{+}, X)^{∧} → Γ(Y, 𝒯_{Y} , A_{Y}^{+}).

and Spec(φ) : (Y, 𝒯_{Y} , A_{Y}^{+}) → (X, 𝒯_{X}, A_{X}^{+})^{∧} is the unique morphism of complete quasiaffinoid schemes that lifts f, whence the proposition. □

Remark 15.4.13. (i) Let \underline{X} := (X, 𝒯_{X}, A_{X}^{+}) be any quasi-affinoid scheme, set (A_{X}, A_{X}^{+}, X) := Γ(\underline{X}), and let A_{X}^{∧} be the separated completion of A_{X}. Let also I ⊂ A_{X} be any finitely generated ideal such that Spec A_{X}/I = Spec A_{X} \ X. By inspecting the proof of proposition 15.4.12we obtain a natural morphism of quasi-affinoid schemes

\underline{X}^{∧} → Spec(Γ(\underline{X})^{∧})

which is an isomorphism if and only if depth_{I}A_{X}^{∧} > 0 (notation of (10.4.29)).

(ii) The morphism of (i) induces by adjunction a morphism of quasi-affinoid rings

Γ(\underline{X})^{∧} → Γ(\underline{X}^{∧})

which is an isomorphism if and only if depth_{I}A_{X}^{∧} > 1.

Definition 15.4.14. Let \underline{A} := (A, A^{+}, U) be any quasi-affinoid ring, and \underline{X} := (X, 𝒯_{X}, A_{X}^{+}) any quasi-affinoid scheme.

(i) The adic spectrum of the affinoid ring (A, A^{+}) is the set

Spa(A, A^{+}) := {v ∈ Cont(A) | v(a) ≤ 1 for every a ∈ A^{+}}

(notation of definition 15.3.12). Hence, Spa(A, A^{+}) is a pro-constructible subset of Cont(A), and we endow Spa(A, A^{+}) with the topology induced by Cont(A). We also let

Spa(A, A^{+})_{a} := Spa(A, A^{+}) ∩ Cont(A)_{a} Spa(A, A^{+})_{na} := Spa(A, A^{+}) ∩ Cont(A)_{na}

(notation of remark 15.3.13(vi)).

(ii) The adic spectrum of \underline{A} is the set

Spa \underline{A} := Spa(A, A^{+}) ∩ Spv U.

We endow this subset with the topology induced by Spa(A, A^{+}).

(iii) The adic spectrum of \underline{X} is the topological space

Spa \underline{X} := Spa Γ(\underline{X})

and we set as well

Spa (\underline{X})_{a} := Spa Γ(\underline{X})_{a} and Spa (\underline{X})_{na} := Spa Γ(\underline{X})_{na}.

(iv) A rational subset of Spa(A, A^{+}, U) is a subset of the form R ∩ Spa(A, A^{+}, U), where R is a rational subset of Cont(A) (see (15.3.16)). Likewise, a rational subset of Spa \underline{X} is a subset of the form R ∩ Spa \underline{X}, where R is a rational subset of Cont(𝒪_{X}(X), 𝒯_{X}).

Remark 15.4.15. (i) Let f : (A, A^{+}) → (B, B^{+}) be any morphism of affinoid rings. A simple inspection of definition 15.4.14(i) shows that the map Cont(f) restricts to a continuous map

Spa f : Spa(B, B^{+}) → Spa(A, A^{+}).

(ii) More generally, a morphism g : (A, A^{+}, U) → (B, B^{+}, V) of quasi-affinoid rings induces a continuous map

Spa g : Spa(B, B^{+}, V) → Spa(A, A^{+}, U)

that agrees with the restriction of the map Spa(B, B^{+}) → Spa(A, A^{+}) attached to g as in (i).

(iii) Likewise, any morphism φ : (X, 𝒯_{X}, A_{X}^{+}) → (Y, 𝒯_{Y} , A_{Y}^{+}) of quasi-affinoid schemes induces a continuous map

Spa φ := Spa Γ(φ) : Spa(X, 𝒯_{X}, A_{X}^{+}) → Spa(Y, 𝒯_{Y} , A_{Y}^{+}).

(iv) In light of theorem 15.3.15(i), it is easily seen that v(x) ≤ 1 for every x ∈ A^{∘} and every v ∈ Cont(A) of rank ≤ 1. Especially, the subset of Spa(A, A^{+}, U) consisting of rank one valuations is independent of the ring of integral elements A^{+}.

Example 15.4.16. For any f-adic ring A, let A′ ⊂ A be the smallest subring of integral elements (see remark 15.4.2(iv)). It is easily seen that Spa(A, A′) = Cont(A). Also, every continuous map f : A → B of f-adic rings yields a morphism of affinoid rings f : (A, A′) → (B, B′), and clearly Spa φ = Cont(f).

(ii) On the other hand, every f-adic map f : A → B of f-adic rings induces a continuous map Spa f : Spa(B, B^{∘}) → Spa(A, A^{∘}) (lemma 8.3.24(iii.a)).

Lemma 15.4.17. Let \underline{A} := (A, A^{+}, U) be any quasi-affinoid ring.

(i) The natural morphisms \underline{A} → \underline{A}_{loc} → \underline{A}^{h} → \underline{A}^{∧} (notation of remark 15.4.10(ii,iii)) induce homeomorphisms

Spa \underline{A}^{∧} →^{∼} Spa \underline{A}^{h} →^{∼} Spa \underline{A}_{loc} →^{∼} Spa \underline{A}.

(ii) The unit of the adjunction (Spec, Γ) of proposition 15.4.3(i) induces a homeomorphism

Spa(Spec \underline{A}) →^{∼} Spa \underline{A}.

(iii) Let \underline{X} := (X, 𝒯_{X}, A_{X}^{+}) be any quasi-affinoid scheme, and \underline{X}^{∧} the completion of \underline{X} (see proposition 15.4.12). The completion map \underline{X} → \underline{X}^{∧} induces a homeomorphism

Spa \underline{X}^{∧} →^{∼} Spa \underline{X}.

(iv) Let φ : Spa \underline{A}^{∧} →^{∼} Spa \underline{A} be the homeomorphism of (i), and R ⊂ Spa \underline{A}^{∧} any rational subset. Then φ(R) is a rational subset of Spa \underline{A}.

Proof. (i): Let A_{loc} (resp. A^{h}) be the topological localization (resp. henselization) of A; from corollary 8.4.15(i) and proposition 15.3.19(iv), we deduce that the natural maps A → A_{loc} → A^{h} → A^{∧} induce homeomorphisms

Cont(A^{∧}) →^{∼} Cont(A^{h}) →^{∼} Cont(A_{loc}) →^{∼} Cont(A).

Moreover, it was already remarked that the localization map induces a natural identification \underline{A}_{loc} →^{∼} A_{loc} ⊗_{A} (A, A^{+}), and likewise for the topological henselization and the completion of \underline{A}; the assertion follows easily (details left to the reader).

(ii): Set (U, 𝒯_{U}, A_{U}^{+}) := Spec \underline{A} and A_{U} := 𝒪_{U}(U); the assertion boils down to the following:

Claim 15.4.18. The natural morphism of quasi-affinoid rings ρ : (A, A^{+}, U) → (A_{U}, A_{U}^{+}, U) induces a homeomorphism Spa ρ : Spa(A_{U}, A_{U}^{+}, U) →^{∼} Spa(A, A^{+}, U).

Proof of the claim. Let us first check that ρ induces a homeomorphism

(15.4.19)  Cont(A_{U}) ∩ Spv U →^{∼} Cont(A) ∩ Spv U.

Indeed, since the topology of both of these spaces are induced from the inclusion into Spv U, it suffices to check that (15.4.19) is surjective. Hence, let v : A → Γ_{v∘} be any continuous valuation with support given by a prime ideal 𝔭 ⊂ A, and suppose that 𝔭 ∈ U; then there exists a unique prime ideal 𝔭′ ⊂ A_{U} such that ρ^{−1}𝔭′ = 𝔭. Set B := A/𝔭 and C := A_{U}/𝔭′; the map v factors through a continuous valuation \overline{v} : B → Γ_{v∘} (for the quotient topology on B induced by the projection A → B) and C is naturally identified with a subring of Frac B, so \overline{v} extends uniquely to a valuation w : C → Γ_{v∘}. It remains only to show that w is continuous, for the quotient topology on C induced by the projection A_{U} → C. By remark 15.3.13(ii) it suffices to prove the continuity of w at the point 0 ∈ C; however, ρ is an open map (proposition 8.3.30(i)), so the same holds for the induced injective map B → C, and the assertion follows.

Since A_{U}^{+} is the integral closure of the image of A^{+} in A_{U}, the claim is an immediate consequence of the foregoing : details left to the reader. ♢

(iii): Let A_{X} := 𝒪_{X}(X), and set (A_{X}^{∧}, A_{X}^{+∧}, X^{∧}) := (A_{X}, A_{X}^{+}, X)^{∧}. Define the family ℱ of ideals of A_{X}^{∧} as in the proof of proposition 15.4.12, and denote by J the minimal element of ℱ; also, endow A′ := A_{X}^{∧}/J with the quotient topology induced via the projection A_{X}^{∧} → A′. Then \underline{X}^{∧} = Spec(A′ ⊗_{A_{X}^{∧}} (A_{X}^{∧}, A_{X}^{+∧}, X^{∧})). Taking into account (i) and (ii), we are then reduced to checking that the natural projection π : (A_{X}^{∧}, A_{X}^{+∧}, X^{∧}) → A′ ⊗_{A_{X}^{∧}} (A_{X}^{∧}, A_{X}^{+∧}, X^{∧}) induces a homeomorphism

Spa π : Spa(A′ ⊗_{A_{X}^{∧}} (A_{X}^{∧}, A_{X}^{+∧}, X^{∧})) →^{∼} Spa(A_{X}^{∧}, A_{X}^{+∧}, X^{∧}).

However, Spa π is a closed immersion, so it suffices to show that it is a bijection. To this aim, let v : A_{X}^{∧} → Γ_{v∘} be any continuous valuation whose support 𝔭 lies in X^{∧}; we come down to checking that 𝔭 ∈ ℱ. However, 𝔭 is obviously a topologically closed ideal of A_{X}^{∧}. Next, set B := A_{X}^{∧}/𝔭 and X_{𝔭}^{∧} := X ×_{Spec A_{X}} Spec B; since B is a domain and X_{𝔭}^{∧} ≠ ∅, the restriction map B → 𝒪_{Spec B}(X_{𝔭}^{∧}) is injective, whence the contention.

(iv): Say that R = R_{A^{∧}}(\frac{a_{1}}{a_{0}} , …, \frac{a_{n}}{a_{0}} ) ∩ Spa \underline{A}^{∧} for a sequence of elements a_{0}, …, a_{n} ∈ A^{∧} that generates an open ideal J. Let also B be a ring of definition of A, and I ⊂ B an ideal of adic definition. To begin with, we remark :

Claim 15.4.20. (i) Let T be a quasi-compact subset of Spa \underline{A}, and t an element of A^{∘∘} such that v(t) ≠ 0 for every v ∈ T. Then there exists k ∈ ℕ such that v(x) < v(t) for every v ∈ T and every x ∈ I^{k}.

(ii) There exists q ∈ ℕ such that every sequence b_{0}, …, b_{n} with b_{i} ∈ a_{i} + I^{q}B^{∧} for i = 0, …, n, generates an open ideal of A^{∧}, and R′ := R_{A^{∧}}(\frac{b_{1}}{b_{0}} , …, \frac{b_{n}}{b_{0}} ) ∩ Spa \underline{A}^{∧} = R.

Proof of the claim. (i): Let x_{1}, …, x_{m} be a finite system of generators of I. For every r ∈ ℕ set

T_{r} := \{v ∈ Spa \underline{A} | v(x_{i}^{r}) ≤ v(t) for every i = 1, …, m\}.

Then T_{r} is open in Spa \underline{A} for every r ∈ ℕ, and T ⊂ ⋃_{r∈ℕ} T_{r}. Thus, T ⊂ T_{r} for some r ∈ ℕ, and the assertion holds with k := r + 1.

(ii): Pick elements c_{1}, …, c_{m} ∈ J ∩B^{∧} that generate an open ideal J_{0} of B^{∧}, and r ∈ ℕ such that I^{r} ⊂ J_{0}. Then, every sequence d_{1}, …, d_{m} of elements of B^{∧} with d_{i} ∈ c_{i} + I^{r}B^{∧} for i = 1, …, m, also generates J_{0} ([126, Th.8.4]). Next, for every i = 1, …, m pick x_{i0}, …, x_{in} ∈ A^{∧} such that c_{i} = ∑_{j=0}^{n} a_{j}x_{ij}, and let p ∈ ℕ be large enough so that I^{p}x_{ij} ⊂ I^{r}B^{∧} for every i = 1, …, m and every j = 0, …, n; it follows already that every sequence b_{0}, …, b_{n} of elements of A^{∧} with b_{i} ∈ a_{i} + I^{p}B^{∧} for i = 0, …, n generates an open ideal of A^{∧}.

Next, we apply part (i) of the claim to the rational subsets R_{i} := R_{A^{∧}}(\frac{a_{1}}{a_{i}} , …, \frac{a_{n}}{a_{i}} ) ∩ Spa \underline{A}^{∧}, to find an integer q ≥ p such that v(x) < v(a_{i}) for every x ∈ I^{q}B^{∧}, every i = 0, …, n and every v ∈ R_{i}. We claim that this q will do. Indeed, let v ∈ Spa \underline{A}^{∧}, and say that v ∈ R = R_{0}; since b_{i} − a_{i} ∈ I^{q}B^{∧}, we have v(b_{i} − a_{i}) < v(a_{0}) for i = 0, …, n. This implies that

v(b_{i}) = v(a_{i} + (b_{i} − a_{i})) ≤ max(v(a_{i}), v(b_{i} − a_{i})) ≤ v(a_{0}) = v(a_{0} + (b_{0} − a_{0})) = v(b_{0})

for every i = 0, …, n, which shows that R ⊂ R′. Next, suppose v ∉ R, and consider first the case where v(a_{i}) = 0 for every i = 0, …, n. Then v is non-analytic, so that v(b_{0} − a_{0}) = 0, whence v(b_{0}) = 0, and finally v ∉ R′. Lastly, suppose that v(a_{i}) ≠ 0 for some i ≤ n, and choose j ≤ n so that v ∈ R_{j}. We must have v(a_{0}) < v(a_{j}), since v ∉ R; also, by construction v(b_{i} − a_{i}) < v(a_{j}) for every i = 0, …, n. Then

v(b_{0}) = v(a_{0} + (b_{0} − a_{0})) ≤ max(v(a_{0}), v(b_{0} − a_{0})) < v(a_{j}) = v(a_{j} + (b_{j} − a_{j})) = v(b_{j})

which yields again v ∉ R′, and the claim follows. ♢

Now, by claim 15.4.20(ii) there exist b_{0}, …, b_{n} in A such that R = R_{A^{∧}}(\frac{b_{1}}{b_{0}} , …, \frac{b_{n}}{b_{0}} )∩Spa \underline{A}^{∧}. Also, by claim 15.4.20(i) there exists k ∈ ℕ such that v(x) < v(b_{0}) for every v ∈ R and every x ∈ I^{k}. Pick a finite system b_{n+1}, …, b_{p} of generators of I^{k}; it follows that φ(R) = R_{A}(\frac{b_{1}}{b_{0}} , …, \frac{b_{p}}{b_{0}} ) whence the contention. □

Remark 15.4.21. By combining lemma 15.4.17(iv) and proposition 15.3.19(i) one may obtain another, more constructive, proof of lemma 15.4.17(i) (and of proposition 15.3.19(iv)). This is the route followed in [99].

Proposition 15.4.22. Let (A, A^{+}, U) be any quasi-affinoid ring. The following holds :

(i) Spa(A, A^{+}, U) is a constructible open subset of Spa(A, A^{+}), and both these topological spaces are spectral.

(ii) The rational subsets are a basis of quasi-compact open subsets of Spa(A, A^{+}) that is

closed under finite intersections. (iii) For any morphism f : (A, A^{+}, U) → (B, B^{+}, V) of quasi-affinoid rings, we have :

(a) If f is f-adic, Spa f is a spectral map. More precisely, if R is a rational subset of Spa(A, A^{+}, U), then (Spa f)^{−1}(R) is a rational subset of Spa(B, B^{+}, V).

(b) Suppose that B is topologically local. Then f is f-adic if and only if Spa f restricts

to a map Spa(B, B^{+})_{a} → Spa(A, A^{+})_{a}. (iv) For any quasi-affinoid scheme (X, 𝒯_{X}, A_{X}^{+}), the topological space Spa(X, 𝒯_{X}, A_{X}^{+}) is

spectral.

(v) For any f-adic morphism φ : (X, 𝒯_{X}, A_{X}^{+}) → (Y, 𝒯_{Y} , A_{Y}^{+}) of quasi-affinoid schemes, the induced map Spa φ is spectral.

Proof. Assertion (i) for Spa(A, A^{+}) follows from theorem 15.3.15(ii) and corollary 8.1.42. According to (15.3.16), the rational subsets of Cont(A) form a basis of quasi-compact open subsets for the latter space; invoking again corollary 8.1.42, we obtain (ii) as well.

Next, let I ⊂ A be any finitely generated ideal such that Spec A/I = Spec A \ U, and pick a finite system of generators a_{1}, …, a_{n} for I. Then

Spa(A, A^{+}, U) = Spa(A, A^{+}) ∩ ⋃_{i=1}^{n} R_{A}(\frac{a_{1}}{a_{i}} , …, \frac{a_{n}}{a_{i}} ).

Since I is open in A, this identity presents Spa(A, A^{+}, U) as a finite union of rational subsets of Spa(A, A^{+}), and the latter are open and quasi-compact, by (ii), so the proof of (i) is complete.

(iii.a) follows immediately from corollary 15.3.17(i). The proofs of (iv) and (v) are similar, and the details shall be left to the reader.

(iii.b): Set φ := Spec f; in light of lemma 8.3.29(iv) and corollary 15.3.21, we need only check that φ^{−1}X_{A}^{∘∘} = X_{B}^{∘∘}, if Spa f restricts to a map Spa(B, B^{+})_{a} → Spa(A, A^{+})_{a}.

Thus, suppose that there exists 𝔭 ∈ X_{B} \ X_{B}^{∘∘} with φ(𝔭) ∈ X_{A}^{∘∘}. Then 𝔭 is a non-open ideal of B, so there exists a rank one continuous analytic valuation v of B whose support contains 𝔭 (lemma 15.3.25); clearly the support of Cont(f)(v) contains φ(𝔭), and it is therefore open, i.e. Cont(f)(v) ∈ Cont(A)_{na}. To conclude it suffices to remark :

Claim 15.4.23. Let R be any f-adic ring, v ∈ Cont(R)_{a} any element such that Γ_{v} has rank one. Then v ∈ Spa(R, R^{∘}).

Proof of the claim. Since v is analytic, there exists a ∈ R such that 0 < v(a) < 1. Let b ∈ R^{∘} be any element, and suppose that v(b) > 1; then there exists n ∈ ℕ large enough, such that v(b^{n}a) > 1, which is absurd, since b^{n}a ∈ R^{∘∘} (remark 8.3.10(iv)). □

Proposition 15.4.24. Let (A, A^{+}, U) be any quasi-affinoid ring, I ⊂ A^{+} an ideal, g_{•} := (g_{i} | i ∈ Σ) and h_{•} := (h_{j} | j ∈ Σ′) two systems of elements of A such that :

(a) The system (g_{•}, h_{•}) generates an open ideal of A.

(b) Spec A/IA is the set of open prime ideals of A.

(c) U is dense in Spec A. Then for every f ∈ A the following conditions are equivalent :

(d) For every v ∈ Spa(A, A^{+}, U) with v(f) ≠ 0 there exists either i ∈ Σ such that v(f) ≤ v(g_{i}) or else j ∈ Σ′ such that v(f) < v(h_{j}).

(e) There exists a polynomial P ∈ 𝒫_{I} such that P (f, g_{•}, h_{•}) = 0 (notation of (9.2.39)).

Proof. From theorem 15.3.15(i) and lemmata 15.4.17(i), 9.2.40we already see that (e)⇒(d).

(d)⇒(e): Let (A′, A^{′+}, U′) be the topological localization of (A, A^{+}, U), and set I′ := IA^{′+}. Notice that U′ is dense in Spec A′ (proposition 8.1.47(iii)), and Spec A′/I′A′ is the set of open prime ideals of A′, since the localization map A → A′ is f-adic (lemma 8.3.29(iv)). We may then consider also the set 𝒫_{I′} ⊂ A^{′+}[Z, X_{•}, Y_{•}] defined as in (9.2.39), and we remark

Claim 15.4.25. Condition (e) is equivalent to :

(f) There exists a polynomial Q ∈ 𝒫_{I′} such that Q(f, g_{•}, h_{•}) = 0 in A^{′+}.

Proof of the claim. Obviously (e)⇒(f). Conversely, let A_{0} ⊂ A^{+} be a subring of definition, and J any ideal of adic definition of A_{0}, so that A^{′+} = (1 + J)^{−1}A^{+} and I′ = (1 + J)^{−1}I. If (f) holds, we deduce that there exists a polynomial G ∈ A^{+}[Z, X_{•}, Y_{•}] of the form

G(Z, X_{•}, Y_{•}) = (1 − a) · Z^{n} + ∑_{k=1}^{n} Z^{n−k}P_{k}(X_{•}, Y_{•})

for some a ∈ J and some P_{1}, …, P_{n} ∈ A^{+}[X_{•}, Y_{•}] fulfilling the conditions of (9.2.39), and such that G(f, g_{•}, h_{•}) = 0 in A. Since A is f-adic, condition (a) implies that the system (g_{i}^{n}, h_{j}^{n} | i ∈ Σ, j ∈ Σ′) generates an open ideal of A (details left to the reader); then we may find N ∈ ℕ large enough, finite subsets Σ_{0} ⊂ Σ′, Σ′_{0} ⊂ Σ′, and systems (b_{i} | i ∈ Σ_{0}), (c_{j} | j ∈ Σ′_{0}) of elements of A such that

a^{N}f^{n} = ∑_{i∈Σ_{0}} b_{i}g_{i}^{n} + ∑_{j∈Σ′_{0}} c_{j}h_{j}^{n}.

In light of (b), we may next find M ∈ ℕ sufficiently large, so that a^{M}b_{i} ∈ A^{+} and a^{M}c_{j} ∈ I for every i ∈ Σ_{0} and j ∈ Σ′_{0}. Summing up, we see that a^{M+N}f^{n} = H(g_{•}, h_{•}) in A for some homogeneous polynomial H(X_{•}, Y_{•}) ∈ A^{+}[X_{•}, Y_{•}] of degree n and with H(0, Y_{•}) ∈ I[Y_{•}]. Then set u := 1 + a + ⋯ + a^{M+N−1}, so that u · (1 − a) = 1 − a^{M+N}; it follows that the polynomial

P := u · G + a^{M+N}Z^{n} − H

fulfills condition (e). ♢

Now, if (e) fails for some f ∈ A, claim 15.4.25 and lemma 9.2.40 show that there exists w ∈ Spv A′ whose support is a minimal prime ideal of A′, and with w(f) ≠ 0 and

w(a) ≤ 1  w(b) < 1  for every a ∈ A^{′+} and b ∈ A^{′∘∘}
w(g_{i}) < w(f)  w(h_{j}) ≤ w(f)  for every i ∈ Σ and j ∈ Σ′.

Since U′ is dense in Spec A′, we have w ∈ Spv U′ (proposition 8.1.47(iii)); if w is continuous, it follows that w ∈ Spa(A′, A^{′+}, U′) = Spa(A, A^{+}, U) (lemma 15.4.17(i)), and (d) fails for w. In case w is not continuous, set L := IA and v := w^{cΓ_{v}(L)} (notation of definition 15.3.3).

Claim 15.4.26. (i) v(L) ≠ {0}.

(ii) v(f) ≠ 0.

Proof of the claim. (i): Since v is an L-admissible specialization of w, it suffices to show that w(L) ≠ {0}. But if the latter fails, then the support of w is an open prime ideal, so w is continuous (remark 15.3.13(vi)), contradicting our assumption.

(ii): Suppose that v(f) = 0; then v(g_{i}) = v(h_{j}) = 0 as well, for every i ∈ Σ and j ∈ Σ′. Hence the support of v is an open ideal of A, and especially, it contains L, contradicting (i). ♢

Claim 15.4.26(ii), theorem 15.3.15(i) and lemma 15.4.17(i) imply that v is an analytic point of Spa(A, A^{+}) and condition (d) fails for v; lastly, notice that Cont(A)_{a} ⊂ Spv U, so v ∈ Spa(A, A^{+}, U) whence the proposition. □

Corollary 15.4.27. Let \underline{A} := (A, A^{+}, U) be any quasi-affinoid ring such that U is dense in Spec A, and f ∈ A any element. We have :

(i) f ∈ A^{+} if and only if v(f) ≤ 1 for every v ∈ Spa \underline{A}. (ii) f ∈ A^{∘∘} if and only if v(f) < 1 for every v ∈ Spa \underline{A}. (iii) Let also J ⊂ A^{+} be any finitely generated ideal whose radical is A^{∘∘}. Then the follow-

ing conditions are equivalent :

(a) |f|_{J}^{∗} = 0 (notation of example 9.1.9).

(b) v(f) = 0 for every v ∈ Spa \underline{A}.

Proof. (i): We may assume that v(f) ≤ 1 for every v ∈ Spa \underline{A}, and we show that f ∈ A^{+}. To this aim, we apply proposition 15.4.24 with g_{•} = {1}, h_{•} the empty subset and I := A^{∘∘}; we deduce that there exists a monic polynomial P (Z) ∈ A^{+}[Z] with P (f) = 0, i.e. f is integral over A^{+}, whence the contention, since A^{+} is integrally closed in A.

(ii): Again, we may assume that v(f) < 1 for every v ∈ Spa \underline{A}, and we check that f ∈ A^{∘∘}. To this aim, we apply proposition 15.4.24 with h_{•} = {1}, g_{•} the empty subset and I := A^{∘∘}; we deduce that there exist n ∈ ℕ and a_{1}, …, a_{n} ∈ A^{∘∘} with f^{n} + f^{n−1}a_{1} + ⋯ + a_{n} = 0. By (i), we know already that f ∈ A^{+}, so that f^{n−1}a_{1} + ⋯ + a_{n} ∈ A^{∘∘}, and combining with remark 8.3.10(iv), the claim follows.

(iii.a)⇒(iii.b): Let a_{1}, …, a_{r} be any finite system of generators of J, and set

γ_{v} := max(v(a_{1}), …, v(a_{r})) ∈ Γ_{v∘} for every v ∈ Spa(A, A^{+}, U).

The assumption implies that for every n ∈ ℕ there exists k ∈ ℕ such that f^{k} ∈ J^{kn}, hence v(f^{k}) ≤ γ_{v}^{kn}, and therefore v(f) ≤ γ_{v}^{n} for every n ∈ ℕ and every v ∈ Spa \underline{A} (details left to the reader). However γ_{v} is final in Γ_{v} for every such v (lemma 15.3.14(i)), whence the assertion.

(iii.b)⇒(iii.a): For every n ∈ ℕ, we apply the criterion of proposition 15.4.24 with g_{•} equal to the empty set, h_{•}^{(n)} := {1}, and I := J^{n}. We deduce that for every n ∈ ℕ there exist k ∈ ℕ and elements a_{i}^{(n)} ∈ J^{ni} for i = 1, …, k, such that f^{k+1} + a_{1}^{(n)}f^{k} + ⋯ + a_{k}^{(n)} = 0. Therefore :

|f^{k+1}|_{J}^{∗} ≤ max(|a_{i}^{(n)}|_{J}^{∗} · |f^{k+1−i}|_{J}^{∗} | i = 1, …, k)

(lemma 9.1.8(ii)) whence (|f|_{J}^{∗})^{i} ≤ |a_{i}^{(n)}|_{J}^{∗} ≤ ρ^{ni} for some i ≤ k, (lemma 9.1.8(iii)), and consequently |f|_{J}^{∗} ≤ ρ^{n} for every n ∈ ℕ, whence (iii.a). □

Corollary 15.4.28. Let \underline{X} := (X, 𝒯_{X}, A_{X}^{+}) be any quasi-affinoid scheme, and (X^{∧}, 𝒯_{X}^{∧}, A_{X^{∧}}^{+}) the completion of \underline{X}. Denote by A_{X}^{∧} (resp. A_{X}^{∧+}) the separated completion of A_{X} := 𝒪_{X}(X) (resp. of A_{X}^{+}). Then we have :

(i) The kernel 𝒥 of the natural map π : A_{X}^{∧} → 𝒪_{X^{∧}}(X^{∧}) is contained in (A_{X}^{∧})^{∘∘}.

(ii) π^{−1}A_{X^{∧}}^{+} = A_{X}^{∧+}.

Proof. (i): To begin with, we remark, quite generally :

Claim 15.4.29. Let A be any ring, I ⊂ A a finitely generated ideal, and set

X_{A} := Spec A  Z := Spec A/I  U := X_{A} \ Z.

Then the following conditions are equivalent :

(a) U is dense in X_{A}.

(b) The support of the A-module I^{n}/I^{n+1} is Z, for every n ∈ ℕ.

Proof of the claim. Since U is constructible in X_{A}, condition (a) is equivalent to :

(c) I_{𝔮} = A_{𝔮} for every minimal prime ideal 𝔮 of A (proposition 8.1.47(iii)). Now, if A = 0 there is nothing to prove, so assume that A ≠ 0. Suppose that (c) holds. Clearly the support of I^{n}/I^{n+1} lies in Z, so we consider any 𝔭 ∈ Z, and we have to show that I_{𝔭}^{n}/I_{𝔭}^{n+1} ≠ 0 for every n ∈ ℕ. However, if the latter fails for some n ∈ ℕ, Nakayama’s lemma implies that I_{𝔭}^{n} = 0; then I_{𝔮}^{n} = 0 for every minimal prime ideal 𝔮 of A contained in 𝔭, contradicting (c).

Conversely, if (c) fails for some minimal prime ideal 𝔮, it follows that I_{𝔮} is a nilpotent ideal of A_{𝔮}. Say that I_{𝔮}^{n} ≠ 0 and I_{𝔮}^{n+1} = 0 for some n ∈ ℕ; then 𝔮 lies in the support of I^{n}/I^{n+1}, hence 𝔮 ∈ Z, and therefore U cannot be dense in X_{A}. ♢

Set Y := Spec A_{X}, Y := Spec A_{X}^{∧}, X′ := X ×_{Y} Y′, and let I ⊂ A_{X} be any ideal such that A_{X}^{∘∘} ⊂ I and Y \ X = Spec A_{X}/I. Then Y′ \ X′ = Spec A_{X}^{∧}/I′, where I′ := IA_{X}^{∧}, and since I is open in A_{X}, the natural maps I^{n}/I^{n+1} → I^{′n}/I^{′n+1} are bijective for every n ∈ ℕ. Moreover, by assumption X is dense in Y; taking into account claim 15.4.29, we deduce that X′ is dense in Y′. Pick any finitely generated ideal J of A_{X}^{∧} whose radical is A_{X}^{∧∘∘}; in view of corollary 15.4.27(ii,iii), we see that

⋂_{v∈Spa (A_{X}^{∧},A_{X}^{∧+},X′)} Ker v = {a ∈ A_{X}^{∧} | |a|_{J}^{∗} = 0} ⊂ (A_{X}^{∧})^{∘∘}.

But we have already noticed that 𝒥 is contained in the support of every v ∈ Spa(A_{X}^{∧}, A_{X}^{∧+}, X′) (see the proof of lemma 15.4.17(iii)), whence (i).

(ii) follows immediately from (i) and corollary 15.4.27(i) and lemma 15.4.17(iii) (details left to the reader; more directly, one can also argue as in the proof of theorem 16.5.13(ii)). □

Proposition 15.4.30. Let (X, 𝒯_{X}, A_{X}^{+}) be any topologically local quasi-affinoid scheme, and f ∈ 𝒪_{X}(X) any element. The following conditions are equivalent :

(a) f ∈ 𝒪_{X}(X)^{×}.

(b) v(f) ≠ 0 for every v ∈ Spa(X, 𝒯_{X}, A_{X}^{+}).

Proof. Clearly, it suffices to check that (b)⇒(a). Hence, suppose that (b) holds, and set

A_{X} := 𝒪_{X}(X)  Y := Spec A_{X}  Y^{∘∘} := Spec A_{X}/A_{X}^{∘∘} · A_{X}.

Let x ∈ X be any point; we need to show that the image f(x) of f does not vanish in κ(x). If x ∈ Y^{∘∘}, let v be the trivial valuation with support equal to x; clearly v ∈ Spa(X, 𝒯_{X}, A_{X}^{+}), so (b) says that v(f) ≠ 0, whence the contention, in this case.

Next, if x ∈ X \ Y^{∘∘}, we proceed as in the proof of proposition 15.4.22(iii.b) : we pick first a minimal specialization y of x in X \ Y^{∘∘}, then a maximal point 𝔭 of the non-empty closed subset {y}^{c} ∩ Y^{∘∘} of Y (where {y}^{c} denotes the topological closure of {y} in Y). The image C of A_{X,𝔭} in κ(y) is a one-dimensional local domain, so we may find a valuation ring V of κ(y) with value group of rank one, that dominates C (claim 15.3.27). Let v ∈ Spv A_{X} be the valuation corresponding to V; arguing as in loc. cit. we see that v ∈ Spa(X, 𝒯_{X}, A_{X}^{+}), hence v(f) ≠ 0, so f(y) ≠ 0 and finally f(x) ≠ 0 as well. □

15.5. Adic spaces. In this section we shall endow the adic spectra introduced in section 15.3 with certain natural presheaves of topological rings, and we shall show that, under suitable conditions, these presheaves are sheaves of rings or of topological rings. The first step is the following universal construction :

15.5.1. Let \underline{A} := (A, A^{+}, U) be any quasi-affinoid ring, Λ a (small) set, S := (s_{λ} | λ ∈ Λ) a system of elements of A, and T := (T_{λ} | λ ∈ Λ) a family of subsets of A such that T_{λ} generates an open ideal of A, for every λ ∈ Λ. We consider the f-adic ring A[X_{λ} | λ ∈ Λ]_{T} provided by proposition 8.3.37(ii), and its ideal J generated by the system (1 − s_{λ}X_{λ} | λ ∈ Λ). We set

A(\frac{T}{S} ) := A[X_{•}]_{T} /J

and we endow this ring with the quotient topology induced by A[X_{•}]_{T} via the natural projection. In other words, A(\frac{T}{S}) is an f-adic topological ring whose underlying A-algebra is naturally identified with the localization A[s_{λ}^{−1} | λ ∈ Λ]. From proposition 8.3.37(iii,iv) we see that – under this identification – the subset \{t/s_{λ} | λ ∈ Λ, t ∈ T_{λ}\} is power bounded in A(\frac{T}{S}). Moreover, the localization map h : A → A(\frac{T}{S}) is f-adic, and enjoys the following universal property. For every f-adic ring B and every continuous map f : A → B such that :

• f(s_{λ}) ∈ B^{×} for every λ ∈ Λ

• the subset {f(t)/f(s_{λ}) | λ ∈ Λ, t ∈ T_{λ}} is power bounded in B

there exists a unique continuous map g : A(\frac{T}{S}) → B such that g ∘ h = f. Furthermore, if A_{0} ⊂ A is a subring of definition, A_{0}[\frac{t}{s_{λ}} | λ ∈ Λ, t ∈ T_{λ}] is a subring of definition of A(\frac{T}{S}).

Next, notice that C := A^{+}[\frac{t}{s_{λ}} | λ ∈ Λ, t ∈ T_{λ}] is an open subring of A(\frac{T}{S})^{∘}, and denote by C′ the integral closure of C in A(\frac{T}{S}). Moreover, set X := Spec A[s_{λ}^{−1}|λ ∈ Λ], and notice that U′ := U ∩ X contains the analytic locus of X, since h is f-adic (lemma 8.3.29(iv)). We obtain therefore a quasi-affinoid ring

\underline{A}(\frac{T}{S} ) := (A(\frac{T}{S} ), C′, U′)

and the localization map yields a natural f-adic morphism of quasi-affinoid rings

(15.5.2)  \underline{A} → \underline{A}(\frac{T}{S} ).

In case Λ = {λ} has only one element, T_{λ} = {f_{0}, …, f_{n}} is a finite subset of A which generates an open ideal, and S = {s} for some s ∈ A, we also denote this quasi-affinoid ring by :

\underline{A}(\frac{f_{0}, …, f_{n}}{s} ).

15.5.3. We consider now the Yoneda embedding (see (1.2.4))

h : q.Afd.Sch_{loc} → q.Afd.Sch_{loc}^{∧} \underline{X} ↦ h\underline{X}

(notation of remark 15.4.10(v)). Let \underline{X} be any topologically local quasi-affinoid scheme, and U ⊂ Spa \underline{X} any open subset. We attach to U the sub-presheaf h_{U} ⊂ h\underline{X} given by the rule :

h_{U}(\underline{Y}) := \{φ ∈ h\underline{X}(\underline{Y}) | Im Spa(φ) ⊂ U\} for every \underline{Y} ∈ Ob(q.Afd.Sch_{loc}).

Definition 15.5.4. (i) In the situation of (15.5.3), we say that U is a quasi-affinoid open subset of Spa \underline{X}, if the presheaf h_{U} is representable by some \underline{Y} ∈ Ob(q.Afd.Sch_{loc}).

(ii) We say that U is an affinoid open subset of Spa \underline{X}, if \underline{Y} as in (i) is an affinoid scheme.

Theorem 15.5.5. Every rational subset of Spa \underline{X} is a quasi-affinoid open subset.

Proof. Let R be such a rational subset, and say that

Γ(\underline{X}) = (A_{X}, A_{X}^{+}, X) and R = R_{A_{X}}(\frac{f_{1}}{f_{0}} , …, \frac{f_{n}}{f_{0}} ) ∩ Spa \underline{X}

for a sequence (f_{0}, f_{1}, …, f_{n}) of elements of A_{X} that generates an open ideal. We consider the quasi-affinoid rings

\underline{A}_{R} := Γ(\underline{X})(\frac{f_{0}, …, f_{n}}{f_{0}} )

defined as in (15.5.1). Also, we let

i_{R} : Γ(\underline{X}) → \underline{A}_{R,loc}

be the composition of the natural morphism Γ(\underline{X}) → \underline{A}_{R} of (15.5.2) with the topological localization \underline{A}_{R} → \underline{A}_{R,loc}. Now, let \underline{Y} be any topologically local quasi-affinoid scheme, and φ ∈ h_{R}(\underline{Y}) any element. Set (A_{Y} , A_{Y}^{+}, Y) := Γ(\underline{Y}), and let φ^{♭} : A_{X} → A_{Y} be the continuous map associated with φ.

Claim 15.5.6. φ^{♭}(f_{0}) ∈ A_{Y}^{×} and φ^{♭}(f_{i})/φ^{♭}(f_{0}) ∈ A_{Y}^{+} for every i = 1, …, n.

Proof of the claim. Indeed, for every v ∈ Spa \underline{Y} we have by assumption v ∘ φ^{♭} ∈ R, so that v(φ^{♭}(f_{0})) ≠ 0 and v(φ^{♭}(f_{i})) ≤ v(φ^{♭}(f_{0})) for i = 1, …, n. Then proposition 15.4.30 implies that φ^{♭}(f_{0}) ∈ A_{Y}^{×}, so we may write v(φ^{♭}(f_{i})/φ^{♭}(f_{0})) ≤ 1 for every i = 1, …, n and every v ∈ Spa \underline{Y}. To conclude, it then suffices to invoke corollary 15.4.27(i). ♢

From claim 15.5.6, remark 8.3.9(ii) and the discussion of (15.5.1) we see that φ^{♭} factors uniquely through a continuous ring homomorphism g : A_{R} → A_{Y}, and g(f_{i}/f_{0}) ∈ A_{Y}^{+} for every i = 1, …, n. Thus, g defines a morphism of quasi-affinoid rings

g : \underline{A}_{R} → Γ(\underline{Y})

and summing up, we easily conclude that φ factors uniquely through Spec(i_{R}) and the morphism of topologically local quasi-affinoid schemes

ψ := Spec(g_{loc}) : \underline{Y} → \underline{R} := Spec (\underline{A}_{R,loc}).

Lastly, taking into account lemma 15.4.17(i) it is easily seen that Spa(i_{R}) is an injective map inducing a homeomorphism Spa \underline{R} →^{∼} R, and the theorem follows. □

15.5.7. In the situation of (15.5.3), let U ⊂ Spa \underline{X} be a quasi-affinoid open subset, and \underline{Y} any topologically local quasi-affinoid scheme that represents h_{U}. Then the morphism 1\underline{Y} corresponds to a well defined section φ_{\underline{Y} /\underline{X}} ∈ h_{U}(\underline{Y}), i.e. to a morphism φ_{\underline{Y} /\underline{X}} : \underline{Y} → \underline{X} of quasi-affinoid schemes with Im(Spa φ_{\underline{Y} /\underline{X}}) ⊂ U.

Corollary 15.5.8. With the notation of (15.5.7), the following holds :

(i) The map Spa φ_{\underline{Y} /\underline{X}} : Spa \underline{Y} → Spa \underline{X} induces a homeomorphism Spa \underline{Y} →^{∼} U.

(ii) Especially, every quasi-affinoid open subset is quasi-compact.

(iii) More precisely, Spa φ_{\underline{Y} /\underline{X}} induces homeomorphisms

(Spa \underline{Y})_{a} →^{∼} U ∩ Spa(\underline{X})_{a} (Spa \underline{Y})_{na} →^{∼} U ∩ Spa(\underline{X})_{na}.

(iv) The morphism φ_{\underline{Y} /\underline{X}} is f-adic.

Proof. (i): Let (R_{i} | i ∈ I) be a system of rational subsets of Spa \underline{X} such that U = ⋃_{i∈I} R_{i}, and for every i, j ∈ I set R_{ij} := R_{i} ∩ R_{j} and choose topologically local quasi-affinoid schemes \underline{Z}_{i} and \underline{Z}_{ij} that represent h_{R_{i}} and h_{R_{ij}}. The inclusions R_{i} ⊂ U and R_{ij} ⊂ U are then represented by morphisms ψ_{i} : \underline{Z}_{i} → \underline{Y} and respectively ψ_{ij} : \underline{Z}_{ij} → \underline{Y} of quasi-affinoid schemes such that φ_{\underline{Z}_{i}/\underline{X}} = φ_{\underline{Y} /\underline{X}} ∘ ψ_{i} and φ_{\underline{Z}_{ij}/\underline{X}} = φ_{\underline{Y} /\underline{X}} ∘ ψ_{ij}, for every i, j ∈ I. For every i ∈ I set U_{i} := Im Spa ψ_{i} ⊂ Spa \underline{Y}; it was observed in the proof of theorem 15.5.5 that Spa φ_{\underline{Z}_{i}/\underline{X}} induces a homeomorphism Spa \underline{Z}_{i} →^{∼} R_{i}; it follows easily that Spa ψ_{i} and Spa φ_{\underline{Y} /\underline{X}} induce homeomorphisms Spa \underline{Z}_{i} →^{∼} U_{i} and respectively U_{i} →^{∼} R_{i}. It is also easily seen that U_{i} ∩ U_{j} = Im Spa ψ_{ij}, so the map Spa φ_{\underline{Y} /\underline{X}} restricts to a homeomorphism U′ := ⋃_{i∈I} U_{i} →^{∼} U. To conclude, it suffices to check that U′ = Spa \underline{Y}. However, suppose that v ∈ Spa \underline{Y} \ U′; we may find a rational subset R′ of Spa \underline{Y} that contains v, and we may assume that Spa φ_{\underline{Y} /\underline{X}}(R′) ⊂ R_{i} for some i ∈ I. Pick a topologically local quasi-affinoid scheme \underline{Y}′ that represents the subsheaf h_{R′} of h\underline{Y}, and let φ′ : \underline{Y}′ → \underline{Y} be the morphism of quasi-affinoid schemes that represents the inclusion R′ ⊂ Spa \underline{Y}. Then φ_{\underline{Y} /\underline{X}} ∘ φ′ : \underline{Y}′ → \underline{X} lies in h_{R_{i}}(\underline{Y}′), so it corresponds to a unique morphism β : \underline{Y}′ → \underline{Z}_{i} such that φ_{\underline{Y} /\underline{X}} ∘ φ′ = φ_{\underline{Z}_{i}/\underline{X}} ∘ β = φ_{\underline{Y} /\underline{X}} ∘ ψ_{i} ∘ β. Since φ_{\underline{Y} /\underline{X}} induces an injective morphism h\underline{Y} → h\underline{X} (whose image is h_{U}), we deduce that φ′ = ψ_{i} ∘ β. Especially, U_{i} contains the image of Spa φ′; but we know that the latter coincides with R′, a contradiction.

(ii) is an immediate consequence of (i).

(iii): By virtue of remark 15.3.13(vii), it suffices to show that Spa φ_{\underline{Y} /\underline{X}} maps (Spa \underline{Y})_{a} into (Spa X)_{a}. Thus, let v ∈ (Spa \underline{Y}), pick i ∈ I such that w := Spa φ_{\underline{Y} /\underline{X}}(v) ∈ R_{i}, and let u := (Spa φ_{\underline{Z}_{i}/\underline{X}})^{−1}(w). Suppose that w ∈ (Spa X)_{na}; by inspecting the proof of theorem 15.5.5, it is easily seen that φ_{\underline{Z}_{i}/\underline{X}} is f-adic, so u ∈ Spa(\underline{Z}_{i})_{na}, by corollary 15.3.17(ii). By the foregoing, we also know that Spa ψ_{i}(u) = v, and then v ∈ (Spa \underline{Y})_{na}, again by remark 15.3.13(vii), whence the assertion.

(iv) follows immediately from (iii) and proposition 15.4.22(iii.b). □

Remark 15.5.9. (i) In the situation of (15.5.3), we can also consider the Yoneda imbeddings for the categories of topologically henselian and of complete, separated quasi-affinoid schemes

h′ : q.Afd.Sch_{hens} → q.Afd.Sch_{hens}^{∧} h′′ : q.Afd.Sch_{comp} → q.Afd.Sch_{comp}^{∧}

and if \underline{X} is topologically henselian (resp. complete and separated) we set

h′_{U}(\underline{Y}′) := h_{U}(\underline{Y}′) ⊂ h′ ; \underline{X}(\underline{Y}′) (resp. h′′_{U}(\underline{Y}′′) := h_{U}(\underline{Y}′′) ⊂ h′′ ; \underline{X}(\underline{Y}′′) )

for every topologically henselian quasi-affinoid scheme \underline{Y}′ and every complete and separated quasi-affinoid scheme \underline{Y}′′. Suppose that U is quasi-affinoid, and let φ_{\underline{Y} /\underline{X}} : \underline{Y} → \underline{X} be a morphism of topologically local quasi-affinoid schemes representing the sub-presheaf h_{U} ⊂ h\underline{X}, as in (15.5.7); from remark 15.4.10(v) and lemma 15.4.17(i) we see that if \underline{X} is topologically henselian, the topological henselization φ_{\underline{Y} /\underline{X}}^{h} : \underline{Y}^{h} → \underline{X} of φ_{\underline{Y} /\underline{X}} represents h′_{U}. Likewise, taking into account proposition 15.4.12and lemma 15.4.17(iii), we conclude that if \underline{X} is complete and separated, the completion φ_{\underline{Y} /\underline{X}}^{∧} : \underline{Y}^{∧} → \underline{X} of φ_{\underline{Y} /\underline{X}} represents h′′_{U}.

(ii) Moreover, we may consider the Yoneda imbedding for the opposite of the category of topologically local affinoid rings :

h′′′ : Afd.Ring_{loc}^{o} → Afd.Ring_{loc}^{o∧}

and if \underline{A} := (A, A^{+}) is any topologically local affinoid ring, and U ⊂ Spa \underline{A} any open subset, we may form again the sub-presheaf h′′′_{U} :⊂ h′′′ ; \underline{A} by the rule : h′′′_{U}(\underline{B}) := h_{U}(Spec \underline{B}) for every topologically local affinoid ring \underline{B}. We may regard U as an open subset of Spa(Spec \underline{A}), and we say that U is a quasi-affinoid open subset of Spa \underline{A} if it is such, as an open subset of Spa Spec \underline{A}. Since Spec is fully faithful on the full subcategory Afd.Ring, it follows that if U is quasi-affinoid, h′′′_{U} is representable : namely, a representing affinoid ring is Γ(\underline{Y}), where \underline{Y} is any topologically local quasi-affinoid scheme that represents the presheaf h_{U}. Lastly, we may in the same way consider variants for topologically henselian (resp. complete and separated) affinoid rings : the reader may spell out the details. (However, the method does not produce representing objects on the whole category of quasi-affinoid rings, essentially because there is – in this generality – no counterpart of proposition 15.4.30for quasi-affinoid rings).

(iii) It follows from lemma 15.4.6and corollary 15.5.8(iv) that the intersection of two quasiaffinoid open subsets U, U′ ⊂ Spa \underline{X} is quasi-affinoid. Indeed, if \underline{Y} and \underline{Y}′ are topologically local quasi-affinoid schemes that represent h_{U} and respectively h_{U′}, it is easily seen that

(\underline{Y} ×\underline{X} \underline{Y}′)_{loc}

represents h_{U} ∩ h_{U′} = h_{U∩U′} (see example 15.4.8).

(iv) In the same vein, if f : \underline{X}′ → \underline{X} is any f-adic morphism of topologically local quasiaffinoid schemes, and U ⊂ Spa \underline{X} is any quasi-affinoid open subset, then U′ := (Spa f)^{−1}U is a quasi-affinoid open subset of Spa \underline{X}′. Indeed, if \underline{Y} represents h_{U}, then h_{U′} is represented by (\underline{Y} ×\underline{X} \underline{X}′)_{loc}, where the fibre product is given as in example 15.4.8.

(v) Lastly, let U be a quasi-affinoid open subset of Spa \underline{X} such that the inclusion h_{U} ⊂ h\underline{X} is represented by the morphism of topologically local quasi-affinoid schemes φ_{\underline{Y} /\underline{X}} : \underline{Y} → \underline{X}, and V a quasi-affinoid open subset of Spa \underline{Y} such that the inclusion h_{V} ⊂ h\underline{Y} is represented by the morphism of topologically local affinoid schemes φ_{\underline{Z}/\underline{Y}} : \underline{Z} → \underline{Y}, and set U′ := Spa φ_{\underline{Y} /\underline{X}}(V) ⊂ U. Then φ_{\underline{Y} /\underline{X}} induces an isomorphism h\underline{Y} →^{∼} h_{U} that identifies h_{V} with the sub-presheaf h_{U′} of h_{U}. Especially, U′ is a quasi-affinoid open subset of Spa X, and the inclusion h_{U′} ⊂ h\underline{X} is represented by φ_{\underline{Y} /\underline{X}} ∘ φ_{\underline{Z}/\underline{Y}}.

Lemma 15.5.10. In the situation of remark 15.5.9(v), suppose furthermore that U is a rational subset of Spa \underline{X} and V is a rational subset of Spa \underline{Y}. Then V is a rational subset of Spa \underline{X}.

Proof. Say that Γ(\underline{X}) = (A, A^{+}, X), and U = R_{A}(\frac{f_{1}}{f_{0}} , …, \frac{f_{n}}{f_{0}} ) ∩ Spa \underline{X}, with (f_{0}, …, f_{n}) a sequence of elements of A that generates an open ideal. Set (A_{U}, A_{U}^{+}) := \underline{A}(\frac{f_{1}}{f_{0}} , …, \frac{f_{n}}{f_{0}} ), where \underline{A} := (A, A^{+}), so that A_{U} = A[f_{0}^{−1}]. By inspecting the proof of theorem 15.5.5, we get an isomorphism of quasi-affinoid schemes

\underline{Y} →^{∼} Spec (A_{U}, A_{U}^{+}, X ∩ Spec A_{U})_{loc}

Since X is quasi-compact, and Spec A_{U} is an open affine subset of Spec A, we have 𝒪_{X}(X ∩ Spec A_{U}) = 𝒪_{X}(X)[f_{0}^{−1}] = A_{U} ([59, Ch.I, Prop.9.2.1]). Taking into account (15.4.11), we deduce that Γ(\underline{Y}) = (B, B^{+}, X ∩ Spec A_{U}), where B := A_{U,loc} and B^{+} := A_{U,loc}^{+}. Consequently, we have V = R_{B}(\frac{g_{1}}{g_{0}} , …, \frac{g_{m}}{g_{0}} ) ∩ Spa \underline{Y} for a sequence g_{•} := (g_{0}, …, g_{m}) of elements of B that generates an open ideal J. Since B is a localization of A, we may also find b ∈ B^{×}, such that bg_{i} ∈ A′ := Im(A → B); after replacing g_{•} by the sequence (bg_{0}, …, bg_{n}), we may then assume that g_{0}, …, g_{m} ∈ A′, and for every i = 0, …, m we pick h_{i} ∈ A whose image in B agrees with g_{i}. Moreover, let A_{0} ⊂ A be a ring of definition, and I ⊂ A_{0} a finitely generated ideal of adic definition; according to claim 15.4.20(i), there exists k ∈ ℕ such that v(a) < v(h_{0}) for every a ∈ I^{k} and every v ∈ V. Pick a finite system x_{1}, …, x_{r} of generators for I^{k} and set h_{i+m} := x_{i} for every i = 1, …, r; by construction, we easily see that

V = U ∩ R where R := R_{A}(\frac{h_{1}}{h_{0}} , …, \frac{h_{m+r}}{h_{0}} )

and the system (h_{i} | i = 1, …, m + r) generates an open ideal of A, so R is a rational subset of Spa \underline{X}, and finally, the same holds for V. □

15.5.11. Let \underline{X} := (X, 𝒯_{X}, A_{X}^{+}) be any topologically local quasi-affinoid scheme; in view of remark 15.5.9(iii) we may associate with \underline{X} a site

(𝒬(\underline{X}), J_{𝒬} )

where 𝒬(\underline{X}) is the category of all quasi-affinoid open subsets of Spa \underline{X}; for every U, U′ ∈ Ob(𝒬(\underline{X})), the set Hom_{𝒬(\underline{X})} (U, U′) contains exactly one morphism if U ⊂ U′, and is empty otherwise. For every such U, the sieves covering U for the topology J_{𝒬} are precisely those generated by the families (U_{λ} | λ ∈ Λ) of quasi-affinoid open subsets of Spa \underline{X} such that ⋃_{λ∈Λ} U_{λ} = U. We consider nine presheaves of topological 𝒪_{X}(X)-algebras on 𝒬(\underline{X}), related by natural morphisms of presheaves :

\xymatrix{𝒪_{Spa \underline{X}}^{loc+} \ar@{^{(}->}[r] \ar[d] & 𝒪_{Spa \underline{X}}^{loc ∘} \ar@{^{(}->}[r] \ar[d] & 𝒪_{Spa \underline{X}}^{loc} \ar[d] \\ 𝒪_{Spa \underline{X}}^{h+} \ar@{^{(}->}[r] \ar[d] & 𝒪_{Spa \underline{X}}^{h ∘} \ar@{^{(}->}[r] \ar[d] & 𝒪_{Spa \underline{X}}^{h} \ar[d] \\ 𝒪_{Spa \underline{X}}^{∧+} \ar@{^{(}->}[r] & 𝒪_{Spa \underline{X}}^{∧∘} \ar@{^{(}->}[r] & 𝒪_{Spa \underline{X}}^{∧}}

and constructed as follows. By theorem 15.5.5, for every U ∈ Ob(𝒬(\underline{X})) the sub-presheaf h_{U} of h\underline{X} is representable (notation of (15.5.3)); we choose a topologically local quasi-affinoid scheme \underline{X}_{U} := (X_{U}, 𝒯_{U}, A_{U}^{+}) that represents this presheaf, and we set

𝒪_{Spa \underline{X}}^{loc}(U) := 𝒪_{X_{U}} (X_{U})  𝒪_{Spa \underline{X}}^{loc ∘}(U) := 𝒪_{X_{U}} (X_{U})^{∘}  𝒪_{Spa \underline{X}}^{loc+}(U) := A_{U}^{+}.

Any inclusion U ⊂ U′ of quasi-affinoid open subsets induces a morphism h_{U} → h_{U′} of presheaves, which is represented by a unique morphism \underline{X}_{U} → \underline{X}_{U′} of topologically local quasi-affinoid schemes (see remark 1.2.8(i)), whence well defined continuous homomorphisms

𝒪_{Spa \underline{X}}^{loc+}(U′) → 𝒪_{Spa \underline{X}}^{loc+}(U) 𝒪_{Spa \underline{X}}^{loc}(U′) → 𝒪_{Spa \underline{X}}^{loc}(U)

of topological 𝒪_{X}(X)-algebras, and clearly the rules U ↦ 𝒪_{Spa \underline{X}}^{loc+}(U) and U ↦ 𝒪_{Spa \underline{X}}^{loc}(U) yield two functors from 𝒬(\underline{X})^{o} to the category of topologically local f-adic 𝒪_{X}(X)-algebras. Likewise, in view of corollary 15.5.8(iv) and lemma 8.3.24(iii.a) we also get a continuous map 𝒪_{Spa \underline{X}}^{loc ∘}(U′) → 𝒪_{Spa \underline{X}}^{loc ∘}(U), whence the presheaf 𝒪_{Spa \underline{X}}^{loc ∘}.

Similarly, let \underline{X}_{U}^{h} := (X_{U}^{h}, 𝒯_{U}^{h}, A_{U}^{h+}) (resp. \underline{X}_{U}^{∧} := (X_{U}^{∧}, 𝒯_{U}^{∧}, A_{U}^{∧+})) be a topologically henselian (resp. complete and separated) quasi-affinoid scheme representing the sub-presheaf h′_{U} of h′_{\underline{X}^{h}} (resp. h′′_{U} of h′′_{\underline{X}^{∧}}) as in remark 15.5.9(i); we define

𝒪_{Spa \underline{X}}^{h}(U) := 𝒪_{X_{U}^{h}} (X_{U}^{h})  𝒪_{Spa \underline{X}}^{h ∘}(U) := 𝒪_{X_{U}^{h}} (X_{U}^{h})^{∘}  𝒪_{Spa \underline{X}}^{h+}(U) := A_{U}^{h+}
𝒪_{Spa \underline{X}}^{∧}(U) := 𝒪_{X_{U}^{∧}}(X_{U}^{∧})  𝒪_{Spa \underline{X}}^{∧∘}(U) := 𝒪_{X_{U}^{∧}}(X_{U}^{∧})^{∘}  𝒪_{Spa \underline{X}}^{∧+}(U) := A_{U}^{∧+}

and arguing as in the foregoing, it is easily seen that these rules yield well defined presheaves of topologically henselian (resp. complete and separated) f-adic 𝒪_{X}(X)-algebras. All these presheaves depend on the choices of representing quasi-affinoid schemes, but for any two such sets of choices there exists a unique isomorphism of presheaves of topological 𝒪_{X}(X)-algebras between the corresponding presheaves. We notice as well that the rule

U ↦ 𝒪_{Spa\underline{X}}^{loc}(U)^{∘∘}

defines a sub-presheaf 𝒪_{Spa \underline{X}}^{loc ∘∘} ⊂ 𝒪_{Spa \underline{X}}^{loc} that is a presheaf of ideals in both 𝒪_{Spa \underline{X}}^{loc+} and 𝒪_{Spa \underline{X}}^{loc ∘}. Likewise we get presheaves of topologically nilpotent sections 𝒪_{Spa \underline{X}}^{h ∘∘} ⊂ 𝒪_{Spa \underline{X}}^{h} (resp. 𝒪_{Spa \underline{X}}^{∧ ∘∘} ⊂ 𝒪_{Spa \underline{X}}^{∧}) that are presheaves of ideals in 𝒪_{Spa \underline{X}}^{h ∘} and 𝒪_{Spa \underline{X}}^{h+} (resp. in 𝒪_{Spa \underline{X}}^{∧ ∘} and 𝒪_{Spa \underline{X}}^{∧+}).

15.5.12. With the notation of (15.5.11), let x ∈ Spa \underline{X} be any point; recall that x is the equivalence class of a continuous valuation v_{x} : A := 𝒪_{X}(X) → Γ_{x}; if R ⊂ Spa \underline{X} is any rational subset containing x, the valuation v_{x} extends uniquely to a continuous valuation v_{x,R}^{∧} on 𝒪_{Spa \underline{X}}^{∧}(R) with value group Γ_{x} (claim 15.3.20(i)). Then, by restricting along the natural maps 𝒪_{Spa \underline{X}}^{loc}(R) → 𝒪_{Spa \underline{X}}^{h}(R) → 𝒪_{Spa \underline{X}}^{∧}(R) we get corresponding valuations v_{x,R}^{h}, v_{x,R}^{loc}, and – after taking colimits – unique valuations on the stalks of these presheaves

(15.5.13)  \xymatrix{𝒪_{Spa \underline{X},x}^{loc} \ar[r] \ar[dr]_{|·|_{x}^{loc}} & 𝒪_{Spa \underline{X},x}^{h} \ar[r] \ar[d]^{|·|_{x}^{h}} & 𝒪_{Spa \underline{X},x}^{∧} \ar[dl]^{|·|_{x}^{∧}} \\  & Γ_{x}.}

We denote by κ(x^{loc}), κ(x^{h}) and κ(x^{∧}) the residue fields of | · |_{x}^{loc}, | · |_{x}^{h} and respectively | · |_{x}^{∧}, and recall that the valuations of (15.5.13) induce natural residual valuations on these fields : see remark 9.1.4(v). There follow natural inclusion of valued fields

κ(x^{loc}) → κ(x^{h}) → κ(x^{∧})

and taking into account claim 15.3.20(ii) we see that the images of κ(x^{loc}) and of κ(x^{h}) in κ(x^{∧}) are both dense for the valuation topology of the latter field. Especially, these inclusions induce natural identifications of the completions of these fields for their respective valuation topologies (theorem 8.2.8(iii)), and we denote this common completion by

(κ(x)^{∧}, | · |_{x}^{∧})

which is a valued field with value group Γ_{x}, by proposition 9.1.16(iii,v).

Lemma 15.5.14. In the situation of (15.5.12), we have :

(i) The stalks 𝒪_{Spa \underline{X},x}^{∧}, 𝒪_{Spa \underline{X},x}^{h} and 𝒪_{Spa \underline{X},x}^{loc} are local rings, and κ(x^{loc}), κ(x^{h}), κ(x^{∧}) are their respective residue fields.

(ii) The stalks 𝒪_{Spa \underline{X},x}^{∧+}, 𝒪_{Spa \underline{X},x}^{h+} and 𝒪_{Spa \underline{X},x}^{loc+} are local rings, and the induced diagrams

\xymatrix{𝒪_{Spa \underline{X},x}^{loc+} \ar[r] \ar[d] & 𝒪_{Spa \underline{X},x}^{loc} \ar[d]^{π_{x}^{loc}} & 𝒪_{Spa \underline{X},x}^{h+} \ar[r] \ar[d] & 𝒪_{Spa \underline{X},x}^{h} \ar[d]^{π_{x}^{h}} & 𝒪_{Spa \underline{X},x}^{∧+} \ar[r] \ar[d] & 𝒪_{Spa \underline{X},x}^{∧} \ar[d]^{π_{x}^{∧}} \\ κ(x^{loc})^{+} \ar[r] & κ(x^{loc}) & κ(x^{h})^{+} \ar[r] & κ(x^{h}) & κ(x^{∧})^{+} \ar[r] & κ(x^{∧})}

are cartesian (here κ(x^{loc})^{+} is the valuation ring of the residual valuation on κ(x^{loc}), and likewise for κ(x^{h})^{+} and κ(x^{∧})^{+}).

(iii) The stalk 𝒪_{Spa \underline{X},x}^{loc ∘∘} (resp. 𝒪_{Spa \underline{X},x}^{h ∘∘}, resp. 𝒪_{Spa \underline{X},x}^{∧ ∘∘}) is the radical of the ideal A^{∘∘}·𝒪_{Spa \underline{X},x}^{loc+}

of 𝒪_{Spa \underline{X},x}^{loc+} (resp. A^{∘∘} · 𝒪_{Spa \underline{X},x}^{h+} of 𝒪_{Spa \underline{X},x}^{h+}, resp. A^{∘∘} · 𝒪_{Spa \underline{X},x}^{∧+} of 𝒪_{Spa \underline{X},x}^{∧+}). (iv) The pairs (𝒪_{Spa \underline{X},x}^{h+}, 𝒪_{Spa \underline{X},x}^{h ∘∘}) and (𝒪_{Spa \underline{X},x}^{∧+}, 𝒪_{Spa \underline{X},x}^{∧ ∘∘}) are henselian.

(v) Suppose that x is analytic, and let

𝔭^{loc} ⊂ κ(x^{loc})^{+}  𝔭^{h} ⊂ κ(x^{h})^{+}  𝔭^{∧} ⊂ κ(x^{∧})^{+}

be the (unique) prime ideals of height one. Then :

(a) 𝒪_{Spa \underline{X},x}^{loc ∘∘} = π_{x}^{loc −1}(𝔭^{loc}) 𝒪_{Spa \underline{X},x}^{h ∘∘} = π_{x}^{h −1}(𝔭^{h}) 𝒪_{Spa \underline{X},x}^{∧∘∘} = π_{x}^{∧ −1}(𝔭^{∧}).

(b) The pairs (κ(x^{h})^{+}, 𝔭^{h}) and (κ(x^{∧})^{+}, 𝔭^{∧}) are henselian.

(c) The local rings 𝒪_{Spa \underline{X},x}^{h} and 𝒪_{Spa \underline{X},x}^{∧} are henselian.

(d) The valuation rings κ(x^{h})_{𝔭^{h}}^{+} and κ(x^{∧})_{𝔭^{∧}}^{+} are henselian. (vi) Suppose that x is non-analytic. Then :

(a) 𝒪_{Spa \underline{X},x}^{loc ∘∘} (resp. 𝒪_{Spa \underline{X},x}^{h ∘∘}, resp. 𝒪_{Spa \underline{X},x}^{∧ ∘∘}) is also an ideal of 𝒪_{Spa \underline{X},x}^{loc} (resp. of 𝒪_{Spa \underline{X},x}^{h}, resp. of 𝒪_{Spa \underline{X},x}^{∧}), and it is the radical of the ideal generated by A^{∘∘}.

(b) The pairs (𝒪_{Spa \underline{X},x}^{h}, 𝒪_{Spa \underline{X},x}^{h ∘∘}) and (𝒪_{Spa \underline{X},x}^{∧}, 𝒪_{Spa \underline{X},x}^{∧∘∘}) are henselian.

Proof. Let f ∈ 𝒪_{Spa \underline{X},x}^{∧}, and pick any rational subset R in Spa\underline{X} with x ∈ R, and such that f extends to a section f_{R} ∈ B := 𝒪_{Spa \underline{X}}^{∧}(R). Let \underline{Y}^{∧} := (Y^{∧}, 𝒯_{Y}^{∧}, A_{Y}^{∧+}) be a complete and separated quasi-affinoid scheme representing the subsheaf h′′_{R} of h′′_{\underline{X}^{∧}}, so that B = 𝒪_{Y^{∧}}(Y^{∧}).

(i): Suppose now that |f|_{x}^{∧} ≠ 0; then clearly v_{x,R}^{∧}(f_{R}) ≠ 0. We set U := R_{B}(\frac{f_{R}}{f_{R}} ) ∩ Spa \underline{Y}^{∧}. In view of lemma 15.4.17(iii) and corollary 15.5.8(i), we may then identify U with an open subset of Spa X containing x, and we pick a rational subset U′ ⊂ U of Spa \underline{X} with x ∈ U′; by construction the image of f_{R} under the restriction map ρ_{U′} : 𝒪_{Spa \underline{X}}^{∧}(R) → 𝒪_{Spa \underline{X}}^{∧}(U′) is an invertible element. This shows that Ker | · |_{x}^{∧} is the unique maximal ideal of 𝒪_{Spa \underline{X},x}^{∧}, and it also easily implies that the projection π_{x}^{∧} : 𝒪_{Spa \underline{X},x}^{∧} → κ(x^{∧}) is surjective (details left to the reader). The same argument applies to 𝒪_{Spa \underline{X},x}^{h} and 𝒪_{Spa \underline{X},x}^{loc}, whence the contention.

(ii): Suppose next that |f|_{x}^{∧} ≤ 1. Then we let U := R_{B}(\frac{f_{R}}{1} ) ∩ Spa \underline{Y}^{∧}, which again we identify naturally with an open subset of Spa \underline{X} containing x, and we pick a rational subset U′ ⊂ U of Spa \underline{X} with x ∈ U′; then ρ_{U′}(f_{R}) ∈ 𝒪_{Spa \underline{X}}^{∧+}(U′). This shows that f ∈ 𝒪_{Spa \underline{X},x}^{∧+}, whence the assertion concerning the third square diagram. The same argument applies to the other two diagrams. It follows easily that the preimage in 𝒪_{Spa \underline{X},x}^{∧+} of the maximal ideal of κ(x^{∧})^{+} is the unique maximal ideal of 𝒪_{Spa \underline{X},x}^{∧+} (details left to the reader); especially, the latter is a local ring. Again, the same applies to the other two rings.

(iii): It follows easily from corollary 15.5.8(iv) that the natural map A → 𝒪_{Spa \underline{X}}^{loc}(U) is f-adic, for every rational subset U ⊂ Spa \underline{X}. Hence, A^{∘∘} ·𝒪_{Spa\underline{X}}^{loc+}(U) is an open ideal of 𝒪_{Spa\underline{X}}^{loc+}(U) consisting of topologically nilpotent section, and therefore its radical equals 𝒪_{Spa\underline{X}}^{loc ∘∘}(U). It suffices then to observe that the radical of A^{∘∘}·𝒪_{Spa\underline{X},x}^{loc+} is the colimit of the filtered system of the radicals of the ideals (A^{∘∘} · 𝒪_{Spa\underline{X}}^{loc+}(U) | x ∈ U). The same applies to the other ideals in (iii).

(iv): By proposition 8.4.2(iii), both (𝒪_{Spa \underline{X}}^{h+}(U), 𝒪_{Spa \underline{X}}^{h ∘∘}(U)) and (𝒪_{Spa \underline{X}}^{∧+}(U), 𝒪_{Spa \underline{X}}^{∧∘∘}(U)) are henselian pairs for every rational subset U ⊂ Spa \underline{X} containing x. The assertion follows, after taking colimits over the filtered family of such rational subsets.

(v.a): First, recall that the existence of a rank one prime ideal 𝔭^{loc} ⊂ κ(x^{loc})^{+} is ensured by lemma 15.3.14(v); it follows easily that 𝔭^{loc} = κ(x^{loc})^{∘∘}, and taking into account remark 15.3.13(i) we deduce already that π_{x}^{loc}(𝒪_{Spa \underline{X},x}^{loc ∘∘}) ⊂ 𝔭^{loc}. Conversely, let f ∈ 𝒪_{Spa \underline{X},x}^{loc} \ 𝒪_{Spa \underline{X},x}^{loc ∘∘} be any element; denote by 𝒰 the set of all rational subsets of Spa \underline{X} containing x, and for every U ∈ 𝒰 set U′ := U ∩R_{A}(\frac{1}{f}). The assumption on f implies that U′ is a non-empty constructible subset of Spa \underline{X} for every U ∈ 𝒰 (corollary 15.4.27(ii)), and therefore T := ⋂_{U∈𝒰} U′ ≠ ∅ (proposition 8.1.23(ii.a)). However, T lies in the set Spa \underline{X}(x) of all generization of x in Spa \underline{X} (remark 8.1.45(i)), and must then contain the unique maximal generization of x; the latter means precisely that π_{x}^{loc}(f) ∉ 𝔭^{loc}, as required.

Next, since | · |_{x}^{loc} and | · |_{x}^{∧} have the same value group, the existence of a (unique) height one prime ideal 𝔭^{∧} ⊂ κ(v^{∧})^{+} follows from the existence of 𝔭^{loc} and remark 9.1.13(vii); then, since v_{x,R}^{∧} is continuous, we deduce likewise that π_{x}^{∧}(𝒪_{Spa \underline{X},x}^{∧∘∘}) ⊂ 𝔭^{∧} for every rational subset R ⊂ Spa \underline{X}. If now f ∈ 𝒪_{Spa \underline{X},x}^{∧} \ 𝒪_{Spa \underline{X},x}^{∧∘∘}, pick R as in the foregoing and \underline{Y} representing h′′_{R}, so that x ∈ R, and f extends to a section f_{R} ∈ B := 𝒪_{Spa \underline{Y}}^{∧} (Y); let 𝒰 be the set of all rational subsets of Spa \underline{Y} containing x, and for every U ∈ 𝒰 set U′ := U ∩ R_{B}(\frac{1}{f_{R}} ). Due to corollary 15.5.8(i) and lemma 15.4.17(iii), the image of T := ⋂_{U∈𝒰} U′ in Spa \underline{X} lies in Spa \underline{X}(x), and is not empty, and arguing as in the foregoing we deduce again that π_{x}^{∧}(f) ∉ 𝔭^{∧}. The same argument applies to 𝔭^{h}.

(v.b): Let us remark more generally :

Claim 15.5.15. Let A be any ring, J ⊂ I ⊂ A two ideals. Then the pair (A, I) is henselian if and only if the same holds for both (A/J, I/J) and (A, J).

Proof of the claim. From [75, Rem.5.1.10(iv,v)] we see that if (A, I) is henselian, the same holds for the pairs (A, J) and (A/J, I/J). Conversely, let A → B be a finite ring homomorphism; if (A, J) is henselian, the projection B → B/JB restricts to a bijection Idemp (B) →^{∼} Idemp (B/JB) on the respective subsets of idempotent elements. Then, if also (A/J, I/J) is henselian, the induced map Idemp (B/JB) → Idemp (B/IB) is bijective as well. Summing up, we conclude that (A, I) is henselian. ♢

In view of (iv) and claim 15.5.15, we are easily reduced to showing that the projections 𝒪_{Spa \underline{X},x}^{h+} → κ(x^{h})^{+} and 𝒪_{Spa \underline{X},x}^{∧+} → κ(x^{∧})^{+} are surjective. The latter follows from (ii) and from the surjectivity of π_{x}^{h} and π_{x}^{∧}, which has already been remarked in the proof of (i).

(v.c): Let 𝔪^{h} := Ker π_{x}^{h}; the proof of (i) shows that 𝔪^{h} is the maximal ideal of 𝒪_{Spa \underline{X},x}^{h}, and (ii) implies that 𝔪^{h} is also an ideal in 𝒪_{Spa \underline{X},x}^{h+}; moreover, in light of (iv) and [75, Rem.5.1.10(iv)] the pair (𝒪_{Spa \underline{X},x}^{h+}, 𝔪^{h}) is henselian. Then the assertion follows from [75, Rem.5.1.10(ii)]. The same argument applies to 𝒪_{Spa \underline{X},x}^{∧}.

(v.d): It is easily seen that the localization map κ(x^{h})^{+} → κ(x^{h})_{𝔭^{h}}^{+} restricts to a bijection from 𝔭^{h} to the maximal ideal of κ(x^{h})_{𝔭^{h}}^{+}; then the assertion follows from [75, Rem.5.1.10(ii)]. The same argument applies to κ(x^{∧})_{𝔭^{∧}}^{+}.

(vi.a): By (iii) we know already that 𝒪_{Spa \underline{X},x}^{∧∘∘} lies in the radical of A^{∘∘} · 𝒪_{Spa \underline{X},x}^{∧}; hence, suppose that f^{n} ∈ A^{∘∘} · 𝒪_{Spa \underline{X},x}^{∧} for some n ∈ ℕ, and say that f^{n} = ∑_{i=1}^{k} a_{i}g_{i} for some a_{1}, …, a_{k} ∈ A^{∘∘} and g_{1}, …, g_{k} ∈ 𝒪_{Spa \underline{X},x}^{∧}. Notice that since x is non-analytic, we have π_{x}^{∧}(a_{i}) = 0 for i = 1, …, k, whence π_{x}^{∧}(f^{n}) = 0, and therefore π_{x}^{∧}(f^{n}g_{i}) = 0 as well for i = 1, …, k; then (ii) implies that f^{n}g_{i} ∈ 𝒪_{Spa \underline{X},x}^{∧+}, so f^{2n} = ∑_{i=1}^{k} a_{i} · (f^{n}g_{i}) ∈ A^{∘∘} · 𝒪_{Spa \underline{X},x}^{∧+}. The same argument applies to 𝒪_{Spa \underline{X},x}^{loc ∘∘} and 𝒪_{Spa \underline{X},x}^{h,∘∘}.

(vi.b) follows directly from (vi.a), (iv) and [75, Rem.5.1.10(ii)]. □

15.5.16. Let \underline{A} := (A, A^{+}, U) be any quasi-affinoid ring, and J ⊂ A a finitely generated ideal such that U = Spec A \ Spec A/J. The Zariski-Riemann spectrum of \underline{A} is the subset of Spv A

ZR(\underline{A}) := Spa(\underline{A}) ∩ Spv(A, J)

(notation of definition 15.3.6(i)) that we endow with the topology induced by the inclusion into Spv A. A rational subset of ZR(\underline{A}) is a subset of the form R ∩ ZR(\underline{A}), where R is a rational subset of Spv(A, J) (see definition 15.3.6(ii)). Notice that neither ZR(\underline{A}) nor the class of its rational subsets depend on the choice of J (lemma 15.3.9(i)).

Proposition 15.5.17. With the notation of (15.5.16), the following holds :

(i) ZR(\underline{A}) = \{v ∈ Spa(\underline{A}) | v has no proper primary specializations in Spa(\underline{A}) \}.

(ii) ZR(\underline{A}) is a pro-constructible subset of Spv(A, J). In particular, it is a spectral topo-

logical space. (iii) The rational subsets of ZR(\underline{A}) are constructible in ZR(\underline{A}), and form a basis of the

topology of ZR(\underline{A}) that is closed under finite intersections. (iv) The unit of adjunction \underline{A} → Γ ∘ Spec \underline{A} induces a homeomorphism

ZR(Γ ∘ Spec \underline{A}) →^{∼} ZR(\underline{A}).

Especially, ZR(\underline{A}) depends only on the quasi-affinoid scheme Spec \underline{A}.

(v) The homeomorphism Spa(\underline{A}_{loc}) →^{∼} Spa(\underline{A}) of lemma 15.4.17(i) restricts to a homeo-

morphism :

ZR(\underline{A}_{loc}) →^{∼} ZR(\underline{A}).

Proof. (i): Recall that Cont(A) is closed under specializations in Spv A (remark 15.3.13(iv)); it follows easily that Spa(A, A^{+}) is closed under primary specializations. Recall as well that a proper specialization (v, w) in Spv A is J-admissible if and only if w is a primary specialization of v and the supports of v and w lie in U (see (15.3.1)); hence, if v ∈ Spa \underline{A}, the proper specialization (v, w) is J-admissible if and only if w is a primary specialisation of v that lies in Spa \underline{A}. Then the assertion follows from lemma 15.3.4(iii).

(ii): Let A_{0} ⊂ A be a subring of definition, and I ⊂ A_{0} a finitely generated ideal of adic definition; recall that Spa(\underline{A}) is a pro-constructible subset of Cont(A) (proposition 15.4.22(i)) and the inclusion map Cont(A) → Spv(A, IA) is a closed immersion (theorem 15.3.15(ii)) so Spv(\underline{A}) is a pro-constructible subset of Spv(A, IA) (corollary 8.1.50(i)). Moreover, the radical of J contains IA, hence the inclusion map Spv(A, J) → Spv(A, IA) admits a spectral retraction (lemma 15.3.9(i))

(15.5.18)  r : Spv(A, IA) → Spv(A, J).

Namely, for every v ∈ Spv(A, IA), the valuation r(v) is the unique J-admissible specialization of v that does not admit further proper J-admissible specializations. We claim that

ZR(\underline{A}) = r(Spa \underline{A}).

Indeed, clearly ZR(\underline{A}) = r(ZR(\underline{A})) ⊂ r(Spa \underline{A}). The converse inclusion follows from the proof of (i). Then, the assertion follows from corollary 8.1.50(i).

(iii) follows immediately from (ii) and theorem 15.3.7(ii).

(iv) follows directly from (i) and lemmata 15.4.17(ii) and 9.2.25(i).

(v): Clearly the homeomorphism Spa \underline{A}_{loc} →^{∼} Spa \underline{A} preserves primary specializations, so the assertion follows from (i). □

Remark 15.5.19. In [157], Temkin associates with every separated morphism f : Y → X of quasi-compact and quasi-separated schemes, a spectral space that he calls the Riemann-Zariski space of f, and denotes RZ_{Y} (X), together with a continuous map RZ_{Y} (X) → X. His construction generalizes earlier work by Zariski, and is related as follows to our ZR(\underline{A}). Let U → Spec A^{+} be the composition of the open immersion U → Spec A with the natural morphism Spec A → Spec A^{+}; then we may consider the induced map φ : RZ_{U}(Spec A^{+}) → Spec A^{+}, and there is a natural homeomorphism φ^{−1}(Spec A^{+}/A^{∘∘}) →^{∼} ZR(\underline{A}).

15.5.20. Keep the notation of (15.5.16), and for some integer n ≥ −1, let R_{•} := (R_{i} | i = 0, …, n) be a finite system of open subsets of ZR(\underline{A}) such that ZR(\underline{A}) = ⋃_{i=0}^{n} R_{i} (the case n = −1 occurs for the empty covering of the empty Zariski-Riemann spectrum); let also A′ ⊂ A be a subring, and J′ an ideal of A′. Then we say that R_{•} is an (A′, J′)-standard covering of ZR(\underline{A}) if there exists a finite system f_{•} := (f_{0}, …, f_{n}) of elements of A′ such that :

• R_{i} = R_{A}(\frac{f_{0}}{f_{i}} , …, \frac{f_{n}}{f_{i}} ) ∩ ZR(\underline{A}) for i = 0, …, n

• J′ is contained in the radical of the ideal of A generated by the system f_{•}. Especially, each R_{i} is a rational subset of ZR(\underline{A}). By construction, the retraction r of (15.5.18) restricts to a surjective continuous map

s : Spa \underline{A} → ZR(\underline{A})

and if f_{•} fulfills the foregoing conditions, lemma 15.3.5(i) says that

s^{−1}R_{i} = R_{A}(\frac{f_{0}}{f_{i}} , …, \frac{f_{n}}{f_{i}} ) ∩ Spa \underline{A} for every i = 0, …, n.

Especially, the system (R_{A}(\frac{f_{0}}{f_{i}} , …, \frac{f_{n}}{f_{i}} ) ∩ Spa \underline{A} | i = 0, …, n) is a covering of Spa \underline{A}. A covering of Spa \underline{A} of this type shall also be called a (A′, J′)-standard covering. In case A′ = A and J′ = J, we shall write simply standard covering for this class of coverings of ZR(\underline{A}) and of Spa \underline{A}. The following lemma refines [100, Lemma 2.6].

Lemma 15.5.21. Let j : A′ → A be the inclusion of a subring, and J′ ⊂ A′ a finitely generated ideal with J′A ⊂ J, and such that Spec(j) restricts to an isomorphism of schemes

(15.5.22)  U →^{∼} Spec(A′) \ Spec(A′/J′).

Then every open covering of ZR(\underline{A}) or Spa \underline{A} is refined by an (A′, J′)-standard covering.

Proof. We consider first the case where A′ = A and J′ = J. Let U_{•} := (U_{λ} | λ ∈ Λ) be an open covering of ZR(\underline{A}); we need to find a standard covering (R_{i} | i = 0, …, n) such that, for every i = 0, …, n there exists λ ∈ Λ with R_{i} ⊂ U_{λ}. To this aim, we suppose first that A is topologically local; in light of proposition 15.5.17(i,ii) we may assume that Λ is a finite set, and each U_{λ} is rational, say

U_{λ} = R_{A}(\frac{g_{λ,1}}{g_{λ,0}} , …, \frac{g_{λ,n_{λ}}}{g_{λ,0}} ) ∩ ZR(\underline{A})

for a finite system g_{λ,•} := (g_{λ,j} | j = 0, …, n_{λ}) of elements of A that generates an ideal whose radical contains J. Define S as the set of all the sequences of integers j_{•} := (j_{λ} | λ ∈ Λ) such that 0 ≤ j_{λ} ≤ n_{λ} for every λ ∈ Λ. Let also T ⊂ S be the subset of all sequences j_{•} such that j_{λ} = 0 for at least one λ ∈ Λ. Set g_{j_{•}} := ∏_{λ∈Λ} g_{λ,j_{λ}} for every j_{•} ∈ S. Notice that :

R_{k_{•}} := R_{A}(\frac{g_{j_{•}}}{g_{k_{•}}} | j_{•} ∈ S) = ⋂_{λ∈Λ} R_{A}(\frac{g_{λ,1}}{g_{λ,k_{λ}}} , …, \frac{g_{λ,n_{λ}}}{g_{λ,k_{λ}}} ) for every k_{•} ∈ S.

Moreover, each g_{λ,•} defines a standard open covering of ZR(\underline{A}). It follows easily that

(15.5.23)  ZR(\underline{A}) = ⋃_{k_{•}∈T}(R_{k_{•}} ∩ ZR(\underline{A}))

and furthermore, for every k_{•} ∈ T, the subset R_{k_{•}} ∩ ZR(\underline{A}) is contained in some U_{λ}.

Claim 15.5.24. R_{k_{•}} ∩ ZR(\underline{A}) = R_{A}(\frac{g_{j•}}{g_{k•}} | j_{•} ∈ T ) ∩ ZR(\underline{A}) for every k_{•} ∈ T.

Proof of the claim. Indeed, the inclusion R_{k_{•}} ⊂ R′_{k_{•}} := R_{A}(\frac{g_{j•}}{g_{k•}} | j_{•} ∈ T ) is obvious. Conversely, let v ∈ R′_{k_{•}} ∩ ZR(A); we see from (15.5.23) that for every j_{•} ∈ S there exists i_{•} ∈ T such that v(g_{j_{•}}) ≤ v(i_{•}) ≠ 0, whence v(g_{j_{•}}) ≤ v(k_{•}) ≠ 0, so v ∈ R_{k_{•}}. ♢

In light of (15.5.23) and claim 15.5.24, to conclude it now suffices to show :

Claim 15.5.25. J is contained in the radical of the ideal J′ of A generated by (g_{k_{•}} | k_{•} ∈ T).

Proof of the claim. First, we check that J′ is an open ideal, using the criterion of lemma 15.3.25. Indeed, by claim 15.4.23, every continuous analytic rank one valuation of A lies in Spa \underline{A}, and from (15.5.23) and claim 15.5.24 we see that for every such valuation v there exists k_{•} ∈ T such that v(g_{k_{•}}) ≠ 0, whence the assertion. By lemma 8.3.29(v), it follows that SpecA/J′ ⊂ X_{A}^{∘∘} := Spec A/A^{∘∘}A. Hence, let 𝔭 ∈ X_{A}^{∘∘} \ SpecA/J′; it suffices to show that J′ ⊄ 𝔭. To this aim, denote by v_{𝔭} the trivial valuation of A supported at 𝔭 (see remark 9.1.4(vii)); since 𝔭 is an open ideal (lemma 8.3.29(i)), the valuation v_{𝔭} is continuous (remark 15.3.13(vi)), and then obviously v ∈ Spa \underline{A}. As in the foregoing, we deduce that v_{𝔭}(g_{k_{•}}) ≠ 0 for some k_{•} ∈ T, i.e. g_{k_{•}} ∉ 𝔭, whence the claim. ♢

• Next, let A be any f-adic ring; let us choose a subring of definition A_{0} of A such that A_{0} ⊂ A^{+}, and let I_{0} ⊂ A_{0} be a finitely generated ideal of definition. We apply the foregoing to the topological localization (A, A^{+}, U)_{loc} : notice that proposition 15.5.17(v) naturally identifies U_{•} with an open covering of ZR(\underline{A}_{loc}). By the foregoing, there exists therefore a finite system f_{0}, …, f_{n} of elements of A_{loc} = (1 + I_{0})^{−1}A such that JA_{loc} lies in the radical of ∑_{i=0}^{n} A_{loc}f_{i}, and the standard covering (R_{A_{loc}}(\frac{f_{0}}{f_{i}} , ⋯, \frac{f_{n}}{f_{i}}) ∩ ZR(\underline{A}_{loc})) refines U_{•}. After clearing some denominators, we may assume that f_{0}, …, f_{n} ∈ A, and it follows easily that there exists s ∈ 1 + I_{0} such that sJ lies in the radical of ∑_{i=0}^{n} Af_{i}. Since J is an open ideal, we have I_{0}^{k} ⊂ J for some k ∈ ℕ; then, after replacing k by a larger integer, we also get sI_{0}^{k} ⊂ ∑_{i=0}^{n} Af_{i}. Next, after replacing k by a still larger integer, we can assume that sI_{0}^{k} ⊂ ∑_{i=0}^{n} A_{0}f_{i}. Let also g_{1}, …, g_{m} be a finite system of generators of I_{0}; since A_{0} ⊂ A^{+}, it follows that

max(v(g_{1})^{k}, …, v(g_{m})^{k}) ≤ max(v(f_{0}), …, v(f_{n})) for every v ∈ Spa(A, A^{+})

and since J is in the radical of ∑_{i=0}^{n} Af_{i}, we deduce :

max(v(g_{1})^{k+1}, …, v(g_{m})^{k+1}) < max(v(f_{0}), …, v(f_{n})) for every v ∈ Spa(A, A^{+}, U).

Since s ∈ 1 + I_{0}, it is easily seen that the radical of ∑_{i=0} Af_{i} + ∑_{j=1}^{m} Ag_{j} contains J. Then, consider the standard covering R′_{•} of ZR(\underline{A}) associated with the sequence (f_{•}, g_{•}^{k+1}) : every element of R′_{•} of the form R_{A}(\frac{f_{•}}{f_{i}} , \frac{g_{•}^{k+1}}{f_{i}}) (for any i = 0, …, n) equals R_{A}(\frac{f_{•}}{f_{i}}), and on the other

hand R_{A}(\frac{f_{•}}{g_{j}^{k+1}} , \frac{g_{•}^{k+1}}{g_{j}^{k+1}} ) = ∅ for every j = 1, …, m. Hence, R′_{•} refines U_{•}.

• Lastly, consider any open covering (U′_{λ} | λ ∈ Λ) of Spa \underline{A}, and set U_{λ} := U′_{λ} ∩ ZR(\underline{A}) for every λ ∈ Λ; clearly (U_{λ} | λ ∈ Λ) is an open covering of ZR(\underline{A}), which can be refined by a standard covering (R_{0}, …, R_{n}); but then the system (s^{−1}R_{0}, …, s^{−1}R_{n}) is a standard covering of Spa \underline{A}, and notice that if R_{i} ⊂ U_{λ}, we get

s^{−1}R_{i} ⊂ s^{−1}U_{λ} ⊂ U′_{λ}

where the last inclusion holds, since U′_{λ} is open in Spa \underline{A} and s^{−1}U_{λ} is the set of all primary generizations of the elements of U_{λ} in Spa \underline{A}. Thus, (s^{−1}R_{0}, …, s^{−1}R_{n}) refines the covering (U′_{λ} | λ ∈ Λ), as required.

• We consider next the case of a general pair (A′, J′). By the foregoing, the given covering U_{•} of ZR(\underline{A}) or Spa \underline{A} is refined by the standard covering associated with a sequence g_{0}, …, g_{n} of elements of A. Set I := ∑_{i=0}^{n} Ag_{i}, and let h_{1}, …, h_{m} be a finite system of generators of J′; under the stated assumptions, j induces an isomorphism A′[h_{k}^{−1}] →^{∼} A[h_{k}^{−1}] for every i = 1, …, m, so there exists N ∈ ℕ such that f_{ki} := h_{k}^{N}g_{i} ∈ A′ for every k = 1, …, m and every i = 0, …, n. Let I′ ⊂ A′ be the ideal generated by f_{••} := (f_{ki} | 1 ≤ k ≤ m, 0 ≤ i ≤ n); notice that Spec(A/I′A) = Spec(A/I) ∪ Spec(A/J′A), and by assumption Spec(A/I) ⊂ Spec(A/J) ⊂ Spec(A/J′A). So, Spec(A/I′A) ⊂ Spec(A/J′A), and notice that

Spec(A/I′A) = (Spec j)^{−1}(Spec A′/I′) and Spec(A/J′A) = (Spec j)^{−1}(Spec A′/J′).

In view of the isomorphism (15.5.22), it follows that Spec(A′/I′) ⊂ Spec(A′/J′), i.e. the radical of I′ contains J′. Moreover, by the same token we have :

Spa \underline{A} = ⋃_{j=1}^{m} Spa(A, A^{+}) ∩ R_{A}(\frac{h_{1}^{N}}{h_{j}^{N}} , …, \frac{h_{m}^{N}}{h_{j}^{N}} ).

Taking into account remark 9.2.4(i), we conclude easily that the (A′, J′)-standard covering associated with the sequence f_{••} is the sought refinement of U_{•}. □

Remark 15.5.26. The following constructions shall be exploited in the proofs of both theorem 15.5.27and theorem 15.5.35.

(i) Let \underline{X} be a topologically local quasi-affinoid scheme, set (A, A^{+}, X) := Γ(\underline{X}), and choose a subring of definition A_{0} ⊂ A and a finitely generated ideal I_{0} ⊂ A_{0} of adic definition. Consider a sequence f_{•} := (f_{0}, …, f_{n}) of elements of A that generates an ideal J ⊂ A such that X ∩ Spec A/J = ∅, and let R_{•} := (R_{0}, …, R_{n}) be the standard covering of Spa \underline{X} associated with f_{•}. For every subset Λ ⊂ {0, …, n}, let A_{Λ} := A[f_{i}^{−1} | i ∈ Λ] and X_{Λ} := X ∩ Spec A_{Λ}; by inspecting the definitions we see that

R_{Λ} := ⋂_{i∈Λ} R_{i} = Spa \underline{A}_{Λ} with \underline{A}_{Λ} := (A_{Λ}, A_{Λ}^{+}, X_{Λ})

where A_{Λ}^{+} is the integral closure of A^{+}[f_{k}/f_{i} | (k, i) ∈ {0, …, n} × Λ] in A_{Λ}; moreover, A_{0,Λ} := A_{0}[f_{k}/f_{i} | (k, i) ∈ {0, …, n} × Λ] is a subring of definition of A_{Λ}, and the natural ring homomorphism A_{0} → A_{0,Λ} is adic (here we let R_{∅} := Spa \underline{X}). Furthermore, since X is quasi-compact, we have 𝒪_{X}(X_{Λ}) = A_{Λ} ([59, Ch.I, Prop.9.2.1]), and therefore

\underline{A}_{Λ} = Γ ∘ Spec (\underline{A}_{Λ}) for every Λ ⊂ \{0, …, n\}.

(ii) Denote by (A_{0,Λ}^{h}, I_{0,Λ}^{h}) the henselization of the pair (A_{0,Λ}, I_{0}A_{0,Λ}), and endow A_{0,Λ}^{h} with the I_{0,Λ}^{h}-adic topology; we deduce a unique isomorphism of topological A-algebras :

ω_{Λ}^{h} : A_{Λ}^{h} := A_{Λ} ⊗_{A_{0,Λ}} A_{0,Λ}^{h} →^{∼} 𝒪_{Spa \underline{X}}^{h}(R_{Λ}) for every Λ ⊂ \{0, …, n\}

where the topology on A_{Λ}^{h} is the unique one such that the natural map A_{0,Λ}^{h} → A_{Λ}^{h} is open (proposition 8.3.34(ii)). Likewise, let A_{Λ}^{∧} be the separated completion of A_{Λ}; we deduce a natural continuous homomorphism of A-algebras

ω_{Λ}^{∧} : A_{Λ}^{∧} → 𝒪_{Spa \underline{X}}^{∧}(R_{Λ}) for every Λ ⊂ \{0, …, n\}.

Notice also that for every Λ′ ⊂ Λ there exist unique continuous homomorphisms of A-algebras and respectively A_{0,Λ}-algebras

μ_{Λ′,Λ} : A_{Λ′} → A_{Λ} μ_{0,Λ′,Λ}^{h} : A_{0,Λ′}^{h} → A_{0,Λ}^{h}

and set μ_{Λ′,Λ}^{h} := μ_{Λ′,Λ} ⊗_{A} μ_{0,Λ′,Λ}^{h}. Let also μ_{Λ′,Λ}^{∧} : A_{Λ′}^{∧} → A_{Λ}^{∧} be the completion of μ_{Λ′,Λ}, and

ρ_{Λ′,Λ}^{h} : 𝒪_{Spa \underline{X}}^{h}(R_{Λ′}) → 𝒪_{Spa \underline{X}}^{h}(R_{Λ}) ρ_{Λ′,Λ}^{∧} : 𝒪_{Spa \underline{X}}^{∧}(R_{Λ′}) → 𝒪_{Spa \underline{X}}^{∧}(R_{Λ})

the restriction homomorphisms associated with the inclusion R_{Λ} ⊂ R_{Λ′}.

(iii) We claim that the diagrams of A-algebras:

\xymatrix{A_{Λ′}^{h} \ar[r]^{ω_{Λ′}^{h}} \ar[d]_{μ_{Λ′,Λ}^{h}} & 𝒪_{Spa \underline{X}}^{h}(R_{Λ′}) \ar[d]^{ρ_{Λ′,Λ}^{h}} & A_{Λ′}^{∧} \ar[r]^{ω_{Λ′}^{∧}} \ar[d]_{μ_{Λ′,Λ}^{∧}} & 𝒪_{Spa \underline{X}}^{∧}(R_{Λ′}) \ar[d]^{ρ_{Λ′,Λ}^{∧}} \\ A_{Λ}^{h} \ar[r]^{ω_{Λ}^{h}} & 𝒪_{Spa \underline{X}}^{h}(R_{Λ}) & A_{Λ}^{∧} \ar[r]^{ω_{Λ}^{∧}} & 𝒪_{Spa \underline{X}}^{∧}(R_{Λ})}

commute for every Λ′ ⊂ Λ ⊂ {0, …, n}. Indeed, a simple inspection shows that μ_{Λ′,Λ}^{h} is continuous. On the other hand, all the arrows in the diagram are A-algebra homomorphisms, therefore also A_{Λ′}-algebra homomorphisms, and by the universal property of the topological henselization there is a unique continuous A_{Λ′}-algebra homomorphism A_{Λ′}^{h} → 𝒪_{Spa \underline{X}}^{h}(R_{Λ}), whence the contention for the left diagram. For the right diagram one argues similarly, using the universal property of the separated completions.

(iv) Endow A[Y] and B := A_{0}[T_{0}, …, T_{n}] with the standard ℕ-gradings such that gr_{k}A[Y] = AY^{k} and gr_{k}B is the free A-module generated by the monomials T_{0}^{ν_{0}} ⋯ T_{n}^{ν_{n}} of total degree ν_{0} + ⋯ + ν_{n} = k, for every k ∈ ℕ. Set as well

S := Spec A_{0} ℙ_{S}^{n} := Proj B

and recall that ℙ_{S}^{n} admits the following finite affine open covering (see (10.6.1)). For every i = 0, …, n we have the subring B_{i} := A_{0}[T_{0}/T_{i}, …, T_{n}/T_{i}] of B[1/T_{i}], and Ω_{i} := Spec B_{i} is naturally identified with the open subscheme of ℙ_{S}^{n} consisting of all homogeneous prime ideals of B that do not contain T_{i}; then

ℙ_{S}^{n} = Ω_{0} ∪ ⋯ ∪ Ω_{n}.

Next, according to (10.6.5), the homomorphism of ℕ-graded A_{0}-algebras

h : B → A[Y]  T_{j} ↦ f_{j}Y  for every j = 0, …, n

induces a morphism of A_{0}-schemes

φ : Spec A \ Spec A/J → ℙ_{S}^{n}.

Explicitly, φ^{−1}Ω_{i} = Spec A_{{i}} for every i = 0, …, n, and the restriction φ^{−1}Ω_{i} → Ω_{i} of φ corresponds to the homomorphism of A_{0}-algebras

h_{i} : B_{i} → A_{{i}}  T_{j}/T_{i} ↦ f_{j}/f_{i}  for every j = 0, …, n.

(v) Now, quite generally, let β : X → Y be any quasi-compact and quasi-separated morphism of schemes. Then β_{∗}𝒪_{X} is a quasi-coherent 𝒪_{Y}-module ([59, Ch.I, Prop.9.2.1]), hence ℐ := Ker β^{♯} : 𝒪_{Y} → β_{∗}𝒪_{X} is a quasi-coherent sheaf of ideals of 𝒪_{Y}, and therefore Spec 𝒪_{Y}/ℐ is a well defined closed subscheme of Y, called the schematic image of ψ. From the definition, it is clear that the construction of the schematic image is local on Y : namely, if Y′ ⊂ Y is any open subscheme, and X′ := β^{−1}Y′, then the schematic image of β_{|X′} : X′ → Y′ is the intersection of Y′ with the schematic image of β.

Thus, if V is the schematic image of our φ, we get a commutative diagram of schemes

\xymatrix{X \ar[r]^{ψ} \ar[d] & V \ar[d]^{ι} \\ Spec A \ Spec A/J \ar[r]^{φ} \ar[ur] & ℙ_{S}^{n}}

whose left vertical arrow is an open immersion, and with ι a closed immersion. Lastly, set V_{∅} := V and for every subset Λ ⊂ {0, …, n} let V_{Λ} := ι^{−1}(⋂_{i∈Λ} Ω_{i}); then V_{Λ} is naturally identified with Spec A_{0,Λ}, for every non-empty Λ.

(vi) Furthermore, let U be the analytic locus of S; recall that the morphism Spec A → S (associated with the inclusion map A_{0} → A) identifies U with the analytic locus of Spec A (lemma 8.3.29(iii)), and U is an open subset of X, under this identification. Then ℙ_{U}^{n} := ℙ_{S}^{n} ×_{S} U is an open subset of ℙ_{S}^{n}, and we claim that ψ restricts to an isomorphism of schemes

ψ_{|U} : U →^{∼} V ∩ ℙ_{U}^{n}.

Indeed, ψ_{|U} is a closed immersion, by [59, Ch.I, Cor.5.4.6]. On the other hand, ψ_{|U} is also the schematic closure of the restriction U → ℙ_{U}^{n} of φ, whence the claim.

Theorem 15.5.27. For any topologically local quasi-affinoid scheme \underline{X} := (X, 𝒯_{X}, A^{+}), the presheaves 𝒪_{Spa \underline{X}}^{h} and 𝒪_{Spa \underline{X}}^{h+} are sheaves of rings on (𝒬(\underline{X}), J_{𝒬} ).

Proof. Notice first that – by virtue of corollary 15.4.27(i) – if 𝒪_{Spa \underline{X}}^{h} is a sheaf, the same holds for 𝒪_{Spa \underline{X}}^{h+}, so it suffices to check the assertion for 𝒪_{Spa \underline{X}}^{h}. To this aim, for every U ∈ Ob(𝒬(\underline{X})) and every 𝒮 ∈ J_{𝒬} (U), let h_{𝒮} be the corresponding sub-presheaf of the presheaf h_{U} on 𝒬(\underline{X}) represented by U. According to claim 4.1.16(ii,iii), it suffices to show that the natural map

r_{U} : 𝒪_{Spa \underline{X}}^{h}(U) → colim_{𝒮 ∈J_{𝒬} (U)} Hom_{𝒬(\underline{X})^{∧}} (h_{𝒮} , 𝒪_{Spa \underline{X}}^{h})

is a bijection for every such U. However, let φ_{\underline{T} /\underline{X}} : \underline{T} → \underline{X} be any morphism of topologically local quasi-affinoid schemes representing the inclusion h_{U} ⊂ h\underline{X} (see (15.5.7)); by remark 15.5.9(v) it follows that φ_{\underline{T} /\underline{X}} induces an equivalence of categories

𝒬(\underline{T}) →^{∼} 𝒬(\underline{X})/U V ↦ Spa φ_{\underline{T} /\underline{X}}(V) ⊂ U

so we may replace \underline{X} by \underline{T}, and reduce to checking that r_{Spa \underline{X}} is an isomorphism. Next, lemma 15.5.21 implies that the sieves of 𝒬(\underline{X}) generated by the standard coverings of Spa \underline{X} form a coinitial subset of J_{𝒬} (Spa \underline{X}). Hence, let f_{•} and R_{•} be as in remark 15.5.26(i); taking into account the discussion of (4.1.6), we are then reduced to checking that the natural map

𝒪_{Spa \underline{X}}^{h}(Spa \underline{X}) → Equal(∏_{i=0}^{n} [figure: 𝒪_{Spa \underline{X}}^{h}(R_{i})]∏_{i,j=0}^{n} 𝒪_{Spa \underline{X}}^{h}(R_{\{i,j\}}))

is a bijection. Now, say that (A, A^{+}, X) = Γ(\underline{X}). Recall that the henselization morphism

λ : \underline{X}^{h} := (X^{h}, 𝒯_{X}^{h}, A^{h+}) → \underline{X}

is f-adic and induces a homeomorphism Spa λ : Spa \underline{X}^{h} →^{∼} Spa \underline{X}, as well as an isomorphism of schemes X^{h} →^{∼} X ×_{A} A^{h}; it follows easily that the system ((Spa λ)^{−1}R_{i} | i = 0, …, n) is a rational covering of Spa \underline{X}^{h}, and moreover there is a unique isomorphism of presheaves of topological A-algebras

𝒪_{Spa \underline{X}^{h}}^{h} →^{∼} (Spa λ)^{−1}𝒪_{Spa \underline{X}}^{h}.

We may then replace \underline{X} by \underline{X}^{h}, and assume from start that \underline{X} is topologically henselian.

In this situation, let A_{0} ⊂ A be a subring of definition, and I_{0} ⊂ A_{0} an ideal of adic definition; consider the system of quasi-affinoid rings (\underline{A}_{Λ} | Λ ⊂ \{0, …, n\}), their rings of definition A_{0,Λ} ⊂ A_{Λ}, and the systems of their topological henselizations (A_{Λ}^{h} | Λ ⊂ {0, …, n}) and (A_{0,Λ}^{h} | Λ ⊂ {0, …, n}) constructed in remark 15.5.26(i). Define also S, V, ψ : X → V and V_{Λ} for every subset Λ ⊂ {0, …, n} as in remark 15.5.26(iv,v); set S′ := Spec A_{0}/I_{0}, V′_{Λ} := V_{Λ} ×_{S} S′ for every such Λ, and consider the cartesian diagram of schemes

\xymatrix{V′ \ar[r]^{τ_{V}} \ar[d]_{π′} & V \ar[d]^{π} \\ S′ \ar[r]^{τ_{S}} & S}

where π is the restriction to V of the projection ℙ_{S}^{n} → S and τ_{S} : S′ → S is the closed immersion. For every scheme X, let X_{ét} be the étale site of X, and 𝒪_{X_{ét}} the structure sheaf of rings on X_{ét}. We define the sheaf on V′_{ét}

ℱ := τ_{V}^{∗} ψ_{∗}𝒪_{X_{ét}}.

Claim 15.5.28. (i) For every Λ ⊂ {0, …, n} there exists a commutative diagram of rings

𝒟_{Λ} : \xymatrix{A_{0,Λ}^{h} \ar[r]^{ω_{0,Λ}} \ar[d] & Γ(V′_{Λ}, τ_{V}^{∗} 𝒪_{V,ét}) \ar[d] \\ A_{Λ}^{h} \ar[r]^{ω_{Λ}} & Γ(V′_{Λ}, ℱ)}

whose left vertical arrow is the inclusion map, and whose right vertical arrow is deduced from ψ^{♯} : 𝒪_{V,ét} → ψ_{∗}𝒪_{X,ét}. Moreover, ω_{0,Λ} is an isomorphism of A_{0}-algebras for every Λ ≠ ∅, and ω_{Λ} is an isomorphism of A-algebras for every Λ.

(ii) For every Λ′ ⊂ Λ, let ρ_{Λ′,Λ} : Γ(V′_{Λ′}, ℱ) → Γ(V′_{Λ}, ℱ) be the restriction homomorphism associated with the inclusion V′_{Λ} ⊂ V′_{Λ′}. Then we have a commutative diagram

\xymatrix{A_{Λ′}^{h} \ar[r]^{ω_{Λ′}} \ar[d]_{μ_{Λ′,Λ}^{h}} & Γ(V′_{Λ′}, ℱ) \ar[d]^{ρ_{Λ′,Λ}} \\ A_{Λ}^{h} \ar[r]^{ω_{Λ}} & Γ(V′_{Λ}, ℱ).}

Proof of the claim. (i): We consider first the case where Λ = ∅, and notice that V_{∅} = V, A_{0,∅}^{h} = A_{0} and A_{∅}^{h} = A. Under the current assumptions, the pair (A_{0}, I_{0}) is henselian; by [10, Exp.XII, Th.5.1(i) and Prop.6.5(i)] we then get for every sheaf 𝒢 on V_{ét} natural isomorphisms

Γ(V, 𝒢) →^{∼} Γ(S, π_{∗}𝒢) →^{∼} Γ(S′, τ_{S}^{∗}π_{∗}𝒢) →^{∼} Γ(S′, π′_{∗}τ_{V}^{∗} 𝒢) →^{∼} Γ(V′, τ_{V}^{∗} 𝒢).

Taking 𝒢 := ψ_{∗}𝒪_{X_{ét}}, we then obtain the sought isomorphism ω_{∅}

A = Γ(X, 𝒪_{X_{ét}}) →^{∼} Γ(V, ψ_{∗}𝒪_{X_{ét}}) →^{∼} Γ(V′, ℱ).

Likewise, the map ω_{0,∅} is defined as the composition

A_{0} = Γ(S, 𝒪_{S}) → Γ(S, π_{∗}𝒪_{V}) →^{∼} Γ(V′, τ_{V}^{∗} 𝒪_{V})

from which the commutativity of 𝒟_{∅} is straightforward. Next, let Λ ≠ ∅, so that V_{Λ} is an affine scheme; set V_{Λ}^{h} := Spec A_{0,Λ}^{h} and X_{Λ} := X ∩ Spec A_{Λ}. We let ψ_{Λ} : X_{Λ} → V_{Λ} be the restriction of ψ, and we consider the cartesian diagram of schemes

\xymatrix{ & X_{Λ}^{h} \ar[r]^{τ′′_{X_{Λ}}} \ar[d]_{ψ_{Λ}^{h}} & X_{Λ} \ar[d]^{ψ_{Λ}} \\ V′_{Λ} \ar[r]^{τ′_{V_{Λ}}} & V_{Λ}^{h} \ar[r]^{τ′′_{V_{Λ}}} & V_{Λ}}

where τ′_{V_{Λ}} and τ′′_{V_{Λ}} are induced by the natural projection A_{0,Λ}^{h} → A_{0,Λ}^{h}/I_{0,Λ}^{h} →^{∼} A_{0,Λ}/I_{0}A_{0,Λ}, and respectively the henselization map A_{0,Λ} → A_{0,Λ}^{h}. Then ω_{Λ} is defined as the composition of isomorphisms of A-algebras

A_{Λ}^{h} = Γ(X_{Λ}^{h}, 𝒪_{X_{Λ,ét}^{h}}) →^{∼} Γ(V_{Λ}^{h}, ψ_{Λ∗}^{h} ∘ τ_{X_{Λ}}^{′′∗}𝒪_{X_{Λ,ét}}) →^{∼} Γ(V_{Λ}^{h}, τ_{V_{Λ}}^{′′∗} ∘ ψ_{Λ∗}𝒪_{X_{Λ,ét}})
→^{∼} Γ(V′_{Λ}, τ_{V_{Λ}}^{′∗} ∘ τ_{V_{Λ}}^{′′∗} ∘ ψ_{Λ∗}𝒪_{X_{Λ,ét}})
→^{∼} Γ(V′_{Λ}, ℱ)

where the bijectivity of the third map is due to [10, Exp.XII, Prop.6.5(i)], and that of the second one is due to the fact that τ′′_{V_{Λ}} is the limit of a cofiltered system of affine étale V_{Λ}-schemes. Likewise, ω_{0,Λ} is obtained as the composition

A_{0,Λ}^{h} = Γ(V_{Λ}^{h}, 𝒪_{V_{Λ}^{h}}) →^{∼} Γ(V′_{Λ}, τ_{V_{Λ}}^{′∗}𝒪_{V_{Λ}^{h}}) →^{∼} Γ(V′_{Λ}, τ_{V}^{∗} 𝒪_{V}).

Then the commutativity of 𝒟_{Λ} follows by a direct inspection.

(ii): We consider the diagram of A_{0}-algebras

\xymatrix{A_{0,Λ′}^{h} \ar[r]^{ω_{0,Λ′}} \ar[d]_{μ_{0,Λ′,Λ}^{h}} & Γ(V′_{Λ′}, τ_{V}^{∗} 𝒪_{V,ét}) \ar[r] \ar[d] & Γ(V′_{Λ′}, ℱ) \ar[d]_{ρ_{Λ′,Λ}} & Γ(V_{Λ′}, ψ_{∗}𝒪_{X,ét}) = A_{Λ′} \ar[l] \ar[d]^{μ_{Λ′,Λ}} \\ A_{0,Λ}^{h} \ar[r]^{ω_{0,Λ′}} & Γ(V′_{Λ}, τ_{V}^{∗} 𝒪_{V,ét}) \ar[r] & Γ(V′_{Λ}, ℱ) & Γ(V_{Λ}, ψ_{∗}𝒪_{X,ét}) = A_{Λ} \ar[l]}

whose central square subdiagram is induced by the natural morphism 𝒪_{V,ét} → ψ_{∗}𝒪_{X,ét}, and the right square subdiagram is induced by the unit of adjunction ψ_{∗}𝒪_{X,ét} → τ_{V ∗}τ_{V}^{∗} ψ_{∗}𝒪_{X,ét} for the adjoint pair (τ_{V}^{∗} , τ_{V ∗}), and where μ_{Λ′,Λ} and μ_{0,Λ′,Λ}^{h} are defined as in remark 15.5.26(ii). Therefore, both these subdiagrams commute; the left square subdiagram commutes as well, due to the universal property of henselization; taking into account (i), the assertion follows (details left to the reader). ♢

Lastly, we have a natural map

A ⟶^{α} Equal(∏_{i=0}^{n} [figure: A_{{i}}^{h}; β; γ]∏_{i,j=0}^{n} A_{{i,j}}^{h})

induced by the maps μ_{∅,{i}}^{h}, where β and γ are deduced from the maps μ_{{i},{i,j}}^{h}. From claim 15.5.28 we deduce that α is an isomorphism, and taking into account remark 15.5.26(iii), the theorem follows. □

Remark 15.5.29. Let \underline{X} be any topologically local quasi-affinoid scheme.

(i) As explained in [59, Ch.0, §3.2.2], every sheaf on 𝒬(\underline{X}) admits a natural extension to a sheaf ℱ′ on the topological space Spa \underline{X}. Namely, if U ⊂ Spa \underline{X} is any open subset, one sets

ℱ′(U) := lim_{V ∈(𝒬(\underline{X})/U)^{o}} ℱ (V)

where 𝒬(\underline{X})/U denotes the full subcategory of 𝒬(\underline{X}) whose objects are the quasi-affinoid open subsets of Spa \underline{X} contained in U. We shall use the notation 𝒪_{Spa \underline{X}}^{h} and 𝒪_{Spa \underline{X}}^{h+} also to refer to the sheaves on Spa \underline{X} that extend naturally the respective sheaves on 𝒬(\underline{X}).

(ii) Simple examples show that, in most cases, the presheaves 𝒪_{Spa \underline{X}}^{loc} and 𝒪_{Spa \underline{X}}^{loc+} are not sheaves on the site 𝒬(\underline{X}). On the other hand, in several situations of interest the presheaves 𝒪_{Spa \underline{X}}^{∧} and 𝒪_{Spa \underline{X}}^{∧+} are sheaves of topological rings. In Huber’s work [98], this sheaf property is proven under two different types of assumptions, corresponding roughly to the two main classes of non-archimedean analytic spaces that are encountered in applications : the “generic fibres” of locally noetherian formal schemes, and the rigid analytic varieties locally of finite type over a complete rank one valued field. In the literature, one finds also some attempts to unify these cases (and the few other known ones) under a single axiomatic framework; hereafter we shall adopt the approach (though, not the terminology) proposed in [71], that is based on the notions introduced in section 15.2. Later, we shall prove the sheaf property also in the case where 𝒪_{X}(X) admits a perfectoid ring of definition : this is a completely different situation, for which we shall need to develop an ad hoc method.

Definition 15.5.30. (i) Let A be any f-adic ring. We say that A is f-adic analytically noetherian (resp. f-adic universally analytically noetherian), if it admits an analytically noetherian (resp. universally analytically noetherian) ring of definition (see definition 15.2.1(i)).

(ii) Let \underline{A} := (A, A^{+}) be any affinoid ring. We say that \underline{A} is analytically noetherian (resp. universally analytically noetherian) if A is f-adic analytically noetherian (resp. f-adic universally analytically noetherian).

(iii) Let \underline{A} := (A, A^{+}, U) be any quasi-affinoid ring. We say that \underline{A} is analytically noetherian (resp. universally analytically noetherian) if the same holds for (A, A^{+}).

(iv) Let \underline{X} be any quasi-affinoid scheme. We say that \underline{X} is analytically noetherian (resp. universally analytically noetherian) if the same holds for Γ(\underline{X}).

15.5.31. Let A be any f-adic ring, and n ∈ ℕ any integer. According to proposition 8.3.37(i,ii), there exists a unique f-adic topology 𝒯_{n} on A[T_{1}, …, T_{n}] such that, if A_{0} is any subring of definition of A, then A_{0}[T_{1}, …, T_{n}] is a subring of definition of A[T_{1}, …, T_{n}], and if I ⊂ A_{0} is an ideal of adic definition, then I[T_{1}, …, T_{n}] is an ideal of adic definition for A_{0}[T_{1}, …, T_{n}].

Lemma 15.5.32. With the notation of (15.5.31), the following holds :

(i) A is f-adic analytically noetherian (resp. f-adic universally analytically noetherian) if and only if every ring of definition of A is analytically noetherian (resp. universally analytically noetherian).

(ii) A is f-adic universally analytically noetherian if and only if (A[T_{1}, …, T_{n}], 𝒯_{n}) is f-

adic analytically noetherian for every n ∈ ℕ. (iii) Let B ⊂ A be any open subring, and endow B with the topology induced from A. Then

A is f-adic analytically noetherian (resp. f-adic universally analytically noetherian) if and only if the same holds for B.

(iv) Let f : A → A′ be a surjective ring homomorphism, and endow A′ with the topology induced by A via f. Then, if A is f-adic analytically noetherian (resp. f-adic universally analytically noetherian), the same holds for A′.

(v) If \underline{A} is any f-adic analytically noetherian (resp. f-adic universally analytically noetherian) quasi-affinoid ring, Spec \underline{A} is an analytically noetherian (resp. universally analytically noetherian) quasi-affinoid scheme.

(vi) Let \underline{X} be a topologically local and universally analytically noetherian quasi-affinoid scheme, and \underline{Y} → \underline{X} a morphism of topologically local quasi-affinoid schemes that represents a rational subset of Spa \underline{X}. Then \underline{Y} is universally analytically noetherian.

Proof. (i): Let A_{0}, A_{1} ⊂ A be two rings of definition, and let us show that A_{0} is analytically noetherian if and only if the same holds for A_{1}. To this aim, suppose first that A_{0} ⊂ A_{1}, and let I ⊂ A_{0} be any finitely generated ideal of adic definition, so that IA_{1} is an ideal of adic definition for A_{1}. Lemma 8.3.29(iii) says that the inclusion map A_{0} → A_{1} identifies the analytic locus U_{0} of Spec A_{0} with the analytic locus U_{1} of Spec A_{1}, so U_{0} is a noetherian scheme if and only if the same holds for U_{1}. Next, suppose that A_{0} is analytically noetherian; let M_{1} be any A_{1}-module of finite type, x_{1}, …, x_{n} any system of generators for M_{1}, and set M_{0} := A_{0}x_{1} + ⋯ + A_{0}x_{n} ⊂ M_{1}. By proposition 8.3.18(ii), there exists n ∈ ℕ such that I^{n}A_{1} ⊂ A_{0}, whence I^{n}M_{1} ⊂ M_{0}. On the other hand, by assumption there exists r ∈ ℕ such that Ann_{M_{0}}(I^{k}) = Ann_{M_{0}}(I^{r}) for every k ≥ r. Thus, let x ∈ Ann_{M_{1}}(I^{k}A_{1}) for some k ≥ r; we get I^{n}x ∈ Ann_{M_{1}}(I^{k}A_{1}) ∩ M_{0} = Ann_{M_{0}}(I^{k}), so I^{n+r}x = 0, as asserted, in this case.

Suppose next that A_{1} is analytically noetherian, and let M_{0} be any A_{0}-module of finite type; the short exact sequence of A_{0}-modules 0 → A_{0} → A_{1} → A_{1}/A_{0} → 0 induces an exact sequence

T_{1} := Tor_{1}^{A_{0}}(A_{1}/A_{0}, M_{0}) → M_{0} ⟶^{φ} M_{1} := A_{1} ⊗_{A_{0}} M_{0}

and notice that I^{n}T_{1} = 0. By assumption, there exists r ∈ ℕ such that Ann_{M_{1}}(I^{k}A_{1}) = Ann_{M_{1}}(I^{r}A_{1}) for every k ≥ r. Suppose then that x ∈ Ann_{M_{0}}(I^{k}) for some k ≥ r; it follows easily that I^{r}x ⊂ Ker φ, whence I^{r+n}x = 0, as required.

Lastly, if A_{0} and A_{1} are arbitrary rings of definition for A, the same holds for A_{2} := A_{0} · A_{1} (corollary 8.3.19(i)). Suppose that A_{0} is analytically noetherian; by the foregoing, it follows that the same holds for A_{2}, and then by the same token it holds for A_{1} as well.

The assertion for the universally noetherian case is an immediate consequence.

Assertion (ii) follows directly from (i) : details left to the reader.

(iii) follows from (i) and corollary 8.3.19(ii).

(iv): First, we know that A′ is f-adic, by example 8.3.27(iii), and if A_{0} ⊂ A is a ring of definition, f(A_{0}) is a ring of definition of A′. But if A_{0} is analytically noetherian (resp. universally analytically noetherian), the same holds for f(A_{0}), by lemma 15.2.4(ii). The assertion follows.

(v): Say that \underline{A} = (A, A^{+}, U), and let ρ : A → A_{U} := 𝒪_{Spec A}(U) be the restriction map. By definition, the topology 𝒯_{ρ(A)} on ρ(A) induced by the inclusion into A_{U} agrees with the quotient topology induced by the surjection A → ρ(A). From (iv) we deduce that (ρ(A), 𝒯_{ρ(A)}) is f-adic analytically noetherian (resp. f-adic universally analytically noetherian). Then, the assertion follows from (iii).

(vi) follows easily from (v) and lemma 15.2.4(ii), after inspecting the proof of theorem 15.5.5. □

Lemma 15.5.33. Let \underline{X} := (X, 𝒯_{X}, A_{X}^{+}) be any analytically noetherian quasi-affinoid scheme. Then the natural morphism Γ(\underline{X})^{∧} → Γ(\underline{X}^{∧}) is an isomorphism (see remark 15.4.13(ii)).

Proof. Let A_{X}^{∧} be the separated completion of A_{X} := 𝒪_{X}(X), and set Y := Spec A_{X}^{∧} ×_{Spec A_{X}} X; in light of proposition 15.2.10(ii) and corollary 10.3.8, the natural map A_{X}^{∧} → 𝒪_{Y} (Y) is an isomorphism. Then the assertion follows from remark 15.4.13(ii). □

15.5.34. For any topologically local quasi-affinoid scheme \underline{X}, we may consider the site

(ℛ(\underline{X}), J_{ℛ} )

where ℛ(\underline{X}) is the full subcategory of 𝒬(\underline{X}) whose objects are the rational subsets of Spa \underline{X}. The sieves covering a given object R of ℛ(\underline{X}) for the topology J_{ℛ} are those generated by the families R_{•} := (R_{λ} | λ ∈ Λ) of rational subsets of Spa \underline{X} such that ⋃_{λ∈Λ} R_{λ} = R. For every such R_{•}, after fixing a total ordering on Λ we get an augmented alternating Čech complex C_{alt}^{•}(R_{•}, 𝒪_{Spa \underline{X}}^{∧}), whose degree n term, for every n ≥ 0, is a product of topological rings of the form 𝒪_{Spa \underline{X}}^{∧}(R_{λ_{0}} ∩⋯∩R_{λ_{n}}), with λ_{0} < ⋯ < λ_{n} ranging over all strictly increasing sequences of elements of Λ (see remark 10.2.7(i). We endow C_{alt}^{n}(R_{•}, 𝒪_{Spa \underline{X}}^{∧}) with the corresponding product topology, and we notice that this topology is independent of the chosen ordering for Λ, and the differentials d^{•} of the Čech complex are continuous maps. With this notation, we have:

Theorem 15.5.35. For any topologically local and universally analytically noetherian quasiaffinoid scheme \underline{X} := (X, 𝒯_{X}, A_{X}^{+}), and every R ∈ Ob(ℛ(\underline{X})), the following holds :

(i) The presheaves 𝒪_{Spa \underline{X}}^{∧+} and 𝒪_{Spa \underline{X}}^{∧} are sheaves of topological rings on (ℛ(\underline{X}), J_{ℛ} ). (ii) For every finite covering 𝔘 of R consisting of rational subsets, the differentials of

C_{alt}^{•}(𝔘, 𝒪_{Spa \underline{X}}^{∧}) are strict and its cohomology has the discrete topology.

Proof. (i): Arguing as in the proof of theorem 15.5.27, we reduce to checking the assertion for 𝒪_{Spa \underline{X}}^{∧}. We remark :

Claim 15.5.36. For any given R ∈ Ob(ℛ(\underline{X})), let φ_{\underline{Y} /\underline{X}} : \underline{Y} → \underline{X} be any morphism of quasiaffinoid schemes representing the sub-presheaf h_{R} of h\underline{X}; then the morphism φ_{\underline{Y} /\underline{X}} induces an equivalence of categories

ℛ(\underline{Y}) →^{∼} ℛ(\underline{X})/R V ↦ Spa φ_{\underline{Y} /\underline{X}}(V) ⊂ R.

Moreover, \underline{Y} is still universally analytically noetherian.

Proof of the claim. These assertions follow from lemmata 15.5.10and 15.5.32(vi). ♢

Now, say that Γ(\underline{X}) = (A, A^{+}, X) and let R_{•} := (R_{0}, …, R_{n}) be the standard covering of Spa \underline{X} attached to a given sequence f_{•} := (f_{0}, …, f_{n}) of elements of A; define also the rational subset R_{Λ} as in remark 15.5.26(i), for every subset Λ ⊂ {0, …, n}. In light of claim 15.5.36 and remark 5.5.2(i), we are reduced to showing that the natural map

(15.5.37)  𝒪_{Spa \underline{X}}^{∧}(Spa \underline{X}) → Equal(∏_{i=0}^{n} [figure: 𝒪_{Spa \underline{X}}^{∧}(R_{i})]∏_{i,j=0}^{n} 𝒪_{Spa \underline{X}}^{∧}(R_{\{i,j\}}))

is an isomorphism of topological rings. To this aim, we pick a subring of definition A_{0} ⊂ A, a finitely generated ideal I_{0} ⊂ A_{0} of adic definition, we set S := Spec A_{0}, we denote by J ⊂ A the ideal generated by f_{•}, and we consider the system (\underline{A}_{Λ} | Λ ⊂ \{0, …, n\}) of quasi-affinoid rings, and their rings of definition A_{0,Λ} ⊂ A_{Λ} constructed in remark 15.5.26(i). Let also A_{Λ}^{∧} be the separated completion of A_{Λ}, and notice that the map

ω_{Λ}^{∧} : A_{Λ}^{∧} → 𝒪_{Spa \underline{X}}^{∧}(R_{Λ})

of remark 15.5.26(ii) is an isomorphism of topological A-algebras for every Λ ⊂ {0, …, n}, by lemma 15.5.33. Define S, V, ψ : X → V and V_{Λ} for every subset Λ ⊂ {0, …, n} as in remark 15.5.26(v); set also S′ := Spec A_{0}/I_{0} and let V^{∧} be the completion of V along S′ ×_{S} V, and π : V^{∧} → V the induced morphism of locally ringed spaces. We define

ℱ := π^{∗}ψ_{∗}𝒪_{X}  and V_{Λ}^{∧} := π^{−1}V_{Λ}  for every Λ ⊂ {0, …, n}

and we regard ℱ as a presheaf of topological abelian groups, as follows. The natural morphism 𝒪_{V} → ψ_{∗}𝒪_{X} induces a morphism

φ : 𝒪_{V^{∧}} → ℱ

and for every open subset U ⊂ V^{∧} we endow ℱ (U) with the unique group topology such that the resulting map φ_{U} : 𝒪_{V^{∧}}(U) → ℱ (U) is continuous and open. It follows easily that for every inclusion U′ ⊂ U of open subsets of V^{∧}, the restriction map ℱ (U) → ℱ (U′) is continuous.

Claim 15.5.38. (i) ℱ (V_{Λ}^{∧}) is a topological ring for every Λ ⊂ {0, …, n}, and there exists a natural isomorphism of topological A-algebras

ω_{Λ} : A_{Λ}^{∧} →^{∼} ℱ (V_{Λ}^{∧}).

(ii) For every Λ′ ⊂ Λ, let ρ_{Λ′,Λ} : ℱ (V_{Λ′}^{∧}) → ℱ (V_{Λ}^{∧}) be the restriction homomorphism associated with the inclusion V_{Λ}^{∧} ⊂ V_{Λ′}^{∧}. Then we have a commutative diagram

\xymatrix{A_{Λ′}^{∧} \ar[r]^{ω_{Λ′}} \ar[d]_{μ_{Λ′,Λ}^{∧}} & ℱ (V_{Λ′}^{∧}) \ar[d]^{ρ_{Λ′,Λ}} \\ A_{Λ}^{∧} \ar[r]^{ω_{Λ}} & ℱ (V_{Λ}^{∧}).}

Proof of the claim. (i): We consider first the case where Λ = ∅, and recall that V_{∅} = V. Notice also that ψ_{∗}𝒪_{X} is a quasi-coherent 𝒪_{V} -module, by virtue of [59, Ch.I, Cor.9.2.2]. Since V is a projective S-scheme, corollary 15.2.34(iii) and remark 15.2.38yield a natural isomorphism

ω_{∅} : A^{∧} →^{∼} A_{0}^{∧} ⊗_{A_{0}} A →^{∼} A_{0}^{∧} ⊗_{A_{0}} Γ(V, ψ_{∗}𝒪_{X}) →^{∼} ℱ (V^{∧})

fitting into a commutative diagram

\xymatrix{A_{0}^{∧} \ar[r] \ar[d] & A^{∧} \ar[d]^{ω_{∅}} \\ 𝒪_{V^{∧}}(V^{∧}) \ar[r]^{φ_{V^{∧}}} & ℱ (V^{∧})}

whose left vertical arrow is the I_{0}-adic completion of the natural map A_{0} → 𝒪_{V} (V), and whose top horizontal arrow is the inclusion map. It follows already that ω_{∅} is a continuous map, and it thus remains only to check that ω_{∅} is an open map. To this aim, it suffices to check that the same holds for the left vertical arrow A_{0}^{∧} → 𝒪_{V^{∧}}(V^{∧}). However, 𝒪_{V} (V) is an A_{0}- module of analytically finite type, by theorem 15.2.22, and the support of 𝒪_{V} (V)/A_{0} lies in the non-analytic locus of S, by virtue of remark 15.5.26(vi); by remark 15.2.2(iii) we deduce that I_{0}^{t} · 𝒪_{V} (V) ⊂ A_{0} for some t ∈ ℕ, whence I_{0}^{t} · 𝒪_{V^{∧}}(V^{∧}) ⊂ A_{0}^{∧}. Since the topology of 𝒪_{V^{∧}}(V^{∧}) is I_{0}-adic (corollary 15.2.34(ii)), the assertion follows.

In case Λ is not empty, V_{Λ} is affine, and a direct inspection yields a natural isomorphism of topological rings A_{0,Λ}^{∧} →^{∼} 𝒪_{V^{∧}}(V_{Λ}^{∧}). We may then appeal to proposition 15.2.20(iii) to see that the natural map

(15.5.39)  A_{Λ}^{∧} →^{∼} A_{0,Λ}^{∧} ⊗_{A_{0,Λ}} A_{Λ} →^{∼} A_{0,Λ}^{∧} ⊗_{A_{0,Λ}} Γ(V_{Λ}, ψ_{∗}𝒪_{X}) → ℱ (V_{Λ}^{∧})

is also an isomorphism of rings, fitting into another commutative diagram

(15.5.40)  \xymatrix{A_{0,Λ}^{∧} \ar[r] \ar[d] & A_{Λ}^{∧} \ar[d] \\ 𝒪_{V^{∧}}(V_{Λ}^{∧}) \ar[r]^{φ_{V^{∧}_{Λ}}} & ℱ (V_{Λ}^{∧})}

whose top horizontal arrow is again the inclusion map, so ℱ (V_{Λ}^{∧}) is also a topological ring with the topology defined in the foregoing, and (15.5.39) is even an isomorphism of topological rings, if Λ ≠ ∅.

(ii): The abelian groups appearing in the diagram are all endowed with natural A_{Λ′}^{∧}-module structures, and it is easily seen that all the arrows are both A_{0,Λ′}^{∧}-linear and A_{Λ′}-linear maps, whence the assertion. ♢

Next, consider the affine open covering V_{•}^{∧} := (V_{{i}}^{∧} | i = 0, …, n) of V^{∧}, and endow the terms of the augmented alternating Čech complex C_{alt}^{•}(V_{•}^{∧}, ℱ) with the product topologies, as in (15.5.34). Combining claim 15.5.38 and remark 15.5.26(iii) we deduce an isomorphism of complexes of topological abelian groups

(15.5.41)  C_{alt}^{•}(V_{•}^{∧}, ℱ) →^{∼} C_{alt}^{•}(R_{•}, 𝒪_{Spa \underline{X}}^{∧}).

Hence, to conclude it suffices to show :

Claim 15.5.42. The complex C_{alt}^{•}(V_{•}^{∧}, ℱ) has strict differentials, and its cohomology has the discrete topology.

Proof of the claim. Diagram (15.5.40) implies that φ_{V_{Λ}^{∧}} is injective for every Λ ≠ ∅; it follows easily that the same holds for φ_{V^{∧}}. Hence the induced map of alternating Čech complexes

C_{alt}^{•}(V_{•}^{∧}, φ) : C_{alt}^{•}(V_{•}^{∧}, 𝒪_{V^{∧}}) → C_{alt}^{•}(V_{•}^{∧}, ℱ)

is injective in every degree, and by construction C_{alt}^{i}(V_{•}^{∧}, φ) is an open map for every i ∈ ℤ, so we are reduced to showing that the differentials d^{i} of C_{alt}^{•}(V_{•}^{∧}, 𝒪_{V^{∧}}) are strict and induce open maps C_{alt}^{i}(V_{•}^{∧}, 𝒪_{V^{∧}}) → Ker d^{i+1} for every i ∈ ℤ. If i < −1, there is nothing to show. For i = −1, the assertion follows from remark 5.5.2(i). In the remaining cases, the assertion comes down to the following. For every i, t ∈ ℕ there exists s ∈ ℕ such that

(15.5.43)  I_{0}^{s} · C_{alt}^{i+1}(V_{•}^{∧}, 𝒪_{V^{∧}}) ∩ Ker d^{i+1} ⊂ I_{0}^{t} · Im d^{i}.

For every k ∈ ℕ, denote by d_{k}^{•} the differentials of theČech complex C_{alt}^{•}(V_{•}^{∧}, I_{0}^{k}𝒪_{V^{∧}}); condition (15.5.43) is equivalent to

Ker d_{s}^{i+1} ⊂ Im d_{t}^{i}.

Taking into account theorem 15.1.37(i), we are therefore reduced to showing that for every i, t ∈ ℕ there exists an integer s ≥ t such that the natural map

H^{i+1}(V^{∧}, I_{0}^{s}𝒪_{V^{∧}}) → H^{i+1}(V^{∧}, I_{0}^{t}𝒪_{V^{∧}})

vanishes. Taking into account corollaries 15.2.30(i) and 15.2.34(i), we are further reduced to showing that for every i, t ∈ ℕ there exists s ∈ ℕ such that I^{s} · H^{i+1}(V, I_{0}^{t}𝒪_{V}) = 0. Notice that the A_{0}-module H^{i+1}(V, I_{0}^{t}𝒪_{V}) is analytically of finite type (theorem 15.2.22 and remark 15.2.38(i)); in light of remark 15.2.2(iii) it then suffices to check that the support of H^{i+1}(V, I_{0}^{t}𝒪_{V}) lies in the non-analytic locus of S. The latter follows easily from remark 15.5.26(vi) : details left to the reader. ♢

(ii): Arguing as in the proof of (i), we reduce easily to the case where R = Spa \underline{X}. Say that 𝔘 = (U_{i} | i ∈ I) for some set I, and pick any standard covering 𝔘′ := (U′_{i} | i ∈ I′) of Spa \underline{X} that refines 𝔘; after fixing total orderings for I and I′ we may consider the double complex

C_{alt}^{••}(𝔘, 𝔘′) := Hom_{ℤ_{X}}^{•}(R_{••}^{alt}(𝔘, 𝔘′), 𝒪_{Spa \underline{X}}^{∧})

(notation of remark 10.2.14(ii)) which vanishes in every bidegree (p, q) with either p < −1 or q < −1. For every n ∈ ℕ, define I_{alt}^{n+1} and I_{alt}^{′n+1} as in (10.2.4), and for every \underline{t} ∈ I_{alt}^{n+1} and

every \underline{t}′ ∈ I_{alt}^{′n+1}, set 𝔘′ ; \underline{t} := (U′_{i′} ∩ U\underline{t} | i′ ∈ I′) and 𝔘\underline{t}′ := (U_{i} ∩ U\underline{t}′ | i ∈ I). We notice that :

• The complex C_{alt}^{•,−1}(𝔘, 𝔘′) coincides with C_{alt}^{•}(𝔘, 𝒪_{Spa \underline{X}}^{∧}).
• The complex C_{alt}^{−1,•}(𝔘, 𝔘′) coincides with C_{alt}^{•}(𝔘′, 𝒪_{Spa \underline{X}}^{∧}).

• For every n ∈ ℕ, the complex C_{alt}^{•,n}(𝔘, 𝔘′) is the product ∏_{\underline{t}′∈I_{alt}^{′n+1}} C_{alt}^{•}(𝔘\underline{t}′, 𝒪_{Spa \underline{X}}^{∧}).

• For every n ∈ ℕ, the complex C_{alt}^{n,•}(𝔘, 𝔘′) is the product ∏_{\underline{t}∈I_{alt}^{n+1}} C_{alt}^{•}(𝔘′ ; \underline{t}, 𝒪_{Spa \underline{X}}^{∧}).

Since 𝔘′ ; \underline{t} is a standard covering of U\underline{t}, the proof of (i) shows that the differentials of the complex C_{alt}^{•}(𝔘′ ; \underline{t}, 𝒪_{Spa \underline{X}}^{∧}) induce open maps δ_{v}^{n,•} as in (8.6.11), for every n ∈ ℕ. Likewise, the same holds for the induced maps δ_{v}^{−1,•}. Moreover, since 𝔘′ refines 𝔘, for every n ∈ ℕ and every \underline{t}′ ∈ I_{alt}^{′n+1} there exists i ∈ I such that U′_{\underline{t}′} ⊂ U_{i}, in which case remark 10.2.13 yields a homotopy h^{•} from the identity automorphism of C_{alt}^{•}(𝔘\underline{t}′, 𝒪_{Spa \underline{X}}^{∧}) to the zero map, and a simple inspection shows that h^{i} is a continuous map, for every i ∈ ℤ. Then, lemma 8.6.15(i) says that the differentials of C_{alt}^{•}(𝔘\underline{t}′, 𝒪_{Spa \underline{X}}^{∧}) induce open maps δ_{h}^{•,n} as well, for every n ∈ ℕ. Lastly, we invoke corollary 8.6.12to conclude. □

15.5.44. Let \underline{X} := (X, 𝒯_{X}, A_{X}^{+}) be a quasi-affinoid scheme fulfilling the assumptions of theorem 15.5.35, so that 𝒪_{Spa \underline{X}}^{∧} is a sheaf of topological rings on the site (ℛ(\underline{X}), J_{ℛ} ). Following [59, Ch.0, §3.2.2] (and see also remark 15.5.29(i)), we may extend (uniquely up to unique isomorphism) both 𝒪_{Spa \underline{X}}^{∧} and 𝒪_{Spa \underline{X}}^{∧+} to sheaves of topological groups on the topological space Spa \underline{X}, and we shall denote these extensions with the same names. Then the pair (Spa \underline{X}, 𝒪_{Spa \underline{X}}^{∧}) is a topologically ringed space. Moreover, lemma 15.5.14says that Spa \underline{X} is a locally ringed space, so there exists a unique morphism of locally ringed spaces

(15.5.45)  σ\underline{X} : (Spa \underline{X}, 𝒪_{Spa \underline{X}}^{∧}) → Spec A

such that the corresponding map 𝒪_{Spec A} → σ_{\underline{X}∗}𝒪_{Spa \underline{X}}^{∧} yields on global sections the natural map A → A^{∧} := 𝒪_{Spa \underline{X}}^{∧}(Spa \underline{X}) ([58, Ch.I, Prop.1.6.3]). By inspecting loc.cit. it is easily seen that σ\underline{X} is the composition

Spa \underline{X} ⊂ Cont A ⟶^{σ_{A}} Spec A

where σ_{A} is the restriction of the support map (see remark 9.2.4(iii)). Especially, the image of σ\underline{X} lies in X, whence a well defined morphism of locally ringed spaces

(Spa \underline{X}, 𝒪_{Spa \underline{X}}^{∧}) → (X, 𝒪_{X}).

Corollary 15.5.46. With the notation of (15.5.44), the morphism σ\underline{X} induces an isomorphism

A^{∧} ⊗_{A} H^{i}(X, 𝒪_{X}) →^{∼} H^{i}(Spa \underline{X}, 𝒪_{Spa \underline{X}}^{∧}) for every i ∈ ℕ.

Proof. Let us show first the following special case :

Claim 15.5.47. In the situation of the corollary, suppose moreover that \underline{X} is affinoid. Then H^{i}(Spa \underline{X}, 𝒪_{Spa \underline{X}}^{∧}) = 0 for every i > 0.

Proof of the claim. In light of theorem 10.2.24(i) and lemma 15.5.21, it suffices to show that for every rational subset R of Spa \underline{X}, and every standard covering 𝔘 of R, the alternating Čech complex C_{alt}^{•}(𝔘, 𝒪_{Spa \underline{X}}^{∧}) is acyclic. However, notice that any such R is represented by an affinoid (topologically local) scheme; then, arguing as in the proof of theorem 15.5.35(i), we are easily reduced to the case where R = Spa \underline{X}. Then, after choosing a subring of definition for A we may define the scheme V, its formal completion V^{∧}, the affine open coverings V_{•} and V_{•}^{∧}, the projection π : V^{∧} → V and the morphism of schemes ψ : X → V as in the proof of theorem 15.5.35. In light of theorem 15.1.37(i) and of the isomorphism (15.5.41), we are then reduced to showing that H^{i}(V^{∧}, π^{∗}ψ_{∗}𝒪_{X}) = 0 for every i > 0. Then, corollary 15.2.34(iii) further reduces to checking that H^{i}(V, ψ_{∗}𝒪_{X}) = 0 for every i > 0; but since X is affine, the morphism ψ is affine, so that H^{i}(V, ψ_{∗}𝒪_{X}) ≃ H^{i}(X, 𝒪_{X}), whence the contention. ♢

Now, say that X = Spec A \ Spec A/J for a finitely generated ideal J ⊂ A, and pick a finite system of generators f_{•} := (f_{0}, …, f_{n}) for J and a subring of definition A_{0} ⊂ A. Let also R_{•} be the standard covering of Spa \underline{X} associated with f_{•}, and notice that R_{i} is represented by an affinoid scheme, for every i = 0, …, n. Set as well U_{i} := Spec A[f_{i}^{−1}] for i = 0, …, n, and notice that σ\underline{X}(R_{i}) ⊂ U_{i} for i = 0, …, n, and X = U_{0} ∪⋯∪U_{n}. There follows a commutative diagram (notation of (10.2.18))

\xymatrix{A_{0}^{∧} ⊗_{A_{0}} H_{alt}^{i}(U_{•}, 𝒪_{X}) \ar[r]^{α} \ar[d] & H_{alt}^{i}(R_{•}, 𝒪_{Spa \underline{X}}^{∧}) \ar[d] \\ A_{0}^{∧} ⊗_{A_{0}} H^{i}(X, 𝒪_{X}) \ar[r] & H^{i}(Spa \underline{X}, 𝒪_{Spa \underline{X}}^{∧})}

where the vertical arrows are isomorphisms, by virtue of claim 15.5.47and corollary 10.2.21(ii). Since A^{∧} = A_{0}^{∧} ⊗_{A_{0}} A, it then suffices to check that α is an isomorphism. On the other hand, as in the proof of theorem 15.5.35, we may associate with the sequence f_{•} a morphism of schemes ψ : X → V, the formal completion V^{∧} of V with its natural projection π : V^{∧} → V, and an affine covering V_{•} := (V_{i} | i = 0, …, n) of V such that ψ^{−1}V_{i} = U_{i} for i = 0, …, n; we set V_{i}^{∧} := π^{−1}V_{i} for i = 0, …, n, and ℱ := π^{∗}ψ_{∗}𝒪_{X}. There follows a commutative diagram

\xymatrix{A_{0}^{∧} ⊗_{A_{0}} H_{alt}^{i}(V_{•}, ψ_{∗}𝒪_{X}) \ar[r]^{β} \ar[d] & H_{alt}^{i}(V_{•}^{∧}, ℱ) \ar[d] \\ A_{0}^{∧} ⊗_{A_{0}} H^{i}(V, ψ_{∗}𝒪_{X}) \ar[r] & H^{i}(V^{∧}, ℱ)}

whose vertical arrows are again isomorphisms, by virtue of proposition 15.2.20(iv), and the same holds for the bottom horizontal arrow, by corollary 15.2.34(iii). To conclude, it suffices to observe that H_{alt}^{i}(U_{•}, 𝒪_{X}) = H_{alt}^{i}(V_{•}, ψ_{∗}𝒪_{X}), and that the isomorphism of Čech complexes (15.5.41) identifies α with β. □

Example 15.5.48. We present two counterexamples that show how theorem 15.5.35 can fail for affinoid schemes that are not analytically noetherian. To begin with, let K be a rank one complete valued field, and fix an element π ≠ 0 in the maximal ideal of K^{+}. For every K^{+}[T]- module J without π-torsion, let us set

A_{J} := K^{+}[T] ⊕ J A_{J,K} := A_{J} ⊗_{K^{+}} K

and endow A_{J} with the π-adic topology and with the K^{+}[T]-algebra structure such that the inclusion map K^{+}[T] → A is a ring homomorphism and J is an ideal with J^{2} = 0 (i.e. (x, y)·(x′, y′) := (xx′, xy′+x′y) for every (x, y), (x′, y′) ∈ A). Clearly A is an adic topological ring, and we endow A_{J,K} with the f-adic topology such that A_{J} is a ring of definition. Let also A_{J}^{∧} and A_{J,K}^{∧} be the completions of A_{J} and respectively A_{J,K}, and A_{J,K}^{∧+} the integral closure of A_{J}^{∧} in A_{J,K}^{∧}, and set

\underline{X} := Spec(A_{J,K}^{∧}, A_{J,K}^{∧+}).

• Take first J := K^{+}[T, T^{−1}]/K^{+}[T]. We claim that in this case the presheaf 𝒪_{Spa \underline{X}}^{∧} on ℛ(\underline{X}) is not separated. Indeed, consider the standard covering associated with the sequence (π, T); it consists of the rational subsets

U_{0} := \{v ∈ Spa \underline{X} | v(T) ≤ v(π)\} and U_{1} := \{v ∈ Spa \underline{X} | v(π) ≤ v(T)\}

and 𝒪_{Spa \underline{U}_{0}}^{∧}(U_{0}) = B_{0}^{∧} ⊗_{K^{+}} K, where B_{0}^{∧} is the completion of B_{0} := A_{J}[T/π] ⊂ A_{J,K}, where the latter is endowed with its π-adic topology. But notice that for every k, n > 0 the element T^{−k} of J can be written in B_{0} as

T^{−k} = T^{−k−n} · (T/π)^{n} · π^{n} ∈ π^{n}B_{0}.

Since n is arbitrary, it follows that T^{−k} lies in the kernel of the restriction map ρ_{0} : A_{J,K}^{∧} → 𝒪_{Spa \underline{X}}^{∧}(U_{0}), and since k is arbitrary, we get : J ⊂ Ker ρ_{0}. Likewise, 𝒪_{Spa \underline{U}_{0}}^{∧}(U_{1}) = B_{1}^{∧} ⊗_{K^{+}} K, where B_{1}^{∧} is the completion of B_{1} := A_{J}[π/T] ⊂ A_{J}[T^{−1}], where the latter is endowed with its π-adic topology. Clearly A_{J}[T^{−1}] = K^{+}[T, T^{−1}], whence J ⊂ Ker (ρ_{1} : A_{J,K}^{∧} → 𝒪_{Spa \underline{X}}^{∧}(U_{1})) as well, and the claim follows.

• Next, we take J to be the ℕ-graded K^{+}[T]-module such that

gr_{n}J := T^{−n}K^{+}[T]/K^{+}[T] for every n ∈ ℕ.

It follows that 𝒪_{Spa \underline{X}}^{∧}(U_{0}) = C_{0} ⊗_{K^{+}} K, where C_{0} is the π-adic completion of K^{+}[T/π] ⊕ J′, with J′ the ℕ-graded K^{+}[T/π]-module such that

gr_{n}J′ := ∑_{d=0}^{n−1} K^{+}T^{−n}(T/π)^{d} for every n ∈ ℕ.

Especially, 𝒪_{Spa \underline{X}}^{∧}(U_{0}) contains the π-adic completion J^{′∧} of J′. On the other hand, it is clear that A_{J}[T^{−1}] = K[T, T^{−1}], so the kernel of the restriction map ρ_{01} : 𝒪_{Spa \underline{X}}^{∧}(U_{0}) → 𝒪_{Spa \underline{X}}^{∧}(U_{0}∩ U_{1}) contains J^{′∧}. Now, consider the series

P := ∑_{n∈ℕ} T^{−n} where T^{−n} ∈ gr_{2n}J′ for every n ∈ ℕ.

Then T^{−n} = T^{−2n} · (T/π)^{n} · π^{n} ∈ π^{n}gr_{2n}J′, so the series P converges π-adically to a well defined element of J^{′∧}, and ρ_{01}(P) = 0 = ρ_{10}(0) (where ρ_{10} : 𝒪_{Spa \underline{X}}^{∧}(U_{1}) → 𝒪_{Spa \underline{X}}^{∧}(U_{0} ∩ U_{1}) is the other restriction map). We shall show that the series P does not lie in the image of ρ_{0}, so the presheaf 𝒪_{Spa \underline{X}}^{∧} is separated but is not a sheaf. Indeed, recall that the completion J^{∧} of J lies in the product of the completions gr_{n}J^{∧} of its graded summands, and likewise for J^{′∧} (remark 8.5.2(iii)); moreover, the induced map J^{∧} → J^{′∧} is the restriction of the product of the corresponding maps gr_{n}J^{∧} → gr_{n}J^{′∧}. It is easily seen that the latter maps are injective, so if P were in the image of ρ_{0}, it would be represented by the sequence (P_{n} | n ∈ ℕ) where P_{2k+1} = 0 and P_{2k} = T^{−k} ∈ gr_{2k}J^{∧} for every k ∈ ℕ. But then the series ∑_{n∈ℕ} P_{n} must converge π-adically in J^{∧} (proposition 8.5.3(ii)); the latter means that for every t ∈ ℕ there exists i ∈ ℕ such that T^{−k} ∈ π^{t}gr_{2k}J for every k ≥ i. But T^{−k} is not divisible by π in gr_{2k}J, for any k ∈ ℕ, a contradiction.

Remark 15.5.49. The earliest counterexample showing that in some cases the presheaf 𝒪_{Spa \underline{X}}^{∧} is not a sheaf, occurs already in [100], where it is attributed to Rost. Constructions similar to that of example 15.5.48, and several others as well, may also be found in [45] and [129], which introduce an interesting general class of “stably uniform” Tate affinoid rings \underline{A} for which the authors can prove that the presheaf 𝒪_{Spa \underline{A}}^{∧} is a sheaf.

15.6. Special loci of quasi-affinoid schemes. The results of this section will be used in chapter 16, for the study of perfectoid spaces.

Definition 15.6.1. Let \underline{X} := (X, 𝒯_{X}, A_{X}^{+}) be any quasi-affinoid scheme, V ⊂ X^{+} := Spec A_{X}^{+} any open subset, and set A_{X} := 𝒪_{X}(X). Let moreover β\underline{X} : X → X^{+} be the restriction of the morphism of schemes γ\underline{X} : Spec A_{X} → X^{+} induced by the inclusion map A_{X}^{+} → A_{X}.

(i) We say that \underline{X} spreads over V, if β\underline{X} restricts to an isomorphism of schemes β−1 ; \underline{X}V →^{∼} V. (ii) Let \underline{Y} := (Y, 𝒯_{Y} , A_{Y}^{+}) be another quasi-affinoid scheme, and φ : \underline{Y} → \underline{X} an f-adic

morphism of quasi-affinoid schemes. Then the induced ring homomorphism φ_{Y}^{♭} : A_{Y} → A_{X} restricts to an f-adic map φ_{Y}^{♭+} : A_{Y}^{+} → A_{X}^{+}, and we set

φ^{+} := Spec φ_{Y}^{♭+} : Y^{+} → X^{+}.

We say that φ spreads over V, if \underline{Y} spreads over (φ^{+})^{−1}(V) (in which case, we also say that \underline{Y} spreads over V, if no ambiguity is likely to arise).

(iii) We shall also denote by Ω\underline{X} ⊂ X^{+} the largest open subset such that \underline{X} spreads over Ω\underline{X}.

Remark 15.6.2. (i) Let \underline{X} := (X, 𝒯_{X}, A_{X}^{+}) be any quasi-affinoid scheme; then \underline{X} spreads over the analytic locus of Spec A_{X}^{+}, by virtue of lemma 8.3.29(iii).

(ii) Keep the notation of definition 15.6.1(ii), let ψ : \underline{Z} → \underline{Y} be another morphism of quasiaffinoid schemes, and suppose that φ spreads over V and ψ spreads over (φ^{+})^{−1}(V); then clearly φ ∘ ψ spreads over V.

Lemma 15.6.3. With the notation of definition 15.6.1, we have :

(i) Let W ⊂ X be an open subset such that β\underline{X} restricts to an open immersion W → X^{+}. Then β\underline{X}(W) ⊂ Ω\underline{X}.

(ii) γ\underline{X} restricts to an isomorphism of schemes γ−1 ; \underline{X}Ω\underline{X} →^{∼} Ω\underline{X}.

Proof. (i): First, let us notice the following

Claim 15.6.4. 𝒪_{X^{+}}(V) is integrally closed in 𝒪_{X}(β−1 ; \underline{X}V), for every open subset V ⊂ X^{+}.

Proof of the claim. Suppose first that V = Spec A_{X}^{+}[f^{−1}] for some f ∈ A_{X}^{+}; since β_{\underline{X}∗}𝒪_{X} is a quasi-coherent 𝒪_{X^{+}}-module, we have 𝒪_{X}(β−1 ; \underline{X}V) = A_{X}[f^{−1}], and since A_{X}^{+} is integrally closed in A_{X}, the assertion holds in this case. For a general open subset V, we may find an affine open covering V = ⋃_{i∈I} V_{i} where each V_{i} is of the form Spec A_{X}^{+}[f_{i}^{−1}] for some f_{i} ∈ A_{X}^{+}, and we get a commutative diagram of rings

\xymatrix{𝒪_{X^{+}}(V) \ar[r] \ar[d] & R := ∏_{i∈I} 𝒪_{X^{+}}(V_{i}) \ar[d] \ar[r]^{ρ_{1}} \ar[r]_{ρ_{2}} & ∏_{i,j∈I} 𝒪_{X^{+}}(V_{i} ∩ V_{j}) \ar[d] \\ 𝒪_{X}(β−1 ; \underline{X}V) \ar[r] & R′ := ∏_{i∈I} 𝒪_{X}(β−1 ; \underline{X}V_{i}) \ar[r] \ar[r] & ∏_{i,j∈I} 𝒪_{X}(β−1 ; \underline{X}(V_{i} ∩ V_{j})).}

Now, if h ∈ 𝒪_{X}(β−1 ; \underline{X}V) is integral over 𝒪_{X^{+}}(V), obviously the image h′ of h in R′ is integral over R; but from the foregoing case, it is easily seen that R is integrally closed in R′, so h′ ∈ R, and since the vertical arrows are injective maps, we deduce easily that ρ_{1}(h′) = ρ_{2}(h′), whence h′ ∈ 𝒪_{X^{+}}(V), and the assertion follows. ♢

Now, by assumption, β\underline{X}(W) is an open subset of X^{+}, and β\underline{X} restricts to a separated morphism β−1 ; \underline{X}β\underline{X}(W) → β\underline{X}(W); the latter admits a section whose image is the open subset W ⊂ β−1 ; \underline{X}β\underline{X}(W). Taking into account [59, Ch.I, Cor.5.4.6], we see that W is an open and closed subset of β−1 ; \underline{X}β\underline{X}(W). There follow ring homomorphisms

𝒪_{X^{+}}(β\underline{X}(W)) → 𝒪_{X}(β−1 ; \underline{X}β\underline{X}(W)) ⟶^{ρ} 𝒪_{X}(W)

whose composition is an isomorphism, and the kernel of ρ is generated by an idempotent element e ∈ 𝒪_{X}(β−1 ; \underline{X}β\underline{X}(W)). However, claim 15.6.4 implies that e lies in the image of 𝒪_{X^{+}}(β\underline{X}(W)), so e = 0, hence W = β−1 ; \underline{X}β\underline{X}(W), and the assertion follows immediately.

Claim 15.6.5. Let f : X → Y and g : Y → Z be two morphisms of schemes, such that h := g ∘ f is an open immersion and the image of f is schematically dense. Then we have :

(i) f is an open immersion.

(ii) If g is separated, then g^{−1}h(X) = f(X).

Proof of the claim. (i): We know from [59, Ch.I, Cor.5.3.13] that the morphism f is an immersion, and since its image is schematically dense, it must be an open immersion.

(ii): Since h is an open immersion, we may replace Z by h(X) and Y by g^{−1}h(X), after which we have h = 1_{X}, and f is a section of g. Since g is separated, [59, Ch.I, Cor.5.4.6] then tells us that f is a closed immersion in this case, and combining with (i), the claim follows. ♢

(ii): Consider now the commutative diagram of schemes

\xymatrix{β−1 ; \underline{X}Ω\underline{X} \ar[r]^{f} \ar@{-->}[dr]_{h} & γ−1 ; \underline{X}Ω\underline{X} \ar[d]^{g} \\  & Ω\underline{X}}

where f is the restriction of β∘ ; \underline{X} and g is the restriction of γ\underline{X}. Thus, h is the restriction of β\underline{X}, and it is an isomorphism, by definition of Ω\underline{X}; moreover, clearly f has schematically dense image. By claim 15.6.5, it follows that f is a surjective open immersion, i.e. it is an isomorphism, and then the same holds for g as well. □

Proposition 15.6.6. In the situation of definition 15.6.1, suppose that \underline{X} spreads over V. Then:

(i) The topological localization j\underline{X} : \underline{X}_{loc} → \underline{X}, the topological henselization j′ ; \underline{X} : \underline{X}^{h} → \underline{X} and the completion j′′ ; \underline{X} : \underline{X}^{∧} → \underline{X} of \underline{X} spread over V.

(ii) Let also f_{0}, …, f_{n} be a sequence of elements of A_{X} that generate an ideal I such that (β−1 ; \underline{X}V) ∩ Spec A_{X}/I = ∅, and set \underline{B} := Γ(\underline{X})(\frac{f_{0},...,f_{n}}{f_{0}} ) (notation of (15.5.1)). The induced morphism \underline{Y} := Spec \underline{B} → \underline{X} spreads over V.

(iii) Let \underline{Y} → \underline{X} and \underline{Y}′ → \underline{X} be two f-adic morphisms of quasi-affinoid schemes that

spread over V. The fibre product \underline{Y} ×\underline{X} \underline{Y}′ → \underline{X} spreads over V (see example 15.4.8). (iv) Let U ⊂ β−1 ; \underline{X}V be a quasi-compact open subset containing the analytic subset of \underline{X}.

The induced morphism φ : U ×_{X} \underline{X} → \underline{X} spreads over V (see example 15.4.4).

Proof. (i): Say that \underline{X}_{loc} = (X′, 𝒯_{X′}, A_{X′}^{+}), and let j+ ; \underline{X} : X′^{+} → X^{+} be the morphism induced by the topological localization A_{X}^{+} → A_{X′}^{+}. By inspecting (8.4.8), and taking into account the natural identifications (15.4.11), we get a cartesian diagram of schemes :

\xymatrix{X′ \ar[r]^{j\underline{X}} \ar[d]_{β_{\underline{X}_{loc}}} & X \ar[d]^{β\underline{X}} \\ X′^{+} \ar[r]^{j+ ; \underline{X}} & X^{+}.}

The assertion for j\underline{X} is an immediate consequence. The same argument applies to j′ ; \underline{X}. Next, say that \underline{X}^{∧} = (X^{∧}, 𝒯_{X^{∧}}, A_{X^{∧}}^{+}), and Γ(\underline{X})^{∧} = (A_{X}^{∧}, A_{X}^{∧+}, X′). According to corollary 15.4.28(i), the kernel 𝒥 of the surjective map A_{X}^{∧} → A_{X^{∧}} := 𝒪_{X^{∧}}(X^{∧}) lies in A_{X}^{∧+}, and we set Y := Spec A_{X}^{∧+}/𝒥; there follows a commutative diagram of schemes

\xymatrix{ & X^{∧} \ar[r] \ar[dl]_{β_{\underline{X}∧}} \ar[d]^{φ} & X \ar[d]^{β\underline{X}} \\ X^{∧+} \ar[r]^{ν} & Y \ar[r] & X^{+}}

whose square subdiagram is cartesian. With this notation, φ_{∗}𝒪_{X^{∧}} is a quasi-coherent 𝒪_{Y} - algebra, and A_{X^{∧}}^{+} = 𝒜 (Y), where 𝒜 is the integral closure of the image of 𝒪_{Y} in φ_{∗}𝒪_{X^{∧}}. Let V′ ⊂ Y and V′′ ⊂ X^{∧+} be the preimages of V; under our assumptions, the morphism φ^{♭} : 𝒪_{Y} → φ_{∗}𝒪_{X^{∧}} restricts to an isomorphism 𝒪_{Y |V′} →^{∼} (φ_{∗}𝒪_{X^{∧}})_{|V′}, so that 𝒜_{|V′} = (φ_{∗}𝒪_{X^{∧}})_{|V′}. The latter implies that the morphism γ\underline{X}^{∧} restricts to an isomorphism γ_{\underline{X}^{∧}}^{−1}V′′ →^{∼} V′′, and ν restricts to an isomorphism V′′ →^{∼} V′; lastly, since the image of X in X^{+} contains V, it is clear that the image of X^{∧} in Y contains V′, and then the image of X^{∧} in X^{∧+} contains V′′, so j′′ ; \underline{X} spreads over V.

(ii): Set B := A_{X}[f_{0}^{−1}], B′ := A_{X}^{+}[f_{1}/f_{0}, …, f_{n}/f_{0}] ⊂ B, say that \underline{Y} = (Y, 𝒯_{Y} , B^{+}), and let Y^{+} := Spec B^{+}. There follows a commutative diagram of schemes

\xymatrix{Y \ar[r]^{i} \ar[d] & Spec B \ar[r]^{ρ} \ar[d] & Spec B′ \ar[d] \\ X \ar[r] & Spec A_{X} \ar[r] & X^{+}}

whose left square subdiagram is cartesian. Denote by V′ ⊂ Spec B′ and V′′ ⊂ Y^{+} the preimages of V, and let g ∈ A_{X}^{+} be any element such that Spec A_{X}^{+}[g^{−1}] ⊂ V; according to lemma 15.6.3(ii), the induced map A_{X}^{+}[g^{−1}] → A_{X}[g^{−1}] is an isomorphism. Especially, we may regard f_{0}, …, f_{n} as elements of C := A_{X}^{+}[g^{−1}], and by assumption ∑_{i=0}^{n} f_{i}C = C. It follows easily that f_{0} is invertible in the subring B′[g^{−1}] = C[f_{1}/f_{0}, …, f_{n}/f_{0}] of B[g^{−1}], i.e. B′[g^{−1}] = B[g^{−1}], and since g is arbitrary, it follows that ρ restricts to an isomorphism ρ^{−1}V′ →^{∼} V′. Recalling now that B^{+} is the integral closure of B′ in 𝒪_{Y} (Y) = B, we deduce that γ\underline{Y} restricts as well to an isomorphism γ−1 ; \underline{Y}V′′ →^{∼} V′′. Lastly, since β\underline{X}(X) contains V, clearly the image of β\underline{Y} contains V′′. The assertion follows easily (details left to the reader).

(iii): Say that \underline{Y} = (Y, 𝒯_{Y} , A_{Y}^{+}), \underline{Y}′ = (Y′, 𝒯_{Y′}, A_{Y′}^{+}), and set Y^{+} := Spec A_{Y}^{+}, Y^{′+} := Spec A_{Y′}^{+}. Set also \underline{Z} := \underline{Y} ×\underline{X} \underline{Y}′ and (A_{Z}, A_{Z}^{+}, Z) := Γ(\underline{Z}), so that Z = Y ×_{X} Y′. Denote

φ : Z → Z′ := Y^{+} ×_{X^{+}} Y^{′+} and π : Z′ → X^{+}

the induced morphisms. With this notation, φ_{∗}𝒪_{Z} is a sheaf of quasi-coherent 𝒪_{Z′}-algebras, and A_{Z}^{+} = 𝒜 (Z′), where 𝒜 is the integral closure of the image of 𝒪_{Z′} in φ_{∗}𝒪_{Z}. Set V′ := π^{−1}V; under the current assumptions it is clear that the associated morphism φ^{♭} : 𝒪_{Z′} → φ_{∗}𝒪_{Z} restricts to an isomorphism 𝒪_{Z′|V′} →^{∼} (φ_{∗}𝒪_{Z})_{|V′}, whence 𝒜_{|V′} = (φ_{∗}𝒪_{Z})_{|V′}, which means that the natural morphism ν : Z^{+} := Spec A_{Z}^{+} → Z′ restricts to an isomorphism ν^{−1}V′ →^{∼} V′; lastly, since the image of φ contains V′, it is clear that the image of β\underline{Z} contains ν^{−1}V′, whence the assertion.

(iv): Set \underline{U} := U ×_{X} \underline{X}, and let h : U → X^{+} be the composition of the open immersion U → β−1 ; \underline{X}V and the restriction β−1 ; \underline{X}V → X^{+} of β\underline{X}. Then h = φ^{+} ∘ β\underline{U}, and clearly β\underline{U} has schematically dense image; by claim 15.6.5 we deduce that β\underline{U} is an open immersion, whence the contention. □

15.6.7. Let \underline{X} and V be as in definition 15.6.1, and suppose that \underline{X} is topologically local and spreads over V. We denote by

ℛ_{V} (\underline{X})

the site whose underlying category is the full subcategory of ℛ(\underline{X}) whose objects are the rational subsets R such that the sub-presheaf h_{R} ⊂ h_{Spa \underline{X}} is represented by a morphism \underline{Y} → \underline{X} of quasi-affinoid schemes that spreads over V. Notice that, in view of proposition 15.6.6(iii), if R, R′ ∈ Ob(ℛ_{V} (\underline{X})), then also R ∩ R′ ∈ Ob(ℛ_{V} (\underline{X})). As usual, a family (R_{i} → R | i ∈ I) of morphisms of ℛ_{V} (\underline{X}) generates a covering sieve of ℛ_{V} (\underline{X})/R if and only if ⋃_{i∈I} R_{i} = R.

In light of proposition 15.6.6(iii), we see that all fibre products are representable in ℛ_{V} (\underline{X}), and the inclusion functor ι : ℛ_{V} (\underline{X}) → ℛ(\underline{X}) commutes with fibre products. By lemma 4.2.4, it follows that ι is continuous, i.e the induced functor on presheaves ι^{∧} : ℛ(\underline{X})^{∧} → ℛ_{V} (\underline{X})^{∧} restricts to a functor on the respective subcategories of sheaves

ι̃_{∗} : ℛ(\underline{X})^{∼} → ℛ_{V} (\underline{X})^{∼}.

We will denote as well by ι̃_{∗} the corresponding functor on abelian sheaves. For every presheaf ℱ on ℛ(\underline{X}) or ℛ_{V} (\underline{X}), let us denote as usual by ℱ^{a} the sheaf associated to ℱ.

Proposition 15.6.8. In the situation of (15.6.7), let 𝒢 be a presheaf, 𝒜 an abelian presheaf, and ℱ an abelian sheaf on ℛ(\underline{X}). The following holds :

(i) The natural morphism (ι^{∧}𝒢)^{a} → ι^{∧}(𝒢^{a}) is an isomorphism.

(ii) The natural morphism ι_{∗}ℱ → Rι̃_{∗}ℱ is an isomorphism in D(ℤ_{ℛ_{V} (\underline{X})} -Mod).

(iii) The natural map Ȟ^{i}(ℛ_{V} (\underline{X}), ι^{∧}𝒜) → Ȟ^{i}(ℛ(\underline{X}), 𝒜) is bijective for every i ∈ ℕ.

Proof. (i): Define ℱ^{+} as in the proof of theorem 4.1.13; in view of claim 4.1.16 it suffices to check that the the natural morphism (ι^{∧}𝒢)^{+} → ι^{∧}(𝒢^{+}) is an isomorphism. For the injectivity, let U ∈ Ob(ℛ_{V} (\underline{X})) be any rational subset, and s, t ∈ (ι^{∧}𝒢)^{+}(U) any two sections whose images agree in 𝒢^{+}(U). Then we may find a covering family (U_{λ} → U | λ ∈ Λ) of ℛ_{V} (\underline{X})/U such that s and t are the classes of compatible systems (s_{λ} | λ ∈ Λ), (t_{λ} | λ ∈ Λ) with s_{λ}, t_{λ} ∈ 𝒢 (U_{λ}) for every λ ∈ Λ, and the assumption means that for every λ ∈ Λ there exists a covering family (U_{λ,i} → U_{λ} | i ∈ I_{λ}) of ℛ(\underline{X})/U_{λ} such that s_{λ|U_{λ,i}} = t_{λ|U_{λ,i}} for every λ ∈ Λ and every i ∈ I_{λ}. But by proposition 15.6.6(ii) and lemma 15.5.21, the covering U_{λ•} can be refined by a covering (U′_{λ,i′} → U_{λ} | i′ ∈ I′_{λ}) of ℛ_{V} (\underline{X})/U_{λ}, for every λ ∈ Λ. Then s_{λ|U′_{λ,i′}} = t_{λ|U′_{λ,i′}} for every λ ∈ Λ and every i′ ∈ I′_{λ}, so that s = t. Lastly, let s ∈ 𝒢^{+}(U) be any section; by definition we may find a covering family (U_{λ} → U | λ ∈ Λ) of ℛ(\underline{X})/U such that s is represented by a compatible system (s_{λ} | λ ∈ Λ) with s_{λ} ∈ 𝒢 (U_{λ}) for every λ ∈ Λ. But again, the covering U_{λ} can be refined by a covering of ℛ_{V} (\underline{X}), and we deduce that s is the image of a section of (ι^{∧}𝒢)^{+}(U).

(ii): The assertion means that R^{i}ι̃_{∗}ℱ = 0 for every i > 0. However, recall that R^{i}ι̃_{∗}ℱ is the sheaf associated to the presheaf that assigns to every U ∈ Ob(ℛ_{V} (\underline{X})) the abelian group H^{i}(ℛ(\underline{X})/U, ℱ_{|U}). Hence, let us fix such a rational subset U, and let s ∈ H^{i}(ℛ(\underline{X})/U, ℱ_{|U}) be any element; it suffices to exhibit a covering (U_{λ} → U | λ ∈ Λ) of U in ℛ_{V} (\underline{X}) such that the image of s in H^{i}(ℛ(\underline{X})/U_{λ}, ℱ_{|U_{λ}}) vanishes for every λ ∈ Λ. However, notice that there exists a covering (U′_{λ′} → U | λ′ ∈ Λ′) of U in ℛ(\underline{X}) such that the image of s in H^{i}(ℛ(\underline{X})/U′_{λ′}, ℱ_{|U′_{λ′}}) vanishes for every λ′ ∈ Λ′ (this is clear, since these cohomology groups are computed by an injective resolution of ℱ, which is exact in degrees > 0). But by proposition 15.6.6(ii) and lemma 15.5.21, the covering U′_{•} can be refined by a covering U_{•} consisting of objects of ℛ_{V} (\underline{X}), and clearly such U_{•} will do.

(iii): This is an immediate consequence of the fact – already remarked in the foregoing – that every covering family of ℛ(\underline{X}) can be refined by a covering family of ℛ_{V} (\underline{X}) : the details shall be left to the reader. □

15.6.9. Consider a quasi-affinoid ring \underline{A} := (A, A^{+}, U) such that A is an adic (and f-adic) topological ring; let A_{U} := 𝒪_{U}(U) and \underline{U} := (U, 𝒯_{U}, A_{U}^{+}) := Spec \underline{A}, and set

X_{A} := Spec A  X_{A}^{+} := Spec A_{U}^{+}  X_{A}^{∘} := Spec A_{U}^{∘}.

The induced map A → A_{U} is f-adic, so its image lies in A_{U}^{∘} (lemma 8.3.24(iii)), and we get a commutative diagram of schemes :

\xymatrix{U \ar[r]^{i} \ar[d]_{j} & X_{A} \\ Spec A_{U} \ar[r]^{j′} & X_{A}^{∘} \ar[u]_{j′′} \ar[r]^{t} & X_{A}^{+}.}

Let also Z ⊂ X_{A}^{+} be the support of the A_{U}^{+}-module A_{U}^{∘}/A_{U}^{+}, and Z′ := X_{A}^{∘} \ j′ ∘ j(U).

Lemma 15.6.10. With the notation of (15.6.9), the following holds :

(i) i, j and j′ ∘ j are open immersions.

(ii) The images of j and j′ are schematically dense.

(iii) The open subset j′ ∘ j(U) contains the analytic locus of X_{A}^{∘}.

(iv) X_{A}^{+} \ Ω\underline{U} is the topological closure of Z ∪ t(Z′).

Proof. (ii): The assertion for j′ is clear. Next, let ℐ be the kernel of the induced morphism j^{♭} : 𝒪_{Spec A_{U}} → j_{∗}𝒪_{U}; since U is quasi-compact, j_{∗}𝒪_{U} is a quasi-coherent 𝒪_{Spec A_{U}} -module, so the same holds for ℐ, and we are reduced to checking that Γ(Spec A_{U}, ℐ) = 0, which is clear, since j^{♭} induces an isomorphism on global sections.

(i): For i there is nothing to prove, and the assertion for j follows from [60, Ch.II, Prop.5.1.2]. The assertion for j′ ∘ j follows from claim 15.6.5(i).

(iii): Notice that j′ maps the analytic locus of Spec A_{U} isomorphically onto that of X_{A}^{∘} (lemma 8.3.29(iii)); likewise, it is easily seen that j′′ ∘ j′ restricts to an isomorphism from the analytic locus of Spec A_{U} onto that of X_{A}. Summing up, we conclude that j′′ restricts to an isomorphism from the analytic locus of X_{A}^{∘} onto that of X_{A}, whence the assertion.

(iv): Consider the commutative diagram of schemes

\xymatrix{β−1 ; \underline{U}Ω\underline{U} \ar[r]^{f} \ar@{-->}[dr]_{h} & t^{−1}Ω\underline{U} \ar[d]^{g} \\  & Ω\underline{U}}

where f is the restriction of j′∘j, and g is the restriction of t. Thus, h is the restriction of β\underline{U}, and it is an isomorphism, by definition of Ω\underline{U}; moreover, clearly f has schematically dense image. By claim 15.6.5, it follows that f is a surjective open immersion, i.e. it is an isomorphism, and then the same holds for g as well. Now, if x ∈ Ω\underline{U} is any point, it follows immediately that x ∉ Z; moreover g^{−1}(x) is the unique point of t^{−1}(x), and this point lies in j′ ∘ j(U). This shows that Z ∪ t(Z′) ⊂ X_{A}^{+} \ Ω\underline{U}. Conversely, suppose that x ∈ X_{A}^{+} lies neither in Ω\underline{U} nor the topological closure of Z; then there exists an open neighborhood V of x in X_{A}^{+} with V ∩ Z = ∅, and it follows that t restricts to an isomorphism t^{−1}V →^{∼} V. In view of (i), the restriction β−1 ; \underline{U}V → t^{−1}V of j′ ∘ j is still an open immersion, so the same holds for the restriction β−1 ; \underline{U}V → V of β\underline{U}, and consequently β\underline{U}(U) ∩ V = β\underline{U}(β−1 ; \underline{U}V) ⊂ Ω\underline{U}, according to lemma 15.6.3(i). We conclude that x ∉ β\underline{U}(U), and since x ∈ t(V), we finally get x ∈ t(Z′), whence (iv). □

15.7. Etale coverings of quasi-affinoid schemes. Let Sch be the category of schemes (in the universe U). For every scheme X, let also X_{ét} be the full subcategory of the category Sch/X, whose objects are the étale morphisms Y → X with Y quasi-compact and quasi-separated; notice that every morphism in X_{ét} is étale. We endow X_{ét} with its standard étale topology J_{X,ét}, whose covering families are all the systems (f_{i} : Y_{i} → Y | i ∈ I) of families of morphisms of X-schemes such that ⋃_{i∈I} f_{i}(Y_{i}) = Y. The site (X_{ét}, J_{X,ét}) is called the étale site of X, and is also often denoted simply by X_{ét}, when no ambiguities are likely to arise. It is easily seen that X_{ét} is a lex-site if and only if X is a quasi-compact and quasi-separated scheme (the details are left to the reader). Recall that we have a fibration Cov → Sch whose fibre category over every Y ∈ Ob(Sch) is the category of finite étale Y -schemes (see (13.1)). From the natural functor

X_{ét} → Sch (Y → X) ↦ Y

we deduce a fibration

Cov_{X_{ét}} := X_{ét} ×_{Sch} Cov → X_{ét}.

By faithfully flat descent, it is easily seen that Cov_{X_{ét}} is an ind-finite stack over the site X_{ét}.

For every k ∈ ℕ let [k] := {0, …, n}, and set as well [−1] := ∅; we denote by N the full subcategory of the category Set with Ob(N) := {[k] | k ≥ −1}. Then, let N_{X_{ét}} be the constant presheaf of categories on X_{ét} with value N, i.e. the presheaf such that N_{X_{ét}}(Y) := N for every Y ∈ Ob(X_{ét}), and N_{X}(f) := 1_{N} for every morphism f of X_{ét}. We get an X_{ét}-cartesian functor

ω_{X} : ℱ ib(N_{X_{ét}}) → Cov_{X_{ét}} (Y, [k]) ↦ ([k] × Y → Y).

Lemma 15.7.1. The functor ω_{X} is i-covering for i = 0, 1, 2.

Proof. Let f : Y → X be a finite étale covering, and x ∈ X; in order to check the assertion for i = 0, we need to exhibit an étale morphism g : X′ → X and k ∈ ℕ from an affine scheme X′ such that x ∈ g(X′), with an isomorphism of X′-schemes X′ ×_{X} Y →^{∼} [k]×X′. To this aim, let ξ : Spec κ → X be a geometric point localized at the point x (see definition 4.9.17(i)); we argue by induction on the cardinality c of the set f^{−1}(ξ) (see definition 4.9.17(iv)). If c = 0, we have x ∉ f(Y), and then there exists an affine open neighborhood X′ of x in X with X′∩f(Y) = ∅; in this case we may take for g the open inclusion X′ → X, and k := 0.

Suppose next that c > 0, and that the assertion is already known for all schemes X, all x ∈ X, and all finite étale coverings f : Y → X such that the cardinality of f^{−1}(ξ) is < c, for any geometric point ξ localized at x. Notice that the diagonal morphism Δ_{Y/X} : Y → Y ×_{X} Y is a finite étale morphism, hence its image is open and closed in Y ×_{X} Y. Thus, the Y -scheme Y ×_{X} Y is isomorphic to the disjoint union of Y and a finite étale covering h : Z → Y. The geometric point ξ lifts to a geometric point ξ_{Y} : Spec κ → Y, and it is then easily seen that h^{−1}(ξ_{Y}) = c − 1. By inductive assumption, we find an étale morphism g_{Y} : Y′ → Y from an affine scheme Y′, such that g_{Y} (Y′) contains the support of ξ_{Y}, and such that the Y′-scheme Y′ ×_{Y} Z is isomorphic to [c − 2] × Y′. We may then take X′ := Y′ and g := f ∘ g_{Y}.

Next, to check the assertion for i = 1, let k, k′ ≥ −1 be two integers, g : [k] × X → [k′] × X a morphism of X-schemes, and x ∈ X; it suffices to find an affine open neighborhood U of x in X, and a map of sets φ : [k] → [k′] such that the restriction g_{|U} : [k] × U → [k′] × U of g agrees with φ × U. To this aim, it suffices to invoke [66, Ch.IV, Cor.17.4.7] : the details are left to the reader. Lastly, it is clear from the definitions that ω_{X} is a 2-covering functor. □

15.7.2. Denote by Sch^{qcqs} the full subcategory of Sch whose objects are the quasi-compact and quasi-separated schemes, and by Sch_{ét}^{qcqs} the full subcategory of Morph(Sch^{qcqs}) whose objects are the étale morphisms of schemes. The restriction

t : Sch_{ét}^{qcqs} → Sch^{qcqs}

of the target functor t : Morph(Sch^{qcqs}) → Sch^{qcqs} is a fibration, and for every X ∈ Ob(Sch^{qcqs}), the fibre category t^{−1}X is the previously defined category X_{ét}, which we endow with its étale topology J_{X,ét}; we get therefore a well defined fibred site :

(Sch_{ét}^{qcqs}, t, J_{•,ét}).

For every category I and every functor F : I → Sch^{qcqs} we let

F_{ét} := I ×_{(Sch^{qcqs},F)} (Sch_{ét}^{qcqs}, t, J_{•,ét})

(notation of (4.5.11)). Thus, F_{ét} is a fibred lex-site over I, whose fibre category over every i ∈ Ob(I) is naturally identified with the lex-site (F i)_{ét}. Moreover, we have a natural functor :

F_{ét} → Sch (i, Y → F i) ↦ Y

from which we deduce a fibration over F_{ét} :

Cov_{F_{ét}} := F_{ét} ×_{Sch} Cov → F_{ét}

whose restriction to the fibre category (F i)_{ét} is naturally identified with the fibration Cov_{(F i)_{ét}}, for every i ∈ Ob(I). Explicitly, the objects of Cov_{F_{ét}} are the data (i, f : Y → F i, g : Z → Y) where f is an étale morphism, g is a finite étale morphism, and i ∈ Ob(I). Let N_{F_{ét}} be the constant presheaf of categories on F_{ét} with value N; as in (15.7), we then have a cartesian functor of F_{ét}-fibrations :

(15.7.3) ℱ ib(N_{F_{ét}}) → Cov_{F_{ét}}  (i, f : Y → F i, [k]) ↦ (i, f : Y → F i, [k] × Y → Y).

Clearly the restriction of (15.7.3) to each fibre category (F i)_{ét} is naturally identified with the functor ω_{F i} of (15.7); in view of lemma 15.7.1 and proposition 5.7.1(i) we deduce that also (15.7.3) is i-covering for i = 0, 1, 2.

15.7.4. Let \underline{X} := (X, 𝒯_{X}, A_{X}^{+}) be a topologically henselian quasi-affinoid scheme; we attach to \underline{X} a site (𝒬(\underline{X}), J_{𝒬} ) of quasi-affinoid open subsets, as in (15.5.11). We now define as follows a fibred lex-site over the category 𝒬(\underline{X}). For every U ∈ Ob(𝒬(\underline{X})) choose a topologically henselian quasi-affinoid scheme \underline{X}_{U}^{h} := (X_{U}^{h}, 𝒯_{U}^{h}, A_{U}^{h+}) representing the sub-presheaf h′_{U} of h′ ; \underline{X} (notation of remark 15.5.9(i)). Every inclusion U ⊂ U′ of quasi-affinoid subsets of Spa \underline{X} induces a morphism

i_{UU′} : \underline{X}_{U}^{h} → \underline{X}_{U′}^{h}

of quasi-affinoid schemes (cp. (15.5.11)), and for every further inclusion U′ ⊂ U′′ of quasiaffinoid open subsets of Spa \underline{X} we have i_{U′U′′} ∘ i_{UU′} = i_{UU′′}, so we get a well defined functor

\underline{X}_{•}^{h} : 𝒬(\underline{X}) → Sch^{qcqs}

whence an associated fibred site as in (15.7.2)

π\underline{X} : \underline{X}_{•,ét}^{h} := 𝒬(\underline{X}) ×_{Sch}^{qcqs} (Sch_{ét}^{qcqs}, t, J_{•,ét}) → 𝒬(\underline{X}).

Notice that \underline{X}_{•,ét}^{h} is a fibred lex-site, since \underline{X}_{U,ét}^{h} is quasi-compact and quasi-separated for every U ∈ Ob(𝒬(\underline{X})). We let (\underline{X}_{•,ét}^{h}, J) be the associated total site.

• The identity functor 1_{𝒬(\underline{X})} : 𝒬(\underline{X}) → 𝒬(\underline{X}) is naturally identified with the fibration ℱ ib(F_{𝟙}) → 𝒬(\underline{X}), where F_{𝟙} : 𝒬(\underline{X})^{o} → Cat is the constant pseudo-functor with value 𝟙. The category 𝟙 admits a unique topology 𝒯_{𝟙}, and the resulting pair (𝟙, 𝒯_{𝟙}) is obviously a lex-site, so we have the fibred lex-site

(𝒬(\underline{X}), 1_{𝒬(\underline{X})} , J_{•}^{∗})

where J_{U}^{∗} is the unique topology on the fibre 1_{𝒬(\underline{X})}^{−1} (U) →^{∼} 𝟙, for every U ∈ Ob(𝒬(\underline{X})), and we let (𝒬(\underline{X}), J_{𝒬}^{∗}) be the resulting total site.

• Denote by

e : 𝒬(\underline{X}) → \underline{X}_{•,ét}^{h}.

the functor that assigns to every U ∈ Ob(𝒬(\underline{X})) the final object X_{U}^{h} of the fibre category X_{U,ét}^{h}, and to every inclusion U ⊂ U′ the morphism i_{UU′}. It is easily seen that e is a 𝒬(\underline{X})- cartesian functor. Moreover, e restricts to a morphism of sites X_{U,ét}^{h} → (𝟙, 𝒯_{𝟙}) for every U ∈ Ob(𝒬(\underline{X})), so e is a morphism of fibred lex-sites, and also a morphism of total sites

e : (\underline{X}_{•,ét}^{h}, J) → (𝒬(\underline{X}), J_{𝒬}^{∗}).

• Lastly, for every U ∈ Ob(𝒬(\underline{X})) we have an inclusion functor i_{U} : \underline{X}_{U,ét}^{h} → \underline{X}_{•,ét}^{h} which identifies \underline{X}_{U,ét}^{h} with the fibre category over U. This is a weak morphism of sites

i_{U} : (\underline{X}_{•,ét}^{h}, J) → \underline{X}_{U,ét}^{h}

and for U = Spa \underline{X} it is even a morphism of sites (example 4.5.8). Let now ℰ be a stack on the étale site X_{ét}; we consider the fibration :

ℰ_{/𝒬} := St(e)_{∗} ∘ St(i_{Spa \underline{X}})^{∗}ℰ .

Remark 15.7.5. (i) The fibres of the fibration ℰ_{/𝒬} can be described as follows. let c_{X} : 𝒬(\underline{X}) → Sch^{qcqs} be the constant functor with value X; we consider as well the fibred lex-site

𝒬(\underline{X}) × X_{ét} = 𝒬(\underline{X}) ×_{(Sch^{qcqs},c_{X})} (Sch_{ét}^{qcqs}, t, J_{•,ét}) → 𝒬(\underline{X})

(see (4.5.11)) whose fibre categories are all naturally identified with X_{ét}. Then the rule U ↦ (i_{U,Spa \underline{X}} : X_{U}^{h} → X) defines a natural transformation \underline{X}_{•}^{h} ⇒ c_{X} which induces, after choosing a cleavage for the fibration Sch_{ét}^{qcqs}, a morphism of fibred lex-sites over 𝒬(\underline{X}) :

j\underline{X} : \underline{X}_{•,ét}^{h} → 𝒬(\underline{X}) × X_{ét}

(see (4.5.12)) which is also a morphism on the respective total sites

j\underline{X} : (\underline{X}_{•,ét}^{h}, J) → (𝒬(\underline{X}) × X_{ét}, J′).

Moreover, let l\underline{X} : X_{ét} → 𝒬(\underline{X}) × X_{ét} be the inclusion functor that identifies X_{ét} with the fibre category over the final object of 𝒬(\underline{X}). By example 4.5.8, this functor is a morphism of sites

l\underline{X} : (𝒬(\underline{X}) × X_{ét}, J′) → X_{ét}

and notice that i_{Spa \underline{X}} = j\underline{X} ∘ l\underline{X}. Furthermore, the projection p\underline{X} : 𝒬(\underline{X}) × X_{ét} → X_{ét} is left adjoint to l\underline{X}, and from corollary 5.7.2(ii) it follows easily that p\underline{X} is a weak morphism of sites for the topology J′. Combining with proposition 5.4.29(ii,iii), we deduce a pseudo-natural equivalence of pseudo-functors :

St(l\underline{X})^{∗} →^{∼} St(p\underline{X})_{∗}.

Especially, we get an equivalence of stacks :

St(i_{Spa \underline{X}})^{∗}ℰ →^{∼} St(j\underline{X})^{∗}St(p\underline{X})_{∗}(ℰ) = St(j\underline{X})^{∗}(𝒬(\underline{X}) × ℰ).

In light of proposition 5.7.3, we deduce a natural equivalence of stacks on \underline{X}_{U,ét}^{h} :

St(i_{U})_{∗} ∘ St(i_{Spa \underline{X}})^{∗}ℰ →^{∼} ℰ_{U} := St((i_{U,Spa \underline{X}})_{ét})^{∗}ℰ for every U ∈ Ob(𝒬(\underline{X})) .

Summing up, we conclude that the fibre category over U of the fibration ℰ_{/𝒬} is naturally equivalent to the fibre category (ℰ_{U})\underline{X}h_{U} of the fibration ℰ_{U}, over the final object \underline{X}_{U}^{h} of the site \underline{X}_{U,ét}^{h}.

(ii) By construction, ℰ_{/𝒬} is a stack on the site (𝒬(\underline{X}), J_{𝒬}^{∗}); but in fact we have :

Theorem 15.7.6. If ℰ is ind-finite, the fibration ℰ_{/𝒬} is a stack on the site (𝒬(\underline{X}), J_{𝒬} ).

Proof. It suffices to check that the natural functor

ℰ_{/𝒬} (U) → 2-colim_{𝒮 ∈J_{𝒬} (U)^{o}} Cart_{𝒬(\underline{X})} (𝒮 , ℰ_{/𝒬} ) for every U ∈ Ob(𝒬(\underline{X}))

is an equivalence. Since J_{𝒬} (U) is cofiltered for the order given by inclusion of sieves, this 2-colimit is represented by the colimit of the same system of categories (example 3.3.13(iv)); then arguing as in the proof of theorem 15.5.27 we may assume that U = Spa \underline{X}, and replace J_{𝒬} (Spa \underline{X}) by its coinitial subset consisting of the sieves generated by the standard coverings. Thus, let f_{•} := (f_{0}, …, f_{n}) and R_{•} be as in remark 15.5.26(i); for every k ∈ ℕ let [k] := {0, …, k}, and set as well [−1] := ∅. We consider the category Σ_{2,n}^{+} whose objects are all

the maps [j] → [n] for j = −1, 0, 1, 2; for two such objects [j] ⟶^{φ} [n] ⟵^{φ′} [k], the morphisms μ : φ → φ′ are the injective non-decreasing maps μ : [j] → [k] such that φ′ ∘ μ = φ. The unique map φ_{∅} : [−1] → [n] is clearly the initial object of Σ_{2,n}^{+}. Let also Σ_{2,n} ⊂ Σ_{2,n}^{+} be the full subcategory whose objects are the morphisms [j] → [n] with j ≥ 0. Consider the functor

Φ^{+} : Σ_{2,n}^{+o} → 𝒬(\underline{X})  :  ([j] ⟶^{φ} [n]) ↦ R_{φ([j])}

(here R_{Λ} ⊂ Spa \underline{X} is defined as in remark 15.5.26(i), for every subset Λ ⊂ [n]), and let Φ : Σ_{2,n}^{o} → 𝒬(\underline{X}) be the restriction of Φ^{+}. Taking into account the discussion of (3.5.18), we come down to checking that the natural functor

ℰ_{/𝒬} (Spa \underline{X}) → 2-lim_{Σ_{2,n}} ℰ_{/𝒬} (−) ∘ Φ^{o}

is an equivalence (notation of (3.2.4)). Following (4.5.11), we deduce two other fibred lex-sites

𝒰_{f_{•}}^{h} := Σ_{2,n}^{+o} ×_{𝒬(\underline{X})} (\underline{X}_{•,ét}^{h}, J) → Σ_{2,n}^{+o} \underline{Σ}_{2,n}^{+o} := Σ_{2,n}^{+o} ×_{𝒬(\underline{X})} (𝒬(\underline{X}), 1_{𝒬(\underline{X})} , J_{•}^{∗})

and according to (4.5.11) and (5.7) and proposition 5.4.29(ii), the projections Φ^{+} and

π_{𝒰} : 𝒰_{f_{•}}^{h} → \underline{X}_{•,ét}^{h}

are cocontinuous weak morphisms of sites for the topologies of the respective total sites. Furthermore, the functor e ∘ Φ^{+} : Σ_{2,n}^{+o} → X_{•,ét}^{h} factors uniquely through a Σ_{2,n}^{+o}-cartesian functor

e_{𝒰} : Σ_{2,n}^{+o} → 𝒰_{f_{•}}^{h}

and the projection π_{𝒰} . Just like for e, the functor e_{𝒰} is a morphism of the respective total sites. Set ℱ := St(π_{𝒰} )_{∗} ∘ St(i_{Spa \underline{X}})^{∗}ℰ; a direct inspection yields an equivalence of stacks

𝒢 := St(e_{𝒰} )_{∗}ℱ →^{∼} St(Φ^{+})_{∗}(ℰ_{/𝒬} )

over the total site of \underline{Σ}_{2,n}^{+o}, and we are reduced to checking that the natural functor

(15.7.7)  𝒢 (φ_{∅}) → 2-lim_{Σ_{2,n}} 𝒢 (−)

is an equivalence. To this aim, we proceed as in remark 15.7.5(i) : from the constant functor c_{X} : Σ_{2,n}^{+o} → Sch^{qcqs} we deduce the fibred site

Σ_{2,n}^{+o} × X_{ét} = Σ_{2,n}^{+o} ×_{(Sch^{qcqs},c_{X})} (Sch_{ét}^{qcqs}, t, J_{•,ét}) → Σ_{2,n}^{+o}

whose fibre categories are naturally identified with X_{ét}, endowed with its étale topology. Let j\underline{X} : \underline{X}_{•,ét}^{h} → 𝒬(\underline{X}) × X_{ét} be as in remark 15.7.5(i); we deduce a morphism of fibred lex-sites

j_{Σ} := Σ_{2,n}^{+o} ×_{𝒬(\underline{X})} j\underline{X} : 𝒰_{f_{•}}^{h} → Σ_{2,n}^{+o} × X_{ét}

which as usual induces a morphism on total sites, and by the same token, the projection

Σ_{2,n}^{+o} ×_{𝒬(\underline{X})} p\underline{X} : Σ_{2,n}^{+o} × X_{ét} → X_{ét}

is a weak morphism of sites, for the topology of the total site, and is left adjoint to the inclusion functor

l_{Σ} := Σ_{2,n}^{+o} ×_{𝒬(\underline{X})} l\underline{X} : X_{ét} → Σ_{2,n}^{+o} × X_{ét}.

By proposition 5.7.3we have then an equivalence of stacks over the total site of 𝒰_{f_{•}}^{h} :

ℱ →^{∼} St(j_{Σ})^{∗}(Σ_{2,n}^{+o} × ℰ).

Next, let A_{0} ⊂ A := 𝒪_{Spa \underline{X}}^{h}(Spa \underline{X}) be a subring of definition, and I_{0} ⊂ A_{0} an ideal of adic definition; set S := Spec A_{0} and S′ := Spec A_{0}/I_{0}. As in remark 15.5.26(iv,v), the sequence f_{•} induces a morphism of schemes X → ℙ_{S}^{n}, whose schematic image we denote by V, and we let ψ : X → V be the resulting morphism of S-schemes; moreover, for every non-empty subset Λ ⊂ [n] we have an open subset V_{Λ} ⊂ V which is naturally identified with Spec A_{0}[f_{k}/f_{i} | (k, i) ∈ [n] × Λ], and we set as well V_{∅} := V. Then, for every Λ ⊂ [n] let V_{Λ}^{h} be the henselization of V_{Λ} along its closed subscheme V′_{Λ} := S′ ×_{S} V_{Λ}; we have natural identifications

X_{Λ}^{h} →^{∼} X ×_{V} V_{Λ}^{h} for every Λ ⊂ [n]

and we denote by ψ_{Λ}^{h} : X_{Λ}^{h} → V_{Λ}^{h} the induced projection, and by q_{Λ} : V_{Λ}^{h} → V the composition of the henselization map V_{Λ}^{h} → V_{Λ} with the open immersion V_{Λ} → V. We consider the functors

G : Σ_{2,n}^{+o} → Sch^{qcqs}  :  ([j] ⟶^{φ} [n]) ↦ V_{φ([j])}^{h}
G′ : Σ_{2,n}^{+o} → Sch^{qcqs}  :  ([j] ⟶^{φ} [n]) ↦ V′_{φ([j])}

which to every morphism μ : φ → φ′ as in the foregoing assign the natural morphisms

(15.7.8)  V_{φ′([k])}^{h} → V_{φ([j])}^{h}  and respectively V′_{φ′([k])} → V′_{φ([j])}.

As in (15.7.2) we deduce fibred lex-sites

𝒱_{f_{•}}^{h} := Σ_{2,n}^{+o} ×_{(Sch^{qcqs},G)} (Sch_{ét}^{qcqs}, t, J_{•,ét}) → Σ_{2,n}^{+o}
𝒱′_{f_{•}} := Σ_{2,n}^{+o} ×_{(Sch^{qcqs},G′)} (Sch_{ét}^{qcqs}, t, J_{•,ét}) → Σ_{2,n}^{+o}.

The morphisms (ψ_{Λ}^{h} | Λ ⊂ [n]), (q_{Λ} | Λ ⊂ [n]), the closed immersions (V′_{Λ} → V_{Λ}^{h} | Λ ⊂ [n]) and V′ → V, and the open immersions (V′_{Λ} → V′ | Λ ⊂ [n]) define natural transformations \underline{X}_{•}^{h}∘Φ^{+} ⇒ G, G ⇒ c_{V}, G′ ⇒ G, c_{V′} ⇒ c_{V} and respectively G′ ⇒ c_{V′}, where c_{V} : Σ_{2,n}^{+o} → Sch is the constant functor with value V, and likewise for c_{V′}. After fixing cleavages, these natural transformations induce morphisms of fibred lex-sites

𝒰_{f_{•}}^{h} ⟶^{ψ_{𝒰}} 𝒱_{f_{•}}^{h} ⟵^{τ_{𝒱}} 𝒱′_{f_{•}} ⟶^{q′} Σ_{2,n}^{+o} × V′_{ét} ⟶^{τ_{Σ}} Σ_{2,n}^{+o} × V_{ét} ⟵^{q} 𝒱_{f_{•}}^{h}

(see (4.5.12)), which are also morphisms of sites for the topologies of the respective total sites (proposition 4.5.9). Define functors e_{𝒱} : Σ_{2,n}^{+o} → 𝒱_{f_{•}}^{h} and e_{𝒱 ′} : Σ_{2,n}^{+o} → 𝒱′_{f_{•}} by the rules :

φ ↦ (φ, V_{φ([j])}^{h})  and respectively : φ ↦ (φ, V′_{φ[j]})  for every φ : [j] → [n]

and which assign to every morphism μ : φ → ψ as in the foregoing, the morphisms (15.7.8). Just as for e and e_{𝒰} , both e_{𝒱} and e_{𝒱 ′} are morphisms of the respective total sites, and we get a commutative diagram of morphisms of sites :

(15.7.9)  [figure: V′_{ét}; τ_{ét}; V_{ét}^{h}; X_{ét}; ψ_{ét}; Σ_{2,n}^{+o} × V′_{ét}; τ_{Σ}; l_{V′}; Σ_{2,n}^{+o} × V_{ét}; l_{V}; Σ_{2,n}^{+o} × X_{ét}; ψ_{Σ}; l_{Σ}; 𝒱′_{f_{•}}; τ_{𝒱}; e_{𝒱 ′}; q′; 𝒱_{f_{•}}^{h}; e_{𝒱}; q; 𝒰_{f_{•}}^{h}; ψ_{𝒰}; e_{𝒰}; j_{Σ}; _{\underline{Σ}{+o \atop 2,n}}; _{\underline{Σ}{+o \atop 2,n}}; \underline{Σ}_{2,n}^{+o}.]

where ψ_{Σ} is likewise deduced from ψ. Set ℱ′ := St(ψ_{𝒰} )_{∗}ℱ. There follows an equivalence :

𝒢 →^{∼} St(e_{𝒱} )_{∗}ℱ′.

We regard the left square of the bottom row and the right square of the central row in (15.7.9) as oriented squares of sites, with orientations given by 1_{e_{𝒱 ′}} , respectively 1_{q∘ψ_{𝒰}} . We remark :

Claim 15.7.10. For every ind-finite stack 𝒜 on 𝒱_{f_{•}}^{h}, the base change transformation

Υ(St(1_{e_{𝒱 ′}} )_{∗}^{γ})_{𝒜} : St(e_{𝒱} )_{∗}(𝒜) → St(e_{𝒱 ′})_{∗}St(τ_{𝒱} )^{∗}(𝒜)

is a natural equivalence (notation of (2.3.8) and (5.5.23)).

Proof of the claim. This follows by combining corollary 5.7.26 and [74, Th.1’ and Cor.1]. Notice that the proof of loc.cit. relies on proposition 1 of the same article; for the latter, a complete proof is available in [105, Exp.XX, Prop.6.3.2]. More precisely, in the latter reference, the required results are stated only for stacks in groupoids; however, a direct inspection shows that the proofs work verbatim for arbitrary ind-finite stacks. ♢

Notice as well the pseudo-natural equivalences of pseudo-functors :

(15.7.11)  St(e_{𝒰} )_{∗} →^{∼} St(e_{𝒱} )_{∗} ∘ St(ψ_{𝒰} )_{∗}  St(τ_{𝒱} )^{∗} ∘ St(q)^{∗} →^{∼} St(q′)^{∗} ∘ St(τ_{Σ})^{∗}.

Claim 15.7.12. The base change transformation

Υ(St(1_{q∘ψ_{𝒰}} )_{∗}^{γ}) : St(q)^{∗} ∘ St(ψ_{Σ})_{∗} → St(ψ_{𝒰} )_{∗} ∘ St(j_{Σ})^{∗}

is a pseudo-natural equivalence.

Proof of the claim. For every subset Λ ⊂ [n] we consider the oriented square of lex-sites :

𝒟_{Λ}  :  [figure: X_{Λ,ét}^{h}; j_{Λ}; ψ_{Λ}; 1_{ψ∘j′_{Λ}}; X_{ét}; ψ; V_{Λ,ét}^{h}; q_{Λ}; V_{ét}.]

According to corollary 5.7.26, it suffices to check that the base change transformation

Υ(St(1_{ψ∘j′_{Λ}})_{∗}^{γ}) : St(q_{Λ})^{∗} ∘ St(ψ)_{∗} → St(ψ_{Λ})_{∗} ∘ St(j_{Λ})^{∗}

is a pseudo-natural equivalence for every such Λ. However, V_{Λ}^{h} is the limit of a cofiltered system of quasi-compact and quasi-separated V -schemes (V_{Λ,i} | i ∈ I) with affine transition morphisms, and X_{Λ}^{h} is the limit of the induced cofiltered system (X_{Λ,i} := X ×_{V} V_{Λ,i} | i ∈ I); hence V_{Λ,ét}^{h} and X_{Λ,ét}^{h} represent the 2-limits of the induced systems of lex-sites V_{•,ét} := (V_{i,ét} | i ∈ I) and X_{Λ,•,ét} := (X_{Λ,i,ét} | i ∈ I). We regard I as a category (see example 1.1.6(iii)), and we may assume that I admits a final object i_{0} such that V_{Λ,i_{0}} = V; then V_{Λ,•,ét} and X_{Λ,•,ét} yield fibred lex-sites \underline{V}_{Λ,•,ét} and \underline{X}_{Λ,•,ét} over I, and the system of projections (X_{Λ,i} → V_{Λ,i} | i ∈ I) induces a morphism of fibred lex-sites \underline{X}_{Λ,•,ét} → \underline{V}_{Λ,•,ét}. Then the assertion follows from corollary 5.7.30. ♢

Now, from proposition 5.6.34 we see that Σ_{2,n}^{+o} × ℰ is an ind-finite stack, and then the same holds for ℱ (theorem 5.6.33), and also for ℱ′, again by proposition 5.6.34. Set

ℰ′ := St(τ_{ét})^{∗} ∘ St(ψ_{ét})_{∗}(ℰ).

From (15.7.11) and claims 15.7.10and 15.7.12, we deduce an equivalence of stacks :

𝒢 →^{∼} St(e_{𝒱 ′})_{∗} ∘ St(q′)^{∗} ∘ St(τ_{Σ})^{∗} ∘ St(ψ_{Σ})_{∗}(Σ_{2,n}^{+o} × ℰ)

→^{∼} St(e_{𝒱 ′})_{∗} ∘ St(q′)^{∗} ∘ St(l_{V′})^{∗}(ℰ′).

Once again, by corollary 5.7.2(ii) the projection Σ_{2,n}^{+o} × V′_{ét} → V′_{ét} is a weak morphism of sites, for the topology of the total site, and is left adjoint to l_{V}, so we get an equivalence

St(l_{V′})^{∗}(ℰ′) →^{∼} Σ_{2,n}^{+o} × ℰ′

(proposition 5.4.29(ii,iii)). Similarly, the functor q′ admits a left adjoint

s : 𝒱′_{f_{•}} → Σ_{2,n}^{+o} × V′_{ét} ([j] ⟶^{φ} [n], U ⟶^{f} V_{φ([j])}) ↦ (φ, U ⟶^{f} V_{φ([j])} → V′)

and as explained in remark 5.7.34, the functor s is cocontinuous and is a weak morphism of sites, for the topologies of the total sites, whence a pseudo-natural equivalence :

St(q′)^{∗} →^{∼} St(s)_{∗}.

Summing up, we obtain an equivalence of stacks :

(15.7.13)  𝒢 →^{∼} St(e_{𝒱 ′})_{∗} ∘ St(s)_{∗}(Σ_{2,n}^{+o} × ℰ′).

Lastly, consider the functor

Ψ^{+} : Σ_{2,n}^{+o} → V′_{ét} ([j] ⟶^{φ} [n]) ↦ V′_{φ([j])}

and let Ψ : Σ_{2,n}^{o} → V′_{ét} be the restriction of Ψ^{+}. By inspecting the definitions, we find that the equivalence (15.7.13) identifies the functor (15.7.7) with the similar natural functor

ℰ′(V′) → 2-lim_{Σ_{2,n}} ℰ′(−) ∘ Ψ^{o}.

The latter is an equivalence, since ℰ′ is a stack on V′_{ét}, whence the theorem. □

15.7.14. Let f : \underline{X}′ → \underline{X} be an adic morphism of topologically henselian quasi-affinoid schemes, and (U_{1}, …, U_{n}) a finite covering of Spa \underline{X} consisting of quasi-affinoid open subsets; recall that U′_{i} := (Spa f)^{−1}U_{i} is a quasi-affinoid subset of Spa \underline{U}, for every i = 1, …, n (remark 15.5.9(iv)). As in (15.7.4), define the corresponding functors

\underline{X}_{•}^{h} : 𝒬(\underline{X}) → Sch^{qcqs} \underline{X}_{•}^{′h} : 𝒬(\underline{X}′) → Sch^{qcqs}

and for every Λ ⊂ \{1, …, n\}, set U_{Λ} := ⋂_{i∈Λ} U_{i} (with U_{∅} := Spa \underline{X}) and U′_{Λ} := (Spa f)^{−1}U_{Λ}. To ease notation, let X_{Λ} (resp. X′_{Λ}) be the scheme underlying the quasi-affinoid scheme \underline{X}_{U_{Λ}}^{h} (resp. \underline{X}_{U′_{Λ}}^{′h}). Recall that \underline{X}_{U′_{Λ}}^{′h} is the topological henselization of \underline{Z}_{Λ} := \underline{X}′ ×\underline{X} \underline{X}_{U_{Λ}}^{h}; the composition of the henselization morphism \underline{X}_{U′_{Λ}}^{′h} → \underline{Z}_{Λ} and the projection \underline{Z}_{Λ} → \underline{X}_{U_{Λ}}^{h} is then a morphism of quasi-affinoid schemes

f_{Λ} : \underline{X}_{U′_{Λ}}^{′h} → \underline{X}_{U_{Λ}}^{h}

so we have a commutative diagram of schemes :

\xymatrix{X′_{Λ} \ar[r]^{f_{Λ}} \ar[d]_{p′_{Λ}} & X_{Λ} \ar[d]^{p_{Λ}} \\ X′ \ar[r]^{f} & X}  for every Λ ⊂ \{1, …, n\}

where p_{Λ} and p′_{Λ} are the natural projections. Let also ℰ be an ind-finite stack on X_{ét}, and set ℰ′ := St(f_{ét})^{∗}ℰ ℰ_{Λ} := St(p_{Λ,ét})^{∗}ℰ ℰ′_{Λ} := St(f_{Λ,ét})^{∗}ℰ_{Λ} for every Λ ⊂ {1, …, n}. We consider the units of adjunction :

η : ℰ → St(f_{ét})_{∗}ℰ′  and η_{Λ} : ℰ_{Λ} → St(f_{Λ,ét})_{∗}ℰ′_{Λ}  for every Λ ⊂ {1, …, n}.

Lemma 15.7.15. In the situation of (15.7.14), suppose that the functor η_{Λ,X_{Λ}} : ℰ_{Λ,X_{Λ}} → ℰ′_{Λ,X′_{Λ}}

is an equivalence for every non-empty Λ ⊂ {1, …, n}. Then η_{X} : ℰ_{X} → ℰ′_{X′} is an equivalence.

Proof. Define the category Σ_{2,n}^{+}, its full subcategory Σ_{2,n}, the fibred site \underline{Σ}_{2,n}^{+} and the site (𝒬(\underline{X}), J_{𝒬}^{∗}) as in the proof of theorem 15.7.6. We consider the functor

Φ^{+} : Σ_{2,n}^{+o} → 𝒬(\underline{X})  :  ([j] ⟶^{φ} [n]) ↦ U_{φ([j])}

and its restriction Φ : Σ_{2,n}^{o} → 𝒬(\underline{X}). We attach likewise to the covering (U′_{1}, …, U′_{n}) of Spa \underline{X}′ the corresponding functors Φ^{′+} : Σ_{2,n}^{+o} → 𝒬(\underline{X}′) and its restriction Φ′ to Σ_{2,n}^{o}. Then, from Ψ^{+} := \underline{X}_{•}^{h} ∘ Φ^{+} and Ψ^{′+} := \underline{X}_{•}^{′h} ∘ Φ^{′+} we get fibred sites :

𝒰 := Σ_{2,n}^{+o} ×_{(Sch,Ψ^{+})} (Sch_{ét}^{qcqs}, t, J_{•,ét}) 𝒰′ := Σ_{2,n}^{+o} ×_{(Sch,Ψ^{′+})} (Sch_{ét}^{qcqs}, t, J_{•,ét}).

Next, let e : Σ_{2,n}^{+o} → 𝒰 be the functor that assigns to every ([j] ⟶^{φ} [n]) ∈ Ob(Σ_{2,n}^{+o}) the final object X_{φ([j])} of the site X_{φ([j]),ét}; let also i : X_{ét} → 𝒰 be the inclusion functor which identifies X_{ét} with the fibre category of the fibration 𝒰 → Σ_{2,n}^{+o}, over the final object [−1] ∈ Ob(Σ_{2,n}^{+o}). Likewise we define the functor e′ : Σ_{2,n}^{+o} → 𝒰′ and i′ : X′_{ét} → 𝒰′.

With this notation, Φ^{+} is a weak morphism of sites (𝒬(\underline{X}), J_{𝒬}^{∗}) → Σ_{2,n}^{+o} for the topology of the total site of \underline{Σ}_{2,n}^{+o}, and likewise for Φ^{′+}. Also, both i and e are morphisms of sites for the topologies of the total sites, and likewise for i′ and e′. Furthermore, the rule : ([j] ⟶^{φ} [n]) ↦ f_{φ([j])} for every φ ∈ Ob(Σ_{2,n}^{+o}) defines a natural transformation

\underline{X}_{•}^{′h} ∘ Φ^{+} ⇒ \underline{X}_{•}^{h} ∘ Φ^{′+}

which, after choosing a cleavage for the fibration Sch_{ét}^{qcqs} → Sch^{qcqs}, determines a morphism of fibred sites φ : 𝒰′ → 𝒰 , which is again a morphism of sites for the topologies of the respective total sites. A direct inspection then yields a commutative diagram of morphisms of sites :

(15.7.14)  [figure: Σ_{2,n}^{+o}; 𝒰′; e′; i_{\underline{X}′}; g; X′_{ét}; f_{ét}; Σ_{2,n}^{+o}; 𝒰; e; i\underline{X}; X_{ét}.]

By theorem 15.7.6, we associate with ℰ a stack ℰ_{/𝒬} on (𝒬(\underline{X}), J_{𝒬} ); likewise, since ℰ′ is also ind-finite (theorem 5.6.33), we get the stack ℰ′_{/𝒬} on (𝒬(\underline{X}′), J′_{𝒬} ). Then, arguing as in the proof of theorem 15.7.6we get equivalences of fibrations over Σ_{2,n}^{+o} :

St(Φ^{+})_{∗}(ℰ_{/𝒬} ) →^{∼} ℰ_{/Σ} := St(e)_{∗} ∘ St(i\underline{X})^{∗}ℰ St(Φ^{′+})_{∗}(ℰ′_{/𝒬} ) →^{∼} ℰ′_{/Σ} := St(e′)_{∗} ∘ St(i\underline{X}′)^{∗}ℰ′.

On the other hand, from diagram (15.7.14) we deduce an equivalence of stacks

St(g)^{∗} ∘ St(i\underline{X})^{∗}(ℰ) →^{∼} St(i\underline{X}′)^{∗}(ℰ′)

whence, by adjunction, a morphism of stacks :

St(i\underline{X})^{∗}(ℰ) → St(g)_{∗} ∘ St(i\underline{X}′)^{∗}(ℰ′)

and after composing with St(e)_{∗}, we arrive at a morphism of fibrations over Σ_{2,n}^{+o} :

ω : ℰ_{/Σ} → ℰ′_{/Σ}.

Now, pick cleavages for ℰ_{/Σ} and ℰ′_{/Σ}, and let c and c′ be the associated pseudo-functors; then ω corresponds to a pseudo-natural transformation ω_{•} : c ⇒ c′. Arguing as in remark 15.7.5(i), we see that c_{φ} is equivalent to ℰ_{φ([j])} for every ([j] ⟶^{φ} [n]) ∈ Ob(Σ_{2,n}^{+o}), and likewise for c′_{φ}, so we have for every such φ an essentially commutative diagram of categories :

\xymatrix{c_{φ} \ar[r]^{ω_{φ}} \ar[d] & c′_{φ} \ar[d] \\ ℰ_{φ([j])} \ar[r]^{ω_{φ}^{∗}} & ℰ′_{φ([j])}}

whose vertical arrows are equivalences. Lastly, since ℰ_{/𝒬} and ℰ′_{/𝒬} are stacks for the topologies J_{𝒬} and respectively J′_{𝒬} , we know that the natural functors

ℰ_{X} → 2-lim_{Σ_{2,n}} c ℰ′_{X′} → 2-lim_{Σ_{2,n}} c′

are equivalences, hence ω_{∅} : ℰ_{X} → ℰ′_{X′} is identified, up to equivalences of categories and isomorphisms of functors, with the induced functor

2-lim_{Σ_{2,n}} ω_{•} : 2-lim_{Σ_{2,n}} c → 2-lim_{Σ_{2,n}} c′.

To conclude the proof, it then suffices to check that for every φ : [j] → [n], the functor ω_{φ}^{∗} is isomorphic to the functor η_{Λ,X_{Λ}}, with Λ := φ([j]). To this aim, we notice that the right square subdiagram of (15.7.14) can be regarded as an oriented square of links, after choosing adjoints and fixing adjunctions for its four arrows; its orientation (from the bottom left corner to the upper right corner) is given by the identity 1_{i\underline{X}∘g} : i\underline{X} ∘ g ⇒ f_{ét} ∘ i\underline{X}′. Then, proposition 5.7.3 says that the base change transformation

Υ(St(1_{i\underline{X}∘g})_{∗}^{γ}) : St(f_{ét})^{∗} ∘ St(i\underline{X})_{∗} → St(i\underline{X}′)_{∗} ∘ St(g)^{∗}

is a pseudo-natural equivalence. The assertion then follows, after combining with proposition 2.3.14: the details shall be left to the reader. □

15.7.16. Resume the situation of (15.7.4), and let also \underline{X}^{∧} := (X^{∧}, 𝒯_{X}^{∧}, A_{X^{∧}}^{+}) be the completion of \underline{X}, and denote by π_{X} : \underline{X}^{∧} → \underline{X} the completion morphism of quasi-affinoid schemes (given by the counit of the adjunction of proposition 15.4.12). The morphism π_{X} induces a morphism of étale sites π_{X,ét} : X_{ét}^{∧} → X_{ét}, and for every stack ℰ on X_{ét} we let

ℰ^{∧} := St(π_{X,ét})^{∗}(ℰ).

Theorem 15.7.17. With the notation of (15.7.16), if ℰ is ind-finite, the unit of adjunction

ℰ_{X} → ℰ_{X^{∧}}^{∧} = St(π_{X,ét})_{∗}(ℰ^{∧})_{X}

is an equivalence of categories.

Proof. To begin with, let f : \underline{Y} → \underline{X} be any f-adic morphism of quasi-affinoid schemes; since π_{X} : \underline{X}^{∧} → \underline{X} is also f-adic, the fibre product \underline{Y}′ := \underline{X}^{∧} ×\underline{X} \underline{Y} is well defined (example 15.4.8), and we let \underline{X}^{∧} ⟵^{p_{1}} \underline{Y}′ ⟶^{p_{2}} \underline{Y} be the natural projections. The morphism f^{∧} : \underline{Y}^{∧} → \underline{X}^{∧} induced by f, and the completion morphism π_{Y} : \underline{Y}^{∧} → \underline{Y} determine a unique morphism of quasi-affinoid schemes

g : \underline{Y}^{∧} → \underline{Y}′ such that p_{1} ∘ g = f^{∧} and p_{2} ∘ g = π_{Y} .

Moreover, since \underline{Y}^{∧} is complete, by adjunction g factors uniquely through a morphism of quasiaffinoid schemes h : \underline{Y}^{∧} → \underline{Y}^{′∧} and the completion morphism π_{Y′} : \underline{Y}^{′∧} → \underline{Y}′.

Claim 15.7.18. In the foregoing situation, h is an isomorphism of quasi-affinoid schemes.

Proof of the claim. It suffices to show that the morphisms p_{2} ∘ π_{Y′} and π_{Y} satisfy the same universal property. Thus, let \underline{Z} be a complete quasi-affinoid schemes, and k : \underline{Z} → \underline{Y} any morphism of quasi-affinoid schemes; the composition f ∘ k : \underline{Z} → \underline{X} factors uniquely through a morphism l : \underline{Z} → \underline{X}^{∧} and π_{X}, and there exists a unique morphism l′ : \underline{Z} → \underline{Y}′ such that p_{1} ∘ l′ = l and p_{2} ∘ l′ = k. Again, l′ factors uniquely through a morphism l^{′∧} : \underline{Z} → \underline{Y}^{′∧} and π_{Y′}. It follows easily that l^{′∧} is the unique morphism such that p_{2} ∘ π_{Y′} ∘ l^{′∧} = k, whence the contention. ♢

Claim 15.7.19. In order to prove the theorem, we may assume that \underline{X} is an affinoid scheme.

Proof of the claim. Let U_{•} := (U_{1}, …, U_{n}) be an open covering of Spa \underline{X} consisting of finitely many affinoid open subsets. (To find such U_{•}, let A_{X} := 𝒪_{X}(X), and J ⊂ A_{X} a finitely generated ideal such that Spec A_{X}/J = Spec A_{X}\X; then we may take a finite set of generators f_{•} := (f_{1}, …, f_{n}) of J, and let U_{•} be the standard covering associated with f_{•}, as in (15.5.20).)

For every Λ ⊂ \{1, …, n\}, define the affinoid open subset U_{Λ} ⊂ Spa \underline{X} as in (15.7.14), and notice that U_{Λ}^{∧} := (Spa π_{X})^{−1}U_{Λ} is an affinoid open subset of Spa \underline{X}^{∧}. To ease notation, let also \underline{X}_{Λ} := \underline{X}_{U_{Λ}}^{h} and \underline{X}′_{Λ} := \underline{X}_{U_{Λ}^{∧}}^{∧h}, for every such Λ. Let p_{Λ} : \underline{X}_{Λ} → \underline{X} be the natural projection, and π_{Λ} : \underline{X}′_{Λ} → \underline{X}_{Λ} the morphism of affinoid schemes deduced from π_{X}, as in (15.7.14), and set ℰ_{Λ} := St(p_{Λ,ét})^{∗}ℰ and ℰ′_{Λ} := St(π_{Λ,ét})^{∗}ℰ_{Λ}, for every Λ ⊂ {1, …, n}. By lemma 15.7.15, in order to prove the theorem it suffices to show that the unit of adjunction

η_{Λ} : ℰ_{Λ,X_{Λ}} → ℰ′_{Λ,X′_{Λ}}

is an equivalence for every non-empty Λ ⊂ \{1, …, n\}. To this aim, let q_{Λ} : (\underline{X}_{Λ})^{∧} → \underline{X}_{Λ} and q′_{Λ} : (\underline{X}′_{Λ})^{∧} → \underline{X}′_{Λ} be the completion morphisms, and denote by (X_{Λ})^{∧} (resp. (X′_{Λ})^{∧}) the scheme underlying (\underline{X}_{Λ})^{∧} ((\underline{X}′_{Λ})^{∧}); since (\underline{X}′_{Λ})^{∧} is also the completion of \underline{X}^{∧} ×\underline{X} \underline{X}_{Λ} (remark 15.4.10(vi) and corollary 8.4.15(i)), claim 15.7.18yields an isomorphism of affinoid schemes

(15.7.20)  h : (\underline{X}_{Λ})^{∧} →^{∼} (\underline{X}′_{Λ})^{∧}  such that π_{Λ} ∘ q′_{Λ} ∘ h = q_{Λ}.

Set ℰ_{Λ}^{∧} := St(q_{Λ,ét})^{∗}ℰ_{Λ} and ℰ_{Λ}^{′∧} := St(q′_{Λ,ét})^{∗}ℰ′_{Λ}, and denote

η_{Λ}^{∧} : ℰ_{Λ,X_{Λ}} → ℰ_{Λ,(X_{Λ})^{∧}}^{∧} and η_{Λ}^{′∧} : ℰ′_{Λ,X_{Λ}} → ℰ_{Λ,(X′_{Λ})^{∧}}^{′∧}

the units of adjunction. Now, up to equivalences of categories and isomorphisms of functors, the composition η_{Λ}^{′∧} ∘ η_{Λ} is naturally identified with the counit

ℰ_{Λ,X_{Λ}} → (St((π_{Λ} ∘ q′_{Λ})_{ét})^{∗}ℰ_{Λ})_{(X′_{Λ})^{∧}}

and the latter is an equivalence if and only if the same holds for η_{Λ}^{∧}, due to (15.7.20). Thus, if both η_{Λ}^{∧} and η_{Λ}^{′∧} are equivalences, the same will follow for η_{Λ}. Summing up, the theorem will follow for the quasi-affinoid scheme \underline{X}, once we have shown that the theorem holds for the affinoid schemes \underline{X}_{Λ} and \underline{X}′_{Λ}, for every non-empty Λ ⊂ \{1, …, n\}, whence the claim. ♢

For every scheme Y and every finite group G, let G_{Y} be the group scheme G × Y over Y; recall that a (right) G-torsor on Y_{ét} is the datum of a surjective étale morphism of schemes Z → Y, together with a morphism ρ : Z ×_{Y} G_{Y} → Z that induces an action of the group G_{Y} (T) of T -sections of G_{Y} on the set Z(T) of T -sections of Z, for every Y -scheme T, and such that ρ and the projection Z ×_{Y} G_{Y} → Z induce an isomorphism

Z ×_{Y} G_{Y} →^{∼} Z ×_{Y} Z.

A morphism of G_{Y} -torsors (Z, ρ) → (Z′, ρ′) is a morphism of schemes Z → Z′ compatible in the obvious way with the morphisms ρ and ρ′. Clearly the G-torsors over Y_{ét} form a category :

Tors(Y_{ét}, G).

According to [105, Exp.XX, Prop.6.3.2], in order to prove the theorem, it suffices to check that :

(i) for every sheaf ℱ on X_{ét}, the natural map H^{0}(X_{ét}, ℱ)→H^{0}(X_{ét}^{∧}, π̃_{X,ét}^{∗}ℱ) is bijective (ii) For every finite group G, and every finite morphism of schemes Y → X, the natural functor Tors(Y_{ét}, G) → Tors((X^{∧} ×_{X} Y)_{ét}, G) is an equivalence of categories.

Moreover, according to [10, Exp.XII, Prop.6.5(i)], condition (i) is equivalent to :

(iii) For every finite morphism of schemes Y → X, the natural map H^{0}(Y_{ét}, ℤ/2ℤ) → H^{0}((X^{∧} ×_{X} Y)_{ét}, ℤ/2ℤ) is bijective.

However, for every scheme Z, the group H^{0}(Z_{ét}, ℤ/2ℤ) is naturally identified with the automorphism group of the trivial ℤ/2ℤ-torsor on Z_{ét}, so (iii) follows from (ii), for G := ℤ/2ℤ.

Now, for every finite morphism of schemes f : Y → X, let \underline{Y} := Y ×_{X} \underline{X}, and recall that the morphism of quasi-affinoid schemes f : \underline{Y} → \underline{X} is f-adic (see example 15.4.9(v)), hence \underline{Y}′ := \underline{X}^{∧} ×\underline{X} \underline{Y} is well defined, and according to claim 15.7.18we have an isomorphism h : \underline{Y}^{∧} →^{∼} \underline{Y}^{′∧} of quasi-affinoid schemes. Let G be any finite group; we deduce an essentially commutative diagram :

\xymatrix{Tors(Y_{ét}, G) \ar[r] \ar[d] & Tors(Y′_{ét}, G) \ar[d] \\ Tors(Y_{ét}^{∧}, G) \ar[r] & Tors(Y_{ét}^{′∧}, G)}

where Y′, Y^{∧}, Y^{′∧} are the schemes underlying respectively \underline{Y}′, \underline{Y}^{∧}, \underline{Y}^{′∧}. The vertical arrows are induced by π_{Y} : Y^{∧} → Y and π_{Y′} : Y^{′∧} → Y′, and the bottom horizontal arrow is induced by h, so it is an equivalence. Hence, condition (ii) will follow for f : Y → X, if we show that the vertical arrows of the diagram are equivalences. Moreover, by comparing the respective universal properties, we easily obtain a natural isomorphism of affinoid schemes \underline{Y}′ →^{∼} (X^{∧} ×_{X} Y) ×_{X^{∧}} \underline{X}^{∧}; since \underline{X}^{∧} is topologically henselian, example 15.4.9(vi) implies that the same holds for \underline{Y}′. With claim 15.7.19, we are then reduced to checking that for every topologically henselian affinoid scheme (X, 𝒯_{X}, A_{X}^{+}) and every finite group G, the completion morphism induces an equivalence Tors(X_{ét}, G) → Tors(X_{ét}^{∧}, G).

Let A_{0} ⊂ A := 𝒪_{X}(X) be a subring of definition; let also (A_{i} | i ∈ I) be the filtered system of finite type A_{0}-subalgebras of A. For every i ∈ I, endow A_{i} and A_{i}^{+} := A_{i} ∩ A_{X}^{+} with the topologies 𝒯_{i} and 𝒯_{i}^{+} induced by the inclusions into A. Let A_{0}^{∧} be the completion of A_{0}; then the completions of A_{i} and A_{i}^{+} are respectively A_{i}^{∧} := A_{0}^{∧} ⊗_{A_{0}} A_{i} and A_{i}^{∧+} := A_{0}^{∧} ⊗_{A_{0}} A_{i}^{+} for every i ∈ I, and the completion of A is A^{∧} := A_{0}^{∧} ⊗_{A_{0}} A, so A^{∧} is the colimit of the induced filtered system of A_{0}^{∧}-algebras (A_{i}^{∧} | i ∈ I). Set as well X_{i} := Spec A_{i} and X_{i}^{∧} := Spec A_{i}^{∧}, and let 𝒯_{i}^{∧} be the topology of A_{i}^{∧}, for every i ∈ I; then \underline{X}_{i} := (X_{i}, 𝒯_{i}, A_{i}^{+}) is a topologically henselian affinoid scheme (proposition 8.4.2(iii)), and its completion \underline{X}_{i}^{∧} is (X_{i}^{∧}, 𝒯_{i}^{∧}, A_{i}^{∧+}).

From [65, Ch.IV, Th.8.8.2, Th.8.10.5, Th.11.2.6] it follows easily that the natural functors

2-colim_{i∈I} Tors(X_{i,ét}, G) → Tors(X_{ét}, G)  2-colim_{i∈I} Tors(X_{i,ét}^{∧}, G) → Tors(X_{ét}^{∧}, G)

are equivalences of categories. Hence it suffices to check that for every i ∈ I the functor

Tors(X_{i,ét}, G) → Tors(X_{i,ét}^{∧}, G)

is an equivalence. We may therefore assume from start that A is an A_{0}-algebra of finite type, say A = A_{0}[t_{1}, …, t_{n}] for a finite sequence t_{•} := (t_{1}, …, t_{n}) of elements of A. For every k = 0, …, n, we shall say that the sequence t_{•} is adequate up to k if for every i = 1, …, k, either one of the following conditions holds :

(a) t_{i} ∈ A^{+}

(b) t_{i} ∈ A^{×} and 1/t_{i} ∈ A^{+}.

Claim 15.7.21. In order to prove the theorem, we may assume that t_{•} is adequate up to n.

Proof of the claim. Suppose that the theorem is known for every f-adic ring A generated over some topologically henselian subring of definition A_{0} by a sequence of elements (t_{1}, …, t_{n}) adequate up to n. We argue by descending induction on k ≤ n, that the theorem then also holds for every f-adic ring generated over any such subring A_{0} by a sequence (t_{1}, …, t_{n}) that is only adequate up to k. If k = n, there is nothing to show; let then k ≤ n and t_{•} a sequence adequate up to k − 1, and suppose that we have already proved our assertion for all sequences adequate up to k. Let \underline{A} be the affinoid ring (A, A^{+}); we consider the affinoid rings

\underline{B} := \underline{A} (\frac{1, t_{k}}{1} )  \underline{C} := \underline{A} (\frac{1, t_{k}}{t_{k}} )  \underline{D} := \underline{B} ⊗\underline{A} \underline{C}

(notation of (15.5.1) and example 15.4.5(i)). Explicitly, we have \underline{B} = (B, B^{+}) and \underline{C} = (C, C^{+}), where the ring underlying the topological ring B (resp. C) is A (resp. A[1/t_{k}]) and B^{+} (resp. C^{+}) is the integral closure in B (resp. in C) of A^{+}[t_{k}] (resp. of A^{+}[1/t_{k}]); moreover, B (resp. C) admits the ring of definition B_{0} := A_{0}[t_{k}] (resp. C_{0} := A_{0}[1/t_{k}]). Recall that the topological henselization B^{h} of B is B_{0}^{h} ⊗_{B_{0}} B, with B_{0}^{h} the topological henselization of B_{0}. Hence the image in B^{h} of the sequence t_{•} generates the f-adic B_{0}^{h}-algebra B^{h}, and is adequate up to k. Likewise we see that the image of t_{•} generates the C_{0}^{h}-algebra C^{h}, and then the same follows for the image of t_{•} in the topological ring underlying \underline{D}^{h} : the details are left to the reader. By inductive assumption, the theorem then holds for the affinoid schemes Spec \underline{B}^{h}, Spec \underline{C}^{h} and Spec \underline{D}^{h}. But notice that \underline{B}^{h} and \underline{C}^{h} represent two affinoid open subsets U′ and U′′ of Spa \underline{X}, and \underline{D}^{h} represents U′ ∩ U′′; more precisely, (U′, U′′) is the standard covering of Spa \underline{A} associated with the sequence (1, t_{k}) (see (15.5.20)); arguing as in the proof of claim 15.7.19, we deduce that the theorem holds for Spec \underline{A}, as required. ♢

Henceforth, we assume that t_{•} is an adequate sequence up to n; let S := {i ≤ n | t_{i} ∈ A^{+}} and T := {1, …, n} \ S, and set t′_{i} := t_{i} for every i ∈ S and t′_{i} := 1/t_{i} for every i ∈ T. Moreover, set t := ∏_{i∈T} t′_{i}. The subring A′_{0} := A_{0}[t′_{1}, …, t′_{n}] is open and bounded in A, hence it is a subring of definition of A, and after replacing A_{0} by A′_{0} we may assume that A = A_{0}[t^{−1}] for an element t ∈ A_{0} ∩ A^{×}. Let I_{0} ⊂ A_{0} be a finite type ideal of adic definition, and denote by B the topological ring whose underlying ring is the same as A, and whose topology is the (t, I_{0})-adic topology as defined in [75, Def.5.4.10(ii)] (invoking remark 8.3.2(i), it is easily seen that the (t, I_{0})-adic topology is a ring topology on A).

Since t ∈ A^{×}, the ideal t^{n}I_{0} is open in A for every n ∈ ℕ, therefore the identity is a continuous map A → B, and its completion is a morphism of topological rings :

f : A^{∧} → B^{∧}.

Let also g : A → A^{∧} and h : B → B^{∧} be the completion maps, I_{0}^{∧} the topological closure of the image of I_{0} in A^{∧}, and 𝒯 the (t, I_{0}^{∧})-adic topology on A^{∧}. We notice :

Claim 15.7.22. The map f is also a morphism (A^{∧}, 𝒯 ) → B^{∧} of topological rings, and induces an isomorphism of topological rings :

f^{∧} : (A^{∧}, 𝒯 )^{∧} →^{∼} B^{∧}.

Proof of the claim. Let J_{0}^{∧} be the topological closure of the image of I_{0} in B^{∧}; it is clear that f is a continuous map (A^{∧}, 𝒯 ) → B^{∧}, since f(t^{n}I_{0}^{∧}) ⊂ t^{n}J_{0}^{∧} for every n ∈ ℕ. In order to show that f^{∧} is an isomorphism, we apply the criterion of theorem 8.2.8(iii) : first, obviously the image of f is dense in B^{∧}, since the image of A in B^{∧} is already dense. It remains to check that 𝒯 agrees with the topology induced from B^{∧} via f^{∧}; we show more precisely :

t^{n}I_{0}^{∧} = f^{−1}(t^{n}J_{0}^{∧}) for every n ∈ ℕ.

Indeed, since both t^{n}I_{0}^{∧} and f^{−1}(t^{n}J_{0}^{∧}) are open subgroups of (A^{∧}, 𝒯 ), it suffices to check that g^{−1}(t^{n}I_{0}^{∧}) = g^{−1}(f^{−1}(t^{n}J_{0}^{∧})); but g^{−1}(t^{n}I_{0}^{∧}) = t^{n}g^{−1}(I_{0}^{∧}) = t^{n}I_{0}, and g^{−1}(f^{−1}(t^{n}J_{0}^{∧})) = h^{−1}(t^{n}J_{0}^{∧}) = t^{n}h^{−1}(J_{0}^{∧}) = t^{n}I_{0}. ♢

Since h = f ∘ g, it suffices to check that f and h induce equivalences of categories

Tors((Spec A^{∧})_{ét}, G) →^{∼} Tors((Spec B^{∧})_{ét}, G) ⟵^{∼} Tors((Spec A)_{ét}, G).

Now, the pair (A_{0}, I_{0}) is henselian by assumption, and since A_{0}^{∧} is complete for the I_{0}^{∧}-adic topology, also the pair (A_{0}^{∧}, I_{0}^{∧}) is henselian; it then follows that the pairs (A_{0}, tI_{0}) and (A_{0}^{∧}, tI_{0}^{∧}) are henselian [75, Rem.5.1.10(iv)]. By [75, Prop.5.4.53] and claim 15.7.22, we deduce that f and h induce equivalences of categories of finite étale morphisms

(15.7.23)  Cov(Spec A^{∧}) →^{∼} Cov(Spec B^{∧}) ⟵^{∼} Cov(Spec A)

(notation of (13.1)). Now, for any scheme Y, the G-torsors over Y_{ét} are the objects φ : Z → Y of Cov(Spec A^{∧}) endowed with a (right) G-action and such that φ is a surjective morphism. Such a morphism has an open and closed image in Y, hence it is surjective if and only if it is an epimorphism (in the category Cov(Y)). The equivalences (15.7.23) induce formally equivalences for the corresponding categories of objects endowed with G-action (details left to the reader), and obviously respect epimorphisms, whence the contention. □

Remark 15.7.24. For every finite group G, let 𝒞_{G} be the category with Ob(𝒞_{G}) = {o} and such that the monoid Hom_{𝒞_{G}}(o, o) is isomorphic to G. For every scheme X, let also 𝒞_{G,X_{ét}} be the constant presheaf on X_{ét} with value 𝒞_{G}. By arguing as in the proof of lemma 15.7.1, it is easily seen that the associated stack 𝒞_{G,X_{ét}}^{a} is naturally equivalent to the stack of G-torsors on X_{ét} : the details shall be left to the reader.

15.7.25. For every U ∈ Ob(𝒬(\underline{X})), choose a complete and separated quasi-affinoid scheme \underline{X}_{U}^{∧} := (X_{U}^{∧}, 𝒯_{U}^{∧}, A_{U}^{∧+}) representing the sub-presheaf h′′_{U} ⊂ h′′_{\underline{X}^{∧}} (notation of remark 15.5.9(i)). Any inclusion U ⊂ U′ of quasi-affinoid subsets of Spa \underline{X} induces a morphism j_{UU′} : \underline{X}_{U}^{∧} → \underline{X}_{U′}^{∧} of quasi-affinoid schemes (cp. (15.5.11)), and for any further inclusion U′ ⊂ U′′ we have

j_{U′U′′} ∘ j_{UU′} = j_{UU′′}

whence a well defined functor

\underline{X}_{•}^{∧} : 𝒬(\underline{X}) → Sch U ↦ X_{U}^{∧}

and we consider the fibration over 𝒬(\underline{X})

Cov∧ ; \underline{X} := Fib(\underline{X}_{•}^{∧})^{∗}(Cov)

where the fibration Cov → Sch is defined as in (15.7). With this notation, we can now state :

Corollary 15.7.26. The fibration Cov∧ ; \underline{X} is a stack on the site (𝒬(\underline{X}), J_{𝒬} ).

Proof. We have already remarked that the stack Cov_{X_{ét}} is ind-finite on X_{ét}, hence the fibration (Cov_{X_{ét}})_{/𝒬} is a stack on (𝒬(\underline{X}), J_{𝒬} ), by theorem 15.7.6. But from theorem 15.7.17 we easily deduce an equivalence of fibrations (Cov_{X_{ét}})_{/𝒬} →^{∼} Cov∧ ; \underline{X}, whence the assertion. □

Remark 15.7.27. A variant of corollary 15.7.26appears in [114, Th.2.6.9].

15.7.28. We wish to give an explicit description, up to natural equivalence of categories, of the stalks of the fibration Cov∧ ; \underline{X}. Hence, let x ∈ Spa \underline{X} be any point; recall that the stalk of Cov∧ ; \underline{X} over x is the category

Cov∧ ; \underline{X}(x) := 2-colim_{U∈𝒬(\underline{X},x)^{o}} Cov(X_{U}^{∧})

where 𝒬(\underline{X}, x) denotes the set of all quasi-affinoid open neighborhoods of x in Spa \underline{X}, which is cofiltered by inclusion. Pick a finitely generated ideal J ⊂ A_{X} := 𝒪_{X}(X) such that X = Spec A_{X} \ Spec A_{X}/J, and let f_{0}, …, f_{n} be a finite system of generators of J. Set U_{i} := R_{A_{X}}(\frac{f_{0},...,f_{n}}{f_{i}} ) ∩ Spa \underline{X} for i = 0, …, n. Then we have

U_{0} ∪ ⋯ ∪ U_{n} = Spa \underline{X}

and we may assume that x ∈ U_{0}. Notice that h′′_{U_{0}} is represented by a quasi-affinoid scheme \underline{X}_{U_{0}}^{∧} := (X_{U_{0}}^{∧}, 𝒯_{U_{0}}^{∧}, A_{U_{0}}^{∧+}) with X_{U_{0}}^{∧} = Spec 𝒪_{Spa \underline{X}}^{∧}(U_{0}). Then, since X_{U_{0}}^{∧} is affine, every rational subset of Spa \underline{X}_{U_{0}}^{∧} containing x is likewise represented by a quasi-affinoid scheme whose underlying scheme is affine, and the system of such rational subsets is naturally identified with a final subset of 𝒬(\underline{X}, x). Taking into account lemma 13.1.6, we deduce a natural equivalence

Cov∧ ; \underline{X}(x) →^{∼} Cov(Spec 𝒪_{Spa \underline{X},x}^{∧}).

• Next, if x is analytic, combining with lemma 15.5.14(i,v.c), proposition 9.1.17(iii) and [66, Ch.IV, Prop.18.5.15] we arrive at a natural equivalence

Cov∧ ; \underline{X}(x) →^{∼} Cov(Spec κ(x^{∧})) →^{∼} Cov(Spec κ(x)^{∧})

where κ(x^{∧}) is the residue field of the natural valuation | · |_{x}^{∧} on 𝒪_{Spa \underline{X},x}^{∧}, and κ(x)^{∧} is the completion of κ(x^{∧}) for its valuation topology (see (15.5.12)).

• Lastly, if x is non-analytic, lemma 15.5.14(vi.b) and [75, Th.5.5.7(iii)] yield the natural equivalence

Cov∧ ; \underline{X}(x) →^{∼} Cov(Spec 𝒪_{Spa \underline{X},x}^{∧}/A_{X}^{∘∘}𝒪_{Spa \underline{X},x}^{∧}).

16. Perfectoid rings and perfectoid spaces

In this chapter we develop a generalization of Scholze’s theory of perfectoid rings and perfectoid spaces.

16.1. Distinguished elements and transversal pairs. Let p be a prime number, (A, 𝒯 ) a complete and separated topological ring whose topology 𝒯 is linear and coarser than the p-adic topology. Remark 9.4.12(ii) endows the topological monoid E(A) with a natural structure of perfect topological 𝔽_{p}-algebra, such that :

• For every topologically nilpotent ideal I ⊂ A containing pA, and which is either closed for the topology 𝒯 , or else finitely generated and closed for the p-adic topology of A, the projection π_{I} : A → A/I induces an isomorphism of topological rings

E(π_{I}) : E(A) →^{∼} E(A/I)

where A/I is endowed with the quotient topology induced by 𝒯 via π_{I}.

• Moreover, if (A′, 𝒯 ′) is any other topological ring fulfilling the above conditions, and f : A → A′ is any continuous ring homomorphism, the map E(f) : E(A) → E(A′) is a morphism of topological rings.

In light of this canonical identification of E(A) with E(A/I), we shall henceforth write slightly abusively

\overline{u}_{A/I} : E(A) → A/I

for the map that would be denoted \overline{u}_{A/I} ∘ E(π_{I}), or equivalently, π_{I} ∘ \overline{u}_{A}, with the notation of section 9.4. Another basic construction of this chapter shall be the topological ring :

A(A, 𝒯 ) := W (E(A))

where W (−) denotes the ring of Witt vectors as in (9.3.11). From remark 9.4.9(i) and lemma 9.3.33(ii,iii) we also know that the topologies of E(A) and A(A, 𝒯 ) are linear, complete and separated. Unless we have to deal with several different topologies on A, we shall usually omit mentioning explicitly 𝒯 , and write simply A(A). According to proposition 9.3.44(ii), the ring A(A) is also complete and separated for its p-adic topology, and the same holds for A, by lemma 8.3.12; moreover we see from (9.3.41) that p is a regular element of A(A), and by remark 9.3.28(i) the ghost component ω_{0} descends to an isomorphism of topological rings

A(A) ⊗_{ℤ} 𝔽_{p} →^{∼} E(A)

provided we endow A(A) ⊗_{ℤ} 𝔽_{p} with the quotient topology induced from A(A) via the projection A(A) → A(A) ⊗_{ℤ} 𝔽_{p}. The map ω_{0} also admits a continuous set-theoretic multiplicative splitting, the Teichmüller mapping of (9.3.34), which will be here denoted :

τ_{A} : E(A) → A(A).

Lemma 16.1.1. With the notation of (16.1), the following holds :

(i) There exists a unique ring homomorphism u_{A} : A(A) → A that makes commute the diagram :

\xymatrix{A(A) \ar[r]^{u_{A}} \ar[d]_{_{ω_{0}}} & A \ar[d]^{π_{pA}} \\ E(A) \ar[r]^{_{\overline{u}_{A/pA}}} & A/pA.}

(ii) The map u_{A} is continuous for the topology 𝒯 and the topology of A(A, 𝒯_{A}).

(iii) A(A)^{×} = u_{A}^{−1}(A^{×}).

(iv) u_{A} ∘ τ_{A} = \overline{u}_{A}.

Proof. We have already remarked that A is complete and separated for its p-adic topology, and the filtration (J_{n} := p^{n}A | n ∈ ℕ) of A trivially fulfills the condition of lemma 9.3.4; therefore (i) follows immediately from proposition 9.3.52(iii). More precisely, (9.3.53) translates as the following explicit expression for u_{A} :

u(\underline{a}) = ∑_{n∈ℕ} p^{n} · \overline{u}_{A}(a_{n}^{p^{−n}}) for every \underline{a} := (a_{n} | n ∈ ℕ) ∈ A(A)

from which (ii) follows easily : details left to the reader.

(iii): The inclusion A(A)^{×} ⊂ u_{A}^{−1}(A^{×}) is obvious. For the converse, let α ∈ A(A) be any element such that u_{A}(α) ∈ A^{×}; then ω_{0}(α) ∈ \overline{u}_{A}^{−1}(A^{×}). By remark 9.4.5(iii) we deduce that ω_{0}(α) ∈ E(A)^{×}, so it remains only to notice that ω_{0}^{−1}(E(A)^{×}) = A(A)^{×}, due to proposition 9.3.44(iii).

(iv) is clear from the foregoing explicit formula for u_{A}. □

Furthermore, A(A) is endowed with an automorphism σ_{A} : A(A) → A(A) that lifts the Frobenius map of E(A), i.e. such that the horizontal arrows of the diagram :

(16.1.2)  \xymatrix{A(A) \ar[r]^{σ_{A}} \ar[d]_{_{ω_{0}}} & A(A) \ar[d]_{_{ω_{0}}} \\ E(A) \ar[r]^{Φ_{E(A)}} \ar[u]_{τ_{A}} & E(A) \ar[u]_{τ_{A}}}

commute with the downward arrows (proposition 9.3.22(iii)). The horizontal arrows commute also with the upward ones, due to (9.3.43).

16.1.3. Keep the notation of (16.1), and recall that τ_{A} induces a continuous map Spec τ_{A} : Spec A(A) → Spec E(A) as in proposition 9.3.66(ii); in light of lemma 16.1.1(iv) and remark 8.1.14(ii) we deduce a continuous and spectral map

Spec \overline{u}_{A} := (Spec τ_{A}) ∘ (Spec u_{A}) : Spec A → Spec E(A) 𝔭 ↦ \overline{u}_{A}^{−1}𝔭.

Moreover, due to proposition 9.3.66(iv), every morphism φ : A → A′ of topological algebras that fulfill the conditions of (16.1) induces a commutative diagram of topological spaces :

(16.1.4)  \xymatrix{Spec A′ \ar[r]^{^{Spec \overline{u}^{A′}}} \ar[d]_{Spec φ} & Spec E(A′) \ar[d]^{Spec E(φ)} \\ Spec A \ar[r]^{_{Spec \overline{u}_{A}}} & Spec E(A).}

Lemma 16.1.5. In the situation of (16.1.3), let 𝔭 ∈ Spec A be any prime ideal, set 𝔮 := Spec \overline{u}_{A}(𝔭) ∈ Spec E(A), and let π_{𝔭} : A → κ(𝔭), π_{𝔮} : E(A) → κ(𝔮) be the projections. If 𝔭 is a closed subset in the p-adic topology of A, then \overline{u}_{A} induces a morphism of multiplicative monoids

\overline{u}_{𝔭} : κ(𝔮) → κ(𝔭)  such that  \overline{u}_{𝔭} ∘ π_{𝔮} = π_{𝔭} ∘ \overline{u}_{A}.

Proof. Since u_{A} is continuous for the p-adic topologies, u_{A}^{−1}𝔭 is a closed subet for the p-adic topology of A(A), so the assertion follows immediately from proposition 9.3.66(iii). □

With the following definition, we single out certain elements of A(A) that shall play a special role in the study of perfectoid rings.

Definition 16.1.6. Let (E, 𝒯 ) be any topological 𝔽_{p}-algebra.

(i) An element (a_{n} | n ∈ ℕ) ∈ W (E) is called distinguished if a_{0} is topologically nilpotent in E, and a_{1} ∈ E^{×}.

(ii) An ideal of W (E) is called distinguished if it is generated by a distinguished element.

Remark 16.1.7. Let E be any topological 𝔽_{p}-algebra, \underline{a} := (a_{n} | n ∈ ℕ) and \underline{b} := (b_{n} | n ∈ ℕ) two elements of W (E), such that \underline{a} is distinguished and set \underline{c} := \underline{a} · \underline{b}.

(i) Suppose that the topology of E is linear, complete and separated. Then \underline{c} is distinguished if and only if \underline{b} is invertible in W (E). Indeed, say that \underline{c} = (c_{n} | n ∈ ℕ). Then remark 9.3.8 says that c_{0} = a_{0}b_{0}, especially c_{0} is topologically nilpotent. Also, c_{1} = a_{0}^{p}b_{1} + a_{1}b_{0}^{p}, and notice that b_{0} ∈ E^{×}, if and only if \underline{b} ∈ W (E)^{×} (proposition 9.3.44(iii)); on the other hand, since the topology of E is linear, a_{0}^{p}b_{1} is topologically nilpotent (remark 8.3.9(v)). Since the topology of E is linear, complete and separated, it follows that c_{1} is invertible if and only if the same holds for b_{0} (details left to the reader), whence the assertion.

(ii) If E is reduced, then every distinguished ideal of W (E) is a closed subset for the p-adic topology of W (E) and a free W (E)-module of rank one (proposition 9.3.47(i,v)).

(iii) Let E be as in (i), and I, J ⊂ W (E) any two distinguished ideals such that I ⊂ J; then (i) implies that I = J. By the same token, we see that every generator of I is distinguished.

Indeed, let \underline{a}_{1} and \underline{a}_{2} be any two generators of I; then there exists \underline{u} := (u_{n} | n ∈ ℕ) ∈ W (E) such that \underline{a}_{2} = \underline{u} · \underline{a}_{1}. Let N ⊂ E be the nilpotent ideal, and endow E_{red} := E/N with the quotient topology induced from E; it follows from (ii) that the image of \underline{u} is invertible in W (E_{red}), therefore the image of u_{0} is invertible in E_{red}, so u_{0} is invertible in E, and finally \underline{u} is invertible in W (E) (proposition 9.3.44(iii)).

Example 16.1.8. With the notation of example 9.3.78, notice that the only distinguished elements of W (A, 𝒯_{T}) = ℤ_{p}{T^{1/p^{∞}}} are of the form pu, where u ∈ W (A, 𝒯_{T})^{×} is an arbitrary element; especially, the only distinguished ideal of ℤ_{p}{T^{1/p^{∞}}} is the principal ideal generated by p. On the other hand, the distinguished elements of W (A^{∧}, 𝒯_{T}^{∧}) are all those of the form pu + aT^{λ}, where a (resp. u) is an arbitrary element (resp. invertible element) of W (A^{∧}, 𝒯_{T}^{∧}), and λ ∈ ℕ[1/p] is any strictly positive number.

The following result shall be applied to produce useful distinguished elements in various situations.

Lemma 16.1.9. Let E be an 𝔽_{p}-algebra, ℐ ⊂ W (E) an ideal, \underline{α} := (α_{n} | n ∈ ℕ) ∈ ℐ, and β_{•} := (β_{1}, …, β_{k}) a finite system of elements of E. Denote by J ⊂ E (resp. by 𝒥 ⊂ W (E)) the ideal generated by β_{•} (resp. by τ_{E}(β_{1}), …, τ_{E}(β_{k})), and by u : W (E) → A := W (E)/ℐ the projection. Suppose that :

(a) α_{0} lies in the Jacobson radical of E and α_{1} ∈ E^{×}.

(b) The Frobenius endomorphism Φ_{E} of E is surjective.

(i) Then the following conditions are equivalent :

(c) pA ⊂ u(𝒥).

(d) There exists an element \underline{α}′ := (α′_{n} | n ∈ ℕ) ∈ ℐ such that α′_{0} ∈ J and the image of α′_{1} is invertible in E/J.

(e) There exists \underline{α}′ := (α′_{n} | n ∈ ℕ) ∈ ℐ such that α′_{0} ∈ J and α′_{1} ∈ E^{×}. (ii) Suppose furthermore that ℐ = \underline{α}W (E). Then (c),(d) and (e) are also equivalent to :

(f) α_{0} ∈ J.

(g) pW (E) + 𝒥 = \underline{α}W (E) + 𝒥.

Proof. (c)⇒(d): Assumption (c) implies that there exist \underline{γ}_{1}, …, \underline{γ}_{k} ∈ W (E) such that

\underline{α}′ := p + \underline{γ}_{1} · τ_{E}(β_{1}) + ⋯ + \underline{γ}_{k} · τ_{E}(β_{k}) ∈ ℐ

and we claim that (d) holds for this \underline{α}′. Indeed, α′_{0} = ∑_{i=1}^{k} ω_{0}(\underline{γ}_{i}) · β_{i} ∈ J, and clearly the image of \underline{α}′ in W (E/J) equals p, i.e. \underline{α}′_{1} ≡ 1 (mod J).

(d)⇒(c): Let \underline{α}′ be as in (d); then there exists λ ∈ E such that λ · α_{1} ∈ 1 + J, and assumption (b) implies that λ = γ^{p} for some γ ∈ E. Recall that \underline{α}′′ := τ_{E}(γ) · \underline{α}′ = (γ^{p^{n}} · α′_{n} | n ∈ ℕ) (proposition 9.3.36(i)); we may then replace \underline{α}′ by \underline{α}′′, after which we may assume that α′_{1} ≡ 1 (mod J). Say that α′_{0} = ∑_{i=1}^{k} c_{i}β_{i} for some c_{1}, …, c_{k} ∈ E, and set

\underline{δ} := \underline{α}′ − (τ_{E}(c_{1}) · τ_{E}(β_{1}) + ⋯ + τ_{E}(c_{k}) · τ_{E}(β_{k})).

Notice that δ_{0} = 0, and the images of \underline{α}′ and \underline{δ} agree in W (E/J); in particular, δ_{1} ≡ 1 (mod J). Invoking (b) again, we may find γ ∈ E such that γ^{p} = δ_{1}, and after replacing \underline{α}′ by τ_{E}(γ^{p−1})·\underline{α}′, and c_{i} by γ^{p−1}c_{i} for i = 1, …, k, we may further assume that δ_{1} ≡ 1 (mod Φ_{E}(J)). In this case, say that δ_{1} = 1+∑_{i=1}^{k} d_{i}^{p}β_{i}^{p} for some d_{1}, …, d_{k} ∈ E; taking into account remark 9.3.8(ii) and (9.3.41) we see that

\underline{δ} ≡ p + p · (τ_{E}(d_{1}) · τ_{E}(β_{1}) + ⋯ + τ_{E}(d_{k}) · τ_{E}(β_{k})) (mod p^{2}W (E)).

Summing up, we conclude that there exists \underline{x} ∈ W (E) such that p + p^{2}\underline{x} − \underline{α}′ ∈ 𝒥. Hence, p · (1 + p · u(\underline{x})) ∈ u(𝒥). Lastly, we have 1 + p · \underline{x} ∈ W (E)^{×}, since p is topologically nilpotent in W (E), whence (c).

Obviously, (e)⇒(d), so we suppose that (d) holds, and we show (e). Arguing as in the foregoing, we may assume that α′_{1} ≡ 1 (mod J). Next, using (b) we find γ ∈ E such that γ^{p} = α′_{1}, and after replacing α′ by τ_{E}(γ^{p−1}) · \underline{α}′, we may even assume that α′_{1} ≡ 1 (mod Φ_{E}(J)).

Likewise, we may find δ ∈ E such that δ^{p} = α_{1}^{−1}, and after replacing \underline{α} by τ_{E}(δ) · \underline{α} we may assume that α_{1} = 1. Now, write α′_{1} = 1 − x^{p} for some x ∈ J, and set

\underline{α}′′ := \underline{α}′ + τ_{E}(x) · \underline{α}.

In light of remark 9.3.8(ii), we easily see that α′′_{0} ∈ J and α′′_{1} − 1 ∈ α_{0}E. Since α_{0} lies in the Jacobson radical of E, it follows that α′′_{1} ∈ E^{×}, whence (e).

Lastly, suppose that ℐ = \underline{α}W (E). In this case, we already know that (f)⇒(e). Conversely, if (e) holds, remark 9.3.8(ii) shows that there exists \underline{β} := (β_{n} | n ∈ ℕ) ∈ W (E) with

α_{0}β_{0} ∈ J and α_{0}^{p}β_{1} + α_{1}β_{0}^{p} ∈ E^{×}.

Then, since α_{0} is in the Jacobson radical of E, we deduce that α_{1}β_{0}^{p} ∈ E^{×}, so also β_{0} ∈ E^{×}, and (f) follows. Next, if (c) holds, then W (E)/(𝒥 + \underline{α}W (E)) = A/u(𝒥) is an 𝔽_{p}-algebra, i.e. p ∈ 𝒥 + \underline{α}W (E); on the other hand, (b) implies that pW (E) = Ker(ω_{0} : W (E) → E) (see (9.3.41)). Hence, W (E)/(𝒥 + \underline{α}W (E)) = E/(α_{0}E + J) = E/J, since we already know that (c)⇒(f); but we have as well W (E)/(𝒥 + pW (E)) = E/J, whence (g). Conversely, if (g) holds, we see that A/u(𝒥) is an 𝔽_{p}-algebra, whence (c). □

Remark 16.1.10. Keep the notation of lemma 16.1.9, and let S := 1 + pW (E) + 𝒥 and R := S^{−1}W (E). Suppose that for some v ∈ R we have

p ∈ v · \underline{α} + 𝒥 R (resp. \underline{α}_{0} ∈ vp + 𝒥 R) .

Then we claim that v ∈ R^{×}. Indeed, it suffices to show that v ∉ 𝔪, for every 𝔪 ∈ Max R. However, we have a natural identification : Max R →^{∼} Max W (E)/(pW (E) + 𝒥) = Max E/J, and since Φ_{E} is surjective, it follows easily that the residue field k(𝔪) is perfect for every such 𝔪, in which case W (k(𝔪)) is a discrete valuation ring of mixed characteristic (0, p) (proposition 9.3.47(iii)); moreover, the projection W (E) → k(𝔪) factors through E, so we get a commutative diagram of rings :

\xymatrix{W (E) \ar[r]^{W (π)} \ar[d] & W (k(𝔪)) \ar[d] \\ E \ar[r]^{π} & k(𝔪)}

whose vertical arrows are the 0-th ghost components. It suffices then to check that x := W (π)(v) ∈ W (k(𝔪))^{×}. Now, if p ∈ v · \underline{α} + 𝒥, we get p = x · W (π)(\underline{α}) in W (k(𝔪)), and notice that π(α_{0}) = 0 and π(α_{1}) ∈ k(𝔪)^{×}, due to assumption (a) of lemma 16.1.9; hence both p and W (π)(\underline{α}) generate the maximal ideal of W (k(𝔪)), whence the assertion in this case. Likewise, if \underline{α} ∈ vp + 𝒥, we get W (π)(\underline{α}) = vp in W (k(𝔪)), and again the assertion follows.

16.1.11. Additionally, often we will be dealing with pairs (\underline{a}, 𝒦 ) consisting of an ideal 𝒦 in a given 𝔽_{p}-algebra E, and an element \underline{a} ∈ W (E) (usually \underline{a} shall be a distinguished element); then we shall say that (\underline{a}, 𝒦 ) is a transversal pair if we have

W (𝒦 ) ∩ \underline{a}W (E) = \underline{a}W (𝒦 ).

This property plays a crucial role in several questions about perfectoid rings, and the following proposition collects a few example of such pairs. One more case is given by theorem 16.3.36(iv).

Proposition 16.1.12. Let E be any perfect 𝔽_{p}-algebra, and \underline{a} := (a_{n} | n ∈ ℕ) ∈ W (E) any element. The following holds :

(i) Suppose that a_{1} ∈ E^{×}, and let J_{1}, J_{2} ⊂ E be any two ideals such that (a) J_{1}^{⟨1⟩}E = J_{1} and J_{2}^{⟨1⟩}E = J_{2} (notation of (9.3.68))

(b) there exist n, m ∈ ℕ such that a_{0}^{n}J_{1} ⊂ J_{2} and a_{0}^{m}J_{2} ⊂ J_{1}.

Then the pair (\underline{a}, J_{1}) is transversal if and only if the same holds for the pair (\underline{a}, J_{2}). (ii) Suppose that E = ∑_{n∈ℕ} a_{n}E. Then the pair (\underline{a}, I^{⌊r⌋}E) is transversal, for every ideal

I ⊂ E and every r ∈ ℝ_{+} (notation of remark 9.3.70(i)). (iii) Let J ⊂ E be any ideal such that J^{⟨1⟩}E = J. Then the pair (p, J) is transversal.

Proof. (iii) is immediate from (9.3.45) and remark 9.3.70(iv).

(i): First, we show that the assertion holds in the special case where n = 0 and m = 1. To begin with, from lemma 9.3.27(i) and our assumption on a_{1} we obtain

\underline{a} = τ_{E}(a_{0}) + p\underline{u} for some \underline{u} ∈ W (E)^{×}.

Notice as well that \underline{a} is a regular element of W (E), by virtue of proposition 9.3.47(i). Now, suppose first that the pair (\underline{a}, J_{1}) is transversal, and let x ∈ W (E) be any element such that \underline{a} · x ∈ W (J_{2}); then τ_{E}(a_{0}) · \underline{a} · x ∈ W (J_{1}) (proposition 9.3.36(i)), so τ_{A}(a_{0}) · x ∈ W (J_{1}), by assumption. Hence p\underline{u} · x = (\underline{a} − τ_{A}) · x ∈ W (J_{2}), so px ∈ W (J_{2}) ∩ pW (E) = pW (J_{2}), where the last equality follows from (i.a) and (iii); therefore x ∈ W (J_{2}), as p is regular in W (E).

Likewise, suppose that (\underline{a}, J_{2}) is a transversal pair, and let x ∈ W (E) be any element such that \underline{a} · x ∈ W (J_{1}); then \underline{a} · x ∈ W (J_{2}), so x ∈ W (J_{2}), and τ_{E}(a_{0}) · x ∈ W (J_{2}) (again, by proposition 9.3.36(i)). It follows that p\underline{u} · x = (\underline{a} − τ_{E}(a_{0})) · x ∈ W (J_{2}), whence px ∈ W (J_{2}) ∩ pW (E) = pW (J_{2}), so x ∈ W (J_{2}), as required.

Next, let (J_{1}, J_{2}) be any pair of ideals fulfilling conditions (i.a) and (i.b). Notice that we have (a_{0}^{i}J_{1})^{⟨1⟩}E = a_{0}^{i}J_{1} for every i ∈ ℕ (lemma 9.3.69(iii)), and the foregoing case implies that assertion (i) holds for the pair (a_{0}^{i}J_{1}, a_{0}^{i+1}J_{1}), for every i ∈ ℕ. It follows that assertion (i) also holds for the pair (J_{1}, a_{0}^{n}J_{1}); consequently we are reduced to showing assertion (i) for the pair (J_{2}, a_{0}^{n}J_{1}), and notice that a_{0}^{m+n}J_{2} ⊂ a_{0}^{n}J_{1}. Thus, we may replace J_{1} by a_{0}^{n}J_{1}, m by n + m, and assume from start that J_{1} ⊂ J_{2} and a_{0}^{m}J_{2} ⊂ J_{1}. Now, we have

a_{0} · (J_{1} + a_{0}^{i}J_{2})^{⟨1⟩}E = (a_{0}J_{1} + a_{0}^{i+1}J_{2})^{⟨1⟩}E ⊂ (J_{1} + a_{0}^{i+1}J_{2})^{⟨1⟩}E for every i ∈ ℕ

so that – again by the foregoing case – we know that assertion (i) holds for the pair

((J_{1} + a_{0}^{i}J_{2})^{⟨1⟩}E, (J′ + a_{0}^{i+1}J_{2})^{⟨1⟩}E) for every i ∈ ℕ

and then it also holds for the pair ((J_{1} + a_{0}^{m}J_{2})^{⟨1⟩}E, (J_{1} + J_{2})^{⟨1⟩}E); but (J_{1} + a_{0}^{m}J_{2})^{⟨1⟩}E = J_{1} and (J_{1} + J_{2})^{⟨1⟩}E = J_{2}, whence the contention.

(ii): Indeed, let \underline{b} ∈ W (A) be any element, and suppose that \underline{a} · \underline{b} ∈ W (I^{⌊r⌋}E). Define ℛ_{E} as in lemma 9.3.85; by lemma 9.3.85(iv) and proposition 9.3.82(iii), we deduce that

|\underline{a}|_{1} · |\underline{b}|_{1} = |\underline{a} · \underline{b}|_{1} ≤ |I|^{r} for every | · | in ℛ_{E} .

But since the system \{a_{n} | n ∈ ℕ\} generates E, it is easily seen that |\underline{a}|_{1} = 1 for every | · | in ℛ_{E}. We conclude that |\underline{b}|_{1} ≤ |I|^{r} for every such | · |, and then the assertion follows, again by invoking lemma 9.3.85(iv). □

Corollary 16.1.13. Let E be a perfect 𝔽_{p}-algebra, b_{•} := (b_{1}, …, b_{n}) a finite system of elements of E, and I ⊂ E (resp. 𝒥 ⊂ W (E)) the ideal generated by b_{•} (resp. by (τ_{E}(b_{1}) …, τ_{E}(b_{n}))). Let also ℐ := pW (E) + 𝒥, and \underline{a} := (a_{n} | n ∈ ℕ) ∈ W (E), and suppose that :

(a) E = ∑_{n∈ℕ} a_{n}E.

(b) E is complete and separated for its I-adic topology. We have :

(i) W (E) is complete and separated for both its ℐ -adic and its 𝒥 -adic topology.

(ii) The 𝒥 -adic topology of W (E) induces the 𝒥 -adic topology on \underline{a}W (E), and \underline{a}W (E) is a closed ideal for the 𝒥 -adic topology of W (E).

(iii) Suppose moreover that a_{0} ∈ I and a_{1} ∈ E^{×}. Then the ℐ -adic topology of W (E) induces the ℐ -adic topology on \underline{a}W (E).

Proof. (i): By proposition 9.3.77(ii) and lemma 9.3.33(ii), W (E) is complete and separated for its ℐ -adic topology, hence also for its 𝒥 -adic topology (lemma 8.3.12).

(ii): Taking into account proposition 9.3.77(i) and lemma 9.3.69(iv), we easily see that the 𝒥 -adic topology on W (E) agrees with the linear topology defined by the cofiltered system of ideals (W (I^{⌊q⌋}) | q ∈ ℚ_{+}). Then, propositions 16.1.12(ii) and 9.3.47(i) imply that scalar multiplication by \underline{a} is a W (E)-linear isomorphism

W (E) →^{∼} \underline{a}W (E)

that identifies the 𝒥 -adic topology of \underline{a}W (E) with the one induced by the 𝒥 -adic topology of W (E). In view of (i), we deduce that \underline{a}W (E) is complete and separated for the subspace topology induced from the 𝒥 -adic topology of W (E), whence the assertion.

(iii): Notice that I lies in the Jacobson radical of E, due to (b); hence, ℐ = \underline{a}W (E) + 𝒥, by lemma 16.1.9(ii). Now, let m ∈ ℕ \ {0}; by virtue of (i) there exists n ∈ ℕ such that for every \underline{w} ∈ W (E) with \underline{a} · \underline{w} ∈ 𝒥^{n} we have \underline{w} ∈ 𝒥^{m}, and clearly we may assume that n ≥ m. Then, let \underline{w} ∈ W (E) be any element such that \underline{a} · \underline{w} ∈ ℐ^{n}; thus, we have \underline{a} · \underline{w} = ∑_{i=0}^{n} \underline{a}^{i} · \underline{w}_{i} where \underline{w}_{i} ∈ ℐ^{n−i} for i = 0, …, n. Therefore, \underline{a} · (\underline{w} − ∑_{i=1}^{n} \underline{a}^{i−1} · \underline{w}_{i}) ∈ 𝒥^{n}, so that \underline{w} − ∑_{i=1}^{n} \underline{a}^{i−1} · \underline{w}_{i} ∈ 𝒥^{m}, by the foregoing, and finally \underline{w} ∈ ℐ^{m−1}, whence the assertion. □

16.2. P-rings. In this section we make a preliminary study of an auxiliary class of topological rings containing the perfectoid rings that shall be introduced in section 16.3.

Definition 16.2.1. A complete and separated topological ring (A, 𝒯 ) is called a P-ring if there exist a prime integer p and an ideal I ⊂ A such that the following conditions hold :

(a) I is finitely generated and 𝒯 agrees with the I-adic topology.

(b) pA ⊂ I^{2} and the Frobenius endomorphism of A/I^{2} is a surjection

Φ_{A/I^{2}} : A/I^{2} → A/I^{2} a ↦ a^{p}.

Any ideal I of A fulfilling conditions (a) and (b) is called an ideal of definition of A. Moreover, we say that I is special, if the following holds :

(c) There exists a finite set of generators a_{1}, …, a_{n} of I such that pA ⊂ I^{(p)} := ∑_{i=1}^{n} Aa_{i}^{p}.

It turns out that if I is a special ideal of definition of the P-ring A, the ideal I^{(p)} depends only on I (and not on the choice of a system of generators for I), so this notation is not abusive. Indeed, we have more generally :

Lemma 16.2.2. Let A be any ring, p a prime integer, and (a_{1}, …, a_{n}), (b_{1}, …, b_{m}) two sequences of elements of A. Suppose that

I := ∑_{i=1}^{n} Aa_{i} ⊂ J := ∑_{i=1}^{m} Ab_{i}  and  p ∈ I^{(p)} := ∑_{i=1}^{n} Aa_{i}^{p}.

Then :

(i) I^{(p)} ⊂ J^{(p)} := ∑_{i=1}^{m} Ab_{i}^{p}.
(ii) Especially, if I = J, we have I^{(p)} = J^{(p)}.

Proof. It is easily seen that (i)⇒(ii), so we need only check (i). To this aim, it suffices to show that I_{𝔪}^{(p)} ⊂ J_{𝔪}^{(p)} for every maximal ideal 𝔪 ⊂ A, so we may replace A by A_{𝔪} and assume from start that A is local. Now, suppose first that J = A; then it is easily seen that J^{(p)} = A as well, so the assertion is clear in this case. Hence, we may also assume that J is contained in the maximal ideal of A. By assumption, for every i = 1, …, n we may write a_{i} = ∑_{j=1}^{m} x_{ij}b_{j} for certain x_{i1}, …, x_{im} ∈ A; it follows easily that a_{i}^{p} − ∑_{j=1}^{m} x_{ij}^{p}b_{j}^{p} ∈ pJ^{p} (details left to the reader). Summing up, we see that I^{(p)} ⊂ J^{(p)} + pJ^{p} ⊂ J^{(p)} + I^{(p)}J^{(p)}, and the assertion follows by Nakayama’s lemma. □

Lemma 16.2.3. Let A be any P-ring, and I any ideal of definition of A. We have :

(i) If A ≠ 0, there exists a unique prime integer p, independent of I, such that condition (b) of definition 16.2.1is fulfilled. (ii) I is contained in the Jacobson radical of A. (iii) There exists a special ideal of definition J such that J^{(p)} = I. (iv) For p as in (i), the Frobenius endomorphism Φ_{A/pA} : A/pA → A/pA is a surjective

ring homomorphism, and there exist π ∈ A and u ∈ A^{×} such that p = uπ^{p}.

(v) For p as in (i), the ring A is p-adically complete and separated.

Proof. (ii) is standard : if a ∈ I, then 1 − ab is invertible in A for every b ∈ A, since A is I-adically complete and separated (the inverse is given by the convergent series ∑_{k∈ℕ}(ab)^{n}); the claim follows.

(i): Let J be another ideal of definition of A, and p, q ∈ ℤ two prime integers with p ∈ I^{2} and q ∈ J^{2}. By (ii), both p and q lie in the Jacobson radical of A; as A ≠ 0, we get p = q.

(iii): Let (f_{λ} | λ ∈ Λ) be a finite system of generators (indexed by the finite set Λ) for I; by assumption, we may find g_{λ} ∈ A such that f_{λ} − g_{λ}^{p} ∈ I^{2} for every λ ∈ Λ. From (ii) and Nakayama’s lemma, we deduce that the system (g_{λ}^{p} | λ ∈ Λ) generates I, so we can take for J the ideal generated by the system (g_{λ} | λ ∈ Λ).

(iv): Let (g_{λ} | λ ∈ Λ) be as in the proof of (iii). For any a ∈ A we construct inductively a sequence (b_{n} | n ∈ ℕ) of elements of A, such that

a − ∑_{k=0}^{n} b_{k}^{p} ∈ I^{n+1}  and b_{n} ∈ I^{n}  for every n ∈ ℕ.

Indeed, for n = 0, since Φ_{A/I} is surjective we may find b_{0} ∈ A such that a − b_{0}^{p} ∈ I. Next, suppose that n ≥ 0 and b_{0}, …, b_{n} ∈ A are already given such that c_{n} := a − ∑_{k=0}^{n} b_{k}^{p} ∈ I^{n+1}; we may find a system (d\underline{λ} | \underline{λ} ∈ Λ^{n+1}) of elements of A such that

c_{n} = ∑_{\underline{λ}∈Λ^{n+1}} gp ; \underline{λ}d\underline{λ} where g\underline{λ} := ∏_{i=1}^{n+1} g_{λ_{i}} for every \underline{λ} := (λ_{1}, …, λ_{n+1}) ∈ Λ^{n+1}.

In turn, for each \underline{λ} ∈ Λ^{n+1} we may write d\underline{λ} = ep ; \underline{λ} + e′ ; \underline{λ} for elements e\underline{λ} ∈ A and e′ ; \underline{λ} ∈ I, so that

c_{n} = c′_{n} + c′′_{n}  where c′_{n} := ∑_{\underline{λ}∈Λ^{n+1}} gp ; \underline{λ}ep ; \underline{λ}  c′′_{n} := ∑_{\underline{λ}∈Λ^{n+1}} gp ; \underline{λ}e′ ; \underline{λ} ∈ I^{n+2}.

Thus, b_{n+1} := ∑_{\underline{λ}∈Λ^{n+1}} g\underline{λ}e\underline{λ} ∈ I^{n+1}, and since p ∈ I^{2}, a simple computation shows that c′_{n} − b_{n+1}^{p} ∈ I^{n+2}, so finally a − ∑_{k=0}^{n+1} b_{k}^{p} ∈ I^{n+2}. Notice that the series ∑_{k∈ℕ} b_{k}^{p} converges to a in the I-adic topology of A, and set b := ∑_{k∈ℕ} b_{k}; it follows easily that a − b^{p} ∈ pA, whence the surjectivity of Φ_{A/pA}. Lastly, by assumption we may write p = ∑_{i=1}^{r} a_{i}b_{i} for finitely many elements a_{1}, b_{1}, …, a_{r}, b_{r} ∈ I; by the foregoing, for every i = 1, …, r we may further find f_{i}, g_{i}, f′_{i}, g′_{i} ∈ A such that a_{i} = f_{i}^{p} + pf′_{i}, b_{i} = g_{i}^{p} + pg′_{i}. Then f_{i}^{p}, g_{i}^{p} ∈ I, and (ii) implies that f_{i} and g_{i} lie in the Jacobson radical rad(A) of A, for every i ≤ r. Set π := ∑_{i=1}^{r} f_{i}g_{i}; summing up, we find p − π^{p} = pc for some c ∈ rad(A), so p · (1 − c) = π^{p}, and to conclude it suffices to remark that 1 − c ∈ A^{×}.

(v) is a special case of lemma 8.3.12. □

Remark 16.2.4. Suppose that A is a complete and separated topological 𝔽_{p}-algebra. Then it follows easily from lemma 16.2.3(iv) that A is a P-ring if and only if its Frobenius endomorphism Φ_{A} is surjective, and there exists a finitely generated ideal I ⊂ A such that the topology of A agrees with the I-adic topology.

Proposition 16.2.5. Let A be a ring, p a prime integer, I ⊂ A a finitely generated ideal such that pA ⊂ I, and denote by 𝒯_{I} (resp. 𝒯_{p}) the I-adic (resp. p-adic) topology on A. We have :

(i) If (A, 𝒯_{I}) is a P-ring, the same holds for (A, 𝒯_{p}).

(ii) If (A, 𝒯_{p}) is a P-ring, the same holds for the completion (A_{I}^{∧}, 𝒯_{I}^{∧}) of (A, 𝒯_{I}).

Proof. (i) follows immediately from lemma 16.2.3(iv,v).

(ii): Let (f_{1}, …, f_{k}) be a finite system of generators of I; by lemma 16.2.3(iv) there exists π ∈ A such that pA = π^{p}A, and the Frobenius endomorphism Φ_{A/pA} of A/pA is surjective, so for every i = 1, …, k we may find g_{i} ∈ A such that f_{i} − g_{i}^{p} ∈ pA, and we let J ⊂ A be the ideal generated by the finite system (g_{1}, …, g_{k}, π). Set N := (p − 1)(k + 1) + 1, and notice that J^{N} ⊂ I ⊂ J^{p}. Especially, 𝒯_{I} agrees with the J-adic topology on A, and moreover p ∈ J^{2}. Furthermore, 𝒯_{I}^{∧} agrees with the JA_{I}^{∧}-adic topology on A_{I}^{∧} (remark 8.3.3(iv) and lemma 8.3.32(iv)), and the natural map A/J^{2} → A_{I}^{∧}/J^{2}A_{I}^{∧} is an isomorphism (remark 8.3.3(ii,iv)); since Φ_{A/pA} is surjective by assumption, the same holds for Φ_{A/J^{2}}, and the assertion follows.

□

16.2.6. Henceforth we fix a prime integer p, and we shall assume that the topology of every P-ring that appears throughout the rest of this chapter, is coarser than the p-adic topology. Let (A, 𝒯 ) be any P-ring and I ⊂ A an ideal of definition. Notice that if I is special (definition 16.2.1), the Frobenius endomorphism Φ_{A/I} of A/I factors uniquely as the composition of a morphism of topological rings

\overline{Φ}_{A/I} : A/I → A/I^{(p)}

and the natural projection A/I^{(p)} → A/I.

Lemma 16.2.7. In the situation of (16.2.6), we have :

(i) The maps u_{A} : A(A) → A and \overline{u}_{A/pA} : E(A) → A/pA are open and surjective.

(ii) Ker u_{A} contains a distinguished element of A(A) (see definition 16.1.6).

(iii) If \underline{α} := (α_{n} | n ∈ ℕ) is any distinguished element in Ker u_{A}, we have

\overline{u}_{A}(α_{0}) = pu for some u ∈ A^{×}.

Proof. (i): Let I ⊂ A be any ideal of definition of A, so that the topology of A/pA agrees with the I/pA-adic topology; by lemma 16.2.3(iv) we know already that Φ_{A/pA} is surjective, and then it is easily seen that there exists n ∈ ℕ such that (I/pA)^{n} ⊂ Φ_{A/pA}(I). Especially, Φ_{A/pA} is an open map; then, the assertion for \overline{u}_{A/pA} follows from remark 9.4.9(iii). By virtue of [126, Th.8.4], we deduce the surjectivity of u_{A}. Next, let (a_{1}, …, a_{r}) be a finite system of generators for I, and denote by \overline{a}_{i} the image of a_{i} in A/pA, for every i = 1, …, k. Since \overline{u}_{A/pA} is surjective, we may find \overline{α}_{1}, …, \overline{α}_{k} ∈ E(A) such that \overline{u}_{A/pA}(\overline{α}_{i}) = \overline{a}_{i} for i = 1, …, k, and we set α_{i} := τ_{A}(\overline{α}_{i}) for i = 1, …, k (notation of (16.1)). Since I is topologically nilpotent in A, it is easily seen that \overline{α}_{i} is topologically nilpotent in E(A), and since τ_{A} is continuous, we deduce that α_{i} is topologically nilpotent in A(A), for every i = 1, …, k. Let ℐ ⊂ A(A) be the ideal generated by the system (α_{1}, …, α_{k}); it follows easily that ℐ is topologically nilpotent as well. On the other hand, by construction we have a_{i} − u_{A}(α_{i}) ∈ pA ⊂ I^{2} for every i = 1, …, k, so that u_{A}(ℐ) = I, by Nakayama’s lemma. Lastly, let 𝒥 ⊂ A(A) be any open ideal; then ℐ^{n} ⊂ 𝒥 for every sufficiently large n ∈ ℕ, so I^{n} ⊂ u_{A}(ℐ), and the assertion follows.

(ii): We notice the following :

Claim 16.2.8. There exist w ∈ A^{×} and t ∈ E(A) such that p = w · \overline{u}_{A}(t) in A.

Proof of the claim. By lemma 16.2.3(iv), we may write p = v · x^{p} for some v ∈ A^{×} and x ∈ A. Since \overline{u}_{A/pA} is surjective, we may find s ∈ E(A) such that x ≡ \overline{u}_{A}(s) (mod pA). In view of lemma 9.3.4(i) we deduce that p ≡ v · \overline{u}_{A}(s)^{p} (mod p^{2}A), i.e. p = v · \overline{u}_{A}(t) + p^{2}b for t := s^{p} and some b ∈ A; thus p · (1 − pb) = v · \overline{u}_{A}(t). But since A is p-adically complete and separated (lemma 16.2.3(v)), 1 − pb ∈ A^{×} (remark 8.3.10(v)), so we get the sought identity with w := v · (1 − pb)^{−1}. ♢

Now, take w and t as in claim 16.2.8. Since u_{A} is surjective, we may find β := (b_{n} | n ∈ ℕ) ∈ A(A) with u_{A}(β) = w. By proposition 9.3.36(i) we have β · τ_{A}(t) = (t^{p^{n}}b_{n} | n ∈ ℕ), and (9.3.42) and remark 9.3.8(ii) show that

γ := β · τ_{A}(t) − p = (tb_{0}, t^{p}b_{1}, …) − (0, 1, …) = (tb_{0}, t^{p}b_{1} − 1, …).

But notice that t is topologically nilpotent in E(A); since the topology of E(A) is linear, it follows that tb_{0} and t^{p}b_{1} are topologically nilpotent in E(A), and since E(A) is complete and separated, 1 − t^{p}b_{1} ∈ E(A)^{×}, so γ is distinguished, and it lies in the kernel of u_{A}, due to lemma 16.1.1(iv).

(iii): Indeed, from (9.3.41) and lemma 9.3.27(i) we get \underline{α} = τ_{A}(α_{0}) + p · \underline{β}, where β := (α_{n+1}^{1/p} | n ∈ ℕ), and since α_{1} ∈ E(A)^{×}, we have \underline{β} ∈ A(A)^{×} (proposition 9.3.44(iii)); now, \overline{u}_{A}(α_{0}) = −p · u_{A}(\underline{β}), whence the assertion, in light of lemma 16.1.1(iv). □

Proposition 16.2.9. In the situation of (16.2.6), the following conditions are equivalent : (a) E(A) is a P-ring.

(b) There exists an ideal of definition I of A such that Ker Φ_{A/I} is a finitely generated ideal of A/I.

(c) There exists a special ideal of definition J of A such that \overline{Φ}_{A/J} is an isomorphism.

Proof. For any ideal of definition I of A, recall that E(A/I) is the limit of the inverse system (A_{n} | n ∈ ℕ) with A_{n} := A/I for every n ∈ ℕ, and with transition maps φ_{n} : A_{n+1} → A_{n} given by the Frobenius endomorphism Φ_{A/I} of A/I. For every n ∈ ℕ we shall denote by

\overline{u}_{n} : E(A) → A_{n}

the composition of the canonical identification E(A) →^{∼} E(A/I) as in (16.1), with the natural projection E(A/I) → A_{n}. Especially \overline{u}_{0} = \overline{u}_{A/I}; also, each \overline{u}_{n} is surjective, since the same holds for Φ_{A/I}. Set 𝒥_{0} := Ker \overline{u}_{A/I}, and let Φ_{E(A)} be the Frobenius endomorphism of E(A); then the family of ideals

𝒥_{n} := Φ_{E(A)}^{n}(𝒥_{0}) = Ker \overline{u}_{n} for every n ∈ ℕ

is a fundamental system of open neighborhoods of 0 for the topology 𝒯_{E(A)} of E(A). Especially, every element of 𝒥_{0} is topologically nilpotent, and since E(A) is complete and separated (remark 9.4.9(i)), we easily deduce that 𝒥_{0} is contained in the Jacobson radical of E(A) (cp. the proof of lemma 16.2.3(ii)).

(c)⇒(a): Let J be as in (c), and take I := J in the foregoing. Pick a finite system (a_{1}, …, a_{r}) of generators of J, and let π : A → A/J^{(p)} be the projection. A simple inspection of the definition yields the identity :

\overline{u}_{A/J^{(p)}} = \overline{Φ}_{A/J} ∘ \overline{u}_{1}.

Since \overline{Φ}_{A/J} is an isomorphism, we deduce that \overline{u}_{A/J^{(p)}} is a surjection and we may find

α_{1}, …, α_{r} ∈ E(A) with \overline{u}_{A/J^{(p)}}(α_{i}) = π(a_{i}) for every i = 1, …, r.

Let 𝒥′_{0} ⊂ E(A) be the ideal generated by the system (α_{1}, …, α_{r}), and set

𝒥′_{n} := Φ_{E(A)}^{n}(𝒥′_{0}) for every n ∈ ℕ.

Claim 16.2.10. 𝒥′_{0} = 𝒥_{0}, and the 𝒥_{0}-adic topology on E(A) agrees with 𝒯_{E(A)}.

Proof of the claim. A simple inspection shows that 𝒥′_{0} ⊂ 𝒥_{0}, and consequently 𝒥′_{n} ⊂ 𝒥_{n} for every n ∈ ℕ. Moreover, \overline{u}_{A/J^{(p)}}(𝒥′_{0}) = J/J^{(p)}, and therefore

\overline{u}_{1}(𝒥′_{0}) = \overline{Φ}_{A/J}^{−1} ∘ \overline{u}_{A/J^{(p)}}(𝒥′_{0}) = \overline{Φ}_{A/J}^{−1}(J/J^{(p)}) = Ker φ_{0} = 𝒥_{0}/𝒥_{1}.

Especially, the natural map 𝒥′_{0}/𝒥′_{1} → 𝒥_{0}/𝒥_{1} is surjective, and since Φ_{E(A)} is an automorphism, we deduce that the induced map 𝒥′_{n}/𝒥′_{n+1} → 𝒥_{n}/𝒥_{n+1} is surjective for every n ∈ ℕ. The filtration (𝒥_{n} | n ∈ ℕ) is separated on E(A), so the identity 𝒥_{0} = 𝒥′_{0} will follow from [34, Ch.III, §2, n.8, Cor.2], after we show that E(A) is complete for the topology 𝒯 defined by the filtration (𝒥′_{n} | n ∈ ℕ). However, since 𝒥′_{0} is finitely generated, for every n ∈ ℕ there exists k(n) ∈ ℕ such that 𝒥′_{k(n)} ⊂ 𝒥_{0}^{′k(n)} ⊂ 𝒥′_{n}, so the topology 𝒯 agrees with the 𝒥′_{0}-adic topology, and 𝒥′_{0} is topologically nilpotent for the topology 𝒯_{E(A)}, since it is contained in 𝒥_{0}. Then, the assertion follows from lemma 8.3.12. By the same token, we see that the 𝒥_{0}-adic topology agrees with 𝒯_{E(A)}. ♢

The assertion follows immediately from claim 16.2.10and remark 16.2.4.

(a)⇒(b): Let I ⊂ A and ℐ ⊂ E(A) be two ideals of definition; then there exists n ∈ ℕ such that 𝒥_{n} ⊂ ℐ. Clearly ℐ′ := Φ_{E(A)}^{−n}(ℐ) is still an ideal of definition, and 𝒥_{0} ⊂ ℐ′. After replacing ℐ by ℐ′, we may therefore assume that 𝒥_{0} ⊂ ℐ. Write \overline{u}_{A/I}(ℐ) = I′/I for an ideal I′ ⊂ A; we have :

Claim 16.2.11. ℐ = Ker \overline{u}_{A/I′} and I′ is an ideal of definition of A.

Proof of the claim. We have just remarked that Ker \overline{u}_{A/I} ⊂ ℐ, and the stated identity is an easy consequence. Next, since both ℐ and I are finitely generated, the same holds for I′, and obviously pA ⊂ I^{′2}. Also ℐ is topologically nilpotent in E(A), and the topology of A/I is discrete, hence I′/I is a nilpotent ideal of A/I, and it follows easily that the I-adic topology on A agrees with the I′-adic topology. ♢

Due to claim 16.2.11, we may replace I by I′, and assume from start that ℐ = 𝒥_{0}. We may write \overline{u}_{1}(𝒥_{0}) = J_{1}/I for an ideal J_{1} ⊂ A containing I, and since 𝒥_{0} and I are finitely generated, the same holds for J_{1}. It suffices now to notice that

(16.2.12)  J_{1}/I = Ker φ_{0}.

(b)⇒(c): By assumption, there exists a finitely generated ideal J_{1} ⊂ A containing I and such that (16.2.12) holds. Furthermore, by lemma 16.2.3(iii) we may write I = J′^{(p)} for some ideal of definition J′ ⊂ A. Set J := J′ + J_{1}. From (16.2.12) we get J_{1}^{(p)} ⊂ I, therefore J^{(p)} = J′^{(p)} + J_{1}^{(p)} = I. We conclude that Φ_{A/J} induces an isomorphism A/J_{1} →^{∼} A/I = A/J^{(p)}; but the latter factors through \overline{Φ}_{A/J} and the projection A/J_{1} → A/J, so \overline{Φ}_{A/J} is an isomorphism as well (and J = J_{1}). Lastly, since 𝒥_{0} is topologically nilpotent, J_{1}/I is a nilpotent ideal, so the same holds for J/J′, and consequently the J-adic topology agrees with the J′-adic topology; so J is an ideal of definition, whence (b). □

Let A be any P-ring; by lemma 16.2.7(ii), the kernel of u_{A} contains a distinguished element \underline{a}, and we shall see later that the case where actually \underline{a} generates the ideal Ker u_{A} is the most interesting for our purposes. For the moment, let us just notice that, if the latter condition holds, the datum (E := E(A), \underline{a}) suffices to recover A: indeed, lemma 16.2.7(i) says that in that case u_{A} induces an isomorphism of topological rings W (E)/\underline{a}W (E) →^{∼} A (where W (E)/\underline{a}W (E) is endowed with the quotient topology induced from W (E) via the natural projection). It is then convenient to insert hereafter a preliminary study of pairs (E, \underline{a}) of this type.

16.2.13. Let (E, 𝒯_{E}) be a perfect topological 𝔽_{p}-algebra, \underline{a} := (a_{n} | n ∈ ℕ) ∈ W (E) any element, and suppose that 𝒯_{E} agrees with the a_{0}-adic topology. Denote by E^{∧} the separated completion of E, and let also E_{d} (resp. E_{d}^{∧}) be the ring E (resp. E^{∧}) endowed with its discrete topology. To ease notation, set

A(E) := W (E)/\underline{a}W (E)

which we view as a topological ring, with the quotient topology induced from W (E). Define likewise the topological rings A(E_{d}), A(E^{∧}) and A(E_{d}^{∧}). The natural diagram of continuous ring homomorphisms

\xymatrix{E_{d} \ar[r] \ar[d] & E_{d}^{∧} \ar[d] \\ E \ar[r] & E^{∧}}

induces a commutative diagram of topological rings

(16.2.14)  \xymatrix{A(E_{d}) \ar[r] \ar[d] & A(E_{d}^{∧}) \ar[d] \\ A(E) \ar[r]^{j_{A}} & A(E^{∧}).}

Moreover, notice that there are natural isomorphisms of topological rings

(16.2.15)  A(E) ⊗_{ℤ} 𝔽_{p} →^{∼} E/a_{0}E  A(E^{∧}) ⊗_{ℤ} 𝔽_{p} →^{∼} E^{∧}/a_{0}E^{∧}

(where, again the sources and the targets are endowed with the quotient topologies induced from A(E), E, A(E^{∧}), and respectively E^{∧}). The composition of the first of these maps with the projection A(E) → A(E) ⊗_{ℤ} 𝔽_{p} is the continuous ring homomorphism

π_{A} : A(E) → E/a_{0}E ((b_{n} | n ∈ ℕ) mod \underline{a}W (E)) ↦ (b_{0} mod a_{0}E)

and likewise we can describe the corresponding map π_{A}^{∧} : A(E^{∧}) → E^{∧}/a_{0}E^{∧}. Lastly, denote

β : E(A(E)) → E^{∧}

the morphism of topological monoids obtained by composing E(π_{A}) with the isomorphism of topological rings E(E/a_{0}E) →^{∼} E^{∧} provided by corollary 9.4.14.

Proposition 16.2.16. In the situation of (16.2.13), suppose that a_{0}E + a_{1}E = E. Then the following holds :

(i) The image of \underline{a} is a distinguished element of W (E^{∧}).

(ii) All the arrows of (16.2.14) are isomorphisms of topological rings.

(iii) A(E) is a P-ring, and its topology agrees with the p-adic topology.

(iv) β is an isomorphism of topological rings (for the topological ring structure on A(E) provided by (16.1)), and we have a commutative diagram of topological rings

\xymatrix{A(A(E)) \ar[r]^{W (β)} \ar[d]_{u_{A(E)}} & W (E^{∧}) \ar[d]^{π_{W}^{∧}} \\ A(E) \ar[r]^{j_{A}} & A(E^{∧})}

where π_{W}^{∧} is the natural projection.

Proof. (i): By remark 8.3.10(v), the image of a_{0} lies in the Jacobson radical of E^{∧}, and since a_{0}E +a_{1}E = E, it follows easily that a_{1} ∈ E^{×} (details left to the reader), whence the assertion.

(ii): Let ℐ ⊂ W (E) (resp. ℐ^{∧} ⊂ W (E^{∧})) be the ideal generated by the system (p, τ_{E}(a_{0})) (resp. (p, τ_{E^{∧}}(a_{0}))); proposition 9.3.77(ii) says that the topology of W (E) (resp. of W (E^{∧})) agrees with the ℐ -adic (resp. ℐ^{∧}-adic) topology, whereas the topologies of both W (E_{d}) and W (E_{d}^{∧}) are p-adic. Let \overline{ℐ} (resp. \overline{ℐ} ^{∧}) be the image of ℐ in A(E) (resp. in A(E^{∧})); it follows easily that the topology of A(E) (resp. of A(E^{∧})) agrees with the \overline{ℐ} -adic (resp. \overline{ℐ} ^{∧}- adic) topology, whereas the topologies of both A(E_{d}) and A(E_{d}^{∧}) are still p-adic. However, set \underline{b} := (a_{1}^{1/p}, a_{2}^{1/p}, …), and notice that

(16.2.17)  \underline{a} = τ_{E}(a_{0}) + p · \underline{b}  in W (E)

((9.3.41) and lemma 9.3.27(i)), which implies that \overline{ℐ} = pA(E). Likewise, \overline{ℐ} ^{∧} = pA(E^{∧}). Summing up, this proves already that the vertical arrows of (16.2.14) are both isomorphisms.

Next, proposition 9.3.47(v) implies that A(E) is complete and separated for the p-adic topology, hence also for the ℐ -adic topology. Recall as well that the family of ideals

Ker (W (E) → W_{n}(E/a_{0}^{k}E)) for every k, n ∈ ℕ

is a fundamental system of open neighborhoods of 0 ∈ W (E) for the ℐ -adic topology, therefore the natural map

A(E) → lim_{k,n∈ℕ} W_{n}(E/a_{0}^{k}E)/\underline{a}W_{n}(E/a_{0}^{k}E)

is an isomorphism of topological rings. By the same token, the same description applies to A(E^{∧}), so we conclude that also the horizontal arrows of (16.2.14) are isomorphisms.

(iii): We know already that the topology of A(E) is complete and separated, and agrees with the p-adic topology. On the other hand, notice that \underline{b} is invertible in W (E^{∧}), by (i) and proposition 9.3.44(iii); since the images of τ_{E^{∧}}(a_{0}) and −p · \underline{b} agree in A(E^{∧}), we deduce that the p-adic filtration agrees with the τ_{E}(a_{0})-adic filtration on A(E^{∧}), and therefore also on A(E), since we already know that j_{A} is an isomorphism. Set J := τ_{E}(a_{0}^{1/p}) · A(E); it follows that the topology of A(E) agrees with the J-adic topology, and moreover p ∈ J^{p} ⊂ J^{2}. Lastly, the Frobenius endomorphism is surjective on A(E)/J^{2}, due to the isomorphism (16.2.15), whence the assertion.

(iv): From (iii) and the discussion of (16.1) we see that β is an isomorphism of topological rings; also, under the identifications (16.2.15), the morphism j_{A}⊗_{ℤ}𝔽_{p} corresponds to the natural map ι : E/a_{0}E →^{∼} E^{∧}/a_{0}E^{∧}. Therefore we have

π_{A}^{∧} ∘ j_{A} ∘ u_{A(E)} = ι ∘ π_{A} ∘ u_{A(E)} = ι ∘ π_{A} ∘ \overline{u}_{A(E)} ∘ ω_{0}

where ω_{0} : A(A(E)) → E(A(E)) is the 0-th ghost component map. Then, since A(E^{∧}) is complete and separated for the p-adic topology, proposition 9.3.52(iii) reduces to checking

Claim 16.2.18. π_{A}^{∧} ∘ π_{W}^{∧} ∘ W (β) = ι ∘ π_{A} ∘ \overline{u}_{A(E)} ∘ ω_{0}.

Proof of the claim. Notice the identities :

β ∘ ω_{0} = ω_{0}^{∧} ∘ W (β) π_{A}^{∧} ∘ π_{W}^{∧} = π_{E^{∧}} ∘ ω_{0}^{∧}

(where ω_{0}^{∧} : W (E^{∧}) → E^{∧} is the 0-th ghost component map, and π_{E^{∧}} : E^{∧} → E^{∧}/a_{0}E^{∧} is the projection) in light of which, it suffices to show that

π_{E^{∧}} ∘ β = ι ∘ π_{A} ∘ \overline{u}_{A(E)}.

The latter follows by a simple inspection of the definition of β : details left to the reader. □

16.2.19. For any prime p, consider the following categories :

• The category ℰ whose objects are all the pairs (E, ℐ) where E is both a perfect topological 𝔽_{p}-algebra and a P-ring, and ℐ ⊂ W (E) is a distinguished ideal. The morphisms (E, ℐ) → (E′, ℐ′) are the continuous ring homomorphisms f : E → E′ such that W (f)(ℐ) ⊂ ℐ′.

• The category 𝒜 of all P-rings A such that Ker u_{A} is a distinguished ideal of W (A). The morphisms in 𝒜 are all the continuous ring homomorphisms.

• The full subcategory ℰ_{p} of ℰ whose objects are all the pairs (E, ℐ) such that the topology of E agrees with the ω_{0}(ℐ)-adic topology, and the full subcategory 𝒜_{p} of 𝒜 consisting of those objects A whose topology agrees with the p-adic topology.

Then proposition 16.2.16(iii,iv) yields an equivalence of categories

(16.2.20)  ℰ_{p} →^{∼} 𝒜_{p} : (E, ℐ) ↦ A(E, ℐ) := W (E)/ℐ

with quasi-inverse given by the rule : A ↦ (E(A), Ker u_{A}). In the following, we aim to extend this equivalence to the whole of ℰ. The first step is :

Proposition 16.2.21. With the notation of (16.2.19), let (E, ℐ) be any object of ℰ. Then : (i) A(E, ℐ) is a P-ring. (ii) Especially, ℐ is closed in the topology 𝒯_{W (E)}.

Proof. Clearly, it suffices to check (i). To this aim, let (a_{n} | n ∈ ℕ) ∈ ℐ be any distinguished element, I ⊂ E be any ideal of definition, (x_{0}, …, x_{r}) a finite system of generators for I, denote by 𝒥 ⊂ W (E) the ideal generated by the system (τ_{E}(x_{0}), …, τ_{E}(x_{r})), and set 𝒥′ := 𝒥 + pW (E). By proposition 9.3.77(ii), we know that the topology of A(E) agrees with the 𝒥′-adic topology. Now, by assumption a_{0} is topologically nilpotent, hence a_{0}^{k} ∈ I for every sufficiently large k ∈ ℕ; since the Frobenius endomorphism Φ_{E} is an automorphism of the topological ring E, we may then replace I by Φ_{E}^{−t}I for some suitably large t ∈ ℕ, after which we may assume that a_{0} ∈ I^{(p)}, and recall that I^{(p)} is generated by (x_{0}^{p}, …, x_{r}^{p}). Then, lemma 16.1.9(ii) implies that

(16.2.22)  p ∈ 𝒥^{2}A(E).

and therefore 𝒥′A(E) = 𝒥A(E); especially, the topology of A(E) agrees with the 𝒥 -adic topology. By corollary 16.1.13(i,ii), ℐ is a closed subset for the 𝒥 -adic topology on W (E), and the 𝒥 -adic topology of W (E) is complete and separated, so the same holds for the 𝒥 -adic topology of A(E). Moreover, the Frobenius endomorphism of A(E)/𝒥A(E) is surjective, by virtue of (16.2.15); taking into account (16.2.22), the proposition follows. □

To define a quasi-inverse for our extension of (16.2.20), we shall also need the following:

Proposition 16.2.23. Let A be any P-ring. We have :

(i) If Ker u_{A} is the topological closure of a finitely generated ideal, E(A) is a P-ring. (ii) The following conditions are equivalent :

(a) Ker u_{A} is generated by any distinguished element contained in it.

(b) Ker u_{A} is a principal ideal.

(c) Ker u_{A} is the topological closure of a principal ideal.

Proof. (i): For any topological space X, and any subset S ⊂ X, denote by S^{c} the topological closure of S in X. By assumption, there exists a finitely generated ideal ℐ ⊂ A(A) such that ℐ^{c} = Ker u_{A}. Let now ω_{0} : A(A) → E(A) be the 0-th ghost component; we notice :

Claim 16.2.24. (i) ω_{0}(Ker u_{A}) = Ker (\overline{u}_{A/pA} : E(A) → A/pA).

(ii) (Ker \overline{u}_{A/pA})^{c} = Ker \overline{u}_{A/(pA)^{c}}.

Proof of the claim. (i): Say that x ∈ Ker \overline{u}_{A/pA}, and pick any y ∈ A(A) such that ω_{0}(y) = x; then u_{A}(y) = pa for some a ∈ A. We pick z ∈ A(A) such that u_{A}(z) = a; then y−pz ∈ Ker u_{A} and ω_{0}(y − pz) = x, whence the contention.

(ii): The ideal K := (Ker \overline{u}_{A/pA})^{c} contains Ker \overline{u}_{A/pA}, therefore

K = \overline{u}_{A/pA}^{−1}(\overline{u}_{A/pA}(K))

so \overline{u}_{A/pA}(K) is a closed ideal of A/pA, by lemma 16.2.7(i), hence (pA)^{c}/pA ⊂ \overline{u}_{A/pA}(K) and consequently K = \overline{u}_{A/pA}^{−1}((pA)^{c}/pA). On the other hand, \overline{u}_{A/(pA)^{c}} is the composition of \overline{u}_{A/pA} with the projection A/pA → A/(pA)^{c}, whence the claim. ♢

Claims 8.3.25and 16.2.24(i) imply that ω_{0}(ℐ)^{c} = (Ker \overline{u}_{A/pA})^{c}. From claim 16.2.24(ii) we deduce that Ker \overline{u}_{A/(pA)^{c}} is the topological closure in E(A) of a finitely generated ideal. Next, let I ⊂ A be any ideal of definition, and 𝒥 ⊂ E(A) any finitely generated ideal such that \overline{u}_{A/(pA)^{c}}(𝒥) = I/(pA)^{c}; it follows easily that 𝒥 +Ker \overline{u}_{A/(pA)^{c}} = Ker \overline{u}_{A/I} (details left to the reader), and then also Ker \overline{u}_{A/I} is the topological closure of a finitely generated ideal of E(A). Lastly, we have a commutative diagram of continuous and surjective ring homomorphisms

[figure: E(A); ^{\overline{u}^{A/I}}; v; A/I; A/I; Φ_{A/I}; A/I]  where v := \overline{u}_{A/I} ∘ Φ_{E(A)}^{−1}

whence v(Ker \overline{u}_{A/I}) = Ker Φ_{A/I}. Since the topology of A/I is discrete, claim 8.3.25 implies that Ker Φ_{A/I} is a finitely generated ideal, so the assertion follows from proposition 16.2.9.

(ii): Clearly (a)⇒(b), and since Ker u_{A} is a closed subset of W (A), we have as well (b)⇒(c). Thus, it remains only to check that (c)⇒(a). To this aim, let again I ⊂ A be any ideal of definition, denote by J ⊂ A the radical of I, and fix any element \underline{b} := (b_{n} | n ∈ ℕ) ∈ A(A) such that Ker u_{A} is the topological closure of \underline{b}A(A) in A(A). We notice that A/J is a perfect 𝔽_{p}-algebra, and its quotient topology induced by the projection π : A → A/J is discrete. We have a commutative diagram of continuous and surjective ring homomorphisms :

\xymatrix{A(A) \ar[r]^{u_{A}} \ar[d]_{f} & A \ar[d]^{π} \\ W (A/J) \ar[r]^{_{ω_{0}}} & A/J}

whose bottom horizontal arrow is the 0-th ghost map (see (9.3.2)), and with f := W (φ), where φ : E(A) → A/J is the composition of \overline{u}_{A/pA} : E(A) → A/pA with the natural projection A/pA → A/J (details left to the reader).

Claim 16.2.25. f(Ker u_{A}) = pW (A/J).

Proof of the claim. It is clear that f(Ker u_{A}) ⊂ pW (A/J), so we need only show the converse inclusion. Now, according to lemma 16.2.7(ii), we may find a distinguished element \underline{a} := (a_{n} | n ∈ ℕ) in Ker u_{A}. Since a_{0} is topologically nilpotent in E(A), the element \overline{u}_{A/pA}(a_{0}) is topologically nilpotent in A/pA, and therefore its image vanishes in A/J; in other words, a_{0} ∈ Ker φ. However, \underline{a} = τ_{A}(a_{0}) + p · \underline{u}, for some \underline{u} ∈ A(A)^{×}, so that

f(\underline{a}) = f ∘ τ_{A}(a_{0}) + p · f(\underline{u}) = τ_{A/J} ∘ φ(a_{0}) + p · f(\underline{u}) = p · f(\underline{u})

(see (9.3.35)) whence the claim. ♢

In light of claims 8.3.25and 16.2.25and example 9.3.48(i) we deduce that f(\underline{b}) · W (A/J) is a dense subset of pW (A/J), for the p-adic topology on W (A/J); especially, we have

pW (A/J) = p^{2}W (A/J) + f(\underline{b}) · W (A/J).

Then, since p lies in the Jacobson radical of W (A/J), Nakayama’s lemma shows that f(\underline{b}) · W (A/J) = pW (A/J), which in turn implies that φ(b_{0}) = 0 and φ(b_{1}) is invertible in A/J.

Claim 16.2.26. \underline{b} is a distinguished element of A(A).

Proof of the claim. First, since J is contained in the Jacobson radical of A (remark 8.3.10(v)) and φ(b_{1}) ∈ (A/J)^{×}, we deduce that \overline{u}_{A}(b_{1}) ∈ A^{×}, and therefore b_{1} ∈ E(A)^{×}, by remark 9.4.5(iii). Likewise, we have \overline{u}_{A}(b_{0}) ∈ JA, which easily implies that b_{0} is topologically nilpotent in E(A) (details left to the reader). ♢

Now, from (i) and assumption (ii.c) we see that E(A) is a P-ring; from claim 16.2.26 and proposition 16.2.21(ii) it then follows that \underline{b}A(A) is a closed ideal of A(A), and therefore it must coincide with Ker u_{A}. Lastly, let \underline{a} be any other distinguished element of A(A) contained in Ker u_{A}; taking into account remark 16.1.7(i) and again claim 16.2.26, we conclude that \underline{a}A(A) = \underline{b}A(A), whence (ii.a). □

16.3. Perfectoid rings. We are now ready to introduce our generalizations of Scholze’s perfectoid rings that will intervene in the proof of the log regular version of almost purity.

Definition 16.3.1. Let A be any P-ring. We say that A is perfectoid if it fulfills any of the three equivalent conditions of proposition 16.2.23(ii).

Example 16.3.2. (i) Let (A, 𝒯 ) be any topological 𝔽_{p}-algebra. Then A is perfectoid if and only if it is perfect, 𝒯 is complete and separated, and there exists a finitely generated ideal I ⊂ A such that 𝒯 agrees with the I-adic topology. Indeed, under these assumptions, A is a P-ring (remark 16.2.4), and the map \overline{u}_{A} : E(A) → A is an isomorphism of topological rings that identifies u_{A} with the 0-th ghost map ω_{0} : W (A) → A, whose kernel is the principal ideal pW (A), by (9.3.45), so A is perfectoid. Conversely, clearly p ∈ Ker u_{A}, and p is obviously a distinguished element of A(A), so if A is a perfectoid 𝔽_{p}-algebra we must have Ker u_{A} = pW (A); moreover, u_{A} is an open ring homomorphism (lemma 16.2.7(i)), so it induces an isomorphism E(A) →^{∼} A of topological rings, especially (A, 𝒯 ) is a perfect topological 𝔽_{p}-algebra, as claimed.

(ii) Let E be any perfectoid 𝔽_{p}-algebra, and \underline{a} := (a_{n} | n ∈ ℕ) any distinguished element of W (E). Then A(E) := W (E)/\underline{a}W (E) is perfectoid for the quotient topology induced by W (E). Indeed, A(E) is a P-ring, by proposition 16.2.21(i); next, lemma 8.3.12 implies that E is complete and separated for the a_{0}-adic topology, and therefore proposition 16.2.16(ii,iv) applies, and shows that Ker u_{A(E)} is a principal ideal, whence the contention. Moreover, in this case the perfectoid ring E can be recovered from A(E) : indeed, we have a natural isomorphism of topological rings

E(A(E)) →^{∼} E

obtained as follows. First, since the 0-th ghost component W (E) → E is an open and surjective map, the same holds for the projection α : A(E) → A(E)/pA(E) →^{∼} E/a_{0}E, where E/a_{0}E is endowed with the quotient topology induced by E. Then, α induces an isomorphism E(α) : E(A(E)) →^{∼} E(E/a_{0}E) of topological rings, by (16.1). But since a_{0} is topologically nilpotent, the projection β : E(E) → E(E/a_{0}E) is an isomorphism of topological rings as well (theorem 9.4.10), and the same holds for the map \overline{u}_{E} : E(E) → E, since E is perfect. Thus, the sought isomorphism is the composition \overline{u}_{E} ∘ β^{−1} ∘ E(α).

Let us also point out the following complement to proposition 16.2.9:

Corollary 16.3.3. Let A be any perfectoid ring. Then \overline{Φ}_{A/J} is an isomorphism for every special ideal of definition J of A.

Proof. Set E := E(A); we may assume that there exists a distinguished element \underline{a} := (a_{n} | n ∈ ℕ) ∈ W (E) such that A is the ring A(E) as defined in (16.2.13), and u_{A} : W (E) → A(E) is the projection. Moreover, we may find a finite system \underline{x}_{•} := (\underline{x}_{0}, …, \underline{x}_{r}) of elements of W (E), such that J is the image in A(E) of the ideal generated by \underline{x}_{•}. We may write

(16.3.4) \underline{x}_{i} = τ_{E}(t_{i}) + p · \underline{y}_{i}  where t_{i} ∈ E and \underline{y}_{i} ∈ W (E) for every i = 0, …, r.

In view of lemma 9.3.4(i), we deduce that

(16.3.5)  \underline{x}_{i}^{p} ≡ τ_{E}(t_{i})^{p} (mod p^{2}W (E))  for i = 0, …, r.

Let 𝒥 ⊂ W (E) be the ideal generated by τ_{E}(t_{0}), …, τ_{E}(t_{r}); we notice :

Claim 16.3.6. p ∈ 𝒥^{(p)}A(E) and 𝒥 A(E) = J.

Proof of the claim. It follows easily from (16.3.5) that there exists \underline{z} ∈ W (E) with

p · u_{A}(1 + p · \underline{z}) ∈ 𝒥^{(p)}A(E)

(recall that 𝒥^{(p)} is the ideal generated by τ_{E}(t_{0})^{p}, …, τ_{E}(t_{r})^{p} : see definition 16.2.1). However, p is topologically nilpotent in W (E), therefore 1+p·\underline{z} ∈ W (E)^{×}, so p ∈ 𝒥^{(p)}A(E). The other assertion follows directly from the first one and (16.3.4) : details left to the reader. ♢

Let I ⊂ E be the ideal generated by t_{0}, …, t_{r}, and set E_{0} := E/a_{0}E; taking into account lemma 16.1.9(ii) and claim 16.3.6, we deduce that a_{0} ∈ I^{(p)}. Also, it is clear that the natural isomorphism (16.2.15) maps the image of J (resp. of J^{(p)}) onto IE_{0} (resp. I^{(p)}E_{0}). But it also follows that E_{0}/IE_{0} = E/I and E_{0}/I^{(p)}E_{0} = E/I^{(p)}; lastly, since E is perfect, the Frobenius endomorphism Φ_{E} maps I onto I^{(p)}, whence the contention. □

Remark 16.3.7. (i) We may now say that the category 𝒜 of (16.2.19) is the category of perfectoid rings, and we shall denote it henceforth by

Perf.

More generally, if A is any perfectoid ring, we shall write

A-Perf

for the category A/Perf of perfectoid A-algebras, whose objects are the continuous ring homomorphisms A → B with B perfectoid. The objects of the category ℰ of (16.2.19) are the pairs (E, ℐ) consisting of a perfectoid 𝔽_{p}-algebra and a distinguished ideal of W (E). Taking into account example 16.3.2, we see that (16.2.19) generalizes verbatim to an equivalence

ℰ →^{∼} Perf : (E, ℐ) ↦ A(E, ℐ) := W (E)/ℐ

with a natural quasi-inverse given by the functor E, which restricts to equivalences

E : A-Perf →^{∼} E(A)-Perf for every perfectoid ring A.

Hereafter, we shall study the stability of the class of perfectoid rings under some standard operations. The following observations (iii) and (iv) show stability under completion with respect to the J-adic topology corresponding to an ideal J ⊂ A of finite type with pA ⊂ J.

(ii) Let A be any perfectoid ring, set E := E(A), and let α_{•} := (α_{n} | n ∈ ℕ) be any distinguished element in Ker u_{A}. Notice that \overline{u}_{A/pA} : E → A/pA induces an isomorphism of topological rings :

ω : E/α_{0}E →^{∼} A/pA

for the quotient topologies induced by the projections A → A/pA and E → E/α_{0}. Indeed, \overline{u}_{A/pA} is open and surjective (lemma 16.2.7(i)), and α_{•} generates Ker u_{A}, so α_{0} generates the kernel of u_{A/pA}, by virtue of lemma 16.1.1(i), whence the contention.

(iii) The isomorphism ω of (ii) induces a natural bijection :

{ideals 𝒥 ⊂ E | α_{0}E ⊂ 𝒥 } ↔ {ideals J ⊂ A | pA ⊂ J}.

Namely, 𝒥 ⊂ E and J ⊂ A correspond under this bijection, if and only if ω(𝒥 /αE) = J/pA. Then, clearly J is finitely generated if and only if the same holds for 𝒥. Let 𝒯_{J} (resp. 𝒯_{𝒥} ) denote the J-adic (resp. 𝒥 -adic) topology on A (resp. on E); taking into account remark 9.4.9(ii), we deduce that if J is finitely generated, then the topology of E(A, 𝒯_{J}) agrees with 𝒯_{𝒥} . Combining with example 16.3.2(i), lemma 8.3.32(iv), example 9.3.48(ii) and remark 9.4.9(v), we conclude that the completion E(A, 𝒯_{J})^{∧} of E(A, 𝒯_{J}) is perfectoid.

(iv) In the situation of (iii), suppose that J and 𝒥 are finitely generated; notice that, since α_{0} ∈ 𝒥, the image of α_{•} is distinguished in W (E(A, 𝒯_{J})^{∧}), and let ℐ := α_{•}W (E(A, 𝒯_{J})^{∧}). We claim that there is a natural isomorphism of topological rings

W (E(A, 𝒯_{J})^{∧})/ℐ →^{∼} (A_{J}, 𝒯_{J})^{∧}

for the quotient topology on W (E(A, 𝒯_{J})^{∧})/ℐ induced by W (E(A, 𝒯_{J})^{∧}). Especially, this shows that (A_{J}, 𝒯_{J})^{∧} is perfectoid. For the proof, fix a finite system (β_{1}, …, β_{k}) of generators of 𝒥, and let 𝒥_{W} ⊂ W (E) be the ideal generated by the system (p, τ_{E}(β_{1}), …, τ_{E}(β_{k})); by proposition 9.3.77(ii), the topology 𝒯_{W} of W (E, 𝒯_{𝒥} ) agrees with the 𝒥_{W} -adic topology. By construction we have u_{A}(𝒥_{W})/pA = J/pA, and consequently u_{A}(𝒥_{W}) = J, so the quotient topology induced by 𝒯_{W} on A via u_{A} agrees with 𝒯_{J}. Lastly, in view of (iii) and proposition 16.2.21(ii) we know that ℐ is a closed ideal of W (E(A, 𝒯_{J})^{∧}). Taking into account proposition 8.2.13(v) and lemma 9.3.33(iv), the assertion follows.

We can summarize these observations in the following :

Proposition 16.3.8. Let A be a ring, p a prime integer, I ⊂ A a finitely generated ideal such that pA ⊂ I, and denote by 𝒯_{I} (resp. 𝒯_{p}) the I-adic (resp. p-adic) topology on A. We have :

(i) If (A, 𝒯_{I}) is perfectoid, the same holds for (A, 𝒯_{p}).

(ii) If (A, 𝒯_{p}) is perfectoid, the same holds for the completion (A_{I}, 𝒯_{I})^{∧} of (A, 𝒯_{I}).

Proof. (i) follows immediately from proposition 16.2.5(i).

(ii): It suffices to apply remark 16.3.7(iv) to the ideal I ⊂ A and the corresponding ideal 𝒥 := \overline{u}_{A/pA}^{−1}(I/pA) of E(A). □

The next observation establishes the stability of Perf under completed tensor products :

Proposition 16.3.9. Let A_{0} be a perfectoid ring and A_{1}, A_{2} two perfectoid A_{0}-algebras. Then: (i) The topological ring A_{3} := A_{1} ˆ⊗_{A_{0}}A_{2} is perfectoid. (ii) There exists a natural isomorphism E(A_{3}) →^{∼} E(A_{1}) ˆ⊗_{E(A_{0})}E(A_{2}) in Perf. (iii) Especially, all finite coproducts are representable in the category Perf.

Proof. (i): Set E_{i} := E(A_{i}) for i = 0, 1, 2, and pick a distinguished element \underline{α} ∈ W (E_{0}) that generates the kernel of u_{A_{0}}; then the image of \underline{α} in W (E_{i}) is still distinguished, and we know that \underline{α}W (E_{i}) is the kernel of u_{A_{i}} also for i = 1, 2. Moreover, it is clear from remark 9.3.56(iii) that the 𝔽_{p}-algebra E_{3} := E_{1} ˆ⊗_{E_{0}}E_{2} is perfectoid, and then the same holds for the topological ring A(E_{3}) := W (E_{3})/\underline{α}W (E_{3}), by virtue of example 16.3.2(ii). Lastly, the isomorphism (9.3.58) induces an isomorphism of topological rings

A(E_{3}) →^{∼} A_{1} ˆ⊗_{A_{0}}A_{2}

whence the assertion.

(ii): Example 16.3.2(ii) also identifies naturally E_{3} with E(A(E_{3})), whence the contention. (iii) is clear, since complete tensor products represent these coproducts, by (8.3.7). □

16.3.10. For any ring A and any ideal I ⊂ A, let 𝒯_{I} be the I-adic topology on A, and denote (A_{I}^{∧}, 𝒯_{I}^{∧}) the separated completion of (A, 𝒯_{I}). Notice that if I′ ⊂ A is another ideal with I′ ⊂ I, the identity map of A yields a continuous map (A, 𝒯_{I′}) → (A, 𝒯_{I}), whose completion is a natural continuous ring homomorphism

(16.3.11)  (A_{I′}^{∧}, 𝒯_{I′}^{∧}) → (A_{I}^{∧}, 𝒯_{I}^{∧}).

Now, suppose that I, J ⊂ A are any two ideals; we deduce a commutative diagram

(16.3.12)  \xymatrix{(A_{I∩J}^{∧}, 𝒯_{I∩J}^{∧}) \ar[r]^{φ_{I}} \ar[d]_{φ_{J}} & (A_{I}^{∧}, 𝒯_{I}^{∧}) \ar[d]^{β_{I}} \\ (A_{J}^{∧}, 𝒯_{J}^{∧}) \ar[r]^{β_{J}} & (A_{I+J}^{∧}, 𝒯_{I+J}^{∧})}

whose arrows are the continuous ring homomorphisms (16.3.11). Moreover, for every k ∈ ℕ let A_{I,k}^{∧} := A_{I}^{∧}/J^{k}A_{I}^{∧}, and endow A_{I,k}^{∧} with the quotient topology 𝒯_{I,k} induced by 𝒯_{I}^{∧} via the natural projection A_{I}^{∧} → A_{I,k}^{∧}. We set

(16.3.13)  (A_{I,J}^{∧}, 𝒯_{I,J}^{∧}) := lim_{k∈ℕ} (A_{I,k}^{∧}, 𝒯_{I,k}).

For every k ∈ ℕ, let 𝒯_{d,k} be the discrete topology on A/(I + J)^{k}; the system of continuous projections ((A_{I}^{∧}, 𝒯_{I}^{∧}) → (A/(I + J)^{k}, 𝒯_{d}) | k ∈ ℕ) factors uniquely through a system of continuous ring homomorphisms ((A_{I,k}^{∧}, 𝒯_{I,k}) → (A/(I + J)^{k}, 𝒯_{d}) | k ∈ ℕ), whose limit is a natural continuous ring homomorphism

(16.3.14)  (A_{I,J}^{∧}, 𝒯_{I,J}^{∧}) → (A_{I+J}^{∧}, 𝒯_{I+J}^{∧}).

Lemma 16.3.15. In the situation of (16.3.10), we have :

(i) (16.3.14) is an open and surjective map.

(ii) Suppose that A is either a noetherian ring or a perfect 𝔽_{p}-algebra, and that both I and J are finitely generated. Then :

(a) (16.3.12) is a cartesian diagram of topological rings and a cocartesian diagram of topological A-modules.

(b) (16.3.14) is an isomorphism of topological rings.

Proof. For every n ∈ ℕ, we have a natural diagram of rings

(16.3.16)  \xymatrix{A/(I^{n} ∩ J^{n}) \ar[r] \ar[d] & A/I^{n} \ar[d] \\ A/J^{n} \ar[r] & A/(I^{n} + J^{n})}

whence a complex of A-modules

Σ_{n}  :  0 → A/(I^{n} ∩ J^{n}) → A/I^{n} ⊕ A/J^{n} → A/(I^{n} + J^{n}) → 0

and it is easily seen that Σ_{n} is exact for every n ∈ ℕ. Therefore, (16.3.16) is a cartesian diagram of discrete topological rings (details left to the reader); let 𝒯′_{d,n} be the discrete topology on A/(I^{n} ∩ J^{n}) for every n ∈ ℕ, and set

(A′, 𝒯 ′) := lim_{n∈ℕ} (A/(I^{n} ∩ J^{n}), 𝒯′_{d,n}).

Notice as well that (I + J)^{2n−1} ⊂ I^{n} + J^{n} ⊂ (I + J)^{n} for every n ∈ ℕ, so the (I + J)- adic topology on A agrees with the linear topology defined by the descending system of ideals (I^{n} +J^{n} | n ∈ ℕ). In light of example 1.5.15(ii), we deduce that the limit of the system of these diagrams is a cartesian diagram of topological rings

(16.3.17)  \xymatrix{(A′, 𝒯 ′) \ar[r]^{φ′_{I}} \ar[d]_{φ′_{J}} & (A_{I}^{∧}, 𝒯_{I}^{∧}) \ar[d]^{β′_{I}} \\ (A_{J}^{∧}, 𝒯_{J}^{∧}) \ar[r]^{β′_{J}} & (A_{I+J}^{∧}, 𝒯_{I+J}^{∧}).}

Moreover, since the inverse system (A/(I^{n} ∩ J^{n}) | n ∈ ℕ) has surjective transition maps, the limit of the system of exact sequences (Σ_{n} | n ∈ ℕ) is still exact (see [163, Lemma 3.5.3]), and therefore (16.3.17) is also a cocartesian diagram of A-modules. Furthermore, by proposition 8.2.13(i,v) we have an induced morphism of exact complexes

\xymatrix{Σ′_{n} \ar[d]_{σ_{n}} & 0 \ar[r] & (I^{n} ∩ J^{n})^{∧} \ar[r] \ar[d] & I^{n}^{∧} ⊕ J^{n}^{∧} \ar[r] \ar[d] & (I^{n} + J^{n})^{∧} \ar[r] \ar[d] & 0 \\ Σ & 0 \ar[r] & A′ \ar[r] & A_{I}^{∧} ⊕ A_{J}^{∧} \ar[r]^{β′_{I,J}} & A_{I+J}^{∧} \ar[r] & 0}

where :

• (I^{n} ∩J^{n})^{∧} is the topological closure of I^{n} ∩J^{n} in (A′, 𝒯 ′), and likewise for I^{n}^{∧} ⊕J^{n}^{∧} and (I^{n} + J^{n})^{∧}.

• Coker σ_{n} is naturally isomorphic to Σ_{n}, for every n ∈ ℕ.

• β′_{I,J} is the sum of β′_{I} and β′_{J}. Especially, β′_{I,J} is a continuous, open and surjective map, for the product topology 𝒯_{I}^{∧}×𝒯_{J}^{∧} on A_{I}^{∧} ⊕A_{J}^{∧}, and it follows easily that (16.3.17) is a cocartesian diagram of topological A-modules.

(i): The system of natural maps (A_{I}^{∧} → A_{I,k}^{∧} ← A/J^{k} | k ∈ ℕ) yields ring homomorphisms

A_{I}^{∧} ⟶^{ψ_{I}} (A_{I}^{∧})_{J}^{∧} ⟵^{ψ_{J}} A_{J}^{∧}.

There follows a map of abelian groups γ : A_{I}^{∧} ⊕ A_{J}^{∧} → (A_{I}^{∧})_{J}^{∧} which is continuous for the topology 𝒯_{I}^{∧} × 𝒯_{J}^{∧} and a simple inspection shows that (16.3.14) ∘ γ equals β′_{I,J}. Now, let K ⊂ (A_{I}^{∧})_{J}^{∧} be any open ideal; then K′ := γ^{−1}K is an open subgroup of A_{I}^{∧} ⊕A_{J}^{∧} and therefore β′_{I,J}K′ is an open subgroup of A_{I+J}^{∧} contained in the image K′′ of K under (16.3.14), so K′′ is open as well, whence (i).

(ii.a): The system of projections (A/(I ∩ J)^{n} → A/(I^{n} ∩ J^{n}) | n ∈ ℕ) yields a natural continuous ring homomorphism

ν : (A_{I∩J}^{∧}, 𝒯_{I∩J}^{∧}) → (A′, 𝒯 ′)

and a simple inspection shows that φ′_{I} ∘ ν = φ_{I} and φ′_{J} ∘ ν = φ_{J}. To conclude the proof of (ii.a), it then suffices to check :

Claim 16.3.18. Under the assumptions of (ii), the map ν is an isomorphism of topological rings.

Proof of the claim. Since (IJ)^{2n} ⊂ (I ∩ J)^{2n} ⊂ (IJ)^{n} for every n ∈ ℕ, the (I ∩ J)-adic and IJ-adic topologies coincide on A. On the other hand, in case A is noetherian, the Artin-Rees lemma ([126, Th.8.5]) implies that for every n ∈ ℕ there exists m ∈ ℕ such that I^{m} ∩ J^{n} ⊂ I^{n}J^{n} ⊂ (I ∩ J)^{n}, from which the claim follows easily.

Thus, suppose A is perfect and both I and J are finitely generated, so the same holds for IJ, and therefore lemma 9.3.69(iv) says that the IJ-adic topology on A agrees with the linear topology defined by the cofiltered system of ideals ((IJ)^{⟨n⟩}A | n ∈ ℕ). Likewise, the linear topology on A defined by the cofiltered system of ideals (I^{n} ∩ J^{n} | n ∈ ℕ) agrees with the one defined by the cofiltered system of ideals (I^{⟨n⟩}A ∩ J^{⟨n⟩}A | n ∈ ℕ). So, it suffices to prove that

I^{⟨pn⟩}A ∩ J^{⟨pn⟩}A ⊂ (IJ)^{⟨n⟩}A for every n ∈ ℕ.

However, if x ∈ I^{⟨pn⟩}A ∩ J^{⟨pn⟩}A, we may write x = x^{1/p} · x^{(p−1)/p} ∈ I^{⟨n⟩}J^{⟨(p−1)n/p⟩}A ⊂ (IJ)^{⟨n⟩}A whence the contention. ♢

(ii.b): For every h, k ∈ ℕ, let A_{h,k} := A/(I^{h} + J^{k}), and endow A_{h,k} with the discrete topology 𝒯_{h,k}; moreover, set

(C_{k}, 𝒯_{C,k}) := lim_{h∈ℕ} (A_{h,k}, 𝒯_{h,k}) for every k ∈ ℕ

(where the transition maps A_{h+1,k} → A_{h,k} are the natural projections). Since I and J are both finitely generated, for every h, k ∈ ℕ there exists n ∈ ℕ such that (I + J)^{n} ⊂ I^{h} + J^{k}, so the natural map

(16.3.19)  (A_{I+J}^{∧}, 𝒯_{I+J}^{∧}) → lim_{h,k∈ℕ} (A_{h,k}, 𝒯_{h,k}) →^{∼} lim_{k∈ℕ} (C_{k}, 𝒯_{C,k})

is an isomorphism of topological rings (see example 1.5.15(ii)). On the other hand, the projection A → A/J^{k} extends uniquely to a continuous map φ_{k} : (A_{I}^{∧}, 𝒯_{I}^{∧}) → (C_{k}, 𝒯_{C,k}), and the latter factors uniquely through a continuous ring homomorphism

\overline{φ}_{k} : (A_{I,k}^{∧}, 𝒯_{I,k}) → (C_{k}, 𝒯_{C,k}) for every ∈ ℕ.

With this notation, a simple inspection shows that (16.3.13) and (16.3.19) identify (16.3.14) with the limit of the system of maps (\overline{φ}_{k} | k ∈ ℕ). However, proposition 8.2.13(i,v) says that φ_{k} is surjective and its kernel is the topological closure (J^{k}A_{I}^{∧})^{c} of J^{k}A_{I}^{∧} in (A_{I}^{∧}, 𝒯_{I}^{∧}), for every k ∈ ℕ. Thus, in order to show the lemma, it suffices to check that the linear topology on A_{I}^{∧} defined by the system of ideals ((J^{k}A_{I}^{∧})^{c} | k ∈ ℕ) agrees with the JA_{I}^{∧}-adic topology. In case A is noetherian, this is clear, since in that case J^{k}A_{I}^{∧} is already closed in A_{I}^{∧} ([126, Th.8.11]). For the case where A is perfect, we remark :

Claim 16.3.20. Let (B, 𝒯_{B}) be a perfect, complete and separated topological 𝔽_{p}-algebra, I, J ⊂ B two ideals of finite type, such that 𝒯_{B} agrees with the I-adic topology. For every λ ∈ ℕ[1/p] let J^{⟨λ⟩}B^{c} be the topological closure of J^{⟨λ⟩}B in B (notation of (9.3.68)). Then

J^{⟨λ⟩}B^{c} ⊂ ⋂_{n∈ℕ}(J + I^{n})^{⟨λ⟩}B ⊂ J^{⟨λ′⟩}B for every λ′ < λ.

Proof of the claim. We show first that J^{⟨λ⟩c}B ⊂ J^{⟨λ′⟩}B. Indeed, pick a finite system of generators (b_{1}, …, b_{k}) (resp. (b_{k+1}, …, b_{s})) for J (resp. for I), and let x ∈ J^{⟨λ⟩}B^{c} be any element; we can write

x = ∑_{n∈ℕ} x_{n}  with x_{n} ∈ J^{⟨λ⟩}B ∩ I^{⟨n⟩}B  for every n ∈ ℕ.

Choose a strictly positive ε ∈ ℕ[1/p] such that λ′′ := (1 − ε) · λ > λ′, and notice that

x_{n} = x_{n}^{1−ε} · x_{n}^{ε} ∈ J^{⟨λ′′⟩} · I^{⟨nε⟩}B for every n ∈ ℕ.

By definition, for every n ∈ ℕ there exist :

• a finite set S_{n} ⊂ ℕ[1/p]^{⊕k} with μ_{1} + ⋯ + μ_{k} = λ′′ for every μ := (μ_{1}, …, μ_{k}) ∈ S_{n}

• a system (a_{μ} | μ ∈ S_{n}) of elements of B such that

x_{n}^{1−ε} = ∑_{μ∈S_{n}} a_{μ}b^{μ} where b^{μ} := b_{1}^{μ_{1}} ⋯ b_{k}^{μ_{k}} for every μ ∈ S_{n}.

Now, choose N ∈ ℕ such that λ′′−λ′ ≥ kp^{−N} and define \overline{μ} and μ^{∗} as in the proof of proposition 9.3.77, so that \overline{μ} ∈ S := \{ν ∈ p^{−N}ℕ^{⊕k} | λ′′ ≥ ν_{1} + ⋯ + ν_{k} > λ′\} for every n ∈ ℕ and every μ ∈ S_{n}. It follows that

x = ∑_{ν∈S} b^{ν}c_{ν} where c_{ν} := ∑_{n∈ℕ} x_{n}^{ε} · ∑_{\begin{matrix}μ∈S_{n} \\ \overline{μ}=ν \end{matrix}} a_{μ}b^{μ^{∗}}.

Clearly b^{ν} ∈ J^{⟨λ′⟩}B, and by lemma 9.3.69(iv) the series c_{ν} converges in the I-adic topology of B for every ν ∈ S; also it is easily seen that S is a finite set, whence the contention.

Next, for any n ∈ ℕ, the ideal (J + I^{n})^{⟨λ⟩}B is generated by all monomials of the form

b^{μ} := b_{1}^{μ_{1}} ⋯ b_{s}^{μ_{s}} such that λ_{1} + n^{−1} · λ_{2} = λ where λ_{1} := ∑_{i=1}^{k} μ_{i} λ_{2} := ∑_{i=k+1}^{s} μ_{i}

and where μ := (μ_{1}, …, μ_{s}) is any sequence of elements of ℕ[1/p]. Fix also λ′′ ∈ ℕ[1/p] with λ′ < λ′′ < λ. Now, for any such μ, we have either λ_{1} ≥ λ′′, in which case b^{μ} ∈ J^{⟨λ′′⟩}B, or else λ_{2} > n · (λ − λ′′). We conclude that

⋂_{n∈ℕ}(J + I^{n})^{⟨λ⟩}B ⊂ ⋂_{n∈ℕ}(J^{⟨λ′′⟩}B + I^{⟨n·(λ−λ′′)⟩}B) = J^{⟨λ′′⟩}B^{c}

where the last identity follows from lemma 9.3.69(iv). But we know already that J^{⟨λ′′⟩}B^{c} ⊂ J^{⟨λ′⟩}B, whence the claim. ♢

To conclude, it suffices now to apply claim 16.3.20 with B := A_{I}^{∧} and invoke lemma 9.3.69(iv) (details left to the reader). □

Theorem 16.3.21. In the situation of (16.3.10), let 𝒯_{p} be the p-adic topology of A, and suppose: (a) (A, 𝒯_{p}) is perfectoid.

(b) p^{N} ∈ I ∩ J for every sufficiently large N ∈ ℕ.

(c) I and J are finitely generated ideals of A. Then we have :

(i) (16.3.12) is a cartesian diagram of rings and a cocartesian diagram of A-modules.

(ii) (16.3.14) is an isomorphism of topological rings.

Proof. (i): Let π ∈ A be as in lemma 16.2.3(iv); quite generally, if K ⊂ A is any finitely generated ideal containing p^{N} (for some N ∈ ℕ), then it is easily seen that the K-adic topology on A agrees with the (K + πA)-adic topology. In view of our assumptions (b) and (c) we may therefore replace I and J by I + πA and respectively J + πA, and assume from start that π ∈ I ∩ J. Fix a finite system (\overline{b}_{1}, …, \overline{b}_{n}) (resp. (\overline{b}_{n+1}, …, \overline{b}_{m})) of generators of the ideal I/pA (resp. of J/pA) of A/pA, and let also \underline{α} := (α_{n} | n ∈ ℕ) be a distinguished element of A := A(A, 𝒯_{p}) that generates Ker u_{A}. Pick elements β_{1}, …, β_{m} ∈ E := E(A) such that \overline{u}_{A/pA}(β_{i}) = \overline{b}_{i} for i = 1, …, m, denote by ℐ_{E} (resp. 𝒥_{E}) the ideal of E generated by the system (α_{0}, β_{1}, …, β_{n}) (resp. (α_{0}, β_{n+1}, …, β_{m})), and let (E_{ℐ}^{∧} , 𝒯_{ℐ}^{∧}) (resp. (E_{𝒥}^{∧} , 𝒯_{𝒥}^{∧}), resp. (E_{ℐ ∩𝒥}^{∧} , 𝒯_{ℐ ∩𝒥}^{∧} ), resp. (E_{ℐ +𝒥}^{∧} , 𝒯_{ℐ +𝒥}^{∧} )) be the ℐ_{E}-adic (resp. 𝒥_{E}-adic, resp. (ℐ ∩𝒥)-adic, resp. (ℐ + 𝒥)-adic) completion of E. By lemma 16.3.15(ii), we get a cartesian diagram of topological rings

ℰ  :  \xymatrix{(E_{ℐ ∩𝒥}^{∧} , 𝒯_{ℐ ∩𝒥}^{∧} ) \ar[r] \ar[d] & (E_{ℐ}^{∧} , 𝒯_{ℐ}^{∧}) \ar[d] \\ (E_{𝒥}^{∧} , 𝒯_{𝒥}^{∧}) \ar[r] & (E_{ℐ +𝒥}^{∧} , 𝒯_{ℐ +𝒥}^{∧} )}

which is also cocartesian as a diagram of topological abelian groups.

Claim 16.3.22. W (ℰ) is still a cartesian diagram of topological rings, and a cocartesian diagram of A-modules.

Proof of the claim. The cartesian property follows from remark 9.3.28(ii). To prove the cocartesian property, it suffices to show that the map of abelian groups deduced from W (ℰ)

τ : W (E_{ℐ}^{∧}) ⊕ W (E_{𝒥}^{∧} ) → W (E_{ℐ +𝒥}^{∧} )

is surjective. Endow both target and source of τ with their p-adic filtrations, so that τ becomes a map of filtered abelian groups, and for every k ∈ ℕ, denote by gr^{k}τ the map induced by τ on the respective k-graded subquotients. Since both target and source of τ are p-adically complete and separated (proposition 9.3.44(iii)), it then suffices to check that gr^{k}τ is surjective for every k ∈ ℕ ([34, Ch.III, §2, n.8, Cor.2]). Since multiplication by p is an injective endomorphism for both target and source of τ, we are then further reduced to the case where k = 0, in which case gr^{0}τ is naturally identified with the map E_{ℐ}^{∧} ⊕ E_{𝒥}^{∧} → E_{ℐ +𝒥}^{∧} deduced from ℰ. But the latter map is indeed surjective, since ℰ is cocartesian. ♢

Now, remark 16.3.7(iii) says that the topology of E(A, 𝒯_{I}) agrees with the ℐ_{E}-adic topology, so the completion E_{I}^{∧} of E(A, 𝒯_{I}) is isomorphic to (E_{ℐ}^{∧} , 𝒯_{ℐ}^{∧}). Likewise, the completion E_{J}^{∧} of E(A, 𝒯_{J}) is isomorphic to (E_{𝒥}^{∧} , 𝒯_{𝒥}^{∧}), and the completion E_{I+J}^{∧} of E(A, 𝒯_{I+J}) is isomorphic to (E_{ℐ +𝒥}^{∧} , 𝒯_{ℐ +𝒥}^{∧} ).

Next, set 𝒦 := ℐ 𝒥 + α_{0}E ⊂ E; since ℐ and 𝒥 are finitely generated and they both contain α_{0}, it is easily seen that the (ℐ ∩𝒥)-adic topology on E agrees with the 𝒦 -adic topology. Likewise, topology 𝒯_{I∩J} on A agrees with the (IJ)-adic topology. Moreover, 𝒦 is a finitely generated ideal, by construction p ∈ IJ, and \overline{u}_{A/pA}(𝒦 ) = (IJ)/pA, hence remark 16.3.7(iii) also tells us that (E_{ℐ ∩𝒥}^{∧} , 𝒯_{ℐ ∩𝒥}^{∧} ) is isomorphic to the completion E_{I∩J}^{∧} of E(A, 𝒯_{I∩J}).

Notice now that, since the image of A is dense in A_{I∩J}^{∧}, the map φ_{I} in (16.3.12) is the unique continuous homomorphism of topological A-algebras from A_{I∩J}^{∧} to A_{I}^{∧}, and the other maps in (16.3.12) enjoy corresponding uniqueness properties. Combining with remark 16.3.7(iv), we conclude that W (ℰ) ⊗_{A} A is naturally identified with (16.3.12), and the image of \underline{α} is still distinguished in each of the Witt rings appearing in W (ℰ). Due to of claim 16.3.22, we already see that (16.3.12) is a cocartesian diagram of A-modules; moreover, it will follow that it is a cartesian diagram of rings, once we know that

(16.3.23)  Tor_{1}^{A}(W (E_{ℐ +𝒥}^{∧} ), A) = 0.

However A = A/\underline{α}A, and \underline{α} is a regular element of W (E_{ℐ +𝒥}^{∧} ) (remark 16.1.7(ii)), so (16.3.23) holds by a standard calculation.

(ii): We know already that (16.3.14) is open and surjective (lemma 16.3.15(i)), so it suffices to show that this map is a ring isomorphism. However, in light of remark 16.3.7(iv) we have a diagram of continuous maps

𝒟 : \xymatrix{(A_{I}^{∧})_{J}^{∧} \ar[r] \ar[d] & A(E_{ℐ}^{∧} , α_{0}E_{ℐ}^{∧})_{J}^{∧} \ar[r] & A((E_{ℐ}^{∧})_{𝒥}^{∧} , α_{0}(E_{ℐ}^{∧})_{𝒥}^{∧} ) \ar[d] \\ A_{I+J}^{∧} \ar[rr] & & A(E_{ℐ +𝒥}^{∧} , α_{0}E_{ℐ +𝒥}^{∧} )}

whose left vertical arrow equals (16.3.14), and whose remaining arrows are ring isomorphisms. Thus, it suffices to check that 𝒟 commutes, i.e. that the two continuous ring homomorphisms f, g : (A_{I}^{∧})_{J}^{∧} → A(E_{ℐ +𝒥}^{∧} , α_{0}E_{ℐ +𝒥}^{∧} ) deduced from 𝒟 coincide. However, let i : A → (A_{I}^{∧})_{J}^{∧} denote the completion map; a simple inspection shows that f ∘ i = g ∘ i, and since the topology of A(E_{ℐ +𝒥}^{∧} , α_{0}E_{ℐ +𝒥}^{∧} ) is separated and i has dense image, the assertion follows. □

16.3.24. Let A, \overline{D}, \overline{A} be three perfectoid rings, and set E := E(A), \overline{E} := E(\overline{D}). Suppose that \overline{A} is a discrete topological 𝔽_{p}-algebra, so that E(\overline{A}) = \overline{A} (see also corollary 16.3.63(iii)), let \overline{φ}_{A} : \overline{D} → \overline{A} and π_{A} : A → \overline{A} be two continuous ring homomorphisms, and suppose as well that π_{A} is surjective. Define the topological rings D_{A} and D_{E} as the fibre products in the resulting cartesian diagrams of topological rings

\xymatrix{D_{A} \ar[r]^{φ_{A}} \ar[d] & A \ar[d]^{π_{A}} & D_{E} \ar[r]^{φ_{E}} \ar[d] & E \ar[d]^{π_{E}} \\ \overline{D} \ar[r]^{_{\overline{φ}_{A}}} & \overline{A} & \overline{E} \ar[r]^{_{\overline{φ}_{E}}} & \overline{A}.}

where π_{E} := E(π_{A}) and \overline{φ}_{E} := E(\overline{φ}_{A}).

Proposition 16.3.25. In the situation of (16.3.24), the rings D_{A} and D_{E} are perfectoid, and there exists a natural isomorphism of topological rings

ω : E(D_{A}) →^{∼} D_{E}  such that  φ_{E} ∘ ω = E(φ_{A}).

Proof. To begin with, we notice that D_{A} (resp. D_{E}) is complete and separated, since \overline{A} is separated and both \overline{D} and A (resp. \overline{E} and E) are complete and separated. Let (α_{n} | n ∈ ℕ) (resp. (α′_{n} | n ∈ ℕ)) be a distinguished element in Ker u_{A} (resp. in Ker u\overline{D}), and pick a finitely generated ideal of adic definition J_{E} ⊂ Ker π_{E} (resp. J\overline{E} ⊂ Ker \overline{φ}_{E}) for E (resp. for \overline{E}).

Claim 16.3.26. We may assume that α_{0}^{1/p^{2}}E ⊂ J_{E} and α_{0}^{′1/p^{2}} \overline{E} ⊂ J\overline{E}.

Proof of the claim. For the first stated inclusion, notice that α_{0}^{1/p^{n}} ∈ Ker π_{E} for every n ∈ ℕ, so J_{E} + α_{0}^{1/p^{n}}E is still an ideal of adic definition of E contained in Ker π_{E}, for every such n. The same argument applies to J\overline{E}. ♢

By virtue of claim 16.3.26 we may find a system of generators β_{•} := (β_{1}, …, β_{k}) for J_{E} with β_{1} = α_{0}^{1/p^{2}}, and we set b_{i} := \overline{u}_{A}(β_{i}) for i = 1, …, k. It follows that b_{•} := (b_{1}, …, b_{k}) is a system of generators for an ideal J_{A} of adic definition of A. Likewise, pick a system of generators β′_{•} := (β′_{1}, …, β′_{h}) for J\overline{E} with β′_{1} = α_{0}^{′1/p^{2}}, and set b′_{i} := \overline{u}\overline{D}(β′_{1}) for i = 1, …, h. Then b′_{•} := (b′_{1}, …, b′_{h}) is a system of generators for an ideal J\overline{D} of adic definition of \overline{D}. Notice that J_{A} × J\overline{D} ⊂ D_{A} (resp. J_{E} × J\overline{E} ⊂ D_{E}), and the topology of D_{A} (resp. of D_{E}) is the linear topology defined by the system of ideals (J_{A}^{n} × J n ; \overline{D} | n ∈ ℕ) (resp. (J_{E}^{n} × J n ; \overline{E} | n ∈ ℕ)). Denote by K_{E} ⊂ D_{E} the ideal generated by (β_{•} × {0}, {0} × β′_{•}); it is easily seen that

J_{E}^{n+1} × J n+1 ; \overline{E} ⊂ K_{E}^{n} ⊂ J_{E}^{n} × J n ; \overline{E} for every n ∈ ℕ

so the topology of D_{E} agrees with its K_{E}-adic topology, and therefore D_{E} is a perfectoid 𝔽_{p}- algebra. Likewise, let K_{A} ⊂ D_{A} be the ideal generated by (b_{•} × {0}, {0} × b′_{•}); by the same token, we have

J_{A}^{n+1} × J n+1 ; \overline{D} ⊂ K_{A}^{n} ⊂ J_{A}^{n} × J n ; \overline{D} for every n ∈ ℕ

so the topology of D_{A} agrees with the K_{A}-adic topology. Moreover, by construction we have pA ⊂ J_{A}^{p^{2}} and p\overline{D} ⊂ J_{E}^{p^{2}} (lemma 16.2.7(iii)); since p^{2} − 1 ≥ 2, we deduce that pD_{A} ⊂ K_{A}^{2}, and we see already that D_{A} is a P-ring. Next, by remark 9.3.28(ii), we deduce a short exact sequence of W (D_{A})-modules

𝒲 : 0 → W (D_{A}) → W (A) ⊕ W (\overline{D}) → W (\overline{A}) → 0.

On the other hand, by lemma 16.2.7(ii) we may find a distinguished element \underline{α}′′ := (α′′_{n} | n ∈ ℕ) in Ker u_{D_{A}} and to ease notation we set D′_{A} := W (D_{A})/\underline{α}′′W (D_{A}); since the image of \underline{α}′′ is still distinguished in W (\overline{A}), remark 16.1.7(ii) implies that Tor_{1}^{W (D_{A})}(D′_{A}, W (\overline{A})) = 0, so the sequence

𝒲 ⊗_{W (D_{A})} D′_{A} : 0 → D′_{A} → A ⊕ \overline{D} → \overline{A} → 0

is still exact, and consequently u_{D_{A}} induces a ring isomorphism D′_{A} → D_{A}, i.e. D_{A} is perfectoid. Lastly, notice that u_{A} ⊗_{ℤ} 𝔽_{p} and u\overline{D} induce isomorphisms of topological 𝔽_{p}-algebras

D_{A}/pD_{A} →^{∼} \overline{D}/p\overline{D} ×\overline{A} A/pA →^{∼} \overline{D}/α′′_{0}\overline{D} ×\overline{E} E/α′′_{0}E →^{∼} D_{E}/α′′_{0}D_{E}.

Combining with theorem 9.4.10we deduce an isomorphism

E(D_{A}) →^{∼} E(D_{A}/pD_{A}) →^{∼} E(D_{E}/α′′_{0}D_{E}) →^{∼} D_{E}

which fulfills the stated condition, by a simple inspection. □

Definition 16.3.27. Let A be a perfectoid ring and 𝒥 ⊂ 𝒦 ⊂ E := E(A) any two ideals. Let also \underline{α} := (α_{n} | n ∈ ℕ) ∈ Ker u_{A} be any distinguished element and β ∈ E any element.

(i) We say that the inclusion of 𝒥 in 𝒦 is β-taut if we have

β · Φ_{E}^{−1}(𝒦 ^{p}) ⊂ 𝒥 .

(ii) We say that 𝒥 is β-taut if the identity map of 𝒥 is a β-taut inclusion.

(iii) We say that 𝒥 is strictly β-taut if it is β^{λ}-taut for some λ ∈ ℤ[1/p] with 0 ≤ λ < 1. (iv) 𝒥 is taut (resp. strictly taut) if it is α_{0}-taut (resp. strictly α_{0}-taut), and the inclusion

𝒥 ⊂ 𝒦 is taut (resp. strictly taut), if it is α_{0}-taut (resp. strictly α_{0}-taut).

(v) We denote by {𝒥 } ⊂ A the topological closure of the ideal generated by the system (\overline{u}_{A}(x) | x ∈ 𝒥) (notation of (9.3.34)).

Remark 16.3.28. With the notation of definition 16.3.27, the following holds.

(i) It follows easily from remark 16.1.7(i) that the definition of taut and strictly taut ideals does not depend on the choice of \underline{α}.

(ii) By inspecting the definition of angular powers (see (9.3.68)), it is easily seen that 𝒥 is 1-taut if and only if 𝒥 = 𝒥^{⟨1⟩}E.

(iii) It follows from (ii) and lemma 9.3.69(ii.c) that for every ideal 𝒥 of E and every λ ∈ ℕ[1/p], the ideal 𝒥^{⟨λ⟩}E is strictly taut. Moreover, say that α_{0} ∈ 𝒥^{⟨ε⟩}E for some ε ∈ ℕ[1/p]; then the inclusion 𝒥^{⟨λ+ε⟩}E ⊂ 𝒥^{⟨λ⟩}E is taut for every λ ∈ ℕ[1/p], by lemma 9.3.69(ii.b).

(iv) If 𝒥_{1}, 𝒥_{2} ⊂ E are two β-taut (resp. strictly β-taut, resp. taut, resp. strictly taut) ideals, then the same holds for 𝒥_{1} ∩ 𝒥_{2}. Indeed, suppose that both ideals are β-taut; then we have :

β · Φ_{E}^{−1}((𝒥_{1} ∩ 𝒥_{2})^{p}) ⊂ β · Φ_{E}^{−1}(𝒥_{1}^{p} ∩ 𝒥_{2}^{p})
= β · (Φ_{E}^{−1}(𝒥_{1}^{p}) ∩ Φ_{E}^{−1}(𝒥_{2}^{p}))
⊂ (β · Φ_{E}^{−1}(𝒥_{1}^{p})) ∩ (β · Φ_{E}^{−1}(𝒥_{2}^{p}))
⊂ 𝒥_{1} ∩ 𝒥_{2}

whence the claim. In the same vein, if 𝒥_{1} ⊂ 𝒥_{2} is a β-taut inclusion, and 𝒦 is any β-taut ideal of E, then the inclusion 𝒦 ∩ 𝒥_{1} ⊂ 𝒦 ∩ 𝒥_{2} is β-taut as well (details left to the reader); the same holds if β-taut is replaced by strictly β-taut, taut or strictly taut.

(v) If 𝒥 is β-taut, then the same holds for the topological closure 𝒥^{c} of 𝒥 in E. Likewise, if 𝒥 ⊂ 𝒦 is a β-taut inclusion, then the same holds for the inclusion 𝒥^{c} ⊂ 𝒦 ^{c} of the respective topological closures; moreover, both 𝒥 and 𝒦 are β-taut. Then clearly the same assertions hold with β-taut replaced by strictly β-taut, taut, or strictly taut (details left to the reader). In the same vein, notice that

{𝒥^{c}} = {𝒥 } for every ideal 𝒥 ⊂ E .

(vi) Let 𝒥_{1}, 𝒥_{2}, 𝒥_{3} ⊂ E be three ideals, n, m ∈ ℕ any two integers, and suppose that

β_{1} · Φ_{E}^{−n}(𝒥_{1}^{p^{n}}) ⊂ 𝒥_{2} and β_{2} · Φ_{E}^{−m}(𝒥_{2}^{p^{m}}) ⊂ 𝒥_{3} for some β_{1}, β_{2} ∈ E.

Then it is easily seen that β_{1}β_{2} · Φ_{E}^{−n−m}(𝒥_{1}^{p^{n+m}}) ⊂ 𝒥_{3}. Especially, if 𝒥 is β-taut, we have

(16.3.29)  β^{n} · Φ_{E}^{−n}(𝒥^{p^{n}}) ⊂ 𝒥  for every n ∈ ℕ.

(vii) For i = 1, 2, let β_{i} ∈ E be any element, and 𝒥_{i} a β_{i}-taut ideal; then it is easily seen that 𝒥_{1}𝒥_{2} is β_{1}β_{2}-taut. More generally, if 𝒥 ⊂ 𝒦 is a β_{1}-taut inclusion of ideals of E, and 𝒥′ ⊂ E is any other β_{2}-taut ideal, then the inclusion 𝒥 𝒥′ ⊂ 𝒦 𝒥′ is β_{1}β_{2}-taut.

(viii) Suppose that 𝒥 is an open ideal of E. Then the system (\overline{u}_{A}(x) | x ∈ 𝒥) generates an open ideal J of A, and therefore {𝒥 } = J is open as well in A. Indeed, let I ⊂ A be any ideal of definition, pick a finite system (\overline{a}_{1}, …, \overline{a}_{k}) of generators of I/pA, let α_{1}, …, α_{k} be elements of E such that \overline{u}_{A/pA}(α_{i}) = \overline{a}_{i} for i = 1, …, k, and denote by ℐ ⊂ E the ideal generated by the system (α_{1}, …, α_{k}). Arguing as in the proof of lemma 16.2.7(i), we see that the system (\overline{u}_{A}(α_{i}) | i = 1, …, k) generates I, and ℐ^{n} ⊂ 𝒥 for every sufficiently large n ∈ ℕ, so that I^{n} ⊂ J for every such n, whence the assertion.

16.3.30. Keep the notation of definition 16.3.27, and let 𝒥 ⊂ 𝒦 be a taut inclusion of ideals of E. Notice that α_{0}𝒦 ⊂ 𝒥, so 𝒦 /𝒥 is an E/α_{0}E-module, and then it can be viewed as an A/pA-module, via the isomorphism ω of remark 16.3.7(ii). Likewise, from lemma 16.2.7(iii) we easily deduce that p{𝒦 } ⊂ {𝒥 }, so {𝒦 }/{𝒥 } is an A/pA-module as well.

Theorem 16.3.31. Let A be any perfectoid ring, and set E := E(A). The following holds : (i) Every taut inclusion 𝒥_{1} ⊂ 𝒥_{2} of ideals of E induces an A/pA-linear map

\overline{τ} : 𝒥_{2}/𝒥_{1} → \{𝒥_{2}\}/\{𝒥_{1}\} : (x mod 𝒥_{1}) ↦ (\overline{u}_{A}(x) mod \{𝒥_{1}\}).

(ii) If both 𝒥_{1} and 𝒥_{2} are closed in the topology of E, the map \overline{τ} of (i) is an isomorphism.

Proof. Let \underline{α} := (α_{n} | n ∈ ℕ) ∈ Ker u_{A} be any distinguished element.

(i): Let x, y ∈ 𝒥_{2} be any two elements, and set δ := τ_{A}(x + y) − τ_{A}(x) − τ_{A}(y). In light of lemma 16.1.1(iv), we have to check that u_{A}(δ) lies in {𝒥_{1}}. However, proposition 9.3.62 expresses δ as a series of the form ∑_{n∈ℕ} p^{n}b_{n}, where each b_{n} is in turn a finite sum of terms of the form c_{n,σ}τ_{A}(β_{n,σ}) (for σ ranging over a certain finite set Σ_{n}), and with β_{n,σ} ∈ Φ_{E}^{−n}(𝒥_{2}^{p^{n}}) and c_{n,σ} ∈ ℤ_{p} for every n ∈ ℕ and every σ ∈ Σ_{n}. Thus, we come down to checking that p^{n} · \overline{u}_{A}(β_{n,σ}) ∈ \{𝒥_{1}\} for every n ∈ ℕ and σ ∈ Σ_{n}. In light of lemma 16.2.7(iii), this holds if and only if \overline{u}_{A}(α_{0}^{n}·β_{n,σ}) ∈ \{𝒥_{1}\}, which in turns will follow, once we know that α_{0}^{n}·β_{n,σ} ∈ 𝒥_{1}. But the latter is clear from remark 16.3.28(vi) (with β := α_{0} : details left to the reader).

Lastly, for any x ∈ 𝒥_{2} and β ∈ E we have τ_{A}(βx) = τ_{A}(β) · τ_{A}(x), and the isomorphism ω of remark 16.3.7(ii) maps the class of β in E/α_{0}E to the class of \overline{u}_{A}(β) in A/pA; this shows that \overline{τ}(\overline{a} · x) = \overline{a} · \overline{τ}(x) for every \overline{a} ∈ A/pA and every x ∈ 𝒥_{2}/𝒥_{1}, whence (i).

Next, choose a strictly positive ε ∈ ℕ[1/p] such that α_{0} ∈ 𝒥^{⟨ε⟩}E, and for every r ∈ ℝ_{+} define the ideal 𝒥^{⌊r⌋}E as in remark 9.3.70(i); we point out the following special case of (ii) :

Claim 16.3.32. Let 𝒥 be any ideal of definition of E, and r, r′ ∈ ℝ_{+} two real numbers such that ε ≥ r′ − r ≥ 0. Then (ii) holds for the taut inclusion 𝒥^{⌊r′⌋}E ⊂ 𝒥^{⌊r⌋}E.

Proof of the claim. Let us set W ⌊s⌋ := W (𝒥^{⌊s⌋}E) for every s ∈ ℕ[1/p] (notation of remark 9.3.28(iv)); from (9.3.73) we see that τ_{A} induces a natural isomorphism

𝒥^{⌊r⌋}E/𝒥^{⌊r′⌋}E →^{∼} W ⌊r⌋/(pW ⌊r⌋ + W ⌊r′⌋)

and we are reduced to showing that the restriction of u_{A} to W ⌊r⌋ induces an isomorphism

(16.3.33)  W ⌊r⌋/(pW ⌊r⌋ + W ⌊r′⌋) →^{∼} {𝒥^{⌊r⌋}E}/{𝒥^{⌊r′⌋}E}.

However, remark 16.3.28(viii) already implies that (16.3.33) is surjective, and its kernel is the image of W ⌊r⌋ ∩ \underline{α}W (E). In light of proposition 16.1.12(ii) we then come down to showing :

\underline{α}W ⌊r⌋ ⊂ pW ⌊r⌋ + W ⌊r′⌋.

But due to our choice of ε we get α_{0}𝒥^{⌊r⌋}E ⊂ 𝒥^{⌊r′⌋}E, so the latter inclusion follows again from (9.3.73). ♢

Claim 16.3.34. Let B be any ring, f : M → M′ and g : M′ → M′′ two continuous maps of topological B-modules, and suppose that

(a) the topologies of M and M′′ are discrete and that of M′ is separated

(b) f has dense image and g ∘ f is injective. Then f is an isomorphism of topological B-modules.

Proof of the claim. By assumption, the topological closure of f(M) in M′ equals M′; by claim 8.3.25 it follows that the topological closure of g ∘ f(M) in M′′ equals the topological closure of g(M′). But since the topology of M′′ is discrete, this just means that g ∘ f(M) = g(M′). We may therefore replace M′′ by g(M′), and assume from start that g is surjective and g ∘ f is an isomorphism. Then we may even assume that M′′ = M and g ∘ f = 1_{M}. Let us endow Im f and Ker g with the topologies induced from the inclusion into M′; then the addition law of M′ restricts to a continuous and bijective B-linear map

h : Im f ⊕ Ker g → M′

(where the direct sum is endowed with the product topology). However, the inverse of h is the map given by the rule : m′ ↦ (f ∘ g(m′), m′ − f ∘ g(m′)) for every m′ ∈ M′. Clearly, this map is also continuous, so h is an isomorphism of topological B-modules. Now, since M′ is separated, the same must hold for Ker g; especially, L := h(Im f ⊕ 0) is a closed subset of M′. On the other hand, Im f = L; since f has dense image, we must then have L = M′, i.e. Ker g = 0, whence the claim. ♢

We may now complete the proof of (ii): fix an ideal of definition 𝒥 of E with α_{0} ∈ 𝒥, and for every ideal 𝒦 ⊂ E set

Fil^{n}𝒦 := 𝒦 ∩ 𝒥^{⌊n⌋}E  Fil^{n}{𝒦 } := {Fil^{n}𝒦 }  for every n ∈ ℕ.

If 𝒦 ′ ⊂ 𝒦 is any inclusion of ideals of E, the image of Fil^{•}𝒦 in 𝒦 /𝒦 ′ defines a filtration Fil^{•}(𝒦 /𝒦 ′) of 𝒦 /𝒦 ′, and likewise we get a filtration Fil^{•}({𝒦 }/{𝒦 ′}) on {𝒦 }/{𝒦 ′}. Notice that the inclusion Fil^{n+1}𝒦 ⊂ Fil^{n}𝒦 is taut for every n ∈ ℕ and every taut ideal 𝒦 , and denote by gr^{•}𝒦 (resp. gr^{•}{𝒦 }) the graded A/pA-module associated to the filtration Fil^{•}𝒦 (resp. to Fil^{•}{𝒦 }). Likewise we define gr^{•}(𝒦 /𝒦 ′) and gr^{•}({𝒦 }/{𝒦 ′}) for an inclusion of ideals 𝒦 ′ ⊂ 𝒦 . We remark :

Claim 16.3.35. If 𝒦 is taut, the map \overline{τ} of (i) induces an isomorphism

gr^{n}𝒦 →^{∼} gr^{n}{𝒦 } for every n ∈ ℕ.

Proof of the claim. Let us endow any ideal of E (resp. of A) with the topology induced from the inclusion into E (resp. into A), and for any inclusion ℐ ⊂ ℐ′ (resp. I ⊂ I′) of ideals of E (resp. of A), let us endow ℐ′/ℐ (resp. I′/I) with the corresponding quotient topology. Notice that the inclusions Fil^{n+1}𝒦 ⊂ Fil^{n}𝒦 and Fil^{n+1}{𝒦 } ⊂ Fil^{n}{𝒦 } are both taut (remark 16.3.28(iv)); from (i) we deduce a commutative diagram of A/pA-modules :

[figure: gr^{n}𝒦; gr^{n}{𝒦 }; gr^{n}E; gr^{n}A]

whose left vertical arrow is injective, and whose bottom horizontal arrow is already known to be an isomorphism, by claim 16.3.32. Moreover, all these maps are continuous, for the topologies that we have just defined on these modules, and furthermore, the top horizontal arrow has dense image. Also, since 𝒥 is open, the same holds for 𝒥^{⌊r′⌋}E, and therefore the two modules on the bottom row have both the discrete topology. Likewise, 𝒦 ∩𝒥^{⌊r′⌋}E is open in 𝒦 ∩𝒥^{⌊r′⌋}E, so also the source of the map on the top row is a discrete A/pA-module. Lastly, since {𝒦 ∩ 𝒥^{⌊r′⌋}E} is a closed ideal, the target of the same map is a separated A/pA-module. Thus, all the conditions of claim 16.3.34are fulfilled, and it follows that the top horizontal arrow is an isomorphism. ♢ There follows, for every n ∈ ℕ a commutative diagram with exact rows

[figure: 0; gr^{n}𝒥_{1}; gr^{n}𝒥_{2}; gr^{n}(𝒥_{2}/𝒥_{1}); 0; 0; gr^{n}{𝒥_{1}}; gr^{n}{𝒥_{2}}; gr^{n}({𝒥_{2}}/{𝒥_{1}}); 0]

and claim 16.3.35 shows that the leftmost and central vertical arrows are both isomorphisms, so the same holds for the rightmost vertical arrow. Now, notice that the filtration Fil^{•}(𝒥_{2}/𝒥_{1}) defines a separated and complete topology on 𝒥_{2}/𝒥_{1}, since 𝒥_{2} and 𝒥_{1} are both closed ideals in E. The same holds for the topology on {𝒥_{2}}/{𝒥_{1}} determined by the filtration Fil^{•}({𝒥_{2}}/{𝒥_{1}}), since {𝒥_{2}} and {𝒥_{1}} are closed ideals in A. Then (ii) follows directly from [34, Ch.III, §2, n.8, Cor.3]. □

Theorem 16.3.36. Let A be any perfectoid ring, 𝒦 a taut ideal of E := E(A), and W (𝒦 )^{c} the topological closure of W (𝒦 ) in A(A) (notation of remark 9.3.28(iv)). We have :

(i) {𝒦 } = u_{A}(W (𝒦 )^{c}).

(ii) 𝒦 is an open ideal if and only if the same holds for {𝒦 }.

(iii) If 𝒦 ′ ⊂ 𝒦 is another taut ideal, and both 𝒦 and 𝒦 ′ are closed, then 𝒦 = 𝒦 ′ if

and only if {𝒦 } = {𝒦 ′}. (iv) Suppose that 𝒦 is 1-taut and closed, and let \underline{α} be any distinguished element of Ker u_{A}.

Then the pair (\underline{α}, 𝒦 ) is transversal (see (16.1.11)).

Proof. Pick any distinguished element \underline{α} := (α_{n} | n ∈ ℕ) of Ker u_{A}.

(i): First we prove that u_{A}(W (𝒦 )^{c}) ⊂ \{𝒦 \}. Indeed, let \underline{a} := (a_{n} | n ∈ ℕ) ∈ W (𝒦 ) be any element; taking into account (9.3.46), we are easily reduced to showing that

u_{A}(p^{n} · τ_{A}(a_{n}^{p^{−n}})) ∈ {𝒦 } for every n ∈ ℕ.

In light of lemmata 16.2.7(iii) and 16.1.1(iv), it then suffices to check that \overline{u}_{A}(α_{0}^{n}·a_{n}^{p^{−n}}) ∈ \{𝒦 \}, and the latter is clear from (16.3.29) (applied with β := α_{0}). Next, fix an ideal of definition 𝒥 of E containing α_{0}, and define the filtrations Fil^{•}𝒦 on 𝒦 and Fil^{•}{𝒦 } on {𝒦 } as in the proof of theorem 16.3.31(ii); by (9.3.29) and remark 16.3.28(v) we may assume that 𝒦 is a closed ideal of E, in which case theorem 16.3.31(ii) shows that the map \overline{u}_{A} induces an isomorphism gr^{•}𝒦 →^{∼} gr^{•}{𝒦 } on the respective associated graded A/pA-modules. Now, let x ∈ {𝒦 } be any element; it follows easily that we may write

x = u_{A}(∑_{n∈ℕ} τ_{A}(β_{n})) where β_{n} ∈ Fil^{n}𝒦 for every n ∈ ℕ

and the series converges in the topology of A(A). But clearly τ_{A}(β_{n}) ∈ W (𝒦 ) for every n ∈ ℕ, whence the sought converse inclusion.

(ii): In light or remark 16.3.28(viii) we may assume that {𝒦 } is open, and we need to show that the same holds for 𝒦 ; then, by lemma 8.3.21(ii.b), we may also assume that 𝒦 is a closed ideal. The inclusion of 𝒦 into E is a map of filtered E-modules Fil^{•}𝒦 → Fil^{•}E, and likewise we get a map of filtered A-modules Fil^{•}{𝒦 } → Fil^{•}A. The assumption on {𝒦 } implies that there exists n ∈ ℕ such that the map of associated graded modules gr^{k}{𝒦 } → gr^{k}A is an isomorphism for every k ≥ n. Then theorem 16.3.31(ii) implies that the same holds for the corresponding map gr^{k}𝒦 → gr^{k}E. However, both 𝒦 and E are complete and separated for their filtrations, so Fil^{n}𝒦 = Fil^{n}E = 𝒥^{⟨n⟩} ([34, Ch.III, §2, n.8, Cor.3]), i.e. 𝒦 is open.

(iii) is similar : we may assume that {𝒦 } = {𝒦 ′}, and we consider the induced map of filtered E-modules Fil^{•}𝒦 ′ → Fil^{•}𝒦 . Arguing as in the proof of (ii), we see that the latter induces an isomorphism on the associated graded E-modules, so 𝒦 = 𝒦 ′, again by [34, Ch.III, §2, n.8, Cor.3].

(iv): Denote by (α_{0}𝒦 )^{c} the topological closure of \underline{α}𝒦 in E. We shall show first that

Claim 16.3.37. W (𝒦 ) ∩ Ker u_{A} = W ((α_{0}𝒦 )^{c}) ∩ Ker u_{A} + \underline{α}W (𝒦 ).

Proof of the claim. Indeed, let \underline{ω} := (ω_{n} | n ∈ ℕ) ∈ W (𝒦 ) ∩ Ker u_{A} be any element; due to remarks 9.3.70(iv) and 16.3.28(ii), we may write

\underline{ω} = τ_{A}(ω_{0}) + p · \underline{ω}′ for some \underline{ω}′ ∈ W (𝒦 )

whence 0 = u_{A}(\underline{ω}) = \overline{u}_{A}(ω_{0}) + p · u_{A}(\underline{ω}′), and especially, the image of \overline{u}_{A}(ω_{0}) vanishes in {𝒦 }/{(α_{0}𝒦 )^{c}} (lemma 16.2.7(iii)). In view of theorem 16.3.31(ii), we deduce that ω_{0} ∈ (α_{0}𝒦 )^{c}. Write also \underline{α} = τ_{A}(α_{0}) + p\underline{u} for some \underline{u} ∈ A(A)^{×}; we obtain

\underline{ω} = τ_{A}(ω_{0}) + (\underline{u}^{−1}\underline{α} − τ_{A}(α_{0})) · \underline{ω}′)

and it suffices to notice that \underline{u}^{−1}\underline{α} · \underline{ω}′ ∈ \underline{α}W (𝒦 ) ⊂ W (𝒦 ) ∩ Ker u_{A}, and consequently τ_{A}(ω_{0}) − τ_{A}(α_{0}) · \underline{ω}′ ∈ W ((α_{0}𝒦 )^{c}) ∩ Ker u_{A}. ♢

Notice next that if 𝒦 is 1-taut, then the same holds for α_{0}𝒦 , and then also for (α_{0}𝒦 )^{c} (remark 16.3.28(v)). Thus, we may apply claim 16.3.37to (α_{0}^{n}𝒦 )^{c}, and get

W ((α_{0}^{n}𝒦 )^{c}) ∩ Ker u_{A} = W ((α_{0}^{n+1}𝒦 )^{c}) ∩ Ker u_{A} + \underline{α}W ((α_{0}^{n}𝒦 )^{c}) for every n ∈ ℕ.

Now, let x_{0} ∈ W (𝒦 ) ∩ \underline{α}W (E) be any element; we may then find inductively identities

x_{n} = \underline{α} · y_{n} + x_{n+1} with x_{n+1} ∈ W ((α^{n+1}𝒦 )^{c}) ∩ Ker u_{A} and y_{n} ∈ W ((α_{0}^{n}𝒦 )^{c})

for every n ∈ ℕ. Therefore :

x_{0} = \underline{α} · (y_{0} + y_{1} + ⋯ + y_{n}) + x_{n+1} for every n ∈ ℕ.

Lastly, since 𝒦 is 1-taut, the ideal W (𝒦 ) is closed in the topology of W (E) (see remark 9.3.28(iv)), and since α_{0} is topologically nilpotent, for every open ideal I of W (E) there exists n ∈ ℕ such that W ((α_{0}^{n}𝒦 )^{c}) ⊂ I. Consequently, the series ∑_{n∈ℕ} y_{n} converges to an element y ∈ W (𝒦 ), and lim_{n→+∞} x_{n} = 0, so finally x = \underline{α} · y, which concludes the proof. □

Corollary 16.3.38. Let A be any perfectoid ring, 𝒦 ⊂ E := E(A) an ideal such that Φ_{E}(𝒦 ) = 𝒦 , and endow A/{𝒦 } (resp. E/𝒦 ^{c}) with the quotient topology induced via the projection π_{A} : A → A/{𝒦 } (resp. π_{E} : E → E/𝒦 ^{c}). Then A/{𝒦 } is perfectoid, and there exists a natural isomorphism of topological rings

ω : E(A/{𝒦 }) →^{∼} E/𝒦 ^{c}  such that  ω ∘ E(π_{A}) = π_{E}.

Proof. The topological 𝔽_{p}-algebra E/𝒦 ^{c} is complete and separated, and clearly Φ_{E}(𝒦 ^{c}) = 𝒦 ^{c}, so E/𝒦 ^{c} is perfectoid, and we have

W (𝒦 ^{c}) = Ker (W (π_{E}) : W (E) → W (E/𝒦 ^{c})).

It follows that the kernel of the induced projection π′_{A} : A → A ⊗_{W (E)} W (E/𝒦 ^{c}) is naturally identified with u_{A}(W (𝒦 ^{c})). Moreover, since W (E/𝒦 ^{c}) is separated (lemma 9.3.33(ii)), W (𝒦 ^{c}) is a closed subset of W (E), and taking into account theorem 16.3.36(i) we deduce that Ker π′_{A} = {𝒦 }, so we get a cocartesian diagram

𝒟  :  \xymatrix{W (E) \ar[r]^{W (π_{E})} \ar[d]_{u_{A}} & W (E/𝒦 ) \ar[d]^{u′} \\ A \ar[r]^{π_{A}} & A/\{𝒦 \}}

and notice that both u_{A} and π_{A} are open and surjective (lemma 16.2.7(i)), so the same holds for u′, and therefore A/{𝒦 } is perfectoid (proposition 16.2.21(i)). Let also π : A → A/pA be the proejction; taking into account remark 9.4.5(ii), there follows a commutative diagram

E(𝒟 ⊗_{ℤ} 𝔽_{p})  :  \xymatrix{E \ar[r]^{π_{E}} \ar[d]_{E(π)} & E/𝒦 \ar[d]^{E(u′)} \\ E(A/pA) \ar[r]^{E(π_{A}⊗_{ℤ}𝔽_{p})} & E(A/(\{𝒦 \} + pA))}

whose vertical arrows are both isomorphisms. The assertion follows easily. □

Remark 16.3.39. (i) In the situation of corollary 16.3.38, let (α_{n} | n ∈ ℕ) be any distinguished element of Ker u_{A}, and suppose that 𝒦 ⊂ Ann_{E}(α_{0}). Then the inclusion 0 ⊂ 𝒦 ^{c} is taut, and therefore theorem 16.3.31(ii) yields a natural W (E)-linear identification

𝒦 ^{c} →^{∼} \{𝒦 \} β ↦ \overline{u}_{A}(β).

Especially, {𝒦 } ⊂ Ann_{A}(p).

(ii) Let ℐ , 𝒦 ⊂ E be two ideals such the topology of E agrees with the ℐ -adic topology, and suppose that

𝒦 ℐ = 0 and Φ_{E}(𝒦 ) = 𝒦 .

Notice that 𝒦 = 𝒦 ^{⟨1⟩}, and α_{0}^{n} ∈ ℐ for some n ∈ ℕ, from which it follows easily that 𝒦 ⊂ Ann_{E}(α_{0}). Moreover, 𝒦 ∩ ℐ = 0; indeed, if x ∈ 𝒦 ∩ ℐ, then x^{2} = 0, hence x = 0. Thus, 𝒦 + ℐ^{n} = 𝒦 ⊕ ℐ^{n} for every n ∈ ℕ, and therefore

⋂_{n∈ℕ}(𝒦 + ℐ^{n}) = 𝒦 ⊕ ⋂_{n∈ℕ} ℐ^{n} = 𝒦

i.e. 𝒦 is closed in the topology of E, so corollary 16.3.38 says that E/𝒦 and A/{𝒦 } are perfectoid for their quotient topologies, and there exists an isomorphism of topological rings

ω : E(A/{𝒦 }) →^{∼} E/𝒦 such that ω ∘ E(π_{A}) = π_{E}

where π_{A} : A → A/{𝒦 } and π_{E} : E → E/𝒦 are the projections.

Corollary 16.3.40. Let A be any perfectoid ring, and a_{•} := (a_{1}, …, a_{k}) a finite system of elements of E := E(A). Denote by J ⊂ E (resp. 𝒥 ⊂ W (E)) the ideal generated by a_{•} (resp. by τ_{E}(a_{1}), …, τ_{E}(a_{k})), and define the ideal [a_{•}]^{⟨λ⟩} ⊂ W (E) as in remark 9.3.70(iii), for every λ ∈ ℕ[1/p]. We regard A as a W (E)-algebra, via the map u_{A}. Then we have :

(i) [a_{•}]^{⟨λ⟩}A ⊂ {J^{⟨λ⟩}E} ⊂ [a_{•}]^{⟨λ′⟩}A for every λ′ < λ in ℕ[1/p].

(ii) The following conditions are equivalent : (a) J is open in E

(b) 𝒥 A is open in A

(c) [a_{•}]^{⟨λ⟩}A is open in A for every λ ∈ ℕ[1/p]

(d) There exists a strictly positive λ ∈ ℕ[1/p] such that [a_{•}]^{⟨λ⟩}A is open in A and if these conditions hold, then [a_{•}]^{⟨λ⟩}A = {J^{⟨λ⟩}E} for every λ ∈ ℕ[1/p].

(iii) If I is any ideal of definition of E, the following holds : (a) {I^{⟨λ⟩}E} is a topologically nilpotent open ideal of A for every λ > 0 in ℕ[1/p].

(b) ⋂_{n∈ℕ}{J^{⟨λ⟩}E + I^{⟨n⟩}E} ⊂ {J^{⟨λ′⟩}E} for every λ′ < λ in ℕ[1/p].

Proof. (i): The first inclusion is clear. For the second, notice that {J^{⟨λ⟩}E} = W (J^{⟨λ⟩})^{c} · A by theorem 16.3.36(i) and remark 16.3.28(iii,v). On the other hand, pick any λ′′ ∈ ℕ[1/p] such that λ′ < λ′′ < λ; we have W (J^{⟨λ⟩})^{c} = ∏_{n∈ℕ}(J^{⟨p^{n}λ⟩}E)^{c} (see remark 9.3.28(iv) and lemma 9.3.69(ii.a)), and moreover (J^{⟨p^{n}λ⟩}E)^{c} ⊂ J^{⟨p^{n}λ′′⟩}E for every n ∈ ℕ (claim 16.3.20), hence

W (J^{⟨λ⟩})^{c} ⊂ W (J^{⟨λ′′⟩}) ⊂ [a_{•}]^{⟨λ′⟩}

where the last inclusion follows from proposition 9.3.77(i).

(ii): Taking into account lemma 9.3.69(ii.a,iv), it is easily seen that (ii.b)⇔(ii.c)⇔(ii.d). Combining with (i), we conclude that (ii.b) holds if and only if {J^{⟨1⟩}E} is open in A. However, from theorem 16.3.36(ii) and remark 16.3.28(iii) we see as well that {J^{⟨1⟩}E} is open in A if and only if J^{⟨1⟩}E is open in E. Lastly, J^{⟨1⟩}E is open in E if and only if the same holds for J, due to lemma 9.3.69(ii.a,iv). Thus (ii.a)⇔(ii.b). Next, we remark :

Claim 16.3.41. For every λ ∈ ℕ[1/p] and every n ∈ ℕ, the ideal generated by the system (\overline{u}_{A}(x) | x ∈ J^{⟨λ⟩}E) is contained in p^{n}A + [a_{•}]^{⟨λ⟩}A.

Proof of the claim. The claim follows easily from proposition 9.3.62 and lemma 16.1.1(iv) (details left to the reader). ♢

Now, if 𝒥 A is open in A, the same holds for [a_{•}]^{⟨λ⟩}A, for every λ ∈ ℕ[1/p], in which case the latter contains p^{n} for some sufficiently large n ∈ ℕ; taking into account claim 16.3.41, we get the second assertion of (ii).

(iii.a): It follows easily from (ii) that {I^{⟨λ⟩}} is open, and combining lemma 9.3.69(ii.a,iv) with (i), it is easily seen that {I^{⟨λ⟩}E} is topologically nilpotent : details left to the reader.

(iii.b): Fix N ∈ ℕ such that λ′ < (1 − p^{−N})λ, let x ∈ J^{⟨λ⟩}E, y ∈ I^{⟨n⟩} be any two elements, and α ∈ ℕ[1/p] any rational number ≤ 1; notice that z_{α} := τ_{A}(x^{1−α} · y^{α}) lies in {I^{⟨n/p^{N} ⟩}E} if α ≥ p^{−N}, and lies in {J^{⟨λ′⟩}E} otherwise. However, proposition 9.3.62 says that τ_{A}(x + y) can be written as a p-adically convergent series whose terms are of the form u_{α} · z_{α} for certain u_{α} ∈ W (E); since (ii) implies that {I^{⟨n/p^{N} ⟩}E} is an open ideal in A, we deduce

⋂_{n∈ℕ}{J^{⟨λ⟩}E + I^{⟨n⟩}E} ⊂ ⋂_{n∈ℕ}({J^{⟨λ′⟩}E} + {I^{⟨n/p^{N} ⟩}E}) = {J^{⟨λ′⟩}E}

where the last equality holds because {J^{⟨λ′⟩}E} is a closed ideal, taking into account (iii.a). □

Theorem 16.3.42. Let f : A → A′ be a continuous ring homomorphism of perfectoid rings; set C := Im f and D := Im E(f). The following holds :

(i) f is surjective (resp. bijective, resp. adic, resp. open, resp. open and injective) if and only if the same holds for E(f).

(ii) C is open in A′ if and only if D is open in E(A′).

(iii) Suppose that C is open in A′, then we have :

(a) A^{′∘∘} ⊂ C.

(b) C is perfectoid both for the quotient topology induced by the projection A → C, and for the subspace topology induced by the inclusion map C → A′.

Proof. Set E := E(A) and E′ := E(A′). Also, denote by 𝒯_{D} and 𝒯′_{D} (resp. 𝒯_{C} and 𝒯′_{C}) the topologies on D (resp. on C) induced respectively by the projection E → D and the open inclusion D → E′ (resp. induced by the projection A → C and the open inclusion C → A′).

Let α_{•} := (α_{n} | n ∈ ℕ) be a distinguished element in the kernel of u_{A} : W (E) → A, and recall that the image of α_{•} in W (E′) is a distinguished element in the kernel of u_{A′}. We denote by Fil^{•}A and Fil^{•}A′ the p-adic filtrations on A and respectively A′; likewise, let Fil^{•}E and Fil^{•}E′ be the α_{0}-adic filtrations on E and respectively E′. Moreover, we let A^{•} be the complex concentrated in degrees 0 and 1, with A^{0} := A, A^{1} := A′, and d^{0} := f. Likewise, we define the complex E^{•} := (E ⟶^{E(f)} E′) in degrees 0 and 1. With the foregoing filtrations, A^{•} and E^{•} are filtered complexes of ℤ-modules, and we wish to consider the associated spectral sequences, as described in (7.2.12). Explicitly, we have Z(A^{•})_{r}^{pq} = 0 whenever p + q ≠ 0, 1, and :

Z(A^{•})_{r}^{p,−p} := {a ∈ Fil^{p}A | f(a) ∈ Fil^{p+r}A′}
Z(A^{•})_{r}^{p,−p+1} := Fil^{p}A′ for every p ∈ ℤ and r ∈ ℕ

and correspondingly for Z(E^{•})_{r}^{pq}. Since \overline{u}_{A}(Fil^{p}E) ⊂ Fil^{p}A for every p ∈ ℤ, and likewise for \overline{u}_{A′}, it is then clear that \overline{u}_{A} and \overline{u}_{A′} restrict to well defined maps

\overline{u}_{r}^{pq} : Z(E^{•})_{r}^{pq} → Z(A^{•})_{r}^{pq} for every p, q ∈ ℤ and r ∈ ℕ.

Moreover, we have B(A^{•})_{r}^{pq} = 0 whenever p + q ≠ 0, 1, and :

B(A^{•})_{r}^{p,−p} := Fil^{p+1}A ∩ Z(A^{•})_{r}^{p,−p}
B(A^{•})_{r}^{p,−p+1} := (Fil^{p+1}A′ + f(Fil^{p−r+1}A)) ∩ Fil^{p}A′ for every p ∈ ℤ and r ∈ ℕ

and correspondingly for B(E^{•})_{r}^{pq}.

Claim 16.3.43. \overline{u}_{r}^{a,b}(B(E^{•})_{r}^{a,b}) ⊂ B(A^{•})_{r}^{a,b} for every a, b ∈ ℤ and r ∈ ℕ.

Proof of the claim. The assertion is trivial for a + b ≠ 0, 1. If b = −a, the assertion follows directly from our explicit descriptions of B(E^{•})_{r}^{a,−a} and B(A^{•})_{r}^{a,−a}. Next, notice that B(E^{•})_{0}^{a,−a+1} = Fil^{a+1}E′, and likewise for B(A^{•})_{0}^{a,−a+1}; the assertion follows then when r = 0 and b = −a + 1. For the case where r > 0 and b = −a + 1, lemma 7.2.13shows that

B(E^{•})_{r}^{a,−a+1} = Fil^{a+1}E′ + f(Z(E^{•})_{r−1}^{a−r+1,−a+r−1})

and likewise for B(A^{•})_{r}^{a,−a+1}. Hence, let x ∈ Fil^{p+1}E′ and y ∈ Z(E^{•})_{r−1}^{a−r+1,−a+r−1} and set z := x+E(f)(y); we need to check that \overline{u}_{A′}(z) ∈ Fil^{a+1}A′ +f(Z(A^{•})_{r−1}^{a−r+1,−a+r−1}). However, proposition 9.3.62 says that \overline{u}_{A′}(z) = \overline{u}_{A′}(x) + \overline{u}_{A′} ∘ E(f)(y) + w, where w is the limit of a p-adically convergent series whose terms are ℤ_{p}-linear combinations of products of the form p^{n} · \overline{u}_{A′}(x^{σ_{0}} · E(f)(y)^{σ_{1}}), with n ∈ ℕ \ \{0\}, σ_{0}, σ_{1} ∈ ℚ_{+}, and σ_{0} + σ_{1} = 1. Now :

\overline{u}_{A′} ∘ E(f)(y) = f(\overline{u}_{A}(y)) ∈ f(Z(A^{•})_{r−1}^{a−r+1,−a+r−1}) and \overline{u}_{A′}(x) ∈ Fil^{a+1}A′.

Moreover, since f(\overline{u}_{A}(y)) ∈ Fil^{a}A′, we see that p^{m} divides \overline{u}_{A′}(x^{σ_{0}} · E(f)(y)^{σ_{1}}) = \overline{u}_{A′}(x)^{σ_{0}} · f(\overline{u}_{A}(y))^{σ_{1}} in A′, for every integer m ≤ σ_{0} · (a + 1) + σ_{1} · a = a + σ_{0}. Thus, w ∈ Fil^{a+1}A′, whence the contention. ♢

Next, notice that E(A^{•})_{0}^{p,−p} = gr^{p}A and E(A^{•})_{0}^{p,−p+1} = gr^{p}A′ for every p ∈ ℤ, and likewise for E(E^{•})_{0}^{pq}, whenever p + q = 0, 1. Especially, E(A^{•})_{r}^{a,b} is a ℤ/pℤ-module for every a, b ∈ ℤ and r ∈ ℕ. Invoking proposition 9.3.62again, we deduce that the composition

Z(E^{•})_{r}^{pq} ⟶^{\overline{u}_{r}^{pq}} Z(A^{•})_{r}^{pq} → E(A^{•})_{r}^{pq}

is an additive map, and combining with claim 16.3.43, it follows that the latter factors through a well defined group homomorphism

v_{r}^{pq} : E(E^{•})_{r}^{pq} → E(A^{•})_{r}^{pq} for every p, q ∈ ℤ and r ∈ ℕ.

Lastly, a simple inspection shows that the system of maps v_{•}^{••} yields a morphism of spectral sequences E(E^{•})_{•}^{••} → E(A^{•})_{•}^{••}. Since v_{0}^{pq} is an isomorphism for every p, q ∈ ℤ, it follows that v_{•}^{••} is an isomorphism of spectral sequences.

Let us endow Coker f with the filtration induced by Fil^{•}A′ : namely

Fil^{p}(Coker f) := Im(Fil^{p}A′ → Coker f) for every p ∈ ℤ.

Likewise, we define a filtration Fil^{•}Coker E(f) on Coker E(f). Let moreover 𝒯_{A′,p} be the p-adic topology on A′, and 𝒯_{E′,α_{0}} the α_{0}-adic topology on E′.

Claim 16.3.44. C is open in (A′, 𝒯_{A′,p}) (resp. D is open in (E′, 𝒯_{E′,α_{0}})) if and only if there exists i ∈ ℕ such that gr^{i}Coker f = 0 (resp. such that gr^{i}Coker E(f) = 0).

Proof of the claim. Clearly, if C is open in (A′, 𝒯_{A′,p}), then we find i ∈ ℕ with gr^{i}Coker f = 0. Conversely, if the latter condition holds, set M := Fil^{i}A′ + C ⊂ A′; we deduce that M = pM + C, whence M = C, by [126, Th.8.4] and lemma 16.2.3(v); i.e. Fil^{i}A′ ⊂ C, so C is open in (A′, 𝒯_{A′,p}). Since E and E′ are complete and separated for their α_{0}-adic topologies (see (16.2.19)), the same argument shows the assertion for D. ♢

According to (7.2.17), the spectral sequences E(A^{•})_{•}^{••} and E(E^{•})_{•}^{••} admit natural abutments, and by inspecting the constructions, we deduce natural isomorphisms :

E(A^{•})_{∞}^{a,1−a} →^{∼} gr^{a}Coker f  E(E^{•})_{∞}^{a,1−a} →^{∼} gr^{a}Coker E(f)  for every a ∈ ℤ.

Moreover, by proposition 7.2.18 they converge in degree 1. Since these spectral sequences are isomorphic, combining with claim 16.3.44we conclude that C is open in (A′, 𝒯_{A′,p}) if and only if D is open in (E′, 𝒯_{E′,α_{0}}).

(ii): By the foregoing, in order to prove the assertion we may assume that C is open in (A′, 𝒯_{A′,p}) and D is open in (E′, 𝒯_{α_{0},E′}). Hence, α_{0}^{n}E′ ⊂ D for some n ∈ ℕ, and since D and E′ are perfect rings, it follows easily that α_{0}E′ ⊂ D, i.e. gr^{1}Coker E(f) = 0. By the foregoing, we deduce that gr^{1}Coker f = 0 as well, i.e. pA′ ⊂ C. Endow E′/α_{0}E′ and A′/pA′ with the quotient topologies 𝒯_{E′/α_{0}E′} and 𝒯_{A′/pA′} induced by E′ and A′, and recall that \overline{u}_{A′/pA′} induces an isomorphism of topological rings

ω : (E′/α_{0}E′, 𝒯_{E′/α_{0}E′}) →^{∼} (A′/pA′, 𝒯_{A′/pA′})

(remark 16.3.7(ii)); clearly ω(C/pA′) = D/α_{0}E′, and especially C/pA′ is open in A′/pA′ if and only if D/α_{0}E′ is open in E′/α_{0}E′. The contention follows immediately.

(iii.a): In view of (ii), the subring D is open in E′; since D is perfect, it follows easily that E^{′∘∘} ⊂ D. Endow D/α_{0}E′ (resp. C/pA′) with the topology 𝒯′_{D/α_{0}E′} (resp. 𝒯′_{C/pA′}) induced by 𝒯′_{D} via the projection D → D/α_{0}E′ (resp. by 𝒯′_{C} via the projection C → C/pA′). Then we have (D, 𝒯′_{D})^{∘∘} = E^{′∘∘}, and notice that

(D/α_{0}E′, 𝒯′_{D/α_{0}E′})^{∘∘} = (D, 𝒯′_{D})^{∘∘}/α_{0}E′ and (C/pA′, 𝒯′_{C/pA′})^{∘∘} = (C, 𝒯′_{C})^{∘∘}/pA′.

Moreover, according to lemma 8.2.3(i), the topology 𝒯′_{D/α_{0}E′} agrees with the topology induced by 𝒯_{E′/α_{0}E′} via the inclusion map D/α_{0}E′ → E′/α_{0}E′, and likewise for 𝒯′_{C/pA′}; it follows that the foregoing isomorphism ω restricts to an isomorphism of topological rings

(C/pA′, 𝒯′_{C/pA′}) →^{∼} (D/α_{0}E′, 𝒯′_{D/α_{0}E′}).

Since (D/α_{0}E′, 𝒯′_{D/α_{0}E′})^{∘∘} = E^{′∘∘}/α_{0}E′ = (E′/α_{0}, 𝒯_{E′/α_{0}})^{∘∘}, we get (C/pA′, 𝒯′_{C/pA′})^{∘∘} = (A′/pA′, 𝒯_{A′/pA′})^{∘∘} = A^{′∘∘}/pA′, and finally A^{′∘∘} = (C, 𝒯′_{C})^{∘∘}.

(i): If f is bijective, then clearly the same holds for E(f); conversely, if E(f) is bijective, then the same holds for

W (E)/α_{•}W (E) ⊗_{W (E)} W (E(f)) : W (E)/α_{•}W (E) → W (E′)/α_{•}W (E′).

But the image of α_{•} is still distinguished in W (E′), and lies in Ker u_{A′}, hence the latter map is naturally identified with f; especially, f is bijective.

• If either f or E(f) is surjective, then pA′ ⊂ C and α_{0}E′ ⊂ D by the proof of (ii), and moreover the isomorphism E′/α_{0}E′ →^{∼} A′/pA′ induced by \overline{u}_{A/pA} restricts to an isomorphism D/α_{0}E′ →^{∼} C/pA′. Then clearly C = A′ if and only if D = E′.

• Next, let 𝒥 ⊂ E be any ideal of definition, (β_{1}, …, β_{k}) a finite system of generators for 𝒥, and J ⊂ A the ideal generated by (\overline{u}_{A}(β_{1}), …, \overline{u}_{A}(β_{k})). By corollary 16.3.40(ii), the ideal JA′ is open in A′ if and only if 𝒥 E′ is open in E′. Moreover, 𝒥 E′ (resp. JA′) is topologically nilpotent in E′ (resp. in A′), since the same holds for 𝒥 (resp. for J); thus, 𝒥 E′ is an ideal of adic definition for E′ if and only if JA′ is an ideal of adic definition for A′, i.e. f is adic if and only if E(f) is adic.

• If either f of E(f) is open, we know already that C is open in A′ and D is open in E′, by (ii). Also, by the foregoing we know already that f is adic if and only if the same holds for E(f); combining with proposition 8.3.18(v), we conclude that f is open if and only if the same holds for E(f).

• Lastly, if either f or E(f) is open and injective, by the foregoing we know already that both f and E(f) are open, and then A^{′∘∘} ⊂ C and E^{′∘∘} ⊂ D, by (iii.a). Endow \overline{A}′ := A′/A^{′∘∘}, \overline{E}′ := E′/E^{′∘∘}, \overline{C} := C/A^{′∘∘} and \overline{D} := D/E^{′∘∘} with their discrete topologies; then the foregoing isomorphism ω induces a ring isomorphism \overline{E}′ →^{∼} \overline{A}′, restricting to a ring isomorphism \overline{D} →^{∼} \overline{C}. Moreover, we get natural isomorphisms of topological rings :

(C, 𝒯′_{C}) →^{∼} \overline{C} ×_{\overline{A}_{′}} A′ and (D, 𝒯′_{D}) →^{∼} \overline{D} ×_{\overline{E}_{′}} E′.

Taking into account proposition 16.3.25, it follows that the projection A → (C, 𝒯′_{C}) is an isomorphism of topological rings if and only if the same holds for the projection E → (D, 𝒯′_{D}); i.e. f is open and injective if and only if the same holds for E(f).

(iii.b): The map E(f) is a composition of continuous ring homomorphisms :

E ⟶^{g_{1}} (D, 𝒯_{D}) ⟶^{g_{2}} (D, 𝒯′_{D}) ⟶^{g_{3}} E′

where g_{1} is open and surjective, g_{2} is bijective, and g_{3} is an open injective map. Since Ker g_{1} = Ker E(f) is a closed ideal of E, we see that (D, 𝒯_{D}) is a perfect, separated and complete topological ring (proposition 8.2.13(v)) whose topology is I-adic, for an ideal I ⊂ D of finite type, i.e. (D, 𝒯_{D}) is perfectoid, and the same holds for (D, 𝒯′_{D}), by virtue of corollaries 8.3.20(i) and 8.3.19(iii). Since the category of perfectoid A-algebras is equivalent to that perfectoid E-algebras (remark 16.3.7(i)) we deduce a corresponding factorization of f :

A ⟶^{f_{1}} C_{1} ⟶^{f_{2}} C_{2} ⟶^{f_{3}} A′

where C_{1} and C_{2} are perfectoid A-algebras with isomorphisms of perfectoid E-algebras

E(C_{1}) →^{∼} (D, 𝒯_{D}) and E(C_{2}) →^{∼} (D, 𝒯′_{D}).

Moreover, f_{1} is open and surjective, f_{2} is bijective, and f_{3} is open and injective, by (i). But then C_{1} is isomorphic to (C, 𝒯_{C}) and C_{2} is isomorphic to (C, 𝒯′_{C}), whence the contention. □

16.3.45. Let A_{•} := (A_{λ} | λ ∈ Λ) be a filtered system of perfectoid rings, with adic transition maps, and for every λ ∈ Λ set E_{λ} := E(A_{λ}). By virtue of theorem 16.3.42(i), the transition maps of the induced system E_{•} := (E_{λ} | λ ∈ Λ) are also adic. Endow the colimit A of A_{•} (resp. E of E_{•}) of the unique adic topology 𝒯_{A} (resp. 𝒯_{E}) such that the universal co-cone (τ_{A,λ} : A_{λ} → A | λ ∈ Λ) (resp. (τ_{E,λ} : E_{λ} → E | λ ∈ Λ)) consists of adic maps. We have :

Corollary 16.3.46. In the situation of (16.3.45), the separated completion A^{∧} of (A, 𝒯_{A}) (resp. E^{∧} of (E, 𝒯_{E})) is perfectoid, and there exists a unique isomorphism of topological rings

ω : E(A^{∧}) →^{∼} E^{∧}  such that  ω ∘ E(τ_{A,λ}^{∧}) = τ_{E,λ}^{∧}  for every λ ∈ Λ.

Proof. We may assume that Λ admits an initial element λ_{0}, and let \underline{α} := (α_{n} | n ∈ ℕ) be a distinguished element in Ker(u_{A_{λ_{0}}}). It suffices to show that there exists a unique isomorphism of topological rings

ω′ : A^{∧} →^{∼} A_{λ_{0}} ⊗_{W (E_{λ_{0}})}W (E^{∧}) such that ω′∘τ_{A,λ}^{∧} = A_{λ_{0}} ⊗_{W (E_{λ_{0}})}W (τ_{E,λ}^{∧}) for every λ ∈ Λ

(under the natural identification A_{λ} →^{∼} A_{0} ⊗_{W (E_{λ_{0}})} W (E_{λ})). Let β_{1}, …, β_{k} be a finite sequence of elements of E_{λ_{0}} that generates an ideal of adic definition. Let 𝒲 be the colimit of the induced system (W (E_{λ}) | λ ∈ Λ); denote by ℐ_{0} ⊂ W (A_{λ_{0}}) the ideal generated by the sequence (p, τ_{A_{λ_{0}}}(β_{1}), …, τ_{A_{λ_{0}}}(β_{k})), so that the topology of W (A_{λ_{0}}) is ℐ_{0}-adic (proposition 9.3.77(ii)). Let also 𝒲 ^{∧} be the separated completion of 𝒲 ; by corollary 9.3.79, the natural map

𝒲 ^{∧} → W (E^{∧})

is an isomorphism of topological rings. On the other hand, we have a short exact sequence :

0 → \underline{α}𝒲 → 𝒲 ⟶^{u} A → 0

and the topology of A is induced by 𝒲 via u. By proposition 8.2.13(iii), it follows that the kernel of the induced map u^{∧} : 𝒲 ^{∧} → A^{∧} is the topological closure of the image of \underline{α}𝒲 ; the latter is also the topological closure of \underline{α}𝒲 ^{∧}, and moreover it is easily seen that A^{∧} is a P-ring. Hence, Ker u^{∧} = \underline{α}𝒲 ^{∧} (proposition 16.2.23(ii)), whence the corollary. □

In the situation of theorem 16.3.36, let 𝒦 and 𝒦 ′ be any two closed and taut ideals of E; we would like to show that 𝒦 = 𝒦 ′ if and only if {𝒦 } = {𝒦 ′}, and taking into account theorem 16.3.36(iii), it would suffice to prove that {𝒦 ∩ 𝒦 ′} = {𝒦 } ∩ {𝒦 ′}. We do not know whether the latter identity always holds, but we have at least the following :

Theorem 16.3.47. Let A be any perfectoid ring, 𝒦 and 𝒦 ′ two taut and closed ideals of E := E(A), and (α_{n} | n ∈ ℕ) any distinguished element of Ker u_{A}. Suppose that either one of the following conditions holds :

(a) α_{0} ∈ 𝒦 ′

(b) 𝒦 ′ is strictly taut and α_{0}^{n} ⊂ 𝒦 ′ for some sufficiently large n ∈ ℕ.

Then {𝒦 ∩ 𝒦 ′} = {𝒦 } ∩ {𝒦 ′}.

Proof. Suppose that (a) holds; in this case, both inclusions 𝒦 ′ ⊂ E and 𝒦 ′ ∩ 𝒦 ⊂ 𝒦 are taut, by remark 16.3.28(iv). There follows a commutative diagram of A/pA-modules

[figure: 𝒦 /(𝒦 ′ ∩ 𝒦 ); {𝒦 }/{𝒦 ′ ∩ 𝒦 }; E/𝒦 ′; A/{𝒦 ′}]

both of whose horizontal arrows are isomorphisms, by theorem 16.3.31(ii), and whose left vertical arrow is clearly injective. Then the right vertical arrow is injective as well, whence the contention, in this case.

In case (b) holds, pick ε ∈ ℕ[1/p] with 0 < ε < 1 and such that 𝒦 ′ is α_{0}^{1−ε}-taut. We set

ℐ_{n} := {x ∈ E | α_{0}^{nε}x ∈ ℐ } for every n ∈ ℕ and every ideal ℐ ⊂ E .

Claim 16.3.48. With the foregoing notation we have :

(i) 𝒦′_{n} is a closed ideal of E, for every n ∈ ℕ. (ii) The inclusion 𝒦′_{n} ⊂ 𝒦′_{n+1} is taut for every n ∈ ℕ. (iii) 𝒦′_{0} = 𝒦 ′ and 𝒦′_{r} = E for every sufficiently large r ∈ ℕ.

Proof of the claim. Since 𝒦 ′ is closed, it is easily seen that (i) holds. Say that α_{0}^{n} ∈ 𝒦 ′; then clearly 𝒦′_{k} = E for every integer k ≥ ε^{−1}n. Next, directly from the definition we see that, for any two ideals ℐ , 𝒥 ⊂ E and any n ∈ ℕ we have

𝒥 · ℐ_{n} ⊂ (𝒥 ℐ)_{n}  α_{0}^{ε} · ℐ_{n+1} ⊂ ℐ_{n}  (ℐ_{n})^{p} ⊂ (ℐ^{p})_{np}  (Φ_{E}^{−1}ℐ)_{n} = Φ_{E}^{−1}(ℐ_{np}).

Therefore :

α_{0} · Φ_{E}^{−1}(𝒦′_{n+1})^{p} ⊂ α_{0} · Φ_{E}^{−1}((𝒦 ^{′p})_{(n+1)p})
= α_{0} · (Φ_{E}^{−1}𝒦 ^{′p})_{n+1}
⊂ α_{0}^{1−ε} · (Φ_{E}^{−1}𝒦 ^{′p})_{n}
⊂ (α_{0}^{1−ε} · Φ_{E}^{−1}𝒦 ^{′p})_{n}
⊂ 𝒦′_{n}

as required. ♢

By claim 16.3.48we have 𝒦′_{r} = E for some r ∈ ℕ, and for every n ∈ ℕ we get a commutative diagram of A/pA-modules

[figure: {𝒦 ∩ 𝒦′_{n+1}}/{𝒦 ∩ 𝒦′_{n}}; {𝒦′_{n+1}}/{𝒦′_{n}}; (𝒦 ∩ 𝒦′_{n+1})/(𝒦 ∩ 𝒦′_{n}); 𝒦′_{n+1}/𝒦′_{n}]

whose vertical arrows are isomorphisms, by theorem 16.3.31(ii), and whose bottom horizontal arrow is injective. Then the top horizontal arrow is injective as well, so that

{𝒦 ∩ 𝒦′_{n+1}} ∩ {𝒦′_{n}} ⊂ {𝒦 ∩ 𝒦′_{n}} for every n ∈ ℕ.

We may now prove that {𝒦 } ∩ {𝒦′_{n}} = {𝒦 ∩ 𝒦′_{n}}, by descending induction on n. Indeed, the assertion is obvious in case n ≥ r. Suppose that the identity has already been shown for some n > 0; then we get

{𝒦 } ∩ {𝒦′_{n−1}} = {𝒦 } ∩ {𝒦′_{n}} ∩ {𝒦′_{n−1}} = {𝒦 ∩ 𝒦′_{n}} ∩ {𝒦′_{n−1}} ⊂ {𝒦 ∩ 𝒦′_{n−1}}

and the converse inclusion is obvious, so the sought identity holds for n − 1. Letting n = 0, we get the assertion. □

In the same vein, let us also point out :

Lemma 16.3.49. In the situation of theorem 16.3.47, let λ_{1}, λ_{2} ∈ ℕ[1/p] be two rationals such that λ_{1} + λ_{2} ≤ 1, and 𝒥_{1}, 𝒥_{2} ⊂ E two ideals such that 𝒥_{i} is α_{0}^{λ_{i}}-taut for i = 1, 2. Then

({𝒥_{1}}{𝒥_{2}})^{c} = {𝒥_{1}𝒥_{2}}.

Proof. The inclusion {𝒥_{1}}{𝒥_{2}} ⊂ {𝒥_{1}𝒥_{2}} is obvious from the definitions, so it suffices to check that for every x ∈ 𝒥_{1}𝒥_{2} we have \overline{u}_{A}(x) ∈ (\{𝒥_{1}\}\{𝒥_{2}\})^{c}. However, every such x can be written as a finite sum ∑_{i=1}^{n} y_{i}z_{i}, with y_{1}, …, y_{n} ∈ 𝒥_{1} and z_{1}, …, z_{n} ∈ 𝒥_{2}; by proposition 9.3.62 we know that \overline{u}_{A}(x) is the limit of a p-adically convergent series of the form ∑_{j∈ℕ} w_{j}, where in turn each w_{j} is a finite ℤ_{p}-linear combination of terms of the form

w\underline{μ} ; j := p^{j} · \overline{u}_{A}((y_{1}z_{1})^{μ_{1}} ⋯ (y_{n}z_{n})^{μ_{n}}) for certain μ_{1}, …, μ_{n} ∈ p^{−j}ℕ such that ∑_{i=1}^{n} μ_{i} = 1.

Thus, we are reduced to checking that any such term lies in \{𝒥_{1}\}\{𝒥_{2}\}. Now, set y\underline{μ} := y_{1}^{μ_{1}} ⋯ y_{n}^{μ_{n}}, and define likewise z\underline{μ}, for every \underline{μ} := (μ_{1}, …, μ_{n}) ∈ p^{−j}ℕ; under our assumptions we have p = \overline{u}_{A}(α_{0}^{λ_{1}+λ_{2}}) · a for some a ∈ A (lemma 16.2.7(iii)), hence

w\underline{μ} ; j = a^{j} · \overline{u}_{A}(α_{0}^{jλ_{1}} · y\underline{μ}) · \overline{u}_{A}(α_{0}^{jλ_{2}} · z\underline{μ})

and notice that α_{0}^{jλ_{1}} · y\underline{μ} ∈ α_{0}^{jλ_{1}} · Φ_{E}^{−j}(𝒥_{1}^{p^{j}}) ⊂ 𝒥_{1} by (16.3.29). Likewise, α_{0}^{jλ_{2}} · z\underline{μ} ∈ 𝒥_{2}, and the assertion follows. □

Corollary 16.3.50. Let A be a perfectoid ring, 𝒦 , 𝒦 ′ ⊂ E(A) two taut and closed ideals, and suppose that either 𝒦 or 𝒦 ′ fulfills conditions (a) and (b) of theorem 16.3.47. Then we have 𝒦 ⊂ 𝒦 ′ if and only if {𝒦 } ⊂ {𝒦 ′}.

Proof. As already announced, this follows straightforwardly from remark 16.3.28(iv) and theorems 16.3.47and 16.3.36(iii). □

Corollary 16.3.51. In the situation of corollary 16.3.40, let a′_{•} := (a′_{1}, …, a′_{t}) be another finite system of elements of E, and J′ ⊂ E the ideal generated by a′_{•}. Define the ideal [a′_{•}]^{⟨β⟩} ⊂ W (E) as in remark 9.3.70(iii), for every β ∈ ℕ[1/p]. Moreover, let λ, μ ∈ ℕ[1/p] be any two rational numbers. We have :

(i) The following conditions are equivalent : (a) J^{⟨λ⟩}E ⊂ J′^{⌊μ⌋}E.

(b) [a_{•}]^{⟨λ⟩}A ⊂ [a′_{•}]^{⟨μ−ε⟩}A for every ε ∈ ℕ[1/p] with 0 < ε ≤ μ. (ii) Suppose that either J or J′ is open. Then the following conditions are equivalent :

(a) J^{⟨λ⟩}E ⊂ J′^{⟨μ⟩}E.

(b) [a_{•}]^{⟨λ⟩}A ⊂ [a′_{•}]^{⟨μ⟩}A. (iii) For every l, m ∈ ℕ[1/p] such that lμ + mλ < λμ we have

[a_{•}]^{⟨λ⟩}A ∩ [a′_{•}]^{⟨μ⟩}A ⊂ [a_{•}]^{⟨l⟩} · [a′_{•}]^{⟨m⟩}A.

Proof. (i): Suppose that (i.a) holds; we deduce :

[a_{•}]^{⟨λ⟩}A ⊂ {J^{⟨λ⟩}E} ⊂ {J′^{⟨μ−ε/p⟩}E} ⊂ [a′_{•}]^{⟨μ−ε⟩}A for every ε as in (i.b)

where the first and last inclusions follow from corollary 16.3.40(i).

Conversely, suppose that (i.b) holds; fix a finite system b_{•} := (b_{1}, …, b_{r}) of generators for an ideal of definition I of the perfectoid ring E, and for every n ∈ ℕ, let b_{•}^{n} be the finite system consisting of the products of the form b_{j(1)} ⋯ b_{j(n)}, where j ranges over all the mappings {1, …, n} → {1, …, r}. For every n ∈ ℕ, we may then form the system (a′_{•}, b_{•}^{n}) which is the union of the systems a′_{•} and b_{•}^{n}, and for every λ ∈ ℕ[1/p] consider the ideal [a′_{•}, b_{•}^{n}]^{⟨λ⟩} ⊂ W (E) attached as in remark 9.3.70(iii) to the system (a′_{•}, b_{•}^{n}). With this notation, corollary 16.3.40(ii) yields :

[a_{•}]^{⟨λ⟩}A ⊂ [a′_{•}, b_{•}^{n}]^{⟨μ−ε/p⟩}A = {(J′ + I^{n})^{⟨μ−ε/p⟩}E} for every n ∈ ℕ and ε as in (i.a).

However, since the topology of A is coarser than the p-adic topology, claim 16.3.41implies that [a_{•}]^{⟨λ⟩}A is a dense subset of {J^{⟨λ⟩}E}; but {(J′ + I^{n})^{⟨μ−ε/p⟩}E} is a closed ideal, so

(16.3.52)  {J^{⟨λ⟩}E} ⊂ {(J′ + I^{n})^{⟨μ−ε/p⟩}E}  for every n ∈ ℕ and ε as in (i.a).

Now, the topological closure J^{⟨λ⟩}E^{c} of J^{⟨λ⟩}E is (strictly) taut, and {J^{⟨λ⟩}E} = {J^{⟨λ⟩}E^{c}} (remark 16.3.28(v)); moreover, (J′ + I^{n})^{⟨μ−ε/p⟩}E is an open ideal of W (E), hence, from corollary 16.3.50and (16.3.52) we derive

J^{⟨λ⟩}E^{c} ⊂ (J′ + I^{n})^{⟨μ−ε/p⟩}E for every n ∈ ℕ and ε as in (i.a)

and combining with claim 16.3.20we conclude that J^{⟨λ⟩}E ⊂ J′^{⟨μ−ε⟩}E for every ε as in (i.a).

(ii): From lemma 9.3.69(ii.a,iv) it follows easily that J is open if and only if the same holds for J^{⟨λ⟩}E, and likewise, J′ is open if and only if the same holds for J′^{⟨μ⟩}E. Hence, suppose that J is open; if condition (ii.a) holds, then it follows that J′ is open as well, and if (ii.b) holds, we use corollary 16.3.40(ii) to deduce first that [a_{•}]^{⟨λ⟩}A is open, so the same holds for [a′_{•}]^{⟨μ⟩}A, and then also for J′, again by corollary 16.3.40(ii). In conclusion, for the proof of (ii) we may assume that J′ is open. Now, suppose that (ii.a) holds; then we get

[a_{•}]^{⟨λ⟩}A ⊂ {J^{⟨λ⟩}E} ⊂ {J′^{⟨μ⟩}E} = [a′_{•}]^{⟨μ⟩}A

where the first inclusion follows from corollary 16.3.40(i), and the last identity follows from corollary 16.3.40(ii). Conversely, suppose that (ii.b) holds; since the topology of A is coarser that the p-adic topology, claim 16.3.41 implies that [a_{•}]^{⟨λ⟩}A is a dense subset of {J^{⟨λ⟩}E}, and since {J′^{⟨μ⟩}E} is a closed ideal, corollary 16.3.40(ii) yields {J^{⟨λ⟩}E} ⊂ {J′^{⟨μ⟩}E}. Then, arguing as in the proof of (i), we easily deduce that (ii.a) holds.

(iii): We may assume that λ, μ > 0, since otherwise there is nothing to prove. Define b_{•}^{n} as in the foregoing, for every n ∈ ℕ; combining corollary 16.3.40(i) and theorem 16.3.47, we get

[a_{•}, b_{•}^{n}]^{⟨λ⟩}A∩[a′_{•}, b_{•}^{n}]^{⟨μ⟩}A = {(J +I^{n})^{⟨λ⟩}E}∩{(J′+I^{n})^{⟨μ⟩}E} = {(J +I^{n})^{⟨λ⟩}E∩(J′+I^{n})^{⟨μ⟩}E}

for every n ∈ ℕ. Now, pick α ∈ ℕ[1/p] such that l/λ < 1 − α and m/μ < α and set l′ := (1 − α)λ, m′ := αμ. If x ∈ (J + I^{n})^{⟨λ⟩}E ∩ (J′ + I^{n})^{⟨μ⟩}E, we may write

x = x^{α} · x^{1−α} ∈ (J + I^{n})^{⟨l′⟩} · (J′ + I^{n})^{⟨m′⟩}E.

Next, to ease notation, for any finite sequence ν_{•} := (ν_{1}, …, ν_{m}) of elements of ℕ[1/p] and any sequence (x_{1}, …, x_{m}) of elements of E, set

x_{•}^{ν_{•}} := x_{1}^{ν_{1}} ⋯ x_{m}^{ν_{m}} and |ν_{•}| := ν_{1} + ⋯ + ν_{m}.

We remark that the ideal (J + I^{n})^{⟨l′⟩} · (J′ + I^{n})^{⟨m′⟩}E is generated by all products

a_{•}^{ν_{•}} · a′_{•}^{ν′_{•}} · b_{•}^{β_{•}} · b_{•}^{β′_{•}}  where |ν_{•}| + n^{−1} · |β_{•}| = l′  |ν′_{•}| + n^{−1} · |β′_{•}| = m′.

Fix l′′, m′′ ∈ ℕ[1/p] such that l < l′′ < l′ and m < m′′ < m′, and set ε := (l′ −l′′)+(m′ −m′′); it is easily seen that each of the above products lies either in J^{⟨l′′⟩} · J′^{⟨m′′⟩}E or else in I^{⟨nε⟩}E. Moreover, we can write l′′ = p^{−N}f, m′′ := p^{−N}g for suitable f, g, N ∈ ℕ, and summing up, we conclude that, for every δ < p^{−N}, the ideal [a_{•}]^{⟨λ⟩}A ∩ [a′_{•}]^{⟨μ⟩}A is contained in :

⋂_{n∈ℕ}{J^{⟨l′′⟩} · J′^{⟨m′′⟩}E + I^{⟨nε⟩}E} = ⋂_{n∈ℕ}{(J^{f}J′^{g})^{⟨1/p^{N} ⟩}E + I^{⟨nε⟩}E} ⊂ {(J^{f}J′^{g})^{⟨δ⟩}E}

where the first equality follows from lemma 9.3.69(ii.a,iii) the last inclusion follows from corollary 16.3.40(iii.b). Let us then choose δ, δ′ ∈ ℕ[1/p] with p^{−N} > δ > δ′ and δ′ close enough to p^{−N}, so that δ′ · f > l and δ′ · g > m; using corollary 16.3.40(i) we get

{(J^{f}J′^{g})^{⟨δ⟩}E} ⊂ [a_{•}^{f}a′_{•}^{g}]^{⟨δ′⟩}A ⊂ [a_{•}]^{⟨l⟩} · [a′_{•}]^{⟨m⟩}A

whence the contention. □

As an application, we can complement as follows theorem 16.3.21(i) :

Proposition 16.3.53. In the situation of theorem 16.3.21, we have :

(i) (16.3.12) is a cartesian diagram of topological rings.

(ii) All the topological rings appearing in (16.3.12) are perfectoid.

Proof. We know already that (16.3.17) is a cartesian diagram of topological rings. Moreover, it is easily seen that the I-adic topology on A agrees with the (I +pA)-adic topology, and likewise for the J-adic and (I + J)-adic topologies, so all the topological rings appearing in (16.3.17) except possibly for (A′, 𝒯 ′) are perfectoid (proposition 16.3.8(ii)). By the same token, the IJadic topology on A agrees with the (IJ + pA)-adic topology, so the proposition will follow, once we have shown the following

Claim 16.3.54. The IJ-adic topology on A agrees with the linear topology 𝒯 defined by the cofiltered system of ideals (I^{n} ∩ J^{n} | n ∈ ℕ).

Proof of the claim. Set I′ := I + pA and J′ := J + pA, and notice that I^{k+N} ⊂ I′^{N+k} ⊂ I^{k+1} for every k ∈ ℕ, and likewise for J. Therefore, 𝒯 agrees with the topology defined by the cofiltered system of ideals (I′^{n} ∩ J′^{n} | n ∈ ℕ), and the IJ-adic topology on A agrees with the I′J′-adic topology. Thus, we can replace I, J by I′, J′, and assume from start that p ∈ I ∩ J. Now, let (α_{n} | n ∈ ℕ) be any distinguished element in Ker u_{A}. Fix finite systems β_{•} := (β_{1}, …, β_{k}) and β′_{•} := (β′_{1}, …, β′_{r}) of elements of E(A) such that \overline{u}_{A/pA}(β_{•}) (resp. \overline{u}_{A/pA}(β′_{•})) is a system of generators for I/pA (resp. for J/pA). In light of lemma 16.2.7(iii), it follows that the sequence (\overline{u}_{A}(α_{0}), \overline{u}_{A}(β_{1}), …, \overline{u}_{A}(β_{k})) is a system of generators for I, and (\overline{u}_{A}(α_{0}), \overline{u}_{A}(β′_{1}), …, \overline{u}_{A}(β′_{r})) is a system of generators for J. On the other hand, lemma 9.3.69(iv) implies that the I-adic (resp. J-adic) topology on A agrees with the linear topology defined by the cofiltered system of ideals ([α_{0}, β_{•}]^{⟨λ⟩}A | λ ∈ ℕ[1/p]) (resp. ([α_{0}, β′_{•}]^{⟨λ⟩}A | λ ∈ ℕ[1/p])). Then, the claim follows from corollary 16.3.51(iii). □

16.3.55. Let A be a ring, and a_{•} := (a_{1}, …, a_{k}) a finite system of elements of A; denote by I the ideal generated by a_{•}, and suppose that

(16.3.56)  p ∈ I^{t}  where t := k(p − 1) + 1.

Set J := I^{(p)}, the ideal generated by the system (a_{1}^{p}, …, a_{k}^{p}), and notice that

(16.3.57)  I^{t} ⊂ J.

Therefore J is well defined independently of the choice of a_{•} (lemma 16.2.2). Denote by gr_{I}^{•}A (resp. gr_{J}^{•}A) the graded ring associated with the I-adic (resp. J-adic) filtration on A. Notice that both these two rings are 𝔽_{p}-algebras.

Remark 16.3.58. Let A be any P-ring, I any ideal of definition of A, and (a_{1}, …, a_{k}) a system of generators of I. By virtue of lemma 16.2.3(iii) and its proof, we may find a sequence of ideals (I_{n} | n ∈ ℕ) such that

• I_{n+1}^{(p)} = I_{n} for every n ∈ ℕ

• I_{n} admits a system of generators consisting of k elements of A. Then I_{n} is still an ideal of definition of A for every n ∈ ℕ, and p ∈ I_{n}^{t} for every sufficiently large n ∈ ℕ, where t is defined as in (16.3.55). We may thus regard the following result as a refinement of corollary 16.3.3.

Proposition 16.3.59. In the situation of (16.3.55), the following holds :

(i) The Frobenius endomorphism of gr_{I}^{•}A factors through a graded ring homomorphism

Φ_{I} : gr_{I}^{•}A → gr_{J}^{•}A  (x mod I^{n+1}) ↦ (x^{p} mod J^{n+1})  for every n ∈ ℕ and x ∈ I^{n}.

(ii) Suppose that A is perfectoid for its p-adic topology. Then Φ_{I} is an isomorphism.

Proof. (i): To begin with, we remark :

Claim 16.3.60. For every n ∈ ℕ and every x, y ∈ I^{n}, we have (x + y)^{p} − x^{p} − y^{p} ∈ J^{n+1}.

Proof of the claim. The difference in the claim is a sum of terms that lie in pI^{pn}. Taking into account (16.3.56), we see that pI^{pn} ⊂ I^{t+pn}, so it remains only to check that

I^{t+pn} ⊂ J^{n+1}.

We argue by induction on n, and notice that the case where n = 0 is (16.3.57). Hence, suppose that r > 0, and that the sought inclusion has already been checked for n := r − 1. The ideal I^{t+pr} is generated by all elements of the type b := ∏_{j=1}^{t+pr} a_{φ(j)}, where φ is any mapping {1, …, t + pr} → {1, …, k}. Since r ≥ 0, it is easily seen that there exists at least one index i ∈ {1, …, k} such that φ^{−1}(i) has cardinality ≥ p. For such index i, we have b = a_{i}^{p} · c, where c ∈ I^{t+pn}, and the inductive assumption says that c ∈ J^{n+1}, whence b ∈ J^{r+1}, as required. ♢

Let us now check that if x ∈ I^{n}, then x^{p} ∈ J^{n}. Indeed, the assertion is clear if x is a monomial ∏_{j=1}^{n} a_{φ(j)} for some mapping φ : {1, …, n} → {1, …, k}. However, we may write x = x_{1} + ⋯ + x_{r} for some r ∈ ℕ and a sequence x_{1}, …, x_{r} of such monomials. Lastly, claim 16.3.60 and an easy induction on r shows that x^{p} − ∑_{j=1}^{r} x_{j}^{p} ∈ J^{n+1}, whence the contention.

Thus, the map Φ_{I} is well defined, and claim 16.3.60also shows that Φ_{I}(x+y) = Φ_{I}(x)+Φ_{I}(y) for every n ∈ ℕ and every x, y ∈ gr_{I}^{n}A. For any such x and y it is also clear that Φ_{I}(xy) = Φ_{I}(x) · Φ_{I}(y), so Φ_{I} is a ring homomorphism.

(ii): Set E := E(A), and let \underline{α} := (α_{n} | n ∈ ℕ) be a distinguished element in Ker u_{A}; arguing as in the proof of lemma 16.2.7(i), we may assume that for every i = 1, …, k we have a_{i} = \overline{u}_{A}(β_{i}) for some β_{i} ∈ E. Denote by ℐ ⊂ E the ideal generated by (β_{1}, …, β_{k}), and set 𝒥 := ℐ^{(p)}, the ideal generated by (β_{1}^{p}, …, β_{k}^{p}); from (16.3.56) and lemma 16.1.9(ii) we get

(16.3.61)  α_{0} ∈ ℐ^{t} ⊂ 𝒥 .

Let gr_{ℐ}^{•} and gr_{𝒥}^{•} be the graded rings associated with the ℐ -adic and 𝒥 -adic filtrations on E.

Claim 16.3.62. (i) For every n ∈ ℕ we have :

(a) The inclusion ℐ^{n+1} ⊂ ℐ^{n} is α_{0}^{1/p}-taut and the inclusion 𝒥^{n+1} ⊂ 𝒥^{n} is taut.

(b) {ℐ^{n}} = I^{n} and {𝒥^{n}} = J^{n}. (ii) The map \overline{u}_{A} : E(A) → A induces graded ring isomorphisms

gr_{ℐ}^{•} →^{∼} gr_{I}^{•} gr_{𝒥}^{•} →^{∼} gr_{J}^{•}.

Proof of the claim. (i.a): Clearly 𝒥 = Φ_{E}(ℐ), therefore it suffices to prove the assertion for 𝒥; however α_{0} · Φ_{E}^{−1}(𝒥^{np}) = α_{0}ℐ^{np} ⊂ ℐ^{t+np} by (16.3.61), and arguing as in the proof of claim 16.3.60, we see that ℐ^{t+np} ⊂ 𝒥^{n+1}, whence the claim.

(i.b): We consider first the assertion for ℐ^{n}. By construction, it is clear that a_{i} ∈ {ℐ } for every i = 1, …, k, hence I^{n} ⊂ {ℐ^{n}}. On the other hand, since A is endowed with its p-adic topology, E carries the α_{0}-adic topology (see (16.2.19)), therefore ℐ^{n} is open in E, and consequently \{ℐ^{n}\} is the ideal generated by the system (\overline{u}_{A}(x) | x ∈ ℐ^{n}) (remark 16.3.28(viii)). Thus, we come down to checking that \overline{u}_{A}(x) ∈ I^{n} for every x ∈ ℐ^{n}. This is clear in case x is a monomial of the form ∏_{j=1}^{n} β_{φ(j)} for some mapping φ : {1, …, n} → {1, …, k}. In general, x shall be a finite sum x_{1} + ⋯ + x_{s}, where x_{1}, …, x_{s} are monomials of this type. According to proposition 9.3.62, τ_{A}(x) can then be expressed as a p-adically convergent series ∑_{r∈ℕ} p^{r} · y_{r}, where each y_{r} is in turn a finite sum of terms of the form w · τ_{A}(z), for certain w ∈ ℤ_{p} and z ∈ Φ_{E}^{−r}(ℐ^{n})^{p^{r}}. In view of (i;A) and remark 16.3.28(vi), we have

α_{0}^{r/p} · Φ_{E}^{−r}(ℐ^{n})^{p^{r}} ⊂ ℐ^{n+r} for every r, n ∈ ℕ.

Pick π ∈ A as in lemma 16.2.3(iv); recalling lemma 16.2.7(iii), we deduce that

p^{r} · \overline{u}_{A}(z) ∈ π^{(p−1)r} · \{ℐ^{n+r}\} for every r ∈ ℕ and every z ∈ Φ_{E}^{−r}(ℐ^{n})^{p^{r}} .

Since {ℐ^{n+r}} is closed in the p-adic topology of A for every n, r ∈ ℕ, it follows that {ℐ^{n}} ⊂ I^{n} + π^{(p−1)r}{ℐ^{n+r}} for every r ∈ ℕ. But clearly π^{(p−1)r} ∈ I^{n} for every large enough r ∈ ℕ, so {ℐ^{n}} ⊂ I^{n}, as required.

To deal with 𝒥^{n}, we repeat the foregoing argument, to get : {𝒥^{n}} ⊂ J^{n} + {𝒥^{n+r}} for every r ∈ ℕ. However, clearly 𝒥^{n+r} ⊂ ℐ^{n+r}; taking into account the previous case, we conclude that {𝒥^{n}} ⊂ J^{n} + I^{s} for every s ∈ ℕ. But we may find s ∈ ℕ large enough, so that I^{s} ⊂ J^{n}, so at last {𝒥^{n}} ⊂ J^{n}, whence the claim.

(ii) follows immediately from (i) and theorem 16.3.31(ii). ♢ Claim 16.3.62(ii) yields a commutative diagram of graded rings

\xymatrix{gr_{ℐ}^{•} \ar[r] \ar[d] & gr_{I}^{•}A \ar[d]^{Φ_{I}} \\ _{gr• 𝒥} \ar[r] & gr_{J}^{n}A}

whose horizontal arrows are isomorphisms, and whose left vertical arrow is induced by Φ_{E}. Now, since E is perfect, the left vertical arrow is an isomorphism, so the same holds for the right vertical arrow, as stated. □

Corollary 16.3.63. Let A be a perfectoid ring, and β ∈ E(A) any element.

(i) A is a reduced ring.

(ii) If β ≠ 0, then \overline{u}_{A}(β) ≠ 0.

(iii) If the topology of A is discrete, then A is an 𝔽_{p}-algebra.

Proof. (i): Suppose that A contains a nilpotent element x ≠ 0, and let n > 1 be the smallest integer such that x^{n} = 0; set y := x^{n−1}, so that y ≠ 0 and y^{p} = 0. By remark 16.3.58 we can find an ideal of definition I of A fulfilling condition (16.3.56); we set J := I^{(p)} and let Φ_{I} : gr_{I}^{•}A →^{∼} gr_{J}^{•}A be the isomorphism of proposition 16.3.59(ii). Let also r ∈ ℕ be the largest integer such that y ∈ I^{r}, and denote by \overline{y} ∈ gr_{I}^{r}A the class of y, so that \overline{y} ≠ 0; but clearly Φ_{I}(\overline{y}) = 0, a contradiction.

(ii) and (iii) are immediate consequences of (i). □

Theorem 16.3.64. In the situation of (16.3.55), let 𝒯_{I} be the I-adic topology on A, and suppose that (A, 𝒯_{I}) is a P-ring. Define also the graded ring homomorphism Φ_{I} as in proposition 16.3.59(i). Then the following conditions are equivalent :

(a) (A, 𝒯_{I}) is perfectoid.

(b) The map Φ_{I} is an isomorphism.

Proof. Suppose that (a) holds, and let 𝒯_{p} be the p-adic topology on A; then (A, 𝒯_{p}) is perfectoid, by proposition 16.3.8(i), in which case proposition 16.3.59(ii) says that (b) holds.

Next, suppose that (b) holds; then proposition 16.2.9 and example 16.3.2(i) say that E := E(A) is a perfectoid ring. Moreover, \overline{u}_{A/pA} is surjective (lemma 16.2.7(i)), so for every i = 1, …, k we may find β_{i} ∈ E such that \overline{u}_{A/pA}(β_{i}) equals the image of a_{i} in A/pA. Denote by 𝒥 ⊂ E the ideal generated by the system (β_{1}, …, β_{k}); from claim 16.2.10 we see that 𝒥 is an ideal of definition of E, and \overline{u}_{A/pA} induces a ring isomorphism

(16.3.65)  E/𝒥 →^{∼} A/I.

Denote by 𝒥_{W} ⊂ W (E) the ideal generated by (τ_{E}(β_{1}), …, τ_{E}(β_{k})); by construction we have u_{A}(τ_{E}(β_{i})) − a_{i} ∈ pA for every i = 1, …, k, and notice that p ∈ I^{t} ⊂ I^{2}, so 𝒥_{W} A + I^{2} = I and Nakayama’s lemma yields

(16.3.66)  𝒥_{W} A = I.

Furthermore, recall that ℐ := Ker u_{A} contains a distinguished element \underline{α} of W (E) (lemma 16.2.7(ii)); on the other hand, (16.3.66) and (16.3.56) imply that p ∈ 𝒥_{W}^{t} A, in which case we may apply lemma 16.1.9(i) to get an element \underline{α}′ := (α′_{n} | n ∈ ℕ) ∈ ℐ such that α′_{0} ∈ 𝒥^{t} and α′_{1} ∈ E^{×}. Especially, \underline{α}′ is a distinguished element of W (E), and so the ring

B := W (E)/\underline{α}′W (E)

is perfectoid, for the quotient topology arising from the natural projection W (E) → B (example 16.3.2(ii)). By construction, u_{A} factors through a surjective and continuous ring homomorphism

v : B → A.

Moreover, set I_{B} := 𝒥_{W} B ⊂ B, and notice that on the one hand, u_{B} ∘ τ_{E}(α′_{0}) = pt for some t ∈ B^{×} (lemma 16.2.7(iii)), and on the other hand we have u_{B} ∘ τ_{E}(α′_{0}) ∈ I_{B}^{t}, so

(16.3.67)  p ∈ I_{B}^{t}.

Combining with proposition 9.3.77(ii), we deduce that I_{B} is an ideal of definition of B, and taking into account (16.3.66), we also get

(16.3.68)  v(I_{B}) = I.

Set J_{B} := I_{B}^{(p)}; from (16.3.67), (16.3.68) and propositions 16.3.59(ii) and 16.3.8(i) we then get a commutative diagram of graded ring homomorphisms

(16.3.69)  \xymatrix{gr_{I_{B}}^{•}B \ar[r] \ar[d] & gr_{I}^{•}A \ar[d] \\ gr_{J_{B}}^{•}B \ar[r] & gr_{J}^{•}A}

both of whose vertical arrows are isomorphisms, and whose horizontal arrows are induced by v. Since B is complete and separated for its I_{B}-adic topology, and A is complete and separated for its I-adic topology, the theorem will follow from :

Claim 16.3.70. The map v induces a ring isomorphism v_{n} : B/I_{B}^{p^{n}} →^{∼} A/I^{p^{n}} for every n ∈ ℕ.

Proof of the claim. We argue by induction on n. For n = 0, notice that the projection W (E) → B/I_{B} factors through E, and by construction, the kernel of the resulting surjection is 𝒥. Then the assertion follows from (16.3.65). Next, suppose that n ≥ 0, and that v_{n} is an isomorphism. In this case, v induces isomorphisms gr_{I_{B}}^{r}B →^{∼} gr_{I}^{r}A for every r < p^{n}. Due to (16.3.69), we then deduce that v also induces isomorphisms gr_{J_{B}}^{r}B →^{∼} gr_{J}^{r}A for every r < p^{n}, from which it follows easily that v induces an isomorphism B/J_{B}^{p^{n}} →^{∼} A/J^{p^{n}}. However, clearly J ⊂ I^{p} and J_{B} ⊂ I_{B}^{p}, hence J_{B}^{p^{n}} ⊂ I_{B}^{p^{n+1}} and J^{p^{n}} ⊂ I^{p^{n+1}}, so finally v_{n+1} is an isomorphism. □

As a corollary of theorem 16.3.64we obtain the following criterion :

Corollary 16.3.71. Let A be a ring, t ∈ ℕ any integer, β := (b_{n} | n ∈ ℕ) an element of E(A); suppose that A is complete and separated for its b_{0}-adic topology 𝒯_{0}, and p = ab_{0} for some a ∈ A. Let also J := ⋃_{n∈ℕ} b_{n}A. Consider the following conditions :

(a) A/J is a perfect 𝔽_{p}-algebra.

(b) The p-th power map of A/b_{t}^{2}A induces a bijection b_{t+1}A/b_{t+1}^{2}A →^{∼} b_{t}A/b_{t}^{2}A.

(c) Ann_{A}(b_{0}) = Ann_{A}(b_{n}) for every n ∈ ℕ.

(d) The Frobenius endomorphism Φ_{A/b_{i}A} of A/b_{i}A induces an isomorphism

\overline{Φ}_{A/b_{i}A} : A/b_{i+1}A →^{∼} A/b_{i}A for every i ∈ ℕ.

(e) (A, 𝒯_{0}) is a perfectoid ring. Then (e) is equivalent to the conjunction of (a),(b) and (c), and also to the conjunction of (c) and (d). Moreover, if (e) holds, then A/Ann_{A}(b_{0}) is perfectoid for the b_{0}-adic topology, and the map \overline{u}_{A} : E(A) → A induces an isomorphism of E(A)-modules :

(16.3.72)  Ann_{E(A)}(β) →^{∼} Ann_{A}(b_{0}).

Proof. Suppose first that (a),(b),(c) hold. For every i, j > 0, consider the commutative diagram

\xymatrix{b_{i}^{j−1}A/b_{i}^{j}A \ar[r]^{f_{i,j}} \ar[d]_{φ_{i,j−1}} & b_{i}^{j}A/b_{i}^{j+1}A \ar[d]^{φ_{i,j}} \\ b_{i−1}^{j−1}A/b_{i−1}^{j}A \ar[r]^{f_{i−1,j}} & b_{i−1}^{j}A/b_{i−1}^{j+1}A}

where φ_{i,j} is induced by the p-th power map of A/b_{i−1}^{j+1}A for every i, j ∈ ℕ with i > 0, and f_{i,j} is induced by scalar multiplication by b_{i}, for every i, j ∈ ℕ with j > 0. We notice :

Claim 16.3.73. (i) f_{i,j} is an isomorphism for every i, j ∈ ℕ with j > 1.

(ii) φ_{i,1} is an isomorphism for every i > t.

Proof of the claim. (i): The surjectivity of f_{i,j} is clear. For the injectivity, suppose that x = b_{i}^{j−1}y and b_{i}x = b_{i}^{j+1}z for some y, z ∈ A. It follows that b_{i}^{j}y = b_{i}^{j+2}z, so b_{i}^{j}(y−b_{i}^{2}z) = 0, and therefore also b_{i}(y − b_{i}^{2}z) = 0, by (c). Since j > 1, we deduce that x = b_{i}^{j}z, whence the assertion.

(ii): We argue by induction on i. The case i = t+1 is (b). Suppose then that i > t and that the surjectivity of φ_{i,1} is already known. This means that for every x ∈ A there exist y, z ∈ A such that b_{i−1}x = b_{i−1}y^{p} + b_{i−1}^{2}z, i.e. that b_{i−1}(x − y^{p} − b_{i−1}z) = 0. By assumption (c), we deduce that b_{i}(x − y^{p} − b_{i−1}z) = 0. Set z′ := b_{i}^{p−1}z; then b_{i}(x − y^{p} − b_{i}z′) = 0, i.e. b_{i}x = b_{i}y_{p} + b_{i}^{2}z′, whence the surjectivity of φ_{i+1,1}.

The injectivity of φ_{i,1} means the following. For every x ∈ A such that b_{i−1}x^{p} ∈ b_{i−1}^{2}A we have b_{i}x ∈ b_{i}^{2}A. Now, let a ∈ A, and suppose that b_{i}a^{p} ∈ b_{i}^{2}A; then b_{i−1}(b_{i}^{p−1}a^{p}) = b_{i}^{2p−1}a^{p} ∈ b_{i−1}^{2}A, so that b_{i+1}^{2p−1}a = b_{i}b_{i+1}^{p−1}a ∈ b_{i}^{2}A. Say that b_{i+1}^{2p−1}a = b_{i}^{2}y; then

b_{i+2}^{2p^{2}−p−1}(b_{i+2}a − b_{i+2}^{p+1}y) = 0.

By assumption (c), it follows that b_{i+2}^{p−1}(b_{i+2}a−b_{i+2}^{p+1}y) = 0 as well, and therefore b_{i+1}a ∈ b_{i+1}^{2}A, hence φ_{i+1,1} is injective. ♢

From claim 16.3.73 it follows immediately that φ_{i,j} is an isomorphism for every i > t and every j > 0. Notice now that (A, 𝒯_{0}) is a P-ring, and I := b_{i}A is an ideal of definition of A, for every i > 0, due to assumptions (a) and (b); moreover, any such I fulfills the conditions of (16.3.55). In light of theorem 16.3.64, it then suffices to show that \overline{Φ}_{A/b_{i}A} is bijective for every i ≥ t. To this aim, from the snake lemma and assumption (a), we are easily reduced to checking that Φ_{A/b_{i−1}A} induces a bijection J/b_{i}A →^{∼} J/b_{i−1}A for every i > t. Then, since J is the union of its subideals b_{k}A (for all k ∈ ℕ), we are further reduced to checking that Φ_{A/b_{i−1}A} induces a bijection b_{k}A/b_{i}A →^{∼} b_{k−1}A/b_{i−1}A for every k ≥ i. By considering the filtration

0 ⊂ b_{i+1}A/b_{i}A ⊂ b_{i+2}A/b_{i} ⊂ ⋯ ⊂ b_{k}A/b_{i}A

an easy induction then reduces to the case where k = i+1. Lastly, a similar induction argument, using the filtration 0 ⊂ b_{i+1}^{p−1}/b_{i} ⊂ b_{i+1}^{p−2}/b_{i} ⊂ ⋯ ⊂ b_{i+1}/b_{i} further reduces the assertion to the surjectivity of φ_{i,j} for every j = 1, …, p − 1, which is already known, by the foregoing.

Conversely, if (e) holds, then we get both (b) and (d), by virtue of proposition 16.3.59; then, notice that the direct limit of the system of maps (\overline{Φ}_{A/b_{i}A} | i ≥ t) is the Frobenius endomorphism of A/J, whence (a). Next, let x ∈ A with b_{0}x = 0; it follows that (b_{n}x)^{p^{n}} = 0 for every n ∈ ℕ, and since A is reduced (corollary 16.3.63(i)), we get b_{n}x = 0, whence (c).

Lastly, suppose that (c) and (d) hold; then we get a commutative diagram whose horizontal arrows are restrictions of Φ_{A/b_{0}A}, and are therefore bijections :

\xymatrix{b_{t+1}^{2}A/b_{1}A \ar[r]^{∼} \ar[d]_{j_{t+1}} & b_{t}^{2}A/b_{0}A \ar[d]^{j_{t}} \\ b_{t+1}A/b_{1}A \ar[r]^{∼} & b_{t}A/b_{0}A}

and whose vertical arrows are the natural injections; hence, we get an induced isomorphism Coker j_{t+1} →^{∼} Coker j_{t}, whence (b). It has already been remarked that (d) implies (a), so the proof of the stated equivalences is complete.

Next, suppose that A is perfectoid, so all conditions (a),…,(e) hold, and let us notice that

(16.3.74)  Ann_{A}(b_{0}) ∩ b_{n}A = 0  for every n ∈ ℕ.

Indeed, if x ∈ A and we have b_{0} · (b_{n}x) = 0, then b_{n}x = 0 by (c). This means that 𝒯_{0} induces the discrete topology on Ann_{A}(b_{0}), so \overline{A} := A/Ann_{A}(b_{0}) is complete and separated for its b_{0}-adic topology. Consider now the commutative diagram :

\xymatrix{A/b_{t+1}A \ar[r]^{^{\overline{Φ}^{A/b_{i}A}}} \ar[d]_{f_{1,t+1}} & A/b_{t}A \ar[d]^{f_{1,t}} \\ b_{t+1}A/b_{t+1}^{2}A \ar[r] & b_{t}A/b_{t}^{2}A}

whose left (resp. right) vertical arrow is induced by multiplication by b_{t+1} (resp. by b_{t}), and whose top (resp. bottom) horizontal arrow is induced by the Frobenius endomorphism of A/b_{t}A (resp. of A/b_{t}^{2}A). Due to (b),(d) and (16.3.74), we see that \overline{Φ}_{A/b_{t}A} restricts to a bijection

Ann_{A}(b_{t+1}) = Ker f_{1,t+1} →^{∼} Ker f_{1,t} = Ann_{A}(b_{t})

and combining with (c) we deduce that the Frobenius endomorphism of A/b_{i}A restricts to a bijection Ann_{A}(b_{0}) →^{∼} Ann_{A}(b_{0}) for every i ∈ ℕ. Then, using (d) we conclude that the Frobenius endomorphism of \overline{A}/b_{i}\overline{A} induces an isomorphism \overline{Φ}_{\overline{A}/b_{i}\overline{A}} : \overline{A}/b_{i+1}\overline{A} →^{∼} \overline{A}/b_{i}\overline{A} for every i ∈ ℕ. Lastly, it is easily seen that Ann\overline{A}(b_{i}) = 0 for every i ∈ ℕ. Summing up, we have shown that the ring \overline{A} fulfills conditions (c) and (d), so it is perfectoid for its b_{0}-adic topology, by the foregoing. Now, by virtue of (16.3.74), we may naturally identify Ann_{A}(b) with its image in A/b_{0}A, and likewise one easily sees that Ann_{E(A)}(β) ∩ βE(A) = 0, hence we may identify Ann_{E(A)}(β) with its image in E(A)/βE(A). By claim 16.3.62(ii), we have a commutative diagram of A_{0}/b_{0}A-modules

\xymatrix{E(A)/βE(A) \ar[r] \ar[d] & βE(A)/β^{2}E(A) \ar[d] \\ A/b_{0}A \ar[r] & b_{0}A_{0}/b_{0}^{2}A_{0}}

whose top (resp. bottom) horizontal arrow is induced by multiplication by β (resp. by b_{0}), and whose vertical arrows are induced by \overline{u}_{A}. To conclude, it suffices to remark that the kernel of the top (resp. bottom) horizontal arrow is Ann_{E(A)}(β) (resp. Ann_{A}(b_{0})). □

Let us point out the following variant of corollary 16.3.71:

Corollary 16.3.75. Let A be ring, b ∈ A an element such that p ∈ b^{p}A, and such that the b-adic topology 𝒯 of A is complete and separated. Then (A, 𝒯 ) is perfectoid if and only if the following two conditions hold :

(a) The Frobenius endomorphism Φ_{A/b^{p}A} of A/b^{p}A induces a bijection A/bA →^{∼} A/b^{p}A.

(b) Ann_{A}(b^{p}) = Ann_{A}(b^{p−1}).

Proof. If A is perfectoid, then \overline{u}_{A/pA} : E(A) → A/pA is surjective (lemma 16.2.7(i)), so there exists (b_{n} | n ∈ ℕ) ∈ E(A) such that b_{0} − b ∈ pA_{0}, and it follows easily that b_{0}/b ∈ A_{0}^{×}. Thus, we may replace b by b_{0}, in which case the assumptions of corollary 16.3.71are fulfilled, and we deduce that both (a) and (b) hold.

Conversely, if (a) holds, then A is a P-ring, and if also (b) holds, arguing as in the foregoing we may again assume that b = b_{0} for some (b_{n} | n ∈ ℕ) ∈ E(A); then we have, more generally:

Claim 16.3.76. If (a) holds and there exists t ∈ ℕ such that Ann_{A}(b^{p}) = Ann_{A}(b^{p}/b_{t}), then A is perfectoid.

Proof of the claim. In light of theorem 16.3.64, it suffices to show that the Frobenius endomorphism of A/b_{t}A induces bijections

Φ_{i} : gr_{i}^{(t+1)} := b_{t+1}^{i}A/b_{t+1}^{i+1}A →^{∼} gr_{i}^{(t)} := b_{t}^{i}A/b_{t}^{i+1}A for every i ∈ ℕ.

However, (a) easily implies that Φ_{i} is an isomorphism for every i = 0, …, p^{t+1} − 1. We then argue by induction on i ≥ p^{t+1} − 1. Thus suppose that Φ_{i} is an isomorphism for such i; we consider the commutative diagram

\xymatrix{gr_{i}^{(t+1)} \ar[r]^{Φ_{i}} \ar[d]_{_{\overline{b}_{t+1}}} & gr_{i}^{(t)} \ar[d]^{_{\overline{b}_{t}}} \\ _{gr{(t+1) \atop i+1}} \ar[r]^{Φ_{i+1}} & _{gr{(t) \atop i+1}}}

whose vertical arrows are induced by scalar multiplication by b_{t+1} and respectively b_{t}. Clearly both vertical arrows are surjections; thus, we are reduced to checking that \overline{b}_{t} is injective. Hence, say that for some a ∈ A, the class of b_{t}^{i}a in gr_{i}^{(t)} lies in the kernel of \overline{b}_{t}; this means that there exists c ∈ A with b_{t}^{i+1}a = b_{t}^{i+2}c, so that b_{t}^{i+1}(a − b_{t}c) = 0, and since i + 1 ≥ p^{t+1}, our assumption easily implies that b_{t}^{i}(a − b_{t}c) = 0 as well. So, the class of b_{t}^{i}a vanishes in gr_{i}^{(t)}, as required. □

Proposition 16.3.8shows that the completion of a perfectoid ring for the topology defined by a p-adically open and finitely generated ideal, is again perfectoid for the p-adic topology. We now wish to prove that the same still holds for a general finitely generated ideal.

16.3.77. Namely, let A be a perfectoid ring, I ⊂ A (resp. ℐ ⊂ E := E(A)) a finitely generated ideal such that the topology of A (resp. of E) agrees with the I-adic topology 𝒯_{I} (resp. with the ℐ -adic topology 𝒯_{ℐ} ). Let also β_{•} := (β_{1}, …, β_{n}) be any finite system of elements of E and 𝒥 ⊂ E (resp. J ⊂ A) the ideal generated by β_{•} (resp. by \overline{u}_{A}(β_{1}), …, \overline{u}_{A}(β_{n})). Denote by 𝒯_{J} (resp. 𝒯_{𝒥} ) the J-adic on A (resp. the 𝒥 -adic topology on E), and let

(A_{J}^{∧}, 𝒯_{J}^{∧}) and (E_{𝒥}^{∧}, 𝒯_{𝒥}^{∧})

respectively the completion of (A, 𝒯_{J}) and of (E, 𝒯_{𝒥} ). Lastly, let 𝒯_{I}^{∧} (resp. 𝒯_{ℐ}^{∧}) be the IA_{J}^{∧}-adic topology on A_{J}^{∧} (resp. the ℐ E_{𝒥}^{∧}-adic topology on E_{𝒥}^{∧}).

Theorem 16.3.78. In the situation of (16.3.77), the following holds :

(i) (A_{J}^{∧}, 𝒯_{I}^{∧}) is perfectoid.

(ii) There is a natural isomorphism of topological rings E(A_{J}^{∧}, 𝒯_{I}^{∧}) →^{∼} (E_{𝒥}^{∧}, 𝒯_{ℐ}^{∧}).

Proof. Let J_{A} ⊂ A := W (E) be the ideal generated by τ_{E}(β_{1}), …, τ_{E}(β_{n}), set J′_{A} := J_{A}+pA and denote by A_{J}^{∧} (resp. A_{J′}^{∧}) the J_{A}-adic (resp. J′_{A}-adic) completion of A. Moreover, pick any finite system of generators (γ_{1}, …, γ_{k}) for ℐ, let I_{A} ⊂ A be the ideal generated by the system (p, τ_{E}(γ_{1}), …, τ_{E}(γ_{k})), and denote by 𝒯_{I,A}^{∧} indifferently the I_{A}-adic topologies on A_{J′}^{∧} and on A_{J}^{∧}, and by 𝒯_{J′,A}^{∧} the J′_{A}-adic topology on A_{J′}^{∧}. With this notation, we have natural isomorphisms of topological A-algebras

(16.3.79)  W (E_{𝒥}^{∧}, 𝒯_{𝒥}^{∧}) →^{∼} (A_{J′}^{∧}, 𝒯_{J′,A}^{∧})  and W (E_{𝒥}^{∧}, 𝒯_{ℐ}^{∧}) →^{∼} (A_{J′}^{∧}, 𝒯_{I,A}^{∧})

(proposition 9.3.77(ii) and lemma 9.3.33(iv)). On the other hand, since the J_{A}-adic topology is finer than the J′_{A}-adic topology, the completion map A → A_{J′}^{∧} factors through a unique map of A-algebras

(16.3.80)  A_{J}^{∧} → A_{J′}^{∧}.

Claim 16.3.81. The map (16.3.80) is a ring isomorphism.

Proof of the claim. Quite generally, consider any abelian group M, endowed with two topologies 𝒯 and 𝒯 ′ defined by cofiltered systems of subgroups (M_{λ} | λ ∈ Λ) and respectively (M′_{μ} | μ ∈ Λ′). Endow each quotient Q_{λ} := M/M_{λ} with the quotient topology 𝒯′_{λ} induced by 𝒯 ′ via the projection map M → Q_{λ}, and suppose moreover that 𝒯 is finer than 𝒯 ′ and (Q_{λ}, 𝒯′_{λ}) is separated and complete for every λ ∈ Λ. Then it follows that the natural map

(M, 𝒯 )^{∧} → (M, 𝒯 ′)^{∧}

(that extends the completion map M → (M, 𝒯 ′)^{∧}), is an isomorphism of abelian groups. Indeed, clearly the family (M_{λ} + M′_{μ} | λ ∈ Λ, μ ∈ Λ′) is still a fundamental system of open neighborhoods of zero in M for the topology 𝒯 ′, so we have natural isomorphisms

(M, 𝒯 ′)^{∧} →^{∼} lim_{(λ,μ)∈Λ×Λ′} M/(M_{λ} + M′_{μ}) →^{∼} lim_{λ∈Λ} lim_{μ∈Λ′} M/(M_{λ} + M′_{μ}) →^{∼} lim_{λ∈Λ} (Q_{λ}, 𝒯 ′)^{∧}

(where the second isomorphism follows from example 1.5.15(ii)) but (Q_{λ}, 𝒯 ′)^{∧} = Q_{λ} under the current assumptions, whence the assertion.

Recall now that the family (W (𝒥^{⟨λ⟩}) | λ ∈ ℕ[1/p]) is a fundamental system of open neighborhoods of zero in for the J_{A}-adic topology on A (proposition 9.3.77(i) and lemma 9.3.69(iv)). In view of the foregoing, the claim will then follow, once we have shown that the quotients A/W (𝒥^{⟨λ⟩}) are complete and separated for the J′_{A}-adic topology. Moreover, it is clear that J′_{A}-adic topology on A/W (𝒥^{⟨λ⟩}) agrees with the p-adic topology, so – taking into account proposition 9.3.44(ii) – we come down to checking that W (𝒥^{⟨λ⟩}) is a closed ideal for the p-adic topology 𝒯_{p,A} on A. But notice that if 𝒯_{d} denotes the discrete topology on E, then W (E, 𝒯_{d}) = (A, 𝒯_{p,A}), so the assertion is a special case of remark 9.3.28(iv). ♢

Claim 16.3.82. (E_{𝒥}^{∧}, 𝒯_{ℐ}^{∧}) is perfectoid.

Proof of the claim. Since E is a perfect topological ring, the same holds for (E, 𝒯_{𝒥} ), hence also for (E_{𝒥}^{∧}, 𝒯_{𝒥}^{∧}) (remark 9.4.9(v) and example 9.3.48(ii)), and by the same token, (E_{𝒥}^{∧}, 𝒯_{ℐ}^{∧}) is a perfect topological 𝔽_{p}-algebra. By example 16.3.2(i), it then remains only to check that the topology 𝒯_{ℐ}^{∧} is complete and separated. However, since E is complete and separated for the ℐ -adic topology, E_{𝒥}^{∧} is complete and separated for the (ℐ + 𝒥)-adic topology (lemma 16.3.15(ii.b)) and then the assertion follows from lemma 8.3.12. ♢

Let \underline{α} := (α_{n} | n ∈ ℕ) ∈ Ker u_{A} be any distinguished element; clearly \underline{α} is still distinguished in W (E_{𝒥}^{∧}, 𝒯_{ℐ}^{∧}), hence W (E_{𝒥}^{∧}, 𝒯_{ℐ}^{∧}) ⊗_{A} A is perfectoid (example 16.3.2(ii)). From (16.3.79) and claim 16.3.81we deduce that both (i) and (ii) will follow, once we have shown

Claim 16.3.83. There is a natural isomorphism of topological rings :

(A_{J}^{∧}, 𝒯_{I,A}^{∧}) ⊗_{A} A →^{∼} (A_{J}^{∧}, 𝒯_{I}^{∧}).

Proof of the claim. By construction, the J_{A}-adic topology induces the J-adic topology on A, via the projection u_{A} : A → A. Therefore, u_{A} extends to a surjective map

u_{A}^{∧} : A_{J}^{∧} → A_{J}^{∧}

whose kernel is the topological closure of \underline{α}A in the J_{A}-adic topology of A_{J}^{∧} (proposition 8.2.13(i,v) and lemma 8.3.32(iv)). On the other hand, by proposition 9.3.77(ii) and lemma 16.2.7(i), the I_{A}-adic topology on A induces the I-adic topology on A via u_{A}, so the topology 𝒯_{I,A}^{∧} induces the topology 𝒯_{I}^{∧} on A_{J}^{∧}, via u_{A}^{∧}. Thus, we are reduced to checking that \underline{α}A_{J}^{∧} is a closed ideal in the J_{A}-adic topology of A_{J}^{∧}. By claim 16.3.81, it then further suffices to show that \underline{α}A_{J′}^{∧} is closed for the topology 𝒯_{J′,A}^{∧}. But in light of (16.3.79), the latter assertion follows from corollary 16.1.13(ii). □

As an application we get the following generalization of theorem 16.3.21(ii) :

Corollary 16.3.84. Let (A, 𝒯 ) be a perfectoid ring, β_{•} := (β_{1}, …, β_{k}) and γ_{•} := (γ_{1}, …, γ_{l}) two finite sequences of elements of E := E(A), and denote by I (resp. J) the ideal of A generated by the system (\overline{u}_{A}(β_{1}), …, \overline{u}_{A}(β_{k})) (resp. by \overline{u}_{A}(γ_{1}), …, \overline{u}_{A}(γ_{l})). Then the resulting map (16.3.14) is an isomorphism.

Proof. Let A′ be the I-adic completion of A, and for every k ∈ ℕ, denote by (J^{k})^{c} the topological closure of the ideal J^{k}A′ relative to the IA′-adic topology; arguing as in the proof of lemma 16.3.15(ii.b), we are easily reduced to showing that the JA′-adic topology on A′ agrees with the linear topology defined by the system of ideals ((J^{k})^{c} | k ∈ ℕ). However, let K be an ideal of definition of A, and and endow A′ with its KA′-adic topology 𝒯 ′; by theorem 16.3.78, the topological ring (A′, 𝒯 ′) is also perfectoid. Moreover, for every i ≤ l, denote by γ′_{i} ∈ E′ := E(A′) the image of γ_{i} under the homomorphism E → E′ induced by the completion map A → A′; then clearly JA′ is the ideal generated by the system (\overline{u}_{A′}(γ′_{1}), …, \overline{u}_{A′}(γ′_{l})). Hence, we may replace A by A′, after which we may assume that A is complete and separated for its I-adic topology. To conclude, it then suffices to show :

Claim 16.3.85. In the situation of the corollary, suppose moreover that A is complete and separated for the I-adic topology. Then the J-adic topology of A agrees with the linear topology induced by the system of ideals ((J^{k})^{c} | k ∈ ℕ).

Proof of the claim. From lemma 9.3.69(iv) we know already that the J-adic topology on A agrees with the linear topology defined by the system of ideals ([γ_{•}]^{⟨λ⟩}A | λ ∈ ℕ[1/p]) so it suffices to check that

([γ_{•}]^{⟨λ⟩}A)^{c} ⊂ [γ_{•}]^{⟨λ′⟩}A for every λ, λ′ in ℕ[1/p] with λ′ < λ.

Now, any element of ([γ_{•}]^{⟨λ⟩}A)^{c} can be written as an I-adically convergent series ∑_{n∈ℕ} x_{n}, where x_{n} ∈ [γ_{•}]^{⟨λ⟩}A∩[β_{•}]^{⟨n⟩}A for every n ∈ ℕ (again by lemma 9.3.69(iv)). Fix λ′′, ε ∈ ℕ[1/p] with λ′ < λ′′ < λ, and 0 < ε < 1 − λ′′/λ; by corollary 16.3.51(iii) we get

x_{n} ∈ [γ_{•}]^{⟨λ′′⟩} · [β_{•}]^{⟨nε⟩}A for every n ∈ ℕ.

Now we argue as in the proof of claim 16.3.20: by definition, for every n ∈ ℕ there exist • a finite set S_{n} ⊂ ℕ[1/p]^{⊕l} with μ_{1} + ⋯ + μ_{l} = λ′′ for every μ := (μ_{1}, …, μ_{l}) ∈ S_{n}

• a system (a_{μ} | μ ∈ S_{n}) of elements of [β_{•}]^{⟨nε⟩}A such that

x_{n} = ∑_{μ∈S_{n}} a_{μ} · \overline{u}_{A}(γ_{•}^{μ}) where γ_{•}^{μ} := γ_{1}^{μ_{1}} ⋯ γ_{l}^{μ_{l}} for every μ ∈ S_{n}.

Choose N ∈ ℕ such that λ′′ − λ′ > lp^{−N}, and define \overline{μ} and μ^{∗} as in the proof of proposition 9.3.77, so that \overline{μ} ∈ S := \{ν ∈ p^{−N}ℕ^{⊕l} | λ′′ ≥ ν_{1} + ⋯ + ν_{l} > λ′\} for every n ∈ ℕ and every μ ∈ S_{n}. It follows that

x = ∑_{ν∈S} c_{ν} · \overline{u}_{A}(γ_{•}^{ν}) where c_{ν} := ∑_{n∈ℕ} ∑_{\begin{matrix}μ∈S_{n} \\ \overline{μ}=ν \end{matrix}} a_{μ} · \overline{u}_{A}(γ_{•}^{μ^{∗}}).

Clearly \overline{u}_{A}(γ_{•}^{ν}) ∈ [γ_{•}]^{⟨λ′⟩}A, and by lemma 9.3.69(iv) the series c_{ν} converges in the I-adic topology of A for every ν ∈ S; also it is easily seen that S is a finite set, whence the contention.

□

16.4. Homological theory of perfectoid rings. This section gathers some results on the homological properties of perfectoid rings. The first result is a simplification of the criterion of theorem 16.3.64, valid in the case where a regular sequence generates a special ideal of definition of the P-ring A.

Theorem 16.4.1. Let A be a P-ring, and (a_{1}, …, a_{k}) a regular sequence of elements of A that generates a special ideal of definition I of A. Then the following conditions are equivalent :

(a) The Frobenius endomorphism of A/pA induces an isomorphism A/I →^{∼} A/I^{(p)}.

(b) A is perfectoid.

Proof. We know already that (b)⇒(a), by corollary 16.3.3. Suppose that (a) holds. Then E := E(A) is perfectoid (proposition 16.2.9 and example 16.3.2(i)). Arguing as in the proof of theorem 16.3.64, we may then find elements β_{1}, …, β_{k} in E, which generate an ideal of definition 𝒥 of E, and such that \overline{u}_{A/pA}(β_{i}) equals the image of a_{i} in A/pA, for i = 1, …, k; moreover, \overline{u}_{A/pA} induces an isomorphism (16.3.65).

Next, since p ∈ I^{(p)}, we may find – again as in the proof of theorem 16.3.64– a distinguished element \underline{α}′ ∈ Ker u_{A} such that α′_{0} ∈ Φ_{E}(𝒥). We let B := W (E)/\underline{α}′W (E), and we notice, as in loc. cit. that B is perfectoid for the quotient topology inherited from W (E), and u_{A} factors through a surjective ring homomorphism v : B → A. We define the ideal I_{B} ⊂ B as in the proof of theorem 16.3.64, so (16.3.68) holds as well in the current situation, and arguing as in loc. cit. we also see that p ∈ I_{B}^{(p)}. Set B_{0} := B/I_{B} and A_{0} := A/I; there follows a commutative diagram

(16.4.2)  \xymatrix{Sym_{B_{0}}^{•}(I_{B}/I_{B}^{2}) \ar[r] \ar[d] & gr_{I_{B}}^{•}B \ar[d] \\ Sym_{A_{0}}^{•}(I/I^{2}) \ar[r] & gr_{I}^{•}A}

where gr_{I}^{•}A is the graded ring associated to the I-adic filtration on A, and likewise for gr_{I_{B}}^{•}B. The top horizontal arrow of this diagram is surjective, and by assumption, the bottom horizontal arrow is an isomorphism. Furthermore, we notice :

Claim 16.4.3. The maps \overline{u}_{A/pA}, \overline{u}_{B/pB} and v induce ring isomorphisms

E/𝒥^{2} →^{∼} B/I_{B}^{2} →^{∼} A/I^{2}.

Proof of the claim. We have a commutative diagram

[figure: E/𝒥; _{\overline{Φ}_{E/𝒥}}; A/I; _{\overline{Φ}_{A/I}}; E/𝒥^{(p)}; A/I^{(p)}]

whose horizontal arrows are induced by \overline{u}_{A/pA}. Now, the vertical arrows are isomorphisms, and we have already remarked that the top horizontal arrow is also an isomorphism. Thus, the same holds for the bottom horizontal arrow; since \overline{u}_{A/pA}(𝒥^{2}) = I^{2}/pA, we deduce already that \overline{u}_{A/pA} induces an isomorphism E/𝒥^{2} →^{∼} A/I^{2}. Similarly, since B is perfectoid, \overline{u}_{B/pB} induces an isomorphism E/α′_{0}E →^{∼} B/pB; however, α′_{0} ∈ 𝒥^{2}, p ∈ I_{B}^{2}, and \overline{u}_{B/pB}(𝒥^{2}) = I_{B}^{2}/pB, so \overline{u}_{B/pB} induces an isomorphism E/𝒥^{2} →^{∼} B/I_{B}^{2}. By construction, it is then clear that v induces an isomorphism B/I_{B}^{2} →^{∼} A/I^{2}. ♢

From claim 16.4.3 we see that v induces an isomorphism B_{0} →^{∼} A_{0}, and we get as well isomorphisms of E-modules

𝒥 /𝒥^{2} →^{∼} I_{B}/I_{B}^{2} →^{∼} I/I^{2}.

Thus, the left vertical arrow of (16.4.2) is an isomorphism; therefore the same holds for the top horizontal arrow, and hence also for the right vertical arrow. Since B is complete and separated for its I_{B}-adic topology, and A is complete and separated for its I-adic topology, we conclude that v is an isomorphism of topological rings, whence (b). □

16.4.4. For every n, r ∈ ℕ, set (see also example 4.8.55(iii))

P_{r} := ℕ[1/p]^{⊕r} R_{r,n} := ℤ[P_{r}] = ℤ[T_{1}^{1/p^{∞}}, …, T_{r}^{1/p^{∞}}]

so, actually R_{r,n} is independent of n, but we also consider the ring homomorphism

v_{r,n} : R_{r,n} → R_{r,0}  T_{i}^{a} ↦ T_{i}^{a/p^{n}}  for every a ∈ ℕ[1/p] and i = 1, …, r

for every r, n ∈ ℕ. Let also T_{•}P_{r} ⊂ P_{r} be the ideal generated by ℕ^{⊕r}, and set

I_{r,n} := (T_{•}P_{r}) · R_{r,n}  I_{r,n}^{⌈s⌉} := (T_{•}P_{r})^{⌈s⌉} · R_{r,n}  for every s ∈ ℝ_{+}

(notation of remark 9.3.70(i)). Thus I_{r,n}^{⌈s⌉} is an ideal of R_{r,n}, for every s ∈ ℝ_{+} and every r, n ∈ ℕ. Next, let A be any ring, u_{0} : R_{r,0} → A any ring homomorphism, and for every n ∈ ℕ set u_{n} := u_{0} ∘ v_{r,n} : R_{r,n} → A, denote by f^{(n)} the sequence (u_{n}(T_{1}), …, u_{n}(T_{r})) of elements of A; for n = 0, we shall also write just f instead of f^{(0)} and denote by J ⊂ A the ideal generated by f. We may then state :

Lemma 16.4.5. In the situation of (16.4.4), suppose furthermore that

Tor_{i}^{R_{r,0}}(R_{r,0}/I_{r,0}^{⌈s⌉}, A) = 0 for every i > 0 and every s ∈ ℝ_{+}.

Then the following holds :

(i) The ring A satisfies condition (a)_{f^{(n)}}^{un} of (7.8.21) with step ≤ r, for every n ∈ ℕ.

(ii) Ann_{A}(J^{k}) = Ann_{A}(I_{r,0}^{⌈0⌉} · A) for every k > 0.

(iii) Ann_{A}(J) ∩ I_{r,0}^{⌈0⌉} · A = 0.

Proof. (i): In case r = 0, there is nothing to prove, so we assume henceforth that r > 0. For every n ∈ ℕ, endow A with the R_{r,n}-module structure induced by u_{n}.

Claim 16.4.6. Tor_{i}^{R_{r,n}}(R_{r,n}/I_{r,n}^{⌈s⌉}, A) = 0 for every i > 0 and every s ∈ ℝ_{+}.

Proof of the claim. The base change v_{r,n} : R_{r,n} → R_{r,0} yields a spectral sequence

E_{ij}^{2} := Tor_{i}^{R_{r,0}}(Tor_{j}^{R_{r,n}}(R_{r,n}/I_{r,n}^{⌈s⌉}, R_{r,0}), A) ⇒ Tor_{i}^{R_{r,n}}(R_{r,n}/I_{r,n}^{⌈s⌉}, A).

But since v_{r,n} is an isomorphism and v_{r,n}(I_{r,n}^{⌈s⌉}) = I_{r,0}^{⌈s/p^{n}⌉}, we see that E_{ij}^{2} = 0 for every j > 0, and E_{i0}^{2} = Tor_{i}^{R_{r,0}}(R_{r,0}/I_{r,0}^{⌈s/p^{n}⌉}, A). Thus, E_{i0}^{2} = 0 for i > 0, whence the claim. ♢

In view of claim 16.4.6, we may replace u_{0} by u_{n}, and reduce to checking condition (a)_{f}^{un} for A. To this aim, define A_{r}, I_{r} and the ring homomorphism β_{f} : A_{r} → A as in (7.8.21); endow also R_{r,0} with the A_{r}-module structure induced by the inclusion map A_{r} → R_{r,0}, and consider the change of rings spectral sequence

E(n)_{pq}^{2} := Tor_{p}^{R_{r,0}}(Tor_{q}^{A_{r}}(A_{r}/I_{r}^{n}, R_{r,0}), A) ⇒ Tor_{p+q}^{A_{r}}(A_{r}/I_{r}^{n}, A) for every n ∈ ℕ.

It is easily seen that R_{r,0} is a free A_{r}-module, hence E(n)_{pq}^{2} = 0 whenever q > 0, and E(n)_{p0}^{2} = R_{r,0}/I_{r,0}^{n}. There follows a natural isomorphism

Tor_{p}^{R_{r,0}}(R_{r,0}/I_{r,0}^{n}, A) →^{∼} Tor_{p}^{A_{r}}(A_{r}/I_{r}^{n}, A) for every p, n ∈ ℕ.

Now, recall that I_{r,0}^{n+r} ⊂ I_{r,0}^{⌈n+r−1⌉} ⊂ I_{r,0}^{n} for every n ∈ ℕ (lemma 9.3.69(iv)); then our assumption implies immediately that the inverse system (Tor_{p}^{A_{r}}(A_{r}/I_{r}^{n}, A) | n ∈ ℕ) is uniformly essentially zero, with step ≤ r.

(iii): We have just seen that condition (a)_{f^{(n)}}^{un} holds for every n, with step bounded by r. For every n ∈ ℕ, let J_{n} ⊂ A be the ideal generated by f^{(n)}; according to remark 7.8.34(ii), it follows that there exists a sequence of integers (d(i) | i ∈ ℕ) such that inverse system (H_{i+1}(f^{(n)}, J_{n}^{k}) | k ∈ ℕ) is uniformly essentially zero, with step bounded by d(i), for every i, n ∈ ℕ. Especially, for d := d(r) we get

Ann_{A}(J_{n}) ∩ J_{n}^{d} = 0 for every n ∈ ℕ.

By a simple induction argument, we deduce that Ann_{A}(J_{n}^{k+1}) ∩ J_{n}^{d} = 0 for every n, k ∈ ℕ. But clearly, for every n ∈ ℕ there exists k ∈ ℕ such that J_{n}^{k+1} ⊂ J, whence

Ann_{A}(J) ∩ J_{n}^{d} = 0 for every n ∈ ℕ

which yields easily the contention.

(ii) is an easy consequence of (iii). □

Proposition 16.4.7. Let f : A → B be any homomorphism of perfect 𝔽_{p}-algebras, I ⊂ A any ideal. Then the natural morphism

A/I^{⌈s⌉}A ⊗^{L}_{A} B → A/I^{⌈s⌉}A ⊗_{A} B[0]

is an isomorphism in D(A-Mod), for every s ∈ ℝ_{+} (notation of remark 9.3.70(i)).

Proof. We reduce easily to the case where I is finitely generated, and f = E^{∗}(f_{0}) for a map f_{0} : A_{0} → B_{0} of 𝔽_{p}-algebras of finite type (where A_{0} and B_{0} are endowed with their discrete topologies : notation of (9.4.7)). Then we may also assume that I = I_{0}A for some finitely generated ideal I_{0} ⊂ A_{0}, and it suffices to show :

Claim 16.4.8. In the foregoing situation, for every λ, λ′ ∈ ℕ[1/p] with λ′ < λ, the inclusion I^{⟨λ⟩}A ⊂ I^{⟨λ′⟩}A induces the zero map

Tor_{i}^{A}(A/I^{⟨λ⟩}A, B) → Tor_{i}^{A}(A/I^{⟨λ′⟩}A, B) for every i > 0.

Proof of the claim. Under the current assumptions, A is the colimit of the system of 𝔽_{p}-algebras (A_{n} | n ∈ ℕ) such that A_{n} := A_{0} for every n ∈ ℕ, with transition maps φ_{n} : A_{n} → A_{n+1} given by the Frobenius map Φ_{A_{0}}, and likewise for B. Moreover, say that λ = ap^{k} and λ′ = bp^{k} for some a, b, k ∈ ℕ, and for every n ∈ ℕ define ideals of A_{n} by the rules :

I_{n} := \{ \begin{matrix}0 & for n < k \\ I_{0}^{ap^{n−k}} & for n ≥ k \end{matrix} and I′_{n} := \{ \begin{matrix}0 & for n < k \\ I_{0}^{bp^{n−k}} & for n ≥ k. \end{matrix}

Then it is easily seen that φ_{n}(I_{n}) ⊂ I_{n+1}, and likewise for I′_{n}, for every n ∈ ℕ. Furthermore, the colimit of the system (I_{n} | n ∈ ℕ) (resp. (I′_{n} | n ∈ ℕ)) is I^{⟨λ⟩}A (resp. I^{⟨λ′⟩}A). By claim 7.11.37(i), we are therefore reduced to showing that the map induced by the inclusion I_{n} ⊂ I′_{n}

Tor_{i}^{A_{0}}(A_{0}/I_{n}, B_{0}) → Tor_{i}^{A_{0}}(A_{0}/I′_{n}, B_{0})

vanishes, for every sufficiently large n ∈ ℕ. Since a > b, the latter assertion follows from claim 7.9.18(ii) (applied with M := A and N := B). □

16.4.9. We are going now to apply the general setup of (16.4.4) to the case where A is a perfectoid ring, and

f := (\overline{u}_{A}(β_{1}), …, \overline{u}_{A}(β_{r}))

for a given sequence β_{•} := (β_{1}, …, β_{r}) of elements of E := E(A). To ease notation, we shall simply write P instead of P_{r} and R instead of R_{r,0}. Also, we shall write I, I^{⌈s⌉} ⊂ R instead of I_{r,0}, and respectively I_{r,0}^{⌈s⌉}, for every s ∈ ℝ_{+}. Likewise, we shall write u instead of u_{0}; hence u : R → A is the unique ring homomorphism such that u(T_{i}) = \overline{u}_{A}(β_{i}) for i = 1, …, r.

Proposition 16.4.10. (i) With the notation of (16.4.9), the natural morphism

R/I^{⌈s⌉} ⊗^{L}_{R} A → A/I^{⌈s⌉}A[0]

is an isomorphism in D(R-Mod), for every s ∈ ℝ_{+}.

(ii) In particular, A satisfies condition (a)_{f^{(n)}}^{un} of (7.8.21) with step ≤ r, for every n ∈ ℕ.

Proof. In view of lemma 16.4.5(i), it suffices to check (i). Notice that R_{0} := R ⊗_{ℤ} 𝔽_{p} is a perfect 𝔽_{p}-algebra, and I^{⌈s⌉}R_{0} = I_{0}^{⌈s⌉}R_{0}, where I_{0} ⊂ R_{0} is the ideal generated by (T_{1}, …, T_{r}). In light of proposition 16.4.7, we deduce

(16.4.11)  Tor_{i}^{R_{0}}(R_{0}/I^{⌈s⌉}R_{0}, E) = 0  for every s ∈ ℝ_{+} and every i > 0.

On the other hand, we have a standard 2-spectral sequence

E_{pq}^{2} := Tor_{p}^{R_{0}}(Tor_{q}^{R}(R/I^{⌈s⌉}, R_{0}), E) ⇒ Tor_{p+q}^{R}(R/I^{⌈s⌉}, E)

([163, Th.5.6.6]). However, by construction I^{⌈s⌉} is a direct summand of the free ℤ-module R, so R/I^{⌈s⌉} is also a free ℤ-module, and then a standard calculation shows that E_{pq}^{2} = 0 whenever q > 0. From (16.4.11) we see as well that E_{p0}^{2} = 0 for every p > 0, so we conclude

Tor_{i}^{R}(R/I^{⌈s⌉}, E) = 0 for every s ∈ ℝ_{+} and every i > 0.

Next, an easy induction argument using the short exact sequences of W (E)-modules

0 → E ≃ p^{n}W_{n+1}(E) → W_{n+1}(E) → W_{n}(E) → 0 for every n ∈ ℕ

shows more generally that

(16.4.12) Tor_{i}^{R}(R/I^{⌈s⌉}, W_{n}(E)) = 0  for every s ∈ ℝ_{+}, every i > 0 and every n ∈ ℕ.

For every n ∈ ℕ, set J_{n} := p_{P}^{−n}(T_{•}P), where p_{P} denotes the p-Frobenius automorphism of P (see definition 4.8.40(ii)), and consider the projection

π_{n}^{(a,b)} : R/J_{n}^{a}R → R/J_{n}^{b}R for every a, b, n ∈ ℕ with a ≥ b.

Claim 16.4.13. With the foregoing notation, we have :

(i) Tor_{i}^{R}(π_{n}^{(a,b)}, W_{m}(E)) = 0 for every a, b, i, m, n ∈ ℕ with a − b ≥ r and i > 0.

(ii) For every a, n ∈ ℕ there exists a complex of free R-modules of finite type L_{a,n}^{•} with a

quasi-isomorphism L_{a,n}^{•} → R/J_{n}^{a}[0] of complexes of R-modules. (iii) Tor_{i}^{R}(π_{n}^{(a,b)}, W (E)) = 0 for every a, b, i, n ∈ ℕ with a − b ≥ 2r and i > 0.

Proof of the claim. If r = 0 there is nothing to prove, so we may assume that r > 0.

(i): Notice that J_{n} is generated by a system of r elements of P, for every n ∈ ℕ; in view of lemma 9.3.69(ii.c,iv), we deduce

J_{n}^{a} ⊂ J_{n}^{⟨a⟩} ⊂ (T_{•}P)^{⌈(b+r−1)p^{−n}⌉} ⊂ J_{n}^{⟨b+r−1⟩} ⊂ J_{n}^{b}

so the assertion follows from (16.4.12).

(ii): The ideal J_{n}R is generated by the system (T_{1}^{1/p^{n}}, …, T_{r}^{1/p^{n}}) of elements of the subring S_{n} := ℤ[T_{1}^{1/p^{n}}, …, T_{r}^{1/p^{n}}]; since S_{n} is noetherian, we may find a complex L_{a,n}^{′•} of free S_{n}- modules of finite type, with a quasi-isomorphism L_{a,n}^{′•} → S_{n}/J_{n}^{a}S_{n}[0], and since the inclusion map S_{n} → R is flat, the complex L_{a,n}^{•} := L_{a,n}^{′•} ⊗_{S_{n}} R will do.

(iii): For given a, n ∈ ℕ, let L_{a,n}^{•} be as in (ii); it is easily seen that the natural map

lim_{m∈ℕ} L_{a,n}^{•} ⊗_{R} W_{m}(E) → L_{a,n}^{•} ⊗_{R} W (E)

is an isomorphism in C(R-Mod). To ease notation, set 𝒯_{i,n,m}^{a} := Tor_{i}^{R}(R/J_{n}^{a}R, W_{m}(E)) for every i, m, n ∈ ℕ; in light of [163, Th.3.5.8] we deduce natural short exact sequences

0 → R^{1} lim_{m∈ℕ} 𝒯_{i−1,n,m}^{a} → Tor_{i}^{R}(R/J_{n}^{a}R, W (E)) → lim_{m∈ℕ} 𝒯_{i,n,m}^{a} → 0

for every i, a, n ∈ ℕ. Then the assertion follows from (i). ♢

Now, for any s ∈ ℝ_{+}, let S_{s} be the set of all pairs (a, n) ∈ ℕ × ℕ such that ap^{−n} > s; we define a partial ordering on S_{s} by the rule : (a, n) ≥ (b, m) if and only if n ≥ m and ap^{m} ≤ bp^{n}. With this notation, notice that J_{m}^{b} ⊂ J_{n}^{a} whenever (a, n) ≥ (b, m), and I^{⌈s⌉} is the union of its filtered system of subideals (J_{n}^{a}R | (a, n) ∈ S_{s}). Taking into account claim 16.4.13(iii) we deduce that

(16.4.14)  Tor_{i}^{R}(R/I^{⌈s⌉}, W (E)) = 0  for every s ∈ ℝ_{+} and every i > 0.

Next, from remark 16.1.7(ii) we get a short exact sequence of R-modules

0 → W (E) ⟶^{\underline{α}} W (E) → A → 0

where \underline{α} is any distinguished element in Ker u_{A}. Then, by the induced long exact Tor-sequence, (16.4.14) implies already that Tor_{i}^{R}(R/I^{⌈s⌉}, A) vanishes for every i > 1. By the same token, we also see that Tor_{1}^{R}(R/I^{⌈s⌉}, A) is isomorphic to the kernel of scalar multiplication by \underline{α} on the R-module W (E)/I^{⌈s⌉}W (E); to conclude the proof, it then suffices to remark :

Claim 16.4.15. \underline{α}W (E) ∩ I^{⌈s⌉}W (E) = \underline{α}I^{⌈s⌉}W (E).

Proof of the claim. Denote by ℐ ⊂ E the ideal generated by β_{•}; from proposition 9.3.77(i) we easily deduce that I^{⌈s⌉}W (E) = ⋃_{s′>s} W (ℐ^{⌊s′⌋}), and in view of proposition 16.1.12(ii) we get

\underline{α}W (E) ∩ I^{⌈s⌉}W (E) = ⋃_{s′>s}(\underline{α}W (E) ∩ W (ℐ^{⌊s′⌋})) = ⋃_{s′>s} \underline{α}W (ℐ^{⌊s′⌋}) = \underline{α}I^{⌈s⌉}W (E)

as stated. □

Corollary 16.4.16. In the situation of (16.4.9), set J := IA, and denote by A_{J}^{∧} the J-adic completion of A. Let also C_{•} be any bounded above complex of flat A-modules such that for every i ∈ ℤ there exists c(i) ∈ ℕ with J^{c(i)} · H_{i}C_{•} = 0. Then the natural morphism

C_{•} → A_{J}^{∧} ⊗_{A} C_{•}

is an isomorphism in D(A-Mod).

Proof. It is an immediate consequence of proposition 16.4.10(ii) and corollary 8.6.34. □

Proposition 16.4.17. In the situation of (16.4.9), suppose that the sequence β_{•} generates an open ideal of E. Then the following conditions are equivalent :

(a) The sequence β_{•} is completely secant in E.

(b) The sequence b_{•} := (\overline{u}_{A}(β_{1}), …, \overline{u}_{A}(β_{r})) is completely secant in A.

Proof. Set as usual J := IA, and let A_{J}^{∧} be the J-adic completion of A. In view of corollary 16.4.16, we see that the sequence b_{•} is completely secant in A if and only if its image is completely secant in A_{J}^{∧}. Moreover, if 𝒥 ⊂ E denotes the ideal generated by β_{•}, we have a natural ring isomorphism E(A_{J}^{∧}) →^{∼} E_{𝒥}^{∧} (theorem 16.3.78(ii)), and by the same token the sequence β_{•} is completely secant in the perfectoid ring E if and only if the same holds for its image in E_{𝒥}^{∧} . Thus, we may replace A by A_{J}^{∧}, and assume from start that A is J-adically complete and separated. Next, let (α_{n} | n ∈ ℕ) be any distinguished element in Ker_{,}u_{A}; since 𝒥 is open, we have α_{0}^{n} ∈ 𝒥 for some integer n ∈ ℕ, and then we may find k ∈ ℕ such that α_{0} is contained in the ideal generated by the sequence β′_{•} := (β_{1}^{1/p^{k}}, …, β_{r}^{1/p^{k}}). In light of corollary 7.8.8, we may then replace β_{•} with β′_{•}, and assume that α_{0} ∈ 𝒥. Notice now that {𝒥^{n}} = J^{n} for every ∈ ℕ (remark 16.3.28(viii)), and the inclusion 𝒥^{n+1} ⊂ 𝒥^{n} is taut for every n ∈ ℕ, so theorem 16.3.31(ii) yields a natural isomorphism

\overline{τ} : ⨁_{n∈ℕ} 𝒥^{n}/𝒥^{n+1} →^{∼} ⨁_{n∈ℕ} J^{n}/J^{n+1}.

Moreover, a simple inspection shows that \overline{τ} is an isomorphism of A/pA-algebras; it follows immediately that the sequence β_{•} is E-quasi-regular if and only if the sequence b_{•} is A-quasi-regular (see remark 7.10.44(i)). To conclude, it suffices now to invoke proposition 7.10.47. □

Proposition 16.4.18. Let f : A → A′ be any continuous map of perfectoid rings, I (resp. ℐ) a finite type ideal of definition of A (resp. of E(A)), and denote by 𝒯_{I} (resp. 𝒯_{ℐ} ) the IA′-adic topology of A′ (resp. the ℐ E(A′)-adic topology of E(A′)). Then f : A → (A′, 𝒯_{I}) is adically flat if and only if the same holds for E(f) : E(A) → (E(A′), 𝒯_{𝒥} ) (see definition 8.3.23(iii)).

Proof. Set E := E(A), E′ := E(A′), and let \underline{α} := (α_{n} | n ∈ ℕ) be a distinguished element of Ker(u_{A}); by lemma 8.3.12 and proposition 16.3.8, the topological ring (A′, 𝒯_{I}) is perfectoid, so we may assume that f is adic, in which case the same holds for E(f) : E → E′, by theorem 16.3.42(i). Let also β_{•} := (β_{1}, …, β_{n}) be a sequence of elements of E that generates ℐ; we may assume that α_{0} ∈ ℐ, and by corollary 16.3.40(i,ii) we may also assume that I is generated by f := (\overline{u}_{A}(β_{1}), …, \overline{u}_{A}(β_{k})). Set moreover γ_{•} := (p, τ_{A}(β_{1}), …, τ_{A}(β_{k})) and δ_{•} := (\underline{α}, τ_{A}(β_{1}), …, τ_{A}(β_{k})) (notation of (16.1)), and denote by ∂_{γ} (resp. ∂_{δ}) the W (E)- linear form W (E)^{⊕k+1} → W (E) such that ∂_{γ}(w_{0}, …, w_{k}) := pw_{0} + ∑_{i=1}^{k} τ_{A}(β_{i})w_{i} (resp. ∂_{γ}(w_{0}, …, w_{k}) := \underline{α}w_{0} + ∑_{i=1}^{k} τ_{A}(β_{i})w_{i}) for every (w_{0}, …, w_{k}) ∈ W (E)^{⊕k+1}. Let 𝒥 ⊂ W (E) be the ideal generated by the sequence γ_{•}, and set S := 1 + 𝒥; according to remark 16.1.10, there exists a S^{−1}W (E)-linear automorphism f of S^{−1}W (E)^{⊕k+1} with S^{−1}∂_{δ} = S^{−1}∂_{γ} ∘ f; the latter induces an isomorphism of differential graded S^{−1}W (E)-algebras :

(16.4.19)  K_{•}(γ_{•}) →^{∼} K_{•}(δ_{•})

(notation of (7.8)). Recall furthermore that p and \underline{α} are regular elements of W (E) (proposition 9.3.47(i)); there follow natural identifications in D(W (E)) :

K_{•}(p) →^{∼} W (E)/pW (E)[0] →^{∼} E[0] K_{•}(\underline{α}) →^{∼} W (E)/\underline{α}W (E)[0] →^{∼} A[0]

whence W (E)-linear isomorphisms :

H_{i}(γ_{•}, W (E)) →^{∼} H_{i}(f, A)  H_{i}(δ_{•}, W (E)) →^{∼} H_{i}(β_{•}, E)  for every i ∈ ℕ.

Notice that H_{i}(γ_{•}, W (E)) is a W (E)/𝒥 -module for every i ∈ ℕ (lemma 7.8.2(i)), hence the localization map H_{i}(γ_{•}, W (E)) → H_{i}(γ_{•}, S^{−1}W (E)) is an isomorphism, and likewise for H_{i}(δ_{•}, W (E)), since δ_{•} generates 𝒥 (lemma 16.1.9(ii)). Combining with (16.4.19), we then deduce a commutative diagram :

\xymatrix{H_{i}(γ_{•}, W (E)) \ar[r]^{∼} \ar[d] & H_{i}(δ_{•}, W (E)) \ar[d] \\ H_{i}(f, A) \ar[r]^{ω_{i,f}} & H_{i}(β_{•}, E)}  for every i ∈ ℕ

all whose arrows are isomorphisms of W (E)-modules. Next, denote by β′_{•} (resp. f′) the image of the sequence β_{•} in E′ (resp. of the sequence f in A′); we get likewise an isomorphism of W (E′)-modules ω_{i,f′} : H_{i}(f′, A′) →^{∼} H_{i}(β′_{•}, E′) for every i ∈ ℕ. Lastly, since α_{0} ∈ ℐ, the map \overline{u}_{A′} : E′ → A′ induces an isomorphism of W (E′)-algebras E′/ℐ E′ →^{∼} A′/IA′, and notice that H_{i}(f′, A′) (resp. H_{i}(β′_{•}, E′)) is an A′/IA′-module (resp. an E′/ℐ E′-module) for every i ∈ ℕ. By inspecting the constructions, we then obtain a commutative diagram of W (E′)-modules :

\xymatrix{A′/IA′ ⊗_{A} H_{i}(f, A) \ar[r]^{A′/IA′⊗_{A}ω_{i,f}} \ar[d] & E′/ℐ E′ ⊗_{E} H_{i}(β_{•}, E) \ar[d] \\ H_{i}(f′, A′) \ar[r]^{ω_{i,f′}} & H_{i}(β′_{•}, E′).}

The proposition now follows, by combining with proposition 14.2.49(i,iii). □

Remark 16.4.20. (i) In the situation of proposition 16.4.18, we also have that f : A → (A′, 𝒯_{I}) is adically faithfully flat if and only if the same holds for E(f) : E(A) → (E(A′), 𝒯_{𝒥} ). Indeed, notice that the closed immersion Spec A/I → Spec A/I^{n} is a homeomorphism for every n ∈ ℕ, and likewise for the map Spec E(A)/ℐ → Spec E(A)/ℐ^{n}. However, f is adically faithfully flat if and only if it is adically flat and the induced map h_{n} : Spec A′/I^{n}A′ → Spec A′/I^{n} is surjective for every n ∈ ℕ, and the latter condition then holds if and only if h_{1} is surjective. Likewise, E(f) is adically faithfully flat if and only if it is adically flat and the induced map k_{1} : Spec E(A′)/ℐ E(A′) → Spec E(A)/ℐ is surjective. Thus, we come down to checking that h_{1} is surjective if and only the same holds for k_{1}, but this is clear, because arguing as in the proof of proposition 16.4.18we may assume that \overline{u}_{A} and \overline{u}_{A′} induce isomorphisms E(A)/ℐ →^{∼} A/I and E(A′)/ℐ E(A′) →^{∼} A′/IA′, and these isomorphisms identify h_{1} with k_{1}.

(ii) For every perfectoid ring A, let A-Perf_{ad.fl} be the full subcategory of A-Perf whose objects are the adic and adically flat maps A → B of perfectoid rings. Proposition 16.4.18 shows that the functor E of remark 16.3.7(i) restricts to an equivalence of categories

A-Perf_{ad.fl} →^{∼} E(A)-Perf_{ad.fl}.

(iii) Let ℐ ⊂ E(A) be an ideal of adic definition, set E_{0} := E(A)/ℐ, and denote by 𝒜 (E_{0}) the category whose objects are all the flat ring homomorphisms f : E_{0} → C such that the following commutative diagram is cocartesian :

(16.4.21)  \xymatrix{E_{0} \ar[r]^{f} \ar[d]_{Φ_{E_{0}}} & C \ar[d]^{Φ_{C}} \\ E_{0} \ar[r]^{f} & C}

where the vertical arrows are the Frobenius endomorphisms. Now, let g : E(A) → E be any object of E(A)-Perf_{ad.fl}; we get a commutative diagram :

\xymatrix{E_{0} \ar[r]^{^{\overline{Φ}^{E(A)}}} \ar[d] & E(A)/ℐ^{p} \ar[r] \ar[d] & E_{0} \ar[d] \\ E/ℐ E \ar[r]^{_{\overline{Φ}_{E}}} & E/ℐ^{p}E \ar[r] & E/ℐ E}

where \overline{Φ}_{E(A)} (resp. \overline{Φ}_{E}) is induced by the Frobenius endomorphism of E(A) (resp. of E). Since E(A) and E are perfect 𝔽_{p}-algebras, we easily see that both square subdiagrams of this diagram are cocartesian, whence an object E_{0} ⊗_{E(A)} g : E_{0} → E/ℐ E of 𝒜 (E_{0}). We have thus a well defined functor :

(16.4.22)  E(A)-Perf_{ad.fl} → 𝒜 (E_{0}).

Proposition 16.4.23. The functor (16.4.22) is an equivalence of categories.

Proof. We construct an explicit quasi-inverse as follows. Let f : E_{0} → C be any object of 𝒜 (E_{0}), and set E_{n} := E(A)/ℐ^{n+1} for every n ∈ ℕ; we exhibit inductively a compatible system of flat ring homomorphisms (f_{n} : E_{n} → C_{n} | n ∈ ℕ) with isomorphisms

φ_{n} : C_{n}/ℐ^{n}C_{n} →^{∼} C_{n−1} such that φ_{n} ∘ (E_{n−1} ⊗_{E_{n}} f_{n}) = f_{n−1} for every n > 0

and such that the following commutative diagram is cocartesian :

𝒟_{n}  :  \xymatrix{E_{n} \ar[r]^{f_{n}} \ar[d]_{Φ_{En}} & C_{n} \ar[d]^{Φ_{Cn}} \\ E_{n} \ar[r]^{f_{n}} & C_{n}.}

Indeed, set C_{0} := C, and f_{0} := f. Suppose next that n ∈ ℕ, and that f_{n} and φ_{n} have already been given; according to [75, Lemma 6.5.13(i)], we have 𝕃_{C_{n}/E_{n}} = 0 in D(C-Mod), so the existence of a flat ring homomorphism f_{n+1} : E_{n+1} → C_{n+1} with the sought isomorphism φ_{n+1} is ensured by [75, Cor.3.2.11(i)]. Diagram 𝒟_{n+1} induces a map of E_{n+1}-algebras ψ_{n+1} : E_{n+1,Φ} ⊗_{E_{n+1}} C_{n+1} → C_{n+1}, where E_{n+1,Φ} denotes the ring E_{n+1}, regarded as a E_{n+1}-algebra via the map Φ_{E_{n+1}}. Since (16.4.21) is cocartesian, E_{1} ⊗_{E_{n+1}} ψ_{n+1} is an isomorphism, hence the same for ℐ^{k}/ℐ^{k+1} ⊗_{E_{n+1}} ψ_{n+1}, for every k = 0, …, n + 1; but since both C_{n+1} and D_{n+1} := E_{n+1,Φ} ⊗_{E_{n+1}} C_{n+1} are flat E_{n+1}-algebras, the latter map is naturally identified with the map ℐ^{k}D_{n+1}/ℐ^{k+1}D_{n+1} → ℐ^{k}C_{n+1}/ℐ^{k+1}C_{n+1} induced by ψ_{n+1} ([126, Th.22.3]). It follows easily that ψ_{n+1} is an isomorphism, i.e. 𝒟_{n+1} is cocartesian, as required, and this completes the construction of C_{n}. Let E := lim_{n∈ℕ} C_{n}; according to [59, Ch.0, Prop.7.2.7] (and its proof), E is complete and separated for its ℐ -adic topology, and the natural map E/ℐ^{n+1}E → C_{n} is an isomorphism for every n ∈ ℕ. Moreover, the limit of the system (f_{n} | n ∈ ℕ) yields an adic ring homomorphism g : E → E. Lastly, for every n ∈ ℕ we consider the commutative diagram

\xymatrix{E_{n} \ar[r]^{π_{E,n}} \ar[d]_{f_{n}} & E/Φ_{E}^{−1}(ℐ^{n+1}) \ar[r]^{^{\overline{Φ}^{En}}} \ar[d] & E_{n} \ar[d]^{f_{n}} \\ C_{n} \ar[r]^{π_{E,n}} & E/Φ_{E}^{−1}(ℐ^{n+1})E \ar[r]^{_{\overline{Φ}_{Cn}}} & C_{n}}

where π_{E,n} and π_{E,n} are the natural projections, and \overline{Φ}_{E_{n}} (resp. \overline{Φ}_{C_{n}}) is induced by the Frobenius endomorphism of E_{n} (resp. of C_{n}), so that the composition of the top (resp. bottom) horizontal arrows is Φ_{E_{n}} (resp. Φ_{C_{n}}). Since 𝒟_{n} is cocartesian, and since π_{E_{,}n} is surjective, it is easily seen that the right square subdiagram is cocartesian; but \overline{Φ}_{E_{n}} is an isomorphism, so the same holds for \overline{Φ}_{C_{n}}. On the other hand, (Φ_{E}^{−1}(ℐ^{n+1})E | n ∈ ℕ) is a fundamental system of open ideals of E; therefore, the limit of the system of maps (\overline{Φ}_{C_{n}} | n ∈ ℕ) is an automorphism of E; clearly the latter is the Frobenius automorphism, so E is a perfectoid 𝔽_{p}-algebra.

In order to check the functoriality of the rule : (f : E_{0} → C) ↦ (g : E(A) → E), consider any other system (f′_{n} : E_{n} → C′_{n} | n ∈ ℕ) of ring homomorphisms, with isomorphisms

φ′_{n} : C′_{n}/ℐ^{n}C′_{n} →^{∼} C′_{n−1} such that φ′_{n} ∘ (E_{n−1} ⊗_{E_{n}} f′_{n}) = f′_{n−1} for every n > 0

and let h_{0} : C = C_{0} → C′_{0} be a given map of E_{0}-algebras. We construct inductively a system h_{•} := (h_{n} : C_{n} → C′_{n} | n ∈ ℕ) where h_{n} is a map of E_{n}-algebras, and φ′_{n+1}∘(E_{n}⊗_{E_{n+1}}h_{n+1}) = h_{n} ∘ φ_{n+1} for every n ∈ ℕ. The map h_{0} is already given. Suppose now that n ∈ ℕ, and that h_{n} has already been exhibited. Notice, since f_{n+1} is flat, the projection C_{n+1} → C_{n} induces E_{n}-linear isomorphisms

ℐ E_{n+1} ⊗_{E_{n+1}} C_{n} →^{∼} ℐ E_{n+1} ⊗_{E_{n+1}} C_{n+1} →^{∼} ℐ C_{n+1}

whence a E_{n}-linear map

l_{n+1} : ℐ C_{n+1} →^{∼} ℐ E_{n+1} ⊗_{E_{n+1}} C_{n} → ℐ E_{n+1} ⊗_{E_{n+1}} C′_{n} →^{∼} ℐ E_{n+1} ⊗_{E_{n+1}} C′_{n+1} → ℐ C′_{n+1}.

Then, since 𝕃_{C_{n}/E_{n}} = 0, by [75, Prop.3.2.16] there exists a unique ring homomorphism h_{n+1} : C_{n+1} → C′_{n+1} that makes commute the diagram :

\xymatrix{0 \ar[r] & ℐ C_{n+1} \ar[r] \ar[d]_{l_{n+1}} & C_{n+1} \ar[r] \ar[d]^{h_{n+1}} & C_{n} \ar[r] \ar[d]^{h_{n}} & 0 \\ 0 \ar[r] & ℐ C′_{n+1} \ar[r] & C′_{n+1} \ar[r] & C′_{n} \ar[r] & 0}

and this completes the construction of the system h_{•}. The sought functoriality follows immediately. Furthermore, for any object g′ : E(A) → E′ of E(A)-Perf_{ad.fl}, set C_{0} := E′/ℐ E′, and let (E_{n} → C_{n} | n ∈ ℕ) be the system of ring homomorphisms associated to E_{0} ⊗_{E(A)} g′ by the foregoing inductive procedure; by the above, we obtain a compatible system of maps (h′_{n} : C_{n} → E′/ℐ^{n+1}E′ | n ∈ ℕ), and h′_{0} is an isomorphism. By a simple induction, we deduce that h′_{n} is an isomorphism for every n ∈ ℕ, whence an isomorphism of E(A)-algebras lim_{n∈ℕ} C_{n} →^{∼} E′; this shows that our functor is indeed a quasi-inverse for (16.4.22). □

16.4.24. Let A be a perfectoid ring, set E := E(A), and denote by \underline{α} := (α_{n} | n ∈ ℕ) any distinguished element of Ker u_{A}. Set I_{1} := Ann_{E}(α_{0}), let I_{2} ⊂ E be the radical of the ideal α_{0}E, and I_{3} := I_{1} + I_{2}; let also E_{i} := E/I_{i} for i = 1, 2, 3. We claim that the resulting diagram of ring homomorphisms

ℰ : \xymatrix{E \ar[r] \ar[d] & E_{1} \ar[d] \\ E_{2} \ar[r] & E_{3}}

is cartesian. Indeed, it is clear the induced map E → E_{1} ×_{E_{3}} E_{2} is surjective; for the injectivity, suppose that x ∈ I_{1} ∩ I_{2}; then x^{n} ∈ α_{0}A for some n ∈ ℕ, and α_{0}x = 0, so that x^{n+1} = 0, and therefore x = 0, since E is perfect. Notice also that the α_{0}E-adic topology agrees with the discrete topology on E_{2} and E_{3}, and by arguing as in remark 16.3.39(ii) we see that I_{1} is a closed ideal for the α_{0}-adic topology of E. Moreover, since E is perfect, it is easily seen that I_{1} is a radical ideal; then it follows that the same holds for I_{3}, since we have already observed that the latter equals I_{1} ⊕ I_{2}. Summing up, we deduce that E_{i}, endowed with its α_{0}-adic topology, is a perfectoid ring for i = 1, 2, 3. Consequently, the same holds for the rings

A_{i} := A ⊗_{𝕎(E)} W (E_{i}) i = 1, 2, 3

and E(A_{i}) = E_{i}, provided we endow each A_{i} with its p-adic topology (proposition 16.2.21(i)). Furthermore, arguing as in the proof of claim 16.3.22 we easily see that the resulting diagram of ring homomorphisms :

A(ℰ) : \xymatrix{A \ar[r] \ar[d] & A_{1} \ar[d] \\ A_{2} \ar[r] & A_{3}}

is still cartesian. Set as well

X_{E} := Spec E X_{A} := Spec A and X_{E_{k}} := Spec E_{k} X_{A_{k}} := Spec A_{k} for k = 1, 2, 3

and denote the resulting closed immersions by

i_{E_{k}} : X_{E_{k}} → X_{E}  i_{A_{k}} : X_{A_{k}} → X_{A}  for k = 1, 2, 3.

According to (16.1.3), we get moreover a continuous and spectral map

φ := (Spec \overline{u}_{A}) : X_{A} → X_{E}

Remark 16.4.25. In the situation of (16.4.24), we notice that :

(i) The image of α_{0} is regular on E_{1}, hence the induced isomorphism E →^{∼} E_{1} ×_{E_{3}} E_{2} identifies I_{1} with 0 ×_{E_{3}} E_{2}, i.e. with Ker (E_{2} → E_{3}).

(ii) Therefore, the image of \overline{u}_{A}(α_{0}) is regular in A_{1} (proposition 16.4.17), and consequently, also p is regular in A_{1}.

(iii) For i = 2, 3, the image of \underline{α} in W (E_{i}) equals pu, for some u ∈ W (E_{i})^{×}, so u_{A_{2}} and u_{A_{3}} induce isomorphisms of topological rings

E_{2} →^{∼} A_{2} E_{3} →^{∼} A_{3}

that identify the projection E_{2} → E_{1} in ℰ with the map A_{2} → A_{3} in A(ℰ).

(iv) Consequently, the isomorphism A →^{∼} A_{1} ×_{A_{3}} A_{2} induced by A(ℰ) identifies Ann_{A}(p) with 0 ×_{A_{3}} A_{2}, i.e. with Ker (A_{2} → A_{3}), and summing up, we get a natural W (E)-linear identification

Ann_{E}(α_{0}) →^{∼} Ann_{A}(p) : β ↦ \overline{u}_{A}(β).

(v) Let I be an ideal of adic definition of A, and J ⊂ A another ideal such that IJ = 0. Since A is reduced (corollary 16.3.63(i)), we deduce that I^{n} ∩ J = 0 for every n ∈ ℕ, whence J + I^{n} = J ⊕ I^{n} for every such n. It follows that

⋂_{n∈ℕ}(J + I^{n}) = ⋂_{n∈ℕ}(J ⊕ I^{n}) = J ⊕ ⋂_{n∈ℕ} I^{n} = J

which means that J is closed in the topology of A.

Proposition 16.4.26. With the notation of (16.4.24), the following holds :

(i) The map φ is surjective.

(ii) For every quasi-coherent 𝒪_{X_{A}}-module ℱ and every integer i > 0 we have

R^{i}φ_{∗}ℱ = 0.

Proof. (i): Consider any 𝔮 ∈ X_{E}, set

J := ∑_{x∈𝔮} \overline{u}_{A}(x) · A S := \{\overline{u}_{A}(x) | x ∈ E \ 𝔮\}

and notice that S is a multiplicative subset of A. Let 𝔭 ∈ X_{A} be any prime ideal; it is easily seen that φ(𝔭) = 𝔮 if and only if S ∩ 𝔭 = ∅ and J ⊂ 𝔭. Thus, we come down to showing that JA_{S} ≠ A_{S}. Suppose this fails; then there exist y ∈ E \ 𝔮 and sequences x_{•} := (x_{1}, …, x_{k}), a_{•} := (a_{1}, …, a_{k}) of elements of 𝔮 and respectively A, such that

\overline{u}_{A}(y) = ∑_{i=1}^{k} \overline{u}_{A}(x_{i}) · a_{i}.

Especially, [y]^{⟨1⟩}A ⊂ [x_{•}]^{⟨1⟩}A. Set 𝒥 := ∑_{i=1}^{k} Ex_{i} ⊂ E; by corollary 16.3.51(i) we deduce

y ∈ 𝒥^{⟨1−1/p⟩}E ⊂ 𝔮

a contradiction.

(ii): Let f ∈ E any element, and set [f] := \overline{u}_{A}(f); in light of (9.3.67) we get

(16.4.27)  φ^{−1}(Spec E_{f}) = Spec A_{[f]}.

Now, recall that R^{i}φ_{∗}ℱ is naturally isomorphic to the sheaf associated to the presheaf

U ↦ H^{i}(φ^{−1}U, ℱ) for every open subset U ⊂ X_{E}.

Then the assertion follows immediately from (16.4.27). □

16.4.28. In the situation of (16.4.24), let \underline{α} := (α_{n} | n ∈ ℕ) be any distinguished element of Ker u_{A}; notice the commutative diagram of topological spaces

\xymatrix{X_{A}^{0} := Spec A/pA \ar[r]^{φ_{0}} \ar[d]_{ι_{A}} & X_{E}^{0} := Spec E/α_{0}E \ar[d]^{ι_{E}} \\ X_{A} \ar[r]^{φ} & X_{E}}

whose vertical arrows are the inclusion maps, and with φ_{0} := Spec ω, where ω : E/α_{0}E →^{∼} A/pA is the ring isomorphism deduced from u_{A} ⊗_{ℤ} 𝔽_{p} as in remark 16.3.7(ii). Denote also

𝒪_{X_{E}}^{t} ⊂ 𝒪_{X_{E}}^{t} and 𝒪_{X_{A}}^{t} ⊂ 𝒪_{X_{A}}^{t}

respectively the subsheaf of α_{0}-torsion sections and the subsheaf of p-torsion sections. Notice that 𝒪_{X_{E}}^{t} is independent of the choice of \underline{α} (remark 16.1.7(i)); more precisely, corollary 16.3.63(i) implies that

𝒪_{X_{E}}^{t} = \underline{Γ}_{X_{E}^{0}}𝒪_{X_{E}} = Ker (𝒪_{X_{E}} → i_{E_{1}∗}𝒪_{X_{E_{1}}})
𝒪_{X_{A}}^{t} = \underline{Γ}_{X_{A}^{0}}𝒪_{X_{A}} = Ker (𝒪_{X_{A}} → i_{A_{1}∗}𝒪_{X_{A_{1}}}).

Lemma 16.4.29. In the situation of (16.4.28), the following holds :

(i) The corresponding isomorphism of sheaves φ_{0}^{♭} : 𝒪_{X_{E}^{0}} →^{∼} φ_{0∗}𝒪_{X_{A}^{0}} lifts to a map of

sheaves of monoids

φ^{♭} : 𝒪_{X_{E}} → φ_{∗}𝒪_{X_{A}}.

(ii) φ^{♭} restricts to an isomorphism of sheaves of abelian groups

𝒪_{X_{E}}^{t} →^{∼} φ_{∗}𝒪_{X_{A}}^{t}.

(iii) For every open subset U ⊂ X_{E} we have

φ_{U}^{♭}(x + y) = φ_{U}^{♭}(x) + φ_{U}^{♭}(y) for every x ∈ 𝒪_{X_{E}}^{t}(U) and y ∈ 𝒪_{X_{E}}(U) .

(iv) For every closed subset W_{A} ⊂ X_{A}^{0}, the 𝒪_{X_{A}}-module \underline{Γ}_{W_{A}}𝒪_{X_{A}} is quasi-coherent.

Proof. (i): For every f ∈ E, set D(f) := Spec E_{f} and [f] := \overline{u}_{A}(f), and let

φ_{f}^{♭} := (\overline{u}_{A})_{f} : E_{f} → A_{[f]}

(notation of (4.8.33)). Now, let g ∈ E be any other element, and suppose that D(g) ⊂ D(f); there follows a commutative diagram

\xymatrix{E \ar[d]_{_{\overline{u}_{A}}} \ar[r] & E_{f} \ar[d]_{φ_{f}^{♭}} \ar[r] & E_{g} \ar[d]^{φ_{g}^{♭}} \\ A \ar[r] & A_{[f]} \ar[r] & A_{[g]}}

whose horizontal arrows are the localization maps. Since all localizations are epimorphisms in the category of monoids, we deduce that the diagram commutes for every such f, g. Especially, φ_{f}^{♭} depends only on D(f) (and not on f). Then the rule D(f) ↦ φ_{f}^{♭} extends to a morphism of sheaves of monoids on X_{E} (see [59, Ch.0, §3.2.3]), and by construction the resulting diagram

\xymatrix{𝒪_{X_{E}} \ar[r]^{φ^{♭}} \ar[d] & φ_{∗}𝒪_{X_{A}} \ar[d] \\ ι_{E∗}𝒪_{X_{E}^{0}} \ar[r]^{ι_{E∗}(φ_{0}^{♭})} & ι_{E∗}φ_{0∗}𝒪_{X_{A}^{0}}}

commutes.

(iii): It suffices to check the stated identity for U of the form Spec E_{f}, where f ∈ E is any element. Then, in light of (16.4.30), we easily reduce to the case where x, y ∈ Im (E → E_{f}). Then, we are further reduced to checking that \overline{u}_{A}(x + y) = \overline{u}_{A}(x) + \overline{u}_{A}(y) for every x, y ∈ E such that α_{0} · x = 0. The latter follows easily from proposition 9.3.62 and lemma 16.2.7(iii) : details left to the reader.

(ii): Lemma 16.2.7(iii) easily implies that φ^{♭} send 𝒪_{X_{E}} into φ_{∗}𝒪_{X_{A}}. In light of (iii), it then suffices to check that the map φ_{f}^{♭} restricts to an isomorphism Ann_{E_{f}} (α_{0}) →^{∼} Ann_{A_{[f]}}(p) for every f ∈ E. However

(16.4.30)  Ann_{E_{f}} (α_{0}) = Ann_{E}(α_{0})_{f}  and Ann_{A_{[f]}}(p) = Ann_{A}(p)_{[f]}

so the assertion follows from remark 16.4.25(iv).

(iv): Set W_{E} := φ_{0}(W_{A}); from (ii) we see that φ^{♭} induces an isomorphism

\underline{Γ}_{W_{A}}𝒪_{X_{A}} = \underline{Γ}_{W_{A}} ∘ \underline{Γ}_{X_{A}^{0}}𝒪_{X_{A}} →^{∼} \underline{Γ}_{W_{A}}(ι_{A∗}φ_{0}^{−1}ι_{E}^{−1}\underline{Γ}_{X_{E}^{0}}𝒪_{X_{E}}) = ι_{A∗}φ_{0}^{−1}ι_{E}^{−1}\underline{Γ}_{W_{E}}𝒪_{X_{E}}

of 𝒪_{X_{A}}-modules (lemma 10.4.13(v)). Since the functor ι_{A∗} sends quasi-coherent 𝒪_{X_{A}^{0}}-modules to quasi-coherent 𝒪_{X_{A}}-modules ([59, Ch.I, Cor.9.2.2]), we are then reduced to checking that ι_{E}^{−1}\underline{Γ}_{W_{E}}𝒪_{X_{E}} is a quasi-coherent 𝒪_{X_{E}^{0}}-module. However, it was already remarked in (16.4.28) that \underline{Γ}_{W_{E}}𝒪_{X_{E}} = \underline{Γ}_{W_{E}}𝒪_{X_{E}}^{t} is an ι_{E∗}𝒪_{X_{E}^{0}}-module, so we are further reduced to checking that \underline{Γ}_{W_{E}}𝒪_{X_{E}}^{t} is a quasi-coherent 𝒪_{X_{E}}-module.

Thus, let f ∈ E be any element, and s ∈ \underline{Γ}_{W_{E}}𝒪_{X_{E}}^{t}(D(f)) any section; pick also a subset S ⊂ E such that U_{E} := X_{E} \ W_{E} = ⋃_{g∈S} D(g). By assumption, s_{|D(fg)} = 0 for every g ∈ S, and there exist n ∈ ℕ and t ∈ E such that s = f^{−n}t; it follows that for every g ∈ S there exists an integer m_{g} ∈ ℕ such that (fg)^{m_{g}}t = 0 in E. Since E is perfect, we deduce that fgt = 0 for every such g, hence (ft)_{|U_{E}} = 0, so finally s ∈ (Γ_{W_{E}}𝒪_{X_{E}}^{t})_{f}, whence the contention. □

Remark 16.4.31. (i) Keep the notation of lemma 16.4.29. Notice that the rule A ↦ (φ, φ^{♭}) is functorial in A. Namely, if g : A′ → A′′ is any continuous ring homomorphism of perfectoid rings, and (φ′, φ^{′♭}) (φ′′, φ^{′′♭}) the pairs attached to A′ and respectively A′′ as in lemma 16.4.29, then we have commutative diagrams of topological spaces and of sheaves of monoids :

\xymatrix{X_{A′′} \ar[r]^{φ′′} \ar[d]_{ψ_{A}} & X_{E′′} \ar[d]^{ψ_{E}} & 𝒪_{X_{E′}} \ar[r]^{ψ_{E}^{♭}} \ar[d]_{φ^{′♭}} & ψ_{E∗}𝒪_{X_{E′′}} \ar[d]^{ψ_{E∗}φ^{′′♭}} \\ X_{A′} \ar[r]^{φ′} & X_{E′} & φ′_{∗}𝒪_{X_{A′}} \ar[r]^{φ′_{∗}ψ_{A}^{♭}} & ψ_{E∗}φ′′_{∗}𝒪_{X_{A′′}} = φ′_{∗}ψ_{A∗}𝒪_{X_{A′′}}}

where X_{A′} := Spec A′, X_{E′} := Spec E(A′) and similarly for X_{A′′} and X_{E′′}, and ψ_{A} := Spec g, ψ_{E} := Spec E(g). The verification shall be left to the reader.

(ii) The short exact sequence of 𝒪_{X_{E}}-modules

ℰ : 0 → 𝒪_{X_{E}}^{t} → 𝒪_{X_{E}} → i_{E_{1}∗}𝒪_{X_{E_{1}}} → 0

induces a commutative ladder for every open subset Ω ⊂ X_{E} :

\xymatrix{0 \ar[r] & H^{0}(Ω, 𝒪_{X_{E}}^{t}) \ar[r] \ar[d]_{β^{t}} & H^{0}(Ω, 𝒪_{X_{E}}) \ar[r] \ar[d]^{β} & H^{0}(Ω, i_{E_{1}∗}𝒪_{X_{E_{1}}}) \ar[d]^{β_{1}} \ar[r] & H^{1}(Ω, 𝒪_{X_{E}}^{t}) \ar[d]^{γ} \\ 0 \ar[r] & H^{0}(φ^{−1}Ω, 𝒪_{X_{A}}^{t}) \ar[r] & H^{0}(φ^{−1}Ω, 𝒪_{X_{A}}) \ar[r] & H^{0}(φ^{−1}Ω, i_{A_{1}∗}𝒪_{X_{A_{1}}}) \ar[r] & H^{1}(φ^{−1}Ω, 𝒪_{X_{A}}^{t})}

whose rows are the initial terms of the long exact cohomology sequence attached to ℰ, and where β is induced by φ^{♭}. The map β_{1} is induced by i_{E_{1}∗}φ_{1}^{♭}, where φ_{1}^{♭} : 𝒪_{X_{E_{1}}} → φ_{1}^{∗}𝒪_{X_{A_{1}}} is the morphism associated with the perfectoid ring A_{1} and the induced continuous map φ_{1} : X_{E_{1}} → X_{A_{1}} as in lemma 16.4.29(i). Lastly, β^{t} and γ are deduced from the restriction 𝒪_{X_{E}}^{t} → φ_{∗}𝒪_{X_{A}}^{t} of φ^{♭}, which is an isomorphism of abelian sheaves (lemma 16.4.29(ii,iii)); furthermore, these subsheaves are supported on the closed subsets X_{E}^{0} and respectively X_{A}^{0}, and φ restricts to a homeomorphism φ_{0} : X_{E}^{0} →^{∼} X_{A}^{0}, so β^{t} and γ are isomorphisms of abelian groups. The commutativity of the left square subdiagram is clear, and that of the central one follows from (i). To show the commutativity of the right square subdiagram, pick any \overline{s} ∈ H^{0}(Ω, i_{E_{1}∗}𝒪_{X_{E}}), and find a covering Ω = ⋃_{i∈I} Ω_{i} consisting of open subsets, such that \overline{s}_{|Ω_{i}} is the image of some s_{i} ∈ H^{0}(Ω_{i}, 𝒪_{X_{E}}), for every i ∈ I. Set Ω_{ij} := Ω_{i} ∩ Ω_{j} for every i, j ∈ I, and let s_{ij} := s_{i|Ω_{ij}} − s_{j|Ω_{ij}}, and notice that s_{ij} ∈ H^{0}(Ω_{ij}, 𝒪_{X_{E}}^{t}) for every such i, j. Then the image of \overline{s} in H^{1}(Ω, 𝒪_{X_{E}}^{t}) is the class of the Čech cocycle (s_{ij} | i, j ∈ I), and the latter maps to the class of the cocycle (φ_{Ω_{ij}}^{♭}(s_{ij}) | i, j ∈ I) in H^{1}(φ^{−1}Ω, 𝒪_{X_{A}}^{t}). On the other hand, set \overline{t} := β(\overline{s}); for every i ∈ I, the restriction of \overline{t} to φ^{−1}Ω_{i} is the image of φ_{Ω_{i}}^{♭}(s_{i}) ∈ H^{0}(Ω_{i}, 𝒪_{X_{A}}), hence the image of \overline{t} in H^{1}(φ^{−1}Ω, 𝒪_{X_{A}}^{t}) is the class of the cocycle (t_{ij} | i ∈ I) with t_{ij} := t_{i|φ^{−1}Ω_{ij}} − t_{j|φ^{−1}Ω_{ij}} for every i, j ∈ I. Lemma 16.4.29(iii) implies that t_{ij} = φ_{Ω_{ij}}^{♭}(s_{ij}) for every i, j ∈ I, whence the assertion.

The following result shows that if A is a perfectoid ring, proposition 8.3.30 holds even in case the open subset U of Spec A is not quasi-compact.

Lemma 16.4.32. Let A be a perfectoid ring, and U ⊂ X_{A} := Spec A an open subset containing the analytic locus of X_{A}. We have :

(i) There exists a unique topology 𝒯_{U} on A_{U} := 𝒪_{X_{A}}(U) such that (A_{U}, 𝒯_{U}) is f-adic and the restriction map ρ_{U} : A → A_{U} is open.

(ii) (A_{U}, 𝒯_{U}) is complete and separated.

(iii) Let f : A → B be any morphism of f-adic topological rings, such that the image of Spec f lies in U. Then the resulting map f_{U} : (A_{U}, 𝒯_{U}) → B is continuous.

Proof. (i): We pick a system of elements (β_{1}, …, β_{r}) of E(A) such that the system (f_{i} := \overline{u}_{A}(β_{i}) | i = 1, …, r) generates an ideal I of adic definition of A. Like in the proof of proposition 8.3.30(i), there exists a unique topology 𝒯_{U} on A_{U} such that (A_{U}, 𝒯_{U}) is a topological group for its additive group structure, and we need to check the following. For every a ∈ A_{U} and every n ∈ ℕ there exists k ∈ ℕ such that

a · ρ_{U}(I^{k}) ⊂ ρ_{U}(I^{n}).

Now, for i = 1, …, r, let a_{i} ∈ Spec A_{f_{i}} be the image of a under the restriction map ρ_{i} : A_{U} → A_{f_{i}}; we may find m ∈ ℕ and b_{1}, …, b_{r} ∈ A such that a_{i} = f_{i}^{−m}b_{i}, hence f_{i}^{m}a − b_{i} ∈ Ker ρ_{i} for i = 1, …, r. Pick a family (g_{λ} | λ ∈ Λ) of elements of A such that U = ⋃_{λ∈Λ} V_{λ}, where V_{λ} := Spec A_{g_{λ}} for every λ ∈ Λ. Notice that (f_{i}^{m}a − b_{i})_{|V_{λ}} lies in the kernel of the restriction map 𝒪_{U}(V_{λ}) → 𝒪_{U}(V_{λ} ∩ Spec A_{f_{i}}), for every i = 1, …, r and every λ ∈ Λ. Thus, for every such λ and i there exist t_{λ}, s_{λ} ∈ ℕ and c_{i,λ} ∈ A such that

g_{λ}^{s_{λ}} · (f_{i}^{m}a − b_{i})_{|V_{λ}} = ρ_{λ}(c_{i,λ})  and f_{i}^{t_{λ}}c_{i,λ} = 0  in A

where ρ_{λ} : A → A_{g_{λ}} is the localization map. From corollary 16.3.63(i) we then deduce that f_{i}c_{i,λ} = 0, so

(f_{i}^{m+1}a − f_{i}b_{i})_{|V_{λ}} = 0 for every λ ∈ Λ and every i = 1, …, r

i.e. f_{i}^{m+1}a = ρ_{U}(f_{i}b_{i}) for every i = 1, …, r. From this, we may argue as in the proof of proposition 8.3.30(i), to derive the assertion.

(ii): It suffices to show that Ker ρ_{U} is a closed ideal of A. However, clearly I · Ker ρ_{U} = 0, so the assertion follows from remark 16.4.25(v).

(iii) is proven as proposition 8.3.30(iii). □

We wish next to show that the formula for the Teichmüller representative of a sum obtained in proposition 9.3.62, is still valid in f-adic rings of the type considered in lemma 16.4.32.

16.4.33. Indeed, resume the situation of (16.4.28), and let U ⊂X_{E} be an open subset containing X_{E} \ Spec E/E^{∘∘}; from (16.4.27) we easily see that X_{A} \ Spec A/A^{∘∘} ⊂ φ^{−1}U, so

E_{U} := 𝒪_{X_{E}}(U) and A_{U} := 𝒪_{X_{A}}(φ^{−1}U)

admit the f-adic topologies 𝒯_{U} and respectively 𝒯_{φ^{−1}U} furnished by lemma 16.4.32.

Proposition 16.4.34. With the notation of (16.4.33), the following holds :

(i) The map φ_{U}^{♭} : (E_{U}, 𝒯_{U}) → (A_{U}, 𝒯_{φ^{−1}U}) of lemma 16.4.29(i) is continuous.

(ii) For every \underline{x} := (x_{0}, …, x_{k}) ∈ E_{U}^{k+1} we have the identity

φ_{U}^{♭}(x_{0} + ⋯ + x_{k}) = ∑_{n∈ℕ} p^{n} · ∑_{σ∈Σ_{n}^{(k)}} c_{σ} · φ_{U}^{♭}(\underline{x}^{σ})

where Σ_{n}^{(k)} is defined as in (9.3.61) for every n ∈ ℕ, and c_{σ} ∈ ℤ_{p} is provided by proposition 9.3.62, for every σ ∈ ⋃_{n∈ℕ} Σ_{n}^{(k)}; the convergence of the series is relative to the topology 𝒯_{φ^{−1}U}.

Proof. (Recall that \underline{x}^{σ} := x_{0}^{σ_{0}} ⋯ x_{k}^{σ_{k}} for every σ := (σ_{0}, …, σ_{k}) ∈ ℕ[1/p]^{k+1}).

(ii): Let (α_{n} | n ∈ ℕ) be any distinguished element in Ker u_{A}; since α_{0} is topologically nilpotent in E, its image is topologically nilpotent in E_{U}, hence there exists c ∈ ℕ such that α_{0}^{c} · x_{i} ∈ Im(E → E_{U}) for i = 0, …, k. In light of lemma 16.2.7(iii), we deduce that

y_{n}(\underline{x}) := ∑_{σ∈Σ_{n}^{(k)}} p^{n} · c_{σ} · φ_{U}^{♭}(x^{σ}) ∈ p^{n−c} · Im(A → A_{U}) for every integer n ≥ c.

Taking into account lemma 16.4.32(ii), it follows that the series ∑_{n∈ℕ} y_{n}(\underline{x}) converges for the topology 𝒯_{φ^{−1}U} to a well determined element y ∈ A_{U}, and it remains only to check that y = φ_{U}^{♭}(x_{0} + ⋯ + x_{k}). To this aim, pick a family (g_{λ} | λ ∈ Λ) of elements of E, such that U = ⋃_{λ∈Λ} V_{λ}, where V_{λ} := Spec E_{g_{λ}} for every λ ∈ Λ. Then, for every such λ we may find n_{λ} ∈ ℕ such that (g_{λ}^{n_{λ}} · x_{i})_{|V_{λ}} ∈ Im(E → E_{g_{λ}}) for i = 0, …, k, and it suffices to check that ([g_{λ}^{n_{λ}}] · φ_{U}^{♭}(x_{0} + ⋯ + x_{k}))_{|φ^{−1}V_{λ}} = ([g_{λ}^{n_{λ}}] · y)_{|φ^{−1}V_{λ}}, where [g_{λ}^{n_{λ}}] := \overline{u}_{A}(g_{λ}^{n_{λ}}) for every such λ. However, we have

[g_{λ}^{n_{λ}}] · φ_{U}^{♭}(x_{0} + ⋯ + x_{k}) = φ_{U}^{♭}(g_{λ}^{n_{λ}} · x_{0} + ⋯ + g_{λ}^{n_{λ}} · x_{k}) and [g_{λ}^{n_{λ}}] · y = ∑_{n∈ℕ} y_{n}(g_{λ}^{n_{λ}} · \underline{x})

where g_{λ}^{n_{λ}} · \underline{x} := (g_{λ}^{n_{λ}} · x_{0}, …, g_{λ}^{n_{λ}} · x_{k}). Hence, we may replace \underline{x} by g_{λ}^{n_{λ}} · \underline{x}, after which we may assume that x_{i} ∈ Im(ρ_{U} : E → E_{U}) for i = 0, …, k. In this case, say that x_{i} = ρ_{U}(z_{i}) for i = 0, …, k and for certain z_{0}, …, z_{k} ∈ E; we have

φ_{U}^{♭}(x_{0} + ⋯ + x_{k}) = ρ_{φ^{−1}U} ∘ φ_{X_{E}}^{♭}(z_{0} + ⋯ + z_{k}) = ρ_{φ^{−1}U} ∘ \overline{u}_{A}(z_{0} + ⋯ + z_{k})

where ρ_{φ^{−1}U} : A → A_{U} is the corresponding restriction map; on the other hand, since ρ_{φ^{−1}U} is continuous for the topology 𝒯_{φ^{−1}U}, we have as well

\begin{matrix}y = & ∑_{n∈ℕ} p^{n} · ∑_{σ∈Σ_{n}^{(k)}} c_{σ} · ρ_{φ^{−1}U} ∘ φ_{X_{E}}^{♭}(\underline{z}^{σ}) = ρ_{φ^{−1}U}(∑_{n∈ℕ} p^{n} · ∑_{σ∈Σ_{n}^{(k)}} c_{σ} · φ_{X_{E}}^{♭}(\underline{z}^{σ})) \\ = & ρ_{φ^{−1}U}(∑_{n∈ℕ} p^{n} · ∑_{σ∈Σ_{n}^{(k)}} c_{σ} · \overline{u}_{A}(\underline{z}^{σ})) = ρ_{φ^{−1}U} ∘ u_{A}(∑_{n∈ℕ} p^{n} · ∑_{σ∈Σ_{n}^{(k)}} c_{σ} · τ_{A}(\underline{z}^{σ})) \end{matrix}

so the sought identity follows from proposition 9.3.62.

(i): Fix x_{0} ∈ E_{U}; the proof of (ii) shows that for every k ∈ ℕ there exists n ∈ ℕ such that y_{r}(x_{0} + x) ∈ p^{k} · Im (A → A_{U}) for every r ≥ n and every x ∈ Im (E → E_{U}). On the other hand, clearly the mappings y_{i} are continuous for every i ∈ ℕ, relative to the topologies 𝒯_{U} and 𝒯_{φ^{−1}U}. The assertion follows immediately. □

16.4.35. Keep the notation of (16.4.28), let U_{E} ⊂ X_{E} be any open subset, and set U_{A} := φ^{−1}U_{E}, Z_{E} := X_{E} \ U_{E}, Z_{A} := X_{A} \ U_{A}. Denote by j_{E} : U_{E} → X_{E} and j_{A} : U_{A} → X_{A} the open immersions; the map φ^{♭} of lemma 16.4.29(i) induces a morphism of sheaves of monoids

ψ : j_{E∗}𝒪_{U_{E}} ⟶^{j_{E∗}j_{E}^{∗}(φ^{♭})} j_{E∗}j_{E}^{∗}φ_{∗}𝒪_{X_{A}} ⟶^{∼} φ_{∗}j_{A∗}𝒪_{U_{A}}.

Moreover, let

\overline{𝒪} _{X_{E}} := Im (𝒪_{X_{E}} → j_{E∗}𝒪_{U_{E}}) \overline{𝒪} _{X_{A}} := Im (𝒪_{X_{A}} → j_{A∗}𝒪_{U_{A}}).

Denote by π_{E} : j_{E∗}𝒪_{U_{E}} → R^{1}\underline{Γ}_{Z_{E}}𝒪_{X_{E}} and π_{A} : j_{A∗}𝒪_{U_{A}} → R^{1}\underline{Γ}_{Z_{A}}𝒪_{X_{A}} the projections, set q := \overline{u}_{A}(α_{0}), and for every c ∈ ℕ[1/p] define

α_{0}^{−c}𝒪_{X_{E}} := Ker α_{0}^{c} · π_{E}  p^{−c}𝒪_{X_{A}} := Ker q^{c} · π_{A}
𝒬_{U_{E}}^{c} := π_{E}(α_{0}^{−c}𝒪_{X_{E}})  𝒬_{U_{A}}^{c} := π_{A}(p^{−c}𝒪_{X_{A}}).

Also, set

ℛ^{c} := Coker (φ_{∗}\overline{𝒪} _{X_{A}} → φ_{∗}(p^{−c}𝒪_{X_{A}})) and let λ : ℛ^{c} → φ_{∗}𝒬_{U_{A}}^{c}

be the natural morphism of 𝒪_{X_{A}}-modules; lemma 16.2.7(iii) easily implies that ψ restricts to a morphism of sheaves of sets

ψ^{c} : α_{0}^{−c}𝒪_{X_{E}} → φ_{∗}(p^{−c}𝒪_{X_{A}}) for every c ∈ ℕ[1/p].

Proposition 16.4.36. With the notation of (16.4.35), suppose moreover that :

(a) Spec E[α_{0}^{−1}] ⊂ U_{E}.

(b) U_{E} is quasi-compact.

(c) c ≤ p/(p − 1). Then we have :

(i) The map ψ^{c} descends to a morphism of sheaves of sets

\overline{ψ}^{c} : 𝒬_{U_{E}}^{c} → ℛ^{c} ⟶^{λ} φ_{∗}𝒬_{U_{A}}^{c}.

(ii) There exist unique morphisms of sheaves of sets

\overline{p}_{E}^{c} : 𝒬_{U_{E}}^{c/p} → 𝒬_{U_{E}}^{c}  and  \overline{p}_{A}^{c} : 𝒬_{U_{A}}^{c/p} → 𝒬_{U_{A}}^{c}

fitting into commutative diagrams

\xymatrix{α_{0}^{−c/p}𝒪_{X_{E}} \ar[r] \ar[d] & α_{0}^{−c}𝒪_{X_{E}} \ar[d] & p^{−c/p}𝒪_{X_{A}} \ar[r] \ar[d] & p^{−c}𝒪_{X_{A}} \ar[d] \\ 𝒬_{U_{E}}^{c/p} \ar[r]^{_{\overline{p}_{c}_{E}}} & 𝒬_{U_{E}}^{c} & 𝒬_{U_{A}}^{c/p} \ar[r]^{_{\overline{p}_{c}_{A}}} & 𝒬_{U_{A}}^{c}}

whose vertical arrows are the projections, and whose top horizontal arrows are the restrictions of the p-Frobenius endomorphisms of the sheaves of monoids j_{E∗}𝒪_{U_{E}} and respectively j_{A∗}𝒪_{U_{A}} (with composition law given by multiplication).

(iii) Moreover, we have a commutative diagram of sheaves of sets :

\xymatrix{𝒬_{U_{E}}^{c/p} \ar[r]^{^{\overline{p}^{c}^{E}}} \ar[d]_{_{\overline{ψ}_{c/p}}} & 𝒬_{U_{E}}^{c} \ar[d]^{_{\overline{ψ}_{c}}} \\ φ_{∗}𝒬_{U_{A}}^{c/p} \ar[r]^{φ_{∗}(\overline{p}_{A}^{c})} & φ_{∗}𝒬_{U_{A}}^{c}.}

(iv) If c ≤ 1, the map \overline{ψ}^{c} is an isomorphism of sheaves of abelian groups.

Proof. (i): Let f ∈ E be any element, and set V := Spec E_{f}; we easily come down to showing

Claim 16.4.37. If c ≤ p/(p − 1), we have

ψ_{V}^{c} (x + y) − ψ_{V}^{c} (x) ∈ \overline{𝒪} _{X_{A}}(φ^{−1}V) for every x ∈ α_{0}^{−c}𝒪_{X_{E}}(V) and y ∈ \overline{𝒪} _{X_{E}}(V) .

Proof of the claim. Since U_{E} is quasi-compact, j_{E∗}𝒪_{X_{E}} is a quasi-coherent 𝒪_{X_{E}}-module ([59, Ch.I, Cor.9.2.2]), hence the same holds for α_{0}^{−c}𝒪_{X_{E}}, and consequently

α_{0}^{−c}𝒪_{X_{E}}(V) = α_{0}^{−c}𝒪_{X_{E}}(X_{E})_{f}.

We may therefore find n ∈ ℕ such that f^{n}x (resp. f^{n}y) is the restriction to V of a global section x′ ∈ α_{0}^{−c}𝒪_{X_{E}}(X_{E}) (resp. of an element y′ ∈ E), and it suffices to show that

ψ_{X_{E}}^{c}(x′ + y′) − ψ_{X_{E}}^{c}(x′) ∈ Im (A → j_{A∗}𝒪_{U_{A}}(X_{A}) = 𝒪_{X_{A}}(U_{A})).

Hence, we may replace V by X_{E}, and assume from start that

x ∈ 𝒪_{X_{E}}(U_{E}) and α_{0}^{c} · x, y ∈ Im (E → 𝒪_{X_{E}}(U_{E}))

and we have to show that φ_{U_{E}}^{♭}(x+y)−φ_{U_{E}}^{♭}(x) ∈ Im (A → 𝒪_{X_{A}}(U_{A})). Notice that the assertion is independent of the topology 𝒯 of A such that (A, 𝒯 ) is perfectoid; by virtue of proposition 16.3.8(i) we may then assume that 𝒯 is the p-adic topology on A, and therefore the topology of E agrees with the α_{0}-adic topology. In this case, assumption (a), proposition 16.4.34(ii) and lemma 16.2.7(iii) imply that φ_{U_{E}}^{♭}(x + y) − φ_{U_{E}}^{♭}(x) − φ_{U_{E}}^{♭}(y) is a p-adically convergent series ∑_{n>0} z_{n}, where z_{n} is a ℤ_{p}-linear combination of terms of the form φ_{U_{E}}^{♭}(α_{0}^{n} · x^{σ} · y^{1−σ}), with 1 − σ, σ ∈ p^{−n}ℕ \ p^{1−n}ℕ, for every n ∈ ℕ. Hence, it suffices to check that n ≥ p · σ/(p − 1) for every such σ and every integer n > 0. We leave the easy verification to the reader. ♢

(ii): The uniqueness of \overline{p}_{U_{A}}^{c} is clear. For the existence, let V be as in the foregoing; we come down to showing that

(x + y)^{p} − x^{p} ∈ \overline{𝒪} _{X_{A}}(V) for every x ∈ p^{−c/p}𝒪_{X_{A}}(V) and every y ∈ \overline{𝒪} _{X_{A}}(V) .

However, (x+y)^{p} −x^{p} = y^{p} +∑_{i=1}^{p−1} \binom{p}{i}·x^{i}y^{p−i}, and px^{i} ∈ \overline{𝒪} _{X_{A}}(V) for every i = 1, …, p−1, since c ≤ p/(p − 1), whence the contention. Assertion (iii) shall be left to the reader.

(iv): Let V be as in the foregoing; in order to prove that \overline{ψ}^{c} is a morphism of abelian sheaves when c ≤ 1, we come down to showing that

ψ_{V}^{1} (x + y) − ψ_{V}^{1} (x) − ψ_{V}^{1} (y) ∈ \overline{𝒪} _{X_{A}}(φ^{−1}V) for every x, y ∈ α_{0}^{−1}𝒪_{X_{E}}(V)

and arguing as in the proof of claim 16.4.37, we reduce to the case where V = X_{E} and x, y are sections of 𝒪_{X_{E}}(U_{E}) such that α_{0}x, α_{0}y ∈ Im (E → 𝒪_{X_{E}}(U_{E})). Then, it suffices to check that

φ_{U_{E}}^{♭}(α_{0}^{n} · y^{σ} · x^{1−σ}) ∈ Im(A → 𝒪_{X_{A}}(U_{A})) whenever n > 0 and 1 − σ, σ ∈ ℕ[1/p]

which is clear. Next notice that, by remark 16.4.31, the pair (φ, φ^{♭}) restricts to isomorphisms of schemes

(φ_{2}, φ_{2}^{♭}) : X_{A_{2}} →^{∼} X_{E_{2}}  (φ_{3}, φ_{3}^{♭}) : X_{A_{3}} →^{∼} X_{E_{3}}  such that φ_{2}(Z_{A}) = Z_{E}.

Claim 16.4.38. Assertion (iv) holds for U_{E} = Spec E[α_{0}^{−1}].

Proof of the claim. We have commutative diagrams

\xymatrix{U_{E_{1}} := Spec E_{1}[α_{0}^{−1}] \ar[r] \ar[d]_{j_{E_{1}}} & U_{E} \ar[d]^{j_{E}} & U_{A_{1}} := Spec A_{1}[p^{−1}] \ar[r] \ar[d]_{j_{A_{1}}} & U_{A} \ar[d]^{j_{A}} \\ X_{E_{1}} \ar[r]^{i_{E_{1}}} & X_{E} & X_{A_{1}} \ar[r]^{i_{A_{1}}} & X_{A}}

whose top horizontal arrows are isomorphisms of schemes, whence commutative diagrams of abelian sheaves

\xymatrix{\overline{𝒪} _{X_{E}} \ar[r]^{∼} \ar[d] & i_{E_{1}∗}𝒪_{X_{E_{1}}} \ar[d] & \overline{𝒪} _{X_{A}} \ar[r]^{∼} \ar[d] & i_{A_{1}∗}𝒪_{X_{A_{1}}} \ar[d] \\ j_{E∗}𝒪_{U_{E}} \ar[r]^{∼} & i_{E_{1}∗} ∘ j_{E_{1}∗}𝒪_{U_{E_{1}}} & j_{A∗}𝒪_{U_{A}} \ar[r]^{∼} & i_{A_{1}∗} ∘ j_{A_{1}∗}𝒪_{U_{A_{1}}}}

whose left (resp. right) vertical arrows are the inclusion maps (resp. are induced by the inclusion maps 𝒪_{X_{E_{1}}} → j_{E_{1}∗}𝒪_{U_{E_{1}}} and 𝒪_{X_{A_{1}}} → j_{A_{1}∗}𝒪_{U_{A_{1}}}). Moreover, denote by φ_{1} : X_{A_{1}} → X_{E_{1}} the continuous map associated with A_{1}, as in (16.4.24), and by

ψ_{1} : j_{E_{1}∗}𝒪_{U_{E_{1}}} → φ_{1∗}j_{A_{1}∗}𝒪_{U_{A_{1}}} (resp. \overline{ψ}_{1}^{c} : 𝒬_{U_{E_{1}}}^{c} → φ_{1∗}𝒬_{U_{A_{1}}}^{c} )

the corresponding morphism of sheaves of monoids as in (16.4.35) (resp. morphism of sheaves of sets, as in (i)); then i_{E_{1}} ∘ φ_{1} = φ ∘ i_{A_{1}} (remark 16.4.31), and the foregoing isomorphisms identify ψ with i_{E_{1}∗}ψ_{1}. So finally, \overline{ψ}^{c} is identified with i_{E_{1}∗}\overline{ψ}_{1}^{c} for every c ≤ p/(p − 1), and we may replace from start A by A_{1}, after which, we may assume that α_{0} (resp. p) is regular in E (resp. in A). In this case, for every c ≤ 1 we have a commutative diagram of sheaves :

\xymatrix{𝒬_{U_{E}}^{c} \ar[r]^{^{\overline{ψ}^{c}^{1}}} \ar[d] & φ_{∗}𝒬_{U_{A}}^{c} \ar[d] \\ 𝒪_{X_{E}}/α_{0}^{c}𝒪_{X_{E}} \ar[r] & φ_{∗}(𝒪_{X_{A}}/p^{c}𝒪_{X_{A}})}

whose left (resp. right) vertical arrow is the isomorphism induced by scalar multiplication by α_{0}^{c} (resp. by q^{c}) and whose bottom horizontal arrow is the isomorphism induced by φ_{0}^{♭} (notation of (16.4.28)). The claim follows. ♢

Claim 16.4.39. We may assume that the natural maps 𝒪_{X_{E}} → \overline{𝒪} _{X_{E}} and 𝒪_{X_{A}} → \overline{𝒪} _{X_{A}} are isomorphisms.

Proof of the claim. Assumption (a) implies that Z_{E} ⊂ X_{E_{2}} and Z_{A} ⊂ X_{A_{2}}, therefore

𝒥_{E} := Ker (𝒪_{X_{E}} → \overline{𝒪}_{X_{E}}) = \underline{Γ}_{Z_{E}}𝒪_{X_{E}}^{t} 𝒥_{A} := Ker (𝒪_{X_{A}} → \overline{𝒪}_{X_{A}}) = \underline{Γ}_{Z_{A}}𝒪_{X_{A}}^{t}

(notation of (16.4.28)), and lemma 16.4.29(ii,iii) implies that φ^{♭} restricts to an isomorphism of abelian sheaves

(16.4.40)  𝒥_{E} →^{∼} φ_{∗}𝒥_{A}.

Notice that J := Γ(X_{E}, 𝒥_{E}) is an ideal of both E and E_{2}, and moreover Φ_{E}(J) = J. Indeed, J consists of all x ∈ E whose images vanish in E_{f}, for every f ∈ E such that Spec E_{f} ⊂ U_{E}; since E_{f} is still a perfect 𝔽_{p}-algebra, it follows easily that x ∈ J if and only if x^{p} ∈ J, whence the assertion. Thus, both \overline{E} := E/J and \overline{E}_{2} := E_{2}/J are still perfect 𝔽_{p}-algebras, and are even perfectoid, with the quotient topologies induced by the projections E → \overline{E} and E_{2} → \overline{E}_{2}. Moreover, it is easily seen that the diagram ℰ of (16.4.24) induces a natural identification

\overline{E} →^{∼} E_{1} ×_{E_{3}} \overline{E}_{2}.

Let h_{E} : X\overline{E} := Spec \overline{E} → X_{E} be the closed immersion, j\overline{E} : U\overline{E} := U_{E} ∩ X\overline{E} → X\overline{E} the open immersion, and consider the corresponding quasi-coherent 𝒪_{X\overline{E}}-module 𝒬_{U\overline{E}}^{c} as in (16.4.35). Since U_{E} is quasi-compact, 𝒥_{E} is a quasi-coherent 𝒪_{X_{E}}-module (see (10.4.16)), so 𝒥_{E} = Ker (𝒪_{X_{E}} → h_{E∗}𝒪_{X\overline{E}}); it follows that the induced map 𝒪_{X\overline{E}} → j_{\overline{E}∗}𝒪_{U\overline{E}} is a monomorphism, and h_{E} restricts to an isomorphism of open subschemes U\overline{E} →^{∼} U_{E}, so the resulting map j_{E∗}𝒪_{U_{E}} → h_{E∗}j_{\overline{E}∗}𝒪_{U\overline{E}} is an isomorphism, and we get a natural identification

ω_{E} : 𝒬_{U_{E}}^{c} →^{∼} h_{E∗}𝒬_{U\overline{E}}^{c}.

Next, consider the cartesian diagram A(ℰ) of (16.4.24), and recall that the bottom horizontal arrow A_{2} → A_{3} of A(ℰ) is naturally isomorphic to the bottom horizontal arrow E_{2} → E_{3} of ℰ (remark 16.4.24(iii)). There follows a commutative diagram

\xymatrix{A \ar[r] \ar[d] & \overline{A} \ar[r]^{^{\overline{π}^{1}}} \ar[d]_{_{\overline{π}_{2}}} & A_{1} \ar[d] \\ E_{2} \ar[r] & _{\overline{E}_{2}} \ar[r] & E_{3}}

whose two square subdiagrams are cartesian. Especially, J is naturally identified with the kernel of the projection A → \overline{A}; moreover, we have an isomorphism of topological rings

E(\overline{A}) →^{∼} \overline{E}

that identifies E(\overline{π}_{1}) and E(\overline{π}_{2}) with the projections \overline{E} → E_{1} and \overline{E} → \overline{E}_{2}, and furthermore, \overline{A} is perfectoid (proposition 16.3.25). Let also

h_{A} : X\overline{A} := Spec \overline{A} → X_{A} and j\overline{A} : U\overline{A} := U_{A} ∩ X\overline{A} → X\overline{A}

be respectively the closed and the open immersion, and consider the corresponding quasicoherent 𝒪_{X\overline{A}}-module 𝒬_{U\overline{A}}^{c} defined as in (16.4.35); in light of (16.4.40), we see that 𝒥_{A} = Ker (𝒪_{X_{A}} → h_{A∗}𝒪_{X\overline{A}}), and h_{A} restricts to an isomorphism of open subschemes U\overline{A} →^{∼} U_{A}, so the resulting map j_{A∗}𝒪_{U_{A}} → h_{A∗}j_{\overline{A}∗}𝒪_{U\overline{A}} is an isomorphism, and we get a natural identification

ω_{A} : 𝒬_{U_{A}}^{c} →^{∼} h_{A∗}𝒬_{U\overline{A}}^{c}.

Lastly, let \overline{φ} : X\overline{A} → X\overline{E} be the continuous map provided by (16.4.24), and \overline{ψ}c ; \overline{A} : 𝒬_{U\overline{E}}^{c} → \overline{φ}_{∗}𝒬_{U\overline{A}}^{c} the morphism of abelian sheaves attached to \overline{A} and U\overline{E} as in (i); a direct inspection yields a commutative diagram

\xymatrix{𝒬_{U_{E}}^{c} \ar[r]^{^{\overline{ψ}^{c}}} \ar[d]_{ω_{E}} & φ_{∗}𝒬_{U_{A}}^{c} \ar[d]^{φ_{∗}ω_{A}} \\ h_{E∗}𝒬_{U\overline{E}}^{c} \ar[r]^{h_{E∗}\overline{ψ}c ; \overline{A}} & h_{E∗}\overline{φ}_{∗}𝒬_{U\overline{A}}^{c} = φ_{∗}h_{A∗}𝒬_{U\overline{A}}^{c}}

whence the claim. ♢

Henceforth we assume that 𝒪_{X_{E}} = \overline{𝒪} _{X_{E}} an 𝒪_{X_{A}} = \overline{𝒪} _{X_{A}}. Set U′_{E} := Spec E[α_{0}^{−1}], so that U′_{A} := φ^{−1}U′_{E} is the open subset Spec A[p^{−1}] of X_{A}, and let j′_{E} : U′_{E} → X_{E} and j′_{A} : U′_{A} → X_{A} be the open immersions. Set also Z′_{E} := X_{E} \ U′_{E} and Z′_{A} := X_{A} \ U′_{A}. Notice that

i_{E_{1}∗}𝒪_{X_{E_{1}}} = Im(𝒪_{X_{E}} → j′_{E∗}𝒪_{U′_{E}}) and i_{A_{1}∗}𝒪_{X_{A_{1}}} = Im(𝒪_{X_{A}} → j′_{A∗}𝒪_{U′_{A}})

and denote by \overline{ψ}^{′c} : 𝒬_{U′_{E}}^{c} → 𝒬_{U′_{A}}^{c} the map of abelian sheaves associated as in (i) with A and U′_{E}. Since U′_{E} ⊂ U_{E}, the natural maps j_{E∗}𝒪_{U_{E}} → j′_{E∗}𝒪_{U′_{E}} and j_{A∗}𝒪_{U_{A}} → j′_{A∗}𝒪_{U′_{A}} induce morphisms ρ_{E} : R^{1}\underline{Γ}_{Z_{E}}𝒪_{X_{E}} → R^{1}\underline{Γ}_{Z′_{E}}𝒪_{X_{E}} and ρ_{A} : R^{1}\underline{Γ}_{Z_{A}}𝒪_{X_{A}} → R^{1}\underline{Γ}_{Z′_{A}}𝒪_{X_{A}}, and notice that the image of ρ_{E} (resp. of ρ_{A}) lies in \underline{Γ}_{Z_{E}}R^{1}\underline{Γ}_{Z′_{E}}𝒪_{X_{E}} (resp. in \underline{Γ}_{Z_{A}}R^{1}\underline{Γ}_{Z′_{A}}𝒪_{X_{A}}); there follows a commutative diagram

\xymatrix{𝒬_{U_{E}}^{c} \ar[r]^{^{\overline{ψ}^{c}}} \ar[d]_{ρ_{E}^{c}} & φ_{∗}𝒬_{U_{A}}^{c} \ar[d]^{φ_{∗}(ρ_{A}^{c})} \\ \underline{Γ}_{Z_{E}}𝒬_{U′_{E}}^{c} \ar[r]^{_{\underline{Γ}_{Z_{E}} \overline{ψ}_{′c}}} & \underline{Γ}_{Z_{E}}φ_{∗}𝒬_{U′_{A}}^{c} = φ_{∗}\underline{Γ}_{Z_{A}}𝒬_{U′_{A}}^{c}}

where ρ_{E}^{c} (resp. ρ_{A}^{c}) is the restriction of ρ_{E} (resp. of ρ_{A}). From the snake lemma, we obtain natural identifications

Ker ρ_{E}^{c} →^{∼} Ker (α_{0}^{−c}𝒪_{X_{E}} → j′_{E∗}𝒪_{U′_{E}})/Ker (𝒪_{X_{E}} → i_{E_{1}∗}𝒪_{X_{E_{1}}}) = (\underline{Γ}_{X_{E_{2}}}α_{0}^{−c}𝒪_{X_{E}})/𝒪_{X_{E}}^{t}
Ker ρ_{A}^{c} →^{∼} Ker (p^{−c}𝒪_{X_{A}} → j′_{A∗}𝒪_{U′_{A}})/Ker (𝒪_{X_{A}} → i_{A_{1}∗}𝒪_{X_{A_{1}}}) = (\underline{Γ}_{X_{A_{2}}}p^{−c}𝒪_{X_{A}})/𝒪_{X_{A}}^{t}

(notation of (16.4.28)).

Claim 16.4.41. The map \overline{ψ}^{c} restricts to an isomorphism of abelian sheaves

Ker ρ_{E}^{c} →^{∼} Ker φ_{∗}(ρ_{A}^{c}).

Proof of the claim. Notice that

\underline{Γ}_{X_{E_{2}}}α_{0}^{−c}𝒪_{X_{E}} = j_{E∗}j_{E}^{∗}𝒪_{X_{E}}^{t} and \underline{Γ}_{X_{A_{2}}}p^{−c}𝒪_{X_{A}} = j_{A∗}j_{A}^{∗}𝒪_{X_{A}}^{t}.

In view of lemma 16.4.29(ii), we deduce that φ^{♭} induces an isomorphism of abelian sheaves

Ker ρ_{E}^{c} →^{∼} ℛ′ := φ_{∗}(\underline{Γ}_{X_{E_{2}}}α_{0}^{−c}𝒪_{X_{E}})/φ_{∗}(𝒪_{X_{A}}^{t}).

Now, proposition 16.4.26(ii) implies that φ_{∗} is an exact functor on the category of quasicoherent 𝒪_{X_{A}}-modules, and since both U_{A} and U′_{A} are quasi-compact (see (16.1.3)), the 𝒪_{X_{A}}- modules j_{A∗}j_{A}^{∗}𝒪_{X_{A}}^{t} and 𝒪_{X_{A}}^{t} are quasi-coherent ([59, Ch.I, Cor.9.2.2]) so λ restricts to an isomorphism ℛ′ →^{∼} φ_{∗}Ker ρ_{A}^{c} = Ker φ_{∗}(ρ_{A}^{c}), whence the claim. ♢

Now, claim 16.4.38 implies that \underline{Γ}_{Z_{E}} \overline{ψ}^{′c} is an isomorphism, and in light of claim 16.4.41 we see that \overline{ψ}^{c} is a monomorphism, so we are reduced to showing that \underline{Γ}_{Z_{E}} \overline{ψ}^{′c} induces an epimorphism Im ρ_{E}^{c} → Im φ_{∗}(ρ_{A}^{c}). To this aim, let x ∈ X_{E} be any point, and s_{x} ∈ Im (φ_{∗}ρ_{A}^{c})_{x} = φ_{∗}(Im ρ_{A}^{c})_{x} any element; by proposition 16.4.26(ii), the functor φ_{∗} is exact on quasi-coherent 𝒪_{X_{A}}-modules, so both maps

φ_{∗}(p^{−c}𝒪_{X_{A}})_{x} → φ_{∗}(𝒬_{U_{A}}^{c})_{x} → φ_{∗}(Im ρ_{A}^{c})_{x}

are surjective, hence we may find an open neighborhood Ω ⊂ X_{E} of x such that s_{x} lifts to a section s ∈ 𝒪_{X_{A}}(U_{A} ∩ φ^{−1}Ω) and q^{c} · s extends to a section of 𝒪_{X_{A}}(φ^{−1}Ω) (where q is as in (16.4.35)). Denote by t_{x} ∈ (\underline{Γ}_{Z_{E}}𝒬_{U′_{E}}^{c})_{x} the preimage of s_{x}. Notice that

\underline{Γ}_{Z_{E}}R^{1}\underline{Γ}_{Z′_{E}}𝒪_{X_{E}} = (j_{E∗}j_{E}^{∗}(i_{E_{1}∗}𝒪_{X_{E_{1}}}))/i_{E_{1}∗}𝒪_{X_{E_{1}}}

so we may replace Ω by a smaller open neighborhood of x, and assume that t_{x} is the image of a section \overline{t} ∈ i_{E_{1}∗}𝒪_{X_{E_{1}}}(U_{E} ∩ Ω). Denote by \overline{s} ∈ i_{A_{1}∗}𝒪_{X_{A_{1}}}(U_{A} ∩ φ^{−1}Ω) the image of s, and set

\overline{u} := i_{E_{1}∗}φ_{1}^{♭}(\overline{t}) − \overline{s} ∈ j_{A_{∗}}j_{A}^{∗}i_{A_{1}∗}𝒪_{X_{A_{1}}}(φ^{−1}Ω).

By construction, the image of \overline{u}_{x} vanishes in (φ_{∗}R^{1}\underline{Γ}_{Z′_{A}}𝒪_{X_{A}})_{x}, thus \overline{u}_{x} ∈ φ_{∗}(i_{A_{1}∗}𝒪_{X_{A_{1}}})_{x}. By proposition 16.4.26(ii), the natural map φ_{∗}(𝒪_{X_{A}})_{x} → φ_{∗}(i_{A_{1}∗}𝒪_{X_{A_{1}}})_{x} is surjective, so we may replace Ω by a smaller open neighborhood of x, and assume that \overline{u} is the image of a local section u ∈ 𝒪_{X_{A}}(φ^{−1}Ω). However, s and s′ := s + u_{|U_{A}∩φ^{−1}Ω} have the same image in φ_{∗}(𝒬_{U_{A}}^{c})_{x}, so we may replace s by s′, and assume that i_{E_{1}∗}φ_{1}^{♭}(\overline{t}) = \overline{s}. In this situation, remark 16.4.31(ii) implies that \overline{t} lifts to a section t ∈ 𝒪_{X_{E}}(U_{E} ∩ Ω) such that φ^{♭}(t) = s, and we denote by v ∈ R^{1}\underline{Γ}_{Z_{E}}𝒪_{X_{E}}(Ω) the image of t. Clearly, the image of v in (\underline{Γ}_{Z_{E}}R^{1}\underline{Γ}_{Z′_{E}}𝒪_{X_{E}})_{x} agrees with t_{x}, hence it remains only to check that v_{x} ∈ 𝒬_{U_{E},x}^{c}. But notice that α_{0}^{c} · t_{x} = 0, so α_{0}^{c} · v_{x} ∈ (j_{E∗}j_{E}^{∗}𝒪_{X_{E}}^{t})_{x}. Pick any ε ∈ ℕ[1/p] such that 0 < ε ≤ min(p/(p − 1) − c, 1); by virtue of corollary 16.3.63(i), it follows that α_{0}^{c+ε} · v_{x} = 0, i.e. α_{0}^{c} · v_{x} ∈ 𝒬_{U_{E},x}^{ε}. Lastly, from (i) we get

\overline{ψ}^{ε}(α_{0}^{c}v_{x}) = q^{c} · \overline{ψ}^{c+ε}(v_{x})

and \overline{ψ}^{c+ε}(v_{x}) is the image of s in φ_{∗}(𝒬_{U_{A}}^{c+ε})_{x}. By construction, the image of q^{c} · s vanishes in φ_{∗}(𝒬_{U_{A}}^{c+ε})_{x}, and therefore α_{0}^{c} · v_{x} = 0, since we have already remarked that \overline{ψ}^{ε} is a monomorphism. The assertion follows. □

16.5. Perfectoid quasi-affinoid rings. We wish now to merge the theory of perfectoid rings with that of affinoid rings of section 15.3. First, let us make the following :

Definition 16.5.1. (i) We say that a quasi-affinoid ring (A, A^{+}, U) is perfectoid if A is a perfectoid ring.

(ii) We say that a quasi-affinoid scheme \underline{X} is perfectoid, if there exists a perfectoid quasiaffinoid ring \underline{A} and an isomorphism \underline{X} →^{∼} Spec \underline{A}.

(iii) We denote by

q.Afd.Ring_{perf}  and q.Afd.Sch_{perf}

the subcategories of q.Afd.Ring and respectively q.Afd.Sch whose objects are the perfectoid quasi-affinoid rings (resp. schemes), and whose morphisms are the f-adic morphisms of quasiaffinoid rings (resp. schemes).

16.5.2. Clearly we have a well defined functor

q.Afd.Ring_{perf} → q.Afd.Sch_{perf}^{o} \underline{A} ↦ Spec \underline{A}

and we will construct a right adjoint. To this aim, we need a few preliminaries : we consider any perfectoid quasi-affinoid ring \underline{A} := (A, A^{+}, U_{A}) and set X_{A} := Spec A, Z_{A} := X_{A} \ U_{A}. Let also E := E(A), define the continuous map φ : X_{A} → X_{E} := Spec E as in (16.4.24), and set Z_{E} := φ(Z_{A}) and U_{E} := X_{E} \ Z_{E}. Then it is easily seen that U_{E} contains the analytic locus of X_{E}, so any choice of ring of integral elements E^{+} ⊂ E will give another perfectoid quasi-affinoid ring \underline{E} := (E, E^{+}, U_{E}), as well as the attached perfectoid quasi-affinoid schemes

(U_{A}, 𝒯_{U_{A}}, A_{U}^{+}) := Spec \underline{A} (U_{E}, 𝒯_{U_{E}}, E_{U}^{+}) := Spec \underline{E}.

More precisely, let \underline{α} := (α_{n} | n ∈ ℕ) be any distinguished element in Ker u_{A}; since φ maps Spec A/pA bijectively onto Spec E/α_{0}E, and maps Spec A[p^{−1}] onto Spec E[α_{0}^{−1}], it is easily seen that U_{E} is the unique open subset of X_{E} containing the analytic locus, and such that

(16.5.3)  φ^{−1}(U_{E}) = U_{A}.

Recall that, according to lemma 16.4.32, the topologies 𝒯_{U_{A}} on A_{U} := 𝒪_{X_{A}}(U_{A}) and 𝒯_{U_{E}} on E_{U} := 𝒪_{X_{E}}(U_{E}) are f-adic, complete and separated, and the restriction maps A → A_{U} and E → E_{U} are open. Especially the image \overline{A} of A (resp. \overline{E} of E) is open in A_{U} (resp. in E_{U}), and we endow this subring with the quotient topology induced by the projection π_{A} : A → \overline{A} (resp. π_{E} : E → \overline{E}). Moreover, we have a continuous map of (multiplicative) monoids

φ_{U}^{♭} : E_{U} → A_{U}

(proposition 16.4.34(i)). Let also

j_{E}^{∘} : \overline{E} → E_{U}^{∘} and j_{A}^{∘} : \overline{A} → A_{U}^{∘}

be the inclusion maps, and we endow A_{U}^{∘} (resp. E_{U}^{∘}) with the topology induced by the inclusion into A_{U} (resp. into E_{U}), so that j_{A}^{∘} and j_{E}^{∘} are open maps.

Proposition 16.5.4. In the situation of (16.5.2), we have :

(i) For every x ∈ A_{U}, the following conditions are equivalent : (a) x is power bounded in A_{U}.

(b) The subset {x^{p^{n}} | n ∈ ℕ} is bounded in A_{U}.

(c) The image of x in R^{1}Γ_{Z_{A}}𝒪_{X_{A}} is annihilated by A^{∘∘}.

(d) x · A_{U}^{∘∘} ⊂ A_{U}^{∘∘}.

(e) v(x) ≤ 1 for every analytic rank one valuation v ∈ Spa \underline{A}. (ii) \overline{A} and \overline{E} are perfectoid rings, and there exists a unique isomorphism

\overline{ω} : \overline{E} →^{∼} E(\overline{A})  such that  \overline{ω} ∘ π_{E} = E(π_{A}).

(iii) The maps π_{A} and π_{E} restrict to bijections E^{∘∘} →^{∼} E_{U}^{∘∘} and A^{∘∘} →^{∼} A_{U}^{∘∘}.

(iv) The map φ_{U}^{♭} restricts to a continuous map of topological monoids

φ_{U}^{♭∘} : E_{U}^{∘} → A_{U}^{∘}

which induces an isomorphism of perfect 𝔽_{p}-algebras

\overline{φ}_{U}^{♭∘} : E_{U}^{∘}/E_{U}^{∘∘} →^{∼} A_{U}^{∘}/A_{U}^{∘∘}.

Proof. We consider first the following special case :

Claim 16.5.5. In the situation of (16.5.2), let β ∈ E be any regular element such that α_{0} ∈ βE, set b := \overline{u}_{A}(β) ∈ A, and suppose moreover that bA is an ideal of adic definition for A, and U_{A} is the analytic locus of X_{A}. Then conditions (i.a)–(i.d) are equivalent in this case, and

A^{∘∘} = A_{U}^{∘∘} = ⋃_{λ∈ℕ[1/p]\\{0\}} b^{λ}A with b^{λ} := \overline{u}_{A}(β^{λ}) for every λ ∈ ℕ[1/p].

Proof of the claim. First, notice that under the assumptions of the claim, p ∈ bA (lemma 16.2.7(iii)) and b^{λ} is a regular element of A for every λ ∈ ℕ[1/p] (proposition 16.4.17). Especially, the localization map A → A[b^{−1}] is injective, and U_{A} = Spec A[b^{−1}], so that A_{U} = A[b^{−1}], R^{1}Γ_{Z_{A}}𝒪_{X_{A}} = A[b^{−1}]/A, and condition (i.c) implies :

(i.f) b^{λ}x ∈ A for every λ ∈ ℕ[1/p] \ {0}. We show first the equivalence of (i.a), (i.b) and (i.f). Indeed, obviously (i.f)⇒(i.a)⇒(i.b), so we may assume that (i.b) holds, and we set

ρ := inf{γ ∈ ℕ[1/p] | b^{γ}x^{p^{n}} ∈ A for every n ∈ ℕ}.

Notice that (i.b) implies that ρ < +∞. We claim that ρ = 0. Indeed suppose, by way of contradiction, that ρ > 0 and pick any γ ∈ ℕ[1/p] such that ρ > γ > max(ρ/p, ρ − 1/p). Then b^{1/p}b^{γ}x^{p^{n}} ∈ A, and hence b^{γ}x^{p^{n}} ∈ b^{−1/p}A for every n ∈ ℕ; on the other hand, we also see that b^{pγ}x^{p^{n}} ∈ A for every n ∈ ℕ. Now, by corollary 16.3.3, the Frobenius endomorphism of A/pA induces an isomorphism A/b^{1/p}A → A/bA; since b is regular, there follows a bijective map

b^{−1/p}A/A →^{∼} b^{−1}A/A (x mod A) ↦ (x^{p} mod A).

We deduce that b^{γ}x^{p^{n}} ∈ A for every n ∈ ℕ, i.e. γ ≤ ρ, which is absurd. This shows that (i.f) holds, whence the assertion. Next, say that x ∈ A[b^{−1}]^{∘∘}; since bA is open in A[b^{−1}], we must have x^{p^{n}} ∈ bA for some n ∈ ℕ, so if set y := b^{−1/p^{n}}x, we get y^{p^{n}} ∈ A, and therefore the subset {y^{p^{n}} | n ∈ ℕ} is bounded in A_{U}. Pick any integer r > n; by the foregoing, it follows that b^{1/p^{r}}y ∈ A, and finally x ∈ b^{1/p^{n}−1/p^{r}}A. This also implies the equivalence of (i.c),(i.d) and (i.f), so the proof of the claim is complete. ♢

For the general case, pick a finite system β_{•} := (β_{0}, …, β_{k}) of elements of E with β_{0} = α_{0}, and such that β_{•} generates an ideal ℐ of adic definition for E. Then the ideal I ⊂ A generated by (\overline{u}_{A}(β_{0}), …, \overline{u}_{A}(β_{k})) is of adic definition for A (see remark 16.3.7(iii)). We notice :

Claim 16.5.6. The ideal A^{∘∘} is generated by (\overline{u}_{A}(β_{i}^{λ}) | i = 0, …, k, λ ∈ ℕ[1/p] \ \{0\}).

Proof of the claim. Clearly \overline{u}_{A}(β_{i}^{λ}) ∈ A^{∘∘} for every i = 0, …, k and every strictly positive λ ∈ ℕ[1/p]. Conversely, let c ∈ A^{∘∘} be any element; then c^{p^{n}} ∈ I for every sufficiently large n ∈ ℕ. By remark 16.3.7(iii) we may find γ ∈ E such that \overline{u}_{A}(γ) − c ∈ pA, and the image of γ^{p^{n}} in E/α_{0}E lies in ℐ /α_{0}E, so γ^{p^{n}} ∈ ℐ, hence γ ∈ ℐ^{⟨1/p^{n}⟩}E, and therefore c ∈ ℐ^{⟨1/p^{n}⟩}A + pA = ℐ^{⟨1/p^{n}⟩}A (notation of remark 9.3.70(i)), whence the claim. ♢

(i.c)⇒(i.a): By assumption, \overline{u}_{A}(β_{i}^{λ}) · x lies in \overline{A}, for every i = 0, …, k and every strictly positive λ ∈ ℕ[1/p]; let n > 0 be any integer, and pick t ∈ ℕ such that p^{t} ≥ n; then \overline{u}_{A}(β_{i}^{n/p^{t}}) · x^{n} = (\overline{u}_{A}(β^{1/p^{t}}) · x)^{n} lies as well in \overline{A}, and we conclude that \overline{u}_{A}(β_{i}) · x^{n} ∈ \overline{A} for every n ∈ ℕ, whence the assertion.

Obviously, (i.a)⇒(i.b). To show that (i.b)⇒(i.c), let 𝒯_{A,p} be the p-adic topology on A, and 𝒯_{U,p} the unique f-adic topology on A_{U} such that the restriction map (A, 𝒯_{A,p}) → (A_{U}, 𝒯_{U,p}) is open (lemma 16.4.32(i)). Suppose first that the topology of A agrees with 𝒯_{A,p}, and let

A_{1} := A/Ann_{A}(p)  Z′ := Spec A/pA  U′ := X_{A} \ Z′.

Denote as well by x_{|U′} ∈ 𝒪_{X_{A}}(U′) = A[1/p] the image of x. We recall that p is regular in A_{1} (remark 16.4.25(ii)), and we endow A_{1} with its p-adic topology 𝒯_{A_{1},p}, and A[1/p] with the unique f-adic topology 𝒯_{U′,p} such that the inclusion map (A_{1}, 𝒯_{A_{1},p}) → (A[1/p], 𝒯_{U′,p}) is open, so that the subset \{x_{|U′}^{p^{n}} | n ∈ ℕ\} is bounded in A[1/p]. Set q := \overline{u}_{A}(α_{0}); by claim 16.5.5 and lemma 16.2.7(iii), it then follows that q^{λ} ·x_{|U′} lies in A_{1} for every λ ∈ ℕ[1/p]\{0}. Denote also by \overline{x} ∈ R^{1}Γ_{Z}𝒪_{X_{A}} the image of x; we deduce that the image of q^{λ} · \overline{x} vanishes in R^{1}Γ_{Z′}𝒪_{X_{A}}, for every such λ. To conclude in this case, it suffices to remark :

Claim 16.5.7. The kernel of the natural map R^{1}Γ_{Z_{A}}𝒪_{X_{A}} → R^{1}Γ_{Z′}𝒪_{X_{A}} is annihilated by q^{λ} for every λ ∈ ℕ[1/p] \ {0}.

Proof of the claim. By the snake lemma, this kernel is a quotient of Γ_{Z′}j_{A∗}𝒪_{U_{A}}, where j_{A} : U_{A} → X_{A} is the open immersion. However, Γ_{Z′}j_{∗}𝒪_{U_{A}} = H^{0}(U_{A}, \underline{Γ}_{Z′}𝒪_{X_{A}}). Since \underline{Γ}_{Z′}𝒪_{X_{A}} is a quasi-coherent 𝒪_{X_{A}}-module (lemma 10.4.17(i)), we are reduced to checking that Ann_{A}(p) is annihilated by q^{λ} for every λ ∈ ℕ[1/p] \ {0}. The latter holds by corollary 16.3.63(i). ♢

Next, let A be any perfectoid ring; let j_{E} : U_{E} → X_{E} be the open immersion, and for every c ∈ ℕ[1/p] consider the 𝒪_{X_{E}}-module 𝒬_{U_{E}}^{c} and the 𝒪_{X_{A}}-module 𝒬_{U_{A}}^{c} as in (16.4.35). We set

𝒬_{U_{E}}^{0} := ⋂_{c>0} 𝒬_{U_{E}}^{c} and 𝒬_{U_{A}}^{0} := ⋂_{c>0} 𝒬_{U_{A}}^{c}.

Condition (i.b) implies that the set {x^{p^{n}} | n ∈ ℕ} is bounded also for the topology 𝒯_{U,p} of A_{U}, and since (A, 𝒯_{A,p}) is perfectoid (proposition 16.3.8(i)), the foregoing case shows that \overline{x} lies in 𝒬_{U_{A}}^{0}(X_{A}). By proposition 16.4.36(iv), the map φ^{♭} of lemma 16.4.29(i) induces an isomorphism

\overline{ψ}^{0} : 𝒬_{U_{E}}^{0} →^{∼} φ_{∗}𝒬_{U_{A}}^{0}

of abelian sheaves, fitting into a commutative diagram

(16.5.8)  \xymatrix{𝒬_{U_{E}}^{0} \ar[r]^{^{\overline{ψ}^{0}}} \ar[d]_{_{\overline{p}_{E}}} & φ_{∗}𝒬_{U_{A}}^{0} \ar[d]^{φ_{∗}(\overline{p}_{A})} \\ 𝒬_{U_{E}}^{0} \ar[r]^{_{\overline{ψ}_{0}}} & φ_{∗}𝒬_{U_{A}}^{0}}

where \overline{p}_{E} (resp. \overline{p}_{A}) is induced by the p-Frobenius endomorphism of the sheaf of monoids j_{E∗}𝒪_{U_{E}} (resp. j_{A∗}𝒪_{U_{A}}). With this notation, condition (i.b) says that for some n ∈ ℕ we have

(16.5.9)  \overline{u}_{A}(β_{i}^{n}) · \overline{p}_{A}^{r}(\overline{x}) = 0  for every i = 0, …, k and every r ∈ ℕ.

Let \overline{γ} ∈ 𝒬_{U_{E}}^{0}(X_{E}) be the preimage of \overline{x}; then β_{i}^{λ} · \overline{γ} is the preimage of the class of \overline{u}_{A}(β_{i}^{λ}) · x in 𝒬_{U_{A}}^{0}(X_{A}), for every i = 0, …, k and every λ ∈ ℕ[1/p]. By virtue of claim 16.5.6, we are then reduced to checking that β_{i}^{λ} · \overline{γ} = 0 for every such i and λ, and (16.5.9) yields already

β_{i}^{n} · \overline{p}_{E}^{r}(\overline{γ}) = 0 for every i = 0, …, k and every r ∈ ℕ.

Now, we may write λ = t · p^{−r} for some integers t ≥ n and r ∈ ℕ; we deduce

\overline{p}_{E}^{r}(β_{i}^{λ} · \overline{γ}) = β_{i}^{t} · \overline{p}_{E}^{r}(\overline{γ}_{E}) = 0.

Lastly, since E is perfect, it is easily seen that \overline{p}_{E} is an isomorphism, whence the assertion. (ii): This was already remarked in the proof of claim 16.4.39. Let us add the following :

Claim 16.5.10. The map π_{E} and π_{A} restrict to bijections E^{∘∘} →^{∼} \overline{E}^{∘∘} and A^{∘∘} →^{∼} \overline{A}^{∘∘}.

Proof of the claim. First, notice that the image of β_{•} in \overline{E} generates an ideal of adic definition for \overline{E}; in view of (ii) and claim 16.5.6, we deduce immediately that \overline{A}^{∘∘} = π_{A}(A^{∘∘}) and \overline{E}^{∘∘} = π_{E}(E^{∘∘}). Next, if I is any ideal of adic definition for A, clearly we have I · Ker π_{A} = 0, whence I ∩ Ker π_{A} = 0, since A is reduced (corollary 16.3.63); but A^{∘∘} is the union of all ideals of adic definition for A, so we conclude that A^{∘∘}∩Ker π_{A} = 0. Likewise, we see that E^{∘∘}∩Ker π_{E} = 0, whence the claim. ♢

(iii): Let x ∈ A_{U}^{∘∘} be any element; denote by \overline{x} ∈ R^{1}Γ_{Z_{A}}𝒪_{X_{A}} the image of x; from (i) we see that \overline{x} ∈ Q_{A} := 𝒬_{U_{A}}^{0}(X_{A}), and since \overline{A} is open in A_{U}, there exists n ∈ ℕ such that x^{p^{n}} ∈ \overline{A}, therefore \overline{p}_{A}^{n}(\overline{x}) = 0. On the other hand, we have already remarked that the maps \overline{ψ}^{0} and \overline{p}_{E} appearing in (16.5.8) are isomorphisms, so the same holds for \overline{p}_{A}, and consequently \overline{x} = 0, i.e. x ∈ \overline{A}; then the assertion follows from claim 16.5.10. Next, from claim 16.5.6 it is easily seen that the isomorphism E/α_{0}E →^{∼} A/pA of remark 16.3.7(ii) restricts to a natural identification

(16.5.11)  E^{∘∘}/α_{0}E →^{∼} A^{∘∘}/pA

therefore Z_{E} ⊂ X_{E} \ Spec E/E^{∘∘}, so the foregoing applies as well to the perfectoid ring E and its closed subset Z_{E}, and the proof of (iii) is thus complete.

Now, taking into account (iii), we easily see that (i.d)⇒(i.c). Conversely, if (i.c) holds, we get x·A^{∘∘} ·A^{∘∘} ⊂ A_{U}^{∘∘}, but claim 16.5.6implies that A^{∘∘} ·A^{∘∘} = A^{∘∘}, whence (i.d). Lastly, from remark 15.4.15(iv) it is clear that (i.a)⇒(i.e). Conversely, suppose (i.e) holds; then we have as well v(xa) < 1 for every a ∈ A_{U}^{∘∘} and every rank one analytic valuation v ∈ Spa \underline{A}. Taking into account lemma 15.3.14(v), it follows that v(xa) < 1 for every a ∈ A_{U}^{∘∘} and every v ∈ (Spa \underline{A})_{a}. On the other hand, we have v(xa) = 0 for every v ∈ (Spa \underline{A})_{na}, and we conclude that xa ∈ A_{U}^{∘∘}, by corollary 15.4.27(ii). This shows that (i.e)⇒(i.d), and completes the proof of (i).

(iv): Set also Q_{E} := 𝒬_{U_{E}}^{0}(X_{E}); in light of (16.5.11), it is easily seen that the isomorphism \overline{ψ}^{0} restricts to a natural identification

Ann_{Q_{E}}(E^{∘∘}) →^{∼} Ann_{Q_{A}}(A^{∘∘})

and taking into account (i) we deduce that φ_{U}^{♭}(E_{U}^{∘}) ⊂ A_{U}^{∘}, so the sought map φ_{U}^{♭∘} is well defined, and it is continuous, by proposition 16.4.34(i); combining with (iii), we also see that φ_{U}^{♭∘} induces bijections

E_{U}^{∘}/\overline{E} →^{∼} A_{U}^{∘}/\overline{A} \overline{E}/E_{U}^{∘∘} →^{∼} \overline{A}/A_{U}^{∘∘}.

On the other hand, we remark :

Claim 16.5.12. (i) For every x, y ∈ E_{U}^{∘} we have φ_{U}^{♭}(x + y) − φ_{U}^{♭}(x) − φ_{U}^{♭}(y) ∈ A_{U}^{∘∘}. (ii) φ_{U}^{♭∘}(E_{U}^{∘∘}) ⊂ A_{U}^{∘∘}.

Proof of the claim. (i): By proposition 16.4.34(ii), the difference in the claim can be written as a p-adically convergent series ∑_{n∈ℕ\{0}} p^{n} · c_{n}, where each summand c_{n} is a finite ℤ_{p}-linear combination of terms of the form φ_{U}^{♭}(x^{λ}y^{1−λ}), where λ, 1 − λ ∈ ℕ[1/p]. It is easily seen that x^{λ}y^{1−λ} ∈ E_{U}^{∘} for every such λ, and obviously p ∈ A_{U}^{∘∘}, whence the contention.

(ii) is obvious, since φ_{U}^{♭∘} is a continuous morphism of multiplicative monoids. ♢ From claim 16.5.12it follows that φ_{U}^{♭∘} descends to a map \overline{φ}_{U}^{♭∘} : E_{U}^{∘}/E_{U}^{∘∘} → A_{U}^{∘}/A_{U}^{∘∘} of abelian

groups, which must then be bijective, by the foregoing. By remark 8.3.10(iv), both the source and target of \overline{φ}_{U}^{♭∘} are endowed with natural quotient ring structures, and it is easily seen that E_{U}^{∘}/E_{U}^{∘∘} is a perfect 𝔽_{p}-algebra (details left to the reader); to conclude the proof it then suffices to notice that, by construction, \overline{φ}_{U}^{♭∘} is also a morphism of multiplicative monoids. □

Theorem 16.5.13. In the situation of (16.5.2), the following holds :

(i) E_{U}^{∘} and A_{U}^{∘} are both perfectoid, and there exists an isomorphism of topological rings

ω^{∘} : E_{U}^{∘} →^{∼} E(A_{U}^{∘})  such that  ω^{∘} ∘ j_{E}^{∘} = E(j_{A}^{∘}).

(ii) The rule D ↦ E(D) establishes a bijection from the set 𝒮_{A} of open integrally closed

subrings of A_{U}^{∘} to the set 𝒮_{E} of open integrally closed subrings of E_{U}^{∘}. (iii) Moreover, every open integrally closed subring of A_{U}^{∘} is perfectoid (for the topology

induced by A_{U}^{∘}).

Proof. (i): First, notice that E_{U}^{∘∘} (resp. A_{U}^{∘∘}) is a bounded open ideal of E_{U}^{∘} (resp. of A_{U}^{∘}), by virtue of proposition 16.5.4(iii); it follows that E_{U}^{∘} and A_{U}^{∘} are bounded, so their topologies are both adic and f-adic (corollary 8.3.19(iii)). Moreover, since E_{U} and A_{U} are complete and separated (lemma 16.4.32(ii)), the same holds for E_{U}^{∘} and A_{U}^{∘}. Furthermore, since E_{U} is a perfect 𝔽_{p}- algebra, proposition 16.5.4(i) easily implies that the same holds for E_{U}^{∘}, so the latter is perfectoid (example 16.3.2(i)). Next, let I ⊂ A be any ideal of definition; proposition 16.5.4(iii) implies that I_{U} := IA_{U}^{∘} ⊂ A^{∘∘}, so I_{U} is a finitely generated ideal of adic definition for A_{U}^{∘} (corollary 8.3.19(iii)), and clearly p ∈ I_{U}^{2}. Furthermore, I_{U} ⊂ \overline{A} (by claim 16.5.10), and the Frobenius endomorphism Φ_{A_{U} /I_{U}} of A_{U}/I_{U} induces surjective endomorphisms of both \overline{A}/I_{U} and A_{U}/\overline{A} (proposition 16.5.4(iv)); thus, Φ_{A_{U} /I_{U}} is surjective, which shows that A_{U} is a P-ring. Next, endow A_{U}^{∘}/pA_{U}^{∘} with the quotient topology induced by the projection π_{U} : A_{U}^{∘} → A_{U}^{∘}/pA_{U}^{∘}, and let φ_{U}^{♭∘} : E_{U}^{∘} → A_{U}^{∘} and \overline{ω} : \overline{E} →^{∼} E(\overline{A}) be respectively the continuous morphism of topological monoids and the isomorphism of topological rings provided by proposition 16.5.4(iii,iv); in light of proposition 16.4.34(ii) it is easily seen that π_{U} ∘ φ_{U}^{♭∘} : E_{U}^{∘} → A_{U}^{∘}/pA_{U}^{∘} is a continuous ring homomorphism fitting into a commutative diagram

\xymatrix{\overline{E} \ar[r]^{\overline{u}_{\overline{A}/p\overline{A}}∘\overline{ω}} \ar[d]_{j_{E}^{∘}} & \overline{A}/p\overline{A} \ar[d]^{j_{A}^{∘}⊗_{ℤ}𝔽_{p}} \\ E_{U}^{∘} \ar[r]^{π_{U} ∘φ_{U}^{♭∘}} & A_{U}^{∘}/pA_{U}^{∘}.}

By proposition 9.3.52(iii), there follows a continuous ring homomorphism v_{U} : W (E_{U}^{∘}) → A_{U}^{∘} fitting into a commutative diagram

\xymatrix{W (\overline{E}) \ar[r]^{u\overline{A}∘W (\overline{ω})} \ar[d]_{W (j_{E}^{∘})} & \overline{A} \ar[d]^{j_{A}^{∘}} \\ W (E_{U}^{∘}) \ar[r]^{v_{U}} & A_{U}^{∘}.}

Since both u\overline{A} and j_{A}^{∘} are open, the same holds for v_{U}, and since \overline{A} is perfectoid (proposition 16.5.4(iii)), the kernel of u\overline{A} ∘ W (\overline{ω}) is a distinguished ideal, so it remains only to check Claim 16.5.14. v_{U} induces a ring isomorphism

\overline{v}_{U} : W (E_{U}^{∘}) ⊗_{W (\overline{E})} \overline{A} →^{∼} A_{U}^{∘}.

Proof of the claim. Let Φ_{E_{U}^{∘}} be the Frobenius endomorphism of E_{U}^{∘}, and notice that

Φ_{E_{U}^{∘}} (E_{U}^{∘∘}) = E_{U}^{∘∘}.

Moreover, W (E_{U}^{∘∘}) is a closed ideal of both W (\overline{E}) and W (E_{U}^{∘}) (remark 9.3.28(iv)); there follows a commutative ladder with exact rows :

ℒ : [figure: 0; W (E_{U}^{∘∘}); W (\overline{E}); W (\overline{E}/E_{U}^{∘∘}); 0; 0; W (E_{U}^{∘∘}); W (E_{U}^{∘}); W (E_{U}^{∘}/E_{U}^{∘∘}); 0.]

Let \underline{α} be any distinguished element of Ker u_{A}; since the image of \underline{α} is a regular element in both W (E_{U}^{∘}/E_{U}^{∘∘}) and W (\overline{E}/E_{U}^{∘∘}), we have

Tor_{1}^{W (E)}(W (E_{U}^{∘}/E_{U}^{∘∘}), A) = 0 Tor_{1}^{W (E)}(W (\overline{E}/E_{U}^{∘∘}), A) = 0

so the rows of the ladder ℒ ⊗_{W (\overline{E})} \overline{A} are still exact. Furthermore, since the topologies of \overline{E}/E_{U}^{∘∘} and E_{U}^{∘}/E_{U}^{∘∘} are discrete, the isomorphism \overline{φ}_{U}^{♭∘} of proposition 16.5.4(iv) induces identifications

W (\overline{E}/E_{U}^{∘∘}) ⊗_{W (\overline{E})} \overline{A} →^{∼} \overline{A}/A_{U}^{∘∘} W (E_{U}^{∘}/E_{U}^{∘∘}) ⊗_{W (\overline{E})} \overline{A} →^{∼} A_{U}^{∘}/A_{U}^{∘∘}.

Summing up, we obtain the commutative diagram with exact rows :

[figure: 0; A_{U}^{∘∘}; \overline{A}; w; \overline{A}/A_{U}^{∘∘}; 0; 0; A_{U}^{∘∘}; W (E_{U}^{∘}) ⊗_{W (\overline{E})} \overline{A}; _{\overline{v}_{U}}; A_{U}^{∘}/A_{U}^{∘∘}; 0; 0; A_{U}^{∘∘}; A_{U}^{∘}; A_{U}^{∘}/A_{U}^{∘∘}; 0]

and to conclude, it suffices to check that \overline{v}_{U} ∘ w = j_{A}^{∘}. To this aim, it suffices to show that \overline{v}_{U} ∘ w ∘ u\overline{A} = j_{A}^{∘} ∘ u\overline{A} : W (E(\overline{A})) → A_{U}^{∘}. However, by construction we have

\overline{v}_{U} ∘ w ∘ u\overline{A} = v_{U} ∘ W (j_{E}^{∘} ∘ \overline{ω}^{−1}) and j_{A}^{∘} ∘ u\overline{A} ∘ W (\overline{ω}) = v_{U} ∘ W (j_{E}^{∘})

whence the claim. ♢

(ii): Let D ⊂ A_{U}^{∘} be any open integrally closed subring; it is easily seen that A_{U}^{∘∘} ⊂ D, so A_{U}^{∘∘} is an ideal of D (remark 8.3.10(iv)) and we set \overline{D} := D/A_{U}^{∘∘} ⊂ C_{A} := A_{U}^{∘}/A_{U}^{∘∘}.

Claim 16.5.15. With the foregoing notation, the rule : D ↦ \overline{D} establishes a bijection from the set of open integrally closed subrings of A_{U}^{∘} to the set of integrally closed subrings of C_{A}.

Proof of the claim. Let us check that \overline{D} is integrally closed in C_{A}. Indeed, let \overline{x} ∈ C_{A}, and P (T) ∈ D[T] a monic polynomial such that P (\overline{x}) = 0 in C_{A}; if x ∈ A_{U}^{∘} is any representative for the class \overline{x}, we get P (x) ∈ A_{U}^{∘∘}. Set Q(T) := P (T) − P (x); then Q(T) ∈ D[T] and Q(x) = 0, so x ∈ D, and therefore \overline{x} ∈ \overline{D}, which shows the contention. Conversely, if \overline{D} ⊂ C_{A} is any integrally closed subring, let us show that the preimage D ⊂ A_{U}^{∘} of \overline{D} is integrally closed in A_{U}^{∘}. Indeed, say that x ∈ A_{U}^{∘} is integral over D; then the class \overline{x} ∈ C_{A} of x is integral over \overline{D}, hence \overline{x} ∈ \overline{D}, and the claim follows. ♢

Claim 16.5.15 also implies that if D ⊂ E_{U}^{∘} is any open and integrally closed subring, then E_{U}^{∘∘} ⊂ D, and the rule D ↦ D/E_{U}^{∘∘} establishes a bijection between the set of all open integrally closed subrings of E_{U}^{∘} and the set of all integrally closed subrings of C_{E} := E_{U}^{∘}/E_{U}^{∘∘}. However, proposition 16.5.4(iv) yields a natural ring isomorphism \overline{φ}_{U}^{♭∘} : C_{A} →^{∼} C_{E}, whence a bijection 𝒮_{A} →^{∼} 𝒮_{E}, and it remains to check that this bijection is realized by the rule D ↦ E(D). To this aim, notice that every integrally closed subring of C_{A} is a perfect 𝔽_{p}-algebra; then the assertion follows easily from proposition 16.3.25, which also gives (iii). □

16.5.16. In the situation of (16.5.2), pick a finitely generated ideal J_{E} ⊂ E with α_{0} ∈ J_{E} and Z_{E} = Spec E/J_{E}. From (16.5.3), it follows easily that Z_{A} = Spec A/J_{E}^{⟨1⟩}A (notation of remark 9.3.70(i)). Also, we easily see that J_{E}^{⌈0⌉} is the radical of J_{E}; on the other hand, \overline{u}_{A} induces an isomorphism E/J_{E}^{⌈0⌉} →^{∼} A/J_{E}^{⌈0⌉}A, so A/J_{E}^{⌈0⌉}A is reduced, i.e. J_{E}^{⌈0⌉}A is the radical of J_{E}^{⟨1⟩}A.

Corollary 16.5.17. With the notation of (16.5.16), suppose moreover that A = A^{+}. Then :

J_{E}^{⌈0⌉}A_{U}^{+} = J_{E}^{⌈0⌉}\overline{A}.

Proof. Notice that it suffices to show that J_{E}^{⌈0⌉}A_{U}^{+} ⊂ \overline{A}. Indeed, suppose that the latter holds; since we have as well (J_{E}^{⌈0⌉})^{2} = J_{E}^{⌈0⌉}, we deduce that J_{E}^{⌈0⌉}A_{U}^{+} ⊂ J_{E}^{⌈0⌉}\overline{A}, and the converse inclusion is obvious. Notice that E_{U}^{+} is the integral closure of \overline{E} in E_{U}, so that theorem 16.8.2(ii) implies that the assertion holds for E_{U}^{+}, i.e.

(16.5.18)  J_{E}^{⌈0⌉}E_{U}^{+} ⊂ \overline{E}.

Next, recall that proposition 16.5.4(iv) yields a ring isomorphism \overline{φ}_{U}^{♭∘} : E_{U}^{∘}/E^{∘∘} →^{∼} A_{U}^{∘}/A_{U}^{∘∘} restricting to an isomorphism \overline{E}/E_{U}^{∘∘} →^{∼} \overline{A}/A_{U}^{∘∘}. Moreover, claim 16.5.15shows that the subring E_{U}^{+}/E_{U}^{∘∘} is the integral closure of \overline{E}/E_{U}^{∘∘} in E_{U}^{∘}/E^{∘∘} and likewise, A_{U}^{+}/A_{U}^{∘∘} is the integral closure of \overline{A}/A_{U}^{∘∘} in A_{U}/A_{U}^{∘∘}. Hence, A_{U}^{+}/A_{U}^{∘∘} = \overline{φ}_{U}^{♭∘}(E_{U}^{+}/E_{U}^{∘∘}), and taking into account (16.5.18), the assertion follows easily. □

16.5.19. Keep the notation of (16.5.2). By virtue of lemma 15.6.10(i,iii) we may identify naturally U_{A} with an open subset of X_{A}^{∘} := Spec A_{U}^{∘}, and we have a well defined quasi-affinoid ring \underline{A}_{U}^{∘} := (A_{U}^{∘}, A_{U}^{+}, U_{A}), which is perfectoid, by theorem 16.5.13(i). Set

(U_{A}, 𝒯_{U_{A}}^{∘}, A_{U}^{+}) := Spec \underline{A}_{U}^{∘}.

since, by definition, the continuous maps A → A_{U}^{∘} → A_{U} are both f-adic, we see that the topology 𝒯_{U_{A}}^{∘} agrees with 𝒯_{U_{A}}; i.e. we have a natural identification

Spec \underline{A}_{U}^{∘} →^{∼} Spec \underline{A}.

Lastly, it is clear that \underline{A}_{U}^{∘} depends only on Spec \underline{A}, and we claim that the rule Spec \underline{A} ↦ \underline{A}_{U}^{∘} extends to a well defined functor

Γ^{∘} : q.Afd.Sch_{perf} → q.Afd.Ring_{perf}^{o}

with a natural isomorphism ε\underline{X} : Spec∘Γ^{∘}(\underline{X}) →^{∼} \underline{X} for every object \underline{X} of q.Afd.Sch_{perf}. Indeed, if \underline{X} := (X, 𝒯_{X}, A_{X}^{+}) and \underline{Y} := (Y, 𝒯_{Y} , A_{Y}^{+}) are any two perfectoid quasi-affinoid schemes, and ψ : \underline{X} → \underline{Y} any f-adic morphism of quasi-affinoid schemes, then the corresponding map ψ^{♭} : 𝒪_{Y} (Y) → 𝒪_{X}(X) induces a morphism ψ^{♭∘} : (𝒪_{Y} (Y)^{∘}, A_{Y}^{+}) → (𝒪_{X}(X)^{∘}, A_{X}^{+}) of affinoid rings (lemma 8.3.24(iii.a)), which is obviously also f-adic, and the resulting diagram

\xymatrix{Spec 𝒪_{X}(X) \ar[r]^{Spec ψ^{♭}} \ar[d]_{j_{X}} & Spec 𝒪_{Y} (Y) \ar[d]^{j_{Y}} \\ Spec 𝒪_{X}(X)^{∘} \ar[r]^{Spec ψ^{♭∘}} & Spec 𝒪_{Y} (Y)^{∘}}

shows that Spec ψ^{♭∘} restricts to a morphism j_{X}(X) → j_{Y} (Y), whence a well defined morphism of perfectoid quasi-affinoid rings :

Γ^{∘}(ψ) : Γ^{∘}(\underline{Y}) → Γ^{∘}(\underline{X}).

Moreover, if \underline{A} := (A, A^{+}, U) is any perfectoid quasi-affinoid ring, the natural map A → 𝒪_{U}(U)^{∘} induces a morphism of quasi-affinoid rings η\underline{A} : \underline{A} → Γ^{∘} ∘ Spec(\underline{A}) which is f-adic, and therefore it is a morphism of perfectoid quasi-affinoid rings. Lastly, it is easily seen that the pair (ε_{•}, η_{•}) fulfills the triangular conditions of (1.1.13), so Γ^{∘} is right adjoint to Spec.

16.5.20. We also want to upgrade the functor E to well defined functors

E : q.Afd.Ring_{perf} → q.Afd.Ring_{perf} E : q.Afd.Sch_{perf} → q.Afd.Sch_{perf}.

Namely, let \underline{A} := (A, A^{+}, U) be any perfectoid quasi-affinoid ring, and set E := E(A); by applying theorem 16.5.13(ii) to the perfectoid quasi-affinoid ring (A, A^{+}, Spec A), we see that E^{+} := E(A^{+}) is a subring of integral elements of E, so we get a perfectoid quasi-affinoid ring

E(\underline{A}) := (E, E^{+}, E(U))

where E(U) ⊂ Spec E is defined as in (16.5.2) : namely, it is the unique open subset of Spec E containing the analytic locus and such that φ^{−1}E(U) = U, where φ : Spec A → Spec E is the continuous map given by (16.4.24). Moreover, if \underline{α} is any distinguished element in Ker u_{A}, endow A_{0} := A/pA and E_{0} := E/α_{0}E with the ring topologies induced by A and respectively E; from remark 16.3.7(ii) we see as well that the map u_{A} : A → E induces a natural isomorphism of quasi-affinoid rings (notation of example 15.4.7(i)) :

(16.5.21)  A_{0} ⊗_{A} \underline{A} →^{∼} E_{0} ⊗_{E} E(\underline{A}).

Furthermore, for every morphism f : \underline{A} → \underline{B} := (B, B^{+}, V) of perfectoid quasi-affinoid rings, it follows easily from (16.1.4) that the morphism of schemes Spec E(f) : Spec E(B) → Spec E maps E(V) into E(U). Moreover, the ring homomorphism E(f) : E → E(B) is adic (theorem 16.3.42(i)), whence a well defined morphism of perfectoid quasi-affinoid rings

E(f) : E(\underline{A}) → E(\underline{B}).

Let also \underline{B}-q.Afd.Ring_{perf} := \underline{B}/q.Afd.Ring_{perf} for every perfectoid quasi-affinoid ring \underline{B}; just as in remark 16.3.7(i), this new functor E restricts to an equivalence

\underline{A}-q.Afd.Ring_{perf} →^{∼} E(\underline{A})-q.Afd.Ring_{perf}

which admits a natural quasi-inverse functor

A : E(\underline{A})-q.Afd.Ring_{perf} →^{∼} \underline{A}-q.Afd.Ring_{perf} (E, E^{+}, V) ↦ (A(E), A(E^{+}), A(V))

with A(E) := W (E) ⊗_{W (A)} A and correspondingly for A(E^{+}), and where A(V) := φ^{−1}V. Likewise, for every perfectoid quasi-affinoid scheme \underline{U} we define

E(\underline{U}) := Spec ∘ E ∘ Γ^{∘}(\underline{U}) and \underline{U}-q.Afd.Sch_{perf} := q.Afd.Sch_{perf}/\underline{U}.

Then, the resulting functor E on quasi-affinoid schemes restricts to an equivalence

\underline{U}-q.Afd.Sch_{perf} →^{∼} E(\underline{U})-q.Afd.Sch_{perf}

with quasi-inverse given by the rule : \underline{V} ↦ Spec ∘ A ∘ Γ^{∘}(\underline{V}) (details left to the reader). Lastly, say that \underline{U} = Γ^{∘}(\underline{A}), and set \underline{U}_{0} := Spec (A_{0} ⊗_{A} \underline{A}) and E(\underline{U})_{0} := Spec (E_{0} ⊗_{E} \underline{E}); then we deduce from (16.5.21) a natural isomorphism of quasi-affinoid schemes

(16.5.22)  \underline{U}_{0} ×\underline{U} \underline{V} →^{∼} E(\underline{U})_{0} ×_{E(\underline{U})} E(\underline{V})  for every \underline{V} ∈ Ob(\underline{U}-q.Afd.Sch_{perf}).

(see example 15.4.8).

16.5.23. Next, we wish to extend the results of section 16.4 to the perfectoid quasi-affinoid case. Consider any perfectoid quasi-affinoid ring \underline{A} := (A, A^{+}, U_{A}), and let (E, E^{+}, U_{E}) := E(\underline{A}) (notation of (16.5.20)). Let also \overline{A} and \overline{E} be the images of the restriction maps A → A_{U} := 𝒪_{U_{A}}(U_{A}) and respectively E → E_{U} := 𝒪_{U_{E}}(U_{E}). We endow \overline{A} and \overline{E} with the quotient topologies induced by the projections A → \overline{A} and respectively E → \overline{E}. Pick any finitely generated ideal of definition I of E; we get a descending filtration on E_{U} and A_{U} by the rules :

Fil^{s}E_{U} := I^{⟨s⟩}\overline{E} Fil^{s}A_{U} := I^{⟨s⟩}\overline{A}
Fil^{−s}E_{U} := \{x ∈ E_{U} | I^{⟨s⟩}x ⊂ \overline{E}\} Fil^{−s}A_{U} := \{x ∈ A_{U} | I^{⟨s⟩}x ⊂ \overline{A}\}

for every s ∈ ℕ[1/p]. As in example 9.1.9(i) we associate with I an angular order function

ν : E_{U} → ℝ ∪ {+∞} x ↦ sup{s ∈ ℤ[1/p] | x ∈ Fil^{s}E_{U}}.

Notice that, since the topology of E_{U} is separated, we have ν(x) = +∞ if and only if x = 0. Then, we fix ρ ∈]0, 1[, and set

|x|_{I} := ρ^{ν(x)} for every x ∈ E_{U}

(with the convention that ρ^{+∞} := 0).

Lemma 16.5.24. With the notation of (16.5.23), we have :

(i) The mapping | · |_{I} is the asymptotic Samuel function on E_{U} associated with I (see example 9.1.9(i)). Especially, it is a real-valued power-multiplicative norm on E_{U}.

(ii) φ_{U}^{♭}(Fil^{s}E_{U}) ⊂ Fil^{s}A_{U} for every s ∈ ℤ[1/p] (notation of proposition 16.4.34(i)).

Proof. (i): Let | · |_{I}^{∗} be the asymptotic Samuel function attached to I (and the real number ρ), let x ∈ E_{U} be any element, and s ∈ ℤ[1/p] any rational number; consider the following conditions:

(a) |x|_{I} ≤ ρ^{s}.

(b) For every t < s in ℤ[1/p] we have x ∈ Fil^{t}E_{U}.

(c) For every t < s in ℤ[1/p] we may find n ∈ ℕ such that p^{n}t ∈ ℤ and x^{p^{n}} ∈ I^{p^{n}t}, where the integral power I^{p^{n}t} is defined as in example 9.1.9(i).

(d) |x|_{I}^{∗} ≤ ρ^{s}. It is easily seen that each of these conditions is equivalent to the following one, whence the assertion.

(ii): Suppose first that s ≥ 0; in this case, by definition we have φ_{U}^{♭}(I^{⟨s⟩}) ⊂ Fil^{s}A_{U}. On the other hand, proposition 16.4.34(ii) implies that φ_{U}^{♭}(Fil^{s}E_{U}) lies in the topological closure of the \overline{A}-submodule of A_{U} generated by φ_{U}^{♭}(I^{⟨s⟩}), and since Fil^{s}A_{U} is an open \overline{A}-submodule of A_{U}, the assertion follows.

Lastly, suppose that s < 0, and let x ∈ Fil^{s}E_{U} be any element; the condition means that I^{⟨s⟩}x ⊂ \overline{E}, whence φ_{U}^{♭}(x) · φ_{U}^{♭}(I^{⟨s⟩}) ⊂ \overline{A}, and finally φ_{U}^{♭}(x) ∈ Fil^{s}A_{U}, as stated. □

16.5.25. The norm | · |_{I} in turns induces a mapping

| · |_{1} : W (E_{U}) → ℝ_{+} ∪ {+∞} (a_{n} | n ∈ ℕ) ↦ sup_{n∈ℕ} |a_{n}|_{I}^{p^{−n}}

as in (9.3.80), such that the subset

W (E_{U}, 1) := \{\underline{a} ∈ W (E_{U}) | |\underline{a}|_{1} ∈ ℝ_{+}\}

is a subring of W (E_{U}), and the restriction of |·|_{1} is a real-valued norm on W (E_{U}, 1) (proposition 9.3.82(ii)). We endow W (E_{U}, 1) with the topology defined by the norm | · |_{1}. Then clearly W (E_{U}, 1) is a separated topological ring, and it is independent of the choice of I (see example 9.1.9(iv)). Furthermore, for every s ∈ ℤ[1/p] we set

W ⟨s⟩ := {(x_{n} | n ∈ ℕ) ∈ W (E_{U}) | x_{n} ∈ Fil^{p^{n}s}E_{U} for every n ∈ ℕ} ⊂ W (E_{U}, 1).

Recall that the f-adic topologies of E_{U} and A_{U} agree with the ℤ-linear topologies defined by Fil^{•}E_{U} and respectively Fil^{•}A_{U} (lemma 9.3.69(iv) and corollary 16.3.40(ii)). Moreover, Fil^{s}A_{U} and Fil^{s}E_{U} are bounded subsets of A_{U} and respectively E_{U}, for every s ∈ ℤ[1/p]. It follows easily that for every \underline{a} := (a_{n} | n ∈ ℕ) ∈ W (E_{U}, 1), the series

∑_{n∈ℕ} p^{n} · φ_{U}^{♭}(a_{n}^{1/p^{n}})

converges to a unique element u_{U}(\underline{a}) of A_{U}. Moreover, just as in definition 16.3.27(v), for any \overline{E}-submodule 𝒦 of E_{U} we denote by \{𝒦 \} the topological closure in A_{U} of the \overline{A}-submodule generated by (φ_{U}^{♭}(x) | x ∈ 𝒦 ). We also define filtrations on 𝒦 and {𝒦 } by the rules :

Fil^{s}𝒦 := 𝒦 ∩ Fil^{s}E_{U}  and Fil^{s}{𝒦 } := {Fil^{s}𝒦 }  for every s ∈ ℤ[1/p].

Proposition 16.5.26. With the notation of (16.5.25), we have :

(i) The resulting mapping u_{U} : W (E_{U}, 1) → A_{U} is a morphism of topological rings. (ii) u_{U}(W ⟨s⟩) = {Fil^{s}E_{U}} ⊂ Fil^{s}A_{U} for every s ∈ ℤ[1/p]. (iii) pW (E_{U}) ∩ W ⟨s⟩ = pW ⟨s⟩ for every s ∈ ℤ[1/p]. (iv) The topological ring W (E_{U}, 1) is complete and separated.

Proof. (i): The continuity of u_{U} follows from the continuity of φ_{U}^{♭} (proposition 16.4.34(i)) and a simple inspection of the definition. Next, we remark :

Claim 16.5.27. For every a ∈ E_{U} and every \underline{x} ∈ W (E_{U}, 1) we have

τ_{E_{U}} (a) · \underline{x} ∈ W (E_{U}, 1) and u_{U}(τ_{E_{U}} (a) · \underline{x}) = φ_{U}^{♭}(a) · u_{U}(\underline{x}).

Proof of the claim. Say that \underline{x} = (x_{n} | n ∈ ℕ); by proposition 9.3.36(i) we have τ_{E_{U}} (a) · \underline{x} = (a^{p^{n}}x_{n} | n ∈ ℕ), whence the first assertion. By the same token, since φ_{U}^{♭} is a continuous morphism of multiplicative monoids, we may compute

u_{U}(τ_{E_{U}} (a) · \underline{x}) = ∑_{n∈ℕ} p^{n} · φ_{U}^{♭}(ax_{n}^{1/p^{n}}) = φ_{U}^{♭}(a) · ∑_{n∈ℕ} p^{n} · φ_{U}^{♭}(x_{n}^{1/p^{n}}) = φ_{U}^{♭}(a) · u_{U}(\underline{x})

as stated. ♢ To conclude, we have to check that

u_{U}(\underline{x} + \underline{y}) = u_{U}(\underline{x}) + u_{U}(\underline{y}) and u_{U}(\underline{x} · \underline{y}) = u_{U}(\underline{x}) · u_{U}(\underline{y})

for every \underline{x} := (x_{n} | n ∈ ℕ), \underline{y} := (y_{n} | n ∈ ℕ) ∈ W (E_{U}, 1). However, pick s ∈ ℤ[1/p] such that \underline{x}, \underline{y} ∈ W ⟨s⟩, and notice that u_{U} restricts to a map W ⟨s⟩ → Fil^{s}A_{U}; set \underline{x}^{(k)} := ∑_{n=0}^{k} p^{n} · τ_{E_{U}} (x_{n}) and define likewise \underline{y}^{(k)} for every k ∈ ℕ. The sequences (\underline{x}^{(k)} | n ∈ ℕ) and (\underline{y}^{(k)} | n ∈ ℕ) lie in W ⟨s⟩ and converge p-adically to \underline{x} and respectively \underline{y}; moreover, the p-adic topology is separated on Fil^{s}A_{U}, so it suffices to check the sought identities with \underline{x} and \underline{y} replaced by \underline{x}^{(k)} and \underline{y}^{(k)}, for every k ∈ ℕ. We may thus assume that there exists k ∈ ℕ such that x_{n} = y_{n} = 0 for every n > k. Then, we argue as in the proof of proposition 16.4.34(ii) : we pick a family (g_{λ} | λ ∈ Λ) of elements of E such that U_{E} = ⋃_{λ∈Λ} Spec E[g_{λ}^{−1}], and we set

h_{λ} := \overline{u}_{A}(g_{λ}) ∈ A  and t_{λ} := τ_{E}(g_{λ}) ∈ W (E)  for every λ ∈ Λ

so that U_{A} = ⋃_{λ∈Λ} Spec A[h_{λ}^{−1}], and it suffices to check that the sought identities hold in A[h_{λ}^{−1}] = \overline{A}[h_{λ}^{−1}], for every λ ∈ Λ. Now, for every λ ∈ Λ we may find n_{λ} ∈ ℕ such that g_{λ}^{p^{i}n_{λ}} · x_{i}, g_{λ}^{p^{i}n_{λ}} · y_{i} ∈ \overline{E} for every i = 0, …, k, and therefore

\underline{x}_{λ} := t_{λ}^{n_{λ}} · \underline{x}, \underline{y}_{λ} := t_{λ}^{n_{λ}} · \underline{y} ∈ W (\overline{E})

(proposition 9.3.36(i)). Lastly, we consider the commutative diagram

[figure: W (\overline{E}); u\overline{A}; \overline{A}; \overline{A}[h_{λ}^{−1}]; W (E_{U}, 1); u_{U}; A_{U}; A[h_{λ}^{−1}]]

whose two vertical arrows are the natural inclusions, and whose unmarked horizontal arrows are the localization maps; taking into account claim 16.5.27, we may compute for every λ ∈ Λ:

\begin{matrix}φ_{U}^{♭}(t_{λ}^{n_{λ}}) · u_{U}(\underline{x} + \underline{y}) = & u_{U}(\underline{x}_{λ} + \underline{y}_{λ}) \\ = & u\overline{A}(\underline{x}_{λ} + \underline{y}_{λ}) \\ = & u\overline{A}(\underline{x}_{λ}) + u\overline{A}(\underline{y}_{λ}) \\ = & u_{U}(\underline{x}_{λ}) + u_{U}(\underline{y}_{λ}) \\ = & φ_{U}^{♭}(t_{λ}^{n_{λ}}) · (u_{U}(\underline{x}) + u_{U}(\underline{y})) \end{matrix}

as required. Likewise one shows the other sought identity.

(ii): Notice first that if s ≥ 0, then W ⟨s⟩ is the ideal W (I^{⟨s⟩}\overline{E}) of W (\overline{E}) (notation of remark 9.3.28(iv)); since I^{⟨s⟩}\overline{E} is a taut and open ideal of \overline{E}, the ideal W (I^{⟨s⟩}\overline{E}) is closed in W (\overline{E}), and from theorem 16.3.36(i) it follows that

u_{U}(W ⟨s⟩) = u\overline{A}(W (I^{⟨s⟩}\overline{E})) = \{Fil^{s}E_{U}\}

as required. Especially u_{U}(W ⟨s⟩) is an open \overline{A}-submodule of A_{U} for every s ≥ 0; but then obviously the same holds also more generally for every s ∈ ℤ[1/p]. Next, for a general s ∈ ℤ[1/p] notice that every element of W ⟨s⟩ can be written as a p-adically convergent series ∑_{n∈ℕ} p^{n} · τ_{E_{U}} (x_{n}), for a unique sequence (x_{n} | n ∈ ℕ) of elements of Fil^{s}E_{U}; to any such element, the map u_{U} assigns the p-adically convergent series ∑_{n∈ℕ} p^{n} · φ_{U}^{♭}(x_{n}), which clearly lies in {Fil^{s}E_{U}}, from which we see that u_{U}(W ⟨s⟩) is dense in {Fil^{s}E_{U}}. However, we have just seen that u_{U}(W ⟨s⟩) is a closed subset of A_{U} for every such s, whence the assertion.

(iii): The intersection pW (E_{U}) ∩ W ⟨s⟩ consists of all elements \underline{x} := (x_{n} | n ∈ ℕ) such that x_{0} = 0 and x_{n} ∈ Fil^{sp^{n}}E_{U} for every n ≥ 1. For such \underline{x}, we have y_{n} := x_{n+1}^{1/p} ∈ Fil^{sp^{n}}E_{U} for every n ∈ ℕ, whence \underline{y} := (y_{n} | n ∈ ℕ) ∈ W ⟨s⟩ and p · \underline{y} = \underline{x}, whence the contention.

(iv): Since W (\overline{E}) is an open subring of W (E_{U}, 1), it suffices to show that W (\overline{E}) is complete and separated for the topology 𝒯 induced by the norm | · |_{1}. To this aim, let x_{1}, …, x_{n} be any finite system of generators of I\overline{E}, and denote by ℐ ⊂ W (\overline{E}) the ideal generated by τ\overline{E}(x_{1}), …, τ\overline{E}(x_{n}). Taking into account proposition 9.3.77(i) and lemma 9.3.69(iv), it is easily seen that 𝒯 agrees with the 𝒥 -adic topology on W (\overline{E}). However, let also 𝒯\overline{E} be the I\overline{E}- adic topology on \overline{E} and 𝒯_{W (\overline{E})} the topology of W (\overline{E}, 𝒯_{E}) (as in definition 9.3.14); then 𝒯_{W (\overline{E})} agrees with the (pW (\overline{E}) + ℐ)-adic topology (proposition 9.3.77(ii)). On the other hand, since 𝒯\overline{E} is complete and separated, the same holds for 𝒯_{W (\overline{E})} (lemma 9.3.33(ii)). To conclude, we may now appeal to lemma 8.3.12. □

16.5.28. In the situation of (16.5.23), let 𝒦 , 𝒥 ⊂ E_{U} be any two \overline{E}-submodules; we let 𝒦 𝒥 ⊂ E_{U} be the \overline{E}-submodule generated by the system (xy | x ∈ 𝒦 , y ∈ 𝒥). Especially, we have well defined \overline{E}-submodules 𝒦 ^{n} of E_{U}, defined inductively by the rule : 𝒦 ^{0} := \overline{E} and 𝒦 ^{n+1} := 𝒦 ^{n}𝒦 for every n ∈ ℕ. Next, suppose that 𝒥 ⊂ 𝒦 , and let β ∈ E be any element; as in definition 16.3.27, we shall say that the inclusion of 𝒥 in 𝒦 is β-taut if

β · Φ_{E_{U}}^{−1} (𝒦 ^{p}) ⊂ 𝒥

where Φ_{E_{U}} is the Frobenius automorphism of E_{U}. For β = α_{0} (where (α_{n} | n ∈ ℕ) is a fixed distinguished element in Ker u_{A}), we just say that the inclusion is taut. Likewise, we say that 𝒦 is β-taut (resp. taut) if the identity map of 𝒦 is a β-taut (resp. taut) inclusion. Just as in (16.3.30), for any taut inclusion 𝒥 ⊂ 𝒦 , the quotients 𝒦 /𝒥 and {𝒦 }/{𝒥 } are both \overline{A}/p\overline{A}-modules. Also, just as in remark 16.3.28(v), if the inclusion 𝒥 ⊂ 𝒦 is taut, then both 𝒥 and 𝒦 are taut. With this terminology, we have the following extension of theorem 16.3.31:

Lemma 16.5.29. In the situation of (16.5.25), the following holds :

(i) Every taut inclusion 𝒦_{1} ⊂ 𝒦_{2} of \overline{E}-submodules of E_{U} induces an \overline{A}/p\overline{A}-linear map

𝒦_{2}/𝒦_{1} → {𝒦_{2}}/{𝒦_{1}} : (x mod 𝒦_{1}) ↦ (φ_{U}^{♭}(x) mod 𝒦_{1}).

(ii) If 𝒦_{1} and 𝒦_{2} are topologically closed in E_{U}, the map of (i) is an isomorphism.

(iii) For every taut \overline{E}-submodule 𝒦 of E_{U} and every t ∈ ℤ[1/p] we have

{𝒦 } ∩ {Fil^{t}E_{U}} = Fil^{t}{𝒦 }.

Proof. (i): The proof is the same as that of theorem 16.3.31(i) : the only difference is that instead of using proposition 9.3.62one must appeal to the more general proposition 16.4.34(ii).

Next, we prove the following special case of (ii) :

Claim 16.5.30. Pick any strictly positive ε ∈ ℕ[1/p] such that α_{0} ∈ I^{⟨ε⟩}E. Then we have :

(i) For every s, t ∈ ℤ[1/p] such that t − ε ≤ s ≤ t, the inclusion Fil^{t}E_{U} ⊂ Fil^{s}E_{U} is taut, and the map of (i) is an isomorphism

τ_{s,t} : Fil^{s}E_{U}/Fil^{t}E_{U} →^{∼} {Fil^{s}E_{U}}/{Fil^{t}E_{U}}.

(ii) Moreover, the inclusion {Fil^{s}E_{U}} ⊂ Fil^{s}A_{U} induces an injection

μ_{s,t} : {Fil^{s}E_{U}}/{Fil^{t}E_{U}} → Fil^{s}A_{U}/Fil^{t}A_{U}.

Proof of the claim. (i): First notice that, since \overline{A} is perfectoid and E(\overline{A}) = \overline{E}, the assertion in case s ≥ 0 is a special case of theorem 16.3.31(ii). Thus, we assume henceforth that s < 0. Next, suppose that t > 0; in this case, we know already that τ_{0,t} is an isomorphism, and therefore the 5-lemma implies that τ_{s,t} is an isomorphism if and only if the same holds for τ_{s,0} (details left to the reader). Hence, we may further assume that t ≤ 0 as well. Then, notice that the map τ_{s,t} has dense image, for the quotient topology on the target; however, {Fil^{t}E_{U}} is an open E_{U}-submodule of {Fil^{t}E_{U}}, so this quotient topology is discrete, and thus τ_{s,t} is surjective. To conclude, it then suffices to check that μ_{s,t} ∘ τ_{s,t} is injective. Hence, let x ∈ Fil^{s}E_{U} be any element, and suppose that φ_{U}^{♭}(x) ∈ Fil^{t}A_{U}; we need to show that x ∈ Fil^{t}E_{U}, i.e. that for every b ∈ I^{⟨−t⟩} we have z := bx ∈ \overline{E}. However, for such z we have

φ_{U}^{♭}(z) = φ_{U}^{♭}(b) · φ_{U}^{♭}(x) ∈ I^{⟨−t⟩} · Fil^{t}A_{U} ⊂ \overline{A}.

Notice that z ∈ Fil^{s−t}E_{U}, and let \overline{z} ∈ Fil^{s−t}E_{U}/Fil^{0}E_{U} = Fil^{s−t}E_{U}/\overline{E} be the class of z. Since s − t ≥ −ε, and since α_{0} ∈ I^{⟨ε⟩}E, we see that Fil^{s−t}E_{U}/\overline{E} ⊂ Ann_{E_{U} /\overline{E}}(α_{0}); on the other hand, proposition 16.4.36(iv) implies that φ_{U}^{♭} induces an isomorphism

Ann_{E_{U} /\overline{E}}(α_{0}) →^{∼} Ann_{A_{U} /\overline{A}}(p)

whence the contention. Assertion (ii) is an immediate consequence. ♢ (iii): Let ε, s, t ∈ ℤ[1/p] be as in claim 16.5.30(i), and notice that the composition

M := \frac{Fil^{s}𝒦}{Fil^{t}𝒦} ⟶^{β} M′ := \frac{Fil^{s}\{𝒦 \}}{Fil^{t}\{𝒦 \}} ⟶^{γ} \frac{\{𝒦 \} ∩ \{Fil^{s}E_{U}\}}{\{𝒦 \} ∩ \{Fil^{t}E_{U}\}} → M′′ := \frac{\{Fil^{s}E_{U}\}}{\{Fil^{t}E_{U}\}}

factors through an injective map M → Fil^{s}E_{U}/Fil^{t}E_{U} and the isomorphism τ_{s,t}. Hence, β is injective; moreover, β has dense image, and the quotient topologies on M and M′′ are discrete, so β is an isomorphism, by claim 16.3.34. We deduce that γ is injective.

Claim 16.5.31. Assertion (iii) holds if there exists s ∈ ℤ[1/p] such that 𝒦 ⊂ Fil^{s}E_{U}.

Proof of the claim. Notice that 𝒦 ∩Fil^{r}E_{U} is still taut for every r ∈ ℤ[1/p]. Now, if t < s there is nothing to prove. In case t ≥ s, let n ∈ ℕ be the unique integer such that t − nε − s ∈ [0, ε[; arguing by induction on n, we are easily reduced to the case where n = 0, i.e. we may assume that t − ε < s ≤ t. In this case, clearly γ is an isomorphism, and the sought identity follows immediately. ♢

Now, for a general 𝒦 , clearly {𝒦 } is the topological closure of ⋃_{s∈ℤ[1/p]}{𝒦 ∩ Fil^{s}E_{U}} in A_{U}, and since {Fil^{t}E_{U}} is open in A_{U}, we deduce that {𝒦 } ∩ {Fil^{t}E_{U}} is the topological closure of ⋃_{s∈ℤ[1/p]}({𝒦 ∩Fil^{s}E_{U}}∩{Fil^{t}E_{U}}). But for every s ≤ t, we have {𝒦 ∩Fil^{s}E_{U}}∩ {Fil^{t}E_{U}} = {𝒦 ∩ Fil^{t}E_{U}}, by virtue of claim 16.5.31, whence (iii).

We may now complete the proof of (ii) arguing as in the proof of theorem 16.3.31(ii) : namely, we may assume that α_{0} ∈ I, and for every \overline{E}-submodule 𝒦 of E_{U} we set

gr^{n}𝒦 := Fil^{n}𝒦 /Fil^{n+1}𝒦  gr^{n}{𝒦 } := Fil^{n}{𝒦 }/Fil^{n+1}{𝒦 }  for every n ∈ ℤ.

Moreover, for every n ∈ ℤ we let Fil^{n}(𝒦_{2}/𝒦_{1}) (resp. Fil^{n}({𝒦_{2}}/{𝒦_{1}})) be the image of Fil^{n}𝒦_{2} in 𝒦_{2}/𝒦_{1} (resp. of Fil^{b}{𝒦_{2}} in {𝒦_{2}}/{𝒦_{1}}) and we denote again by gr^{•}(𝒦_{2}/𝒦_{1}) and gr^{•}({𝒦_{2}}/{𝒦_{1}}) the respective associated graded modules. There follows for every n ∈ ℤ a commutative diagram with exact rows

[figure: 0; gr^{n}𝒦_{1}; gr^{n}𝒦_{2}; gr^{n}(𝒦_{2}/𝒦_{1}); 0; 0; gr^{n}{𝒦_{1}}; gr^{n}{𝒦_{2}}; gr^{n}({𝒦_{2}}/{𝒦_{1}}); 0]

and since both 𝒦_{1} and 𝒦_{2} are taut, we know already from the proof of (iii) that the leftmost and central vertical arrows are isomorphisms, hence the same holds for the rightmost vertical arrow. Now, notice that the filtration Fil^{•}(𝒦_{2}/𝒦_{1}) defines a separated and complete topology on 𝒦_{2}/𝒦_{1}, since 𝒦_{2} and 𝒦_{1} are both closed \overline{E}-submodules of E. The same holds for the topology on {𝒦_{2}}/{𝒦_{1}} determined by the filtration Fil^{•}({𝒦_{2}}/{𝒦_{1}}), since {𝒦_{2}} and {𝒦_{1}} are closed \overline{A}-submodules in A. Then (ii) follows directly from [34, Ch.III, §2, n.8, Cor.3]. □

Theorem 16.5.32. In the situation of (16.5.25), we have :

(i) The continuous ring homomorphism u_{U} is surjective and open.

(ii) Ker u_{U} = \underline{α}W (E_{U}, 1).

Proof. (ii): Clearly \underline{α} ∈ Keru_{U}; thus, it suffices to check that u_{U} induces an isomorphism

\overline{u}_{s} : W ⟨s⟩/\underline{α}W ⟨s⟩ →^{∼} \{Fil^{s}E_{U}\} for every s ∈ ℤ[1/p]

and by virtue of proposition 16.5.26(ii) we know already that \overline{u}_{s} is surjective for every such s. Now, if s ≥ 0, we have W ⟨s⟩ = W (I^{⟨s⟩}\overline{E}) ⊂ 𝕎(\overline{E}), and the assertion follows from theorem 16.3.36(iv). For the general case, we may assume that α_{0} ∈ I, and then an easy induction argument then reduces to checking :

Claim 16.5.33. Suppose that α_{0} ∈ I. Then, if \overline{u}_{s+1} is injective, the same holds for \overline{u}_{s}.

Proof of the claim. We consider the commutative diagram

(16.5.34)  \xymatrix{W ⟨s + 1⟩/\underline{α}W ⟨s + 1⟩ \ar[r]^{j} \ar[d]_{_{\overline{u}_{s+1}}} & W ⟨s⟩/\underline{α}W ⟨s⟩ \ar[d]^{_{\overline{u}_{s}}} \\ \{Fil^{s+1}E_{U}\} \ar[r]^{i} & \{Fil^{s}E_{U}\}}

where i is the inclusion map, and j is induced by the inclusion W ⟨s + 1⟩ ⊂ W ⟨s⟩. Since \overline{u}_{s+1} is injective by assumption, the same holds for j. Next, let us write α = τ_{E}(α_{0}) + p · \underline{u} for some invertible element \underline{u} of W (E); from proposition 9.3.36(ii) and our assumption on α_{0}, we deduce that τ_{E}(α_{0})·W ⟨s⟩ ⊂ W ⟨s+1⟩, hence the cokernel of j is annihilated by p. Taking into account proposition 16.5.26(iii), it follows that Coker j is naturally identified with the cokernel of the natural map j′ : (Fil^{s+1}E_{U})/α_{0}(Fil^{s+1}E_{U}) → (Fil^{s}E_{U})/α_{0}(Fil^{s}E_{U}). But by the same token we see that α_{0}(Fil^{s}E_{U}) ⊂ Fil^{s+1}E_{U}, so Coker j′ = Fil^{s}E_{U}/Fil^{s+1}E_{U}. Moreover, a simple inspection of the definitions shows that the map Coker j′ → Coker i resulting from (16.5.34) is the same as the isomorphism τ_{s,s+1} of claim 16.5.30(i). Then the assertion follows from the 5-lemma. ♢

(i): By proposition 16.5.26(ii) we know already that u_{U} is an open map. Next, we remark :

Claim 16.5.35. To prove that u_{U} is surjective, we may assume that the topology of E is α_{0}-adic.

Proof of the claim. Notice that the image R of u_{U} contains u_{U}(W (\overline{E})) = u\overline{A}(W (\overline{E})) = \overline{A} (lemma 16.2.7(i)). Especially, R is an open subring of A_{U}; by the same token, R contains the \overline{A}-subalgebra R′ of A_{U} generated by φ_{U}^{♭}(E_{U}), and proposition 16.5.26(ii) implies that R′ is dense in R. However, R′ is also open in A_{U}, so R = R′. Furthermore, a simple inspection shows that the definition of φ_{U}^{♭} depends only on the ring A and the open subset U (and is independent of the topology of A or of E). This shows that the image of u_{U} is independent of the topology of E_{U}. Lastly, let 𝒯_{α_{0}} be the α_{0}-adic topology of E; it follows easily from example 16.3.2(i) and lemma 8.3.12that the topological ring (E, 𝒯_{α_{0}}) is perfectoid, and U_{E} contains the analytic locus of Spec E, relative to the topology 𝒯_{α_{0}}, whence the claim. ♢

Henceforth, we assume that the topology of E_{U} is α_{0}-adic. Set J_{1} := Ann\overline{E}(α_{0}), let J_{2} ⊂ \overline{E} be the radical of the ideal α_{0}\overline{E}, and J_{3} := J_{1} + J_{2}; let \overline{X}_{E} := Spec \overline{E}, \overline{X}_{A} := Spec \overline{A}, and for i = 1, 2, 3 set \overline{E}_{i} := \overline{E}/J_{i}, \overline{A}_{i} := W (\overline{E}_{i}) ⊗_{W (\overline{A})} \overline{A}, \overline{X}_{E,i} := Spec \overline{E}_{i} and \overline{X}_{A,i} := Spec \overline{A}_{i}. Lastly, let j_{E,i} : \overline{X}_{E,i} → \overline{X}_{E} and j_{A,i} : \overline{X}_{A,i} → \overline{X}_{A} be the induced closed immersions and set E_{U,i} := j_{E,i∗}𝒪_{\overline{X}_{E,i}}(U_{E}), A_{U,i} := j_{A,i∗}𝒪_{\overline{X}_{A,i}}(U_{A}) for i = 1, 2, 3. According to (16.4.24) we have cartesian diagrams of quasi-coherent 𝒪_{\overline{X}_{E}}-modules

\xymatrix{𝒪_{\overline{X}_{E}} \ar[r] \ar[d] & j_{E,1∗}𝒪_{\overline{X}_{E,1}} \ar[d] & 𝒪_{\overline{X}_{A}} \ar[r] \ar[d] & j_{A,1∗}𝒪_{\overline{X}_{A,1}} \ar[d] \\ j_{E,2∗}𝒪_{\overline{X}_{E,2}} \ar[r] & j_{E,3∗}𝒪_{\overline{X}_{E,3}} & j_{A,2∗}𝒪_{\overline{X}_{A,2}} \ar[r] & j_{A,3∗}𝒪_{\overline{X}_{A,3}}}

whence cartesian diagrams of ring homomorphisms

(16.5.36)  \xymatrix{E_{U} \ar[r] \ar[d] & E_{U,1} \ar[d] & A_{U} \ar[r] \ar[d] & A_{U,1} \ar[d] \\ E_{U,2} \ar[r] & E_{U,3} & A_{U,2} \ar[r] & A_{U,3}.}

We endow each E_{U,i} with its α_{0}-adic topology 𝒯_{i}; then 𝒯_{i} is the discrete topology for i = 2, 3, and the topological ring (E_{U,i}, 𝒯_{i}) is perfectoid for i = 1, 2, 3 (see (16.4.24)). We also define the power-multiplicative norms | · |_{I,i} : E_{U,i} → ℝ_{+} associated with the ideal IE_{U,i} as in (16.5.23), and we get consequently the normed topological rings W (E_{U,i}, 1) as in (16.5.25).

Claim 16.5.37. The induced diagrams of rings

𝒲 (E) : \xymatrix{W (E_{U}) \ar[r] \ar[d] & W (E_{U,1}) \ar[d] \\ W (E_{U,2}) \ar[r] & W (E_{U,3})} 𝒲 (E, 1) : \xymatrix{W (E_{U}, 1) \ar[r] \ar[d] & W (E_{U,1}, 1) \ar[d] \\ W (E_{U,2}, 1) \ar[r] & W (E_{U,3}, 1)}

are cartesian.

Proof of the claim. The assertion is clear for the left diagram. Next, set

E_{U}^{+} := {x ∈ E_{U} | |x|_{I} ≤ 1}

and define likewise the subring E_{U,i}^{+} ⊂ E_{U,i} for i = 1, 2. For every x ∈ E_{U} and i = 1, 2, denote by x^{(i)} ∈ E_{U,i} the image of x; since we already know that diagram 𝒲 (E) is cartesian, the assertion for 𝒲 (E, 1) comes down to checking that if x ∈ E_{U} and |x^{(i)}|_{I,i} ≤ ρ^{−s} for i = 1, 2 and some s ∈ ℕ[1/p], then |x|_{I} ≤ ρ^{−s}. The latter is equivalent in turn to the following. Suppose that for every strictly positive ε ∈ ℕ[1/p] and every b ∈ I^{⟨s+ε⟩} we have b^{(i)}x^{(i)} ∈ \overline{E}_{i}; then z := bx ∈ E_{U}^{+}. Now, for such x and b pick y_{i} ∈ \overline{E} with y_{i}^{(i)} = b^{(i)}x^{(i)} for i = 1, 2; then (z − y_{1})(z − y_{2}) ∈ J_{1} ∩ J_{2} = 0, and especially, z is integral over \overline{E}. But \overline{E} ⊂ E_{U}^{+}, and E_{U}^{+} is integrally closed in E_{U} (remark 9.1.4(vi)), whence the claim. ♢

Claim 16.5.38. The diagram of rings 𝒲 (E, 1) ⊗_{W (E)} A is cartesian.

Proof of the claim. Consider the complex in C^{[0,1]}(W (E)-Mod)

K^{•} : 0 → W (E_{U,1}, 1) ⊕ W (E_{U,2}, 1) → W (E_{U,3}, 1) → 0

whose differential is deduced from 𝒲 (E, 1), so that H^{0}K^{•} = W (E_{U}, 1). Taking into account proposition 9.3.47(i), we have a short exact sequence

0 → K^{•} ⟶^{\underline{α}·1_{K^{•}}} K^{•} → K^{•}/\underline{α}K^{•} → 0

and by the same token, scalar multiplication is injective on H^{0}K^{•}, so we get a long exact cohomology sequence

0 → H^{0}(K^{•})/\underline{α} · H^{0}(K^{•}) ⟶^{f\underline{α}} H^{0}(K^{•}/\underline{α}K^{•}) → H^{1}K^{•} ⟶^{\underline{α}} H^{1}K^{•}

and the claim comes down to checking that f\underline{α} is an isomorphism, or equivalently, that scalar multiplication by \underline{α} is injective on H^{1}K^{•}. However, notice that H^{1}K^{•} is a quotient of the W (E)-module W (E_{U,3}, 1), and the latter is annihilated by τ_{E}(α_{0}). Since \underline{α} = τ_{E}(α_{0}) + p\underline{u} for an invertible element \underline{u} of W (E), we are reduced to checking that scalar multiplication by p is injective on H^{1}K^{•}, i.e. we may assume that \underline{α} = p, and then it suffices to show that f_{p} is an isomorphism. However, it follows easily from proposition 16.5.26(iii) that

(16.5.39)  pW (E_{U,i}) ∩ W (E_{U,i}, 1) = pW (E_{U,i}, 1)  for i = 1, 2, 3

and likewise for W (E_{U}, 1). Therefore, H^{0}(K^{•})/p·H^{0}(K^{•}) = E_{U}, and K^{•}/pK^{•} is the complex

0 → E_{U,1} ⊕ E_{U,2} → E_{U,2} → 0

deduced from the cartesian diagram (16.5.36). The claim follows. ♢ From claim 16.5.38we deduce a commutative diagram of rings

\xymatrix{A_{U} \ar[rrr] \ar[ddd] &  &  & A_{U,i} \ar[ddd] \\  & W (E_{U}, 1) ⊗_{W (E)} A \ar[r] \ar[d] \ar[ul]_{u_{U}} & W (E_{U,1}, 1) ⊗_{W (E)} A \ar[d] \ar@{-->}[ur]^{u_{U,1}} \\  & W (E_{U,2}, 1) ⊗_{W (E)} A \ar[r] \ar[dl]_{u_{U,2}} & W (E_{U,3}, 1) ⊗_{W (E)} A \ar[dr]^{u_{U,3}} \\ A_{U,2} \ar[rrr] &  &  & A_{U,3}}

whose two square subdiagrams are both cartesian. Recall that the topologies of E_{U,2} and E_{U,3} are discrete; taking into account theorem 16.5.13(i) it follows that E_{U,i} = E_{U,i}^{∘} = A_{U,i}^{∘} = A_{U,i} and W (E_{U,i}, 1) = W (E_{U,i}) for i = 2, 3, which easily implies that u_{U,2} and u_{U,3} are isomorphisms. Thus, to conclude the proof it suffices to show that u_{U,1} is an isomorphism. Hence, we may replace E by \overline{E}_{1} and assume from start that α_{0} is a regular element of E and the topology of E is α_{0}-adic. Let R be the image of u_{U}; under the current assumptions, p is a regular element of A (remark 16.4.25(ii)), and therefore

A ⊂ R ⊂ A_{U} ⊂ A[1/p]

whence R[1/p] = A[1/p]. Moreover, we notice :

Claim 16.5.40. The inclusion R ⊂ A induces an injective map R/pR → A_{U}/pA_{U}.

Proof of the claim. From (ii) and (16.5.39) we get a natural identification

(16.5.41)  R/pR →^{∼} W (E_{U}, 1)/(\underline{α}W (E_{U}, 1) + pW (E_{U}, 1)) →^{∼} E_{U}/α_{0}E_{U}

and from the short exact sequence of 𝒪_{U_{E}}-modules

0 → 𝒪_{U_{E}} ⟶^{α_{0}·} 𝒪_{U_{E}} → 𝒪_{U_{E}}/α_{0}𝒪_{U_{E}} → 0

we see that the natural map

E_{U}/α_{0}E_{U} → E_{U}^{0} := H^{0}(U_{E} ∩ Spec E/α_{0}E, 𝒪_{U_{E}}/α_{0}𝒪_{U_{E}})

is injective. Likewise, we see that the same holds for the natural map

A_{U}/pA_{U} → A_{U}^{0} := H^{0}(U_{A} ∩ Spec A/pA, 𝒪_{U_{A}}/p𝒪_{U_{A}}).

On the other hand, recall that the ring homomorphism u_{A} : W (E) → A induces an isomorphism of schemes φ_{0} : Spec A/pA →^{∼} Spec E/α_{0}E (see (16.4.28)), whence a ring isomorphism φ_{0,U}^{♭} : E_{U}^{0} →^{∼} A_{U}^{0} and a simple inspection of the definitions shows that the resulting diagram

\xymatrix{E_{U}/α_{0}E_{U} \ar[r]^{u_{U} ⊗_{W (E)}A/pA} \ar[d] & A_{U}/pA_{U} \ar[d] \\ E_{U}^{0} \ar[r]^{φ_{0,U}^{♭}} & A_{U}^{0}}

commutes. But it is easily seen that the isomorphism (16.5.41) identifies u_{U} ⊗_{W (E)} A/pA with the map R/pR → A_{U}/pA_{U} induced by the inclusion R ⊂ A, whence the claim. ♢

Now, let a ∈ A_{U} be any element, and pick n ∈ ℕ such that p^{n}a ∈ A; then there exists \underline{x} ∈ W (E) such that u_{U}(\underline{x}) = u_{A}(\underline{x}) = p^{n}a. To conclude, we shall show, by induction on i = 0, …, n that p^{n−i}a ∈ R. Indeed, we have just shown that p^{n}a ∈ R. Suppose that i ≥ 0 and p^{n−i}a ∈ R; if i = n, we are done, and if i < n, the image of p^{n−i}a vanishes in A_{U}/pA_{U}, hence also in R/pR, i.e. there exists b ∈ R with pb = p^{n−i}a; since p is regular in R, it follows that b = p^{n−i−1}a, as required. □

Corollary 16.5.42. In the situation of (16.5.25), we have :

(i) {Fil^{s}E_{U}} = Fil^{s}A_{U} for every s ∈ ℤ[1/p].

(ii) The map φ_{U}^{♭} induces a ring isomorphism E_{U}/α_{0}E_{U} →^{∼} A_{U}/pA_{U}.

Proof. (i): Let us first remark that

{Fil^{s−n}E_{U}} ∩ Fil^{s}A_{U} = {Fil^{s}E_{U}} for every n ∈ ℕ and every s ∈ ℤ[1/p].

Indeed, the case where n = 1 follows immediately from claim 16.5.30(ii), and then the general case follows by a simple induction on n ∈ ℕ : details left to the reader. But theorem 16.5.32(i) shows that A_{U} = ⋃_{n∈ℕ}{Fil^{s−n}E_{U}}, whence the assertion.

(ii) follows directly from theorem 16.5.32(i) and (16.5.41). □

As an application, we point out the following criterion :

Corollary 16.5.43. Let A be any perfectoid ring, U_{E} ⊂ V_{E} ⊂ X_{E} := Spec E(A) two constructible open subsets that contain the analytic locus. Set also U_{A} := φ^{−1}U_{E} and V_{A} := φ^{−1}V_{E}, where φ : X_{A} := Spec A → X_{E} is the continuous map of (16.4.24). Then we have :

(i) U_{E} is an affine scheme if and only if the same holds for U_{A}.

(ii) More generally, the inclusion map U_{E} → V_{E} is an affine morphism of schemes if and only if the same holds for the inclusion map U_{A} → V_{A}.

Proof. (i): Set E_{U} := 𝒪_{X_{E}}(U_{E}) and A_{U} := 𝒪_{X_{A}}(U_{A}); by lemma 15.6.10(i), we get commutative diagrams of schemes

\xymatrix{U_{E} \ar[r]^{j_{E}} \ar[dr]_{i_{E}} & Spec E_{U} \ar[d]^{j′_{E}} & U_{A} \ar[r]^{j_{A}} \ar[dr]_{i_{A}} & Spec A_{U} \ar[d]^{j′_{A}} \\  & X_{E} &  & X_{A}}

where i_{E}, i_{A}, j_{E} and j_{A} are open immersions. We have to show that j_{E} is an isomorphism if and only if the same holds for j_{A}. To this aim, set Z_{E} := X_{E} \U_{E} and Z_{A} := X_{A} \U_{A}; equivalently, we have to check that j_{E}^{′−1}Z_{E} = ∅ if and only if j_{A}^{′−1}Z_{A} = ∅. Now, since U_{E} contains the analytic locus of X_{E}, we may assume that Z_{E} ⊂ X_{E}^{0} := Spec E/α_{0}E, where (α_{n} | n ∈ ℕ) is any fixed distinguished element of Ker u_{A}, and likewise Z_{A} ⊂ X_{A}^{0} := Spec A/pA. On the other hand, by corollary 16.5.42(ii) we have a commutative diagram of schemes

\xymatrix{Spec A_{U}/pA_{U} \ar[r] \ar[d] & Spec E_{U}/α_{O}E_{U} \ar[d] \\ X_{A}^{0} \ar[r]^{φ_{0}} & X_{E}^{0}}

whose horizontal arrows are isomorphisms induced by φ and by the map φ_{U}^{♭} : E_{U} → A_{U} of proposition 16.4.34(i), and whose vertical arrows are the restrictions of j′_{E} and j′_{A}. Lastly, by construction we have φ_{0}^{−1}Z_{E} = Z_{A}, whence the contention.

(ii) is analogous : the open immersion U_{E} → V_{E} is affine if and only if for every affine open subset W ⊂ V_{E}, the intersection W ∩ U_{E} is still affine. The latter condition means that

Spec 𝒪_{X_{E}}(W ∩ U_{E}) ×_{W} Z_{E} = ∅ for every such W

which in turns is equivalent to the condition :

(16.5.44)  j_{E}^{′−1}(Z_{E} ∩ V_{E}) = ∅.

Likewise, we easily see that the inclusion U_{A} ⊂ V_{A} is affine if and only if j_{A}^{′−1}(Z_{A} ∩ V_{A}) = ∅, and since Z_{E} ⊂ X_{E}^{0}, this latter condition is equivalent to (16.5.44): details left to the reader. □

Lemma 16.5.45. In the situation of (16.5.23), let 𝒦 be any \overline{E}-submodule of E_{U}, and set :

W (𝒦 ) := {(x_{n} | n ∈ ℕ) | x_{n} ∈ 𝒦 ^{p^{n}} for every n ∈ ℕ} W (𝒦 , 1) := W (𝒦 )∩W (E_{U}, 1).

(notation of (16.5.28)). Denote also by (𝒦 ^{n})^{c} the topological closure of 𝒦 ^{n} in E_{U}, for every n ∈ ℕ, and by W (𝒦 , 1)^{c} the topological closure of W (𝒦 , 1) in W (E_{U}, 1). We have :

(i) W (𝒦 ) and W (𝒦 , 1) are W (\overline{E})-submodules of respectively W (E_{U}) and W (E_{U}, 1). (ii) W (𝒦 , 1)^{c} = M := {(x_{n} | n ∈ ℕ) ∈ W (E_{U}, 1) | x_{n} ∈ (𝒦 ^{p^{n}})^{c} for every n ∈ ℕ}. (iii) If 𝒦 is taut, u_{U}(W (𝒦 , 1)^{c}) = {𝒦 }. (iv) If 𝒦 is 1-taut and topologically closed in E_{U}, then u_{U} induces an isomorphism

W (𝒦 , 1)/\underline{α}W (𝒦 , 1) →^{∼} \{𝒦 \}.

Proof. (i): This is analogous to remark 9.3.28: details left to the reader.

(ii): Let \underline{x} := (x_{n} | n ∈ ℕ) ∈ M be any element, and fix any ε ∈ ℝ_{>0}. We shall exhibit inductively a sequence \underline{y} := (y_{n} | n ∈ ℕ) of elements of E_{U} such that

|\underline{y}|_{1} ≤ ε and \underline{x} + \underline{y} ∈ W (𝒦 , 1).

Indeed, let n ∈ ℕ, and suppose that we have already exhibited y_{i} ∈ (𝒦 ^{p^{i}})^{c} for i = 0, …, n − 1 such that \underline{y}^{(n)} := ∑_{i=0}^{n−1} p^{i} · τ_{E_{U}} (y_{i}) has norm < ε and such that if we set (z_{i}^{(n)} | i ∈ ℕ) := \underline{x} + \underline{y}^{(n)}, we have z_{i}^{(n)} ∈ 𝒦 ^{p^{i}} for every i = 0, …, n − 1. It follows easily that z_{i}^{(n)} ∈ (𝒦 ^{p^{i}})^{c} for every i ∈ ℕ. We may then pick y_{n} ∈ (𝒦 ^{p^{n}})^{c} such that |y_{n}|_{I} < ε^{p^{n}} and z_{n}^{(n)} + y_{n} ∈ 𝒦 ^{p^{n}}. Set \underline{y}^{(n+1)} := \underline{y}^{(n)} + p^{n} · τ_{E_{U}} (y_{n}) and (z_{i}^{(n+1)} | i ∈ ℕ) := \underline{x} + \underline{y}^{(n+1)}. In light of lemma 9.3.27(i) and claim 9.3.31 it is easily seen that z_{i}^{(n+1)} = z_{i}^{(n)} for every i = 0, …, n − 1, and z_{n}^{(n+1)} = z_{n}^{(n)} + y_{n}. Then, it is clear that the element \underline{y} := ∑_{n∈ℕ} p^{n} · τ_{E_{U}} (y_{n}) will do.

This shows that M ⊂ W (𝒦 , 1)^{c}. The converse inclusion is obvious, since ∏_{n∈ℕ}(𝒦 ^{p^{n}})^{c} is a closed subset of W (E_{U}) that contains W (𝒦 , 1) and the topology of W (E_{U}, 1) is finer than the one induced by W (E_{U}).

(iii): Recall that φ_{U}^{♭}(α_{0}) = \overline{u}\overline{A}(α_{0}) = pv for an element v ∈ \overline{A}^{×}. Then, since 𝒦 is taut, we see that p^{n} · φ_{U}^{♭}(x^{1/p^{n}}) = v^{−n}φ_{U}^{♭}(α_{0}^{n}x^{1/p^{n}}) ∈ {𝒦 } for every x ∈ 𝒦 ^{p^{n}}. It follows easily that u_{U}(\underline{x}) ∈ \{𝒦 \} for every \underline{x} ∈ W (𝒦 ), and thus u_{U}(W (𝒦 , 1)^{c}) ⊂ \{𝒦 \}.

For the converse inclusion, pick a strictly positive ε ∈ ℕ[1/p] such that α_{0} ∈ I^{⟨ε⟩}; it was observed in the proof of lemma 16.5.29(iii) that φ_{U}^{♭} induces an \overline{A}/p\overline{A}-linear isomorphism

(Fil^{s}𝒦 )/(Fil^{s+ε}𝒦 ) →^{∼} (Fil^{s}{𝒦 })/(Fil^{s+ε}{𝒦 }) for every s ∈ ℤ[1/p]

and moreover Fil^{s}{𝒦 } = {𝒦 } ∩ {Fil^{s}E_{U}} = {𝒦 } ∩ Fil^{s}A_{U} for every s ∈ A_{U} (lemma 16.5.29(iii) and corollary 16.5.42), hence the filtration Fil^{•}{𝒦 } is exhaustive and separated on {𝒦 }. Thus, say that x ∈ Fil^{s}{𝒦 } for some s ∈ ℤ[1/p]; it follows easily that we may find a sequence (y_{n} | n ∈ ℕ) of elements of E_{U} such that y_{n} ∈ Fil^{s+nε}𝒦 for every n ∈ ℕ, and the series ∑_{n∈ℕ} φ_{U}^{♭}(y_{n}) converges to x in the topology of A_{U}. Clearly τ_{E_{U}} (y_{n}) ∈ W (𝒦 ) and |τ_{E_{U}} (y_{n})|_{1} ≤ ρ^{s+nε} for every n ∈ ℕ, so the series ∑_{n∈ℕ} τ_{E_{U}} (y_{n}) converges in W (E_{U}, 1) to an element \underline{z} ∈ W (𝒦 )^{c}, and finally u_{U}(\underline{z}) = ∑_{n∈ℕ} u_{U} ∘ τ_{E_{U}} (y_{n}) = x.

(iv): The assumptions on 𝒦 imply that 𝒦 ^{p^{n}} is topologically closed in E_{U}, for every n ∈ ℕ; combining with (ii) and (iii), it follows already that the induced map W (𝒦 , 1)/\underline{α}W (𝒦 , 1) → {𝒦 } is surjective. Next, say that α_{0} ∈ I^{⟨ε⟩} for some element ε > 0 of ℤ[1/p]; notice that Fil^{s}𝒦 is still 1-taut and topologically closed in E_{U} for every s ∈ ℤ[1/p]; taking into account lemma 16.5.29(iii), we may therefore reduce to the case where 𝒦 ⊂ Fil^{−nε}E_{U} for some n ∈ ℕ. We argue by induction on n : indeed, the assertion for n = 0 is already known, by theorem 16.3.36(iv). Thus, suppose that n > 0, and that the assertion is already known for 𝒦 ∩ Fil^{(1−n)ε}E_{U}; let \underline{x} := (x_{n} | n ∈ ℕ) ∈ W (𝒦 , 1) be an element in the kernel of u_{U}. This means that ∑_{n∈ℕ} p^{i} · φ_{U}^{♭}(x_{n}^{1/p^{n}}) = 0 in A_{U}, and we need to check that \underline{x} ∈ \underline{α}W (𝒦 , 1). Now, since 𝒦 is 1-taut, we have \underline{z} := (x_{n+1}^{1/p} | n ∈ ℕ) ∈ W (𝒦 , 1) as well, so that

φ_{U}^{♭}(x_{0}) = − ∑_{n∈ℕ} p^{n+1} · φ_{U}^{♭}(x_{n+1}^{1/p^{n+1}}) ∈ Fil^{(1−n)ε}A_{U}.

In view of claim 16.5.30(ii), we deduce that x_{0} ∈ Fil^{(1−n)ε}𝒦 . Moreover, \underline{x} = τ_{E_{U}} (x_{0}) + p · \underline{z}, and recall that \underline{α} = τ_{E}(α_{0}) + p · \underline{u} for an invertible element \underline{u} ∈ W (E); we conclude that \underline{x} is the sum of

τ_{E_{U}} (x_{0}) − \underline{u}^{−1} · τ_{E}(α_{0}) · \underline{z} ∈ W (Fil^{(1−n)ε}𝒦 , 1) and \underline{u}^{−1} · \underline{α} · \underline{z} ∈ \underline{α}W (𝒦 , 1).

Especially, τ_{E_{U}} (x_{0}) − \underline{u}^{−1} · τ_{E}(α_{0}) · \underline{z} lies as well in the kernel of u_{U}; by inductive assumption it is therefore an element of \underline{α}W (Fil^{(1−n)ε}𝒦 , 1), and the proof is concluded. □

Proposition 16.5.46. Let A be any complete and separated topological ring, whose topology is linear and coarser than the p-adic topology, and v ∈ Cont(A) any valuation. We have :

(i) The mapping v ∘ \overline{u}_{A} is a continuous valuation of the topological ring E := E(A) (see remark 9.4.12(ii)).

(ii) The rule : v ↦ v ∘ \overline{u}_{A} defines continuous maps

Cont(\overline{u}_{A}) : Cont(A) → Cont(E) Cont^{+}(\overline{u}_{A}) : Cont^{+}(A) → Cont^{+}(E).

(iii) The diagrams of continuous maps

\xymatrix{Cont(A) \ar[r]^{Cont(\overline{u}_{A})} \ar[d] & Cont(E) \ar[d] & Cont^{+}(A) \ar[r]^{Cont^{+}(\overline{u}_{A})} \ar[d] & Cont^{+}(E) \ar[d] \\ Spec A \ar[r]^{_{Spec \overline{u}_{A}}} & Spec E & Spec A \ar[r]^{_{Spec \overline{u}_{A}}} & Spec E}

commute, where Spec \overline{u}_{A} is defined as in (16.1) and the vertical arrows in the left (resp. right) diagram are the restrictions of the support maps σ_{A} and σ_{E} (resp. of the center maps σ_{A}^{+} and σ_{E}^{+}) : see remark 9.2.4(iii,v).

(iv) If v′ ∈ Cont(A) is a primary (resp. secondary) specialization of v, then Cont(\overline{u}_{A})(v′)

is a primary (resp. secondary) specialization of Cont(\overline{u}_{A})(v).

(v) The map Cont(\overline{u}_{A}) restricts to a surjection from the set of secondary specializations of v onto the set of secondary specializations of Cont(\overline{u}_{A})(v).

(vi) If A is a P-ring, Cont(\overline{u}_{A}) also restricts to a surjection from the set of secondary generizations of v onto the set of secondary generizations of Cont(\overline{u}_{A})(v).

Proof. (i): Since both v and \overline{u}_{A} are continuous morphisms of pointed monoids, the same holds for v ∘ \overline{u}_{A}. Thus, it remains only to check that

v(\overline{u}_{A}(a_{•} + b_{•})) ≤ max(v(a_{0}), v(b_{0})) for every a_{•}, b_{•} ∈ E.

However, recall that

\overline{u}_{A}(a_{•} + b_{•}) = lim_{n→+∞}(a_{n} + b_{n})^{p^{n}}

(remark 9.4.12(ii)). Since the pointed monoid Γ_{v∘} is separated for its topology 𝒯_{Γ_{v}} (notation of definition 15.3.12(i)), it follows that v(\overline{u}_{A}(a_{•} + b_{•})) is the unique limit point of the sequence (v(a_{n} + b_{n})^{p^{n}} | n ∈ ℕ). However :

v(a_{n} + b_{n})^{p^{n}} ≤ max(v(a_{n})^{p^{n}}, v(b_{n})^{p^{n}}) = max(v(a_{0}), v(b_{0}))

whence the assertion.

(ii): Let a_{•}, b_{•} ∈ E be any two elements, and set U := {v ∈ Cont(E) | v(a_{•}) ≤ v(b_{•}) ≠ 0}; directly from the definition we find :

Cont(\overline{u}_{A})^{−1}U = \{w ∈ Cont(A) | w(a_{0}) ≤ w(b_{0}) ≠ 0\}

whence the assertion.

(iii) and (iv) are immediate from the definitions.

(v): Set w := Cont(\overline{u}_{A})(v) and notice that 𝔭 := σ_{A}(v) is a closed subset in the p-adic topology of A; we remark the following :

Claim 16.5.47. Denote by \overline{v} (resp. \overline{w}) the residual valuation of v (resp. of w). Then we have :

(i) The map \overline{u}_{𝔭} : κ(w) → κ(v) of lemma 16.1.5 restricts to a local morphism of multiplicative monoids \overline{u}_{v}^{+} : (κ(w), \overline{w})^{+} → (κ(v), \overline{v})^{+}.

(ii) Let also \overline{κ}(v) (resp. \overline{κ}(w)) be the residue field of κ(v)^{+} (resp. of κ(w)^{+}). There exists a unique ring homomorphism \overline{φ}_{v} : \overline{κ}(w) → \overline{κ}(v) fitting into a commutative diagram

\xymatrix{κ(w)^{+} \ar[r]^{^{\overline{u}^{+}^{v}}} \ar[d] & κ(v)^{+} \ar[d] \\ \overline{κ}(w) \ar[r]^{_{\overline{φ}_{v}}} & \overline{κ}(v)}

whose vertical arrows are the projections.

Proof of the claim. (i): Indeed, a simple inspection of the definitions yields a commutative diagram of monoids

\xymatrix{κ(w) \ar[rr]^{^{\overline{u}^{𝔭}}} \ar[dr]_{\overline{w}} &  & κ(v) \ar[dl]^{\overline{v}} \\  & Γ_{v}}

whence the assertion : details left to the reader.

(ii): Let \overline{π}_{v} : κ(v)^{+} → \overline{κ}(v) be the projection; since \overline{u}_{v}^{+} is local, we are easily reduced to checking that the composition ψ := \overline{π}_{v} ∘ \overline{u}_{v}^{+} is a ring homomorphism. However, ψ is obviously a morphism of multiplicative monoids, so it suffices to show that

\overline{v}(\overline{u}_{v}^{+}(x_{1} + x_{2}) − \overline{u}_{v}^{+}(x_{1}) − \overline{u}_{v}^{+}(x_{2})) < 1 for every x_{1}, x_{2} ∈ κ(w)^{+}.

However, let π_{w} : E → κ(w) and π_{v} : A → κ(v) be the projections; we may write

x_{1} = π_{w}(β_{1})/π_{w}(γ)  and x_{2} = π_{w}(β_{2})/π_{w}(γ)  for some β_{1}, β_{2}, γ ∈ E

with w(β_{1}), w(β_{2}) ≤ w(γ) ≠ 0. Set x_{3} := x_{1} + x_{2} and β_{3} := β_{1} + β_{2}; then

\overline{u}_{v}^{+}(x_{i}) = π_{v} ∘ \overline{u}_{A}(β_{i})/π_{v} ∘ \overline{u}_{A}(γ) for i = 1, 2, 3.

Let y := \overline{u}_{A}(β_{3}) − \overline{u}_{A}(β_{1}) − \overline{u}_{A}(β_{2}); we are then reduced to checking that v(y) < v(\overline{u}_{A}(γ)). However, proposition 9.3.62 says that y = ∑_{n∈ℕ} p^{n+1}z_{n} where each z_{n} is a finite ℤ_{p}-linear combination of terms of the form \overline{u}_{A}(β_{1}^{λ}β_{2}^{1−λ}) with λ, 1 − λ ∈ ℕ[1/p]. Clearly w(β_{1}^{λ}β_{2}^{1−λ}) ≤ w(γ) for every such λ, so that

v(p^{n+1}z_{n}) ≤ v(p)^{n+1} · v(\overline{u}_{A}(γ)) for every n ∈ ℕ.

Since v is a continuous valuation and v(p) is final in Γ_{v}, the contention follows easily : details left to the reader. ♢

Now we argue as in the proof of lemma 9.2.25(iii). Namely, let w′ be a secondary specialization of w; then κ(w′) = κ(w), and κ(w′)^{+} ⊂ κ(w)^{+}. The image W of κ(w′)^{+} in \overline{κ}(w) is a valuation ring, and by corollary 9.1.25 there exists a valuation ring V of \overline{κ}(v) that dominates \overline{φ}_{v}(W). The preimage of V in κ(v)^{+} is a valuation ring of κ(v), and the corresponding valuation v′ is a secondary specialization of v. Lastly, v′ is continuous, by remark 15.3.13(iv), and by construction we have Cont(\overline{u}_{A})(v′) = w′.

(vi): Clearly \overline{u}_{A} induces an injective morphism of ordered groups i : Γ_{w} → Γ_{v}, and Spec i is surjective, by remark 9.1.2(v). Now, let w′ be a secondary generization of w := Cont(\overline{u}_{A})(v), so that w′ = w_{Δ} for a convex subgroup Δ ⊂ Γ_{w}, and we may write Δ = i^{−1}Δ′ for some convex subgroup Δ′ ⊂ Γ_{v}. If Δ′ ≠ Γ_{v′}, the projection Γ_{v∘} → (Γ_{v}/Δ′)_{∘} is continuous (remark 15.3.13(iii)), so v′ := v_{Δ′} is a secondary specialization of v in Cont(A) with Cont(\overline{u}_{A})(v′) = w′. Hence, we may assume that w′ is a trivial valuation. It follows easily that E^{∘∘} ⊂ Ker w′. On the other hand, we may find finitely many elements β_{1}, …, β_{k} ∈ E^{∘∘} such that the system \overline{u}_{A}(β_{1}), …, \overline{u}_{A}(β_{k}) generates an open ideal of A (see the proof of lemma 16.2.7(i)). Summing up, we see that v′ is an analytic valuation of A, especially v′ ∈ Cont(A), and the proof is concluded. □

Remark 16.5.48. Keep the notation of proposition 16.5.46; for every v ∈ Cont(A) we have the commutative diagram of topological monoids

\xymatrix{E \ar[r]^{^{\overline{u}^{A}}} \ar[d]_{E(v)} & A \ar[d]^{v} \\ E(Γ_{v∘}) \ar[r]^{_{\overline{u}_{Γv∘}}} & Γ_{v∘}}

(where Γ_{v∘} is endowed with its natural topology 𝒯_{Γ_{v}} as in definition 15.3.12(i)) and under the natural identification E(Γ_{v∘}) →^{∼} E(Γ_{v})_{∘}, the map \overline{u}_{Γv∘} corresponds to (\overline{u}_{Γ_{v}})_{∘}. Then E(Γ_{v}) inherits from Γ_{v} a unique ordering such that \overline{u}_{Γ_{v}} is an injective morphism of ordered abelian groups. Moreover, since the p-Frobenius endomorphism of Γ_{v} is injective, it is easily seen that the topology of E(Γ_{v∘}, 𝒯_{Γ_{v}}) is the same as the topology 𝒯_{E(Γ_{v})} provided by definition 15.3.12(i) (details left to the reader). In view of proposition 16.5.46(i), it follows that E(v) is a continuous valuation on E(v), and we have

Cont(\overline{u}_{A})(v) = E(v) in Cont(E).

Proposition 16.5.49. Let A be any perfectoid ring. We have :

(i) A is a valuation ring if and only if the same holds for E := E(A).

(ii) Suppose that A is a valuation ring, and let

v_{A} : A → Γ_{A∘} := (Frac A)/A^{×}  and  v_{E} : E → Γ_{E∘} := (Frac E)/E^{×}

be the valuations of A and E (see remark 9.1.13(iv)). Then :

(a) v_{A} ∈ Cont(A) and v_{E} ∈ Cont(E).

(b) The map \overline{u}_{A} induces an isomorphism of ordered abelian groups :

γ_{A} : Γ_{E} →^{∼} Γ_{A}

fitting into a commutative diagram

\xymatrix{E \ar[r]^{^{\overline{u}^{A}}} \ar[d]_{v_{E}} & A \ar[d]^{v_{A}} \\ Γ_{E∘} \ar[r]^{γ_{A∘}} & Γ_{A∘}.}

(c) More precisely, if (α_{n} | n ∈ ℕ) is any distinguished element of Ker u_{A}, the map \overline{u}_{A} induces a group isomorphism

Frac(E)^{×}/(1 + α_{0}E) →^{∼} Frac(A)^{×}/(1 + pA).

(iii) Let V be a valuation ring, and 𝒯_{V} a linear, complete, separated and f-adic topology on V that is coarser than the p-adic topology. Suppose that :

(a) The value group of V is p-divisible.

(b) The Frobenius endomorphism of V/pV is surjective.

Then (V, 𝒯_{V}) is a perfectoid ring.

Proof. (i): We remark :

Claim 16.5.50. If E is a valuation ring, then for every a ∈ A there exist t ∈ A^{×} and e ∈ E such that a = t · \overline{u}_{A}(e).

Proof of the claim. Fix any distinguished element (α_{n} | n ∈ ℕ) of Ker u_{A}, and recall that p = w · \overline{u}_{A}(α_{0}) for some w ∈ A^{×} (lemma 16.2.7(iii)). Since the p-adic topology is separated on A, there exists n ∈ ℕ such that a ∈ p^{n}A \ p^{n+1}A, so that a = p^{n}b for some b ∈ A \ pA. Then there exists β ∈ E such that b − \overline{u}_{A}(β) = pz for some z ∈ A (lemma 16.2.7(i)), and notice that β ∉ α_{0}E, since otherwise we would have \overline{u}_{A}(β) ∈ pA, whence b ∈ pA, a contradiction. Thus, α_{0} = β · γ for some element γ of the maximal ideal of E, and we may write

x = p^{n}b = p^{n} · (\overline{u}_{A}(β) + pz) = w^{n} · \overline{u}_{A}(α_{0}^{n}β) · (1 + \overline{u}_{A}(γ) · wz).

We need to check that u := 1 + \overline{u}_{A}(γ) · wz is invertible in A; to this aim, since p lies in the Jacobson radical of A, it suffices to show that the image of u is invertible in A/pA, and we are further reduced to checking that the image \overline{γ} of \overline{u}_{A}(γ) lies in the Jacobson radical of A/pA. However, u_{A} induces a ring isomorphism ω : E/α_{0}E →^{∼} A/pA (remark 16.3.7(ii)), and ω^{−1}(\overline{γ}) is the class of γ, whence the contention. ♢

Now, suppose that E is a valuation ring; directly from claim 16.5.50and corollary 16.3.63(ii) we deduce that A is a domain. Then, let a_{1}, a_{2} ∈ A be any two elements, and write a_{i} = t_{i} · \overline{u}_{A}(e_{i}) with t_{i} ∈ A^{×} and e_{i} ∈ E for i = 1, 2; we may assume that e_{1} ∈ e_{2}E, whence a_{1} ∈ a_{2}A, which easily implies that A is a valuation ring.

Conversely, suppose that A is a valuation ring; if the field of fractions K of A has characteristic 0, we know already from lemma 9.4.16 that E is a valuation ring. If the characteristic of K equals p, the ring E is isomorphic to A, whence the assertion.

(ii.a): In light of proposition 9.1.16(i,ii) we see that either the topology 𝒯_{A} of A is discrete, or else it agrees with the valuation topology of A. In either case, it is clear that v_{A} ∈ Cont(A). The same applies to E, since the latter is perfectoid as well.

(ii.b): From claim 16.5.50we immediately see that γ_{A} is surjective, and the injectivity follows from remark 16.5.48.

(ii.c): By (ii.b) and the snake lemma, we are reduced to showing that \overline{u}_{A} induces a group isomorphism

E^{×}/(1 + α_{0}E) →^{∼} A^{×}/(1 + pA).

However, \overline{u}_{A} induces a ring isomorphism E/α_{0}E →^{∼} A/pA (remark 16.3.7(ii)), whence a group isomorphism (E/α_{0}E)^{×} →^{∼} (A/pA)^{×}, and it remains only to observe that the natural maps

E^{×}/(1 + α_{0}E) → (E/α_{0}E)^{×} A^{×}/(1 + pA) → (A/pA)^{×}

are isomorphisms, since α_{0}E (resp. pA) lies in the Jacobson radical of E (resp. of A).

(iii): In light of proposition 9.1.16(i), we see that 𝒯_{V} is either the discrete topology or the valuation topology on V. If 𝒯_{V} is discrete and coarser than the p-adic topology, then V must be an 𝔽_{p}-algebra, so in this case the assertion is a special case of example 16.3.2(i). If 𝒯_{V} is the valuation topology, let us endow K := Frac V with its valuation topology 𝒯_{K}; then (K, 𝒯_{K}) is a Tate ring, by proposition 9.1.16(ii), hence V admits an ideal of adic definition of the form I := aV, for a non-zero topologically nilpotent element a ∈ V (corollary 8.3.20(ii)). Since the value group of V is p-divisible, and p is topologically nilpotent in V, we may assume that p ∈ I^{p}, in which case, assumption (iii.b) implies that the Frobenius endomorphism of V/pV induces an isomorphism V/I →^{∼} V/I^{p}. By the same token, we easily see that (V, 𝒯_{V}) is a P-ring, so the assertion follows from theorem 16.4.1. □

Remark 16.5.51. Let A, A′ be two perfectoid valuation rings with valuations v : A → Γ_{A∘}, v′ : A′ → Γ_{A′∘}, and f : A → A′ a continuous ring homomorphism; according to proposition 16.5.49(i), both E := E(A) and E′ := E(A′) are valuation rings, and we denote by v_{E} : E → Γ_{E∘}, v_{E′} : E′ → Γ_{E′∘} their respective valuations. Then it follows easily from proposition 16.5.49(ii.c) that f is injective if and only if the same holds for E(f) : E → E′. Moreover, if f is injective there exist unique group homomorphisms φ_{E} : Γ_{E} → Γ_{E′}, φ_{A} : Γ_{A} → Γ_{A′} fitting into a commutative diagram :

\xymatrix{Γ_{E∘} \ar[d]_{γ_{A∘}} \ar[rrr]^{φ_{E∘}} & E \ar[l]_{v_{E}} \ar[r]^{E(f)} \ar[d]_{_{\overline{u}_{A}}} & E′ \ar[r]^{v_{E′}} \ar[d]^{_{\overline{u}_{A′}}} & Γ_{E′∘} \ar[d]^{γ_{A′∘}} \\ Γ_{A∘} \ar[rrr]_{φ_{A∘}} & A \ar[l]_{v_{A}} \ar[r]^{f} & A′ \ar[r]^{v_{A′}} & Γ_{A′∘}}

where γ_{A} and γ_{A′} are the group isomorphisms provided by proposition 16.5.49(ii.b) : details left to the reader.

Theorem 16.5.52. Let A be any perfectoid ring, and set E := E(A). We have :

(i) The induced map Cont(\overline{u}_{A}) of proposition 16.5.46(ii) is a homeomorphism.

(ii) For every rational subset R of Cont(E), the preimage Cont(\overline{u}_{A})^{−1}R is a rational

subset of Cont(A) (see (15.3.16)). (iii) Cont(\overline{u}_{A}) restricts to bijections

Cont(A)_{a} →^{∼} Cont(E)_{a}  and  Cont(A)_{na} →^{∼} Cont(E)_{na}.

(iv) Let w ∈ Cont(A) be any valuation, and set v := Cont(\overline{u}_{A})(w). Then \overline{u}_{A} induces group isomorphisms

γ_{w} : Γ_{v} →^{∼} Γ_{w}  and  cΓ_{v} →^{∼} cΓ_{w}.

Proof. (ii): If A is perfectoid, both Cont(A) and Cont(E) are spectral spaces, and their rational subsets form a basis of constructible open subsets. Thus, let (e_{i} | i = 0, …, n) be any finite system of elements of E that generate an open ideal, and set R := R_{E}(\frac{e_{1}}{e_{0}} , …, \frac{e_{n}}{e_{0}} ) ∩ Cont(E). Let also a_{i} := \overline{u}_{A}(e_{i}) for every i = 0, …, n. Then

Cont(\overline{u}_{A})^{−1}R = R_{A}(\frac{a_{1}}{a_{0}} , …, \frac{a_{n}}{a_{0}} ) ∩ Cont(A).

However, the system (a_{i} | i = 0, …, n) generates an open ideal of A (corollary 16.3.40(ii)), whence the assertion.

(iii): Let \underline{α} := (α_{n} | n ∈ ℕ) be any distinguished element of Ker u_{A}; since α_{0} is topologically nilpotent, it lies in every open prime ideal of E(A/pA). Likewise, pA is contained in every open prime ideal of A, and due to lemmata 16.2.7(i,iii) and 16.1.1(iv), the map \overline{u}_{A} induces an isomorphism

ω : E/α_{0}E →^{∼} A/pA

of topological rings, so that Cont(\overline{u}_{A}) agrees with Cont(ω) on the closed subset Cont(E/α_{0}E) of Cont(E), and induces a homeomorphism

Cont(E/α_{0}E) →^{∼} Cont(A/pA).

It is also clear that Cont(ω) maps Cont(E/α_{0}E)_{na} = Cont(E)_{na} homeomorphically onto Cont(A/pA)_{na} = Cont(A)_{na}, so we need only to check that the restriction

C(A) := Cont(A) ∩ R_{A}(\frac{p}{p}) → C(E) := Cont(E) ∩ R_{E}(\frac{α_{0}}{α_{0}} )

of Cont(\overline{u}_{A}) is a bijection. Set C^{+}(A) := C(A) ∩ Spv^{+}A and C^{+}(E) := C(E) ∩ Spv^{+}E. Clearly Cont(\overline{u}_{A}) restricts to a mapping C^{+}(A) → C^{+}(E), and we shall first exhibit an inverse mapping

σ : C^{+}(E) → C^{+}(A)

for Cont(\overline{u}_{A})_{|C^{+}(A)}. To this aim, let v ∈ C^{+}(E) be any valuation; then v factors through a ring homomorphism π : E → V := κ(v)^{+} and the residual valuation \overline{v} : V → Γ_{v∘} of v. Since E is a perfect 𝔽_{p}-algebra, the field κ(v) is perfect, hence V is a perfect 𝔽_{p}-algebra as well : indeed, it is clear that x^{p} ∈ V if and only if x ∈ V, for every x ∈ κ(v). Let us endow V with its valuation topology 𝒯_{V}, so that \overline{v} is continuous for the topology 𝒯_{V} (remark 15.3.13(i)).

Claim 16.5.53. The completion (V^{∧}, 𝒯_{V}^{∧}) of (V, 𝒯_{V}) is a perfectoid valuation ring.

Proof of the claim. We know that V^{∧} is a valuation ring, by proposition 9.1.16(iii), and its topology 𝒯_{V}^{∧} is adic with a principal ideal of adic definition, since the same holds for 𝒯_{V} (lemma 15.3.14(ii)). Moreover, it is easily seen that the Frobenius endomorphism Φ_{V} of V is a homeomorphism for the topology 𝒯_{V}, hence it induces a homeomorphism Φ_{V}^{∧} on V^{∧}; but clearly Φ_{V}^{∧} is just the Frobenius endomorphism of V^{∧}, so the latter is a perfect topological 𝔽_{p}-algebra, and the assertion follows (see example 16.3.2(i)). ♢

Let π^{∧} : E → V^{∧} be the composition of π with the completion map V → V^{∧}; since π^{∧} is a continuous map of perfectoid 𝔽_{p}-algebras (remark 15.3.13(i)), the image of \underline{α} under the map W (π^{∧}) is still distinguished in W (V^{∧}), so claim 16.5.53and proposition 16.5.49(i,ii) say that

A_{V} := W (V^{∧}) ⊗_{W (E)} A

is a perfectoid valuation ring whose valuation v_{A_{V}}, is continuous, and the map \overline{u}_{A_{V}} induces an isomorphism of ordered abelian groups from the value group of V^{∧} onto that of A_{V}

γ_{V^{∧}} : Γ_{V^{∧}} →^{∼} Γ_{A_{V}} such that v_{A_{V}} ∘ \overline{u}_{A_{V}} = γ_{V^{∧}∘} ∘ \overline{v}^{∧}

where \overline{v}^{∧} is the valuation of V^{∧}. Moreover, we get a continuous map of perfectoid rings

φ := W (π^{∧}) ⊗_{W (E)} A : A → A_{V}

and w := v_{A_{V}} ∘ φ is a continuous valuation of A such that

Cont(\overline{u}_{A})(w) = w ∘ \overline{u}_{A} = v_{A_{V}} ∘ \overline{u}_{V^{∧}} ∘ π^{∧} = γ_{V^{∧}∘} ∘ \overline{v}^{∧} ∘ π^{∧} = γ_{V^{∧}∘} ∘ v.

Since v(α_{0}) ≠ 0, we have w(p) ≠ 0 as well, so we obtain the sought map σ, by setting σ(v) := w, and we already see that σ is a right inverse for Cont(\overline{u}_{A})_{|C^{+}(A)}. We notice that since v(E) ∩ Γ_{v} generates the group Γ_{v}, the subset w ∘ \overline{u}_{A}(E) ∩ Γ_{A_{V}} generates the group Γ_{A_{V}}, so that \overline{u}_{A} induces an isomorphism

(16.5.54)  Γ_{v} →^{∼} Γ_{σ(v)}  for every v ∈ C^{+}(E).

Next, we check that σ is a left inverse as well. To this aim, let v′ ∈ C^{+}(A) be any valuation; then v′ factors through a ring homomorphism π′ : A → V′ := κ(v′)^{+} and the residual valuation \overline{v}′ : V′ → Γ_{v′∘}. By lemma 15.3.14(ii), the map π′ is continuous for the valuation topology 𝒯_{V′} of V′, and 𝒯_{V′} is adic with a principal ideal of adic definition. Since pV′ ≠ 0 and p is topologically nilpotent in A, the ideal pV′ is open and topologically nilpotent in V′, so 𝒯_{V′} agrees with the p-adic topology on V′. Let (V^{′∧}, 𝒯_{V′}^{∧}) be the completion of (V′, 𝒯_{V′}); then V^{′∧} is a valuation ring whose valuation \overline{v}^{′∧} : V^{′∧} → Γ_{v′∘} extends \overline{v}′ (proposition 9.1.16(iii,v)), and we let π^{′∧} : A → V^{′∧} be the composition of π′ with the completion map V′ → V^{′∧}. Then V′_{E} := E(V^{′∧}) is a valuation ring whose valuation is

\overline{v}′_{E} := \overline{v}^{′∧} ∘ \overline{u}_{V^{′∧}} : V′_{E} → Γ_{v′∘}

(lemma 9.4.16) and E(π^{′∧}) : E → V′_{E} is a continuous ring homomorphism. Since π^{′∧} ∘ \overline{u}_{A} = \overline{u}_{V^{′∧}} ∘ E(π^{′∧}), we deduce that

(16.5.55)  v′_{E} := \overline{v}′_{E} ∘ E(π^{′∧}) = Cont(\overline{u}_{A})(v′)  in Cont(E).

Claim 16.5.56. (i) The topology of V′_{E} agrees with the valuation topology 𝒯_{V′_{E}}.

(ii) (V′_{E}, 𝒯_{V′_{E}}) is a perfectoid ring.

Proof of the claim. (i): Since the quotient topology induced by 𝒯_{V′} on V′/pV′ is discrete, the topology of V′_{E} is linear, complete and separated (remark 9.4.9(i)), so it suffices to check that this topology is not discrete (proposition 9.1.16(i)). To this aim, since α_{0} is topologically nilpotent in E, it suffices to check that ν := E(π^{′∧})(α_{0}) ≠ 0. However :

\overline{u}_{V^{′∧}}(ν) = π^{′∧} ∘ \overline{u}_{A}(α_{0}) = π^{′∧}(pu) for some u ∈ A^{×}

(lemma 16.2.7(iii)), and since v′(p) ≠ 0, we must have π^{′∧}(p) ≠ 0, whence the claim.

(ii): We have just seen that ν is a non-zero topologically nilpotent element of V′_{E}, so 𝒯_{V′_{E}} agrees with the ν-adic topology on V′_{E}; then the assertion follows from example 16.3.2(i). ♢

Arguing as in the foregoing we see that the image of \underline{α} in W (V′_{E}) is still distinguished, so

A_{V′} := W (V′_{E}) ⊗_{W (E)} A

is a perfectoid valuation ring whose value group is naturally identified with the value group Γ_{v′_{E}} of v′_{E} (proposition 16.5.49(i,ii.b) and claim 16.5.56(ii)); by the same token, we get a natural isomorphism

β : E(A_{V′}) →^{∼} V′_{E} such that π_{V′_{E}} ∘ W (β) = u_{A_{V′}}

where π_{V′_{E}} : W (V′_{E}) → A_{V′} is the projection (proposition 16.2.16(iv)). Moreover, in light of the commutative diagram

\xymatrix{W (E) \ar[r]^{u_{A}} \ar[d]_{W (E(π^{′∧}))} & A \ar[d]^{π^{′∧}} \\ W (V′_{E}) \ar[r]^{u_{V^{′∧}}} & V^{′∧}}

we get a unique homomorphism of A-algebras

ψ : A_{V′} → V^{′∧} such that u_{V^{′∧}} = ψ ∘ π_{V′_{E}}.

There follows a diagram :

[figure: A_{V′}; v′_{A}; ψ; V′_{E}; \overline{u}_{A_{V′}} ∘β^{−1}; ^{\overline{u}^{V^{′∧}}}; _{\overline{v}_{′}_{E}}; V^{′∧}; _{\overline{v}_{′∧}}; Γ_{v′_{E}∘}; Γ_{v′_{E}∘}; j; Γ_{v′∘}]

whose two square subdiagrams commute, where v′_{A} is the valuation of A_{V′} and j the inclusion map of Γ_{v′_{E}∘} into Γ_{v′∘}. Let also φ′ : A → A_{V′} be the structure map of the A-algebra A_{V′}.

Claim 16.5.57. (i) The top subdiagram commutes as well.

(ii) The external subdiagram also commutes, i.e. \overline{v}^{′∧} ∘ ψ = j ∘ v′_{A}.

(iii) The topology of A_{V′} agrees with its valuation topology.

(iv) ψ is an isomorphism of topological rings and Γ_{v′_{E}} = Γ_{v′}.

Proof of the claim. (i): Indeed, taking into account lemma 16.1.1(iv) we may compute :

\begin{matrix}ψ ∘ \overline{u}_{A_{V′}} = & ψ ∘ u_{A_{V′}} ∘ τ_{E(A_{V′})} \\ = & ψ ∘ π_{V′_{E}} ∘ W (β) ∘ τ_{E(A_{V′})} \\ = & u_{V′} ∘ W (β) ∘ τ_{E(A_{V′})} \\ = & u_{V′} ∘ τ_{V′_{E}} ∘ β \\ = & \overline{u}_{V′} ∘ β. \end{matrix}

(ii): Let x ∈ A_{V′} be any element; since the left square subdiagram commutes, we may find y ∈ V′_{E} such that \overline{u}_{A_{V′}} ∘ β^{−1}(y) · u = x for some u ∈ A_{V′}^{×}, and therefore ψ(x) = \overline{u}_{V^{′∧}}(y) · ψ(u). Obviously, ψ(u) is invertible in V^{′∧}, so that \overline{v}^{′∧} ∘ ψ(x) = \overline{v}^{′∧} ∘ \overline{u}_{V^{′∧}}(y) = j ∘ \overline{v}′_{E}(y) = j ∘ v′_{A}(x), as stated.

(iii): In light of proposition 9.1.16(i), it suffices to check that the topology of A_{V′} is not discrete. However, notice that by construction φ′ is a continuous map; thus, if the topology of A_{V′} were discrete, Ker φ′ would be an open ideal, and then the same would hold for Ker π^{′∧}. But this is absurd, since v′ is analytic.

(iv): As an immediate consequence of (ii), we already see that ψ is injective. Then it is clear that φ′ factors uniquely through π′ and an injective map V′ → A_{V′} whose composition with ψ equals the completion map. Thus, the image of ψ is a dense subring of V^{′∧}; combining with (ii), we deduce already the stated equality of value groups. Then by (iii) the topology of A_{V′} agrees with the topology induced by V^{′∧} via ψ. Since A_{V′} is complete and separated, theorem 8.2.8(iii) implies that ψ is an isomorphism. ♢

From claim 16.5.57(ii) we get the following equalities in Cont(A) :

σ ∘ Cont(\overline{u}_{A})(v′) = σ(\overline{v}^{′∧} ∘ \overline{u}_{V^{′∧}} ∘ E(π_{E}^{′∧}))
= σ(\overline{v}′_{E} ∘ E(π^{′∧}))
= v′_{A} ∘ φ′
= \overline{v}^{′∧} ∘ ψ ∘ φ′
= \overline{v}^{′∧} ∘ π^{′∧}
= v′

as required. Let κ(v′_{E})^{∧+} be the completion of κ(v′_{E})^{+} for its valuation topology, π′_{E} : E → κ(v′_{E})^{+} the natural ring homomorphism (whose composition with the valuation κ(v′_{E})^{+} → Γ_{v′∘} yields v′_{E}), and π_{E}^{′∧} : E → κ(v′_{E})^{∧+} the completion of π′_{E}; we notice as well :

Claim 16.5.58. There exists a ring isomorphism

ω_{E}^{∧} : κ(v′_{E})^{∧+} →^{∼} V′_{E} such that ω_{E}^{∧} ∘ π_{E}^{′∧} = E(π^{′∧}).

Proof of the claim. The map E(π^{′∧}) factors through π′_{E} and a unique injective ring homomorphism ω_{E} : κ(v′_{E})^{+} → V′_{E} such that \overline{v}′_{E} ∘ ω_{E} : κ(v′_{E})^{+} → Γ_{v′∘} is the residual valuation. Thus, the valuation topology of κ(v′_{E})^{+} agrees with the topology induced from V′_{E} via ω_{E}, so the latter extends to a continuous ring homomorphism ω_{E}^{∧} : κ(v′_{E})^{∧+} → V′_{E} such that \overline{v}′_{E}∘ω_{E}^{∧} is the valuation of κ(v′_{E})^{∧+}. The map ω_{E}^{∧} is injective (proposition 8.2.13(i)), and clearly ω_{E}^{∧} ∘π_{E}^{′∧} = E(π^{′∧}), so it remains only to check that ω_{E}^{∧} is bijective. To this aim, set V′′ := W (κ(v′_{E})^{∧+}) ⊗_{W (E)} A; by the foregoing we know already that V′′ is a perfectoid valuation ring with value group Γ_{v′}, and it suffices to show that

ω_{A}^{∧} := W (ω_{E}^{∧}) ⊗_{W (E)} A : V′′ → A_{V′}

is an isomorphism. However, set π_{A}^{′∧} := W (π_{E}^{′∧}) ⊗_{W (E)} A; by remark 16.5.51, we know already that ω_{A}^{∧} is injective, and moreover we have a commutative diagram

[figure: A; π_{A}^{′∧}; φ′; V′′; v′′; ω_{A}^{∧}; Γ_{v′∘}; A_{V′}; v′_{A}; Γ_{v′∘}]

where v′′ is the valuation of V′′. It follows that the valuation topology of V′′ is induced by the valuation topology of A_{V′} via ω_{A}^{∧}. Furthermore, we deduce that v′′ ∘ π_{A}^{′∧} = v′_{A} ∘ φ′ = v′, so π_{A}^{′∧} factors through π′ : A → V′ and an injective map V′ → V′′ whose composition with ω_{A}^{∧} has dense image, by virtue of claim 16.5.57(iv). Since V′′ is complete, it follows that ω_{A}^{∧} is surjective, whence the claim. ♢

Next, let A^{+} be the smallest subring of integral elements of A (remark 15.4.2(iv)); by theorem 16.5.13(ii,iii), the ring A^{+} is perfectoid, and E(A^{+}) is naturally identified with the smallest ring E^{+} of integral elements of E. Moreover, we get a commutative diagram

\xymatrix{Cont(A) \ar[r]^{Cont(\overline{u}_{A})} \ar[d]_{Cont(j_{A})} & Cont(E) \ar[d]^{Cont(j_{E})} \\ Cont(A^{+}) \ar[r]^{Cont(\overline{u}_{A^{+}})} & Cont(E^{+})}

where j_{A} : A^{+} → A and j_{E} : E^{+} → E are the inclusion maps. Since A^{+} is the integral closure in A of the ℤ-subalgebra generated by A^{∘∘}, it is easily seen that Cont(A^{+}) = Cont^{+}(A^{+}), and likewise for Cont(E^{+}). Therefore, C(A^{+}) = C^{+}(A^{+}) and C(E^{+}) = C^{+}(E^{+}), and the foregoing shows that Cont(\overline{u}_{A^{+}}) restricts to a bijection

C(E^{+}) →^{∼} C(A^{+}).

Lastly, proposition 15.3.18(ii) says that Cont(j_{A}) restricts to a bijection from the analytic valuations of A to those of A^{+}, and likewise for Cont(j_{E}), whence (iii).

(iv): Let π_{A} : A → A/pA and π_{E} : E → E/α_{0}E be the projections; if w(p) = 0, then w = Cont(π_{A})(w′) for a unique w′ ∈ Cont(A/pA), and v = Cont(π_{E})(v′), where v′ := Cont(ω)(w′). Since π_{A} and π_{E} are surjective, obviously Γ_{w} = Γ_{w′} and Γ_{v} = Γ_{v′}, and likewise for the characteristic subgroups; since ω is a ring isomorphism, it induces a group isomorphism Γ_{v′} →^{∼} Γ_{w′} that identifies the respective characteristic subgroups, whence the assertion, in this case. Next, suppose that w ∈ C(A), and set w′ := Cont(j_{A})(w), v′ := Cont(\overline{u}_{A^{+}})(w′); then w′ ∈ C^{+}(A^{+}), and from (iii) and (16.5.54) we see that \overline{u}_{A^{+}} induces an isomorphism γ_{w′} : Γ_{w′} →^{∼} Γ_{v′}. Moreover, we have a commutative diagram of groups

\xymatrix{Γ_{w′} \ar[r]^{γ_{w′}} \ar[d]_{i_{w}} & Γ_{v′} \ar[d]^{i_{v}} \\ Γ_{w} \ar[r]^{γ_{w}} & Γ_{v}}

where i_{w} and i_{v} are the injective maps induced by j_{A} and j_{E}. In order to prove that γ_{w} is an isomorphism, it suffices therefore to check that i_{w} and i_{v} are surjective. To this aim, it suffices to show that S_{A^{+}} := w(A^{+}) ∩ Γ_{w} generates Γ_{w} and S_{E^{+}} := v(A^{+}) ∩ Γ_{v} generates Γ_{v}. However, notice that pA ⊂ A^{+} and α_{0}E ⊂ E^{+}; since w(p) ≠ 0 and v(α_{0}) ≠ 0, it follows easily that S_{A^{+}} and S_{A} := w(A) ∩ Γ_{w} generate the same subgroup of Γ_{w}, and likewise for S_{E^{+}}. To conclude, it suffices to remark that S_{A} generates Γ_{w}, and likewise for Γ_{v}.

Next, obviously γ_{w} restricts to an injective group homomorphism cΓ_{v} → cΓ_{w}. To show that this map is surjective, consider any a ∈ A such that w(a) > 1; since \overline{u}_{A/pA} is surjective (lemma 16.2.7(i)), we may write a = \overline{u}_{A}(β) + pb for some β ∈ E and b ∈ A, and since pb ∈ A^{∘∘}, we have w(pb) < 1, so that w(a) = v(β), whence the contention.

(i): We argue as in the proof of proposition 15.3.19 : by (ii) and (iii) we know already that Cont(\overline{u}_{A}) is a spectral and bijective map of spectral spaces, so it suffices to check that Cont(\overline{u}_{A}) is a closed map, and by proposition 8.1.47(i) and corollary 8.1.50(i) we are further reduced to showing that Cont(\overline{u}_{A}) is specializing. Hence, let w ∈ Cont(A) be any valuation, and v′ ∈ Cont(A) any specialization of v := Cont(\overline{u}_{A})(w); we may find a valuation v′′ of E that is both a secondary specialization of v and a primary generization of v′, and v′′ ∈ Cont(E), by remark 15.3.13(iv). By proposition 16.5.46(v), there exists a secondary specialization w′′ of w such that Cont(\overline{u}_{A})(w′′) = v′′. By (iv) we know that w′′ and v′′ have the same value groups and characteristic subgroups, so we may find a primary specialization w′ of w such that Cont(\overline{u}_{A})(w′) = v′, whence the assertion. □

16.5.59. Let \underline{X} be any perfectoid quasi-affinoid scheme, and with the notation of (16.5.19) and (16.5.20), let us set

\underline{A} := (A_{X}^{∘}, A_{X}^{+}, X) := Γ^{∘}(\underline{X}) and \underline{E} := (E_{X}^{∘}, E_{X}^{+}, X_{E}) := E(\underline{A}).

Especially, E_{X}^{∘} = E(A_{X}^{∘}) and E_{X}^{+} = E(A_{X}^{+}); it is then clear that Cont(\overline{u}_{A_{X}^{∘}}) maps the subset Spa (A_{X}^{∘}, A_{X}^{+}) into Spa (E_{X}^{∘}, E_{X}^{+}). Conversely, let v : A_{X}^{∘} → Γ_{v} be any continuous valuation such that v ∘ \overline{u}_{A_{X}^{∘}} ∈ Spa (E_{X}^{∘}, E_{X}^{+}), and \underline{α} any distinguished element of Ker u_{A_{X}^{∘}}; for every a ∈ A_{X}^{+} there exists b ∈ A_{X}^{+} and e ∈ E_{X}^{+} such that a = \overline{u}_{A_{X}^{∘}}(e) + α_{0}b, and since α_{0}b ∈ A_{X}^{∘∘}, we deduce that v(a) ≤ 1, so v ∈ Spa (A_{X}^{∘}, A_{X}^{+}). Combining with proposition 16.5.46(iii), we deduce that Cont(\overline{u}_{A_{X}^{∘}}) restricts to a homeomorphism

Spa (\overline{u}\underline{A}) : Spa \underline{A} →^{∼} Spa \underline{E}

and taking into account the natural identifications Spa \underline{A} →^{∼} Spa \underline{X} and Spa \underline{E} →^{∼} Spa E(\underline{X}), we get as well an induced homeomorphism

Spa(\overline{u}\underline{X}) : Spa \underline{X} →^{∼} Spa E(\underline{X}).

Moreover, it is easily seen that every morphism f : \underline{Y} → \underline{X} of perfectoid quasi-affinoid schemes yields a commutative diagram

(16.5.60)  \xymatrix{Spa \underline{Y} \ar[r]^{Spa(\overline{u}\underline{Y} )} \ar[d]_{Spa f} & Spa E(\underline{Y}) \ar[d]^{Spa E(f)} \\ _{Spa \underline{X}} \ar[r]^{Spa(\overline{u}\underline{X})} & Spa E(\underline{X}).}

Corollary 16.5.61. In the situation of (16.5.59), let x ∈ Spa \underline{X} be any element and set y := Spa (\overline{u}\underline{X})(x). Denote also by π_{x}^{∧+} : A_{X}^{+} → κ(x)^{∧+} and π_{y}^{∧+} : E_{X}^{+} → κ(y)^{∧+} the natural maps (notation of (15.5.12)), and endow κ(x)^{∧+} and κ(y)^{∧+} with the unique ring topologies 𝒯_{x}^{∧} and 𝒯_{y}^{∧} such that π_{x}^{∧+} and π_{y}^{∧+} are adic ring homomorphisms. We have :

(i) The topological rings (κ(x)^{∧+}, 𝒯_{x}^{∧}) and (κ(y)^{∧+}, 𝒯_{y}^{∧}) are perfectoid valuation rings. (ii) There exists a unique isomorphism of topological rings :

ω^{∧+} : E(κ(x)^{∧+}) →^{∼} κ(y)^{∧+}  such that  ω^{∧+} ∘ E(π_{x}^{∧+}) = π_{y}^{∧+}.

Proof. Suppose first that p ∈ Ker π_{x}^{+}; in this case x corresponds to a continuous ring homomorphism \overline{π}_{x} : A_{X}^{∘}/pA_{X}^{∘} → κ(x) and y corresponds to the continuous ring homomorphism

\overline{π}_{y} := \overline{π}_{x} ∘ φ : E_{X}^{∘}/α_{0}E_{X}^{∘} → κ(x), where α_{0} is as in (16.5.59) and φ : E_{X}^{∘}/α_{0}E_{X}^{∘} →^{∼} A_{X}^{∘}/pA_{X}^{∘}

is the isomorphism of topological rings induced by \overline{u}_{A_{X}^{∘}}. There follows a unique isomorphism of valued fields ω : κ(y) →^{∼} κ(x) fitting into the commutative diagram

\xymatrix{E_{X}^{∘}/α_{0}E_{X}^{∘} \ar[r]^{φ} \ar[d]_{_{\overline{π}_{y}}} & A_{X}^{∘}/pA_{X}^{∘} \ar[d]^{_{\overline{π}_{x}}} \\ κ(y) \ar[r]^{ω} & κ(x).}

Let us now endow κ(x)^{+} and κ(y)^{+} with the unique ring topologies such that the natural maps π_{x}^{+} : A_{X}^{+} → κ(x)^{+} and π_{y}^{+} : E_{X}^{+} → κ(y)^{+} are adic; then clearly ω restricts to an isomorphism ω^{+} : κ(y)^{+} →^{∼} κ(x)^{+} of topological rings, and assertion (ii) follows straightforwardly in this case. Moreover, since E is a perfect 𝔽_{p}-algebra, the same holds for its quotient E/Ker π_{y}, and then also for κ(y) = Frac (E/Ker π_{y}), and for κ(y)^{∧} as well (example 9.3.48(ii)). By remark 9.4.9(v), it follows easily that κ(y)^{∧+} is a perfect adic and f-adic topological ring (details left to the reader), and hence it is perfectoid (example 16.3.2(i)); the same then holds for κ(x)^{∧+}.

Next, if π_{x}^{+}(p) ≠ 0, the point x is analytic, and the same then holds for y. In this case, it is easily seen that the topologies 𝒯_{x}^{∧} and 𝒯_{y}^{∧} are separated and not discrete, so they coincide with the respective valuation topologies (proposition 9.1.16(i)). Recall that the inclusion map i : A_{X}^{+} → A_{X}^{∘} induces a homeomorphism Cont(i)_{a} : Cont(A_{X}^{∘})_{a} →^{∼} Cont(A_{X}^{+})_{a} and likewise for Cont(E_{X}^{∘}) (proposition 15.3.18(ii)); moreover, if x′ := Cont(i)(x), we have a unique isomorphism of valued fields κ(x′) →^{∼} κ(x) fitting into the commutative diagram

\xymatrix{A_{X}^{+} \ar[rr]^{i} \ar[d]_{π_{x′}^{+}} &  & A_{X}^{∘} \ar[d]^{π_{x}^{∘}} \\ κ(x′)^{+} \ar@{^{(}->}[r] & κ(x′) \ar[r]^{∼} & κ(x)}

where π_{x}^{∘} and π_{x′}^{+} are the natural maps, and likewise for y and its image y′ ∈ Cont(E_{X}^{+}). Hence we may regard x and y as elements of Cont(A_{X}^{+}) and respectively Cont(E_{X}^{+}), and in this case assertions (i) and (ii) follow from claims 16.5.57(iv) and 16.5.58. □

16.6. Graded perfectoid rings. This section is mainly dedicated to the construction and investigation of certain angular Rees algebras that shall be useful in the study of blowing up morphisms of perfectoid formal schemes.

Proposition 16.6.1. Let (Γ, +, 0) be a monoid, (A, \underline{B}) a Γ-graded structure on the topological ring (A, 𝒯 ), and i_{0} : gr_{0}B → A the inclusion map. Suppose that A is a P-ring. Then we have :

(i) If the p-Frobenius endomorphism p_{Γ} of Γ is injective, there exists a finitely generated graded ideal J of B such that JA is an ideal of definition of A.

(ii) Suppose that i_{0} is c-adic and p_{Γ} is injective. Then gr_{0}B is a P-ring, and if A is perfec-

toid, the same holds for gr_{0}B. (iii) If A is a perfectoid ring, Ker u_{A} is generated by a distinguished element in A(gr_{0}B).

Proof. (i): By lemma 16.2.3(iv), we may write p = b^{p}u for some u ∈ A^{×} and b ∈ A, and by the same token, u = x^{p} + py for some x, y ∈ A, so that p = a^{p} + p^{2}w, with a := bx and w := u^{−1}y. Let now (a_{γ} | γ ∈ Γ) be the sequence attached to a, as in remark 8.5.2(iii), and c (resp. d) the limit of the Cauchy net (c_{S} | S ⊂ Γ) (resp. (d_{S} | S ⊂ Γ)) provided by lemma 8.5.6(i), where S ranges over the finite subsets of Γ, and a_{S}^{p} = c_{S} + p · d_{S} for every such S (with a_{S} defined as in remark 8.5.2(iii)). In light of proposition 8.5.3(ii), we deduce that a^{p} = c + pd, whence p = c + pd + p^{2}w. Moreover, since the p-Frobenius map of Γ is injective, we have π′_{pγ}(c_{S}) = a_{γ}^{p} for every S ⊂ Γ and every γ ∈ S; therefore π′_{pγ}(c) = a_{γ}^{p} for every γ ∈ Γ, and especially, π′_{0}(c) = a_{0}^{p}. Consequently

p = a_{0}^{p} + p · d_{0} + p^{2}w_{0}

where (d_{γ} | γ ∈ Γ) and (w_{γ} | γ ∈ Γ) are the sequences attached to d and respectively w. Next, notice that a is topologically nilpotent in A, hence a_{γ} is topologically nilpotent in B, for every γ ∈ Γ (lemma 8.5.6(ii)), and then the same holds for d_{0}. Thus v := 1 − d_{0} − pw_{0} is invertible in A, with inverse given by the convergent series ∑_{n∈ℕ}(d_{0} + pw_{0})^{n}, and since gr_{0}B is closed in A, we conclude that v ∈ (gr_{0}B)^{×}. Summing up, we arrive at the identity

pv = a_{0}^{p} with v ∈ (gr_{0}B)^{×} and a_{0} ∈ gr_{0}B.

Lastly, by proposition 8.5.11(i.a) we know already that A admits an ideal of adic definition of the form J′A, for some graded ideal J′ ⊂ B, and we set J := J′ + a_{0}B; then it is easily seen that JA is an ideal of definition for the P-ring A.

(ii): We remark :

Claim 16.6.2. In the situation of the proposition, suppose that p_{Γ} is injective. Then we have : (i) The Frobenius endomorphism of B/pB restricts to surjective maps

gr_{γ}B ⊗_{ℤ} 𝔽_{p} → gr_{pγ}B ⊗_{ℤ} 𝔽_{p} for every γ ∈ Γ.

(ii) gr_{γ}B = 0 for every γ ∈ Γ \ E(Γ).

Proof of the claim. (i): Say that b ∈ gr_{γ}B; by lemma 16.2.3(iv) there exists x ∈ A such that x^{p} − b ∈ pA. Let (x_{γ} | γ ∈ Γ) be the system attached to x as in remark 8.5.2(iii); it was remarked in the proof of (i) that the system (x_{γ}^{p} | γ ∈ ℕ) is a Cauchy net in B, and if y ∈ A denotes the limit of this net, then x^{p} − y ∈ pA, so that y − b ∈ pA as well. Consequently,

b = π′_{γ}(b) ≡ π′_{γ}(y) = x_{p^{−1}γ}^{p} (mod p · gr_{γ}B)

if γ ∈ pΓ, and otherwise b ∈ p · gr_{γ}B. Assertion (i) follows already. Next, using part (i) of the claim it is easily seen that, if γ ∈ Γ \ E(Γ), we have gr_{γ}B ⊗_{ℤ} 𝔽_{p} = 0, whence (ii), since the p-adic topology is separated on B. ♢

Claim 16.6.2 says especially that the Frobenius endomorphism of gr_{0}B ⊗_{ℤ} 𝔽_{p} is surjective. On the other hand, we already know by proposition 8.5.11(iii) that the topology of gr_{0}B is adic with a finitely generated ideal of adic definition I. Moreover, the proof of (i) shows that there exists a_{0} ∈ gr_{0}B such that p = a_{0}^{p}u for some u ∈ (gr_{0}B)^{×}; then, it is easily seen that gr_{0}B is a P-ring with ideal of definition I + a_{0} · gr_{0}B.

Lastly, suppose that A is perfectoid, and to ease notation set B_{0} := gr_{0}B. By the foregoing, we know already that B_{0} is a P-ring, and then i_{0} is adic, by lemma 8.3.24(i.c). Pick any ideal of definition I_{0} of B_{0}, and say that I_{0} is generated by k elements, for some k ∈ ℕ; by remark 16.3.58, we can assume that p ∈ I_{0}^{k(p−1)+1}, and therefore I := I_{0}A is an open ideal of A that fulfills the same condition. Set J_{0} := I_{0}^{(p)} and J := I^{(p)} = J_{0}A; it follows that the Frobenius endomorphism of A/pA induces an isomorphism Φ_{I} : gr_{I}^{•}A → gr_{J}A (propositions 16.3.8(i) and 16.3.59(ii)).

Claim 16.6.3. The natural maps

gr_{I}^{n}B := I_{0}^{n}B/I_{0}^{n+1}B → gr_{I}^{n}A gr_{J}^{n}B := J_{0}^{n}B/J_{0}^{n+1}B → gr_{J}^{n}A

are bijective for every n ∈ ℕ.

Proof of the claim. By claim 8.3.26we have (I^{n}B)^{c} = I^{n}A, so that I^{n} = I^{n}B + I^{n+1}, whence the surjectivity of the first map. For the injectivity, we need to show that I_{0}^{n+1}B = I_{0}^{n}B ∩ I^{n+1}. Hence, let x ∈ I_{0}^{n}B ∩ I^{n+1}; then x = ∑_{i=1}^{r} x_{r}a_{r} for some r ∈ ℕ and certain x_{i} ∈ I_{0}^{n+1} and a_{i} ∈ A (i = 1, …, r). Say that x ∈ gr_{S}B := ⨁_{γ∈S} gr_{γ}B for some finite subset S ⊂ Γ, and let π′_{S} : A → gr_{S}B be the induced projection (the sum of the canonical γ-projections as in remark 8.5.2(iii), for γ ranging over the subset S). Then x = π′_{S}(x) = ∑_{i=0}^{r} x_{i} · π′_{S}(a), whence the contention. The same argument applies to the second map. ♢

From claim 16.6.3we conclude that the map Φ_{I} : gr_{I}^{•}B → gr_{J}^{•}B is also an isomorphism. But then, since p_{Γ} is injective, clearly the direct summand gr_{0}Φ_{I} := Φ_{I_{0}} : gr_{I_{0}}^{•}B_{0} → gr_{J_{0}}^{•}B_{0} is still an isomorphism, and therefore B_{0} is perfectoid, by theorem 16.3.64.

(iii): By proposition 16.3.8(i), we may assume that 𝒯 is the p-adic topology of A. Let j_{A} : A → B′ be as in remark 8.5.2(iii); clearly p^{n}B′ ∩ B_{0} = p^{n}B_{0} and p^{n}A ⊂ j_{A}^{−1}(p^{n}B′), so B_{0} ∩ p^{n}A = p^{n}B_{0} for every n ∈ ℕ, and therefore the topology of B_{0} agrees with its p-adic topology. It follows that i_{0} is an adic map, and therefore B_{0} is even perfectoid, by (ii); then any distinguished element of Ker u_{B_{0}} generates Ker u_{A}, by proposition 16.2.23(ii). □

16.6.4. Let (Γ, 0, +) be a p-perfect monoid, (A, \underline{B}) a perfectoid ring with Γ-graded structure, and set (E, \underline{B}_{E}) := E(A, \underline{B}). Let also β_{1}, …, β_{r} (resp. γ_{1}, …, γ_{r}) be a finite system of elements of E (resp. of Γ), such that β_{i} ∈ gr_{γ_{i}} \underline{B}_{E} for every i = 1, …, r. Set f := (\overline{u}_{A}(β_{1}), …, \overline{u}_{A}(β_{r})) and define the ring R_{r,0} and its ideals I_{r,0}^{⌈s⌉} for every s ∈ ℝ_{+}, as in (16.4.4); we endow \underline{B} with the R_{r,0}-module structure induced by the unique ring homomorphism u : R_{r,0} → \underline{B} such that u(T_{i}^{1/p^{n}}) := \overline{u}_{A}(β_{i}^{1/p^{n}}) for every i = 1, …, r and every n ∈ ℕ.

Proposition 16.6.5. In the situation of (16.6.4), suppose moreover that the translation map Γ → Γ : γ ↦ γ + γ_{i} is injective for every i = 1, …, r. Then we have :

(i) Tor_{i}^{R_{r,0}}(R_{r,0}/I_{r,0}^{⌈s⌉}, \underline{B}) = 0 for every i > 0 and every s ∈ ℝ_{+}.

(ii) The ring \underline{B} satisfies condition (a)_{f}^{un} of (7.8.21).

Proof. (i): Denote by Γ_{0} ⊂ Γ the submonoid generated by (γ_{1}/p^{n}, …, γ_{r}/p^{n} | n ∈ ℕ), and let \underline{B}_{0} := Γ_{0} ×_{Γ} \underline{B}. Recall that R_{r,0} := ℤ[P_{r}], where P_{r} := ℕ[1/p]^{⊕r}, and let φ : P_{r} → Γ_{0} be the unique morphism of monoids such that φ(e_{i}) := γ_{i} for i = 1, …, r, where (e_{1}, …, e_{r}) is the standard minimal system of generators of ℕ^{⊕r}. With this notation, R_{r,0} is a P_{r}-graded ℤalgebra, and we set \underline{S} := (R_{r,0})_{/Γ_{0}} (see definition 7.6.1(v)). Then clearly u induces a morphism of Γ_{0}-graded ℤ-algebras \underline{S} → \underline{B}_{0}, and \underline{B} can be regarded naturally as a Γ-graded \underline{S}_{/Γ}-module. We remark, quite generally :

Claim 16.6.6. Let (Δ, 0, +) be a monoid, Δ_{0} ⊂ Δ a submonoid such that the translation maps Δ → Δ : δ ↦ δ + δ_{0} are injective for every δ_{0} ∈ Δ_{0}. Let also \underline{S} = (S, gr_{•}S) be a Δ_{0}-graded ℤ-algebra, N a Δ_{0}-graded \underline{S}-module, and M a Δ-graded \underline{S}_{/Δ}-module. Then we have :

(i) The product M′ := ∏_{δ∈Δ} gr_{δ}M carries a natural S-module structure such that the natural map M → M′ is S-linear.

(ii) The induced map τ_{i} : Tor_{i}^{S}(N, M) → Tor_{i}^{S}(N, M′) is injective, for every i ∈ ℕ.

Proof of the claim. (i): We define a ℤ-bilinear map

μ_{δ_{0}} : gr_{δ_{0}}S × M′ → M′ for every δ_{0} ∈ Δ_{0}

by the rule : (s, (m_{δ} | δ ∈ Δ)) ↦ (m′_{δ} | δ ∈ Δ), where m′_{δ} := s·m_{γ} if there exists a (necessarily unique) γ ∈ Δ with γ + δ_{0} = δ, and otherwise m′_{δ} := 0. The sum of the maps μ_{δ_{0}} yields a well defined ℤ-bilinear map

μ : S × M′ → M′

and it is easily seen that μ defines an S-module structure on M′ with the sought property.

(ii): In view of remark 7.6.18(i), we may find a resolution L_{•} → N by free S-modules such that L_{i} is a Δ_{0}-graded \underline{S}-module, and the differential d_{i+1} : L_{i+1} → L_{i} is a morphism of Δ_{0}- graded \underline{S}-modules, for every i ∈ ℕ. Then, for every i ∈ ℕ the tensor product P_{i} := L_{i} ⊗_{S} M carries a natural structure of Δ-graded \underline{S}_{/Δ}-module : namely

gr_{δ}P_{i} := Im( ⨁_{(δ_{0},γ)∈D(δ)} gr_{δ_{0}}L_{i} ⊗_{ℤ} gr_{γ}M → P_{i}) for every δ ∈ Δ

where D(δ) is the set of all pairs (δ_{0}, γ) ∈ Δ_{0} × Δ such that γ + δ_{0} = δ. With this definition, it is then easily seen that the differential d_{i+1}^{P} := d_{i+1} ⊗_{S} 1_{M} of P_{•} is a morphism of Δ-graded \underline{S}_{/Δ}-modules, whence an induced Δ-graded \underline{S}_{/Δ}-module structure on Tor_{i}^{S}(N, M), for every i ∈ ℕ. Moreover, we get a ℤ-bilinear map

ψ : L_{i} × M′ → P′_{i} := ∏_{δ∈Δ} gr_{δ}P_{i} for every i ∈ ℕ

namely, the unique one whose restriction gr_{δ_{0}}L_{i} × M′ → P′_{i} for every δ_{0} ∈ Δ_{0} is given by the rule : (l, (m_{δ} | δ ∈ Δ)) ↦ (l ⊗ m′_{δ} | δ ∈ Δ), where m′_{δ} := m_{γ} if γ + δ_{0} = δ, and m′_{δ} := 0 if there is no such γ. Then it is easily seen that ψ is even S-bilinear, so it induces an S-linear map

L_{i} ⊗_{S} M′ → P′_{i} for every i ∈ ℕ.

Furthermore, the resulting diagram of S-linear maps

\xymatrix{L_{i+1} ⊗_{S} M′ \ar[r] \ar[d]_{d_{i+1}⊗_{S}1_{M^{∧}}} & P′_{i+1} \ar[d]^{∏_{δ∈Δ} gr_{δ}d_{i+1}^{P}} \\ L_{i} ⊗_{S} M′ \ar[r] & P′_{i}}

commutes for every i ∈ ℕ, and we deduce an S-linear map

Tor_{i}^{S}(N, M′) → ∏_{δ∈Δ} gr_{δ}Tor_{i}^{S}(N, M) for every i ∈ ℕ

whose composition with τ_{i} is the natural injection. The claim follows. ♢

Now, set B′ := ∏_{γ∈Γ} gr_{γ}B, endow B′ with the linear topology defined as in remark 8.5.2(ii), and recall that the inclusion map j : B → B′ factors through the inclusion map B → A and a continuous map j_{A} : A → B′ (remark 8.5.2(iii)). Endow B′ with the S-module structure provided by claim 16.6.6(i); a direct inspection of the proof of loc.cit. shows that scalar multiplication by any homogeneous element s of S defines a continuous endomorphism of B′, and on the other hand we have j(s · b) = s · j(b) for every such s and every b ∈ B; since j is also a continuous map, we deduce that the same identity holds more generally for every b ∈ A, i.e. j_{A} is an S-linear map. Lastly, notice that the ideal I_{r,0}^{⌈s⌉} of \underline{S} is Γ_{0}-graded, so R_{r,0}/I_{r,0}^{⌈s⌉} is a Γ_{0}-graded \underline{S}-module for every s ∈ ℝ_{+}. By claim 16.6.6, it follows that the composition of the two natural maps

Tor_{i}^{R_{r,0}}(R_{r,0}/I_{r,0}^{⌈s⌉}, \underline{B}) → Tor_{i}^{R_{r,0}}(R_{r,0}/I_{r,0}^{⌈s⌉}, A) → Tor_{i}^{R_{r,0}}(R_{r,0}/I_{r,0}^{⌈s⌉}, B′)

is injective for every i ∈ ℕ; on the other hand, the middle term vanishes for i > 0, due to proposition 16.4.10(i), whence the contention.

(ii) follows from (i) and lemma 16.4.5(i). □

16.6.7. We consider now a perfectoid quasi-affinoid ring (A, A^{+}, U_{A}), and let (E, E^{+}, U_{E}) := E(A, A^{+}, U_{A}) (see (16.5.20)). Set as well X_{A} := Spec A, X_{E} := Spec E, and define as usual the f-adic rings A_{U} := 𝒪_{X_{A}}(U_{A}) and E_{U} := 𝒪_{X_{E}}(U_{E}), so that the restriction maps A → A_{U} and E → E_{U} are open, hence the image \overline{A} of A (resp. \overline{E} of E) is open in A_{U} (resp. in E_{U}), and we endow it with the topology induced by the projection A → \overline{A} (resp. E → \overline{E}). Suppose we are given a family (𝒦_{γ} | γ ∈ Γ) consisting of topologically closed 1-taut bounded \overline{E}-submodules of E_{U}, indexed by a p-perfect monoid (Γ, +, 0), and such that (notation of (16.5.28)) :

(a) 1 ∈ 𝒦_{0}
(b) 𝒦_{pγ} = 𝒦_{γ}^{p} for every γ ∈ Γ
(c) 𝒦_{γ} · 𝒦_{μ} ⊂ 𝒦_{γ+μ} for every γ, μ ∈ Γ

(notice that, in terms of the descending filtration Fil^{•}E_{U} defined in (16.5.23), a subset of E_{U} is bounded if and only if it lies in Fil^{s}E_{U}, for some s ∈ ℤ[1/p]). Set

ℰ := ⨁_{γ∈Γ} 𝒦_{γ} 𝒜 := ⨁_{γ∈Γ}{𝒦_{γ}}.

It is easily seen that the multiplication of E_{U} (resp. of A_{U}) induces a Γ-graded \overline{E}-algebra structure (resp. \overline{A}-algebra structure) on ℰ (resp. on 𝒜). Choose ideals of definition J for A and 𝒥 for E, and endow ℰ (resp. 𝒜) with its 𝒥 -adic (resp. J-adic) topology. Let also ℰ^{∧} (resp. 𝒜^{∧}) be the separated completion of ℰ (resp. of 𝒜).

Lemma 16.6.8. The pairs

(ℰ^{∧}, ℰ) and (𝒜^{∧}, 𝒜)

are topological rings with Γ-graded structures.

Proof. Indeed, the assertion comes down to checking that 𝒦_{γ} (resp. {𝒦_{γ}}) is a closed subset of ℰ^{∧} (resp. of 𝒜^{∧}) for every γ ∈ Γ, and notice that the topological closure of 𝒦_{γ} in ℰ^{∧} (resp. of {𝒦_{γ}} in 𝒜^{∧}) is the 𝒥 -adic (resp. J-adic) completion of 𝒦_{γ} (resp. of {𝒦_{γ}}). Now, by assumption 𝒦_{γ} is bounded in E_{U}, so {𝒦_{γ}} is bounded in A_{U}, for every γ ∈ Γ; it follows that the topology 𝒯_{γ} on 𝒦_{γ} (resp. on {𝒦_{γ}}) induced by the inclusion into E_{U} (resp. into A_{U}) is coarser than the 𝒥 -adic (resp. J-adic) topology; on the other hand, 𝒯_{δ} is complete and separated, since the same holds for the topology of E_{U} (resp. of A_{U}). Then lemma 8.3.12 implies that 𝒦_{γ} (resp. {𝒦_{γ}}) is complete and separated for its 𝒥 -adic (resp. J-adic) topology, whence the contention. □

Proposition 16.6.9. With the notation of (16.6.7), the following holds :

(i) The rings ℰ^{∧}, and 𝒜^{∧} are perfectoid.

(ii) There is a natural isomorphism of topological rings with Γ-graded structures

E(𝒜^{∧}, 𝒜) →^{∼} (ℰ^{∧}, ℰ).

Proof. Taking into account remark 9.4.9(v) and example 9.3.48(ii), it is easily seen that ℰ^{∧} is a perfect topological E-algebra, hence it is perfectoid (see example 16.3.2(i)). Set

(𝒲 ^{∧}, 𝒲 ) := W (ℰ^{∧}, ℰ) (\overline{𝒲} ^{∧}, \overline{𝒲} ) := (𝒲 ^{∧}, 𝒲 ) ⊗_{W (E)} A.

To prove both assertions, it will therefore suffice to exhibit isomorphisms

(16.6.10)  (\overline{𝒲} ^{∧}, \overline{𝒲} ) →^{∼} (𝒜^{∧}, 𝒜)

of topological A-algebras with Γ-graded structures. We shall first exhibit a continuous map

u_{𝒜} : (𝒲 ^{∧}, 𝒲 ) → (𝒜^{∧}, 𝒜).

To this aim, define W (𝒦_{γ}) as in lemma 16.5.45, and notice that condition (b) of (16.6.7) yields:

gr_{γ}𝒲 = W (𝒦_{γ}) for every γ ∈ Γ

and the multiplication law of 𝒲 is the unique bilinear map 𝒲 × 𝒲 → 𝒲 whose restriction

W (𝒦_{γ}) × W (𝒦_{μ}) → W (𝒦_{δ+γ})

is induced by the multiplication law of W (E_{U}), for every γ, μ ∈ Γ. Also, since 𝒦_{γ} is closed and 1-taut, lemma 16.5.45(iv) says that u_{U} restricts to a well defined surjective map

gr_{γ}u_{𝒜} : W (𝒦_{γ}) → gr_{γ}𝒜 for every γ ∈ Γ

whose kernel is \underline{α}W (𝒦_{γ}), for any distinguished element \underline{α} of Ker u_{A}. Summing up, we conclude that the direct sum of the system of maps (gr_{γ}u_{𝒜} | γ ∈ Γ) is a well defined surjective ring homomorphism gr_{•}u_{𝒜} that makes commute the diagram

(16.6.11)  \xymatrix{W (E) \ar[r]^{u_{A}} \ar[d] & A \ar[d] \\ 𝒲 \ar[r]^{gr_{•}u_{𝒜}} & 𝒜}

and with kernel equal to \underline{α}𝒲 . Moreover, it follows easily from propositions 9.3.77(ii) and 8.5.11(i.b) that the left vertical map of (16.6.11) is an adic ring homomorphism, and the same obviously holds for the right vertical map. Since u_{A} is an open map, we deduce that gr_{•}u_{𝒜} is continuous and open as well; especially, it extends to a well defined continuous surjective ring homomorphism u_{𝒜} as sought, whose kernel is the topological closure of \underline{α}𝒲 in 𝒲 ^{∧} (proposition 8.2.13). However, recall that \underline{α}𝒲 ^{∧} is a closed ideal of 𝒲 ^{∧} (proposition 16.2.21(ii)), and obviously \underline{α}𝒲 is dense in it, hence the kernel of u_{𝒜} is \underline{α}𝒲 ^{∧}, as required. Lastly, since gr_{•}u_{𝒜} is an open map, the same holds for u_{𝒜} , and the proof is concluded. □

Example 16.6.12. In the situation of (16.6.7), let φ_{U}^{♭} : E_{U} → A_{U} be the continuous morphism of topological monoids furnished by proposition 16.4.34(i). Let also β_{•} := (β_{λ} | λ ∈ Λ) be any bounded system of elements of E_{U}. We consider the unique morphism of monoids

φ_{E} : P := ℕ[1/p]^{(Λ)} → E_{U}  e_{λ} ↦ β_{λ}  for every λ ∈ Λ

where e_{•} := (e_{λ} | λ ∈ Λ) is the standard minimal set of generators of ℕ^{(Λ)}. Let also I ⊂ P be the ideal generated by e_{•}. The composition φ_{A} : φ_{U}^{♭} ∘ φ_{E} : P → A is also a morphism of monoids, and we let I^{⟨δ⟩}\overline{E} (resp. I^{⟨δ⟩}\overline{A}) be the \overline{E}-submodule of E_{U} generated by φ_{E}(I^{⟨δ⟩}) (resp. the \overline{A}-submodule of A_{U} generated by φ_{A}(I^{⟨δ⟩})), for every δ ∈ Δ := ℕ[1/p] (notation of (9.3.68)). We denote by gr_{δ}ℰ (resp. gr_{δ}𝒜) the topological closure of I^{⟨δ⟩}\overline{E} in E_{U} (resp. of I^{⟨δ⟩}\overline{A} in A_{U}) for every δ ∈ Δ (notation of remark 9.3.70(i)). We set

ℰ (β_{•}) := ⨁_{δ∈Δ} gr_{δ}ℰ 𝒜 (β_{•}) := ⨁_{δ∈Δ} gr_{δ}𝒜 .

Hence, ℰ (β_{•}) is a Δ-graded E-algebra, and 𝒜 (β_{•}) is a Δ-graded A-algebras. We endow ℰ (β_{•}) with the 𝒥 -adic topology and 𝒜 (β_{•}) with the J-adic topology, and let ℰ (β_{•})^{∧} and 𝒜 (β_{•})^{∧} be the respective separated completion. It is easily seen that gr_{δ}ℰ is 1-taut and bounded in E_{U}, and moreover gr_{pδ}ℰ = (gr_{δ}ℰ)^{p} for every δ ∈ Δ; furthermore, gr_{δ}𝒜 = {gr_{δ}ℰ } for every such δ. Then proposition 16.6.9 shows that (ℰ (β_{•})^{∧}, ℰ (β_{•})), and (𝒜 (β_{•})^{∧}, 𝒜 (β_{•})) are perfectoid rings with Δ-graded structures, and we have a natural isomorphism of topological rings with Δ-graded structures :

E(𝒜 (β_{•})^{∧}, 𝒜 (β_{•})) →^{∼} (ℰ (β_{•})^{∧}, ℰ (β_{•})).

Example 16.6.13. In the situation of (16.5.23), let Γ := ℤ[1/p]; the family (Fil^{γ}E_{U} | γ ∈ Γ) fulfills conditions (a)–(c) of (16.6.7), and taking into account corollary 16.5.42and proposition 16.6.9, we get perfectoid rings with Γ-graded structures (ℰ_{U}^{∧}, ℰ_{U}) and (𝒜_{U}^{∧}, 𝒜_{U}), by setting

ℰ_{U} := ⨁_{γ∈Γ} Fil^{γ}E_{U} 𝒜_{U} := ⨁_{γ∈Γ} Fil^{γ}A_{U}

as well as a natural isomorphism of topological rings with Γ-graded structures

(16.6.14)  E(𝒜_{U}^{∧}, 𝒜_{U}) →^{∼} (ℰ_{U}^{∧}, ℰ_{U}).

Moreover, by inspecting the proof of proposition 16.6.9 it is easily seen that – under the identification (16.6.14) – the restriction gr_{γ}ℰ_{U} → gr_{γ}𝒜_{U} of the map \overline{u}_{𝒜_{U}^{∧}} : ℰ_{U}^{∧} → 𝒜_{U}^{∧} agrees with the restriction Fil^{γ}E_{U} → Fil^{γ}A_{U} of the map φ_{U}^{♭} : E_{U} → A_{U}, for every γ ∈ Γ.

16.6.15. Let Γ be a p-perfect monoid, (A^{∧}, A) a perfectoid ring with Γ-graded structure, set (E^{∧}, E) := E(A^{∧}, A) and let \underline{α} ∈ W (gr_{0}E) be a distinguished element of Ker u_{A} (see proposition 16.6.1(iii)). We denote by \overline{u}_{A} : E → A the restriction of the map u_{A^{∧}} : E^{∧} → A^{∧}. For every integer n > 0 let

λ_{n} := ∑_{i=0}^{n−1} p^{−i} S_{n} := {(i, j) ∈ ℕ[1/p]^{⊕2} | λ_{n} ≤ i < n and 1 > j > 1 − i/n}

and for every γ ∈ Γ and every e ∈ gr_{γ}E, define the gr_{0}A-module

ℐ (e, n, γ) := ∑_{(i,j)∈S_{n}} gr_{(1−j)γ}A · \overline{u}_{A}(α_{0}^{i} · e^{j}).

Proposition 16.6.16. With the notation of (16.6.15), let γ ∈ Γ and a ∈ gr_{γ}A be any elements. Then, for every integer n > 0 there exists e ∈ gr_{γ}E such that

a − \overline{u}_{A}(e) ∈ ℐ (e, n, γ) + p^{n} · gr_{γ}A.

Proof. Notice that the assertion is independent of the topology 𝒯_{A^{∧}} of A^{∧}, so we may assume that 𝒯_{A^{∧}} is the p-adic topology of A^{∧} (proposition 16.3.8(i)). We argue by induction on n ∈ ℕ. For n = 1, we have S_{1} = ∅, and the assertion means that there exists e ∈ gr_{γ}E such that a − \overline{u}_{A}(e) ∈ p · gr_{γ}A. Hence, let π : A → A/pA be the projection; it suffices to remark :

Claim 16.6.17. The map π ∘ \overline{u}_{A} : E → A/pA is surjective.

Proof of the claim. The assertion follows easily from claim 16.6.2(i). ♢

Thus, suppose that the proposition has already been proven for some n ≥ 1, for every γ ∈ Γ and every a ∈ gr_{γ}A. We notice :

Claim 16.6.18. (i) For every δ ∈ Γ, every b ∈ gr_{δ}A and every k ∈ ℕ there exist a sequence (β_{0}, …, β_{k}) of elements of gr_{δ}E, and an element d ∈ gr_{δ}A such that

(16.6.19)  b = ∑_{i=0}^{k} p^{i} · \overline{u}_{A}(β_{i}) + p^{k+1}d.

(ii) There exist elements c ∈ gr_{γ}A, e_{n} ∈ gr_{γ}E, a finite set Λ, a system of positive integers (k_{λ} | λ ∈ Λ), a mapping Λ → S_{n} : λ ↦ (i_{λ}, j_{λ}), a system (f_{λ} | λ ∈ Λ) with f_{λ} ∈ gr_{(1−j_{λ})γ}E for every λ ∈ Λ, such that

(16.6.20)  a = \overline{u}_{A}(e_{n}) + ∑_{λ∈Λ} k_{λ} · \overline{u}_{A}(f_{λ} · α_{0}^{i_{λ}} · e_{n}^{j_{λ}}) + p^{n}c.

(iii) There exists f ∈ gr_{0}E such that p^{n} − \overline{u}_{A}(α_{0} · f)^{n} ∈ p^{n+1}gr_{0}A for every n ∈ ℕ.

Proof of the claim. (i): We argue by induction on k, and notice that the assertion for k = 0 follows from the case n = 1 of the proposition, which is already known. Thus, suppose that k ∈ ℕ, and that we have already found a sequence (β_{0}, …, β_{k}) such that (16.6.19) holds. Then we may also find β_{k+1} ∈ gr_{δ}E and d′ ∈ gr_{δ}A such that d = \overline{u}_{A}(β_{k+1}) + pd′, and replacing this expression for d in (16.6.19) we get the sought identity for k + 1.

(ii): By inductive assumption, there exist a finite subset T ⊂ S_{n} and for every (i, j) ∈ T an element b_{(i,j)} ∈ gr_{(1−j)γ}A such that

a = \overline{u}_{A}(e_{n}) + ∑_{(i,j)∈T} b_{(i,j)} · \overline{u}_{A}(α_{0}^{i} · e_{n}^{j}) + p^{n}d for some e_{n} ∈ gr_{γ}E and d ∈ gr_{γ}A.

By (i), we may then find, for every (i, j) ∈ T, a system (f_{(i,j),0}, …, f_{(i,j),n−1}) of elements of gr_{(1−j)γ}E and an element c_{(i,j)} ∈ gr_{(1−j)γ}A such that b_{(i,j)} = ∑_{i=0}^{n−1} p^{i} · \overline{u}_{A}(f_{(i,j),i}) + p^{n}c_{(i,j)}. Hence, the sought identity holds with

c := d+∑_{(i,j)∈T} c_{(i,j)}·\overline{u}_{A}(α_{0}^{i}·e_{n}^{j})  Λ := T ×\{0, …, n−1\}  k_{(t,s)} := p^{s} for every (t, s) ∈ Λ

and with the mapping Λ → S_{n} : (t, k) ↦ t.

(iii): By lemma 16.2.7(iii), there exists x ∈ A^{∧} such that p = \overline{u}_{A}(α_{0})·x. Let π_{0}^{∧} : A^{∧} → gr_{0}A be the canonical 0-projection (see remark 8.5.2(iii)); it follows that p = \overline{u}_{A}(α_{0}) · π_{0}^{∧}(x), so we may assume that x ∈ gr_{0}A. By (i), we have y_{0} := x − \overline{u}_{A}(f) ∈ p · gr_{0}A for some f ∈ E, and we check, by induction on n, that such f will do. Indeed, for n = 0 the assertion is trivial, and for n = 1 we have

p = \overline{u}_{A}(α_{0}) · (y_{0} + \overline{u}_{A}(f)) = \overline{u}_{A}(α_{0} · f) + y_{0} · \overline{u}_{A}(α_{0})

and y_{1} := y_{0} · \overline{u}_{A}(α_{0}) = py_{0} · π_{0}^{∧}(x^{−1}) ∈ p^{2} · gr_{0}A, again by lemma 16.2.7(iii). Suppose now that n ≥ 1, and that we have already shown that y_{n} := p^{n} − \overline{u}_{A}(α_{0} · f)^{n} ∈ p^{n+1}gr_{0}A. We compute

\begin{matrix}py_{n} = & p^{n+1} − p · \overline{u}_{A}(α_{0} · f)^{n} \\ = & p^{n+1} − (y_{1} + \overline{u}_{A}(α_{0} · f)) · \overline{u}_{A}(α_{0} · f)^{n} \\ = & p^{n+1} − \overline{u}_{A}(α_{0} · f)^{n+1} − y_{1} · \overline{u}_{A}(α_{0} · f)^{n} \end{matrix}

Since both py_{n} and y_{1} · \overline{u}_{A}(α_{0} · f)^{n} = p^{n}y_{1} · \overline{u}_{A}(f)^{n} · π_{0}^{∧}(x^{−n}) lie in p^{n+2}gr_{0}A, the assertion follows for n + 1. ♢

Pick any identity (16.6.20)as provided by claim 16.6.18(ii), and any f ∈ gr_{0}E such that claim 16.6.18(iii) holds; by claim 16.6.18(i) we may also find g ∈ gr_{γ}E such that c−\overline{u}_{A}(g) ∈ p·gr_{γ}A, and notice that, after replacing the set Λ by ⋃_{λ∈Λ}{λ} × {1, …, k_{λ}}, we may also assume that k_{λ} = 1 for every λ ∈ Λ, so that

a = \overline{u}_{A}(e_{n}) + ∑_{λ∈Λ} \overline{u}_{A}(f_{λ} · α_{0}^{i_{λ}} · e_{n}^{j_{λ}}) + \overline{u}_{A}(g · α_{0}^{n} · f^{n}) + p^{n+1}d for some d ∈ gr_{γ}A.

We let S′_{n} := {(i, j) ∈ ℕ[1/p]^{⊕2} | λ_{n+1} ≤ i < n + 1, nj + i ≥ 1 + n and j < 1} and we set

δ := ∑_{λ∈Λ} f_{λ} · α_{0}^{i_{λ}} · e_{n}^{j_{λ}} and e_{n+1} := e_{n} + δ + g · α_{0}^{n} · f^{n}.

Claim 16.6.21. There exists a finite subset Λ′ with a mapping Λ′ → S′_{n} : λ ↦ (i′_{λ}, j′_{λ}), and for every λ ∈ Λ′ an element z_{λ} ∈ gr_{(1−j′_{λ})γ}A such that

a = \overline{u}_{A}(e_{n+1}) + ∑_{λ∈Λ′} z_{λ} · \overline{u}_{A}(α_{0}^{i′_{λ}} · e_{n}^{j′_{λ}}) + p^{n+1}d′ for some d′ ∈ gr_{γ}A.

Proof of the claim. For every r = 1, …, n, let Λ′_{r} be the subset of all sequences \underline{σ} := (σ_{0}, σ_{1}, σ_{λ} | λ ∈ Λ) of elements of p^{−r}ℕ \ p^{1−r}ℕ such that σ_{0} + σ_{1} + ∑_{λ∈Λ} σ_{λ} = 1, and set Λ′ := ⋃_{r=1}^{n} Λ′_{r}. Let also x := a − \overline{u}_{A}(e_{n+1}); according to proposition 9.3.62, there exist a mapping Λ′ → ℤ_{p} : \underline{σ} ↦ z′ ; \underline{σ} and an element d′ ∈ gr_{γ}A such that

\begin{matrix}x = & ∑_{r=1}^{n} p^{r} · ∑_{\underline{σ}∈Λ′_{r}} z′ ; \underline{σ} · \overline{u}_{A}(e_{n}^{σ_{0}} · g^{σ_{1}} · f^{nσ_{1}} · α_{0}^{nσ_{1}} · ∏_{λ∈Λ} f_{λ}^{σ_{λ}} · α_{0}^{σ_{λ}i_{λ}} · e_{n}^{σ_{λ}j_{λ}}) + p^{n+1}d′ \\ = & ∑_{r=1}^{n} p^{r} · ∑_{\underline{σ}∈Λ′_{r}} z′ ; \underline{σ} · \overline{u}_{A}(f ′ ; \underline{σ} · α_{0}^{i′′ ; \underline{σ}} · e_{n}^{j′ ; \underline{σ}}) + p^{n+1}d′ \end{matrix}

where

f ′ ; \underline{σ} := g^{σ_{1}}·f^{nσ_{1}}·∏_{λ∈Λ} f_{λ}^{σ_{λ}}  i′′ ; \underline{σ} := nσ_{1}+∑_{λ∈Λ} σ_{λ}i_{λ}  j′ ; \underline{σ} := σ_{0}+∑_{λ∈Λ} σ_{λ}j_{λ}  for every \underline{σ} ∈ Λ′.

Let us remark that

(16.6.22) j′ ; \underline{σ} + \frac{i′′ ; \underline{σ}}{n} = σ_{0} + σ_{1} + ∑_{λ∈Λ} σ_{λ} · (j_{λ} + \frac{i_{λ}}{n} ) ≥ σ_{0} + σ_{1} + ∑_{λ∈Λ} σ_{λ} = 1 for every \underline{σ} ∈ Λ′.

Moreover, we notice that

(16.6.23) 1 > j′ ; \underline{σ}  and n > i′′ ; \underline{σ} ≥ \frac{λ_{n}}{p^{r}}  for every r = 1, …, n and every \underline{σ} ∈ Λ′_{r}.

Indeed, the first inequality is immediate, since σ_{0} < 1; the second inequality is clear if σ_{λ} ≠ 0 for at least one index λ ∈ Λ, since in this case σ_{λ} ≥ p^{−r} and i′′ ; \underline{σ} < nσ_{1} + n ∑_{λ∈Λ} σ_{λ} < n. If σ_{λ} = 0 for every λ ∈ Λ, we must have 1 > σ_{1} > 0, since otherwise we would get σ_{0}, σ_{1} ∈ ℕ, which is absurd; hence, in this case 1 > σ_{1} ≥ p^{−r}, so n > i′′ ; \underline{σ} ≥ n/p^{r} ≥ λ_{n}/p^{r}.

Next, for every r = 1, …, n and every \underline{σ} ∈ Λ′_{r}, we may find z\underline{σ} ∈ gr_{(1−j′ ; \underline{σ})γ}A such that

p^{r} · z′ ; \underline{σ} · \overline{u}_{A}(f ′ ; \underline{σ}) = z\underline{σ} · \overline{u}_{A}(α_{0}^{r})

(cp. the proof of claim 16.6.18(iii)). Hence, if we set

i′ ; \underline{σ} := i′′ ; \underline{σ} + r for every r = 1, …, n and every \underline{σ} ∈ Λ′_{r}

we get an expression for a of the sought type, and it remains only to check that (i′ ; \underline{σ}, j′ ; \underline{σ}) ∈ S′_{n} for every \underline{σ} ∈ Λ′. However, from (16.6.23) we see that i′ ; \underline{σ} ≥ 1 + λ_{n}/p = λ_{n+1} if \underline{σ} ∈ Λ′_{1}, and i′ ; \underline{σ} ≥ 2 + λ_{n}/p^{2} > λ_{n+1} if \underline{σ} ∈ Λ′_{r} for some r > 1. Lastly, from (16.6.22) we get i′′ ; \underline{σ} + j′ ; \underline{σ} · n ≥ n for every \underline{σ} ∈ Λ′, whence i′ ; \underline{σ} + j′ ; \underline{σ} · n ≥ n + 1, and the proof is complete. ♢

Let I (resp. J) be the ideal of E generated by the system (α_{0}^{n}, e_{n}) (resp. by (α_{0}^{n}, e_{n} + δ)) and set q := min(j + i/n | (i, j) ∈ Λ); then δ ∈ i.c.(I, E, q), and the radical of I equals the radical of J. Since q > 1, lemma 9.3.74(ii) shows therefore that

I^{⟨1⟩}E = J^{⟨1⟩}E

and consequently e_{n}^{t} ∈ J^{⟨t⟩}E for every t ∈ ℕ[1/p]. Notice that (α_{0}^{n}, e_{n+1}) is another system of generators for J. Since the topology of A is p-adic, it follows that for every t ∈ ℕ[1/p] there exist a finite subset T_{t} ⊂ ℕ[1/p] with 0 ≤ σ ≤ t for every σ ∈ T_{t}, and a system (h′_{σ} | σ ∈ T_{t}) of elements of A^{∧} such that

\overline{u}_{A}(e_{n}^{t}) = ∑_{σ∈T_{t}} h′_{σ} · \overline{u}_{A}(α_{0}^{nσ} · e_{n+1}^{t−σ})

(corollary 16.3.40(ii)); after replacing h′_{σ} with its canonical σγ-projection, we may also assume that h′_{σ} ∈ gr_{σγ}A for every σ ∈ T_{t}. Now, pick an expression for a as in claim 16.6.21; to conclude the proof, it suffices to show that \overline{u}_{A}(α_{0}^{i′_{λ}} · e_{n}^{j′_{λ}}) ∈ ℐ (e_{n+1}, n + 1, γ) + p^{n+1} · gr_{j′_{λ}γ}A for every λ ∈ Λ′. To ease notation, set (s, t) := (i′_{λ}, j′_{λ}); we see that

\overline{u}_{A}(α_{0}^{s} · e_{n}^{t}) = ∑_{σ∈T_{t}} h′_{σ} · \overline{u}_{A}(α_{0}^{s+nσ} · e_{n+1}^{t−σ}).

Now, since (s, t) ∈ S′_{n}, we have s + nσ ≥ s ≥ λ_{n+1} for every σ ∈ T′_{t}. Moreover :

s + nσ + (t − σ) · (n + 1) = s + nt + t − σ ≥ n + 1 + t − σ.

Recall also that s + nt ≥ n + 1; therefore, if s + nσ < n + 1, we must have t − σ > 0, so the pair (s + nσ, t − σ) lies in S_{n+1}; lastly, if s + nσ ≥ n + 1, the term \overline{u}_{A}(α_{0}^{s+nσ} · e_{n+1}^{t−σ}) lies in p^{n+1} · gr_{(t−σ)γ}A, whence h′_{σ} · \overline{u}_{A}(α_{0}^{s+nσ} · e_{n+1}^{t−σ}) ∈ p^{n+1} · gr_{tγ}A and the assertion follows. □

Corollary 16.6.24. Let A be a perfectoid ring, and a ∈ A such that p^{k} ∈ Aa for some k ∈ ℕ. Then there exist t ∈ A^{×}, s ∈ A, and e ∈ E(A) such that a = t · \overline{u}_{A}(e) and p = s · \overline{u}_{A}(e^{1/p^{k}}).

Proof. In view of proposition 16.3.8(ii), we may assume that the topology of A is p-adic, and then the topology of E := E(A) is α_{0}-adic, where \underline{α} := (α_{n} | n ∈ ℕ) ∈ W (E) is a distinguished element of Ker(u_{A}) (remark 16.3.7(iii)). Also, the case k = 0 is trivial, so we may assume that k > 0. We apply proposition 16.6.16 to A (graded with by the trivial monoid Γ = {0}), its element a, and with n := p^{4k}. With the notation of (16.6.15), notice that for every (i, j) ∈ S_{n} we have either i ∈ [p^{k}, n[, in which case i ≥ k + 1 and \overline{u}_{A}(α_{0}^{i}) ∈ p^{k+1}A (lemma 16.2.7(iii)), or else i ∈ [λ_{n}, p^{k}[, in which case j > 1 − p^{−3k}. Consequently, we get e ∈ E and b, c ∈ A with :

a − \overline{u}_{A}(e) ∈ \overline{u}_{A}(α_{0}^{1+p^{−k}}e^{1−p^{−3k}})b + p^{k+1}c.

By assumption, p^{k} = ad for some d ∈ A, so that a − p^{k+1}c = a(1 − cdp), and notice that x := 1 − cdp ∈ A^{×}; hence a = x^{−1} · \overline{u}_{A}(e^{1−p^{−3k}}) · (\overline{u}_{A}(α_{0}^{1+p^{−k}}) + \overline{u}_{A}(e^{p^{−3k}})), and in particular :

p^{k} ∈ \overline{u}_{A}(e^{1−p^{−3k}})A.

Notice also that p^{k}A = \overline{u}_{A}(α_{0}^{k}); combining with corollary 16.3.51(ii), it follows that

α_{0}^{k} ∈ e^{1−p^{−3k}}E and therefore α_{0}^{kp^{−2k}} ∈ e^{p^{−2k}−p^{−5k}}E.

Clearly r := p^{−k} − kp^{−2k} > 0, and p^{−2k} − p^{−5k} − p^{−3k} ≥ 0, so that :

α_{0}^{1+p^{−k}}e^{1−p^{−3k}} ∈ α_{0}^{1+r}e

and finally we get, for some y ∈ E :

ax = \overline{u}_{A}(e) · z where z := 1 + \overline{u}_{A}(α_{0}^{1+r}y).

Clearly z ∈ A^{×}, so the first sought identity holds with t := z/x. It follows that \overline{u}_{A}(α_{0}^{k}) ∈ \overline{u}_{A}(e)A, so that α_{0}^{k} ∈ eE, again by corollary 16.3.51(ii); thus, α_{0} ∈ e^{1/p^{k}}E, whence the second identity. □

Remark 16.6.25. (i) One can show by a direct computation that the element e_{n} appearing in the proof of proposition 16.6.16 is integral over the ideal generated by e_{n+1} and α_{0}^{n}; as a consequence, the proof of loc. cit. can be made in principle completely effective, so that one can extract from it an algorithm that exhibits an element e with the sought properties, for any given a ∈ gr_{γ}A.

(ii) In [149], Scholze proves theorem 16.5.52 (for his perfectoid rings, that are a special case of our perfectoid quasi-affinoid rings) by means of his “approximation lemma” (see [149, Lemma 6.5]). With proposition 16.6.16, we can give an alternative proof of theorem 16.5.52that generalizes to quasi-affinoid perfectoid rings, along the lines of Scholze’s original argument. The first step is the following extension of proposition 16.5.46to quasi-affinoid perfectoid rings.

Lemma 16.6.26. In the situation of (16.5.23), let v be any continuous valuation on A_{U}. Then : (i) The mapping v ∘ φ_{U}^{♭} is a continuous valuation of E_{U}. (ii) The rule : v ↦ v ∘ φ_{U}^{♭} defines a continuous map

Cont(φ_{U}^{♭}) : Cont(A_{U}) → Cont(E_{U}).

Proof. (i): Since both v and φ_{U}^{♭} are morphisms of pointed monoids, the same holds for v ∘ φ_{U}^{♭}. Thus, it remains only to check that

v(φ_{U}^{♭}(x + y)) ≤ M(x, y) := max(v ∘ φ_{U}^{♭}(x), v ∘ φ_{U}^{♭}(y)) for every x, y ∈ E_{U}.

However, from proposition 16.4.34(ii) we know that φ_{U}^{♭}(x + y) is the sum of φ_{U}^{♭}(x) + φ_{U}^{♭}(y) and a p-adically convergent series, whose n-th term can be written as p^{n} · z_{n}, where z_{n} is in turn a finite ℤ_{p}-linear combination of elements of the form φ_{U}^{♭}(x^{σ}y^{1−σ}), with σ ranging over the elements of Σ_{n} := p^{−n}ℕ ∩ [0, 1]. It follows that

v(z_{n}) ≤ max(σ · v(φ_{U}^{♭}(x)) + (1 − σ) · v(φ_{U}^{♭}(y)) | σ ∈ Σ_{n}) ≤ M(x, y).

On the other hand, v(p) is cofinal in the value group of v, since the latter is continuous (see (15.3.1)). The assertion is an immediate consequence.

The proof of (ii) is similar to that of proposition 16.5.46(ii) : details left to the reader. □

Next, we remark the following corollary of proposition 16.6.16, that generalizes and improves slightly Scholze’s “approximation lemma” in [149, Lemma 6.5]. The same improvement appears in [114, Cor.3.6.7].

Corollary 16.6.27. Resume the notation of (16.5.23). For every γ ∈ ℤ[1/p], every a ∈ Fil^{γ}A_{U}, every m ∈ ℕ and every β ∈ ℕ[1/p] with β < p/(p − 1) there exists e ∈ Fil^{γ}E_{U} such that

v(a − φ_{U}^{♭}(e)) ≤ v(p)^{β} · max(v(φ_{U}^{♭}(e)), v(p)^{m}) for every v ∈ Cont(A_{U}).

Proof. (Notice that the value group Γ_{v} is p-divisible, due to theorem 16.5.52(iv), but even discounting loc.cit., the term v(p)^{β} is still well defined as an element of (ℚ ⊗_{ℤ} Γ_{v})_{∘}, so the above inequality makes sense in any case.) Fix v, m, β as in the corollary, and a distinguished element \underline{α} := (α_{n} | n ∈ ℕ) ∈ Ker u_{A}, and define the sequence of rational numbers (λ_{n+1} | n ∈ ℕ) as in (16.6.15), as well as the subset S_{n} ⊂ ℕ[1/p]^{⊕2} and the \overline{A}-submodule ℐ (e, n, γ) of Fil^{γ}A_{U}, for every e ∈ Fil^{γ}E_{U} and every integer n > 0. Notice that

(16.6.28)  lim_{n→+∞} λ_{n} = p/(p − 1).

We apply proposition 16.6.16 to the perfectoid rings with ℤ[1/p]-graded structure ℰ_{U} and 𝒜_{U} given by example 16.6.13 : for every integer n > 0 we then get e_{n} ∈ Fil^{γ}E_{U}, d_{n} ∈ Fil^{γ}A_{U}, a finite subset T_{n} ⊂ S_{n}, and for every (i, j) ∈ T_{n} an element b_{i,j,n} ∈ Fil^{(1−j)γ}A_{U} such that

a = φ_{U}^{♭}(e_{n}) + ∑_{(i,j)∈T_{n}} b_{i,j,n} · φ_{U}^{♭}(α_{0}^{i} · e_{n}^{j}) + p^{n}d_{n}.

Pick also ε ∈ ℕ[1/p] \ {0} such that α_{0} ∈ Fil^{ε}E_{U}, and k ∈ ℕ large enough so that γ + kε > 0.

Claim 16.6.29. We have v(p^{n}d_{n}) ≤ v(p)^{β+m} for every sufficiently large n ∈ ℕ.

Proof of the claim. We have p^{k}d_{n} ∈ Fil^{γ+kε}A_{U}, and notice that Fil^{δ}A_{U} ⊂ A_{U}^{∘∘} for every δ > 0. It follows that v(p^{k}d_{n}) < 1, so the claim holds for every n ≥ k + m + β. ♢

Recall that pu = φ_{U}^{♭}(α_{0}) for some u ∈ \overline{A}^{×}; suppose first that there exists n_{0} ∈ ℕ such that v(φ_{U}^{♭}(e_{n})) ≤ v(p)^{m} for every n ≥ n_{0}. Then we have

v(b_{i,j,n} · φ_{U}^{♭}(α_{0}^{i} · e_{n}^{j})) ≤ v(b_{i,j,n}) · v(u)^{i} · v(p)^{i+mj} for every n ≥ n_{0} and every (i, j) ∈ T_{n}

and on the other hand b_{i,j,n} · φ_{u}^{♭}(α_{0}^{(1−j)k}) ∈ Fil^{(1−j)(γ+kε)} ⊂ A_{U}^{∘∘}, and v(u)^{i} · v(x) < 1 for every x ∈ A_{U}^{∘∘}, whence

v(u)^{i} · v(b_{i,j,n} · φ_{u}^{♭}(α_{0}^{(1−j)k})) < 1 for every n ∈ ℕ and every (i, j) ∈ T_{n}.

By the same token, we have v(u)^{−(1−j)k} · v(p)^{δ} < 1 for every δ > 0, whence

v(p)^{δ} ≤ v(φ_{U}^{♭}(α_{0}^{(1−j)k})) for every δ > (1 − j)k.

Summing up we obtain, for every (i, j) ∈ T_{n}, the inequality

v(b_{i,j,n} · φ_{U}^{♭}(α_{0}^{i} · e_{n}^{j})) ≤ v(p)^{β+m} whenever i + mj − (1 − j)k > m + β.

The latter condition is the same as the inequality i > (1 − j)(m + k) + β. However, notice that 1 − j < i/n, so this inequality holds provided i > i · (m + k)/n + β, i.e. when

(16.6.30)  i · (1 − (m + k)/n) > β.

Lastly, from (16.6.28) we see that β < λ_{n} for every sufficiently large n, and since i ≥ λ_{n}, we may find n large enough so that (16.6.30) is verified for every (i, j) ∈ T_{n}; taking into account claim 16.6.29, we get the corollary, in this case.

Thus, we may assume that there exists an infinite subset Σ ⊂ ℕ such that v(p)^{m} < v(φ_{U}^{♭}(e_{s})) for every s ∈ Σ, and taking into account claim 16.6.29, we are reduced to checking that

v(b_{i,j,s} · φ_{U}^{♭}(α_{0}^{i} · e_{s}^{j})) ≤ v(p)^{β} · v(φ_{U}^{♭}(e_{s})) for some s ∈ Σ and every (i, j) ∈ T_{s}.

However, arguing as in the foregoing we easily see that

v(φ_{U}^{♭}(α_{0}^{i})) ≤ v(p)^{δ} for every δ < i.

Hence, after replacing β by some β′ ∈ ℕ[1/p] with p/(p − 1) > β′ > β, we reduce to checking that there exists s ∈ Σ such that

v(b_{i,j,s}) · v(p)^{i−β} ≤ v(p)^{(1−j)m} for every (i, j) ∈ T_{s}.

On the other hand, we have b_{i,j,n} · φ_{U}^{♭}(α_{0}^{(1−j)k}) ∈ A_{U}^{∘∘}, and v(u)^{(1−j)k} · v(x) < 1 for every x ∈ A_{U}^{∘∘}, whence

v(b_{i,j,n}) · v(p)^{(1−j)k} < 1 for every n ∈ ℕ and every (i, j) ∈ T_{n}.

Thus, it suffices to show that there exists s ∈ Σ such that

i − β > (1 − j)k + (1 − j)m = (1 − j)(m + k) for every (i, j) ∈ T_{s}.

This condition is equivalent to (16.6.30), so we may argue as in the foregoing to conclude. □

16.6.31. Now, in the situation of corollary 16.6.27, let

C(f_{•}) := Cont(A_{U}) ∩ R_{A_{U}} (\frac{f_{1}}{f_{0}} , ⋯, \frac{f_{n}}{f_{0}} ) and S(f_{•}) := C(f_{•}) ∩ Spa \underline{A}

for a sequence f_{•} := (f_{0}, …, f_{n}) of elements of A_{U} that generates an open ideal. There exist a_{0}, …, a_{n} ∈ A_{U} and an integer m > 0 such that p^{m−1} = ∑_{i=0}^{n} a_{i}f_{i}, and after replacing each a_{i} by p^{n}a_{i} for a suitable n ∈ ℕ, and m by m + n, we may even assume that a_{0}, …, a_{n} ∈ \overline{A} (recall that \overline{A} is the image of A in A_{U}, and \overline{E} is the image of E in E_{U}) whence

(16.6.32)  v(p)^{m} ≤ max(v(pa_{i}) · v(f_{i}) | i = 0, …, n) ≤ max(v(f_{i}) | i = 0, …, n)

for every v ∈ Cont(A_{U}). It follows that

(16.6.33)  v(f_{0}) ≥ v(p)^{m}  for every v ∈ C(f_{•}).

By corollary 16.6.27we may find e_{0}, …, e_{n} ∈ E_{U} such that for every i = 1, …, n we have :

(16.6.34) v(f_{i} − φ_{U}^{♭}(e_{i})) ≤ v(p) · max(v(φ_{U}^{♭}(e_{i})), v(p)^{m})  for every v ∈ Cont(A_{U}).

We may now state the announced generalization of theorem 16.5.52:

Proposition 16.6.35. With the notation of (16.6.31), we have :

(i) The map Cont(φ_{U}^{♭}) of lemma 16.6.26(ii) is a homeomorphism. (ii) The system e_{•} := (e_{0}, …, e_{n}) generates an open ideal of E_{U}. (iii) C(f_{•}) = Cont(φ_{U}^{♭})^{−1}C(e_{•}), where C(e_{•}) := R_{E_{U}} (\frac{e_{1}}{e_{0}} , ⋯, \frac{e_{n}}{e_{0}} ) ∩ Cont(E_{U}). (iv) φ_{U}^{♭}(e_{0}) ∈ 𝒪_{Spa \underline{A}}^{loc}(S(f_{•}))^{×} and f_{i}/f_{0} − φ_{U}^{♭}(e_{i})/φ_{U}^{♭}(e_{0}) ∈ 𝒪_{Spa \underline{A}}^{loc}(S(f_{•}))^{∘∘} for i =

1, …, n (notation of (15.5.11)).

Proof. (ii): Set φ_{U}^{♭}(e_{•}) := (φ_{U}^{♭}(e_{0}), …, φ_{U}^{♭}(e_{n})); to begin with, we remark :

Claim 16.6.36. (ii) holds if and only if the system φ_{U}^{♭}(e_{•}) generates an open ideal of A_{U}.

Proof of the claim. Let b_{1}, …, b_{r} be a finite system of elements of \overline{E} that generates an ideal of definition. Then we may find k ∈ ℕ large enough so that b_{i}^{k}e_{j} ∈ \overline{E} for every i = 1, …, r and every j = 1, …, n. Clearly the system b_{•}^{k}e_{•} := (b_{i}^{k}e_{j} | i ≤ r, j ≤ n) generates an open ideal of E_{U} if and only if the same holds for e_{•}; likewise; φ^{♭}(e_{•}) generates an open ideal of A_{U} if and only if the same holds for φ_{U}^{♭}(b_{•}^{k}e_{•}). So we may replace e_{•} by b_{•}^{k}e_{•}, and assume from start that e_{i} ∈ \overline{E} for i = 1, …, n, in which case φ_{U}^{♭}(e_{i}) = \overline{u}\overline{A}(e_{i}) ∈ \overline{A} for i = 1, …, n. Now, let I_{E} ⊂ \overline{E} be the ideal generated by e_{•}; then I_{E} is open in \overline{E} if and only if Z := Spec \overline{E}/I_{E} lies in the non-analytic locus X of Spec \overline{E}; likewise, I_{E}E_{U} is open in E_{U} if and only if Z′ := Spec E_{U}/I_{E}E_{U} lies in the non-analytic locus X′ of Spec E_{U} (lemma 8.3.29(v)). On the other hand, let j : \overline{E} → E_{U} be the inclusion map; then Z′ = (Spec j)^{−1}Z, and lemma 8.3.29(iii) implies that Z lies in X if and only if Z′ lies in X′. Summing up, we see that I_{E} is open in \overline{E} if and only if I_{E}E_{U} is open in E_{U}. Likewise, let I_{A} ⊂ \overline{A} be the ideal generated by φ_{U}^{♭}(e_{•}); then I_{A} is open in \overline{A} if and only if I_{A}A_{U} is open in A_{U}. Thus, we are further reduced to showing that e_{•} generates an open ideal of \overline{E} if and only if φ_{U}^{♭}(e_{•}) generates an open ideal of \overline{A}. The latter assertion follows from corollary 16.3.40(ii). ♢

By claim 16.6.36, it suffices to show that the system φ_{U}^{♭}(e_{•}) generates an open ideal of A_{U}. To this aim, we apply the criterion of lemma 15.3.25, which reduces to checking that for every v ∈ Cont(A_{U})_{a} there exists i ∈ {0, …, n} such that v(φ_{U}^{♭}(e_{i})) ≠ 0. However, from (16.6.34) we deduce that for every i = 0, …, n and every such v we have :

• either v(f_{i}) = v(φ_{U}^{♭}(e_{i}))

• or else v(p) > 0 and v(f_{i}), v(φ_{U}^{♭}(e_{i})) < v(p)^{m}.

Now, since the system f_{•} generates an open ideal of A_{U}, lemma 15.3.25 implies that v(f_{i}) ≠ 0 for some i ≤ n. If v(p) = 0 we deduce that v(φ_{U}^{♭}(e_{i})) ≠ 0, as required.

Thus, suppose v(p) > 0; from (16.6.32) we know that v(f_{i}) ≥ v(p)^{m} > 0 for some i ≤ n, whence v(φ_{U}^{♭}(e_{i})) = v(f_{i}) ≠ 0, again as needed.

(iii): In view of (16.6.33) and the foregoing, it follows that

(16.6.37)  v(f_{0}) = v(φ_{U}^{♭}(e_{0})) ≥ v(p)^{m}  for every v ∈ C(f_{•}).

Now, let v ∈ C(f_{•}) and i ∈ {0, …, n}; if v(f_{i}) ≥ v(p)^{m}, the foregoing also implies that

v(φ_{U}^{♭}(e_{i})) = v(f_{i}) ≤ v(f_{0}) = v(φ_{U}^{♭}(e_{0})).

If v(f_{i}) < v(p)^{m}, likewise we get v(φ_{U}^{♭}(e_{i})) < v(p)^{m} ≤ v(φ_{U}^{♭}(e_{0})), so v ∈ Cont(φ_{U}^{♭})^{−1}C(e_{•}).

Lastly, let v ∈ Cont(A_{U}) \ C(f_{•}); if v(p) = 0, we know that v(f_{i}) = v(φ_{U}^{♭}(e_{i})) for every i = 0, …, n, whence v ∉ Cont(φ_{U}^{♭})^{−1}C(e_{•}). Otherwise, (16.6.32) says that v(f_{j}) ≥ v(p)^{m} for some j ≤ n, whence v(φ_{U}^{♭}(e_{j})) ≥ v(p)^{m}. If now v(f_{0}) < v(p)^{m}, we get v(φ_{U}^{♭}(e_{0})) < v(p)^{m}, so v ∉ Cont(φ_{U}^{♭})^{−1}C(e_{•}). Hence, suppose v(φ_{U}^{♭}(e_{0})) = v(f_{0}) ≥ v(p)^{m} > 0. Then there exists k ≤ n such that v(f_{k}) > v(f_{0}), whence v(f_{k}) = v(φ_{U}^{♭}(e_{k})), and again we conclude that v ∉ Cont(φ_{U}^{♭})^{−1}C(e_{•}).

(i): First, let us show that the continuous map Cont(φ_{U}^{♭}) is bijective. Indeed, a simple inspection yields a commutative diagram of continuous maps

\xymatrix{Cont(A_{U}) \ar[r]^{Cont(φ_{U}^{♭})} \ar[d] & Cont(E_{U}) \ar[d] \\ Cont(\overline{A}) \ar[r]^{Cont(\overline{u}\overline{A})} & Cont(\overline{E})}

where the vertical arrows are induced by the open inclusion maps \overline{A} → A_{U} and \overline{E} → E_{U} (notation of (16.5.2)). Therefore, combining theorem 16.5.52(iii) and proposition 15.3.18(ii) we already see that Cont(φ_{U}^{♭}) restricts to a bijection Cont(A_{U})_{a} →^{∼} Cont(E_{U})_{a}. On the other hand, proposition 16.5.4(iv) implies easily that Cont(φ_{U}^{♭}) restricts as well to a bijection Cont(A_{U})_{na} →^{∼} Cont(E_{U})_{na}, whence the contention. Lastly, (ii) and (iii) imply easily that the topology of Cont(A_{U}) is induced by the topology of Cont(E_{U}), via the map Cont(φ_{U}^{♭}); the assertion follows.

(iv): Since f_{0} is invertible in 𝒪_{Spa \underline{A}}^{loc}(S(f_{•})), combining (16.6.37) and proposition 15.4.30we see that the same holds for φ_{U}^{♭}(e_{0}). For the second assertion, in view of corollary 15.4.27(ii) it suffices to show :

Claim 16.6.38. v(f_{i}/f_{0} − φ_{U}^{♭}(e_{i})/φ_{U}^{♭}(e_{0})) < 1 for every i = 1, …, n and every v ∈ C(f_{•}).

Proof of the claim. From (16.6.34) we easily deduce that

v(1/f_{0} − 1/φ_{U}^{♭}(e_{0})) ≤ v(p/f_{0}) for every v ∈ C(f_{•})

whence v(f_{i}/f_{0} − f_{i}/φ_{U}^{♭}(e_{0})) ≤ v(pf_{i}/f_{0}) < 1 for every v ∈ C(f_{•}) and every i = 1, …, n. We are then reduced to checking that v(f_{i}/φ_{U}^{♭}(e_{0}) − φ_{U}^{♭}(e_{i})/φ_{U}^{♭}(e_{0})) < 1, or equivalently, that v(f_{i}−φ_{U}^{♭}(e_{i})) < v(φ_{U}^{♭}(e_{0})) for every such v and i. Now, if v(f_{i}) < v(p)^{m}, we know already that v(φ_{U}^{♭}(e_{i})) < v(p)^{m}, whence v(f_{i} − φ_{U}^{♭}(e_{i})) < v(p)^{m}, and the assertion follows from (16.6.37) in this case. Otherwise, we have v(f_{i}) = v(φ_{U}^{♭}(e_{i})), whence v(φ_{U}^{♭}(e_{i})) ≤ v(f_{0}) = v(φ_{U}^{♭}(e_{0})), and thus v(f_{i} − v(φ_{U}^{♭}(e_{i}))) ≤ v(p) · max(φ_{U}^{♭}(e_{0}), v(p)^{m}) = v(p · φ_{U}^{♭}(e_{0})) < v(φ_{U}^{♭}(e_{0})). □

16.6.39. Let Γ, Δ be two monoids and (A, \underline{B}) a topological ring with Δ-graded structure. Then we may form the complete topological ring with Δ ⊕ Γ-graded structure

(A, \underline{B})[Γ]^{∧}

by combining the constructions of example 8.5.7(i) and remark 8.5.2(iv). Explicitly, a direct inspection shows that (A, \underline{B})[Γ]^{∧} is the pair (A[Γ]^{∧}, \underline{B}^{c}[Γ]), where \underline{B}^{c} is the Δ-graded ring such that gr_{δ}B^{c} is the topological closure of gr_{δ}B in the completion A^{∧} of A, and A[Γ]^{∧} is the completion of A[Γ], for the unique topology on the latter ring such that the inclusion map A → A[Γ] is adic. Notice that if A is complete and separated, then B^{c} = B, hence in this case (A, \underline{B})[Γ]^{∧} = (A[Γ]^{∧}, \underline{B}[Γ]).

Lemma 16.6.40. In the situation of (16.6.39), suppose furthermore that A is perfectoid, and Γ, Δ are both p-perfect, and let

A ⟶^{j_{A}} A[Γ]^{∧} ⟵^{i_{A}} Γ  and  E(A) ⟶^{j_{E}} E(A)[Γ]^{∧} ⟵^{i_{E}} Γ

be the natural inclusion maps. Then we have :

(i) A[Γ]^{∧} is perfectoid.

(ii) There exists a natural isomorphism of topological rings with Δ ⊕ Γ-graded structures

ω : E(A, \underline{B})[Γ]^{∧} →^{∼} E((A, \underline{B})[Γ]^{∧})

fitting into a commutative diagram

\xymatrix{E(A) \ar[r]^{j_{E}} \ar[dr]_{E(j_{A})} & E(A)[Γ]^{∧} \ar[d]^{ω} & Γ \ar[dl]^{E(i_{A})} \ar[l]_{i_{E}} \\  & E(A[Γ]^{∧}).}

Proof. (Here Γ is identified with the source E(Γ) of the morphism E(i_{A}), via \overline{u}_{Γ}.)

(i): Indeed, by remark 16.3.58 we may find an ideal of definition I ⊂ A fulfilling the conditions of (16.3.55), and we let J := I^{(p)}; by proposition 16.3.59(ii), the corresponding map Φ_{I} : gr_{I}^{•}A → gr_{J}^{•}A is a ring isomorphism. Set I′ := IA[Γ]^{∧} and J′ := JA[Γ]^{∧}; by construction, the topology of A[Γ]^{∧} agrees with the I′-adic topology, and the natural map (A/I^{2})[Γ] → A[Γ]^{∧}/I^{′2} is an isomorphism (remark 8.3.3(ii) and lemma 8.3.32(iv)). Since Γ is p-perfect, it follows easily that the Frobenius endomorphism of A[Γ]^{∧}/I^{′2} is surjective, so A[Γ]^{∧} is a P-ring. On the other hand, notice the natural identifications :

gr_{I′}^{•}(A[Γ]^{∧}) →^{∼} (gr_{I}^{•}A)[Γ] gr_{J′}^{•}(A[Γ]^{∧}) →^{∼} (gr_{J}^{•}A)[Γ]

which identify the ring homomorphism Φ_{I′} of proposition 16.3.59(i) with the map induced by Φ_{I} and the p-Frobenius endomorphism of Γ. Especially, Φ_{I′} is an isomorphism, so the assertion follows from theorem 16.3.64.

(ii): According to proposition 16.6.1(i), we may find a Δ-graded ideal I_{B} of B such that I_{A} := I_{B}A is an ideal of definition of A; then notice that the topological closure I_{B}^{∧} of I_{B} in A lies in I_{A} and I_{B}^{∧} ∩ B is the topological closure I_{B}^{c} of I_{B} in B, therefore I_{B}^{∧} = I_{A} and I_{B}^{c} = B ∩ I_{A}, which says especially that I_{B}^{c} is open in B. Set A_{0} := A/I_{A} and B_{0} := B/I_{B}^{c}; since I_{B}^{c} is a Δ-graded ideal as well (remark 8.5.2(i)), we may form the quotient of (A, \underline{B})[Γ]^{∧} by the open ideal I_{B}^{c}[Γ] of B[Γ] as in example 8.5.4(), and get a natural isomorphism :

(16.6.41)  (A, \underline{B})[Γ]^{∧} ⊗_{B} B_{0} →^{∼} (A_{0}[Γ], B_{0}[Γ]).

However, the projections (A, \underline{B}) → (A_{0}, \underline{B}_{0}) and (A, \underline{B})[Γ]^{∧} → (A, \underline{B})[Γ]^{∧} ⊗_{B} B_{0} induce morphisms (see (9.4.24))

(16.6.42)  E(A, \underline{B}) → E(A_{0}, \underline{B}_{0})  E((A, \underline{B})[Γ]^{∧}) → E((A, \underline{B})[Γ]^{∧} ⊗_{B} B_{0}).

On the other hand, the projections A → A_{0} and A[Γ]^{∧} → A_{0}[Γ] induce isomorphisms

E(A) →^{∼} E_{0} := E(A_{0}) and E(A[Γ]^{∧}) →^{∼} E(A_{0}[Γ])

of topological rings (see (16.1)). Denote

A_{0} ⟶^{j_{A_{0}}} A_{0}[Γ]^{∧} ⟵^{i_{A_{0}}} Γ and E_{0} ⟶^{j_{E_{0}}} E_{0}[Γ]^{∧} ⟵^{i_{E_{0}}} Γ

the natural inclusion maps; taking into account example 8.5.7(iii), it follows that (16.6.42) are isomorphisms of topological rings with graded structures, and in view of (16.6.41) it suffices to exhibit an isomorphism ω_{0} : (E_{0}[Γ], E(\underline{B}_{0})[Γ])^{∧} →^{∼} E(A_{0}[Γ], B_{0}[Γ]) such that

ω_{0} ∘ j_{E_{0}} = E(j_{A_{0}}) and ω_{0} ∘ i_{E_{0}} = E(i_{A_{0}})

(where again, Γ is identified with the source E(Γ) of E(i_{A_{0}}) via \overline{u}_{Γ}). To this aim, set

J := Ker (\overline{u}_{A_{0}}[Γ] : E_{0}[Γ] → A_{0}[Γ])

and notice that, since A_{0} is a discrete topological ring, the topology of E_{0}[Γ] agrees with the Jadic topology (remark 9.4.9(ii)); moreover, J is finitely generated, since A is perfectoid (details left to the reader). By corollary 9.4.14and remark 9.4.9(v) we deduce a natural isomorphism of topological rings

ω_{0} : E_{0}[Γ]^{∧} →^{∼} E(A_{0}[Γ]) such that \overline{u}_{A_{0}[Γ]} ∘ ω_{0} = \overline{u}_{A_{0}}[Γ]^{∧}

where \overline{u}_{A_{0}}[Γ]^{∧} : E_{0}[Γ]^{∧} → A_{0}[Γ] is the completion of \overline{u}_{A_{0}}[Γ], and a simple inspection shows that ω_{0} is a morphism (E_{0}[Γ], E(\underline{B}_{0})[Γ])^{∧} → E(A_{0}[Γ], \underline{B}_{0}[Γ]), which then must be an isomorphism, again by example 8.5.7(iii). Let us check that ω_{0} ∘ i_{E_{0}} = E(i_{A_{0}}). Indeed, we have

\overline{u}_{A_{0}[Γ]} ∘ ω_{0} ∘ i_{E_{0}} = \overline{u}_{A_{0}}[Γ]^{∧} ∘ i_{E_{0}} = i_{A_{0}} = \overline{u}_{A_{0}[Γ]} ∘ E(i_{A_{0}})

whence the sought identity, by adjunction. A similar argument proves the remaining sought identity. □

We conclude this section with a sample of other graded perfectoid rings that are analogous to certain constructions found in [149], and might be interesting for other purposes, but shall not be needed in the rest of this treatise.

Example 16.6.43. (i) Let (Δ, 0, +) be a monoid, (A, \underline{B}) a topological ring with Δ-graded structure, and consider two morphisms of Δ-graded monoids

Γ_{2} ⟵^{φ} Γ_{1} ⟶^{ψ} B^{∗}

where B^{∗} is the Δ-graded monoid defined as in (9.4.21). Hence, B[Γ_{2}] is a Δ ⊕ Γ_{2}-graded ℤ-algebra, and we may form the Δ-graded ℤ-algebra B[Γ_{2}]_{/Δ} corresponding to the morphism

η : Δ ⊕ Γ_{2} → Δ  (δ, γ) ↦ δ + |γ|  for every δ ∈ Δ and γ ∈ Γ_{2}

where | · | : Γ_{2} → Δ is the Δ-grading of Γ_{2}. It follows easily that

I := Ker (B[Γ_{2}] → Γ_{2} ⊗_{Γ_{1}} B)

is a Δ-graded ideal of B[Γ_{2}]_{/Δ}, and by combining the constructions of examples 8.5.4(i) and 8.5.7(i,ii) we may define the topological ring with Δ-graded structure

Γ_{2} ⊗_{Γ_{1}} (A, \underline{B}) := (A, \underline{B})[Γ_{2}]_{/Δ}/I.

Explicitly, this is the pair (A′, \underline{B}′), such that A′ (resp. B′) is the maximal separated quotient of Γ_{2} ⊗_{Γ_{1}} A (resp. of Γ_{2} ⊗_{Γ_{1}} B), where the latter is endowed with the unique linear topology such that the natural map

A → Γ_{2} ⊗_{Γ_{1}} A (resp. B → Γ_{2} ⊗_{Γ_{1}} B)

is adic. Then, gr_{δ}B′ is the topological closure of the image of gr_{δ}(B[Γ_{2}]_{/Δ}) in A′, for each δ ∈ Δ. By construction, we have a natural morphism of topological rings with Δ-graded structures (resp. of monoids)

j_{(A,\underline{B})} : (A, \underline{B}) → Γ_{2} ⊗_{Γ_{1}} (A, \underline{B}) (resp. i_{(A,\underline{B})} : Γ_{2} → Γ_{2} ⊗_{Γ_{1}} A ).

(ii) Suppose now that Δ is p-perfect, and the topology of A is complete, separated and coarser than the p-adic topology. Then the topological ring with Δ-graded structure E(A, \underline{B}) is well defined as in (9.4.24). Moreover, from the morphisms φ and ψ we derive two morphisms of monoids

E(Γ_{2}) ⟵^{φ_{E}} E(Γ_{1}) ⟶^{ψ_{E}} E(B^{∗}) where φ_{E} := E(φ) and ψ_{E} := E(ψ)

so the considerations of (i) can be repeated on E(A, \underline{B}), and we get a topological ring with E(Δ)-graded structure

E(Γ_{2}) ⊗_{E(Γ_{1})} E(A, \underline{B}).

Proposition 16.6.44. In the situation of example 16.6.43, suppose moreover that A is perfectoid, and Δ, Γ_{1}, Γ_{2} are p-perfect. We have :

(i) The completion (Γ_{2} ⊗_{Γ_{1}} A)^{∧} of Γ_{2} ⊗_{Γ_{1}} A is perfectoid.

(ii) There exists a natural isomorphism of topological rings with Δ-graded structures :

ω : E((Γ_{2} ⊗_{Γ_{1}} (A, \underline{B}))^{∧}) →^{∼} (Γ_{2} ⊗_{Γ_{1}} E(A, \underline{B}))^{∧}

(notation of (9.4.24)) fitting into the commutative diagrams

\xymatrix{E(A, \underline{B}) \ar[r]^{E(j_{(A,\underline{B})})} \ar[dr]_{j_{E(A,\underline{B})}} & E((Γ_{2} ⊗_{Γ_{1}} (A, \underline{B}))^{∧}) \ar[d]^{ω} & Γ_{2} \ar[r]^{E(i_{(A,\underline{B})})} \ar[dr]_{i_{E(A,\underline{B})}} & E((Γ_{2} ⊗_{Γ_{1}} A)^{∧}) \ar[d]^{ω} \\  & (Γ_{2} ⊗_{Γ_{1}} E(A, \underline{B}))^{∧} &  & (Γ_{2} ⊗_{Γ_{1}} E(A))^{∧}.}

Proof. (Here we identify Γ_{1} and Γ_{2} with E(Γ_{1}) and E(Γ_{2}), via the isomorphisms \overline{u}_{Γ_{1}} and \overline{u}_{Γ_{2}}.) Let us form as well the topological ring with Δ-graded structure

(A′, \underline{D}) := ((A, \underline{B})[Γ_{2}]^{∧})_{/Δ}.

Explicitly, A′ is the completion of A[Γ_{2}], where the latter is endowed with the unique linear topology such that the natural map A → A[Γ_{2}] is adic; then gr_{δ}D is the topological closure in A′ of gr_{δ}(B[Γ_{2}]_{/Δ}), for every δ ∈ Δ. Here, as in example 16.6.43(i), the Δ-graded ℤ-algebra B[Γ_{2}]_{/Δ} is obtained from the Δ ⊕ Γ_{2}-graded ℤ-algebra B[Γ_{2}] via the morphism η. With this notation, from proposition 8.2.13(i,v) and a simple inspection we get a natural identification

(16.6.45)  (Γ_{2} ⊗_{Γ_{1}} (A, \underline{B}))^{∧} →^{∼} (A′, \underline{D})/ID.

Set (E, \underline{B}_{E}) := E(A, \underline{B}); from lemma 16.6.40 and proposition 9.4.22(iii), we know already that A′ is perfectoid, and there exists an isomorphism

(16.6.46)  (E′, \underline{D}_{E}) := E(A′, \underline{D}) →^{∼} ((E, \underline{B}_{E})[Γ_{2}]^{∧})_{/Δ}

of topological rings with Δ-graded structures. Set

ℐ := Ker (B_{E}[Γ_{2}] → Γ_{2} ⊗_{Γ_{1}} B_{E}).

Claim 16.6.47. (i) Φ_{E′}(ℐ E′) = ℐ E′.

(ii) {ℐ E′} = (IA′)^{c}.

Proof of the claim. Let \overline{ψ}_{E} := E(i) ∘ ψ_{E} : Γ_{1} → E where i : B^{∗} → A is the natural map. Clearly, the ideals IA′ and ℐ E′ are generated respectively by the systems

𝒮_{A} := (φ(γ) − ψ(γ) | γ ∈ Γ_{1}) and 𝒮_{E} := (φ_{E}(γ) − \overline{ψ}_{E}(γ) | γ ∈ Γ_{1}).

Since Γ_{1} is perfect, assertion (i) follows already, and for (ii) we come down to showing that the topological closure of the ideal generated by the system \overline{u}_{A′}(𝒮_{E}) equals the topological closure of the ideal generated by 𝒮_{A}. Now, proposition 9.3.62 says that each generator \overline{u}_{A′}(φ_{E}(γ) − \overline{ψ}_{E}(γ)) can be written as a p-adically convergent series ∑_{n∈ℕ} p^{n}c_{γ,n} where

(16.6.48)  c_{γ,0} = \overline{u}_{A′}(φ_{E}(γ)) − \overline{u}_{A′}(\overline{ψ}_{E}(γ))

and for every n > 0, the term c_{γ,n} is a finite ℤ_{p}-linear combination of elements of the form

\overline{u}_{A′}(φ_{E}(γ)^{s} · \overline{ψ}_{E}(γ)^{s′}) − \overline{u}_{A′}(φ_{E}(γ)^{s′} · \overline{ψ}_{E}(γ)^{s})

where s, s′ ∈ p^{−n}ℕ and s + s′ = 1. However, lemma 16.6.40(ii) implies that

\overline{u}_{A′}(φ_{E}(γ)) = φ(γ)  and \overline{u}_{A′}(\overline{ψ}_{E}(γ)) = ψ(γ)  for every γ ∈ Γ_{1}.

Thus {ℐ E′} lies in the topological closure of the ideal generated by the elements

t_{γ,s} := φ(γ)^{s} · ψ(γ)^{s′} − φ(γ)^{s′} · ψ(γ)^{s} for every γ ∈ Γ_{1} and s, s′ ∈ ℕ[1/p] with s + s′ = 1.

To ease notation, set λ := φ(γ) and μ := ψ(γ); we notice that

(λ^{s} − μ^{s}) · (μ^{1−s} + λ^{1−s}) − (λ − μ) = t_{γ,s}

and since λ^{s} − μ^{s}, λ − μ ∈ 𝒮_{A}, we conclude that t_{γ,s} ∈ IA′. By the same token, from (16.6.48) we get c_{0} = λ − μ, so finally (IA′)^{c} = {ℐ E′} + p(IA′)^{c}. From this, an easy induction yields

(IA′)^{c} = {ℐ E′} + p^{n}(IA′)^{c} for every n ∈ ℕ

whence the claim, since {ℐ E′} is a closed ideal. ♢

From claim 16.6.47 and corollary 16.3.38 we deduce that A′/(IA′)^{c} is perfectoid and there exists an isomorphism of topological rings ω′ : E(A′/(IA′)^{c}) →^{∼} E′/(ℐ E′)^{c} such that

ω′ ∘ E(π_{A′}) = π_{E′}

where π_{A′} : A′ → A′/(IA′)^{c} and π_{E′} : E′ → E′/(ℐ E′)^{c} are the projections. This completes the proof of (i), and we also see that the projection (E′, \underline{D}_{E}) → (E′, \underline{D}_{E})/ℐ \underline{D}_{E} factors through a morphism

(16.6.49)  E((A′, \underline{D})/ID) → (E′, \underline{D}_{E})/ℐ \underline{D}_{E}

which must be an isomorphism, by example 8.5.7(iii). On the other hand, from (16.6.46) we get a natural identification

(16.6.50)  (E′, \underline{D}_{E})/ℐ \underline{D}_{E} →^{∼} (Γ_{2} ⊗_{Γ_{1}} E(A, \underline{B}))^{∧}.

Lastly, combining (16.6.49), (16.6.50) and (16.6.45) we get the sought isomorphism ω. The commutativity of the diagrams in (ii) follows by direct inspection, taking into account the corresponding properties of the isomorphism (16.6.46) stated in lemma 16.6.40(ii) : details left to the reader. □

16.6.51. Let Λ be any (small) set and (Δ, +, 0, ≤) any ordered abelian group; we set

Δ^{+} = {δ ∈ Δ | δ ≥ 0}  P := Δ^{+(Λ)}  Q := Δ^{+(Λ×Λ)}

(cp. definition 9.1.1(i)). Denote by I ⊂ P the ideal generated by the canonical system of generators e_{•} := (e_{λ} | λ ∈ Λ) for P, i.e. e_{λ} := (δ_{λμ} | μ ∈ Λ), where δ_{λμ} := 1 for λ = μ, and δ_{λμ} := 0 otherwise, for every λ, μ ∈ Λ. Likewise, we let

(f_{λ}, f′_{λ} | λ ∈ Λ) (resp. (e_{λ,λ′} | λ, λ′ ∈ Λ))

be the canonical system of generators of P × P (resp. of Q).

• With this notation, we define Δ-gradings on P, P × P and Q by the rules :

|δe_{λ}|_{P} := δ  |δf_{λ} + δ′f′_{λ}|_{P ×P} := δ  |δe_{λ,λ′}|_{Q} := δ

for every λ, λ′ ∈ Λ and every δ, δ′ ∈ Δ (see definition 4.8.8(i)), and we set

I^{⟨δ⟩} := {x ∈ P | |x|_{P} ≥ δ} for every δ ∈ Δ.

Notice that, in case Δ = ℤ[1/p], the ideals I^{⟨δ⟩} of P are the same as the ones introduced in (9.3.68), so the notation does not conflict with loc.cit. Lastly, we define

Γ_{I}^{+} := ⋃_{δ∈Δ^{+}} I^{⟨δ⟩} × {δ} Γ_{I} := ⋃_{δ∈Δ} I^{⟨δ⟩} × {δ}

and notice that Γ_{I}^{+} (resp. Γ_{I}) is a submonoid of P × Δ^{+} (resp. of P × Δ), so it inherits a natural Δ-grading, given by the projection on the factor Δ. We have morphisms of Δ-graded monoids

(16.6.52)  \xymatrix{Q \ar[r]^{ψ} \ar[r]_{ψ′} & P × P \ar[r]^{φ} & Γ_{I}^{+}}

such that

φ(δf_{λ} + δ′f′_{λ}) := ((δ + δ′)e_{λ}, δ)  ψ(δe_{λ,λ′}) := δ(f_{λ} + f′_{λ′})  ψ′(δe_{λ,λ′}) := δ(f′_{λ} + f_{λ′})

for every λ, λ′ ∈ Λ and every δ, δ′ ∈ Δ.

Lemma 16.6.53. With the notation of (16.6.51), we have :

(i) (16.6.52) is a presentation for Γ_{I}^{+}, i.e. φ identifies Γ_{I}^{+} with the coequalizer of ψ and ψ′. (ii) Define a Δ-grading of P × Δ^{+} by the rule : (x, δ) ↦ |x|_{P} − δ for every x ∈ P and

δ ∈ Δ^{+}. Then the mapping

P × Δ^{+} → Γ_{I} (x, δ) ↦ (x, |x|_{P} − δ)

is an isomorphism of Δ-graded monoids.

Proof. (i): Indeed, say that

g := ∑_{λ∈Λ}(r_{λ}f_{λ} + r′_{λ}f′_{λ}) and h := ∑_{λ∈Λ}(s_{λ}f_{λ} + s′_{λ}f′_{λ})

are two elements of P × P whose images agree in Γ_{I}^{+}, so that r_{λ}, r′_{λ}, s_{λ}, s′_{λ} ∈ Δ^{+} and

(16.6.54)  r_{λ} + r′_{λ} = s_{λ} + s′_{λ}  for every λ ∈ Λ and  ∑_{λ∈Λ} r_{λ} = ∑_{λ∈Λ} s_{λ}.

Let ∼ be the equivalence relation defined by the pair (ψ, ψ′); we have to check that g ∼ h. Now, consider any λ ∈ Λ, and suppose that r_{λ} ≤ s_{λ}; then we may write g = g′ + r_{λ}f_{λ}, h = h′ + r_{λ}f_{λ} for elements g′, h′ ∈ P × P such that φ(g′) = φ(h′); it then suffices to check that g′ ∼ h′, and we are reduced to the case where r_{λ} = 0. In case r_{λ} > s_{λ}, we argue symmetrically, to reduce to the case where s_{λ} = 0. Likewise, we can reduce to the case where either r′_{λ} = 0 or s′_{λ} = 0. Repeating the argument for every λ ∈ Λ, and taking (16.6.54) into account, we may suppose from start that

g = ∑_{λ∈A} r_{λ}f_{λ} + ∑_{μ∈B} r_{μ}f′_{μ} h = ∑_{λ∈A} r_{λ}f′_{λ} + ∑_{μ∈B} r_{μ}f_{μ}

for some finite subsets A, B ∈ Λ with A ∩ B = ∅, and a system (r_{λ} | λ ∈ A ∪ B) of elements of Δ^{+} \ {0} such that

(16.6.55)  ∑_{μ∈B} r_{μ} = ∑_{λ∈A} r_{λ}.

Next, we argue by induction on the cardinality c of A ∪ B. If c = 0, there is nothing to prove. Suppose then that c > 0 and that the assertion is already known whenever the cardinality of A ∪ B is strictly smaller than c. Notice that A ≠ ∅, since otherwise (16.6.55) would imply that B = ∅, contradicting the assumption that c > 0; likewise, we must have B ≠ ∅. Thus, pick any λ ∈ A and μ ∈ B, set A′ := A \ {λ}, B′ := B \ {μ}, and suppose first that r_{λ} ≤ r_{μ}; we set

g′ := ∑_{λ∈A′} r_{λ}f_{λ} + ∑_{μ∈B′} r_{μ}f′_{μ} + (r_{μ} − r_{λ}) · f′_{μ} h′ := ∑_{λ∈A′} r_{λ}f′_{λ} + ∑_{μ∈B′} r_{μ}f_{μ} + (r_{μ} − r_{λ}) · f_{μ}

so that g = g′ +r_{λ} ·(f_{λ} +f′_{μ}) and h = h′ +r_{λ} ·(f′_{λ} +f_{μ}). However, r_{λ} ·(f_{λ} +f′_{μ}) ∼ r_{λ} ·(f′_{λ} +f_{μ}), so we are reduced to checking that g′ ∼ h′. The latter is known by our inductive assumption. A symmetric argument will do, in case r_{λ} > r_{μ}.

(ii) shall be left to the reader. □

16.6.56. In the situation of (16.6.51), take Δ := ℤ[1/p] with its standard ordering, so that Δ^{+} = ℕ[1/p]. In this case, it is easily seen that P, Γ_{I}^{+} and Γ_{I} are p-perfect monoids. In this paragraph, it will be convenient to switch to a multiplicative notation for the composition laws of these monoids : in other words, we shall hereafter replace (P, +, 0) by (exp P, ·, 1), and likewise for Γ_{I}^{+} and Γ_{I} (notation of (6.1)). Correspondingly, the product δ · x of an element δ ∈ Δ^{+} and an element x ∈ P shall be replaced by the exponential x^{δ}.

Let also A be any perfectoid ring, set E := E(A) and denote by \underline{A} the topological ring with Δ-graded structure (A, A, Δ), such that gr_{0}A = A and gr_{δ}A = 0 for every δ ≠ 0 in Δ; define likewise the topological ring with Δ-graded structure \underline{E} := (E, E, Δ). Choose an ideal of definition J for A, and set 𝒥 := \overline{u}_{A/pA}^{−1}(J/pA), which is an ideal of definition for E. Let β_{•} := (β_{λ} | λ ∈ Λ) be any system of elements of E; then β_{•} determines a unique morphism of monoids

φ_{E} : P → E  such that e_{λ}^{δ} ↦ β_{λ}^{δ}  for every λ ∈ Λ and δ ∈ Δ

and composing with the map \overline{u}_{A} : E → A we get also a morphism φ_{A} : P → A. Moreover, notice that I^{⟨0⟩} = P, so we have as well a morphism i_{0} : P → Γ_{I}^{+}, and we can define the rings

ℰ_{+} := Γ_{I}^{+} ⊗_{P} E ⊂ ℰ := Γ_{I} ⊗_{P} E and 𝒜_{+} := Γ_{I}^{+} ⊗_{P} A ⊂ 𝒜 := Γ_{I} ⊗_{P} A

as well as the topological rings with Δ-graded structures

\begin{aligned}(ℰ_{+}^{∧}, \underline{ℰ} _{+}^{c}) := & (Γ_{I}^{+} ⊗_{P} \underline{E})^{∧} & ⊂ & (ℰ^{∧}, \underline{ℰ} _{+}^{c}) := (Γ_{I} ⊗_{P} \underline{E})^{∧} \\ (𝒜_{+}^{∧}, \underline{𝒜} _{+}^{c}) := & (Γ_{I}^{+} ⊗_{P} \underline{A})^{∧} & ⊂ & (𝒜 , \underline{𝒜} ^{c}) := (Γ_{I} ⊗_{P} \underline{A})^{∧} \end{aligned}

(notation of (6.1.33) and example 16.6.43). We endow ℰ and ℰ_{+} with their 𝒥 -adic topologies, and 𝒜, 𝒜_{+} with their J-adic topologies; then ℰ_{+}^{∧} (resp. ℰ^{∧}) is the completion of ℰ_{+} (resp. of ℰ) and 𝒜_{+}^{∧} (resp. 𝒜^{∧}) is the completion of 𝒜_{+} (resp. of 𝒜). Furthermore, the rings 𝒜 and ℰ inherit natural Δ-gradings from Γ_{I}, such that

I^{⟨δ⟩} ⊗_{P} E = gr_{δ}ℰ  I^{⟨δ⟩} ⊗_{P} A = gr_{δ}𝒜  for every δ ∈ Δ

and by restricting to Δ^{+}, we get corresponding Δ^{+}-gradings on 𝒜_{+} and ℰ_{+}. Then, a direct inspection shows that gr_{δ}ℰ^{c} (resp. gr_{δ}𝒜^{c}) is the separated completion of gr_{δ}ℰ (resp. of gr_{δ}𝒜), for every δ ∈ Δ. Let i_{Γ} : Γ_{I}^{+} → Γ_{I} be the inclusion map; as in example 16.6.43(i), we point out the commutative diagram of monoids and of topological rings with Δ-graded structures :

\xymatrix{\underline{E} \ar[r]^{j_{E}^{+}} \ar[dr]_{j_{E}} & (ℰ_{+}^{∧}, \underline{ℰ} _{+}^{c}) \ar[d]|{(i_{Γ}⊗_{P} E)^{∧}} & Γ_{I}^{+} \ar[r]^{i_{A}^{+}} \ar[d]^{i_{Γ}} \ar[l]_{i_{E}^{+}} & (𝒜_{+}^{∧}, \underline{𝒜} _{+}^{c}) \ar[d]|{(i_{Γ}⊗_{P} A)^{∧}} & \underline{A} \ar[l]_{j_{A}^{+}} \ar[dl]^{j_{A}} \\  & (ℰ^{∧}, \underline{ℰ}^{c}) & Γ_{I} \ar[r]^{i_{A}} \ar[l]_{i_{E}} & (𝒜^{∧}, \underline{𝒜} ^{c}).}

Remark 16.6.57. In the situation of (16.6.56), suppose furthermore that the system β_{•} generates an open ideal of E. Then, combining corollaries 8.6.40(ii) and 16.3.40(ii) together with proposition 16.6.5(ii), we deduce that Γ_{I}^{+} ⊗_{P} E and Γ_{I} ⊗_{P} E are already 𝒥 -adically complete and separated, and Γ_{I}^{+} ⊗_{P} A and Γ_{I} ⊗_{P} A are J-adically complete and separated.

Corollary 16.6.58. With the notation of (16.6.56), the rings ℰ_{+}^{∧}, ℰ^{∧}, 𝒜_{+}^{∧} and 𝒜^{∧} are perfectoid, and we have natural isomorphisms of topological rings with Δ-graded structures

ω_{+} : E(𝒜_{+}^{∧}, \underline{𝒜} _{+}^{c}) →^{∼} (ℰ_{+}^{∧}, \underline{ℰ} _{+}^{c}) ω : E(𝒜^{∧}, \underline{𝒜} ^{c}) →^{∼} (ℰ^{∧}, \underline{ℰ}^{c})

fitting into the commutative diagrams

\xymatrix{\underline{E} \ar[r]^{E(j_{A}^{+})} \ar[dr]_{j_{E}^{+}} & E(𝒜_{+}^{∧}, \underline{𝒜} _{+}^{c}) \ar[d]_{ω_{+}} & Γ_{I}^{+} \ar[l]_{E(i_{A}^{+})} \ar[dl]^{i_{E}^{+}} & \underline{E} \ar[r]^{E(j_{A})} \ar[dr]_{j_{E}} & E(𝒜^{∧}, \underline{𝒜} ^{c}) \ar[d]_{ω} & Γ_{I} \ar[l]_{E(i_{A})} \ar[dl]^{i_{E}} \\  & (ℰ_{+}^{∧}, \underline{ℰ} _{+}^{c}) &  &  & (ℰ^{∧}, \underline{ℰ}^{c}).}

Proof. Clearly we have E(\underline{A}) = \underline{E}. The corollary then becomes a special case of proposition 16.6.44, after we have shown

Claim 16.6.59. φ_{E} ∘ \overline{u}_{P} = E(\overline{u}_{A} ∘ φ_{E}).

Proof of the claim. Clearly φ_{E} ∘ \overline{u}_{P} = \overline{u}_{E} ∘ E(φ_{E}), so we are reduced to checking the identity \overline{u}_{E} = E(\overline{u}_{A}), and the latter holds by remark 9.4.5(ii). □

16.6.60. From lemma 16.6.53(i) we deduce a corresponding coequalizer presentation for ℰ_{+} in the category of Δ-graded ℤ-algebras :

\xymatrix{ℤ[Q] \ar[r] \ar[r] & ℤ[P × P] ⊗_{ℤ[P]} E = E[P] \ar[r] & ℤ[Γ_{I}^{+}] ⊗_{ℤ[P]} E = ℰ_{+}}

where E and ℤ[P × P] are regarded as ℤ[P]-algebras via φ_{E} and respectively via the morphism of monoids P → P × P given by the rule e_{λ}^{δ} ↦ f_{λ}^{′δ} for every λ ∈ Λ and every δ ∈ Δ^{+}. Likewise we may present 𝒜_{+}, and there follow natural isomorphisms

ℰ_{+} →^{∼} E[P]/ℐ_{E} 𝒜_{+} →^{∼} A[P]/ℐ_{A}

where ℐ_{E} ⊂ E[P] is the ideal generated by the system

(β_{λ}^{r} · e_{μ}^{r} − β_{μ}^{r} · e_{λ}^{r} | λ, μ ∈ Λ; r ∈ ℕ[1/p]).

and ℐ_{A} ⊂ A[P] is the ideal generated by the system

(\overline{u}_{A}(β_{λ}^{r}) · e_{μ}^{r} − \overline{u}_{A}(β_{μ}^{r}) · e_{λ}^{r} | λ, μ ∈ Λ; r ∈ ℕ[1/p]).

Especially, notice that ℐ_{E} (resp. ℐ_{A}) is a graded ideal of the Δ-graded ring E[P]_{/Δ} (resp. A[P]_{/Δ}), arising via the grading | · |_{P} : P → Δ, and

gr_{δ}ℰ_{+} = (gr_{δ}E[P]_{/Δ})/gr_{δ}ℐ_{E} = E[gr_{δ}P]/gr_{δ}ℐ_{E} for every δ ∈ Δ.

We can then form the topological rings with Δ-graded structure

(ℰ_{+}^{s}, \underline{ℰ} _{+}^{s}) := (\underline{E}[P]_{/Δ})/ℐ_{E} (𝒜_{+}^{s}, \underline{𝒜} _{+}^{s}) := (\underline{A}[P]_{/Δ})/ℐ_{A}

that are the maximal separated quotients ℰ_{+}^{s} of ℰ_{+} and 𝒜_{+}^{s} of 𝒜_{+}, endowed with the Δ-graded ℤ-algebra structures

\underline{ℰ} _{+}^{s} := ⨁_{δ∈Δ^{+}}(I^{⟨δ⟩} ⊗_{P} E)^{s} and \underline{𝒜} _{+}^{s} := ⨁_{δ∈Δ^{+}}(I^{⟨δ⟩} ⊗_{P} A)^{s}

where (I^{⟨δ⟩} ⊗_{P} E)^{s} denotes the maximal separated quotient of I^{⟨δ⟩} ⊗_{P} E, for every δ ∈ Δ, and likewise for (I^{⟨δ⟩} ⊗_{P} A)^{s}. It is now straightforward that the completion maps ℰ_{+}^{s} → ℰ_{+}^{∧} and 𝒜_{+}^{s} → 𝒜_{+}^{∧} factor through natural isomorphisms

(ℰ_{+}^{s}, \underline{ℰ} _{+}^{s})^{∧} →^{∼} (ℰ_{+}^{∧}, \underline{ℰ} _{+}^{c}) and (𝒜_{+}^{s}, \underline{𝒜} _{+}^{s})^{∧} →^{∼} (𝒜_{+}^{∧}, \underline{𝒜} _{+}^{c}).

16.7. Perfectoid spaces. Some basic verifications concerning perfectoid spaces will employ a certain amount of formal algebraic geometry, that we collect hereafter.

16.7.1. Quite generally, let R be any ℚ_{+}-graded ring. We associate with R a projective scheme

Proj R

as follows. For every γ ∈ ℚ_{>0}, let R^{(γ)} := ⨁_{k∈ℕ} R_{kγ}, which we regard as an ℕ-graded ring, whose k-graded direct summand is R_{kγ}, for every k ∈ ℕ. Set Y^{(γ)} := Proj R^{(γ)} for every such γ. If n > 0 is any integer, the discussion of (10.6.11) yields a natural isomorphism

Y^{(γ)} ⟶^{Ω_{n}^{(γ)}} Y^{(nγ)}

of R_{0}-schemes, and it is easily seen that

Ω_{p}^{(nγ)} ∘ Ω_{n}^{(γ)} = Ω_{pn}^{(γ)} for every integers n, p > 0.

Thus, the rule γ ↦ Y^{(γ)} yields a well defined filtered system of isomorphisms, and we let Proj R be any choice of an R_{0}-scheme representing the colimit of this system; e.g. Proj R := Y^{(γ_{0})}, for some fixed γ_{0} ∈ ℚ_{>0}, together with the cocone (Ω^{(γ)} : Y^{(γ)} → Proj R | γ ∈ ℚ_{>0}) resulting from the filtered system of morphisms Ω_{•}^{(•)}.

16.7.2. To ease notation, we shall let Y := Proj R. For every γ ∈ ℚ_{>0}, every n ∈ ℕ, and every f ∈ R_{nγ} we have the open subset D_{+}(f) ⊂ Y^{(γ)} defined as in (10.6.1), and we denote also D_{+}(f) ⊂ Y the image of this open subset under Ω^{(γ)}. It is easily seen that the resulting open subset of Y is independent of the choice of γ. We also set 𝒪_{Y} (0) := 𝒪_{Y} and

𝒪_{Y} (γ) := Ω_{∗}^{(|γ|)}𝒪_{Y^{(|γ|)}}(γ/|γ|) for every γ ∈ ℚ \ {0}

and we notice that – according to (10.6.16) – the quasi-coherent 𝒪_{Y} -module 𝒪_{Y} (γ) restricts to an invertible 𝒪_{Y} -module on the open subset

U_{γ}(R) := ⋃_{f∈R_{|γ|}} D_{+}(f) for every γ ∈ ℚ \ {0}.

Remark 16.7.3. Let Γ ⊂ ℚ_{+} be any submonoid, and suppose that R = ℚ_{+}×_{Γ}R′ for a Γ-graded ring R′. Then it is easily seen that 𝒪_{Y} (γ) = 0 and U_{γ}(R) = ∅ for every γ ∈ ℚ \ Γ^{gp}.

16.7.4. In view of the isomorphisms (10.6.14), we get natural identifications

(16.7.5)  𝒪_{Y} (nγ) →^{∼} Ω_{∗}^{(γ)}𝒪_{Y^{(γ)}}(n)  for every n ∈ ℤ and every γ ∈ ℚ_{>0}.

For every γ, γ′ ∈ ℚ there exists as well a natural morphism

(16.7.6)  𝒪_{Y} (γ) ⊗_{𝒪_{Y}} 𝒪_{Y} (γ′) → 𝒪_{Y} (γ + γ′)

that generalizes (10.6.21). Namely, pick a common divisor, i.e. a rational number δ ∈ ℚ_{>0} and integers p, q ∈ ℕ such that pδ = γ and qδ = γ′; we have first a natural identification

𝒪_{Y} (γ) ⊗_{𝒪_{Y}} 𝒪_{Y} (γ′) →^{∼} Ω_{∗}^{(δ)}𝒪_{Y^{(δ)}}(p) ⊗_{𝒪_{Y}} Ω_{∗}^{(δ)}𝒪_{Y^{(δ)}}(q) →^{∼} Ω_{∗}^{(δ)}(𝒪_{Y^{(δ)}}(p) ⊗_{𝒪_{Y}^{(δ)}} 𝒪_{Y^{(δ)}}(q))

which we combine with the map

Ω_{∗}^{(δ)}(𝒪_{Y^{(δ)}}(p) ⊗_{𝒪_{Y}^{(δ)}} 𝒪_{Y^{(δ)}}(q)) → Ω_{∗}^{(δ)}𝒪_{Y^{(δ)}}(p + q) →^{∼} 𝒪_{Y} (γ + γ′)

deduced from (10.6.21) and the inverse of the isomorphism (16.7.5), to obtain (16.7.6), and it is easily seen that the resulting map is independent of the choice of δ : details left to the reader. Especially, (16.7.6) restricts to an isomorphism on U_{gcd(γ,γ′)}(R), where gcd(γ, γ′) is the greatest common divisor of γ and γ′ (here we let gcd(0, 0) := 0; actually, it is not difficult to show that (16.7.6) is an isomorphism on the open subset U_{γ}(R) ∪ U_{γ′}(R)). Furthermore, let

π^{(γ)} : Y^{(γ)} → Spec R_{0} and π : Y → Spec R_{0}

be the structure morphisms; the morphism (10.6.22) induces a natural map

π^{∗}R_{γ}^{∼} →^{∼} Ω_{∗}^{(γ)}π^{(γ)∗}R_{1}^{(γ)} → Ω_{∗}^{(γ)}𝒪_{Y^{(γ)}}(1) = 𝒪_{Y} (γ) for every γ ∈ Γ \ {0}

which restricts to an epimorphism on U_{γ}(R).

16.7.7. Let R′ be another ℚ_{+}-graded ring, and φ : R → R′ a morphism of ℚ_{+}-graded rings; set Y′ := Proj R′, Y^{′(γ)} := Proj R^{′(γ)}, and denote by Ω^{′(γ)} : Y^{(γ)} → Y′ and Ω_{n}^{′(γ)} : Y^{(γ)} → Y^{′(nγ)} the induced morphism, for every γ ∈ ℚ_{>0} and every integer n > 0. The restriction of φ

φ^{(γ)} : R^{(γ)} → R^{′(γ)} for every γ ∈ ℚ_{>0}

is a morphism of ℕ-graded rings, whence a morphism Proj φ^{(γ)} : G(φ^{(γ)}) → Y^{(γ)} (notation of (10.6.5)). Moreover, it is easily seen that

Ω_{n}^{′(γ)}G(φ^{(γ)}) = G(φ^{(nγ)}) for every γ ∈ ℚ_{>0} and every integer n > 0.

Furthermore, the resulting diagram of schemes

\xymatrix{G(φ^{(γ)}) \ar[r]^{Proj φ^{(γ)}} \ar[d]_{Ω_{n}^{′(γ)}} & Y^{(γ)} \ar[d]^{Ω_{n}^{(γ)}} \\ G(φ^{(nγ)}) \ar[r]^{Proj φ^{(nγ)}} & Y^{(nγ)}}

commutes for every such γ and n. Thus, we may set

G(φ) := Ω^{(γ)}(G(φ^{(γ)})) for any γ ∈ ℚ_{>0}

and the colimit of the system (Proj φ^{(γ)} | γ ∈ ℚ_{+}) is a well defined morphism

Proj φ : G(φ) → Y.

Remark 16.7.8. (i) Set R_{+} := ⨁_{γ∈ℚ_{>0}} R_{γ}, and define likewise R′_{+}. Let also f′ ∈ R′_{+} be any homogeneous element; we claim that D_{+}(f′) ⊂ G(φ) if and only if f′ lies in the radical J of the ideal of R′ generated by φ(R_{+}). Indeed, say that f′ ∈ R′_{γ} for some γ ∈ ℚ_{>0}; if D_{+}(f′) ⊂ G(φ), then D_{+}(f′) ⊂ G(φ^{(γ)}) as well, so f′ lies in the radical of the ideal of R^{′(γ)} generated by φ^{(γ)}(R_{+}^{(γ)}) (see the discussion of (10.6.5)), and therefore f′ lies in J as well. Conversely, if f′ ∈ J, we may find n, k ∈ ℕ, and elements ν(i) ∈ ℚ_{>0}, a_{i} ∈ R_{ν(i)}, f′_{i} ∈ R′_{γ−ν(i)} for i = 1, …, k, such that f^{′n} = ∑_{i=1}^{k} f_{i} · φ(a_{i}). Then, pick any common divisor δ of γ, ν(1), …, ν(k) in ℚ_{>0}; it follows that f′ lies in the radical of the ideal of R^{′(δ)} generated by φ^{(δ)}(R_{+}^{(δ)}), whence D_{+}(f′) ⊂ G(φ^{(δ)}), and finally D_{+}(f′) ∈ G(φ).

(ii) Especially, if φ(R_{+}) generates the ideal R′_{+} of R′, then G(φ) = Y′.

16.7.9. We shall apply the foregoing constructions to the situation contemplated in (16.4.4) : we let (A, u_{0}) be a pair consisting of a ring A, and a ring homomorphism

u_{0} : R_{r,0} := ℤ[T_{1}^{1/p^{∞}}, …, T_{r}^{1/p^{∞}}] → A.

Also, we denote by T_{•}P_{r} ⊂ P_{r} := ℕ[1/p]^{⊕r} the ideal generated by ℕ^{⊕r}, we set Γ := ℕ[1/p], and we consider the Γ-graded angular Rees algebra

R⟨A⟩ := ⨁_{γ∈Γ} T_{•}^{⟨γ⟩}A.

Hence, for every γ ∈ Γ, the direct summand R⟨A⟩_{γ} is the ideal of A generated by all products of the form u_{0}(T_{1}^{a_{1}} ⋯ T_{r}^{a_{r}}), where (a_{1}, …, a_{r}) ∈ P_{r} is any sequence of exponents such that a_{1} + ⋯ + a_{r} = γ. The angular blowing up of the ideal T_{•}A_{0} is the morphism

Y := Proj (R⟨A⟩_{/ℚ_{+}}) → X := Spec A

(notation of definition 7.6.1(v)). Notice that U_{γ}(R⟨A⟩_{/ℚ_{+}}) = Y for every γ ∈ Γ \ {0}.

Theorem 16.7.10. In the situation of (16.7.9), suppose furthermore that the ring A fulfills the condition of lemma 16.4.5. Then, for every s ∈ ℝ_{+} we have :

(i) The natural maps T_{•}^{⌈s⌉}A → T_{•}^{⌈s⌉} · H^{0}(Y, 𝒪_{Y}) → H^{0}(Y, T_{•}^{⌈s⌉}𝒪_{Y}) are isomorphisms. (ii) H^{p}(Y, T_{•}^{⌈s⌉}𝒪_{Y}) = 0 for every p > 0. (iii) The ring R⟨A⟩ fulfills the condition of lemma 16.4.5. (iv) For every t ∈ ℝ_{+} the natural map

T_{•}^{⌈s⌉}A ⊗_{A} T_{•}^{⌈t⌉}A → T_{•}^{⌈t+s⌉}A

is an isomorphism.

Proof. Notice first that T_{•}^{⌈s⌉} · H^{0}(Y, 𝒪_{Y}) and H^{0}(Y, T_{•}^{⌈s⌉}𝒪_{Y}) are A-submodules of H^{0}(Y, 𝒪_{Y}); hence, the map T_{•}^{⌈s⌉} · H^{0}(Y, 𝒪_{Y}) → H^{0}(Y, T_{•}^{⌈s⌉}𝒪_{Y}) is injective, and in order to show (i), it suffices to prove that the composition T_{•}^{⌈s⌉}A → H^{0}(Y, T_{•}^{⌈s⌉}𝒪_{Y}) is an isomorphism.

Now, for every k ∈ ℕ we set Γ_{k} := {n/p^{k} | n ∈ ℕ} ⊂ Γ, and consider the Γ_{k}-graded ring

S_{k} := ⨁_{γ∈Γ_{k}} I_{k}^{p^{k}γ}A and the ℚ_{+}-graded ring S′_{k} := (S_{k})_{/ℚ_{+}}

where I_{k} ⊂ R_{r,0} denotes the ideal generated by (T_{1}^{1/p^{k}}, …, T_{k}^{1/p^{k}}). The natural inclusion map is a morphism of ℚ_{+}-graded rings S′_{k} → S′_{k+1} for every k ∈ ℕ, and the colimit of the resulting system (S′_{k} | k ∈ ℕ) is R(A)_{/ℚ_{+}}. Set as well

Y_{k} := Proj S′_{k} for every k ∈ ℕ.

There follows a system of affine morphisms of X-schemes (Y_{k} | k ∈ ℕ), and in view of [65, Ch.IV, Prop.8.2.3] it is easily seen that its limit is Y. For every k ∈ ℕ, let π_{k} : Y → Y_{k} be the natural projection. Notice that the direct summand S′_{k,γ} of S′ is an ideal of A, and

⋃_{k∈ℕ} S′_{k,γ} = T_{•}^{⟨γ⟩}A for every γ ∈ Γ.

We deduce natural identifications as in remark 10.6.39(iv) for every γ ∈ Γ and every k ∈ ℕ

𝒪_{Y_{k}}(γ) →^{∼} S′_{k,γ}𝒪_{Y_{k}}  colim_{i∈ℕ} π_{i}^{−1}𝒪_{Y_{i}}(γ) →^{∼} 𝒪_{Y} (γ) →^{∼} T_{•}^{⟨γ⟩}𝒪_{Y}

whence, by proposition 10.1.10(ii), a natural isomorphism of A-modules :

(16.7.11) H^{p}(Y, T_{•}^{⟨γ⟩}𝒪_{Y}) →^{∼} colim_{i∈ℕ} H^{p}(Y_{i}, 𝒪_{Y_{i}}(γ))  for every p ∈ ℕ and every γ ∈ Γ.

By the same token, we have as well a natural isomorphism

(16.7.12) H^{p}(Y, T_{•}^{⌈s⌉}𝒪_{Y}) →^{∼} colim_{γ>s} H^{p}(Y, T_{•}^{⟨γ⟩}𝒪_{Y})  for every p ∈ ℕ and every s ∈ ℝ_{+}.

Claim 16.7.13. There exists an integer c > 0 such that for every k ∈ ℕ and every γ ∈ Γ_{k}, the inclusion map i : S′_{k,γ+c/p^{k}}𝒪_{Y_{k}} → S′_{k,γ}𝒪_{Y_{k}} induces the zero map in cohomology :

0 = H^{p}(Y_{k}, i) : H^{p}(Y_{k}, 𝒪_{Y_{k}}(γ + c/p^{k})) → H^{p}(Y_{k}, 𝒪_{Y_{k}}(γ)) for every p > 0

and moreover the kernel and cokernel of the natural morphism of inverse systems

(S′_{k,n/p^{k}} → H^{0}(Y_{k}, 𝒪_{Y_{k}}(n/p^{k})) | n ∈ ℕ)

is uniformly essentially zero with step ≤ c.

Proof of the claim. Recall that Y_{k} is the colimit of the filtered system (Y_{k}^{(γ)} | γ ∈ Γ_{k}) defined as in (16.7.1). By inspecting the construction, it is easily seen that Z_{k} := Y_{k}^{(1/p^{k})} is the blowing up morphism of the ideal of 𝒪_{X} generated by the sequence f^{(k)} := (u_{0}(T_{1}^{1/p^{k}}), …, u_{0}(T_{k}^{1/p^{k}})). Say now that γ = a/p^{k}; according to (16.7.5), we have a natural identification

𝒪_{Y_{k}}(γ + d/p^{k}) →^{∼} Ω_{∗}^{(1/p^{k})}𝒪_{Z_{k}}(a + d) for every d ∈ ℕ.

On the other hand, lemma 16.4.5(i) implies that the ring A satisfies condition (a)_{f^{(k)}}^{un} with step ≤ r. Then, by theorem 10.6.50and remark 10.6.57, there exists a constant c ∈ ℕ independent of k and a, such that the inclusion map 𝒪_{Z_{k}}(a) → 𝒪_{Z_{k}}(a+c) induces the zero map in cohomology

H^{p}(Z_{k}, 𝒪_{Z_{k}}(a + c)) → H^{p}(Z_{k}, 𝒪_{Z_{k}}(a)) for every p > 0

and moreover, the kernel and cokernel of the morphism of inverse systems

(I_{k}^{n}A → H^{0}(Z_{k}, 𝒪_{Z_{k}}(n)) | n ∈ ℕ)

are uniformly essentially zero with step ≤ c. Both assertions of the claim are an immediate consequence. ♢

Now, in view of (16.7.12), in order to prove (ii), it suffices to show that for every γ > s there exists γ′ ∈ ℚ_{+} with s < γ′ < γ, and such that the natural map

H^{p}(Y, T_{•}^{⟨γ⟩}𝒪_{Y}) → H^{p}(Y, T_{•}^{⟨γ′⟩}𝒪_{Y})

vanishes. To this aim, pick any k ∈ ℕ such that c/p^{k} < γ − s, where c ∈ ℕ is as in claim 16.7.13; by virtue of (16.7.11), we get the sought vanishing with γ′ := γ − c/p^{k}. A similar argument proves also assertion (i) : details left to the reader.

(iii): It suffices to check that for every γ′ < γ in ℕ[1/p] the inclusion I_{r,0}^{⟨γ⟩} ⊂ I_{r,0}^{⟨γ′⟩} induces the zero morphism

Tor_{i}^{R_{r,0}}(R_{r,0}/I_{r,0}^{⟨γ⟩}, R⟨A⟩) → Tor_{i}^{R_{r,0}}(R_{r,0}/I_{r,0}^{⟨γ′⟩}, R⟨A⟩) for every i > 0.

However, for every k ∈ ℕ set A_{r,k} := ℤ[T_{1}^{1/p^{k}}, …, T_{r}^{1/p^{k}}] and let J_{r,k} ⊂ A_{r,k} be the ideal generated by (T_{1}^{1/p^{k}}, …, T_{r}^{1/p^{k}}); we regard A as an A_{r,k}-algebra for every k ∈ ℕ, via restriction of scalars along the inclusion map A_{r,k} → R_{r,0}, and we let R_{k}(A) be the Rees A-algebra associated with the J_{r,k}A-adic filtration of A. Now, say that γ = c/p^{n} and γ′ = c′/p^{n} for some c, c′, n ∈ ℕ; in view of claim 7.11.37(i) it suffices to show that for every i > 0 there exists k_{0} ∈ ℕ such that the inclusion J_{r,n+k}^{cp^{k}} ⊂ J_{r,n+k}^{c′p^{k}} induces the zero morphism

Tor_{i}^{A_{r,n+k}}(A_{r,n+k}/J_{r,n+k}^{cp^{k}}, R_{n+k}(A)) → Tor_{i}^{A_{r,n+k}}(A_{r,n+k}/J_{r,n+k}^{c′p^{k}}, R_{n+k}(A))

for every k ≥ k_{0}. However, for every k ∈ ℕ let f_{•}^{(k)} := (u_{0}(T_{1}^{1/p^{k}}), …, u_{0}(T_{r}^{1/p^{k}})), and denote by g_{•}^{(k)} the image of f_{•}^{(k)} under the natural ring homomorphism A → R_{k}(A). By lemma 16.4.5(i) we know that A fulfills condition (a)_{f_{•}^{(k)}}^{un} with step ≤ r; then corollary 7.9.22(ii) and remark 7.9.27 imply that R_{k}(A) fulfills condition (a)_{g_{•}^{(k)}}^{un} with a step independent of k; explic-

itly, the latter means that the inverse system (Tor_{i}^{A_{r,k}}(A_{r,k}/J_{r,k}^{t}, R_{k}(A)) | t ∈ ℕ) is uniformly essentially zero for every k ∈ ℕ and every i > 0, with a step independent of k. The assertion is an immediate consequence.

(iv): It suffices to show that the natural map

(16.7.14)  T_{•}^{⌈s⌉}A ⊗_{A} T_{•}^{⟨t⟩}A → T_{•}^{⌈t+s⌉}A

is an isomorphism for every t ∈ ℝ_{+}. However, (iii) easily implies that the natural map

T_{•}^{⌈s⌉}R_{r,0} ⊗_{R_{r,0}} T_{•}^{⟨t⟩}A → T_{•}^{⌈t+s⌉}A

is an isomorphism, and the latter factors through (16.7.14) and a surjection T_{•}^{⌈s⌉}R_{r,0} ⊗_{R_{r,0}} T_{•}^{⟨t⟩}A → T_{•}^{⌈s⌉}A ⊗_{A} T_{•}^{⟨t⟩}A, whence the contention. □

16.7.15. In the situation of (16.7.9), suppose that A fulfills the condition of lemma 16.4.5, and is endowed with a ℚ_{+}-grading which makes it an A_{0}-subalgebra of A_{0}[ℚ_{+}] (where A_{0} ⊂ A is the subring of homogeneous elements of degree zero), and moreover suppose that the image of u_{0} lies in A_{0}. In this case, the foregoing considerations apply to A_{0} as well, and we get a well defined Γ-graded subring R⟨A_{0}⟩ of R⟨A⟩, as well the angular blowing up morphism

Y_{0} := Proj (R⟨A_{0}⟩_{/ℚ_{+}}) → X_{0} := Spec A_{0}.

Notice also that R⟨A⟩ carries a natural (Γ × ℚ_{+})-graded ring structure : namely,

gr_{(γ,t)}R⟨A⟩ := T_{•}^{⟨γ⟩}A_{t} for every (γ, t) ∈ Γ × ℚ_{+}

which is an ideal in A_{0}, for every such (γ, t), and we set

R⟨A_{t}⟩ := ⨁_{γ∈Γ} T_{•}^{⟨γ⟩}A_{t} = A_{t}R⟨A_{0}⟩ for every t ∈ ℚ_{+}.

Corollary 16.7.16. In the situation of (16.7.15), for every s ∈ ℝ_{+} and every t ∈ ℚ_{+} we have :

(i) The natural maps T_{•}^{⌈s⌉}A_{t} → T_{•}^{⌈s⌉}A_{t} · H^{0}(Y_{0}, 𝒪_{Y_{0}}) → H^{0}(Y_{0}, T_{•}^{⌈s⌉}A_{t}𝒪_{Y_{0}}) are isomorphisms.

(ii) H^{p}(Y_{0}, T_{•}^{⌈s⌉}A_{t}𝒪_{Y_{0}}) = 0 for every p > 0.

Proof. Let i : A_{0} → A and R⟨i⟩ : R⟨A_{0}⟩ → R⟨A⟩ be the inclusion maps; since i(R⟨A_{0}⟩_{+}) generates the ideal R⟨A⟩_{+} of R⟨A⟩, then i induces a morphism of schemes Proj R⟨i⟩ : Y → Y_{0} (remark 16.7.8(ii)) fitting into the commutative diagram

\xymatrix{Y \ar[r]^{Proj R⟨i⟩} \ar[d]_{π} & Y_{0} \ar[d]^{π_{0}} \\ X \ar[r]^{Spec i} & X_{0}.}

To ease notation, we let φ_{X} := Spec i and φ_{Y} := Proj R⟨i⟩. Notice that

T_{•}^{⌈s⌉}R⟨A⟩_{(f)} = T_{•}^{⌈s⌉}(⨁_{t∈ℚ_{+}} R⟨A_{t}⟩)_{(f)} = ⨁_{t∈ℚ_{+}} T_{•}^{⌈s⌉}A_{t}R⟨A_{0}⟩_{(f)}

for every homogeneous element f ∈ R⟨A_{0}⟩, which shows that

φ_{Y ∗}T_{•}^{⌈s⌉}𝒪_{Y} = ⨁_{t∈ℚ_{+}} T_{•}^{⌈s⌉}A_{t}𝒪_{Y_{0}}.

Since φ_{Y} is an affine morphism, we deduce

H^{p}(Y, T_{•}^{⌈s⌉}𝒪_{Y}) = H^{p}(Y_{0}, φ_{Y ∗}T_{•}^{⌈s⌉}𝒪_{Y}) = ⨁_{t∈ℚ_{+}} H^{p}(Y_{0}, T_{•}^{⌈s⌉}A_{t}𝒪_{Y_{0}})

and combining with theorem 16.7.10we get the corollary. □

Example 16.7.17. (i) According to proposition 16.4.10(i), theorem 16.7.10 applies especially to the case where A is perfectoid. In this case there exist unique β_{1}, …, β_{r} ∈ E := E(A) such that u_{0}(T_{i}^{1/p^{k}}) = \overline{u}_{A}(β_{i}^{1/p^{k}}) for every i = 1, …, r and every k ∈ ℕ, so we may regard as well E as an R_{r,0}-algebra via the unique ring homomorphism

R_{r,0} → E  :  T_{i} ↦ β_{i}  for every i = 1, …, r.

(ii) More generally, proposition 16.6.5shows that theorem 16.7.10applies to the case where A is the graded subring of a perfectoid ring (A^{∧}, A) with Δ-graded structure, where Δ is an integral p-perfect monoid, provided the elements β_{1}, …, β_{r} as in (i) are homogeneous elements of the Δ-graded subring of E(A).

(iii) A ring A as in (ii) is given by example 16.6.12 : we take a perfectoid ring A_{0}, a second finite family β′_{•} := (β′_{1}, …, β′_{r′}) of elements of E_{0} := E(A_{0}), we set I := β′_{•}E_{0}, and let A_{t} be the topological closure in A_{0} of I^{⟨t⟩}A_{0} for every t ∈ Δ := ℕ[1/p]. Then, clearly corollary 16.7.16shall apply to the ℚ_{+}-graded ring A_{/ℚ_{+}}.

(iv) Let A_{0}, E_{0}, β′_{•} and A be as in (iii), and suppose that also the sequence β_{•} lies in E(A_{0}). Suppose moreover that I is an ideal of adic definition for E_{0}. In this case, the ideal I^{⟨t⟩}A_{0} is also open in A_{0} (corollary 16.3.40(ii)), and therefore it coincides with A_{t} for every t ∈ ℕ[1/p]. Especially, gr_{(γ,t)}R⟨A⟩ = T_{•}^{⟨γ⟩}I^{⟨t⟩}A_{0} for every (γ, t) ∈ Γ × ℕ[1/p].

(v) In the situation of (iv), suppose furthermore that the ideal T_{•}E_{0} ⊂ E_{0} generated by β_{•} is open. Notice that T_{•}^{⌈0⌉}E_{0} is the radical of T_{•}E_{0} (details left to the reader). It follows that for every s ∈ ℕ[1/p]\{0} there exists s′ > 0 such that I^{⟨s⟩}E_{0} ⊂ T_{•}^{⟨s′⟩}E_{0}, whence I^{⟨s⟩}A_{0} ⊂ T_{•}^{⟨s′⟩}A_{0} (corollary 16.3.51(ii)). Therefore :

I^{⟨t+s⟩}A_{0} ⊂ I^{⟨t⟩}T_{•}^{⌈0⌉}A_{0} ⊂ I^{⟨t⟩}A_{0} for every t, s ∈ ℕ[1/p] with s > 0.

Corollary 16.7.18. In the situation of (16.7.9), suppose that A is perfectoid, and let I be any finitely generated ideal of adic definition of E := E(A). With the notation of example 16.7.17(v), for every t ∈ ℝ_{+} we have :

(i) The natural maps I^{⌈t⌉}A → I^{⌈t⌉} · H^{0}(Y, 𝒪_{Y}) → H^{0}(Y, I^{⌈t⌉}𝒪_{Y}) are isomorphisms.

(ii) H^{p}(Y, I^{⌈t⌉}𝒪_{Y}) = 0 for every p > 0.

Proof. With the notation of example 16.7.17(v), notice that

H^{p}(Y, I^{⌈t⌉}T_{•}^{⌈0⌉}𝒪_{Y}) = colim_{s>t} H^{p}(Y, I^{⟨s⟩}T_{•}^{⌈0⌉}𝒪_{Y}).

On the other hand, example 16.7.17(v) implies that I^{⌈t⌉}A = I^{⌈t⌉}T_{•}^{⌈0⌉}A for every t ∈ ℝ_{+}. To conclude, it suffices to invoke corollary 16.7.16 as in example 16.7.17(iii,iv) : i.e. with A_{0} and Y_{0} replaced by the current A and respectively Y, and with A replaced by the ℚ_{+}-graded ring A′_{/ℚ_{+}}, where A′_{s} := I^{⟨s⟩}A for every s ∈ ℕ[1/p]. □

Definition 16.7.19. We say that a topological ring (A, 𝒯 ) is a formal P-ring (resp. a formally perfectoid ring) if it is adic with a finitely generated ideal of adic definition, and the separated completion of (A, 𝒯 ) is a P-ring (resp. a perfectoid ring).

Remark 16.7.20. Let (A, 𝒯 ) be any topological ring.

(i) Suppose that (A, 𝒯 ) is a P-ring. Then, any ideal of definition of the P-ring A is also an ideal of adic definition of the adic ring A. Remark 16.2.4implies that the converse holds if and only if A is an 𝔽_{p}-algebra. Indeed, if I is any ideal of adic definition of A, then any power I^{n} is also an ideal of adic definition, but if p does not vanish in A, then one such power will not contain p, whence the contention.

(ii) Let φ : (A, 𝒯 ) → (A′, 𝒯 ′) be an adic morphism of topological rings, and suppose that φ is a weakly étale ring homomorphism. Then [75, Lemma 3.1.2(i)] implies that φ is adically weakly étale (see definition 8.3.23(iii)).

Theorem 16.7.21. Let f : A → B be a c-adically weakly étale morphism of topological rings.

(i) If A and B are f-adic and A is a formal P-ring (resp. a formally perfectoid ring), the same holds for B.

(ii) If f is c-adically faithfully flat, the following holds : (a) If B is f-adic and A and B are complete and separated, A is f-adic and f is adic.

(b) If B is a formal P-ring (resp. a formally perfectoid ring), the same holds for A.

Proof. (i): In light of lemma 8.3.32(iii,iv), we may assume that A and B are complete and separated, in which case A is a P-ring (resp. a perfectoid ring), f is adically weakly étale (lemma 8.3.24(i.c)), and we have to show that B is a P-ring (resp. is perfectoid).

Suppose first that A is a P-ring, and let I be any ideal of definition of A. It suffices to check that IB is an ideal of definition for B. However, the natural map B/I^{2}B → B^{∧}/I^{2}B^{∧} is an isomorphism, by remark 8.3.3(ii), so we come down to checking that the Frobenius endomorphism of B/I^{2}B is surjective. However, the induced ring homomorphism A/I^{2} → B/I^{2}B is weakly étale by assumption; since the Frobenius endomorphism of A/I^{2} is surjective by assumption, the assertion then follows easily from [75, Th.3.5.13(ii)].

Next, suppose that A is perfectoid. Due to remark 16.3.58, we may assume that p ∈ I^{t}, where t ∈ ℕ is defined as in (16.3.55), in terms of the length of a system of generators for I. Set J := I^{(p)}; taking into account (i) and theorem 16.3.64, we are then reduced to checking that the morphism of graded rings

Φ_{IB} : gr_{IB}^{•}B → gr_{JB}^{•}B

is an isomorphism. However, since f ⊗_{A} A/I^{n} is flat for every n ∈ ℕ, we get induced graded ring isomorphisms

φ_{I} : B ⊗_{A} gr_{I}^{•}A →^{∼} gr_{IB}^{•}B φ_{J} : B ⊗_{A} gr_{J}^{•}A →^{∼} gr_{JB}^{•}B

and φ_{I} identifies the map

gr_{I}^{•}f : gr_{I}^{•}A → gr_{IB}^{•}B

with the base change f ⊗_{A} gr_{I}^{•}A. Especially, since f ⊗_{A} A/I is weakly étale, the same holds for gr_{I}^{•}f ([75, Lemma 3.1.2(i)]). Moreover, denote by (gr_{I}^{•}A)_{(Φ)} the gr_{I}^{•}A-algebra whose underlying ring is gr_{I}^{•}A and whose structure map is the Frobenius endomorphism Φ_{gr_{I}^{•}A}, and define likewise (gr_{IB}^{•}B)_{(Φ)}; by [75, Th.3.5.13(ii)], the maps φ_{J} and gr_{J}^{•}f induce an isomorphism

ψ_{J} : B ⊗_{A} (gr_{J}^{•}A)_{(Φ)} →^{∼} gr_{JB}^{•}B ⊗_{gr_{J}^{•} A} (gr_{J}^{•}A)_{(Φ)} →^{∼} (gr_{JB}^{•}B)_{(Φ)} b ⊗ a ↦ b^{p} · gr_{I}f(a).

Summing up, we get a commutative diagram of ring homomorphisms

(16.7.22)  \xymatrix{B ⊗_{A} gr_{I}^{•}A \ar[r]^{B⊗_{A}Φ_{I}} \ar[d]_{φ_{I}} & B ⊗_{A} (gr_{J}^{•}A)_{(Φ)} \ar[d]^{ψ_{J}} \\ gr_{IB}^{•}B \ar[r]^{Φ_{IB}} & (gr_{JB}^{•}B)_{(Φ)}}

whose vertical arrows are isomorphisms. Lastly, since A is a formally perfectoid ring, Φ_{I} is an isomorphism as well (theorem 16.3.64), whence the assertion.

(ii.a): Let J be any finitely generated ideal of adic definition of B, and (I_{λ} | λ ∈ Λ) a cofiltered system of ideals that defines the linear topology of A. By assumption, (I_{λ}B)^{c} is open in B, hence I_{λ}B = (I_{λ}B)^{c} for every λ ∈ Λ (lemma 8.3.21(ii.b)); then the induced map

f_{λ} : A_{λ} := A/I_{λ} → B_{λ} := B/I_{λ}B

is weakly étale and faithfully flat for every λ ∈ Λ. Also by assumption, we may find μ ∈ Λ such that I_{μ}B ⊂ J^{2}. Hence, JB_{μ} is contained in the nilradical nil(B_{μ}) of B_{μ}, and it follows easily from claim 14.3.27(i) that

nil(B_{μ}) = nil(A_{μ}) · B_{μ}.

Therefore, we may find a finitely generated ideal K′ ⊂ nil(A_{μ}) such that JB_{μ} ⊂ K′B_{μ}. Pick a finitely generated ideal K ⊂ A such that KA_{μ} = K′. It follows easily that J ⊂ KB + J^{2}. However, B is complete and J is topologically nilpotent in B, so J lies in the Jacobson radical of B, and consequently J ⊂ KB, by Nakayama’s lemma. On the other hand, by construction, we may find n ∈ ℕ such that K′^{n} = 0, whence K^{n} ⊂ I_{μ}, and therefore K^{n}B ⊂ J^{2}. Summing up, we conclude that KB is an ideal of adic definition of B. For every n ∈ ℕ, pick ν(n) ∈ Λ such that I_{ν(n)}B ⊂ J^{n}; then I_{ν(n)}B_{λ} ⊂ K^{n}B_{λ} for every λ ∈ Λ, so that I_{ν(n)}A_{λ} ⊂ K^{n}A_{λ}, as f_{λ} is faithfully flat. We deduce that the topological closure (K^{n})^{c} of K^{n} contains I_{ν(n)}, and especially, it is an open ideal of A, for every n ∈ ℕ. On the other hand, for every λ ∈ Λ we may find k(λ) ∈ ℕ such that J^{2k(λ)} ⊂ I_{λ}B, so that K^{n·k(λ)}B_{ν} ⊂ I_{λ}B_{ν} for every ν ≥ λ. It follows that K^{n·k(λ)}A_{ν} ⊂ I_{λ}A_{ν}, again since f_{ν} is faithfully flat; then, since I_{λ} is closed in A, we deduce that K^{n·k(λ)} ⊂ I_{λ}, for every λ ∈ Λ. Summing up, we see that the topology of A is c-adic, hence f-adic, by lemma 8.3.21(ii.a). Then f is adic, by lemma 8.3.24(i.c).

(ii.b): In light of lemma 8.3.32(iii), we may replace f by f^{∧}, and assume from start that A and B are complete and B is a P-ring (resp. is perfectoid), and we need to show that the same holds for A. However, (ii.a) already says that A is f-adic, and f is adic. Suppose first that B is a P-ring, and pick any finitely generated ideal of adic definition for A; without loss of generality we may assume that J is an ideal of definition for the P-ring B, and that IB ⊂ J. Set A_{n} := A/I^{n+1} and B_{n} := B/I^{n+1}B for every n ∈ ℕ, and notice that JB_{0} is nilpotent ideal of B_{0}; arguing as in the proof of (ii.a), we deduce that there exists a finitely generated ideal

(16.7.23)  K_{0} ⊂ nil(A_{0})

such that JB_{0} ⊂ K_{0}B_{0}. Then, the preimage K of K_{0} in A is a finitely generated ideal such that JB ⊂ KB; hence p^{2} ∈ K^{2}B_{n} for every n ∈ ℕ. Since the induced map A_{n} → B_{n} is faithfully flat, it follows that p ∈ K^{2}A_{n} for every n ∈ ℕ, so p lies in the topological closure of K^{2} in A; but (K^{2})^{c} = K^{2}, since K is open in A by construction. Lastly, (16.7.23) implies that K^{n} ⊂ I for some sufficiently large n ∈ ℕ; thus, K is also an ideal of adic definition of A, and we are reduced to checking that the Frobenius endomorphism Φ_{A/J^{2}} of A/J^{2} is surjective. However, by lemma 16.2.3(iv) the Frobenius endomorphism Φ_{B/J^{2}B} is surjective; by [75, Th.3.5.13(ii)], it follows that Φ_{A/J^{2}} ⊗_{A/J^{2}} B/J^{2}B is also surjective. But the induced map A/J^{2} → B/J^{2}B is faithfully flat, whence the contention.

Lastly, suppose that B is perfectoid; by the foregoing, we know already that A is a P-ring, and by remark 16.3.58 we may find an ideal of definition I of A such that p ∈ I^{t}, where t ∈ ℕ is defined as in (16.3.55). We set J := I^{(p)}; arguing as in the proof of (i), we get a commutative diagram (16.7.22), whose vertical arrows are isomorphisms. Now, notice that, since f is adic, IB is an ideal of definition of B, so the bottom horizontal arrow of (16.7.22) is an isomorphism; hence the same holds for B⊗_{A}Φ_{I} = (B/IB)⊗_{A/I}Φ_{I}. But by assumption, f induces a faithfully flat map A/I → B/IB, so Φ_{I} is an isomorphism, and therefore A is perfectoid, by theorem 16.3.64. □

16.7.24. Let \underline{U} := (U, 𝒯_{A}, A_{U}^{+}) be any perfectoid quasi-affinoid scheme, and set \underline{U}_{E} := E(\underline{U}) (notation of (16.5.20)), so that \underline{U}_{E} = (E(U), 𝒯_{E}, E_{U}^{+}), where E_{U}^{+} := E(A_{U}^{+}). Let as well

A_{U} := 𝒪_{U}(U) E_{U} := 𝒪_{E(U)}(E(U))

and pick a perfectoid subring of definition A of A_{U} such that A_{U}^{+} is the integral closure of A ∩ A_{U}^{+} in A_{U} and such that the induced map U → Spec A is an open immersion (by theorem 16.5.13(i) and lemma 15.6.10(i), the ring A_{U}^{∘} is one such subring of definition), and recall that E := E(A) is a ring of definition of E_{U}. We denote as usual by φ_{U}^{♭} : E_{U} → A_{U} the continuous map of monoids provided by proposition 16.4.34(i), and recall that φ_{U}^{♭} restricts to a continuous map \overline{u}_{A} : E → A. Now, let e_{0}, …, e_{n} be a finite system of elements of E_{U} that generates an open ideal; according to claim 16.6.36, the system (a_{i} := φ_{U}^{♭}(e_{i}) | i = 0, …, n) generates an open ideal of A_{U}, so we may consider the rational subset

R := R_{A_{U}} (\frac{a_{1}}{a_{0}} , …, \frac{a_{n}}{a_{0}} ) ∩ Spa \underline{U}

and the corresponding topological rings 𝒪_{Spa U}^{∧}(R) and 𝒪_{Spa U}^{∧+}(R), defined as in (15.5.11). Recall the construction of 𝒪_{Spa U}^{∧}(R) : first, there is a natural f-adic topology on the localization A_{R} := A_{U}[1/a_{0}] such that the localization map A_{U} → A_{R} is f-adic, and the subring

B_{R} := A[a_{i}/a_{0} | i = 1, …, n] ⊂ A_{R}

is a ring of definition of A_{R}; hence A_{R} induces on B_{R} the unique linear topology such that the natural map A → B_{R} is adic. We let A_{R}^{+} be the integral closure of A_{U}^{+}[a_{i}/a_{0} | i = 1, …, n] in A_{R}, set U_{R} := U ∩ Spec A_{R}, and define

\underline{A}_{R} := (A_{R}, A_{R}^{+}, U_{R}) and \underline{U}_{R}^{∧} := (Spec \underline{A}_{R})^{∧}.

Then (up to natural isomorphism) we have

Γ(\underline{U}_{R}^{∧}) = (𝒪_{Spa U}^{∧}(R), 𝒪_{Spa U}^{∧+}(R), U_{R}^{∧}) where U_{R}^{∧} := U_{R} ×_{Spec A_{R}} Spec 𝒪_{Spa U}^{∧}(R).

Notice that A_{R} contains also the subrings

\begin{matrix}C_{R} := & A[φ_{U}^{♭}(e_{i}^{1/p^{k}})/φ_{U}^{♭}(e_{0}^{1/p^{k}}) | (i, k) ∈ \{1, …, n\} × ℕ] \\ D_{R} := & A_{U}^{+}[φ_{U}^{♭}(e_{i}^{1/p^{k}})/φ_{U}^{♭}(e_{0}^{1/p^{k}}) | (i, k) ∈ \{1, …, n\} × ℕ] \end{matrix}

and endow C_{R} and D_{R} with the topologies induced by the inclusion into A_{R}. Let also C_{R}^{+} := C_{R} ∩ A_{R}^{+}. The first observation is the following :

Proposition 16.7.25. With the notation of (16.7.24), the following holds :

(i) The natural morphism \underline{A}_{R}^{∧} → Γ(\underline{U}_{R}^{∧}) is an isomorphism.

(ii) The datum \underline{C}_{R} := (C_{R}, C_{R}^{+}, U_{R}) is a quasi-affinoid ring, and the natural morphism

Spec \underline{A}_{R} → Spec \underline{C}_{R} is an isomorphism. (iii) \underline{C}_{R}^{∧} is a perfectoid quasi-affinoid ring, and \underline{U}_{R}^{∧} is a perfectoid quasi-affinoid scheme. (iv) Suppose moreover that β\underline{U}(U_{R}) ⊂ Ω\underline{U} (notation of definition 15.6.1). Then the datum

\underline{D}_{R} := (D_{R}, D_{R}, U_{R}) is a quasi-affinoid ring, and \underline{D}_{R}^{∧} is perfectoid.

Proof. (ii): We consider the natural morphisms of schemes

U_{R} ⟶^{f} Spec A_{R} ⟶^{g} Spec C_{R} ⟶^{h} Spec A

whose composition is also the composition of the open immersion U_{R} → U with the natural morphism i : U → Spec A. By assumption, i is an open immersion, and its image contains the analytic locus of Spec A (lemma 8.3.29(iii)). It follows that h ∘ g ∘ f is an open immersion; moreover, by construction, both f and g have schematically dense images. From claim 15.6.5, we deduce that g ∘ f is an open immersion, and f(U_{R}) = (h ∘ g)^{−1}i(U_{R}) = (h ∘ g)^{−1}i(U). Now, U contains the analytic locus of Spec A, and the natural map A → A_{R} is f-adic; taking into account lemma 8.3.29(iv), it follows that f(U_{R}) contains the analytic locus of Spec A_{R}. Lastly, g identifies the analytic locus of Spec A_{R} with that of Spec C_{R} (lemma 8.3.29(iii)), so g ∘ f(U_{R}) contains the analytic locus of Spec C_{R}. Summing up, this shows that \underline{C}_{R} is a quasiaffinoid ring. Then, a simple inspection shows that the inclusion map C_{R} → A_{R} induces an isomorphism Spec \underline{A}_{R} → Spec \underline{C}_{R}.

(i): To begin with, we remark :

Claim 16.7.26. C_{R} and D_{R} are subrings of definition of A_{R}.

Proof of the claim. Clearly C_{R} and D_{R} are open in A_{R}, and D_{R} ⊂ C_{R}; by virtue of proposition 8.3.18(ii), we have then only to check that C_{R} is bounded in A_{R}. To this aim, since B_{R} is open and bounded in A_{R}, it suffices to show that there exists an open ideal J of B_{R} such that J · C_{R} ⊂ B_{R}. However, pick any finite system f_{1}, …, f_{r} of generators for an ideal of definition of E; since the system e_{0}, …, e_{n} generates an open ideal of E_{U}, we may find an integer k ∈ ℕ large enough so that the f_{i}^{r}e_{j} ∈ E for every i = 1, …, r and every j = 0, …, n, and then it is clear that the system (f_{i}^{r}e_{j} | i ≤ r, j ≤ n) generates an open ideal of E. By claim 16.6.36, it follows that the system (φ_{U}^{♭}(f_{i}^{r}e_{j}) | i ≤ r, j ≤ n) generates an open ideal J_{0} of A, and therefore J := J_{0}B_{R} is open in B_{R}. To conclude, it suffices to show that φ_{U}^{♭}(f_{i}^{r}e_{j}e_{s}^{ν}e_{0}^{−ν}) ∈ B_{R} for every i = 1, …, r every j, s = 0, …, n, and every ν ∈ ℕ[1/p]. Moreover, we may easily reduce to the case where 0 < ν < 1. Then we see that

f_{i}^{r}e_{j}e_{s}^{ν}e_{0}^{−ν} = \frac{e_{j}}{e_{0}} · (f_{i}^{rν}e_{s}^{ν}) · (f_{i}^{r(1−ν)}e_{0}^{1−ν})

where both f_{i}^{rν}e_{s}^{ν} and f_{i}^{r(1−ν)}e_{0}^{1−ν} lie in E, since the latter is a perfect 𝔽_{p}-algebra. The assertion follows. ♢

Pick a finite system β_{•} := (β_{1}, …, β_{s}) of elements of E that generates an ideal of adic definition, and set f_{i} := \overline{u}_{A}(β_{i}) for i = 1, …, s, so that the system f_{•} := (f_{1}, …, f_{s}) generates an ideal of adic definition of A. By claim 16.7.26, it then follows that the system f_{•} also generates an ideal of adic definition J for C_{R}. Denote by C_{R}^{∧} the separated completion of C_{R}; we notice :

Claim 16.7.27. The ring C_{R} fulfills condition (a)_{f}^{un} of (7.8.21), and C_{R}^{∧} is perfectoid.

Proof of the claim. Set Δ := ℕ[1/p]; recall that example 16.6.12 attaches to the sequence e_{•} := (e_{0}, …, e_{n}) of elements of E_{U}, two perfectoid rings with Δ-graded structure

(ℰ (e_{•})^{∧}, ℰ (e_{•})) and (𝒜 (e_{•})^{∧}, 𝒜 (e_{•})).

Moreover, gr_{0}ℰ (e_{•}) = E, gr_{0}𝒜 (e_{•}) = A, and we have a natural isomorphism

E(𝒜 (e_{•})^{∧}, 𝒜 (e_{•})) →^{∼} (ℰ (e_{•})^{∧}, ℰ (e_{•})).

Furthermore, since e_{•} generates an open ideal of E_{U}, the E-module gr_{δ}ℰ (e_{•}) (resp. the Amodule gr_{δ}𝒜 (e_{•})) is the submodule of E_{U} (resp. of A_{U}) generated by the elements of the form e_{0}^{ν_{0}} ⋯ e_{n}^{ν_{n}} (resp. φ_{U}^{♭}(e_{0}^{ν_{0}} ⋯ e_{n}^{ν_{n}})), where (ν_{0}, …, ν_{n}) ranges over all elements of Δ^{⊕n+1} such that ν_{0} + ⋯ + ν_{n} = δ (see remark 16.3.28(viii)). Especially, we may regard β_{•} as a sequence of elements of gr_{0}ℰ (e_{•}), whose image under the map \overline{u}_{𝒜 (e_{•})} : ℰ (e_{•}) → 𝒜 (e_{•}) is (naturally identified with) the sequence f_{•}. Next, let us endow 𝒜′ := 𝒜 (e_{•})[a_{0}^{−1}] with its J𝒜′-adic topology; since we are inverting an element a_{0} ∈ gr_{1}𝒜 (e_{•}), the A-algebra 𝒜′ is also Δ-graded, and a simple inspection yields a natural isomorphism of C_{R} →^{∼} gr_{0}𝒜′ of topological rings, which identifies the sequences f_{•} in these two rings. Lastly, by virtue of proposition 16.6.5(ii), we deduce that 𝒜 (e_{•}) satisfies condition (a)_{f}^{un} of (7.8.21), and then the same holds for gr_{0}𝒜′, whence the first assertion.

In order to show that C_{R}^{∧} is perfectoid, notice that J^{n}𝒜′ = ⨁_{δ∈Δ} J^{n}gr_{δ}𝒜′ for every n ∈ ℕ. It follows easily that the maximal separated quotient 𝒜′′ of 𝒜′ is also a Δ-graded ring (so that the projection 𝒜′ → 𝒜′′ is a homomorphism of Δ-graded rings), and gr_{0}𝒜′′ is the maximal separated quotient of gr_{0}𝒜′ : details left to the reader. Hence (𝒜′′, 𝒜′′) is a topological ring with Δ-pre-graded structure, and therefore (𝒞^{∧}, 𝒞) := (𝒜′′, 𝒜′′)^{∧} is a topological ring with Δgraded structure (proposition 8.5.3(iii)), and the induced map C_{R}^{∧} → gr_{0}𝒞 is an isomorphism of topological rings. On the other hand, since 𝒜 (e_{•})^{∧} is perfectoid, and since the localization map 𝒜 (e_{•}) → 𝒜′ is an adic ring homomorphism, theorem 16.7.21(i) implies that 𝒞^{∧} is perfectoid as well. To conclude, it suffices now to invoke proposition 16.6.1(ii). ♢

As the morphism i : U → Spec A is an open immersion whose image contains the analytic locus of Spec A, we may find an open ideal I ⊂ A such that Spec A \ U = Spec A/I, and we pick a finite system g_{•} := (g_{1}, …, g_{r}) of generators of I. On the other hand, i factors through the natural open immersion j : U → Spec A_{U} and the morphism φ : Spec A_{U} → Spec A induced by the inclusion map A → A_{U}; by claim 15.6.5(ii), it follows that φ^{−1}i(U) = j(U), whence Spec A_{U}/IA_{U} = Spec A_{U} \ j(U), and therefore

Z := Spec A_{R}/IA_{R} = Spec A_{R} \ U_{R}.

In view of claim 16.7.26, remark 15.4.13(ii) and proposition 8.3.33(iii), we are then reduced to showing that

depth_{Z}A_{R} ⊗_{C_{R}} C_{R}^{∧} > 1.

However, set Q^{•} := Cone(C_{R}[0] → C_{R}^{∧}[0]); clearly we have depth_{IA_{U}} A_{U} > 1, whence depth_{Z}A_{R} > 1, and thus it suffices to check that

RΓ_{Z}(A_{R} ⊗_{C_{R}} Q^{•}) = 0.

In light of proposition 10.4.32(i), we are then reduced to showing that the Koszul complex K^{•}(g_{•}, A_{R} ⊗_{C_{R}} Q^{•}) is acyclic. We remark :

Claim 16.7.28. The natural map A_{R} ⊗^{L}_{C_{R}} Q^{•} → A_{R} ⊗_{C_{R}} Q^{•} is an isomorphism in D(C_{R}-Mod).

Proof of the claim. We have a natural morphism of distinguished triangles

[figure: A_{R}[0]; A_{R}[0] ⊗^{L}_{C_{R}} C_{R}^{∧}[0]; A_{R} ⊗^{L}_{C_{R}} Q^{•}; A_{R}[1]; A_{R}[0]; A_{R} ⊗_{C_{R}} C_{R}^{∧}[0]; A_{R}[0] ⊗_{C_{R}} Q^{•}; A_{R}[1]]

in light of which, we are reduced to checking that T_{i} := Tor_{i}^{C_{R}}(A_{R}, C_{R}^{∧}) = 0 for every i > 0. However, clearly the natural map T_{i} → Tor_{i}^{C_{R}}(A_{R}/C_{R}, C_{R}^{∧}) is an isomorphism for i > 1, and is injective for i = 1, and moreover A_{R}/C_{R} = ⋃_{n∈ℕ} Ann_{A_{R}/C_{R}}(J^{n}). Hence, the assertion follows from claims 8.6.36and 16.7.27. ♢

Taking into account lemma 7.8.2(v) and claim 16.7.28, we are further reduced to showing

that K_{•}(g_{•}) ⊗^{L}_{C_{R}} A_{R} ⊗^{L}_{C_{R}} Q^{•} is acyclic, and to this aim it suffices to check that the same holds for the complex K_{•}(g_{•}) ⊗_{C_{R}} Q^{•}. The latter follows easily from lemma 7.8.2(ii), claim 16.7.27 and corollary 8.6.34(i).

(iii): From claim 16.7.27we know already that \underline{C}_{R}^{∧} is perfectoid, and from (i) and (ii) we get isomorphisms of quasi-affinoid schemes

\underline{U}_{R}^{∧} →^{∼} Spec (\underline{A}_{R}^{∧}) →^{∼} Spec (\underline{C}_{R}^{∧})

whence the contention.

(iv): We consider the natural morphisms of schemes :

U_{R} ⟶^{f} Spec A_{R} ⟶^{g} Spec D_{R} → Spec A_{U}^{+}

whose composition is the same as the restriction of β\underline{U} to U_{R}, hence it is an open immersion. Notice also that β\underline{U} is in turn the composition of the open immersion i : U → Spec A and the natural morphism Spec A → Spec A_{U}^{+}; the latter identifies the analytic locus of Spec A with that of Spec A_{U}^{+} (lemma 8.3.29(iii)), hence the image of β\underline{U} contains the analytic locus of Spec A_{U}^{+}. We may then argue as in the proof of (ii) to conclude that g ∘ f is an open immersion and g ∘f(U_{R}) contains the analytic locus of Spec D_{R}. Together with claim 16.7.26, this already shows that \underline{D}_{R} is a quasi-affinoid ring. Moreover, we also know that A_{U}^{+} is a perfectoid ring (theorem 16.5.13(iii)), and our assumptions imply that \underline{A}′ := (A_{U}^{+}, A_{U}^{+}, U) is another perfectoid quasi-affinoid ring with Spec \underline{A}′ = \underline{U}. We may then argue as in the proof of claim 16.7.27 to see that the separated completion D_{R}^{∧} of D_{R} is perfectoid, and the proof is complete. □

Remark 16.7.29. (i) Let \underline{X} be any perfectoid quasi-affinoid scheme; taking into account proposition 16.6.35(iii) we deduce that the continuous map Spa(\overline{u}\underline{X}) of (16.5.59) induces a natural isomorphism of sites (notation of (15.5.34)) :

E : ℛ(\underline{X}) →^{∼} ℛ(E(\underline{X})) R ↦ Spa(\overline{u}\underline{X})(R).

(ii) Moreover, proposition 16.7.25(iii) says that for every R ∈ Ob(ℛ(\underline{X})) the sub-presheaf h′′_{R} of h′′ ; \underline{X} is represented by a perfectoid quasi-affinoid scheme \underline{Y} (notation of remark 15.5.9(i)), and taking into account (16.5.60) together with the equivalence of categories

\underline{X}-q.Afd.Sch_{perf} →^{∼} E(\underline{X})-q.Afd.Sch_{perf}

described in (16.5.20), we deduce easily that E(\underline{Y}) represents the sub-presheaf h′′_{E(R)} of h′′_{E(\underline{X})} (details left to the reader).

Theorem 16.7.30. For every perfectoid quasi-affinoid scheme \underline{X}, the following holds :

(i) The presheaves 𝒪_{Spa\underline{X}}^{∧}, 𝒪_{Spa\underline{X}}^{∧∘} and 𝒪_{Spa\underline{X}}^{∧+} are sheaves of topological rings on the site ℛ(\underline{X}) (notation of (15.5.34)).

(ii) For every rational subset R of Spa \underline{X} and every finite covering 𝔘 of R consisting of rational subsets, the augmented alternating Čech complex C_{alt}^{•}(𝔘, 𝒪_{Spa \underline{X}}^{∧}) has strict differentials, and its cohomology has the discrete topology.

Proof. (i): By virtue of corollary 15.4.27(i) and proposition 16.5.4(i), if 𝒪_{Spa \underline{X}}^{∧} is a sheaf, the same holds for 𝒪_{Spa \underline{X}}^{∧+} and 𝒪_{Spa \underline{X}}^{∧∘}, so it suffices to check the assertion for 𝒪_{Spa \underline{X}}^{∧}. Now, say that \underline{X} = (X, 𝒯_{X}, A_{X}^{+}). Notice first that, for every rational subset R of Spa \underline{X}, and every morphism φ_{\underline{Y} /\underline{X}} : \underline{Y} → \underline{X} representing the sub-presheaf h_{R} of h\underline{X}, the quasi-affinoid scheme \underline{Y}^{∧} is also perfectoid (proposition 16.7.25(iii)). Moreover, lemmata 15.4.17(iv) and 15.5.10and proposition 16.7.25(i) imply that φ_{\underline{Y} /\underline{X}} induces equivalences of categories

(16.7.31)  ℛ(\underline{Y}^{∧}) →^{∼} ℛ(\underline{Y}) →^{∼} ℛ(\underline{X})/R.

Hence, let A_{X} := 𝒪_{X}(X) and A := A_{X}^{∘}, so that Γ^{∘}(\underline{X}) = (A, A_{X}^{+}, X) (notation of (16.5.19)); according to lemma 15.5.21, every open covering of Spa \underline{X} can be refined by the standard covering R_{•} := (R_{0}, …, R_{n}) attached to a given sequence a_{•} := (a_{0}, …, a_{n}) of elements of A that generates an (open) ideal J ⊂ A such that Spec A/J ⊂ Spec A \ X; for every subset Λ ⊂ \{0, …, n\} we define the rational subset R_{Λ} ⊂ R and the quasi-affinoid ring \underline{A}_{X,Λ} := (A_{X,Λ}, A_{X,Λ}^{+}, X_{Λ}) as in remark 15.5.26(i), so that R_{Λ} = Spa \underline{A}_{X,Λ} for every such Λ. As in the proof of theorem 15.5.27, we are then reduced to checking that the natural map

(16.7.32)  𝒪_{Spa \underline{X}}^{∧}(Spa \underline{X}) → Equal(∏_{i=0}^{n} [figure: 𝒪_{Spa \underline{X}}^{∧}(R_{i})]∏_{i,j=0}^{n} 𝒪_{Spa \underline{X}}^{∧}(R_{\{i,j\}}))

is an isomorphism of topological rings. To this aim, set E := E(A); by proposition 16.6.35, we may assume that there exists a sequence e_{•} := (e_{0}, …, e_{n}) of elements of E such that a_{i} = \overline{u}_{A}(e_{i}) for i = 0, …, n. Then, for every subset Λ ⊂ \{0, …, n\}, we consider the subring

C_{R,Λ} := A[\overline{u}_{A}(e_{j}^{1/p^{k}})/\overline{u}_{A}(e_{i}^{1/p^{k}}) | (j, i, k) ∈ \{0, …, n\} × Λ × ℕ] ⊂ A_{X,Λ}.

Claim 16.7.33. For every Λ ⊂ {0, …, n} the following holds :

(i) C_{R,Λ} is a ring of definition of A_{X,Λ}.

(ii) The natural map ω_{Λ}^{∧} : A_{X,Λ}^{∧} → 𝒪_{Spa \underline{X}}^{∧}(R_{Λ}) of remark 15.5.26(ii) is an isomorphism of topological A-algebras.

Proof of the claim. Both assertions are clear when Λ = {i}, for any i = 0, …, n, due to proposition 16.7.25(i) and claim 16.7.26. We reduce to this case as follows. Let r be the cardinality of Λ, and set

Σ := {e_{i_{1}} ⋯ e_{i_{r}} | 0 ≤ i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{r} ≤ n}.

It is easily seen that the family (\overline{u}_{A}(e) | e ∈ Σ) generates an open ideal of A whose radical equals the radical of J. Set also e_{Λ} := ∏_{i∈Λ} e_{i} ∈ Σ, and notice that

R_{Λ} = R_{A}(\frac{\overline{u}_{A}(e)}{\overline{u}_{A}(e_{Λ})} | e ∈ Σ) C_{R,Λ} = A[\overline{u}_{A}(e^{1/p^{k}})/\overline{u}_{A}(e_{Λ}^{1/p^{k}}) | e ∈ Σ, k ∈ ℕ].

After replacing e_{0} by e_{Λ}, and the system (e_{0}, …, e_{n}) by Σ, we are therefore reduced to the case where R = R_{0}, to which proposition 16.7.25(i) and claim 16.7.26apply. ♢

In view of claim 16.7.33, we may now proceed as in remark 15.5.26(iv) : we set

B := A[T_{0}^{1/p^{∞}}, …, T_{n}^{1/p^{∞}}]

and consider the homomorphism of ℕ[1/p]-graded A-algebras

u : B → A_{X}[Y^{1/p^{∞}}]  T_{i}^{1/p^{k}} ↦ \overline{u}_{A}(e_{i}^{1/p^{k}}) · Y^{1/p^{k}}  for i = 0, …, n and every k ∈ ℕ.

Set also S := Spec A_{X}^{∘}; according to (16.7.7), the map u induces a well defined morphism of S-schemes

Proj (u) : X → Proj B.

Recall also that the inclusion map A[T_{0}, …, T_{n}] → B induces an affine morphism of schemes ψ : Proj B → ℙ_{S}^{n}. Let ℙ_{S}^{n} = Ω_{0} ∪ ⋯ ∪ Ω_{n} be the standard affine covering by complements of hyperplanes, as in remark 15.5.26(iv), set Ω′_{i} := ψ^{−1}Ω_{i} and U_{i} := (Proj u)^{−1}Ω′_{i} for i = 0, …, n; there follows an affine open covering of Proj B and an open covering of X :

Proj B = Ω′_{0} ∪ ⋯ ∪ Ω′_{n} X = U_{0} ∪ ⋯ ∪ U_{n}.

Set also U_{Λ} := ⋂_{i∈Λ} U_{i} for every non-empty Λ ⊂ {0, …, n}; by remark 15.5.26(i) we have

(16.7.34)  𝒪_{X}(U_{Λ}) = A_{X,Λ}  for every non-empty Λ ⊂ {0, …, n}.

Let V be the schematic image of Proj u (see remark 15.5.26(v)); a simple inspection shows that

V = Proj 𝒜 (e_{•})

where 𝒜 (e_{•}) is the ℕ[1/p]-graded algebra associated with the sequence e_{•} via example 16.6.12 (cp. the proof of claim 16.7.27). Especially, we see that

V_{Λ} := V ∩ ⋂_{i∈Λ} Ω′_{i} = Spec C_{R,Λ} for every non-empty subset Λ ⊂ {0, …, n}.

Fix a finitely generated ideal I of adic definition for E, and for every t ∈ ℝ_{+}, denote by C_{t}^{•} the augmented alternatingČech complex of I^{⌈t⌉}𝒪_{V}, relative to the open covering (V_{i} | i = 0, …, n). Notice that if Λ ≠ ∅, the scheme V_{Λ} is affine, hence we have I^{⌈t⌉}C_{R,Λ} = H^{0}(V_{Λ}, I^{⌈t⌉}𝒪_{V}). Moreover, by corollary 16.7.18 (and theorem 10.2.28(ii)), the complex C_{t}^{•} is acyclic, and the natural map I^{⌈t⌉}A → C_{t}^{−1} is an isomorphism, for every t ∈ ℝ_{+}. It follows that

I^{⌈s⌉}C_{t}^{i} = C_{t+s}^{i} for every i ∈ ℤ and every s, t ∈ ℝ_{+}

and we endow C_{t}^{i} with the I-adic topology, so that (C_{s+t}^{i} | s ∈ ℝ_{+}) is a fundamental system of open submodules of C_{t}^{i}, for every i ∈ ℤ and every t ∈ ℝ_{+}.

Furthermore, denote by A_{X}^{•} the augmented alternating Čech complex of 𝒪_{X} relative to the open covering (U_{i} | i = 0, …, n), and for every i ∈ ℤ endow A_{X}^{i} with the topology deduced from the topologies of the rings A_{X,Λ}, via the identifications (16.7.34), as in (15.5.34). There follows a natural monomorphism of complexes of topological A-modules

C_{t}^{•} → A_{X}^{•} for every t ∈ ℝ_{+}

and the image of C_{t}^{i} is open in A_{X}^{i}, for every i ∈ ℤ. Denote by (A_{X}^{∧•}, d_{X}^{•}) and (C_{t}^{∧•}, d_{t}^{•}) the complexes obtained by termwise completion of A_{X}^{•} and C_{t}^{•}; for every t ∈ ℝ_{+} the induced map

C_{t}^{∧•} → A_{X}^{∧•}

is still a monomorphism. By the foregoing, we have a natural isomorphism of complexes

C_{t}^{∧•} →^{∼} lim_{s∈ℝ_{+}} C_{t}^{•}/C_{s+t}^{•} for every t ∈ ℝ_{+}

and moreover C_{t}^{•}/C_{s+t}^{•} is acyclic for every s, t ∈ ℝ_{+}. By virtue of [163, Th.3.5.8], it follows that C_{t}^{∧•} is acyclic for every t ∈ ℝ_{+}. Next, notice that the induced morphism

A_{X}^{•}/C_{t}^{•} → A_{X}^{∧•}/C_{t}^{∧•}

is an isomorphism of complexes, for every t ∈ ℝ_{+}; we conclude that

H^{n}A_{X}^{∧•} = H^{n}(A_{X}^{∧•}/C_{t}^{∧•}) = H^{n}(A_{X}^{•}/C_{t}^{•}) = H^{n}A_{X}^{•} for every n ∈ ℤ.

On the other hand, claim 16.7.33(ii) and remark 15.5.26(iii) yield an isomorphism of complexes of topological A-modules

(16.7.35)  A_{X}^{∧•} →^{∼} C_{alt}^{•}(R_{•}, 𝒪_{Spa \underline{X}}^{∧}).

Since H^{0}A_{X}^{•} = 0, we conclude that (16.7.32) is a ring isomorphism. Lastly, let E be the equalizer appearing in (16.7.32), and endow it with the topology induced by the inclusion into ∏_{i=0}^{n} 𝒪_{Spa \underline{X}}^{∧}(R_{i}); we have to check that the resulting map 𝒪_{Spa \underline{X}}^{∧}(Spa \underline{X}) → E is open. In view of the isomorphism (16.7.35), the assertion will follow from the following :

Claim 16.7.36. The differentials of the complex (A_{X}^{∧•}, d_{X}^{•}) are strict and its cohomology has the discrete topology.

Proof of the claim. For every i ∈ ℤ, the family (Ker d_{t}^{i} | t ∈ ℝ_{+}) yields a fundamental system of open submodules of Ker d_{X}^{i}. But we have already noticed that C_{t}^{∧•} is acyclic, so this system is the image of (C_{t}^{∧i} | t ∈ ℝ_{+}), which is a fundamental system of open subgroups of A_{X}^{∧i}, whence the contention. ♢

(ii): We may argue as in the proof of theorem 15.5.35(ii) : we choose a standard covering 𝔘′ that refines 𝔘 and we apply lemma 8.6.15(i) and corollary 8.6.12 to the double complex C_{alt}^{•}(𝔘, 𝔘′) (details left to the reader). □

16.7.37. Let \underline{X} := (X, 𝒯_{X}, A_{X}^{+}) be any perfectoid quasi-affinoid scheme; in light of theorem 16.7.30, we may now argue as in (15.5.44) to first extend 𝒪_{Spa \underline{X}}^{∧} to a sheaf of topological rings on the topological space Spa \underline{X}, and then show that the stalks of the latter sheaf are local rings. There follows a well defined morphism of locally ringed spaces

σ\underline{X} : (Spa \underline{X}, 𝒪_{Spa \underline{X}}^{∧}) → (X, 𝒪_{X})

whose induced map on global sections is the identity map 𝒪_{X}(X) → 𝒪_{Spa \underline{X}}^{∧}(Spa \underline{X}), and whose underlying continuous map Spa \underline{X} → X is the restriction of the support map of remark 9.2.4(iii). Also corollary 15.5.46extends to the perfectoid case :

Corollary 16.7.38. With the notation of (16.7.37), the morphism σ\underline{X} induces an isomorphism

H^{i}(X, 𝒪_{X}) →^{∼} H^{i}(Spa \underline{X}, 𝒪_{Spa \underline{X}}^{∧}) for every i ∈ ℕ.

Proof. We proceed as in the proof of corollary 15.5.46: first we show the following

Claim 16.7.39. In the situation of the corollary, suppose moreover that \underline{X} is affinoid. Then H^{i}(Spa \underline{X}, 𝒪_{Spa \underline{X}}^{∧}) = 0 for every i > 0.

Proof of the claim. Arguing as in the proof of claim 15.5.47, we reduce to checking that for every standard covering 𝔘 of Spa X the Čech cohomology H^{i} := H_{alt}^{i}(𝔘, 𝒪_{Spa \underline{X}}^{∧}) vanishes for every i > 0; but the proof of theorem 16.7.30(i) shows that H^{i} is naturally isomorphic to the Čech cohomology H_{alt}^{i}(U_{•}, X), relative to a certain affine open covering U_{•} of X. Then the assertion follows from theorem 10.2.28. ♢

Now, set A_{X} := 𝒪_{X}(X), say that X = Spec A_{X} \ Spec A_{X}/J for some finitely generated ideal J ⊂ A_{X}, and pick a finite system of generators f_{•} := (f_{0}, …, f_{n}) for J. Let R_{•} := (R_{0}, …, R_{n}) be the standard covering of Spa \underline{X} associated with f_{•}, and set as well U_{i} := Spec A_{X}[f_{i}^{−1}] for every i = 0, …, n, so that U_{•} := (U_{0}, …, U_{n}) is an affine open covering of X, and moreover R_{i} = σ−1 ; \underline{X}(U_{i}) for every i = 0, …, n. There follows a commutative diagram

\xymatrix{H_{alt}^{i}(U_{•}, 𝒪_{X}) \ar[r] \ar[d] & H_{alt}^{i}(R_{•}, 𝒪_{Spa \underline{X}}^{∧}) \ar[d] \\ H^{i}(X, 𝒪_{X}) \ar[r] & H^{i}(Spa \underline{X}, 𝒪_{Spa \underline{X}}^{∧})}

where the vertical arrows are isomorphisms, by virtue of claim 16.7.39and corollary 10.2.21(ii). We are then reduced to checking that the top horizontal arrow is an isomorphism; but as already pointed out, the latter assertion was shown in the proof of theorem 16.7.30(i). □

16.7.40. Let again \underline{X} be as in (16.7.37); set A_{X} := 𝒪_{X}(X), and let V ⊂ X^{+} := Spec A_{X}^{+} be any quasi-compact open subset containing the analytic locus of X^{+}, and such that \underline{X} spreads over V (see definition 15.6.1(i)). Let also J_{V} ⊂ A_{X}^{+} be the unique radical ideal such that Spec A_{X}^{+}/J_{V} = X^{+} \ V. We consider the sheaf of ideals

𝒥_{V} ⊂ 𝒪_{Spa \underline{X}}^{∧+} on ℛ(\underline{X})

defined as the sheaf associated to the presheaf given by the rule : R ↦ J_{V} 𝒪_{Spa \underline{X}}^{∧+}(R) for every rational subset R ⊂ Spa \underline{X}.

Theorem 16.7.41. In the situation of (16.7.40), we have :

(i) 𝒥_{V} (R) = J_{V} 𝒪_{Spa \underline{X}}^{∧+}(R) for every R ∈ Ob(ℛ_{V} (\underline{X})) (notation of (15.6.7)).

(ii) H^{i}(X, 𝒥_{V}) = 0 for every i > 0.

Proof. To begin with, we remark :

Claim 16.7.42. Let R ∈ Ob(ℛ_{V} (\underline{X})), and \underline{Y} any topologically local quasi-affinoid scheme representing the sub-presheaf h_{R} of h_{Spa \underline{X}}. Say that \underline{Y}^{∧} := (Y^{∧}, 𝒯_{Y}^{∧}, A_{Y}^{∧+}). We have :

(i) The completion \underline{Y}^{∧} spreads over the preimage W of V in Y^{∧+} := Spec A_{Y}^{∧+}.

(ii) Let J_{W} ⊂ A_{Y}^{∧+} be the unique radical ideal such that Spec A_{Y}^{∧+}/J_{W} = Y^{∧+} \ W. Then

J_{W} = J_{V} A_{Y}^{∧+}.

Proof of the claim. (i): Since \underline{Y} spreads over V, the assertion follows from proposition 15.6.6(i).

(ii): We may find a finitely generated open ideal J_{E} ⊂ E^{+} such that J_{E}^{⌈0⌉}A_{X}^{+} = J_{V} (see (16.5.16)); then we have Spec A_{Y}^{∧+}/J_{V} A_{Y}^{∧+} = Y^{∧+} \ W, and on the other hand, arguing as in (16.5.16) we see that J_{E}^{⌈0⌉}A_{Y}^{∧+} is a radical ideal of A_{Y}^{∧+}, whence the claim. ♢

Claim 16.7.43. In order to prove the theorem it suffices to show that for every \underline{X} and V as in (16.7.40) we have Ȟ^{0}(ℛ(\underline{X}), J_{V} 𝒪_{Spa \underline{X}}^{∧+}) = J_{V} and Ȟ^{i}(ℛ(\underline{X}), J_{V} 𝒪_{Spa \underline{X}}^{∧+}) = 0 for every i > 0.

Proof of the claim. Indeed, in light of proposition 15.6.8(i,ii), both (i) and (ii) will follow once we show that the presheaf J_{V} 𝒪_{Spa \underline{X}}^{∧+} is a sheaf on the site ℛ_{V} (\underline{X}), and H^{i}(ℛ_{V} (\underline{X}), J_{V} 𝒪_{Spa \underline{X}}^{∧+}) = 0 for every i > 0. Next, in view of theorem 10.2.24, in order to prove the latter assertions it suffices to show that for every R ∈ Ob(ℛ_{V} (\underline{X})) we have Ȟ^{0}(ℛ_{V} (\underline{X})/R, J_{V} 𝒪_{Spa \underline{X}}^{∧+}) = J_{V} 𝒪_{Spa \underline{X}}^{∧+}(R) and Ȟ^{i}(ℛ_{V} (\underline{X})/R, J_{V} 𝒪_{Spa \underline{X}}^{∧+}) = 0 for every i > 0. Proposition 15.6.8(iii) then further reduces to showing both identities, with ℛ_{V} (\underline{X}) replaced by ℛ(\underline{X}). Lastly, for such a rational subset R, define \underline{Y} and W as in claim 16.7.42(i); in light of claim 16.7.42(ii), and taking into account the equivalences (16.7.31), we may then replace \underline{X} by \underline{Y}^{∧}, and V by W, and reduce to prove the foregoing identities for R = Spa \underline{X}, whence the claim. ♢

We consider first the case where the induced morphism β\underline{X} : X → X^{+} := Spec A_{X}^{+} is an open immersion, so that Ω\underline{X} is the image of β\underline{X} (notation of definition 15.6.1), and we take V := Ω\underline{X}. By theorem 16.5.13(iii) we have the perfectoid quasi-affinoid ring

Γ^{+}(\underline{X}) := (A_{X}^{+}, A_{X}^{+}, V)

and clearly Spec (Γ^{+}(\underline{X})) is isomorphic to \underline{X}. In view of lemma 15.5.21, we deduce that every open covering of Spa \underline{X} can be refined by the standard covering 𝔘_{•} := (𝔘_{0}, …, 𝔘_{n}) associated to some sequence (a_{0}, …, a_{n}) of elements of A_{X}^{+} that generates an ideal K with Spec A_{X}^{+}/K ⊂ X^{+} \V, and notice that 𝔘_{•} is a covering family of the site ℛ_{V} (\underline{X}), due to proposition 15.6.6(ii). We shall show more precisely that the augmented alternating Čech complex C_{alt}^{•}(𝔘_{•}, J_{V} 𝒪_{Spa \underline{X}}^{∧+}) is acyclic. To this aim, set E^{+} := E(A_{X}^{+}); in view of proposition 16.6.35, we may assume that there exists a sequence e_{•} := (e_{0}, …, e_{n}) of elements of E^{+} such that a_{i} = \overline{u}_{A_{X}^{+}}(e_{i}) for i = 0, …, n, and we let K_{E} ⊂ E^{+} be the open ideal generated by e_{•} (proposition 16.6.35(ii)). Notice that J_{E}^{⌈0⌉} and K_{E}^{⌈0⌉} are the radicals of J_{E} and respectively K_{E}; we deduce that

(16.7.44)  J_{E}^{⌈0⌉} = J_{E}^{⌈0⌉}J_{E}^{⌈0⌉} ⊂ K_{E}^{⌈0⌉}J_{E}^{⌈0⌉} ⊂ J_{E}^{⌈0⌉}.

We consider the ℕ[1/p]-graded A_{X}^{+}-algebra 𝒜 associated as in example 16.6.12to the sequence e_{•}, i.e. such that

𝒜_{0} := A_{X}^{+}  and 𝒜_{γ} := [e_{•}]^{⟨γ⟩}A_{X}^{+}  for every γ ∈ ℕ[1/p] \ {0}

(with multiplication induced by that of A_{X}^{+}). We have then a ring homomorphism as in (16.7.9)

φ : ℤ[T_{0}^{1/p^{∞}}, …, T_{n}^{1/p^{∞}}] → 𝒜  T_{i}^{1/p^{r}} ↦ \overline{u}_{A_{X}^{+}}(e_{i}^{1/p^{r}})  for i = 0, …, n and every r ∈ ℕ

whose image lies in 𝒜_{0} = A_{X}^{+}, whence corresponding angular Rees algebras R⟨A_{X}^{+}⟩ and R⟨𝒜 ⟩. Set Y_{0} := Proj R⟨A_{X}^{+}⟩; the map φ induces a morphism of schemes

Proj φ : Y_{0} → ℙ^{n} := Proj ℤ[T_{0}^{1/p^{∞}}, …, T_{n}^{1/p^{∞}}].

Let ℙ^{n} = Ω_{0} ∪ ⋯ ∪ Ω_{n} be the standard affine open covering by complements of hyperplanes, set U_{i} := (Proj φ)^{−1}Ω_{i} for i = 0, …, n, and let

C^{•} := C_{alt}^{•}(U_{•}, 𝒪_{Y_{0}})

be the augmented alternating Čech complex relative to the covering U_{•} := (U_{0}, …, U_{n}). Taking into account (16.7.44) and corollary 16.7.16(as well as theorem 10.2.28(ii)), we deduce that

C_{alt}^{•}(U_{•}, J_{V} 𝒪_{Y_{0}}) = J_{V} C^{•}

and moreover this second augmented alternating Čech complex is acyclic. Furthermore, let I_{E} be any finitely generated ideal of adic definition for E^{+}; corollary 16.7.18shows likewise that

C_{alt}^{•}(U_{•}, I_{E}^{⌈t⌉}𝒪_{Y_{0}}) = I_{E}^{⌈t⌉}C^{•} for every t ∈ ℝ_{+}

and also these complexes are acyclic. We may describe the terms of C^{•} as in the proof of theorem 16.7.30 : for every i = 0, …, n, the A_{X}^{+}-module C^{i} is a direct sum ⨁_{Λ⊂{0,...,n}} 𝒪_{Y_{0}}(U_{Λ}), where Λ ranges over the set of subsets of cardinality equal to i + 1, and U_{Λ} := ⋂_{j∈Λ} U_{j} for every such Λ. Then

𝒪_{Y_{0}}(U_{Λ}) = A_{X}^{+}[\overline{u}_{A_{X}^{+}}(e_{i}^{1/p^{∞}})/\overline{u}_{A_{X}^{+}}(e_{j}^{1/p^{∞}}) | (i, j) ∈ \{0, …, n\} × Λ] ⊂ A_{X}^{+}[a_{j}^{−1} | j ∈ Λ]

for every non-empty Λ ⊂ {0, …, n}. If we set likewise 𝔘_{Λ} := ⋂_{j∈Λ} 𝔘_{j} for every such Λ, we see that 𝒪_{Y_{0}}(U_{Λ}) ⊂ 𝒪_{Spa \underline{X}}^{∧+}(𝔘_{Λ}). Set as usual A_{X} := 𝒪_{X}(X); in light of claim 16.7.33 we see that there exists for every Λ ≠ ∅ a (unique) f-adic topology on A_{X}[a_{j}^{−1} | j ∈ Λ] for which the localization map A_{X} → A_{X}[a_{j}^{−1} | j ∈ Λ] is f-adic and 𝒪_{Y_{0}}(U_{Λ}) is a subring of definition. Especially, the induced topology on 𝒪_{Y_{0}}(U_{Λ}) is adic, and the natural map A_{X}^{+} → 𝒪_{Y_{0}}(U_{Λ}) is an adic ring homomorphism. Let us endow C^{i} with the corresponding product topology, for every i = 0, …, n; then (I_{E}^{⌈t⌉}C^{i} | t ∈ ℝ_{+}) is a fundamental system of open submodules of C^{i}, and hence also of J_{V} C^{i}, for every such i. Moreover, we know that the natural maps J_{V} A_{X}^{+} → J_{V} C^{−1} and I_{E}^{⌈t⌉}A_{X}^{+} → I_{E}^{⌈t⌉}C^{−1} are isomorphisms (again, by corollaries 16.7.16(i) and 16.7.18(i)), so (I_{E}^{⌈t⌉}C^{−1} | t ∈ ℝ_{+}) is a fundamental system of open submodules of J_{V} C^{−1}. Taking into account [163, Th.3.5.8], we conclude that the termwise completion of J_{V} C^{•} is still acyclic. Furthermore, the latter is isomorphic to J_{V} C^{∧•}, where C^{∧•} is the termwise completion of C^{•} : indeed, J_{V} C^{∧i} is dense and open in the completion of J_{V} C^{i} for every i = 0, …, n, so the assertion is clear.

Claim 16.7.45. Set W_{Λ} := U_{Λ} ∩ Spec 𝒪_{Y_{0}}(U_{Λ})[a_{j}^{−1} | j ∈ Λ] for every non-empty Λ ⊂ {0, …, n}. The datum (𝒪_{Y_{0}}(U_{Λ}), 𝒪_{Y_{0}}(U_{Λ}), W_{Λ}) is a quasi-affinoid ring, and its completion is a perfectoid quasi-affinoid ring

\underline{R}_{Λ} := (𝒪_{Y_{0}}(U_{Λ})^{∧}, 𝒪_{Y_{0}}(U_{Λ})^{∧}, W_{Λ}^{∧} := W_{Λ} ×_{U_{Λ}} Spec 𝒪_{Y_{0}}(U_{Λ})^{∧}).

Proof of the claim. Arguing as in the proof of claim 16.7.33, we may assume that Λ = {i}, for some i ∈ {0, …, n}, in which case it suffices to invoke proposition 16.7.25(iv). ♢

With the notation of claim 16.7.45, we see that

Γ(Spec \underline{R}_{Λ}) = (𝒪_{Y_{0}}(U_{Λ})^{∧}[a_{j}^{−1} | j ∈ Λ], 𝒪_{Y_{0}}(U_{Λ})^{∧+}, W_{Λ}^{∧})

where 𝒪_{Y_{0}}(U_{Λ})^{∧}[a_{j}^{−1} | j ∈ Λ] carries the unique f-adic topology such that the localization map 𝒪_{Y_{0}}(U_{Λ})^{∧} → 𝒪_{Y_{0}}(U_{Λ})^{∧}[a_{j}^{−1} | j ∈ Λ] is open (proposition 8.3.30(ii)), and 𝒪_{Y_{0}}(U_{Λ})^{∧+} is the integral closure of 𝒪_{Y_{0}}(U_{Λ})^{∧} in 𝒪_{Y_{0}}(U_{Λ})^{∧}[a_{j}^{−1} | j ∈ Λ]. In other words, 𝒪_{Y_{0}}(U_{Λ})^{∧}[a_{j}^{−1} | j ∈ Λ] is the completion of 𝒪_{Y_{0}}(U_{Λ})[a_{j}^{−1} | j ∈ Λ] = A_{X}^{+}[a_{j}^{−1} | j ∈ Λ], where the latter is endowed with the unique f-adic topology such that the localization map 𝒪_{Y_{0}}(U_{Λ}) → A_{X}^{+}[a_{j}^{−1} | j ∈ Λ] is open (proposition 8.3.33). It follows that 𝒪_{Y_{0}}(U_{Λ})^{∧+} is also the completion of the integral closure of 𝒪_{Y_{0}}(U_{Λ}) in A_{X}^{+}[a_{j}^{−1} | j ∈ Λ] (lemma 8.3.6), which is the same as the completion of the integral closure of A_{X}^{+}[a_{i}/a_{j} | (i, j) ∈ {0, …, n} × Λ] in A_{X}^{+}[a_{j}^{−1} | j ∈ Λ]. This means that

Γ(Spec \underline{R}_{Λ}) = (𝒪_{Spa \underline{X}}^{∧}(𝔘_{Λ}), 𝒪_{Spa \underline{X}}^{∧+}(𝔘_{Λ}), W_{Λ}^{∧}).

We have already observed that J_{V} 𝒪_{Spa \underline{X}}^{∧+}(𝔘_{Λ}) is a radical ideal of 𝒪_{Spa \underline{X}}^{∧+}(𝔘_{Λ}); combining with corollary 16.5.17we conclude that

J_{V} 𝒪_{Spa \underline{X}}^{∧+}(𝔘_{Λ}) = J_{V} 𝒪_{Y_{0}}(U_{Λ})^{∧} for every non-empty Λ ⊂ \{0, …, n\}

which implies that the natural map of complexes J_{V} C^{∧•} → C_{alt}^{•}(𝔘_{•}, J_{V} 𝒪_{Spa \underline{X}}^{∧+}) is an isomorphism, so the proof of the theorem is complete in this case.

For the general case, let us set

(X_{E}, 𝒯_{E}, E_{X}^{+}) := E(\underline{X})  X′ := β−1 ; \underline{X}(V)  A_{X′} := 𝒪_{X′}(X′)  \underline{X}′ := X′ ×_{X} \underline{X}

(see example 15.4.4). Say that \underline{X}′ = (X′, 𝒯_{X′}, A_{X′}^{+}); then \underline{X}′ spreads over the preimage V′ of V in X^{′+} := Spec A_{X′}^{+} (proposition 15.6.6(iv)); on the other hand, by construction the image of β\underline{X}′ : X′ → X^{′+} lies in V′, so β\underline{X}′ is an open immersion and V′ = Ω\underline{X}′. Therefore we have as well the perfectoid quasi-affinoid ring Γ(\underline{X}′) := (A_{X}^{+}, A_{X}^{+}, V′), and Spec Γ^{+}(\underline{X}′) is isomorphic to \underline{X}′; especially, the latter is a perfectoid quasi-affinoid scheme of the type considered in the foregoing case. Next, since the natural morphism π : \underline{X}′ → \underline{X} is f-adic, proposition 15.4.22(iii.a) says that Spa π : Spa \underline{X}′ → Spa \underline{X} induces a functor

ℛ(\underline{X}) → ℛ(\underline{X}′) R ↦ (Spa π)^{−1}R

which restricts to a functor

ℛ_{V} (\underline{X}) → ℛ_{V′}(\underline{X}′).

Indeed, let R be any rational subset of Spa \underline{X}, and say that the corresponding sub-presheaf h_{R} of h_{Spa \underline{X}} is represented by an f-adic morphism \underline{Y} → \underline{X} of quasi-affinoid schemes. Then (Spa π)^{−1}R is represented by the induced morphism \underline{Y}′ := (\underline{Y} ×\underline{X} \underline{X}′)_{loc} → \underline{X}′ (see remark 15.5.9(iii)). Now, if \underline{Y} spreads over V, proposition 15.6.6(i,iii,iv) implies that \underline{Y}′ spreads over V′, whence the contention. Furthermore, let J_{V′} ⊂ A_{X′}^{+} be the unique radical ideal such that Spec A_{X′}^{+}/J_{V′} = X^{′+} \ V′; we have :

Claim 16.7.46. For every R ∈ Ob(ℛ_{V} (\underline{X})), the induced map

(16.7.47)  J_{V} 𝒪_{Spa \underline{X}}^{∧+}(R) → J_{V′}𝒪_{Spa \underline{X}′}^{∧+}((Spa π)^{−1}R)

is an isomorphism.

Proof of the claim. We consider first the case where R = Spa \underline{X} : arguing as in the proof of claim 16.7.42(ii) we easily see that J_{V′} = J_{V} A_{X′}^{+}. Therefore, let \overline{A}_{X}^{+} be the image of A_{X}^{+} in A_{X′}^{+}; combining with corollary 16.5.17, it follows that

J_{V′} = J_{V} \overline{A}_{X}^{+}

which already shows that the natural ring homomorphism ρ : A_{X}^{+} → A_{X′}^{+} restricts to a surjection J_{V} → J_{V′}. Next, notice that the support of Ker ρ lies in the closed subset X^{+} \V, and therefore J_{V} ∩Ker ρ is contained in the nilradical of A_{X}^{+}; but A_{X}^{+} is reduced (corollary 16.3.63(i)), whence the claim, in this case. Next, let R ∈ Ob(ℛ_{V} (\underline{X})) be a general rational subset; to ease notation, set R′ := (Spa \underline{π})^{−1}R, and let \underline{Z} be any topologically local quasi-affinoid scheme representing the sub-presheaf h_{R′} of h_{Spa} \underline{X}′. Define \underline{Y}^{∧}, Y^{∧}, Y^{∧+}, W ⊂ Y^{∧+} and J_{W} as in claim 16.7.42. Let also Y′ := β_{\underline{Y}^{∧}}^{−1}(W) and \underline{Y}′ := Y′ ×_{Y^{∧}} \underline{Y}^{∧}. Say that \underline{Y}′ = (Y′, 𝒯_{Y′}, A_{Y′}^{+}), and let moreover W′ be the preimage of W in Y^{′+} := Spec A_{Y′}^{+}, and J_{W′} ⊂ A_{Y′}^{+} the unique radical ideal such that Spec A_{Y′}^{+}/J_{W′} = Y^{′+} \ W′; arguing as in the foregoing, we see that \underline{Y}′ is a perfectoid quasi-affinoid scheme that spreads over W′, and a direct inspection of the constructions shows that \underline{Y}′ is the completion of \underline{Z}. Thus, the image of Spa \underline{Y}^{∧} (resp. of Spa \underline{Y}′) in Spa \underline{X} (resp. in Spa \underline{X}′) agrees with R (resp. with R′), and taking into account claim 16.7.42(ii) we conclude that (16.7.47) is naturally identified with the map

J_{W} → J_{W′}

induced by the projection \underline{Y}′ → \underline{Y}^{∧}. By the previous case, we know already that this map is bijective, whence the claim. ♢

We may now conclude the proof of the theorem : indeed, we have already remarked that every covering family of ℛ(\underline{X}) can be refined by a standard covering 𝔘_{•} := (𝔘_{0}, …, 𝔘_{n}) which is also a covering family of ℛ_{V} (\underline{X}). It follows that 𝔘′_{•} := ((Spa π)^{−1}𝔘_{i} | i = 0, …, n) is a standard covering of Spa \underline{X}′ which is also a covering family of ℛ_{V′}(\underline{X}′). By the previous case, we know already that Ȟ^{0}(𝔘′_{•}, J_{V′}𝒪_{Spa \underline{X}′}^{∧+}) = J_{V′} and Ȟ^{i}(𝔘′_{•}, J_{V′}𝒪_{Spa \underline{X}′}^{∧+}) = 0 for every i > 0. In view of claim 16.7.46, it follows that Ȟ^{0}(𝔘_{•}, J_{V} 𝒪_{Spa \underline{X}′}^{∧+}) = J_{V} and Ȟ^{i}(𝔘_{•}, J_{V} 𝒪_{Spa \underline{X}′}^{∧+}) = 0 for every i > 0, and taking into account claim 16.7.43, the theorem follows. □

16.8. Almost purity. To begin with, let p be a prime integer; we consider a perfect 𝔽_{p}-algebra V and a radical ideal 𝔪 ⊂ V. Since V is perfect, (V, 𝔪) is clearly a basic setup, in the sense of [75, §2.1.1]. Our first observation is the following :

Lemma 16.8.1. In the situation of (16.8), let A ⊂ B be an inclusion of perfect V -algebras, and 𝔪_{0} ⊂ 𝔪 a subideal whose radical is 𝔪. We set

A′ := {x ∈ B | 𝔪 · x ⊂ A} B′ := {x ∈ B | there exists n ∈ ℕ with 𝔪_{0}^{n} · x ⊂ A}.

Then A′ and B′ are perfect V -algebras, and A′ is integrally closed in B′.

Proof. It is easily seen that A′ and B′ are V -subalgebras of B, and obviously A ⊂ A′ ⊂ B′. Next, let x ∈ B, and say that x^{p} ∈ A′; thus, 𝔪 · x^{p} ⊂ A, and therefore (ax)^{p} ∈ A for every a ∈ 𝔪. Since A is perfect, it follows that ax ∈ A for every such a, so that x ∈ A′; this show that A′ is perfect. Likewise one checks that B′ is perfect.

Next, let x ∈ B′, and suppose that x^{n}+a_{1}x^{n−1}+⋯+a_{n} = 0 for some a_{1}, …, a_{n} ∈ A′. Then A′[x] = A′ +A′x+⋯+A′x^{n−1} ⊂ B′. By assumption, there exists k ∈ ℕ with 𝔪_{0}^{k} ·x ⊂ A, and 𝔪_{0}A′ ⊂ 𝔪A′ ⊂ A; it follows easily that 𝔪_{0}^{nk}A′[x] ⊂ A. Now, let a ∈ 𝔪; by assumption, there exists M ∈ ℕ such that a^{p^{M}} ∈ 𝔪_{0}^{nk}, whence (ax)^{p^{M}} ∈ A, and since A is perfect, we conclude that ax ∈ A for every such a, i.e. x ∈ A′. This shows that A′ is integrally closed in B′. □

We may now state the following generalization of [75, Th.3.5.28] :

Theorem 16.8.2. In the situation of (16.8), set S := Spec V and Z := Spec V/𝔪. Then : (i) The pair (S, Z) is almost pure, relative to the basic setup (V, 𝔪). (ii) If Z is constructible in S, the pair (S, Z) is normal, for the basic setup (V, 𝔪).

Proof. (ii): Denote by j : U → S the open immersion; the kernel 𝒦 of the natural map 𝒪_{S} → j_{∗}𝒪_{U} is the largest 𝒪_{S}-submodule such that 𝔪𝒦 = 0, so the induced morphism 𝒪_{S}^{a} → j_{∗}𝒪_{U}^{a} is a monomorphism. It remains to check that the image of 𝒪_{S}^{a} is integrally closed in j_{∗}𝒪_{U}^{a}, and since Z is constructible, we are reduced to showing that V^{a} is integrally closed in 𝒪_{S}(U)^{a} ([59, Ch.I, Prop.9.2.1]). The latter follows from lemma 16.8.1and [75, Lemma 8.2.28].

(i): Let (Z_{λ} | λ ∈ Λ) be the cofiltered system of closed constructible subsets of S containing Z, and for every λ ∈ Λ let 𝔪_{λ} ⊂ 𝔪 be the unique radical ideal such that Spec V/𝔪_{λ} = Z_{λ}. Set also U := S \ Z and U_{λ} := S \ Z_{λ} for every λ ∈ Λ. Then 𝔪_{λ} = 𝔪_{λ}^{2} and notice that both 𝔪_{λ} (for every λ ∈ Λ) and 𝔪 trivially fulfill condition (B) of [75, §2.1.6], since V is perfect. We then get an essentially commutative diagram

\xymatrix{(𝒪_{S}, 𝔪𝒪_{S})^{a}-Ét_{afr} \ar[r]^{ρ} \ar[d] & 𝒪_{U}-Ét \ar[d] \\ (𝒪_{S}, 𝔪_{λ}𝒪_{S})^{a}-Ét_{afr} \ar[r]^{ρ_{λ}} & 𝒪_{U_{λ}}-Ét} for every λ ∈ Λ.

However, on the one hand, proposition 14.2.3 and corollary 14.2.5(ii,iv,v) imply that the category (𝒪_{S}, 𝔪𝒪_{S})^{a}-Ét_{afr} is equivalent to the 2-limit of the system ((𝒪_{S}, 𝔪_{λ}𝒪_{S})^{a}-Ét_{afr} | λ ∈ Λ); on the other hand, it is clear that 𝒪_{U}-Ét is equivalent to the 2-limit of the system (𝒪_{U_{λ}}-Ét | λ ∈ Λ). Thus, we are reduced to checking that ρ_{λ} is an equivalence for every λ ∈ Λ, and we may therefore assume from start that Z is constructible. In view of (ii) and lemma 14.4.5(ii), we are then reduced to showing that for every finite étale 𝒪_{U}-algebra ℬ there exists an étale 𝒪_{S}^{a}-algebra 𝒜 of almost finite rank such that ρ(𝒜) is isomorphic to ℬ. Now, by applying proposition 14.3.30 with X := S and 𝔪_{0} = 𝔪 := V (so, we consider here the “classical limit” almost ring structure) we obtain a quasi-coherent 𝒪_{S}-algebra ℛ_{0} which is finitely presented as an 𝒪_{S}-module, with an isomorphism ℬ →^{∼} ℛ_{0|U} of 𝒪_{U}-algebras.

Let Φ_{S} : S → S be the Frobenius endomorphism, and consider the directed system of quasi-coherent 𝒪_{S}-algebras (ℛ_{n} | n ∈ ℕ) with ℛ_{n} := Φ_{S}^{−n}ℛ_{0} for every n ∈ ℕ; explicitly, ℛ_{n}(U′) = ℛ_{0}(U′) for every open subset U′ ⊂ S, and the 𝒪_{S}(U′)-algebra structure on ℛ_{n}(U′) is given by the composition of the Frobenius endomorphism 𝒪_{S}(U′) → 𝒪_{S}(U′) with the structure morphism 𝒪_{S}(U′) → ℛ_{0}(U′). The transition maps φ_{n} : ℛ_{n} → ℛ_{n+1} are given on each such U′ by the Frobenius endomorphism ℛ_{0}(U′) → ℛ_{0}(U′), for every n ∈ ℕ. We let

ℛ := colim_{n∈ℕ} ℛ_{n}.

Since V is perfect, [75, Th.3.5.13] (applied again in the “classical limit” case with 𝔪 = V) implies that φ_{n|U} is an isomorphism for every n ∈ ℕ, so the natural morphism ℛ_{0} → ℛ induces an isomorphism of 𝒪_{U}-algebras

ℬ →^{∼} ℛ_{|U}.

Claim 16.8.3. There exists a subideal 𝔪_{0} ⊂ 𝔪 whose radical is 𝔪, and with

𝔪_{0} · Ker φ_{0} = 𝔪_{0} · Coker φ_{0} = 0.

Proof of the claim. Since Z is constructible, there exist finitely many elements x_{1}, …, x_{n} ∈ 𝔪 such that U = ⋃_{i=1}^{n} Spec V [x_{i}^{−1}]. Notice that both ℛ_{0} and ℛ_{1} are finitely presented 𝒪_{S}- modules : indeed, this holds for assumption for ℛ_{0}, and then it follows also for ℛ_{1}, since Φ_{S} is an automorphism of S. Thus, since moreover φ_{0|U} is an isomorphism, after replacing each x_{i} by a power x_{i}^{N} for some suitable N ∈ ℕ, we may assume already that x_{i} · Coker φ_{0} = 0 for every i = 1, …, n. Next, let R_{i} := ℛ_{i}(S) for i = 0, 1, and denote by f : R_{0} → R_{1} the Frobenius map; the natural map

V [x_{i}^{−1}] ⊗_{V} Hom_{V} (R_{1}, R_{0}) → Hom_{V [x_{i}^{−1}]}(R_{1}[x_{i}^{−1}], R_{0}[x_{i}^{−1}])

is bijective ([75, Lemma 2.4.29(i.a)]) and V [x_{i}^{−1}] ⊗_{V} f is an isomorphism, hence we may find for every i = 1, …, n a V -linear map g_{i} : R_{1} → R_{0} such that V [x_{i}^{−1}] ⊗_{V} (g ∘ f) = 1_{R_{0}}. Then there exists N ∈ ℕ such that x_{i}^{N} · (g ∘ f) = x_{i}^{N} · 1_{R_{0}} for every i = 1, …, n, whence x_{i}^{N} · Ker f = 0 for every i = 1, …, n. The claim follows. ♢

Let 𝔪_{0} be as in claim 16.8.3, and for every n ∈ ℕ set 𝔪_{n} := Φ_{V}^{−n}(𝔪_{0}), where Φ_{V} : V → V denotes the Frobenius automorphism; then 𝔪_{n} · Ker φ_{n} = 𝔪_{n} · Coker φ_{n} = 0 for every n ∈ ℕ. Notice that

𝔪_{n}^{2} ⊂ 𝔪_{n} · 𝔪_{n+1} ⋯ 𝔪_{n+k} for every n, k ∈ ℕ

and therefore 𝔪_{n}^{2} annihilates the kernel and cokernel of the transition map ℛ_{n} → ℛ_{n+k}, for every n, k ∈ ℕ; we conclude that

(16.8.4)  𝔪_{n}^{2} · Ker (ℛ_{n} → ℛ) = 𝔪_{n}^{2} · Coker (ℛ_{n} → ℛ) = 0  for every n ∈ ℕ.

Especially, 𝒜 := ℛ^{a} is an almost finitely presented and uniformly almost finitely generated 𝒪_{S}^{a}-module ([75, Cor.2.3.13]). Next, pick again elements x_{1}, …, x_{n} ∈ 𝔪 such that U is the union of the affine open subsets U_{i} := Spec V [x_{i}^{−1}] for i = 1, …, n; since ℬ is a projective 𝒪_{U}-module of finite rank, for every i = 1, …, n there exist N_{i} ∈ ℕ and 𝒪_{U}-linear morphisms

ℬ_{|U_{i}} ⟶^{f_{i}} 𝒪_{U_{i}}^{⊕N_{i}} ⟶^{g_{i}} ℬ_{|U_{i}} such that g_{i} ∘ f_{i} = 1_{ℬ|U_{i}} .

Then, arguing as in the proof of claim 16.8.3we find for every i = 1, …, n, an integer M_{i} ∈ ℕ and 𝒪_{S}-linear morphisms

ℛ_{0} ⟶^{f_{0,i}} 𝒪_{S}^{⊕N_{i}} ⟶^{g_{0,i}} ℛ_{0} such that g_{0,i} ∘ f_{0,i} = x_{i}^{M_{i}} · 1_{ℛ_{0}}

whence, for every k ∈ ℕ, and every i = 1, …, n, a pair of 𝒪_{S}-linear morphisms

ℛ_{k} ⟶^{f_{k,i}} 𝒪_{S}^{⊕N_{i}} ⟶^{g_{k,i}} ℛ_{k} such that g_{k,i} ∘ f_{k,i} = x_{i}^{M_{i}/p^{k}} · 1_{ℛ_{0}} .

It follows easily that x_{i}^{M_{i}/p^{k}} · Tor_{j}^{𝒪_{S}} (ℛ_{k}, 𝒩 ) = 0 for every j > 0, every i = 1, …, n, every k ∈ ℕ and every quasi-coherent 𝒪_{S}-module 𝒩 . After taking colimits, we conclude that 𝒜 is a flat 𝒪_{S}^{a}-module; then 𝒜 is also an almost projective 𝒪_{S}-module of finite rank, in light of [75, Prop.2.4.18(ii) and Rem.4.3.10(i)]. Next, let e be the diagonal idempotent of the étale 𝒪_{U}- algebra ℬ (see remark 14.3.10(i)). Pick a finite system of elements ε_{1}, …, ε_{k} ∈ 𝔪 such that 𝔪 is the radical of the ideal ∑_{i=1}^{k} V ε_{i}; we may then find N ∈ ℕ and for every i = 1, …, k an element e_{i} ∈ R_{0} ⊗_{V} R_{0} such that

(16.8.5)  (ε_{i}^{N} · e_{i})_{|Spec V [ε^{−1}]} = (ε_{i}^{N} · e)_{|Spec V [ε^{−1}]}  and μ_{0}(e_{i}) = ε_{i}^{N}

where μ_{n} : ℛ_{n} ⊗_{𝒪_{S}} ℛ_{n} → ℛ_{n} is the multiplication map, for every n ∈ ℕ. Next, let a_{1}, …, a_{n} be a finite system of generators for the V -module R_{0}; we may find N′ ∈ ℕ such that

ε_{i}^{N′} · (a_{j} ⊗ 1 − 1 ⊗ a_{j}) · e_{i} = 0 for every i = 1, …, k and every j = 1, …, n

and after replacing e_{i} by ε_{i}^{N′}e_{i} and N by N + N′ we may therefore assume that (16.8.5) still holds, and additionally :

e_{i} · Ker μ_{0} = 0 for every i = 1, …, k.

The same e_{1}, …, e_{k} can also be regarded as elements of R_{n} ⊗_{V} R_{n}, and we deduce that

(16.8.6) μ_{n}(e_{i}) = ε_{i}^{N/p^{n}}  and e_{i} · Ker μ_{n} = 0  for i = 1, …, k and every n ∈ ℕ.

Set R := ℛ(S) and denote by e_{n,i} ∈ R ⊗_{V} R the image of e_{i} under the natural morphism ℛ_{n} → ℛ, for i = 1, …, k. Say that ε_{i}^{N′′} ∈ 𝔪_{0} for i = 1, …, k (where 𝔪_{0} is as in claim 16.8.3); then ε_{i}^{N′′/p^{n}} ∈ 𝔪_{n} for every n ∈ ℕ and i = 1, …, k. Recall that the kernel of the multiplication map μ : ℛ ⊗_{𝒪_{S}} ℛ → ℛ is generated, as a ℛ ⊗_{𝒪_{S}} ℛ-module, by the system (a ⊗ 1 − 1 ⊗ a | a ∈ R); in light of (16.8.4) and (16.8.6) we deduce that

ε_{i}^{2N′′/p^{n}} · e_{n,i} · Ker μ = 0 for i = 1, …, k

where μ : ℛ ⊗_{𝒪_{S}} ℛ → ℛ is the multiplication map. After replacing e_{n,i} by ε_{i}^{2N′′/p^{n}} · e_{n,i} and N by N + 2N′′, we then obtain ℛ ⊗_{𝒪_{S}} ℛ-linear morphisms

ν_{n,i} : ℛ → ℛ ⊗_{𝒪_{S}} ℛ  x ↦ (x ⊗ 1) · e_{n,i}  for every n ∈ ℕ and i = 1, …, k

such that μ ∘ ν_{n,i} = ε_{i}^{N/p^{n}} · 1_{ℛ} . Set R := ℛ(S) for every n ∈ ℕ; it follows easily that ε_{i}^{N/p^{n}}·Ext_{R⊗_{V} R}^{j}(R, M) = 0 for every i = 1, …, k, every n ∈ ℕ, every j > 0 and every R⊗_{V} Rmodule M, so 𝒜 is an almost projective 𝒜 ⊗_{𝒪_{S}} 𝒜 -module, and the proof is concluded. □

Notice next that, in the situation of (16.8), the ring V is reduced, hence every flat almost finitely presented V^{a}-module has almost finite rank (lemma 14.3.4(ii)). We can generalize this observation as follows :

Proposition 16.8.7. Let (V, 𝔪) be a basic setup as in (16.8), and consider ideals

I ⊂ J ⊂ 𝔪

with J finitely generated. Let P be any flat almost finitely presented (V/I)^{a}-module. We have : (i) P has almost finite rank. (ii) If Spec V/𝔪 is a constructible subset of S := Spec V, then P has finite rank.

Proof. In light of corollary 14.2.5(iv) we may assume that 𝔪 is the radical of a finitely generated subideal 𝔪_{0} containing J, in which case Spec V/𝔪 is constructible in S, and it suffices to show assertion (ii). Then we may even replace J by 𝔪_{0}, and assume that J^{⌈0⌉}V = 𝔪 (notation of remark 9.3.70(i)).

Claim 16.8.8. We may assume that J lies in the Jacobson radical of V.

Proof of the claim. Indeed, let x_{1}, …, x_{k} be a finite system of generators of J; the induced map

V → (1 + J)^{−1}V × ∏_{i=1}^{k} V [x_{i}^{−1}]

is faithfully flat, and clearly V [x_{i}^{−1}] ⊗_{V} P is a projective V [x_{i}^{−1}] ⊗_{V} V/I-module of finite rank for every i = 1, …, k, so we are reduced to checking that (1 + J)^{−1}V ⊗_{V} P is an almost projective (1 + J)^{−1}V ⊗_{V} (V/I)^{a}-module of finite rank, whence the claim. ♢

Henceforth we assume that J lies in the Jacobson radical of V. Now, suppose that we have Λ_{(V/J)^{a}}^{r}(P/JP) = 0 for some r ∈ ℕ; then J · Λ_{(V/I)^{a}}^{r}P = Λ_{(V/I)^{a}}^{r}P, and since J is tight and Λ_{(V/I)}^{i}P is almost finitely generated, we deduce that Λ_{(V/I)^{a}}^{r}P = 0, by [75, Lemma 5.1.7]. Thus, we are reduced to checking the assertion in case I = J. Next, notice that there exists n ∈ ℕ such that (J^{⟨1⟩})^{n}V = J^{⟨n⟩}V ⊂ J (lemma 9.3.69(ii.a,iv)), so J^{⟨1⟩}V is also a tight ideal contained in the Jacobson radical of V, and by the same token we are reduced to checking that Q := P/J^{⟨1⟩}P is a (V/J^{⟨1⟩})^{a}-module of finite rank. To this aim, let us consider the S-scheme

Y := Proj R where R := ⨁_{γ∈ℕ[1/p]} J^{⟨γ⟩}V

as in (16.7.9), as well as the closed immersion

i : Y_{0} := Y ×_{S} Spec V/J^{⟨1⟩}V = Proj R/J^{⟨1⟩}R → Y.

Let also 𝒯 be the discrete topology on V, and notice that (V, 𝒯 ) is a perfectoid ring. Notice furthermore that (J^{⟨1⟩}V)^{a} = (J^{⌈1⌉}V)^{a}; in light of example 16.7.9(i), theorem 16.7.10, and the short exact sequence of quasi-coherent 𝒪_{Y} -modules

0 → J^{⟨1⟩}𝒪_{Y} → 𝒪_{Y} → i_{∗}𝒪_{Y_{0}} → 0

we then deduce an isomorphism of V^{a}-algebras

(V/J^{⟨1⟩}V)^{a} →^{∼} H^{0}(Y_{0}, 𝒪_{Y_{0}})^{a}

(details left to the reader). Say that J is generated by a finite system x_{1}, …, x_{k}; then Y admits a closed immersion Y → ℙ_{S}^{k−1} := Proj V [T_{1}^{1/p^{∞}}, …, T_{k}^{1/p^{∞}}] (see example 4.8.55(iii)), whence an affine covering Y = U_{1} …, U_{k} consisting of preimages of complements of hyperplanes of ℙ_{S}^{k}, as usual. Set U_{0,i} := U_{i} ×_{Y} Y_{0} for i = 1, …, k; we get an injective ring homomorphism

H^{0}(Y_{0}, 𝒪_{Y_{0}}) → A := ∏_{i=1}^{k} H^{0}(U_{0,i}, 𝒪_{Y_{0}})

and on the other hand, natural identifications of V -algebras

A_{i} := gr_{0}R[x_{i}^{−1}] →^{∼} H^{0}(U_{i}, 𝒪_{Y})  A_{i}/J^{⟨1⟩}A_{i} →^{∼} H^{0}(U_{0,i}, 𝒪_{Y_{0}})  i = 1, …, k.

Since Q is a flat (V/J^{⟨1⟩}V)^{a}-module, the same holds for Λ_{(V/J^{⟨1⟩}V)^{a}}^{r}Q, for every r ∈ ℕ; hence the induced map

Λ_{(V/J^{⟨1⟩}V)^{a}}^{r}Q → A^{a} ⊗_{V^{a}} Λ_{(V/J^{⟨1⟩}V)^{a}}^{r}Q = Λ_{A^{a}}^{r}(A^{a} ⊗_{V^{a}} Q)

is a monomorphism, and we are reduced to checking that A_{i}^{a} ⊗_{V^{a}} Q is an (A_{i}/J^{⟨1⟩}A_{i})^{a}-module of finite rank for every i = 1, …, k. Now, let (B_{i}, JB_{i}) be the henselization of the pair (A_{i}, JA_{i}). The ideal 𝔪B_{i} is generated by the system (x_{i}^{1/p^{n}} | n ∈ ℕ) consisting of regular elements of B_{i}; by [75, Th.2.1.12(ii.b)] we deduce that 𝔪B_{i} is a B_{i}-module of homological dimension ≤ 1. Lastly, since B_{i}/J^{⟨1⟩}B_{i} = A_{i}/J^{⟨1⟩}A_{i}, we may apply [75, Th.5.5.7(i)] to find a flat almost finitely presented B_{i}^{a}-module Q′ with an isomorphism Q′/J^{⟨1⟩}Q′ →^{∼} A_{i}^{a}⊗_{V^{a}} Q; since the localization B_{i} → B_{i}[x_{i}^{−1}] is a monomorphism, and since B_{i}[x_{i}^{−1}] ⊗_{B_{i}} Q′ is a projective B_{i}[x_{i}^{−1}]-module of finite rank, the B_{i}^{a}-module Q′ has finite rank as well. But then obviously the same holds for the (A_{i}/J^{⟨1⟩}A)^{a}-module A_{i}^{a} ⊗_{V^{a}} Q, as required. □

16.8.9. Let A be a perfectoid ring, E := E(A), and \underline{α} ∈ W (E) a distinguished element in the kernel of u_{A} : W (E) → A. Recall that \overline{u}_{A} : E → A induces an isomorphism of topological rings ω : E/α_{0}E →^{∼} A/pA (remark 16.3.7(ii)); hence, for every ideal I ⊂ A containing pA, there exists a unique ideal I_{E} ⊂ E containing α_{0}E such that ω(I_{E}/α_{0}E) = I/pA.

Definition 16.8.10. In the situation of (16.8.9), let (A, 𝔪) and (E, 𝔫) be two basic setups. We say that (A, 𝔪) and (E, 𝔫) are compatible if there exists an ideal of definition I ⊂ A such that ω induces a bijection (𝔫 + I_{E})/I_{E} →^{∼} (𝔪 + I)/I.

Remark 16.8.11. With the notation of definition 16.8.10, let M, N be two A/I-modules, and h : M → N an A-linear map. By assumption, \overline{u}_{A} induces a ring isomorphism ω_{I} : E/I_{E} →^{∼} A/I, and the compatibility of (A, 𝔪) and (E, 𝔫) means that ω_{I} is an isomorphism

(E/I_{E}, (𝔫 + I_{E})/I_{E}) →^{∼} (A/I, (𝔪 + I)/I)

in the category ℬ of basic setups as in [75, §3.5]. Hence h^{a} is an isomorphism of (A, 𝔪)^{a}- modules if and only if ω_{I}^{∗}(h^{a}) : ω_{I}^{∗}(M^{a}) → ω_{I}^{∗}(N^{a}) is an isomorphism of (E, 𝔫)^{a}-modules.

Lemma 16.8.12. In the situation of (16.8.9), consider two compatible basic setups (A, 𝔪) and (E, 𝔫), and let also f : B → B′ be any continuous map in A-Perf. Then the morphism f^{a} : B^{a} → B^{′a} is an isomorphism in (A, 𝔪)^{a}-Alg if and only if E(f)^{a} : E(B)^{a} → E(B′)^{a} is an isomorphism in (E, 𝔫)^{a}-Alg.

Proof. By assumption, there exists an ideal of definition I ⊂ A such that ω induces a bijection (𝔪 + I)/I →^{∼} (𝔫 + I_{E})/I_{E}, and arguing as in the proof of lemma 16.2.7(i), we find a sequence β_{1}, …, β_{k} of elements of E such that I is generated by \overline{u}_{A}(β_{1}), …, \overline{u}_{A}(β_{k}); then, after replacing each β_{i} by β_{i}^{1/p^{n}} for some large n ∈ ℕ, we may also assume that p ∈ I^{t}, with t := k(p − 1) + 1. In light of lemma 16.1.9(ii), it follows easily that I_{E} is generated by β_{1}, …, β_{k}, and α_{0} ∈ I_{E}^{t}. Denote by 𝒯 and 𝒯 ′ the I-adic topologies of B and B′; by lemma 8.3.12 and proposition 16.3.8, the topological rings (B, 𝒯 ) and (B′, 𝒯 ′) are still perfectoid. Now, let gr^{•}B (resp. gr^{•}B′) be the graded ring associated with the I-adic filtration on B (resp. on B′); likewise, let gr^{•}E(B) (resp. gr^{•}E(B′)) be the graded ring associated with the I_{E}-adic filtration on E(B) (resp. on E(B′)). By claim 16.3.62(ii), the maps \overline{u}_{B} and \overline{u}_{B′} induce graded ring isomorphisms

ϑ : gr^{•}E(B) →^{∼} gr^{•}B ϑ′ : gr^{•}E(B′) →^{∼} gr^{•}B′.

On the other hand, f and E(f) induce graded ring homomorphisms

gr^{•}(f) : gr^{•}B → gr^{•}B′ gr^{•}E(f) : gr^{•}E(B) → gr^{•}E(B′)

such that :

(16.8.13)  gr^{•}(f) ∘ ϑ = ϑ′ ∘ gr^{•}E(f).

Now, if f^{a} is an isomorphism, then the same holds for gr^{•}(f)^{a}; conversely, if gr^{•}(f)^{a} is an isomorphism, then we easily see that f_{n}^{a} := (A/I^{n} ⊗_{A} f)^{a} : (B/I^{n}B)^{a} → (B′/I^{n}B′)^{a} is an isomorphism for every n ∈ ℕ, and therefore f^{a} = lim_{n∈ℕ} f_{n}^{a} is an isomorphism (recall that the functor (−)^{a} commutes with limits, since it is a right adjoint). Likewise we see that E(f)^{a} is an isomorphism if and only if the same holds for gr^{•}E(f)^{a}. Thus, we are reduced to checking that gr^{•}(f)^{a} is an isomorphism of (A, 𝔪)^{a}-algebras if and only if gr^{•}E(f)^{a} is an isomorphism of (E, 𝔫)^{a}-algebras. However, notice that gr^{•}B and gr^{•}B′ are A/I-modules, and let ω_{I} : E/I_{E} →^{∼} A/I be the ring isomorphism deduced from \overline{u}_{A}; we may then regard ϑ as an isomorphism gr^{•}E(B) →^{∼} ω_{I}^{∗}(gr^{•}B) of E/I_{E}-modules, and likewise for ϑ′. Also, gr^{•}(f) is an isomorphism of (A, 𝔪)^{a}-algebras if and only if ω_{I}^{∗}(gr^{•}(f)) is an isomorphism of (E, 𝔫)^{a}- algebras, by remark 16.8.11. Taking into account (16.8.13), the assertion follows. □

16.8.14. Let now A be a perfectoid ring, and 𝔪 ⊂ A an open radical ideal. It follows easily that A^{∘∘} ⊂ 𝔪; moreover, if 𝔪_{E} ⊂ E := E(A) is associated with 𝔪 as in (16.8.9), then \overline{u}_{A} induces a ring isomorphism E/𝔪_{E} →^{∼} A/𝔪.

Proposition 16.8.15. In the situation of (16.8.14), let also f : B → B′ be any continuous map in A-Perf. The following holds :

(i) (A, 𝔪) and (E, 𝔪_{E}) are basic setups fulfilling condition (B) of [75, §2.1.6].

(ii) More precisely, the multiplication maps 𝔪̃ := 𝔪 ⊗_{A} 𝔪 → 𝔪 and 𝔪̃_{E} := 𝔪_{E} ⊗_{E} 𝔪_{E} →

𝔪_{E} are isomorphisms. (iii) The morphism f^{a} : B^{a} → B^{′a} is an isomorphism in (A, 𝔪)^{a}-Alg if and only if E(f)^{a} :

E(B)^{a} → E(B′)^{a} is an isomorphism in (E, 𝔪_{E})^{a}-Alg. (iv) 𝔪B is a radical ideal of B.

Proof. (i): Let I be any ideal of definition of A. Since A/𝔪 is reduced, the same holds for E/𝔪_{E}, so 𝔪_{E} is a radical ideal of E, hence 𝔪_{E}^{2} = 𝔪_{E}, since E is perfect. Now, let \underline{α} ∈ Ker u_{A} be any distinguished element; by remark 16.3.7(ii), the map \overline{u}_{A} induces a ring isomorphism E/α_{0}E →^{∼} A/pA, and it follows easily that 𝔪^{2} + pA = 𝔪. On the other hand, we have pA ⊂ I^{2} ⊂ 𝔪^{2}, so 𝔪^{2} = 𝔪. Next, it is clear that 𝔪_{E} fulfills condition (B), in particular 𝔪_{E} is generated by its p-th powers, therefore the same follows for the ideal 𝔪_{E}/α_{0}E of E/α_{0}E, and then also for the ideal 𝔪/pA of A/pA. Combining with lemma 16.2.3(iv), we deduce that 𝔪 is generated by its p-powers. If l ≠ p is any prime integer, we have l ∈ A^{×}, hence 𝔪/l𝔪 = 0, and taking into account [75, Claim 2.1.9], it follows that 𝔪 fulfills condition (B).

(ii): Let us write 𝔪 as the union of a filtered system (𝔪_{λ} | λ ∈ Λ) of open radical subideals such that Spec A/𝔪_{λ} is constructible in Spec A for every λ ∈ Λ; clearly it suffices to show that the corresponding map 𝔪_{λ} ⊗_{A} 𝔪_{λ} → 𝔪_{λ} is an isomorphism for every λ ∈ Λ. We may therefore assume that 𝔪 is the radical of an open ideal generated by finitely many elements a_{1}, …, a_{r} of A; moreover, we may also assume that there exist α_{1}, …, α_{r} ∈ E with a_{i} = \overline{u}_{A}(α_{i}) for i = 1, …, r (see remark 16.3.7(iii)). In this situation, we have a ring homomorphism u_{0} : R_{r,0} → A such that u_{0}(T_{i}^{1/p^{k}}) = \overline{u}_{A}(α_{i}^{1/p^{k}}) for i = 0, …, r (notation of (16.7.9)), and it is easily seen that 𝔪 = T^{⌈0⌉}A. Then the assertion follows from proposition 16.4.10(i) and theorem 16.7.10(iv).

(iii): It is clear that the basic setups (A, 𝔪) and (E, 𝔪_{E}) are compatible, in the sense of definition 16.8.10; then the assertion follows from lemma 16.8.12.

(iv): The map \overline{u}_{B} induces a ring isomorphism :

E(B)/𝔪_{E}E(B) →^{∼} B/𝔪B

and we need to check that B/𝔪B is reduced; so we are reduced to showing that the same holds for E(B)/𝔪_{E}E(B), i.e. that 𝔪_{E}E(B) is a radical ideal. But since E(B) is perfect, this is equivalent to proving that Φ_{E(B)}(𝔪_{E}B) = 𝔪_{E}B (where Φ_{E(B)} denotes the Frobenius automorphism of E(B)); the latter is clear, since we have already shown that 𝔪_{E} is a radical ideal of E. □

16.8.16. Let A be a perfectoid ring, E := E(A), and 𝔪 ⊂ A an open radical ideal; let also B be any perfectoid A-algebra, and 𝔪_{E} ⊂ E the unique open ideal such that \overline{u}_{A} : E → A induces a ring isomorphism E/𝔪_{E} →^{∼} A/𝔪 (see (16.8.9)). According to proposition 16.8.15(i), we have a well defined (A, 𝔪)^{a}-algebra B^{a} and an (E, 𝔪_{E})^{a}-algebra E(B)^{a}. We denote by

η_{B} : B → B_{∗}^{a} and η_{E(B)} : E(B) → E(B)_{∗}^{a}

the units of adjunction, and endow B_{∗}^{a} (resp. E(B)_{∗}^{a}) with the unique ring topology such that η_{B} (resp. η_{E(B)}) is an adic morphism.

Corollary 16.8.17. With the notation of (16.8.16), the topological rings B_{∗}^{a} and E(B)_{∗}^{a} are perfectoid, and there exists an isomorphism of E-algebras

ω : E(B)_{∗}^{a} →^{∼} E(B_{∗}^{a})  such that  ω ∘ η_{E(B)} = E(η_{B}).

Proof. We consider first the case where the structure map A → B is adic, and Z := Spec A/𝔪 is constructible in X := Spec A. Then, set U := Spec B ×_{X} (X \ Z), B_{U} := 𝒪_{U}(U), and let \overline{B} ⊂ B_{U} be the image of the restriction map ρ : B → B_{U}. Notice that B is reduced, by corollary 16.3.63(i); it follows easily that

(16.8.18)  𝔪 · Ker ρ = 0

(the details are left to the reader). We remark :

Claim 16.8.19. If Z is constructible in X, we have a natural identification of B-algebras :

B_{∗}^{a} →^{∼} C := \{x ∈ B_{U} | 𝔪 · x ⊂ \overline{B}\}.

Proof of the claim. Let a_{1}, …, a_{n} ∈ A be a finite system of elements that generate an ideal whose radical equals 𝔪; then :

B_{U} = \xymatrix{Equal (∏_{i=1}^{n} B[a_{i}^{−1}] \ar[r] \ar[r] & ∏_{i,j=1}^{n} B[a_{i}^{−1}, a_{j}^{−1}])} .

Now, obviously B[a_{i}^{−1}]_{∗}^{a} = B[a_{i}^{−1}], and likewise for B[a_{i}^{−1}, a_{j}^{−1}], for every i, j = 1, …, n; since the functor of almost elements is left exact, we deduce that B_{U∗}^{a} = B_{U}. Therefore, ρ induces a map of B-algebras ρ_{∗}^{a} : B_{∗}^{a} → B_{U}, and notice that ρ^{a} is a monomorphism, due to (16.8.18), so the same holds for ρ_{∗}^{a} (proposition 1.3.18(i)). Obviously \overline{B}^{a} = B^{a}, hence 𝔪·ρ_{∗}^{a}(B_{∗}^{a}) = 𝔪\overline{B}, i.e. the image of ρ_{∗}^{a} lies in the subring C. Moreover, ρ induces an isomorphism C^{a} →^{∼} B^{a}, whence by adjunction, a unique map of B-algebras C → B_{∗}^{a} whose composition with ρ_{∗}^{a} agrees with the inclusion map C → B_{U}. This shows that ρ_{∗}^{a} maps B_{∗}^{a} onto C, whence the claim. ♢

Recall that B_{U} carries a natural f-adic topology for which \overline{B} is a subring of definition (lemma 16.4.32(i)), and 𝔪\overline{B} is an open ideal of \overline{B}, since the structure map A → B is adic; from claim 16.8.19it then follows that B_{∗}^{a} is a bounded subring of B_{U}, and therefore its topology as defined in (16.8.16) agrees with the topology induced by the inclusion into B_{U} (proposition 8.3.18(i,ii)). From proposition 16.5.4(iii) we also see that (B_{∗}^{a})^{∘∘} = B_{U}^{∘∘} = B^{∘∘}.

Claim 16.8.20. If the structure map A → B is adic, and Z is constructible in X, we have : (i) B_{∗}^{a}/B_{U}^{∘∘} is a perfect 𝔽_{p}-algebra and it is integrally closed in B_{U}^{∘}/B_{U}^{∘∘}. (ii) B_{∗}^{a} is perfectoid and integrally closed in B_{U}^{∘}.

Proof of the claim. (i): Notice that A/A^{∘∘}, \overline{B}/B^{∘∘} and B_{U}^{∘}/B_{U}^{∘∘} are perfect 𝔽_{p}-algebras; also, 𝔪/A^{∘∘} is a radical ideal of A/A^{∘∘}, and since Z is constructible, there exists a finitely generated ideal 𝔪_{0} ⊂ A/A^{∘∘} whose radical is 𝔪/A^{∘∘}. With this notation, B_{U}^{∘}/B_{U}^{∘∘} is an A/A^{∘∘}-algebra, and for every x ∈ B_{U}^{∘}/B_{U}^{∘∘} there exists k ∈ ℕ such that 𝔪_{0}^{k} · x ⊂ \overline{B}/B^{∘∘}. Then the assertion follows from lemma 16.8.1.

(ii) follows from (i), claim 16.5.15, and theorem 16.5.13(iii). ♢ Next, set Z_{E} := Spec E/𝔪_{E}, X_{E} := Spec E, and U_{E} := Spec E(B) ×_{X_{E}} (X_{E} \ Z_{E}). The

foregoing argument applies also to E(B)_{∗}^{a}, and shows that the latter is a perfectoid subring of E_{U} := 𝒪_{U_{E}}(U_{E}); moreover, we get cartesian diagrams of topological rings :

\xymatrix{B_{∗}^{a} \ar[r]^{i_{B}} \ar[d] & B_{U}^{∘} \ar[d] & E(B)_{∗}^{a} \ar[r]^{ι_{E}} \ar[d] & E_{U}^{∘} \ar[d] \\ B_{∗}^{a}/B_{U}^{∘∘} \ar[r] & B_{U}^{∘}/B_{U}^{∘∘} & E(B)_{∗}^{a}/E_{U}^{∘∘} \ar[r] & E_{U}^{∘}/E_{U}^{∘∘}.}

Furthermore, according to proposition 16.5.4(iv), the morphism of topological monoids φ_{U}^{♭∘} : E_{U}^{∘} → B_{U}^{∘} induces a ring isomorphism \overline{φ}_{U}^{♭∘} : E_{U}^{∘}/E_{U}^{∘∘} →^{∼} B_{U}^{∘}/B_{U}^{∘∘}, which in turn restricts to a ring isomorphism \overline{E}(B)/E_{U}^{∘∘} →^{∼} \overline{B}/B_{U}^{∘∘}, where \overline{E}(B) denotes the image of E(B) into E_{U}. Notice also that claim 16.8.19yields natural identifications :

\begin{matrix}B_{∗}^{a}/B_{U}^{∘∘} →^{∼} & \{x ∈ B_{U}^{∘}/B_{U}^{∘∘} | 𝔪 · x ⊂ \overline{B}/B_{U}^{∘∘}\} \\ E(B)_{∗}^{a}/E_{U}^{∘∘} →^{∼} & \{x ∈ E_{U}^{∘}/E_{U}^{∘∘} | 𝔪_{E} · x ⊂ \overline{E}(B)/E_{U}^{∘∘}\}. \end{matrix}

But clearly \overline{φ}_{U}^{♭∘}(𝔪_{E} · x) = 𝔪 · \overline{φ}_{U}^{♭∘}(x) for every x ∈ E_{U}^{∘}/E_{U}^{∘∘}, so \overline{φ}_{U}^{♭∘} restricts as well to a ring isomorphism

E(B)_{∗}^{a}/E_{U}^{∘∘} →^{∼} B_{∗}^{a}/B_{U}^{∘∘}.

Summing up, and invoking proposition 16.3.25, we obtain a unique isomorphism of topological rings ω : E(B)_{∗}^{a} →^{∼} E(B_{∗}^{a}) fitting into the commutative diagram :

\xymatrix{E(B)_{∗}^{a} \ar[r]^{ι_{E}} \ar[d]_{ω} & E_{U}^{∘} \ar[d]^{ω^{∘}} \\ E(B_{∗}^{a}) \ar[r]^{E(ι_{B})} & E(B_{U}^{∘})}

where ω^{∘} is the isomorphism provided by theorem 16.5.13(i). Lastly, denote by ρ^{∘} : B → B_{U}^{∘} and ρ_{E}^{∘} : E(B) → E_{U}^{∘} the continuous ring homomorphisms induced by the restriction maps; the isomorphism ω^{∘} is characterized by the identity :

ω^{∘} ∘ ρ_{E}^{∘} = E(ρ^{∘}).

Since ρ^{∘} and ρ_{E}^{∘} factor through η_{B} and respectively η_{E(B)}, we conclude that ω ∘ η_{E(B)} = E(η_{B}), as stated. This completes the proof of the corollary in case Z is constructible.

Next, we consider the case where Z can be an arbitrary closed subset of X, and the structure map A → B is still adic. Let us write 𝔪 as the filtered union of a system (𝔪_{λ} | λ ∈ Λ) of open radical subideals such that Spec A/𝔪_{λ} is constructible in X for every λ ∈ Λ; correspondingly, we obtain a filtered system (𝔪_{E,λ} | λ ∈ Λ) of ideals in E whose union is 𝔪_{E}. For every such λ let B_{λ} (resp. E(B)_{λ}) be the image of B (resp. of E(B)) in the category (A, 𝔪_{λ})^{a}-Alg (resp. (E, 𝔪_{E,λ})^{a}-Alg); we endow B_{λ∗}^{a} (resp. E(B)_{λ∗}^{a}) with the unique adic topology such that the unit of adjunction η_{B,λ} : B → B_{λ∗}^{a} (resp. η_{E(B),λ} : E(B) → E(B)_{λ∗}^{a}) is adic. After replacing Λ by a cofinal subset, we may assume that Λ admits an initial element λ_{0}. The localization functors (A, 𝔪)^{a}-Alg → (A, 𝔪_{λ})^{a}-Alg → (A, 𝔪_{μ})^{a}-Alg induce natural maps of B-algebras

f_{λ} : B_{∗}^{a} → B_{λ∗}^{a}  and f_{λμ} : B_{λ∗}^{a} → B_{μ∗}^{a}  for every λ, μ ∈ Λ with λ ≥ μ

which are clearly continuous. Moreover, clearly the images f_{λ}^{a} and f_{λμ}^{a} of f_{λ} and f_{λμ} in (A, 𝔪_{λ_{0}})^{a}-Alg are isomorphism, hence

𝔪_{λ_{0}}·Ker f_{λ} = 𝔪_{λ_{0}}·Coker f_{λ} = 0  𝔪_{λ_{0}}·Ker f_{λμ} = 𝔪_{λ_{0}}·Coker f_{λμ} = 0  for every λ ≥ μ.

Claim 16.8.21. The maps f_{λ} and f_{λμ} induce bijections

𝔪_{λ_{0}}B_{∗}^{a} →^{∼} 𝔪_{λ_{0}}B_{λ∗}^{a} →^{∼} 𝔪_{λ_{0}}B_{μ∗}^{a} for every λ ≥ μ.

Proof of the claim. By the foregoing, we see already that f_{λ} (resp. f_{λμ}) maps 𝔪_{λ_{0}}B_{∗}^{a} onto 𝔪_{λ_{0}}B_{λ∗}^{a} (resp. 𝔪_{λ_{0}}B_{λ∗}^{a} onto 𝔪_{λ_{0}}B_{μ∗}^{a}). Moreover, by the previous case we also know that B_{λ∗}^{a} is perfectoid, and under our current assumptions 𝔪_{λ_{0}}B_{λ∗}^{a} is an open ideal of B_{λ∗}^{a}, hence B_{λ∗}^{a}/𝔪_{λ_{0}}B_{λ∗}^{a} = B_{λ∗}^{a} ˆ⊗_{A}A/𝔪_{λ_{0}} is a discrete perfectoid 𝔽_{p}-algebra (proposition 16.3.9(i)); especially, the latter is a reduced ring (corollary 16.3.63(i)). Now, let x ∈ 𝔪_{λ_{0}}B_{λ∗}^{a} ∩ Ker f_{λμ}; then 𝔪_{λ_{0}} · x = 0, whence x^{2} = 0, and finally x = 0, which shows that f_{λμ} is injective on 𝔪_{λ_{0}}B_{λ∗}^{a} for every λ ∈ Λ. Next, from corollary 14.2.5(i) we know that B_{∗}^{a} is the limit of the cofiltered system of B-algebras (B_{λ∗}^{a} | λ ∈ Λ), hence the natural map

𝔪_{λ_{0}}B_{∗}^{a} → L := lim_{λ∈Λ} 𝔪_{λ_{0}}B_{λ∗}^{a}

is injective. But we have also just seen that all the projections L → 𝔪_{λ_{0}}B_{λ∗}^{a} are isomorphisms, so the maps 𝔪_{λ_{0}}B_{∗}^{a} → 𝔪_{λ_{0}}B_{λ∗}^{a} are injective as well. ♢

From claim 16.8.21and [163, Lemma 3.5.3] we deduce that the natural map

B_{∗}^{a}/𝔪_{λ_{0}}B_{∗}^{a} → lim_{λ∈Λ} B_{λ∗}^{a}/𝔪_{λ_{0}}B_{λ∗}^{a}

is a ring isomorphism; since we have already observed that each quotient B_{λ∗}^{a}/𝔪_{λ_{0}}B_{λ∗}^{a} is a perfect 𝔽_{p}-algebra, it follows easily that the same holds for B_{∗}^{a}/𝔪_{λ_{0}}B_{∗}^{a}; especially, the latter is a discrete perfectoid ring. We then get for every λ ∈ Λ a cartesian diagram of rings

𝒟_{λ} : \xymatrix{B_{∗}^{a} \ar[r]^{f_{λ}} \ar[d] & B_{λ∗}^{a} \ar[d] \\ B_{∗}^{a}/𝔪_{λ_{0}}B_{∗}^{a} \ar[r] & B_{λ∗}^{a}/𝔪_{λ_{0}}B_{λ∗}^{a}}

which is also cartesian in the category of topological rings, by lemma 8.3.11, and proposition 16.3.25 implies that B_{∗}^{a} is perfectoid. The same argument applies as well to the perfectoid E-algebra E(B), and it yields a cartesian diagram of perfectoid rings :

𝒟_{E,λ} : \xymatrix{E(B)_{∗}^{a} \ar[r]^{f_{E,λ}} \ar[d] & E(B)_{λ∗}^{a} \ar[d] \\ E(B)_{∗}^{a}/𝔪_{E,λ_{0}}E(B)_{∗}^{a} \ar[r] & E(B)_{λ∗}^{a}/𝔪_{E,λ}E(B)_{λ∗}^{a}.}

By the same token, the system (f_{E,λ} | λ ∈ Λ) induces a ring isomorphism

E(B)_{∗}^{a}/𝔪_{E,λ_{0}}E(B)_{∗}^{a} → lim_{λ∈Λ} E(B)_{λ∗}^{a}/𝔪_{E,λ_{0}}E(B)_{λ∗}^{a}.

Furthermore, by the previous case we already have an isomorphism of topological rings

ω_{λ} : E(B)_{λ∗}^{a} →^{∼} E(B_{λ∗}^{a})  such that ω_{λ} ∘ η_{E(B),λ} = E(η_{B,λ})  for every λ ∈ Λ.

We may then identify naturally the bottom row of 𝒟_{λ} with that of 𝒟_{E,λ}, and by invoking again proposition 16.3.25we obtain a natural isomorphism of topological rings ω : E(B)_{∗}^{a} →^{∼} E(B_{∗}^{a}) fitting into the commutative diagrams

\xymatrix{E(B)_{∗}^{a} \ar[r]^{f_{E,λ}} \ar[d]_{ω} & E(B)_{λ∗}^{a} \ar[d]^{ω_{λ}} \\ E(B_{∗}^{a}) \ar[r]^{E(f_{λ})} & E(B_{λ∗}^{a})} for every λ ∈ Λ.

Then the sought identity ω ∘ η_{E(B)} = E(η_{B}) follows from the corresponding ones for the isomorphisms ω_{λ}, after taking limits (details left to the reader). This completes the proof, in case the topologies of A and B are p-adic.

For the general case, let 𝒯_{A,p} (resp. 𝒯_{B,p}) be the p-adic topology on A (resp. on B); then (A, 𝒯_{A,p}) and (B, 𝒯_{B,p}) are perfectoid topological rings as well (proposition 16.3.8(i)), and the rings underlying E((B, 𝒯_{B,p})_{∗}^{a}) and E(B, 𝒯_{B,p})_{∗}^{a} coincide with the rings underlying E(B_{∗}^{a}) and respectively E(B)_{∗}^{a}, so the previous case already yields a ring isomorphism ω as sought. It remains to check that B_{∗}^{a} (resp. E(B)_{∗}^{a}) is perfectoid for the topology such that η_{B} (resp. η_{E(B)}) is adic, and that ω is also an isomorphism of topological rings for these topologies. However, let I ⊂ B be any finitely generated ideal of definition; the natural map B^{a} → lim_{n∈ℕ}(B/I^{n})^{a} is an isomorphism, hence the same holds for the induced map B_{∗}^{a} → lim_{n∈ℕ}(B/I^{n})_{∗}^{a}, and especially, B_{∗}^{a} is complete and separated for the linear topology for which (J_{n} := Ker B_{∗}^{a} → (B/I^{n})_{∗}^{a} | n ∈ ℕ) is a fundamental system of open submodules (corollary 8.2.16(i)). Clearly I^{n}B_{∗}^{a} ⊂ J_{n} for every n ∈ ℕ, so B_{∗}^{a} is also complete and separated for its I-adic topology (lemma 8.3.12). Together with proposition 16.3.8(ii), this proves that B_{∗}^{a} is perfectoid, and the same applies to E(B)_{∗}^{a}. Lastly, since η_{B} is adic, the same holds for E(η_{B}) (corollary 16.3.8), but then the identity ω ∘ η_{E(B)} = E(η_{B}) implies easily that ω is an isomorphism of topological rings. □

16.8.22. In the situation of (16.8.16), denote also by ε_{B} : B_{!!}^{a} → B and ε_{E(B)} : E(B)_{!!}^{a} → E(B) the counits of adjunction, and endow B_{!!}^{a} (resp. E(B)_{!!}^{a}) with the unique topology such that the natural map A → B_{!!}^{a} (resp. E → E(B)_{!!}^{a}) is adic.

Corollary 16.8.23. With the notation of (16.8.22), the topological rings B_{!!}^{a} and E(B)_{!!}^{a} are perfectoid and there exists an isomorphism of E-algebras

ω : E(B)_{!!}^{a} →^{∼} E(B_{!!}^{a})  such that  E(ε_{B}) ∘ ω = ε_{E(B)}.

Proof. Notice that, by virtue of proposition 16.8.15(ii) we have cartesian diagrams of rings

𝒟 : \xymatrix{B_{!!}^{a} \ar[r]^{ε_{B}} \ar[d] & B \ar[d] \\ A/𝔪 \ar[r] & B/𝔪B.} 𝒟_{E} : \xymatrix{E(B)_{!!}^{a} \ar[r]^{ε_{E(B)}} \ar[d] & E(B) \ar[d] \\ E/𝔪_{E} \ar[r] & E(B)/𝔪_{E}E(B).}

Moreover, clearly we may assume that the structure map A → B is an adic ring homomorphism, in which case the same holds for the map E → E(B) (theorem 16.3.42(i)), and the bottom rows of 𝒟 and 𝒟_{E} consist of discrete perfectoid 𝔽_{p}-algebras (cp. the proof of claim 16.8.21). Then 𝒟 and 𝒟_{E} are cartesian in the category of topological rings as well (lemma 8.3.11), and the assertion follows from proposition 16.3.25. □

Lemma 16.8.24. In the situation of (16.8.16), let f : B → C be a morphism of perfectoid A-algebras, (b_{λ} | λ ∈ Λ) a system of elements of B, and for every n ∈ ℕ denote by I_{n} ⊂ B the ideal generated by the system (b_{λ}^{n} | λ ∈ Λ). Suppose that p^{k}B ⊂ I_{1} for some k ∈ ℕ, and that (f ⊗_{B} B/I_{1})^{a} : (B/I_{1})^{a} → (C/I_{1}C)^{a} is a flat morphism of (A, 𝔪)^{a}-algebras. Then the same holds for (f ⊗_{B} B/I_{n})^{a}, for every n ∈ ℕ.

Proof. Set J_{n} := p^{n}B + I_{n} for every n ∈ ℕ; since p^{k}B ⊂ I_{1}, it is easily seen that for every n ∈ ℕ there exists n′ ∈ ℕ such that J_{n′} ⊂ I_{n} ⊂ J_{n} (details left to the reader). Hence, it suffices to show that (f ⊗_{B} B/J_{n})^{a} is flat for every n ∈ ℕ. Next, let \underline{α} ∈ Ker u_{B} be a distinguished element, and pick for every λ ∈ Λ elements e_{λ} ∈ E(B) and c_{λ} ∈ B such that

b_{λ} = \overline{u}_{B}(e_{λ}) + \overline{u}_{B}(α_{0}) · c_{λ}.

For every n ∈ ℕ, denote by I′_{n} ⊂ B the ideal generated by (\overline{u}_{B}(e_{λ}^{n}) | λ ∈ Λ), and set J′_{n} := \overline{u}_{B}(α_{0}^{n}) · B + I′_{n}; it follows easily that

J_{2n−1} ⊂ J′_{n}  and J′_{2n−1} ⊂ J_{n}  for every n ∈ ℕ

so (f ⊗_{B} B/J′_{1})^{a} is flat, and we are further reduced to checking that (f ⊗_{B} B/J′_{n})^{a} is flat for every n ∈ ℕ. For every n ∈ ℕ, denote by I_{E,n} ⊂ E(B) the ideal generated by (e_{λ}^{n} | λ ∈ Λ), and set J_{E,n} := α_{0}^{n}E(B) + I_{E,n}; notice the natural identifications

(16.8.25)  B/J′_{1} = E(B)/J_{E,1}  and C/J′_{1}C = E(C)/J_{E,1}E(C).

Denote also by Φ_{B/J′_{1}} and Φ_{C/J′_{1}C} the Frobenius endomorphism of B/J′_{1} and respectively C/J′_{1}C. We remark :

Claim 16.8.26. The induced diagram of E-algebras

\xymatrix{B/J′_{1} \ar[r]^{f⊗_{B}B/J′_{1}} \ar[d]_{Φ_{B/J′_{1}}} & C/J′_{1}C \ar[d]^{Φ_{C/J′_{1}C}} \\ B/J′_{1} \ar[r]^{f⊗_{B}B/J′_{1}} & C/J′_{1}C}

is cocartesian.

Proof of the claim. Indeed, since E(B) is perfect, it is easily seen that Φ_{B/J′_{1}} is surjective, and its kernel is the ideal generated by the system {α_{0}^{1/p}} ∪ {e_{λ}^{1/p} | λ ∈ Λ}, under the identifications (16.8.25); likewise, Φ_{C/J′_{1}C} is surjective and its kernel is generated by the image in C/J′_{1}C of the same system, whence the claim. ♢

For every n ∈ ℕ, set 𝔫_{n} := \{\underline{a} ∈ W_{n}E | a_{0}, …, a_{n−1} ∈ 𝔪\}; since 𝔪_{E} fulfills condition (B) of [75, §2.1.6], proposition 14.7.11 says that 𝔫_{n} is the unique ideal 𝔫_{n} ⊂ W_{n}E with 𝔫_{n}^{2} = 𝔫_{n} and \overline{ω}_{i}(𝔫_{n}) = 𝔪_{E} for i = 0, …, n − 1. From claim 16.8.26 and theorem 14.7.26(i) it follows that the induced morphism of (W_{n}E, 𝔫_{n})^{a}-algebras

W_{n}(f ⊗_{B} B/J′_{1})^{a} : W_{n}(E(B)/J_{E,1})^{a} → W_{n}(E(C)/J_{E,1}E(C))^{a}

is flat for every n ∈ ℕ. Moreover, the Frobenius automorphisms of E(B) and E(C) induce isomorphisms E(B)/J_{E,1} →^{∼} E(B)/J_{E,p^{k}} and E(C)/J_{E,1}E(C) →^{∼} E(C)/J_{E,p^{k}}E(C) for every k ∈ ℕ, whence a commutative diagram of W_{n}E-algebras

\xymatrix{W_{n}(E(B)/J_{E,1}) \ar[r]^{W_{n}(f⊗_{B}E(B)/J_{E,1})} \ar[d] & W_{n}(E(C)/J_{E,1}E(C)) \ar[d] \\ W_{n}(E(B)/J_{E,p^{k}}) \ar[r]^{W_{n}(f⊗_{B}E(B)/J_{E,p^{k}})} & W_{n}(E(C)/J_{E,p^{k}}E(C))} for every n, k ∈ ℕ

whose vertical arrows are isomorphisms, and where the rings on the bottom row are regarded as W_{n}E-algebras via restriction of scalars along the automorphism W_{n}(Φ_{E}) : W_{n}E → W_{n}E. Since we have

W_{n}(Φ_{E})(𝔫_{n}) = 𝔫_{n} for every n ∈ ℕ

it follows that W_{n}(f ⊗_{B} E(B)/J_{E,p^{k}})^{a} is a flat morphism of (W_{n}E, 𝔫_{n})^{a}-algebras, for every n, k ∈ ℕ. We remark, quite generally :

Claim 16.8.27. Let R be any ring such that the Frobenius endomorphism Φ_{R/pR} of R/pR is surjective, (x_{λ} | λ ∈ Λ) any set of elements of R, and I ⊂ R the ideal generated by this set. Then, for every n ∈ ℕ, the kernel of the projection π_{n} : W_{n}R → W_{n}R/I is generated by

(V_{R}^{i}(x_{λ}) | λ ∈ Λ, i = 0, …, n − 1).

Proof of the claim. Arguing by induction on n ∈ ℕ, we are reduced to checking that the ideal V_{n}I := V_{n}R ∩ Ker π_{n+1} is generated by (V^{n}x_{λ} | λ ∈ Λ), for every n ∈ ℕ. However, according to claim 9.3.31we have an isomorphism of W_{n+1}R-modules : I →^{∼} V_{n}I, where I is regarded as a W_{n+1}R-module via restriction of scalars along the ring homomorphism \overline{ω}_{n} : W_{n+1}R → R. Thus, we come down to showing that \overline{ω}_{n} is a surjection for every n ∈ ℕ. We argue by induction on n: the assertion is trivial for n = 0, so suppose that \overline{ω}_{n} is already known to be surjective for some n ≥ 0; since Φ_{R/pR} is surjective, the same holds for Φ_{R/pR}^{n}, and taking into account (9.3.3) together with our inductive assumption, we easily deduce that \overline{ω}_{n+1} is surjective as well. ♢

In light of claim 16.8.27we see that

W_{n}(E(B)/J_{E,p^{k}}) = W (E(B))/(p^{n}W (E(B)) + ℐ_{k})

where ℐ_{k} is the ideal generated by (p^{i} ·τ_{E(B)}(e_{λ}^{p^{k−i}}) | λ ∈ Λ, i = 0, …, k). Taking into account lemma 16.1.1(iv), we deduce that

A ⊗_{W (E)} W_{n}(E(B)/J_{E,p^{k}}) = B/(p^{n}B + I′′_{k}) for every n, k ∈ ℕ

where I′′_{k} is the ideal generated by (p^{i} · \overline{u}_{B}(e_{λ}^{p^{k−i}}) | λ ∈ Λ, i = 0, …, k). Notice that

(16.8.28)  J′_{p^{2k−1}} ⊂ I′′_{2k−1} ⊂ J′_{k}  for every k ∈ ℕ.

We have as well A ⊗_{W (E)} W_{n}(f ⊗_{E(B)} E(B)/J_{E,p^{k}}) = f ⊗_{B} B/(p^{n}B + I′′_{k}), and furthermore, the ring homomorphism W_{n}E → A/p^{n}A induced by u_{A} maps 𝔫_{n} onto 𝔪/p^{n}A; therefore, (f ⊗_{B} B/(p^{n}B + I′′_{k}))^{a} is a flat morphism of (A, 𝔪)^{a}-algebras, for every n, k ∈ ℕ. Combining with (16.8.28), the assertion follows. □

Proposition 16.8.29. In the situation of (16.8.16), let f^{a} : B^{a} → C be an étale morphism of (A, 𝔪)^{a}-algebras such that C is an almost finitely presented B^{a}-module, and endow C_{∗} with the unique ring topology such that the induced map B → C_{∗} is adic. Then C_{∗} is perfectoid.

Proof. Let I ⊂ B be any ideal of definition, and I_{E} ⊂ E(B) the corresponding ideal such that \overline{u}_{A} induces an isomorphism E(B)/I_{E} →^{∼} B/I (see remark 16.3.7(iii)); then f^{a} ⊗_{B} B/I : E(B)/I_{E} → C/IC is an étale morphism of (E, 𝔪_{E})^{a}-algebras. By [75, Th.5.3.27], the latter lifts uniquely to an étale morphism g : E(B)^{a} → C′ of (E, 𝔪_{E})^{a}-algebras such that C′ is almost finitely presented as an E(B)^{a}-module. Since E(B) is perfect, [75, Th.3.5.13] implies that the Frobenius morphism

(Φ_{E}, Φ_{C′}) : ((E(B), 𝔪_{E}), C′) → ((E(B), 𝔪_{E}), C′)

is cartesian in the fibred category ℬ^{a}-Alg of [75, §3.5.3]. Since the functor (−)_{∗} on ℬ^{a}-algebras is cartesian (see [75, §3.5.4]), it follows that the morphism

(Φ_{E}, Φ_{C′_{∗}}) : ((E(B), 𝔪_{E}), C′_{∗}) → ((E(B), 𝔪_{E}), C′_{∗})

is cartesian as well, i.e. C′_{∗} is a perfect 𝔽_{p}-algebra. Let us endow C′_{∗} with its I_{E}-adic topology. We remark :

Claim 16.8.30. C′_{∗} is perfectoid.

Proof of the claim. Since E(B)^{a} is complete and separated for its I_{E}-adic topology, [75, Claim 5.3.25], implies that the natural morphism C′ → lim_{n∈ℕ} C′/I_{E}^{n}C′ is an isomorphism, so the same holds for the induced map C′_{∗} → lim_{n∈ℕ}(C′/I_{E}^{n}C′)_{∗}. Especially, C′_{∗} is complete and separated for the linear topology defined by the system of ideals (J_{n} := Ker (C′_{∗} → (C′/I_{E}^{n}C′)_{∗} | n ∈ ℕ) (corollary 8.2.16(i)). Clearly I_{E}^{n}C′_{∗} ⊂ J_{n} for every n ∈ ℕ, so C′_{∗} is also complete and separated for its I_{E}-adic topology (lemma 8.3.12), whence the claim. ♢

By claim 16.8.30 it follows that there exists a perfectoid A-algebra D with an isomorphism of perfectoid E-algebras C′_{∗} →^{∼} E(D). We consider the composition

h : E(B) ⟶^{η_{E(B)}} E(B)_{∗}^{a} ⟶^{g_{∗}} C′_{∗} →^{∼} E(D)

where η_{E(B)} is the unit of adjunction; since h is an adic ring homomorphism, there exists a unique adic map of A-algebras k : B → D with E(k) = h (theorem 16.3.42(i)). By construction, we have also an isomorphism (C/IC)^{a} →^{∼} (D/ID)^{a} of (B/I)^{a}-algebras; especially, (k ⊗_{B} B/I)^{a} is a flat ring homomorphism, and therefore the same holds for (k ⊗_{B} B/I^{n})^{a} : (B/I^{n})^{a} → (D/I^{n}D)^{a}, for every n ∈ ℕ (lemma 16.8.24). By virtue of [75, Cor.3.2.11(ii,iii)] and remark 14.1.91, it follows that there exists a unique isomorphism of (B/I^{n})^{a}-algebras

ω_{n} : C/I^{n}C →^{∼} (D/I^{n}D)^{a} for every n ∈ ℕ.

Notice that the uniqueness of the morphisms ω_{n} implies that ω_{n+1} ⊗_{B} B/I^{n} = ω_{n} for every n ∈ ℕ. However, D is complete and separated for its I-adic topology, since h is adic; the same holds for C, since the latter is an almost projective B-module of almost finite presentation ([75, Claim 5.3.25]). Hence, the limit of the inverse system (ω_{n} | n ∈ ℕ) is an isomorphism ω : C →^{∼} D^{a} of B^{a}-algebras, whence the isomorphism ω_{∗} : C_{∗} →^{∼} D_{∗}^{a} of B-algebras, and it suffices to invoke corollary 16.8.17to conclude. □

16.8.31. Let now \underline{A} := (A, A^{+}, U) be any perfectoid quasi-affinoid ring, and set

\underline{U} := Spec \underline{A}  \underline{U}_{E} := E(\underline{U})  X_{A} := Spec A  Z_{A} := X_{A} \ U

(notation of (16.5.20)). We let 𝔪 ⊂ A be the unique radical ideal such that Spec A/𝔪 = Z_{A}. Also, set E := E(A), X_{E} := Spec E, and let 𝔪_{E} ⊂ E be the unique ideal such that \overline{u}_{A} induces a ring isomorphism E/𝔪_{E} →^{∼} A/𝔪 (see remark 16.3.7(iii)). According to proposition 16.8.15(i), both pairs (A, 𝔪) and (E, 𝔪_{E}) are basic setups.

Theorem 16.8.32. With the notation of (16.8.31), the pair (X_{A}, Z_{A}) is normal and almost pure, relative to the basic setup (A, 𝔪).

Proof. Let j_{A} : U → X_{A} be the open immersion; since j_{A∗}𝒪_{U} is a quasi-coherent 𝒪_{X_{A}}-algebra ([59, Ch.I, Prop.9.4.2(i)]), in order to check that the pair (X_{A}, Z_{A}) is normal, it suffices to show that the induced map A → 𝒪_{X_{A}}(U) is almost injective and the image of A^{a} is integrally closed in 𝒪_{X_{A}}(U)^{a}. These assertions follow from claim 16.8.20(ii).

Next, pick any ideal of definition I ⊂ A, set E := E(A), let I_{E} ⊂ E be the unique ideal of definition such that \overline{u}_{A} induces a ring isomorphism E/I_{E} →^{∼} A/I, and consider the following diagram of categories :

𝒟_{A}  :  [figure: Cov(U); (A, 𝔪)^{a}-Ét_{afr}; j_{A}^{∗}; i_{A}^{∗}; (A/I, 𝔪/I)^{a}-Ét_{afr}; Cov(U_{E}); w_{A}; (E, 𝔪_{E})^{a}-Ét_{afr}; i_{E}^{∗}; j_{E}^{∗}; (E/I_{E}, 𝔪_{E}/I_{E})^{a}-Ét_{afr}]

where j_{A}^{∗} (resp. i_{A}^{∗}) is induced by the open (resp. closed) immersion j_{A} : U → X_{A} (resp. i_{A} : Z_{A} → X_{A}) and likewise for j_{E}^{∗} and i_{E}^{∗}. We need to show that j_{A}^{∗} is an equivalence. However, according to [75, Th.5.5.7(iii)], both i_{A}^{∗} and i_{E}^{∗} are equivalences, and the same holds for j_{E}^{∗}, by virtue of theorem 16.8.2(i). Then, choose an arbitrary quasi-inverse functor s_{E} for j_{E}^{∗} and t_{A} for i_{A}^{∗}; we let the dotted arrow be the functor w_{A} := j_{A}^{∗} ∘ t_{A} ∘ i_{E}^{∗} ∘ s_{E} : Cov(U_{E}) → Cov(U), so that 𝒟_{A} is essentially commutative, and we are reduced to checking that w_{A} is an equivalence.

Claim 16.8.33. We may assume that \underline{A} = Γ^{∘}(\underline{U}) (notation of (16.5.19)).

Proof of the claim. Say that Γ^{∘}(\underline{U}) = (A_{U}^{∘}, A_{U}^{+}, U), set X_{U}^{∘} := Spec A_{U}^{∘}, and denote by 𝔪_{U} ⊂ A_{U}^{∘} the unique radical ideal such that Spec A_{U}^{∘}/𝔪_{U} = X_{U}^{∘} \ U. Likewise, say that Γ^{∘}(\underline{U}_{E}) = (E_{U}^{∘}, E_{U}^{+}, U_{E}), set X_{E}^{∘} := Spec E_{U}^{∘} and denote by 𝔪_{E,U} ⊂ E_{U}^{∘} the unique radical ideal such that Spec E_{U}^{∘}/𝔪_{E,U} = X_{E}^{∘} \ U_{E}. It follows easily that

𝔪_{U} ⊂ 𝔪A_{U}^{∘} 𝔪_{U,E} ⊂ 𝔪_{E}E_{U}^{∘}.

Moreover, set I_{U} := IA_{U}^{∘} and I_{E,U} := I_{E}E_{U}^{∘}; the map \overline{u}_{A_{U}^{∘}} : E_{U}^{∘} → A_{U}^{∘} induces a ring isomorphism E_{U}^{∘}/I_{E,U} →^{∼} A_{U}^{∘}/I_{U}. Also, as in the foregoing, the projection A_{U}^{∘} → A_{U}^{∘}/I_{U} and the open immersion j_{E,U} : U_{E} → X_{E}^{∘} induce equivalences of categories

i_{U}^{∗} : (A_{U}^{∘}, 𝔪_{U})^{a}-Ét_{afr} →^{∼} (A_{U}^{∘}/I_{U}, 𝔪_{U}/I_{U})^{a}-Ét_{afr} j_{E,U}^{∗} : (E_{U}^{∘}, 𝔪_{E,U})^{a}-Ét_{afr} →^{∼} Cov(U_{E}).

After choosing quasi-inverse functors t_{U} for i_{U}^{∗} and s_{E,U} for j_{E,U}^{∗}, there follows an essentially commutative diagram of functors

\xymatrix{(A, 𝔪)^{a}-Ét_{afr} \ar[d]^{j_{A}^{∗}} \ar[dd] & (A/I, 𝔪/I)^{a}-Ét_{afr} \ar[l]_{t_{A}} \ar[dd] & (E, 𝔪_{E})^{a}-Ét_{afr} \ar[l]_{i_{E}^{∗}} \ar[dd] \\ Cov(U) &  & Cov(U_{E}) \ar[u]^{s_{E}} \ar[d]_{s_{E,U}} \\ (A_{U}^{∘}, 𝔪_{U})^{a}-Ét_{afr} \ar[u]_{j_{U}^{∗}} & (A_{U}^{∘}/I_{U}, 𝔪_{U}/I_{U})^{a}-Ét_{afr} \ar[l]_{t_{U}} & (E_{U}^{∘}, 𝔪_{E,U})^{a}-Ét_{afr} \ar[l]_{i_{E,U}^{∗}}}

whose unmarked vertical arrows are induced by the natural ring homomorphisms A → A_{U}^{∘}, A/I → A_{U}^{∘}/I_{U} and E → E_{U}^{∘}, and where j_{U}^{∗} is induced by the open immersion j_{U} : U → X_{U}^{∘}. The claim follows easily. ♢

The restriction to the site ℛ(\underline{U}) of the stack Cov∧ ; \underline{U} of corollary 15.7.26 along the inclusion ℛ(\underline{U}) ⊂ 𝒬(\underline{U}) is a stack that we denote again

π\underline{U} : Cov∧ ; \underline{U} → ℛ(\underline{U}).

Recall the construction : for every rational subset R ⊂ Spa \underline{U} we choose a complete and separated quasi-affinoid scheme \underline{U}_{R}^{∧} := (U_{R}^{∧}, 𝒯_{R}^{∧}, A_{R}^{∧+}) representing the sub-presheaf h′′_{R} of h′′ ; \underline{U}; then π−1 ; \underline{U}(R) := Cov(U_{R}^{∧}). We shall consider similarly the quasi-affinoid schemes

\underline{X}_{A} := Spec (A, A^{+}, X_{A}) and \underline{Y}_{A} := Spec A/I ⊗_{A} (A, A^{+}, X_{A})

(notation of example 15.4.7(i)) and the fibrations

π\underline{X}_{A}^{a} : Cov_{(\underline{X}_{A},𝔪)^{a}}^{∧} → ℛ(\underline{U}) π\underline{Y} _{A}^{a} : Cov_{(\underline{Y}_{A},𝔪)^{a}}^{∧} → ℛ(\underline{U})

defined as follows. For any R ∈ Ob(ℛ(\underline{U})), we know that (A_{R}^{∧∘}, A_{R}^{∧+}, U_{R}^{∧}) := Γ^{∘}(\underline{U}_{R}^{∧}) is a perfectoid quasi-affinoid ring (proposition 16.7.25(iii) and (16.5.19)), and we set X_{R}^{∘} := Spec A_{R}^{∧∘}; then, let 𝔪_{R} ⊂ A_{R}^{∧∘} be the unique radical ideal such that X_{R}^{∘} \ U_{R}^{∧} = Spec A_{R}^{∧∘}/𝔪_{R}. Set also I_{R} := IA_{R}^{∧∘}; by proposition 16.8.15(i), the pair (A_{R}^{∧∘}, 𝔪_{R}) is a basic setup, and we let

π_{\underline{X}_{A}^{a}}^{−1}(R) := (A_{R}^{∧∘}, 𝔪_{R})^{a}-Ét_{afp} π_{\underline{Y} _{A}^{a}}^{−1}(R) := (A_{R}^{∧∘}/I_{R}, 𝔪_{R}/I_{R})^{a}-Ét_{afp}.

If R′ ⊂ R is another rational subset, the induced inclusion of sub-presheaves h′′_{R′} ⊂ h′′_{R} corresponds to a morphism \underline{U}_{R′}^{∧} → \underline{U}_{R}^{∧} of quasi-affinoid schemes, which in turn yields a morphism Γ^{∘}(\underline{U}_{R}^{∧}) → Γ^{∘}(\underline{U}_{R′}^{∧}) of perfectoid quasi-affinoid rings, and it follows easily that

𝔪_{R′} ⊂ 𝔪_{R}A_{R′}^{∧∘}

whence, finally, induced functors

j_{R′R}^{∗} : π_{\underline{X}_{A}^{a}}^{−1}(R) → π_{\underline{X}_{A}^{a}}^{−1}(R′) j_{R′R}^{′∗} : π_{\underline{Y} _{A}^{a}}^{−1}(R) → π_{\underline{Y} _{A}^{a}}^{−1}(R′).

Moreover, for every further inclusion of rational subset R′′ ⊂ R′ we have a natural isomorphisms of functors

j_{R′′R′}^{∗} ∘ j_{R′R}^{∗} →^{∼} j_{R′′R}^{∗} j_{R′′R′}^{′∗} ∘ j_{R′R}^{′∗} →^{∼} j_{R′′R}^{′∗}

and the systems of such functors and isomorphisms of functors amount to well defined pseudofunctors ℛ(\underline{U})^{o} → Cat, whence the sought fibrations π\underline{X}_{A}^{a} and π\underline{Y} _{A}^{a}. Furthermore, the system of open immersions j_{R} : U_{R}^{∧} → X_{R}^{∘} and the projections A_{R}^{∧∘} → A_{R}^{∧∘}/I_{R} induce pseudo-natural transformations

(A_{R}^{∧∘}/I_{R}, 𝔪_{R}/I_{R})^{a}-Ét_{afp} ⟵^{i_{R}^{∗}} (A_{R}^{∧∘}, 𝔪_{R})^{a}-Ét_{afp} ⟶^{j_{R}^{∗}} Cov(U_{R}^{∧}) for every R ∈ Ob(ℛ(\underline{U}))

whence morphisms of fibrations

Cov_{(\underline{Y}_{A},𝔪)^{a}}^{∧} ⟵^{i∗ ; \underline{U}} Cov_{(\underline{X}_{A},𝔪)^{a}}^{∧} ⟶^{j∗ ; \underline{U}} Cov∧ ; \underline{U}.

Likewise, by repeating the foregoing with the quasi-affinoid schemes

\underline{X}_{E} := Spec (E, E^{+}, X_{E}) \underline{Y}_{E} := Spec E/I_{E} ⊗_{E} (E, E^{+}, X_{E})

we get the following fibrations and morphisms of fibrations over the site ℛ(\underline{U}_{E}) :

(16.8.34)  Cov_{(\underline{Y}_{E},𝔪_{E})^{a}}^{∧} ⟵^{i_{\underline{U}_{E}}^{∗}} Cov_{(\underline{X}_{E},𝔪_{E})^{a}}^{∧} ⟶^{j_{\underline{U}_{E}}^{∗}} Cov_{\underline{U}_{E}}^{∧}.

However, in view of the isomorphism of sites E : ℛ(\underline{U}) →^{∼} ℛ(\underline{U}_{E}) of remark 16.7.29(i), we may also regard (16.8.34) as fibrations and morphisms of fibrations over the site ℛ(\underline{U}). Moreover, by combining remark 16.7.29(ii) and (16.5.22), we obtain natural isomorphisms

(16.8.35)  \underline{Y}_{A} ×_{\underline{X}_{A}} \underline{U}_{R}^{∧} →^{∼} \underline{Y}_{E} ×_{\underline{X}_{E}} \underline{U}_{E(R)}^{∧}  for every R ∈ Ob(ℛ(\underline{U}))

and a simple inspection shows that the resulting ring isomorphism A_{R}^{∘}/I_{R} →^{∼} A_{E(R)}^{∘}/I_{E(R)} maps 𝔪_{R}/I_{R} isomorphically onto 𝔪_{E(R)}/I_{E(R)}. Summing up, we get a natural fibrewise equivalence of fibrations over ℛ(\underline{U}) :

(16.8.36)  Cov_{(\underline{Y}_{E},𝔪_{E})^{a}}^{∧} →^{∼} Cov_{(\underline{Y}_{A},𝔪)^{a}}^{∧}.

Furthermore, both i∗ ; \underline{U} and i_{\underline{U}_{E}}^{∗} are fibrewise equivalences, by virtue of [75, Th.5.5.7(iii)], and the same holds for j_{\underline{U}_{E}}^{∗}, by theorem 16.8.2(i). We may then find morphisms of fibrations

s_{\underline{U}_{E}} : Cov_{\underline{U}_{E}}^{∧} → Cov_{(\underline{X}_{E},𝔪_{E})^{a}}^{∧} and t\underline{U} : Cov_{(\underline{Y}_{A},𝔪)^{a}}^{∧} → Cov_{(\underline{X}_{A},𝔪)^{a}}^{∧}

that are quasi-inverse functors for j_{\underline{U}_{E}}^{∗} and i∗ ; \underline{U} (corollary 3.1.28(i)). The composition of these functors yields a morphism of stacks

w\underline{U} : Cov_{\underline{U}_{E}}^{∧} ⟶^{s_{\underline{U}_{E}}} Cov_{(\underline{X}_{E},𝔪_{E})^{a}}^{∧} ⟶^{i_{\underline{U}_{E}}^{∗}} Cov_{(\underline{Y}_{E},𝔪_{E})^{a}}^{∧} →^{∼} Cov_{(\underline{Y}_{A},𝔪)^{a}} ⟶^{t\underline{U}} Cov_{(\underline{X}_{A},𝔪)^{a}}^{∧} ⟶^{j∗ ; \underline{U}} Cov∧ ; \underline{U}

such that (w\underline{U})_{Spa \underline{U}} : Cov(U_{E}) → Cov(U) is isomorphic to w_{A}.

For every x ∈ Spa \underline{U}, let x_{E} := Spa(u\underline{X})(x) ∈ Spa \underline{U}_{E}; by propositions 5.2.9and 5.6.37, we are then reduced to checking that the induced functor on stalks

w\underline{U}(x) : Cov_{\underline{U}_{E}}^{∧}(x_{E}) → Cov∧ ; \underline{U}(x)

is an equivalence, for every such x. To this aim, suppose first that x is analytic; in this case, the completed residue field κ(x)^{∧} is a Tate valued field (see (15.5.12)), so that the analytic locus of Spec κ(x)^{∧+} is {η_{x}}, where η_{x} is the generic point. We have a well defined quasi-affinoid ring

\underline{κ}(x)^{∧∘} := (κ(x)^{∧∘}, κ(x)^{∧+}, \{η_{x}\})

and the natural map A → κ(x)^{∧∘} yields a morphism of quasi-affinoid rings π_{x}^{∧} : \underline{A} → \underline{κ}(x)^{∧∘}. Likewise we get a morphism of quasi-affinoid rings

π_{x_{E}}^{∧} : E(\underline{A}) → \underline{κ}(x_{E})^{∧∘} := (κ(x_{E})^{∧∘}, κ(x_{E})^{∧+}, \{η_{x_{E}}\}).

Moreover, by virtue of corollary 16.5.61, both \underline{κ}(x)^{∧∘} and \underline{κ}(x_{E})^{∧∘} are perfectoid, and there exists a unique isomorphism of quasi-affinoid rings

(16.8.37)  ω : E(\underline{κ}(x)^{∧∘}) →^{∼} \underline{κ}(x_{E})^{∧∘}  such that ω ∘ E(π_{x}^{∧}) = π_{x_{E}}^{∧}.

We may then repeat the foregoing constructions with \underline{A} replaced by \underline{κ}(x)^{∧∘} and \underline{κ}(x_{E})^{∧∘} : we get first the quasi-affinoid schemes

\underline{U}(x) := Spec \underline{κ}(x)^{∧∘} \underline{U}_{E}(x_{E}) := Spec \underline{κ}(x_{E})^{∧∘}.

Then we set \underline{A}(x) := (κ(x)^{∧∘}, κ(x)^{∧+}, Spec κ(x)^{∧∘}), and

\underline{X}_{A}(x) := Spec \underline{A}(x) \underline{Y}_{A}(x) := Spec (A/I ⊗_{A} \underline{A}(x))

and define likewise \underline{X}_{E}(x_{E}) and \underline{Y}_{E}(x_{E}). Also, let 𝔪_{x} ⊂ κ(x)^{∧∘} and 𝔪_{E,x} ⊂ κ(x_{E})^{∧∘} be the maximal ideals. To these quasi-affinoid schemes we attach as in the foregoing fibrations over the sites ℛ(\underline{U}(x)) and ℛ(\underline{U}_{E}(x_{E})). However, notice that the points of Spa \underline{U}(x) are (the equivalence classes of) the continuous valuations v : κ(x)^{∘} → Γ_{v∘} with Ker v = 0 and v(κ(x)^{+}) ⊂ Γ_{v∘}^{+}; it is easily seen that there is only one such valuation, i.e. the one corresponding to the point x of Spa \underline{U}. Likewise, we have Spa \underline{U}_{E} = \{x_{E}\}; hence, these fibrations amount to categories, or equivalently, they can be identified naturally with their stalks over the unique point x of Spa \underline{U} and the unique point x_{E} of Spa \underline{U}_{E}. Then, a simple inspection yields an essentially commutative diagram of categories :

𝒟′_{A}  :  \xymatrix{Cov_{(\underline{Y}_{A},𝔪)^{a}}^{∧}(x) \ar[d] & Cov_{(\underline{X}_{A},𝔪)^{a}}^{∧}(x) \ar[l]_{i∗ ; \underline{U} (x)} \ar[r]^{j∗ ; \underline{U} (x)} \ar[d] & Cov∧ ; \underline{U}(x) \ar[d] \\ _{Cov{∧ \atop (\underline{Y}_{A}(x),𝔪_{x})^{a}}} & _{Cov{∧ \atop (\underline{X}_{A}(x),𝔪_{x})^{a}}} \ar[l]_{i_{\underline{U}(x)}^{∗}} \ar[r]^{j_{\underline{U}(x)}^{∗}} & _{Cov{∧ \atop \underline{U}(x)}}}

whose right-most vertical arrow is an equivalence, by virtue of (15.7.28). Likewise we have an essentially commutative diagram 𝒟′_{E} for the corresponding fibrations over the site ℛ(\underline{U}_{E}); moreover, in light of (16.8.37) we have as well an essentially commutative diagram

\xymatrix{Cov_{(\underline{Y}_{E},𝔪_{E})^{a}}^{∧}(x_{E}) \ar[r] \ar[d] & Cov_{(\underline{Y}_{A},𝔪)^{a}}^{∧}(x) \ar[d] \\ _{Cov{∧ \atop (\underline{Y}_{E}(x_{E}),𝔪_{E,x})^{a}}} \ar[r] & _{Cov{∧ \atop (\underline{Y}_{A}(x),𝔪_{x})^{a}}}}

whose vertical arrows are the same as the left-most vertical arrows of 𝒟′_{A} and 𝒟′_{E}, and whose horizontal arrows are induced by the equivalences (16.8.36). Summing up, we are reduced to checking the following :

Claim 16.8.38. The functor j_{\underline{U}(x)}^{∗} is an equivalence.

Proof of the claim. By unwinding the definitions, we see that Cov_{\underline{U}(x)}^{∧} is the category of étale κ(x)^{∧}-algebras, Cov_{(\underline{X}_{A}(x),𝔪_{x})^{a}}^{∧} is the category of étale (κ(x)^{∧∘}, 𝔪_{x})^{a}-algebras of almost finite rank, and j_{\underline{U}(x)}^{∗} is the natural restriction functor. In light of lemma 14.4.5(ii) and remark 14.4.3(i), it follows already that j_{\underline{U}(x)}^{∗} is fully faithful. Next, let E be any finite separable field extension of κ(x)^{∧}, and E^{∘} the integral closure of κ(x)^{∧∘} in E; it suffices to check that E^{∘a} is an étale κ(x)^{∧∘a}-algebra of finite rank. However, notice that κ(x)^{∧∘} is a valuation ring of rank one with field of fractions κ(x)^{∧}; moreover, it is perfectoid for the ring topology such that the natural map A → κ(x)^{∧∘} is adic (corollary 16.5.61(i)), and therefore it is deeply ramified, by [75, Prop.6.6.6] and lemma 16.2.3(iv). Let also E^{s} be any separable closure of E, and E^{s∘} the integral closure of E^{∘} in E^{s}; by [75, Prop.6.6.2 and Rem.6.1.12(iv)] it follows that (Ω_{E^{s∘}/κ(x)^{∧∘}})^{a} = 0, and combining with [75, Th.6.3.23] we deduce that (Ω_{E^{∘}/κ(x)^{∧∘}})^{a} = 0 (cp. the proof of [75, Prop.6.6.2]). By invoking again [75, Th.6.3.23], it follows that 𝒟_{E^{∘}/κ(x)^{∧∘}} = E^{∘a}. Then the assertion follows from [75, Prop.6.3.8 and Lemma 4.1.27]. ♢

Lastly, in case x is non-analytic, set

Y (x) := Spec (A/I ⊗_{A} 𝒪_{Spa \underline{U},x}^{∧}) Y_{E}(x_{E}) := Spec (E/I_{E} ⊗_{E} 𝒪_{Spa \underline{U}_{E},x_{E}}^{∧})

According to (15.7.28) we have natural equivalences of categories

Ψ : Cov∧ ; \underline{U}(x) →^{∼} Cov(Y (x)) Ψ_{E} : Cov∧ ; \underline{U}(x_{E}) →^{∼} Cov(Y_{E}(x_{E})).

Recall the construction of these equivalences : an object of Cov\underline{U}(x) is a pair (R, φ) where R is a rational subset of Spa \underline{U} containing x, and φ : V → U_{R}^{∧} is a finite étale morphism of schemes; to such an object, the equivalence Ψ assigns the finite étale morphism Y (x) ×_{U_{R}^{∧}} φ. Similarly we may describe Ψ_{E}. We get likewise well defined functors

Cov_{(X_{A},𝔪_{A})^{a}}^{∧}(x) ⟶^{Ψ′} Cov(Y (x)) ⟵^{Ψ′′} Cov_{(\underline{Y}_{A},𝔪_{A})^{a}}^{∧}(x).

Namely, an object of Cov_{(X_{A},𝔪_{A})^{a}}^{∧}(x) is a pair (R, f : (A_{R}^{∧∘}, 𝔪_{R})^{a} → B^{a}) where R is as in the foregoing and f is an étale covering of almost rings, and notice that 𝔪_{R}𝒪_{Spa \underline{U},x}^{∧} = 𝒪_{Spa \underline{U},x}^{∧}, so f_{x} := A/I ⊗_{A} 𝒪_{Spa \underline{U},x}^{∧} ⊗_{A_{R}^{∧∘}} f is a homomorphism of (usual) rings; then the functor Ψ′ assigns to the pair (R, f) the finite étale covering Spec f_{x} of Y (x). Similarly we define Ψ′′, and it is easily seen that the resulting diagram of categories :

𝒟′′_{A}  :  \xymatrix{Cov_{(\underline{Y}_{A},𝔪)^{a}}^{∧}(x) \ar@{-->}[dr]_{Ψ′′} & Cov_{(\underline{X}_{A},𝔪)^{a}}^{∧}(x) \ar[l]_{i∗ ; \underline{U} (x)} \ar[r]^{j∗ ; \underline{U} (x)} \ar[d]^{Ψ′} & Cov∧ ; \underline{U}(x) \ar@{-->}[dl]^{Ψ} \\  & Cov(Y (x))}

is essentially commutative. Correspondingly, we have an essentially commutative diagram 𝒟′′_{E} for the stalks over x_{E}, and notice that all the arrows of 𝒟′′_{E} are equivalences (details left to the reader). On the other hand, the system of isomorphisms (16.8.35) induces an isomorphism of schemes

Y_{A}(x) →^{∼} Y_{E}(x_{E})

whence an essentially commutative diagram

\xymatrix{Cov_{(\underline{Y}_{E},𝔪_{E})^{a}}^{∧}(x_{E}) \ar[r]^{Ψ′′_{E}} \ar[d] & Cov(Y_{E}(x_{E})) \ar[d] \\ Cov_{(\underline{Y}_{A},𝔪)^{a}}^{∧}(x) \ar[r]^{Ψ′′} & Cov(Y (x))}

whose left vertical arrow is induced by the equivalence (16.8.36), and the right vertical arrow is an equivalence as well. We conclude that Ψ′′ is an equivalence; then the same holds for j∗ ; \underline{U}(x), and finally also for w\underline{U}(x), as sought. The proof is complete. □

Remark 16.8.39. Theorem 16.8.32 extends results of [149] (for perfectoid algebras over perfectoid fields), as well as [114, Th.5.5.9] (for perfectoid Tate rings A with p ∈ A^{×} ∩ A^{∘∘}), and [151, Th.7.4.5].

16.8.40. Almost purity in the formally perfectoid case. We show now how to extend theorem 16.8.32to the case of formally perfectoid rings. We begin with some preliminary observations:

Lemma 16.8.41. Let A be an adic topological ring that admits a finitely generated ideal I of adic definition, A^{∧} the completion of A, and 𝔪 ⊂ A an open ideal. The following holds :

(i) 𝔪 is a radical ideal of A if and only if 𝔪A^{∧} is a radical ideal of A^{∧}.

(ii) 𝔪 = 𝔪^{2} if and only if 𝔪A^{∧} = 𝔪^{2}A^{∧}.

(iii) 𝔪 fulfills condition (B) of [75, §2.1.6] if and only if the same holds for 𝔪A^{∧}.

Proof. By assumption, I^{n} ⊂ 𝔪 for some integer n > 0; after replacing I by I^{n}, we may then assume that I ⊂ 𝔪.

(i): Let 𝔫 be the radical ideal of 𝔪; since I ⊂ 𝔪, it is easily seen that 𝔫/I is the radical ideal of 𝔪/I in the quotient A/I. Since the completion map A → A^{∧} induces an isomorphism A/I →^{∼} A^{∧}/IA^{∧}, it follows that 𝔫A^{∧}/IA^{∧} is the radical ideal of 𝔪A^{∧}/IA^{∧}; again, this easily implies that 𝔫A^{∧} is the radical ideal of 𝔪A^{∧}, whence the assertion.

(ii): We have 𝔪 = 𝔪^{2} ⇔ 𝔪/I^{2} = 𝔪^{2}/I^{2}, and since the completion map induces an isomor-

phism A/I^{2} →^{∼} A^{∧}/I^{2}A^{∧}, we have 𝔪/I^{2} = 𝔪^{2}/I^{2} ⇔ 𝔪A^{∧}/I^{2}A^{∧} = 𝔪^{2}/I^{2}A^{∧}. Again, the

latter condition holds if and only if 𝔪A^{∧} = 𝔪^{2}A^{∧}, whence the assertion.

(iii): In view of (ii), it is clear that if 𝔪 fulfills condition (B), then so does 𝔪A^{∧}. Conversely, suppose that 𝔪A^{∧} fulfills condition (B); by (ii), it follows already that 𝔪 = 𝔪^{2}, and it remains to check that for every integer k > 0, the ideal 𝔪 is generated by the system x_{•}^{k} := (x^{k} | x ∈ 𝔪). However, let a_{1}, …, a_{n} be a finite system of generators of I, and choose N ∈ ℕ such that I^{N} lies in the ideal generated by a_{•}^{k} := (a_{1}^{k}, …, a_{n}^{k}); by assumption 𝔪A^{∧} is generated by the system b_{•}^{k} := (b^{k} | b ∈ 𝔪A^{∧}), hence the same holds for 𝔪A^{∧}/I^{N}A^{∧}. The completion map induces an isomorphism 𝔪/I^{N} →^{∼} 𝔪A^{∧}/I^{N}A^{∧} and maps x_{•}^{k} onto the family b_{•}^{k}, so that 𝔪/I^{N} is generated by x_{•}^{k}, and therefore the same holds for 𝔪, since the family x_{•}^{k} contains a_{•}^{k}. □

Proposition 16.8.42. Let A be a formally perfectoid ring, I ⊂ A a finitely generated ideal of adic definition, 𝔪 ⊂ A a radical ideal with I ⊂ 𝔪, and U := Spec A \ Spec A/𝔪. We have :

(i) (A, 𝔪) is a basic setup fulfilling condition (B) (see [75, §2.1.1, §2.1.6]).

(ii) For every subideal J ⊂ I, every almost finitely generated almost projective (A/J)^{a}- module P has almost finite rank, for the almost structure induced by (A, 𝔪).

(iii) In the situation of (ii), if moreover U is quasi-compact, then P has finite rank.

Proof. (i) follows from lemma 16.8.41and proposition 16.8.15(i).

(ii): After replacing I by I + pA, we may assume that p ∈ I. Set X := Spec A/J and U′ := X \ Spec A/I. Let S ⊂ A be the multiplicative subset 1 + I, and B := S^{−1}(A/J). Let also 𝒫 be the quasi-coherent 𝒪_{X}-module associated with the A/J-module P_{!}. Then X = U′ ∪ Spec B, and U′ is quasi-compact, so that 𝒫_{|U′} is a locally free 𝒪_{U′}-module of finite rank. Thus, it suffices to check that P_{B} := P ⊗_{A} B is almost projective B^{a}-module of almost finite rank. Now, the ideal I_{B} := I/J of B is obviously tight, and by construction I_{B} ⊂ rad(B); moreover, the localization A/J → B induces a ring isomorphism A/I →^{∼} B/I_{B}, whence an isomorphism of (A/I)^{a}-modules P_{0} := P/IP →^{∼} P_{B}/I_{B}P_{B}. By lemma 14.3.17(ii) we are then reduced to checking that P_{0} is an (A/I)^{a}-module of almost finite rank. To this aim, let E := E(A), and pick a distinguished element \underline{α} ∈ Ker u_{A}; by remark 16.3.7(ii), the map \overline{u}_{A} induces a ring isomorphism ω : E/α_{0}E →^{∼} A/pA, and we let 𝔪_{E} and I_{E} be the unique ideals of E with

α_{0} ∈ 𝔪_{E}  α_{0} ∈ I  𝔪/pA →^{∼} 𝔪_{E}/α_{0}E  I/pA →^{∼} I_{E}/α_{0}E.

In particular, I_{E} is finitely generated, and (E, 𝔪_{E}) is a basic setup, by proposition 16.8.15(i). Via the isomorphism ω, we may then regard P_{0} as an almost finitely generated almost projective (E/I_{E})^{a}-module (for the almost structure given by (E, 𝔪_{E})), and clearly it suffices to show that P_{0} is of almost finite rank, when regarded as an (E/I_{E})^{a}-module. The latter follows from proposition 16.8.7(i).

(iii): Arguing as in the proof of (ii), we reduce to showing that if U is quasi-compact, then P_{0} is of finite rank, when regarded as a (E/I_{E})^{a}-module, and according to proposition 16.8.7(ii), the latter will follow, once we show that Z := Spec E/𝔪_{E} is constructible in Spec E. However, (Spec A/pA) \ (Spec A/𝔪) = U ∩ Spec A/pA is quasi-compact, hence the same holds for W := (Spec E/α_{0}E) \ Z. Thus, (Spec E) \ Z = W ∪ SpecE[α_{0}^{−1}] is constructible in Spec E, and then the same follows for Z. □

16.8.43. Let A be a formally perfectoid ring, I ⊂ A a finitely generated ideal of adic definition, and 𝔪 ⊂ A a radical ideal with I ⊂ 𝔪. Set X := Spec A, Z := Spec A/𝔪, and U := X \ Z. According to proposition 16.8.42(i), we may consider the almost structure associated with the basic setup (A, 𝔪). We then have :

Theorem 16.8.44. In the situation of (16.8.43), let also r ∈ ℕ. The following holds :

(i) The pair (X, Z) is almost pure.

(ii) Let ℳ be any almost projective 𝒪_{X}^{a}-module of almost finite rank. Then ℳ is a faithfully flat 𝒪_{X}^{a}-module (resp. is an 𝒪_{X}^{a}-module of finite rank ≤ r) if and only if the same holds for the 𝒪_{U}-module ℳ_{|U}.

Proof. (i): Suppose first that Z is constructible in X. Let A^{h} and A^{∧} be respectively the topological henselization and the completion of A, and set X^{h} := Spec A^{h} and X^{∧} := Spec A^{∧}. Let also U^{h} := X^{h} ×_{X} U and U^{∧} := X^{∧} ×_{X} U. In light of theorem 16.8.32, it suffices to check that both square subdiagrams of the following induced diagram are 2-cartesian :

\xymatrix{𝒪_{X}^{a}-Ét_{afr} \ar[r] \ar[d] & 𝒪_{X^{h}}^{a}-Ét_{afr} \ar[d] \ar[r] & 𝒪_{X^{∧}}^{a}-Ét_{afr} \ar[d] \\ 𝒪_{U}-Ét_{fr} \ar[r] & 𝒪_{U^{h}}-Ét_{fr} \ar[r] & 𝒪_{U^{∧}}-Ét_{fr}.}

Now, by [75, Prop.5.5.6(i)] it follows that the natural functor

𝒪_{X}^{a}-Ét_{afr} → 𝒪_{X^{h}}^{a}-Ét_{afr} ×_{𝒪_{U^{h}}-Ét_{fr}}^{2} 𝒪_{U}-Ét_{fr}

is fully faithful. Thus, let (ℰ^{h}, ℰ_{U}, ξ) be the datum of an object ℰ^{h} of 𝒪_{X^{h}}^{a}-Ét_{afr}, an object ℰ_{U} of 𝒪_{U}-Ét_{fr}, and an isomorphism ξ in 𝒪_{U^{h}}-Ét_{fr} between the images of ℰ^{h} and ℰ_{U}. By loc.cit., there exists a quasi-coherent 𝒪_{X}^{a}-algebra ℰ whose restriction to X^{h} and to U is isomorphic to ℰ^{h} and respectively ℰ_{U}, and in view of proposition 16.8.42(iii), it remains to check that ℰ is étale and almost finitely presented as an 𝒪_{X}^{a}-module. Since the induced morphism U ⊔X^{h} → X is quasi-compact and faithfully flat, the assertion follows from lemmata 14.3.5(i) and 14.3.26(i). This shows that the left square subdiagram of the foregoing diagram is 2-cartesian.

Next, since both of the natural maps A/I → A^{h}/IA^{h} and A/I → A^{∧}/IA^{∧} are isomorphisms, [75, Th.5.5.7(iii)] implies that the top horizontal arrow of the right square subdiagram is an equivalence. To conclude the proof in this case, it then suffices to show :

Claim 16.8.45. The base change functor 𝒪_{U^{h}}-Ét_{fr} → 𝒪_{U^{∧}}-Ét_{fr} is an equivalence.

Proof of the claim. According to proposition 8.3.30(i) there exists a unique f-adic topology on B := 𝒪_{U^{h}}(U^{h}) (resp. on C := 𝒪_{U^{∧}}(U^{∧})) such that the restriction map A^{h} → B (resp. A^{∧} → C) is open. It follows easily that the induced map B → C is continuous and adic, since the same holds for the completion map A → A^{∧}. Set A^{h+} := A^{h∘} ⊂ A^{h} and A^{∧+} := A^{∧∘} ⊂ A^{∧}; in view of lemma 8.3.24(iii.a), the integral closure in B (resp. in C) of the image of A^{h+} (resp. of A^{∧+}) is a subring of integral elements, that we denote B^{+} (resp. C^{+}). Recall also that the natural morphism of schemes Spec B → X^{h} (resp. X^{∧} → Spec C) induces an isomorphism U_{B} := U^{h} ×_{X^{h}} Spec B →^{∼} U^{h} (resp. U_{C} := U^{∧} ×_{X^{∧}} Spec C →^{∼} U^{∧}); combining with lemma 15.4.6, we deduce that the resulting diagram

\xymatrix{Spec(C, C^{+}, U_{C}) \ar[r] \ar[d]_{φ} & Spec(B, B^{+}, U_{B}) \ar[d] \\ Spec(A^{∧}, A^{∧+}, U^{∧}) \ar[r] & Spec(A^{h}, A^{h+}, U^{h})}

is cartesian in the category of quasi-affinoid schemes. Furthermore, the quasi-affinoid scheme Spec(C, C^{+}, U_{C}) is complete, by virtue of lemma 16.4.32(ii). By arguing with universal properties, it follows easily that the morphism φ identifies Spec(C, C^{+}, U_{C}) with the completion of Spec(A^{∧}, A^{∧+}, U^{∧}) : the details are left to the reader. Then the claim follows from theorem 15.7.17and proposition 16.8.42(ii). ♢

Lastly, we consider the case where Z is an arbitrary closed subset in X. Then, let (I_{λ} | λ ∈ Λ) be the filtered system of all finitely generated ideals of A containing I and contained in 𝔪. For every λ ∈ Λ, let also 𝔪_{λ} be the radical of I_{λ}; then Z_{λ} := Spec A/𝔪_{λ} = Spec A/I_{λ} and U_{λ} := X \ Z_{λ} are constructible subsets of X, and (A, 𝔪_{λ}) is a basic setup fulfilling condition (B), for every such λ (proposition 16.8.42(i)). Both natural functors :

(A, 𝔪)^{a}-Ét_{afr} → 2-lim_{λ∈Λ} (A, 𝔪_{λ})^{a}-Ét_{afr} 𝒪_{U}-Ét_{fr} → 2-lim_{λ∈Λ} 𝒪_{U_{λ}}-Ét_{fr}

are equivalences of categories (corollary 14.2.5(ii,iv,v)), hence we are reduced to prove the theorem for the pair (X, Z_{λ}), relative to the almost structure (A, 𝔪_{λ}), for every λ ∈ Λ. The latter is already known by the previous case, so the proof is concluded.

(ii): Again, we assume first that Z is constructible, in which case ℳ has finite rank (proposition 16.8.42(iii)). Let A_{loc} be the topological localization of A (see (8.4.8)); set also X_{loc} := Spec A_{loc}. Then A_{loc} is a formally perfectoid ring, and the localization A → A_{loc} induces an isomorphism of the respective completions A^{∧} →^{∼} A_{loc}^{∧} (corollary 8.4.15(i)), and an isomorphism of A-schemes π : U_{loc} := X_{loc} ×_{X} U →^{∼} U. Set M := ℳ (X), M_{loc} := A_{loc} ⊗_{A} M, and let ℳ_{loc} be the almost projective 𝒪_{X_{loc}}^{a}-module of finite rank corresponding to M_{loc}. Then π induces a natural identification of ℳ_{|U} with (ℳ_{loc})_{|U_{loc}}, so that the latter is a faithfully flat 𝒪_{U_{loc}}-module. We notice :

Claim 16.8.46. In the situation of (16.8.31), let ℱ be any almost projective 𝒪_{X_{A}}^{a}-module of finite rank. Then ℱ is a faithfully flat 𝒪_{X_{A}}^{a}-module (resp. is an 𝒪_{X_{A}}^{a}-module of finite rank ≤ r) if and only if the same holds for the 𝒪_{U}-module ℱ_{|U}.

Proof of the claim. Since the pair (X, Z) is normal (theorem 16.8.32), the assertion follows immediately from lemma 14.3.18. ♢

From claim 16.8.46, it follows that A_{loc}^{∧} ⊗_{A_{loc}} M_{loc} is a faithfully flat (A_{loc}^{∧})^{a}-module of finite rank; consequently M_{loc}/IM_{loc} is a faithfully flat A_{loc}^{a}/IA_{loc}^{a}-module of finite rank. By construction, I_{loc}^{a} ⊂ rad(A_{loc}^{a}) is a tight ideal of A_{loc}^{a} (relative to the basic setup (A, 𝔪)), hence M_{loc} is a faithfully flat A_{loc}^{a}-module (lemma 14.3.17(i)). Lastly, let Y be the disjoint union of U and Spec A_{loc}; we have an obvious faithfully flat morphism of A-schemes f : Y → X, and the foregoing implies that f^{∗}ℳ is a faithfully flat 𝒪_{Y}^{a} -module, so ℳ is a faithfully flat 𝒪_{X}^{a}-module, by lemma 14.3.5(i).

Next, for a general Z, define the filtered system of subideals (𝔪_{λ} | λ ∈ Λ) of 𝔪, and the corresponding filtered system of open constructible subsets (U_{λ} | λ ∈ Λ) of U, as in the proof of (i). Then if ℳ_{|U} is a faithfully flat 𝒪_{U}-module (resp. if the rank of ℳ is ≤ r), obviously the same holds for the 𝒪_{U_{λ}}-module ℳ_{|U_{λ}}. For every λ ∈ Λ, denote by M_{λ} the image of M in the category of (A, 𝔪_{λ})^{a}-modules; by the foregoing case, M_{λ} is a faithfully flat (A, 𝔪_{λ})^{a}-module (resp. has rank ≤ r) for every such λ. Then M is a faithfully flat (A, 𝔪)^{a}-module (resp. has rank ≤ r), by corollary 14.2.5(iii,iv). □

Remark 16.8.47. A ring A is Witt-perfect (relative to a given prime integer p) in the sense of [52, Def.1.8] if and only if the p-adic completion of A is a P -ring in our sense (for its p-adic topology). In the case where 𝔪 is the radical of I := Ap, with p regular in A, and A integrally closed in A[1/p], theorem 16.8.44is essentially the purity theorem in [52, Th.2.9]. This includes cases studied by Faltings using the technique of normalized length.

16.9. Perfectoid Tate rings and perfectoid Abhyankar’s lemma. In this section we introduce some special classes of perfectoid quasi-affinoid rings that will play an important role in the proof of Hochster’s direct summand conjecture. The main results are theorems 16.9.17and 16.9.42, that generalize important results from Y.André’s papers [5] and [6]. The following definition of perfectoid Tate ring is equivalent to the one found in [70].

Definition 16.9.1. Let (A, 𝒯_{A}) be a Tate ring (see definition 8.3.8(vi)).

(i) We say that A is a perfectoid Tate ring (resp. formally perfectoid Tate ring) if it has a subring of definition A_{0} that is perfectoid (resp. formally perfectoid), for the topology induced by 𝒯_{A} via the inclusion map A_{0} → A.

(ii) We say that A is uniform, if A^{∘} is a bounded subset of A (see definition 8.3.8(ii)). We let

Tate and u.Tate

be the full subcategories of ℤ-TopAlg (see definition 8.3.1(iii)) whose objects are the Tate rings and respectively the uniform Tate rings.

Lemma 16.9.2. (i) Let (A, 𝒯_{A}) be a Tate ring. The following conditions are equivalent : (a) A is uniform.

(b) The completion A^{∧} of A is uniform.

(c) 𝒯_{A} agrees with the topology given by a power-multiplicative semi-norm ∥ · ∥ : A → ℝ.

(ii) Moreover, if the power-multiplicative semi-norm ∥ · ∥ defines the topology 𝒯_{A}, we have :

A^{∘} = {a ∈ A | ∥a∥ ≤ 1}  and  A^{∘∘} = {a ∈ A | ∥a∥ < 1}.

(iii) The inclusion functor u.Tate → Tate admits a left adjoint

u : Tate → u.Tate A ↦ u(A).

We call u(A) the uniformization of the Tate ring A.

Proof. (i.a)⇒(i.b): Indeed, if A^{∘} is a bounded subset of A, then its image B in A^{∧} is bounded in the latter ring; but then the same holds for the topological closure B^{c} of B in A^{∧} (remark 8.3.9(iii)). However, B^{c} = (A^{∧})^{∘} (corollary 8.4.15(ii) and proposition 8.2.13(iii)) whence the contention.

(i.b)⇒(i.a): Let j : A → A^{∧} be the completion map; we have j^{−1}((A^{∧})^{∘}) = A^{∘}, by corollaries 8.4.15(ii) and 8.2.17(ii). Since the topology of A is induced by that of A^{∧} via j, the assertion follows easily.

(i.a)⇒(i.c): Fix ρ ∈]0, 1[ and pick t ∈ A^{×} ∩ A^{∘∘}. Notice that A^{∘} is a subring of definition of A (proposition 8.3.18(ii)). Denote by ν : A → ℤ ∪ {+∞} the order function associated with the ideal tA^{∘} of A^{∘} (see example 9.1.9(i)), and set |a| := ρ^{ν(a)} for every a ∈ A. Moreover, let ∥ · ∥ : A → ℝ be the asymptotic Samuel function of tA^{∘}, i.e. ∥a∥ := lim_{n→+∞} |a^{n}|^{1/n} for every

a ∈ A. Notice that ν(t^{k}a) = k + ν(a) for every k ∈ ℤ and a ∈ A, whence :

(16.9.3)  ∥t^{k}a∥ = ρ^{k} · ∥a∥  for every k ∈ ℤ and a ∈ A.

By construction, ∥a∥ ≤ 1 for every a ∈ A^{∘}. On the other hand, if ∥a∥ > r > 1, then there exists k ∈ ℕ such that |a^{n}| > r^{n} for every n ≥ k, and it follows easily that a ∉ A^{∘}. Combining with (16.9.3), we conclude that t^{k}A^{∘} = {a ∈ A | ∥a∥ ≤ ρ^{k}} for every k ∈ ℕ, whence (i.c).

(i.c)⇒(i.a): Clearly {a ∈ A | ∥a∥ < 1} = A^{∘∘}. Pick t ∈ A^{×} ∩ A^{∘∘}, so that r := ∥t∥ ∈]0, 1[; then t^{k}A^{∘∘} is an open ideal of A^{∘} for every k ∈ ℕ, and it lies in the subset {a ∈ A | ∥a∥ < r^{k}}, whence (i.a).

(ii) follows by simple inspection of the definitions.

(iii): Let A_{0} ⊂ A be a subring of definition, t ∈ A^{×} ∩ A^{∘∘}, and ν : A → ℤ ∪ {+∞} the order function associated with the ideal tA_{0} of A_{0}. For given ρ ∈]0, 1[, the map | · |_{A} : A → ℝ such that |a|_{A} := ρ^{ν(a)} for every a ∈ A is a semi-norm, and the topology defined by | · |_{A} agrees with 𝒯_{A}. Then, let ∥ · ∥_{A} : A → ℝ be the asymptotic Samuel function of tA_{0}. Recall that ∥ · ∥_{A} is a multiplicative semi-norm, and let 𝒯_{A}^{u} be the topology on A defined by ∥ · ∥_{A}. Lemma 9.1.8(iii) implies that the identity map of A is a continuous ring homomorphism (A, 𝒯_{A}) → (A, 𝒯_{A}^{u}); also, the discussion of example 9.1.9 shows that 𝒯_{A}^{u} depends only on the original topology 𝒯_{A} (and is independent of all auxiliary choices). Clearly A^{+} := {a ∈ A | ∥a∥_{A} ≤ 1} is an open subring of (A, 𝒯_{A}^{u}) and arguing as in the foregoing we see that (16.9.3) holds as well for the semi-norm ∥ · ∥_{A}, so that t^{k}A^{+} = {a ∈ A | ∥a∥ ≤ ρ^{k}} for every k ∈ ℤ; especially, 𝒯_{A}^{u} induces the t-adic topology on tA^{+}, and (A, 𝒯_{A}^{u})^{∘} = A^{+}, so (A, 𝒯_{A}^{u}) is a uniform Tate ring. Now, let (B, 𝒯_{B}) be any uniform Tate ring, and f : (A, 𝒯_{A}) → (B, 𝒯_{B}) a continuous ring homomorphism. By lemma 8.3.29(iv), the map f is f-adic, hence it restricts to an adic ring homomorphism f_{0} : A_{0} → B^{∘} (lemma 8.3.24(iii.a,iii.b)), and the proof of (i) shows that 𝒯_{B} agrees with the topology defined by the asymptotic Samuel function ∥ · ∥_{B} : B → ℝ of the ideal f(t) · B^{∘}. With this notation, it follows easily that f is a morphism of normed rings (A, | · |_{A}) → (B, ∥ · ∥_{B}) (see (9.1.6)), and then lemma 9.1.8(iv) shows that f is also a morphism of uniform semi-normed rings (A, ∥ · ∥_{A}) → (B, ∥ · ∥_{B}). In view of (i), we deduce that f : (A, 𝒯_{A}^{u}) → (B, 𝒯_{B}) is a continuous ring homomorphism of uniform Tate rings. Thus the sought functor is obtained by the rule : (A, 𝒯_{A}) ↦ (A, 𝒯_{A}^{u}). □

Example 16.9.4. Let A be a Tate ring, A_{0} ⊂ A a subring of definition, and suppose there exists an integer n > 1 such that {x ∈ A | x^{n} ∈ A_{0}} ⊂ A_{0}. Then it follows easily that A^{∘∘} ⊂ A_{0}, and since A^{∘∘} is an ideal of A^{∘}, we deduce that A is uniform.

16.9.5. Let (A, 𝒯_{A}) be a perfectoid Tate ring, and A_{0} ⊂ A a perfectoid ring of definition, so that A = A_{0}[t^{−1}], for any t ∈ A^{×} ∩ A_{0}^{∘∘}; moreover, tA_{0} is an ideal of adic definition for A_{0} (corollary 8.3.20(iv)). Let also A^{+} be the integral closure of A_{0} in A; it follows that U := Spec A is the analytic locus of Spec A_{0}, and

\underline{U} := (U, 𝒯_{A}, A^{+})

is a perfectoid affinoid scheme (see definition 16.5.1(i)). Then, according to (16.5.20), theorem 16.5.13(ii) and corollary 16.5.43(i), we get a perfectoid affinoid scheme

(U_{E}, 𝒯_{E}, E_{U}^{+}) := E(\underline{U}) with E_{U} := 𝒪_{U_{E}}(U_{E})

so that E_{U} is a complete f-adic ring, and its open subring E_{U}^{+} is perfectoid and naturally isomorphic to E(A^{+}). Let (α_{n} | n ∈ ℕ) be a distinguished element in Ker u_{A^{∘}}; then the isomorphism of topological rings E_{U}^{∘}/α_{0}E_{U}^{∘} →^{∼} A^{∘}/pA^{∘} of remark 16.3.7(ii) induces an isomorphism

\overline{ω} : E_{U}^{∘}/E_{U}^{∘∘} →^{∼} A^{∘}/A^{∘∘}.

Recall also that the composition E_{U}^{∘} → A of \overline{u}_{A^{∘}} : E_{U}^{∘} → A^{∘} with the localization A^{∘} → A, factors through a continuous morphism of monoids

φ_{U}^{♭} : E_{U} → A

(lemma 16.4.29(i) and proposition 16.4.34(i)). Let π_{A} : A^{∘} → A^{∘}/A^{∘∘} and π_{E} : E_{U}^{∘} → E_{U}^{∘}/E_{U}^{∘∘} be the projections; denote by 𝒟_{A} and 𝒟_{E} the set of perfectoid subrings of adic definition of A and respectively E_{U}, and by 𝒫_{A} the set of perfect subrings of A^{∘}/A^{∘∘}.

Lemma 16.9.6. In the situation of (16.9.5), the following holds :

(i) (A, 𝒯_{A}) is a uniform Tate ring.

(ii) (E_{U}, 𝒯_{E}) is a perfectoid Tate ring, and U_{E} is the analytic locus of Spec E_{U}^{∘}.

(iii) The rules : B ↦ π_{A}^{−1}B and B ↦ (\overline{ω} ∘ π_{E})^{−1}B establish natural bijections

𝒟_{E} ←^{∼} 𝒫_{A} →^{∼} 𝒟_{A}.

Proof. (i) follows immediately from theorem 16.5.13(i).

(ii): It is easily seen that tA^{∘} is an ideal of adic definition for A^{∘}, and A^{∘}[t^{−1}] = A, hence U is the analytic locus of Spec A^{∘}, and therefore U_{E} is the analytic locus of Spec E_{U}^{∘}. Pick m ∈ ℕ such that p^{m}/t ∈ A^{∘}; by applying corollary 16.6.27 to the quasi-affinoid ring (A^{∘}, A^{∘}, U), we deduce that there exists t′ ∈ E_{U}^{∘} such that

v(t − \overline{u}_{A^{∘}}(t′)) ≤ v(p) · max(v(\overline{u}_{A^{∘}}(t′)), v(p^{m})) for every v ∈ Cont(A^{∘})

where \overline{u}_{A^{∘}} : E_{U}^{∘} → A^{∘} is the natural morphism of topological monoids as in (9.4). It follows that v(t) = v(\overline{u}_{A^{∘}}(t′)) for every v ∈ Cont(A^{∘}). We have v(t) < 1 for every v ∈ Cont(A^{∘}) and v(t) ≠ 0 for every v ∈ Cont(A), since t ∈ A^{∘∘} ∩ A^{×} (theorem 15.3.15(i)); hence w(t′) < 1 for every w ∈ Cont(E_{U}^{∘}) (theorem 16.5.52(i)), and therefore t′ ∈ E_{U}^{∘∘} (corollary 15.4.27(ii)). Also, combining with (16.5.59) we deduce that w(t′) ≠ 0 for every w ∈ Spa E(\underline{U}), hence t′ ∈ E_{U}^{×} (proposition 15.4.30).

(iii): For every A′ ∈ 𝒟_{A}, the inclusion map A′ → A^{∘} induces an open injective continuous ring homomorphism E′ := E(A′) → E_{U}^{∘}, and moreover we have A^{∘∘} ⊂ A′ and E_{U}^{∘∘} ⊂ E′, by theorem 16.3.42(i,iii.a). As in (16.9.5), we deduce a ring isomorphism E′/E_{U}^{∘∘} →^{∼} A′/A^{∘∘}, which shows that A′/A^{∘∘} ∈ 𝒫_{A}. Conversely, let B ∈ 𝒫_{A}; according to proposition 16.3.25, the open subring π_{A}^{−1}B = B ×_{A^{∘}/A^{∘∘}} A^{∘} of A^{∘} is perfectoid for the topology induced by the inclusion into A^{∘}, and then π_{A}^{−1}B ∈ 𝒟_{A}. Likewise one proves that the map 𝒫_{A} → 𝒟_{E} is a bijection. □

16.9.7. Let now (A, 𝒯_{A}) be a perfectoid Tate ring, A_{0} ⊂ A a perfectoid subring of definition, (B, 𝒯_{B}) an f-adic ring, and f : A → B a continuous ring homomorphism. Define the perfectoid affinoid schemes \underline{U} and (U_{E}, 𝒯_{E}, E_{U}^{+}) and the f-adic ring E_{U} as in (16.9.5); by lemma 16.9.6(ii) we find t′ ∈ E_{U}^{×}∩E_{U}^{∘∘}, and we let t := \overline{u}_{A^{∘}}(t′), which is a topologically nilpotent unit of A. Then more generally, the element t^{γ} := φ_{U}^{♭}(t^{′γ}) ∈ A is well defined for every γ ∈ ℤ[1/p]. Notice that s := f(t) is a topologically nilpotent unit of B, hence B is a Tate ring; especially, the nonanalytic loci of both Spec A and Spec B are empty, so that f is an f-adic ring homomorphism (lemma 8.3.29(iv)). By lemma 8.3.24(iii.c) there exists a subring of definition B_{0} ⊂ B such that f restricts to a continuous ring homomorphism f_{0} : A_{0} → B_{0}. Then sB_{0} is an ideal of adic definition of B_{0}, and B = B_{0}[s^{−1}] (corollary 8.3.20(ii.b)). Let s^{γ} := f(t^{γ}) for every γ ∈ ℤ[1/p]. We fix ρ ∈]0, 1[ and consider the map

| · |_{B} : B → ℝ ∪ {+∞} b ↦ inf{ρ^{γ} | b ∈ s^{γ}B_{0}}

(cp. (16.5.23)). Let also ∥ · ∥_{B} : B → ℝ ∪ {+∞} be the asymptotic Samuel function of sB_{0}, as defined in example 9.1.9(i). Recall furthermore that A^{∘} is perfectoid for the topology induced by A (theorem 16.5.13(i)), and the pair (A^{∘}, A^{∘∘}) is a basic setup verifying condition (B) of [75, §2.1.6] (proposition 16.8.15(i)). Hence, the (A^{∘}, A^{∘∘})^{a}-algebra B^{a} is well defined, and in view of proposition 16.8.15(ii) we see that :

B_{∗}^{a} = Hom_{A^{∘}}(A^{∘∘}, B) = Hom_{A}(A ⊗_{A^{∘}} A^{∘∘}, B) = Hom_{A}(A, B) = B.

Lemma 16.9.8. (i) For every b ∈ B we have : lim_{n→+∞} |b^{n}|_{B}^{1/n} = ∥b∥_{B}.

(ii) ∥s^{γ}b∥_{B} = ρ^{γ} · ∥b∥_{B} for every γ ∈ ℤ[1/p] and every b ∈ B.

(iii) B^{∘∘} = {b ∈ B | ∥b∥_{B} < 1}.

(iv) (B^{∘})_{∗}^{a} = {b ∈ B | ∥b∥_{B} ≤ 1}.

Proof. (i): Recall that ∥b∥_{B} := lim_{n→+∞} ρ^{ν(b^{n})/n}, where ν(b) := inf{k ∈ ℤ | b ∈ s^{k}B_{0}}. For every

b ∈ B, set μ(b) := sup{γ ∈ ℤ[1/p] | b ∈ s^{γ}B_{0}}, so that |b|_{B} = ρ^{μ(b)}. Clearly we have

ν(b^{n}) < μ(b^{n}) < ν(b^{n}) + 1 for every b ∈ B and n ∈ ℕ.

It follows that lim_{n→+∞}(|b^{n}|_{B}/ρ^{ν(b^{n})})^{1/n} = 1, whence the contention.

(ii): It suffices to observe that |s^{γ}b|_{B} = ρ^{γ} · |b|_{B} for every such b and γ, and apply (i).

(iii): It suffices to remark that ∥b∥_{B} < 1 if and only if x^{n} ∈ sB_{0} for some n ∈ ℕ.

(iv): Notice that (B^{∘})_{∗}^{a} ⊂ B_{∗}^{a} = B. On the other hand, we have (B^{∘})_{∗}^{a} = Hom_{A^{∘}}(A^{∘∘}, B^{∘}). It then follows that (B^{∘})_{∗}^{a} = {b ∈ B | bA^{∘∘} ⊂ B^{∘}} = {b ∈ B | bA^{∘∘} ⊂ B^{∘∘}}. Especially, s^{γ}b ∈ B^{∘∘} for every b ∈ (B^{∘})_{∗}^{a} and γ ∈ ℕ[1/p] \ {0}, whence ρ^{γ} · ∥b∥_{B} < 1 for every such γ, in view of (ii) and (iii); i.e. ∥b∥_{B} ≤ 1. Conversely, if ∥b∥_{B} ≤ 1, we get ∥f(a)b∥_{B} ≤ ∥b∥_{B} ·∥f(a)∥_{B} < 1 for every a ∈ A^{∘∘}, due to (iii); then f(a)b ∈ B^{∘∘} for every such a, again due to (iii), and finally, b ∈ (B^{∘})_{∗}^{a}. □

16.9.9. Keep the situation of (16.9.7), and set C := (B^{∘})_{∗}^{a}; lemma 16.9.8(ii,iv) implies that

s^{γ}C = {b ∈ B | ∥b∥_{B} ≤ ρ^{γ}} for every γ ∈ ℤ[1/p].

In light of lemma 16.9.2(iii), we may now conclude that C is the subring of power bounded elements of the uniformization u(B) of (B, 𝒯_{B}), and the topology of u(B) is the f-adic topology such that C ⊂ B is a subring of definition, and sC is an ideal of adic definition. Notice that f : (A, 𝒯_{A}) → u(B) is still an f-adic ring homomorphism. Moreover, from lemmata 16.9.2(ii) and 16.9.8(iii) we see that

(16.9.10)  u(B)^{∘∘} = B^{∘∘}.

Definition 16.9.11. (i) With the notation of (16.9.7), we say that B is a pre-perfectoid Aalgebra, if u(B) is a formally perfectoid Tate ring.

Remark 16.9.12. (i) With the notation of (16.9.9), endow C with its sC-adic topology 𝒯_{C}; from corollary 8.4.15(ii) and theorem 16.5.13(i) we see that B is a pre-perfectoid A-algebra if and only if (C, 𝒯_{C})^{∧} is a perfectoid ring.

(ii) Let D ⊂ C be any subring that is p-root closed in C (see definition 9.8.24(i)). From (16.9.10) we see that :

D is open in (B, 𝒯_{B}) ⇔ B^{∘∘} ⊂ D ⇔ u(B)^{∘∘} ⊂ D ⇔ D is open in u(B) .

Suppose then that D is open in (B, 𝒯_{B}), and denote by 𝒯_{D} the topology of D induced by 𝒯_{C}; then we claim that B is a pre-perfectoid A-algebra if and only if (D, 𝒯_{D})^{∧} perfectoid. Indeed, suppose that B is a pre-perfectoid A-algebra; since B^{∘∘} ⊂ D, we see that D/B^{∘∘} is a subring of C/B^{∘∘}, and in view of (16.9.10) we get natural identifications :

C/B^{∘∘} = C/u(B)^{∘∘} = \overline{C} := (C, 𝒯_{C})^{∧}/(u(B)^{∧})^{∘∘}.

Then, (i) implies that C/B^{∘∘} is a perfect ring, and D/B^{∘∘} is a perfect subring, since D is proot closed in C. In light of (i), lemma 16.9.6(iii) and proposition 8.2.13(i,ii), we conclude that (D, 𝒯_{D})^{∧} = D/B^{∘∘} ×\overline{C} (C, 𝒯_{C})^{∧} is a perfectoid subring of definition of u(B)^{∧}. Conversely, suppose that (D, 𝒯_{D})^{∧} is perfectoid; we know already that D is open in u(B), and then (D, 𝒯_{D})^{∧} is open in u(B)^{∧}, and its topology is induced by the inclusion into u(B)^{∧} (proposition 8.2.13(i,ii)); hence, B is a pre-perfectoid A-algebra.

(iii) Lastly, let D ⊂ B be an open subring, endow D with the topology induced by B, and suppose that D is formally perfectoid (definition 16.7.19); then D is p-root closed in B. Indeed, under these assumptions, the topology of D^{∧} is adic, so D^{∧} is a perfectoid subring of definition of B^{∧}, hence (B^{∧})^{∘∘} ⊂ D^{∧}, by virtue of lemma 16.9.6(iii); especially, s^{γ} ∈ D for every γ ∈ ℕ[1/p], and s^{−γ}p ∈ D for every sufficiently small γ ∈ ℕ[1/p] (corollary 8.2.17(ii)). It then suffices to invoke the criterion of theorem 16.4.1, together with lemma 9.8.26(i).

Example 16.9.13. Let us return to the situation of (16.6.31) : we consider a quasi-affinoid perfectoid ring \underline{A} := (A, A^{+}, U_{A}) and a sequence f_{•} := (f_{0}, …, f_{n}) of elements of A_{U} := 𝒪_{U_{A}}(U_{A}) that generates an open ideal (for the f-adic topology on A_{U} as in lemma 16.4.32(i)).

(i) Let (E, E^{+}, U_{E}) := E(\underline{A}) (notation of (16.5.20)), and E_{U} := 𝒪_{U_{E}}(U_{E}). As recalled in (16.9.5), the map \overline{u}_{A} : E → A extends to a continuous morphism of topological monoids φ_{U}^{♭} : E_{U} → A_{U}; we may then find m ∈ ℕ and e_{0}, …, e_{n} ∈ E_{U} verifying the inequalities (16.6.33) and (16.6.34) for every i = 1, …, n. Let also A_{0} ⊂ A_{U} be any subring of definition, and I_{0} ⊂ A_{0} an ideal of adic definition; recall that the I_{0}-adic completion B(f_{•})^{∧} of B(f_{•}) := A_{0}[f_{1}/f_{0}, …, f_{n}/f_{0}] ⊂ A_{U}[1/f_{0}] is a subring of definition of 𝒪_{Spa \underline{A}}^{∧}(S(f_{•})), with S(f_{•}) := R_{A_{U}} (\frac{f_{1}}{f_{0}} , ⋯, \frac{f_{n}}{f_{0}} ) ∩ Spa \underline{A}. Moreover, the sequence φ_{U}^{♭}(e_{0}), …, φ_{U}^{♭}(e_{n}) generates an open ideal of A_{U} as well, according to proposition 16.6.35(ii) and claim 16.6.36, and S(f_{•}) = R_{A_{U}} (\frac{φ_{U}^{♭} (e_{1})}{φ_{U}^{♭} (e_{0})}, ⋯, \frac{φ_{U}^{♭} (e_{n})}{φ_{U}^{♭} (e_{0})}) ∩ Spa \underline{A}, by proposition 16.6.35(iii). It follows that the I_{0}-

adic completion B(e_{•})^{∧} of B(e_{•}) := A_{0}[\frac{φ_{U}^{♭} (e_{1})}{φ_{U}^{♭} (e_{0})}, …, \frac{φ_{U}^{♭} (e_{n})}{φ_{U}^{♭} (e_{0})}] is another subring of definition of 𝒪_{Spa \underline{A}}^{∧}(S(f_{•})). Therefore, the p-root closures D(f_{•}) and D(e_{•}) of B(f_{•})^{∧} and B(e_{•})^{∧} in 𝒪_{Spa \underline{A}}^{∧}(S(f_{•})) both contain 𝒪_{Spa \underline{A}}^{∧}(S(f_{•}))^{∘∘}. Combining with proposition 16.6.35(iv), we conclude that D(f_{•}) = D(e_{•}).

(ii) In the situation of (i), notice that if f_{0}, …, f_{n} ∈ A, we may take e_{0}, …, e_{n} ∈ E, and by construction we shall then have f_{i} − \overline{u}_{A}(e_{i}) ∈ pA for i = 0, …, n.

Theorem 16.9.14. In the situation of (16.9.7), pick γ ∈ ℕ[1/p] \ {0} with pB_{0} ⊂ s^{pγ}B_{0}, and suppose that the Frobenius endomorphism of B_{0}/s^{pγ}B_{0} is surjective. Then B is pre-perfectoid.

Proof. We exhibit an increasing chain of subrings

B_{0} ⊂ B_{1} ⊂ B_{2} ⊂ ⋯

of the p-root closure of B_{0} in B, verifying the following conditions for every i ∈ ℕ : (a) The Frobenius endomorphism Φ_{i} of B_{i}/s^{pγ}B_{i} is surjective.

(b) Let \overline{Φ}_{i} : B_{i}/s^{γ}B_{i} → B_{i}/s^{pγ}B_{i} be the ring homomorphism induced by Φ_{i}. Then the

kernel of the induced map j_{i} : B_{i}/s^{γ}B_{i} → B_{i+1}/s^{γ}B_{i+1} contains Ker \overline{Φ}_{i}.

To this aim, we argue by induction on i ∈ ℕ. Hence, suppose that for a given i ∈ ℕ we have already exhibited a subring B_{i} fulfilling condition (a); we need to exhibit B_{i+1} so that condition (b) holds as well for B_{i}, and moreover such that condition (a) holds for B_{i+1}. Now, since (a) holds for B_{i}, for every x ∈ Ker \overline{Φ}_{i} we may find a sequence (a(x, n) | n ∈ ℕ) of elements of B_{i}, such that the image of a(x, 0) in B_{i}/s^{γ}B_{i} agrees with x, and :

a(x, n + 1)^{p} − a(x, n) ∈ s^{pγ}B_{i} for every n ∈ ℕ.

It follows easily that a(x, n)^{p^{n+1}} ∈ s^{pγ}B_{i}, and therefore b(x, n) := a(x, n) · s^{−γ/p^{n}} lies in the p-root closure of B_{i} in B, for every n ∈ ℕ. Let us then set

B_{i+1} := B_{i}[b(x, n) | (x, n) ∈ Ker(\overline{Φ}_{i}) × ℕ] ⊂ B.

Since the image of b(x, 0) · s^{γ} in B_{i}/s^{γ}B_{i} agrees with the image of x, we see that j_{i}(x) = 0 for every x ∈ Ker \overline{Φ}_{i}, hence condition (b) holds for B_{i}. Lastly, notice that

b(x, n + 1)^{p} − b(x, n) ∈ B_{i} for every n ∈ ℕ.

Since B_{i}/s^{pγ}B_{i} ⊂ Im \overline{Φ}_{i+1}, it follows easily that condition (a) holds for B_{i+1}, as required.

Set B_{∞} := ⋃_{i∈ℕ} B_{i} = B_{∞}; by construction, the Frobenius endomorphism of B_{∞} induces an isomorphism B_{∞}/s^{γ}B_{∞} →^{∼} B_{∞}/s^{pγ}B_{∞}, hence B_{∞} is the p-root closure of B_{0} in B, by lemma 9.8.26(i). Next, let B_{∞}^{∧} be the completion of B_{∞} for its s-adic topology; the Frobenius endomorphism of B_{∞}^{∧} induces an isomorphism B_{∞}^{∧}/s^{γ}B_{∞}^{∧} →^{∼} B_{∞}^{∧}/s^{pγ}B_{∞}^{∧}. Clearly B_{∞}^{∧} is a Pring; by theorem 16.4.1, the ring B_{∞}^{∧} is then perfectoid, and the assertion follows from remark 16.9.12(ii). □

Corollary 16.9.15. In the situation of (16.9.7), let I ⊂ B be any ideal, and endow \overline{B} := B/I with the f-adic topology 𝒯\overline{B} induced by 𝒯_{B} via the projection B → \overline{B} (see example 8.3.27(iii)). Then, if B is a pre-perfectoid A-algebra, the same holds for \overline{B}.

Proof. Let 𝒯_{B}^{u} be the topology of u(B), and denote by 𝒯 u ; \overline{B} the f-adic topology induced by 𝒯_{B}^{u} via the projection B → \overline{B}. If C is any uniform Tate ring, and f : (\overline{B}, 𝒯_{B}) → C any continuous ring homomorphism, it is easily seen that f factors through the identity map 1\overline{B} : (\overline{B}, 𝒯_{B}) → (\overline{B}, 𝒯 u ; \overline{B}), which is continuous, and the continuous ring homomorphism f : (\overline{B}, 𝒯 u ; \overline{B}) → C. By the universal property of uniformization, this implies that 1\overline{B} induces an isomorphism of uniform Tate rings

u(\overline{B}, 𝒯_{B}) →^{∼} u(\overline{B}, 𝒯 u ; \overline{B})

(whose underlying ring homomorphism is of course again the identity map of \overline{B}). Thus, it suffices to check that (\overline{B}, 𝒯 u ; \overline{B}) is pre-perfectoid. However, since B is pre-perfectoid, C := u(B)^{∘} is a perfectoid ring, hence for every γ ∈ ℕ[1/p]\{0} such that pC ⊂ s^{pγ}C, the Frobenius endomorphism of C/s^{pγ}C is surjective. By construction, the image \overline{C} of C in \overline{B} is a subring of definition for the topology 𝒯 u ; \overline{B}, and it follows that the Frobenius endomorphism of \overline{C}/s^{pγ}\overline{C} is again surjective; then the assertion follows from theorem 16.9.14. □

16.9.16. Let A be a formally perfectoid Tate ring, A_{0} ⊂ A a formally perfectoid subring of definition, t ∈ A_{0}^{∘∘} ∩ A^{×}, and P ∈ A_{0}[X] \ A_{0} a monic polynomial. We let \overline{P} be the image of P in \overline{A}_{0} := A_{0}/A_{0}^{∘∘}, we define B_{0} := A_{0}[X^{1/p^{∞}}] := ⋃_{n∈ℕ} A_{0}[X^{1/p^{n}}], B := A ⊗_{A_{0}} B_{0}, and set

C_{0,n} := A_{0}[X^{1/p^{n}}]/(P) for every n ∈ ℕ C_{0} := B_{0}/P B_{0} and C := B/P B.

Clearly the natural map of A_{0}-algebras C_{0,m} → C_{0,n} is injective for every n, m ∈ ℕ with n ≥ m, and C_{0,n} is a free A_{0}-module of rank p^{n} · deg_{X}(P) for every such n. Hence C_{0} = ⋃_{n∈ℕ} C_{0,n} is a faithfully flat A_{0}-algebra, and the localization C_{0} → C is injective. Lastly, let D ⊂ C be the p-root closure of C_{0} in C (see definition 9.8.24(ii)), and endow D with its t-adic topology.

The following result and its proof are a generalization and refinement of [5, Th.2.5.2] (modulo powers of p, as in [5, Rem.2.6.1(4)]); see also [23, Th.2.3]. We also borrowed from a set of lecture notes of B.Bhatt the idea of stating the theorem for a general monic polynomial P : see [24, Th.9.4.3] (whereas André’s original theorem would be the special case with P = X − a, for any a ∈ A_{0}).

Theorem 16.9.17. (i) In the situation of (16.9.16), D is a faithfully flat A_{0}-algebra. (ii) Moreover, D is a formally perfectoid ring for its t-adic topology. (iii) Furthermore, there exists an isomorphism of \overline{A}_{0}-algebras (notation of (9.8.11)) :

(\overline{A}_{0}[X]/(\overline{P}))^{perf} →^{∼} \overline{A}_{0} ⊗_{A_{0}} D.

Proof. Clearly D[1/t] = C is a faithfully flat A-algebra, and the image of t is regular in D; by virtue of [75, Lemma 5.2.1], in order to show (i), it then suffices to check that D/tD is a faithfully flat A_{0}/tA_{0}-algebra. Recall that the completion A_{0}^{∧} is a subring of definition for the completion A^{∧} of A (proposition 8.3.33(ii)); set C′_{0} := A_{0}^{∧} ⊗_{A_{0}} C_{0}, C′ := A^{∧} ⊗_{A} C, and let D′ be the p-root closure of C′_{0} in C′. Endow also C_{0} (resp. C′_{0}) with its t-adic topology, and C (resp. C′) with the unique f-adic topology such that C_{0} (resp. C′_{0}) is a subring of definition; let 𝒯_{D} (resp. 𝒯_{D′}) be the topology induced by C on D (resp. by C′ on D′). Taking into account proposition 8.3.33, it is easily seen that the natural maps C_{0} → C′_{0} and C → C′ induce isomorphisms on the respective completions; by virtue of lemma 9.8.26(ii), the same then holds for the induced continuous ring homomorphism (D, 𝒯_{D}) → (D′, 𝒯_{D′}).

Claim 16.9.18. Let D^{∧} be the completion of (D, 𝒯_{D}). Then the completion map u : D → D^{∧} induces an isomorphism D/t^{k}D →^{∼} D^{∧}/t^{k}D^{∧} for every k ∈ ℕ.

Proof of the claim. Let also C_{0}^{∧} be the completion of C_{0}; since t^{k}C_{0} is an open A_{0}-submodule of D, the map u induces an isomorphism of A_{0}-modules \overline{u} : D/t^{k}C_{0} →^{∼} D^{∧}/t^{k}C_{0}^{∧}. Hence, \overline{u}(t^{k}D/t^{k}C_{0}) = t^{k} · \overline{u}(D/t^{k}C_{0}) = t^{k}D^{∧}/t^{k}C_{0}^{∧}, whence the assertion. ♢

By applying claim 16.9.18 to both D^{∧} and the completion D^{′∧} of (D′, 𝒯_{D′}), we reduce to checking that D′/tD′ is a faithfully flat A_{0}/tA_{0}-algebra, and that D′ is formally perfectoid for its t-adic topology. Thus, we may replace A by A^{∧}, and assume that A is a perfectoid Tate ring.

(ii): By construction, the Frobenius endomorphism of C_{0} is a surjection; hence C is a preperfectoid A-algebra, by theorem 16.9.14. Then the assertion follows from remark 16.9.12(ii).

(i): Define the affinoid schemes \underline{U} := (U, 𝒯_{A}, A^{+}) and \underline{U}_{E} := (U_{E}, 𝒯_{E}, E_{U}^{+}) as in (16.9.5); recall that E_{0} := E(A_{0}) is a subring of definition of the Tate perfectoid ring E_{U} := 𝒪_{U_{E}}(U_{E}) (lemma 16.9.6(ii,iii)). We may also assume that t := \overline{u}_{A_{0}}(t′) ∈ A for some t′ ∈ E_{U}^{×} ∩ E_{0}^{∘∘}.

We endow B_{0} with its t-adic topology, and B with the f-adic topology such that B_{0} is a subring of definition. We have then a short exact sequence of flat A_{0}-modules

Σ  :  0 → P B_{0} → B_{0} ⟶^{π_{0}} C_{0} → 0.

As in (16.9.7), we let t^{γ} := φ_{U}^{♭}(t^{′γ}) ∈ A for every γ ∈ ℤ[1/p]. With this notation, the sequence A_{0}/t^{γ}A_{0}⊗_{A_{0}}Σ is still short exact, for every γ ∈ ℕ[1/p], which means that t^{γ}B_{0}∩P B_{0} = t^{γ}P B_{0} for every such γ. Especially, the topology of B_{0} induces the t-adic topology on the ideal P B_{0}, and after taking completions, we still get a short exact sequence :

Σ^{∧}  :  0 → P B_{0}^{∧} → B_{0}^{∧} ⟶^{π_{0}^{∧}} C_{0}^{∧} → 0.

Recall that the completion C^{∧} of C is naturally identified with C_{0}^{∧}[t^{−1}], and C_{0}^{∧} is a subring of definition for C^{∧} (proposition 8.3.33(ii,iii)). Likewise, B_{0}^{∧} is a subring of definition of B^{∧} = B_{0}^{∧}[t^{−1}]. We consider the affinoid ring \underline{B}^{∧} := (B^{∧}, B^{∧∘}), and the rational subset

R_{n} := R_{B^{∧}}(\frac{P}{t^{n}} ) ∩ Spa \underline{B}^{∧} for every n ∈ ℕ.

Endow the subring B_{n} := B_{0}^{∧}[P/t^{n}] ⊂ B^{∧} with its t-adic topology, and let 𝒯_{n} be the f-adic topology on B^{∧} such that B_{n} is a subring of definition; then the completion B_{n}^{∧} of B_{n} is a subring of definition of 𝒪_{Spa \underline{B}^{∧}}^{∧}(R_{n}) = (B^{∧}, 𝒯_{n})^{∧}, for every n ∈ ℕ. Notice that the projection π_{0}^{∧} extends to a continuous open ring homomorphism π^{∧} : B^{∧} → C^{∧}, and for every n ∈ ℕ, the image of the induced continuous map

Spa π^{∧} : Spa(C^{∧}, C^{∧∘}) → Spa \underline{B}^{∧}

lies in R_{n}. It follows that π^{∧} factors uniquely through a continuous ring homomorphism

ψ_{n} : 𝒪_{Spa \underline{B}^{∧}}^{∧}(R_{n}) → C^{∧} for every n ∈ ℕ

(see remark 15.5.9(i)). Moreover, we have R_{n+1} ⊂ R_{n} for every n ∈ ℕ, and clearly the composition of ψ_{n+1} with the restriction map 𝒪_{Spa \underline{B}^{∧}}^{∧}(R_{n}) → 𝒪_{Spa \underline{B}^{∧}}^{∧}(R_{n+1}) agrees with ψ_{n}. Furthermore, a direct inspection shows that ψ_{n}(B_{n}^{∧}) = C_{0}^{∧} for every n ∈ ℕ. Summing up, we deduce a commutative diagram of rings :

(16.9.19)  \xymatrix{B′ := colim_{n∈ℕ} B_{n}^{∧} \ar[r] \ar[d]_{α} & 𝒪 := colim_{n∈ℕ} 𝒪_{Spa \underline{B}^{∧}}^{∧}(R_{n}) \ar[d]^{β} \\ C_{0}^{∧} \ar[r] & C^{∧}}

whose vertical (resp. horizontal) arrows are surjections (resp. injections).

Claim 16.9.20. The diagram (16.9.19) is cartesian.

Proof of the claim. The assertion means that the induced map Ker α → Ker β is bijective. However, this map is clearly injective, so it remains to check its surjectivity. To this aim, notice that for every n ∈ ℕ and every g_{n} ∈ 𝒪_{Spa \underline{B}^{∧}}^{∧}(R_{n}) there exists N ∈ ℕ such that

t^{N}g_{n} = ∑_{i=0}^{∞} b_{i} · (\frac{P}{t^{n}} )^{i} with b_{i} ∈ B_{0}^{∧} for every i ∈ ℕ and lim_{i→+∞} b_{i} = 0.

With this notation, it follows easily that g_{n} ∈ Ker ψ_{n} if and only if b_{0} = 0. Suppose then that ψ_{n}(g_{n}) = 0; the image g_{n+N} of g_{n} in 𝒪_{Spa \underline{B}^{∧}}^{∧}(R_{n+N}) can be written as :

g_{n+N} = ∑_{i=1}^{∞} t^{N(i−1)}b_{i} · (\frac{P}{t^{n+N}} )^{i}

and we have t^{N(i−1)}b_{i} ∈ B_{0}^{∧} for every i ≥ 1, and lim_{i→+∞}t^{N(i−1)}b_{i} = 0. Hence g_{n+N} ∈ B_{n+N}^{∧}, and

clearly g_{n+N} ∈ Ker ψ_{n+N}, whence the contention. ♢

Let 𝒟 be the p-root closure of B′ in 𝒪, and endow 𝒟 with its t-adic topology; it follows easily from claim 16.9.20that 𝒟 = β^{−1}D. Moreover, the induced ring homomorphism β_{|𝒟} : 𝒟 → D is surjective, and its kernel is Ker β. The latter is clearly t-divisible, hence β_{|𝒟} induces an isomorphism of A_{0}-algebras

𝒟/t^{n}𝒟 →^{∼} D/t^{n}D for every n ∈ ℕ.

Thus, it suffices to check that the natural ring homomorphism A_{0} → 𝒟 is adically faithfully flat. To this aim, let also 𝒟_{n} be the p-root closure of B_{n}^{∧} in 𝒪_{Spa \underline{B}^{∧}}^{∧}(R_{n}), for every n ∈ ℕ, and endow 𝒟_{n} with its t-adic topology; in view of lemma 9.8.26(iii.b) we are reduced to checking that 𝒟_{n} is an adically faithfully flat A_{0}-algebra for every n ∈ ℕ.

Now, notice that B_{0}^{∧} is a perfectoid ring, by virtue of lemma 16.6.40, and we have a natural identification of topological rings :

E(B_{0}^{∧}) →^{∼} E_{0}[X^{1/p^{∞}}]^{∧}  such that \overline{u}_{B_{0}^{∧}}(X^{γ}) = X^{γ}  for every γ ∈ ℕ[1/p]

where E_{0}[X^{1/p^{∞}}]^{∧} denotes the t′-adic completion of E_{0}[X^{1/p^{∞}}]. Next, let (α_{n} | n ∈ ℕ) be a distinguished element in the kernel of u_{A_{0}} : W (E_{0}) → A_{0}; after replacing t′ by t^{′γ} for some sufficiently small γ ∈ ℕ[1/p] \ {0}, we may assume that α_{0}E_{0} ⊂ t′E_{0}, in which case the isomorphism ω of remark 16.3.7(ii) induces ring isomorphisms

(16.9.21)  E_{0}/t′E_{0} →^{∼} A_{0}/tA_{0}  E(B_{0}^{∧})/t′E(B_{0}^{∧}) →^{∼} B_{0}^{∧}/tB_{0}^{∧}.

On the other hand, set U_{B} := Spec B^{∧}; then (B_{0}^{∧}, B_{0}^{∧}, U_{B}) is a perfectoid quasi-affinoid ring, and by example 16.9.13we may then find e_{n} ∈ E(B_{0}^{∧}) such that the 𝒟_{n} is the p-root closure in 𝒪_{Spa \underline{B}^{∧}}^{∧}(R_{n}) of the t-adic completion of B_{0}^{∧}[\overline{u}_{B_{0}^{∧}}(e_{n})/t^{n}], and moreover – after replacing again t′ by t^{′γ} for some small γ ∈ ℕ[1/p] \ \{0\} – we may assume that P − \overline{u}_{B_{0}^{∧}}(e_{n}) ∈ tB_{0}^{∧}. Hence, there exist a monic polynomial Q_{n} ∈ E_{0}[X] and g_{n} ∈ E_{0}[X^{1/p^{∞}}]^{∧} with

(16.9.22)  e_{n} = Q_{n} + t′g_{n}.

According to proposition 16.7.25(iii), the subring

D_{n} := B_{0}^{∧}[\overline{u}_{B_{0}^{∧}}(e_{n}^{1/p^{k}})/t^{n/p^{k}} | k ∈ ℕ] ⊂ B^{∧}

is formally perfectoid for the topology 𝒯′_{n} induced by 𝒯_{n}. Then 𝒯′_{n} is the t-adic topology, and combining with remark 16.9.12(iii) and lemma 9.8.26(ii), we deduce that

(D_{n}, 𝒯′_{n})^{∧} = 𝒟_{n}.

Thus, we are further reduced to checking that the A_{0}-algebra D_{n,k} := B_{0}^{∧}[\overline{u}_{B_{0}^{∧}}(e_{n}^{1/p^{k}})/t^{n/p^{k}}] is adically faithfully flat for every n, k ∈ ℕ with n > 0 and n/p^{k} ≤ 1. Since t is a regular element of D_{n,k}, the local flatness criterion ([126, Th.22.3]) further reduces to showing that D_{n,k}/(t^{n/p^{k}}) is a faithfully flat A_{0}/(t^{n/p^{k}})-algebra, for every such n and k. To this aim, notice that (16.9.21) induces an isomorphism E_{n,k} := E_{0}/(t^{′n/p^{k}}) →^{∼} A_{0}/(t^{n/p^{k}}), and B_{0}^{∧}/(t^{n/p^{k}}) is isomorphic to the A_{0}-algebra E_{n,k}[X^{1/p^{∞}}] (as n/p^{k} ≤ 1). Also, the image of \overline{u}_{B_{0}^{∧}}(e_{n}^{1/p^{k}}) in E_{n,k}[X^{1/p^{∞}}] is the regular element Q_{n}^{1/p^{k}}, due to (16.9.22). Thus, the sequence (t^{n/p^{k}}, \overline{u}_{B_{0}^{∧}}(e_{n}^{1/p^{k}})) is regular in the ring B_{0}^{∧}; in particular, this sequence is completely secant (proposition 7.8.7), so the natural map

B_{0}^{∧}[Y]/(\overline{u}_{B_{0}^{∧}}(e_{n}^{1/p^{k}}) − t^{n/p^{k}}Y) → D_{n,k}

is an isomorphism of A_{0}-algebras (lemma 7.8.17). We are then furter reduced to checking that for every n, k, r ∈ ℕ with n > 0, r ≥ k and n/p^{k} ≤ 1, the E_{n,k}-algebra

E_{n,k}[X^{1/p^{r}}, Y]/(Q_{n}^{1/p^{k}})

is faithfully flat. The latter follows from [65, Ch.IV, Prop.11.3.7].

(iii): The proof of (i) yields an isomorphism of \overline{A}_{0}-algebras :

\overline{A}_{0} ⊗_{A_{0}} D →^{∼} colim_{n∈ℕ} \overline{A}_{0} ⊗_{A_{0}} D_{n}

with D_{n} = B_{0}^{∧}[Y^{1/p^{∞}}]/((t^{n}Y)^{1/p^{k}} − \overline{u}_{B_{0}^{∧}}(e_{n}^{1/p^{k}}) | k ∈ ℕ) for every n ∈ ℕ, and the transition map D_{n} → D_{n+1} is the map of B_{0}^{∧}-algebras such that Y^{1/p^{k}} ↦ (tY)^{1/p^{k}} for every k, n ∈ ℕ. Set \overline{B}_{0} := \overline{A}_{0}⊗_{A_{0}} B_{0}; it follows that \overline{A}_{0}⊗_{A_{0}} D_{n} = \overline{B}_{0}[Y^{1/p^{∞}}]/(\overline{P}^{1/p^{k}} | k ∈ ℕ) for every n ∈ ℕ, and the induced map \overline{A}_{0} ⊗_{A_{0}} D_{n} → \overline{A}_{0} ⊗_{A_{0}} D_{n+1} is the map of \overline{B}_{0}-algebras such that Y^{1/p^{k}} ↦ 0 for every n, k ∈ ℕ. Therefore, we get an isomorphism \overline{A}_{0} ⊗_{A_{0}} D →^{∼} \overline{B}_{0}/(\overline{P}^{1/p^{k}} | k ∈ ℕ), whence the assertion. □

16.9.23. Let (A, a) be a pair consisting of a ring A and an element a ∈ A such that p^{n}A ⊂ aA for some n ∈ ℕ, and denote by 𝒞 (A, a) the category of all formally perfectoid A-algebras B whose topology is a-adic, and whose a-torsion is bounded, i.e. there exists k ∈ ℕ with Ann_{B}(a^{k}) = Ann_{B}(a^{k+1}). The morphisms of 𝒞 (A, a) are all the maps of A-algebras. Moreover, let P ∈ A[X] be a non-constant monic polynomial; we associate with P the functor

F_{P} : 𝒞 (A, a) → Set B ↦ {(β_{n} | n ∈ ℕ) ∈ E(B) | P (β_{0}) = 0}.

Every morphism f : B → B′ of 𝒞 (A, a) induces a map E(f) : E(B) → E(B′), and we let F_{P} (f) : F_{P} B → F_{P} B′ be the restriction of E(f).

Lemma 16.9.24. With the notation of (16.9.23), let B ∈ Ob(𝒞 (A, a)), set B_{0} := B/Ann_{B}(a), B_{1} := B/B^{∘∘}, and B_{2} := B_{0} ⊗_{B} B_{1}. Then we have :

(i) B_{1} and B_{2} are perfect 𝔽_{p}-algebras, and B_{0} is formally perfectoid for its a-adic topology. (ii) The image of a is regular in B_{0}, and B_{0} is p-root closed in B_{0}[a^{−1}]. (iii) The resulting commutative diagram of rings :

\xymatrix{B \ar[r] \ar[d] & B_{0} \ar[d] \\ B_{1} \ar[r] & B_{2}}

is cartesian (and cocartesian).

(iv) B_{0}, B_{1}, B_{2} ∈ Ob(𝒞 (A, a)), and F_{P} B = F_{P} B_{0} ×_{F_{P} B_{2}} F_{P} B_{1}.

Proof. We begin with the following observation :

Claim 16.9.25. Let M be any A-module; endow M with its a-adic topology, and denote by M^{∧} the completion of M. Then, Ann_{M^{∧}}(a^{k}) is the topological closure of the image of Ann_{M}(a^{k}) in M^{∧}, for every k ∈ ℕ.

Proof of the claim. Set N := a^{k}M, and consider the induced exact sequence :

0 → Ann_{M}(a^{k}) → M ⟶^{f} N → 0.

Endow N with its a-adic topology, and Ann_{M}(a^{k}) with the topology induced by M; after taking a-adic completions, we get the exact sequence :

0 → Ann_{M}(a^{k})^{∧} → M^{∧} ⟶^{f^{∧}} N^{∧} → 0

and Ann_{M}(a^{k})^{∧} is the topological closure in M^{∧} of the image of Ann_{M}(a^{k}) in M^{∧} (proposition 8.2.13(i,iii,v)). Moreover, N^{∧} = a^{k}M^{∧}, and under this identification, f^{∧} corresponds to the scalar multiplication by a^{k} on M^{∧} (remark 8.3.3(iv)), whence the assertion. ♢

Now, notice that Ann_{B}(a^{i}) ∩ a^{k}B = a^{k} · Ann_{B}(a^{k+i}) for every i, k ∈ ℕ; if Ann_{B}(a^{k}) = Ann_{B}(a^{k+1}), it follows that Ann_{B}(a^{i}) ∩ a^{k}B = a^{k} · Ann_{B}(a^{k}) = 0 for every i > 0. In particular, the a-adic topology of B induces the discrete topology on Ann_{B}(a^{i}), for every i ∈ ℕ, and taking into account claim 16.9.25, we see that the natural map Ann_{B}(a^{i}) → Ann_{B^{∧}}(a^{i}) is an isomorphism, for every i ∈ ℕ. Next, according to corollary 16.6.24, there exists β := (b_{k} | k ∈ ℕ) ∈ E(B^{∧}) such that a/b_{0} ∈ (B^{∧})^{×} and p ∈ b_{n}B^{∧}; then the a-adic topology of B^{∧} agrees with its b_{k}-adic topology, for every k ∈ ℕ, and by corollary 16.3.71, B_{0}^{∧} := B^{∧}/Ann_{B^{∧}}(b_{n}) is perfectoid for its b_{n}-adic topology; moreover, B^{∧} is reduced (corollary 16.3.63(i)), so Ann_{B^{∧}}(b_{n}) = Ann_{B^{∧}}(b_{0}). By claim 16.9.25 and proposition 8.2.13(i,v), B_{0}^{∧} is then naturally identified with the a-adic completion of B_{0}, i.e. B_{0} is formally perfectoid for its a-adic topology. Recall as well that (B^{∧})^{∘∘} = (B^{∘∘})^{∧}, and B^{∘∘} is a radical ideal of B (remark 8.3.10(iv) and corollary 8.4.15(ii)); since B^{∧} is perfectoid, it follows easily that B_{1} is a perfect 𝔽_{p}-algebra. Also, obviously the a-adic topology is discrete on B_{2}; in light of proposition 16.3.9(i) we deduce that B_{2} is a perfect 𝔽_{p}-algebra, and this completes the proof of (i).

(ii): The ring B^{∧} is reduced (corollary 16.3.63(i)), so Ann_{B^{∧}}(a) = Ann_{B^{∧}}(a^{k}) for every k > 0, and in view of the foregoing, we deduce that Ann_{B}(a) = Ann_{B}(a^{k}) for every k > 0, so the image of a is regular in B_{0}. Let us endow B_{0}[a^{−1}] with the f-adic topology such that B_{0} is a subring of definition, and aB_{0} ⊂ B_{0} is an ideal of adic definition; then, let C be the p-root closure of B_{0} in B_{0}[a^{−1}], endow B_{0} and C with the topologies induced by B_{0}[a^{−1}], and denote by B_{0}^{∧}, C^{∧} and B_{0}[a^{−1}]^{∧} the respective completions. Notice that the quotient topology on C/B_{0} is discrete, so the completion map C/B_{0} → (C/B_{0})^{∧} is an isomorphism; we need to check that C = B_{0}, and taking into account proposition 8.2.13(i,v), we are then reduced to showing that B_{0}^{∧} = C^{∧}. To this aim, let (b_{k} | k ∈ ℕ) be as in the proof of (i); we have already noticed that the topology of B^{∧} is b_{n+1}-adic, so the same holds for the topology of B_{0}^{∧}, and clearly B_{0}^{∧}[a^{−1}] = B_{0}^{∧}[b_{n+1}^{−1}]. On the other hand, C^{∧} is the p-root closure of B_{0}^{∧} in B_{0}[a^{−1}]^{∧} (lemma 9.8.26(ii)), and B_{0}[a^{−1}]^{∧} = B_{0}^{∧}[a^{−1}] (proposition 8.3.33(iii)), so finally C^{∧} is the p-root closure of B_{0}^{∧} in B_{0}^{∧}[b_{n+1}^{−1}]. Since B_{0}^{∧} is perfectoid, the assertion then follows from lemma 9.8.26(i) and theorem 16.4.1.

(iii): By the foregoing, we see that the natural map

Ann_{B}(a) ∩ B^{∘∘} → J := Ann_{B^{∧}}(b_{0}) ∩ (B^{∧})^{∘∘}

is injective. However J = 0 : indeed, if x ∈ J, we have x^{n} = b_{0}c for some n ∈ ℕ and some c ∈ B^{∧}, and on the other hand b_{0}x = 0, whence x^{n+1} = 0, so x = 0, as B^{∧} is reduced (corollary 16.3.63(i)). So finally Ann_{B}(a) ∩ B^{∘∘} = 0, and the assertion follows easily.

(iv): Since B_{1} and B_{2} are perfect, they are reduced, so their a-torsion is bounded. The same holds trivially for the a-torsion of B_{0}, due to (ii). This shows that B_{0}, B_{1}, B_{2} ∈ Ob(𝒞 (A, a)). The remaining identity follows easily from the definitions. □

Proposition 16.9.26. In the situation of (16.9.23), suppose that A ∈ Ob(𝒞 (A, a)). Then the functor F_{P} is represented by a faithfully flat A-algebra D_{P} ∈ Ob(𝒞 (A, a)).

Proof. Let J be the radical of the ideal aA, and I := Ann_{A}(a); set A_{1} := A/J and A_{2} := A/(I + J). By lemma 16.9.24we get a cartesian diagram of rings :

\xymatrix{A \ar[r] \ar[d] & A_{0} \ar[d] \\ A_{1} \ar[r] & A_{2}}

whose arrows are the natural projections, where A_{1} and A_{2} are perfect 𝔽_{p}-algebras, and A_{0} is formally perfectoid for its a-adic topology; moreover, a is regular in A_{0}. Let \overline{P} denote the image of P in A_{2}[X]; according to theorem 16.9.17, the polynomial P acquires a root in a faithfully flat A_{0}-algebra D_{0} that is formally perfectoid for its a-adic topology, and such that we have an isomorphism A_{2} ⊗_{A_{0}} D_{0} →^{∼} D_{2} := (A_{2}[X]/(\overline{P}))^{perf} of \overline{A}_{0}-algebras. Let then also Q ∈ A_{1}[X] be the image of P, and D_{1} := (A_{1}[X]/(Q))^{perf}; we notice :

Claim 16.9.27. (i) The natural map A_{2} ⊗_{A_{1}} D_{1} → D_{2} is an isomorphism.

(ii) D_{1} is a faithfully flat A_{1}-algebra.

Proof of the claim. (i): The functor (−)^{perf} of (9.8.11) commutes with all colimits, since it is a left adjoint; in particular, the natural map A_{2}^{perf} ⊗_{A_{1}^{perf}} D_{1} → D_{2} is an isomorphism. But

A_{2} = A_{2}^{perf} and A_{1} = A_{1}^{perf}, whence the assertion.

(ii): Let j_{0} : A_{1} → D_{1,0} be the structure map, and say that Q = X^{k} + a_{1}X^{k−1} + ⋯ + a_{k} with a_{1}, …, a_{k} ∈ A_{1}; it suffices to show that the composition j_{n} : A_{1} → D_{1,n} of j_{0} with φ_{n−1} ∘ ⋯ ∘ φ_{0} : D_{1,0} → D_{1,n} is faithfully flat for every n ∈ ℕ. But since A_{1} is a perfect 𝔽_{p}-algebra, j_{n} identifies D_{1,n} with the A_{1}-algebra A_{1}[X]/(Q_{n}), where Q_{n} := X^{k} + a_{1}^{1/p^{n}}X^{k−1} + ⋯ + a_{k}^{1/p^{n}}, whence the assertion. ♢

In view of claim 16.9.27 and [75, Prop.3.4.21] the A-algebra D_{P} := D_{0} ×_{D_{2}} D_{1} is flat, and the projections D_{P} → D_{i} induce isomorphisms A_{i} ⊗_{A} D_{P} →^{∼} D_{i} of A_{i}-algebras, for i = 0, 1, 2. Hence, Ann_{D_{P}} (a^{n}) = D_{P} ⊗_{A} Ann_{A}(a^{n}) for every n ∈ ℕ, and since the a-torsion of A is bounded, the same then holds for the a-torsion of D_{P}. We endow D_{P} with the topology 𝒯 of the fibre product of the formally perfectoid ring D_{0} with the discrete topological 𝔽_{p}-algebra D_{1}, over the discrete topological 𝔽_{p}-algebra D_{2}. It is easily seen that 𝒯 coincides with the a-adic topology, and the completion (D, 𝒯 )^{∧} of (D, 𝒯 ) is the fibre product D_{0}^{∧} ×_{D_{2}} D_{1}, where D_{0}^{∧} denotes the completion of D_{0}. Hence, D_{P} is formally perfectoid, by proposition 16.3.25; this shows that D_{P} ∈ Ob(𝒞 (A, a)). In order to check that D_{P} is faithfully flat, let M be any non-zero A-module, and set M_{i} := A_{i} ⊗_{A} M for i = 0, 1; it suffices to show that D_{P} ⊗_{A} M ≠ 0 ([126, Th.7.2]), and according to [75, Lemma 3.4.18], we have either M_{0} ≠ 0 or M_{1} ≠ 0, so either D_{0} ⊗_{A} M ≠ 0 or D_{1} ⊗_{A} M ≠ 0, since D_{i} is a faithfully flat A_{i}-algebra for i = 0, 1, whence the assertion.

It remains to check that D_{P} represents the functor F_{P}. To this aim, recall first that D_{0} is the p-root closure of C_{0} := A_{0}[X^{1/p^{∞}}]/(P) in C_{0}[a^{−1}]; hence the image of δ := (X^{1/p^{n}} | n ∈ ℕ) in E(D_{0}) is an element δ_{0} ∈ F_{P} D_{0}. Likewise, the image of δ in E(D_{1}) is an element δ_{1} ∈ F_{P} D_{1}, and clearly the images of δ_{0} and δ_{1} coincide in F_{P} D_{2}, so the pair (δ_{0}, δ_{1}) is a well defined element δ_{P} ∈ F_{P} D_{P} (lemma 16.9.24(iv)). Now, let B be any object of 𝒞 (A, a), and define B_{0}, B_{1}, B_{2} as in lemma 16.9.24; we easily get a natural identification :

Hom_{A-Alg}(D_{P} , B) →^{∼} Hom_{A_{0}-Alg}(D_{0}, B_{0}) ×_{Hom_{A_{2}-Alg}(D_{2},B_{2})} Hom_{A_{1}-Alg}(D_{1}, B_{1})

(detail left to the reader). Taking into account lemma 16.9.24(iv), we are then reduced to checking that for i = 0, 1, and every β_{i} ∈ F_{P} B_{i}, there exists a unique map of A_{i}-algebras f_{i} : D_{i} → B_{i} such that F_{P} (f_{i})(δ_{i}) = β_{i}. This is clear for i = 1, since B_{1} is perfect and D_{1} is the perfection of A_{1}[X]/(Q). Lastly, say that β_{0} = (b_{n} | n ∈ ℕ), so that b_{n+1}^{p} = b_{n} for every n ∈ ℕ. Then there is a unique map of A_{0}-algebras g : A_{0}[X^{1/p^{∞}}] → B_{0} such that g(X^{1/p^{n}}) = b_{n} for every n ∈ ℕ; since P (b_{0}) = 0, the map g factors uniquely through a map h : C_{0} → B_{0}. But according to lemma 16.9.25(ii), B_{0} is p-root closed in B_{0}[a^{−1}]; in view of remark 9.8.25(ii), the map h then extends to a map f_{0} : D_{0} → B_{0}, which is necessarily the unique map whose restriction to C_{0} agrees with h, since A_{0}[a^{−1}] ⊗_{A_{0}} h = A_{0}[a^{−1}] ⊗_{A_{0}} f_{0}, and since the localization B_{0} → B_{0}[a^{−1}] is injective. □

Remark 16.9.28. (i) In the situation of proposition 16.9.26, let f : (A, a) → (A′, a′) be any morphism of 𝒞 (A, a), and denote by f(P) the image of P in A′[X]. Then we can consider as well the category 𝒞 (A′, a′), the corresponding functor F_{f(P)} : 𝒞 (A′, a) → Set, and the representing A′-algebra D_{f(P)}. Let also δ ∈ F_{P} D_{P} and δ′ ∈ F_{f(P)}D_{f(P)} be the respective universal elements. Notice that A′ ⊗_{A} D_{P} is formally perfectoid for its a′-adic topology (proposition 16.3.9(i)), and Ann_{A′⊗_{A}D_{P}} (a^{′n}) = D_{P} ⊗_{A} Ann_{A′}(a^{′n}) for every n ∈ ℕ, since D_{P} is a faithfully flat A-algebra; hence A′ ⊗_{A} D_{P} ∈ Ob(𝒞 (A′, a′)), and by comparing the respective universal properties, it follows easily that there exists a unique isomorphism of A′-algebras

ω : D_{f(P)} →^{∼} A′ ⊗_{A} D_{P} such that (F_{f(P)}ω)(δ′) = 1 ⊗ δ.

(ii) We may also consider the following variant of the situation of (16.9.23). Let A be a ring, I ⊂ A an ideal of finite type, and suppose that A is perfectoid for its I-adic topology; let also P ∈ A[X] be a non-constant monic polynomial. We associate with P the functor

G_{P} : A-Perf → Set B ↦ {(β_{n} | n ∈ ℕ) ∈ E(B) | P (β_{0}) = 0}

defined just as F_{P}. Then we claim that G_{P} is represented by an adic and adically faithfully flat map A → D_{P}^{∧} of perfectoid rings (see definition 8.3.23(i,iii)). Indeed, recall that there exists a ∈ A^{∘∘} such that pA = a^{p}A (lemma 16.2.3(iv)), and A is perfectoid for its a-adic topology (which is just the p-adic topology : proposition 16.3.8(i)), hence we may consider the category 𝒞 (A, a) and the functor F_{P} : 𝒞 (A, a) → Set; moreover, A ∈ Ob(𝒞 (A, a)) (corollary 16.3.75) so F_{P} is represented by a faithfully flat A-algebra D_{P} ∈ Ob(𝒞 (A, a)), and its I-adic completion D_{P}^{∧} is perfectoid, by proposition 16.3.8(ii). Now, if A → B is any continuous map of perfectoid rings, then B ∈ Ob(𝒞 (A, a)), by proposition 16.3.8(i) and corollary 16.3.75, and on the other hand, every map of A-algebras D_{P} → B extends uniquely to a continuous map D_{P}^{∧} → B; it follows easily that D_{P}^{∧} represents G_{P}.

The following result strenghtens [25, Th.7.12].

Proposition 16.9.29. For every perfectoid 𝔽_{p}-algebra E there exists an adic and adically faithfully flat map E → D of perfectoid rings such that the following holds. For every non-constant monic polynomial P ∈ W (D)[X] and every distinguished ideal ℐ ⊂ W (D) there exists \underline{w} ∈ W (D) such that P (\underline{w}) ∈ ℐ.

Proof. Let I ⊂ E be an adic ideal of definition; let ℱ be the set of all non-constant monic polynomials in W (E)[X], and 𝒢 the set of all distinguished ideals of W (E). For every (P, ℐ) ∈ ℱ × 𝒢, let A_{ℐ} := W (E)/ℐ, define the functor G_{P} : A_{ℐ} → Set as in remark 16.9.28(ii) (so here we identify P with its image in A_{ℐ} [X]), and denote by D_{P,ℐ}^{∧} the adic and adically faithfully flat perfectoid A_{ℐ} -algebra that represents G_{P}. We have a natural identification E →^{∼} E(A_{ℐ} ) for every ℐ ∈ 𝒢 (example 16.3.2(ii)), so E(D_{P,ℐ}^{∧} ) is naturally an E-algebra, for every (P, ℐ) ∈ ℱ × 𝒢; moreover, the structure map E → E(D_{P,ℐ}^{∧} ) is adic and adically faithfully flat, by remark 16.4.20(i). We denote by ℰ (A) the coproduct of the family of Ealgebras (E(D_{P,ℐ}^{∧} ) | (P, ℐ) ∈ ℱ × 𝒢), and we let ℰ (A)^{∧} be the I-adic completion of ℰ (A). Since each E(D_{P,ℐ}^{∧} ) is a perfect ring, it is easily seen that the same holds for ℰ (A), and then the induced map E → ℰ (A)^{∧} is an adic and adically flat ring homomorphism of perfectoid 𝔽_{p}-algebras (remark 9.4.9(v)). Next, by transfinite induction we define, for every ordinal, a perfectoid E-algebra as follows : ℰ_{0}(A) := E, ℰ_{α+1}(A) := ℰ (ℰ_{α}(A))^{∧} for every successor ordinal α + 1, and ℰ_{α}(A) is the I-adic completion of colim_{β<α} ℰ_{β}(A), for every limit ordinal α. We observe :

Claim 16.9.30. Let (Λ, ≤) be a partially ordered set such that the following holds : for every finite or countable subset S ⊂ Λ there exists λ ∈ Λ such that λ ≥ μ for every μ ∈ S. We have :

(i) Let R be a ring, J ⊂ R an ideal, M_{•} := (M_{λ} | λ ∈ Λ) a system of J-adically complete and separated R-modules. Then the colimit M of M_{•} is J-adically complete and separated.

(ii) Let R_{•} := (R_{λ} | λ) be a system of rings, and R the colimit of R_{•}. Then the natural map τ : colim_{λ∈Λ} W (R_{λ}) → W (R) is a ring isomorphism.

Proof of the claim. (i): To check that M is separated, let x ∈ M such that x ∈ J^{n}M for every n ∈ ℕ; then x is represented by an element y ∈ M_{λ}, for some λ ∈ Λ, and the condition means that for every n ∈ ℕ there exists λ(n) ∈ Λ such that λ(n) ≥ λ, and the image of y in M_{λ(n)} lies in I^{n}M_{λ(n)}. By assumption, there exists μ ∈ Λ such that μ ≥ λ(n) for every n ∈ ℕ, so the image y_{μ} of y in M_{μ} lies in ⋂_{n∈ℕ} J^{n}M_{μ}. Since M_{μ} is separated, we get y_{μ} = 0, whence x = 0, as required. Next, let x_{•} := (x_{n} | n ∈ ℕ) be a Cauchy sequence in M; after replacing x_{•} by a subsequence, we may assume that x_{n+1} − x_{n} ∈ J^{n}M for every n ∈ ℕ. Each x_{n} is represented by an element y_{n} ∈ M_{μ(n)} for some μ(n) ∈ Λ; then we find μ ∈ Λ such that μ ≥ μ(n) for every n ∈ ℕ, and after replacing each y_{n} by its image in M_{μ}, we may assume that x_{•} is the image of a sequence y_{•} in M_{μ}. For every n ∈ ℕ and every λ ∈ Λ with λ ≥ μ, let us denote by y_{n,λ} the image of y_{n} in M_{λ}. With this notation, for every n ∈ ℕ we can then find λ(n) ∈ Λ such that λ(n) ≥ μ, and such that y_{n+1,λ(n)} − y_{n,λ(n)} ∈ J^{n}M_{λ(n)}. Pick λ ∈ Λ such that λ ≥ λ(n) for every n ∈ ℕ; then y_{n+1,λ} − y_{n,λ} ∈ J^{n}M_{λ} for every n ∈ ℕ, so the sequence (y_{n,λ} | n ∈ ℕ) is Cauchy in M_{λ}, and therefore it admits a limit y. Clearly, the image of y in M is a limit for x_{•}.

(ii): Let λ ∈ Λ, and \underline{w} := (w_{n} | n ∈ ℕ) ∈ W (R_{λ}) whose image in 𝒲 := colim_{λ∈Λ} W (R_{λ}) lies in Ker(τ); then for each n ∈ ℕ there exists λ(n) ∈ Λ such that λ(n) ≥ λ, and the image of w_{n} vanishes in R_{λ(n)}. Pick μ ∈ Λ such that μ ≥ λ(n) for every n ∈ ℕ; then clearly the image of \underline{w} vanishes in W (R_{μ}), so also in 𝒲 . This shows that τ is injective. Lastly, let \underline{u} := (u_{n} | n ∈ ℕ) ∈ W (R); then for every n ∈ ℕ there exists λ(n) ∈ Λ and an element in R_{λ(n)} whose image in R agrees with u_{n}. Pick again λ ∈ Λ with λ ≥ λ(n) for every n ∈ ℕ; then clearly \underline{u} is the image of an element of W (R_{λ}), so τ is surjective. ♢

Let ω_{1} be the smallest uncountable ordinal; we claim that D := ℰ_{ω_{1}}(A) will do. Indeed, let P ∈ W (D)[X] be a non-constant monic polynomial, and \underline{α} := (α_{n} | n ∈ ℕ) ∈ W (D) a distinguished element; according to claim 16.9.30(i), D represents the direct limit of the system of E-algebras (ℰ_{β}(A) | β < ω_{1}), so we may assume that P ∈ ℰ_{β}(A)[X] for some β < ω_{1}. Moreover, W (D) = colim_{β<ω_{1}} W (ℰ_{β}(A)), by claim 16.9.30(ii), so that we may also assume that \underline{α} ∈ W (ℰ_{β}(A)). Since the topology of D is I-adic, we have α_{0}^{k} ∈ ID for some k ∈ ℕ, and α_{1} ∈ D^{×}; then, up to replacing β by a larger ordinal, we may assume as well that α_{0}^{k} ∈ Iℰ_{β}(A) and α_{1} ∈ ℰ_{β}(A)^{×}, so that \underline{α} is distinguished in W (ℰ_{β}(A)). By construction, there exists therefore \underline{w} ∈ W (ℰ_{β+1}(A)) such that P (\underline{w}) ∈ \underline{α}W (ℰ_{β+1}(A)), and since β + 1 < ω_{1}, the proposition follows. □

16.9.31. Let A be a perfectoid Tate ring, A_{0} ⊂ A a perfectoid subring of definition, and (A_{0}, 𝔪) a basic setup in the sense of [75, §2.1.1], such that 𝔪 is the filtered union of a countable system of principal subideals. If 𝔪 is an open radical ideal of A_{0}, then we have attached to 𝔪 an open radical ideal 𝔪_{E} of E_{0} := E(A_{0}/pA_{0}), so that (E_{0}, 𝔪_{E}) is a basic setup as well (see proposition 16.8.15). One can show that here 𝔪 is still a radical ideal; however, it is not necessarily open, but nevertheless we can construct a useful basic setup (E_{0}, 𝔪_{E}) as follows. By assumption, 𝔪 is the filtered union of a countable system of principal ideals; hence, we may find a generating system (a_{n} | n ∈ ℕ) for 𝔪 such that for every n ∈ ℕ there exists an element c_{n} ∈ A with a_{n} = c_{n}a_{n+1}^{p}. For every x ∈ A_{0}, let \overline{x} ∈ A_{0}/pA_{0} be the class of x; since A_{0} is perfectoid, for every n ∈ ℕ there exists (\overline{x}_{n,k} | k ∈ ℕ) ∈ E_{0} with \overline{x}_{n,0} = \overline{c}_{n} (recall that \overline{x}_{n,k+1}^{p} = \overline{x}_{n,k} for every k ∈ ℕ). Set

α_{n,k} := \overline{a}_{n+k} · \overline{x}_{n,k} · \overline{x}_{n+1,k−1} ⋯ \overline{x}_{n+k−1,1} for every n ∈ ℕ.

Thus, α_{n,0} = \overline{a}_{n} and it is easily seen that α_{n,•} := (α_{n,k} | k ∈ ℕ) ∈ E_{0} for every n ∈ ℕ; moreover, α_{n,•} ∈ α_{n+1,•}^{p}E_{0} for every n ∈ ℕ. Let then 𝔪_{E} ⊂ E_{0} be the ideal generated by the system (α_{n,•} | n ∈ ℕ); it follows that (E_{0}, 𝔪_{E}) is a basic setup.

Remark 16.9.32. (i) In the situation of (16.9.31), we have :

(16.9.33)  ⋂_{n∈ℕ}(\overline{u}_{A_{0}}(𝔪_{E}) · A_{0} + p^{n}A_{0}) = ⋂_{n∈ℕ}(𝔪 + p^{n}A_{0}).

Indeed, on the one hand, \overline{u}_{A_{0}}(α_{n,•}) is the limit of the p-adically convergent sequence

((a_{n+k}x_{n,k} ⋯ x_{n+k−1,1})^{p^{k}} | k ∈ ℕ)

hence it lies in the p-adic topological closure of 𝔪. On the other hand, by construction for every x ∈ 𝔪 there exists y ∈ \overline{u}_{A_{0}}(𝔪_{E}) such that x − y ∈ pA_{0}; then x^{p^{n}} − y^{p^{n}} ∈ p^{n}A_{0} for every n ∈ ℕ (lemma 9.3.4(i)), and since 𝔪 is generated by (x^{p^{n}} | x ∈ 𝔪), we deduce the converse inclusion.

(ii) Quite generally, let (V, 𝔪) be a basic setup in the sense of [75, §2.1.1], such that 𝔪 is a filtered countable union of principal subideals; then it is easily seen that we may write 𝔪 = ⋃_{n∈ℕ} V a_{n} for a system (a_{n} | n ∈ ℕ) of elements of A such that V a_{n} ⊂ 𝔪 · a_{n+1} for every n ∈ ℕ : the details are left to the reader. Hence, for every n ∈ ℕ let c_{n} ∈ 𝔪 such that a_{n} = c_{n}a_{n+1}. We consider the inductive system L_{•} := (L_{n} | n ∈ ℕ) with L_{n} := V for every n ∈ ℕ, with V -linear transition map φ_{n} : L_{n} → L_{n+1} such that φ_{n}(1) := c_{n} for every n ∈ ℕ. The inductive limit L of L_{•} is the set of equivalence classes [x, n] of all pairs (x, n) with n ∈ ℕ and x ∈ L_{n}. We have an obvious V -linear surjection

ψ : L → 𝔪 [x, n] ↦ xa_{n}.

Notice that if [x, n] ∈ Ker ψ, then a_{n} · [x, n] = [a_{n}x, n] = 0; on the other hand, for every k ≥ n there exists y ∈ V such that [x, n] = [y, k], whence a_{k} · [x, n] = a_{k} · [y, k] = 0 as well. This shows that 𝔪·Ker ψ = 0. Therefore, 𝔪⊗_{V} ψ : 𝔪⊗_{V} L → 𝔪̃ := 𝔪⊗_{V} 𝔪 is an isomorphism. But by construction, L is a flat V -module, hence the natural map 𝔪⊗_{V} L → 𝔪L is an isomorphism; lastly, for every [x, n] ∈ L we have [x, n] = c_{n}[x, n + 1], so 𝔪L = L. Summing up, 𝔪 ⊗_{V} ψ is naturally identified with the isomorphism :

L →^{∼} 𝔪̃ [x, n] ↦ x ⊗ a_{n}.

(iii) Furthermore, the functor E induces a functor on almost algebras

E^{a} : (A_{0}, 𝔪)^{a}-Alg → (E_{0}, 𝔪_{E})^{a}-Alg R^{a} ↦ E(R/pR)^{a}.

Indeed, let f : R → S be a homomorphism of A_{0}-algebras such that f^{a} : R^{a} → S^{a} is an isomorphism of (A_{0}, 𝔪)^{a}-algebras, and set f_{0} := f ⊗_{ℤ} ℤ/pℤ; we need to check that E(f_{0})^{a} : E(R/pR)^{a} → E(S/pS)^{a} is an isomorphism of (E_{0}, 𝔪_{E})^{a}-algebras. However, E(f_{0}) is the limit of the system of rings (R_{n} | n ∈ ℕ), with R_{n} := R/pR for every n ∈ ℕ, and with transition map R_{n+1} → R_{n} given by the Frobenius endomorphism Φ_{R/pR}. Now, if g : A_{0} → R is the structure map of the A_{0}-algebra R, then R_{0} is naturally an E_{0}-algebra, via the composition g′ : E_{0} → R_{0} of the projection \overline{u}_{A_{0}/pA_{0}} : E_{0} → A_{0}/pA_{0} and g ⊗_{ℤ} ℤ/pℤ : A_{0}/pA_{0} → R_{0}. Let Φ_{E_{0}} be the Frobenius automorphism of E_{0}; then for every n ∈ ℕ, the ring R_{n} is an E_{0}-algebra with structure morphism g′ ∘ Φ_{E_{0}}^{−n} : E_{0} → R_{n}, i.e. R_{n} = (R_{0})_{(Φ_{E_{0}}^{−n})}, and Φ_{R/pR} : R_{n+1} → R_{n} is a homomorphism of E_{0}-algebras, for these E_{0}-algebra structures. Likewise, E(S/pS) is the limit of a system of E_{0}-algebras (S_{n} | n ∈ ℕ), and f induces a system of maps of E_{0}- algebras f_{n} := f_{0} : R_{n} → S_{n}, whose inverse limit is E(f_{0}). Since f^{a} is an isomorphism, and since Φ_{E_{0}} : (E_{0}, 𝔪_{E}) →^{∼} (E_{0}, 𝔪_{E}) is an isomorphism of basic setups, it is easily seen that f_{n}^{a} : R_{n}^{a} → S_{n}^{a} is an isomorphism of (E_{0}, 𝔪_{E})^{a}-algebras, for every n ∈ ℕ; on the other hand, the functor (−)^{a} commutes with limits, since it has a left adjoint, whence the contention.

16.9.34. Let A be a Tate ring, A_{0} ⊂ A a subring of definition, p ∈ ℕ a prime integer, and A_{1} ⊂ A the p-root closure of A_{0} in A (see definition 9.8.24(ii)). Moreover, suppose there exists b ∈ A_{0}^{∘∘} ∩ A^{×} with pA_{0} ⊂ b^{p}A_{0}, and let A_{0}^{∧} and A_{1}^{∧} be the b-adic completions of A_{0} and A_{1}. Notice that the Frobenius endomorphism Φ_{A_{i}} of A_{i}/b^{p}A_{i} induces a ring homomorphism

\overline{Φ}_{A_{i}} : A_{i}/bA_{i} → A_{i}/b^{p}A_{i} for i = 0, 1.

Furthermore, let (A_{0}, 𝔪) be a basic setup, such that 𝔪 is the filtered union of a countable system of principal subideals, and denote by \overline{𝔪} the image of 𝔪 in \overline{A}_{0} := A_{0}/pA_{0}. Then 𝔪 is generated by the system (x^{p} | x ∈ 𝔪) ([75, Prop.2.17]), and therefore the Frobenius endomorphism Φ of \overline{A}_{0} is a morphism of basic setups in the sense of [75, §3.5]

Φ : (\overline{A}_{0}, \overline{𝔪}) → (\overline{A}_{0}, \overline{𝔪}).

As detailed in [75, §3.5.7], the latter induces a pull-back functor

(\overline{A}_{0}, \overline{𝔪})^{a}-Alg → (\overline{A}_{0}, \overline{𝔪})^{a}-Alg B ↦ B_{(Φ)}.

Definition 16.9.35. In the situation of (16.9.34), we say that the basic setup (A_{0}, 𝔪) is almost perfectoid if A_{0} is complete and separated, and the Frobenius endomorphism of A_{0}/b^{p}A_{0} induces an isomorphism of (A_{0}, 𝔪)^{a}-algebras \overline{Φ}_{A_{0}}^{a} : (A_{0}/bA_{0})^{a} →^{∼} (A_{0}/b^{p}A_{0})_{(Φ)}^{a}.

Remark 16.9.36. The following proposition 16.9.37(i) implies in particular that definition 16.9.35 is independent of the choice of the element b ∈ A_{0} such that p ∈ b^{p}A_{0} : if b′ ∈ A_{0} is another such element, and if the b-adic topology on A_{0} agrees with the b′-adic topology, then \overline{Φ}_{A_{0}}^{a} is an isomorphism if and only if the same holds for the corresponding morphism of (A_{0}, 𝔪)^{a}-algebras (A_{0}/b′A_{0})^{a} →^{∼} (A_{0}/b^{′p}A_{0})_{(Φ)}^{a}. See also lemma 17.5.10.

Proposition 16.9.37. (i) In the situation of (16.9.34), the following conditions are equivalent : (a) The basic setup (A_{0}^{∧}, 𝔪A_{0}^{∧}) is almost perfectoid.

(b) The basic setup (A_{1}^{∧}, 𝔪A_{1}^{∧}) is almost perfectoid, and the inclusion map j : A_{0} → A_{1} induces isomorphisms of (A_{0}, 𝔪)^{a}-algebras j^{a} : A_{0}^{a} →^{∼} A_{1}^{a}.

(c) The topological closure \overline{C} ⊂ A_{0} of the subring C := ℤ[a^{p} | a ∈ A_{0}] contains 𝔪, and j

induces an isomorphism of (A_{0}, 𝔪)^{a}-algebras j^{a} : A_{0}^{a} →^{∼} A_{1}^{a}. (ii) If A_{0} is integrally closed (resp. p-root closed) in A, then A_{0∗}^{a} is integrally closed (resp.

p-root closed) in A_{0∗}^{a}[b^{−1}].

(iii) If (A_{0}, 𝔪) is an almost perfectoid basic setup, the same holds for (A_{0∗}^{a}, 𝔪A_{0∗}^{a}).

(iv) If (A_{0}, 𝔪) is an almost perfectoid basic setup, and the unit of adjunction A_{0} → (A_{0})_{∗}^{a} is an isomorphism, then A_{0} = A_{1}.

Proof. By simple inspection we see that (i.b)⇒(i.a). Conversely, suppose that (i.a) holds, and let also j^{∧} : A_{0}^{∧} → A_{1}^{∧} be the b-adic completion of j; in order to show (i.b), it suffices to check :

Claim 16.9.38. Ker \overline{Φ}_{A_{0}}^{a} = 0 ⇔ j^{a} is an isomorphism ⇔ (j^{∧})^{a} is an isomorphism.

Proof of the claim. Suppose first that Ker \overline{Φ}_{A_{0}}^{a} = 0. We can construct A_{1} as the increasing union of the ascending chain (A_{0,n} | n ∈ ℕ) of subrings of A defined inductively as follows: A_{0,0} := A_{0}, and A_{0,n} := A_{0,n−1}[Σ_{n}], with Σ_{n} := {x ∈ A | x^{p} ∈ A_{0,n−1}}, for every n > 0. Thus, we are reduced to showing that the inclusion A_{0} → A_{0,n} induces an isomorphism A_{0}^{a} →^{∼} A_{0,n}^{a} for every n ∈ ℕ. However, if for some n > 0 the inclusion j_{n−1} : A_{0,n−1} → A_{0,n} induces an isomorphism j_{n−1}^{a} : A_{0,n−1}^{a} →^{∼} A_{0,n}^{a}, then the same holds for j_{n} : A_{0,n} → A_{0,n+1} : indeed, let x ∈ Σ_{n+1}; since x^{p} ∈ A_{0,n}, by assumption we have (ax)^{p} ∈ A_{0,n−1} for every a ∈ 𝔪, whence ax ∈ Σ_{n} ⊂ A_{0,n}. Thus, we are further reduced to checking that j_{0} induces an isomorphism j_{0}^{a} : A_{0}^{a} →^{∼} A_{0,1}^{a}, i.e. that (Coker j_{0})^{a} = 0. We argue as in the proof of lemma 9.8.26(i) : let x ∈ A_{1} \ A_{0}, and denote by m ∈ ℕ the smallest integer such that b^{m}x · 𝔪 ⊄ A_{0}. Hence, y := b^{mp}x^{p} ∈ A_{0}, and (b^{m+1}x)^{p} = b^{p}y^{p} ∈ b^{p}A_{0}. By assumption, for every a ∈ 𝔪 we then have z ∈ A_{0} such that ab^{m+1}x = bz, whence ab^{m}x ∈ A_{0}, contradicting the choice of m.

Conversely, suppose that j^{a} is an isomorphism, and let x ∈ A_{0} be an element whose class \overline{x} ∈ A_{0}/bA_{0} lies in Ker \overline{Φ}_{A_{0}}; then x^{p} ∈ b^{p}A_{0}, so (x/b)^{p} ∈ A_{0}, and our assumption implies that 𝔪 · x ∈ bA_{0}, i.e. 𝔪 · \overline{x} = 0.

Next, if j^{a} is an isomorphism, the same holds for j_{k}^{a} := A_{0}^{a}/b^{k}A_{0} ⊗_{A_{0}^{a}} j^{a} : A_{0}^{a}/b^{k}A_{0}^{a} → A_{1}^{a}/b^{k}A_{1}^{a}, for every k ∈ ℕ, and then also for the limit of the inverse system (j_{k}^{a} | k ∈ ℕ). But the functor (−)^{a} commutes with limits, since it has a left adjoint, so (j^{∧})^{a} is an isomorphism.

Conversely, if (j^{∧})^{a} is an isomorphism, the induced map A_{0}^{∧}[b^{−1}]/A_{0}^{∧} → A_{1}^{∧}[b^{−1}]/A_{1}^{∧} is an almost isomorphism of A_{0}-modules. But proposition 8.3.33(iii) easily implies that the natural map A_{0}[b^{−1}]/A_{0} → A_{0}^{∧}[b^{−1}]/A_{0}^{∧} is bijective, and similarly for A_{1}; hence j induces isomorphisms A_{0}^{a}[b^{−1}]/A_{0}^{a} →^{∼} A_{1}^{a}[b^{−1}]/A_{1}^{a} and A_{0}[b^{−1}] →^{∼} A_{1}[b^{−1}]. By the 5-lemma, it follows easily that j^{a} is an isomorphism. ♢

Next, let 𝔪\overline{C} ⊂ \overline{C} be the ideal generated by (a^{p} | a ∈ 𝔪). Since 𝔪 is the filtered union of a countable system of principal subideals, the same holds for \overline{𝔪}_{C}; by the same token, since 𝔪 = 𝔪^{2}, for every a ∈ 𝔪 there exist b, c ∈ 𝔪 such that a = bc, whence a^{p} = b^{p}c^{p}, which shows that 𝔪\overline{C} = 𝔪2 ; \overline{C}. Hence, (\overline{C}, 𝔪\overline{C}) is a basic setup; moreover, we have already remarked that 𝔪 = 𝔪\overline{C}A_{0}, hence the inclusion \overline{C} → A_{0} is a morphism of basic setups (\overline{C}, 𝔪\overline{C}) → (A_{0}, 𝔪).

Notice then that the \overline{C}-module A_{0}/\overline{C} is the limit of the inverse system

(M_{n} := A_{0}/(b^{n}A_{0} + C) | n ∈ ℕ).

Hence, 𝔪 ⊂ \overline{C} if and only if the (\overline{C}, 𝔪\overline{C})^{a}-module (A_{0}/\overline{C})^{a} vanishes, if and only if M_{n}^{a} = 0 for every n ∈ ℕ. Now, for given n ∈ ℕ, consider the b^{p}-adic filtration on M_{n}, and let gr_{•}M_{n} be the associated graded \overline{C}-module; clearly gr_{i}M_{n} is a quotient of M_{p}, for every i ∈ ℕ. Hence, 𝔪 ⊂ \overline{C} if and only if M_{p}^{a} = 0; the latter in turns holds if and only if Coker \overline{Φ}_{A_{0}}^{a} = 0. This already shows that (a),(b)⇒(c). Conversely, if (c) holds, in order to deduce (a), it suffices to check that Ker \overline{Φ}_{A_{0}}^{a} = 0; since j^{a} is an isomorphism, the latter holds by claim 16.9.38.

(ii): Notice first that since b · 1_{A_{0}} is an injective map, the same holds for (b · 1_{A_{0}})_{∗}^{a} = b · 1_{A_{0∗}^{a}}, since the functor (−)_{∗}^{a} commutes with all limits. Thus, A_{0∗}^{a} is naturally a subring of A_{0∗}^{a}[b^{−1}]. Now, let x ∈ A_{0∗}^{a}[b^{−1}] be integral over A_{0∗}^{a}; hence there exist n, k ∈ ℕ and y, a_{1}, …, a_{k} ∈ A_{0∗}^{a} such that x = y/b^{n} and

(y/b^{n})^{k} + a_{1} · (y/b^{n})^{k−1} + ⋯ + a_{k} = 0.

Recall that A_{0∗}^{a} = Hom_{A_{0}}(𝔪̃, A_{0}), with 𝔪̃ := 𝔪 ⊗_{A_{0}} 𝔪. Then we get (y(t)/b^{n})^{k} + a_{1}(t) · (y(t)/b^{n})^{k−1} + ⋯ + a_{k}(t) = 0 for every t ∈ 𝔪̃, and if A_{0} is integrally closed in A, it follows that y(t)/b^{n} ∈ A_{0} for every such t, i.e. x ∈ A_{0∗}^{a}. One shows likewise that if A_{0} is p-root closed in A, then A_{0∗}^{a} is p-root closed in A_{0∗}^{a}[b^{−1}].

(iii): By assumption, b · 1_{A_{0}} is an injective map, so the same holds for b · 1_{A_{0∗}^{a}}, because the functor (−)_{∗}^{a} commutes with limits. This shows that A_{0∗}^{a} has no b-torsion; next, we need to check that A_{0∗}^{a} is b-adically complete and separated. This follows from the more general :

Claim 16.9.39. Let (V, 𝔪_{V}) be a basic setup, A a V -algebra, I ⊂ A an ideal of finite type, M an A-module whose I-adic topology is complete and separated. Then the same holds for the I-adic topology of M_{∗}^{a}.

Proof of the claim. Both functors (−)^{a} and (−)_{∗} commute with limits, since they are right adjoints, hence M_{∗}^{a} is the limit of the inverse system ((M/I^{n}M)_{∗}^{a} | n ∈ ℕ). For every n ∈ ℕ, let M_{n} be the image of the natural map π_{n} : M_{∗}^{a} → (M/I^{n}M)_{∗}^{a}; it follows that M_{∗}^{a} is also the limit of the induced inverse system (M_{n} | n ∈ ℕ), and therefore M_{∗}^{a} is complete and separated for the linear topology that admits the fundamental system of open submodules (Ker π_{n} = (I^{n}M)_{∗}^{a} | n ∈ ℕ) (corollary 8.2.16(i)). Since I^{n}M_{∗}^{a} ⊂ (I^{n}M)_{∗}^{a} for every n ∈ ℕ, the claim now follows from lemma 8.3.12. ♢

It remains to check that the Frobenius endomorphism of R := A_{0∗}^{a} induces an isomorphism (R/bR)^{a} →^{∼} (R/b^{p}R)_{(Φ)}^{a}; the latter is clear, since the same holds by assumption for the Frobenius endomorphism of A_{0}, and the natural map A_{0} → A_{0∗}^{a} is an almost isomorphism.

(iv): Under the stated assumptions, j induces an isomorphism A_{0} →^{∼} (A_{0})_{∗}^{a} →^{∼} (A_{1})_{∗}^{a}, by (i), and on the other hand, (A_{1})_{∗}^{a} is p-root closed in A_{0∗}^{a}[b^{−1}] = A, by (ii), hence A_{0} is p-root closed in A, i.e. A_{0} = A_{1}. □

16.9.40. Let A be a perfectoid Tate ring, A_{0} ⊂ A a perfectoid subring of definition, and b ∈ A_{0}^{∘∘} ∩ A^{×}, so that the topology of A_{0} is b-adic. Let also g_{•} := (g_{n} | n ∈ ℕ) ∈ E_{0} := E(A_{0}), and set g := g_{0} = \overline{u}_{A_{0}}(g_{•}). Hence, g^{γ} is well defined in A_{0} for every γ ∈ ℕ[1/p], and set

𝔪 := ⋃_{n∈ℕ} g_{n}A_{0}  A′_{0} := Im(A_{0} → A_{0}[1/g])  and A_{1} := {a ∈ A_{0}[1/g] | 𝔪 · a ⊂ A′_{0}}.

Clearly (A_{0}, 𝔪) is a basic setup fulfilling the conditions of (16.9.34).

Lemma 16.9.41. With the notation of (16.9.40), let A_{0}^{a} be the (A_{0}, 𝔪)^{a}-algebra associated with A_{0}. Then the natural map A_{0} → A_{1} factors through the unit of adjunction A_{0} → A_{0∗}^{a} and an isomorphism of A_{0}-algebras A_{0∗}^{a} →^{∼} A_{1}.

Proof. We claim that the induced surjection π : A_{0} → A′_{0} induces an isomorphism of (A_{0}, 𝔪)^{a}- algebras π^{a} : A_{0}^{a} →^{∼} A_{0}^{′a}. Indeed, if a ∈ Ker π, there exists n ∈ ℕ such that g^{n}a = 0 in A_{0}; then (g^{n/p^{k}}a)^{p^{k}} = 0 in A_{0}, and consequently g^{n/p^{k}}a = 0 in A_{0} for every k ∈ ℕ, due to corollary 16.3.63(i), whence the contention. Since the natural map A_{0} → A_{1} factors through π and the inclusion i : A′_{0} → A_{1}, and since π^{a} is an isomorphism we are reduced to checking that i factors through the unit of adjunction A′_{0} → A_{0∗}^{′a} and an isomorphism A_{0∗}^{′a} →^{∼} A_{1}.

Next, we claim that the surjection 𝔪 → 𝔪A′_{0} is bijective. Indeed, say that x ∈ 𝔪 and π(x) = 0; we may write x = g^{γ}y for some γ ∈ ℕ[1/p] \ {0}, and then clearly π(y) = 0. But by the foregoing, it follows that g^{γ}y = 0, whence the contention.

Set 𝔪̃ := 𝔪 ⊗_{A_{0}} 𝔪; by the foregoing, the isomorphism π^{a} induces identifications :

A_{0∗}^{a} = Hom_{A_{0}}(𝔪̃, A_{0}) →^{∼} A_{0∗}^{′a} = Hom_{A_{0}}(𝔪̃, A′_{0}) = Hom_{A′_{0}}(𝔪A′_{0} ⊗_{A′_{0}} 𝔪A′_{0}, A′_{0}).

Notice that the multiplication map 𝔪A′_{0} ⊗_{A′_{0}} 𝔪A′_{0} → 𝔪A′_{0} is the colimit of the system of isomorphisms (g^{γ}A′_{0}⊗_{A′_{0}}g^{γ}A′_{0} →^{∼} g^{2γ}A′_{0} | γ ∈ ℕ[1/p]\{0}), and it is therefore an isomorphism. So, finally, we get a natural identification

A_{0∗}^{a} →^{∼} Hom_{A′_{0}}(𝔪A′_{0}, A′_{0}) = A_{1}

from which the lemma follows easily : details left to the reader. □

The following result generalizes the “perfectoid Abhyankar’s lemma” of [6].

Theorem 16.9.42. In the situation of (16.9.40), let B be a finite étale A_{0}[1/g]-algebra, B_{1} the integral closure of A_{1} in B, and endow A_{1} and B_{1} with their b-adic topologies. For the almost structure given by (A_{0}, 𝔪), we have :

(i) The basic setup (B_{1},𝔪B_{1}) is almost perfectoid, and B_{1} is p-root closed in B_{1}[1/b]. (ii) (B_{1}/b^{n}B_{1})^{a} is an étale (A_{0}/b^{n}A_{0})^{a}-algebra of finite rank, for every n ∈ ℕ. (iii) If B is a faithfully flat A_{0}[1/g]-algebra, then (B_{1}/b^{n}B_{1})^{a} is a faithfully flat (A_{0}/b^{n}A_{0})^{a}-

algebra, for every n ∈ ℕ. (iv) A_{1} is complete and separated, and is integrally closed in A_{0}[1/g].

(v) The unit of adjunction B_{1} → B_{1∗}^{a} is an isomorphism.

Proof. Arguing as in (16.9.7), we may assume that pA_{0} ⊂ b^{p}A_{0}, and b = \overline{u}_{A_{0}}(b_{•}) for some b_{•} ∈ E(A_{0}), so that b^{γ} is well defined in A for every γ ∈ ℤ[1/p]. We let I := A_{0}b + A_{0}g, and U := Spec A_{0} \ Spec A/I; consider the quasi-affinoid ring \underline{A}_{0} := (A_{0}, A_{0}, U), and the rational subsets of X := Spa \underline{A}_{0}

R_{n} := R_{A_{0}}(\frac{b^{n}}{g} ) ∩ X  R′_{n} := R_{A_{0}}(\frac{g}{b^{n}} ) ∩ X  for every n ∈ ℕ

(definition 15.4.14(iv)). Set A_{n} := 𝒪_{X}^{∧+}(R_{n}) and A′_{n} := 𝒪_{X}^{∧+}(R′_{n}) for every n ∈ ℕ; since R_{n} ⊂ R_{n+1} for every n ∈ ℕ, we deduce a well defined inverse system of A_{0}-algebras (A_{n} | n ∈ ℕ).

Claim 16.9.43. (i) The natural map A_{0} → 𝒪_{X}^{∧+}(X) induces an isomorphism A_{0}^{a} →^{∼} 𝒪_{X}^{∧+}(X)^{a} of (A_{0}, 𝔪)^{a}-algebras.

(ii) For every i ∈ ℕ, the following holds :

(a) The induced inverse system (A_{n}/b^{i}A_{n} | n ∈ ℕ) is almost essentially constant, relative to the basic setup (A_{0}, 𝔪) (see definition 14.2.11(iii)).

(b) The natural cone (ρ_{n} : A_{0} → A_{n} | n ∈ ℕ) induces a universal cone ((A_{0}/b^{i}A_{0})^{a} → (A_{n}/b^{i}A_{n})^{a} | n ∈ ℕ) in the category of (A_{0}, 𝔪)^{a}-algebras.

Proof of the claim. (i): By construction, 𝒪_{X}^{∧+}(X) is the integral closure of the image of A_{0} in A_{U} := 𝒪_{U}(U). On the other hand, A_{0∗}^{a} is integrally closed in A_{U}^{∘} (claim 16.8.20(ii)), and recall that A_{U}^{∘} is integrally closed in A_{U} (remark 8.3.10(iv)). It follows that the unit of adjunction η : A_{0} → A_{0∗}^{a} factors through the inclusion map 𝒪_{X}^{∧+}(X) → A_{0∗}^{a}. Since η^{a} is an isomorphism, the assertion follows.

(ii): In light of (i) and theorem 16.7.41, the following sequence of (A_{0}, 𝔪)^{a}-modules is short exact for every n ∈ ℕ :

0 → A_{0}^{a} → A_{n}^{a} ⊕ A_{n}^{′a} → 𝒪_{X}^{+}(R_{n} ∩ R′_{n})^{a} → 0.

Moreover, clearly the image of b is a regular element in 𝒪_{X}^{+}(R_{n} ∩ R′_{n}); by the snake lemma, we easily deduce, for every n, i ∈ ℕ, a short exact sequence of (A_{0}, 𝔪)^{a}-modules

0 → A_{0}^{a}/b^{i}A_{0}^{a} ⟶^{r_{n}} A_{0}/b^{i}A_{0} ⊗_{A_{0}} (A_{n} ⊕ A′_{n})^{a} ⟶^{s_{n}} A_{0}/b^{i}A_{0} ⊗_{A_{0}} 𝒪_{X}^{+}(R_{n} ∩ R′_{n})^{a} → 0.

Let us show that for every γ ∈ ℕ[1/p] \ {0}, every n ∈ ℕ with nγ ≥ i and every a ∈ A_{0} with ρ_{n}(a) ∈ b^{i}A_{n}, we have g^{2γ}a ∈ b^{i}A_{0}. Indeed, let \overline{a} ∈ A_{0}/b^{i}A_{0} be the image of a; then r_{n}(\overline{a}) = (0, \overline{x}) for some \overline{x} ∈ A′_{n}, and we notice that g^{γ} · \overline{x} = b^{nγ} · (g/b^{n})^{γ} · \overline{x}, whence g^{γ} · \overline{x} = 0, since (g/b^{n})^{γ} ∈ A′_{n}. It follows that r_{n}(g^{γ} · \overline{a}) = 0, whence g^{2γ} · \overline{a} = 0, as required. Lastly, let us check that for every γ ∈ ℕ[1/p] \ \{0\}, every n ∈ ℕ with nγ ≥ i and every \overline{x} ∈ A_{n}/b^{i}A_{n} there exists \overline{a} ∈ A_{0}/b^{i}A_{0} with ρ_{n}(\overline{a}) = g^{2γ} · \overline{x}. To this aim, set \overline{y} := s_{n}(\overline{x}, 0); again, we notice that g^{γ} · \overline{y} = b^{nγ} · (g/b^{n})^{γ} · \overline{y} = 0, so that (g^{γ}\overline{x}, 0) ∈ Ker s_{n} = Im r_{n}, whence the contention. This completes the proof of (ii.a). Assertion (ii.b) follows from the proof of (ii.a) and lemma 14.2.14(iii). ♢

Recall that 𝒪_{X}^{∧}(R_{n}) = A_{n}[1/g], and set B_{n} := A_{n}[1/g] ⊗_{A_{0}} B for every n ∈ ℕ; recall as well that A_{n} is perfectoid for its b-adic topology (proposition 16.7.25(iii) and theorem 16.5.13(iii)). Since the functor (−)^{a} commutes with limits, from claim 16.9.43 we deduce that also the cone (ρ_{n} | n ∈ ℕ) is universal, i.e. we have an isomorphism of (A_{0}, 𝔪)^{a}-algebras :

(16.9.44)  A_{0}^{a} →^{∼} lim_{i∈ℕ} lim_{n∈ℕ} A_{n}^{a}/b^{i}A_{n}^{a} →^{∼} lim_{n∈ℕ} A_{n}^{a}.

Claim 16.9.45. 𝔪A_{n} is an open radical ideal of A_{n}, for every n ∈ ℕ.

Proof of the claim. Clearly 𝔪A_{n} is an open ideal of A_{n}, since (b^{n}/g)^{γ} ∈ A_{n} for every γ ∈ ℕ[1/p]. Moreover, set J := ⋃_{γ∈ℕ[1/p]\{0}} b^{γ}A_{n}; by corollary 16.3.71, the quotient A_{n}/J is a perfect 𝔽_{p}-algebra; it follows easily that 𝔪/J is a radical ideal of A_{n}/J, whence the claim. ♢

Say that B is a projective A_{0}[1/g]-module of rank ≤ r; from claim 16.9.45 and theorem 16.8.44, we deduce that for every n ∈ ℕ there exists an étale A_{n}^{a}-algebra C_{n} of rank ≤ r, unique up to unique isomorphism, with an isomorphism of A_{n}[1/g]-algebras C_{n}[1/g] →^{∼} B_{n}. From the uniqueness property of C_{n}, there follows a unique isomorphism of A_{n}^{a}-algebras :

(16.9.46)  A_{n}^{a} ⊗_{A_{n+1}^{a}} C_{n+1} →^{∼} C_{n}  for every n ∈ ℕ.

Denote by D (resp. by E) the inverse limit of the resulting system (C_{n} | n ∈ ℕ) of A_{0}^{a}- algebras (resp. (B_{n} | n ∈ ℕ) of A_{0}-algebras), and for every i ∈ ℕ let as well D_{i} be the limit of the induced system (C_{n}/b^{i}C_{n} | n ∈ ℕ) of A_{0}^{a}/b^{i}A_{0}^{a}-algebras. By proposition 14.2.32(ii,iii), (C_{n}/b^{i}C_{n} | n ∈ ℕ) is almost essentially constant, and the projection D_{i} → C_{n}/b^{i}C_{n} induces an isomorphism of A_{0}^{a}/b^{i}A_{0}^{a}-algebras :

(16.9.47)  A_{n}^{a} ⊗_{A_{0}^{a}} D_{i} →^{∼} C_{n}/b^{i}C_{n}  for every n ∈ ℕ.

Notice that b and g are regular elements of A_{n}, and then they are also regular elements of C_{n∗}, since C_{n} is a flat A_{n}^{a}-algebra; in view of the induced isomorphism

(16.9.48)  D_{∗} →^{∼} lim_{n∈ℕ} C_{n∗}

(proposition 1.3.25(iii)) it follows easily that b and g are regular elements of D_{∗}.

Claim 16.9.49. (i) For every n ∈ ℕ, the ring C_{n∗} is perfectoid for its b-adic topology. (ii) D_{∗} is p-root closed in D_{∗}[1/b] and is integrally closed in D_{∗}[1/g]. (iii) The A_{0}^{a}/b^{i}A_{0}-algebra D_{i} is étale of rank ≤ r for every i ∈ ℕ. (iv) If B is a faithfully flat A_{0}[1/g]-algebra, then D_{i} is a faithfully flat A_{0}^{a}/b^{i}A_{0}-algebra for

every i ∈ ℕ.

(v) B and D_{∗} are integrally closed in E.

(vi) The basic setup (D_{∗}, 𝔪D_{∗}) is almost perfectoid for the b-adic topology of D_{∗}, and the projection D → D_{i} factors through an isomorphism D/b^{i}D →^{∼} D_{i} for every i ∈ ℕ.

Proof of the claim. (i) follows from proposition 16.8.29 and claim 16.9.45, since we have already recalled that A_{n} is perfectoid for its b-adic topology. Together with corollary 16.3.75 and lemma 9.8.26(i) we deduce that C_{n∗} is p-root closed in C_{n∗}[1/b], for every n ∈ ℕ. Then lemma 9.8.26(iii.a) and the isomorphism (16.9.48) imply that D_{∗} is p-root closed in the limit L of the system (C_{n,∗}[1/b] | n ∈ ℕ). However, the induced map D_{∗} → L factors through an injective map D_{∗}[1/b] → L, so D_{∗} is also p-root closed in D_{∗}[1/b].

Similarly, C_{n∗} is integrally closed in C_{n∗}[1/g] = B_{n} for every n ∈ ℕ, due to claim 16.8.20(ii); it follows easily that D_{∗} is integrally closed in E. But the induced map D_{∗} → E factors through an injective map D_{∗}[1/g] → E, so D_{∗} is also integrally closed in D_{∗}[1/g].

(iii) follows from theorem 14.2.39(iii,iv).

(iv): If B is a faithfully flat A_{0}[1/g]-algebra, C_{n} is a faithfully flat A_{n}^{a}-algebra for every n ∈ ℕ (theorem 16.8.44(ii)); then the assertion follows from theorem 14.2.39(ii)).

(v): We have already observed that D_{∗} is integrally closed in E. Next, by construction, B_{n} = B ⊗_{A_{0}[1/g]} A_{n}[1/g] for every n ∈ ℕ. Since B is a projective A_{0}[1/g]-module of finite rank, there follows a natural identification :

E →^{∼} B ⊗_{A_{0}[1/g]} lim_{n∈ℕ} A_{n}[1/g].

Since B is an étale A_{0}[1/g]-algebra, by proposition 14.3.25 we are then reduced to showing that A_{0}[1/g] is integrally closed in the limit L of the system of A_{0}-algebras (A_{n}[1/g] | n ∈ ℕ). However, invoking again claim 16.8.20(ii) we see that A_{n∗}^{a} is integrally closed in A_{n}[1/g] for every n ∈ ℕ, hence the limit L′ of the system (A_{n∗}^{a} | n ∈ ℕ) is integrally closed in L, and therefore L′[1/g] is integrally closed in L. But in view of (16.9.44), we have a natural identification of A_{0}-algebras : A_{0}[1/g] →^{∼} A_{0∗}^{a}[1/g] →^{∼} L′[1/g], whence the assertion.

(vi): Since C_{n} is the limit of the inverse system (C_{n}/b^{i}C_{n} | i ∈ ℕ) for every n ∈ ℕ, we obtain a natural isomorphism of A_{0}^{a}-algebras :

D →^{∼} lim_{i∈ℕ} D_{i}.

On the other hand, in view of (16.9.47) we have a natural isomorphism of A_{0}^{a}-algebras

(D_{i+1} ⊗_{A_{0}^{a}} A_{0}^{a}/b^{i}A_{0}^{a}) ⊗_{A_{0}^{a}} A_{n}^{a} →^{∼} C_{n}/b^{i}C_{n} for every n, i ∈ ℕ

whence, again by theorem 14.2.39(i), an induced isomorphism of A_{0}^{a}-algebras :

(16.9.50)  D_{i+1} ⊗_{A_{0}^{a}} A_{0}^{a}/b^{i}A_{0}^{a} →^{∼} D_{i}  for every i ∈ ℕ.

For every i ∈ ℕ, let π_{i} : D → D_{i} be the projection; notice that I := Ker π_{1} = bD + Ker π_{i+1} for every i ∈ ℕ, in light of the isomorphism D_{i+1}/bD_{i+1} →^{∼} D_{1} deduced from (16.9.50). By [75, Lemma 5.3.5(i)], it follows that I = ⋂_{i∈ℕ}(bD + \overline{I^{i+1}}) = \overline{bD} (where \overline{J} denotes the closure of J, for every ideal J ⊂ D : see [75, Def.5.3.1(iii)]). By [75, Lemma 5.3.8(iii)] we deduce that I = bD, and more generally b^{i}D = Ker π_{i} for every i ∈ ℕ. On the other hand, π_{i} is an epimorphism for every i ∈ ℕ, by lemma 14.2.24(i), so it induces an isomorphism D/b^{i}D →^{∼} D_{i}, for every i ∈ ℕ. Hence, the natural morphism D → lim_{i∈ℕ} D/b^{i}D is an isomorphism of A_{0}^{a}- algebras, so we have an induced isomorphism of A_{1}-algebras D_{∗} →^{∼} lim_{i∈ℕ}(D/b^{i}D)_{∗} (lemma 16.9.41). As we have already noticed, b is regular in D_{∗}, so for every i ∈ ℕ we get a short exact

sequence of A_{0}^{a}-modules 0 → D ⟶^{b^{i}·1_{D∗}} D → D/b^{i}D → 0 and since the functor (−)_{∗} is left exact, we deduce an injective A_{1}-linear map

D_{∗}/b^{i}D_{∗} → (D/b^{i}D)_{∗} for every i ∈ ℕ

from which it follows easily that also the induced map D_{∗} → lim_{i∈ℕ} D_{∗}/b^{i}D_{∗} is an isomorphism, hence D_{∗} is b-adically complete and separated. In order to show that (D_{∗}, 𝔪D_{∗}) is almost perfectoid, it remains only to check that the Frobenius endomorphism of D_{∗}/pD_{∗} induces an isomorphism D/bD →^{∼} (D/b^{p}D)_{(Φ)} of A_{0}^{a}/bA_{0}^{a}-algebras (where Φ denotes the Frobenius endomorphism of A_{0}/pA_{0}). But we have already seen that D/b^{i}D is the inverse limit of the system (C_{n}/b^{i}C_{n} | n ∈ ℕ), for every i ∈ ℕ, hence we are reduced to checking that the Frobenius endomorphism of C_{n∗}/pC_{n∗} induces an isomorphism C_{n}/bC_{n} →^{∼} (C_{n}/b^{p}C_{n})_{(Φ)} of A_{0}^{a}/bA_{0}^{a}-algebras, for every n ∈ ℕ. The latter is clear, since we know already that C_{n∗} is perfectoid (corollary 16.3.3). ♢

Next, recall that for every n ∈ ℕ we have A_{0}^{a}-algebras A_{n,0}, …, A_{n,r}, and an isomorphism of A_{0}^{a}-algebras A_{n}^{a} →^{∼} A_{n,0} × ⋯ × A_{n,r} such that C_{n,k} := A_{n,k} ⊗_{A_{n}^{a}} C_{n} is an A_{n,k}-module of constant rank k, for every k = 0, …, r ([75, Prop.4.3.27]). For every n ∈ ℕ and k = 0, …, r consider the map of sets :

χ^{(n,k)} : C_{n,k∗} → A_{n,k∗}^{a}[X] c ↦ χ_{c}(X)

which associates with every c ∈ C_{n,k∗} the characteristic polynomial χ_{c}(X) of the A_{n,k}^{a}-linear endomorphism c · 1_{C_{n,k}} of C_{n,k} (see [75, §4.4.29]); by [75, Prop.4.4.30] we have χ_{c}(c) = 0 for every such c. By virtue of (16.9.46), we see that the restriction map A_{n+1} → A_{n} induces isomorphisms of A_{0}^{a}-algebras

A_{n}^{a} ⊗_{A_{n+1}^{a}} A_{n+1,k} →^{∼} A_{n,k} for every k = 0, …, r

and from the compatibility of traces with ring extensions ([75, Prop.4.1.8(ii)]), we deduce for every n ∈ ℕ and k = 0, …, r a commutative diagram :

\xymatrix{C_{n+1,k∗} \ar[r]^{χ^{(n+1,k)}} \ar[d] & A_{n+1,k∗}^{a}[X] \ar[d] \\ C_{n,k∗} \ar[r]^{χ^{(n,k)}} & A_{n,k∗}^{a}[X]}

whose vertical arrows are induced by the induced projection A_{n+1,k} → A_{n,k}. For every k = 0, …, r, let A^{(k)} be the limit of the induced inverse system of A_{0}^{a}-algebras (A_{n,k} | n ∈ ℕ), and set D^{(k)} := A^{(k)} ⊗_{A_{0}^{a}} D; in view of (16.9.44), there follows an isomorphism of A_{0}^{a}-algebras D →^{∼} D^{(0)} × ⋯ × D^{(r)}. Notice as well that χ_{c}(X) is a monic polynomial of degree k, for every n ∈ ℕ, every k = 0, …, r and and every c ∈ C_{n,k∗}; in light of (16.9.48), it follows easily that the limit of the system of maps (χ^{(n,k)} | n ∈ ℕ) is a well defined map of sets

χ^{(k)} : D_{∗}^{(k)} → A_{∗}^{(k)}[X]  d ↦ χ_{d}(X)  for every k = 0, …, r.

Moreover, χ_{d}(X) is a monic polynomial with χ_{d}(d) = 0 for every k = 0, …, r and every d ∈ D_{∗}^{(k)}. Taking into account lemma 16.9.41, it follows that D_{∗} is an integral A_{1}-algebra. Since both D_{∗} and B are integrally closed subrings of E (claim 16.9.49(v)), and since A_{1} ⊂ A_{0}[1/g], we deduce that D_{∗} ⊂ B, and hence D_{∗} is also the integral closure of A_{1} in B, i.e. D_{∗} = B_{1}. Then assertions (i), (ii) and (iii) of the theorem follow from claim 16.9.49.

(iv): By construction, A_{1} is an open bounded subring of A, hence it is complete and separated for the topology induced by A, which agrees with the b-adic topology (proposition 8.3.18(i,ii)). In order to show that A_{1} is integrally closed in A_{0}[1/g], consider the case where B = A_{0}[1/g]; then C_{n} = A_{0,n} for every n ∈ ℕ, so that B_{1} = D_{∗} = A_{0∗}^{a} = A_{1}, whence the assertion. □

Remark 16.9.51. (i) In the situation of theorem 16.9.42, let g′_{•} := (g′_{n} | n ∈ ℕ) ∈ E_{0} be another element, and g′ := g′_{0} = \overline{u}_{A_{0}}(g′_{•}). Let also A′′_{0} := Im(A_{0} → A_{0}[1/(gg′)]) and set

𝔪′ := ⋃_{n∈ℕ} g′_{n}A_{0}  𝔪′′ := 𝔪 · 𝔪′  A_{2} := {a ∈ A_{0}[1/(gg′)] | 𝔪′′ · a ⊂ A′′_{0}}.

Clearly (A_{0}, 𝔪′) and (A_{0}, 𝔪′′) are two other basic setups; for every A_{0}-module M, let (M, 𝔪)^{a} ∈ (A_{0}, 𝔪)^{a}-Mod (resp. (M, 𝔪′)^{a} ∈ (A_{0}, 𝔪′)^{a}-Mod, resp. (M, 𝔪′′)^{a} ∈ (A_{0}, 𝔪′′)^{a}-Mod) be the image of M. We may apply theorem 16.9.42 with (A_{0}, 𝔪) and B replaced respectively by (A_{0}, 𝔪′′), and B[1/g′] : hence, let B_{2} be the integral closure of A_{2} in B[1/g′]; then B_{2} = (B_{2}, 𝔪′′)_{∗}^{a} is an almost perfectoid A_{0}-algebra relative to (A_{0}, 𝔪′′), and (B_{2}/b^{n}B_{2}, 𝔪′′)^{a} is an étale (A_{0}/b^{n}A_{0}, 𝔪′′)^{a}-algebra of finite rank, for every n ∈ ℕ. The localizations A_{0}[1/g] → A_{0}[1/(gg′)] and B → B[1/g′] clearly restrict to ring homomorphisms A_{1} → A_{2} and B_{1} → B_{2}.

Now, (A_{0}, 𝔪′′)^{a} is integrally closed in (A_{0}[1/(gg′)], 𝔪′′)^{a} (theorem 16.9.42(iv) and lemma 16.9.41), hence the same holds for (A_{0}[1/g], 𝔪′′)^{a}; since B is a finite étale A_{0}[1/g]-algebra, it follows that (B, 𝔪′′)^{a} is integrally closed in (B[1/g′], 𝔪′′)^{a} (proposition 14.3.25). Taking into account theorem 16.9.42(v), remark 14.2.7(ii) and [75, Lemma 8.2.28] we deduce :

B_{2} = (B_{2}, 𝔪′′)_{∗}^{a} = i.c.((A_{0}, 𝔪′′)^{a}, (B[1/g′], 𝔪′′)^{a})_{∗}
= i.c.((A_{0}, 𝔪′′)^{a}, (B, 𝔪′′)^{a})_{∗}
= (i.c.((A_{0}, 𝔪′′)_{∗}^{a}, (B, 𝔪′′)_{∗}^{a}), 𝔪′′)_{∗}^{a}
= (i.c.((A_{1}, 𝔪′)_{∗}^{a}, (B, 𝔪′)_{∗}^{a}), 𝔪′′)_{∗}^{a}
= ((i.c.(A_{1}, B), 𝔪′)_{∗}^{a}, 𝔪′′)_{∗}^{a}
= ((B_{1}, 𝔪′)_{∗}^{a}, 𝔪′′)_{∗}^{a}
= (B_{1}, 𝔪′′)_{∗}^{a}
= (B_{1}, 𝔪′)_{∗}^{a}.

(ii) As a special case of (i), we may take g′_{•} := b_{•} such that b := b_{0} ∈ A^{×} ∩A_{0}^{∘∘}. Then, endow B_{1} with its b-adic topology, so that B_{1} is a subring of definition of the Tate ring B_{1}[1/b], and by theorem 16.9.42(i) it is p-root closed in B_{1}[1/b], hence B_{1}[1/b]^{∘∘} ⊂ B_{1} (example 16.9.4). It follows easily that (B_{1}[1/b]^{∘}, 𝔪′)^{a} = (B_{1}, 𝔪′)^{a}, so we have inclusions of rings :

B_{1} ⊂ B^{∘} ⊂ (B_{1}, 𝔪′)_{∗}^{a} = (B_{1}[1/b]^{∘}, 𝔪′)_{∗}^{a} ⊂ B_{1}[1/b].

On the other hand, it is easily seen that (B_{1}, 𝔪′)_{∗}^{a} ⊂ B_{1}[1/b]^{∘}, so B_{2} = (B_{1}, 𝔪′)_{∗}^{a} = B_{1}[1/b]^{∘}.

(iii) In the situation of theorem 16.9.42, suppose that A is an 𝔽_{p}-algebra; then A_{0} = E_{0} is a perfect 𝔽_{p}-algebra, and it follows easily that the same holds for A_{1}. Then, set X := Spec A_{1} and Z := Spec A_{1}/gA_{1}; by theorem 16.8.2, the pair (X, Z) is normal and almost pure, hence B_{1}^{a} is an étale A_{1}^{a}-algebra of finite rank (proposition 14.4.8).

(iv) If A is not an 𝔽_{p}-algebra, we know neither whether A_{1} is perfectoid, nor whether B_{1}^{a} is an étale A_{1}^{a}-algebra. However, by theorem 16.9.42(ii) and lemma 14.3.15(ii), for every n ∈ ℕ we have a well defined map

χ_{B_{1}/b^{n}B_{1}} : (B_{1}/b^{n}B_{1})_{∗}^{a} → (A_{0}/b^{n}A_{0})_{∗}^{a}[T] b ↦ det((1 + bT) · 1_{B_{1}^{a}/b^{n}B_{1}^{a}})

and clearly the inverse system of maps (χ_{B_{1}/b^{n}B_{1}} | n ∈ ℕ) amounts to a well defined map

χ_{B_{1}} : B_{1} → A_{1}[T].

By the same token, we may associate to the finite étale A_{0}[1/g]-algebra B the map

χ_{B} : B → A_{0}[1/g, T] b ↦ det((1 + bT) · 1_{B})

and we claim that

χ_{B}(b) = χ_{B_{1}}(b) for every b ∈ B_{1}.

For the proof, define the inverse system of A_{0}-algebras A_{•} := (A_{n} | n ∈ ℕ), and for every n ∈ ℕ the étale A_{n}^{a}-algebra C_{n} of rank ≤ r, as in the proof of theorem 16.9.42; recall that A_{1} (resp. B_{1}) is the limit of A_{•} (resp. of (C_{n∗} | n ∈ ℕ)). It follows easily that χ_{B_{1}} is also the inverse limit of the system of similar maps (χ_{C_{n}} | n ∈ ℕ). On the other hand, A_{0}[1/g] is integrally closed in the limit L of the induced inverse system (A_{n}[1/g] | n ∈ ℕ), and B is the integral closure of A_{0}[1/g] in the inverse system E of the induced inverse system (C_{n}[1/g] | n ∈ ℕ); also, the natural map E → B ⊗_{A_{0}[1/g]} L is an isomorphism (see the proof of claim 16.9.49). Thus, both χ_{B} and χ_{B_{1}} are restrictions of the corresponding map χ_{E} : E → L, whence the assertion.

(v) Furthermore, set R := B_{1} ⊗_{A_{1}} B_{1}, let R^{∧} be the b-adic completion of R, and

e_{B/A_{0}[1/g]} ∈ R[1/g] = B ⊗_{A_{0}[1/g]} B

the diagonal idempotent of the A_{0}[1/g]-algebra B; we notice :

Corollary 16.9.52. In the situation of theorem 16.9.42, we have :

(i) The following conditions are equivalent:

(a) g^{γ} · e_{B/A_{0}[1/g]} ∈ R′ := Im(R → R[1/g]) for every γ ∈ ℕ[1/p] \ {0}.

(b) For every γ ∈ ℕ[1/p] \ {0} there exists N_{γ} ∈ ℕ such that b^{N_{γ}}g^{γ} · e_{B/A_{0}[1/g]} ∈ R′.

(c) B_{1}^{a} is an étale A_{1}^{a}-algebra.

(d) B_{1}^{a} is a weakly unramified A_{1}^{a}-algebra.

(e) B_{1}^{a} is an almost finitely generated A_{1}^{a}-module.

(f) B_{1}^{a} is an almost finite projective A_{1}^{a}-module.

(g) B_{1}^{a}[1/b] is a weakly unramified A_{1}^{a}[1/b]-algebra.

(h) B_{1}^{a}[1/b] is an almost finitely generated A_{1}^{a}[1/b]-module. (ii) There exists n ∈ ℕ such that g^{n} annihilates the kernel and cokernel of the completion

map R → R^{∧}. Also, Ann_{R^{∧}}(g)^{a} = 0.

Proof. Obviously (c)⇒(a),(d); also (d)⇒(g), (a)⇒(b) and (f)⇒(e)⇒(h).

(a)⇒(c),(f): The assumption implies that e_{B/A_{0}[1/g]} lies in the image of the induced map

R_{∗}^{a} → B ⊗_{A_{0}[1/g]} B.

Then the assertion follows from proposition 14.3.21and theorem 16.9.42(iv)).

(f)⇒(c): We consider the commutative diagram of A_{1}^{a}-algebras :

(16.9.53)  \xymatrix{R^{a} \ar[r]^{φ} \ar[d]_{μ^{a}} & lim_{n∈ℕ} R^{a}/b^{n}R^{a} \ar[d] \\ B_{1}^{a} \ar[r] & lim_{n∈ℕ} B_{1}^{a}/b^{n}B_{1}^{a}}

where μ^{a} is the multiplication law, and the right vertical arrow is the inverse limit of the system of multiplication laws (R^{a}/b^{n}R → B_{1}^{a}/b^{n}B_{1}^{a} | n ∈ ℕ). According to theorem 16.9.42(i), B_{1} is b-adically complete and separated; then by lemma 14.2.24(iii) the morphism φ is an isomorphism. By theorem 16.9.42(ii), for each n ∈ ℕ we have a diagonal idempotent e_{n} ∈ (R/b^{n}R)_{∗}, and it is easily seen that the system (e_{n} | n ∈ ℕ) corresponds, under the isomorphism φ, to an idempotent element e_{B_{1}^{a}/A_{1}^{a}} ∈ R_{∗}^{a} such that μ(_{B_{1}^{a}/A_{1}^{a}}) = 1 and x · e_{B_{1}^{a}/A_{1}^{a}} = 0 for every x ∈ (Ker μ^{a})_{∗}. Then the assertion follows from [75, Prop.3.1.4].

(d)⇒(a): Let J := ⋃_{n∈ℕ} Ann_{R}(g^{n}) and \overline{R} := R/J. By construction, g is a regular element of B_{1}, so that JB_{1} = 0, for the R-algebra structure on B_{1} given by the multiplication map μ : R → B_{1}. On the other hand, by assumption μ^{a} is a flat morphism of A_{1}^{a}-algebras; hence μ^{a} ⊗_{R} \overline{R} : \overline{R} → B_{1} is still flat. Also, the localisation R → R[1/g] factors through a monomorphism \overline{R} → R[1/g] of A_{1}^{a}-algebras, and since B is an étale A_{0}[1/g]-algebra, B = B_{1}[1/g] is a projective R[1/g]-module of finite rank. By [75, Prop.2.4.19] it follows that B_{1} is an almost finite projective \overline{R}-module, and therefore Ker (μ^{a} ⊗_{R} \overline{R}) is generated by an idempotent almost element d ∈ \overline{R}_{∗} ([75, Rem.3.1.8]). For every γ ∈ ℕ[1/p] \ \{0\}, we then find e_{γ} ∈ R whose image in (R/J)_{∗} agrees with g^{γ} · (1 − d). It follows easily that the image of e_{γ} in R[1/g] agrees with g^{γ}e_{B/A_{0}[1/g]}, whence the contention.

(g)⇒(b): The assumption means that μ^{a} ⊗_{A_{1}} A_{1}[1/b] : R[1/b]^{a} → B_{1}[1/b]^{a} is a flat morphism, hence the same holds for μ^{a} ⊗_{R} \overline{R}[1/b], and arguing as in the foregoing we find for every γ ∈ ℕ[1/p] \ {0} an integer N_{γ} ∈ ℕ and e_{γ} ∈ R whose image in R[1/(gb)] agrees with the image of g^{γ}b^{N_{γ}}e_{B/A_{0}[1/g]}. Notice that, since B_{1}[1/g] = B is a flat A_{1}[1/g]-algebra, and since b is regular in A_{1}, it follows that b is regular also in R[1/g], hence the localization R[1/g] → R[1/(gb)] is injective, and we deduce that the image of e_{γ} agrees with g^{γ}b^{N_{γ}}e_{B/A_{0}[1/g]} already in R[1/g], as required.

(b)⇒(a): Define the inverse system of A_{0}-algebras (A_{n} | n ∈ ℕ) as in the proof of theorem 16.9.42, and for every n ∈ ℕ let C_{n} be the unique étale A_{n}^{a}-algebra up to unique isomorphism, such that C_{n}[1/g] is isomorphic to A_{n}[1/g] ⊗_{A_{0}} B. By claim 16.9.43(ii), for every i ∈ ℕ the induced inverse system (A_{n}/b^{i}A_{n} | n ∈ ℕ) is almost essentially constant relative to the basic setup (A_{0}, 𝔪). In view of (16.9.46) the same follows for the inverse system (C_{n}^{a}/b^{i}C_{n}^{a} | n ∈ ℕ), for every i ∈ ℕ; but the proof of theorem 16.9.42 also shows that the inverse limit of the latter system is naturally isomorphic to B_{1}^{a}/b^{i}B_{1}^{a}, for every i ∈ ℕ. Thus, set R_{n} := C_{n} ⊗_{A_{n}^{a}} C_{n} for every n ∈ ℕ; we deduce that for every i ∈ ℕ and there exists n(i, γ) ∈ ℕ such that g^{γ} annihilates the kernel of the induced morphism

R^{a}/b^{i}R^{a} → R_{n(i,γ)}/b^{i}R_{n(i,γ)}.

Notice also that, since g is regular in A_{n}, and C_{n} is a flat A_{n}^{a}-algebra, then g is regular in C_{n}, and the localization R_{n} → R_{n}[1/g] is a monomorphism of A_{n}^{a}-modules, for every n ∈ ℕ. Moreover, the image of e_{B/A_{0}[1/g]} in R_{n}[1/g] is the diagonal idempotent of the étale A_{n}[1/g]- algebra C_{n}[1/g], and the latter agrees with the image of the diagonal idempotent e_{n} ∈ R_{n∗} of the étale A_{n}^{a}-algebra C_{n}. Summing up, we conclude that for every γ ∈ ℕ[1/p] \ {0}, the image of e_{γ} vanishes in R_{n(N_{γ},β)}/b^{N_{γ}}R_{n(N_{γ},γ)}, and hence g^{2γ}e_{γ} vanishes in R/b^{N_{γ}}R, i.e. g^{2γ}e_{γ} = b^{N_{γ}}e′_{γ} for some e′_{γ} ∈ R. Since b is regular in R[1/g], it follows that the image of e′_{γ} in R[1/g] agrees with g^{3γ}e_{B/A_{0}[1/g]}, as required.

(h)⇒(b): By assumption, for every γ ∈ ℕ[1/p] \ {0} there exists n(γ) ∈ ℕ and an A_{1}-linear map f_{γ} : A_{1}^{⊕n(γ)} → B_{1} such that

(16.9.54)  g^{γ} · Coker(A_{1}[1/b] ⊗_{A_{1}} f_{γ}) = 0.

Then, let E_{γ} be the fibre product in the cartesian diagram of A_{1}-modules :

𝒟_{γ}  :  \xymatrix{E_{γ} \ar[r]^{f′_{γ}} \ar[d] & B_{1} \ar[d]^{g^{γ}1_{B_{1}}} \\ A_{1}^{⊕n(γ)} \ar[r]^{f_{γ}} & B_{1}.}

By virtue of (16.9.54), the image of A_{1}[1/b] ⊗_{A_{1}} g^{γ}1_{B_{1}} lies in the image of A_{1}[1/b] ⊗_{A_{1}} f_{γ}; on the other hand, the diagram A_{1}[1/b] ⊗_{A_{1}} 𝒟_{γ} is still cartesian, so the map A_{1}[1/b] ⊗_{A_{1}} f′_{γ} is surjective. Recall that b is regular in both A_{1} and B_{1}, therefore also in E_{γ}; then endow A_{1}[1/b] (resp. B_{1}[1/b]) with the f-adic topology 𝒯_{A_{1}} (resp. 𝒯_{B_{1}}) such that A_{1} (resp. B_{1}) is a subring of definition, with ideal of definition bA_{1} (resp. b_{1}B_{1}); since A_{1} and B_{1} are b-adically complete and separated (theorem 16.9.42(i)), then 𝒯_{A_{1}} and 𝒯_{B_{1}} are complete and separated (proposition 8.3.33). Endow also A_{1}[1/b]^{⊕n(γ)} with the product topology 𝒯_{A_{1}} × ⋯ × 𝒯_{A_{1}}; it is easily seen that all the maps of A_{1}[1/b] ⊗_{A_{1}} 𝒟_{γ} are continuous for these topologies, hence A_{1}[1/b] ⊗_{A_{1}} E_{γ} is a closed subset of A_{1}[1/b] ⊗_{A_{1}} (A_{1}^{⊕n(γ)} ⊕ B_{1}), and is therefore complete and separated for the topology induced by the inclusion in this A_{1}-module; moreover, since A_{1}^{⊕n(γ)} is open in A_{1}[1/b]^{⊕n(γ)} and B_{1} is open in B_{1}[1/b], we see that E_{γ} is open in A_{1}[1/b] ⊗_{A_{1}} E_{γ}. By the Banach open mapping theorem ([100, Lemma 2.4]), there exists therefore N_{γ} ∈ ℕ such that b^{N_{γ}}B_{1} ⊂ Im(f′_{γ}). It follows easily that

(16.9.55)  g^{γ}b^{N_{γ}}B_{1} ⊂ Im(f_{γ}).

By virtue of lemma 14.2.24(iv), we deduce that

(16.9.56)  g^{γ}b^{N_{γ}}Coker φ = 0

where φ is as in (16.9.53). On the other hand, since A_{1}[1/(gb)] ⊗_{A_{1}} f_{γ} : A[1/g]^{⊕n(γ)} → B[1/b] is surjective, and since B[1/b] is a projective A[1/g]-module, there exists an A[1/g]-linear map h : B[1/b] → A[1/g]^{⊕n(γ)} such that f_{γ} ∘ h = 1_{B[1/b]}. Let ε_{1}, …, ε_{n(γ)} be the canonical basis of the free A[1/g]-module A[1/g]^{⊕n(γ)}, and pick k ∈ ℕ such that h ∘ f_{γ}(b^{k}g^{k}ε_{i}) ∈ A_{1}^{⊕n(γ)} for every i = 1, …, n(γ). In view of (16.9.55), we deduce an A_{1}-linear map :

h′ : B_{1} → A_{1}^{⊕n(γ)} x ↦ b^{N_{γ}+k}g^{k+γ}h(x)

and by construction we have f_{γ} ∘ h′ = b^{N_{γ}+k}g^{k+γ}1_{B_{1}}. Then lemma 14.2.24(iv) implies that

(16.9.57)  b^{N_{γ}+k}g^{k+γ}Ker φ = 0.

Now, let e_{•} := (e_{n} | n ∈ ℕ) ∈ lim_{n∈ℕ}(R^{a}/b^{n}R^{a})_{∗} be the compatible system of diagonal idempotents of the étale (A_{0}/b^{n}A_{0})^{a}-algebras B_{1}^{a}/b^{n}B_{1}^{a}, for every n ∈ ℕ; by (16.9.56) there exists e′ ∈ R such that φ(e′) = g^{2γ}b^{N_{γ}}e_{•}. Then we have

φ(x · e′) = 0 for every x ∈ Ker μ.

Combining with (16.9.57), it follows that b^{N_{γ}+k}g^{k+γ}x · e′ = 0 in R, for every x ∈ Ker μ. But we have already observed that b is regular in R[1/g]; we then easily conclude that the image of e′ in R[1/g] agrees with g^{2γ}b^{N_{γ}}e_{B_{1}/A_{0}[1/g]}.

(ii): Pick n ∈ ℕ such that g^{n}e_{B/A_{0}[1/g]} lies in the image of the localization B_{1} ⊗_{A_{1}} B_{1} → B ⊗_{A_{0}[1/g]} B. Arguing as in the proof of claim 14.3.23, we then find k ∈ ℕ and A_{1}-linear maps B_{1} → A_{1}^{⊕k} → B_{1} whose composition is g^{k} · 1_{B_{1}}. Recall now that R^{∧} is the limit of the inverse system (B_{1} ⊗_{A_{1}} B_{1}/b^{i}B_{1} | i ∈ ℕ); then the first assertion follows from lemma 14.2.24(iv).

In order to show the second assertion of (ii), let C_{n} be the étale A_{n}^{a}-algebra of finite rank as in the proof of theorem 16.9.42, for every n ∈ ℕ. Since A_{n} is b-adically complete and separated, the almost projective A_{n}^{a}-module of finite rank S_{n} := C_{n} ⊗_{A_{n}} C_{n} is b-adically complete and separated as well, for every n ∈ ℕ, by [75, Claim 5.3.25], and moreover Ann_{S_{n}}(g) = 0, since Ann_{A_{n}}(g) = 0; then we are easily reduced to checking :

Claim 16.9.58. (R^{∧})^{a} represents the limit of the inverse system of A_{0}-algebras (S_{n} | n ∈ ℕ).

Proof of the claim. It suffices to show that the A_{0}^{a}/b^{k}A_{0}^{a}-algebra (B_{1}/b^{k}B_{1} ⊗_{A_{1}/b^{k}A_{1}} B_{1}/b^{k}B_{1})^{a} represents the limit of the induced inverse system (C_{n}/b^{k}C_{n} ⊗_{A_{n}^{a}/b^{k}A_{n}^{a}} C_{n}/b^{k}C_{n} | n ∈ ℕ), for every k ∈ ℕ. However, the proof of theorem 16.9.42 shows that the inverse system (C_{n}/b^{k}C_{n} | n ∈ ℕ) is almost essentially constant for every k ∈ ℕ, and the same holds fo the inverse system (A_{n}^{a}/b^{k}A_{n}^{a} | n ∈ ℕ), by claim 16.9.43(ii). Then the assertion follows from proposition 14.2.38. □

We conclude with a discussion of some further constructions that we do not need (though, see remark 17.5.17), but may be useful for other questions, and are related to some recent work of Y.André and others as well.

16.9.59. Let A be a perfectoid Tate ring, A_{0} ⊂ A a perfectoid subring of definition, p ∈ ℕ a prime integer, and b ∈ A_{0}^{∘∘} ∩ A^{×} with pA_{0} ⊂ b^{p}A_{0}. We consider the categories :

A_{0}-cp.Adic  A_{0}-Tate  A_{0}-Perf.Adic  A_{0}-Perf.Tate

such that :

• the objects of A_{0}-cp.Adic are the adic ring homomorphisms A_{0} → B_{0} of complete separated topological rings, and the morphisms are the maps of A_{0}-algebras

• A_{0}-Tate is the full subcategory of A_{0}-cp.Adic whose objects are the A_{0}-algebras B_{0} such that the localisation B_{0} → B_{0}[b^{−1}] is injective

• A_{0}-Perf.Adic (resp. A_{0}-Perf.Tate) is the full subcategory of A_{0}-cp.Adic (resp. of

A_{0}-Tate) whose objects are the perfectoid A_{0}-algebras. Notice that every morphism in these categories is adic. We wish to exhibit right adjoints for the inclusion functors

(16.9.60)  A_{0}-Perf.Adic → A_{0}-cp.Adic  A_{0}-Perf.Tate → A_{0}-Tate.

Thus, consider any B_{0} ∈ Ob(A_{0}-cp.Adic), set \overline{A}_{0} := A_{0}/pA_{0}, \overline{B}_{0} := B_{0}/pB_{0}, and let \overline{β}_{•} := (\overline{β}_{n} | n ∈ ℕ) ∈ E_{0} := E(\overline{A}_{0}) be any element such that \overline{β}_{0} ∈ \overline{A}_{0} is the class of b. Endow E(\overline{B}_{0}) with its \overline{β}_{•}-adic topology 𝒯_{E(\overline{B}_{0})}; let also 𝒯_{\overline{B}_{0}} be the b-adic topology of \overline{B}_{0}. Then the topology of E(\overline{B}_{0}, 𝒯_{\overline{B}_{0}}) is complete, separated, adic, and coarser than 𝒯_{E(\overline{B}_{0})} (remark 9.4.9(ii)). By lemma 8.3.12, it follows that 𝒯_{E(\overline{B}_{0})} is complete and separated. Then the A_{0}-algebra

B_{0}^{♮} := A_{0} ⊗_{W (E_{0})} W (E(\overline{B}_{0}), 𝒯_{E(\overline{B}_{0})})

is a perfectoid ring for its b-adic topology (example 16.3.2(ii)). The structure map A_{0} → B_{0} of B_{0}, and the map u_{B_{0}} : W (E(\overline{B}_{0})) → B_{0} induce an adic map

ε_{B_{0}} : B_{0}^{♮} → B_{0} 1 ⊗ (x_{n} | n ∈ ℕ) ↦ ∑_{n∈ℕ} p^{n} · \overline{u}_{A}(x_{n}^{p^{−n}})

and notice that ε_{B_{0}} is an isomorphism if B_{0} is perfectoid. Next, let β_{•} := (β_{n} | n ∈ ℕ) ∈ E(A_{0}) be the preimage of \overline{β}_{•} under the isomorphism E(A_{0}) →^{∼} E_{0} induced by the projection A_{0} → \overline{A}_{0}. Hence β_{0} − b ∈ pA_{0}, so that β_{0}/b ∈ A_{0}^{×}; if B_{0} ∈ Ob(A_{0}-Tate), then the image of b is regular in B_{0}, so the same holds for the image of β_{0}, and finally, the image of β_{•} is a regular element of E(\overline{B}_{0}) (proposition 16.4.17); by the same token, the image of β_{0} in B_{0}^{♮} is then regular, and hence the same holds for the image of b. Hence, in this case B_{0}^{♮} is an object of A_{0}-Perf.Tate. Moreover, every morphism g : B_{0} → C_{0} of A_{0}-Tate induces an adic map of A_{0}-algebras

g^{♮} : B_{0}^{♮} → C_{0}^{♮} 1 ⊗ (\overline{x}_{n} | n ∈ ℕ) ↦ 1 ⊗ (E(\overline{g})(\overline{x}_{n}) | n ∈ ℕ)

with \overline{g} := g ⊗_{ℤ} ℤ/pℤ : \overline{B}_{0} → \overline{C}_{0}. We have therefore well defined functors

(−)^{♮} : A_{0}-cp.Adic → A_{0}-Perf.Adic (−)^{♮} : A_{0}-Tate → A_{0}-Perf.Tate.

Now, let C ∈ Ob(A_{0}-Perf); to every adic map f : C → B_{0}^{♮} let us attach f^{∗} := ε_{B_{0}}∘f : C → B_{0}, and to every adic map h : C → B_{0} let us attach h^{∗} := h^{♮} ∘ ε_{C}^{−1} : C → B_{0}^{♮}. By a simple inspection we get h^{∗∗} = h. In order to check that f^{∗∗} = f, it suffices to show that ε_{B_{0}^{♮}} = ε_{B_{0}}^{♮}.

Thus, let \overline{x}_{•} := (x_{n} | n ∈ ℕ) ∈ W (E(\overline{B}_{0}^{♮})) be any element; via the natural identification E(B_{0}^{♮}) →^{∼} E(\overline{B}_{0}^{♮}), we may view \overline{x}_{•} as a system x_{••} := (x_{n,k} | n, k ∈ ℕ) of elements of B_{0}^{♮} such that x_{n,k+1}^{p} = x_{n,k} for every n, k ∈ ℕ; then we have :

ε_{B_{0}^{♮}}(1 ⊗ x_{••}) = ∑_{n∈ℕ} p^{n}x_{n,n} ε_{B_{0}}^{♮}(1 ⊗ \overline{x}_{•}) = 1 ⊗ (E(\overline{ε}_{B_{0}})(\overline{x}_{n}) | n ∈ ℕ).

But we have ∑_{n∈ℕ} p^{n} · τ(E(\overline{ε}_{B_{0}})(\overline{x}_{n})^{p^{−n}}) = (E(\overline{ε}_{B_{0}})(\overline{x}_{n}) | n ∈ ℕ) in W (E(B_{0}^{♮})), by virtue of (9.3.46), where τ denotes the Teichmüller representative. Hence, let \overline{y}_{•} := (\overline{y}_{k} | k ∈ ℕ) ∈ E(\overline{B}_{0}^{♮}) be any element, which we identify with a unique element (y_{k} | k ∈ ℕ) ∈ E(B_{0}^{♮}); we come down to showing that 1 ⊗ τ(\overline{y}_{•}) = y_{0} = \overline{u}_{B_{0}^{♮}}(y_{•}) in B_{0}^{♮}. But we have 1 ⊗ τ(\overline{y}_{•}) = u_{B_{0}^{♮}}(τ(\overline{y}_{•})), so the assertion follows from lemma 16.1.1(iv). This shows that the rule f ↦ f^{∗} yields the sought adjunctions between (16.9.60) and the functors (−)^{♮}.

16.9.61. In the situation of (16.9.59), let also (A_{0}, 𝔪) be a basic setup, such that 𝔪 is the filtered union of a countable system of principal subideals, and consider as well the categories

(A_{0}, 𝔪)^{a}-Perf.Adic (A_{0}, 𝔪)^{a}-Perf.Tate

with the same objects as A_{0}-Perf.Adic (resp. as A_{0}-Perf.Tate), and whose morphisms B_{0} → C_{0} are the morphisms B_{0}^{a} → C_{0}^{a} of (A_{0}, 𝔪)^{a}-algebras. We have obvious forgetful functors :

(16.9.62)  A_{0}-Perf.Adic →(A_{0}, 𝔪)^{a}-Perf.Adic A_{0}-Perf.Tate →(A_{0}, 𝔪)^{a}-Perf.Tate  B_{0} ↦ B_{0}^{a}.

As explained in (16.9.31), we may attach to 𝔪 an ideal 𝔪_{E} ⊂ E_{0}, and remark 16.9.32(i) implies that (E_{0}, 𝔪_{E}) is a basic setup compatible with (A_{0}, 𝔪), in the sense of definition 16.8.10. For every B_{0} ∈ Ob(A_{0}-cp.Adic), we then let B_{0}^{a} (resp. E(B_{0})^{a}) be the (A_{0}, 𝔪)^{a}-algebra attached to B (resp. the (E_{0}, 𝔪_{E})^{a}-algebras attached to E(B_{0})).

Theorem 16.9.63. Let B_{0} ∈ Ob(A_{0}-cp.Adic), and endow C := (B_{0})_{!!}^{a} with its b-adic topology, and D := E(B_{0})_{!!}^{a} with its \overline{β}_{•}-adic topology, as in (16.9.59). We have :

(i) (B_{0})_{∗}^{a} ∈ Ob(A_{0}-cp.Adic), and if B_{0} ∈ Ob(A_{0}-Tate), then (B_{0})_{∗}^{a} ∈ Ob(A_{0}-Tate). (ii) There exists a natural isomorphism E(B_{0∗}^{a}) →^{∼} (E(B_{0}))_{∗}^{a} of E_{0}-algebras. (iii) Suppose moreover that the basic setup (B_{0}, 𝔪B_{0}) is almost perfectoid; then we have :

(a) ε_{B_{0}}^{a} : (B_{0}^{♮})^{a} → B_{0}^{a} is an isomorphism.

(b) The completions C^{∧} and D^{∧} of C and D are perfectoid.

(c) C^{∧}/Ann_{C^{∧}}(b) ∈ Ob(A_{0}-Perf.Tate).

(d) There exists a natural isomorphism E(C^{∧}) →^{∼} D^{∧}. (iv) The natural morphisms (C^{∧})^{a} → B_{0}^{a} and (D^{∧})^{a} → E(B_{0})^{a} are isomorphisms.

(v) If B_{0} is perfectoid, the same holds for C^{∧} and D^{∧}.

(vi) The functors (16.9.62) admit both left and right adjoints.

Proof. (i): If b · 1_{B_{0}} is an injective map, the same holds for b · 1_{B_{0∗}^{a}}, because the functor (−)_{∗}^{a} commutes with limits; the assertion then follows from claim 16.9.39.

(ii): By remark 16.9.32(iii), the natural map B_{0} → B_{0∗}^{a} induces an isomorphism E(B_{0})_{∗}^{a} →^{∼} E(B_{0∗}^{a})_{∗}^{a}; in light of (i), we may then replace B_{0} by B_{0∗}^{a}, and assume from start that B_{0} = B_{0∗}^{a}, in which case we show that the natural map E(B_{0}) → E(B_{0})_{∗}^{a} is an isomorphism. To this aim, consider the system of sets and maps of sets S_{••} := (S_{ij}, φ_{ij}^{v}, φ_{ij}^{h} | i, j ∈ ℕ) with S_{ij} := (B_{0}/b^{j}B_{0})_{∗}^{a} for every i, j ∈ ℕ, and where :

• φ_{ij}^{h} : S_{i+1,j} → S_{ij} is given by the rule : x ↦ x^{p} for every x ∈ (B_{0}/b^{j}B_{0})_{∗}^{a}

• φ_{ij}^{v} : S_{i,j+1} → S_{ij} is induced by the projection B_{0}/b^{j+1}B_{0} → B_{0}/b^{j}B_{0}.

We compute the inverse limit L of the system S_{••} in two different ways :

L = lim_{i∈ℕ} lim_{j∈ℕ} S_{ij} →^{∼} lim_{j∈ℕ} lim_{i∈ℕ} S_{ij}.

Notice that, for fixed i ∈ ℕ, the system S_{i•} := (S_{ij}, φ_{ij}^{v} | j ∈ ℕ) consists of rings and ring homomorphisms, and since the functor (−)_{∗}^{a} commutes with limits, the limit of S_{i•} represents L_{i} := B_{0∗}^{a} = B; then, the system of maps (φ_{ij}^{h} | j ∈ ℕ) is a natural transformation S_{i+1•} → S_{i•} whose limit φ_{i}^{h} : L_{i+1} → L_{i} is obviously the map given by the rule : x ↦ x^{p} for every x ∈ B. Thus L := lim_{i∈ℕ}(S_{i•}, φ_{i}^{h}) represents the set underlying E(B). Next, for every j ∈ ℕ, let L′_{j} be the limit of the system of sets S_{•j} := (S_{ij}, φ_{ij}^{h} | i ∈ ℕ); again, the system (φ_{ij}^{v} | i ∈ ℕ) is a natural transformation S_{•j+1} → S_{•j}, inducing a map φ_{j}^{v} : L′_{j+1} → L_{j} and L is also lim_{j∈ℕ}(L′_{j}, φ_{j}^{v}). Notice now that L′_{0} is the set underlying E(B_{0}/bB_{0})_{∗}^{a}, since the functor (−)_{∗}^{a} commutes with limits. We are therefore reduced to checking that φ_{j}^{v} is bijective for every j ∈ ℕ. To this aim, it suffices to exhibit a system of maps (s_{ij} : S_{i+1,j} → S_{i,j+1} | i ∈ ℕ) such that

s_{ij} ∘ φ_{i+1,j}^{v} = φ_{i,j+1}^{h}  φ_{ij}^{v} ∘ s_{ij} = φ_{ij}^{h}  for every i, j ∈ ℕ

(details left to the reader). This in turn is achieved by the following more general :

Claim 16.9.64. Let (V, 𝔪_{V}) be a basic setup such that 𝔪̃_{V} := 𝔪_{V} ⊗_{V} 𝔪_{V} is a flat V -module, R a V -algebra, I ⊂ R an ideal with pI = 0 and x^{p} = 0 for every x ∈ I. Denote by π : R → R/I the projection. Then there exists a map s that makes commute the diagram of sets :

\xymatrix{R_{∗}^{a} \ar[r]^{Φ} \ar[d]_{π_{∗}^{a}} & R_{∗}^{a} \ar[d]^{π_{∗}^{a}} \\ (R/I)_{∗}^{a} \ar[r]^{\overline{Φ}} \ar@{-->}[ur]^{s} & (R/I)_{∗}^{a}}

where Φ is the map given by the rule : x ↦ x^{p} for every x ∈ R_{∗}^{a}, and likewise for \overline{Φ}.

Proof of the claim. For every V -algebra S, denote by σ_{S} : R/I ⊗_{V} S → R ⊗_{V} S the map defined as follows. For every \overline{x} ∈ R/I ⊗_{V} S, pick x ∈ R ⊗_{V} S such that π(x) = \overline{x}, and set σ_{S}(x) := x^{p}; it is easily seen that this map is independent of the choice of representatives, and moreover the rule : S ↦ σ_{S} yields a homogeneous polynomial law σ : R/I ⇝ R of degre p (see definition 9.5.1(ii)). Thus, σ factors through the universal homogeneous polynomial law λ_{R/I}^{p} : R/I ⇝ Γ_{V}^{p} (R/I) and a unique R-linear map σ^{∗} : Γ_{V}^{p} (R/I) → R (see (9.5.19)). On the other hand, since 𝔪̃_{V} is a flat V -module, we have a V -linear isomorphism ω : 𝔪̃_{V} →^{∼} Γ_{V}^{p} (𝔪̃_{V}) ([75, (2.1.11)]); then the map given by the rule :

Hom_{V} (𝔪̃_{V} , R/I) → Hom_{V} (𝔪̃_{V} , R) φ ↦ (𝔪̃_{V} ⟶^{ω} Γ_{V}^{p} (𝔪̃_{V}) ⟶^{Γ_{V}^{p} (φ)} Γ_{V}^{p} (R/I) ⟶^{σ^{∗}} R)

fulfills the required conditions : details left to the reader. ♢

(iii.a): Let B_{1} be the p-root closure of B_{0} in B_{0}[b^{−1}], endow B_{1} with its b-adic topology, and let B_{1}^{∧} be the completion of B_{1}. Since b · 1_{B_{1}} is injective, the same holds for (b · 1_{B_{1}})^{∧} = b · 1_{B_{1}^{∧}} (proposition 8.2.13(i)). Then, according to claim 16.9.38and proposition 16.9.37(i), the basic setup (B_{1}^{∧}, 𝔪B_{1}^{∧}) is almost perfectoid and the induced map i : B_{0} → B_{1}^{∧} is an almost isomorphism. Moreover, B_{1}^{∧} is p-root closed in B_{1}^{∧}[b^{−1}], by lemma 9.8.26(ii) and proposition 8.3.33(iii). We consider the commutative diagram

\xymatrix{B_{0}^{♮} \ar[r]^{ε_{B_{0}}} \ar[d]_{i^{♮}} & B_{0} \ar[d]^{i} \\ (B_{1}^{∧})^{♮} \ar[r]^{ε_{B^{∧}_{1}}} & B_{1}^{∧}.}

By remark 16.9.32(iii), i induces an isomorphism E(i)^{a} : E(B_{0})^{a} → E(B_{1}^{∧})^{a} of (E, 𝔪_{E})^{a}- algebras. Let 𝔫 ⊂ A_{0} be the ideal generated by \overline{u}_{A_{0}}(𝔪_{E}); it is easily seen that (A_{0}, 𝔫) is a basic setup, and then i^{♮} := A_{0} ⊗_{W (E_{0})} W (E(i)) is an almost isomorphism relative to the almost structure furnished by (A_{0}, 𝔫). Now, i^{♮} is the inverse limit of the system of maps (i_{n} := i^{♮} ⊗_{ℤ} ℤ/p^{n} | n ∈ ℕ), and in view of (16.9.33) we deduce that i_{n} is an almost isomorphism also relative to the original almost structure (A_{0}, 𝔪), for every n ∈ ℕ, and finally, the same holds for i^{♮}. Summing up, we are therefore reduced to checking that ε_{B_{1}^{∧}}^{a} is an isomorphism of (A_{0}, 𝔪)^{a}-algebras, so we may assume from start that B_{0} is p-root closed. Denote by gr_{•}B_{1} the graded ring associated with the b-adic filtration on B_{0}, and likewise define gr_{•}B_{0}^{♮}; since both B_{0} and B_{0}^{♮} are b-adically complete and separated, we are further reduced to showing that the induced map gr_{•}ε_{B_{0}} : gr_{•}B_{0}^{♮} → gr_{•}B_{0} is an almost isomorphism, and since b is regular in both B_{0} and B_{0}^{♮}, it suffices to prove that the same holds for

gr_{0}ε_{B_{0}} : B_{0}^{♮}/bB_{0}^{♮} → B_{0}/bB_{0}.

Claim 16.9.65. Define β_{•} ∈ E(A_{0}) as in (16.9.59). Then we have :

(i) The Frobenius endomorphism of B_{0}/β_{n}B_{0} induces an injective map

B_{0}/β_{n+1}B_{0} → B_{0}/β_{n}B_{0} for every n ∈ ℕ.

(ii) The kernel of the projection E(B_{0}) → B_{0}/bB_{0} is β_{•}E(B_{0}).

(iii) We have a natural identification B_{0}^{♮}/bB_{0}^{♮} →^{∼} E(B_{0})/β_{•}E(B_{0}).

Proof of the claim. (i) follows straightforwardly from the assumption that B_{0} is p-root closed in B_{0}[b^{−1}], and (ii) follows easily from (i) : the details are left to the reader.

(iii): Let α_{•} ∈ W (E_{0}) be a distinguished element in the kernel of u_{A} : W (E_{0}) → A_{0}; we have p ∈ b^{p} by assumption, and α_{0} ∈ β_{•}^{p}E_{0}, by lemma 16.1.9(ii). But the natural isomorphism

E(B_{0})/α_{0}E(B_{0}) →^{∼} E_{0}/α_{0}E_{0} ⊗_{E_{0}} E(B_{0}) →^{∼} A_{0} ⊗_{W (E_{0})} E(B_{0}) →^{∼} B_{0}^{♮}/pB_{0}^{♮}

maps the class of β_{•} to the class of b, whence the contention. ♢

By claim 16.9.65(ii), the projection \overline{u}_{B_{0}/pB_{0}} : E(B_{0}) →^{∼} E(B_{0}/pB_{0}) → B_{0}/bB_{0} induces an injective map

j : E(B_{0})/β_{•}E(B_{0}) → B_{0}/bB_{0}

and claim 16.9.65(iii) yields a natural identification of j with gr_{0}ε_{B_{0}}. Thus, it remains only to check that \overline{u}_{B_{0}/pB_{0}}^{a} is an epimorphism of (E_{0}, 𝔪_{E})^{a}-algebras. However, recall that E(B_{0}) is the limit of the system of rings (R_{n} | n ∈ ℕ), with R_{n} := B_{0}/bB_{0} for every n ∈ ℕ, and with transition maps given by the Frobenius endomorphism Φ_{B_{0}/bB_{0}}. Thus, \overline{u}_{B_{0}/pB_{0}} is the inverse limit of the system of maps (Φ_{B_{0}/bB_{0}}^{n} : R_{n} → B_{0}/bB_{0} | n ∈ ℕ), each of which is almost surjective, since (B_{0}, 𝔪B_{0}) is almost perfectoid (as usual, we regard (Φ_{B_{0}/bB_{0}}^{n})^{a} as a morphism of (E_{0}, 𝔪_{E})^{a}-algebras R_{n,(Φ_{E_{0}}^{−n})}^{a} → B_{0}^{a}/bB_{0}^{a} : cp. remark 16.9.32(iii)). To conclude, it suffices

now to invoke lemma 14.2.24(i).

(iii.b): We consider first the assertion for C^{∧}: set \overline{A}_{0} := A_{0}/bA_{0}, and \overline{𝔪} := 𝔪\overline{A}_{0}; by assumption, the Frobenius endomorphism of B_{0}/b^{p}B_{0} induces an isomorphism \overline{Φ}_{B_{0}}^{a} : (B_{0}/bB_{0})^{a} →^{∼} (B_{0}/b^{p}B_{0})_{(Φ)}^{a} of (\overline{A}_{0}, \overline{𝔪})^{a}-algebras, whence the isomorphism of \overline{A}_{0}-algebras

(\overline{Φ}_{B_{0}})_{!!}^{a} : (B_{0}/bB_{0})_{!!}^{a} →^{∼} ((B_{0}/b^{p}B_{0})_{(Φ)}^{a})_{!!}

where (−)_{!!} denotes the left adjoint of the localization functor (−)^{a} : \overline{A}_{0}-Alg → (\overline{A}_{0}, \overline{𝔪})^{a}-Alg. On the other hand, by a direct inspection of the construction of these left adjoints, one gets natural identifications

C/bC →^{∼} (B_{0}/bB_{0})_{!!}^{a} C/b^{p}C →^{∼} ((B_{0}/b^{p}B_{0})_{(Φ)}^{a})_{!!}

where, however, the construction of C := (B_{0})_{!!}^{a} refers to the left adjoint of the corresponding localization (−)^{a} : A_{0}-Alg → (A_{0}, 𝔪)^{a}-Alg. It is then easily seen that, under these identifications, the map (\overline{Φ}_{B_{0}})_{!!}^{a} corresponds to the ring homomorphism \overline{Φ}_{C} : C/bC → C/b^{p}C induced by the Frobenius endomorphism of C/b^{p}C. The latter is therefore an isomorphism, and so the same holds for the corresponding ring homomorphism \overline{Φ}_{C^{∧}} : C^{∧}/bC^{∧} → C^{∧}/b^{p}C^{∧}. Then, according to corollary 16.3.75, in order to check that C^{∧} is perfectoid, it remains only to show that Ann_{C^{∧}}(b^{p}) = Ann_{C^{∧}}(b^{p−1}), and in light of remark 8.6.39(iii), we are reduced to proving that Ann_{C}(b^{p}) = Ann_{C}(b^{p−1}). Let μ : 𝔪̃ → A_{0} be the A_{0}-linear map such that μ(x ⊗ y) := xy for every x, y ∈ 𝔪; we notice :

Claim 16.9.66. Ann_{C}(b^{n}) is a quotient of Ker(𝔪̃/b^{n}𝔪̃ ⟶^{A_{0}/b^{n}A_{0}⊗_{A_{0}}μ} A_{0}/b^{n}A_{0}), for all n ∈ ℕ.

Proof of the claim. We consider the commutative diagram of A_{0}-modules with exact rows :

\xymatrix{0 \ar[r] & 𝔫 \ar[r] \ar[d] & A_{0} ⊕ (𝔪̃ ⊗_{A_{0}} B_{0}) \ar[r] \ar[d] & (B_{0})_{!!}^{a} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & 𝔫 \ar[r] & A_{0} ⊕ (𝔪̃ ⊗_{A_{0}} B_{0}) \ar[r] & (B_{0})_{!!}^{a} \ar[r] & 0}

whose vertical arrows are scalar multiplication by b^{n}, and where 𝔫 denotes the image of the map 𝔪̃ → A_{0} ⊕ (𝔪̃ ⊗_{A_{0}} B_{0}) given by the rule : z ↦ (μ(z), z ⊗ 1) for every z ∈ 𝔪̃. By assumption, b is a regular element in both A_{0} and B_{0}, and 𝔪̃ is a flat A_{0}-module, by [75, Prop.2.1.7(i)]. Hence, the central vertical arrow is injective, and the assertion follows easily from the snake lemma. ♢

In view of claim 16.9.66, we are reduced to showing that b · Ker(A_{0}/b^{n}A_{0} ⊗_{A_{0}} μ) = 0 for every n ∈ ℕ. To this aim, write 𝔪 = ⋃_{n∈ℕ} A_{0}a_{i} for a system (a_{i} | i ∈ ℕ) of elements of A_{0} such that for every i ∈ ℕ we have a_{i} = c_{i}a_{i+1}^{p} for some c_{i} ∈ A_{0} (see (16.9.31)); by remark 16.9.32(ii), the A_{0}-module 𝔪̃ is then isomorphic to the inductive limit L of a system (F_{i} | i ∈ ℕ) of free rank one A_{0}-modules F_{i} = A_{0}e_{i}, with transition maps f_{i} : F_{i} → F_{i+1} such that f_{i}(e_{i}) = c_{i}a_{i+1}^{p−1}e_{i+1} for every i ∈ ℕ. Hence L is the set of equivalence classes [x, i] of pairs (x, i) with i ∈ ℕ and x ∈ F_{i}, and with this notation, μ corresponds to the A_{0}-linear map L → A_{0} given by the rule : [ye_{i}, i] ↦ ya_{i} for every i ∈ ℕ and y ∈ A_{0}. Hence, let [ye_{i}, i] ∈ L and suppose that ya_{i} ∈ b^{n}A_{0}; we need to show that [ye_{i}, i] ∈ b^{n−1}L. Pick k ∈ ℕ with p^{k} ≥ n; by a simple induction, we see that :

[ye_{i}, i] = y_{k}[e_{i+k}, i + k] with y_{k} := yc_{i}c_{i+1}^{p} ⋯ c_{i+k−1}^{p^{k−1}}a_{i+k}^{p^{k}−1}.

We are then reduced to checking that y_{k} ∈ b^{n−1}A_{0}, and by theorem 16.3.64, the latter is equivalent to the condition : y_{k}^{p^{k}} ∈ b^{(n−1)p^{k}}A_{0}. But we have :

y_{k}^{p^{k}} = y^{p^{k}}c_{i}c_{i+1}^{p} ⋯ c_{i+k−1}^{p^{k−1}}a_{i}^{p^{k}−1} = yc_{i}c_{i+1}^{p} ⋯ c_{i+k−1}^{p^{k−1}} · (ya_{i})^{p^{k}−1} ∈ b^{n(p^{k}−1)}A_{0} ⊂ b^{(n−1)p^{k}}A_{0}.

Next, we consider the assertion concerning D^{∧} : the discussion of (16.9.59) already shows that E(B_{0}) is a perfectoid ring, when endowed with its \overline{β}_{•}-adic topology, and moreover the image of \overline{β}_{•} is regular in E(B_{0}). Hence, the assertion for D^{∧} follows from the foregoing case, after replacing the basic setup (A_{0}, 𝔪) by (E, 𝔪_{E}).

(iii.c): Arguing as in the proof of corollary 16.3.75, we may assume that b = b_{0} for some (b_{n} | n ∈ ℕ) ∈ E(A_{0}); notice also that C^{∧} is reduced, due to (iii.b) and corollary 16.3.63(i), hence Ann_{C^{∧}}(b) = ⋃_{n∈ℕ} Ann_{C^{∧}}(b^{n}), so scalar multiplication by b is an injective endomorphism of C^{∧}/Ann_{C^{∧}}(b). Then the assertion follows from corollary 16.3.71.

(iv): The counit of adjunction for the pair of functors ((−)_{!!}, (−)^{a}) is an almost isomorphism ε_{B_{0}} : C → B_{0}, hence the same for its b-adic completion (cp. the proof of claim 16.9.38). Likewise, recall that E(B_{0}) is \overline{β}_{•}-adically complete and separated (see (16.9.59)); then the same argument shows that the counit of adjunction ε_{E(B_{0})} : D → E(B_{0}) induces, after taking \overline{β}_{•}-adic completions, an almost isomorphism D^{∧} → E(B_{0}).

(v): We may assume that b = β_{0}, where β_{•} ∈ E(A_{0}) is as in (16.9.59); also, arguing as in the proof of (iii.b), we see that \overline{Φ}_{C^{∧}} is an isomorphism, and then claim 16.3.76 and remark 8.6.39(iii) reduce to showing that Ann_{C}(b^{p}) = Ann_{C}(b^{p}/β_{1}) = Ann_{C}(β_{1}^{p^{2}−1}). But, arguing as in the proof of claim 16.9.66we obtain, for every n ∈ ℕ, an exact sequence of A_{0}-modules

𝔪̃ ⊗_{A_{0}} Ann_{B_{0}}(b^{n}) → Ann_{C}(b^{n}) → M

where M is a quotient of Ker(A_{0}/b^{n}A_{0} ⊗_{A_{0}} μ). The proof of (iii.b) then shows that bM = 0, and on the other hand β_{1} · Ann_{B_{0}}(b^{n}) = 0, by corollary 16.3.71; thus β_{1}^{p+1} · Ann_{C}(b^{n}) = 0, and since p^{2} − 1 ≥ p + 1, the assertion follows.

(vi): Let X ∈ Ob(A_{0}-Perf.Tate) and Y ∈ Ob((A_{0}, 𝔪)^{a}-Perf.Tate), and set Z := (Y_{!!}^{a})^{∧}; we know already from (i) and (iii.c) that Y_{∗}^{a} ∈ Ob(A_{0}-Tate) and Z/Ann_{Z}(b) ∈ Ob(A_{0}-Perf.Tate). Then, in view of the discussion of (16.9.59) we get natural bijections

Hom_{(A_{0},𝔪)^{a}-Perf.Tate}(X^{a}, Y^{a}) →^{∼} Hom_{A_{0}-Tate}(X, Y_{∗}^{a}) →^{∼} Hom_{A_{0}-Perf.Tate}(X, (Y_{∗}^{a})^{♮})

Hom_{(A_{0},𝔪)^{a}-Perf.Tate}(Y^{a}, X^{a}) →^{∼} Hom_{A_{0}--Alg}(Y_{!!}^{a}, X) →^{∼} Hom_{A_{0}-Perf.Tate}(Z/Ann_{Z}(b), X)

so the rule Y ↦ (Y_{∗}^{a})^{♮} (resp. Y ↦ Z/Ann_{Z}(b)) yields the sought right (resp. left) adjoint for the functor A_{0}-Perf.Tate → (A_{0}, 𝔪)^{a}-Perf.Tate. The same argument shows that the rule Y ↦ (Y_{∗}^{a})^{♮} also provides a right adjoint for the forgetful functor A_{0}-Perf.Adic → (A_{0}, 𝔪)^{a}-Perf.Adic, and using (v) we easily see that the rule Y ↦ Z yields a left adjoint for the same functor.

(iii.d): To begin with, let us notice :

Claim 16.9.67. The functors (16.9.62) induce equivalences between (A_{0}, 𝔪)^{a}-Perf.Adic (resp. (A_{0}, 𝔪)^{a}-Perf.Tate) and the localization of A_{0}-Perf.Adic (resp. A_{0}-Perf.Tate) obtained by inverting all almost isomorphisms.

Proof of the claim. Recall that ε_{B_{0}} : B_{0}^{♮} → B_{0} is an isomorphism if B_{0} is perfectoid (see (16.9.59)); we deduce for every Y ∈ Ob(A_{0}-Perf.Adic) an isomorphism ((Y_{∗}^{a})^{♮})^{a} →^{∼} (Y_{∗}^{a})^{a} →^{∼} Y^{a} in the category (A_{0}, 𝔪)^{a}-Perf.Adic. Combining with (vi) and proposition 1.1.20(iii), we deduce that the right adjoints to both of the forgetful functors (16.9.62) are fully faithful. Then the claim follows from proposition 1.6.13. ♢

From (vi) and remark 16.9.32(iii) we get a diagram of categories :

\xymatrix{A_{0}-Perf.Adic \ar[r]^{E(−)} \ar[d]^{(−)^{a}} & E_{0}-Perf.Adic \ar[d]_{(−)^{a}} \\ (A_{0}, 𝔪)^{a}-Perf.Adic \ar[r]^{E(−)} \ar[u] & (E_{0}, 𝔪_{E})^{a}-Perf.Adic \ar[u]}

whose upward arrows are left adjoints to the downward arrows, and whose top horizontal arrow is an equivalence; in light of claim 16.9.67, it follows easily that the bottom horizontal arrow is an equivalence as well. Clearly the horizontal arrows commute with the downward arrows; then the horizontal arrows also commute with the upward arrows, up to isomorphism of functors (example 1.2.7(ii)). Hence, if R ∈ Ob(A_{0}-Perf.Adic), we have a natural ring isomorphism

ω_{R} : E((R_{!!}^{a})^{∧}) →^{∼} (E(R)_{!!}^{a})^{∧}.

Next, if (B_{0}, 𝔪B_{0}) is almost perfectoid, from (iv) and remark 16.9.32(iii) we deduce an almost isomorphism E((B_{0})_{!!}^{a})^{∧}) → E(B_{0}), whence a ring isomorphism

τ_{B_{0}} : (E((B_{0!!}^{a})^{∧})_{!!}^{a})^{∧} →^{∼} (E(B_{0})_{!!}^{a})^{∧}.

Lastly, R := (B_{0!!}^{a})^{∧} is perfectoid, by (v), whence the isomorphism

τ_{B_{0}} ∘ ω_{R} : E((R_{!!}^{a})^{∧}) →^{∼} (E(B_{0})_{!!}^{a})^{∧}.

Thus, we are reduced to checking that the natural map (((B_{0!!}^{a})^{∧})_{!!}^{a})^{∧} → (B_{0!!}^{a})^{∧} is an isomorphism. This in turn will follow, once we have shown that the natural map ((B_{0!!}^{a})^{∧})_{!!}^{a} → B_{0!!}^{a} is an isomorphism. But the latter is a straightforward consequence of (iv). □

17. Applications

17.1. Model algebras. We let (K, | · |) be a valued field of characteristic 0 with value group Γ_{K} of rank one, such that the residue field κ of K^{+} has characteristic p > 0.

Definition 17.1.1. The category MA_{K} of model K^{+}-algebras consists of all the pairs (A, Γ), where (Γ, +) is an integral monoid, and A is a Γ-graded K^{+}-algebra A, fulfilling the following conditions : (MA1) gr_{γ}A is a torsion-free K^{+}-module, with dim_{K} gr_{γ}A ⊗_{K^{+}} K = 1, for every γ ∈ Γ. (MA2) gr_{α}A · gr_{β}A ≠ 0 for every α, β ∈ Γ, and (gr_{γ}A)^{n} = gr_{nγ}A for every n ∈ ℕ and γ ∈ Γ. (MA3) gr_{0}A = K^{+}. (MA4) Γ is saturated and Γ^{gp} is a ℤ[1/p]-module. The morphisms (A, Γ) → (A′, Γ′) in MA_{K} are the pairs (f, φ), where φ : Γ → Γ′ is a morphism of monoids, and f : A → Γ ×_{Γ′} A′ is a morphism of Γ-graded K^{+}-algebras (see (7.6.19)).

Example 17.1.2. Let M be an integral monoid, N → M an exact and injective morphism of monoids, N → K^{+}\{0} a morphism of monoids, and suppose that :

• M is divisible, i.e. the k-Frobenius endomorphism of M is surjective for every k > 0 • M^{gp}/N^{gp} is a ℚ-vector space.

Let (Γ, +) be the image of M in M^{gp}/N^{gp}, and denote by I the nilradical of the K^{+}-algebra A := M ⊗_{N} K^{+}; then A is a Γ-graded K^{+}-algebra, and I is a Γ-graded ideal (proposition 7.6.28(ii)). We claim that I ⊗_{K^{+}} K = 0 and (A/I, Γ) is a model K^{+}-algebra. Indeed, (MA4) is immediate, and it is easily seen that gr_{nγ}A = (gr_{γ}A)^{n} for every γ ∈ Γ and n ∈ ℕ. Moreover, since the map N → M is exact, the kernel of the map M → M^{gp}/N^{gp} equals N, so (A/I)_{0} = K^{+}, i.e. (MA3) holds as well. The remaining assertions can be checked after tensoring with K : namely, we have to show that the Γ-graded K-algebra A_{K} := M ⊗_{N} K is reduced, with dim_{K} gr_{γ}A_{K} = 1 for every γ ∈ Γ, and gr_{α}A_{K} · gr_{β}A_{K} ≠ 0 for every α, β ∈ Γ. However, the morphism N → K^{+} extends uniquely to a group homomorphism N^{gp} → K^{×}, and A_{K} = (N^{−1}M) ⊗_{N^{gp}} K; hence, we may assume that N is a group, in which case Γ is the set-theoretic quotient of M by the translation action of N (lemma 4.8.31(iii)). In this case, choose a representative γ^{∗} ∈ M for every γ ∈ Γ; it follows easily that (γ^{∗} ⊗1 | γ ∈ Γ) is a basis of the K-vector space A_{K}, whence (MA1), and (α^{∗} ⊗ 1) · (β^{∗} ⊗ 1) is a non-zero multiple of (α+β)^{∗}⊗1, which yields (MA2). Lastly, since the nilradical I_{K} of A_{K} is Γ-graded (proposition 7.6.28(ii)), we also deduce that I_{K} = 0, as sought.

Remark 17.1.3. (i) The category MA_{K} admits a tensor product, defined by the rule :

(A, Γ) ⊗ (A′, Γ′) := (A ⊗_{K^{+}} A′, Γ ⊕ Γ′).

(ii) Let (A, Γ) be any model K^{+}-algebra, and suppose that Γ = Γ_{1} ⊕ Γ_{2} is a given decomposition of Γ as direct sum of monoids. Then it is easily seen that Γ_{1} and Γ_{2} are integral and saturated, and they fulfill axiom (MA4). There follows a morphism of model K^{+}-algebras :

(17.1.4)  (Γ_{1} ×_{Γ} A, Γ_{1}) ⊗ (Γ_{2} ×_{Γ} A, Γ_{2}) → (A, Γ)

(iii) In the situation of (ii), set A_{K} := A ⊗_{K^{+}} K. Then it is easily seen that (17.1.4) induces an isomorphism of K-algebras

(Γ_{1} ×_{Γ} A_{K}) ⊗_{K} (Γ_{2} ×_{Γ} A_{K}) →^{∼} A_{K}.

In general, (17.1.4) need not be an isomorphism. However, (17.1.4) is an isomorphism, if either Γ_{1} or Γ_{2} is a torsion abelian group. Indeed, in any case the induced maps

gr_{α}A ⊗_{K^{+}} gr_{β}A → gr_{α+β}A a_{1} ⊗ a_{2} ↦ a_{1}a_{2}

will be injective for every α ∈ Γ_{1} and β ∈ Γ_{2}. To check surjectivity, let a ∈ gr_{α+β}A be any element, and say that nβ = 0; then a^{n} ∈ gr_{nα}A, and by (MA2) we know that there exist a_{1} ∈ gr_{α}A and x ∈ K^{+} such that a^{n} = a_{1}^{n}x. It follows that a = a_{1}a_{2} for some a_{2} ∈ gr_{β}A_{K} such that a_{2}^{n} ∈ K^{+}. Then a_{2} ∈ gr_{β}A, whence the claim.

(iv) Let (E, | · |_{E}) be any valued field extension of (K, | · |), such that | · |_{E} is also a valuation of rank one. Then we have an obvious base change functor :

MA_{K} → MA_{E} : (A, Γ) ↦ (A ⊗_{K^{+}} E^{+}, Γ).

17.1.5. Let (A, Γ) be any model K^{+}-algebra. For any subset Δ ⊂ Γ, set

A_{Δ} := Δ ×_{Γ} A  A_{Δ,K} := A_{Δ} ⊗_{K^{+}} K  Δ[1/p] := ⋃_{n∈ℕ}{γ ∈ Γ | p^{n}γ ∈ Δ}.

Clearly, if Δ is a submonoid of Γ, then A_{Δ} is a K^{+}-subalgebra of A, and (A_{Δ}, Δ) is a subobject of (A, Γ), provided Δ satisfies (MA4). On the other hand, if Δ is saturated, it is easily seen that the same holds for Δ[1/p], and then the latter does satisfy (MA4).

Also, if Δ is an ideal of Γ, then clearly A_{Δ} is an ideal of A.

Set K^{∗} := K^{+} \{0} and let A_{γ}^{∗} := gr_{γ}A\{0} for every γ ∈ Γ; for any submonoid Δ ⊂ Γ, we deduce a sequence of morphisms of integral monoids :

(17.1.6)  1 → K^{∗} ⟶^{φ_{Δ}} A_{Δ}^{∗} → Δ → 1  where A_{Δ}^{∗} := ⨁_{γ∈Δ} A_{γ}^{∗}

(and where the direct sum is formed in the category of K^{∗}-modules), such that (17.1.6)^{gp} is a short exact complex of abelian groups; then (MA2) implies that φ_{Δ} is saturated (proposition 6.2.31). Also, (MA1) implies that each A_{γ}^{∗} is a filtered union of free cyclic K^{∗}-modules, hence φ_{Δ} is also integral. Furthermore, (MA1) and (MA2) imply that A_{Δ}^{∗gp} is Δ^{gp}-graded, and

(17.1.7)  gr_{γ}A_{Δ}^{∗gp} = A_{γ}^{∗} ⊗_{K^{∗}} K^{×}  for every γ ∈ Δ.

(details left to the reader). We shall just write A^{∗} and A_{K} instead of A_{Γ}^{∗}, and respectively A_{Γ,K}.

17.1.8. Let now (A, Γ) be any model K^{+}-algebra; set S := Spec K^{+}, X := Spec A, and let \overline{x} be any geometric point of X, localized on the closed subset Z := X ×_{S} Spec κ. Let 𝒯_{A,p} be the p-adic topology on A. Suppose furthermore, that K^{+} is deeply ramified (see [75, Def.6.6.1]), and let (K^{+}, 𝔪_{K}) be the standard setup associated with K^{+} (see [75, §6.1.15]); then we have the corresponding sheaf 𝒪_{X(\overline{x})}^{a} of K^{+a}-algebras on X(\overline{x}), and we may state the following almost purity theorem :

Theorem 17.1.9. With the notation of (17.1.8), the following holds :

(i) (A, 𝒯_{A,p}) is a formally perfectoid ring. (ii) The pair (X(\overline{x}), Z(\overline{x})) is almost pure.

Proof. (i): Let π ∈ K^{+} be any element such that π^{p}K^{+} = pK^{+}; denote by A^{∧} the completion of (A, 𝒯_{A,p}), and set I := πA^{∧}. Then the topology of A^{∧} agrees with the I-adic topology (remark 8.3.3(iv)), and the image of π is a regular element in A^{∧}, by proposition 8.2.13(i) (the details shall be left to the reader). Moreover we have pA^{∧} = I^{(p)} (notation of definition 16.2.1); by theorem 16.4.1, it then suffices to check that the Frobenius endomorphism Φ_{A/pA} of A^{∧}/pA^{∧} = A/pA induces an isomorphism A/πA →^{∼} A/pA. We are then further reduced to showing that Φ_{A/pA} restricts to a bijection

gr_{γ}A/π · gr_{γ}A →^{∼} gr_{pγ}A/p · gr_{pγ}A for every γ ∈ Γ.

However, since Γ_{K} is of rank one, there exists a sequence (a_{n} | n ∈ ℕ) of elements of gr_{γ}A such that K^{+}a_{n} ⊂ K^{+}a_{n+1} for every n ∈ ℕ and gr_{γ}A = ⋃_{n∈ℕ} K^{+}a_{n}. Then gr_{pγ}A = ⋃_{n∈ℕ} K^{+}a_{n}^{p}. By [75, Prop.6.6.6] the Frobenius endomorphism of K^{+} induces an isomorphism K^{+}/πK^{+} →^{∼} K^{+}/pK^{+}. It follows that Φ_{A/pA} induces a bijection (K^{+}a_{n})/π(K^{+}a_{n}) →^{∼} (K^{+}a_{n}^{p})/p(K^{+}a_{n}^{p}) for every n ∈ ℕ, whence the contention.

(ii) follows from (i), and theorems 16.7.21(i) and 16.8.44. □

17.1.10. The inclusion map A^{∗} → A is a morphism of (multiplicative) monoids, hence induces a log structure on X := Spec A (see (12.1.15)). We shall denote

𝕊(A, Γ) := (X, (A_{X}^{∗})^{log})

the resulting log scheme. Clearly, every morphism (f, φ) : (A, Γ) → (A′, Γ′) of model algebras induces a morphism of log schemes

𝕊(f, φ) : 𝕊(A′, Γ′) → 𝕊(A, Γ).

Especially, by lemma 12.1.13(iv), the map φ_{Γ} yields a saturated morphism

(17.1.11)  𝕊(A, Γ) → 𝕊(K^{+}) := 𝕊(K^{+}, {1}).

Also, if (K, | · |) → (E, | · |_{E}) is an extension of rank one valued fields, the inclusion A^{∗} ⊂ (A ⊗_{K^{+}} E^{+})^{∗} induces a morphism of log schemes

𝕊(A ⊗_{K^{+}} E^{+}, Γ) → 𝕊(A, Γ)

for any model K^{+}-algebra (A, Γ), and we remark that the resulting diagram of log schemes

\xymatrix{𝕊(A ⊗_{K^{+}} E^{+}, Γ) \ar[r] \ar[d] & 𝕊(A, Γ) \ar[d] \\ 𝕊(E^{+}) \ar[r] & 𝕊(K^{+})}

is cartesian. Indeed, since (12.1.16) is right exact, it suffices to check that the natural map

A^{∗} ⊗_{K^{∗}} E^{∗} → (A ⊗_{K^{+}} E^{+})^{∗}

is an isomorphism, which is clear.

Remark 17.1.12. (i) Let (A, Γ) be any model K^{+}-algebra, and suppose that Δ_{0} ⊂ Γ is a fine and saturated submonoid, such that Δ_{0}^{gp} is torsion-free. In this case, Δ_{0}^{gp} is a free abelian group of finite rank, hence (17.1.6)^{gp} admits a splitting σ : Δ_{0}^{gp} → A^{∗gp}. Then, using (17.1.7) and (MA1), it is easily seen that the rule γ ↦ σ(γ) extends to an isomorphism of Δ_{0}-graded K-algebras

K[Δ_{0}] →^{∼} A_{Δ_{0},K}.

(ii) In the situation of (i), suppose furthermore, that K is algebraically closed. Pick any x ∈ K^{×} such that γ := |x| ≠ 1, and let ⟨γ⟩ ⊂ Γ_{K} be the subgroup generated by γ; we define a group homomorphism ⟨γ⟩ → K^{×} by the rule : γ^{k} ↦ x^{k} for every k ∈ ℤ. Since K^{×} is divisible, the latter map extends to a group homomorphism

(17.1.13)  Γ_{K} → K^{×}

and since Γ_{K} is a group of rank one, it is easily seen that (17.1.13) is a right inverse for the valuation map | · | : K^{×} → Γ_{K}, whence a decomposition :

K^{×} →^{∼} (K^{+})^{×} ⊕ Γ_{K}.

On the other hand, set A_{Δ_{0},K}^{∗} := A_{Δ_{0}}^{∗} ⊗_{K^{∗}} K^{×}; from (i) we deduce an isomorphism of Δ_{0}- graded monoids :

A_{Δ_{0},K}^{∗} →^{∼} Δ_{0} ⊕ K^{×}.

Combining these two isomorphisms, we deduce a surjection τ : A_{Δ_{0},K}^{∗} → (K^{+})^{×} which is a left inverse to the inclusion (K^{+})^{×} → A_{0}^{∗}. Then, for every γ ∈ Δ_{0}, let us set C_{γ} := A_{γ}^{∗} ∩ Ker τ; there follows a (non-canonical) isomorphism of Δ-graded monoids :

A_{Δ_{0}}^{∗} →^{∼} (K^{+})^{×} ⊕ C where C := ⨁_{γ∈Δ_{0}} C_{γ} ⊂ A_{Δ_{0}}^{∗} and C^{gp} ≃ Δ_{0}^{gp} ⊕ Γ_{K}

(details left to the reader).

(iii) Suppose that Γ^{gp} is torsion-free, and K is still algebraically closed. Let us set :

Δ_{n} := {γ ∈ Γ | p^{n}γ ∈ Δ_{0}} for every n ∈ ℕ.

It is easily seen that Δ_{n} is still fine and saturated, and since K^{×} is divisible, we may extend inductively the splitting σ of (i) to a system of homomorphisms

σ_{n} : Δ_{n}^{gp} → A^{∗gp} such that σ_{n+1|Δ_{n}^{gp}} = σ_{n} for every n ∈ ℕ

whence a compatible system of isomorphisms

(17.1.14)  K[Δ_{n}] →^{∼} A_{Δ_{n},K}  for every n ∈ ℕ.

Proceeding as in (ii), we deduce a compatible system of isomorphisms of Δ_{n}-graded monoids

A_{Δ_{n}}^{∗} →^{∼} (K^{+})^{×} ⊕ C^{(n)} such that C^{(n)} ⊂ C^{(n+1)} for every n ∈ ℕ.

In this situation, notice that the p-Frobenius automorphism of Γ^{gp} restricts to an isomorphism

Δ_{n+1} →^{∼} Δ_{n} for every n ∈ ℕ.

Likewise, since Γ_{K} is p-divisible, taking p-th powers induces isomorphisms

C^{(n+1)} →^{∼} C^{(n)} for every n ∈ ℕ.

Definition 17.1.15. Let (B, Δ) be model K^{+}-algebra. We say that (B, Δ) is small, if the following conditions hold :

(a) gr_{γ}B is a finitely generated K^{+}-module for every γ ∈ Δ.

(b) Δ = Δ_{0}[1/p] for some fine and saturated submonoid Δ_{0}.

(c) Δ_{0} ×_{Δ} B is a finitely generated K^{+}-algebra.

Remark 17.1.16. Notice that condition (a) of definition 17.1.15 and axiom (MA1) imply that gr_{γ}B is a free K^{+}-module of rank one, for every small model algebra (B, Δ) and every γ ∈ Δ. Moreover, actually conditions (b) and (c) (together with axioms (MA1) and (MA2)) imply condition (a). Indeed, (c), (MA1) and proposition 7.6.11(ii) imply that gr_{γ}B is a free K^{+}- module of rank one, for every γ ∈ Δ_{0}. Now, in case nγ ∈ Δ_{0}, (MA2) implies that gr_{nγ}B is generated by an element of the form z = x_{1} ⋯ x_{n}, for certain x_{1}, …, x_{n} ∈ gr_{γ}B. Suppose that x_{1} = ay for some a ∈ K^{+} and y ∈ gr_{γ}B; then z is divisible by a in gr_{nγ}B, so a ∈ (K^{+})^{×}, i.e. x_{1} generates gr_{γ}B. In view of (b), for every γ ∈ Δ we may find k ≥ 0 such that p^{k}γ ∈ Δ_{0}, so (a) follows.

Remark 17.1.17. Let (B, Δ) be any small model algebra, so that conditions (b) and (c) of definition 17.1.15are satisfied for some submonoid Δ_{0} ⊂ Δ.

(i) Choose a decomposition Δ_{0}^{×} = G ⊕ H, where G is a free abelian group, and H is the torsion subgroup of Δ_{0}; we have an isomorphism

Δ_{0} ≃ Λ ⊕ H with Λ := Δ_{0}^{♯} ⊕ G

(lemma 6.2.10), which induces a decomposition :

Δ ≃ Λ[1/p] ⊕ H

inducing, in turn, an isomorphism of model K^{+}-algebras :

(B, Δ) →^{∼} (B_{Λ[1/p]}, Λ[1/p]) ⊗ (B_{H}, H)

(remark 17.1.3(iii)). Notice that both B_{Λ} and B_{H} are finitely generated K^{+}-algebras (proposition 7.6.11(i)). In other words, every small model algebra can be written as the tensor product of two small model algebras (B′, Δ′) and (B′′, Δ′′), such that Δ′^{gp} is torsion-free, and Δ′′ is a finite abelian group whose order is not divisible by p.

(ii) For every n ∈ ℕ, set Δ_{n} := {γ ∈ Δ | p^{n}γ ∈ Δ_{0}}. Then B_{n} := Δ_{n} ×_{Δ} B is a finitely generated K^{+}-algebra for every n ∈ ℕ. Indeed, in view of (i), it suffices to check the assertion in case Δ^{gp} is a torsion-free abelian group. Now, pick a system x_{1}, …, x_{k} of homogeneous generators of the K^{+}-algebra B_{0}; by (MA2), for every i = 1, …, k there exist a homogeneous element y_{i} ∈ B_{n} and u_{i} ∈ (K^{+})^{×} such that u_{i}y_{i}^{p^{n}} = x_{i}. Let z ∈ B_{n} be any homogeneous element; then z^{p^{n}} = vx_{1}^{t_{1}} ⋯ x_{k}^{t_{k}} for some v ∈ K and t_{1}, …, t_{k} ∈ ℕ. Set y := y_{1}^{t_{1}} ⋯ y_{k}^{t_{k}} and u := u_{1}^{t_{1}} ⋯ u_{k}^{t_{k}} ∈ K^{×}; then y^{p^{n}}uv = z^{p^{n}}, and since Δ^{gp} is torsion-free, we deduce that yw = z for some w ∈ K^{+}, i.e. the system y_{1}, …, y_{k} generates the K^{+}-algebra B_{n}.

(iii) Suppose that Δ is a finite group whose order is not divisible by p. Then we claim that B is a finite étale K^{+}-algebra. Indeed, in view of remark 17.1.3(iii), it suffices to verify the assertion for Δ a cyclic finite group, say of order n, with (n, p) = 1; in the latter case, (MA2) implies that B ≃ K^{+}[X]/(X^{n} − u) for some u ∈ (K^{+})^{×}, whence the contention.

(iv) Let F ⊂ Δ be any face. Then (B_{F} , F) is a small model algebra as well. Indeed, by (i), it suffices to consider the case where Δ^{gp} is a torsion-free abelian group. In this case, set F_{0} := F ∩ Δ_{0}; it is easily seen that F = F_{0}[1/p], and F_{0} is a fine and saturated monoid, by lemma 6.1.20(ii) and corollary 6.2.33(ii). Moreover, B_{F_{0}} = F_{0} ×_{F} B_{Δ_{0}} is a finitely generated K^{+}-algebra, by proposition 7.6.11(i), whence the contention.

(v) Suppose moreover, that K is algebraically closed, and let (B, Δ) be any small model K^{+}-algebra. Then we have a (non-canonical) isomorphism :

K[Δ] →^{∼} B_{K}.

To exhibit such an isomorphism, we may – in light of (i) – assume that Δ^{gp} is either torsionfree, or a finite group of order not divisible by p. In the latter case, the assertion follows easily from (iii). In case Δ^{gp} is torsion-free, the sought isomorphism is the colimit of the system of isomorphisms (17.1.14).

Lemma 17.1.18. Let (A, Γ) be a model K^{+}-algebra, and denote by ℱ (Γ) the filtered family of all fine and saturated submonoids of Γ. We have :

(i) A = colim_{Δ∈ℱ (Γ)} A_{Δ}.

(ii) Suppose that Γ^{gp} is a torsion-free abelian group. Then A is a normal domain.

Proof. (i) is an immediate consequence of corollary 6.4.1(ii).

(ii): We show first the following :

Claim 17.1.19. Suppose that Γ^{gp} is a torsion-free abelian group. Then A_{K} is a normal domain.

Proof of the claim. In view of (i), it suffices to show that A_{Δ,K} is a normal domain, when Δ ⊂ Γ is fine and saturated. The latter assertion follows from remark 17.1.12(i) and theorem 6.4.18(iii). ♢

Let A^{ν} be the integral closure of A in A_{K}; in view of claim 17.1.19we are reduced to showing that A = A^{ν}. By proposition 7.6.29(ii), A^{ν} is Γ-graded. Suppose now that x ∈ A^{ν}; we need to show that x ∈ A, and we may assume that x ∈ gr_{γ}A^{ν} for some γ ∈ Γ. Hence, let

x^{n} + a_{1}x^{n−1} + ⋯ + a_{n} = 0 with a_{1}, …, a_{n} ∈ A

be an integral equation for x over A. If we replace each a_{i} by its homogeneous component in degree iγ, we still obtain an integral equation for x, so we may assume that a_{i} ∈ gr_{iγ}A for every i = 1, …, n. Then, (MA2) implies that, for every i = 1, …, n there exist u_{i} ∈ K^{+} and b_{i} ∈ gr_{γ}A, such that |u_{i}| = 1 and a_{i} = u_{i}b_{i}^{i}. Also, from (MA1) we deduce that there exists b ∈ gr_{γ}A such that b_{1}, …, b_{n} ∈ K^{+}b. Set y := b^{−1}x ∈ gr_{0}A ⊗_{K^{+}} K; clearly y is integral over the subring grA_{0}. Lastly, (MA3) shows that y ∈ K^{+}, whence x ∈ gr_{γ}A, which proves the contention. □

Proposition 17.1.20. Let (A, Γ) be a model K^{+}-algebra, and suppose that Γ_{K} is divisible. Then (A, Γ) is the filtered union of its small model K^{+}-subalgebras.

Proof. In view of lemma 17.1.18(i), we see that (A, Γ) is the colimit of the filtered system of its subobjects (A_{Δ[1/p]}, Δ[1/p]), for Δ ranging over the fine and saturated submonoids of Γ. We may then assume from start that Γ = Γ_{0}[1/p] for some fine and saturated submonoid Γ_{0}.

Next, in view of remark 17.1.17(i), we may consider separately the cases where Γ_{0}^{gp} is a torsion-free abelian group, and where Γ = Γ_{0} is a finite abelian group.

Suppose first that Γ_{0}^{gp} is torsion-free, and let \underline{γ} := (γ_{1}, …, γ_{n}) be a finite system of generators for Γ_{0}. For every i = 1, …, n, choose a_{i} ∈ A_{γ_{i}}^{∗}, and let B(\underline{γ}, \underline{a}) ⊂ A be the K^{+}-subalgebra generated by \underline{a} := (a_{1}, …, a_{n}). Clearly the grading of A induces a Γ_{0}-grading on B(\underline{γ}, \underline{a}), and gr_{β}B(\underline{γ}, \underline{a}) is a finitely generated K^{+}-module for every β ∈ Γ_{0} (proposition 7.6.11(ii)); by virtue of (MA1), we know that gr_{β}B(\underline{γ}, \underline{a}) is then even a free rank one K^{+}-module, for every β ∈ Γ_{0}. Furthermore, \underline{a} generates a fine submonoid of A^{∗}, so – by proposition 6.6.35(ii) – there exists an integer k > 0 such that

(gr_{kβ}B(\underline{γ}, \underline{a}))^{n} = gr_{nkβ}B(\underline{γ}, \underline{a}) for every β ∈ Γ_{0} and every integer n > 0.

Now, for any β ∈ Γ, let t > 0 be an integer such that p^{t}β ∈ Γ_{0}, and pick a generator b of the K^{+}-module gr_{kp^{t}β}B(\underline{γ}, \underline{a}); in light of (MA2) we may find x ∈ K^{+} and c ∈ gr_{β}A, such that b = c^{kp^{t}}x. Since Γ_{K} is divisible, we may write x = y^{kp^{t}}u for some y, u ∈ K^{+} with |u| = 1.

It is easily seen that the K^{+}-submodule of gr_{β}A generated by cy does not depend on the choices of c, y, t and k; hence we denote gr_{β}C(\underline{γ}, \underline{a}) this submodule; a simple inspection shows that the resulting Γ-graded K^{+}-module

C(\underline{γ}, \underline{a}) := ⨁_{β∈Γ} gr_{β}C(\underline{γ}, \underline{a})

is actually a Γ-graded K^{+}-subalgebra of A, for which (MA2) holds, and therefore (C(\underline{γ}, \underline{a}), Γ) is a small model algebra. Lastly, it is clear that the family of all such (C(\underline{γ}, \underline{a}), Γ), for \underline{γ} ranging over all finite sets of generators of Γ, and \underline{a} ranging over all the finite sequences of elements of A as above, form a cofiltered system of subobjects, whose colimit is (A, Γ). This concludes the proof of the proposition in this case.

Next, suppose that Γ = Γ_{0} is a finite abelian group. In view of remark 17.1.3(iii), we are reduced to the case where Γ is cyclic, say Γ = ℤ/nℤ. Fix any generator γ of Γ, and for every a ∈ gr_{γ}A, let C(a) be the K^{+}-subalgebra of A generated by a; it is easily seen that the colimit of the filtered family ((C(a), Γ) | a ∈ gr_{γ}A) equals (A, Γ). (Details left to the reader.) □

17.1.21. Suppose now that K is algebraically closed, let (B, Δ) be a small model K^{+}-algebra, with Δ^{gp} torsion-free, and write Δ = Δ_{0}[1/p] for some fine and saturated submonoid Δ_{0} such that the K^{+}-algebra B_{0} := B_{Δ_{0}} is finitely generated. Let us set

Δ_{n} := {γ ∈ Δ | p^{n}γ ∈ Δ_{0}}  B_{n} := B_{Δ_{n}}  Y_{n} := Spec B_{n}  for every n ∈ ℕ.

We wish to construct a ladder of log schemes

(17.1.22)  \xymatrix{⋯ \ar[r]^{g_{n+1}} & (Y_{n+1}, \underline{M}_{n+1}) \ar[r]^{g_{n}} \ar[d]_{φ_{n+1}} & (Y_{n}, \underline{M}_{n}) \ar[r]^{g_{n−1}} \ar[d]_{φ_{n}} & ⋯ \ar[r]^{g_{0}} & (Y_{0}, \underline{M}_{0}) \ar[d]^{φ_{0}} \\ ⋯ \ar[r]^{h_{n+1}} & (S, \underline{N}_{n+1}) \ar[r]^{h_{n}} & (S, \underline{N}_{n}) \ar[r]^{h_{n−1}} & ⋯ \ar[r]^{h_{0}} & (S, \underline{N}_{0})}

such that, for every n ∈ ℕ :

• φ_{n} is smooth and saturated, and \underline{M}_{n} admits a chart, given by a Δ_{n}-graded fine monoid P^{(n)}, such that P_{0}^{(n)} is sharp, and the inclusion map P_{0}^{(n)} → P^{(n)} is flat and saturated, and gives a chart for φ_{n}

• the morphism of schemes underlying h_{n} (resp. g_{n}) is the identity of S (resp. is induced by the inclusion B_{n} ⊂ B_{n+1})

• the morphism g_{n} admits a chart, given by an injective map P^{(n)} → Δ_{n} ×_{Δ_{n+1}} P^{(n+1)}

of Δ_{n}-graded monoids, whose restriction P_{0}^{(n)} → P_{0}^{(n+1)} gives a chart for h_{n}. This will be achieved in several steps, as follows :

• First, let b_{1}, …, b_{k} be a finite system of generators for B_{0}. By remark 17.1.12(ii), we have a decomposition B_{0}^{∗} = (K^{+})^{×} ⊕ C for some submonoid C ⊂ B_{0}^{∗}, and we may suppose that each b_{i} lies in C. Let P^{(0)} ⊂ B_{0}^{∗} be the submonoid generated by b_{1}, …, b_{k}. The restriction of the surjection B_{0}^{∗} → Δ_{0} is then still a surjection π : P^{(0)} → Δ_{0}. Notice that, for every x ∈ Ker π^{gp}, we have either x ∈ K^{+} or x^{−1} ∈ K^{+}; especially, we may find a finite system x_{1}, …, x_{n} of generators for Ker π^{gp}, such that x_{i} ∈ K^{+} for every i = 1, …, n. Now, let Σ ⊂ P^{(0)gp} be the submonoid generated by x_{1}, …, x_{n}; after replacing P^{(0)} by P^{(0)} · Σ, we may assume that Ker π^{gp} = P_{0}^{(0)gp}, where P_{0}^{(0)} := Ker π^{gp} ∩ P^{(0)} ⊂ K^{+} is a fine submonoid.

• Next, by theorem 6.6.45, we may find a finitely generated submonoid Σ′ ⊂ P_{0}^{(0)gp} ∩ K^{+} such that the induced morphism P_{0}^{(0)} · Σ′ → P^{(0)} · Σ′ is flat. Clearly

P_{0}^{(0)} · Σ′ = Ker π^{gp} ∩ (P^{(0)} · Σ′)

hence, we may replace P^{(0)} by P^{(0)} · Σ′, and assume that the morphism P_{0}^{(0)} → P^{(0)} is also flat.

• We claim that the map P_{0}^{(0)} → P^{(0)} is also saturated. We shall apply the criterion of proposition 6.2.31 : indeed, for given γ ∈ Δ_{0} and integer n > 0, let x (resp. y) be a generator of the P_{0}^{(0)}-module P_{γ}^{(0)} (resp. P_{nγ}^{(0)}). Then x (resp. y) is also a generator of the K^{+}-module gr_{γ}B (resp. gr_{nγ}B), and (MA2) implies that there exists u ∈ (K^{+})^{×} such that uy = x^{n}; but since x, y ∈ C, we must have u = 1 therefore (P_{γ}^{(0)})^{n} = P_{nγ}^{(0)}, as required.

• Next, we claim that the induced map of Δ_{0}-graded K^{+}-algebras

(17.1.23)  P^{(0)} ⊗_{P_{0}^{(0)}} K^{+} → B_{0}

is an isomorphism. Indeed, since P^{(0)} contains a set of generators for the K^{+}-algebra B, the map (17.1.23) is obviously surjective. However, for every γ ∈ Δ_{0}, the P_{0}^{(0)}-module P_{γ}^{(0)} is free of rank one (remark 6.2.5(iv)), therefore (P^{(0)} ⊗_{P_{0}^{(0)}} K^{+})_{γ} is a free K^{+}-module of rank one. It follows easily that (17.1.23) is also injective.

• Now, denote by \underline{N}_{0} (resp. \underline{M}_{0}) the fine log structure on S (resp. on X_{0}) deduced from the inclusion map P_{0}^{(0)} → K^{+} (resp. P^{(0)} → B_{0}). By lemma 12.1.13(iv), the inclusion P_{0}^{(0)} → P^{(0)} yields a chart for a morphism φ_{0} : (Y_{0}, \underline{M}_{0}) → (S, \underline{N}_{0}) that is saturated, as sought. Lastly, since (17.1.23) is an isomorphism, theorem 12.3.37shows that φ_{0} is also smooth.

• Next, according to remark 17.1.12(iii), we have a compatible system of decompositions

B_{n}^{∗} = (K^{+})^{×} ⊕ C^{(n)} for every n ∈ ℕ

with C^{(0)} = C, and such that the p-Frobenius induces an isomorphism τ_{n} : C^{(n+1)} →^{∼} C^{(n)} for every n ∈ ℕ. Hence, define inductively an increasing sequence of submonoids

P^{(0)} ⊂ P^{(1)} ⊂ P^{(2)} ⊂ ⋯ ⊂ B^{∗} by letting P^{(n+1)} := τ_{n}^{−1}P^{(n)} for every n ∈ ℕ.

Clearly, the grading of B^{∗} restricts to a Δ_{n}-grading on P^{(n)}, and induces a Δ-grading on P. We deduce isomorphisms of Δ_{n+1}-graded monoids :

(17.1.24)  P^{(n+1)} →^{∼} Δ_{n+1} ×_{Δ_{n}} P^{(n)}  for every n ∈ ℕ.

Especially, the induced maps of monoids P_{0}^{(n)} → P^{(n)} are still flat and saturated, and P_{0}^{(n)} is still sharp (remark 17.1.26). The inclusion maps P^{(n)} → B_{n} and P_{0}^{(n)} → K^{+} determine an isomorphism of Δ_{n}-graded K^{+}-algebras

(17.1.25)  P^{(n)} ⊗_{P_{0}^{(n)}} K^{+} →^{∼} B_{n}  for every n ∈ ℕ

as well as fine log structures \underline{M}_{n} on Y_{n}, and \underline{N}_{n} on S, whence a morphism of log schemes φ_{n} : (Y_{n}, \underline{M}_{n}) → (S, \underline{N}_{n}) which is again smooth and saturated. This completes the construction of (17.1.22).

Remark 17.1.26. (i) With the notation of (17.1.21), notice that, by construction, P^{(n)gp} ⊂ C^{(n)gp}; especially, since Δ^{gp} is torsion-free, the same holds for P^{(n)gp}. Likewise, the construction shows that the induced map P_{0}^{(n)gp} → Γ_{K} is always injective; especially, P_{0}^{(n)gp} is a free abelian group of finite rank, and P_{0}^{(n)} is sharp. It follows that the P_{0}^{(n)}-module P_{γ}^{(n)} admits a unique generator g_{γ}, for every γ ∈ Δ_{n}, and the saturation condition for the inclusion P_{0}^{(n)} → P^{(n)} translates as the system of identities :

g_{γ}^{k} = g_{kγ} for every k ∈ ℕ and every γ ∈ Δ_{n}.

(ii) Moreover, set P := ⋃_{n∈ℕ} P^{(n)}. Notice that the colimit of the maps (17.1.25) is an isomorphism of Δ-graded K^{+}-algebras

(17.1.27)  P ⊗_{P_{0}} K^{+} →^{∼} B.

Furthermore, since the natural map P^{(n)}/P_{0}^{(n)} → Δ is an isomorphism for every n ∈ ℕ, we deduce that the grading of P induces an isomorphism :

(17.1.28)  P/P_{0} →^{∼} Δ.

(iii) We also obtain a commutative diagram of log schemes :

(17.1.29)  \xymatrix{𝕊(B_{n}, Δ_{n}) \ar[r] \ar[d] & (Y_{n}, \underline{M}_{n}) \ar[d]^{φ_{n}} \\ 𝕊(K^{+}) \ar[r] & (S, \underline{N}_{n})}  for every n ∈ ℕ

whose left vertical arrow is the saturated morphism (17.1.11), and whose bottom (resp. top) arrow is the identity of S (resp. of Y_{n}) on the underlying schemes, and is induced by the inclusion map P_{n}^{(0)} → K^{∗} (resp. P^{(n)} → B_{n}^{∗}). It is easily seen that (17.1.29) is cartesian : indeed, since the functor (12.1.16) is right exact, it suffices to check that the natural map

P^{(n)} ⊗_{P_{0}^{(n)}} K^{∗} → B_{n}^{∗}

is an isomorphism, which is clear from (17.1.25).

(iv) Notice that the inclusion map i_{0} : P_{0}^{(0)} → P^{(0)} is also local; then corollary 6.4.7 yields a decomposition

ϑ_{0} : P^{(0)} →^{∼} P^{(0)×} ⊕ P^{(0)♯}

such that ϑ_{0} ∘ i_{0} = ι_{0} ∘ λ_{0}^{♯}, where ι_{0} : P^{(0)♯} → P^{(0)×} ⊕ P^{(0)♯} is the natural inclusion map. By means of the isomorphisms (17.1.24), we may then inductively construct decompositions

ϑ_{n} : P^{(n)} →^{∼} P^{(n)×} ⊕ P^{(n)♯} for every n ∈ ℕ

fitting into a commutative diagram

\xymatrix{P_{0}^{(n)} \ar[r]^{i_{n}} \ar[d]_{i_{n}^{♯}} & P^{(n)} \ar[r]^{j_{n}} \ar[d]_{ϑ_{n}} & P^{(n+1)} \ar[d]^{ϑ_{n+1}} \\ P^{(n)♯} \ar[r]^{ι_{n}} & P^{(n)×} ⊕ P^{(n)♯} \ar[r]^{j_{n}^{×}⊕j_{n}^{♯}} & P^{(n+1)×} ⊕ P^{(n+1)♯}}

where i_{n}, j_{n} and ι_{n} are the natural inclusion maps. Now, set

B′_{n} := K^{+}[P^{(n)×}]  B′′_{n} := P^{(n)♯} ⊗_{P_{0}^{(n)}} K^{+}  for every n ∈ ℕ

and let Δ′_{n} (resp. Δ′′_{n}) be the image of P^{(n)×} (resp. of ϑ_{n}^{−1}P^{(n)♯}) in Δ_{n}; since the grading of P^{(n)} induces an isomorphism P^{(n)}/P_{0}^{(n)} →^{∼} Δ_{n}, we see that Δ_{n} = Δ′_{n} ⊕ Δ′′_{n}, and by construction, for every n ∈ ℕ we have isomorphisms

(B_{n}, Δ_{n}) →^{∼} (B′_{n}, Δ′_{n}) ⊗ (B′′_{n}, Δ′′_{n})

of model K^{+}-algebras, which identify the inclusions B_{n} → B_{n+1} with the tensor product of the induced inclusion maps B′_{n} → B′_{n+1} and B′′_{n} → B′′_{n+1}. Furthermore, set Y′_{n} := Spec B′_{n} and Y′′_{n} := Spec B′′_{n} for every n ∈ ℕ. The induced map of monoids P^{(n)♯} → B′′_{n} determines a fine log structure \underline{M}′′_{n} on Y′′_{n}, the map i_{n}^{♯} gives a chart for a smooth and saturated morphism of log schemes

φ′′_{n} : (Y′′_{n}, \underline{M}′′_{n}) → (S, \underline{N}_{n})

and there follows an isomorphism of (S, \underline{N}_{n})-schemes (lemma 12.1.4)

(17.1.30)  (Y_{n}, \underline{M}_{n}) →^{∼} Y′_{n} ×_{S} (Y′′_{n}, \underline{M}′′_{n})  for every n ∈ ℕ.

(v) Starting with (17.1.31), we shall consider the strict henselization B^{sh} of B at a given geometric point \overline{x} of Spec B ⊗_{K^{+}} κ. In this situation, let x ∈ Y := Spec B be the support of \overline{x}, and 𝔭_{x} ⊂ B the corresponding prime ideal. Let also P be as in (ii), denote by β : P → B the natural map deduced from (17.1.27), and set 𝔭 := β^{−1}𝔭_{x}. Moreover, set 𝔭_{n} := 𝔭 ∩ P^{(n)} and Q^{(n)} := P_{𝔭_{n}}^{(n)} for every n ∈ ℕ; clearly 𝔭_{n} ⊂ 𝔭_{n+1} (so the isomorphism (17.1.24) maps 𝔭_{n+1} onto 𝔭_{n}), and therefore

Q^{(n)} ⊂ Q^{(n+1)} for every n ∈ ℕ.

Furthermore, the image of Q^{(n)} in Δ_{n}^{gp} is a localization Γ_{n} of Δ_{n}, especially it is still saturated, and the maps (17.1.24) extend to isomorphisms of Γ_{n+1}-graded monoids

Q^{(n+1)} →^{∼} Γ_{n+1} ×_{Γ_{n}} Q^{(n)} for every n ∈ ℕ.

It is also clear that the p-Frobenius of Γ_{n+1} factors through an isomorphism Γ_{n+1} →^{∼} Γ_{n}, for every n ∈ ℕ. Set F_{n} := P^{(n)}\𝔭_{n} for every n ∈ ℕ; notice that, by construction, F_{n}∩P_{0}^{(n)} = {1}, hence F_{n} ∩ P_{γ}^{(n)} is either empty or else it contains exactly one element, namely the generator g_{γ} of P_{γ}^{(n)}, by virtue of (i). It follows that Q_{0}^{(n)} = P_{0}^{(n)} for every n ∈ ℕ, and therefore the inclusion map Q_{0}^{(n)} → Q^{(n)} is still flat and saturated (lemma 6.2.12(ii)). Let Γ := ⋃_{n∈ℕ} Γ_{n}; we conclude that Q := P_{𝔭} is a Γ-graded monoid with Q_{0} = P_{0}, and we may define

B_{𝔭} := Q ⊗_{Q_{0}} K^{+}.

The foregoing shows that (B_{𝔭}, Γ) is still a small model algebra, and we also obtain a ladder of log schemes with the properties listed in (17.1.21) : namely, set B_{𝔭,n} := B_{𝔭,Γ_{n}} = Q^{(n)} ⊗_{Q_{0}^{(n)}} K^{+} for every n ∈ ℕ, and endow Y_{𝔭,n} := Spec B_{𝔭,n} with the log structure \underline{M}_{𝔭,n} determined by the induced map Q^{(n)} → B_{𝔭,n}. Clearly the geometric point \overline{x} lifts uniquely to a geometric point \overline{x}_{𝔭} of Y_{𝔭} := Spec B_{𝔭}, and the localization map B → B_{𝔭} induces an isomorphism

Y_{𝔭}(\overline{x}_{𝔭}) →^{∼} Y (\overline{x}).

Hence, for the study of the scheme Y (\overline{x}), it shall be usually possible to replace the original small algebra B by its localization B_{𝔭} thus constructed. In so doing, we gain one more property: indeed, notice that the new chart Q^{(n)} → B_{𝔭,n} is local at the geometric point \overline{x}, for every n ∈ ℕ. Now, choose a compatible system of decompositions

(Y_{𝔭,n}, \underline{M}_{𝔭,n}) →^{∼} Y′_{𝔭,n} ×_{S} (Y′′_{𝔭,n}, \underline{M}′′_{𝔭,n})

as in (iv), and denote by \overline{x}_{𝔭,n} (resp. \overline{x}′′_{𝔭,n}) the image of \overline{x}_{𝔭} in Y_{𝔭,n} (resp. in Y′′_{𝔭,n}), for every n ∈ ℕ. By inspecting the construction, it is easily seen that the chart Q^{(n)♯} → B′′_{𝔭,n} := Q^{(n)♯} ⊗_{Q_{0}^{(n)}} K^{+} is also local at \overline{x}′′_{𝔭,n}.

17.1.31. Let us return to the situation of (17.1.21). Fix a geometric point \overline{x} of Y := Spec B, localized on Y ×_{S} Spec κ, and for every n ∈ ℕ, let \overline{x}_{n} be the image of \overline{x} in Y_{n}. Set

B^{sh} := 𝒪_{Y (\overline{x}),\overline{x}} and B_{n}^{sh} := 𝒪_{Y_{n}(\overline{x}_{n}),\overline{x}_{n}} for every n ∈ ℕ.

For the next step, we shall apply the method of (14.5.60), to construct a normalized length function for B^{sh}-modules. To this aim, we need an auxiliary model K^{+}-algebra, defined as follows. First, notice that the grading P → Δ extends to a morphism of monoids

π_{ℚ} : P_{ℚ} → Δ_{ℚ}

(notation of (6.3.20)); i.e. P_{ℚ} is a Δ_{ℚ}-graded monoid. Since K^{×} is a divisible group, the inclusion map P_{0} → K^{∗} extends to a group homomorphism P_{ℚ,0}^{gp} → K^{×}, and it is easily seen that the latter restricts to a morphism of monoids

P_{ℚ,0} → K^{∗}.

So finally, we may set

A := P_{ℚ} ⊗_{P_{ℚ,0}} K^{+}.

Taking into account (17.1.27), we deduce an isomorphism of Δ-graded K^{+}-algebras

(17.1.32)  B →^{∼} A_{Δ}.

Moreover, arguing as in remark 17.1.12(iii), we get an isomorphism

(17.1.33)  K[Δ_{ℚ}] →^{∼} A_{K}

fitting into a commutative diagram of K-algebras

\xymatrix{K[Δ] \ar[r]^{∼} \ar[d] & B_{K} \ar[d] \\ K[Δ_{ℚ}] \ar[r]^{∼} & A_{K}}

whose top horizontal arrow is the isomorphism of remark 17.1.17(v), and whose right (resp. left) vertical arrow is deduced from (17.1.32) (resp. is induced by the natural inclusion Δ ⊂ Δ_{ℚ}).

Lemma 17.1.34. With the notation of (17.1.31), we have :

(i) P_{ℚ,γ} is a free P_{ℚ,0}-module of rank one, for every γ ∈ Δ_{ℚ}.

(ii) P_{ℚ,0} is a sharp monoid, and the inclusion map P_{ℚ,0} → P_{ℚ} is flat and saturated.

Proof. (i): We can write P_{ℚ} as the colimit of the system of monoids

(P_{[n]}; μ_{P,n,m} : P_{[n]} → P_{[nm]} | n, m ∈ ℕ)

such that P_{[n]} := P^{(0)} for every n ∈ ℕ, and μ_{P,n,m} is the m-Frobenius map of P^{(0)}, for every n, m ∈ ℕ. Likewise, we may write Δ_{ℚ} as colimit of a system of Frobenius endomorphisms (Δ_{0,[n]}; μ_{Δ,n,m} | n, m ∈ ℕ), and π_{ℚ} is the colimit of the corresponding system of maps (π_{[n]} : P_{[n]} → Δ_{0,[n]} | n ∈ ℕ) where π_{[n]} := π for every n ∈ ℕ. Hence, for any γ ∈ Δ_{0} there exists n ∈ ℕ such that γ is the image of some γ_{n} ∈ Δ_{0,[n]}, and

P_{ℚ,γ} = colim_{k∈ℕ} P_{[nk],kγ_{n}} = colim_{k∈ℕ} P_{kγ_{n}}.

However, we know that P_{kγ_{n}} is free of rank one, generated by a unique element g_{kγ_{n}}; moreover, the transition maps μ_{P,n,k} send g_{γ_{n}} onto g_{kγ_{n}}, for every k ∈ ℕ (remark 17.1.26). This implies that P_{ℚ,γ} is generated by the image of g_{γ_{n}}, whence (i).

(ii): The flatness follows from (i). Since the inclusion map P_{0}^{(0)} → P^{(0)} is saturated, it is easily seen that the same holds for the inclusion P_{ℚ,0} → P_{ℚ}. Lastly, the sharpness of P_{ℚ,0} is likewise deduced from the sharpness of P_{0}^{(0)} (details left to the reader). □

17.1.35. Lemma 17.1.34(ii) implies that the pair (A, Δ_{ℚ}) fulfills axiom (MA2), and we have thus our sought auxiliary model K^{+}-algebra. Since P_{ℚ,0} is sharp, the P_{ℚ,0}-module P_{ℚ,γ} admits a unique generator g_{γ}, for every γ ∈ Δ_{ℚ}. The image g_{γ} ⊗1 of g_{γ} in A is a generator of the direct summand A_{γ}, which is a free K^{+}-module of rank one; hence, there exists a unique a_{γ} ∈ K such that γ ⊗ a_{γ} gets mapped to g_{γ} ⊗ 1, under the isomorphism (17.1.33).

After choosing an order-preserving isomorphism (14.5.63), we may define a function

f_{A} : Δ_{ℚ} → ℝ : γ ↦ log |a_{γ}|.

The inclusion A_{γ} · A_{δ} ⊂ A_{γ+δ} translates as the inequality

f_{A}(γ) + f_{A}(δ) ≥ f_{A}(γ + δ) for every γ, δ ∈ Δ_{ℚ}.

Likewise, the saturation condition of axiom (MA2) translates as the identity

f_{A}(nγ) = n · f_{A}(γ) for every γ ∈ Δ_{ℚ} and every n ∈ ℕ.

We also fix a (Banach) norm

Δ_{ℝ}^{gp} → ℝ γ ↦ ∥γ∥.

We can then state the following

Lemma 17.1.36. With the notation of (17.1.35), we have :

(i) There exists a Δ_{0}^{gp}-rational subdivision Θ of the convex polyhedral cone (Δ_{ℝ}^{gp}, Δ_{ℝ}), such that

f_{A}(γ + δ) = f_{A}(γ) + f_{A}(δ) for every σ ∈ Θ and every γ, δ ∈ σ ∩ Δ_{ℚ}.

(ii) Especially, the function f_{A} is of Lipschitz type, i.e. there exists a real constant C_{A} > 0 such that

|f_{A}(γ) − f_{A}(γ′)| ≤ C_{A} · ∥γ − γ′∥ for every γ, γ′ ∈ Δ_{ℚ}^{gp}.

Proof. (Here, Δ_{ℝ}^{gp} := Δ_{0}^{gp} ⊗_{ℤ} ℝ and Δ_{ℝ} ⊂ Δ_{ℝ}^{gp} is the cone spanned by Δ_{0}.) Combining proposition 6.3.28and lemma 6.3.33, we find a Δ_{0}^{gp}-rational subdivision Θ of Δ_{ℝ} such that

P_{ℚ,γ+δ} = P_{ℚ,γ} + P_{ℚ,δ} for every σ ∈ Θ and every γ, δ ∈ σ ∩ Δ_{ℚ}.

This means that g_{γ} · g_{δ} = g_{γ+δ} for every σ ∈ Θ and every γ, δ ∈ σ ∩ Δ_{ℚ}, whence (i).

(ii): Clearly, for every σ ∈ Θ we may find a constant C_{σ} such that the stated inequality holds – with C_{A} replaced by C_{σ} – for every γ, γ′ ∈ σ∩Δ_{ℚ}. It is easily seen that C_{A} := max(C_{σ} | σ ∈ Θ) will do (details left to the reader). □

17.1.37. Fix n ∈ ℕ, pick a subdivision Θ of (Δ_{ℝ}^{gp}, Δ_{ℝ}) as in lemma 17.1.36, and let Θ^{s} ⊂ Θ be the subset of all σ ∈ Θ that span Δ_{ℝ}^{gp}. Also, for every γ ∈ Δ_{ℚ}^{gp}, denote by [γ] the class of γ in Δ_{ℚ}^{gp}/Δ_{n}^{gp}. For any subset Σ ⊂ Δ_{ℝ}^{gp} we let

A_{Σ} := A_{Σ∩Δ_{ℚ}}

(where the right-hand side is defined as in (17.1.5)). In light of lemma 6.3.26 we obtain a decomposition of B_{n} = A_{Δ_{n}} = A_{[0]} as sum of K^{+}-subalgebras :

B_{n} = ∑_{σ∈Θ^{s}} A_{[0]∩σ}

and a corresponding decomposition of the B_{n}-module A_{[γ]} as sum of A_{[0]∩σ}-modules

A_{[γ]} = ∑_{σ∈Θ^{s}} A_{[γ]∩σ} for every γ ∈ Δ_{ℚ}^{gp}.

Lemma 17.1.38. With the notation of (17.1.37) and (17.1.21), there exists N ∈ ℕ such that for every n ∈ ℕ and every γ ∈ Δ_{ℚ}^{gp}, the B_{n}-module A_{[γ]} admits a system of generators of cardinality at most N.

Proof. According to proposition 6.3.22(ii), for every σ ∈ Θ^{s} and every γ ∈ Δ_{ℚ}^{gp}, the subset Σ_{σ,γ} := Δ_{n}^{gp} ∩ (σ − γ) is a finitely generated (Δ_{n}^{gp} ∩ σ)-module; hence, let us fix a finite system of generators G_{σ,γ} for Σ_{σ,γ}. In light of lemma 17.1.36(i) we see that the finite set

G′_{σ,γ} := {g_{γ+δ} ⊗ 1 | δ ∈ G_{σ,γ}} ⊂ A

generates the A_{[0]∩σ}-module A_{[γ]∩σ}. Consequently, the finite set ⋃_{σ∈Θ^{s}} G′_{σ,γ} generates the B_{n}- module A_{[γ]}. On the other hand, since 𝒮_{Δ_{n}^{gp},σ} is a finitely generated Δ_{n}^{gp}-module (proposition 6.3.35(ii)) it is clear that the cardinality of G_{σ,γ} is bounded by a constant N_{σ} that is independent of γ, and then A_{[γ]} is generated by at most N_{n} := ∑_{σ∈Θ^{s}} N_{σ} elements. It remains to show that the estimate for N_{n} is independent of n. However, notice that, for every σ ∈ Θ^{s}, the automorphism of Δ_{ℚ}^{gp} given by multiplication by p^{n}, induces a natural bijection

𝒮_{Δ_{n}^{gp},σ} →^{∼} 𝒮_{Δ_{0}^{gp},σ}

that sends each Δ_{n}-module Δ_{n}^{gp} ∩ (σ − v) onto the Δ_{0}-module

Δ_{0}^{gp} ∩ (σ − p^{n}v) = Δ_{0} ⊗_{Δ_{n}} (Δ_{n}^{gp} ∩ (σ − v))

(where the extension of scalars Δ_{n} → Δ_{0} is the isomorphism given by the rule : γ ↦ p^{n}γ for every γ ∈ Δ_{n} ∩ σ). Thus, we see that N := N_{0} will already do. □

17.1.39. Let us choose N as in lemma 17.1.38, and endow the set ℳ_{N}(B_{n}^{a}) with the uniform structure defined in (14.5.85) (relative to the standard setup attached to K^{+}); we consider the mapping

Δ_{ℚ}^{gp} → ℳ_{N}(B_{n}^{a}) γ ↦ A_{[γ]}^{a}.

Lemma 17.1.40. Keep the notation of (6.3.34) and (14.5.85), and let also C_{A} > 0 be the constant appearing in lemma 17.1.36(ii). Then :

(A_{[γ]}^{a}, A_{[γ′]}^{a}) ∈ E_{C_{A}·∥γ−γ′∥·2}

for every Σ ∈ 𝒮_{Δ_{n}^{gp},Δ_{ℝ}} and every γ, γ′ ∈ Ω(Δ_{ℝ}, Σ) ∩ Δ_{ℚ}^{gp}.

Proof. By inspecting the definitions, we get an isomorphism of B_{n}-modules

(17.1.41)  A_{[γ]} →^{∼} ⨁_{δ∈Σ}{δ ⊗ a ∈ K[Δ_{n}^{gp}] | log |a| ≥ f_{A}(γ + δ)}

for every Σ ∈ 𝒮_{Δ_{n}^{gp},Δ_{ℝ}} and every γ ∈ Ω(Δ_{ℝ}, Σ) ∩ Δ_{ℚ}^{gp}. Hence, suppose more generally that Σ, Σ′ are two elements of 𝒮_{Δ_{n}^{gp},Δ_{ℝ}} such that Σ ⊂ Σ′, and let γ ∈ Ω(Δ_{ℝ}, Σ) ∩ Δ_{ℚ}^{gp}, γ′ ∈ Ω(Δ_{ℝ}, Σ′) ∩ Δ_{ℚ}^{gp} be any two elements; from lemma 17.1.36(ii) we get, for any b ∈ K^{+} with log |b| ≥ C_{A}∥γ′ − γ∥, a B_{n}-linear map

τ_{b,γ′−γ} : A_{[γ]} → A_{[γ′]}

which, under the identifications (17.1.41), corresponds to the map given by the rule : δ ⊗ a ↦ δ ⊗ ba for every δ ∈ Σ and every a ∈ K such that log |a| ≥ f(γ + δ). In case Σ = Σ′, also τ_{b,γ−γ′} is well defined, and clearly

τ_{b,γ′−γ} ∘ τ_{b,γ−γ′} = b^{2} · 1_{A_{[γ′]}} τ_{b,γ−γ′} ∘ τ_{b,γ′−γ} = b^{2} · 1_{A_{[γ]}}

whence the contention. □

Remark 17.1.42. Let γ_{1}, γ_{2}, γ′_{1}, γ′_{2} ∈ Δ_{ℚ}^{gp} be four elements, and set γ_{3} := γ_{1}+γ_{2}, γ′_{3} := γ′_{1}+γ′_{2}; suppose that

Δ_{n}^{gp} ∩ (Δ_{ℝ} − γ_{i}) ⊂ Δ_{n}^{gp} ∩ (Δ_{ℝ} − γ′_{i}) for i = 1, 2, 3.

With the notation of the proof of lemma 17.1.40, for every b_{1}, b_{2} ∈ K^{+} such that

log |b_{i}| ≥ C_{A}∥γ′_{i} − γ_{i}∥ (i = 1, 2)

we obtain a commutative diagram of B_{n}-linear maps

\xymatrix{A_{[γ_{1}]} ⊗_{B_{n}} A_{[γ_{2}]} \ar[r] \ar[d]_{τ_{b_{1},γ′_{1}−γ_{1}}⊗τ_{b_{2},γ′_{2}−γ_{2}}} & A_{[γ_{3}]} \ar[d]^{τ_{b_{1}b_{2},γ′_{3}−γ_{3}}} \\ A_{[γ′_{1}]} ⊗_{B_{n}} A_{[γ′_{2}]} \ar[r] & A_{[γ′_{3}]}}

whose horizontal arrows are the restrictions of the multiplication map of the B_{n}-algebra B : the detailed verification shall be left as an exercise for the reader.

Theorem 17.1.43. In the situation of (17.1.31), the K^{+}-algebra B^{sh} is ind-measurable.

Proof. First, notice that, for every k ≥ 0, the inclusion map B_{n} → B_{n+k} induces a radicial morphism B_{n} ⊗_{K^{+}} κ → B_{n+k} ⊗_{K^{+}} κ, and therefore also an isomorphism of B_{n+k}-algebras

(17.1.44)  B_{n}^{sh} ⊗_{B_{n}} B_{n+k} →^{∼} B_{n+k}^{sh}

([66, Ch.IV, Prop.18.8.10]). Notice as well, that B_{n}^{sh} is a measurable K^{+}-algebra (see definition 14.5.3(ii)), hence we have a well defined normalized length function λ_{n} on isomorphism classes of B_{n}^{sh}-modules, characterized as in theorem 14.5.43. Thus, our task is to exhibit a sequence (d_{n} | n ∈ ℕ) of normalizing factors fulfilling conditions (a) and (b) of definition 14.5.64, for the directed system (B_{n}^{sh} | n ∈ ℕ), whose colimit is B^{sh}. To this aim, let M be any object of B_{n}^{sh}-Mod_{coh,{s}}; in view of (17.1.44), we need to estimate

λ_{k+n}(B_{k+n} ⊗_{B_{n}} M) = λ_{k+n} ( ⨁_{[γ]∈Δ_{k+n}^{gp}/Δ_{n}^{gp}} A_{[γ]} ⊗_{B_{n}} M) .

To this aim, let us introduce the function

l_{M} : Δ_{ℚ}^{gp} → ℝ γ ↦ λ_{n}(A_{[γ]} ⊗_{B_{n}} M).

Claim 17.1.45. Let Σ be as in lemma 17.1.40, set J := Ann_{B_{0}^{sh}}(M/𝔪M), and suppose that M admits a generating set of cardinality k. Then the restriction of l_{M} to Ω(Δ_{ℝ}, Σ) ∩ Δ_{ℚ}^{gp} is of Lipschitz type, i.e. there exists a real constant C′_{A} > 0 independent of n, Σ and M such that

|l_{M}(γ) − l_{M}(γ′)| ≤ C′_{A} · k · length_{B_{n}^{sh}}(B_{n}^{sh}/JB_{n}^{sh}) · ∥γ − γ′∥

for every γ, γ′ ∈ Ω(Δ_{ℝ}, Σ) ∩ Δ_{ℚ}^{gp}.

Proof of the claim. Let I ⊂ B_{0}^{sh} be any finitely generated 𝔪_{B_{n}^{sh}}-primary ideal contained in the annihilator of M. Also, let N′ be the cardinality of a finite set of generators for the B_{n}^{sh}-module M, and N the constant provided by lemma 17.1.38; clearly A_{[γ]} ⊗_{B_{n}} M admits a system of generators of cardinality ≤ NN′, for every γ ∈ Δ_{ℚ}^{gp}. Furthermore, an argument as in the proof of [75, Lemma 2.3.7(iv)] shows that the induced mapping

ℳ_{N}(B_{n}) → ℳ_{NN′}(B_{n}^{sh}/IB_{n}^{sh}) L ↦ L ⊗_{B_{0}} M

is of Lipschitz type; more precisely, it sends the entourage E_{r} into E_{2r}, for every r ∈ ℝ_{>0} (details left to the reader). Then the claim follows by combining lemmata 14.5.86 and 17.1.40.

♢

From proposition 6.3.35(iv), we see that, for every Σ ∈ 𝒮_{Δ_{n}^{gp},Δ_{ℝ}}, the subset Ω(Δ_{ℝ}, Σ) ∩ Δ_{ℚ}^{gp} is dense in Ω(Δ_{ℝ}, Σ); taking into account claim 17.1.45, it follows that l_{M} extends (uniquely) to a function l_{M,ℝ} : Δ_{ℝ}^{gp} → ℝ, whose restriction to every Ω(Δ_{ℝ}, Σ) is continuous and still satisfies the same Lipschitz type estimate. Notice that l_{M,ℝ} descends to a function on the torus

\overline{l}_{M} : T_{n} := Δ_{ℝ}^{gp}/Δ_{n}^{gp} → ℝ.

Hence, set r := rk_{ℤ}Δ_{0}^{gp}, let us fix a basis e_{1}, …, e_{r} for the free ℤ-module Δ_{0}^{gp}, and define

Ω_{n} := \{∑_{i=1}^{r} t_{i}e_{i} | − \frac{1}{2p^{n}} ≤ t_{i} < \frac{1}{2p^{n}} for i = 1, …, r\}

so Ω_{n} ⊂ Δ_{ℝ}^{gp} is a fundamental domain for the lattice Δ_{n}^{gp}; in view of theorem 14.5.43(ii.b), we reach the following identity :

λ_{k+n}(B_{k+n} ⊗_{B_{n}} M) = \frac{1}{[κ(B_{k+n}^{sh}) : κ(B_{n}^{sh})]} · ∑_{γ∈Δ_{k+n}^{gp}∩Ω_{n}} l_{M}(γ).

Now, set

(17.1.46)  d_{n} := p^{nr} · [κ(B_{n}^{sh}) : κ(B_{0}^{sh})]^{−1}  for every n ∈ ℕ.

We claim that (d_{n} | n ∈ ℕ) is a suitable sequence of normalizing factors for B^{sh}. Indeed, recall that Ω(Δ_{ℝ}, Σ) is linearly constructible for every Σ ∈ 𝒮_{Δ_{0}^{gp},Δ_{ℝ}} (proposition 6.3.35(iii)), especially, the bounded function l_{M} is continuous outside a subset of Δ_{ℝ}^{gp} of measure zero, hence it is integrable on every bounded measurable subset of Δ_{ℝ}. It follows that \overline{l}_{M} is integrable relative to the invariant measure dμ_{n} on T_{n} of total volume equal to 1; lastly, a simple inspection (comparison of Riemann and Lebesgue integrals) yields

λ(B ⊗_{B_{n}} M) := lim_{k→+∞} d_{k+n}^{−1} · λ_{k+n}(B_{k+n} ⊗_{B_{n}} M) = d_{n}^{−1} ∫_{T_{n}} \overline{l}_{M}dμ_{n}

which shows that condition (a) of definition 14.5.64holds for this choice of normalizing factors.

In order to check condition (b), set Δ_{ℝ}(ρ) := {v ∈ Δ_{ℝ} | ∥v∥ ≤ ρ} for every ρ > 0. Notice that the automorphism of Δ_{ℝ}^{gp} given by multiplication by p^{n} restricts to a bijection

Ω(Δ_{ℝ}, Δ_{n}) →^{∼} Ω(Δ_{ℝ}, Δ_{0}) for every n ∈ ℕ.

Then, by claim 6.3.40, we see that there exists ρ_{0} such that

(17.1.47)  Δ_{ℝ}(p^{−n}ρ_{0}) ⊂ Ω_{n} ∩ Ω(Δ_{ℝ}, Δ_{n})  for all n ∈ ℕ.

Especially, for every ρ ≤ ρ_{0} we may regard Δ_{ℝ}(p^{−n}ρ) as a measurable subset of T_{n}, whose measure Vol(ρ) is strictly positive and independent of n.

Claim 17.1.48. Let I ⊂ B_{0}^{sh} be any finitely generated 𝔪_{B_{0}^{sh}}-primary ideal. Then there exists a real constant C_{I} > 0 such that

d_{n}^{−1} · length_{B_{n}^{sh}}(B_{n}^{sh}/(I + 𝔪)B_{n}^{sh}) ≤ C_{I} for every n ∈ ℕ.

Proof of the claim. On the one hand, we may write B_{n}^{sh} as the direct sum of the B_{0}^{sh}-modules A_{[γ]∩Δ_{ℚ}} ⊗_{B_{0}} B_{0}^{sh}, for γ ranging over the elements of Δ_{n}^{gp}/Δ_{0}^{gp}. There are p^{rn} such direct summands, and each of them admits a generating system of cardinality ≤ N, where N is the constant provided by lemma 17.1.38. It follows that

length_{B_{0}^{sh}}(B_{n}^{sh}/(I + 𝔪)B_{n}^{sh}) ≤ Np^{rn} · length_{B_{0}^{sh}}(B_{0}^{sh}/(I + 𝔪B_{0}^{sh})).

On the other hand, say that l := length_{B_{n}^{sh}}(B_{n}^{sh}/(I+𝔪)B_{n}^{sh}); this means that may find a filtration of B_{n}^{sh}/(I + 𝔪)B_{n}^{sh} of length l, consisting of B_{n}^{sh}-submodules, whose graded subquotients are all isomorphic to κ(B_{n}^{sh}). Given such a filtration, we easily obtain a filtration of B_{n}^{sh}/(I +𝔪)B_{n}^{sh} of length l · [κ(B_{n}^{sh}) : κ(B_{0}^{sh})], consisting of B_{0}^{sh}-submodules, whose graded subquotients are all isomorphic to κ(B_{0}^{sh}). Therefore

l · [κ(B_{n}^{sh}) : κ(B_{0}^{sh})] = length_{B_{0}^{sh}}(B_{n}^{sh}/(I + 𝔪)B_{n}^{sh})

whence the claim. ♢

Now, fix ε > 0, and suppose there is given a finitely generated 𝔪_{B_{0}^{sh}}-primary ideal I ⊂ B_{0}^{sh}, and a surjection of finitely presented B_{n}^{sh}/IB_{n}^{sh}-modules M → M′, generated by k elements, such that

d_{n}^{−1} · (λ_{n}(M) − λ_{n}(M′)) > ε.

Set C(I, k) := 2kC′_{A}C_{I}, where C′_{A} and C_{I} are as in claims 17.1.45and 17.1.48; since

l_{M}(0) − l_{M′}(0) = λ_{n}(M) − λ_{n}(M′)

we may estimate :

|λ(B^{sh} ⊗_{B_{n}^{sh}} M) − λ(B^{sh} ⊗_{B_{n}^{sh}} M′)| ≥ d_{n}^{−1} · ∫_{Δ_{ℝ}(p^{−n}ρ)}(\overline{l}_{M} − \overline{l}_{M′})dμ_{n}
≥ Vol(ρ) · (d_{n}^{−1}(l_{M}(0) − l_{M′}(0)) − C(I, k) · p^{−n}ρ)
≥ Vol(ρ) · (ε − C(I, k) · ρ)

for every ρ ≤ ρ_{0}. Therefore, if we set

ρ(k, ε, I) := min{ρ_{0}, 2^{−1}C(I, k)^{−1}ε}

it is easily seen that the sought condition (b) holds with δ(k, ε, I) := 2^{−1} ·Vol(ρ(k, ε, I))·ε. □

17.2. Almost purity : the log regular case. In this section, we prove an almost purity theorem for certain towers of regular log schemes.

17.2.1. Let \underline{M}_{0} be a log structure on the Zariski site of a local scheme X_{0}, such that (X_{0}, \underline{M}_{0}) is regular at the closed point x_{0} ∈ X_{0}, and say that X_{0} = Spec B_{0} for a local ring B_{0} which is necessarily noetherian, normal and Cohen-Macaulay (corollary 12.5.13). Let also β_{0} : P → B_{0} be a chart for \underline{M}_{0} that is sharp at x_{0}. Especially, P is a fine and saturated monoid, and A := B_{0}/𝔪_{P} B_{0} is a regular local ring; we denote by 𝔪_{A} (resp. 𝔪_{B_{0}}) the maximal ideal of A (resp. of B_{0}). We assume furthermore that :

(a) The characteristic p of the residue field κ(x_{0}) of A is positive.

(b) The Frobenius endomorphism Φ_{B_{0}} of B_{0}/pB_{0} is a finite ring homomorphism. Notice that (b) implies that B_{0}/pB_{0} is excellent (theorem 9.7.26(i)) and also that Ω_{A/ℤ}^{1} ⊗_{A} κ(x_{0}) is a finite dimensional κ(x_{0})-vector space. Moreover, (X_{0}, \underline{M}_{0}) is a regular log scheme, by theorem 12.5.31.

Example 17.2.2. Let (B, β) be a pair consisting of (a) a local ring B whose residue field κ_{B} has positive characteristic

(b) a local morphism of monoids β : P → B from a fine, sharp and saturated monoid P. Denote by 𝔪_{B} the maximal ideal of B, by \underline{M} the logarithmic structure on the Zariski site of X := Spec B induced by β, and suppose that (X, \underline{M}) is regular at the closed point of X.

(i) If moreover, B is complete for the 𝔪_{B}-adic topology and κ_{B} is perfect, then the Frobenius endomorphism of B/pB is a finite map. Indeed, in this case, for a suitable d ∈ ℕ we have a surjective ring homomorphism

R[[P × ℕ^{⊕d}]] → B

where R is either κ_{B}, or a discrete valuation ring with an isomorphism R/pR →^{∼} κ_{B} (remark 12.5.12). The assertion is an immediate consequence.

(ii) It follows that for any such pair (B, β) we may find a local ring B_{0} and a local and flat ring homomorphism f : B → B_{0} such that the log scheme

(X_{0}, \underline{M}_{0}) := Spec B_{0} ×_{X} (X, \underline{M})

and its chart β_{0} := β ∘ f : P → B_{0} fulfill the conditions of (17.2.1). Indeed, according to [61, Ch.0, Prop.10.3.1] there exists a flat and local B-algebra C such that κ_{B} ⊗_{B} C is a perfect field, and we take B_{0} to be the completion of C. By proposition 12.5.30(ii), the resulting log scheme (X_{0}, \underline{M}_{0}) is regular at the closed point of X_{0}, so the assertion follows from (i).

17.2.3. In the situation of (17.2.1), set

P^{(n)} := {γ ∈ P_{ℚ} | γ^{p^{n}} ∈ P } for every n ∈ ℕ.

The p^{n}-Frobenius map of P^{(n)} identifies P^{(n)} with its submonoid P; in other words, the inclusion map P → P^{(n)} is naturally identified with the p^{n}-Frobenius endomorphism of P. Fix a sequence (f_{1}, …, f_{r}) of elements of B_{0} whose image in A is maximal in the sense of remark 9.6.35(iii), in which case we say that (f_{1}, …, f_{r}) is maximal in B_{0}. Notice that dim_{κ(x_{0})} 𝔪_{A}/𝔪_{A}^{2} = dim A, since A is regular; it follows that

r = dim A + dim_{κ(x_{0})} Ω_{κ(x_{0})/ℤ}^{1}

by virtue of the short exact sequence (9.6.5), if p ∈ 𝔪_{A}^{2}, and otherwise, by virtue of proposition 9.6.17. Denote also by \overline{f}_{i} ∈ A the image of f_{i}, for every i = 1, …, r. According to corollary 9.6.34, the ring

A_{n} := A[T_{1}, …, T_{r}]/(T_{1}^{p^{n}} − \overline{f}_{1}, …, T_{r}^{p^{n}} − \overline{f}_{r})

is regular. For every n ∈ ℕ, we set :

B′_{n} := P^{(n)} ⊗_{P} B_{0}  B′′_{n} := B_{0}[T_{1}, …, T_{r}]/(T_{1}^{p^{n}} − f_{1}, …, T_{r}^{p^{n}} − f_{r})  B_{n} := B′_{n} ⊗_{B_{0}} B′′_{n}.

Lemma 17.2.4. The induced maps

B_{n} → B_{n+1}  and  B_{n}/pB_{n} → B_{n+1}/pB_{n+1}

are injective, for every n ∈ ℕ.

Proof. The natural map B_{n} → B_{n+1} factors as the composition

B′_{n} ⊗_{B_{0}} B′′_{n} → B′_{n+1} ⊗_{B_{0}} B′′_{n} → B′_{n+1} ⊗_{B_{0}} B′′_{n+1} for every n ∈ ℕ.

Notice that B_{0} = B′′_{0}, and B′′_{n+1} is a free B′′_{n}-module (of rank p^{r}) for every n ∈ ℕ; we are then reduced to checking that the maps B′_{n} → B′_{n+1} and B′_{n}/pB′_{n} → B′_{n+1}/pB′_{n+1} are injective for every n ∈ ℕ. However, set G := P^{(n+1)}/P^{(n)}, and notice that P^{(n+1)} is a G-graded monoid, with P_{0}^{(n+1)} = P^{(n)}, hence B′_{n+1} is a G-graded B′_{n}-algebra with (B′_{n+1})_{0} = B′_{n} for every n ∈ ℕ. The assertion follows. □

17.2.5. In view of lemma 17.2.4, we may set

B′′ := ⋃_{n∈ℕ} B′′_{n}  B := ⋃_{n∈ℕ} B_{n}  P^{(∞)} := ⋃_{n∈ℕ} P^{(n)}

and clearly

B = P^{(∞)} ⊗_{P} B′′ for every n ∈ ℕ

from which we see that B is naturally a P^{(∞)}/P^{(n)}-graded B_{n}-algebra, for every n ∈ ℕ. Also, the induced morphism Spec B_{n+1}/pB_{n+1} → Spec B_{n}/pB_{n} is radicial and surjective, so Spec B_{n}/pB_{n} is a local scheme for every n ∈ ℕ; on the other hand, the map B_{0} → B_{n} is finite, so every point of Spec B_{n} specializes to a point of Spec B_{n}/pB_{n}. We conclude that B_{n} is a local ring, and we denote by 𝔪_{B_{n}} its maximal ideal, for every n ∈ ℕ. Let \underline{M}_{n} be the log structure on the Zariski site of X_{n} := Spec B_{n} deduced from the natural map β_{n} : P^{(n)} → B_{n}; notice that B_{n}/𝔪_{P^{(n)}}B_{n} = A_{n} is a regular local ring of dimension equal to dim A. Since we have as well dim P^{(n)} = dim P, it follows that (X_{n}, \underline{M}_{n}) is regular at the closed point x_{n} ∈ X_{n}. Then theorem 12.5.31 shows that (X_{n}, \underline{M}_{n}) is a regular log scheme. Thus, we have obtained a tower of finite morphisms of regular log schemes

(17.2.6)  ⋯ → (X_{n+1}, \underline{M}_{n+1}) → (X_{n}, \underline{M}_{n}) → ⋯ → (X_{0}, \underline{M}_{0})

which we call the maximal tower associated with the chart β_{0} : P → B_{0} and the maximal sequence (f_{1}, …, f_{r}). The limit of the tower (17.2.6) is a log scheme (X, \underline{M}) whose log structure admits a chart P^{(∞)} → B which is sharp at the closed point.

Remark 17.2.7. Keep the notation of (17.2.5), and let s ∈ ℕ be any integer; from remark 9.6.35(iii) it is easily seen that the sequence ((X_{n+s}, \underline{M}_{n+s}) | n ∈ ℕ), obtained by removing from (17.2.6) the first s terms, is the maximal tower associated with the chart β_{n} and the maximal sequence (f_{1}^{1/p^{s}}, …, f_{r}^{1/p^{s}}).

17.2.8. Let now y ∈ (X_{0}, \underline{M}_{0})_{tr} be any point, and n ∈ ℕ any integer. The chart β_{0} extends uniquely to a morphism of monoids β_{0}^{gp} : P^{gp} → B_{0,y} := 𝒪_{X_{0},y}, and we have a natural isomorphism

B_{n} ⊗_{B_{0}} B_{0,y} →^{∼} P^{(n)gp} ⊗_{P^{gp}} B_{0,y} ⊗_{B_{0}} B′′_{n}.

Especially, B_{n} ⊗_{B_{0}} B_{0,y} is a free B_{0,y}-module of rank p^{nd}, where

d := dim P + r = dim P + dim A + dim_{κ(x_{0})} Ω_{κ(x_{0})/ℤ}^{1}.

However, since (X_{0}, \underline{M}_{0}) is regular, we have dim A + dim P = dim B_{0}. Summing up, we find

d = dim B_{0} + dim_{κ(x_{0})} Ω_{κ(x_{0})/ℤ}^{1}.

Next, let z ∈ Spec B_{0}/pB_{0} ⊂ X_{0} be any point, and 𝔭_{z} ⊂ B_{0} the corresponding prime ideal; as in the foregoing, the chart β_{0} extends uniquely to a morphism β_{𝔭} : P_{𝔭} → B_{0,z} := 𝒪_{X_{0},z}, where 𝔭 := β_{0}^{−1}𝔭_{z}. As already remarked, the point z lifts uniquely to a point z_{n} ∈ X_{n}, and on the other hand, there exists a unique prime ideal 𝔭^{(n)} ⊂ P^{(n)} containing 𝔭, and the inclusion map j_{𝔭} : P_{𝔭} → P_{𝔭^{(n)}}^{(n)} is naturally identified with the p^{n}-Frobenius endomorphism of P_{𝔭}. By lemma 6.2.10, there exists an isomorphism of monoids P_{𝔭} →^{∼} G × Q, with G := P_{𝔭}^{×} and Q := P_{𝔭}^{♯}; we may then find a corresponding decomposition P_{𝔭^{(n)}}^{(n)} = G^{(n)} × Q^{(n)} that identifies j_{𝔭} with the product of maps of monoids G → G^{(n)} and Q → Q^{(n)}. Summing up, there follows an isomorphism of B_{0}-algebras

𝒪_{X_{n},z_{n}} →^{∼} P_{𝔭^{(n)}}^{(n)} ⊗_{P_{𝔭}} B_{0,z} ⊗_{B_{0}} B′′_{n} →^{∼} (Q^{(n)} ⊗_{Q} B_{0,z}) ⊗_{B_{0,z}} (G^{(n)} ⊗_{G} B_{0,z}) ⊗_{B_{0}} B′′_{n}.

Fix a basis g_{1}, …, g_{s} of the free ℤ-module G, and set f_{r+i} := β_{𝔭}(g_{i}) for i = 1, ⋯, s; clearly

(G^{(n)} ⊗_{G} B_{0,z}) ⊗_{B_{0}} B′′_{n} = B_{0,z}[T_{1}, …, T_{r+s}]/(T_{1}^{p^{n}} − f_{1}, …, T_{r+s}^{p^{n}} − f_{r+s}).

On the other hand, set A_{z} := B_{0,z}/𝔪_{Q}B_{0,z}; we have

dim P = dim Q + s  (corollary 6.4.12(i))
dim B_{0,z} = dim Q + dim A_{z}  (since (X_{0}, \underline{M}_{0}) is regular)
dim B_{0,z} = d − dim_{κ(z)} Ω_{κ(z)/ℤ}^{1}  (proposition 9.7.20).

Therefore r + s = dim A_{z} + dim_{κ(z)} Ω_{κ(z)/ℤ}^{1}. However, A_{z_{n}} := 𝒪_{X_{n},z_{n}}/𝔪_{Q^{(n)}}𝒪_{X_{n},z_{n}} is a regular local ring, since (X_{n}, \underline{M}_{n}) is regular, and by inspecting the construction we see that

A_{z_{n}} = A_{z}[T_{1}, …, T_{r+s}]/(T_{1}^{p^{n}} − f_{1}, …, T_{r+s}^{p^{n}} − f_{r+s})

hence the image of the system (f_{1}, …, f_{r+s}) yields a basis of either Ω_{A_{z}/ℤ}^{1} ⊗_{A_{z}} κ(z) or Ω_{A_{z}}, depending on whether or not p ∈ 𝔪_{A_{z}}^{2} (corollary 9.6.34). In conclusion, we see that the induced sequence of morphisms of log schemes

⋯ → (X_{n+1}(z_{n}), \underline{M}_{n+1}(z_{n})) → (X_{n}(z_{n}), \underline{M}_{n}(z_{n})) → ⋯ → (X_{0}(z), \underline{M}_{0}(z))

(notation of (12.7.11)) is the maximal tower associated with the induced chart Q → 𝒪_{X_{0},z} and the maximal sequence (f_{1}, …, f_{r+s}).

Remark 17.2.9. (i) Let \overline{x}_{0} be any geometric point of X_{0}, localized at x_{0}, and denote by B_{0}^{sh} the strict henselization of B_{0} at the point \overline{x}_{0}. In light of [75, Th.3.5.13(ii)], we see that the log scheme (X_{0}(\overline{x}_{0}), \underline{M}_{0}(\overline{x}_{0})) still fulfills the conditions of (17.2.1). Moreover, the image in B_{0}^{sh} of the maximal sequence (f_{1}, …, f_{n}) is still maximal (remark 9.6.35(iv)). It follows easily that the induced tower of regular log schemes

⋯ → X_{0}(\overline{x}_{0}) ×_{X_{0}} (X_{n+1}, \underline{M}_{n+1}) → X_{0}(\overline{x}_{0}) ×_{X_{0}} (X_{n}, \underline{M}_{n}) → ⋯ → (X_{0}(\overline{x}_{0}), \underline{M}_{0}(\overline{x}_{0}))

is (naturally isomorphic to) the maximal tower associated with the induced chart P_{0} → B_{0}^{sh} and the maximal sequence (f_{1}, …, f_{n}) of B_{0}^{sh}.

(ii) In the same vein, let I ⊂ B_{0} be any proper ideal, B_{0,I}^{∧} the I-adic completion of the local ring B_{0}, and set X_{0}^{∧} := Spec B_{0,I}^{∧}; according to claim 9.7.21(i), the log scheme (X_{0}^{∧}, \underline{M}_{0}^{∧}) := X_{0}^{∧} ×_{X_{0}} (X_{0}, \underline{M}_{0}) still fulfills the conditions of (17.2.1), and the image in B_{0,I}^{∧} of the maximal sequence (f_{1}, …, f_{n}) of B_{0} is still maximal (remark 9.6.35(iv)). It follows that the induced tower of log schemes

⋯ → X_{0}^{∧} ×_{X_{0}} (X_{n+1}, \underline{M}_{n+1}) → X_{0}^{∧} ×_{X_{0}} (X_{n}, \underline{M}_{n}) → ⋯ → (X_{0}^{∧}, \underline{M}_{0}^{∧})

is (naturally isomorphic to) the maximal tower associated with the induced chart P_{0} → B_{0,I}^{∧} and the maximal sequence (f_{1}, …, f_{n}) of B_{0,I}^{∧}.

17.2.10. Let Φ_{B_{n}} : B_{n}/pB_{n} → B_{n}/pB_{n} be the Frobenius endomorphism of B_{n}/pB_{n}; taking into account lemma 17.2.4, we see that Φ_{B_{n+1}} factors through a unique ring homomorphism

Ψ_{B_{n+1}} : B_{n+1}/pB_{n+1} → B_{n}/pB_{n} for every n ∈ ℕ.

Lemma 17.2.11. The map Ψ_{B_{n+1}} is surjective for every n ∈ ℕ.

Proof. Let us start out with the following general :

Claim 17.2.12. Let φ : R → S be an injective, finite and radicial ring homomorphism. Then φ is an isomorphism if and only if Ω_{S/R}^{1} = 0.

Proof of the claim. We may assume that Ω_{S/R}^{1} = 0, and we show that φ is an isomorphism. To this aim, it suffices to show that

φ ⊗_{R} R/𝔪 : R′ := R/𝔪 → S′ := S/𝔪S

is an isomorphism for every maximal ideal 𝔪 ⊂ R. However, Ω_{S′/R′}^{1} = Ω_{S/R}^{1} ⊗_{R} R′ = 0, so we may replace R by R′ and S by S′, and assume from start that R is a field. In this case, S is a local unramified R-algebra, so S must be a finite separable field extension of R, by [66, Ch.IV, Cor.17.4.2]. But S is also a radicial extension of R, so S = R. ♢

Claim 17.2.13. Let p > 0 be a prime integer, R a local 𝔽_{p}-algebra whose Frobenius endomorphism Φ_{R} is a finite map, k_{R} the residue field of R, and Σ ⊂ R any subset. Set R_{0} := R/ΣR, and let (g_{1}, …, g_{n}) be a finite sequence of elements of R such that dg_{1}, …, dg_{n} is a system of generators for the k_{R}-vector space Ω_{R_{0}/ℤ}^{1} ⊗_{R} k_{R}. Then R = R^{p}[Σ, g_{1}, …, g_{n}].

Proof of the claim. Set S := R^{p}[Σ, g_{1}, …, g_{n}]. The inclusion map S → R is clearly radicial, and it is finite, since Φ_{R} is finite; hence claim 17.2.12 reduces to checking that Ω_{R/S}^{1} vanishes. By Nakayama’s lemma, it then suffices to show that Ω_{R/S}^{1} ⊗_{R} k_{R} = 0. However, let M (resp. N) be the R-submodule of Ω := Ω_{R/ℤ}^{1} generated by {da | a ∈ Σ} (resp. by {dg_{1}, …, dg_{n}}); then Ω_{R/S}^{1} = Ω/(M + N) ([63, Ch.0, Th.20.5.7(i)]) and on the other hand, the induced map (Ω/M)⊗_{R} R_{0} → Ω_{R_{0}/ℤ}^{1} is an isomorphism ([63, Ch.0, Th.20.5.12(i)]). We deduce a right exact sequence of R-modules

N ⟶^{j} Ω_{R_{0}/ℤ}^{1} ⊗_{R} k_{R} → Ω_{R/S}^{1} ⊗_{R} k_{R} → 0.

But our choice of the sequence (g_{1}, …, g_{n}) implies that j is surjective, whence the contention.

♢

Finally, notice that the image of Ψ_{B_{n+1}} is the subring (B_{n}/pB_{n})^{p}[β_{n}(P^{(n)}), f_{1}^{1/p^{n}}, …, f_{r}^{1/p^{n}}]. For every n > 0, consider the exact sequence

Ω_{A/ℤ}^{1} ⊗_{A} κ(x_{n}) ⟶^{j} Ω_{A_{n}/ℤ}^{1} ⊗_{A_{n}} κ(x_{n}) → Ω_{A_{n}/A}^{1} ⊗_{A_{n}} κ(x_{n}) → 0

([63, Ch.0, Th.20.5.7(i)]); the image of j is generated by the image of the generating system df_{•} := (df_{1}, …, df_{r}) of the κ(x_{0})-vector space Ω_{A/ℤ}^{1} ⊗_{A} κ(x_{0}). However, since f_{i} admits a p-th root in A_{n} for every i = 1, …, r, it is easily seen that the image of df_{•} vanishes in Ω_{A_{n}/ℤ}^{1}. Hence Ω_{A_{n}/ℤ}^{1} ⊗_{A_{n}} κ(x_{n}) = Ω_{A_{n}/A}^{1} ⊗_{A_{n}} κ(x_{n}) is generated by the image of the system (df_{1}^{1/p^{n}}, …, df_{r}^{1/p^{n}}). Therefore, to prove the lemma, it suffices to apply claim 17.2.13 to R := B_{n}/pB_{n}, Σ := β_{n}(𝔪_{P^{(n)}}) and the sequence (f_{1}^{1/p^{n}}, …, f_{r}^{1/p^{n}}). □

Theorem 17.2.14. With the notation of (17.2.5), we have :

(i) If B_{0} is an 𝔽_{p}-algebra, Ψ_{B_{n}} is an isomorphism for every n ∈ ℕ.

(ii) If B_{0} is not an 𝔽_{p}-algebra, then there exist π ∈ B_{1} and u ∈ B_{1}^{×} such that π^{p} = pu. (iii) For every π ∈ B_{1} as in (ii) and every integer n > 0, we have Ker Ψ_{B_{n}} = πB_{n}/pB_{n}. (iv) Let 𝒯_{p,B} be the p-adic topology of B. Then (B, 𝒯_{p,B}) is a formally perfectoid ring.

Proof. (i) is immediate from lemma 17.2.11, since in that case Φ_{B_{n}} is the Frobenius endomorphism of B_{n}, so it is injective, since the latter is a domain.

(iii): Suppose we have found π as in (ii), and let x ∈ B_{n} whose image in B_{n}/pB_{n} lies in Ker Ψ_{B_{n}}; this means that x^{p} = py holds in B_{n} for some y ∈ B_{n}; hence (x/π)^{p} ∈ B_{n}, and since B_{n} is a normal ring (corollary 12.5.13), we deduce that x/π ∈ B_{n}, i.e. x ∈ πB_{n}.

(ii): Assume first that p ∈ 𝔪_{A}^{2}. Then we may write

p = ∑_{i=1}^{t} b_{i}b′_{i} + ∑_{j=1}^{s} b′′_{j} · β_{0}(x_{j})

for certain b_{1}, b′_{1}, …, b_{t}, b′_{t} ∈ 𝔪_{B_{0}}, b′′_{1}, …, b′′_{s} ∈ B_{0}, and x_{1}, …, x_{s} ∈ 𝔪_{P}. In light of lemma 17.2.11, we may then write

b_{i} = c_{i}^{p} + pd_{i} b′_{i} = c′_{i}^{p} + pd′_{i} where c_{i}, c′_{i}, d_{i}, d′_{i} ∈ 𝔪_{B_{1}} for every i = 1, …, t

and likewise, b′′_{j} = c′′_{j}^{p} + pd′′_{j} where c′′_{j} ∈ B_{1} for j = 1, …, s. Moreover, by construction, β_{0} extends to a morphism of monoids β_{1} : P^{(1)} → B_{1}, and we may write x_{j} = y_{j}^{p} with y_{j} ∈ 𝔪_{P^{(1)}} for j = 1, …, s. A simple computation then yields

(17.2.15)  p · (1 + e) = ∑_{i=1}^{t} c_{i}^{p}c′_{i}^{p} + ∑_{j=1}^{s} c′′_{j}^{p} · β_{1}(y_{j})^{p}  for some e ∈ 𝔪_{B_{1}}.

However, the right-hand side of (17.2.15) can be written in the form g^{p}+ph for some g, h ∈ 𝔪_{B_{1}} (details left to the reader); clearly 1 + e − h ∈ B_{1}^{×}, whence the contention, in this case.

Next, suppose that p ∉ 𝔪_{A}^{2}. In this case, recall that

dim_{κ(x_{0})^{1/p}} Ω_{A} = 1 + dim_{κ(x_{0})} Ω_{A/ℤ}^{1} ⊗_{A} κ(x_{0}).

Therefore, after reordering the sequence (f_{1}, …, f_{r}), we may assume that df_{1}, …, df_{r−1} is a basis of the κ(x_{0})-vector space Ω_{A/ℤ}^{1} ⊗_{A} κ(x_{0}). Set

B′ := (P^{(1)} ⊗_{P} B_{0})[T_{1}, …, T_{r−1}]/(T_{1}^{p} − f_{1}, …, T_{r−1}^{p} − f_{r−1}).

Clearly B_{1} is a faithfully flat B′-algebra, hence B′/pB′ is a B_{0}/pB_{0}-subalgebra of B_{1}/pB_{1}, so the natural map B_{0}/pB_{0} → B′/pB′ is injective (lemma 17.2.4), and just as in (17.2.5), we deduce that the Frobenius endomorphism of B′/pB′ factors through a ring homomorphism Ψ_{B′} : B′/pB′ → B_{0}/pB_{0}, and arguing as in the foregoing, we also see that B′ is a local ring. Moreover, by applying claim 17.2.13 with R := B′/pB′, Σ := β_{0}(𝔪_{P}) and the sequence (f_{1}, …, f_{r−1}) we conclude – as in the proof of lemma 17.2.11 – that Ψ_{B′} is surjective. Hence, denote by 𝔪_{B′} the maximal ideal of B′; it follows that there exist g ∈ 𝔪_{B′} and h ∈ B′ such that f_{r} = g^{p} + ph in B′. Set

A′ := B′/𝔪_{P^{(1)}}B′ = A[T_{1}, …, T_{r−1}]/(T_{1}^{p} − f_{1}, …, T_{r−1}^{p} − f_{r−1}).

Then A′ is a regular local ring, by corollary 9.6.34, applied to the sequence (\overline{f}_{1}, …, \overline{f}_{r−1}) consisting of the images of the elements f_{i} in A. By the same criterion – applied to the similar sequence (\overline{f}_{1}, …, \overline{f}_{r}) – we see that A′[T]/(T^{p} − f_{r}) is regular as well. So once again the same corollary – applied to the element \overline{f}_{r} of A′ – says that the element d(f_{r}) of Ω_{A′} does not vanish. However

d(f_{r}) = d(g^{p}) + d(ph) = pg^{p−1}d(g) + hd(p) + pd(h) = hd(p) in Ω_{A′}

(see (9.6.16)). We conclude that h ∈ B′^{×}. Lastly, notice that f_{r} admits a p-th root f_{r}^{1/p} in B_{1}, hence ph = f_{r} − g^{p} = (f_{r}^{1/p} − g)^{p} + pe in B_{1} for some e ∈ 𝔪_{B_{1}} (details left to the reader). Since h − e ∈ B_{1}^{×}, we are done.

(iv): If B_{0} is an 𝔽_{p}-algebra, then 𝒯_{p,B} is the discrete topology, and B is a perfect 𝔽_{p}-algebra, whence the assertion in this case. Suppose then that B_{0} is not an 𝔽_{p}-algebra, in which case the topology of the completion B^{∧} of (B, 𝒯_{p,B}) is still p-adic (remark 8.3.3(iv)). Pick π ∈ B_{1} as in (ii), set I := πB^{∧}, and notice that π is a regular element of B^{∧}, by proposition 8.2.13(i) : the details shall be left to the reader. In light of (iii), it is easily seen that B^{∧} is a P-ring, I is a special ideal of definition, and the Frobenius endomorphism of B^{∧} : pB^{∧} = B/pB induces an isomorphism B^{∧}/I →^{∼} B^{∧}/I^{(p)}. The assertion then follows from theorem 16.4.1. □

17.2.16. Let β : P^{(∞)} → B be the chart of the log structure \underline{M} on X from (17.2.5), and notice that the inclusion map P^{(n)} → P^{(∞)} induces bijections Spec P^{(∞)} →^{∼} Spec P^{(n)} for every n ∈ ℕ (lemma 6.4.59(i)); especially, Spec P^{(∞)} is a finite set. Let I ⊂ B be any non-zero ideal; we say that I is a branch ideal, if there exist radical ideals J ⊂ B and 𝔯 ⊂ P^{(∞)} such that

(17.2.17)  p ∈ J  and I = J ∩ 𝔯B.

Remark 17.2.18. (i) Notice that, in case B_{0} is an 𝔽_{p}-algebra, a branch ideal is just any non-zero radical ideal of B.

(ii) Let I ⊂ B be a branch ideal, and pick radical ideals J ⊂ B and 𝔯 ⊂ P^{(∞)} such that (17.2.17) holds. Set 𝔯^{(n)} := 𝔯 ∩ P^{(n)} for every n ∈ ℕ; from corollary 12.5.20(ii) we know that 𝔯^{(n)}B_{n} is a radical ideal of B_{n}, hence 𝔯B is a radical ideal of B, and then the same holds for I.

(iii) In the situation of (ii), let ℨ ⊂ Spec P^{(∞)} be a subset such that 𝔯 = ⋂_{𝔭∈ℨ} 𝔭 (lemma 6.1.16), and set 𝔭^{(n)} := 𝔭 ∩ P^{(n)} for every 𝔭 ∈ ℨ and every n ∈ ℕ, so 𝔯^{(n)} = ⋂_{𝔭∈ℨ} 𝔭^{(n)} for every n ∈ ℕ. In view of proposition 12.5.16 and lemmata 12.5.1 and 6.1.37, we see that 𝔯^{(n)}B = ⋂_{𝔭∈ℨ} 𝔭^{(n)}B_{n}, and therefore

(17.2.19)  𝔯B = ⋂_{𝔭∈ℨ} 𝔭B.

(iv) Furthermore, any branch ideal I ⊂ B is the radical of I_{0}B, for some ideal I_{0} ⊂ B_{0}. Indeed, pick J and 𝔯 such that (17.2.17) holds; since the projection Spec B/pB → (X_{0})_{/p} is radicial and surjective, it is clear that J is the radical of J_{0}B for some ideal J_{0} ⊂ B_{0}, and on the other hand, 𝔯B is the radical of 𝔯^{(0)}B.

Proposition 17.2.20. With the notation of (17.2.16), let I ⊂ B be any branch ideal. We have : (i) I^{2} = I, and I fulfills condition (B) of [75, §2.1.6]. (ii) Let I_{1}, I_{2} ⊂ B be any two branch ideals. Then I_{1} ∩ I_{2} and I_{1} + I_{2} are also branch

ideals, and I_{1}I_{2} = I_{1} ∩ I_{2}.

Proof. (i): By assumption, I = J ∩ 𝔯B for radical ideals J ⊂ B and 𝔯 ⊂ P^{(∞)} with p ∈ J.

Claim 17.2.21. (𝔯B)^{2} = 𝔯B and J^{2} = J, and the ideals 𝔯B and J fulfill condition (B).

Proof of the claim. Since the p-Frobenius endomorphism of P^{(∞)} is an automorphism, the first stated identity is clear. Next, set \overline{J} := J/pB. By lemma 17.2.11, the Frobenius endomorphism of B/pB is surjective; since \overline{J} is a radical ideal, we deduce that \overline{J}^{2} = \overline{J}. On the other hand, from theorem 17.2.14(ii) we see that p ∈ J^{2}, from which also the second identity follows easily.

Next, it is clear that 𝔯B fulfills condition (B), and for J we apply [75, Claim 2.1.9] which reduces checking that J/pJ is generated by the p-th powers of its elements. However, the foregoing already shows that J/pB is generated by the p-th powers of its elements; combining with theorem 17.2.14(ii), the contention follows easily. ♢

Taking into account remark 17.2.18(i), we see that claim 17.2.21 already implies the assertion, in case B_{0} is an 𝔽_{p}-algebra. Thus, we may assume that B_{0} is not an 𝔽_{p}-algebra, in which case – again by claim 17.2.21– it suffices to show that J ∩ 𝔯B = 𝔯 · J, or equivalently, that :

Tor_{1}^{B}(B/𝔯B, B/J) = 0.

Now, the chart β yields a ring homomorphism C := ℤ[P^{(∞)}] → B, and we have a change of rings spectral sequence for Tor-functors (see [163, Th.5.6.6]) :

E_{pq}^{2} := Tor_{p}^{B}(Tor_{q}^{C}(C/𝔯C, B), B/J) ⇒ Tor_{p+q}^{C}(C/𝔯C, B/J).

Claim 17.2.22. Tor_{q}^{C}(C/𝔯C, B) = 0 for every q > 0.

Proof of the claim. Set 𝔯^{(n)} := 𝔯 ∩ P^{(n)} and C_{n} := ℤ[P^{(n)}] for every n ∈ ℕ. We have natural isomorphisms :

Tor_{q}^{C}(C/𝔯C, B) →^{∼} colim_{n∈ℕ} Tor_{q}^{C_{n}}(C_{n}/𝔯_{n}C, B_{n}) for every q, n ∈ ℕ

so we are reduced to showing that Tor_{q}^{C_{n}}(C_{n}/𝔯_{n}C, B_{n}) = 0 for every q, n ∈ ℕ with q > 0. The latter follows from propositions 12.5.16(ii) and 6.1.40(i). ♢

From claim 17.2.22 we see that E_{pq}^{2} = 0 for every q > 0 and every p ∈ ℕ. Thus, we deduce a natural isomorphism :

Tor_{p}^{B}(B/𝔯B, B/J) →^{∼} Tor_{p}^{C}(C/𝔯C, B/J) for every p ∈ ℕ.

By the same token, the projection C → \overline{C} := C/pC induces a change of rings spectral sequence

F_{ij}^{2} := Tor\overline{C} ; i(Tor_{j}^{C}(C/𝔯C, \overline{C}), B/J) ⇒ Tor_{i+j}^{C}(C/𝔯C, B/J)

(notice that B/J is a \overline{C}-algebra, since p ∈ J) and Tor_{j}^{C}(C/𝔯C, \overline{C}) = 0 for every j > 0, since C/𝔯C has no p-torsion. Hence F_{ij}^{2} = 0 whenever j > 0, and we get an isomorphism :

Tor\overline{C} ; i(\overline{C}/𝔯\overline{C}, B/J) →^{∼} Tor_{i}^{C}(C/𝔯C, B/J) for every i ∈ ℕ.

Notice now that \overline{C} is a perfect 𝔽_{p}-algebra, since P^{(∞)} is p-divisbile, and with I\overline{C} := 𝔯\overline{C} we have \overline{C}/𝔯\overline{C} = \overline{C}/I⌈0⌉ ; \overline{C}\overline{C}. Then Tor\overline{C} ; i(\overline{C}/𝔯\overline{C}, B/J) = 0 for every i > 0, by proposition 16.4.7, whence the contention.

(ii): It is clear that I_{1} ∩ I_{2} is a branch ideal; we have inclusions

(I_{1} ∩ I_{2})^{2} ⊂ I_{1}I_{2} ⊂ I_{1} ∩ I_{2}

and from (i) we know that the first ideal in this chain coincides with the third, so I_{1}I_{2} = I_{1} ∩ I_{2}.

Next, for k = 1, 2 let us write I_{k} = J_{k} ∩r_{k}B, where J_{k} ⊂ B and 𝔯_{k} ⊂ P^{(∞)} are radical ideals and p ∈ J_{k}. Set Z_{k} := Spec B/J_{k}, Z′_{k} := Spec B/𝔯_{k}B for k = 1, 2; clearly

Z := Spec B/(I_{1} + I_{2}) = (Z_{1} ∪ Z′_{1}) ∩ (Z_{2} ∪ Z′_{2}).

Now let J ⊂ B be the largest ideal such that Spec B/J = (Z_{1} ∩ Z_{2}) ∪ (Z_{1} ∩ Z′_{2}) ∪ (Z′_{1} ∩ Z_{2}). It follows easily that

Z = (Spec B/J) ∪ (Spec B/(𝔯_{1} ∪ 𝔯_{2})B).

Now, J and 𝔯_{1} ∪ 𝔯_{2} are radical ideals, and p ∈ J, so we deduce that the radical of I_{1} + I_{2} is a branch ideal, and it remains only to show that I_{1} + I_{2} is a radical ideal. However, notice the inclusions

(𝔯_{1}J_{1} + 𝔯_{2}J_{2})^{2} ⊂ I := (J_{1} + J_{2})(J_{1} + 𝔯_{2}B)(𝔯_{1}B + J_{2})(𝔯_{1}𝔯_{2}B) ⊂ (𝔯_{1}J_{1} + 𝔯_{2}J_{2}).

Especially, 𝔯_{1}J_{1} + 𝔯_{2}J_{2} is contained in the radical I′ of I, Since we already know that 𝔯_{k}J_{k} = I_{k} for k = 1, 2, it suffices to check that each of the four factors of the middle term in this chain of inclusions is a branch ideal : indeed, in this case the same holds for I, and therefore I = I′, so finally I = I_{1} + I_{2}.

• First, set J := J_{1} + J_{2}, and let \overline{J} ⊂ B/pB be the image of J; since p ∈ J, in order to see that J is a branch ideal, it suffices to check that J is a radical ideal; then it suffices to prove that the same holds for \overline{J}, and the latter assertion will follow, if we show that Φ_{B}^{−1}\overline{J} = \overline{J}, where Φ_{B} : B/pB → B/pB is the Frobenius endomorphism. Now, say that x ∈ B/pB and x^{p} ∈ \overline{J}; then x^{p} = x_{1} + x_{2} for some x_{i} ∈ \overline{J}_{i} := J_{i}/pB (i = 1, 2), and since Φ_{B} is surjective and \overline{J}_{i} is a radical ideal, we can write x_{i} = y_{i}^{p} for some y_{i} ∈ \overline{J}_{i}. Consequently x^{p} = (y_{1} + y_{2})^{p}, so x − y_{1} − y_{2} is nilpotent; but clearly \overline{J} contains the nilradical of B/pB, whence the claim.

• Next we consider J′ := J_{1} + 𝔯_{2}B; as in the previous case, since p ∈ J, we see that J′ is a branch ideal, provided J′/pB is a radical ideal of B/pB. However, say that x ∈ B/pB and x^{p} = y + z, for some y ∈ \overline{J}_{1} and z ∈ 𝔯_{2}B/pB; arguing as in the foregoing case, we may write y = u^{p} for some u ∈ \overline{J}_{1}. Likewise, since both Φ_{B} and the Frobenius endomorphism of P^{(∞)} are surjective, we see that z = w^{p} for some w ∈ 𝔯_{2}B/pB, so x − u − w is nilpotent, and especially, it lies in \overline{J}_{1}, whence the claim. The same argument applies as well to the factor 𝔯_{1}B + J_{2}.

• Lastly, since 𝔯_{1} and 𝔯_{2} are both radical ideals of P^{(∞)}, the same holds for 𝔯_{1}𝔯_{2}, so 𝔯_{1}𝔯_{2}B is a branch ideal. □

Proposition 17.2.20(i) says that the pair (B, I) is a basic setup, in the sense of [75, (2.1.1)], so we may consider the associated categories of almost modules and almost algebras.

17.2.23. Henceforth we shall restrict to the case where B_{0} is not an 𝔽_{p}-algebra, so B_{0} is a local integral domain whose field of fractions has characteristic zero, and whose residue field has characteristic p. For every n ∈ ℕ, set

(X_{n})_{/p} := Spec B_{n}/pB_{n}  V_{n} := X_{n} \ (X_{n})_{/p}  V_{n,tr} := (X_{n}, \underline{M})_{tr} ∩ V_{n}.

We consider now a finite morphism φ_{0} : Y_{0} → X_{0} with Y_{0} a normal scheme, such that φ_{0} maps each connected component of Y_{0} onto X_{0}, and such that the restriction φ_{0}^{−1}V_{0,tr} → V_{0,tr} is a finite étale covering. For every n ∈ ℕ, we let Y_{n} be the normalization of X_{n} in φ^{−1}V_{0,tr} ×_{X_{0}} X_{n}, and denote by φ_{n} : Y_{n} → X_{n} the resulting finite morphism ([126, Lemma 1, p.262]), by

φ : Y → X

the limit of the system (φ_{n} | n ∈ ℕ), and by W_{n} ⊂ X_{n} the étale locus of φ_{n}, i.e. the largest open subset such that the restriction φ_{n}^{−1}W_{n} → W_{n} of φ_{n} is an étale covering (lemma 13.1.7(iii) and claim 13.1.8). Let also (X_{n,𝔭} | 𝔭 ∈ Spec P^{(n)}) be the logarithmic stratification of (X_{n}, \underline{M}_{n}) (see (12.5.35)) and set V_{n,𝔭} := X_{n,𝔭} ∩ V_{n} for every n ∈ ℕ and every 𝔭 ∈ Spec P^{(n)}. We call (V_{n,𝔭} | 𝔭 ∈ Spec P^{(n)}) the logarithmic stratification of V_{n}.

Lemma 17.2.24. With the notation of (17.2.23), the following holds :

(i) V_{n} \ W_{n} is a union of strata of the logarithmic stratification of V_{n}, for every n ∈ ℕ. (ii) X_{n} ×_{X_{m}} W_{m} = W_{n} for every sufficiently large m ∈ ℕ and every n ≥ m. (iii) For m as in (ii), set W := X ×_{X_{m}} W_{m}, and let I ⊂ B be the largest ideal such that

Spec B/I = X \W. Then I is a branch ideal of B.

Proof. (i): Since every X_{n,𝔭} is irreducible (corollary 12.5.37(iii)), the same holds for each V_{n,𝔭}. Hence, suppose that V_{n}\W_{n} contains a point z of V_{n,𝔭}, for some 𝔭 ∈ Spec P^{(n)}; we have to show that in this case V_{n,𝔭} ∩ W_{n} = ∅, and since W_{n} is an open subset of X_{n}, it suffices to check that the generic point η of V_{n,𝔭} does not lie in W_{n}. However, let \overline{η} be any geometric point localized at η; then η lies in W_{n} if and only if the induced morphism φ_{n} ×_{X_{n}} X_{n}(\overline{η}) is étale (claim 13.1.9). Let also \overline{z} be a geometric point localized at z, and denote by \underline{M}_{n,\overline{z}} (resp. \underline{M}_{n,\overline{η}}) the stalk at \overline{z} (resp. \overline{η}) of the logarithmic structure of (X_{n}(\overline{z}), \underline{M}_{n}(\overline{z})) (resp. of (X_{n}(\overline{η}), \underline{M}_{n}(\overline{η}))). Any choice of a strict specialization morphism s : X_{n}(\overline{η}) → X_{n}(\overline{z}) induces a strict specialization map \overline{σ} : \underline{M}_{n,\overline{z}} → \underline{M}_{n,\overline{η}} (see (4.9.23)) that extends the specialization map σ : \underline{M}_{n,z} → \underline{M}_{n,η}. Since the natural maps \underline{M}_{n,z}^{♯} → \underline{M}_{n,\overline{z}}^{♯} and \underline{M}_{n,η}^{♯} → \underline{M}_{n,\overline{η}}^{♯} are isomorphisms (see (12.1.9)), and σ^{♯gp} is an isomorphism (since z and η lie in the same stratum), the same holds for \overline{σ}^{♯gp}, and therefore the induced map \underline{M}_{n,\overline{η}}^{gp∨} → \underline{M}_{n,\overline{z}}^{gp∨} is bijective. Pick any geometric point ξ of X_{n}(\overline{z}) localized at the maximal point, and lift ξ to a geometric point ξ_{η} of X_{n}(\overline{η}); in light of (13.3.54), we conclude that s induces an isomorphism

π_{1}((X_{n}(\overline{η}), \underline{M}_{n}(\overline{η}))_{ét}, ξ_{η}) →^{∼} π_{1}((X_{n}(\overline{z}), \underline{M}_{n}(\overline{z}))_{ét}, ξ).

Therefore, φ_{n} ×_{X_{n}} X_{n}(\overline{η}) is étale if and only if it is a trivial covering, if and only if the same holds for φ_{n} ×_{X_{n}} X_{n}(\overline{z}), if and only if z ∈ W_{n}, whence the assertion.

(ii): For every n ∈ ℕ, set

(17.2.25)  ℨ_{n} := {𝔭 ∈ Spec P^{(n)} | V_{n,𝔭} ≠ ∅ and V_{n,𝔭} ∩ W_{n} = ∅}.

The continuous map ω_{n} : Spec P^{(n+1)} → Spec P^{(n)} induced by the inclusion P^{(n)} → P^{(n+1)} sends ℨ_{n+1} into ℨ_{n}, for every n ∈ ℕ. On the other hand, ω_{n} is also a bijection of finite sets (lemmata 6.1.20(iii) and 6.4.59(i)); we conclude that ω_{n} restricts to a bijection ℨ_{n+1} →^{∼} ℨ_{n} for every sufficiently large integer n ∈ ℕ. For every n, m ∈ ℕ with n ≥ m, let g_{n,m} : X_{n} → X_{m} be the transition morphism in the maximal tower (17.2.6), in light of (i), it follows that

g_{n,m}^{−1}(V_{m} ∩ W_{m}) = V_{n} ∩ W_{n} for every sufficiently large m ∈ ℕ and every n ≥ m.

On the other hand, the restriction (X_{n})_{/p} → (X_{m})_{/p} of g_{n,m} is radicial and surjective for every such n, m ∈ ℕ, so the underlying continuous map is a homeomorphism; since the underlying topological spaces are noetherian, we see as well that

g_{n,m}^{−1}((X_{m})_{/p} ∩ W_{m}) = (X_{n})_{/p} ∩ W_{n} for every sufficiently large m ∈ ℕ and every n ≥ m.

Summing up, the assertion follows.

(iii): The assertion follows easily from (i) and (17.2.19). □

17.2.26. Let I ⊂ B be a fixed branch ideal, and consider a pair (X, Z) with Z ⊂ Spec B/I and such that Z is constructible in X. Clearly such a pair is normal (see definition 14.4.1(ii)), and we aim to show that (X, Z) is almost pure for the almost structure given by the basic setup (B, I) supplied by proposition 17.2.20. This will be achieved in several steps.

To begin with, set U := X \ Z, and let 𝒜 be any étale almost finitely presented 𝒪_{U}^{a}-algebra. Set also U_{I} := X \ Spec B/I; then 𝒜_{|U_{I}} is a finite étale 𝒪_{U_{I}}-algebra, U_{I} = X ×_{X_{0}} U_{I,0} for some open subset U_{I,0} ⊂ X_{0} (remark 17.2.18(iv)), and a simple inspection shows that

V_{0,tr} ⊂ U_{I,0}.

Let ψ : U_{I} → U_{I,0} be the natural projection; by [66, Ch.IV, Prop.17.7.8(ii)] and [65, Ch.IV, Th.8.8.2(ii)], and by virtue of remark 17.2.7, we may assume – after replacing (X_{0}, \underline{M}_{0}) by (X_{n}, \underline{M}_{n}) for some sufficiently large n ∈ ℕ – that there exists a coherent étale 𝒪_{U_{I,0}}-algebra 𝒜_{0} with an isomorphism ψ^{∗}𝒜_{0} →^{∼} 𝒜_{|U_{I}} of 𝒪_{U_{I}}-algebras. Denote by Y_{0} the normalization of X_{0} in Spec 𝒜_{0}(U_{I,0}); the resulting morphism φ_{0} : Y_{0} → X_{0} is finite (lemma 9.7.4(i)) and induces an isomorphism (φ_{0∗}𝒪_{Y_{0}})_{|U_{I,0}} →^{∼} 𝒜_{0}. Especially, φ_{0} is a morphism of the type contemplated in (17.2.23), and the resulting morphism φ : Y → X induces an isomorphism (φ_{∗}𝒪_{Y})_{|U_{I}} →^{∼} 𝒜_{|U_{I}}. However, we have as well 𝒜 = (𝒜_{|U_{I}})^{ν} (lemma 14.4.5(i)), and since Y is normal, there follows an isomorphism of 𝒪_{U}^{a}-algebras (φ_{∗}𝒪_{Y}^{a})_{|U} →^{∼} 𝒜. Then proposition 14.4.8 and [75, Lemma 8.2.28] show that (X, Z) is almost pure if and only if the resulting 𝒪_{X}^{a}-algebra φ_{∗}𝒪_{Y}^{a} is weakly unramified, for every such φ_{0}.

17.2.27. Keep the notation of (17.2.26), and suppose now that B_{0} is strictly henselian; especially, the N-torsion subgroup μ_{N} of B_{0}^{×} has cardinality equal to N, for every N > 0 such that (N, p) = 1. Let k > 0 be any integer and write k = p^{s} · q, with s, q ∈ ℕ and (q, p) = 1; set Q := P, let ν : P → Q be the k-Frobenius map, define

C′_{0} := Q ⊗_{P} B_{0} C_{0} := C′_{0} ⊗_{B_{0}} B′′_{s}

and endow X′_{0} := Spec C_{0} with the log structure \underline{M}′_{0} deduced from the natural map Q → C_{0}. Notice that C_{0} is strictly henselian and the Frobenius endomorphism of C_{0}/pC_{0} is still a finite map (claim 9.7.21(i)); also, since the induced map A → C′_{0}/𝔪_{Q}C′_{0} is an isomorphism, the image of the sequence (f_{1}, …, f_{r}) in C_{0} is still maximal, in the sense of (17.2.3), and therefore C_{0}/𝔪_{Q}C_{0} is still a regular local ring, by corollary 9.6.34. Arguing as in (17.2.5), we deduce that (X′_{0}, \underline{M}′_{0}) is a regular log scheme, and we may consider the maximal tower

((X′_{n}, \underline{M}′_{n}) | n ∈ ℕ)

associated with the chart Q → C_{0} and the maximal sequence (f_{1}^{1/p^{s}}, …, f_{r}^{1/p^{s}}) (see remark 9.6.35(iii)). So, X′_{n} = Spec C_{n} for a finite C_{0}-algebra C_{n}, and \underline{M}′_{n} is given by a chart Q^{(n)} → C_{n}, where Q^{(n)} is a submonoid of Q_{ℚ} containing Q, for every n ∈ ℕ. For every n ∈ ℕ, we set V′_{n,tr} := V_{0,tr} ×_{X_{0}} X′_{n}, we let Y′_{n} be the normalization of X′_{n} in Y_{0} ×_{X_{0}} V′_{n,tr}, we denote by

φ′_{n} : Y′_{n} → X′_{n} and h_{n} : (X′_{n}, \underline{M}′_{n}) → (X_{n}, \underline{M}_{n})

the resulting finite morphisms, and by φ′ : Y′ → X′ (resp. h : (X′, \underline{M}′) → (X, \underline{M})) the limit of the system of morphisms (φ′_{n} | n ∈ ℕ) (resp. the limit of the system (h_{n} | n ∈ ℕ)). Recall also that the induced morphism V′_{0,tr} → V_{0,tr} is a torsor for the finite abelian group

G := Hom_{ℤ}(Q^{gp}, μ_{k})

(see (13.3.32)). Set

C := ⋃_{n∈ℕ} C_{n} Q^{(∞)} := ⋃_{n∈ℕ} Q^{(n)}.

With this notation, we have

Q^{(∞)} = Q ⊗_{P} P^{(∞)} and C = Q^{(∞)} ⊗_{P} B′′.

Let also write Y = Spec D (resp. Y′ = Spec D′) for a B-algebra D (resp. for a C-algebra D′). The morphism h : (X′, \underline{M}′) → (X, \underline{M}) shall be called the standard covering of (X, \underline{M}) of degree k.

Lemma 17.2.28. In the situation of (17.2.27), set U′ := X′ \ h^{−1}Z. The induced diagram

\xymatrix{𝒪_{X}^{a}-Ét_{fr} \ar[r]^{ρ} \ar[d] & 𝒪_{U}^{a}-Ét_{fr} \ar[d] \\ 𝒪_{X′}^{a}-Ét_{fr} \ar[r]^{ρ′} & 𝒪_{U′}^{a}-Ét_{fr}}

is 2-cartesian (notation of definition 14.4.2).

Proof. A simple inspection shows that h_{0} factors through a morphism (X′_{0}, \underline{M}′_{0}) → (X_{s}, \underline{M}_{s}), and indeed we have natural isomorphisms of (X′, \underline{M}′)-schemes

(X′_{n}, \underline{M}′_{n}) →^{∼} (X′_{0}, \underline{M}′_{0}) ×_{(X_{s},\underline{M}_{s})} (X_{n+s}, \underline{M}_{n+s}) for every n ∈ ℕ

that identify the transition morphisms (X′_{n+1}, \underline{M}′_{n+1}) → (X′_{n}, \underline{M}′_{n}) with the base change of the corresponding morphisms for the tower ((X_{n+s}, \underline{M}_{n+s}) | n ∈ ℕ). In view of remark 17.2.7, we may then replace (X_{0}, \underline{M}_{0}) by (X_{s}, \underline{M}_{s}), and P by P^{(s)}, and assume that (k, p) = 1, so also (o(G), p) = 1.

Suppose now that (φ′_{∗}𝒪_{Y′}^{a})_{|U′} is in the essential image of the restriction functor ρ′; by proposition 14.4.8, this means that D^{′a} is an étale almost finitely presented (C, IC)^{a}-algebra. Then, taking into account the discussion of (17.2.26), and recalling that ρ and ρ′ are fully faithful (lemma 14.4.5(ii)), we are reduced to checking that D^{a} is an étale almost finitely presented (B, I)^{a}-algebra.

However, the action of G on V′_{0,tr} is inherited by C and D′, and clearly C^{G} = B and D^{′G} = D; by corollary 14.6.28(ii), it suffices therefore to show that the action of G on D^{′a} is horizontal. Denote

G → Aut(C) : χ ↦ ρ_{χ}

the action of G, let χ ∈ G be any element, and J_{χ} ⊂ C the ideal generated by all elements of the form ρ_{χ}(c) − c, for c ranging over all elements of C. By the discussion in (13.3.32) we get

ρ_{χ}((q ⊗ y) ⊗ b) = χ(q) · (q ⊗ y) ⊗ b for every q ∈ Q, y ∈ P^{(∞)} and b ∈ B′′.

Hence, J_{χ} is the ideal generated by all elements of the form (1 − χ(q)) · q ⊗ y ⊗ 1, for all q ⊗ y ∈ Q^{(∞)}. However, since χ(q) ∈ μ_{k} and (k, p) = 1, it is easily seen that 1 − χ(q) either vanishes, or else it is invertible in B_{0}. Thus, denote by 𝔮_{χ} ⊂ Q^{(∞)} the ideal generated by all elements of the form q ⊗ y, with χ(q) ≠ 1; it follows that

J_{χ} = 𝔮_{χ} · C.

Claim 17.2.29. 𝔮_{χ} is a radical ideal.

Proof of the claim. Indeed, say that (q ⊗ y)^{n} ∈ 𝔮_{χ}, so there exist elements q_{1} ⊗ y_{1} and x_{1} of Q^{(∞)} such that (q ⊗ y)^{n} = (q_{1} ⊗ y_{1}) · x_{1} and χ(q_{1}) ≠ 1. We may assume that n = p^{t} for some integer t ∈ ℕ, and since Q^{(∞)} is uniquely p-divisible, we may write q_{1} ⊗ y_{1} = (q_{2} ⊗ y_{2})^{n} and x_{1} = x_{2}^{n} for some elements q_{2} ⊗ y_{2} and x_{2} of Q^{(∞)}, and then q ⊗ y = (q_{2} ⊗ y_{2}) · x_{2}; however, clearly χ(q_{2}) ≠ 1, whence the claim. ♢

In view of claim 17.2.29, we may write 𝔮_{χ} = 𝔮_{1} ∩⋯∩𝔮_{s}, for certain 𝔮_{1}, …, 𝔮_{s} ∈ Spec Q^{(∞)} (lemma 6.1.16). Set 𝔔 := {𝔮_{1}, …, 𝔮_{s}}; we also know that (𝔮 ∩ Q^{(n)}) · C_{n} is a prime ideal of C_{n}, for every 𝔮 ∈ 𝔔 and every n ∈ ℕ (corollary 12.5.20(i)), therefore 𝔮C is a prime ideal of C, for every 𝔮 ∈ 𝔔. Especially, the induced map

(17.2.30)  C/J_{χ} → ∏_{𝔮∈𝔔} C/𝔮C

is injective. Since, by assumption, D^{′a} is a flat C^{a}-algebra, it follows that the map of almost modules D^{′a} ⊗_{C} (17.2.30) is a monomorphism. Summing up, we are reduced to checking that χ acts trivially on (D′/𝔮D′)^{a}, for every 𝔮 ∈ 𝔔. However, set

X′_{𝔮} := Spec C/𝔮C and U′_{𝔮} := X′_{𝔮} ×_{X} U_{I} for every 𝔮 ∈ 𝔔.

Suppose first that U′_{𝔮} = ∅; in that case, set 𝔭 := 𝔮 ∩ P^{(∞)}, and notice that 𝔮 is the radical of the ideal 𝔭·Q^{(∞)}, so 𝔮C is the radical of 𝔭C, and therefore Spec (C/𝔭C)×_{X} U_{I} = ∅. However, the induced morphism Spec C/𝔭C → Spec B/𝔭B is surjective, so Spec (B/𝔭B) ∩ U_{I} = ∅; since 𝔭B is a prime ideal of B, the latter means that I ⊂ 𝔭B, whence (B/𝔭B)^{a} = 0, so (D′/𝔮D′)^{a} vanishes as well, and the assertion is trivial. If U′_{𝔮} ≠ ∅, set Y′_{𝔮} := Y′ ×_{X′} X′_{𝔮}, let φ′_{𝔮} : Y′_{𝔮} → X′_{𝔮} be the induced morphism, and define

𝒟 := φ_{∗}𝒪_{Y}  𝒟′ := φ′_{∗}𝒪_{Y′}  𝒟′_{𝔮} := φ′_{𝔮∗}𝒪_{Y′_{𝔮}}.

On the one hand, by assumption 𝒟_{𝔮}^{′a} is a flat 𝒪_{X′_{𝔮}}^{a}-algebra; on the other hand, X′_{𝔮} is reduced and irreducible, hence the restriction map Γ(X′_{𝔮}, 𝒪_{X′_{𝔮}}) → Γ(U′_{𝔮}, 𝒪_{X′_{𝔮}}) is injective. Consequently, the restriction map

(D′/𝔮D′)^{a} = Γ(X′_{𝔮}, 𝒟_{𝔮}^{′a}) → Γ(U′_{𝔮}, 𝒟_{𝔮}^{′a})

is a monomorphism; so, we are reduced to checking that χ acts trivially on Γ(U′_{𝔮}, 𝒟_{𝔮}^{′a}). To this aim, we remark that the open subset U′_{I} := X′ ×_{X} U_{I} is stable under the action of G, and the restriction 𝒟′_{|U′_{I}} of 𝒟′ to the open subset U′_{I} is isomorphic to (h^{∗}𝒟)_{|U′_{I}} ([66, Ch.IV, Prop.17.5.8(iii)]), so the action of G on 𝒟′_{|U′_{I}} is horizontal, and the assertion follows easily. □

17.2.31. In the situation of (17.2.27), define ℨ_{n} as in (17.2.25), for every n ∈ ℕ. As already remarked, in view of lemma 17.2.24(i,ii) we may assume that the map Spec P^{(n+1)} → Spec P^{(n)} sends ℨ_{n+1} bijectively onto ℨ_{n}, for every n ∈ ℕ. Set V′_{0} := V_{0} ×_{X_{0}} X′_{0}, and let

(V′_{0,𝔭} | 𝔭 ∈ Spec Q) and (V_{0,𝔭} | 𝔭 ∈ Spec P)

be the logarithmic stratifications of V′_{0} and respectively V_{0}, defined as in (17.2.23); notice that the map ν^{∗} : Spec Q → Spec P induced by ν is bijective (lemma 6.4.59(i)), and clearly

V′_{0,𝔭} = h_{0}^{−1}V_{0,ν^{∗}(𝔭)} for every 𝔭 ∈ Spec Q.

Let ℨ′_{0} := ν^{∗−1}ℨ_{0}, and for every 𝔭 ∈ ℨ′_{0}, let η′_{𝔭} (resp. η_{𝔭}) denote the generic point of V′_{0,𝔭} (resp. of V_{0,ν^{∗}(𝔭)}), pick geometric points \overline{η}′_{𝔭} and ξ′_{𝔭} localized respectively at η′_{𝔭} and at a point of V′_{0,tr}(\overline{η}′_{𝔭}) := V′_{0,tr} ×_{X′_{0}} X′_{0}(\overline{η}′_{𝔭}). Denote by \overline{η}_{𝔭} the image of \overline{η}′_{𝔭} in V_{0,ν^{∗}(𝔭)}, and by ξ_{𝔭} the image of ξ′_{𝔭} in V_{0,tr}(\overline{η}_{𝔭}) := V_{0,tr} ×_{X_{0}} X_{0}(\overline{η}_{𝔭}). According to (13.3.31) there follows, for every integer N > 0, a commutative diagram of groups

\xymatrix{π_{1}(V′_{0,tr}(\overline{η}′_{𝔭})_{ét}, ξ′_{𝔭}) \ar[r] \ar[d] & π_{1}(V_{0,tr}(\overline{η}_{𝔭})_{ét}, ξ_{𝔭}) \ar[d] \\ \underline{M}_{0,η′_{𝔭}}^{′gp∨} ⊗_{ℤ} μ_{N} \ar[r] & \underline{M}_{0,η_{𝔭}}^{gp∨} ⊗_{ℤ} μ_{N}}

(where μ_{N} is the N-torsion subgroup of κ(ξ_{𝔭})^{×}) whose top arrow is induced by the natural morphism V′_{0,tr}(\overline{η}′_{𝔭}) → V_{0,tr}(\overline{η}_{𝔭}), and whose bottom arrow is induced by ν^{gp∨} : Q^{gp∨} → P^{gp∨}, i.e. by the k-Frobenius map of P^{gp∨}, for every 𝔭 ∈ ℨ′_{0}. Now, the restriction

φ_{𝔭} : Y_{0}(\overline{η}_{𝔭}) := Y_{0} ×_{X_{0}} X_{0}(\overline{η}_{𝔭}) → X_{0}(\overline{η}_{𝔭})

of φ is a tamely ramified covering, hence the action of π_{1}(V_{0,tr}(\overline{η}_{𝔭})_{ét}, ξ_{𝔭}) on F_{𝔭} := φ_{𝔭}^{−1}(ξ_{𝔭}) factors through a group homomorphism

ρ_{𝔭} : \underline{M}_{0,η_{𝔭}}^{gp∨} ⊗_{ℤ} μ_{N} → Aut(F_{𝔭})

for some sufficiently large N ∈ ℕ (theorem 13.3.45). We may then find k ∈ ℕ such that the image of k · P^{gp∨} in \underline{M}_{0,η_{𝔭}}^{gp∨} lies in the kernel of ρ_{𝔭}, for every 𝔭 ∈ ℨ′_{0}. Especially, for this choice of k, the image of \underline{M}_{0,η′_{𝔭}}^{′gp∨} ⊗_{ℤ} μ_{N} acts trivially on Aut(F_{𝔭}) via ρ_{𝔭}, for every such 𝔭. Consequently, π_{1}(V′_{0,tr}(\overline{η}′_{𝔭})_{ét}, ξ′_{𝔭}) acts trivially on the fibres φ_{0}^{′−1}(ξ′_{𝔭}) (by virtue of (3.6.22)); after applying lemma 17.2.24(i) to the morphism φ′_{0}, we conclude that the étale locus of φ′_{0} contains the whole of V′_{0}.

Theorem 17.2.32. In the situation of (17.2.26), the pair (X, Z) is almost pure, relative to the basic setup (B, I).

Proof. Fix a geometric point \overline{x} of X localized at the closed point x of X, and let B^{sh} denote the strict henselization of B at x. In view of proposition 14.4.10, in order to prove that (X, Z) is almost pure, it suffices to show that the pair (X(\overline{x}), Z(\overline{x})) is almost pure relative to the basic setup (B, I), or – equivalently – relative to the basic setup (B^{sh}, IB^{sh}). However, it is easily seen that IB^{sh} is also a branch ideal of B^{sh}, if the latter is endowed with the chart P^{(∞)} → B^{sh} deduced from the given chart of \underline{M}. Taking into account remark 17.2.9(i), we may then replace B by B^{sh}, and assume that X_{0} and X are strictly local.

Now, consider a finite morphism φ_{0} : Y_{0} → X_{0} and the resulting morphism φ : Y → X as in (17.2.23). The discussion of (17.2.26) shows that (X, Z) is almost pure if and only if φ_{∗}𝒪_{Y}^{a} is an étale 𝒪_{X}^{a}-algebra, for every such φ_{0}. Set V := Spec B[1/p]; in view of (17.2.31) and lemma 17.2.28, we may then assume that V_{0} ⊂ U_{0}, and therefore (φ_{∗}𝒪_{Y})_{|V} is a finite étale 𝒪_{X}-algebra. On the other hand, let I′ ⊂ B be the radical of the ideal I + pB, and set Z′ := Z \V; clearly, I′ is a branch ideal. It then suffices to show that φ_{∗}𝒪_{Y}^{a} is an étale 𝒪_{X}^{a}-algebra, for the almost structure given by the new setup (B, I′). Thus, we may assume that p ∈ I, in which case the assertion follows from theorems 17.2.14(iii) and 16.8.44. □

Theorem 17.2.33. In the situation of (17.2.5), the ring B is ind-measurable.

Proof. (See remark 14.5.69(i) for the definition of ind-measurable ring.) One argues as in the proof of theorem 17.1.43, with some simplifications. We have to exhibit a sequence (d_{n} | n ∈ ℕ) of normalizing factors fulfilling conditions (a) and (b) of definition 14.5.64, where the λ_{n} occurring in loc.cit. is meant to be the usual length function for finitely generated B_{n}-modules supported at the closed point x_{n} of X_{n}. Now, fix n ∈ ℕ, set

T_{ℝ} := P_{ℝ}^{gp}/P^{(n)gp}

and endow T_{ℝ} with its invariant measure dμ_{n} of total volume equal to 1. For every γ ∈ P_{ℝ}^{gp}, let [γ] ∈ T_{ℝ} be the equivalence class of γ; notice that the P^{(n)}-module

(17.2.34)  S_{[γ]} := γP^{(n)gp} ∩ P_{ℚ}

is finitely generated (proposition 6.3.22(ii)) and depends only on the class [γ], and for any given finitely generated B_{n}-module M supported at x_{n}, consider the function

l_{M} : T_{ℝ} → ℕ [γ] ↦ λ_{n}(S_{[γ]} ⊗_{P^{(n)}} M).

Let e_{1}, …, e_{r} be a basis of the free ℤ-module P^{gp}, and define Ω_{n} ⊂ P_{ℝ}^{gp} as in the proof of theorem 17.1.43, so that Ω_{n} is a fundamental domain for the lattice P^{(n)gp}, and 0 lies in the interior of Ω_{n}. Denote also by Σ ⊂ T_{ℝ} the image of P_{ℝ} ∩ Ω_{n}.

Claim 17.2.35. There is a partition of T_{ℝ} into finitely many measurable subsets Θ_{1}, …, Θ_{t}, independent of M, such that :

(i) For every γ, λ ∈ Ω_{n}, the classes [γ], [λ] ∈ T_{ℝ} lie in the same Θ_{i} if and only if γ^{−1}S_{[γ]} = λ^{−1}S_{[λ]}.

(ii) Especially, l_{M} restricts to a constant function on each Θ_{i}.

(iii) Let Θ ∈ {Θ_{1}, …, Θ_{t}} be the subset containing [0] ∈ T_{ℝ}; then Θ ∩ Σ has measure > 0.

Proof of the claim. According to proposition 6.3.35(i,iii), the set 𝒮 := {γ^{−1}S_{[γ]} | γ ∈ Ω_{n}} is finite, and for every non-empty S ∈ 𝒮, the set {γ ∈ Ω_{n} | γ^{−1}S_{[γ]} = S} is the intersection of Ω_{n} with a ℚ-linearly constructible subset. It follows that the same must hold also in case S = ∅. The image in T_{ℝ} of any such ℚ-linearly constructible subset is obviously measurable, whence (i). Moreover, in view of our choice of Ω, assertion (ii) follows easily from claim 6.3.40. ♢

Let m ≥ n be any integer; since B′′_{m} is a free B′′_{n}-module of rank p^{r(m−n)} (notation of (17.2.3)), we may compute :

λ_{m}(B_{m} ⊗_{B_{n}} M) = \frac{p^{r(m−n)}}{[κ(x_{m}) : κ(x_{n})]} · ∑_{[γ]∈P^{(m)gp}/P^{(n)gp}} l_{M}([γ]).

However, lemma 17.2.11easily implies that κ(x_{n+1})^{p} = κ(x_{n}) for every n ∈ ℕ, whence

[κ(x_{n+1}) : κ(x_{n})] = p^{e_{n}} where e_{n} := Ω_{κ(x_{n})/ℤ}^{1}

by virtue of [63, Ch.0, Th.21.4.5]. But by the same token, the field κ(x_{m}) is isomorphic to κ(x_{n}) for every m ≥ n, so e_{n} is actually independent of n, and we get

[κ(x_{m}) : κ(x_{n})] = p^{e_{0}(m−n)} for every m ≥ n.

Therefore, set

d_{n} := p^{n·dim B_{0}} for every n ∈ ℕ.

We claim that (d_{n} | n ∈ ℕ) is a suitable sequence of normalizing factors for B. Indeed, claim 17.2.35(ii) says that l_{M} is a measurable function on T_{ℝ}, and the foregoing, together with the discussion of (17.2.8) implies that :

λ(B ⊗_{B_{n}} M) := lim_{m→+∞} d_{m}^{−1} · λ_{m}(B_{m} ⊗_{B_{n}} M) = d_{n}^{−1} ∫_{T_{ℝ}} l_{M}dμ_{n}

(recall that dim P = rk_{ℤ}P^{gp}, by corollary 6.4.12(i)), so condition (a) holds for this choice of factors. Next, fix ε > 0, let N → N′ be a surjection of finitely generated B_{n}-modules supported at x_{n}, and suppose that d_{n}^{−1}(λ_{n}(N) − λ_{n}(N′)) ≥ ε. Since λ_{n}(N) = l_{N}(0) (and likewise for N′), we deduce that

λ(B ⊗_{B_{n}} N) − λ(B ⊗_{B_{n}} N′) ≥ ε · ∫_{Θ} dμ_{n}

where Θ ∈ {Θ_{1}, …, Θ_{t}} is the unique subset of T_{ℝ} such that [0] ∈ Θ, so the volume of Θ is > 0, by claim 17.2.35(iii). This shows that condition (b) holds as well, and concludes the proof of the theorem. □

17.3. The direct summand conjecture. In the following paragraphs we prove a generalization of the direct summand conjecture for finite injective extensions of log-regular rings.

To begin with, let A be a ring, and consider a short exact sequence of A-modules

Σ  :  0 → M′ ⟶^{f} M ⟶^{g} M′′ → 0.

We say that Σ is universally exact if the sequence X ⊗_{A} Σ is short exact for every A-module X.

Lemma 17.3.1. (i) In the situation of (17.3), the following conditions are equivalent : (a) The sequence Σ is universally exact.

(b) The sequence X ⊗_{A} Σ is short exact for every finitely presented A-module X.

(c) The sequence Σ^{∨} := Hom_{ℤ}(Σ, ℚ/ℤ) splits.

(d) Σ^{∨} is universally exact.

(e) The sequence Hom_{A}(X, Σ) is exact for every finitely presented A-module X.

(f) Σ is the colimit of a filtered system of split short exact sequences of A-modules. (ii) If moreover M′′ is finitely presented, then conditions (a)–(f) are also equivalent to :

(g) Σ is a split short exact sequence of A-modules. (iii) If furthermore, A is local and noetherian, and M, M′, M′′ are finitely generated, then

conditions (a)–(g) are also equivalent to :

(h) The sequence X ⊗_{A} Σ is short exact for every A-module X of finite length.

Proof. Obviously (f)⇒(a)⇒(b). To see that (b)⇒(a), write any given A-module X as the filtered colimit of a system (X_{λ} | λ ∈ Λ) of finitely presented A-modules; then H^{i}(X ⊗_{A} Σ) is the colimit of the induced filtered system of A-modules (H^{i}(X_{λ} ⊗_{A} Σ) | λ ∈ Λ), for every i ∈ ℤ. But (b) says that each X_{λ} ⊗_{A} Σ is exact, whence (a).

(a)⇒(c): Clearly Σ^{∨} is short exact, since ℚ/ℤ is an injective ℤ-module. For every A-module X, recall that X^{∨} := Hom_{ℤ}(X, ℚ/ℤ) is naturally an A-module (see (14.2.26)); notice that for every pair of A-modules X, Y we have a natural isomorphism of A-modules :

ω_{X,Y} : Hom_{A}(X^{∨}, Y^{∨}) →^{∼} (X^{∨} ⊗_{A} Y)^{∨} φ ↦ (h ⊗ y ↦ φ(h)(y)).

There follows a commutative diagram :

\xymatrix{Hom_{A}(M^{′∨}, M^{∨}) \ar[r]^{Hom_{A}(M^{′∨},f^{∨})} \ar[d]_{ω_{M′,M}} & Hom_{A}(M^{′∨}, M^{′∨}) \ar[d]^{ω_{M′,M′}} \\ (M^{′∨} ⊗_{A} M)^{∨} \ar[r]^{(M^{′∨}⊗_{A}f)^{∨}} & (M^{′∨} ⊗_{A} M′)^{∨}.}

By assumption, M^{′∨} ⊗_{A} f is injective, hence (M^{′∨} ⊗_{A} f)^{∨} is surjective, so the same holds for Hom_{A}(M^{′∨}, f^{∨}), and (c) follows easily.

Obviously (c)⇒(d). In order to show that (d)⇒(e), let X be any finitely presented A-module; by assumption, X ⊗_{A} g^{∨} is an injection. On the other hand, for every pair of A-modules X, Y we have a natural A-linear map

σ_{X,Y} : X ⊗_{A} Y^{∨} → Hom_{A}(X, Y)^{∨} x ⊗ t ↦ (h ↦ t ∘ h(x))

and it is easily seen that σ_{X,Y} is an isomorphism if X is finitely presented (details left to the reader). By naturality of σ_{X,•}, it follows that Hom_{A}(X, g^{∨})^{∨} is injective, hence Hom_{A}(X, g^{∨}) is surjective, whence (e).

(e)⇒(f): Write M′′ as the colimit of a filtered system (M′′_{λ} | λ ∈ Λ) of finitely presented Amodules, and let (j_{λ} : M_{λ} → M | λ ∈ Λ) be the universal co-cone; for every λ, set Σ_{λ} := j_{λ}^{∗}Σ. Then Σ is the colimit of the resulting system of short exact sequences (Σ_{λ} | λ ∈ Λ), and we are reduced to checking that each Σ_{λ} is split. However, (e) implies that for every λ there exists an A-linear map h_{λ} : M′′_{λ} → M such that g ∘ h_{λ} = j_{λ}; the assertion follows straightforwardly.

Next, if M′′ is finitely presented, (e) implies that Hom_{A}(M′′, g) is surjective, whence (g). (iii): Let (A, 𝔪) be a local and noetherian ring, and let X be any finitely presented A-module;

if (h) holds, then for every n ∈ ℕ the map f_{n} := X/𝔪^{n}X ⊗_{A} f is injective. Let t ∈ Ker (X ⊗_{A} f) \ {0}; since the 𝔪-adic topology of X ⊗_{A} M′ is separated ([126, Th.8.10(i)]), there exists n ∈ ℕ such that the image \overline{t} of t in X/𝔪^{n}X ⊗_{A}M′ does not vanish, so f_{n}(\overline{t}) ≠ 0 by assumption, and therefore f(t) ≠ 0, a contradiction. So X ⊗_{A} f is injective for every such X, whence (b). □

17.3.2. Let ℳ be a regular log-structure on either the Zariski or the étale site of a noetherian and affine scheme X; say that X = Spec A for a noetherian ring A. Let also f : A → B be a finite injective ring homomorphism; we want to show that there exists an A-linear map g : B → A such that g ∘ f = 1_{A}, or equivalently, that the short exact sequence of A-modules

Σ_{f}  :  0 → A ⟶^{f} B → Q := Coker f → 0

splits. The latter holds if and only if the class [Σ_{f}] ∈ Ext_{A}^{1}(Q, A) of Σ_{f} vanishes. Let us recall:

Lemma 17.3.3. (i) Let A be a noetherian ring, and M, N two A-modules, with M of finite type; then every flat A-algebra A′ induces an isomorphism :

A′ ⊗_{A} Ext_{A}^{1}(M, N) →^{∼} Ext_{A′}^{1}(A′ ⊗_{A} M, A′ ⊗_{A} N).

(ii) In the situation of (17.3), suppose that A is noetherian and M′′ is of finite type. Then the following conditions are equivalent :

(a) Σ is a split short exact sequence of A-modules.

(b) For every 𝔭 ∈ Spec A there exists a faithfully flat A_{𝔭}-algebra A′ such that A′ ⊗_{A} Σ is a split short exact sequence of A′-module.

Proof. (i): Pick a resolution P^{•} → M by projective A-modules of finite type; then Ext_{A}^{1}(M, N) is computed by H^{1}Hom_{A}(P^{•}, N), and likewise Ext_{A′}^{1}(A′ ⊗_{A} M, A′ ⊗_{A} N) is computed by H^{1}Hom_{A′}(A′ ⊗_{A} P^{•}, A′ ⊗_{A} N). But since A is noetherian and P^{i} is an A-module of finite type for every i, the induced morphism of complexes

A′ ⊗_{A} Hom_{A}(P^{•}, N) → Hom_{A′}(A′ ⊗_{A} P^{•}, A′ ⊗_{A} N)

is an isomorphism, whence the assertion.

(ii): It is easily seen that the class [A′ ⊗_{A} Σ] ∈ Ext_{A′}^{1}(A′ ⊗_{A} Q, A′) corresponds – under the isomorphism of (i) – to 1 ⊗ [Σ] ∈ A′ ⊗_{A} Ext_{A}^{1}(Q, A′). The assertion follows immediately. □

In view of lemma 17.3.3(ii) we are reduced to checking that f_{𝔭} : A_{𝔭} → B_{𝔭} admits an A_{𝔭}-linear section for every 𝔭 ∈ Spec A; hence we may assume that (A, 𝔪) is a local noetherian ring. Likewise, let A^{sh} be a strict henselization of A (at the closed point 𝔪); since the structure map A → A^{sh} is faithfully flat, lemma 17.3.3(ii) implies that [Σ_{f}] = 0 if and only if [A^{sh} ⊗_{A} Σ_{f}] = 0 in Ext_{A^{sh}}^{1}(A^{sh} ⊗_{A} Q, A^{sh}); also, A^{sh} is noetherian, by [66, Ch.IV, Prop.18.8.8(iv)]. Thus, it suffices to check that A^{sh} ⊗_{A} f admits an A^{sh}-linear section, and therefore we may assume that ℳ admits a chart β : P → A that is sharp at the point 𝔪.

17.3.4. Arguing likewise, we may replace A by its 𝔪-adic completion, and suppose therefore that A is a complete local noetherian ring. Moreover, A is normal and Cohen-Macaulay, by corollary 12.5.13. Furthermore, the proof of theorem 12.5.10 show that there exists r ∈ ℕ and a surjective ring homomorphism

φ : V [[Q]] → A where Q := P × ℕ^{⊕r}

and where :

• (V, 𝔪_{V}) is a coefficient ring of A, i.e. either a field, or a complete discrete valuation ring with A = φ(V) + 𝔪 and such that 𝔪_{V} = pV for a prime integer p

• The kernel of φ is trivial if V is a field, and otherwise it is principal, generated by a power series ∑_{q∈Q} a_{q} · q with a_{0} ∈ 𝔪_{V} \ 𝔪_{V}^{2}

• The chart β is the composition of φ with the obvious inclusion map P → Q → V [[Q]]. Let k′ be an algebraically closed field extension of the residue field k of A; by [126, Th.29.1] there exists a flat ring homomorphism ψ : V → V′, where V′ = k′ if V = k, and otherwise V′ is a complete discrete valuation ring with maximal ideal pV′ and residue field k′. The induced ring extension ψ[Q] : V [Q] → V′[Q] is again faithfully flat, hence it induces a faithfully flat ring homomorphism V [[Q]] → V′[[Q]] ([126, Th.22.4]). Set A′ := V′[[Q]] ⊗_{V [[Q]]} A; the induced map A → A′ is faithfully flat, and A′ is the quotient of V′[[Q]] by the principal ideal generated by the image of Ker φ. Hence, A′ is a complete local noetherian ring, and by theorems 12.5.10 and 12.5.31, the induced map of monoids Q → A′ is a chart for a regular log structure on Spec A′. By lemma 17.3.3(ii), we may then replace A, B, P by respectively A′, A′ ⊗_{A} B, Q, and assume that k is algebraically closed and 𝔪 = 𝔪_{P} A (with 𝔪_{P} = P \ {0} : see definition 6.1.10(i)).

17.3.5. Let 𝔭_{1}, …, 𝔭_{k} be the finitely many minimal prime ideals of B; since A is reduced, f(A)∩𝔭_{1}∩⋯∩𝔭_{k} = 0, hence the composition of f with the natural map B → B/𝔭_{1}×⋯×B/𝔭_{k} is still finite and injective, and its kernel contains the ideal ∏_{i=1}^{k} f^{−1}𝔭_{i}. Then there exists i ∈ \{1, …, k\} such that f^{−1}𝔭_{i} = 0, so that the composition \overline{f}_{i} : A → B/𝔭_{i} of f with the projection B → B/𝔭_{i} is still injective; clearly it suffices to check that \overline{f}_{i} admits an A-linear section, so we may assume that B is a domain.

Let K be the field of fractions of A, set B_{K} := K ⊗_{A} B, let d := dim_{K} B_{K}, and denote by Tr_{B_{K}/K} : B_{K} → K the trace form; since A is integrally closed in K, the composition of Tr_{B_{K}/K} with the localization B → B_{K} factors through an A-linear map (lemma 14.3.15(i))

Tr_{B/A} : B → A.

Then Tr_{B/A} ∘ f = d · 1_{A}. In particular, if A is a ℚ-algebra, the map d^{−1} · Tr_{B/A} is a section of f as sought. Thus, we may assume that k has characteristic p > 0. Let us set

P^{(n)} := {γ ∈ P_{ℚ} | γ^{p^{n}} ∈ P }  P^{(∞)} := ⋃_{n∈ℕ} P^{(n)}  and Γ := (P^{(∞)}/P)^{gp}.

Then P^{(∞)} is naturally a Γ-graded monoid. More precisely, for every γ ∈ (P^{(∞)})^{gp}, denote by [γ] ∈ Γ the class of γ; then we have :

gr_{[γ]}P^{(∞)} = P^{(∞)} ∩ γP^{gp} for every γ ∈ (P^{(∞)})^{gp}.

Theorem 17.3.6. In the situation of (17.3.2), suppose that A is an 𝔽_{p}-algebra. Then f admits an A-linear section.

Proof. After the foregoing reductions, we may assume that A = k[[P]] for a sharp, fine and saturated monoid P, and that B is a domain. For every n ∈ ℕ, set

A_{n} := k[[P^{(n)}]]  B_{n} := A_{n} ⊗_{A} B  and A_{∞} := ⋃_{n∈ℕ} A_{n}.

Notice that A_{0} is a direct summand of the A_{0}-module A_{n}, hence the inclusion map i_{n} : A → A_{n} has a section σ_{n} : A_{n} → A for every n ∈ ℕ; we are then reduced to checking that the induced map f_{n} : A_{n} → B_{n} has an A_{n}-linear section s_{n} : B_{n} → A_{n}, for some n ∈ ℕ : indeed, in this case let j_{n} : B → B_{n} be the induced inclusion map, and set s := σ_{n} ∘ s_{n} ∘ j_{n} : B → A. We get:

s ∘ f = σ_{n} ∘ s_{n} ∘ j_{n} ∘ f = σ_{n} ∘ s_{n} ∘ f_{n} ∘ i_{n} = σ_{n} ∘ i_{n} = 1_{A}

as required. On the other hand, let K_{n} be the field of fractions of A_{n}, and denote by B′_{n} the maximal reduced quotient of B_{n}, for every n ∈ ℕ; since A_{n} is reduced, the induced map f′_{n} : A_{n} → B′_{n} is still injective, so it suffices to exhibit a section for f′_{n}. Moreover, E_{n} := K_{n} ⊗_{A_{n}} B′_{n} is a finite field extension of K_{n} (since K_{n} is a purely inseparable field extension of K), and the inclusion map K_{n} → K_{n+1} induces a field extension E_{n} → E_{n+1} for every such n. Notice also that K_{∞} := ⋃_{n∈ℕ} K_{n} is a perfect field; hence E_{∞} := ⋃_{n∈ℕ} E_{n} is the maximal reduced quotient of K_{∞} ⊗_{A} B, and is a finite separable field extensions of K_{∞}. It follows easily that for n ∈ ℕ sufficiently large, E_{n} is already a separable and finite field extension of K_{n}. Summing up, we may replace A and B by A_{n} and B′_{n} for n large enough, and assume that f is generically étale, i.e. there exists g ∈ A \ {0} such that A[1/g] ⊗_{A} f is a finite, injective and étale map. Then, notice that A_{∞} is a perfect 𝔽_{p}-algebra, and let 𝔫 ⊂ A_{∞} be the radical of the ideal A_{∞}g; let also C_{∞} be the integral closure of A_{∞} in A_{∞}[1/g] ⊗_{A} B. By theorems 16.8.2 and 16.8.44(ii), and proposition 14.4.8, the (A_{∞}, 𝔫)^{a}-algebra C_{∞}^{a} is étale, faithfully flat and of almost finite rank. Now, let X be any A-module of finite length; by virtue of lemma 17.3.1(ii,iii), it suffices to check that the map X ⊗_{A} f : X → X ⊗_{A} B is injective. However, since A is a direct summand of the A-module A_{∞}, the induced map h : X → X ⊗_{A} A_{∞} is injective, and since C_{∞}^{a} is a faithfully flat (A_{∞}, 𝔫)^{a}-algebra, the kernel of the induced map l : X ⊗_{A} A_{∞} → X ⊗_{A} C_{∞} is annihilated by 𝔫. Clearly l ∘ h factors through X ⊗_{A} f; hence :

𝔫 · Ker(X ⊗_{A} f) = 0

(we regard Ker(X⊗_{A}f) as a subset of Ker l, via h). Now, as P^{(∞)} is Γ-graded, A_{∞} is naturally a Γ-graded A-algebra, and X⊗_{A}A_{∞} is a Γ-graded (A_{∞}, gr_{•}A_{∞})-module (see definition 7.6.1(ii)); by definition :

gr_{[γ]}A_{∞} := A ⊗_{ℤ[P]} ℤ[gr_{[γ]}P^{(∞)}] for every γ ∈ P^{(∞)}.

According to proposition 6.4.14(ii), there exist λ_{1}, …, λ_{k} ∈ P^{∨} := Hom_{Mnd}(P, ℕ) such that ℝλ_{1}, …, ℝλ_{k} are the extremal rays of P_{ℝ}^{∧} (see (6.3.15)).

Claim 17.3.7. If γ ∈ P^{(∞)}, and 0 ≤ λ_{i}(γ) < 1 for every i = 1, …, k, then gr_{[γ]}P^{(∞)} = γP.

Proof of the claim. By proposition 6.4.14(iv), we have

P^{(∞)} = {γ ∈ (P^{(∞)})^{gp} | λ_{i}(γ) ≥ 0 for every i = 1, …, k}.

Hence gr_{[γ]}P^{(∞)} is the subset of all γ · β, with β ∈ P^{gp} and λ_{i}(γ · β) ≥ 0 for i = 1, …, k. But λ_{i}(β) ∈ ℤ for every such i and β; our assumption on λ_{i}(γ) then implies that λ_{i}(β) ∈ ℕ for every such i and β, whence β ∈ P (again, by proposition 6.4.14(iv)), and the claim follows. ♢

We have g = ∑_{γ∈P} g_{γ} · γ for a system (g_{γ} | γ ∈ P) of elements of k. Pick β ∈ P with g_{β} ≠ 0, and N ∈ ℕ large enough, so that λ_{i}(β^{1/p^{N}}) = p^{−N}λ_{i}(β) < 1 for i = 1, …, k; by claim 17.3.7, we then have gr_{[β^{1/p^{N}}]}P^{(∞)} = β^{1/p^{N}} P. Clearly g^{1/p^{N}} = ∑_{γ∈P} g_{γ}^{1/p^{N}} · γ^{1/p^{N}}; let us also write g^{1/p^{N}} = ∑_{[γ]∈Γ} g_{[γ]}, with g_{[γ]} ∈ gr_{[γ]}A_{∞} for every [γ] ∈ Γ. It follows that

g_{[β^{1/p^{N}}]} = g_{β}^{1/p^{N}} · β^{1/p^{N}} · (1 + a) for some a ∈ 𝔪. Let now x ∈ Ker X ⊗_{A} f; we have x ∈ gr_{0}(X ⊗_{A} A_{∞}), and g^{1/p^{N}} x = 0; then g_{[γ]}x = 0 for every [γ] ∈ Γ, and especially :

x ⊗ g_{β}^{1/p^{N}} · β^{1/p^{N}} = 0 in gr_{[β^{1/p^{N}}]}(X ⊗_{A} A_{∞}) = X ⊗_{A} A_{[β^{1/p^{N}}]} = X ⊗_{ℤ[P]} ℤ[β^{1/p^{N}} P]

and since g_{β} ≠ 0, we have x⊗β^{1/p^{N}} = 0 in X⊗_{ℤ[P]}ℤ[β^{1/p^{N}} P], so finally x = 0, as required. □

Remark 17.3.8. Theorem 17.3.6also has a proof along the lines of [93, Rem.6.2], which proves the direct summand conjecture for regular 𝔽_{p}-algebras. This alternative argument still relies on claim 17.3.7, but avoids lemma 17.3.1and the reduction to the case where f is generically étale.

17.3.9. By theorem 17.3.6and the foregoing reductions, we may assume that

A = V [[P]]/(ϑ)

for a fine, sharp and saturated monoid P ≠ 0, a complete discrete valuation ring (V, 𝔪_{V}) with 𝔪_{V} = pV, whose residue field k := V/pV is algebraically closed, and a power series ϑ := ∑_{γ∈P} a_{γ} · γ with a_{0} ∈ pV \ p^{2}V. Then 𝔪_{P} A = 𝔪, the maximal ideal of A. We set

A_{n} := A ⊗_{ℤ[P]} ℤ[P^{(n)}]  and B_{n} := A_{n} ⊗_{A} B  for every n ∈ ℕ.

Notice that the induced map A_{n} → A_{n+1} is injective for every n ∈ ℕ, since ℤ[P^{(n)}] is a direct summand of the ℤ[P^{(n)}]-module ℤ[P^{(n+1)}]; hence, let as well A_{∞} := ⋃_{n∈ℕ} A_{n}. Moreover, notice that A/𝔪_{P} A = V/a_{0}V = k, and Ω_{k/ℤ}^{1} = 0, since k is perfect; then, the integer r of (17.2.3) vanishes in the current situation, hence theorem 17.2.14(iv) tells us that A_{∞} is a formally perfectoid ring, for its p-adic topology, and clearly it is also naturally Γ-graded.

Let λ_{1}, …, λ_{k} ∈ P^{∨} := Hom_{Mnd}(P, ℕ) such that ℝλ_{1}, …, ℝλ_{k} are the extremal rays of P_{ℝ}^{∨} (see the proof of theorem 17.3.6), and β_{1}, …, β_{l} ∈ P a finite system of generators of the ℚ_{+}-module ℚ_{+} ⊗_{ℕ} P (see (3.7.17)); set

λ := λ_{1} + ⋯ + λ_{k}  β := β_{1} ⋯ β_{l}  and L := max(λ(β_{1}), …, λ(β_{l})).

Lemma 17.3.10. In the situation of (17.3.9), let J ⊂ A_{∞} be a Γ-graded ideal, g ∈ A_{∞}, and let e, n ∈ ℕ such that

g ∈ 𝔪^{e}A_{∞} \ 𝔪^{e+1}A_{∞}  g ∈ J^{p^{n}}  and  1 ≥ ε(e, n) := (eL + λ(β))/p^{n}.

Then there exists γ ∈ P^{(∞)} such that :

gr_{[γ]}J = gr_{[γ]}A_{∞}  and  λ(γ) < ε(e, n).

Proof. We may write g = g_{1} + ⋯ + g_{r} for a finite sequence g_{1}, …, g_{r} ∈ J^{p^{n}} such that each g_{i} is in turn of the form g_{i} = ∏_{j=1}^{p^{n}} a_{ij} ⊗ γ_{ij} with :

a_{i1}, …, a_{i,p^{n}} ∈ A  γ_{i1}, …, γ_{i,p^{n}} ∈ P^{(∞)}  a_{ij} ⊗ γ_{ij} ∈ gr_{[γ_{ij}]}J for every j = 1, …, p^{n}.

It is clear that there exists e′ ≤ e and i ∈ {1, …, r} such that g_{i} ∈ 𝔪^{e′}A_{∞} \ 𝔪^{e′+1}A_{∞}. Up to reordering the p^{n} factors of g_{i}, we may then assume that there exists c ∈ {0, …, p^{n}} such that λ(γ_{ij}) < ε(e, n) for j = 1, …, c, and λ(γ_{ij}) ≥ ε(e, n) for j = c + 1, …, p^{n}. Suppose now that the lemma fails; then we must have a_{ij} ∈ 𝔪 for every i = 1, …, c, and therefore

(17.3.11)  ∏_{j=c+1}^{p^{n}} a_{ij} ⊗ γ_{ij} ∉ 𝔪^{e′−c+1}A_{∞}.

For every j = c + 1, …, p^{n}, write γ_{ij} = ∏_{s=1}^{l} β_{s}^{d_{ijs}} for certain d_{ij1}, …, d_{ijl} ∈ ℚ_{+}; then there exist unique d′_{is} ∈ ℕ and d′′_{is} ∈ [0, 1[ with

∑_{j=c+1}^{p^{n}} d_{ijs} = d′_{is} + d′′_{is} for every s = 1, …, l

and (17.3.11) then implies that ∑_{s=1}^{l} d′_{is} ≤ e′ − c, whence :

∑_{j=c+1}^{p^{n}} λ(γ_{ij}) < (e′ − c) · L + λ(β′) ≤ p^{n} · ε(e, n) − cL.

However, ∑_{j=c+1}^{p^{n}} λ(γ_{ij}) ≥ (p^{n} − c) · ε(e, n), and since ε(e, n) ≤ 1 ≤ L, this is absurd. □

Theorem 17.3.12. In the situation of (17.3.2), the map f admits an A-linear section.

Proof. In view of the foregoing reductions, we may assume that A and A_{∞} are as in (17.3.9) and B is a domain; recall also that B[1/g] is an étale and faithfully flat A[1/g]-algebra. Let :

A′_{n} := A_{∞}[X^{1/p^{n}}]/(X − g) for every n ∈ ℕ and A′_{∞} := ⋃_{n∈ℕ} A′_{n}.

Let also D be the p-root closure of A′_{∞} in A′_{∞}[1/p], and endow D with its p-adic topology; by theorem 16.9.17, we know that D is a faithfully flat A_{∞}-algebra, and that D is a formally perfectoid ring. Denote by D^{∧} the p-adic completion of D, and set B′ := D^{∧}⊗_{A}B[1/g]. Hence, B′ is a finite étale D^{∧}[1/g]-algebra. Let now 𝔫 ⊂ D^{∧} be the ideal generated by (g^{1/p^{n}} | n ∈ ℕ); then (D^{∧}, 𝔫) is a basic setup, and we set D_{1}^{∧} := (D^{∧}, 𝔫)_{∗}^{a} ⊂ D^{∧}[1/g] (see lemma 16.9.41); lastly, let B′_{1} be the integral closure of D_{1}^{∧} in B′. By theorem 16.9.42(ii,iii), for every n ∈ ℕ, the (D/p^{n}D)^{a}-algebra (B′_{1}/p^{n}B′_{1})^{a} is faithfully flat and étale of finite rank, for the almost structure given by the basic setup (D_{1}^{∧}, 𝔫). After these preliminaries, let X be any A-module of finite length; by lemma 17.3.1(ii,iii), it suffices to show that the map X ⊗_{A} f is injective. However, let x ∈ Ker(X ⊗_{A} f), and pick m ∈ ℕ large enough, so that p^{m}X = 0. Since A_{∞} is a Γgraded A-algebra with gr_{0}A_{∞} = A, the induced map φ : X → X_{∞} := X ⊗_{A} A_{∞}/p^{m}A_{∞} is injective, and X_{∞} is naturally a Γ-graded (A_{∞}, gr_{•})-module, with gr_{0}X_{∞} = X. Especially, the annihilator J of φ(x) = x ⊗ 1 ∈ gr_{0}X_{∞} is a Γ-graded ideal of A_{∞}. Next, since D is a faithfully flat A_{∞}-algebra, the induced map φ′ : X_{∞} → X_{∞} ⊗_{A_{∞}} D/p^{m}D is again injective, and Ann_{D}(φ′(x ⊗ 1)) = JD. Lastly, since (B′_{1}/p^{m}B′_{1})^{a} is a faithfully flat (D/p^{m}D)^{a}-algebra, the kernel of the induced map φ′′ : X_{∞} ⊗_{A_{∞}} D/p^{m}D → X_{∞} ⊗_{A_{∞}} B′_{1}/p^{m}B′_{1} is annihilated by 𝔫. Clearly φ′′ ∘ φ′ ∘ φ factors through X ⊗_{A} f, hence φ′(x ⊗ 1) ∈ Ker φ′′, and consequently :

𝔫 ⊂ JD.

Thus, g^{1/p^{n}} ∈ JD ∩ A_{∞} = J for every n ∈ ℕ ([126, Th.7.5(ii)]), i.e. g ∈ J^{p^{n}} for every n ∈ ℕ. Notice that the 𝔪-adic topology is separated on A_{∞}; hence, let e ∈ ℕ such that g ∈ 𝔪^{e}A_{∞} \ 𝔪^{e+1}A_{∞}; by lemma 17.3.10, for every n ∈ ℕ there exists γ_{n} ∈ P^{(∞)} such that λ(γ_{n}) < ε(e, n) and gr_{[γ_{n}]}J = gr_{[γ_{n}]}A_{∞}. By claim 17.3.7, it follows that for every sufficiently large n ∈ ℕ, we have gr_{[γ_{n}]}J = A ⊗_{ℤ[P]} ℤ[γ_{n} + P], a free A-module of rank one, with generator 1 ⊗ γ_{n}. Finally, we get x ⊗ γ_{n} = 0 in gr_{[γ_{n}]}X_{∞} = X ⊗_{ℤ[P]} ℤ[γ_{n} + P] for every such n; thus, x = 0, and the proof is concluded. □

17.4. Diagonal idempotents of generically étale maps. Let f : A → B be a finite and injective ring homomorphism of noetherian rings; set X := Spec A, denote by ℬ the quasi-coherent 𝒪_{X}-algebra associated with B, and by μ : ℬ ⊗_{𝒪_{X}} ℬ → ℬ the multiplication morphism. For every affine open subset U ⊂ X such that ℬ_{|U} is an étale 𝒪_{U}-algebra, we have a unique diagonal idempotent e_{f,U} ∈ ℬ(U) ⊗_{𝒪_{X}(U)} ℬ(U) such that :

μ_{U}(e_{f,U}) = 1 and x · e_{f,U} = 0 for every x ∈ Ker (ℬ(U) ⊗_{𝒪_{X}(U)} ℬ(U) ⟶^{μ_{U}} ℬ(U)) .

We suppose moreover that f is generically étale, i.e. that the union U_{f} ⊂ X of all such affine open subsets is dense in X; the uniqueness property of the diagonal idempotents then implies that there exists a unique section

e_{f} ∈ Γ(U_{f}, ℬ ⊗_{𝒪_{X}} ℬ)

such that e_{f|U} = e_{f,U} for every affine open subset U ⊂ U_{f}. Moreover, ℬ_{|U_{f}} is a locally free 𝒪_{U_{f}} -module of finite rank, hence we have a well defined 𝒪_{U_{f}} -linear trace map

tr_{ℬ/𝒪_{X}} : ℬ_{|U_{f}} → 𝒪_{U_{f}}

that assigns to every affine open subset U ⊂ U_{f} and every b ∈ ℬ(U) the trace of the 𝒪_{X}(U)- linear endomorphism b · 1_{ℬ(U)} . The corresponding trace form

t_{ℬ/𝒪_{X}} := tr_{ℬ/𝒪_{X}} ∘ μ : (ℬ ⊗_{𝒪_{X}} ℬ)_{|U_{f}} → 𝒪_{U_{f}}

is a perfect pairing ([75, Th.4.1.14]), hence it induces an 𝒪_{U_{f}} -linear isomorphism

ω_{ℬ/𝒪_{X}} : ℬ_{|U_{f}} →^{∼} (ℬ^{∨})_{|U_{f}} where ℬ^{∨} := ℋ om_{𝒪_{X}} (ℬ, 𝒪_{X}).

Remark 17.4.1. Let f : A → B be as in (17.4), p ∈ ℕ a prime integer, and suppose that A is an 𝔽_{p}-algebra. Let also h ∈ A such that the localization f_{h} : A_{h} → B_{h} of f is étale. It is easily seen that the natural map (notation of (9.8.11))

B^{perf} ⊗_{A^{perf}} B^{perf} → (B ⊗_{A} B)^{perf}

is bijective. By assumption, h^{n}e_{f} is in the image of the localization B ⊗_{A} B → B_{h} ⊗_{A_{h}} B_{h}, for some n ∈ ℕ. The image e_{f}^{perf} of e_{f} in (B ⊗_{A} B)_{h}^{perf} is the diagonal idempotent of the étale map A_{h}^{perf} ⊗_{A} f : A_{h}^{perf} → B_{h}^{perf} = A_{h}^{perf} ⊗_{A} B. Since (e_{f}^{perf})^{p} = e_{f}^{perf}, we conclude that h^{n/p^{k}}e_{f}^{perf} lies in the image of the localization (B ⊗_{A} B)^{perf} → (B ⊗_{A} B)_{h}^{perf}, for every k ∈ ℕ.

Definition 17.4.2. (i) A log ring (A, P, β) is the datum of a ring A, an integral saturated monoid P whose associated abelian group P^{gp} is torsion-free, and a morphism of monoids β : P → A from P to the multiplicative monoid (A, ·), furnishing a chart for a log structure 𝒫 on the Zariski site of Spec A (see definition 12.1.17(i)), which we call the log structure of (A, P, β).

(ii) We say that the log ring (A, P, β) is regular if A is noetherian, P is fine and saturated, and the log structure of (A, P, β) is regular. In this case, P^{gp} is a free abelian group of finite rank, and A is a product of finitely many normal domains (corollary 12.5.13).

17.4.3. Let (A, P, β) be any log ring, f : A → B a ring homomorphism as in (17.4), and set X := Spec A. We consider the monoid P^{(∞)} associated with P as in (17.3.5), and the abelian group Γ := (P^{(∞)}/P)^{gp}. We will likewise use the monoid P_{ℚ} := ℚ_{+} ⊗_{ℕ} P, and the abelian group Γ_{ℚ} := (P_{ℚ}/P)^{gp}, and for every γ ∈ P_{ℚ}, we let [γ] ∈ Γ_{ℚ} be the class of γ. Then the ring

A_{∞} := A ⊗_{ℤ[P]} ℤ[P_{ℚ}]

is naturally a Γ_{ℚ}-graded A-algebra, with graded summands :

A_{[γ]} := A ⊗_{ℤ[P]} ℤ[gr_{[γ]}P_{ℚ}] for every γ ∈ P_{ℚ}.

Likewise, set B_{∞} := A_{∞} ⊗_{A} B; both B_{∞} and its dual A_{∞}-module B_{∞}^{∨} := Hom_{A_{∞}}(B_{∞}, A_{∞}) inherit natural Γ_{ℚ}-graded structures whose graded summands are respectively

B_{[γ]} := A_{[γ]} ⊗_{A} B  and B_{[γ]}^{∨} := Hom_{A}(B, A_{[γ]})  for every γ ∈ P_{ℚ}.

Let 𝒪_{X_{∞}} and ℬ_{∞} be the quasi-coherent 𝒪_{X}-algebras associated with A_{∞} and B_{∞}. Hence, ℬ_{∞|U_{f}} is an étale 𝒪_{X_{∞}|U_{f}} -algebra, whose trace morphism induces the isomorphism

ω_{ℬ_{∞}/𝒪_{X∞}} := 𝒪_{X_{∞}|U_{f}} ⊗_{𝒪_{U_{f}}} ω_{ℬ/𝒪_{X}} : ℬ_{∞|U_{f}} →^{∼} ℬ_{∞|U_{f}}^{∨} .

Also, the corresponding diagonal idempotent e_{f,∞} ∈ Γ(U_{f}, ℬ_{∞} ⊗_{𝒪_{X∞}} ℬ_{∞}) is the image of e_{f}, under the natural morphism

(17.4.4)  ℬ ⊗_{𝒪_{X}} ℬ → 𝒪_{X_{∞}} ⊗_{𝒪_{X}} ℬ ⊗_{𝒪_{X}} ℬ →^{∼} ℬ_{∞} ⊗_{𝒪_{X∞}} ℬ_{∞}.

More precisely, since U_{f} is quasi-compact, the A_{∞}-module Γ(U_{f}, ℬ_{∞} ⊗_{𝒪_{X∞}} ℬ_{∞}) is also Γ_{ℚ}- graded, and on the other hand, gr_{0}P_{ℚ} = P, since P is saturated; then gr_{0}B_{∞} = B, and the map (17.4.4) identifies Γ(U_{f}, ℬ ⊗_{𝒪_{X}} ℬ) with gr_{0}Γ(U_{f}, ℬ_{∞} ⊗_{𝒪_{X∞}} ℬ_{∞}).

17.4.5. Let \underline{f} := (f, P, β) be the datum of a log ring (A, P, β) and a ring homomorphism f : A → B as in (17.4); with the notation of (17.4.3), we attach to \underline{f} the composition

(17.4.6)  B_{∞}^{∨} ⊗_{A_{∞}} B_{∞}^{∨} ⟶^{ρ} Γ(U_{f}, ℬ_{∞}^{∨} ⊗_{𝒪_{X∞}} ℬ_{∞}^{∨}) ⟶^{α} Γ(U_{f}, ℬ_{∞} ⊗_{𝒪_{X∞}} ℬ_{∞})

where ρ is the restriction map of the quasi-coherent 𝒪_{X}-module ℬ_{∞}^{∨} ⊗_{𝒪_{X∞}} ℬ_{∞}^{∨}, and α is induced by the isomorphism ω_{ℬ_{∞}/𝒪_{X∞}}^{−1} ⊗_{𝒪_{X∞|U∞}} ω_{ℬ_{∞}/𝒪_{X∞}}^{−1} . Set as well :

R(B, Δ) := ⨁_{[γ]∈Δ} B_{[γ]}^{∨} ⊗_{A} B_{[1/γ]}^{∨} for every subset Δ ⊂ Γ_{ℚ}.

Since (17.4.6) is a morphism of graded A_{∞}-modules, it restricts to an A-linear map

ϑ\underline{f} : R(B, Γ_{ℚ}) → gr_{0}Γ(U_{f}, ℬ_{∞} ⊗_{𝒪_{X∞}} ℬ_{∞}) →^{∼} Γ(U_{f}, ℬ ⊗_{𝒪_{X}} ℬ).

Remark 17.4.7. (i) In the situation of (17.4.5), let g : A → A′ be a ring homomorphism; set B′ := A′ ⊗_{A} B, let f′ : A′ → B′ be the induced map, and suppose that \underline{f}′ := (f′, P, g ∘ β) is also a datum as in (17.4.5). Set X′ := Spec A′, and let h : X′ → X be the morphism of schemes induced by g; also, let ℬ′ be the quasi-coherent 𝒪_{X′}-algebra arising from B′. For every [γ] ∈ Γ_{ℚ} we get a natural A-linear map

(17.4.8)  B_{[γ]}^{∨} → Hom_{A}(B, A′_{[γ]}) →^{∼} B_{[γ]}^{′∨} := Hom_{A′}(B′, A′_{[γ]})

and a direct inspection of the definitions yields a commutative diagram :

[figure: R(B, Γ_{ℚ}); ϑ\underline{f}; Γ(U_{f}, ℬ ⊗_{𝒪_{X}} ℬ); R(B′, Γ_{ℚ}); ϑ_{\underline{f}′}; Γ(U_{f′}, ℬ′ ⊗_{𝒪_{X′}} ℬ′); ρ; Γ(h^{−1}U_{f}, ℬ′ ⊗_{𝒪_{X′}} ℬ′)]

whose left (resp. right) vertical arrow is induced by the maps (17.4.8) (resp. by the natural morphisms of quasi-coherent 𝒪_{X}-algebras 𝒪_{X} → h_{∗}𝒪_{X′} and ℬ → h_{∗}ℬ′), and where ρ is the restriction map. Also, it is clear that the right vertical arrow maps e_{f} to ρ(e_{f′}).

(ii) Let R be any ring, S and S′ two finite R-algebras whose underlying R-modules are projective (of finite rank). Then all the trace maps tr_{S/R}, tr_{S′/R} and tr_{S⊗_{R}S′/R} are well defined, and [75, Lemma 4.1.3] implies that :

tr_{S⊗_{R}S′/R} = tr_{S/R} ⊗_{R} tr_{S′/R} : S ⊗_{R} S′ → R.

We deduce the following alternative description of ϑ\underline{f}. First, we have a natural A_{∞}-linear map

(17.4.9)  B_{∞}^{∨} ⊗_{A_{∞}} B_{∞}^{∨} → (B_{∞} ⊗_{A_{∞}} B_{∞})^{∨} := Hom_{A}(B ⊗_{A} B, A_{∞})

given by the rule : φ ⊗ φ′ ↦ (b ⊗ b′ ↦ φ(b) · φ′(b′)) for every φ, φ′ ∈ B_{∞}^{∨} and every b, b′ ∈ B. Next, the trace map of the 𝒪_{X_{∞}}-algebra ℬ_{∞} ⊗_{𝒪_{X}} ℬ_{∞} induces an 𝒪_{X_{∞}}-linear map

ω_{ℬ_{∞}⊗_{𝒪_{X∞}} ℬ_{∞}/𝒪_{X∞}} : ℬ_{∞} ⊗_{𝒪_{X}} ℬ_{∞} → (ℬ_{∞} ⊗_{𝒪_{X}} ℬ_{∞})^{∨}

whose restriction to the open subset U_{f} is an isomorphism. Then it is easily seen that (17.4.6) equals the composition of (17.4.9) with the map

(B_{∞} ⊗_{A_{∞}} B_{∞})^{∨} ⟶^{ρ′} Γ(U_{f}, (ℬ_{∞} ⊗_{𝒪_{X}} ℬ_{∞})^{∨}) ⟶^{α′} Γ(U_{f}, ℬ_{∞} ⊗_{𝒪_{X}} ℬ_{∞})

where ρ′ is the restriction map of the quasi-coherent 𝒪_{X}-module (ℬ_{∞} ⊗_{𝒪_{X}} ℬ_{∞})^{∨}, and α′ is induced by the isomorphism (ω_{ℬ_{∞}⊗_{𝒪_{X∞}} ℬ_{∞}/𝒪_{X∞}} )_{|U_{f}}. Moreover, (B_{∞} ⊗_{A_{∞}} B_{∞})^{∨} is naturally a Γ_{ℚ}-graded A_{∞}-module, with a natural identification :

gr_{0}(B_{∞} ⊗_{A_{∞}} B_{∞})^{∨} →^{∼} (B ⊗_{A} B)^{∨} := Hom_{A}(B ⊗_{A} B, A)

and (17.4.9) is a morphism of Γ_{ℚ}-graded modules. It follows that ϑ\underline{f} is the composition

R(B, Γ_{ℚ}) ⟶^{ξ\underline{f}} (B ⊗_{A} B)^{∨} ⟶^{ξ′ ; \underline{f}} Γ(U_{f}, ℬ ⊗_{𝒪_{X}} ℬ)

where ξ\underline{f} is induced by the restriction of (17.4.9), and ξ′ ; \underline{f} is the restriction of α′ ∘ ρ′. In turn, ξ′ ; \underline{f} can be described as the composition of the restriction map (B ⊗_{A} B)^{∨} → Γ(U_{f}, (ℬ ⊗_{𝒪_{X}} ℬ)^{∨}) of the quasi-coherent 𝒪_{X}-module (ℬ ⊗_{𝒪_{X}} ℬ)^{∨} arising from (B ⊗_{A} B)^{∨}, and the isomorphism Γ(U_{f}, (ℬ ⊗_{𝒪_{X}} ℬ)^{∨}) →^{∼} Γ(U_{f}, ℬ ⊗_{𝒪_{X}} ℬ) induced by the isomorphism (ℬ ⊗_{𝒪_{X}} ℬ)_{|U_{f}}^{∨} →^{∼} (ℬ ⊗_{𝒪_{X}} ℬ)_{U_{f}} deduced as usual from the trace map of ℬ ⊗_{𝒪_{X}} ℬ.

(iii) With the notation of (ii), notice as well that ξ′ ; \underline{f} is injective if A is reduced. Indeed, if φ : B ⊗_{A} B → A restricts on U_{f} to the zero section of (ℬ ⊗_{𝒪_{X}} ℬ)^{∨}, then for every b, b′ ∈ B the element φ(b ⊗ b′) ∈ A restricts on U_{f} to the zero section of 𝒪_{X}; since A is reduced and U_{f} is dense in X, this means that φ(b ⊗ b′) = 0 for every such b, b′, whence the contention.

Lemma 17.4.10. (i) In the situation of (17.4.5), let Δ ⊂ Γ_{ℚ} be any subset. Then there exists a finite subset Σ ⊂ Δ such that ξ\underline{f}(R(B, Δ)) = ξ\underline{f}(R(B, Σ)) (where ξ\underline{f} is defined as in remark 17.4.7(ii)). Moreover, Σ depends only on P and Δ (and neither on β nor f).

(ii) Suppose that P = P_{1} × P_{2}, where P_{2} = ℕ^{⊕r} × ℤ^{⊕s}, for some r, s ∈ ℕ. For i = 1, 2, let Δ_{i} ⊂ Γ_{i,ℚ} := (P_{i,ℚ}/P_{i})^{gp} be any subset, so that Δ_{1} × Δ_{2} ⊂ Γ_{ℚ} = Γ_{1,ℚ} ⊕ Γ_{2,ℚ}. Then

ξ\underline{f}(R(B, Δ_{1} × Δ_{2})) = ξ\underline{f}(R(B, Δ_{1} × \{0\})).

Proof. (i): On the one hand, the multiplication law of A_{∞} induces by restriction A-linear maps

μ_{[γ]} : A_{[γ]} ⊗_{A} A_{[1/γ]} → gr_{0}A_{∞} = A (a ⊗ γλ) ⊗ (a′ ⊗ λ′/γ) ↦ aa′ · β(λλ′)

(notice that if γ, λ, λ′ ∈ P_{ℚ}^{gp} and we have both γλ ∈ gr_{[γ]}P_{ℚ} and λ′/γ ∈ gr_{[1/γ]}P_{ℚ}, then λλ′ ∈ gr_{0}P_{ℚ} = P). On the other hand, we have an obvious isomorphism of P -modules :

Q_{γ} := P^{gp} ∩ γ^{−1}P_{ℚ} →^{∼} gr_{[γ]}P_{ℚ}  λ ↦ γλ  for every γ ∈ P_{ℚ}^{gp}.

Especially, if Q_{γ} = Q_{δ} for some γ, δ ∈ P_{γ}^{gp}, we deduce an A-linear isomorphism

ω_{γ,δ} : A_{[γ]} →^{∼} A_{[δ]}  a ⊗ γλ ↦ a ⊗ δλ  for every a ∈ A and γλ ∈ gr_{[γ]}P_{ℚ}

which induces an A-linear isomorphism

ω_{γ,δ∗} : B_{[γ]}^{∨} →^{∼} B_{[δ]}^{∨} (φ : B → A_{[γ]}) ↦ (ω_{γ,δ} ∘ φ : B → A_{[δ]}).

Furthermore, if we have both Q_{γ} = Q_{δ} and Q_{1/γ} = Q_{1/δ} for some γ, δ ∈ P_{γ}^{gp}, a direct inspection yields a commutative diagram :

(17.4.11)  \xymatrix{A_{[γ]} ⊗_{A} A_{[1/γ]} \ar[rr]^{ω_{γ,δ}⊗_{A}ω_{1/γ,1/δ}} \ar[dr]_{μ_{[γ]}} &  & A_{[δ]} ⊗_{A} A_{[1/δ]} \ar@{-->}[dl]^{μ_{[δ]}} \\  & A.}

For such γ and δ, there follows a commutative diagram :

\xymatrix{B_{[γ]}^{∨} ⊗_{A} B_{[1/γ]}^{∨} \ar[rr]^{ω_{γ,δ∗}⊗_{A}ω_{1/γ,1/δ∗}} \ar@{-->}[dr] &  & B_{[δ]}^{∨} ⊗_{A} B_{[1/δ]}^{∨} \ar@{-->}[dl] \\  & (B ⊗_{A} B)^{∨}}

whose downward arrows are the restrictions of ξ\underline{f} : the details shall be left to the reader. Pick a section of the projection P_{ℚ}^{gp} → Γ_{ℚ} :

Γ_{ℚ} → P_{ℚ}^{gp} ⊂ P_{ℝ}^{gp} [γ] ↦ [γ]^{∗}

whose image is contained in a bounded subset of the finite dimensional ℝ-vector space P_{ℝ}^{gp}. Summing up, we see that if Q_{[γ]^{∗}} = Q_{[δ]^{∗}} and Q_{[1/γ]^{∗}} = Q_{[1/δ]^{∗}}, then ξ\underline{f}(B_{[γ]}^{∨} ⊗ B_{[1/γ]}^{∨}) = ξ\underline{f}(B_{[δ]}^{∨} ⊗ B_{[1/δ]}^{∨}). To conclude, it suffices now to invoke proposition 6.3.35(i).

(ii): For every (γ_{1}, γ_{2}) ∈ P_{ℚ}^{gp}, let [γ_{i}] ∈ Γ_{i,ℚ} be the class of γ_{i} for i = 1, 2, and [γ_{1}, γ_{2}] ∈ Γ_{ℚ} the class of (γ_{1}, γ_{2}); also, for each x ∈ ℚ, denote by x^{†} the smallest element of (x + ℤ) ∩ ℚ_{+}. It is easily seen that

gr_{[γ_{1},γ_{2}]}P_{ℚ} = (gr_{[γ_{1}]}P_{1,ℚ}) × (γ_{2}^{†}P_{2}) where γ_{2}^{†} := (γ_{2,1}^{†}, …, γ_{2,r}^{†})

whence a natural identification :

(17.4.12)  B_{[γ_{1},0]}^{∨} ⊗_{ℤ[P]} ℤ[γ_{2}^{†}P_{2}] →^{∼} B_{[γ_{1},γ_{2}]}^{∨}.

Notice that for every x ∈ ℚ the rational number x^{†} + (−x)^{†} equals 0 if x ∈ ℤ, and otherwise it equals 1. Especially :

γ_{2}^{†} + (−γ_{2})^{†} ∈ P_{2} for every γ_{2} ∈ P_{2,ℚ}^{gp}.

Therefore the multiplication law of ℤ[P_{2,ℚ}] induces by restriction an A-linear map :

μ_{γ_{2}} : ℤ[γ_{2}^{†}P_{2}] ⊗_{ℤ[P]} ℤ[(−γ_{2})^{†}P_{2}] → ℤ[P_{2}].

We then get a commutative diagram

\xymatrix{(B_{[γ_{1},0]}^{∨} ⊗_{ℤ[P_{2}]} ℤ[γ_{2}^{†}P_{2}]) ⊗_{A} (B_{[1/γ_{1},0]}^{∨} ⊗_{ℤ[P_{2}]} ℤ[(−γ_{2})^{†}P_{2}]) \ar[r] \ar[d]_{ξ_{\underline{f},[γ_{1},0]}⊗_{ℤ[P_{2}]}ℤ[γ_{2}^{†}P_{2}]⊗_{ℤ[P_{2}]}ℤ[(−γ_{2})^{†}P_{2}]} & B_{[γ_{1},γ_{2}]}^{∨} ⊗_{A} B_{[1/γ_{1},−γ_{2}]}^{∨} \ar[d]^{ξ_{\underline{f},[γ_{1},γ_{2}]}} \\ (B ⊗_{A} B)^{∨} ⊗_{ℤ[P_{2}]} ℤ[γ_{2}^{†}P_{2}] ⊗_{ℤ[P_{2}]} ℤ[(−γ_{2})^{†}P_{2}] \ar[r] & (B ⊗_{A} B)^{∨}}

where ξ_{\underline{f},[γ_{1},0]} : B_{[γ_{1},0]}^{∨} ⊗_{A} B_{[1/γ_{1},0]}^{∨} → (B ⊗_{A} B)^{∨} is the restriction of ξ\underline{f}, and likewise for ξ_{\underline{f},[γ_{1},γ_{2}]}. Also, the top horizontal arrow is the isomorphism induced by (17.4.12), and the bottom horizontal arrow is (B ⊗_{A} B)^{∨} ⊗_{ℤ[P_{2}]} μ_{γ_{2}}. Summing up, we conclude that :

ξ\underline{f}(B_{[γ_{1},γ_{2}]}^{∨} ⊗_{A} B_{[1/γ_{1},−γ_{2}]}^{∨}) ⊂ ξ\underline{f}(B_{[γ_{1},0]}^{∨} ⊗_{A} B_{[1/γ_{1},0]}^{∨}) for every [γ_{1}, γ_{2}] ∈ P_{ℚ}^{gp}

whence the assertion. □

Example 17.4.13. Let V be a noetherian regular ring, P a fine and saturated monoid such that P^{gp} is torsion-free, ψ : V [P] → A a smooth ring homomorphism, and β : P → A the composition of ψ with the natural inclusion map α : P → V [P]. Set S := Spec V, X_{0} := Spec V [P] and X := Spec A; then α is a chart for a log structure 𝒫_{0} on the Zariski site of X_{0}, and the induced morphism of log schemes (X_{0}, 𝒫_{0}) → (S, 𝒪_{S}^{×}) is smooth, by virtue of proposition 12.3.34. Since (S, 𝒪_{S}^{×}) is trivially a regular log scheme, it follows that the same holds for (X_{0}, 𝒫_{0}), by theorem 12.5.28; then, again by theorem 12.5.28and corollary 12.3.27, the log scheme (X, 𝒫) := X ×_{X_{0}} (X_{0}, 𝒫_{0}) is regular, and β is a chart for 𝒫. Summing up, this shows that the datum (A, P, β) is a regular log ring.

Lemma 17.4.14. Let A be a domain, B an A-algebra of finite type, C a B-algebra of finite type, M a B-module of finite type, and N a C-module of finite type. Set also K := Frac(A).

(i) There exists f ∈ A \ {0} such that the following holds for every A-algebra A′. Set B′ := A′ ⊗_{A} B, C′ := A′ ⊗_{A} C, M′ := B′ ⊗_{B} M and N′ := C′ ⊗_{C} N; then the natural map

A′ ⊗_{A} Hom_{B}(M, N)_{f} → Hom_{B′}(M′, N′)_{f}

is an isomorphism.

(ii) For every B-linear map φ : M → N there exists f ∈ A \ {0} such that (Coker φ)_{f} is a free A-module.

(iii) In the situation of (ii), let x ∈ N. Then x ∈ Im (K ⊗_{A} φ) if and only if there exists a dense subset U ⊂ Spec A such that 1 ⊗ x ∈ Im(κ(𝔭) ⊗_{A} φ) for every 𝔭 ∈ U.

Proof. (i): Pick n ∈ ℕ and a B-linear surjection ψ : B^{⊕n} → M; let also M′ ⊂ B^{⊕n} be a finitely generated B-submodule such that K ⊗_{A} M′ = Ker (K ⊗_{A} ψ). Set M′′ := B^{⊕n}/M′, and denote by \overline{ψ} : M′′ → M the B-linear surjection induced by ψ. By [65, Ch.IV, Lemma 8.9.4.1], there exists f ∈ A \ {0} such that the A_{f}-module M′′_{f} is flat, and by construction Ker(ψ_{f} : M′′_{f} → M_{f}) is a torsion A-module; then Ker ψ_{f} = 0, hence M_{f} is a finitely presented B_{f}-module. We may thus replace A, B, C, M, N by A_{f}, B_{f}, C_{f}, M_{f}, N_{f}, and assume that M is a B-module of finite presentation. Now, let

B^{⊕m} ⟶^{φ} B^{⊕n} → M

be a finite presentation of M; there follow short exact sequences of B-modules :

Σ : 0 → Hom_{B}(M, N) → N^{⊕n} → T := Im(φ^{∨} ⊗_{B} N) → 0
Σ′ : 0 → T → N^{⊕m} → T′ := Coker(φ^{∨} ⊗_{B} N) → 0

where φ^{∨} : B^{⊕n} → B^{⊕m} is the transpose of φ. Notice that T and T′ are C-modules of finite type. By invoking again [65, Ch.IV, Lemma 8.9.4.1], we find f ∈ A \ {0} such that T_{f} and T′_{f} are flat A_{f}-modules; after replacing again A, B, C, M, N and φ by their respective localizations, we may therefore assume that T and T′ are flat A-modules. In this case, the induced sequences A′⊗_{A}Σ and A′⊗_{A}Σ′ are again short exact, for every A-algebra A′, whence a left exact sequence

0 → A′ ⊗_{A} Hom_{B}(M, N) → N^{′⊕n} ⟶^{φ^{′∨}⊗_{B′}N′} N^{′⊕m} with φ′ := φ ⊗_{B} B′.

But we have a natural identification : Ker(φ^{′∨} ⊗_{B′} N′) →^{∼} Hom_{B′}(M′, N′), and the resulting isomorphism A′ ⊗_{A} Hom_{B}(M, N) →^{∼} Hom_{B′}(M′, N′) is the natural map of (i).

(ii): Arguing as in the proof of (i), we reduce easily to the case where B and C are finitely presented A-algebras, and M (resp. N) is a finitely presented B-module (resp. C-module). Then we may find a ℤ-subalgebra of finite type A_{0} ⊂ A, an A_{0}-algebra of finite type B_{0}, a B_{0}-algebra of finite type C_{0}, a B_{0}-module M_{0} of finite type, a C_{0}-module of finite type N_{0}, and a B_{0}-linear map φ_{0} : M_{0} → N_{0} with isomorphisms

A ⊗_{A_{0}} B_{0} →^{∼} B  A ⊗_{A_{0}} C_{0} →^{∼} C  B ⊗_{B_{0}} M_{0} →^{∼} M  C ⊗_{C_{0}} N_{0} →^{∼} N.

which identify B ⊗_{B_{0}} φ_{0} with φ. It suffices then to exhibit f ∈ A_{0} such that (Coker φ_{0})_{f} is a free A_{0,f}-module; the latter follows easily from [126, Th.24.1].

(iii): If x ∈ Im (K ⊗_{A} φ), we have x ∈ Im φ_{f} for some f ∈ A \ {0}, and then x ⊗ 1 ∈ Im(κ(𝔭) ⊗_{A} φ) for every 𝔭 ∈ Spec A_{f}. Conversely, suppose that U ⊂ Spec A is a dense subset such that x ⊗ 1 ∈ Im(κ(𝔭) ⊗_{A} φ) for every 𝔭 ∈ U. We pick f ∈ A \ {0} as in (ii), so that (Coker φ)_{f} is a free A-module. Clearly U ∩ Spec A_{f} is still dense in Spec A_{f}, hence we may replace A, B, C, M, N and φ by A_{f}, B_{f}, C_{f}, M_{f}, N_{f} and φ_{f}, and assume that Coker φ is a free A-module, say with basis (e_{i} | i ∈ I). Suppose now that x ∉ Im (K ⊗_{A} φ), so that the image \overline{x} ∈ Coker φ of x does not vanish. Then there exists a finite subset I_{0} ⊂ I and a system (f_{i} | i ∈ I_{0}) of non-zero elements of A, with \overline{x} = ∑_{i∈I_{0}} f_{i}e_{i}. Pick j ∈ I_{0}, and any 𝔭 ∈ U ∩Spec A_{f_{j}}; then Coker (κ(𝔭)⊗_{A} φ) is a free κ(𝔭)-vector space with basis (1⊗e_{i} | i ∈ I), and 1⊗x = ∑_{i∈I_{0}} \overline{f}_{i} ⊗e_{i}, where \overline{f}_{i} ∈ κ(𝔭) denotes the image of f_{i}, for every i ∈ I. Especially, \overline{f}_{j} ≠ 0, whence 1 ⊗ x ≠ 0, a contradiction. □

17.4.15. Let (P, +, 0) be a fine and saturated monoid such that P^{gp} is torsion-free, and fix a Banach norm ∥ · ∥ on the finite dimensional ℝ-vector space P_{ℝ}^{gp} := ℝ ⊗_{ℤ} P^{gp}. Let also

P_{ℝ}^{gp}(ρ) := {γ ∈ P_{ℝ}^{gp} | ∥γ∥ ≤ ρ} for all ρ ∈ ℝ_{+}.

Define also P^{(∞)} and P_{ℚ} as in (17.4.3), as well as the polyhedral cone P_{ℝ} := ℝ_{+} ⊗_{ℤ} P ⊂ P_{ℝ}^{gp}. Moreover, for every m ∈ ℕ \ {0}, let

P_{(m)} := {γ ∈ P_{ℚ} | mγ ∈ P^{(∞)}}.

Lemma 17.4.16. With the notation of (17.4.15), there exists an integer m > 0 with (p, m) = 1 such that the following holds. For every δ > 0 there exists a real number ε > 0 such that for every β_{1}, β_{2} ∈ P_{ℝ}^{gp} with ∥β_{1} + β_{2}∥ < ε we may find β′_{1}, β′_{2} ∈ P_{(m)}^{gp} with :

(i) β′_{1} + β′_{2} = 0 (ii) ∥β′_{i} − β_{i}∥ < δ for i = 1, 2 (iii) (P_{ℝ} − β_{i}) ∩ P^{gp} ⊂ (P_{ℝ} − β′_{i}) ∩ P^{gp} for i = 1, 2.

Proof. Since P^{(∞)gp} is p-divisible, P_{(m)}^{gp} = P_{(pm)}^{gp} for every integer m > 0, so we may always arrange that (p, m) = 1, if all the other conditions are already fulfilled. Moreover, let ρ > 0 such that every class of P_{ℝ}^{gp}/P^{gp} admits a representative β ∈ P_{ℝ}^{gp} with ∥β∥ ≤ ρ. Then it is easily seen that, after replacing β_{i} by β_{i} + (−1)^{i}γ for i = 1, 2, with a suitable γ ∈ P^{gp}, we may assume that β_{1} ∈ P_{ℝ}^{gp}(ρ). Now, for every γ ∈ P_{ℝ}^{gp}, let \overline{Ω}(γ) be the topological closure of the subset Ω(P_{ℝ}, P^{gp} ∩ (P_{ℝ} − γ)) in P_{ℝ}^{gp} (notation of (6.3.34)); in view of condition (i) and of proposition 6.3.35(v), condition (iii) is implied by :

(17.4.17)  β′_{1} ∈ \overline{Ω}(β_{1}) ∩ (−\overline{Ω}(β_{2})).

Moreover, suppose that

(17.4.18)  ∥β′_{1} − β_{1}∥ ≤ δ/2.

Then

∥β′_{2} − β_{2}∥ = ∥β′_{2} + β_{1} − β_{1} − β_{2}∥ ≤ ∥ − β′_{1} + β_{1}∥ + ∥β_{1} + β_{2}∥ < δ/2 + ε.

Hence condition (ii) will hold, provided ε ≤ δ/2. Thus, we have to exhibit m > 0 and ε > 0 such that, for every β_{1}, β_{2} ∈ P_{ℝ}^{gp} with ∥β_{1}∥ ≤ ρ and ∥β_{1} + β_{2}∥ < ε, there exists β′_{1} ∈ P_{(m)}^{gp} fulfilling (17.4.17) and (17.4.18). Then, by proposition 6.3.35(i) and lemma 6.3.42, it suffices to find β′_{1} ∈ P_{ℝ}^{gp} fulfilling these two latter conditions. By way of contradiction, suppose that such β′_{1} cannot always be found : this means that there exists a sequence \underline{β} := ((β_{1,k}, β_{2,k}) | k ∈ ℕ) of pairs of elements in P_{ℝ}^{gp}, with β_{1,k} ∈ P_{ℝ}^{gp}(ρ) for every k ∈ ℕ, and such that

(a) ∥β_{1,k} + β_{2,k}∥ < 2^{−k} for every k ∈ ℕ

(b) \overline{Ω}(β_{1,k}) ∩ (−\overline{Ω}(β_{2,k})) ∩ (P_{ℝ}^{gp}(δ/2) + β_{1,k}) = ∅ for every k ∈ ℕ.

However, by proposition 6.3.35(i), after replacing \underline{β} by a subsequence we may assume that both \overline{Ω}(β_{1,k}) and \overline{Ω}(β_{2,k}) are independent of k. Since P_{ℝ}^{gp}(ρ) is a compact subset, we may also assume that the sequence (β_{1,k} | k ∈ ℕ) converges to an element β′_{1} ∈ P_{ℝ}^{gp}(ρ); then clearly (β_{2,k} | k ∈ ℕ) converges to β′_{2} := −β′_{1}. Since β_{i,k} ∈ \overline{Ω}(β_{i,k}) for i = 1, 2 and every k ∈ ℕ, we also have β′_{i} ∈ \overline{Ω}(β_{i,k}) for i = 1, 2. Lastly, we have ∥β′_{1} − β_{1,k}∥ < δ/2 for every sufficiently large k ∈ ℕ; this contradicts (b), and the claim follows. □

Theorem 17.4.19. In the situation of (17.4.5), suppose that (A, P, β) is regular. We have : (i) e_{f} ∈ ϑ\underline{f}(R(B, Γ_{ℚ})). (ii) Let J ⊂ A be the radical ideal such that Spec A/J = X \ U_{f}. Then Je_{f} ⊂ ϑ\underline{f}(R(B, Γ)).

Proof. For every datum \underline{f} := (f, β, P) as in (17.4.5), let X(\underline{f}) := Γ(U_{f}, ℬ ⊗_{𝒪_{X}} ℬ); denote also by Y(\underline{f}) ⊂ X(\underline{f}) the image of ϑ\underline{f}, and set Z(\underline{f}) := Ae_{f} ⊂ X(\underline{f}). We notice :

Claim 17.4.20. Let A′ be a noetherian ring, g : A → A′ a flat ring homomorphism; set X′ := Spec A′, f′ := A′ ⊗_{A} f and \underline{f}′ := (f′, P, g ∘ β). Then U_{f′} = X′ ×_{X} U_{f}, and we have a natural isomorphism

A′ ⊗_{A} X(\underline{f}) →^{∼} X(\underline{f}′)

that identifies Y(\underline{f}′) and Z(\underline{f}′) respectively with A′ ⊗_{A} Y(\underline{f}) and A′ ⊗_{A} Z(\underline{f}).

Proof of the claim. The first assertion follows from [66, Ch.IV, Prop.17.7.1]. Next, set B′ := A′ ⊗_{A} B; since B is an A-module of finite type, B′ is an A′-module of finite presentation. Hence, the natural map A′ ⊗_{A} B^{∨} → B^{′∨} := Hom_{A′}(B′, A′) is an isomorphism (details left to the reader). Then the second assertion follows by a direct inspection of the constructions. ♢

Now, assertion (i) comes down to the inclusion Z(\underline{f}) ⊂ Y(\underline{f}), and it suffices to check that Z(\underline{f})_{𝔭} ⊂ Y(\underline{f})_{𝔭} in X(\underline{f})_{𝔭}, for every 𝔭 ∈ Spec A. Let j_{(𝔭)} : A → A_{𝔭} be the localization map, set f_{𝔭} := A_{𝔭} ⊗_{A} f : A_{𝔭} → B_{𝔭}, and \underline{f}_{𝔭} := (f_{𝔭}, P, β ∘ j_{(𝔭)}); by claim 17.4.20, we are then reduced to checking that Z(\underline{f}_{𝔭}) ⊂ Y(\underline{f}_{𝔭}) in X(\underline{f}_{𝔭}), for every such 𝔭. Thus, in order to show (i), we may assume that (A, 𝔪) is a local noetherian ring. Likewise we may reduce assertion (ii) to the case where (A, 𝔪) is local. Next, let A^{∧} be the 𝔪-adic completion of A; since the completion map j : A → A^{∧} is faithfully flat, we are reduced to checking that A^{∧} ⊗_{A} Z(\underline{f}) ⊂ A^{∧} ⊗_{A} Y(\underline{f}) in A^{∧} ⊗_{A} X(\underline{f}). Moreover, the log structure attached to the chart j ∘ β : P → A^{∧} is still regular (theorems 12.5.10 and 12.5.31); after invoking again claim 17.4.20, we may thus assume that (A, 𝔪) is a complete noetherian local ring in order to prove (i), and a similar argument reduces the proof of (ii) as well to the complete case.

Next, let 𝔭 := β^{−1}(A^{×}); then β extends uniquely to a morphism of monoids β′ : P_{𝔭} → A (notation of remark 6.1.15(i)); moreover, the log structures on the Zariski site of X induced by β and by β′ are naturally isomorphic, so the datum \underline{f}′ := (f, P_{𝔭}, β′) still fulfills the conditions of (17.4.5) and (A, P_{𝔭}, β′) is regular. Moreover, (P_{𝔭})^{gp} = P^{gp} and (P_{𝔭})_{ℚ}^{gp} = P_{ℚ}^{gp}, hence (P_{𝔭})_{ℚ} is still naturally Γ-graded. Furthermore, for every γ ∈ P_{ℚ}^{gp} we have

gr_{[γ]}((P_{𝔭})_{ℚ}) = (gr_{[γ]}(P_{ℚ}))_{𝔭} whence : A ⊗_{ℤ[P_{𝔭}]} ℤ[gr_{[γ]}((P_{𝔭})_{ℚ})] = A_{[γ]}.

Hence, the A-modules X(\underline{f}′), Y(\underline{f}′), Z(\underline{f}′) are naturally identified with X(\underline{f}), Y(\underline{f}), Z(\underline{f}). Thus, in order to prove (i), it suffices to show that Z(\underline{f}′) ⊂ Y(\underline{f}′); after replacing \underline{f} by \underline{f}′, we may therefore assume that the chart β is local at the closed point of X (see definition 12.1.17(vi)). Arguing likewise we reduce as well the proof of (ii) to the case where β is local.

Recall now that P admits a decomposition P →^{∼} Q × P^{×}, where Q is a fine, sharp and saturated monoid, and P^{×} ⊂ P is the abelian group of invertible elements of P (lemma 6.2.10). Let α : Q → A be the restriction of β; by direct inspection, we see that the inclusion Q → P induces an isomorphism between the log structures on Spec A associated with the charts (P, β) and (Q, α) : see (12.1.6); hence, the datum \underline{f}′′ := (f, Q, α) fulfills again the conditions of (17.4), and (A, Q, α) is regular. In light of lemma 17.4.10(ii), we may then replace \underline{f} by \underline{f}′′, and thus assume that β is a sharp chart at the closed point of X (see definition 12.1.17(vi)).

After these preliminaries, we may then assume, as in (17.3.4), that there exists r ∈ ℕ, a coefficient ring (V, 𝔪_{V}) of A, and a surjective ring homomorphism

φ : V [[P × ℕ^{⊕r}]] → A

and β is the composition of φ with the natural inclusion map P → V [[P × ℕ^{⊕r}]].

Claim 17.4.21. The theorem holds if A contains a field of positive characteristic.

Proof of the claim. In this case, V is the residue field of A, and φ is an isomorphism. The induced morphism of monoids β′ : P × ℕ^{⊕r} → V [[P × ℕ^{⊕r}]] → A is the chart for another regular log structure on the Zariski site of X (theorems 12.5.10and 12.5.31); in view of lemma 17.4.10(ii) we may then replace β by β′, and assume that A = V [[P]] and 𝔪 = 𝔪_{P} · A (with 𝔪_{P} := P \ {1}). Next, let k be an algebraic closure of V; the induced map V [P] → k[P] is faithfully flat, so the same holds for its completion A → k[[P]] ([126, Th.22.4]); moreover, the inclusion map P → k[[P]] is again a chart for a regular log structure on the Zariski site of Spec k[[P]]. In view of claim 17.4.20, we may then replace V by k and B by k[[P]] ⊗_{A} B, and assume that A = k[[P]]. In this case, notice that, with the notation of (9.8.11), we have :

A^{perf} = Γ ×_{Γ_{ℚ}} A_{∞}.

Let now h ∈ J. The isomorphism (9.8.12) yields a natural identification of A^{perf}-algebras :

Γ ×_{Γ_{ℚ}} B_{∞}[h^{−1}] →^{∼} B^{perf}[h^{−1}]

and then remark 17.4.1shows that h^{1/p^{n}}e_{f} lies in the image of the induced map

(17.4.22) B^{perf} ⊗_{A^{perf}} B^{perf} = (B ⊗_{A} B)^{perf} → B_{∞} ⊗_{A_{∞}} B_{∞}[h^{−1}] → Γ(U_{f}, ℬ_{∞} ⊗_{𝒪_{X∞}} ℬ_{∞})

for every n ∈ ℕ. On the other hand, B^{perf} is an integral A^{perf}-algebra, and A^{perf} is a normal domain, hence the trace map of the finite étale map A^{perf}[h^{−1}] → B^{perf}[h^{−1}] restricts to an A^{perf}-linear map B^{perf} → A^{perf} (lemma 14.3.15(i)) which in turns induces an A^{perf}-linear map

ω_{B^{perf}/A^{perf}} : B^{perf} → Hom_{A^{perf}}(B^{perf}, A^{perf}) → B_{∞}^{∨}

such that

A_{∞}[h^{−1}] ⊗_{A^{perf}} ω_{B^{perf}/A^{perf}} = Γ(Spec A[h^{−1}], ω_{ℬ_{∞}/𝒪_{X∞}} ).

A simple inspection then shows that (17.4.22)is the composition of ω_{B^{perf}/A^{perf}} ⊗_{A^{perf}} ω_{B^{perf}/A^{perf}} with (17.4.6). Summing up, this proves that h^{1/p^{n}}e_{f} lies in the image of the map (17.4.6) for every n ∈ ℕ, and especially, he_{f} ∈ ϑ\underline{f}(R(B, Γ)), whence assertion (ii).

In order to show (i), say that h = ∑_{γ∈P} h_{γ} · γ for a system (h_{γ} | γ ∈ P) of elements of k, and pick γ_{0} ∈ P with h_{γ_{0}} ≠ 0. According to claim 17.3.7, there exists n ∈ ℕ such that gr_{[γ_{0}^{1/p^{n}}]}P_{ℚ} = γ_{0}^{1/p^{n}}P; then we have an A-linear isomorphism :

(17.4.23) X(\underline{f}) →^{∼} gr_{[γ_{0}^{1/p^{n}}]}Γ(U_{f}, ℬ_{∞} ⊗_{𝒪_{X∞}} ℬ_{∞}) = X(\underline{f}) ⊗_{A} A_{[γ_{0}^{1/p^{n}}]}  x ↦ x ⊗ γ_{0}^{1/p^{n}}.

We have already remarked that h^{1/p^{n}}e_{f} lies in the image of (17.4.6); let us write h^{1/p^{n}} = ∑_{[γ]∈Γ} h_{[γ]}, with h_{[γ]} ∈ A_{[γ]} for every [γ] ∈ Γ; then h^{1/p^{n}} = ∑_{γ∈P} h_{γ}^{1/p^{n}} · γ^{1/p^{n}}, and h_{[γ_{0}^{1/p^{n}}]} =

h_{γ_{0}}^{1/p^{n}} · γ_{0}^{1/p^{n}} · a for some a ∈ 1 + 𝔪. Hence, h_{[γ_{0}^{1/p^{n}}]}e_{f} is in the image of gr_{[γ_{0}^{1/p^{n}}]}(B_{∞}^{∨} ⊗_{A_{∞}} B_{∞}^{∨}),

and so the same holds for e_{f} ⊗ γ_{0}^{1/p^{n}}. Thus, there exists a finite subset Σ ⊂ Γ and an element of ⨁_{[γ]∈Σ} B_{[γ]}^{∨} ⊗_{A} B_{[γ_{0}^{1/p^{n}}/γ]}^{∨} whose image under the map (17.4.6) equals e_{f} ⊗ γ_{0}^{1/p^{n}}. Then, lemma

17.4.16yields for every [γ] ∈ Σ an element [γ′] ∈ Σ_{ℚ} such that :

γ^{−1}P_{ℚ} ∩ P^{gp} ⊂ γ^{′−1}P_{ℚ} ∩ P^{gp} and (γ/γ_{0}^{1/p^{n}})P_{ℚ} ∩ P^{gp} ⊂ γ′P_{ℚ} ∩ P^{gp}.

With Σ′ := {[γ′] | [γ] ∈ Σ}, we finally deduce a commutative diagram of A-modules :

\xymatrix{⨁_{[γ]∈Σ} B_{[γ]}^{∨} ⊗_{A} B_{[γ_{0}^{1/p^{n}}/γ]}^{∨} \ar[r] \ar[d] & X(\underline{f}) ⊗_{A} A_{[γ_{0}^{1/p^{n}}]} \\ R(B, Σ′) \ar[r] & X(\underline{f}) \ar[u]}

whose right vertical arrow is the isomorphism (17.4.23) and whose bottom horizontal arrow is the restriction of ϑ\underline{f}; it follows easily that e_{f} ∈ ϑ\underline{f}(R(B, Σ′)), as required. ♢

Suppose next that A is a ℚ-algebra, so that V is a field of characteristic zero, and φ is an isomorphism. We pick a complete discrete valuation ring W with residue field V, and a surjective map of monoids π : ℕ^{⊕r′} → P; for s := r + r′ we get a surjective map of W - algebras W [ℕ^{⊕s}] → A_{0} := V [P × ℕ^{⊕r}] whose completion is a surjective map W [[ℕ^{⊕s}]] → A. According to [32, §3.6, Th.12], the completion map W [ℕ^{⊕s}] → W [[ℕ^{⊕s}]] is the colimit of a filtered system (R_{λ} | λ ∈ Λ) of smooth W [ℕ^{⊕s}]-algebras; set A_{λ} := R_{λ} ⊗_{W [ℕ^{⊕s}]} A_{0} for every λ ∈ Λ. Then A is the colimit of the filtered system of smooth A_{0}-algebras A_{•} := (A_{λ} | λ ∈ Λ). For every λ ∈ Λ, let X′_{λ} be the connected component of Spec A_{λ} containing the image of X, and let A′_{λ} be the quotient of A_{λ} with Spec A′_{λ} = X′_{λ}. It is easily seen that the colimit of the induced system A′_{•} := (A′_{λ} | λ ∈ Λ) is still A; after replacing A_{•} by A′_{•}, we may then assume that A_{λ} is a domain for every λ ∈ Λ. Next, after replacing Λ by a cofinal subset, we may assume that for every λ ∈ Λ there exists an open subset U_{λ} ⊂ X_{λ} such that U_{f} = X ×_{X_{λ}} U_{λ}, and U_{μ} = X_{μ}×_{X_{λ}} U_{λ} for every μ ∈ Λ with μ ≥ λ. We may then find λ ∈ Λ and a finite morphism of schemes φ_{λ} : Y_{λ} → X_{λ} with an isomorphism of X-schemes X ×_{X_{λ}} Y_{λ} →^{∼} Spec B. Then there exists μ ∈ Λ with μ ≥ λ such that U_{μ} ×_{X_{λ}} Y_{λ} is a finite étale U_{μ}-scheme; set Y_{μ} := X_{μ} ×_{X_{λ}} Y_{λ}, and B_{μ} := 𝒪_{Y_{μ}}(Y_{μ}). By construction, the morphism Y_{μ} → X_{μ} is surjective; since A_{μ} is a domain, it follows easily that the induced ring homomorphism f_{μ} : A_{μ} → B_{μ} is injective. Let β_{μ} : P → A_{0} → A_{μ} be the natural map; by example 17.4.13, the datum (A_{μ}, P, β_{μ}) is a regular log ring, and \underline{f}_{μ} := (f_{μ}, P, β_{μ}) is a datum as in (17.4.5). In light of remark 17.4.7(i), it then suffices to prove assertions (i) and (ii) for the map ϑ_{\underline{f}_{μ}}; the latter is covered by the following :

Claim 17.4.24. The theorem holds if there exists a field K of characteristic zero and a smooth ring homomorphism ψ : K[P] → A, such that β is the composition of ψ and the inclusion map P → K[P] (see example 17.4.13).

Proof of the claim. The field K is the colimit of the filtered system of its smooth ℤ-subalgebras (V_{λ} | λ ∈ Λ). Then we may find λ ∈ Λ, a smooth V_{λ}[P]-algebra A_{λ}, and a finite, injective and generically finite ring homomorphism f_{λ} : A_{λ} → B_{λ} with isomorphisms

K ⊗_{V_{λ}} A_{λ} →^{∼} A A ⊗_{A_{λ}} B_{λ} →^{∼} B

(details left to the reader). Let α : P → V_{λ}[P] → A_{λ} be the natural map; since V_{λ} is a regular noetherian ring, the datum (A_{λ}, P, α) is a regular log ring, by example 17.4.13, and in light of remark 17.4.7(i), it suffices to show assertions (i) and (ii) for the map ϑ_{\underline{f}_{λ}} corresponding to the datum \underline{f}_{λ} := (f_{λ}, P, α). Now, for every 𝔭 ∈ Spec V_{λ}, every V_{λ}-module M, let M(𝔭) := κ(𝔭) ⊗_{A_{λ}} M; also, for every homomorphism of V_{λ}-modules h : M → N let likewise h_{(𝔭)} := κ(𝔭) ⊗_{V_{λ}} h : M(𝔭) → N(𝔭); moreover, let α_{(𝔭)} : P → A_{λ}(𝔭) be the composition of α with the natural map A_{λ} → A_{λ}(𝔭). Let also a ∈ A_{λ} \ {0} such that Spec A_{λ}[a^{−1}] ⊂ U_{f_{λ}}. According to [65, Ch.IV, Prop.9.5.3], the set Z := {𝔭 ∈ Spec V_{λ} | Spec A_{λ}[a^{−1}](𝔭) is dense in Spec A_{λ}(𝔭)} is constructible in Spec V_{λ}; furthermore, Z contains the generic point of Spec V_{λ}, so it is dense in Spec V_{λ}. For every 𝔭 ∈ Z, the datum \underline{f}_{λ,(𝔭)} := (f_{λ,(𝔭)}, P, α_{(𝔭)}) fulfills therefore the conditions of (17.4), and moreover (A_{λ}(𝔭), P, α_{(𝔭)}) is still a regular log ring. Set X_{λ} := Spec A_{λ}, and denote by ℬ_{λ} the quasi-coherent 𝒪_{X_{λ}}-algebra arising from B_{λ}. By remark 17.4.7(i), for Δ either equal to Γ_{ℚ} or to Γ we have a commutative diagram :

\xymatrix{R(B_{λ}, Δ) \ar[r]^{ϑ′_{Δ}} \ar[d] & B_{λ} ⊗_{A_{λ}} B_{λ}[a^{−1}] \ar[d] \\ R(B_{λ}(𝔭, Δ)) \ar[r]^{ϑ_{Δ}^{′(𝔭)}} & B_{λ} ⊗_{A_{λ}} B_{λ}(𝔭)[a^{−1}]}

whose right vertical arrow maps e_{f_{λ}} to e_{f_{λ,(𝔭)}}, and where ϑ′_{Δ} is the composition of the restriction of ϑ_{\underline{f}_{λ}} to R(B, Δ) with the restriction map Γ(U_{f_{λ}}, ℬ_{λ} ⊗_{𝒪_{X_{λ}}} ℬ_{λ}) → B_{λ} ⊗_{A_{λ}} B_{λ}[a^{−1}], and likewise ϑ_{Δ}^{′(𝔭)} is deduced from ϑ_{\underline{f}_{λ},(𝔭)}. Lemma 17.4.10(i) yields a finite subset Σ ⊂ Δ such that

ϑ_{\underline{f}_{λ,(𝔭)}}(R(B(𝔭), Σ)) = ϑ_{\underline{f}_{λ,(𝔭)}}(R(B(𝔭), Δ)) for every 𝔭 ∈ Z.

On the other hand, in view of lemma 17.4.14(i), after replacing V_{λ} by a suitable localization V_{λ}[t^{−1}] (with t ≠ 0), and Z by Z ∩ Spec V [t^{−1}], we may assume that the natural maps

(B_{λ}^{∨})_{[γ]}(𝔭) → (B_{λ}(𝔭))_{[γ]}^{∨} (B_{λ}^{∨})_{[1/γ]}(𝔭) → (B_{λ}(𝔭))_{[1/γ]}^{∨}

are isomorphisms, for every 𝔭 ∈ Z and every [γ] ∈ Σ. Summing up, we may assume that the image of ϑ_{Δ}^{′(𝔭)} equals the image of ϑ′_{Δ,(𝔭)}, for every 𝔭 ∈ Z. Lastly, recall that the set Z′ of all 𝔭 ∈ Spec V_{λ} such that κ(𝔭) is a finite field is dense in the constructible topology of Spec V_{λ}, since V_{λ} is a ℤ-algebra of finite type ([65, Ch.IV, Cor.10.4.6]). Hence, Z ∩ Z′ is a dense subset of Spec V_{λ}, and by claim 17.4.21, the image of ϑ_{Γ_{ℚ}}^{′(𝔭)} (resp. of ϑ_{Γ}^{′(𝔭)}) contains e_{f_{λ,(𝔭)}} = 1 ⊗ e_{f_{λ}} (resp. be_{f_{λ,(𝔭)}} = 1 ⊗ be_{f_{λ}}) for every 𝔭 ∈ Z ∩ Z′ (resp. and every b ∈ J). By lemma 17.4.14(iii) it follows that the image of ϑ′_{Γ_{ℚ}} contains e_{f} (resp. that the image of ϑ′_{Γ} contains be_{f} for every b ∈ J). The claim is an immediate consequence. ♢

It remains to deal with the case where A does not contain a field, hence A = V [[P]]/(h), for a discrete valuation ring (V, 𝔪_{V}) whose residue field k has positive characteristic, such that 𝔪_{V} = pV, and with some h := ∑_{γ∈P} h_{γ} · γ such that h_{0} ∈ pV \ p^{2}V. Arguing as in (17.3.4), and taking into account claim 17.4.20, we may then further reduce to the case where k is algebraically closed. In this situation, let g ∈ J \ {0}, and set

A_{Γ} := Γ ×_{Γ_{ℚ}} A_{∞} and A′_{Γ} := A_{Γ}[X^{1/p^{∞}}]/(X − g) = A_{Γ}[g^{1/p^{∞}}].

We get a basic setup (A_{Γ}, 𝔫), where 𝔫 ⊂ A_{Γ} is the ideal generated by (g^{δ} | δ ∈ ℕ[1/p] \ {0}). Moreover, let D be the p-root closure of A′_{Γ} in A′_{Γ}[1/p], and D^{∧} the p-adic completion of D; also, set D_{1}^{∧} := (D^{∧})_{∗}^{a}, the ring of almost elements of the (A_{Γ}, 𝔫)^{a}-algebra (D^{∧})^{a}, and denote by B′_{1} the integral closure of D_{1}^{∧} in B′ := D^{∧} ⊗_{A} B[1/g]. Pick N ∈ ℕ and e ∈ B ⊗_{A} B whose image in B ⊗_{A} B[1/g] equals g^{N}e_{f}, and let e′ ∈ B′_{1} ˆ⊗_{D^{∧}}B′_{1} be the image of e.

Claim 17.4.25. For every δ ∈ ℕ[1/p] \ {0} we have g^{δ}e′ ∈ g^{N}(B′_{1} ˆ⊗_{D^{∧}}B′_{1}).

Proof of the claim. For every n ∈ ℕ, the (D/p^{n}D)^{a}-algebra (B′_{1}/p^{n}B′_{1})^{a} is étale of finite rank, and we let ε_{n} ∈ (B′_{1}/p^{n}B′_{1} ⊗_{D} B′_{1}/p^{n}B′)_{∗}^{a} be the corresponding diagonal idempotent. The system (ε_{n} | n ∈ ℕ) corresponds to a unique element

ε ∈ (B′_{1} ˆ⊗_{D^{∧}}B′_{1})_{∗}^{a}

where B′_{1} ˆ⊗_{D^{∧}}B′_{1} denotes the p-adic completion of B′_{1} ⊗_{D^{∧}} B′_{1}. Let μ : B′_{1} ⊗_{D^{∧}} B′_{1} → B′_{1} be the multiplication law of B′_{1}, and μ̂ the p-adic completion of μ; by construction, we see that :

(17.4.26) μ̂(ε) = 1  and ε · (1 ⊗ x − x ⊗ 1) = 0 for every x ∈ B′_{1} = (B′_{1})_{∗}^{a}.

By corollary 16.9.52(ii), the completion map B′_{1} ⊗_{D^{∧}} B′_{1} → B′_{1} ˆ⊗_{D^{∧}}B′_{1} induces an isomorphism

B ⊗_{A} B ⊗_{A} D^{∧}[1/g] →^{∼} B′_{1} ⊗_{D^{∧}} B′_{1}[1/g] →^{∼} B′_{1} ˆ⊗_{D^{∧}}B′_{1}[1/g]

and it follows easily from (17.4.26) that the image of e_{f} ⊗ 1 in B′_{1} ˆ⊗_{D^{∧}}B′_{1}[1/g] agrees with the image of ε. But g is a regular element of (B′_{1} ˆ⊗_{D^{∧}}B′_{1})_{∗}^{a}, due to corollary 16.9.52(ii), hence the image of e in (B′_{1} ˆ⊗_{D^{∧}}B′_{1})_{∗}^{a} agrees with g^{N}ε, whence the claim. ♢

Recall that D_{1}^{∧} is integrally closed in D^{∧}[1/g] (theorem 16.9.42(iv)), and A is a normal domain; then the trace maps for the étale ring homomorphisms D^{∧}[1/g] → B′ and A[1/g] → B[1/g] restrict to a D_{1}^{∧}-linear map and respectively an A-linear map

tr_{B′_{1}/D_{1}^{∧}} : B′_{1} → D_{1}^{∧} tr_{B/A} : B → A

(lemma 14.3.15(i)) whence an A-linear map and a D_{1}^{∧}-linear map

τ : B → B^{∨} := Hom_{A}(B, A) b ↦ (b′ ↦ tr_{B/A}(bb′))
τ′ : B′_{1} → Hom_{A}(B, D_{1}^{∧})  b ↦ (b′ ↦ tr_{B′_{1}/D_{1}^{∧}}(bb′)).

Let also i : A → A_{Γ} and i_{D} : D^{∧} → D_{1}^{∧} be the natural maps; since i_{D}^{a} is an isomorphism of A_{Γ}^{a}-modules, we get by adjunction a unique A_{Γ}-linear map j_{D} : 𝔫̃ ⊗_{A_{Γ}} D_{1}^{∧} → D^{∧} such that i_{D} ∘ j_{D} = μ ⊗_{A_{Γ}} D_{1}^{∧}, where as usual 𝔫̃ := 𝔫 ⊗_{A_{Γ}} 𝔫, and μ : 𝔫̃ → A is the multiplication map : a ⊗ a′ ↦ aa′. Then, for every δ, δ′ ∈ ℕ[1/p] \ {0} we let j_{δ+δ′} : D_{1}^{∧} → D^{∧} be the A_{Γ}-linear map such that j_{δ+δ′}(d) := j_{D}(g^{δ} ⊗ g^{δ′} ⊗ d) for every d ∈ D_{1}^{∧}; it is easily seen that this map depends only on the sum δ + δ′. Hence :

i_{D} ∘ j_{δ} = g^{δ} · 1_{D_{1}^{∧}}  and j_{δ+δ′} = g^{δ} · j_{δ′}  for every δ, δ′ ∈ ℕ[1/p] \ {0}

and moreover for every such δ we get a commutative diagram of A_{Γ}-modules :

\xymatrix{A_{Γ} \ar[r]^{g^{δ}·1_{A_{Γ}}} \ar[d] & A_{Γ} \ar[d] \\ D_{1}^{∧} \ar[r]^{j_{δ}} & D^{∧}}

whose vertical arrows are the structure maps of the A_{Γ}-algebras D_{1}^{∧} and D^{∧}. Set

B_{Γ}^{∨} := Hom_{A}(B, A_{Γ})  B_{1}^{′∨} := Hom_{A}(B, D_{1}^{∧})  C := Hom_{A}(B, D^{∧})

and let i_{∗} : B^{∨} → B_{Γ}^{∨} (resp. j_{δ∗} : B_{1}^{′∨} → C) be the A-linear (resp. A_{Γ}-linear) map such that i_{∗}(φ) := i ∘ φ for every A-linear map φ : B → A (resp. j_{δ∗}(φ) := j_{δ} ∘ φ for every A-linear map φ : B → D_{1}^{∧}). To ease notation, set as well

B_{Γ,i}^{∨} := Hom_{A}(B, A_{Γ}/p^{i}A_{Γ})  and C_{i} := Hom_{A}(B, D/p^{i}D)  for every i ∈ ℕ.

We then obtain a commutative diagram for every δ ∈ ℕ[1/p] \ {0} and every i ∈ ℕ :

\xymatrix{B ⊗_{A} B \ar[r]^{τ⊗_{A}τ} \ar[d] & B^{∨} ⊗_{A} B^{∨} \ar[r]^{g^{δ}·i_{∗}⊗_{A}g^{δ}·i_{∗}} \ar[d] & B_{Γ}^{∨} ⊗_{A_{Γ}} B_{Γ}^{∨} \ar[r] \ar[d] & B_{Γ,i}^{∨} ⊗_{A_{Γ}} B_{Γ,i}^{∨} \ar[d] \\ B′_{1} ˆ⊗_{D^{∧}}B′_{1} \ar[r]^{τ′ ˆ⊗_{D^{∧}}τ′} & B_{1}^{′∨} ˆ⊗_{D^{∧}}B_{1}^{′∨} \ar[r]^{j_{δ∗} ˆ⊗_{D^{∧}}j_{δ∗}} & C ˆ⊗_{D^{∧}}C \ar[r] & C_{i} ⊗_{D} C_{i}.}

For given δ, pick δ′, δ′′ ∈ ℕ[1/p] \ {0} with δ = δ′ + δ′′; then

(j_{δ∗} ∘ τ′)ˆ⊗_{D^{∧}}(j_{δ∗} ∘ τ′)(e′) = (j_{δ′∗} ∘ τ′)ˆ⊗_{D^{∧}}(j_{δ′∗} ∘ τ′)(g^{2δ′′}e′).

By claim 17.4.25, we deduce that the image of (j_{δ∗} ∘ τ′)ˆ⊗_{D^{∧}}(j_{δ∗} ∘ τ′)(e′) in C_{i} ⊗_{D} C_{i} is divisible by g^{N}. Recall now that D is a faithfully flat A_{Γ}-algebra (theorem 16.9.17); since A is noetherian, and B is a finite A-algebra, we deduce a natural isomorphism

B_{Γ,i}^{∨} ⊗_{A_{Γ}} D →^{∼} C_{i} for every i ∈ ℕ.

Set e′′ := (i_{∗} ∘ τ) ⊗_{A} (i_{∗} ∘ τ)(e); it follows that the image of ge′′ in B_{Γ,i}^{∨} ⊗_{A_{Γ}} B_{Γ,i}^{∨} is divisible by g^{N}, for every i ∈ ℕ. Next, for every i ∈ ℕ and every subset Δ ⊂ Γ_{ℚ}, let :

R(B, Δ, i) := ⨁_{[γ]∈Δ} B_{[γ],i}^{∨} ⊗_{A} B_{[1/γ],i}^{∨} with B_{[γ],i}^{∨} := Hom_{A}(B, A_{[γ]}/p^{i}A_{[γ]}) for every [γ] ∈ Γ_{ℚ}

and notice that we have an A-linear surjection R(B, Γ, i) → gr_{0}(B_{Γ,i}^{∨} ⊗_{A_{Γ}} B_{Γ,i}^{∨}), for the natural Γ-grading on B_{Γ,i}^{∨} ⊗_{A_{Γ}} B_{Γ,i}^{∨}. Summing up, for every i ∈ ℕ there exists e_{i} ∈ R(B, Γ, i) such that the image of ge′′ in gr_{0}(B_{Γ,i}^{∨} ⊗_{A_{Γ}} B_{Γ,i}^{∨}) equals the image of g^{N}e_{i}. On the other hand, as in remark 17.4.7(ii) we obtain for every i ∈ ℕ a commutative diagram of A-linear maps

\xymatrix{A/p^{i}A ⊗_{A} R(B, Γ) \ar[r] \ar[d] & A/p^{i}A ⊗_{A} gr_{0}(B_{Γ}^{∨} ⊗_{A_{Γ}} B_{Γ}^{∨}) \ar[d] \ar[r] & A/p^{i}A ⊗_{A} (B ⊗_{A} B)^{∨} \ar[d] \\ R(B, Γ, i) \ar[r] & gr_{0}(B_{Γ,i}^{∨} ⊗_{A_{Γ}} B_{Γ,i}^{∨}) \ar[r] & Hom_{A}(B ⊗_{A} B, A/p^{i}A)}

whose left vertical arrow is induced by the projections A_{[γ]} → A_{[γ]}/p^{i}A_{[γ]} for every [γ] ∈ Γ_{ℚ}, and such that the composition of the top horizontal arrows is A/p^{i}A ⊗_{A} ξ\underline{f}. Denote by ξ_{\underline{f},i} the composition of the bottom horizontal arrows; arguing as in the proof of lemma 17.4.10(i) we find a finite subset Σ ⊂ Γ such that ξ\underline{f}(R(B, Γ)) = ξ\underline{f}(R(B, Σ)) and ξ_{\underline{f},i}(R(B, Γ, i)) = ξ_{\underline{f},i}(R(B, Σ, i)) for every i ∈ ℕ. We notice:

Claim 17.4.27. Let R be a noetherian ring, M, N two R-modules of finite type, and I ⊂ R an ideal. Set H^{i} := Hom_{R}(M, N/I^{i}N) for every i ∈ ℕ. Then there exists c ∈ ℕ such that

Im(H^{i+c} → H^{i}) = Im(Hom_{R}(M, N) → H^{i}) for every i ∈ ℕ.

Proof of the claim. Pick a finite presentation R^{⊕q} ⟶^{φ} R^{⊕p} → M of the R-module M; let φ^{∨} : R^{⊕p} → R^{⊕q} be the transpose of φ, set φ_{N}^{∨} := φ^{∨} ⊗_{A} N : N^{⊕q} → N^{⊕p} and N′ := Im φ_{N}^{∨}; by the Artin-Rees lemma ([126, Th.8.5]) there exists c ∈ ℕ such that N′ ∩ I^{i+c}N^{⊕q} = I^{i}(N′ ∩ I^{c}N^{⊕q}) for every i ∈ ℕ. We consider then the induced commutative diagram :

\xymatrix{0 \ar[r] & Hom_{R}(M, N) \ar[r] \ar[d] & N^{⊕p} \ar[r]^{φ_{N}^{∨}} \ar[d] & N^{⊕q} \ar[d] \\ 0 \ar[r] & H^{i+c} \ar[r] \ar[d] & N^{⊕p}/I^{i+c}N^{⊕p} \ar[r]^{φ_{N/I^{i+c}N}^{∨}} \ar[d] & N^{⊕q}/I^{i+c}N^{⊕q} \ar[d] \\ 0 \ar[r] & H^{i} \ar[r] & N^{⊕p}/I^{i}N^{⊕p} \ar[r]^{φ_{N/I^{i}N}^{∨}} & N^{⊕q}/I^{i}N^{⊕q}}

whose horizontal rows are left exact sequences. Let then \overline{h} ∈ Ker φ_{N/I^{i+c}N}^{∨}, and pick a representative h ∈ N^{⊕p} for the class \overline{h}. Then φ_{N}^{∨}(h) ∈ N′ ∩ I^{i+c}N^{⊕q}, so there exists k ∈ ℕ such that φ_{N}^{∨}(h) = ∑_{j=1}^{k} a_{j}φ_{N}^{∨}(x_{j}) for certain a_{1}, …, a_{k} ∈ I^{i} and x_{1}, …, x_{k} ∈ N^{⊕p}. Set h′ := h − ∑_{j=1}^{k} a_{j}x_{j}. Then h′ ∈ Ker φ_{N}^{∨}, and the image of h′ in N^{⊕p}/I^{i}N^{⊕p} agrees with the image of \overline{h}, whence the claim. ♢

By claim 17.4.27, for every i ∈ ℕ there exists e′_{i} ∈ R(B, Σ) such that ξ_{\underline{f},i}(e_{i}) agrees with the image of 1 ⊗ ξ\underline{f}(e′_{i}) ∈ A/p^{i}A ⊗_{A} (B ⊗_{A} B)^{∨}. Now, let M ⊂ (B ⊗_{A} B)^{∨} be the A-submodule generated by the system (ξ\underline{f}(e′_{i}) | i ∈ ℕ); we conclude that the image of ge′′ in (B ⊗_{A} B)^{∨} lies in the submodule g^{N}M + p^{i}(B ⊗_{A} B)^{∨}, for every i ∈ ℕ; but then it lies already in g^{N}M, since g^{N}M is a closed subset for the p-adic topology of the finitely generated A-module (B ⊗_{A} B)^{∨} ([126, Th.8.10(i)]). Lastly, this shows that the image of ge′′ in X(\underline{f}) lies in g^{N} · ϑ\underline{f}(R(B, Γ)); but this image equals g^{N+1}e_{f}, whence ge_{f} ∈ ϑ\underline{f}(R(B, Γ)), which achieves the proof of (ii).

To show (i), let e′′_{i} be the image of e′′ in B_{Γ,i}^{∨} ⊗_{A_{Γ}} B_{Γ,i}^{∨} for every i ∈ ℕ, and set

I_{i} := {a ∈ A_{Γ} | ae′′_{i} ∈ g^{N}(B_{Γ,i}^{∨}⊗_{A_{Γ}} B_{Γ,i}^{∨})} and I_{D,i} := {d ∈ D | de′′_{i} ∈ g^{N}(C_{i}⊗_{D} C_{i})}.

As already observed in the foregoing, g^{δ} ∈ I_{D,i} for every δ ∈ ℕ[1/p]\{0}; then, since D is a flat A_{Γ}-algebra, we have I_{D,i} = I_{i}D for every i ∈ ℕ (details left to the reader), so that I_{D,i}^{k} = I_{i}^{k}D for every i, k ∈ ℕ. Since D is a faithfully flat A_{Γ}-algebra, it follows that I_{i}^{k} = I_{D,i}^{k} ∩ A_{Γ} for every i, k ∈ ℕ ([126, Th.7.5(ii)]), whence g ∈ I_{i}^{k} for every i, k ∈ ℕ, and notice as well that I_{i} is a Γ-graded ideal of A_{Γ}. Let λ_{1}, …, λ_{k} ∈ P^{∨} := Hom_{Mnd}(P_{ℝ}, ℝ_{+}) such that ℝλ_{1}, …, ℝλ_{k} are the extremal rays of P_{ℝ}^{∨}, and set λ := λ_{1} + ⋯ + λ_{k}; by lemma 17.3.10there exists a sequence (γ_{n} | n ∈ ℕ) of elements of P^{(∞)} with lim_{n→+∞} λ(γ_{n}) = 0 and such that gr_{[γ_{n}]}I_{i} = A_{[γ_{n}]} for every n ∈ ℕ, and by claim 17.3.7 we may assume that gr_{[γ_{n}]}P^{(∞)} = γ_{n}P for every n ∈ ℕ, so that A_{[γ_{n}]} = A · (1 ⊗ γ_{n}), and therefore

(1 ⊗ γ_{n}) · e′′_{i} ∈ g^{N} · gr_{[γ_{n}]}(B_{Γ,i}^{∨} ⊗_{A_{Γ}} B_{Γ,i}^{∨}).

On the other hand, lemma 17.4.16yields for every [γ] ∈ Γ an element [γ′] ∈ Σ_{ℚ} such that :

γ^{−1}P_{ℚ} ∩ P^{gp} ⊂ γ^{′−1}P_{ℚ} ∩ P^{gp} and (γ/γ_{n})P_{ℚ} ∩ P^{gp} ⊂ γ′P_{ℚ} ∩ P^{gp}.

We then get a commutative diagram of A-modules :

\xymatrix{R(B, Γ_{ℚ}, i) \ar[r] & gr_{0}(B_{∞,i}^{∨} ⊗_{A_{Γ}} B_{∞,i}^{∨}) \ar[r] & Hom_{A}(B ⊗_{A} B, A/p^{i}A) \ar[d] \\ ⨁_{[γ]∈Γ} B_{[γ],i}^{∨} ⊗_{A} B_{[γ_{n}/γ],i}^{∨} \ar[u] \ar[r] & gr_{[γ_{n}]}(B_{Γ,i}^{∨} ⊗_{A_{Γ}} B_{Γ,i}^{∨}) \ar[r] & Hom_{A}(B ⊗_{A} B, A_{[γ_{n}]}/p^{i}A_{[γ_{n}]})}

whose right vertical arrow is an isomorphism induced by scalar multiplication by 1 ⊗ γ_{n}. Moreover, the composition of the top horizontal arrows is ξ_{\underline{f},i}. With this notation, we have e′′_{i} ∈ gr_{0}(B_{Γ,i}^{∨} ⊗_{A_{Γ}} B_{Γ,i}^{∨}), and it follows easily that the image of e′′_{i} lies in g^{N} · ξ_{\underline{f},i}(R(B, Γ_{ℚ}, i)), for every i ∈ ℕ. Arguing as in the proof of lemma 17.4.10(i), we then find a finite subset Δ ⊂ Γ_{ℚ} such that the image of e′′_{i} in Hom_{A}(B ⊗_{A} B, A/p^{i}A) lies in the A-submodule g^{N} ·ξ_{\underline{f},i}(R(B, Δ, i)), for every i ∈ ℕ. By invoking again claim 17.4.27, we deduce that for every i ∈ ℕ, the image of e′′ in (B ⊗_{A} B)^{∨} lies in the A-submodule g^{N} ·ξ\underline{f}(R(B, Δ))+p^{i} ·(B ⊗_{A} B)^{∨}. Arguing as in the foregoing proof of (ii), we conclude that the image of e′′ in (B ⊗_{A} B)^{∨} lies already in g^{N} · ξ\underline{f}(R(B, Δ)), and finally e_{f} ∈ ϑ\underline{f}(R(B, Γ_{ℚ})), as stated. □

17.4.28. Keep the notation of (17.4.5), and for every m ∈ ℕ, let Γ_{m} := {γ ∈ Γ_{ℚ} | mγ = 0}. By theorem 17.4.19, if (A, P, β) is regular, there exists m ∈ ℕ such that e_{f} ∈ ϑ\underline{f}(R(B, Γ_{m})). It turns out that a suitable integer m with this property can be exhibited purely in terms of the combinatorics of the monoid P. Indeed, let P^{∨} be the dual of P (see (6.4.13)); since P is fine, P^{∨} is fine, sharp and saturated (proposition 6.4.14(i,ii)), and we let F_{1}, …, F_{k} be the onedimensional faces of P^{∨}. Then F_{i} ≃ ℕ (theorem 6.4.18(ii)), and we denote by λ_{i} the unique generator of F_{i}, for every i = 1, …, k. Set Λ := {λ_{1}, …, λ_{k}}, and denote by ℰ the set of all subsets Σ ⊂ Λ such that {λ^{gp} ⊗_{ℤ} ℚ : P_{ℚ}^{gp} → ℚ | λ ∈ Σ} is a basis of (P_{ℚ}^{gp})^{∨}. For every Σ ∈ ℰ, let L_{Σ} ⊂ P^{∨gp} be the subgroup generated by (λ^{gp} | λ ∈ Σ), and denote by m_{Σ} the exponent of the finite abelian group P^{∨gp}/L_{Σ}. Lastly, denote by m_{P} the least common multiple of the finite system of integers (m_{Σ} | Σ ∈ ℰ). We may then state :

Corollary 17.4.29. In the situation of theorem 17.4.19, we have e_{f} ∈ ϑ\underline{f}(R(B, Γ_{m_{p}})).

Proof. For every Σ ∈ ℰ, let λ_{Σ} : P_{ℚ}^{gp} →^{∼} ℚ^{Σ} be the ℚ-linear isomorphism given by the rule :

γ ↦ (λ_{ℚ}^{gp}(γ) | λ ∈ Σ) for every γ ∈ P_{ℚ}^{gp}.

Notice that P^{gp} ⊂ λ_{Σ}^{−1}(ℤ^{Σ}) for every such B, and m_{Σ} equals the exponent of the finite abelian group λ_{Σ}^{−1}(ℤ^{Σ})/P^{gp}. With this notation, we have :

Claim 17.4.30. For every β ∈ P_{ℚ}^{gp} there exist Σ ∈ ℰ and an element β^{†} ∈ λ_{Σ}^{−1}(ℤ^{Σ}) such that

(17.4.31)  βP_{ℚ} ∩ P^{gp} ⊂ β^{†}P_{ℚ} ∩ P^{gp}  and β^{−1}P_{ℚ} ∩ P^{gp} ⊂ β^{†−1}P_{ℚ} ∩ P^{gp}.

Proof of the claim. Notice that, for every β ∈ P_{ℝ}^{gp}, we have

βP_{ℝ} ∩ P^{gp} = {x ∈ P^{gp} | λ^{gp}(x) ≥ λ_{ℝ}^{gp}(β) for every λ ∈ Λ}.

For every such β and every λ ∈ Λ, denote by N_{β,λ} (resp. N′_{β,λ}) the largest (resp. smallest) integer which is smaller than (resp. larger than) or equal to λ_{ℝ}^{gp}(β). Recalling that λ^{gp}(P^{gp}) ⊂ ℤ for every λ ∈ Λ, it is easily seen that (17.4.31) holds if and only if

(17.4.32)  λ_{ℚ}^{gp}(β^{†}) ∈ [N_{β,λ}, N′_{β,λ}]  for every λ ∈ Λ.

Set Σ_{β} := {λ ∈ Λ | λ_{ℝ}^{gp}(β) ∈ ℤ}. Now, if β ∈ P_{ℚ}^{gp} and Σ_{β} spans (P_{ℚ}^{gp})^{∨}, obviously the claim holds with β^{†} := β. Thus, a simple induction argument reduces to showing that, if β ∈ P_{ℚ}^{gp} and Σ_{β} does not span (P_{ℚ}^{gp})^{∨}, there exists γ ∈ P_{ℚ}^{gp} such that (17.4.32) holds, and such that Σ_{γ} strictly contains Σ_{β}. To this aim, let l : P_{ℝ}^{gp} → ℝ^{Λ} be the ℝ-linear map such that

l(γ) := (λ_{ℝ}^{gp}(γ) | λ ∈ Λ) for every γ ∈ P_{ℝ}^{gp}

and set

K := ∏_{λ∈Λ}[N_{β,γ}, N′_{β,γ}] ⊂ ℝ^{Λ} M := ⋂_{λ∈Σ_{β}} Ker λ_{ℚ}^{gp} ⊂ P_{ℚ}^{gp}.

Notice that l is a closed immersion, K is a compact subset of ℝ^{Λ}, and by assumption, M ≠ 0. Clearly β ∈ l^{−1}K, and we pick any non-zero μ ∈ M. It follows that the subset

{a ∈ ℝ | β + aμ ∈ l^{−1}K}

is non-empty and compact, so it admits a largest element a_{0}. Set γ := β + a_{0}μ; by construction, Σ_{β} ⊂ Σ_{γ}, and it is easily seen that there must exist some λ ∈ Λ \ Σ_{β} such that λ_{ℝ}^{gp}(γ) ∈ {N_{β,λ}, N′_{β,λ}}, since otherwise we could find a real number a > a_{0} with β + aμ ∈ l^{−1}K. Thus Σ_{γ} strictly contains Σ_{β}. Lastly, since both β and μ lie in P_{ℚ}^{gp} and λ_{ℝ}^{gp}(β) ∉ ℤ, we must have λ_{ℝ}^{gp}(a_{0}μ) ∈ ℚ \ ℤ, and since λ_{ℝ}^{gp}(μ) ∈ ℚ, we conclude that a_{0} ∈ ℚ, hence γ ∈ P_{ℚ}^{gp}, whence the claim. ♢

Now, for every [β] ∈ Γ_{ℚ} pick β^{†} ∈ P_{ℚ}^{gp} fulfilling the condition of claim 17.4.30, and set Δ := {[β^{†}] | [β] ∈ Γ_{ℚ}}. Notice that Δ ⊂ Γ_{m_{P}}. Arguing as in the proof of lemma 17.4.10(ii) we easily deduce that ϑ\underline{f}(R(B, Δ)) = ϑ\underline{f}(R(B, Γ_{ℚ})), whence the corollary. □

Example 17.4.33. (i) Suppose that dim P = 2. Then we may find a basis (f_{1}, f_{2}) of P^{gp} and integers a, b ∈ ℕ such that b > a, (a, b) = 1 and P is the submonoid of P^{gp} generated by f_{1} and af_{1} + bf_{2} (example 6.4.19(ii)). With this notation, a simple calculation shows that m_{P} = b (details left to the reader).

(ii) In the situation of theorem 17.4.19, suppose that P = ℕ^{⊕r} × ℤ^{⊕s} for some r, s ∈ ℕ; by corollary 12.5.19, the latter holds if and only if A is a regular ring. In this case, the integer m_{P} of (17.4.28) equals 1, and therefore corollary 17.4.29says that e_{f} ∈ ϑ\underline{f}(B^{∨} ⊗_{A} B^{∨}). However, the latter follows also more directly from lemma 17.4.10(ii). Moreover, in case A is regular, the submodule ϑ\underline{f}(B^{∨} ⊗_{A} B^{∨}) can be interpreted in terms of the different ideal of the finite ring extension f : A → B. We conclude this section with a review of different ideals in the more general context of quasi-coherent algebras on schemes; especially, remark 17.4.44explains the connection with theorem 17.4.19.

17.4.34. Let X be a reduced scheme, 𝒜 a quasi-coherent and finite 𝒪_{X}-algebra and j : U → X a quasi-compact open immersion with dense image, such that j^{∗}𝒜 is a locally free 𝒪_{U}- module. Denote by

Max X

the set of maximal points of X, and notice that Max X ⊂ U, by proposition 8.1.47(i). In this situation, there is a trace form as in remark 14.3.10(iii) :

t_{j^{∗}𝒜} : j^{∗}𝒜 ⊗_{𝒪_{U}} j^{∗}𝒜 → 𝒪_{U}

whence a pairing

t_{𝒜 ,U} : 𝒜 ⊗_{𝒪_{X}} j_{∗}j^{∗}𝒜 → j_{∗}(j^{∗}𝒜 ⊗_{𝒪_{U}} j^{∗}𝒜) ⟶^{j_{∗}t_{j^{∗}𝒜}} j_{∗}𝒪_{U}

which in turns yields a morphism of 𝒪_{X}-modules

τ_{𝒜 ,U} : j_{∗}j^{∗}𝒜 → ℋ om_{𝒪_{X}} (𝒜 , j_{∗}𝒪_{U}).

Since X is reduced, the natural morphism 𝒪_{X} → j_{∗}𝒪_{U} is injective, and therefore the same holds for the induced 𝒪_{X}-linear morphism

𝒜^{∨} → ℋ om_{𝒪_{X}} (𝒜 , j_{∗}𝒪_{U})

so we may define

𝒟_{𝒜 ,U}^{−1} := τ_{𝒜 ,U}^{−1} (𝒜^{∨}).

We call 𝒟_{𝒜 ,U}^{−1} ⊂ j_{∗}j^{∗}𝒜 the inverse different of 𝒜 relative to the open subset U.

Remark 17.4.35. (i) In the situation of (17.4.34), notice that the natural isomorphism

ℋ om_{𝒪_{X}} (𝒜 , j_{∗}𝒪_{U}) →^{∼} j_{∗}(j^{∗}𝒜)^{∨}

(see (10.1.19)) identifies τ_{𝒜 ,U} with j_{∗}τ_{j^{∗}𝒜} , where τ_{j^{∗}𝒜} : j^{∗}𝒜 → j^{∗}𝒜^{∨} is the 𝒪_{U}-linear morphism deduced from the trace pairing t_{j^{∗}𝒜} .

(ii) Moreover, set ℬ := Im (𝒜 → j_{∗}j^{∗}𝒜). Since the natural map 𝒪_{X} → j_{∗}𝒪_{U} is a monomorphism, it is easily seen that the the epimorphism 𝒜 → ℬ induces an isomorphism ℬ^{∨} →^{∼} 𝒜^{∨}, and taking (i) into account, we deduce that the latter induces an isomorphism

𝒟_{𝒜 ,U}^{−1} →^{∼} 𝒟_{ℬ,U}^{−1} .

(iii) Suppose next that U′ ⊂ U is another open subset, such that the open immersion j′ : U′ → X is also quasi-compact with dense image. A direct inspection of the definitions yields a commutative diagram

\xymatrix{j_{∗}j^{∗}𝒜 \ar[r]^{τ_{𝒜 ,U}} \ar[d] & ℋ om_{𝒪_{X}} (𝒜 , j_{∗}𝒪_{U}) \ar[d] \\ j′_{∗}j′^{∗}𝒜 \ar[r]^{τ_{𝒜 ,U′}} & ℋ om_{𝒪_{X}} (𝒜 , j′_{∗}𝒪_{U′})}

whose vertical arrows are monomorphisms, under the current assumptions. It follows easily that

𝒟_{𝒜 ,U}^{−1} = 𝒟_{𝒜 ,U′}^{−1} ∩ j_{∗}j^{∗}𝒜 .

(iv) In the situation of (iii), suppose that 𝒜 is generically étale, and let U′ ⊂ U be the largest open subset such that 𝒜_{|U′} is an étale 𝒪_{U′}-algebra. Then the inclusion U′ → U is a dense and quasi-compact open immersion. For the proof, we may assume that U is affine, say U = Spec R, and that j^{∗}𝒜 is the quasi-coherent 𝒪_{U}-algebra associated with an R-algebra A which is a projective R-module of finite rank; then we easily reduce to the case where A is a free R-module of finite rank. In this case, let (e_{i} | i = 1, …, r) be a basis of the R-module A; consider the matrix D := (t_{j^{∗}𝒜} (e_{i}, e_{j}) | i, j = 1, …, r), and set d := det(D). We have U′ = Spec R[d^{−1}], a quasi-compact dense open subset of U ([75, Th.4.1.14]), as required.

(v) In the situation of (iii), suppose additionally that j^{∗}𝒜 is an étale 𝒪_{U}-algebra. Then t_{j^{∗}𝒜} and t_{j′^{∗}𝒜} are perfect pairings ([75, Th.4.1.14]). Taking (i) into account, we conclude that τ_{𝒜 ,U} is an isomorphism, and the same applies to τ_{𝒜 ,U′} , so the inclusion map j_{∗}j^{∗}𝒜 → j′_{∗}j′^{∗}𝒜 restricts to a natural isomorphism

𝒟_{𝒜 ,U}^{−1} →^{∼} 𝒟_{𝒜 ,U′}^{−1} .

More generally, if j : U → X and j′ : U′ → X are any two quasi-compact dense open immersions such that j^{∗}𝒜 and j^{′∗}𝒜 are respectively an étale 𝒪_{U}-algebra and an étale 𝒪_{U′}- algebra, then both 𝒟_{𝒜 ,U}^{−1} and 𝒟_{𝒜 ,U′}^{−1} are naturally identified with 𝒟_{𝒜 ,U∩U′}^{−1} , and in this sense we may say that the inverse different of a finite and generically étale 𝒪_{X}-algebra is independent of the open subset U. We shall therefore henceforth denote just by 𝒟_{𝒜}^{−1} the inverse different of 𝒜.

Lemma 17.4.36. In the situation of (17.4.34), denote by 𝒜^{ν} the normalization of j^{∗}𝒜 over X (see remark 14.4.3(ii)). The following holds :

(i) If X is normal, 𝒜 ⊂ 𝒟_{𝒜 ,U}^{−1} .

(ii) Suppose that 𝒜 is a generically étale 𝒪_{X}-algebra. Then :

(a) There exists a dense quasi-compact open subset U′ ⊂ U such that the map τ_{𝒜 ,U′} restricts to an 𝒪_{X}-linear isomorphism 𝒟_{𝒜}^{−1} →^{∼} 𝒜^{∨}.

(b) If X is locally coherent, 𝒟_{𝒜}^{−1} is a reflexive 𝒪_{X}-module of finite type.

(c) If X is normal, 𝒜^{ν} ⊂ 𝒟_{𝒜}^{−1}. (iii) Suppose that X is normal, and j^{∗}𝒜 is an étale 𝒪_{U}-algebra. Let also j′ : U′ → X be a

quasi-compact open immersion with dense image, with U′ ⊂ U, and 𝒜^{′ν} the integral closure of 𝒪_{X} in j^{′∗}𝒜. Then the natural map 𝒜^{ν} → 𝒜^{′ν} is an isomorphism.

Proof. (i) follows immediately from lemma 14.3.15(i).

(ii.a) follows immediately from the discussion of remark (17.4.35)(iv,v), and (ii.b) follows from (ii.a) and lemma 11.3.3.

(ii.c): Taking into account remark 17.4.35(ii), we may assume without loss of generality that 𝒜 = Im (𝒜 → j_{∗}j^{∗}𝒜), in which case the natural map 𝒜 → 𝒜^{ν} is a monomorphism. Moreover, the assertion is clearly local on X, hence we may assume that X is affine, in which case 𝒜^{ν} is the union of the filtered family of its 𝒜 -subalgebras ℬ that are quasi-coherent and finite 𝒪_{X}-algebras. For any such ℬ, the 𝒪_{X}-module ℬ/𝒜 is supported on X \ U. It follows that (ℬ/𝒜)^{∨} = 0, so the transpose map ℬ^{∨} → 𝒜^{∨} is a monomorphism as well; by the same token, we see that τ_{𝒜 ,U} = τ_{ℬ,U} (notation of (17.4.34)), so 𝒟_{ℬ}^{−1} ⊂ 𝒟_{𝒜}^{−1}, whence ℬ ⊂ 𝒟_{𝒜}^{−1}, by virtue of (i). Since ℬ is arbitrary, the assertion follows.

(iii): We may assume that X is local, say X = Spec R for a local and normal domain R, and let K be the field of fractions of R; then A^{ν} := 𝒜^{ν}(X) (resp. A^{′ν}) is the integral closure of R in 𝒜 (U) (resp. in 𝒜 (U′)); but under the stated assumptions, both A^{ν} and A^{′ν} are also the integral closure of R in K ⊗_{R} 𝒜 (X), whence the assertion. □

Remark 17.4.37. Let X be a reduced and quasi-separated scheme, such that Max X is finite.

(i) Let also ℱ be a quasi-coherent 𝒪_{X}-module of finite type. Then we claim that there exists a dense and quasi-compact open immersion j : U → X such that j^{∗}ℱ is a locally free 𝒪_{U}- module. Indeed, for any maximal point η ∈ X, pick an affine open subset V_{η} ⊂ X such that η is the unique maximal point of V_{η}; hence V_{η} = Spec A for an integral domain A and ℱ_{|V} = M^{∼} for an A-module M of finite type. Set K := Frac A; then P := M ⊗_{A} K is a finite dimensional K-vector space, and we may find a element f ∈ A\{0}, a free A_{f}-module Q of finite rank and an A_{f}-linear map φ : Q → M ⊗_{A} A_{f} such that Ker φ ⊗_{A} K = 0 = Coker φ ⊗_{A} K. However, the natural map Q → Q ⊗_{A_{f}} K is injective, so we see that Ker φ = 0, and on the other hand Coker φ is an A_{f}-module of finite type, so we may find g ∈ A\{0} such that φ_{g} is surjective. Set U_{η} := Spec A_{fg}; by construction ℱ_{|U_{η}} is a locally free 𝒪_{U_{η}}-module, and since η is arbitrary, the assertion follows easily.

(ii) Let 𝒜 be a quasi-coherent and finite 𝒪_{X}-algebra, and suppose that 𝒜_{η} is an étale 𝒪_{X,η}- algebra, for every maximal point η of X. Then we claim that there exists an dense quasicompact open immersion j : U → X such that j^{∗}𝒜 is an étale 𝒪_{U}-algebra. Indeed, by (i), we may already assume that 𝒜 is a locally free 𝒪_{X}-module, and then we may argue as in remark 17.4.35(iv).

17.4.38. Keep the situation of (17.4.34), let X′ be another reduced scheme, and g : X′ → X a quasi-compact and quasi-separated morphism of schemes which restricts to a map

Max X′ → Max X.

Set

U′ := g^{−1}U 𝒜′ := g^{∗}𝒜

and denote by j′ : U′ → X′ the resulting open immersion. It is easily seen that j′ is quasicompact and has dense image, and clearly j′^{∗}𝒜′ is a locally free 𝒪_{U′}-module, so there is a well defined inverse different 𝒟_{𝒜′,U′}^{−1} relative to U′. Moreover, we have a commutative diagram

(17.4.39)  [figure: g^{∗}j_{∗}j^{∗}𝒜; g^{∗}τ_{𝒜 ,U}; a; g^{∗}ℋ om_{𝒪_{X}} (𝒜 , j_{∗}𝒪_{U}); b; g^{∗}(𝒜^{∨}); c; j′_{∗}j′^{∗}𝒜′; τ_{𝒜′,U′}; ℋ om_{𝒪_{X′}} (𝒜′, j′_{∗}𝒪_{U′}); 𝒜^{′∨}]

([75, Lemma 4.1.13(iii)]). It follows that the left vertical arrow restricts to an 𝒪_{X′}-linear map

(17.4.40)  g^{∗}𝒟_{𝒜 ,U}^{−1} → 𝒟_{𝒜′,U′}^{−1} .

Lemma 17.4.41. The map (17.4.40) is an isomorphism in the following cases :

(a) if 𝒜 is a locally free 𝒪_{X}-module of finite type and j^{∗}𝒜 is an étale 𝒪_{U}-algebra

(b) or if g is a flat morphism.

Proof. For case (a), notice that when 𝒜 is locally free, the map c is an isomorphism, and if j^{∗}𝒜 is an étale 𝒪_{U}-algebra, the discussion of remark 17.4.35(v) identifies (17.4.40) with the map c.

If g is flat, the map a in (17.4.39) is an isomorphism (corollary 10.3.8). The isomorphism of remark 17.4.35(i) identifies b with the composition

g^{∗}j_{∗}ℋ om_{𝒪_{X}} (j^{∗}𝒜 , 𝒪_{U}) ⟶^{d} j′_{∗}g^{∗}ℋ om_{𝒪_{X}} (j^{∗}𝒜 , 𝒪_{U}) ⟶^{e} j′_{∗}ℋ om_{𝒪_{U′}} (j′^{∗}𝒜′, 𝒪_{U′})

where d is an isomorphism, since g is flat (corollary 10.3.8), and the same holds for e, since j^{∗}𝒜 is locally free of finite type (proposition 10.3.3(ii)). To conclude, it then suffices to show that c is an isomorphism as well. The assertion is local on X, hence we may assume that X is affine; in this case, we may find a finitely presented 𝒪_{X}-module 𝒢 and an 𝒪_{X}-linear epimorphism φ : 𝒢 → 𝒜 such that j^{∗}φ is an isomorphism. Set 𝒢′ := g^{∗}𝒢; there follows a commutative diagram

\xymatrix{g^{∗}(𝒜^{∨}) \ar[r]^{g^{∗}(φ^{∨})} \ar[d] & g^{∗}(𝒢^{∨}) \ar[d] \\ 𝒜′^{∨} \ar[r]^{(φ^{∗}g)^{∨}} & 𝒢′^{∨}}

whose right vertical arrows is an isomorphism, by virtue of [75, Lemma 2.4.29(i)]. Moreover Ker (φ^{∨}) = 0 and Coker (φ^{∨}) ⊂ (Ker φ)^{∨}, and since Max X ⊂ U, we see that Ker φ_{η} = 0 for every η ∈ Max X, whence (Ker φ)^{∨} = 0, since X is reduced. Thus, the top horizontal arrow of the foregoing diagram is also an isomorphism; the same argument applies as well to the bottom horizontal arrow, and the assertion follows. □

17.4.42. In the situation of (17.4.38), suppose now that X and X′ are locally coherent and 𝒜 is generically étale. Taking into account lemma 17.4.36(ii.b), we see that (17.4.40) factors as a composition

g^{∗}𝒟_{𝒜}^{−1} ⟶^{β} (g^{∗}𝒟_{𝒜}^{−1})^{∨∨} ⟶^{γ} 𝒟_{𝒜′}^{−1}

where β is the natural map as in (11.3).

Proposition 17.4.43. With the notation of (17.4.42), suppose moreover that :

(a) both X and X′ are locally noetherian normal schemes

(b) g maps the points of codimension one of X′ to points of codimension ≤ 1 of X. Then γ is an isomorphism.

Proof. Set ℬ := Im (𝒜 → j_{∗}j^{∗}𝒜) and ℬ′ := g^{∗}ℬ. It is easily seen that the image of ℬ′ in j′_{∗}j′^{∗}ℬ′ agrees with the image of 𝒜′; in view of remark 17.4.35(ii) we deduce that the natural maps 𝒜 → ℬ and 𝒜′ → ℬ′ induce isomorphisms

𝒟_{𝒜}^{−1} →^{∼} 𝒟_{ℬ}^{−1} 𝒟_{𝒜′}^{−1} →^{∼} 𝒟_{ℬ′}^{−1}.

We may then replace 𝒜 by ℬ, and assume that the natural map 𝒜 → j_{∗}j^{∗}𝒜 is a monomorphism. By lemmata 11.3.3 and 17.4.36(ii.b), both the source and target of γ are reflexive 𝒪_{X′}- modules. Thus, denote by V ⊂ X′ the subset of all points such that γ_{x} is an isomorphism; then V is an open subset of X′ ([126, Th.4.10]) and the inclusion map V → X′ is quasicompact, since X′ is locally noetherian. In view of proposition 11.3.8(ii), it suffices to check that V contains all the points of codimension one of X′. Thus, let x′ be such a point, and set x := g(x′); by assumption (b), the point x has codimension ≤ 1 in X, so 𝒪_{X,x} is either a field or a discrete valuation ring and (j_{∗}j^{∗}𝒜)_{x} is a torsion-free 𝒪_{X,x}-module of finite type. Then the same holds for 𝒜_{x}, so the latter is a free 𝒪_{X,x}-module of finite rank. By lemma 17.4.41, it follows that (17.4.40)_{x′} is an isomorphism. It is also clear that β_{x′} is an isomorphism, and since γ_{x′} = (17.4.40)_{x′} ∘ β_{x′}^{−1}, the contention follows. □

Remark 17.4.44. (i) Let us now consider a domain A, and a finite, injective, and generically étale ring homomorphism f : A → B. As in (17.4), we let X := Spec A, and denote by ℬ the quasi-coherent 𝒪_{X}-algebra arising from B. Let also K be the field of fractions of A; set B_{K} := K ⊗_{A} B, and let \overline{B} be the image of B in B_{K}. We may then regard the diagonal idempotent e_{f} of the map f as an element in B_{K} ⊗_{K} B_{K}, and the associated trace form as a Kbilinear map B_{K} ⊗_{K} B_{K} → K, inducing an isomorphism ω_{f} : B_{K} →^{∼} B_{K}^{∨}. With this notation, the inverse different 𝒟_{ℬ}^{−1} corresponds to ω_{f}^{−1}(\overline{B}^{∨}) ⊂ B_{K}, and the subset

𝒟_{ℬ} := \{b ∈ B | b𝒟_{ℬ}^{−1} ⊂ \overline{B}\}

is an ideal of B called the different of f. (If A and B are noetherian normal domains, then 𝒟_{ℬ}^{−1} is the inverse of 𝒟_{ℬ} in the group of reflexive fractional ideals of B : see example 6.4.38.)

(ii) Suppose moreover that B is a projective A-module. Then we have a natural isomorphism B ⊗_{A} B^{∨} →^{∼} Hom_{A}(B, B), and we denote by ζ_{f} the unique element of B ⊗_{A} B^{∨} corresponding to 1_{B} : B → B under this isomorphism. According to [75, Rem.4.1.17], we have the identity :

(B_{K} ⊗_{K} ω_{f})(e_{f}) = ζ_{f}

and it follows easily that :

(B ⊗_{A} 𝒟_{ℬ} ) · e_{f} ⊂ Im(B ⊗_{A} B → B_{K} ⊗_{K} B_{K}).

(iii) If B is not a projective A-module, the inclusion of (ii) can fail; however, if A is a regular and noetherian domain, theorem 17.4.19tells us that we have at least :

(𝒟_{ℬ} ⊗_{A} 𝒟_{ℬ} ) · e_{f} ⊂ Im(B ⊗_{A} B → B_{K} ⊗_{K} B_{K}).

17.5. Big Cohen-Macaulay algebras. We shall use the following auxiliary construction. For every abelian group (G, 0, +), set

G^{⋄} := G^{ℕ}/G^{(ℕ)}.

Hence, the elements of G^{⋄} are represented by sequences (g_{n} | n ∈ ℕ) of elements of G, and such a sequence represents the zero element of the quotient group G^{⋄} ⇔ the set {n ∈ ℕ | g_{n} ≠ 0} is finite. Especially, G = 0 ⇔ G^{⋄} = 0. Moreover, for every group homomorphism φ : G → H, we have φ^{ℕ}(G^{(ℕ)}) ⊂ H^{(ℕ)}, hence φ induces a group homomorphism

φ^{⋄} : G^{⋄} → H^{⋄}

and the rules : G ↦ G^{⋄} and φ :↦ φ^{⋄} define an endofunctor of the category of abelian groups. Notice as well that (φ+ψ)^{⋄} = φ^{⋄} +ψ^{⋄} for every pair of group homomorphisms φ, ψ : G → H. Lastly, we have an injective natural transformation that maps G diagonally into G^{⋄}

Δ_{G} : G → G^{⋄} : g ↦ (g, g, …).

• If (R, +, ·, 0, 1) is a ring, then R^{(ℕ)} is an ideal of the R-algebra R^{ℕ} (the product in the category of rings of copies of R indexed by ℕ), hence R^{⋄} is naturally an R-algebra via the map Δ_{R}, and for every ring homomorphism φ : R → S, the map φ^{⋄} : R^{⋄} → S^{⋄} is a ring homomorphism such that

Δ_{S} ∘ φ = φ^{⋄} ∘ Δ_{R}.

• Moreover, if M is any R-module, then M^{ℕ} is obviously an R^{ℕ}-module, and since R^{(ℕ)} · M^{ℕ} = M^{(ℕ)}, it follows that M^{⋄} is naturally an R^{⋄}-module, and for every R-linear map f : M → N, the map f^{⋄} : M^{⋄} → N^{⋄} is R^{⋄}-linear.

Lemma 17.5.1. (i) The functor (−)^{⋄} : R-Mod → R^{⋄}-Mod is exact and faithful. (ii) We have (IM)^{⋄} = IM^{⋄} for every ideal I ⊂ R of finite type, and every R-module M.

Proof. (i): Since we have M = 0 ⇔ M^{⋄} = 0, it is clear that (−)^{⋄} is faithful. Next, let f : M → N be an R-linear map, and g_{•} := (g_{n} | n ∈ ℕ) ∈ M^{ℕ}; denote by \overline{g}_{•} ∈ M^{⋄} the class of g_{•}, and suppose that f^{⋄}(\overline{g}_{•}) = 0. This means that there exists k ∈ ℕ such that f(g_{n}) = 0 for every n ≥ k; then clearly \overline{g}_{•} is in the image of the natural map (Ker f)^{⋄} → M^{⋄}. In particular, if φ is injective, the same holds for φ^{⋄}, and it is also clear that if φ is surjective, the same holds for φ^{⋄}. It follows that the natural surjection N → Coker f induces a surjection N^{⋄} → (Coker f)^{⋄}, whose kernel is (Im f)^{⋄}, and it is easily seen that (Im f)^{⋄} = Im(f^{⋄}); hence (Coker f)^{⋄} represents Coker(f^{⋄}), whence the exactness of (−)^{⋄}.

(ii): Let a_{1}, …, a_{k} a finite system of generators of I; there follows an R-linear surjection f : M^{⊕k} → IM whose composition with the inclusion map j : IM → M is given by the rule : (m_{1}, …, m_{k}) ↦ a_{1}m_{1} + ⋯ + a_{k}m_{k}. Hence j^{⋄} ∘ f^{⋄} = (j ∘ f)^{⋄} : (M^{⋄})^{⊕k} → M^{⋄} is given by the same rule, and by the foregoing f^{⋄} : (M^{⋄})^{⊕k} → (IM)^{⋄} is a surjection, whence the claim. □

17.5.2. Let now (V, 𝔪) be a basic setup (in the sense of [75, §2.1.1]), such that 𝔪 is the filtered union of a countable system of principal subideals. We consider the subset

S ⊂ V^{ℕ}

of all (a_{n} | n ∈ ℕ) such that V a_{n} ⊂ V a_{n+1} for every n ∈ ℕ, and 𝔪 ⊂ ⋃_{n∈ℕ} V a_{n}.

Remark 17.5.3. (i) Since 𝔪 = 𝔪^{2}, it is easily seen that S is a multiplicative system. (ii) Also, for every V -module M we have

Ker(j_{M} : M^{⋄} → S^{−1}M^{⋄}) = {x ∈ M^{⋄} | 𝔪x = 0} and 𝔪 · Coker(j_{M}) = 0.

Especially, j_{M}^{a} : (M^{⋄})^{a} → (S^{−1}M^{⋄})^{a} is an isomorphism of (V, 𝔪)^{a}-modules.

(iii) Notice that S contains the multiplicative subset T := (e_{k,•} | k ∈ ℕ), where e_{k,•} := (e_{k,n} | n ∈ ℕ), with e_{k,n} := 0 for every n = 0, …, k − 1 and e_{k,n} := 1 for every n ≥ k. Notice moreover that e_{k,•} is invertible in V^{⋄} for every k ∈ ℕ, and for every V -module M the projection M^{ℕ} → M^{⋄} factors through the localization M^{ℕ} → T^{−1}M^{ℕ} and an isomorphism

T^{−1}M^{ℕ} →^{∼} M^{⋄}.

Hence, we also get an isomorphism of V^{⋄}-modules

S^{−1}M^{⋄} →^{∼} S^{−1}M^{ℕ} for every V -module M.

Lemma 17.5.4. In the situation of (17.5.2), let A → V be a ring homomorphism. We have : (i) M^{a} = 0 if and only if S^{−1}M^{⋄} = 0. (ii) The functor V -Mod → V^{⋄}-Mod : M ↦ S^{−1}M^{⋄} factors via an exact faithful functor

S^{−1}(−)^{⋄} : V^{a}-Mod → V^{⋄}-Mod M^{a} ↦ S^{−1}M^{⋄}.

(iii) The unit of adjunction S^{−1}M^{⋄} → (S^{−1}M^{⋄})_{∗}^{a} is an isomorphism of V -modules.

(iv) If A is a coherent ring, then S^{−1}M^{⋄} is a flat A-module if and only if M^{a} is flat over A

in the sense of definition 14.2.40.

(v) If A is noetherian, then S^{−1}M^{⋄} is a faithfully flat A-module if and only if M^{a} is faithfully flat over A in the sense of definition 14.2.40.

Proof. The proof of (i) shall be left to the reader.

(ii) is an immediate consequence of (i) and lemma 17.5.1(i).

(iii): Let x ∈ S^{−1}M^{⋄} be an element such that 𝔪 · x = 0, and write x = s_{•}^{−1}m_{•} for some s_{•} := (s_{n} | n ∈ ℕ) ∈ S and m_{•} := (m_{n} | n ∈ ℕ) ∈ M^{⋄}. Let us also write 𝔪 = ⋃_{i∈ℕ} V a_{n} for a sequence (a_{i} | i ∈ ℕ) of elements of V such that V a_{i} ⊂ 𝔪 · a_{i+1} for every i ∈ ℕ. Then, for every i ∈ ℕ there exist ρ(i) ∈ ℕ, c_{i} ∈ 𝔪, and t_{i,•} := (t_{i,n} | n ∈ ℕ) ∈ S such that

a_{i} = c_{i}a_{i+1}  and a_{i}m_{n}t_{i,n} = 0  for every n ≥ ρ(i)

and we may assume that ρ(i + 1) > ρ(i) for every i ∈ ℕ. Moreover, since 𝔪 ⊂ ⋃_{n∈ℕ} V t_{i,n}, we may also assume that a_{i} ∈ V t_{i,n} for every n ≥ ρ(i), in which case we get :

a_{i}^{2}m_{n} = 0 for every i ∈ ℕ and n ≥ ρ(i).

Hence, let b_{•} := (b_{n} | n ∈ ℕ) ∈ V^{ℕ} be the element such that b_{n} := 0 for every n < ρ(0), and b_{n} := a_{i}^{2} for every i ∈ ℕ and every n = ρ(i), …, ρ(i + 1) − 1. It follows easily that b_{•} ∈ S and b_{•} · m_{•} = 0 in M^{⋄}, so finally x = 0. This shows that the unit of adjunction η_{S^{−1}M^{⋄}} : S^{−1}M^{⋄} → (S^{−1}M^{⋄})_{∗}^{a} is injective. Next, let y ∈ (S^{−1}M^{⋄})_{∗}^{a}. By remark 16.9.32(ii), the V -module 𝔪̃ is the inductive limit of a system (L_{n} | n ∈ ℕ) with L := V for every n ∈ ℕ, and with V -linear transition maps φ_{n} : L_{n} → L_{n+1} such that φ_{n}(1) = c_{n} for every n ∈ ℕ; hence y is represented by a system (y^{(n)} | n ∈ ℕ) of elements of S^{−1}M^{⋄} such that

(17.5.5)  c_{n}y^{(n+1)} = y^{(n)}  for every n ∈ ℕ.

Claim 17.5.6. Let Σ ⊂ S be any countable subset. Then S ∩ ⋂_{s_{•}∈Σ} V^{⋄}s_{•} ≠ ∅.

Proof of the claim. Say that Σ = {s_{n,•} | n ∈ ℕ}. Let the sequence (a_{n} | n ∈ ℕ) of generators of 𝔪 be as in the foregoing; then, for every n, k ∈ ℕ there exists ρ(n, k) ∈ ℕ such that

a_{k} ∈ V s_{n,j} for every j ≥ ρ(n, k).

Hence, for every k ∈ ℕ, we can find ρ′(k) ∈ ℕ such that

a_{k} ∈ V s_{n,j} for every n = 0, …, k and every j ≥ ρ′(k)

and moreover we may assume that ρ′(k + 1) > ρ′(k) for every k ∈ ℕ. Then, let x_{•} := (x_{n} | n ∈ ℕ) ∈ V^{ℕ} be the sequence such that x_{n} := 0 for every n = 0, …, ρ′(0) − 1, and x_{n} := a_{k} whenever ρ′(k) ≤ n < ρ′(k + 1), for every k ∈ ℕ. It is easily seen that x_{•} ∈ S ∩ V^{⋄}s_{n,•} for every n ∈ ℕ, whence the claim. ♢

Now, say that y^{(n)} = s_{n,•}^{−1}m_{n,•}, with s_{n,•} ∈ S and m_{n,•} ∈ M^{⋄} for every n ∈ ℕ; by claim 17.5.6, we may assume that all the elements s_{n,•} are equal, in which case (17.5.5) amounts to the identities :

m_{n,•} = c_{n}m_{n+1,•} in S^{−1}M^{⋄} for every n ∈ ℕ.

The latter in turns means that for every n ∈ ℕ there exists t_{n,•} ∈ S such that t_{n,•}m_{n,•} = c_{n}t_{n,•}m_{n+1,•} in M^{⋄}. By invoking again claim 17.5.6, we may also assume that all the t_{n,•} are equal; then set m′_{n,•} := t_{0,•}m_{n,•} for every n ∈ ℕ, so that m′_{n,•} = c_{n}m′_{n+1,•} in M^{⋄} for every n ∈ ℕ. The latter means that for every n ∈ ℕ there exists ρ(n) ∈ ℕ such that

m_{n,j} = c_{n}m_{n+1,j} for every j ≥ ρ(n)

and we may assume that ρ(n + 1) > ρ(n) for every n ∈ ℕ. Then, let t_{•} := (t_{n} | n ∈ ℕ) ∈ V^{ℕ} (resp. l_{•} := (l_{n} | n ∈ ℕ) ∈ M^{ℕ}) be the element with t_{j} := 0 (resp. l_{j} := 0) for j = 0, …, ρ(0)−1 and t_{j} := a_{n} (resp. l_{j} := m_{n,j}) whenever ρ(n) ≤ j < ρ(n+1), for every n ∈ ℕ.

Claim 17.5.7. t_{•}m_{n,•} = a_{n}l_{•} in M^{⋄} for every n ∈ ℕ.

Proof of the claim. It suffices to show that for every k ≥ n and every j = ρ(k), …, ρ(k +1)−1 we have a_{k}m_{n,j} = a_{n}m_{k,j}. The latter follows by a simple inspection. ♢

Set z := t_{•}^{−1}l_{•}; claim 17.5.7 implies that m_{n,•} = a_{n}y in S^{−1}M^{⋄} for every n ∈ ℕ; hence η_{S^{−1}M^{⋄}}(z) = y, whence the assertion.

(iv): Suppose that M^{a} is flat over A, and let I ⊂ A be an ideal of finite type; in light of lemma 17.5.1(ii), it suffices to check that the natural map

f : I ⊗_{A} S^{−1}M^{⋄} → S^{−1}(IM)^{⋄}

is an isomorphism. Now, it is easily seen that for every A-module N of finite type, the natural map N ⊗_{A} S^{−1}M^{⋄} → S^{−1}(N ⊗_{A} M)^{⋄} is surjective; on the other hand, since A is coherent, there exists k ∈ ℕ and a surjective A-linear map A^{⊕k} → I, whose kernel is an A-module N of finite type. Thus, we have the following commutative diagram

\xymatrix{0 \ar[r] & N ⊗_{A} S^{−1}M^{⋄} \ar[r] \ar[d] & A^{⊕k} ⊗_{A} S^{−1}M^{⋄} \ar[r] \ar[d] & I ⊗_{A} S^{−1}M^{⋄} \ar[r] \ar[d]^{f} & 0 \\ 0 \ar[r] & S^{−1}(N ⊗_{A} M)^{⋄} \ar[r] & S^{−1}(M^{⊕k})^{⋄} \ar[r] & S^{−1}(IM)^{⋄} \ar[r] & 0}

whose first and third vertical arrows are surjections, and whose central vertical arrow is an isomorphism. Moreover, the top horizontal sequence is right exact, and since M^{a} is flat over A, combining (i) and lemma 17.5.1(i) it is easily seen that the bottom horizontal arrow is a short exact sequence. Then, by the snake lemma, Ker f = 0, whence the contention.

Conversely, suppose that S^{−1}M^{⋄} is a flat A-module, and consider any short exact sequence Σ := (0 → N′ → N → N′′ → 0) of A-modules; we need to show that the induced sequence M^{a} ⊗_{A} Σ is still exact. To this aim, notice that Σ is the filtered colimit of a system of short exact sequences (0 → N′_{λ} → N_{λ} → N′′_{λ} → 0 | λ ∈ Λ) such that N_{λ} and N′′_{λ} are finitely presented A-modules, for every λ ∈ Λ. Thus, we may assume that N and N′′ is finitely presented, in which case the same holds for N′, since A is coherent, and we consider the diagram :

\xymatrix{Σ ⊗_{A} S^{−1}M^{⋄} \ar[d] & 0 \ar[r] & N′ ⊗_{A} S^{−1}M^{⋄} \ar[r] \ar[d] & N ⊗_{A} S^{−1}M^{⋄} \ar[r] \ar[d] & N′′ ⊗_{A} S^{−1}M^{⋄} \ar[r] \ar[d] & 0 \\ S^{−1}((Σ ⊗_{A} M)^{⋄}) & 0 \ar[r] & S^{−1}(N′ ⊗_{A} M)^{⋄} \ar[r] & S^{−1}(N ⊗_{A} M)^{⋄} \ar[r] & S^{−1}(N′′ ⊗_{A} M)^{⋄} \ar[r] & 0.}

Since N, N′ and N′′ are finitely presented, lemma 17.5.1(i) easily implies that all the vertical arrows are isomorphisms. By assumption, the top horizontal row is short exact, hence the same holds for the bottom one, and therefore the sequence of (V, 𝔪)^{a}-modules ((Σ ⊗_{A} M)^{⋄})^{a} is short exact; then we conclude by invoking again lemma 17.5.1(i).

(v): Suppose that M^{a} is faithfully flat over A; in view of (iv), it suffices to show that for every A-module N ≠ 0 of finite type, we have S^{−1}M^{⋄} ⊗_{A} N ≠ 0. But then N is finitely presented, since A is noetherian, hence the natural map S^{−1}M^{⋄} ⊗_{A} N → S^{−1}(M ⊗_{A} N)^{⋄} is an isomorphism, and by assumption M^{a} ⊗_{A} N ≠ 0; then the assertion follows from (i). One argues similarly for the converse : details left to the reader. □

Proposition 17.5.8. In the situation of (17.5.2), let I ⊂ V be any ideal of finite type; then both M^{⋄} and S^{−1}M^{⋄} are I-adically complete V -modules.

Proof. Let x_{••} := (x_{k,•} | k ∈ ℕ) be a Cauchy sequence for the I-adic topology in M^{⋄}. Hence, x_{k,•} is the class of a sequence (x_{k,i} | i ∈ ℕ), for every i ∈ ℕ, and after replacing x_{••} by a subsequence, we may assume that x_{k,•} − x_{k+1,•} ∈ I^{k}M^{⋄} for every k ∈ ℕ. In light of lemma 17.5.1(ii), the latter means that for every k ∈ ℕ there exists n(k) ∈ ℕ such that x_{k,i} − x_{k+1,i} ∈ I^{k}M for every i ≥ n(k), and clearly we may assume that n(k + 1) > n(k) for every k ∈ ℕ. Now, let y_{•} := (y_{i} | i ∈ ℕ) be the following element of M^{⋄}. For every i ≤ n(0) we let y_{i} := x_{0,i}; then, for every k ∈ ℕ and n(k) < i ≤ n(k + 1) we let y_{i} := x_{k+1,i}. For every k ≥ j and n(k) < i ≤ n(k + 1), we get :

y_{i} − x_{j,i} = (x_{k+1,i} − x_{k,i}) + ⋯ + (x_{j+1,i} − x_{j,i}) ∈ I^{k}M + ⋯ + I^{j}M = I^{j}M.

Hence, y_{•} − x_{j,•} ∈ I^{j}M^{⋄} for every j ∈ ℕ, so y_{•} is a limit of x_{••}.

Next, let x_{••} := (x_{k,•} | k ∈ ℕ) be a Cauchy sequence for the I-adic topology in S^{−1}M^{⋄}; again, we may assume that y_{k,•} := x_{k+1,•} − x_{k,•} ∈ I^{k}S^{−1}M^{⋄} for every k ∈ ℕ, and then there exist s_{k,•} ∈ S and z_{k,•} ∈ I^{k}M^{⋄} with y_{k,•} = s_{k,•}^{−1} · z_{k,•} for every k ∈ ℕ. Also, we may assume :

(17.5.9)  s_{k+1,•} ∈ V^{ℕ}s_{k,•}  for every k ∈ ℕ.

Now, by assumption there exists a sequence (a_{n} | n ∈ ℕ) of elements of V such that V a_{n} ⊂ V a_{n+1} for every n ∈ ℕ, and ⋃_{n∈ℕ} V a_{n} = 𝔪. For every k ∈ ℕ we define inductively a sequence of integers (n(k, i) | i ∈ ℕ) as follows. For i = 0, let n(k, 0) be the smallest of the integers n ∈ ℕ such that a_{0} ∈ V s_{k,n}; then, for every i > 0 let n(k, i) ∈ ℕ be the smallest of the integers n > n(k, i − 1) such that a_{i} ∈ V s_{k,n}. Notice that (17.5.9) implies

n(k + 1, i) ≥ n(k, i) for every k, i ∈ ℕ.

Lastly, let t_{•} be the sequence such that t_{n} := 0 for every n < n(0, 0), and t_{n} := a_{i} for every n, i ∈ ℕ with n(i, i) ≤ n < n(i + 1, i + 1). We have t_{•} ∈ S, and t_{i} ∈ V s_{k,i} for every k ∈ ℕ and every i ≥ n(k, k); thus, the class of s_{k,•} in V^{⋄} divides the class of t_{•}, and we may replace s_{k,•} by t_{•} for every k ∈ ℕ. By the foregoing case, the series ∑_{k∈ℕ} z_{k,•} converges in M^{⋄}, and let z_{•} be a limit; then t_{•}^{−1}z_{•} is a limit for x_{••} in S^{−1}M^{⋄}. □

Lemma 17.5.10. In the situation of (16.9.34), let us endow S^{−1}A_{0}^{⋄} with its b-adic topology, and let A_{0}^{∧} be the b-adic completion of A_{0}; then the basic setup (A_{0}^{∧}, 𝔪A_{0}^{∧}) is almost perfectoid if and only if the maximal separated quotient (S^{−1}A_{0}^{⋄})^{sep} of S^{−1}A_{0}^{⋄} is perfectoid.

Proof. Clearly S^{−1}\overline{Φ}_{A_{0}}^{⋄} : S^{−1}(A_{0}/bA_{0})^{⋄} → S^{−1}(A_{0}/b^{p}A_{0})^{⋄} is the ring endomorphism induced by Frobenius endomorphism of S^{−1}(A_{0}/b^{p}A_{0})^{⋄}. On the other hand, by lemma 17.5.4(ii), \overline{Φ}_{A_{0}}^{a} is an isomorphism if and only if the same holds for S^{−1}\overline{Φ}_{A_{0}}^{⋄}. Notice also that C := (S^{−1}A_{0}^{⋄})^{sep} is b-adically complete and separated, by virtue of propositions 17.5.8 and 8.2.13(v). Then let \overline{Φ}_{C} : C/bC → C/b^{p}C be the ring homomorphism induced by the Frobenius endomorphism of C/b^{p}C; in view of theorem 16.4.1, we are reduced to showing :

Claim 17.5.11. \overline{Φ}_{C} is an isomorphism if and only if the same holds for S^{−1}\overline{Φ}_{A_{0}}^{⋄}.

Proof of the claim. By lemma 17.5.1(ii), the projection A_{0} → A_{0}/bA_{0} induces a ring isomorphism S^{−1}A_{0}^{⋄}/(b) →^{∼} S^{−1}(A_{0}/bA_{0})^{⋄}, and likewise S^{−1}A_{0}^{⋄}/(b^{p}) →^{∼} S^{−1}(A_{0}/b^{p}A_{0})^{⋄}. Therefore, the projections S^{−1}A_{0}^{⋄} → C/bC and S^{−1}A_{0}^{⋄} → C/b^{p}C factor through ring isomorphisms

S^{−1}(A_{0}/bA_{0})^{⋄} →^{∼} C/bC S^{−1}(A_{0}/b^{p}A_{0})^{⋄} →^{∼} C/b^{p}C

whence the claim. □

Remark 17.5.12. As an alternative to our constructions of G^{⋄} and S^{−1}M^{⋄}, one could use an ultraproduct with respect to a non-principal ultrafilter of ℕ: this would lead to analogous results. In a similar vein, see remark 17.5.47.

17.5.13. Let A be a Tate ring, A_{0} ⊂ A a subring of definition, and p ∈ ℕ a prime integer; suppose that there exists b ∈ A_{0}^{∘∘} ∩ A^{×} with pA_{0} ⊂ b^{p}A_{0}, and let A_{0}^{∧} be the b-adic completion of A_{0}. Let also g_{•} := (g_{n} | n ∈ ℕ) ∈ E(A_{0}), and set g := g_{0} ∈ A_{0}, so that g^{γ} is well defined in A_{0} for every γ ∈ ℕ[1/p]. We let 𝔪 := ⋃_{γ∈ℕ[1/p]\{0}} A_{0}g^{γ}, and we suppose that the basic setup (A_{0}^{∧}, 𝔪A_{0}^{∧}) is almost perfectoid (see definition 16.9.35).

Proposition 17.5.14. In the situation of (17.5.13), suppose moreover that A_{0} is complete and separated, and let A_{0}^{ν} be the integral closure of the image of A_{0} in A_{0}[1/g]. Then we have :

(i) The induced morphism A_{0}^{a} → (A_{0}^{ν})^{a} is an isomorphism of (A_{0}, 𝔪)^{a}-algebras.

(ii) If moreover, the unit of adjunction A_{0} → A_{0∗}^{a} is an isomorphism, the same holds for the natural map A_{0} → A_{0}^{ν}.

Proof. (i): Let us check first that the kernel of the localization A_{0} → A_{0}[1/g] is annihilated by 𝔪. Hence, let x ∈ A_{0} such that g^{n}x = 0 for some n ∈ ℕ, and denote by \overline{x}_{•} ∈ B := (S^{−1}A_{0}^{⋄})^{sep} the image of the constant sequence (x, x, x, …) ∈ A_{0}^{ℕ} (notation of lemma 17.5.10). Since B is reduced (corollary 16.3.63), we have g^{γ} · \overline{x} = 0 in B for every γ ∈ ℕ[1/p] \ \{0\}. The latter means that for all i ∈ ℕ and γ ∈ ℕ[1/p] \ {0} there exists s_{•} := (s_{n} | n ∈ ℕ) ∈ S such that s_{•} · g^{γ} · \overline{x} ∈ b^{i}A_{0}^{⋄}. In turn, this means that for every such i and γ there exists s_{•} and n ∈ ℕ such that s_{m}g^{γ}x ∈ b^{i}A_{0} for every m ≥ n; equivalently, for every i ∈ ℕ and γ ∈ ℕ[1/p] \ {0} we have g^{γ}x ∈ b^{i}A_{0}, and then g^{γ}x = 0 in A_{0} for every such γ, as required.

Next, we show that the cokernel of the map A_{0} → A_{0}^{ν} is annihilated by 𝔪. Indeed, let a/g^{n} ∈ A_{0}^{ν} for some a ∈ A_{0} and n ∈ ℕ, and denote by \overline{a}_{•} ∈ B the image of the constant sequence (a, a, a, …) ∈ A_{0}^{ℕ}. Hence \overline{a}_{•}/g^{n} ∈ B[g^{−1}] lies in the integral closure C of the image \overline{B} of B in B[1/g]. Since B is perfectoid, the induced map B^{a} → C^{a} is an isomorphism of (A_{0}, 𝔪)^{a}-algebras (theorem 16.9.42(iv)), hence g^{γ} · (\overline{a}_{•}/g^{n}) ∈ \overline{B} for every γ ∈ ℕ[1/p] \ \{0\}; then there exists y_{•} := (y_{n} | n ∈ ℕ) ∈ B such that

g^{γ+γ′} · \overline{a}_{•} = g^{n+γ′} · y_{•} for every γ, γ′ ∈ ℕ[1/p].

Hence, for every i ∈ ℕ there exists s_{•} ∈ S such that s_{•}(g^{γ+γ′} · \overline{a}_{•} − g^{n+γ′} · y_{•}) ∈ b^{i}A_{0}^{⋄}. The latter implies that for every i ∈ ℕ and every γ ∈ ℕ[1/p] \ {0} the image z_{γ,i} of g^{γ}a in A_{0}/b^{i}A_{0} lies in J_{i} := g^{n}(A_{0}/b^{i}A_{0}). For every i ∈ ℕ, let K_{i} := Ker g^{n}1_{A_{0}/b^{i}A_{0}}; we get an exact sequence

K := lim_{i∈ℕ} K_{i} → A_{0} = lim_{i∈ℕ} A_{0}/b^{i}A_{0} ⟶^{α} J := lim_{i∈ℕ} J_{i} → lim_{n∈ℕ}^{1}K_{i}

and notice that K = Ker g^{n}1_{A_{0}} lies in the kernel of the localization A_{0} → A_{0}[1/g], hence K^{a} = 0, by the foregoing. Moreover, the system of inclusions (J_{i} → A_{0}/b^{i}A_{0} | i ∈ ℕ) induces a map β : J → A_{0} such that β ∘ α = g^{n}1_{A_{0}}.

Claim 17.5.15. The inverse system (K_{i} | i ∈ ℕ) is almost essentially zero.

Proof of the claim. Since A_{0} is almost perfectoid, the Frobenius endomorphism of A_{0}/b^{p}A_{0} induces a map A_{0}/bA_{0} → A_{0}/b^{p}A_{0} whose kernel is almost zero. By a simple induction on j, it follows that for every j ∈ ℕ and every x ∈ A_{0} such that x^{p} ∈ b^{pj}A_{0}, we have g^{γ}x ∈ b^{j}A_{0} for every γ ∈ ℕ[1/p] \ {0}. Now, fix i, k ∈ ℕ and let x ∈ A_{0} whose class in A_{0}/b^{p^{k}i}A_{0} lies in K_{p^{k}i}, i.e. g^{n}x ∈ b^{p^{k}i}A_{0}. Then (g^{n/p^{k}}x)^{p^{k}} ∈ b^{p^{k}i}A_{0}, and an easy induction on j yields

(g^{(j+1)n/p^{k}}x)^{p^{k−j}i} ∈ b^{p^{k−j}i}A_{0} for every j = 0, …, k.

For j = k, it follows that g^{(k+1)n/p^{k}}x ∈ b^{i}A_{0}. This shows that for every i ∈ ℕ and every β ∈ ℕ[1/p] \ {0} there exists k ∈ ℕ such that the image of K_{p^{k}i} in K_{i} is annhilated by g^{β}, whence the contention. ♢

From proposition 14.2.25(iii) and claim 17.5.15 we get lim_{i∈ℕ}^{1} K_{i}^{a} = 0, hence α is almost surjective. By construction, z_{γ,•} := (z_{γ,i} | i ∈ ℕ) ∈ lim_{i∈ℕ} J_{i}; so we find for every γ ∈ ℕ[1/p] \ {0} an element a′ ∈ A such that α(a′) = g^{γ}z_{γ,•}, whence g^{n}a′ = β ∘ (a′) = g^{γ}β(z_{γ,•}) = g^{2γ}a. The image of a′ in A_{0}^{ν} therefore equals g^{2γ}(a/g^{n}), and the proof of (i) is complete.

(ii): We have already seen that 𝔪 annihilates the kernel of the localization j : A_{0} → A_{0}[1/g]; on the other hand, A_{0∗}^{a} has no 𝔪-torsion elements, so j is injective under the assumptions of (ii). By (i), the inclusion map A_{0} → A_{0}^{ν} induces an isomorphism A_{0} →^{∼} A_{0∗}^{a} →^{∼} (A_{0}^{ν})_{∗}^{a}, and notice that (A_{0}^{ν})_{∗}^{a} is a subring of A_{0}[1/g] containing A_{0}^{ν}; the assertion follows immediately. □

Proposition 17.5.16. In the situation of (17.5.13), let h ∈ A_{0}, and set B := A_{0}[h^{1/p^{∞}}] = A_{0}[X^{1/p^{∞}}]/(X − h). Let moreover C be the p-root closure of B in B[1/b], and C^{∧} the b-adic completion of C. Then C^{a} is a faithfully flat (A_{0}, 𝔪)^{a}-algebra, and the basic setup (C^{∧}, 𝔪C^{∧}) is almost perfectoid for the b-adic topology of C^{∧}.

Proof. For every A_{0}-module M, let Δ_{M} : M → M^{ℕ} be the diagonal map, and for every n ∈ ℕ, let π_{M,n} : M^{ℕ} → M be the projection on the n-th factor. Let us regard A_{0}^{ℕ} as an A_{0}-algebra via Δ_{A_{0}} : A_{0} → A_{0}^{ℕ}, and denote by C′ the p-root closure of B′ := B ⊗_{A_{0}} A_{0}^{ℕ} in B′[1/b]. By remark 9.8.25(ii), the induced map B[1/b] → B′[1/b] restricts to a ring homomorphism j_{C} : C → C′. By the same token, B[1/p] ⊗_{A_{0}} π_{A_{0},n} : B′[1/b] → B[1/b] restricts to a map φ_{C,n} : C′ → C such that φ_{C,n} ∘ j_{C} = 1_{C}, for every n ∈ ℕ. For every such M and every n ∈ ℕ we then get maps of A_{0}-modules :

T_{0} := Tor_{1}^{A_{0}}(M, C) ⟶^{α} T := Tor_{1}^{A_{0}^{ℕ}}(M^{ℕ}, C′) ⟶^{β_{n}} T_{0}

where α is induced by (Δ_{A_{0}}, Δ_{M}, j_{C}), and β_{n} is induced by (π_{A_{0},n}, π_{M,n}, φ_{C,n}). On the other hand, S^{−1}A_{0}^{ℕ} is a formally perfectoid ring for its b-adic topology, by lemma 17.5.10and remark 17.5.3(iii), hence S^{−1}C′ is a faithfully flat S^{−1}A_{0}^{ℕ}-algebra, by theorem 16.9.17(i) and lemma 9.8.26(iv). Hence, S^{−1}T = 0. Now, let x ∈ T_{0}; then there exists s_{•} := (s_{n} | n ∈ ℕ) ∈ S with s_{•} · α(x) = 0 in T, whence s_{n} · β_{n} ∘ α(x) = s_{n}x = 0 in T_{0} for every n ∈ ℕ, so T_{0}^{a} = 0. This shows that C^{a} is a flat A_{0}^{a}-algebra. Next, consider, for every n ∈ ℕ, the commutative diagram :

\xymatrix{M \ar[r]^{Δ_{M}} \ar[d]_{γ} & M^{ℕ} \ar[d]^{δ} \\ C ⊗_{A_{0}} M \ar[r]^{j_{C}⊗_{A_{0}}Δ_{M}} & C′ ⊗_{A_{0}^{ℕ}} M^{ℕ}}

and let y ∈ Ker γ; then Δ_{M}(y) ∈ Ker δ, and since S^{−1}C′ is a faithfully flat S^{−1}A_{0}^{ℕ}-algebra, it follows that there exists s_{•} ∈ S with s_{•} · Δ_{M}(y) = 0, i.e. s_{n}y = 0 for every n ∈ ℕ, so that (Ker γ)^{a} = 0, which proves that C^{a} is a faithfully flat A_{0}^{a}-algebra.

Next, since b is a regular element of C, it is also regular in C^{∧}; also, we already know that the Frobenius endomorphism of C^{∧} induces an injection \overline{Φ}_{C} : C/bC → C/b^{p}C (lemma 9.8.26(i)); on the other hand, for any n ∈ ℕ, consider the commutative diagram :

\xymatrix{C/bC \ar[r]^{A_{0}/bA_{0}⊗_{A_{0}}j_{C}} \ar[d]_{_{\overline{Φ}_{C}}} & C′/bC′ \ar[r]^{A_{0}/bA_{0}⊗_{A_{0}}φ_{C,n}} \ar[d]_{_{\overline{Φ}_{C′}}} & C/bC \ar[d]^{_{\overline{Φ}_{C}}} \\ C/b^{p}C \ar[r]^{A_{0}/b^{p}A_{0}⊗_{A_{0}}j_{C}} & C′/b^{p}C′ \ar[r]^{A_{0}/b^{p}A_{0}⊗_{A_{0}}φ_{C,n}} & C/b^{p}C.}

By theorem 16.9.17(ii), the map S^{−1}\overline{Φ}_{C′} is an isomorphism. Now, let x ∈ C/b^{p}C, and pick s_{•} := (s_{n} | n ∈ ℕ) ∈ S and y_{•} ∈ C′/bC′ such that \overline{Φ}_{C′}(y_{•}) = s_{•} · (A_{0}/b^{p}A_{0} ⊗_{A_{0}} j_{C})(x). Set z_{n} := (A_{0}/bA_{0} ⊗_{A_{0}} φ_{C,n})(y_{•}) for every n ∈ ℕ; then \overline{Φ}_{C}(z_{n}) = s_{n}x for every n ∈ ℕ. This shows that \overline{Φ}_{C} is an almost isomorphism, so (C^{∧}, 𝔪C^{∧}) is an almost perfectoid basic setup. □

Remark 17.5.17. (i) In the situation of (17.5.13), suppose moreover that there exists a perfectoid Tate ring R, with a perfectoid subring of definition R_{0}, and an adic ring homomorphism R_{0} → A_{0}, such that g_{•} is the image of an element g′_{•} := (g′_{n} | n ∈ ℕ) of E(R_{0}). Then we get a basic setup (R_{0}, 𝔪_{R}) with 𝔪_{R} := ⋃_{γ∈ℕ[1/p]\{0}} g_{0}^{′γ}R_{0}, and we can form as in (16.9.59) the topological R-algebra A_{0}^{♮} := R⊗_{W (E_{0}(R_{0}))}W (E(A_{0}/pA_{0})), which is perfectoid and admits a natural adic map A_{0}^{♮} → A_{0}; the latter is an almost isomorphism relative to the basic setup (R_{0}, 𝔪_{R}), by theorem 16.9.63(iii.a). Let A_{0}^{ν} (resp. (A_{0}^{♮})^{ν}) be the integral closure of A_{0} in A_{0}[1/g] (resp. of A_{0}^{♮} in A_{0}^{♮}[1/g′_{0}]). Then, on the one hand, the induced morphism ((A_{0}^{♮})^{ν})^{a} → (A_{0}^{ν})^{a} is an isomorphism of (R_{0}, 𝔪_{R})^{a}-algebras ([75, Lemma 8.2.28]); on the other hand, the same holds for the morphism (A_{0}^{♮})^{a} → ((A_{0}^{♮})^{ν})^{a}, due to theorem 16.9.42(iv). In this way we thus obtain a shorter proof of proposition 17.5.14, under the stated further assumptions.

(ii) We can also give an alternative proof of the first assertion of proposition 17.5.16 along the same lines, under the assumptions of (i). Indeed, first let us remark that, since the map A_{0}^{♮} → A_{0} is an almost isomorphism, there exists an almost element l ∈ (A_{0}^{♮})_{∗}^{a} whose image in (A_{0})_{∗}^{a} agrees with h; by definition, l : 𝔪_{R} ⊗_{R} 𝔪_{R} → A_{0}^{♮} is an R-linear map, and we set l_{γ+δ} := l(g_{0}^{′γ} ⊗ g_{0}^{′δ}) for every γ, δ ∈ ℕ[1/p] \ {0}. It is easily seen that l_{γ+δ} depends only on the sum γ + δ, hence we get a compatible system (l_{γ} | γ ∈ ℕ[1/p] \ {0}) of elements of A_{0}^{♮} such that g_{0}^{′δ} · l_{γ} = l_{γ+δ} for every such γ and δ. Let D_{γ} (resp. E_{γ}) be the p-root closure of A_{0,γ}^{♮} := A_{0}^{♮}[l_{γ}^{1/p^{∞}}] in A_{0,γ}^{♮}[1/b] (resp. of A_{0,γ} := A_{0}[(g^{γ}h)^{1/p^{∞}}] in A_{0,γ}[1/b]) for every γ ∈ ℕ[1/p] \ {0}; by theorem 16.9.17(i), D_{γ} is a faithfully flat A_{0}^{♮}-algebra; moreover, for every γ, δ ∈ ℕ[1/p] \ {0}, the natural map A_{0,γ+δ}^{♮} → A_{0,γ}^{♮} induces a map of A_{0}^{♮}-algebras D_{γ+δ} → D_{γ} (remark 9.8.25(ii)). Furthermore, for every such γ we have a natural map i_{γ} : A_{0,γ}^{♮} → A_{0,γ}, whence an induced map j_{γ} : D_{γ} → E_{γ}. Since p ∈ b^{p}A_{0}, the localization A_{0,γ} → A_{0,γ}[1/b] induces an isomorphism A_{0,γ}[1/p] →^{∼} A_{0,γ}[1/b, 1/p], and therefore E_{γ} is also the weak normalization of A_{0,γ} in A_{0,γ}[1/b] (proposition 9.8.28); likewise we see that D_{γ} is the weak normalization of A_{0,γ}^{♮} in A_{0,γ}^{♮}[1/b]. On the other hand, clearly i_{γ}^{a} is an isomorphism; taking into account 14.3.12(iii), we conclude that the same holds for j_{γ}^{a}. Lastly, the compatible system of ring homomorphisms (A_{0}[(g^{γ}h)^{1/p^{∞}}] → A_{0}[h^{1/p^{∞}}] | γ ∈ ℕ[1/p] \ {0}) yields a compatible system of maps (E_{γ} → B | γ ∈ ℕ[1/p] \ {0}), and it is easily seen that the induced co-cone of A_{0}^{a}-algebras (E_{γ}^{a} → B^{a} | γ ∈ ℕ[1/p] \ {0}) is universal. Summing up, B^{a} is the colimit of the filtered system of faithfully flat A_{0}^{a}-algebras (D_{γ}^{a} | γ ∈ ℕ[1/p] \ {0}), so it is a faithfully flat A_{0}^{a}-algebra (lemma 14.3.7(ii)).

(iii) Likewise, by applying the functor (−)^{♮} in a similar fashion, one may show that the basic setup (C^{∧}, 𝔪C^{∧}) is almost perfectoid : we leave the details as an exercise for the reader.

17.5.18. Let (A, 𝔫) be a noetherian local ring. The aim of this section is to exhibit an A-algebra B such that depth_{A}B = dim A (notation of remark 10.4.29). Quite generally, we say that an A-module M is big Cohen-Macaulay (or briefly, that M is big CM) if depth_{A}M = dim A; hence our sought B is a big CM A-algebra.

Remark 17.5.19. Set d := dim A, and let f := (f_{1}, …, f_{d}) a system of parameters for A, i.e. a system of generators for an 𝔫-primary ideal of A. Let also M be an A-module.

(i) By proposition 10.4.32(i), M is a big CM A-module if and only if H^{i}(f, M) = 0 for every i = 0, …, d − 1, and H^{d}(f, M) ≠ 0, or equivalently, if and only if f is completely secant on M (see definition 7.8.6), and H_{0}(f, M) = M/fM ≠ 0 (lemma 7.8.2(v)). It follows easily that M is a big CM A-module if and only if H^{i}(f, M) = 0 for i = 0, …, d − 1, and M/𝔫M ≠ 0.

(ii) Notice also that a big CM A-module is the same as a Cohen-Macaulay module whose support contains an irreducible component of Spec A of dimension d (see definition 11.3.60(ii)). The additional big adjective can then be somewhat justified, by remarking that the condition on the support often forces an A-algebra B to be of non-finite type : the question is discussed in B.Bhatt’s article [21], which exhibits a non-trivial cohomological obstruction to the existence of finite CM A-algebras, when A has positive characteristic. On the other hand, it is conjectured that every complete local noetherian domain admits a faithful Cohen-Macaulay module of finite type : see [142, Conj.12]; moreover, sometimes a big CM-algebra can be of finite type : e.g. this is the case for a Nagata universally catenary local domain (A, 𝔫) of dimension 2. Indeed, the normalization B of A in its field of fractions is a finite A-algebra, hence noetherian, catenary and of dimension 2; so it is a Cohen-Macaulay ring, by Serre’s criterion for normality ([126, Th.23.8]), and the support of B is Spec A.

(iii) Let f : (A′, 𝔫′) → (A, 𝔫) be a local ring homomorphism of local noetherian rings with dim A′ = dim A, and such that 𝔫 is the radical of 𝔫′A. Let B be an A-algebra, and B′ an A′-algebra. We have :

(a) B is a big CM A-algebra if and only if, by restriction of scalars, B is a big CM A′- algebra (lemma 10.4.17(ii)).

(b) If f is faithfully flat, then B′ is a big CM A′-algebra if and only if A ⊗_{A′} B′ is a big

CM A-algebra (corollary 10.4.37). (iv) Let B_{•} := (B_{λ} | λ ∈ Λ) be a filtered system of big CM A-algebras. Then the colimit

B of B_{•} is also a big CM A-algebra. Indeed, since the functor H_{i}(f, −) commutes with filtered colimits, we have H_{i}(f, B) = 0 for i = 0, …, d − 1, and if we had B/𝔫B = 0, we would have 1 = 0 in B/𝔫B, so 1 ∈ 𝔫B_{λ} for some λ ∈ Λ, but we know from (i) that B_{λ}/𝔫B_{λ} ≠ 0, whence the assertion.

Lemma 17.5.20. Let (A, 𝔪_{A}) and (B, 𝔪_{B}) be local noetherian rings, f : A → B a local ring homomorphism of complete intersection (see definition 9.7.30(ii)), and M a big CM A-module. Then the 𝔪_{B}-adic completion B ˆ⊗_{A}M of B ⊗_{A} M is a big CM B-module.

Proof. We establish first the following special cases :

Claim 17.5.21. (i) The 𝔪_{A}-adic completion M^{∧} of M is a big CM A-module.

(ii) If f is surjective, with kernel generated by a regular sequence g := (g_{1}, …, g_{r}) of A, and M is 𝔪_{A}-adically complete and separated, then B ⊗_{A} M is a big CM B-module.

Proof of the claim. (i): Let f be any system of parameters for A. Clearly the sequence f is M-quasi-regular if and only if it is M^{∧}-quasi-regular (see definition 7.8.13). Then the assertion follows from proposition 7.8.15 and remark 17.5.19(i). (The assertion is already essentially contained in [15, Th.1.7], which shows, by a different argument, that if a given system of parameters for A is M-regular, then every system of parameters for A is M^{∧}-regular.)

(ii): By [63, Ch.0, Prop.16.4.1] we may find a sequence g′ := (g_{r+1}, …, g_{d}) such that g′′ := (g_{1}, …, g_{d}) is a system of parameters for A, and then H_{i}(g′′, M) = 0 for every i > 0, by remark 17.5.19(i). Consequently, g′′ is a regular sequence on M (proposition 7.8.15), so that g′ is regular on M/gM, and thus H_{i}(g′, B ⊗_{A} M) = 0 for every i > 0, again by proposition 7.8.15; in view of remark 17.5.19(i), we are then reduced to checking that g′ is a system of parameters for B. But the latter follows immediately from [126, Th.14.1]. ♢

Now, let us pick a Cohen factorization A ⟶^{g} (C, 𝔪_{C}) ⟶^{h} B^{∧} of f, such that Ker h is generated by a regular sequence of 𝔪_{C}. Since g is flat, and \overline{C} := C/𝔪_{A}C is regular, we have :

dim C = dim A + dim \overline{C} = depth_{A}M + depth\overline{C} \overline{C} = depth_{C}(C ⊗_{A} M)

by [126, Th.15.1] and theorem 10.4.38. Combining with claim 17.5.21(i), we see that the 𝔪_{C}- adic completion C ˆ⊗_{A}M of C ⊗_{A} M is a CM C-module. Then, claim 17.5.21(ii) says that N := B^{∧} ⊗_{C} (C ˆ⊗_{A}M) is a CM B^{∧}-module, so also a CM B-module, by remark 17.5.19(iii.a). Lastly, it is easily seen that the 𝔪_{B}-adic completion of N is naturally isomorphic to B ˆ⊗_{A}M, so the latter is a big CM B-module, again by claim 17.5.21(i). □

To explain how we shall produce big CM-algebras, let us make the following :

Definition 17.5.22. Let (A, 𝔫) be a noetherian local ring, V an A-algebra, 𝔪 ⊂ V an ideal such that 𝔪 = 𝔪^{2}, so that (V, 𝔪) is a basic setup, in the sense of [75, §2.1.1], and suppose moreover that 𝔪̃ := 𝔪 ⊗_{V} 𝔪 is a flat V -module. Let also M be a V -module; we say that M is big almost CM over A if M_{!}^{a} is a big Cohen-Macaulay A-module.

With this terminology, our strategy relies on the following observation, which is a replacement of Hochster’s techniques of algebra modifications, that have been employed in previous papers, to turn almost CM algebras into CM algebras (see e.g. [94], [5], [7], [153]).

Lemma 17.5.23. In the situation of definition 17.5.22, set d := dim A, and suppose that the basic setup (V, 𝔪) fulfills the condition of (17.5.2). The following conditions are equivalent :

(a) M is big almost CM over A.

(b) (R^{i}Γ_{{𝔫}}M)^{a} = 0 for every i = 0, …, d − 1, and (R^{d}Γ_{{𝔫}}M)^{a} ≠ 0.

(c) M^{⋄} is big almost CM over A.

(d) S^{−1}M^{⋄} is a big Cohen-Macaulay A-module.

Proof. (Here, S ⊂ V^{ℕ} is the multiplicative subset associated with 𝔪, as in (17.5.2).) Let Z := Spec V/𝔫V; ; in view of lemma 10.4.13(v), we have a natural identification

R^{i}Γ_{{𝔫}}M →^{∼} R^{i}Γ_{Z}M for every i ∈ ℕ

whence a natural V -module structure on R^{i}Γ_{{𝔫}}M, so the (V, 𝔪)^{a}-module (R^{i}Γ_{{𝔫}}M)^{a} is well defined, and the lemma will follow from the more general :

Claim 17.5.24. Let I ⊂ V be a finitely generated ideal, d ∈ ℕ, and Z := Spec V/I ⊂ Spec V. The following conditions are equivalent :

(a) depth_{I}(M_{!}^{a}) = d (notation of (10.4.29)).

(b) (R^{i}Γ_{Z}M)^{a} = 0 for every i < d and (R^{d}Γ_{Z}M)^{a} ≠ 0.
(c) depth_{I}((M^{⋄})_{!}^{a}) = d.
(d) depth_{I}(S^{−1}M^{⋄}) = d.

Proof of the claim. (a)⇔(b): Since 𝔪̃ is a flat V -module, we have natural identifications :

(R^{i}Γ_{Z}M)_{!}^{a} →^{∼} 𝔪̃ ⊗_{V} R^{i}Γ_{Z}M →^{∼} R^{i}Γ_{Z}(𝔪̃ ⊗_{V} M) →^{∼} R^{i}Γ_{Z}(M_{!}^{a}) for every i ∈ ℕ

provided by lemma 10.4.17(iv), whence the contention.

(a)⇔(c)⇔(d): Let f be a finite system of generators of I, and K^{•}(f, M) the Koszul complex of M relative to the sequence f. Let S^{−1}K^{•}(f, M_{!})^{⋄} be the complex obtained by applying the functor S^{−1}(−)^{⋄} termwise to K^{•}(f, M_{!}), and define likewise K^{•}(M)^{⋄} and K^{•}(f, M)_{!}^{a}. By direct inspection (and since 𝔪̃ is a flat V -module), we easily obtain natural identifications :

K^{•}(f, M)_{!}^{a} →^{∼} K^{•}(f, M_{!}^{a}) K^{•}(f, M)^{⋄} →^{∼} K^{•}(f, M^{⋄}) S^{−1}K^{•}(f, M)^{⋄} →^{∼} K^{•}(f, S^{−1}M^{⋄}).

In view of lemmata 17.5.1(i) and 17.5.4(ii), we deduce for every i ∈ ℕ natural isomorphisms :

H^{i}(f, M)_{!}^{a} →^{∼} H^{i}(f, M_{!}^{a})  H^{i}(f, M)^{⋄} →^{∼} H^{i}(f, M^{⋄})  S^{−1}H^{i}(f, M)^{⋄} →^{∼} H^{i}(f, S^{−1}M^{⋄})

and moreover, H^{i}(f, M)^{a} = 0 if and only if S^{−1}H^{i}(f, M)^{⋄} = 0. Combining with proposition 10.4.32(i), the assertion follows. □

Hence, we shall seek a suitable basic setup (V, 𝔪) as in definition 17.5.22, fulfilling the additional condition of (17.5.2), together with a V -algebra C whose underlying V -module is big almost CM over A; in this situation, lemma 17.5.23 will ensure that S^{−1}C^{⋄} is a big CM A-algebra. Proposition 17.5.25 and theorem 17.5.28 below illustrate the method in the easier case where A is an 𝔽_{p}-algebra, for some prime integer p.

Proposition 17.5.25. Let p be a prime integer, (A, 𝔫) a noetherian local 𝔽_{p}-algebra, and a ∈ A a non-nilpotent element. Let also A^{perf} be the perfection of A, defined as in (9.8.11), and J ⊂ A^{perf} the ideal generated by (a^{1/p^{n}} | n ∈ ℕ). We have :

(i) J(A^{perf}/𝔫A^{perf}) ≠ 0.

(ii) If A is excellent and equidimensional, and A[a^{−1}] is a Cohen-Macaulay ring, then :

J · R^{i}Γ_{{𝔫}}A^{perf} = 0 for i = 0, …, dim A − 1.

Proof. (i): Let 𝔮 ⊂ A be a minimal prime ideal that does not contain a, and set B := A/𝔮; clearly B^{perf} is a quotient of A^{perf}, hence it suffices to check that J(B^{perf}/𝔫B^{perf}) ≠ 0. Thus, we may replace A by B, and assume that A is a domain. If a ∈ A^{×}, the assertion is trivial; suppose then that a ∉ A^{×}, in which case a lies in a prime ideal 𝔭 ⊂ A of height one ([126, Th.13.5]). Set C := A_{𝔭}; clearly C^{perf} is a localization of A^{perf}, hence it suffices to check that J(C^{perf}/𝔫C^{perf}) ≠ 0. Thus, we may replace A by C, and assume that A is a one-dimensional domain, in which case there exists a one-dimensional valuation ring V of Frac A that dominates A (claim 15.3.27). It is easily seen that V^{perf} is a one-dimensional valuation ring of Frac (A^{perf}) dominating A^{perf} (the details shall be left to the reader). We have to show that JA^{perf} ⊄ 𝔫A^{perf}, and it suffices to check that JV^{perf} ⊄ 𝔫V^{perf}. However, since 𝔫 is an ideal of finite type, 𝔫V^{perf} is a principal ideal lying in the maximal ideal 𝔪_{V}^{perf} of V^{perf}; on the other hand, the value group of V^{perf} is of rank one and p-divisible, and a ∈ 𝔪_{V}^{perf}, so JV^{perf} = 𝔪_{V}^{perf}, whence the assertion.

(ii): Let us first notice :

Claim 17.5.26. Let i ∈ ℕ such that a^{n}R^{i}Γ_{{𝔫}}A = 0 for some n ∈ ℕ; then J · R^{i}Γ_{{𝔫}}A^{perf} = 0.

Proof of the claim. Let Φ_{A} : A → A be the Frobenius endomorphism, and for every k ∈ ℕ and every A-module M, let M_{(k)} be the A-module obtained from M by restriction of scalars along the map Φ_{A}^{k} : A → A. Clearly 𝔫 is the radical of the ideal Φ_{A}^{k}(𝔫), for every k ∈ ℕ, hence we have a natural identification of A-modules :

H_{k}^{i} := R^{i}Γ_{{𝔫}}(A_{(k)}) = (R^{i}Γ_{{𝔫}}A)_{(k)} for every i, k ∈ ℕ

(lemma 10.4.13(v)). It follows that a^{n/k}H_{k}^{i} = 0 for every k ∈ ℕ. But by definition, A^{perf} is the direct limit of the system of A-algebras (A_{(k)} | k ∈ ℕ), hence R^{i}Γ_{{𝔫}}A^{perf} is the direct limit of the system of A-modules (H_{k}^{i} | k ∈ ℕ) (proposition 10.4.9), whence the claim. ♢

In light of claim 17.5.26, it suffices to check that for every i = 0, …, dim A − 1 we have a^{n}R^{i}Γ_{{𝔫}}A = 0 for some n ∈ ℕ. Let (A^{∧}, 𝔫^{∧}) be the completion of A; by assumption, the induced morphism of schemes Spec A^{∧} → Spec A is faithfully flat and regular, hence A^{∧}[a^{−1}] is again a Cohen-Macaulay ring ([126, Th.32.2(i)]), and dim A^{∧} = dim A; moreover, A^{∧} is equidimensional, by [126, Th.31.7]. Furthermore, since 𝔫^{∧} = 𝔫A^{∧}, we have a natural identification

R^{i}Γ_{{𝔫^{∧}}}A^{∧} = A^{∧} ⊗_{A} R^{i}Γ_{{𝔫}}A for every i ∈ ℕ

(lemma 10.4.17(iii)). Thus, it suffices to check that for every i = 0, …, dim A^{∧} − 1 there exists n ∈ ℕ such that a^{n}R^{i}Γ_{{𝔫^{∧}}}A^{∧} = 0; i.e. we may replace A by A^{∧}, and assume from start that A is complete, and especially, A admits a dualizing complex ω^{•} (remark 11.3.45(iv)). Set X := Spec A, and define the map c_{X} : |X| → ℤ as in (11.3.49); since A is equidimensional, we have c_{X}(𝔫) − c_{X}(𝔭) = dim A for every minimal prime ideal 𝔭 ⊂ A (lemma 11.3.50). Thus, after replacing ω^{•} by a suitable shift ω^{•}[c], we may assume that c_{X}(𝔭) = 0 for every minimal prime ideal 𝔭 ⊂ A, and consequently c_{X}(x) = dim 𝒪_{X,x} for every x ∈ X.

Claim 17.5.27. H^{i}(ω^{•}) = 0 for every i < 0, and A[a^{−1}] ⊗_{A} H^{i}(ω^{•}) = 0 for every i > 0.

Proof of the claim. Let x ∈ X be any point; by proposition 11.3.37, the complex ω^{•}(x) is dualizing on X(x), and denote by 𝒟_{x} the corresponding dualizing functor. Recall that 𝒟_{x}(𝒪_{X(x)}) = ω_{x}^{•}, and that the natural map

𝒪_{X(x)} → 𝒟_{x} ∘ 𝒟_{x}(𝒪_{X(x)}) = 𝒟_{x}(ω^{•}(x))

is an isomorphism. Let also A_{x}^{∧} be the completion of the local ring A_{x} := 𝒪_{X,x}; by corollary 11.5.17(iii) there follow natural isomorphisms :

D(H_{{x}}^{c(x)−i}𝒪_{X(x)}) →^{∼} D(H_{{x}}^{c(x)−i}𝒟_{x}(ω^{•}(x))) →^{∼} A_{x}^{∧} ⊗_{A} H^{i}(ω^{•}(x)) for every i ∈ ℤ

where the functor D is defined as in (11.5.16). By assumption, if x ∈ Spec A[a^{−1}], we have H_{{x}}^{c(x)−i}𝒪_{X(x)} = 0 for every i > 0, so that H^{i}(ω^{•}(x)) = 0 for every such i and every such x,

since A_{x}^{∧} is a faithfully flat A_{x}-algebra. Lastly, we have H_{{x}}^{c(x)−i}𝒪_{X(x)} = 0 for every i < 0 and every x ∈ X, by proposition 10.4.7(ii.b), hence H^{i}(ω^{•}(x)) = 0 for every such i and x. ♢

By corollary 11.5.17(ii) we have a natural isomorphism :

H_{{𝔫}}^{i}(A) →^{∼} H_{{𝔫}}^{i}(𝒟(ω^{•})) →^{∼} D(H^{c(𝔫)−i}(ω^{•})) for every i ∈ ℤ

and c(𝔫) = dim A. Since H^{j}(ω^{•}) is a finitely generated A-module for every j ∈ ℤ, claim 17.5.27 implies that there exists n ∈ ℕ such that a^{n} · H^{j}(ω^{•}) = 0 for every j > 0. Since D is an A-linear functor, it follows that a^{n} · H_{{𝔫}}^{i}(A) = 0 for every i < dim A, as required. □

Theorem 17.5.28. Let p be a prime integer, and (A, 𝔫) a local noetherian 𝔽_{p}-algebra. Then there exists a perfect big CM A-algebra.

Proof. We first notice :

Claim 17.5.29. We may assume that A is an excellent and equidimensional local ring.

Proof of the claim. Let 𝔭 be a minimal prime ideal of the completion A^{∧} of A, such that A′ := A^{∧}/𝔭 has dimension equal to dim A; then the natural map A → A′ fulfills the conditions of remark 17.5.19(ii); thus it suffices to exhibit a big CM A′-algebra, and A′ is an excellent local domain ([64, Ch.IV, Sch.7.8.3(iii)]). ♢

Thus, we assume now that A is as in claim 17.5.29. Then, by [64, Ch.IV, Sch.7.8.3(iv)], the subset U of points 𝔭 ∈ X := Spec A such that A_{𝔭} is Cohen-Macaulay, is open in X, and trivially contains all the maximal points of X, hence it is dense in X. We may therefore find a ∈ A such that ∅ ≠ Spec A[a^{−1}] ⊂ U, so that a is not nilpotent, and A[a^{−1}] is a Cohen-Macaulay ring. Define A^{perf} and J as in proposition 17.5.25. Clearly J^{2} = J, and the basic setup (A^{perf}, J) fulfills the condition of (17.5.2). By proposition 17.5.25(ii), we then have

(A^{perf}/𝔫A^{perf})^{a} ≠ 0  and (R^{i}Γ_{{𝔫}}A^{perf})^{a} = 0  for every i = 0, …, dim A − 1.

It follows that (A^{perf})_{!}^{a}/𝔫(A^{perf})_{!}^{a} ≠ 0. Recall that J^{∼} := J ⊗_{A^{perf}} J is a flat A^{perf}-module ([75, Prop.2.1.7(i)]), and let S_{a} ⊂ (A^{perf})^{ℕ} be the multiplicative subset associated with J, as in (17.5.2); then by lemmata 17.5.23 and 17.5.4(i,ii) we conclude that S_{a}^{−1}(A^{perf})^{⋄} is a big CM A-algebra, and since A^{perf} is perfect, it is easily seen that the same holds for S_{a}^{−1}(A^{perf})^{⋄}. □

It is useful to generalize theorem 17.5.28 to non-local noetherian 𝔽_{p}-algebras. To this aim, let us make the following :

Definition 17.5.30. Let A be a noetherian ring, M an A-module, and B an A-algebra. (i) We say that A is locally equidimensional if A_{𝔭} is equidimensional for every 𝔭 ∈ Spec A. (ii) We say that M is a big locally CM A-module, if M_{𝔭} is a big CM A_{𝔭}-module for every

𝔭 ∈ Spec A. Likewise, B is a big locally CM A-algebra, if it is big locally CM as an A-module.

(iii) We say that a ring homomorphism f : A → B is maximizing, if the continuous map Spec f is maximing (see definition 8.1.46(ii)).

Lemma 17.5.31. Let A be a catenary noetherian ring (see definition 9.7.3(iii)). The following conditions are equivalent :

(a) A is locally equidimensional.

(b) A_{𝔪} is equidimensional for every maximal ideal 𝔪 ⊂ A.

(c) For every two irreducible components Z and Z′ of Spec A, and every maximal point 𝔭 of the topological space Z ∩ Z′, the ring A_{𝔭} is equidimensional.

Proof. Trivially (a)⇒(b), and (a)⇒(c).

(b)⇒(a): Indeed, say that 𝔭 ∈ Spec A, and let 𝔮 and 𝔮′ be two minimal prime ideal contained in 𝔭; we need to check that two every saturated chain of prime ideals

(17.5.32)  𝔮 ⊂ ⋯ ⊂ 𝔭  and 𝔮′ ⊂ ⋯ ⊂ 𝔭

have the same length. To this aim, pick any maximal ideal 𝔪 of A containing 𝔭, and complete the two chains with the same saturated chain 𝔭 ⊂ ⋯ ⊂ 𝔪; the resulting chains 𝔮 ⊂ ⋯ ⊂ 𝔪 and 𝔮′ ⊂ ⋯ ⊂ 𝔪 are saturated, so by assumption they both have length equal to dim A_{𝔪}, whence the assertion.

(c)⇒(b): Let 𝔪 ⊂ A be a maximal ideal, and 𝔮, 𝔮′ two minimal prime ideals of A contained in 𝔪; set Z := Spec A/𝔮 and Z′ := Spec A/𝔮′. Then, 𝔪 ∈ Z ∩ Z′, so let 𝔭 be any maximal generization of 𝔪 in Z ∩ Z′; pick any two saturated chains as in (17.5.32), and complete them again with the same saturated chain 𝔭 ⊂ ⋯ ⊂ 𝔪, say of length l_{0}. Since A_{𝔭} is equidimensional, the chains (17.5.32) have the same length l_{1}; now, let 𝔮 ⊂ ⋯ 𝔪 and 𝔮′ ⊂ ⋯ 𝔪 be any other pair of saturated chains, of lengths l and respectively l′; since A is catenary, we must have l = l_{0} + l_{1} = l′, so A_{𝔪} is equidimensional. □

Corollary 17.5.33. Let A be an excellent and locally equidimensional 𝔽_{p}-algebra. Then there exists a big locally CM A-algebra B.

Proof. The subset U of points 𝔭 ∈ X := Spec A such that A_{𝔭} is Cohen-Macaulay is open ([64, Ch.IV, Sch.7.8.3(iv)]), and trivially contains the subset Min(A) of all minimal prime ideals of A, hence it is dense in X. Say that X \U = Spec A/I for some ideal I ⊂ A; then I is not contained in any minimal ideal of A, and so there exists a ∈ I \ ⋃_{𝔭∈Min(A)} 𝔭 ([12, Prop.1.11(i)]). Let A^{perf} be the perfection of A (see (9.8.11)), and J ⊂ A^{perf} the ideal generated by (a^{1/p^{n}} | n ∈ ℕ). Let moreover S_{a} ⊂ (A^{perf})^{ℕ} be the multiplicative subset associated with J, as in (17.5.2). We claim that B := S_{a}^{−1}(A^{perf})^{⋄} will do. Indeed, let 𝔭 ⊂ A be any prime ideal, k(𝔭) its residue field, j : A → A_{𝔭} the localization, and S_{j(a)} ⊂ (A_{𝔭}^{perf})^{ℕ} the corresponding multiplicative subset, associated with the ideal JA_{𝔭}.

Claim 17.5.34. S_{j(a)}^{−1}(A_{𝔭}^{perf})^{⋄} is a big CM A_{𝔭}-algebra; especially, S_{j(a)}^{−1}(A_{𝔭}^{perf})^{⋄} ⊗_{A_{𝔭}} k(𝔭) ≠ 0.

Proof of the claim. By [64, Ch.IV, Sch.7.8.3(ii)], A_{𝔭} is excellent and equidimensional. Then the assertion follows from the proof of theorem 17.5.28. ♢

The map j induces a ring homomorphism B → S_{j(a)}^{−1}(A_{𝔭}^{perf})^{⋄}; with claim 17.5.34we then get B ⊗_{A} k(𝔭) ≠ 0 for every 𝔭 ∈ Spec A. In view of (10.4.30), we are then reducing to showing :

Claim 17.5.35. depth_{𝔭}B = dim A_{𝔭} for every 𝔭 ∈ Spec A.

Proof of the claim. Set I := 𝔭A^{perf}, Z := Spec A/𝔭 and Z′ := Spec A^{perf}/I; by lemma 10.4.13(v), we have a natural identification :

R^{i}Γ_{Z}B →^{∼} R^{i}Γ_{Z′}B for every i ∈ ℕ.

In view of claim 17.5.24, we are then reduced to checking that :

depth_{I}((A^{perf})_{!}^{a}) = dim A_{𝔭}

for the almost structure attached to the basic setup (A^{perf}, J). However, recall that (A^{perf})_{𝔮} = (A_{𝔮})^{perf} for every 𝔮 ∈ Spec A (see (9.8.11)); also, ((A^{perf})_{!}^{a})_{𝔮} agrees with (A_{𝔮}^{perf})_{!}^{a}, for every such 𝔮, where now (A_{𝔮}^{perf})^{a} is regarded as an (A_{𝔮}^{perf}, JA_{𝔮}^{perf})^{a}-module, and (−)_{!} refers to the left adjoint of the localization A_{𝔮}^{perf}-Mod → (A_{𝔮}^{perf}, JA_{𝔮}^{perf})^{a}-Mod. It follows that :

depth_{A_{𝔮}}(A_{𝔮}^{perf})_{!}^{a} = dim A_{𝔮} for every 𝔮 ∈ Spec A

by virtue of lemma 17.5.23and claim 17.5.34. We conclude by invoking again (10.4.30). □

Remark 17.5.36. (i) Theorem 17.5.28was first proved by Hochster and Huneke in [95]; more precisely, they showed that for every excellent local domain A of equicharacteristic p > 0, the integral closure of A in the algebraic closure of Frac(A) is a big CM A-algebra; this result has been recently extended to excellent local domains of mixed characteristic, in [26]. Our construction yields a different big CM A-algebra.

(ii) Our construction can be easily upgraded to a functor on the category

ℬ_{p}

whose objects are all the excellent and locally equidimensional 𝔽_{p}-algebras, and whose morphisms are the maximizing ring homomorphisms (see definition 17.5.30(iii)). Namely, for every A ∈ Ob(ℬ_{p}), let Min(A) be the set of minimal primes of A, and endow A^{∗} := A \ ⋃_{𝔭∈Min(A)} 𝔭 with the partial preordering such that a ≤ b if and only if bA ⊂ aA, and for every a ∈ A^{∗}, define the multiplicative subset S_{a} ⊂ (A^{perf})^{ℕ} as in the proof of theorem 17.5.28. Clearly (A^{∗}, ≤) is a filtered category (see example 1.1.6(iii)), and we have S_{a} ⊂ S_{b} for every a, b ∈ A^{∗} with a ≤ b. Then, denote by E_{p}(A) the colimit of the induced filtered system of A-algebras (S_{a}^{−1}(A^{perf})^{⋄} | a ∈ A^{∗}). The proof of corollary 17.5.33 shows that A^{∗} admits a cofinal subset Σ ⊂ A^{∗} such that S_{a}^{−1}(A^{perf})^{⋄} is a big locally CM A-algebra for every a ∈ Σ; then the same holds for E_{p}(A) (remark 17.5.19(iv)). Now, let f : A → B be a morphism of ℬ_{p}; clearly f extends uniquely to a ring homomorphism f^{perf} : A^{perf} → B^{perf}, and restricts to a map f^{∗} : A^{∗} → B^{∗}, as f is maximizing. Moreover, (f^{perf})^{ℕ}(S_{a}) ⊂ S_{f(a)} for every a ∈ A^{∗}. Hence, we obtain an induced map S^{−1}(A^{perf})^{⋄} → S_{f(a)}^{−1}(B^{perf})^{⋄} → E_{p}(B) for every such a, and the colimit of these maps is a ring homomorphism E_{p}(f) : E_{p}(A) → E_{p}(B). It is easily seen that the rules : A ↦ E_{p}(A) and f ↦ E_{p}(f) yield the sought functor E_{p} : ℬ_{p} → 𝔽_{p}-Alg.

(iii) Moreover, consider the functor

(−)^{ν} : ℬ_{p} → ℬ_{p} A ↦ A^{ν} := ∏_{𝔭∈Min A}(A/𝔭)^{ν}

where, for every 𝔭 ∈ Min A, we denote by (A/𝔭)^{ν} the integral closure of A/𝔭 in its field of fractions. Clearly, the natural map j_{A} : A → A^{ν} is a morphism of ℬ_{0}. It is easily seen that every morphism f : A → B in ℬ_{p} induces a well defined morphism f^{ν} : A^{ν} → B^{ν}, such that f^{ν} ∘ j_{A} = j_{B} ∘ f; this completes the construction of the functor (−)^{ν}, and shows that the rule : A ↦ j_{A} yields a natural transformation j_{•} : 1_{ℬ_{p}} ⇒ (−)^{ν}. With this notation, we have :

Proposition 17.5.37. (i) The functor E_{p} factors through the functor (−)^{ν}.

(ii) The functor E_{p} commutes with finite products.

Proof. (i): We need to check that E_{p}(j_{A}) is an isomorphism for every A ∈ Ob(ℬ_{p}); to this aim, let A_{red} be the maximal reduced quotient of A, and notice that j_{A} factors through the quotient map j′_{A} : A → A_{red} and a unique map j′′_{A} : A_{red} → A^{ν}, and both maps are morphisms in ℬ_{p}. Now, clearly j′_{A} induces an isomorphism (j′_{A})^{perf} : A^{perf} →^{∼} (A_{red})^{perf} that identifies S_{a} with S_{j′_{A}(a)} for every a ∈ A^{∗} (notation of remark 17.5.36(ii)). Then, under these identifications, E_{p}(A_{red}) is the colimit of the functor

F : ((A_{red})^{∗}, ≤) → 𝔽_{p}-Alg a ↦ S_{a}^{−1}(A^{perf})^{⋄}

and E_{p}(A) is the colimit of the functor F ∘ (j′_{A})^{∗} : (A^{∗}, ≤) → 𝔽_{p}-Alg. But notice that the map (j′_{A})^{∗} : A^{∗} → (A_{red})^{∗} is surjective, so it is a cofinal functor of filtered categories (lemma 1.5.7(i)), and thus E_{p}(j′_{A}) is an isomorphism (proposition 1.5.2). Next, we remark :

Claim 17.5.38. (i) There exists c ∈ A_{red}^{∗} such that j′′_{A}(c) · A^{ν} ⊂ j′′_{A}(A_{red}).

(ii) For every c as in (i), and every n ∈ ℕ, we have c^{1/p^{n}}(A^{ν})^{perf} ⊂ A^{perf}.

Proof of the claim. (i): We further write j′′_{A} as the composition of injective maps

A_{red} ⟶^{i_{1}} B := ∏_{𝔭∈Min A}A/𝔭 ⟶^{i_{2}} A^{ν}.

It then suffices to find c_{1} ∈ A_{red}^{∗} and c_{2} ∈ B^{∗} such that c_{1}B ⊂ i_{1}(A_{red}) and c_{2}A^{ν} ⊂ i_{2}(B), since in that case c_{3} := i_{1}(c_{1})c_{2} ∈ i_{1}(A_{red}) ∩ B^{∗} = i_{1}(A_{red}^{∗}), and i_{2}(c_{3})A^{ν} ⊂ j′′_{A}(A_{red}). Now, let Z_{1}, …, Z_{k} be the irreducible components of Spec A_{red}, and set Z := ⋃_{1≤i<j≤k} Z_{i} ∩ Z_{j}; pick any ideal J ⊂ A_{red} such that Z = Spec A/J. Hence, J does not lie in any minimal prime ideal of A_{red}, so J ∩ A_{red}^{∗} ≠ ∅ ([12, Prop.1.11(i)]); let then d ∈ J ∩ A_{red}^{∗}. We have Z ∩ Spec A_{red}[d^{−1}] = ∅, so i_{1} induces an isomorphism A_{red}[d^{−1}] →^{∼} B[d^{−1}]. Since B is a finite A_{red}-algebra, it follows that d^{n}B ⊂ i_{1}(A_{red}), for some n ∈ ℕ, so c_{1} := d^{n} will do. Next, notice that B^{∗} = ∏_{𝔭∈Min A}(A/𝔭)^{∗} and (A^{ν})^{∗} = ∏_{𝔭∈Min A}((A/𝔭)^{ν})^{∗}, so in order to exhibit c_{2} it suffices to find for every 𝔭 ∈ Min A some c_{𝔭} ∈ (A/𝔭)^{∗} = (A/𝔭) \ {0} with c_{𝔭}(A/𝔭)^{ν} ⊂ A/𝔭. To this aim, it suffices to notice that, as A is excellent, (A/𝔭)^{ν} is a finite A/𝔭-subalgebra of Frac(A/𝔭).

(ii) shall be left to the reader. ♢ Pick c ∈ A_{red}^{∗} as in claim 17.5.38(i); set Σ := {ca | a ∈ A_{red}^{∗}} ⊂ A_{red}^{∗}, and let c_{•} ∈ (A_{red})^{⋄}

be the class of (c^{1/p^{n}} | n ∈ ℕ). With this notation, claim 17.5.38(i) states that the image of Σ is cofinal in the partially ordered set ((A^{ν})^{∗}, ≤). Moreover, j′′_{A} induces an injection A_{red}^{perf} = A^{perf} → (A^{ν})^{perf}; combining with claim 17.5.38(ii) we deduce that j′′_{A} induces an isomorphism

(A^{perf})^{⋄}[c_{•}^{−1}] →^{∼} ((A^{ν})^{perf})^{⋄}[c_{•}^{−1}]

and on the other hand, c_{•} ∈ S_{b} for every b ∈ Σ. Summing up, we conclude that E_{p}(j′′_{A}) is naturally identified with the colimit of the system of maps

(17.5.39)  S_{b}^{−1}((A^{ν})^{perf})^{⋄} → S_{j′′_{A}(b)}^{−1}((A^{ν})^{perf})^{⋄}  for every b ∈ Σ

induced by the inclusion S_{b} ⊂ S_{j′′_{A}(b)}, for every such b (here, S_{b} lies in (A^{perf})^{ℕ}, whereas S_{j′′_{A}(b)} lies in ((A^{ν})^{perf})^{ℕ}). In order to check that E_{p}(j′′_{A}) is an isomorphism, it then suffices to prove that the same holds for the maps (17.5.39); the latter follows at once from :

Claim 17.5.40. c_{•} · s_{•} ∈ S_{b}, for every s_{•} ∈ S_{j_{A}(b)}.

Proof of the claim. Say that s_{•} = (s_{n} | n ∈ ℕ) ∈ ((A^{ν})^{perf})^{ℕ} and b = ac for some a ∈ (A^{perf})^{∗}; we need to check that b^{1/p^{k}} ∈ ⋃_{n∈ℕ} c^{1/p^{n}}s_{n}A^{perf} for every k ∈ ℕ. However, by assumption j′′_{A}(b^{1/p^{k}}) ∈ ⋃_{n∈ℕ} s_{n}(A^{ν})^{perf}, so say that j′′_{A}(b^{1/p^{k}}) = s_{n}x for some n ∈ ℕ and x ∈ (A^{ν})^{perf}; also, we may assume that n > k. By claim 17.5.38(ii), we get c^{1/p^{n}}x ∈ A^{perf}, and it follows easily that (ac^{2})^{1/p^{k}} = (bc)^{1/p^{k}} ∈ c^{1/p^{n}}s_{n}A^{perf}. Then we get b^{2/p^{k}} = (ac)^{2/p^{k}} ∈ ⋃_{n∈ℕ} c^{1/p^{n}}s_{n}A^{perf} for every k ∈ ℕ, whence the claim. ♢

(ii): First, it is easily seen that the finite products in ℬ_{p} are represented by the usual (cartesian) products of rings (details left to the reader). Hence, say that A = A_{1} × ⋯ × A_{n}, for some A_{1}, …, A_{n} ∈ Ob(ℬ_{p}). It is easily seen that A^{∗} = A_{1}^{∗} × ⋯ × A_{n}^{∗}. It is also clear that (A^{perf})^{⋄} = ∏_{i=1}^{n}(A_{i}^{perf})^{⋄} (lemma 17.5.1(i)). Moreover, for each a_{•} := (a_{1}, …, a_{n}) ∈ A^{∗}, we have S_{a_{•}} = S_{a_{1}} × ⋯ × S_{a_{n}}, and thus S_{a_{•}}^{−1} = ∏_{i=1}^{n} S_{a_{i}}^{−1}(A_{i}^{perf})⋄. Since finite products commute with filtered colimits in the category of rings, the assertion follows. □

17.5.41. It turns out that the construction of big CM algebras over ℚ-algebras can be reduced to the case of algebras of positive characteristic. Indeed, consider first a locally equidimensional ℚ-algebra A of finite type. Let Δ(A) be the set of all ℤ-subalgebras B ⊂ A of finite type with ℚ ⊗_{ℤ} B = A. Let also Max ℤ be the maximal spectrum of ℤ, i.e. the set of prime integers.

Lemma 17.5.42. With the notation of (17.5.41), the following holds :

(i) Every connected component of Spec A is equidimensional.

(ii) For every B ∈ Δ(A) there exists a finite subset S_{B} ⊂ Max ℤ such that B/pB is excellent and locally equidimensional, for every p ∈ Max (ℤ) \ S_{B}.

Proof. (i): It suffices to check that if Z and Z′ are two irreducible components ot Spec A with Z ∩ Z′ ≠ ∅, then dim Z = dim Z′. Hence, pick any maximal ideal 𝔪 of A that lies in Z ∩ Z′; by assumption, A_{𝔪} is equidimensional, and Z(𝔪), Z′(𝔪) are two irreducible components of Spec A_{𝔪} (notation of definition 8.1.44(iii)), so dim Z(𝔪) = dim Z′(𝔪). On the other hand, dim Z = dim Z(𝔪), and likewise for Z′ ([64, Ch.IV, Prop.5.2.1]), whence the assertion.

(ii): Since the localization B → ℚ ⊗_{ℤ} B is injective, the unique morphism of schemes φ : Spec B → Spec ℤ is flat. Let T_{1}, …, T_{l} be the connected components of Spec A, and for i = 1, …, l, denote by \overline{T}_{i} the topological closure of T_{i} in Spec B; since φ is generizing, it follows easily that Spec B = \overline{T}_{1} ∪ ⋯ ∪ \overline{T}_{l}. On the other hand, for every i = 1, …, l, we have also T_{i} = Spec A/e_{i}A, for a unique idempotent element e_{i} ∈ A; we may then find d ∈ ℤ \ {0} such that e_{1}, …, e_{l} ∈ B[d^{−1}]. Set U := Spec B[d^{−1}]; it follows easily that the connected components of U are \overline{T}_{1} ∩ U, …, \overline{T}_{l} ∩ U. After replacing B by B[d^{−1}], we may then assume that B = B_{1} × ⋯ × B_{l}, and Spec ℚ ⊗_{ℤ} B_{i} is connected for every i = 1, …, l. Thus, it suffices to prove the assertion for A replaced by ℚ ⊗_{ℤ} B_{i} for every i = 1, …, l, and we may therefore assume from start that Spec A is connected, so A is equidimensional, by (i).

Let Z_{1}, …, Z_{k} be the irreducible components of Spec A, and denote by \overline{Z}_{i} the topological closure of Z_{i} in Spec B, for every i = 1, …, k; it follows easily that \overline{Z}_{1}, …, \overline{Z}_{k} are the irreducible components of Spec B, and we endow them with their reduced closed subscheme structures. Then the restriction φ_{i} : \overline{Z}_{i} → Spec ℤ of φ is a flat morphism of schemes, for every i = 1, …, k. According to lemma 11.4.1(i), for every p ∈ Max ℤ and every i = 1, …, k, every irreducible component of φ_{i}^{−1}(p) has dimension equal to dim Z_{i}, and we know already that the latter equals d := dim A. Summing up, for every p ∈ Max ℤ, every irreducible component of Spec B/pB has dimension d; combining with [64, Ch.IV, Prop.5.2.1], we deduce that for every maximal ideal 𝔪 ⊂ B/pB, the localization (B/pB)_{𝔪} is equidimensional of dimension d. Lastly, B/pB is an excellent ring, by [64, Ch.IV, Sch.7.8.3(ii)], so we may invoke lemma 17.5.31to conclude. □

For every B ∈ Δ(A), pick a finite subset S_{B} ⊂ Max ℤ fulfilling the condition of lemma 17.5.42(ii), and set

F_{0}(A, B) := ∏_{p∈Max(ℤ)\S_{B}} E_{p}(B/pB) and I_{0}(A, B) := ⨁_{p∈Max(ℤ)\S_{B}} E_{p}(B/pB)

where E_{p} is the functor exhibited in remark 17.5.36(ii). Notice then that F_{0}(A, B) is naturally a B-algebra, and I_{0}(A, B) is an ideal of F_{0}(A, B); then the quotient

E_{0}(A, B) := F_{0}(A, B)/I_{0}(A, B)

is again endowed with a natural B-algebra structure. Moreover, if B, B′ ∈ Δ(A), we may find d ∈ ℤ \ {0} such that B[d^{−1}] = B′[d^{−1}], and hence B/pB = B′/pB′ for every p ∈ Max(ℤ) \ Spec ℤ/dℤ. There follows a natural identification

E_{0}(A, B) →^{∼} E_{0}(A, B′)

in which of which, we shall denote this ring simply by E_{0}(A). Since A = ⋃_{B∈Δ(A)} B, we conclude that E_{0}(A) carries a natural A-algebra structure.

17.5.43. Next, let f : A → A′ be a maximizing ring homomorphism of locally equidimensional ℚ-algebras of finite type. Clearly, we may find B ∈ Δ(A) and B′ ∈ Δ(A′) such that f restricts to a ring homomorphism g : B → B′. If h : C → C′ is another such restriction with C ∈ Δ(A) and C′ ∈ Δ′, then for any d ∈ ℤ \ {0} such that B[d^{−1}] = C[d^{−1}] and B′[d^{−1}] = C′[d^{−1}] we have ℤ[d^{−1}] ⊗_{ℤ} g = ℤ[d^{−1}] ⊗_{ℤ} h, so the induced maps 𝔽_{p} ⊗_{ℤ} g : B/pB → B′/pB′ and 𝔽_{p} ⊗_{ℤ} h : C/pC → C′/pC′ are naturally identified for every p ∈ Max(ℤ) \ Spec ℤ/dℤ. Moreover, we have :

Lemma 17.5.44. With the notation of (17.5.43), there exists a finite subset S_{g} ⊂ Max ℤ such that 𝔽_{p} ⊗_{ℤ} g is maximizing for every p ∈ Max(ℤ) \ S_{g}.

Proof. Notice first that B and B′ are flat ℤ-algebra, so all the maximal points of X := Spec B and X′ := Spec B′ lie in Spec A and respectively Spec A′, so φ := Spec g is maximizing. Let Z_{1}, …, Z_{k} (resp. Z′_{1}, …, Z′_{l}) be the irreducible components of X (resp. of X′), which we endow with their reduced closed subscheme structures; it follows that for every i = 1, …, l there exists a unique t(i) ∈ {1, …, k} such that φ restricts to a dominant morphism of schemes φ_{i} : Z′_{i} → Z_{t(i)}. By [64, Ch.IV, Lemme 6.9.2], we may then find h ∈ B \ {0} such that, with U := Spec B[h^{−1}], the morphism φ_{i} restricts to a flat morphism

φ_{i|U} : φ_{i}^{−1}(U ∩ Z_{t(i)}) → U ∩ Z_{t(i)} for every i = 1, …, l.

Endow likewise the irreducible components T_{1}, …, T_{n} of Spec B/hB with their reduced closed subscheme structures; by the same token, we may find d ∈ ℤ \ {0} such that, with W := Spec ℤ[d^{−1}], the unique morphism of schemes ψ_{i} : T_{i} → Spec ℤ restricts to a flat morphism

ψ_{i}^{−1}(W) → W for every i = 1, …, n.

Let η ∈ Spec ℤ be the generic point; on the one hand, we have

dim ψ_{i}^{−1}(η) < dim A for every i = 1, …, n

and on the other hand, lemma 11.4.1(i) says that for every p ∈ W we have :

dim ψ_{i}^{−1}(p) = dim ψ_{i}^{−1}(η) for every i = 1, …, n and dim B/pB = dim A.

Summing up, we conclude that for every p ∈ W ∩Max(ℤ), all the maximal points of Spec B/pB lie in U. Arguing likewise with the irreducible components of Spec B′/hB′, we get an open subset W′ ⊂ Spec ℤ such that, for every p ∈ W ∩ Max(ℤ), all the maximal points of Spec B′/pB′ lie in φ^{−1}U. But since each φ_{i|U} is flat, for every p ∈ Max(ℤ) the restriction of φ :

φ^{−1}(U) ∩ Spec B′/pB′ → U ∩ Spec B/pB

is maximizing, so the lemma holds with S_{g} := Max(ℤ) \ (W ∩ W′). □

Using lemma 17.5.44, we may attach to any f : A → A′ as in (17.5.43) a well defined map

E_{0}(f) : E_{0}(A) → E_{0}(A′).

Namely, pick B, B′ and g : B → B′ as in (17.5.43), and finite subsets S_{B}, S_{B′}, S_{g} ⊂ Max ℤ as in lemmata 17.5.42and 17.5.44; set M := Max(ℤ) \ (S_{B} ∪ S_{B′} ∪ S_{g}) and

F_{0}(f, g) := ∏_{p∈M} E_{p}(𝔽_{p} ⊗_{ℤ} g) : ∏_{p∈M} E_{p}(B/pB) → ∏_{p∈M} E_{p}(B′/pB′).

Clearly F_{0}(f, g) maps the ideal ⨁_{p∈M} E_{p}(B/pB) into ⨁_{p∈M} E_{p}(B′/pB′), so it induces a unique map E_{0}(f) as sought. It is also easily seen that if A′′ is any other locally equidimensional ℚ-algebra of finite type, and f′ : A′ → A′′ is any maximizing ring homomorphism, then E_{0}(f′ ∘ f) = E_{0}(f′) ∘ E_{0}(f) : details left to the reader. Summing up, we have a well defined functor

E_{0} : ℬ_{0} → ℤ-Alg

where ℬ_{0} is the category whose objects are the locally equidimensional ℚ-algebras of finite type, and whose morphisms are the maximizing ring homomorphisms. Furthermore, just as in remark 17.5.36(iii), we have a normalization functor

(−)^{ν} : ℬ_{0} → ℬ_{0} A ↦ ∏_{𝔭∈Min A}(A/𝔭)^{ν}

(where (A/𝔭)^{ν} denotes the integral closure of A/𝔭 in Frac(A/𝔭), for every 𝔭 ∈ Min A), as well as a natural transformation j_{•} : 1_{ℬ_{0}} ⇒ (−)^{ν}. With this notation, we may now state :

Theorem 17.5.45. (i) E_{0}(A) is a big locally CM A-algebra, for every A ∈ Ob(ℬ_{0}). (ii) The functor E_{0} commutes with finite products. (iii) The functor E_{0} factors through the functor (−)^{ν}.

Proof. (ii): It is easily seen that the products in ℬ_{0} are represented by the usual (cartesian) products of rings; then, say that A = A_{1} × ⋯ × A_{k} for some A_{1}, …, A_{k} ∈ Ob(ℬ_{0}), and pick B_{i} ∈ Δ(A_{i}) for i = 1, …, k. Clearly B := B_{1} × ⋯ × B_{k} ∈ Δ(A), and in light of proposition 17.5.37(ii), we easily deduce that E_{0}(A, B) = ∏_{i=1}^{k} E_{0}(A_{i}, B_{i}), whence the assertion.

(i): In view of (ii) and lemma 17.5.42, we may assume that A is equidimensional. Then, let 𝔭 ⊂ A be any prime ideal; set Z := Spec A/𝔭, and c := dim A_{𝔭}. We show :

Claim 17.5.46. depth_{𝔭}(E_{0}(A)) = c.

Proof of the claim. Pick any B ∈ Δ(A), and let \overline{Z} be the topological closure of Z in Spec B, which we endow with its reduced closed subscheme structure. Hence, \overline{Z} = Spec B/𝔮, with 𝔮 = B ∩ 𝔭, and let f := (f_{1}, …, f_{r}) be any finite system of generators of 𝔮. In view of (10.4.30) and proposition 10.4.32(i), it suffices to check that H^{i}(f, E_{0}(A, B)) = 0 for every i < c and H^{c}(f, E_{0}(A, B)) ≠ 0. However, pick also a finite subset S_{B} ⊂ Max ℤ as in lemma 17.5.42(ii); moreover, notice that the image of \overline{Z} in Spec ℤ is constructible and dense ([63, Ch.IV, Th.1.8.4]), so its complement S_{Z} is a finite set of closed points. Set S := S_{B} ∪ S_{Z}; a direct inspection yields a natural short exact sequence of complexes of B-modules :

0 → ⨁_{p∈Max(ℤ)\S} K^{•}(f, E_{p}(B/pB)) → ∏_{p∈Max(ℤ)\S} K^{•}(f, E_{p}(B/pB)) → K^{•}(f, E_{0}(A, B)) → 0

so we are reduced to checking that for every p ∈ Max(ℤ) \ S we have :

H^{i}(f, E_{p}(B/pB)) = 0 for every i < c, and H^{c}(f, E_{p}(B/pB)) ≠ 0.

But since E_{p}(B/pB) is a big locally CM B/pB-algebra, invoking again (10.4.30) and proposition 10.4.32(i), we are further reduced to showing that \overline{Z} ∩ Spec B/pB has codimension c in Spec B/pB, for every p ∈ Max(ℤ) \ S. But the unique morphisms of schemes φ : Spec B → Spec ℤ and ψ : \overline{Z} → Spec ℤ are flat, and the generic fibre of φ (resp. ψ) is the equidimensional scheme Spec A (resp. Z), so φ^{−1}(p) is as well equidimensional of dimension dim A, for every p ∈ Max(ℤ) \ S, by lemma 11.4.1(i). By the same token, ψ^{−1}(p) is equidimensional of dimension dim Z, for every such p, whence the assertion. ♢

From claim 17.5.46 and (10.4.30), we already see that depth_{A_{𝔭}}E_{0}(A)_{𝔭} ≥ dim A_{𝔭} for every prime ideal 𝔭 ⊂ A. By the same token, if 𝔭 ⊊ 𝔮 are two such prime ideals, we have depth_{A_{𝔮}}E_{0}(A)_{𝔮} > dim A_{𝔭}, so, again by virtue of claim 17.5.46 and (10.4.30), we must have depth_{A_{𝔭}}E_{0}(A)_{𝔭} = dim A_{𝔭} for every 𝔭 ∈ Spec A, as required.

(iii): Arguing as in the proof of proposition 17.5.37(i), for every A ∈ Ob(ℬ_{0}) we write j_{A} : A → A^{ν} as the composition of the projection j′_{A} : A → A_{red} onto the maximal reduced quotient, and the natural injective map j′′_{A} : A_{red} → A^{ν}. It is easily seen that both maps are morphisms of ℬ_{0}, and we then need to check that E_{0}(j′_{A}) and E_{0}(j′′_{A}) are isomorphisms. However, clearly B_{red} ∈ Δ(A_{red}), for every B ∈ Δ(A); for such B, pick a finite subset S_{B} ⊂ Max ℤ as in lemma 17.5.42(ii). In order to prove that E_{0}(j′_{A}) is an isomorphism, it suffices to check that the induced map j′_{B,p} : B/pB → B_{red}/pB_{red} induces an isomorphism E_{p}(j′_{B,p}) : E_{p}(B/pB) →^{∼} E_{p}(B_{red}/pB_{red}) for every p ∈ Max(ℤ) \ S_{B}; but j′_{B,p} induces an isomorphism

(B/pB)_{red} →^{∼} (B_{red}/pB_{red})_{red}

so the assertion follows from proposition 17.5.37(i). Next, set B^{ν} := ∏_{𝔭∈Max B}(B/𝔭)^{ν}, and consider the commutative diagram of schemes :

[figure: Spec B^{ν}; π; ψ; Spec B_{red}; φ; Spec ℤ; Spec ℤ.]

We have a maximal open subset U ⊂ Spec B_{red} such that π restricts to an isomorphism π^{−1}U →^{∼} U, and since B_{red} is reduced, U is dense in Spec B_{red}. Let η ∈ Spec ℤ be the generic point; it follows that U ∩ φ^{−1}(η) is a dense open subset of φ^{−1}(η), and then there exists an open subset V ⊂ Spec ℤ such that U ∩ φ^{−1}(x) is dense in φ^{−1}(x) for every x ∈ V ([65, Ch.IV, Prop.9.5.3]). Therefore, for every x ∈ V, the subset π^{−1}U ∩ ψ^{−1}(x) is open and dense in ψ^{−1}(x), and π induces an isomorphism

π^{−1}U ∩ ψ^{−1}(x) →^{∼} U ∩ φ^{−1}(x) for every x ∈ V .

On the other hand, since the ℚ-schemes φ^{−1}(η) and ψ^{−1}(η) are respectively geometrically reduced and geometrically normal, there exists a dense open subset W ⊂ Spec ℤ such that φ^{−1}(x) and ψ^{−1}(x) are respectively a geometrically reduced and a geometrically normal κ(x)- scheme, for every x ∈ W ([65, Ch.IV, Prop.9.9.4]). Summing up, we conclude that, for every x ∈ V ∩ W, the fibre ψ^{−1}(x) is the normalization of the affine reduced scheme φ^{−1}(x). By proposition 17.5.37(ii), we conclude that the natural map j′′_{B,p} : B_{red}/pB_{red} → B^{ν}/pB^{ν} induces an isomorphism E_{p}(j′′_{B,p}) for every p ∈ Max(ℤ)∩V ∩W. Therefore, j′′_{A} induces an isomorphism E_{0}(A_{red}, B_{red}) →^{∼} E_{0}(A^{ν}, B^{ν}), as required. □

Remark 17.5.47. For the definition of E_{0}(A), we could have used instead an ultraproduct with respect to a non-principal ultrafilter of ℕ : this would have amounted to replacing I_{0}(A, B) by a larger ideal (and thus, E_{0}(A) by a smaller quotient); the resulting functor would have similar properties (and some others as well). The use of ultraproducts in commutative algebra is expounded in Schoutens’s monograph [152], and our construction can be viewed as a simplification of the construction of big CM algebras in characteristic zero via ultraproducts, presented in [152, (6.4.2)].

17.5.48. The functor E_{0} can be extended naturally to the category

ℬ′_{0}

whose objects are all locally equidimensional ℚ-algebras of essentially finite type, and whose morphisms are again the maximizing ring homomorphisms. Indeed, for A ∈ Ob(ℬ′_{0}), let Δ′(A) be the set of ℚ-subalgebras of finite type B ⊂ A, such that A is a localization of B. Notice that Δ′(A) is a filtered set, for the partial order given by inclusion of ℚ-subalgebras; moreover, A = ⋃_{B∈Δ′(A)} B. Let also Δ′′(A) := {B ∈ Δ′(A) | B is locally equidimensional}. We notice:

Lemma 17.5.49. With the notation of (17.5.48), we have :

(i) For every B ∈ Δ′(A), the natural map Spec A → Spec B restricts to a bijection Min A →^{∼} Min B on the respective sets of minimal prime ideals.

(ii) Δ′′(A) is a cofinal subset of Δ′(A).

Proof. (i): This is clear, since the natural morphism of schemes Spec A → Spec B is flat, injective on the underlying topological spaces, and with dense image.

(ii): Let B ∈ Δ′(A), so that A = S^{−1}B for some multiplicative subset S ⊂ B, and say that Min A = {𝔭_{1}, …, 𝔭_{n}}. Set 𝔮_{i} := B ∩ 𝔭_{i} for every i = 1, …, n. Also, for 1 ≤ i < j ≤ n, set B_{ij} := B/(𝔮_{i} + 𝔮_{j}) and A_{ij} := A/(𝔭_{i} + 𝔭_{j}) = S^{−1}B_{ij}. For every such i, j and every s ∈ S, the induced map Spec A_{ij} → Spec B_{ij}[s^{−1}] restricts to an injective map of finite sets Min A_{ij} → Min B_{ij}[s^{−1}], and the system of such maps yields a natural identification : Min A_{ij} →^{∼} lim_{s∈S} Min B_{ij}[s^{−1}]. Hence, for a suitable s ∈ S we get bijections

(17.5.50)  Min A_{ij} →^{∼} Min B_{ij}[s^{−1}]  for every 1 ≤ i < j ≤ n.

Now, both A and B are excellent rings ([64, Ch.IV, Sch.7.8.3(ii)]), and A fulfills condition (c) of lemma 17.5.31, since it is locally equidimensional; but in light of (17.5.50), we see that B[s^{−1}] fulfills the same condition as well, so B[s^{−1}] ∈ Δ′′(A), by lemma 17.5.31. □

In view of lemma 17.5.49(i), for every pair B_{1} ⊂ B_{2} of elements of Δ′(A), the inclusion map B_{1} → B_{2} is trivially maximizing; with lemma 17.5.49(ii), we then get a well defined ring

E′_{0}(A) := colim_{B∈Δ′′(A)} E_{0}(B)

and the B-algebra structures on each E_{0}(B) assemble into a natural A-algebra structure for E′_{0}(A). Then, with theorem 17.5.45 and remark 17.5.19(iv), we see that E′_{0}(A) is a big locally CM A-algebra. If A ∈ Ob(ℬ_{0}), clearly A is a final object of Δ′′(A), so E′_{0}(A) = E_{0}(A) in this case. Furthermore, let f : A → A′ be any morphism of ℬ′_{0}; for every B ∈ Δ′′(A), we may pick B′ ∈ Δ′′(A′) such that f restricts to a map of ℚ-algebras g : B → B′. Since f is maximizing, from lemma 17.5.49(i) it follows easily that the same holds for g. We may therefore compose E_{0}(g) : E_{0}(B) → E_{0}(B′) with the natural map E_{0}(B′) → E_{0}(A′), to get a ring homomorphism h_{B} : E_{0}(B) → E′_{0}(A), and it is easily seen that h_{B} depends only on the restriction B → A′ of f (and is independent of the choice of B′ : details left to the reader). The colimit of the system of maps (h_{B} | B ∈ Δ′′(A)) is a well defined ring homomorphism

E′_{0}(f) : E′_{0}(A) → E′_{0}(A′)

and the rules : A ↦ E′_{0}(A) and f ↦ E′_{0}(f) yield the sought functor E′ : ℬ′_{0} → ℤ-Alg. Lastly, from theorem 17.5.45 we easily deduce that E′_{0} commutes with finite products, and factors through the corresponding normalization functor (−)^{ν} : ℬ′_{0} → ℬ′_{0} : details left to the reader.

17.5.51. We turn now to consider the case of a noetherian local ring (A, 𝔫) with residue field k := A/𝔫 of characteristic p > 0. Arguing as in the proof of theorem 17.5.28, we can assume that A is a complete domain, with field of fractions of zero characteristic. By [126, Th.29.4], there exists a complete discrete valuation ring R ⊂ A whose maximal ideal is pR, and such that the induced map R/pR → k is an isomorphism; moreover, if p, x_{1}, …, x_{n} is any system of parameters of A, then the inclusion of R into A extends to a finite injective map :

R[[X_{1}, …, X_{n}]] → A  X_{i} ↦ x_{i}  for i = 1, …, n

whose image we call A_{0}. Let 𝔫_{0} ⊂ A_{0} be the maximal ideal; clearly the natural map A_{0}/𝔫_{0} → k is an isomorphism. Pick also g ∈ A_{0} \ {0} such that A[g^{−1}] is faithfully flat étale over A_{0}[g^{−1}].

• We consider first the case where [k : k^{p}] < +∞, and we pick x_{n+1}, …, x_{m} ∈ A_{0} whose classes in k form a p-basis (i.e. a basis of the k^{p}-vector space k). Set x_{0} := p, and

A_{∞} := A_{0}[T_{0}^{1/p^{∞}}, …, T_{m}^{1/p^{∞}}]/(x_{0} − T_{0}, …, x_{m} − T_{m})

(notation as in example 4.8.55(iii)). Notice that the sequence (p, x_{n+1}, …, x_{m}) is maximal in the sense of remark 9.6.35(iii); hence A_{∞} is formally perfectoid for its p-adic topology, according to theorem 17.2.14(iv). Next, if g ∈ {p, x_{1}, …, x_{m}}, set B_{∞} := A_{∞}, and otherwise, let

B_{∞} := A_{∞}[T_{m+1}^{1/p^{∞}}]/(g − T_{m+1}).

In either case, let C be the p-root closure of B_{∞} in B_{∞}[1/p], and C^{∧} the p-adic completion of C; then C^{∧} is perfectoid for its p-adic topology, and C is a faithfully flat A_{∞}-algebra (theorem 16.9.17). Let D be the integral closure of C^{∧} in C^{∧} ⊗_{A_{0}} A[1/g], and D^{∧} the p-adic completion of D. Denote also by 𝔪 ⊂ C the ideal generated by (g^{1/p^{n}} | n ∈ ℕ); clearly (C, 𝔪) is a basic setup (in the sense of [75, §2.1.1]), and we let (C^{∧})^{a} and D^{a} be the (C, 𝔪)^{a}-algebras arising from C^{∧} and respectively D.

Lemma 17.5.52. (i) D^{a} is faithfully flat over A_{0}, in the sense of definition 14.2.40. (ii) The basic setup (D^{∧}, 𝔪D^{∧}) is almost perfectoid.

Proof. Denote as usual by (C^{∧})_{∗}^{a} the ring of almost elements of (C^{∧})^{a}, and let D_{1} be the integral closure of (C^{∧})_{∗}^{a} in C^{∧} ⊗_{A_{0}} A[1/g]. Recall that the natural morphism D^{a} → D_{1}^{a} of (C^{∧}, 𝔪)^{a}-algebras is an isomorphism ([75, Lemma 8.2.28]); taking into account theorem 16.9.42(i,iii), it follows that D^{a} is a p-adically complete C^{a}-algebra, that (D/p^{n}D)^{a} is a faithfully flat (C/p^{n}C)^{a}-algebra, for every n ∈ ℕ, and that (ii) holds. By virtue of proposition 14.2.43, in order to prove that D^{a} is flat over A_{0}, it then suffices to check that (D/p^{n}D)^{a} is flat over A_{0}/p^{n}A_{0}, for every n ∈ ℕ. But it is easily seen that A_{∞} is a flat A_{0}-algebra, whence the assertion. By lemma 14.2.41(ii), we then come down to checking that (D/𝔫_{0}D)^{a} ≠ 0. Since we already know that (D/pD)^{a} is a faithfully flat (C/pC)^{a}-algebra, and since p ∈ 𝔫_{0}, we are further reduced to checking that (C/𝔫_{0}C)^{a} ≠ 0. In turns, this amounts to showing that g^{1/p^{m}} ∉ 𝔫_{0}C for some m ∈ ℕ. If the latter fails, we get :

(17.5.53)  g ∈ 𝔫_{0}^{n}C  for every n ∈ ℕ.

Now, recall that C is a faithfully flat A_{∞}-algebra, and A_{∞} is a faithfully flat A_{0}-algebra, so C is a faithfully flat A_{0}-algebra. Then, from (17.5.53) we deduce that g ∈ ⋂_{n∈ℕ} 𝔫_{0}^{n}. But the 𝔫_{0}-adic topology of A_{0} is separated ([126, Th.8.10(i)]), hence g = 0, a contradiction. □

Denote by S ⊂ C^{ℕ} the multiplicative subset associated with 𝔪, as in (17.5.2); we endow E := S^{−1}D^{⋄} with its p-adic topology, and denote by E^{sep} the maximal separated quotient of E. Notice that D contains the integral closure of A_{0} in A[1/g], which contains A, hence D is naturally an A-algebra, and therefore the same holds for E and E^{sep}. We may then state :

Theorem 17.5.54. E and E^{sep} are big CM A-algebras, and E^{sep} is perfectoid.

Proof. In order to show that E^{sep} is perfectoid, it suffices to invoke lemmata 17.5.52(ii) and 17.5.10. Next, in light of remark 17.5.19(iii), it suffices to show that E and E^{sep} are big CM A_{0}-algebras. By lemmata 17.5.52(i) and 17.5.4(v), E is a faithfully flat A_{0}-algebra; then, by proposition 14.2.43(i), the same holds for the A_{0}-algebra E^{sep}, since the latter is the p-adic completion of E (proposition 17.5.8). Especially, E/𝔫_{0}E →^{∼} E^{sep}/𝔫_{0}E^{sep} ≠ 0. Now the theorem follows from lemma 10.4.20, proposition 10.4.32(i), and remark 17.5.19(i). □

• Next, we consider the case where [k : k^{p}] can be infinite. Define the regular local ring (A_{0}, 𝔫_{0}), and pick g ∈ A_{0} as in the foregoing case; choose also a family (b_{λ} | λ ∈ Λ) of elements of A_{0} whose image (\overline{b}_{λ} | λ ∈ Λ) in k is a p-basis. Again, we set x_{0} := p, and

A_{∞} := A_{0}[T_{0}^{1/p^{∞}}, …, T_{n}^{1/p^{∞}}][T_{λ}^{1/p^{∞}} | λ ∈ Λ]/(x_{0} − T_{0}, …, x_{n} − T_{n}, b_{λ} − T_{λ} | λ ∈ Λ).

We endow A_{∞} with its 𝔫_{0}-adic topology, and let A_{∞}^{∧} be the completion of A_{∞}.

Lemma 17.5.55. A_{∞}^{∧} is perfectoid for its p-adic topology.

Proof. With the notation of (9.8.11), set

B_{0} := k[[T_{0}, …, T_{n}]] ⊗_{k} k^{perf} B_{∞} := k[[T_{0}^{1/p^{∞}}, …, T_{n}^{1/p^{∞}}]] ⊗_{k} k^{perf}

and let \overline{𝔫}_{0} ⊂ B_{0} be the ideal generated by T_{0}, …, T_{n}. Endow B_{∞} with its \overline{𝔫}_{0}-adic topology, and let B_{∞}^{∧} be the completion of B_{∞}. Then B_{∞} is naturally an A_{0}-algebra, via the projection A_{0} → k, and by simple inspection we get an isomorphism of topological A_{0}-algebras

A_{∞}/pA_{∞} →^{∼}B_{∞}/T_{0}B_{∞} T_{i}^{1/p^{k}} ↦ T_{i}^{1/p^{k}} (i = 0, …, n) T_{λ}^{1/p^{k}} ↦ 1 ⊗ \overline{b}_{λ}^{1/p^{k}} for every λ ∈ Λ

for the respective quotient topologies. Notice that the sequence (p, x_{1}, …, x_{n}) is A_{0}-regular, hence also A_{∞}-regular, since A_{∞} is a faithfully flat A_{0}-algebra; by proposition 7.8.15, we deduce that the 𝔫_{0}-topology of A_{∞} induces the 𝔫_{0}-topology on the ideal pA_{∞}, and therefore the natural map A_{∞}^{∧}/pA_{∞}^{∧} → (A_{∞}/pA_{∞})^{∧} is an isomorphism of topological rings (proposition 8.2.13(i,v)). Likewise, the sequence (T_{0}, …, T_{n}) is B_{0}-regular, hence also B_{∞}-regular, since B_{∞} is a faithfully flat B_{0}-algebra; arguing as in the foregoing, we deduce that the natural map B_{∞}^{∧}/T_{0}B_{∞}^{∧} → (B_{∞}/T_{0}B_{∞})^{∧} is as well an isomorphism of topological rings. However, by direct inspection we see that B_{0}^{perf} is a dense subring of B_{∞}^{∧} (here it is important to work with the \overline{𝔫}_{0}-adic topology), and the \overline{𝔫}_{0}-adic topology of B_{∞}^{∧} induces the \overline{𝔫}_{0}-adic topology on B_{0}^{perf}; hence B_{∞}^{∧} is also the completion C of B_{0}^{perf} (theorem 8.2.8(iii)), and therefore it is a perfect topological 𝔽_{p}-algebra (remark 9.4.9(v) and corollary 9.4.14). Summing up, we see that A_{∞}^{∧}/pA_{∞}^{∧} is isomorphic to C/T_{0}C, and clearly the Frobenius endomorphism of C induces a bijection C/T_{0}^{1/p}C →^{∼} C/T_{0}C; therefore the Frobenius endomorphism of A_{∞}^{∧} induces a bijection A_{∞}^{∧}/p^{1/p}A_{∞}^{∧} →^{∼} A_{∞}^{∧}/pA_{∞}^{∧}. Moreover, A_{∞}^{∧} is complete and separated for its p-adic topology (lemma 8.3.12); then the lemma follows from corollary 16.3.75. □

17.5.56. We may now proceed as in the previous case : if g ∈ {x_{0}, …, x_{n}}∪{b_{λ} | λ ∈ Λ}, set B_{∞} := A_{∞}, and otherwise set B_{∞} := A_{∞}^{∧}[X^{1/p^{∞}}]/(g − X). Let also C be the p-root closure of B_{∞} in B_{∞}[1/p], and C^{∧} the p-adic completion of C. Then, again by theorem 16.9.17, C^{∧} is perfectoid and C is a faithfully flat A_{∞}^{∧}-algebra. Denote by 𝔪 ⊂ C^{∧} the ideal generated by (g^{1/p^{n}} | n ∈ ℕ), by D be the integral closure of C^{∧} in C^{∧} ⊗_{A_{0}} A[1/g], and by D^{∧} the p-adic completion of D. Arguing as in (17.5.51) we see that (D/p^{n}D)^{a} is a faithfully flat (C/p^{n}C)^{a}- algebra for every n ∈ ℕ (relative to the basic setup (C^{∧}, 𝔪)), and (D^{∧}, 𝔪D^{∧}) is an almost perfectoid basic setup. Moreover, A_{∞}^{∧} is the 𝔫_{0}-adic completion of a faithfully flat A_{0}-algebra, hence it is a faithfully flat A_{0}-algebra (proposition 14.2.43(i)). Thus, C is a faithfully flat A_{0}- algebra. After these observations, the proof of lemma 17.5.52 can be repeated in our current situation, so that D^{a} is still faithfully flat over A_{0} (in the sense of definition 14.2.40). Lastly, we define again E := S^{−1}D^{⋄}, and let E^{sep} be the maximal p-adically separated quotient of E; now the proof of theorem 17.5.54 applies again verbatim, and we finally conclude that E and E^{sep} are big CM A-algebras, and that E^{sep} is perfectoid for its p-adic topology.

17.5.57. Next, we would like to upgrade the constructions of the previous paragraphs to a functor on a suitable category of local noetherian domains. Hence, let (A, 𝔫) be a local noetherian domain, complete for its 𝔫-adic topology, with field of fractions of zero characteristic, and residue field k of characteristic p > 0, and suppose first that [k : k^{p}] < +∞. Let Σ ⊂ A \ {0} be a finite subset with p ∈ Σ, and Σ′ ⊂ Σ a non-empty subset; we consider the category

𝒞_{Σ,Σ′}^{A}

whose objects are all pairs (f, Δ), where f : A → B is a ring homomorphism, and Δ := (β_{a,•} | a ∈ Σ) is a family of elements of E(B) (notation of remark 9.4.12(ii)), fulfilling the following conditions :

(a) the projection β_{a,0} ∈ B of β_{a,•} equals f(a), for every a ∈ Σ (especially, β_{p,1}^{p} = p)

(b) let 𝔪 ⊂ B be the ideal generated by the system (δ_{n} := ∏_{a∈Σ′} β_{a,n} | n ∈ ℕ); we call (B, 𝔪) the basic setup attached to (f, Δ), and denote by (B, 𝔪)^{a} the (B, 𝔪)^{a}-algebra arising from B; then the unit of adjunction B → (B, 𝔪)_{∗}^{a} is an isomorphism

(c) B is p-torsion free, and complete and separated for its p-adic topology

(d) B is δ_{0}-torsion free, integrally closed in B[δ_{0}^{−1}], and p-root closed in B[p^{−1}]. For two such pairs (f : A → B, Δ) and (f′ : A → B′, Δ′ := (β′_{a,•} | a ∈ Σ)), the morphisms (f, Δ) → (f′, Δ′) in 𝒞_{Σ,Σ′}^{A} are the ring homomorphisms h : B → B′ with :

f′ = h ∘ f and E(h)(β_{a,•}) = β′_{a,•} for every a ∈ Σ.

We shall say that the object (f : A → B, Δ) of 𝒞_{Σ,Σ′}^{A} is almost perfectoid, if the basic setup (B, 𝔪) attached to (f, Δ) is almost perfectoid for the p-adic topology of B.

Remark 17.5.58. Let Σ_{1} ⊂ Σ_{2} be two finite subsets of A \ {0} with p ∈ Σ_{1}, and Σ′_{1} ⊂ Σ_{1}, Σ′_{2} ⊂ Σ_{2} two subsets, with ∅ ≠ Σ′_{1} ⊂ Σ′_{2}. For every object (f : A → B, Δ := (β_{a,•} | a ∈ Σ_{2})) of 𝒞_{Σ_{2},Σ′_{2}}^{A} set Δ_{|Σ_{1}} := (β_{a,•} | a ∈ Σ_{2}). By remark 14.2.7(ii), we get a well defined functor :

𝒞_{Σ_{2},Σ′_{2}}^{A} → 𝒞_{Σ_{1},Σ′_{1}}^{A} (f, Δ) ↦ (f, Δ)_{|Σ_{1},Σ′_{1}} := (f, Δ_{|Σ_{1}}).

Proposition 17.5.59. Let Σ_{1} ⊂ Σ_{2} be an inclusion of finite subsets of A \ {0} with p ∈ Σ_{1}, and Σ′_{1} ⊂ Σ_{1}, Σ′_{2} ⊂ Σ_{2} two non-empty subsets, with Σ′_{1} ⊂ Σ′_{2}. If the category 𝒞_{Σ_{1},Σ′_{1}}^{A} admits an almost perfectoid initial object, the same holds for the category 𝒞_{Σ_{2},Σ′_{2}}^{A}.

Proof. Let (f_{1} : A → D_{1}, Δ_{1}) be a given almost perfectoid initial object of 𝒞_{Σ_{1},Σ′}^{A}, with Δ_{1} := (β_{x,•} | x ∈ Σ_{1}), and let 𝔪_{1} ⊂ D_{1} be the ideal generated by (δ_{n} := ∏_{x∈Σ′_{1}} β_{x,n} | n ∈ ℕ).

We suppose first that Σ′ := Σ′_{2} = Σ′_{1}, and we are easily reduced to the case where Σ_{2} = Σ_{1} ∪ {a} for some a ∈ A. We set

D′_{2} := D_{1}[T^{1/p^{∞}}]/(a − T).

Let D′′_{2} be the p-root closure of D′_{2} in D′_{2}[1/p], and D_{2}^{′′∧} the p-adic completion of D′′_{2}. For every x ∈ Σ_{1}, denote by β′_{x,•} ∈ E(D′_{2}) the image of β_{x,•}, and let likewise

β′_{a,•} := (\overline{T}^{1/p^{n}} | n ∈ ℕ) ∈ E(D′_{2})

where \overline{T}^{1/p^{n}} ∈ D′_{2} denotes the image of T^{1/p^{n}}, for every n ∈ ℕ. Set D_{2} := (D_{2}^{′′∧}, 𝔪_{1}D_{2}^{′′∧})_{∗}^{a}, and let f_{2} : A → D_{2} be the structure map of the A-algebra D_{2}. Lastly, let β′′_{x,•} ∈ E(D_{2}) be the image of β′_{x,•}, for every x ∈ Σ_{2}, and set Δ_{2} := (β′′_{x,•} | x ∈ Σ_{2}). We show more precisely :

Claim 17.5.60. (f_{2}, Δ_{2}) is an almost perfectoid initial object of the category 𝒞_{Σ_{2},Σ′}^{A}.

Proof of the claim. By proposition 17.5.16, the basic setup (D_{2}^{′′∧}, 𝔪_{1}D_{2}^{′′∧}) is almost perfectoid, hence the same holds for (D_{2}, 𝔪_{1}D_{2}), by proposition 16.9.37(iii); then D_{2} is p-root closed in D_{2}[1/p], by proposition 16.9.37(iv), and is integrally closed in D_{2}[1/δ_{0}], by virtue of proposition 17.5.14(ii). This shows that (f_{2}, Δ_{2}) is an almost perfectoid object of 𝒞_{Σ_{2},Σ′}^{A}.

Next, let (f : A → D, Δ := (δ_{x,•} | x ∈ Σ_{2})) be any other object of 𝒞_{Σ_{2},Σ′}^{A}; by assumption, there exists a unique morphism h_{0} : (f_{1}, Δ_{1}) → (f, Δ′)_{|Σ_{1},Σ′} in 𝒞_{Σ_{1},Σ′}^{A} (notation of remark 17.5.58). Clearly h_{0} extends to a unique map h_{1} : D′_{2} → D such that E(h_{1})(β′_{x,•}) = δ_{x,•} for every x ∈ Σ_{2}, and arguing as in the proof of proposition 17.5.62, we see that A_{0}[1/p] ⊗_{1_{0}} h_{1} : D′_{2}[1/p] → D[1/p] restricts to a map of A-algebras h_{2} : D′′_{2} → D. Since D is p-adically complete and separated, the completion of h_{2} is a map h_{2}^{∧} : D_{2}^{′′∧} → D, and since D = D_{∗}^{a}, the latter induces a map of A-algebras h := (h_{2}^{∧})_{∗}^{a} : D_{2} → D; it is then easily seen that h is the unique morphism (f_{2}, Δ_{2}) → (f, Δ) in 𝒞_{Σ_{2},Σ′}^{A}. ♢

It remains to consider the case where Σ := Σ_{1} = Σ_{2}. Then, let 𝔪_{2} ⊂ D_{1} be the ideal generated by (γ_{n} := ∏_{x∈Σ′_{2}} β_{x,n} | n ∈ ℕ); endow D_{2} := (D_{1}, 𝔪_{2})_{∗}^{a} with its p-adic topology, and let η : D_{1} → D_{2} be the unit of adjunction. Since the basic setup (D_{1}, 𝔪_{1}) is almost perfectoid, and since 𝔪_{2} ⊂ 𝔪_{1}, it is clear that the basic setup (D_{1}, 𝔪_{2}) is almost perfectoid. Then the same holds for the basic setup (D_{2}, 𝔪_{2}D_{2}), according to proposition 16.9.37(iii). By proposition 17.5.14(ii), it follows that D_{2} is γ_{0}-torsion free and integrally closed in D_{2}[γ_{0}^{−1}]. For every x ∈ Σ, let β′_{x,•} ∈ E(D_{2}) be the image of β_{x,•}; set Δ_{2} := (β′_{x,•} | x ∈ Σ) and f_{2} := η ∘ f_{1}. The foregoing already shows that (f_{2}, Δ_{2}) is an object of 𝒞_{Σ,Σ′_{2}}^{A}. Lastly, let (f′ : A → D′, Δ′) be any object of 𝒞_{Σ,Σ′_{2}}^{A}; by assumption, there exists a unique morphism h : (f_{1}, Δ_{1}) → (f′, Δ′)_{|Σ,Σ′_{1}} in 𝒞_{Σ,Σ′_{1}}^{A}, and since D′ = (D′, 𝔪_{2})_{∗}^{a}, it follows easily that (h, 𝔪_{2})_{∗}^{a} : D_{2} → D′ is the unique morphism (f_{2}, Δ_{2}) → (f′, Δ′) in 𝒞_{Σ,Σ′_{2}}^{A}. □

17.5.61. Resume the notation of (17.5.51), set x_{0} := p, and define either t := m if g ∈ {x_{0}, …, x_{m}}, or else t := m + 1 and x_{t} := g; we take

Σ_{0} := {x_{0}, …, x_{t}} Σ′_{0} := {x_{t}}.

Let also f_{0} : A → D_{∗}^{a} := (D, 𝔪D)_{∗}^{a} be the structure map of the A-algebra D_{∗}^{a}, and for i = 0, …, t, set β_{x_{i},•} := (\overline{T} _{i}^{1/p^{n}} | n ∈ ℕ) ∈ E(D_{∗}^{a}), where \overline{T} _{i}^{1/p^{n}} ∈ D_{∗}^{a} denotes the image of T_{i}^{1/p^{n}}, for every such i and every n ∈ ℕ. Set Δ_{0} := (β_{a,•} | a ∈ Σ_{0}). Lastly, let Σ′_{1} ⊂ Σ_{1} ⊂ A \ {0} be two finite subsets with Σ_{0} ⊂ Σ_{1} and Σ′_{0} ⊂ Σ′_{1}.

Proposition 17.5.62. In the situation of (17.5.61), we have :

(i) The pair (f_{0}, Δ_{0}) is an almost perfectoid initial object of the category 𝒞_{Σ_{0},Σ′_{0}}^{A}.

(ii) Let (f_{1} : A → D_{1}, Δ_{1}) be an initial object of 𝒞_{Σ_{1},Σ′_{1}}^{A}, and (D_{1}, 𝔪_{1}) the basic setup attached to (f_{1}, Δ_{1}). Then (D_{1}, 𝔪_{1})^{a} is faithfully flat over A_{0}, in the sense of definition 14.2.40.

Proof. (i): From theorem 16.9.42(i,v) and lemma 16.9.41 we see that (f_{0}, Δ_{0}) is an almost perfectoid object of 𝒞_{Σ_{0},Σ′_{0}}^{A}. Now, define B_{∞}, C and C^{∧} as in (17.5.51), and let

(f′ : A → D′, Δ′ := (β′_{a,•} | a ∈ Σ_{0}))

be any other object of 𝒞_{Σ_{0},Σ′}^{A}; clearly there exists a unique homomorphism h_{0} : B_{∞} → D′ of A_{0}-algebras such that E(h_{0})(β_{a,•}) = β′_{a,•} for every a ∈ Σ_{0}. Since D′ is p-root closed in D′[1/p], by proposition 9.8.28(i,ii) and remark 9.8.16(i), the map

A_{0}[1/p] ⊗_{A_{0}} h_{0} : B_{∞}[1/p] → D′[1/p]

restricts to a map of A_{0}-algebras h_{1} : C → D′, and clearly h_{1} is the unique map of A_{0}- algebras extending h_{0}. Next, h_{1} extends uniquely to a map of A_{0}-algebras h_{1}^{∧} : C^{∧} → D′, since by assumption D′ is p-adically complete and separated. Set h_{2} := A[1/g] ⊗_{A_{0}} h_{1}^{∧} :

A[1/g] ⊗_{A_{0}} C^{∧} → D′[1/g]; by assumption, the localization D′ → D′[1/g] is injective, and D′ is integrally closed in D′[1/g], so h_{2} restricts to a map of A-algebras h_{3} : D → D′. Lastly, since D′ = (D′)_{∗}^{a}, we get an induced map of A-algebras h := h_{3∗}^{a} : D_{∗}^{a} → D′, and by construction it is easily seen that h is the unique morphism (f_{0}, Δ_{0}) → (f′, Δ′) in 𝒞_{Σ_{0},Σ′_{0}}^{A}.

(ii): Suppose first that Σ′_{1} = Σ′_{0}. Recall that D_{1} is p-adically complete and separated; taking into account proposition 14.2.43, in order to show that (D_{1}, 𝔪_{1})^{a} is flat over A_{0}, it then suffices to check that (D_{1}/p^{n}D_{1})^{a} is flat over A_{0}/p^{n}A_{0} for every n ∈ ℕ. But lemma 17.5.52(i) implies that (D/p^{n}D)^{a} is flat over A_{0}/p^{n}A_{0} for every such n; hence we come down to the following :

Claim 17.5.63. In the situation of proposition 17.5.59, suppose that Σ′ := Σ′_{1} = Σ′_{2}, and let (f_{1} : A → D_{1}, Δ_{1}) (resp. (f_{2} : A → D_{2}, Δ_{2})) be an initial object of 𝒞_{Σ_{1},Σ′}^{A} (resp. of 𝒞_{Σ_{2},σ′}^{A}). Let also (D_{1}, 𝔪_{1}) be the basic setup attached to (f_{1}, Δ_{1}). Then the unique morphism (f_{1}, Δ_{1}) → (f_{2}, Δ_{2})_{Σ_{1},Σ′} in 𝒞_{Σ_{1},Σ′}^{A} induces a faithfully flat morphism of (D_{1}, 𝔪_{1})^{a}-algebras

(D_{1}/p^{n}D_{1})^{a} → (D_{2}/p^{n}D_{2})^{a} for every n ∈ ℕ.

Proof of the claim. We are easily reduced to the case where Σ_{2} = Σ_{1}∪{a} for some a ∈ A\{0}, and then the assertion follows from proposition 17.5.16, after inspecting the proof of proposition 17.5.59. ♢

By lemma 14.2.41(ii), there remains only to show that (D_{1}/𝔫_{0}D_{1})^{a} ≠ 0. Since p ∈ 𝔫_{0}, it suffices to check that (D_{1}/pD_{1})^{a} is faithfully flat over A_{0}/pA_{0}; since the morphism (D/pD)^{a} → (D_{1}/pD_{1})^{a} is faithfully flat (claim 17.5.63), lemma 14.2.41(iii) further reduces to checking that (D/pD)^{a} is faithfully flat over A_{0}/pA_{0}, which follows from lemma 17.5.52(i).

To deal with the general case, let (f′_{1} : A → D′_{1}, Δ′_{1} := (β′_{a,•} | a ∈ Σ_{1})) be an initial object of 𝒞_{Σ_{1},Σ′_{0}}^{A}, and 𝔪′_{1} ⊂ D′_{1} the ideal generated by (β′_{n} := ∏_{a∈Σ′_{1}} β′_{a,n} | n ∈ ℕ). By the foregoing, the (C, 𝔪)^{a}-algebra (D′_{1}, 𝔪D′_{1})^{a} is faithfully flat over A_{0}, and by inspecting the proof of proposition 17.5.59, we see that the A-algebra D_{1} is isomorphic to (D′_{1}, 𝔪′_{1})_{∗}^{a}. Hence, the two functors

A_{0}-Mod → (D_{1}, 𝔪_{1})^{a}-Mod  N ↦ (D_{1}, 𝔪_{1})^{a} ⊗_{A_{0}} N  N ↦ (D′_{1}, 𝔪′_{1})^{a} ⊗_{A_{0}} N

are isomorphic (notation of definition 14.2.40). However, the functor (D′_{1}, 𝔪′_{1})^{a} ⊗_{A_{0}} − is the composition of the exact functor

A_{0}-Mod → (D′_{1}, 𝔪D′_{1})^{a}-Mod N ↦ (D′_{1}, 𝔪D′_{1})^{a} ⊗_{A_{0}} N

and the natural functor (D′_{1}, 𝔪D′_{1})^{a}-Mod → (D′_{1}, 𝔪′_{1})^{a}-Mod →^{∼} (D_{1}, 𝔪_{1})^{a}-Mod, which is obviously exact, so the same holds for the functor (D′_{1}, 𝔪′_{1})^{a} ⊗_{A_{0}} −, and hence also for (D_{1}, 𝔪_{1})^{a}⊗_{A_{0}} −, i.e. (D_{1}, 𝔪_{1})^{a} is flat over A_{0}. Again, there remains to show that the (D_{1}, 𝔪_{1})^{a}- module (D_{1}/𝔫_{0}D_{1})^{a} is not zero, or equivalently, that the same holds for the (D′_{1}, 𝔪′_{1})^{a}-module (D′_{1}/𝔫_{0}D′_{1})^{a}. Suppose that the latter fails; then we have β′_{n} ∈ 𝔫_{0}D′_{1} for every n ∈ ℕ. Set b := ∏_{a∈Σ′_{1}} a ∈ A \ {0}; it follows that f′_{1}(b) ∈ 𝔫_{0}^{k}D′_{1} for every k ∈ ℕ.

Claim 17.5.64. (i) There exists c ∈ A \ {0} such that bc ∈ A_{0}.

(ii) f′_{1} induces an injective map A_{0}/𝔫_{0}^{k} → D′_{1}/𝔫_{0}^{k}D′_{1} for every k ∈ ℕ.

Proof of the claim. (i): Since A is a finite A_{0}-algebra, we may find a monic polynomial P ∈ A_{0}[X] such that P (b) = 0, and since A is a domain, we may assume that P (0) ≠ 0. Then it is easily seen that P (0) = bc for some c ∈ A \ {0}.

(ii): We know already that (D′_{1}, 𝔪′_{1})^{a} is faithfully flat over A_{0}, hence (D′_{1}/𝔫_{0}^{k}D′_{1})^{a} is faithfully flat over A_{0}/𝔫_{0}^{k} for every k ∈ ℕ. Then the assertion follows from lemma 14.2.41(i). ♢

Let c be as in claim 17.5.64(i); hence bc ∈ A_{0} \ {0}, and f′_{1}(bc) ∈ 𝔫_{0}^{k}D′_{1} for every k ∈ ℕ. By claim 17.5.64(ii), it follows that bc ∈ ⋂_{k∈ℕ} 𝔫_{0}^{k}; but the 𝔫_{0}-adic topology of A_{0} is separated ([126, Th.8.10(i)]), so bc = 0, a contradiction. □

17.5.65. Next, let (A, 𝔫) be any local noetherian domain, complete for its 𝔫-adic topology, with Frac A of zero characteristic, and residue field of characteristic p > 0. Let Σ′ ⊂ Σ be two subsets of A \ {0} such that Σ′ is finite and non-empty, and p ∈ Σ; we consider the category

𝒟_{Σ,Σ′}^{A}

whose objects are all pairs (f, Δ), where f : A → B is a ring homomorphism, and Δ := (β_{a,•} | a ∈ Σ) is a family of elements of E(B), fulfilling conditions (a)–(d) of (17.5.57), as well as the following :

(e) B is complete and separated for its 𝔫-adic topology. If Σ_{1} ⊂ Σ_{2} is an inclusion of subsets of A \ {0}, and Σ′_{1} ⊂ Σ_{1}, Σ′_{2} ⊂ Σ_{2} are two non-empty finite subsets with Σ′_{1} ⊂ Σ′_{2}, then clearly, remark 17.5.58holds verbatim in the present situation, i.e. we get a well defined restriction functor :

𝒟_{Σ_{2},Σ′_{2}}^{A} → 𝒟_{Σ_{1},Σ′_{1}}^{A} (f, Δ) ↦ (f, Δ)_{|Σ_{1},Σ′_{1}} := (f, Δ_{|Σ_{1}}).

Let moreover the subring A_{0} ⊂ A be as in (17.5.51); we shall say that an object (f : A → B, Δ) of 𝒟_{Σ,Σ′}^{A} is almost flat (resp. almost faithfully flat) over A_{0}, if B^{a} is flat (resp. faithfully flat) over A_{0} (as usual, we refer here to the almost structure associated with the basic setup (B, 𝔪) attached to (f, Δ)). We have the following variant of proposition 17.5.59:

Proposition 17.5.66. Let Σ_{1} ⊂Σ_{2} be two subsets of A\{0} with p ∈ Σ_{1}, and Σ′_{1} ⊂ Σ_{1}, Σ′_{2} ⊂ Σ_{2} two finite subsets, with ∅ ≠ Σ′_{1} ⊂ Σ′_{2}. If the category 𝒟_{Σ_{1},Σ′_{1}}^{A} admits an almost perfectoid initial object that is almost flat (resp. almost faithfully flat) over A_{0}, the same holds for 𝒟_{Σ_{2},Σ′_{2}}^{A}.

Proof. Let (f_{1} : A → D_{1}, Δ_{1} := (β_{a,•} | a ∈ Σ_{1})) be an almost perfectoid initial object of 𝒟_{Σ_{1},Σ′_{1}}^{A}. Consider first the case where Σ′ := Σ′_{2} = Σ′_{1}. Say that Σ_{2} \ Σ_{1} = (a_{λ} | λ ∈ Λ); we set :

D′_{2} := D_{1}[T_{λ}^{1/p^{∞}} | λ ∈ Λ]/(a_{λ} − T_{λ} | λ ∈ Λ).

Let D′′_{2} be the p-root closure of D′_{2} in D′_{2}[1/p]; endow D′′_{2} with its p-adic topology, and let D_{2}^{′′∧} be the completion of D′′_{2}. Let moreover (D_{1}, 𝔪_{1}) be the basic setup attached to (f_{1}, Δ_{1}).

Claim 17.5.67. (i) The basic setup (D_{2}^{′′∧}, 𝔪_{1}D_{2}^{′′∧}) is almost perfectoid.

(ii) (D′′_{2})^{a} is a faithfully flat D_{1}^{a}-algebra.

Proof of the claim. Let 𝒫 be the set of all finite subsets of Λ, and for every Ω ∈ 𝒫, set

D′_{2,Ω} := D_{1}[T_{λ}^{1/p^{∞}} | λ ∈ Ω]/(a_{λ} − T_{λ} | λ ∈ Ω)

and let D′′_{2,Ω} be the p-root closure of D′_{2,Ω} in D′_{2,Ω}[1/p]; endow D′′_{2,Ω} of its p-adic topology, and denote by D_{2,Ω}^{′′∧} the completion of D′′_{2,Ω}. Clearly, every inclusion Ω ⊂ Ω′ of elements of 𝒫 induces an injective map of D_{1}-algebras D′_{2,Ω} → D′_{2,Ω′}, which induces in turn a map of D_{1}-algebras D′′_{2,Ω} → D′′_{2,Ω′} (remark 9.8.16(i) and proposition 9.8.28). Clearly D′_{2} is the colimit of the filtered system of D_{1}-algebras (D′_{2,Ω} | Ω ∈ 𝒫), and by lemma 9.8.17, D′′_{2} is the colimit of the filtered system (D′′_{2,Ω} | Ω ∈ 𝒫). Recall that, by construction, p admits a p-th root p^{1/p} in D_{1}. Let Φ be the Frobenius endomorphism of D_{1}/pD_{1}, and suppose now that the Frobenius endomorphism of D′′_{2,Ω} induces an isomorphism (D′′_{2,Ω}/p^{1/p}D′′_{2,Ω})^{a} →^{∼} (D′′_{2,Ω}/pD′′_{2,Ω})_{(Φ)}^{a}, for every Ω ∈ 𝒫 (notation as in definition 16.9.35); then the Frobenius endomorphism of D′′_{2} induces an isomorphism (D′′_{2}/p^{1/p}D′′_{2})^{a} →^{∼} (D′′_{2}/pD′′_{2})_{(Φ)}^{a}. We are thus reduced to checking that the basic setup (D_{2,Ω}^{′′∧}, 𝔪_{1}D_{2,Ω}^{′′∧}) is almost perfectoid, and that (D′′_{2,Ω})^{a} is a faithfully flat D_{1}^{a}- algebra, for every Ω ∈ 𝒫. The latter follows from proposition 17.5.16, by an easy induction on the cardinality of Ω : the details shall be left to the reader. ♢

Set D_{2} := (D_{2}^{′′∧}, 𝔪_{1}D_{2}^{′′∧})_{∗}^{a}; the basic setup (D_{2}, 𝔪_{1}D_{2}) is almost perfectoid for the p-adic topology of D_{2}, by claim 17.5.67(i) and proposition 16.9.37(iii). Moreover, if D_{1}^{a} is flat (resp. faithfully flat) over A_{0}, the same holds for (D′′_{2})^{a}, by claim 17.5.67(ii) and lemma 14.2.41(iii); consequently, the same holds as well for both (D_{2}^{′′∧})^{a} and D_{2}^{a}, by proposition 14.2.43(i).

Endow D_{2} with its 𝔫-adic topology, let D_{2}^{∧} be the completion of D_{2}, and E_{2} := (D_{2}^{∧}, 𝔪_{1}D_{2}^{∧})_{∗}^{a}. Notice that the 𝔫-adic topology agrees with the 𝔫_{0}-adic topology, on every A-module M.

Claim 17.5.68. (i) E_{2} has no p-torsion and is 𝔫-adically complete and separated.

(ii) Let R be any D_{1}/pD_{1}-algebra, and denote by R^{∧} and (R_{(Φ)})^{∧} the 𝔫-adic completions of R and of R_{(Φ)}. Then (R^{∧})_{(Φ)} = (R_{(Φ)})^{∧}.

(iii) The basic setup (E_{2}, 𝔪_{1}E_{2}) is almost perfectoid for the p-adic topology of E_{2}.

Proof of the claim. (i): For the first assertion, it suffices to check that the endomorphism p·1_{(D_{2}^{∧})^{a}} is a monomorphism, since the functor (−)_{∗} is left exact. The latter will follow from lemma 14.2.45, after we check that the 𝔫_{0}-adic topology of D_{2}^{a} induces the 𝔫_{0}-adic topology on the ideal pD_{2}^{a} (see [75, §5.3] for generalities on topological almost algebras and modules). However, by the Artin-Rees lemma, there exists c ∈ ℕ such that 𝔫_{0}^{n+c}A_{0} ∩ pA_{0} ⊂ 𝔫_{0}^{n}pA_{0} for every n ∈ ℕ. But we have already noticed that D_{2}^{a} is flat over A_{0}, hence 𝔫_{0}^{n+c}D_{2}^{a} ∩ pD_{2}^{a} ⊂ 𝔫_{0}^{n}pD_{2}^{a} for every n ∈ ℕ, as required. The second assertion follows from claim 16.9.39.

(ii): By definition, 𝔫 · R_{(Φ)} = J · R, where J ⊂ A is the ideal generated by (a^{p} | a ∈ 𝔫); since 𝔫 is finitely generated, there exists N ∈ ℕ such that 𝔫^{N} ⊂ J ⊂ 𝔫, so the J-adic topology agrees with the 𝔫-adic topology on R, whence the assertion.

(iii): First, E_{2} is p-adically complete and separated, by (i) and lemma 8.3.12. Next, pick c ∈ ℕ as in the proof of (i); for every n ∈ ℕ and every x ∈ 𝔫_{0}^{n+c}D_{2} ∩ p^{1/p}D_{2} we have p^{(p−1)/p}x ∈ 𝔫_{0}^{n+c}D_{2} ∩ pD_{2}, whence 𝔪_{1} · p^{(p−1)/p}x ⊂ 𝔫_{0}^{n}pD_{2}, and therefore 𝔪_{1} · x ⊂ 𝔫_{0}^{n}p^{1/p}D_{2}. This shows that 𝔫_{0}^{n+c}D_{2}^{a} ∩ p^{1/p}D_{2}^{a} ⊂ 𝔫_{0}^{n}p^{1/p}D_{2}^{a} for every n ∈ ℕ, so the 𝔫_{0}-adic topology of D_{2} induces the 𝔫_{0}-adic topology on the ideal p^{1/p}D_{2}^{a}. By assumption, the Frobenius endomorphism of D_{2}/pD_{2} induces an isomorphism D_{2}^{a}/p^{1/p}D_{2}^{a} →^{∼} (D_{2}^{a}/pD_{2}^{a})_{(Φ)}; the latter persists after taking 𝔫_{0}-adic completions, and in view of (ii) and lemma 14.2.45, we conclude that the Frobenius endomorphism of E_{2}/pE_{2} induces an isomorphism E_{2}^{a}/p^{1/p}E_{2}^{a} →^{∼} (E_{2}^{a}/pE_{2}^{a})_{(Φ)}, as needed. ♢

For every a ∈ Σ_{1}, let β′_{a,•} ∈ E(D_{2}) be the image of β_{a,•}, and set likewise

β′_{a_{λ},•} := (\overline{T} _{λ}^{1/p^{n}} | n ∈ ℕ) ∈ E(D_{2}) for every λ ∈ Λ

where \overline{T} _{λ}^{1/p^{n}} ∈ D_{2} denotes the image of T_{λ}^{1/p^{n}}, for every such λ and every n ∈ ℕ. Moreover, for every a ∈ Σ_{2}, let β′′_{a,•} ∈ E(E_{2}) be the image of β′_{a,•}, and set Δ_{2} := (β′′_{a,•} | a ∈ Σ_{2}). Let also f_{2} : A → E_{2} be the structure map of the A-algebra E_{2}; by claim 17.5.68(i) the pair (f_{2} : A → E_{2}, Δ_{2}) fulfills conditions (a),(b),(c) of (17.5.57) and condition (e) of (17.5.65). Combining with propositions 16.9.37(iv) and 17.5.14(ii) we deduce that also condition (d) of (17.5.57) holds for this pair, i.e. (f_{2}, Δ_{2}) is an almost perfectoid object of 𝒟_{Σ_{2},Σ′}^{A}; moreover, if D_{2}^{a} is flat (resp. faithfully flat) over A_{0}, the same holds for E_{2}^{a}, by virtue of proposition 14.2.43.

Lastly, let (g : A → F, Δ := (δ_{a,•} | a ∈ Σ_{2})) be any other object of 𝒟_{Σ_{2},Σ′}^{A}; arguing as in the proof of claim 17.5.60, we see that there exists a unique map h : D_{2} → F of A-algebras such that E(h′)(β′_{a,•}) = δ_{a,•} for every a ∈ Σ_{2}. Let h^{∧} : D_{2}^{∧} → F be the 𝔫-adic completion of h′; it follows easily that h := (h^{∧})_{∗}^{a} : E_{2} → F is the unique morphism (f_{2}, Δ_{2}) → (g, Δ) in 𝒟_{Σ_{2},Σ′}^{A}.

It remains consider the case where Σ := Σ_{1} = Σ_{2}. Then, let 𝔪_{2} ⊂ D_{1} be the ideal generated by (∏_{a∈Σ′_{2}} β_{a,n} | n ∈ ℕ); let D_{2}^{∧} be the 𝔫-adic completion of D_{2} := (D_{1}, 𝔪_{2})_{∗}^{a}, and set E_{2} := (D_{2}^{∧}, 𝔪_{2}D_{2}^{∧})_{∗}^{a}. Arguing as in the proof of proposition 17.5.59, we see that the basic setup (D_{2}, 𝔪_{2}D_{2}) is almost perfectoid for the p-adic topology of D_{2}. Moreover, arguing as in the proof of proposition 17.5.62(ii), we see that if (D_{1}, 𝔪_{1})^{a} is flat over A_{0}, the same holds for (D_{1}, 𝔪_{2})^{a}, so also for (D_{2}, 𝔪_{2}D_{2})^{a}, and then also for (D_{2}^{∧}, 𝔪_{2}D_{2}^{∧})^{a}, by proposition 16.9.37. Furthermore, if (D_{1}, 𝔪_{1})^{a} is faithfully flat over A_{0}, then arguing again as in the proof of proposition 17.5.62(ii), we check that the same holds for (D_{2}, 𝔪_{2}D_{2})^{a}, and thus, also for (D_{2}^{∧}, 𝔪_{2}D_{2}^{∧})^{a}. Summing up, we conclude that if (D_{1}, 𝔪_{1})^{a} is flat (resp. faithfully flat) over A_{0}, the same holds for (E_{2}, 𝔪_{2}E_{2})^{a}. Then, arguing as in the proof of claim 17.5.68we see that E_{2} has no p-torsion and is 𝔫-adically complete and separated, and the basic setup (E_{2}, 𝔪_{1}E_{2}) is almost perfectoid for the p-adic topology of E_{2}. For every a ∈ Σ, let β′_{a,•} ∈ E(E_{2}) be the image of β_{a,•}, and set Δ_{2} := (β′_{a,•} | a ∈ Σ); let also f_{2} : A → E_{2} be the structure map of the A-algebra E_{2}; as in the previous case, we conclude that (f_{2}, Δ_{2}) is an almost perfectoid object of 𝒟_{Σ,Σ′_{2}}^{A}, and it is easily seen that it is the sought initial object : details left to the reader. □

17.5.69. Resume the notation of (17.5.56), and let D^{∧} be the 𝔫-adic completion of D, and D′ := (D^{∧}, 𝔪D^{∧})_{∗}^{a}. Set also x_{0} := p, and let either t := n if g ∈ {x_{0}, …, x_{n}} ∪ {b_{λ} | λ ∈ Λ}, or else t := n + 1 and x_{t} := g; we take

Σ′_{0} := {x_{t}} Σ_{0} := {x_{0}, …, x_{t}} ∪ {b_{λ} | λ ∈ Λ}.

For i = 0, …, t and for every λ ∈ Λ, let β_{x_{i},•} := (\overline{T} _{i}^{1/p^{k}} | k ∈ ℕ) ∈ E(D′) and β_{b_{λ},•} := (\overline{T} _{λ}^{1/p^{k}} | k ∈ ℕ) ∈ E(D′), where as usual \overline{T} _{i}^{1/p^{k}}, \overline{T} _{λ}^{1/p^{k}} ∈ D′ denote the image of T_{i}^{1/p^{k}} and T_{λ}^{1/p^{k}}, for every such i, every λ, and every k ∈ ℕ. Lastly, set Δ_{0} := (β_{a,•} | a ∈ Σ_{0}), and let f_{0} : A → D′ be the structure map of the A-algebra D′.

Proposition 17.5.70. The pair (f_{0}, Δ_{0}) is an almost perfectoid initial object of 𝒟_{Σ_{0},Σ′_{0}}^{A}, and is almost faithfully flat over A_{0}.

Proof. Let D_{1} be the integral closure of (C^{∧}, 𝔪C^{∧})_{∗}^{a} in C^{∧} ⊗_{A_{0}} A[1/g], and recall that the inclusion map j : D → D_{1} induces an isomorphism D^{a} →^{∼} D_{1}^{a} of (C, 𝔪)^{a}-algebras ([75, Lemma 8.2.28]), so its 𝔫-adic completion j^{∧} : D^{∧} → D_{1}^{∧} induces an isomorphism of Calgebras (j^{∧})_{∗}^{a} : D′ →^{∼} (D_{1}^{∧})_{∗}^{a}. By theorem 16.9.42(i,iii), (D_{1}, 𝔪D_{1}) is an almost perfectoid basic setup, for the p-adic topology of D_{1}, and D_{1}^{a}/p^{n}D_{1}^{a} is a faithfully flat C^{a}/p^{n}C^{a}-algebra, for every n ∈ ℕ; hence D_{1}^{a}/p^{n}D_{1}^{a} is faithfully flat over A_{0}/p^{n}A_{0}, for every n ∈ ℕ, and therefore D_{1}^{a} is faithfully flat over A_{0} (proposition 14.2.43(i)). We may now argue as in the proof of claim 17.5.68(iii), to conclude that ((D_{1}^{∧})_{∗}^{a}, 𝔪(D_{1}^{∧})_{∗}^{a}) is an almost perfectoid basic setup, for the p-adic topology of D_{1}, so the same holds for (D′, 𝔪D′). This already shows that the pair (f_{0}, Δ_{0}) fulfills conditions (a),(b),(c) of (17.5.57) and condition (e) of (17.5.65). Combining with propositions 16.9.37(iv) and 17.5.14(ii) we deduce that also condition (d) of (17.5.57) holds for this pair, i.e. (f_{0}, Δ_{0}) is an almost perfectoid object of 𝒟_{Σ_{0},Σ′_{0}}^{A}. Moreover, since D_{1}^{a} is faithfully flat over A_{0}, the same holds for (D_{1}^{∧})^{a}, again by proposition 14.2.43(i), so also for D^{′a}. Lastly, one verifies as in the proof of proposition 17.5.62(i) that (f_{0}, Δ_{0}) is an initial object of 𝒟_{Σ_{0},Σ′_{0}}^{A} : details left to the reader. □

17.5.71. Consider now the category

𝒜

whose objects are all local noetherian domains (A, 𝔫), complete for their 𝔫-adic topology, with Frac A of zero characteristic, residue field k of characteristic p > 0 and with [k : k^{p}] < +∞. The morphisms (A, 𝔫) → (A′, 𝔫′) of 𝒜 are all the injective ring homomorphisms A → A′.

For every object (A, 𝔫) of 𝒜, let Ω(A) be the set of all pairs (Σ, Σ′) of finite non-empty subsets of A \ {0} with Σ′ ⊂ Σ and p ∈ Σ, such that the corresponding category 𝒞_{Σ,Σ′}^{A} admits an initial object (notation of (17.5.57)), and for every (Σ, Σ′) ∈ Ω(A), fix such an initial object

(f_{Σ,Σ′} : A → D_{Σ,Σ′}, Δ_{Σ,Σ′}).

We let ≤ be the partial ordering on Ω(A) such that (Σ_{1}, Σ′_{1}) ≤ (Σ_{2}, Σ′_{2}) if and only if Σ_{1} ⊂ Σ_{2} and Σ′_{1} ⊂ Σ′_{2}. Clearly, (Ω(A), ≤) is a filtered partially ordered set, and with the notation of remark 17.5.58, we have a unique morphism

(f_{Σ_{1},Σ′_{1}}, Δ_{Σ_{1},Σ′_{1}}) → (f_{Σ_{2},Σ′_{2}}, Δ_{Σ_{2},Σ′_{2}})_{|Σ_{1},Σ′_{1}} in 𝒞_{Σ_{1},Σ′_{2}}^{A}, whenever (Σ_{1}, Σ′_{1}) ≤ (Σ_{2}, Σ′_{2}).

In other words, we get a well defined filtered system of A-algebras (D_{Σ,Σ′} | (Σ, Σ′) ∈ Ω(A)).

For every (Σ, Σ′) ∈ Ω(A), let (D_{Σ,Σ′}, 𝔪_{Σ,Σ′}) be the basic setup attached to (f_{Σ,Σ′}, Δ_{Σ,Σ′}), and consider the subset S_{Σ,Σ′} ⊂ D_{Σ,Σ′}^{ℕ} of all (a_{n} | n ∈ ℕ) such that a_{n} ∈ D_{Σ,Σ′}a_{n+1} for every n ∈ ℕ, and 𝔪_{Σ,Σ′} ⊂ ⋃_{n∈ℕ} D_{Σ,Σ′}a_{n}. We let Θ(A) ⊂ Ω(A) be the subset of all (Σ, Σ′) such that

E_{Σ,Σ′} := S_{Σ,Σ′}^{−1}D_{Σ,Σ′}^{⋄}

is a big CM A-algebra (see (17.5.2)).

Lemma 17.5.72. Θ(A) is a cofinal subset of Ω(A), for every (A, 𝔫) ∈ Ob(𝒜).

Proof. Choose a subring (A_{0}, 𝔫_{0}) ⊂ (A, 𝔫) as in (17.5.51); then propositions 17.5.59 and 17.5.62 show that there exists (Σ_{0}, Σ′_{0}) ∈ Ω(A) such that (D_{Σ,Σ′}, 𝔪_{Σ,Σ′})^{a} is faithfully flat over A_{0}, for every (Σ, Σ′) ≥ (Σ_{0}, Σ′_{0}). Then, arguing as in the proof of theorem 17.5.54 we see that E_{Σ,Σ′} is a big CM A-algebra for every such (Σ, Σ′). □

Notice that whenever (Σ_{1}, Σ′_{1}) ≤ (Σ_{2}, Σ′_{2}), the induced map D_{Σ_{1},Σ′_{1}}^{ℕ} → D_{Σ_{2},Σ′_{2}}^{ℕ} sends S_{Σ_{1},Σ_{2}} into S_{Σ_{2},Σ′_{2}}. Let then E(A) be the colimit of the induced filtered system (E_{Σ,Σ′} | (Σ, Σ′) ∈ Ω(A)). By lemma 17.5.72 and remark 17.5.19(iv), E(A) is a big CM A-algebra. Next, let g : (A, 𝔫) → (A′, 𝔫′) be any morphism of 𝒜, and (Σ_{1}, Σ′_{1}) ∈ Ω(A), (Σ_{2}, Σ′_{2}) ∈ Ω(A′) with

g(Σ_{1}) ⊂ Σ_{2} and g(Σ′_{1}) ⊂ Σ′_{2}.

For every object (f, Δ := (β_{a,•} | a ∈ Σ_{2})) of 𝒞_{Σ_{2},Σ′_{2}}^{A′}, set Δ_{|Σ_{1}} := (β_{g(a),•} | a ∈ Σ_{1}); since g is injective, the pair (f ∘ g, Δ_{|Σ_{1}}) fulfills conditions (a),(c) and (d) of (17.5.57), defining the objects of 𝒞_{Σ_{1},Σ′_{1}}^{A}, and with remark 14.2.7(ii), we easily check that also condition (b) holds for this pair. We get therefore a well defined functor :

𝒞_{Σ_{2},Σ′_{2}}^{A′} → 𝒞_{Σ_{1},Σ′_{1}}^{A} (f, Δ) ↦ (f, Δ)_{|Σ_{1},Σ′_{1}} := (f ∘ g, Δ_{|Σ_{1}}).

Especially, notice that for every (Σ, Σ′) ∈ Ob(A) we have (gΣ, gΣ′) ∈ Ob(A′), whence a natural morphism in 𝒞_{Σ,Σ′}^{A} :

(f_{Σ,Σ′}, Δ_{Σ,Σ′}) → (f_{gΣ,gΣ′}, Δ_{gΣ,gΣ′})_{|Σ,Σ′}.

Moreover, the induced map D_{Σ,Σ′}^{ℕ} → D_{gΣ,gΣ′}^{ℕ} sends S_{Σ,Σ′} into S_{gΣ,gΣ′}, so we get a map of A-algebras :

E_{Σ,Σ′} → E_{gΣ,gΣ′} for every (Σ, Σ′) ∈ Ω(A).

Finally, the colimit of this system of maps yields a well defined map E(g) : E(A) → E(A′), and a simple inspection shows that the rules : (A, 𝔫) ↦ E(A) and g ↦ E(g) define a functor

E : 𝒜 → ℤ-Alg.

17.5.73. Using the categories 𝒟_{Σ,Σ′}^{A} of (17.5.65) we may obtain the following variant of the functor E. Consider the category

𝒜′

whose objects are all local noetherian domains (A, 𝔫), complete for their 𝔫-adic topology, with Frac A of zero characteristic, and residue field k of characteristic p > 0. The morphisms (A, 𝔫) → (A′, 𝔫′) of 𝒜′ are all the injective and local ring homomorphisms A → A′.

For every object (A, 𝔫) of 𝒜, let Ω′(A) be the set of all pairs (Σ, Σ′) of non-empty subsets of A \ {0} where Σ′ is finite, and Σ′ ∪ {p} ⊂ Σ, and such that the corresponding category 𝒟_{Σ,Σ′}^{A} admits an initial object. For every (Σ, Σ′) ∈ Ω(A), fix such an initial object

(f_{Σ,Σ′} : A → D_{Σ,Σ′}, Δ_{Σ,Σ′}).

We define a partial ordering ≤ on Ω′(A) just as for Ω(A) in (17.5.71). Clearly, (Ω′(A), ≤) is a filtered partially ordered set, and with the notation of (17.5.65), we have a unique morphism

(f_{Σ_{1},Σ′_{1}}, Δ_{Σ_{1},Σ′_{1}}) → (f_{Σ_{2},Σ′_{2}}, Δ_{Σ_{2},Σ′_{2}})_{|Σ_{1},Σ′_{1}} in 𝒟_{Σ_{1},Σ′_{2}}^{A}, whenever (Σ_{1}, Σ′_{1}) ≤ (Σ_{2}, Σ′_{2}) .

In other words, we get a well defined filtered system of A-algebras (D_{Σ,Σ′} | (Σ, Σ′) ∈ Ω′(A)).

For every (Σ, Σ′) ∈ Ω′(A), let (D_{Σ,Σ′}, 𝔪_{Σ,Σ′}) be the basic setup attached to (f_{Σ,Σ′}, Δ_{Σ,Σ′}), and define the multiplicative subset S_{Σ,Σ′} ⊂ D_{Σ,Σ′}^{ℕ} as in (17.5.71). We let Θ′(A) ⊂ Ω′(A) be the subset of all (Σ, Σ′) such that E′_{Σ,Σ′} := S_{Σ,Σ′}^{−1}D_{Σ,Σ′}^{⋄} is a big CM A-algebra. Using propositions 17.5.66 and 17.5.70, and arguing as in the proof of lemma 17.5.72, we see that Θ′(A) is a cofinal subset of Ω′(A), for every (A, 𝔫) ∈ Ob(𝒜′). Then, the colimit E′(A) of the induced filtered system (E′_{Σ,Σ′} | (Σ, Σ′) ∈ Ω′(A)) is a big CM A-algebra (remark 17.5.19(iv)). Next, let g : (A, 𝔫) → (A′, 𝔫′) be any morphism of 𝒜′, and (Σ_{1}, Σ′_{1}) ∈ Ω′(A), (Σ_{2}, Σ′_{2}) ∈ Ω′(A′) with

g(Σ_{1}) ⊂ Σ_{2} and g(Σ′_{1}) ⊂ Σ′_{2}.

For every object (f, Δ := (β_{a,•} | a ∈ Σ_{2})) of 𝒟_{Σ_{2},Σ′_{2}}^{A′}, set again Δ_{|Σ_{1}} := (β_{g(a),•} | a ∈ Σ_{1}); since g is injective, arguing as in the foregoing we see that the pair (f ∘ g, Δ_{|Σ_{1}}) fulfills conditions (a),(b), (c),(d) of (17.5.57). Moreover, since g is local, lemma 8.3.12implies that also condition (e) of (17.5.65) is fulfilled. We get therefore a well defined functor :

𝒟_{Σ_{2},Σ′_{2}}^{A′} → 𝒟_{Σ_{1},Σ′_{1}}^{A} (f, Δ) ↦ (f, Δ)_{|Σ_{1},Σ′_{1}} := (f ∘ g, Δ_{|Σ_{1}}).

From this, arguing as in the foregoing case, we obtain a natural map of A-algebras E′(g) : E′(A) → E′(A′), whence a well defined functor :

E′ : 𝒜′ → ℤ-Alg.

17.5.74. Next, we wish to establish a weak funtoriality property of big CM algebras, for not necessarily injective maps of local rings. The first step is the following :

Proposition 17.5.75. Let A be noetherian ring, I ⊂ A an ideal, f_{1}, …, f_{r} ∈ I a finite sequence such that the following holds for every 𝔭 ∈ Σ := Ass_{A}(A/I) (see definition 10.5.1(iii)) :

(a) I_{𝔭} is generated by the image of the sequence f_{•}.

(b) The image of the sequence f_{•} is regular in I_{𝔭}. Then we have :

(i) There exists g ∈ A \ ⋃_{𝔭∈Σ} 𝔭 such that gI ⊂ J := ∑_{i=1}^{r} f_{i}A.

(ii) For every g as in (i), the sequence f′_{•} := (f_{1}/g, …, f_{r}/g) is quasi-regular in the sub-

ring A′ := A[f_{1}/g, …, f_{r}/g] of A[1/g]. (iii) With the notation of (ii), let J′ ⊂ A′ be the ideal generated by f′_{•}. The natural map

A → A′ induces an isomorphism A/I →^{∼} A′/J′.

Proof. (i): Let L := {x ∈ A | xI ⊂ J}; since I is finitely generated, it is easily seen that L_{𝔭} = {x ∈ A_{𝔭} | xI_{𝔭} ⊂ J_{𝔭}} for every 𝔭 ∈ Spec A. Especially, L_{𝔭} = A_{𝔭} for every 𝔭 ∈ Σ, by assumption (a), i.e. L ⊄ 𝔭 for every such 𝔭; then the assertion follows from [12, Prop.1.11(i)].

(iii): Clearly the natural map A → A′ induces a surjection φ : A → A′/J′, and we need to check that Ker φ = I. However, if x ∈ I, then gx = ∑_{i=1} a_{i}f_{i} for some a_{1}, …, a_{r} ∈ A, whence x = ∑_{i=1}^{r} a_{i}f_{i}/g ∈ J′ in A′, so I ⊂ Ker φ. Let then \overline{φ} : A/I → A′/J′, be the map induced by φ; notice that A[1/g] = A′[1/g], and IA[1/g] = J′A′[1/g], so that the localization \overline{φ}_{g} of \overline{φ} is an isomorphism. Especially, \overline{φ}_{𝔭} is an isomorphism for every 𝔭 ∈ Σ, and it suffices then to invoke the following :

Claim 17.5.76. Let M and N be two A-modules, and h : M → N an A-linear map. Then h is injective if and only if h_{𝔭} : M_{𝔭} → N_{𝔭} is injective for every 𝔭 ∈ Ass_{A}(M).

Proof of the claim. To ease notation, set Ω := Ass_{A}(M), and consider the commutative diagram

\xymatrix{M \ar[r]^{h} \ar[d] & N \ar[d] \\ ∏_{𝔭∈Ω} M_{𝔭} \ar[r]^{∏_{𝔭∈Ω} h_{𝔭}} & ∏_{𝔭∈Ω} N_{𝔭}}

whose left vertical arrow is injective, by lemma 10.5.3(iii), and whose bottom horizontal arrow is injective if and only if h_{𝔭} is injective for every 𝔭 ∈ Ω. The assertion is an immediate consequence. ♢

(ii): Set A_{r} := A/I[T_{1}, …, T_{r}], let I_{r} ⊂ A_{r} be the ideal generated by T_{1}, …, T_{r}, and denote by gr_{•}A_{r} the graded A/I-algebra associated with the I_{r}-adic filtration of A_{r}. Also, for every i = 1, …, r, let \overline{T}_{i} ∈ gr_{1}A_{r} (resp. x_{i} ∈ J′/J^{′2}) be the image of T_{i} (resp. of f_{i}/g); we consider the map of graded A/I-algebras

ψ_{•} : gr_{•}A_{r} → ⨁_{n∈ℕ} J^{′n}/J^{′n+1} \overline{T}_{1} ↦ x_{1}, …, \overline{T}_{r} ↦ x_{r}.

In view of (iii), we need to check that ψ is an isomorphism, and since ψ is surjective, it suffices to show that its restriction ψ_{n} : gr_{n}A_{r} → J^{′n}/J^{′n+1} is injective for every n ∈ ℕ. However, by assumption (b), the image of the sequence f_{•} is regular in I_{𝔭} = J′_{𝔭}, for every 𝔭 ∈ Σ, hence ψ_{n,𝔭} is an isomorphism for every n ∈ ℕ and every 𝔭 ∈ Σ (proposition 7.8.15). Then the assertion follows from claim 17.5.76. □

Remark 17.5.77. Let A be a noetherian ring, and 𝔭 ⊂ A a prime ideal such that A_{𝔭} is a regular local ring. Then Ass_{A}(A/𝔭) = {𝔭}, and we may find g_{1}, …, g_{r} ∈ 𝔭 whose images in A_{𝔭} are a regular system of parameters (just take any regular system of parameters for A_{𝔭}, and multiply by a suitable element of A\𝔭 to clear denominators : details left to the reader). Hence, assumptions (a) and (b) of proposition 17.5.75hold for I := 𝔭 and f_{•} := g_{•}.

17.5.78. Consider now a sequence of local ring homomorphisms

A_{0} ⟶^{f_{0}} A_{1} ⟶^{f_{1}} ⋯ ⟶^{f_{n−1}} A_{n}

of some finite length n, where A_{0}, …, A_{n} are complete, local noetherian domains.

Proposition 17.5.79. In the situation of (17.5.78), there exists a commutative diagram of complete local noetherian domains, and local ring homomorphisms :

(17.5.80)  \xymatrix{A_{0} \ar[r]^{f_{0}} \ar[ddr]_{h_{0}} & A_{1} \ar[d]^{j_{1}} \ar[r]^{f_{1}} & ⋯ \ar[r]^{f_{n−1}} & A_{n} \ar[d]^{j_{n}} \\  & B_{1} \ar[r]^{g_{1}} & ⋯ \ar[r]^{g_{n−1}} & B_{n} \\  & C_{1} \ar[r]^{h_{1}} \ar[u]_{π_{1}} & ⋯ \ar[r]^{h_{n−1}} & C_{n} \ar[u]_{π_{n}}}

such that h_{0}, …, h_{n−1} are injective maps, and the following holds for every i = 1, …, n : (i) the map j_{i} is finite and injective (ii) the map π_{i} is surjective, and Ker π_{i} is generated by a regular sequence of C_{i}.

Proof. Let us show first how to reduce to the case where n = 1. Indeed, suppose that n > 1, and B_{1} and C_{1} are given as required, with local ring homomorphisms h_{0} : A_{0} → C_{0}, j_{1} : A_{1} → B_{1} and π_{1} : C_{1} → B_{1} such that h_{0} is injective, and conditions (i) and (ii) hold for i = 1. By induction on n, we further suppose that the proposition is already known for every sequence A_{0} → ⋯ → A_{n−1} of local ring homomorphisms of complete local noetherian domains.

• We construct by induction on i = 1, …, n, a commutative diagram :

(17.5.81)  \xymatrix{A_{1} \ar[d]^{j′_{1}} \ar[r]^{f_{1}} & ⋯ \ar[r]^{f_{n−1}} & A_{n} \ar[d]^{j′_{n}} \\ A′_{1} \ar[r]^{f′_{1}} & ⋯ \ar[r]^{f′_{n−1}} & A′_{n}}

where A′_{1}, …, A′_{n} are complete local noetherian domains, j′_{1}, …, j′_{n} are local, finite and injective ring homomorphisms, and f′_{1}, …, f′_{n−1} are local ring homomorphisms. For i = 1, we take A′_{1} := B_{1} and j′_{1} := j_{1}; suppose next that i > 1, and A′_{j} has already been given for every j < i, together with the required commutative diagram of maps j′_{1}, …, j′_{i−1} and f′_{1}, …, f′_{i−2}. Since j′_{i−1} is finite and injective, the induced map Spec A′_{i−1} → Spec A_{i−1} is surjective, so the same holds for its base change Spec (A_{i} ⊗_{A_{i−1}} A′_{i−1}) → Spec A_{i}, and hence we may pick 𝔭 ∈ Spec(A_{i} ⊗_{A_{i−1}} A′_{i−1}) whose image in Spec A_{i} is the generic point; we set A′_{i} := (A_{i} ⊗_{A_{i−1}} A′_{i−1})/𝔭, and let j_{i} : A_{i} → A′_{i} be the induced map. By construction, Spec(j_{i}) is dominant with closed image, hence it is surjective, so Ker j_{i} = 0, since A_{i} is a domain. Moreover, since A_{i} is complete and separated, j_{i} is local, and A′_{i} is a finite A_{i}-algebra, so it is a complete local noetherian domain, as required. Lastly, let k_{i−1}, k_{i} and k′_{i−1} be the residue fields of A_{i−1}, A_{i} and A′_{i−1}; to see that the natural map f′_{i−1} : A′_{i−1} → A′_{i} is local, it suffices to remark that the residue field of A′_{i} is a quotient of k_{i} ⊗_{k_{i−1}} k′_{i−1}.

• Next, we apply the inductive assumption to the sequence of local ring homomorphisms

(17.5.82)  C_{1} ⟶^{f′_{1}∘π_{1}} A′_{2} ⟶^{f′_{2}} ⋯ ⟶^{f′_{n−1}} A′_{n}.

Thus, we find a commutative diagram of complete local noetherian domains and local ring homomorphisms :

\xymatrix{C_{1} \ar[r]^{f′_{1}∘π_{1}} \ar[ddr]_{h_{1}} & A′_{2} \ar[d]^{j′′_{2}} \ar[r]^{f′_{2}} & ⋯ \ar[r]^{f′_{n−1}} & A′_{n} \ar[d]^{j′′_{n}} \\  & B_{2} \ar[r]^{g_{2}} & ⋯ \ar[r]^{g_{n−1}} & B_{n} \\  & C_{2} \ar[r]^{h_{2}} \ar[u]_{π_{2}} & ⋯ \ar[r]^{h_{n−1}} & C_{n} \ar[u]_{π_{n}}}

where h_{1}, …, h_{n−1} are injective, and the map j′′_{i} (resp. π_{i}) fulfills the corresponding condition (i) (resp. (ii)) of the proposition, for every i = 2, …, n. Then we obtain the sought diagram with g_{1} := j′′_{2} ∘ f′_{1} and j_{i} := j′′_{i} ∘ j′_{i} for every i = 2, …, n.

• We consider now the case n = 1, so we are given a single local ring homomorphism f_{0} : A_{0} → A_{1} of complete local noetherian domains. We notice :

Claim 17.5.83. There exist a complete local noetherian domain D_{0}, and local ring homomorphisms φ_{0} : A_{0} → D_{0}, ψ_{0} : D_{0} → A_{1} such that φ_{0} is injective, ψ_{0} is surjective, and ψ_{0}∘φ_{0} = f_{0}.

Proof of the claim. By proposition 9.7.31(i) there exists a complete local noetherian ring D′_{0}, a faithfully flat ring homomorphism φ′_{0} : A_{0} → D′_{0}, and a surjective map ψ′_{0} : D′_{0} → A_{1} with ψ′_{0} ∘ φ′_{0} = f_{0}. Then, let 𝔭 be any minimal prime ideal of D′_{0} contained in Ker ψ′_{0}, and set D_{0} := D′_{0}/𝔭; since φ′_{0} is flat, we have φ_{0}^{′−1}𝔭 = 0, so the induced map φ_{0} : A_{0} → D_{0} is still injective, and ψ′_{0} factors through a surjection ψ_{0} : D_{0} → A_{1}, whence the claim. ♢

Let φ_{0} : A_{0} → D_{0} and ψ_{0} : D_{0} → A_{1} be as in claim 17.5.83, and set 𝔭 := Ker ψ_{0}; lemma 9.7.4(v) says that D_{0} is excellent, so D_{0,𝔭} is quasi-excellent, by lemma 9.7.4(ii,iv).

Especially, D_{0,𝔭} is a Nagata ring, and the regular locus of Spec D_{0,𝔭} is an open subset ([66, Ch.IV, Th.6.12.4]); by corollary 9.6.56, we may then find a regular local domain (D_{1}, 𝔪_{1}), and an injective local ring homomorphism D_{0,𝔭} → D_{1} of essentially finite type, inducing an isomorphism Frac(D_{0}) →^{∼} Frac(D_{1}) and a finite field extension Frac(D_{0}/𝔭) → D_{1}/𝔪_{1}. Pick y_{1}, …, y_{n} ∈ D_{1} and a prime ideal 𝔮 ⊂ D_{2} := D_{0}[y_{1}, …, y_{n}] such that D_{1} = D_{2,𝔮}; we remark :

Claim 17.5.84. We can assume that the induced injective map A_{1} →^{∼} D_{0}/𝔭 → D_{2}/𝔮 is finite.

Proof of the claim. The image \overline{y}_{i} ∈ D_{1}/𝔪_{1} = Frac(D_{2}/𝔮) of y_{i} is algebraic over Frac(D_{0}/𝔭) for every i = 1, …, n, so we may find x ∈ D_{0} \ 𝔭 such that the images of xy_{1}, …, xy_{n} in Frac(D_{2}/𝔮) are integral over D_{0}/𝔭. Set D′_{2} := D_{0}[xy_{1}, …, xy_{n}] and 𝔮′ := 𝔮 ∩ D′_{2}; then clearly D′_{2,𝔮′} = D_{1}, so we may replace y_{i} by xy_{i} for i = 1, …, n, and assume that \overline{y}_{i} is integral over D_{0}/𝔭, for every i = 1, …, n, whence the claim. ♢

• Now we apply proposition 17.5.75 and remark 17.5.77 to A := D_{2}, I := 𝔮, and to a finite sequence x_{1}, …, x_{d} of elements of 𝔮 whose images in the regular local ring D_{1} are a regular system of parameters : we obtain g ∈ D_{2} \𝔮 with g ·𝔮 ⊂ ∑_{i=1}^{d} x_{i}D_{2}, and such that the sequence t_{1} := x_{1}/g, …, t_{d} := x_{d}/g is quasi-regular in the subring D′_{2} := D_{2}[t_{1}, …, t_{d}] ⊂ D_{2}[1/g]; moreover, if 𝔮′ ⊂ D′_{2} denotes the ideal generated by t_{1}, …, t_{d}, the inclusion map D_{2} → D′_{2} induces an isomorphism

D_{2}/𝔮 →^{∼} B_{1} := D′_{2}/𝔮′.

Since A_{1} is complete, claim 17.5.84 implies that D_{2}/𝔮 is a complete local noetherian domain, so the same holds for B_{1}, and the resulting map g_{1} : A_{1} → B_{1} is injective and finite; then, let 𝔪 ⊂ B_{1} be the maximal ideal, 𝔪′ ⊂ D′_{2} its preimage, and C_{1} the 𝔪′-adic completion of D′_{2}. Since C_{1} is a flat D′_{2}-algebras, the image t in C_{1} of the sequence t_{1}, …, t_{d} is still quasi-regular, hence it is regular, by proposition 7.8.15and [126, Th.8.10]. Also, the localization D′_{2} → D′_{2,𝔪′} is injective, since D_{2} is a domain, and the same holds for the completion map D′_{2,𝔪′} → C_{1}, since the latter is faithfully flat; summing up, we deduce that the structure map h_{0} : A_{0} → C_{1} of the resulting A_{0}-algebra C_{1} is injective. To conclude the proof, it now suffices to remark :

Claim 17.5.85. C_{1} is a domain, and h_{0} is a local ring homomorphism.

Proof of the claim. In order to show that h_{0} is local, it suffices to check that the same holds for the natural map A_{0} → D′_{2,𝔪′}. The latter comes down to verifying that the preimage of 𝔪′ under the natural map A_{0} → D′_{2} is the maximal ideal of A_{0}. But in turns, this is the same as the preimage of 𝔪 under the composition A_{0} → A_{1} → D_{2}/𝔮 →^{∼} B_{1}, which is local, whence the assertion. Next, let gr_{•}C_{1} be the graded ring associated with the 𝔮′-adic filtration on C_{1}; hence gr_{0}C_{1} is the 𝔪′-adic completion of B_{1}. But B_{1} is 𝔪′-adically complete and separated, so gr_{0}C_{1} = B_{1} and since t is quasi-regular, the natural map of graded B_{1}-algebras

B_{1}[T_{1}, …, T_{d}] → gr_{•}C_{1} T_{1} ↦ t_{1}, …, T_{d} ↦ t_{d}

is an isomorphism. Since the 𝔮′-adic filtration is separated on C_{1}, the assertion follows. □

17.5.86. We will also need the following variant of the situation contemplated in (17.5.78) : we consider a sequence, of some finite length n, of ring homomorphisms

A_{0} ⟶^{f_{0}} A_{1} ⟶^{f_{1}} ⋯ ⟶^{f_{n−1}} A_{n}

where A_{0}, …, A_{n} are quasi-excellent domains, and f_{0}, …, f_{n−1} are of finite type.

Proposition 17.5.87. In the situation of (17.5.86), there exists a commutative diagram (17.5.80) of quasi-excellent domains, such that h_{0}, …, h_{n−1} are injective maps of finite type, and for every i = 1, …, n the following holds :

(i) the map j_{i} is finite and injective

(ii) the map π_{i} is surjective, and Ker π_{i} is generated by a quasi-regular sequence of C_{i}.

Proof. We argue as in the proof of proposition 17.5.79 : first, we show how to reduce to the case where n = 1. Hence, suppose that n > 1, and B_{1} and C_{1} are given as required, with local ring homomorphisms h_{0} : A_{0} → C_{0}, j_{1} : A_{1} → B_{1} and π_{1} : C_{1} → B_{1} such that h_{0} is injective, and conditions (i) and (ii) of proposition 17.5.79 hold for i = 1. By induction on n, we further suppose that the proposition is already known for every sequence A_{0} → ⋯ → A_{n−1} consisting of maps of finite type, and quasi-excellent domains A_{0}, …, A_{n−1}.

• We construct by induction on i = 1, …, n, quasi-excellent domains A′_{1}, …, A′_{n}, and a commutative diagram (17.5.81), where j′_{1}, …, j′_{n} are injective and finite ring homomorphisms, and f′_{1}, …, f′_{n−1} are of finite type. For i = 1, we take A′_{1} := B_{1} and j′_{1} := j_{1}; suppose next that i > 1, and A′_{j} has already been given for every j < i, together with the required commutative diagram of maps j′_{1}, …, j′_{i−1} and f′_{1}, …, f′_{i−2}. Just as in the proof of proposition 17.5.79, we may then find 𝔭 ∈ Spec(A_{i} ⊗_{A_{i−1}} A′_{i−1}) whose image in Spec A_{i} is the generic point; we set A′_{i} := (A_{i} ⊗_{A_{i−1}} A′_{i−1})/𝔭, and let j_{i} : A_{i} → A′_{i} and f′_{i−1} : A′_{i−1} → A′_{i} be the induced maps.

• Next, we apply the inductive assumption to the sequence (17.5.82), to construct the sought commutative diagram, as in the proof of proposition 17.5.79.

• We consider now the case n = 1, so we are given a single map of finite type f_{0} : A_{0} → A_{1} of quasi-excellent domains. Then we pick a polynomial A_{0}-algebra D_{0} := A_{0}[T_{1}, …, T_{k}] with a surjection of A_{0}-algebras ψ_{0} : D_{0} → A_{1}. Set 𝔭 := Ker ψ_{0}; then D_{0,𝔭} is quasi-excellent, by lemma 9.7.4(ii,iv). Especially, D_{0,𝔭} is a Nagata ring (lemma 9.7.4(i)), and the regular locus of Spec D_{0,𝔭} is an open subset ([66, Ch.IV, Th.6.12.4]); by corollary 9.6.56, we may then find a regular local domain (D_{1}, 𝔪_{1}), and an injective local ring homomorphism D_{0,𝔭} → D_{1} of essentially finite type, inducing an isomorphism Frac(D_{0}) →^{∼} Frac(D_{1}) and a finite field extension Frac(D_{0}/𝔭) → D_{1}/𝔪_{1}. Pick y_{1}, …, y_{n} ∈ D_{1} and a prime ideal 𝔮 ⊂ D_{2} := D_{0}[y_{1}, …, y_{n}] such that D_{1} = D_{2,𝔮}. Arguing as in the proof of claim 17.5.84, we can assume that the induced injective map A_{1} →^{∼} D_{0}/𝔭 → D_{2}/𝔮 is finite.

• Now we apply proposition 17.5.75 and remark 17.5.77 to A := D_{2}, I := 𝔮, and to a finite sequence x_{1}, …, x_{d} of elements of 𝔮 whose images in the regular local ring D_{1} are a regular system of parameters : we obtain g ∈ D_{2} \𝔮 with g ·𝔮 ⊂ ∑_{i=1}^{d} x_{i}D_{2}, and such that the sequence t_{1} := x_{1}/g, …, t_{d} := x_{d}/g is quasi-regular in the subring C_{1} := D_{2}[t_{1}, …, t_{d}] ⊂ D_{2}[1/g]; moreover, if 𝔮′ ⊂ C_{1} denotes the ideal generated by t_{1}, …, t_{d}, the inclusion map D_{2} → C_{1} induces an isomorphism

D_{2}/𝔮 →^{∼} B_{1} := C_{1}/𝔮′.

By construction, the resulting map g_{1} : A_{1} → B_{1} is injective and finite, and the natural map A_{0} → C_{1} is injective of finite type, as required. □

Corollary 17.5.88. In the situation of (17.5.86), suppose moreover that A_{0} is a ℚ-algebra of essentially finite type. Then there exists a commutative diagram of ring homomorphisms :

\xymatrix{A_{0} \ar[r]^{f_{0}} \ar[d] & A_{1} \ar[r]^{f_{1}} \ar[d] & ⋯ \ar[r]^{f_{n−1}} & A_{n} \ar[d] \\ D_{0} \ar[r] & D_{1} \ar[r] & ⋯ \ar[r] & D_{n}}

such that D_{i} is a big locally CM A_{i}-algebra for every i = 0, …, n.

Proof. Pick a commutative diagram (17.5.80) fulfilling the conditions of proposition 17.5.87. Especially, h_{0}, …, h_{n} are maximizing, since they are injective maps of domains. Then, with the functor E′_{0} of (17.5.48), we get a commutative diagram of ring homomorphisms :

(17.5.89)  \xymatrix{A_{0} \ar[r]^{h_{0}} \ar[d] & C_{1} \ar[r]^{h_{1}} \ar[d] & ⋯ \ar[r]^{h_{n−1}} & C_{n} \ar[d] \\ D′_{0} \ar[r] & D′_{1} \ar[r] & ⋯ \ar[r] & D′_{n}}

such that D′_{0} is a big locally CM A_{0}-algebra, and D′_{i} is a big locally CM C_{i}-algebra, for i = 1, …, n. Next, by construction, for every i = 1, …, n the kernel 𝔭_{i} of π_{i} is a prime ideal generated by a quasi-regular sequence f_{i} := (f_{i,1}, …, f_{i,r_{i}}) of C_{i}; then, the image of the sequence f_{i} is regular in C_{i,𝔭_{i}} (proposition 7.8.15), so it can be completed to a system of parameters for C_{i,𝔭_{i}} ([63, Ch.0, Prop.16.4.1]) and hence 0 = dim B_{i,𝔭_{i}} = dim C_{i,𝔭_{i}} − r_{i} ([126, Th.14.1(i)]), i.e. 𝔭_{i} is a prime of height r_{i}, for every i = 1, …, n. Then, by corollary 10.4.36, the ring D_{i} := B_{i} ⊗_{C_{i}} D′_{i} is a big locally CM B_{i}-algebra, for i = 1, …, n; set as well D_{0} := D′_{0}. It remains to check that D_{i} is a big CM A_{i}-algebra, for i = 1, …, n, after restriction of scalars along the map j_{i}. Thus, for any such i, let 𝔮 be any prime ideal of A_{i}; clearly D_{i,𝔮} is a locally CM B_{i,𝔮}-algebra, and we need to show that depth_{A_{i,𝔮}}D_{i,𝔮} = dim A_{i,𝔮}. Since B_{i,𝔮} is a finite A_{i,𝔮}- algebra, taking into account [126, §9, Lemma 2], lemma 10.4.17(ii), and (10.4.30), we come down to checking that the height of every maximal ideal of B_{i,𝔮} equals the height of 𝔮. The latter holds by [64, Ch.IV, Prop.5.6.1]. □

We are now ready to state :

Theorem 17.5.90. In the situation of (17.5.78), there exists a commutative diagram of ring homomorphisms :

\xymatrix{A_{0} \ar[r]^{f_{0}} \ar[d] & A_{1} \ar[r]^{f_{1}} \ar[d] & ⋯ \ar[r]^{f_{n−1}} & A_{n} \ar[d] \\ D_{0} \ar[r] & D_{1} \ar[r] & ⋯ \ar[r] & D_{n}}

such that D_{i} is a big CM A_{i}-algebra for every i = 0, …, n.

Proof. Suppose first that the residue field k_{i} of A_{i} is of characteristic p > 0, for i = 0, …, n. Then, we pick a commutative diagram as in proposition 17.5.79. Especially, h_{0}, …, h_{n−1} are injective and local maps, so if Frac A_{0} has characteristic zero, the same holds for Frac C_{i}, for every i = 1, …, n, in which case the discussion of (17.5.73) yields a commutative diagram (17.5.89) such that D′_{0} is a big CM A_{0}-algebra, and D′_{i} is a big CM C_{i}-algebra for every i = 1, …, n. Let 𝔪_{0}, 𝔫_{1}, …, 𝔫_{n} be the maximal ideals of A_{0}, C_{1}, …, C_{n}, and endow D′_{0} (resp. D′_{i} for i = 1, …, n) with its 𝔪_{0}-adic (resp. 𝔫_{i}-adic) topology; by claim 17.5.21(i), the completion D_{0} of D′_{0} (resp. D_{i}^{′∧} of D′_{i}, for i = 1, …, n) is still a CM D_{0}-algebra (resp. a CM C_{i}-algebra). Moreover, since f_{0}, …, f_{n−1} are local maps, the maps D′_{0} → D′_{1} → ⋯ → D′_{n} are continuous, so they induce a sequence of ring homomorphisms D_{0} → D_{1}^{′∧} → ⋯ → D_{n}^{′∧}. After setting D_{i} := B_{i} ⊗_{C_{i}} D_{i}^{′∧} for i = 1, …, n, we get a commutative diagram as sought, such that D_{i} is a big CM B_{i}-algebra for every i = 1, …, n, by claim 17.5.21(ii). Next, by restriction of scalars along the finite injective map j_{i}, the ring D_{i} is a big CM A_{i}-algebra for i = 1, …, n (remark 17.5.19(iii)), whence the theorem, in this case.

Next, if A_{0} is an 𝔽_{p}-algebra, the same holds for C_{1}, …, C_{n}, and then we may argue likewise, using the functor E_{p} from remark 17.5.36(ii) (recall that all complete noetherian local rings are excellent : see lemma 9.7.4(v)).

Lastly, we consider the case where k_{0}, …, k_{n} are of zero characteristic, in which case, notice that H_{1}𝕃_{k_{i}/ℚ} = 0 for i = 0, …, n, by proposition 9.6.1(ii). Then, invoking theorem 9.7.41, we obtain a commutative diagram of rings :

\xymatrix{A′_{0} \ar[d]^{i_{0}} \ar[r]^{f′_{0}} & ⋯ \ar[r]^{f′_{n−1}} & A′_{n} \ar[d]^{i_{n}} \\ A_{0} \ar[r]^{f_{0}} & ⋯ \ar[r]^{f_{n−1}} & A_{n}}

such that A′_{0}, …, A′_{n} are local ℚ-algebras of essentially finite type, the maps f′_{0}, …, f′_{n−1} are local, and the maps i_{0}, …, i_{n} are local and of complete intersection. We show :

Claim 17.5.91. There exists a commutative diagram of rings :

\xymatrix{A′_{0} \ar[r]^{f′_{0}} \ar[d] & A′_{1} \ar[r]^{f′_{1}} \ar[d] & ⋯ \ar[r]^{f′_{n−1}} & A′_{n} \ar[d] \\ D′_{0} \ar[r] & D′_{1} \ar[r] & ⋯ \ar[r] & D′_{n}}

such that D′_{i} is a big CM A′_{i}-algebra for every i = 0, …, n.

Proof of the claim. We may find a further commutative diagram of rings :

\xymatrix{A′′_{0} \ar[d]^{i′_{0}} \ar[r]^{f′′_{0}} & ⋯ \ar[r]^{f′′_{n−1}} & A′′_{n} \ar[d]^{i′_{n}} \\ A′_{0} \ar[r]^{f′_{0}} & ⋯ \ar[r]^{f′_{n−1}} & A′_{n}}

such that A′′_{j} is a ℚ-algebra of finite type for j = 0, …, n, and i′_{j} is a localization map, so that A′_{j} = (A′′_{j})_{𝔮_{j}} for some prime ideal 𝔮_{j} ⊂ A′′_{j}. Then we must have 𝔮_{j} = f_{j}^{′′−1}(𝔮_{j+1}) for every j = 0, …, n − 1, since the maps f′_{0}, …, f′_{n} are local. Notice that A′′_{0} is excellent ([64, Ch.IV, Sch.7.8.3(ii)]); then, by corollary 17.5.88we have a commutative diagram

\xymatrix{A′′_{0} \ar[r]^{f′′_{0}} \ar[d] & A′′_{1} \ar[r]^{f′′_{1}} \ar[d] & ⋯ \ar[r]^{f′′_{n−1}} & A′′_{n} \ar[d] \\ D′′_{0} \ar[r] & D′′_{1} \ar[r] & ⋯ \ar[r] & D′′_{n}}

such that D′′_{i} is a big locally CM A′′_{i}-algebra for every i = 0, …, n. Therefore, D′_{i} := (D′′_{i})_{𝔮_{i}} is a big CM A′_{i}-algebra for i = 0, …, n, and the map D′′_{i} → D′′_{i+1} induces a map D′_{i} → D′_{i+1} for every i = 0, …, n − 1. ♢

Let then D′_{0}, …, D′_{n} be as in claim 17.5.91; according to lemma 17.5.20, the ring D_{i} := A_{i} ˆ⊗_{A′_{i}}D′_{i} is a big CM A_{i}-algebra for i = 0, …, n, and since each map f_{i} is local, we may argue as in the foregoing, to obtain the sought sequence of maps D_{0} → ⋯ → D_{n}. □

Corollary 17.5.92. Every local noetherian ring (A, 𝔪) admits a big CM algebra.

Proof. If the residue field k of A has characteristic p > 0, the assertion just summarizes theorems 17.5.28 and 17.5.54. If k has characteristic zero, we may argue as in the proof of claim 17.5.29, to reduce to the case where A is a complete local domain, and the assertion is then the special case of theorem 17.5.90where n = 0. □

Remark 17.5.93. The existence of big CM algebras over rings of equicharacteristic zero was first established by Hochster and Huneke in [95, Th.8.1]; moreover, a later paper by the same authors proves for such algebras a weak functoriality property which is stronger than our theorem 17.5.90 ([96, Th.3.9]). A similar result was proved by Aschenbrenner and Schoutens in [11, §7], using a technique that has been further developed in [57]. Theorem 17.5.90 for local rings of residue characteristic p > 0 has also been proved by Y.André via a different method which yields weakly functorial perfectoid big CM algebras ([7]); the recent preprint [26] gives another proof of this using p-adic completions of absolute integral closures.

17.5.94. Further refinements. We conclude by explaining a variant of our constructions, which has some further useful properties, and especially, it generalizes to mixed characteristic [56, Th.7.8]. Consider a complete local noetherian ring (A, 𝔫) of dimension d, with residue field k := A/𝔫 of characteristic p > 0. Suppose moreover that B is a big CM A-algebra, perfectoid for its 𝔫-adic topology; let β_{•} := (β_{n} | n ∈ ℕ) be any element of E(B) \ {0}, and denote by 𝔪_{β} ⊂ B the ideal generated by the system (β_{n} | n ∈ ℕ). Then (B, 𝔪_{β}) is a basic setup in the sense of [75, §2.1.1], and we denote as usual by S_{β} ⊂ B^{ℕ} the multiplicative subset associated with (B, 𝔪_{β}) as in (17.5.2).

Lemma 17.5.95. In the situation of (17.5.94), endow S_{β}^{−1}B^{⋄} with its 𝔫-adic topology. Then the maximal separated quotient (S_{β}^{−1}B^{⋄})^{sep} of S_{β}^{−1}B^{⋄} is a perfectoid big CM A-algebra.

Proof. Recall that C := (S_{β}^{−1}B^{⋄})^{sep} is the 𝔫-adic completion of S_{β}^{−1}B^{⋄} (proposition 17.5.8). In order to show that C is a big CM A-algebra, it then suffices to check that the same holds for S_{β}^{−1}B^{⋄} (lemma 17.5.20), and since B is a big CM A-algebra, lemma 17.5.23 reduces to verify that (RΓ_{{𝔫}}^{d}B)^{a} ≠ 0, for the almost structure relative to the basic setup (B, 𝔪_{β}). Arguing as in remark 17.5.19(i), we are then further reduced to checking that (B/𝔫B)^{a} ≠ 0, i.e. that 𝔪_{β}B ⊄ 𝔫B. If the latter fails, we have β_{k} ∈ 𝔫B for every k ∈ ℕ, whence β_{0}^{k} ∈ 𝔫^{k} for every such k; since the 𝔫-adic topology is separated on B, we then get β_{0} = 0, so β_{•} = 0 in E(B), a contradiction. Next, since B is a P-ring for its 𝔫-adic topology, it is easily seen that the same holds for B^{ℕ}, and then also for its quotient B^{⋄}, and for C. Now, by remark 16.3.58 we may find an ideal I ⊂ B of adic definition, fulfilling the conditions of (16.3.55), and for any given finite sequence b_{1}, …, b_{k} of generators of I, we let J be the ideal generated by b_{1}^{p}, …, b_{k}^{p}; let moreover gr_{I}^{•}B and gr_{J}^{•}B be the graded rings associated with the I-adic and respectively the J-adic filtrations of B, and define likewise gr_{I}^{•}C and gr_{J}^{•}C. According to proposition 16.3.59, the p-th power map on B (resp. on C) induces an isomorphism Φ_{B,I} : gr_{I}^{•}B →^{∼} gr_{J}^{•}B (resp. a ring homomorphism Φ_{C,I} : gr_{I}^{•}C → gr_{J}^{•}C); on the other hand, by lemma 17.5.1 we get natural identifications :

gr_{I}^{•}C →^{∼} gr_{I}^{•}(S_{β}^{−1}B^{⋄}) →^{∼} S_{β}^{−1}(gr_{I}^{•}B)^{⋄} gr_{J}^{•}C →^{∼} gr_{J}^{•}(S_{β}^{−1}B^{⋄}) →^{∼} S_{β}^{−1}(gr_{J}^{•}B)^{⋄}

and Φ_{C,I} corresponds to S_{β}^{−1}Φ_{I,B}^{⋄}, under these identifications. Hence, Φ_{C,I} is an isomorphism, so C is perfectoid, by theorem 16.3.64. □

Theorem 17.5.96. For every complete local noetherian ring (A, 𝔫) with residue field A/𝔫 of characteristic p > 0, and every perfectoid big CM A-algebra B_{0} there exists a ring homomorphism B_{0} → B such that the following holds :

(i) B is a big CM A-algebra, and is perfectoid for its 𝔫-adic topology.

(ii) B is a local domain, whose maximal ideal is the radical of 𝔫B.

(iii) Set E := E(B). Then, for every non-constant monic polynomial P ∈ W (E)[X] and every distinguished ideal ℐ ⊂ W (E) there exists \underline{w} ∈ W (E) such that P (\underline{w}) ∈ ℐ.

(iv) Especially, B is normal, and Frac(B) is an algebraically closed field.

(v) For every β_{•} ∈ E \ {0}, define the basic setup (B, 𝔪_{β}) as in (17.5.94). Then the unit of adjunction η_{B} : B → (B, 𝔪_{β})_{∗}^{a} is an isomorphism.

Proof. By proposition 16.3.8(ii), the 𝔫-adic completion of B_{0} is perfectoid for its 𝔫-adic topology, and is still a big CM A-algebra (lemma 17.5.20); hence we may assume from start that B_{0} is perfectoid for its 𝔫-adic topology. Set E_{0} := E(B_{0}). By proposition 16.9.29, we have an adic, adically faithfully flat, and perfectoid E_{0}-algebra E_{1} such that for every non-constant monic polynomial P ∈ W (E_{1})[X] and every distinguished ideal ℐ ⊂ W (E_{1}) there exists \underline{w} ∈ W (E_{1}) with P (\underline{w}) ∈ ℐ. Set B_{1} := B_{0} ⊗_{W (E_{0})} W (E_{1}); the induced map B_{0} → B_{1} is then adically faithfully flat (proposition 16.4.18), so B_{1}/𝔫B_{1} is a faithfully flat B_{0}/𝔫B_{0}-algebra, and especially B_{1}/𝔫B_{1} ≠ 0. Moreover depth_{A}B_{1} ≥ depth_{A}B_{0} (corollary 14.2.50(ii) and proposition 16.4.10(ii)). By remark 17.5.19(i), this shows that B_{1} is a big CM A-algebra. Notice that, due to the foregoing property of E_{1}, the map \overline{u}_{B_{1}} : E_{1} → B_{1} is surjective. Choose any minimal prime ideal 𝔭 ⊂ B_{1}, and endow S := \overline{u}_{B_{1}}^{−1}(B_{1} \ 𝔭) with the preordering such that β_{•} ≤ β′_{•} if and only if β′_{•}E_{1} ⊂ β_{•}E_{1}, for every β_{•}, β′_{•} ∈ S. Then clearly (S, ≤) is a filtered category (see example 1.1.6(iii)), and for every β_{•} := (β_{n} | n ∈ ℕ), define the basic setup (B_{1}, 𝔪_{β}) and the multiplicative subset S_{β} ⊂ B_{1}^{ℕ} as in (17.5.94), and the perfectoid big CM A-algebra (S_{β}^{−1}B_{1}^{⋄})^{sep} as in lemma 17.5.95. Let B_{2} be the colimit of the filtered system of B_{1}-algebras ((S_{β}^{−1}B_{1}^{⋄})^{sep} | β_{•} ∈ S). Then the 𝔫-adic completion B_{2}^{∧} of B_{2} is again a perfectoid big CM A-algebra, by remark 17.5.19(iv), lemma 17.5.20and corollary 16.3.46.

Claim 17.5.97. (i) The kernel of the induced map j : B_{1} → B_{2} contains 𝔭.

(ii) For every β ∈ S we have a commutative diagram of B_{1}-modules :

\xymatrix{B_{1} \ar[r]^{j} \ar[d]_{η_{B_{1}}} & B_{2} \ar[d]^{η_{B_{2}}} \\ (B_{1}, 𝔪_{β})_{∗}^{a} \ar[r]^{j_{∗}^{a}} \ar[ur] & (B_{2}, 𝔪_{β})_{∗}^{a}}

whose vertical arrows are the units of adjunctions.

Proof of the claim. (i): Recall that B_{1} is reduced (corollary 16.3.63(i)), hence B_{1,𝔭} is a field, and in particular, the kernel of the localization map B_{1} → B_{1,𝔭} equals 𝔭. The latter means that for every b ∈ 𝔭 there exists β_{0} ∈ B_{1} \ 𝔭 such that bβ_{0} = 0 in B_{1}. We have already observed that \overline{u}_{B_{1}} is surjective; then pick β_{•} ∈ S such that \overline{u}_{B_{1}}(β_{•}) = β_{0}. We regard β_{•} := (β_{n} | n ∈ ℕ) as an element of B_{1}^{ℕ}, in which case β_{•} ∈ S_{β}; but notice that β_{•} · j(b) is the class of (β_{n}b | n ∈ ℕ), and we have β_{n}b = 0 for every n ∈ ℕ, since (β_{n}b)^{p^{n}} = β_{0}g^{p^{n}} = 0, and since B_{1} is reduced. Thus, j(b) = 0, as stated.

(ii): According to lemma 17.5.4(iii), the unit of adjunction S_{β}^{−1}B_{1}^{⋄} → (S_{β}^{−1}B_{1}^{⋄})_{∗}^{a} is an isomorphism (relative to the basic setup (B_{1}, 𝔪_{β})), hence the structure map B_{1} → S_{β}^{−1}B_{1}^{⋄} factors through η_{B_{1}}, and then the same holds for j. ♢

Next, let \overline{𝔮} be any minimal prime ideal of B_{2}^{∧}/𝔫B_{2}^{∧}, and 𝔮 ⊂ B_{2}^{∧} the preimage of \overline{𝔮}. Let B_{3} be the 𝔫-adic completion of the localization B_{2,𝔮}^{∧}. Clearly B_{3} is a P-ring, for its 𝔫-adic topology. By remark 16.3.58 we may find an ideal I ⊂ B_{2}^{∧} of adic definition, fulfilling the conditions of (16.3.55), and for any given finite sequence b_{1}, …, b_{k} of generators of I, we let J be the ideal generated by b_{1}^{p}, …, b_{k}^{p}. Let moreover gr_{I}^{•}B_{2}^{∧} and gr_{J}^{•}B_{2}^{∧} be the graded rings associated with the I-adic and respectively the J-adic filtrations of B, and define likewise gr_{I}^{•}B_{3} and gr_{J}^{•}B_{3}. According to proposition 16.3.59, the p-th power map on B induces an isomorphism Φ_{I} : gr_{I}^{•}B_{2}^{∧} →^{∼} gr_{J}^{•}B_{2}^{∧} and a ring homomorphism Φ_{IB_{3}} : gr_{I}^{•}B_{3} → gr_{J}^{•}B_{3}. Clearly gr_{I}^{j}B_{3} = (gr_{I}^{j}B_{2}^{∧})_{𝔮} and gr_{J}^{j}B_{3} = (gr_{J}^{j}B_{2}^{∧})_{𝔮}, for every j ∈ ℕ, and Φ_{IB_{3}} corresponds to the localization (Φ_{I})_{𝔮} under these identifications. Hence Φ_{IB_{3}} is an isomorphism, so B_{3} is perfectoid, by theorem 16.3.64. Let us check that B_{3} is a big CM A-algebra. By lemma 17.5.20, it suffices to show that the same holds for B_{2,𝔮}^{∧}. But clearly depth_{A}B_{2,𝔮}^{∧} ≥ depth_{A}B_{2}^{∧}, and moreover B_{2,𝔮}^{∧}/𝔫B_{2,𝔮}^{∧} ≠ 0, since B_{2}^{∧}/𝔫B_{2}^{∧} ≠ 0, and since the preimage of 𝔮 in A equals 𝔫. Then the assertion follows from remark 17.5.19(i). Summing up, to every big CM A-algebra B_{0} that is perfectoid for its 𝔫-adic topology, we have attached three more such algebras; we introduce the notation :

ℬ(B_{0})_{1} := B_{1} ℬ(B_{0})_{2} := B_{2}^{∧} ℬ(B_{0}) := B_{3}.

Let also 𝔭(B_{0}) ⊂ ℬ_{0}(B_{0})_{1} be the minimal ideal 𝔭. Then we define by transfinite induction, for every ordinal α an A-algebra ℬ_{α}(B_{0}), as follows. First, we set ℬ_{0} := B_{0}; next, ℬ_{α+1} := ℬ(ℬ_{α}) for every successor ordinal α + 1, and ℬ_{α} is the 𝔫-adic completion of colim_{β<α} ℬ_{β}, for every limit ordinal α. Taking into account remark 17.5.19(iv) and lemma 17.5.20, we see easily that ℬ_{α} is a perfectoid big CM A-algebra, for every ordinal α. Lastly, let ω_{1} be the smallest uncountable ordinal; we claim that B := ℬ_{ω_{1}} will do. Indeed, arguing as in the proof of proposition 16.9.29 we see that B = colim_{β<α} ℬ_{β}, and that (iii) holds. Next, let x, y ∈ B with xy = 0; we find β < ω_{1} such that x and y are the images of elements x′ and y′ of ℬ_{β}, and we may assume that x′y′ = 0. But then the image of either x′ or y′ in ℬ(ℬ_{β})_{1} lies in 𝔭(ℬ_{β})_{1}, and then the image of either x′ or y′ vanishes in ℬ(ℬ_{β})_{2}, by claim 17.5.97(i), so either x = 0 or y = 0; this shows that B is a domain. Moreover, by construction ℬ(ℬ_{α}) is a local ring, and its maximal ideal is the radical of 𝔫ℬ(ℬ_{α}), for every ordinal α; assertion (ii) follows easily. Assertion (iv) follows easily from (ii) and (iii) : the details shall be left to the reader.

(v): From corollary 16.3.46 and claim 16.9.30, we deduce that E(B) is the colimit of the system (E(ℬ_{α}) | α < ω_{1}). Let now β_{•} ∈ E, so that β_{•} is the image of an element β′_{•} ∈ E(ℬ_{α}) for some ordinal α < ω_{1}, and the ideal 𝔪_{β} ⊂ B equals 𝔪B, where 𝔪 ⊂ ℬ_{α} is the ideal generated by (β′_{n} | n ∈ ℕ); let x ∈ (B, 𝔪_{β})_{∗}^{a}. We notice :

Claim 17.5.98. (B, 𝔪_{β})_{∗}^{a} = colim_{α≤γ<ω_{1}}(ℬ_{γ}, 𝔪ℬ_{γ})_{∗}^{a}.

Proof of the claim. Recall that for any basic setup (V, 𝔪) and any V -module M we have M_{∗}^{a} = Hom_{V} (𝔪̃, M), with 𝔪̃ := 𝔪 ⊗_{V} 𝔪. Moreover, for every ring homomorphism V → W we have 𝔪W ⊗_{W} 𝔪W = W ⊗_{V} 𝔪̃ ([75, Rem.2.1.4(ii)]). Thus, it suffices to check that B_{∗}^{a} = colim_{α≤γ<ω_{1}} ℬ_{γ,∗}^{a} for the almost structure given by the basic setup (ℬ_{α}, 𝔪). However, according to [75, Th.2.1.12(ii.a)] we have a ℬ_{α}-linear presentation

ℬ_{α}^{(ℕ)} ⟶^{f} ℬ_{α}^{(ℕ)} → 𝔪̃ → 0.

Let (e_{i} | i ∈ ℕ) be the canonical basis of the free ℬ_{α}-module ℬ_{α}^{(ℕ)}, so that for every i ∈ ℕ we have a finite subset S_{i} ⊂ ℕ and a system (a_{ij} | j ∈ S_{i}) of elements of ℬ_{α} such that f(e_{i}) = ∑_{j∈S_{i}} a_{ij}e_{j}. With this notation, the datum of a ℬ_{α}-linear map φ : 𝔪̃ → B is the same as that of a system b_{•} := (b_{i} | i ∈ ℕ) of elements of B with ∑_{j∈S_{i}} a_{ij}b_{j} = 0 for every i ∈ ℕ. Since B = colim_{γ<ω_{1}} ℬ_{γ}, for every i ∈ ℕ the element b_{i} is the class [c_{i}, γ_{i}] of a pair consisting of an ordinal γ_{i} < ω_{1} and some c_{i} ∈ ℬ_{γ_{i}}; pick any ordinal γ < ω_{1} such that γ ≥ γ_{i} for every i ∈ ℕ, and γ ≥ α. After replacing each c_{i} by its image in ℬ_{γ}, we may assume that γ_{i} = γ for every i ∈ ℕ. Likewise, for every i ∈ ℕ there exists an ordinal δ_{i} < ω_{1} with δ_{i} ≥ γ, and such that the image of ∑_{j∈S_{i}} a_{ij}c_{j} vanishes in ℬ_{δ_{i}}. We pick again δ < ω_{1} such that δ ≥ δ_{i} for every i ∈ ℕ, and let d_{i} ∈ ℬ_{δ} be the image of c_{i} in ℬ_{δ}, for every i ∈ ℕ; then the system (d_{i} | i ∈ ℕ) corresponds to a ℬ_{α}-linear map ψ : 𝔪̃ → ℬ_{δ,∗}^{a} whose composition with the natural map ℬ_{δ,∗}^{a} → B_{∗}^{a} agrees with φ. This shows that the natural map t : colim_{α≤γ<ω_{1}} ℬ_{γ,∗}^{a} → B_{∗}^{a} is surjective. Lastly, notice that φ is the zero map if and only if b_{i} = 0 for every i ∈ ℕ; if b_{i} = [c_{i}, γ_{i}] as in the foregoing, we may then find δ < ω_{1} with δ ≥ γ_{i} and δ ≥ α, and such that the image of c_{i} vanishes in ℬ_{δ} for every i ∈ ℕ. This shows that t is injective, and concludes the proof. ♢

From claim 17.5.98, the almost element x is the image of some y ∈ (ℬ_{γ}, 𝔪ℬ_{γ})_{∗}^{a}, for some γ < ω_{1} with γ ≥ α, in which case the image of y in (ℬ_{γ+1}, 𝔪ℬ_{γ+1})_{∗}^{a} is also the image of some z ∈ ℬ(ℬ_{γ})_{2}, by claim 17.5.97(ii); if z′ ∈ B is the image of z, we see that η_{B}(z′) = x. This shows that η_{B} is surjective, and we also know that its kernel is annihilated by 𝔪_{β}; since B is a domain, it then follows that η_{B} is an isomorphism. □

Remark 17.5.99. (i) With the notation of theorem 17.5.96, let β_{•} := (β_{n} | n ∈ ℕ) ∈ E(B) be any non-zero element; notice that every principal ideal of B is a free B-module, since B is a domain (theorem 17.5.96(iv)), hence 𝔪_{β} is a flat B-module, since it is the filtered union of principal ideals, and therefore 𝔪_{β} = 𝔪_{β} ⊗_{B} 𝔪_{β} ([75, Rem.2.14(i)]). It follows that

(17.5.100)  (B, 𝔪_{β})_{∗}^{a} = Hom_{B}(𝔪_{β}, B) = ⋂_{n∈ℕ}{x ∈ Frac(B) | β_{n}x ∈ B}.

(ii) Recall that an integral domain A is said to be completely integrally closed if the following holds for every x ∈ K := Frac(A). The subring A[x] ⊂ K lies in a finitely generated Asubmodule of K if and only if x ∈ A. Then, we claim that condition (v) of theorem 17.5.96 is equivalent to the assertion that B is completely integrally closed. Indeed, let β_{•} be as in (i), and x ∈ (B, 𝔪_{β})_{∗}^{a}; in light of (17.5.100), we see that B[x] ⊂ β_{0}^{−1}B. Hence, if B is completely integrally closed, we have B = (B, 𝔪_{β})_{∗}^{a}. Conversely, suppose the latter condition holds, and let x ∈ Frac(B) be an element such that B[x] lies in a B-module of finite type; this means that B[x] ⊂ b^{−1}B for some b ∈ B \ {0}, so that bx^{p^{n}} ∈ B for every n ∈ ℕ. But by virtue of theorem 17.5.96(iii), there exists β_{•} ∈ E(B) with β_{0} = b; then, with (17.5.100), we deduce that x ∈ (B, 𝔪_{β})_{∗}^{a}, so x ∈ B, by assumption.

17.6. Monomial conjectures. The existence of big CM modules and algebras can be applied to settle some other long-standing conjectures concerning systems of parameters in local noetherian rings. In this section we prove some of these conjectures. The first result is the following:

Theorem 17.6.1 (Monomial conjecture). Let (A, 𝔪) be a local noetherian ring, and x_{1}, …, x_{d} a system of parameters for A. Then we have :

x_{1}^{t} ⋯ x_{d}^{t} ∉ Ax_{1}^{t+1} + ⋯ + Ax_{d}^{t+1} for every t ∈ ℕ.

Proof. We reproduce the argument from [92, Prop.3.5]. Let R := ℤ[T_{1}, …, T_{d}], and for every t ∈ ℕ, denote by I_{t} ⊂ R the ideal generated by T_{1}^{t}, …, T_{d}^{t}. We have a short exact sequence of R-modules :

0 → R/I_{1} ⟶^{φ_{t}} R/I_{t+1} → Q := R/(I_{t+1} + Rp_{t}) → 0

where φ_{t} is induced by the endomorphism R → R such that q ↦ qT_{1}^{t} ⋯ T_{d}^{t} for every q ∈ R. Let also f : R → A be the ring homomorphism such that f(T_{i}) = x_{i} for i = 1, …, d. Then, every A-module becomes an R-module, after restriction of scalars along f. We notice :

Claim 17.6.2. For every big CM A-module M we have Tor_{1}^{R}(Q, M) = 0.

Proof of the claim. Notice the natural ring isomorphism R/I_{1} →^{∼} ℤ, and let gr_{•}Q be the graded ℤ-module associated with the I_{1}-adic filtration of Q; we easily see that I^{k}Q = 0 for every sufficiently large k ∈ ℕ; then the long Tor exact sequences induced by the short exact sequences of R-modules :

(17.6.3)  0 → I_{1}^{k+1}Q → I^{k}Q → gr_{k}Q → 0  for every k ∈ ℕ

easily reduce to checking that Tor_{i}^{R}(gr_{k}Q, M) = 0 for every k ∈ ℕ and every i > 0. Moreover, it is easily seen that gr_{k}Q is a free ℤ-module, for every k ∈ ℕ (details left to the reader), hence we are further reduced to checking that Tor_{1}^{R}(R/I_{1}, Q) = 0 for every i > 0. The latter follows from remark 17.5.19(i) and lemma 7.8.14. ♢

By corollary 17.5.92, we may find a big CM A-module M. Then it suffices to check :

Claim 17.6.4. For every big CM A-module M, the map φ_{t} induces an injective map

φ_{t} ⊗_{A} M : M/I_{1}M → M/I_{t+1}M  \overline{m} ↦ \overline{x_{1}^{t} ⋯ x_{d}^{t}m}  for every t ∈ ℕ.

Proof of the claim. It follows immediately by combining claim 17.6.2 with the long Tor exact sequence attached to (17.6.3). □

Remark 17.6.5. (i) In the situation of theorem 17.6.1, set X := Spec A and U_{i} := Spec A[x_{i}^{−1}] for i = 1, …, d. Then, U_{•} := (U_{i} | i = 1, …, d) is an affine covering of X \ {𝔪}, and proposition 10.4.18yields a natural isomorphism

RΓ_{{𝔪}}𝒪_{X} →^{∼} C_{alt}^{•}(U_{•}, 𝒪_{X})[−1] in D(A-Mod)

where C_{alt}^{•}(U_{•}, 𝒪_{X}) denotes the alternating augmented Čech complex of the structure sheaf 𝒪_{X}, relative to the covering U_{•}. Set as well y := x_{1} ⋯ x_{d}, and y_{i} := x_{1} ⋯ x_{i−1}x_{i+1} ⋯ x_{d} for i = 1, …, d; there follows an A-linear isomorphism

H_{{𝔪}}^{d}(A) →^{∼} Coker(∂ : ⨁_{i=1,...,d} A[y_{i}^{−1}] → A[y^{−1}])

where ∂ is the sum of the localization maps A[y_{i}^{−1}] → A[y^{−1}]. Especially, the class of y^{−1} in Coker ∂ corresponds to an element β ∈ H_{{𝔪}}^{d}(A), and a simple inspection shows that theorem 17.6.1is equivalent to the assertion that β ≠ 0 : the details shall be left to the reader.

(ii) Moreover, it was already known, by some early work of Hochster, that the monomial conjecture is equivalent to the direct summand conjecture (for finite extensions of regular local rings) : see [93, Th.6.1].

Theorem 17.6.6 (Strong monomial conjecture). Let (A, 𝔪) be a noetherian local ring, 𝔭 ⊂ A a prime ideal, and x_{1}, …, x_{d} a system of parameters for A, such that :

(a) 𝔭 has height one, and dim A/𝔭 = d − 1.

(b) x_{1} ∈ 𝔭.

Then we have :  x_{1}^{t+1}x_{2}^{t} ⋯ x_{d}^{t} ∉ 𝔭 · (Ax_{1}^{t+1} + ⋯ + Ax_{d}^{t+1})  for every t ∈ ℕ.

Proof. We first notice :

Claim 17.6.7. We may assume that (A, 𝔪) is a complete local domain.

Proof of the claim. Let A^{∧} and (A/𝔭)^{∧} be the 𝔪-adic completions of A and A/𝔭; recall that the 𝔪-adic topology of A induces the 𝔪-adic topology on 𝔭A ([126, Th.8.6]), and 𝔭A^{∧} is a closed ideal for the 𝔪-adic topology of A^{∧} ([126, Th.8.10(i)]), hence the natural map A^{∧}/𝔭A^{∧} → (A/𝔭)^{∧} is a ring isomorphism (proposition 8.2.13(i,v)). Especially, dim A^{∧}/𝔭A^{∧} = d − 1, i.e., there exists a maximal point 𝔭′ of the closed subset Spec A^{∧}/𝔭A^{∧} ⊂ Spec A^{∧}, such that dim A^{∧}/𝔮 = d − 1. Moreover, by the going-down theorem for flat extensions, for every such 𝔭′ we have 𝔭 = 𝔭′ ∩ A ([126, Th.9.5]), and furthermore, 𝔭′ has height one ([126, Th.15.1(ii)]). Pick any minimal prime ideal 𝔮 ⊂ A^{∧} contained in 𝔭′. By the same token, 𝔮 is a closed ideal for the 𝔪-adic topology of A^{∧}, and therefore A′ := A^{∧}/𝔮 is still a complete local ring; since dim A^{∧}/𝔮 = d − 1, we must have dim A′ = d, and 𝔭′A′ has height one. Clearly it suffices to prove the theorem for the ring A′, its prime ideal 𝔭′A′, and the image in A′ of the sequence x_{1}, …, x_{d}, which is still a system of parameters for A′. ♢

Henceforth, we assume that (A, 𝔪) is a complete local domain, so the same holds for A/𝔭, and we may find a commutative diagram of rings :

\xymatrix{A \ar[r] \ar[d]_{f} & A/𝔭 \ar[d]^{g} \\ B_{0} \ar[r] & B_{1}}

such that B_{0} is a big CM A-algebra, and B_{1} is a big CM A/𝔭-algebra (theorem 17.5.90). For a given t ∈ ℕ, set y := x_{1}^{t+1}x_{2}^{t} ⋯ x_{d}^{t}, and I := Ax_{1}^{t+1} + ⋯ + Ax_{d}^{t+1}. Clearly it suffices to check :

Claim 17.6.8. f(y) ∉ (I𝔭 + I^{2})B_{0}.

Proof of the claim. The sequence x_{1}, …, x_{d} is completely secant in B_{0} (remark 17.5.19(i)), so the same holds for the sequence x_{1}^{t+1}, …, x_{d}^{t+1} (corollary 7.8.8(ii)), and then the latter is also B_{0}-quasi-regular (proposition 7.8.15). The latter means that the A-linear map

ω : ⨁_{i=1}^{d}(B_{0}/IB_{0}) → M := IB_{0}/I^{2}B_{0} (\overline{b}_{1}, …, \overline{b}_{d}) ↦ \overline{x_{1}^{t+1}b_{1} + ⋯ + x_{d}^{t+1}b_{d}}

is an isomorphism. We have f(y) ∈ IB_{0}, and clearly ω(\overline{f(x_{2}^{t} ⋯ x_{d}^{t})}, 0, …, 0) is the class \overline{f(y)} of f(y) in M. Now, if the claim fails, we have \overline{f(y)} ∈ 𝔭M, and therefore

(17.6.9) (\overline{f(x_{2}^{t} ⋯ x_{d}^{t})}, 0, …, 0) ∈ ⨁_{i=1}^{d} 𝔭(B_{0}/IB_{0})  i.e. f(x_{2}^{t} ⋯ x_{d}^{t}) ∈ (𝔭 + I)B_{0}.

Now, notice that the image \overline{x}_{2}, …, \overline{x}_{d} of the sequence x_{2}, …, x_{d} is a system of parameters for A/𝔭, since dim A/𝔭 = d − 1. However, from (17.6.9) we get

g(\overline{x}_{2}^{t} ⋯ \overline{x}_{d}^{t}) ∈ IB_{1}

which contradicts claim 17.6.4, since B_{1} is a big CM A/𝔭-algebra. □

Remark 17.6.10. (i) In the situation of theorem 17.6.6, notice that dim A/x_{1}A = d − 1 ([126, Th.14.1]), hence 𝔭 must be a maximal point of the closed subset Spec A/x_{1}A ⊂ Spec A.

(ii) Conversely, if A is equidimensional and catenary, and 𝔭 is a maximal point of the closed subset Spec A/x_{1}A, then conditions (a) and (b) of theorem 17.6.6are clearly fulfilled.

(iii) The hypothesis on dim A/𝔭 in condition (a) of theorem 17.6.6 is incorrectly omitted from the statement of the strong monomial conjecture in the survey article [142], where it is listed as conjecture 19. To see that this hypothesis cannot be omitted, consider a local noetherian domain (A, 𝔪) of dimension d, with a prime ideal 𝔭 of height one, such that dim A/𝔭 ≤ d − 2. Pick any x_{1} ∈ 𝔭 \ {0}; then let Σ ⊂ Spec A be the subset of all prime ideals 𝔮 of A such that x_{1} ∈ 𝔮 and dim A/𝔮 = d − 1. Clearly every element of Σ must have height one, so Σ lies in the finite set of maximal points of Spec A/x_{1}A. Moreover, for every 𝔮 ∈ Σ we have 𝔭 ⊈ 𝔮, since dim A/𝔭 < d−1. Hence, there exists x_{2} ∈ 𝔭\⋃_{𝔮∈Σ} 𝔮 ([12, Prop.1.11(i)]). Let I := Ax_{1} +Ax_{2}; we have dim A/I ≥ d − 2 by [126, Th.13.6(ii)], and on the other hand, by construction there exists no prime ideal 𝔮 ⊂ A such that I ⊂ 𝔮 and dim A/𝔮 = d − 1, so dim A/I = d − 2. Now, pick x_{3}, …, x_{d} ∈ 𝔪 whose images in A/I are a system of parameters for the latter local ring. It follows easily that x_{1}, …, x_{d} is a system of parameters for A, and it is easily seen that x_{1}^{2}x_{2} ⋯ x_{d} ∈ 𝔭 · (Ax_{1}^{2} + ⋯ + Ax_{d}^{2}).

References

[1] L.Alonso Tarrío, A.Jeremías López, J.Lipman, Local homology and cohomology on schemes. Ann. Sci. E.N.S. 30 (1997) pp.1–39.

[2] A.B.Altman, R.T.Hoobler, S.L.Kleiman, A note on the base change map for cohomology. Comp.Math. 27 (1973) pp.25–38.

[3] M.André, Homologie des algèbres commutatives. Springer Grundl. Math. Wiss. 206 (1974).

[4] M.André, Localisation de la lissité formelle. Manuscr. Math. 13 (1974) pp.297–307.

[5] Y.André, La conjecture du facteur direct. Publ. Math. IHES 127 (2018) pp.71–93.

[6] Y.André, Le lemme d’Abhyankar perfectoïde. Publ. Math. IHES 127 (2018) pp.1–70.

[7] Y.André, Weak functoriality of Cohen-Macaulay algebras. J. Amer. Math Soc. 33 (2020) pp.363–380.

[8] M.Artin et al., Théorie des topos et cohomologie étale des schémas (SGA4) – tome 1. Springer Lect. Notes Math. 269 (1972).

[9] M.Artin et al., Théorie des topos et cohomologie étale des schémas (SGA4) – tome 2. Springer Lect. Notes Math. 270 (1972).

[10] M.Artin et al., Théorie des topos et cohomologie étale des schémas (SGA4) – tome 3. Springer Lect. Notes Math. 305 (1973).

[11] M.Aschenbrenner, H.Schoutens, Lefschetz extensions, tight closure and big Cohen-Macaulay algebras. Israel J. Math. 161 (2007) pp.221–310.

[12] M.Atiyah, I.G.MacDonald, Introduction to Commutative Algebra. Addison-Wesley Series in Math. (1969).

[13] L.Avramov, Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology. Ann. of Math. 150 (1999) pp.455–487.

[14] L.L.Avramov, H-B.Foxby, B.Herzog, Structure of local homomorphisms. J. of Algebra 164 (1994) pp.124-145.

[15] J.Bartijn, J.R.Strooker, Modifications monomiales. Springer Lect. Notes Math. 1029 (1983) pp.192– 217.

[16] H.Bass, M.P.Murthy, Grothendieck groups and Picard groups of abelian group rings. Ann. of Math. 86 (1967) pp.16–73.

[17] A.Beilinson, p-adic periods and derived deRham cohomology. J. Amer. Math. Soc. 25 (2012) pp.715–738.

[18] A.Beilinson, J.Bernstein, P.Deligne, O.Gabber, Faisceaux pervers (new edition). Astérisque 100 (2018).

[19] B.M.Bennett, On the characteristic functions of a local ring. Ann. of Math. 91 (1970) pp.25–87.

[20] P.Berthelot et al., Théorie des Intersections et Théorèmes de Riemann-Roch (SGA6). Springer Lect. Notes Math. 225 (1971).

[21] B.Bhatt, On the non-existence of small Cohen-Macaulay algebras. J. Algebra 411 (2014) pp.1–11.

[22] B.Bhatt, Almost direct summands. Nagoya Math. J. 214 (2014) pp.195–204.

[23] B.Bhatt, On the direct summand conjecture and its derived variant. Invent. Math. 212 (2018) pp.297–317.

[24] B.Bhatt, Lecture notes for a class on perfectoid spaces. Unpublished notes, available at :

    http://www-personal.umich.edu/~bhattb/teaching/mat679w17/lectures.pdf

[25] B.Bhatt, P.Scholze, Prisms and prismatic cohomology. Preprint on arXiv:1905.08229 (2019).

[26] B.Bhatt, Cohen-Macaulayness of absolute integral closures. Preprint on arXiv:2008.08070 (2020).

[27] J.-E. Björk, Analytic 𝒟-Modules and Applications. Kluwer Math. and Its Appl. 247 (1993).

[28] F.Borceux, Handbook of Categorical Algebra I. Basic Category Theory. Cambridge Univ. Press Encycl. of Math. and Its Appl. 50 (1994).

[29] F.Borceux, Handbook of Categorical Algebra II. Categories and structures. Cambridge Univ. Press Encycl. of Math. and Its Appl. 51 (1994).

[30] F.Borceux, Handbook of Categorical Algebra III. Categories of Sheaves. Cambridge Univ. Press Encycl. of Math. and Its Appl. 52 (1994).

[31] J.Borger, The basic geometry of Witt vectors – I. Algebra and Number Theory 5 (2011) pp.231–285.

[32] S.Bosch, W.Lütkebohmert, M.Raynaud, Neron Models. Springer Ergebn. der Math. 21 (1990).

[33] S.Bosch, U.Güntzer, R.Remmert, Non-Archimedean analysis. Springer Grundl. Math. Wiss. 261 (1984).

[34] N.Bourbaki, Algèbre Commutative – Chapitres 1 à 9. Hermann (1961).

[35] N.Bourbaki, Algèbre Commutative – Chapitre 10. Masson (1998).

[36] N.Bourbaki, Théorie des Ensembles. Hermann (1963).

[37] N.Bourbaki, Algèbre Homologique. Masson (1980).

[38] N.Bourbaki, Algèbre – Chapitres 1 à 3. Hermann (1970).

[39] N.Bourbaki, Algèbre – Chapitres 4 à 7. Masson (1981).

[40] N.Bourbaki, Espaces Vectoriels Topologiques. Hermann (1966).

[41] N.Bourbaki, Topologie Générale. Hermann (1971).

[42] M.Brodmann, Asymptotic stability of Ass(M/I^{n}M). Proc. Amer. Math. Soc. 74 (1979) pp.16–18.

[43] W.Bruns, J.Herzog, Cohen-Macaulay rings – revised edition. Cambridge Univ. Press (1998).

[44] M.Buckley, Fibred 2-categories and bicategories. J. Pure Appl. Alg.. 218 (2014) pp.1034–1074.

[45] K.Buzzard, A.Verberkmoes, Stably uniform affinoids are sheafy. J. Reine Angew. Math. 740 (2018) pp.25–40.

[46] O.Caramello, Denseness conditions, morphisms and equivalences of toposes. Preprint on

    arXiv:1906.08737 (2019).

[47] G.Chenevier, The p-adic analytic space of pseudocharacters of a profinite group, and pseudorepresentations over arbitrary rings. London Math. Soc. Lecture Notes Series 414 (2014) pp.221–285.

[48] L.G.Chouinard II, Krull semigroups and divisor class groups. Can. J. Math. 33 (1981) pp.1459–1468.

[49] B.Conrad, Deligne’s notes on Nagata compactifications. J. Ramanujan Math. Soc. 22 (2007) pp.205–257; erratum 24 (2009) pp.427–428.

[50] B.Conrad, Grothendieck Duality and Base Change. Springer Lect. Notes Math. 1750 (2000).

[51] H.Dao, A.De Stefani, E.Grifo, C.Huneke, L.Núñez-Betancourt, Symbolic powers of ideals. Springer Proc. Math Stat. 222 (2018) pp.387–432.

[52] C.Davis, K.S.Kedlaya, Almost purity and overconvergent Witt vectors. J.Algebra 422 (2015) pp.373–412.

[53] C.Davis, A.Langer, T.Zink, Overconvergent Witt vectors. J. reine angew. Math. 668 (2012) pp.1–34.

[54] P.Deligne, Théorie de Hodge – II. Publ. Math. IHES 40 (1971) pp.5-57.

[55] M.Demazure, A.Grothendieck et al., Schémas en Groupes (SGA3) – tome 1. Springer Lect. Notes Math. 151 (1970).

[56] G.D.Dietz Big Cohen-Macaulay algebras and seeds. Trans. Amer. Math. Soc. 359 (2007) pp.5959–5989.

[57] G.D.Dietz, R.G.G. Big Cohen-Macaulay and seed algebras in equal characteristic zero via ultraproducts. J. Commut. Algebra 11 (2019) pp.511–533.

[58] J.Dieudonné, A.Grothendieck, Éléments de Géométrie Algébrique – Chapitre I. Springer Grundl. Math. Wiss. 166 (1971).

[59] J.Dieudonné, A.Grothendieck, Éléments de Géométrie Algébrique – Chapitre I. Publ. Math. IHES 4 (1960).

[60] J.Dieudonné, A.Grothendieck, Éléments de Géométrie Algébrique – Chapitre II. Publ. Math. IHES 8 (1961).

[61] J.Dieudonné, A.Grothendieck, Éléments de Géométrie Algébrique – Chapitre III, partie 1. Publ. Math. IHES 11 (1961).

[62] J.Dieudonné, A.Grothendieck, Éléments de Géométrie Algébrique – Chapitre III, partie 2. Publ. Math. IHES 17 (1963).

[63] J.Dieudonné, A.Grothendieck, Éléments de Géométrie Algébrique – Chapitre IV, partie 1. Publ. Math. IHES 20 (1964).

[64] J.Dieudonné, A.Grothendieck, Éléments de Géométrie Algébrique – Chapitre IV, partie 2. Publ. Math. IHES 24 (1965).

[65] J.Dieudonné, A.Grothendieck, Éléments de Géométrie Algébrique – Chapitre IV, partie 3. Publ. Math. IHES 28 (1966).

[66] J.Dieudonné, A.Grothendieck, Éléments de Géométrie Algébrique – Chapitre IV, partie 4. Publ. Math. IHES 32 (1967).

[67] P.Dutton, Prime ideals attached to a module. Quart. J. Math. Oxford 29 (1978) pp.403–413.

[68] N.Epstein, J.Shapiro, Strong Krull primes and flat modules. J. Pure Appl. Alg. 218 (2014) pp.1712–1729.

[69] G.Faltings, Almost étale extensions. Astérisque 279 (2002) pp.185–270.

[70] J.-M. Fontaine, Perfectoïdes, presque pureté et monodromie-poids (d’après Peter Scholze). Séminaire Bourbaki, vol. 2011/2012. Astérisque 352 (2013), pp. 509–534.

[71] K.Fujiwara, O.Gabber, F.Kato, On Hausdorff completions of commutative rings in rigid geometry. J. of Algebra 332 (2011) pp.293–321.

[72] K.Fujiwara, F.Kato, Foundations of Rigid Geometry I. EMS Monographs in Math., European Math. Society (2018).

[73] W.Fulton, Introduction to Toric Varieties. Princeton Univ. Press Ann. of Math. Studies 131 (1993).

[74] O.Gabber, Affine analog of the proper base change theorem. Isr. J. of Math. 87 (1994) pp.325–335.

[75] O.Gabber, L.Ramero, Almost ring theory. Springer Lect. Notes Math. 1800 (2003).

[76] P.Gabriel, Des catégories abéliennes. Bull. Soc. Math. France 90 (1962) pp.323–449.

[77] P.Gabriel, M.Zisman, Calculus of fractions and homotopy theory. Springer Ergebn. der Math. 35 (1967).

[78] J.Giraud, Cohomologie non abélienne. Springer Grundl. Math. Wiss. 179 (1971).

[79] R.Godement, Théorie des faisceaux. Hermann (1958).

[80] P.G.Goerss, J.F.Jardine, Simplicial Homotopy Theory. Birkhäuser Modern Classics (2009).

[81] J.W.Gray, Formal Category Theory : Adjointness for 2-Categories. Springer Lecture Notes Math. 391 (1974).

[82] A.Grothendieck et al., Revêtements Étales et Groupe Fondamental (SGA1). Documents Math. Soc. Math. France 3 (2003).

[83] A.Grothendieck, Local cohomology. Springer Lect. Notes Math. 41 (1967).

[84] A.Grothendieck, Sur quelques points d’algébre homologique. Tohoku Math.J. 9 (1957) pp.119-221.

[85] A.Grothendieck, Cohomologie locale des faisceaux cohérents et théorémes de Lefschetz locaux et globaux (SGA2). Documents Math. Soc. Math. France 4 (2005).

[86] L.Gruson, M.Raynaud, Critères de platitude et de projectivité. Invent. Math. 13 (1971) pp.1–89.

[87] R.Hartshorne, Residues and duality. Springer Lect. Notes Math. 20 (1966).

[88] R.Hartshorne, Affine duality and cofiniteness. Invent. Math. 9 (1969) pp.145–164.

[89] R.Hartshorne, Generalized divisors on Gorenstein schemes. K-Theory 8 (1994) pp.287–339.

[90] M.Hochster, Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. 142 (1969) pp.43–60.

[91] M.Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes. Ann. of Math. 96 (1972) pp.318–337.

[92] M.Hochster, Topics in the homological theory of modules over commutative rings. CBMS Regional Conference Series in Math. 24 (1975).

[93] M.Hochster, Canonical elements in local cohomology modules, and the direct summand conjectures. J. of Algebra 84 (1983) pp.503–553.

[94] M.Hochster, Big Cohen-Macaulay algebras in dimension three via Heitmann’s theorem. J. of Algebra 254 (2002) pp.395–408.

[95] M.Hochster, C.Huneke, Infinite integral extensions and big Cohen-Macaulay algebras. Ann. of Math. 135 (1992) pp.53–89.

[96] M.Hochster, C.Huneke, Applications of the existence of big Cohen-Macaulay algebras. Adv. Math. 113 (1995) pp.45–117.

[97] G.Horrocks, Vector bundles on the punctured spectrum of a local ring. Proc. London Math. Soc. 14 (1964) pp.689–713.

[98] R.Huber, Bewertungsspektrum und rigide Geometrie. Regensburger Math. Schriften 23 (1993).

[99] R.Huber, Continuous valuations. Math. Zeitschr. 212 (1993) pp.455–477.

[100] R.Huber, A generalization of formal schemes and rigid analytic varieties. Math. Zeithschr. 217 (1994) pp.513–551.

[101] R.Huber, Étale Cohomology of Rigid Analytic Varieties and Adic Spaces. Aspects of Math. 30 (1996).

[102] C. Huneke, I. Swanson, Integral Closure of Ideals, Rings, and Modules. London Math. Soc. Lect. Notes 336 (2006).

[103] L.Illusie, Complexe cotangent et déformations I. Springer Lect. Notes Math. 239 (1971).

[104] L.Illusie, Complexe de de Rham-Witt et cohomologie crystalline. Ann. Sci. E.N.S. 4^{e} série 12 (1979) pp.501–661.

[105] L.Illusie, Y.Laszlo, F.Orgogozo, Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents. Astérisque 363–364 (2014).

[106] C.U.Jensen, Les Foncteurs Dérivés de lim_{⟵} et leurs applications en Théorie des Modules. Springer Lect. Notes Math. 254 (1972).

[107] A.J.de Jong et al., The Stacks Project. Available at : https://stacks.math.columbia.edu/

[108] A.Joyal, M.Tierney, Notes on simplicial homotopy theory. Quaderns of the CRM Barcelona 47 (2008).

[109] W. van der Kallen, Descent for the K-Theory of Polynomial Rings. Math. Zeitschr. 191 (1986) pp.405– 416.

[110] M.Kashiwara, P.Schapira, Categories and sheaves. Springer Grundl. Math. Wiss. 332 (2006).

[111] K.Kato, Logarithmic structures of Fontaine-Illusie. Algebraic analysis, geometry, and number theory – Johns Hopkins Univ. Press (1989) pp.191–224.

[112] K.Kato, Toric singularities. Amer. J. of Math. 116 (1994) pp.1073-1099.

[113] K.S.Kedlaya, Nonarchiemdean geometry of Witt vectors. Nagoya Math. J. 209 (2013) pp.111-165

[114] K.S.Kedlaya, R.Liu, Relative p-adic Hodge theory : foundations. Astérisque 371 (2015).

[115] J.L.Kelley, General Topology. Springer Grad. Texts Math. 27 (1975).

[116] G.Kempf, F.Knudsen, D.Mumford, B.Saint-Donat, Toroidal embeddings – I. Springer Lect. Notes Math. 339 (1973).

[117] R.Kiehl, Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie. Invent. Math. 2 (1967), pp.191–214.

[118] L.Kindler, Pullback of regular singular stratified bundles and restriction to curves. Math. Res. Lett. 22 (2015), pp.1733–1748

[119] F.Knudsen, D.Mumford, The projectivity of the moduli space of stable curves – I : Preliminaries on “det” and “Div”. Math. Scand. 39 (1976) pp.19–55.

[120] D.Lazard, Autour de la platitude. Bull. Soc. Math. France 97 (1969) pp.81–128.

[121] M.Lazard, Commutative Formal Groups. Springer Lect. Notes Math. 443 (1975).

[122] J.Lipman, Rational singularities with applications to algebraic surfaces and unique factorization. Publ.Math. IHES 36 (1969) pp.195–279.

[123] J.Lipman, Desingularization of two-dimensional schemes. Ann. of. Math. 107 (1978), pp.151–207

[124] W.S.Massey, A basic course in algebraic topology. Springer Grad. Texts Math. 127 (1991).

[125] H.Matsumura, Commutative algebra – Second edition. Benjamin/Cummings Math. Lect. Notes (1980).

[126] H.Matsumura, Commutative ring theory. Cambridge Univ. Press (1986).

[127] W.Messing, T.Zink, De Jong’s Theorem on Homomorphisms of p-divisible Groups. preprint available

    at : https://www.math.uni-bielefeld.de/~zink/z_publ.html

[128] A.Micali, Sur les algèbres universelles. Ann. Inst. Fourier 14 (1964) pp. 33-87.

[129] T.Mihara, On Tate Acyclicity and Uniformity of Berkovich Spectra and Adic Spectra. Israel J. of Math 216 (2016) pp.61–105.

[130] S.Mochizuki, Extending families of curves over log regular schemes. J. Reine Angew. Math. 511 (1999) pp.43–71.

[131] D.Mumford, Geometric invariant theory. Springer Ergebn. der Math. 34 (1965).

[132] D.Mumford, Lectures on curves on algebraic surfaces. Princeton Univ. Press (1966).

[133] M.Nagata, Local rings. Interscience Publ. (1962).

[134] M.Nagata, Finitely generated rings over a valuation ring. J.Math. Kyoto Univ. (1966) pp.163–169.

[135] W.Niziol, Toric singularities : log-blow-ups and global resolutions. J. of Alg. Geom. 15 (2006) pp.1–29.

[136] J.Ohm, R.L.Pendleton, Rings with noetherian spectrum. Duke Math. J. 35 (1968) pp.631–639.

[137] J.-P.Olivier, Going up along absolutely flat morphisms. J.Pure Appl. Algebra 30 (1983) pp.47–59.

[138] D.Popescu, General Néron desingularization and approximation. Nagoya Math J. 104 (1986) pp.85–115.

[139] D.Quillen, Homology of commutative rings. unpublished notes MIT (1968).

[140] D.Quillen, Higher algebraic K-theory – I. Springer Lect. Notes Math. 341 (1973) pp.85–147.

[141] M.Raynaud, Anneaux locaux henséliens. Springer Lect. Notes Math. 169 (1970).

[142] P.Roberts, The homological conjectures, Progress in commutative algebra 1, de Gruyter (2012) pp.199– 230.

[143] N.Roby, Lois polynômes et lois formelles en théorie des modules. Ann. Sci. E.N.S. 3^{e} série 80 (1963) pp.213–348.

[144] J.-E.Roos, Sur les foncteurs dérivés de lim_{⟵}. Applications. C.R. Acad. Sci. Paris 252 (1961) pp.3702–3704.

[145] P.Samuel, O.Zariski, Commutative algebra, Vol.II. Springer Grad. Texts Math. 29 (1975).

[146] D.Schäppi, A characterization of categories of coherent sheaves of certain algebraic stacks. Preprint on

    arXiv:1206.2764 (2012).

[147] C.Scheiderer, Quasi-augmented simplicial spaces, with an application to cohomological dimension. J.Pure Appl. Algebra 81 (1992) pp.293–311.

[148] P.Schenzel, Finiteness of relative Rees rings and asymptotic prime divisors. Math. Nachr. 129 (1986) pp. 123–148.

[149] P.Scholze, Perfectoid spaces. Publ. Math. IHES 116 (2012) pp.245–313.

[150] P.Scholze, p-adic Hodge theory for rigid-analytic varieties. Forum of Mathematics, Pi 1 (2013) e1; corrigendum 4 (2016) e6.

[151] P.Scholze, Berkeley Lectures on p-adic geometry. Annals of Math. Studies 207 (2020).

[152] H.Schoutens, The use of Ultraproducts in Commutative Algebra. Springer Lect. Notes Math. 1999 (2010).

[153] K.Shimomoto, Integral perfectoid big Cohen-Macaulay algebras via André’s theorem. Math. Ann. 372 (2018) pp.1167–1188.

[154] N.Spaltenstein, Resolutions of unbounded complexes. Comp. Math. 65 (1988) pp.121–154.

[155] R.P.Stanley, Hilbert functions of graded algebras. Adv. in Math. 28 (1978) pp.57–83.

[156] R.Swan, On seminormality. J. Alg. 67 (1980) pp.210–229.

[157] M.Temkin, Relative Riemann-Zariski spaces. Israel J. of Math. 185 (2011) pp.1–42.

[158] R.W.Thomason, T.Trobaugh, Higher Algebraic K-theory of Schemes and of Derived Categories. Birkhäuser Progr. Math. 88 (1990) (The Grothendieck Festschrift, Vol. III), pp.247–435.

[159] T.Tsuji, Saturated morphisms of logarithmic schemes. Tunisian J. of Math. 1 (2019) pp.185–220.

[160] R.Vyas, A.Yekutieli, Weak proregularity, weak stability, and the non-commutative MGM equivalence. J. of Algebra 513 (2018) pp.265–325.

[161] H.Yanagihara, Some results on weakly normal ring extensions. J.Math. Soc. Japan 35 (1983) pp.649– 661.

[162] A.Yekuteli, Derived categories. Cambridge Univ. Press (2019).

[163] C.Weibel, An introduction to homological algebra. Cambridge Univ. Press (1994).